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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)


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.


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

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  1. Boolos, G.: The Unprovability of Consistency: An Essay in Modal Logic. Cambridge University Press, Cambridge (1979)

    MATH  Google Scholar 

  2. Borga, M.: On some proof theoretical properties of the modal logic GL. Stud. Logica. 42(4), 453–459 (1983)

    MathSciNet  CrossRef  Google Scholar 

  3. Brighton, J.: Cut elimination for GLS using the terminability of its regress process. J. Philos. Log. 45(2), 147–153 (2016)

    MathSciNet  CrossRef  Google Scholar 

  4. Dawson, J.E., Goré, R.: Generic methods for formalising sequent calculi applied to provability logic. In: Fermüller, C.G., Voronkov, A. (eds.) LPAR 2010. LNCS, vol. 6397, pp. 263–277. Springer, Heidelberg (2010).

    CrossRef  MATH  Google Scholar 

  5. Gentzen, G.: Untersuchungen über das logische schließen. II. Math. Zeitschrift 39, 176–210, 405–431 (1935)

    Google Scholar 

  6. Gentzen, G.: Investigations into logical deduction. In: Szabo, M.E. (ed.) The Collected Papers of Gerhard Gentzen, volume 55 of Studies in Logic and the Foundations of Mathematics, pp. 68–131. Elsevier (1969)

    Google Scholar 

  7. Goré, R., Ramanayake, R.: Valentini’s cut-elimination for provability logic resolved. Rev. Symb. Log. 5(2), 212–238 (2012)

    MathSciNet  CrossRef  Google Scholar 

  8. Goré, R., Ramanayake, R.: Cut-elimination for weak Grzegorczyk logic Go. Stud. Log. 102(1), 1–27 (2014)

    MathSciNet  CrossRef  Google Scholar 

  9. Indrzejczak, A.: Tautology elimination, cut elimination, and S5. Logic Log. Philos. 26(4), 461–471 (2017)

    MathSciNet  MATH  Google Scholar 

  10. Mints, G.: Cut elimination for provability logic. In: Collegium Logicum 2005: Cut-Elimination (2005)

    Google Scholar 

  11. Negri, S.: Proof analysis in modal logic. J. Philos. Log. 34(5–6), 507–544 (2005)

    MathSciNet  CrossRef  Google Scholar 

  12. Negri, S.: Proofs and countermodels in non-classical logics. Log. Univers. 8(1), 25–60 (2014)

    MathSciNet  CrossRef  Google Scholar 

  13. Sambin, G., Valentini, S.: The modal logic of provability: the sequential approach. J. Philos. Log. 11, 311–342 (1982)

    MathSciNet  CrossRef  Google Scholar 

  14. Sasaki, K.: Löb’s axiom and cut-elimination theorem. Acad. Math. Sci. Inf. Eng. Nanzan Univ. 1, 91–98 (2001)

    Google Scholar 

  15. Savateev, Y., Shamkanov, D.: Cut elimination for the weak modal Grzegorczyk logic via non-well-founded proofs. In: Iemhoff, R., Moortgat, M., de Queiroz, R. (eds.) WoLLIC 2019. LNCS, vol. 11541, pp. 569–583. Springer, Heidelberg (2019).

    CrossRef  MATH  Google Scholar 

  16. Solovay, R.: Provability interpretations of modal logic. Israel J. Math. 25, 287–304 (1976)

    MathSciNet  CrossRef  Google Scholar 

  17. Valentini, S.: The modal logic of provability: cut-elimination. J. Philos. Log. 12, 471–476 (1983)

    MathSciNet  CrossRef  Google Scholar 

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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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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.

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