An adequate theory of partial computable functions should provide a basis for defining computational complexity measures and should justify the principle of computational induction for reasoning about programs on the basis of their recursive calls. There is no practical account of these notions in type theory, and consequently such concepts are not available in applications of type theory where they are greatly needed. It is also not clear how to provide a practical and adequate account in programming logics based on set theory. This paper provides a practical theory supporting all these concepts in the setting of constructive type theories. We first introduce an extensional theory of partial computable functions in type theory. We then add support for intensional reasoning about programs by explicitly reflecting the essential properties of the underlying computation system. We use the resulting intensional reasoning tools to justify computational induction and to define computationa...

In the context of possibly in nite computations yielding finite or infinite (binary) outputs, the space 2 6! =2 ∗ ∪ 2! appears to be one of the most fundamental spaces in Computer Science. Though underconsidered, next to 2! , this space can be viewed (Section 3.5.2) as the simplest compact space native to computer science. In this paper we study some of its properties involving topology and computability. Though 2 6! can be considered as a computable metric space in the sense of computable analysis, a direct and self-contained study, based on its peculiar properties related to words, is much illuminating. It is well known that computability for maps 2! → 2! reduces to continuity with recursive