A Modular First Formalisation of Combinatorial Design Theory

Edmonds, Chelsea; Paulson, Lawrence C.

doi:10.1007/978-3-030-81097-9_1

Cited by 6 publications

(7 citation statements)

References 10 publications

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“…In Isabelle, locales have proven to be valuable towards these efforts. This was again demonstrated in this formalisation, by the ease at which our existing incidence system locales from past work [9] could be extended and adapted. Similarly, they enable the transfer of results between vector space definitions.…”

Section: Integrating Multiple Formal Librariesmentioning

confidence: 83%

“…Aside from numerous graph theory libraries, of which the most extensive is presented by Noschinksi [21], there are limited formalisations of incidence set systems in Isabelle, including a small matroid library [18], and our previous work on design theory [9], which this work extends. Furthermore, no existing formalisation of Fisher's inequality or formalisations of linear algebraic proofs for combinatorics are known to be available in any system.…”

Section: Related Formalisation Workmentioning

confidence: 99%

“…This formalisation builds on prior work [9] which emphasises use of locales, Isabelle's module system [3], to establish a flexible and easily extendable mathematical structure hierarchy. Locales are an important element of the Isar proof language, providing persistent contexts which can be used across numerous theories drawing on similar structures.…”

Section: Isabelle and Localesmentioning

confidence: 99%

“…To enable the formalisation of Fisher's inequality in its multiple forms, a number of extensions are required to the existing design theory library [9]. In particular, constant intersect set systems are crucial to reasoning for the non-uniform Fisher's inequality.…”

Section: Constant Intersect Designsmentioning

confidence: 99%

“…Currently no formalisations of combinatorial proofs using a linear algebraic methodology exist. This paper presents the formalisation of linear algebraic representations of set systems, building on our prior work in design theory [9] and Isabelle/HOL's linear algebra libraries such as matrices in [25]. We develop alternate formal reasoning techniques for set systems using these representations, and the formalisation of two particularly notable proof methods: the rank argument and linear bound method.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

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

Edmonds¹,

Paulson²

2022

Preprint

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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.

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Section: Integrating Multiple Formal Librariesmentioning

confidence: 83%

Section: Related Formalisation Workmentioning

confidence: 99%

Section: Isabelle and Localesmentioning

confidence: 99%

Section: Constant Intersect Designsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

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

Edmonds¹,

Paulson²

2022

Preprint

Self Cite

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Large-Scale Formal Proof for the Working Mathematician—Lessons Learnt from the ALEXANDRIA Project

Paulson

2023

Lecture Notes in Computer Science

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Formalising Szemerédi’s Regularity Lemma and Roth’s Theorem on Arithmetic Progressions in Isabelle/HOL

2022

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We have formalised Szemerédi’s Regularity Lemma and Roth’s Theorem on Arithmetic Progressions, two major results in extremal graph theory and additive combinatorics, using the proof assistant Isabelle/HOL. For the latter formalisation, we used the former to first show the Triangle Counting Lemma and the Triangle Removal Lemma: themselves important technical results. Here, in addition to showcasing the main formalised statements and definitions, we focus on sensitive points in the proofs, describing how we overcame the difficulties that we encountered.

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A Modular First Formalisation of Combinatorial Design Theory

Cited by 6 publications

References 10 publications

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

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

Large-Scale Formal Proof for the Working Mathematician—Lessons Learnt from the ALEXANDRIA Project

Formalising Szemerédi’s Regularity Lemma and Roth’s Theorem on Arithmetic Progressions in Isabelle/HOL

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