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Ctopology.v
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Ctopology.v
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Require Export Polynomial.
Require Import Setoid.
(************************************)
(* First, we define a topology on C *)
(************************************)
Declare Scope topology_scope.
Delimit Scope topology_scope with T.
Open Scope topology_scope.
(* we define a subset of C as a function from C to {True, False} *)
(* so c is in A if A(c) = True *)
Definition Cset := C -> Prop.
Definition union (A B : Cset) : Cset :=
fun c => A c \/ B c.
Definition intersection (A B : Cset) : Cset :=
fun c => A c /\ B c.
Definition complement (A : Cset) : Cset :=
fun c => not (A c).
Definition setminus (A B : Cset) : Cset :=
intersection A (complement B).
Definition is_in (a : C) (A : Cset) : Prop := A a.
Definition subset (A B : Cset) : Prop :=
forall c, A c -> B c.
Definition eq_set (A B : Cset) : Prop :=
forall c, A c <-> B c.
Definition ϵ_disk (a : C) (ϵ : R) : Cset :=
fun c => Cmod (c - a) < ϵ.
Definition open_square (cen : C) (s : R) : Cset :=
fun c => Rabs (fst c - fst cen) < s /\ Rabs (snd c - snd cen) < s.
Definition open_rect (cen : C) (s1 s2 : R) : Cset :=
fun c => Rabs (fst c - fst cen) < s1 /\ Rabs (snd c - snd cen) < s2.
Definition bounded (A : Cset) : Prop :=
exists ϵ, subset A (ϵ_disk C0 ϵ).
Definition image (f : C -> C) (A : Cset) : Cset :=
fun c => (exists c', A c' /\ f c' = c).
Definition preimage (f : C -> C) (B : Cset) : Cset :=
fun c => B (f c).
Definition continuous_on (f : C -> C) (A : Cset) : Prop :=
forall c, A c -> continuous_at f c.
Infix "∪" := union (at level 50, left associativity) : topology_scope.
Infix "∩" := intersection (at level 40, left associativity) : topology_scope.
Infix "⊂" := subset (at level 0) : topology_scope.
Infix "⩦" := eq_set (at level 0) : topology_scope.
Infix "\" := setminus (at level 0) : topology_scope.
Infix "@" := image (at level 0) : topology_scope.
Notation "f *{ A }" := (preimage f A) (at level 0) : topology_scope.
Notation "A `" := (complement A) (at level 0) : topology_scope.
Infix "∈" := is_in (at level 0) : topology_scope.
Notation "B( a , ϵ )" := (ϵ_disk a ϵ) (at level 30, no associativity) : topology_scope.
Definition open (A : Cset) : Prop :=
forall c, A c -> exists ϵ, ϵ > 0 /\ B(c,ϵ) ⊂ A.
Definition closed (A : Cset) : Prop :=
open (A`).
Definition empty_set : Cset :=
fun _ => False.
Definition C_ : Cset :=
fun _ => True.
(** showing that all the above def's are preserved by eq_set *)
Lemma eq_set_refl : forall (A : Cset), A ⩦ A.
Proof. easy. Qed.
Lemma eq_set_symm : forall (A B : Cset), A ⩦ B -> B ⩦ A.
Proof. easy. Qed.
Lemma eq_set_trans : forall (A B C : Cset),
A ⩦ B -> B ⩦ C -> A ⩦ C.
Proof. intros.
unfold eq_set in *; intros.
split; intros.
apply H0; apply H; easy.
apply H; apply H0; easy.
Qed.
Add Parametric Relation : Cset eq_set
reflexivity proved by eq_set_refl
symmetry proved by eq_set_symm
transitivity proved by eq_set_trans
as eq_set_equiv_rel.
Add Parametric Morphism : subset
with signature eq_set ==> eq_set ==> iff as subset_mor.
Proof. unfold subset, eq_set; split; intros;
apply H0; apply H1; apply H; easy.
Qed.
Add Parametric Morphism : union
with signature eq_set ==> eq_set ==> eq_set as union_mor.
Proof. unfold union, eq_set; split; intros; destruct H1;
try (left; apply H; easy); right; apply H0; easy.
Qed.
Add Parametric Morphism : intersection
with signature eq_set ==> eq_set ==> eq_set as intersection_mor.
Proof. unfold intersection, eq_set; split; intros; split;
try (apply H; apply H1); apply H0; apply H1.
Qed.
Add Parametric Morphism : bounded
with signature eq_set ==> iff as bounded_mor.
Proof. unfold bounded; split; intros;
destruct H0; exists x0.
rewrite <- H; easy.
rewrite H; easy.
Qed.
