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 Times_eq_image_sum:
fixes S :: "'a :: comm_monoid_add set" and T :: "'b :: comm_monoid_add set"
shows "S × T = {u + v |u v. u ∈ (λx. (x, 0)) ` S ∧ v ∈ Pair 0 ` T}"
by force
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})"
apply (case_tac "m=0", force)
apply (auto simp: scaleR_left_mono)
apply (rule_tac x="inverse m *⇩R (x-c)" in rev_image_eqI, auto simp: pos_le_divideR_eq le_diff_eq scaleR_left_mono_neg)
apply (metis diff_le_eq inverse_inverse_eq order.not_eq_order_implies_strict pos_le_divideR_eq positive_imp_inverse_positive)
apply (rule_tac x="inverse m *⇩R (x-c)" in rev_image_eqI, auto simp: not_le neg_le_divideR_eq diff_le_eq)
using le_diff_eq scaleR_le_cancel_left_neg
apply fastforce
done
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 "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 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:
obtains A where "countable A" "⋀a. a ∈ A ⟹ x ∈ a" "⋀a. a ∈ A ⟹ open a"
"⋀S. open S ⟹ x ∈ S ⟹ (∃a∈A. a ⊆ S)"
using first_countable_basis[of x]
apply atomize_elim
apply (elim exE)
apply (rule_tac x="range A" in exI, auto)
done
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 A' where A':
"countable A'"
"⋀a. a ∈ A' ⟹ x ∈ a"
"⋀a. a ∈ A' ⟹ open a"
"⋀S. open S ⟹ x ∈ S ⟹ ∃a∈A'. a ⊆ S"
by (rule first_countable_basisE) blast
define A where [abs_def]:
"A = (λN. ⋂((λn. from_nat_into A' 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 A'(4)[OF open_UNIV] by (auto simp: A_def intro: A' from_nat_into)
next
let ?int = "λN. ⋂(from_nat_into A' ` 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∈A'" "a ⊆ S" using A' by blast
then show "∃a∈A. a ⊆ S" using a A'
by (intro bexI[where x=a]) (auto simp: A_def intro: image_eqI[where x="{to_nat_on A' 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"
with A(2, 3)[of a] B(2, 3)[of b] show "x ∈ a × b" and "open (a × b)"
unfolding mem_Times_iff
by (auto intro: open_Times)
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 ‹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 "istopology L ⟷
L {} ∧ (∀S T. L S ⟶ L T ⟶ L (S ∩ T)) ∧ (∀K. Ball K L ⟶ L (⋃K))"
typedef '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›
lemma 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
subsubsection ‹Closed sets›
definition "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 "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 U T) S ∧ openin (subtopology U u) S
⟹ openin (subtopology U (T ∪ u)) S"
by (simp add: openin_subtopology) blast
subsubsection ‹The standard Euclidean topology›
definition 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)
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 closed_closedin 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, 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 ≠ {})"
proof -
{ fix A B x x'
assume s_sub: "s ⊆ A ∪ B"
and disj: "A ∩ B ∩ s = {}"
and x: "x ∈ s" "x ∈ B" and x': "x' ∈ s" "x' ∈ A"
and cl: "closed A" "closed B"
assume "∀e1. (∀T. closed T ⟶ e1 ≠ s ∩ T) ∨ (∀e2. e1 ∩ e2 = {} ⟶ s ⊆ e1 ∪ e2 ⟶ (∀T. closed T ⟶ e2 ≠ s ∩ T) ∨ e1 = {} ∨ e2 = {})"
then have "⋀C D. s ∩ C = {} ∨ s ∩ D = {} ∨ s ∩ (C ∩ (s ∩ D)) ≠ {} ∨ ¬ s ⊆ s ∩ (C ∪ D) ∨ ¬ closed C ∨ ¬ closed D"
by (metis (no_types) Int_Un_distrib Int_assoc)
moreover have "s ∩ (A ∩ B) = {}" "s ∩ (A ∪ B) = s" "s ∩ B ≠ {}"
using disj s_sub x by blast+
ultimately have "s ∩ A = {}"
using cl by (metis inf.left_commute inf_bot_right order_refl)
then have False
using x' by blast
} note * = this
show ?thesis
apply (simp add: connected_closed closedin_closed)
apply (safe; simp)
apply blast
apply (blast intro: *)
done
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_closedin 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 ball :: "'a::metric_space ⇒ real ⇒ 'a set"
where "ball x e = {y. dist x y < e}"
definition cball :: "'a::metric_space ⇒ real ⇒ 'a set"
where "cball x e = {y. dist x y ≤ e}"
definition 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 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 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 "op + 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 "op + 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 euclidean_dist_l2:
fixes x y :: "'a :: euclidean_space"
shows "dist x y = setL2 (λi. dist (x ∙ i) (y ∙ i)) Basis"
unfolding dist_norm norm_eq_sqrt_inner setL2_def
by (subst euclidean_inner) (simp add: power2_eq_square inner_diff_left)
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)
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 (in euclidean_space) eucl_less (infix "<e" 50)
where "eucl_less a b ⟷ (∀i∈Basis. a ∙ i < b ∙ i)"
definition box_eucl_less: "box a b = {x. a <e x ∧ x <e b}"
definition "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 setL2_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 setL2_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 -
show ?th1
unfolding subset_eq and Ball_def and mem_box
by (auto intro: order_trans)
show ?th2
unfolding subset_eq and Ball_def and mem_box
by (auto intro: le_less_trans less_le_trans order_trans less_imp_le)
{
assume as: "box c d ⊆ cbox a b" "∀i∈Basis. c∙i < d∙i"
then have "box c d ≠ {}"
unfolding box_eq_empty by auto
fix i :: 'a
assume i: "i ∈ Basis"
{
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 as2: "a∙i > c∙i"
{
fix j :: 'a
assume j: "j ∈ Basis"
then have "c ∙ j < ?x ∙ j ∧ ?x ∙ j < d ∙ j"
apply (cases "j = i")
using as(2)[THEN bspec[where x=j]] i
apply (auto simp: as2)
done
}
then have "?x∈box c d"
using i unfolding mem_box by auto
moreover
have "?x ∉ cbox a b"
unfolding mem_box
apply auto
apply (rule_tac x=i in bexI)
using as(2)[THEN bspec[where x=i]] and as2 i
apply auto
done
ultimately have False using as by auto
}
then have "a∙i ≤ c∙i" by (rule ccontr) auto
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 as2: "b∙i < d∙i"
{
fix j :: 'a
assume "j∈Basis"
then have "d ∙ j > ?x ∙ j ∧ ?x ∙ j > c ∙ j"
apply (cases "j = i")
using as(2)[THEN bspec[where x=j]]
apply (auto simp: as2)
done
}
then have "?x∈box c d"
unfolding mem_box by auto
moreover
have "?x∉cbox a b"
unfolding mem_box
apply auto
apply (rule_tac x=i in bexI)
using as(2)[THEN bspec[where x=i]] and as2 using i
apply auto
done
ultimately have False using as by auto
}
then have "b∙i ≥ d∙i" by (rule ccontr) auto
ultimately
have "a∙i ≤ c∙i ∧ d∙i ≤ b∙i" by auto
} note part1 = this
show ?th3
unfolding subset_eq and Ball_def and mem_box
apply (rule, rule, rule, rule)
apply (rule part1)
unfolding subset_eq and Ball_def and mem_box
prefer 4
apply auto
apply (erule_tac x=xa in allE, erule_tac x=xa in allE, fastforce)+
done
{
assume as: "box c d ⊆ box a b" "∀i∈Basis. c∙i < d∙i"
fix i :: 'a
assume i:"i∈Basis"
from as(1) have "box c d ⊆ cbox a b"
using box_subset_cbox[of a b] by auto
then have "a∙i ≤ c∙i ∧ d∙i ≤ b∙i"
using part1 and as(2) using i by auto
} note * = this
show ?th4
unfolding subset_eq and Ball_def and mem_box
apply (rule, rule, rule, rule)
apply (rule *)
unfolding subset_eq and Ball_def and mem_box
prefer 4
apply auto
apply (erule_tac x=xa in allE, simp)+
done
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 ?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 ‹cbox a b = box c d› box_ne_empty box_sing
apply (fastforce simp add:)
done
next
assume ?rhs
then show ?lhs
by force
qed
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 ?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) ‹box a b = box c d›
apply auto
apply (meson euclidean_eqI less_eq_real_def not_less)+
done
next
assume ?rhs
then show ?lhs
by force
qed
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
text ‹Intervals in general, including infinite and mixtures of open and closed.›
definition "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_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
subsection ‹Connectedness›
lemma connected_local:
"connected S ⟷
¬ (∃e1 e2.
openin (subtopology euclidean S) e1 ∧
openin (subtopology euclidean S) e2 ∧
S ⊆ e1 ∪ e2 ∧
e1 ∩ e2 = {} ∧
e1 ≠ {} ∧
e2 ≠ {})"
unfolding connected_def openin_open
by safe blast+
lemma exists_diff:
fixes P :: "'a set ⇒ bool"
shows "(∃S. P (- S)) ⟷ (∃S. P S)"
(is "?lhs ⟷ ?rhs")
proof -
have ?rhs if ?lhs
using that by blast
moreover have "P (- (- S))" if "P S" for S
proof -
have "S = - (- S)" by simp
with that show ?thesis by metis
qed
ultimately show ?thesis by metis
qed
lemma connected_clopen: "connected S ⟷
(∀T. openin (subtopology euclidean S) T ∧
closedin (subtopology euclidean S) T ⟶ T = {} ∨ T = S)" (is "?lhs ⟷ ?rhs")
proof -
have "¬ connected S ⟷
(∃e1 e2. open e1 ∧ open (- e2) ∧ S ⊆ e1 ∪ (- e2) ∧ e1 ∩ (- e2) ∩ S = {} ∧ e1 ∩ S ≠ {} ∧ (- e2) ∩ S ≠ {})"
unfolding connected_def openin_open closedin_closed
by (metis double_complement)
then have th0: "connected S ⟷
¬ (∃e2 e1. closed e2 ∧ open e1 ∧ S ⊆ e1 ∪ (- e2) ∧ e1 ∩ (- e2) ∩ S = {} ∧ e1 ∩ S ≠ {} ∧ (- e2) ∩ S ≠ {})"
(is " _ ⟷ ¬ (∃e2 e1. ?P e2 e1)")
by (simp add: closed_def) metis
have th1: "?rhs ⟷ ¬ (∃t' t. closed t'∧t = S∩t' ∧ t≠{} ∧ t≠S ∧ (∃t'. open t' ∧ t = S ∩ t'))"
(is "_ ⟷ ¬ (∃t' t. ?Q t' t)")
unfolding connected_def openin_open closedin_closed by auto
have "(∃e1. ?P e2 e1) ⟷ (∃t. ?Q e2 t)" for e2
proof -
have "?P e2 e1 ⟷ (∃t. closed e2 ∧ t = S∩e2 ∧ open e1 ∧ t = S∩e1 ∧ t≠{} ∧ t ≠ S)" for e1
by auto
then show ?thesis
by metis
qed
then have "∀e2. (∃e1. ?P e2 e1) ⟷ (∃t. ?Q e2 t)"
by blast
then show ?thesis
by (simp add: th0 th1)
qed
subsection ‹Limit points›
definition (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_set_avoid:
fixes a :: "'a::metric_space"
assumes fS: "finite S"
shows "∃d>0. ∀x∈S. x ≠ a ⟶ d ≤ dist a x"
proof (induct rule: finite_induct[OF fS])
case 1
then show ?case by (auto intro: zero_less_one)
next
case (2 x F)
from 2 obtain d where d: "d > 0" "∀x∈F. x ≠ a ⟶ d ≤ dist a x"
by blast
show ?case
proof (cases "x = a")
case True
with d show ?thesis by auto
next
case False
let ?d = "min d (dist a x)"
from False d(1) have dp: "?d > 0"
by auto
from d have d': "∀x∈F. x ≠ a ⟶ ?d ≤ dist a x"
by auto
with dp False show ?thesis
by (auto intro!: exI[where x="?d"])
qed
qed
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>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[rule_format, 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 "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 "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 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 ‹Connected components, considered as a connectedness relation or a set›
definition "connected_component s x y ≡ ∃t. connected t ∧ t ⊆ s ∧ x ∈ t ∧ y ∈ t"
abbreviation "connected_component_set s x ≡ Collect (connected_component s x)"
lemma connected_componentI:
"connected t ⟹ t ⊆ s ⟹ x ∈ t ⟹ y ∈ t ⟹ connected_component s x y"
by (auto simp: connected_component_def)
lemma connected_component_in: "connected_component s x y ⟹ x ∈ s ∧ y ∈ s"
by (auto simp: connected_component_def)
lemma connected_component_refl: "x ∈ s ⟹ connected_component s x x"
by (auto simp: connected_component_def) (use connected_sing in blast)
lemma connected_component_refl_eq [simp]: "connected_component s x x ⟷ x ∈ s"
by (auto simp: connected_component_refl) (auto simp: connected_component_def)
lemma connected_component_sym: "connected_component s x y ⟹ connected_component s y x"
by (auto simp: connected_component_def)
lemma connected_component_trans:
"connected_component s x y ⟹ connected_component s y z ⟹ connected_component s x z"
unfolding connected_component_def
by (metis Int_iff Un_iff Un_subset_iff equals0D connected_Un)
lemma connected_component_of_subset:
"connected_component s x y ⟹ s ⊆ t ⟹ connected_component t x y"
by (auto simp: connected_component_def)
lemma connected_component_Union: "connected_component_set s x = ⋃{t. connected t ∧ x ∈ t ∧ t ⊆ s}"
by (auto simp: connected_component_def)
lemma connected_connected_component [iff]: "connected (connected_component_set s x)"
by (auto simp: connected_component_Union intro: connected_Union)
lemma connected_iff_eq_connected_component_set:
"connected s ⟷ (∀x ∈ s. connected_component_set s x = s)"
proof (cases "s = {}")
case True
then show ?thesis by simp
next
case False
then obtain x where "x ∈ s" by auto
show ?thesis
proof
assume "connected s"
then show "∀x ∈ s. connected_component_set s x = s"
by (force simp: connected_component_def)
next
assume "∀x ∈ s. connected_component_set s x = s"
then show "connected s"
by (metis ‹x ∈ s› connected_connected_component)
qed
qed
lemma connected_component_subset: "connected_component_set s x ⊆ s"
using connected_component_in by blast
lemma connected_component_eq_self: "connected s ⟹ x ∈ s ⟹ connected_component_set s x = s"
by (simp add: connected_iff_eq_connected_component_set)
lemma connected_iff_connected_component:
"connected s ⟷ (∀x ∈ s. ∀y ∈ s. connected_component s x y)"
using connected_component_in by (auto simp: connected_iff_eq_connected_component_set)
lemma connected_component_maximal:
"x ∈ t ⟹ connected t ⟹ t ⊆ s ⟹ t ⊆ (connected_component_set s x)"
using connected_component_eq_self connected_component_of_subset by blast
lemma connected_component_mono:
"s ⊆ t ⟹ connected_component_set s x ⊆ connected_component_set t x"
by (simp add: Collect_mono connected_component_of_subset)
lemma connected_component_eq_empty [simp]: "connected_component_set s x = {} ⟷ x ∉ s"
using connected_component_refl by (fastforce simp: connected_component_in)
lemma connected_component_set_empty [simp]: "connected_component_set {} x = {}"
using connected_component_eq_empty by blast
lemma connected_component_eq:
"y ∈ connected_component_set s x ⟹ (connected_component_set s y = connected_component_set s x)"
by (metis (no_types, lifting)
Collect_cong connected_component_sym connected_component_trans mem_Collect_eq)
lemma closed_connected_component:
assumes s: "closed s"
shows "closed (connected_component_set s x)"
proof (cases "x ∈ s")
case False
then show ?thesis
by (metis connected_component_eq_empty closed_empty)
next
case True
show ?thesis
unfolding closure_eq [symmetric]
proof
show "closure (connected_component_set s x) ⊆ connected_component_set s x"
apply (rule connected_component_maximal)
apply (simp add: closure_def True)
apply (simp add: connected_imp_connected_closure)
apply (simp add: s closure_minimal connected_component_subset)
done
next
show "connected_component_set s x ⊆ closure (connected_component_set s x)"
by (simp add: closure_subset)
qed
qed
lemma connected_component_disjoint:
"connected_component_set s a ∩ connected_component_set s b = {} ⟷
a ∉ connected_component_set s b"
apply (auto simp: connected_component_eq)
using connected_component_eq connected_component_sym
apply blast
done
lemma connected_component_nonoverlap:
"connected_component_set s a ∩ connected_component_set s b = {} ⟷
a ∉ s ∨ b ∉ s ∨ connected_component_set s a ≠ connected_component_set s b"
apply (auto simp: connected_component_in)
using connected_component_refl_eq
apply blast
apply (metis connected_component_eq mem_Collect_eq)
apply (metis connected_component_eq mem_Collect_eq)
done
lemma connected_component_overlap:
"connected_component_set s a ∩ connected_component_set s b ≠ {} ⟷
a ∈ s ∧ b ∈ s ∧ connected_component_set s a = connected_component_set s b"
by (auto simp: connected_component_nonoverlap)
lemma connected_component_sym_eq: "connected_component s x y ⟷ connected_component s y x"
using connected_component_sym by blast
lemma connected_component_eq_eq:
"connected_component_set s x = connected_component_set s y ⟷
x ∉ s ∧ y ∉ s ∨ x ∈ s ∧ y ∈ s ∧ connected_component s x y"
apply (cases "y ∈ s", simp)
apply (metis connected_component_eq connected_component_eq_empty connected_component_refl_eq mem_Collect_eq)
apply (cases "x ∈ s", simp)
apply (metis connected_component_eq_empty)
using connected_component_eq_empty
apply blast
done
lemma connected_iff_connected_component_eq:
"connected s ⟷ (∀x ∈ s. ∀y ∈ s. connected_component_set s x = connected_component_set s y)"
by (simp add: connected_component_eq_eq connected_iff_connected_component)
lemma connected_component_idemp:
"connected_component_set (connected_component_set s x) x = connected_component_set s x"
apply (rule subset_antisym)
apply (simp add: connected_component_subset)
apply (metis connected_component_eq_empty connected_component_maximal
connected_component_refl_eq connected_connected_component mem_Collect_eq set_eq_subset)
done
lemma connected_component_unique:
"⟦x ∈ c; c ⊆ s; connected c;
⋀c'. x ∈ c' ∧ c' ⊆ s ∧ connected c'
⟹ c' ⊆ c⟧
⟹ connected_component_set s x = c"
apply (rule subset_antisym)
apply (meson connected_component_maximal connected_component_subset connected_connected_component contra_subsetD)
by (simp add: connected_component_maximal)
lemma joinable_connected_component_eq:
"⟦connected t; t ⊆ s;
connected_component_set s x ∩ t ≠ {};
connected_component_set s y ∩ t ≠ {}⟧
⟹ connected_component_set s x = connected_component_set s y"
apply (simp add: ex_in_conv [symmetric])
apply (rule connected_component_eq)
by (metis (no_types, hide_lams) connected_component_eq_eq connected_component_in connected_component_maximal subsetD mem_Collect_eq)
lemma Union_connected_component: "⋃(connected_component_set s ` s) = s"
apply (rule subset_antisym)
apply (simp add: SUP_least connected_component_subset)
using connected_component_refl_eq
by force
lemma complement_connected_component_unions:
"s - connected_component_set s x =
⋃(connected_component_set s ` s - {connected_component_set s x})"
apply (subst Union_connected_component [symmetric], auto)
apply (metis connected_component_eq_eq connected_component_in)
by (metis connected_component_eq mem_Collect_eq)
lemma connected_component_intermediate_subset:
"⟦connected_component_set u a ⊆ t; t ⊆ u⟧
⟹ connected_component_set t a = connected_component_set u a"
apply (case_tac "a ∈ u")
apply (simp add: connected_component_maximal connected_component_mono subset_antisym)
using connected_component_eq_empty by blast
proposition connected_Times:
assumes S: "connected S" and T: "connected T"
shows "connected (S × T)"
proof (clarsimp simp add: connected_iff_connected_component)
fix x y x' y'
assume xy: "x ∈ S" "y ∈ T" "x' ∈ S" "y' ∈ T"
with xy obtain U V where U: "connected U" "U ⊆ S" "x ∈ U" "x' ∈ U"
and V: "connected V" "V ⊆ T" "y ∈ V" "y' ∈ V"
using S T ‹x ∈ S› ‹x' ∈ S› by blast+
show "connected_component (S × T) (x, y) (x', y')"
unfolding connected_component_def
proof (intro exI conjI)
show "connected ((λx. (x, y)) ` U ∪ Pair x' ` V)"
proof (rule connected_Un)
have "continuous_on U (λx. (x, y))"
by (intro continuous_intros)
then show "connected ((λx. (x, y)) ` U)"
by (rule connected_continuous_image) (rule ‹connected U›)
have "continuous_on V (Pair x')"
by (intro continuous_intros)
then show "connected (Pair x' ` V)"
by (rule connected_continuous_image) (rule ‹connected V›)
qed (use U V in auto)
qed (use U V in auto)
qed
corollary connected_Times_eq [simp]:
"connected (S × T) ⟷ S = {} ∨ T = {} ∨ connected S ∧ connected T" (is "?lhs = ?rhs")
proof
assume L: ?lhs
show ?rhs
proof (cases "S = {} ∨ T = {}")
case True
then show ?thesis by auto
next
case False
have "connected (fst ` (S × T))" "connected (snd ` (S × T))"
using continuous_on_fst continuous_on_snd continuous_on_id
by (blast intro: connected_continuous_image [OF _ L])+
with False show ?thesis
by auto
qed
next
assume ?rhs
then show ?lhs
by (auto simp: connected_Times)
qed
subsection ‹The set of connected components of a set›
definition components:: "'a::topological_space set ⇒ 'a set set"
where "components s ≡ connected_component_set s ` s"
lemma components_iff: "s ∈ components u ⟷ (∃x. x ∈ u ∧ s = connected_component_set u x)"
by (auto simp: components_def)
lemma componentsI: "x ∈ u ⟹ connected_component_set u x ∈ components u"
by (auto simp: components_def)
lemma componentsE:
assumes "s ∈ components u"
obtains x where "x ∈ u" "s = connected_component_set u x"
using assms by (auto simp: components_def)
lemma Union_components [simp]: "⋃(components u) = u"
apply (rule subset_antisym)
using Union_connected_component components_def apply fastforce
apply (metis Union_connected_component components_def set_eq_subset)
done
lemma pairwise_disjoint_components: "pairwise (λX Y. X ∩ Y = {}) (components u)"
apply (simp add: pairwise_def)
apply (auto simp: components_iff)
apply (metis connected_component_eq_eq connected_component_in)+
done
lemma in_components_nonempty: "c ∈ components s ⟹ c ≠ {}"
by (metis components_iff connected_component_eq_empty)
lemma in_components_subset: "c ∈ components s ⟹ c ⊆ s"
using Union_components by blast
lemma in_components_connected: "c ∈ components s ⟹ connected c"
by (metis components_iff connected_connected_component)
lemma in_components_maximal:
"c ∈ components s ⟷
c ≠ {} ∧ c ⊆ s ∧ connected c ∧ (∀d. d ≠ {} ∧ c ⊆ d ∧ d ⊆ s ∧ connected d ⟶ d = c)"
apply (rule iffI)
apply (simp add: in_components_nonempty in_components_connected)
apply (metis (full_types) components_iff connected_component_eq_self connected_component_intermediate_subset connected_component_refl in_components_subset mem_Collect_eq rev_subsetD)
apply (metis bot.extremum_uniqueI components_iff connected_component_eq_empty connected_component_maximal connected_component_subset connected_connected_component subset_emptyI)
done
lemma joinable_components_eq:
"connected t ∧ t ⊆ s ∧ c1 ∈ components s ∧ c2 ∈ components s ∧ c1 ∩ t ≠ {} ∧ c2 ∩ t ≠ {} ⟹ c1 = c2"
by (metis (full_types) components_iff joinable_connected_component_eq)
lemma closed_components: "⟦closed s; c ∈ components s⟧ ⟹ closed c"
by (metis closed_connected_component components_iff)
lemma components_nonoverlap:
"⟦c ∈ components s; c' ∈ components s⟧ ⟹ (c ∩ c' = {}) ⟷ (c ≠ c')"
apply (auto simp: in_components_nonempty components_iff)
using connected_component_refl apply blast
apply (metis connected_component_eq_eq connected_component_in)
by (metis connected_component_eq mem_Collect_eq)
lemma components_eq: "⟦c ∈ components s; c' ∈ components s⟧ ⟹ (c = c' ⟷ c ∩ c' ≠ {})"
by (metis components_nonoverlap)
lemma components_eq_empty [simp]: "components s = {} ⟷ s = {}"
by (simp add: components_def)
lemma components_empty [simp]: "components {} = {}"
by simp
lemma connected_eq_connected_components_eq: "connected s ⟷ (∀c ∈ components s. ∀c' ∈ components s. c = c')"
by (metis (no_types, hide_lams) components_iff connected_component_eq_eq connected_iff_connected_component)
lemma components_eq_sing_iff: "components s = {s} ⟷ connected s ∧ s ≠ {}"
apply (rule iffI)
using in_components_connected apply fastforce
apply safe
using Union_components apply fastforce
apply (metis components_iff connected_component_eq_self)
using in_components_maximal
apply auto
done
lemma components_eq_sing_exists: "(∃a. components s = {a}) ⟷ connected s ∧ s ≠ {}"
apply (rule iffI)
using connected_eq_connected_components_eq apply fastforce
apply (metis components_eq_sing_iff)
done
lemma connected_eq_components_subset_sing: "connected s ⟷ components s ⊆ {s}"
by (metis Union_components components_empty components_eq_sing_iff connected_empty insert_subset order_refl subset_singletonD)
lemma connected_eq_components_subset_sing_exists: "connected s ⟷ (∃a. components s ⊆ {a})"
by (metis components_eq_sing_exists connected_eq_components_subset_sing empty_iff subset_iff subset_singletonD)
lemma in_components_self: "s ∈ components s ⟷ connected s ∧ s ≠ {}"
by (metis components_empty components_eq_sing_iff empty_iff in_components_connected insertI1)
lemma components_maximal: "⟦c ∈ components s; connected t; t ⊆ s; c ∩ t ≠ {}⟧ ⟹ t ⊆ c"
apply (simp add: components_def ex_in_conv [symmetric], clarify)
by (meson connected_component_def connected_component_trans)
lemma exists_component_superset: "⟦t ⊆ s; s ≠ {}; connected t⟧ ⟹ ∃c. c ∈ components s ∧ t ⊆ c"
apply (cases "t = {}", force)
apply (metis components_def ex_in_conv connected_component_maximal contra_subsetD image_eqI)
done
lemma components_intermediate_subset: "⟦s ∈ components u; s ⊆ t; t ⊆ u⟧ ⟹ s ∈ components t"
apply (auto simp: components_iff)
apply (metis connected_component_eq_empty connected_component_intermediate_subset)
done
lemma in_components_unions_complement: "c ∈ components s ⟹ s - c = ⋃(components s - {c})"
by (metis complement_connected_component_unions components_def components_iff)
lemma connected_intermediate_closure:
assumes cs: "connected s" and st: "s ⊆ t" and ts: "t ⊆ closure s"
shows "connected t"
proof (rule connectedI)
fix A B
assume A: "open A" and B: "open B" and Alap: "A ∩ t ≠ {}" and Blap: "B ∩ t ≠ {}"
and disj: "A ∩ B ∩ t = {}" and cover: "t ⊆ A ∪ B"
have disjs: "A ∩ B ∩ s = {}"
using disj st by auto
have "A ∩ closure s ≠ {}"
using Alap Int_absorb1 ts by blast
then have Alaps: "A ∩ s ≠ {}"
by (simp add: A open_Int_closure_eq_empty)
have "B ∩ closure s ≠ {}"
using Blap Int_absorb1 ts by blast
then have Blaps: "B ∩ s ≠ {}"
by (simp add: B open_Int_closure_eq_empty)
then show False
using cs [unfolded connected_def] A B disjs Alaps Blaps cover st
by blast
qed
lemma closedin_connected_component: "closedin (subtopology euclidean s) (connected_component_set s x)"
proof (cases "connected_component_set s x = {}")
case True
then show ?thesis
by (metis closedin_empty)
next
case False
then obtain y where y: "connected_component s x y"
by blast
have *: "connected_component_set s x ⊆ s ∩ closure (connected_component_set s x)"
by (auto simp: closure_def connected_component_in)
have "connected_component s x y ⟹ s ∩ closure (connected_component_set s x) ⊆ connected_component_set s x"
apply (rule connected_component_maximal, simp)
using closure_subset connected_component_in apply fastforce
using * connected_intermediate_closure apply blast+
done
with y * show ?thesis
by (auto simp: Topology_Euclidean_Space.closedin_closed)
qed
subsection ‹Frontier (also known as boundary)›
definition "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_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_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 ‹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 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›
lemma 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.›
lemma 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)
lemma 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
lemma 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)
lemma 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:
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 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
subsection ‹Infimum Distance›
definition "infdist x A = (if A = {} then 0 else INF a:A. dist x a)"
lemma bdd_below_infdist[intro, simp]: "bdd_below (dist x`A)"
by (auto intro!: zero_le_dist)
lemma infdist_notempty: "A ≠ {} ⟹ infdist x A = (INF a:A. dist x a)"
by (simp add: infdist_def)
lemma infdist_nonneg: "0 ≤ infdist x A"
by (auto simp: infdist_def intro: cINF_greatest)
lemma infdist_le: "a ∈ A ⟹ infdist x A ≤ dist x a"
by (auto intro: cINF_lower simp add: infdist_def)
lemma infdist_le2: "a ∈ A ⟹ dist x a ≤ d ⟹ infdist x A ≤ d"
by (auto intro!: cINF_lower2 simp add: infdist_def)
lemma infdist_zero[simp]: "a ∈ A ⟹ infdist a A = 0"
by (auto intro!: antisym infdist_nonneg infdist_le2)
lemma infdist_triangle: "infdist x A ≤ infdist y A + dist x y"
proof (cases "A = {}")
case True
then show ?thesis by (simp add: infdist_def)
next
case False
then obtain a where "a ∈ A" by auto
have "infdist x A ≤ Inf {dist x y + dist y a |a. a ∈ A}"
proof (rule cInf_greatest)
from ‹A ≠ {}› show "{dist x y + dist y a |a. a ∈ A} ≠ {}"
by simp
fix d
assume "d ∈ {dist x y + dist y a |a. a ∈ A}"
then obtain a where d: "d = dist x y + dist y a" "a ∈ A"
by auto
show "infdist x A ≤ d"
unfolding infdist_notempty[OF ‹A ≠ {}›]
proof (rule cINF_lower2)
show "a ∈ A" by fact
show "dist x a ≤ d"
unfolding d by (rule dist_triangle)
qed simp
qed
also have "… = dist x y + infdist y A"
proof (rule cInf_eq, safe)
fix a
assume "a ∈ A"
then show "dist x y + infdist y A ≤ dist x y + dist y a"
by (auto intro: infdist_le)
next
fix i
assume inf: "⋀d. d ∈ {dist x y + dist y a |a. a ∈ A} ⟹ i ≤ d"
then have "i - dist x y ≤ infdist y A"
unfolding infdist_notempty[OF ‹A ≠ {}›] using ‹a ∈ A›
by (intro cINF_greatest) (auto simp: field_simps)
then show "i ≤ dist x y + infdist y A"
by simp
qed
finally show ?thesis by simp
qed
lemma in_closure_iff_infdist_zero:
assumes "A ≠ {}"
shows "x ∈ closure A ⟷ infdist x A = 0"
proof
assume "x ∈ closure A"
show "infdist x A = 0"
proof (rule ccontr)
assume "infdist x A ≠ 0"
with infdist_nonneg[of x A] have "infdist x A > 0"
by auto
then have "ball x (infdist x A) ∩ closure A = {}"
apply auto
apply (metis ‹x ∈ closure A› closure_approachable dist_commute infdist_le not_less)
done
then have "x ∉ closure A"
by (metis ‹0 < infdist x A› centre_in_ball disjoint_iff_not_equal)
then show False using ‹x ∈ closure A› by simp
qed
next
assume x: "infdist x A = 0"
then obtain a where "a ∈ A"
by atomize_elim (metis all_not_in_conv assms)
show "x ∈ closure A"
unfolding closure_approachable
apply safe
proof (rule ccontr)
fix e :: real
assume "e > 0"
assume "¬ (∃y∈A. dist y x < e)"
then have "infdist x A ≥ e" using ‹a ∈ A›
unfolding infdist_def
by (force simp: dist_commute intro: cINF_greatest)
with x ‹e > 0› show False by auto
qed
qed
lemma in_closed_iff_infdist_zero:
assumes "closed A" "A ≠ {}"
shows "x ∈ A ⟷ infdist x A = 0"
proof -
have "x ∈ closure A ⟷ infdist x A = 0"
by (rule in_closure_iff_infdist_zero) fact
with assms show ?thesis by simp
qed
lemma tendsto_infdist [tendsto_intros]:
assumes f: "(f ⤏ l) F"
shows "((λx. infdist (f x) A) ⤏ infdist l A) F"
proof (rule tendstoI)
fix e ::real
assume "e > 0"
from tendstoD[OF f this]
show "eventually (λx. dist (infdist (f x) A) (infdist l A) < e) F"
proof (eventually_elim)
fix x
from infdist_triangle[of l A "f x"] infdist_triangle[of "f x" A l]
have "dist (infdist (f x) A) (infdist l A) ≤ dist (f x) l"
by (simp add: dist_commute dist_real_def)
also assume "dist (f x) l < e"
finally show "dist (infdist (f x) A) (infdist l A) < e" .
