The higher-order recursive path ordering

Abstract

This paper extends the termination proof techniques based on reduction orderings to a higher-order setting, by adapting the recursive path ordering definition to terms of a typed lambda-calculus generated by a signature of polymorphic higher-order function symbols. The obtained ordering is well-founded, compatible with -reductions and with polymorphic typing, monotonic with respect to the function symbols, and stable under substitution. It can therefore be used to prove the strong normalization property of higher-order calculi in which constants can be defined by higher-order rewrite rules. 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. Several other non-trivial examples are given which examplify the expressive power of the ordering.

Simplifying Algebraic Functional Systems

“…But the latter is immediate with HORPO [4], using a precedence @ > F pair. CPO The latest recursive path ordering, CPO, is defined only for monotonic systems where all symbols have a data type as output type.…”

“…Algebraic Functional Systems (AFSs) were first defined in [3], but we follow (roughly) the more common definitions of [4]. Rather than using type declarations for variables in an environment, we annotate variables with their types directly in terms.…”

“…Haskell will commonly have such a form. Due to the repeated occurrence of the · symbol, no version of a higher-order path ordering [4,1] can handle this system. To avoid this problem, we might consider · as a stronger construction, much like function application; this is done in Nipkow's Higher-Order Rewriting Systems (HRSs) [10], where terms are built using only abstraction and application.…”

“…While this area is still far behind its first order counterpart, various techniques for proving termination have been developed, such as monotone algebras [11], path orderings [4,1] and dependency pairs [12,9,8]. Since 2010 the annual termination competition [13] has a higher order category.…”

Definitions by rewriting in the calculus of constructions

“…• Jouannaud and Rubio [22] extended to the higherorder case the use of Dershowitz's recursive path ordering. The obtained ordering can be seen as a recursive version of the General Schema and has been extended by Walukiewicz [30] to the Calculus of Constructions with object-level rewriting.…”