We introduce an extension of the Calculus of Construction with inductive and co-inductive types that preserves strong normalisation for a lazy computation relation. This extension considerably enlarges the expressiveness of the language, enabling a direct translation of recursive programs, while keeping a relatively simple collection of typing rules. 1 Introduction The last twenty five years have seen an increasing development of different proof environments based on type theory. Several type theories have been proposed as a foundation of such proof environments [15, 6, 16], trying to find an accurate compromise between two criteria. On the one hand, we search for extensions of type theory that preserve its conceptual simplicity of type theory (a few primitive constructions, a small number of typing rules) and meta-theoretical properties ensuring its soundness and a direct mechanisation (strong normalisation, decidability of type-checking, etc). On the other hand, we would like to pro...