Theory Topology_Euclidean_Space

theory Topology_Euclidean_Space
imports Indicator_Function Countable_Set FuncSet Linear_Algebra Norm_Arith
```(*  Author:     L C Paulson, University of Cambridge
Author:     Amine Chaieb, University of Cambridge
Author:     Robert Himmelmann, TU Muenchen
Author:     Brian Huffman, Portland State University
*)

section ‹Elementary topology in Euclidean space›

theory Topology_Euclidean_Space
imports
"HOL-Library.Indicator_Function"
"HOL-Library.Countable_Set"
"HOL-Library.FuncSet"
Linear_Algebra
Norm_Arith
begin

(* FIXME: move elsewhere *)

lemma halfspace_Int_eq:
"{x. a ∙ x ≤ b} ∩ {x. b ≤ a ∙ x} = {x. a ∙ x = b}"
"{x. b ≤ a ∙ x} ∩ {x. a ∙ x ≤ b} = {x. a ∙ x = b}"
by auto

definition (in monoid_add) support_on :: "'b set ⇒ ('b ⇒ 'a) ⇒ 'b set"
where "support_on s f = {x∈s. f x ≠ 0}"

lemma in_support_on: "x ∈ support_on s f ⟷ x ∈ s ∧ f x ≠ 0"

lemma support_on_simps[simp]:
"support_on {} f = {}"
"support_on (insert x s) f =
(if f x = 0 then support_on s f else insert x (support_on s f))"
"support_on (s ∪ t) f = support_on s f ∪ support_on t f"
"support_on (s ∩ t) f = support_on s f ∩ support_on t f"
"support_on (s - t) f = support_on s f - support_on t f"
"support_on (f ` s) g = f ` (support_on s (g ∘ f))"
unfolding support_on_def by auto

lemma support_on_cong:
"(⋀x. x ∈ s ⟹ f x = 0 ⟷ g x = 0) ⟹ support_on s f = support_on s g"
by (auto simp: support_on_def)

lemma support_on_if: "a ≠ 0 ⟹ support_on A (λx. if P x then a else 0) = {x∈A. P x}"
by (auto simp: support_on_def)

lemma support_on_if_subset: "support_on A (λx. if P x then a else 0) ⊆ {x ∈ A. P x}"
by (auto simp: support_on_def)

lemma finite_support[intro]: "finite S ⟹ finite (support_on S f)"
unfolding support_on_def by auto

(* TODO: is supp_sum really needed? TODO: Generalize to Finite_Set.fold *)
definition (in comm_monoid_add) supp_sum :: "('b ⇒ 'a) ⇒ 'b set ⇒ 'a"
where "supp_sum f S = (∑x∈support_on S f. f x)"

lemma supp_sum_empty[simp]: "supp_sum f {} = 0"
unfolding supp_sum_def by auto

lemma supp_sum_insert[simp]:
"finite (support_on S f) ⟹
supp_sum f (insert x S) = (if x ∈ S then supp_sum f S else f x + supp_sum f S)"
by (simp add: supp_sum_def in_support_on insert_absorb)

lemma supp_sum_divide_distrib: "supp_sum f A / (r::'a::field) = supp_sum (λn. f n / r) A"
by (cases "r = 0")
(auto simp: supp_sum_def sum_divide_distrib intro!: sum.cong support_on_cong)

(*END OF SUPPORT, ETC.*)

lemma image_affinity_interval:
fixes c :: "'a::ordered_real_vector"
shows "((λx. m *⇩R x + c) ` {a..b}) =
(if {a..b}={} then {}
else if 0 ≤ m then {m *⇩R a + c .. m  *⇩R b + c}
else {m *⇩R b + c .. m *⇩R a + c})"
(is "?lhs = ?rhs")
proof (cases "m=0")
case True
then show ?thesis
by force
next
case False
show ?thesis
proof
show "?lhs ⊆ ?rhs"
by (auto simp: scaleR_left_mono scaleR_left_mono_neg)
show "?rhs ⊆ ?lhs"
proof (clarsimp, intro conjI impI subsetI)
show "⟦0 ≤ m; a ≤ b; x ∈ {m *⇩R a + c..m *⇩R b + c}⟧
⟹ x ∈ (λx. m *⇩R x + c) ` {a..b}" for x
apply (rule_tac x="inverse m *⇩R (x-c)" in rev_image_eqI)
using False apply (auto simp: le_diff_eq pos_le_divideRI)
using diff_le_eq pos_le_divideR_eq by force
show "⟦¬ 0 ≤ m; a ≤ b;  x ∈ {m *⇩R b + c..m *⇩R a + c}⟧
⟹ x ∈ (λx. m *⇩R x + c) ` {a..b}" for x
apply (rule_tac x="inverse m *⇩R (x-c)" in rev_image_eqI)
apply (auto simp: diff_le_eq neg_le_divideR_eq)
using diff_eq_diff_less_eq linordered_field_class.sign_simps(11) minus_diff_eq not_less scaleR_le_cancel_left_neg by fastforce
qed
qed
qed

lemma countable_PiE:
"finite I ⟹ (⋀i. i ∈ I ⟹ countable (F i)) ⟹ countable (Pi⇩E I F)"
by (induct I arbitrary: F rule: finite_induct) (auto simp: PiE_insert_eq)

lemma open_sums:
fixes T :: "('b::real_normed_vector) set"
assumes "open S ∨ open T"
shows "open (⋃x∈ S. ⋃y ∈ T. {x + y})"
using assms
proof
assume S: "open S"
show ?thesis
proof (clarsimp simp: open_dist)
fix x y
assume "x ∈ S" "y ∈ T"
with S obtain e where "e > 0" and e: "⋀x'. dist x' x < e ⟹ x' ∈ S"
by (auto simp: open_dist)
then have "⋀z. dist z (x + y) < e ⟹ ∃x∈S. ∃y∈T. z = x + y"
then show "∃e>0. ∀z. dist z (x + y) < e ⟶ (∃x∈S. ∃y∈T. z = x + y)"
using ‹0 < e› ‹x ∈ S› by blast
qed
next
assume T: "open T"
show ?thesis
proof (clarsimp simp: open_dist)
fix x y
assume "x ∈ S" "y ∈ T"
with T obtain e where "e > 0" and e: "⋀x'. dist x' y < e ⟹ x' ∈ T"
by (auto simp: open_dist)
then have "⋀z. dist z (x + y) < e ⟹ ∃x∈S. ∃y∈T. z = x + y"
then show "∃e>0. ∀z. dist z (x + y) < e ⟶ (∃x∈S. ∃y∈T. z = x + y)"
using ‹0 < e› ‹y ∈ T› by blast
qed
qed

subsection ‹Topological Basis›

context topological_space
begin

definition%important "topological_basis B ⟷
(∀b∈B. open b) ∧ (∀x. open x ⟶ (∃B'. B' ⊆ B ∧ ⋃B' = x))"

lemma topological_basis:
"topological_basis B ⟷ (∀x. open x ⟷ (∃B'. B' ⊆ B ∧ ⋃B' = x))"
unfolding topological_basis_def
apply safe
apply fastforce
apply fastforce
apply (erule_tac x=x in allE, simp)
apply (rule_tac x="{x}" in exI, auto)
done

lemma topological_basis_iff:
assumes "⋀B'. B' ∈ B ⟹ open B'"
shows "topological_basis B ⟷ (∀O'. open O' ⟶ (∀x∈O'. ∃B'∈B. x ∈ B' ∧ B' ⊆ O'))"
(is "_ ⟷ ?rhs")
proof safe
fix O' and x::'a
assume H: "topological_basis B" "open O'" "x ∈ O'"
then have "(∃B'⊆B. ⋃B' = O')" by (simp add: topological_basis_def)
then obtain B' where "B' ⊆ B" "O' = ⋃B'" by auto
then show "∃B'∈B. x ∈ B' ∧ B' ⊆ O'" using H by auto
next
assume H: ?rhs
show "topological_basis B"
using assms unfolding topological_basis_def
proof safe
fix O' :: "'a set"
assume "open O'"
with H obtain f where "∀x∈O'. f x ∈ B ∧ x ∈ f x ∧ f x ⊆ O'"
by (force intro: bchoice simp: Bex_def)
then show "∃B'⊆B. ⋃B' = O'"
by (auto intro: exI[where x="{f x |x. x ∈ O'}"])
qed
qed

lemma topological_basisI:
assumes "⋀B'. B' ∈ B ⟹ open B'"
and "⋀O' x. open O' ⟹ x ∈ O' ⟹ ∃B'∈B. x ∈ B' ∧ B' ⊆ O'"
shows "topological_basis B"
using assms by (subst topological_basis_iff) auto

lemma topological_basisE:
fixes O'
assumes "topological_basis B"
and "open O'"
and "x ∈ O'"
obtains B' where "B' ∈ B" "x ∈ B'" "B' ⊆ O'"
proof atomize_elim
from assms have "⋀B'. B'∈B ⟹ open B'"
with topological_basis_iff assms
show  "∃B'. B' ∈ B ∧ x ∈ B' ∧ B' ⊆ O'"
using assms by (simp add: Bex_def)
qed

lemma topological_basis_open:
assumes "topological_basis B"
and "X ∈ B"
shows "open X"
using assms by (simp add: topological_basis_def)

lemma topological_basis_imp_subbasis:
assumes B: "topological_basis B"
shows "open = generate_topology B"
proof (intro ext iffI)
fix S :: "'a set"
assume "open S"
with B obtain B' where "B' ⊆ B" "S = ⋃B'"
unfolding topological_basis_def by blast
then show "generate_topology B S"
by (auto intro: generate_topology.intros dest: topological_basis_open)
next
fix S :: "'a set"
assume "generate_topology B S"
then show "open S"
by induct (auto dest: topological_basis_open[OF B])
qed

lemma basis_dense:
fixes B :: "'a set set"
and f :: "'a set ⇒ 'a"
assumes "topological_basis B"
and choosefrom_basis: "⋀B'. B' ≠ {} ⟹ f B' ∈ B'"
shows "∀X. open X ⟶ X ≠ {} ⟶ (∃B' ∈ B. f B' ∈ X)"
proof (intro allI impI)
fix X :: "'a set"
assume "open X" and "X ≠ {}"
from topological_basisE[OF ‹topological_basis B› ‹open X› choosefrom_basis[OF ‹X ≠ {}›]]
obtain B' where "B' ∈ B" "f X ∈ B'" "B' ⊆ X" .
then show "∃B'∈B. f B' ∈ X"
by (auto intro!: choosefrom_basis)
qed

end

lemma topological_basis_prod:
assumes A: "topological_basis A"
and B: "topological_basis B"
shows "topological_basis ((λ(a, b). a × b) ` (A × B))"
unfolding topological_basis_def
proof (safe, simp_all del: ex_simps add: subset_image_iff ex_simps(1)[symmetric])
fix S :: "('a × 'b) set"
assume "open S"
then show "∃X⊆A × B. (⋃(a,b)∈X. a × b) = S"
proof (safe intro!: exI[of _ "{x∈A × B. fst x × snd x ⊆ S}"])
fix x y
assume "(x, y) ∈ S"
from open_prod_elim[OF ‹open S› this]
obtain a b where a: "open a""x ∈ a" and b: "open b" "y ∈ b" and "a × b ⊆ S"
by (metis mem_Sigma_iff)
moreover
from A a obtain A0 where "A0 ∈ A" "x ∈ A0" "A0 ⊆ a"
by (rule topological_basisE)
moreover
from B b obtain B0 where "B0 ∈ B" "y ∈ B0" "B0 ⊆ b"
by (rule topological_basisE)
ultimately show "(x, y) ∈ (⋃(a, b)∈{X ∈ A × B. fst X × snd X ⊆ S}. a × b)"
by (intro UN_I[of "(A0, B0)"]) auto
qed auto
qed (metis A B topological_basis_open open_Times)

subsection ‹Countable Basis›

locale%important countable_basis =
fixes B :: "'a::topological_space set set"
assumes is_basis: "topological_basis B"
and countable_basis: "countable B"
begin

lemma open_countable_basis_ex:
assumes "open X"
shows "∃B' ⊆ B. X = ⋃B'"
using assms countable_basis is_basis
unfolding topological_basis_def by blast

lemma open_countable_basisE:
assumes "open X"
obtains B' where "B' ⊆ B" "X = ⋃B'"
using assms open_countable_basis_ex
by atomize_elim simp

lemma countable_dense_exists:
"∃D::'a set. countable D ∧ (∀X. open X ⟶ X ≠ {} ⟶ (∃d ∈ D. d ∈ X))"
proof -
let ?f = "(λB'. SOME x. x ∈ B')"
have "countable (?f ` B)" using countable_basis by simp
with basis_dense[OF is_basis, of ?f] show ?thesis
by (intro exI[where x="?f ` B"]) (metis (mono_tags) all_not_in_conv imageI someI)
qed

lemma countable_dense_setE:
obtains D :: "'a set"
where "countable D" "⋀X. open X ⟹ X ≠ {} ⟹ ∃d ∈ D. d ∈ X"
using countable_dense_exists by blast

end

lemma (in first_countable_topology) first_countable_basisE:
fixes x :: 'a
obtains 𝒜 where "countable 𝒜" "⋀A. A ∈ 𝒜 ⟹ x ∈ A" "⋀A. A ∈ 𝒜 ⟹ open A"
"⋀S. open S ⟹ x ∈ S ⟹ (∃A∈𝒜. A ⊆ S)"
proof -
obtain 𝒜 where 𝒜: "(∀i::nat. x ∈ 𝒜 i ∧ open (𝒜 i))" "(∀S. open S ∧ x ∈ S ⟶ (∃i. 𝒜 i ⊆ S))"
using first_countable_basis[of x] by metis
show thesis
proof
show "countable (range 𝒜)"
by simp
qed (use 𝒜 in auto)
qed

lemma (in first_countable_topology) first_countable_basis_Int_stableE:
obtains 𝒜 where "countable 𝒜" "⋀A. A ∈ 𝒜 ⟹ x ∈ A" "⋀A. A ∈ 𝒜 ⟹ open A"
"⋀S. open S ⟹ x ∈ S ⟹ (∃A∈𝒜. A ⊆ S)"
"⋀A B. A ∈ 𝒜 ⟹ B ∈ 𝒜 ⟹ A ∩ B ∈ 𝒜"
proof atomize_elim
obtain ℬ where ℬ:
"countable ℬ"
"⋀B. B ∈ ℬ ⟹ x ∈ B"
"⋀B. B ∈ ℬ ⟹ open B"
"⋀S. open S ⟹ x ∈ S ⟹ ∃B∈ℬ. B ⊆ S"
by (rule first_countable_basisE) blast
define 𝒜 where [abs_def]:
"𝒜 = (λN. ⋂((λn. from_nat_into ℬ n) ` N)) ` (Collect finite::nat set set)"
then show "∃𝒜. countable 𝒜 ∧ (∀A. A ∈ 𝒜 ⟶ x ∈ A) ∧ (∀A. A ∈ 𝒜 ⟶ open A) ∧
(∀S. open S ⟶ x ∈ S ⟶ (∃A∈𝒜. A ⊆ S)) ∧ (∀A B. A ∈ 𝒜 ⟶ B ∈ 𝒜 ⟶ A ∩ B ∈ 𝒜)"
proof (safe intro!: exI[where x=𝒜])
show "countable 𝒜"
unfolding 𝒜_def by (intro countable_image countable_Collect_finite)
fix A
assume "A ∈ 𝒜"
then show "x ∈ A" "open A"
using ℬ(4)[OF open_UNIV] by (auto simp: 𝒜_def intro: ℬ from_nat_into)
next
let ?int = "λN. ⋂(from_nat_into ℬ ` N)"
fix A B
assume "A ∈ 𝒜" "B ∈ 𝒜"
then obtain N M where "A = ?int N" "B = ?int M" "finite (N ∪ M)"
by (auto simp: 𝒜_def)
then show "A ∩ B ∈ 𝒜"
by (auto simp: 𝒜_def intro!: image_eqI[where x="N ∪ M"])
next
fix S
assume "open S" "x ∈ S"
then obtain a where a: "a∈ℬ" "a ⊆ S" using ℬ by blast
then show "∃a∈𝒜. a ⊆ S" using a ℬ
by (intro bexI[where x=a]) (auto simp: 𝒜_def intro: image_eqI[where x="{to_nat_on ℬ a}"])
qed
qed

lemma (in topological_space) first_countableI:
assumes "countable 𝒜"
and 1: "⋀A. A ∈ 𝒜 ⟹ x ∈ A" "⋀A. A ∈ 𝒜 ⟹ open A"
and 2: "⋀S. open S ⟹ x ∈ S ⟹ ∃A∈𝒜. A ⊆ S"
shows "∃𝒜::nat ⇒ 'a set. (∀i. x ∈ 𝒜 i ∧ open (𝒜 i)) ∧ (∀S. open S ∧ x ∈ S ⟶ (∃i. 𝒜 i ⊆ S))"
proof (safe intro!: exI[of _ "from_nat_into 𝒜"])
fix i
have "𝒜 ≠ {}" using 2[of UNIV] by auto
show "x ∈ from_nat_into 𝒜 i" "open (from_nat_into 𝒜 i)"
using range_from_nat_into_subset[OF ‹𝒜 ≠ {}›] 1 by auto
next
fix S
assume "open S" "x∈S" from 2[OF this]
show "∃i. from_nat_into 𝒜 i ⊆ S"
using subset_range_from_nat_into[OF ‹countable 𝒜›] by auto
qed

instance prod :: (first_countable_topology, first_countable_topology) first_countable_topology
proof
fix x :: "'a × 'b"
obtain 𝒜 where 𝒜:
"countable 𝒜"
"⋀a. a ∈ 𝒜 ⟹ fst x ∈ a"
"⋀a. a ∈ 𝒜 ⟹ open a"
"⋀S. open S ⟹ fst x ∈ S ⟹ ∃a∈𝒜. a ⊆ S"
by (rule first_countable_basisE[of "fst x"]) blast
obtain B where B:
"countable B"
"⋀a. a ∈ B ⟹ snd x ∈ a"
"⋀a. a ∈ B ⟹ open a"
"⋀S. open S ⟹ snd x ∈ S ⟹ ∃a∈B. a ⊆ S"
by (rule first_countable_basisE[of "snd x"]) blast
show "∃𝒜::nat ⇒ ('a × 'b) set.
(∀i. x ∈ 𝒜 i ∧ open (𝒜 i)) ∧ (∀S. open S ∧ x ∈ S ⟶ (∃i. 𝒜 i ⊆ S))"
proof (rule first_countableI[of "(λ(a, b). a × b) ` (𝒜 × B)"], safe)
fix a b
assume x: "a ∈ 𝒜" "b ∈ B"
show "x ∈ a × b"
by (simp add: 𝒜(2) B(2) mem_Times_iff x)
show "open (a × b)"
by (simp add: 𝒜(3) B(3) open_Times x)
next
fix S
assume "open S" "x ∈ S"
then obtain a' b' where a'b': "open a'" "open b'" "x ∈ a' × b'" "a' × b' ⊆ S"
by (rule open_prod_elim)
moreover
from a'b' 𝒜(4)[of a'] B(4)[of b']
obtain a b where "a ∈ 𝒜" "a ⊆ a'" "b ∈ B" "b ⊆ b'"
by auto
ultimately
show "∃a∈(λ(a, b). a × b) ` (𝒜 × B). a ⊆ S"
by (auto intro!: bexI[of _ "a × b"] bexI[of _ a] bexI[of _ b])
qed

class second_countable_topology = topological_space +
assumes ex_countable_subbasis:
"∃B::'a::topological_space set set. countable B ∧ open = generate_topology B"
begin

lemma ex_countable_basis: "∃B::'a set set. countable B ∧ topological_basis B"
proof -
from ex_countable_subbasis obtain B where B: "countable B" "open = generate_topology B"
by blast
let ?B = "Inter ` {b. finite b ∧ b ⊆ B }"

show ?thesis
proof (intro exI conjI)
show "countable ?B"
by (intro countable_image countable_Collect_finite_subset B)
{
fix S
assume "open S"
then have "∃B'⊆{b. finite b ∧ b ⊆ B}. (⋃b∈B'. ⋂b) = S"
unfolding B
proof induct
case UNIV
show ?case by (intro exI[of _ "{{}}"]) simp
next
case (Int a b)
then obtain x y where x: "a = UNION x Inter" "⋀i. i ∈ x ⟹ finite i ∧ i ⊆ B"
and y: "b = UNION y Inter" "⋀i. i ∈ y ⟹ finite i ∧ i ⊆ B"
by blast
show ?case
unfolding x y Int_UN_distrib2
by (intro exI[of _ "{i ∪ j| i j.  i ∈ x ∧ j ∈ y}"]) (auto dest: x(2) y(2))
next
case (UN K)
then have "∀k∈K. ∃B'⊆{b. finite b ∧ b ⊆ B}. UNION B' Inter = k" by auto
then obtain k where
"∀ka∈K. k ka ⊆ {b. finite b ∧ b ⊆ B} ∧ UNION (k ka) Inter = ka"
unfolding bchoice_iff ..
then show "∃B'⊆{b. finite b ∧ b ⊆ B}. UNION B' Inter = ⋃K"
by (intro exI[of _ "UNION K k"]) auto
next
case (Basis S)
then show ?case
by (intro exI[of _ "{{S}}"]) auto
qed
then have "(∃B'⊆Inter ` {b. finite b ∧ b ⊆ B}. ⋃B' = S)"
unfolding subset_image_iff by blast }
then show "topological_basis ?B"
unfolding topological_space_class.topological_basis_def
by (safe intro!: topological_space_class.open_Inter)
qed
qed

end

sublocale second_countable_topology <
countable_basis "SOME B. countable B ∧ topological_basis B"
using someI_ex[OF ex_countable_basis]
by unfold_locales safe

instance prod :: (second_countable_topology, second_countable_topology) second_countable_topology
proof
obtain A :: "'a set set" where "countable A" "topological_basis A"
using ex_countable_basis by auto
moreover
obtain B :: "'b set set" where "countable B" "topological_basis B"
using ex_countable_basis by auto
ultimately show "∃B::('a × 'b) set set. countable B ∧ open = generate_topology B"
by (auto intro!: exI[of _ "(λ(a, b). a × b) ` (A × B)"] topological_basis_prod
topological_basis_imp_subbasis)
qed

instance second_countable_topology ⊆ first_countable_topology
proof
fix x :: 'a
define B :: "'a set set" where "B = (SOME B. countable B ∧ topological_basis B)"
then have B: "countable B" "topological_basis B"
using countable_basis is_basis
by (auto simp: countable_basis is_basis)
then show "∃A::nat ⇒ 'a set.
