Quelle  Jordan_Curve.thy

(*  Title:      HOL/Analysis/Jordan_Curve.thy
  Authors: LC Paulson, based on material from HOL Light
*)

section ‹The Jordan Curve Theorem and Applications›

theory Jordan_Curve
  imports Arcwise_Connected Further_Topology
begin

subsection‹Janiszewski's theorem›

lemma Janiszewski_weak:
  fixes a b::complex
  assumes "compact S" "compact T" and conST: "connected(S ∩ T)"
      and ccS: "connected_component (- S) a b" and ccT: "connected_component (- T) a b"
    shows "connected_component (- (S ∪ T)) a b"
proof -
  have [simp]: "a ∉ S" "a ∉ T" "b ∉ S" "b ∉ T"
    by (meson ComplD ccS ccT connected_component_in)+
  have clo: "closedin (top_of_set (S ∪ T)) S" "closedin (top_of_set (S ∪ T)) T"
    by (simp_all add: assms closed_subset compact_imp_closed)
  obtain g where contg: "continuous_on S g"
             and g: "∧x. x ∈ S ==> exp (i* of_real (g x)) = (x - a) /🪙R cmod (x - a) / ((x - b) /🪙R cmod (x - b))"
    using ccS ‹compact S›
    apply (simp add: Borsuk_maps_homotopic_in_connected_component_eq [symmetric])
    apply (subst (asm) homotopic_circlemaps_divide)
    apply (auto simp: inessential_eq_continuous_logarithm_circle)
    done
  obtain h where conth: "continuous_on T h"
             and h: "∧x. x ∈ T ==> exp (i* of_real (h x)) = (x - a) /🪙R cmod (x - a) / ((x - b) /🪙R cmod (x - b))"
    using ccT ‹compact T›
    apply (simp add: Borsuk_maps_homotopic_in_connected_component_eq [symmetric])
    apply (subst (asm) homotopic_circlemaps_divide)
    apply (auto simp: inessential_eq_continuous_logarithm_circle)
    done
  have "continuous_on (S ∪ T) (λx. (x - a) /🪙R cmod (x - a))" "continuous_on (S ∪ T) (λx. (x - b) /🪙R cmod (x - b))"
    by (intro continuous_intros; force)+
  moreover have "(λx. (x - a) /🪙R cmod (x - a)) ` (S ∪ T) ⊆ sphere 0 1" "(λx. (x - b) /🪙R cmod (x - b)) ` (S ∪ T) ⊆ sphere 0 1"
    by (auto simp: divide_simps)
  moreover have "∃g. continuous_on (S ∪ T) g ∧
                     (∀x∈S ∪ T. (x - a) /🪙R cmod (x - a) / ((x - b) /🪙R cmod (x - b)) = exp (i*complex_of_real (g x)))"
  proof (cases "S ∩ T = {}")
    case True
    then have "continuous_on (S ∪ T) (λx. if x ∈ S then g x else h x)"
      using continuous_on_cases_local [OF clo contg conth]
      by (meson disjoint_iff)
    then show ?thesis
      by (rule_tac x="(λx. if x ∈ S then g x else h x)" in exI) (auto simp: g h)
  next
    case False
    have diffpi: "∃n. g x = h x + 2* of_int n*pi" if "x ∈ S ∩ T" for x
    proof -
      have "exp (i* of_real (g x)) = exp (i* of_real (h x))"
        using that by (simp add: g h)
      then obtain n where "complex_of_real (g x) = complex_of_real (h x) + 2* of_int n*complex_of_real pi"
        apply (simp add: exp_eq)
        by (metis complex_i_not_zero distrib_left mult.commute mult_cancel_left)
      then show ?thesis
        using of_real_eq_iff by (fastforce intro!: exI [where x=n])
    qed
    have contgh: "continuous_on (S ∩ T) (λx. g x - h x)"
      by (intro continuous_intros continuous_on_subset [OF contg] continuous_on_subset [OF conth]) auto
    moreover have disc:
          "∃e>0. ∀y. y ∈ S ∩ T ∧ g y - h y ≠ g x - h x ⟶ e ≤ norm ((g y - h y) - (g x - h x))"
          if "x ∈ S ∩ T" for x
    proof -
      obtain nx where nx: "g x = h x + 2* of_int nx*pi"
        using ‹x ∈ S ∩ T› diffpi by blast
      have "2*pi ≤ norm (g y - h y - (g x - h x))" if y: "y ∈ S ∩ T" and neq: "g y - h y ≠ g x - h x" for y
      proof -
        obtain ny where ny: "g y = h y + 2* of_int ny*pi"
          using ‹y ∈ S ∩ T› diffpi by blast
        { assume "nx ≠ ny"
          then have "1 ≤ ∣real_of_int ny - real_of_int nx∣"
            by linarith
          then have "(2*pi)*1 ≤ (2*pi)*∣real_of_int ny - real_of_int nx∣"
            by simp
          also have "... = ∣2*real_of_int ny*pi - 2*real_of_int nx*pi∣"
            by (simp add: algebra_simps abs_if)
          finally have "2*pi ≤ ∣2*real_of_int ny*pi - 2*real_of_int nx*pi∣" by simp
        }
        with neq show ?thesis
          by (simp add: nx ny)
      qed
      then show ?thesis
        by (rule_tac x="2*pi" in exI) auto
    qed
    ultimately have "(λx. g x - h x) constant_on S ∩ T"
      using continuous_discrete_range_constant [OF conST contgh] by blast
    then obtain z where z: "∧x. x ∈ S ∩ T ==> g x - h x = z"
      by (auto simp: constant_on_def)
    obtain w where "exp(i * of_real(h w)) = exp (i * of_real(z + h w))"
      using disc z False
      by auto (metis diff_add_cancel g h of_real_add)
    then have [simp]: "exp (i* of_real z) = 1"
      by (metis cis_conv_exp cis_mult exp_not_eq_zero mult_cancel_right1)
    show ?thesis
    proof (intro exI conjI)
      show "continuous_on (S ∪ T) (λx. if x ∈ S then g x else z + h x)"
        by (intro continuous_intros continuous_on_cases_local [OF clo contg] conth) (use z in force)
    qed (auto simp: g h algebra_simps exp_add)
  qed
  ultimately have "homotopic_with_canon (λx. True) (S ∪ T) (sphere 0 1)
                          (λx. (x - a) /🪙R cmod (x - a)) (λx. (x - b) /🪙R cmod (x - b))"
    by (subst homotopic_circlemaps_divide) (auto simp: inessential_eq_continuous_logarithm_circle)
  moreover have "compact (S ∪ T)"
    using assms by blast
  ultimately show ?thesis
    using assms Borsuk_maps_homotopic_in_connected_component_eq by fastforce
qed


