From Hammer Require Import Hammer.



























Require Export Finite_sets.
Require Export Constructive_sets.
Require Export Classical.
Require Export Classical_sets.
Require Export Powerset.
Require Export Powerset_facts.
Require Export Powerset_Classical_facts.
Require Export Gt.
Require Export Lt.

Section Finite_sets_facts.
Variable U : Type.

Lemma finite_cardinal :
forall X:Ensemble U, Finite U X ->  exists n : nat, cardinal U X n.
Proof. hammer_hook "Finite_sets_facts" "Finite_sets_facts.finite_cardinal".  
induction 1 as [| A _ [n H]].
exists 0; auto with sets.
exists (S n); auto with sets.
Qed.

Lemma cardinal_finite :
forall (X:Ensemble U) (n:nat), cardinal U X n -> Finite U X.
Proof. hammer_hook "Finite_sets_facts" "Finite_sets_facts.cardinal_finite".  
induction 1; auto with sets.
Qed.

Theorem Add_preserves_Finite :
forall (X:Ensemble U) (x:U), Finite U X -> Finite U (Add U X x).
Proof. hammer_hook "Finite_sets_facts" "Finite_sets_facts.Add_preserves_Finite".  
intros X x H'.
elim (classic (In U X x)); intro H'0; auto with sets.
rewrite (Non_disjoint_union U X x); auto with sets.
Qed.

Theorem Singleton_is_finite : forall x:U, Finite U (Singleton U x).
Proof. hammer_hook "Finite_sets_facts" "Finite_sets_facts.Singleton_is_finite".  
intro x; rewrite <- (Empty_set_zero U (Singleton U x)).
change (Finite U (Add U (Empty_set U) x)); auto with sets.
Qed.

Theorem Union_preserves_Finite :
forall X Y:Ensemble U, Finite U X -> Finite U Y -> Finite U (Union U X Y).
Proof. hammer_hook "Finite_sets_facts" "Finite_sets_facts.Union_preserves_Finite".  
intros X Y H; induction H as [|A Fin_A Hind x].
rewrite (Empty_set_zero U Y). trivial.
intros.
rewrite (Union_commutative U (Add U A x) Y).
rewrite <- (Union_add U Y A x).
rewrite (Union_commutative U Y A).
apply Add_preserves_Finite.
apply Hind. assumption.
Qed.

Lemma Finite_downward_closed :
forall A:Ensemble U,
Finite U A -> forall X:Ensemble U, Included U X A -> Finite U X.
Proof. hammer_hook "Finite_sets_facts" "Finite_sets_facts.Finite_downward_closed".  
intros A H'; elim H'; auto with sets.
intros X H'0.
rewrite (less_than_empty U X H'0); auto with sets.
intros; elim Included_Add with U X A0 x; auto with sets.
destruct 1 as [A' [H5 H6]].
rewrite H5; auto with sets.
Qed.

Lemma Intersection_preserves_finite :
forall A:Ensemble U,
Finite U A -> forall X:Ensemble U, Finite U (Intersection U X A).
Proof. hammer_hook "Finite_sets_facts" "Finite_sets_facts.Intersection_preserves_finite".  
intros A H' X; apply Finite_downward_closed with A; auto with sets.
Qed.

Lemma cardinalO_empty :
forall X:Ensemble U, cardinal U X 0 -> X = Empty_set U.
Proof. hammer_hook "Finite_sets_facts" "Finite_sets_facts.cardinalO_empty".  
intros X H; apply (cardinal_invert U X 0); trivial with sets.
Qed.

Lemma inh_card_gt_O :
forall X:Ensemble U, Inhabited U X -> forall n:nat, cardinal U X n -> n > 0.
Proof. hammer_hook "Finite_sets_facts" "Finite_sets_facts.inh_card_gt_O".  
induction 1 as [x H'].
intros n H'0.
elim (gt_O_eq n); auto with sets.
intro H'1; generalize H'; generalize H'0.
rewrite <- H'1; intro H'2.
rewrite (cardinalO_empty X); auto with sets.
intro H'3; elim H'3.
Qed.