(∀i. x ∈ A i ∧ open (A i)) ∧ (∀S. open S ∧ x ∈ S ⟶ (∃i. A i ⊆ S))"
by (intro first_countableI[of "{b∈B. x ∈ b}"])
(fastforce simp: topological_space_class.topological_basis_def)+
qed

instance nat :: second_countable_topology
proof
show "∃B::nat set set. countable B ∧ open = generate_topology B"
by (intro exI[of _ "range lessThan ∪ range greaterThan"]) (auto simp: open_nat_def)
qed

lemma countable_separating_set_linorder1:
shows "∃B::('a::{linorder_topology, second_countable_topology} set). countable B ∧ (∀x y. x < y ⟶ (∃b ∈ B. x < b ∧ b ≤ y))"
proof -
obtain A::"'a set set" where "countable A" "topological_basis A" using ex_countable_basis by auto
define B1 where "B1 = {(LEAST x. x ∈ U)| U. U ∈ A}"
then have "countable B1" using ‹countable A› by (simp add: Setcompr_eq_image)
define B2 where "B2 = {(SOME x. x ∈ U)| U. U ∈ A}"
then have "countable B2" using ‹countable A› by (simp add: Setcompr_eq_image)
have "∃b ∈ B1 ∪ B2. x < b ∧ b ≤ y" if "x < y" for x y
proof (cases)
assume "∃z. x < z ∧ z < y"
then obtain z where z: "x < z ∧ z < y" by auto
define U where "U = {x<..<y}"
then have "open U" by simp
moreover have "z ∈ U" using z U_def by simp
ultimately obtain V where "V ∈ A" "z ∈ V" "V ⊆ U" using topological_basisE[OF ‹topological_basis A›] by auto
define w where "w = (SOME x. x ∈ V)"
then have "w ∈ V" using ‹z ∈ V› by (metis someI2)
then have "x < w ∧ w ≤ y" using ‹w ∈ V› ‹V ⊆ U› U_def by fastforce
moreover have "w ∈ B1 ∪ B2" using w_def B2_def ‹V ∈ A› by auto
ultimately show ?thesis by auto
next
assume "¬(∃z. x < z ∧ z < y)"
then have *: "⋀z. z > x ⟹ z ≥ y" by auto
define U where "U = {x<..}"
then have "open U" by simp
moreover have "y ∈ U" using ‹x < y› U_def by simp
ultimately obtain "V" where "V ∈ A" "y ∈ V" "V ⊆ U" using topological_basisE[OF ‹topological_basis A›] by auto
have "U = {y..}" unfolding U_def using * ‹x < y› by auto
then have "V ⊆ {y..}" using ‹V ⊆ U› by simp
then have "(LEAST w. w ∈ V) = y" using ‹y ∈ V› by (meson Least_equality atLeast_iff subsetCE)
then have "y ∈ B1 ∪ B2" using ‹V ∈ A› B1_def by auto
moreover have "x < y ∧ y ≤ y" using ‹x < y› by simp
ultimately show ?thesis by auto
qed
moreover have "countable (B1 ∪ B2)" using ‹countable B1› ‹countable B2› by simp
ultimately show ?thesis by auto
qed

lemma countable_separating_set_linorder2:
shows "∃B::('a::{linorder_topology, second_countable_topology} set). countable B ∧ (∀x y. x < y ⟶ (∃b ∈ B. x ≤ b ∧ b < y))"
proof -
obtain A::"'a set set" where "countable A" "topological_basis A" using ex_countable_basis by auto
define B1 where "B1 = {(GREATEST x. x ∈ U) | U. U ∈ A}"
then have "countable B1" using ‹countable A› by (simp add: Setcompr_eq_image)
define B2 where "B2 = {(SOME x. x ∈ U)| U. U ∈ A}"
then have "countable B2" using ‹countable A› by (simp add: Setcompr_eq_image)
have "∃b ∈ B1 ∪ B2. x ≤ b ∧ b < y" if "x < y" for x y
proof (cases)
assume "∃z. x < z ∧ z < y"
then obtain z where z: "x < z ∧ z < y" by auto
define U where "U = {x<..<y}"
then have "open U" by simp
moreover have "z ∈ U" using z U_def by simp
ultimately obtain "V" where "V ∈ A" "z ∈ V" "V ⊆ U" using topological_basisE[OF ‹topological_basis A›] by auto
define w where "w = (SOME x. x ∈ V)"
then have "w ∈ V" using ‹z ∈ V› by (metis someI2)
then have "x ≤ w ∧ w < y" using ‹w ∈ V› ‹V ⊆ U› U_def by fastforce
moreover have "w ∈ B1 ∪ B2" using w_def B2_def ‹V ∈ A› by auto
ultimately show ?thesis by auto
next
assume "¬(∃z. x < z ∧ z < y)"
then have *: "⋀z. z < y ⟹ z ≤ x" using leI by blast
define U where "U = {..<y}"
then have "open U" by simp
moreover have "x ∈ U" using ‹x < y› U_def by simp
ultimately obtain "V" where "V ∈ A" "x ∈ V" "V ⊆ U" using topological_basisE[OF ‹topological_basis A›] by auto
have "U = {..x}" unfolding U_def using * ‹x < y› by auto
then have "V ⊆ {..x}" using ‹V ⊆ U› by simp
then have "(GREATEST x. x ∈ V) = x" using ‹x ∈ V› by (meson Greatest_equality atMost_iff subsetCE)
then have "x ∈ B1 ∪ B2" using ‹V ∈ A› B1_def by auto
moreover have "x ≤ x ∧ x < y" using ‹x < y› by simp
ultimately show ?thesis by auto
qed
moreover have "countable (B1 ∪ B2)" using ‹countable B1› ‹countable B2› by simp
ultimately show ?thesis by auto
qed

lemma countable_separating_set_dense_linorder:
shows "∃B::('a::{linorder_topology, dense_linorder, second_countable_topology} set). countable B ∧ (∀x y. x < y ⟶ (∃b ∈ B. x < b ∧ b < y))"
proof -
obtain B::"'a set" where B: "countable B" "⋀x y. x < y ⟹ (∃b ∈ B. x < b ∧ b ≤ y)"
using countable_separating_set_linorder1 by auto
have "∃b ∈ B. x < b ∧ b < y" if "x < y" for x y
proof -
obtain z where "x < z" "z < y" using ‹x < y› dense by blast
then obtain b where "b ∈ B" "x < b ∧ b ≤ z" using B(2) by auto
then have "x < b ∧ b < y" using ‹z < y› by auto
then show ?thesis using ‹b ∈ B› by auto
qed
then show ?thesis using B(1) by auto
qed

subsection%important ‹Polish spaces›

text ‹Textbooks define Polish spaces as completely metrizable.
We assume the topology to be complete for a given metric.›

class polish_space = complete_space + second_countable_topology

subsection ‹General notion of a topology as a value›

definition%important "istopology L ⟷
L {} ∧ (∀S T. L S ⟶ L T ⟶ L (S ∩ T)) ∧ (∀K. Ball K L ⟶ L (⋃K))"

typedef%important 'a topology = "{L::('a set) ⇒ bool. istopology L}"
morphisms "openin" "topology"
unfolding istopology_def by blast

lemma istopology_openin[intro]: "istopology(openin U)"
using openin[of U] by blast

lemma topology_inverse': "istopology U ⟹ openin (topology U) = U"
using topology_inverse[unfolded mem_Collect_eq] .

lemma topology_inverse_iff: "istopology U ⟷ openin (topology U) = U"
using topology_inverse[of U] istopology_openin[of "topology U"] by auto

lemma topology_eq: "T1 = T2 ⟷ (∀S. openin T1 S ⟷ openin T2 S)"
proof
assume "T1 = T2"
then show "∀S. openin T1 S ⟷ openin T2 S" by simp
next
assume H: "∀S. openin T1 S ⟷ openin T2 S"
then have "openin T1 = openin T2" by (simp add: fun_eq_iff)
then have "topology (openin T1) = topology (openin T2)" by simp
then show "T1 = T2" unfolding openin_inverse .
qed

text‹Infer the "universe" from union of all sets in the topology.›

definition "topspace T = ⋃{S. openin T S}"

subsubsection ‹Main properties of open sets›

proposition openin_clauses:
fixes U :: "'a topology"
shows
"openin U {}"
"⋀S T. openin U S ⟹ openin U T ⟹ openin U (S∩T)"
"⋀K. (∀S ∈ K. openin U S) ⟹ openin U (⋃K)"
using openin[of U] unfolding istopology_def mem_Collect_eq by fast+

lemma openin_subset[intro]: "openin U S ⟹ S ⊆ topspace U"
unfolding topspace_def by blast

lemma openin_empty[simp]: "openin U {}"
by (rule openin_clauses)

lemma openin_Int[intro]: "openin U S ⟹ openin U T ⟹ openin U (S ∩ T)"
by (rule openin_clauses)

lemma openin_Union[intro]: "(⋀S. S ∈ K ⟹ openin U S) ⟹ openin U (⋃K)"
using openin_clauses by blast

lemma openin_Un[intro]: "openin U S ⟹ openin U T ⟹ openin U (S ∪ T)"
using openin_Union[of "{S,T}" U] by auto

lemma openin_topspace[intro, simp]: "openin U (topspace U)"
by (force simp: openin_Union topspace_def)

lemma openin_subopen: "openin U S ⟷ (∀x ∈ S. ∃T. openin U T ∧ x ∈ T ∧ T ⊆ S)"
(is "?lhs ⟷ ?rhs")
proof
assume ?lhs
then show ?rhs by auto
next
assume H: ?rhs
let ?t = "⋃{T. openin U T ∧ T ⊆ S}"
have "openin U ?t" by (force simp: openin_Union)
also have "?t = S" using H by auto
finally show "openin U S" .
qed

lemma openin_INT [intro]:
assumes "finite I"
"⋀i. i ∈ I ⟹ openin T (U i)"
shows "openin T ((⋂i ∈ I. U i) ∩ topspace T)"
using assms by (induct, auto simp: inf_sup_aci(2) openin_Int)

lemma openin_INT2 [intro]:
assumes "finite I" "I ≠ {}"
"⋀i. i ∈ I ⟹ openin T (U i)"
shows "openin T (⋂i ∈ I. U i)"
proof -
have "(⋂i ∈ I. U i) ⊆ topspace T"
using ‹I ≠ {}› openin_subset[OF assms(3)] by auto
then show ?thesis
using openin_INT[of _ _ U, OF assms(1) assms(3)] by (simp add: inf.absorb2 inf_commute)
qed

lemma openin_Inter [intro]:
assumes "finite ℱ" "ℱ ≠ {}" "⋀X. X ∈ ℱ ⟹ openin T X" shows "openin T (⋂ℱ)"
by (metis (full_types) assms openin_INT2 image_ident)

subsubsection ‹Closed sets›

definition%important "closedin U S ⟷ S ⊆ topspace U ∧ openin U (topspace U - S)"

lemma closedin_subset: "closedin U S ⟹ S ⊆ topspace U"
by (metis closedin_def)

lemma closedin_empty[simp]: "closedin U {}"

lemma closedin_topspace[intro, simp]: "closedin U (topspace U)"

lemma closedin_Un[intro]: "closedin U S ⟹ closedin U T ⟹ closedin U (S ∪ T)"
by (auto simp: Diff_Un closedin_def)

lemma Diff_Inter[intro]: "A - ⋂S = ⋃{A - s|s. s∈S}"
by auto

lemma closedin_Union:
assumes "finite S" "⋀T. T ∈ S ⟹ closedin U T"
shows "closedin U (⋃S)"
using assms by induction auto

lemma closedin_Inter[intro]:
assumes Ke: "K ≠ {}"
and Kc: "⋀S. S ∈K ⟹ closedin U S"
shows "closedin U (⋂K)"
using Ke Kc unfolding closedin_def Diff_Inter by auto

lemma closedin_INT[intro]:
assumes "A ≠ {}" "⋀x. x ∈ A ⟹ closedin U (B x)"
shows "closedin U (⋂x∈A. B x)"
apply (rule closedin_Inter)
using assms
apply auto
done

lemma closedin_Int[intro]: "closedin U S ⟹ closedin U T ⟹ closedin U (S ∩ T)"
using closedin_Inter[of "{S,T}" U] by auto

lemma openin_closedin_eq: "openin U S ⟷ S ⊆ topspace U ∧ closedin U (topspace U - S)"
apply (auto simp: closedin_def Diff_Diff_Int inf_absorb2)
apply (metis openin_subset subset_eq)
done

lemma openin_closedin: "S ⊆ topspace U ⟹ (openin U S ⟷ closedin U (topspace U - S))"

lemma openin_diff[intro]:
assumes oS: "openin U S"
and cT: "closedin U T"
shows "openin U (S - T)"
proof -
have "S - T = S ∩ (topspace U - T)" using openin_subset[of U S]  oS cT
by (auto simp: topspace_def openin_subset)
then show ?thesis using oS cT
by (auto simp: closedin_def)
qed

lemma closedin_diff[intro]:
assumes oS: "closedin U S"
and cT: "openin U T"
shows "closedin U (S - T)"
proof -
have "S - T = S ∩ (topspace U - T)"
using closedin_subset[of U S] oS cT by (auto simp: topspace_def)
then show ?thesis
using oS cT by (auto simp: openin_closedin_eq)
qed

subsubsection ‹Subspace topology›

definition%important "subtopology U V = topology (λT. ∃S. T = S ∩ V ∧ openin U S)"

lemma istopology_subtopology: "istopology (λT. ∃S. T = S ∩ V ∧ openin U S)"
(is "istopology ?L")
proof -
have "?L {}" by blast
{
fix A B
assume A: "?L A" and B: "?L B"
from A B obtain Sa and Sb where Sa: "openin U Sa" "A = Sa ∩ V" and Sb: "openin U Sb" "B = Sb ∩ V"
by blast
have "A ∩ B = (Sa ∩ Sb) ∩ V" "openin U (Sa ∩ Sb)"
using Sa Sb by blast+
then have "?L (A ∩ B)" by blast
}
moreover
{
fix K
assume K: "K ⊆ Collect ?L"
have th0: "Collect ?L = (λS. S ∩ V) ` Collect (openin U)"
by blast
from K[unfolded th0 subset_image_iff]
obtain Sk where Sk: "Sk ⊆ Collect (openin U)" "K = (λS. S ∩ V) ` Sk"
by blast
have "⋃K = (⋃Sk) ∩ V"
using Sk by auto
moreover have "openin U (⋃Sk)"
using Sk by (auto simp: subset_eq)
ultimately have "?L (⋃K)" by blast
}
ultimately show ?thesis
unfolding subset_eq mem_Collect_eq istopology_def by auto
qed

lemma openin_subtopology: "openin (subtopology U V) S ⟷ (∃T. openin U T ∧ S = T ∩ V)"
unfolding subtopology_def topology_inverse'[OF istopology_subtopology]
by auto

lemma topspace_subtopology: "topspace (subtopology U V) = topspace U ∩ V"
by (auto simp: topspace_def openin_subtopology)

lemma closedin_subtopology: "closedin (subtopology U V) S ⟷ (∃T. closedin U T ∧ S = T ∩ V)"
unfolding closedin_def topspace_subtopology
by (auto simp: openin_subtopology)

lemma openin_subtopology_refl: "openin (subtopology U V) V ⟷ V ⊆ topspace U"
unfolding openin_subtopology
by auto (metis IntD1 in_mono openin_subset)

lemma subtopology_superset:
assumes UV: "topspace U ⊆ V"
shows "subtopology U V = U"
proof -
{
fix S
{
fix T
assume T: "openin U T" "S = T ∩ V"
from T openin_subset[OF T(1)] UV have eq: "S = T"
by blast
have "openin U S"
unfolding eq using T by blast
}
moreover
{
assume S: "openin U S"
then have "∃T. openin U T ∧ S = T ∩ V"
using openin_subset[OF S] UV by auto
}
ultimately have "(∃T. openin U T ∧ S = T ∩ V) ⟷ openin U S"
by blast
}
then show ?thesis
unfolding topology_eq openin_subtopology by blast
qed

lemma subtopology_topspace[simp]: "subtopology U (topspace U) = U"

lemma subtopology_UNIV[simp]: "subtopology U UNIV = U"

lemma openin_subtopology_empty:
"openin (subtopology U {}) S ⟷ S = {}"
by (metis Int_empty_right openin_empty openin_subtopology)

lemma closedin_subtopology_empty:
"closedin (subtopology U {}) S ⟷ S = {}"
by (metis Int_empty_right closedin_empty closedin_subtopology)

lemma closedin_subtopology_refl [simp]:
"closedin (subtopology U X) X ⟷ X ⊆ topspace U"
by (metis closedin_def closedin_topspace inf.absorb_iff2 le_inf_iff topspace_subtopology)

lemma openin_imp_subset:
"openin (subtopology U S) T ⟹ T ⊆ S"
by (metis Int_iff openin_subtopology subsetI)

lemma closedin_imp_subset:
"closedin (subtopology U S) T ⟹ T ⊆ S"

lemma openin_subtopology_Un:
"⟦openin (subtopology X T) S; openin (subtopology X U) S⟧
⟹ openin (subtopology X (T ∪ U)) S"

lemma closedin_subtopology_Un:
"⟦closedin (subtopology X T) S; closedin (subtopology X U) S⟧
⟹ closedin (subtopology X (T ∪ U)) S"

subsubsection ‹The standard Euclidean topology›

definition%important euclidean :: "'a::topological_space topology"
where "euclidean = topology open"

lemma open_openin: "open S ⟷ openin euclidean S"
unfolding euclidean_def
apply (rule cong[where x=S and y=S])
apply (rule topology_inverse[symmetric])
apply (auto simp: istopology_def)
done

declare open_openin [symmetric, simp]

lemma topspace_euclidean [simp]: "topspace euclidean = UNIV"
by (force simp: topspace_def)

lemma topspace_euclidean_subtopology[simp]: "topspace (subtopology euclidean S) = S"

lemma closed_closedin: "closed S ⟷ closedin euclidean S"
by (simp add: closed_def closedin_def Compl_eq_Diff_UNIV)

declare closed_closedin [symmetric, simp]

lemma open_subopen: "open S ⟷ (∀x∈S. ∃T. open T ∧ x ∈ T ∧ T ⊆ S)"
using openI by auto

lemma openin_subtopology_self [simp]: "openin (subtopology euclidean S) S"
by (metis openin_topspace topspace_euclidean_subtopology)

text ‹Basic "localization" results are handy for connectedness.›

lemma openin_open: "openin (subtopology euclidean U) S ⟷ (∃T. open T ∧ (S = U ∩ T))"
by (auto simp: openin_subtopology)

lemma openin_Int_open:
"⟦openin (subtopology euclidean U) S; open T⟧
⟹ openin (subtopology euclidean U) (S ∩ T)"
by (metis open_Int Int_assoc openin_open)

lemma openin_open_Int[intro]: "open S ⟹ openin (subtopology euclidean U) (U ∩ S)"
by (auto simp: openin_open)

lemma open_openin_trans[trans]:
"open S ⟹ open T ⟹ T ⊆ S ⟹ openin (subtopology euclidean S) T"
by (metis Int_absorb1  openin_open_Int)

lemma open_subset: "S ⊆ T ⟹ open S ⟹ openin (subtopology euclidean T) S"
by (auto simp: openin_open)

lemma closedin_closed: "closedin (subtopology euclidean U) S ⟷ (∃T. closed T ∧ S = U ∩ T)"

lemma closedin_closed_Int: "closed S ⟹ closedin (subtopology euclidean U) (U ∩ S)"
by (metis closedin_closed)

lemma closed_subset: "S ⊆ T ⟹ closed S ⟹ closedin (subtopology euclidean T) S"
by (auto simp: closedin_closed)

lemma closedin_closed_subset:
"⟦closedin (subtopology euclidean U) V; T ⊆ U; S = V ∩ T⟧
⟹ closedin (subtopology euclidean T) S"
by (metis (no_types, lifting) Int_assoc Int_commute closedin_closed inf.orderE)

lemma finite_imp_closedin:
fixes S :: "'a::t1_space set"
shows "⟦finite S; S ⊆ T⟧ ⟹ closedin (subtopology euclidean T) S"

lemma closedin_singleton [simp]:
fixes a :: "'a::t1_space"
shows "closedin (subtopology euclidean U) {a} ⟷ a ∈ U"
using closedin_subset  by (force intro: closed_subset)

lemma openin_euclidean_subtopology_iff:
fixes S U :: "'a::metric_space set"
shows "openin (subtopology euclidean U) S ⟷
S ⊆ U ∧ (∀x∈S. ∃e>0. ∀x'∈U. dist x' x < e ⟶ x'∈ S)"
(is "?lhs ⟷ ?rhs")
proof
assume ?lhs
then show ?rhs
unfolding openin_open open_dist by blast
next
define T where "T = {x. ∃a∈S. ∃d>0. (∀y∈U. dist y a < d ⟶ y ∈ S) ∧ dist x a < d}"
have 1: "∀x∈T. ∃e>0. ∀y. dist y x < e ⟶ y ∈ T"
unfolding T_def
apply clarsimp
apply (rule_tac x="d - dist x a" in exI)
by (metis dist_commute dist_triangle_lt)
assume ?rhs then have 2: "S = U ∩ T"
unfolding T_def
by auto (metis dist_self)
from 1 2 show ?lhs
unfolding openin_open open_dist by fast
qed

lemma connected_openin:
"connected S ⟷
~(∃E1 E2. openin (subtopology euclidean S) E1 ∧
openin (subtopology euclidean S) E2 ∧
S ⊆ E1 ∪ E2 ∧ E1 ∩ E2 = {} ∧ E1 ≠ {} ∧ E2 ≠ {})"
apply (simp add: connected_def openin_open disjoint_iff_not_equal, safe)
apply (simp_all, blast+)  (* SLOW *)
done

lemma connected_openin_eq:
"connected S ⟷
~(∃E1 E2. openin (subtopology euclidean S) E1 ∧
openin (subtopology euclidean S) E2 ∧
E1 ∪ E2 = S ∧ E1 ∩ E2 = {} ∧
E1 ≠ {} ∧ E2 ≠ {})"
apply (simp add: connected_openin, safe, blast)
by (metis Int_lower1 Un_subset_iff openin_open subset_antisym)

lemma connected_closedin:
"connected S ⟷
(∄E1 E2.
