Imported Upstream version 8.1~gammaupstream/8.1.gamma

1 files changed, 290 insertions, 293 deletions

diff --git a/theories/Sets/Finite_sets_facts.v b/theories/Sets/Finite_sets_facts.v
index ddbf62e4..91717f9e 100644
--- a/theories/Sets/Finite_sets_facts.v
+++ b/theories/Sets/Finite_sets_facts.v
@@ -24,7 +24,7 @@
 (* in Summer 1995. Several developments by E. Ledinot were an inspiration.  *)
 (****************************************************************************)
 
-(*i $Id: Finite_sets_facts.v 8642 2006-03-17 10:09:02Z notin $ i*)
+(*i $Id: Finite_sets_facts.v 9245 2006-10-17 12:53:34Z notin $ i*)
 
 Require Export Finite_sets.
 Require Export Constructive_sets.
@@ -37,311 +37,308 @@ Require Export Gt.
 Require Export Lt.
 
 Section Finite_sets_facts.
-Variable U : Type.
+  Variable U : Type.
 
-Lemma finite_cardinal :
- forall X:Ensemble U, Finite U X ->  exists n : nat, cardinal U X n.
-Proof.
-induction 1 as [| A _ [n H]].
-exists 0; auto with sets.
-exists (S n); auto with sets.
-Qed.
+  Lemma finite_cardinal :
+    forall X:Ensemble U, Finite U X ->  exists n : nat, cardinal U X n.
+  Proof.
+    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.
-induction 1; auto with sets.
-Qed.
+  Lemma cardinal_finite :
+    forall (X:Ensemble U) (n:nat), cardinal U X n -> Finite U X.
+  Proof.
+    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.
-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.
-Hint Resolve Add_preserves_Finite.
+  Theorem Add_preserves_Finite :
+    forall (X:Ensemble U) (x:U), Finite U X -> Finite U (Add U X x).
+  Proof.
+    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.
-intro x; rewrite <- (Empty_set_zero U (Singleton U x)).
-change (Finite U (Add U (Empty_set U) x)) in |- *; auto with sets.
-Qed.
-Hint Resolve Singleton_is_finite.
+  Theorem Singleton_is_finite : forall x:U, Finite U (Singleton U x).
+  Proof.
+    intro x; rewrite <- (Empty_set_zero U (Singleton U x)).
+    change (Finite U (Add U (Empty_set U) x)) in |- *; 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.
-intros X Y H'; elim H'.
-rewrite (Empty_set_zero U Y); auto with sets.
-intros A H'0 H'1 x H'2 H'3.
-rewrite (Union_commutative U (Add U A x) Y).
-rewrite <- (Union_add U Y A x).
-rewrite (Union_commutative U Y A); 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.
+    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.
-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 Finite_downward_closed :
+    forall A:Ensemble U,
+      Finite U A -> forall X:Ensemble U, Included U X A -> Finite U X.
+  Proof.
+    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.
-intros A H' X; apply Finite_downward_closed with A; 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.
+    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.
+    intros X H; apply (cardinal_invert U X 0); trivial with sets.
+  Qed.
 
-Lemma cardinalO_empty :
- forall X:Ensemble U, cardinal U X 0 -> X = Empty_set U.
-Proof.
-intros X H; apply (cardinal_invert U X 0); trivial with sets.
-Qed.
-Hint Resolve cardinalO_empty.
+  Lemma inh_card_gt_O :
+    forall X:Ensemble U, Inhabited U X -> forall n:nat, cardinal U X n -> n > 0.
+  Proof.
+    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 inh_card_gt_O :
- forall X:Ensemble U, Inhabited U X -> forall n:nat, cardinal U X n -> n > 0.
-Proof.
-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.
+    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 in |- *; intro H'6; elim H'6.
+    intros H'7 H'8; try assumption.
+    elim H'1; auto with sets.
+    unfold pred at 2 in |- *; symmetry  in |- *.
+    apply S_pred with (m := 0).
+    change (n > 0) in |- *.
+    apply inh_card_gt_O with (X := X); auto with sets.
+    apply Inhabited_intro with (x := x0); auto with sets.
+    red in |- *; 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 in |- *; intro H'5; try exact H'5.
+    lapply (Add_inv U X x x0); tauto.
+  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.
-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 in |- *; intro H'6; elim H'6.
-intros H'7 H'8; try assumption.
-elim H'1; auto with sets.
-unfold pred at 2 in |- *; symmetry  in |- *.
-apply S_pred with (m := 0).
-change (n > 0) in |- *.
-apply inh_card_gt_O with (X := X); auto with sets.
-apply Inhabited_intro with (x := x0); auto with sets.
-red in |- *; 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 in |- *; 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.
+    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 with nat.
+    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 with nat.
+    apply H'0 with (Y := X0); auto with sets.
+    apply Simplify_add with (x := x); auto with sets.
+    pattern x at 2 in |- *; 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 in |- *; 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_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.
-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 with nat.
-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 with nat.
-apply H'0 with (Y := X0); auto with sets.
-apply Simplify_add with (x := x); auto with sets.
-pattern x at 2 in |- *; 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 in |- *; 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.
+    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_Empty : forall m:nat, cardinal U (Empty_set U) m -> 0 = m.
-Proof.
-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.
+    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.
+    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 cardinal_unicity :
- forall (X:Ensemble U) (n:nat),
-   cardinal U X n -> forall m:nat, cardinal U X m -> n = m.
-Proof.
-intros; apply cardinal_is_functional with X 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.
+    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 in |- *; 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 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.
-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_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.
+    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.
+    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 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.
-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 in |- *; 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 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.
+    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.
 
-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.
-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.
-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.
-
-Hint Unfold not.
-
-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.
-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.
\ No newline at end of file
+End Finite_sets_facts.