Add Parametric Morphism : open
with signature eq_set ==> iff as open_mor.
Proof. unfold open; split; intros;
apply H in H1; apply H0 in H1; destruct H1 as [e [H2 H3] ];
exists e; split; auto.
rewrite <- H; easy.
rewrite H; easy.
Qed.
(* one quick helper *)
Lemma Cmod_lt_helper : forall (c : C) (ϵ : R),
Cmod c < ϵ -> Rabs (fst c) < ϵ /\ Rabs (snd c) < ϵ.
Proof. intros.
assert (H' := Rmax_Cmod c).
split.
- eapply Rle_lt_trans.
eapply Rle_trans.
apply Rmax_l.
apply H'.
easy.
- eapply Rle_lt_trans.
eapply Rle_trans.
apply Rmax_r.
apply H'.
easy.
Qed.
(** Subset lemmas *)
Lemma subset_cup_l : forall (A B : Cset),
A ⊂ (A ∪ B).
Proof. unfold subset, union; left; easy. Qed.
Lemma subset_cup_r : forall (A B : Cset),
B ⊂ (A ∪ B).
Proof. unfold subset, union; right; easy. Qed.
Lemma subset_cap_l : forall (A B : Cset),
(A ∩ B) ⊂ A.
Proof. unfold subset, intersection; easy. Qed.
Lemma subset_cap_r : forall (A B : Cset),
(A ∩ B) ⊂ B.
Proof. unfold subset, intersection; easy. Qed.
Lemma subset_self : forall (A : Cset),
A ⊂ A.
Proof. easy. Qed.
Lemma subset_transitive : forall (A B C : Cset),
A ⊂ B -> B ⊂ C -> A ⊂ C.
Proof. unfold subset; intros.
apply H0; apply H; easy.
Qed.
#[global] Hint Resolve subset_cup_l subset_cup_r subset_cap_l subset_cap_r subset_self subset_transitive : Csub_db.
Lemma subset_cup_reduce : forall (A B C : Cset),
A ⊂ B \/ A ⊂ C -> A ⊂ (B ∪ C).
Proof. unfold subset, union; intros.
destruct H.
- left; apply H; easy.
- right; apply H; easy.
Qed.
Lemma subset_cap_reduce : forall (A B C : Cset),
A ⊂ B /\ A ⊂ C -> A ⊂ (B ∩ C).
Proof. unfold subset, intersection; intros.
destruct H; split.
apply H; easy.
apply H1; easy.
Qed.
Lemma subset_ball_l : forall (a : C) (ϵ1 ϵ2 : R),
B(a, Rmin ϵ1 ϵ2) ⊂ B(a, ϵ1).
Proof. unfold subset, ϵ_disk; intros.
eapply Rlt_le_trans; eauto.
apply Rmin_l.
Qed.
Lemma subset_ball_r : forall (a : C) (ϵ1 ϵ2 : R),
B(a, Rmin ϵ1 ϵ2) ⊂ B(a, ϵ2).
Proof. unfold subset, ϵ_disk; intros.
eapply Rlt_le_trans; eauto.
apply Rmin_r.
Qed.
Lemma subset_C_ : forall (A : Cset),
A ⊂ C_.
Proof. unfold subset; easy. Qed.
#[global] Hint Resolve subset_cup_reduce subset_cap_reduce subset_ball_l subset_ball_r subset_C_ : Csub_db.
Lemma subset_equal : forall (A B : Cset),
A ⊂ B -> B ⊂ A -> A ⩦ B.
Proof. unfold subset, eq_set; intros.
split; try apply H; apply H0.
Qed.
Lemma subset_not : forall (A B : Cset),
~ (A ⊂ B) -> (exists a, A a /\ B` a).
Proof. intros.
destruct (Classical_Prop.classic (exists a : C, A a /\ (B) ` a)); auto.
assert (H1 : forall a, ~ (A a /\ (B) ` a)).
{ unfold not; intros.
apply H0; exists a; easy. }
assert (H2 : A ⊂ B).
{ unfold subset; intros.
assert (H' := H1 c).
apply Classical_Prop.not_and_or in H'.
destruct H'; try easy.
unfold not, complement in H3.
apply Decidable.dec_not_not; auto.
unfold Decidable.decidable.
apply Classical_Prop.classic. }
easy.
Qed.
(* some image and preimage relationships with subsets *)
Lemma subset_image : forall (A B : Cset) (f : C -> C),
A ⊂ B -> (f @ A) ⊂ (f @ B).
Proof. unfold subset; intros.
destruct H0 as [x [H0 H1] ].
apply H in H0.
exists x; easy.
Qed.
