# Library Coq.Logic.ClassicalDescription

This file provides classical logic and definite description, which is equivalent to providing classical logic and Church's iota operator
Classical logic and definite descriptions implies excluded-middle in Set and leads to a classical world populated with non computable functions. It conflicts with the impredicativity of Set
Require Export Classical.
Require Import ChoiceFacts.
Notation Local inhabited A := A (only parsing).
Axiom constructive_definite_description :
forall (A : Type) (P : A->Prop), (exists ! x : A , P x) -> { x : A | P x }.
The idea for the following proof comes from ChicliPottierSimpson02
Theorem excluded_middle_informative : forall P:Prop, { P } + { ~ P }.
Theorem classical_definite_description :
forall (A : Type) (P : A->Prop), inhabited A ->
{ x : A | (exists ! x : A , P x) -> P x }.
Church's iota operator
Definition iota (A : Type) (i:inhabited A) (P : A->Prop) : A
:= proj1_sig (classical_definite_description P i).
Definition iota_spec (A : Type) (i:inhabited A) (P : A->Prop) :
(exists ! x : A , P x) -> P (iota i P)
:= proj2_sig (classical_definite_description P i).
Axiom of unique "choice" (functional reification of functional relations)
Theorem dependent_unique_choice :
forall (A:Type) (B:A -> Type) (R:forall x:A, B x -> Prop),
(forall x:A, exists ! y : B x , R x y) ->
(exists f : ( forall x:A, B x ), forall x:A, R x (f x)).
Theorem unique_choice :
forall (A B:Type) (R:A -> B -> Prop),
(forall x:A, exists ! y : B , R x y) ->
(exists f : A -> B , forall x:A, R x (f x)).
Compatibility lemmas
Definition dependent_description := dependent_unique_choice.
Definition description := unique_choice.