closedin (subtopology euclidean S) E1 ∧
closedin (subtopology euclidean S) E2 ∧
S ⊆ E1 ∪ E2 ∧ E1 ∩ E2 = {} ∧ E1 ≠ {} ∧ E2 ≠ {})"
(is "?lhs = ?rhs")
proof
assume ?lhs
then show ?rhs
by (auto simp add: connected_closed closedin_closed)
next
assume R: ?rhs
then show ?lhs
proof (clarsimp simp add: connected_closed closedin_closed)
fix A B
assume s_sub: "S ⊆ A ∪ B" "B ∩ S ≠ {}"
and disj: "A ∩ B ∩ S = {}"
and cl: "closed A" "closed B"
have "S ∩ (A ∪ B) = S"
using s_sub(1) by auto
have "S - A = B ∩ S"
using Diff_subset_conv Un_Diff_Int disj s_sub(1) by auto
then have "S ∩ A = {}"
by (metis Diff_Diff_Int Diff_disjoint Un_Diff_Int R cl closedin_closed_Int inf_commute order_refl s_sub(2))
then show "A ∩ S = {}"
by blast
qed
qed

lemma connected_closedin_eq:
"connected S ⟷
~(∃E1 E2.
closedin (subtopology euclidean S) E1 ∧
closedin (subtopology euclidean S) E2 ∧
E1 ∪ E2 = S ∧ E1 ∩ E2 = {} ∧
E1 ≠ {} ∧ E2 ≠ {})"
apply (simp add: connected_closedin, safe, blast)
by (metis Int_lower1 Un_subset_iff closedin_closed subset_antisym)

text ‹These "transitivity" results are handy too›

lemma openin_trans[trans]:
"openin (subtopology euclidean T) S ⟹ openin (subtopology euclidean U) T ⟹
openin (subtopology euclidean U) S"
unfolding open_openin openin_open by blast

lemma openin_open_trans: "openin (subtopology euclidean T) S ⟹ open T ⟹ open S"
by (auto simp: openin_open intro: openin_trans)

lemma closedin_trans[trans]:
"closedin (subtopology euclidean T) S ⟹ closedin (subtopology euclidean U) T ⟹
closedin (subtopology euclidean U) S"
by (auto simp: closedin_closed closed_Inter Int_assoc)

lemma closedin_closed_trans: "closedin (subtopology euclidean T) S ⟹ closed T ⟹ closed S"
by (auto simp: closedin_closed intro: closedin_trans)

lemma openin_subtopology_Int_subset:
"⟦openin (subtopology euclidean u) (u ∩ S); v ⊆ u⟧ ⟹ openin (subtopology euclidean v) (v ∩ S)"
by (auto simp: openin_subtopology)

lemma openin_open_eq: "open s ⟹ (openin (subtopology euclidean s) t ⟷ open t ∧ t ⊆ s)"
using open_subset openin_open_trans openin_subset by fastforce

subsection ‹Open and closed balls›

definition%important ball :: "'a::metric_space ⇒ real ⇒ 'a set"
where "ball x e = {y. dist x y < e}"

definition%important cball :: "'a::metric_space ⇒ real ⇒ 'a set"
where "cball x e = {y. dist x y ≤ e}"

definition%important sphere :: "'a::metric_space ⇒ real ⇒ 'a set"
where "sphere x e = {y. dist x y = e}"

lemma mem_ball [simp]: "y ∈ ball x e ⟷ dist x y < e"

lemma mem_cball [simp]: "y ∈ cball x e ⟷ dist x y ≤ e"

lemma mem_sphere [simp]: "y ∈ sphere x e ⟷ dist x y = e"

lemma ball_trivial [simp]: "ball x 0 = {}"

lemma cball_trivial [simp]: "cball x 0 = {x}"

lemma sphere_trivial [simp]: "sphere x 0 = {x}"

lemma mem_ball_0 [simp]: "x ∈ ball 0 e ⟷ norm x < e"
for x :: "'a::real_normed_vector"

lemma mem_cball_0 [simp]: "x ∈ cball 0 e ⟷ norm x ≤ e"
for x :: "'a::real_normed_vector"

lemma disjoint_ballI: "dist x y ≥ r+s ⟹ ball x r ∩ ball y s = {}"

lemma disjoint_cballI: "dist x y > r + s ⟹ cball x r ∩ cball y s = {}"
by (metis add_mono disjoint_iff_not_equal dist_triangle2 dual_order.trans leD mem_cball)

lemma mem_sphere_0 [simp]: "x ∈ sphere 0 e ⟷ norm x = e"
for x :: "'a::real_normed_vector"

lemma sphere_empty [simp]: "r < 0 ⟹ sphere a r = {}"
for a :: "'a::metric_space"
by auto

lemma centre_in_ball [simp]: "x ∈ ball x e ⟷ 0 < e"
by simp

lemma centre_in_cball [simp]: "x ∈ cball x e ⟷ 0 ≤ e"
by simp

lemma ball_subset_cball [simp, intro]: "ball x e ⊆ cball x e"

lemma mem_ball_imp_mem_cball: "x ∈ ball y e ⟹ x ∈ cball y e"
by (auto simp: mem_ball mem_cball)

lemma sphere_cball [simp,intro]: "sphere z r ⊆ cball z r"
by force

lemma cball_diff_sphere: "cball a r - sphere a r = ball a r"
by auto

lemma subset_ball[intro]: "d ≤ e ⟹ ball x d ⊆ ball x e"

lemma subset_cball[intro]: "d ≤ e ⟹ cball x d ⊆ cball x e"

lemma mem_ball_leI: "x ∈ ball y e ⟹ e ≤ f ⟹ x ∈ ball y f"
by (auto simp: mem_ball mem_cball)

lemma mem_cball_leI: "x ∈ cball y e ⟹ e ≤ f ⟹ x ∈ cball y f"
by (auto simp: mem_ball mem_cball)

lemma cball_trans: "y ∈ cball z b ⟹ x ∈ cball y a ⟹ x ∈ cball z (b + a)"
unfolding mem_cball
proof -
have "dist z x ≤ dist z y + dist y x"
by (rule dist_triangle)
also assume "dist z y ≤ b"
also assume "dist y x ≤ a"
finally show "dist z x ≤ b + a" by arith
qed

lemma ball_max_Un: "ball a (max r s) = ball a r ∪ ball a s"

lemma ball_min_Int: "ball a (min r s) = ball a r ∩ ball a s"

lemma cball_max_Un: "cball a (max r s) = cball a r ∪ cball a s"

lemma cball_min_Int: "cball a (min r s) = cball a r ∩ cball a s"

lemma cball_diff_eq_sphere: "cball a r - ball a r =  sphere a r"
by (auto simp: cball_def ball_def dist_commute)

fixes a :: "'a::real_normed_vector"
shows "(+) b ` ball a r = ball (a+b) r"
apply (intro equalityI subsetI)
apply (force simp: dist_norm)
apply (rule_tac x="x-b" in image_eqI)
apply (auto simp: dist_norm algebra_simps)
done

fixes a :: "'a::real_normed_vector"
shows "(+) b ` cball a r = cball (a+b) r"
apply (intro equalityI subsetI)
apply (force simp: dist_norm)
apply (rule_tac x="x-b" in image_eqI)
apply (auto simp: dist_norm algebra_simps)
done

lemma open_ball [intro, simp]: "open (ball x e)"
proof -
have "open (dist x -` {..<e})"
by (intro open_vimage open_lessThan continuous_intros)
also have "dist x -` {..<e} = ball x e"
by auto
finally show ?thesis .
qed

lemma open_contains_ball: "open S ⟷ (∀x∈S. ∃e>0. ball x e ⊆ S)"
by (simp add: open_dist subset_eq mem_ball Ball_def dist_commute)

lemma openI [intro?]: "(⋀x. x∈S ⟹ ∃e>0. ball x e ⊆ S) ⟹ open S"
by (auto simp: open_contains_ball)

lemma openE[elim?]:
assumes "open S" "x∈S"
obtains e where "e>0" "ball x e ⊆ S"
using assms unfolding open_contains_ball by auto

lemma open_contains_ball_eq: "open S ⟹ x∈S ⟷ (∃e>0. ball x e ⊆ S)"
by (metis open_contains_ball subset_eq centre_in_ball)

lemma openin_contains_ball:
"openin (subtopology euclidean t) s ⟷
s ⊆ t ∧ (∀x ∈ s. ∃e. 0 < e ∧ ball x e ∩ t ⊆ s)"
(is "?lhs = ?rhs")
proof
assume ?lhs
then show ?rhs
apply (metis Int_commute Int_mono inf.cobounded2 open_contains_ball order_refl subsetCE)
done
next
assume ?rhs
then show ?lhs
by (metis (no_types) Int_iff dist_commute inf.absorb_iff2 mem_ball)
qed

lemma openin_contains_cball:
"openin (subtopology euclidean t) s ⟷
s ⊆ t ∧
(∀x ∈ s. ∃e. 0 < e ∧ cball x e ∩ t ⊆ s)"
apply (rule iffI)
apply (auto dest!: bspec)
apply (rule_tac x="e/2" in exI, force+)
done

lemma ball_eq_empty[simp]: "ball x e = {} ⟷ e ≤ 0"
unfolding mem_ball set_eq_iff
apply (metis zero_le_dist order_trans dist_self)
done

lemma ball_empty: "e ≤ 0 ⟹ ball x e = {}" by simp

lemma closed_cball [iff]: "closed (cball x e)"
proof -
have "closed (dist x -` {..e})"
by (intro closed_vimage closed_atMost continuous_intros)
also have "dist x -` {..e} = cball x e"
by auto
finally show ?thesis .
qed

lemma open_contains_cball: "open S ⟷ (∀x∈S. ∃e>0.  cball x e ⊆ S)"
proof -
{
fix x and e::real
assume "x∈S" "e>0" "ball x e ⊆ S"
then have "∃d>0. cball x d ⊆ S" unfolding subset_eq by (rule_tac x="e/2" in exI, auto)
}
moreover
{
fix x and e::real
assume "x∈S" "e>0" "cball x e ⊆ S"
then have "∃d>0. ball x d ⊆ S"
unfolding subset_eq
apply (rule_tac x="e/2" in exI, auto)
done
}
ultimately show ?thesis
unfolding open_contains_ball by auto
qed

lemma open_contains_cball_eq: "open S ⟹ (∀x. x ∈ S ⟷ (∃e>0. cball x e ⊆ S))"
by (metis open_contains_cball subset_eq order_less_imp_le centre_in_cball)

lemma euclidean_dist_l2:
fixes x y :: "'a :: euclidean_space"
shows "dist x y = L2_set (λi. dist (x ∙ i) (y ∙ i)) Basis"
unfolding dist_norm norm_eq_sqrt_inner L2_set_def
by (subst euclidean_inner) (simp add: power2_eq_square inner_diff_left)

lemma norm_nth_le: "norm (x ∙ i) ≤ norm x" if "i ∈ Basis"
proof -
have "(x ∙ i)⇧2 = (∑i∈{i}. (x ∙ i)⇧2)"
by simp
also have "… ≤ (∑i∈Basis. (x ∙ i)⇧2)"
by (intro sum_mono2) (auto simp: that)
finally show ?thesis
unfolding norm_conv_dist euclidean_dist_l2[of x] L2_set_def
by (auto intro!: real_le_rsqrt)
qed

lemma eventually_nhds_ball: "d > 0 ⟹ eventually (λx. x ∈ ball z d) (nhds z)"
by (rule eventually_nhds_in_open) simp_all

lemma eventually_at_ball: "d > 0 ⟹ eventually (λt. t ∈ ball z d ∧ t ∈ A) (at z within A)"
unfolding eventually_at by (intro exI[of _ d]) (simp_all add: dist_commute)

lemma eventually_at_ball': "d > 0 ⟹ eventually (λt. t ∈ ball z d ∧ t ≠ z ∧ t ∈ A) (at z within A)"
unfolding eventually_at by (intro exI[of _ d]) (simp_all add: dist_commute)

lemma at_within_ball: "e > 0 ⟹ dist x y < e ⟹ at y within ball x e = at y"
by (subst at_within_open) auto

lemma atLeastAtMost_eq_cball:
fixes a b::real
shows "{a .. b} = cball ((a + b)/2) ((b - a)/2)"
by (auto simp: dist_real_def field_simps mem_cball)

lemma greaterThanLessThan_eq_ball:
fixes a b::real
shows "{a <..< b} = ball ((a + b)/2) ((b - a)/2)"
by (auto simp: dist_real_def field_simps mem_ball)

subsection ‹Boxes›

abbreviation One :: "'a::euclidean_space"
where "One ≡ ∑Basis"

lemma One_non_0: assumes "One = (0::'a::euclidean_space)" shows False
proof -
have "dependent (Basis :: 'a set)"
apply (rule_tac x="λi. 1" in exI)
using SOME_Basis apply (auto simp: assms)
done
with independent_Basis show False by force
qed

corollary One_neq_0[iff]: "One ≠ 0"
by (metis One_non_0)

corollary Zero_neq_One[iff]: "0 ≠ One"
by (metis One_non_0)

definition%important (in euclidean_space) eucl_less (infix "<e" 50)
where "eucl_less a b ⟷ (∀i∈Basis. a ∙ i < b ∙ i)"

definition%important box_eucl_less: "box a b = {x. a <e x ∧ x <e b}"
definition%important "cbox a b = {x. ∀i∈Basis. a ∙ i ≤ x ∙ i ∧ x ∙ i ≤ b ∙ i}"

lemma box_def: "box a b = {x. ∀i∈Basis. a ∙ i < x ∙ i ∧ x ∙ i < b ∙ i}"
and in_box_eucl_less: "x ∈ box a b ⟷ a <e x ∧ x <e b"
and mem_box: "x ∈ box a b ⟷ (∀i∈Basis. a ∙ i < x ∙ i ∧ x ∙ i < b ∙ i)"
"x ∈ cbox a b ⟷ (∀i∈Basis. a ∙ i ≤ x ∙ i ∧ x ∙ i ≤ b ∙ i)"
by (auto simp: box_eucl_less eucl_less_def cbox_def)

lemma cbox_Pair_eq: "cbox (a, c) (b, d) = cbox a b × cbox c d"
by (force simp: cbox_def Basis_prod_def)

lemma cbox_Pair_iff [iff]: "(x, y) ∈ cbox (a, c) (b, d) ⟷ x ∈ cbox a b ∧ y ∈ cbox c d"
by (force simp: cbox_Pair_eq)

lemma cbox_Complex_eq: "cbox (Complex a c) (Complex b d) = (λ(x,y). Complex x y) ` (cbox a b × cbox c d)"
apply (auto simp: cbox_def Basis_complex_def)
apply (rule_tac x = "(Re x, Im x)" in image_eqI)
using complex_eq by auto

lemma cbox_Pair_eq_0: "cbox (a, c) (b, d) = {} ⟷ cbox a b = {} ∨ cbox c d = {}"
by (force simp: cbox_Pair_eq)

lemma swap_cbox_Pair [simp]: "prod.swap ` cbox (c, a) (d, b) = cbox (a,c) (b,d)"
by auto

lemma mem_box_real[simp]:
"(x::real) ∈ box a b ⟷ a < x ∧ x < b"
"(x::real) ∈ cbox a b ⟷ a ≤ x ∧ x ≤ b"
by (auto simp: mem_box)

lemma box_real[simp]:
fixes a b:: real
shows "box a b = {a <..< b}" "cbox a b = {a .. b}"
by auto

lemma box_Int_box:
fixes a :: "'a::euclidean_space"
shows "box a b ∩ box c d =
box (∑i∈Basis. max (a∙i) (c∙i) *⇩R i) (∑i∈Basis. min (b∙i) (d∙i) *⇩R i)"
unfolding set_eq_iff and Int_iff and mem_box by auto

lemma rational_boxes:
fixes x :: "'a::euclidean_space"
assumes "e > 0"
shows "∃a b. (∀i∈Basis. a ∙ i ∈ ℚ ∧ b ∙ i ∈ ℚ) ∧ x ∈ box a b ∧ box a b ⊆ ball x e"
proof -
define e' where "e' = e / (2 * sqrt (real (DIM ('a))))"
then have e: "e' > 0"
using assms by (auto simp: DIM_positive)
have "∀i. ∃y. y ∈ ℚ ∧ y < x ∙ i ∧ x ∙ i - y < e'" (is "∀i. ?th i")
proof
fix i
from Rats_dense_in_real[of "x ∙ i - e'" "x ∙ i"] e
show "?th i" by auto
qed
from choice[OF this] obtain a where
a: "∀xa. a xa ∈ ℚ ∧ a xa < x ∙ xa ∧ x ∙ xa - a xa < e'" ..
have "∀i. ∃y. y ∈ ℚ ∧ x ∙ i < y ∧ y - x ∙ i < e'" (is "∀i. ?th i")
proof
fix i
from Rats_dense_in_real[of "x ∙ i" "x ∙ i + e'"] e
show "?th i" by auto
qed
from choice[OF this] obtain b where
b: "∀xa. b xa ∈ ℚ ∧ x ∙ xa < b xa ∧ b xa - x ∙ xa < e'" ..
let ?a = "∑i∈Basis. a i *⇩R i" and ?b = "∑i∈Basis. b i *⇩R i"
show ?thesis
proof (rule exI[of _ ?a], rule exI[of _ ?b], safe)
fix y :: 'a
assume *: "y ∈ box ?a ?b"
have "dist x y = sqrt (∑i∈Basis. (dist (x ∙ i) (y ∙ i))⇧2)"
unfolding L2_set_def[symmetric] by (rule euclidean_dist_l2)
also have "… < sqrt (∑(i::'a)∈Basis. e^2 / real (DIM('a)))"
proof (rule real_sqrt_less_mono, rule sum_strict_mono)
fix i :: "'a"
assume i: "i ∈ Basis"
have "a i < y∙i ∧ y∙i < b i"
using * i by (auto simp: box_def)
moreover have "a i < x∙i" "x∙i - a i < e'"
using a by auto
moreover have "x∙i < b i" "b i - x∙i < e'"
using b by auto
ultimately have "¦x∙i - y∙i¦ < 2 * e'"
by auto
then have "dist (x ∙ i) (y ∙ i) < e/sqrt (real (DIM('a)))"
unfolding e'_def by (auto simp: dist_real_def)
then have "(dist (x ∙ i) (y ∙ i))⇧2 < (e/sqrt (real (DIM('a))))⇧2"
by (rule power_strict_mono) auto
then show "(dist (x ∙ i) (y ∙ i))⇧2 < e⇧2 / real DIM('a)"
qed auto
also have "… = e"
using ‹0 < e› by simp
finally show "y ∈ ball x e"
by (auto simp: ball_def)
qed (insert a b, auto simp: box_def)
qed

lemma open_UNION_box:
fixes M :: "'a::euclidean_space set"
assumes "open M"
defines "a' ≡ λf :: 'a ⇒ real × real. (∑(i::'a)∈Basis. fst (f i) *⇩R i)"
defines "b' ≡ λf :: 'a ⇒ real × real. (∑(i::'a)∈Basis. snd (f i) *⇩R i)"
defines "I ≡ {f∈Basis →⇩E ℚ × ℚ. box (a' f) (b' f) ⊆ M}"
shows "M = (⋃f∈I. box (a' f) (b' f))"
proof -
have "x ∈ (⋃f∈I. box (a' f) (b' f))" if "x ∈ M" for x
proof -
obtain e where e: "e > 0" "ball x e ⊆ M"
using openE[OF ‹open M› ‹x ∈ M›] by auto
moreover obtain a b where ab:
"x ∈ box a b"
"∀i ∈ Basis. a ∙ i ∈ ℚ"
"∀i∈Basis. b ∙ i ∈ ℚ"
"box a b ⊆ ball x e"
using rational_boxes[OF e(1)] by metis
ultimately show ?thesis
by (intro UN_I[of "λi∈Basis. (a ∙ i, b ∙ i)"])
(auto simp: euclidean_representation I_def a'_def b'_def)
qed
then show ?thesis by (auto simp: I_def)
qed

corollary open_countable_Union_open_box:
fixes S :: "'a :: euclidean_space set"
assumes "open S"
obtains 𝒟 where "countable 𝒟" "𝒟 ⊆ Pow S" "⋀X. X ∈ 𝒟 ⟹ ∃a b. X = box a b" "⋃𝒟 = S"
proof -
let ?a = "λf. (∑(i::'a)∈Basis. fst (f i) *⇩R i)"
let ?b = "λf. (∑(i::'a)∈Basis. snd (f i) *⇩R i)"
let ?I = "{f∈Basis →⇩E ℚ × ℚ. box (?a f) (?b f) ⊆ S}"
let ?𝒟 = "(λf. box (?a f) (?b f)) ` ?I"
show ?thesis
proof
have "countable ?I"
then show "countable ?𝒟"
by blast
show "⋃?𝒟 = S"
using open_UNION_box [OF assms] by metis
qed auto
qed

lemma rational_cboxes:
fixes x :: "'a::euclidean_space"
assumes "e > 0"
shows "∃a b. (∀i∈Basis. a ∙ i ∈ ℚ ∧ b ∙ i ∈ ℚ) ∧ x ∈ cbox a b ∧ cbox a b ⊆ ball x e"
proof -
define e' where "e' = e / (2 * sqrt (real (DIM ('a))))"
then have e: "e' > 0"
using assms by auto
have "∀i. ∃y. y ∈ ℚ ∧ y < x ∙ i ∧ x ∙ i - y < e'" (is "∀i. ?th i")
proof
fix i
from Rats_dense_in_real[of "x ∙ i - e'" "x ∙ i"] e
show "?th i" by auto
qed
from choice[OF this] obtain a where
a: "∀u. a u ∈ ℚ ∧ a u < x ∙ u ∧ x ∙ u - a u < e'" ..
