Abstract
Some new calculi [1,12,7], referred to by the collective name of λσ- calculus, have been recently introduced to provide an explicit treatment of substitutions in the λ-calculus. They are term rewriting systems, with two sorts: substitution and term. The λ-terms are exactly the ground λσ-terms of sort term containing no substitutions and the β-reduction is decomposed in these calculi, into a starting reduction with a rule called (Beta) followed by a derivation computing explicitly the substitution. These calculi differ by their treatment of substitution. In this paper, we extend the λσ-calculi with a conditional rewriting relation, called cη. This relation coincides, on λ-terms, with the classicalη-reduction of λ-calculus. We prove that the confluent λσ-calculus, augmented bycη, remains confluent and that the ground confluent version [1], extended by cη, is still ground confluent. The result extends readily to Categorical Combinatory Logic. The proof is done by the interpretation method introduced in [9].
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This work has been partially supported by the Eureka Software Factory project.
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Hardin, T. Eta-conversion for the languages of explicit substitutions. AAECC 5, 317–341 (1994). https://doi.org/10.1007/BF01188746
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DOI: https://doi.org/10.1007/BF01188746