The effectiveness of formalizing substantial parts of mathematics largely depends on the availability of relevant background knowledge. The bigger the knowledge library, however, the harder it is to specify what is or should be relevant. Even with today’s size of the libraries available for various proof assistants, importing the whole library is not an option for practical performance reasons. On the other hand, too detailed import mechanisms are prone to dependency problems and pose certain difficulty for the user. In this paper we present the key ideas of a project aimed at generating standardized formalization environments which could be used to facilitate developing new formalizations based on the current content of the Mizar Mathematical Library. Showing the results of this research based on the library designed with great focus on reusability can also be insightful for designers of other formalization systems and libraries.


Computerized proof assistants Mizar Mathematical Library Standardized formal environments 


  1. 1.
    Harrison, J.: HOL Light tutorial. Accessed 18 May 2017
  2. 2.
    Bertot, Y., Castéran, P.: Interactive theorem proving and program development - Coq’Art: the calculus of inductive constructions. In: Texts in Theoretical Computer Science. An EATCS Series. Springer (2004)Google Scholar
  3. 3.
    Nipkow, T., Paulson, L.C., Wenzel, M.: Isabelle/HOL: A Proof Assistant for Higher-Order Logic, vol. 2283. Springer, New York (2002)zbMATHGoogle Scholar
  4. 4.
    Grabowski, A., Korniłowicz, A., Naumowicz, A.: Four decades of Mizar. J. Autom. Reason. 55(3), 191–198 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bancerek, G., Byliński, C., Grabowski, A., Korniłowicz, A., Matuszewski, R., Naumowicz, A., Pąk, K., Urban, J.: Mizar: state-of-the-art and beyond. In: [24], pp. 261–279 (2015)Google Scholar
  6. 6.
    Alama, J.: Mizar-items: exploring fine-grained dependencies in the Mizar Mathematical Library. In: Davenport, J.H., Farmer, W.M., Urban, J., Rabe, F. (eds.) Intelligent Computer Mathematics - 18th Symposium, Calculemus 2011, and Proceedings of 10th International Conference, MKM 2011, Bertinoro, Italy, 18–23 July 2011. Lecture Notes in Computer Science, vol. 6824, pp. 276–277. Springer (2011)Google Scholar
  7. 7.
    Blanchette, J.C., Haslbeck, M., Matichuk, D., Nipkow, T.: Mining the Archive of formal proofs. In: [24], pp. 3–17 (2015)Google Scholar
  8. 8.
    Czuba, S.T.: Schemes. Formaliz. Math. 2(3), 385–391 (1991)Google Scholar
  9. 9.
    Korniłowicz, A.: Jordan curve theorem. Formaliz. Math. 13(4), 481–491 (2005)Google Scholar
  10. 10.
    Trybulec, Z.: Properties of subsets. Formaliz. Math. 1(1), 67–71 (1990)Google Scholar
  11. 11.
    Grabowski, A., Korniłowicz, A., Naumowicz, A.: Mizar in a nutshell. J. Formaliz. Reason. 3(2), 153–245 (2010)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Naumowicz, A.: Enhanced processing of adjectives in Mizar. Stud. Log. Gramm. Rhetor. 18(31), 89–101 (2009)Google Scholar
  13. 13.
    Korniłowicz, A.: On rewriting rules in Mizar. J. Autom. Reason. 50(2), 203–210 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Grabowski, A., Korniłowicz, A., Schwarzweller, C.: Equality in computer proof-assistants. In: 2015 Federated Conference on Computer Science and Information Systems, FedCSIS 2015, Łódź, Poland, 13–16 September 2015, pp. 45–54 (2015)Google Scholar
  15. 15.
    Korniłowicz, A.: Definitional expansions in Mizar. J. Autom. Reason. 55(3), 257–268 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Naumowicz, A.: Automating Boolean set operations in Mizar proof checking with the aid of an external SAT solver. J. Autom. Reason. 55(3), 285–294 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Naumowicz, A.: Interfacing external CA systems for Gröbner bases computation in Mizar proof checking. Int. J. Comput. Math. 87(1), 1–11 (2010)CrossRefzbMATHGoogle Scholar
  18. 18.
    Byliński, C.: Cartesian categories. Formaliz. Math. 3(2), 161–169 (1992)Google Scholar
  19. 19.
    Trybulec, Z., Święczkowska, H.: Boolean properties of sets. Formaliz. Math. 1(1), 17–23 (1990)Google Scholar
  20. 20.
    Naumowicz, A.: Tools for MML environment analysis. In: [24], pp. 348–352 (2015)Google Scholar
  21. 21.
    Bancerek, G., Rudnicki, P.: A compendium of continuous lattices in Mizar. J. Autom. Reason. 29(3–4), 189–224 (2002)CrossRefzbMATHGoogle Scholar
  22. 22.
    Korniłowicz, A.: Cartesian products of relations and relational structures. Formaliz. Math. 6(1), 145–152 (1997)Google Scholar
  23. 23.
    Bancerek, G.: Bases and refinements of topologies. Formaliz. Math. 7(1), 35–43 (1998)Google Scholar
  24. 24.
    Kerber, M., Carette, J., Kaliszyk, C., Rabe, F., Sorge, V. (eds.): Intelligent Computer Mathematics - International Conference, CICM 2015, Proceedings. Washington, DC, USA, 13–17 July 2015. Lecture Notes in Computer Science, vol. 9150. Springer (2015)Google Scholar

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Authors and Affiliations

  1. 1.Institute of InformaticsUniversity of BialystokBialystokPoland

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