Quelle  Path_Connected.thy

(*  Title:      HOL/Analysis/Path_Connected.thy
  Authors: LC Paulson and Robert Himmelmann (TU Muenchen), based on material from HOL Light
*)

section ‹Path-Connectedness›

theory Path_Connected
imports
  Starlike
  T1_Spaces
begin

subsection ‹Paths and Arcs›

definition🍋‹tag important› path :: "(real ==> 'a::topological_space) ==> bool"
  where "path g ≡ continuous_on {0..1} g"

definition🍋‹tag important› pathstart :: "(real ==> 'a::topological_space) ==> 'a"
  where "pathstart g ≡ g 0"

definition🍋‹tag important› pathfinish :: "(real ==> 'a::topological_space) ==> 'a"
  where "pathfinish g ≡ g 1"

definition🍋‹tag important› path_image :: "(real ==> 'a::topological_space) ==> 'a set"
  where "path_image g ≡ g ` {0 .. 1}"

definition🍋‹tag important› reversepath :: "(real ==> 'a::topological_space) ==> real ==> 'a"
  where "reversepath g ≡ (λx. g(1 - x))"

definition🍋‹tag important› joinpaths :: "(real ==> 'a::topological_space) ==> (real ==> 'a) ==> real ==> 'a"
    (infixr ‹+++› 75)
  where "g1 +++ g2 ≡ (λx. if x ≤ 1/2 then g1 (2 * x) else g2 (2 * x - 1))"

definition🍋‹tag important› loop_free :: "(real ==> 'a::topological_space) ==> bool"
  where "loop_free g ≡ ∀x∈{0..1}. ∀y∈{0..1}. g x = g y ⟶ x = y ∨ x = 0 ∧ y = 1 ∨ x = 1 ∧ y = 0"

definition🍋‹tag important› simple_path :: "(real ==> 'a::topological_space) ==> bool"
  where "simple_path g ≡ path g ∧ loop_free g"

definition🍋‹tag important› arc :: "(real ==> 'a :: topological_space) ==> bool"
  where "arc g ≡ path g ∧ inj_on g {0..1}"


subsection🍋‹tag unimportant›\<open>Invariance theorems›

lemma path_eq: "path p ==> (∧t. t ∈ {0..1} ==> p t = q t) ==> path q"
  using continuous_on_eq path_def by blast

lemma path_continuous_image: "path g ==> continuous_on (path_image g) f ==> path(f ∘ g)"
  unfolding path_def path_image_def
  using continuous_on_compose by blast

lemma path_translation_eq:
  fixes g :: "real ==> 'a :: real_normed_vector"
  shows "path((λx. a + x) ∘ g) = path g"
  using continuous_on_translation_eq path_def by blast

lemma path_linear_image_eq:
  fixes f :: "'a::euclidean_space ==> 'b::euclidean_space"
   assumes "linear f" "inj f"
     shows "path(f ∘ g) = path g"
proof -
  from linear_injective_left_inverse [OF assms]
  obtain h where h: "linear h" "h ∘ f = id"
    by blast
  with assms show ?thesis
    by (metis comp_assoc id_comp linear_continuous_on linear_linear path_continuous_image)
qed

lemma pathstart_translation: "pathstart((λx. a + x) ∘ g) = a + pathstart g"
  by (simp add: pathstart_def)

lemma pathstart_linear_image_eq: "linear f ==> pathstart(f ∘ g) = f(pathstart g)"
  by (simp add: pathstart_def)

lemma pathfinish_translation: "pathfinish((λx. a + x) ∘ g) = a + pathfinish g"
  by (simp add: pathfinish_def)

lemma pathfinish_linear_image: "linear f ==> pathfinish(f ∘ g) = f(pathfinish g)"
  by (simp add: pathfinish_def)

lemma path_image_translation: "path_image((λx. a + x) ∘ g) = (λx. a + x) ` (path_image g)"
  by (simp add: image_comp path_image_def)

lemma path_image_linear_image: "linear f ==> path_image(f ∘ g) = f ` (path_image g)"
  by (simp add: image_comp path_image_def)

lemma reversepath_translation: "reversepath((λx. a + x) ∘ g) = (λx. a + x) ∘ reversepath g"
  by (rule ext) (simp add: reversepath_def)

lemma reversepath_linear_image: "linear f ==> reversepath(f ∘ g) = f ∘ reversepath g"
  by (rule ext) (simp add: reversepath_def)

lemma joinpaths_translation:
    "((λx. a + x) ∘ g1) +++ ((λx. a + x) ∘ g2) = (λx. a + x) ∘ (g1 +++ g2)"
  by (rule ext) (simp add: joinpaths_def)

lemma joinpaths_linear_image: "linear f ==> (f ∘ g1) +++ (f ∘ g2) = f ∘ (g1 +++ g2)"
  by (rule ext) (simp add: joinpaths_def)

lemma loop_free_translation_eq:
  fixes g :: "real ==> 'a::euclidean_space"
  shows "loop_free((λx. a + x) ∘ g) = loop_free g"
  by (simp add: loop_free_def)

lemma simple_path_translation_eq:
  fixes g :: "real ==> 'a::euclidean_space"
  shows "simple_path((λx. a + x) ∘ g) = simple_path g"
  by (simp add: simple_path_def loop_free_translation_eq path_translation_eq)

lemma loop_free_linear_image_eq:
  fixes f :: "'a::euclidean_space ==> 'b::euclidean_space"
  assumes "linear f" "inj f"
    shows "loop_free(f ∘ g) = loop_free g"
  using assms inj_on_eq_iff [of f] by (auto simp: loop_free_def)

lemma simple_path_linear_image_eq:
  fixes f :: "'a::euclidean_space ==> 'b::euclidean_space"
  assumes "linear f" "inj f"
    shows "simple_path(f ∘ g) = simple_path g"
  using assms
  by (simp add: loop_free_linear_image_eq path_linear_image_eq simple_path_def)

lemma simple_pathI [intro?]:
  assumes "path p"
  assumes "∧x y. 0 ≤ x ==> x < y ==> y ≤ 1 ==> p x = p y ==> x = 0 ∧ y = 1"
  shows   "simple_path p"
  unfolding simple_path_def loop_free_def
proof (intro ballI conjI impI)
  fix x y assume "x ∈ {0..1}" "y ∈ {0..1}" "p x = p y"
  thus "x = y ∨ x = 0 ∧ y = 1 ∨ x = 1 ∧ y = 0"
    by (metis assms(2) atLeastAtMost_iff linorder_less_linear)
qed fact+

lemma arcD: "arc p ==> p x = p y ==> x ∈ {0..1} ==> y ∈ {0..1} ==> x = y"
  by (auto simp: arc_def inj_on_def)

lemma arc_translation_eq:
  fixes g :: "real ==> 'a::euclidean_space"
  shows "arc((λx. a + x) ∘ g) ⟷ arc g"
  by (auto simp: arc_def inj_on_def path_translation_eq)

lemma arc_linear_image_eq:
  fixes f :: "'a::euclidean_space ==> 'b::euclidean_space"
   assumes "linear f" "inj f"
     shows  "arc(f ∘ g) = arc g"
  using assms inj_on_eq_iff [of f]
  by (auto simp: arc_def inj_on_def path_linear_image_eq)


subsection🍋‹tag unimportant›\<open>Basic lemmas about paths›

lemma path_of_real: "path complex_of_real" 
  unfolding path_def by (intro continuous_intros)

lemma path_const: "path (λt. a)" for a::"'a::real_normed_vector"
  unfolding path_def by (intro continuous_intros)

lemma path_minus: "path g ==> path (λt. - g t)" for g::"real==>'a::real_normed_vector"
  unfolding path_def by (intro continuous_intros)

lemma path_add: "[path f; path g] ==> path (λt. f t + g t)" for f::"real==>'a::real_normed_vector"
  unfolding path_def by (intro continuous_intros)

lemma path_diff: "[path f; path g] ==> path (λt. f t - g t)" for f::"real==>'a::real_normed_vector"
  unfolding path_def by (intro continuous_intros)

lemma path_mult: "[path f; path g] ==> path (λt. f t * g t)" for f::"real==>'a::real_normed_field"
  unfolding path_def by (intro continuous_intros)

lemma pathin_iff_path_real [simp]: "pathin euclideanreal g ⟷ path g"
  by (simp add: pathin_def path_def)

lemma continuous_on_path: "path f ==> t ⊆ {0..1} ==> continuous_on t f"
  using continuous_on_subset path_def by blast

lemma inj_on_imp_loop_free: "inj_on g {0..1} ==> loop_free g"
  by (simp add: inj_onD loop_free_def)

lemma arc_imp_simple_path: "arc g ==> simple_path g"
  by (simp add: arc_def inj_on_imp_loop_free simple_path_def)

lemma arc_imp_path: "arc g ==> path g"
  using arc_def by blast

lemma arc_imp_inj_on: "arc g ==> inj_on g {0..1}"
  by (auto simp: arc_def)

lemma simple_path_imp_path: "simple_path g ==> path g"
  using simple_path_def by blast

lemma loop_free_cases: "loop_free g ==> inj_on g {0..1} ∨ pathfinish g = pathstart g"
  by (force simp: inj_on_def loop_free_def pathfinish_def pathstart_def)

lemma simple_path_cases: "simple_path g ==> arc g ∨ pathfinish g = pathstart g"
  using arc_def loop_free_cases simple_path_def by blast

lemma simple_path_imp_arc: "simple_path g ==> pathfinish g ≠ pathstart g ==> arc g"
  using simple_path_cases by auto

lemma arc_distinct_ends: "arc g ==> pathfinish g ≠ pathstart g"
  unfolding arc_def inj_on_def pathfinish_def pathstart_def
  by fastforce

lemma arc_simple_path: "arc g ⟷ simple_path g ∧ pathfinish g ≠ pathstart g"
  using arc_distinct_ends arc_imp_simple_path simple_path_cases by blast

lemma simple_path_eq_arc: "pathfinish g ≠ pathstart g ==> (simple_path g = arc g)"
  by (simp add: arc_simple_path)

lemma path_image_const [simp]: "path_image (λt. a) = {a}"
  by (force simp: path_image_def)

lemma path_image_nonempty [simp]: "path_image g ≠ {}"
  unfolding path_image_def image_is_empty box_eq_empty
  by auto

lemma pathstart_in_path_image[intro]: "pathstart g ∈ path_image g"
  unfolding pathstart_def path_image_def
  by auto

lemma pathfinish_in_path_image[intro]: "pathfinish g ∈ path_image g"
  unfolding pathfinish_def path_image_def
  by auto

lemma connected_path_image[intro]: "path g ==> connected (path_image g)"
  unfolding path_def path_image_def
  using connected_continuous_image connected_Icc by blast

lemma compact_path_image[intro]: "path g ==> compact (path_image g)"
  unfolding path_def path_image_def
  using compact_continuous_image connected_Icc by blast

lemma reversepath_reversepath[simp]: "reversepath (reversepath g) = g"
  unfolding reversepath_def
  by auto

lemma pathstart_reversepath[simp]: "pathstart (reversepath g) = pathfinish g"
  unfolding pathstart_def reversepath_def pathfinish_def
  by auto

lemma pathfinish_reversepath[simp]: "pathfinish (reversepath g) = pathstart g"
  unfolding pathstart_def reversepath_def pathfinish_def
  by auto

lemma reversepath_o: "reversepath g = g ∘ (-)1"
  by (auto simp: reversepath_def)

lemma pathstart_join[simp]: "pathstart (g1 +++ g2) = pathstart g1"
  unfolding pathstart_def joinpaths_def pathfinish_def
  by auto

lemma pathfinish_join[simp]: "pathfinish (g1 +++ g2) = pathfinish g2"
  unfolding pathstart_def joinpaths_def pathfinish_def
  by auto

lemma path_image_reversepath[simp]: "path_image (reversepath g) = path_image g"
  by (metis cancel_comm_monoid_add_class.diff_cancel diff_zero image_comp 
      image_diff_atLeastAtMost path_image_def reversepath_o)

lemma path_reversepath [simp]: "path (reversepath g) ⟷ path g"
  by (metis continuous_on_compose continuous_on_op_minus image_comp image_ident path_def path_image_def path_image_reversepath reversepath_o reversepath_reversepath)

lemma arc_reversepath:
  assumes "arc g" shows "arc(reversepath g)"
proof -
  have injg: "inj_on g {0..1}"
    using assms
    by (simp add: arc_def)
  have **: "∧x y::real. 1-x = 1-y ==> x = y"
    by simp
  show ?thesis
    using assms  by (clarsimp simp: arc_def intro!: inj_onI) (simp add: inj_onD reversepath_def **)
qed

lemma loop_free_reversepath:
  assumes "loop_free g" shows "loop_free(reversepath g)"
  using assms by (simp add: reversepath_def loop_free_def Ball_def) (smt (verit))

lemma simple_path_reversepath: "simple_path g ==> simple_path (reversepath g)"
  by (simp add: loop_free_reversepath simple_path_def)

lemmas reversepath_simps =
  path_reversepath path_image_reversepath pathstart_reversepath pathfinish_reversepath

lemma path_join[simp]:
  assumes "pathfinish g1 = pathstart g2"
  shows "path (g1 +++ g2) ⟷ path g1 ∧ path g2"
  unfolding path_def pathfinish_def pathstart_def
proof safe
  assume cont: "continuous_on {0..1} (g1 +++ g2)"
  have g1: "continuous_on {0..1} g1 ⟷ continuous_on {0..1} ((g1 +++ g2) ∘ (λx. x / 2))"
    by (intro continuous_on_cong refl) (auto simp: joinpaths_def)
  have g2: "continuous_on {0..1} g2 ⟷ continuous_on {0..1} ((g1 +++ g2) ∘ (λx. x / 2 + 1/2))"
    using assms
    by (intro continuous_on_cong refl) (auto simp: joinpaths_def pathfinish_def pathstart_def)
  show "continuous_on {0..1} g1" and "continuous_on {0..1} g2"
    unfolding g1 g2
    by (auto intro!: continuous_intros continuous_on_subset[OF cont] simp del: o_apply)
next
  assume g1g2: "continuous_on {0..1} g1" "continuous_on {0..1} g2"
  have 01: "{0 .. 1} = {0..1/2} ∪ {1/2 .. 1::real}"
    by auto
  {
    fix x :: real
    assume "0 ≤ x" and "x ≤ 1"
    then have "x ∈ (λx. x * 2) ` {0..1 / 2}"
      by (intro image_eqI[where x="x/2"]) auto
  }
  note 1 = this
  {
    fix x :: real
    assume "0 ≤ x" and "x ≤ 1"
    then have "x ∈ (λx. x * 2 - 1) ` {1 / 2..1}"
      by (intro image_eqI[where x="x/2 + 1/2"]) auto
  }
  note 2 = this
  show "continuous_on {0..1} (g1 +++ g2)"
    using assms
    unfolding joinpaths_def 01
    apply (intro continuous_on_cases closed_atLeastAtMost g1g2[THEN continuous_on_compose2] continuous_intros)
    apply (auto simp: field_simps pathfinish_def pathstart_def intro!: 1 2)
    done
qed


subsection🍋‹tag unimportant› ‹Path Images›

lemma bounded_path_image: "path g ==> bounded(path_image g)"
  by (simp add: compact_imp_bounded compact_path_image)

lemma closed_path_image:
  fixes g :: "real ==> 'a::t2_space"
  shows "path g ==> closed(path_image g)"
  by (metis compact_path_image compact_imp_closed)

lemma connected_simple_path_image: "simple_path g ==> connected(path_image g)"
  by (metis connected_path_image simple_path_imp_path)

lemma compact_simple_path_image: "simple_path g ==> compact(path_image g)"
  by (metis compact_path_image simple_path_imp_path)

lemma bounded_simple_path_image: "simple_path g ==> bounded(path_image g)"
  by (metis bounded_path_image simple_path_imp_path)

lemma closed_simple_path_image:
  fixes g :: "real ==> 'a::t2_space"
  shows "simple_path g ==> closed(path_image g)"
  by (metis closed_path_image simple_path_imp_path)

lemma connected_arc_image: "arc g ==> connected(path_image g)"
  by (metis connected_path_image arc_imp_path)

lemma compact_arc_image: "arc g ==> compact(path_image g)"
  by (metis compact_path_image arc_imp_path)

lemma bounded_arc_image: "arc g ==> bounded(path_image g)"
  by (metis bounded_path_image arc_imp_path)

lemma closed_arc_image:
  fixes g :: "real ==> 'a::t2_space"
  shows "arc g ==> closed(path_image g)"
  by (metis closed_path_image arc_imp_path)

lemma path_image_join_subset: "path_image (g1 +++ g2) ⊆ path_image g1 ∪ path_image g2"
  unfolding path_image_def joinpaths_def
  by auto

lemma subset_path_image_join:
  assumes "path_image g1 ⊆ S" and "path_image g2 ⊆ S"
  shows "path_image (g1 +++ g2) ⊆ S"
  using path_image_join_subset[of g1 g2] and assms
  by auto

lemma path_image_join:
  assumes "pathfinish g1 = pathstart g2"
  shows "path_image(g1 +++ g2) = path_image g1 ∪ path_image g2"
proof -
  have "path_image g1 ⊆ path_image (g1 +++ g2)"
  proof (clarsimp simp: path_image_def joinpaths_def)
    fix u::real
    assume "0 ≤ u" "u ≤ 1"
    then show "g1 u ∈ (λx. g1 (2 * x)) ` ({0..1} ∩ {x. x * 2 ≤ 1})"
      by (rule_tac x="u/2" in image_eqI) auto
  qed
  moreover 
  have 🍋: "g2 u ∈ (λx. g2 (2 * x - 1)) ` ({0..1} ∩ {x. ¬ x * 2 ≤ 1})" 
    if "0 < u" "u ≤ 1" for u
    using that assms
    by (rule_tac x="(u+1)/2" in image_eqI) (auto simp: field_simps pathfinish_def pathstart_def)
  have "g2 0 ∈ (λx. g1 (2 * x)) ` ({0..1} ∩ {x. x * 2 ≤ 1})"
    using assms
    by (rule_tac x="1/2" in image_eqI) (auto simp: pathfinish_def pathstart_def)
  then have "path_image g2 ⊆ path_image (g1 +++ g2)"
    by (auto simp: path_image_def joinpaths_def intro!: 🍋)
  ultimately show ?thesis
    using path_image_join_subset by blast
qed

lemma not_in_path_image_join:
  assumes "x ∉ path_image g1" and "x ∉ path_image g2"
  shows "x ∉ path_image (g1 +++ g2)"
  using assms and path_image_join_subset[of g1 g2]
  by auto

lemma pathstart_compose: "pathstart(f ∘ p) = f(pathstart p)"
  by (simp add: pathstart_def)

lemma pathfinish_compose: "pathfinish(f ∘ p) = f(pathfinish p)"
  by (simp add: pathfinish_def)

lemma path_image_compose: "path_image (f ∘ p) = f ` (path_image p)"
  by (simp add: image_comp path_image_def)

lemma path_compose_join: "f ∘ (p +++ q) = (f ∘ p) +++ (f ∘ q)"
  by (rule ext) (simp add: joinpaths_def)

lemma path_compose_reversepath: "f ∘ reversepath p = reversepath(f ∘ p)"
  by (rule ext) (simp add: reversepath_def)

lemma joinpaths_eq:
  "(∧t. t ∈ {0..1} ==> p t = p' t) ==>
   (∧t. t ∈ {0..1} ==> q t = q' t)
   ==> t ∈ {0..1} ==> (p +++ q) t = (p' +++ q') t"
  by (auto simp: joinpaths_def)

lemma loop_free_inj_on: "loop_free g ==> inj_on g {0<..<1}"
  by (force simp: inj_on_def loop_free_def)

lemma simple_path_inj_on: "simple_path g ==> inj_on g {0<..<1}"
  using loop_free_inj_on simple_path_def by auto


subsection🍋‹tag unimportant›\<open>Simple paths with the endpoints removed›

lemma simple_path_endless:
  assumes "simple_path c"
  shows "path_image c - {pathstart c,pathfinish c} = c ` {0<..<1}" (is "?lhs = ?rhs")
proof
  show "?lhs ⊆ ?rhs"
    using less_eq_real_def by (auto simp: path_image_def pathstart_def pathfinish_def)
  show "?rhs ⊆ ?lhs"
    using assms 
    apply (simp add: image_subset_iff path_image_def pathstart_def pathfinish_def simple_path_def loop_free_def Ball_def)
    by (smt (verit))
qed

lemma connected_simple_path_endless:
  assumes "simple_path c"
  shows "connected(path_image c - {pathstart c,pathfinish c})"
proof -
  have "continuous_on {0<..<1} c"
    using assms by (simp add: simple_path_def continuous_on_path path_def subset_iff)
  then have "connected (c ` {0<..<1})"
    using connected_Ioo connected_continuous_image by blast
  then show ?thesis
    using assms by (simp add: simple_path_endless)
qed

lemma nonempty_simple_path_endless:
    "simple_path c ==> path_image c - {pathstart c,pathfinish c} ≠ {}"
  by (simp add: simple_path_endless)

lemma simple_path_continuous_image:
  assumes "simple_path f" "continuous_on (path_image f) g" "inj_on g (path_image f)"
  shows   "simple_path (g ∘ f)"
  unfolding simple_path_def
proof
  show "path (g ∘ f)"
    using assms unfolding simple_path_def by (intro path_continuous_image) auto
  from assms have [simp]: "g (f x) = g (f y) ⟷ f x = f y" if "x ∈ {0..1}" "y ∈ {0..1}" for x y
    unfolding inj_on_def path_image_def using that by fastforce
  show "loop_free (g ∘ f)"
    using assms(1) by (auto simp: loop_free_def simple_path_def)
qed

subsection🍋‹tag unimportant›\<open>The operations on paths›

lemma path_image_subset_reversepath: "path_image(reversepath g) ≤ path_image g"
  by simp

lemma path_imp_reversepath: "path g ==> path(reversepath g)"
  by simp

lemma half_bounded_equal: "1 ≤ x * 2 ==> x * 2 ≤ 1 ⟷ x = (1/2::real)"
  by simp

lemma continuous_on_joinpaths:
  assumes "continuous_on {0..1} g1" "continuous_on {0..1} g2" "pathfinish g1 = pathstart g2"
    shows "continuous_on {0..1} (g1 +++ g2)"
  using assms path_def path_join by blast

lemma path_join_imp: "[path g1; path g2; pathfinish g1 = pathstart g2] ==> path(g1 +++ g2)"
  by simp

lemma arc_join:
  assumes "arc g1" "arc g2"
          "pathfinish g1 = pathstart g2"
          "path_image g1 ∩ path_image g2 ⊆ {pathstart g2}"
    shows "arc(g1 +++ g2)"
proof -
  have injg1: "inj_on g1 {0..1}" and injg2: "inj_on g2 {0..1}" 
     and g11: "g1 1 = g2 0" and sb: "g1 ` {0..1} ∩ g2 ` {0..1} ⊆ {g2 0}"
    using assms
    by (auto simp: arc_def pathfinish_def pathstart_def path_image_def)
  { fix x and y::real
    assume xy: "g2 (2 * x - 1) = g1 (2 * y)" "x ≤ 1" "0 ≤ y" " y * 2 ≤ 1" "¬ x * 2 ≤ 1"
    then have "g1 (2 * y) = g2 0"
      using sb by force
    then have False
      using xy inj_onD injg2 by fastforce
   } note * = this
  have "inj_on (g1 +++ g2) {0..1}"
    using inj_onD [OF injg1] inj_onD [OF injg2] *
    by (simp add: inj_on_def joinpaths_def Ball_def) (smt (verit))
  then show ?thesis
    using arc_def assms path_join_imp by blast
qed

lemma simple_path_join_loop:
  assumes "arc g1" "arc g2"
          "pathfinish g1 = pathstart g2"  "pathfinish g2 = pathstart g1"
          "path_image g1 ∩ path_image g2 ⊆ {pathstart g1, pathstart g2}"
        shows "simple_path(g1 +++ g2)"
proof -
  have injg1: "inj_on g1 {0..1}" and injg2: "inj_on g2 {0..1}"
    using assms by (auto simp add: arc_def)
  have g12: "g1 1 = g2 0"
   and g21: "g2 1 = g1 0"
   and sb:  "g1 ` {0..1} ∩ g2 ` {0..1} ⊆ {g1 0, g2 0}"
    using assms
    by (simp_all add: arc_def pathfinish_def pathstart_def path_image_def)
  { fix x and y::real
    assume g2_eq: "g2 (2 * x - 1) = g1 (2 * y)"
      and xyI: "x ≠ 1 ∨ y ≠ 0"
      and xy: "x ≤ 1" "0 ≤ y" " y * 2 ≤ 1" "¬ x * 2 ≤ 1" 
    then consider "g1 (2 * y) = g1 0" | "g1 (2 * y) = g2 0"
      using sb by force
    then have False
    proof cases
      case 1
      then have "y = 0"
        using xy g2_eq by (auto dest!: inj_onD [OF injg1])
      then show ?thesis
        using xy g2_eq xyI by (auto dest: inj_onD [OF injg2] simp flip: g21)
    next
      case 2
      then have "2*x = 1"
        using g2_eq g12 inj_onD [OF injg2] atLeastAtMost_iff xy(1) xy(4) by fastforce
      with xy show False by auto
    qed
  } note * = this 
  have "loop_free(g1 +++ g2)"
    using inj_onD [OF injg1] inj_onD [OF injg2] *
    by (simp add: loop_free_def joinpaths_def Ball_def) (smt (verit))
  then show ?thesis
    by (simp add: arc_imp_path assms simple_path_def)
qed

