Theory HOL-Analysis.Homotopy

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

section ‹Homotopy of Maps›

theory Homotopy
  imports Path_Connected Product_Topology Uncountable_Sets
begin

definition✐‹tag important› homotopic_with
where
 "homotopic_with P X Y f g ≡
   (∃h. continuous_map (prod_topology (top_of_set {0..1::real}) X) Y h ∧
       (∀x. h(0, x) = f x) ∧
       (∀x. h(1, x) = g x) ∧
       (∀t ∈ {0..1}. P(λx. h(t,x))))"

text‹‹p›, ‹q› are functions ‹X → Y›, and the property ‹P› restricts all intermediate maps.
We often just want to require that ‹P› fixes some subset, but to include the case of a loop homotopy,
it is convenient to have a general property ‹P›.›

abbreviation homotopic_with_canon ::
  "[('a::topological_space ⇒ 'b::topological_space) ⇒ bool, 'a set, 'b set, 'a ⇒ 'b, 'a ⇒ 'b] ⇒ bool"
where
 "homotopic_with_canon P S T p q ≡ homotopic_with P (top_of_set S) (top_of_set T) p q"

lemma split_01: "{0..1::real} = {0..1/2} ∪ {1/2..1}"
  by force

lemma split_01_prod: "{0..1::real} × X = ({0..1/2} × X) ∪ ({1/2..1} × X)"
  by force

lemma image_Pair_const: "(λx. (x, c)) ` A = A × {c}"
  by auto

lemma fst_o_paired [simp]: "fst ∘ (λ(x,y). (f x y, g x y)) = (λ(x,y). f x y)"
  by auto

lemma snd_o_paired [simp]: "snd ∘ (λ(x,y). (f x y, g x y)) = (λ(x,y). g x y)"
  by auto

lemma continuous_on_o_Pair: "⟦continuous_on (T × X) h; t ∈ T⟧ ⟹ continuous_on X (h ∘ Pair t)"
  by (fast intro: continuous_intros elim!: continuous_on_subset)

lemma continuous_map_o_Pair: 
  assumes h: "continuous_map (prod_topology X Y) Z h" and t: "t ∈ topspace X"
  shows "continuous_map Y Z (h ∘ Pair t)"
  by (intro continuous_map_compose [OF _ h] continuous_intros; simp add: t)

subsection✐‹tag unimportant›‹Trivial properties›

text ‹We often want to just localize the ending function equality or whatever.›
text✐‹tag important› ‹%whitespace›
proposition homotopic_with:
  assumes "⋀h k. (⋀x. x ∈ topspace X ⟹ h x = k x) ⟹ (P h ⟷ P k)"
  shows "homotopic_with P X Y p q ⟷
           (∃h. continuous_map (prod_topology (subtopology euclideanreal {0..1}) X) Y h ∧
              (∀x ∈ topspace X. h(0,x) = p x) ∧
              (∀x ∈ topspace X. h(1,x) = q x) ∧
              (∀t ∈ {0..1}. P(λx. h(t, x))))"
  unfolding homotopic_with_def
  apply (rule iffI, blast, clarify)
  apply (rule_tac x="λ(u,v). if v ∈ topspace X then h(u,v) else if u = 0 then p v else q v" in exI)
  apply simp
  by (smt (verit, best) SigmaE assms case_prod_conv continuous_map_eq topspace_prod_topology)

lemma homotopic_with_mono:
  assumes hom: "homotopic_with P X Y f g"
    and Q: "⋀h. ⟦continuous_map X Y h; P h⟧ ⟹ Q h"
  shows "homotopic_with Q X Y f g"
  using hom unfolding homotopic_with_def
  by (force simp: o_def dest: continuous_map_o_Pair intro: Q)

lemma homotopic_with_imp_continuous_maps:
    assumes "homotopic_with P X Y f g"
    shows "continuous_map X Y f ∧ continuous_map X Y g"
proof -
  obtain h :: "real × 'a ⇒ 'b"
    where conth: "continuous_map (prod_topology (top_of_set {0..1}) X) Y h"
      and h: "∀x. h (0, x) = f x" "∀x. h (1, x) = g x"
    using assms by (auto simp: homotopic_with_def)
  have *: "t ∈ {0..1} ⟹ continuous_map X Y (h ∘ (λx. (t,x)))" for t
    by (rule continuous_map_compose [OF _ conth]) (simp add: o_def continuous_map_pairwise)
  show ?thesis
    using h *[of 0] *[of 1] by (simp add: continuous_map_eq)
qed

lemma homotopic_with_imp_continuous:
    assumes "homotopic_with_canon P X Y f g"
    shows "continuous_on X f ∧ continuous_on X g"
  by (meson assms continuous_map_subtopology_eu homotopic_with_imp_continuous_maps)

lemma homotopic_with_imp_property:
  assumes "homotopic_with P X Y f g"
  shows "P f ∧ P g"
proof
  obtain h where h: "⋀x. h(0, x) = f x" "⋀x. h(1, x) = g x" and P: "⋀t. t ∈ {0..1::real} ⟹ P(λx. h(t,x))"
    using assms by (force simp: homotopic_with_def)
  show "P f" "P g"
    using P [of 0] P [of 1] by (force simp: h)+
qed

lemma homotopic_with_equal:
  assumes "P f" "P g" and contf: "continuous_map X Y f" and fg: "⋀x. x ∈ topspace X ⟹ f x = g x"
  shows "homotopic_with P X Y f g"
  unfolding homotopic_with_def
proof (intro exI conjI allI ballI)
  let ?h = "λ(t::real,x). if t = 1 then g x else f x"
  show "continuous_map (prod_topology (top_of_set {0..1}) X) Y ?h"
  proof (rule continuous_map_eq)
    show "continuous_map (prod_topology (top_of_set {0..1}) X) Y (f ∘ snd)"
      by (simp add: contf continuous_map_of_snd)
  qed (auto simp: fg)
  show "P (λx. ?h (t, x))" if "t ∈ {0..1}" for t
    by (cases "t = 1") (simp_all add: assms)
qed auto

lemma homotopic_with_imp_subset1:
     "homotopic_with_canon P X Y f g ⟹ f ` X ⊆ Y"
  by (meson continuous_map_subtopology_eu homotopic_with_imp_continuous_maps)

lemma homotopic_with_imp_subset2:
     "homotopic_with_canon P X Y f g ⟹ g ` X ⊆ Y"
  by (meson continuous_map_subtopology_eu homotopic_with_imp_continuous_maps)

lemma homotopic_with_imp_funspace1:
     "homotopic_with_canon P X Y f g ⟹ f ∈ X → Y"
  using homotopic_with_imp_subset1 by blast

lemma homotopic_with_imp_funspace2:
     "homotopic_with_canon P X Y f g ⟹ g ∈ X → Y"
  using homotopic_with_imp_subset2 by blast

lemma homotopic_with_subset_left:
     "⟦homotopic_with_canon P X Y f g; Z ⊆ X⟧ ⟹ homotopic_with_canon P Z Y f g"
  unfolding homotopic_with_def by (auto elim!: continuous_on_subset ex_forward)

lemma homotopic_with_subset_right:
     "⟦homotopic_with_canon P X Y f g; Y ⊆ Z⟧ ⟹ homotopic_with_canon P X Z f g"
  unfolding homotopic_with_def by (auto elim!: continuous_on_subset ex_forward)

subsection‹Homotopy with P is an equivalence relation›

text ‹(on continuous functions mapping X into Y that satisfy P, though this only affects reflexivity)›

lemma homotopic_with_refl [simp]: "homotopic_with P X Y f f ⟷ continuous_map X Y f ∧ P f"
  by (metis homotopic_with_equal homotopic_with_imp_continuous_maps homotopic_with_imp_property)

lemma homotopic_with_symD:
    assumes "homotopic_with P X Y f g"
      shows "homotopic_with P X Y g f"
proof -
  let ?I01 = "subtopology euclideanreal {0..1}"
  let ?j = "λy. (1 - fst y, snd y)"
  have 1: "continuous_map (prod_topology ?I01 X) (prod_topology euclideanreal X) ?j"
    by (intro continuous_intros; simp add: continuous_map_subtopology_fst prod_topology_subtopology)
  have *: "continuous_map (prod_topology ?I01 X) (prod_topology ?I01 X) ?j"
  proof -
    have "continuous_map (prod_topology ?I01 X) (subtopology (prod_topology euclideanreal X) ({0..1} × topspace X)) ?j"
      by (simp add: continuous_map_into_subtopology [OF 1] image_subset_iff flip: image_subset_iff_funcset)
    then show ?thesis
      by (simp add: prod_topology_subtopology(1))
  qed
  show ?thesis
    using assms
    apply (clarsimp simp: homotopic_with_def)
    subgoal for h
      by (rule_tac x="h ∘ (λy. (1 - fst y, snd y))" in exI) (simp add: continuous_map_compose [OF *])
    done
qed

lemma homotopic_with_sym:
   "homotopic_with P X Y f g ⟷ homotopic_with P X Y g f"
  by (metis homotopic_with_symD)

proposition homotopic_with_trans:
    assumes "homotopic_with P X Y f g"  "homotopic_with P X Y g h"
    shows "homotopic_with P X Y f h"
proof -
  let ?X01 = "prod_topology (subtopology euclideanreal {0..1}) X"
  obtain k1 k2
    where contk1: "continuous_map ?X01 Y k1" and contk2: "continuous_map ?X01 Y k2"
      and k12: "∀x. k1 (1, x) = g x" "∀x. k2 (0, x) = g x"
      "∀x. k1 (0, x) = f x" "∀x. k2 (1, x) = h x"
      and P:   "∀t∈{0..1}. P (λx. k1 (t, x))" "∀t∈{0..1}. P (λx. k2 (t, x))"
    using assms by (auto simp: homotopic_with_def)
  define k where "k ≡ λy. if fst y ≤ 1/2
                             then (k1 ∘ (λx. (2 *⇩R fst x, snd x))) y
                             else (k2 ∘ (λx. (2 *⇩R fst x -1, snd x))) y"
  have keq: "k1 (2 * u, v) = k2 (2 * u -1, v)" if "u = 1/2"  for u v
    by (simp add: k12 that)
  show ?thesis
    unfolding homotopic_with_def
  proof (intro exI conjI)
    show "continuous_map ?X01 Y k"
      unfolding k_def
    proof (rule continuous_map_cases_le)
      show fst: "continuous_map ?X01 euclideanreal fst"
        using continuous_map_fst continuous_map_in_subtopology by blast
      show "continuous_map ?X01 euclideanreal (λx. 1/2)"
        by simp
      show "continuous_map (subtopology ?X01 {y ∈ topspace ?X01. fst y ≤ 1/2}) Y
               (k1 ∘ (λx. (2 *⇩R fst x, snd x)))"
        apply (intro fst continuous_map_compose [OF _ contk1] continuous_intros continuous_map_into_subtopology continuous_map_from_subtopology | simp)+
        by (force simp: prod_topology_subtopology)
      show "continuous_map (subtopology ?X01 {y ∈ topspace ?X01. 1/2 ≤ fst y}) Y
               (k2 ∘ (λx. (2 *⇩R fst x -1, snd x)))"
        apply (intro fst continuous_map_compose [OF _ contk2] continuous_intros continuous_map_into_subtopology continuous_map_from_subtopology | simp)+
        by (force simp: prod_topology_subtopology)
      show "(k1 ∘ (λx. (2 *⇩R fst x, snd x))) y = (k2 ∘ (λx. (2 *⇩R fst x -1, snd x))) y"
        if "y ∈ topspace ?X01" and "fst y = 1/2" for y
        using that by (simp add: keq)
    qed
    show "∀x. k (0, x) = f x"
      by (simp add: k12 k_def)
    show "∀x. k (1, x) = h x"
      by (simp add: k12 k_def)
    show "∀t∈{0..1}. P (λx. k (t, x))"
    proof 
      fix t show "t∈{0..1} ⟹ P (λx. k (t, x))"
        by (cases "t ≤ 1/2") (auto simp: k_def P)
    qed
  qed
qed

lemma homotopic_with_id2: 
  "(⋀x. x ∈ topspace X ⟹ g (f x) = x) ⟹ homotopic_with (λx. True) X X (g ∘ f) id"
  by (metis comp_apply continuous_map_id eq_id_iff homotopic_with_equal homotopic_with_symD)

subsection‹Continuity lemmas›

lemma homotopic_with_compose_continuous_map_left:
  "⟦homotopic_with p X1 X2 f g; continuous_map X2 X3 h; ⋀j. p j ⟹ q(h ∘ j)⟧
   ⟹ homotopic_with q X1 X3 (h ∘ f) (h ∘ g)"
  unfolding homotopic_with_def
  apply clarify
  subgoal for k
    by (rule_tac x="h ∘ k" in exI) (rule conjI continuous_map_compose | simp add: o_def)+
  done

lemma homotopic_with_compose_continuous_map_right:
  assumes hom: "homotopic_with p X2 X3 f g" and conth: "continuous_map X1 X2 h"
    and q: "⋀j. p j ⟹ q(j ∘ h)"
  shows "homotopic_with q X1 X3 (f ∘ h) (g ∘ h)"
proof -
  obtain k
    where contk: "continuous_map (prod_topology (subtopology euclideanreal {0..1}) X2) X3 k"
      and k: "∀x. k (0, x) = f x" "∀x. k (1, x) = g x" and p: "⋀t. t∈{0..1} ⟹ p (λx. k (t, x))"
    using hom unfolding homotopic_with_def by blast
  have hsnd: "continuous_map (prod_topology (subtopology euclideanreal {0..1}) X1) X2 (h ∘ snd)"
    by (rule continuous_map_compose [OF continuous_map_snd conth])
  let ?h = "k ∘ (λ(t,x). (t,h x))"
  show ?thesis
    unfolding homotopic_with_def
  proof (intro exI conjI allI ballI)
    have "continuous_map (prod_topology (top_of_set {0..1}) X1)
     (prod_topology (top_of_set {0..1::real}) X2) (λ(t, x). (t, h x))"
      by (metis (mono_tags, lifting) case_prod_beta' comp_def continuous_map_eq continuous_map_fst continuous_map_pairedI hsnd)
    then show "continuous_map (prod_topology (subtopology euclideanreal {0..1}) X1) X3 ?h"
      by (intro conjI continuous_map_compose [OF _ contk])
    show "q (λx. ?h (t, x))" if "t ∈ {0..1}" for t
      using q [OF p [OF that]] by (simp add: o_def)
  qed (auto simp: k)
qed

corollary homotopic_compose:
  assumes "homotopic_with (λx. True) X Y f f'" "homotopic_with (λx. True) Y Z g g'"
  shows "homotopic_with (λx. True) X Z (g ∘ f) (g' ∘ f')"
  by (metis assms homotopic_with_compose_continuous_map_left homotopic_with_compose_continuous_map_right homotopic_with_imp_continuous_maps homotopic_with_trans)

proposition homotopic_with_compose_continuous_right:
    "⟦homotopic_with_canon (λf. p (f ∘ h)) X Y f g; continuous_on W h; h ∈ W → X⟧
     ⟹ homotopic_with_canon p W Y (f ∘ h) (g ∘ h)"
  by (simp add: homotopic_with_compose_continuous_map_right image_subset_iff_funcset)

proposition homotopic_with_compose_continuous_left:
     "⟦homotopic_with_canon (λf. p (h ∘ f)) X Y f g; continuous_on Y h; h ∈ Y → Z⟧
      ⟹ homotopic_with_canon p X Z (h ∘ f) (h ∘ g)"
  by (simp add: homotopic_with_compose_continuous_map_left image_subset_iff_funcset)

lemma homotopic_from_subtopology:
   "homotopic_with P X X' f g ⟹ homotopic_with P (subtopology X S) X' f g"
  by (metis continuous_map_id_subt homotopic_with_compose_continuous_map_right o_id)

lemma homotopic_on_emptyI:
  assumes "P f" "P g"
  shows "homotopic_with P trivial_topology X f g"
  by (metis assms continuous_map_on_empty empty_iff homotopic_with_equal topspace_discrete_topology)

lemma homotopic_on_empty:
   "(homotopic_with P trivial_topology X f g ⟷ P f ∧ P g)"
  using homotopic_on_emptyI homotopic_with_imp_property by metis

lemma homotopic_with_canon_on_empty: "homotopic_with_canon (λx. True) {} t f g"
  by (auto intro: homotopic_with_equal)

lemma homotopic_constant_maps:
   "homotopic_with (λx. True) X X' (λx. a) (λx. b) ⟷
    X = trivial_topology ∨ path_component_of X' a b" (is "?lhs = ?rhs")
proof (cases "X = trivial_topology")
  case False
  then obtain c where c: "c ∈ topspace X"
    by fastforce
  have "∃g. continuous_map (top_of_set {0..1::real}) X' g ∧ g 0 = a ∧ g 1 = b"
    if "x ∈ topspace X" and hom: "homotopic_with (λx. True) X X' (λx. a) (λx. b)" for x
  proof -
    obtain h :: "real × 'a ⇒ 'b"
      where conth: "continuous_map (prod_topology (top_of_set {0..1}) X) X' h"
        and h: "⋀x. h (0, x) = a" "⋀x. h (1, x) = b"
      using hom by (auto simp: homotopic_with_def)
    have cont: "continuous_map (top_of_set {0..1}) X' (h ∘ (λt. (t, c)))"
      by (rule continuous_map_compose [OF _ conth] continuous_intros | simp add: c)+
    then show ?thesis
      by (force simp: h)
  qed
  moreover have "homotopic_with (λx. True) X X' (λx. g 0) (λx. g 1)"
    if "x ∈ topspace X" "a = g 0" "b = g 1" "continuous_map (top_of_set {0..1}) X' g"
    for x and g :: "real ⇒ 'b"
    unfolding homotopic_with_def
    by (force intro!: continuous_map_compose continuous_intros c that)
  ultimately show ?thesis
    using False
    by (metis c path_component_of_set pathin_def)
qed (simp add: homotopic_on_empty)

proposition homotopic_with_eq:
   assumes h: "homotopic_with P X Y f g"
       and f': "⋀x. x ∈ topspace X ⟹ f' x = f x"
       and g': "⋀x. x ∈ topspace X ⟹ g' x = g x"
       and P:  "(⋀h k. (⋀x. x ∈ topspace X ⟹ h x = k x) ⟹ P h ⟷ P k)"
   shows "homotopic_with P X Y f' g'"
  by (smt (verit, ccfv_SIG) assms homotopic_with)

lemma homotopic_with_prod_topology:
  assumes "homotopic_with p X1 Y1 f f'" and "homotopic_with q X2 Y2 g g'"
    and r: "⋀i j. ⟦p i; q j⟧ ⟹ r(λ(x,y). (i x, j y))"
  shows "homotopic_with r (prod_topology X1 X2) (prod_topology Y1 Y2)
                          (λz. (f(fst z),g(snd z))) (λz. (f'(fst z), g'(snd z)))"
proof -
  obtain h
    where h: "continuous_map (prod_topology (subtopology euclideanreal {0..1}) X1) Y1 h"
      and h0: "⋀x. h (0, x) = f x"
      and h1: "⋀x. h (1, x) = f' x"
      and p: "⋀t. ⟦0 ≤ t; t ≤ 1⟧ ⟹ p (λx. h (t,x))"
    using assms unfolding homotopic_with_def by auto
  obtain k
    where k: "continuous_map (prod_topology (subtopology euclideanreal {0..1}) X2) Y2 k"
      and k0: "⋀x. k (0, x) = g x"
      and k1: "⋀x. k (1, x) = g' x"
      and q: "⋀t. ⟦0 ≤ t; t ≤ 1⟧ ⟹ q (λx. k (t,x))"
    using assms unfolding homotopic_with_def by auto
  let ?hk = "λ(t,x,y). (h(t,x), k(t,y))"
  show ?thesis
    unfolding homotopic_with_def
  proof (intro conjI allI exI)
    show "continuous_map (prod_topology (subtopology euclideanreal {0..1}) (prod_topology X1 X2))
                         (prod_topology Y1 Y2) ?hk"
      unfolding continuous_map_pairwise case_prod_unfold
      by (rule conjI continuous_map_pairedI continuous_intros continuous_map_id [unfolded id_def]
          continuous_map_fst_of [unfolded o_def] continuous_map_snd_of [unfolded o_def]
          continuous_map_compose [OF _ h, unfolded o_def]
          continuous_map_compose [OF _ k, unfolded o_def])+
  next
    fix x
    show "?hk (0, x) = (f (fst x), g (snd x))" "?hk (1, x) = (f' (fst x), g' (snd x))"
      by (auto simp: case_prod_beta h0 k0 h1 k1)
  qed (auto simp: p q r)
qed


lemma homotopic_with_product_topology:
  assumes ht: "⋀i. i ∈ I ⟹ homotopic_with (p i) (X i) (Y i) (f i) (g i)"
    and pq: "⋀h. (⋀i. i ∈ I ⟹ p i (h i)) ⟹ q(λx. (λi∈I. h i (x i)))"
  shows "homotopic_with q (product_topology X I) (product_topology Y I)
                          (λz. (λi∈I. (f i) (z i))) (λz. (λi∈I. (g i) (z i)))"
proof -
  obtain h
    where h: "⋀i. i ∈ I ⟹ continuous_map (prod_topology (subtopology euclideanreal {0..1}) (X i)) (Y i) (h i)"
      and h0: "⋀i x. i ∈ I ⟹ h i (0, x) = f i x"
      and h1: "⋀i x. i ∈ I ⟹ h i (1, x) = g i x"
      and p: "⋀i t. ⟦i ∈ I; t ∈ {0..1}⟧ ⟹ p i (λx. h i (t,x))"
    using ht unfolding homotopic_with_def by metis
  show ?thesis
    unfolding homotopic_with_def
  proof (intro conjI allI exI)
    let ?h = "λ(t,z). λi∈I. h i (t,z i)"
    have "continuous_map (prod_topology (subtopology euclideanreal {0..1}) (product_topology X I))
                         (Y i) (λx. h i (fst x, snd x i))" if "i ∈ I" for i
    proof -
      have §: "continuous_map (prod_topology (top_of_set {0..1}) (product_topology X I)) (X i) (λx. snd x i)"
        using continuous_map_componentwise continuous_map_snd that by fastforce
      show ?thesis
        unfolding continuous_map_pairwise case_prod_unfold
        by (intro conjI that § continuous_intros continuous_map_compose [OF _ h, unfolded o_def])
    qed
    then show "continuous_map (prod_topology (subtopology euclideanreal {0..1}) (product_topology X I))
         (product_topology Y I) ?h"
      by (auto simp: continuous_map_componentwise case_prod_beta)
    show "?h (0, x) = (λi∈I. f i (x i))" "?h (1, x) = (λi∈I. g i (x i))" for x
      by (auto simp: case_prod_beta h0 h1)
    show "∀t∈{0..1}. q (λx. ?h (t, x))"
      by (force intro: p pq)
  qed
qed

text‹Homotopic triviality implicitly incorporates path-connectedness.›
lemma homotopic_triviality:
  shows  "(∀f g. continuous_on S f ∧ f ∈ S → T ∧
                 continuous_on S g ∧ g ∈ S → T
                 ⟶ homotopic_with_canon (λx. True) S T f g) ⟷
          (S = {} ∨ path_connected T) ∧
          (∀f. continuous_on S f ∧ f ∈ S → T ⟶ (∃c. homotopic_with_canon (λx. True) S T f (λx. c)))"
          (is "?lhs = ?rhs")
proof (cases "S = {} ∨ T = {}")
  case True then show ?thesis
    by (auto simp: homotopic_on_emptyI simp flip: image_subset_iff_funcset)
next
  case False show ?thesis
  proof
    assume LHS [rule_format]: ?lhs
    have pab: "path_component T a b" if "a ∈ T" "b ∈ T" for a b
    proof -
      have "homotopic_with_canon (λx. True) S T (λx. a) (λx. b)"
        by (simp add: LHS image_subset_iff that)
      then show ?thesis
        using False homotopic_constant_maps [of "top_of_set S" "top_of_set T" a b]
        by (metis path_component_of_canon_iff topspace_discrete_topology topspace_euclidean_subtopology)
    qed
    moreover
    have "∃c. homotopic_with_canon (λx. True) S T f (λx. c)" if "continuous_on S f" "f ∈ S → T" for f
      using False LHS continuous_on_const that by blast
    ultimately show ?rhs
      by (simp add: path_connected_component)
  next
    assume RHS: ?rhs
    with False have T: "path_connected T"
      by blast
    show ?lhs
    proof clarify
      fix f g
      assume "continuous_on S f" "f ∈ S → T" "continuous_on S g" "g ∈ S → T"
      obtain c d where c: "homotopic_with_canon (λx. True) S T f (λx. c)" and d: "homotopic_with_canon (λx. True) S T g (λx. d)"
        using RHS ‹continuous_on S f› ‹continuous_on S g› ‹f ∈ S → T› ‹g ∈ S → T› by presburger
      with T have "path_component T c d"
        by (metis False ex_in_conv homotopic_with_imp_subset2 image_subset_iff path_connected_component)
      then have "homotopic_with_canon (λx. True) S T (λx. c) (λx. d)"
        by (simp add: homotopic_constant_maps)
      with c d show "homotopic_with_canon (λx. True) S T f g"
        by (meson homotopic_with_symD homotopic_with_trans)
    qed
  qed
qed


subsection‹Homotopy of paths, maintaining the same endpoints›


definition✐‹tag important› homotopic_paths :: "['a set, real ⇒ 'a, real ⇒ 'a::topological_space] ⇒ bool"
  where
     "homotopic_paths S p q ≡
       homotopic_with_canon (λr. pathstart r = pathstart p ∧ pathfinish r = pathfinish p) {0..1} S p q"

lemma homotopic_paths:
   "homotopic_paths S p q ⟷
      (∃h. continuous_on ({0..1} × {0..1}) h ∧
          h ∈ ({0..1} × {0..1}) → S ∧
          (∀x ∈ {0..1}. h(0,x) = p x) ∧
          (∀x ∈ {0..1}. h(1,x) = q x) ∧
          (∀t ∈ {0..1::real}. pathstart(h ∘ Pair t) = pathstart p ∧
                        pathfinish(h ∘ Pair t) = pathfinish p))"
  by (auto simp: homotopic_paths_def homotopic_with pathstart_def pathfinish_def)

proposition homotopic_paths_imp_pathstart:
     "homotopic_paths S p q ⟹ pathstart p = pathstart q"
  by (metis (mono_tags, lifting) homotopic_paths_def homotopic_with_imp_property)

proposition homotopic_paths_imp_pathfinish:
     "homotopic_paths S p q ⟹ pathfinish p = pathfinish q"
  by (metis (mono_tags, lifting) homotopic_paths_def homotopic_with_imp_property)

lemma homotopic_paths_imp_path:
     "homotopic_paths S p q ⟹ path p ∧ path q"
  using homotopic_paths_def homotopic_with_imp_continuous_maps path_def continuous_map_subtopology_eu by blast

lemma homotopic_paths_imp_subset:
     "homotopic_paths S p q ⟹ path_image p ⊆ S ∧ path_image q ⊆ S"
  by (metis (mono_tags) continuous_map_subtopology_eu homotopic_paths_def homotopic_with_imp_continuous_maps path_image_def)

proposition homotopic_paths_refl [simp]: "homotopic_paths S p p ⟷ path p ∧ path_image p ⊆ S"
  by (simp add: homotopic_paths_def path_def path_image_def)

proposition homotopic_paths_sym: "homotopic_paths S p q ⟹ homotopic_paths S q p"
  by (metis (mono_tags) homotopic_paths_def homotopic_paths_imp_pathfinish homotopic_paths_imp_pathstart homotopic_with_symD)

proposition homotopic_paths_sym_eq: "homotopic_paths S p q ⟷ homotopic_paths S q p"
  by (metis homotopic_paths_sym)

