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
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"
by (simp add: support_on_def)
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
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)
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"
by (metis ‹y ∈ T› diff_add_cancel dist_add_cancel2)
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"
by (metis ‹x ∈ S› add_diff_cancel_left' add_diff_eq diff_diff_add dist_norm)
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'"
by (simp add: topological_basis_def)
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 (simp add: 𝒜 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)
(simp_all add: B generate_topology.Basis subset_eq)
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 {}"
by (simp add: closedin_def)
lemma closedin_topspace[intro, simp]: "closedin U (topspace U)"
by (simp add: closedin_def)
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))"
by (simp add: openin_closedin_eq)
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"
by (simp add: subtopology_superset)
lemma subtopology_UNIV[simp]: "subtopology U UNIV = U"
by (simp add: subtopology_superset)
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"
by (simp add: closedin_def topspace_subtopology)
lemma openin_subtopology_Un:
"⟦openin (subtopology X T) S; openin (subtopology X U) S⟧
⟹ openin (subtopology X (T ∪ U)) S"
by (simp add: openin_subtopology) blast
lemma closedin_subtopology_Un:
"⟦closedin (subtopology X T) S; closedin (subtopology X U) S⟧
⟹ closedin (subtopology X (T ∪ U)) S"
by (simp add: closedin_subtopology) blast
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"
by (simp add: topspace_subtopology)
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)"
by (simp add: closedin_subtopology Int_ac)
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"
by (simp add: finite_imp_closed closed_subset)
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)
apply (clarsimp simp add: less_diff_eq)
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+)
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"
by (simp add: ball_def)
lemma mem_cball [simp]: "y ∈ cball x e ⟷ dist x y ≤ e"
by (simp add: cball_def)
lemma mem_sphere [simp]: "y ∈ sphere x e ⟷ dist x y = e"
by (simp add: sphere_def)
lemma ball_trivial [simp]: "ball x 0 = {}"
by (simp add: ball_def)
lemma cball_trivial [simp]: "cball x 0 = {x}"
by (simp add: cball_def)
lemma sphere_trivial [simp]: "sphere x 0 = {x}"
by (simp add: sphere_def)
lemma mem_ball_0 [simp]: "x ∈ ball 0 e ⟷ norm x < e"
for x :: "'a::real_normed_vector"
by (simp add: dist_norm)
lemma mem_cball_0 [simp]: "x ∈ cball 0 e ⟷ norm x ≤ e"
for x :: "'a::real_normed_vector"
by (simp add: dist_norm)
lemma disjoint_ballI: "dist x y ≥ r+s ⟹ ball x r ∩ ball y s = {}"
using dist_triangle_less_add not_le by fastforce
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"
by (simp add: dist_norm)
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"
by (simp add: subset_eq)
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"
by (simp add: subset_eq)
lemma subset_cball[intro]: "d ≤ e ⟹ cball x d ⊆ cball x e"
by (simp add: subset_eq)
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"
by (simp add: set_eq_iff) arith
lemma ball_min_Int: "ball a (min r s) = ball a r ∩ ball a s"
by (simp add: set_eq_iff)
lemma cball_max_Un: "cball a (max r s) = cball a r ∪ cball a s"
by (simp add: set_eq_iff) arith
lemma cball_min_Int: "cball a (min r s) = cball a r ∩ cball a s"
by (simp add: set_eq_iff)
lemma cball_diff_eq_sphere: "cball a r - ball a r = sphere a r"
by (auto simp: cball_def ball_def dist_commute)
lemma image_add_ball [simp]:
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
lemma image_add_cball [simp]:
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 (simp add: openin_open)
apply (metis Int_commute Int_mono inf.cobounded2 open_contains_ball order_refl subsetCE)
done
next
assume ?rhs
then show ?lhs
apply (simp add: openin_euclidean_subtopology_iff)
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 (simp add: openin_contains_ball)
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 (simp add: not_less)
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 (simp add: dependent_finite)
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)"
by (simp add: power_divide)
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"
by (simp add: countable_PiE countable_rat)
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)"
by (simp add: power_divide)
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"
by (simp add: countable_PiE countable_rat)
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"
by (auto simp: inner_add_left)
}
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"
by (auto simp: inner_add_left)
}
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
apply (simp add: subset_box)
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
apply (simp add: subset_box)
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
apply (clarsimp simp add: mem_box)
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]
by (metis image_cong uminus_add_conv_diff)
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 (simp add: islimpt_def subset_eq)
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"
by (simp add: islimpt_iff_eventually eventually_conj_iff)
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)"
by (simp add: interior_def open_Union)
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"]
by (simp add: open_subset_interior)
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)"
by (simp add: interior_closure)
lemma closed_closure[simp, intro]: "closed (closure S)"
by (simp add: closure_interior closed_Compl)
lemma closure_subset: "S ⊆ closure S"
by (simp add: closure_def)
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"
by (simp add: closure_interior)
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)"
by (simp_all add: open_Int)
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"
by (simp add: closure_interior)
lemma interior_complement: "interior (- S) = - closure S"
by (simp add: closure_interior)
lemma interior_diff: "interior(S - T) = interior S - closure T"
by (simp add: Diff_eq interior_complement)
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 (simp add: closure_interior interior_def)
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 (clarsimp simp add: islimpt_approachable)
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 (simp add: closedin_closed, safe)
apply (simp add: closed_limpt islimpt_subset)
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)"
by (simp add: Int_commute openin_open_Int)
show "x ∈ ball x (e / 2) ∩ S"
by (simp add: ‹0 < e› ‹x ∈ S›)
show "openin (subtopology euclidean T) (ball y (e/2) ∩ T)"
by (simp add: Int_commute openin_open_Int)
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
by (simp add: openin_Times)
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)"
by (simp add: frontier_def closed_Diff)
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"
by (simp_all add: closure_mono)
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"
by (simp add: Int_commute frontier_closures)
ultimately show ?thesis
by (simp add: Int_Un_distrib Int_assoc Int_left_commute assms frontier_closures)
qed
lemma frontier_straddle:
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 {} = {}"
by (simp add: frontier_def)
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 (clarsimp simp add: set_eq_iff)
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)"
by (simp add: filter_eq_iff)
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 (simp add: Lim_within, clarify)
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 (simp add: Lim_at_infinity, clarify)
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]
by (simp add: sequentially_imp_eventually_within)
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
by (simp_all add: f_def)
have f: "f n ∈ S ∧ (f n ≠ x) ∧ dist (f n) x < inverse(2 ^ n)" for n
proof (induction n)
case 0 show ?case
by (simp add: y)
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]])
apply (simp add: field_simps)
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
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"
by (simp add: Lim dist_norm)
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"
by (simp add: mult_left_mono g)
also have "… ≤ ¦f x¦ * (¦B¦ + 1)"
by (simp add: mult_left_mono)
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"
by (simp_all add: open_Compl)
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"
by (simp add: eventually_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›
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
by auto (metis add.commute add_le_cancel_right dist_commute dist_triangle_le)
lemma bounded_iff: "bounded S ⟷ (∃a. ∀x∈S. norm x ≤ a)"
unfolding bounded_any_center [where a=0]
by (simp add: dist_norm)
lemma bdd_above_norm: "bdd_above (norm ` X) ⟷ bounded X"
by (simp add: bounded_iff bdd_above_def)
lemma bounded_norm_comp: "bounded ((λx. norm (f x)) ` S) = bounded (f ` S)"
by (simp add: bounded_iff)
lemma boundedI:
assumes "⋀x. x ∈ S ⟹ norm x ≤ B"
shows "bounded S"
using assms bounded_iff by blast
lemma bounded_empty [simp]: "bounded {}"
by (simp add: bounded_def)
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"
by (simp add: f(1))
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 (simp add: bounded_def)
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 (simp add: add.commute add_pos_nonneg)
apply (erule subset_trans)
apply (clarsimp simp add: cball_def)
by (metis add_le_cancel_right add_strict_increasing dist_commute dist_triangle_le zero_less_one)
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 (simp add: bounded_iff)
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 (simp add: bounded_pos)
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))"
by (simp add: norm_sgn)
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]
by (rule order_trans[OF dist_triangle add_mono])
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"
by (auto intro!: norm_triangle_le add_mono)
} 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])
apply (simp add: closure_minimal)
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})"
by (simp_all add: open_Diff)
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"
by (simp add: islimpt_Un islimpt_finite)
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 (simp add: islimpt_eq_acc_point, safe)
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 (simp add: islimpt_eq_infinite_ball, safe)
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..})"
by (simp add: image_Un)
then have "l' islimpt (f ` {N..})"
by (simp add: islimpt_Un_finite)
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)"
by (simp add: bounded_UN)
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
by (simp add: Int_commute)
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})"
by (simp add: open_Diff)
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"
by (simp add: closure_subset open_Compl)
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" "x ∈ S" "S ⊆ A"
then have "eventually (λx. x ∈ A) (nhds x)"
by (auto simp: eventually_nhds)
with V have "eventually (λx. x ∈ V ∩ A) (inf (nhds x) F)"
by (auto simp: eventually_inf)
with x show "V ∩ A ≠ {}"
by (auto simp del: Int_iff simp add: trivial_limit_def)
qed
then have "x ∈ V"
using ‹V ∈ A› A(1) by simp
}
with ‹x∈U› have "x ∈ U ∩ ⋂A" by auto
with ‹U ∩ ⋂A = {}› show False by auto
qed
definition%important "countably_compact U ⟷
(∀A. countable A ⟶ (∀a∈A. open a) ⟶ U ⊆ ⋃A ⟶ (∃T⊆A. finite T ∧ U ⊆ ⋃T))"
lemma countably_compactE:
assumes "countably_compact s" and "∀t∈C. open t" and "s ⊆ ⋃C" "countable C"
obtains C' where "C' ⊆ C" and "finite C'" and "s ⊆ ⋃C'"
using assms unfolding countably_compact_def by metis
lemma countably_compactI:
assumes "⋀C. ∀t∈C. open t ⟹ s ⊆ ⋃C ⟹ countable C ⟹ (∃C'⊆C. finite C' ∧ s ⊆ ⋃C')"
shows "countably_compact s"
using assms unfolding countably_compact_def by metis
lemma compact_imp_countably_compact: "compact U ⟹ countably_compact U"
by (auto simp: compact_eq_heine_borel countably_compact_def)
lemma countably_compact_imp_compact:
assumes "countably_compact U"
and ccover: "countable B" "∀b∈B. open b"
and basis: "⋀T x. open T ⟹ x ∈ T ⟹ x ∈ U ⟹ ∃b∈B. x ∈ b ∧ b ∩ U ⊆ T"
shows "compact U"
using ‹countably_compact U›
unfolding compact_eq_heine_borel countably_compact_def
proof safe
fix A
assume A: "∀a∈A. open a" "U ⊆ ⋃A"
assume *: "∀A. countable A ⟶ (∀a∈A. open a) ⟶ U ⊆ ⋃A ⟶ (∃T⊆A. finite T ∧ U ⊆ ⋃T)"
moreover define C where "C = {b∈B. ∃a∈A. b ∩ U ⊆ a}"
ultimately have "countable C" "∀a∈C. open a"
unfolding C_def using ccover by auto
moreover
have "⋃A ∩ U ⊆ ⋃C"
proof safe
fix x a
assume "x ∈ U" "x ∈ a" "a ∈ A"
with basis[of a x] A obtain b where "b ∈ B" "x ∈ b" "b ∩ U ⊆ a"
by blast
with ‹a ∈ A› show "x ∈ ⋃C"
unfolding C_def by auto
qed
then have "U ⊆ ⋃C" using ‹U ⊆ ⋃A› by auto
ultimately obtain T where T: "T⊆C" "finite T" "U ⊆ ⋃T"
using * by metis
then have "∀t∈T. ∃a∈A. t ∩ U ⊆ a"
by (auto simp: C_def)
then obtain f where "∀t∈T. f t ∈ A ∧ t ∩ U ⊆ f t"
unfolding bchoice_iff Bex_def ..
