Abstract. We generalize the various path orderings and the conditions under which they work, and describe an implementation of this general ordering. We look at methods for proving termination of orthogonal systems and give a new solution to a problem of Zantema's. 1

. A graded sequence of examples---presented in a uniform framework---spotlights stages in the development of methods for proving termination of rewrite systems. Let T be the set of all terms over some vocabulary.Arewrite system over T is a ##nite or in#nite# set of rules, eachoftheforml ! r, where l and r are terms containing variables ranging over T . A rule l ! r applies to a term t in T if a subterm s of t matches the left-hand side l with some substitution # of terms in T for variables appearing in l. The rule is applied by replacing the redex s in t with the corresponding right-hand side r# of the rule, to which the same substitution # of terms for variables has been applied. We write t !u to indicate that the term t in T rewrites in this waytothetermu in T by a single application of some rule. Note that more than one rule can apply to t and rules can apply at more than one subterm s. Rewrite systems have long been used as decision procedures for validity in equational theories,...