Editing Calculus of constructions (section)

===Terms===

A ''term'' in the calculus of constructions is constructed using the following rules:

* <math>\mathbf{T}</math> is a term (also called ''type'');
* <math>\mathbf{P}</math> is a term (also called ''prop'', the type of all propositions);
* Variables (<math>x, y, \ldots</math>) are terms;
* If <math>A</math> and <math>B</math> are terms, then so is <math>(A B)</math>;
* If <math>A</math> and <math>B</math> are terms and <math>x</math> is a variable, then the following are also terms:
** <math>(\lambda x:A. B)</math>,
** <math>(\forall x:A. B)</math>.

In other words, the term syntax, in [[Backus–Naur form]], is then:
:<math>e ::= \mathbf{T} \mid \mathbf{P} \mid x \mid e \, e \mid \lambda x\mathbin{:}e.e\mid \forall x\mathbin{:}e.e</math>

The calculus of constructions has five kinds of objects:
# ''proofs'', which are terms whose types are ''propositions'';
# ''propositions'', which are also known as ''small types'';
# ''predicates'', which are functions that return propositions;
# ''large types'', which are the types of predicates (<math>\mathbf{P}</math> is an example of a large type);
# <math>\mathbf{T}</math> itself, which is the type of large types.