Lemma card_soustr_1 :
forall (X:Ensemble U) (n:nat),
cardinal U X n ->
forall x:U, In U X x -> cardinal U (Subtract U X x) (pred n).
Proof. hammer_hook "Finite_sets_facts" "Finite_sets_facts.card_soustr_1".  
intros X n H'; elim H'.
intros x H'0; elim H'0.
clear H' n X.
intros X n H' H'0 x H'1 x0 H'2.
elim (classic (In U X x0)).
intro H'4; rewrite (add_soustr_xy U X x x0).
elim (classic (x = x0)).
intro H'5.
absurd (In U X x0); auto with sets.
rewrite <- H'5; auto with sets.
intro H'3; try assumption.
cut (S (pred n) = pred (S n)).
intro H'5; rewrite <- H'5.
apply card_add; auto with sets.
red; intro H'6; elim H'6.
intros H'7 H'8; try assumption.
elim H'1; auto with sets.
unfold pred at 2; symmetry .
apply S_pred with (m := 0).
change (n > 0).
apply inh_card_gt_O with (X := X); auto with sets.
apply Inhabited_intro with (x := x0); auto with sets.
red; intro H'3.
apply H'1.
elim H'3; auto with sets.
rewrite H'3; auto with sets.
elim (classic (x = x0)).
intro H'3; rewrite <- H'3.
cut (Subtract U (Add U X x) x = X); auto with sets.
intro H'4; rewrite H'4; auto with sets.
intros H'3 H'4; try assumption.
absurd (In U (Add U X x) x0); auto with sets.
red; intro H'5; try exact H'5.
lapply (Add_inv U X x x0); tauto.
Qed.

Lemma cardinal_is_functional :
forall (X:Ensemble U) (c1:nat),
cardinal U X c1 ->
forall (Y:Ensemble U) (c2:nat), cardinal U Y c2 -> X = Y -> c1 = c2.
Proof. hammer_hook "Finite_sets_facts" "Finite_sets_facts.cardinal_is_functional".  
intros X c1 H'; elim H'.
intros Y c2 H'0; elim H'0; auto with sets.
intros A n H'1 H'2 x H'3 H'5.
elim (not_Empty_Add U A x); auto with sets.
clear H' c1 X.
intros X n H' H'0 x H'1 Y c2 H'2.
elim H'2.
intro H'3.
elim (not_Empty_Add U X x); auto with sets.
clear H'2 c2 Y.
intros X0 c2 H'2 H'3 x0 H'4 H'5.
elim (classic (In U X0 x)).
intro H'6; apply f_equal.
apply H'0 with (Y := Subtract U (Add U X0 x0) x).
elimtype (pred (S c2) = c2); auto with sets.
apply card_soustr_1; auto with sets.
rewrite <- H'5.
apply Sub_Add_new; auto with sets.
elim (classic (x = x0)).
intros H'6 H'7; apply f_equal.
apply H'0 with (Y := X0); auto with sets.
apply Simplify_add with (x := x); auto with sets.
pattern x at 2; rewrite H'6; auto with sets.
intros H'6 H'7.
absurd (Add U X x = Add U X0 x0); auto with sets.
clear H'0 H' H'3 n H'5 H'4 H'2 H'1 c2.
red; intro H'.
lapply (Extension U (Add U X x) (Add U X0 x0)); auto with sets.
clear H'.
intro H'; red in H'.
elim H'; intros H'0 H'1; red in H'0; clear H' H'1.
absurd (In U (Add U X0 x0) x); auto with sets.
lapply (Add_inv U X0 x0 x); [ intuition | apply (H'0 x); apply Add_intro2 ].
Qed.

Lemma cardinal_Empty : forall m:nat, cardinal U (Empty_set U) m -> 0 = m.
Proof. hammer_hook "Finite_sets_facts" "Finite_sets_facts.cardinal_Empty".  
intros m Cm; generalize (cardinal_invert U (Empty_set U) m Cm).
elim m; auto with sets.
intros; elim H0; intros; elim H1; intros; elim H2; intros.
elim (not_Empty_Add U x x0 H3).
Qed.

Lemma cardinal_unicity :
forall (X:Ensemble U) (n:nat),
cardinal U X n -> forall m:nat, cardinal U X m -> n = m.
Proof. hammer_hook "Finite_sets_facts" "Finite_sets_facts.cardinal_unicity".  
intros; apply cardinal_is_functional with X X; auto with sets.
Qed.

Lemma card_Add_gen :
forall (A:Ensemble U) (x:U) (n n':nat),
cardinal U A n -> cardinal U (Add U A x) n' -> n' <= S n.
Proof. hammer_hook "Finite_sets_facts" "Finite_sets_facts.card_Add_gen".  
intros A x n n' H'.
elim (classic (In U A x)).
intro H'0.
rewrite (Non_disjoint_union U A x H'0).
intro H'1; cut (n = n').
intro E; rewrite E; auto with sets.
apply cardinal_unicity with A; auto with sets.
intros H'0 H'1.
cut (n' = S n).
intro E; rewrite E; auto with sets.
apply cardinal_unicity with (Add U A x); auto with sets.
Qed.