with T show "∃T⊆A. finite T ∧ U ⊆ ⋃T"
unfolding C_def by (intro exI[of _ "f`T"]) fastforce
qed
proposition countably_compact_imp_compact_second_countable:
"countably_compact U ⟹ compact (U :: 'a :: second_countable_topology set)"
proof (rule countably_compact_imp_compact)
fix T and x :: 'a
assume "open T" "x ∈ T"
from topological_basisE[OF is_basis this] obtain b where
"b ∈ (SOME B. countable B ∧ topological_basis B)" "x ∈ b" "b ⊆ T" .
then show "∃b∈SOME B. countable B ∧ topological_basis B. x ∈ b ∧ b ∩ U ⊆ T"
by blast
qed (insert countable_basis topological_basis_open[OF is_basis], auto)
lemma countably_compact_eq_compact:
"countably_compact U ⟷ compact (U :: 'a :: second_countable_topology set)"
using countably_compact_imp_compact_second_countable compact_imp_countably_compact by blast
subsubsection‹Sequential compactness›
definition%important seq_compact :: "'a::topological_space set ⇒ bool"
where "seq_compact S ⟷
(∀f. (∀n. f n ∈ S) ⟶ (∃l∈S. ∃r::nat⇒nat. strict_mono r ∧ ((f ∘ r) ⤏ l) sequentially))"
lemma seq_compactI:
assumes "⋀f. ∀n. f n ∈ S ⟹ ∃l∈S. ∃r::nat⇒nat. strict_mono r ∧ ((f ∘ r) ⤏ l) sequentially"
shows "seq_compact S"
unfolding seq_compact_def using assms by fast
lemma seq_compactE:
assumes "seq_compact S" "∀n. f n ∈ S"
obtains l r where "l ∈ S" "strict_mono (r :: nat ⇒ nat)" "((f ∘ r) ⤏ l) sequentially"
using assms unfolding seq_compact_def by fast
lemma closed_sequentially:
assumes "closed s" and "∀n. f n ∈ s" and "f ⇢ l"
shows "l ∈ s"
proof (rule ccontr)
assume "l ∉ s"
with ‹closed s› and ‹f ⇢ l› have "eventually (λn. f n ∈ - s) sequentially"
by (fast intro: topological_tendstoD)
with ‹∀n. f n ∈ s› show "False"
by simp
qed
lemma seq_compact_Int_closed:
assumes "seq_compact s" and "closed t"
shows "seq_compact (s ∩ t)"
proof (rule seq_compactI)
fix f assume "∀n::nat. f n ∈ s ∩ t"
hence "∀n. f n ∈ s" and "∀n. f n ∈ t"
by simp_all
from ‹seq_compact s› and ‹∀n. f n ∈ s›
obtain l r where "l ∈ s" and r: "strict_mono r" and l: "(f ∘ r) ⇢ l"
by (rule seq_compactE)
from ‹∀n. f n ∈ t› have "∀n. (f ∘ r) n ∈ t"
by simp
from ‹closed t› and this and l have "l ∈ t"
by (rule closed_sequentially)
with ‹l ∈ s› and r and l show "∃l∈s ∩ t. ∃r. strict_mono r ∧ (f ∘ r) ⇢ l"
by fast
qed
lemma seq_compact_closed_subset:
assumes "closed s" and "s ⊆ t" and "seq_compact t"
shows "seq_compact s"
using assms seq_compact_Int_closed [of t s] by (simp add: Int_absorb1)
lemma seq_compact_imp_countably_compact:
fixes U :: "'a :: first_countable_topology set"
assumes "seq_compact U"
shows "countably_compact U"
proof (safe intro!: countably_compactI)
fix A
assume A: "∀a∈A. open a" "U ⊆ ⋃A" "countable A"
have subseq: "⋀X. range X ⊆ U ⟹ ∃r x. x ∈ U ∧ strict_mono (r :: nat ⇒ nat) ∧ (X ∘ r) ⇢ x"
using ‹seq_compact U› by (fastforce simp: seq_compact_def subset_eq)
show "∃T⊆A. finite T ∧ U ⊆ ⋃T"
proof cases
assume "finite A"
with A show ?thesis by auto
next
assume "infinite A"
then have "A ≠ {}" by auto
show ?thesis
proof (rule ccontr)
assume "¬ (∃T⊆A. finite T ∧ U ⊆ ⋃T)"
then have "∀T. ∃x. T ⊆ A ∧ finite T ⟶ (x ∈ U - ⋃T)"
by auto
then obtain X' where T: "⋀T. T ⊆ A ⟹ finite T ⟹ X' T ∈ U - ⋃T"
by metis
define X where "X n = X' (from_nat_into A ` {.. n})" for n
have X: "⋀n. X n ∈ U - (⋃i≤n. from_nat_into A i)"
using ‹A ≠ {}› unfolding X_def by (intro T) (auto intro: from_nat_into)
then have "range X ⊆ U"
by auto
with subseq[of X] obtain r x where "x ∈ U" and r: "strict_mono r" "(X ∘ r) ⇢ x"
by auto
from ‹x∈U› ‹U ⊆ ⋃A› from_nat_into_surj[OF ‹countable A›]
obtain n where "x ∈ from_nat_into A n" by auto
with r(2) A(1) from_nat_into[OF ‹A ≠ {}›, of n]
have "eventually (λi. X (r i) ∈ from_nat_into A n) sequentially"
unfolding tendsto_def by (auto simp: comp_def)
then obtain N where "⋀i. N ≤ i ⟹ X (r i) ∈ from_nat_into A n"
by (auto simp: eventually_sequentially)
moreover from X have "⋀i. n ≤ r i ⟹ X (r i) ∉ from_nat_into A n"
by auto
moreover from ‹strict_mono r›[THEN seq_suble, of "max n N"] have "∃i. n ≤ r i ∧ N ≤ i"
by (auto intro!: exI[of _ "max n N"])
ultimately show False
by auto
qed
qed
qed
lemma compact_imp_seq_compact:
fixes U :: "'a :: first_countable_topology set"
assumes "compact U"
shows "seq_compact U"
unfolding seq_compact_def
proof safe
fix X :: "nat ⇒ 'a"
assume "∀n. X n ∈ U"
then have "eventually (λx. x ∈ U) (filtermap X sequentially)"
by (auto simp: eventually_filtermap)
moreover
have "filtermap X sequentially ≠ bot"
by (simp add: trivial_limit_def eventually_filtermap)
ultimately
obtain x where "x ∈ U" and x: "inf (nhds x) (filtermap X sequentially) ≠ bot" (is "?F ≠ _")
using ‹compact U› by (auto simp: compact_filter)
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 s where "s n i = (SOME j. i < j ∧ X j ∈ A (Suc n))" for n i
{
fix n i
have "∃a. i < a ∧ X a ∈ A (Suc n)"
proof (rule ccontr)
assume "¬ (∃a>i. X a ∈ A (Suc n))"
then have "⋀a. Suc i ≤ a ⟹ X a ∉ A (Suc n)"
by auto
then have "eventually (λx. x ∉ A (Suc n)) (filtermap X sequentially)"
by (auto simp: eventually_filtermap eventually_sequentially)
moreover have "eventually (λx. x ∈ A (Suc n)) (nhds x)"
using A(1,2)[of "Suc n"] by (auto simp: eventually_nhds)
ultimately have "eventually (λx. False) ?F"
by (auto simp: eventually_inf)
with x show False
by (simp add: eventually_False)
qed
then have "i < s n i" "X (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. X (r n)) ⇢ x"
proof (rule topological_tendstoI)
fix S
assume "open S" "x ∈ S"
with A(3) have "eventually (λi. A i ⊆ S) sequentially"
by auto
moreover
{
fix i
assume "Suc 0 ≤ i"
then have "X (r i) ∈ A i"
by (cases i) (simp_all add: r_def s)
}
then have "eventually (λi. X (r i) ∈ A i) sequentially"
by (auto simp: eventually_sequentially)
ultimately show "eventually (λi. X (r i) ∈ S) sequentially"
by eventually_elim auto
qed
ultimately show "∃x ∈ U. ∃r. strict_mono r ∧ (X ∘ r) ⇢ x"
using ‹x ∈ U› by (auto simp: convergent_def comp_def)
qed
lemma countably_compact_imp_acc_point:
assumes "countably_compact s"
and "countable t"
and "infinite t"
and "t ⊆ s"
shows "∃x∈s. ∀U. x∈U ∧ open U ⟶ infinite (U ∩ t)"
proof (rule ccontr)
define C where "C = (λF. interior (F ∪ (- t))) ` {F. finite F ∧ F ⊆ t }"
note ‹countably_compact s›
moreover have "∀t∈C. open t"
by (auto simp: C_def)
moreover
assume "¬ (∃x∈s. ∀U. x∈U ∧ open U ⟶ infinite (U ∩ t))"
then have s: "⋀x. x ∈ s ⟹ ∃U. x∈U ∧ open U ∧ finite (U ∩ t)" by metis
have "s ⊆ ⋃C"
using ‹t ⊆ s›
unfolding C_def
apply (safe dest!: s)
apply (rule_tac a="U ∩ t" in UN_I)
apply (auto intro!: interiorI simp add: finite_subset)
done
moreover
from ‹countable t› have "countable C"
unfolding C_def by (auto intro: countable_Collect_finite_subset)
ultimately
obtain D where "D ⊆ C" "finite D" "s ⊆ ⋃D"
by (rule countably_compactE)
then obtain E where E: "E ⊆ {F. finite F ∧ F ⊆ t }" "finite E"
and s: "s ⊆ (⋃F∈E. interior (F ∪ (- t)))"
by (metis (lifting) finite_subset_image C_def)
from s ‹t ⊆ s› have "t ⊆ ⋃E"
using interior_subset by blast
moreover have "finite (⋃E)"
using E by auto
ultimately show False using ‹infinite t›
by (auto simp: finite_subset)
qed
lemma countable_acc_point_imp_seq_compact:
fixes s :: "'a::first_countable_topology set"
assumes "∀t. infinite t ∧ countable t ∧ t ⊆ s ⟶
(∃x∈s. ∀U. x∈U ∧ open U ⟶ infinite (U ∩ t))"
shows "seq_compact s"
proof -
{
fix f :: "nat ⇒ 'a"
assume f: "∀n. f n ∈ s"
have "∃l∈s. ∃r. strict_mono r ∧ ((f ∘ r) ⤏ l) sequentially"
proof (cases "finite (range f)")
case True
obtain l where "infinite {n. f n = f l}"
using pigeonhole_infinite[OF _ True] by auto
then obtain r :: "nat ⇒ nat" where "strict_mono r" and fr: "∀n. f (r n) = f l"
using infinite_enumerate by blast
then have "strict_mono r ∧ (f ∘ r) ⇢ f l"
by (simp add: fr o_def)
with f show "∃l∈s. ∃r. strict_mono r ∧ (f ∘ r) ⇢ l"
by auto
next
case False
with f assms have "∃x∈s. ∀U. x∈U ∧ open U ⟶ infinite (U ∩ range f)"
by auto
then obtain l where "l ∈ s" "∀U. l∈U ∧ open U ⟶ infinite (U ∩ range f)" ..
from this(2) have "∃r. strict_mono r ∧ ((f ∘ r) ⤏ l) sequentially"
using acc_point_range_imp_convergent_subsequence[of l f] by auto
with ‹l ∈ s› show "∃l∈s. ∃r. strict_mono r ∧ ((f ∘ r) ⤏ l) sequentially" ..
