Editing Calculus of constructions (section)

===Defining data types===

The basic data types used in computer science can be defined within the calculus of constructions:

; Booleans : <math>\forall A: \mathbf{P} . A \Rightarrow A \Rightarrow A</math>
; Naturals : <math>\forall A: \mathbf{P} . 
(A \Rightarrow A) \Rightarrow A \Rightarrow A</math>
; Product <math>A \times B</math> : <math>A \wedge B</math>
; Disjoint union <math>A + B</math> : <math>A \vee B</math>

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.<ref name=":0">{{Cite web|title=Standard Library {{!}} The Coq Proof Assistant|url=https://coq.inria.fr/stdlib/Coq.Logic.ClassicalFacts.html|access-date=2020-08-08|website=coq.inria.fr}}</ref>