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(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
(*i $Id: ClassicalChoice.v 6401 2004-12-05 16:44:57Z herbelin $ i*)
(** This file provides classical logic and functional choice *)
(** This file extends ClassicalDescription.v with the axiom of choice.
As ClassicalDescription.v, it implies the double-negation of
excluded-middle in Set and implies a strongly classical
world. Especially it conflicts with impredicativity of Set, knowing
that true<>false in Set.
*)
Require Export ClassicalDescription.
Require Export RelationalChoice.
Require Import ChoiceFacts.
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)).
Proof.
intros A B.
apply description_rel_choice_imp_funct_choice.
exact (description A B).
exact (relational_choice A B).
Qed.
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