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Source-Level Proof Reconstruction for Interactive Theorem Proving

  • Lawrence C. Paulson
  • Kong Woei Susanto
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4732)

Abstract

Interactive proof assistants should verify the proofs they receive from automatic theorem provers. Normally this proof reconstruction takes place internally, forming part of the integration between the two tools. We have implemented source-level proof reconstruction: resolution proofs are automatically translated to Isabelle proof scripts. Users can insert this text into their proof development or (if they wish) examine it manually. Each step of a proof is justified by calling Hurd’s Metis prover, which we have ported to Isabelle. A recurrent issue in this project is the treatment of Isabelle’s axiomatic type classes.

Keywords

Type Variable Inference Rule Type Class Type Information Interactive Proof 
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.

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References

  1. 1.
    Avigad, J., Donnelly, K., Gray, D., Raff, P.: A formally verified proof of the prime number theorem. ACM Transactions on Computational Logic (in press)Google Scholar
  2. 2.
    Gordon, M., Wadsworth, C.P., Milner, R.: Edinburgh LCF. LNCS, vol. 78. Springer, Heidelberg (1979)zbMATHGoogle Scholar
  3. 3.
    Harrison, J.: HOL Light: A tutorial introduction. In: Srivas, M., Camilleri, A. (eds.) FMCAD 1996. LNCS, vol. 1166, pp. 265–269. Springer, Heidelberg (1996)CrossRefGoogle Scholar
  4. 4.
    Hurd, J.: Integrating Gandalf and HOL. In: Bertot, Y., Dowek, G., Hirschowitz, A., Paulin, C., Théry, L. (eds.) TPHOLs 1999. LNCS, vol. 1690, pp. 311–321. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  5. 5.
    Hurd, J.: First-order proof tactics in higher-order logic theorem provers. In: Archer, M., Di Vito, B., Muñoz, C. (eds.) Design and Application of Strategies/Tactics in Higher Order Logics, number NASA/CP-2003-212448 in NASA Technical Reports, pp. 56–68 (September 2003)Google Scholar
  6. 6.
    Hurd, J.: Metis performance benchmarks (2004), http://gilith.com/software/metis/performance.html
  7. 7.
    Melham, T.F.: The HOL logic extended with quantification over type variables. Formal Methods in System Design 3(1-2), 7–24 (1994)CrossRefGoogle Scholar
  8. 8.
    Meng, J., Paulson, L.C.: Lightweight relevance filtering for machine-generated resolution problems. In: Sutcliffe, G., Schmidt, R., Schulz, S. (eds.) FLoC 2006 Workshop on Empirically Successful Computerized Reasoning. CEUR Workshop Proceedings, vol. 192, pp. 53–69 (2006)Google Scholar
  9. 9.
    Meng, J., Paulson, L.C.: Translating higher-order problems to first-order clauses. In: Sutcliffe, G., Schmidt, R., Schulz, S. (eds.) FLoC 2006 Workshop on Empirically Successful Computerized Reasoning. CEUR Workshop Proceedings, vol. 192, pp. 70–80 (2006)Google Scholar
  10. 10.
    Meng, J., Paulson, L.C.: Lightweight relevance filtering for machine-generated resolution problems. Journal of Applied Logic (in press)Google Scholar
  11. 11.
    Meng, J., Quigley, C., Paulson, L.C.: Automation for interactive proof: First prototype. Information and Computation 204(10), 1575–1596 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Nipkow, T.: Order-sorted polymorphism in Isabelle. In: Huet, G., Plotkin, G. (eds.) Logical Environments, pp. 164–188. Cambridge University Press, Cambridge (1993)Google Scholar
  13. 13.
    Nipkow, T.: Structured Proofs in Isar/HOL. In: Geuvers, H., Wiedijk, F. (eds.) TYPES 2002. LNCS, vol. 2646, pp. 259–278. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  14. 14.
    Nipkow, T., Paulson, L.C., Wenzel, M.: Isabelle/HOL. LNCS, vol. 2283. Springer, Heidelberg (2002)zbMATHGoogle Scholar
  15. 15.
    Norrish, M., Slind, K.: The HOL system description (2007), On the Internet at http://hol.sourceforge.net/
  16. 16.
    Paulson, L.C.: The foundation of a generic theorem prover. Journal of Automated Reasoning 5(3), 363–397 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Paulson, L.C.: Organizing numerical theories using axiomatic type classes. Journal of Automated Reasoning 33(1), 29–49 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Riazanov, A., Voronkov, A.: The design and implementation of VAMPIRE. AI Communications 15(2), 91–110 (2002)zbMATHGoogle Scholar
  19. 19.
    Schulz, S.: System description: E 0.81. In: Basin, D., Rusinowitch, M. (eds.) IJCAR 2004. LNCS (LNAI), vol. 3097, pp. 223–228. Springer, Heidelberg (2004)Google Scholar
  20. 20.
    Sutcliffe, G., Schmidt, R., Schulz, S. (eds.): FLoC’06 Workshop on Empirically Successful Computerized Reasoning. CEUR Workshop Proceedings, vol. 192 (2006)Google Scholar
  21. 21.
    Sutcliffe, G., Zimmer, J., Schulz, S.: TSTP data-exchange formats for automated theorem proving tools. In: Zhang, W., Sorge, V. (eds.) Distributed Constraint Problem Solving and Reasoning in Multi-Agent Systems. Frontiers in Artificial Intelligence and Applications, vol. 112, pp. 201–215. IOS Press, Amsterdam (2004)Google Scholar
  22. 22.
    Wadler, P., Blott, S.: How to make ad-hoc polymorphism less ad hoc. In: 16th AnnualSymposium on Principles of Programming Languages, pp. 60–76. ACM Press, New York (1989)CrossRefGoogle Scholar
  23. 23.
    Weidenbach, C.: Combining superposition, sorts and splitting. In: Robinson, A., Voronkov, A. (eds.) Handbook of Automated Reasoning. ch. 27, vol. II, pp. 1965–2013. Elsevier Science, Amsterdam (2001)CrossRefGoogle Scholar
  24. 24.
    Wenzel, M.: Type classes and overloading in higher-order logic. In: Gunter, E.L., Felty, A.P. (eds.) TPHOLs 1997. LNCS, vol. 1275, pp. 307–322. Springer, Heidelberg (1997)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Lawrence C. Paulson
    • 1
  • Kong Woei Susanto
    • 1
  1. 1.Computer Laboratory, University of CambridgeEngland

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