Intersection and Union Type Assignment and Polarised λ̄μμ̃

Intersection and union type assignment systems are powerful tools for reasoning programs that completely characterise many semantic properties such as strong normalisation. At the same time, they are known to be subtle, particularly in the presence of computational effect. To address the difficulty, this paper develops an approach based on polarities and refinement intersection type systems. We introduce a simply-typed polarised calculus, in which a type is either positive or negative, based on Curien and Herbelin’s calculus λ̄μμ̃. This polarised calculus interacts well with intersection and union types: by adding intersection on positive types and union on negative types, we obtain a sound and complete type system. One can design an intersection and union type system for another calculus, guided by a translation to the polarised calculus. To demonstrate usefulness of the approach, we derive the first intersection and union type system for the call-by-value λ̄μμ̃, which satisfies expected properties.