qed
qed
text‹Some other lemmas about sequences.›
lemma sequentially_offset:
assumes "eventually (λi. P i) sequentially"
shows "eventually (λi. P (i + k)) sequentially"
using assms by (rule eventually_sequentially_seg [THEN iffD2])
lemma seq_offset_neg:
"(f ⤏ l) sequentially ⟹ ((λi. f(i - k)) ⤏ l) sequentially"
apply (erule filterlim_compose)
apply (simp add: filterlim_def le_sequentially eventually_filtermap eventually_sequentially, arith)
done
lemma seq_harmonic: "((λn. inverse (real n)) ⤏ 0) sequentially"
using LIMSEQ_inverse_real_of_nat by (rule LIMSEQ_imp_Suc)
subsection ‹More properties of closed balls›
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 mem_interior_cball: "x ∈ interior S ⟷ (∃e>0. cball x e ⊆ S)"
apply (simp add: interior_def, safe)
apply (force simp: open_contains_cball)
apply (rule_tac x="ball x e" in exI)
apply (simp add: subset_trans [OF ball_subset_cball])
done
lemma islimpt_ball:
fixes x y :: "'a::{real_normed_vector,perfect_space}"
shows "y islimpt ball x e ⟷ 0 < e ∧ y ∈ cball x e"
(is "?lhs ⟷ ?rhs")
proof
show ?rhs if ?lhs
proof
{
assume "e ≤ 0"
then have *: "ball x e = {}"
using ball_eq_empty[of x e] by auto
have False using ‹?lhs›
unfolding * using islimpt_EMPTY[of y] by auto
}
then show "e > 0" by (metis not_less)
show "y ∈ cball x e"
using closed_cball[of x e] islimpt_subset[of y "ball x e" "cball x e"]
ball_subset_cball[of x e] ‹?lhs›
unfolding closed_limpt by auto
qed
show ?lhs if ?rhs
proof -
from that have "e > 0" by auto
{
fix d :: real
assume "d > 0"
have "∃x'∈ball x e. x' ≠ y ∧ dist x' y < d"
proof (cases "d ≤ dist x y")
case True
then show "∃x'∈ball x e. x' ≠ y ∧ dist x' y < d"
proof (cases "x = y")
case True
then have False
using ‹d ≤ dist x y› ‹d>0› by auto
then show "∃x'∈ball x e. x' ≠ y ∧ dist x' y < d"
by auto
next
case False
have "dist x (y - (d / (2 * dist y x)) *⇩R (y - x)) =
norm (x - y + (d / (2 * norm (y - x))) *⇩R (y - x))"
unfolding mem_cball mem_ball dist_norm diff_diff_eq2 diff_add_eq[symmetric]
by auto
also have "… = ¦- 1 + d / (2 * norm (x - y))¦ * norm (x - y)"
using scaleR_left_distrib[of "- 1" "d / (2 * norm (y - x))", symmetric, of "y - x"]
unfolding scaleR_minus_left scaleR_one
by (auto simp: norm_minus_commute)
also have "… = ¦- norm (x - y) + d / 2¦"
unfolding abs_mult_pos[of "norm (x - y)", OF norm_ge_zero[of "x - y"]]
unfolding distrib_right using ‹x≠y› by auto
also have "… ≤ e - d/2" using ‹d ≤ dist x y› and ‹d>0› and ‹?rhs›
by (auto simp: dist_norm)
finally have "y - (d / (2 * dist y x)) *⇩R (y - x) ∈ ball x e" using ‹d>0›
by auto
moreover
have "(d / (2*dist y x)) *⇩R (y - x) ≠ 0"
using ‹x≠y›[unfolded dist_nz] ‹d>0› unfolding scaleR_eq_0_iff
by (auto simp: dist_commute)
moreover
have "dist (y - (d / (2 * dist y x)) *⇩R (y - x)) y < d"
unfolding dist_norm
apply simp
unfolding norm_minus_cancel
using ‹d > 0› ‹x≠y›[unfolded dist_nz] dist_commute[of x y]
unfolding dist_norm
apply auto
done
ultimately show "∃x'∈ball x e. x' ≠ y ∧ dist x' y < d"
apply (rule_tac x = "y - (d / (2*dist y x)) *⇩R (y - x)" in bexI)
apply auto
done
qed
next
case False
then have "d > dist x y" by auto
show "∃x' ∈ ball x e. x' ≠ y ∧ dist x' y < d"
proof (cases "x = y")
case True
obtain z where **: "z ≠ y" "dist z y < min e d"
using perfect_choose_dist[of "min e d" y]
using ‹d > 0› ‹e>0› by auto
show "∃x'∈ball x e. x' ≠ y ∧ dist x' y < d"
unfolding ‹x = y›
using ‹z ≠ y› **
apply (rule_tac x=z in bexI)
apply (auto simp: dist_commute)
done
next
case False
then show "∃x'∈ball x e. x' ≠ y ∧ dist x' y < d"
using ‹d>0› ‹d > dist x y› ‹?rhs›
apply (rule_tac x=x in bexI, auto)
done
qed
qed
}
then show ?thesis
unfolding mem_cball islimpt_approachable mem_ball by auto
qed
qed
lemma closure_ball_lemma:
fixes x y :: "'a::real_normed_vector"
assumes "x ≠ y"
shows "y islimpt ball x (dist x y)"
proof (rule islimptI)
fix T
assume "y ∈ T" "open T"
then obtain r where "0 < r" "∀z. dist z y < r ⟶ z ∈ T"
unfolding open_dist by fast
define k where "k = min 1 (r / (2 * dist x y))"
define z where "z = y + scaleR k (x - y)"
have z_def2: "z = x + scaleR (1 - k) (y - x)"
unfolding z_def by (simp add: algebra_simps)
have "dist z y < r"
unfolding z_def k_def using ‹0 < r›
by (simp add: dist_norm min_def)
then have "z ∈ T"
using ‹∀z. dist z y < r ⟶ z ∈ T› by simp
have "dist x z < dist x y"
unfolding z_def2 dist_norm
apply (simp add: norm_minus_commute)
apply (simp only: dist_norm [symmetric])
apply (subgoal_tac "¦1 - k¦ * dist x y < 1 * dist x y", simp)
apply (rule mult_strict_right_mono)
apply (simp add: k_def ‹0 < r› ‹x ≠ y›)
apply (simp add: ‹x ≠ y›)
done
then have "z ∈ ball x (dist x y)"
by simp
have "z ≠ y"
unfolding z_def k_def using ‹x ≠ y› ‹0 < r›
by (simp add: min_def)
show "∃z∈ball x (dist x y). z ∈ T ∧ z ≠ y"
using ‹z ∈ ball x (dist x y)› ‹z ∈ T› ‹z ≠ y›
by fast
qed
lemma closure_ball [simp]:
fixes x :: "'a::real_normed_vector"
shows "0 < e ⟹ closure (ball x e) = cball x e"
apply (rule equalityI)
apply (rule closure_minimal)
apply (rule ball_subset_cball)
apply (rule closed_cball)
apply (rule subsetI, rename_tac y)
apply (simp add: le_less [where 'a=real])
apply (erule disjE)
apply (rule subsetD [OF closure_subset], simp)
apply (simp add: closure_def, clarify)
apply (rule closure_ball_lemma)
apply (simp add: zero_less_dist_iff)
done
lemma interior_cball [simp]:
fixes x :: "'a::{real_normed_vector, perfect_space}"
shows "interior (cball x e) = ball x e"
proof (cases "e ≥ 0")
case False note cs = this
from cs have null: "ball x e = {}"
using ball_empty[of e x] by auto
moreover
{
fix y
assume "y ∈ cball x e"
then have False
by (metis ball_eq_empty null cs dist_eq_0_iff dist_le_zero_iff empty_subsetI mem_cball subset_antisym subset_ball)
}
then have "cball x e = {}" by auto
then have "interior (cball x e) = {}"
using interior_empty by auto
ultimately show ?thesis by blast
next
case True note cs = this
have "ball x e ⊆ cball x e"
using ball_subset_cball by auto
moreover
{
fix S y
assume as: "S ⊆ cball x e" "open S" "y∈S"
then obtain d where "d>0" and d: "∀x'. dist x' y < d ⟶ x' ∈ S"
unfolding open_dist by blast
then obtain xa where xa_y: "xa ≠ y" and xa: "dist xa y < d"
using perfect_choose_dist [of d] by auto
have "xa ∈ S"
using d[THEN spec[where x = xa]]
using xa by (auto simp: dist_commute)
then have xa_cball: "xa ∈ cball x e"
using as(1) by auto
then have "y ∈ ball x e"
proof (cases "x = y")
case True
then have "e > 0" using cs order.order_iff_strict xa_cball xa_y by fastforce
then show "y ∈ ball x e"
using ‹x = y › by simp
next
case False
have "dist (y + (d / 2 / dist y x) *⇩R (y - x)) y < d"
unfolding dist_norm
using ‹d>0› norm_ge_zero[of "y - x"] ‹x ≠ y› by auto
then have *: "y + (d / 2 / dist y x) *⇩R (y - x) ∈ cball x e"
using d as(1)[unfolded subset_eq] by blast
have "y - x ≠ 0" using ‹x ≠ y› by auto
hence **:"d / (2 * norm (y - x)) > 0"
unfolding zero_less_norm_iff[symmetric] using ‹d>0› by auto
have "dist (y + (d / 2 / dist y x) *⇩R (y - x)) x =
norm (y + (d / (2 * norm (y - x))) *⇩R y - (d / (2 * norm (y - x))) *⇩R x - x)"
by (auto simp: dist_norm algebra_simps)
also have "… = norm ((1 + d / (2 * norm (y - x))) *⇩R (y - x))"
by (auto simp: algebra_simps)
also have "… = ¦1 + d / (2 * norm (y - x))¦ * norm (y - x)"
using ** by auto
also have "… = (dist y x) + d/2"
using ** by (auto simp: distrib_right dist_norm)
finally have "e ≥ dist x y +d/2"
using *[unfolded mem_cball] by (auto simp: dist_commute)
then show "y ∈ ball x e"
unfolding mem_ball using ‹d>0› by auto
qed
}
then have "∀S ⊆ cball x e. open S ⟶ S ⊆ ball x e"
by auto
ultimately show ?thesis
using interior_unique[of "ball x e" "cball x e"]
using open_ball[of x e]
by auto
qed
lemma interior_ball [simp]: "interior (ball x e) = ball x e"
by (simp add: interior_open)
lemma frontier_ball [simp]:
fixes a :: "'a::real_normed_vector"
shows "0 < e ⟹ frontier (ball a e) = sphere a e"
by (force simp: frontier_def)
lemma frontier_cball [simp]:
fixes a :: "'a::{real_normed_vector, perfect_space}"
shows "frontier (cball a e) = sphere a e"
by (force simp: frontier_def)
lemma cball_eq_empty [simp]: "cball x e = {} ⟷ e < 0"
apply (simp add: set_eq_iff not_le)
apply (metis zero_le_dist dist_self order_less_le_trans)
done
lemma cball_empty [simp]: "e < 0 ⟹ cball x e = {}"
by (simp add: cball_eq_empty)
lemma cball_eq_sing:
fixes x :: "'a::{metric_space,perfect_space}"
shows "cball x e = {x} ⟷ e = 0"
proof (rule linorder_cases)
assume e: "0 < e"
obtain a where "a ≠ x" "dist a x < e"
using perfect_choose_dist [OF e] by auto
then have "a ≠ x" "dist x a ≤ e"
by (auto simp: dist_commute)
with e show ?thesis by (auto simp: set_eq_iff)
qed auto
lemma cball_sing:
fixes x :: "'a::metric_space"
shows "e = 0 ⟹ cball x e = {x}"
by (auto simp: set_eq_iff)
lemma ball_divide_subset: "d ≥ 1 ⟹ ball x (e/d) ⊆ ball x e"
apply (cases "e ≤ 0")
apply (simp add: ball_empty divide_simps)
apply (rule subset_ball)
apply (simp add: divide_simps)
done
lemma ball_divide_subset_numeral: "ball x (e / numeral w) ⊆ ball x e"
using ball_divide_subset one_le_numeral by blast
lemma cball_divide_subset: "d ≥ 1 ⟹ cball x (e/d) ⊆ cball x e"
apply (cases "e < 0")
apply (simp add: divide_simps)
apply (rule subset_cball)
apply (metis div_by_1 frac_le not_le order_refl zero_less_one)
done
lemma cball_divide_subset_numeral: "cball x (e / numeral w) ⊆ cball x e"
using cball_divide_subset one_le_numeral by blast
subsection ‹Boundedness›
definition (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: "b > 0" "∀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)
{
fix x
assume "x ∈ S"
then have "norm x ≤ b"
using b by auto
then have "norm (f x) ≤ B * b"
using B(2)
apply (erule_tac x=x in allE)
apply (metis B(1) B(2) order_trans mult_le_cancel_left_pos)
done
}
then show ?thesis
unfolding bounded_pos
apply (rule_tac x="b*B" in exI)
using b B by (auto simp: mult.commute)
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_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
subsection‹Some theorems on sups and infs using the notion "bounded".›
lemma bounded_real: "bounded (S::real set) ⟷ (∃a. ∀x∈S. ¦x¦ ≤ a)"
by (simp add: bounded_iff)
lemma bounded_imp_bdd_above: "bounded S ⟹ bdd_above (S :: real set)"
by (auto simp: bounded_def bdd_above_def dist_real_def)
(metis abs_le_D1 abs_minus_commute diff_le_eq)
lemma bounded_imp_bdd_below: "bounded S ⟹ bdd_below (S :: real set)"
by (auto simp: bounded_def bdd_below_def dist_real_def)
(metis abs_le_D1 add.commute diff_le_eq)
lemma bounded_inner_imp_bdd_above:
assumes "bounded s"
shows "bdd_above ((λx. x ∙ a) ` s)"
by (simp add: assms bounded_imp_bdd_above bounded_linear_image bounded_linear_inner_left)
lemma bounded_inner_imp_bdd_below:
assumes "bounded s"
shows "bdd_below ((λx. x ∙ a) ` s)"
by (simp add: assms bounded_imp_bdd_below bounded_linear_image bounded_linear_inner_left)
lemma bounded_has_Sup:
fixes S :: "real set"
assumes "bounded S"
and "S ≠ {}"
shows "∀x∈S. x ≤ Sup S"
and "∀b. (∀x∈S. x ≤ b) ⟶ Sup S ≤ b"
proof
show "∀b. (∀x∈S. x ≤ b) ⟶ Sup S ≤ b"
using assms by (metis cSup_least)
qed (metis cSup_upper assms(1) bounded_imp_bdd_above)
lemma Sup_insert:
fixes S :: "real set"
shows "bounded S ⟹ Sup (insert x S) = (if S = {} then x else max x (Sup S))"
by (auto simp: bounded_imp_bdd_above sup_max cSup_insert_If)
lemma Sup_insert_finite:
fixes S :: "'a::conditionally_complete_linorder set"
shows "finite S ⟹ Sup (insert x S) = (if S = {} then x else max x (Sup S))"
by (simp add: cSup_insert sup_max)
lemma bounded_has_Inf:
fixes S :: "real set"
assumes "bounded S"
and "S ≠ {}"
shows "∀x∈S. x ≥ Inf S"
and "∀b. (∀x∈S. x ≥ b) ⟶ Inf S ≥ b"
proof
show "∀b. (∀x∈S. x ≥ b) ⟶ Inf S ≥ b"
using assms by (metis cInf_greatest)
qed (metis cInf_lower assms(1) bounded_imp_bdd_below)
lemma Inf_insert:
fixes S :: "real set"
shows "bounded S ⟹ Inf (insert x S) = (if S = {} then x else min x (Inf S))"
by (auto simp: bounded_imp_bdd_below inf_min cInf_insert_If)
lemma Inf_insert_finite:
fixes S :: "'a::conditionally_complete_linorder set"
shows "finite S ⟹ Inf (insert x S) = (if S = {} then x else min x (Inf S))"
by (simp add: cInf_eq_Min)
lemma finite_imp_less_Inf:
fixes a :: "'a::conditionally_complete_linorder"
shows "⟦finite X; x ∈ X; ⋀x. x∈X ⟹ a < x⟧ ⟹ a < Inf X"
by (induction X rule: finite_induct) (simp_all add: cInf_eq_Min Inf_insert_finite)
lemma finite_less_Inf_iff:
fixes a :: "'a :: conditionally_complete_linorder"
shows "⟦finite X; X ≠ {}⟧ ⟹ a < Inf X ⟷ (∀x ∈ X. a < x)"
by (auto simp: cInf_eq_Min)
lemma finite_imp_Sup_less:
fixes a :: "'a::conditionally_complete_linorder"
shows "⟦finite X; x ∈ X; ⋀x. x∈X ⟹ a > x⟧ ⟹ a > Sup X"
by (induction X rule: finite_induct) (simp_all add: cSup_eq_Max Sup_insert_finite)
lemma finite_Sup_less_iff:
fixes a :: "'a :: conditionally_complete_linorder"
shows "⟦finite X; X ≠ {}⟧ ⟹ a > Sup X ⟷ (∀x ∈ X. a > x)"
by (auto simp: cSup_eq_Max)
subsection ‹Compactness›
subsubsection ‹Bolzano-Weierstrass property›
lemma 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 "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
lemma 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 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
lemma 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
lemma 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›
lemma 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
lemma 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
lemma compact_def: ―‹this is the definition of compactness in HOL Light›
"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›
lemma 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
lemma 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"
lemma 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_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 compact_components:
fixes s :: "'a::heine_borel set"
shows "⟦compact s; c ∈ components s⟧ ⟹ compact c"
by (meson bounded_subset closed_components in_components_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)
instance real :: heine_borel
proof
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 euclidean_space ⊆ heine_borel
proof
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 setL2_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 prod :: (heine_borel, heine_borel) heine_borel
proof
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 ‹Intersecting chains of compact sets›
proposition bounded_closed_chain:
fixes ℱ :: "'a::heine_borel set set"
assumes "B ∈ ℱ" "bounded B" and ℱ: "⋀S. S ∈ ℱ ⟹ closed S" and "{} ∉ ℱ"
and chain: "⋀S T. S ∈ ℱ ∧ T ∈ ℱ ⟹ S ⊆ T ∨ T ⊆ S"
shows "⋂ℱ ≠ {}"
proof -
have "B ∩ ⋂ℱ ≠ {}"
proof (rule compact_imp_fip)
show "compact B" "⋀T. T ∈ ℱ ⟹ closed T"
by (simp_all add: assms compact_eq_bounded_closed)
show "⟦finite 𝒢; 𝒢 ⊆ ℱ⟧ ⟹ B ∩ ⋂𝒢 ≠ {}" for 𝒢
proof (induction 𝒢 rule: finite_induct)
case empty
with assms show ?case by force
next
case (insert U 𝒢)
then have "U ∈ ℱ" and ne: "B ∩ ⋂𝒢 ≠ {}" by auto
then consider "B ⊆ U" | "U ⊆ B"
using ‹B ∈ ℱ› chain by blast
then show ?case
proof cases
case 1
then show ?thesis
using Int_left_commute ne by auto
next
case 2
have "U ≠ {}"
using ‹U ∈ ℱ› ‹{} ∉ ℱ› by blast
moreover
have False if "⋀x. x ∈ U ⟹ ∃Y∈𝒢. x ∉ Y"
proof -
have "⋀x. x ∈ U ⟹ ∃Y∈𝒢. Y ⊆ U"
by (metis chain contra_subsetD insert.prems insert_subset that)
then obtain Y where "Y ∈ 𝒢" "Y ⊆ U"
by (metis all_not_in_conv ‹U ≠ {}›)
moreover obtain x where "x ∈ ⋂𝒢"
by (metis Int_emptyI ne)
ultimately show ?thesis
by (metis Inf_lower subset_eq that)
qed
with 2 show ?thesis
by blast
qed
qed
qed
then show ?thesis by blast
qed
corollary compact_chain:
fixes ℱ :: "'a::heine_borel set set"
assumes "⋀S. S ∈ ℱ ⟹ compact S" "{} ∉ ℱ"
"⋀S T. S ∈ ℱ ∧ T ∈ ℱ ⟹ S ⊆ T ∨ T ⊆ S"
shows "⋂ ℱ ≠ {}"
proof (cases "ℱ = {}")
case True
then show ?thesis by auto
next
case False
show ?thesis
by (metis False all_not_in_conv assms compact_imp_bounded compact_imp_closed bounded_closed_chain)
qed
lemma compact_nest:
fixes F :: "'a::linorder ⇒ 'b::heine_borel set"
assumes F: "⋀n. compact(F n)" "⋀n. F n ≠ {}" and mono: "⋀m n. m ≤ n ⟹ F n ⊆ F m"
shows "⋂range F ≠ {}"
proof -
have *: "⋀S T. S ∈ range F ∧ T ∈ range F ⟹ S ⊆ T ∨ T ⊆ S"
by (metis mono image_iff le_cases)
show ?thesis
apply (rule compact_chain [OF _ _ *])
using F apply (blast intro: dest: *)+
done
qed
text‹The Baire property of dense sets›
theorem Baire:
fixes S::"'a::{real_normed_vector,heine_borel} set"
assumes "closed S" "countable 𝒢"
and ope: "⋀T. T ∈ 𝒢 ⟹ openin (subtopology euclidean S) T ∧ S ⊆ closure T"
shows "S ⊆ closure(⋂𝒢)"
proof (cases "𝒢 = {}")
case True
then show ?thesis
using closure_subset by auto
next
let ?g = "from_nat_into 𝒢"
case False
then have gin: "?g n ∈ 𝒢" for n
by (simp add: from_nat_into)
show ?thesis
proof (clarsimp simp: closure_approachable)
fix x and e::real
assume "x ∈ S" "0 < e"
obtain TF where opeF: "⋀n. openin (subtopology euclidean S) (TF n)"
and ne: "⋀n. TF n ≠ {}"
and subg: "⋀n. S ∩ closure(TF n) ⊆ ?g n"
and subball: "⋀n. closure(TF n) ⊆ ball x e"
and decr: "⋀n. TF(Suc n) ⊆ TF n"
proof -
have *: "∃Y. (openin (subtopology euclidean S) Y ∧ Y ≠ {} ∧
S ∩ closure Y ⊆ ?g n ∧ closure Y ⊆ ball x e) ∧ Y ⊆ U"
if opeU: "openin (subtopology euclidean S) U" and "U ≠ {}" and cloU: "closure U ⊆ ball x e" for U n
proof -
obtain T where T: "open T" "U = T ∩ S"
using ‹openin (subtopology euclidean S) U› by (auto simp: openin_subtopology)
with ‹U ≠ {}› have "T ∩ closure (?g n) ≠ {}"
using gin ope by fastforce
then have "T ∩ ?g n ≠ {}"
using ‹open T› open_Int_closure_eq_empty by blast
then obtain y where "y ∈ U" "y ∈ ?g n"
using T ope [of "?g n", OF gin] by (blast dest: openin_imp_subset)
moreover have "openin (subtopology euclidean S) (U ∩ ?g n)"
using gin ope opeU by blast
ultimately obtain d where U: "U ∩ ?g n ⊆ S" and "d > 0" and d: "ball y d ∩ S ⊆ U ∩ ?g n"
by (force simp: openin_contains_ball)
show ?thesis
proof (intro exI conjI)
show "openin (subtopology euclidean S) (S ∩ ball y (d/2))"
by (simp add: openin_open_Int)
show "S ∩ ball y (d/2) ≠ {}"
using ‹0 < d› ‹y ∈ U› opeU openin_imp_subset by fastforce
have "S ∩ closure (S ∩ ball y (d/2)) ⊆ S ∩ closure (ball y (d/2))"
using closure_mono by blast
also have "... ⊆ ?g n"
using ‹d > 0› d by force
finally show "S ∩ closure (S ∩ ball y (d/2)) ⊆ ?g n" .
have "closure (S ∩ ball y (d/2)) ⊆ S ∩ ball y d"
proof -
have "closure (ball y (d/2)) ⊆ ball y d"
using ‹d > 0› by auto
then have "closure (S ∩ ball y (d/2)) ⊆ ball y d"
by (meson closure_mono inf.cobounded2 subset_trans)
then show ?thesis
by (simp add: ‹closed S› closure_minimal)
qed
also have "... ⊆ ball x e"
using cloU closure_subset d by blast
finally show "closure (S ∩ ball y (d/2)) ⊆ ball x e" .
show "S ∩ ball y (d/2) ⊆ U"
using ball_divide_subset_numeral d by blast
qed
qed
let ?Φ = "λn X. openin (subtopology euclidean S) X ∧ X ≠ {} ∧
S ∩ closure X ⊆ ?g n ∧ closure X ⊆ ball x e"
have "closure (S ∩ ball x (e / 2)) ⊆ closure(ball x (e/2))"
by (simp add: closure_mono)
also have "... ⊆ ball x e"
using ‹e > 0› by auto
finally have "closure (S ∩ ball x (e / 2)) ⊆ ball x e" .
moreover have"openin (subtopology euclidean S) (S ∩ ball x (e / 2))" "S ∩ ball x (e / 2) ≠ {}"
using ‹0 < e› ‹x ∈ S› by auto
ultimately obtain Y where Y: "?Φ 0 Y ∧ Y ⊆ S ∩ ball x (e / 2)"
using * [of "S ∩ ball x (e/2)" 0] by metis
show thesis
proof (rule exE [OF dependent_nat_choice [of ?Φ "λn X Y. Y ⊆ X"]])
show "∃x. ?Φ 0 x"
using Y by auto
show "∃Y. ?Φ (Suc n) Y ∧ Y ⊆ X" if "?Φ n X" for X n
using that by (blast intro: *)
qed (use that in metis)
qed
have "(⋂n. S ∩ closure (TF n)) ≠ {}"
proof (rule compact_nest)
show "⋀n. compact (S ∩ closure (TF n))"
by (metis closed_closure subball bounded_subset_ballI compact_eq_bounded_closed closed_Int_compact [OF ‹closed S›])
show "⋀n. S ∩ closure (TF n) ≠ {}"
by (metis Int_absorb1 opeF ‹closed S› closure_eq_empty closure_minimal ne openin_imp_subset)
show "⋀m n. m ≤ n ⟹ S ∩ closure (TF n) ⊆ S ∩ closure (TF m)"
by (meson closure_mono decr dual_order.refl inf_mono lift_Suc_antimono_le)
qed
moreover have "(⋂n. S ∩ closure (TF n)) ⊆ {y ∈ ⋂𝒢. dist y x < e}"
proof (clarsimp, intro conjI)
fix y
assume "y ∈ S" and y: "∀n. y ∈ closure (TF n)"
then show "∀T∈𝒢. y ∈ T"
by (metis Int_iff from_nat_into_surj [OF ‹countable 𝒢›] set_mp subg)
show "dist y x < e"
by (metis y dist_commute mem_ball subball subsetCE)
qed
ultimately show "∃y ∈ ⋂𝒢. dist y x < e"
by auto
qed
qed
subsubsection ‹Completeness›
lemma (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
lemma (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
lemma nat_approx_posE:
fixes e::real
assumes "0 < e"
obtains n :: nat where "1 / (Suc n) < e"
proof atomize_elim
have "1 / real (Suc (nat ⌈1/e⌉)) < 1 / ⌈1/e⌉"
by (rule divide_strict_left_mono) (auto simp: ‹0 < e›)
also have "1 / ⌈1/e⌉ ≤ 1 / (1/e)"
by (rule divide_left_mono) (auto simp: ‹0 < e› ceiling_correct)
also have "… = e" by simp
finally show "∃n. 1 / real (Suc n) < e" ..
qed
lemma 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 compact_cball[simp]:
fixes x :: "'a::heine_borel"
shows "compact (cball x e)"
using compact_eq_bounded_closed bounded_cball closed_cball
by blast
lemma compact_frontier_bounded[intro]:
fixes S :: "'a::heine_borel set"
shows "bounded S ⟹ compact (frontier S)"
unfolding frontier_def
using compact_eq_bounded_closed
by blast
lemma compact_frontier[intro]:
fixes S :: "'a::heine_borel set"
shows "compact S ⟹ compact (frontier S)"
using compact_eq_bounded_closed compact_frontier_bounded
by blast
corollary compact_sphere [simp]:
fixes a :: "'a::{real_normed_vector,perfect_space,heine_borel}"
shows "compact (sphere a r)"
using compact_frontier [of "cball a r"] by simp
corollary bounded_sphere [simp]:
fixes a :: "'a::{real_normed_vector,perfect_space,heine_borel}"
shows "bounded (sphere a r)"
by (simp add: compact_imp_bounded)
corollary closed_sphere [simp]:
fixes a :: "'a::{real_normed_vector,perfect_space,heine_borel}"
shows "closed (sphere a r)"
by (simp add: compact_imp_closed)
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‹Relations among convergence and absolute convergence for power series.›
lemma summable_imp_bounded:
fixes f :: "nat ⇒ 'a::real_normed_vector"
shows "summable f ⟹ bounded (range f)"
by (frule summable_LIMSEQ_zero) (simp add: convergent_imp_bounded)
lemma summable_imp_sums_bounded:
"summable f ⟹ bounded (range (λn. sum f {..<n}))"
by (auto simp: summable_def sums_def dest: convergent_imp_bounded)
lemma power_series_conv_imp_absconv_weak:
fixes a:: "nat ⇒ 'a::{real_normed_div_algebra,banach}" and w :: 'a
assumes sum: "summable (λn. a n * z ^ n)" and no: "norm w < norm z"
shows "summable (λn. of_real(norm(a n)) * w ^ n)"
proof -
obtain M where M: "⋀x. norm (a x * z ^ x) ≤ M"
using summable_imp_bounded [OF sum] by (force simp: bounded_iff)
then have *: "summable (λn. norm (a n) * norm w ^ n)"
by (rule_tac M=M in Abel_lemma) (auto simp: norm_mult norm_power intro: no)
show ?thesis
apply (rule series_comparison_complex [of "(λn. of_real(norm(a n) * norm w ^ n))"])
apply (simp only: summable_complex_of_real *)
apply (auto simp: norm_mult norm_power)
done
qed
subsection ‹Bounded closed nest property (proof does not use Heine-Borel)›
lemma bounded_closed_nest:
fixes s :: "nat ⇒ ('a::heine_borel) set"
assumes "∀n. closed (s n)"
and "∀n. s n ≠ {}"
and "∀m n. m ≤ n ⟶ s n ⊆ s m"
and "bounded (s 0)"
shows "∃a. ∀n. a ∈ s n"
proof -
from assms(2) obtain x where x: "∀n. x n ∈ s n"
using choice[of "λn x. x ∈ s n"] by auto
from assms(4,1) have "seq_compact (s 0)"
by (simp add: bounded_closed_imp_seq_compact)
then obtain l r where lr: "l ∈ s 0" "strict_mono r" "(x ∘ r) ⇢ l"
using x and assms(3) unfolding seq_compact_def by blast
have "∀n. l ∈ s n"
proof
fix n :: nat
have "closed (s n)"
using assms(1) by simp
moreover have "∀i. (x ∘ r) i ∈ s i"
using x and assms(3) and lr(2) [THEN seq_suble] by auto
then have "∀i. (x ∘ r) (i + n) ∈ s n"
using assms(3) by (fast intro!: le_add2)
moreover have "(λi. (x ∘ r) (i + n)) ⇢ l"
using lr(3) by (rule LIMSEQ_ignore_initial_segment)
ultimately show "l ∈ s n"
by (rule closed_sequentially)
qed
then show ?thesis ..
