ATP Cross-Verification of the Mizar MPTP Challenge Problems

  • Josef Urban
  • Geoff Sutcliffe
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4790)


Mizar is a proof assistant used for formalization and mechanical verification of mathematics. The main use of Mizar is in the development of the Mizar Mathematical Library (MML), in which proofs are verified by the Mizar proof checker. The Mizar proof checker has a quite complex implementation, and also lacks the ability to print out detailed atomic proof steps in a format that is easy to verify by an independent proof-checking tool. This can raise concerns about the correctness of the MML. This paper describes how a Mizar-to-ATP translation (the MPTP system), ATP verification tools (the GDV system), and Automated Theorem Proving (ATP) systems, have been used for an independent cross-verification of a part of the MML.


Natural Deduction Proof Assistant Automate Theorem Prove Mizar Mathematical Library Automate Theorem Prove System 
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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  1. [BDH+99]
    Bertot, Y., Dowek, G., Hirschowitz, A., Paulin, C., Théry, L. (eds.): TPHOLs 1999. LNCS, vol. 1690. Springer, Heidelberg (1999)zbMATHGoogle Scholar
  2. [CS03]
    Claessen, K., Sorensson, N.: New Techniques that Improve MACE-style Finite Model Finding. In: Baumgartner, P., Fermueller, C. (eds.) Proceedings of the CADE-19 Workshop: Model Computation - Principles, Algorithms, Applications (2003)Google Scholar
  3. [Dav81]
    Davis, M.: Obvious logical inferences. In: Hayes, P.J. (ed.) IJCAI, pp. 530–531. William Kaufmann (1981)Google Scholar
  4. [DFS06]
    Denney, E., Fischer, B., Schumann, J.: An empirical evaluation of automated theorem provers in software certification. International Journal on Artificial Intelligence Tools 15(1), 81–108 (2006)CrossRefGoogle Scholar
  5. [FS06]
    Furbach, U., Shankar, N. (eds.): IJCAR 2006. LNCS (LNAI), vol. 4130, pp. 17–20. Springer, Heidelberg (2006)Google Scholar
  6. [Hal04]
    Hales, T.C.: Formalizing the proof of the kepler conjecture. In: Slind, K., Bunker, A., Gopalakrishnan, G.C. (eds.) TPHOLs 2004. LNCS, vol. 3223, p. 117. Springer, Heidelberg (2004)Google Scholar
  7. [Har96]
    Harrison, J.: A Mizar Mode for HOL. In: von Wright, J., Harrison, J., Grundy, J. (eds.) TPHOLs 1996. LNCS, vol. 1125, pp. 203–220. Springer, Heidelberg (1996)Google Scholar
  8. [Jas34]
    Jaskowski, S.: On the rules of suppositions. Studia Logica 1 (1934)Google Scholar
  9. [McL06]
    McLaughlin, S.: An interpretation of Isabelle/HOL in HOL Light. In: Furbach and Shankar [FS06], pp. 192–204Google Scholar
  10. [MS00]
    McCune, W., Shumsky, O.: System description: IVY. In: McAllester, D. (ed.) Automated Deduction - CADE-17. LNCS, vol. 1831, pp. 401–405. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  11. [NB02]
    Naumowicz, A., Bylinski, C.: Basic elements of computer algebra in MIZAR. Mechanized Mathematics and Its Applications 2 (2002)Google Scholar
  12. [NB04]
    Naumowicz, A., Bylinski, C.: Improving mizar texts with properties and requirements. In: Asperti, A., Bancerek, G., Trybulec, A. (eds.) MKM 2004. LNCS, vol. 3119, pp. 290–301. Springer, Heidelberg (2004)Google Scholar
  13. [OS06]
    Obua, S., Skalberg, S.: Importing HOL into Isabelle/HOL. In: Furbach and Shankar [FS06], pp. 298–302Google Scholar
  14. [Pel99]
    Pelletier, F.J.: A brief history of natural deduction. History and Philosophy of Logic 20, 1–31 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  15. [RT99]
