Algorithmic Equality in Heyting Arithmetic Modulo

Abstract

Deduction Modulo is a formalism that aims at distinguish reasoning from computation in proofs. A theory modulo is formed with a set of axioms and a congruence defined by rewrite rules: the reasoning part of the theory is given by the axioms, the computational part by the congruence. In deduction modulo, we can in particular build theories without any axiom, called purely computational theories. What is interesting in building such theories - purely defined by a set of rewrite rules - is the possibility, in some cases to simplify the proofs (typically equality between two closed terms), and also the algorithmic aspect of these proofs.