(* 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_inclus.v                               *)
(****************************************************************************)
(*              Formal Language Theory                                      *)
(*                                                                          *)
(*              Judicael Courant - Jean-Christophe Filliatre                *)
(*                                                                          *)
(*              Developped in V5.8  June-July 1993                          *)
(*              Ported     to V5.10 October 1994                            *)
(****************************************************************************)
(*									*)
(*  INCLUSION  								*)
(*    On definit le predicat (inclus E F) par (dans x E)->(dans x F).	*)
(*    On montre ensuite facilement les resultats suivants :		*)
(*    -- inclus empty A							*)
(*    -- inclus A A							*)
(*    -- (inclus a b)->(inclus c d)					*)
(*		->(inclus (prodcart a c) (prodcart b d)			*)
(*									*)
Definition inclus (A B : Ensf) : Prop := forall x : Elt, dans x A -> dans x B.
Hint Unfold inclus.
unfold inclus in |- *; intros.
absurd (dans x0 empty); auto.
Qed.
Hint Resolve empty_inclus.
auto.
Qed.
Hint Resolve refl_inclus.
auto.
Qed.
unfold inclus in |- *.
intros.
cut
(exists x1 : Elt,
(exists x2 : Elt, dans x1 a /\ dans x2 c /\ x = couple x1 x2)).
2: apply coupl3; auto.
intro H2; elim H2; clear H2.
intros x1 H2; elim H2; clear H2.
intros x2 H2; elim H2; clear H2.
intros H2 H3; elim H3; clear H3.
intros H3 H4.
rewrite H4.
auto.
Qed.
Hint Resolve cart_inclus.
auto.
Qed.
Hint Resolve inclus_add.
unfold inclus in |- *.
intros e a x H.
cut (x = e); auto.
intro H0.
rewrite H0; auto.
Qed.
Hint Resolve singl_inclus_add.
auto.
Qed.

Lemma add_inclus :
forall (x : Elt) (a b : Ensf), dans x b -> inclus a b -> inclus (add x a) b.