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Type classes for mathematics in type theory†

Published online by Cambridge University Press:  01 July 2011

BAS SPITTERS  and
EELIS VAN DER WEEGEN
BAS SPITTERS
Affiliation:
Radboud University Nijmegen, Nijmegen, The Netherlands Email: spitters@cs.ru.nl; eelis@eelis.net
EELIS VAN DER WEEGEN
Affiliation:
Radboud University Nijmegen, Nijmegen, The Netherlands Email: spitters@cs.ru.nl; eelis@eelis.net
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Abstract

The introduction of first-class type classes in the Coq system calls for a re-examination of the basic interfaces used for mathematical formalisation in type theory. We present a new set of type classes for mathematics and take full advantage of their unique features to make practical a particularly flexible approach that was formerly thought to be unfeasible. Thus, we address traditional proof engineering challenges as well as new ones resulting from our ambition to build upon this development a library of constructive analysis in which any abstraction penalties inhibiting efficient computation are reduced to a minimum.

The basis of our development consists of type classes representing a standard algebraic hierarchy, as well as portions of category theory and universal algebra. On this foundation, we build a set of mathematically sound abstract interfaces for different kinds of numbers, succinctly expressed using categorical language and universal algebra constructions. Strategic use of type classes lets us support these high-level theory-friendly definitions, while still enabling efficient implementations unhindered by gratuitous indirection, conversion or projection.

Algebra thrives on the interplay between syntax and semantics. The Prolog-like abilities of type class instance resolution allow us to conveniently define a quote function, thus facilitating the use of reflective techniques.

Type
Paper
Information
Mathematical Structures in Computer Science , Volume 21 , Special Issue 4: Interactive Theorem Proving and the Formalisation of Mathematics , August 2011 , pp. 795 - 825
Copyright
Copyright © Cambridge University Press 2011

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