Advertisement

Automated Certification of Implicit Induction Proofs

  • Sorin Stratulat
  • Vincent Demange
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7086)

Abstract

Theorem proving is crucial for the formal validation of properties about user specifications. With the help of the Coq proof assistant, we show how to certify properties about conditional specifications that are proved using automated proof techniques like those employed by the Spike prover, a rewrite-based implicit induction proof system. The certification methodology is based on a new representation of the implicit induction proofs for which the underlying induction principle is an instance of Noetherian induction governed by an induction ordering over equalities. We propose improvements of the certification process and show that the certification time is reasonable even for industrial-size applications. As a case study, we automatically prove and certify more than 40% of the lemmas needed for the validation of a conformance algorithm for the ABR protocol.

Keywords

Inference System Function Symbol Main Lemma Induction Principle Conditional Equality 
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.
    Armando, A., Rusinowitch, M., Stratulat, S.: Incorporating decision procedures in implicit induction. J. Symb. Comput. 34(4), 241–258 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Baader, F., Nipkow, T.: Term Rewriting and All That. Cambridge University Press (1998)Google Scholar
  3. 3.
    Barthe, G., Stratulat, S.: Validation of the JavaCard Platform with Implicit Induction Techniques. In: Nieuwenhuis, R. (ed.) RTA 2003. LNCS, vol. 2706, pp. 337–351. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  4. 4.
    Berger, A., Bonomi, F., Fendick, K.: Proposed TM baseline text on an ABR conformance definition. Technical Report 95-0212R1, ATM Forum Traffic Management Group (1995)Google Scholar
  5. 5.
    Bouhoula, A., Kounalis, E., Rusinowitch, M.: Automated mathematical induction. Journal of Logic and Computation 5(5), 631–668 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Contejean, E., Courtieu, P., Forest, J., Pons, O., Urbain, X.: Certification of Automated Termination Proofs. In: Konev, B., Wolter, F. (eds.) FroCos 2007. LNCS (LNAI), vol. 4720, pp. 148–162. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  7. 7.
    Courant, J.: Proof reconstruction. Research Report RR96-26, LIP (1996); Preliminary versionGoogle Scholar
  8. 8.
    Delahaye, D.: A Tactic Language for the System Coq. In: Parigot, M., Voronkov, A. (eds.) LPAR 2000. LNCS (LNAI), vol. 1955, pp. 85–95. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  9. 9.
    ITU-T. Traffic control and congestion control in B ISDN. Recommandation I.371.1 (1997)Google Scholar
  10. 10.
    Kaliszyk, C.: Validation des preuves par récurrence implicite avec des outils basés sur le calcul des constructions inductives. Master’s thesis, Université Paul Verlaine - Metz (2005)Google Scholar
  11. 11.
    Klein, G., Andronick, J., Elphinstone, K., Heiser, G., Cock, D., Derrin, P., Elkaduwe, D., Engelhardt, K., Kolanski, R., Norrish, M., Sewell, T., Tuch, H., Winwood, S.: seL4: Formal verification of an operating-system kernel. Communications of the ACM 53(6), 107–115 (2010)CrossRefGoogle Scholar
  12. 12.
    Leroy, X., Doligez, D., Frisch, A., Garrigue, J., Rémy, D., Vouillon, J.: The Objective Caml system - release 3.12. Documentation and user’s manual. INRIAGoogle Scholar
  13. 13.
    Nahon, F., Kirchner, C., Kirchner, H., Brauner, P.: Inductive proof search modulo. Annals of Mathematics and Artificial Intelligence 55(1–2), 123–154 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Rabadan, C., Klay, F.: Un nouvel algorithme de contrôle de conformité pour la capacité de transfert ‘Available Bit Rate’. Technical Report NT/CNET/5476, CNET (1997)Google Scholar
  15. 15.
    Rusinowitch, M., Stratulat, S., Klay, F.: Mechanical verification of a generic incremental ABR conformance algorithm. Technical Report 3794, INRIA (1999)Google Scholar
  16. 16.
    Rusinowitch, M., Stratulat, S., Klay, F.: Mechanical Verification of an Ideal Incremental ABR Conformance Algorithm. In: Emerson, E.A., Sistla, A.P. (eds.) CAV 2000. LNCS, vol. 1855, pp. 344–357. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  17. 17.
    Rusinowitch, M., Stratulat, S., Klay, F.: Mechanical verification of an ideal incremental ABR conformance algorithm. J. Autom. Reasoning 30(2), 53–177 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Shankar, N., Owre, S., Rushby, J.M., Stringer-Calvert, D.W.J.: PVS prover guide - version 2.4. SRI International (November 2001)Google Scholar
  19. 19.
    Stratulat, S.: A general framework to build contextual cover set induction provers. J. Symb. Comput. 32(4), 403–445 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Stratulat, S.: Automatic ‘Descente Infinie’ Induction Reasoning. In: Beckert, B. (ed.) TABLEAUX 2005. LNCS (LNAI), vol. 3702, pp. 262–276. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  21. 21.
    Stratulat, S.: ‘Descente Infinie’ induction-based saturation procedures. In: SYNASC 2007: Proceedings of the Ninth International Symposium on Symbolic and Numeric Algorithms for Scientific Computing, Washington, DC, USA, pp. 17–24. IEEE Computer Society (2007)Google Scholar
  22. 22.
    Stratulat, S.: Combining Rewriting with Noetherian Induction to Reason on Non-Orientable Equalities. In: Voronkov, A. (ed.) RTA 2008. LNCS, vol. 5117, pp. 351–365. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  23. 23.
    Stratulat, S.: Integrating Implicit Induction Proofs into Certified Proof Environments. In: Méry, D., Merz, S. (eds.) IFM 2010. LNCS, vol. 6396, pp. 320–335. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  24. 24.
    Stratulat, S., Demange, V.: Validating implicit induction proofs using certified proof environments. In: Poster Session of 2010 Grande Region Security and Reliability Day, Saarbrucken (March 2010)Google Scholar
  25. 25.
    The Coq Development Team. The Coq reference manual - version 8.2 (2009), http://coq.inria.fr/doc

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Sorin Stratulat
    • 1
  • Vincent Demange
    • 1
  1. 1.LITA, Paul Verlaine-Metz UniversityMetzFrance

Personalised recommendations