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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 definite description, which is equivalent to providing classical logic and Church's iota operator *)
(** Classical logic and definite descriptions implies excluded-middle in [Set] and leads to a classical world populated with non computable functions. It conflicts with the impredicativity of [Set] *)
Set Implicit Arguments.
Require Export Classical.   (* Axiomatize classical reasoning *)
Require Export Description. (* Axiomatize constructive form of Church's iota *)
Local Notation inhabited A := A (only parsing).
(** The idea for the following proof comes from [ChicliPottierSimpson02] *)

(* Specifications: *)
Theorem excluded_middle_informative : forall P:Prop, {P} + {~ P}.
Theorem classical_definite_description :
forall (A : Type) (P : A->Prop), inhabited A ->
{ x : A | (exists! x : A, P x) -> P x }.