lemma reversepath_joinpaths:
    "pathfinish g1 = pathstart g2 ==> reversepath(g1 +++ g2) = reversepath g2 +++ reversepath g1"
  unfolding reversepath_def pathfinish_def pathstart_def joinpaths_def
  by (rule ext) (auto simp: mult.commute)


subsection🍋‹tag unimportant›\<open>Some reversed and "if and only if" versions of joining theorems›

lemma path_join_path_ends:
  fixes g1 :: "real ==> 'a::metric_space"
  assumes "path(g1 +++ g2)" "path g2"
    shows "pathfinish g1 = pathstart g2"
proof (rule ccontr)
  define e where "e = dist (g1 1) (g2 0)"
  assume Neg: "pathfinish g1 ≠ pathstart g2"
  then have "0 < dist (pathfinish g1) (pathstart g2)"
    by auto
  then have "e > 0"
    by (metis e_def pathfinish_def pathstart_def)
  then have "∀e>0. ∃d>0. ∀x'∈{0..1}. dist x' 0 < d ⟶ dist (g2 x') (g2 0) < e"
    using ‹path g2› atLeastAtMost_iff zero_le_one unfolding path_def continuous_on_iff
    by blast
  then obtain d1 where "d1 > 0"
       and d1: "∧x'. [x'∈{0..1}; norm x' < d1] ==> dist (g2 x') (g2 0) < e/2"
    by (metis ‹0 🚫› half_gt_zero_iff norm_conv_dist)
  obtain d2 where "d2 > 0"
       and d2: "∧x'. [x'∈{0..1}; dist x' (1/2) < d2]
                      ==> dist ((g1 +++ g2) x') (g1 1) < e/2"
    using assms(1) ‹e > 0› unfolding path_def continuous_on_iff
    apply (drule_tac x="1/2" in bspec, simp)
    apply (drule_tac x="e/2" in spec, force simp: joinpaths_def)
    done
  have int01_1: "min (1/2) (min d1 d2) / 2 ∈ {0..1}"
    using ‹d1 > 0› ‹d2 > 0› by (simp add: min_def)
  have dist1: "norm (min (1 / 2) (min d1 d2) / 2) < d1"
    using ‹d1 > 0› ‹d2 > 0› by (simp add: min_def dist_norm)
  have int01_2: "1/2 + min (1/2) (min d1 d2) / 4 ∈ {0..1}"
    using ‹d1 > 0› ‹d2 > 0› by (simp add: min_def)
  have dist2: "dist (1 / 2 + min (1 / 2) (min d1 d2) / 4) (1 / 2) < d2"
    using ‹d1 > 0› ‹d2 > 0› by (simp add: min_def dist_norm)
  have [simp]: "¬ min (1 / 2) (min d1 d2) ≤ 0"
    using ‹d1 > 0› ‹d2 > 0› by (simp add: min_def)
  have "dist (g2 (min (1 / 2) (min d1 d2) / 2)) (g1 1) < e/2"
       "dist (g2 (min (1 / 2) (min d1 d2) / 2)) (g2 0) < e/2"
    using d1 [OF int01_1 dist1] d2 [OF int01_2 dist2] by (simp_all add: joinpaths_def)
  then have "dist (g1 1) (g2 0) < e/2 + e/2"
    using dist_triangle_half_r e_def by blast
  then show False
    by (simp add: e_def [symmetric])
qed

lemma path_join_eq [simp]:
  fixes g1 :: "real ==> 'a::metric_space"
  assumes "path g1" "path g2"
    shows "path(g1 +++ g2) ⟷ pathfinish g1 = pathstart g2"
  using assms by (metis path_join_path_ends path_join_imp)

lemma simple_path_joinE:
  assumes "simple_path(g1 +++ g2)" and "pathfinish g1 = pathstart g2"
  obtains "arc g1" "arc g2"
          "path_image g1 ∩ path_image g2 ⊆ {pathstart g1, pathstart g2}"
proof -
  have *: "∧x y. [0 ≤ x; x ≤ 1; 0 ≤ y; y ≤ 1; (g1 +++ g2) x = (g1 +++ g2) y]
               ==> x = y ∨ x = 0 ∧ y = 1 ∨ x = 1 ∧ y = 0"
    using assms by (simp add: simple_path_def loop_free_def)
  have "path g1"
    using assms path_join simple_path_imp_path by blast
  moreover have "inj_on g1 {0..1}"
  proof (clarsimp simp: inj_on_def)
    fix x y
    assume "g1 x = g1 y" "0 ≤ x" "x ≤ 1" "0 ≤ y" "y ≤ 1"
    then show "x = y"
      using * [of "x/2" "y/2"] by (simp add: joinpaths_def split_ifs)
  qed
  ultimately have "arc g1"
    using assms  by (simp add: arc_def)
  have [simp]: "g2 0 = g1 1"
    using assms by (metis pathfinish_def pathstart_def)
  have "path g2"
    using assms path_join simple_path_imp_path by blast
  moreover have "inj_on g2 {0..1}"
  proof (clarsimp simp: inj_on_def)
    fix x y
    assume "g2 x = g2 y" "0 ≤ x" "x ≤ 1" "0 ≤ y" "y ≤ 1"
    then show "x = y"
      using * [of "(x+1) / 2" "(y+1) / 2"]
      by (force simp: joinpaths_def split_ifs field_split_simps)
  qed
  ultimately have "arc g2"
    using assms  by (simp add: arc_def)
  have "g2 y = g1 0 ∨ g2 y = g1 1"
       if "g1 x = g2 y" "0 ≤ x" "x ≤ 1" "0 ≤ y" "y ≤ 1" for x y
      using * [of "x / 2" "(y + 1) / 2"] that
      by (auto simp: joinpaths_def split_ifs field_split_simps)
  then have "path_image g1 ∩ path_image g2 ⊆ {pathstart g1, pathstart g2}"
    by (fastforce simp: pathstart_def pathfinish_def path_image_def)
  with ‹arc g1› ‹arc g2› show ?thesis using that by blast
qed

lemma simple_path_join_loop_eq:
  assumes "pathfinish g2 = pathstart g1" "pathfinish g1 = pathstart g2"
  shows "simple_path(g1 +++ g2) ⟷
             arc g1 ∧ arc g2 ∧ path_image g1 ∩ path_image g2 ⊆ {pathstart g1, pathstart g2}"
  by (metis assms simple_path_joinE simple_path_join_loop)

lemma arc_join_eq:
  assumes "pathfinish g1 = pathstart g2"
    shows "arc(g1 +++ g2) ⟷
           arc g1 ∧ arc g2 ∧ path_image g1 ∩ path_image g2 ⊆ {pathstart g2}"
           (is "?lhs = ?rhs")
proof
  assume ?lhs then show ?rhs 
    using reversepath_simps assms
    by (smt (verit, best) Int_commute arc_reversepath arc_simple_path in_mono insertE pathstart_join 
          reversepath_joinpaths simple_path_joinE subsetI)
next
  assume ?rhs then show ?lhs
    using assms
    by (fastforce simp: pathfinish_def pathstart_def intro!: arc_join)
qed

lemma arc_join_eq_alt:
  "pathfinish g1 = pathstart g2
   ==> arc(g1 +++ g2) ⟷ arc g1 ∧ arc g2 ∧ path_image g1 ∩ path_image g2 = {pathstart g2}"
  using pathfinish_in_path_image by (fastforce simp: arc_join_eq)


subsubsection🍋‹tag unimportant›\<open>Symmetry and loops›

lemma path_sym:
  "[pathfinish p = pathstart q; pathfinish q = pathstart p] ==> path(p +++ q) ⟷ path(q +++ p)"
  by auto

lemma simple_path_sym:
  "[pathfinish p = pathstart q; pathfinish q = pathstart p]
     ==> simple_path(p +++ q) ⟷ simple_path(q +++ p)"
  by (metis (full_types) inf_commute insert_commute simple_path_joinE simple_path_join_loop)

lemma path_image_sym:
  "[pathfinish p = pathstart q; pathfinish q = pathstart p]
     ==> path_image(p +++ q) = path_image(q +++ p)"
  by (simp add: path_image_join sup_commute)

lemma simple_path_joinI:
  assumes "arc p1" "arc p2" "pathfinish p1 = pathstart p2"
  assumes "path_image p1 ∩ path_image p2
        ⊆ insert (pathstart p2) (if pathstart p1 = pathfinish p2 then {pathstart p1} else {})"
  shows   "simple_path (p1 +++ p2)"
  by (smt (verit, best) Int_commute arc_imp_simple_path arc_join assms insert_commute simple_path_join_loop simple_path_sym)

lemma simple_path_join3I:
  assumes "arc p1" "arc p2" "arc p3"
  assumes "path_image p1 ∩ path_image p2 ⊆ {pathstart p2}"
  assumes "path_image p2 ∩ path_image p3 ⊆ {pathstart p3}"
  assumes "path_image p1 ∩ path_image p3 ⊆ {pathstart p1} ∩ {pathfinish p3}"
  assumes "pathfinish p1 = pathstart p2" "pathfinish p2 = pathstart p3"
  shows   "simple_path (p1 +++ p2 +++ p3)"
  using assms by (intro simple_path_joinI arc_join) (auto simp: path_image_join)

subsection🍋‹tag unimportant›\<open>The joining of paths is associative›

lemma path_assoc:
  "[pathfinish p = pathstart q; pathfinish q = pathstart r]
     ==> path(p +++ (q +++ r)) ⟷ path((p +++ q) +++ r)"
  by simp

lemma simple_path_assoc:
  assumes "pathfinish p = pathstart q" "pathfinish q = pathstart r"
    shows "simple_path (p +++ (q +++ r)) ⟷ simple_path ((p +++ q) +++ r)"
proof (cases "pathstart p = pathfinish r")
  case True show ?thesis
  proof
    assume "simple_path (p +++ q +++ r)"
    with assms True show "simple_path ((p +++ q) +++ r)"
      by (fastforce simp add: simple_path_join_loop_eq arc_join_eq path_image_join
                    dest: arc_distinct_ends [of r])
  next
    assume 0: "simple_path ((p +++ q) +++ r)"
    with assms True have q: "pathfinish r ∉ path_image q"
      using arc_distinct_ends
      by (fastforce simp add: simple_path_join_loop_eq arc_join_eq path_image_join)
    have "pathstart r ∉ path_image p"
      using assms
      by (metis 0 IntI arc_distinct_ends arc_join_eq_alt empty_iff insert_iff
              pathfinish_in_path_image pathfinish_join simple_path_joinE)
    with assms 0 q True show "simple_path (p +++ q +++ r)"
      by (auto simp: simple_path_join_loop_eq arc_join_eq path_image_join
               dest!: subsetD [OF _ IntI])
  qed
next
  case False
  { fix x :: 'a
    assume a: "path_image p ∩ path_image q ⊆ {pathstart q}"
              "(path_image p ∪ path_image q) ∩ path_image r ⊆ {pathstart r}"
              "x ∈ path_image p" "x ∈ path_image r"
    have "pathstart r ∈ path_image q"
      by (metis assms(2) pathfinish_in_path_image)
    with a have "x = pathstart q"
      by blast
  }
  with False assms show ?thesis
    by (auto simp: simple_path_eq_arc simple_path_join_loop_eq arc_join_eq path_image_join)
qed

lemma arc_assoc:
     "[pathfinish p = pathstart q; pathfinish q = pathstart r]
      ==> arc(p +++ (q +++ r)) ⟷ arc((p +++ q) +++ r)"
by (simp add: arc_simple_path simple_path_assoc)


subsection‹Subpath›

definition🍋‹tag important› subpath :: "real ==> real ==> (real ==> 'a) ==> real ==> 'a::real_normed_vector"
  where "subpath a b g ≡ λx. g((b - a) * x + a)"

lemma path_image_subpath_gen:
  fixes g :: "_ ==> 'a::real_normed_vector"
  shows "path_image(subpath u v g) = g ` (closed_segment u v)"
  by (auto simp add: closed_segment_real_eq path_image_def subpath_def)

lemma path_image_subpath:
  fixes g :: "real ==> 'a::real_normed_vector"
  shows "path_image(subpath u v g) = (if u ≤ v then g ` {u..v} else g ` {v..u})"
  by (simp add: path_image_subpath_gen closed_segment_eq_real_ivl)

lemma path_image_subpath_commute:
  fixes g :: "real ==> 'a::real_normed_vector"
  shows "path_image(subpath u v g) = path_image(subpath v u g)"
  by (simp add: path_image_subpath_gen closed_segment_eq_real_ivl)

lemma path_subpath [simp]:
  fixes g :: "real ==> 'a::real_normed_vector"
  assumes "path g" "u ∈ {0..1}" "v ∈ {0..1}"
    shows "path(subpath u v g)"
proof -
  have "continuous_on {u..v} g" "continuous_on {v..u} g"
    using assms continuous_on_path by fastforce+
  then have "continuous_on {0..1} (g ∘ (λx. ((v-u) * x+ u)))"
    by (intro continuous_intros; simp add: image_affinity_atLeastAtMost [where c=u])
  then show ?thesis
    by (simp add: path_def subpath_def)
qed

lemma pathstart_subpath [simp]: "pathstart(subpath u v g) = g(u)"
  by (simp add: pathstart_def subpath_def)

lemma pathfinish_subpath [simp]: "pathfinish(subpath u v g) = g(v)"
  by (simp add: pathfinish_def subpath_def)

lemma subpath_trivial [simp]: "subpath 0 1 g = g"
  by (simp add: subpath_def)

lemma subpath_reversepath: "subpath 1 0 g = reversepath g"
  by (simp add: reversepath_def subpath_def)

lemma reversepath_subpath: "reversepath(subpath u v g) = subpath v u g"
  by (simp add: reversepath_def subpath_def algebra_simps)

lemma subpath_translation: "subpath u v ((λx. a + x) ∘ g) = (λx. a + x) ∘ subpath u v g"
  by (rule ext) (simp add: subpath_def)

lemma subpath_image: "subpath u v (f ∘ g) = f ∘ subpath u v g"
  by (rule ext) (simp add: subpath_def)

lemma affine_ineq:
  fixes x :: "'a::linordered_idom"
  assumes "x ≤ 1" "v ≤ u"
    shows "v + x * u ≤ u + x * v"
proof -
  have "(1-x)*(u-v) ≥ 0"
    using assms by auto
  then show ?thesis
    by (simp add: algebra_simps)
qed

lemma sum_le_prod1:
  fixes a::real shows "[a ≤ 1; b ≤ 1] ==> a + b ≤ 1 + a * b"
  by (metis add.commute affine_ineq mult.right_neutral)

lemma simple_path_subpath_eq:
  "simple_path(subpath u v g) ⟷
     path(subpath u v g) ∧ u≠v ∧
     (∀x y. x ∈ closed_segment u v ∧ y ∈ closed_segment u v ∧ g x = g y
                ⟶ x = y ∨ x = u ∧ y = v ∨ x = v ∧ y = u)"
    (is "?lhs = ?rhs")
proof 
  assume ?lhs
  then have p: "path (λx. g ((v - u) * x + u))"
        and sim: "(∧x y. [x∈{0..1}; y∈{0..1}; g ((v - u) * x + u) = g ((v - u) * y + u)]
                  ==> x = y ∨ x = 0 ∧ y = 1 ∨ x = 1 ∧ y = 0)"
    by (auto simp: simple_path_def loop_free_def subpath_def)
  { fix x y
    assume "x ∈ closed_segment u v" "y ∈ closed_segment u v" "g x = g y"
    then have "x = y ∨ x = u ∧ y = v ∨ x = v ∧ y = u"
      using sim [of "(x-u)/(v-u)" "(y-u)/(v-u)"] p
      by (auto split: if_split_asm simp add: closed_segment_real_eq image_affinity_atLeastAtMost)
        (simp_all add: field_split_simps)
  } moreover
  have "path(subpath u v g) ∧ u≠v"
    using sim [of "1/3" "2/3"] p
    by (auto simp: subpath_def)
  ultimately show ?rhs
    by metis
next
  assume ?rhs
  then
  have d1: "∧x y. [g x = g y; u ≤ x; x ≤ v; u ≤ y; y ≤ v] ==> x = y ∨ x = u ∧ y = v ∨ x = v ∧ y = u"
   and d2: "∧x y. [g x = g y; v ≤ x; x ≤ u; v ≤ y; y ≤ u] ==> x = y ∨ x = u ∧ y = v ∨ x = v ∧ y = u"
   and ne: "u < v ∨ v < u"
   and psp: "path (subpath u v g)"
    by (auto simp: closed_segment_real_eq image_affinity_atLeastAtMost)
  have [simp]: "∧x. u + x * v = v + x * u ⟷ u=v ∨ x=1"
    by algebra
  show ?lhs using psp ne
    unfolding simple_path_def loop_free_def subpath_def
    by (fastforce simp add: algebra_simps affine_ineq mult_left_mono crossproduct_eq dest: d1 d2)
qed

lemma arc_subpath_eq:
  "arc(subpath u v g) ⟷ path(subpath u v g) ∧ u≠v ∧ inj_on g (closed_segment u v)"
  by (smt (verit, best) arc_simple_path closed_segment_commute ends_in_segment(2) inj_on_def pathfinish_subpath pathstart_subpath simple_path_subpath_eq)


lemma simple_path_subpath:
  assumes "simple_path g" "u ∈ {0..1}" "v ∈ {0..1}" "u ≠ v"
  shows "simple_path(subpath u v g)"
  using assms
  unfolding simple_path_subpath_eq
  by (force simp:  simple_path_def loop_free_def closed_segment_real_eq image_affinity_atLeastAtMost)

lemma arc_simple_path_subpath:
    "[simple_path g; u ∈ {0..1}; v ∈ {0..1}; g u ≠ g v] ==> arc(subpath u v g)"
  by (force intro: simple_path_subpath simple_path_imp_arc)

lemma arc_subpath_arc:
    "[arc g; u ∈ {0..1}; v ∈ {0..1}; u ≠ v] ==> arc(subpath u v g)"
  by (meson arc_def arc_imp_simple_path arc_simple_path_subpath inj_onD)

lemma arc_simple_path_subpath_interior:
    "[simple_path g; u ∈ {0..1}; v ∈ {0..1}; u ≠ v; ∣u-v∣ < 1] ==> arc(subpath u v g)"
  by (force simp: simple_path_def loop_free_def intro: arc_simple_path_subpath)

lemma path_image_subpath_subset:
    "[u ∈ {0..1}; v ∈ {0..1}] ==> path_image(subpath u v g) ⊆ path_image g"
  by (metis atLeastAtMost_iff atLeastatMost_subset_iff path_image_def path_image_subpath subset_image_iff)

lemma join_subpaths_middle: "subpath (0) ((1 / 2)) p +++ subpath ((1 / 2)) 1 p = p"
  by (rule ext) (simp add: joinpaths_def subpath_def field_split_simps)


subsection🍋‹tag unimportant›\<open>There is a subpath to the frontier›

lemma subpath_to_frontier_explicit:
    fixes S :: "'a::metric_space set"
    assumes g: "path g" and "pathfinish g ∉ S"
    obtains u where "0 ≤ u" "u ≤ 1"
                "∧x. 0 ≤ x ∧ x < u ==> g x ∈ interior S"
                "(g u ∉ interior S)" "(u = 0 ∨ g u ∈ closure S)"
proof -
  have gcon: "continuous_on {0..1} g"     
    using g by (simp add: path_def)
  moreover have "bounded ({u. g u ∈ closure (- S)} ∩ {0..1})"
    using compact_eq_bounded_closed by fastforce
  ultimately have com: "compact ({0..1} ∩ {u. g u ∈ closure (- S)})"
    using closed_vimage_Int
    by (metis (full_types) Int_commute closed_atLeastAtMost closed_closure compact_eq_bounded_closed vimage_def)
  have "1 ∈ {u. g u ∈ closure (- S)}"
    using assms by (simp add: pathfinish_def closure_def)
  then have dis: "{0..1} ∩ {u. g u ∈ closure (- S)} ≠ {}"
    using atLeastAtMost_iff zero_le_one by blast
  then obtain u where "0 ≤ u" "u ≤ 1" and gu: "g u ∈ closure (- S)"
                  and umin: "∧t. [0 ≤ t; t ≤ 1; g t ∈ closure (- S)] ==> u ≤ t"
    using compact_attains_inf [OF com dis] by fastforce
  then have umin': "∧t. [0 ≤ t; t ≤ 1; t < u] ==> g t ∈ S"
    using closure_def by fastforce
  have 🍋: "g u ∈ closure S" if "u ≠ 0"
  proof -
    have "u > 0" using that ‹0 ≤ u› by auto
    { fix e::real assume "e > 0"
      obtain d where "d>0" and d: "∧x'. [x' ∈ {0..1}; dist x' u ≤ d] ==> dist (g x') (g u) < e"
        using continuous_onE [OF gcon _ ‹e > 0›] ‹0 ≤ _› ‹_ ≤ 1› atLeastAtMost_iff by auto
      have *: "dist (max 0 (u - d / 2)) u ≤ d"
        using ‹0 ≤ u› ‹u ≤ 1› ‹d > 0› by (simp add: dist_real_def)
      have "∃y∈S. dist y (g u) < e"
        using ‹0 🚫› ‹u ≤ 1› ‹d > 0›
        by (force intro: d [OF _ *] umin')
    }
    then show ?thesis
      by (simp add: frontier_def closure_approachable)
  qed
  show ?thesis
  proof
    show "∧x. 0 ≤ x ∧ x < u ==> g x ∈ interior S"
      using ‹u ≤ 1› interior_closure umin by fastforce
    show "g u ∉ interior S"
      by (simp add: gu interior_closure)
  qed (use ‹0 ≤ u› ‹u ≤ 1› 🍋 in auto)
qed

lemma subpath_to_frontier_strong:
    assumes g: "path g" and "pathfinish g ∉ S"
    obtains u where "0 ≤ u" "u ≤ 1" "g u ∉ interior S"
                    "u = 0 ∨ (∀x. 0 ≤ x ∧ x < 1 ⟶ subpath 0 u g x ∈ interior S) ∧ g u ∈ closure S"
proof -
  obtain u where "0 ≤ u" "u ≤ 1"
             and gxin: "∧x. 0 ≤ x ∧ x < u ==> g x ∈ interior S"
             and gunot: "(g u ∉ interior S)" and u0: "(u = 0 ∨ g u ∈ closure S)"
    using subpath_to_frontier_explicit [OF assms] by blast
  show ?thesis
  proof
    show "g u ∉ interior S"
      using gunot by blast
  qed (use ‹0 ≤ u› ‹u ≤ 1› u0 in ‹(force simp: subpath_def gxin)+›)
qed

lemma subpath_to_frontier:
    assumes g: "path g" and g0: "pathstart g ∈ closure S" and g1: "pathfinish g ∉ S"
    obtains u where "0 ≤ u" "u ≤ 1" "g u ∈ frontier S" "path_image(subpath 0 u g) - {g u} ⊆ interior S"
proof -
  obtain u where "0 ≤ u" "u ≤ 1"
             and notin: "g u ∉ interior S"
             and disj: "u = 0 ∨
                        (∀x. 0 ≤ x ∧ x < 1 ⟶ subpath 0 u g x ∈ interior S) ∧ g u ∈ closure S"
                       (is "_ ∨ ?P")
    using subpath_to_frontier_strong [OF g g1] by blast
  show ?thesis
  proof
    show "g u ∈ frontier S"
      by (metis DiffI disj frontier_def g0 notin pathstart_def)
    show "path_image (subpath 0 u g) - {g u} ⊆ interior S"
      using disj
    proof
      assume "u = 0"
      then show ?thesis
        by (simp add: path_image_subpath)
    next
      assume P: ?P
      show ?thesis
      proof (clarsimp simp add: path_image_subpath_gen)
        fix y
        assume y: "y ∈ closed_segment 0 u" "g y ∉ interior S"
        with ‹0 ≤ u› have "0 ≤ y" "y ≤ u" 
          by (auto simp: closed_segment_eq_real_ivl split: if_split_asm)
        then have "y=u ∨ subpath 0 u g (y/u) ∈ interior S"
          using P less_eq_real_def by force
        then show "g y = g u"
          using y by (auto simp: subpath_def split: if_split_asm)
      qed
    qed
  qed (use ‹0 ≤ u› ‹u ≤ 1› in auto)
qed

lemma exists_path_subpath_to_frontier:
    fixes S :: "'a::real_normed_vector set"
    assumes "path g" "pathstart g ∈ closure S" "pathfinish g ∉ S"
    obtains h where "path h" "pathstart h = pathstart g" "path_image h ⊆ path_image g"
                    "path_image h - {pathfinish h} ⊆ interior S"
                    "pathfinish h ∈ frontier S"
proof -
  obtain u where u: "0 ≤ u" "u ≤ 1" "g u ∈ frontier S" "(path_image(subpath 0 u g) - {g u}) ⊆ interior S"
    using subpath_to_frontier [OF assms] by blast
  show ?thesis
  proof
    show "path_image (subpath 0 u g) ⊆ path_image g"
      by (simp add: path_image_subpath_subset u)
    show "pathstart (subpath 0 u g) = pathstart g"
      by (metis pathstart_def pathstart_subpath)
  qed (use assms u in ‹auto simp: path_image_subpath›)
qed

