Combining Proofs and Programs in a Dependently Typed Language ( With technical appendix )

Most dependently-typed p rogramming languages either require that all expressions terminate (e.g. Coq, Agda, and Epigram), or allow infinite loops but are inconsistent when viewed as logics (e.g. Haskell, ATS, Ωmega). Here, we combine these two approaches into a single dependently-typed core language. The language is composed of two fragments t hat share a common syntax and overlapping semantics: a logic that guarantees total correctness, and a call-by-value programming language that guarantees type safety but not termination. The two fragments may interact: logical expressions may b e used as programs; the logic may soundly r eason about potentially nonterminating p rograms; programs can require logical proofs as arguments; and “mobile” program values, including p roofs computed at runtime, may be used as evidence b y the logic. This language allows programmers to work with total and partial functions uniformly, p roviding a smooth path from functional programming t o dependently-typed programming. Categories and Subject D escriptors D.3. 1 [Programming L anguages]: Formal Definitions and T heory