qed
text ‹Decreasing case does not even need compactness, just completeness.›
lemma decreasing_closed_nest:
fixes s :: "nat ⇒ ('a::complete_space) set"
assumes
"∀n. closed (s n)"
"∀n. s n ≠ {}"
"∀m n. m ≤ n ⟶ s n ⊆ s m"
"∀e>0. ∃n. ∀x∈s n. ∀y∈s n. dist x y < e"
shows "∃a. ∀n. a ∈ s n"
proof -
have "∀n. ∃x. x ∈ s n"
using assms(2) by auto
then have "∃t. ∀n. t n ∈ s n"
using choice[of "λn x. x ∈ s n"] by auto
then obtain t where t: "∀n. t n ∈ s n" by auto
{
fix e :: real
assume "e > 0"
then obtain N where N:"∀x∈s N. ∀y∈s N. dist x y < e"
using assms(4) by auto
{
fix m n :: nat
assume "N ≤ m ∧ N ≤ n"
then have "t m ∈ s N" "t n ∈ s N"
using assms(3) t unfolding subset_eq t by blast+
then have "dist (t m) (t n) < e"
using N by auto
}
then have "∃N. ∀m n. N ≤ m ∧ N ≤ n ⟶ dist (t m) (t n) < e"
by auto
}
then have "Cauchy t"
unfolding cauchy_def by auto
then obtain l where l:"(t ⤏ l) sequentially"
using complete_UNIV unfolding complete_def by auto
{
fix n :: nat
{
fix e :: real
assume "e > 0"
then obtain N :: nat where N: "∀n≥N. dist (t n) l < e"
using l[unfolded lim_sequentially] by auto
have "t (max n N) ∈ s n"
using assms(3)
unfolding subset_eq
apply (erule_tac x=n in allE)
apply (erule_tac x="max n N" in allE)
using t
apply auto
done
then have "∃y∈s n. dist y l < e"
apply (rule_tac x="t (max n N)" in bexI)
using N
apply auto
done
}
then have "l ∈ s n"
using closed_approachable[of "s n" l] assms(1) by auto
}
then show ?thesis by auto
qed
text ‹Strengthen it to the intersection actually being a singleton.›
lemma decreasing_closed_nest_sing:
fixes s :: "nat ⇒ 'a::complete_space set"
assumes
"∀n. closed(s n)"
"∀n. s n ≠ {}"
"∀m n. m ≤ n ⟶ s n ⊆ s m"
"∀e>0. ∃n. ∀x ∈ (s n). ∀ y∈(s n). dist x y < e"
shows "∃a. ⋂(range s) = {a}"
proof -
obtain a where a: "∀n. a ∈ s n"
using decreasing_closed_nest[of s] using assms by auto
{
fix b
assume b: "b ∈ ⋂(range s)"
{
fix e :: real
assume "e > 0"
then have "dist a b < e"
using assms(4) and b and a by blast
}
then have "dist a b = 0"
by (metis dist_eq_0_iff dist_nz less_le)
}
with a have "⋂(range s) = {a}"
unfolding image_def by auto
then show ?thesis ..
qed
subsection ‹Continuity›
text‹Derive the epsilon-delta forms, which we often use as "definitions"›
lemma 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
apply auto
apply (metis dist_nz dist_self, blast)
done
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 ― "legacy theorem name"
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_sequentially:
fixes f :: "'a::metric_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)"
(is "?lhs = ?rhs")
proof
assume ?lhs
{
fix x :: "nat ⇒ 'a"
assume x: "∀n. x n ∈ s" "∀e>0. eventually (λn. dist (x n) a < e) sequentially"
fix T :: "'b set"
assume "open T" and "f a ∈ T"
with ‹?lhs› obtain d where "d>0" and d:"∀x∈s. 0 < dist x a ∧ dist x a < d ⟶ f x ∈ T"
unfolding continuous_within tendsto_def eventually_at by auto
have "eventually (λn. dist (x n) a < d) sequentially"
using x(2) ‹d>0› by simp
then have "eventually (λn. (f ∘ x) n ∈ T) sequentially"
proof eventually_elim
case (elim n)
then show ?case
using d x(1) ‹f a ∈ T› by auto
qed
}
then show ?rhs
unfolding tendsto_iff tendsto_def by simp
next
assume ?rhs
then show ?lhs
unfolding continuous_within tendsto_def [where l="f a"]
by (simp add: sequentially_imp_eventually_within)
qed
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_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 o 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 ‹Structural rules for pointwise continuity›
lemma continuous_infdist[continuous_intros]:
assumes "continuous F f"
shows "continuous F (λx. infdist (f x) A)"
using assms unfolding continuous_def by (rule tendsto_infdist)
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)
lemmas continuous_at_inverse = isCont_inverse
subsubsection ‹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 ‹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)
lemmas continuous_at_compose = isCont_o
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) {x ∈ s. f x ∈ 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) {x ∈ S. f x ∈ U})"
(is "?lhs = ?rhs")
proof
assume ?lhs
then show ?rhs
apply (auto 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 ?rhs
then show ?lhs
apply (clarsimp simp add: continuous_on_iff)
apply (drule_tac x = "ball (f x) e ∩ T" in spec)
apply (clarsimp simp add: ope openin_euclidean_subtopology_iff [of S])
by (metis (no_types, hide_lams) assms dist_commute dist_self image_subset_iff)
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) {x ∈ S. f x ∈ 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) {x ∈ s. f x ∈ 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) {x ∈ S. f x ∈ U})"
proof -
have *: "U ⊆ T ⟹ {x ∈ S. f x ∈ T ∧ f x ∉ U} = S - {x ∈ S. f x ∈ U}" for U
using assms by blast
show ?thesis
apply (simp add: continuous_on_open_gen [OF assms], safe)
apply (drule_tac [!] x="T-U" in spec)
apply (force simp: closedin_def *)
apply (force simp: openin_closedin_eq *)
done
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) {x ∈ S. f x ∈ 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) {x. x ∈ S ∧ f x ∈ T}"
using assms continuous_on_closed by blast
subsection ‹Half-global and completely global cases.›
lemma continuous_openin_preimage_gen:
assumes "continuous_on s f" "open t"
shows "openin (subtopology euclidean s) {x ∈ s. f x ∈ t}"
proof -
have *: "∀x. x ∈ s ∧ f x ∈ t ⟷ x ∈ s ∧ f x ∈ (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) {x ∈ s. f x ∈ t}"
proof -
have *: "∀x. x ∈ s ∧ f x ∈ t ⟷ x ∈ s ∧ f x ∈ (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) {x. x ∈ S ∧ f x ∈ 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) {x. x ∈ S ∧ f x ∈ t})"
apply safe
apply (simp add: continuous_closedin_preimage)
apply (fastforce simp add: continuous_on_closed closedin_closed)
done
lemma continuous_open_preimage:
assumes "continuous_on s f"
and "open s"
and "open t"
shows "open {x ∈ s. f x ∈ t}"
proof-
obtain T where T: "open T" "{x ∈ s. f x ∈ t} = s ∩ T"
using continuous_openin_preimage_gen[OF assms(1,3)] unfolding openin_open by auto
then show ?thesis
using open_Int[of s T, OF assms(2)] by auto
qed
lemma continuous_closed_preimage:
assumes "continuous_on s f"
and "closed s"
and "closed t"
shows "closed {x ∈ s. f x ∈ t}"
proof-
obtain T where "closed T" "{x ∈ s. f x ∈ t} = s ∩ T"
using continuous_closedin_preimage[OF assms(1,3)]
unfolding closedin_closed by auto
then show ?thesis using closed_Int[of s T, OF assms(2)] by auto
qed
lemma continuous_open_preimage_univ:
"open s ⟹ (⋀x. continuous (at x) f) ⟹ open {x. f x ∈ s}"
using continuous_open_preimage[of UNIV f s] open_UNIV continuous_at_imp_continuous_on by auto
lemma continuous_closed_preimage_univ:
"closed s ⟹ (⋀x. continuous (at x) f) ⟹ closed {x. f x ∈ s}"
using continuous_closed_preimage[of UNIV f s] closed_UNIV continuous_at_imp_continuous_on by auto
lemma continuous_open_vimage: "open s ⟹ (⋀x. continuous (at x) f) ⟹ open (f -` s)"
unfolding vimage_def by (rule continuous_open_preimage_univ)
lemma continuous_closed_vimage: "closed s ⟹ (⋀x. continuous (at x) f) ⟹ closed (f -` s)"
unfolding vimage_def by (rule continuous_closed_preimage_univ)
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 (vimage f T)"
using assms ‹open T› by (metis continuous_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 ‹Equality of continuous functions on closure and related results.›
lemma continuous_closedin_preimage_constant:
fixes f :: "_ ⇒ 'b::t1_space"
shows "continuous_on s f ⟹ closedin (subtopology euclidean s) {x ∈ s. f x = a}"
using continuous_closedin_preimage[of s f "{a}"] by auto
lemma continuous_closed_preimage_constant:
fixes f :: "_ ⇒ 'b::t1_space"
shows "continuous_on s f ⟹ closed s ⟹ closed {x ∈ s. f x = a}"
using continuous_closed_preimage[of s f "{a}"] by auto
lemma continuous_constant_on_closure:
fixes f :: "_ ⇒ 'b::t1_space"
assumes "continuous_on (closure S) f"
and "⋀x. x ∈ S ⟹ f x = a"
and "x ∈ closure S"
shows "f x = a"
using continuous_closed_preimage_constant[of "closure S" f a]
assms closure_minimal[of S "{x ∈ closure S. f x = a}"] closure_subset
unfolding subset_eq
by auto
lemma image_closure_subset:
assumes "continuous_on (closure s) f"
and "closed t"
and "(f ` s) ⊆ t"
shows "f ` (closure s) ⊆ t"
proof -
have "s ⊆ {x ∈ closure s. f x ∈ t}"
using assms(3) closure_subset by auto
moreover have "closed {x ∈ closure s. f x ∈ t}"
using continuous_closed_preimage[OF assms(1)] and assms(2) by auto
ultimately have "closure s = {x ∈ closure s . f x ∈ t}"
using closure_minimal[of s "{x ∈ closure s. f x ∈ t}"] by auto
then show ?thesis by auto
qed
lemma continuous_on_closure_norm_le:
fixes f :: "'a::metric_space ⇒ 'b::real_normed_vector"
assumes "continuous_on (closure s) f"
and "∀y ∈ s. norm(f y) ≤ b"
and "x ∈ (closure s)"
shows "norm (f x) ≤ b"
proof -
have *: "f ` s ⊆ cball 0 b"
using assms(2)[unfolded mem_cball_0[symmetric]] by auto
show ?thesis
using image_closure_subset[OF assms(1) closed_cball[of 0 b] *] assms(3)
unfolding subset_eq
apply (erule_tac x="f x" in ballE)
apply (auto simp: dist_norm)
done
qed
lemma isCont_indicator:
fixes x :: "'a::t2_space"
shows "isCont (indicator A :: 'a ⇒ real) x = (x ∉ frontier A)"
proof auto
fix x
assume cts_at: "isCont (indicator A :: 'a ⇒ real) x" and fr: "x ∈ frontier A"
with continuous_at_open have 1: "∀V::real set. open V ∧ indicator A x ∈ V ⟶
(∃U::'a set. open U ∧ x ∈ U ∧ (∀y∈U. indicator A y ∈ V))" by auto
show False
proof (cases "x ∈ A")
assume x: "x ∈ A"
hence "indicator A x ∈ ({0<..<2} :: real set)" by simp
hence "∃U. open U ∧ x ∈ U ∧ (∀y∈U. indicator A y ∈ ({0<..<2} :: real set))"
using 1 open_greaterThanLessThan by blast
then guess U .. note U = this
hence "∀y∈U. indicator A y > (0::real)"
unfolding greaterThanLessThan_def by auto
hence "U ⊆ A" using indicator_eq_0_iff by force
hence "x ∈ interior A" using U interiorI by auto
thus ?thesis using fr unfolding frontier_def by simp
next
assume x: "x ∉ A"
hence "indicator A x ∈ ({-1<..<1} :: real set)" by simp
hence "∃U. open U ∧ x ∈ U ∧ (∀y∈U. indicator A y ∈ ({-1<..<1} :: real set))"
using 1 open_greaterThanLessThan by blast
then guess U .. note U = this
hence "∀y∈U. indicator A y < (1::real)"
unfolding greaterThanLessThan_def by auto
hence "U ⊆ -A" by auto
hence "x ∈ interior (-A)" using U interiorI by auto
thus ?thesis using fr interior_complement unfolding frontier_def by auto
qed
next
assume nfr: "x ∉ frontier A"
hence "x ∈ interior A ∨ x ∈ interior (-A)"
by (auto simp: frontier_def closure_interior)
thus "isCont ((indicator A)::'a ⇒ real) x"
proof
assume int: "x ∈ interior A"
then obtain U where U: "open U" "x ∈ U" "U ⊆ A" unfolding interior_def by auto
hence "∀y∈U. indicator A y = (1::real)" unfolding indicator_def by auto
hence "continuous_on U (indicator A)" by (simp add: continuous_on_const indicator_eq_1_iff)
thus ?thesis using U continuous_on_eq_continuous_at by auto
next
assume ext: "x ∈ interior (-A)"
then obtain U where U: "open U" "x ∈ U" "U ⊆ -A" unfolding interior_def by auto
then have "continuous_on U (indicator A)"
using continuous_on_topological by (auto simp: subset_iff)
thus ?thesis using U continuous_on_eq_continuous_at by auto
qed
qed
subsection‹ Theorems relating continuity and uniform continuity to closures›
lemma continuous_on_closure:
"continuous_on (closure S) f ⟷
(∀x e. x ∈ closure S ∧ 0 < e
⟶ (∃d. 0 < d ∧ (∀y. y ∈ S ∧ dist y x < d ⟶ dist (f y) (f x) < e)))"
(is "?lhs = ?rhs")
proof
assume ?lhs then show ?rhs
unfolding continuous_on_iff by (metis Un_iff closure_def)
next
assume R [rule_format]: ?rhs
show ?lhs
proof
fix x and e::real
assume "0 < e" and x: "x ∈ closure S"
obtain δ::real where "δ > 0"
and δ: "⋀y. ⟦y ∈ S; dist y x < δ⟧ ⟹ dist (f y) (f x) < e/2"
using R [of x "e/2"] ‹0 < e› x by auto
have "dist (f y) (f x) ≤ e" if y: "y ∈ closure S" and dyx: "dist y x < δ/2" for y
proof -
obtain δ'::real where "δ' > 0"
and δ': "⋀z. ⟦z ∈ S; dist z y < δ'⟧ ⟹ dist (f z) (f y) < e/2"
using R [of y "e/2"] ‹0 < e› y by auto
obtain z where "z ∈ S" and z: "dist z y < min δ' δ / 2"
using closure_approachable y
by (metis ‹0 < δ'› ‹0 < δ› divide_pos_pos min_less_iff_conj zero_less_numeral)
have "dist (f z) (f y) < e/2"
apply (rule δ' [OF ‹z ∈ S›])
using z ‹0 < δ'› by linarith
moreover have "dist (f z) (f x) < e/2"
apply (rule δ [OF ‹z ∈ S›])
using z ‹0 < δ› dist_commute[of y z] dist_triangle_half_r [of y] dyx by auto
ultimately show ?thesis
by (metis dist_commute dist_triangle_half_l less_imp_le)
qed
then show "∃d>0. ∀x'∈closure S. dist x' x < d ⟶ dist (f x') (f x) ≤ e"
by (rule_tac x="δ/2" in exI) (simp add: ‹δ > 0›)
qed
qed
lemma continuous_on_closure_sequentially:
fixes f :: "'a::metric_space ⇒ 'b :: metric_space"
shows
"continuous_on (closure S) f ⟷
(∀x a. a ∈ closure S ∧ (∀n. x n ∈ S) ∧ x ⇢ a ⟶ (f ∘ x) ⇢ f a)"
(is "?lhs = ?rhs")
proof -
have "continuous_on (closure S) f ⟷
(∀x ∈ closure S. continuous (at x within S) f)"
by (force simp: continuous_on_closure Topology_Euclidean_Space.continuous_within_eps_delta)
also have "... = ?rhs"
by (force simp: continuous_within_sequentially)
finally show ?thesis .
qed
lemma uniformly_continuous_on_closure:
fixes f :: "'a::metric_space ⇒ 'b::metric_space"
assumes ucont: "uniformly_continuous_on S f"
and cont: "continuous_on (closure S) f"
shows "uniformly_continuous_on (closure S) f"
unfolding uniformly_continuous_on_def
proof (intro allI impI)
fix e::real
assume "0 < e"
then obtain d::real
where "d>0"
and d: "⋀x x'. ⟦x∈S; x'∈S; dist x' x < d⟧ ⟹ dist (f x') (f x) < e/3"
using ucont [unfolded uniformly_continuous_on_def, rule_format, of "e/3"] by auto
show "∃d>0. ∀x∈closure S. ∀x'∈closure S. dist x' x < d ⟶ dist (f x') (f x) < e"
proof (rule exI [where x="d/3"], clarsimp simp: ‹d > 0›)
fix x y
assume x: "x ∈ closure S" and y: "y ∈ closure S" and dyx: "dist y x * 3 < d"
obtain d1::real where "d1 > 0"
and d1: "⋀w. ⟦w ∈ closure S; dist w x < d1⟧ ⟹ dist (f w) (f x) < e/3"
using cont [unfolded continuous_on_iff, rule_format, of "x" "e/3"] ‹0 < e› x by auto
obtain x' where "x' ∈ S" and x': "dist x' x < min d1 (d / 3)"
using closure_approachable [of x S]
by (metis ‹0 < d1› ‹0 < d› divide_pos_pos min_less_iff_conj x zero_less_numeral)
obtain d2::real where "d2 > 0"
and d2: "∀w ∈ closure S. dist w y < d2 ⟶ dist (f w) (f y) < e/3"
using cont [unfolded continuous_on_iff, rule_format, of "y" "e/3"] ‹0 < e› y by auto
obtain y' where "y' ∈ S" and y': "dist y' y < min d2 (d / 3)"
using closure_approachable [of y S]
by (metis ‹0 < d2› ‹0 < d› divide_pos_pos min_less_iff_conj y zero_less_numeral)
have "dist x' x < d/3" using x' by auto
moreover have "dist x y < d/3"
by (metis dist_commute dyx less_divide_eq_numeral1(1))
moreover have "dist y y' < d/3"
by (metis (no_types) dist_commute min_less_iff_conj y')
ultimately have "dist x' y' < d/3 + d/3 + d/3"
by (meson dist_commute_lessI dist_triangle_lt add_strict_mono)
then have "dist x' y' < d" by simp
then have "dist (f x') (f y') < e/3"
by (rule d [OF ‹y' ∈ S› ‹x' ∈ S›])
moreover have "dist (f x') (f x) < e/3" using ‹x' ∈ S› closure_subset x' d1
by (simp add: closure_def)
moreover have "dist (f y') (f y) < e/3" using ‹y' ∈ S› closure_subset y' d2
by (simp add: closure_def)
ultimately have "dist (f y) (f x) < e/3 + e/3 + e/3"
by (meson dist_commute_lessI dist_triangle_lt add_strict_mono)
then show "dist (f y) (f x) < e" by simp
qed
qed
lemma uniformly_continuous_on_extension_at_closure:
fixes f::"'a::metric_space ⇒ 'b::complete_space"
assumes uc: "uniformly_continuous_on X f"
assumes "x ∈ closure X"
obtains l where "(f ⤏ l) (at x within X)"
proof -
from assms obtain xs where xs: "xs ⇢ x" "⋀n. xs n ∈ X"
by (auto simp: closure_sequential)
from uniformly_continuous_on_Cauchy[OF uc LIMSEQ_imp_Cauchy, OF xs]
obtain l where l: "(λn. f (xs n)) ⇢ l"
by atomize_elim (simp only: convergent_eq_Cauchy)
have "(f ⤏ l) (at x within X)"
proof (safe intro!: Lim_within_LIMSEQ)
fix xs'
assume "∀n. xs' n ≠ x ∧ xs' n ∈ X"
and xs': "xs' ⇢ x"
then have "xs' n ≠ x" "xs' n ∈ X" for n by auto
from uniformly_continuous_on_Cauchy[OF uc LIMSEQ_imp_Cauchy, OF ‹xs' ⇢ x› ‹xs' _ ∈ X›]
obtain l' where l': "(λn. f (xs' n)) ⇢ l'"
by atomize_elim (simp only: convergent_eq_Cauchy)
show "(λn. f (xs' n)) ⇢ l"
proof (rule tendstoI)
fix e::real assume "e > 0"
define e' where "e' ≡ e / 2"
have "e' > 0" using ‹e > 0› by (simp add: e'_def)
have "∀⇩F n in sequentially. dist (f (xs n)) l < e'"
by (simp add: ‹0 < e'› l tendstoD)
moreover
from uc[unfolded uniformly_continuous_on_def, rule_format, OF ‹e' > 0›]
obtain d where d: "d > 0" "⋀x x'. x ∈ X ⟹ x' ∈ X ⟹ dist x x' < d ⟹ dist (f x) (f x') < e'"
by auto
have "∀⇩F n in sequentially. dist (xs n) (xs' n) < d"
by (auto intro!: ‹0 < d› order_tendstoD tendsto_eq_intros xs xs')
ultimately
show "∀⇩F n in sequentially. dist (f (xs' n)) l < e"
proof eventually_elim
case (elim n)
have "dist (f (xs' n)) l ≤ dist (f (xs n)) (f (xs' n)) + dist (f (xs n)) l"
by (metis dist_triangle dist_commute)
also have "dist (f (xs n)) (f (xs' n)) < e'"
by (auto intro!: d xs ‹xs' _ ∈ _› elim)
also note ‹dist (f (xs n)) l < e'›
also have "e' + e' = e" by (simp add: e'_def)
finally show ?case by simp
qed
qed
qed
thus ?thesis ..
qed
lemma uniformly_continuous_on_extension_on_closure:
fixes f::"'a::metric_space ⇒ 'b::complete_space"
assumes uc: "uniformly_continuous_on X f"
obtains g where "uniformly_continuous_on (closure X) g" "⋀x. x ∈ X ⟹ f x = g x"
"⋀Y h x. X ⊆ Y ⟹ Y ⊆ closure X ⟹ continuous_on Y h ⟹ (⋀x. x ∈ X ⟹ f x = h x) ⟹ x ∈ Y ⟹ h x = g x"
proof -
from uc have cont_f: "continuous_on X f"
by (simp add: uniformly_continuous_imp_continuous)
obtain y where y: "(f ⤏ y x) (at x within X)" if "x ∈ closure X" for x
apply atomize_elim
apply (rule choice)
using uniformly_continuous_on_extension_at_closure[OF assms]
by metis
let ?g = "λx. if x ∈ X then f x else y x"
have "uniformly_continuous_on (closure X) ?g"
unfolding uniformly_continuous_on_def
proof safe
fix e::real assume "e > 0"
define e' where "e' ≡ e / 3"
have "e' > 0" using ‹e > 0› by (simp add: e'_def)
from uc[unfolded uniformly_continuous_on_def, rule_format, OF ‹0 < e'›]
obtain d where "d > 0" and d: "⋀x x'. x ∈ X ⟹ x' ∈ X ⟹ dist x' x < d ⟹ dist (f x') (f x) < e'"
by auto
define d' where "d' = d / 3"
have "d' > 0" using ‹d > 0› by (simp add: d'_def)
show "∃d>0. ∀x∈closure X. ∀x'∈closure X. dist x' x < d ⟶ dist (?g x') (?g x) < e"
proof (safe intro!: exI[where x=d'] ‹d' > 0›)
fix x x' assume x: "x ∈ closure X" and x': "x' ∈ closure X" and dist: "dist x' x < d'"
then obtain xs xs' where xs: "xs ⇢ x" "⋀n. xs n ∈ X"
and xs': "xs' ⇢ x'" "⋀n. xs' n ∈ X"
by (auto simp: closure_sequential)
have "∀⇩F n in sequentially. dist (xs' n) x' < d'"
and "∀⇩F n in sequentially. dist (xs n) x < d'"
by (auto intro!: ‹0 < d'› order_tendstoD tendsto_eq_intros xs xs')
moreover
have "(λx. f (xs x)) ⇢ y x" if "x ∈ closure X" "x ∉ X" "xs ⇢ x" "⋀n. xs n ∈ X" for xs x
using that not_eventuallyD
by (force intro!: filterlim_compose[OF y[OF ‹x ∈ closure X›]] simp: filterlim_at)
then have "(λx. f (xs' x)) ⇢ ?g x'" "(λx. f (xs x)) ⇢ ?g x"
using x x'
by (auto intro!: continuous_on_tendsto_compose[OF cont_f] simp: xs' xs)
then have "∀⇩F n in sequentially. dist (f (xs' n)) (?g x') < e'"
"∀⇩F n in sequentially. dist (f (xs n)) (?g x) < e'"
by (auto intro!: ‹0 < e'› order_tendstoD tendsto_eq_intros)
ultimately
have "∀⇩F n in sequentially. dist (?g x') (?g x) < e"
proof eventually_elim
case (elim n)
have "dist (?g x') (?g x) ≤
dist (f (xs' n)) (?g x') + dist (f (xs' n)) (f (xs n)) + dist (f (xs n)) (?g x)"
by (metis add.commute add_le_cancel_left dist_commute dist_triangle dist_triangle_le)
also
{
have "dist (xs' n) (xs n) ≤ dist (xs' n) x' + dist x' x + dist (xs n) x"
by (metis add.commute add_le_cancel_left dist_triangle dist_triangle_le)
also note ‹dist (xs' n) x' < d'›
also note ‹dist x' x < d'›
also note ‹dist (xs n) x < d'›
finally have "dist (xs' n) (xs n) < d" by (simp add: d'_def)
}
with ‹xs _ ∈ X› ‹xs' _ ∈ X› have "dist (f (xs' n)) (f (xs n)) < e'"
by (rule d)
also note ‹dist (f (xs' n)) (?g x') < e'›
also note ‹dist (f (xs n)) (?g x) < e'›
finally show ?case by (simp add: e'_def)
qed
then show "dist (?g x') (?g x) < e" by simp
qed
qed
moreover have "f x = ?g x" if "x ∈ X" for x using that by simp
moreover
{
fix Y h x
assume Y: "x ∈ Y" "X ⊆ Y" "Y ⊆ closure X" and cont_h: "continuous_on Y h"
and extension: "(⋀x. x ∈ X ⟹ f x = h x)"
{
assume "x ∉ X"
have "x ∈ closure X" using Y by auto
then obtain xs where xs: "xs ⇢ x" "⋀n. xs n ∈ X"
by (auto simp: closure_sequential)
from continuous_on_tendsto_compose[OF cont_h xs(1)] xs(2) Y
have hx: "(λx. f (xs x)) ⇢ h x"
by (auto simp: set_mp extension)
then have "(λx. f (xs x)) ⇢ y x"
using ‹x ∉ X› not_eventuallyD xs(2)
by (force intro!: filterlim_compose[OF y[OF ‹x ∈ closure X›]] simp: filterlim_at xs)
with hx have "h x = y x" by (rule LIMSEQ_unique)
} then
have "h x = ?g x"
using extension by auto
}
ultimately show ?thesis ..
qed
lemma bounded_uniformly_continuous_image:
fixes f :: "'a :: heine_borel ⇒ 'b :: heine_borel"
assumes "uniformly_continuous_on S f" "bounded S"
shows "bounded(image f S)"
by (metis (no_types, lifting) assms bounded_closure_image compact_closure compact_continuous_image compact_eq_bounded_closed image_cong uniformly_continuous_imp_continuous uniformly_continuous_on_extension_on_closure)
subsection‹Quotient maps›
lemma quotient_map_imp_continuous_open:
assumes t: "f ` s ⊆ t"
and ope: "⋀u. u ⊆ t
⟹ (openin (subtopology euclidean s) {x. x ∈ s ∧ f x ∈ u} ⟷
openin (subtopology euclidean t) u)"
shows "continuous_on s f"
proof -
have [simp]: "{x ∈ s. f x ∈ f ` s} = s" by auto
show ?thesis
using ope [OF t]
apply (simp add: continuous_on_open)
by (metis (no_types, lifting) "ope" openin_imp_subset openin_trans)
qed
lemma quotient_map_imp_continuous_closed:
assumes t: "f ` s ⊆ t"
and ope: "⋀u. u ⊆ t
⟹ (closedin (subtopology euclidean s) {x. x ∈ s ∧ f x ∈ u} ⟷
closedin (subtopology euclidean t) u)"
shows "continuous_on s f"
proof -
have [simp]: "{x ∈ s. f x ∈ f ` s} = s" by auto
show ?thesis
using ope [OF t]
apply (simp add: continuous_on_closed)
by (metis (no_types, lifting) "ope" closedin_imp_subset closedin_subtopology_refl closedin_trans openin_subtopology_refl openin_subtopology_self)
qed
lemma open_map_imp_quotient_map:
assumes contf: "continuous_on s f"
and t: "t ⊆ f ` s"
and ope: "⋀t. openin (subtopology euclidean s) t
⟹ openin (subtopology euclidean (f ` s)) (f ` t)"
shows "openin (subtopology euclidean s) {x ∈ s. f x ∈ t} =
openin (subtopology euclidean (f ` s)) t"
proof -
have "t = image f {x. x ∈ s ∧ f x ∈ t}"
using t by blast
then show ?thesis
using "ope" contf continuous_on_open by fastforce
qed
lemma closed_map_imp_quotient_map:
assumes contf: "continuous_on s f"
and t: "t ⊆ f ` s"
and ope: "⋀t. closedin (subtopology euclidean s) t
⟹ closedin (subtopology euclidean (f ` s)) (f ` t)"
shows "openin (subtopology euclidean s) {x ∈ s. f x ∈ t} ⟷
openin (subtopology euclidean (f ` s)) t"
(is "?lhs = ?rhs")
proof
assume ?lhs
then have *: "closedin (subtopology euclidean s) (s - {x ∈ s. f x ∈ t})"
using closedin_diff by fastforce
have [simp]: "(f ` s - f ` (s - {x ∈ s. f x ∈ t})) = t"
using t by blast
show ?rhs
using ope [OF *, unfolded closedin_def] by auto
next
assume ?rhs
with contf show ?lhs
by (auto simp: continuous_on_open)
qed
lemma continuous_right_inverse_imp_quotient_map:
assumes contf: "continuous_on s f" and imf: "f ` s ⊆ t"
and contg: "continuous_on t g" and img: "g ` t ⊆ s"
and fg [simp]: "⋀y. y ∈ t ⟹ f(g y) = y"
and u: "u ⊆ t"
shows "openin (subtopology euclidean s) {x. x ∈ s ∧ f x ∈ u} ⟷
openin (subtopology euclidean t) u"
(is "?lhs = ?rhs")
proof -
have f: "⋀z. openin (subtopology euclidean (f ` s)) z ⟹
openin (subtopology euclidean s) {x ∈ s. f x ∈ z}"
and g: "⋀z. openin (subtopology euclidean (g ` t)) z ⟹
openin (subtopology euclidean t) {x ∈ t. g x ∈ z}"
using contf contg by (auto simp: continuous_on_open)
show ?thesis
proof
have "{x ∈ t. g x ∈ g ` t ∧ g x ∈ s ∧ f (g x) ∈ u} = {x ∈ t. f (g x) ∈ u}"
using imf img by blast
also have "... = u"
using u by auto
finally have [simp]: "{x ∈ t. g x ∈ g ` t ∧ g x ∈ s ∧ f (g x) ∈ u} = u" .
