Orderings for term-rewriting systems

Methods of proving that a term-rewriting system terminates are presented. They are based on the intuitive notion of 'simplification orderings', orderings in which any term that is syntactically simpler than another is smaller than the other. As a consequence of Kruskal's Tree Theorem, any nonterminating system must be self-embedding in the sense that it allows for the derivation of some term from a simpler one; thus termination is guaranteed if every rule in the system is a reduction in some simplification ordering. Most of the orderings that have been used for proving termination are indeed simplication orderings; using this notion often allows for much easier proofs. A particularly useful class of simplification orderings, the 'recursive path orderings', is defined. Examples of the use of simplication orderings in termination proofs are given.