qed
}
then show ?thesis
unfolding seq_compact_def by auto
qed
lemma seq_compact_eq_countably_compact:
fixes U :: "'a :: first_countable_topology set"
shows "seq_compact U ⟷ countably_compact U"
using
countable_acc_point_imp_seq_compact
countably_compact_imp_acc_point
seq_compact_imp_countably_compact
by metis
lemma seq_compact_eq_acc_point:
fixes s :: "'a :: first_countable_topology set"
shows "seq_compact s ⟷
(∀t. infinite t ∧ countable t ∧ t ⊆ s --> (∃x∈s. ∀U. x∈U ∧ open U ⟶ infinite (U ∩ t)))"
using
countable_acc_point_imp_seq_compact[of s]
countably_compact_imp_acc_point[of s]
seq_compact_imp_countably_compact[of s]
by metis
lemma seq_compact_eq_compact:
fixes U :: "'a :: second_countable_topology set"
shows "seq_compact U ⟷ compact U"
using seq_compact_eq_countably_compact countably_compact_eq_compact by blast
proposition bolzano_weierstrass_imp_seq_compact:
fixes s :: "'a::{t1_space, first_countable_topology} set"
shows "∀t. infinite t ∧ t ⊆ s ⟶ (∃x ∈ s. x islimpt t) ⟹ seq_compact s"
by (rule countable_acc_point_imp_seq_compact) (metis islimpt_eq_acc_point)
subsubsection‹Totally bounded›
lemma cauchy_def: "Cauchy s ⟷ (∀e>0. ∃N. ∀m n. m ≥ N ∧ n ≥ N ⟶ dist (s m) (s n) < e)"
unfolding Cauchy_def by metis
proposition seq_compact_imp_totally_bounded:
assumes "seq_compact s"
shows "∀e>0. ∃k. finite k ∧ k ⊆ s ∧ s ⊆ (⋃x∈k. ball x e)"
proof -
{ fix e::real assume "e > 0" assume *: "⋀k. finite k ⟹ k ⊆ s ⟹ ¬ s ⊆ (⋃x∈k. ball x e)"
let ?Q = "λx n r. r ∈ s ∧ (∀m < (n::nat). ¬ (dist (x m) r < e))"
have "∃x. ∀n::nat. ?Q x n (x n)"
proof (rule dependent_wellorder_choice)
fix n x assume "⋀y. y < n ⟹ ?Q x y (x y)"
then have "¬ s ⊆ (⋃x∈x ` {0..<n}. ball x e)"
using *[of "x ` {0 ..< n}"] by (auto simp: subset_eq)
then obtain z where z:"z∈s" "z ∉ (⋃x∈x ` {0..<n}. ball x e)"
unfolding subset_eq by auto
show "∃r. ?Q x n r"
using z by auto
qed simp
then obtain x where "∀n::nat. x n ∈ s" and x:"⋀n m. m < n ⟹ ¬ (dist (x m) (x n) < e)"
by blast
then obtain l r where "l ∈ s" and r:"strict_mono r" and "((x ∘ r) ⤏ l) sequentially"
using assms by (metis seq_compact_def)
from this(3) have "Cauchy (x ∘ r)"
using LIMSEQ_imp_Cauchy by auto
then obtain N::nat where "⋀m n. N ≤ m ⟹ N ≤ n ⟹ dist ((x ∘ r) m) ((x ∘ r) n) < e"
unfolding cauchy_def using ‹e > 0› by blast
then have False
using x[of "r N" "r (N+1)"] r by (auto simp: strict_mono_def) }
then show ?thesis
by metis
qed
subsubsection‹Heine-Borel theorem›
proposition seq_compact_imp_heine_borel:
fixes s :: "'a :: metric_space set"
assumes "seq_compact s"
shows "compact s"
proof -
from seq_compact_imp_totally_bounded[OF ‹seq_compact s›]
obtain f where f: "∀e>0. finite (f e) ∧ f e ⊆ s ∧ s ⊆ (⋃x∈f e. ball x e)"
unfolding choice_iff' ..
define K where "K = (λ(x, r). ball x r) ` ((⋃e ∈ ℚ ∩ {0 <..}. f e) × ℚ)"
have "countably_compact s"
using ‹seq_compact s› by (rule seq_compact_imp_countably_compact)
then show "compact s"
proof (rule countably_compact_imp_compact)
show "countable K"
unfolding K_def using f
by (auto intro: countable_finite countable_subset countable_rat
intro!: countable_image countable_SIGMA countable_UN)
show "∀b∈K. open b" by (auto simp: K_def)
next
fix T x
assume T: "open T" "x ∈ T" and x: "x ∈ s"
from openE[OF T] obtain e where "0 < e" "ball x e ⊆ T"
by auto
then have "0 < e / 2" "ball x (e / 2) ⊆ T"
by auto
from Rats_dense_in_real[OF ‹0 < e / 2›] obtain r where "r ∈ ℚ" "0 < r" "r < e / 2"
by auto
from f[rule_format, of r] ‹0 < r› ‹x ∈ s› obtain k where "k ∈ f r" "x ∈ ball k r"
by auto
from ‹r ∈ ℚ› ‹0 < r› ‹k ∈ f r› have "ball k r ∈ K"
by (auto simp: K_def)
then show "∃b∈K. x ∈ b ∧ b ∩ s ⊆ T"
proof (rule bexI[rotated], safe)
fix y
assume "y ∈ ball k r"
with ‹r < e / 2› ‹x ∈ ball k r› have "dist x y < e"
by (intro dist_triangle_half_r [of k _ e]) (auto simp: dist_commute)
with ‹ball x e ⊆ T› show "y ∈ T"
by auto
next
show "x ∈ ball k r" by fact
qed
qed
qed
proposition compact_eq_seq_compact_metric:
"compact (s :: 'a::metric_space set) ⟷ seq_compact s"
using compact_imp_seq_compact seq_compact_imp_heine_borel by blast
proposition compact_def:
"compact (S :: 'a::metric_space set) ⟷
(∀f. (∀n. f n ∈ S) ⟶ (∃l∈S. ∃r::nat⇒nat. strict_mono r ∧ (f ∘ r) ⇢ l))"
unfolding compact_eq_seq_compact_metric seq_compact_def by auto
subsubsection ‹Complete the chain of compactness variants›
proposition compact_eq_bolzano_weierstrass:
fixes s :: "'a::metric_space set"
shows "compact s ⟷ (∀t. infinite t ∧ t ⊆ s --> (∃x ∈ s. x islimpt t))"
(is "?lhs = ?rhs")
proof
assume ?lhs
then show ?rhs
using heine_borel_imp_bolzano_weierstrass[of s] by auto
next
assume ?rhs
then show ?lhs
unfolding compact_eq_seq_compact_metric by (rule bolzano_weierstrass_imp_seq_compact)
qed
proposition bolzano_weierstrass_imp_bounded:
"∀t. infinite t ∧ t ⊆ s ⟶ (∃x ∈ s. x islimpt t) ⟹ bounded s"
using compact_imp_bounded unfolding compact_eq_bolzano_weierstrass .
subsection ‹Metric spaces with the Heine-Borel property›
text ‹
A metric space (or topological vector space) is said to have the
Heine-Borel property if every closed and bounded subset is compact.
›
class heine_borel = metric_space +
assumes bounded_imp_convergent_subsequence:
"bounded (range f) ⟹ ∃l r. strict_mono (r::nat⇒nat) ∧ ((f ∘ r) ⤏ l) sequentially"
proposition bounded_closed_imp_seq_compact:
fixes s::"'a::heine_borel set"
assumes "bounded s"
and "closed s"
shows "seq_compact s"
proof (unfold seq_compact_def, clarify)
fix f :: "nat ⇒ 'a"
assume f: "∀n. f n ∈ s"
with ‹bounded s› have "bounded (range f)"
by (auto intro: bounded_subset)
obtain l r where r: "strict_mono (r :: nat ⇒ nat)" and l: "((f ∘ r) ⤏ l) sequentially"
using bounded_imp_convergent_subsequence [OF ‹bounded (range f)›] by auto
from f have fr: "∀n. (f ∘ r) n ∈ s"
by simp
have "l ∈ s" using ‹closed s› fr l
by (rule closed_sequentially)
show "∃l∈s. ∃r. strict_mono r ∧ ((f ∘ r) ⤏ l) sequentially"
using ‹l ∈ s› r l by blast
qed
lemma compact_eq_bounded_closed:
fixes s :: "'a::heine_borel set"
shows "compact s ⟷ bounded s ∧ closed s"
(is "?lhs = ?rhs")
proof
assume ?lhs
then show ?rhs
using compact_imp_closed compact_imp_bounded
by blast
next
assume ?rhs
then show ?lhs
using bounded_closed_imp_seq_compact[of s]
unfolding compact_eq_seq_compact_metric
by auto
qed
lemma compact_Inter:
fixes ℱ :: "'a :: heine_borel set set"
assumes com: "⋀S. S ∈ ℱ ⟹ compact S" and "ℱ ≠ {}"
shows "compact(⋂ ℱ)"
using assms
by (meson Inf_lower all_not_in_conv bounded_subset closed_Inter compact_eq_bounded_closed)
lemma compact_closure [simp]:
fixes S :: "'a::heine_borel set"
shows "compact(closure S) ⟷ bounded S"
by (meson bounded_closure bounded_subset closed_closure closure_subset compact_eq_bounded_closed)
lemma not_compact_UNIV[simp]:
fixes s :: "'a::{real_normed_vector,perfect_space,heine_borel} set"
shows "~ compact (UNIV::'a set)"
by (simp add: compact_eq_bounded_closed)
text‹Representing sets as the union of a chain of compact sets.›
lemma closed_Union_compact_subsets:
fixes S :: "'a::{heine_borel,real_normed_vector} set"
assumes "closed S"
obtains F where "⋀n. compact(F n)" "⋀n. F n ⊆ S" "⋀n. F n ⊆ F(Suc n)"
"(⋃n. F n) = S" "⋀K. ⟦compact K; K ⊆ S⟧ ⟹ ∃N. ∀n ≥ N. K ⊆ F n"
proof
show "compact (S ∩ cball 0 (of_nat n))" for n
using assms compact_eq_bounded_closed by auto
next
show "(⋃n. S ∩ cball 0 (real n)) = S"
by (auto simp: real_arch_simple)
next
fix K :: "'a set"
assume "compact K" "K ⊆ S"
then obtain N where "K ⊆ cball 0 N"
by (meson bounded_pos mem_cball_0 compact_imp_bounded subsetI)
then show "∃N. ∀n≥N. K ⊆ S ∩ cball 0 (real n)"
by (metis of_nat_le_iff Int_subset_iff ‹K ⊆ S› real_arch_simple subset_cball subset_trans)
qed auto
instance%important real :: heine_borel
proof%unimportant
fix f :: "nat ⇒ real"
assume f: "bounded (range f)"
obtain r :: "nat ⇒ nat" where r: "strict_mono r" "monoseq (f ∘ r)"
unfolding comp_def by (metis seq_monosub)
then have "Bseq (f ∘ r)"
unfolding Bseq_eq_bounded using f by (force intro: bounded_subset)
with r show "∃l r. strict_mono r ∧ (f ∘ r) ⇢ l"
using Bseq_monoseq_convergent[of "f ∘ r"] by (auto simp: convergent_def)
qed
lemma compact_lemma_general:
fixes f :: "nat ⇒ 'a"
fixes proj::"'a ⇒ 'b ⇒ 'c::heine_borel" (infixl "proj" 60)
fixes unproj:: "('b ⇒ 'c) ⇒ 'a"
assumes finite_basis: "finite basis"
assumes bounded_proj: "⋀k. k ∈ basis ⟹ bounded ((λx. x proj k) ` range f)"
assumes proj_unproj: "⋀e k. k ∈ basis ⟹ (unproj e) proj k = e k"
assumes unproj_proj: "⋀x. unproj (λk. x proj k) = x"
shows "∀d⊆basis. ∃l::'a. ∃ r::nat⇒nat.
