On Specifications, Subset Types and Interpretation of Proposition in Type Theory

Type theory may be used as a programming language with an integrated programming logic. An implementation of the theory should then give an environment allowing you to specify, construct and verify programs interactively. We have used NUPRL [10], which is such an environment for type theory, to develop some examples. NUPRL’s logic is based on Martin-Löf’s type theory [7] and provides you with a proof system, a function which extracts programs from proofs, a program evaluator, a library system and a possibility to define your own syntax and proof tactics for (partial) automatization of proofs. The usual way to specify a program in type theory is by identifying the specification with a type and a program satisfying the specification is then an element of this type. For instance a specification of a function which, given a non-empty list, returns any member of that list may be given by the proposition: