The Implicit Calculus of Constructions

In this paper, we introduce a new type system, the Implicit Calculus of Constructions, which is a Curry-style variant of the Calculus of Constructions that we extend by adding an intersection type binder-called the implicit dependent product. Unlike the usual approach of Type Assignment Systems, the implicit product can be used at every place in the universe hierarchy. We study syntactical properties of this calculus such as the βη-subject reduction property, and we show that the implicit product induces a rich subtyping relation over the type system in a natural way. We also illustrate the specificities of this calculus by revisiting the impredicative encodings of the Calculus of Constructions, and we show that their translation into the implicit calculus helps to reflect the computational meaning of the underlying terms in a more accurate way.