Definitional Expansions in Mizar

In memoriam of Andrzej Trybulec, a pioneer of computerized formalization


The Mizar Verifier uses definitional expansions for controlling proof structures. In this paper we propose another use of definitional expansions—enriching verified inferences by expansions of definitions of formulae included in the inferences and increasing the number of premises accessible by Checker. This introduces more knowledge to the reasoning, which helps to draw more conclusions. Some statistics about influence of such expansions on the Mizar Mathematical Library are presented.


  1. 1.

    Alama, J., Kohlhase, M., Mamane, L., Naumowicz, A., Rudnicki, P., Urban, J.: Licensing the Mizar Mathematical Library. In: Davenport, J.H. et al. (eds.) Proceedings of Calculemus/MKM 2011, LNCS, vol. 6824, pp. 149–163. Springer-Verlag, Berlin, Heidelberg. doi: 10.1007/978-3-642-22673-1_11 (2011)

  2. 2.

    Bancerek, G.: Information retrieval and rendering with MML query. In: Borwein, J. et al. (eds.) Mathematical Knowledge Management, LNCS, vol. 4108, pp. 266–279. Springer Berlin Heidelberg. doi: 10.1007/11812289_21 (2006)

  3. 3.

    Bancerek, G., Urban, J.: Integrated semantic browsing of the Mizar Mathematical Library for authoring Mizar articles. In: Asperti, A. et al. (eds.) MKM 2004, Bialowieza, Poland, September 2004, Proceedings, LNCS, vol. 3119, pp. 44–57. Springer. doi: 10.1007/978-3-540-27818-4_4 (2004)

  4. 4.

    Barker-Plummer, D.: Gazing: An approach to the problem of definition and lemma use. J. Autom. Reasoning 8(3), 311–344 (1992)

    MATH  MathSciNet  Article  Google Scholar 

  5. 5.

    Bishop, M., Andrews, P.B.: Selectively instantiating definitions. In: CADE-15, Lindau, Germany, July, 1998, Proceedings, LNCS, vol. 1421, pp. 365–380. Springer (1998)

  6. 6.

    Bledsoe, W.W.: The UT interactive prover. Memo ATP-17B, Mathematics Department. University of Texas (1983)

  7. 7.

    Bylinski, C., Alama, J.: New developments in parsing Mizar. In: Jeuring, J. et al. (eds.) Intelligent Computer Mathematics 11th International Conference LNAI vol. 7362, pp. 427–431 Springer-Verlag Berlin Heidelberg (2012), doi: 10.1007/978-3-642-31374-5_30

  8. 8.

    Cairns, P., Gow, J.: Using and parsing the Mizar language. Electronic Notes in Theoretical Computer Science 93, 60–69 (2004). doi: 10.1016/j.entcs.2003.12.028.

  9. 9.

    Davis, M.: Obvious logical inferences. In: Proceedings of the Seventh International Joint Conference on Artificial Intelligence, pp. 530–531 (1981)

  10. 10.

    Giunchiglia, F., Walsh, T.: Theorem proving with definitions. In: Proceedings of AISB 89, pp. 433–435 (1989)

  11. 11.

    Grabowski, A., Naumowicz, A.: Mizar in a nutshell. J. Formal. Reasoning, Special Issue: User Tutorials I 3(2), 153–245 (2010)

    MATH  MathSciNet  Google Scholar 

  12. 12.

    Grabowski, A., Schwarzweller, C.: Translating mathematical vernacular into knowledge repositories. In: Proceedings of MKM’05, pp. 49–64. Springer-Verlag, Berlin, Heidelberg. doi: 10.1007/11618027_4 (2006)

  13. 13.

    Grabowski, A., Schwarzweller, C.: Revisions as an essential tool to maintain mathematical repositories. In: Proceedings of Calculemus ’07 / MKM ’07, pp. 235–249. Springer-Verlag, Berlin, Heidelberg. doi: 10.1007/978-3-540-73086-6_20(2007)

  14. 14.

    Grabowski, A., Schwarzweller, C.: Towards automatically categorizing mathematical knowledge. In: Ganzha, M. et al. (eds.) FedCSIS 2012, Wroclaw, Poland, September 2012, Proceedings, pp. 63–68 (2012)

  15. 15.

    Grätzer, G.: General Lattice Theory. Academic Press, New York (1978)

    Book  Google Scholar 

  16. 16.