proposition homotopic_paths_trans [trans]:
  assumes "homotopic_paths S p q" "homotopic_paths S q r"
  shows "homotopic_paths S p r"
  using assms homotopic_paths_imp_pathfinish homotopic_paths_imp_pathstart unfolding homotopic_paths_def
  by (smt (verit, ccfv_SIG) homotopic_with_mono homotopic_with_trans)

proposition homotopic_paths_eq:
     "⟦path p; path_image p ⊆ S; ⋀t. t ∈ {0..1} ⟹ p t = q t⟧ ⟹ homotopic_paths S p q"
  by (smt (verit, best) homotopic_paths homotopic_paths_refl)

proposition homotopic_paths_reparametrize:
  assumes "path p"
      and pips: "path_image p ⊆ S"
      and contf: "continuous_on {0..1} f"
      and f01 :"f ∈ {0..1} → {0..1}"
      and [simp]: "f(0) = 0" "f(1) = 1"
      and q: "⋀t. t ∈ {0..1} ⟹ q(t) = p(f t)"
    shows "homotopic_paths S p q"
proof -
  have contp: "continuous_on {0..1} p"
    by (metis ‹path p› path_def)
  then have "continuous_on {0..1} (p ∘ f)"
    by (meson assms(4) contf continuous_on_compose continuous_on_subset image_subset_iff_funcset)
  then have "path q"
    by (simp add: path_def) (metis q continuous_on_cong)
  have piqs: "path_image q ⊆ S"
    by (smt (verit, ccfv_threshold) Pi_iff assms(2) assms(4) assms(7) image_subset_iff path_defs(4))
  have fb0: "⋀a b. ⟦0 ≤ a; a ≤ 1; 0 ≤ b; b ≤ 1⟧ ⟹ 0 ≤ (1 - a) * f b + a * b"
    using f01 by force
  have fb1: "⟦0 ≤ a; a ≤ 1; 0 ≤ b; b ≤ 1⟧ ⟹ (1 - a) * f b + a * b ≤ 1" for a b
    by (intro convex_bound_le) (use f01 in auto)
  have "homotopic_paths S q p"
  proof (rule homotopic_paths_trans)
    show "homotopic_paths S q (p ∘ f)"
      using q by (force intro: homotopic_paths_eq [OF  ‹path q› piqs])
  next
    show "homotopic_paths S (p ∘ f) p"
      using pips [unfolded path_image_def]
      apply (simp add: homotopic_paths_def homotopic_with_def)
      apply (rule_tac x="p ∘ (λy. (1 - (fst y)) *⇩R ((f ∘ snd) y) + (fst y) *⇩R snd y)"  in exI)
      apply (rule conjI contf continuous_intros continuous_on_subset [OF contp] | simp)+
      by (auto simp: fb0 fb1 pathstart_def pathfinish_def)
  qed
  then show ?thesis
    by (simp add: homotopic_paths_sym)
qed

lemma homotopic_paths_subset: "⟦homotopic_paths S p q; S ⊆ t⟧ ⟹ homotopic_paths t p q"
  unfolding homotopic_paths by fast


text‹ A slightly ad-hoc but useful lemma in constructing homotopies.›
lemma continuous_on_homotopic_join_lemma:
  fixes q :: "[real,real] ⇒ 'a::topological_space"
  assumes p: "continuous_on ({0..1} × {0..1}) (λy. p (fst y) (snd y))" (is "continuous_on ?A ?p")
      and q: "continuous_on ({0..1} × {0..1}) (λy. q (fst y) (snd y))" (is "continuous_on ?A ?q")
      and pf: "⋀t. t ∈ {0..1} ⟹ pathfinish(p t) = pathstart(q t)"
    shows "continuous_on ({0..1} × {0..1}) (λy. (p(fst y) +++ q(fst y)) (snd y))"
proof -
  have §: "(λt. p (fst t) (2 * snd t)) = ?p ∘ (λy. (fst y, 2 * snd y))"
          "(λt. q (fst t) (2 * snd t - 1)) = ?q ∘ (λy. (fst y, 2 * snd y - 1))"
    by force+
  show ?thesis
    unfolding joinpaths_def
  proof (rule continuous_on_cases_le)
    show "continuous_on {y ∈ ?A. snd y ≤ 1/2} (λt. p (fst t) (2 * snd t))" 
         "continuous_on {y ∈ ?A. 1/2 ≤ snd y} (λt. q (fst t) (2 * snd t - 1))"
         "continuous_on ?A snd"
      unfolding §
      by (rule continuous_intros continuous_on_subset [OF p] continuous_on_subset [OF q] | force)+
  qed (use pf in ‹auto simp: mult.commute pathstart_def pathfinish_def›)
qed

text‹ Congruence properties of homotopy w.r.t. path-combining operations.›

lemma homotopic_paths_reversepath_D:
      assumes "homotopic_paths S p q"
      shows   "homotopic_paths S (reversepath p) (reversepath q)"
  using assms
  apply (simp add: homotopic_paths_def homotopic_with_def, clarify)
  apply (rule_tac x="h ∘ (λx. (fst x, 1 - snd x))" in exI)
  apply (rule conjI continuous_intros)+
  apply (auto simp: reversepath_def pathstart_def pathfinish_def elim!: continuous_on_subset)
  done

proposition homotopic_paths_reversepath:
     "homotopic_paths S (reversepath p) (reversepath q) ⟷ homotopic_paths S p q"
  using homotopic_paths_reversepath_D by force


proposition homotopic_paths_join:
    "⟦homotopic_paths S p p'; homotopic_paths S q q'; pathfinish p = pathstart q⟧ ⟹ homotopic_paths S (p +++ q) (p' +++ q')"
  apply (clarsimp simp: homotopic_paths_def homotopic_with_def)
  apply (rename_tac k1 k2)
  apply (rule_tac x="(λy. ((k1 ∘ Pair (fst y)) +++ (k2 ∘ Pair (fst y))) (snd y))" in exI)
  apply (intro conjI continuous_intros continuous_on_homotopic_join_lemma; force simp: joinpaths_def pathstart_def pathfinish_def path_image_def)
  done

proposition homotopic_paths_continuous_image:
    "⟦homotopic_paths S f g; continuous_on S h; h ∈ S → t⟧ ⟹ homotopic_paths t (h ∘ f) (h ∘ g)"
  unfolding homotopic_paths_def
  by (simp add: homotopic_with_compose_continuous_map_left pathfinish_compose pathstart_compose image_subset_iff_funcset)


subsection‹Group properties for homotopy of paths›

text✐‹tag important›‹So taking equivalence classes under homotopy would give the fundamental group›

proposition homotopic_paths_rid:
  assumes "path p" "path_image p ⊆ S"
  shows "homotopic_paths S (p +++ linepath (pathfinish p) (pathfinish p)) p"
proof -
  have §: "continuous_on {0..1} (λt::real. if t ≤ 1/2 then 2 *⇩R t else 1)"
    unfolding split_01
    by (rule continuous_on_cases continuous_intros | force simp: pathfinish_def joinpaths_def)+
  show ?thesis
    apply (rule homotopic_paths_sym)
    using assms unfolding pathfinish_def joinpaths_def
    by (intro § continuous_on_cases continuous_intros homotopic_paths_reparametrize [where f = "λt. if t ≤ 1/2 then 2 *⇩R t else 1"]; force)
qed

proposition homotopic_paths_lid:
   "⟦path p; path_image p ⊆ S⟧ ⟹ homotopic_paths S (linepath (pathstart p) (pathstart p) +++ p) p"
  using homotopic_paths_rid [of "reversepath p" S]
  by (metis homotopic_paths_reversepath path_image_reversepath path_reversepath pathfinish_linepath
        pathfinish_reversepath reversepath_joinpaths reversepath_linepath)

lemma homotopic_paths_rid':
  assumes "path p" "path_image p ⊆ s" "x = pathfinish p"
  shows "homotopic_paths s (p +++ linepath x x) p"
  using homotopic_paths_rid[of p s] assms by simp

lemma homotopic_paths_lid':
   "⟦path p; path_image p ⊆ s; x = pathstart p⟧ ⟹ homotopic_paths s (linepath x x +++ p) p"
  using homotopic_paths_lid[of p s] by simp

proposition homotopic_paths_assoc:
   "⟦path p; path_image p ⊆ S; path q; path_image q ⊆ S; path r; path_image r ⊆ S; pathfinish p = pathstart q;
     pathfinish q = pathstart r⟧
    ⟹ homotopic_paths S (p +++ (q +++ r)) ((p +++ q) +++ r)"
  apply (subst homotopic_paths_sym)
  apply (rule homotopic_paths_reparametrize
           [where f = "λt. if  t ≤ 1/2 then inverse 2 *⇩R t
                           else if  t ≤ 3 / 4 then t - (1 / 4)
                           else 2 *⇩R t - 1"])
  apply (simp_all del: le_divide_eq_numeral1 add: subset_path_image_join)
  apply (rule continuous_on_cases_1 continuous_intros | auto simp: joinpaths_def)+
  done

proposition homotopic_paths_rinv:
  assumes "path p" "path_image p ⊆ S"
    shows "homotopic_paths S (p +++ reversepath p) (linepath (pathstart p) (pathstart p))"
proof -
  have p: "continuous_on {0..1} p" 
    using assms by (auto simp: path_def)
  let ?A = "{0..1} × {0..1}"
  have "continuous_on ?A (λx. (subpath 0 (fst x) p +++ reversepath (subpath 0 (fst x) p)) (snd x))"
    unfolding joinpaths_def subpath_def reversepath_def path_def add_0_right diff_0_right
  proof (rule continuous_on_cases_le)
    show "continuous_on {x ∈ ?A. snd x ≤ 1/2} (λt. p (fst t * (2 * snd t)))"
         "continuous_on {x ∈ ?A. 1/2 ≤ snd x} (λt. p (fst t * (1 - (2 * snd t - 1))))"
         "continuous_on ?A snd"
      by (intro continuous_on_compose2 [OF p] continuous_intros; auto simp: mult_le_one)+
  qed (auto simp: algebra_simps)
  then show ?thesis
    using assms
    apply (subst homotopic_paths_sym_eq)
    unfolding homotopic_paths_def homotopic_with_def
    apply (rule_tac x="(λy. (subpath 0 (fst y) p +++ reversepath(subpath 0 (fst y) p)) (snd y))" in exI)
    apply (force simp: mult_le_one path_defs joinpaths_def subpath_def reversepath_def)
    done
qed

proposition homotopic_paths_linv:
  assumes "path p" "path_image p ⊆ S"
    shows "homotopic_paths S (reversepath p +++ p) (linepath (pathfinish p) (pathfinish p))"
  using homotopic_paths_rinv [of "reversepath p" S] assms by simp


subsection‹Homotopy of loops without requiring preservation of endpoints›

definition✐‹tag important› homotopic_loops :: "'a::topological_space set ⇒ (real ⇒ 'a) ⇒ (real ⇒ 'a) ⇒ bool"  where
 "homotopic_loops S p q ≡
     homotopic_with_canon (λr. pathfinish r = pathstart r) {0..1} S p q"

lemma homotopic_loops:
   "homotopic_loops S p q ⟷
      (∃h. continuous_on ({0..1::real} × {0..1}) h ∧
          image h ({0..1} × {0..1}) ⊆ S ∧
          (∀x ∈ {0..1}. h(0,x) = p x) ∧
          (∀x ∈ {0..1}. h(1,x) = q x) ∧
          (∀t ∈ {0..1}. pathfinish(h ∘ Pair t) = pathstart(h ∘ Pair t)))"
  by (simp add: homotopic_loops_def pathstart_def pathfinish_def homotopic_with)

proposition homotopic_loops_imp_loop:
     "homotopic_loops S p q ⟹ pathfinish p = pathstart p ∧ pathfinish q = pathstart q"
using homotopic_with_imp_property homotopic_loops_def by blast

proposition homotopic_loops_imp_path:
     "homotopic_loops S p q ⟹ path p ∧ path q"
  unfolding homotopic_loops_def path_def
  using homotopic_with_imp_continuous_maps continuous_map_subtopology_eu by blast

proposition homotopic_loops_imp_subset:
     "homotopic_loops S p q ⟹ path_image p ⊆ S ∧ path_image q ⊆ S"
  unfolding homotopic_loops_def path_image_def
  by (meson continuous_map_subtopology_eu homotopic_with_imp_continuous_maps)

proposition homotopic_loops_refl:
     "homotopic_loops S p p ⟷
      path p ∧ path_image p ⊆ S ∧ pathfinish p = pathstart p"
  by (simp add: homotopic_loops_def path_image_def path_def)

proposition homotopic_loops_sym: "homotopic_loops S p q ⟹ homotopic_loops S q p"
  by (simp add: homotopic_loops_def homotopic_with_sym)

proposition homotopic_loops_sym_eq: "homotopic_loops S p q ⟷ homotopic_loops S q p"
  by (metis homotopic_loops_sym)

proposition homotopic_loops_trans:
   "⟦homotopic_loops S p q; homotopic_loops S q r⟧ ⟹ homotopic_loops S p r"
  unfolding homotopic_loops_def by (blast intro: homotopic_with_trans)

proposition homotopic_loops_subset:
   "⟦homotopic_loops S p q; S ⊆ t⟧ ⟹ homotopic_loops t p q"
  by (fastforce simp: homotopic_loops)

proposition homotopic_loops_eq:
   "⟦path p; path_image p ⊆ S; pathfinish p = pathstart p; ⋀t. t ∈ {0..1} ⟹ p(t) = q(t)⟧
          ⟹ homotopic_loops S p q"
  unfolding homotopic_loops_def path_image_def path_def pathstart_def pathfinish_def
  by (auto intro: homotopic_with_eq [OF homotopic_with_refl [where f = p, THEN iffD2]])

proposition homotopic_loops_continuous_image:
   "⟦homotopic_loops S f g; continuous_on S h; h ∈ S → t⟧ ⟹ homotopic_loops t (h ∘ f) (h ∘ g)"
  unfolding homotopic_loops_def
  by (simp add: homotopic_with_compose_continuous_map_left pathfinish_def pathstart_def image_subset_iff_funcset)


subsection‹Relations between the two variants of homotopy›

proposition homotopic_paths_imp_homotopic_loops:
    "⟦homotopic_paths S p q; pathfinish p = pathstart p; pathfinish q = pathstart p⟧ ⟹ homotopic_loops S p q"
  by (auto simp: homotopic_with_def homotopic_paths_def homotopic_loops_def)

proposition homotopic_loops_imp_homotopic_paths_null:
  assumes "homotopic_loops S p (linepath a a)"
    shows "homotopic_paths S p (linepath (pathstart p) (pathstart p))"
proof -
  have "path p" by (metis assms homotopic_loops_imp_path)
  have ploop: "pathfinish p = pathstart p" by (metis assms homotopic_loops_imp_loop)
  have pip: "path_image p ⊆ S" by (metis assms homotopic_loops_imp_subset)
  let ?A = "{0..1::real} × {0..1::real}"
  obtain h where conth: "continuous_on ?A h"
             and hs: "h ∈ ?A → S"
             and h0[simp]: "⋀x. x ∈ {0..1} ⟹ h(0,x) = p x"
             and h1[simp]: "⋀x. x ∈ {0..1} ⟹ h(1,x) = a"
             and ends: "⋀t. t ∈ {0..1} ⟹ pathfinish (h ∘ Pair t) = pathstart (h ∘ Pair t)"
    using assms by (auto simp: homotopic_loops homotopic_with image_subset_iff_funcset)
  have conth0: "path (λu. h (u, 0))"
    unfolding path_def
  proof (rule continuous_on_compose [of _ _ h, unfolded o_def])
    show "continuous_on ((λx. (x, 0)) ` {0..1}) h"
      by (force intro: continuous_on_subset [OF conth])
  qed (force intro: continuous_intros)
  have pih0: "path_image (λu. h (u, 0)) ⊆ S"
    using hs by (force simp: path_image_def)
  have c1: "continuous_on ?A (λx. h (fst x * snd x, 0))"
  proof (rule continuous_on_compose [of _ _ h, unfolded o_def])
    show "continuous_on ((λx. (fst x * snd x, 0)) ` ?A) h"
      by (force simp: mult_le_one intro: continuous_on_subset [OF conth])
  qed (force intro: continuous_intros)+
  have c2: "continuous_on ?A (λx. h (fst x - fst x * snd x, 0))"
  proof (rule continuous_on_compose [of _ _ h, unfolded o_def])
    show "continuous_on ((λx. (fst x - fst x * snd x, 0)) ` ?A) h"
      by (auto simp: algebra_simps add_increasing2 mult_left_le intro: continuous_on_subset [OF conth])
  qed (force intro: continuous_intros)
  have [simp]: "⋀t. ⟦0 ≤ t ∧ t ≤ 1⟧ ⟹ h (t, 1) = h (t, 0)"
    using ends by (simp add: pathfinish_def pathstart_def)
  have adhoc_le: "c * 4 ≤ 1 + c * (d * 4)" if "¬ d * 4 ≤ 3" "0 ≤ c" "c ≤ 1" for c d::real
  proof -
    have "c * 3 ≤ c * (d * 4)" using that less_eq_real_def by auto
    with ‹c ≤ 1› show ?thesis by fastforce
  qed
  have *: "⋀p x. ⟦path p ∧ path(reversepath p);
                  path_image p ⊆ S ∧ path_image(reversepath p) ⊆ S;
                  pathfinish p = pathstart(linepath a a +++ reversepath p) ∧
                   pathstart(reversepath p) = a ∧ pathstart p = x⟧
                  ⟹ homotopic_paths S (p +++ linepath a a +++ reversepath p) (linepath x x)"
    by (metis homotopic_paths_lid homotopic_paths_join
              homotopic_paths_trans homotopic_paths_sym homotopic_paths_rinv)
  have 1: "homotopic_paths S p (p +++ linepath (pathfinish p) (pathfinish p))"
    using ‹path p› homotopic_paths_rid homotopic_paths_sym pip by blast
  moreover have "homotopic_paths S (p +++ linepath (pathfinish p) (pathfinish p))
                                   (linepath (pathstart p) (pathstart p) +++ p +++ linepath (pathfinish p) (pathfinish p))"
    using homotopic_paths_lid [of "p +++ linepath (pathfinish p) (pathfinish p)" S]
    by (metis 1 homotopic_paths_imp_path homotopic_paths_imp_subset homotopic_paths_sym pathstart_join)
  moreover 
  have "homotopic_paths S (linepath (pathstart p) (pathstart p) +++ p +++ linepath (pathfinish p) (pathfinish p))
                                   ((λu. h (u, 0)) +++ linepath a a +++ reversepath (λu. h (u, 0)))"
    unfolding homotopic_paths_def homotopic_with_def
  proof (intro exI strip conjI)
    let ?h = "λy. (subpath 0 (fst y) (λu. h (u, 0)) +++ (λu. h (Pair (fst y) u)) 
               +++ subpath (fst y) 0 (λu. h (u, 0))) (snd y)" 
    have "continuous_on ?A ?h"
      by (intro continuous_on_homotopic_join_lemma; simp add: path_defs joinpaths_def subpath_def conth c1 c2)
    moreover have "?h ∈ ?A → S"
      using hs
      unfolding joinpaths_def subpath_def
      by (force simp: algebra_simps mult_le_one mult_left_le  intro: adhoc_le)
  ultimately show "continuous_map (prod_topology (top_of_set {0..1}) (top_of_set {0..1}))
                         (top_of_set S) ?h"
    by (simp add: subpath_reversepath image_subset_iff_funcset)
  qed (use ploop in ‹simp_all add: reversepath_def path_defs joinpaths_def o_def subpath_def conth c1 c2›)
  moreover have "homotopic_paths S ((λu. h (u, 0)) +++ linepath a a +++ reversepath (λu. h (u, 0)))
                                   (linepath (pathstart p) (pathstart p))"
    by (rule *; simp add: pih0 pathstart_def pathfinish_def conth0; simp add: reversepath_def joinpaths_def)
  ultimately show ?thesis
    by (blast intro: homotopic_paths_trans)
qed

proposition homotopic_loops_conjugate:
  fixes S :: "'a::real_normed_vector set"
  assumes "path p" "path q" and pip: "path_image p ⊆ S" and piq: "path_image q ⊆ S"
      and pq: "pathfinish p = pathstart q" and qloop: "pathfinish q = pathstart q"
    shows "homotopic_loops S (p +++ q +++ reversepath p) q"
proof -
  have contp: "continuous_on {0..1} p"  using ‹path p› [unfolded path_def] by blast
  have contq: "continuous_on {0..1} q"  using ‹path q› [unfolded path_def] by blast
  let ?A = "{0..1::real} × {0..1::real}"
  have c1: "continuous_on ?A (λx. p ((1 - fst x) * snd x + fst x))"
  proof (rule continuous_on_compose [of _ _ p, unfolded o_def])
    show "continuous_on ((λx. (1 - fst x) * snd x + fst x) ` ?A) p"
      by (auto intro: continuous_on_subset [OF contp] simp: algebra_simps add_increasing2 mult_right_le_one_le sum_le_prod1)
  qed (force intro: continuous_intros)
  have c2: "continuous_on ?A (λx. p ((fst x - 1) * snd x + 1))"
  proof (rule continuous_on_compose [of _ _ p, unfolded o_def])
    show "continuous_on ((λx. (fst x - 1) * snd x + 1) ` ?A) p"
      by (auto intro: continuous_on_subset [OF contp] simp: algebra_simps add_increasing2 mult_left_le_one_le)
  qed (force intro: continuous_intros)

  have ps1: "⋀a b. ⟦b * 2 ≤ 1; 0 ≤ b; 0 ≤ a; a ≤ 1⟧ ⟹ p ((1 - a) * (2 * b) + a) ∈ S"
    using sum_le_prod1
    by (force simp: algebra_simps add_increasing2 mult_left_le intro: pip [unfolded path_image_def, THEN subsetD])
  have ps2: "⋀a b. ⟦¬ 4 * b ≤ 3; b ≤ 1; 0 ≤ a; a ≤ 1⟧ ⟹ p ((a - 1) * (4 * b - 3) + 1) ∈ S"
    apply (rule pip [unfolded path_image_def, THEN subsetD])
    apply (rule image_eqI, blast)
    apply (simp add: algebra_simps)
    by (metis add_mono affine_ineq linear mult.commute mult.left_neutral mult_right_mono
              add.commute zero_le_numeral)
  have qs: "⋀a b. ⟦4 * b ≤ 3; ¬ b * 2 ≤ 1⟧ ⟹ q (4 * b - 2) ∈ S"
    using path_image_def piq by fastforce
  have "homotopic_loops S (p +++ q +++ reversepath p)
                          (linepath (pathstart q) (pathstart q) +++ q +++ linepath (pathstart q) (pathstart q))"
    unfolding homotopic_loops_def homotopic_with_def
  proof (intro exI strip conjI)
    let ?h = "(λy. (subpath (fst y) 1 p +++ q +++ subpath 1 (fst y) p) (snd y))" 
    have "continuous_on ?A (λy. q (snd y))"
      by (force simp: contq intro: continuous_on_compose [of _ _ q, unfolded o_def] continuous_on_id continuous_on_snd)
    then have "continuous_on ?A ?h"
      using pq qloop
      by (intro continuous_on_homotopic_join_lemma) (auto simp: path_defs joinpaths_def subpath_def c1 c2)
    then show "continuous_map (prod_topology (top_of_set {0..1}) (top_of_set {0..1})) (top_of_set S) ?h"
      by (auto simp: joinpaths_def subpath_def  ps1 ps2 qs)
    show "?h (1,x) = (linepath (pathstart q) (pathstart q) +++ q +++ linepath (pathstart q) (pathstart q)) x"  for x
      using pq by (simp add: pathfinish_def subpath_refl)
  qed (auto simp: subpath_reversepath)
  moreover have "homotopic_loops S (linepath (pathstart q) (pathstart q) +++ q +++ linepath (pathstart q) (pathstart q)) q"
  proof -
    have "homotopic_paths S (linepath (pathfinish q) (pathfinish q) +++ q) q"
      using ‹path q› homotopic_paths_lid qloop piq by auto
    hence 1: "⋀f. homotopic_paths S f q ∨ ¬ homotopic_paths S f (linepath (pathfinish q) (pathfinish q) +++ q)"
      using homotopic_paths_trans by blast
    hence "homotopic_paths S (linepath (pathfinish q) (pathfinish q) +++ q +++ linepath (pathfinish q) (pathfinish q)) q"
      by (smt (verit, best) ‹path q› homotopic_paths_imp_path homotopic_paths_imp_subset homotopic_paths_lid 
          homotopic_paths_rid homotopic_paths_trans pathstart_join piq qloop)
    thus ?thesis
      by (metis (no_types) qloop homotopic_loops_sym homotopic_paths_imp_homotopic_loops homotopic_paths_imp_pathfinish homotopic_paths_sym)
  qed
  ultimately show ?thesis
    by (blast intro: homotopic_loops_trans)
qed

lemma homotopic_paths_loop_parts:
  assumes loops: "homotopic_loops S (p +++ reversepath q) (linepath a a)" and "path q"
  shows "homotopic_paths S p q"
proof -
  have paths: "homotopic_paths S (p +++ reversepath q) (linepath (pathstart p) (pathstart p))"
    using homotopic_loops_imp_homotopic_paths_null [OF loops] by simp
  then have "path p"
    using ‹path q› homotopic_loops_imp_path loops path_join path_join_path_ends path_reversepath by blast
  show ?thesis
  proof (cases "pathfinish p = pathfinish q")
    case True
    obtain pipq: "path_image p ⊆ S" "path_image q ⊆ S"
      by (metis Un_subset_iff paths ‹path p› ‹path q› homotopic_loops_imp_subset homotopic_paths_imp_path loops
           path_image_join path_image_reversepath path_imp_reversepath path_join_eq)
    have "homotopic_paths S p (p +++ (linepath (pathfinish p) (pathfinish p)))"
      using ‹path p› ‹path_image p ⊆ S› homotopic_paths_rid homotopic_paths_sym by blast
    moreover have "homotopic_paths S (p +++ (linepath (pathfinish p) (pathfinish p))) (p +++ (reversepath q +++ q))"
      by (simp add: True ‹path p› ‹path q› pipq homotopic_paths_join homotopic_paths_linv homotopic_paths_sym)
    moreover have "homotopic_paths S (p +++ (reversepath q +++ q)) ((p +++ reversepath q) +++ q)"
      by (simp add: True ‹path p› ‹path q› homotopic_paths_assoc pipq)
    moreover have "homotopic_paths S ((p +++ reversepath q) +++ q) (linepath (pathstart p) (pathstart p) +++ q)"
      by (simp add: ‹path q› homotopic_paths_join paths pipq)
    ultimately show ?thesis
      by (metis ‹path q› homotopic_paths_imp_path homotopic_paths_lid homotopic_paths_trans path_join_path_ends pathfinish_linepath pipq(2))
  next
    case False
    then show ?thesis
      using ‹path q› homotopic_loops_imp_path loops path_join_path_ends by fastforce
  qed
qed


subsection✐‹tag unimportant›‹Homotopy of "nearby" function, paths and loops›

lemma homotopic_with_linear:
  fixes f g :: "_ ⇒ 'b::real_normed_vector"
  assumes contf: "continuous_on S f"
      and contg:"continuous_on S g"
      and sub: "⋀x. x ∈ S ⟹ closed_segment (f x) (g x) ⊆ t"
    shows "homotopic_with_canon (λz. True) S t f g"
  unfolding homotopic_with_def
  apply (rule_tac x="λy. ((1 - (fst y)) *⇩R f(snd y) + (fst y) *⇩R g(snd y))" in exI)
  using sub closed_segment_def
     by (fastforce intro: continuous_intros continuous_on_subset [OF contf] continuous_on_compose2 [where g=f]
            continuous_on_subset [OF contg] continuous_on_compose2 [where g=g])

