Realizability, Covers, and Sheaves II. Applications to the Second-Order Lambda-Calculus

We present a general method for proving properties of typed A-terms. This method is obtained by introducing a semantic notion of realizability which uses the notion of a cover algebra (as in abstract sheaf theory, a cover algebra being a Grothendieck topology in the case of a preorder). For this, we introduce a new class of semantic structures equipped with preorders, called pre-applicative structures. These structures need not be extensional. In this framework, a general realizability theorem can be shown. Applying this theorem to the special case of the term model, yields a general theorem for proving properties of typed A-terms, in particular, strong normalization and confluence. This approach clarifies the reducibility method by showing that the closure conditions on candidates of reducibility can be viewed as sheaf conditions. Part I1 of this paper applies the above approach to the second-order (polymorphic) A-calculus x'~"~ (with types + and V). 'This research was partially supported by ONR Grant N00014-88-K-0593.