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(* 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.
unfold inclus in |- *.
intros.
cut (x = x0 \/ dans x0 a).
2: apply dans_add; auto.
intro H2; elim H2; clear H2.
intro H2; rewrite <- H2; auto.
auto.
Qed.
Hint Resolve add_inclus.
auto.
Qed.

Lemma union_inclus :
forall a b c : Ensf, inclus a c -> inclus b c -> inclus (union a b) c.