lemma exists_path_subpath_to_frontier_closed:
    fixes S :: "'a::real_normed_vector set"
    assumes S: "closed S" and g: "path g" and g0: "pathstart g ∈ S" and g1: "pathfinish g ∉ S"
    obtains h where "path h" "pathstart h = pathstart g" "path_image h ⊆ path_image g ∩ S"
                    "pathfinish h ∈ frontier S"
  by (smt (verit, del_insts) Diff_iff Int_iff S closure_closed exists_path_subpath_to_frontier 
      frontier_def g g0 g1 interior_subset singletonD subset_eq)


subsection ‹Shift Path to Start at Some Given Point›

definition🍋‹tag important› shiftpath :: "real ==> (real ==> 'a::topological_space) ==> real ==> 'a"
  where "shiftpath a f = (λx. if (a + x) ≤ 1 then f (a + x) else f (a + x - 1))"

lemma shiftpath_alt_def: "shiftpath a f = (λx. if x ≤ 1-a then f (a + x) else f (a + x - 1))"
  by (auto simp: shiftpath_def)

lemma pathstart_shiftpath: "a ≤ 1 ==> pathstart (shiftpath a g) = g a"
  unfolding pathstart_def shiftpath_def by auto

lemma pathfinish_shiftpath:
  assumes "0 ≤ a"
    and "pathfinish g = pathstart g"
  shows "pathfinish (shiftpath a g) = g a"
  using assms
  unfolding pathstart_def pathfinish_def shiftpath_def
  by auto

lemma endpoints_shiftpath:
  assumes "pathfinish g = pathstart g"
    and "a ∈ {0 .. 1}"
  shows "pathfinish (shiftpath a g) = g a"
    and "pathstart (shiftpath a g) = g a"
  using assms
  by (simp_all add: pathstart_shiftpath pathfinish_shiftpath)

lemma closed_shiftpath:
  assumes "pathfinish g = pathstart g"
    and "a ∈ {0..1}"
  shows "pathfinish (shiftpath a g) = pathstart (shiftpath a g)"
  using endpoints_shiftpath[OF assms]
  by auto

lemma path_shiftpath:
  assumes "path g"
    and "pathfinish g = pathstart g"
    and "a ∈ {0..1}"
  shows "path (shiftpath a g)"
proof -
  have *: "{0 .. 1} = {0 .. 1-a} ∪ {1-a .. 1}"
    using assms(3) by auto
  have **: "∧x. x + a = 1 ==> g (x + a - 1) = g (x + a)"
    by (smt (verit, best) assms(2) pathfinish_def pathstart_def)
  show ?thesis
    unfolding path_def shiftpath_def *
  proof (rule continuous_on_closed_Un)
    have contg: "continuous_on {0..1} g"
      using ‹path g› path_def by blast
    show "continuous_on {0..1-a} (λx. if a + x ≤ 1 then g (a + x) else g (a + x - 1))"
    proof (rule continuous_on_eq)
      show "continuous_on {0..1-a} (g ∘ (+) a)"
        by (intro continuous_intros continuous_on_subset [OF contg]) (use ‹a ∈ {0..1}› in auto)
    qed auto
    show "continuous_on {1-a..1} (λx. if a + x ≤ 1 then g (a + x) else g (a + x - 1))"
    proof (rule continuous_on_eq)
      show "continuous_on {1-a..1} (g ∘ (+) (a - 1))"
        by (intro continuous_intros continuous_on_subset [OF contg]) (use ‹a ∈ {0..1}› in auto)
    qed (auto simp: "**" add.commute add_diff_eq)
  qed auto
qed

lemma shiftpath_shiftpath:
  assumes "pathfinish g = pathstart g"
    and "a ∈ {0..1}"
    and "x ∈ {0..1}"
  shows "shiftpath (1 - a) (shiftpath a g) x = g x"
  using assms
  unfolding pathfinish_def pathstart_def shiftpath_def
  by auto

lemma path_image_shiftpath:
  assumes a: "a ∈ {0..1}"
    and "pathfinish g = pathstart g"
  shows "path_image (shiftpath a g) = path_image g"
proof -
  { fix x
    assume g: "g 1 = g 0" "x ∈ {0..1::real}" and gne: "∧y. y∈{0..1} ∩ {x. ¬ a + x ≤ 1} ==> g x ≠ g (a + y - 1)"
    then have "∃y∈{0..1} ∩ {x. a + x ≤ 1}. g x = g (a + y)"
    proof (cases "a ≤ x")
      case False
      then show ?thesis
        apply (rule_tac x="1 + x - a" in bexI)
        using g gne[of "1 + x - a"] a by (force simp: field_simps)+
    next
      case True
      then show ?thesis
        using g a  by (rule_tac x="x - a" in bexI) (auto simp: field_simps)
    qed
  }
  then show ?thesis
    using assms
    unfolding shiftpath_def path_image_def pathfinish_def pathstart_def
    by (auto simp: image_iff)
qed

lemma loop_free_shiftpath:
  assumes "loop_free g" "pathfinish g = pathstart g" and a: "0 ≤ a" "a ≤ 1"
    shows "loop_free (shiftpath a g)"
  unfolding loop_free_def
proof (intro conjI impI ballI)
  show "x = y ∨ x = 0 ∧ y = 1 ∨ x = 1 ∧ y = 0"
    if "x ∈ {0..1}" "y ∈ {0..1}" "shiftpath a g x = shiftpath a g y" for x y
    using that a assms unfolding shiftpath_def loop_free_def
    by (smt (verit, ccfv_threshold) atLeastAtMost_iff)
qed

lemma simple_path_shiftpath:
  assumes "simple_path g" "pathfinish g = pathstart g" and a: "0 ≤ a" "a ≤ 1"
  shows "simple_path (shiftpath a g)"
  using assms loop_free_shiftpath path_shiftpath simple_path_def by fastforce


subsection ‹Straight-Line Paths›

definition🍋‹tag important› linepath :: "'a::real_normed_vector ==> 'a ==> real ==> 'a"
  where "linepath a b = (λx. (1 - x) *🪙R a + x *🪙R b)"

lemma pathstart_linepath[simp]: "pathstart (linepath a b) = a"
  unfolding pathstart_def linepath_def
  by auto

lemma pathfinish_linepath[simp]: "pathfinish (linepath a b) = b"
  unfolding pathfinish_def linepath_def
  by auto

lemma linepath_inner: "linepath a b x ∙ v = linepath (a ∙ v) (b ∙ v) x"
  by (simp add: linepath_def algebra_simps)

lemma Re_linepath': "Re (linepath a b x) = linepath (Re a) (Re b) x"
  by (simp add: linepath_def)

lemma Im_linepath': "Im (linepath a b x) = linepath (Im a) (Im b) x"
  by (simp add: linepath_def)

lemma linepath_0': "linepath a b 0 = a"
  by (simp add: linepath_def)

lemma linepath_1': "linepath a b 1 = b"
  by (simp add: linepath_def)

lemma continuous_linepath_at[intro]: "continuous (at x) (linepath a b)"
  unfolding linepath_def
  by (intro continuous_intros)

lemma continuous_on_linepath [intro,continuous_intros]: "continuous_on s (linepath a b)"
  using continuous_linepath_at
  by (auto intro!: continuous_at_imp_continuous_on)

lemma path_linepath[iff]: "path (linepath a b)"
  unfolding path_def
  by (rule continuous_on_linepath)

lemma path_image_linepath[simp]: "path_image (linepath a b) = closed_segment a b"
  unfolding path_image_def segment linepath_def
  by auto

lemma reversepath_linepath[simp]: "reversepath (linepath a b) = linepath b a"
  unfolding reversepath_def linepath_def
  by auto

lemma linepath_0 [simp]: "linepath 0 b x = x *🪙R b"
  by (simp add: linepath_def)

lemma linepath_cnj: "cnj (linepath a b x) = linepath (cnj a) (cnj b) x"
  by (simp add: linepath_def)

lemma arc_linepath:
  assumes "a ≠ b" shows [simp]: "arc (linepath a b)"
proof -
  {
    fix x y :: "real"
    assume "x *🪙R b + y *🪙R a = x *🪙R a + y *🪙R b"
    then have "(x - y) *🪙R a = (x - y) *🪙R b"
      by (simp add: algebra_simps)
    with assms have "x = y"
      by simp
  }
  then show ?thesis
    unfolding arc_def inj_on_def
    by (fastforce simp: algebra_simps linepath_def)
qed

lemma simple_path_linepath[intro]: "a ≠ b ==> simple_path (linepath a b)"
  by (simp add: arc_imp_simple_path)

lemma linepath_trivial [simp]: "linepath a a x = a"
  by (simp add: linepath_def real_vector.scale_left_diff_distrib)

lemma linepath_refl: "linepath a a = (λx. a)"
  by auto

lemma subpath_refl: "subpath a a g = linepath (g a) (g a)"
  by (simp add: subpath_def linepath_def algebra_simps)

lemma linepath_of_real: "(linepath (of_real a) (of_real b) x) = of_real ((1 - x)*a + x*b)"
  by (simp add: scaleR_conv_of_real linepath_def)

lemma of_real_linepath: "of_real (linepath a b x) = linepath (of_real a) (of_real b) x"
  by (metis linepath_of_real mult.right_neutral of_real_def real_scaleR_def)

lemma inj_on_linepath:
  assumes "a ≠ b" shows "inj_on (linepath a b) {0..1}"
  using arc_imp_inj_on arc_linepath assms by blast

lemma linepath_le_1:
  fixes a::"'a::linordered_idom" shows "[a ≤ 1; b ≤ 1; 0 ≤ u; u ≤ 1] ==> (1 - u) * a + u * b ≤ 1"
  using mult_left_le [of a "1-u"] mult_left_le [of b u] by auto

lemma linepath_in_path:
  shows "x ∈ {0..1} ==> linepath a b x ∈ closed_segment a b"
  by (auto simp: segment linepath_def)

lemma linepath_image_01: "linepath a b ` {0..1} = closed_segment a b"
  by (auto simp: segment linepath_def)

lemma linepath_in_convex_hull:
  fixes x::real
  assumes "a ∈ convex hull S"
    and "b ∈ convex hull S"
    and "0≤x" "x≤1"
  shows "linepath a b x ∈ convex hull S"
  by (meson assms atLeastAtMost_iff convex_contains_segment convex_convex_hull linepath_in_path subset_eq)

lemma Re_linepath: "Re(linepath (of_real a) (of_real b) x) = (1 - x)*a + x*b"
  by (simp add: linepath_def)

lemma Im_linepath: "Im(linepath (of_real a) (of_real b) x) = 0"
  by (simp add: linepath_def)

lemma
  assumes "x ∈ closed_segment y z"
  shows in_closed_segment_imp_Re_in_closed_segment: "Re x ∈ closed_segment (Re y) (Re z)" (is ?th1)
    and in_closed_segment_imp_Im_in_closed_segment: "Im x ∈ closed_segment (Im y) (Im z)" (is ?th2)
proof -
  from assms obtain t where t: "t ∈ {0..1}" "x = linepath y z t"
    by (metis imageE linepath_image_01)
  have "Re x = linepath (Re y) (Re z) t" "Im x = linepath (Im y) (Im z) t"
    by (simp_all add: t Re_linepath' Im_linepath')
  with t(1) show ?th1 ?th2
    using linepath_in_path[of t "Re y" "Re z"] linepath_in_path[of t "Im y" "Im z"] by simp_all
qed

lemma linepath_in_open_segment: "t ∈ {0<..<1} ==> x ≠ y ==> linepath x y t ∈ open_segment x y"
  unfolding greaterThanLessThan_iff by (metis in_segment(2) linepath_def)

lemma in_open_segment_imp_Re_in_open_segment:
  assumes "x ∈ open_segment y z" "Re y ≠ Re z"
  shows   "Re x ∈ open_segment (Re y) (Re z)"
proof -
  from assms obtain t where t: "t ∈ {0<..<1}" "x = linepath y z t"
    by (metis greaterThanLessThan_iff in_segment(2) linepath_def)
  have "Re x = linepath (Re y) (Re z) t"
    by (simp_all add: t Re_linepath')
  with t(1) show ?thesis
    using linepath_in_open_segment[of t "Re y" "Re z"] assms by auto
qed

lemma in_open_segment_imp_Im_in_open_segment:
  assumes "x ∈ open_segment y z" "Im y ≠ Im z"
  shows   "Im x ∈ open_segment (Im y) (Im z)"
proof -
  from assms obtain t where t: "t ∈ {0<..<1}" "x = linepath y z t"
    by (metis greaterThanLessThan_iff in_segment(2) linepath_def)
  have "Im x = linepath (Im y) (Im z) t"
    by (simp_all add: t Im_linepath')
  with t(1) show ?thesis
    using linepath_in_open_segment[of t "Im y" "Im z"] assms by auto
qed

lemma bounded_linear_linepath:
  assumes "bounded_linear f"
  shows   "f (linepath a b x) = linepath (f a) (f b) x"
proof -
  interpret f: bounded_linear f by fact
  show ?thesis by (simp add: linepath_def f.add f.scale)
qed

lemma bounded_linear_linepath':
  assumes "bounded_linear f"
  shows   "f ∘ linepath a b = linepath (f a) (f b)"
  using bounded_linear_linepath[OF assms] by (simp add: fun_eq_iff)

lemma linepath_cnj': "cnj ∘ linepath a b = linepath (cnj a) (cnj b)"
  by (simp add: linepath_def fun_eq_iff)

lemma differentiable_linepath [intro]: "linepath a b differentiable at x within A"
  by (auto simp: linepath_def)

lemma has_vector_derivative_linepath_within:
    "(linepath a b has_vector_derivative (b - a)) (at x within S)"
  by (force intro: derivative_eq_intros simp add: linepath_def has_vector_derivative_def algebra_simps)

lemma linepath_real_ge_left:
  fixes x y :: real
  assumes "x ≤ y" "t ≥ 0"
  shows   "linepath x y t ≥ x"
proof -
  have "x + 0 ≤ x + t *🪙R (y - x)"
    using assms by (intro add_left_mono) auto
  also have "… = linepath x y t"
    by (simp add: linepath_def algebra_simps)
  finally show ?thesis by simp
qed

lemma linepath_real_le_right:
  fixes x y :: real
  assumes "x ≤ y" "t ≤ 1"
  shows   "linepath x y t ≤ y"
proof -
  have "y + 0 ≥ y + (1 - t) *🪙R (x - y)"
    using assms by (intro add_left_mono) (auto intro: mult_nonneg_nonpos)
  also have "y + (1 - t) *🪙R (x - y) = linepath x y t"
    by (simp add: linepath_def algebra_simps)
  finally show ?thesis by simp
qed

lemma linepath_translate: "(+) c ∘ linepath a b = linepath (a + c) (b + c)"
  by (auto simp: linepath_def algebra_simps)


subsection🍋‹tag unimportant›\<open>Segments via convex hulls›

lemma segments_subset_convex_hull:
    "closed_segment a b ⊆ (convex hull {a,b,c})"
    "closed_segment a c ⊆ (convex hull {a,b,c})"
    "closed_segment b c ⊆ (convex hull {a,b,c})"
    "closed_segment b a ⊆ (convex hull {a,b,c})"
    "closed_segment c a ⊆ (convex hull {a,b,c})"
    "closed_segment c b ⊆ (convex hull {a,b,c})"
by (auto simp: segment_convex_hull linepath_of_real  elim!: rev_subsetD [OF _ hull_mono])

lemma midpoints_in_convex_hull:
  assumes "x ∈ convex hull s" "y ∈ convex hull s"
    shows "midpoint x y ∈ convex hull s"
  using assms closed_segment_subset_convex_hull csegment_midpoint_subset by blast

lemma not_in_interior_convex_hull_3:
  fixes a :: "complex"
  shows "a ∉ interior(convex hull {a,b,c})"
        "b ∉ interior(convex hull {a,b,c})"
        "c ∉ interior(convex hull {a,b,c})"
  by (auto simp: card_insert_le_m1 not_in_interior_convex_hull)

lemma midpoint_in_closed_segment [simp]: "midpoint a b ∈ closed_segment a b"
  using midpoints_in_convex_hull segment_convex_hull by blast

lemma midpoint_in_open_segment [simp]: "midpoint a b ∈ open_segment a b ⟷ a ≠ b"
  by (simp add: open_segment_def)

lemma continuous_IVT_local_extremum:
  fixes f :: "'a::euclidean_space ==> real"
  assumes contf: "continuous_on (closed_segment a b) f"
      and ab: "a ≠ b" "f a = f b"
  obtains z where "z ∈ open_segment a b"
                  "(∀w ∈ closed_segment a b. (f w) ≤ (f z)) ∨
                   (∀w ∈ closed_segment a b. (f z) ≤ (f w))"
proof -
  obtain c where "c ∈ closed_segment a b" and c: "∧y. y ∈ closed_segment a b ==> f y ≤ f c"
    using continuous_attains_sup [of "closed_segment a b" f] contf by auto
  moreover
  obtain d where "d ∈ closed_segment a b" and d: "∧y. y ∈ closed_segment a b ==> f d ≤ f y"
    using continuous_attains_inf [of "closed_segment a b" f] contf by auto
  ultimately show ?thesis
    by (smt (verit) UnE ab closed_segment_eq_open empty_iff insert_iff midpoint_in_open_segment that)
qed

text‹An injective map into R is also an open map w.r.T. the universe, and conversely. ›
proposition injective_eq_1d_open_map_UNIV:
  fixes f :: "real ==> real"
  assumes contf: "continuous_on S f" and S: "is_interval S"
    shows "inj_on f S ⟷ (∀T. open T ∧ T ⊆ S ⟶ open(f ` T))"
          (is "?lhs = ?rhs")
proof safe
  fix T
  assume injf: ?lhs and "open T" and "T ⊆ S"
  have "∃U. open U ∧ f x ∈ U ∧ U ⊆ f ` T" if "x ∈ T" for x
  proof -
    obtain δ where "δ > 0" and δ: "cball x δ ⊆ T"
      using ‹open T› ‹x ∈ T› open_contains_cball_eq by blast
    show ?thesis
    proof (intro exI conjI)
      have "closed_segment (x-δ) (x+δ) = {x-δ..x+δ}"
        using ‹0 🚫δ› by (auto simp: closed_segment_eq_real_ivl)
      also have "… ⊆ S"
        using δ ‹T ⊆ S› by (auto simp: dist_norm subset_eq)
      finally have "f ` (open_segment (x-δ) (x+δ)) = open_segment (f (x-δ)) (f (x+δ))"
        using continuous_injective_image_open_segment_1
        by (metis continuous_on_subset [OF contf] inj_on_subset [OF injf])
      then show "open (f ` {x-δ<..
        using ‹0 🚫δ› by (simp add: open_segment_eq_real_ivl)
      show "f x ∈ f ` {x - δ<..
        by (auto simp: ‹δ > 0›)
      show "f ` {x - δ<..⊆ f ` T"
        using δ by (auto simp: dist_norm subset_iff)
    qed
  qed
  with open_subopen show "open (f ` T)"
    by blast
next
  assume R: ?rhs
  have False if xy: "x ∈ S" "y ∈ S" and "f x = f y" "x ≠ y" for x y
  proof -
    have "open (f ` open_segment x y)"
      using R
      by (metis S convex_contains_open_segment is_interval_convex open_greaterThanLessThan open_segment_eq_real_ivl xy)
    moreover
    have "continuous_on (closed_segment x y) f"
      by (meson S closed_segment_subset contf continuous_on_subset is_interval_convex that)
    then obtain ξ where "ξ ∈ open_segment x y"
                    and ξ: "(∀w ∈ closed_segment x y. (f w) ≤ (f ξ)) ∨
                            (∀w ∈ closed_segment x y. (f ξ) ≤ (f w))"
      using continuous_IVT_local_extremum [of x y f] ‹f x = f y› ‹x ≠ y› by blast
    ultimately obtain e where "e>0" and e: "∧u. dist u (f ξ) < e ==> u ∈ f ` open_segment x y"
      using open_dist by (metis image_eqI)
    have fin: "f ξ + (e/2) ∈ f ` open_segment x y" "f ξ - (e/2) ∈ f ` open_segment x y"
      using e [of "f ξ + (e/2)"] e [of "f ξ - (e/2)"] ‹e > 0› by (auto simp: dist_norm)
    show ?thesis
      using ξ ‹0 🚫› fin open_closed_segment by fastforce
  qed
  then show ?lhs
    by (force simp: inj_on_def)
qed


subsection🍋‹tag unimportant› ‹Bounding a point away from a path›

lemma not_on_path_ball:
  fixes g :: "real ==> 'a::heine_borel"
  assumes "path g"
    and z: "z ∉ path_image g"
  shows "∃e > 0. ball z e ∩ path_image g = {}"
proof -
  have "closed (path_image g)"
    by (simp add: ‹path g› closed_path_image)
  then obtain a where "a ∈ path_image g" "∀y ∈ path_image g. dist z a ≤ dist z y"
    by (auto intro: distance_attains_inf[OF _ path_image_nonempty, of g z])
  then show ?thesis
    by (rule_tac x="dist z a" in exI) (use dist_commute z in auto)
qed

lemma not_on_path_cball:
  fixes g :: "real ==> 'a::heine_borel"
  assumes "path g"
    and "z ∉ path_image g"
  shows "∃e>0. cball z e ∩ (path_image g) = {}"
  by (smt (verit, ccfv_threshold) open_ball assms centre_in_ball inf.orderE inf_assoc
      inf_bot_right not_on_path_ball open_contains_cball_eq)

subsection ‹Path component›

text ‹Original formalization by Tom Hales›

definition🍋‹tag important› "path_component S x y ≡
  (∃g. path g ∧ path_image g ⊆ S ∧ pathstart g = x ∧ pathfinish g = y)"

abbreviation🍋‹tag important›
  "path_component_set S x ≡ Collect (path_component S x)"

lemmas path_defs = path_def pathstart_def pathfinish_def path_image_def path_component_def

lemma path_component_mem:
  assumes "path_component S x y"
  shows "x ∈ S" and "y ∈ S"
  using assms
  unfolding path_defs
  by auto

lemma path_component_refl:
  assumes "x ∈ S"
  shows "path_component S x x"
  using assms
  unfolding path_defs
  by (metis (full_types) assms continuous_on_const image_subset_iff path_image_def)

lemma path_component_refl_eq: "path_component S x x ⟷ x ∈ S"
  by (auto intro!: path_component_mem path_component_refl)

lemma path_component_sym: "path_component S x y ==> path_component S y x"
  unfolding path_component_def
  by (metis (no_types) path_image_reversepath path_reversepath pathfinish_reversepath pathstart_reversepath)

lemma path_component_trans:
  assumes "path_component S x y" and "path_component S y z"
  shows "path_component S x z"
  using assms
  unfolding path_component_def
  by (metis path_join pathfinish_join pathstart_join subset_path_image_join)

lemma path_component_of_subset: "S ⊆ T ==> path_component S x y ==> path_component T x y"
  unfolding path_component_def by auto

lemma path_component_linepath:
    fixes S :: "'a::real_normed_vector set"
    shows "closed_segment a b ⊆ S ==> path_component S a b"
  unfolding path_component_def by fastforce

subsubsection🍋‹tag unimportant› ‹Path components as sets›

lemma path_component_set:
  "path_component_set S x =
    {y. (∃g. path g ∧ path_image g ⊆ S ∧ pathstart g = x ∧ pathfinish g = y)}"
  by (auto simp: path_component_def)

lemma path_component_subset: "path_component_set S x ⊆ S"
  by (auto simp: path_component_mem(2))

lemma path_component_eq_empty: "path_component_set S x = {} ⟷ x ∉ S"
  using path_component_mem path_component_refl_eq
    by fastforce

lemma path_component_mono:
     "S ⊆ T ==> (path_component_set S x) ⊆ (path_component_set T x)"
  by (simp add: Collect_mono path_component_of_subset)

lemma path_component_eq:
   "y ∈ path_component_set S x ==> path_component_set S y = path_component_set S x"
by (metis (no_types, lifting) Collect_cong mem_Collect_eq path_component_sym path_component_trans)


subsection ‹Path connectedness of a space›

definition🍋‹tag important› "path_connected S ⟷
  (∀x∈S. ∀y∈S. ∃g. path g ∧ path_image g ⊆ S ∧ pathstart g = x ∧ pathfinish g = y)"

lemma path_connectedin_iff_path_connected_real [simp]:
     "path_connectedin euclideanreal S ⟷ path_connected S"
  by (simp add: path_connectedin path_connected_def path_defs image_subset_iff_funcset) 

lemma path_connected_component: "path_connected S ⟷ (∀x∈S. ∀y∈S. path_component S x y)"
  unfolding path_connected_def path_component_def by auto

lemma path_connected_component_set: "path_connected S ⟷ (∀x∈S. path_component_set S x = S)"
  unfolding path_connected_component path_component_subset
  using path_component_mem by blast

lemma path_component_maximal:
     "[x ∈ T; path_connected T; T ⊆ S] ==> T ⊆ (path_component_set S x)"
  by (metis path_component_mono path_connected_component_set)

lemma convex_imp_path_connected:
  fixes S :: "'a::real_normed_vector set"
  assumes "convex S"
  shows "path_connected S"
  unfolding path_connected_def
  using assms convex_contains_segment by fastforce

lemma path_connected_UNIV [iff]: "path_connected (UNIV :: 'a::real_normed_vector set)"
  by (simp add: convex_imp_path_connected)

lemma path_component_UNIV: "path_component_set UNIV x = (UNIV :: 'a::real_normed_vector set)"
  using path_connected_component_set by auto