(* showing subset relationships between squares, circles, and rectangles *)
Lemma circle_in_square : forall (cen : C) (s : R),
(ϵ_disk cen s) ⊂ (open_square cen s).
Proof. unfold subset, ϵ_disk, open_square; intros.
apply Cmod_lt_helper in H.
easy.
Qed.
Lemma square_in_circle : forall (cen : C) (ϵ : R),
(open_square cen (ϵ / √ 2)) ⊂ (ϵ_disk cen ϵ).
Proof. unfold subset, ϵ_disk, open_square; intros.
destruct H.
destruct (Rle_lt_dec ϵ 0).
- assert (H' : ϵ / √ 2 <= 0).
{ unfold Rdiv.
rewrite <- (Rmult_0_r ϵ).
apply Rmult_le_compat_neg_l; auto; left.
apply Rinv_0_lt_compat; apply Rlt_sqrt2_0. }
assert (H1 := Rabs_pos ((fst c - fst cen))).
lra.
- assert (H' : 0 < ϵ / √ 2).
unfold Rdiv; apply Rmult_lt_0_compat; auto.
apply Rinv_0_lt_compat; apply Rlt_sqrt2_0.
assert (H1 : (ϵ / √ 2 = Rabs (ϵ / √ 2))%R).
{ unfold Rabs.
destruct (Rcase_abs (ϵ / √ 2)); lra. }
rewrite H1 in *.
apply Rsqr_lt_abs_1 in H; apply Rsqr_lt_abs_1 in H0.
assert (H2 : ((ϵ / √ 2)² = ϵ² / 2)%R).
{ unfold Rsqr, Rdiv.
R_field_simplify; try easy.
apply sqrt2_neq_0. }
rewrite H2 in *.
unfold Cmod.
rewrite <- sqrt_Rsqr; try lra.
apply sqrt_lt_1_alt. split.
apply Rplus_le_le_0_compat; apply pow2_ge_0.
rewrite double_var.
apply Rplus_lt_compat;
unfold Rsqr in *; simpl; lra.
Qed.
Lemma square_is_rectangle : forall (cen : C) (s : R),
(open_square cen s) ⩦ (open_rect cen s s).
Proof. intros; easy. Qed.
Lemma square_in_rect_at_point : forall (cen c : C) (s1 s2 : R),
s1 > 0 -> s2 > 0 ->
(open_rect cen s1 s2) c ->
exists s, s > 0 /\ (open_square c s) ⊂ (open_rect cen s1 s2).
Proof. intros.
exists (Rmin (s1 - (Rabs (fst c - fst cen))) (s2 - (Rabs (snd c - snd cen)))).
destruct H1 as [H1 H2].
repeat split.
apply Rmin_pos; try lra.
all : destruct H3.
- eapply Rlt_le_trans in H3; try apply Rmin_l.
replace (fst c0 - fst cen)%R with ((fst c0 - fst c) + (fst c - fst cen))%R by lra.
eapply Rle_lt_trans; try apply Rabs_triang; lra.
- eapply Rlt_le_trans in H4; try apply Rmin_r.
replace (snd c0 - snd cen)%R with ((snd c0 - snd c) + (snd c - snd cen))%R by lra.
eapply Rle_lt_trans; try apply Rabs_triang; lra.
Qed.
Lemma square_contains_center : forall (cen : C) (s : R),
s > 0 -> open_square cen s cen.
Proof. intros.
unfold open_square.
do 2 rewrite Rminus_eq_0, Rabs_R0; easy.
Qed.
(** some lemmas about open/closed sets *)
Lemma emptyset_open : open empty_set.
Proof. easy. Qed.
Lemma C_open : open C_.
Proof. unfold open, C_, is_in; intros.
exists 1; split; try lra.
easy.
Qed.
Lemma ball_open : forall (a : C) (ϵ : R),
ϵ > 0 -> open (B(a,ϵ)).
Proof. unfold open; intros.
unfold ϵ_disk in H0.
exists (ϵ - Cmod (c - a))%R.
split; try lra.
unfold subset; intros.
unfold ϵ_disk in *.
assert (H2 : Cmod (c0 - c) + Cmod (c - a) < ϵ). { lra. }
apply Cmod_triangle_diff in H2.
easy.
Qed.
Lemma closed_ball_complement_open : forall (c : C) (r : R),
open (fun c' => Cmod (c' - c) > r).
Proof. unfold open; intros.
exists (Cmod (c0 - c) - r)%R.
split; try lra.
unfold subset, ϵ_disk; intros.
assert (H' := Cmod_triangle (c1 - c) (c0 - c1)).
replace (c1 - c + (c0 - c1)) with (c0 - c) in H' by lca.
rewrite Cmod_switch in H0; lra.
