A Constructive Logic of Multiple Subtyping

Abstract

We show how a higher order logic, the calculus of constructions, can be used to give a simple, first principles treatment of record calculi, polymorphism, and subtyping. The development follows the constructive idiom of extracting implementations of equationally specified programs from proofs of their termination, with a logic for reasoning about programs, and a semantics that comes as a gratuity. In this framework, records are finitely specified functions where equality is decidable over the domain, with types that are a particular kind of logical assertion. By proving that a record specification satisfies its type, we can extract its implementation. While program extraction serves as a sort of compiler, proof normalization serves as an interpreter; the latter serves to ensure in some sense the coherence of the translation embedded in the former.