Abstract

This paper extends the termination proof techniques based on reduction orderings to a higherorder setting, by defining a family of recursive path orderings for terms of a typed lambda-calculus generated by a signature of polymorphic higher-order function symbols. These relations can be generated from two given well-founded orderings, on the function symbols and on the type constructors. The obtained orderings on terms are well-founded, monotonic, stable under substitution and include β-reductions. They can be used to prove the strong normalization property of higher-order calculi in which constants can be defined by higher-order rewrite rules using firstorder pattern matching. For example, the polymorphic version of Gödel’s recursor for the natural numbers is easily oriented. And indeed, our ordering is polymorphic, in the sense that a single comparison allows to prove the termination property of all monomorphic instances of a polymorphic rewrite rule. Many non-trivial examples are given which exemplify the expressive power of these orderings. All have been checked by our implementation. This paper is an extended and improved version of [Jouannaud and Rubio 1999]. Polymorphic algebras have been made more expressive than in our previous framework. The intuitive notion of a polymorphic higher-order ordering has now been made precise. The higher-order recursive

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