Qed.
Lemma closed_ball_closed : forall (a : C) (ϵ : R),
closed (fun c' => Cmod (c' - a) <= ϵ).
Proof. intros; unfold closed.
assert (H' : (fun c' : C => Cmod (c' - a) <= ϵ) ` ⩦ (fun c' : C => Cmod (c' - a) > ϵ)).
{ apply subset_equal; unfold subset, complement; intros; lra. }
rewrite H'.
apply closed_ball_complement_open.
Qed.
Lemma rect_open : forall (cen : C) (s1 s2 : R),
s1 > 0 -> s2 > 0 -> open (open_rect cen s1 s2).
Proof. unfold open; intros.
apply square_in_rect_at_point in H1; auto.
destruct H1 as [s [H1 H2] ].
exists s; split; auto.
eapply subset_transitive; try apply H2.
apply circle_in_square.
Qed.
Lemma cup_open : forall (A B : Cset),
open A -> open B -> open (A ∪ B).
Proof. unfold open; intros.
unfold union in H1.
destruct H1.
- destruct ((H c) H1) as [ϵ [H2 H3] ].
exists ϵ.
split; eauto with Csub_db.
- destruct ((H0 c) H1) as [ϵ [H2 H3] ].
exists ϵ.
split; eauto with Csub_db.
Qed.
Lemma cap_open : forall (A B : Cset),
open A -> open B -> open (A ∩ B).
Proof. unfold open; intros.
unfold intersection in H1.
destruct H1.
destruct ((H c) H1) as [ϵ1 [H3 H4] ].
destruct ((H0 c) H2) as [ϵ2 [H5 H6] ].
exists (Rmin ϵ1 ϵ2).
split.
apply Rmin_Rgt_r; easy.
eauto with Csub_db.
Qed.
(** lemmas about preimage *)
(** some lemmas showing basic properties *)
Lemma complement_involutive : forall (A : Cset),
(A`)` ⩦ A.
Proof. unfold complement; intros.
apply subset_equal.
- unfold subset; intros.
apply Classical_Prop.NNPP; easy.
- unfold subset; intros.
unfold not. intros.
easy.
Qed.
Lemma bounded_cup : forall (A B : Cset),
bounded A -> bounded B -> bounded (A ∪ B).
Proof. intros.
destruct H as [ϵ1 H].
destruct H0 as [ϵ2 H0].
exists (Rmax ϵ1 ϵ2).
unfold subset, ϵ_disk in *; intros.
destruct H1.
- apply H in H1.
eapply Rlt_le_trans; eauto.
apply Rmax_l.
- apply H0 in H1.
eapply Rlt_le_trans; eauto.
apply Rmax_r.
Qed.
(************************)
(* Defining compactness *)
(************************)
Definition Ccover := Cset -> Prop.
Definition WF_cover (G : Ccover) : Prop :=
forall A A', (A ⩦ A' /\ G A) -> G A'.
Definition WFify_cover (G : Ccover) : Ccover :=
fun A => exists A', A ⩦ A' /\ G A'.
Definition open_cover (G : Ccover) : Prop :=
forall A, G A -> open A.
Definition subcover (G1 G2 : Ccover) : Prop :=
forall A, G1 A -> G2 A.
Definition eq_cover (G1 G2 : Ccover) : Prop :=
forall A, G1 A <-> G2 A.
(* used in list_to_cover *)
Fixpoint In' (A : Cset) (l : list Cset) : Prop :=
match l with
| [] => False
| (A' :: l) => A ⩦ A' \/ In' A l
end.
Definition list_to_cover (l : list Cset) : Ccover :=
fun A => In' A l.
Definition finite_cover (G : Ccover) : Prop :=
exists l, eq_cover G (list_to_cover l).
Definition big_cup (G : Ccover) : Cset :=
fun c => (exists A, G A /\ A c).
Definition big_cap (G : Ccover) : Cset :=
fun c => (forall A, G A -> A c).
(* the star of the show! *)
Definition compact (A : Cset) : Prop :=
forall G, open_cover G -> WF_cover G -> A ⊂ (big_cup G) ->
(exists G', finite_cover G' /\ subcover G' G /\ A ⊂ (big_cup G')).
(* showing some basic wf lemmas *)
Lemma WF_WFify : forall (G : Ccover),
WF_cover (WFify_cover G).
Proof. intros. unfold WF_cover, WFify_cover. intros.
destruct H as [H [A0 [H0 H1 ] ] ].
exists A0; split; try easy.
symmetry in H.
eapply eq_set_trans;
try apply H; easy.
Qed.