have "∀i. ∃y. y ∈ ℚ ∧ x ∙ i < y ∧ y - x ∙ i < e'" (is "∀i. ?th i")
proof
fix i
from Rats_dense_in_real[of "x ∙ i" "x ∙ i + e'"] e
show "?th i" by auto
qed
from choice[OF this] obtain b where
b: "∀u. b u ∈ ℚ ∧ x ∙ u < b u ∧ b u - x ∙ u < e'" ..
let ?a = "∑i∈Basis. a i *⇩R i" and ?b = "∑i∈Basis. b i *⇩R i"
show ?thesis
proof (rule exI[of _ ?a], rule exI[of _ ?b], safe)
fix y :: 'a
assume *: "y ∈ cbox ?a ?b"
have "dist x y = sqrt (∑i∈Basis. (dist (x ∙ i) (y ∙ i))⇧2)"
unfolding L2_set_def[symmetric] by (rule euclidean_dist_l2)
also have "… < sqrt (∑(i::'a)∈Basis. e^2 / real (DIM('a)))"
proof (rule real_sqrt_less_mono, rule sum_strict_mono)
fix i :: "'a"
assume i: "i ∈ Basis"
have "a i ≤ y∙i ∧ y∙i ≤ b i"
using * i by (auto simp: cbox_def)
moreover have "a i < x∙i" "x∙i - a i < e'"
using a by auto
moreover have "x∙i < b i" "b i - x∙i < e'"
using b by auto
ultimately have "¦x∙i - y∙i¦ < 2 * e'"
by auto
then have "dist (x ∙ i) (y ∙ i) < e/sqrt (real (DIM('a)))"
unfolding e'_def by (auto simp: dist_real_def)
then have "(dist (x ∙ i) (y ∙ i))⇧2 < (e/sqrt (real (DIM('a))))⇧2"
by (rule power_strict_mono) auto
then show "(dist (x ∙ i) (y ∙ i))⇧2 < e⇧2 / real DIM('a)"
qed auto
also have "… = e"
using ‹0 < e› by simp
finally show "y ∈ ball x e"
by (auto simp: ball_def)
next
show "x ∈ cbox (∑i∈Basis. a i *⇩R i) (∑i∈Basis. b i *⇩R i)"
using a b less_imp_le by (auto simp: cbox_def)
qed (use a b cbox_def in auto)
qed

lemma open_UNION_cbox:
fixes M :: "'a::euclidean_space set"
assumes "open M"
defines "a' ≡ λf. (∑(i::'a)∈Basis. fst (f i) *⇩R i)"
defines "b' ≡ λf. (∑(i::'a)∈Basis. snd (f i) *⇩R i)"
defines "I ≡ {f∈Basis →⇩E ℚ × ℚ. cbox (a' f) (b' f) ⊆ M}"
shows "M = (⋃f∈I. cbox (a' f) (b' f))"
proof -
have "x ∈ (⋃f∈I. cbox (a' f) (b' f))" if "x ∈ M" for x
proof -
obtain e where e: "e > 0" "ball x e ⊆ M"
using openE[OF ‹open M› ‹x ∈ M›] by auto
moreover obtain a b where ab: "x ∈ cbox a b" "∀i ∈ Basis. a ∙ i ∈ ℚ"
"∀i ∈ Basis. b ∙ i ∈ ℚ" "cbox a b ⊆ ball x e"
using rational_cboxes[OF e(1)] by metis
ultimately show ?thesis
by (intro UN_I[of "λi∈Basis. (a ∙ i, b ∙ i)"])
(auto simp: euclidean_representation I_def a'_def b'_def)
qed
then show ?thesis by (auto simp: I_def)
qed

corollary open_countable_Union_open_cbox:
fixes S :: "'a :: euclidean_space set"
assumes "open S"
obtains 𝒟 where "countable 𝒟" "𝒟 ⊆ Pow S" "⋀X. X ∈ 𝒟 ⟹ ∃a b. X = cbox a b" "⋃𝒟 = S"
proof -
let ?a = "λf. (∑(i::'a)∈Basis. fst (f i) *⇩R i)"
let ?b = "λf. (∑(i::'a)∈Basis. snd (f i) *⇩R i)"
let ?I = "{f∈Basis →⇩E ℚ × ℚ. cbox (?a f) (?b f) ⊆ S}"
let ?𝒟 = "(λf. cbox (?a f) (?b f)) ` ?I"
show ?thesis
proof
have "countable ?I"
then show "countable ?𝒟"
by blast
show "⋃?𝒟 = S"
using open_UNION_cbox [OF assms] by metis
qed auto
qed

lemma box_eq_empty:
fixes a :: "'a::euclidean_space"
shows "(box a b = {} ⟷ (∃i∈Basis. b∙i ≤ a∙i))" (is ?th1)
and "(cbox a b = {} ⟷ (∃i∈Basis. b∙i < a∙i))" (is ?th2)
proof -
{
fix i x
assume i: "i∈Basis" and as:"b∙i ≤ a∙i" and x:"x∈box a b"
then have "a ∙ i < x ∙ i ∧ x ∙ i < b ∙ i"
unfolding mem_box by (auto simp: box_def)
then have "a∙i < b∙i" by auto
then have False using as by auto
}
moreover
{
assume as: "∀i∈Basis. ¬ (b∙i ≤ a∙i)"
let ?x = "(1/2) *⇩R (a + b)"
{
fix i :: 'a
assume i: "i ∈ Basis"
have "a∙i < b∙i"
using as[THEN bspec[where x=i]] i by auto
then have "a∙i < ((1/2) *⇩R (a+b)) ∙ i" "((1/2) *⇩R (a+b)) ∙ i < b∙i"
}
then have "box a b ≠ {}"
using mem_box(1)[of "?x" a b] by auto
}
ultimately show ?th1 by blast

{
fix i x
assume i: "i ∈ Basis" and as:"b∙i < a∙i" and x:"x∈cbox a b"
then have "a ∙ i ≤ x ∙ i ∧ x ∙ i ≤ b ∙ i"
unfolding mem_box by auto
then have "a∙i ≤ b∙i" by auto
then have False using as by auto
}
moreover
{
assume as:"∀i∈Basis. ¬ (b∙i < a∙i)"
let ?x = "(1/2) *⇩R (a + b)"
{
fix i :: 'a
assume i:"i ∈ Basis"
have "a∙i ≤ b∙i"
using as[THEN bspec[where x=i]] i by auto
then have "a∙i ≤ ((1/2) *⇩R (a+b)) ∙ i" "((1/2) *⇩R (a+b)) ∙ i ≤ b∙i"
}
then have "cbox a b ≠ {}"
using mem_box(2)[of "?x" a b] by auto
}
ultimately show ?th2 by blast
qed

lemma box_ne_empty:
fixes a :: "'a::euclidean_space"
shows "cbox a b ≠ {} ⟷ (∀i∈Basis. a∙i ≤ b∙i)"
and "box a b ≠ {} ⟷ (∀i∈Basis. a∙i < b∙i)"
unfolding box_eq_empty[of a b] by fastforce+

lemma
fixes a :: "'a::euclidean_space"
shows cbox_sing [simp]: "cbox a a = {a}"
and box_sing [simp]: "box a a = {}"
unfolding set_eq_iff mem_box eq_iff [symmetric]
by (auto intro!: euclidean_eqI[where 'a='a])
(metis all_not_in_conv nonempty_Basis)

lemma subset_box_imp:
fixes a :: "'a::euclidean_space"
shows "(∀i∈Basis. a∙i ≤ c∙i ∧ d∙i ≤ b∙i) ⟹ cbox c d ⊆ cbox a b"
and "(∀i∈Basis. a∙i < c∙i ∧ d∙i < b∙i) ⟹ cbox c d ⊆ box a b"
and "(∀i∈Basis. a∙i ≤ c∙i ∧ d∙i ≤ b∙i) ⟹ box c d ⊆ cbox a b"
and "(∀i∈Basis. a∙i ≤ c∙i ∧ d∙i ≤ b∙i) ⟹ box c d ⊆ box a b"
unfolding subset_eq[unfolded Ball_def] unfolding mem_box
by (best intro: order_trans less_le_trans le_less_trans less_imp_le)+

lemma box_subset_cbox:
fixes a :: "'a::euclidean_space"
shows "box a b ⊆ cbox a b"
unfolding subset_eq [unfolded Ball_def] mem_box
by (fast intro: less_imp_le)

lemma subset_box:
fixes a :: "'a::euclidean_space"
shows "cbox c d ⊆ cbox a b ⟷ (∀i∈Basis. c∙i ≤ d∙i) ⟶ (∀i∈Basis. a∙i ≤ c∙i ∧ d∙i ≤ b∙i)" (is ?th1)
and "cbox c d ⊆ box a b ⟷ (∀i∈Basis. c∙i ≤ d∙i) ⟶ (∀i∈Basis. a∙i < c∙i ∧ d∙i < b∙i)" (is ?th2)
and "box c d ⊆ cbox a b ⟷ (∀i∈Basis. c∙i < d∙i) ⟶ (∀i∈Basis. a∙i ≤ c∙i ∧ d∙i ≤ b∙i)" (is ?th3)
and "box c d ⊆ box a b ⟷ (∀i∈Basis. c∙i < d∙i) ⟶ (∀i∈Basis. a∙i ≤ c∙i ∧ d∙i ≤ b∙i)" (is ?th4)
proof -
let ?lesscd = "∀i∈Basis. c∙i < d∙i"
let ?lerhs = "∀i∈Basis. a∙i ≤ c∙i ∧ d∙i ≤ b∙i"
show ?th1 ?th2
by (fastforce simp: mem_box)+
have acdb: "a∙i ≤ c∙i ∧ d∙i ≤ b∙i"
if i: "i ∈ Basis" and box: "box c d ⊆ cbox a b" and cd: "⋀i. i ∈ Basis ⟹ c∙i < d∙i" for i
proof -
have "box c d ≠ {}"
using that
unfolding box_eq_empty by force
{ let ?x = "(∑j∈Basis. (if j=i then ((min (a∙j) (d∙j))+c∙j)/2 else (c∙j+d∙j)/2) *⇩R j)::'a"
assume *: "a∙i > c∙i"
then have "c ∙ j < ?x ∙ j ∧ ?x ∙ j < d ∙ j" if "j ∈ Basis" for j
using cd that by (fastforce simp add: i *)
then have "?x ∈ box c d"
unfolding mem_box by auto
moreover have "?x ∉ cbox a b"
using i cd * by (force simp: mem_box)
ultimately have False using box by auto
}
then have "a∙i ≤ c∙i" by force
moreover
{ let ?x = "(∑j∈Basis. (if j=i then ((max (b∙j) (c∙j))+d∙j)/2 else (c∙j+d∙j)/2) *⇩R j)::'a"
assume *: "b∙i < d∙i"
then have "d ∙ j > ?x ∙ j ∧ ?x ∙ j > c ∙ j" if "j ∈ Basis" for j
using cd that by (fastforce simp add: i *)
then have "?x ∈ box c d"
unfolding mem_box by auto
moreover have "?x ∉ cbox a b"
using i cd * by (force simp: mem_box)
ultimately have False using box by auto
}
then have "b∙i ≥ d∙i" by (rule ccontr) auto
ultimately show ?thesis by auto
qed
show ?th3
using acdb by (fastforce simp add: mem_box)
have acdb': "a∙i ≤ c∙i ∧ d∙i ≤ b∙i"
if "i ∈ Basis" "box c d ⊆ box a b" "⋀i. i ∈ Basis ⟹ c∙i < d∙i" for i
using box_subset_cbox[of a b] that acdb by auto
show ?th4
using acdb' by (fastforce simp add: mem_box)
qed

lemma eq_cbox: "cbox a b = cbox c d ⟷ cbox a b = {} ∧ cbox c d = {} ∨ a = c ∧ b = d"
(is "?lhs = ?rhs")
proof
assume ?lhs
then have "cbox a b ⊆ cbox c d" "cbox c d ⊆ cbox a b"
by auto
then show ?rhs
by (force simp: subset_box box_eq_empty intro: antisym euclidean_eqI)
next
assume ?rhs
then show ?lhs
by force
qed

lemma eq_cbox_box [simp]: "cbox a b = box c d ⟷ cbox a b = {} ∧ box c d = {}"
(is "?lhs ⟷ ?rhs")
proof
assume L: ?lhs
then have "cbox a b ⊆ box c d" "box c d ⊆ cbox a b"
by auto
then show ?rhs
using L box_ne_empty box_sing apply (fastforce simp add:)
done
qed force

lemma eq_box_cbox [simp]: "box a b = cbox c d ⟷ box a b = {} ∧ cbox c d = {}"
by (metis eq_cbox_box)

lemma eq_box: "box a b = box c d ⟷ box a b = {} ∧ box c d = {} ∨ a = c ∧ b = d"
(is "?lhs ⟷ ?rhs")
proof
assume L: ?lhs
then have "box a b ⊆ box c d" "box c d ⊆ box a b"
by auto
then show ?rhs
using box_ne_empty(2) L
apply auto
apply (meson euclidean_eqI less_eq_real_def not_less)+
done
qed force

lemma subset_box_complex:
"cbox a b ⊆ cbox c d ⟷
(Re a ≤ Re b ∧ Im a ≤ Im b) ⟶ Re a ≥ Re c ∧ Im a ≥ Im c ∧ Re b ≤ Re d ∧ Im b ≤ Im d"
"cbox a b ⊆ box c d ⟷
(Re a ≤ Re b ∧ Im a ≤ Im b) ⟶ Re a > Re c ∧ Im a > Im c ∧ Re b < Re d ∧ Im b < Im d"
"box a b ⊆ cbox c d ⟷
(Re a < Re b ∧ Im a < Im b) ⟶ Re a ≥ Re c ∧ Im a ≥ Im c ∧ Re b ≤ Re d ∧ Im b ≤ Im d"
"box a b ⊆ box c d ⟷
(Re a < Re b ∧ Im a < Im b) ⟶ Re a ≥ Re c ∧ Im a ≥ Im c ∧ Re b ≤ Re d ∧ Im b ≤ Im d"
by (subst subset_box; force simp: Basis_complex_def)+

lemma Int_interval:
fixes a :: "'a::euclidean_space"
shows "cbox a b ∩ cbox c d =
cbox (∑i∈Basis. max (a∙i) (c∙i) *⇩R i) (∑i∈Basis. min (b∙i) (d∙i) *⇩R i)"
unfolding set_eq_iff and Int_iff and mem_box
by auto

lemma disjoint_interval:
fixes a::"'a::euclidean_space"
shows "cbox a b ∩ cbox c d = {} ⟷ (∃i∈Basis. (b∙i < a∙i ∨ d∙i < c∙i ∨ b∙i < c∙i ∨ d∙i < a∙i))" (is ?th1)
and "cbox a b ∩ box c d = {} ⟷ (∃i∈Basis. (b∙i < a∙i ∨ d∙i ≤ c∙i ∨ b∙i ≤ c∙i ∨ d∙i ≤ a∙i))" (is ?th2)
and "box a b ∩ cbox c d = {} ⟷ (∃i∈Basis. (b∙i ≤ a∙i ∨ d∙i < c∙i ∨ b∙i ≤ c∙i ∨ d∙i ≤ a∙i))" (is ?th3)
and "box a b ∩ box c d = {} ⟷ (∃i∈Basis. (b∙i ≤ a∙i ∨ d∙i ≤ c∙i ∨ b∙i ≤ c∙i ∨ d∙i ≤ a∙i))" (is ?th4)
proof -
let ?z = "(∑i∈Basis. (((max (a∙i) (c∙i)) + (min (b∙i) (d∙i))) / 2) *⇩R i)::'a"
have **: "⋀P Q. (⋀i :: 'a. i ∈ Basis ⟹ Q ?z i ⟹ P i) ⟹
(⋀i x :: 'a. i ∈ Basis ⟹ P i ⟹ Q x i) ⟹ (∀x. ∃i∈Basis. Q x i) ⟷ (∃i∈Basis. P i)"
by blast
note * = set_eq_iff Int_iff empty_iff mem_box ball_conj_distrib[symmetric] eq_False ball_simps(10)
show ?th1 unfolding * by (intro **) auto
show ?th2 unfolding * by (intro **) auto
show ?th3 unfolding * by (intro **) auto
show ?th4 unfolding * by (intro **) auto
qed

lemma UN_box_eq_UNIV: "(⋃i::nat. box (- (real i *⇩R One)) (real i *⇩R One)) = UNIV"
proof -
have "¦x ∙ b¦ < real_of_int (⌈Max ((λb. ¦x ∙ b¦)`Basis)⌉ + 1)"
if [simp]: "b ∈ Basis" for x b :: 'a
proof -
have "¦x ∙ b¦ ≤ real_of_int ⌈¦x ∙ b¦⌉"
by (rule le_of_int_ceiling)
also have "… ≤ real_of_int ⌈Max ((λb. ¦x ∙ b¦)`Basis)⌉"
by (auto intro!: ceiling_mono)
also have "… < real_of_int (⌈Max ((λb. ¦x ∙ b¦)`Basis)⌉ + 1)"
by simp
finally show ?thesis .
qed
then have "∃n::nat. ∀b∈Basis. ¦x ∙ b¦ < real n" for x :: 'a
by (metis order.strict_trans reals_Archimedean2)
moreover have "⋀x b::'a. ⋀n::nat.  ¦x ∙ b¦ < real n ⟷ - real n < x ∙ b ∧ x ∙ b < real n"
by auto
ultimately show ?thesis
by (auto simp: box_def inner_sum_left inner_Basis sum.If_cases)
qed

subsection ‹Intervals in general, including infinite and mixtures of open and closed›

definition%important "is_interval (s::('a::euclidean_space) set) ⟷
(∀a∈s. ∀b∈s. ∀x. (∀i∈Basis. ((a∙i ≤ x∙i ∧ x∙i ≤ b∙i) ∨ (b∙i ≤ x∙i ∧ x∙i ≤ a∙i))) ⟶ x ∈ s)"

lemma is_interval_1:
"is_interval (s::real set) ⟷ (∀a∈s. ∀b∈s. ∀ x. a ≤ x ∧ x ≤ b ⟶ x ∈ s)"
unfolding is_interval_def by auto

lemma is_interval_inter: "is_interval X ⟹ is_interval Y ⟹ is_interval (X ∩ Y)"
unfolding is_interval_def
by blast

lemma is_interval_cbox [simp]: "is_interval (cbox a (b::'a::euclidean_space))" (is ?th1)
and is_interval_box [simp]: "is_interval (box a b)" (is ?th2)
unfolding is_interval_def mem_box Ball_def atLeastAtMost_iff
by (meson order_trans le_less_trans less_le_trans less_trans)+

lemma is_interval_empty [iff]: "is_interval {}"
unfolding is_interval_def  by simp

lemma is_interval_univ [iff]: "is_interval UNIV"
unfolding is_interval_def  by simp

lemma mem_is_intervalI:
assumes "is_interval s"
and "a ∈ s" "b ∈ s"
and "⋀i. i ∈ Basis ⟹ a ∙ i ≤ x ∙ i ∧ x ∙ i ≤ b ∙ i ∨ b ∙ i ≤ x ∙ i ∧ x ∙ i ≤ a ∙ i"
shows "x ∈ s"
by (rule assms(1)[simplified is_interval_def, rule_format, OF assms(2,3,4)])

lemma interval_subst:
fixes S::"'a::euclidean_space set"
assumes "is_interval S"
and "x ∈ S" "y j ∈ S"
and "j ∈ Basis"
shows "(∑i∈Basis. (if i = j then y i ∙ i else x ∙ i) *⇩R i) ∈ S"
by (rule mem_is_intervalI[OF assms(1,2)]) (auto simp: assms)

lemma mem_box_componentwiseI:
fixes S::"'a::euclidean_space set"
assumes "is_interval S"
assumes "⋀i. i ∈ Basis ⟹ x ∙ i ∈ ((λx. x ∙ i) ` S)"
shows "x ∈ S"
proof -
from assms have "∀i ∈ Basis. ∃s ∈ S. x ∙ i = s ∙ i"
by auto
with finite_Basis obtain s and bs::"'a list"
where s: "⋀i. i ∈ Basis ⟹ x ∙ i = s i ∙ i" "⋀i. i ∈ Basis ⟹ s i ∈ S"
and bs: "set bs = Basis" "distinct bs"
by (metis finite_distinct_list)
from nonempty_Basis s obtain j where j: "j ∈ Basis" "s j ∈ S"
by blast
define y where
"y = rec_list (s j) (λj _ Y. (∑i∈Basis. (if i = j then s i ∙ i else Y ∙ i) *⇩R i))"
have "x = (∑i∈Basis. (if i ∈ set bs then s i ∙ i else s j ∙ i) *⇩R i)"
using bs by (auto simp: s(1)[symmetric] euclidean_representation)
also have [symmetric]: "y bs = …"
using bs(2) bs(1)[THEN equalityD1]
by (induct bs) (auto simp: y_def euclidean_representation intro!: euclidean_eqI[where 'a='a])
also have "y bs ∈ S"
using bs(1)[THEN equalityD1]
apply (induct bs)
apply (auto simp: y_def j)
apply (rule interval_subst[OF assms(1)])
apply (auto simp: s)
done
finally show ?thesis .
