9 research outputs found
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Formalising Mathematics â in Praxis; A Mathematicianâs First Experiences with Isabelle/HOL and the Why and How of Getting Started
AbstractThis is an account of a mathematicianâs first experiences with the proof assistant (interactive theorem prover) Isabelle/HOL, including a discussion on the rationale behind formalising mathematics and the choice of Isabelle/HOL in particular, some instructions for new users, some technical and conceptual observations focussing on some of the first difficulties encountered, and some thoughts on the use and potential of proof assistants for mathematics.</jats:p
A formalisation of the BalogâSzemerĂ©diâGowers theorem in Isabelle/HOL
We describe our formalisation in the interactive theorem prover Isabelle/HOL of the BalogâSzemerĂ©diâGowers Theorem, a profound result in additive combinatorics which played a central role in Gowersâs proof deriving the first effective bounds for SzemerĂ©diâs 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, Isabelleâs 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 BalogâSzemerĂ©diâGowers Theorem in any proof assistant to our knowledge
Formalising SzemerĂ©diâs regularity lemma and Rothâs theorem on arithmetic progressions in Isabelle/HOL
We have formalised SzemerĂ©diâs Regularity Lemma and Rothâs 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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Formalising SzemerĂ©diâs Regularity Lemma and Rothâs Theorem on Arithmetic Progressions in Isabelle/HOL
AbstractWe have formalised SzemerĂ©diâs Regularity Lemma and Rothâs 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.</jats:p
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Formalising SzemerĂ©diâs Regularity Lemma and Rothâs Theorem on Arithmetic Progressions in Isabelle/HOL
Acknowledgements: Many thanks to Timothy Gowers and Yufei Zhao for valuable advice, and to YaĂ«l Dillies and Bhavik Mehta for fruitful discussions. The referees scrutinised the manuscript with care and made numerous helpful suggestions.AbstractWe have formalised SzemerĂ©diâs Regularity Lemma and Rothâs 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.</jats:p
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Formalizing Ordinal Partition Relations Using Isabelle/HOL
This is an overview of a formalisation project in the proof assistant
Isabelle/HOL of a number of research results in infinitary combinatorics and
set theory (more specifically in ordinal partition relations) by
Erd\H{o}s--Milner, Specker, Larson and Nash-Williams, leading to Larson's proof
of the unpublished result by E.C. Milner asserting that for all , \omega^\omega\arrows(\omega^\omega, m). This material has been
recently formalised by Paulson and is available on the Archive of Formal
Proofs; here we discuss some of the most challenging aspects of the
formalisation process. This project is also a demonstration of working with
Zermelo-Fraenkel set theory in higher-order logic
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Irrationality and Transcendence Criteria for Infinite Series in Isabelle/HOL
We give an overview of our formalizations in the proof assistant Isabelle/HOL of certain irrationality and transcendence criteria for infinite series from three different research papers: by ErdĆs and Straus (1974), HanÄl (2002), and HanÄl and Rucki (2005). Our formalizations in Isabelle/HOL can be found on the Archive of Formal Proofs. Here we describe selected aspects of the formalization and discuss what this reveals about the use and potential of Isabelle/HOL in formalizing modern mathematical research, particularly in these parts of number theory and analysis