Lemma WF_finitecover : forall (l : list Cset),
WF_cover (list_to_cover l).
Proof. induction l as [| A].
- unfold WF_cover, list_to_cover; intros.
destruct H.
easy.
- unfold WF_cover, list_to_cover in *; intros.
destruct H; destruct H0.
left; rewrite <- H, <- H0; easy.
right; apply (IHl A0); easy.
Qed.
Lemma WFify_is_projection : forall (G : Ccover),
WF_cover G ->
eq_cover G (WFify_cover G).
Proof. unfold eq_cover; intros; split; intros.
exists A; easy.
destruct H0 as [A0 [H0 H1] ].
apply (H A0 A); easy.
Qed.
(* showing that all the def's are preserved by eq_cover *)
Lemma eq_cover_refl : forall (G : Ccover), eq_cover G G.
Proof. easy. Qed.
Lemma eq_cover_symm : forall (G1 G2 : Ccover), eq_cover G1 G2 -> eq_cover G2 G1.
Proof. easy. Qed.
Lemma eq_cover_trans : forall (G1 G2 G3 : Ccover),
eq_cover G1 G2 -> eq_cover G2 G3 -> eq_cover G1 G3.
Proof. intros.
unfold eq_cover in *; intros.
split; intros.
apply H0; apply H; easy.
apply H; apply H0; easy.
Qed.
Add Parametric Relation : Ccover eq_cover
reflexivity proved by eq_cover_refl
symmetry proved by eq_cover_symm
transitivity proved by eq_cover_trans
as eq_cover_equiv_rel.
Add Parametric Morphism : subcover
with signature eq_cover ==> eq_cover ==> iff as subcover_mor.
Proof. intros. unfold subcover, eq_cover; split; intros;
apply H0; apply H1; apply H; easy.
Qed.
Add Parametric Morphism : open_cover
with signature eq_cover ==> iff as opencover_mor.
Proof. unfold eq_cover, open_cover; split; intros;
apply H0; apply H; easy.
Qed.
Add Parametric Morphism : big_cup
with signature eq_cover ==> eq_set as bigcup_mor.
Proof. unfold eq_cover, big_cup; split; intros;
destruct H0 as [A [H0 H1] ]; exists A; split; auto; apply H; auto.
Qed.
Add Parametric Morphism : big_cap
with signature eq_cover ==> eq_set as bigcap_mor.
Proof. unfold eq_cover, big_cap; split; intros;
apply H0; apply H; apply H1.
Qed.
(* must also show that compactness is preserved by eq_set *)
Add Parametric Morphism : compact
with signature eq_set ==> iff as compact_mor.
Proof. intros. unfold compact; split; intros.
- rewrite <- H in *.
apply H0 in H3; try easy.
destruct H3 as [G' [H3 [H4 H5] ] ].
rewrite H in H5.
exists G'; easy.
- rewrite H in *.
apply H0 in H3; try easy.
destruct H3 as [G' [H3 [H4 H5] ] ].
rewrite <- H in H5.
exists G'; easy.
Qed.
(** now some lemmas *)
Lemma list_to_cover_reduce : forall (A A0 : Cset) (l : list Cset),
list_to_cover (A :: l) A0 <->
A ⩦ A0 \/ list_to_cover l A0.
Proof. intros; split; intros;
destruct H; try (left; easy); right; easy.
Qed.
Lemma open_cover_reduce : forall (l : list Cset) (A : Cset),
open_cover (list_to_cover (A :: l)) ->
open A /\ open_cover (list_to_cover l).
Proof. intros; split; unfold open_cover in H.
apply H.
apply list_to_cover_reduce; left; easy.
unfold open_cover; intros.
apply H.
apply list_to_cover_reduce; right; easy.
Qed.
Lemma subcover_reduce : forall (l : list Cset) (A : Cset) (G : Ccover),
subcover (list_to_cover (A :: l)) G ->
G A /\ subcover (list_to_cover l) G.
Proof. intros; split.
apply H.
apply list_to_cover_reduce; left; easy.
unfold subcover; intros.
apply H.
apply list_to_cover_reduce; right; easy.
Qed.
Lemma finite_cover_subset : forall (l : list Cset) (G : Ccover),
WF_cover G -> subcover G (list_to_cover l) ->
finite_cover G.
Proof. induction l as [| A].
- intros.
exists []; split; intros; auto; easy.
- intros.
destruct (IHl (fun A' => G A' /\ ~ (eq_set A' A))).
+ unfold WF_cover in *; intros; split;
destruct H1 as [H1 [H2 H3] ].
apply (H A0 A'); easy.
unfold not; intros; apply H3.
rewrite H1; easy.