qed

lemma cbox01_nonempty [simp]: "cbox 0 One ≠ {}"
by (simp add: box_ne_empty inner_Basis inner_sum_left sum_nonneg)

lemma box01_nonempty [simp]: "box 0 One ≠ {}"
by (simp add: box_ne_empty inner_Basis inner_sum_left)

lemma empty_as_interval: "{} = cbox One (0::'a::euclidean_space)"
using nonempty_Basis box01_nonempty box_eq_empty(1) box_ne_empty(1) by blast

lemma interval_subset_is_interval:
assumes "is_interval S"
shows "cbox a b ⊆ S ⟷ cbox a b = {} ∨ a ∈ S ∧ b ∈ S" (is "?lhs = ?rhs")
proof
assume ?lhs
then show ?rhs  using box_ne_empty(1) mem_box(2) by fastforce
next
assume ?rhs
have "cbox a b ⊆ S" if "a ∈ S" "b ∈ S"
using assms unfolding is_interval_def
using that by blast
with ‹?rhs› show ?lhs
by blast
qed

lemma is_real_interval_union:
"is_interval (X ∪ Y)"
if X: "is_interval X" and Y: "is_interval Y" and I: "(X ≠ {} ⟹ Y ≠ {} ⟹ X ∩ Y ≠ {})"
for X Y::"real set"
proof -
consider "X ≠ {}" "Y ≠ {}" | "X = {}" | "Y = {}" by blast
then show ?thesis
proof cases
case 1
then obtain r where "r ∈ X ∨ X ∩ Y = {}" "r ∈ Y ∨ X ∩ Y = {}"
by blast
then show ?thesis
using I 1 X Y unfolding is_interval_1
by (metis (full_types) Un_iff le_cases)
qed (use that in auto)
qed

lemma is_interval_translationI:
assumes "is_interval X"
shows "is_interval ((+) x ` X)"
unfolding is_interval_def
proof safe
fix b d e
assume "b ∈ X" "d ∈ X"
"∀i∈Basis. (x + b) ∙ i ≤ e ∙ i ∧ e ∙ i ≤ (x + d) ∙ i ∨
(x + d) ∙ i ≤ e ∙ i ∧ e ∙ i ≤ (x + b) ∙ i"
hence "e - x ∈ X"
by (intro mem_is_intervalI[OF assms ‹b ∈ X› ‹d ∈ X›, of "e - x"])
(auto simp: algebra_simps)
thus "e ∈ (+) x ` X" by force
qed

lemma is_interval_uminusI:
assumes "is_interval X"
shows "is_interval (uminus ` X)"
unfolding is_interval_def
proof safe
fix b d e
assume "b ∈ X" "d ∈ X"
"∀i∈Basis. (- b) ∙ i ≤ e ∙ i ∧ e ∙ i ≤ (- d) ∙ i ∨
(- d) ∙ i ≤ e ∙ i ∧ e ∙ i ≤ (- b) ∙ i"
hence "- e ∈ X"
by (intro mem_is_intervalI[OF assms ‹b ∈ X› ‹d ∈ X›, of "- e"])
(auto simp: algebra_simps)
thus "e ∈ uminus ` X" by force
qed

lemma is_interval_uminus[simp]: "is_interval (uminus ` x) = is_interval x"
using is_interval_uminusI[of x] is_interval_uminusI[of "uminus ` x"]
by (auto simp: image_image)

lemma is_interval_neg_translationI:
assumes "is_interval X"
shows "is_interval ((-) x ` X)"
proof -
have "(-) x ` X = (+) x ` uminus ` X"
by (force simp: algebra_simps)
also have "is_interval …"
by (metis is_interval_uminusI is_interval_translationI assms)
finally show ?thesis .
qed

lemma is_interval_translation[simp]:
"is_interval ((+) x ` X) = is_interval X"
using is_interval_neg_translationI[of "(+) x ` X" x]
by (auto intro!: is_interval_translationI simp: image_image)

lemma is_interval_minus_translation[simp]:
shows "is_interval ((-) x ` X) = is_interval X"
proof -
have "(-) x ` X = (+) x ` uminus ` X"
by (force simp: algebra_simps)
also have "is_interval … = is_interval X"
by simp
finally show ?thesis .
qed

lemma is_interval_minus_translation'[simp]:
shows "is_interval ((λx. x - c) ` X) = is_interval X"
using is_interval_translation[of "-c" X]

subsection ‹Limit points›

definition%important (in topological_space) islimpt:: "'a ⇒ 'a set ⇒ bool"  (infixr "islimpt" 60)
where "x islimpt S ⟷ (∀T. x∈T ⟶ open T ⟶ (∃y∈S. y∈T ∧ y≠x))"

lemma islimptI:
assumes "⋀T. x ∈ T ⟹ open T ⟹ ∃y∈S. y ∈ T ∧ y ≠ x"
shows "x islimpt S"
using assms unfolding islimpt_def by auto

lemma islimptE:
assumes "x islimpt S" and "x ∈ T" and "open T"
obtains y where "y ∈ S" and "y ∈ T" and "y ≠ x"
using assms unfolding islimpt_def by auto

lemma islimpt_iff_eventually: "x islimpt S ⟷ ¬ eventually (λy. y ∉ S) (at x)"
unfolding islimpt_def eventually_at_topological by auto

lemma islimpt_subset: "x islimpt S ⟹ S ⊆ T ⟹ x islimpt T"
unfolding islimpt_def by fast

lemma islimpt_approachable:
fixes x :: "'a::metric_space"
shows "x islimpt S ⟷ (∀e>0. ∃x'∈S. x' ≠ x ∧ dist x' x < e)"
unfolding islimpt_iff_eventually eventually_at by fast

lemma islimpt_approachable_le: "x islimpt S ⟷ (∀e>0. ∃x'∈ S. x' ≠ x ∧ dist x' x ≤ e)"
for x :: "'a::metric_space"
unfolding islimpt_approachable
using approachable_lt_le [where f="λy. dist y x" and P="λy. y ∉ S ∨ y = x",
THEN arg_cong [where f=Not]]
by (simp add: Bex_def conj_commute conj_left_commute)

lemma islimpt_UNIV_iff: "x islimpt UNIV ⟷ ¬ open {x}"
unfolding islimpt_def by (safe, fast, case_tac "T = {x}", fast, fast)

lemma islimpt_punctured: "x islimpt S = x islimpt (S-{x})"
unfolding islimpt_def by blast

text ‹A perfect space has no isolated points.›

lemma islimpt_UNIV [simp, intro]: "x islimpt UNIV"
for x :: "'a::perfect_space"
unfolding islimpt_UNIV_iff by (rule not_open_singleton)

lemma perfect_choose_dist: "0 < r ⟹ ∃a. a ≠ x ∧ dist a x < r"
for x :: "'a::{perfect_space,metric_space}"
using islimpt_UNIV [of x] by (simp add: islimpt_approachable)

lemma closed_limpt: "closed S ⟷ (∀x. x islimpt S ⟶ x ∈ S)"
unfolding closed_def
apply (subst open_subopen)
apply (metis ComplE ComplI)
done

lemma islimpt_EMPTY[simp]: "¬ x islimpt {}"
by (auto simp: islimpt_def)

lemma finite_ball_include:
fixes a :: "'a::metric_space"
assumes "finite S"
shows "∃e>0. S ⊆ ball a e"
using assms
proof induction
case (insert x S)
then obtain e0 where "e0>0" and e0:"S ⊆ ball a e0" by auto
define e where "e = max e0 (2 * dist a x)"
have "e>0" unfolding e_def using ‹e0>0› by auto
moreover have "insert x S ⊆ ball a e"
using e0 ‹e>0› unfolding e_def by auto
ultimately show ?case by auto
qed (auto intro: zero_less_one)

lemma finite_set_avoid:
fixes a :: "'a::metric_space"
assumes "finite S"
shows "∃d>0. ∀x∈S. x ≠ a ⟶ d ≤ dist a x"
using assms
proof induction
case (insert x S)
then obtain d where "d > 0" and d: "∀x∈S. x ≠ a ⟶ d ≤ dist a x"
by blast
show ?case
proof (cases "x = a")
case True
with ‹d > 0 ›d show ?thesis by auto
next
case False
let ?d = "min d (dist a x)"
from False ‹d > 0› have dp: "?d > 0"
by auto
from d have d': "∀x∈S. x ≠ a ⟶ ?d ≤ dist a x"
by auto
with dp False show ?thesis
by (metis insert_iff le_less min_less_iff_conj not_less)
qed
qed (auto intro: zero_less_one)

lemma islimpt_Un: "x islimpt (S ∪ T) ⟷ x islimpt S ∨ x islimpt T"

lemma discrete_imp_closed:
fixes S :: "'a::metric_space set"
assumes e: "0 < e"
and d: "∀x ∈ S. ∀y ∈ S. dist y x < e ⟶ y = x"
shows "closed S"
proof -
have False if C: "⋀e. e>0 ⟹ ∃x'∈S. x' ≠ x ∧ dist x' x < e" for x
proof -
from e have e2: "e/2 > 0" by arith
from C[rule_format, OF e2] obtain y where y: "y ∈ S" "y ≠ x" "dist y x < e/2"
by blast
let ?m = "min (e/2) (dist x y) "
from e2 y(2) have mp: "?m > 0"
by simp
from C[OF mp] obtain z where z: "z ∈ S" "z ≠ x" "dist z x < ?m"
by blast
from z y have "dist z y < e"
by (intro dist_triangle_lt [where z=x]) simp
from d[rule_format, OF y(1) z(1) this] y z show ?thesis
by (auto simp: dist_commute)
qed
then show ?thesis
by (metis islimpt_approachable closed_limpt [where 'a='a])
qed

lemma closed_of_nat_image: "closed (of_nat ` A :: 'a::real_normed_algebra_1 set)"
by (rule discrete_imp_closed[of 1]) (auto simp: dist_of_nat)

lemma closed_of_int_image: "closed (of_int ` A :: 'a::real_normed_algebra_1 set)"
by (rule discrete_imp_closed[of 1]) (auto simp: dist_of_int)

lemma closed_Nats [simp]: "closed (ℕ :: 'a :: real_normed_algebra_1 set)"
unfolding Nats_def by (rule closed_of_nat_image)

lemma closed_Ints [simp]: "closed (ℤ :: 'a :: real_normed_algebra_1 set)"
unfolding Ints_def by (rule closed_of_int_image)

lemma closed_subset_Ints:
fixes A :: "'a :: real_normed_algebra_1 set"
assumes "A ⊆ ℤ"
shows   "closed A"
proof (intro discrete_imp_closed[OF zero_less_one] ballI impI, goal_cases)
case (1 x y)
with assms have "x ∈ ℤ" and "y ∈ ℤ" by auto
with ‹dist y x < 1› show "y = x"
by (auto elim!: Ints_cases simp: dist_of_int)
qed

subsection ‹Interior of a Set›

definition%important "interior S = ⋃{T. open T ∧ T ⊆ S}"

lemma interiorI [intro?]:
assumes "open T" and "x ∈ T" and "T ⊆ S"
shows "x ∈ interior S"
using assms unfolding interior_def by fast

lemma interiorE [elim?]:
assumes "x ∈ interior S"
obtains T where "open T" and "x ∈ T" and "T ⊆ S"
using assms unfolding interior_def by fast

lemma open_interior [simp, intro]: "open (interior S)"

lemma interior_subset: "interior S ⊆ S"
by (auto simp: interior_def)

lemma interior_maximal: "T ⊆ S ⟹ open T ⟹ T ⊆ interior S"
by (auto simp: interior_def)

lemma interior_open: "open S ⟹ interior S = S"
by (intro equalityI interior_subset interior_maximal subset_refl)

lemma interior_eq: "interior S = S ⟷ open S"
by (metis open_interior interior_open)

lemma open_subset_interior: "open S ⟹ S ⊆ interior T ⟷ S ⊆ T"
by (metis interior_maximal interior_subset subset_trans)

lemma interior_empty [simp]: "interior {} = {}"
using open_empty by (rule interior_open)

lemma interior_UNIV [simp]: "interior UNIV = UNIV"
using open_UNIV by (rule interior_open)

lemma interior_interior [simp]: "interior (interior S) = interior S"
using open_interior by (rule interior_open)

lemma interior_mono: "S ⊆ T ⟹ interior S ⊆ interior T"
by (auto simp: interior_def)

lemma interior_unique:
assumes "T ⊆ S" and "open T"
assumes "⋀T'. T' ⊆ S ⟹ open T' ⟹ T' ⊆ T"
shows "interior S = T"
by (intro equalityI assms interior_subset open_interior interior_maximal)

lemma interior_singleton [simp]: "interior {a} = {}"
for a :: "'a::perfect_space"
apply (rule interior_unique, simp_all)
using not_open_singleton subset_singletonD
apply fastforce
done

lemma interior_Int [simp]: "interior (S ∩ T) = interior S ∩ interior T"
by (intro equalityI Int_mono Int_greatest interior_mono Int_lower1
Int_lower2 interior_maximal interior_subset open_Int open_interior)

lemma mem_interior: "x ∈ interior S ⟷ (∃e>0. ball x e ⊆ S)"
using open_contains_ball_eq [where S="interior S"]

lemma eventually_nhds_in_nhd: "x ∈ interior s ⟹ eventually (λy. y ∈ s) (nhds x)"
using interior_subset[of s] by (subst eventually_nhds) blast

lemma interior_limit_point [intro]:
fixes x :: "'a::perfect_space"
assumes x: "x ∈ interior S"
shows "x islimpt S"
using x islimpt_UNIV [of x]
unfolding interior_def islimpt_def
apply (clarsimp, rename_tac T T')
apply (drule_tac x="T ∩ T'" in spec)
apply (auto simp: open_Int)
done

lemma interior_closed_Un_empty_interior:
assumes cS: "closed S"
and iT: "interior T = {}"
shows "interior (S ∪ T) = interior S"
proof
show "interior S ⊆ interior (S ∪ T)"
by (rule interior_mono) (rule Un_upper1)
show "interior (S ∪ T) ⊆ interior S"
proof
fix x
assume "x ∈ interior (S ∪ T)"
then obtain R where "open R" "x ∈ R" "R ⊆ S ∪ T" ..
show "x ∈ interior S"
proof (rule ccontr)
assume "x ∉ interior S"
with ‹x ∈ R› ‹open R› obtain y where "y ∈ R - S"
unfolding interior_def by fast
from ‹open R› ‹closed S› have "open (R - S)"
by (rule open_Diff)
from ‹R ⊆ S ∪ T› have "R - S ⊆ T"
by fast
from ‹y ∈ R - S› ‹open (R - S)› ‹R - S ⊆ T› ‹interior T = {}› show False
unfolding interior_def by fast
qed
qed
qed

lemma interior_Times: "interior (A × B) = interior A × interior B"
proof (rule interior_unique)
show "interior A × interior B ⊆ A × B"
by (intro Sigma_mono interior_subset)
show "open (interior A × interior B)"
by (intro open_Times open_interior)
fix T
assume "T ⊆ A × B" and "open T"
then show "T ⊆ interior A × interior B"
proof safe
fix x y
assume "(x, y) ∈ T"
then obtain C D where "open C" "open D" "C × D ⊆ T" "x ∈ C" "y ∈ D"
using ‹open T› unfolding open_prod_def by fast
then have "open C" "open D" "C ⊆ A" "D ⊆ B" "x ∈ C" "y ∈ D"
using ‹T ⊆ A × B› by auto
then show "x ∈ interior A" and "y ∈ interior B"
by (auto intro: interiorI)
qed
qed

lemma interior_Ici:
fixes x :: "'a :: {dense_linorder,linorder_topology}"
assumes "b < x"
shows "interior {x ..} = {x <..}"
proof (rule interior_unique)
fix T
assume "T ⊆ {x ..}" "open T"
moreover have "x ∉ T"
proof
assume "x ∈ T"
obtain y where "y < x" "{y <.. x} ⊆ T"
using open_left[OF ‹open T› ‹x ∈ T› ‹b < x›] by auto
with dense[OF ‹y < x›] obtain z where "z ∈ T" "z < x"
by (auto simp: subset_eq Ball_def)
with ‹T ⊆ {x ..}› show False by auto
qed
ultimately show "T ⊆ {x <..}"
by (auto simp: subset_eq less_le)
qed auto

lemma interior_Iic:
fixes x :: "'a ::{dense_linorder,linorder_topology}"
assumes "x < b"
shows "interior {.. x} = {..< x}"
proof (rule interior_unique)
fix T
assume "T ⊆ {.. x}" "open T"
moreover have "x ∉ T"
proof
assume "x ∈ T"
obtain y where "x < y" "{x ..< y} ⊆ T"
using open_right[OF ‹open T› ‹x ∈ T› ‹x < b›] by auto
with dense[OF ‹x < y›] obtain z where "z ∈ T" "x < z"
by (auto simp: subset_eq Ball_def less_le)
with ‹T ⊆ {.. x}› show False by auto
qed
ultimately show "T ⊆ {..< x}"
by (auto simp: subset_eq less_le)
qed auto

subsection ‹Closure of a Set›

definition%important "closure S = S ∪ {x | x. x islimpt S}"

lemma interior_closure: "interior S = - (closure (- S))"
by (auto simp: interior_def closure_def islimpt_def)

lemma closure_interior: "closure S = - interior (- S)"

lemma closed_closure[simp, intro]: "closed (closure S)"

lemma closure_subset: "S ⊆ closure S"

lemma closure_hull: "closure S = closed hull S"
by (auto simp: hull_def closure_interior interior_def)

lemma closure_eq: "closure S = S ⟷ closed S"
unfolding closure_hull using closed_Inter by (rule hull_eq)

lemma closure_closed [simp]: "closed S ⟹ closure S = S"
by (simp only: closure_eq)

lemma closure_closure [simp]: "closure (closure S) = closure S"
unfolding closure_hull by (rule hull_hull)

lemma closure_mono: "S ⊆ T ⟹ closure S ⊆ closure T"
unfolding closure_hull by (rule hull_mono)

lemma closure_minimal: "S ⊆ T ⟹ closed T ⟹ closure S ⊆ T"
unfolding closure_hull by (rule hull_minimal)

lemma closure_unique:
assumes "S ⊆ T"
and "closed T"
and "⋀T'. S ⊆ T' ⟹ closed T' ⟹ T ⊆ T'"
shows "closure S = T"
using assms unfolding closure_hull by (rule hull_unique)

lemma closure_empty [simp]: "closure {} = {}"
using closed_empty by (rule closure_closed)

lemma closure_UNIV [simp]: "closure UNIV = UNIV"
using closed_UNIV by (rule closure_closed)

lemma closure_Un [simp]: "closure (S ∪ T) = closure S ∪ closure T"

lemma closure_eq_empty [iff]: "closure S = {} ⟷ S = {}"
using closure_empty closure_subset[of S] by blast

lemma closure_subset_eq: "closure S ⊆ S ⟷ closed S"
using closure_eq[of S] closure_subset[of S] by simp

lemma open_Int_closure_eq_empty: "open S ⟹ (S ∩ closure T) = {} ⟷ S ∩ T = {}"
using open_subset_interior[of S "- T"]
using interior_subset[of "- T"]
by (auto simp: closure_interior)

lemma open_Int_closure_subset: "open S ⟹ S ∩ closure T ⊆ closure (S ∩ T)"
proof
fix x
assume *: "open S" "x ∈ S ∩ closure T"
have "x islimpt (S ∩ T)" if **: "x islimpt T"
proof (rule islimptI)
fix A
assume "x ∈ A" "open A"
with * have "x ∈ A ∩ S" "open (A ∩ S)"
with ** obtain y where "y ∈ T" "y ∈ A ∩ S" "y ≠ x"
by (rule islimptE)
then have "y ∈ S ∩ T" "y ∈ A ∧ y ≠ x"
by simp_all
then show "∃y∈(S ∩ T). y ∈ A ∧ y ≠ x" ..
