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).
General traits.
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 terms to terms, as well as terms to types, types to types, and types to terms.
The CoC is strongly normalizing, although it is impossible to prove this property within the CoC since it implies consistency, which by Gödel's incompleteness theorem is impossible to prove from within the system itself.
Usage.
The CoC has been 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.
The basics of the calculus of constructions.
The calculus of constructions can be considered an extension of the Curry–Howard isomorphism. The Curry–Howard isomorphism associates a term in the simply typed lambda calculus with each natural-deduction proof in intuitionistic propositional logic. The calculus of constructions extends this isomorphism to proofs in the full intuitionistic predicate calculus, which includes proofs of quantified statements (which we will also call "propositions").
Terms.
A "term" in the calculus of constructions is constructed using the following rules:
In other words, the term syntax, in BNF, is then:
The calculus of constructions has five kinds of objects:
Judgments.
The calculus of constructions allows proving typing judgments:
Which can be read as the implication
The valid judgments for the calculus of constructions are derivable from a set of inference rules. In the following, we use formula_20 to mean a sequence of type assignments 
formula_21; formula_22 to mean terms; and formula_23 to mean either formula_2 or formula_1. We shall write formula_26 to mean the result of substituting the term formula_27 for the free variable formula_9 in the term formula_5.
An inference rule is written in the form
which means
Inference rules for the calculus of constructions.
1. formula_33
2. formula_33
3. formula_35
4. formula_36
5. formula_37
6. formula_38
Defining logical operators.
The calculus of constructions has very few basic operators: the only logical operator for forming propositions is formula_39. However, this one operator is sufficient to define all the other logical operators:
Defining data types.
The basic data types used in computer science can be defined within the calculus of constructions:
Note that Booleans and Naturals are defined in the same way as in Church encoding. However, additional problems arise from propositional extensionality and proof irrelevance.