The Rooster and the Butterflies

  • Assia Mahboubi
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7961)


This paper describes a machine-checked proof of the Jordan-Hölder theorem for finite groups. This purpose of this description is to discuss the representation of the elementary concepts of finite group theory inside type theory. The design choices underlying these representations were crucial to the successful formalization of a complete proof of the Odd Order Theorem with the Coq system.


Normal Subgroup Group Type Coset Space Group Morphism Domain Type 
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  1. 1.
    The Coq Proof Assistant,
  2. 2.
    Aschbacher, M.: Finite Group Theory. Cambridge Studies in Advanced Mathematics. Cambridge University Press (2000)Google Scholar
  3. 3.
    Bender, H., Glauberman, G.: Local analysis for the Odd Order Theorem. London Mathematical Society, LNS, vol. 188. Cambridge University Press (1994)Google Scholar
  4. 4.
    Bertot, Y., Castéran, P.: Interactive theorem proving and program development: Coq’Art: The calculus of inductive constructions. Springer, Berlin (2004)CrossRefGoogle Scholar
  5. 5.
    Bertot, Y., Gonthier, G., Ould Biha, S., Pasca, I.: Canonical big operators. In: Mohamed, O.A., Muñoz, C., Tahar, S. (eds.) TPHOLs 2008. LNCS, vol. 5170, pp. 86–101. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  6. 6.
    de Bruijn, N.G.: The mathematical language AUTOMATH, its usage, and some of its extensions. In: Laudet, M., Lacombe, D., Nolin, L., Schützenberger, M. (eds.) Symposium on Automatic Demonstration. Lecture Notes in Mathematics, vol. 125, pp. 29–61. Springer, Heidelberg (1970)CrossRefGoogle Scholar
  7. 7.
    Feit, W., Thompson, J.G.: Solvability of groups of odd order. Pacific Journal of Mathematics 13(3), 775–1029 (1963)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Garillot, F.: Generic Proof Tools and Finite Group Theory. PhD thesis, École polytechnique (2011)Google Scholar
  9. 9.
    Gonthier, G.: Point-free, set-free concrete linear algebra. In: van Eekelen, M., Geuvers, H., Schmaltz, J., Wiedijk, F. (eds.) ITP 2011. LNCS, vol. 6898, pp. 103–118. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  10. 10.
    Gonthier, G., Asperti, A., Avigad, J., Bertot, Y., Cohen, C., Garillot, F., Roux, S.L., Mahboubi, A., O’Connor, R., Biha, S.O., Pasca, I., Rideau, L., Solovyev, A., Tassi, E., Théry, L.: A machine-checked proof of the odd order theorem. To appear in the Proceedings of the ITP 2013 Conference (2013)Google Scholar
  11. 11.
    Gonthier, G., Mahboubi, A.: An introduction to small scale reflection in Coq. Journal of Formalized Reasoning 3(2), 95–152 (2010)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Gonthier, G., Mahboubi, A., Rideau, L., Tassi, E., Théry, L.: A Modular Formalisation of Finite Group Theory. In: Schneider, K., Brandt, J. (eds.) TPHOLs 2007. LNCS, vol. 4732, pp. 86–101. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  13. 13.
    Gonthier, G., Mahboubi, A., Tassi, E.: A Small Scale Reflection Extension for the Coq system. Rapport de recherche RR-6455, INRIA (2012)Google Scholar
  14. 14.
    Hedberg, M.: A coherence theorem for Martin-Löf’s type theory. Journal of Functional Programming 8(4), 413–436 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Hölder, O.: Zurückführung einer beliebigen algebraischen Gleichung auf eine Kette von Gleichungen. Mathematische Annalen 34(1), 26–56 (1889)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Isaacs, I.: Character Theory of Finite Groups. AMS Chelsea Pub. Series (1976)Google Scholar
  17. 17.
    Jordan, C.: Traité des substitutions et des équations algébriques. Gauthier-Villars, Paris (1870)Google Scholar
  18. 18.
    Kurzweil, H., Stellmacher, B.: The Theory of Finite Groups: An Introduction. Universitext Series. Springer (2010)Google Scholar
  19. 19.
    Mahboubi, A., Tassi, E.: Canonical structures for the working coq user. To appear in the Proceedings of the ITP 2013 Conference (2013)Google Scholar
  20. 20.
    Peterfalvi, T.: Character Theory for the Odd Order Theorem. London Mathematical Society, LNS, vol. 272. Cambridge University Press (2000)Google Scholar
  21. 21.
    The Univalent Foundations Program. Homotopy type theory: Univalent foundations of mathematics. Technical report, Institute for Advanced Study (2013)Google Scholar
  22. 22.
    Wiedijk, F.: The “de Bruijn factor”,
  23. 23.
    Zassenhaus, H.: Zum satz von Jordan-Hölder-Schreier. Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 10(1), 106–108 (1934)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Assia Mahboubi
    • 1
  1. 1.Inria Joint CentreMicrosoft ResearchFrance

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