Proving Properties of Typed Lambda Terms Using Realizability, Covers, and Sheaves
The main purpose of this paper is to take apart the reducibility method in order to understand how its pieces fit together, and in particular, to recast the conditions on candidates of reducibility as sheaf conditions. There has been a feeling among experts on this subject that it should be possible to present the reducibility method using more semantic means, and that a deeper understanding would then be gained. This paper gives mathematical substance to this feeling, by presenting a generalization of the reducibility method based on a semantic notion of realizability which uses the notion of a cover algebra (as in abstract sheaf theory). A key technical ingredient is the introduction of 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. Kleene's recursive realizability and a variant of Kreisel's modified realizability both fit into this framework. We are then able to prove a meta-theorem which shows that if a property of realizers satisfies some simple conditions, then it holds for the semantic interpretations of all terms. Applying this theorem to the special case of the term model, yields a general theorem for proving properties of typed λ-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. The above approach is applied to the simply-typed λ-calculus (with types →, x, +, and ┴), and to the second-order (polymorphic) λ-calculus (with types → and ∀2), for which it yields a new theorem.