theorem Janiszewski:
  fixes a b :: complex
  assumes "compact S" "closed T" and conST: "connected (S ∩ T)"
      and ccS: "connected_component (- S) a b" and ccT: "connected_component (- T) a b"
    shows "connected_component (- (S ∪ T)) a b"
proof -
  have "path_component(- T) a b"
    by (simp add: ‹closed T› ccT open_Compl open_path_connected_component)
  then obtain g where g: "path g" "path_image g ⊆ - T" "pathstart g = a" "pathfinish g = b"
    by (auto simp: path_component_def)
  then obtain C where C: "compact C" "connected C" "a ∈ C" "b ∈ C" "C ∩ T = {}"
    by fastforce
  obtain r where "0 < r" and r: "C ∪ S ⊆ ball 0 r"
    by (metis ‹compact C› ‹compact S› bounded_Un compact_imp_bounded bounded_subset_ballD)
  have "connected_component (- (S ∪ (T ∩ cball 0 r ∪ sphere 0 r))) a b"
  proof (rule Janiszewski_weak [OF ‹compact S›])
    show comT': "compact ((T ∩ cball 0 r) ∪ sphere 0 r)"
      by (simp add: ‹closed T› closed_Int_compact compact_Un)
    have "S ∩ (T ∩ cball 0 r ∪ sphere 0 r) = S ∩ T"
      using r by auto
    with conST show "connected (S ∩ (T ∩ cball 0 r ∪ sphere 0 r))"
      by simp
    show "connected_component (- (T ∩ cball 0 r ∪ sphere 0 r)) a b"
      using conST C r
      apply (simp add: connected_component_def)
      apply (rule_tac x=C in exI)
      by auto
  qed (simp add: ccS)
  then obtain U where U: "connected U" "U ⊆ - S" "U ⊆ - T ∪ - cball 0 r" "U ⊆ - sphere 0 r" "a ∈ U" "b ∈ U"
    by (auto simp: connected_component_def)
  show ?thesis
    unfolding connected_component_def
  proof (intro exI conjI)
    show "U ⊆ - (S ∪ T)"
      using U r ‹0 🚫› ‹a ∈ C› connected_Int_frontier [of U "cball 0 r"]
      apply simp
      by (metis ball_subset_cball compl_inf disjoint_eq_subset_Compl disjoint_iff_not_equal inf.orderE inf_sup_aci(3) subsetCE)
  qed (auto simp: U)
qed

lemma Janiszewski_connected:
  fixes S :: "complex set"
  assumes ST: "compact S" "closed T" "connected(S ∩ T)"
    and notST: "connected (- S)" "connected (- T)"
  shows "connected(- (S ∪ T))"
  using Janiszewski [OF ST] by (metis IntD1 IntD2 notST compl_sup connected_iff_connected_component)


subsection‹The Jordan Curve theorem›

lemma exists_double_arc:
  fixes g :: "real ==> 'a::real_normed_vector"
  assumes "simple_path g" "pathfinish g = pathstart g" "a ∈ path_image g" "b ∈ path_image g" "a ≠ b"
  obtains u d where "arc u" "arc d" "pathstart u = a" "pathfinish u = b"
                    "pathstart d = b" "pathfinish d = a"
                    "(path_image u) ∩ (path_image d) = {a,b}"
                    "(path_image u) ∪ (path_image d) = path_image g"
proof -
  obtain u where u: "0 ≤ u" "u ≤ 1" "g u = a"
    using assms by (auto simp: path_image_def)
  define h where "h ≡ shiftpath u g"
  have "simple_path h"
    using ‹simple_path g› simple_path_shiftpath ‹0 ≤ u› ‹u ≤ 1› assms(2) h_def by blast
  have "pathstart h = g u"
    by (simp add: ‹u ≤ 1› h_def pathstart_shiftpath)
  have "pathfinish h = g u"
    by (simp add: ‹0 ≤ u› assms h_def pathfinish_shiftpath)
  have pihg: "path_image h = path_image g"
    by (simp add: ‹0 ≤ u› ‹u ≤ 1› assms h_def path_image_shiftpath)
  then obtain v where v: "0 ≤ v" "v ≤ 1" "h v = b"
    using assms by (metis (mono_tags, lifting) atLeastAtMost_iff imageE path_image_def)
  show ?thesis
  proof
    show "arc (subpath 0 v h)"
      by (metis (no_types) ‹pathstart h = g u› ‹simple_path h› arc_simple_path_subpath ‹a ≠ b› atLeastAtMost_iff zero_le_one order_refl pathstart_def u(3) v)
    show "arc (subpath v 1 h)"
      by (metis (no_types) ‹pathfinish h = g u› ‹simple_path h› arc_simple_path_subpath ‹a ≠ b› atLeastAtMost_iff zero_le_one order_refl pathfinish_def u(3) v)
    show "pathstart (subpath 0 v h) = a"
      by (metis ‹pathstart h = g u› pathstart_def pathstart_subpath u(3))
    show "pathfinish (subpath 0 v h) = b"  "pathstart (subpath v 1 h) = b"
      by (simp_all add: v(3))
    show "pathfinish (subpath v 1 h) = a"
      by (metis ‹pathfinish h = g u› pathfinish_def pathfinish_subpath u(3))
    show "path_image (subpath 0 v h) ∩ path_image (subpath v 1 h) = {a, b}"
    proof
      have "loop_free h"
        using ‹simple_path h› simple_path_def by blast
      then show "path_image (subpath 0 v h) ∩ path_image (subpath v 1 h) ⊆ {a, b}"
        using v ‹pathfinish (subpath v 1 h) = a›
        apply (clarsimp simp add: loop_free_def path_image_subpath Ball_def)
        by (smt (verit))
      show "{a, b} ⊆ path_image (subpath 0 v h) ∩ path_image (subpath v 1 h)"
        using v ‹pathstart (subpath 0 v h) = a› ‹pathfinish (subpath v 1 h) = a›
        by (auto simp: path_image_subpath image_iff Bex_def)
    qed
    show "path_image (subpath 0 v h) ∪ path_image (subpath v 1 h) = path_image g"
      using v path_image_subpath pihg path_image_def
      by (metis (full_types) image_Un ivl_disj_un_two_touch(4))
  qed
qed


