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(* 02110-1301 USA                                                     *)
(* Contribution to the Coq Library   V6.3 (July 1999)                    *)
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(* This contribution was updated for Coq V5.10 by the COQ workgroup.        *)
(* January 1995                                                             *)
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(*                                                                          *)
(*                      Rem Theorem in Baire space                          *)
(*                                                                          *)
(*                           Henk Barendregt                                *)
(*                                                                          *)
(*                           Bastad, June 1993                              *)
(*                               Coq V5.8                                   *)
(*                                                                          *)
(*                                                                          *)
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(*                                  Rem.v                                   *)
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(* Needs classical logic *)
(* Baire Space *)
Definition B := nat -> nat.
Definition Top := B -> Prop.
Definition inclusion (X Y : Top) := forall z : B, X z -> Y z.
Definition equal (X Y : Top) := inclusion X Y /\ inclusion Y X.
Definition complement (X : Top) (z : B) := ~ X z.
Definition union (X Y : Top) (z : B) := X z \/ Y z.
Definition inter (X Y : Top) (z : B) := X z /\ Y z.
Definition neighbour (f : B) (n : nat) (g : B) :=
forall m : nat, m <= n -> f m = g m.
Definition open (X : Top) :=
forall f : B, X f -> exists n : nat, inclusion (neighbour f n) X.
Definition closed (X : Top) := open (complement X).
Definition dense (X : Top) :=
forall (f : B) (n : nat), exists g : B, X g /\ neighbour f n g.
Definition closure (X : Top) (f : B) :=
forall n : nat, exists g : B, X g /\ neighbour f n g.
unfold neighbour in |- *; auto.
Qed.
Hint Resolve refl_neighbour.
unfold neighbour in |- *.
intros.
rewrite (H m); trivial.
apply H0; trivial.
Qed.
unfold closed, closure in |- *.
unfold open, complement in |- *.
intros X f complement_clX.
generalize
(not_all_ex_not nat (fun n : nat => exists g : B, X g /\ neighbour f n g)
complement_clX).
simple induction 1; intros n H1.
exists n.
unfold inclusion in |- *; intros g fng.
unfold not in |- *; intro H2.
apply H1.
elim (H2 n).
simple induction 1; intros.
exists x; split; trivial.
apply trans_neighbour with g; trivial.
Qed.
Hint Resolve closedc_clX.
unfold inclusion in |- *; intros X f Xf.
unfold closure in |- *; intro n.
exists f; split; trivial.
Qed.
unfold equal, inclusion in |- *; intro X; split.
unfold inter, union, closure in |- *; intros.
split.
intro n; exists z; split; auto.
left; trivial.
unfold inter, union, closure in |- *; intros.
elim H; intros.
elim H1; trivial.
unfold complement in |- *; intros.
elim (H2 H0).
Qed.

Lemma density : forall X : Top, dense (union X (complement (closure X))).