# Theory Isolated

```theory Isolated
imports "HOL-Analysis.Elementary_Metric_Spaces"

begin

subsection ‹Isolate and discrete›

definition (in topological_space) isolated_in:: "'a ⇒ 'a set ⇒ bool"  (infixr "isolated'_in" 60)
where "x isolated_in S ⟷ (x∈S ∧ (∃T. open T ∧ T ∩ S = {x}))"

definition (in topological_space) discrete:: "'a set ⇒ bool"
where "discrete S ⟷ (∀x∈S. x isolated_in S)"

definition (in metric_space) uniform_discrete :: "'a set ⇒ bool" where
"uniform_discrete S ⟷ (∃e>0. ∀x∈S. ∀y∈S. dist x y < e ⟶ x = y)"

lemma discreteI: "(⋀x. x ∈ X ⟹ x isolated_in X ) ⟹ discrete X"
unfolding discrete_def by auto

lemma discreteD: "discrete X ⟹ x ∈ X ⟹ x isolated_in X "
unfolding discrete_def by auto

lemma uniformI1:
assumes "e>0" "⋀x y. ⟦x∈S;y∈S;dist x y<e⟧ ⟹ x =y "
shows "uniform_discrete S"
unfolding uniform_discrete_def using assms by auto

lemma uniformI2:
assumes "e>0" "⋀x y. ⟦x∈S;y∈S;x≠y⟧ ⟹ dist x y≥e "
shows "uniform_discrete S"
unfolding uniform_discrete_def using assms not_less by blast

lemma isolated_in_islimpt_iff:"(x isolated_in S) ⟷ (¬ (x islimpt S) ∧ x∈S)"
unfolding isolated_in_def islimpt_def by auto

lemma isolated_in_dist_Ex_iff:
fixes x::"'a::metric_space"
shows "x isolated_in S ⟷ (x∈S ∧ (∃e>0. ∀y∈S. dist x y < e ⟶ y=x))"
unfolding isolated_in_islimpt_iff islimpt_approachable by (metis dist_commute)

lemma discrete_empty[simp]: "discrete {}"
unfolding discrete_def by auto

lemma uniform_discrete_empty[simp]: "uniform_discrete {}"
unfolding uniform_discrete_def by (simp add: gt_ex)

lemma isolated_in_insert:
fixes x :: "'a::t1_space"
shows "x isolated_in (insert a S) ⟷ x isolated_in S ∨ (x=a ∧ ¬ (x islimpt S))"
by (meson insert_iff islimpt_insert isolated_in_islimpt_iff)

lemma isolated_inI:
assumes "x∈S" "open T" "T ∩ S = {x}"
shows   "x isolated_in S"
using assms unfolding isolated_in_def by auto

lemma isolated_inE:
assumes "x isolated_in S"
obtains T where "x ∈ S" "open T" "T ∩ S = {x}"
using assms that unfolding isolated_in_def by force

lemma isolated_inE_dist:
assumes "x isolated_in S"
obtains d where "d > 0" "⋀y. y ∈ S ⟹ dist x y < d ⟹ y = x"
by (meson assms isolated_in_dist_Ex_iff)

lemma isolated_in_altdef:
"x isolated_in S ⟷ (x∈S ∧ eventually (λy. y ∉ S) (at x))"
proof
assume "x isolated_in S"
from isolated_inE[OF this]
obtain T where "x ∈ S" and T:"open T" "T ∩ S = {x}"
by metis
have "∀⇩F y in nhds x. y ∈ T"
apply (rule eventually_nhds_in_open)
using T by auto
then have  "eventually (λy. y ∈ T - {x}) (at x)"
unfolding eventually_at_filter by eventually_elim auto
then have "eventually (λy. y ∉ S) (at x)"
by eventually_elim (use T in auto)
then show " x ∈ S ∧ (∀⇩F y in at x. y ∉ S)" using ‹x ∈ S› by auto
next
assume "x ∈ S ∧ (∀⇩F y in at x. y ∉ S)"
then have "∀⇩F y in at x. y ∉ S" "x∈S" by auto
from this(1) have "eventually (λy. y ∉ S ∨ y = x) (nhds x)"
unfolding eventually_at_filter by eventually_elim auto
then obtain T where T:"open T" "x ∈ T" "(∀y∈T. y ∉ S ∨ y = x)"
unfolding eventually_nhds by auto
with ‹x ∈ S› have "T ∩ S = {x}"
by fastforce
with ‹x∈S› ‹open T›
show "x isolated_in S"
unfolding isolated_in_def by auto
qed

lemma discrete_altdef:
"discrete S ⟷ (∀x∈S. ∀⇩F y in at x. y ∉ S)"
unfolding discrete_def isolated_in_altdef by auto

(*
TODO.