theorem🍋‹tag unimportant› Jordan_curve:
  fixes c :: "real ==> complex"
  assumes "simple_path c" and loop: "pathfinish c = pathstart c"
  obtains inner outer where
                "inner ≠ {}" "open inner" "connected inner"
                "outer ≠ {}" "open outer" "connected outer"
                "bounded inner" "¬ bounded outer" "inner ∩ outer = {}"
                "inner ∪ outer = - path_image c"
                "frontier inner = path_image c"
                "frontier outer = path_image c"
proof -
  have "path c"
    by (simp add: assms simple_path_imp_path)
  have hom: "(path_image c) homeomorphic (sphere(0::complex) 1)"
    by (simp add: assms homeomorphic_simple_path_image_circle)
  with Jordan_Brouwer_separation have "¬ connected (- (path_image c))"
    by fastforce
  then obtain inner where inner: "inner ∈ components (- path_image c)" and "bounded inner"
    using cobounded_has_bounded_component [of "- (path_image c)"]
    using ‹¬ connected (- path_image c)› ‹simple_path c› bounded_simple_path_image by force
  obtain outer where outer: "outer ∈ components (- path_image c)" and "¬ bounded outer"
    using cobounded_unbounded_components [of "- (path_image c)"]
    using ‹path c› bounded_path_image by auto
  show ?thesis
  proof
    show "inner ≠ {}"
      using inner in_components_nonempty by auto
    show "open inner"
      by (meson ‹simple_path c› compact_imp_closed compact_simple_path_image inner open_Compl open_components)
    show "connected inner"
      using in_components_connected inner by blast
    show "outer ≠ {}"
      using outer in_components_nonempty by auto
    show "open outer"
      by (meson ‹simple_path c› compact_imp_closed compact_simple_path_image outer open_Compl open_components)
    show "connected outer"
      using in_components_connected outer by blast
    show inner_outer: "inner ∩ outer = {}"
      by (meson ‹¬ bounded outer› ‹bounded inner› ‹connected outer› bounded_subset components_maximal in_components_subset inner outer)
    show fro_inner: "frontier inner = path_image c"
      by (simp add: Jordan_Brouwer_frontier [OF hom inner])
    show fro_outer: "frontier outer = path_image c"
      by (simp add: Jordan_Brouwer_frontier [OF hom outer])
    have False if m: "middle ∈ components (- path_image c)" and "middle ≠ inner" "middle ≠ outer" for middle
    proof -
      have "frontier middle = path_image c"
        by (simp add: Jordan_Brouwer_frontier [OF hom] that)
      obtain middle: "open middle" "connected middle" "middle ≠ {}"
        by (metis fro_inner frontier_closed in_components_maximal m open_Compl open_components)
      obtain a0 b0 where "a0 ∈ path_image c" "b0 ∈ path_image c" "a0 ≠ b0"
        using simple_path_image_uncountable [OF ‹simple_path c›]
        by (metis Diff_cancel countable_Diff_eq countable_empty insert_iff subsetI subset_singleton_iff)
      obtain a b g where ab: "a ∈ path_image c" "b ∈ path_image c" "a ≠ b"
                     and g: "arc g" "pathstart g = a" "pathfinish g = b"
                     and pag_sub: "path_image g - {a,b} ⊆ middle"
      proof (rule dense_accessible_frontier_point_pairs [OF ‹open middle› ‹connected middle›, of "path_image c ∩ ball a0 (dist a0 b0)" "path_image c ∩ ball b0 (dist a0 b0)"])
        show "openin (top_of_set (frontier middle)) (path_image c ∩ ball a0 (dist a0 b0))"
             "openin (top_of_set (frontier middle)) (path_image c ∩ ball b0 (dist a0 b0))"
          by (simp_all add: ‹frontier middle = path_image c› openin_open_Int)
        show "path_image c ∩ ball a0 (dist a0 b0) ≠ path_image c ∩ ball b0 (dist a0 b0)"
          using ‹a0 ≠ b0› ‹b0 ∈ path_image c› by auto
        show "path_image c ∩ ball a0 (dist a0 b0) ≠ {}"
          using ‹a0 ∈ path_image c› ‹a0 ≠ b0› by auto
        show "path_image c ∩ ball b0 (dist a0 b0) ≠ {}"
          using ‹b0 ∈ path_image c› ‹a0 ≠ b0› by auto
      qed (use arc_distinct_ends arc_imp_simple_path simple_path_endless that in fastforce)
      obtain u d where "arc u" "arc d"
                   and "pathstart u = a" "pathfinish u = b" "pathstart d = b" "pathfinish d = a"
                   and ud_ab: "(path_image u) ∩ (path_image d) = {a,b}"
                   and ud_Un: "(path_image u) ∪ (path_image d) = path_image c"
        using exists_double_arc [OF assms ab] by blast
      obtain x y where "x ∈ inner" "y ∈ outer"
        using ‹inner ≠ {}› ‹outer ≠ {}› by auto
      have "inner ∩ middle = {}" "middle ∩ outer = {}"
        using components_nonoverlap inner outer m that by blast+
      have "connected_component (- (path_image u ∪ path_image g ∪ (path_image d ∪ path_image g))) x y"
      proof (rule Janiszewski)
        show "compact (path_image u ∪ path_image g)"
          by (simp add: ‹arc g› ‹arc u› compact_Un compact_arc_image)
        show "closed (path_image d ∪ path_image g)"
          by (simp add: ‹arc d› ‹arc g› closed_Un closed_arc_image)
        show "connected ((path_image u ∪ path_image g) ∩ (path_image d ∪ path_image g))"
          using ud_ab
