2 research outputs found
Type Classes for Mathematics in Type Theory
The introduction of first-class type classes in the Coq system calls for
re-examination of the basic interfaces used for mathematical formalization 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 formerly thought infeasible. Thus, we address both 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 abstraction
penalties inhibiting efficient computation are reduced to a minimum.
The base 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
Working with Mathematical Structures in Type Theory
Abstract. We address the problem of representing mathematical structures in a proof assistant which: 1) is based on a type theory with dependent types, telescopes and a computational version of Leibniz equality; 2) implements coercive subtyping, accepting multiple coherent paths between type families; 3) implements a restricted form of higher order unification and type reconstruction. We show how to exploit the previous quite common features to reduce the “syntactic ” gap between pen&paper and formalised algebra. However, to reach our goal we need to propose unification and type reconstruction heuristics that are slightly different from the ones usually implemented. We have implemented them in Matita.