Bridging Curry and Church's typing style

There are two versions of type assignment in the λ-calculus: Church-style, in which the type of each variable is fixed, and Currystyle (also called “domain free”), in which it is not. As an example, in Church-style typing, λx:A.x is the identity function on type A, and it has type A → A but not B → B for a type B different from A. In Currystyle typing, λx.x is a general identity function with type C → C for every type C. In this paper, we will show how to interpret in a Currystyle system every Pure Type System (PTS) in the Church-style without losing any typing information. We will also prove a kind of conservative extension result for this interpretation, a result which implies that for most consistent PTSs of the Church-style, the corresponding Curry-style system is consistent. We will then show how to interpret in a system of the Church-style (a modified PTS, stronger than a PTS) every PTS-like system in the Curry style.