Formalising Fisher's Inequality: Formal Linear Algebraic Proof Techniques in Combinatorics

TL;DR

This paper develops formal linear algebraic proof techniques within Isabelle/HOL to formalise Fisher's inequality in combinatorics, addressing challenges in formalising combinatorial proofs.

Contribution

It introduces formal linear algebraic methods for incidence structures and applies them to the first formal proof of Fisher's inequality, enhancing formalisation tools for combinatorics.

Findings

Formalisation of incidence matrices in Isabelle/HOL

Application of linear algebra bounds and rank arguments

Extension to variations of Fisher's inequality

Abstract

The formalisation of mathematics is continuing rapidly, however combinatorics continues to present challenges to formalisation efforts, such as its reliance on techniques from a wide range of other fields in mathematics. This paper presents formal linear algebraic techniques for proofs on incidence structures in Isabelle/HOL, and their application to the first formalisation of Fisher's inequality. In addition to formalising incidence matrices and simple techniques for reasoning on linear algebraic representations, the formalisation focuses on the linear algebra bound and rank arguments. These techniques can easily be adapted for future formalisations in combinatorics, as we demonstrate through further application to proofs of variations on Fisher's inequality.