A Proof of Weak Termination Providing the Right Way to Terminate

Abstract

We 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 rewriting tree of any ground term, and generated with two mechanisms: abstraction, schematizing normalization of subterms, and narrowing, schematizing rewriting steps. Then, we show how, for any ground term, a normalizing rewriting strategy can be extracted from the proof trees, even if the ground term admits infinite rewriting derivations.