          by (metis Un_insert_left g connected_arc_image insert_absorb pathfinish_in_path_image pathstart_in_path_image sup_bot_left sup_commute sup_inf_distrib1)
        show "connected_component (- (path_image u ∪ path_image g)) x y"
          unfolding connected_component_def
        proof (intro exI conjI)
          have "connected ((inner ∪ (path_image c - path_image u)) ∪ (outer ∪ (path_image c - path_image u)))"
          proof (rule connected_Un)
            show "connected (inner ∪ (path_image c - path_image u))"
              using connected_intermediate_closure [OF ‹connected inner›]
              by (metis Diff_subset closure_Un_frontier dual_order.refl fro_inner sup.mono sup_ge1)
            show "connected (outer ∪ (path_image c - path_image u))"
              using connected_intermediate_closure [OF ‹connected outer›]
              by (simp add: Diff_eq closure_Un_frontier fro_outer sup_inf_distrib1)
            have "(inner ∩ outer) ∪ (path_image c - path_image u) ≠ {}"
              using ‹arc d› ‹pathfinish d = a› ‹pathstart d = b› arc_imp_simple_path nonempty_simple_path_endless ud_Un ud_ab by fastforce
            then show "(inner ∪ (path_image c - path_image u)) ∩ (outer ∪ (path_image c - path_image u)) ≠ {}"
              by auto
          qed
          then show "connected (inner ∪ outer ∪ (path_image c - path_image u))"
            by (metis sup.right_idem sup_assoc sup_commute)
          have "inner ⊆ - path_image u" "outer ⊆ - path_image u"
            using in_components_subset inner outer ud_Un by auto
          moreover have "inner ⊆ - path_image g" "outer ⊆ - path_image g"
            using ‹inner ∩ middle = {}› ‹inner ⊆ - path_image u›
            using ‹middle ∩ outer = {}› ‹outer ⊆ - path_image u› pag_sub ud_ab by fastforce+
          moreover have "path_image c - path_image u ⊆ - path_image g"
            using in_components_subset m pag_sub ud_ab by fastforce
          ultimately show "inner ∪ outer ∪ (path_image c - path_image u) ⊆ - (path_image u ∪ path_image g)"
            by force
          show "x ∈ inner ∪ outer ∪ (path_image c - path_image u)"
            by (auto simp: ‹x ∈ inner›)
          show "y ∈ inner ∪ outer ∪ (path_image c - path_image u)"
            by (auto simp: ‹y ∈ outer›)
        qed
        show "connected_component (- (path_image d ∪ path_image g)) x y"
          unfolding connected_component_def
        proof (intro exI conjI)
          have "connected ((inner ∪ (path_image c - path_image d)) ∪ (outer ∪ (path_image c - path_image d)))"
          proof (rule connected_Un)
            show "connected (inner ∪ (path_image c - path_image d))"
              using connected_intermediate_closure [OF ‹connected inner›] fro_inner
              by (simp add: closure_Un_frontier sup.coboundedI2)
            show "connected (outer ∪ (path_image c - path_image d))"
              using connected_intermediate_closure [OF ‹connected outer›]
              by (simp add: closure_Un_frontier fro_outer sup.coboundedI2)
            have "(inner ∩ outer) ∪ (path_image c - path_image d) ≠ {}"
              using ‹arc u› ‹pathfinish u = b› ‹pathstart u = a› arc_imp_simple_path nonempty_simple_path_endless ud_Un ud_ab by fastforce
            then show "(inner ∪ (path_image c - path_image d)) ∩ (outer ∪ (path_image c - path_image d)) ≠ {}"
              by auto
          qed
          then show "connected (inner ∪ outer ∪ (path_image c - path_image d))"
            by (metis sup.right_idem sup_assoc sup_commute)
          have "inner ⊆ - path_image d" "outer ⊆ - path_image d"
            using in_components_subset inner outer ud_Un by auto
          moreover have "inner ⊆ - path_image g" "outer ⊆ - path_image g"
            using ‹inner ∩ middle = {}› ‹inner ⊆ - path_image d›
            using ‹middle ∩ outer = {}› ‹outer ⊆ - path_image d› pag_sub ud_ab by fastforce+
          moreover have "path_image c - path_image d ⊆ - path_image g"
            using in_components_subset m pag_sub ud_ab by fastforce
          ultimately show "inner ∪ outer ∪ (path_image c - path_image d) ⊆ - (path_image d ∪ path_image g)"
            by force
          show "x ∈ inner ∪ outer ∪ (path_image c - path_image d)"
            by (auto simp: ‹x ∈ inner›)
          show "y ∈ inner ∪ outer ∪ (path_image c - path_image d)"
            by (auto simp: ‹y ∈ outer›)
        qed
      qed
      then have "connected_component (- (path_image u ∪ path_image d ∪ path_image g)) x y"
        by (simp add: Un_ac)
      moreover have "¬(connected_component (- (path_image c)) x y)"
        by (metis (no_types, lifting) ‹¬ bounded outer› ‹bounded inner› ‹x ∈ inner› ‹y ∈ outer› componentsE connected_component_eq inner mem_Collect_eq outer)
      ultimately show False
        by (auto simp: ud_Un [symmetric] connected_component_def)
    qed
    then have "components (- path_image c) = {inner,outer}"
      using inner outer by blast
    then have "Union (components (- path_image c)) = inner ∪ outer"
      by simp
    then show "inner ∪ outer = - path_image c"
      by auto
  qed (auto simp: ‹bounded inner› ‹¬ bounded outer›)
qed


