Pragmatic quotient types in coq

Authors:
Cyril Cohen
Affiliations:
Department of Computer Science and Engineering, University of Gothenburg, Sweden
Venue:
ITP'13 Proceedings of the 4th international conference on Interactive Theorem Proving
Year:
2013

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Abstract

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.