Characterizing Strongly Normalizing Terms of a Calculus with Generalized Applications via Intersection Types

An intersection type assignment system for the extension ΛJ of the untyped λ-calculus, introduced by Joachimski and Matthes, is given and proven to characterize the strongly normalizing terms of ΛJ. Since ΛJ’s generalized applications naturally allow permutative/commuting conversions, this is the first analysis of a term rewrite system with permutative conversions by help of intersection types. Two proofs are given for the fact that the typable terms are strongly normalizing: One by the computability predicates method à la Tait and one showing directly that strongly normalizing typable terms are closed under (generalized) application and substitution. It is also shown that a straightforward extension of the type assignment for λ-calculus fails to capture the strongly normalizing terms.