lemma homotopic_paths_linear:
  fixes g h :: "real ⇒ 'a::real_normed_vector"
  assumes "path g" "path h" "pathstart h = pathstart g" "pathfinish h = pathfinish g"
          "⋀t. t ∈ {0..1} ⟹ closed_segment (g t) (h t) ⊆ S"
    shows "homotopic_paths S g h"
  using assms
  unfolding path_def
  apply (simp add: closed_segment_def pathstart_def pathfinish_def homotopic_paths_def homotopic_with_def)
  apply (rule_tac x="λy. ((1 - (fst y)) *⇩R (g ∘ snd) y + (fst y) *⇩R (h ∘ snd) y)" in exI)
  apply (intro conjI subsetI continuous_intros; force)
  done

lemma homotopic_loops_linear:
  fixes g h :: "real ⇒ 'a::real_normed_vector"
  assumes "path g" "path h" "pathfinish g = pathstart g" "pathfinish h = pathstart h"
          "⋀t x. t ∈ {0..1} ⟹ closed_segment (g t) (h t) ⊆ S"
    shows "homotopic_loops S g h"
  using assms
  unfolding path_defs homotopic_loops_def homotopic_with_def
  apply (rule_tac x="λy. ((1 - (fst y)) *⇩R g(snd y) + (fst y) *⇩R h(snd y))" in exI)
  by (force simp: closed_segment_def intro!: continuous_intros intro: continuous_on_compose2 [where g=g] continuous_on_compose2 [where g=h])

lemma homotopic_paths_nearby_explicit:
  assumes §: "path g" "path h" "pathstart h = pathstart g" "pathfinish h = pathfinish g"
      and no: "⋀t x. ⟦t ∈ {0..1}; x ∉ S⟧ ⟹ norm(h t - g t) < norm(g t - x)"
    shows "homotopic_paths S g h"
  using homotopic_paths_linear [OF §] by (metis linorder_not_le no norm_minus_commute segment_bound1 subsetI)

lemma homotopic_loops_nearby_explicit:
  assumes §: "path g" "path h" "pathfinish g = pathstart g" "pathfinish h = pathstart h"
      and no: "⋀t x. ⟦t ∈ {0..1}; x ∉ S⟧ ⟹ norm(h t - g t) < norm(g t - x)"
    shows "homotopic_loops S g h"
  using homotopic_loops_linear [OF §] by (metis linorder_not_le no norm_minus_commute segment_bound1 subsetI)

lemma homotopic_nearby_paths:
  fixes g h :: "real ⇒ 'a::euclidean_space"
  assumes "path g" "open S" "path_image g ⊆ S"
    shows "∃e. 0 < e ∧
               (∀h. path h ∧
                    pathstart h = pathstart g ∧ pathfinish h = pathfinish g ∧
                    (∀t ∈ {0..1}. norm(h t - g t) < e) ⟶ homotopic_paths S g h)"
proof -
  obtain e where "e > 0" and e: "⋀x y. x ∈ path_image g ⟹ y ∈ - S ⟹ e ≤ dist x y"
    using separate_compact_closed [of "path_image g" "-S"] assms by force
  show ?thesis
    using e [unfolded dist_norm] ‹e > 0›
    by (fastforce simp: path_image_def intro!: homotopic_paths_nearby_explicit assms exI)
qed

lemma homotopic_nearby_loops:
  fixes g h :: "real ⇒ 'a::euclidean_space"
  assumes "path g" "open S" "path_image g ⊆ S" "pathfinish g = pathstart g"
    shows "∃e. 0 < e ∧
               (∀h. path h ∧ pathfinish h = pathstart h ∧
                    (∀t ∈ {0..1}. norm(h t - g t) < e) ⟶ homotopic_loops S g h)"
proof -
  obtain e where "e > 0" and e: "⋀x y. x ∈ path_image g ⟹ y ∈ - S ⟹ e ≤ dist x y"
    using separate_compact_closed [of "path_image g" "-S"] assms by force
  show ?thesis
    using e [unfolded dist_norm] ‹e > 0›
    by (fastforce simp: path_image_def intro!: homotopic_loops_nearby_explicit assms exI)
qed


subsection‹ Homotopy and subpaths›

lemma homotopic_join_subpaths1:
  assumes "path g" and pag: "path_image g ⊆ S"
      and u: "u ∈ {0..1}" and v: "v ∈ {0..1}" and w: "w ∈ {0..1}" "u ≤ v" "v ≤ w"
    shows "homotopic_paths S (subpath u v g +++ subpath v w g) (subpath u w g)"
proof -
  have 1: "t * 2 ≤ 1 ⟹ u + t * (v * 2) ≤ v + t * (u * 2)" for t
    using affine_ineq ‹u ≤ v› by fastforce
  have 2: "t * 2 > 1 ⟹ u + (2*t - 1) * v ≤ v + (2*t - 1) * w" for t
    by (metis add_mono_thms_linordered_semiring(1) diff_gt_0_iff_gt less_eq_real_def mult.commute mult_right_mono ‹u ≤ v› ‹v ≤ w›)
  have t2: "⋀t::real. t*2 = 1 ⟹ t = 1/2" by auto
  have "homotopic_paths (path_image g) (subpath u v g +++ subpath v w g) (subpath u w g)"
  proof (cases "w = u")
    case True
    then show ?thesis
      by (metis ‹path g› homotopic_paths_rinv path_image_subpath_subset path_subpath pathstart_subpath reversepath_subpath subpath_refl u v)
  next
    case False
    let ?f = "λt. if  t ≤ 1/2 then inverse((w - u)) *⇩R (2 * (v - u)) *⇩R t
                               else inverse((w - u)) *⇩R ((v - u) + (w - v) *⇩R (2 *⇩R t - 1))"
    show ?thesis
    proof (rule homotopic_paths_sym [OF homotopic_paths_reparametrize [where f = ?f]])
      show "path (subpath u w g)"
        using assms(1) path_subpath u w(1) by blast
      show "path_image (subpath u w g) ⊆ path_image g"
        by (meson path_image_subpath_subset u w(1))
      show "continuous_on {0..1} ?f"
        unfolding split_01
        by (rule continuous_on_cases continuous_intros | force simp: pathfinish_def joinpaths_def dest!: t2)+
      show "?f ∈ {0..1} → {0..1}"
        using False assms
        by (force simp: field_simps not_le mult_left_mono affine_ineq dest!: 1 2)
      show "(subpath u v g +++ subpath v w g) t = subpath u w g (?f t)" if "t ∈ {0..1}" for t 
        using assms
        unfolding joinpaths_def subpath_def by (auto simp: divide_simps add.commute mult.commute mult.left_commute)
    qed (use False in auto)
  qed
  then show ?thesis
    by (rule homotopic_paths_subset [OF _ pag])
qed

lemma homotopic_join_subpaths2:
  assumes "homotopic_paths S (subpath u v g +++ subpath v w g) (subpath u w g)"
  shows "homotopic_paths S (subpath w v g +++ subpath v u g) (subpath w u g)"
  by (metis assms homotopic_paths_reversepath_D pathfinish_subpath pathstart_subpath reversepath_joinpaths reversepath_subpath)

lemma homotopic_join_subpaths3:
  assumes hom: "homotopic_paths S (subpath u v g +++ subpath v w g) (subpath u w g)"
      and "path g" and pag: "path_image g ⊆ S"
      and u: "u ∈ {0..1}" and v: "v ∈ {0..1}" and w: "w ∈ {0..1}"
    shows "homotopic_paths S (subpath v w g +++ subpath w u g) (subpath v u g)"
proof -
  obtain wvg: "path (subpath w v g)" "path_image (subpath w v g) ⊆ S" 
     and wug: "path (subpath w u g)" "path_image (subpath w u g) ⊆ S"
     and vug: "path (subpath v u g)" "path_image (subpath v u g) ⊆ S"
    by (meson ‹path g› pag path_image_subpath_subset path_subpath subset_trans u v w)
  have "homotopic_paths S (subpath u w g +++ subpath w v g) 
                          ((subpath u v g +++ subpath v w g) +++ subpath w v g)"
    by (simp add: hom homotopic_paths_join homotopic_paths_sym wvg)
  also have "homotopic_paths S …  (subpath u v g +++ subpath v w g +++ subpath w v g)"
    using wvg vug ‹path g›
    by (metis homotopic_paths_assoc homotopic_paths_sym path_image_subpath_commute path_subpath
        pathfinish_subpath pathstart_subpath u v w)
  also have "homotopic_paths S … (subpath u v g +++ linepath (pathfinish (subpath u v g)) (pathfinish (subpath u v g)))"
    using wvg vug ‹path g›
    by (metis homotopic_paths_join homotopic_paths_linv homotopic_paths_refl path_image_subpath_commute
        path_subpath pathfinish_subpath pathstart_join pathstart_subpath reversepath_subpath u v)
  also have "homotopic_paths S … (subpath u v g)"
    using vug ‹path g› by (metis homotopic_paths_rid path_image_subpath_commute path_subpath u v)
  finally have "homotopic_paths S (subpath u w g +++ subpath w v g) (subpath u v g)" .
  then show ?thesis
    using homotopic_join_subpaths2 by blast
qed

proposition homotopic_join_subpaths:
   "⟦path g; path_image g ⊆ S; u ∈ {0..1}; v ∈ {0..1}; w ∈ {0..1}⟧
    ⟹ homotopic_paths S (subpath u v g +++ subpath v w g) (subpath u w g)"
  by (smt (verit, del_insts) homotopic_join_subpaths1 homotopic_join_subpaths2 homotopic_join_subpaths3)

text‹Relating homotopy of trivial loops to path-connectedness.›

lemma path_component_imp_homotopic_points:
  assumes "path_component S a b"
  shows "homotopic_loops S (linepath a a) (linepath b b)"
proof -
  obtain g :: "real ⇒ 'a" where g: "continuous_on {0..1} g" "g ∈ {0..1} → S" "g 0 = a" "g 1 = b"
    using assms by (auto simp: path_defs)
  then have "continuous_on ({0..1} × {0..1}) (g ∘ fst)"
    by (fastforce intro!: continuous_intros)+
  with g show ?thesis
    by (auto simp: homotopic_loops_def homotopic_with_def path_defs Pi_iff)
qed

lemma homotopic_loops_imp_path_component_value:
  "⟦homotopic_loops S p q; 0 ≤ t; t ≤ 1⟧ ⟹ path_component S (p t) (q t)"
  apply (clarsimp simp: homotopic_loops_def homotopic_with_def path_defs)
  apply (rule_tac x="h ∘ (λu. (u, t))" in exI)
  apply (fastforce elim!: continuous_on_subset intro!: continuous_intros)
  done

lemma homotopic_points_eq_path_component:
   "homotopic_loops S (linepath a a) (linepath b b) ⟷ path_component S a b"
  using homotopic_loops_imp_path_component_value path_component_imp_homotopic_points by fastforce

lemma path_connected_eq_homotopic_points:
  "path_connected S ⟷
      (∀a b. a ∈ S ∧ b ∈ S ⟶ homotopic_loops S (linepath a a) (linepath b b))"
  by (auto simp: path_connected_def path_component_def homotopic_points_eq_path_component)


subsection‹Simply connected sets›

text✐‹tag important›‹defined as "all loops are homotopic (as loops)›

definition✐‹tag important› simply_connected where
  "simply_connected S ≡
        ∀p q. path p ∧ pathfinish p = pathstart p ∧ path_image p ⊆ S ∧
              path q ∧ pathfinish q = pathstart q ∧ path_image q ⊆ S
              ⟶ homotopic_loops S p q"

lemma simply_connected_empty [iff]: "simply_connected {}"
  by (simp add: simply_connected_def)

lemma simply_connected_imp_path_connected:
  fixes S :: "_::real_normed_vector set"
  shows "simply_connected S ⟹ path_connected S"
  by (simp add: simply_connected_def path_connected_eq_homotopic_points)

lemma simply_connected_imp_connected:
  fixes S :: "_::real_normed_vector set"
  shows "simply_connected S ⟹ connected S"
  by (simp add: path_connected_imp_connected simply_connected_imp_path_connected)

lemma simply_connected_eq_contractible_loop_any:
  fixes S :: "_::real_normed_vector set"
  shows "simply_connected S ⟷
            (∀p a. path p ∧ path_image p ⊆ S ∧ pathfinish p = pathstart p ∧ a ∈ S
                  ⟶ homotopic_loops S p (linepath a a))"
        (is "?lhs = ?rhs")
proof
  assume ?rhs then show ?lhs
    unfolding simply_connected_def
    by (metis pathfinish_in_path_image subsetD  homotopic_loops_trans homotopic_loops_sym)
qed (force simp: simply_connected_def)

lemma simply_connected_eq_contractible_loop_some:
  fixes S :: "_::real_normed_vector set"
  shows "simply_connected S ⟷
                path_connected S ∧
                (∀p. path p ∧ path_image p ⊆ S ∧ pathfinish p = pathstart p
                    ⟶ (∃a. a ∈ S ∧ homotopic_loops S p (linepath a a)))"
     (is "?lhs = ?rhs")
proof
  assume ?lhs
  then show ?rhs
  using simply_connected_eq_contractible_loop_any by (blast intro: simply_connected_imp_path_connected)
next
  assume ?rhs
  then show ?lhs
    by (meson homotopic_loops_trans path_connected_eq_homotopic_points simply_connected_eq_contractible_loop_any)
qed

lemma simply_connected_eq_contractible_loop_all:
  fixes S :: "_::real_normed_vector set"
  shows "simply_connected S ⟷
         S = {} ∨
         (∃a ∈ S. ∀p. path p ∧ path_image p ⊆ S ∧ pathfinish p = pathstart p
                ⟶ homotopic_loops S p (linepath a a))"
  by (meson ex_in_conv homotopic_loops_sym homotopic_loops_trans simply_connected_def simply_connected_eq_contractible_loop_any)

lemma simply_connected_eq_contractible_path:
  fixes S :: "_::real_normed_vector set"
  shows "simply_connected S ⟷
           path_connected S ∧
           (∀p. path p ∧ path_image p ⊆ S ∧ pathfinish p = pathstart p
            ⟶ homotopic_paths S p (linepath (pathstart p) (pathstart p)))"
     (is "?lhs = ?rhs")
proof
  assume ?lhs
  then show ?rhs
    unfolding simply_connected_imp_path_connected
    by (metis simply_connected_eq_contractible_loop_some homotopic_loops_imp_homotopic_paths_null)
next
  assume  ?rhs
  then show ?lhs
    using homotopic_paths_imp_homotopic_loops simply_connected_eq_contractible_loop_some by fastforce
qed

lemma simply_connected_eq_homotopic_paths:
  fixes S :: "_::real_normed_vector set"
  shows "simply_connected S ⟷
          path_connected S ∧
          (∀p q. path p ∧ path_image p ⊆ S ∧
                path q ∧ path_image q ⊆ S ∧
                pathstart q = pathstart p ∧ pathfinish q = pathfinish p
                ⟶ homotopic_paths S p q)"
         (is "?lhs = ?rhs")
proof
  assume ?lhs
  then have pc: "path_connected S"
        and *:  "⋀p. ⟦path p; path_image p ⊆ S;
                       pathfinish p = pathstart p⟧
                      ⟹ homotopic_paths S p (linepath (pathstart p) (pathstart p))"
    by (auto simp: simply_connected_eq_contractible_path)
  have "homotopic_paths S p q"
        if "path p" "path_image p ⊆ S" "path q"
           "path_image q ⊆ S" "pathstart q = pathstart p"
           "pathfinish q = pathfinish p" for p q
  proof -
    have "homotopic_paths S p (p +++ reversepath q +++ q)"
      using that
      by (smt (verit, best) homotopic_paths_join homotopic_paths_linv homotopic_paths_rid homotopic_paths_sym 
          homotopic_paths_trans pathstart_linepath)
    also have "homotopic_paths S … ((p +++ reversepath q) +++ q)"
      by (simp add: that homotopic_paths_assoc)
    also have "homotopic_paths S … (linepath (pathstart q) (pathstart q) +++ q)"
      using * [of "p +++ reversepath q"] that
      by (simp add: homotopic_paths_assoc homotopic_paths_join path_image_join)
    also have "homotopic_paths S … q"
      using that homotopic_paths_lid by blast
    finally show ?thesis .
  qed
  then show ?rhs
    by (blast intro: pc *)
next
  assume ?rhs
  then show ?lhs
    by (force simp: simply_connected_eq_contractible_path)
qed

proposition simply_connected_Times:
  fixes S :: "'a::real_normed_vector set" and T :: "'b::real_normed_vector set"
  assumes S: "simply_connected S" and T: "simply_connected T"
    shows "simply_connected(S × T)"
proof -
  have "homotopic_loops (S × T) p (linepath (a, b) (a, b))"
       if "path p" "path_image p ⊆ S × T" "p 1 = p 0" "a ∈ S" "b ∈ T"
       for p a b
  proof -
    have "path (fst ∘ p)"
      by (simp add: continuous_on_fst Path_Connected.path_continuous_image [OF ‹path p›])
    moreover have "path_image (fst ∘ p) ⊆ S"
      using that by (force simp: path_image_def)
    ultimately have p1: "homotopic_loops S (fst ∘ p) (linepath a a)"
      using S that
      by (simp add: simply_connected_eq_contractible_loop_any pathfinish_def pathstart_def)
    have "path (snd ∘ p)"
      by (simp add: continuous_on_snd Path_Connected.path_continuous_image [OF ‹path p›])
    moreover have "path_image (snd ∘ p) ⊆ T"
      using that by (force simp: path_image_def)
    ultimately have p2: "homotopic_loops T (snd ∘ p) (linepath b b)"
      using T that
      by (simp add: simply_connected_eq_contractible_loop_any pathfinish_def pathstart_def)
    show ?thesis
      using p1 p2 unfolding homotopic_loops
      apply clarify
      subgoal for h k
        by (rule_tac x="λz. (h z, k z)" in exI) (force intro: continuous_intros simp: path_defs)
      done
  qed
  with assms show ?thesis
    by (simp add: simply_connected_eq_contractible_loop_any pathfinish_def pathstart_def)
qed


subsection‹Contractible sets›

definition✐‹tag important› contractible where
 "contractible S ≡ ∃a. homotopic_with_canon (λx. True) S S id (λx. a)"

proposition contractible_imp_simply_connected:
  fixes S :: "_::real_normed_vector set"
  assumes "contractible S" shows "simply_connected S"
proof (cases "S = {}")
  case True then show ?thesis by force
next
  case False
  obtain a where a: "homotopic_with_canon (λx. True) S S id (λx. a)"
    using assms by (force simp: contractible_def)
  then have "a ∈ S"
    using False homotopic_with_imp_funspace2 by fastforce
  have "∀p. path p ∧
            path_image p ⊆ S ∧ pathfinish p = pathstart p ⟶
            homotopic_loops S p (linepath a a)"
    using a apply (clarsimp simp: homotopic_loops_def homotopic_with_def path_defs)
    apply (rule_tac x="(h ∘ (λy. (fst y, (p ∘ snd) y)))" in exI)
    apply (intro conjI continuous_on_compose continuous_intros; force elim: continuous_on_subset)
    done
  with ‹a ∈ S› show ?thesis
    by (auto simp: simply_connected_eq_contractible_loop_all False)
qed

corollary contractible_imp_connected:
  fixes S :: "_::real_normed_vector set"
  shows "contractible S ⟹ connected S"
  by (simp add: contractible_imp_simply_connected simply_connected_imp_connected)

lemma contractible_imp_path_connected:
  fixes S :: "_::real_normed_vector set"
  shows "contractible S ⟹ path_connected S"
  by (simp add: contractible_imp_simply_connected simply_connected_imp_path_connected)

lemma nullhomotopic_through_contractible:
  fixes S :: "_::topological_space set"
  assumes f: "continuous_on S f" "f ∈ S → T"
      and g: "continuous_on T g" "g ∈ T → U"
      and T: "contractible T"
    obtains c where "homotopic_with_canon (λh. True) S U (g ∘ f) (λx. c)"
proof -
  obtain b where b: "homotopic_with_canon (λx. True) T T id (λx. b)"
    using assms by (force simp: contractible_def)
  have "homotopic_with_canon (λf. True) T U (g ∘ id) (g ∘ (λx. b))"
    by (metis b continuous_map_subtopology_eu g homotopic_with_compose_continuous_map_left image_subset_iff_funcset)
  then have "homotopic_with_canon (λf. True) S U (g ∘ id ∘ f) (g ∘ (λx. b) ∘ f)"
    by (simp add: f homotopic_with_compose_continuous_map_right image_subset_iff_funcset)
  then show ?thesis
    by (simp add: comp_def that)
qed

lemma nullhomotopic_into_contractible:
  assumes f: "continuous_on S f" "f ∈ S → T"
      and T: "contractible T"
    obtains c where "homotopic_with_canon (λh. True) S T f (λx. c)"
  by (rule nullhomotopic_through_contractible [OF f, of id T]) (use assms in auto)

lemma nullhomotopic_from_contractible:
  assumes f: "continuous_on S f" "f ∈ S → T"
      and S: "contractible S"
    obtains c where "homotopic_with_canon (λh. True) S T f (λx. c)"
  by (auto simp: comp_def intro: nullhomotopic_through_contractible [OF continuous_on_id _ f S])

lemma homotopic_through_contractible:
  fixes S :: "_::real_normed_vector set"
  assumes "continuous_on S f1" "f1 ∈ S → T"
          "continuous_on T g1" "g1 ∈ T → U"
          "continuous_on S f2" "f2 ∈ S → T"
          "continuous_on T g2" "g2 ∈ T → U"
          "contractible T" "path_connected U"
   shows "homotopic_with_canon (λh. True) S U (g1 ∘ f1) (g2 ∘ f2)"
proof -
  obtain c1 where c1: "homotopic_with_canon (λh. True) S U (g1 ∘ f1) (λx. c1)"
    by (rule nullhomotopic_through_contractible [of S f1 T g1 U]) (use assms in auto)
  obtain c2 where c2: "homotopic_with_canon (λh. True) S U (g2 ∘ f2) (λx. c2)"
    by (rule nullhomotopic_through_contractible [of S f2 T g2 U]) (use assms in auto)
  have "S = {} ∨ (∃t. path_connected t ∧ t ⊆ U ∧ c2 ∈ t ∧ c1 ∈ t)"
  proof (cases "S = {}")
    case True then show ?thesis by force
  next
    case False
    with c1 c2 have "c1 ∈ U" "c2 ∈ U"
      using homotopic_with_imp_continuous_maps
       by (metis PiE equals0I homotopic_with_imp_funspace2)+
    with ‹path_connected U› show ?thesis by blast
  qed
  then have "homotopic_with_canon (λh. True) S U (λx. c2) (λx. c1)"
    by (auto simp: path_component homotopic_constant_maps)
  then show ?thesis
    using c1 c2 homotopic_with_symD homotopic_with_trans by blast
qed

lemma homotopic_into_contractible:
  fixes S :: "'a::real_normed_vector set" and T:: "'b::real_normed_vector set"
  assumes f: "continuous_on S f" "f ∈ S → T"
    and g: "continuous_on S g" "g ∈ S → T"
    and T: "contractible T"
  shows "homotopic_with_canon (λh. True) S T f g"
  using homotopic_through_contractible [of S f T id T g id]
  by (simp add: assms contractible_imp_path_connected)

lemma homotopic_from_contractible:
  fixes S :: "'a::real_normed_vector set" and T:: "'b::real_normed_vector set"
  assumes f: "continuous_on S f" "f ∈ S → T"
    and g: "continuous_on S g" "g ∈ S → T"
    and "contractible S" "path_connected T"
  shows "homotopic_with_canon (λh. True) S T f g"
  using homotopic_through_contractible [of S id S f T id g]
  by (simp add: assms contractible_imp_path_connected)

subsection‹Starlike sets›

definition✐‹tag important› "starlike S ⟷ (∃a∈S. ∀x∈S. closed_segment a x ⊆ S)"

lemma starlike_UNIV [simp]: "starlike UNIV"
  by (simp add: starlike_def)

lemma convex_imp_starlike:
  "convex S ⟹ S ≠ {} ⟹ starlike S"
  unfolding convex_contains_segment starlike_def by auto

lemma starlike_convex_tweak_boundary_points:
  fixes S :: "'a::euclidean_space set"
  assumes "convex S" "S ≠ {}" and ST: "rel_interior S ⊆ T" and TS: "T ⊆ closure S"
  shows "starlike T"
proof -
  have "rel_interior S ≠ {}"
    by (simp add: assms rel_interior_eq_empty)
  with ST obtain a where a: "a ∈ rel_interior S" and "a ∈ T" by blast
  have "⋀x. x ∈ T ⟹ open_segment a x ⊆ rel_interior S"
    by (rule rel_interior_closure_convex_segment [OF ‹convex S› a]) (use assms in auto)
  then have "∀x∈T. a ∈ T ∧ open_segment a x ⊆ T"
    using ST by (blast intro: a ‹a ∈ T› rel_interior_closure_convex_segment [OF ‹convex S› a])
  then show ?thesis
    unfolding starlike_def using bexI [OF _ ‹a ∈ T›]
    by (simp add: closed_segment_eq_open)
qed

lemma starlike_imp_contractible_gen:
  fixes S :: "'a::real_normed_vector set"
  assumes S: "starlike S"
      and P: "⋀a T. ⟦a ∈ S; 0 ≤ T; T ≤ 1⟧ ⟹ P(λx. (1 - T) *⇩R x + T *⇩R a)"
    obtains a where "homotopic_with_canon P S S (λx. x) (λx. a)"
proof -
  obtain a where "a ∈ S" and a: "⋀x. x ∈ S ⟹ closed_segment a x ⊆ S"
    using S by (auto simp: starlike_def)
  have "⋀t b. 0 ≤ t ∧ t ≤ 1 ⟹
              ∃u. (1 - t) *⇩R b + t *⇩R a = (1 - u) *⇩R a + u *⇩R b ∧ 0 ≤ u ∧ u ≤ 1"
    by (metis add_diff_cancel_right' diff_ge_0_iff_ge le_add_diff_inverse pth_c(1))
  then have "(λy. (1 - fst y) *⇩R snd y + fst y *⇩R a) ` ({0..1} × S) ⊆ S"
    using a [unfolded closed_segment_def] by force
  then have "homotopic_with_canon P S S (λx. x) (λx. a)"
    using ‹a ∈ S›
    unfolding homotopic_with_def
    apply (rule_tac x="λy. (1 - (fst y)) *⇩R snd y + (fst y) *⇩R a" in exI)
    apply (force simp: P intro: continuous_intros)
    done
  then show ?thesis
    using that by blast
qed

lemma starlike_imp_contractible:
  fixes S :: "'a::real_normed_vector set"
  shows "starlike S ⟹ contractible S"
  using starlike_imp_contractible_gen contractible_def by (fastforce simp: id_def)

lemma contractible_UNIV [simp]: "contractible (UNIV :: 'a::real_normed_vector set)"
  by (simp add: starlike_imp_contractible)

lemma starlike_imp_simply_connected:
  fixes S :: "'a::real_normed_vector set"
  shows "starlike S ⟹ simply_connected S"
  by (simp add: contractible_imp_simply_connected starlike_imp_contractible)

lemma convex_imp_simply_connected:
  fixes S :: "'a::real_normed_vector set"
  shows "convex S ⟹ simply_connected S"
  using convex_imp_starlike starlike_imp_simply_connected by blast

lemma starlike_imp_path_connected:
  fixes S :: "'a::real_normed_vector set"
  shows "starlike S ⟹ path_connected S"
  by (simp add: simply_connected_imp_path_connected starlike_imp_simply_connected)

lemma starlike_imp_connected:
  fixes S :: "'a::real_normed_vector set"
  shows "starlike S ⟹ connected S"
  by (simp add: path_connected_imp_connected starlike_imp_path_connected)

lemma is_interval_simply_connected_1:
  fixes S :: "real set"
  shows "is_interval S ⟷ simply_connected S"
  by (meson convex_imp_simply_connected is_interval_connected_1 is_interval_convex_1 simply_connected_imp_connected)

