Infinite intersection types

A type theory with infinitary intersection and union types for an extension of the λ-calculus is introduced. Types are viewed as upper closed subsets of a Scott domain and intersection and union type constructors are interpreted as the set-theoretic intersection and union, respectively, even when they are not finite. The assignment of types to λ-terms extends naturally the basic type assignment system. We prove soundness and completeness using a generalization of Abramsky’s finitary logic of domains. Finally, we apply the framework to applicative transition systems, obtaining a sound a complete infinitary intersection type assignment system for the lazy λ-calculus.