Confluence properties of weak and strong calculi of explicit substitutions

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Abstract
Categorical combinators [Curien 1986/1993; Hardin 1989; Yokouchi 1989] and more recently λσ-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]: λσ-calculus is not confluent (it is confluent on ground terms only). This unfortunate result is “repaired” by presenting a confluent version of λσ-calculus, named the λ Env -caldulus in Hardin and Le´vy [1989], called here the confluent λσ-calculus.