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Some new calculi [1, 12, 8], 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 classical η-reduction of λ-calculus. We prove that the confluent λσ-calculus, augmented by cη, remains confluent and that the ground confluent version[1], extended by cη, is still ground confluent. The proof is done by the interpretation method introduced in [10].

Eta-conversion for the languages of explicit substitutions