The Connuence of the S E -calculus via a Generalized Interpretation Method

The last fteen years have seen an explosion in work on explicit substitution, most of which is done in the style of the-calculus. In KR95a], we extended the-calculus with explicit substitutions by turning de Bruijn's meta-operators into object-operators ooering a style of explicit substitution that diiers from that of. The resulting calculus , s, remains as close as possible to the-calculus from an intuitive point of view and, while preserving strong normalisation ((KR95a]), is extended in this paper to a connuent calculus on open terms: the s e-caculus. Since the establishment of the results of this paper 1 , another calculus, , came into being in MH95] which preserves strong normal-isation and is itself connuent on open terms. However, we believe that s e still deserves attention because, while ooering a new style to work with explicit substitutions, it is able to simulate one step of classical-reduction, whereas is not. To prove connuence we introduce a generalization of the interpretation method (cf. Har89] and CHL92]) to a technique which uses weak normal forms (instead of strong ones). This technique is general enough to apply to many reduction systems and we consider it as a powerful tool to obtain connuence. Strong normalisation of the corresponding calculus of substitutions s e , is left as a challenging problem to the rewrite community but its weak normalisation is established via an eeective strategy.