Other than

uniform_discrete S ⟶ discrete S
uniform_discrete S ⟶ closed S

, we should be able to prove

discrete S ∧ closed S ⟶ uniform_discrete S

but the proof (based on Tietze Extension Theorem) seems not very trivial to me. Informal proofs can be found in

http://topology.auburn.edu/tp/reprints/v30/tp30120.pdf
http://msp.org/pjm/1959/9-2/pjm-v9-n2-p19-s.pdf
*)

lemma uniform_discrete_imp_closed:
"uniform_discrete S ⟹ closed S"
by (meson discrete_imp_closed uniform_discrete_def)

lemma uniform_discrete_imp_discrete:
"uniform_discrete S ⟹ discrete S"
by (metis discrete_def isolated_in_dist_Ex_iff uniform_discrete_def)

lemma isolated_in_subset:"x isolated_in S ⟹ T ⊆ S ⟹ x∈T ⟹ x isolated_in T"
unfolding isolated_in_def by fastforce

lemma discrete_subset[elim]: "discrete S ⟹ T ⊆ S ⟹ discrete T"
unfolding discrete_def using islimpt_subset isolated_in_islimpt_iff by blast

lemma uniform_discrete_subset[elim]: "uniform_discrete S ⟹ T ⊆ S ⟹ uniform_discrete T"
by (meson subsetD uniform_discrete_def)

lemma continuous_on_discrete: "discrete S ⟹ continuous_on S f"
unfolding continuous_on_topological by (metis discrete_def islimptI isolated_in_islimpt_iff)

lemma uniform_discrete_insert: "uniform_discrete (insert a S) ⟷ uniform_discrete S"
proof
assume asm:"uniform_discrete S"
let ?thesis = "uniform_discrete (insert a S)"
have ?thesis when "a∈S" using that asm by (simp add: insert_absorb)
moreover have ?thesis when "S={}" using that asm by (simp add: uniform_discrete_def)
moreover have ?thesis when "a∉S" "S≠{}"
proof -
obtain e1 where "e1>0" and e1_dist:"∀x∈S. ∀y∈S. dist y x < e1 ⟶ y = x"
using asm unfolding uniform_discrete_def by auto
define e2 where "e2 ≡ min (setdist {a} S) e1"
have "closed S" using asm uniform_discrete_imp_closed by auto
then have "e2>0"
by (smt (verit) ‹0 < e1› e2_def infdist_eq_setdist infdist_pos_not_in_closed that)
moreover have "x = y" if "x∈insert a S" "y∈insert a S" "dist x y < e2" for x y
proof (cases "x=a ∨ y=a")
case True then show ?thesis
by (smt (verit, best) dist_commute e2_def infdist_eq_setdist infdist_le insertE that)
next
case False then show ?thesis
using e1_dist e2_def that by force
qed
ultimately show ?thesis unfolding uniform_discrete_def by meson
qed
ultimately show ?thesis by auto

lemma discrete_compact_finite_iff:
fixes S :: "'a::t1_space set"
shows "discrete S ∧ compact S ⟷ finite S"
proof
assume "finite S"
then have "compact S" using finite_imp_compact by auto
moreover have "discrete S"
unfolding discrete_def using isolated_in_islimpt_iff islimpt_finite[OF ‹finite S›] by auto
ultimately show "discrete S ∧ compact S" by auto
next