strict_mono r ∧ (∀e>0. eventually (λn. ∀i∈d. dist (f (r n) proj i) (l proj i) < e) sequentially)"
proof safe
fix d :: "'b set"
assume d: "d ⊆ basis"
with finite_basis have "finite d"
by (blast intro: finite_subset)
from this d show "∃l::'a. ∃r::nat⇒nat. strict_mono r ∧
(∀e>0. eventually (λn. ∀i∈d. dist (f (r n) proj i) (l proj i) < e) sequentially)"
proof (induct d)
case empty
then show ?case
unfolding strict_mono_def by auto
next
case (insert k d)
have k[intro]: "k ∈ basis"
using insert by auto
have s': "bounded ((λx. x proj k) ` range f)"
using k
by (rule bounded_proj)
obtain l1::"'a" and r1 where r1: "strict_mono r1"
and lr1: "∀e > 0. eventually (λn. ∀i∈d. dist (f (r1 n) proj i) (l1 proj i) < e) sequentially"
using insert(3) using insert(4) by auto
have f': "∀n. f (r1 n) proj k ∈ (λx. x proj k) ` range f"
by simp
have "bounded (range (λi. f (r1 i) proj k))"
by (metis (lifting) bounded_subset f' image_subsetI s')
then obtain l2 r2 where r2:"strict_mono r2" and lr2:"((λi. f (r1 (r2 i)) proj k) ⤏ l2) sequentially"
using bounded_imp_convergent_subsequence[of "λi. f (r1 i) proj k"]
by (auto simp: o_def)
define r where "r = r1 ∘ r2"
have r:"strict_mono r"
using r1 and r2 unfolding r_def o_def strict_mono_def by auto
moreover
define l where "l = unproj (λi. if i = k then l2 else l1 proj i)"
{
fix e::real
assume "e > 0"
from lr1 ‹e > 0› have N1: "eventually (λn. ∀i∈d. dist (f (r1 n) proj i) (l1 proj i) < e) sequentially"
by blast
from lr2 ‹e > 0› have N2:"eventually (λn. dist (f (r1 (r2 n)) proj k) l2 < e) sequentially"
by (rule tendstoD)
from r2 N1 have N1': "eventually (λn. ∀i∈d. dist (f (r1 (r2 n)) proj i) (l1 proj i) < e) sequentially"
by (rule eventually_subseq)
have "eventually (λn. ∀i∈(insert k d). dist (f (r n) proj i) (l proj i) < e) sequentially"
using N1' N2
by eventually_elim (insert insert.prems, auto simp: l_def r_def o_def proj_unproj)
}
ultimately show ?case by auto
qed
qed
lemma compact_lemma:
fixes f :: "nat ⇒ 'a::euclidean_space"
assumes "bounded (range f)"
shows "∀d⊆Basis. ∃l::'a. ∃ r.
strict_mono r ∧ (∀e>0. eventually (λn. ∀i∈d. dist (f (r n) ∙ i) (l ∙ i) < e) sequentially)"
by (rule compact_lemma_general[where unproj="λe. ∑i∈Basis. e i *⇩R i"])
(auto intro!: assms bounded_linear_inner_left bounded_linear_image
simp: euclidean_representation)
instance%important euclidean_space ⊆ heine_borel
proof%unimportant
fix f :: "nat ⇒ 'a"
assume f: "bounded (range f)"
then obtain l::'a and r where r: "strict_mono r"
and l: "∀e>0. eventually (λn. ∀i∈Basis. dist (f (r n) ∙ i) (l ∙ i) < e) sequentially"
using compact_lemma [OF f] by blast
{
fix e::real
assume "e > 0"
hence "e / real_of_nat DIM('a) > 0" by (simp add: DIM_positive)
with l have "eventually (λn. ∀i∈Basis. dist (f (r n) ∙ i) (l ∙ i) < e / (real_of_nat DIM('a))) sequentially"
by simp
moreover
{
fix n
assume n: "∀i∈Basis. dist (f (r n) ∙ i) (l ∙ i) < e / (real_of_nat DIM('a))"
have "dist (f (r n)) l ≤ (∑i∈Basis. dist (f (r n) ∙ i) (l ∙ i))"
apply (subst euclidean_dist_l2)
using zero_le_dist
apply (rule L2_set_le_sum)
done
also have "… < (∑i∈(Basis::'a set). e / (real_of_nat DIM('a)))"
apply (rule sum_strict_mono)
using n
apply auto
done
finally have "dist (f (r n)) l < e"
by auto
}
ultimately have "eventually (λn. dist (f (r n)) l < e) sequentially"
by (rule eventually_mono)
}
then have *: "((f ∘ r) ⤏ l) sequentially"
unfolding o_def tendsto_iff by simp
with r show "∃l r. strict_mono r ∧ ((f ∘ r) ⤏ l) sequentially"
by auto
qed
lemma bounded_fst: "bounded s ⟹ bounded (fst ` s)"
unfolding bounded_def
by (metis (erased, hide_lams) dist_fst_le image_iff order_trans)
lemma bounded_snd: "bounded s ⟹ bounded (snd ` s)"
unfolding bounded_def
by (metis (no_types, hide_lams) dist_snd_le image_iff order.trans)
instance%important prod :: (heine_borel, heine_borel) heine_borel
proof%unimportant
fix f :: "nat ⇒ 'a × 'b"
assume f: "bounded (range f)"
then have "bounded (fst ` range f)"
by (rule bounded_fst)
then have s1: "bounded (range (fst ∘ f))"
by (simp add: image_comp)
obtain l1 r1 where r1: "strict_mono r1" and l1: "(λn. fst (f (r1 n))) ⇢ l1"
using bounded_imp_convergent_subsequence [OF s1] unfolding o_def by fast
from f have s2: "bounded (range (snd ∘ f ∘ r1))"
by (auto simp: image_comp intro: bounded_snd bounded_subset)
obtain l2 r2 where r2: "strict_mono r2" and l2: "((λn. snd (f (r1 (r2 n)))) ⤏ l2) sequentially"
using bounded_imp_convergent_subsequence [OF s2]
unfolding o_def by fast
have l1': "((λn. fst (f (r1 (r2 n)))) ⤏ l1) sequentially"
using LIMSEQ_subseq_LIMSEQ [OF l1 r2] unfolding o_def .
have l: "((f ∘ (r1 ∘ r2)) ⤏ (l1, l2)) sequentially"
using tendsto_Pair [OF l1' l2] unfolding o_def by simp
have r: "strict_mono (r1 ∘ r2)"
using r1 r2 unfolding strict_mono_def by simp
show "∃l r. strict_mono r ∧ ((f ∘ r) ⤏ l) sequentially"
using l r by fast
qed
subsubsection ‹Completeness›
proposition (in metric_space) completeI:
assumes "⋀f. ∀n. f n ∈ s ⟹ Cauchy f ⟹ ∃l∈s. f ⇢ l"
shows "complete s"
using assms unfolding complete_def by fast
proposition (in metric_space) completeE:
assumes "complete s" and "∀n. f n ∈ s" and "Cauchy f"
obtains l where "l ∈ s" and "f ⇢ l"
using assms unfolding complete_def by fast
lemma compact_imp_complete:
fixes s :: "'a::metric_space set"
assumes "compact s"
shows "complete s"
proof -
{
fix f
assume as: "(∀n::nat. f n ∈ s)" "Cauchy f"
from as(1) obtain l r where lr: "l∈s" "strict_mono r" "(f ∘ r) ⇢ l"
using assms unfolding compact_def by blast
note lr' = seq_suble [OF lr(2)]
{
fix e :: real
assume "e > 0"
from as(2) obtain N where N:"∀m n. N ≤ m ∧ N ≤ n ⟶ dist (f m) (f n) < e/2"
unfolding cauchy_def
using ‹e > 0›
apply (erule_tac x="e/2" in allE, auto)
done
from lr(3)[unfolded lim_sequentially, THEN spec[where x="e/2"]]
obtain M where M:"∀n≥M. dist ((f ∘ r) n) l < e/2"
using ‹e > 0› by auto
{
fix n :: nat
assume n: "n ≥ max N M"
have "dist ((f ∘ r) n) l < e/2"
using n M by auto
moreover have "r n ≥ N"
using lr'[of n] n by auto
then have "dist (f n) ((f ∘ r) n) < e / 2"
using N and n by auto
ultimately have "dist (f n) l < e"
using dist_triangle_half_r[of "f (r n)" "f n" e l]
by (auto simp: dist_commute)
}
then have "∃N. ∀n≥N. dist (f n) l < e" by blast
}
then have "∃l∈s. (f ⤏ l) sequentially" using ‹l∈s›
unfolding lim_sequentially by auto
}
then show ?thesis unfolding complete_def by auto
qed
proposition compact_eq_totally_bounded:
"compact s ⟷ complete s ∧ (∀e>0. ∃k. finite k ∧ s ⊆ (⋃x∈k. ball x e))"
(is "_ ⟷ ?rhs")
proof
assume assms: "?rhs"
then obtain k where k: "⋀e. 0 < e ⟹ finite (k e)" "⋀e. 0 < e ⟹ s ⊆ (⋃x∈k e. ball x e)"
by (auto simp: choice_iff')
show "compact s"
proof cases
assume "s = {}"
then show "compact s" by (simp add: compact_def)
next
assume "s ≠ {}"
show ?thesis
unfolding compact_def
proof safe
fix f :: "nat ⇒ 'a"
assume f: "∀n. f n ∈ s"
define e where "e n = 1 / (2 * Suc n)" for n
then have [simp]: "⋀n. 0 < e n" by auto
define B where "B n U = (SOME b. infinite {n. f n ∈ b} ∧ (∃x. b ⊆ ball x (e n) ∩ U))" for n U
{
fix n U
assume "infinite {n. f n ∈ U}"
then have "∃b∈k (e n). infinite {i∈{n. f n ∈ U}. f i ∈ ball b (e n)}"
using k f by (intro pigeonhole_infinite_rel) (auto simp: subset_eq)
then obtain a where
"a ∈ k (e n)"
"infinite {i ∈ {n. f n ∈ U}. f i ∈ ball a (e n)}" ..
then have "∃b. infinite {i. f i ∈ b} ∧ (∃x. b ⊆ ball x (e n) ∩ U)"
by (intro exI[of _ "ball a (e n) ∩ U"] exI[of _ a]) (auto simp: ac_simps)
from someI_ex[OF this]
have "infinite {i. f i ∈ B n U}" "∃x. B n U ⊆ ball x (e n) ∩ U"
unfolding B_def by auto
}
note B = this
define F where "F = rec_nat (B 0 UNIV) B"
{
fix n
have "infinite {i. f i ∈ F n}"
by (induct n) (auto simp: F_def B)
}
then have F: "⋀n. ∃x. F (Suc n) ⊆ ball x (e n) ∩ F n"
using B by (simp add: F_def)
then have F_dec: "⋀m n. m ≤ n ⟹ F n ⊆ F m"
using decseq_SucI[of F] by (auto simp: decseq_def)
obtain sel where sel: "⋀k i. i < sel k i" "⋀k i. f (sel k i) ∈ F k"
proof (atomize_elim, unfold all_conj_distrib[symmetric], intro choice allI)
fix k i
have "infinite ({n. f n ∈ F k} - {.. i})"
using ‹infinite {n. f n ∈ F k}› by auto
from infinite_imp_nonempty[OF this]
show "∃x>i. f x ∈ F k"
by (simp add: set_eq_iff not_le conj_commute)
qed
define t where "t = rec_nat (sel 0 0) (λn i. sel (Suc n) i)"
have "strict_mono t"
unfolding strict_mono_Suc_iff by (simp add: t_def sel)
moreover have "∀i. (f ∘ t) i ∈ s"
using f by auto
moreover
{
fix n
have "(f ∘ t) n ∈ F n"
by (cases n) (simp_all add: t_def sel)
}
note t = this
have "Cauchy (f ∘ t)"
proof (safe intro!: metric_CauchyI exI elim!: nat_approx_posE)
fix r :: real and N n m
assume "1 / Suc N < r" "Suc N ≤ n" "Suc N ≤ m"
then have "(f ∘ t) n ∈ F (Suc N)" "(f ∘ t) m ∈ F (Suc N)" "2 * e N < r"
using F_dec t by (auto simp: e_def field_simps of_nat_Suc)
with F[of N] obtain x where "dist x ((f ∘ t) n) < e N" "dist x ((f ∘ t) m) < e N"
by (auto simp: subset_eq)
with dist_triangle[of "(f ∘ t) m" "(f ∘ t) n" x] ‹2 * e N < r›
show "dist ((f ∘ t) m) ((f ∘ t) n) < r"
by (simp add: dist_commute)
qed
ultimately show "∃l∈s. ∃r. strict_mono r ∧ (f ∘ r) ⇢ l"
using assms unfolding complete_def by blast
qed
qed
qed (metis compact_imp_complete compact_imp_seq_compact seq_compact_imp_totally_bounded)
lemma cauchy_imp_bounded:
assumes "Cauchy s"
shows "bounded (range s)"
proof -
from assms obtain N :: nat where "∀m n. N ≤ m ∧ N ≤ n ⟶ dist (s m) (s n) < 1"
unfolding cauchy_def by force
then have N:"∀n. N ≤ n ⟶ dist (s N) (s n) < 1" by auto
moreover
have "bounded (s ` {0..N})"
using finite_imp_bounded[of "s ` {1..N}"] by auto
then obtain a where a:"∀x∈s ` {0..N}. dist (s N) x ≤ a"
unfolding bounded_any_center [where a="s N"] by auto
ultimately show "?thesis"
unfolding bounded_any_center [where a="s N"]
apply (rule_tac x="max a 1" in exI, auto)
apply (erule_tac x=y in allE)
apply (erule_tac x=y in ballE, auto)
done
qed
instance heine_borel < complete_space
proof
fix f :: "nat ⇒ 'a" assume "Cauchy f"
then have "bounded (range f)"
by (rule cauchy_imp_bounded)
then have "compact (closure (range f))"
unfolding compact_eq_bounded_closed by auto
then have "complete (closure (range f))"
by (rule compact_imp_complete)
moreover have "∀n. f n ∈ closure (range f)"
using closure_subset [of "range f"] by auto
ultimately have "∃l∈closure (range f). (f ⤏ l) sequentially"
using ‹Cauchy f› unfolding complete_def by auto
then show "convergent f"
unfolding convergent_def by auto
qed
instance euclidean_space ⊆ banach ..
