Automating Sufficient Completeness Check for Conditional and Constrained TRS

International audienceWe present a procedure for checking sufficient completeness for conditional and constrained term rewriting systems containing axioms for constructors which may be constrained (by e.g. equalities, disequalities, ordering, membership...). Such axioms allow to specify complex data structures like e.g. sets, sorted lists or powerlists. Our approach is integrated in a framework for inductive theorem proving based on tree grammars with constraints, a formalism which permits an exact representation of languages of ground constructor terms in normal form. The key technique used in the procedure is a generalized form of narrowing where, given a term, instead of unifying it with left members of rewrite rules, we instantiate it, at selected variables, following the productions of a constrained tree grammar, and test whether it can be rewritten. Our procedure is sound and complete and has been successfully applied to several examples, yielding very natural proofs and, in case of negative answer, a counter example suggesting how to complete the specification. Moreover, it is a decision procedure when the TRS is unconditional but constrained, for a large class of constrained constructor axioms