assume "discrete S ∧ compact S"
then show "finite S"
by (meson discrete_def Heine_Borel_imp_Bolzano_Weierstrass isolated_in_islimpt_iff order_refl)
qed

lemma uniform_discrete_finite_iff:
fixes S :: "'a::heine_borel set"
shows "uniform_discrete S ∧ bounded S ⟷ finite S"
proof
assume "uniform_discrete S ∧ bounded S"
then have "discrete S" "compact S"
using uniform_discrete_imp_discrete uniform_discrete_imp_closed compact_eq_bounded_closed
by auto
then show "finite S" using discrete_compact_finite_iff by auto
next
assume asm:"finite S"
let ?thesis = "uniform_discrete S ∧ bounded S"
have ?thesis when "S={}" using that by auto
moreover have ?thesis when "S≠{}"
proof -
have "∀x. ∃d>0. ∀y∈S. y ≠ x ⟶ d ≤ dist x y"
using finite_set_avoid[OF ‹finite S›] by auto
then obtain f where f_pos:"f x>0"
and f_dist: "∀y∈S. y ≠ x ⟶ f x ≤ dist x y"
if "x∈S" for x
by metis
define f_min where "f_min ≡ Min (f ` S)"
have "f_min > 0"
unfolding f_min_def
by (simp add: asm f_pos that)
moreover have "∀x∈S. ∀y∈S. f_min > dist x y ⟶ x=y"
using f_dist unfolding f_min_def
by (metis Min_le asm finite_imageI imageI le_less_trans linorder_not_less)
ultimately have "uniform_discrete S"
unfolding uniform_discrete_def by auto
moreover have "bounded S" using ‹finite S› by auto
ultimately show ?thesis by auto
qed
ultimately show ?thesis by blast
qed

lemma uniform_discrete_image_scale:
assumes "uniform_discrete S" and dist:"∀x∈S. ∀y∈S. dist x y = c * dist (f x) (f y)"
shows "uniform_discrete (f ` S)"
proof -
have ?thesis when "S={}" using that by auto
moreover have ?thesis when "S≠{}" "c≤0"
proof -
obtain x1 where "x1∈S" using ‹S≠{}› by auto
have ?thesis when "S-{x1} = {}"
using ‹x1 ∈ S› subset_antisym that uniform_discrete_insert by fastforce
moreover have ?thesis when "S-{x1} ≠ {}"
proof -
obtain x2 where "x2∈S-{x1}" using ‹S-{x1} ≠ {}› by auto
then have "x2∈S" "x1≠x2" by auto
then have "dist x1 x2 > 0" by auto
moreover have "dist x1 x2 = c * dist (f x1) (f x2)"
by (simp add: ‹x1 ∈ S› ‹x2 ∈ S› dist)
moreover have "dist (f x2) (f x2) ≥ 0" by auto
ultimately have False using ‹c≤0› by (simp add: zero_less_mult_iff)
then show ?thesis by auto
qed
ultimately show ?thesis by auto
qed
moreover have ?thesis when "S≠{}" "c>0"
proof -
obtain e1 where "e1>0" and e1_dist:"∀x∈S. ∀y∈S. dist y x < e1 ⟶ y = x"
using ‹uniform_discrete S› unfolding uniform_discrete_def by auto
define e where "e ≡ e1/c"
have "x1 = x2" when "x1 ∈ f ` S" "x2 ∈ f ` S" and d: "dist x1 x2 < e" for x1 x2
by (smt (verit) ‹0 < c› d dist divide_right_mono e1_dist e_def imageE nonzero_mult_div_cancel_left that)
moreover have "e>0" using ‹e1>0› ‹c>0› unfolding e_def by auto
