Programs and types

The first two sections of this paper motivate and outline a constructive theory of (data) types which we developed for formal program verification. The executable component of the theory provides a very high level programming language with a rich type structure. A theory of this generality appears necessary to manage complex programming and formal reasoning about it. The logical component, influencea by AUTOMATH and LCF and based on Martin-Löf\´s ITT, appears strong enough to formalize constructive mathematics; hence a theory or this generality is probably sufficient for program development and verification. The last two sections of the paper illustrate the richness of the theory and the benefits of generality by describing with it different "denotational" semantics for programs. Because the theory is constructive, these abstract semantics are also computational.