    Rudnicki, P., Trybulec, A.: On equivalents of well-foundedness. J. Autom. Reasoning 23(3-4), 197–234 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  16. [Rud87]
    Rudnicki, P.: Obvious inferences. J. Autom. Reasoning 3(4), 383–393 (1987)zbMATHCrossRefMathSciNetGoogle Scholar
  17. [Rud92]
    Rudnicki, P.: An overview of the Mizar project. In: 1992 Workshop on Types for Proofs and Programs, Chalmers University of Technology, Bastad, pp. 311–332 (1992)Google Scholar
  18. [RV02]
    Riazanov, A., Voronkov, A.: The design and implementation of VAMPIRE. Journal of AI Communications 15(2-3), 91–110 (2002)zbMATHGoogle Scholar
  19. [Sch02]
    Schulz, S.: E – a brainiac theorem prover. Journal of AI Communications 15(2-3), 111–126 (2002)zbMATHGoogle Scholar
  20. [SS98]
    Sutcliffe, G., Suttner, C.B.: The TPTP problem library: CNF release v1.2.1. Journal of Automated Reasoning 21(2), 177–203 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  21. [Sut06]
    Sutcliffe, G.: Semantic Derivation Verification. International Journal on Artificial Intelligence Tools 15(6), 1053–1070 (2006)CrossRefGoogle Scholar
  22. [Sym99]
    Syme, D.: Three tactic theorem proving. In: Bertot et al. [BDH+99], pp. 203–220Google Scholar
  23. [SZS04]
    Sutcliffe, G., Zimmer, J., Schulz, S.: TSTP Data-Exchange Formats for Automated Theorem Proving Tools. In: Sorge, V., Zhang, W. (eds.) Distributed and Multi-Agent Reasoning. Frontiers in Artificial Intelligence and Applications, IOS Press, Amsterdam (2004)Google Scholar
  24. [Urb04]
    Urban, J.: MPTP - motivation, implementation, first experiments. Journal of Automated Reasoning 33(3-4), 319–339 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  25. [Urb06a]
    Urban, J.: MPTP 0.2: Design, implementation, and initial experiments. J. Autom. Reasoning 37(1-2), 21–43 (2006)zbMATHCrossRefGoogle Scholar
  26. [Urb06b]
    Urban, J.: XML-izing Mizar: making semantic processing and presentation of MML easy. In: Kohlhase, M. (ed.) MKM 2005. LNCS (LNAI), vol. 3863, pp. 346–360. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  27. [Urb07]
    Urban, J.: MaLARea: a metasystem for automated reasoning in large theories. In: Sutcliffe, G., Urban, J., Schulz, S. (eds.) ESARLT: Empirically Successful Automated Reasoning in Large Theories. CEUR Workshop Proceedings. CEUR, vol. 257, pp. 45–58 (2007)Google Scholar
  28. [UTSP07]
    Urban, J., Trac, S., Sutcliffe, G., Puzis, Y.: Combining Mizar and TPTP semantic presentation tools. In: Proceedings of the Mathematical User-Interfaces Workshop 2007 (2007),
  29. [WBH+02]
    Weidenbach, C., Brahm, U., Hillenbrand, T., Keen, E., Theobald, C., Topic, D.: SPASS version 2.0. In: CADE, pp. 275–279 (2002)Google Scholar
  30. [Wen99]
    Wenzel, M.: Isar - a generic interpretative approach to readable formal proof documents. In: Bertot et al. [BDH+99], pp. 167–184Google Scholar
  31. [Wie00]
    Wiedijk, F.: CHECKER - notes on the basic inference step in Mizar (2000), available at
  32. [Zam99]
    Zammit, V.: On the implementation of an extensible declarative proof language. In: Bertot et al. [BDH+99], pp. 185–202Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Josef Urban
    • 1
  • Geoff Sutcliffe
    • 2
  1. 1.Dep’t of Theoretical Computer Science, Charles University in Prague 
  2. 2.Dep’t of Computer Science, University of Miami 

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