corollary🍋‹tag unimportant› Jordan_disconnected:
  fixes c :: "real ==> complex"
  assumes "simple_path c" "pathfinish c = pathstart c"
  shows "¬ connected(- path_image c)"
  using Jordan_curve [OF assms]
  by (metis Jordan_Brouwer_separation assms homeomorphic_simple_path_image_circle zero_less_one)


corollary Jordan_inside_outside:
  fixes c :: "real ==> complex"
  assumes "simple_path c" "pathfinish c = pathstart c"
    shows "inside(path_image c) ≠ {} ∧
          open(inside(path_image c)) ∧
          connected(inside(path_image c)) ∧
          outside(path_image c) ≠ {} ∧
          open(outside(path_image c)) ∧
          connected(outside(path_image c)) ∧
          bounded(inside(path_image c)) ∧
          ¬ bounded(outside(path_image c)) ∧
          inside(path_image c) ∩ outside(path_image c) = {} ∧
          inside(path_image c) ∪ outside(path_image c) =
          - path_image c ∧
          frontier(inside(path_image c)) = path_image c ∧
          frontier(outside(path_image c)) = path_image c"
proof -
  obtain inner outer
    where *: "inner ≠ {}" "open inner" "connected inner"
             "outer ≠ {}" "open outer" "connected outer"
             "bounded inner" "¬ bounded outer" "inner ∩ outer = {}"
             "inner ∪ outer = - path_image c"
             "frontier inner = path_image c"
             "frontier outer = path_image c"
    using Jordan_curve [OF assms] by blast
  then have inner: "inside(path_image c) = inner"
    by (metis dual_order.antisym inside_subset interior_eq interior_inside_frontier)
  have outer: "outside(path_image c) = outer"
    using ‹inner ∪ outer = - path_image c› ‹inside (path_image c) = inner›
          outside_inside ‹inner ∩ outer = {}› by auto
  show ?thesis
    using * by (auto simp: inner outer)
qed

subsubsection‹Triple-curve or "theta-curve" theorem›

text‹Proof that there is no fourth component taken from
  Kuratowski's Topology vol 2, para 61, II.›