qed
with * show "x ∈ closure (S ∩ T)"
unfolding closure_def by blast
qed

lemma closure_complement: "closure (- S) = - interior S"

lemma interior_complement: "interior (- S) = - closure S"

lemma interior_diff: "interior(S - T) = interior S - closure T"

lemma closure_Times: "closure (A × B) = closure A × closure B"
proof (rule closure_unique)
show "A × B ⊆ closure A × closure B"
by (intro Sigma_mono closure_subset)
show "closed (closure A × closure B)"
by (intro closed_Times closed_closure)
fix T
assume "A × B ⊆ T" and "closed T"
then show "closure A × closure B ⊆ T"
apply (simp add: closed_def open_prod_def, clarify)
apply (rule ccontr)
apply (drule_tac x="(a, b)" in bspec, simp, clarify, rename_tac C D)
apply (drule_tac x=C in spec)
apply (drule_tac x=D in spec, auto)
done
qed

lemma closure_openin_Int_closure:
assumes ope: "openin (subtopology euclidean U) S" and "T ⊆ U"
shows "closure(S ∩ closure T) = closure(S ∩ T)"
proof
obtain V where "open V" and S: "S = U ∩ V"
using ope using openin_open by metis
show "closure (S ∩ closure T) ⊆ closure (S ∩ T)"
proof (clarsimp simp: S)
fix x
assume  "x ∈ closure (U ∩ V ∩ closure T)"
then have "V ∩ closure T ⊆ A ⟹ x ∈ closure A" for A
by (metis closure_mono subsetD inf.coboundedI2 inf_assoc)
then have "x ∈ closure (T ∩ V)"
by (metis ‹open V› closure_closure inf_commute open_Int_closure_subset)
then show "x ∈ closure (U ∩ V ∩ T)"
by (metis ‹T ⊆ U› inf.absorb_iff2 inf_assoc inf_commute)
qed
next
show "closure (S ∩ T) ⊆ closure (S ∩ closure T)"
by (meson Int_mono closure_mono closure_subset order_refl)
qed

lemma islimpt_in_closure: "(x islimpt S) = (x∈closure(S-{x}))"
unfolding closure_def using islimpt_punctured by blast

lemma connected_imp_connected_closure: "connected S ⟹ connected (closure S)"
by (rule connectedI) (meson closure_subset open_Int open_Int_closure_eq_empty subset_trans connectedD)

lemma limpt_of_limpts: "x islimpt {y. y islimpt S} ⟹ x islimpt S"
for x :: "'a::metric_space"
apply (drule_tac x="e/2" in spec)
apply (auto simp: simp del: less_divide_eq_numeral1)
apply (drule_tac x="dist x' x" in spec)
apply (auto simp: zero_less_dist_iff simp del: less_divide_eq_numeral1)
apply (erule rev_bexI)
apply (metis dist_commute dist_triangle_half_r less_trans less_irrefl)
done

lemma closed_limpts:  "closed {x::'a::metric_space. x islimpt S}"
using closed_limpt limpt_of_limpts by blast

lemma limpt_of_closure: "x islimpt closure S ⟷ x islimpt S"
for x :: "'a::metric_space"
by (auto simp: closure_def islimpt_Un dest: limpt_of_limpts)

lemma closedin_limpt:
"closedin (subtopology euclidean T) S ⟷ S ⊆ T ∧ (∀x. x islimpt S ∧ x ∈ T ⟶ x ∈ S)"
apply (rule_tac x="closure S" in exI, simp)
apply (force simp: closure_def)
done

lemma closedin_closed_eq: "closed S ⟹ closedin (subtopology euclidean S) T ⟷ closed T ∧ T ⊆ S"
by (meson closedin_limpt closed_subset closedin_closed_trans)

lemma connected_closed_set:
"closed S
⟹ connected S ⟷ (∄A B. closed A ∧ closed B ∧ A ≠ {} ∧ B ≠ {} ∧ A ∪ B = S ∧ A ∩ B = {})"
unfolding connected_closedin_eq closedin_closed_eq connected_closedin_eq by blast

text ‹If a connnected set is written as the union of two nonempty closed sets, then these sets
have to intersect.›

lemma connected_as_closed_union:
assumes "connected C" "C = A ∪ B" "closed A" "closed B" "A ≠ {}" "B ≠ {}"
shows "A ∩ B ≠ {}"
by (metis assms closed_Un connected_closed_set)

lemma closedin_subset_trans:
"closedin (subtopology euclidean U) S ⟹ S ⊆ T ⟹ T ⊆ U ⟹
closedin (subtopology euclidean T) S"
by (meson closedin_limpt subset_iff)

lemma openin_subset_trans:
"openin (subtopology euclidean U) S ⟹ S ⊆ T ⟹ T ⊆ U ⟹
openin (subtopology euclidean T) S"
by (auto simp: openin_open)

lemma openin_Times:
"openin (subtopology euclidean S) S' ⟹ openin (subtopology euclidean T) T' ⟹
openin (subtopology euclidean (S × T)) (S' × T')"
unfolding openin_open using open_Times by blast

lemma Times_in_interior_subtopology:
fixes U :: "('a::metric_space × 'b::metric_space) set"
assumes "(x, y) ∈ U" "openin (subtopology euclidean (S × T)) U"
obtains V W where "openin (subtopology euclidean S) V" "x ∈ V"
"openin (subtopology euclidean T) W" "y ∈ W" "(V × W) ⊆ U"
proof -
from assms obtain e where "e > 0" and "U ⊆ S × T"
and e: "⋀x' y'. ⟦x'∈S; y'∈T; dist (x', y') (x, y) < e⟧ ⟹ (x', y') ∈ U"
by (force simp: openin_euclidean_subtopology_iff)
with assms have "x ∈ S" "y ∈ T"
by auto
show ?thesis
proof
show "openin (subtopology euclidean S) (ball x (e/2) ∩ S)"
show "x ∈ ball x (e / 2) ∩ S"
by (simp add: ‹0 < e› ‹x ∈ S›)
show "openin (subtopology euclidean T) (ball y (e/2) ∩ T)"
show "y ∈ ball y (e / 2) ∩ T"
by (simp add: ‹0 < e› ‹y ∈ T›)
show "(ball x (e / 2) ∩ S) × (ball y (e / 2) ∩ T) ⊆ U"
by clarify (simp add: e dist_Pair_Pair ‹0 < e› dist_commute sqrt_sum_squares_half_less)
qed
qed

lemma openin_Times_eq:
fixes S :: "'a::metric_space set" and T :: "'b::metric_space set"
shows
"openin (subtopology euclidean (S × T)) (S' × T') ⟷
S' = {} ∨ T' = {} ∨ openin (subtopology euclidean S) S' ∧ openin (subtopology euclidean T) T'"
(is "?lhs = ?rhs")
proof (cases "S' = {} ∨ T' = {}")
case True
then show ?thesis by auto
next
case False
then obtain x y where "x ∈ S'" "y ∈ T'"
by blast
show ?thesis
proof
assume ?lhs
have "openin (subtopology euclidean S) S'"
apply (subst openin_subopen, clarify)
apply (rule Times_in_interior_subtopology [OF _ ‹?lhs›])
using ‹y ∈ T'›
apply auto
done
moreover have "openin (subtopology euclidean T) T'"
apply (subst openin_subopen, clarify)
apply (rule Times_in_interior_subtopology [OF _ ‹?lhs›])
using ‹x ∈ S'›
apply auto
done
ultimately show ?rhs
by simp
next
assume ?rhs
with False show ?lhs
qed
qed

lemma closedin_Times:
"closedin (subtopology euclidean S) S' ⟹ closedin (subtopology euclidean T) T' ⟹
closedin (subtopology euclidean (S × T)) (S' × T')"
unfolding closedin_closed using closed_Times by blast

lemma bdd_below_closure:
fixes A :: "real set"
assumes "bdd_below A"
shows "bdd_below (closure A)"
proof -
from assms obtain m where "⋀x. x ∈ A ⟹ m ≤ x"
by (auto simp: bdd_below_def)
then have "A ⊆ {m..}" by auto
then have "closure A ⊆ {m..}"
using closed_real_atLeast by (rule closure_minimal)
then show ?thesis
by (auto simp: bdd_below_def)
qed

subsection ‹Frontier (also known as boundary)›

definition%important "frontier S = closure S - interior S"

lemma frontier_closed [iff]: "closed (frontier S)"

lemma frontier_closures: "frontier S = closure S ∩ closure (- S)"
by (auto simp: frontier_def interior_closure)

lemma frontier_Int: "frontier(S ∩ T) = closure(S ∩ T) ∩ (frontier S ∪ frontier T)"
proof -
have "closure (S ∩ T) ⊆ closure S" "closure (S ∩ T) ⊆ closure T"
then show ?thesis
by (auto simp: frontier_closures)
qed

lemma frontier_Int_subset: "frontier(S ∩ T) ⊆ frontier S ∪ frontier T"
by (auto simp: frontier_Int)

lemma frontier_Int_closed:
assumes "closed S" "closed T"
shows "frontier(S ∩ T) = (frontier S ∩ T) ∪ (S ∩ frontier T)"
proof -
have "closure (S ∩ T) = T ∩ S"
using assms by (simp add: Int_commute closed_Int)
moreover have "T ∩ (closure S ∩ closure (- S)) = frontier S ∩ T"
ultimately show ?thesis
by (simp add: Int_Un_distrib Int_assoc Int_left_commute assms frontier_closures)
qed

fixes a :: "'a::metric_space"
shows "a ∈ frontier S ⟷ (∀e>0. (∃x∈S. dist a x < e) ∧ (∃x. x ∉ S ∧ dist a x < e))"
unfolding frontier_def closure_interior
by (auto simp: mem_interior subset_eq ball_def)

lemma frontier_subset_closed: "closed S ⟹ frontier S ⊆ S"
by (metis frontier_def closure_closed Diff_subset)

lemma frontier_empty [simp]: "frontier {} = {}"

lemma frontier_subset_eq: "frontier S ⊆ S ⟷ closed S"
proof -
{
assume "frontier S ⊆ S"
then have "closure S ⊆ S"
using interior_subset unfolding frontier_def by auto
then have "closed S"
using closure_subset_eq by auto
}
then show ?thesis using frontier_subset_closed[of S] ..
qed

lemma frontier_complement [simp]: "frontier (- S) = frontier S"
by (auto simp: frontier_def closure_complement interior_complement)

lemma frontier_Un_subset: "frontier(S ∪ T) ⊆ frontier S ∪ frontier T"
by (metis compl_sup frontier_Int_subset frontier_complement)

lemma frontier_disjoint_eq: "frontier S ∩ S = {} ⟷ open S"
using frontier_complement frontier_subset_eq[of "- S"]
unfolding open_closed by auto

lemma frontier_UNIV [simp]: "frontier UNIV = {}"
using frontier_complement frontier_empty by fastforce

lemma frontier_interiors: "frontier s = - interior(s) - interior(-s)"
by (simp add: Int_commute frontier_def interior_closure)

lemma frontier_interior_subset: "frontier(interior S) ⊆ frontier S"
by (simp add: Diff_mono frontier_interiors interior_mono interior_subset)

lemma connected_Int_frontier:
"⟦connected s; s ∩ t ≠ {}; s - t ≠ {}⟧ ⟹ (s ∩ frontier t ≠ {})"
apply (simp add: frontier_interiors connected_openin, safe)
apply (drule_tac x="s ∩ interior t" in spec, safe)
apply (drule_tac [2] x="s ∩ interior (-t)" in spec)
apply (auto simp: disjoint_eq_subset_Compl dest: interior_subset [THEN subsetD])
done

lemma closure_Un_frontier: "closure S = S ∪ frontier S"
proof -
have "S ∪ interior S = S"
using interior_subset by auto
then show ?thesis
using closure_subset by (auto simp: frontier_def)
qed

subsection%unimportant ‹Filters and the ``eventually true'' quantifier›

definition indirection :: "'a::real_normed_vector ⇒ 'a ⇒ 'a filter"  (infixr "indirection" 70)
where "a indirection v = at a within {b. ∃c≥0. b - a = scaleR c v}"

text ‹Identify Trivial limits, where we can't approach arbitrarily closely.›

lemma trivial_limit_within: "trivial_limit (at a within S) ⟷ ¬ a islimpt S"
proof
assume "trivial_limit (at a within S)"
then show "¬ a islimpt S"
unfolding trivial_limit_def
unfolding eventually_at_topological
unfolding islimpt_def
apply (rename_tac T, rule_tac x=T in exI)
apply (clarsimp, drule_tac x=y in bspec, simp_all)
done
next
assume "¬ a islimpt S"
then show "trivial_limit (at a within S)"
unfolding trivial_limit_def eventually_at_topological islimpt_def
by metis
qed

lemma trivial_limit_at_iff: "trivial_limit (at a) ⟷ ¬ a islimpt UNIV"
using trivial_limit_within [of a UNIV] by simp

lemma trivial_limit_at: "¬ trivial_limit (at a)"
for a :: "'a::perfect_space"
by (rule at_neq_bot)

lemma trivial_limit_at_infinity:
"¬ trivial_limit (at_infinity :: ('a::{real_normed_vector,perfect_space}) filter)"
unfolding trivial_limit_def eventually_at_infinity
apply clarsimp
apply (subgoal_tac "∃x::'a. x ≠ 0", clarify)
apply (rule_tac x="scaleR (b / norm x) x" in exI, simp)
apply (cut_tac islimpt_UNIV [of "0::'a", unfolded islimpt_def])
apply (drule_tac x=UNIV in spec, simp)
done

lemma not_trivial_limit_within: "¬ trivial_limit (at x within S) = (x ∈ closure (S - {x}))"
using islimpt_in_closure by (metis trivial_limit_within)

lemma not_in_closure_trivial_limitI:
"x ∉ closure s ⟹ trivial_limit (at x within s)"
using not_trivial_limit_within[of x s]
by safe (metis Diff_empty Diff_insert0 closure_subset contra_subsetD)

lemma filterlim_at_within_closure_implies_filterlim: "filterlim f l (at x within s)"
if "x ∈ closure s ⟹ filterlim f l (at x within s)"
by (metis bot.extremum filterlim_filtercomap filterlim_mono not_in_closure_trivial_limitI that)

lemma at_within_eq_bot_iff: "at c within A = bot ⟷ c ∉ closure (A - {c})"
using not_trivial_limit_within[of c A] by blast

text ‹Some property holds "sufficiently close" to the limit point.›

lemma trivial_limit_eventually: "trivial_limit net ⟹ eventually P net"
by simp

lemma trivial_limit_eq: "trivial_limit net ⟷ (∀P. eventually P net)"

subsection ‹Limits›

proposition Lim: "(f ⤏ l) net ⟷ trivial_limit net ∨ (∀e>0. eventually (λx. dist (f x) l < e) net)"
by (auto simp: tendsto_iff trivial_limit_eq)

text ‹Show that they yield usual definitions in the various cases.›

proposition Lim_within_le: "(f ⤏ l)(at a within S) ⟷
(∀e>0. ∃d>0. ∀x∈S. 0 < dist x a ∧ dist x a ≤ d ⟶ dist (f x) l < e)"
by (auto simp: tendsto_iff eventually_at_le)

proposition Lim_within: "(f ⤏ l) (at a within S) ⟷
(∀e >0. ∃d>0. ∀x ∈ S. 0 < dist x a ∧ dist x a  < d ⟶ dist (f x) l < e)"
by (auto simp: tendsto_iff eventually_at)

corollary Lim_withinI [intro?]:
assumes "⋀e. e > 0 ⟹ ∃d>0. ∀x ∈ S. 0 < dist x a ∧ dist x a < d ⟶ dist (f x) l ≤ e"
shows "(f ⤏ l) (at a within S)"
apply (rule ex_forward [OF assms [OF half_gt_zero]], auto)
done

proposition Lim_at: "(f ⤏ l) (at a) ⟷
(∀e >0. ∃d>0. ∀x. 0 < dist x a ∧ dist x a < d  ⟶ dist (f x) l < e)"
by (auto simp: tendsto_iff eventually_at)

proposition Lim_at_infinity: "(f ⤏ l) at_infinity ⟷ (∀e>0. ∃b. ∀x. norm x ≥ b ⟶ dist (f x) l < e)"
by (auto simp: tendsto_iff eventually_at_infinity)

corollary Lim_at_infinityI [intro?]:
assumes "⋀e. e > 0 ⟹ ∃B. ∀x. norm x ≥ B ⟶ dist (f x) l ≤ e"
shows "(f ⤏ l) at_infinity"
apply (rule ex_forward [OF assms [OF half_gt_zero]], auto)
done

lemma Lim_eventually: "eventually (λx. f x = l) net ⟹ (f ⤏ l) net"
by (rule topological_tendstoI) (auto elim: eventually_mono)

lemma Lim_transform_within_set:
fixes a :: "'a::metric_space" and l :: "'b::metric_space"
shows "⟦(f ⤏ l) (at a within S); eventually (λx. x ∈ S ⟷ x ∈ T) (at a)⟧
⟹ (f ⤏ l) (at a within T)"
apply (clarsimp simp: eventually_at Lim_within)
apply (drule_tac x=e in spec, clarify)
apply (rename_tac k)
apply (rule_tac x="min d k" in exI, simp)
done

lemma Lim_transform_within_set_eq:
fixes a l :: "'a::real_normed_vector"
shows "eventually (λx. x ∈ s ⟷ x ∈ t) (at a)
⟹ ((f ⤏ l) (at a within s) ⟷ (f ⤏ l) (at a within t))"
by (force intro: Lim_transform_within_set elim: eventually_mono)

lemma Lim_transform_within_openin:
fixes a :: "'a::metric_space"
assumes f: "(f ⤏ l) (at a within T)"
and "openin (subtopology euclidean T) S" "a ∈ S"
and eq: "⋀x. ⟦x ∈ S; x ≠ a⟧ ⟹ f x = g x"
shows "(g ⤏ l) (at a within T)"
proof -
obtain ε where "0 < ε" and ε: "ball a ε ∩ T ⊆ S"
using assms by (force simp: openin_contains_ball)
then have "a ∈ ball a ε"
by simp
show ?thesis
by (rule Lim_transform_within [OF f ‹0 < ε› eq]) (use ε in ‹auto simp: dist_commute subset_iff›)
qed

lemma continuous_transform_within_openin:
fixes a :: "'a::metric_space"
assumes "continuous (at a within T) f"
and "openin (subtopology euclidean T) S" "a ∈ S"
and eq: "⋀x. x ∈ S ⟹ f x = g x"
shows "continuous (at a within T) g"
using assms by (simp add: Lim_transform_within_openin continuous_within)

text ‹The expected monotonicity property.›

lemma Lim_Un:
assumes "(f ⤏ l) (at x within S)" "(f ⤏ l) (at x within T)"
shows "(f ⤏ l) (at x within (S ∪ T))"
using assms unfolding at_within_union by (rule filterlim_sup)

lemma Lim_Un_univ:
"(f ⤏ l) (at x within S) ⟹ (f ⤏ l) (at x within T) ⟹
S ∪ T = UNIV ⟹ (f ⤏ l) (at x)"
by (metis Lim_Un)

text ‹Interrelations between restricted and unrestricted limits.›

lemma Lim_at_imp_Lim_at_within: "(f ⤏ l) (at x) ⟹ (f ⤏ l) (at x within S)"
by (metis order_refl filterlim_mono subset_UNIV at_le)

lemma eventually_within_interior:
assumes "x ∈ interior S"
shows "eventually P (at x within S) ⟷ eventually P (at x)"
(is "?lhs = ?rhs")
proof
from assms obtain T where T: "open T" "x ∈ T" "T ⊆ S" ..
{
assume ?lhs
then obtain A where "open A" and "x ∈ A" and "∀y∈A. y ≠ x ⟶ y ∈ S ⟶ P y"
by (auto simp: eventually_at_topological)
with T have "open (A ∩ T)" and "x ∈ A ∩ T" and "∀y ∈ A ∩ T. y ≠ x ⟶ P y"
by auto
then show ?rhs
by (auto simp: eventually_at_topological)
next
assume ?rhs
then show ?lhs
by (auto elim: eventually_mono simp: eventually_at_filter)
}
qed

lemma at_within_interior: "x ∈ interior S ⟹ at x within S = at x"
unfolding filter_eq_iff by (intro allI eventually_within_interior)

lemma Lim_within_LIMSEQ:
fixes a :: "'a::first_countable_topology"
assumes "∀S. (∀n. S n ≠ a ∧ S n ∈ T) ∧ S ⇢ a ⟶ (λn. X (S n)) ⇢ L"
shows "(X ⤏ L) (at a within T)"
using assms unfolding tendsto_def [where l=L]

lemma Lim_right_bound:
fixes f :: "'a :: {linorder_topology, conditionally_complete_linorder, no_top} ⇒
'b::{linorder_topology, conditionally_complete_linorder}"
assumes mono: "⋀a b. a ∈ I ⟹ b ∈ I ⟹ x < a ⟹ a ≤ b ⟹ f a ≤ f b"
and bnd: "⋀a. a ∈ I ⟹ x < a ⟹ K ≤ f a"
shows "(f ⤏ Inf (f ` ({x<..} ∩ I))) (at x within ({x<..} ∩ I))"
proof (cases "{x<..} ∩ I = {}")
case True
then show ?thesis by simp
next
case False
show ?thesis
proof (rule order_tendstoI)
fix a
assume a: "a < Inf (f ` ({x<..} ∩ I))"
{
fix y
assume "y ∈ {x<..} ∩ I"
with False bnd have "Inf (f ` ({x<..} ∩ I)) ≤ f y"
by (auto intro!: cInf_lower bdd_belowI2)
with a have "a < f y"
by (blast intro: less_le_trans)
}
then show "eventually (λx. a < f x) (at x within ({x<..} ∩ I))"
by (auto simp: eventually_at_filter intro: exI[of _ 1] zero_less_one)
next
fix a
assume "Inf (f ` ({x<..} ∩ I)) < a"
from cInf_lessD[OF _ this] False obtain y where y: "x < y" "y ∈ I" "f y < a"
by auto
then have "eventually (λx. x ∈ I ⟶ f x < a) (at_right x)"
unfolding eventually_at_right[OF ‹x < y›] by (metis less_imp_le le_less_trans mono)
then show "eventually (λx. f x < a) (at x within ({x<..} ∩ I))"
unfolding eventually_at_filter by eventually_elim simp
qed
qed

text ‹Another limit point characterization.›

lemma limpt_sequential_inj:
fixes x :: "'a::metric_space"
shows "x islimpt S ⟷
(∃f. (∀n::nat. f n ∈ S - {x}) ∧ inj f ∧ (f ⤏ x) sequentially)"
(is "?lhs = ?rhs")
proof
assume ?lhs
then have "∀e>0. ∃x'∈S. x' ≠ x ∧ dist x' x < e"
by (force simp: islimpt_approachable)
then obtain y where y: "⋀e. e>0 ⟹ y e ∈ S ∧ y e ≠ x ∧ dist (y e) x < e"
by metis
define f where "f ≡ rec_nat (y 1) (λn fn. y (min (inverse(2 ^ (Suc n))) (dist fn x)))"
have [simp]: "f 0 = y 1"
"f(Suc n) = y (min (inverse(2 ^ (Suc n))) (dist (f n) x))" for n
have f: "f n ∈ S ∧ (f n ≠ x) ∧ dist (f n) x < inverse(2 ^ n)" for n
proof (induction n)
case 0 show ?case
next
case (Suc n) then show ?case
apply (auto simp: y)
by (metis half_gt_zero_iff inverse_positive_iff_positive less_divide_eq_numeral1(1) min_less_iff_conj y zero_less_dist_iff zero_less_numeral zero_less_power)
qed
show ?rhs
proof (rule_tac x=f in exI, intro conjI allI)
show "⋀n. f n ∈ S - {x}"
using f by blast
have "dist (f n) x < dist (f m) x" if "m < n" for m n
using that
proof (induction n)
case 0 then show ?case by simp
next
case (Suc n)
then consider "m < n" | "m = n" using less_Suc_eq by blast
then show ?case
proof cases
assume "m < n"
have "dist (f(Suc n)) x = dist (y (min (inverse(2 ^ (Suc n))) (dist (f n) x))) x"
by simp
also have "… < dist (f n) x"
by (metis dist_pos_lt f min.strict_order_iff min_less_iff_conj y)
also have "… < dist (f m) x"
using Suc.IH ‹m < n› by blast
finally show ?thesis .