assume ?lhs
then have *: "openin (subtopology euclidean (g ` t)) (g ` t ∩ {x ∈ s. f x ∈ u})"
by (meson img openin_Int openin_subtopology_Int_subset openin_subtopology_self)
show ?rhs
using g [OF *] by simp
next
assume rhs: ?rhs
show ?lhs
apply (rule f)
by (metis fg image_eqI image_subset_iff imf img openin_subopen openin_subtopology_self openin_trans rhs)
qed
qed
lemma continuous_left_inverse_imp_quotient_map:
assumes "continuous_on s f"
and "continuous_on (f ` s) g"
and "⋀x. x ∈ s ⟹ g(f x) = x"
and "u ⊆ f ` s"
shows "openin (subtopology euclidean s) {x. x ∈ s ∧ f x ∈ u} ⟷
openin (subtopology euclidean (f ` s)) u"
apply (rule continuous_right_inverse_imp_quotient_map)
using assms
apply force+
done
subsection ‹A function constant on a set›
definition constant_on (infixl "(constant'_on)" 50)
where "f constant_on A ≡ ∃y. ∀x∈A. f x = y"
lemma constant_on_subset: "⟦f constant_on A; B ⊆ A⟧ ⟹ f constant_on B"
unfolding constant_on_def by blast
lemma injective_not_constant:
fixes S :: "'a::{perfect_space} set"
shows "⟦open S; inj_on f S; f constant_on S⟧ ⟹ S = {}"
unfolding constant_on_def
by (metis equals0I inj_on_contraD islimpt_UNIV islimpt_def)
lemma constant_on_closureI:
fixes f :: "_ ⇒ 'b::t1_space"
assumes cof: "f constant_on S" and contf: "continuous_on (closure S) f"
shows "f constant_on (closure S)"
using continuous_constant_on_closure [OF contf] cof unfolding constant_on_def
by metis
text ‹Making a continuous function avoid some value in a neighbourhood.›
lemma continuous_within_avoid:
fixes f :: "'a::metric_space ⇒ 'b::t1_space"
assumes "continuous (at x within s) f"
and "f x ≠ a"
shows "∃e>0. ∀y ∈ s. dist x y < e --> f y ≠ a"
proof -
obtain U where "open U" and "f x ∈ U" and "a ∉ U"
using t1_space [OF ‹f x ≠ a›] by fast
have "(f ⤏ f x) (at x within s)"
using assms(1) by (simp add: continuous_within)
then have "eventually (λy. f y ∈ U) (at x within s)"
using ‹open U› and ‹f x ∈ U›
unfolding tendsto_def by fast
then have "eventually (λy. f y ≠ a) (at x within s)"
using ‹a ∉ U› by (fast elim: eventually_mono)
then show ?thesis
using ‹f x ≠ a› by (auto simp: dist_commute zero_less_dist_iff eventually_at)
qed
lemma continuous_at_avoid:
fixes f :: "'a::metric_space ⇒ 'b::t1_space"
assumes "continuous (at x) f"
and "f x ≠ a"
shows "∃e>0. ∀y. dist x y < e ⟶ f y ≠ a"
using assms continuous_within_avoid[of x UNIV f a] by simp
lemma continuous_on_avoid:
fixes f :: "'a::metric_space ⇒ 'b::t1_space"
assumes "continuous_on s f"
and "x ∈ s"
and "f x ≠ a"
shows "∃e>0. ∀y ∈ s. dist x y < e ⟶ f y ≠ a"
using assms(1)[unfolded continuous_on_eq_continuous_within, THEN bspec[where x=x],
OF assms(2)] continuous_within_avoid[of x s f a]
using assms(3)
by auto
lemma continuous_on_open_avoid:
fixes f :: "'a::metric_space ⇒ 'b::t1_space"
assumes "continuous_on s f"
and "open s"
and "x ∈ s"
and "f x ≠ a"
shows "∃e>0. ∀y. dist x y < e ⟶ f y ≠ a"
using assms(1)[unfolded continuous_on_eq_continuous_at[OF assms(2)], THEN bspec[where x=x], OF assms(3)]
using continuous_at_avoid[of x f a] assms(4)
by auto
text ‹Proving a function is constant by proving open-ness of level set.›
lemma continuous_levelset_openin_cases:
fixes f :: "_ ⇒ 'b::t1_space"
shows "connected s ⟹ continuous_on s f ⟹
openin (subtopology euclidean s) {x ∈ s. f x = a}
⟹ (∀x ∈ s. f x ≠ a) ∨ (∀x ∈ s. f x = a)"
unfolding connected_clopen
using continuous_closedin_preimage_constant by auto
lemma continuous_levelset_openin:
fixes f :: "_ ⇒ 'b::t1_space"
shows "connected s ⟹ continuous_on s f ⟹
openin (subtopology euclidean s) {x ∈ s. f x = a} ⟹
(∃x ∈ s. f x = a) ⟹ (∀x ∈ s. f x = a)"
using continuous_levelset_openin_cases[of s f ]
by meson
lemma continuous_levelset_open:
fixes f :: "_ ⇒ 'b::t1_space"
assumes "connected s"
and "continuous_on s f"
and "open {x ∈ s. f x = a}"
and "∃x ∈ s. f x = a"
shows "∀x ∈ s. f x = a"
using continuous_levelset_openin[OF assms(1,2), of a, unfolded openin_open]
using assms (3,4)
by fast
text ‹Some arithmetical combinations (more to prove).›
lemma open_scaling[intro]:
fixes s :: "'a::real_normed_vector set"
assumes "c ≠ 0"
and "open s"
shows "open((λx. c *⇩R x) ` s)"
proof -
{
fix x
assume "x ∈ s"
then obtain e where "e>0"
and e:"∀x'. dist x' x < e ⟶ x' ∈ s" using assms(2)[unfolded open_dist, THEN bspec[where x=x]]
by auto
have "e * ¦c¦ > 0"
using assms(1)[unfolded zero_less_abs_iff[symmetric]] ‹e>0› by auto
moreover
{
fix y
assume "dist y (c *⇩R x) < e * ¦c¦"
then have "norm ((1 / c) *⇩R y - x) < e"
unfolding dist_norm
using norm_scaleR[of c "(1 / c) *⇩R y - x", unfolded scaleR_right_diff_distrib, unfolded scaleR_scaleR] assms(1)
assms(1)[unfolded zero_less_abs_iff[symmetric]] by (simp del:zero_less_abs_iff)
then have "y ∈ op *⇩R c ` s"
using rev_image_eqI[of "(1 / c) *⇩R y" s y "op *⇩R c"]
using e[THEN spec[where x="(1 / c) *⇩R y"]]
using assms(1)
unfolding dist_norm scaleR_scaleR
by auto
}
ultimately have "∃e>0. ∀x'. dist x' (c *⇩R x) < e ⟶ x' ∈ op *⇩R c ` s"
apply (rule_tac x="e * ¦c¦" in exI, auto)
done
}
then show ?thesis unfolding open_dist by auto
qed
lemma minus_image_eq_vimage:
fixes A :: "'a::ab_group_add set"
shows "(λx. - x) ` A = (λx. - x) -` A"
by (auto intro!: image_eqI [where f="λx. - x"])
lemma open_negations:
fixes S :: "'a::real_normed_vector set"
shows "open S ⟹ open ((λx. - x) ` S)"
using open_scaling [of "- 1" S] by simp
lemma open_translation:
fixes S :: "'a::real_normed_vector set"
assumes "open S"
shows "open((λx. a + x) ` S)"
proof -
{
fix x
have "continuous (at x) (λx. x - a)"
by (intro continuous_diff continuous_ident continuous_const)
}
moreover have "{x. x - a ∈ S} = op + a ` S"
by force
ultimately show ?thesis
by (metis assms continuous_open_vimage vimage_def)
qed
lemma open_affinity:
fixes S :: "'a::real_normed_vector set"
assumes "open S" "c ≠ 0"
shows "open ((λx. a + c *⇩R x) ` S)"
proof -
have *: "(λx. a + c *⇩R x) = (λx. a + x) ∘ (λx. c *⇩R x)"
unfolding o_def ..
have "op + a ` op *⇩R c ` S = (op + a ∘ op *⇩R c) ` S"
by auto
then show ?thesis
using assms open_translation[of "op *⇩R c ` S" a]
unfolding *
by auto
qed
lemma interior_translation:
fixes S :: "'a::real_normed_vector set"
shows "interior ((λx. a + x) ` S) = (λx. a + x) ` (interior S)"
proof (rule set_eqI, rule)
fix x
assume "x ∈ interior (op + a ` S)"
then obtain e where "e > 0" and e: "ball x e ⊆ op + a ` S"
unfolding mem_interior by auto
then have "ball (x - a) e ⊆ S"
unfolding subset_eq Ball_def mem_ball dist_norm
by (auto simp: diff_diff_eq)
then show "x ∈ op + a ` interior S"
unfolding image_iff
apply (rule_tac x="x - a" in bexI)
unfolding mem_interior
using ‹e > 0›
apply auto
done
next
fix x
assume "x ∈ op + a ` interior S"
then obtain y e where "e > 0" and e: "ball y e ⊆ S" and y: "x = a + y"
unfolding image_iff Bex_def mem_interior by auto
{
fix z
have *: "a + y - z = y + a - z" by auto
assume "z ∈ ball x e"
then have "z - a ∈ S"
using e[unfolded subset_eq, THEN bspec[where x="z - a"]]
unfolding mem_ball dist_norm y group_add_class.diff_diff_eq2 *
by auto
then have "z ∈ op + a ` S"
unfolding image_iff by (auto intro!: bexI[where x="z - a"])
}
then have "ball x e ⊆ op + a ` S"
unfolding subset_eq by auto
then show "x ∈ interior (op + a ` S)"
unfolding mem_interior using ‹e > 0› by auto
qed
subsection ‹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
subsubsection‹Relating linear images to open/closed/interior/closure.›
proposition open_surjective_linear_image:
fixes f :: "'a::real_normed_vector ⇒ 'b::euclidean_space"
assumes "open A" "linear f" "surj f"
shows "open(f ` A)"
unfolding open_dist
proof clarify
fix x
assume "x ∈ A"
have "bounded (inv f ` Basis)"
by (simp add: finite_imp_bounded)
with bounded_pos obtain B where "B > 0" and B: "⋀x. x ∈ inv f ` Basis ⟹ norm x ≤ B"
by metis
obtain e where "e > 0" and e: "⋀z. dist z x < e ⟹ z ∈ A"
by (metis open_dist ‹x ∈ A› ‹open A›)
define δ where "δ ≡ e / B / DIM('b)"
show "∃e>0. ∀y. dist y (f x) < e ⟶ y ∈ f ` A"
proof (intro exI conjI)
show "δ > 0"
using ‹e > 0› ‹B > 0› by (simp add: δ_def divide_simps)
have "y ∈ f ` A" if "dist y (f x) * (B * real DIM('b)) < e" for y
proof -
define u where "u ≡ y - f x"
show ?thesis
proof (rule image_eqI)
show "y = f (x + (∑i∈Basis. (u ∙ i) *⇩R inv f i))"
apply (simp add: linear_add linear_sum linear.scaleR ‹linear f› surj_f_inv_f ‹surj f›)
apply (simp add: euclidean_representation u_def)
done
have "dist (x + (∑i∈Basis. (u ∙ i) *⇩R inv f i)) x ≤ (∑i∈Basis. norm ((u ∙ i) *⇩R inv f i))"
by (simp add: dist_norm sum_norm_le)
also have "... = (∑i∈Basis. ¦u ∙ i¦ * norm (inv f i))"
by simp
also have "... ≤ (∑i∈Basis. ¦u ∙ i¦) * B"
by (simp add: B sum_distrib_right sum_mono mult_left_mono)
also have "... ≤ DIM('b) * dist y (f x) * B"
apply (rule mult_right_mono [OF sum_bounded_above])
using ‹0 < B› by (auto simp: Basis_le_norm dist_norm u_def)
also have "... < e"
by (metis mult.commute mult.left_commute that)
finally show "x + (∑i∈Basis. (u ∙ i) *⇩R inv f i) ∈ A"
by (rule e)
qed
qed
then show "∀y. dist y (f x) < δ ⟶ y ∈ f ` A"
using ‹e > 0› ‹B > 0›
by (auto simp: δ_def divide_simps mult_less_0_iff)
qed
qed
corollary open_bijective_linear_image_eq:
fixes f :: "'a::euclidean_space ⇒ 'b::euclidean_space"
assumes "linear f" "bij f"
shows "open(f ` A) ⟷ open A"
proof
assume "open(f ` A)"
then have "open(f -` (f ` A))"
using assms by (force simp: linear_continuous_at linear_conv_bounded_linear continuous_open_vimage)
then show "open A"
by (simp add: assms bij_is_inj inj_vimage_image_eq)
next
assume "open A"
then show "open(f ` A)"
by (simp add: assms bij_is_surj open_surjective_linear_image)
qed
corollary interior_bijective_linear_image:
fixes f :: "'a::euclidean_space ⇒ 'b::euclidean_space"
assumes "linear f" "bij f"
shows "interior (f ` S) = f ` interior S" (is "?lhs = ?rhs")
proof safe
fix x
assume x: "x ∈ ?lhs"
then obtain T where "open T" and "x ∈ T" and "T ⊆ f ` S"
by (metis interiorE)
then show "x ∈ ?rhs"
by (metis (no_types, hide_lams) assms subsetD interior_maximal open_bijective_linear_image_eq subset_image_iff)
next
fix x
assume x: "x ∈ interior S"
then show "f x ∈ interior (f ` S)"
by (meson assms imageI image_mono interiorI interior_subset open_bijective_linear_image_eq open_interior)
qed
lemma interior_injective_linear_image:
fixes f :: "'a::euclidean_space ⇒ 'a::euclidean_space"
assumes "linear f" "inj f"
shows "interior(f ` S) = f ` (interior S)"
by (simp add: linear_injective_imp_surjective assms bijI interior_bijective_linear_image)
lemma interior_surjective_linear_image:
fixes f :: "'a::euclidean_space ⇒ 'a::euclidean_space"
assumes "linear f" "surj f"
shows "interior(f ` S) = f ` (interior S)"
by (simp add: assms interior_injective_linear_image linear_surjective_imp_injective)
lemma interior_negations:
fixes S :: "'a::euclidean_space set"
shows "interior(uminus ` S) = image uminus (interior S)"
by (simp add: bij_uminus interior_bijective_linear_image linear_uminus)
text ‹Also bilinear functions, in composition form.›
lemma bilinear_continuous_at_compose:
"continuous (at x) f ⟹ continuous (at x) g ⟹ bounded_bilinear h ⟹
continuous (at x) (λx. h (f x) (g x))"
unfolding continuous_at
using Lim_bilinear[of f "f x" "(at x)" g "g x" h]
by auto
lemma bilinear_continuous_within_compose:
"continuous (at x within s) f ⟹ continuous (at x within s) g ⟹ bounded_bilinear h ⟹
continuous (at x within s) (λx. h (f x) (g x))"
by (rule Limits.bounded_bilinear.continuous)
lemma bilinear_continuous_on_compose:
"continuous_on s f ⟹ continuous_on s g ⟹ bounded_bilinear h ⟹
continuous_on s (λx. h (f x) (g x))"
by (rule Limits.bounded_bilinear.continuous_on)
text ‹Preservation of compactness and connectedness under continuous function.›
lemma compact_eq_openin_cover:
"compact S ⟷
(∀C. (∀c∈C. openin (subtopology euclidean S) c) ∧ S ⊆ ⋃C ⟶
(∃D⊆C. finite D ∧ S ⊆ ⋃D))"
proof safe
fix C
assume "compact S" and "∀c∈C. openin (subtopology euclidean S) c" and "S ⊆ ⋃C"
then have "∀c∈{T. open T ∧ S ∩ T ∈ C}. open c" and "S ⊆ ⋃{T. open T ∧ S ∩ T ∈ C}"
unfolding openin_open by force+
with ‹compact S› obtain D where "D ⊆ {T. open T ∧ S ∩ T ∈ C}" and "finite D" and "S ⊆ ⋃D"
by (meson compactE)
then have "image (λT. S ∩ T) D ⊆ C ∧ finite (image (λT. S ∩ T) D) ∧ S ⊆ ⋃(image (λT. S ∩ T) D)"
by auto
then show "∃D⊆C. finite D ∧ S ⊆ ⋃D" ..
next
assume 1: "∀C. (∀c∈C. openin (subtopology euclidean S) c) ∧ S ⊆ ⋃C ⟶
(∃D⊆C. finite D ∧ S ⊆ ⋃D)"
show "compact S"
proof (rule compactI)
fix C
let ?C = "image (λT. S ∩ T) C"
assume "∀t∈C. open t" and "S ⊆ ⋃C"
then have "(∀c∈?C. openin (subtopology euclidean S) c) ∧ S ⊆ ⋃?C"
unfolding openin_open by auto
with 1 obtain D where "D ⊆ ?C" and "finite D" and "S ⊆ ⋃D"
by metis
let ?D = "inv_into C (λT. S ∩ T) ` D"
have "?D ⊆ C ∧ finite ?D ∧ S ⊆ ⋃?D"
proof (intro conjI)
from ‹D ⊆ ?C› show "?D ⊆ C"
by (fast intro: inv_into_into)
from ‹finite D› show "finite ?D"
by (rule finite_imageI)
from ‹S ⊆ ⋃D› show "S ⊆ ⋃?D"
apply (rule subset_trans, clarsimp)
apply (frule subsetD [OF ‹D ⊆ ?C›, THEN f_inv_into_f])
apply (erule rev_bexI, fast)
done
qed
then show "∃D⊆C. finite D ∧ S ⊆ ⋃D" ..
qed
qed
lemma connected_continuous_image:
assumes "continuous_on s f"
and "connected s"
shows "connected(f ` s)"
proof -
{
fix T
assume as:
"T ≠ {}"
"T ≠ f ` s"
"openin (subtopology euclidean (f ` s)) T"
"closedin (subtopology euclidean (f ` s)) T"
have "{x ∈ s. f x ∈ T} = {} ∨ {x ∈ s. f x ∈ T} = s"
using assms(1)[unfolded continuous_on_open, THEN spec[where x=T]]
using assms(1)[unfolded continuous_on_closed, THEN spec[where x=T]]
using assms(2)[unfolded connected_clopen, THEN spec[where x="{x ∈ s. f x ∈ T}"]] as(3,4) by auto
then have False using as(1,2)
using as(4)[unfolded closedin_def topspace_euclidean_subtopology] by auto
}
then show ?thesis
unfolding connected_clopen by auto
qed
lemma connected_linear_image:
fixes f :: "'a::euclidean_space ⇒ 'b::real_normed_vector"
assumes "linear f" and "connected s"
shows "connected (f ` s)"
using connected_continuous_image assms linear_continuous_on linear_conv_bounded_linear by blast
text ‹Continuity implies uniform continuity on a compact domain.›
subsection ‹Continuity implies uniform continuity on a compact domain.›
text‹From the proof of the Heine-Borel theorem: Lemma 2 in section 3.7, page 69 of
J. C. Burkill and H. Burkill. A Second Course in Mathematical Analysis (CUP, 2002)›
lemma Heine_Borel_lemma:
assumes "compact S" and Ssub: "S ⊆ ⋃𝒢" and op: "⋀G. G ∈ 𝒢 ⟹ open G"
obtains e where "0 < e" "⋀x. x ∈ S ⟹ ∃G ∈ 𝒢. ball x e ⊆ G"
proof -
have False if neg: "⋀e. 0 < e ⟹ ∃x ∈ S. ∀G ∈ 𝒢. ¬ ball x e ⊆ G"
proof -
have "∃x ∈ S. ∀G ∈ 𝒢. ¬ ball x (1 / Suc n) ⊆ G" for n
using neg by simp
then obtain f where "⋀n. f n ∈ S" and fG: "⋀G n. G ∈ 𝒢 ⟹ ¬ ball (f n) (1 / Suc n) ⊆ G"
by metis
then obtain l r where "l ∈ S" "strict_mono r" and to_l: "(f ∘ r) ⇢ l"
using ‹compact S› compact_def that by metis
then obtain G where "l ∈ G" "G ∈ 𝒢"
using Ssub by auto
then obtain e where "0 < e" and e: "⋀z. dist z l < e ⟹ z ∈ G"
using op open_dist by blast
obtain N1 where N1: "⋀n. n ≥ N1 ⟹ dist (f (r n)) l < e/2"
using to_l apply (simp add: lim_sequentially)
using ‹0 < e› half_gt_zero that by blast
obtain N2 where N2: "of_nat N2 > 2/e"
using reals_Archimedean2 by blast
obtain x where "x ∈ ball (f (r (max N1 N2))) (1 / real (Suc (r (max N1 N2))))" and "x ∉ G"
using fG [OF ‹G ∈ 𝒢›, of "r (max N1 N2)"] by blast
then have "dist (f (r (max N1 N2))) x < 1 / real (Suc (r (max N1 N2)))"
by simp
also have "... ≤ 1 / real (Suc (max N1 N2))"
apply (simp add: divide_simps del: max.bounded_iff)
using ‹strict_mono r› seq_suble by blast
also have "... ≤ 1 / real (Suc N2)"
by (simp add: field_simps)
also have "... < e/2"
using N2 ‹0 < e› by (simp add: field_simps)
finally have "dist (f (r (max N1 N2))) x < e / 2" .
moreover have "dist (f (r (max N1 N2))) l < e/2"
using N1 max.cobounded1 by blast
ultimately have "dist x l < e"
using dist_triangle_half_r by blast
then show ?thesis
using e ‹x ∉ G› by blast
qed
then show ?thesis
by (meson that)
qed
lemma compact_uniformly_equicontinuous:
assumes "compact S"
and cont: "⋀x e. ⟦x ∈ S; 0 < e⟧
⟹ ∃d. 0 < d ∧
(∀f ∈ ℱ. ∀x' ∈ S. dist x' x < d ⟶ dist (f x') (f x) < e)"
and "0 < e"
obtains d where "0 < d"
"⋀f x x'. ⟦f ∈ ℱ; x ∈ S; x' ∈ S; dist x' x < d⟧ ⟹ dist (f x') (f x) < e"
proof -
obtain d where d_pos: "⋀x e. ⟦x ∈ S; 0 < e⟧ ⟹ 0 < d x e"
and d_dist : "⋀x x' e f. ⟦dist x' x < d x e; x ∈ S; x' ∈ S; 0 < e; f ∈ ℱ⟧ ⟹ dist (f x') (f x) < e"
using cont by metis
let ?𝒢 = "((λx. ball x (d x (e / 2))) ` S)"
have Ssub: "S ⊆ ⋃ ?𝒢"
by clarsimp (metis d_pos ‹0 < e› dist_self half_gt_zero_iff)
then obtain k where "0 < k" and k: "⋀x. x ∈ S ⟹ ∃G ∈ ?𝒢. ball x k ⊆ G"
by (rule Heine_Borel_lemma [OF ‹compact S›]) auto
moreover have "dist (f v) (f u) < e" if "f ∈ ℱ" "u ∈ S" "v ∈ S" "dist v u < k" for f u v
proof -
obtain G where "G ∈ ?𝒢" "u ∈ G" "v ∈ G"
using k that
by (metis ‹dist v u < k› ‹u ∈ S› ‹0 < k› centre_in_ball subsetD dist_commute mem_ball)
then obtain w where w: "dist w u < d w (e / 2)" "dist w v < d w (e / 2)" "w ∈ S"
by auto
with that d_dist have "dist (f w) (f v) < e/2"
by (metis ‹0 < e› dist_commute half_gt_zero)
moreover
have "dist (f w) (f u) < e/2"
using that d_dist w by (metis ‹0 < e› dist_commute divide_pos_pos zero_less_numeral)
ultimately show ?thesis
using dist_triangle_half_r by blast
qed
ultimately show ?thesis using that by blast
qed
corollary compact_uniformly_continuous:
fixes f :: "'a :: metric_space ⇒ 'b :: metric_space"
assumes f: "continuous_on S f" and S: "compact S"
shows "uniformly_continuous_on S f"
using f
unfolding continuous_on_iff uniformly_continuous_on_def
by (force intro: compact_uniformly_equicontinuous [OF S, of "{f}"])
subsection ‹Topological stuff about the set of Reals›
lemma open_real:
fixes s :: "real set"
shows "open s ⟷ (∀x ∈ s. ∃e>0. ∀x'. ¦x' - x¦ < e --> x' ∈ s)"
unfolding open_dist dist_norm by simp
lemma islimpt_approachable_real:
fixes s :: "real set"
shows "x islimpt s ⟷ (∀e>0. ∃x'∈ s. x' ≠ x ∧ ¦x' - x¦ < e)"
unfolding islimpt_approachable dist_norm by simp
lemma closed_real:
fixes s :: "real set"
shows "closed s ⟷ (∀x. (∀e>0. ∃x' ∈ s. x' ≠ x ∧ ¦x' - x¦ < e) ⟶ x ∈ s)"
unfolding closed_limpt islimpt_approachable dist_norm by simp
lemma continuous_at_real_range:
fixes f :: "'a::real_normed_vector ⇒ real"
shows "continuous (at x) f ⟷ (∀e>0. ∃d>0. ∀x'. norm(x' - x) < d --> ¦f x' - f x¦ < e)"
unfolding continuous_at
unfolding Lim_at
unfolding dist_norm
apply auto
apply (erule_tac x=e in allE, auto)
apply (rule_tac x=d in exI, auto)
apply (erule_tac x=x' in allE, auto)
apply (erule_tac x=e in allE, auto)
done
lemma continuous_on_real_range:
fixes f :: "'a::real_normed_vector ⇒ real"
shows "continuous_on s f ⟷
(∀x ∈ s. ∀e>0. ∃d>0. (∀x' ∈ s. norm(x' - x) < d ⟶ ¦f x' - f x¦ < e))"
unfolding continuous_on_iff dist_norm by simp
text ‹Hence some handy theorems on distance, diameter etc. of/from a set.›
lemma distance_attains_sup:
assumes "compact s" "s ≠ {}"
shows "∃x∈s. ∀y∈s. dist a y ≤ dist a x"
proof (rule continuous_attains_sup [OF assms])
{
fix x
assume "x∈s"
have "(dist a ⤏ dist a x) (at x within s)"
by (intro tendsto_dist tendsto_const tendsto_ident_at)
}
then show "continuous_on s (dist a)"
unfolding continuous_on ..
qed
text ‹For \emph{minimal} distance, we only need closure, not compactness.›
lemma distance_attains_inf:
fixes a :: "'a::heine_borel"
assumes "closed s" and "s ≠ {}"
obtains x where "x∈s" "⋀y. y ∈ s ⟹ dist a x ≤ dist a y"
proof -
from assms obtain b where "b ∈ s" by auto
let ?B = "s ∩ cball a (dist b a)"
have "?B ≠ {}" using ‹b ∈ s›
by (auto simp: dist_commute)
moreover have "continuous_on ?B (dist a)"
by (auto intro!: continuous_at_imp_continuous_on continuous_dist continuous_ident continuous_const)
moreover have "compact ?B"
by (intro closed_Int_compact ‹closed s› compact_cball)
ultimately obtain x where "x ∈ ?B" "∀y∈?B. dist a x ≤ dist a y"
by (metis continuous_attains_inf)
with that show ?thesis by fastforce
qed
subsection ‹Cartesian products›
lemma bounded_Times:
assumes "bounded s" "bounded t"
shows "bounded (s × t)"
proof -
obtain x y a b where "∀z∈s. dist x z ≤ a" "∀z∈t. dist y z ≤ b"
using assms [unfolded bounded_def] by auto
then have "∀z∈s × t. dist (x, y) z ≤ sqrt (a⇧2 + b⇧2)"
by (auto simp: dist_Pair_Pair real_sqrt_le_mono add_mono power_mono)
then show ?thesis unfolding bounded_any_center [where a="(x, y)"] by auto
qed
lemma mem_Times_iff: "x ∈ A × B ⟷ fst x ∈ A ∧ snd x ∈ B"
by (induct x) simp
lemma seq_compact_Times: "seq_compact s ⟹ seq_compact t ⟹ seq_compact (s × t)"
unfolding seq_compact_def
apply clarify
apply (drule_tac x="fst ∘ f" in spec)
apply (drule mp, simp add: mem_Times_iff)
apply (clarify, rename_tac l1 r1)
apply (drule_tac x="snd ∘ f ∘ r1" in spec)
apply (drule mp, simp add: mem_Times_iff)
apply (clarify, rename_tac l2 r2)
apply (rule_tac x="(l1, l2)" in rev_bexI, simp)
apply (rule_tac x="r1 ∘ r2" in exI)
apply (rule conjI, simp add: strict_mono_def)
apply (drule_tac f=r2 in LIMSEQ_subseq_LIMSEQ, assumption)
apply (drule (1) tendsto_Pair) back
apply (simp add: o_def)
done
lemma compact_Times:
assumes "compact s" "compact t"
shows "compact (s × t)"
proof (rule compactI)
fix C
assume C: "∀t∈C. open t" "s × t ⊆ ⋃C"
have "∀x∈s. ∃a. open a ∧ x ∈ a ∧ (∃d⊆C. finite d ∧ a × t ⊆ ⋃d)"
proof
fix x
assume "x ∈ s"
have "∀y∈t. ∃a b c. c ∈ C ∧ open a ∧ open b ∧ x ∈ a ∧ y ∈ b ∧ a × b ⊆ c" (is "∀y∈t. ?P y")
proof
fix y
assume "y ∈ t"
with ‹x ∈ s› C obtain c where "c ∈ C" "(x, y) ∈ c" "open c" by auto
then show "?P y" by (auto elim!: open_prod_elim)
qed
then obtain a b c where b: "⋀y. y ∈ t ⟹ open (b y)"
and c: "⋀y. y ∈ t ⟹ c y ∈ C ∧ open (a y) ∧ open (b y) ∧ x ∈ a y ∧ y ∈ b y ∧ a y × b y ⊆ c y"
by metis
then have "∀y∈t. open (b y)" "t ⊆ (⋃y∈t. b y)" by auto
with compactE_image[OF ‹compact t›] obtain D where D: "D ⊆ t" "finite D" "t ⊆ (⋃y∈D. b y)"
by metis
moreover from D c have "(⋂y∈D. a y) × t ⊆ (⋃y∈D. c y)"
by (fastforce simp: subset_eq)
ultimately show "∃a. open a ∧ x ∈ a ∧ (∃d⊆C. finite d ∧ a × t ⊆ ⋃d)"
using c by (intro exI[of _ "c`D"] exI[of _ "⋂(a`D)"] conjI) (auto intro!: open_INT)
qed
then obtain a d where a: "⋀x. x∈s ⟹ open (a x)" "s ⊆ (⋃x∈s. a x)"
and d: "⋀x. x ∈ s ⟹ d x ⊆ C ∧ finite (d x) ∧ a x × t ⊆ ⋃d x"
unfolding subset_eq UN_iff by metis
moreover
from compactE_image[OF ‹compact s› a]
obtain e where e: "e ⊆ s" "finite e" and s: "s ⊆ (⋃x∈e. a x)"
by auto
moreover
{
from s have "s × t ⊆ (⋃x∈e. a x × t)"
by auto
also have "… ⊆ (⋃x∈e. ⋃d x)"
using d ‹e ⊆ s› by (intro UN_mono) auto
finally have "s × t ⊆ (⋃x∈e. ⋃d x)" .