theorem split_inside_simple_closed_curve:
  fixes c :: "real ==> complex"
  assumes "simple_path c1" and c1: "pathstart c1 = a" "pathfinish c1 = b"
      and "simple_path c2" and c2: "pathstart c2 = a" "pathfinish c2 = b"
      and "simple_path c"  and c: "pathstart c = a" "pathfinish c = b"
      and "a ≠ b"
      and c1c2: "path_image c1 ∩ path_image c2 = {a,b}"
      and c1c: "path_image c1 ∩ path_image c = {a,b}"
      and c2c: "path_image c2 ∩ path_image c = {a,b}"
      and ne_12: "path_image c ∩ inside(path_image c1 ∪ path_image c2) ≠ {}"
  obtains "inside(path_image c1 ∪ path_image c) ∩ inside(path_image c2 ∪ path_image c) = {}"
          "inside(path_image c1 ∪ path_image c) ∪ inside(path_image c2 ∪ path_image c) ∪
           (path_image c - {a,b}) = inside(path_image c1 ∪ path_image c2)"
proof -
  let ?Θ = "path_image c"  let ?Θ1 = "path_image c1"  let ?Θ2 = "path_image c2"
  have sp: "simple_path (c1 +++ reversepath c2)" "simple_path (c1 +++ reversepath c)" "simple_path (c2 +++ reversepath c)"
    using assms by (auto simp: simple_path_join_loop_eq arc_simple_path simple_path_reversepath)
  then have op_in12: "open (inside (?Θ1 ∪ ?Θ2))"
     and op_out12: "open (outside (?Θ1 ∪ ?Θ2))"
     and op_in1c: "open (inside (?Θ1 ∪ ?Θ))"
     and op_in2c: "open (inside (?Θ2 ∪ ?Θ))"
     and op_out1c: "open (outside (?Θ1 ∪ ?Θ))"
     and op_out2c: "open (outside (?Θ2 ∪ ?Θ))"
     and co_in1c: "connected (inside (?Θ1 ∪ ?Θ))"
     and co_in2c: "connected (inside (?Θ2 ∪ ?Θ))"
     and co_out12c: "connected (outside (?Θ1 ∪ ?Θ2))"
     and co_out1c: "connected (outside (?Θ1 ∪ ?Θ))"
     and co_out2c: "connected (outside (?Θ2 ∪ ?Θ))"
     and pa_c: "?Θ - {pathstart c, pathfinish c} ⊆ - ?Θ1"
               "?Θ - {pathstart c, pathfinish c} ⊆ - ?Θ2"
     and pa_c1: "?Θ1 - {pathstart c1, pathfinish c1} ⊆ - ?Θ2"
                "?Θ1 - {pathstart c1, pathfinish c1} ⊆ - ?Θ"
     and pa_c2: "?Θ2 - {pathstart c2, pathfinish c2} ⊆ - ?Θ1"
                "?Θ2 - {pathstart c2, pathfinish c2} ⊆ - ?Θ"
     and co_c: "connected(?Θ - {pathstart c,pathfinish c})"
     and co_c1: "connected(?Θ1 - {pathstart c1,pathfinish c1})"
     and co_c2: "connected(?Θ2 - {pathstart c2,pathfinish c2})"
     and fr_in: "frontier(inside(?Θ1 ∪ ?Θ2)) = ?Θ1 ∪ ?Θ2"
              "frontier(inside(?Θ2 ∪ ?Θ)) = ?Θ2 ∪ ?Θ"
              "frontier(inside(?Θ1 ∪ ?Θ)) = ?Θ1 ∪ ?Θ"
     and fr_out: "frontier(outside(?Θ1 ∪ ?Θ2)) = ?Θ1 ∪ ?Θ2"
              "frontier(outside(?Θ2 ∪ ?Θ)) = ?Θ2 ∪ ?Θ"
              "frontier(outside(?Θ1 ∪ ?Θ)) = ?Θ1 ∪ ?Θ"
    using Jordan_inside_outside [of "c1 +++ reversepath c2"]
    using Jordan_inside_outside [of "c1 +++ reversepath c"]
    using Jordan_inside_outside [of "c2 +++ reversepath c"] assms
              apply (simp_all add: path_image_join closed_Un closed_simple_path_image open_inside open_outside)
      apply (blast | metis connected_simple_path_endless)+
    done
  have inout_12: "inside (?Θ1 ∪ ?Θ2) ∩ (?Θ - {pathstart c, pathfinish c}) ≠ {}"
    by (metis (no_types, lifting) c c1c ne_12 Diff_Int_distrib Diff_empty Int_empty_right Int_left_commute inf_sup_absorb inf_sup_aci(1) inside_no_overlap)
  have pi_disjoint:  "?Θ ∩ outside(?Θ1 ∪ ?Θ2) = {}"
  proof (rule ccontr)
    assume "?Θ ∩ outside (?Θ1 ∪ ?Θ2) ≠ {}"
    then show False
      using connectedD [OF co_c, of "inside(?Θ1 ∪ ?Θ2)" "outside(?Θ1 ∪ ?Θ2)"]
      using c c1c2 pa_c op_in12 op_out12 inout_12
      apply clarsimp
      by (smt (verit, ccfv_threshold) Diff_Int_distrib Diff_cancel Diff_empty Int_assoc inf_sup_absorb inf_sup_aci(1) outside_no_overlap)
  qed
  have out_sub12: "outside(?Θ1 ∪ ?Θ2) ⊆ outside(?Θ1 ∪ ?Θ)" "outside(?Θ1 ∪ ?Θ2) ⊆ outside(?Θ2 ∪ ?Θ)"
    by (metis Un_commute pi_disjoint outside_Un_outside_Un)+
  have pa1_disj_in2: "?Θ1 ∩ inside (?Θ2 ∪ ?Θ) = {}"
  proof (rule ccontr)
    assume ne: "?Θ1 ∩ inside (?Θ2 ∪ ?Θ) ≠ {}"
    have 1: "inside (?Θ ∪ ?Θ2) ∩ ?Θ = {}"
      by (metis (no_types) Diff_Int_distrib Diff_cancel inf_sup_absorb inf_sup_aci(3) inside_no_overlap)
    have 2: "outside (?Θ ∪ ?Θ2) ∩ ?Θ = {}"
      by (metis (no_types) Int_empty_right Int_left_commute inf_sup_absorb outside_no_overlap)
    have "path_image c1 ∩ outside (path_image c2 ∪ path_image c) = {}"