+ unfold subcover in *; intros.
destruct H1.
apply H0 in H1.
apply list_to_cover_reduce in H1.
destruct H1; try easy.
unfold not in H2.
symmetry in H1.
apply H2 in H1; easy.
+ destruct (Classical_Prop.classic (G A)).
* exists (A :: x).
unfold eq_cover; split; intros; simpl.
destruct (Classical_Prop.classic (A ⩦ A0)); try (left; easy).
right; apply H1; split; auto.
unfold not; intros; apply H4; easy.
apply list_to_cover_reduce in H3.
destruct H3.
apply (H A A0); easy.
apply H1 in H3; easy.
* exists x.
unfold eq_cover; intros; split; intros.
apply H1; split; auto.
unfold not; intros; apply H2.
apply (H A0 A); easy.
apply H1 in H3; easy.
Qed.
Lemma big_cup_extend_l : forall (l : list Cset) (A : Cset),
(A ∪ (big_cup (list_to_cover l))) ⩦ (big_cup (list_to_cover (A :: l))).
Proof. intros.
unfold union, big_cup, list_to_cover; intros.
apply subset_equal.
- unfold subset; intros.
destruct H.
+ exists A; split; try left; easy.
+ destruct H as [A0 [H H0] ].
exists A0. split; try right; easy.
- unfold subset; intros.
destruct H as [A0 [ [H | H] H0] ]; subst.
left; apply H; easy.
right; exists A0; split; easy.
Qed.
Lemma big_cap_extend_l : forall (l : list Cset) (A : Cset),
(A ∩ (big_cap (list_to_cover l))) ⩦ (big_cap (list_to_cover (A :: l))).
Proof. intros.
unfold intersection, big_cap, list_to_cover; intros.
apply subset_equal.
- unfold subset; intros.
destruct H; destruct H0; subst; try easy.
apply H0; easy.
apply H1; apply H0.
- unfold subset; intros.
split.
apply H; left; easy.
intros; apply H; right; apply H0.
Qed.
Lemma app_union : forall (l1 l2 : list Cset),
(big_cup (list_to_cover (l1 ++ l2))) ⩦
((big_cup (list_to_cover l1)) ∪ (big_cup (list_to_cover l2))).
Proof. induction l1 as [| A].
- intros; simpl.
apply subset_equal; auto with Csub_db.
unfold subset; intros.
destruct H; try easy.
destruct H; easy.
- intros.
rewrite <- app_comm_cons.
rewrite <- big_cup_extend_l.
rewrite IHl1.
rewrite <- big_cup_extend_l.
split; intros.
+ destruct H.
left; left; easy.
destruct H. left; right; easy.
right; easy.
+ destruct H. destruct H.
left; easy.
right; left; easy.
right; right; easy.
Qed.
Lemma In'_app_or : forall (l1 l2 : list Cset) (A : Cset),
In' A (l1 ++ l2) -> In' A l1 \/ In' A l2.
Proof. induction l1 as [| A0].
- intros; right; easy.
- intros.
rewrite <- app_comm_cons in H.
destruct H.
left; left; easy.
apply IHl1 in H.
destruct H.
left; right; easy.
right; easy.
Qed.
Lemma In'_map : forall {X} (l : list X) (f : X -> Cset) (A : Cset),
In' A (map f l) -> exists (x : X), (f x) ⩦ A /\ In x l.
Proof. induction l; firstorder (subst; auto).
Qed.
Lemma arb_cup_open : forall (G : Ccover),
open_cover G -> open (big_cup G).
Proof. unfold open_cover, open, big_cup in *; intros.
destruct H0 as [A [H0 H1] ].
destruct (H A H0 c) as [ϵ [H2 H3] ]; auto.
exists ϵ.
split; auto.
eapply subset_transitive; eauto.
unfold subset; intros.
exists A; split; easy.
Qed.
Lemma ltc_open : forall (l : list Cset),
open_cover (list_to_cover l) -> open (big_cap (list_to_cover l)).
Proof. induction l as [| h].
- intros.
unfold list_to_cover, big_cap, open; intros.
exists 1; split; try easy; lra.
- intros.
apply open_cover_reduce in H.
rewrite <- big_cap_extend_l.
apply cap_open; try easy.
apply IHl; easy.
Qed.
Lemma fin_cap_open : forall (G : Ccover),
open_cover G -> finite_cover G -> open (big_cap G).
Proof. intros.
unfold finite_cover in H0.
destruct H0 as [l H0].
rewrite H0.
apply ltc_open.
rewrite <- H0; easy.
Qed.
(* we have not yet defined enough setup to define preimage for functions whose domains differ
from C_. This is fine when proving FTA, since polynomials are continuous everywhere.
