Reducibility: A ubiquitous method in lambda calculus with intersection types

Abstract

A general reducibility method is developed for proving reduction properties of lambda terms typeable in intersection type systems with and without the universal type Ω. Sufficient conditions for its application are derived. This method leads to uniform proofs of confluence, standardization, weak head normalization of terms typeable in the system with the universal type Ω. The method extends Tait's reducibility method for the proof of strong normalization of the simply typed lambda calculus, Krivine's extension of the same method for the strong normalization of intersection type system without Ω, and Statman-Mitchell's logical relation method for the proof of confluence of β-η-reduction on the simply typed lambda terms. As a consequence, the confluence and the standardization of all (untyped) lambda terms is obtained. © 2002 Published by Elsevier Science B.V.