A proof of weak termination providing the right way to terminate

Colloque avec actes et comité de lecture. internationale.International audienceWe give an inductive method for proving weak innermost termination of rule-based programs, from which we automatically infer, for each successful proof, a finite strategy for data evaluation. We first present the proof principle, using an explicit induction on the termination property, to prove that any input data has at least one finite evaluation. For that, we observe proof trees built from the rewrite system, schematizing the innermost derivations of any ground term. These proof trees are issued from patterns representing sets of ground terms. They are built using two mechanisms, namely abstraction, introducing variables that represent ground terms in normal form, and narrowing, schematizing rewriting on ground terms. Then, we show how, for any ground term, a rewriting strategy to compute one of its normal forms can be extracted from the proof trees, even if the ground term admits infinite rewriting derivations