Inductive Theorem Proving in Non-terminating Rewriting Systems and Its Application to Program Transformation

We present a framework for proving inductive theorems of first-order equational theories, using techniques of implicit induction developed in the field of term rewriting. In this framework, we make use of automated confluence provers, which have recently been developed intensively, as well as a novel condition of sufficient completeness, called local sufficient completeness. The condition is a key to automated proof of inductive theorems of term rewriting systems that include non-terminating functions. We also apply the technique to showing the correctness of program transformation that is realised as an equivalence transformation of term rewriting systems.