# Library Coq.Logic.ClassicalEpsilon

This file provides classical logic and indefinite description under the form of Hilbert's epsilon operator
Hilbert's epsilon operator and classical logic 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.
Axiom constructive_indefinite_description :
forall (A : Type) (P : A->Prop),
(exists x , P x) -> { x : A | P x }.
Lemma constructive_definite_description :
forall (A : Type) (P : A->Prop),
(exists ! x , P x) -> { x : A | P x }.
Theorem excluded_middle_informative : forall P:Prop, { P } + { ~ P }.
Theorem classical_indefinite_description :
forall (A : Type) (P : A->Prop), inhabited A ->
{ x : A | (exists x , P x) -> P x }.
Hilbert's epsilon operator
Definition epsilon (A : Type) (i:inhabited A) (P : A->Prop) : A
:= proj1_sig (classical_indefinite_description P i).
Definition epsilon_spec (A : Type) (i:inhabited A) (P : A->Prop) :
(exists x , P x) -> P (epsilon i P)
:= proj2_sig (classical_indefinite_description P i).
Open question: is classical_indefinite_description constructively provable from relational_choice and constructive_definite_description (at least, using the fact that functional_choice is provable from relational_choice and unique_choice, we know that the double negation of classical_indefinite_description is provable (see relative_non_contradiction_of_indefinite_desc).
A proof that if P is inhabited, epsilon a P does not depend on the actual proof that the domain of P is inhabited (proof idea kindly provided by Pierre Castéran)
Lemma epsilon_inh_irrelevance :
forall (A:Type) (i j : inhabited A) (P:A->Prop),
(exists x , P x) -> epsilon i P = epsilon j P.
Opaque epsilon.

### Weaker lemmas (compatibility lemmas)

Theorem 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)).