lemma path_connected_imp_connected:
  assumes "path_connected S"
  shows "connected S"
proof (rule connectedI)
  fix e1 e2
  assume as: "open e1" "open e2" "S ⊆ e1 ∪ e2" "e1 ∩ e2 ∩ S = {}" "e1 ∩ S ≠ {}" "e2 ∩ S ≠ {}"
  then obtain x1 x2 where obt:"x1 ∈ e1 ∩ S" "x2 ∈ e2 ∩ S"
    by auto
  then obtain g where g: "path g" "path_image g ⊆ S" and pg: "pathstart g = x1" "pathfinish g = x2"
    using assms[unfolded path_connected_def,rule_format,of x1 x2] by auto
  have *: "connected {0..1::real}"
    by (auto intro!: convex_connected)
  have "{0..1} ⊆ {x ∈ {0..1}. g x ∈ e1} ∪ {x ∈ {0..1}. g x ∈ e2}"
    using as(3) g(2)[unfolded path_defs] by blast
  moreover have "{x ∈ {0..1}. g x ∈ e1} ∩ {x ∈ {0..1}. g x ∈ e2} = {}"
    using as(4) g(2)[unfolded path_defs]
    unfolding subset_eq
    by auto
  moreover have "{x ∈ {0..1}. g x ∈ e1} ≠ {} ∧ {x ∈ {0..1}. g x ∈ e2} ≠ {}"
    by (smt (verit, ccfv_threshold) IntE atLeastAtMost_iff empty_iff pg mem_Collect_eq obt pathfinish_def pathstart_def)
  ultimately show False
    using *[unfolded connected_local not_ex, rule_format,
      of "{0..1} ∩ g -` e1" "{0..1} ∩ g -` e2"]
    using continuous_openin_preimage_gen[OF g(1)[unfolded path_def] as(1)]
    using continuous_openin_preimage_gen[OF g(1)[unfolded path_def] as(2)]
    by auto
qed

lemma open_path_component:
  fixes S :: "'a::real_normed_vector set"
  assumes "open S"
  shows "open (path_component_set S x)"
  unfolding open_contains_ball
  by (metis assms centre_in_ball convex_ball convex_imp_path_connected equals0D openE 
      path_component_eq path_component_eq_empty path_component_maximal)

lemma open_non_path_component:
  fixes S :: "'a::real_normed_vector set"
  assumes "open S"
  shows "open (S - path_component_set S x)"
  unfolding open_contains_ball
proof
  fix y
  assume y: "y ∈ S - path_component_set S x"
  then obtain e where e: "e > 0" "ball y e ⊆ S"
    using assms openE by auto
  show "∃e>0. ball y e ⊆ S - path_component_set S x"
  proof (intro exI conjI subsetI DiffI notI)
    show "∧x. x ∈ ball y e ==> x ∈ S"
      using e by blast
    show False if "z ∈ ball y e" "z ∈ path_component_set S x" for z
      by (metis (no_types, lifting) Diff_iff centre_in_ball convex_ball convex_imp_path_connected  
          path_component_eq path_component_maximal subsetD that y e)
  qed (use e in auto)
qed

lemma connected_open_path_connected:
  fixes S :: "'a::real_normed_vector set"
  assumes "open S"
    and "connected S"
  shows "path_connected S"
  unfolding path_connected_component_set
proof (rule, rule, rule path_component_subset, rule)
  fix x y
  assume "x ∈ S" and "y ∈ S"
  show "y ∈ path_component_set S x"
  proof (rule ccontr)
    assume "¬ ?thesis"
    moreover have "path_component_set S x ∩ S ≠ {}"
      using ‹x ∈ S› path_component_eq_empty path_component_subset[of S x]
      by auto
    ultimately
    show False
      using ‹y ∈ S› open_non_path_component[OF ‹open S›] open_path_component[OF ‹open S›]
      using ‹connected S›[unfolded connected_def not_ex, rule_format,
        of "path_component_set S x" "S - path_component_set S x"]
      by auto
  qed
qed

lemma path_connected_continuous_image:
  assumes contf: "continuous_on S f"
    and "path_connected S"
  shows "path_connected (f ` S)"
  unfolding path_connected_def
proof clarsimp
  fix x y 
  assume x: "x ∈ S" and y: "y ∈ S" 
  with ‹path_connected S›
  show "∃g. path g ∧ path_image g ⊆ f ` S ∧ pathstart g = f x ∧ pathfinish g = f y"
    unfolding path_defs path_connected_def
    using continuous_on_subset[OF contf]
    by (smt (verit, ccfv_threshold) continuous_on_compose2 image_eqI image_subset_iff)
qed

lemma path_connected_translationI:
  fixes a :: "'a :: topological_group_add"
  assumes "path_connected S" shows "path_connected ((λx. a + x) ` S)"
  by (intro path_connected_continuous_image assms continuous_intros)

lemma path_connected_translation:
  fixes a :: "'a :: topological_group_add"
  shows "path_connected ((λx. a + x) ` S) = path_connected S"
proof -
  have "∀x y. (+) (x::'a) ` (+) (0 - x) ` y = y"
    by (simp add: image_image)
  then show ?thesis
    by (metis (no_types) path_connected_translationI)
qed

lemma path_connected_segment [simp]:
    fixes a :: "'a::real_normed_vector"
    shows "path_connected (closed_segment a b)"
  by (simp add: convex_imp_path_connected)

lemma path_connected_open_segment [simp]:
    fixes a :: "'a::real_normed_vector"
    shows "path_connected (open_segment a b)"
  by (simp add: convex_imp_path_connected)

lemma homeomorphic_path_connectedness:
  "S homeomorphic T ==> path_connected S ⟷ path_connected T"
  unfolding homeomorphic_def homeomorphism_def by (metis path_connected_continuous_image)

lemma path_connected_empty [simp]: "path_connected {}"
  unfolding path_connected_def by auto

lemma path_connected_singleton [simp]: "path_connected {a}"
  unfolding path_connected_def pathstart_def pathfinish_def path_image_def
  using path_def by fastforce

lemma path_connected_Un:
  assumes "path_connected S"
    and "path_connected T"
    and "S ∩ T ≠ {}"
  shows "path_connected (S ∪ T)"
  unfolding path_connected_component
proof (intro ballI)
  fix x y
  assume x: "x ∈ S ∪ T" and y: "y ∈ S ∪ T"
  from assms obtain z where z: "z ∈ S" "z ∈ T"
    by auto
  with x y show "path_component (S ∪ T) x y"
    by (smt (verit) assms(1,2) in_mono mem_Collect_eq path_component_eq path_component_maximal 
        sup.bounded_iff sup.cobounded2 sup_ge1)
qed

lemma path_connected_UNION:
  assumes "∧i. i ∈ A ==> path_connected (S i)"
    and "∧i. i ∈ A ==> z ∈ S i"
  shows "path_connected (∪i∈A. S i)"
  unfolding path_connected_component
proof clarify
  fix x i y j
  assume *: "i ∈ A" "x ∈ S i" "j ∈ A" "y ∈ S j"
  then have "path_component (S i) x z" and "path_component (S j) z y"
    using assms by (simp_all add: path_connected_component)
  then have "path_component (∪i∈A. S i) x z" and "path_component (∪i∈A. S i) z y"
    using *(1,3) by (meson SUP_upper path_component_of_subset)+
  then show "path_component (∪i∈A. S i) x y"
    by (rule path_component_trans)
qed

lemma path_component_path_image_pathstart:
  assumes p: "path p" and x: "x ∈ path_image p"
  shows "path_component (path_image p) (pathstart p) x"
proof -
  obtain y where x: "x = p y" and y: "0 ≤ y" "y ≤ 1"
    using x by (auto simp: path_image_def)
  show ?thesis
    unfolding path_component_def 
  proof (intro exI conjI)
    have "continuous_on ((*) y ` {0..1}) p"
      by (simp add: continuous_on_path image_mult_atLeastAtMost_if p y)
    then have "continuous_on {0..1} (p ∘ ((*) y))"
      using continuous_on_compose continuous_on_mult_const by blast
    then show "path (λu. p (y * u))"
      by (simp add: path_def)
    show "path_image (λu. p (y * u)) ⊆ path_image p"
      using y mult_le_one by (fastforce simp: path_image_def image_iff)
  qed (auto simp: pathstart_def pathfinish_def x)
qed

lemma path_connected_path_image: "path p ==> path_connected(path_image p)"
  unfolding path_connected_component
  by (meson path_component_path_image_pathstart path_component_sym path_component_trans)

lemma path_connected_path_component [simp]:
  "path_connected (path_component_set S x)"
  by (smt (verit) mem_Collect_eq path_component_def path_component_eq path_component_maximal 
      path_connected_component path_connected_path_image pathstart_in_path_image)

lemma path_component: 
  "path_component S x y ⟷ (∃t. path_connected t ∧ t ⊆ S ∧ x ∈ t ∧ y ∈ t)"
    (is "?lhs = ?rhs")
proof 
  assume ?lhs then show ?rhs
    by (metis path_component_def path_connected_path_image pathfinish_in_path_image pathstart_in_path_image)
next
  assume ?rhs then show ?lhs
    by (meson path_component_of_subset path_connected_component)
qed

lemma path_component_path_component [simp]:
   "path_component_set (path_component_set S x) x = path_component_set S x"
  by (metis (full_types) mem_Collect_eq path_component_eq_empty path_component_refl path_connected_component_set path_connected_path_component)

lemma path_component_subset_connected_component:
   "(path_component_set S x) ⊆ (connected_component_set S x)"
proof (cases "x ∈ S")
  case True show ?thesis
    by (simp add: True connected_component_maximal path_component_refl path_component_subset path_connected_imp_connected)
next
  case False then show ?thesis
    using path_component_eq_empty by auto
qed


subsection🍋‹tag unimportant›\<open>Lemmas about path-connectedness›

lemma path_connected_linear_image:
  fixes f :: "'a::real_normed_vector ==> 'b::real_normed_vector"
  assumes "path_connected S" "bounded_linear f"
  shows "path_connected(f ` S)"
  by (auto simp: linear_continuous_on assms path_connected_continuous_image)

lemma is_interval_path_connected: "is_interval S ==> path_connected S"
  by (simp add: convex_imp_path_connected is_interval_convex)

lemma path_connected_Ioi[simp]: "path_connected {a<..}" for a :: real
  by (simp add: convex_imp_path_connected)

lemma path_connected_Ici[simp]: "path_connected {a..}" for a :: real
  by (simp add: convex_imp_path_connected)

lemma path_connected_Iio[simp]: "path_connected {.. for a :: real
  by (simp add: convex_imp_path_connected)

lemma path_connected_Iic[simp]: "path_connected {..a}" for a :: real
  by (simp add: convex_imp_path_connected)

lemma path_connected_Ioo[simp]: "path_connected {a<.. for a b :: real
  by (simp add: convex_imp_path_connected)

lemma path_connected_Ioc[simp]: "path_connected {a<..b}" for a b :: real
  by (simp add: convex_imp_path_connected)

lemma path_connected_Ico[simp]: "path_connected {a.. for a b :: real
  by (simp add: convex_imp_path_connected)

lemma path_connectedin_path_image:
  assumes "pathin X g" shows "path_connectedin X (g ` ({0..1}))"
  unfolding pathin_def
proof (rule path_connectedin_continuous_map_image)
  show "continuous_map (subtopology euclideanreal {0..1}) X g"
    using assms pathin_def by blast
qed (auto simp: is_interval_1 is_interval_path_connected)

lemma path_connected_space_subconnected:
     "path_connected_space X ⟷
      (∀x ∈ topspace X. ∀y ∈ topspace X. ∃S. path_connectedin X S ∧ x ∈ S ∧ y ∈ S)"
  by (metis path_connectedin path_connectedin_topspace path_connected_space_def)


lemma connectedin_path_image: "pathin X g ==> connectedin X (g ` ({0..1}))"
  by (simp add: path_connectedin_imp_connectedin path_connectedin_path_image)

lemma compactin_path_image: "pathin X g ==> compactin X (g ` ({0..1}))"
  unfolding pathin_def
  by (rule image_compactin [of "top_of_set {0..1}"]) auto

lemma linear_homeomorphism_image:
  fixes f :: "'a::euclidean_space ==> 'b::euclidean_space"
  assumes "linear f" "inj f"
  obtains g where "homeomorphism (f ` S) S g f"
proof -
  obtain g where "linear g" "g ∘ f = id"
    using assms linear_injective_left_inverse by blast
  then have "homeomorphism (f ` S) S g f"
    using assms unfolding homeomorphism_def
    by (auto simp: eq_id_iff [symmetric] image_comp linear_conv_bounded_linear linear_continuous_on)
  then show thesis ..
qed

lemma linear_homeomorphic_image:
  fixes f :: "'a::euclidean_space ==> 'b::euclidean_space"
  assumes "linear f" "inj f"
  shows "S homeomorphic f ` S"
  by (meson homeomorphic_def homeomorphic_sym linear_homeomorphism_image [OF assms])

lemma path_connected_Times:
  assumes "path_connected s" "path_connected t"
    shows "path_connected (s × t)"
proof (simp add: path_connected_def Sigma_def, clarify)
  fix x1 y1 x2 y2
  assume "x1 ∈ s" "y1 ∈ t" "x2 ∈ s" "y2 ∈ t"
  obtain g where "path g" and g: "path_image g ⊆ s" and gs: "pathstart g = x1" and gf: "pathfinish g = x2"
    using ‹x1 ∈ s› ‹x2 ∈ s› assms by (force simp: path_connected_def)
  obtain h where "path h" and h: "path_image h ⊆ t" and hs: "pathstart h = y1" and hf: "pathfinish h = y2"
    using ‹y1 ∈ t› ‹y2 ∈ t› assms by (force simp: path_connected_def)
  have "path (λz. (x1, h z))"
    using ‹path h›
    unfolding path_def
    by (intro continuous_intros continuous_on_compose2 [where g = "Pair _"]; force)
  moreover have "path (λz. (g z, y2))"
    using ‹path g›
    unfolding path_def
    by (intro continuous_intros continuous_on_compose2 [where g = "Pair _"]; force)
  ultimately have 1: "path ((λz. (x1, h z)) +++ (λz. (g z, y2)))"
    by (metis hf gs path_join_imp pathstart_def pathfinish_def)
  have "path_image ((λz. (x1, h z)) +++ (λz. (g z, y2))) ⊆ path_image (λz. (x1, h z)) ∪ path_image (λz. (g z, y2))"
    by (rule Path_Connected.path_image_join_subset)
  also have "… ⊆ (∪x∈s. ∪x1∈t. {(x, x1)})"
    using g h ‹x1 ∈ s› ‹y2 ∈ t› by (force simp: path_image_def)
  finally have 2: "path_image ((λz. (x1, h z)) +++ (λz. (g z, y2))) ⊆ (∪x∈s. ∪x1∈t. {(x, x1)})" .
  show "∃g. path g ∧ path_image g ⊆ (∪x∈s. ∪x1∈t. {(x, x1)}) ∧
            pathstart g = (x1, y1) ∧ pathfinish g = (x2, y2)"
    using 1 2 gf hs
    by (metis (no_types, lifting) pathfinish_def pathfinish_join pathstart_def pathstart_join)
qed

lemma is_interval_path_connected_1:
  fixes s :: "real set"
  shows "is_interval s ⟷ path_connected s"
  using is_interval_connected_1 is_interval_path_connected path_connected_imp_connected by blast


subsection🍋‹tag unimportant›\<open>Path components›

lemma Union_path_component [simp]:
   "Union {path_component_set S x |x. x ∈ S} = S"
  using path_component_subset path_component_refl by blast

lemma path_component_disjoint:
   "disjnt (path_component_set S a) (path_component_set S b) ⟷
    (a ∉ path_component_set S b)"
  unfolding disjnt_iff
  using path_component_sym path_component_trans by blast

lemma path_component_eq_eq:
   "path_component S x = path_component S y ⟷
        (x ∉ S) ∧ (y ∉ S) ∨ x ∈ S ∧ y ∈ S ∧ path_component S x y"
    (is "?lhs = ?rhs")
proof 
  assume ?lhs then show ?rhs
    by (metis (no_types) path_component_mem(1) path_component_refl)
next
  assume ?rhs then show ?lhs
  proof
    assume "x ∉ S ∧ y ∉ S" then show ?lhs
      by (metis Collect_empty_eq_bot path_component_eq_empty)
  next
    assume S: "x ∈ S ∧ y ∈ S ∧ path_component S x y" show ?lhs
      by (rule ext) (metis S path_component_trans path_component_sym)
  qed
qed

lemma path_component_unique:
  assumes "x ∈ C" "C ⊆ S" "path_connected C"
          "∧C'. [x ∈ C'; C' ⊆ S; path_connected C'] ==> C' ⊆ C"
        shows "path_component_set S x = C"
  by (smt (verit, best) Collect_cong assms path_component path_component_of_subset path_connected_component_set)

lemma path_component_intermediate_subset:
  "path_component_set U a ⊆ T ∧ T ⊆ U
        ==> path_component_set T a = path_component_set U a"
  by (metis (no_types) path_component_mono path_component_path_component subset_antisym)

lemma complement_path_component_Union:
  fixes x :: "'a :: topological_space"
  shows "S - path_component_set S x =
         ∪({path_component_set S y| y. y ∈ S} - {path_component_set S x})"
proof -
  have *: "(∧x. x ∈ S - {a} ==> disjnt a x) ==> ∪S - a = ∪(S - {a})"
    for a::"'a set" and S
    by (auto simp: disjnt_def)
  have "∧y. y ∈ {path_component_set S x |x. x ∈ S} - {path_component_set S x}
            ==> disjnt (path_component_set S x) y"
    using path_component_disjoint path_component_eq by fastforce
  then have "∪{path_component_set S x |x. x ∈ S} - path_component_set S x =
             ∪({path_component_set S y |y. y ∈ S} - {path_component_set S x})"
    by (meson *)
  then show ?thesis by simp
qed


subsection‹Path components›

definition path_component_of
  where "path_component_of X x y ≡ ∃g. pathin X g ∧ g 0 = x ∧ g 1 = y"

abbreviation path_component_of_set
  where "path_component_of_set X x ≡ Collect (path_component_of X x)"

definition path_components_of :: "'a topology ==> 'a set set"
  where "path_components_of X ≡ path_component_of_set X ` topspace X"

lemma pathin_canon_iff: "pathin (top_of_set T) g ⟷ path g ∧ g ∈ {0..1} → T"
  by (simp add: path_def pathin_def image_subset_iff_funcset)

lemma path_component_of_canon_iff [simp]:
  "path_component_of (top_of_set T) a b ⟷ path_component T a b"
  by (simp add: path_component_of_def pathin_canon_iff path_defs image_subset_iff_funcset)

lemma path_component_in_topspace:
   "path_component_of X x y ==> x ∈ topspace X ∧ y ∈ topspace X"
  by (auto simp: path_component_of_def pathin_def continuous_map_def)

lemma path_component_of_refl:
   "path_component_of X x x ⟷ x ∈ topspace X"
  by (metis path_component_in_topspace path_component_of_def pathin_const)

lemma path_component_of_sym:
  assumes "path_component_of X x y"
  shows "path_component_of X y x"
  using assms
  apply (clarsimp simp: path_component_of_def pathin_def)
  apply (rule_tac x="g ∘ (λt. 1 - t)" in exI)
  apply (auto intro!: continuous_map_compose simp: continuous_map_in_subtopology continuous_on_op_minus)
  done

lemma path_component_of_sym_iff:
   "path_component_of X x y ⟷ path_component_of X y x"
  by (metis path_component_of_sym)

lemma continuous_map_cases_le:
  assumes contp: "continuous_map X euclideanreal p"
    and contq: "continuous_map X euclideanreal q"
    and contf: "continuous_map (subtopology X {x. x ∈ topspace X ∧ p x ≤ q x}) Y f"
    and contg: "continuous_map (subtopology X {x. x ∈ topspace X ∧ q x ≤ p x}) Y g"
    and fg: "∧x. [x ∈ topspace X; p x = q x] ==> f x = g x"
  shows "continuous_map X Y (λx. if p x ≤ q x then f x else g x)"
proof -
  have "continuous_map X Y (λx. if q x - p x ∈ {0..} then f x else g x)"
  proof (rule continuous_map_cases_function)
    show "continuous_map X euclideanreal (λx. q x - p x)"
      by (intro contp contq continuous_intros)
    show "continuous_map (subtopology X {x ∈ topspace X. q x - p x ∈ euclideanreal closure_of {0..}}) Y f"
      by (simp add: contf)
    show "continuous_map (subtopology X {x ∈ topspace X. q x - p x ∈ euclideanreal closure_of (topspace euclideanreal - {0..})}) Y g"
      by (simp add: contg flip: Compl_eq_Diff_UNIV)
  qed (auto simp: fg)
  then show ?thesis
    by simp
qed

lemma continuous_map_cases_lt:
  assumes contp: "continuous_map X euclideanreal p"
    and contq: "continuous_map X euclideanreal q"
    and contf: "continuous_map (subtopology X {x. x ∈ topspace X ∧ p x ≤ q x}) Y f"
    and contg: "continuous_map (subtopology X {x. x ∈ topspace X ∧ q x ≤ p x}) Y g"
    and fg: "∧x. [x ∈ topspace X; p x = q x] ==> f x = g x"
  shows "continuous_map X Y (λx. if p x < q x then f x else g x)"
proof -
  have "continuous_map X Y (λx. if q x - p x ∈ {0<..} then f x else g x)"
  proof (rule continuous_map_cases_function)
    show "continuous_map X euclideanreal (λx. q x - p x)"
      by (intro contp contq continuous_intros)
    show "continuous_map (subtopology X {x ∈ topspace X. q x - p x ∈ euclideanreal closure_of {0<..}}) Y f"
      by (simp add: contf)
    show "continuous_map (subtopology X {x ∈ topspace X. q x - p x ∈ euclideanreal closure_of (topspace euclideanreal - {0<..})}) Y g"
      by (simp add: contg flip: Compl_eq_Diff_UNIV)
  qed (auto simp: fg)
  then show ?thesis
    by simp
qed

lemma path_component_of_trans:
  assumes "path_component_of X x y" and "path_component_of X y z"
  shows "path_component_of X x z"
  unfolding path_component_of_def pathin_def
proof -
  let ?T01 = "top_of_set {0..1::real}"
  obtain g1 g2 where g1: "continuous_map ?T01 X g1" "x = g1 0" "y = g1 1"
    and g2: "continuous_map ?T01 X g2" "g2 0 = g1 1" "z = g2 1"
    using assms unfolding path_component_of_def pathin_def by blast
  let ?g = "λx. if x ≤ 1/2 then (g1 ∘ (λt. 2 * t)) x else (g2 ∘ (λt. 2 * t -1)) x"
  show "∃g. continuous_map ?T01 X g ∧ g 0 = x ∧ g 1 = z"
  proof (intro exI conjI)
    show "continuous_map (subtopology euclideanreal {0..1}) X ?g"
    proof (intro continuous_map_cases_le continuous_map_compose, force, force)
      show "continuous_map (subtopology ?T01 {x ∈ topspace ?T01. x ≤ 1/2}) ?T01 ((*) 2)"
        by (auto simp: continuous_map_in_subtopology continuous_map_from_subtopology)
      have "continuous_map
             (subtopology (top_of_set {0..1}) {x. 0 ≤ x ∧ x ≤ 1 ∧ 1 ≤ x * 2})
             euclideanreal (λt. 2 * t - 1)"
        by (intro continuous_intros) (force intro: continuous_map_from_subtopology)
      then show "continuous_map (subtopology ?T01 {x ∈ topspace ?T01. 1/2 ≤ x}) ?T01 (λt. 2 * t - 1)"
        by (force simp: continuous_map_in_subtopology)
      show "(g1 ∘ (*) 2) x = (g2 ∘ (λt. 2 * t - 1)) x" if "x ∈ topspace ?T01" "x = 1/2" for x
        using that by (simp add: g2(2) mult.commute continuous_map_from_subtopology)
    qed (auto simp: g1 g2)
  qed (auto simp: g1 g2)
qed

lemma path_component_of_mono:
   "[path_component_of (subtopology X S) x y; S ⊆ T] ==> path_component_of (subtopology X T) x y"
  unfolding path_component_of_def
  by (metis subsetD pathin_subtopology)

lemma path_component_of:
  "path_component_of X x y ⟷ (∃T. path_connectedin X T ∧ x ∈ T ∧ y ∈ T)"
    (is "?lhs = ?rhs")
proof 
  assume ?lhs then show ?rhs
    by (metis atLeastAtMost_iff image_eqI order_refl path_component_of_def path_connectedin_path_image zero_le_one)
next
  assume ?rhs then show ?lhs
    by (metis path_component_of_def path_connectedin)
qed

lemma path_component_of_set:
   "path_component_of X x y ⟷ (∃g. pathin X g ∧ g 0 = x ∧ g 1 = y)"
  by (auto simp: path_component_of_def)

lemma path_component_of_subset_topspace:
   "Collect(path_component_of X x) ⊆ topspace X"
  using path_component_in_topspace by fastforce

lemma path_component_of_eq_empty:
   "Collect(path_component_of X x) = {} ⟷ (x ∉ topspace X)"
  using path_component_in_topspace path_component_of_refl by fastforce

lemma path_connected_space_iff_path_component:
   "path_connected_space X ⟷ (∀x ∈ topspace X. ∀y ∈ topspace X. path_component_of X x y)"
  by (simp add: path_component_of path_connected_space_subconnected)