next
assume "m = n" then show ?case
by simp (metis dist_pos_lt f half_gt_zero_iff inverse_positive_iff_positive min_less_iff_conj y zero_less_numeral zero_less_power)
qed
qed
then show "inj f"
by (metis less_irrefl linorder_injI)
show "f ⇢ x"
apply (rule tendstoI)
apply (rule_tac c="nat (ceiling(1/e))" in eventually_sequentiallyI)
apply (rule less_trans [OF f [THEN conjunct2, THEN conjunct2]])
by (meson le_less_trans mult_less_cancel_left not_le of_nat_less_two_power)
qed
next
assume ?rhs
then show ?lhs
by (fastforce simp add: islimpt_approachable lim_sequentially)
qed

(*could prove directly from islimpt_sequential_inj, but only for metric spaces*)
lemma islimpt_sequential:
fixes x :: "'a::first_countable_topology"
shows "x islimpt S ⟷ (∃f. (∀n::nat. f n ∈ S - {x}) ∧ (f ⤏ x) sequentially)"
(is "?lhs = ?rhs")
proof
assume ?lhs
from countable_basis_at_decseq[of x] obtain A where A:
"⋀i. open (A i)"
"⋀i. x ∈ A i"
"⋀S. open S ⟹ x ∈ S ⟹ eventually (λi. A i ⊆ S) sequentially"
by blast
define f where "f n = (SOME y. y ∈ S ∧ y ∈ A n ∧ x ≠ y)" for n
{
fix n
from ‹?lhs› have "∃y. y ∈ S ∧ y ∈ A n ∧ x ≠ y"
unfolding islimpt_def using A(1,2)[of n] by auto
then have "f n ∈ S ∧ f n ∈ A n ∧ x ≠ f n"
unfolding f_def by (rule someI_ex)
then have "f n ∈ S" "f n ∈ A n" "x ≠ f n" by auto
}
then have "∀n. f n ∈ S - {x}" by auto
moreover have "(λn. f n) ⇢ x"
proof (rule topological_tendstoI)
fix S
assume "open S" "x ∈ S"
from A(3)[OF this] ‹⋀n. f n ∈ A n›
show "eventually (λx. f x ∈ S) sequentially"
by (auto elim!: eventually_mono)
qed
ultimately show ?rhs by fast
next
assume ?rhs
then obtain f :: "nat ⇒ 'a" where f: "⋀n. f n ∈ S - {x}" and lim: "f ⇢ x"
by auto
show ?lhs
unfolding islimpt_def
proof safe
fix T
assume "open T" "x ∈ T"
from lim[THEN topological_tendstoD, OF this] f
show "∃y∈S. y ∈ T ∧ y ≠ x"
unfolding eventually_sequentially by auto
qed
qed

lemma Lim_null:
fixes f :: "'a ⇒ 'b::real_normed_vector"
shows "(f ⤏ l) net ⟷ ((λx. f(x) - l) ⤏ 0) net"

lemma Lim_null_comparison:
fixes f :: "'a ⇒ 'b::real_normed_vector"
assumes "eventually (λx. norm (f x) ≤ g x) net" "(g ⤏ 0) net"
shows "(f ⤏ 0) net"
using assms(2)
proof (rule metric_tendsto_imp_tendsto)
show "eventually (λx. dist (f x) 0 ≤ dist (g x) 0) net"
using assms(1) by (rule eventually_mono) (simp add: dist_norm)
qed

lemma Lim_transform_bound:
fixes f :: "'a ⇒ 'b::real_normed_vector"
and g :: "'a ⇒ 'c::real_normed_vector"
assumes "eventually (λn. norm (f n) ≤ norm (g n)) net"
and "(g ⤏ 0) net"
shows "(f ⤏ 0) net"
using assms(1) tendsto_norm_zero [OF assms(2)]
by (rule Lim_null_comparison)

lemma lim_null_mult_right_bounded:
fixes f :: "'a ⇒ 'b::real_normed_div_algebra"
assumes f: "(f ⤏ 0) F" and g: "eventually (λx. norm(g x) ≤ B) F"
shows "((λz. f z * g z) ⤏ 0) F"
proof -
have *: "((λx. norm (f x) * B) ⤏ 0) F"
by (simp add: f tendsto_mult_left_zero tendsto_norm_zero)
have "((λx. norm (f x) * norm (g x)) ⤏ 0) F"
apply (rule Lim_null_comparison [OF _ *])
apply (simp add: eventually_mono [OF g] mult_left_mono)
done
then show ?thesis
by (subst tendsto_norm_zero_iff [symmetric]) (simp add: norm_mult)
qed

lemma lim_null_mult_left_bounded:
fixes f :: "'a ⇒ 'b::real_normed_div_algebra"
assumes g: "eventually (λx. norm(g x) ≤ B) F" and f: "(f ⤏ 0) F"
shows "((λz. g z * f z) ⤏ 0) F"
proof -
have *: "((λx. B * norm (f x)) ⤏ 0) F"
by (simp add: f tendsto_mult_right_zero tendsto_norm_zero)
have "((λx. norm (g x) * norm (f x)) ⤏ 0) F"
apply (rule Lim_null_comparison [OF _ *])
apply (simp add: eventually_mono [OF g] mult_right_mono)
done
then show ?thesis
by (subst tendsto_norm_zero_iff [symmetric]) (simp add: norm_mult)
qed

lemma lim_null_scaleR_bounded:
assumes f: "(f ⤏ 0) net" and gB: "eventually (λa. f a = 0 ∨ norm(g a) ≤ B) net"
shows "((λn. f n *⇩R g n) ⤏ 0) net"
proof
fix ε::real
assume "0 < ε"
then have B: "0 < ε / (abs B + 1)" by simp
have *: "¦f x¦ * norm (g x) < ε" if f: "¦f x¦ * (¦B¦ + 1) < ε" and g: "norm (g x) ≤ B" for x
proof -
have "¦f x¦ * norm (g x) ≤ ¦f x¦ * B"
also have "… ≤ ¦f x¦ * (¦B¦ + 1)"
also have "… < ε"
by (rule f)
finally show ?thesis .
qed
show "∀⇩F x in net. dist (f x *⇩R g x) 0 < ε"
apply (rule eventually_mono [OF eventually_conj [OF tendstoD [OF f B] gB] ])
apply (auto simp: ‹0 < ε› divide_simps * split: if_split_asm)
done
qed

text‹Deducing things about the limit from the elements.›

lemma Lim_in_closed_set:
assumes "closed S"
and "eventually (λx. f(x) ∈ S) net"
and "¬ trivial_limit net" "(f ⤏ l) net"
shows "l ∈ S"
proof (rule ccontr)
assume "l ∉ S"
with ‹closed S› have "open (- S)" "l ∈ - S"
with assms(4) have "eventually (λx. f x ∈ - S) net"
by (rule topological_tendstoD)
with assms(2) have "eventually (λx. False) net"
by (rule eventually_elim2) simp
with assms(3) show "False"
qed

text‹Need to prove closed(cball(x,e)) before deducing this as a corollary.›

lemma Lim_dist_ubound:
assumes "¬(trivial_limit net)"
and "(f ⤏ l) net"
and "eventually (λx. dist a (f x) ≤ e) net"
shows "dist a l ≤ e"
using assms by (fast intro: tendsto_le tendsto_intros)

lemma Lim_norm_ubound:
fixes f :: "'a ⇒ 'b::real_normed_vector"
assumes "¬(trivial_limit net)" "(f ⤏ l) net" "eventually (λx. norm(f x) ≤ e) net"
shows "norm(l) ≤ e"
using assms by (fast intro: tendsto_le tendsto_intros)

lemma Lim_norm_lbound:
fixes f :: "'a ⇒ 'b::real_normed_vector"
assumes "¬ trivial_limit net"
and "(f ⤏ l) net"
and "eventually (λx. e ≤ norm (f x)) net"
shows "e ≤ norm l"
using assms by (fast intro: tendsto_le tendsto_intros)

text‹Limit under bilinear function›

lemma Lim_bilinear:
assumes "(f ⤏ l) net"
and "(g ⤏ m) net"
and "bounded_bilinear h"
shows "((λx. h (f x) (g x)) ⤏ (h l m)) net"
using ‹bounded_bilinear h› ‹(f ⤏ l) net› ‹(g ⤏ m) net›
by (rule bounded_bilinear.tendsto)

text‹These are special for limits out of the same vector space.›

lemma Lim_within_id: "(id ⤏ a) (at a within s)"
unfolding id_def by (rule tendsto_ident_at)

lemma Lim_at_id: "(id ⤏ a) (at a)"
unfolding id_def by (rule tendsto_ident_at)

lemma Lim_at_zero:
fixes a :: "'a::real_normed_vector"
and l :: "'b::topological_space"
shows "(f ⤏ l) (at a) ⟷ ((λx. f(a + x)) ⤏ l) (at 0)"
using LIM_offset_zero LIM_offset_zero_cancel ..

text‹It's also sometimes useful to extract the limit point from the filter.›

abbreviation netlimit :: "'a::t2_space filter ⇒ 'a"
where "netlimit F ≡ Lim F (λx. x)"

lemma netlimit_within: "¬ trivial_limit (at a within S) ⟹ netlimit (at a within S) = a"
by (rule tendsto_Lim) (auto intro: tendsto_intros)

lemma netlimit_at [simp]:
fixes a :: "'a::{perfect_space,t2_space}"
shows "netlimit (at a) = a"
using netlimit_within [of a UNIV] by simp

lemma lim_within_interior:
"x ∈ interior S ⟹ (f ⤏ l) (at x within S) ⟷ (f ⤏ l) (at x)"
by (metis at_within_interior)

lemma netlimit_within_interior:
fixes x :: "'a::{t2_space,perfect_space}"
assumes "x ∈ interior S"
shows "netlimit (at x within S) = x"
using assms by (metis at_within_interior netlimit_at)

lemma netlimit_at_vector:
fixes a :: "'a::real_normed_vector"
shows "netlimit (at a) = a"
proof (cases "∃x. x ≠ a")
case True then obtain x where x: "x ≠ a" ..
have "¬ trivial_limit (at a)"
unfolding trivial_limit_def eventually_at dist_norm
apply clarsimp
apply (rule_tac x="a + scaleR (d / 2) (sgn (x - a))" in exI)
apply (simp add: norm_sgn sgn_zero_iff x)
done
then show ?thesis
by (rule netlimit_within [of a UNIV])
qed simp

text‹Useful lemmas on closure and set of possible sequential limits.›

lemma closure_sequential:
fixes l :: "'a::first_countable_topology"
shows "l ∈ closure S ⟷ (∃x. (∀n. x n ∈ S) ∧ (x ⤏ l) sequentially)"
(is "?lhs = ?rhs")
proof
assume "?lhs"
moreover
{
assume "l ∈ S"
then have "?rhs" using tendsto_const[of l sequentially] by auto
}
moreover
{
assume "l islimpt S"
then have "?rhs" unfolding islimpt_sequential by auto
}
ultimately show "?rhs"
unfolding closure_def by auto
next
assume "?rhs"
then show "?lhs" unfolding closure_def islimpt_sequential by auto
qed

lemma closed_sequential_limits:
fixes S :: "'a::first_countable_topology set"
shows "closed S ⟷ (∀x l. (∀n. x n ∈ S) ∧ (x ⤏ l) sequentially ⟶ l ∈ S)"
by (metis closure_sequential closure_subset_eq subset_iff)

lemma closure_approachable:
fixes S :: "'a::metric_space set"
shows "x ∈ closure S ⟷ (∀e>0. ∃y∈S. dist y x < e)"
apply (auto simp: closure_def islimpt_approachable)
apply (metis dist_self)
done

lemma closure_approachable_le:
fixes S :: "'a::metric_space set"
shows "x ∈ closure S ⟷ (∀e>0. ∃y∈S. dist y x ≤ e)"
unfolding closure_approachable
using dense by force

lemma closure_approachableD:
assumes "x ∈ closure S" "e>0"
shows "∃y∈S. dist x y < e"
using assms unfolding closure_approachable by (auto simp: dist_commute)

lemma closed_approachable:
fixes S :: "'a::metric_space set"
shows "closed S ⟹ (∀e>0. ∃y∈S. dist y x < e) ⟷ x ∈ S"
by (metis closure_closed closure_approachable)

lemma closure_contains_Inf:
fixes S :: "real set"
assumes "S ≠ {}" "bdd_below S"
shows "Inf S ∈ closure S"
proof -
have *: "∀x∈S. Inf S ≤ x"
using cInf_lower[of _ S] assms by metis
{
fix e :: real
assume "e > 0"
then have "Inf S < Inf S + e" by simp
with assms obtain x where "x ∈ S" "x < Inf S + e"
by (subst (asm) cInf_less_iff) auto
with * have "∃x∈S. dist x (Inf S) < e"
by (intro bexI[of _ x]) (auto simp: dist_real_def)
}
then show ?thesis unfolding closure_approachable by auto
qed

lemma closure_Int_ballI:
fixes S :: "'a :: metric_space set"
assumes "⋀U. ⟦openin (subtopology euclidean S) U; U ≠ {}⟧ ⟹ T ∩ U ≠ {}"
shows "S ⊆ closure T"
proof (clarsimp simp: closure_approachable dist_commute)
fix x and e::real
assume "x ∈ S" "0 < e"
with assms [of "S ∩ ball x e"] show "∃y∈T. dist x y < e"
by force
qed

lemma closed_contains_Inf:
fixes S :: "real set"
shows "S ≠ {} ⟹ bdd_below S ⟹ closed S ⟹ Inf S ∈ S"
by (metis closure_contains_Inf closure_closed)

lemma closed_subset_contains_Inf:
fixes A C :: "real set"
shows "closed C ⟹ A ⊆ C ⟹ A ≠ {} ⟹ bdd_below A ⟹ Inf A ∈ C"
by (metis closure_contains_Inf closure_minimal subset_eq)

lemma atLeastAtMost_subset_contains_Inf:
fixes A :: "real set" and a b :: real
shows "A ≠ {} ⟹ a ≤ b ⟹ A ⊆ {a..b} ⟹ Inf A ∈ {a..b}"
by (rule closed_subset_contains_Inf)
(auto intro: closed_real_atLeastAtMost intro!: bdd_belowI[of A a])

lemma not_trivial_limit_within_ball:
"¬ trivial_limit (at x within S) ⟷ (∀e>0. S ∩ ball x e - {x} ≠ {})"
(is "?lhs ⟷ ?rhs")
proof
show ?rhs if ?lhs
proof -
{
fix e :: real
assume "e > 0"
then obtain y where "y ∈ S - {x}" and "dist y x < e"
using ‹?lhs› not_trivial_limit_within[of x S] closure_approachable[of x "S - {x}"]
by auto
then have "y ∈ S ∩ ball x e - {x}"
unfolding ball_def by (simp add: dist_commute)
then have "S ∩ ball x e - {x} ≠ {}" by blast
}
then show ?thesis by auto
qed
show ?lhs if ?rhs
proof -
{
fix e :: real
assume "e > 0"
then obtain y where "y ∈ S ∩ ball x e - {x}"
using ‹?rhs› by blast
then have "y ∈ S - {x}" and "dist y x < e"
unfolding ball_def by (simp_all add: dist_commute)
then have "∃y ∈ S - {x}. dist y x < e"
by auto
}
then show ?thesis
using not_trivial_limit_within[of x S] closure_approachable[of x "S - {x}"]
by auto
qed
qed

lemma tendsto_If_within_closures:
assumes f: "x ∈ s ∪ (closure s ∩ closure t) ⟹
(f ⤏ l x) (at x within s ∪ (closure s ∩ closure t))"
assumes g: "x ∈ t ∪ (closure s ∩ closure t) ⟹
(g ⤏ l x) (at x within t ∪ (closure s ∩ closure t))"
assumes "x ∈ s ∪ t"
shows "((λx. if x ∈ s then f x else g x) ⤏ l x) (at x within s ∪ t)"
proof -
have *: "(s ∪ t) ∩ {x. x ∈ s} = s" "(s ∪ t) ∩ {x. x ∉ s} = t - s"
by auto
have "(f ⤏ l x) (at x within s)"
by (rule filterlim_at_within_closure_implies_filterlim)
(use ‹x ∈ _› in ‹auto simp: inf_commute closure_def intro: tendsto_within_subset[OF f]›)
moreover
have "(g ⤏ l x) (at x within t - s)"
by (rule filterlim_at_within_closure_implies_filterlim)
(use ‹x ∈ _› in
‹auto intro!: tendsto_within_subset[OF g] simp: closure_def intro: islimpt_subset›)
ultimately show ?thesis
by (intro filterlim_at_within_If) (simp_all only: *)
qed

subsection ‹Boundedness›

(* FIXME: This has to be unified with BSEQ!! *)
definition%important (in metric_space) bounded :: "'a set ⇒ bool"
where "bounded S ⟷ (∃x e. ∀y∈S. dist x y ≤ e)"

lemma bounded_subset_cball: "bounded S ⟷ (∃e x. S ⊆ cball x e ∧ 0 ≤ e)"
unfolding bounded_def subset_eq  by auto (meson order_trans zero_le_dist)

lemma bounded_any_center: "bounded S ⟷ (∃e. ∀y∈S. dist a y ≤ e)"
unfolding bounded_def

lemma bounded_iff: "bounded S ⟷ (∃a. ∀x∈S. norm x ≤ a)"
unfolding bounded_any_center [where a=0]

lemma bdd_above_norm: "bdd_above (norm ` X) ⟷ bounded X"

lemma bounded_norm_comp: "bounded ((λx. norm (f x)) ` S) = bounded (f ` S)"

lemma boundedI:
assumes "⋀x. x ∈ S ⟹ norm x ≤ B"
shows "bounded S"
using assms bounded_iff by blast

lemma bounded_empty [simp]: "bounded {}"

lemma bounded_subset: "bounded T ⟹ S ⊆ T ⟹ bounded S"
by (metis bounded_def subset_eq)

lemma bounded_interior[intro]: "bounded S ⟹ bounded(interior S)"
by (metis bounded_subset interior_subset)

lemma bounded_closure[intro]:
assumes "bounded S"
shows "bounded (closure S)"
proof -
from assms obtain x and a where a: "∀y∈S. dist x y ≤ a"
unfolding bounded_def by auto
{
fix y
assume "y ∈ closure S"
then obtain f where f: "∀n. f n ∈ S"  "(f ⤏ y) sequentially"
unfolding closure_sequential by auto
have "∀n. f n ∈ S ⟶ dist x (f n) ≤ a" using a by simp
then have "eventually (λn. dist x (f n) ≤ a) sequentially"
have "dist x y ≤ a"
apply (rule Lim_dist_ubound [of sequentially f])
apply (rule trivial_limit_sequentially)
apply (rule f(2))
apply fact
done
}
then show ?thesis
unfolding bounded_def by auto
qed

lemma bounded_closure_image: "bounded (f ` closure S) ⟹ bounded (f ` S)"
by (simp add: bounded_subset closure_subset image_mono)

lemma bounded_cball[simp,intro]: "bounded (cball x e)"
apply (rule_tac x=x in exI)
apply (rule_tac x=e in exI, auto)
done

lemma bounded_ball[simp,intro]: "bounded (ball x e)"
by (metis ball_subset_cball bounded_cball bounded_subset)

lemma bounded_Un[simp]: "bounded (S ∪ T) ⟷ bounded S ∧ bounded T"
by (auto simp: bounded_def) (metis Un_iff bounded_any_center le_max_iff_disj)

lemma bounded_Union[intro]: "finite F ⟹ ∀S∈F. bounded S ⟹ bounded (⋃F)"
by (induct rule: finite_induct[of F]) auto

lemma bounded_UN [intro]: "finite A ⟹ ∀x∈A. bounded (B x) ⟹ bounded (⋃x∈A. B x)"
by (induct set: finite) auto

lemma bounded_insert [simp]: "bounded (insert x S) ⟷ bounded S"
proof -
have "∀y∈{x}. dist x y ≤ 0"
by simp
then have "bounded {x}"
unfolding bounded_def by fast
then show ?thesis
by (metis insert_is_Un bounded_Un)
qed

lemma bounded_subset_ballI: "S ⊆ ball x r ⟹ bounded S"
by (meson bounded_ball bounded_subset)

lemma bounded_subset_ballD:
assumes "bounded S" shows "∃r. 0 < r ∧ S ⊆ ball x r"
proof -
obtain e::real and y where "S ⊆ cball y e"  "0 ≤ e"
using assms by (auto simp: bounded_subset_cball)
then show ?thesis
apply (rule_tac x="dist x y + e + 1" in exI)
apply (erule subset_trans)
qed

lemma finite_imp_bounded [intro]: "finite S ⟹ bounded S"
by (induct set: finite) simp_all

lemma bounded_pos: "bounded S ⟷ (∃b>0. ∀x∈ S. norm x ≤ b)"
apply (subgoal_tac "⋀x (y::real). 0 < 1 + ¦y¦ ∧ (x ≤ y ⟶ x ≤ 1 + ¦y¦)")
apply metis
apply arith
done

lemma bounded_pos_less: "bounded S ⟷ (∃b>0. ∀x∈ S. norm x < b)"
apply (safe; rule_tac x="b+1" in exI; force)
done

lemma Bseq_eq_bounded:
fixes f :: "nat ⇒ 'a::real_normed_vector"
shows "Bseq f ⟷ bounded (range f)"
unfolding Bseq_def bounded_pos by auto

lemma bounded_Int[intro]: "bounded S ∨ bounded T ⟹ bounded (S ∩ T)"
by (metis Int_lower1 Int_lower2 bounded_subset)

lemma bounded_diff[intro]: "bounded S ⟹ bounded (S - T)"
by (metis Diff_subset bounded_subset)

lemma not_bounded_UNIV[simp]:
"¬ bounded (UNIV :: 'a::{real_normed_vector, perfect_space} set)"
proof (auto simp: bounded_pos not_le)
obtain x :: 'a where "x ≠ 0"
using perfect_choose_dist [OF zero_less_one] by fast
fix b :: real
assume b: "b >0"
have b1: "b +1 ≥ 0"
using b by simp
with ‹x ≠ 0› have "b < norm (scaleR (b + 1) (sgn x))"
then show "∃x::'a. b < norm x" ..
