Redexes are stable in the λ-calculus

Abstract

In the theory of sequential programming languages, Berry's stability property is an important step claiming that values have unique origins in their calculations. It has been shown that Bohm trees are stable in the λ-calculus, meaning that there is a unique minimum prefix of any lambda term which computes its Bohm tree. Moreover this property is also true for finite prefixes of Bohm trees. The proof relies on Curry's standardization theorem as initially pointed out by Plotkin in his famous articleLCF considered as a programming language. In this paper we will show that the stability property also holds for redexes. Namely for any redex family, there is a unique minimum calculation and a unique redex which computes this family. This property was already known in the study of optimal reductions, but we stress here on stability and want to show that stability is inside the basic objects of calculations. The proof will be based on nice commutations between residuals and creations of new redexes. Our tool for proving this property will be the labelled lambda calculus used in the study of optimal reductions.