Reducibility Proofs in the λ-Calculus

Abstract

Reducibility, despite being quite mysterious and inflexible, has been used to prove a number of properties of the λ-calculus and is well known to offer general proofs which can be applied to a number of instantiations. In this paper, we look at two related but different results in λ-calculi with intersection types. 1. 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. We provide a proposal to partially repair the method. 2. We consider a second result whose purpose is to use reducibility for typed terms in order to show the Church-Rosser of β-developments for the untyped terms and hence the Church-Rosser of β-reduction. In this second result, strong normalisation is not needed. We extend the second result to encompass both βI-and βη-reduction rather than simply β-reduction.