A Proof-Theoretical Investigation of Zantema's Problem

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 rst proof in Type Theory that uses generalised inductive deenitions. 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.