Reducibility proofs in λ-calculi with intersection types

Reducibility has been used to prove a number of properties in the λ-calculus and is well known to offer on one hand very general proofs which can be applied to a number of instantiations, and on the other hand, to be quite mysterious and inflexible. In this paper, we look at two related but different results in λ-calculi with intersection types. We show that one such result (which aims at giving reducibility proofs of Church-Rosser, standardisation and weak normalisation for the untyped λ-calculus) faces serious problems which break the reducibility method and then we provide a proposal to partially repair the method. Then, we consider a second result whose purpose is to use reducibility for typed terms to show Church-Rosser of β-developments for untyped terms (without needing to use strong normalisation), from which Church-Rosser of β-reduction easily follows. We extend the second result to encompass both βIand βη-reduction rather than simply β-reduction.