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(*         *   The Coq Proof Assistant / The Coq Development Team       *)
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(*         *     GNU Lesser General Public License Version 2.1          *)
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(** This file provides a constructive form of indefinite description that
    allows building choice functions; this is weaker than Hilbert's
    epsilon operator (which implies weakly classical properties) but
    stronger than the axiom of choice (which cannot be used outside
    the context of a theorem proof). *)

Require Import ChoiceFacts.

Set Implicit Arguments.

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 }.
Proof.
  intros; apply constructive_indefinite_description; firstorder.
Qed.

Lemma functional_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)).
Proof.
  apply constructive_indefinite_descr_fun_choice.
  exact constructive_indefinite_description.
Qed.