Proofs by Induction in “Fairly” Specified Equational Theories

Abstract

This paper presents a method of proof inspired from the works of Musser, Goguen, Huet and Hullot. The method of proof described here is more general and requires simpler, less constraining hypotheses. As a matter of fact, a specification of an equational theory would be said “fair” if it can be structured into smaller, one-sorted presentations, each of them partitioned in two : the first part expresses the relations between the data type generators, the second one can be formed into a canonical term rewriting system. Thus “fairness” extends the sufficient conditions given by Huet and Hullot for deciding what they call “the Definition Principle”. Moreover, “fairness” is a very easy to respect hypothesis, in so far as it only consists in syntactical conditions. However our method requires explicitly the invocation of an inductive rule of inference, but we show how heuristics can be chosen accordingly in order to gain full advantage from our framework. Finally we outline how this method can be extended in order to automatically transform a given “fair” presentation into another equivalent one.