qed

corollary cobounded_imp_unbounded:
fixes S :: "'a::{real_normed_vector, perfect_space} set"
shows "bounded (- S) ⟹ ~ (bounded S)"
using bounded_Un [of S "-S"]  by (simp add: sup_compl_top)

lemma bounded_dist_comp:
assumes "bounded (f ` S)" "bounded (g ` S)"
shows "bounded ((λx. dist (f x) (g x)) ` S)"
proof -
from assms obtain M1 M2 where *: "dist (f x) undefined ≤ M1" "dist undefined (g x) ≤ M2" if "x ∈ S" for x
by (auto simp: bounded_any_center[of _ undefined] dist_commute)
have "dist (f x) (g x) ≤ M1 + M2" if "x ∈ S" for x
using *[OF that]
then show ?thesis
by (auto intro!: boundedI)
qed

lemma bounded_linear_image:
assumes "bounded S"
and "bounded_linear f"
shows "bounded (f ` S)"
proof -
from assms(1) obtain b where "b > 0" and b: "∀x∈S. norm x ≤ b"
unfolding bounded_pos by auto
from assms(2) obtain B where B: "B > 0" "∀x. norm (f x) ≤ B * norm x"
using bounded_linear.pos_bounded by (auto simp: ac_simps)
show ?thesis
unfolding bounded_pos
proof (intro exI, safe)
show "norm (f x) ≤ B * b" if "x ∈ S" for x
by (meson B b less_imp_le mult_left_mono order_trans that)
qed (use ‹b > 0› ‹B > 0› in auto)
qed

lemma bounded_scaling:
fixes S :: "'a::real_normed_vector set"
shows "bounded S ⟹ bounded ((λx. c *⇩R x) ` S)"
apply (rule bounded_linear_image, assumption)
apply (rule bounded_linear_scaleR_right)
done

lemma bounded_scaleR_comp:
fixes f :: "'a ⇒ 'b::real_normed_vector"
assumes "bounded (f ` S)"
shows "bounded ((λx. r *⇩R f x) ` S)"
using bounded_scaling[of "f ` S" r] assms
by (auto simp: image_image)

lemma bounded_translation:
fixes S :: "'a::real_normed_vector set"
assumes "bounded S"
shows "bounded ((λx. a + x) ` S)"
proof -
from assms obtain b where b: "b > 0" "∀x∈S. norm x ≤ b"
unfolding bounded_pos by auto
{
fix x
assume "x ∈ S"
then have "norm (a + x) ≤ b + norm a"
using norm_triangle_ineq[of a x] b by auto
}
then show ?thesis
unfolding bounded_pos
using norm_ge_zero[of a] b(1) and add_strict_increasing[of b 0 "norm a"]
by (auto intro!: exI[of _ "b + norm a"])
qed

lemma bounded_translation_minus:
fixes S :: "'a::real_normed_vector set"
shows "bounded S ⟹ bounded ((λx. x - a) ` S)"
using bounded_translation [of S "-a"] by simp

lemma bounded_uminus [simp]:
fixes X :: "'a::real_normed_vector set"
shows "bounded (uminus ` X) ⟷ bounded X"
by (auto simp: bounded_def dist_norm; rule_tac x="-x" in exI; force simp: add.commute norm_minus_commute)

lemma uminus_bounded_comp [simp]:
fixes f :: "'a ⇒ 'b::real_normed_vector"
shows "bounded ((λx. - f x) ` S) ⟷ bounded (f ` S)"
using bounded_uminus[of "f ` S"]
by (auto simp: image_image)

lemma bounded_plus_comp:
fixes f g::"'a ⇒ 'b::real_normed_vector"
assumes "bounded (f ` S)"
assumes "bounded (g ` S)"
shows "bounded ((λx. f x + g x) ` S)"
proof -
{
fix B C
assume "⋀x. x∈S ⟹ norm (f x) ≤ B" "⋀x. x∈S ⟹ norm (g x) ≤ C"
then have "⋀x. x ∈ S ⟹ norm (f x + g x) ≤ B + C"
} then show ?thesis
using assms by (fastforce simp: bounded_iff)
qed

lemma bounded_plus:
fixes S ::"'a::real_normed_vector set"
assumes "bounded S" "bounded T"
shows "bounded ((λ(x,y). x + y) ` (S × T))"
using bounded_plus_comp [of fst "S × T" snd] assms
by (auto simp: split_def split: if_split_asm)

lemma bounded_minus_comp:
"bounded (f ` S) ⟹ bounded (g ` S) ⟹ bounded ((λx. f x - g x) ` S)"
for f g::"'a ⇒ 'b::real_normed_vector"
using bounded_plus_comp[of "f" S "λx. - g x"]
by auto

lemma bounded_minus:
fixes S ::"'a::real_normed_vector set"
assumes "bounded S" "bounded T"
shows "bounded ((λ(x,y). x - y) ` (S × T))"
using bounded_minus_comp [of fst "S × T" snd] assms
by (auto simp: split_def split: if_split_asm)

subsection ‹Compactness›

subsubsection ‹Bolzano-Weierstrass property›

proposition heine_borel_imp_bolzano_weierstrass:
assumes "compact s"
and "infinite t"
and "t ⊆ s"
shows "∃x ∈ s. x islimpt t"
proof (rule ccontr)
assume "¬ (∃x ∈ s. x islimpt t)"
then obtain f where f: "∀x∈s. x ∈ f x ∧ open (f x) ∧ (∀y∈t. y ∈ f x ⟶ y = x)"
unfolding islimpt_def
using bchoice[of s "λ x T. x ∈ T ∧ open T ∧ (∀y∈t. y ∈ T ⟶ y = x)"]
by auto
obtain g where g: "g ⊆ {t. ∃x. x ∈ s ∧ t = f x}" "finite g" "s ⊆ ⋃g"
using assms(1)[unfolded compact_eq_heine_borel, THEN spec[where x="{t. ∃x. x∈s ∧ t = f x}"]]
using f by auto
from g(1,3) have g':"∀x∈g. ∃xa ∈ s. x = f xa"
by auto
{
fix x y
assume "x ∈ t" "y ∈ t" "f x = f y"
then have "x ∈ f x"  "y ∈ f x ⟶ y = x"
using f[THEN bspec[where x=x]] and ‹t ⊆ s› by auto
then have "x = y"
using ‹f x = f y› and f[THEN bspec[where x=y]] and ‹y ∈ t› and ‹t ⊆ s›
by auto
}
then have "inj_on f t"
unfolding inj_on_def by simp
then have "infinite (f ` t)"
using assms(2) using finite_imageD by auto
moreover
{
fix x
assume "x ∈ t" "f x ∉ g"
from g(3) assms(3) ‹x ∈ t› obtain h where "h ∈ g" and "x ∈ h"
by auto
then obtain y where "y ∈ s" "h = f y"
using g'[THEN bspec[where x=h]] by auto
then have "y = x"
using f[THEN bspec[where x=y]] and ‹x∈t› and ‹x∈h›[unfolded ‹h = f y›]
by auto
then have False
using ‹f x ∉ g› ‹h ∈ g› unfolding ‹h = f y›
by auto
}
then have "f ` t ⊆ g" by auto
ultimately show False
using g(2) using finite_subset by auto
qed

lemma acc_point_range_imp_convergent_subsequence:
fixes l :: "'a :: first_countable_topology"
assumes l: "∀U. l∈U ⟶ open U ⟶ infinite (U ∩ range f)"
shows "∃r::nat⇒nat. strict_mono r ∧ (f ∘ r) ⇢ l"
proof -
from countable_basis_at_decseq[of l]
obtain A where A:
"⋀i. open (A i)"
"⋀i. l ∈ A i"
"⋀S. open S ⟹ l ∈ S ⟹ eventually (λi. A i ⊆ S) sequentially"
by blast
define s where "s n i = (SOME j. i < j ∧ f j ∈ A (Suc n))" for n i
{
fix n i
have "infinite (A (Suc n) ∩ range f - f`{.. i})"
using l A by auto
then have "∃x. x ∈ A (Suc n) ∩ range f - f`{.. i}"
unfolding ex_in_conv by (intro notI) simp
then have "∃j. f j ∈ A (Suc n) ∧ j ∉ {.. i}"
by auto
then have "∃a. i < a ∧ f a ∈ A (Suc n)"
by (auto simp: not_le)
then have "i < s n i" "f (s n i) ∈ A (Suc n)"
unfolding s_def by (auto intro: someI2_ex)
}
note s = this
define r where "r = rec_nat (s 0 0) s"
have "strict_mono r"
by (auto simp: r_def s strict_mono_Suc_iff)
moreover
have "(λn. f (r n)) ⇢ l"
proof (rule topological_tendstoI)
fix S
assume "open S" "l ∈ S"
with A(3) have "eventually (λi. A i ⊆ S) sequentially"
by auto
moreover
{
fix i
assume "Suc 0 ≤ i"
then have "f (r i) ∈ A i"
by (cases i) (simp_all add: r_def s)
}
then have "eventually (λi. f (r i) ∈ A i) sequentially"
by (auto simp: eventually_sequentially)
ultimately show "eventually (λi. f (r i) ∈ S) sequentially"
by eventually_elim auto
qed
ultimately show "∃r::nat⇒nat. strict_mono r ∧ (f ∘ r) ⇢ l"
by (auto simp: convergent_def comp_def)
qed

lemma sequence_infinite_lemma:
fixes f :: "nat ⇒ 'a::t1_space"
assumes "∀n. f n ≠ l"
and "(f ⤏ l) sequentially"
shows "infinite (range f)"
proof
assume "finite (range f)"
then have "closed (range f)"
by (rule finite_imp_closed)
then have "open (- range f)"
by (rule open_Compl)
from assms(1) have "l ∈ - range f"
by auto
from assms(2) have "eventually (λn. f n ∈ - range f) sequentially"
using ‹open (- range f)› ‹l ∈ - range f›
by (rule topological_tendstoD)
then show False
unfolding eventually_sequentially
by auto
qed

lemma closure_insert:
fixes x :: "'a::t1_space"
shows "closure (insert x s) = insert x (closure s)"
apply (rule closure_unique)
apply (rule insert_mono [OF closure_subset])
apply (rule closed_insert [OF closed_closure])
done

lemma islimpt_insert:
fixes x :: "'a::t1_space"
shows "x islimpt (insert a s) ⟷ x islimpt s"
proof
assume *: "x islimpt (insert a s)"
show "x islimpt s"
proof (rule islimptI)
fix t
assume t: "x ∈ t" "open t"
show "∃y∈s. y ∈ t ∧ y ≠ x"
proof (cases "x = a")
case True
obtain y where "y ∈ insert a s" "y ∈ t" "y ≠ x"
using * t by (rule islimptE)
with ‹x = a› show ?thesis by auto
next
case False
with t have t': "x ∈ t - {a}" "open (t - {a})"
obtain y where "y ∈ insert a s" "y ∈ t - {a}" "y ≠ x"
using * t' by (rule islimptE)
then show ?thesis by auto
qed
qed
next
assume "x islimpt s"
then show "x islimpt (insert a s)"
by (rule islimpt_subset) auto
qed

lemma islimpt_finite:
fixes x :: "'a::t1_space"
shows "finite s ⟹ ¬ x islimpt s"
by (induct set: finite) (simp_all add: islimpt_insert)

lemma islimpt_Un_finite:
fixes x :: "'a::t1_space"
shows "finite s ⟹ x islimpt (s ∪ t) ⟷ x islimpt t"

lemma islimpt_eq_acc_point:
fixes l :: "'a :: t1_space"
shows "l islimpt S ⟷ (∀U. l∈U ⟶ open U ⟶ infinite (U ∩ S))"
proof (safe intro!: islimptI)
fix U
assume "l islimpt S" "l ∈ U" "open U" "finite (U ∩ S)"
then have "l islimpt S" "l ∈ (U - (U ∩ S - {l}))" "open (U - (U ∩ S - {l}))"
by (auto intro: finite_imp_closed)
then show False
by (rule islimptE) auto
next
fix T
assume *: "∀U. l∈U ⟶ open U ⟶ infinite (U ∩ S)" "l ∈ T" "open T"
then have "infinite (T ∩ S - {l})"
by auto
then have "∃x. x ∈ (T ∩ S - {l})"
unfolding ex_in_conv by (intro notI) simp
then show "∃y∈S. y ∈ T ∧ y ≠ l"
by auto
qed

corollary infinite_openin:
fixes S :: "'a :: t1_space set"
shows "⟦openin (subtopology euclidean U) S; x ∈ S; x islimpt U⟧ ⟹ infinite S"
by (clarsimp simp add: openin_open islimpt_eq_acc_point inf_commute)

lemma islimpt_range_imp_convergent_subsequence:
fixes l :: "'a :: {t1_space, first_countable_topology}"
assumes l: "l islimpt (range f)"
shows "∃r::nat⇒nat. strict_mono r ∧ (f ∘ r) ⇢ l"
using l unfolding islimpt_eq_acc_point
by (rule acc_point_range_imp_convergent_subsequence)

lemma islimpt_eq_infinite_ball: "x islimpt S ⟷ (∀e>0. infinite(S ∩ ball x e))"
apply (metis Int_commute open_ball centre_in_ball)
by (metis open_contains_ball Int_mono finite_subset inf_commute subset_refl)

lemma islimpt_eq_infinite_cball: "x islimpt S ⟷ (∀e>0. infinite(S ∩ cball x e))"
apply (meson Int_mono ball_subset_cball finite_subset order_refl)
by (metis open_ball centre_in_ball finite_Int inf.absorb_iff2 inf_assoc open_contains_cball_eq)

lemma sequence_unique_limpt:
fixes f :: "nat ⇒ 'a::t2_space"
assumes "(f ⤏ l) sequentially"
and "l' islimpt (range f)"
shows "l' = l"
proof (rule ccontr)
assume "l' ≠ l"
obtain s t where "open s" "open t" "l' ∈ s" "l ∈ t" "s ∩ t = {}"
using hausdorff [OF ‹l' ≠ l›] by auto
have "eventually (λn. f n ∈ t) sequentially"
using assms(1) ‹open t› ‹l ∈ t› by (rule topological_tendstoD)
then obtain N where "∀n≥N. f n ∈ t"
unfolding eventually_sequentially by auto

have "UNIV = {..<N} ∪ {N..}"
by auto
then have "l' islimpt (f ` ({..<N} ∪ {N..}))"
using assms(2) by simp
then have "l' islimpt (f ` {..<N} ∪ f ` {N..})"
then have "l' islimpt (f ` {N..})"
then obtain y where "y ∈ f ` {N..}" "y ∈ s" "y ≠ l'"
using ‹l' ∈ s› ‹open s› by (rule islimptE)
then obtain n where "N ≤ n" "f n ∈ s" "f n ≠ l'"
by auto
with ‹∀n≥N. f n ∈ t› have "f n ∈ s ∩ t"
by simp
with ‹s ∩ t = {}› show False
by simp
qed

lemma bolzano_weierstrass_imp_closed:
fixes s :: "'a::{first_countable_topology,t2_space} set"
assumes "∀t. infinite t ∧ t ⊆ s --> (∃x ∈ s. x islimpt t)"
shows "closed s"
proof -
{
fix x l
assume as: "∀n::nat. x n ∈ s" "(x ⤏ l) sequentially"
then have "l ∈ s"
proof (cases "∀n. x n ≠ l")
case False
then show "l∈s" using as(1) by auto
next
case True note cas = this
with as(2) have "infinite (range x)"
using sequence_infinite_lemma[of x l] by auto
then obtain l' where "l'∈s" "l' islimpt (range x)"
using assms[THEN spec[where x="range x"]] as(1) by auto
then show "l∈s" using sequence_unique_limpt[of x l l']
using as cas by auto
qed
}
then show ?thesis
unfolding closed_sequential_limits by fast
qed

lemma compact_imp_bounded:
assumes "compact U"
shows "bounded U"
proof -
have "compact U" "∀x∈U. open (ball x 1)" "U ⊆ (⋃x∈U. ball x 1)"
using assms by auto
then obtain D where D: "D ⊆ U" "finite D" "U ⊆ (⋃x∈D. ball x 1)"
by (metis compactE_image)
from ‹finite D› have "bounded (⋃x∈D. ball x 1)"
then show "bounded U" using ‹U ⊆ (⋃x∈D. ball x 1)›
by (rule bounded_subset)
qed

text‹In particular, some common special cases.›

lemma compact_Un [intro]:
assumes "compact s"
and "compact t"
shows " compact (s ∪ t)"
proof (rule compactI)
fix f
assume *: "Ball f open" "s ∪ t ⊆ ⋃f"
from * ‹compact s› obtain s' where "s' ⊆ f ∧ finite s' ∧ s ⊆ ⋃s'"
unfolding compact_eq_heine_borel by (auto elim!: allE[of _ f])
moreover
from * ‹compact t› obtain t' where "t' ⊆ f ∧ finite t' ∧ t ⊆ ⋃t'"
unfolding compact_eq_heine_borel by (auto elim!: allE[of _ f])
ultimately show "∃f'⊆f. finite f' ∧ s ∪ t ⊆ ⋃f'"
by (auto intro!: exI[of _ "s' ∪ t'"])
qed

lemma compact_Union [intro]: "finite S ⟹ (⋀T. T ∈ S ⟹ compact T) ⟹ compact (⋃S)"
by (induct set: finite) auto

lemma compact_UN [intro]:
"finite A ⟹ (⋀x. x ∈ A ⟹ compact (B x)) ⟹ compact (⋃x∈A. B x)"
by (rule compact_Union) auto

lemma closed_Int_compact [intro]:
assumes "closed s"
and "compact t"
shows "compact (s ∩ t)"
using compact_Int_closed [of t s] assms

lemma compact_Int [intro]:
fixes s t :: "'a :: t2_space set"
assumes "compact s"
and "compact t"
shows "compact (s ∩ t)"
using assms by (intro compact_Int_closed compact_imp_closed)

lemma compact_sing [simp]: "compact {a}"
unfolding compact_eq_heine_borel by auto

lemma compact_insert [simp]:
assumes "compact s"
shows "compact (insert x s)"
proof -
have "compact ({x} ∪ s)"
using compact_sing assms by (rule compact_Un)
then show ?thesis by simp
qed

lemma finite_imp_compact: "finite s ⟹ compact s"
by (induct set: finite) simp_all

lemma open_delete:
fixes s :: "'a::t1_space set"
shows "open s ⟹ open (s - {x})"

lemma openin_delete:
fixes a :: "'a :: t1_space"
shows "openin (subtopology euclidean u) s
⟹ openin (subtopology euclidean u) (s - {a})"
by (metis Int_Diff open_delete openin_open)

text‹Compactness expressed with filters›

lemma closure_iff_nhds_not_empty:
"x ∈ closure X ⟷ (∀A. ∀S⊆A. open S ⟶ x ∈ S ⟶ X ∩ A ≠ {})"
proof safe
assume x: "x ∈ closure X"
fix S A
assume "open S" "x ∈ S" "X ∩ A = {}" "S ⊆ A"
then have "x ∉ closure (-S)"
by (auto simp: closure_complement subset_eq[symmetric] intro: interiorI)
with x have "x ∈ closure X - closure (-S)"
by auto
also have "… ⊆ closure (X ∩ S)"
using ‹open S› open_Int_closure_subset[of S X] by (simp add: closed_Compl ac_simps)
finally have "X ∩ S ≠ {}" by auto
then show False using ‹X ∩ A = {}› ‹S ⊆ A› by auto
next
assume "∀A S. S ⊆ A ⟶ open S ⟶ x ∈ S ⟶ X ∩ A ≠ {}"
from this[THEN spec, of "- X", THEN spec, of "- closure X"]
show "x ∈ closure X"
qed

corollary closure_Int_ball_not_empty:
assumes "S ⊆ closure T" "x ∈ S" "r > 0"
shows "T ∩ ball x r ≠ {}"
using assms centre_in_ball closure_iff_nhds_not_empty by blast

lemma compact_filter:
"compact U ⟷ (∀F. F ≠ bot ⟶ eventually (λx. x ∈ U) F ⟶ (∃x∈U. inf (nhds x) F ≠ bot))"
proof (intro allI iffI impI compact_fip[THEN iffD2] notI)
fix F
assume "compact U"
assume F: "F ≠ bot" "eventually (λx. x ∈ U) F"
then have "U ≠ {}"
by (auto simp: eventually_False)

define Z where "Z = closure ` {A. eventually (λx. x ∈ A) F}"
then have "∀z∈Z. closed z"
by auto
moreover
have ev_Z: "⋀z. z ∈ Z ⟹ eventually (λx. x ∈ z) F"
unfolding Z_def by (auto elim: eventually_mono intro: set_mp[OF closure_subset])
have "(∀B ⊆ Z. finite B ⟶ U ∩ ⋂B ≠ {})"
proof (intro allI impI)
fix B assume "finite B" "B ⊆ Z"
with ‹finite B› ev_Z F(2) have "eventually (λx. x ∈ U ∩ (⋂B)) F"
by (auto simp: eventually_ball_finite_distrib eventually_conj_iff)
with F show "U ∩ ⋂B ≠ {}"
by (intro notI) (simp add: eventually_False)
qed
ultimately have "U ∩ ⋂Z ≠ {}"
using ‹compact U› unfolding compact_fip by blast
then obtain x where "x ∈ U" and x: "⋀z. z ∈ Z ⟹ x ∈ z"
by auto

have "⋀P. eventually P (inf (nhds x) F) ⟹ P ≠ bot"
unfolding eventually_inf eventually_nhds
proof safe
fix P Q R S
assume "eventually R F" "open S" "x ∈ S"
with open_Int_closure_eq_empty[of S "{x. R x}"] x[of "closure {x. R x}"]
have "S ∩ {x. R x} ≠ {}" by (auto simp: Z_def)
moreover assume "Ball S Q" "∀x. Q x ∧ R x ⟶ bot x"
ultimately show False by (auto simp: set_eq_iff)
qed
with ‹x ∈ U› show "∃x∈U. inf (nhds x) F ≠ bot"
by (metis eventually_bot)
next
fix A
assume A: "∀a∈A. closed a" "∀B⊆A. finite B ⟶ U ∩ ⋂B ≠ {}" "U ∩ ⋂A = {}"
define F where "F = (INF a:insert U A. principal a)"
have "F ≠ bot"
unfolding F_def
proof (rule INF_filter_not_bot)
fix X
assume X: "X ⊆ insert U A" "finite X"
with A(2)[THEN spec, of "X - {U}"] have "U ∩ ⋂(X - {U}) ≠ {}"
by auto
with X show "(INF a:X. principal a) ≠ bot"
by (auto simp: INF_principal_finite principal_eq_bot_iff)
qed
moreover
have "F ≤ principal U"
unfolding F_def by auto
then have "eventually (λx. x ∈ U) F"
by (auto simp: le_filter_def eventually_principal)
moreover
assume "∀F. F ≠ bot ⟶ eventually (λx. x ∈ U) F ⟶ (∃x∈U. inf (nhds x) F ≠ bot)"
ultimately obtain x where "x ∈ U" and x: "inf (nhds x) F ≠ bot"
by auto

{ fix V assume "V ∈ A"
then have "F ≤ principal V"
unfolding F_def by (intro INF_lower2[of V]) auto
then have V: "eventually (λx. x ∈ V) F"
by (auto simp: le_filter_def eventually_principal)
have "x ∈ closure V"
unfolding closure_iff_nhds_not_empty
proof (intro impI allI)
fix S A
assume "open S" ```