lemma contractible_empty [simp]: "contractible {}"
  by (simp add: contractible_def homotopic_on_emptyI)

lemma contractible_convex_tweak_boundary_points:
  fixes S :: "'a::euclidean_space set"
  assumes "convex S" and TS: "rel_interior S ⊆ T" "T ⊆ closure S"
  shows "contractible T"
  by (metis assms closure_eq_empty contractible_empty empty_subsetI 
      starlike_convex_tweak_boundary_points starlike_imp_contractible subset_antisym)

lemma convex_imp_contractible:
  fixes S :: "'a::real_normed_vector set"
  shows "convex S ⟹ contractible S"
  using contractible_empty convex_imp_starlike starlike_imp_contractible by blast

lemma contractible_sing [simp]:
  fixes a :: "'a::real_normed_vector"
  shows "contractible {a}"
  by (rule convex_imp_contractible [OF convex_singleton])

lemma is_interval_contractible_1:
  fixes S :: "real set"
  shows  "is_interval S ⟷ contractible S"
  using contractible_imp_simply_connected convex_imp_contractible is_interval_convex_1
    is_interval_simply_connected_1 by auto

lemma contractible_Times:
  fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set"
  assumes S: "contractible S" and T: "contractible T"
  shows "contractible (S × T)"
proof -
  obtain a h where conth: "continuous_on ({0..1} × S) h"
             and hsub: "h ∈ ({0..1} × S) → S"
             and [simp]: "⋀x. x ∈ S ⟹ h (0, x) = x"
             and [simp]: "⋀x. x ∈ S ⟹  h (1::real, x) = a"
    using S by (force simp: contractible_def homotopic_with)
  obtain b k where contk: "continuous_on ({0..1} × T) k"
             and ksub: "k ∈ ({0..1} × T) → T"
             and [simp]: "⋀x. x ∈ T ⟹ k (0, x) = x"
             and [simp]: "⋀x. x ∈ T ⟹  k (1::real, x) = b"
    using T by (force simp: contractible_def homotopic_with)
  show ?thesis
    apply (simp add: contractible_def homotopic_with)
    apply (rule exI [where x=a])
    apply (rule exI [where x=b])
    apply (rule exI [where x = "λz. (h (fst z, fst(snd z)), k (fst z, snd(snd z)))"])
    using hsub ksub
    apply (fastforce intro!: continuous_intros continuous_on_compose2 [OF conth] continuous_on_compose2 [OF contk])
    done
qed


subsection‹Local versions of topological properties in general›

definition✐‹tag important› locally :: "('a::topological_space set ⇒ bool) ⇒ 'a set ⇒ bool"
where
 "locally P S ≡
        ∀w x. openin (top_of_set S) w ∧ x ∈ w
              ⟶ (∃U V. openin (top_of_set S) U ∧ P V ∧ x ∈ U ∧ U ⊆ V ∧ V ⊆ w)"

lemma locallyI:
  assumes "⋀w x. ⟦openin (top_of_set S) w; x ∈ w⟧
                  ⟹ ∃U V. openin (top_of_set S) U ∧ P V ∧ x ∈ U ∧ U ⊆ V ∧ V ⊆ w"
    shows "locally P S"
using assms by (force simp: locally_def)

lemma locallyE:
  assumes "locally P S" "openin (top_of_set S) w" "x ∈ w"
  obtains U V where "openin (top_of_set S) U" "P V" "x ∈ U" "U ⊆ V" "V ⊆ w"
  using assms unfolding locally_def by meson

lemma locally_mono:
  assumes "locally P S" "⋀T. P T ⟹ Q T"
    shows "locally Q S"
by (metis assms locally_def)

lemma locally_open_subset:
  assumes "locally P S" "openin (top_of_set S) t"
    shows "locally P t"
  by (smt (verit, ccfv_SIG) assms order.trans locally_def openin_imp_subset openin_subset_trans openin_trans)

lemma locally_diff_closed:
    "⟦locally P S; closedin (top_of_set S) t⟧ ⟹ locally P (S - t)"
  using locally_open_subset closedin_def by fastforce

lemma locally_empty [iff]: "locally P {}"
  by (simp add: locally_def openin_subtopology)

lemma locally_singleton [iff]:
  fixes a :: "'a::metric_space"
  shows "locally P {a} ⟷ P {a}"
proof -
  have "∀x::real. ¬ 0 < x ⟹ P {a}"
    using zero_less_one by blast
  then show ?thesis
    unfolding locally_def
    by (auto simp: openin_euclidean_subtopology_iff subset_singleton_iff conj_disj_distribR)
qed

lemma locally_iff:
    "locally P S ⟷
     (∀T x. open T ∧ x ∈ S ∩ T ⟶ (∃U. open U ∧ (∃V. P V ∧ x ∈ S ∩ U ∧ S ∩ U ⊆ V ∧ V ⊆ S ∩ T)))"
  by (smt (verit) locally_def openin_open)

lemma locally_Int:
  assumes S: "locally P S" and T: "locally P T"
      and P: "⋀S T. P S ∧ P T ⟹ P(S ∩ T)"
  shows "locally P (S ∩ T)"
  unfolding locally_iff
proof clarify
  fix A x
  assume "open A" "x ∈ A" "x ∈ S" "x ∈ T"
  then obtain U1 V1 U2 V2 
    where "open U1" "P V1" "x ∈ S ∩ U1" "S ∩ U1 ⊆ V1 ∧ V1 ⊆ S ∩ A" 
          "open U2" "P V2" "x ∈ T ∩ U2" "T ∩ U2 ⊆ V2 ∧ V2 ⊆ T ∩ A"
    using S T unfolding locally_iff by (meson IntI)
  then have "S ∩ T ∩ (U1 ∩ U2) ⊆ V1 ∩ V2" "V1 ∩ V2 ⊆ S ∩ T ∩ A" "x ∈ S ∩ T ∩ (U1 ∩ U2)"
    by blast+
  moreover have "P (V1 ∩ V2)"
    by (simp add: P ‹P V1› ‹P V2›)
  ultimately show "∃U. open U ∧ (∃V. P V ∧ x ∈ S ∩ T ∩ U ∧ S ∩ T ∩ U ⊆ V ∧ V ⊆ S ∩ T ∩ A)"
    using ‹open U1› ‹open U2› by blast
qed


lemma locally_Times:
  fixes S :: "('a::metric_space) set" and T :: "('b::metric_space) set"
  assumes PS: "locally P S" and QT: "locally Q T" and R: "⋀S T. P S ∧ Q T ⟹ R(S × T)"
  shows "locally R (S × T)"
    unfolding locally_def
proof (clarify)
  fix W x y
  assume W: "openin (top_of_set (S × T)) W" and xy: "(x, y) ∈ W"
  then obtain U V where "openin (top_of_set S) U" "x ∈ U"
                        "openin (top_of_set T) V" "y ∈ V" "U × V ⊆ W"
    using Times_in_interior_subtopology by metis
  then obtain U1 U2 V1 V2
         where opeS: "openin (top_of_set S) U1 ∧ P U2 ∧ x ∈ U1 ∧ U1 ⊆ U2 ∧ U2 ⊆ U"
           and opeT: "openin (top_of_set T) V1 ∧ Q V2 ∧ y ∈ V1 ∧ V1 ⊆ V2 ∧ V2 ⊆ V"
    by (meson PS QT locallyE)
  then have "openin (top_of_set (S × T)) (U1 × V1)"
    by (simp add: openin_Times)
  moreover have "R (U2 × V2)"
    by (simp add: R opeS opeT)
  moreover have "U1 × V1 ⊆ U2 × V2 ∧ U2 × V2 ⊆ W"
    using opeS opeT ‹U × V ⊆ W› by auto 
  ultimately show "∃U V. openin (top_of_set (S × T)) U ∧ R V ∧ (x,y) ∈ U ∧ U ⊆ V ∧ V ⊆ W"
    using opeS opeT by auto 
qed


proposition homeomorphism_locally_imp:
  fixes S :: "'a::metric_space set" and T :: "'b::t2_space set"
  assumes S: "locally P S" and hom: "homeomorphism S T f g"
      and Q: "⋀S S'. ⟦P S; homeomorphism S S' f g⟧ ⟹ Q S'"
    shows "locally Q T"
proof (clarsimp simp: locally_def)
  fix W y
  assume "y ∈ W" and "openin (top_of_set T) W"
  then obtain A where T: "open A" "W = T ∩ A"
    by (force simp: openin_open)
  then have "W ⊆ T" by auto
  have f: "⋀x. x ∈ S ⟹ g(f x) = x" "f ` S = T" "continuous_on S f"
   and g: "⋀y. y ∈ T ⟹ f(g y) = y" "g ` T = S" "continuous_on T g"
    using hom by (auto simp: homeomorphism_def)
  have gw: "g ` W = S ∩ f -` W"
    using ‹W ⊆ T› g by force
  have "openin (top_of_set S) (g ` W)"
    using ‹openin (top_of_set T) W› continuous_on_open f gw by auto
  then obtain U V
    where osu: "openin (top_of_set S) U" and uv: "P V" "g y ∈ U" "U ⊆ V" "V ⊆ g ` W"
    by (metis S ‹y ∈ W› image_eqI locallyE)
  have "V ⊆ S" using uv by (simp add: gw)
  have fv: "f ` V = T ∩ {x. g x ∈ V}"
    using ‹f ` S = T› f ‹V ⊆ S› by auto
  have contvf: "continuous_on V f"
    using ‹V ⊆ S› continuous_on_subset f(3) by blast
  have "openin (top_of_set (g ` T)) U"
    using ‹g ` T = S› by (simp add: osu)
  then have "openin (top_of_set T) (T ∩ g -` U)"
    using ‹continuous_on T g› continuous_on_open [THEN iffD1] by blast
  moreover have "∃V. Q V ∧ y ∈ (T ∩ g -` U) ∧ (T ∩ g -` U) ⊆ V ∧ V ⊆ W"
  proof (intro exI conjI)
    show "f ` V ⊆ W"
      using uv using Int_lower2 gw image_subsetI mem_Collect_eq subset_iff by auto
    then have contvg: "continuous_on (f ` V) g"
      using ‹W ⊆ T› continuous_on_subset [OF g(3)] by blast
    have "V ⊆ g ` f ` V"
      by (metis ‹V ⊆ S› hom homeomorphism_def homeomorphism_of_subsets order_refl)
    then have homv: "homeomorphism V (f ` V) f g"
      using ‹V ⊆ S› f by (auto simp: homeomorphism_def contvf contvg)
    show "Q (f ` V)"
      using Q homv ‹P V› by blast
    show "y ∈ T ∩ g -` U"
      using T(2) ‹y ∈ W› ‹g y ∈ U› by blast
    show "T ∩ g -` U ⊆ f ` V"
      using g(1) image_iff uv(3) by fastforce
  qed
  ultimately show "∃U. openin (top_of_set T) U ∧ (∃v. Q v ∧ y ∈ U ∧ U ⊆ v ∧ v ⊆ W)"
    by meson
qed

lemma homeomorphism_locally:
  fixes f:: "'a::metric_space ⇒ 'b::metric_space"
  assumes "homeomorphism S T f g"
      and "⋀S T. homeomorphism S T f g ⟹ (P S ⟷ Q T)"
    shows "locally P S ⟷ locally Q T"
  by (smt (verit) assms homeomorphism_locally_imp homeomorphism_symD)

lemma homeomorphic_locally:
  fixes S:: "'a::metric_space set" and T:: "'b::metric_space set"
  assumes hom: "S homeomorphic T"
          and iff: "⋀X Y. X homeomorphic Y ⟹ (P X ⟷ Q Y)"
    shows "locally P S ⟷ locally Q T"
  by (smt (verit, ccfv_SIG) hom homeomorphic_def homeomorphism_locally homeomorphism_locally_imp iff)

lemma homeomorphic_local_compactness:
  fixes S:: "'a::metric_space set" and T:: "'b::metric_space set"
  shows "S homeomorphic T ⟹ locally compact S ⟷ locally compact T"
  by (simp add: homeomorphic_compactness homeomorphic_locally)

lemma locally_translation:
  fixes P :: "'a :: real_normed_vector set ⇒ bool"
  shows "(⋀S. P ((+) a ` S) = P S) ⟹ locally P ((+) a ` S) = locally P S"
  using homeomorphism_locally [OF homeomorphism_translation]
  by (metis (full_types) homeomorphism_image2)

lemma locally_injective_linear_image:
  fixes f :: "'a::euclidean_space ⇒ 'b::euclidean_space"
  assumes f: "linear f" "inj f" and iff: "⋀S. P (f ` S) ⟷ Q S"
  shows "locally P (f ` S) ⟷ locally Q S"
  by (smt (verit) f homeomorphism_image2 homeomorphism_locally iff linear_homeomorphism_image)

lemma locally_open_map_image:
  fixes f :: "'a::real_normed_vector ⇒ 'b::real_normed_vector"
  assumes P: "locally P S"
      and f: "continuous_on S f"
      and oo: "⋀T. openin (top_of_set S) T ⟹ openin (top_of_set (f ` S)) (f ` T)"
      and Q: "⋀T. ⟦T ⊆ S; P T⟧ ⟹ Q(f ` T)"
    shows "locally Q (f ` S)"
proof (clarsimp simp: locally_def)
  fix W y
  assume oiw: "openin (top_of_set (f ` S)) W" and "y ∈ W"
  then have "W ⊆ f ` S" by (simp add: openin_euclidean_subtopology_iff)
  have oivf: "openin (top_of_set S) (S ∩ f -` W)"
    by (rule continuous_on_open [THEN iffD1, rule_format, OF f oiw])
  then obtain x where "x ∈ S" "f x = y"
    using ‹W ⊆ f ` S› ‹y ∈ W› by blast
  then obtain U V
    where "openin (top_of_set S) U" "P V" "x ∈ U" "U ⊆ V" "V ⊆ S ∩ f -` W"
    by (metis IntI P ‹y ∈ W› locallyE oivf vimageI)
  then have "openin (top_of_set (f ` S)) (f ` U)"
    by (simp add: oo)
  then show "∃X. openin (top_of_set (f ` S)) X ∧ (∃Y. Q Y ∧ y ∈ X ∧ X ⊆ Y ∧ Y ⊆ W)"
    using Q ‹P V› ‹U ⊆ V› ‹V ⊆ S ∩ f -` W› ‹f x = y› ‹x ∈ U› by blast
qed


subsection‹An induction principle for connected sets›

proposition connected_induction:
  assumes "connected S"
      and opD: "⋀T a. ⟦openin (top_of_set S) T; a ∈ T⟧ ⟹ ∃z. z ∈ T ∧ P z"
      and opI: "⋀a. a ∈ S
             ⟹ ∃T. openin (top_of_set S) T ∧ a ∈ T ∧
                     (∀x ∈ T. ∀y ∈ T. P x ∧ P y ∧ Q x ⟶ Q y)"
      and etc: "a ∈ S" "b ∈ S" "P a" "P b" "Q a"
    shows "Q b"
proof -
  let ?A = "{b. ∃T. openin (top_of_set S) T ∧ b ∈ T ∧ (∀x∈T. P x ⟶ Q x)}"
  let ?B = "{b. ∃T. openin (top_of_set S) T ∧ b ∈ T ∧ (∀x∈T. P x ⟶ ¬ Q x)}"
  have "?A ∩ ?B = {}"
    by (clarsimp simp: set_eq_iff) (metis (no_types, opaque_lifting) Int_iff opD openin_Int)
  moreover have "S ⊆ ?A ∪ ?B"
    by clarsimp (meson opI)
  moreover have "openin (top_of_set S) ?A"
    by (subst openin_subopen, blast)
  moreover have "openin (top_of_set S) ?B"
    by (subst openin_subopen, blast)
  ultimately have "?A = {} ∨ ?B = {}"
    by (metis (no_types, lifting) ‹connected S› connected_openin)
  then show ?thesis
    by clarsimp (meson opI etc)
qed

lemma connected_equivalence_relation_gen:
  assumes "connected S"
      and etc: "a ∈ S" "b ∈ S" "P a" "P b"
      and trans: "⋀x y z. ⟦R x y; R y z⟧ ⟹ R x z"
      and opD: "⋀T a. ⟦openin (top_of_set S) T; a ∈ T⟧ ⟹ ∃z. z ∈ T ∧ P z"
      and opI: "⋀a. a ∈ S
             ⟹ ∃T. openin (top_of_set S) T ∧ a ∈ T ∧
                     (∀x ∈ T. ∀y ∈ T. P x ∧ P y ⟶ R x y)"
    shows "R a b"
proof -
  have "⋀a b c. ⟦a ∈ S; P a; b ∈ S; c ∈ S; P b; P c; R a b⟧ ⟹ R a c"
    apply (rule connected_induction [OF ‹connected S› opD], simp_all)
    by (meson trans opI)
  then show ?thesis by (metis etc opI)
qed

lemma connected_induction_simple:
  assumes "connected S"
      and etc: "a ∈ S" "b ∈ S" "P a"
      and opI: "⋀a. a ∈ S
             ⟹ ∃T. openin (top_of_set S) T ∧ a ∈ T ∧
                     (∀x ∈ T. ∀y ∈ T. P x ⟶ P y)"
    shows "P b"
  by (rule connected_induction [OF ‹connected S› _, where P = "λx. True"])
     (use opI etc in auto)

lemma connected_equivalence_relation:
  assumes "connected S"
      and etc: "a ∈ S" "b ∈ S"
      and sym: "⋀x y. ⟦R x y; x ∈ S; y ∈ S⟧ ⟹ R y x"
      and trans: "⋀x y z. ⟦R x y; R y z; x ∈ S; y ∈ S; z ∈ S⟧ ⟹ R x z"
      and opI: "⋀a. a ∈ S ⟹ ∃T. openin (top_of_set S) T ∧ a ∈ T ∧ (∀x ∈ T. R a x)"
    shows "R a b"
proof -
  have "⋀a b c. ⟦a ∈ S; b ∈ S; c ∈ S; R a b⟧ ⟹ R a c"
    by (smt (verit, ccfv_threshold) connected_induction_simple [OF ‹connected S›] 
            assms openin_imp_subset subset_eq)
  then show ?thesis by (metis etc opI)
qed

lemma locally_constant_imp_constant:
  assumes "connected S"
      and opI: "⋀a. a ∈ S
             ⟹ ∃T. openin (top_of_set S) T ∧ a ∈ T ∧ (∀x ∈ T. f x = f a)"
    shows "f constant_on S"
proof -
  have "⋀x y. x ∈ S ⟹ y ∈ S ⟹ f x = f y"
    apply (rule connected_equivalence_relation [OF ‹connected S›], simp_all)
    by (metis opI)
  then show ?thesis
    by (metis constant_on_def)
qed

lemma locally_constant:
  assumes "connected S"
  shows "locally (λU. f constant_on U) S ⟷ f constant_on S" (is "?lhs = ?rhs")
proof
  assume ?lhs
  then show ?rhs
    by (smt (verit, del_insts) assms constant_on_def locally_constant_imp_constant locally_def openin_subtopology_self subset_iff)
next
  assume ?rhs then show ?lhs
    by (metis constant_on_subset locallyI openin_imp_subset order_refl)
qed


subsection‹Basic properties of local compactness›

proposition locally_compact:
  fixes S :: "'a :: metric_space set"
  shows
    "locally compact S ⟷
     (∀x ∈ S. ∃u v. x ∈ u ∧ u ⊆ v ∧ v ⊆ S ∧
                    openin (top_of_set S) u ∧ compact v)"
     (is "?lhs = ?rhs")
proof
  assume ?lhs
  then show ?rhs
    by (meson locallyE openin_subtopology_self)
next
  assume r [rule_format]: ?rhs
  have *: "∃u v.
              openin (top_of_set S) u ∧
              compact v ∧ x ∈ u ∧ u ⊆ v ∧ v ⊆ S ∩ T"
          if "open T" "x ∈ S" "x ∈ T" for x T
  proof -
    obtain U V where uv: "x ∈ U" "U ⊆ V" "V ⊆ S" "compact V" "openin (top_of_set S) U"
      using r [OF ‹x ∈ S›] by auto
    obtain e where "e>0" and e: "cball x e ⊆ T"
      using open_contains_cball ‹open T› ‹x ∈ T› by blast
    show ?thesis
      apply (rule_tac x="(S ∩ ball x e) ∩ U" in exI)
      apply (rule_tac x="cball x e ∩ V" in exI)
      using that ‹e > 0› e uv
      apply auto
      done
  qed
  show ?lhs
    by (rule locallyI) (metis "*" Int_iff openin_open)
qed

lemma locally_compactE:
  fixes S :: "'a :: metric_space set"
  assumes "locally compact S"
  obtains u v where "⋀x. x ∈ S ⟹ x ∈ u x ∧ u x ⊆ v x ∧ v x ⊆ S ∧
                             openin (top_of_set S) (u x) ∧ compact (v x)"
  using assms unfolding locally_compact by metis

lemma locally_compact_alt:
  fixes S :: "'a :: heine_borel set"
  shows "locally compact S ⟷
         (∀x ∈ S. ∃U. x ∈ U ∧
                    openin (top_of_set S) U ∧ compact(closure U) ∧ closure U ⊆ S)"
  by (smt (verit, ccfv_threshold) bounded_subset closure_closed closure_mono closure_subset 
      compact_closure compact_imp_closed order.trans locally_compact)

lemma locally_compact_Int_cball:
  fixes S :: "'a :: heine_borel set"
  shows "locally compact S ⟷ (∀x ∈ S. ∃e. 0 < e ∧ closed(cball x e ∩ S))"
        (is "?lhs = ?rhs")
proof
  assume L: ?lhs
  then have "⋀x U V e. ⟦U ⊆ V; V ⊆ S; compact V; 0 < e; cball x e ∩ S ⊆ U⟧
       ⟹ closed (cball x e ∩ S)"
    by (metis compact_Int compact_cball compact_imp_closed inf.absorb_iff2 inf.assoc inf.orderE)
  with L show ?rhs
    by (meson locally_compactE openin_contains_cball)
next
  assume R: ?rhs
  show ?lhs unfolding locally_compact 
  proof
    fix x
    assume "x ∈ S"
    then obtain e where "e>0" and "compact (cball x e ∩ S)"
      by (metis Int_commute compact_Int_closed compact_cball inf.right_idem R)
    moreover have "∀y∈ball x e ∩ S. ∃ε>0. cball y ε ∩ S ⊆ ball x e"
      by (meson Elementary_Metric_Spaces.open_ball IntD1 le_infI1 open_contains_cball_eq)
    moreover have "openin (top_of_set S) (ball x e ∩ S)"
      by (simp add: inf_commute openin_open_Int)
    ultimately show "∃U V. x ∈ U ∧ U ⊆ V ∧ V ⊆ S ∧ openin (top_of_set S) U ∧ compact V"
      by (metis Int_iff ‹0 < e› ‹x ∈ S› ball_subset_cball centre_in_ball inf_commute inf_le1 inf_mono order_refl)
  qed
qed

lemma locally_compact_compact:
  fixes S :: "'a :: heine_borel set"
  shows "locally compact S ⟷
         (∀K. K ⊆ S ∧ compact K
              ⟶ (∃U V. K ⊆ U ∧ U ⊆ V ∧ V ⊆ S ∧
                         openin (top_of_set S) U ∧ compact V))"
        (is "?lhs = ?rhs")
proof
  assume ?lhs
  then obtain u v where
    uv: "⋀x. x ∈ S ⟹ x ∈ u x ∧ u x ⊆ v x ∧ v x ⊆ S ∧
                             openin (top_of_set S) (u x) ∧ compact (v x)"
    by (metis locally_compactE)
  have *: "∃U V. K ⊆ U ∧ U ⊆ V ∧ V ⊆ S ∧ openin (top_of_set S) U ∧ compact V"
          if "K ⊆ S" "compact K" for K
  proof -
    have "⋀C. (∀c∈C. openin (top_of_set K) c) ∧ K ⊆ ⋃C ⟹
                    ∃D⊆C. finite D ∧ K ⊆ ⋃D"
      using that by (simp add: compact_eq_openin_cover)
    moreover have "∀c ∈ (λx. K ∩ u x) ` K. openin (top_of_set K) c"
      using that by clarify (metis subsetD inf.absorb_iff2 openin_subset openin_subtopology_Int_subset topspace_euclidean_subtopology uv)
    moreover have "K ⊆ ⋃((λx. K ∩ u x) ` K)"
      using that by clarsimp (meson subsetCE uv)
    ultimately obtain D where "D ⊆ (λx. K ∩ u x) ` K" "finite D" "K ⊆ ⋃D"
      by metis
    then obtain T where T: "T ⊆ K" "finite T" "K ⊆ ⋃((λx. K ∩ u x) ` T)"
      by (metis finite_subset_image)
    have Tuv: "⋃(u ` T) ⊆ ⋃(v ` T)"
      using T that by (force dest!: uv)
    moreover
    have "openin (top_of_set S) (⋃ (u ` T))"
      using T that uv by fastforce
    moreover
    obtain "compact (⋃ (v ` T))" "⋃ (v ` T) ⊆ S"
      by (metis T UN_subset_iff ‹K ⊆ S› compact_UN subset_iff uv)
    ultimately show ?thesis
      using T by auto 
  qed
  show ?rhs
    by (blast intro: *)
next
  assume ?rhs
  then show ?lhs
    apply (clarsimp simp: locally_compact)
    apply (drule_tac x="{x}" in spec, simp)
    done
qed

lemma open_imp_locally_compact:
  fixes S :: "'a :: heine_borel set"
  assumes "open S"
    shows "locally compact S"
proof -
  have *: "∃U V. x ∈ U ∧ U ⊆ V ∧ V ⊆ S ∧ openin (top_of_set S) U ∧ compact V"
          if "x ∈ S" for x
  proof -
    obtain e where "e>0" and e: "cball x e ⊆ S"
      using open_contains_cball assms ‹x ∈ S› by blast
    have ope: "openin (top_of_set S) (ball x e)"
      by (meson e open_ball ball_subset_cball dual_order.trans open_subset)
    show ?thesis
      by (meson ‹0 < e› ball_subset_cball centre_in_ball compact_cball e ope)
  qed
  show ?thesis
    unfolding locally_compact by (blast intro: *)
qed

lemma closed_imp_locally_compact:
  fixes S :: "'a :: heine_borel set"
  assumes "closed S"
    shows "locally compact S"
proof -
  have *: "∃U V. x ∈ U ∧ U ⊆ V ∧ V ⊆ S ∧ openin (top_of_set S) U ∧ compact V"
          if "x ∈ S" for x
    apply (rule_tac x = "S ∩ ball x 1" in exI, rule_tac x = "S ∩ cball x 1" in exI)
    using ‹x ∈ S› assms by auto
  show ?thesis
    unfolding locally_compact by (blast intro: *)
qed

lemma locally_compact_UNIV: "locally compact (UNIV :: 'a :: heine_borel set)"
  by (simp add: closed_imp_locally_compact)

lemma locally_compact_Int:
  fixes S :: "'a :: t2_space set"
  shows "⟦locally compact S; locally compact T⟧ ⟹ locally compact (S ∩ T)"
  by (simp add: compact_Int locally_Int)

lemma locally_compact_closedin:
  fixes S :: "'a :: heine_borel set"
  shows "⟦closedin (top_of_set S) T; locally compact S⟧
        ⟹ locally compact T"
  unfolding closedin_closed
  using closed_imp_locally_compact locally_compact_Int by blast

lemma locally_compact_delete:
     fixes S :: "'a :: t1_space set"
     shows "locally compact S ⟹ locally compact (S - {a})"
  by (auto simp: openin_delete locally_open_subset)

lemma locally_closed:
  fixes S :: "'a :: heine_borel set"
  shows "locally closed S ⟷ locally compact S"
        (is "?lhs = ?rhs")
proof
  assume ?lhs
  then show ?rhs
    unfolding locally_def
    apply (elim all_forward imp_forward asm_rl exE)
    apply (rename_tac U V)
    apply (rule_tac x = "U ∩ ball x 1" in exI)
    apply (rule_tac x = "V ∩ cball x 1" in exI)
    apply (force intro: openin_trans)
    done
next
  assume ?rhs then show ?lhs
    using compact_eq_bounded_closed locally_mono by blast
qed