lemma complete_UNIV: "complete (UNIV :: ('a::complete_space) set)"
proof (rule completeI)
fix f :: "nat ⇒ 'a" assume "Cauchy f"
then have "convergent f" by (rule Cauchy_convergent)
then show "∃l∈UNIV. f ⇢ l" unfolding convergent_def by simp
qed
lemma complete_imp_closed:
fixes S :: "'a::metric_space set"
assumes "complete S"
shows "closed S"
proof (unfold closed_sequential_limits, clarify)
fix f x assume "∀n. f n ∈ S" and "f ⇢ x"
from ‹f ⇢ x› have "Cauchy f"
by (rule LIMSEQ_imp_Cauchy)
with ‹complete S› and ‹∀n. f n ∈ S› obtain l where "l ∈ S" and "f ⇢ l"
by (rule completeE)
from ‹f ⇢ x› and ‹f ⇢ l› have "x = l"
by (rule LIMSEQ_unique)
with ‹l ∈ S› show "x ∈ S"
by simp
qed
lemma complete_Int_closed:
fixes S :: "'a::metric_space set"
assumes "complete S" and "closed t"
shows "complete (S ∩ t)"
proof (rule completeI)
fix f assume "∀n. f n ∈ S ∩ t" and "Cauchy f"
then have "∀n. f n ∈ S" and "∀n. f n ∈ t"
by simp_all
from ‹complete S› obtain l where "l ∈ S" and "f ⇢ l"
using ‹∀n. f n ∈ S› and ‹Cauchy f› by (rule completeE)
from ‹closed t› and ‹∀n. f n ∈ t› and ‹f ⇢ l› have "l ∈ t"
by (rule closed_sequentially)
with ‹l ∈ S› and ‹f ⇢ l› show "∃l∈S ∩ t. f ⇢ l"
by fast
qed
lemma complete_closed_subset:
fixes S :: "'a::metric_space set"
assumes "closed S" and "S ⊆ t" and "complete t"
shows "complete S"
using assms complete_Int_closed [of t S] by (simp add: Int_absorb1)
lemma complete_eq_closed:
fixes S :: "('a::complete_space) set"
shows "complete S ⟷ closed S"
proof
assume "closed S" then show "complete S"
using subset_UNIV complete_UNIV by (rule complete_closed_subset)
next
assume "complete S" then show "closed S"
by (rule complete_imp_closed)
qed
lemma convergent_eq_Cauchy:
fixes S :: "nat ⇒ 'a::complete_space"
shows "(∃l. (S ⤏ l) sequentially) ⟷ Cauchy S"
unfolding Cauchy_convergent_iff convergent_def ..
lemma convergent_imp_bounded:
fixes S :: "nat ⇒ 'a::metric_space"
shows "(S ⤏ l) sequentially ⟹ bounded (range S)"
by (intro cauchy_imp_bounded LIMSEQ_imp_Cauchy)
lemma frontier_subset_compact:
fixes S :: "'a::heine_borel set"
shows "compact S ⟹ frontier S ⊆ S"
using frontier_subset_closed compact_eq_bounded_closed
by blast
subsection ‹Continuity›
text‹Derive the epsilon-delta forms, which we often use as "definitions"›
proposition continuous_within_eps_delta:
"continuous (at x within s) f ⟷ (∀e>0. ∃d>0. ∀x'∈ s. dist x' x < d --> dist (f x') (f x) < e)"
unfolding continuous_within and Lim_within by fastforce
corollary continuous_at_eps_delta:
"continuous (at x) f ⟷ (∀e > 0. ∃d > 0. ∀x'. dist x' x < d ⟶ dist (f x') (f x) < e)"
using continuous_within_eps_delta [of x UNIV f] by simp
lemma continuous_at_right_real_increasing:
fixes f :: "real ⇒ real"
assumes nondecF: "⋀x y. x ≤ y ⟹ f x ≤ f y"
shows "continuous (at_right a) f ⟷ (∀e>0. ∃d>0. f (a + d) - f a < e)"
apply (simp add: greaterThan_def dist_real_def continuous_within Lim_within_le)
apply (intro all_cong ex_cong, safe)
apply (erule_tac x="a + d" in allE, simp)
apply (simp add: nondecF field_simps)
apply (drule nondecF, simp)
done
lemma continuous_at_left_real_increasing:
assumes nondecF: "⋀ x y. x ≤ y ⟹ f x ≤ ((f y) :: real)"
shows "(continuous (at_left (a :: real)) f) = (∀e > 0. ∃delta > 0. f a - f (a - delta) < e)"
apply (simp add: lessThan_def dist_real_def continuous_within Lim_within_le)
apply (intro all_cong ex_cong, safe)
apply (erule_tac x="a - d" in allE, simp)
apply (simp add: nondecF field_simps)
apply (cut_tac x="a - d" and y=x in nondecF, simp_all)
done
text‹Versions in terms of open balls.›
lemma continuous_within_ball:
"continuous (at x within s) f ⟷
(∀e > 0. ∃d > 0. f ` (ball x d ∩ s) ⊆ ball (f x) e)"
(is "?lhs = ?rhs")
proof
assume ?lhs
{
fix e :: real
assume "e > 0"
then obtain d where d: "d>0" "∀xa∈s. 0 < dist xa x ∧ dist xa x < d ⟶ dist (f xa) (f x) < e"
using ‹?lhs›[unfolded continuous_within Lim_within] by auto
{
fix y
assume "y ∈ f ` (ball x d ∩ s)"
then have "y ∈ ball (f x) e"
using d(2)
apply (auto simp: dist_commute)
apply (erule_tac x=xa in ballE, auto)
using ‹e > 0›
apply auto
done
}
then have "∃d>0. f ` (ball x d ∩ s) ⊆ ball (f x) e"
using ‹d > 0›
unfolding subset_eq ball_def by (auto simp: dist_commute)
}
then show ?rhs by auto
next
assume ?rhs
then show ?lhs
unfolding continuous_within Lim_within ball_def subset_eq
apply (auto simp: dist_commute)
apply (erule_tac x=e in allE, auto)
done
qed
lemma continuous_at_ball:
"continuous (at x) f ⟷ (∀e>0. ∃d>0. f ` (ball x d) ⊆ ball (f x) e)" (is "?lhs = ?rhs")
proof
assume ?lhs
then show ?rhs
unfolding continuous_at Lim_at subset_eq Ball_def Bex_def image_iff mem_ball
apply auto
apply (erule_tac x=e in allE, auto)
apply (rule_tac x=d in exI, auto)
apply (erule_tac x=xa in allE)
apply (auto simp: dist_commute)
done
next
assume ?rhs
then show ?lhs
unfolding continuous_at Lim_at subset_eq Ball_def Bex_def image_iff mem_ball
apply auto
apply (erule_tac x=e in allE, auto)
apply (rule_tac x=d in exI, auto)
apply (erule_tac x="f xa" in allE)
apply (auto simp: dist_commute)
done
qed
text‹Define setwise continuity in terms of limits within the set.›
lemma continuous_on_iff:
"continuous_on s f ⟷
(∀x∈s. ∀e>0. ∃d>0. ∀x'∈s. dist x' x < d ⟶ dist (f x') (f x) < e)"
unfolding continuous_on_def Lim_within
by (metis dist_pos_lt dist_self)
lemma continuous_within_E:
assumes "continuous (at x within s) f" "e>0"
obtains d where "d>0" "⋀x'. ⟦x'∈ s; dist x' x ≤ d⟧ ⟹ dist (f x') (f x) < e"
using assms apply (simp add: continuous_within_eps_delta)
apply (drule spec [of _ e], clarify)
apply (rule_tac d="d/2" in that, auto)
done
lemma continuous_onI [intro?]:
assumes "⋀x e. ⟦e > 0; x ∈ s⟧ ⟹ ∃d>0. ∀x'∈s. dist x' x < d ⟶ dist (f x') (f x) ≤ e"
shows "continuous_on s f"
apply (simp add: continuous_on_iff, clarify)
apply (rule ex_forward [OF assms [OF half_gt_zero]], auto)
done
text‹Some simple consequential lemmas.›
lemma continuous_onE:
assumes "continuous_on s f" "x∈s" "e>0"
obtains d where "d>0" "⋀x'. ⟦x' ∈ s; dist x' x ≤ d⟧ ⟹ dist (f x') (f x) < e"
using assms
apply (simp add: continuous_on_iff)
apply (elim ballE allE)
apply (auto intro: that [where d="d/2" for d])
done
lemma uniformly_continuous_onE:
assumes "uniformly_continuous_on s f" "0 < e"
obtains d where "d>0" "⋀x x'. ⟦x∈s; x'∈s; dist x' x < d⟧ ⟹ dist (f x') (f x) < e"
using assms
by (auto simp: uniformly_continuous_on_def)
lemma continuous_at_imp_continuous_within:
"continuous (at x) f ⟹ continuous (at x within s) f"
unfolding continuous_within continuous_at using Lim_at_imp_Lim_at_within by auto
lemma Lim_trivial_limit: "trivial_limit net ⟹ (f ⤏ l) net"
by simp
lemmas continuous_on = continuous_on_def
lemma continuous_within_subset:
"continuous (at x within s) f ⟹ t ⊆ s ⟹ continuous (at x within t) f"
unfolding continuous_within by(metis tendsto_within_subset)
lemma continuous_on_interior:
"continuous_on s f ⟹ x ∈ interior s ⟹ continuous (at x) f"
by (metis continuous_on_eq_continuous_at continuous_on_subset interiorE)
lemma continuous_on_eq:
"⟦continuous_on s f; ⋀x. x ∈ s ⟹ f x = g x⟧ ⟹ continuous_on s g"
unfolding continuous_on_def tendsto_def eventually_at_topological
by simp
text ‹Characterization of various kinds of continuity in terms of sequences.›
lemma continuous_within_sequentiallyI:
fixes f :: "'a::{first_countable_topology, t2_space} ⇒ 'b::topological_space"
assumes "⋀u::nat ⇒ 'a. u ⇢ a ⟹ (∀n. u n ∈ s) ⟹ (λn. f (u n)) ⇢ f a"
shows "continuous (at a within s) f"
using assms unfolding continuous_within tendsto_def[where l = "f a"]
by (auto intro!: sequentially_imp_eventually_within)
lemma continuous_within_tendsto_compose:
fixes f::"'a::t2_space ⇒ 'b::topological_space"
assumes "continuous (at a within s) f"
"eventually (λn. x n ∈ s) F"
"(x ⤏ a) F "
shows "((λn. f (x n)) ⤏ f a) F"
proof -
have *: "filterlim x (inf (nhds a) (principal s)) F"
using assms(2) assms(3) unfolding at_within_def filterlim_inf by (auto simp: filterlim_principal eventually_mono)
show ?thesis
by (auto simp: assms(1) continuous_within[symmetric] tendsto_at_within_iff_tendsto_nhds[symmetric] intro!: filterlim_compose[OF _ *])
qed
lemma continuous_within_tendsto_compose':
fixes f::"'a::t2_space ⇒ 'b::topological_space"
assumes "continuous (at a within s) f"
"⋀n. x n ∈ s"
