Intersection Types and Termination Properties

Abstract

We use the system of intersection types and the type assignment method to prove ter- mination properties in λ-calculus. In the first part we deal with conservation properties. We give a type assignment proof of the classical conservation theorem for λI calculus and then we extend this method to the notion of the reduction βI and βS of de Groote (9). We also give a direct type assignment proof of the extended conservation property according to which if a term is βI , βS- normalizable then it is β-strongly normalizable. We further extend the conservation theorem by introducing the notion of β -normal form. In the second part we prove that if Ω is not a substring of a λ-term M then M can be typed in the Krivine's system D of intersection types. In that way we obtain a type assignment proof of the Sorensen's Ω-theorem.