lemma locally_compact_openin_Un:
  fixes S :: "'a::euclidean_space set"
  assumes LCS: "locally compact S" and LCT: "locally compact T"
      and opS: "openin (top_of_set (S ∪ T)) S"
      and opT: "openin (top_of_set (S ∪ T)) T"
    shows "locally compact (S ∪ T)"
proof -
  have "∃e>0. closed (cball x e ∩ (S ∪ T))" if "x ∈ S" for x
  proof -
    obtain e1 where "e1 > 0" and e1: "closed (cball x e1 ∩ S)"
      using LCS ‹x ∈ S› unfolding locally_compact_Int_cball by blast
    moreover obtain e2 where "e2 > 0" and e2: "cball x e2 ∩ (S ∪ T) ⊆ S"
      by (meson ‹x ∈ S› opS openin_contains_cball)
    then have "cball x e2 ∩ (S ∪ T) = cball x e2 ∩ S"
      by force
    ultimately have "closed (cball x (min e1 e2) ∩ (S ∪ T))"
      by (metis (no_types, lifting) cball_min_Int closed_Int closed_cball inf_assoc inf_commute)
    then show ?thesis
      by (metis ‹0 < e1› ‹0 < e2› min_def)
  qed
  moreover have "∃e>0. closed (cball x e ∩ (S ∪ T))" if "x ∈ T" for x
  proof -
    obtain e1 where "e1 > 0" and e1: "closed (cball x e1 ∩ T)"
      using LCT ‹x ∈ T› unfolding locally_compact_Int_cball by blast
    moreover obtain e2 where "e2 > 0" and e2: "cball x e2 ∩ (S ∪ T) ⊆ T"
      by (meson ‹x ∈ T› opT openin_contains_cball)
    then have "cball x e2 ∩ (S ∪ T) = cball x e2 ∩ T"
      by force
    moreover have "closed (cball x e1 ∩ (cball x e2 ∩ T))"
      by (metis closed_Int closed_cball e1 inf_left_commute)
    ultimately show ?thesis
      by (rule_tac x="min e1 e2" in exI) (simp add: ‹0 < e2› cball_min_Int inf_assoc)
  qed
  ultimately show ?thesis
    by (force simp: locally_compact_Int_cball)
qed

lemma locally_compact_closedin_Un:
  fixes S :: "'a::euclidean_space set"
  assumes LCS: "locally compact S" and LCT:"locally compact T"
      and clS: "closedin (top_of_set (S ∪ T)) S"
      and clT: "closedin (top_of_set (S ∪ T)) T"
    shows "locally compact (S ∪ T)"
proof -
  have "∃e>0. closed (cball x e ∩ (S ∪ T))" if "x ∈ S" "x ∈ T" for x
  proof -
    obtain e1 where "e1 > 0" and e1: "closed (cball x e1 ∩ S)"
      using LCS ‹x ∈ S› unfolding locally_compact_Int_cball by blast
    moreover
    obtain e2 where "e2 > 0" and e2: "closed (cball x e2 ∩ T)"
      using LCT ‹x ∈ T› unfolding locally_compact_Int_cball by blast
    moreover have "closed (cball x (min e1 e2) ∩ (S ∪ T))"
      by (smt (verit) Int_Un_distrib2 Int_commute cball_min_Int closed_Int closed_Un closed_cball e1 e2 inf_left_commute)
    ultimately show ?thesis
      by (rule_tac x="min e1 e2" in exI) linarith
  qed
  moreover
  have "∃e>0. closed (cball x e ∩ (S ∪ T))" if x: "x ∈ S" "x ∉ T" for x
  proof -
    obtain e1 where "e1 > 0" and e1: "closed (cball x e1 ∩ S)"
      using LCS ‹x ∈ S› unfolding locally_compact_Int_cball by blast
    moreover
    obtain e2 where "e2>0" and "cball x e2 ∩ (S ∪ T) ⊆ S - T"
      using clT x by (fastforce simp: openin_contains_cball closedin_def)
    then have "closed (cball x e2 ∩ T)"
    proof -
      have "{} = T - (T - cball x e2)"
        using Diff_subset Int_Diff ‹cball x e2 ∩ (S ∪ T) ⊆ S - T› by auto
      then show ?thesis
        by (simp add: Diff_Diff_Int inf_commute)
    qed
    with e1 have "closed ((cball x e1 ∩ cball x e2) ∩ (S ∪ T))"
      apply (simp add: inf_commute inf_sup_distrib2)
      by (metis closed_Int closed_Un closed_cball inf_assoc inf_left_commute)
    then have "closed (cball x (min e1 e2) ∩ (S ∪ T))"
      by (simp add: cball_min_Int inf_commute)
    ultimately show ?thesis
      using ‹0 < e2› by (rule_tac x="min e1 e2" in exI) linarith 
  qed
  moreover
  have "∃e>0. closed (cball x e ∩ (S ∪ T))" if x: "x ∉ S" "x ∈ T" for x
  proof -
    obtain e1 where "e1 > 0" and e1: "closed (cball x e1 ∩ T)"
      using LCT ‹x ∈ T› unfolding locally_compact_Int_cball by blast
    moreover
    obtain e2 where "e2>0" and "cball x e2 ∩ (S ∪ T) ⊆ S ∪ T - S"
      using clS x by (fastforce simp: openin_contains_cball closedin_def)
    then have "closed (cball x e2 ∩ S)"
      by (metis Diff_disjoint Int_empty_right closed_empty inf.left_commute inf.orderE inf_sup_absorb)
    with e1 have "closed ((cball x e1 ∩ cball x e2) ∩ (S ∪ T))"
      apply (simp add: inf_commute inf_sup_distrib2)
      by (metis closed_Int closed_Un closed_cball inf_assoc inf_left_commute)
    then have "closed (cball x (min e1 e2) ∩ (S ∪ T))"
      by (auto simp: cball_min_Int)
    ultimately show ?thesis
      using ‹0 < e2› by (rule_tac x="min e1 e2" in exI) linarith
  qed
  ultimately show ?thesis
    by (auto simp: locally_compact_Int_cball)
qed

lemma locally_compact_Times:
  fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set"
  shows "⟦locally compact S; locally compact T⟧ ⟹ locally compact (S × T)"
  by (auto simp: compact_Times locally_Times)

lemma locally_compact_compact_subopen:
  fixes S :: "'a :: heine_borel set"
  shows
   "locally compact S ⟷
    (∀K T. K ⊆ S ∧ compact K ∧ open T ∧ K ⊆ T
          ⟶ (∃U V. K ⊆ U ∧ U ⊆ V ∧ U ⊆ T ∧ V ⊆ S ∧
                     openin (top_of_set S) U ∧ compact V))"
   (is "?lhs = ?rhs")
proof
  assume L: ?lhs
  show ?rhs
  proof clarify
    fix K :: "'a set" and T :: "'a set"
    assume "K ⊆ S" and "compact K" and "open T" and "K ⊆ T"
    obtain U V where "K ⊆ U" "U ⊆ V" "V ⊆ S" "compact V"
                 and ope: "openin (top_of_set S) U"
      using L unfolding locally_compact_compact by (meson ‹K ⊆ S› ‹compact K›)
    show "∃U V. K ⊆ U ∧ U ⊆ V ∧ U ⊆ T ∧ V ⊆ S ∧
                openin (top_of_set S) U ∧ compact V"
    proof (intro exI conjI)
      show "K ⊆ U ∩ T"
        by (simp add: ‹K ⊆ T› ‹K ⊆ U›)
      show "U ∩ T ⊆ closure(U ∩ T)"
        by (rule closure_subset)
      show "closure (U ∩ T) ⊆ S"
        by (metis ‹U ⊆ V› ‹V ⊆ S› ‹compact V› closure_closed closure_mono compact_imp_closed inf.cobounded1 subset_trans)
      show "openin (top_of_set S) (U ∩ T)"
        by (simp add: ‹open T› ope openin_Int_open)
      show "compact (closure (U ∩ T))"
        by (meson Int_lower1 ‹U ⊆ V› ‹compact V› bounded_subset compact_closure compact_eq_bounded_closed)
    qed auto
  qed
next
  assume ?rhs then show ?lhs
    unfolding locally_compact_compact
    by (metis open_openin openin_topspace subtopology_superset top.extremum topspace_euclidean_subtopology)
qed


subsection‹Sura-Bura's results about compact components of sets›

proposition Sura_Bura_compact:
  fixes S :: "'a::euclidean_space set"
  assumes "compact S" and C: "C ∈ components S"
  shows "C = ⋂{T. C ⊆ T ∧ openin (top_of_set S) T ∧
                           closedin (top_of_set S) T}"
         (is "C = ⋂?𝒯")
proof
  obtain x where x: "C = connected_component_set S x" and "x ∈ S"
    using C by (auto simp: components_def)
  have "C ⊆ S"
    by (simp add: C in_components_subset)
  have "⋂?𝒯 ⊆ connected_component_set S x"
  proof (rule connected_component_maximal)
    have "x ∈ C"
      by (simp add: ‹x ∈ S› x)
    then show "x ∈ ⋂?𝒯"
      by blast
    have clo: "closed (⋂?𝒯)"
      by (simp add: ‹compact S› closed_Inter closedin_compact_eq compact_imp_closed)
    have False
      if K1: "closedin (top_of_set (⋂?𝒯)) K1" and
         K2: "closedin (top_of_set (⋂?𝒯)) K2" and
         K12_Int: "K1 ∩ K2 = {}" and K12_Un: "K1 ∪ K2 = ⋂?𝒯" and "K1 ≠ {}" "K2 ≠ {}"
       for K1 K2
    proof -
      have "closed K1" "closed K2"
        using closedin_closed_trans clo K1 K2 by blast+
      then obtain V1 V2 where "open V1" "open V2" "K1 ⊆ V1" "K2 ⊆ V2" and V12: "V1 ∩ V2 = {}"
        using separation_normal ‹K1 ∩ K2 = {}› by metis
      have SV12_ne: "(S - (V1 ∪ V2)) ∩ (⋂?𝒯) ≠ {}"
      proof (rule compact_imp_fip)
        show "compact (S - (V1 ∪ V2))"
          by (simp add: ‹open V1› ‹open V2› ‹compact S› compact_diff open_Un)
        show clo𝒯: "closed T" if "T ∈ ?𝒯" for T
          using that ‹compact S›
          by (force intro: closedin_closed_trans simp add: compact_imp_closed)
        show "(S - (V1 ∪ V2)) ∩ ⋂ℱ ≠ {}" if "finite ℱ" and ℱ: "ℱ ⊆ ?𝒯" for ℱ
        proof
          assume djo: "(S - (V1 ∪ V2)) ∩ ⋂ℱ = {}"
          obtain D where opeD: "openin (top_of_set S) D"
                   and cloD: "closedin (top_of_set S) D"
                   and "C ⊆ D" and DV12: "D ⊆ V1 ∪ V2"
          proof (cases "ℱ = {}")
            case True
            with ‹C ⊆ S› djo that show ?thesis
              by force
          next
            case False show ?thesis
            proof
              show ope: "openin (top_of_set S) (⋂ℱ)"
                using openin_Inter ‹finite ℱ› False ℱ by blast
              then show "closedin (top_of_set S) (⋂ℱ)"
                by (meson clo𝒯 ℱ closed_Inter closed_subset openin_imp_subset subset_eq)
              show "C ⊆ ⋂ℱ"
                using ℱ by auto
              show "⋂ℱ ⊆ V1 ∪ V2"
                using ope djo openin_imp_subset by fastforce
            qed
          qed
          have "connected C"
            by (simp add: x)
          have "closed D"
            using ‹compact S› cloD closedin_closed_trans compact_imp_closed by blast
          have cloV1: "closedin (top_of_set D) (D ∩ closure V1)"
            and cloV2: "closedin (top_of_set D) (D ∩ closure V2)"
            by (simp_all add: closedin_closed_Int)
          moreover have "D ∩ closure V1 = D ∩ V1" "D ∩ closure V2 = D ∩ V2"
            using ‹D ⊆ V1 ∪ V2› ‹open V1› ‹open V2› V12
            by (auto simp: closure_subset [THEN subsetD] closure_iff_nhds_not_empty, blast+)
          ultimately have cloDV1: "closedin (top_of_set D) (D ∩ V1)"
                      and cloDV2:  "closedin (top_of_set D) (D ∩ V2)"
            by metis+
          then obtain U1 U2 where "closed U1" "closed U2"
               and D1: "D ∩ V1 = D ∩ U1" and D2: "D ∩ V2 = D ∩ U2"
            by (auto simp: closedin_closed)
          have "D ∩ U1 ∩ C ≠ {}"
          proof
            assume "D ∩ U1 ∩ C = {}"
            then have *: "C ⊆ D ∩ V2"
              using D1 DV12 ‹C ⊆ D› by auto
            have 1: "openin (top_of_set S) (D ∩ V2)"
              by (simp add: ‹open V2› opeD openin_Int_open)
            have 2: "closedin (top_of_set S) (D ∩ V2)"
              using cloD cloDV2 closedin_trans by blast
            have "⋂ ?𝒯 ⊆ D ∩ V2"
              by (rule Inter_lower) (use * 1 2 in simp)
            then show False
              using K1 V12 ‹K1 ≠ {}› ‹K1 ⊆ V1› closedin_imp_subset by blast
          qed
          moreover have "D ∩ U2 ∩ C ≠ {}"
          proof
            assume "D ∩ U2 ∩ C = {}"
            then have *: "C ⊆ D ∩ V1"
              using D2 DV12 ‹C ⊆ D› by auto
            have 1: "openin (top_of_set S) (D ∩ V1)"
              by (simp add: ‹open V1› opeD openin_Int_open)
            have 2: "closedin (top_of_set S) (D ∩ V1)"
              using cloD cloDV1 closedin_trans by blast
            have "⋂?𝒯 ⊆ D ∩ V1"
              by (rule Inter_lower) (use * 1 2 in simp)
            then show False
              using K2 V12 ‹K2 ≠ {}› ‹K2 ⊆ V2› closedin_imp_subset by blast
          qed
          ultimately show False
            using ‹connected C› [unfolded connected_closed, simplified, rule_format, of concl: "D ∩ U1" "D ∩ U2"]
            using ‹C ⊆ D› D1 D2 V12 DV12 ‹closed U1› ‹closed U2› ‹closed D›
            by blast
        qed
      qed
      show False
        by (metis (full_types) DiffE UnE Un_upper2 SV12_ne ‹K1 ⊆ V1› ‹K2 ⊆ V2› disjoint_iff_not_equal subsetCE sup_ge1 K12_Un)
    qed
    then show "connected (⋂?𝒯)"
      by (auto simp: connected_closedin_eq)
    show "⋂?𝒯 ⊆ S"
      by (fastforce simp: C in_components_subset)
  qed
  with x show "⋂?𝒯 ⊆ C" by simp
qed auto


corollary Sura_Bura_clopen_subset:
  fixes S :: "'a::euclidean_space set"
  assumes S: "locally compact S" and C: "C ∈ components S" and "compact C"
      and U: "open U" "C ⊆ U"
  obtains K where "openin (top_of_set S) K" "compact K" "C ⊆ K" "K ⊆ U"
proof (rule ccontr)
  assume "¬ thesis"
  with that have neg: "∄K. openin (top_of_set S) K ∧ compact K ∧ C ⊆ K ∧ K ⊆ U"
    by metis
  obtain V K where "C ⊆ V" "V ⊆ U" "V ⊆ K" "K ⊆ S" "compact K"
               and opeSV: "openin (top_of_set S) V"
    using S U ‹compact C› by (meson C in_components_subset locally_compact_compact_subopen)
  let ?𝒯 = "{T. C ⊆ T ∧ openin (top_of_set K) T ∧ compact T ∧ T ⊆ K}"
  have CK: "C ∈ components K"
    by (meson C ‹C ⊆ V› ‹K ⊆ S› ‹V ⊆ K› components_intermediate_subset subset_trans)
  with ‹compact K›
  have "C = ⋂{T. C ⊆ T ∧ openin (top_of_set K) T ∧ closedin (top_of_set K) T}"
    by (simp add: Sura_Bura_compact)
  then have Ceq: "C = ⋂?𝒯"
    by (simp add: closedin_compact_eq ‹compact K›)
  obtain W where "open W" and W: "V = S ∩ W"
    using opeSV by (auto simp: openin_open)
  have "-(U ∩ W) ∩ ⋂?𝒯 ≠ {}"
  proof (rule closed_imp_fip_compact)
    show "- (U ∩ W) ∩ ⋂ℱ ≠ {}"
      if "finite ℱ" and ℱ: "ℱ ⊆ ?𝒯" for ℱ
    proof (cases "ℱ = {}")
      case True
      have False if "U = UNIV" "W = UNIV"
      proof -
        have "V = S"
          by (simp add: W ‹W = UNIV›)
        with neg show False
          using ‹C ⊆ V› ‹K ⊆ S› ‹V ⊆ K› ‹V ⊆ U› ‹compact K› by auto
      qed
      with True show ?thesis
        by auto
    next
      case False
      show ?thesis
      proof
        assume "- (U ∩ W) ∩ ⋂ℱ = {}"
        then have FUW: "⋂ℱ ⊆ U ∩ W"
          by blast
        have "C ⊆ ⋂ℱ"
          using ℱ by auto
        moreover have "compact (⋂ℱ)"
          by (metis (no_types, lifting) compact_Inter False mem_Collect_eq subsetCE ℱ)
        moreover have "⋂ℱ ⊆ K"
          using False that(2) by fastforce
        moreover have opeKF: "openin (top_of_set K) (⋂ℱ)"
          using False ℱ ‹finite ℱ› by blast
        then have opeVF: "openin (top_of_set V) (⋂ℱ)"
          using W ‹K ⊆ S› ‹V ⊆ K› opeKF ‹⋂ℱ ⊆ K› FUW openin_subset_trans by fastforce
        then have "openin (top_of_set S) (⋂ℱ)"
          by (metis opeSV openin_trans)
        moreover have "⋂ℱ ⊆ U"
          by (meson ‹V ⊆ U› opeVF dual_order.trans openin_imp_subset)
        ultimately show False
          using neg by blast
      qed
    qed
  qed (use ‹open W› ‹open U› in auto)
  with W Ceq ‹C ⊆ V› ‹C ⊆ U› show False
    by auto
qed


corollary Sura_Bura_clopen_subset_alt:
  fixes S :: "'a::euclidean_space set"
  assumes S: "locally compact S" and C: "C ∈ components S" and "compact C"
      and opeSU: "openin (top_of_set S) U" and "C ⊆ U"
  obtains K where "openin (top_of_set S) K" "compact K" "C ⊆ K" "K ⊆ U"
proof -
  obtain V where "open V" "U = S ∩ V"
    using opeSU by (auto simp: openin_open)
  with ‹C ⊆ U› have "C ⊆ V"
    by auto
  then show ?thesis
    using Sura_Bura_clopen_subset [OF S C ‹compact C› ‹open V›]
    by (metis ‹U = S ∩ V› inf.bounded_iff openin_imp_subset that)
qed

corollary Sura_Bura:
  fixes S :: "'a::euclidean_space set"
  assumes "locally compact S" "C ∈ components S" "compact C"
  shows "C = ⋂ {K. C ⊆ K ∧ compact K ∧ openin (top_of_set S) K}"
         (is "C = ?rhs")
proof
  show "?rhs ⊆ C"
  proof (clarsimp, rule ccontr)
    fix x
    assume *: "∀X. C ⊆ X ∧ compact X ∧ openin (top_of_set S) X ⟶ x ∈ X"
      and "x ∉ C"
    obtain U V where "open U" "open V" "{x} ⊆ U" "C ⊆ V" "U ∩ V = {}"
      using separation_normal [of "{x}" C]
      by (metis Int_empty_left ‹x ∉ C› ‹compact C› closed_empty closed_insert compact_imp_closed insert_disjoint(1))
    have "x ∉ V"
      using ‹U ∩ V = {}› ‹{x} ⊆ U› by blast
    then show False
      by (meson "*" Sura_Bura_clopen_subset ‹C ⊆ V› ‹open V› assms(1) assms(2) assms(3) subsetCE)
  qed
qed blast


subsection‹Special cases of local connectedness and path connectedness›

lemma locally_connected_1:
  assumes
    "⋀V x. ⟦openin (top_of_set S) V; x ∈ V⟧ ⟹ ∃U. openin (top_of_set S) U ∧ connected U ∧ x ∈ U ∧ U ⊆ V"
   shows "locally connected S"
  by (metis assms locally_def)

lemma locally_connected_2:
  assumes "locally connected S"
          "openin (top_of_set S) t"
          "x ∈ t"
   shows "openin (top_of_set S) (connected_component_set t x)"
proof -
  { fix y :: 'a
    let ?SS = "top_of_set S"
    assume 1: "openin ?SS t"
              "∀w x. openin ?SS w ∧ x ∈ w ⟶ (∃u. openin ?SS u ∧ (∃v. connected v ∧ x ∈ u ∧ u ⊆ v ∧ v ⊆ w))"
    and "connected_component t x y"
    then have "y ∈ t" and y: "y ∈ connected_component_set t x"
      using connected_component_subset by blast+
    obtain F where
      "∀x y. (∃w. openin ?SS w ∧ (∃u. connected u ∧ x ∈ w ∧ w ⊆ u ∧ u ⊆ y)) = (openin ?SS (F x y) ∧ (∃u. connected u ∧ x ∈ F x y ∧ F x y ⊆ u ∧ u ⊆ y))"
      by moura
    then obtain G where
       "∀a A. (∃U. openin ?SS U ∧ (∃V. connected V ∧ a ∈ U ∧ U ⊆ V ∧ V ⊆ A)) = (openin ?SS (F a A) ∧ connected (G a A) ∧ a ∈ F a A ∧ F a A ⊆ G a A ∧ G a A ⊆ A)"
      by moura
    then have *: "openin ?SS (F y t) ∧ connected (G y t) ∧ y ∈ F y t ∧ F y t ⊆ G y t ∧ G y t ⊆ t"
      using 1 ‹y ∈ t› by presburger
    have "G y t ⊆ connected_component_set t y"
      by (metis (no_types) * connected_component_eq_self connected_component_mono contra_subsetD)
    then have "∃A. openin ?SS A ∧ y ∈ A ∧ A ⊆ connected_component_set t x"
      by (metis (no_types) * connected_component_eq dual_order.trans y)
  }
  then show ?thesis
    using assms openin_subopen by (force simp: locally_def)
qed

lemma locally_connected_3:
  assumes "⋀t x. ⟦openin (top_of_set S) t; x ∈ t⟧
              ⟹ openin (top_of_set S)
                          (connected_component_set t x)"
          "openin (top_of_set S) v" "x ∈ v"
   shows  "∃u. openin (top_of_set S) u ∧ connected u ∧ x ∈ u ∧ u ⊆ v"
using assms connected_component_subset by fastforce

lemma locally_connected:
  "locally connected S ⟷
   (∀v x. openin (top_of_set S) v ∧ x ∈ v
          ⟶ (∃u. openin (top_of_set S) u ∧ connected u ∧ x ∈ u ∧ u ⊆ v))"
by (metis locally_connected_1 locally_connected_2 locally_connected_3)

lemma locally_connected_open_connected_component:
  "locally connected S ⟷
   (∀t x. openin (top_of_set S) t ∧ x ∈ t
          ⟶ openin (top_of_set S) (connected_component_set t x))"
by (metis locally_connected_1 locally_connected_2 locally_connected_3)

lemma locally_path_connected_1:
  assumes
    "⋀v x. ⟦openin (top_of_set S) v; x ∈ v⟧
              ⟹ ∃u. openin (top_of_set S) u ∧ path_connected u ∧ x ∈ u ∧ u ⊆ v"
   shows "locally path_connected S"
  by (force simp: locally_def dest: assms)

lemma locally_path_connected_2:
  assumes "locally path_connected S"
          "openin (top_of_set S) t"
          "x ∈ t"
   shows "openin (top_of_set S) (path_component_set t x)"
proof -
  { fix y :: 'a
    let ?SS = "top_of_set S"
    assume 1: "openin ?SS t"
              "∀w x. openin ?SS w ∧ x ∈ w ⟶ (∃u. openin ?SS u ∧ (∃v. path_connected v ∧ x ∈ u ∧ u ⊆ v ∧ v ⊆ w))"
    and "path_component t x y"
    then have "y ∈ t" and y: "y ∈ path_component_set t x"
      using path_component_mem(2) by blast+
    obtain F where
      "∀x y. (∃w. openin ?SS w ∧ (∃u. path_connected u ∧ x ∈ w ∧ w ⊆ u ∧ u ⊆ y)) = (openin ?SS (F x y) ∧ (∃u. path_connected u ∧ x ∈ F x y ∧ F x y ⊆ u ∧ u ⊆ y))"
      by moura
    then obtain G where
       "∀a A. (∃U. openin ?SS U ∧ (∃V. path_connected V ∧ a ∈ U ∧ U ⊆ V ∧ V ⊆ A)) = (openin ?SS (F a A) ∧ path_connected (G a A) ∧ a ∈ F a A ∧ F a A ⊆ G a A ∧ G a A ⊆ A)"
      by moura
    then have *: "openin ?SS (F y t) ∧ path_connected (G y t) ∧ y ∈ F y t ∧ F y t ⊆ G y t ∧ G y t ⊆ t"
      using 1 ‹y ∈ t› by presburger
    have "G y t ⊆ path_component_set t y"
      using * path_component_maximal rev_subsetD by blast
    then have "∃A. openin ?SS A ∧ y ∈ A ∧ A ⊆ path_component_set t x"
      by (metis "*" ‹G y t ⊆ path_component_set t y› dual_order.trans path_component_eq y)
  }
  then show ?thesis
    using assms openin_subopen by (force simp: locally_def)
qed

lemma locally_path_connected_3:
  assumes "⋀t x. ⟦openin (top_of_set S) t; x ∈ t⟧
              ⟹ openin (top_of_set S) (path_component_set t x)"
          "openin (top_of_set S) v" "x ∈ v"
   shows  "∃u. openin (top_of_set S) u ∧ path_connected u ∧ x ∈ u ∧ u ⊆ v"
proof -
  have "path_component v x x"
    by (meson assms(3) path_component_refl)
  then show ?thesis
    by (metis assms mem_Collect_eq path_component_subset path_connected_path_component)
qed

proposition locally_path_connected:
  "locally path_connected S ⟷
   (∀V x. openin (top_of_set S) V ∧ x ∈ V
          ⟶ (∃U. openin (top_of_set S) U ∧ path_connected U ∧ x ∈ U ∧ U ⊆ V))"
  by (metis locally_path_connected_1 locally_path_connected_2 locally_path_connected_3)

proposition locally_path_connected_open_path_component:
  "locally path_connected S ⟷
   (∀t x. openin (top_of_set S) t ∧ x ∈ t
          ⟶ openin (top_of_set S) (path_component_set t x))"
  by (metis locally_path_connected_1 locally_path_connected_2 locally_path_connected_3)

lemma locally_connected_open_component:
  "locally connected S ⟷
   (∀t c. openin (top_of_set S) t ∧ c ∈ components t
          ⟶ openin (top_of_set S) c)"
by (metis components_iff locally_connected_open_connected_component)

