Advertisement

Reflecting Proofs in First-Order Logic with Equality

  • Evelyne Contejean
  • Pierre Corbineau
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3632)

Abstract

Our general goal is to provide better automation in interactive proof assistants such as Coq. We present an interpreter of proof traces in first-order multi-sorted logic with equality. Thanks to the reflection ability of Coq, this interpreter is both implemented and formally proved sound — with respect to a reflective interpretation of formulae as Coq properties — inside Coq’s type theory. Our generic framework allows to interpret proofs traces computed by any automated theorem prover, as long as they are precise enough: we illustrate that on traces produced by the CiME tool when solving unifiability problems by ordered completion. We discuss some benchmark results obtained on the TPTP library.

Keywords

Word Problem Critical Pair Sequent Calculus Proof Assistant Interpretation Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Alvarado, C.: Réflexion pour la réécriture dans le calcul des constructions inductives. PhD thesis, Université Paris-Sud (December 2002)Google Scholar
  2. 2.
    Bachmair, L., Dershowitz, N., Plaisted, D.A.: Completion without failure. In: Aït-Kaci, H., Nivat, M. (eds.) Resolution of Equations in Algebraic Structures 2: Rewriting Techniques, ch. 1, pp. 1–30. Academic Press, New York (1989)Google Scholar
  3. 3.
    Bezem, M., Hendriks, D., de Nivelle, H.: Automated proof construction in type theory using resolution. Journal of Automated Reasoning 29(3), 253–275 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Contejean, E., Marché, C., Urbain, X.: CiME3 (2004), http://cime.lri.fr/
  5. 5.
    Corbineau, P.: First-order reasoning in the calculus of inductive constructions. In: Berardi, S., Coppo, M., Damiani, F. (eds.) TYPES 2003. LNCS, vol. 3085, pp. 162–177. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  6. 6.
    Crégut, P.: Une procédure de décision réflexive pour l’arithmétique de Presburger en Coq. Deliverable, Projet RNRT Calife (2001)Google Scholar
  7. 7.
    de Bruijn, N.G.: Lambda calculus with nameless dummies, a tool for automatic formula manipulation, with application to the Church-Rosser theorem. Proc. of the Koninklijke Nederlands Akademie 75(5), 380–392 (1972)Google Scholar
  8. 8.
    Dyckhoff, R.: Contraction-free sequent calculi for intuitionistic logic. The Journal of Symbolic Logic 57(3), 795–807 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Grégoire, B., Leroy, X.: A compiled implementation of strong reduction. In: International Conference on Functional Programming 2002, pp. 235–246. ACM Press, New York (2002)Google Scholar
  10. 10.
    Harrison, J.: Metatheory and reflection in theorem proving: A survey and critique. Technical Report CRC-053, SRI Cambridge, Millers Yard, Cambridge, UK (1995)Google Scholar
  11. 11.
    Hendriks, D.: Proof Reflection in Coq. Journal of Automated Reasoning 29(3-4), 277–307 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Hillenbrand, T., Löchner, B.: The next WALDMEISTER loop. In: Voronkov, A. (ed.) CADE 2002. LNCS (LNAI), vol. 2392, pp. 486–500. Springer, Heidelberg (2002)Google Scholar
  13. 13.
    Hsiang, J., Rusinowitch, M.: On word problems in equational theories. In: Ottmann, T. (ed.) ICALP 1987. LNCS, vol. 267, pp. 54–71. Springer, Heidelberg (1987)Google Scholar
  14. 14.
    Nguyen, Q.-H.: Certifying Term Rewriting Proofs in ELAN. In: van den Brand, M., Verma, R. (eds.) Proceedings of the International Workshop RULE 2001. Electronic Notes in Theoretical Computer Science, vol. 59, Elsevier Science Publishers, Amsterdam (2001)Google Scholar
  15. 15.
    Peterson, G.E.: A technique for establishing completeness results in theorem proving with equality. SIAM Journal on Computing 12(1), 82–100 (1983)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Sutcliffe, G., Suttner, C.: The TPTP Problem Library: CNF Release v1.2.1. Journal of Automated Reasoning 21(2), 177–203 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    The Coq Development Team. The Coq Proof Assistant Reference Manual – Version V8.0 (April 2004), http://coq.inria.fr

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Evelyne Contejean
    • 1
  • Pierre Corbineau
    • 1
  1. 1.PCRI — LRI (CNRS UMR 8623) & INRIA Futurs, Bât. 490Université Paris-SudOrsay CedexFrance

Personalised recommendations