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

Authors Chelsea Edmonds , Lawrence C. Paulson

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Chelsea Edmonds
  • Department of Computer Science and Technology, University of Cambridge, UK
Lawrence C. Paulson
  • Department of Computer Science and Technology, University of Cambridge, UK


Thanks to Wenda Li, for the helpful pointers on working with linear algebra in Isabelle, and Yiannos Stathopoulos, for the suggestions on utilising SErAPIS to navigate results from past formalisations.

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Chelsea Edmonds and Lawrence C. Paulson. Formalising Fisher’s Inequality: Formal Linear Algebraic Proof Techniques in Combinatorics. In 13th International Conference on Interactive Theorem Proving (ITP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 237, pp. 11:1-11:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Automated reasoning
  • Theory of computation → Higher order logic
  • Mathematics of computing → Combinatorics
  • Isabelle/HOL
  • Mathematical Formalisation
  • Fisher’s Inequality
  • Linear Algebra
  • Formal Proof Techniques
  • Combinatorics


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