proposition locally_connected_im_kleinen:
  "locally connected S ⟷
   (∀v x. openin (top_of_set S) v ∧ x ∈ v
       ⟶ (∃u. openin (top_of_set S) u ∧
                x ∈ u ∧ u ⊆ v ∧
                (∀y. y ∈ u ⟶ (∃c. connected c ∧ c ⊆ v ∧ x ∈ c ∧ y ∈ c))))"
   (is "?lhs = ?rhs")
proof
  assume ?lhs
  then show ?rhs
    by (fastforce simp: locally_connected)
next
  assume ?rhs
  have *: "∃T. openin (top_of_set S) T ∧ x ∈ T ∧ T ⊆ c"
       if "openin (top_of_set S) t" and c: "c ∈ components t" and "x ∈ c" for t c x
  proof -
    from that ‹?rhs› [rule_format, of t x]
    obtain u where u:
      "openin (top_of_set S) u ∧ x ∈ u ∧ u ⊆ t ∧
       (∀y. y ∈ u ⟶ (∃c. connected c ∧ c ⊆ t ∧ x ∈ c ∧ y ∈ c))"
      using in_components_subset by auto
    obtain F :: "'a set ⇒ 'a set ⇒ 'a" where
      "∀x y. (∃z. z ∈ x ∧ y = connected_component_set x z) = (F x y ∈ x ∧ y = connected_component_set x (F x y))"
      by moura
    then have F: "F t c ∈ t ∧ c = connected_component_set t (F t c)"
      by (meson components_iff c)
    obtain G :: "'a set ⇒ 'a set ⇒ 'a" where
        G: "∀x y. (∃z. z ∈ y ∧ z ∉ x) = (G x y ∈ y ∧ G x y ∉ x)"
      by moura
     have "G c u ∉ u ∨ G c u ∈ c"
      using F by (metis (full_types) u connected_componentI connected_component_eq mem_Collect_eq that(3))
    then show ?thesis
      using G u by auto
  qed
  show ?lhs
    unfolding locally_connected_open_component by (meson "*" openin_subopen)
qed

proposition locally_path_connected_im_kleinen:
  "locally path_connected S ⟷
   (∀v x. openin (top_of_set S) v ∧ x ∈ v
       ⟶ (∃u. openin (top_of_set S) u ∧
                x ∈ u ∧ u ⊆ v ∧
                (∀y. y ∈ u ⟶ (∃p. path p ∧ path_image p ⊆ v ∧
                                pathstart p = x ∧ pathfinish p = y))))"
   (is "?lhs = ?rhs")
proof
  assume ?lhs
  then show ?rhs
    apply (simp add: locally_path_connected path_connected_def)
    apply (erule all_forward ex_forward imp_forward conjE | simp)+
    by (meson dual_order.trans)
next
  assume ?rhs
  have *: "∃T. openin (top_of_set S) T ∧
               x ∈ T ∧ T ⊆ path_component_set u z"
       if "openin (top_of_set S) u" and "z ∈ u" and c: "path_component u z x" for u z x
  proof -
    have "x ∈ u"
      by (meson c path_component_mem(2))
    with that ‹?rhs› [rule_format, of u x]
    obtain U where U:
      "openin (top_of_set S) U ∧ x ∈ U ∧ U ⊆ u ∧
       (∀y. y ∈ U ⟶ (∃p. path p ∧ path_image p ⊆ u ∧ pathstart p = x ∧ pathfinish p = y))"
       by blast
    show ?thesis
      by (metis U c mem_Collect_eq path_component_def path_component_eq subsetI)
  qed
  show ?lhs
    unfolding locally_path_connected_open_path_component
    using "*" openin_subopen by fastforce
qed

lemma locally_path_connected_imp_locally_connected:
  "locally path_connected S ⟹ locally connected S"
using locally_mono path_connected_imp_connected by blast

lemma locally_connected_components:
  "⟦locally connected S; c ∈ components S⟧ ⟹ locally connected c"
by (meson locally_connected_open_component locally_open_subset openin_subtopology_self)

lemma locally_path_connected_components:
  "⟦locally path_connected S; c ∈ components S⟧ ⟹ locally path_connected c"
by (meson locally_connected_open_component locally_open_subset locally_path_connected_imp_locally_connected openin_subtopology_self)

lemma locally_path_connected_connected_component:
  "locally path_connected S ⟹ locally path_connected (connected_component_set S x)"
by (metis components_iff connected_component_eq_empty locally_empty locally_path_connected_components)

lemma open_imp_locally_path_connected:
  fixes S :: "'a :: real_normed_vector set"
  assumes "open S"
  shows "locally path_connected S"
proof (rule locally_mono)
  show "locally convex S"
    using assms unfolding locally_def
    by (meson open_ball centre_in_ball convex_ball openE open_subset openin_imp_subset openin_open_trans subset_trans)
  show "⋀T::'a set. convex T ⟹ path_connected T"
    using convex_imp_path_connected by blast
qed

lemma open_imp_locally_connected:
  fixes S :: "'a :: real_normed_vector set"
  shows "open S ⟹ locally connected S"
by (simp add: locally_path_connected_imp_locally_connected open_imp_locally_path_connected)

lemma locally_path_connected_UNIV: "locally path_connected (UNIV::'a :: real_normed_vector set)"
  by (simp add: open_imp_locally_path_connected)

lemma locally_connected_UNIV: "locally connected (UNIV::'a :: real_normed_vector set)"
  by (simp add: open_imp_locally_connected)

lemma openin_connected_component_locally_connected:
    "locally connected S
     ⟹ openin (top_of_set S) (connected_component_set S x)"
  by (metis connected_component_eq_empty locally_connected_2 openin_empty openin_subtopology_self)

lemma openin_components_locally_connected:
    "⟦locally connected S; c ∈ components S⟧ ⟹ openin (top_of_set S) c"
  using locally_connected_open_component openin_subtopology_self by blast

lemma openin_path_component_locally_path_connected:
  "locally path_connected S
        ⟹ openin (top_of_set S) (path_component_set S x)"
by (metis (no_types) empty_iff locally_path_connected_2 openin_subopen openin_subtopology_self path_component_eq_empty)

lemma closedin_path_component_locally_path_connected:
  assumes "locally path_connected S"
  shows "closedin (top_of_set S) (path_component_set S x)"
proof -
  have "openin (top_of_set S) (⋃ ({path_component_set S y |y. y ∈ S} - {path_component_set S x}))"
    using locally_path_connected_2 assms by fastforce
  then show ?thesis
    by  (simp add: closedin_def path_component_subset complement_path_component_Union)
qed

lemma convex_imp_locally_path_connected:
  fixes S :: "'a:: real_normed_vector set"
  assumes "convex S"
  shows "locally path_connected S"
proof (clarsimp simp: locally_path_connected)
  fix V x
  assume "openin (top_of_set S) V" and "x ∈ V"
  then obtain T e where  "V = S ∩ T" "x ∈ S" "0 < e" "ball x e ⊆ T"
    by (metis Int_iff openE openin_open)
  then have "openin (top_of_set S) (S ∩ ball x e)" "path_connected (S ∩ ball x e)"
    by (simp_all add: assms convex_Int convex_imp_path_connected openin_open_Int)
  then show "∃U. openin (top_of_set S) U ∧ path_connected U ∧ x ∈ U ∧ U ⊆ V"
    using ‹0 < e› ‹V = S ∩ T› ‹ball x e ⊆ T› ‹x ∈ S› by auto
qed

lemma convex_imp_locally_connected:
  fixes S :: "'a:: real_normed_vector set"
  shows "convex S ⟹ locally connected S"
  by (simp add: locally_path_connected_imp_locally_connected convex_imp_locally_path_connected)


subsection‹Relations between components and path components›

lemma path_component_eq_connected_component:
  assumes "locally path_connected S"
    shows "(path_component S x = connected_component S x)"
proof (cases "x ∈ S")
  case True
  have "openin (top_of_set (connected_component_set S x)) (path_component_set S x)"
  proof (rule openin_subset_trans)
    show "openin (top_of_set S) (path_component_set S x)"
      by (simp add: True assms locally_path_connected_2)
    show "connected_component_set S x ⊆ S"
      by (simp add: connected_component_subset)
  qed (simp add: path_component_subset_connected_component)
  moreover have "closedin (top_of_set (connected_component_set S x)) (path_component_set S x)"
    proof (rule closedin_subset_trans [of S])
  show "closedin (top_of_set S) (path_component_set S x)"
    by (simp add: assms closedin_path_component_locally_path_connected)
  show "connected_component_set S x ⊆ S"
    by (simp add: connected_component_subset)
  qed (simp add: path_component_subset_connected_component)
  ultimately have *: "path_component_set S x = connected_component_set S x"
    by (metis connected_connected_component connected_clopen True path_component_eq_empty)
  then show ?thesis
    by blast
next
  case False then show ?thesis
    by (metis Collect_empty_eq_bot connected_component_eq_empty path_component_eq_empty)
qed

lemma path_component_eq_connected_component_set:
     "locally path_connected S ⟹ (path_component_set S x = connected_component_set S x)"
by (simp add: path_component_eq_connected_component)

lemma locally_path_connected_path_component:
     "locally path_connected S ⟹ locally path_connected (path_component_set S x)"
using locally_path_connected_connected_component path_component_eq_connected_component by fastforce

lemma open_path_connected_component:
  fixes S :: "'a :: real_normed_vector set"
  shows "open S ⟹ path_component S x = connected_component S x"
by (simp add: path_component_eq_connected_component open_imp_locally_path_connected)

lemma open_path_connected_component_set:
  fixes S :: "'a :: real_normed_vector set"
  shows "open S ⟹ path_component_set S x = connected_component_set S x"
by (simp add: open_path_connected_component)

proposition locally_connected_quotient_image:
  assumes lcS: "locally connected S"
      and oo: "⋀T. T ⊆ f ` S
                ⟹ openin (top_of_set S) (S ∩ f -` T) ⟷
                    openin (top_of_set (f ` S)) T"
    shows "locally connected (f ` S)"
proof (clarsimp simp: locally_connected_open_component)
  fix U C
  assume opefSU: "openin (top_of_set (f ` S)) U" and "C ∈ components U"
  then have "C ⊆ U" "U ⊆ f ` S"
    by (meson in_components_subset openin_imp_subset)+
  then have "openin (top_of_set (f ` S)) C ⟷
             openin (top_of_set S) (S ∩ f -` C)"
    by (auto simp: oo)
  moreover have "openin (top_of_set S) (S ∩ f -` C)"
  proof (subst openin_subopen, clarify)
    fix x
    assume "x ∈ S" "f x ∈ C"
    show "∃T. openin (top_of_set S) T ∧ x ∈ T ∧ T ⊆ (S ∩ f -` C)"
    proof (intro conjI exI)
      show "openin (top_of_set S) (connected_component_set (S ∩ f -` U) x)"
      proof (rule ccontr)
        assume **: "¬ openin (top_of_set S) (connected_component_set (S ∩ f -` U) x)"
        then have "x ∉ (S ∩ f -` U)"
          using ‹U ⊆ f ` S› opefSU lcS locally_connected_2 oo by blast
        with ** show False
          by (metis (no_types) connected_component_eq_empty empty_iff openin_subopen)
      qed
    next
      show "x ∈ connected_component_set (S ∩ f -` U) x"
        using ‹C ⊆ U› ‹f x ∈ C› ‹x ∈ S› by auto
    next
      have contf: "continuous_on S f"
        by (simp add: continuous_on_open oo openin_imp_subset)
      then have "continuous_on (connected_component_set (S ∩ f -` U) x) f"
        by (meson connected_component_subset continuous_on_subset inf.boundedE)
      then have "connected (f ` connected_component_set (S ∩ f -` U) x)"
        by (rule connected_continuous_image [OF _ connected_connected_component])
      moreover have "f ` connected_component_set (S ∩ f -` U) x ⊆ U"
        using connected_component_in by blast
      moreover have "C ∩ f ` connected_component_set (S ∩ f -` U) x ≠ {}"
        using ‹C ⊆ U› ‹f x ∈ C› ‹x ∈ S› by fastforce
      ultimately have fC: "f ` (connected_component_set (S ∩ f -` U) x) ⊆ C"
        by (rule components_maximal [OF ‹C ∈ components U›])
      have cUC: "connected_component_set (S ∩ f -` U) x ⊆ (S ∩ f -` C)"
        using connected_component_subset fC by blast
      have "connected_component_set (S ∩ f -` U) x ⊆ connected_component_set (S ∩ f -` C) x"
      proof -
        { assume "x ∈ connected_component_set (S ∩ f -` U) x"
          then have ?thesis
            using cUC connected_component_idemp connected_component_mono by blast }
        then show ?thesis
          using connected_component_eq_empty by auto
      qed
      also have "… ⊆ (S ∩ f -` C)"
        by (rule connected_component_subset)
      finally show "connected_component_set (S ∩ f -` U) x ⊆ (S ∩ f -` C)" .
    qed
  qed
  ultimately show "openin (top_of_set (f ` S)) C"
    by metis
qed

text‹The proof resembles that above but is not identical!›
proposition locally_path_connected_quotient_image:
  assumes lcS: "locally path_connected S"
      and oo: "⋀T. T ⊆ f ` S
                ⟹ openin (top_of_set S) (S ∩ f -` T) ⟷ openin (top_of_set (f ` S)) T"
    shows "locally path_connected (f ` S)"
proof (clarsimp simp: locally_path_connected_open_path_component)
  fix U y
  assume opefSU: "openin (top_of_set (f ` S)) U" and "y ∈ U"
  then have "path_component_set U y ⊆ U" "U ⊆ f ` S"
    by (meson path_component_subset openin_imp_subset)+
  then have "openin (top_of_set (f ` S)) (path_component_set U y) ⟷
             openin (top_of_set S) (S ∩ f -` path_component_set U y)"
  proof -
    have "path_component_set U y ⊆ f ` S"
      using ‹U ⊆ f ` S› ‹path_component_set U y ⊆ U› by blast
    then show ?thesis
      using oo by blast
  qed
  moreover have "openin (top_of_set S) (S ∩ f -` path_component_set U y)"
  proof (subst openin_subopen, clarify)
    fix x
    assume "x ∈ S" and Uyfx: "path_component U y (f x)"
    then have "f x ∈ U"
      using path_component_mem by blast
    show "∃T. openin (top_of_set S) T ∧ x ∈ T ∧ T ⊆ (S ∩ f -` path_component_set U y)"
    proof (intro conjI exI)
      show "openin (top_of_set S) (path_component_set (S ∩ f -` U) x)"
      proof (rule ccontr)
        assume **: "¬ openin (top_of_set S) (path_component_set (S ∩ f -` U) x)"
        then have "x ∉ (S ∩ f -` U)"
          by (metis (no_types, lifting) ‹U ⊆ f ` S› opefSU lcS oo locally_path_connected_open_path_component)
        then show False
          using ** ‹path_component_set U y ⊆ U›  ‹x ∈ S› ‹path_component U y (f x)› by blast
      qed
    next
      show "x ∈ path_component_set (S ∩ f -` U) x"
        by (simp add: ‹f x ∈ U› ‹x ∈ S› path_component_refl)
    next
      have contf: "continuous_on S f"
        by (simp add: continuous_on_open oo openin_imp_subset)
      then have "continuous_on (path_component_set (S ∩ f -` U) x) f"
        by (meson Int_lower1 continuous_on_subset path_component_subset)
      then have "path_connected (f ` path_component_set (S ∩ f -` U) x)"
        by (simp add: path_connected_continuous_image)
      moreover have "f ` path_component_set (S ∩ f -` U) x ⊆ U"
        using path_component_mem by fastforce
      moreover have "f x ∈ f ` path_component_set (S ∩ f -` U) x"
        by (force simp: ‹x ∈ S› ‹f x ∈ U› path_component_refl_eq)
      ultimately have "f ` (path_component_set (S ∩ f -` U) x) ⊆ path_component_set U (f x)"
        by (meson path_component_maximal)
       also have  "… ⊆ path_component_set U y"
        by (simp add: Uyfx path_component_maximal path_component_subset path_component_sym)
      finally have fC: "f ` (path_component_set (S ∩ f -` U) x) ⊆ path_component_set U y" .
      have cUC: "path_component_set (S ∩ f -` U) x ⊆ (S ∩ f -` path_component_set U y)"
        using path_component_subset fC by blast
      have "path_component_set (S ∩ f -` U) x ⊆ path_component_set (S ∩ f -` path_component_set U y) x"
      proof -
        have "⋀a. path_component_set (path_component_set (S ∩ f -` U) x) a ⊆ path_component_set (S ∩ f -` path_component_set U y) a"
          using cUC path_component_mono by blast
        then show ?thesis
          using path_component_path_component by blast
      qed
      also have "… ⊆ (S ∩ f -` path_component_set U y)"
        by (rule path_component_subset)
      finally show "path_component_set (S ∩ f -` U) x ⊆ (S ∩ f -` path_component_set U y)" .
    qed
  qed
  ultimately show "openin (top_of_set (f ` S)) (path_component_set U y)"
    by metis
qed

subsection✐‹tag unimportant›‹Components, continuity, openin, closedin›

lemma continuous_on_components_gen:
 fixes f :: "'a::topological_space ⇒ 'b::topological_space"
  assumes "⋀C. C ∈ components S ⟹
              openin (top_of_set S) C ∧ continuous_on C f"
    shows "continuous_on S f"
proof (clarsimp simp: continuous_openin_preimage_eq)
  fix t :: "'b set"
  assume "open t"
  have *: "S ∩ f -` t = (⋃c ∈ components S. c ∩ f -` t)"
    by auto
  show "openin (top_of_set S) (S ∩ f -` t)"
    unfolding * using ‹open t› assms continuous_openin_preimage_gen openin_trans openin_Union by blast
qed

lemma continuous_on_components:
 fixes f :: "'a::topological_space ⇒ 'b::topological_space"
  assumes "locally connected S " "⋀C. C ∈ components S ⟹ continuous_on C f"
  shows "continuous_on S f"
proof (rule continuous_on_components_gen)
  fix C
  assume "C ∈ components S"
  then show "openin (top_of_set S) C ∧ continuous_on C f"
    by (simp add: assms openin_components_locally_connected)
qed

lemma continuous_on_components_eq:
    "locally connected S
     ⟹ (continuous_on S f ⟷ (∀c ∈ components S. continuous_on c f))"
by (meson continuous_on_components continuous_on_subset in_components_subset)

lemma continuous_on_components_open:
 fixes S :: "'a::real_normed_vector set"
  assumes "open S "
          "⋀c. c ∈ components S ⟹ continuous_on c f"
    shows "continuous_on S f"
using continuous_on_components open_imp_locally_connected assms by blast

lemma continuous_on_components_open_eq:
  fixes S :: "'a::real_normed_vector set"
  shows "open S ⟹ (continuous_on S f ⟷ (∀c ∈ components S. continuous_on c f))"
using continuous_on_subset in_components_subset
by (blast intro: continuous_on_components_open)

lemma closedin_union_complement_components:
  assumes U: "locally connected U"
      and S: "closedin (top_of_set U) S"
      and cuS: "c ⊆ components(U - S)"
    shows "closedin (top_of_set U) (S ∪ ⋃c)"
proof -
  have di: "(⋀S T. S ∈ c ∧ T ∈ c' ⟹ disjnt S T) ⟹ disjnt (⋃ c) (⋃ c')" for c'
    by (simp add: disjnt_def) blast
  have "S ⊆ U"
    using S closedin_imp_subset by blast
  moreover have "U - S = ⋃c ∪ ⋃(components (U - S) - c)"
    by (metis Diff_partition Union_components Union_Un_distrib assms(3))
  moreover have "disjnt (⋃c) (⋃(components (U - S) - c))"
    apply (rule di)
    by (metis di DiffD1 DiffD2 assms(3) components_nonoverlap disjnt_def subsetCE)
  ultimately have eq: "S ∪ ⋃c = U - (⋃(components(U - S) - c))"
    by (auto simp: disjnt_def)
  have *: "openin (top_of_set U) (⋃(components (U - S) - c))"
  proof (rule openin_Union [OF openin_trans [of "U - S"]])
    show "openin (top_of_set (U - S)) T" if "T ∈ components (U - S) - c" for T
      using that by (simp add: U S locally_diff_closed openin_components_locally_connected)
    show "openin (top_of_set U) (U - S)" if "T ∈ components (U - S) - c" for T
      using that by (simp add: openin_diff S)
  qed
  have "closedin (top_of_set U) (U - ⋃ (components (U - S) - c))"
    by (metis closedin_diff closedin_topspace topspace_euclidean_subtopology *)
  then have "openin (top_of_set U) (U - (U - ⋃(components (U - S) - c)))"
    by (simp add: openin_diff)
  then show ?thesis
    by (force simp: eq closedin_def)
qed

lemma closed_union_complement_components:
  fixes S :: "'a::real_normed_vector set"
  assumes S: "closed S" and c: "c ⊆ components(- S)"
    shows "closed(S ∪ ⋃ c)"
proof -
  have "closedin (top_of_set UNIV) (S ∪ ⋃c)"
    by (metis Compl_eq_Diff_UNIV S c closed_closedin closedin_union_complement_components locally_connected_UNIV subtopology_UNIV)
  then show ?thesis by simp
qed

lemma closedin_Un_complement_component:
  fixes S :: "'a::real_normed_vector set"
  assumes u: "locally connected u"
      and S: "closedin (top_of_set u) S"
      and c: " c ∈ components(u - S)"
    shows "closedin (top_of_set u) (S ∪ c)"
proof -
  have "closedin (top_of_set u) (S ∪ ⋃{c})"
    using c by (blast intro: closedin_union_complement_components [OF u S])
  then show ?thesis
    by simp
qed

lemma closed_Un_complement_component:
  fixes S :: "'a::real_normed_vector set"
  assumes S: "closed S" and c: " c ∈ components(-S)"
    shows "closed (S ∪ c)"
  by (metis Compl_eq_Diff_UNIV S c closed_closedin closedin_Un_complement_component
      locally_connected_UNIV subtopology_UNIV)


subsection‹Existence of isometry between subspaces of same dimension›

lemma isometry_subset_subspace:
  fixes S :: "'a::euclidean_space set"
    and T :: "'b::euclidean_space set"
  assumes S: "subspace S"
      and T: "subspace T"
      and d: "dim S ≤ dim T"
  obtains f where "linear f" "f ∈ S → T" "⋀x. x ∈ S ⟹ norm(f x) = norm x"
proof -
  obtain B where "B ⊆ S" and Borth: "pairwise orthogonal B"
             and B1: "⋀x. x ∈ B ⟹ norm x = 1"
             and "independent B" "finite B" "card B = dim S" "span B = S"
    by (metis orthonormal_basis_subspace [OF S] independent_imp_finite)
  obtain C where "C ⊆ T" and Corth: "pairwise orthogonal C"
             and C1:"⋀x. x ∈ C ⟹ norm x = 1"
             and "independent C" "finite C" "card C = dim T" "span C = T"
    by (metis orthonormal_basis_subspace [OF T] independent_imp_finite)
  obtain fb where "fb ` B ⊆ C" "inj_on fb B"
    by (metis ‹card B = dim S› ‹card C = dim T› ‹finite B› ‹finite C› card_le_inj d)
  then have pairwise_orth_fb: "pairwise (λv j. orthogonal (fb v) (fb j)) B"
    using Corth unfolding pairwise_def inj_on_def
    by (blast intro: orthogonal_clauses)
  obtain f where "linear f" and ffb: "⋀x. x ∈ B ⟹ f x = fb x"
    using linear_independent_extend ‹independent B› by fastforce
  have "span (f ` B) ⊆ span C"
    by (metis ‹fb ` B ⊆ C› ffb image_cong span_mono)
  then have "f ` S ⊆ T"
    unfolding ‹span B = S› ‹span C = T› span_linear_image[OF ‹linear f›] .
  have [simp]: "⋀x. x ∈ B ⟹ norm (fb x) = norm x"
    using B1 C1 ‹fb ` B ⊆ C› by auto
  have "norm (f x) = norm x" if "x ∈ S" for x
  proof -
    interpret linear f by fact
    obtain a where x: "x = (∑v ∈ B. a v *⇩R v)"
      using ‹finite B› ‹span B = S› ‹x ∈ S› span_finite by fastforce
    have "norm (f x)^2 = norm (∑v∈B. a v *⇩R fb v)^2" by (simp add: sum scale ffb x)
    also have "… = (∑v∈B. norm ((a v *⇩R fb v))^2)"
    proof (rule norm_sum_Pythagorean [OF ‹finite B›])
      show "pairwise (λv j. orthogonal (a v *⇩R fb v) (a j *⇩R fb j)) B"
        by (rule pairwise_ortho_scaleR [OF pairwise_orth_fb])
    qed
    also have "… = norm x ^2"
      by (simp add: x pairwise_ortho_scaleR Borth norm_sum_Pythagorean [OF ‹finite B›])
    finally show ?thesis
      by (simp add: norm_eq_sqrt_inner)
  qed
  then show ?thesis
    by (meson ‹f ` S ⊆ T› ‹linear f› image_subset_iff_funcset that)
qed

