A Reduction Ordering for Higher - Order

We investigate one of the classical problems of the theory of term rewriting, namely termination. We present an ordering for comparing higher-order terms that can be utilized for testing termination and decreasingness of higher-order conditional term rewriting systems. The ordering relies on a rst-order interpretation of higher-order terms and a suitable extension of the RPO. 1 Motivation Term rewriting systems (TRSs) can be considered as a powerful theoretical model for reasoning about functional and logic programming in an abstract way, independently of a particular programming language. In such an approach to computer programming, logic and functional programs are represented by means of executable speciications essentially consisting of conditional equations. The operational semantics of these speciications is deened by term rewriting and equation solving, respectively. The extension of rst-order logic to higher-order logic by means of (universally quantiied) conditional equations enormously increases the expressive power of the speciications and permits an eecient oper-ationalization. Equations on terms specify (pure) term replacement criteria that can be performed on formulae or expressions. However, equations are ambiguous since they can be used in two diierent directions. Rewriting rules are directed equations which can be applied only in one direction by imposing some orientation on the terms in the equation. Such an orientation should imply that the repeated replacement of subterms in a given expression using the rules eventually stops yielding a simplest term or normal form unable to be further simpliied. If equations represent, for instance, axioms of a theory (i.e. facts about some abstract entity) oriented replacement can be used to prove equality of formulae or expressions. In other words, to solve the so-called word problem of the equational theory we only need to reduce expressions to a common normal form. In the case