lemma path_connected_space_imp_path_component_of:
   "[path_connected_space X; a ∈ topspace X; b ∈ topspace X]
        ==> path_component_of X a b"
  by (simp add: path_connected_space_iff_path_component)

lemma path_connected_space_path_component_set:
   "path_connected_space X ⟷ (∀x ∈ topspace X. Collect(path_component_of X x) = topspace X)"
  using path_component_of_subset_topspace path_connected_space_iff_path_component by fastforce

lemma path_component_of_maximal:
   "[path_connectedin X s; x ∈ s] ==> s ⊆ Collect(path_component_of X x)"
  using path_component_of by fastforce

lemma path_component_of_equiv:
   "path_component_of X x y ⟷ x ∈ topspace X ∧ y ∈ topspace X ∧ path_component_of X x = path_component_of X y"
    (is "?lhs = ?rhs")
proof
  assume ?lhs
  then show ?rhs
    unfolding fun_eq_iff path_component_in_topspace
    by (metis path_component_in_topspace path_component_of_sym path_component_of_trans)
qed (simp add: path_component_of_refl)

lemma path_component_of_disjoint:
     "disjnt (Collect (path_component_of X x)) (Collect (path_component_of X y)) ⟷
      ~(path_component_of X x y)"
  by (force simp: disjnt_def path_component_of_eq_empty path_component_of_equiv)

lemma path_component_of_eq:
   "path_component_of X x = path_component_of X y ⟷
        (x ∉ topspace X) ∧ (y ∉ topspace X) ∨
        x ∈ topspace X ∧ y ∈ topspace X ∧ path_component_of X x y"
  by (metis Collect_empty_eq_bot path_component_of_eq_empty path_component_of_equiv)

lemma path_component_of_aux:
  "path_component_of X x y
        ==> path_component_of (subtopology X (Collect (path_component_of X x))) x y"
    by (meson path_component_of path_component_of_maximal path_connectedin_subtopology)

lemma path_connectedin_path_component_of:
  "path_connectedin X (Collect (path_component_of X x))"
proof -
  have "topspace (subtopology X (path_component_of_set X x)) = path_component_of_set X x"
    by (meson path_component_of_subset_topspace topspace_subtopology_subset)
  then have "path_connected_space (subtopology X (path_component_of_set X x))"
    by (metis mem_Collect_eq path_component_of_aux path_component_of_equiv path_connected_space_iff_path_component)
  then show ?thesis
    by (simp add: path_component_of_subset_topspace path_connectedin_def)
qed

lemma path_connectedin_euclidean [simp]:
   "path_connectedin euclidean S ⟷ path_connected S"
  by (auto simp: path_connectedin_def path_connected_space_iff_path_component path_connected_component)

lemma path_connected_space_euclidean_subtopology [simp]:
   "path_connected_space(subtopology euclidean S) ⟷ path_connected S"
  using path_connectedin_topspace by force

lemma Union_path_components_of:
     "∪(path_components_of X) = topspace X"
  by (auto simp: path_components_of_def path_component_of_equiv)

lemma path_components_of_maximal:
   "[C ∈ path_components_of X; path_connectedin X S; ~disjnt C S] ==> S ⊆ C"
  by (smt (verit, ccfv_SIG) disjnt_iff imageE mem_Collect_eq path_component_of_equiv 
      path_component_of_maximal path_components_of_def)

lemma pairwise_disjoint_path_components_of:
     "pairwise disjnt (path_components_of X)"
  by (auto simp: path_components_of_def pairwise_def path_component_of_disjoint path_component_of_equiv)

lemma complement_path_components_of_Union:
   "C ∈ path_components_of X ==> topspace X - C = ∪(path_components_of X - {C})"
  by (metis Union_path_components_of bot.extremum ccpo_Sup_singleton diff_Union_pairwise_disjoint 
        insert_subsetI pairwise_disjoint_path_components_of)

lemma nonempty_path_components_of:
  assumes "C ∈ path_components_of X" shows "C ≠ {}"
  by (metis assms imageE path_component_of_eq_empty path_components_of_def)

lemma path_components_of_subset: "C ∈ path_components_of X ==> C ⊆ topspace X"
  by (auto simp: path_components_of_def path_component_of_equiv)

lemma path_connectedin_path_components_of:
   "C ∈ path_components_of X ==> path_connectedin X C"
  by (auto simp: path_components_of_def path_connectedin_path_component_of)

lemma path_component_in_path_components_of:
  "Collect (path_component_of X a) ∈ path_components_of X ⟷ a ∈ topspace X"
  by (metis imageI nonempty_path_components_of path_component_of_eq_empty path_components_of_def)

lemma path_connectedin_Union:
  assumes A: "∧S. S ∈ A ==> path_connectedin X S" and "∩A ≠ {}"
  shows "path_connectedin X (∪A)"
proof -
  obtain a where "∧S. S ∈ A ==> a ∈ S"
    using assms by blast
  then have "∧x. x ∈ topspace (subtopology X (∪A)) ==> path_component_of (subtopology X (∪A)) a x"
    unfolding topspace_subtopology path_component_of
    by (metis (full_types) IntD2 Union_iff Union_upper A path_connectedin_subtopology)
  then show ?thesis
    using A unfolding path_connectedin_def
    by (metis Sup_le_iff path_component_of_equiv path_connected_space_iff_path_component)
qed

lemma path_connectedin_Un:
   "[path_connectedin X S; path_connectedin X T; S ∩ T ≠ {}]
    ==> path_connectedin X (S ∪ T)"
  by (blast intro: path_connectedin_Union [of "{S,T}", simplified])

lemma path_connected_space_iff_components_eq:
  "path_connected_space X ⟷
    (∀C ∈ path_components_of X. ∀C' ∈ path_components_of X. C = C')"
  unfolding path_components_of_def
proof (intro iffI ballI)
  assume "∀C ∈ path_component_of_set X ` topspace X.
             ∀C' ∈ path_component_of_set X ` topspace X. C = C'"
  then show "path_connected_space X"
    using path_component_of_refl path_connected_space_iff_path_component by fastforce
qed (auto simp: path_connected_space_path_component_set)

lemma path_components_of_eq_empty:
   "path_components_of X = {} ⟷ X = trivial_topology"
  by (metis image_is_empty path_components_of_def subtopology_eq_discrete_topology_empty)

lemma path_components_of_empty_space:
   "path_components_of trivial_topology = {}"
  by (simp add: path_components_of_eq_empty)

lemma path_components_of_subset_singleton:
  "path_components_of X ⊆ {S} ⟷
        path_connected_space X ∧ (topspace X = {} ∨ topspace X = S)"
proof (cases "topspace X = {}")
  case True
  then show ?thesis
    by (auto simp: path_components_of_empty_space path_connected_space_topspace_empty)
next
  case False
  have "(path_components_of X = {S}) ⟷ (path_connected_space X ∧ topspace X = S)"
    by (metis False Set.set_insert ex_in_conv insert_iff path_component_in_path_components_of 
        path_connected_space_iff_components_eq path_connected_space_path_component_set)
  with False show ?thesis
    by (simp add: path_components_of_eq_empty subset_singleton_iff)
qed

lemma path_connected_space_iff_components_subset_singleton:
   "path_connected_space X ⟷ (∃a. path_components_of X ⊆ {a})"
  by (simp add: path_components_of_subset_singleton)

lemma path_components_of_eq_singleton:
   "path_components_of X = {S} ⟷ path_connected_space X ∧ topspace X ≠ {} ∧ S = topspace X"
  by (metis cSup_singleton insert_not_empty path_components_of_subset_singleton subset_singleton_iff)

lemma path_components_of_path_connected_space:
   "path_connected_space X ==> path_components_of X = (if topspace X = {} then {} else {topspace X})"
  by (simp add: path_components_of_eq_empty path_components_of_eq_singleton)

lemma path_component_subset_connected_component_of:
   "path_component_of_set X x ⊆ connected_component_of_set X x"
proof (cases "x ∈ topspace X")
  case True
  then show ?thesis
    by (simp add: connected_component_of_maximal path_component_of_refl path_connectedin_imp_connectedin path_connectedin_path_component_of)
next
  case False
  then show ?thesis
    using path_component_of_eq_empty by fastforce
qed

lemma exists_path_component_of_superset:
  assumes S: "path_connectedin X S" and ne: "topspace X ≠ {}"
  obtains C where "C ∈ path_components_of X" "S ⊆ C"
    by (metis S ne ex_in_conv path_component_in_path_components_of path_component_of_maximal path_component_of_subset_topspace subset_eq that)

lemma path_component_of_eq_overlap:
   "path_component_of X x = path_component_of X y ⟷
      (x ∉ topspace X) ∧ (y ∉ topspace X) ∨
      Collect (path_component_of X x) ∩ Collect (path_component_of X y) ≠ {}"
  by (metis disjnt_def empty_iff inf_bot_right mem_Collect_eq path_component_of_disjoint path_component_of_eq path_component_of_eq_empty)

lemma path_component_of_nonoverlap:
   "Collect (path_component_of X x) ∩ Collect (path_component_of X y) = {} ⟷
    (x ∉ topspace X) ∨ (y ∉ topspace X) ∨
    path_component_of X x ≠ path_component_of X y"
  by (metis inf.idem path_component_of_eq_empty path_component_of_eq_overlap)

lemma path_component_of_overlap:
   "Collect (path_component_of X x) ∩ Collect (path_component_of X y) ≠ {} ⟷
    x ∈ topspace X ∧ y ∈ topspace X ∧ path_component_of X x = path_component_of X y"
  by (meson path_component_of_nonoverlap)

lemma path_components_of_disjoint:
     "[C ∈ path_components_of X; C' ∈ path_components_of X] ==> disjnt C C' ⟷ C ≠ C'"
  by (auto simp: path_components_of_def path_component_of_disjoint path_component_of_equiv)

lemma path_components_of_overlap:
    "[C ∈ path_components_of X; C' ∈ path_components_of X] ==> C ∩ C' ≠ {} ⟷ C = C'"
  by (auto simp: path_components_of_def path_component_of_equiv)

lemma path_component_of_unique:
   "[x ∈ C; path_connectedin X C; ∧C'. [x ∈ C'; path_connectedin X C'] ==> C' ⊆ C]
        ==> Collect (path_component_of X x) = C"
  by (meson subsetD eq_iff path_component_of_maximal path_connectedin_path_component_of)

lemma path_component_of_discrete_topology [simp]:
  "Collect (path_component_of (discrete_topology U) x) = (if x ∈ U then {x} else {})"
proof -
  have "∧C'. [x ∈ C'; path_connectedin (discrete_topology U) C'] ==> C' ⊆ {x}"
    by (metis path_connectedin_discrete_topology subsetD singletonD)
  then have "x ∈ U ==> Collect (path_component_of (discrete_topology U) x) = {x}"
    by (simp add: path_component_of_unique)
  then show ?thesis
    using path_component_in_topspace by fastforce
qed

lemma path_component_of_discrete_topology_iff [simp]:
  "path_component_of (discrete_topology U) x y ⟷ x ∈ U ∧ y=x"
  by (metis empty_iff insertI1 mem_Collect_eq path_component_of_discrete_topology singletonD)

lemma path_components_of_discrete_topology [simp]:
   "path_components_of (discrete_topology U) = (λx. {x}) ` U"
  by (auto simp: path_components_of_def image_def fun_eq_iff)

lemma homeomorphic_map_path_component_of:
  assumes f: "homeomorphic_map X Y f" and x: "x ∈ topspace X"
  shows "Collect (path_component_of Y (f x)) = f ` Collect(path_component_of X x)"
proof -
  obtain g where g: "homeomorphic_maps X Y f g"
    using f homeomorphic_map_maps by blast
  show ?thesis
  proof
    have "Collect (path_component_of Y (f x)) ⊆ topspace Y"
      by (simp add: path_component_of_subset_topspace)
    moreover have "g ` Collect(path_component_of Y (f x)) ⊆ Collect (path_component_of X (g (f x)))"
      using f g x unfolding homeomorphic_maps_def
      by (metis image_Collect_subsetI image_eqI mem_Collect_eq path_component_of_equiv path_component_of_maximal 
          path_connectedin_continuous_map_image path_connectedin_path_component_of)
    ultimately show "Collect (path_component_of Y (f x)) ⊆ f ` Collect (path_component_of X x)"
      using g x unfolding homeomorphic_maps_def continuous_map_def image_iff subset_iff
      by metis
    show "f ` Collect (path_component_of X x) ⊆ Collect (path_component_of Y (f x))"
    proof (rule path_component_of_maximal)
      show "path_connectedin Y (f ` Collect (path_component_of X x))"
        by (meson f homeomorphic_map_path_connectedness_eq path_connectedin_path_component_of)
    qed (simp add: path_component_of_refl x)
  qed
qed

lemma homeomorphic_map_path_components_of:
  assumes "homeomorphic_map X Y f"
  shows "path_components_of Y = (image f) ` (path_components_of X)"
  unfolding path_components_of_def homeomorphic_imp_surjective_map [OF assms, symmetric]
  using assms homeomorphic_map_path_component_of by fastforce


subsection‹Paths and path-connectedness›

lemma path_connected_space_quotient_map_image:
   "[quotient_map X Y q; path_connected_space X] ==> path_connected_space Y"
  by (metis path_connectedin_continuous_map_image path_connectedin_topspace quotient_imp_continuous_map quotient_imp_surjective_map)

lemma path_connected_space_retraction_map_image:
   "[retraction_map X Y r; path_connected_space X] ==> path_connected_space Y"
  using path_connected_space_quotient_map_image retraction_imp_quotient_map by blast

lemma path_connected_space_prod_topology:
  "path_connected_space(prod_topology X Y) ⟷
        topspace(prod_topology X Y) = {} ∨ path_connected_space X ∧ path_connected_space Y"
proof (cases "topspace(prod_topology X Y) = {}")
  case True
  then show ?thesis
    using path_connected_space_topspace_empty by force
next
  case False
  have "path_connected_space (prod_topology X Y)" 
    if X: "path_connected_space X" and Y: "path_connected_space Y"
  proof (clarsimp simp: path_connected_space_def)
    fix x y x' y'
    assume "x ∈ topspace X" and "y ∈ topspace Y" and "x' ∈ topspace X" and "y' ∈ topspace Y"
    obtain f where "pathin X f" "f 0 = x" "f 1 = x'"
      by (meson X ‹x ∈ topspace X› ‹x' ∈ topspace X› path_connected_space_def)
    obtain g where "pathin Y g" "g 0 = y" "g 1 = y'"
      by (meson Y ‹y ∈ topspace Y› ‹y' ∈ topspace Y› path_connected_space_def)
    show "∃h. pathin (prod_topology X Y) h ∧ h 0 = (x,y) ∧ h 1 = (x',y')"
    proof (intro exI conjI)
      show "pathin (prod_topology X Y) (λt. (f t, g t))"
        using ‹pathin X f› ‹pathin Y g› by (simp add: continuous_map_paired pathin_def)
      show "(λt. (f t, g t)) 0 = (x, y)"
        using ‹f 0 = x› ‹g 0 = y› by blast
      show "(λt. (f t, g t)) 1 = (x', y')"
        using ‹f 1 = x'› ‹g 1 = y'› by blast
    qed
  qed
  then show ?thesis
    by (metis False path_connected_space_quotient_map_image prod_topology_trivial1 prod_topology_trivial2 
        quotient_map_fst quotient_map_snd topspace_discrete_topology)
qed

lemma path_connectedin_Times:
   "path_connectedin (prod_topology X Y) (S × T) ⟷
        S = {} ∨ T = {} ∨ path_connectedin X S ∧ path_connectedin Y T"
  by (auto simp add: path_connectedin_def subtopology_Times path_connected_space_prod_topology)


subsection‹Path components›

lemma path_component_of_subtopology_eq:
  "path_component_of (subtopology X U) x = path_component_of X x ⟷ path_component_of_set X x ⊆ U"  
  (is "?lhs = ?rhs")
proof
  show "?lhs ==> ?rhs"
    by (metis path_connectedin_path_component_of path_connectedin_subtopology)
next
  show "?rhs ==> ?lhs"
    unfolding fun_eq_iff
    by (metis path_connectedin_subtopology path_component_of path_component_of_aux path_component_of_mono)
qed

lemma path_components_of_subtopology:
  assumes "C ∈ path_components_of X" "C ⊆ U"
  shows "C ∈ path_components_of (subtopology X U)"
  using assms path_component_of_refl path_component_of_subtopology_eq topspace_subtopology
  by (smt (verit) imageE path_component_in_path_components_of path_components_of_def)

lemma path_imp_connected_component_of:
   "path_component_of X x y ==> connected_component_of X x y"
  by (metis in_mono mem_Collect_eq path_component_subset_connected_component_of)

lemma finite_path_components_of_finite:
   "finite(topspace X) ==> finite(path_components_of X)"
  by (simp add: Union_path_components_of finite_UnionD)

lemma path_component_of_continuous_image:
  "[continuous_map X X' f; path_component_of X x y] ==> path_component_of X' (f x) (f y)"
  by (meson image_eqI path_component_of path_connectedin_continuous_map_image)

lemma path_component_of_pair [simp]:
   "path_component_of_set (prod_topology X Y) (x,y) =
    path_component_of_set X x × path_component_of_set Y y"   (is "?lhs = ?rhs")
proof (cases "?lhs = {}")
  case True
  then show ?thesis
    by (metis Sigma_empty1 Sigma_empty2 mem_Sigma_iff path_component_of_eq_empty topspace_prod_topology) 
next
  case False
  then have "path_component_of X x x" "path_component_of Y y y"
    using path_component_of_eq_empty path_component_of_refl by fastforce+
  moreover
  have "path_connectedin (prod_topology X Y) (path_component_of_set X x × path_component_of_set Y y)"
    by (metis path_connectedin_Times path_connectedin_path_component_of)
  moreover have "path_component_of X x a" "path_component_of Y y b"
    if "(x, y) ∈ C'" "(a,b) ∈ C'" and "path_connectedin (prod_topology X Y) C'" for C' a b
    by (smt (verit, best) that continuous_map_fst continuous_map_snd fst_conv snd_conv path_component_of path_component_of_continuous_image)+
  ultimately show ?thesis
    by (intro path_component_of_unique) auto
qed

lemma path_components_of_prod_topology:
   "path_components_of (prod_topology X Y) =
    (λ(C,D). C × D) ` (path_components_of X × path_components_of Y)"
  by (force simp add: image_iff path_components_of_def)

lemma path_components_of_prod_topology':
   "path_components_of (prod_topology X Y) =
    {C × D |C D. C ∈ path_components_of X ∧ D ∈ path_components_of Y}"
  by (auto simp: path_components_of_prod_topology)

lemma path_component_of_product_topology:
   "path_component_of_set (product_topology X I) f =
    (if f ∈ extensional I then PiE I (λi. path_component_of_set (X i) (f i)) else {})"
    (is "?lhs = ?rhs")
proof (cases "path_component_of_set (product_topology X I) f = {}")
  case True
  then show ?thesis
    by (smt (verit) PiE_eq_empty_iff PiE_iff path_component_of_eq_empty topspace_product_topology)
next
  case False
  then have [simp]: "f ∈ extensional I"
    by (auto simp: path_component_of_eq_empty PiE_iff path_component_of_equiv)
  show ?thesis
  proof (intro path_component_of_unique)
    show "f ∈ ?rhs"
      using False path_component_of_eq_empty path_component_of_refl by force
    show "path_connectedin (product_topology X I) (if f ∈ extensional I then Π🪙E i∈I. path_component_of_set (X i) (f i) else {})"
      by (simp add: path_connectedin_PiE path_connectedin_path_component_of)
    fix C'
    assume "f ∈ C'" and C': "path_connectedin (product_topology X I) C'" 
    show "C' ⊆ ?rhs"
    proof -
      have "C' ⊆ extensional I"
        using PiE_def C' path_connectedin_subset_topspace by fastforce
      with ‹f ∈ C'› C' show ?thesis
        apply (clarsimp simp: PiE_iff subset_iff)
        by (smt (verit, ccfv_threshold) continuous_map_product_projection path_component_of path_component_of_continuous_image)
    qed   
  qed
qed

lemma path_components_of_product_topology:
  "path_components_of (product_topology X I) =
    {PiE I B |B. ∀i ∈ I. B i ∈ path_components_of(X i)}"  (is "?lhs=?rhs")
proof
  show "?lhs ⊆ ?rhs"
    unfolding path_components_of_def image_subset_iff
    by (smt (verit) image_iff mem_Collect_eq path_component_of_product_topology topspace_product_topology_alt)
next
  show "?rhs ⊆ ?lhs"
  proof
    fix F
    assume "F ∈ ?rhs"
    then obtain B where B: "F = Pi🪙E I B"
      and "∀i∈I. ∃x∈topspace (X i). B i = path_component_of_set (X i) x"
      by (force simp add: path_components_of_def image_iff)
    then obtain f where ftop: "∧i. i ∈ I ==> f i ∈ topspace (X i)"
      and BF: "∧i. i ∈ I ==> B i = path_component_of_set (X i) (f i)"
      by metis
    then have "F = path_component_of_set (product_topology X I) (restrict f I)"
      by (metis (mono_tags, lifting) B PiE_cong path_component_of_product_topology restrict_apply' restrict_extensional)
    then show "F ∈ ?lhs"
      by (simp add: ftop path_component_in_path_components_of)
  qed
qed

subsection ‹Sphere is path-connected›

lemma path_connected_punctured_universe:
  assumes "2 ≤ DIM('a::euclidean_space)"
  shows "path_connected (- {a::'a})"
proof -
  let ?A = "{x::'a. ∃i∈Basis. x ∙ i < a ∙ i}"
  let ?B = "{x::'a. ∃i∈Basis. a ∙ i < x ∙ i}"

  have A: "path_connected ?A"
    unfolding Collect_bex_eq
  proof (rule path_connected_UNION)
    fix i :: 'a
    assume "i ∈ Basis"
    then show "(∑i∈Basis. (a ∙ i - 1)*🪙R i) ∈ {x::'a. x ∙ i < a ∙ i}"
      by simp
    show "path_connected {x. x ∙ i < a ∙ i}"
      using convex_imp_path_connected [OF convex_halfspace_lt, of i "a ∙ i"]
      by (simp add: inner_commute)
  qed
  have B: "path_connected ?B"
    unfolding Collect_bex_eq
  proof (rule path_connected_UNION)
    fix i :: 'a
    assume "i ∈ Basis"
    then show "(∑i∈Basis. (a ∙ i + 1) *🪙R i) ∈ {x::'a. a ∙ i < x ∙ i}"
      by simp
    show "path_connected {x. a ∙ i < x ∙ i}"
      using convex_imp_path_connected [OF convex_halfspace_gt, of "a ∙ i" i]
      by (simp add: inner_commute)
  qed
  obtain S :: "'a set" where "S ⊆ Basis" and "card S = Suc (Suc 0)"
    using obtain_subset_with_card_n[OF assms] by (force simp add: eval_nat_numeral)
  then obtain b0 b1 :: 'a where "b0 ∈ Basis" and "b1 ∈ Basis" and "b0 ≠ b1"
    unfolding card_Suc_eq by auto
  then have "a + b0 - b1 ∈ ?A ∩ ?B"
    by (auto simp: inner_simps inner_Basis)
  then have "?A ∩ ?B ≠ {}"
    by fast
  with A B have "path_connected (?A ∪ ?B)"
    by (rule path_connected_Un)
  also have "?A ∪ ?B = {x. ∃i∈Basis. x ∙ i ≠ a ∙ i}"
    unfolding neq_iff bex_disj_distrib Collect_disj_eq ..
  also have "… = {x. x ≠ a}"
    unfolding euclidean_eq_iff [where 'a='a]
    by (simp add: Bex_def)
  also have "… = - {a}"
    by auto
  finally show ?thesis .
qed

corollary connected_punctured_universe:
  "2 ≤ DIM('N::euclidean_space) ==> connected(- {a::'N})"
  by (simp add: path_connected_punctured_universe path_connected_imp_connected)

proposition path_connected_sphere:
  fixes a :: "'a :: euclidean_space"
  assumes "2 ≤ DIM('a)"
  shows "path_connected(sphere a r)"
proof (cases r "0::real" rule: linorder_cases)
  case greater
  then have eq: "(sphere (0::'a) r) = (λx. (r / norm x) *🪙R x) ` (- {0::'a})"
    by (force simp: image_iff split: if_split_asm)
  have "continuous_on (- {0::'a}) (λx. (r / norm x) *🪙R x)"
    by (intro continuous_intros) auto
  then have "path_connected ((λx. (r / norm x) *🪙R x) ` (- {0::'a}))"
    by (intro path_connected_continuous_image path_connected_punctured_universe assms)
  with eq have "path_connected((+) a ` (sphere (0::'a) r))"
    by (simp add: path_connected_translation)
  then show ?thesis
    by (metis add.right_neutral sphere_translation)
qed auto

lemma connected_sphere:
    fixes a :: "'a :: euclidean_space"
    assumes "2 ≤ DIM('a)"
      shows "connected(sphere a r)"
  using path_connected_sphere [OF assms]
  by (simp add: path_connected_imp_connected)