proposition isometries_subspaces:
  fixes S :: "'a::euclidean_space set"
    and T :: "'b::euclidean_space set"
  assumes S: "subspace S"
      and T: "subspace T"
      and d: "dim S = dim T"
  obtains f g where "linear f" "linear g" "f ` S = T" "g ` T = S"
                    "⋀x. x ∈ S ⟹ norm(f x) = norm x"
                    "⋀x. x ∈ T ⟹ norm(g x) = norm x"
                    "⋀x. x ∈ S ⟹ g(f x) = x"
                    "⋀x. x ∈ T ⟹ f(g x) = x"
proof -
  obtain B where "B ⊆ S" and Borth: "pairwise orthogonal B"
             and B1: "⋀x. x ∈ B ⟹ norm x = 1"
             and "independent B" "finite B" "card B = dim S" "span B = S"
    by (metis orthonormal_basis_subspace [OF S] independent_imp_finite)
  obtain C where "C ⊆ T" and Corth: "pairwise orthogonal C"
             and C1:"⋀x. x ∈ C ⟹ norm x = 1"
             and "independent C" "finite C" "card C = dim T" "span C = T"
    by (metis orthonormal_basis_subspace [OF T] independent_imp_finite)
  obtain fb where "bij_betw fb B C"
    by (metis ‹finite B› ‹finite C› bij_betw_iff_card ‹card B = dim S› ‹card C = dim T› d)
  then have pairwise_orth_fb: "pairwise (λv j. orthogonal (fb v) (fb j)) B"
    using Corth unfolding pairwise_def inj_on_def bij_betw_def
    by (blast intro: orthogonal_clauses)
  obtain f where "linear f" and ffb: "⋀x. x ∈ B ⟹ f x = fb x"
    using linear_independent_extend ‹independent B› by fastforce
  interpret f: linear f by fact
  define gb where "gb ≡ inv_into B fb"
  then have pairwise_orth_gb: "pairwise (λv j. orthogonal (gb v) (gb j)) C"
    using Borth ‹bij_betw fb B C› unfolding pairwise_def bij_betw_def by force
  obtain g where "linear g" and ggb: "⋀x. x ∈ C ⟹ g x = gb x"
    using linear_independent_extend ‹independent C› by fastforce
  interpret g: linear g by fact
  have "span (f ` B) ⊆ span C"
    by (metis ‹bij_betw fb B C› bij_betw_imp_surj_on eq_iff ffb image_cong)
  then have "f ` S ⊆ T"
    unfolding ‹span B = S› ‹span C = T› span_linear_image[OF ‹linear f›] .
  have [simp]: "⋀x. x ∈ B ⟹ norm (fb x) = norm x"
    using B1 C1 ‹bij_betw fb B C› bij_betw_imp_surj_on by fastforce
  have f [simp]: "norm (f x) = norm x" "g (f x) = x" if "x ∈ S" for x
  proof -
    obtain a where x: "x = (∑v ∈ B. a v *⇩R v)"
      using ‹finite B› ‹span B = S› ‹x ∈ S› span_finite by fastforce
    have "f x = (∑v ∈ B. f (a v *⇩R v))"
      using linear_sum [OF ‹linear f›] x by auto
    also have "… = (∑v ∈ B. a v *⇩R f v)"
      by (simp add: f.sum f.scale)
    also have "… = (∑v ∈ B. a v *⇩R fb v)"
      by (simp add: ffb cong: sum.cong)
    finally have *: "f x = (∑v∈B. a v *⇩R fb v)" .
    then have "(norm (f x))⇧2 = (norm (∑v∈B. a v *⇩R fb v))⇧2" by simp
    also have "… = (∑v∈B. norm ((a v *⇩R fb v))^2)"
    proof (rule norm_sum_Pythagorean [OF ‹finite B›])
      show "pairwise (λv j. orthogonal (a v *⇩R fb v) (a j *⇩R fb j)) B"
        by (rule pairwise_ortho_scaleR [OF pairwise_orth_fb])
    qed
    also have "… = (norm x)⇧2"
      by (simp add: x pairwise_ortho_scaleR Borth norm_sum_Pythagorean [OF ‹finite B›])
    finally show "norm (f x) = norm x"
      by (simp add: norm_eq_sqrt_inner)
    have "g (f x) = g (∑v∈B. a v *⇩R fb v)" by (simp add: *)
    also have "… = (∑v∈B. g (a v *⇩R fb v))"
      by (simp add: g.sum g.scale)
    also have "… = (∑v∈B. a v *⇩R g (fb v))"
      by (simp add: g.scale)
    also have "… = (∑v∈B. a v *⇩R v)"
    proof (rule sum.cong [OF refl])
      show "a x *⇩R g (fb x) = a x *⇩R x" if "x ∈ B" for x
        using that ‹bij_betw fb B C› bij_betwE bij_betw_inv_into_left gb_def ggb by fastforce
    qed
    also have "… = x"
      using x by blast
    finally show "g (f x) = x" .
  qed
  have [simp]: "⋀x. x ∈ C ⟹ norm (gb x) = norm x"
    by (metis B1 C1 ‹bij_betw fb B C› bij_betw_imp_surj_on gb_def inv_into_into)
  have g [simp]: "f (g x) = x" if "x ∈ T" for x
  proof -
    obtain a where x: "x = (∑v ∈ C. a v *⇩R v)"
      using ‹finite C› ‹span C = T› ‹x ∈ T› span_finite by fastforce
    have "g x = (∑v ∈ C. g (a v *⇩R v))"
      by (simp add: x g.sum)
    also have "… = (∑v ∈ C. a v *⇩R g v)"
      by (simp add: g.scale)
    also have "… = (∑v ∈ C. a v *⇩R gb v)"
      by (simp add: ggb cong: sum.cong)
    finally have "f (g x) = f (∑v∈C. a v *⇩R gb v)" by simp
    also have "… = (∑v∈C. f (a v *⇩R gb v))"
      by (simp add: f.scale f.sum)
    also have "… = (∑v∈C. a v *⇩R f (gb v))"
      by (simp add: f.scale f.sum)
    also have "… = (∑v∈C. a v *⇩R v)"
      using ‹bij_betw fb B C›
      by (simp add: bij_betw_def gb_def bij_betw_inv_into_right ffb inv_into_into)
    also have "… = x"
      using x by blast
    finally show "f (g x) = x" .
  qed
  have gim: "g ` T = S"
    by (metis (full_types) S T ‹f ` S ⊆ T› d dim_eq_span dim_image_le f(2) g.linear_axioms
        image_iff linear_subspace_image span_eq_iff subset_iff)
  have fim: "f ` S = T"
    using ‹g ` T = S› image_iff by fastforce
  have [simp]: "norm (g x) = norm x" if "x ∈ T" for x
    using fim that by auto
  show ?thesis
    by (rule that [OF ‹linear f› ‹linear g›]) (simp_all add: fim gim)
qed

corollary isometry_subspaces:
  fixes S :: "'a::euclidean_space set"
    and T :: "'b::euclidean_space set"
  assumes S: "subspace S"
      and T: "subspace T"
      and d: "dim S = dim T"
  obtains f where "linear f" "f ` S = T" "⋀x. x ∈ S ⟹ norm(f x) = norm x"
using isometries_subspaces [OF assms]
by metis

corollary isomorphisms_UNIV_UNIV:
  assumes "DIM('M) = DIM('N)"
  obtains f::"'M::euclidean_space ⇒'N::euclidean_space" and g
  where "linear f" "linear g"
                    "⋀x. norm(f x) = norm x" "⋀y. norm(g y) = norm y"
                    "⋀x. g (f x) = x" "⋀y. f(g y) = y"
  using assms by (auto intro: isometries_subspaces [of "UNIV::'M set" "UNIV::'N set"])

lemma homeomorphic_subspaces:
  fixes S :: "'a::euclidean_space set"
    and T :: "'b::euclidean_space set"
  assumes S: "subspace S"
      and T: "subspace T"
      and d: "dim S = dim T"
    shows "S homeomorphic T"
proof -
  obtain f g where "linear f" "linear g" "f ` S = T" "g ` T = S"
                   "⋀x. x ∈ S ⟹ g(f x) = x" "⋀x. x ∈ T ⟹ f(g x) = x"
    by (blast intro: isometries_subspaces [OF assms])
  then show ?thesis
    unfolding homeomorphic_def homeomorphism_def
    apply (rule_tac x=f in exI, rule_tac x=g in exI)
    apply (auto simp: linear_continuous_on linear_conv_bounded_linear)
    done
qed

lemma homeomorphic_affine_sets:
  assumes "affine S" "affine T" "aff_dim S = aff_dim T"
    shows "S homeomorphic T"
proof (cases "S = {} ∨ T = {}")
  case True  with assms aff_dim_empty homeomorphic_empty show ?thesis
    by metis
next
  case False
  then obtain a b where ab: "a ∈ S" "b ∈ T" by auto
  then have ss: "subspace ((+) (- a) ` S)" "subspace ((+) (- b) ` T)"
    using affine_diffs_subspace assms by blast+
  have dd: "dim ((+) (- a) ` S) = dim ((+) (- b) ` T)"
    using assms ab  by (simp add: aff_dim_eq_dim  [OF hull_inc] image_def)
  have "S homeomorphic ((+) (- a) ` S)"
    by (fact homeomorphic_translation)
  also have "… homeomorphic ((+) (- b) ` T)"
    by (rule homeomorphic_subspaces [OF ss dd])
  also have "… homeomorphic T"
    using homeomorphic_translation [of T "- b"] by (simp add: homeomorphic_sym [of T])
  finally show ?thesis .
qed


subsection‹Retracts, in a general sense, preserve (co)homotopic triviality)›

locale✐‹tag important› Retracts =
  fixes S h t k
  assumes conth: "continuous_on S h"
      and imh: "h ` S = t"
      and contk: "continuous_on t k"
      and imk: "k ∈ t → S"
      and idhk: "⋀y. y ∈ t ⟹ h(k y) = y"

begin

lemma homotopically_trivial_retraction_gen:
  assumes P: "⋀f. ⟦continuous_on U f; f ∈ U → t; Q f⟧ ⟹ P(k ∘ f)"
      and Q: "⋀f. ⟦continuous_on U f; f ∈ U → S; P f⟧ ⟹ Q(h ∘ f)"
      and Qeq: "⋀h k. (⋀x. x ∈ U ⟹ h x = k x) ⟹ Q h = Q k"
      and hom: "⋀f g. ⟦continuous_on U f; f ∈ U → S; P f;
                       continuous_on U g; g ∈ U → S; P g⟧
                       ⟹ homotopic_with_canon P U S f g"
      and contf: "continuous_on U f" and imf: "f ∈ U → t" and Qf: "Q f"
      and contg: "continuous_on U g" and img: "g ∈ U → t" and Qg: "Q g"
    shows "homotopic_with_canon Q U t f g"
proof -
  have "continuous_on U (k ∘ f)"
    by (meson contf continuous_on_compose continuous_on_subset contk funcset_image imf)
  moreover have "(k ∘ f) ` U ⊆ S"
    using imf imk by fastforce
  moreover have "P (k ∘ f)"
    by (simp add: P Qf contf imf)
  moreover have "continuous_on U (k ∘ g)"
    by (meson contg continuous_on_compose continuous_on_subset contk funcset_image img)
  moreover have "(k ∘ g) ` U ⊆ S"
    using img imk by fastforce
  moreover have "P (k ∘ g)"
    by (simp add: P Qg contg img)
  ultimately have "homotopic_with_canon P U S (k ∘ f) (k ∘ g)"
    by (simp add: hom image_subset_iff)
  then have "homotopic_with_canon Q U t (h ∘ (k ∘ f)) (h ∘ (k ∘ g))"
    apply (rule homotopic_with_compose_continuous_left [OF homotopic_with_mono])
    using Q conth imh by force+
  then show ?thesis
  proof (rule homotopic_with_eq; simp)
    show "⋀h k. (⋀x. x ∈ U ⟹ h x = k x) ⟹ Q h = Q k"
      using Qeq topspace_euclidean_subtopology by blast
    show "⋀x. x ∈ U ⟹ f x = h (k (f x))" "⋀x. x ∈ U ⟹ g x = h (k (g x))"
      using idhk imf img by fastforce+
  qed 
qed

lemma homotopically_trivial_retraction_null_gen:
  assumes P: "⋀f. ⟦continuous_on U f; f ∈ U → t; Q f⟧ ⟹ P(k ∘ f)"
      and Q: "⋀f. ⟦continuous_on U f; f ∈ U → S; P f⟧ ⟹ Q(h ∘ f)"
      and Qeq: "⋀h k. (⋀x. x ∈ U ⟹ h x = k x) ⟹ Q h = Q k"
      and hom: "⋀f. ⟦continuous_on U f; f ∈ U → S; P f⟧
                     ⟹ ∃c. homotopic_with_canon P U S f (λx. c)"
      and contf: "continuous_on U f" and imf:"f ∈ U → t" and Qf: "Q f"
  obtains c where "homotopic_with_canon Q U t f (λx. c)"
proof -
  have feq: "⋀x. x ∈ U ⟹ (h ∘ (k ∘ f)) x = f x" using idhk imf by auto
  have "continuous_on U (k ∘ f)"
    by (meson contf continuous_on_compose continuous_on_subset contk funcset_image imf)
  moreover have "(k ∘ f) ∈ U → S"
    using imf imk by fastforce
  moreover have "P (k ∘ f)"
    by (simp add: P Qf contf imf)
  ultimately obtain c where "homotopic_with_canon P U S (k ∘ f) (λx. c)"
    by (metis hom)
  then have "homotopic_with_canon Q U t (h ∘ (k ∘ f)) (h ∘ (λx. c))"
    apply (rule homotopic_with_compose_continuous_left [OF homotopic_with_mono])
    using Q conth imh by force+
  then have "homotopic_with_canon Q U t f (λx. h c)"
  proof (rule homotopic_with_eq)
    show "⋀x. x ∈ topspace (top_of_set U) ⟹ f x = (h ∘ (k ∘ f)) x"
      using feq by auto
    show "⋀h k. (⋀x. x ∈ topspace (top_of_set U) ⟹ h x = k x) ⟹ Q h = Q k"
      using Qeq topspace_euclidean_subtopology by blast
  qed auto
  then show ?thesis
    using that by blast
qed

lemma cohomotopically_trivial_retraction_gen:
  assumes P: "⋀f. ⟦continuous_on t f; f ∈ t → U; Q f⟧ ⟹ P(f ∘ h)"
      and Q: "⋀f. ⟦continuous_on S f; f ∈ S → U; P f⟧ ⟹ Q(f ∘ k)"
      and Qeq: "⋀h k. (⋀x. x ∈ t ⟹ h x = k x) ⟹ Q h = Q k"
      and hom: "⋀f g. ⟦continuous_on S f; f ∈ S → U; P f;
                       continuous_on S g; g ∈ S → U; P g⟧
                       ⟹ homotopic_with_canon P S U f g"
      and contf: "continuous_on t f" and imf: "f ∈ t → U" and Qf: "Q f"
      and contg: "continuous_on t g" and img: "g ∈ t → U" and Qg: "Q g"
    shows "homotopic_with_canon Q t U f g"
proof -
  have feq: "⋀x. x ∈ t ⟹ (f ∘ h ∘ k) x = f x" using idhk imf by auto
  have geq: "⋀x. x ∈ t ⟹ (g ∘ h ∘ k) x = g x" using idhk img by auto
  have "continuous_on S (f ∘ h)"
    using contf conth continuous_on_compose imh by blast
  moreover have "(f ∘ h) ∈ S → U"
    using imf imh by fastforce
  moreover have "P (f ∘ h)"
    by (simp add: P Qf contf imf)
  moreover have "continuous_on S (g ∘ h)"
    using contg continuous_on_compose continuous_on_subset conth imh by blast
  moreover have "(g ∘ h) ∈ S → U"
    using img imh by fastforce
  moreover have "P (g ∘ h)"
    by (simp add: P Qg contg img)
  ultimately have "homotopic_with_canon P S U (f ∘ h) (g ∘ h)"
    by (simp add: hom)
  then have "homotopic_with_canon Q t U (f ∘ h ∘ k) (g ∘ h ∘ k)"
    apply (rule homotopic_with_compose_continuous_right [OF homotopic_with_mono])
    using Q contk imk by force+
  then show ?thesis
  proof (rule homotopic_with_eq)
    show "f x = (f ∘ h ∘ k) x" "g x = (g ∘ h ∘ k) x" 
      if "x ∈ topspace (top_of_set t)" for x
      using feq geq that by force+
  qed (use Qeq topspace_euclidean_subtopology in blast)
qed

lemma cohomotopically_trivial_retraction_null_gen:
  assumes P: "⋀f. ⟦continuous_on t f; f ∈ t → U; Q f⟧ ⟹ P(f ∘ h)"
      and Q: "⋀f. ⟦continuous_on S f; f ∈ S → U; P f⟧ ⟹ Q(f ∘ k)"
      and Qeq: "⋀h k. (⋀x. x ∈ t ⟹ h x = k x) ⟹ Q h = Q k"
      and hom: "⋀f g. ⟦continuous_on S f; f ∈ S → U; P f⟧
                       ⟹ ∃c. homotopic_with_canon P S U f (λx. c)"
      and contf: "continuous_on t f" and imf: "f ∈ t → U" and Qf: "Q f"
  obtains c where "homotopic_with_canon Q t U f (λx. c)"
proof -
  have feq: "⋀x. x ∈ t ⟹ (f ∘ h ∘ k) x = f x" using idhk imf by auto
  have "continuous_on S (f ∘ h)"
    using contf conth continuous_on_compose imh by blast
  moreover have "(f ∘ h) ∈ S → U"
    using imf imh by fastforce
  moreover have "P (f ∘ h)"
    by (simp add: P Qf contf imf)
  ultimately obtain c where "homotopic_with_canon P S U (f ∘ h) (λx. c)"
    by (metis hom)
  then have §: "homotopic_with_canon Q t U (f ∘ h ∘ k) ((λx. c) ∘ k)"
  proof (rule homotopic_with_compose_continuous_right [OF homotopic_with_mono])
    show "⋀h. ⟦continuous_map (top_of_set S) (top_of_set U) h; P h⟧ ⟹ Q (h ∘ k)"
      using Q by auto
  qed (use contk imk in force)+
  moreover have "homotopic_with_canon Q t U f (λx. c)"
    using homotopic_with_eq [OF §] feq Qeq by fastforce
  ultimately show ?thesis 
    using that by blast
qed

end

lemma simply_connected_retraction_gen:
  shows "⟦simply_connected S; continuous_on S h; h ` S = T;
          continuous_on T k; k ∈ T → S; ⋀y. y ∈ T ⟹ h(k y) = y⟧
        ⟹ simply_connected T"
apply (simp add: simply_connected_def path_def path_image_def homotopic_loops_def, clarify)
apply (rule Retracts.homotopically_trivial_retraction_gen
        [of S h _ k _ "λp. pathfinish p = pathstart p"  "λp. pathfinish p = pathstart p"])
apply (simp_all add: Retracts_def pathfinish_def pathstart_def image_subset_iff_funcset)
done

lemma homeomorphic_simply_connected:
    "⟦S homeomorphic T; simply_connected S⟧ ⟹ simply_connected T"
  by (auto simp: homeomorphic_def homeomorphism_def intro: simply_connected_retraction_gen)

lemma homeomorphic_simply_connected_eq:
    "S homeomorphic T ⟹ (simply_connected S ⟷ simply_connected T)"
  by (metis homeomorphic_simply_connected homeomorphic_sym)


subsection‹Homotopy equivalence›

subsection‹Homotopy equivalence of topological spaces.›

definition✐‹tag important› homotopy_equivalent_space
             (infix "homotopy'_equivalent'_space" 50)
  where "X homotopy_equivalent_space Y ≡
        (∃f g. continuous_map X Y f ∧
              continuous_map Y X g ∧
              homotopic_with (λx. True) X X (g ∘ f) id ∧
              homotopic_with (λx. True) Y Y (f ∘ g) id)"

lemma homeomorphic_imp_homotopy_equivalent_space:
  "X homeomorphic_space Y ⟹ X homotopy_equivalent_space Y"
  unfolding homeomorphic_space_def homotopy_equivalent_space_def
  apply (erule ex_forward)+
  by (simp add: homotopic_with_equal homotopic_with_sym homeomorphic_maps_def)

lemma homotopy_equivalent_space_refl:
   "X homotopy_equivalent_space X"
  by (simp add: homeomorphic_imp_homotopy_equivalent_space homeomorphic_space_refl)

lemma homotopy_equivalent_space_sym:
   "X homotopy_equivalent_space Y ⟷ Y homotopy_equivalent_space X"
  by (meson homotopy_equivalent_space_def)

lemma homotopy_eqv_trans [trans]:
  assumes 1: "X homotopy_equivalent_space Y" and 2: "Y homotopy_equivalent_space U"
    shows "X homotopy_equivalent_space U"
proof -
  obtain f1 g1 where f1: "continuous_map X Y f1"
                 and g1: "continuous_map Y X g1"
                 and hom1: "homotopic_with (λx. True) X X (g1 ∘ f1) id"
                           "homotopic_with (λx. True) Y Y (f1 ∘ g1) id"
    using 1 by (auto simp: homotopy_equivalent_space_def)
  obtain f2 g2 where f2: "continuous_map Y U f2"
                 and g2: "continuous_map U Y g2"
                 and hom2: "homotopic_with (λx. True) Y Y (g2 ∘ f2) id"
                           "homotopic_with (λx. True) U U (f2 ∘ g2) id"
    using 2 by (auto simp: homotopy_equivalent_space_def)
  have "homotopic_with (λf. True) X Y (g2 ∘ f2 ∘ f1) (id ∘ f1)"
    using f1 hom2(1) homotopic_with_compose_continuous_map_right by metis
  then have "homotopic_with (λf. True) X Y (g2 ∘ (f2 ∘ f1)) (id ∘ f1)"
    by (simp add: o_assoc)
  then have "homotopic_with (λx. True) X X
         (g1 ∘ (g2 ∘ (f2 ∘ f1))) (g1 ∘ (id ∘ f1))"
    by (simp add: g1 homotopic_with_compose_continuous_map_left)
  moreover have "homotopic_with (λx. True) X X (g1 ∘ id ∘ f1) id"
    using hom1 by simp
  ultimately have SS: "homotopic_with (λx. True) X X (g1 ∘ g2 ∘ (f2 ∘ f1)) id"
    by (metis comp_assoc homotopic_with_trans id_comp)
  have "homotopic_with (λf. True) U Y (f1 ∘ g1 ∘ g2) (id ∘ g2)"
    using g2 hom1(2) homotopic_with_compose_continuous_map_right by fastforce
  then have "homotopic_with (λf. True) U Y (f1 ∘ (g1 ∘ g2)) (id ∘ g2)"
    by (simp add: o_assoc)
  then have "homotopic_with (λx. True) U U
         (f2 ∘ (f1 ∘ (g1 ∘ g2))) (f2 ∘ (id ∘ g2))"
    by (simp add: f2 homotopic_with_compose_continuous_map_left)
  moreover have "homotopic_with (λx. True) U U (f2 ∘ id ∘ g2) id"
    using hom2 by simp
  ultimately have UU: "homotopic_with (λx. True) U U (f2 ∘ f1 ∘ (g1 ∘ g2)) id"
    by (simp add: fun.map_comp hom2(2) homotopic_with_trans)
  show ?thesis
    unfolding homotopy_equivalent_space_def
    by (blast intro: f1 f2 g1 g2 continuous_map_compose SS UU)
qed

lemma deformation_retraction_imp_homotopy_equivalent_space:
  "⟦homotopic_with (λx. True) X X (S ∘ r) id; retraction_maps X Y r S⟧
    ⟹ X homotopy_equivalent_space Y"
  unfolding homotopy_equivalent_space_def retraction_maps_def
  using homotopic_with_id2 by fastforce

lemma deformation_retract_imp_homotopy_equivalent_space:
   "⟦homotopic_with (λx. True) X X r id; retraction_maps X Y r id⟧
    ⟹ X homotopy_equivalent_space Y"
  using deformation_retraction_imp_homotopy_equivalent_space by force

lemma deformation_retract_of_space:
  "S ⊆ topspace X ∧
   (∃r. homotopic_with (λx. True) X X id r ∧ retraction_maps X (subtopology X S) r id) ⟷
   S retract_of_space X ∧ (∃f. homotopic_with (λx. True) X X id f ∧ f ` (topspace X) ⊆ S)"
proof (cases "S ⊆ topspace X")
  case True
  moreover have "(∃r. homotopic_with (λx. True) X X id r ∧ retraction_maps X (subtopology X S) r id)
             ⟷ (S retract_of_space X ∧ (∃f. homotopic_with (λx. True) X X id f ∧ f ` topspace X ⊆ S))"
    unfolding retract_of_space_def
  proof safe
    fix f r
    assume f: "homotopic_with (λx. True) X X id f"
      and fS: "f ` topspace X ⊆ S"
      and r: "continuous_map X (subtopology X S) r"
      and req: "∀x∈S. r x = x"
    show "∃r. homotopic_with (λx. True) X X id r ∧ retraction_maps X (subtopology X S) r id"
    proof (intro exI conjI)
      have "homotopic_with (λx. True) X X f r"
        proof (rule homotopic_with_eq)
          show "homotopic_with (λx. True) X X (r ∘ f) (r ∘ id)"
            by (metis continuous_map_into_fulltopology f homotopic_with_compose_continuous_map_left homotopic_with_symD r)
          show "f x = (r ∘ f) x" if "x ∈ topspace X" for x
            using that fS req by auto
        qed auto
      then show "homotopic_with (λx. True) X X id r"
        by (rule homotopic_with_trans [OF f])
    next
      show "retraction_maps X (subtopology X S) r id"
        by (simp add: r req retraction_maps_def)
    qed
  qed (use True in ‹auto simp: retraction_maps_def topspace_subtopology_subset continuous_map_in_subtopology›)
  ultimately show ?thesis by simp
qed (auto simp: retract_of_space_def retraction_maps_def)



subsection‹Contractible spaces›

text‹The definition (which agrees with "contractible" on subsets of Euclidean space)
is a little cryptic because we don't in fact assume that the constant "a" is in the space.
This forces the convention that the empty space / set is contractible, avoiding some special cases. ›

definition contractible_space where
  "contractible_space X ≡ ∃a. homotopic_with (λx. True) X X id (λx. a)"

lemma contractible_space_top_of_set [simp]:"contractible_space (top_of_set S) ⟷ contractible S"
  by (auto simp: contractible_space_def contractible_def)

lemma contractible_space_empty [simp]:
   "contractible_space trivial_topology"
  unfolding contractible_space_def homotopic_with_def
  apply (rule_tac x=undefined in exI)
  apply (rule_tac x="λ(t,x). if t = 0 then x else undefined" in exI)
  apply (auto simp: continuous_map_on_empty)
  done

lemma contractible_space_singleton [simp]:
  "contractible_space (discrete_topology{a})"
  unfolding contractible_space_def homotopic_with_def
  apply (rule_tac x=a in exI)
  apply (rule_tac x="λ(t,x). if t = 0 then x else a" in exI)
  apply (auto intro: continuous_map_eq [where f = "λz. a"])
  done

lemma contractible_space_subset_singleton:
   "topspace X ⊆ {a} ⟹ contractible_space X"
  by (metis contractible_space_empty contractible_space_singleton null_topspace_iff_trivial subset_singletonD subtopology_eq_discrete_topology_sing)

lemma contractible_space_subtopology_singleton [simp]:
   "contractible_space (subtopology X {a})"
  by (meson contractible_space_subset_singleton insert_subset path_connectedin_singleton path_connectedin_subtopology subsetI)

lemma contractible_space:
   "contractible_space X ⟷
        X = trivial_topology ∨
        (∃a ∈ topspace X. homotopic_with (λx. True) X X id (λx. a))"
proof (cases "X = trivial_topology")
  case False
  then show ?thesis
    using homotopic_with_imp_continuous_maps  by (fastforce simp: contractible_space_def)
qed (simp add: contractible_space_empty)

lemma contractible_imp_path_connected_space:
  assumes "contractible_space X" shows "path_connected_space X"
proof (cases "X = trivial_topology")
  case False
  have *: "path_connected_space X"
    if "a ∈ topspace X" and conth: "continuous_map (prod_topology (top_of_set {0..1}) X) X h"
      and h: "∀x. h (0, x) = x" "∀x. h (1, x) = a"
    for a and h :: "real × 'a ⇒ 'a"
  proof -
    have "path_component_of X b a" if "b ∈ topspace X" for b
      unfolding path_component_of_def
    proof (intro exI conjI)
      let ?g = "h ∘ (λx. (x,b))"
      show "pathin X ?g"
        unfolding pathin_def
      proof (rule continuous_map_compose [OF _ conth])
        show "continuous_map (top_of_set {0..1}) (prod_topology (top_of_set {0..1}) X) (λx. (x, b))"
          using that by (auto intro!: continuous_intros)
      qed
    qed (use h in auto)
  then show ?thesis
    by (metis path_component_of_equiv path_connected_space_iff_path_component)
  qed
  show ?thesis
    using assms False by (auto simp: contractible_space homotopic_with_def *)
qed (simp add: path_connected_space_topspace_empty)

lemma contractible_imp_connected_space:
   "contractible_space X ⟹ connected_space X"
  by (simp add: contractible_imp_path_connected_space path_connected_imp_connected_space)

lemma contractible_space_alt:
   "contractible_space X ⟷ (∀a ∈ topspace X. homotopic_with (λx. True) X X id (λx. a))" (is "?lhs = ?rhs")
proof
  assume X: ?lhs
  then obtain a where a: "homotopic_with (λx. True) X X id (λx. a)"
    by (auto simp: contractible_space_def)
  show ?rhs
  proof
    show "homotopic_with (λx. True) X X id (λx. b)" if "b ∈ topspace X" for b
    proof (rule homotopic_with_trans [OF a])
      show "homotopic_with (λx. True) X X (λx. a) (λx. b)"
        using homotopic_constant_maps path_connected_space_imp_path_component_of
        by (metis X a contractible_imp_path_connected_space homotopic_with_sym homotopic_with_trans path_component_of_equiv that)
    qed
  qed
next
  assume R: ?rhs
  then show ?lhs
    using contractible_space_def by fastforce
qed


lemma compose_const [simp]: "f ∘ (λx. a) = (λx. f a)" "(λx. a) ∘ g = (λx. a)"
  by (simp_all add: o_def)

lemma nullhomotopic_through_contractible_space:
  assumes f: "continuous_map X Y f" and g: "continuous_map Y Z g" and Y: "contractible_space Y"
  obtains c where "homotopic_with (λh. True) X Z (g ∘ f) (λx. c)"
proof -
  obtain b where b: "homotopic_with (λx. True) Y Y id (λx. b)"
    using Y by (auto simp: contractible_space_def)
  show thesis
    using homotopic_with_compose_continuous_map_right
           [OF homotopic_with_compose_continuous_map_left [OF b g] f]
    by (force simp: that)
qed

