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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) *)
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(*    //   *    This file is distributed under the terms of the         *)
(*         *     GNU Lesser General Public License Version 2.1          *)
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(** 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.
Set Implicit Arguments.

(* Specifications: *)
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 }.