Confluence and preservation of strong normalisation in an explicit substitutions calculus

Explicit substitutions calculi are formal systems that implement β-reduction by means of an internal substitution operator. In that calculi it is possible to delay the application of a substitution to a term or to consider terms with partially applied substitutions. The λ_σ-calculus of explicit substitutions, proposed by M. Abadi et al. (1991), is a first-order rewriting system that implements substitution and renaming mechanism of λ-calculus. However; λ_σ does not preserve strong normalisation of λ-calculus and it is not a confluent system. Typed variants of λ_σ without composition are strongly normalising but not confluent, while variants with composition are confluent but do not preserve strong normalisation. Neither of them enjoys both properties. In this paper we propose the λ_ζ -calculus. This is, as far as we know, the first confluent calculus of explicit substitutions that preserves strong normalisation