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(* 02110-1301 USA                                                     *)
(* Contribution to the Coq Library   V6.3 (July 1999)                    *)
(* The cartesian product and its properties *)
(* This definition of the ordered pair is slightly different from *)
(* the usual one, since we want it to work in an intuisionistic   *)
(* setting. Works the same, neitherless. The soundness proofs are *)
(* unpleasant.                                                    *)
Definition Couple (E E' : Ens) := Paire (Sing E) (Paire Vide (Sing E')).
unfold Couple in |- *; simpl in |- *.
simple induction 1.
intros HA HB; elim (HA true).
intros x; elim x; simpl in |- *; simple induction 1; intros H3 H4;
elim (H3 true); simpl in |- *; intros xx; elim xx;
simpl in |- *; auto with zfc.
elim (H4 false); simpl in |- *.
simple induction x0; simpl in |- *.
intros.
cut (EQ (Sing B') Vide).
simpl in |- *.
simple induction 1.
intros yy; elim (yy true).
simple induction x1.
apply EQ_tran with A; auto with zfc.
intros; cut (EQ (Sing B') Vide).
simpl in |- *.
simple induction 1.
intros yy; elim (yy true).
simple induction x1.
apply EQ_tran with A; auto with zfc.
intros yy.
elim (HB true); simpl in |- *.
simple induction x0.
change (EQ (Sing A) (Sing B) -> EQ A B) in |- *; intros EE.
apply IN_Sing_EQ.
apply IN_sound_right with (Sing A); auto with zfc.
change (EQ (Paire Vide (Sing A')) (Sing B) -> EQ A B) in |- *.
intros zz.
elimtype F.
apply (not_EQ_Sing_Vide A').
apply EQ_tran with B.
apply IN_Sing_EQ.
apply IN_sound_right with (Paire Vide (Sing A')); auto with zfc.
apply EQ_sym; apply IN_Sing_EQ;
apply IN_sound_right with (Paire Vide (Sing A'));
auto with zfc.
Qed.
unfold Couple in |- *; simpl in |- *.
simple induction 1; intros H1 H2.
elim (H1 false).
intros bb1; elim bb1.
intros HF.
change (EQ (Paire Vide (Sing A')) (Sing B)) in HF.
cut F.
simple induction 1.
apply (not_EQ_Vide_Sing A').
apply EQ_tran with B.
apply IN_Sing_EQ; apply IN_sound_right with (Paire Vide (Sing A'));
auto with zfc.
apply EQ_sym; apply IN_Sing_EQ;
apply IN_sound_right with (Paire Vide (Sing A'));
auto with zfc.
change (EQ (Paire Vide (Sing A')) (Paire Vide (Sing B')) -> EQ A' B') in |- *.
intros HP; cut (EQ (Sing A') (Sing B')).
intros; auto with zfc.
cut (IN (Sing A') (Paire Vide (Sing B'))).
intros HI; elim (Paire_IN Vide (Sing B') (Sing A') HI).
intros; cut F.
simple induction 1.
apply not_EQ_Sing_Vide with A'; assumption.
trivial with zfc.
apply IN_sound_right with (Paire Vide (Sing A')); auto with zfc.
Qed.
(* Here we cheat. It is easier to define the cartesian product using    *)
(* the type theoretical product, i.e. we here use non set-theoretical   *)
(* constructions. We could however use the usual definitions.           *)
Definition Prod (E E' : Ens) : Ens :=
match E, E' with
| sup A f, sup A' f' =>
sup _
(fun c : prod_t A A' =>
match c with
| pair_t a a' => Couple (f a) (f' a')
end)
end.
Hint Resolve Paire_sound_left Paire_sound_right: zfc.
unfold Couple in |- *; intros; auto with zfc.
Qed.
unfold Couple in |- *; intros; auto with zfc.
Qed.

Theorem Couple_IN_Prod :
forall E1 E2 E1' E2' : Ens,
IN E1' E1 -> IN E2' E2 -> IN (Couple E1' E2') (Prod E1 E2).