Formalizing Hall's Marriage Theorem in Lean (2101.00127v1)

Abstract: 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.