Conferences on Intelligent Computer Mathematics

CICM 2015: Intelligent Computer Mathematics pp 211-226 | Cite as

Readable Formalization of Euler’s Partition Theorem in Mizar

  • Karol Pąk
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9150)


We present a case study on formalization of a textbook theorem in a form that is as close to the original textbook presentation as possible. Euler’s partition theorem, listed as #45 at Freek Wiedijk’s list of “Top 100 mathematical theorems”, is taken as the subject of the study. As a result new formal concepts including informal flexary (i.e. flexible arity) addition are created and existing ones are extended to go around existing limitations of the Mizar system, without modification of its core. Such developments bring more flexibility of informal language reasoning into the Mizar system and make it useful for wider audience.


Operations on languages Legibility of proofs Euler’s partition 


  1. 1.
    Andrews, G.E.: Number Theory, Dover edn. W. B. Saunders Company, Philadelphia (1971)MATHGoogle Scholar
  2. 2.
    Bancerek, G.: Countable sets and Hessenberg’s theorem. Formalized Math. 2(1), 65–69 (1991)Google Scholar
  3. 3.
    Bancerek, G., Rudnicki, P.: Information retrieval in MML. In: Asperti, A., Buchberger, B., Davenport, J.H. (eds.) MKM 2003. LNCS, vol. 2594, pp. 119–131. Springer, Heidelberg (2003) CrossRefGoogle Scholar
  4. 4.
    Byliński, C.: Functions and their basic properties. Formalized Math. 1(1), 55–65 (1990)Google Scholar
  5. 5.
    Engelking, R.: General Topology. PWN - Polish Scientific Publishers, Warsaw (1977)MATHGoogle Scholar
  6. 6.
    Euler, L.: Introduction to the Analysis of the Infinite Book I Translated by John D. Blanton. Springer, New York (1988)CrossRefMATHGoogle Scholar
  7. 7.
    Grabowski, A., Korniłowicz, A., Naumowicz, A.: Mizar in a nutshell. J. Formalized Reasoning 3(2), 153–245 (2010)MathSciNetMATHGoogle Scholar
  8. 8.
    Grabowski, A., Schwarzweller, C.: On duplication in mathematical repositories. In: Autexier, S., Calmet, J., Delahaye, D., Ion, P.D.F., Rideau, L., Rioboo, R., Sexton, A.P. (eds.) AISC 2010. LNCS, vol. 6167, pp. 300–314. Springer, Heidelberg (2010) CrossRefGoogle Scholar
  9. 9.
    Grabowski, A., Schwarzweller, C.: Improving representation of knowledge within the mizar library. Stud. Logic Grammar Rhetoric 18(31), 35–50 (2009)Google Scholar
  10. 10.
    Harrison, J.V.: A HOL theory of euclidean space. In: Hurd, J., Melham, T. (eds.) TPHOLs 2005. LNCS, vol. 3603, pp. 114–129. Springer, Heidelberg (2005) CrossRefGoogle Scholar
  11. 11.
    Korniłowicz, A.: Tentative experiments with ellipsis in mizar. In: Campbell, J.A., Jeuring, J., Carette, J., Dos Reis, G., Sojka, P., Wenzel, M., Sorge, V. (eds.) CICM 2012. LNCS, vol. 7362, pp. 453–457. Springer, Heidelberg (2012) CrossRefGoogle Scholar
  12. 12.
    Horozal, F., Rabe, F., Kohlhase, M.: Flexary operators for formalized mathematics. In: Watt, S.M., Davenport, J.H., Sexton, A.P., Sojka, P., Urban, J. (eds.) CICM 2014. LNCS, vol. 8543, pp. 312–327. Springer, Heidelberg (2014) CrossRefGoogle Scholar
  13. 13.
    Korniłowicz, A., Shidama, Y.: Brouwer fixed point theorem for disks on the plane. Formalized Math. 13(2), 333–336 (2005)Google Scholar
  14. 14.
    Leisenring, A.C.: Mathematical Logic and Hilbert’s \(\varepsilon \)-Symbol. Gordon and Breach, New York (1969)MATHGoogle Scholar
  15. 15.
    Pąk, K.: Euler’s partition theorem. Formalized Math. 23(2), 91–98 (2015). doi: 10.2478/forma-2015-0009 Google Scholar
  16. 16.
    Pąk, K.: Flexary operations. Formalized Math. 23(2), 79–90 (2015). doi: 10.2478/forma-2015-0008 Google Scholar
  17. 17.
    Pąk, K.: Methods of lemma extraction in natural deduction proofs. J. Autom. Reasoning 50(2), 217–228 (2013)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Pąk, K.: Topological manifolds. Formalized Math. 22(2), 179–186 (2014)Google Scholar
  19. 19.
    Rudnicki, P., Trybulec, A.A.: Abian’s fixed point theorem. Formalized Math. 6(3), 335–338 (1997)Google Scholar
  20. 20.
    Sylvester, J.J., Franklin, F.: A constructive theory of partitions, arranged in three acts, an interact and an exodion. Amer. J. Math. 5, 251–330 (1882)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Trybulec, W.A.: Non-contiguous substrings and one-to-one finite sequences. Formalized Math. 1(3), 569–573 (1990)Google Scholar
  22. 22.
    Wilf, H.S.: Lectures on Integer Partitions (2000)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Institute of InformaticsUniversity of BialystokBialystokPoland

Personalised recommendations