Formalizing Hall's Marriage Theorem in Lean

We formalize Hall's Marriage Theorem in the Lean theorem prover for inclusion in mathlib, which is a community-driven effort to build a unified mathematics library for Lean. One goal of the mathlib project is to contain all of the topics of a complete undergraduate mathematics education. We pro...

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Bibliographic Details
Main Authors Gusakov, Alena, Mehta, Bhavik, Miller, Kyle A
Format Journal Article
LanguageEnglish
Published 31.12.2020
Subjects
Computer Science - Logic in Computer Science
Mathematics - Combinatorics
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Summary:We formalize Hall's Marriage Theorem in the Lean theorem prover for inclusion in mathlib, which is a community-driven effort to build a unified mathematics library for Lean. One goal of the mathlib project is to contain all of the topics of a complete undergraduate mathematics education. We provide three presentations of the main theorem statement: in terms of indexed families of finite sets, of relations on types, and of matchings in bipartite graphs. We also formalize a version of K\H{o}nig's lemma (in terms of inverse limits) to boost the theorem to the case of countably infinite index sets. We give a description of the design of the recent mathlib library for simple graphs, and we also give a necessary and sufficient condition for a simple graph to carry a function.
DOI:10.48550/arxiv.2101.00127