Advertisement

Enhancement of Mizar Texts with Transitivity Property of Predicates

  • Artur Korniłowicz
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9791)

Abstract

A typical proof step in mathematical reasoning consists of two parts – a formula to be proven and a list of references used to justify the formula. In addition, computer proof-assistants can use specialized procedures and algorithms to strengthen their computational power to verify the correctness of reasonings.

The Mizar system supports several mechanisms to increase automation of some reasoning steps. One of them is registration of chosen properties of predicates and functors when they are defined. We propose strengthening of the Mizar system by processing another common property used in mathematics – transitivity.

References

  1. 1.
    Trybulec, A.: Mizar. In: Wiedijk, F. (ed.) The Seventeen Provers of the World. LNCS (LNAI), vol. 3600, pp. 20–23. Springer, Heidelberg (2006). doi: 10.1007/11542384_4 CrossRefGoogle Scholar
  2. 2.
    Bancerek, G., et al.: 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, Heidelberg (2015). doi: 10.1007/978-3-319-20615-8_17 CrossRefGoogle Scholar
  3. 3.
    Grabowski, A., Korniłowicz, A., Naumowicz, A.: Four decades of Mizar. J. Autom. Reason. 55(3), 191–198 (2015). doi: 10.1007/s10817-015-9345-1 MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Korniłowicz, A.: On rewriting rules in Mizar. J. Autom. Reason. 50(2), 203–210 (2013). doi: 10.1007/s10817-012-9261-6 MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Korniłowicz, A.: Flexary connectives in Mizar. Comput. Lang. Syst. Struct. 44, 238–250 (2015). doi: 10.1016/j.cl.2015.07.002 Google Scholar
  6. 6.
    Naumowicz, A., Byliński, 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). doi: 10.1007/978-3-540-27818-4_21 CrossRefGoogle Scholar
  7. 7.
    Naumowicz, A.: Enhanced processing of adjectives in Mizar. In: Grabowski, A., Naumowicz, A. (eds.) Computer Reconstruction of the Body of Mathematics. Studies in Logic, Grammar and Rhetoric, pp. 89–101. University of Białystok, Białystok (2009)Google Scholar
  8. 8.
    Korniłowicz, A.: Definitional expansions in Mizar. J. Autom. Reason. 55(3), 257–268 (2015). doi: 10.1007/s10817-015-9331-7 MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Grabowski, A., Schwarzweller, C.: Revisions as an essential tool to maintain mathematical repositories. In: Kauers, M., Kerber, M., Miner, R., Windsteiger, W. (eds.) MKM/CALCULEMUS 2007. LNCS (LNAI), vol. 4573, pp. 235–249. Springer, Heidelberg (2007). doi: 10.1007/978-3-540-73086-6_20 CrossRefGoogle Scholar
  10. 10.
    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.) MKM 2011 and Calculemus 2011. LNCS, vol. 6824, pp. 149–163. Springer, Heidelberg (2011). doi: 10.1007/978-3-642-22673-1_11 CrossRefGoogle Scholar
  11. 11.
    Żukowski, S.: Introduction to lattice theory. Formalized Math. 1(1), 215–222 (1990)Google Scholar
  12. 12.
    Trybulec, W.A., Trybulec, M.J.: Homomorphisms and isomorphisms of groups. Quotient group. Formalized Math. 2(4), 573–578 (1991)zbMATHGoogle Scholar
  13. 13.
    Bancerek, G.: Reduction relations. Formalized Math. 5(4), 469–478 (1996)Google Scholar
  14. 14.
    Pąk, K.: Improving legibility of formal proofs based on the close reference principle is NP-hard. J. Autom. Reason. 55(3), 295–306 (2015). doi: 10.1007/s10817-015-9337-1 MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Urban, J.: Order sorted algebras. Formalized Math. 10(3), 179–188 (2002)MathSciNetGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Institute of InformaticsUniversity of BiałystokBiałystokPoland

Personalised recommendations