The CoC provides the basic framework for Coq, while the Co(I)C or even the Calculus of (Co)-Inductive Constructions provides a dramatic extension in power and e

The CoC provides the basic framework for Coq, while the Co(I)C or even the Calculus of (Co)-Inductive Constructions provides a dramatic extension in power and expressivity necessary for "real" programming in Coq.