corollary path_connected_complement_bounded_convex:
    fixes S :: "'a :: euclidean_space set"
    assumes "bounded S" "convex S" and 2: "2 ≤ DIM('a)"
    shows "path_connected (- S)"
proof (cases "S = {}")
  case True then show ?thesis
    using convex_imp_path_connected by auto
next
  case False
  then obtain a where "a ∈ S" by auto
  have 🍋 [rule_format]: "∀y∈S. ∀u. 0 ≤ u ∧ u ≤ 1 ⟶ (1 - u) *🪙R a + u *🪙R y ∈ S"
    using ‹convex S› ‹a ∈ S› by (simp add: convex_alt)
  { fix x y assume "x ∉ S" "y ∉ S"
    then have "x ≠ a" "y ≠ a" using ‹a ∈ S› by auto
    then have bxy: "bounded(insert x (insert y S))"
      by (simp add: ‹bounded S›)
    then obtain B::real where B: "0 < B" and Bx: "norm (a - x) < B" and By: "norm (a - y) < B"
                          and "S ⊆ ball a B"
      using bounded_subset_ballD [OF bxy, of a] by (auto simp: dist_norm)
    define C where "C = B / norm(x - a)"
    let ?Cxa = "a + C *🪙R (x - a)"
    { fix u
      assume u: "(1 - u) *🪙R x + u *🪙R ?Cxa ∈ S" and "0 ≤ u" "u ≤ 1"
      have CC: "1 ≤ 1 + (C - 1) * u"
        using ‹x ≠ a› ‹0 ≤ u› Bx
        by (auto simp add: C_def norm_minus_commute)
      have *: "∧v. (1 - u) *🪙R x + u *🪙R (a + v *🪙R (x - a)) = a + (1 + (v - 1) * u) *🪙R (x - a)"
        by (simp add: algebra_simps)
      have "a + ((1 / (1 + C * u - u)) *🪙R x + ((u / (1 + C * u - u)) *🪙R a + (C * u / (1 + C * u - u)) *🪙R x)) =
            (1 + (u / (1 + C * u - u))) *🪙R a + ((1 / (1 + C * u - u)) + (C * u / (1 + C * u - u))) *🪙R x"
        by (simp add: algebra_simps)
      also have "… = (1 + (u / (1 + C * u - u))) *🪙R a + (1 + (u / (1 + C * u - u))) *??R x"
        using CC by (simp add: field_simps)
      also have "… = x + (1 + (u / (1 + C * u - u))) *🪙R a + (u / (1 + C * u - u)) *🪙R x"
        by (simp add: algebra_simps)
      also have "… = x + ((1 / (1 + C * u - u)) *🪙R a +
              ((u / (1 + C * u - u)) *🪙R x + (C * u / (1 + C * u - u)) *🪙R a))"
        using CC by (simp add: field_simps) (simp add: add_divide_distrib scaleR_add_left)
      finally have xeq: "(1 - 1 / (1 + (C - 1) * u)) *🪙R a + (1 / (1 + (C - 1) * u)) *🪙R (a + (1 + (C - 1) * u) *🪙R (x - a)) = x"
        by (simp add: algebra_simps)
      have False
        using 🍋 [of "a + (1 + (C - 1) * u) *🪙R (x - a)" "1 / (1 + (C - 1) * u)"]
        using u ‹x ≠ a› ‹x ∉ S› ‹0 ≤ u› CC
        by (auto simp: xeq *)
    }
    then have pcx: "path_component (- S) x ?Cxa"
      by (force simp: closed_segment_def intro!: path_component_linepath)
    define D where "D = B / norm(y - a)"  🍋 ‹massive duplication with the proof above›
    let ?Dya = "a + D *🪙R (y - a)"
    { fix u
      assume u: "(1 - u) *🪙R y + u *🪙R ?Dya ∈ S" and "0 ≤ u" "u ≤ 1"
      have DD: "1 ≤ 1 + (D - 1) * u"
        using ‹y ≠ a› ‹0 ≤ u› By
        by (auto simp add: D_def norm_minus_commute)
      have *: "∧v. (1 - u) *🪙R y + u *🪙R (a + v *🪙R (y - a)) = a + (1 + (v - 1) * u) *🪙R (y - a)"
        by (simp add: algebra_simps)
      have "a + ((1 / (1 + D * u - u)) *🪙R y + ((u / (1 + D * u - u)) *🪙R a + (D * u / (1 + D * u - u)) *🪙R y)) =
            (1 + (u / (1 + D * u - u))) *🪙R a + ((1 / (1 + D * u - u)) + (D * u / (1 + D * u - u))) *🪙R y"
        by (simp add: algebra_simps)
      also have "… = (1 + (u / (1 + D * u - u))) *🪙R a + (1 + (u / (1 + D * u - u))) *??R y"
        using DD by (simp add: field_simps)
      also have "… = y + (1 + (u / (1 + D * u - u))) *🪙R a + (u / (1 + D * u - u)) *🪙R y"
        by (simp add: algebra_simps)
      also have "… = y + ((1 / (1 + D * u - u)) *🪙R a +
              ((u / (1 + D * u - u)) *🪙R y + (D * u / (1 + D * u - u)) *🪙R a))"
        using DD by (simp add: field_simps) (simp add: add_divide_distrib scaleR_add_left)
      finally have xeq: "(1 - 1 / (1 + (D - 1) * u)) *🪙R a + (1 / (1 + (D - 1) * u)) *🪙R (a + (1 + (D - 1) * u) *🪙R (y - a)) = y"
        by (simp add: algebra_simps)
      have False
        using 🍋 [of "a + (1 + (D - 1) * u) *🪙R (y - a)" "1 / (1 + (D - 1) * u)"]
        using u ‹y ≠ a› ‹y ∉ S› ‹0 ≤ u› DD
        by (auto simp: xeq *)
    }
    then have pdy: "path_component (- S) y ?Dya"
      by (force simp: closed_segment_def intro!: path_component_linepath)
    have pyx: "path_component (- S) ?Dya ?Cxa"
    proof (rule path_component_of_subset)
      show "sphere a B ⊆ - S"
        using ‹S ⊆ ball a B› by (force simp: ball_def dist_norm norm_minus_commute)
      have aB: "?Dya ∈ sphere a B" "?Cxa ∈ sphere a B"
        using ‹x ≠ a› using ‹y ≠ a› B by (auto simp: dist_norm C_def D_def)
      then show "path_component (sphere a B) ?Dya ?Cxa"
        using path_connected_sphere [OF 2] path_connected_component by blast
    qed
    have "path_component (- S) x y"
      by (metis path_component_trans path_component_sym pcx pdy pyx)
  }
  then show ?thesis
    by (auto simp: path_connected_component)
qed

lemma connected_complement_bounded_convex:
    fixes S :: "'a :: euclidean_space set"
    assumes "bounded S" "convex S" "2 ≤ DIM('a)"
      shows  "connected (- S)"
  using path_connected_complement_bounded_convex [OF assms] path_connected_imp_connected by blast

lemma connected_diff_ball:
    fixes S :: "'a :: euclidean_space set"
    assumes "connected S" "cball a r ⊆ S" "2 ≤ DIM('a)"
      shows "connected (S - ball a r)"
proof (rule connected_diff_open_from_closed [OF ball_subset_cball])
  show "connected (cball a r - ball a r)"
    using assms connected_sphere by (auto simp: cball_diff_eq_sphere)
qed (auto simp: assms dist_norm)

proposition connected_open_delete:
  assumes "open S" "connected S" and 2: "2 ≤ DIM('N::euclidean_space)"
    shows "connected(S - {a::'N})"
proof (cases "a ∈ S")
  case True
  with ‹open S› obtain ε where "ε > 0" and ε: "cball a ε ⊆ S"
    using open_contains_cball_eq by blast
  define b where "b ≡ a + ε *🪙R (SOME i. i ∈ Basis)"
  have "dist a b = ε"
    by (simp add: b_def dist_norm SOME_Basis ‹0 🚫ε› less_imp_le)
  with ε have "b ∈ ∩{S - ball a r |r. 0 < r ∧ r < ε}"
    by auto
  then have nonemp: "(∩{S - ball a r |r. 0 < r ∧ r < ε}) = {} ==> False"
    by auto
  have con: "∧r. r < ε ==> connected (S - ball a r)"
    using ε by (force intro: connected_diff_ball [OF ‹connected S› _ 2])
  have "x ∈ ∪{S - ball a r |r. 0 < r ∧ r < ε}" if "x ∈ S - {a}" for x
     using that ‹0 🚫ε›
     by (intro UnionI [of "S - ball a (min ε (dist a x) / 2)"]) auto
  then have "S - {a} = ∪{S - ball a r | r. 0 < r ∧ r < ε}"
    by auto
  then show ?thesis
    by (auto intro: connected_Union con dest!: nonemp)
next
  case False then show ?thesis
    by (simp add: ‹connected S›)
qed

corollary path_connected_open_delete:
  assumes "open S" "connected S" and 2: "2 ≤ DIM('N::euclidean_space)"
  shows "path_connected(S - {a::'N})"
  by (simp add: assms connected_open_delete connected_open_path_connected open_delete)

corollary path_connected_punctured_ball:
  "2 ≤ DIM('N::euclidean_space) ==> path_connected(ball a r - {a::'N})"
  by (simp add: path_connected_open_delete)

corollary connected_punctured_ball:
  "2 ≤ DIM('N::euclidean_space) ==> connected(ball a r - {a::'N})"
  by (simp add: connected_open_delete)

corollary connected_open_delete_finite:
  fixes S T::"'a::euclidean_space set"
  assumes S: "open S" "connected S" and 2: "2 ≤ DIM('a)" and "finite T"
  shows "connected(S - T)"
  using ‹finite T› S
proof (induct T)
  case empty
  show ?case using ‹connected S› by simp
next
  case (insert x T)
  then have "connected (S-T)" 
    by auto
  moreover have "open (S - T)" 
    using finite_imp_closed[OF ‹finite T›] ‹open S› by auto
  ultimately have "connected (S - T - {x})" 
    using connected_open_delete[OF _ _ 2] by auto
  thus ?case by (metis Diff_insert)
qed

lemma sphere_1D_doubleton_zero:
  assumes 1: "DIM('a) = 1" and "r > 0"
  obtains x y::"'a::euclidean_space"
    where "sphere 0 r = {x,y} ∧ dist x y = 2*r"
proof -
  obtain b::'a where b: "Basis = {b}"
    using 1 card_1_singletonE by blast
  show ?thesis
  proof (intro that conjI)
    have "x = norm x *🪙R b ∨ x = - norm x *🪙R b" if "r = norm x" for x
    proof -
      have xb: "(x ∙ b) *🪙R b = x"
        using euclidean_representation [of x, unfolded b] by force
      then have "norm ((x ∙ b) *🪙R b) = norm x"
        by simp
      with b have "∣x ∙ b∣ = norm x"
        using norm_Basis by (simp add: b)
      with xb show ?thesis
        by (metis (mono_tags, opaque_lifting) abs_eq_iff abs_norm_cancel)
    qed
    with ‹r > 0› b show "sphere 0 r = {r *🪙R b, - r *🪙R b}"
      by (force simp: sphere_def dist_norm)
    have "dist (r *🪙R b) (- r *🪙R b) = norm (r *🪙R b + r *🪙R b)"
      by (simp add: dist_norm)
    also have "… = norm ((2*r) *🪙R b)"
      by (metis mult_2 scaleR_add_left)
    also have "… = 2*r"
      using ‹r > 0› b norm_Basis by fastforce
    finally show "dist (r *🪙R b) (- r *🪙R b) = 2*r" .
  qed
qed

lemma sphere_1D_doubleton:
  fixes a :: "'a :: euclidean_space"
  assumes "DIM('a) = 1" and "r > 0"
  obtains x y where "sphere a r = {x,y} ∧ dist x y = 2*r"
  using sphere_1D_doubleton_zero [OF assms] dist_add_cancel image_empty image_insert
  by (metis (no_types, opaque_lifting) add.right_neutral sphere_translation)

lemma psubset_sphere_Compl_connected:
  fixes S :: "'a::euclidean_space set"
  assumes S: "S ⊂ sphere a r" and "0 < r" and 2: "2 ≤ DIM('a)"
  shows "connected(- S)"
proof -
  have "S ⊆ sphere a r"
    using S by blast
  obtain b where "dist a b = r" and "b ∉ S"
    using S mem_sphere by blast
  have CS: "- S = {x. dist a x ≤ r ∧ (x ∉ S)} ∪ {x. r ≤ dist a x ∧ (x ∉ S)}"
    by auto
  have "{x. dist a x ≤ r ∧ x ∉ S} ∩ {x. r ≤ dist a x ∧ x ∉ S} ≠ {}"
    using ‹b ∉ S› ‹dist a b = r› by blast
  moreover have "connected {x. dist a x ≤ r ∧ x ∉ S}"
    using assms
    by (force intro: connected_intermediate_closure [of "ball a r"])
  moreover have "connected {x. r ≤ dist a x ∧ x ∉ S}"
  proof (rule connected_intermediate_closure [of "- cball a r"])
    show "{x. r ≤ dist a x ∧ x ∉ S} ⊆ closure (- cball a r)"
      using interior_closure by (force intro: connected_complement_bounded_convex)
  qed (use assms connected_complement_bounded_convex in auto)
  ultimately show ?thesis
    by (simp add: CS connected_Un)
qed


subsection‹Every annulus is a connected set›

lemma path_connected_2DIM_I:
  fixes a :: "'N::euclidean_space"
  assumes 2: "2 ≤ DIM('N)" and pc: "path_connected {r. 0 ≤ r ∧ P r}"
  shows "path_connected {x. P(norm(x - a))}"
proof -
  have "{x. P(norm(x - a))} = (+) a ` {x. P(norm x)}"
    by force
  moreover have "path_connected {x::'N. P(norm x)}"
  proof -
    let ?D = "{x. 0 ≤ x ∧ P x} × sphere (0::'N) 1"
    have "x ∈ (λz. fst z *🪙R snd z) ` ?D"
      if "P (norm x)" for x::'N
    proof (cases "x=0")
      case True
      with that show ?thesis
        apply (simp add: image_iff)
        by (metis (no_types) mem_sphere_0 order_refl vector_choose_size zero_le_one)
    next
      case False
      with that show ?thesis
        by (rule_tac x="(norm x, x /🪙R norm x)" in image_eqI) auto
    qed
    then have *: "{x::'N. P(norm x)} = (λz. fst z *🪙R snd z) ` ?D"
      by auto
    have "continuous_on ?D (λz:: real×'N. fst z *🪙R snd z)"
      by (intro continuous_intros)
    moreover have "path_connected ?D"
      by (metis path_connected_Times [OF pc] path_connected_sphere 2)
    ultimately show ?thesis
      by (simp add: "*" path_connected_continuous_image)
  qed
  ultimately show ?thesis
    using path_connected_translation by metis
qed

proposition path_connected_annulus:
  fixes a :: "'N::euclidean_space"
  assumes "2 ≤ DIM('N)"
  shows "path_connected {x. r1 < norm(x - a) ∧ norm(x - a) < r2}"
        "path_connected {x. r1 < norm(x - a) ∧ norm(x - a) ≤ r2}"
        "path_connected {x. r1 ≤ norm(x - a) ∧ norm(x - a) < r2}"
        "path_connected {x. r1 ≤ norm(x - a) ∧ norm(x - a) ≤ r2}"
  by (auto simp: is_interval_def intro!: is_interval_convex convex_imp_path_connected path_connected_2DIM_I [OF assms])

proposition connected_annulus:
  fixes a :: "'N::euclidean_space"
  assumes "2 ≤ DIM('N::euclidean_space)"
  shows "connected {x. r1 < norm(x - a) ∧ norm(x - a) < r2}"
        "connected {x. r1 < norm(x - a) ∧ norm(x - a) ≤ r2}"
        "connected {x. r1 ≤ norm(x - a) ∧ norm(x - a) < r2}"
        "connected {x. r1 ≤ norm(x - a) ∧ norm(x - a) ≤ r2}"
  by (auto simp: path_connected_annulus [OF assms] path_connected_imp_connected)


subsection🍋‹tag unimportant›\<open>Relations between components and path components›

lemma open_connected_component:
  fixes S :: "'a::real_normed_vector set"
  assumes "open S"
  shows "open (connected_component_set S x)"
proof (clarsimp simp: open_contains_ball)
  fix y
  assume xy: "connected_component S x y"
  then obtain e where "e>0" "ball y e ⊆ S"
    using assms connected_component_in openE by blast
  then show "∃e>0. ball y e ⊆ connected_component_set S x"
    by (metis xy centre_in_ball connected_ball connected_component_eq_eq connected_component_in connected_component_maximal)
qed

corollary open_components:
    fixes S :: "'a::real_normed_vector set"
    shows "[open u; S ∈ components u] ==> open S"
  by (simp add: components_iff) (metis open_connected_component)

lemma in_closure_connected_component:
  fixes S :: "'a::real_normed_vector set"
  assumes x: "x ∈ S" and S: "open S"
  shows "x ∈ closure (connected_component_set S y) ⟷ x ∈ connected_component_set S y"
proof -
  have "x islimpt connected_component_set S y ==> connected_component S y x"
    by (metis (no_types, lifting) S connected_component_eq connected_component_refl islimptE mem_Collect_eq open_connected_component x)
  then show ?thesis
    by (auto simp: closure_def)
qed

lemma connected_disjoint_Union_open_pick:
  assumes "pairwise disjnt B"
          "∧S. S ∈ A ==> connected S ∧ S ≠ {}"
          "∧S. S ∈ B ==> open S"
          "∪A ⊆ ∪B"
          "S ∈ A"
  obtains T where "T ∈ B" "S ⊆ T" "S ∩ ∪(B - {T}) = {}"
proof -
  have "S ⊆ ∪B" "connected S" "S ≠ {}"
    using assms ‹S ∈ A› by blast+
  then obtain T where "T ∈ B" "S ∩ T ≠ {}"
    by (metis Sup_inf_eq_bot_iff inf.absorb_iff2 inf_commute)
  have 1: "open T" by (simp add: ‹T ∈ B› assms)
  have 2: "open (∪(B-{T}))" using assms by blast
  have 3: "S ⊆ T ∪ ∪(B - {T})" using ‹S ⊆ ∪B› by blast
  have "T ∩ ∪(B - {T}) = {}" using ‹T ∈ B› ‹pairwise disjnt B›
    by (auto simp: pairwise_def disjnt_def)
  then have 4: "T ∩ ∪(B - {T}) ∩ S = {}" by auto
  from connectedD [OF ‹connected S› 1 2 4 3]
  have "S ∩ ∪(B-{T}) = {}"
    by (auto simp: Int_commute ‹S ∩ T ≠ {}›)
  with ‹T ∈ B› 3 that show ?thesis
    by (metis IntI UnE empty_iff subsetD subsetI)
qed

lemma connected_disjoint_Union_open_subset:
  assumes A: "pairwise disjnt A" and B: "pairwise disjnt B"
      and SA: "∧S. S ∈ A ==> open S ∧ connected S ∧ S ≠ {}"
      and SB: "∧S. S ∈ B ==> open S ∧ connected S ∧ S ≠ {}"
      and eq [simp]: "∪A = ∪B"
    shows "A ⊆ B"
proof
  fix S
  assume "S ∈ A"
  obtain T where "T ∈ B" "S ⊆ T" "S ∩ ∪(B - {T}) = {}"
    using SA SB ‹S ∈ A› connected_disjoint_Union_open_pick [OF B, of A] eq order_refl by blast
  moreover obtain S' where "S' ∈ A" "T ⊆ S'" "T ∩ ∪(A - {S'}) = {}"
    using SA SB ‹T ∈ B› connected_disjoint_Union_open_pick [OF A, of B] eq order_refl by blast
  ultimately have "S' = S"
    by (metis A Int_subset_iff SA ‹S ∈ A› disjnt_def inf.orderE pairwise_def)
  with ‹T ⊆ S'› have "T ⊆ S" by simp
  with ‹S ⊆ T› have "S = T" by blast
  with ‹T ∈ B› show "S ∈ B" by simp
qed

lemma connected_disjoint_Union_open_unique:
  assumes A: "pairwise disjnt A" and B: "pairwise disjnt B"
      and SA: "∧S. S ∈ A ==> open S ∧ connected S ∧ S ≠ {}"
      and SB: "∧S. S ∈ B ==> open S ∧ connected S ∧ S ≠ {}"
      and eq [simp]: "∪A = ∪B"
    shows "A = B"
by (metis subset_antisym connected_disjoint_Union_open_subset assms)

proposition components_open_unique:
 fixes S :: "'a::real_normed_vector set"
  assumes "pairwise disjnt A" "∪A = S"
          "∧X. X ∈ A ==> open X ∧ connected X ∧ X ≠ {}"
    shows "components S = A"
proof -
  have "open S" using assms by blast
  show ?thesis
  proof (rule connected_disjoint_Union_open_unique)
    show "disjoint (components S)"
      by (simp add: components_eq disjnt_def pairwise_def)
  qed (use ‹open S› in ‹simp_all add: assms open_components in_components_connected in_components_nonempty›)
qed


subsection🍋‹tag unimportant›\<open>Existence of unbounded components›

lemma cobounded_unbounded_component:
    fixes S :: "'a :: euclidean_space set"
    assumes "bounded (-S)"
      shows "∃x. x ∈ S ∧ ¬ bounded (connected_component_set S x)"
proof -
  obtain i::'a where i: "i ∈ Basis"
    using nonempty_Basis by blast
  obtain B where B: "B>0" "-S ⊆ ball 0 B"
    using bounded_subset_ballD [OF assms, of 0] by auto
  then have *: "∧x. B ≤ norm x ==> x ∈ S"
    by (force simp: ball_def dist_norm)
  have unbounded_inner: "¬ bounded {x. inner i x ≥ B}"
  proof (clarsimp simp: bounded_def dist_norm)
    fix e x
    show "∃y. B ≤ i ∙ y ∧ ¬ norm (x - y) ≤ e"
      using i
      by (rule_tac x="x + (max B e + 1 + ∣i ∙ x∣) *🪙R i" in exI) (auto simp: inner_right_distrib)
  qed
  have 🍋: "∧x. B ≤ i ∙ x ==> x ∈ S"
    using * Basis_le_norm [OF i] by (metis abs_ge_self inner_commute order_trans)
  have "{x. B ≤ i ∙ x} ⊆ connected_component_set S (B *🪙R i)"
    by (intro connected_component_maximal) (auto simp: i intro: convex_connected convex_halfspace_ge [of B] 🍋)
  then have "¬ bounded (connected_component_set S (B *🪙R i))"
    using bounded_subset unbounded_inner by blast
  moreover have "B *🪙R i ∈ S"
    by (rule *) (simp add: norm_Basis [OF i])
  ultimately show ?thesis
    by blast
qed

lemma cobounded_unique_unbounded_component:
    fixes S :: "'a :: euclidean_space set"
    assumes bs: "bounded (-S)" and "2 ≤ DIM('a)"
        and bo: "¬ bounded(connected_component_set S x)"
                "¬ bounded(connected_component_set S y)"
      shows "connected_component_set S x = connected_component_set S y"
proof -
  obtain i::'a where i: "i ∈ Basis"
    using nonempty_Basis by blast
  obtain B where "B>0" and B: "-S ⊆ ball 0 B"
    using bounded_subset_ballD [OF bs, of 0] by auto
  then have *: "∧x. B ≤ norm x ==> x ∈ S"
    by (force simp: ball_def dist_norm)
  obtain x' y' where x': "connected_component S x x'" "norm x' > B"
  and  y': "connected_component S y y'" "norm y' > B"
    using ‹B>0› bo bounded_pos by (metis linorder_not_le mem_Collect_eq) 
  have x'y': "connected_component S x' y'"
    unfolding connected_component_def
  proof (intro exI conjI)
    show "connected (- ball 0 B :: 'a set)"
      using assms by (auto intro: connected_complement_bounded_convex)
  qed (use x' y' dist_norm * in auto)
  show ?thesis
      using x' y' x'y'
      by (metis connected_component_eq mem_Collect_eq)
qed

lemma cobounded_unbounded_components:
    fixes S :: "'a :: euclidean_space set"
    shows "bounded (-S) ==> ∃c. c ∈ components S ∧ ¬bounded c"
  by (metis cobounded_unbounded_component components_def imageI)

lemma cobounded_unique_unbounded_components:
    fixes S :: "'a :: euclidean_space set"
    shows  "[bounded (- S); c ∈ components S; ¬ bounded c; c' ∈ components S; ¬ bounded c'; 2 ≤ DIM('a)] ==> c' = c"
  unfolding components_iff
  by (metis cobounded_unique_unbounded_component)

lemma cobounded_has_bounded_component:
  fixes S :: "'a :: euclidean_space set"
  assumes "bounded (- S)" "¬ connected S" "2 ≤ DIM('a)"
  obtains C where "C ∈ components S" "bounded C"
  by (meson cobounded_unique_unbounded_components connected_eq_connected_components_eq assms)


subsection‹The ‹inside› and ‹outside› of a Set›

text🍋‹tag important›\<open>The inside comprises the points in a bounded connected component of the set's complement.
  The outside comprises the points in unbounded connected component of the complement.›

definition🍋‹tag important› inside where
  "inside S ≡ {x. (x ∉ S) ∧ bounded(connected_component_set ( - S) x)}"

definition🍋‹tag important› outside where
  "outside S ≡ -S ∩ {x. ¬ bounded(connected_component_set (- S) x)}"

lemma outside: "outside S = {x. ¬ bounded(connected_component_set (- S) x)}"
  by (auto simp: outside_def) (metis Compl_iff bounded_empty connected_component_eq_empty)

lemma inside_no_overlap [simp]: "inside S ∩ S = {}"
  by (auto simp: inside_def)

