Extending a -calculus with Explicit Substitution Which Preserves Strong Normalisation into a Connuent Calculus on Open Terms

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 (Kamareddine & R os, 1995a), 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 (Kamareddine & R os, 1995a), is extended in this paper to a connuent calculus on open terms: the se-caculus. Since the establishment of these results, another calculus, , came into being in (Mu~ noz Hurtado, 1996) which preserves strong normalisation and is itself connuent on open terms. However, we believe that se 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 generalisation of the interpretation method (cf. (Hardin, 1989) and (Curien et al., 1992)) to a technique which uses weak normal forms (instead of strong ones). We consider that this extended method is a useful tool to obtain connuence when strong normalisation of the subcalculus of substitutions is not available. In our case, strong normalisation of the corresponding subcalculus of substitutions se, is still a challenging open problem to the rewrite community but its weak normalisation is established here via an eeective strategy.