Lemma incl_st_card_lt :
forall (X:Ensemble U) (c1:nat),
cardinal U X c1 ->
forall (Y:Ensemble U) (c2:nat),
cardinal U Y c2 -> Strict_Included U X Y -> c2 > c1.
Proof. hammer_hook "Finite_sets_facts" "Finite_sets_facts.incl_st_card_lt".  
intros X c1 H'; elim H'.
intros Y c2 H'0; elim H'0; auto with sets arith.
intro H'1.
elim (Strict_Included_strict U (Empty_set U)); auto with sets arith.
clear H' c1 X.
intros X n H' H'0 x H'1 Y c2 H'2.
elim H'2.
intro H'3; elim (not_SIncl_empty U (Add U X x)); auto with sets arith.
clear H'2 c2 Y.
intros X0 c2 H'2 H'3 x0 H'4 H'5; elim (classic (In U X0 x)).
intro H'6; apply gt_n_S.
apply H'0 with (Y := Subtract U (Add U X0 x0) x).
elimtype (pred (S c2) = c2); auto with sets arith.
apply card_soustr_1; auto with sets arith.
apply incl_st_add_soustr; auto with sets arith.
elim (classic (x = x0)).
intros H'6 H'7; apply gt_n_S.
apply H'0 with (Y := X0); auto with sets arith.
apply sincl_add_x with (x := x0).
rewrite <- H'6; auto with sets arith.
pattern x0 at 1; rewrite <- H'6; trivial with sets arith.
intros H'6 H'7; red in H'5.
elim H'5; intros H'8 H'9; try exact H'8; clear H'5.
red in H'8.
generalize (H'8 x).
intro H'5; lapply H'5; auto with sets arith.
intro H; elim Add_inv with U X0 x0 x; auto with sets arith.
intro; absurd (In U X0 x); auto with sets arith.
intro; absurd (x = x0); auto with sets arith.
Qed.

Lemma incl_card_le :
forall (X Y:Ensemble U) (n m:nat),
cardinal U X n -> cardinal U Y m -> Included U X Y -> n <= m.
Proof. hammer_hook "Finite_sets_facts" "Finite_sets_facts.incl_card_le".  
intros; elim Included_Strict_Included with U X Y; auto with sets arith; intro.
cut (m > n); auto with sets arith.
apply incl_st_card_lt with (X := X) (Y := Y); auto with sets arith.
generalize H0; rewrite <- H2; intro.
cut (n = m).
intro E; rewrite E; auto with sets arith.
apply cardinal_unicity with X; auto with sets arith.
Qed.

Lemma G_aux :
forall P:Ensemble U -> Prop,
(forall X:Ensemble U,
Finite U X ->
(forall Y:Ensemble U, Strict_Included U Y X -> P Y) -> P X) ->
P (Empty_set U).
Proof. hammer_hook "Finite_sets_facts" "Finite_sets_facts.G_aux".  
intros P H'; try assumption.
apply H'; auto with sets.
clear H'; auto with sets.
intros Y H'; try assumption.
red in H'.
elim H'; intros H'0 H'1; try exact H'1; clear H'.
lapply (less_than_empty U Y); [ intro H'3; try exact H'3 | assumption ].
elim H'1; auto with sets.
Qed.

Lemma Generalized_induction_on_finite_sets :
forall P:Ensemble U -> Prop,
(forall X:Ensemble U,
Finite U X ->
(forall Y:Ensemble U, Strict_Included U Y X -> P Y) -> P X) ->
forall X:Ensemble U, Finite U X -> P X.
Proof. hammer_hook "Finite_sets_facts" "Finite_sets_facts.Generalized_induction_on_finite_sets".  
intros P H'0 X H'1.
generalize P H'0; clear H'0 P.
elim H'1.
intros P H'0.
apply G_aux; auto with sets.
clear H'1 X.
intros A H' H'0 x H'1 P H'3.
cut (forall Y:Ensemble U, Included U Y (Add U A x) -> P Y); auto with sets.
generalize H'1.
apply H'0.
intros X K H'5 L Y H'6; apply H'3; auto with sets.
apply Finite_downward_closed with (A := Add U X x); auto with sets.
intros Y0 H'7.
elim (Strict_inclusion_is_transitive_with_inclusion U Y0 Y (Add U X x));
auto with sets.
intros H'2 H'4.
elim (Included_Add U Y0 X x);
[ intro H'14
| intro H'14; elim H'14; intros A' E; elim E; intros H'15 H'16; clear E H'14
| idtac ]; auto with sets.
elim (Included_Strict_Included U Y0 X); auto with sets.
intro H'9; apply H'5 with (Y := Y0); auto with sets.
intro H'9; rewrite H'9.
apply H'3; auto with sets.
intros Y1 H'8; elim H'8.
intros H'10 H'11; apply H'5 with (Y := Y1); auto with sets.
elim (Included_Strict_Included U A' X); auto with sets.
intro H'8; apply H'5 with (Y := A'); auto with sets.
rewrite <- H'15; auto with sets.
intro H'8.
elim H'7.
intros H'9 H'10; apply H'10 || elim H'10; try assumption.
generalize H'6.
rewrite <- H'8.
rewrite <- H'15; auto with sets.
Qed.

End Finite_sets_facts.
