Formalization of some central theorems in combinatorics of finite sets

15 pages•Published: June 4, 2017

Abstract

We present fully formalized proofs of some central theorems from combinatorics. These are Dilworth's decomposition theorem, Mirsky's theorem, Hall's marriage theorem and the Erdős-Szekeres theorem. Dilworth's decomposition theorem is the key result among these. It states that in any finite partially ordered set (poset), the size of a smallest chain cover and a largest antichain are the same. Mirsky's theorem is a dual of Dilworth's decomposition theorem, which states that in any finite poset, the size of a smallest antichain cover and a largest chain are the same. We use Dilworth's theorem in the proofs of Hall's Marriage theorem and the Erdős-Szekeres theorem. The combinatorial objects involved in these theorems are sets and sequences. All the proofs are formalized in the Coq proof assistant. We develop a library of definitions and facts that can be used as a framework for formalizing other theorems on finite posets.

Keyphrases: antichains, chains, dilworth s theorem, formal proofs, hall s theorem, mirsky s theorems, partially ordered sets

In: Thomas Eiter, David Sands, Geoff Sutcliffe and Andrei Voronkov (editors). IWIL Workshop and LPAR Short Presentations, vol 1, pages 43-57.

Links:	https://easychair.org/publications/paper/Nr
	https://doi.org/10.29007/r7fg