(************************************************************************)
(*         *      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)         *)
(************************************************************************)
(****************************************************************************)
(*                                                                          *)
(*                         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.  *)
(****************************************************************************)
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.
#[global]
Hint Resolve Empty_is_finite Union_is_finite: sets.
#[global]
Hint Resolve card_empty card_add: 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.