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Journal of Automated Reasoning

, Volume 55, Issue 3, pp 211–221 | Cite as

Mechanizing Complemented Lattices Within Mizar Type System

  • Adam Grabowski
Open Access
Article

Abstract

Recently some longstanding open lattice theory problems were solved with the help of automated theorem provers. The question which may be posed is how to cope with such results to improve their presentation for human without loss of machine-readability, not only at the proof level, which should be rather straightforward, but also at the stage of rebuilding appropriate data structure. We describe the framework extending already existed in the Mizar library for Boolean algebras to cover more general cases of lattice with complements. The efficiency of this approach was tested e.g. on short axiom systems for Boolean algebras based on negation and disjunction. We also proved Nachbin theorem for spectra of distributive lattices.

Keywords

Formalization of mathematics Mizar Complemented lattices 

References

  1. 1.
    Alama, J., Kohlhase, M., Mamane, L., Naumowicz, A., Rudnicki, P., Urban, J.: Licensing the Mizar Mathematical Library. In: Davenport, J.H., Farmer, W.M., et al. (eds.) Proc. of MKM 2011, LNCS, vol. 6824, pp. 149–163. Springer, Berlin, Heidelberg (2011)Google Scholar
  2. 2.
    Balbes, R., Dwinger, P.: Distributive Lattices. University of Missouri Press (1975)Google Scholar
  3. 3.
    Bancerek, G.: Development of the theory of continuous lattices in Mizar. In: Kerber, M., Kohlhase, M. (eds.) Symbolic Computation and Automatic Reasoning. The Calculemus-2000 Symposium, A K Peters, pp. 65–80 (2000)Google Scholar
  4. 4.
    Dahn, B.I.: Robbins algebras are Boolean: A revision of McCune’s computer-generated solution of Robbins problem. J. Algebra 208, 526–532 (1998)zbMATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Fitelson, B.: Using Mathematica 3.0 to understand the computer proof of the Robbins conjecture. Math. Ed. Res. 7(1) (1998)Google Scholar
  6. 6.
    Geuvers, H., Pollack, R., Wiedijk, F., Zwanenburg, J.: A constructive algebraic hierarchy in Coq. J. Symb. Comput. 34(4), 271–286 (2002)zbMATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Grabowski, A.: Automated discovery of properties of rough sets. Fundamenta Informaticae 128(1–2), 65–79 (2013)zbMATHMathSciNetGoogle Scholar
  8. 8.
    Grabowski, A.: Prime filters and ideals in distributive lattices, MML Id: LATTICEA. Formalized Mathematics 21(3), 213–221 (2013)zbMATHCrossRefGoogle Scholar
  9. 9.
    Grabowski, A.: Robbins algebras vs. Boolean algebras, MML Id: ROBBINS1. Formalized Mathematics 9(4), 681–690 (2001)MathSciNetGoogle Scholar
  10. 10.
    Grabowski, A., Kornilowicz, A., Naumowicz, A.: Mizar in a nutshell. J. Formalized Reason. 3(2), 153–245 (2010)zbMATHMathSciNetGoogle Scholar
  11. 11.
    Grabowski, A., Schwarzweller, Ch.: Revisions as an essential tool to maintain mathematical repositories. In: Proc. of the 14th symposium on Towards Mechanized Mathematical Assistants, MKM 2007, Hagenberg, Austria, LNCS, vol. 4573, pp. 235–249, Springer (2007)Google Scholar
  12. 12.
    Grabowski, A., Schwarzweller, Ch.: Towards automatically categorizing mathematical knowledge. In: Federated Conference on Computer Science and Information Systems – FedCSIS 2012, Wroclaw, Poland, 9–12 September 2012, Proceedings, pp. 63–68 (2012)Google Scholar
  13. 13.
    Grätzer, G.: Lattice Theory: Foundation. Birkhäuser (2011)Google Scholar
  14. 14.
    Huntington, E.V.: Boolean algebra: a correction. Trans. AMS 35(2), 557–558 (1933)MathSciNetGoogle Scholar
  15. 15.
    Huntington, E.V.: New sets of independent postulates for the algebra of logic, with special reference to Whitehead and Russell’s Principia Mathematica. Trans. AMS 35, 274–304 (1933)MathSciNetGoogle Scholar
  16. 16.
    Kornilowicz, A.: On rewriting rules in Mizar. J. Autom. Reason. 50(2), 203–210 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    McCune, W.: Solution of the Robbins problem. J. Autom. Reason. 19, 263–276 (1997)zbMATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    McCune, W., Veroff, R., Fitelson, B., Harris, K., Feist, A., Wos, L.: Short single axioms for Boolean algebra. J. Autom. Reason. 29(1), 1–16 (2002)zbMATHMathSciNetCrossRefGoogle Scholar
  19. 19.
    Naumowicz, A.: Interfacing external CA systems for Grobner bases computation in Mizar proof checking. Int. J. Comput. Math. 87(1), 1–11 (2010)zbMATHCrossRefGoogle Scholar
  20. 20.
    Naumowicz, A., Bylinski, C.: Improving Mizar texts with properties and requirements. In: Asperti, A., Bancerek, G., Trybulec, A. (eds.) Proc. of Third International Conference on Mathematical Knowledge Management, LNCS, vol. 3119, pp. 290–301. Springer (2004)Google Scholar
  21. 21.
    Naumowicz, A., Kornilowicz, A.: A brief overview of Mizar. In: Berghofer, S., Nipkow, T., Urban, C., Wenzel, M. (eds.) Proceedings of TPHOLs’09, LNCS, vol. 5674, pp. 67–72. Springer, Berlin, Heidelberg (2009)Google Scholar
  22. 22.
    Nipkow, T., Wenzel, M., Paulson, L.C.: Isabelle/HOL: a proof assistant for higher order logic. Springer, Berlin (2002)CrossRefGoogle Scholar
  23. 23.
    Pak, K.: Improving legibility of natural deduction proofs is not trivial. Logical Methods Comput. Sci. 10(3), 1–30 (2014)CrossRefGoogle Scholar
  24. 24.
    Pak, K.: Methods of lemma extraction in natural deduction proofs. J. Autom. Reason. 50(2), 217–228 (2013)zbMATHCrossRefGoogle Scholar
  25. 25.
    Trybulec, A., Kornilowicz, A., Naumowicz, A., Kuperberg, K.: Formal mathematics for mathematicians. J. Autom. Reason. 50(2), 119–121 (2013)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Urban, J., Rudnicki, P., Sutcliffe, G.: ATP and presentation service for Mizar formalizations. J. Autom. Reason. 50(2), 229–241 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  27. 27.
    zukowski, S.: Introduction to lattice theory, MML Id: LATTICES. Formalized Mathematics 1(1), 215–222 (1990)Google Scholar

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© The Author(s) 2015

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Institute of InformaticsUniversity of BiałystokBiałystokPoland

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