lemma outside_no_overlap [simp]:
   "outside S ∩ S = {}"
  by (auto simp: outside_def)

lemma inside_Int_outside [simp]: "inside S ∩ outside S = {}"
  by (auto simp: inside_def outside_def)

lemma inside_Un_outside [simp]: "inside S ∪ outside S = (- S)"
  by (auto simp: inside_def outside_def)

lemma inside_eq_outside:
   "inside S = outside S ⟷ S = UNIV"
  by (auto simp: inside_def outside_def)

lemma inside_outside: "inside S = (- (S ∪ outside S))"
  by (force simp: inside_def outside)

lemma outside_inside: "outside S = (- (S ∪ inside S))"
  by (auto simp: inside_outside) (metis IntI equals0D outside_no_overlap)

lemma union_with_inside: "S ∪ inside S = - outside S"
  by (auto simp: inside_outside) (simp add: outside_inside)

lemma union_with_outside: "S ∪ outside S = - inside S"
  by (simp add: inside_outside)

lemma outside_mono: "S ⊆ T ==> outside T ⊆ outside S"
  by (auto simp: outside bounded_subset connected_component_mono)

lemma inside_mono: "S ⊆ T ==> inside S - T ⊆ inside T"
  by (auto simp: inside_def bounded_subset connected_component_mono)

lemma segment_bound_lemma:
  fixes u::real
  assumes "x ≥ B" "y ≥ B" "0 ≤ u" "u ≤ 1"
  shows "(1 - u) * x + u * y ≥ B"
  by (smt (verit) assms convex_bound_le ge_iff_diff_ge_0 minus_add_distrib 
      mult_minus_right neg_le_iff_le)

lemma cobounded_outside:
  fixes S :: "'a :: real_normed_vector set"
  assumes "bounded S" shows "bounded (- outside S)"
proof -
  obtain B where B: "B>0" "S ⊆ ball 0 B"
    using bounded_subset_ballD [OF assms, of 0] by auto
  { fix x::'a and C::real
    assume Bno: "B ≤ norm x" and C: "0 < C"
    have "∃y. connected_component (- S) x y ∧ norm y > C"
    proof (cases "x = 0")
      case True with B Bno show ?thesis by force
    next
      case False 
      have "closed_segment x (((B + C) / norm x) *🪙R x) ⊆ - ball 0 B"
      proof
        fix w
        assume "w ∈ closed_segment x (((B + C) / norm x) *🪙R x)"
        then obtain u where
          w: "w = (1 - u + u * (B + C) / norm x) *🪙R x" "0 ≤ u" "u ≤ 1"
          by (auto simp add: closed_segment_def real_vector_class.scaleR_add_left [symmetric])
        with False B C have "B ≤ (1 - u) * norm x + u * (B + C)"
          using segment_bound_lemma [of B "norm x" "B + C" u] Bno
          by simp
        with False B C show "w ∈ - ball 0 B"
          using distrib_right [of _ _ "norm x"]
          by (simp add: ball_def w not_less)
      qed
      also have "... ⊆ -S"
        by (simp add: B)
      finally have "∃T. connected T ∧ T ⊆ - S ∧ x ∈ T ∧ ((B + C) / norm x) *🪙R x ∈ T"
        by (rule_tac x="closed_segment x (((B+C)/norm x) *🪙R x)" in exI) simp
      with False B
      show ?thesis
        by (rule_tac x="((B+C)/norm x) *🪙R x" in exI) (simp add: connected_component_def)
    qed
  }
  then show ?thesis
    apply (simp add: outside_def assms)
    apply (rule bounded_subset [OF bounded_ball [of 0 B]])
    apply (force simp: dist_norm not_less bounded_pos)
    done
qed

lemma unbounded_outside:
    fixes S :: "'a::{real_normed_vector, perfect_space} set"
    shows "bounded S ==> ¬ bounded(outside S)"
  using cobounded_imp_unbounded cobounded_outside by blast

lemma bounded_inside:
    fixes S :: "'a::{real_normed_vector, perfect_space} set"
    shows "bounded S ==> bounded(inside S)"
  by (simp add: bounded_Int cobounded_outside inside_outside)

lemma connected_outside:
    fixes S :: "'a::euclidean_space set"
    assumes "bounded S" "2 ≤ DIM('a)"
    shows "connected(outside S)"
  apply (clarsimp simp add: connected_iff_connected_component outside)
  apply (rule_tac S="connected_component_set (- S) x" in connected_component_of_subset)
  apply (metis (no_types) assms cobounded_unbounded_component cobounded_unique_unbounded_component connected_component_eq_eq connected_component_idemp double_complement mem_Collect_eq)
  by (simp add: Collect_mono connected_component_eq)

lemma outside_connected_component_lt:
  "outside S = {x. ∀B. ∃y. B < norm(y) ∧ connected_component (- S) x y}"
proof -
  have "∧x B. x ∈ outside S ==> ∃y. B < norm y ∧ connected_component (- S) x y"
    by (metis boundedI linorder_not_less mem_Collect_eq outside)
  moreover
  have "∧x. ∀B. ∃y. B < norm y ∧ connected_component (- S) x y ==> x ∈ outside S"
    by (metis bounded_pos linorder_not_less mem_Collect_eq outside)
  ultimately show ?thesis by auto
qed

lemma outside_connected_component_le:
  "outside S = {x. ∀B. ∃y. B ≤ norm(y) ∧ connected_component (- S) x y}"
  apply (simp add: outside_connected_component_lt Set.set_eq_iff)
  by (meson gt_ex leD le_less_linear less_imp_le order.trans)

lemma not_outside_connected_component_lt:
    fixes S :: "'a::euclidean_space set"
    assumes S: "bounded S" and "2 ≤ DIM('a)"
      shows "- (outside S) = {x. ∀B. ∃y. B < norm(y) ∧ ¬ connected_component (- S) x y}"
proof -
  obtain B::real where B: "0 < B" and Bno: "∧x. x ∈ S ==> norm x ≤ B"
    using S [simplified bounded_pos] by auto
  have cyz: "connected_component (- S) y z" 
    if yz: "B < norm z" "B < norm y" for y::'a and z::'a
  proof -
    have "connected_component (- cball 0 B) y z"
      using assms yz
      by (force simp: dist_norm intro: connected_componentI [OF _ subset_refl] connected_complement_bounded_convex)
    then show ?thesis
      by (metis connected_component_of_subset Bno Compl_anti_mono mem_cball_0 subset_iff)
  qed
  show ?thesis
    apply (auto simp: outside bounded_pos)
    apply (metis Compl_iff bounded_iff cobounded_imp_unbounded mem_Collect_eq not_le)
    by (metis B connected_component_trans cyz not_le)
qed

lemma not_outside_connected_component_le:
  fixes S :: "'a::euclidean_space set"
  assumes S: "bounded S"  "2 ≤ DIM('a)"
  shows "- (outside S) = {x. ∀B. ∃y. B ≤ norm(y) ∧ ¬ connected_component (- S) x y}"
  unfolding not_outside_connected_component_lt [OF assms]
  by (metis (no_types, opaque_lifting) dual_order.strict_trans1 gt_ex pinf(8))

lemma inside_connected_component_lt:
    fixes S :: "'a::euclidean_space set"
    assumes S: "bounded S"  "2 ≤ DIM('a)"
      shows "inside S = {x. (x ∉ S) ∧ (∀B. ∃y. B < norm(y) ∧ ¬ connected_component (- S) x y)}"
  by (auto simp: inside_outside not_outside_connected_component_lt [OF assms])

lemma inside_connected_component_le:
    fixes S :: "'a::euclidean_space set"
    assumes S: "bounded S"  "2 ≤ DIM('a)"
      shows "inside S = {x. (x ∉ S) ∧ (∀B. ∃y. B ≤ norm(y) ∧ ¬ connected_component (- S) x y)}"
  by (auto simp: inside_outside not_outside_connected_component_le [OF assms])

lemma inside_subset:
  assumes "connected U" and "¬ bounded U" and "T ∪ U = - S"
  shows "inside S ⊆ T"
  using bounded_subset [of "connected_component_set (- S) _" U] assms
  by (metis (no_types, lifting) ComplI Un_iff connected_component_maximal inside_def mem_Collect_eq subsetI)

lemma frontier_not_empty:
  fixes S :: "'a :: real_normed_vector set"
  shows "[S ≠ {}; S ≠ UNIV] ==> frontier S ≠ {}"
    using connected_Int_frontier [of UNIV S] by auto

lemma frontier_eq_empty:
  fixes S :: "'a :: real_normed_vector set"
  shows "frontier S = {} ⟷ S = {} ∨ S = UNIV"
using frontier_UNIV frontier_empty frontier_not_empty by blast

lemma frontier_of_connected_component_subset:
  fixes S :: "'a::real_normed_vector set"
  shows "frontier(connected_component_set S x) ⊆ frontier S"
proof -
  { fix y
    assume y1: "y ∈ closure (connected_component_set S x)"
       and y2: "y ∉ interior (connected_component_set S x)"
    have "y ∈ closure S"
      using y1 closure_mono connected_component_subset by blast
    moreover have "z ∈ interior (connected_component_set S x)"
          if "0 < e" "ball y e ⊆ interior S" "dist y z < e" for e z
    proof -
      have "ball y e ⊆ connected_component_set S y"
        using connected_component_maximal that interior_subset 
        by (metis centre_in_ball connected_ball subset_trans)
      then show ?thesis
        using y1 apply (simp add: closure_approachable open_contains_ball_eq [OF open_interior])
        by (metis connected_component_eq dist_commute mem_Collect_eq mem_ball mem_interior subsetD ‹0 🚫› y2)
    qed
    then have "y ∉ interior S"
      using y2 by (force simp: open_contains_ball_eq [OF open_interior])
    ultimately have "y ∈ frontier S"
      by (auto simp: frontier_def)
  }
  then show ?thesis by (auto simp: frontier_def)
qed

lemma frontier_Union_subset_closure:
  fixes F :: "'a::real_normed_vector set set"
  shows "frontier(∪F) ⊆ closure(∪t ∈ F. frontier t)"
proof -
  have "∃y∈F. ∃y∈frontier y. dist y x < e"
       if "T ∈ F" "y ∈ T" "dist y x < e"
          "x ∉ interior (∪F)" "0 < e" for x y e T
  proof (cases "x ∈ T")
    case True with that show ?thesis
      by (metis Diff_iff Sup_upper closure_subset contra_subsetD dist_self frontier_def interior_mono)
  next
    case False
    have 🍋: "closed_segment x y ∩ T ≠ {}" "closed_segment x y - T ≠ {}"
      using ‹y ∈ T› False by blast+
    obtain c where "c ∈ closed_segment x y" "c ∈ frontier T"
       using False connected_Int_frontier [OF connected_segment 🍋] by auto
     with that show ?thesis
       by (smt (verit) dist_norm segment_bound1)
  qed
  then show ?thesis
    by (fastforce simp add: frontier_def closure_approachable)
qed

lemma frontier_Union_subset:
  fixes F :: "'a::real_normed_vector set set"
  shows "finite F ==> frontier(∪F) ⊆ (∪t ∈ F. frontier t)"
  by (metis closed_UN closure_closed frontier_Union_subset_closure frontier_closed)

lemma frontier_of_components_subset:
  fixes S :: "'a::real_normed_vector set"
  shows "C ∈ components S ==> frontier C ⊆ frontier S"
  by (metis Path_Connected.frontier_of_connected_component_subset components_iff)

lemma frontier_of_components_closed_complement:
  fixes S :: "'a::real_normed_vector set"
  shows "[closed S; C ∈ components (- S)] ==> frontier C ⊆ S"
  using frontier_complement frontier_of_components_subset frontier_subset_eq by blast

lemma frontier_minimal_separating_closed:
  fixes S :: "'a::real_normed_vector set"
  assumes "closed S"
      and nconn: "¬ connected(- S)"
      and C: "C ∈ components (- S)"
      and conn: "∧T. [closed T; T ⊂ S] ==> connected(- T)"
    shows "frontier C = S"
proof (rule ccontr)
  assume "frontier C ≠ S"
  then have "frontier C ⊂ S"
    using frontier_of_components_closed_complement [OF ‹closed S› C] by blast
  then have "connected(- (frontier C))"
    by (simp add: conn)
  have "¬ connected(- (frontier C))"
    unfolding connected_def not_not
  proof (intro exI conjI)
    show "open C"
      using C ‹closed S› open_components by blast
    show "open (- closure C)"
      by blast
    show "C ∩ - closure C ∩ - frontier C = {}"
      using closure_subset by blast
    show "C ∩ - frontier C ≠ {}"
      using C ‹open C› components_eq frontier_disjoint_eq by fastforce
    show "- frontier C ⊆ C ∪ - closure C"
      by (simp add: ‹open C› closed_Compl frontier_closures)
    then show "- closure C ∩ - frontier C ≠ {}"
      by (metis C Compl_Diff_eq Un_Int_eq(4) Un_commute ‹frontier C ⊂ S› ‹open C› compl_le_compl_iff frontier_def in_components_subset interior_eq leD sup_bot.right_neutral)
  qed
  then show False
    using ‹connected (- frontier C)› by blast
qed

lemma connected_component_UNIV [simp]:
  fixes x :: "'a::real_normed_vector"
  shows "connected_component_set UNIV x = UNIV"
  using connected_iff_eq_connected_component_set [of "UNIV::'a set"] connected_UNIV
  by auto

lemma connected_component_eq_UNIV:
    fixes x :: "'a::real_normed_vector"
    shows "connected_component_set s x = UNIV ⟷ s = UNIV"
  using connected_component_in connected_component_UNIV by blast

lemma components_UNIV [simp]: "components UNIV = {UNIV :: 'a::real_normed_vector set}"
  by (auto simp: components_eq_sing_iff)

lemma interior_inside_frontier:
    fixes S :: "'a::real_normed_vector set"
    assumes "bounded S"
      shows "interior S ⊆ inside (frontier S)"
proof -
  { fix x y
    assume x: "x ∈ interior S" and y: "y ∉ S"
       and cc: "connected_component (- frontier S) x y"
    have "connected_component_set (- frontier S) x ∩ frontier S ≠ {}"
    proof (rule connected_Int_frontier; simp add: set_eq_iff)
      show "∃u. connected_component (- frontier S) x u ∧ u ∈ S"
        by (meson cc connected_component_in connected_component_refl_eq interior_subset subsetD x)
      show "∃u. connected_component (- frontier S) x u ∧ u ∉ S"
        using y cc by blast
    qed
    then have "bounded (connected_component_set (- frontier S) x)"
      using connected_component_in by auto
  }
  then show ?thesis
    using bounded_subset [OF assms]
    by (metis (no_types, lifting) Diff_iff frontier_def inside_def mem_Collect_eq subsetI)
qed

lemma inside_empty [simp]: "inside {} = ({} :: 'a :: {real_normed_vector, perfect_space} set)"
  by (simp add: inside_def)

lemma outside_empty [simp]: "outside {} = (UNIV :: 'a :: {real_normed_vector, perfect_space} set)"
  using inside_empty inside_Un_outside by blast

lemma inside_same_component:
   "[connected_component (- S) x y; x ∈ inside S] ==> y ∈ inside S"
  using connected_component_eq connected_component_in
  by (fastforce simp add: inside_def)

lemma outside_same_component:
   "[connected_component (- S) x y; x ∈ outside S] ==> y ∈ outside S"
  using connected_component_eq connected_component_in
  by (fastforce simp add: outside_def)

lemma convex_in_outside:
  fixes S :: "'a :: {real_normed_vector, perfect_space} set"
  assumes S: "convex S" and z: "z ∉ S"
    shows "z ∈ outside S"
proof (cases "S={}")
  case True then show ?thesis by simp
next
  case False then obtain a where "a ∈ S" by blast
  with z have zna: "z ≠ a" by auto
  { assume "bounded (connected_component_set (- S) z)"
    with bounded_pos_less obtain B where "B>0" and B: "∧x. connected_component (- S) z x ==> norm x < B"
      by (metis mem_Collect_eq)
    define C where "C = (B + 1 + norm z) / norm (z-a)"
    have "C > 0"
      using ‹0 🚫› zna by (simp add: C_def field_split_simps add_strict_increasing)
    have "∣norm (z + C *🪙R (z-a)) - norm (C *🪙R (z-a))∣ ≤ norm z"
      by (metis add_diff_cancel norm_triangle_ineq3)
    moreover have "norm (C *🪙R (z-a)) > norm z + B"
      using zna ‹B>0› by (simp add: C_def le_max_iff_disj)
    ultimately have C: "norm (z + C *🪙R (z-a)) > B" by linarith
    { fix u::real
      assume u: "0≤u" "u≤1" and ins: "(1 - u) *🪙R z + u *🪙R (z + C *🪙R (z - a)) ∈ S"
      then have Cpos: "1 + u * C > 0"
        by (meson ‹0 🚫› add_pos_nonneg less_eq_real_def zero_le_mult_iff zero_less_one)
      then have *: "(1 / (1 + u * C)) *🪙R z + (u * C / (1 + u * C)) *🪙R z = z"
        by (simp add: scaleR_add_left [symmetric] field_split_simps)
      then have False
        using convexD_alt [OF S ‹a ∈ S› ins, of "1/(u*C + 1)"] ‹C>0› ‹z ∉ S› Cpos u
        by (simp add: * field_split_simps)
    } note contra = this
    have "connected_component (- S) z (z + C *🪙R (z-a))"
    proof (rule connected_componentI [OF connected_segment])
      show "closed_segment z (z + C *🪙R (z - a)) ⊆ - S"
        using contra by (force simp add: closed_segment_def)
    qed auto
    then have False
      using zna B [of "z + C *🪙R (z-a)"] C
      by (auto simp: field_split_simps max_mult_distrib_right)
  }
  then show ?thesis
    by (auto simp: outside_def z)
qed

lemma outside_convex:
  fixes S :: "'a :: {real_normed_vector, perfect_space} set"
  assumes "convex S"
    shows "outside S = - S"
  by (metis ComplD assms convex_in_outside equalityI inside_Un_outside subsetI sup.cobounded2)

lemma outside_singleton [simp]:
  fixes x :: "'a :: {real_normed_vector, perfect_space}"
  shows "outside {x} = -{x}"
  by (auto simp: outside_convex)

lemma inside_convex:
  fixes S :: "'a :: {real_normed_vector, perfect_space} set"
  shows "convex S ==> inside S = {}"
  by (simp add: inside_outside outside_convex)

lemma inside_singleton [simp]:
  fixes x :: "'a :: {real_normed_vector, perfect_space}"
  shows "inside {x} = {}"
  by (auto simp: inside_convex)

lemma outside_subset_convex:
  fixes S :: "'a :: {real_normed_vector, perfect_space} set"
  shows "[convex T; S ⊆ T] ==> - T ⊆ outside S"
  using outside_convex outside_mono by blast

lemma outside_Un_outside_Un:
  fixes S :: "'a::real_normed_vector set"
  assumes "S ∩ outside(T ∪ U) = {}"
  shows "outside(T ∪ U) ⊆ outside(T ∪ S)"
proof
  fix x
  assume x: "x ∈ outside (T ∪ U)"
  have "Y ⊆ - S" if "connected Y" "Y ⊆ - T" "Y ⊆ - U" "x ∈ Y" "u ∈ Y" for u Y
  proof -
    have "Y ⊆ connected_component_set (- (T ∪ U)) x"
      by (simp add: connected_component_maximal that)
    also have "… ⊆ outside(T ∪ U)"
      by (metis (mono_tags, lifting) Collect_mono mem_Collect_eq outside outside_same_component x)
    finally have "Y ⊆ outside(T ∪ U)" .
    with assms show ?thesis by auto
  qed
  with x show "x ∈ outside (T ∪ S)"
    by (simp add: outside_connected_component_lt connected_component_def) meson
qed

lemma outside_frontier_misses_closure:
    fixes S :: "'a::real_normed_vector set"
    assumes "bounded S"
    shows  "outside(frontier S) ⊆ - closure S"
  using assms frontier_def interior_inside_frontier outside_inside by fastforce

lemma outside_frontier_eq_complement_closure:
  fixes S :: "'a :: {real_normed_vector, perfect_space} set"
    assumes "bounded S" "convex S"
      shows "outside(frontier S) = - closure S"
by (metis Diff_subset assms convex_closure frontier_def outside_frontier_misses_closure
          outside_subset_convex subset_antisym)

lemma inside_frontier_eq_interior:
     fixes S :: "'a :: {real_normed_vector, perfect_space} set"
     shows "[bounded S; convex S] ==> inside(frontier S) = interior S"
  unfolding inside_outside outside_frontier_eq_complement_closure
  using closure_subset interior_subset by (auto simp: frontier_def)

lemma open_inside:
    fixes S :: "'a::real_normed_vector set"
    assumes "closed S"
      shows "open (inside S)"
proof -
  { fix x assume x: "x ∈ inside S"
    have "open (connected_component_set (- S) x)"
      using assms open_connected_component by blast
    then obtain e where e: "e>0" and e: "∧y. dist y x < e ⟶ connected_component (- S) x y"
      using dist_not_less_zero
      apply (simp add: open_dist)
      by (metis (no_types, lifting) Compl_iff connected_component_refl_eq inside_def mem_Collect_eq x)
    then have "∃e>0. ball x e ⊆ inside S"
      by (metis e dist_commute inside_same_component mem_ball subsetI x)
  }
  then show ?thesis
    by (simp add: open_contains_ball)
qed

lemma open_outside:
    fixes S :: "'a::real_normed_vector set"
    assumes "closed S"
      shows "open (outside S)"
proof -
  { fix x assume x: "x ∈ outside S"
    have "open (connected_component_set (- S) x)"
      using assms open_connected_component by blast
    then obtain e where e: "e>0" and e: "∧y. dist y x < e ⟶ connected_component (- S) x y"
      using dist_not_less_zero x
      by (auto simp add: open_dist outside_def intro: connected_component_refl)
    then have "∃e>0. ball x e ⊆ outside S"
      by (metis e dist_commute outside_same_component mem_ball subsetI x)
  }
  then show ?thesis
    by (simp add: open_contains_ball)
qed

lemma closure_inside_subset:
  fixes S :: "'a::real_normed_vector set"
  assumes "closed S"
  shows "closure(inside S) ⊆ S ∪ inside S"
  by (metis assms closure_minimal open_closed open_outside sup.cobounded2 union_with_inside)

lemma frontier_inside_subset:
    fixes S :: "'a::real_normed_vector set"
    assumes "closed S"
      shows "frontier(inside S) ⊆ S"
  using assms closure_inside_subset frontier_closures frontier_disjoint_eq open_inside by fastforce

lemma closure_outside_subset:
    fixes S :: "'a::real_normed_vector set"
    assumes "closed S"
      shows "closure(outside S) ⊆ S ∪ outside S"
  by (metis assms closed_open closure_minimal inside_outside open_inside sup_ge2)

lemma closed_path_image_Un_inside:
  fixes g :: "real ==> 'a :: real_normed_vector"
  assumes "path g"
  shows   "closed (path_image g ∪ inside (path_image g))"
  by (simp add: assms closed_Compl closed_path_image open_outside union_with_inside)

lemma frontier_outside_subset:
  fixes S :: "'a::real_normed_vector set"
  assumes "closed S"
  shows "frontier(outside S) ⊆ S"
  unfolding frontier_def
  by (metis Diff_subset_conv assms closure_outside_subset interior_eq open_outside sup_aci(1))

lemma inside_complement_unbounded_connected_empty:
     "[connected (- S); ¬ bounded (- S)] ==> inside S = {}"
  using inside_subset by blast

lemma inside_bounded_complement_connected_empty:
    fixes S :: "'a::{real_normed_vector, perfect_space} set"
    shows "[connected (- S); bounded S] ==> inside S = {}"
  by (metis inside_complement_unbounded_connected_empty cobounded_imp_unbounded)

lemma inside_inside:
    assumes "S ⊆ inside T"
    shows "inside S - T ⊆ inside T"
unfolding inside_def
proof clarify
  fix x
  assume x: "x ∉ T" "x ∉ S" and bo: "bounded (connected_component_set (- S) x)"
  show "bounded (connected_component_set (- T) x)"
  proof (cases "S ∩ connected_component_set (- T) x = {}")
    case True then show ?thesis
      by (metis bounded_subset [OF bo] compl_le_compl_iff connected_component_idemp connected_component_mono disjoint_eq_subset_Compl double_compl)
  next
    case False 
    then obtain y where y: "y ∈ S" "y ∈ connected_component_set (- T) x"
      by (meson disjoint_iff)
    then have "bounded (connected_component_set (- T) y)"
      using assms [unfolded inside_def] by blast
    with y show ?thesis
      by (metis connected_component_eq)
  qed
qed

lemma inside_inside_subset: "inside(inside S) ⊆ S"
  using inside_inside union_with_outside by fastforce

lemma inside_outside_intersect_connected:
      "[connected T; inside S ∩ T ≠ {}; outside S ∩ T ≠ {}] ==> S ∩ T ≠ {}"
  apply (simp add: inside_def outside_def ex_in_conv [symmetric] disjoint_eq_subset_Compl, clarify)
  by (metis compl_le_swap1 connected_componentI connected_component_eq mem_Collect_eq)

lemma outside_bounded_nonempty:
  fixes S :: "'a :: {real_normed_vector, perfect_space} set"
  assumes "bounded S" shows "outside S ≠ {}"
  using assms unbounded_outside by force

