Confluent and Coherent Equational Term Rewriting Systems: Application to Proofs in Abstract Data Types

The well known Knuth and Bendix completion procedure computes a convergent term rewriting system from a given set of equational axioms. We describe here an abstract model of computation to handle the case where some axioms cannot be treated as rewrite rules without loosing the required termination property. We call Equational Term Rewriting Systems such mixted sets of rules and equations. We show that two abstract properties, namely E-confluence and E-coherence are both necessary and sufficient ones to compute with these models. These abstract properties can be checked on critical pairs yielding Huet's classical results on “confluence modulo” as well as a more general version of Peterson and Stickel's without any linearity hypothesis on the equations. These results are finally used to check consitency of an algebraic specification of data type.