A proof-theoretical investigation of Zantema's problem

Abstract

We present a concrete example of how one can extract constructive content from a non-constructive proof. The proof investigated is a termination proof of the string-rewriting system 1100 → 000111. This rewriting system is self-embedding, so the standard termination techniques which rely on Kruskal's Tree Theorem cannot be applied directly. Dershowitz and Hoot [3] have given a classical termination proof using a minimal bad sequence argument. We analyse their proof and give a constructive interpretation of it, which enables us to extract a first proof in Type Theory that uses generalised inductive definitions. By simplifying this constructive proof we obtain a second proof in a theory conservative over primitive recursive arithmetic. This proof is generalised to a theorem about string rewriting systems.