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(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2014 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
(** This file provides a constructive form of indefinite description that
allows to build 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.
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