on recursive equations having a unique solution

Abstract

We give conditions on a left-linear Church-Rosser term rewriting system S allowing to define S-normal forms for infinite terms. We obtain a characterization of the S-equivalence of recursive program schemes (i.e. equivalence in all interpretations which validate S considered as a set of axioms). We give sufficient conditions for a recursive program scheme Σ to be S-univocal i.e. to have only one solution up to S-equivalence (considering Σ as a system of equations). For such schemes, we obtain proofs of S-equivalence which do not use any "induction principle". We also consider (SUE)-equivalence where S satisfies the above conditions and E is a set of bilinear equations such that no E-normal form does exist.