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Cut-Elimination for Provability Logic by Terminating Proof-Search: Formalised and Deconstructed Using Coq

Part of the Lecture Notes in Computer Science book series (LNAI,volume 12842)

Abstract

Recently, Brighton gave another cut-admissibility proof for the standard set-based sequent calculus GLS for modal provability logic GL. One of the two induction measures that Brighton uses is novel: the maximum height of regress trees in an auxiliary calculus called RGL. Tautology elimination is established rather than direct cut-admissibility, and at some points the input derivation appears to be ignored in favour of a derivation obtained by backward proof-search. By formalising the GLS calculus and the proofs in Coq, we show that: (1) the use of the novel measure is problematic under the usual interpretation of the Gentzen comma as set union, and a multiset-based sequent calculus provides a more natural formulation; (2) the detour through tautology elimination is unnecessary; and (3) we can use the same induction argument without regress trees to obtain a direct proof of cut-admissibility that is faithful to the input derivation.

Keywords

  • Provability logic
  • Cut admissibility
  • Interactive theorem proving
  • Proof theory

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Acknowledgements

Work supported by the FWF project P33548, CogniGron research center, and the Ubbo Emmius Funds (University of Groningen). Work supported by the FWF projects I 2982 and P 33548.

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Correspondence to Ian Shillito .

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Goré, R., Ramanayake, R., Shillito, I. (2021). Cut-Elimination for Provability Logic by Terminating Proof-Search: Formalised and Deconstructed Using Coq. In: Das, A., Negri, S. (eds) Automated Reasoning with Analytic Tableaux and Related Methods. TABLEAUX 2021. Lecture Notes in Computer Science(), vol 12842. Springer, Cham. https://doi.org/10.1007/978-3-030-86059-2_18

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  • DOI: https://doi.org/10.1007/978-3-030-86059-2_18

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