Presentation and Manipulation of Mizar Properties in an Isabelle Object Logic

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10383)

Abstract

One of the crucial factors enabling an efficient use of a logical framework is the convenience of entering, manipulating, and presenting object logic constants, statements, and proofs. In this paper, we discuss various elements of the Mizar language and the possible ways how these can be represented in the Isabelle framework in order to allow a suitable way of working in typed set theory. We explain the interpretation of various components declared in each Mizar article environment and create Isabelle attributes and outer syntax that allow simulating them. We further discuss introducing notations for symbols defined in the Mizar Mathematical Library, but also synonyms and redefinitions of such symbols. We also compare the language elements corresponding to the actual proofs, with special care for implicit proof expansions not present in Isabelle. We finally discuss Mizar’s hidden arguments and demonstrate that some of them are not necessary in an Isabelle representation.

References

  1. 1.
    Abrial, J., Butler, M.J., Hallerstede, S., Hoang, T.S., Mehta, F., Voisin, L.: Rodin: an open toolset for modelling and reasoning in Event-B. STTT 12(6), 447–466 (2010)CrossRefGoogle Scholar
  2. 2.
    Alama, J., Kohlhase, M., Mamane, L., Naumowicz, A., Rudnicki, P., Urban, J.: Licensing the Mizar mathematical library. In: Davenport, J.H., Farmer, W.M., Urban, J., Rabe, F. (eds.) CICM 2011. LNCS, vol. 6824, pp. 149–163. Springer, Heidelberg (2011). doi:10.1007/978-3-642-22673-1_11 CrossRefGoogle Scholar
  3. 3.
    Bancerek, G.: On the structure of Mizar types. In: Geuvers, H., Kamareddine, F. (eds.) ENTCS, vol. 85, pp. 69–85. Elsevier (2003)Google Scholar
  4. 4.
    Bancerek, G., Byliński, C., Grabowski, A., Korniłowicz, A., Matuszewski, R., Naumowicz, A., Pa̧k, K., Urban, J.: Mizar: state-of-the-art and beyond. In: Kerber, M., Carette, J., Kaliszyk, C., Rabe, F., Sorge, V. (eds.) CICM 2015. LNCS, vol. 9150, pp. 261–279. Springer, Cham (2015). doi:10.1007/978-3-319-20615-8_17 CrossRefGoogle Scholar
  5. 5.
    Grabowski, A., Korniłowicz, A., Naumowicz, A.: Four decades of Mizar. J. Autom. Reasoning 55(3), 191–198 (2015)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Kaliszyk, C., Pąk, K., Urban, J.: Towards a Mizar environment for Isabelle: foundations and language. In: Avigad, J., Chlipala, A. (eds.) Conference on Certified Programs and Proofs (CPP 2016), pp. 58–65. ACM (2016). doi:10.1145/2854065.2854070
  7. 7.
    Kaliszyk, C., Urban, J.: MizAR 40 for Mizar 40. J. Autom. Reasoning 55(3), 245–256 (2015). doi:10.1007/s10817-015-9330-8 MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Korniłowicz, A.: On rewriting rules in Mizar. J. Autom. Reasoning 50(2), 203–210 (2013)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Korniłowicz, A.: Enhancement of Mizar texts with transitivity property of predicates. In: Kohlhase, M., Johansson, M., Miller, B., de Moura, L., Tompa, F. (eds.) CICM 2016. LNCS, vol. 9791, pp. 157–162. Springer, Cham (2016). doi:10.1007/978-3-319-42547-4_12 CrossRefGoogle Scholar
  10. 10.
    Megill, N.D.: Metamath: A Computer Language for Pure Mathematics. Lulu Press, Morrisville, North Carolina (2007)Google Scholar
  11. 11.
    Obua, S., Fleuriot, J.D., Scott, P., Aspinall, D.: ProofPeer: Collaborative theorem proving. CoRR, abs/1404.6186 (2014)Google Scholar
  12. 12.
    Obua, S., Fleuriot, J., Scott, P., Aspinall, D.: Type inference for ZFH. In: Kerber, M., Carette, J., Kaliszyk, C., Rabe, F., Sorge, V. (eds.) CICM 2015. LNCS, vol. 9150, pp. 87–101. Springer, Cham (2015). doi:10.1007/978-3-319-20615-8_6 CrossRefGoogle Scholar
  13. 13.
    Paulson, L.C.: Isabelle: the next 700 theorem provers. In: Odifreddi, P. (ed.) Logic and Computer Science (1990), pp. 361–386 (1990)Google Scholar
  14. 14.
    Paulson, L.C.: Set theory for verification: I. From foundations to functions. J. Autom. Reasoning 11(3), 353–389 (1993)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Pąk, K.: Improving legibility of formal proofs based on the close reference principle is NP-hard. J. Autom. Reasoning 55(3), 295–306 (2015)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Rabe, F.: A logical framework combining model and proof theory. Math. Struct. Comput. Sci. 23(5), 945–1001 (2013)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Schürmann, C.: The Twelf proof assistant. In: Berghofer, S., Nipkow, T., Urban, C., Wenzel, M. (eds.) TPHOLs 2009. LNCS, vol. 5674, pp. 79–83. Springer, Heidelberg (2009). doi:10.1007/978-3-642-03359-9_7 CrossRefGoogle Scholar
  18. 18.
    Urban, J., Sutcliffe, G.: ATP-based cross-verification of Mizar proofs: method, systems, and first experiments. Math. in Comput. Sci. 2(2), 231–251 (2008)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Wenzel, M., Paulson, L.C., Nipkow, T.: The Isabelle framework. In: Mohamed, O.A., Muñoz, C., Tahar, S. (eds.) TPHOLs 2008. LNCS, vol. 5170, pp. 33–38. Springer, Heidelberg (2008). doi:10.1007/978-3-540-71067-7_7 CrossRefGoogle Scholar
  20. 20.
    Wiedijk, F. (ed.): The Seventeen Provers of the World. LNCS (LNAI), vol. 3600. Springer, Heidelberg (2006). doi:10.1007/11542384 Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Universität InnsbruckInnsbruckAustria
  2. 2.Uniwersytet w BiałymstokuBiałystokPoland

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