      using connectedD [OF co_c1, of "inside(?Θ2 ∪ ?Θ)" "outside(?Θ2 ∪ ?Θ)"]
        pa_c1 op_in2c op_out2c ne c1 c2c 1 2 by (auto simp: inf_sup_aci)
    then have "outside (?Θ2 ∪ ?Θ) ⊆ outside (?Θ1 ∪ ?Θ2)"
      by (metis outside_Un_outside_Un sup_commute)
    with out_sub12
    have "outside(?Θ1 ∪ ?Θ2) = outside(?Θ2 ∪ ?Θ)" by blast
    then have "frontier(outside(?Θ1 ∪ ?Θ2)) = frontier(outside(?Θ2 ∪ ?Θ))"
      by simp
    then show False
      using inout_12 pi_disjoint c c1c c2c fr_out by auto
  qed
  have pa2_disj_in1: "?Θ2 ∩ inside(?Θ1 ∪ ?Θ) = {}"
  proof (rule ccontr)
    assume ne: "?Θ2 ∩ inside (?Θ1 ∪ ?Θ) ≠ {}"
    have 1: "inside (?Θ ∪ ?Θ1) ∩ ?Θ = {}"
      by (metis (no_types) Diff_Int_distrib Diff_cancel inf_sup_absorb inf_sup_aci(3) inside_no_overlap)
    have 2: "outside (?Θ ∪ ?Θ1) ∩ ?Θ = {}"
      by (metis (no_types) Int_empty_right Int_left_commute inf_sup_absorb outside_no_overlap)
    have "outside (?Θ1 ∪ ?Θ) ⊆ outside (?Θ1 ∪ ?Θ2)"
      apply (rule outside_Un_outside_Un)
      using connectedD [OF co_c2, of "inside(?Θ1 ∪ ?Θ)" "outside(?Θ1 ∪ ?Θ)"]
        pa_c2 op_in1c op_out1c ne c2 c1c 1 2 by (auto simp: inf_sup_aci)
    with out_sub12
    have "outside(?Θ1 ∪ ?Θ2) = outside(?Θ1 ∪ ?Θ)"
      by blast
    then have "frontier(outside(?Θ1 ∪ ?Θ2)) = frontier(outside(?Θ1 ∪ ?Θ))"
      by simp
    then show False
      using inout_12 pi_disjoint c c1c c2c fr_out by auto
  qed
  have in_sub_in1: "inside(?Θ1 ∪ ?Θ) ⊆ inside(?Θ1 ∪ ?Θ2)"
    using pa2_disj_in1 out_sub12 by (auto simp: inside_outside)
  have in_sub_in2: "inside(?Θ2 ∪ ?Θ) ⊆ inside(?Θ1 ∪ ?Θ2)"
    using pa1_disj_in2 out_sub12 by (auto simp: inside_outside)
  have in_sub_out12: "inside(?Θ1 ∪ ?Θ) ⊆ outside(?Θ2 ∪ ?Θ)"
  proof
    fix x
    assume x: "x ∈ inside (?Θ1 ∪ ?Θ)"
    then have xnot: "x ∉ ?Θ"
      by (simp add: inside_def)
    obtain z where zim: "z ∈ ?Θ1" and zout: "z ∈ outside(?Θ2 ∪ ?Θ)"
      unfolding outside_inside
      using nonempty_simple_path_endless [OF ‹simple_path c1›] c1 c1c c1c2 pa1_disj_in2 by auto
    obtain e where "e > 0" and e: "ball z e ⊆ outside(?Θ2 ∪ ?Θ)"
      using zout op_out2c open_contains_ball_eq by blast
    have "z ∈ frontier (inside (?Θ1 ∪ ?Θ))"
      using zim by (auto simp: fr_in)
    then obtain w where w1: "w ∈ inside (?Θ1 ∪ ?Θ)" and dwz: "dist w z < e"
      using zim ‹e > 0› by (auto simp: frontier_def closure_approachable)
    then have w2: "w ∈ outside (?Θ2 ∪ ?Θ)"
      by (metis e dist_commute mem_ball subsetCE)
    then have "connected_component (- ?Θ2 ∩ - ?Θ) z w"
      unfolding connected_component_def
      by (metis co_out2c compl_sup inside_Un_outside sup_ge2 zout)
    moreover have "connected_component (- ?Θ2 ∩ - ?Θ) w x"
      unfolding connected_component_def
      using pa2_disj_in1 co_in1c x w1 union_with_outside by fastforce
    ultimately have eq: "connected_component_set (- ?Θ2 ∩ - ?Θ) x =
                         connected_component_set (- ?Θ2 ∩ - ?Θ) z"
      by (metis (mono_tags, lifting) connected_component_eq mem_Collect_eq)
    show "x ∈ outside (?Θ2 ∪ ?Θ)"
      using zout x pa2_disj_in1 by (auto simp: outside_def eq xnot)
  qed
  have in_sub_out21: "inside(?Θ2 ∪ ?Θ) ⊆ outside(?Θ1 ∪ ?Θ)"
  proof
    fix x
    assume x: "x ∈ inside (?Θ2 ∪ ?Θ)"
    then have xnot: "x ∉ ?Θ"
      by (simp add: inside_def)
    obtain z where zim: "z ∈ ?Θ2" and zout: "z ∈ outside(?Θ1 ∪ ?Θ)"
      unfolding outside_inside
      using nonempty_simple_path_endless [OF ‹simple_path c2›] c1c2 c2 c2c pa2_disj_in1 by auto
    obtain e where "e > 0" and e: "ball z e ⊆ outside(?Θ1 ∪ ?Θ)"
      using zout op_out1c open_contains_ball_eq by blast
    have "z ∈ frontier (inside (?Θ2 ∪ ?Θ))"
      using zim by (auto simp: fr_in)
    then obtain w where w2: "w ∈ inside (?Θ2 ∪ ?Θ)" and dwz: "dist w z < e"
      using zim ‹e > 0› by (auto simp: frontier_def closure_approachable)
    then have w1: "w ∈ outside (?Θ1 ∪ ?Θ)"
      by (metis e dist_commute mem_ball subsetCE)
    then have "connected_component (- ?Θ1 ∩ - ?Θ) z w"
      unfolding connected_component_def
      by (metis Compl_Un co_out1c inside_Un_outside sup_ge2 zout)
    moreover have "connected_component (- ?Θ1 ∩ - ?Θ) w x"
      unfolding connected_component_def
