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Reducibility Proofs in the λ-Calculus

Kamareddine F., Rahli V., Wells J. B.

Fundamenta Informaticae

2012

Vol. 121, nr 1/4

121-152

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 &lbeta;-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.

Church–Rosser Made Easy

Kozen D.

Fundamenta Informaticae

2010

Vol. 103, nr 1-4

129-136

The Church–Rosser theorem states that the λ-calculus is confluent under β-reductions. The standard proof of this result is due to Tait and Martin-Löf. In this note, we present an alternative proof based on the notion of acceptable orderings. The technique is easily modified to give confluence of the βη-calculus.