Abstract

Our purpose is to provide a typed framework for working with non terminating computations. The basic system is the Calculus of Constructions. It is extended using an original idea proposed by R.Constable and S.F.Smith and and implemented in Nuprl. From the computational point of view we get an equivalent of Kleene's theorem for partial recursive functions over the integers within an index-free setting. A larger class of algebraic types is dened. Logical aspects need more examination. But we already give a syntactic way for dealing with partial and total objects, leading to the notion of generic proof. 1 Introduction The Calculus of Constructions (CC for short) [8], [9] is a typed high-order functional calculus which provides a nice formalism for constructive proofs in natural deduction style. It can also be seen as a high-level functional programming language. Since F! is embeddable in it, we already know that any fonction over integers is denable on Nat (C : P rop)C ! (C ! C) !...

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