}
ultimately show "∃C'⊆C. finite C' ∧ s × t ⊆ ⋃C'"
by (intro exI[of _ "(⋃x∈e. d x)"]) (auto simp: subset_eq)
qed
text‹Hence some useful properties follow quite easily.›
lemma compact_scaling:
fixes s :: "'a::real_normed_vector set"
assumes "compact s"
shows "compact ((λx. c *⇩R x) ` s)"
proof -
let ?f = "λx. scaleR c x"
have *: "bounded_linear ?f" by (rule bounded_linear_scaleR_right)
show ?thesis
using compact_continuous_image[of s ?f] continuous_at_imp_continuous_on[of s ?f]
using linear_continuous_at[OF *] assms
by auto
qed
lemma compact_negations:
fixes s :: "'a::real_normed_vector set"
assumes "compact s"
shows "compact ((λx. - x) ` s)"
using compact_scaling [OF assms, of "- 1"] by auto
lemma compact_sums:
fixes s t :: "'a::real_normed_vector set"
assumes "compact s"
and "compact t"
shows "compact {x + y | x y. x ∈ s ∧ y ∈ t}"
proof -
have *: "{x + y | x y. x ∈ s ∧ y ∈ t} = (λz. fst z + snd z) ` (s × t)"
apply auto
unfolding image_iff
apply (rule_tac x="(xa, y)" in bexI)
apply auto
done
have "continuous_on (s × t) (λz. fst z + snd z)"
unfolding continuous_on by (rule ballI) (intro tendsto_intros)
then show ?thesis
unfolding * using compact_continuous_image compact_Times [OF assms] by auto
qed
lemma compact_differences:
fixes s t :: "'a::real_normed_vector set"
assumes "compact s"
and "compact t"
shows "compact {x - y | x y. x ∈ s ∧ y ∈ t}"
proof-
have "{x - y | x y. x∈s ∧ y ∈ t} = {x + y | x y. x ∈ s ∧ y ∈ (uminus ` t)}"
apply auto
apply (rule_tac x= xa in exI, auto)
done
then show ?thesis
using compact_sums[OF assms(1) compact_negations[OF assms(2)]] by auto
qed
lemma compact_translation:
fixes s :: "'a::real_normed_vector set"
assumes "compact s"
shows "compact ((λx. a + x) ` s)"
proof -
have "{x + y |x y. x ∈ s ∧ y ∈ {a}} = (λx. a + x) ` s"
by auto
then show ?thesis
using compact_sums[OF assms compact_sing[of a]] by auto
qed
lemma compact_affinity:
fixes s :: "'a::real_normed_vector set"
assumes "compact s"
shows "compact ((λx. a + c *⇩R x) ` s)"
proof -
have "op + a ` op *⇩R c ` s = (λx. a + c *⇩R x) ` s"
by auto
then show ?thesis
using compact_translation[OF compact_scaling[OF assms], of a c] by auto
qed
text ‹Hence we get the following.›
lemma compact_sup_maxdistance:
fixes s :: "'a::metric_space set"
assumes "compact s"
and "s ≠ {}"
shows "∃x∈s. ∃y∈s. ∀u∈s. ∀v∈s. dist u v ≤ dist x y"
proof -
have "compact (s × s)"
using ‹compact s› by (intro compact_Times)
moreover have "s × s ≠ {}"
using ‹s ≠ {}› by auto
moreover have "continuous_on (s × s) (λx. dist (fst x) (snd x))"
by (intro continuous_at_imp_continuous_on ballI continuous_intros)
ultimately show ?thesis
using continuous_attains_sup[of "s × s" "λx. dist (fst x) (snd x)"] by auto
qed
subsection ‹The diameter of a set.›
definition diameter :: "'a::metric_space set ⇒ real" where
"diameter S = (if S = {} then 0 else SUP (x,y):S×S. dist x y)"
lemma diameter_empty [simp]: "diameter{} = 0"
by (auto simp: diameter_def)
lemma diameter_singleton [simp]: "diameter{x} = 0"
by (auto simp: diameter_def)
lemma diameter_le:
assumes "S ≠ {} ∨ 0 ≤ d"
and no: "⋀x y. ⟦x ∈ S; y ∈ S⟧ ⟹ norm(x - y) ≤ d"
shows "diameter S ≤ d"
using assms
by (auto simp: dist_norm diameter_def intro: cSUP_least)
lemma diameter_bounded_bound:
fixes s :: "'a :: metric_space set"
assumes s: "bounded s" "x ∈ s" "y ∈ s"
shows "dist x y ≤ diameter s"
proof -
from s obtain z d where z: "⋀x. x ∈ s ⟹ dist z x ≤ d"
unfolding bounded_def by auto
have "bdd_above (case_prod dist ` (s×s))"
proof (intro bdd_aboveI, safe)
fix a b
assume "a ∈ s" "b ∈ s"
with z[of a] z[of b] dist_triangle[of a b z]
show "dist a b ≤ 2 * d"
by (simp add: dist_commute)
qed
moreover have "(x,y) ∈ s×s" using s by auto
ultimately have "dist x y ≤ (SUP (x,y):s×s. dist x y)"
by (rule cSUP_upper2) simp
with ‹x ∈ s› show ?thesis
by (auto simp: diameter_def)
qed
lemma diameter_lower_bounded:
fixes s :: "'a :: metric_space set"
assumes s: "bounded s"
and d: "0 < d" "d < diameter s"
shows "∃x∈s. ∃y∈s. d < dist x y"
proof (rule ccontr)
assume contr: "¬ ?thesis"
moreover have "s ≠ {}"
using d by (auto simp: diameter_def)
ultimately have "diameter s ≤ d"
by (auto simp: not_less diameter_def intro!: cSUP_least)
with ‹d < diameter s› show False by auto
qed
lemma diameter_bounded:
assumes "bounded s"
shows "∀x∈s. ∀y∈s. dist x y ≤ diameter s"
and "∀d>0. d < diameter s ⟶ (∃x∈s. ∃y∈s. dist x y > d)"
using diameter_bounded_bound[of s] diameter_lower_bounded[of s] assms
by auto
lemma diameter_compact_attained:
assumes "compact s"
and "s ≠ {}"
shows "∃x∈s. ∃y∈s. dist x y = diameter s"
proof -
have b: "bounded s" using assms(1)
by (rule compact_imp_bounded)
then obtain x y where xys: "x∈s" "y∈s"
and xy: "∀u∈s. ∀v∈s. dist u v ≤ dist x y"
using compact_sup_maxdistance[OF assms] by auto
then have "diameter s ≤ dist x y"
unfolding diameter_def
apply clarsimp
apply (rule cSUP_least, fast+)
done
then show ?thesis
by (metis b diameter_bounded_bound order_antisym xys)
qed
lemma diameter_ge_0:
assumes "bounded S" shows "0 ≤ diameter S"
by (metis all_not_in_conv assms diameter_bounded_bound diameter_empty dist_self order_refl)
lemma diameter_subset:
assumes "S ⊆ T" "bounded T"
shows "diameter S ≤ diameter T"
proof (cases "S = {} ∨ T = {}")
case True
with assms show ?thesis
by (force simp: diameter_ge_0)
next
case False
then have "bdd_above ((λx. case x of (x, xa) ⇒ dist x xa) ` (T × T))"
using ‹bounded T› diameter_bounded_bound by (force simp: bdd_above_def)
with False ‹S ⊆ T› show ?thesis
apply (simp add: diameter_def)
apply (rule cSUP_subset_mono, auto)
done
qed
lemma diameter_closure:
assumes "bounded S"
shows "diameter(closure S) = diameter S"
proof (rule order_antisym)
have "False" if "diameter S < diameter (closure S)"
proof -
define d where "d = diameter(closure S) - diameter(S)"
have "d > 0"
using that by (simp add: d_def)
then have "diameter(closure(S)) - d / 2 < diameter(closure(S))"
by simp
have dd: "diameter (closure S) - d / 2 = (diameter(closure(S)) + diameter(S)) / 2"
by (simp add: d_def divide_simps)
have bocl: "bounded (closure S)"
using assms by blast
moreover have "0 ≤ diameter S"
using assms diameter_ge_0 by blast
ultimately obtain x y where "x ∈ closure S" "y ∈ closure S" and xy: "diameter(closure(S)) - d / 2 < dist x y"
using diameter_bounded(2) [OF bocl, rule_format, of "diameter(closure(S)) - d / 2"] ‹d > 0› d_def by auto
then obtain x' y' where x'y': "x' ∈ S" "dist x' x < d/4" "y' ∈ S" "dist y' y < d/4"
using closure_approachable
by (metis ‹0 < d› zero_less_divide_iff zero_less_numeral)
then have "dist x' y' ≤ diameter S"
using assms diameter_bounded_bound by blast
with x'y' have "dist x y ≤ d / 4 + diameter S + d / 4"
by (meson add_mono_thms_linordered_semiring(1) dist_triangle dist_triangle3 less_eq_real_def order_trans)
then show ?thesis
using xy d_def by linarith
qed
then show "diameter (closure S) ≤ diameter S"
by fastforce
next
show "diameter S ≤ diameter (closure S)"
by (simp add: assms bounded_closure closure_subset diameter_subset)
qed
lemma diameter_cball [simp]:
fixes a :: "'a::euclidean_space"
shows "diameter(cball a r) = (if r < 0 then 0 else 2*r)"
proof -
have "diameter(cball a r) = 2*r" if "r ≥ 0"
proof (rule order_antisym)
show "diameter (cball a r) ≤ 2*r"
proof (rule diameter_le)
fix x y assume "x ∈ cball a r" "y ∈ cball a r"
then have "norm (x - a) ≤ r" "norm (a - y) ≤ r"
by (auto simp: dist_norm norm_minus_commute)
then have "norm (x - y) ≤ r+r"
using norm_diff_triangle_le by blast
then show "norm (x - y) ≤ 2*r" by simp
qed (simp add: that)
have "2*r = dist (a + r *⇩R (SOME i. i ∈ Basis)) (a - r *⇩R (SOME i. i ∈ Basis))"
apply (simp add: dist_norm)
by (metis abs_of_nonneg mult.right_neutral norm_numeral norm_scaleR norm_some_Basis real_norm_def scaleR_2 that)
also have "... ≤ diameter (cball a r)"
apply (rule diameter_bounded_bound)
using that by (auto simp: dist_norm)
finally show "2*r ≤ diameter (cball a r)" .
qed
then show ?thesis by simp
qed
lemma diameter_ball [simp]:
fixes a :: "'a::euclidean_space"
shows "diameter(ball a r) = (if r < 0 then 0 else 2*r)"
proof -
have "diameter(ball a r) = 2*r" if "r > 0"
by (metis bounded_ball diameter_closure closure_ball diameter_cball less_eq_real_def linorder_not_less that)
then show ?thesis
by (simp add: diameter_def)
qed
lemma diameter_closed_interval [simp]: "diameter {a..b} = (if b < a then 0 else b-a)"
proof -
have "{a .. b} = cball ((a+b)/2) ((b-a)/2)"
by (auto simp: dist_norm abs_if divide_simps split: if_split_asm)
then show ?thesis
by simp
qed
lemma diameter_open_interval [simp]: "diameter {a<..<b} = (if b < a then 0 else b-a)"
proof -
have "{a <..< b} = ball ((a+b)/2) ((b-a)/2)"
by (auto simp: dist_norm abs_if divide_simps split: if_split_asm)
then show ?thesis
by simp
qed
proposition Lebesgue_number_lemma:
assumes "compact S" "𝒞 ≠ {}" "S ⊆ ⋃𝒞" and ope: "⋀B. B ∈ 𝒞 ⟹ open B"
obtains δ where "0 < δ" "⋀T. ⟦T ⊆ S; diameter T < δ⟧ ⟹ ∃B ∈ 𝒞. T ⊆ B"
proof (cases "S = {}")
case True
then show ?thesis
by (metis ‹𝒞 ≠ {}› zero_less_one empty_subsetI equals0I subset_trans that)
next
case False
{ fix x assume "x ∈ S"
then obtain C where C: "x ∈ C" "C ∈ 𝒞"
using ‹S ⊆ ⋃𝒞› by blast
then obtain r where r: "r>0" "ball x (2*r) ⊆ C"
by (metis mult.commute mult_2_right not_le ope openE real_sum_of_halves zero_le_numeral zero_less_mult_iff)
then have "∃r C. r > 0 ∧ ball x (2*r) ⊆ C ∧ C ∈ 𝒞"
using C by blast
}
then obtain r where r: "⋀x. x ∈ S ⟹ r x > 0 ∧ (∃C ∈ 𝒞. ball x (2*r x) ⊆ C)"
by metis
then have "S ⊆ (⋃x ∈ S. ball x (r x))"
by auto
then obtain 𝒯 where "finite 𝒯" "S ⊆ ⋃𝒯" and 𝒯: "𝒯 ⊆ (λx. ball x (r x)) ` S"
by (rule compactE [OF ‹compact S›]) auto
then obtain S0 where "S0 ⊆ S" "finite S0" and S0: "𝒯 = (λx. ball x (r x)) ` S0"
by (meson finite_subset_image)
then have "S0 ≠ {}"
using False ‹S ⊆ ⋃𝒯› by auto
define δ where "δ = Inf (r ` S0)"
have "δ > 0"
using ‹finite S0› ‹S0 ⊆ S› ‹S0 ≠ {}› r by (auto simp: δ_def finite_less_Inf_iff)
show ?thesis
proof
show "0 < δ"
by (simp add: ‹0 < δ›)
show "∃B ∈ 𝒞. T ⊆ B" if "T ⊆ S" and dia: "diameter T < δ" for T
proof (cases "T = {}")
case True
then show ?thesis
using ‹𝒞 ≠ {}› by blast
next
case False
then obtain y where "y ∈ T" by blast
then have "y ∈ S"
using ‹T ⊆ S› by auto
then obtain x where "x ∈ S0" and x: "y ∈ ball x (r x)"
using ‹S ⊆ ⋃𝒯› S0 that by blast
have "ball y δ ⊆ ball y (r x)"
by (metis δ_def ‹S0 ≠ {}› ‹finite S0› ‹x ∈ S0› empty_is_image finite_imageI finite_less_Inf_iff imageI less_irrefl not_le subset_ball)
also have "... ⊆ ball x (2*r x)"
by clarsimp (metis dist_commute dist_triangle_less_add mem_ball mult_2 x)
finally obtain C where "C ∈ 𝒞" "ball y δ ⊆ C"
by (meson r ‹S0 ⊆ S› ‹x ∈ S0› dual_order.trans subsetCE)
have "bounded T"
using ‹compact S› bounded_subset compact_imp_bounded ‹T ⊆ S› by blast
then have "T ⊆ ball y δ"
using ‹y ∈ T› dia diameter_bounded_bound by fastforce
then show ?thesis
apply (rule_tac x=C in bexI)
using ‹ball y δ ⊆ C› ‹C ∈ 𝒞› by auto
qed
qed
qed
subsection ‹Compact sets and the closure operation.›
lemma closed_scaling:
fixes S :: "'a::real_normed_vector set"
assumes "closed S"
shows "closed ((λx. c *⇩R x) ` S)"
proof (cases "c = 0")
case True then show ?thesis
by (auto simp: image_constant_conv)
next
case False
from assms have "closed ((λx. inverse c *⇩R x) -` S)"
by (simp add: continuous_closed_vimage)
also have "(λx. inverse c *⇩R x) -` S = (λx. c *⇩R x) ` S"
using ‹c ≠ 0› by (auto elim: image_eqI [rotated])
finally show ?thesis .
qed
lemma closed_negations:
fixes S :: "'a::real_normed_vector set"
assumes "closed S"
shows "closed ((λx. -x) ` S)"
using closed_scaling[OF assms, of "- 1"] by simp
lemma compact_closed_sums:
fixes S :: "'a::real_normed_vector set"
assumes "compact S" and "closed T"
shows "closed (⋃x∈ S. ⋃y ∈ T. {x + y})"
proof -
let ?S = "{x + y |x y. x ∈ S ∧ y ∈ T}"
{
fix x l
assume as: "∀n. x n ∈ ?S" "(x ⤏ l) sequentially"
from as(1) obtain f where f: "∀n. x n = fst (f n) + snd (f n)" "∀n. fst (f n) ∈ S" "∀n. snd (f n) ∈ T"
using choice[of "λn y. x n = (fst y) + (snd y) ∧ fst y ∈ S ∧ snd y ∈ T"] by auto
obtain l' r where "l'∈S" and r: "strict_mono r" and lr: "(((λn. fst (f n)) ∘ r) ⤏ l') sequentially"
using assms(1)[unfolded compact_def, THEN spec[where x="λ n. fst (f n)"]] using f(2) by auto
have "((λn. snd (f (r n))) ⤏ l - l') sequentially"
using tendsto_diff[OF LIMSEQ_subseq_LIMSEQ[OF as(2) r] lr] and f(1)
unfolding o_def
by auto
then have "l - l' ∈ T"
using assms(2)[unfolded closed_sequential_limits,
THEN spec[where x="λ n. snd (f (r n))"],
THEN spec[where x="l - l'"]]
using f(3)
by auto
then have "l ∈ ?S"
using ‹l' ∈ S›
apply auto
apply (rule_tac x=l' in exI)
apply (rule_tac x="l - l'" in exI, auto)
done
}
moreover have "?S = (⋃x∈ S. ⋃y ∈ T. {x + y})"
by force
ultimately show ?thesis
unfolding closed_sequential_limits
by (metis (no_types, lifting))
qed
lemma closed_compact_sums:
fixes S T :: "'a::real_normed_vector set"
assumes "closed S" "compact T"
shows "closed (⋃x∈ S. ⋃y ∈ T. {x + y})"
proof -
have "(⋃x∈ T. ⋃y ∈ S. {x + y}) = (⋃x∈ S. ⋃y ∈ T. {x + y})"
by auto
then show ?thesis
using compact_closed_sums[OF assms(2,1)] by simp
qed
lemma compact_closed_differences:
fixes S T :: "'a::real_normed_vector set"
assumes "compact S" "closed T"
shows "closed (⋃x∈ S. ⋃y ∈ T. {x - y})"
proof -
have "(⋃x∈ S. ⋃y ∈ uminus ` T. {x + y}) = (⋃x∈ S. ⋃y ∈ T. {x - y})"
by force
then show ?thesis
using compact_closed_sums[OF assms(1) closed_negations[OF assms(2)]] by auto
qed
lemma closed_compact_differences:
fixes S T :: "'a::real_normed_vector set"
assumes "closed S" "compact T"
shows "closed (⋃x∈ S. ⋃y ∈ T. {x - y})"
proof -
have "(⋃x∈ S. ⋃y ∈ uminus ` T. {x + y}) = {x - y |x y. x ∈ S ∧ y ∈ T}"
by auto
then show ?thesis
using closed_compact_sums[OF assms(1) compact_negations[OF assms(2)]] by simp
qed
lemma closed_translation:
fixes a :: "'a::real_normed_vector"
assumes "closed S"
shows "closed ((λx. a + x) ` S)"
proof -
have "(⋃x∈ {a}. ⋃y ∈ S. {x + y}) = (op + a ` S)" by auto
then show ?thesis
using compact_closed_sums[OF compact_sing[of a] assms] by auto
qed
lemma translation_Compl:
fixes a :: "'a::ab_group_add"
shows "(λx. a + x) ` (- t) = - ((λx. a + x) ` t)"
apply (auto simp: image_iff)
apply (rule_tac x="x - a" in bexI, auto)
done
lemma translation_UNIV:
fixes a :: "'a::ab_group_add"
shows "range (λx. a + x) = UNIV"
apply (auto simp: image_iff)
apply (rule_tac x="x - a" in exI, auto)
done
lemma translation_diff:
fixes a :: "'a::ab_group_add"
shows "(λx. a + x) ` (s - t) = ((λx. a + x) ` s) - ((λx. a + x) ` t)"
by auto
lemma translation_Int:
fixes a :: "'a::ab_group_add"
shows "(λx. a + x) ` (s ∩ t) = ((λx. a + x) ` s) ∩ ((λx. a + x) ` t)"
by auto
lemma closure_translation:
fixes a :: "'a::real_normed_vector"
shows "closure ((λx. a + x) ` s) = (λx. a + x) ` (closure s)"
proof -
have *: "op + a ` (- s) = - op + a ` s"
apply auto
unfolding image_iff
apply (rule_tac x="x - a" in bexI, auto)
done
show ?thesis
unfolding closure_interior translation_Compl
using interior_translation[of a "- s"]
unfolding *
by auto
qed
lemma frontier_translation:
fixes a :: "'a::real_normed_vector"
shows "frontier((λx. a + x) ` s) = (λx. a + x) ` (frontier s)"
unfolding frontier_def translation_diff interior_translation closure_translation
by auto
lemma sphere_translation:
fixes a :: "'n::euclidean_space"
shows "sphere (a+c) r = op+a ` sphere c r"
apply safe
apply (rule_tac x="x-a" in image_eqI)
apply (auto simp: dist_norm algebra_simps)
done
lemma cball_translation:
fixes a :: "'n::euclidean_space"
shows "cball (a+c) r = op+a ` cball c r"
apply safe
apply (rule_tac x="x-a" in image_eqI)
apply (auto simp: dist_norm algebra_simps)
done
lemma ball_translation:
fixes a :: "'n::euclidean_space"
shows "ball (a+c) r = op+a ` ball c r"
apply safe
apply (rule_tac x="x-a" in image_eqI)
apply (auto simp: dist_norm algebra_simps)
done
subsection ‹Separation between points and sets›
lemma separate_point_closed:
fixes s :: "'a::heine_borel set"
assumes "closed s" and "a ∉ s"
shows "∃d>0. ∀x∈s. d ≤ dist a x"
proof (cases "s = {}")
case True
then show ?thesis by(auto intro!: exI[where x=1])
next
case False
from assms obtain x where "x∈s" "∀y∈s. dist a x ≤ dist a y"
using ‹s ≠ {}› by (blast intro: distance_attains_inf [of s a])
with ‹x∈s› show ?thesis using dist_pos_lt[of a x] and‹a ∉ s›
by blast
qed
lemma separate_compact_closed:
fixes s t :: "'a::heine_borel set"
assumes "compact s"
and t: "closed t" "s ∩ t = {}"
shows "∃d>0. ∀x∈s. ∀y∈t. d ≤ dist x y"
proof cases
assume "s ≠ {} ∧ t ≠ {}"
then have "s ≠ {}" "t ≠ {}" by auto
let ?inf = "λx. infdist x t"
have "continuous_on s ?inf"
by (auto intro!: continuous_at_imp_continuous_on continuous_infdist continuous_ident)
then obtain x where x: "x ∈ s" "∀y∈s. ?inf x ≤ ?inf y"
using continuous_attains_inf[OF ‹compact s› ‹s ≠ {}›] by auto
then have "0 < ?inf x"
using t ‹t ≠ {}› in_closed_iff_infdist_zero by (auto simp: less_le infdist_nonneg)
moreover have "∀x'∈s. ∀y∈t. ?inf x ≤ dist x' y"
using x by (auto intro: order_trans infdist_le)
ultimately show ?thesis by auto
qed (auto intro!: exI[of _ 1])
lemma separate_closed_compact:
fixes s t :: "'a::heine_borel set"
assumes "closed s"
and "compact t"
and "s ∩ t = {}"
shows "∃d>0. ∀x∈s. ∀y∈t. d ≤ dist x y"
proof -
have *: "t ∩ s = {}"
using assms(3) by auto
show ?thesis
using separate_compact_closed[OF assms(2,1) *]
apply auto
apply (rule_tac x=d in exI, auto)
apply (erule_tac x=y in ballE)
apply (auto simp: dist_commute)
done
qed
subsection ‹Closure of halfspaces and hyperplanes›
lemma isCont_open_vimage:
assumes "⋀x. isCont f x"
and "open s"
shows "open (f -` s)"
proof -
from assms(1) have "continuous_on UNIV f"
unfolding isCont_def continuous_on_def by simp
then have "open {x ∈ UNIV. f x ∈ s}"
using open_UNIV ‹open s› by (rule continuous_open_preimage)
then show "open (f -` s)"
by (simp add: vimage_def)
qed
lemma isCont_closed_vimage:
assumes "⋀x. isCont f x"
and "closed s"
shows "closed (f -` s)"
using assms unfolding closed_def vimage_Compl [symmetric]
by (rule isCont_open_vimage)
lemma continuous_on_closed_Collect_le:
fixes f g :: "'a::t2_space ⇒ real"
assumes f: "continuous_on s f" and g: "continuous_on s g" and s: "closed s"
shows "closed {x ∈ s. f x ≤ g x}"
proof -
have "closed ((λx. g x - f x) -` {0..} ∩ s)"
using closed_real_atLeast continuous_on_diff [OF g f]
by (simp add: continuous_on_closed_vimage [OF s])
also have "((λx. g x - f x) -` {0..} ∩ s) = {x∈s. f x ≤ g x}"
by auto
finally show ?thesis .
qed
lemma continuous_at_inner: "continuous (at x) (inner a)"
unfolding continuous_at by (intro tendsto_intros)
lemma closed_halfspace_le: "closed {x. inner a x ≤ b}"
by (simp add: closed_Collect_le continuous_on_inner continuous_on_const continuous_on_id)
lemma closed_halfspace_ge: "closed {x. inner a x ≥ b}"
by (simp add: closed_Collect_le continuous_on_inner continuous_on_const continuous_on_id)
lemma closed_hyperplane: "closed {x. inner a x = b}"
by (simp add: closed_Collect_eq continuous_on_inner continuous_on_const continuous_on_id)
lemma closed_halfspace_component_le: "closed {x::'a::euclidean_space. x∙i ≤ a}"
by (simp add: closed_Collect_le continuous_on_inner continuous_on_const continuous_on_id)
lemma closed_halfspace_component_ge: "closed {x::'a::euclidean_space. x∙i ≥ a}"
by (simp add: closed_Collect_le continuous_on_inner continuous_on_const continuous_on_id)
lemma closed_interval_left:
fixes b :: "'a::euclidean_space"
shows "closed {x::'a. ∀i∈Basis. x∙i ≤ b∙i}"
by (simp add: Collect_ball_eq closed_INT closed_Collect_le continuous_on_inner continuous_on_const continuous_on_id)
lemma closed_interval_right:
fixes a :: "'a::euclidean_space"
shows "closed {x::'a. ∀i∈Basis. a∙i ≤ x∙i}"
by (simp add: Collect_ball_eq closed_INT closed_Collect_le continuous_on_inner continuous_on_const continuous_on_id)
lemma continuous_le_on_closure:
fixes a::real
assumes f: "continuous_on (closure s) f"
and x: "x ∈ closure(s)"
and xlo: "⋀x. x ∈ s ==> f(x) ≤ a"
shows "f(x) ≤ a"
using image_closure_subset [OF f]
using image_closure_subset [OF f] closed_halfspace_le [of "1::real" a] assms
by force
lemma continuous_ge_on_closure:
fixes a::real
assumes f: "continuous_on (closure s) f"
and x: "x ∈ closure(s)"
and xlo: "⋀x. x ∈ s ==> f(x) ≥ a"
shows "f(x) ≥ a"
using image_closure_subset [OF f] closed_halfspace_ge [of a "1::real"] assms
by force
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 Lim_component_le:
fixes f :: "'a ⇒ 'b::euclidean_space"
assumes "(f ⤏ l) net"
and "¬ (trivial_limit net)"
and "eventually (λx. f(x)∙i ≤ b) net"
shows "l∙i ≤ b"
by (rule tendsto_le[OF assms(2) tendsto_const tendsto_inner[OF assms(1) tendsto_const] assms(3)])
lemma Lim_component_ge:
fixes f :: "'a ⇒ 'b::euclidean_space"
assumes "(f ⤏ l) net"
and "¬ (trivial_limit net)"
and "eventually (λx. b ≤ (f x)∙i) net"
shows "b ≤ l∙i"
by (rule tendsto_le[OF assms(2) tendsto_inner[OF assms(1) tendsto_const] tendsto_const assms(3)])
lemma Lim_component_eq:
fixes f :: "'a ⇒ 'b::euclidean_space"
assumes net: "(f ⤏ l) net" "¬ trivial_limit net"
and ev:"eventually (λx. f(x)∙i = b) net"
shows "l∙i = b"
using ev[unfolded order_eq_iff eventually_conj_iff]
using Lim_component_ge[OF net, of b i]
using Lim_component_le[OF net, of i b]
by auto
text ‹Limits relative to a union.›
lemma eventually_within_Un:
"eventually P (at x within (s ∪ t)) ⟷
eventually P (at x within s) ∧ eventually P (at x within t)"
unfolding eventually_at_filter
by (auto elim!: eventually_rev_mp)
lemma Lim_within_union:
"(f ⤏ l) (at x within (s ∪ t)) ⟷
(f ⤏ l) (at x within s) ∧ (f ⤏ l) (at x within t)"
unfolding tendsto_def
by (auto simp: eventually_within_Un)
lemma Lim_topological:
"(f ⤏ l) net ⟷
trivial_limit net ∨ (∀S. open S ⟶ l ∈ S ⟶ eventually (λx. f x ∈ S) net)"
unfolding tendsto_def trivial_limit_eq by auto
text ‹Continuity relative to a union.›
lemma continuous_on_Un_local:
"⟦closedin (subtopology euclidean (s ∪ t)) s; closedin (subtopology euclidean (s ∪ t)) t;
continuous_on s f; continuous_on t f⟧
⟹ continuous_on (s ∪ t) f"
unfolding continuous_on closedin_limpt
by (metis Lim_trivial_limit Lim_within_union Un_iff trivial_limit_within)
lemma continuous_on_cases_local:
"⟦closedin (subtopology euclidean (s ∪ t)) s; closedin (subtopology euclidean (s ∪ t)) t;
continuous_on s f; continuous_on t g;
⋀x. ⟦x ∈ s ∧ ~P x ∨ x ∈ t ∧ P x⟧ ⟹ f x = g x⟧
⟹ continuous_on (s ∪ t) (λx. if P x then f x else g x)"
by (rule continuous_on_Un_local) (auto intro: continuous_on_eq)
lemma continuous_on_cases_le:
fixes h :: "'a :: topological_space ⇒ real"
assumes "continuous_on {t ∈ s. h t ≤ a} f"
and "continuous_on {t ∈ s. a ≤ h t} g"
and h: "continuous_on s h"
and "⋀t. ⟦t ∈ s; h t = a⟧ ⟹ f t = g t"
shows "continuous_on s (λt. if h t ≤ a then f(t) else g(t))"
proof -
have s: "s = {t ∈ s. h t ∈ atMost a} ∪ {t ∈ s. h t ∈ atLeast a}"
by force
have 1: "closedin (subtopology euclidean s) {t ∈ s. h t ∈ atMost a}"
by (rule continuous_closedin_preimage [OF h closed_atMost])
have 2: "closedin (subtopology euclidean s) {t ∈ s. h t ∈ atLeast a}"
by (rule continuous_closedin_preimage [OF h closed_atLeast])
show ?thesis
apply (rule continuous_on_subset [of s, OF _ order_refl])
apply (subst s)
apply (rule continuous_on_cases_local)
using 1 2 s assms apply auto
done
qed
lemma continuous_on_cases_1:
fixes s :: "real set"
assumes "continuous_on {t ∈ s. t ≤ a} f"
and "continuous_on {t ∈ s. a ≤ t} g"
and "a ∈ s ⟹ f a = g a"
shows "continuous_on s (λt. if t ≤ a then f(t) else g(t))"
using assms
by (auto simp: continuous_on_id intro: continuous_on_cases_le [where h = id, simplified])
text‹Some more convenient intermediate-value theorem formulations.›
lemma connected_ivt_hyperplane:
assumes "connected s"
and "x ∈ s"
and "y ∈ s"
and "inner a x ≤ b"
and "b ≤ inner a y"
shows "∃z ∈ s. inner a z = b"
proof (rule ccontr)
assume as:"¬ (∃z∈s. inner a z = b)"
let ?A = "{x. inner a x < b}"
let ?B = "{x. inner a x > b}"
have "open ?A" "open ?B"
using open_halfspace_lt and open_halfspace_gt by auto
moreover
have "?A ∩ ?B = {}" by auto
moreover
have "s ⊆ ?A ∪ ?B" using as by auto
ultimately
show False
using assms(1)[unfolded connected_def not_ex,
THEN spec[where x="?A"], THEN spec[where x="?B"]]
using assms(2-5)
by auto
qed
lemma connected_ivt_component:
fixes x::"'a::euclidean_space"
shows "connected s ⟹
x ∈ s ⟹ y ∈ s ⟹
x∙k ≤ a ⟹ a ≤ y∙k ⟹ (∃z∈s. z∙k = a)"
using connected_ivt_hyperplane[of s x y "k::'a" a]
by (auto simp: inner_commute)
subsection ‹Intervals›
lemma open_box[intro]: "open (box a b)"
proof -
have "open (⋂i∈Basis. (op ∙ i) -` {a ∙ i <..< b ∙ i})"
by (auto intro!: continuous_open_vimage continuous_inner continuous_ident continuous_const)
also have "(⋂i∈Basis. (op ∙ 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 x: "∀i∈Basis. a ∙ i ≤ x ∙ i ∧ x ∙ i ≤ b ∙ i"
{
fix i :: 'a
assume "i ∈ Basis"
then have "¦x∙i¦ ≤ ¦a∙i¦ + ¦b∙i¦"
using x[THEN bspec[where x=i]] by auto
}
then have "(∑i∈Basis. ¦x ∙ i¦) ≤ ?b"
apply -
apply (rule sum_mono, auto)
done
then have "norm x ≤ ?b"
using norm_le_l1[of x] by auto
}
then show ?thesis
unfolding cbox_def bounded_iff by auto
qed
lemma bounded_box [simp]:
fixes a :: "'a::euclidean_space"
shows "bounded (box a b)"
using bounded_cbox[of a b]
using box_subset_cbox[of a b]
using 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)
proposition is_interval_compact:
"is_interval S ∧ compact S ⟷ (∃a b. S = cbox a b)" (is "?lhs = ?rhs")
proof (cases "S = {}")
case True
with empty_as_interval show ?thesis by auto
next
case False
show ?thesis
proof
assume L: ?lhs
then have "is_interval S" "compact S" by auto
define a where "a ≡ ∑i∈Basis. (INF x:S. x ∙ i) *⇩R i"
define b where "b ≡ ∑i∈Basis. (SUP x:S. x ∙ i) *⇩R i"
have 1: "⋀x i. ⟦x ∈ S; i ∈ Basis⟧ ⟹ (INF x:S. x ∙ i) ≤ x ∙ i"
by (simp add: cInf_lower bounded_inner_imp_bdd_below compact_imp_bounded L)
have 2: "⋀x i. ⟦x ∈ S; i ∈ Basis⟧ ⟹ x ∙ i ≤ (SUP x:S. x ∙ i)"
by (simp add: cSup_upper bounded_inner_imp_bdd_above compact_imp_bounded L)
have 3: "x ∈ S" if inf: "⋀i. i ∈ Basis ⟹ (INF x:S. x ∙ i) ≤ x ∙ i"
and sup: "⋀i. i ∈ Basis ⟹ x ∙ i ≤ (SUP x:S. x ∙ i)" for x
proof (rule mem_box_componentwiseI [OF ‹is_interval S›])
fix i::'a
assume i: "i ∈ Basis"
have cont: "continuous_on S (λx. x ∙ i)"
by (intro continuous_intros)
obtain a where "a ∈ S" and a: "⋀y. y∈S ⟹ a ∙ i ≤ y ∙ i"
using continuous_attains_inf [OF ‹compact S› False cont] by blast
obtain b where "b ∈ S" and b: "⋀y. y∈S ⟹ y ∙ i ≤ b ∙ i"
using continuous_attains_sup [OF ‹compact S› False cont] by blast
have "a ∙ i ≤ (INF x:S. x ∙ i)"
by (simp add: False a cINF_greatest)
also have "… ≤ x ∙ i"
by (simp add: i inf)
finally have ai: "a ∙ i ≤ x ∙ i" .
