Pragmatic Quotient Types in Coq
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.
KeywordsQuotient types Formalization of mathematics Coq
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- 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
- 4.Gonthier, G., Mahboubi, A., Tassi, E.: A small scale reflection extension for the Coq system. INRIA Technical report, http://hal.inria.fr/inria-00258384
- 5.The Mathematical Components Project: SSReflect extension and libraries, http://www.msr-inria.inria.fr/Projects/math-components/index_html
- 7.Cohen, C.: Formalized algebraic numbers: construction and first order theory. PhD thesis, École polytechnique (2012)Google Scholar
- 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.Hedberg, M.: A coherence theorem for Martin-Löf’s type theory. Journal of Functional Programming, 4–8 (1998)Google Scholar
- 10.Bartzia, I., Strub, P.Y.: A formalization of elliptic curves (2011), http://pierre-yves.strub.nu/research/ec/
- 12.Cohen, C.: Reasoning about big enough numbers in Coq (2012), http://perso.crans.org/cohen/work/bigenough/
- 14.Hofmann, M.: Extensional concepts in intensional type theory. Phd thesis, University of Edinburgh (1995)Google Scholar
- 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