Contents

# Contents

## Idea

The Calculus of Constructions (CoC) is a type theory formal system for constructive proof in natural deduction style. This calculus goes back to Thierry Coquand and Gérard Huet

When supplemented by inductive types, it become the Calculus of Inductive Constructions (CIC). Sometimes coinductive types are included as well and one speaks of the Calculus of (co)inductive Constructions. This is what the Coq software implements.

More in detail, the Calculus of (co)Inductive Constructions is

1. a pure type system, hence

1. a system of natural deduction with dependent types;

2. with the natural-deduction rule for dependent product types specified;

2. with a rule for how to introduce new natural-deduction rules for arbitrary (co)inductive types.

3. and with universes:

a cumulative hierarchy of predicative types of types

and an impredicative type of propositions.

All of the other standard type formations are subsumed by the existence of arbitrary inductive types, notably the empty type, dependent sum types and identity types are special inductive types. Specifying these hence makes the calculus of constructions be an intensional dependent type theory.

## References

Original articles include

Surveys include

A categorical semantics for CoC is discussed in

• Martin Hyland, Andy Pitts, The Theory of Constructions: Categorical Semantics and Topos-theoretic Models , Cont. Math. 92 (1989) pp.137-199. (draft)

For specifics of the implementation in Coq see

Last revised on November 2, 2017 at 10:25:32. See the history of this page for a list of all contributions to it.