lemma outside_compact_in_open:
    fixes S :: "'a :: {real_normed_vector,perfect_space} set"
    assumes S: "compact S" and T: "open T" and "S ⊆ T" "T ≠ {}"
      shows "outside S ∩ T ≠ {}"
proof -
  have "outside S ≠ {}"
    by (simp add: compact_imp_bounded outside_bounded_nonempty S)
  with assms obtain a b where a: "a ∈ outside S" and b: "b ∈ T" by auto
  show ?thesis
  proof (cases "a ∈ T")
    case True with a show ?thesis by blast
  next
    case False
      have front: "frontier T ⊆ - S"
        using ‹S ⊆ T› frontier_disjoint_eq T by auto
      { fix γ
        assume "path γ" and pimg_sbs: "path_image γ - {pathfinish γ} ⊆ interior (- T)"
           and pf: "pathfinish γ ∈ frontier T" and ps: "pathstart γ = a"
        define c where "c = pathfinish γ"
        have "c ∈ -S" unfolding c_def using front pf by blast
        moreover have "open (-S)" using S compact_imp_closed by blast
        ultimately obtain ε::real where "ε > 0" and ε: "cball c ε ⊆ -S"
          using open_contains_cball[of "-S"] S by blast
        then obtain d where "d ∈ T" and d: "dist d c < ε"
          using closure_approachable [of c T] pf unfolding c_def
          by (metis Diff_iff frontier_def)
        then have "d ∈ -S" using ε
          using dist_commute by (metis contra_subsetD mem_cball not_le not_less_iff_gr_or_eq)
        have pimg_sbs_cos: "path_image γ ⊆ -S"
          using ‹c ∈ - S› ‹S ⊆ T› c_def interior_subset pimg_sbs by fastforce
        have "closed_segment c d ≤ cball c ε"
          by (metis ‹0 🚫ε› centre_in_cball closed_segment_subset convex_cball d dist_commute less_eq_real_def mem_cball)
        with ε have "closed_segment c d ⊆ -S" by blast
        moreover have con_gcd: "connected (path_image γ ∪ closed_segment c d)"
          by (rule connected_Un) (auto simp: c_def ‹path γ› connected_path_image)
        ultimately have "connected_component (- S) a d"
          unfolding connected_component_def using pimg_sbs_cos ps by blast
        then have "outside S ∩ T ≠ {}"
          using outside_same_component [OF _ a]  by (metis IntI ‹d ∈ T› empty_iff)
      } note * = this
      have pal: "pathstart (linepath a b) ∈ closure (- T)"
        by (auto simp: False closure_def)
      show ?thesis
        by (rule exists_path_subpath_to_frontier [OF path_linepath pal _ *]) (auto simp: b)
  qed
qed

lemma inside_inside_compact_connected:
    fixes S :: "'a :: euclidean_space set"
    assumes S: "closed S" and T: "compact T" and "connected T" "S ⊆ inside T"
      shows "inside S ⊆ inside T"
proof (cases "inside T = {}")
  case True with assms show ?thesis by auto
next
  case False
  consider "DIM('a) = 1" | "DIM('a) ≥ 2"
    using antisym not_less_eq_eq by fastforce
  then show ?thesis
  proof cases
    case 1 then show ?thesis
             using connected_convex_1_gen assms False inside_convex by blast
  next
    case 2
    have "bounded S"
      using assms by (meson bounded_inside bounded_subset compact_imp_bounded)
    then have coms: "compact S"
      by (simp add: S compact_eq_bounded_closed)
    then have bst: "bounded (S ∪ T)"
      by (simp add: compact_imp_bounded T)
    then obtain r where "0 < r" and r: "S ∪ T ⊆ ball 0 r"
      using bounded_subset_ballD by blast
    have outst: "outside S ∩ outside T ≠ {}"
      by (metis bounded_Un bounded_subset bst cobounded_outside disjoint_eq_subset_Compl unbounded_outside)
    have "S ∩ T = {}" using assms
      by (metis disjoint_iff_not_equal inside_no_overlap subsetCE)
    moreover have "outside S ∩ inside T ≠ {}"
      by (meson False assms(4) compact_eq_bounded_closed coms open_inside outside_compact_in_open T)
    ultimately have "inside S ∩ T = {}"
      using inside_outside_intersect_connected [OF ‹connected T›, of S]
      by (metis "2" compact_eq_bounded_closed coms connected_outside inf.commute inside_outside_intersect_connected outst)
    then show ?thesis
      using inside_inside [OF ‹S ⊆ inside T›] by blast
  qed
qed

lemma connected_with_inside:
    fixes S :: "'a :: real_normed_vector set"
    assumes S: "closed S" and cons: "connected S"
      shows "connected(S ∪ inside S)"
proof (cases "S ∪ inside S = UNIV")
  case True with assms show ?thesis by auto
next
  case False
  then obtain b where b: "b ∉ S" "b ∉ inside S" by blast
  have *: "∃y T. y ∈ S ∧ connected T ∧ a ∈ T ∧ y ∈ T ∧ T ⊆ (S ∪ inside S)" 
    if "a ∈ S ∪ inside S" for a
    using that 
  proof
    assume "a ∈ S" then show ?thesis
      using cons by blast
  next
    assume a: "a ∈ inside S"
    then have ain: "a ∈ closure (inside S)"
      by (simp add: closure_def)
    obtain h where h: "path h" "pathstart h = a" 
                   "path_image h - {pathfinish h} ⊆ interior (inside S)"
                   "pathfinish h ∈ frontier (inside S)"
      using ain b
      by (metis exists_path_subpath_to_frontier path_linepath pathfinish_linepath pathstart_linepath)
    moreover
    have h1S: "pathfinish h ∈ S"  
      using S h frontier_inside_subset by blast
    moreover 
    have "path_image h ⊆ S ∪ inside S"
      using IntD1 S h1S h interior_eq open_inside by fastforce
    ultimately show ?thesis by blast
  qed
  show ?thesis
    apply (simp add: connected_iff_connected_component)
    apply (clarsimp simp add: connected_component_def dest!: *)
    subgoal for x y u u' T t'
      by (rule_tac x = "S ∪ T ∪ t'" in exI) (auto intro!: connected_Un cons)
    done
qed

text‹The proof is virtually the same as that above.›
lemma connected_with_outside:
    fixes S :: "'a :: real_normed_vector set"
    assumes S: "closed S" and cons: "connected S"
      shows "connected(S ∪ outside S)"
proof (cases "S ∪ outside S = UNIV")
  case True with assms show ?thesis by auto
next
  case False
  then obtain b where b: "b ∉ S" "b ∉ outside S" by blast
  have *: "∃y T. y ∈ S ∧ connected T ∧ a ∈ T ∧ y ∈ T ∧ T ⊆ (S ∪ outside S)" if "a ∈ (S ∪ outside S)" for a
  using that proof
    assume "a ∈ S" then show ?thesis
      by (rule_tac x=a in exI, rule_tac x="{a}" in exI, simp)
  next
    assume a: "a ∈ outside S"
    then have ain: "a ∈ closure (outside S)"
      by (simp add: closure_def)
    obtain h where h: "path h" "pathstart h = a" 
                   "path_image h - {pathfinish h} ⊆ interior (outside S)"
                   "pathfinish h ∈ frontier (outside S)"
      using ain b
      by (metis exists_path_subpath_to_frontier path_linepath pathfinish_linepath pathstart_linepath)
    moreover 
    have h1S: "pathfinish h ∈ S"
      using S frontier_outside_subset h(4) by blast
    moreover 
    have "path_image h ⊆ S ∪ outside S"
      using IntD1 S h1S h interior_eq open_outside by fastforce
    ultimately show ?thesis
      by blast
  qed
  show ?thesis
    apply (simp add: connected_iff_connected_component)
    apply (clarsimp simp add: connected_component_def dest!: *)
    subgoal for x y u u' T t'
      by (rule_tac x="(S ∪ T ∪ t')" in exI) (auto intro!: connected_Un cons)
    done
qed

lemma inside_inside_eq_empty [simp]:
    fixes S :: "'a :: {real_normed_vector, perfect_space} set"
    assumes S: "closed S" and cons: "connected S"
    shows "inside (inside S) = {}"
proof -
  have "connected (- inside S)"
    by (metis S connected_with_outside cons union_with_outside)
  then show ?thesis
    by (metis bounded_Un inside_complement_unbounded_connected_empty unbounded_outside union_with_outside)
qed

lemma inside_in_components:
     "inside S ∈ components (- S) ⟷ connected(inside S) ∧ inside S ≠ {}" (is "?lhs = ?rhs")
proof 
  assume R: ?rhs
  then have "∧x. [x ∈ S; x ∈ inside S] ==> ¬ connected (inside S)"
    by (simp add: inside_outside)
  with R show ?lhs
    unfolding in_components_maximal
    by (auto intro: inside_same_component connected_componentI)
qed (simp add: in_components_maximal)

text‹The proof is like that above.›
lemma outside_in_components:
     "outside S ∈ components (- S) ⟷ connected(outside S) ∧ outside S ≠ {}" (is "?lhs = ?rhs")
proof 
  assume R: ?rhs
  then have "∧x. [x ∈ S; x ∈ outside S] ==> ¬ connected (outside S)"
    by (meson disjoint_iff outside_no_overlap)
  with R show ?lhs
    unfolding in_components_maximal
    by (auto intro: outside_same_component connected_componentI)
qed (simp add: in_components_maximal)

lemma bounded_unique_outside:
  fixes S :: "'a :: euclidean_space set"
  assumes "bounded S" "DIM('a) ≥ 2"
  shows "(c ∈ components (- S) ∧ ¬ bounded c) ⟷ c = outside S" 
  using assms
  by (metis cobounded_unique_unbounded_components connected_outside double_compl outside_bounded_nonempty
      outside_in_components unbounded_outside)


subsection‹Condition for an open map's image to contain a ball›

proposition ball_subset_open_map_image:
  fixes f :: "'a::heine_borel ==> 'b :: {real_normed_vector,heine_borel}"
  assumes contf: "continuous_on (closure S) f"
      and oint: "open (f ` interior S)"
      and le_no: "∧z. z ∈ frontier S ==> r ≤ norm(f z - f a)"
      and "bounded S" "a ∈ S" "0 < r"
    shows "ball (f a) r ⊆ f ` S"
proof (cases "f ` S = UNIV")
  case True then show ?thesis by simp
next
  case False
  then have "closed (frontier (f ` S))" "frontier (f ` S) ≠ {}"
    using ‹a ∈ S› by (auto simp: frontier_eq_empty)
  then obtain w where w: "w ∈ frontier (f ` S)"
    and dw_le: "∧y. y ∈ frontier (f ` S) ==> norm (f a - w) ≤ norm (f a - y)"
    by (auto simp add: dist_norm intro: distance_attains_inf [of "frontier(f ` S)" "f a"])
  then obtain ξ where ξ: "∧n. ξ n ∈ f ` S" and tendsw: "ξ <---- w"
    by (metis Diff_iff frontier_def closure_sequential)
    then have "∧n. ∃x ∈ S. ξ n = f x" by force
    then obtain z where zs: "∧n. z n ∈ S" and fz: "∧n. ξ n = f (z n)"
      by metis
    then obtain y K where y: "y ∈ closure S" and "strict_mono (K :: nat ==> nat)" 
                      and Klim: "(z ∘ K) <---- y"
      using ‹bounded S›
      unfolding compact_closure [symmetric] compact_def by (meson closure_subset subset_iff)
    then have ftendsw: "((λn. f (z n)) ∘ K) <---- w"
      by (metis LIMSEQ_subseq_LIMSEQ fun.map_cong0 fz tendsw)
    have zKs: "∧n. (z ∘ K) n ∈ S" by (simp add: zs)
    have fz: "f ∘ z = ξ"  "(λn. f (z n)) = ξ"
      using fz by auto
    then have "(ξ ∘ K) <---- f y"
      by (metis (no_types) Klim zKs y contf comp_assoc continuous_on_closure_sequentially)
    with fz have wy: "w = f y" using fz LIMSEQ_unique ftendsw by auto
    have "r ≤ norm (f y - f a)"
    proof (rule le_no)
      show "y ∈ frontier S"
        using w wy oint by (force simp: imageI image_mono interiorI interior_subset frontier_def y)
    qed
    then have "∧y. [norm (f a - y) < r; y ∈ frontier (f ` S)] ==> False"
      by (metis dw_le norm_minus_commute not_less order_trans wy)
    then have "ball (f a) r ∩ frontier (f ` S) = {}"
      by (metis disjoint_iff_not_equal dist_norm mem_ball)
    moreover
    have "ball (f a) r ∩ f ` S ≠ {}"
      using ‹a ∈ S› ‹0 🚫› centre_in_ball by blast
    ultimately show ?thesis
      by (meson connected_Int_frontier connected_ball diff_shunt_var)
qed


subsubsection‹Special characterizations of classes of functions into and out of R.›

lemma Hausdorff_space_euclidean [simp]: "Hausdorff_space (euclidean :: 'a::metric_space topology)"
proof -
  have "∃U V. open U ∧ open V ∧ x ∈ U ∧ y ∈ V ∧ disjnt U V"
    if "x ≠ y" for x y :: 'a
  proof (intro exI conjI)
    let ?r = "dist x y / 2"
    have [simp]: "?r > 0"
      by (simp add: that)
    show "open (ball x ?r)" "open (ball y ?r)" "x ∈ (ball x ?r)" "y ∈ (ball y ?r)"
      by (auto simp add: that)
    show "disjnt (ball x ?r) (ball y ?r)"
      unfolding disjnt_def by (simp add: disjoint_ballI)
  qed
  then show ?thesis
    by (simp add: Hausdorff_space_def)
qed

proposition embedding_map_into_euclideanreal:
  assumes "path_connected_space X"
  shows "embedding_map X euclideanreal f ⟷
         continuous_map X euclideanreal f ∧ inj_on f (topspace X)"
  proof safe
  show "continuous_map X euclideanreal f"
    if "embedding_map X euclideanreal f"
    using continuous_map_in_subtopology homeomorphic_imp_continuous_map that
    unfolding embedding_map_def by blast
  show "inj_on f (topspace X)"
    if "embedding_map X euclideanreal f"
    using that homeomorphic_imp_injective_map
    unfolding embedding_map_def by blast
  show "embedding_map X euclideanreal f"
    if cont: "continuous_map X euclideanreal f" and inj: "inj_on f (topspace X)"
  proof -
    obtain g where gf: "∧x. x ∈ topspace X ==> g (f x) = x"
      using inv_into_f_f [OF inj] by auto
    show ?thesis
      unfolding embedding_map_def homeomorphic_map_maps homeomorphic_maps_def
    proof (intro exI conjI)
      show "continuous_map X (top_of_set (f ` topspace X)) f"
        by (simp add: cont continuous_map_in_subtopology)
      let ?S = "f ` topspace X"
      have eq: "{x ∈ ?S. g x ∈ U} = f ` U" if "openin X U" for U
        using openin_subset [OF that] by (auto simp: gf)
      have 1: "g ` ?S ⊆ topspace X"
        using eq by blast
      have "openin (top_of_set ?S) {x ∈ ?S. g x ∈ T}"
        if "openin X T" for T
      proof -
        have "T ⊆ topspace X"
          by (simp add: openin_subset that)
        have RR: "∀x ∈ ?S ∩ g -` T. ∃d>0. ∀x' ∈ ?S ∩ ball x d. g x' ∈ T"
        proof (clarsimp simp add: gf)
          have pcS: "path_connectedin euclidean ?S"
            using assms cont path_connectedin_continuous_map_image path_connectedin_topspace by blast
          show "∃d>0. ∀x'∈f ` topspace X ∩ ball (f x) d. g x' ∈ T"
            if "x ∈ T" for x
          proof -
            have x: "x ∈ topspace X"
              using ‹T ⊆ topspace X› ‹x ∈ T› by blast
            obtain u v d where "0 < d" "u ∈ topspace X" "v ∈ topspace X"
                         and sub_fuv: "?S ∩ {f x - d .. f x + d} ⊆ {f u..f v}"
            proof (cases "∃u ∈ topspace X. f u < f x")
              case True
              then obtain u where u: "u ∈ topspace X" "f u < f x" ..
              show ?thesis
              proof (cases "∃v ∈ topspace X. f x < f v")
                case True
                then obtain v where v: "v ∈ topspace X" "f x < f v" ..
                show ?thesis
                proof
                  let ?d = "min (f x - f u) (f v - f x)"
                  show "0 < ?d"
                    by (simp add: ‹f u 🚫 x› ‹f x 🚫 v›)
                  show "f ` topspace X ∩ {f x - ?d..f x + ?d} ⊆ {f u..f v}"
                    by fastforce
                qed (auto simp: u v)
              next
                case False
                show ?thesis
                proof
                  let ?d = "f x - f u"
                  show "0 < ?d"
                    by (simp add: u)
                  show "f ` topspace X ∩ {f x - ?d..f x + ?d} ⊆ {f u..f x}"
                    using x u False by auto
                qed (auto simp: x u)
              qed
            next
              case False
              note no_u = False
              show ?thesis
              proof (cases "∃v ∈ topspace X. f x < f v")
                case True
                then obtain v where v: "v ∈ topspace X" "f x < f v" ..
                show ?thesis
                proof
                  let ?d = "f v - f x"
                  show "0 < ?d"
                    by (simp add: v)
                  show "f ` topspace X ∩ {f x - ?d..f x + ?d} ⊆ {f x..f v}"
                    using False by auto
                qed (auto simp: x v)
              next
                case False
                show ?thesis
                proof
                  show "f ` topspace X ∩ {f x - 1..f x + 1} ⊆ {f x..f x}"
                    using False no_u by fastforce
                qed (auto simp: x)
              qed
            qed
            then obtain h where "pathin X h" "h 0 = u" "h 1 = v"
              using assms unfolding path_connected_space_def by blast
            obtain C where "compactin X C" "connectedin X C" "u ∈ C" "v ∈ C"
            proof
              show "compactin X (h ` {0..1})"
                using that by (simp add: ‹pathin X h› compactin_path_image)
              show "connectedin X (h ` {0..1})"
                using ‹pathin X h› connectedin_path_image by blast
            qed (use ‹h 0 = u› ‹h 1 = v› in auto)
            have "continuous_map (subtopology euclideanreal (?S ∩ {f x - d .. f x + d})) (subtopology X C) g"
            proof (rule continuous_inverse_map)
              show "compact_space (subtopology X C)"
                using ‹compactin X C› compactin_subspace by blast
              show "continuous_map (subtopology X C) euclideanreal f"
                by (simp add: cont continuous_map_from_subtopology)
              have "{f u .. f v} ⊆ f ` topspace (subtopology X C)"
              proof (rule connected_contains_Icc)
                show "connected (f ` topspace (subtopology X C))"
                  using connectedin_continuous_map_image [OF cont]
                  by (simp add: ‹compactin X C› ‹connectedin X C› compactin_subset_topspace inf_absorb2)
                show "f u ∈ f ` topspace (subtopology X C)"
                  by (simp add: ‹u ∈ C› ‹u ∈ topspace X›)
                show "f v ∈ f ` topspace (subtopology X C)"
                  by (simp add: ‹v ∈ C› ‹v ∈ topspace X›)
              qed
              then show "f ` topspace X ∩ {f x - d..f x + d} ⊆ f ` topspace (subtopology X C)"
                using sub_fuv by blast
            qed (auto simp: gf)
            then have contg: "continuous_map (subtopology euclideanreal (?S ∩ {f x - d .. f x + d})) X g"
              using continuous_map_in_subtopology by blast
            have "∃e>0. ∀x ∈ ?S ∩ {f x - d .. f x + d} ∩ ball (f x) e. g x ∈ T"
              using openin_continuous_map_preimage [OF contg ‹openin X T›] x ‹x ∈ T› ‹0 🚫›
              unfolding openin_euclidean_subtopology_iff
              by (force simp: gf dist_commute)
            then obtain e where "e > 0 ∧ (∀x∈f ` topspace X ∩ {f x - d..f x + d} ∩ ball (f x) e. g x ∈ T)"
              by metis
            with ‹0 🚫› have "min d e > 0" "∀u. u ∈ topspace X ⟶ ∣f x - f u∣ < min d e ⟶ u ∈ T"
              using dist_real_def gf by force+
            then show ?thesis
              by (metis (full_types) Int_iff dist_real_def image_iff mem_ball gf)
          qed
        qed
        then obtain d where d: "∧r. r ∈ ?S ∩ g -` T ==>
                d r > 0 ∧ (∀x ∈ ?S ∩ ball r (d r). g x ∈ T)"
          by metis
        show ?thesis
          unfolding openin_subtopology
        proof (intro exI conjI)
          show "{x ∈ ?S. g x ∈ T} = (∪r ∈ ?S ∩ g -` T. ball r (d r)) ∩ f ` topspace X"
            using d by (auto simp: gf)
        qed auto
      qed
      then show "continuous_map (top_of_set ?S) X g"
        by (simp add: "1" continuous_map)
    qed (auto simp: gf)
  qed
qed

subsubsection ‹An injective function into R is a homeomorphism and so an open map.›

lemma injective_into_1d_eq_homeomorphism:
  fixes f :: "'a::topological_space ==> real"
  assumes f: "continuous_on S f" and S: "path_connected S"
  shows "inj_on f S ⟷ (∃g. homeomorphism S (f ` S) f g)"
proof
  show "∃g. homeomorphism S (f ` S) f g"
    if "inj_on f S"
  proof -
    have "embedding_map (top_of_set S) euclideanreal f"
      using that embedding_map_into_euclideanreal [of "top_of_set S" f] assms by auto
    then show ?thesis
      unfolding embedding_map_def topspace_euclidean_subtopology
      by (metis f homeomorphic_map_closedness_eq homeomorphism_injective_closed_map that)
  qed
qed (metis homeomorphism_def inj_onI)

lemma injective_into_1d_imp_open_map:
  fixes f :: "'a::topological_space ==> real"
  assumes "continuous_on S f" "path_connected S" "inj_on f S" "openin (subtopology euclidean S) T"
  shows "openin (subtopology euclidean (f ` S)) (f ` T)"
  using assms homeomorphism_imp_open_map injective_into_1d_eq_homeomorphism by blast

lemma homeomorphism_into_1d:
  fixes f :: "'a::topological_space ==> real"
  assumes "path_connected S" "continuous_on S f" "f ` S = T" "inj_on f S"
  shows "∃g. homeomorphism S T f g"
  using assms injective_into_1d_eq_homeomorphism by blast

subsection🍋‹tag unimportant› ‹Rectangular paths›

definition🍋‹tag unimportant› rectpath where
  "rectpath a1 a3 = (let a2 = Complex (Re a3) (Im a1); a4 = Complex (Re a1) (Im a3)
                      in linepath a1 a2 +++ linepath a2 a3 +++ linepath a3 a4 +++ linepath a4 a1)"

lemma path_rectpath [simp, intro]: "path (rectpath a b)"
  by (simp add: Let_def rectpath_def)

lemma pathstart_rectpath [simp]: "pathstart (rectpath a1 a3) = a1"
  by (simp add: rectpath_def Let_def)

lemma pathfinish_rectpath [simp]: "pathfinish (rectpath a1 a3) = a1"
  by (simp add: rectpath_def Let_def)

lemma simple_path_rectpath [simp, intro]:
  assumes "Re a1 ≠ Re a3" "Im a1 ≠ Im a3"
  shows   "simple_path (rectpath a1 a3)"
  unfolding rectpath_def Let_def using assms
  by (intro simple_path_join_loop arc_join arc_linepath)
     (auto simp: complex_eq_iff path_image_join closed_segment_same_Re closed_segment_same_Im)

lemma path_image_rectpath:
  assumes "Re a1 ≤ Re a3" "Im a1 ≤ Im a3"
  shows "path_image (rectpath a1 a3) =
           {z. Re z ∈ {Re a1, Re a3} ∧ Im z ∈ {Im a1..Im a3}} ∪
           {z. Im z ∈ {Im a1, Im a3} ∧ Re z ∈ {Re a1..Re a3}}" (is "?lhs = ?rhs")
proof -
  define a2 a4 where "a2 = Complex (Re a3) (Im a1)" and "a4 = Complex (Re a1) (Im a3)"
  have "?lhs = closed_segment a1 a2 ∪ closed_segment a2 a3 ∪
                  closed_segment a4 a3 ∪ closed_segment a1 a4"
    by (simp_all add: rectpath_def Let_def path_image_join closed_segment_commute
                      a2_def a4_def Un_assoc)
  also have "… = ?rhs" using assms
    by (auto simp: rectpath_def Let_def path_image_join a2_def a4_def
          closed_segment_same_Re closed_segment_same_Im closed_segment_eq_real_ivl)
  finally show ?thesis .
qed

lemma path_image_rectpath_subset_cbox:
  assumes "Re a ≤ Re b" "Im a ≤ Im b"
  shows   "path_image (rectpath a b) ⊆ cbox a b"
  using assms by (auto simp: path_image_rectpath in_cbox_complex_iff)

lemma path_image_rectpath_inter_box:
  assumes "Re a ≤ Re b" "Im a ≤ Im b"
  shows   "path_image (rectpath a b) ∩ box a b = {}"
  using assms by (auto simp: path_image_rectpath in_box_complex_iff)

lemma path_image_rectpath_cbox_minus_box:
  assumes "Re a ≤ Re b" "Im a ≤ Im b"
  shows   "path_image (rectpath a b) = cbox a b - box a b"
  using assms by (auto simp: path_image_rectpath in_cbox_complex_iff in_box_complex_iff)

end