# Fully Mechanized Proofs of Dilworths Theorem and Mirskys Theorem

## TL;DR

This paper provides fully formalized proofs of Dilworth's and Mirsky's theorems in the Coq proof assistant, along with a reusable library for finite poset theorems, enhancing formal verification in order theory.

## Contribution

It introduces fully mechanized proofs of Dilworth's and Mirsky's theorems in Coq and develops a foundational library for finite poset formalizations.

## Key findings

- Formal proofs of Dilworth's and Mirsky's theorems in Coq
- A reusable library for finite poset theorems
- Enhanced framework for formal order theory

## Abstract

We present two fully mechanized proofs of Dilworths and Mirskys theorems in the Coq proof assistant. Dilworths Theorem states that in any finite partially ordered set (poset), the size of a smallest chain cover and a largest antichain are the same. Mirskys Theorem is a dual of Dilworths Theorem. We formalize the proofs by Perles [2] (for Dilworths Theorem) and by Mirsky [5] (for the dual theorem). We also come up with a library of definitions and facts that can be used as a framework for formalizing other theorems on finite posets.