ultimately show ?thesis unfolding uniform_discrete_def by meson
qed
ultimately show ?thesis by fastforce
qed

definition sparse :: "real ⇒ 'a :: metric_space set ⇒ bool"
where "sparse ε X ⟷ (∀x∈X. ∀y∈X-{x}. dist x y > ε)"

lemma sparse_empty [simp, intro]: "sparse ε {}"
by (auto simp: sparse_def)

lemma sparseI [intro?]:
"(⋀x y. x ∈ X ⟹ y ∈ X ⟹ x ≠ y ⟹ dist x y > ε) ⟹ sparse ε X"
unfolding sparse_def by auto

lemma sparseD:
"sparse ε X ⟹ x ∈ X ⟹ y ∈ X ⟹ x ≠ y ⟹ dist x y > ε"
unfolding sparse_def by auto

lemma sparseD':
"sparse ε X ⟹ x ∈ X ⟹ y ∈ X ⟹ dist x y ≤ ε ⟹ x = y"
unfolding sparse_def by force

lemma sparse_singleton [simp, intro]: "sparse ε {x}"
by (auto simp: sparse_def)

definition setdist_gt where "setdist_gt ε X Y ⟷ (∀x∈X. ∀y∈Y. dist x y > ε)"

lemma setdist_gt_empty [simp]: "setdist_gt ε {} Y" "setdist_gt ε X {}"
by (auto simp: setdist_gt_def)

lemma setdist_gtI: "(⋀x y. x ∈ X ⟹ y ∈ Y ⟹ dist x y > ε) ⟹ setdist_gt ε X Y"
unfolding setdist_gt_def by auto

lemma setdist_gtD: "setdist_gt ε X Y ⟹ x ∈ X ⟹ y ∈ Y ⟹ dist x y > ε"
unfolding setdist_gt_def by auto

lemma setdist_gt_setdist: "ε < setdist A B ⟹ setdist_gt ε A B"
unfolding setdist_gt_def using setdist_le_dist by fastforce

lemma setdist_gt_mono: "setdist_gt ε' A B ⟹ ε ≤ ε' ⟹ A' ⊆ A ⟹ B' ⊆ B ⟹ setdist_gt ε A' B'"
by (force simp: setdist_gt_def)

lemma setdist_gt_Un_left: "setdist_gt ε (A ∪ B) C ⟷ setdist_gt ε A C ∧ setdist_gt ε B C"
by (auto simp: setdist_gt_def)

lemma setdist_gt_Un_right: "setdist_gt ε C (A ∪ B) ⟷ setdist_gt ε C A ∧ setdist_gt ε C B"
by (auto simp: setdist_gt_def)

lemma compact_closed_imp_eventually_setdist_gt_at_right_0:
assumes "compact A" "closed B" "A ∩ B = {}"
shows   "eventually (λε. setdist_gt ε A B) (at_right 0)"
proof (cases "A = {} ∨ B = {}")
case False
hence "setdist A B > 0"
by (metis IntI assms empty_iff in_closed_iff_infdist_zero order_less_le setdist_attains_inf setdist_pos_le setdist_sym)
hence "eventually (λε. ε < setdist A B) (at_right 0)"
using eventually_at_right_field by blast
thus ?thesis
by eventually_elim (auto intro: setdist_gt_setdist)
qed auto

lemma setdist_gt_symI: "setdist_gt ε A B ⟹ setdist_gt ε B A"
by (force simp: setdist_gt_def dist_commute)

lemma setdist_gt_sym: "setdist_gt ε A B ⟷ setdist_gt ε B A"
by (force simp: setdist_gt_def dist_commute)

lemma eventually_setdist_gt_at_right_0_mult_iff:
assumes "c > 0"
shows   "eventually (λε. setdist_gt (c * ε) A B) (at_right 0) ⟷
eventually (λε. setdist_gt ε A B) (at_right 0)"
proof -
have "eventually (λε. setdist_gt (c * ε) A B) (at_right 0) ⟷
eventually (λε. setdist_gt ε A B) (filtermap ((*) c) (at_right 0))"