Section First_inductive_lemma.
Variable U : Type.
Variable C : Cpo U.
Let D : PO U := PO_of_cpo U C.
Inductive Lubs_of_finite_parts (X : Ensemble U) : Ensemble U :=
Lubs_of_finite_parts_def :
forall (bsup : U) (Y : Ensemble U),
Included U Y X ->
Finite U Y -> Lub U D Y bsup -> In U (Lubs_of_finite_parts X) bsup.
intros X K H' L; apply Definition_of_Directed.
red in |- *; simpl in |- *; intros x H'0; elim H'0; auto with sets.
intros bsup Y H'1 H'2 H'3; apply Lub_is_in_Carrier with (X := Y);
auto with sets.
elim (Empty_set_has_lub U C); intros bsup E.
apply Inhabited_intro with (x := bsup).
apply Lubs_of_finite_parts_def with (Y := Empty_set U); auto with sets.
intros x1 x2 H'4; red in H'4.
lapply (H'4 x1); [ intro H'1; elim H'1 | idtac ]; auto with sets.
intros bsup Y H'0 H'2 H'3; try assumption.
lapply (H'4 x2); [ intro H'6; elim H'6 | idtac ]; auto with sets.
intros bsup0 Y0 H'5 H'7 H'8; try assumption.
lapply (Union_preserves_Finite U Y Y0);
[ intro H'12; lapply H'12; [ intro H'13; clear H'12 | clear H'12 ] | idtac ];
auto with sets.
elim (K (Union U Y Y0)); [ intros bsup1 E | idtac | idtac ]; auto with sets.
exists bsup1; split.
apply Lubs_of_finite_parts_def with (Y := Union U Y Y0); auto with sets.
apply Upper_Bound_Couple_intro.
apply Lub_is_in_Carrier with (X := Y); auto with sets.
apply Lub_is_in_Carrier with (X := Y0); auto with sets.
apply Lub_is_in_Carrier with (X := Union U Y Y0); auto with sets.
elim H'3.
intros H'10 H'11; apply H'11.
elim E; intros H'12 H'14;
apply Upper_downward_stable with (B := Union U Y Y0);
auto with sets.
elim H'8.
intros H'10 H'11; apply H'11.
elim E; intros H'12 H'14;
apply Upper_downward_stable with (B := Union U Y Y0);
auto with sets.
Qed.
intros X K bsup E0; elim E0.
intros H'4 H'5; apply Lub_definition; auto with sets.
apply Upper_Bound_definition; auto with sets.
apply Lub_is_in_Carrier with (X := Lubs_of_finite_parts X); auto with sets.
red in |- *; simpl in |- *; intros x H'8; elim H'8.
intros bsup0 Y H'9 H'10 H'11; apply Lub_is_in_Carrier with Y; auto with sets.
intros y H'8; elim H'4.
intros H'9 H'10; apply H'10.
apply Lubs_of_finite_parts_def with (Y := Singleton U y); auto with sets.
red in |- *; simpl in |- *.
intros x H'11; elim H'11; auto with sets.
apply Singleton_is_finite.
intros y H'8; apply H'5.
apply Upper_Bound_definition.
elim H'8; auto with sets.
intros y0 H'9; elim H'9.
intros bsup1 Y H'10 H'11 H'12; elim H'12.
intros H'13 H'14; apply H'14.
apply Upper_downward_stable with (B := X); auto with sets.
Qed.
Hint Resolve Lub_is_LFP.
Hint Resolve Pairs_are_enough_finite_case.

Theorem Pairs_are_enough :
(forall x y : U,
Compatible U D x y -> ex (fun bsup : U => Lub U D (Couple U x y) bsup)) ->
Conditionally_complete U D.