Hybrid Partial-Total Type Theory

In this paper a hybrid type theory HTT is deened which combines the programming language notion of partial type with the logical notion of total type into a single theory. A new partial type constructor A is added to the type theory: objects in A may diverge, but if they converge, they must be members of A. A xed point typing rule is given to allow for typing of xed points. The underlying theory is based on ideas from Feferman's Class Theory and Martin LL of's Intuitionistic Type Theory. The extraction paradigm of constructive type theory is extended to allow direct extraction of arbitrary xed points. Important features of general programming logics such as LCF are preserved, including the typing of all partial functions, a partial ordering < on computations, and a xed point induction principle. The resulting theory is thus intended as a general-purpose programming logic. Rules are presented and soundness of the theory established.