Pragmatic Quotient Types in Coq

  • Cyril Cohen
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7998)


In intensional type theory, it is not always possible to form the quotient of a type by an equivalence relation. However, quotients are extremely useful when formalizing mathematics, especially in algebra. We provide a Coq library with a pragmatic approach in two complementary components. First, we provide a framework to work with quotient types in an axiomatic manner. Second, we program construction mechanisms for some specific cases where it is possible to build a quotient type. This library was helpful in implementing the types of rational fractions, multivariate polynomials, field extensions and real algebraic numbers.


Quotient types Formalization of mathematics Coq 


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  1. 1.
    Barthe, G., Capretta, V., Pons, O.: Setoids in type theory. Journal of Functional Programming 13(2), 261–293 (2003); Special Issue on Logical Frameworks and MetalanguagesGoogle Scholar
  2. 2.
    Sozeau, M.: A New Look at Generalized Rewriting in Type Theory. Journal of Formalized Reasoning 2(1), 41–62 (2009)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Chicli, L., Pottier, L., Simpson, D.: Mathematical quotients and quotient types in Coq. In: Geuvers, H., Wiedijk, F. (eds.) TYPES 2002. LNCS, vol. 2646, pp. 95–107. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  4. 4.
    Gonthier, G., Mahboubi, A., Tassi, E.: A small scale reflection extension for the Coq system. INRIA Technical report,
  5. 5.
    The Mathematical Components Project: SSReflect extension and libraries,
  6. 6.
    Garillot, F., Gonthier, G., Mahboubi, A., Rideau, L.: Packaging Mathematical Structures. In: Berghofer, S., Nipkow, T., Urban, C., Wenzel, M. (eds.) TPHOLs 2009. LNCS, vol. 5674, pp. 327–342. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  7. 7.
    Cohen, C.: Formalized algebraic numbers: construction and first order theory. PhD thesis, École polytechnique (2012)Google Scholar
  8. 8.
    Saibi, A.: Typing algorithm in type theory with inheritance. In: Proceedings of Principles of Programming Languages, POPL 1997, pp. 292–301 (1997)Google Scholar
  9. 9.
    Hedberg, M.: A coherence theorem for Martin-Löf’s type theory. Journal of Functional Programming, 4–8 (1998)Google Scholar
  10. 10.
    Bartzia, I., Strub, P.Y.: A formalization of elliptic curves (2011),
  11. 11.
    Cohen, C., Coquand, T.: A constructive version of laplace’s proof on the existence of complex roots. Journal of Algebra 381, 110–115 (2013)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Cohen, C.: Reasoning about big enough numbers in Coq (2012),
  13. 13.
    Cohen, C.: Construction of real algebraic numbers in Coq. In: Beringer, L., Felty, A. (eds.) ITP 2012. LNCS, vol. 7406, pp. 67–82. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  14. 14.
    Hofmann, M.: Extensional concepts in intensional type theory. Phd thesis, University of Edinburgh (1995)Google Scholar
  15. 15.
    Courtieu, P.: Normalized types. In: Fribourg, L. (ed.) CSL 2001 and EACSL 2001. LNCS, vol. 2142, pp. 554–569. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  16. 16.
    Gonthier, G., Mahboubi, A., Rideau, L., Tassi, E., Théry, L.: A Modular Formalisation of Finite Group Theory. Rapport de recherche RR-6156, INRIA (2007)Google Scholar

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© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Cyril Cohen
    • 1
  1. 1.Department of Computer Science and EngineeringUniversity of GothenburgSweden

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