    Iancu, M., Kohlhase, M., Rabe, F., Urban, J.: The Mizar Mathematical Library in OMDoc Translation and applications. J. Autom. Reasoning 50(2), 191–202 (2013). doi: 10.1007/s10817-012-9271-4

    MATH  MathSciNet  Article  Google Scholar 

  17. 17.

    Jaskowski, S.: On the Rules of Suppositions in Formal Logic. Studia Logica. Nakladem Seminarjum Filozoficznego Wydzialu Matematyczno-Przyrodniczego Uniwersytetu Warszawskiego (1934).

  18. 18.

    Kieffer, S., Avigad, J., Friedman, H.: A language for mathematical knowledge management. In: Grabowski, A. et al. (eds.) Computer Reconstruction of the Body of Mathematics, Studies in Logic, Grammar and Rhetoric, vol. 18(31), pp. 51–66. Bialystok (2009)

  19. 19.

    Kornilowicz, A.: On rewriting rules in Mizar. J. Autom. Reasoning 50(2), 203–210 (2013). doi: 10.1007/s10817-012-9261-6

    MATH  MathSciNet  Article  Google Scholar 

  20. 20.

    Naumowicz, A.: Interfacing external CA systems for Grobner bases computation in Mizar proof checking. Int. J. Comput. Math. 87(1), 1–11 (2010). doi: 10.1080/00207160701864459

    MATH  Article  Google Scholar 

  21. 21.

    Naumowicz, A. In: Watt, S.M. et al. (eds.) : SAT-enhanced Mizar proof checking (2014). doi: 10.1007/978-3-319-08434-3_37

  22. 22.

    Naumowicz, A., Byliński, C.: Improving Mizar texts with properties and requirements. In: Asperti, A. et al. (eds.) MKM 2004 Proceedings, LNCS, vol. 3119, pp. 290–301. doi: 10.1007/978-3-540-27818-4_21 (2004)

  23. 23.

    Naumowicz, A., Kornilowicz, A., et al.: A brief overview of Mizar. In: Berghofer, S. (ed.) Proceedings of TPHOLs’09, LNCS, vol. 5674, pp. 67–72. Springer-Verlag, Berlin, Heidelberg. doi: 10.1007/978-3-642-03359-9_5 (2009)

  24. 24.

    Pak, K.: Methods of lemma extraction in natural deduction proofs. J. Autom. Reasoning 50(2), 217–228 (2013). doi: 10.1007/s10817-012-9267-0

    MATH  Article  Google Scholar 

  25. 25.

    Pak, K.: Improving legibility of natural deduction proofs is not trivial. Logical Methods in Comput. Sc. 10(3), 1–30 (2014). doi: 10.2168/LMCS-10(3:23)2014

    Google Scholar 

  26. 26.

    Trybulec, A., Kornilowicz, A., Naumowicz, A., Kuperberg, K.: Formal mathematics for mathematicians. J. Autom. Reasoning 50(2), 119–121 (2013). doi: 10.1007/s10817-012-9268-z

    MathSciNet  Article  Google Scholar 

  27. 27.

    Urban, J.: XML-izing Mizar: Making semantic processing and presentation of MML easy. In: Kohlhase, M. (ed.) MKM 2005, Bremen, Germany July 2005, LNCS, vol. 3863, pp. 346–360 Springer . doi: 10.1007/11618027_23 (2005)

  28. 28.

    Urban, J., Hoder, K., Voronkov, A.: Evaluation of automated theorem proving on the Mizar Mathematical Library. In: Fukuda, K. et al. (eds.) ICMS 2010, Kobe, Japan. LNCS, vol. 6327, pp. 155–166. Springer. doi: 10.1007/978-3-642-15582-6_30 (2010)

  29. 29.

    In: Wiedijk, F. (ed.) : The Seventeen Provers of the World, Foreword by Dana S. Scott, LNCS, vol. 3600. Springer (2006)

  30. 30.

    Woronowicz, E.: Relations and their basic properties. Formalized Mathematics 1(1), 73–83 (1990).

    Google Scholar 

  31. 31.

    Wos, L.: Automated Reasoning: 33 Basic Research Problems. Prentice-Hall, Englewood Cliffs. N.J (1987)

  32. 32.

    Wos, L.: The problem of definition expansion and contraction. J. Autom. Reasoning 3(4), 433–435 (1987)

    Article  Google Scholar 

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Correspondence to Artur Korniłowicz.

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Korniłowicz, A. Definitional Expansions in Mizar. J Autom Reasoning 55, 257–268 (2015).

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  • Proof assistant
  • Formal verification
  • Definitional expansion
  • Mizar