The ∆-calculus: Syntax and Types

International audienceWe present the ∆-calculus, an explicitly typed λ-calculus with strong pairs, projections and explicit type coercions. The calculus can be parametrized with different intersection type theories T , e.g. the Coppo-Dezani, the Coppo-Dezani-Sallé, the Coppo-Dezani-Venneri and the Barendregt-Coppo-Dezani ones, producing a family of ∆-calculi with related intersection typed systems. We prove the main properties like Church-Rosser, unicity of type, subject reduction, strong normalization, decidability of type checking and type reconstruction. We state the relationship between the intersection type assignment systems à la Curry and the corresponding intersection typed systems à la Church by means of an essence function translating an explicitly typed ∆-term into a pure λ-term one. We finally translate a ∆-term with type coercions into an equivalent one without them; the translation is proved to be coherent because its essence is the identity. The generic ∆-calculus can be parametrized to take into account other intersection type theories as the ones in the Barendregt et al. book