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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: Epsilon.v 10170 2007-10-03 14:41:25Z herbelin $ i*)
+
+(** This file provides indefinite description under the form of
+    Hilbert's epsilon operator; it does not assume classical logic. *)
+
+Require Import ChoiceFacts.
+
+Set Implicit Arguments.
+
+(** Hilbert's epsilon: operator and specification in one statement *)
+
+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 }.
+Proof.
+  apply epsilon_imp_constructive_indefinite_description.
+  exact epsilon_statement.
+Qed.
+
+Lemma small_drinkers'_paradox :
+  forall (A:Type) (P:A -> Prop), inhabited A ->
+    exists x, (exists x, P x) -> P x.
+Proof.
+  apply epsilon_imp_small_drinker.
+  exact epsilon_statement.
+Qed.
+
+Theorem iota_statement :
+  forall (A : Type) (P : A->Prop), inhabited A ->
+  { x : A | (exists! x : A, P x) -> P x }.
+Proof.
+  intros; destruct epsilon_statement with (P:=P); firstorder.
+Qed.
+
+Lemma constructive_definite_description :
+  forall (A : Type) (P : A->Prop), 
+    (exists! x, P x) -> { x : A | P x }.
+Proof.
+  apply iota_imp_constructive_definite_description.
+  exact iota_statement.
+Qed.
+
+(** Hilbert's epsilon operator and its specification *)
+
+Definition epsilon (A : Type) (i:inhabited A) (P : A->Prop) : A
+  := proj1_sig (epsilon_statement P i).
+
+Definition epsilon_spec (A : Type) (i:inhabited A) (P : A->Prop) : 
+  (exists x, P x) -> P (epsilon i P)
+  := proj2_sig (epsilon_statement P i).
+
+(** Church's iota operator and its specification *)
+
+Definition iota (A : Type) (i:inhabited A) (P : A->Prop) : A
+  := proj1_sig (iota_statement P i).
+
+Definition iota_spec (A : Type) (i:inhabited A) (P : A->Prop) : 
+  (exists! x:A, P x) -> P (iota i P)
+  := proj2_sig (iota_statement P i).
+