Proving Inductive Theorems Based on Term Rewriting Systems

Sufficient criteria for an equation to be in the inductive theory of a term rewriting system are given. Inspecting only special critical pairs, we need not require the underlying system to be confluent, not even on ground terms. We are able to deal with equations which — if viewed as rules — are possibly not terminating if added to the given rewrite system; we have to restrict, however, their use in the induction process. Modular use of lemmata, already known inductive theorems, is incorporated into the results. As examples we treat natural number arithmetic, sorting lists of natural numbers, and sorting lists over arbitrary data structures.