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(*         *      The Rocq Prover / The Rocq Development Team           *)
(*  v      *         Copyright INRIA, CNRS and contributors             *)
(* <O___,, * (see version control and CREDITS file for authors & dates) *)
(*   \VV/  **************************************************************)
(*    //   *    This file is distributed under the terms of the         *)
(*         *     GNU Lesser General Public License Version 2.1          *)
(*         *     (see LICENSE file for the text of the license)         *)
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(** This file provides classical logic and functional choice; this especially provides both indefinite descriptions and choice functions but this is weaker than providing epsilon operator and classical logic as the indefinite descriptions provided by the axiom of choice can be used only in a propositional context (especially, they cannot be used to build choice functions outside the scope of a theorem proof) *)
(** This file extends ClassicalUniqueChoice.v with full choice. As ClassicalUniqueChoice.v, it implies the double-negation of excluded-middle in [Set] and leads to a classical world populated with non computable functions. Especially it conflicts with the impredicativity of [Set], knowing that [true<>false] in [Set].  *)
Require Export ClassicalUniqueChoice.
Require Export RelationalChoice.
Set Implicit Arguments.
Definition subset (U:Type) (P Q:U->Prop) : Prop := forall x, P x -> Q x.

(* Specifications: *)
Theorem singleton_choice :
forall (A : Type) (P : A->Prop),
(exists x : A, P x) -> exists P' : A->Prop, subset P' P /\ exists! x, P' x.
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)).