Higher-order rewriting with types and arities

Background. Higher-order rewrite rules are increasingly used in programming languages and logical systems, with three main goals: to set up a general infrastructure for computations, to describe rule-based decision procedures, and to encode other logical systems. As usual, the choice of the syntax in which the rules are expressed plays a crucial role, as well as the definition of rewriting itself. Several proposals for frameworks of higher-order rewriting (we will not consider here the explicit substitutions calculi) have been made by Klop, Khasidashvili, Nipkow, and by Jouannaud and Okada. We will compare them with an example: the encoding of λ-calculus with the β-reduction rule (λx.M)N → M{x 7→ N}. Note that here the substitution {x 7→ N} is defined on a meta-level, that is, the expression M{x 7→ N} is not a λ-term. In an encoding of logic, it is necessary to internalize the definition of substitution. The first proposal for a format of higher-order rewriting are the Combinatory Reduction Systems (CRSs) defined by Klop [?], inspired by the Contraction Schemes of Aczel [?]. The encoding of the β-reduction rule as a CRS is as follows: app(abs([x]Z(x)), Z ′) → Z(Z ′). Here app and abs are function symbols, the first of arity 2 and the second of arity 1. Both Z and Z ′ are so-called metavariables; they stand for the free variables to be instantiated when applying the rule. Like function symbols they come equipped with an arity. Here Z has arity 1 and Z ′ has arity 0. The abstraction operator [ ] is built-in. The rewrite step (λx. yx)v → yv in the untyped λ-calculus is obtained using the following substitution σ: {Z 7→ λu. app(y, u), Z ′ 7→ v}. In general, a substitution τ assigns to a meta-variable U of arity n a so-called substitute of the form λ(x1, . . . , xn). t. The result (written in postfix notation) of applying τ to a term U(t1, . . . , tn)