Partial Computations in Constructive

Constructive type theory as conceived by Per Martin-LL of has a very rich type system, but partial functions cannot be typed. This also makes it impossible to directly write recursive programs. In this paper a constructive type theory Red is deened which includes a partial type constructor A; objects in the type A may diverge, but if they converge, they must be members of A. A xed point typing principle is given to allow typing of recursive functions. The extraction paradigm of type theory, whereby programs are automatically extracted from constructive proofs, is extended to allow extraction of xed points. There is a Scott xed point induction principle for reasoning about these functions. Soundness of the theory is proven. Type theory becomes a more expressive programming logic as a result.