On the relation between Church - style typing and Curry - style typing : Extended Abstract ∗

There are two versions of type assignment in λ-calculus: Church-style, in which the type of each variable is fixed, and Curry-style (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 Curry-style typing, λx.x is a general identity function with type C → C for every type C. In this paper, I will show how to interpret in a Curry-style system every Pure Type System (PTS) in the Church-style. (This generalizes some unpublished work with Garrel Pottinger.) ∗This work was supported in part by a grant from the Natural Sciences and Engineering Research Council of Canada. Some of the work in this talk was joint work with Garrel Pottinger. I will then show how to interpret in a system of the Church-style (a modified PTS) every PTSlike system in the Curry style.