Higher-order substitutions

The lambda sigma -calculus is a concrete lambda -calculus of explicit subst itutions, designed for reasoning about implementations of lambda -calculi. Higher-order abstract syntax is an approach to metaprogramming that explici tly captures the variable-binding aspects of programming language construct s. A new calculus of explicit substitutions for higher-order abstract synta x is introduced, allowing a high-level description of variable binding in o bject languages while also providing substitutions as explicit programmer-m anipulable data objects. The new calculus is termed the lambda sigma beta ( 0)-calculus, since it makes essential use of an extension of beta (0)-unifi cation (described in another paper). Termination and confluence are verifie d for the lambda sigma beta (0)-calculus similarly to that for the ncr-calc ulus, and an efficient implementation is given in terms of first-order rena ming substitutions. The verification of confluence makes use of a verified adaptation of Nipkow's higher-order critical pairs lemma to the forms of re write rules required for the statement of the lambda sigma beta (0)-calculu s. (C) 2001 Academic Press.