Application of typed lambda calculi in the untyped lambda calculus

Abstract

We discuss some properties of typed lambda calculi which can be related and applyed to the proofs of some properties of the untyped lambda calculus. The strong normalization property of the intersection type assignment system is used in order to prove the finitness of developments property of the untyped lambda calculus in Krivine (1990). Similarly, the strong normalization property of the simply typed lambda calculus can be used for the same reason. Typability in various intersection type assignment systems characterizes tambda terms in normal form, normalizing, solvable and unsolvable terms. Hence, its application in the proof of the Genericity Lemma turns out to be appropriate.