A realizability interpretation of Church's simple theory of types

We present a realizability interpretation of an intuitionistic version of Church's Simple Theory of Types (CST) which can be viewed as a formalization of intuitionistic higher-order logic. Although definable in CST we include operators for monotone induction and coinduction and provide simple realizers for them. Realizers are formally represented in an untyped lambda-calculus with pairing and case-construct. We introduce a general notion of interpretation of one instance of the simply typed lambda calculus in another, and define realizability as an instance of such an interpretation. In this way, important syntactic properties of realizability (e.g. being well-behaved w.r.t. substitution) can be proven elegantly on an abstract lambda-calculus level