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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 indefinite description under the form of Hilbert's epsilon operator; it does not assume classical logic. *)
Set Implicit Arguments.
(** Hilbert's epsilon: operator and specification in one statement *)

(* Specifications: *)
Axiom epsilon_statement :
forall (A : Type) (P : A->Prop), inhabited A ->
{ x : A | (exists x, P x) -> P x }.
Lemma constructive_indefinite_description :
forall (A : Type) (P : A->Prop),
(exists x, P x) -> { x : A | P x }.
Lemma small_drinkers'_paradox :
forall (A:Type) (P:A -> Prop), inhabited A ->
exists x, (exists x, P x) -> P x.
Theorem iota_statement :
forall (A : Type) (P : A->Prop), inhabited A ->
{ x : A | (exists! x : A, P x) -> P x }.
Lemma constructive_definite_description :
forall (A : Type) (P : A->Prop),
(exists! x, P x) -> { x : A | P x }.