Could be expanded if we want to make a general topology library *)
Lemma continuous_preimage_open : forall (f : C -> C),
continuous_on f C_ -> (forall A, open A -> open f*{A}).
Proof. intros.
unfold open in *; intros.
unfold preimage in H1.
destruct (H0 (f c) H1) as [ϵ [H2 H3] ].
assert (H' : C_ c). easy.
destruct (H c H' ϵ) as [δ [H4 H5] ]; auto.
exists δ; split; auto.
unfold subset, preimage, ϵ_disk in *; intros.
destruct (Ceq_dec c0 c); subst; try easy.
apply H3; apply H5; try easy.
Qed.
Lemma preimage_open_continuous : forall (f : C -> C),
(forall A, open A -> open f*{A}) -> continuous_on f C_.
Proof. unfold continuous_on; intros.
unfold continuous_at, limit_at_point; intros.
assert (H2 := (H ( B(f c,ϵ) ) (ball_open (f c) ϵ H1))).
unfold open in H2.
destruct (H2 c) as [δ [H3 H4] ].
unfold preimage, ϵ_disk.
replace (f c - f c)%C with C0 by lca.
rewrite Cmod_0; lra.
exists δ; split; auto; intros.
unfold preimage, subset, ϵ_disk in H4.
apply H4; easy.
Qed.
(* we define the preimage of a cover *)
Definition preimage_cover (f : C -> C) (G : Ccover) : Ccover :=
fun A => (exists A', G A' /\ A ⩦ (f*{A'})).
Lemma open_cover_preimage_open : forall (f : C -> C) (G : Ccover),
open_cover G -> continuous_on f C_ ->
open_cover (preimage_cover f G).
Proof. intros.
unfold open_cover, preimage_cover; intros.
destruct H1 as [A' [H1 H2] ]; subst.
apply H in H1.
rewrite H2.
apply (continuous_preimage_open _ H0); easy.
Qed.
Lemma WF_preimage_cover : forall (f : C -> C) (G : Ccover),
WF_cover (preimage_cover f G).
Proof. unfold WF_cover, preimage_cover; intros.
destruct H as [H [A0 [H0 H1] ] ].
exists A0; split; auto.
rewrite <- H; easy.
Qed.
Lemma subset_preimage_pres : forall (f : C -> C) (A : Cset) (G : Ccover),
((f) @ (A)) ⊂ (big_cup G) ->
A ⊂ (big_cup (preimage_cover f G)).
Proof. unfold subset; intros.
unfold big_cup, preimage_cover.
assert (H' : big_cup G (f c)).
{ apply H.
exists c; split; easy. }
unfold big_cup in H'.
destruct H' as [A0 [H1 H2] ].
exists (f*{A0}); split; try easy.
exists A0; split; easy.
Qed.
Lemma extract_finite_image : forall (f : C -> C) (l : list Cset) (A : Cset) (G : Ccover),
WF_cover G ->
subcover (list_to_cover l) (preimage_cover f G) ->
A ⊂ (big_cup (list_to_cover l)) ->
exists l', subcover (list_to_cover l') G /\ (f @ A) ⊂ (big_cup (list_to_cover l')).
Proof. induction l as [| A0].
- intros.
exists []; split; auto.
unfold subcover; easy.
unfold subset in *; intros.
destruct H2 as [c0 [H2 H3] ].
apply H1 in H2.
unfold list_to_cover, big_cup in H2.
destruct H2; easy.
- intros.
apply subcover_reduce in H0.
destruct H0.
apply (IHl (A \ A0)) in H2; auto.
destruct H0 as [A'0 [H0 H3] ].
destruct H2 as [l' [H2 H4] ].
exists (A'0 :: l'); split.
unfold subcover, list_to_cover; intros.
destruct H5; try (apply (H A'0 A1); easy).
apply H2; easy.
unfold subset in *; intros.
apply big_cup_extend_l.
destruct (Classical_Prop.classic (A'0 c)).
left; easy.
right; apply H4.
destruct H5 as [c' [H5 H7] ].
exists c'; repeat split; auto.
unfold complement, not; intros.
apply H6; apply H3 in H8.
unfold preimage in H8; subst; easy.
unfold subset in *; intros.
destruct H3; apply H1 in H3.
apply big_cup_extend_l in H3.
destruct H3; easy.
Qed.
Lemma continuous_image_compact : forall (f : C -> C) (A : Cset),
continuous_on f C_ -> compact A ->
compact (f @ A).