"(x ⤏ a) F "
shows "((λn. f (x n)) ⤏ f a) F"
by (auto intro!: continuous_within_tendsto_compose[OF assms(1)] simp add: assms)
lemma continuous_within_sequentially:
fixes f :: "'a::{first_countable_topology, t2_space} ⇒ 'b::topological_space"
shows "continuous (at a within s) f ⟷
(∀x. (∀n::nat. x n ∈ s) ∧ (x ⤏ a) sequentially
⟶ ((f ∘ x) ⤏ f a) sequentially)"
using continuous_within_tendsto_compose'[of a s f _ sequentially]
continuous_within_sequentiallyI[of a s f]
by (auto simp: o_def)
lemma continuous_at_sequentiallyI:
fixes f :: "'a::{first_countable_topology, t2_space} ⇒ 'b::topological_space"
assumes "⋀u. u ⇢ a ⟹ (λn. f (u n)) ⇢ f a"
shows "continuous (at a) f"
using continuous_within_sequentiallyI[of a UNIV f] assms by auto
lemma continuous_at_sequentially:
fixes f :: "'a::metric_space ⇒ 'b::topological_space"
shows "continuous (at a) f ⟷
(∀x. (x ⤏ a) sequentially --> ((f ∘ x) ⤏ f a) sequentially)"
using continuous_within_sequentially[of a UNIV f] by simp
lemma continuous_on_sequentiallyI:
fixes f :: "'a::{first_countable_topology, t2_space} ⇒ 'b::topological_space"
assumes "⋀u a. (∀n. u n ∈ s) ⟹ a ∈ s ⟹ u ⇢ a ⟹ (λn. f (u n)) ⇢ f a"
shows "continuous_on s f"
using assms unfolding continuous_on_eq_continuous_within
using continuous_within_sequentiallyI[of _ s f] by auto
lemma continuous_on_sequentially:
fixes f :: "'a::metric_space ⇒ 'b::topological_space"
shows "continuous_on s f ⟷
(∀x. ∀a ∈ s. (∀n. x(n) ∈ s) ∧ (x ⤏ a) sequentially
--> ((f ∘ x) ⤏ f a) sequentially)"
(is "?lhs = ?rhs")
proof
assume ?rhs
then show ?lhs
using continuous_within_sequentially[of _ s f]
unfolding continuous_on_eq_continuous_within
by auto
next
assume ?lhs
then show ?rhs
unfolding continuous_on_eq_continuous_within
using continuous_within_sequentially[of _ s f]
by auto
qed
lemma uniformly_continuous_on_sequentially:
"uniformly_continuous_on s f ⟷ (∀x y. (∀n. x n ∈ s) ∧ (∀n. y n ∈ s) ∧
(λn. dist (x n) (y n)) ⇢ 0 ⟶ (λn. dist (f(x n)) (f(y n))) ⇢ 0)" (is "?lhs = ?rhs")
proof
assume ?lhs
{
fix x y
assume x: "∀n. x n ∈ s"
and y: "∀n. y n ∈ s"
and xy: "((λn. dist (x n) (y n)) ⤏ 0) sequentially"
{
fix e :: real
assume "e > 0"
then obtain d where "d > 0" and d: "∀x∈s. ∀x'∈s. dist x' x < d ⟶ dist (f x') (f x) < e"
using ‹?lhs›[unfolded uniformly_continuous_on_def, THEN spec[where x=e]] by auto
obtain N where N: "∀n≥N. dist (x n) (y n) < d"
using xy[unfolded lim_sequentially dist_norm] and ‹d>0› by auto
{
fix n
assume "n≥N"
then have "dist (f (x n)) (f (y n)) < e"
using N[THEN spec[where x=n]]
using d[THEN bspec[where x="x n"], THEN bspec[where x="y n"]]
using x and y
by (simp add: dist_commute)
}
then have "∃N. ∀n≥N. dist (f (x n)) (f (y n)) < e"
by auto
}
then have "((λn. dist (f(x n)) (f(y n))) ⤏ 0) sequentially"
unfolding lim_sequentially and dist_real_def by auto
}
then show ?rhs by auto
next
assume ?rhs
{
assume "¬ ?lhs"
then obtain e where "e > 0" "∀d>0. ∃x∈s. ∃x'∈s. dist x' x < d ∧ ¬ dist (f x') (f x) < e"
unfolding uniformly_continuous_on_def by auto
then obtain fa where fa:
"∀x. 0 < x ⟶ fst (fa x) ∈ s ∧ snd (fa x) ∈ s ∧ dist (fst (fa x)) (snd (fa x)) < x ∧ ¬ dist (f (fst (fa x))) (f (snd (fa x))) < e"
using choice[of "λd x. d>0 ⟶ fst x ∈ s ∧ snd x ∈ s ∧ dist (snd x) (fst x) < d ∧ ¬ dist (f (snd x)) (f (fst x)) < e"]
unfolding Bex_def
by (auto simp: dist_commute)
define x where "x n = fst (fa (inverse (real n + 1)))" for n
define y where "y n = snd (fa (inverse (real n + 1)))" for n
have xyn: "∀n. x n ∈ s ∧ y n ∈ s"
and xy0: "∀n. dist (x n) (y n) < inverse (real n + 1)"
and fxy:"∀n. ¬ dist (f (x n)) (f (y n)) < e"
unfolding x_def and y_def using fa
by auto
{
fix e :: real
assume "e > 0"
then obtain N :: nat where "N ≠ 0" and N: "0 < inverse (real N) ∧ inverse (real N) < e"
unfolding real_arch_inverse[of e] by auto
{
fix n :: nat
assume "n ≥ N"
then have "inverse (real n + 1) < inverse (real N)"
using of_nat_0_le_iff and ‹N≠0› by auto
also have "… < e" using N by auto
finally have "inverse (real n + 1) < e" by auto
then have "dist (x n) (y n) < e"
using xy0[THEN spec[where x=n]] by auto
}
then have "∃N. ∀n≥N. dist (x n) (y n) < e" by auto
}
then have "∀e>0. ∃N. ∀n≥N. dist (f (x n)) (f (y n)) < e"
using ‹?rhs›[THEN spec[where x=x], THEN spec[where x=y]] and xyn
unfolding lim_sequentially dist_real_def by auto
then have False using fxy and ‹e>0› by auto
}
then show ?lhs
unfolding uniformly_continuous_on_def by blast
qed
lemma continuous_closed_imp_Cauchy_continuous:
fixes S :: "('a::complete_space) set"
shows "⟦continuous_on S f; closed S; Cauchy σ; ⋀n. (σ n) ∈ S⟧ ⟹ Cauchy(f ∘ σ)"
apply (simp add: complete_eq_closed [symmetric] continuous_on_sequentially)
by (meson LIMSEQ_imp_Cauchy complete_def)
text‹The usual transformation theorems.›
lemma continuous_transform_within:
fixes f g :: "'a::metric_space ⇒ 'b::topological_space"
assumes "continuous (at x within s) f"
and "0 < d"
and "x ∈ s"
and "⋀x'. ⟦x' ∈ s; dist x' x < d⟧ ⟹ f x' = g x'"
shows "continuous (at x within s) g"
using assms
unfolding continuous_within
by (force intro: Lim_transform_within)
subsubsection%unimportant ‹Structural rules for pointwise continuity›
lemma continuous_infnorm[continuous_intros]:
"continuous F f ⟹ continuous F (λx. infnorm (f x))"
unfolding continuous_def by (rule tendsto_infnorm)
lemma continuous_inner[continuous_intros]:
assumes "continuous F f"
and "continuous F g"
shows "continuous F (λx. inner (f x) (g x))"
using assms unfolding continuous_def by (rule tendsto_inner)
subsubsection%unimportant ‹Structural rules for setwise continuity›
lemma continuous_on_infnorm[continuous_intros]:
"continuous_on s f ⟹ continuous_on s (λx. infnorm (f x))"
unfolding continuous_on by (fast intro: tendsto_infnorm)
lemma continuous_on_inner[continuous_intros]:
fixes g :: "'a::topological_space ⇒ 'b::real_inner"
assumes "continuous_on s f"
and "continuous_on s g"
shows "continuous_on s (λx. inner (f x) (g x))"
using bounded_bilinear_inner assms
by (rule bounded_bilinear.continuous_on)
subsubsection%unimportant ‹Structural rules for uniform continuity›
lemma uniformly_continuous_on_dist[continuous_intros]:
fixes f g :: "'a::metric_space ⇒ 'b::metric_space"
assumes "uniformly_continuous_on s f"
and "uniformly_continuous_on s g"
shows "uniformly_continuous_on s (λx. dist (f x) (g x))"
proof -
{
fix a b c d :: 'b
have "¦dist a b - dist c d¦ ≤ dist a c + dist b d"
using dist_triangle2 [of a b c] dist_triangle2 [of b c d]
using dist_triangle3 [of c d a] dist_triangle [of a d b]
by arith
} note le = this
{
fix x y
assume f: "(λn. dist (f (x n)) (f (y n))) ⇢ 0"
assume g: "(λn. dist (g (x n)) (g (y n))) ⇢ 0"
have "(λn. ¦dist (f (x n)) (g (x n)) - dist (f (y n)) (g (y n))¦) ⇢ 0"
by (rule Lim_transform_bound [OF _ tendsto_add_zero [OF f g]],
simp add: le)
}
then show ?thesis
using assms unfolding uniformly_continuous_on_sequentially
unfolding dist_real_def by simp
qed
lemma uniformly_continuous_on_norm[continuous_intros]:
fixes f :: "'a :: metric_space ⇒ 'b :: real_normed_vector"
assumes "uniformly_continuous_on s f"
shows "uniformly_continuous_on s (λx. norm (f x))"
unfolding norm_conv_dist using assms
by (intro uniformly_continuous_on_dist uniformly_continuous_on_const)
lemma (in bounded_linear) uniformly_continuous_on[continuous_intros]:
fixes g :: "_::metric_space ⇒ _"
assumes "uniformly_continuous_on s g"
shows "uniformly_continuous_on s (λx. f (g x))"
using assms unfolding uniformly_continuous_on_sequentially
unfolding dist_norm tendsto_norm_zero_iff diff[symmetric]
by (auto intro: tendsto_zero)
lemma uniformly_continuous_on_cmul[continuous_intros]:
fixes f :: "'a::metric_space ⇒ 'b::real_normed_vector"
assumes "uniformly_continuous_on s f"
shows "uniformly_continuous_on s (λx. c *⇩R f(x))"
using bounded_linear_scaleR_right assms
by (rule bounded_linear.uniformly_continuous_on)
lemma dist_minus:
fixes x y :: "'a::real_normed_vector"
shows "dist (- x) (- y) = dist x y"
unfolding dist_norm minus_diff_minus norm_minus_cancel ..
lemma uniformly_continuous_on_minus[continuous_intros]:
fixes f :: "'a::metric_space ⇒ 'b::real_normed_vector"
shows "uniformly_continuous_on s f ⟹ uniformly_continuous_on s (λx. - f x)"
unfolding uniformly_continuous_on_def dist_minus .
