Skip to main content
SpringerLink
Log in

The Rooster and the Butterflies

  • Conference paper
Intelligent Computer Mathematics (CICM 2013)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 7961))

Included in the following conference series:

  • 940 Accesses

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. The Coq Proof Assistant, http://coq.inria.fr

  2. Aschbacher, M.: Finite Group Theory. Cambridge Studies in Advanced Mathematics. Cambridge University Press (2000)

    Google Scholar 

  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. Bertot, Y., Castéran, P.: Interactive theorem proving and program development: Coq’Art: The calculus of inductive constructions. Springer, Berlin (2004)

    Book  Google Scholar 

  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)

    Chapter  Google Scholar 

  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)

    Chapter  Google Scholar 

  7. Feit, W., Thompson, J.G.: Solvability of groups of odd order. Pacific Journal of Mathematics 13(3), 775–1029 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  8. Garillot, F.: Generic Proof Tools and Finite Group Theory. PhD thesis, École polytechnique (2011)

    Google Scholar 

  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)

    Chapter  Google Scholar 

  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. Gonthier, G., Mahboubi, A.: An introduction to small scale reflection in Coq. Journal of Formalized Reasoning 3(2), 95–152 (2010)

    MathSciNet  MATH  Google Scholar 

  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)

    Chapter  Google Scholar 

  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. Hedberg, M.: A coherence theorem for Martin-Löf’s type theory. Journal of Functional Programming 8(4), 413–436 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  15. Hölder, O.: Zurückführung einer beliebigen algebraischen Gleichung auf eine Kette von Gleichungen. Mathematische Annalen 34(1), 26–56 (1889)

    Article  MathSciNet  MATH  Google Scholar 

  16. Isaacs, I.: Character Theory of Finite Groups. AMS Chelsea Pub. Series (1976)

    Google Scholar 

  17. Jordan, C.: Traité des substitutions et des équations algébriques. Gauthier-Villars, Paris (1870)

    Google Scholar 

  18. Kurzweil, H., Stellmacher, B.: The Theory of Finite Groups: An Introduction. Universitext Series. Springer (2010)

    Google Scholar 

  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. Peterfalvi, T.: Character Theory for the Odd Order Theorem. London Mathematical Society, LNS, vol. 272. Cambridge University Press (2000)

    Google Scholar 

  21. The Univalent Foundations Program. Homotopy type theory: Univalent foundations of mathematics. Technical report, Institute for Advanced Study (2013)

    Google Scholar 

  22. Wiedijk, F.: The “de Bruijn factor”, http://www.cs.ru.nl/~freek/factor/

  23. Zassenhaus, H.: Zum satz von Jordan-Hölder-Schreier. Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 10(1), 106–108 (1934)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Inria Joint Centre, Microsoft Research, France

    Assia Mahboubi

Authors
  1. Assia Mahboubi
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Computing and Software, McMaster University, 1280 Main Street West, L8S 4K1, Hamilton, ON, Canada

    Jacques Carette

  2. School of Informatics, University of Edinburgh, 10 Crichton Street, EH8 9AB, Edinburgh, Scotland, UK

    David Aspinall

  3. School of Computer Science, University of Birmingham, Edgbaston, B15 2TT, Birmingham, UK

    Christoph Lange

  4. Faculty of Informatics, Department of Computer Graphics and Design, Masaryk University, Botanická 68a, 60200, Brno, Czech Republic

    Petr Sojka

  5. Johannes Kepler University Linz (JKU), RISC, Altenberger Strasse 69, 4040, Linz, Austria

    Wolfgang Windsteiger

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Mahboubi, A. (2013). The Rooster and the Butterflies. In: Carette, J., Aspinall, D., Lange, C., Sojka, P., Windsteiger, W. (eds) Intelligent Computer Mathematics. CICM 2013. Lecture Notes in Computer Science(), vol 7961. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-39320-4_1

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-39320-4_1

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-39319-8

  • Online ISBN: 978-3-642-39320-4

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation