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Proving Termination of Higher-order Rewrite Systems

This paper deals with termination proofs for Higher-Order Rewrite Systems (HRSs), introduced in [Nip9l, Nip93]. This formalism combines the computational aspects of term rewriting and simply typed lambda calculus. Our result is a proof technique for the termination of a HRS, similar to the proof technique "Termination by interpretation in a well-founded monotone algebra" described in [Zan93]. The resulting technique is as follows: Choose a higher-order algebra with operations for each function symbol in the HRS, equipped with some well-founded partial ordering. The operations must be strictly monotonic in this ordering. This choice generates a model for the HRS. If the choice can be made in such a way that for each rule and for each valuation of the free variables in that rule the value of the left hand side is greater than the value of the right hand side, then the HRS is terminating. At the end of the paper two applications of this technique are given, which show that this technique is natural and can easily be applied.