lemma uniformly_continuous_on_add[continuous_intros]:
fixes f g :: "'a::metric_space ⇒ 'b::real_normed_vector"
assumes "uniformly_continuous_on s f"
and "uniformly_continuous_on s g"
shows "uniformly_continuous_on s (λx. f x + g x)"
using assms
unfolding uniformly_continuous_on_sequentially
unfolding dist_norm tendsto_norm_zero_iff add_diff_add
by (auto intro: tendsto_add_zero)
lemma uniformly_continuous_on_diff[continuous_intros]:
fixes f :: "'a::metric_space ⇒ 'b::real_normed_vector"
assumes "uniformly_continuous_on s f"
and "uniformly_continuous_on s g"
shows "uniformly_continuous_on s (λx. f x - g x)"
using assms uniformly_continuous_on_add [of s f "- g"]
by (simp add: fun_Compl_def uniformly_continuous_on_minus)
text ‹Continuity in terms of open preimages.›
lemma continuous_at_open:
"continuous (at x) f ⟷ (∀t. open t ∧ f x ∈ t --> (∃s. open s ∧ x ∈ s ∧ (∀x' ∈ s. (f x') ∈ t)))"
unfolding continuous_within_topological [of x UNIV f]
unfolding imp_conjL
by (intro all_cong imp_cong ex_cong conj_cong refl) auto
lemma continuous_imp_tendsto:
assumes "continuous (at x0) f"
and "x ⇢ x0"
shows "(f ∘ x) ⇢ (f x0)"
proof (rule topological_tendstoI)
fix S
assume "open S" "f x0 ∈ S"
then obtain T where T_def: "open T" "x0 ∈ T" "∀x∈T. f x ∈ S"
using assms continuous_at_open by metis
then have "eventually (λn. x n ∈ T) sequentially"
using assms T_def by (auto simp: tendsto_def)
then show "eventually (λn. (f ∘ x) n ∈ S) sequentially"
using T_def by (auto elim!: eventually_mono)
qed
lemma continuous_on_open:
"continuous_on S f ⟷
(∀T. openin (subtopology euclidean (f ` S)) T ⟶
openin (subtopology euclidean S) (S ∩ f -` T))"
unfolding continuous_on_open_invariant openin_open Int_def vimage_def Int_commute
by (simp add: imp_ex imageI conj_commute eq_commute cong: conj_cong)
lemma continuous_on_open_gen:
fixes f :: "'a::metric_space ⇒ 'b::metric_space"
assumes "f ` S ⊆ T"
shows "continuous_on S f ⟷
(∀U. openin (subtopology euclidean T) U
⟶ openin (subtopology euclidean S) (S ∩ f -` U))"
(is "?lhs = ?rhs")
proof
assume ?lhs
then show ?rhs
apply (clarsimp simp: openin_euclidean_subtopology_iff continuous_on_iff)
by (metis assms image_subset_iff)
next
have ope: "openin (subtopology euclidean T) (ball y e ∩ T)" for y e
by (simp add: Int_commute openin_open_Int)
assume R [rule_format]: ?rhs
show ?lhs
proof (clarsimp simp add: continuous_on_iff)
fix x and e::real
assume "x ∈ S" and "0 < e"
then have x: "x ∈ S ∩ (f -` ball (f x) e ∩ f -` T)"
using assms by auto
show "∃d>0. ∀x'∈S. dist x' x < d ⟶ dist (f x') (f x) < e"
using R [of "ball (f x) e ∩ T"] x
by (fastforce simp add: ope openin_euclidean_subtopology_iff [of S] dist_commute)
qed
qed
lemma continuous_openin_preimage:
fixes f :: "'a::metric_space ⇒ 'b::metric_space"
shows
"⟦continuous_on S f; f ` S ⊆ T; openin (subtopology euclidean T) U⟧
⟹ openin (subtopology euclidean S) (S ∩ f -` U)"
by (simp add: continuous_on_open_gen)
text ‹Similarly in terms of closed sets.›
lemma continuous_on_closed:
"continuous_on S f ⟷
(∀T. closedin (subtopology euclidean (f ` S)) T ⟶
closedin (subtopology euclidean S) (S ∩ f -` T))"
unfolding continuous_on_closed_invariant closedin_closed Int_def vimage_def Int_commute
by (simp add: imp_ex imageI conj_commute eq_commute cong: conj_cong)
lemma continuous_on_closed_gen:
fixes f :: "'a::metric_space ⇒ 'b::metric_space"
assumes "f ` S ⊆ T"
shows "continuous_on S f ⟷
(∀U. closedin (subtopology euclidean T) U
⟶ closedin (subtopology euclidean S) (S ∩ f -` U))"
(is "?lhs = ?rhs")
proof -
have *: "U ⊆ T ⟹ S ∩ f -` (T - U) = S - (S ∩ f -` U)" for U
using assms by blast
show ?thesis
proof
assume L: ?lhs
show ?rhs
proof clarify
fix U
assume "closedin (subtopology euclidean T) U"
then show "closedin (subtopology euclidean S) (S ∩ f -` U)"
using L unfolding continuous_on_open_gen [OF assms]
by (metis * closedin_def inf_le1 topspace_euclidean_subtopology)
qed
next
assume R [rule_format]: ?rhs
show ?lhs
unfolding continuous_on_open_gen [OF assms]
by (metis * R inf_le1 openin_closedin_eq topspace_euclidean_subtopology)
qed
qed
lemma continuous_closedin_preimage_gen:
fixes f :: "'a::metric_space ⇒ 'b::metric_space"
assumes "continuous_on S f" "f ` S ⊆ T" "closedin (subtopology euclidean T) U"
shows "closedin (subtopology euclidean S) (S ∩ f -` U)"
using assms continuous_on_closed_gen by blast
lemma continuous_on_imp_closedin:
assumes "continuous_on S f" "closedin (subtopology euclidean (f ` S)) T"
shows "closedin (subtopology euclidean S) (S ∩ f -` T)"
using assms continuous_on_closed by blast
subsection%unimportant ‹Half-global and completely global cases›
lemma continuous_openin_preimage_gen:
assumes "continuous_on S f" "open T"
shows "openin (subtopology euclidean S) (S ∩ f -` T)"
proof -
have *: "(S ∩ f -` T) = (S ∩ f -` (T ∩ f ` S))"
by auto
have "openin (subtopology euclidean (f ` S)) (T ∩ f ` S)"
using openin_open_Int[of T "f ` S", OF assms(2)] unfolding openin_open by auto
then show ?thesis
using assms(1)[unfolded continuous_on_open, THEN spec[where x="T ∩ f ` S"]]
using * by auto
qed
lemma continuous_closedin_preimage:
assumes "continuous_on S f" and "closed T"
shows "closedin (subtopology euclidean S) (S ∩ f -` T)"
proof -
have *: "(S ∩ f -` T) = (S ∩ f -` (T ∩ f ` S))"
by auto
have "closedin (subtopology euclidean (f ` S)) (T ∩ f ` S)"
using closedin_closed_Int[of T "f ` S", OF assms(2)]
by (simp add: Int_commute)
then show ?thesis
using assms(1)[unfolded continuous_on_closed, THEN spec[where x="T ∩ f ` S"]]
using * by auto
qed
lemma continuous_openin_preimage_eq:
"continuous_on S f ⟷
(∀T. open T ⟶ openin (subtopology euclidean S) (S ∩ f -` T))"
apply safe
apply (simp add: continuous_openin_preimage_gen)
apply (fastforce simp add: continuous_on_open openin_open)
done
lemma continuous_closedin_preimage_eq:
"continuous_on S f ⟷
(∀T. closed T ⟶ closedin (subtopology euclidean S) (S ∩ f -` T))"
apply safe
apply (simp add: continuous_closedin_preimage)
apply (fastforce simp add: continuous_on_closed closedin_closed)
done
lemma continuous_open_preimage:
assumes contf: "continuous_on S f" and "open S" "open T"
shows "open (S ∩ f -` T)"
proof-
obtain U where "open U" "(S ∩ f -` T) = S ∩ U"
using continuous_openin_preimage_gen[OF contf ‹open T›]
unfolding openin_open by auto
then show ?thesis
using open_Int[of S U, OF ‹open S›] by auto
qed
lemma continuous_closed_preimage:
assumes contf: "continuous_on S f" and "closed S" "closed T"
shows "closed (S ∩ f -` T)"
proof-
obtain U where "closed U" "(S ∩ f -` T) = S ∩ U"
using continuous_closedin_preimage[OF contf ‹closed T›]
unfolding closedin_closed by auto
then show ?thesis using closed_Int[of S U, OF ‹closed S›] by auto
qed
lemma continuous_open_vimage: "open S ⟹ (⋀x. continuous (at x) f) ⟹ open (f -` S)"
by (metis continuous_on_eq_continuous_within open_vimage)
lemma continuous_closed_vimage: "closed S ⟹ (⋀x. continuous (at x) f) ⟹ closed (f -` S)"
by (simp add: closed_vimage continuous_on_eq_continuous_within)
lemma interior_image_subset:
assumes "inj f" "⋀x. continuous (at x) f"
shows "interior (f ` S) ⊆ f ` (interior S)"
proof
fix x assume "x ∈ interior (f ` S)"
then obtain T where as: "open T" "x ∈ T" "T ⊆ f ` S" ..
then have "x ∈ f ` S" by auto
then obtain y where y: "y ∈ S" "x = f y" by auto
have "open (f -` T)"
using assms ‹open T› by (simp add: continuous_at_imp_continuous_on open_vimage)
moreover have "y ∈ vimage f T"
using ‹x = f y› ‹x ∈ T› by simp
moreover have "vimage f T ⊆ S"
using ‹T ⊆ image f S› ‹inj f› unfolding inj_on_def subset_eq by auto
ultimately have "y ∈ interior S" ..
with ‹x = f y› show "x ∈ f ` interior S" ..
qed
subsection%unimportant ‹Topological properties of linear functions›
lemma linear_lim_0:
assumes "bounded_linear f"
shows "(f ⤏ 0) (at (0))"
proof -
interpret f: bounded_linear f by fact
have "(f ⤏ f 0) (at 0)"
using tendsto_ident_at by (rule f.tendsto)
then show ?thesis unfolding f.zero .
qed
lemma linear_continuous_at:
assumes "bounded_linear f"
shows "continuous (at a) f"
unfolding continuous_at using assms
apply (rule bounded_linear.tendsto)
apply (rule tendsto_ident_at)
done
lemma linear_continuous_within:
"bounded_linear f ⟹ continuous (at x within s) f"
using continuous_at_imp_continuous_within[of x f s] using linear_continuous_at[of f] by auto
lemma linear_continuous_on:
"bounded_linear f ⟹ continuous_on s f"
using continuous_at_imp_continuous_on[of s f] using linear_continuous_at[of f] by auto
subsection%unimportant ‹Intervals›
text ‹Openness of halfspaces.›
lemma open_halfspace_lt: "open {x. inner a x < b}"
by (simp add: open_Collect_less continuous_on_inner continuous_on_const continuous_on_id)
lemma open_halfspace_gt: "open {x. inner a x > b}"
by (simp add: open_Collect_less continuous_on_inner continuous_on_const continuous_on_id)
lemma open_halfspace_component_lt: "open {x::'a::euclidean_space. x∙i < a}"
by (simp add: open_Collect_less continuous_on_inner continuous_on_const continuous_on_id)
lemma open_halfspace_component_gt: "open {x::'a::euclidean_space. x∙i > a}"
by (simp add: open_Collect_less continuous_on_inner continuous_on_const continuous_on_id)
text ‹This gives a simple derivation of limit component bounds.›
lemma open_box[intro]: "open (box a b)"
proof -
have "open (⋂i∈Basis. ((∙) i) -` {a ∙ i <..< b ∙ i})"
by (auto intro!: continuous_open_vimage continuous_inner continuous_ident continuous_const)
also have "(⋂i∈Basis. ((∙) i) -` {a ∙ i <..< b ∙ i}) = box a b"
by (auto simp: box_def inner_commute)
finally show ?thesis .
qed
instance euclidean_space ⊆ second_countable_topology
proof
define a where "a f = (∑i∈Basis. fst (f i) *⇩R i)" for f :: "'a ⇒ real × real"
then have a: "⋀f. (∑i∈Basis. fst (f i) *⇩R i) = a f"
by simp
define b where "b f = (∑i∈Basis. snd (f i) *⇩R i)" for f :: "'a ⇒ real × real"
then have b: "⋀f. (∑i∈Basis. snd (f i) *⇩R i) = b f"
by simp
define B where "B = (λf. box (a f) (b f)) ` (Basis →⇩E (ℚ × ℚ))"
have "Ball B open" by (simp add: B_def open_box)
moreover have "(∀A. open A ⟶ (∃B'⊆B. ⋃B' = A))"
proof safe
fix A::"'a set"
assume "open A"
show "∃B'⊆B. ⋃B' = A"
apply (rule exI[of _ "{b∈B. b ⊆ A}"])
apply (subst (3) open_UNION_box[OF ‹open A›])
apply (auto simp: a b B_def)
done
qed
ultimately
have "topological_basis B"
unfolding topological_basis_def by blast
moreover
have "countable B"
unfolding B_def
by (intro countable_image countable_PiE finite_Basis countable_SIGMA countable_rat)
ultimately show "∃B::'a set set. countable B ∧ open = generate_topology B"
by (blast intro: topological_basis_imp_subbasis)
qed
instance euclidean_space ⊆ polish_space ..