      using pa1_disj_in2 co_in2c x w2 union_with_outside by fastforce
    ultimately have eq: "connected_component_set (- ?Θ1 ∩ - ?Θ) x =
                           connected_component_set (- ?Θ1 ∩ - ?Θ) z"
      by (metis (no_types, lifting) connected_component_eq mem_Collect_eq)
    show "x ∈ outside (?Θ1 ∪ ?Θ)"
      using zout x pa1_disj_in2 by (auto simp: outside_def eq xnot)
  qed
  show ?thesis
  proof
    show "inside (?Θ1 ∪ ?Θ) ∩ inside (?Θ2 ∪ ?Θ) = {}"
      by (metis Int_Un_distrib in_sub_out12 bot_eq_sup_iff disjoint_eq_subset_Compl outside_inside)
    have *: "outside (?Θ1 ∪ ?Θ) ∩ outside (?Θ2 ∪ ?Θ) ⊆ outside (?Θ1 ∪ ?Θ2)"
    proof (rule components_maximal)
      show out_in: "outside (?Θ1 ∪ ?Θ2) ∈ components (- (?Θ1 ∪ ?Θ2))"
        unfolding outside_in_components co_out12c
        using co_out12c fr_out(1) by force
      have conn_U: "connected (- (closure (inside (?Θ1 ∪ ?Θ)) ∪ closure (inside (?Θ2 ∪ ?Θ))))"
      proof (rule Janiszewski_connected, simp_all)
        show "bounded (inside (?Θ1 ∪ ?Θ))"
          by (simp add: ‹simple_path c1› ‹simple_path c› bounded_inside bounded_simple_path_image)
        have if1: "- (inside (?Θ1 ∪ ?Θ) ∪ frontier (inside (?Θ1 ∪ ?Θ))) = - ?Θ1 ∩ - ?Θ ∩ - inside (?Θ1 ∪ ?Θ)"
          by (metis (no_types, lifting) Int_commute Jordan_inside_outside c c1 compl_sup path_image_join path_image_reversepath pathfinish_join pathfinish_reversepath pathstart_join pathstart_reversepath sp(2) closure_Un_frontier fr_out(3))
        then show "connected (- closure (inside (?Θ1 ∪ ?Θ)))"
          by (metis Compl_Un outside_inside co_out1c closure_Un_frontier)
        have if2: "- (inside (?Θ2 ∪ ?Θ) ∪ frontier (inside (?Θ2 ∪ ?Θ))) = - ?Θ2 ∩ - ?Θ ∩ - inside (?Θ2 ∪ ?Θ)"
          by (metis (no_types, lifting) Int_commute Jordan_inside_outside c c2 compl_sup path_image_join path_image_reversepath pathfinish_join pathfinish_reversepath pathstart_join pathstart_reversepath sp(3) closure_Un_frontier fr_out(2))
        then show "connected (- closure (inside (?Θ2 ∪ ?Θ)))"
          by (metis Compl_Un outside_inside co_out2c closure_Un_frontier)
        have "connected(?Θ)"
          by (metis ‹simple_path c› connected_simple_path_image)
        moreover
        have "closure (inside (?Θ1 ∪ ?Θ)) ∩ closure (inside (?Θ2 ∪ ?Θ)) = ?Θ"
          (is "?lhs = ?rhs")
        proof
          show "?lhs ⊆ ?rhs"
          proof clarify
            fix x
            assume x: "x ∈ closure (inside (?Θ1 ∪ ?Θ))" "x ∈ closure (inside (?Θ2 ∪ ?Θ))"
            then have "x ∉ inside (?Θ1 ∪ ?Θ)"
              by (meson closure_iff_nhds_not_empty in_sub_out12 inside_Int_outside op_in1c)
            with fr_in x show "x ∈ ?Θ"
              by (metis c1c c1c2 closure_Un_frontier pa1_disj_in2 Int_iff Un_iff insert_disjoint(2) insert_subset subsetI subset_antisym)
          qed
          show "?rhs ⊆ ?lhs"
            using if1 if2 closure_Un_frontier by fastforce
        qed
        ultimately
        show "connected (closure (inside (?Θ1 ∪ ?Θ)) ∩ closure (inside (?Θ2 ∪ ?Θ)))"
          by auto
      qed
      show "connected (outside (?Θ1 ∪ ?Θ) ∩ outside (?Θ2 ∪ ?Θ))"
        using fr_in conn_U  by (simp add: closure_Un_frontier outside_inside Un_commute)
      show "outside (?Θ1 ∪ ?Θ) ∩ outside (?Θ2 ∪ ?Θ) ⊆ - (?Θ1 ∪ ?Θ2)"
        by clarify (metis Diff_Compl Diff_iff Un_iff inf_sup_absorb outside_inside)
      show "outside (?Θ1 ∪ ?Θ2) ∩
            (outside (?Θ1 ∪ ?Θ) ∩ outside (?Θ2 ∪ ?Θ)) ≠ {}"
        by (metis Int_assoc out_in inf.orderE out_sub12(1) out_sub12(2) outside_in_components)
    qed
    show "inside (?Θ1 ∪ ?Θ) ∪ inside (?Θ2 ∪ ?Θ) ∪ (?Θ - {a, b}) = inside (?Θ1 ∪ ?Θ2)"
      (is "?lhs = ?rhs")
    proof
      have " path_image c - {a, b} ⊆ inside (path_image c1 ∪ path_image c2)"
        using c1c c2c inside_outside pi_disjoint by fastforce
      then show "?lhs ⊆ ?rhs"
        by (simp add: in_sub_in1 in_sub_in2)
      have "inside (?Θ1 ∪ ?Θ2) ⊆ inside (?Θ1 ∪ ?Θ) ∪ inside (?Θ2 ∪ ?Θ) ∪ (?Θ)"
        using Compl_anti_mono [OF *] by (force simp: inside_outside)
      moreover have "inside (?Θ1 ∪ ?Θ2) ⊆ -{a,b}"
        using c1 union_with_outside by fastforce
      ultimately show "?rhs ⊆ ?lhs" by auto
    qed
  qed
qed

end