Calculus of Constructions

In mathematical logic and computer science, the calculus of constructions (CoC) is a type theory created by Thierry Coquand. It can serve as both a typed programming language and as constructive foundation for mathematics. For this second reason, the CoC and its variants have been the basis for Coq and other proof assistants.

Some of its variants include the calculus of inductive constructions (which adds inductive types), the calculus of (co)inductive constructions (which adds coinduction), and the predicative calculus of inductive constructions (which removes some impredicativity).

The CoC is a higher-order typed lambda calculus, initially developed by Thierry Coquand. It is well known for being at the top of Barendregt's lambda cube. It is possible within CoC to define functions from, say, integers to types, types to types as well as functions from integers to integers.

The CoC is strongly normalizing, although, by Gödel's incompleteness theorem, it is impossible to prove this property within the CoC since it implies inconsistency.

The CoC has developed alongside the Coq proof assistant. As features were added (or possible liabilities removed) to the theory, they became available in Coq.

Variants of the CoC are used in other proof assistants, such as Matita.

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