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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        *)
+(************************************************************************)
+(****************************************************************************)
+(*                                                                          *)
+(*                         Naive Set Theory in Coq                          *)
+(*                                                                          *)
+(*                     INRIA                        INRIA                   *)
+(*              Rocquencourt                        Sophia-Antipolis        *)
+(*                                                                          *)
+(*                                 Coq V6.1                                 *)
+(*									    *)
+(*			         Gilles Kahn 				    *)
+(*				 Gerard Huet				    *)
+(*									    *)
+(*									    *)
+(*                                                                          *)
+(* Acknowledgments: This work was started in July 1993 by F. Prost. Thanks  *)
+(* to the Newton Institute for providing an exceptional work environment    *)
+(* in Summer 1995. Several developments by E. Ledinot were an inspiration.  *)
+(****************************************************************************)
+
+(*i $Id: Finite_sets.v,v 1.6.2.1 2004/07/16 19:31:17 herbelin Exp $ i*)
+
+Require Import Ensembles.
+
+Section Ensembles_finis.
+Variable U : Type.
+
+Inductive Finite : Ensemble U -> Prop :=
+  | Empty_is_finite : Finite (Empty_set U)
+  | Union_is_finite :
+      forall A:Ensemble U,
+        Finite A -> forall x:U, ~ In U A x -> Finite (Add U A x).
+
+Inductive cardinal : Ensemble U -> nat -> Prop :=
+  | card_empty : cardinal (Empty_set U) 0
+  | card_add :
+      forall (A:Ensemble U) (n:nat),
+        cardinal A n -> forall x:U, ~ In U A x -> cardinal (Add U A x) (S n).
+
+End Ensembles_finis.
+
+Hint Resolve Empty_is_finite Union_is_finite: sets v62.
+Hint Resolve card_empty card_add: sets v62.
+
+Require Import Constructive_sets.
+
+Section Ensembles_finis_facts.
+Variable U : Type.
+
+Lemma cardinal_invert :
+ forall (X:Ensemble U) (p:nat),
+   cardinal U X p ->
+   match p with
+   | O => X = Empty_set U
+   | S n =>
+        exists A : _,
+         (exists x : _, X = Add U A x /\ ~ In U A x /\ cardinal U A n)
+   end.
+Proof.
+induction 1; simpl in |- *; auto.
+exists A; exists x; auto.
+Qed.
+
+Lemma cardinal_elim :
+ forall (X:Ensemble U) (p:nat),
+   cardinal U X p ->
+   match p with
+   | O => X = Empty_set U
+   | S n => Inhabited U X
+   end.
+Proof.
+intros X p C; elim C; simpl in |- *; trivial with sets.
+Qed.
+
+End Ensembles_finis_facts.