Proof. intros.
unfold compact; intros.
assert (H4 := (subset_preimage_pres _ _ _ H3)).
apply H0 in H4.
destruct H4 as [G' [H4 [H5 H6] ] ].
destruct H4 as [l H4]; subst.
destruct (extract_finite_image f l A G) as [l' [H7 H8] ]; auto.
all : try (rewrite <- H4; easy).
exists (list_to_cover l'); repeat split; try easy.
exists l'; easy.
apply open_cover_preimage_open; easy.
apply WF_preimage_cover.
Qed.
(*************************************************************************)
(* We now introduce cube_compact and show that is is the same as compact *)
(*************************************************************************)
Definition cube_cover (G : Ccover) : Prop :=
forall A, G A -> exists s cen, s > 0 /\ A ⩦ (open_square cen s).
Definition cube_compact (A : Cset) : Prop :=
forall G, cube_cover G -> WF_cover G -> A ⊂ (big_cup G) ->
(exists G', finite_cover G' /\ subcover G' G /\ A ⊂ (big_cup G')).
(* we use this to turn covers to cube_covers *)
Definition cover_to_cube_cover (G : Ccover) : Ccover :=
fun A => (exists s cen A', s > 0 /\ (A ⩦ (open_square cen s)) /\ G A' /\ (open_square cen s) ⊂ A').
Lemma cube_cover_open_cover : forall (G : Ccover),
cube_cover G -> open_cover G.
Proof. unfold open_cover, cube_cover; intros.
apply H in H0.
destruct H0 as [s [cen [H0 H1] ] ].
rewrite H1, square_is_rectangle.
apply rect_open; easy.
Qed.
Lemma WF_ctcc : forall (G : Ccover),
WF_cover (cover_to_cube_cover G).
Proof. unfold WF_cover; intros.
destruct H.
destruct H0 as [s [cen [A0 [H0 [H1 [H2 H3] ] ] ] ] ].
exists s, cen, A0; split; auto.
split; try easy.
rewrite <- H1; easy.
Qed.
Lemma cube_cover_ctcc : forall (G : Ccover),
cube_cover (cover_to_cube_cover G).
Proof. unfold cube_cover; intros.
destruct H as [s [cen [A0 [H0 [H1 [H2 H3] ] ] ] ] ].
exists s, cen.
split; easy.
Qed.
Lemma ctcc_in_cover : forall (G : Ccover),
open_cover G -> (big_cup G) ⩦ (big_cup (cover_to_cube_cover G)).
Proof. intros; split; intros.
- destruct H0 as [A [H0 H1] ].
assert (H0' := H0).
apply H in H0.
assert (H2 := H1); apply H0 in H1.
destruct H1 as [ϵ [H1 H3] ].
exists (open_square c (ϵ / √ 2)).
split.
unfold cover_to_cube_cover.
exists (ϵ / √ 2)%R, c, A; split.
apply lt_ep_helper in H1; easy.
split; try easy.
split; try easy.
eapply subset_transitive; try apply H3.
apply square_in_circle.
apply square_contains_center.
apply (lt_ep_helper ϵ); easy.
- destruct H0 as [A [H0 H1] ].
destruct H0 as [s [cen [A0 [H0 [H2 [H3 H4] ] ] ] ] ].
exists A0; split; auto.
apply H4; apply H2; easy.
Qed.
Lemma fin_cubecover_gives_fin_cover : forall (l : list Cset) (G : Ccover),
WF_cover G ->
subcover (list_to_cover l) (cover_to_cube_cover G) ->
exists l', subcover (list_to_cover l') G /\
(big_cup (list_to_cover l)) ⊂ (big_cup (list_to_cover l')).
Proof. induction l as [| A].
- intros. exists []; split; try easy.
- intros.
apply subcover_reduce in H0; destruct H0.
apply IHl in H1; auto.
destruct H1 as [l' [H1 H2] ].
destruct H0 as [s [cen [A0 [H0 [H3 [H4 H5] ] ] ] ] ].
exists (A0 :: l').
split.
unfold subcover, list_to_cover; intros.
destruct H6.
apply (H A0 A1); easy.
apply H1; easy.
do 2 rewrite <- big_cup_extend_l.
unfold subset; intros.
destruct H6.
left; apply H5; apply H3; easy.
right; apply H2; easy.
Qed.
Lemma compact_is_cube_compact : forall (A : Cset),
compact A <-> cube_compact A.
Proof. split; intros.
- unfold cube_compact; intros.
apply cube_cover_open_cover in H0.
apply H in H2; easy.
- unfold compact, cube_compact in *; intros.
destruct (H (cover_to_cube_cover G)) as [G' [H3 [H4 H5] ] ].
apply cube_cover_ctcc.
apply WF_ctcc.