Koutsoukou-Argyraki, A., Bakšys, M. and Edmonds, C. orcid.org/0000-0002-8559-9133 (2023) A formalisation of the Balog–Szemerédi–Gowers theorem in Isabelle/HOL. In: CPP 2023: Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs. CPP '23: 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, 16-17 Jan 2023, Boston, MA, USA. Association for Computing Machinery (ACM) , pp. 225-238. ISBN 9798400700262
Abstract
We describe our formalisation in the interactive theorem prover Isabelle/HOL of the Balog–Szemerédi–Gowers Theorem, a profound result in additive combinatorics which played a central role in Gowers’s proof deriving the first effective bounds for Szemerédi’s Theorem. The proof is of great mathematical interest given that it involves an interplay between different mathematical areas, namely applications of graph theory and probability theory to additive combinatorics involving algebraic objects. This interplay is what made the process of the formalisation, for which we had to develop formalisations of new background material in the aforementioned areas, more rich and technically challenging. We demonstrate how locales, Isabelle’s module system, can be employed to handle such interplays in mathematical formalisations. To treat the graph-theoretic aspects of the proof, we make use of a new, more general undirected graph theory library developed by Edmonds, which is both flexible and extensible. In addition to the main theorem, which, following our source, is formulated for difference sets, we also give an alternative version for sumsets which required a formalisation of an auxiliary triangle inequality. We moreover formalise a few additional results in additive combinatorics that are not used in the proof of the main theorem. This is the first formalisation of the Balog–Szemerédi–Gowers Theorem in any proof assistant to our knowledge.
Metadata
Item Type: | Proceedings Paper |
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Authors/Creators: |
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Copyright, Publisher and Additional Information: | © 2023 Owner/Author. This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License. (http://creativecommons.org/licenses/by-nc/4.0/) |
Keywords: | interactive theorem proving; proof assistant; formalisation of mathematics; Isabelle/HOL; additive combinatorics; graph theory; probabilistic method |
Dates: |
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Institution: | The University of Sheffield |
Academic Units: | The University of Sheffield > Faculty of Engineering (Sheffield) > Department of Computer Science (Sheffield) |
Depositing User: | Symplectic Sheffield |
Date Deposited: | 25 Jan 2024 14:36 |
Last Modified: | 25 Jan 2024 14:41 |
Status: | Published |
Publisher: | Association for Computing Machinery (ACM) |
Refereed: | Yes |
Identification Number: | 10.1145/3573105.3575680 |
Related URLs: | |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:207520 |