lemma closed_cbox[intro]:
fixes a b :: "'a::euclidean_space"
shows "closed (cbox a b)"
proof -
have "closed (⋂i∈Basis. (λx. x∙i) -` {a∙i .. b∙i})"
by (intro closed_INT ballI continuous_closed_vimage allI
linear_continuous_at closed_real_atLeastAtMost finite_Basis bounded_linear_inner_left)
also have "(⋂i∈Basis. (λx. x∙i) -` {a∙i .. b∙i}) = cbox a b"
by (auto simp: cbox_def)
finally show "closed (cbox a b)" .
qed
lemma interior_cbox [simp]:
fixes a b :: "'a::euclidean_space"
shows "interior (cbox a b) = box a b" (is "?L = ?R")
proof(rule subset_antisym)
show "?R ⊆ ?L"
using box_subset_cbox open_box
by (rule interior_maximal)
{
fix x
assume "x ∈ interior (cbox a b)"
then obtain s where s: "open s" "x ∈ s" "s ⊆ cbox a b" ..
then obtain e where "e>0" and e:"∀x'. dist x' x < e ⟶ x' ∈ cbox a b"
unfolding open_dist and subset_eq by auto
{
fix i :: 'a
assume i: "i ∈ Basis"
have "dist (x - (e / 2) *⇩R i) x < e"
and "dist (x + (e / 2) *⇩R i) x < e"
unfolding dist_norm
apply auto
unfolding norm_minus_cancel
using norm_Basis[OF i] ‹e>0›
apply auto
done
then have "a ∙ i ≤ (x - (e / 2) *⇩R i) ∙ i" and "(x + (e / 2) *⇩R i) ∙ i ≤ b ∙ i"
using e[THEN spec[where x="x - (e/2) *⇩R i"]]
and e[THEN spec[where x="x + (e/2) *⇩R i"]]
unfolding mem_box
using i
by blast+
then have "a ∙ i < x ∙ i" and "x ∙ i < b ∙ i"
using ‹e>0› i
by (auto simp: inner_diff_left inner_Basis inner_add_left)
}
then have "x ∈ box a b"
unfolding mem_box by auto
}
then show "?L ⊆ ?R" ..
qed
lemma bounded_cbox [simp]:
fixes a :: "'a::euclidean_space"
shows "bounded (cbox a b)"
proof -
let ?b = "∑i∈Basis. ¦a∙i¦ + ¦b∙i¦"
{
fix x :: "'a"
assume "⋀i. i∈Basis ⟹ a ∙ i ≤ x ∙ i ∧ x ∙ i ≤ b ∙ i"
then have "(∑i∈Basis. ¦x ∙ i¦) ≤ ?b"
by (force simp: intro!: sum_mono)
then have "norm x ≤ ?b"
using norm_le_l1[of x] by auto
}
then show ?thesis
unfolding cbox_def bounded_iff by force
qed
lemma bounded_box [simp]:
fixes a :: "'a::euclidean_space"
shows "bounded (box a b)"
using bounded_cbox[of a b] box_subset_cbox[of a b] bounded_subset[of "cbox a b" "box a b"]
by simp
lemma not_interval_UNIV [simp]:
fixes a :: "'a::euclidean_space"
shows "cbox a b ≠ UNIV" "box a b ≠ UNIV"
using bounded_box[of a b] bounded_cbox[of a b] by force+
lemma not_interval_UNIV2 [simp]:
fixes a :: "'a::euclidean_space"
shows "UNIV ≠ cbox a b" "UNIV ≠ box a b"
using bounded_box[of a b] bounded_cbox[of a b] by force+
lemma compact_cbox [simp]:
fixes a :: "'a::euclidean_space"
shows "compact (cbox a b)"
using bounded_closed_imp_seq_compact[of "cbox a b"] using bounded_cbox[of a b]
by (auto simp: compact_eq_seq_compact_metric)
lemma box_midpoint:
fixes a :: "'a::euclidean_space"
assumes "box a b ≠ {}"
shows "((1/2) *⇩R (a + b)) ∈ box a b"
proof -
have "a ∙ i < ((1 / 2) *⇩R (a + b)) ∙ i ∧ ((1 / 2) *⇩R (a + b)) ∙ i < b ∙ i" if "i ∈ Basis" for i
using assms that by (auto simp: inner_add_left box_ne_empty)
then show ?thesis unfolding mem_box by auto
qed
lemma open_cbox_convex:
fixes x :: "'a::euclidean_space"
assumes x: "x ∈ box a b"
and y: "y ∈ cbox a b"
and e: "0 < e" "e ≤ 1"
shows "(e *⇩R x + (1 - e) *⇩R y) ∈ box a b"
proof -
{
fix i :: 'a
assume i: "i ∈ Basis"
have "a ∙ i = e * (a ∙ i) + (1 - e) * (a ∙ i)"
unfolding left_diff_distrib by simp
also have "… < e * (x ∙ i) + (1 - e) * (y ∙ i)"
proof (rule add_less_le_mono)
show "e * (a ∙ i) < e * (x ∙ i)"
using ‹0 < e› i mem_box(1) x by auto
show "(1 - e) * (a ∙ i) ≤ (1 - e) * (y ∙ i)"
by (meson diff_ge_0_iff_ge ‹e ≤ 1› i mem_box(2) mult_left_mono y)
qed
finally have "a ∙ i < (e *⇩R x + (1 - e) *⇩R y) ∙ i"
unfolding inner_simps by auto
moreover
{
have "b ∙ i = e * (b∙i) + (1 - e) * (b∙i)"
unfolding left_diff_distrib by simp
also have "… > e * (x ∙ i) + (1 - e) * (y ∙ i)"
proof (rule add_less_le_mono)
show "e * (x ∙ i) < e * (b ∙ i)"
using ‹0 < e› i mem_box(1) x by auto
show "(1 - e) * (y ∙ i) ≤ (1 - e) * (b ∙ i)"
by (meson diff_ge_0_iff_ge ‹e ≤ 1› i mem_box(2) mult_left_mono y)
qed
finally have "(e *⇩R x + (1 - e) *⇩R y) ∙ i < b ∙ i"
unfolding inner_simps by auto
}
ultimately have "a ∙ i < (e *⇩R x + (1 - e) *⇩R y) ∙ i ∧ (e *⇩R x + (1 - e) *⇩R y) ∙ i < b ∙ i"
by auto
}
then show ?thesis
unfolding mem_box by auto
qed
lemma closure_cbox [simp]: "closure (cbox a b) = cbox a b"
by (simp add: closed_cbox)
lemma closure_box [simp]:
fixes a :: "'a::euclidean_space"
assumes "box a b ≠ {}"
shows "closure (box a b) = cbox a b"
proof -
have ab: "a <e b"
using assms by (simp add: eucl_less_def box_ne_empty)
let ?c = "(1 / 2) *⇩R (a + b)"
{
fix x
assume as:"x ∈ cbox a b"
define f where [abs_def]: "f n = x + (inverse (real n + 1)) *⇩R (?c - x)" for n
{
fix n
assume fn: "f n <e b ⟶ a <e f n ⟶ f n = x" and xc: "x ≠ ?c"
have *: "0 < inverse (real n + 1)" "inverse (real n + 1) ≤ 1"
unfolding inverse_le_1_iff by auto
have "(inverse (real n + 1)) *⇩R ((1 / 2) *⇩R (a + b)) + (1 - inverse (real n + 1)) *⇩R x =
x + (inverse (real n + 1)) *⇩R (((1 / 2) *⇩R (a + b)) - x)"
by (auto simp: algebra_simps)
then have "f n <e b" and "a <e f n"
using open_cbox_convex[OF box_midpoint[OF assms] as *]
unfolding f_def by (auto simp: box_def eucl_less_def)
then have False
using fn unfolding f_def using xc by auto
}
moreover
{
assume "¬ (f ⤏ x) sequentially"
{
fix e :: real
assume "e > 0"
then obtain N :: nat where N: "inverse (real (N + 1)) < e"
using reals_Archimedean by auto
have "inverse (real n + 1) < e" if "N ≤ n" for n
by (auto intro!: that le_less_trans [OF _ N])
then have "∃N::nat. ∀n≥N. inverse (real n + 1) < e" by auto
}
then have "((λn. inverse (real n + 1)) ⤏ 0) sequentially"
unfolding lim_sequentially by(auto simp: dist_norm)
then have "(f ⤏ x) sequentially"
unfolding f_def
using tendsto_add[OF tendsto_const, of "λn::nat. (inverse (real n + 1)) *⇩R ((1 / 2) *⇩R (a + b) - x)" 0 sequentially x]
using tendsto_scaleR [OF _ tendsto_const, of "λn::nat. inverse (real n + 1)" 0 sequentially "((1 / 2) *⇩R (a + b) - x)"]
by auto
}
ultimately have "x ∈ closure (box a b)"
using as box_midpoint[OF assms]
unfolding closure_def islimpt_sequential
by (cases "x=?c") (auto simp: in_box_eucl_less)
}
then show ?thesis
using closure_minimal[OF box_subset_cbox, of a b] by blast
qed
lemma bounded_subset_box_symmetric:
fixes S :: "('a::euclidean_space) set"
assumes "bounded S"
obtains a where "S ⊆ box (-a) a"
proof -
obtain b where "b>0" and b: "∀x∈S. norm x ≤ b"
using assms[unfolded bounded_pos] by auto
define a :: 'a where "a = (∑i∈Basis. (b + 1) *⇩R i)"
have "(-a)∙i < x∙i" and "x∙i < a∙i" if "x ∈ S" and i: "i ∈ Basis" for x i
using b Basis_le_norm[OF i, of x] that by (auto simp: a_def)
then have "S ⊆ box (-a) a"
by (auto simp: simp add: box_def)
then show ?thesis ..
qed
lemma bounded_subset_cbox_symmetric:
fixes S :: "('a::euclidean_space) set"
assumes "bounded S"
obtains a where "S ⊆ cbox (-a) a"
proof -
obtain a where "S ⊆ box (-a) a"
using bounded_subset_box_symmetric[OF assms] by auto
then show ?thesis
by (meson box_subset_cbox dual_order.trans that)
qed
lemma frontier_cbox:
fixes a b :: "'a::euclidean_space"
shows "frontier (cbox a b) = cbox a b - box a b"
unfolding frontier_def unfolding interior_cbox and closure_closed[OF closed_cbox] ..
lemma frontier_box:
fixes a b :: "'a::euclidean_space"
shows "frontier (box a b) = (if box a b = {} then {} else cbox a b - box a b)"
proof (cases "box a b = {}")
case True
then show ?thesis
using frontier_empty by auto
next
case False
then show ?thesis
unfolding frontier_def and closure_box[OF False] and interior_open[OF open_box]
by auto
qed
lemma Int_interval_mixed_eq_empty:
fixes a :: "'a::euclidean_space"
assumes "box c d ≠ {}"
shows "box a b ∩ cbox c d = {} ⟷ box a b ∩ box c d = {}"
unfolding closure_box[OF assms, symmetric]
unfolding open_Int_closure_eq_empty[OF open_box] ..
lemma eucl_less_eq_halfspaces:
fixes a :: "'a::euclidean_space"
shows "{x. x <e a} = (⋂i∈Basis. {x. x ∙ i < a ∙ i})"
"{x. a <e x} = (⋂i∈Basis. {x. a ∙ i < x ∙ i})"
by (auto simp: eucl_less_def)
lemma open_Collect_eucl_less[simp, intro]:
fixes a :: "'a::euclidean_space"
shows "open {x. x <e a}" "open {x. a <e x}"
by (auto simp: eucl_less_eq_halfspaces open_halfspace_component_lt open_halfspace_component_gt)
no_notation
eucl_less (infix "<e" 50)
end