Confluence properties of weak and strong calculi of explicit substitutions

Categorical combinators [Curien 1986/1993; Hardin 1989; Yokouchi 1989] and more recently λ&sgr;-calculus [Abadi 1991; Hardin and Le´vy 1989], have been introduced to provide an explicit treatment of substitutions in the λ-calculus. We reintroduce here the ingredients of these calculi in a self-contained and stepwise way, with a special emphasis on confluence properties. The main new results of the paper with respect to Curien [1986/1993], Hardin [1989], Abadi [1991], and Hardin and Le´vy [1989] are the following: (1) We present a confluent weak calculus of substitutions, where no variable clashes can be feared;(2) We solve a conjecture raised in Abadi [1991]: λ&sgr;-calculus is not confluent (it is confluent on ground terms only). This unfortunate result is “repaired” by presenting a confluent version of λ&sgr;-calculus, named the λEnv-caldulus in Hardin and Le´vy [1989], called here the confluent λ&sgr;-calculus.

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