have "x ∙ i ≤ (SUP x:S. x ∙ i)"
by (simp add: i sup)
also have "(SUP x:S. x ∙ i) ≤ b ∙ i"
by (simp add: False b cSUP_least)
finally have bi: "x ∙ i ≤ b ∙ i" .
show "x ∙ i ∈ (λx. x ∙ i) ` S"
apply (rule_tac x="∑j∈Basis. (if j = i then x ∙ i else a ∙ j) *⇩R j" in image_eqI)
apply (simp add: i)
apply (rule mem_is_intervalI [OF ‹is_interval S› ‹a ∈ S› ‹b ∈ S›])
using i ai bi apply force
done
qed
have "S = cbox a b"
by (auto simp: a_def b_def mem_box intro: 1 2 3)
then show ?rhs
by blast
next
assume R: ?rhs
then show ?lhs
using compact_cbox is_interval_cbox by blast
qed
qed
lemma box_midpoint:
fixes a :: "'a::euclidean_space"
assumes "box a b ≠ {}"
shows "((1/2) *⇩R (a + b)) ∈ box a b"
proof -
{
fix i :: 'a
assume "i ∈ Basis"
then have "a ∙ i < ((1 / 2) *⇩R (a + b)) ∙ i ∧ ((1 / 2) *⇩R (a + b)) ∙ i < b ∙ i"
using assms[unfolded box_ne_empty, THEN bspec[where x=i]] by (auto simp: inner_add_left)
}
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)"
apply (rule add_less_le_mono)
using e unfolding mult_less_cancel_left and mult_le_cancel_left
apply simp_all
using x unfolding mem_box using i
apply simp
using y unfolding mem_box using i
apply simp
done
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)"
apply (rule add_less_le_mono)
using e unfolding mult_less_cancel_left and mult_le_cancel_left
apply simp_all
using x
unfolding mem_box
using i
apply simp
using y
unfolding mem_box
using i
apply simp
done
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 have "∃N::nat. inverse (real (N + 1)) < e"
using real_arch_inverse[of e]
apply (auto simp: Suc_pred')
apply (metis Suc_pred' of_nat_Suc)
done
then obtain N :: nat where N: "inverse (real (N + 1)) < e"
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 and box_midpoint[OF assms]
unfolding closure_def
unfolding 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"
shows "∃a. 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)"
{
fix x
assume "x ∈ s"
fix i :: 'a
assume i: "i ∈ Basis"
then have "(-a)∙i < x∙i" and "x∙i < a∙i"
using b[THEN bspec[where x=x], OF ‹x∈s›]
using Basis_le_norm[OF i, of x]
unfolding inner_simps and a_def
by auto
}
then show ?thesis
by (auto intro: exI[where x=a] simp add: box_def)
qed
lemma bounded_subset_open_interval:
fixes s :: "('a::euclidean_space) set"
shows "bounded s ⟹ (∃a b. s ⊆ box a b)"
by (auto dest!: bounded_subset_box_symmetric)
lemma bounded_subset_cbox_symmetric:
fixes s :: "('a::euclidean_space) set"
assumes "bounded s"
shows "∃a. s ⊆ cbox (-a) a"
proof -
obtain a where "s ⊆ box (-a) a"
using bounded_subset_box_symmetric[OF assms] by auto
then show ?thesis
using box_subset_cbox[of "-a" a] by auto
qed
lemma bounded_subset_cbox:
fixes s :: "('a::euclidean_space) set"
shows "bounded s ⟹ ∃a b. s ⊆ cbox a b"
using bounded_subset_cbox_symmetric[of s] by auto
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 inter_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 diameter_cbox:
fixes a b::"'a::euclidean_space"
shows "(∀i ∈ Basis. a ∙ i ≤ b ∙ i) ⟹ diameter (cbox a b) = dist a b"
by (force simp: diameter_def intro!: cSup_eq_maximum setL2_mono
simp: euclidean_dist_l2[where 'a='a] cbox_def dist_norm)
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 eucl_le_eq_halfspaces:
fixes a :: "'a::euclidean_space"
shows "{x. ∀i∈Basis. x ∙ i ≤ a ∙ i} = (⋂i∈Basis. {x. x ∙ i ≤ a ∙ i})"
"{x. ∀i∈Basis. a ∙ i ≤ x ∙ i} = (⋂i∈Basis. {x. a ∙ i ≤ x ∙ i})"
by auto
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)
lemma closed_Collect_eucl_le[simp, intro]:
fixes a :: "'a::euclidean_space"
shows "closed {x. ∀i∈Basis. a ∙ i ≤ x ∙ i}"
"closed {x. ∀i∈Basis. x ∙ i ≤ a ∙ i}"
unfolding eucl_le_eq_halfspaces
by (simp_all add: closed_INT closed_Collect_le continuous_on_inner continuous_on_const continuous_on_id)
lemma image_affinity_cbox: fixes m::real
fixes a b c :: "'a::euclidean_space"
shows "(λx. m *⇩R x + c) ` cbox a b =
(if cbox a b = {} then {}
else (if 0 ≤ m then cbox (m *⇩R a + c) (m *⇩R b + c)
else cbox (m *⇩R b + c) (m *⇩R a + c)))"
proof (cases "m = 0")
case True
{
fix x
assume "∀i∈Basis. x ∙ i ≤ c ∙ i" "∀i∈Basis. c ∙ i ≤ x ∙ i"
then have "x = c"
apply -
apply (subst euclidean_eq_iff)
apply (auto intro: order_antisym)
done
}
moreover have "c ∈ cbox (m *⇩R a + c) (m *⇩R b + c)"
unfolding True by (auto simp: cbox_sing)
ultimately show ?thesis using True by (auto simp: cbox_def)
next
case False
{
fix y
assume "∀i∈Basis. a ∙ i ≤ y ∙ i" "∀i∈Basis. y ∙ i ≤ b ∙ i" "m > 0"
then have "∀i∈Basis. (m *⇩R a + c) ∙ i ≤ (m *⇩R y + c) ∙ i" and "∀i∈Basis. (m *⇩R y + c) ∙ i ≤ (m *⇩R b + c) ∙ i"
by (auto simp: inner_distrib)
}
moreover
{
fix y
assume "∀i∈Basis. a ∙ i ≤ y ∙ i" "∀i∈Basis. y ∙ i ≤ b ∙ i" "m < 0"
then have "∀i∈Basis. (m *⇩R b + c) ∙ i ≤ (m *⇩R y + c) ∙ i" and "∀i∈Basis. (m *⇩R y + c) ∙ i ≤ (m *⇩R a + c) ∙ i"
by (auto simp: mult_left_mono_neg inner_distrib)
}
moreover
{
fix y
assume "m > 0" and "∀i∈Basis. (m *⇩R a + c) ∙ i ≤ y ∙ i" and "∀i∈Basis. y ∙ i ≤ (m *⇩R b + c) ∙ i"
then have "y ∈ (λx. m *⇩R x + c) ` cbox a b"
unfolding image_iff Bex_def mem_box
apply (intro exI[where x="(1 / m) *⇩R (y - c)"])
apply (auto simp: pos_le_divide_eq pos_divide_le_eq mult.commute inner_distrib inner_diff_left)
done
}
moreover
{
fix y
assume "∀i∈Basis. (m *⇩R b + c) ∙ i ≤ y ∙ i" "∀i∈Basis. y ∙ i ≤ (m *⇩R a + c) ∙ i" "m < 0"
then have "y ∈ (λx. m *⇩R x + c) ` cbox a b"
unfolding image_iff Bex_def mem_box
apply (intro exI[where x="(1 / m) *⇩R (y - c)"])
apply (auto simp: neg_le_divide_eq neg_divide_le_eq mult.commute inner_distrib inner_diff_left)
done
}
ultimately show ?thesis using False by (auto simp: cbox_def)
qed
lemma image_smult_cbox:"(λx. m *⇩R (x::_::euclidean_space)) ` cbox a b =
(if cbox a b = {} then {} else if 0 ≤ m then cbox (m *⇩R a) (m *⇩R b) else cbox (m *⇩R b) (m *⇩R a))"
using image_affinity_cbox[of m 0 a b] by auto
lemma islimpt_greaterThanLessThan1:
fixes a b::"'a::{linorder_topology, dense_order}"
assumes "a < b"
shows "a islimpt {a<..<b}"
proof (rule islimptI)
fix T
assume "open T" "a ∈ T"
from open_right[OF this ‹a < b›]
obtain c where c: "a < c" "{a..<c} ⊆ T" by auto
with assms dense[of a "min c b"]
show "∃y∈{a<..<b}. y ∈ T ∧ y ≠ a"
by (metis atLeastLessThan_iff greaterThanLessThan_iff min_less_iff_conj
not_le order.strict_implies_order subset_eq)
qed
lemma islimpt_greaterThanLessThan2:
fixes a b::"'a::{linorder_topology, dense_order}"
assumes "a < b"
shows "b islimpt {a<..<b}"
proof (rule islimptI)
fix T
assume "open T" "b ∈ T"
from open_left[OF this ‹a < b›]
obtain c where c: "c < b" "{c<..b} ⊆ T" by auto
with assms dense[of "max a c" b]
show "∃y∈{a<..<b}. y ∈ T ∧ y ≠ b"
by (metis greaterThanAtMost_iff greaterThanLessThan_iff max_less_iff_conj
not_le order.strict_implies_order subset_eq)
qed
lemma closure_greaterThanLessThan[simp]:
fixes a b::"'a::{linorder_topology, dense_order}"
shows "a < b ⟹ closure {a <..< b} = {a .. b}" (is "_ ⟹ ?l = ?r")
proof
have "?l ⊆ closure ?r"
by (rule closure_mono) auto
thus "closure {a<..<b} ⊆ {a..b}" by simp
qed (auto simp: closure_def order.order_iff_strict islimpt_greaterThanLessThan1
islimpt_greaterThanLessThan2)
lemma closure_greaterThan[simp]:
fixes a b::"'a::{no_top, linorder_topology, dense_order}"
shows "closure {a<..} = {a..}"
proof -
from gt_ex obtain b where "a < b" by auto
hence "{a<..} = {a<..<b} ∪ {b..}" by auto
also have "closure … = {a..}" using ‹a < b› unfolding closure_Un
by auto
finally show ?thesis .
qed
lemma closure_lessThan[simp]:
fixes b::"'a::{no_bot, linorder_topology, dense_order}"
shows "closure {..<b} = {..b}"
proof -
from lt_ex obtain a where "a < b" by auto
hence "{..<b} = {a<..<b} ∪ {..a}" by auto
also have "closure … = {..b}" using ‹a < b› unfolding closure_Un
by auto
finally show ?thesis .
qed
lemma closure_atLeastLessThan[simp]:
fixes a b::"'a::{linorder_topology, dense_order}"
assumes "a < b"
shows "closure {a ..< b} = {a .. b}"
proof -
from assms have "{a ..< b} = {a} ∪ {a <..< b}" by auto
also have "closure … = {a .. b}" unfolding closure_Un
by (auto simp: assms less_imp_le)
finally show ?thesis .
qed
lemma closure_greaterThanAtMost[simp]:
fixes a b::"'a::{linorder_topology, dense_order}"
assumes "a < b"
shows "closure {a <.. b} = {a .. b}"
proof -
from assms have "{a <.. b} = {b} ∪ {a <..< b}" by auto
also have "closure … = {a .. b}" unfolding closure_Un
by (auto simp: assms less_imp_le)
finally show ?thesis .
qed
subsection ‹Homeomorphisms›
definition "homeomorphism s t f g ⟷
(∀x∈s. (g(f x) = x)) ∧ (f ` s = t) ∧ continuous_on s f ∧
(∀y∈t. (f(g y) = y)) ∧ (g ` t = s) ∧ continuous_on t g"
lemma homeomorphismI [intro?]:
assumes "continuous_on S f" "continuous_on T g"
"f ` S ⊆ T" "g ` T ⊆ S" "⋀x. x ∈ S ⟹ g(f x) = x" "⋀y. y ∈ T ⟹ f(g y) = y"
shows "homeomorphism S T f g"
using assms by (force simp: homeomorphism_def)
lemma homeomorphism_translation:
fixes a :: "'a :: real_normed_vector"
shows "homeomorphism (op + a ` S) S (op + (- a)) (op + a)"
unfolding homeomorphism_def by (auto simp: algebra_simps continuous_intros)
lemma homeomorphism_ident: "homeomorphism T T (λa. a) (λa. a)"
by (rule homeomorphismI) (auto simp: continuous_on_id)
lemma homeomorphism_compose:
assumes "homeomorphism S T f g" "homeomorphism T U h k"
shows "homeomorphism S U (h o f) (g o k)"
using assms
unfolding homeomorphism_def
by (intro conjI ballI continuous_on_compose) (auto simp: image_comp [symmetric])
lemma homeomorphism_symD: "homeomorphism S t f g ⟹ homeomorphism t S g f"
by (simp add: homeomorphism_def)
lemma homeomorphism_sym: "homeomorphism S t f g = homeomorphism t S g f"
by (force simp: homeomorphism_def)
definition homeomorphic :: "'a::topological_space set ⇒ 'b::topological_space set ⇒ bool"
(infixr "homeomorphic" 60)
where "s homeomorphic t ≡ (∃f g. homeomorphism s t f g)"
lemma homeomorphic_empty [iff]:
"S homeomorphic {} ⟷ S = {}" "{} homeomorphic S ⟷ S = {}"
by (auto simp: homeomorphic_def homeomorphism_def)
lemma homeomorphic_refl: "s homeomorphic s"
unfolding homeomorphic_def homeomorphism_def
using continuous_on_id
apply (rule_tac x = "(λx. x)" in exI)
apply (rule_tac x = "(λx. x)" in exI)
apply blast
done
lemma homeomorphic_sym: "s homeomorphic t ⟷ t homeomorphic s"
unfolding homeomorphic_def homeomorphism_def
by blast
lemma homeomorphic_trans [trans]:
assumes "S homeomorphic T"
and "T homeomorphic U"
shows "S homeomorphic U"
using assms
unfolding homeomorphic_def
by (metis homeomorphism_compose)
lemma homeomorphic_minimal:
"s homeomorphic t ⟷
(∃f g. (∀x∈s. f(x) ∈ t ∧ (g(f(x)) = x)) ∧
(∀y∈t. g(y) ∈ s ∧ (f(g(y)) = y)) ∧
continuous_on s f ∧ continuous_on t g)"
(is "?lhs = ?rhs")
proof
assume ?lhs
then show ?rhs
by (fastforce simp: homeomorphic_def homeomorphism_def)
next
assume ?rhs
then show ?lhs
apply clarify
unfolding homeomorphic_def homeomorphism_def
by (metis equalityI image_subset_iff subsetI)
qed
lemma homeomorphicI [intro?]:
"⟦f ` S = T; g ` T = S;
continuous_on S f; continuous_on T g;
⋀x. x ∈ S ⟹ g(f(x)) = x;
⋀y. y ∈ T ⟹ f(g(y)) = y⟧ ⟹ S homeomorphic T"
unfolding homeomorphic_def homeomorphism_def by metis
lemma homeomorphism_of_subsets:
"⟦homeomorphism S T f g; S' ⊆ S; T'' ⊆ T; f ` S' = T'⟧
⟹ homeomorphism S' T' f g"
apply (auto simp: homeomorphism_def elim!: continuous_on_subset)
by (metis subsetD imageI)
lemma homeomorphism_apply1: "⟦homeomorphism S T f g; x ∈ S⟧ ⟹ g(f x) = x"
by (simp add: homeomorphism_def)
lemma homeomorphism_apply2: "⟦homeomorphism S T f g; x ∈ T⟧ ⟹ f(g x) = x"
by (simp add: homeomorphism_def)
lemma homeomorphism_image1: "homeomorphism S T f g ⟹ f ` S = T"
by (simp add: homeomorphism_def)
lemma homeomorphism_image2: "homeomorphism S T f g ⟹ g ` T = S"
by (simp add: homeomorphism_def)
lemma homeomorphism_cont1: "homeomorphism S T f g ⟹ continuous_on S f"
by (simp add: homeomorphism_def)
lemma homeomorphism_cont2: "homeomorphism S T f g ⟹ continuous_on T g"
by (simp add: homeomorphism_def)
lemma continuous_on_no_limpt:
"(⋀x. ¬ x islimpt S) ⟹ continuous_on S f"
unfolding continuous_on_def
by (metis UNIV_I empty_iff eventually_at_topological islimptE open_UNIV tendsto_def trivial_limit_within)
lemma continuous_on_finite:
fixes S :: "'a::t1_space set"
shows "finite S ⟹ continuous_on S f"
by (metis continuous_on_no_limpt islimpt_finite)
lemma homeomorphic_finite:
fixes S :: "'a::t1_space set" and T :: "'b::t1_space set"
assumes "finite T"
shows "S homeomorphic T ⟷ finite S ∧ finite T ∧ card S = card T" (is "?lhs = ?rhs")
proof
assume "S homeomorphic T"
with assms show ?rhs
apply (auto simp: homeomorphic_def homeomorphism_def)
apply (metis finite_imageI)
by (metis card_image_le finite_imageI le_antisym)
next
assume R: ?rhs
with finite_same_card_bij obtain h where "bij_betw h S T"
by auto
with R show ?lhs
apply (auto simp: homeomorphic_def homeomorphism_def continuous_on_finite)
apply (rule_tac x=h in exI)
apply (rule_tac x="inv_into S h" in exI)
apply (auto simp: bij_betw_inv_into_left bij_betw_inv_into_right bij_betw_imp_surj_on inv_into_into bij_betwE)
apply (metis bij_betw_def bij_betw_inv_into)
done
qed
text ‹Relatively weak hypotheses if a set is compact.›
lemma homeomorphism_compact:
fixes f :: "'a::topological_space ⇒ 'b::t2_space"
assumes "compact s" "continuous_on s f" "f ` s = t" "inj_on f s"
shows "∃g. homeomorphism s t f g"
proof -
define g where "g x = (SOME y. y∈s ∧ f y = x)" for x
have g: "∀x∈s. g (f x) = x"
using assms(3) assms(4)[unfolded inj_on_def] unfolding g_def by auto
{
fix y
assume "y ∈ t"
then obtain x where x:"f x = y" "x∈s"
using assms(3) by auto
then have "g (f x) = x" using g by auto
then have "f (g y) = y" unfolding x(1)[symmetric] by auto
}
then have g':"∀x∈t. f (g x) = x" by auto
moreover
{
fix x
have "x∈s ⟹ x ∈ g ` t"
using g[THEN bspec[where x=x]]
unfolding image_iff
using assms(3)
by (auto intro!: bexI[where x="f x"])
moreover
{
assume "x∈g ` t"
then obtain y where y:"y∈t" "g y = x" by auto
then obtain x' where x':"x'∈s" "f x' = y"
using assms(3) by auto
then have "x ∈ s"
unfolding g_def
using someI2[of "λb. b∈s ∧ f b = y" x' "λx. x∈s"]
unfolding y(2)[symmetric] and g_def
by auto
}
ultimately have "x∈s ⟷ x ∈ g ` t" ..
}
then have "g ` t = s" by auto
ultimately show ?thesis
unfolding homeomorphism_def homeomorphic_def
apply (rule_tac x=g in exI)
using g and assms(3) and continuous_on_inv[OF assms(2,1), of g, unfolded assms(3)] and assms(2)
apply auto
done
qed
lemma homeomorphic_compact:
fixes f :: "'a::topological_space ⇒ 'b::t2_space"
shows "compact s ⟹ continuous_on s f ⟹ (f ` s = t) ⟹ inj_on f s ⟹ s homeomorphic t"
unfolding homeomorphic_def by (metis homeomorphism_compact)
text‹Preservation of topological properties.›
lemma homeomorphic_compactness: "s homeomorphic t ⟹ (compact s ⟷ compact t)"
unfolding homeomorphic_def homeomorphism_def
by (metis compact_continuous_image)
text‹Results on translation, scaling etc.›
lemma homeomorphic_scaling:
fixes s :: "'a::real_normed_vector set"
assumes "c ≠ 0"
shows "s homeomorphic ((λx. c *⇩R x) ` s)"
unfolding homeomorphic_minimal
apply (rule_tac x="λx. c *⇩R x" in exI)
apply (rule_tac x="λx. (1 / c) *⇩R x" in exI)
using assms
apply (auto simp: continuous_intros)
done
lemma homeomorphic_translation:
fixes s :: "'a::real_normed_vector set"
shows "s homeomorphic ((λx. a + x) ` s)"
unfolding homeomorphic_minimal
apply (rule_tac x="λx. a + x" in exI)
apply (rule_tac x="λx. -a + x" in exI)
using continuous_on_add [OF continuous_on_const continuous_on_id, of s a]
continuous_on_add [OF continuous_on_const continuous_on_id, of "plus a ` s" "- a"]
apply auto
done
lemma homeomorphic_affinity:
fixes s :: "'a::real_normed_vector set"
assumes "c ≠ 0"
shows "s homeomorphic ((λx. a + c *⇩R x) ` s)"
proof -
have *: "op + a ` op *⇩R c ` s = (λx. a + c *⇩R x) ` s" by auto
show ?thesis
using homeomorphic_trans
using homeomorphic_scaling[OF assms, of s]
using homeomorphic_translation[of "(λx. c *⇩R x) ` s" a]
unfolding *
by auto
qed
lemma homeomorphic_balls:
fixes a b ::"'a::real_normed_vector"
assumes "0 < d" "0 < e"
shows "(ball a d) homeomorphic (ball b e)" (is ?th)
and "(cball a d) homeomorphic (cball b e)" (is ?cth)
proof -
show ?th unfolding homeomorphic_minimal
apply(rule_tac x="λx. b + (e/d) *⇩R (x - a)" in exI)
apply(rule_tac x="λx. a + (d/e) *⇩R (x - b)" in exI)
using assms
apply (auto intro!: continuous_intros
simp: dist_commute dist_norm pos_divide_less_eq mult_strict_left_mono)
done
show ?cth unfolding homeomorphic_minimal
apply(rule_tac x="λx. b + (e/d) *⇩R (x - a)" in exI)
apply(rule_tac x="λx. a + (d/e) *⇩R (x - b)" in exI)
using assms
apply (auto intro!: continuous_intros
simp: dist_commute dist_norm pos_divide_le_eq mult_strict_left_mono)
done
qed
lemma homeomorphic_spheres:
fixes a b ::"'a::real_normed_vector"
assumes "0 < d" "0 < e"
shows "(sphere a d) homeomorphic (sphere b e)"
unfolding homeomorphic_minimal
apply(rule_tac x="λx. b + (e/d) *⇩R (x - a)" in exI)
apply(rule_tac x="λx. a + (d/e) *⇩R (x - b)" in exI)
using assms
apply (auto intro!: continuous_intros
simp: dist_commute dist_norm pos_divide_less_eq mult_strict_left_mono)
done
subsection‹Inverse function property for open/closed maps›
lemma continuous_on_inverse_open_map:
assumes contf: "continuous_on S f"
and imf: "f ` S = T"
and injf: "⋀x. x ∈ S ⟹ g (f x) = x"
and oo: "⋀U. openin (subtopology euclidean S) U ⟹ openin (subtopology euclidean T) (f ` U)"
shows "continuous_on T g"
proof -
from imf injf have gTS: "g ` T = S"
by force
from imf injf have fU: "U ⊆ S ⟹ (f ` U) = {x ∈ T. g x ∈ U}" for U
by force
show ?thesis
by (simp add: continuous_on_open [of T g] gTS) (metis openin_imp_subset fU oo)
qed
lemma continuous_on_inverse_closed_map:
assumes contf: "continuous_on S f"
and imf: "f ` S = T"
and injf: "⋀x. x ∈ S ⟹ g(f x) = x"
and oo: "⋀U. closedin (subtopology euclidean S) U ⟹ closedin (subtopology euclidean T) (f ` U)"
shows "continuous_on T g"
proof -
from imf injf have gTS: "g ` T = S"
by force
from imf injf have fU: "U ⊆ S ⟹ (f ` U) = {x ∈ T. g x ∈ U}" for U
by force
show ?thesis
by (simp add: continuous_on_closed [of T g] gTS) (metis closedin_imp_subset fU oo)
qed
lemma homeomorphism_injective_open_map:
assumes contf: "continuous_on S f"
and imf: "f ` S = T"
and injf: "inj_on f S"
and oo: "⋀U. openin (subtopology euclidean S) U ⟹ openin (subtopology euclidean T) (f ` U)"
obtains g where "homeomorphism S T f g"
proof
have "continuous_on T (inv_into S f)"
by (metis contf continuous_on_inverse_open_map imf injf inv_into_f_f oo)
with imf injf contf show "homeomorphism S T f (inv_into S f)"
by (auto simp: homeomorphism_def)
qed
lemma homeomorphism_injective_closed_map:
assumes contf: "continuous_on S f"
and imf: "f ` S = T"
and injf: "inj_on f S"
and oo: "⋀U. closedin (subtopology euclidean S) U ⟹ closedin (subtopology euclidean T) (f ` U)"
obtains g where "homeomorphism S T f g"
proof
have "continuous_on T (inv_into S f)"
by (metis contf continuous_on_inverse_closed_map imf injf inv_into_f_f oo)
with imf injf contf show "homeomorphism S T f (inv_into S f)"
by (auto simp: homeomorphism_def)
qed
lemma homeomorphism_imp_open_map:
assumes hom: "homeomorphism S T f g"
and oo: "openin (subtopology euclidean S) U"
shows "openin (subtopology euclidean T) (f ` U)"
proof -
from hom oo have [simp]: "f ` U = {y. y ∈ T ∧ g y ∈ U}"
using openin_subset by (fastforce simp: homeomorphism_def rev_image_eqI)
from hom have "continuous_on T g"
unfolding homeomorphism_def by blast
moreover have "g ` T = S"
by (metis hom homeomorphism_def)
ultimately show ?thesis
by (simp add: continuous_on_open oo)
qed
lemma homeomorphism_imp_closed_map:
assumes hom: "homeomorphism S T f g"
and oo: "closedin (subtopology euclidean S) U"
shows "closedin (subtopology euclidean T) (f ` U)"
proof -
from hom oo have [simp]: "f ` U = {y. y ∈ T ∧ g y ∈ U}"
using closedin_subset by (fastforce simp: homeomorphism_def rev_image_eqI)
from hom have "continuous_on T g"
unfolding homeomorphism_def by blast
moreover have "g ` T = S"
by (metis hom homeomorphism_def)
ultimately show ?thesis
by (simp add: continuous_on_closed oo)
qed
subsection ‹"Isometry" (up to constant bounds) of injective linear map etc.›
lemma cauchy_isometric:
assumes e: "e > 0"
and s: "subspace s"
and f: "bounded_linear f"
and normf: "∀x∈s. norm (f x) ≥ e * norm x"
and xs: "∀n. x n ∈ s"
and cf: "Cauchy (f ∘ x)"
shows "Cauchy x"
proof -
interpret f: bounded_linear f by fact
have "∃N. ∀n≥N. norm (x n - x N) < d" if "d > 0" for d :: real
proof -
from that obtain N where N: "∀n≥N. norm (f (x n) - f (x N)) < e * d"
using cf[unfolded Cauchy_def o_def dist_norm, THEN spec[where x="e*d"]] e
by auto
have "norm (x n - x N) < d" if "n ≥ N" for n
proof -
have "e * norm (x n - x N) ≤ norm (f (x n - x N))"
using subspace_diff[OF s, of "x n" "x N"]
using xs[THEN spec[where x=N]] and xs[THEN spec[where x=n]]
using normf[THEN bspec[where x="x n - x N"]]
by auto
also have "norm (f (x n - x N)) < e * d"
using ‹N ≤ n› N unfolding f.diff[symmetric] by auto
finally show ?thesis
using ‹e>0› by simp
qed
then show ?thesis by auto
qed
then show ?thesis
by (simp add: Cauchy_altdef2 dist_norm)
qed
lemma complete_isometric_image:
assumes "0 < e"
and s: "subspace s"
and f: "bounded_linear f"
and normf: "∀x∈s. norm(f x) ≥ e * norm(x)"
and cs: "complete s"
shows "complete (f ` s)"
proof -
have "∃l∈f ` s. (g ⤏ l) sequentially"
if as:"∀n::nat. g n ∈ f ` s" and cfg:"Cauchy g" for g
proof -
from that obtain x where "∀n. x n ∈ s ∧ g n = f (x n)"
using choice[of "λ n xa. xa ∈ s ∧ g n = f xa"] by auto
then have x: "∀n. x n ∈ s" "∀n. g n = f (x n)" by auto
then have "f ∘ x = g" by (simp add: fun_eq_iff)
then obtain l where "l∈s" and l:"(x ⤏ l) sequentially"
using cs[unfolded complete_def, THEN spec[where x=x]]
using cauchy_isometric[OF ‹0 < e› s f normf] and cfg and x(1)
by auto
then show ?thesis
using linear_continuous_at[OF f, unfolded continuous_at_sequentially, THEN spec[where x=x], of l]
by (auto simp: ‹f ∘ x = g›)
qed
then show ?thesis
unfolding complete_def by auto
qed
lemma injective_imp_isometric:
fixes f :: "'a::euclidean_space ⇒ 'b::euclidean_space"
assumes s: "closed s" "subspace s"
and f: "bounded_linear f" "∀x∈s. f x = 0 ⟶ x = 0"
shows "∃e>0. ∀x∈s. norm (f x) ≥ e * norm x"
proof (cases "s ⊆ {0::'a}")
case True
have "norm x ≤ norm (f x)" if "x ∈ s" for x
proof -
from True that have "x = 0" by auto
then show ?thesis by simp
qed
then show ?thesis
by (auto intro!: exI[where x=1])
next
case False
interpret f: bounded_linear f by fact
from False obtain a where a: "a ≠ 0" "a ∈ s"
by auto
from False have "s ≠ {}"
by auto
let ?S = "{f x| x. x ∈ s ∧ norm x = norm a}"
let ?S' = "{x::'a. x∈s ∧ norm x = norm a}"
let ?S'' = "{x::'a. norm x = norm a}"
have "?S'' = frontier (cball 0 (norm a))"
by (simp add: sphere_def dist_norm)
then have "compact ?S''" by (metis compact_cball compact_frontier)
moreover have "?S' = s ∩ ?S''" by auto
ultimately have "compact ?S'"
using closed_Int_compact[of s ?S''] using s(1) by auto
moreover have *:"f ` ?S' = ?S" by auto
ultimately have "compact ?S"
using compact_continuous_image[OF linear_continuous_on[OF f(1)], of ?S'] by auto
then have "closed ?S"
using compact_imp_closed by auto
moreover from a have "?S ≠ {}" by auto
ultimately obtain b' where "b'∈?S" "∀y∈?S. norm b' ≤ norm y"
using distance_attains_inf[of ?S 0] unfolding dist_0_norm by auto
then obtain b where "b∈s"
and ba: "norm b = norm a"
and b: "∀x∈{x ∈ s. norm x = norm a}. norm (f b) ≤ norm (f x)"
unfolding *[symmetric] unfolding image_iff by auto
let ?e = "norm (f b) / norm b"
have "norm b > 0"
using ba and a and norm_ge_zero by auto
moreover have "norm (f b) > 0"
using f(2)[THEN bspec[where x=b], OF ‹b∈s›]
using ‹norm b >0› by simp
ultimately have "0 < norm (f b) / norm b" by simp
moreover
have "norm (f b) / norm b * norm x ≤ norm (f x)" if "x∈s" for x
proof (cases "x = 0")
case True
then show "norm (f b) / norm b * norm x ≤ norm (f x)"
by auto
next
case False
with ‹a ≠ 0› have *: "0 < norm a / norm x"
unfolding zero_less_norm_iff[symmetric] by simp
have "∀x∈s. c *⇩R x ∈ s" for c
using s[unfolded subspace_def] by simp
with ‹x ∈ s› ‹x ≠ 0› have "(norm a / norm x) *⇩R x ∈ {x ∈ s. norm x = norm a}"
by simp
with ‹x ≠ 0› ‹a ≠ 0› show "norm (f b) / norm b * norm x ≤ norm (f x)"
using b[THEN bspec[where x="(norm a / norm x) *⇩R x"]]
unfolding f.scaleR and ba
by (auto simp: mult.commute pos_le_divide_eq pos_divide_le_eq)
qed
ultimately show ?thesis by auto
qed
lemma closed_injective_image_subspace:
fixes f :: "'a::euclidean_space ⇒ 'b::euclidean_space"
assumes "subspace s" "bounded_linear f" "∀x∈s. f x = 0 ⟶ x = 0" "closed s"
shows "closed(f ` s)"
proof -
obtain e where "e > 0" and e: "∀x∈s. e * norm x ≤ norm (f x)"
using injective_imp_isometric[OF assms(4,1,2,3)] by auto
show ?thesis
using complete_isometric_image[OF ‹e>0› assms(1,2) e] and assms(4)
unfolding complete_eq_closed[symmetric] by auto
qed
subsection ‹Some properties of a canonical subspace›
lemma subspace_substandard: "subspace {x::'a::euclidean_space. (∀i∈Basis. P i ⟶ x∙i = 0)}"
by (auto simp: subspace_def inner_add_left)
lemma closed_substandard: "closed {x::'a::euclidean_space. ∀i∈Basis. P i ⟶ x∙i = 0}"
(is "closed ?A")
proof -
let ?D = "{i∈Basis. P i}"
have "closed (⋂i∈?D. {x::'a. x∙i = 0})"
by (simp add: closed_INT closed_Collect_eq continuous_on_inner
continuous_on_const continuous_on_id)
also have "(⋂i∈?D. {x::'a. x∙i = 0}) = ?A"
by auto
finally show "closed ?A" .
