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Formalization of some central theorems in combinatorics of finite sets

15 pagesPublished: 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

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BibTeX entry
@inproceedings{LPAR-21S:Formalization_of_some_central,
  author    = {Abhishek Kr Singh},
  title     = {Formalization of some central theorems in combinatorics of finite sets},
  booktitle = {IWIL Workshop and LPAR Short Presentations},
  editor    = {Thomas Eiter and David Sands and Geoff Sutcliffe and Andrei Voronkov},
  series    = {Kalpa Publications in Computing},
  volume    = {1},
  pages     = {43--57},
  year      = {2017},
  publisher = {EasyChair},
  bibsource = {EasyChair, http://www.easychair.org},
  issn      = {2515-1762},
  url       = {https://easychair.org/publications/paper/Nr},
  doi       = {10.29007/r7fg}}
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