Calculi of Generalised -Reduction and Explicit Substitutions: The Type Free and Simply Typed Versions

Extending the-calculus with either explicit substitution or generalised reduction has been the subject of extensive research recently and still has many open problems. This paper is the rst investigation into the properties of a calculus combining both generalised reduction and explicit substitutions. We present a calculus, gs, that combines a calculus of explicit substitution, s, and a calculus with generalized reduction, g. We believe that gs is a useful extension of the-calculus because it allows postponment of work in two diierent but complementary ways. Moreover, gs (and also s) satisses desirable properties of calculi of explicit substitutions and generalised reductions. In particular, we show that gs preserves strong normalisation, is a conservative extension of g, and simulates-reduction of g and the classical-calculus. Furthermore, we study the simply typed versions of s and gs and show that well typed terms are strongly normalising and that other properties such as typing of subterms and subject reduction hold. Our proof of the preservation of strong normalisation (PSN) is based on the minimal derivation method. It is however much simpler because we prove the commutation of arbitrary internal and external reductions. Moreover, we use one proof to show both the preservation of-strong normalisation in s and the preservation of g-strong normalisation in gs. We remark that the technique of these proofs is not suitable for calculi without explicit substitutions (e.g. the preservation of-strong normalisation in g requires a diierent technique).