qed
lemma dim_substandard:
assumes d: "d ⊆ Basis"
shows "dim {x::'a::euclidean_space. ∀i∈Basis. i ∉ d ⟶ x∙i = 0} = card d" (is "dim ?A = _")
proof (rule dim_unique)
from d show "d ⊆ ?A"
by (auto simp: inner_Basis)
from d show "independent d"
by (rule independent_mono [OF independent_Basis])
have "x ∈ span d" if "∀i∈Basis. i ∉ d ⟶ x ∙ i = 0" for x
proof -
have "finite d"
by (rule finite_subset [OF d finite_Basis])
then have "(∑i∈d. (x ∙ i) *⇩R i) ∈ span d"
by (simp add: span_sum span_clauses)
also have "(∑i∈d. (x ∙ i) *⇩R i) = (∑i∈Basis. (x ∙ i) *⇩R i)"
by (rule sum.mono_neutral_cong_left [OF finite_Basis d]) (auto simp: that)
finally show "x ∈ span d"
by (simp only: euclidean_representation)
qed
then show "?A ⊆ span d" by auto
qed simp
text ‹Hence closure and completeness of all subspaces.›
lemma ex_card:
assumes "n ≤ card A"
shows "∃S⊆A. card S = n"
proof (cases "finite A")
case True
from ex_bij_betw_nat_finite[OF this] obtain f where f: "bij_betw f {0..<card A} A" ..
moreover from f ‹n ≤ card A› have "{..< n} ⊆ {..< card A}" "inj_on f {..< n}"
by (auto simp: bij_betw_def intro: subset_inj_on)
ultimately have "f ` {..< n} ⊆ A" "card (f ` {..< n}) = n"
by (auto simp: bij_betw_def card_image)
then show ?thesis by blast
next
case False
with ‹n ≤ card A› show ?thesis by force
qed
lemma closed_subspace:
fixes s :: "'a::euclidean_space set"
assumes "subspace s"
shows "closed s"
proof -
have "dim s ≤ card (Basis :: 'a set)"
using dim_subset_UNIV by auto
with ex_card[OF this] obtain d :: "'a set" where t: "card d = dim s" and d: "d ⊆ Basis"
by auto
let ?t = "{x::'a. ∀i∈Basis. i ∉ d ⟶ x∙i = 0}"
have "∃f. linear f ∧ f ` {x::'a. ∀i∈Basis. i ∉ d ⟶ x ∙ i = 0} = s ∧
inj_on f {x::'a. ∀i∈Basis. i ∉ d ⟶ x ∙ i = 0}"
using dim_substandard[of d] t d assms
by (intro subspace_isomorphism[OF subspace_substandard[of "λi. i ∉ d"]]) (auto simp: inner_Basis)
then obtain f where f:
"linear f"
"f ` {x. ∀i∈Basis. i ∉ d ⟶ x ∙ i = 0} = s"
"inj_on f {x. ∀i∈Basis. i ∉ d ⟶ x ∙ i = 0}"
by blast
interpret f: bounded_linear f
using f by (simp add: linear_conv_bounded_linear)
have "x ∈ ?t ⟹ f x = 0 ⟹ x = 0" for x
using f.zero d f(3)[THEN inj_onD, of x 0] by auto
moreover have "closed ?t" by (rule closed_substandard)
moreover have "subspace ?t" by (rule subspace_substandard)
ultimately show ?thesis
using closed_injective_image_subspace[of ?t f]
unfolding f(2) using f(1) unfolding linear_conv_bounded_linear by auto
qed
lemma complete_subspace: "subspace s ⟹ complete s"
for s :: "'a::euclidean_space set"
using complete_eq_closed closed_subspace by auto
lemma closed_span [iff]: "closed (span s)"
for s :: "'a::euclidean_space set"
by (simp add: closed_subspace)
lemma dim_closure [simp]: "dim (closure s) = dim s" (is "?dc = ?d")
for s :: "'a::euclidean_space set"
proof -
have "?dc ≤ ?d"
using closure_minimal[OF span_inc, of s]
using closed_subspace[OF subspace_span, of s]
using dim_subset[of "closure s" "span s"]
by simp
then show ?thesis
using dim_subset[OF closure_subset, of s]
by simp
qed
subsection ‹Affine transformations of intervals›
lemma real_affinity_le: "0 < m ⟹ m * x + c ≤ y ⟷ x ≤ inverse m * y + - (c / m)"
for m :: "'a::linordered_field"
by (simp add: field_simps)
lemma real_le_affinity: "0 < m ⟹ y ≤ m * x + c ⟷ inverse m * y + - (c / m) ≤ x"
for m :: "'a::linordered_field"
by (simp add: field_simps)
lemma real_affinity_lt: "0 < m ⟹ m * x + c < y ⟷ x < inverse m * y + - (c / m)"
for m :: "'a::linordered_field"
by (simp add: field_simps)
lemma real_lt_affinity: "0 < m ⟹ y < m * x + c ⟷ inverse m * y + - (c / m) < x"
for m :: "'a::linordered_field"
by (simp add: field_simps)
lemma real_affinity_eq: "m ≠ 0 ⟹ m * x + c = y ⟷ x = inverse m * y + - (c / m)"
for m :: "'a::linordered_field"
by (simp add: field_simps)
lemma real_eq_affinity: "m ≠ 0 ⟹ y = m * x + c ⟷ inverse m * y + - (c / m) = x"
for m :: "'a::linordered_field"
by (simp add: field_simps)
subsection ‹Banach fixed point theorem (not really topological ...)›
theorem banach_fix:
assumes s: "complete s" "s ≠ {}"
and c: "0 ≤ c" "c < 1"
and f: "f ` s ⊆ s"
and lipschitz: "∀x∈s. ∀y∈s. dist (f x) (f y) ≤ c * dist x y"
shows "∃!x∈s. f x = x"
proof -
from c have "1 - c > 0" by simp
from s(2) obtain z0 where z0: "z0 ∈ s" by blast
define z where "z n = (f ^^ n) z0" for n
with f z0 have z_in_s: "z n ∈ s" for n :: nat
by (induct n) auto
define d where "d = dist (z 0) (z 1)"
have fzn: "f (z n) = z (Suc n)" for n
by (simp add: z_def)
have cf_z: "dist (z n) (z (Suc n)) ≤ (c ^ n) * d" for n :: nat
proof (induct n)
case 0
then show ?case
by (simp add: d_def)
next
case (Suc m)
with ‹0 ≤ c› have "c * dist (z m) (z (Suc m)) ≤ c ^ Suc m * d"
using mult_left_mono[of "dist (z m) (z (Suc m))" "c ^ m * d" c] by simp
then show ?case
using lipschitz[THEN bspec[where x="z m"], OF z_in_s, THEN bspec[where x="z (Suc m)"], OF z_in_s]
by (simp add: fzn mult_le_cancel_left)
qed
have cf_z2: "(1 - c) * dist (z m) (z (m + n)) ≤ (c ^ m) * d * (1 - c ^ n)" for n m :: nat
proof (induct n)
case 0
show ?case by simp
next
case (Suc k)
from c have "(1 - c) * dist (z m) (z (m + Suc k)) ≤
(1 - c) * (dist (z m) (z (m + k)) + dist (z (m + k)) (z (Suc (m + k))))"
by (simp add: dist_triangle)
also from c cf_z[of "m + k"] have "… ≤ (1 - c) * (dist (z m) (z (m + k)) + c ^ (m + k) * d)"
by simp
also from Suc have "… ≤ c ^ m * d * (1 - c ^ k) + (1 - c) * c ^ (m + k) * d"
by (simp add: field_simps)
also have "… = (c ^ m) * (d * (1 - c ^ k) + (1 - c) * c ^ k * d)"
by (simp add: power_add field_simps)
also from c have "… ≤ (c ^ m) * d * (1 - c ^ Suc k)"
by (simp add: field_simps)
finally show ?case by simp
qed
have "∃N. ∀m n. N ≤ m ∧ N ≤ n ⟶ dist (z m) (z n) < e" if "e > 0" for e
proof (cases "d = 0")
case True
from ‹1 - c > 0› have "(1 - c) * x ≤ 0 ⟷ x ≤ 0" for x
by (metis mult_zero_left mult.commute real_mult_le_cancel_iff1)
with c cf_z2[of 0] True have "z n = z0" for n
by (simp add: z_def)
with ‹e > 0› show ?thesis by simp
next
case False
with zero_le_dist[of "z 0" "z 1"] have "d > 0"
by (metis d_def less_le)
with ‹1 - c > 0› ‹e > 0› have "0 < e * (1 - c) / d"
by simp
with c obtain N where N: "c ^ N < e * (1 - c) / d"
using real_arch_pow_inv[of "e * (1 - c) / d" c] by auto
have *: "dist (z m) (z n) < e" if "m > n" and as: "m ≥ N" "n ≥ N" for m n :: nat
proof -
from c ‹n ≥ N› have *: "c ^ n ≤ c ^ N"
using power_decreasing[OF ‹n≥N›, of c] by simp
from c ‹m > n› have "1 - c ^ (m - n) > 0"
using power_strict_mono[of c 1 "m - n"] by simp
with ‹d > 0› ‹0 < 1 - c› have **: "d * (1 - c ^ (m - n)) / (1 - c) > 0"
by simp
from cf_z2[of n "m - n"] ‹m > n›
have "dist (z m) (z n) ≤ c ^ n * d * (1 - c ^ (m - n)) / (1 - c)"
by (simp add: pos_le_divide_eq[OF ‹1 - c > 0›] mult.commute dist_commute)
also have "… ≤ c ^ N * d * (1 - c ^ (m - n)) / (1 - c)"
using mult_right_mono[OF * order_less_imp_le[OF **]]
by (simp add: mult.assoc)
also have "… < (e * (1 - c) / d) * d * (1 - c ^ (m - n)) / (1 - c)"
using mult_strict_right_mono[OF N **] by (auto simp: mult.assoc)
also from c ‹d > 0› ‹1 - c > 0› have "… = e * (1 - c ^ (m - n))"
by simp
also from c ‹1 - c ^ (m - n) > 0› ‹e > 0› have "… ≤ e"
using mult_right_le_one_le[of e "1 - c ^ (m - n)"] by auto
finally show ?thesis by simp
qed
have "dist (z n) (z m) < e" if "N ≤ m" "N ≤ n" for m n :: nat
proof (cases "n = m")
case True
with ‹e > 0› show ?thesis by simp
next
case False
with *[of n m] *[of m n] and that show ?thesis
by (auto simp: dist_commute nat_neq_iff)
qed
then show ?thesis by auto
qed
then have "Cauchy z"
by (simp add: cauchy_def)
then obtain x where "x∈s" and x:"(z ⤏ x) sequentially"
using s(1)[unfolded compact_def complete_def, THEN spec[where x=z]] and z_in_s by auto
define e where "e = dist (f x) x"
have "e = 0"
proof (rule ccontr)
assume "e ≠ 0"
then have "e > 0"
unfolding e_def using zero_le_dist[of "f x" x]
by (metis dist_eq_0_iff dist_nz e_def)
then obtain N where N:"∀n≥N. dist (z n) x < e / 2"
using x[unfolded lim_sequentially, THEN spec[where x="e/2"]] by auto
then have N':"dist (z N) x < e / 2" by auto
have *: "c * dist (z N) x ≤ dist (z N) x"
unfolding mult_le_cancel_right2
using zero_le_dist[of "z N" x] and c
by (metis dist_eq_0_iff dist_nz order_less_asym less_le)
have "dist (f (z N)) (f x) ≤ c * dist (z N) x"
using lipschitz[THEN bspec[where x="z N"], THEN bspec[where x=x]]
using z_in_s[of N] ‹x∈s›
using c
by auto
also have "… < e / 2"
using N' and c using * by auto
finally show False
unfolding fzn
using N[THEN spec[where x="Suc N"]] and dist_triangle_half_r[of "z (Suc N)" "f x" e x]
unfolding e_def
by auto
qed
then have "f x = x" by (auto simp: e_def)
moreover have "y = x" if "f y = y" "y ∈ s" for y
proof -
from ‹x ∈ s› ‹f x = x› that have "dist x y ≤ c * dist x y"
using lipschitz[THEN bspec[where x=x], THEN bspec[where x=y]] by simp
with c and zero_le_dist[of x y] have "dist x y = 0"
by (simp add: mult_le_cancel_right1)
then show ?thesis by simp
qed
ultimately show ?thesis
using ‹x∈s› by blast
qed
subsection ‹Edelstein fixed point theorem›
theorem edelstein_fix:
fixes s :: "'a::metric_space set"
assumes s: "compact s" "s ≠ {}"
and gs: "(g ` s) ⊆ s"
and dist: "∀x∈s. ∀y∈s. x ≠ y ⟶ dist (g x) (g y) < dist x y"
shows "∃!x∈s. g x = x"
proof -
let ?D = "(λx. (x, x)) ` s"
have D: "compact ?D" "?D ≠ {}"
by (rule compact_continuous_image)
(auto intro!: s continuous_Pair continuous_ident simp: continuous_on_eq_continuous_within)
have "⋀x y e. x ∈ s ⟹ y ∈ s ⟹ 0 < e ⟹ dist y x < e ⟹ dist (g y) (g x) < e"
using dist by fastforce
then have "continuous_on s g"
by (auto simp: continuous_on_iff)
then have cont: "continuous_on ?D (λx. dist ((g ∘ fst) x) (snd x))"
unfolding continuous_on_eq_continuous_within
by (intro continuous_dist ballI continuous_within_compose)
(auto intro!: continuous_fst continuous_snd continuous_ident simp: image_image)
obtain a where "a ∈ s" and le: "⋀x. x ∈ s ⟹ dist (g a) a ≤ dist (g x) x"
using continuous_attains_inf[OF D cont] by auto
have "g a = a"
proof (rule ccontr)
assume "g a ≠ a"
with ‹a ∈ s› gs have "dist (g (g a)) (g a) < dist (g a) a"
by (intro dist[rule_format]) auto
moreover have "dist (g a) a ≤ dist (g (g a)) (g a)"
using ‹a ∈ s› gs by (intro le) auto
ultimately show False by auto
qed
moreover have "⋀x. x ∈ s ⟹ g x = x ⟹ x = a"
using dist[THEN bspec[where x=a]] ‹g a = a› and ‹a∈s› by auto
ultimately show "∃!x∈s. g x = x"
using ‹a ∈ s› by blast
qed
lemma cball_subset_cball_iff:
fixes a :: "'a :: euclidean_space"
shows "cball a r ⊆ cball a' r' ⟷ dist a a' + r ≤ r' ∨ r < 0"
(is "?lhs ⟷ ?rhs")
proof
assume ?lhs
then show ?rhs
proof (cases "r < 0")
case True
then show ?rhs by simp
next
case False
then have [simp]: "r ≥ 0" by simp
have "norm (a - a') + r ≤ r'"
proof (cases "a = a'")
case True
then show ?thesis
using subsetD [where c = "a + r *⇩R (SOME i. i ∈ Basis)", OF ‹?lhs›] subsetD [where c = a, OF ‹?lhs›]
by (force simp: SOME_Basis dist_norm)
next
case False
have "norm (a' - (a + (r / norm (a - a')) *⇩R (a - a'))) = norm (a' - a - (r / norm (a - a')) *⇩R (a - a'))"
by (simp add: algebra_simps)
also have "... = norm ((-1 - (r / norm (a - a'))) *⇩R (a - a'))"
by (simp add: algebra_simps)
also from ‹a ≠ a'› have "... = ¦- norm (a - a') - r¦"
by (simp add: abs_mult_pos field_simps)
finally have [simp]: "norm (a' - (a + (r / norm (a - a')) *⇩R (a - a'))) = ¦norm (a - a') + r¦"
by linarith
from ‹a ≠ a'› show ?thesis
using subsetD [where c = "a' + (1 + r / norm(a - a')) *⇩R (a - a')", OF ‹?lhs›]
by (simp add: dist_norm scaleR_add_left)
qed
then show ?rhs
by (simp add: dist_norm)
qed
next
assume ?rhs
then show ?lhs
by (auto simp: ball_def dist_norm)
(metis add.commute add_le_cancel_right dist_norm dist_triangle3 order_trans)
qed
lemma cball_subset_ball_iff: "cball a r ⊆ ball a' r' ⟷ dist a a' + r < r' ∨ r < 0"
(is "?lhs ⟷ ?rhs")
for a :: "'a::euclidean_space"
proof
assume ?lhs
then show ?rhs
proof (cases "r < 0")
case True then
show ?rhs by simp
next
case False
then have [simp]: "r ≥ 0" by simp
have "norm (a - a') + r < r'"
proof (cases "a = a'")
case True
then show ?thesis
using subsetD [where c = "a + r *⇩R (SOME i. i ∈ Basis)", OF ‹?lhs›] subsetD [where c = a, OF ‹?lhs›]
by (force simp: SOME_Basis dist_norm)
next
case False
have False if "norm (a - a') + r ≥ r'"
proof -
from that have "¦r' - norm (a - a')¦ ≤ r"
by (simp split: abs_split)
(metis ‹0 ≤ r› ‹?lhs› centre_in_cball dist_commute dist_norm less_asym mem_ball subset_eq)
then show ?thesis
using subsetD [where c = "a + (r' / norm(a - a') - 1) *⇩R (a - a')", OF ‹?lhs›] ‹a ≠ a'›
by (simp add: dist_norm field_simps)
(simp add: diff_divide_distrib scaleR_left_diff_distrib)
qed
then show ?thesis by force
qed
then show ?rhs by (simp add: dist_norm)
qed
next
assume ?rhs
then show ?lhs
by (auto simp: ball_def dist_norm)
(metis add.commute add_le_cancel_right dist_norm dist_triangle3 le_less_trans)
qed
lemma ball_subset_cball_iff: "ball a r ⊆ cball a' r' ⟷ dist a a' + r ≤ r' ∨ r ≤ 0"
(is "?lhs = ?rhs")
for a :: "'a::euclidean_space"
proof (cases "r ≤ 0")
case True
then show ?thesis
using dist_not_less_zero less_le_trans by force
next
case False
show ?thesis
proof
assume ?lhs
then have "(cball a r ⊆ cball a' r')"
by (metis False closed_cball closure_ball closure_closed closure_mono not_less)
with False show ?rhs
by (fastforce iff: cball_subset_cball_iff)
next
assume ?rhs
with False show ?lhs
using ball_subset_cball cball_subset_cball_iff by blast
qed
qed
lemma ball_subset_ball_iff:
fixes a :: "'a :: euclidean_space"
shows "ball a r ⊆ ball a' r' ⟷ dist a a' + r ≤ r' ∨ r ≤ 0"
(is "?lhs = ?rhs")
proof (cases "r ≤ 0")
case True then show ?thesis
using dist_not_less_zero less_le_trans by force
next
case False show ?thesis
proof
assume ?lhs
then have "0 < r'"
by (metis (no_types) False ‹?lhs› centre_in_ball dist_norm le_less_trans mem_ball norm_ge_zero not_less set_mp)
then have "(cball a r ⊆ cball a' r')"
by (metis False‹?lhs› closure_ball closure_mono not_less)
then show ?rhs
using False cball_subset_cball_iff by fastforce
next
assume ?rhs then show ?lhs
apply (auto simp: ball_def)
apply (metis add.commute add_le_cancel_right dist_commute dist_triangle_lt not_le order_trans)
using dist_not_less_zero order.strict_trans2 apply blast
done
qed
qed
lemma ball_eq_ball_iff:
fixes x :: "'a :: euclidean_space"
shows "ball x d = ball y e ⟷ d ≤ 0 ∧ e ≤ 0 ∨ x=y ∧ d=e"
(is "?lhs = ?rhs")
proof
assume ?lhs
then show ?rhs
proof (cases "d ≤ 0 ∨ e ≤ 0")
case True
with ‹?lhs› show ?rhs
by safe (simp_all only: ball_eq_empty [of y e, symmetric] ball_eq_empty [of x d, symmetric])
next
case False
with ‹?lhs› show ?rhs
apply (auto simp: set_eq_subset ball_subset_ball_iff dist_norm norm_minus_commute algebra_simps)
apply (metis add_le_same_cancel1 le_add_same_cancel1 norm_ge_zero norm_pths(2) order_trans)
apply (metis add_increasing2 add_le_imp_le_right eq_iff norm_ge_zero)
done
qed
next
assume ?rhs then show ?lhs
by (auto simp: set_eq_subset ball_subset_ball_iff)
qed
lemma cball_eq_cball_iff:
fixes x :: "'a :: euclidean_space"
shows "cball x d = cball y e ⟷ d < 0 ∧ e < 0 ∨ x=y ∧ d=e"
(is "?lhs = ?rhs")
proof
assume ?lhs
then show ?rhs
proof (cases "d < 0 ∨ e < 0")
case True
with ‹?lhs› show ?rhs
by safe (simp_all only: cball_eq_empty [of y e, symmetric] cball_eq_empty [of x d, symmetric])
next
case False
with ‹?lhs› show ?rhs
apply (auto simp: set_eq_subset cball_subset_cball_iff dist_norm norm_minus_commute algebra_simps)
apply (metis add_le_same_cancel1 le_add_same_cancel1 norm_ge_zero norm_pths(2) order_trans)
apply (metis add_increasing2 add_le_imp_le_right eq_iff norm_ge_zero)
done
qed
next
assume ?rhs then show ?lhs
by (auto simp: set_eq_subset cball_subset_cball_iff)
qed
lemma ball_eq_cball_iff:
fixes x :: "'a :: euclidean_space"
shows "ball x d = cball y e ⟷ d ≤ 0 ∧ e < 0" (is "?lhs = ?rhs")
proof
assume ?lhs
then show ?rhs
apply (auto simp: set_eq_subset ball_subset_cball_iff cball_subset_ball_iff algebra_simps)
apply (metis add_increasing2 add_le_cancel_right add_less_same_cancel1 dist_not_less_zero less_le_trans zero_le_dist)
apply (metis add_less_same_cancel1 dist_not_less_zero less_le_trans not_le)
using ‹?lhs› ball_eq_empty cball_eq_empty apply blast+
done
next
assume ?rhs then show ?lhs by auto
qed
lemma cball_eq_ball_iff:
fixes x :: "'a :: euclidean_space"
shows "cball x d = ball y e ⟷ d < 0 ∧ e ≤ 0"
using ball_eq_cball_iff by blast
lemma finite_ball_avoid:
fixes S :: "'a :: euclidean_space set"
assumes "open S" "finite X" "p ∈ S"
shows "∃e>0. ∀w∈ball p e. w∈S ∧ (w≠p ⟶ w∉X)"
proof -
obtain e1 where "0 < e1" and e1_b:"ball p e1 ⊆ S"
using open_contains_ball_eq[OF ‹open S›] assms by auto
obtain e2 where "0 < e2" and "∀x∈X. x ≠ p ⟶ e2 ≤ dist p x"
using finite_set_avoid[OF ‹finite X›,of p] by auto
hence "∀w∈ball p (min e1 e2). w∈S ∧ (w≠p ⟶ w∉X)" using e1_b by auto
thus "∃e>0. ∀w∈ball p e. w ∈ S ∧ (w ≠ p ⟶ w ∉ X)" using ‹e2>0› ‹e1>0›
apply (rule_tac x="min e1 e2" in exI)
by auto
qed
lemma finite_cball_avoid:
fixes S :: "'a :: euclidean_space set"
assumes "open S" "finite X" "p ∈ S"
shows "∃e>0. ∀w∈cball p e. w∈S ∧ (w≠p ⟶ w∉X)"
proof -
obtain e1 where "e1>0" and e1: "∀w∈ball p e1. w∈S ∧ (w≠p ⟶ w∉X)"
using finite_ball_avoid[OF assms] by auto
define e2 where "e2 ≡ e1/2"
have "e2>0" and "e2 < e1" unfolding e2_def using ‹e1>0› by auto
then have "cball p e2 ⊆ ball p e1" by (subst cball_subset_ball_iff,auto)
then show "∃e>0. ∀w∈cball p e. w ∈ S ∧ (w ≠ p ⟶ w ∉ X)" using ‹e2>0› e1 by auto
qed
subsection‹Various separability-type properties›
lemma univ_second_countable:
obtains ℬ :: "'a::euclidean_space set set"
where "countable ℬ" "⋀C. C ∈ ℬ ⟹ open C"
"⋀S. open S ⟹ ∃U. U ⊆ ℬ ∧ S = ⋃U"
by (metis ex_countable_basis topological_basis_def)
lemma subset_second_countable:
obtains ℬ :: "'a:: euclidean_space set set"
where "countable ℬ"
"{} ∉ ℬ"
"⋀C. C ∈ ℬ ⟹ openin(subtopology euclidean S) C"
"⋀T. openin(subtopology euclidean S) T ⟹ ∃𝒰. 𝒰 ⊆ ℬ ∧ T = ⋃𝒰"
proof -
obtain ℬ :: "'a set set"
where "countable ℬ"
and opeB: "⋀C. C ∈ ℬ ⟹ openin(subtopology euclidean S) C"
and ℬ: "⋀T. openin(subtopology euclidean S) T ⟹ ∃𝒰. 𝒰 ⊆ ℬ ∧ T = ⋃𝒰"
proof -
obtain 𝒞 :: "'a set set"
where "countable 𝒞" and ope: "⋀C. C ∈ 𝒞 ⟹ open C"
and 𝒞: "⋀S. open S ⟹ ∃U. U ⊆ 𝒞 ∧ S = ⋃U"
by (metis univ_second_countable that)
show ?thesis
proof
show "countable ((λC. S ∩ C) ` 𝒞)"
by (simp add: ‹countable 𝒞›)
show "⋀C. C ∈ op ∩ S ` 𝒞 ⟹ openin (subtopology euclidean S) C"
using ope by auto
show "⋀T. openin (subtopology euclidean S) T ⟹ ∃𝒰⊆op ∩ S ` 𝒞. T = ⋃𝒰"
by (metis 𝒞 image_mono inf_Sup openin_open)
qed
qed
show ?thesis
proof
show "countable (ℬ - {{}})"
using ‹countable ℬ› by blast
show "⋀C. ⟦C ∈ ℬ - {{}}⟧ ⟹ openin (subtopology euclidean S) C"
by (simp add: ‹⋀C. C ∈ ℬ ⟹ openin (subtopology euclidean S) C›)
show "∃𝒰⊆ℬ - {{}}. T = ⋃𝒰" if "openin (subtopology euclidean S) T" for T
using ℬ [OF that]
apply clarify
apply (rule_tac x="𝒰 - {{}}" in exI, auto)
done
qed auto
qed
lemma univ_second_countable_sequence:
obtains B :: "nat ⇒ 'a::euclidean_space set"
where "inj B" "⋀n. open(B n)" "⋀S. open S ⟹ ∃k. S = ⋃{B n |n. n ∈ k}"
proof -
obtain ℬ :: "'a set set"
where "countable ℬ"
and op: "⋀C. C ∈ ℬ ⟹ open C"
and Un: "⋀S. open S ⟹ ∃U. U ⊆ ℬ ∧ S = ⋃U"
using univ_second_countable by blast
have *: "infinite (range (λn. ball (0::'a) (inverse(Suc n))))"
apply (rule Infinite_Set.range_inj_infinite)
apply (simp add: inj_on_def ball_eq_ball_iff)
done
have "infinite ℬ"
proof
assume "finite ℬ"
then have "finite (Union ` (Pow ℬ))"
by simp
then have "finite (range (λn. ball (0::'a) (inverse(Suc n))))"
apply (rule rev_finite_subset)
by (metis (no_types, lifting) PowI image_eqI image_subset_iff Un [OF open_ball])
with * show False by simp
qed
obtain f :: "nat ⇒ 'a set" where "ℬ = range f" "inj f"
by (blast intro: countable_as_injective_image [OF ‹countable ℬ› ‹infinite ℬ›])
have *: "∃k. S = ⋃{f n |n. n ∈ k}" if "open S" for S
using Un [OF that]
apply clarify
apply (rule_tac x="f-`U" in exI)
using ‹inj f› ‹ℬ = range f› apply force
done
show ?thesis
apply (rule that [OF ‹inj f› _ *])
apply (auto simp: ‹ℬ = range f› op)
done
qed
proposition separable:
fixes S :: "'a:: euclidean_space set"
obtains T where "countable T" "T ⊆ S" "S ⊆ closure T"
proof -
obtain ℬ :: "'a:: euclidean_space set set"
where "countable ℬ"
and "{} ∉ ℬ"
and ope: "⋀C. C ∈ ℬ ⟹ openin(subtopology euclidean S) C"
and if_ope: "⋀T. openin(subtopology euclidean S) T ⟹ ∃𝒰. 𝒰 ⊆ ℬ ∧ T = ⋃𝒰"
by (meson subset_second_countable)
then obtain f where f: "⋀C. C ∈ ℬ ⟹ f C ∈ C"
by (metis equals0I)
show ?thesis
proof
show "countable (f ` ℬ)"
by (simp add: ‹countable ℬ›)
show "f ` ℬ ⊆ S"
using ope f openin_imp_subset by blast
show "S ⊆ closure (f ` ℬ)"
proof (clarsimp simp: closure_approachable)
fix x and e::real
assume "x ∈ S" "0 < e"
have "openin (subtopology euclidean S) (S ∩ ball x e)"
by (simp add: openin_Int_open)
with if_ope obtain 𝒰 where 𝒰: "𝒰 ⊆ ℬ" "S ∩ ball x e = ⋃𝒰"
by meson
show "∃C ∈ ℬ. dist (f C) x < e"
proof (cases "𝒰 = {}")
case True
then show ?thesis
using ‹0 < e› 𝒰 ‹x ∈ S› by auto
next
case False
then obtain C where "C ∈ 𝒰" by blast
show ?thesis
proof
show "dist (f C) x < e"
by (metis Int_iff Union_iff 𝒰 ‹C ∈ 𝒰› dist_commute f mem_ball subsetCE)
show "C ∈ ℬ"
using ‹𝒰 ⊆ ℬ› ‹C ∈ 𝒰› by blast
qed
qed
qed
qed
qed
proposition Lindelof:
fixes ℱ :: "'a::euclidean_space set set"
assumes ℱ: "⋀S. S ∈ ℱ ⟹ open S"
obtains ℱ' where "ℱ' ⊆ ℱ" "countable ℱ'" "⋃ℱ' = ⋃ℱ"
proof -
obtain ℬ :: "'a set set"
where "countable ℬ" "⋀C. C ∈ ℬ ⟹ open C"
and ℬ: "⋀S. open S ⟹ ∃U. U ⊆ ℬ ∧ S = ⋃U"
using univ_second_countable by blast
define 𝒟 where "𝒟 ≡ {S. S ∈ ℬ ∧ (∃U. U ∈ ℱ ∧ S ⊆ U)}"
have "countable 𝒟"
apply (rule countable_subset [OF _ ‹countable ℬ›])
apply (force simp: 𝒟_def)
done
have "⋀S. ∃U. S ∈ 𝒟 ⟶ U ∈ ℱ ∧ S ⊆ U"
by (simp add: 𝒟_def)
then obtain G where G: "⋀S. S ∈ 𝒟 ⟶ G S ∈ ℱ ∧ S ⊆ G S"
by metis
have "⋃ℱ ⊆ ⋃𝒟"
unfolding 𝒟_def by (blast dest: ℱ ℬ)
moreover have "⋃𝒟 ⊆ ⋃ℱ"
using 𝒟_def by blast
ultimately have eq1: "⋃ℱ = ⋃𝒟" ..
