On the Role of Type Decorations in the Calculus of Inductive Constructions.

In proof systems like Coq [16], proof-checking involves comparing types modulo β-conversion, which is potentially a time-consuming task. Significant speed-ups are achieved by compiling proof terms, see [9]. Since compilation erases some type information, we have to show that convertibility is preserved by type erasure. This article shows the equivalence of the Calculus of Inductive Constructions (formalism of Coq) and its domain-free version where parameters of inductive types are also erased. It generalizes and strengthens significantly a similar result by Barthe and Sørensen [5] on the class of functional Domain-free Pure Type Systems. [ABSTRACT FROM AUTHOR]