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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_produit.v                              *)
(****************************************************************************)
(*              Formal Language Theory                                      *)
(*                                                                          *)
(*              Judicael Courant - Jean-Christophe Filliatre                *)
(*                                                                          *)
(*              Developped in V5.8  June-July 1993                          *)
(*              Ported     to V5.10 October 1994                            *)
(****************************************************************************)
(*									*)
(*  PRODUIT CARTESIEN 							*)
(*    On definit ici le produit cartesien de 2 ensembles. Pour cela     *)
(*    on commence par definir (singleprod x E) qui est l'ensemble	*)
(*    des couples (x,y) pour y dans E, puis (prodcart E F) qui est	*)
(*    l'union des (singleprod x F) pour x dans E.			*)
(*									*)
(* Fixpoint singleprod [x:Elt; A:Ensf] : Ensf := (<Ensf>Case A of (* empty *) empty (* add   *) [y:Elt][e:Ensf] (add (couple x y) (singleprod x e)) end ). *)
Fixpoint singleprod (x : Elt) (A : Ensf) {struct A} : Ensf :=
match A with
| empty => empty
| add y e => add (couple x y) (singleprod x e)
end.
(* Fixpoint prodcart [A:Ensf] : Ensf -> Ensf := [B:Ensf] (<Ensf>Case A of (* empty *) empty (* add   *) [x:Elt][e:Ensf] (union (singleprod x B) (prodcart e B)) end ). *)
Fixpoint prodcart (A : Ensf) : Ensf -> Ensf :=
fun B : Ensf =>
match A with
| empty => empty
| add x e => union (singleprod x B) (prodcart e B)
end.
(*  On montre en premier que si (x,y) est dans (singleprod x0 b) alors	*)
(*  on a x=x0 et y dans b						*)
intros x y x0.
simple induction b.
simpl in |- *.
intro.
apply (dans_empty_imp_P (couple x y)); auto.
intros a b0 H.
simpl in |- *.
intro.
cut (couple x0 a = couple x y :>Elt \/ dans (couple x y) (singleprod x0 b0)).
2: apply dans_add; auto.
intro H1; elim H1; clear H1.
intro H1.
injection H1; intros.
split; auto.
rewrite H2; auto.
intro.
cut (x = x0 :>Elt /\ dans y b0); auto.
intro H2; elim H2; clear H2.
intros.
split; auto.
Qed.
(*  On peut ensuite en deduire que si (x,y) est dans AxB alors 		*)
(*  x est dans A et y est dans B.					*)
intros x y.
simple induction a.
intro b.
simpl in |- *.
intro.
apply (dans_empty_imp_P (couple x y)); auto.
intros a0 b H b0.
simpl in |- *.
intro.
cut
(dans (couple x y) (singleprod a0 b0) \/ dans (couple x y) (prodcart b b0)).
2: apply dans_union; auto.
intro H1; elim H1; clear H1.
intro H1.
cut (x = a0 :>Elt /\ dans y b0).
2: apply dans_singleprod; auto.
intro H2; elim H2; clear H2.
intros.
rewrite H2.
split; auto.
intro.
cut (dans x b /\ dans y b0); auto.
intro H2; elim H2; clear H2.
intros.
split; auto.
Qed.
(*  Plus facile : l'inverse...						*)

Lemma dans_single :
forall (x y : Elt) (a : Ensf),
dans y a -> dans (couple x y) (singleprod x a).