have eq2: "⋃𝒟 = UNION 𝒟 G"
using G eq1 by auto
show ?thesis
apply (rule_tac ℱ' = "G ` 𝒟" in that)
using G ‹countable 𝒟› apply (auto simp: eq1 eq2)
done
qed
lemma Lindelof_openin:
fixes ℱ :: "'a::euclidean_space set set"
assumes "⋀S. S ∈ ℱ ⟹ openin (subtopology euclidean U) S"
obtains ℱ' where "ℱ' ⊆ ℱ" "countable ℱ'" "⋃ℱ' = ⋃ℱ"
proof -
have "⋀S. S ∈ ℱ ⟹ ∃T. open T ∧ S = U ∩ T"
using assms by (simp add: openin_open)
then obtain tf where tf: "⋀S. S ∈ ℱ ⟹ open (tf S) ∧ (S = U ∩ tf S)"
by metis
have [simp]: "⋀ℱ'. ℱ' ⊆ ℱ ⟹ ⋃ℱ' = U ∩ ⋃(tf ` ℱ')"
using tf by fastforce
obtain 𝒢 where "countable 𝒢 ∧ 𝒢 ⊆ tf ` ℱ" "⋃𝒢 = UNION ℱ tf"
using tf by (force intro: Lindelof [of "tf ` ℱ"])
then obtain ℱ' where ℱ': "ℱ' ⊆ ℱ" "countable ℱ'" "⋃ℱ' = ⋃ℱ"
by (clarsimp simp add: countable_subset_image)
then show ?thesis ..
qed
lemma countable_disjoint_open_subsets:
fixes ℱ :: "'a::euclidean_space set set"
assumes "⋀S. S ∈ ℱ ⟹ open S" and pw: "pairwise disjnt ℱ"
shows "countable ℱ"
proof -
obtain ℱ' where "ℱ' ⊆ ℱ" "countable ℱ'" "⋃ℱ' = ⋃ℱ"
by (meson assms Lindelof)
with pw have "ℱ ⊆ insert {} ℱ'"
by (fastforce simp add: pairwise_def disjnt_iff)
then show ?thesis
by (simp add: ‹countable ℱ'› countable_subset)
qed
lemma closedin_compact:
"⟦compact S; closedin (subtopology euclidean S) T⟧ ⟹ compact T"
by (metis closedin_closed compact_Int_closed)
lemma closedin_compact_eq:
fixes S :: "'a::t2_space set"
shows
"compact S
⟹ (closedin (subtopology euclidean S) T ⟷
compact T ∧ T ⊆ S)"
by (metis closedin_imp_subset closedin_compact closed_subset compact_imp_closed)
lemma continuous_imp_closed_map:
fixes f :: "'a::metric_space ⇒ 'b::metric_space"
assumes "closedin (subtopology euclidean S) U"
"continuous_on S f" "image f S = T" "compact S"
shows "closedin (subtopology euclidean T) (image f U)"
by (metis assms closedin_compact_eq compact_continuous_image continuous_on_subset subset_image_iff)
lemma continuous_imp_quotient_map:
fixes f :: "'a::metric_space ⇒ 'b::metric_space"
assumes "continuous_on S f" "image f S = T" "compact S" "U ⊆ T"
shows "openin (subtopology euclidean S) {x. x ∈ S ∧ f x ∈ U} ⟷
openin (subtopology euclidean T) U"
by (metis (no_types, lifting) Collect_cong assms closed_map_imp_quotient_map continuous_imp_closed_map)
subsection‹ Finite intersection property›
text‹Also developed in HOL's toplogical spaces theory, but the Heine-Borel type class isn't available there.›
lemma closed_imp_fip:
fixes S :: "'a::heine_borel set"
assumes "closed S"
and T: "T ∈ ℱ" "bounded T"
and clof: "⋀T. T ∈ ℱ ⟹ closed T"
and none: "⋀ℱ'. ⟦finite ℱ'; ℱ' ⊆ ℱ⟧ ⟹ S ∩ ⋂ℱ' ≠ {}"
shows "S ∩ ⋂ℱ ≠ {}"
proof -
have "compact (S ∩ T)"
using ‹closed S› clof compact_eq_bounded_closed T by blast
then have "(S ∩ T) ∩ ⋂ℱ ≠ {}"
apply (rule compact_imp_fip)
apply (simp add: clof)
by (metis Int_assoc complete_lattice_class.Inf_insert finite_insert insert_subset none ‹T ∈ ℱ›)
then show ?thesis by blast
qed
lemma closed_imp_fip_compact:
fixes S :: "'a::heine_borel set"
shows
"⟦closed S; ⋀T. T ∈ ℱ ⟹ compact T;
⋀ℱ'. ⟦finite ℱ'; ℱ' ⊆ ℱ⟧ ⟹ S ∩ ⋂ℱ' ≠ {}⟧
⟹ S ∩ ⋂ℱ ≠ {}"
by (metis Inf_greatest closed_imp_fip compact_eq_bounded_closed empty_subsetI finite.emptyI inf.orderE)
lemma closed_fip_heine_borel:
fixes ℱ :: "'a::heine_borel set set"
assumes "closed S" "T ∈ ℱ" "bounded T"
and "⋀T. T ∈ ℱ ⟹ closed T"
and "⋀ℱ'. ⟦finite ℱ'; ℱ' ⊆ ℱ⟧ ⟹ ⋂ℱ' ≠ {}"
shows "⋂ℱ ≠ {}"
proof -
have "UNIV ∩ ⋂ℱ ≠ {}"
using assms closed_imp_fip [OF closed_UNIV] by auto
then show ?thesis by simp
qed
lemma compact_fip_heine_borel:
fixes ℱ :: "'a::heine_borel set set"
assumes clof: "⋀T. T ∈ ℱ ⟹ compact T"
and none: "⋀ℱ'. ⟦finite ℱ'; ℱ' ⊆ ℱ⟧ ⟹ ⋂ℱ' ≠ {}"
shows "⋂ℱ ≠ {}"
by (metis InterI all_not_in_conv clof closed_fip_heine_borel compact_eq_bounded_closed none)
lemma compact_sequence_with_limit:
fixes f :: "nat ⇒ 'a::heine_borel"
shows "(f ⤏ l) sequentially ⟹ compact (insert l (range f))"
apply (simp add: compact_eq_bounded_closed, auto)
apply (simp add: convergent_imp_bounded)
by (simp add: closed_limpt islimpt_insert sequence_unique_limpt)
subsection‹Componentwise limits and continuity›
text‹But is the premise really necessary? Need to generalise @{thm euclidean_dist_l2}›
lemma Euclidean_dist_upper: "i ∈ Basis ⟹ dist (x ∙ i) (y ∙ i) ≤ dist x y"
by (metis (no_types) member_le_setL2 euclidean_dist_l2 finite_Basis)
text‹But is the premise @{term ‹i ∈ Basis›} really necessary?›
lemma open_preimage_inner:
assumes "open S" "i ∈ Basis"
shows "open {x. x ∙ i ∈ S}"
proof (rule openI, simp)
fix x
assume x: "x ∙ i ∈ S"
with assms obtain e where "0 < e" and e: "ball (x ∙ i) e ⊆ S"
by (auto simp: open_contains_ball_eq)
have "∃e>0. ball (y ∙ i) e ⊆ S" if dxy: "dist x y < e / 2" for y
proof (intro exI conjI)
have "dist (x ∙ i) (y ∙ i) < e / 2"
by (meson ‹i ∈ Basis› dual_order.trans Euclidean_dist_upper not_le that)
then have "dist (x ∙ i) z < e" if "dist (y ∙ i) z < e / 2" for z
by (metis dist_commute dist_triangle_half_l that)
then have "ball (y ∙ i) (e / 2) ⊆ ball (x ∙ i) e"
using mem_ball by blast
with e show "ball (y ∙ i) (e / 2) ⊆ S"
by (metis order_trans)
qed (simp add: ‹0 < e›)
then show "∃e>0. ball x e ⊆ {s. s ∙ i ∈ S}"
by (metis (no_types, lifting) ‹0 < e› ‹open S› half_gt_zero_iff mem_Collect_eq mem_ball open_contains_ball_eq subsetI)
qed
proposition tendsto_componentwise_iff:
fixes f :: "_ ⇒ 'b::euclidean_space"
shows "(f ⤏ l) F ⟷ (∀i ∈ Basis. ((λx. (f x ∙ i)) ⤏ (l ∙ i)) F)"
(is "?lhs = ?rhs")
proof
assume ?lhs
then show ?rhs
unfolding tendsto_def
apply clarify
apply (drule_tac x="{s. s ∙ i ∈ S}" in spec)
apply (auto simp: open_preimage_inner)
done
next
assume R: ?rhs
then have "⋀e. e > 0 ⟹ ∀i∈Basis. ∀⇩F x in F. dist (f x ∙ i) (l ∙ i) < e"
unfolding tendsto_iff by blast
then have R': "⋀e. e > 0 ⟹ ∀⇩F x in F. ∀i∈Basis. dist (f x ∙ i) (l ∙ i) < e"
by (simp add: eventually_ball_finite_distrib [symmetric])
show ?lhs
unfolding tendsto_iff
proof clarify
fix e::real
assume "0 < e"
have *: "setL2 (λi. dist (f x ∙ i) (l ∙ i)) Basis < e"
if "∀i∈Basis. dist (f x ∙ i) (l ∙ i) < e / real DIM('b)" for x
proof -
have "setL2 (λi. dist (f x ∙ i) (l ∙ i)) Basis ≤ sum (λi. dist (f x ∙ i) (l ∙ i)) Basis"
by (simp add: setL2_le_sum)
also have "... < DIM('b) * (e / real DIM('b))"
apply (rule sum_bounded_above_strict)
using that by auto
also have "... = e"
by (simp add: field_simps)
finally show "setL2 (λi. dist (f x ∙ i) (l ∙ i)) Basis < e" .
qed
have "∀⇩F x in F. ∀i∈Basis. dist (f x ∙ i) (l ∙ i) < e / DIM('b)"
apply (rule R')
using ‹0 < e› by simp
then show "∀⇩F x in F. dist (f x) l < e"
apply (rule eventually_mono)
apply (subst euclidean_dist_l2)
using * by blast
qed
qed
corollary continuous_componentwise:
"continuous F f ⟷ (∀i ∈ Basis. continuous F (λx. (f x ∙ i)))"
by (simp add: continuous_def tendsto_componentwise_iff [symmetric])
corollary continuous_on_componentwise:
fixes S :: "'a :: t2_space set"
shows "continuous_on S f ⟷ (∀i ∈ Basis. continuous_on S (λx. (f x ∙ i)))"
apply (simp add: continuous_on_eq_continuous_within)
using continuous_componentwise by blast
lemma linear_componentwise_iff:
"(linear f') ⟷ (∀i∈Basis. linear (λx. f' x ∙ i))"
apply (auto simp: linear_iff inner_left_distrib)
apply (metis inner_left_distrib euclidean_eq_iff)
by (metis euclidean_eqI inner_scaleR_left)
lemma bounded_linear_componentwise_iff:
"(bounded_linear f') ⟷ (∀i∈Basis. bounded_linear (λx. f' x ∙ i))"
(is "?lhs = ?rhs")
proof
assume ?lhs then show ?rhs
by (simp add: bounded_linear_inner_left_comp)
next
assume ?rhs
then have "(∀i∈Basis. ∃K. ∀x. ¦f' x ∙ i¦ ≤ norm x * K)" "linear f'"
by (auto simp: bounded_linear_def bounded_linear_axioms_def linear_componentwise_iff [symmetric] ball_conj_distrib)
then obtain F where F: "⋀i x. i ∈ Basis ⟹ ¦f' x ∙ i¦ ≤ norm x * F i"
by metis
have "norm (f' x) ≤ norm x * sum F Basis" for x
proof -
have "norm (f' x) ≤ (∑i∈Basis. ¦f' x ∙ i¦)"
by (rule norm_le_l1)
also have "... ≤ (∑i∈Basis. norm x * F i)"
by (metis F sum_mono)
also have "... = norm x * sum F Basis"
by (simp add: sum_distrib_left)
finally show ?thesis .
qed
then show ?lhs
by (force simp: bounded_linear_def bounded_linear_axioms_def ‹linear f'›)
qed
subsection‹Pasting functions together›
subsubsection‹on open sets›
lemma pasting_lemma:
fixes f :: "'i ⇒ 'a::topological_space ⇒ 'b::topological_space"
assumes clo: "⋀i. i ∈ I ⟹ openin (subtopology euclidean S) (T i)"
and cont: "⋀i. i ∈ I ⟹ continuous_on (T i) (f i)"
and f: "⋀i j x. ⟦i ∈ I; j ∈ I; x ∈ S ∩ T i ∩ T j⟧ ⟹ f i x = f j x"
and g: "⋀x. x ∈ S ⟹ ∃j. j ∈ I ∧ x ∈ T j ∧ g x = f j x"
shows "continuous_on S g"
proof (clarsimp simp: continuous_openin_preimage_eq)
fix U :: "'b set"
assume "open U"
have S: "⋀i. i ∈ I ⟹ (T i) ⊆ S"
using clo openin_imp_subset by blast
have *: "{x ∈ S. g x ∈ U} = ⋃{{x. x ∈ (T i) ∧ (f i x) ∈ U} |i. i ∈ I}"
apply (auto simp: dest: S)
apply (metis (no_types, lifting) g mem_Collect_eq)
using clo f g openin_imp_subset by fastforce
show "openin (subtopology euclidean S) {x ∈ S. g x ∈ U}"
apply (subst *)
apply (rule openin_Union, clarify)
apply (metis (full_types) ‹open U› cont clo openin_trans continuous_openin_preimage_gen)
done
qed
lemma pasting_lemma_exists:
fixes f :: "'i ⇒ 'a::topological_space ⇒ 'b::topological_space"
assumes S: "S ⊆ (⋃i ∈ I. T i)"
and clo: "⋀i. i ∈ I ⟹ openin (subtopology euclidean S) (T i)"
and cont: "⋀i. i ∈ I ⟹ continuous_on (T i) (f i)"
and f: "⋀i j x. ⟦i ∈ I; j ∈ I; x ∈ S ∩ T i ∩ T j⟧ ⟹ f i x = f j x"
obtains g where "continuous_on S g" "⋀x i. ⟦i ∈ I; x ∈ S ∩ T i⟧ ⟹ g x = f i x"
proof
show "continuous_on S (λx. f (SOME i. i ∈ I ∧ x ∈ T i) x)"
apply (rule pasting_lemma [OF clo cont])
apply (blast intro: f)+
apply (metis (mono_tags, lifting) S UN_iff subsetCE someI)
done
next
fix x i
assume "i ∈ I" "x ∈ S ∩ T i"
then show "f (SOME i. i ∈ I ∧ x ∈ T i) x = f i x"
by (metis (no_types, lifting) IntD2 IntI f someI_ex)
qed
subsubsection‹Likewise on closed sets, with a finiteness assumption›
lemma pasting_lemma_closed:
fixes f :: "'i ⇒ 'a::topological_space ⇒ 'b::topological_space"
assumes "finite I"
and clo: "⋀i. i ∈ I ⟹ closedin (subtopology euclidean S) (T i)"
and cont: "⋀i. i ∈ I ⟹ continuous_on (T i) (f i)"
and f: "⋀i j x. ⟦i ∈ I; j ∈ I; x ∈ S ∩ T i ∩ T j⟧ ⟹ f i x = f j x"
and g: "⋀x. x ∈ S ⟹ ∃j. j ∈ I ∧ x ∈ T j ∧ g x = f j x"
shows "continuous_on S g"
proof (clarsimp simp: continuous_closedin_preimage_eq)
fix U :: "'b set"
assume "closed U"
have *: "{x ∈ S. g x ∈ U} = ⋃{{x. x ∈ (T i) ∧ (f i x) ∈ U} |i. i ∈ I}"
apply auto
apply (metis (no_types, lifting) g mem_Collect_eq)
using clo closedin_closed apply blast
apply (metis Int_iff f g clo closedin_limpt inf.absorb_iff2)
done
show "closedin (subtopology euclidean S) {x ∈ S. g x ∈ U}"
apply (subst *)
apply (rule closedin_Union)
using ‹finite I› apply simp
apply (blast intro: ‹closed U› continuous_closedin_preimage cont clo closedin_trans)
done
qed
lemma pasting_lemma_exists_closed:
fixes f :: "'i ⇒ 'a::topological_space ⇒ 'b::topological_space"
assumes "finite I"
and S: "S ⊆ (⋃i ∈ I. T i)"
and clo: "⋀i. i ∈ I ⟹ closedin (subtopology euclidean S) (T i)"
and cont: "⋀i. i ∈ I ⟹ continuous_on (T i) (f i)"
and f: "⋀i j x. ⟦i ∈ I; j ∈ I; x ∈ S ∩ T i ∩ T j⟧ ⟹ f i x = f j x"
obtains g where "continuous_on S g" "⋀x i. ⟦i ∈ I; x ∈ S ∩ T i⟧ ⟹ g x = f i x"
proof
show "continuous_on S (λx. f (SOME i. i ∈ I ∧ x ∈ T i) x)"
apply (rule pasting_lemma_closed [OF ‹finite I› clo cont])
apply (blast intro: f)+
apply (metis (mono_tags, lifting) S UN_iff subsetCE someI)
done
next
fix x i
assume "i ∈ I" "x ∈ S ∩ T i"
then show "f (SOME i. i ∈ I ∧ x ∈ T i) x = f i x"
by (metis (no_types, lifting) IntD2 IntI f someI_ex)
qed
lemma tube_lemma:
assumes "compact K"
assumes "open W"
assumes "{x0} × K ⊆ W"
shows "∃X0. x0 ∈ X0 ∧ open X0 ∧ X0 × K ⊆ W"
proof -
{
fix y assume "y ∈ K"
then have "(x0, y) ∈ W" using assms by auto
with ‹open W›
have "∃X0 Y. open X0 ∧ open Y ∧ x0 ∈ X0 ∧ y ∈ Y ∧ X0 × Y ⊆ W"
by (rule open_prod_elim) blast
}
then obtain X0 Y where
*: "∀y ∈ K. open (X0 y) ∧ open (Y y) ∧ x0 ∈ X0 y ∧ y ∈ Y y ∧ X0 y × Y y ⊆ W"
by metis
from * have "∀t∈Y ` K. open t" "K ⊆ ⋃(Y ` K)" by auto
with ‹compact K› obtain CC where CC: "CC ⊆ Y ` K" "finite CC" "K ⊆ ⋃CC"
by (meson compactE)
then obtain c where c: "⋀C. C ∈ CC ⟹ c C ∈ K ∧ C = Y (c C)"
by (force intro!: choice)
with * CC show ?thesis
by (force intro!: exI[where x="⋂C∈CC. X0 (c C)"])
qed
lemma continuous_on_prod_compactE:
fixes fx::"'a::topological_space × 'b::topological_space ⇒ 'c::metric_space"
and e::real
assumes cont_fx: "continuous_on (U × C) fx"
assumes "compact C"
assumes [intro]: "x0 ∈ U"
notes [continuous_intros] = continuous_on_compose2[OF cont_fx]
assumes "e > 0"
obtains X0 where "x0 ∈ X0" "open X0"
"∀x∈X0 ∩ U. ∀t ∈ C. dist (fx (x, t)) (fx (x0, t)) ≤ e"
proof -
define psi where "psi = (λ(x, t). dist (fx (x, t)) (fx (x0, t)))"
define W0 where "W0 = {(x, t) ∈ U × C. psi (x, t) < e}"
have W0_eq: "W0 = psi -` {..<e} ∩ U × C"
by (auto simp: vimage_def W0_def)
have "open {..<e}" by simp
have "continuous_on (U × C) psi"
by (auto intro!: continuous_intros simp: psi_def split_beta')
from this[unfolded continuous_on_open_invariant, rule_format, OF ‹open {..<e}›]
obtain W where W: "open W" "W ∩ U × C = W0 ∩ U × C"
unfolding W0_eq by blast
have "{x0} × C ⊆ W ∩ U × C"
unfolding W
by (auto simp: W0_def psi_def ‹0 < e›)
then have "{x0} × C ⊆ W" by blast
from tube_lemma[OF ‹compact C› ‹open W› this]
obtain X0 where X0: "x0 ∈ X0" "open X0" "X0 × C ⊆ W"
by blast
have "∀x∈X0 ∩ U. ∀t ∈ C. dist (fx (x, t)) (fx (x0, t)) ≤ e"
proof safe
fix x assume x: "x ∈ X0" "x ∈ U"
fix t assume t: "t ∈ C"
have "dist (fx (x, t)) (fx (x0, t)) = psi (x, t)"
by (auto simp: psi_def)
also
{
have "(x, t) ∈ X0 × C"
using t x
by auto
also note ‹… ⊆ W›
finally have "(x, t) ∈ W" .
with t x have "(x, t) ∈ W ∩ U × C"
by blast
also note ‹W ∩ U × C = W0 ∩ U × C›
finally have "psi (x, t) < e"
by (auto simp: W0_def)
}
finally show "dist (fx (x, t)) (fx (x0, t)) ≤ e" by simp
qed
from X0(1,2) this show ?thesis ..
qed
subsection‹Constancy of a function from a connected set into a finite, disconnected or discrete set›
text‹Still missing: versions for a set that is smaller than R, or countable.›
lemma continuous_disconnected_range_constant:
assumes s: "connected s"
and conf: "continuous_on s f"
and fim: "f ` s ⊆ t"
and cct: "⋀y. y ∈ t ⟹ connected_component_set t y = {y}"
shows "∃a. ∀x ∈ s. f x = a"
proof (cases "s = {}")
case True then show ?thesis by force
next
case False
{ fix x assume "x ∈ s"
then have "f ` s ⊆ {f x}"
by (metis connected_continuous_image conf connected_component_maximal fim image_subset_iff rev_image_eqI s cct)
}
with False show ?thesis
by blast
qed
lemma discrete_subset_disconnected:
fixes s :: "'a::topological_space set"
fixes t :: "'b::real_normed_vector set"
assumes conf: "continuous_on s f"
and no: "⋀x. x ∈ s ⟹ ∃e>0. ∀y. y ∈ s ∧ f y ≠ f x ⟶ e ≤ norm (f y - f x)"
shows "f ` s ⊆ {y. connected_component_set (f ` s) y = {y}}"
proof -
{ fix x assume x: "x ∈ s"
then obtain e where "e>0" and ele: "⋀y. ⟦y ∈ s; f y ≠ f x⟧ ⟹ e ≤ norm (f y - f x)"
using conf no [OF x] by auto
then have e2: "0 ≤ e / 2"
by simp
have "f y = f x" if "y ∈ s" and ccs: "f y ∈ connected_component_set (f ` s) (f x)" for y
apply (rule ccontr)
using connected_closed [of "connected_component_set (f ` s) (f x)"] ‹e>0›
apply (simp add: del: ex_simps)
apply (drule spec [where x="cball (f x) (e / 2)"])
apply (drule spec [where x="- ball(f x) e"])
apply (auto simp: dist_norm open_closed [symmetric] simp del: le_divide_eq_numeral1 dest!: connected_component_in)
apply (metis diff_self e2 ele norm_minus_commute norm_zero not_less)
using centre_in_cball connected_component_refl_eq e2 x apply blast
using ccs
apply (force simp: cball_def dist_norm norm_minus_commute dest: ele [OF ‹y ∈ s›])
done
moreover have "connected_component_set (f ` s) (f x) ⊆ f ` s"
by (auto simp: connected_component_in)
ultimately have "connected_component_set (f ` s) (f x) = {f x}"
by (auto simp: x)
}
with assms show ?thesis
by blast
qed
lemma finite_implies_discrete:
fixes s :: "'a::topological_space set"
assumes "finite (f ` s)"
shows "(∀x ∈ s. ∃e>0. ∀y. y ∈ s ∧ f y ≠ f x ⟶ e ≤ norm (f y - f x))"
proof -
have "∃e>0. ∀y. y ∈ s ∧ f y ≠ f x ⟶ e ≤ norm (f y - f x)" if "x ∈ s" for x
proof (cases "f ` s - {f x} = {}")
case True
with zero_less_numeral show ?thesis
by (fastforce simp add: Set.image_subset_iff cong: conj_cong)
next
case False
then obtain z where z: "z ∈ s" "f z ≠ f x"
by blast
have finn: "finite {norm (z - f x) |z. z ∈ f ` s - {f x}}"
using assms by simp
then have *: "0 < Inf{norm(z - f x) | z. z ∈ f ` s - {f x}}"
apply (rule finite_imp_less_Inf)
using z apply force+
done
show ?thesis
by (force intro!: * cInf_le_finite [OF finn])
qed
with assms show ?thesis
by blast
qed
text‹This proof requires the existence of two separate values of the range type.›
lemma finite_range_constant_imp_connected:
assumes "⋀f::'a::topological_space ⇒ 'b::real_normed_algebra_1.
⟦continuous_on s f; finite(f ` s)⟧ ⟹ ∃a. ∀x ∈ s. f x = a"
shows "connected s"
proof -
{ fix t u
assume clt: "closedin (subtopology euclidean s) t"
and clu: "closedin (subtopology euclidean s) u"
and tue: "t ∩ u = {}" and tus: "t ∪ u = s"
have conif: "continuous_on s (λx. if x ∈ t then 0 else 1)"
apply (subst tus [symmetric])
apply (rule continuous_on_cases_local)
using clt clu tue
apply (auto simp: tus continuous_on_const)
done
have fi: "finite ((λx. if x ∈ t then 0 else 1) ` s)"
by (rule finite_subset [of _ "{0,1}"]) auto
have "t = {} ∨ u = {}"
using assms [OF conif fi] tus [symmetric]
by (auto simp: Ball_def) (metis IntI empty_iff one_neq_zero tue)
}
then show ?thesis
by (simp add: connected_closedin_eq)
qed
lemma continuous_disconnected_range_constant_eq:
"(connected S ⟷
(∀f::'a::topological_space ⇒ 'b::real_normed_algebra_1.
∀t. continuous_on S f ∧ f ` S ⊆ t ∧ (∀y ∈ t. connected_component_set t y = {y})
⟶ (∃a::'b. ∀x ∈ S. f x = a)))" (is ?thesis1)
and continuous_discrete_range_constant_eq:
"(connected S ⟷
(∀f::'a::topological_space ⇒ 'b::real_normed_algebra_1.
continuous_on S f ∧
(∀x ∈ S. ∃e. 0 < e ∧ (∀y. y ∈ S ∧ (f y ≠ f x) ⟶ e ≤ norm(f y - f x)))
⟶ (∃a::'b. ∀x ∈ S. f x = a)))" (is ?thesis2)
and continuous_finite_range_constant_eq:
"(connected S ⟷
(∀f::'a::topological_space ⇒ 'b::real_normed_algebra_1.
continuous_on S f ∧ finite (f ` S)
⟶ (∃a::'b. ∀x ∈ S. f x = a)))" (is ?thesis3)
proof -
have *: "⋀s t u v. ⟦s ⟹ t; t ⟹ u; u ⟹ v; v ⟹ s⟧
⟹ (s ⟷ t) ∧ (s ⟷ u) ∧ (s ⟷ v)"
by blast
have "?thesis1 ∧ ?thesis2 ∧ ?thesis3"
apply (rule *)
using continuous_disconnected_range_constant apply metis
apply clarify
apply (frule discrete_subset_disconnected; blast)
apply (blast dest: finite_implies_discrete)
apply (blast intro!: finite_range_constant_imp_connected)
done
then show ?thesis1 ?thesis2 ?thesis3
by blast+
qed
lemma continuous_discrete_range_constant:
fixes f :: "'a::topological_space ⇒ 'b::real_normed_algebra_1"
assumes S: "connected S"
and "continuous_on S f"
and "⋀x. x ∈ S ⟹ ∃e>0. ∀y. y ∈ S ∧ f y ≠ f x ⟶ e ≤ norm (f y - f x)"
obtains a where "⋀x. x ∈ S ⟹ f x = a"
using continuous_discrete_range_constant_eq [THEN iffD1, OF S] assms
by blast
lemma continuous_finite_range_constant:
fixes f :: "'a::topological_space ⇒ 'b::real_normed_algebra_1"
assumes "connected S"
and "continuous_on S f"
and "finite (f ` S)"
obtains a where "⋀x. x ∈ S ⟹ f x = a"
using assms continuous_finite_range_constant_eq
by blast
subsection ‹Continuous Extension›
definition clamp :: "'a::euclidean_space ⇒ 'a ⇒ 'a ⇒ 'a" where
"clamp a b x = (if (∀i∈Basis. a ∙ i ≤ b ∙ i)
then (∑i∈Basis. (if x∙i < a∙i then a∙i else if x∙i ≤ b∙i then x∙i else b∙i) *⇩R i)
else a)"
lemma clamp_in_interval[simp]:
assumes "⋀i. i ∈ Basis ⟹ a ∙ i ≤ b ∙ i"
shows "clamp a b x ∈ cbox a b"
unfolding clamp_def
using box_ne_empty(1)[of a b] assms by (auto simp: cbox_def)
lemma clamp_cancel_cbox[simp]:
fixes x a b :: "'a::euclidean_space"
assumes x: "x ∈ cbox a b"
shows "clamp a b x = x"
using assms
by (auto simp: clamp_def mem_box intro!: euclidean_eqI[where 'a='a])
lemma clamp_empty_interval:
assumes "i ∈ Basis" "a ∙ i > b ∙ i"
shows "clamp a b = (λ_. a)"
using assms
by (force simp: clamp_def[abs_def] split: if_splits intro!: ext)
lemma dist_clamps_le_dist_args:
fixes x :: "'a::euclidean_space"
shows "dist (clamp a b y) (clamp a b x) ≤ dist y x"
proof cases
assume le: "(∀i∈Basis. a ∙ i ≤ b ∙ i)"
then have "(∑i∈Basis. (dist (clamp a b y ∙ i) (clamp a b x ∙ i))⇧2) ≤
(∑i∈Basis. (dist (y ∙ i) (x ∙ i))⇧2)"
by (auto intro!: sum_mono simp: clamp_def dist_real_def abs_le_square_iff[symmetric])
then show ?thesis
by (auto intro: real_sqrt_le_mono
simp: euclidean_dist_l2[where y=x] euclidean_dist_l2[where y="clamp a b x"] setL2_def)
qed (auto simp: clamp_def)
lemma clamp_continuous_at:
fixes f :: "'a::euclidean_space ⇒ 'b::metric_space"
and x :: 'a
assumes f_cont: "continuous_on (cbox a b) f"
shows "continuous (at x) (λx. f (clamp a b x))"
proof cases
assume le: "(∀i∈Basis. a ∙ i ≤ b ∙ i)"
show ?thesis
unfolding continuous_at_eps_delta
proof safe
fix x :: 'a
fix e :: real
assume "e > 0"
moreover have "clamp a b x ∈ cbox a b"
by (simp add: clamp_in_interval le)
moreover note f_cont[simplified continuous_on_iff]
ultimately
obtain d where d: "0 < d"
"⋀x'. x' ∈ cbox a b ⟹ dist x' (clamp a b x) < d ⟹ dist (f x') (f (clamp a b x)) < e"
by force
show "∃d>0. ∀x'. dist x' x < d ⟶
dist (f (clamp a b x')) (f (clamp a b x)) < e"
using le
by (auto intro!: d clamp_in_interval dist_clamps_le_dist_args[THEN le_less_trans])
qed
qed (auto simp: clamp_empty_interval)
lemma clamp_continuous_on:
fixes f :: "'a::euclidean_space ⇒ 'b::metric_space"
assumes f_cont: "continuous_on (cbox a b) f"
shows "continuous_on S (λx. f (clamp a b x))"
using assms
by (auto intro: continuous_at_imp_continuous_on clamp_continuous_at)
lemma clamp_bounded:
fixes f :: "'a::euclidean_space ⇒ 'b::metric_space"
assumes bounded: "bounded (f ` (cbox a b))"
shows "bounded (range (λx. f (clamp a b x)))"
proof cases
assume le: "(∀i∈Basis. a ∙ i ≤ b ∙ i)"
from bounded obtain c where f_bound: "∀x∈f ` cbox a b. dist undefined x ≤ c"
by (auto simp: bounded_any_center[where a=undefined])
then show ?thesis
by (auto intro!: exI[where x=c] clamp_in_interval[OF le[rule_format]]
simp: bounded_any_center[where a=undefined])
qed (auto simp: clamp_empty_interval image_def)
definition ext_cont :: "('a::euclidean_space ⇒ 'b::metric_space) ⇒ 'a ⇒ 'a ⇒ 'a ⇒ 'b"
where "ext_cont f a b = (λx. f (clamp a b x))"
lemma ext_cont_cancel_cbox[simp]:
fixes x a b :: "'a::euclidean_space"
assumes x: "x ∈ cbox a b"
shows "ext_cont f a b x = f x"
using assms
unfolding ext_cont_def
by (auto simp: clamp_def mem_box intro!: euclidean_eqI[where 'a='a] arg_cong[where f=f])
lemma continuous_on_ext_cont[continuous_intros]:
"continuous_on (cbox a b) f ⟹ continuous_on S (ext_cont f a b)"
by (auto intro!: clamp_continuous_on simp: ext_cont_def)
no_notation
eucl_less (infix "<e" 50)
end