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Semantic A-translations and Super-Consistency Entail Classical Cut Elimination

  • Lisa Allali
  • Olivier Hermant
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8312)

Abstract

We show that if a theory R defined by a rewrite system is super-consistent, the classical sequent calculus modulo R enjoys the cut elimination property, which was an open question. For such theories it was already known that proofs strongly normalize in natural deduction modulo R, and that cut elimination holds in the intuitionistic sequent calculus modulo R.

We first define a syntactic and a semantic version of Friedman’s A-translation, showing that it preserves the structure of pseudo-Heyting algebra, our semantic framework. Then we relate the interpretation of a theory in the A-translated algebra and its A-translation in the original algebra. This allows to show the stability of the super-consistency criterion and the cut elimination theorem.

Keywords

Deduction modulo cut elimination A-translation pseudo-Heyting algebra super-consistency 

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References

  1. 1.
    Boespflug, M., Carbonneaux, Q., Hermant, O.: The λΠ-Calculus Modulo as a Universal Proof Language. In: Proof Exchange for Theorem Proving (PxTP), Manchester, UK, pp. 28–43 (June 2012)Google Scholar
  2. 2.
    Bonichon, R.: TaMeD: A tableau method for deduction modulo. In: Basin, D., Rusinowitch, M. (eds.) IJCAR 2004. LNCS (LNAI), vol. 3097, pp. 445–459. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  3. 3.
    Bonichon, R., Hermant, O.: A Semantic Completeness Proof for TaMeD. In: Hermann, M., Voronkov, A. (eds.) LPAR 2006. LNCS (LNAI), vol. 4246, pp. 167–181. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  4. 4.
    Brunel, A., Hermant, O., Houtmann, C.: Orthogonality and boolean algebras for deduction modulo. In: Ong, L. (ed.) TLCA 2011. LNCS, vol. 6690, pp. 76–90. Springer, Heidelberg (2011)Google Scholar
  5. 5.
    Burel, G.: Embedding Deduction Modulo into a Prover. In: Dawar, A., Veith, H. (eds.) CSL 2010. LNCS, vol. 6247, pp. 155–169. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  6. 6.
    Burel, G., Kirchner, C.: Regaining cut admissibility in deduction modulo using abstract completion. Inf. Comput. 208(2), 140–164 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Dowek, G.: Truth values algebras and proof normalization. In: Altenkirch, T., McBride, C. (eds.) TYPES 2006. LNCS, vol. 4502, pp. 110–124. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  8. 8.
    Dowek, G., Hardin, T., Kirchner, C.: HOL-λσ an intentional first-order expression of higher-order logic. Mathematical Structures in Computer Science 11(1), 21–45 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Dowek, G., Hermant, O.: A simple proof that super-consistency implies cut elimination. Notre-Dame Journal of Formal Logic 53(4), 439–456 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Dowek, G., Werner, B.: Proof normalization modulo. The Journal of Symbolic Logic 68(4), 1289–1316 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Dowek, G., Werner, B.: Arithmetic as a theory modulo. In: Giesl, J. (ed.) RTA 2005. LNCS, vol. 3467, pp. 423–437. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  12. 12.
    Friedman, H.: Classically and intuitionistically provably recursive functions. In: Müller, G.H., Scott, D.S. (eds.) MPC 1992. Lecture Notes in Mathematics, vol. 669, pp. 21–27. Springer, Heidelberg (1978)CrossRefGoogle Scholar
  13. 13.
    Gentzen, G.: Die widerspruchsfreiheit der reinen zahlentheorie. Mathematische Annalen 112, 493–565 (1936)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Gödel, K.: Zur intuitionistischen arithmetik und zahlentheorie. Ergebnisse Eines Mathematischen Kolloquiums 4, 34–38 (1933)Google Scholar
  15. 15.
    Guglielmi, A.: A system of interaction and structure. ACM Trans. Comput. Log. 8(1), 1–64 (2007)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Hermant, O.: Semantic cut elimination in the intuitionistic sequent calculus. In: Urzyczyn, P. (ed.) TLCA 2005. LNCS, vol. 3461, pp. 221–233. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  17. 17.
    Hermant, O., Lipton, J.: A constructive semantic approach to cut elimination in type theories with axioms. In: Kaminski, M., Martini, S. (eds.) CSL 2008. LNCS, vol. 5213, pp. 169–183. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  18. 18.
    Jacquel, M., Berkani, K., Delahaye, D., Dubois, C.: Tableaux Modulo Theories using Superdeduction: An Application to the Verification of B Proof Rules with the Zenon Automated Theorem Prover. In: Gramlich, B., Miller, D., Sattler, U. (eds.) IJCAR 2012. LNCS, vol. 7364, pp. 332–338. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  19. 19.
    TeReSe. Term Rewriting Systems. Cambridge Tracts in Theoretical Computer Science, vol. 55. Cambridge University Press (2003)Google Scholar
  20. 20.
    Troelstra, A.S., van Dalen, D.: Constructivism in Mathematics, An Introduction. North-Holland (1988)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Lisa Allali
    • 1
  • Olivier Hermant
    • 2
  1. 1.École Polytechnique, INRIA & Région Ile de FranceFrance
  2. 2.CRIMINES ParisTechFrance

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