(* This program is free software; you can redistribute it and/or      *)
(* modify it under the terms of the GNU Lesser General Public License *)
(* as published by the Free Software Foundation; either version 2.1   *)
(* of the License, or (at your option) any later version.             *)
(*                                                                    *)
(* This program is distributed in the hope that it will be useful,    *)
(* but WITHOUT ANY WARRANTY; without even the implied warranty of     *)
(* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the      *)
(* GNU General Public License for more details.                       *)
(*                                                                    *)
(* You should have received a copy of the GNU Lesser General Public   *)
(* License along with this program; if not, write to the Free         *)
(* Software Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA *)
(* 02110-1301 USA                                                     *)
(* Contribution to the Coq Library   V6.3 (July 1999)                    *)
(****************************************************************************)
(*                 The Calculus of Inductive Constructions                  *)
(*                                                                          *)
(*                                Projet Coq                                *)
(*                                                                          *)
(*                     INRIA                        ENS-CNRS                *)
(*              Rocquencourt                        Lyon                    *)
(*                                                                          *)
(*                                Coq V5.10                                 *)
(*                              Nov 25th 1994                               *)
(*                                                                          *)
(****************************************************************************)
(*                               Ensf_dans.v                                *)
(****************************************************************************)
(*              Formal Language Theory                                      *)
(*                                                                          *)
(*              Judicael Courant - Jean-Christophe Filliatre                *)
(*                                                                          *)
(*              Developped in V5.8  June-July 1993                          *)
(*              Ported     to V5.10 October 1994                            *)
(****************************************************************************)
(*                                                                    *)
(*  APPARTENANCE :                                                    *)
(*    On definit le predicat (dans x E) pour un element x et un       *)
(*    ensemble E.                                                     *)
(*                                                                    *)
Inductive dans : Elt -> Ensf -> Prop :=
| dans_add1 : forall (x : Elt) (e : Ensf), dans x (add x e)
| dans_add2 : forall (x y : Elt) (e : Ensf), dans x e -> dans x (add y e).
Hint Resolve dans_add1 dans_add2.
(*  Quelques resultats concernant l'appartenance...  *)

Lemma dans_add :
forall (x y : Elt) (e : Ensf), dans x (add y e) -> y = x \/ dans x e.