Hereditary Substitution for the λΔ-Calculus

Abstract

Hereditary substitution is a form of type-bounded iterated substitution, first made explicit by Watkins et al. and Adams in order to show normalization of proof terms for various constructive logics. This paper is the first to apply hereditary substitution to show normalization of a type theory corresponding to a non-constructive logic, namely the lambda-Delta calculus as formulated by Rehof. We show that there is a non-trivial extension of the hereditary substitution function of the simply-typed lambda calculus to one for the lambda-Delta calculus. Then hereditary substitution is used to prove normalization.