lemma nullhomotopic_into_contractible_space:
  assumes f: "continuous_map X Y f" and Y: "contractible_space Y"
  obtains c where "homotopic_with (λh. True) X Y f (λx. c)"
  using nullhomotopic_through_contractible_space [OF f _ Y]
  by (metis continuous_map_id id_comp)

lemma nullhomotopic_from_contractible_space:
  assumes f: "continuous_map X Y f" and X: "contractible_space X"
  obtains c where "homotopic_with (λh. True) X Y f (λx. c)"
  using nullhomotopic_through_contractible_space [OF _ f X]
  by (metis comp_id continuous_map_id)

lemma homotopy_dominated_contractibility:
  assumes f: "continuous_map X Y f" and g: "continuous_map Y X g"
    and hom: "homotopic_with (λx. True) Y Y (f ∘ g) id" and X: "contractible_space X"
  shows "contractible_space Y"
proof -
  obtain c where c: "homotopic_with (λh. True) X Y f (λx. c)"
    using nullhomotopic_from_contractible_space [OF f X] .
  have "homotopic_with (λx. True) Y Y (f ∘ g) (λx. c)"
    using homotopic_with_compose_continuous_map_right [OF c g] by fastforce
  then have "homotopic_with (λx. True) Y Y id (λx. c)"
    using homotopic_with_trans [OF _ hom] homotopic_with_symD by blast
  then show ?thesis
    unfolding contractible_space_def ..
qed

lemma homotopy_equivalent_space_contractibility:
   "X homotopy_equivalent_space Y ⟹ (contractible_space X ⟷ contractible_space Y)"
  unfolding homotopy_equivalent_space_def
  by (blast intro: homotopy_dominated_contractibility)

lemma homeomorphic_space_contractibility:
   "X homeomorphic_space Y
        ⟹ (contractible_space X ⟷ contractible_space Y)"
  by (simp add: homeomorphic_imp_homotopy_equivalent_space homotopy_equivalent_space_contractibility)

lemma homotopic_through_contractible_space:
   "continuous_map X Y f ∧
        continuous_map X Y f' ∧
        continuous_map Y Z g ∧
        continuous_map Y Z g' ∧
        contractible_space Y ∧ path_connected_space Z
        ⟹ homotopic_with (λh. True) X Z (g ∘ f) (g' ∘ f')"
  using nullhomotopic_through_contractible_space [of X Y f Z g]
  using nullhomotopic_through_contractible_space [of X Y f' Z g']
  by (smt (verit) continuous_map_const homotopic_constant_maps homotopic_with_imp_continuous_maps
      homotopic_with_symD homotopic_with_trans path_connected_space_imp_path_component_of)

lemma homotopic_from_contractible_space:
   "continuous_map X Y f ∧ continuous_map X Y g ∧
        contractible_space X ∧ path_connected_space Y
        ⟹ homotopic_with (λx. True) X Y f g"
  by (metis comp_id continuous_map_id homotopic_through_contractible_space)

lemma homotopic_into_contractible_space:
   "continuous_map X Y f ∧ continuous_map X Y g ∧
        contractible_space Y
        ⟹ homotopic_with (λx. True) X Y f g"
  by (metis continuous_map_id contractible_imp_path_connected_space homotopic_through_contractible_space id_comp)

lemma contractible_eq_homotopy_equivalent_singleton_subtopology:
   "contractible_space X ⟷
        X = trivial_topology ∨ (∃a ∈ topspace X. X homotopy_equivalent_space (subtopology X {a}))"(is "?lhs = ?rhs")
proof (cases "X = trivial_topology")
  case False
  show ?thesis
  proof
    assume ?lhs
    then obtain a where a: "homotopic_with (λx. True) X X id (λx. a)"
      by (auto simp: contractible_space_def)
    then have "a ∈ topspace X"
      by (metis False continuous_map_const homotopic_with_imp_continuous_maps)
    then have "homotopic_with (λx. True) (subtopology X {a}) (subtopology X {a}) id (λx. a)"
      using connectedin_absolute connectedin_sing contractible_space_alt contractible_space_subtopology_singleton by fastforce
    then have "X homotopy_equivalent_space subtopology X {a}"
      unfolding homotopy_equivalent_space_def using ‹a ∈ topspace X›
      by (metis (full_types) a comp_id continuous_map_const continuous_map_id_subt empty_subsetI homotopic_with_symD
           id_comp insertI1 insert_subset topspace_subtopology_subset)
    with ‹a ∈ topspace X› show ?rhs
      by blast
  next
    assume ?rhs
    then show ?lhs
      by (meson False contractible_space_subtopology_singleton homotopy_equivalent_space_contractibility)
  qed
qed (simp add: contractible_space_empty)

lemma contractible_space_retraction_map_image:
  assumes "retraction_map X Y f" and X: "contractible_space X"
  shows "contractible_space Y"
proof -
  obtain g where f: "continuous_map X Y f" and g: "continuous_map Y X g" and fg: "∀y ∈ topspace Y. f(g y) = y"
    using assms by (auto simp: retraction_map_def retraction_maps_def)
  obtain a where a: "homotopic_with (λx. True) X X id (λx. a)"
    using X by (auto simp: contractible_space_def)
  have "homotopic_with (λx. True) Y Y id (λx. f a)"
  proof (rule homotopic_with_eq)
    show "homotopic_with (λx. True) Y Y (f ∘ id ∘ g) (f ∘ (λx. a) ∘ g)"
      using f g a homotopic_with_compose_continuous_map_left homotopic_with_compose_continuous_map_right by metis
  qed (use fg in auto)
  then show ?thesis
    unfolding contractible_space_def by blast
qed

lemma contractible_space_prod_topology:
   "contractible_space(prod_topology X Y) ⟷
    X = trivial_topology ∨ Y = trivial_topology ∨ contractible_space X ∧ contractible_space Y"
proof (cases "X = trivial_topology ∨ Y = trivial_topology")
  case True
  then have "(prod_topology X Y) = trivial_topology"
    by simp
  then show ?thesis
    by (auto simp: contractible_space_empty)
next
  case False
  have "contractible_space(prod_topology X Y) ⟷ contractible_space X ∧ contractible_space Y"
  proof safe
    assume XY: "contractible_space (prod_topology X Y)"
    with False have "retraction_map (prod_topology X Y) X fst"
      by (auto simp: contractible_space False retraction_map_fst)
    then show "contractible_space X"
      by (rule contractible_space_retraction_map_image [OF _ XY])
    have "retraction_map (prod_topology X Y) Y snd"
      using False XY  by (auto simp: contractible_space False retraction_map_snd)
    then show "contractible_space Y"
      by (rule contractible_space_retraction_map_image [OF _ XY])
  next
    assume "contractible_space X" and "contractible_space Y"
    with False obtain a b
      where "a ∈ topspace X" and a: "homotopic_with (λx. True) X X id (λx. a)"
        and "b ∈ topspace Y" and b: "homotopic_with (λx. True) Y Y id (λx. b)"
      by (auto simp: contractible_space)
    with False show "contractible_space (prod_topology X Y)"
      apply (simp add: contractible_space)
      apply (rule_tac x=a in bexI)
       apply (rule_tac x=b in bexI)
      using homotopic_with_prod_topology [OF a b]
        apply (metis (no_types, lifting) case_prod_Pair case_prod_beta' eq_id_iff)
       apply auto
      done
  qed
  with False show ?thesis
    by auto
qed


lemma contractible_space_product_topology:
  "contractible_space(product_topology X I) ⟷
    (product_topology X I) = trivial_topology ∨ (∀i ∈ I. contractible_space(X i))"
proof (cases "(product_topology X I) = trivial_topology")
  case False
  have 1: "contractible_space (X i)"
    if XI: "contractible_space (product_topology X I)" and "i ∈ I"
    for i
  proof (rule contractible_space_retraction_map_image [OF _ XI])
    show "retraction_map (product_topology X I) (X i) (λx. x i)"
      using False by (simp add: retraction_map_product_projection ‹i ∈ I›)
  qed
  have 2: "contractible_space (product_topology X I)"
    if "x ∈ topspace (product_topology X I)" and cs: "∀i∈I. contractible_space (X i)"
    for x :: "'a ⇒ 'b"
  proof -
    obtain f where f: "⋀i. i∈I ⟹ homotopic_with (λx. True) (X i) (X i) id (λx. f i)"
      using cs unfolding contractible_space_def by metis
    have "homotopic_with (λx. True)
                         (product_topology X I) (product_topology X I) id (λx. restrict f I)"
      by (rule homotopic_with_eq [OF homotopic_with_product_topology [OF f]]) (auto)
    then show ?thesis
      by (auto simp: contractible_space_def)
  qed
  show ?thesis
    using False 1 2 by (meson equals0I subtopology_eq_discrete_topology_empty)
qed auto


lemma contractible_space_subtopology_euclideanreal [simp]:
  "contractible_space(subtopology euclideanreal S) ⟷ is_interval S"
  (is "?lhs = ?rhs")
proof
  assume ?lhs
  then have "path_connectedin (subtopology euclideanreal S) S"
    using contractible_imp_path_connected_space path_connectedin_topspace path_connectedin_absolute
    by (simp add: contractible_imp_path_connected) 
  then show ?rhs
    by (simp add: is_interval_path_connected_1)
next
  assume ?rhs
  then have "convex S"
    by (simp add: is_interval_convex_1)
  show ?lhs
  proof (cases "S = {}")
    case False
    then obtain z where "z ∈ S"
      by blast
    show ?thesis
      unfolding contractible_space_def homotopic_with_def
    proof (intro exI conjI allI)
      note § = convexD [OF ‹convex S›, simplified]
      show "continuous_map (prod_topology (top_of_set {0..1}) (top_of_set S)) (top_of_set S)
                           (λ(t,x). (1 - t) * x + t * z)"
        using  ‹z ∈ S› 
        by (auto simp: case_prod_unfold intro!: continuous_intros §)
    qed auto
  qed (simp add: contractible_space_empty)
qed


corollary contractible_space_euclideanreal: "contractible_space euclideanreal"
proof -
  have "contractible_space (subtopology euclideanreal UNIV)"
    using contractible_space_subtopology_euclideanreal by blast
  then show ?thesis
    by simp
qed


abbreviation✐‹tag important› homotopy_eqv :: "'a::topological_space set ⇒ 'b::topological_space set ⇒ bool"
             (infix "homotopy'_eqv" 50)
  where "S homotopy_eqv T ≡ top_of_set S homotopy_equivalent_space top_of_set T"

lemma homeomorphic_imp_homotopy_eqv: "S homeomorphic T ⟹ S homotopy_eqv T"
  unfolding homeomorphic_def homeomorphism_def homotopy_equivalent_space_def
  by (metis continuous_map_subtopology_eu homotopic_with_id2 openin_imp_subset openin_subtopology_self topspace_euclidean_subtopology)

lemma homotopy_eqv_inj_linear_image:
  fixes f :: "'a::euclidean_space ⇒ 'b::euclidean_space"
  assumes "linear f" "inj f"
    shows "(f ` S) homotopy_eqv S"
  by (metis assms homeomorphic_sym homeomorphic_imp_homotopy_eqv linear_homeomorphic_image)

lemma homotopy_eqv_translation:
    fixes S :: "'a::real_normed_vector set"
    shows "(+) a ` S homotopy_eqv S"
  using homeomorphic_imp_homotopy_eqv homeomorphic_translation homeomorphic_sym by blast

lemma homotopy_eqv_homotopic_triviality_imp:
  fixes S :: "'a::real_normed_vector set"
    and T :: "'b::real_normed_vector set"
    and U :: "'c::real_normed_vector set"
  assumes "S homotopy_eqv T"
      and f: "continuous_on U f" "f ∈ U → T"
      and g: "continuous_on U g" "g ∈ U → T"
      and homUS: "⋀f g. ⟦continuous_on U f; f ∈ U → S;
                         continuous_on U g; g ∈ U → S⟧
                         ⟹ homotopic_with_canon (λx. True) U S f g"
    shows "homotopic_with_canon (λx. True) U T f g"
proof -
  obtain h k where h: "continuous_on S h" "h ∈ S → T"
               and k: "continuous_on T k" "k ∈ T → S"
               and hom: "homotopic_with_canon (λx. True) S S (k ∘ h) id"
                        "homotopic_with_canon (λx. True) T T (h ∘ k) id"
    using assms by (force simp: homotopy_equivalent_space_def image_subset_iff_funcset)
  have "homotopic_with_canon (λf. True) U S (k ∘ f) (k ∘ g)"
  proof (rule homUS)
    show "continuous_on U (k ∘ f)" "continuous_on U (k ∘ g)"
      using continuous_on_compose continuous_on_subset f g k by (metis funcset_image)+
  qed (use f g k in ‹(force simp: o_def)+› )
  then have "homotopic_with_canon (λx. True) U T (h ∘ (k ∘ f)) (h ∘ (k ∘ g))"
    by (simp add: h homotopic_with_compose_continuous_map_left image_subset_iff_funcset)
  moreover have "homotopic_with_canon (λx. True) U T (h ∘ k ∘ f) (id ∘ f)"
    by (rule homotopic_with_compose_continuous_right [where X=T and Y=T]; simp add: hom f)
  moreover have "homotopic_with_canon (λx. True) U T (h ∘ k ∘ g) (id ∘ g)"
    by (rule homotopic_with_compose_continuous_right [where X=T and Y=T]; simp add: hom g)
  ultimately show "homotopic_with_canon (λx. True) U T f g"
    unfolding o_assoc
    by (metis homotopic_with_trans homotopic_with_sym id_comp) 
qed

lemma homotopy_eqv_homotopic_triviality:
  fixes S :: "'a::real_normed_vector set"
    and T :: "'b::real_normed_vector set"
    and U :: "'c::real_normed_vector set"
  assumes "S homotopy_eqv T"
    shows "(∀f g. continuous_on U f ∧ f ∈ U → S ∧
                   continuous_on U g ∧ g ∈ U → S
                   ⟶ homotopic_with_canon (λx. True) U S f g) ⟷
           (∀f g. continuous_on U f ∧ f ∈ U → T ∧
                  continuous_on U g ∧ g ∈ U → T
                  ⟶ homotopic_with_canon (λx. True) U T f g)"
      (is "?lhs = ?rhs")
proof
  assume ?lhs
  then show ?rhs
    by (metis assms homotopy_eqv_homotopic_triviality_imp)
next
  assume ?rhs
  moreover
  have "T homotopy_eqv S"
    using assms homotopy_equivalent_space_sym by blast
  ultimately show ?lhs
    by (blast intro: homotopy_eqv_homotopic_triviality_imp)
qed


lemma homotopy_eqv_cohomotopic_triviality_null_imp:
  fixes S :: "'a::real_normed_vector set"
    and T :: "'b::real_normed_vector set"
    and U :: "'c::real_normed_vector set"
  assumes "S homotopy_eqv T"
      and f: "continuous_on T f" "f ∈ T → U"
      and homSU: "⋀f. ⟦continuous_on S f; f ∈ S → U⟧
                      ⟹ ∃c. homotopic_with_canon (λx. True) S U f (λx. c)"
  obtains c where "homotopic_with_canon (λx. True) T U f (λx. c)"
proof -
  obtain h k where h: "continuous_on S h" "h ∈ S → T"
               and k: "continuous_on T k" "k ∈ T → S"
               and hom: "homotopic_with_canon (λx. True) S S (k ∘ h) id"
                        "homotopic_with_canon (λx. True) T T (h ∘ k) id"
    using assms by (force simp: homotopy_equivalent_space_def image_subset_iff_funcset)
  obtain c where "homotopic_with_canon (λx. True) S U (f ∘ h) (λx. c)"
  proof (rule exE [OF homSU])
    show "continuous_on S (f ∘ h)"
      by (metis continuous_on_compose continuous_on_subset f h funcset_image)
  qed (use f h in force)
  then have "homotopic_with_canon (λx. True) T U ((f ∘ h) ∘ k) ((λx. c) ∘ k)"
    by (rule homotopic_with_compose_continuous_right [where X=S]) (use k in auto)
  moreover have "homotopic_with_canon (λx. True) T U (f ∘ id) (f ∘ (h ∘ k))"
    by (rule homotopic_with_compose_continuous_left [where Y=T])
       (use f in ‹auto simp: hom homotopic_with_symD›)
  ultimately show ?thesis
    using that homotopic_with_trans by (fastforce simp: o_def)
qed

lemma homotopy_eqv_cohomotopic_triviality_null:
  fixes S :: "'a::real_normed_vector set"
    and T :: "'b::real_normed_vector set"
    and U :: "'c::real_normed_vector set"
  assumes "S homotopy_eqv T"
    shows "(∀f. continuous_on S f ∧ f ∈ S → U
                ⟶ (∃c. homotopic_with_canon (λx. True) S U f (λx. c))) ⟷
           (∀f. continuous_on T f ∧ f ∈ T → U
                ⟶ (∃c. homotopic_with_canon (λx. True) T U f (λx. c)))"
by (rule iffI; metis assms homotopy_eqv_cohomotopic_triviality_null_imp homotopy_equivalent_space_sym)

text ‹Similar to the proof above›
lemma homotopy_eqv_homotopic_triviality_null_imp:
  fixes S :: "'a::real_normed_vector set"
    and T :: "'b::real_normed_vector set"
    and U :: "'c::real_normed_vector set"
  assumes "S homotopy_eqv T"
      and f: "continuous_on U f" "f ∈ U → T"
      and homSU: "⋀f. ⟦continuous_on U f; f ∈ U → S⟧
                      ⟹ ∃c. homotopic_with_canon (λx. True) U S f (λx. c)"
    shows "∃c. homotopic_with_canon (λx. True) U T f (λx. c)"
proof -
  obtain h k where h: "continuous_on S h" "h ∈ S → T"
               and k: "continuous_on T k" "k ∈ T → S"
               and hom: "homotopic_with_canon (λx. True) S S (k ∘ h) id"
                        "homotopic_with_canon (λx. True) T T (h ∘ k) id"
    using assms by (force simp: homotopy_equivalent_space_def image_subset_iff_funcset)
  obtain c::'a where "homotopic_with_canon (λx. True) U S (k ∘ f) (λx. c)"
  proof (rule exE [OF homSU [of "k ∘ f"]])
    show "continuous_on U (k ∘ f)"
      using continuous_on_compose continuous_on_subset f k by (metis funcset_image)
  qed (use f k in force)
  then have "homotopic_with_canon (λx. True) U T (h ∘ (k ∘ f)) (h ∘ (λx. c))"
    by (rule homotopic_with_compose_continuous_left [where Y=S]) (use h in auto)
  moreover have "homotopic_with_canon (λx. True) U T (id ∘ f) ((h ∘ k) ∘ f)"
    by (rule homotopic_with_compose_continuous_right [where X=T])
       (use f in ‹auto simp: hom homotopic_with_symD›)
  ultimately show ?thesis
    using homotopic_with_trans by (fastforce simp: o_def)
qed

lemma homotopy_eqv_homotopic_triviality_null:
  fixes S :: "'a::real_normed_vector set"
    and T :: "'b::real_normed_vector set"
    and U :: "'c::real_normed_vector set"
  assumes "S homotopy_eqv T"
    shows "(∀f. continuous_on U f ∧ f ∈ U → S
                  ⟶ (∃c. homotopic_with_canon (λx. True) U S f (λx. c))) ⟷
           (∀f. continuous_on U f ∧ f ∈ U → T
                  ⟶ (∃c. homotopic_with_canon (λx. True) U T f (λx. c)))"
by (rule iffI; metis assms homotopy_eqv_homotopic_triviality_null_imp homotopy_equivalent_space_sym)

lemma homotopy_eqv_contractible_sets:
  fixes S :: "'a::real_normed_vector set"
    and T :: "'b::real_normed_vector set"
  assumes "contractible S" "contractible T" "S = {} ⟷ T = {}"
    shows "S homotopy_eqv T"
proof (cases "S = {}")
  case True with assms show ?thesis
    using homeomorphic_imp_homotopy_eqv by fastforce
next
  case False
  with assms obtain a b where "a ∈ S" "b ∈ T"
    by auto
  then show ?thesis
    unfolding homotopy_equivalent_space_def
    apply (rule_tac x="λx. b" in exI, rule_tac x="λx. a" in exI)
    apply (intro assms conjI continuous_on_id' homotopic_into_contractible; force)
    done
qed

lemma homotopy_eqv_empty1 [simp]:
  fixes S :: "'a::real_normed_vector set"
  shows "S homotopy_eqv ({}::'b::real_normed_vector set) ⟷ S = {}" (is "?lhs = ?rhs")
proof
  assume ?lhs then show ?rhs
    by (metis continuous_map_subtopology_eu empty_iff equalityI homotopy_equivalent_space_def image_subset_iff subsetI)
qed (use homeomorphic_imp_homotopy_eqv in force)

lemma homotopy_eqv_empty2 [simp]:
  fixes S :: "'a::real_normed_vector set"
  shows "({}::'b::real_normed_vector set) homotopy_eqv S ⟷ S = {}"
  using homotopy_equivalent_space_sym homotopy_eqv_empty1 by blast

lemma homotopy_eqv_contractibility:
  fixes S :: "'a::real_normed_vector set" and T :: "'b::real_normed_vector set"
  shows "S homotopy_eqv T ⟹ (contractible S ⟷ contractible T)"
  by (meson contractible_space_top_of_set homotopy_equivalent_space_contractibility)

lemma homotopy_eqv_sing:
  fixes S :: "'a::real_normed_vector set" and a :: "'b::real_normed_vector"
  shows "S homotopy_eqv {a} ⟷ S ≠ {} ∧ contractible S"
  by (metis contractible_sing empty_not_insert homotopy_eqv_contractibility homotopy_eqv_contractible_sets homotopy_eqv_empty2)

lemma homeomorphic_contractible_eq:
  fixes S :: "'a::real_normed_vector set" and T :: "'b::real_normed_vector set"
  shows "S homeomorphic T ⟹ (contractible S ⟷ contractible T)"
by (simp add: homeomorphic_imp_homotopy_eqv homotopy_eqv_contractibility)

lemma homeomorphic_contractible:
  fixes S :: "'a::real_normed_vector set" and T :: "'b::real_normed_vector set"
  shows "⟦contractible S; S homeomorphic T⟧ ⟹ contractible T"
  by (metis homeomorphic_contractible_eq)


subsection✐‹tag unimportant›‹Misc other results›

lemma bounded_connected_Compl_real:
  fixes S :: "real set"
  assumes "bounded S" and conn: "connected(- S)"
    shows "S = {}"
proof -
  obtain a b where "S ⊆ box a b"
    by (meson assms bounded_subset_box_symmetric)
  then have "a ∉ S" "b ∉ S"
    by auto
  then have "∀x. a ≤ x ∧ x ≤ b ⟶ x ∈ - S"
    by (meson Compl_iff conn connected_iff_interval)
  then show ?thesis
    using ‹S ⊆ box a b› by auto
qed

corollary bounded_path_connected_Compl_real:
  fixes S :: "real set"
  assumes "bounded S" "path_connected(- S)" shows "S = {}"
  by (simp add: assms bounded_connected_Compl_real path_connected_imp_connected)

lemma bounded_connected_Compl_1:
  fixes S :: "'a::{euclidean_space} set"
  assumes "bounded S" and conn: "connected(- S)" and 1: "DIM('a) = 1"
    shows "S = {}"
proof -
  have "DIM('a) = DIM(real)"
    by (simp add: "1")
  then obtain f::"'a ⇒ real" and g
  where "linear f" "⋀x. norm(f x) = norm x" and fg: "⋀x. g(f x) = x" "⋀y. f(g y) = y"
    by (rule isomorphisms_UNIV_UNIV) blast
  with ‹bounded S› have "bounded (f ` S)"
    using bounded_linear_image linear_linear by blast
  have "bij f" by (metis fg bijI')
  have "connected (f ` (-S))"
    using connected_linear_image assms ‹linear f› by blast
  moreover have "f ` (-S) = - (f ` S)"
    by (simp add: ‹bij f› bij_image_Compl_eq)
  finally have "connected (- (f ` S))"
    by simp
  then have "f ` S = {}"
    using ‹bounded (f ` S)› bounded_connected_Compl_real by blast
  then show ?thesis
    by blast
qed

lemma connected_card_eq_iff_nontrivial:
  fixes S :: "'a::metric_space set"
  shows "connected S ⟹ uncountable S ⟷ ¬(∃a. S ⊆ {a})"
  by (metis connected_uncountable finite.emptyI finite.insertI rev_finite_subset singleton_iff subsetI uncountable_infinite)

lemma connected_finite_iff_sing:
  fixes S :: "'a::metric_space set"
  assumes "connected S"
  shows "finite S ⟷ S = {} ∨ (∃a. S = {a})" 
  using assms connected_uncountable countable_finite by blast

subsection✐‹tag unimportant›‹ Some simple positive connection theorems›

proposition path_connected_convex_diff_countable:
  fixes U :: "'a::euclidean_space set"
  assumes "convex U" "¬ collinear U" "countable S"
    shows "path_connected(U - S)"
proof (clarsimp simp: path_connected_def)
  fix a b
  assume "a ∈ U" "a ∉ S" "b ∈ U" "b ∉ S"
  let ?m = "midpoint a b"
  show "∃g. path g ∧ path_image g ⊆ U - S ∧ pathstart g = a ∧ pathfinish g = b"
  proof (cases "a = b")
    case True
    then show ?thesis
      by (metis DiffI ‹a ∈ U› ‹a ∉ S› path_component_def path_component_refl)
  next
    case False
    then have "a ≠ ?m" "b ≠ ?m"
      using midpoint_eq_endpoint by fastforce+
    have "?m ∈ U"
      using ‹a ∈ U› ‹b ∈ U› ‹convex U› convex_contains_segment by force
    obtain c where "c ∈ U" and nc_abc: "¬ collinear {a,b,c}"
      by (metis False ‹a ∈ U› ‹b ∈ U› ‹¬ collinear U› collinear_triples insert_absorb)
    have ncoll_mca: "¬ collinear {?m,c,a}"
      by (metis (full_types) ‹a ≠ ?m› collinear_3_trans collinear_midpoint insert_commute nc_abc)
    have ncoll_mcb: "¬ collinear {?m,c,b}"
      by (metis (full_types) ‹b ≠ ?m› collinear_3_trans collinear_midpoint insert_commute nc_abc)
    have "c ≠ ?m"
      by (metis collinear_midpoint insert_commute nc_abc)
    then have "closed_segment ?m c ⊆ U"
      by (simp add: ‹c ∈ U› ‹?m ∈ U› ‹convex U› closed_segment_subset)
    then obtain z where z: "z ∈ closed_segment ?m c"
                    and disjS: "(closed_segment a z ∪ closed_segment z b) ∩ S = {}"
    proof -
      have False if "closed_segment ?m c ⊆ {z. (closed_segment a z ∪ closed_segment z b) ∩ S ≠ {}}"
      proof -
        have closb: "closed_segment ?m c ⊆
                 {z ∈ closed_segment ?m c. closed_segment a z ∩ S ≠ {}} ∪ {z ∈ closed_segment ?m c. closed_segment z b ∩ S ≠ {}}"
          using that by blast
        have *: "countable {z ∈ closed_segment ?m c. closed_segment z u ∩ S ≠ {}}"
          if "u ∈ U" "u ∉ S" and ncoll: "¬ collinear {?m, c, u}" for u
        proof -
          have **: False if x1: "x1 ∈ closed_segment ?m c" and x2: "x2 ∈ closed_segment ?m c"
                            and "x1 ≠ x2" "x1 ≠ u"
                            and w: "w ∈ closed_segment x1 u" "w ∈ closed_segment x2 u"
                            and "w ∈ S" for x1 x2 w
          proof -
            have "x1 ∈ affine hull {?m,c}" "x2 ∈ affine hull {?m,c}"
              using segment_as_ball x1 x2 by auto
            then have coll_x1: "collinear {x1, ?m, c}" and coll_x2: "collinear {?m, c, x2}"
              by (simp_all add: affine_hull_3_imp_collinear) (metis affine_hull_3_imp_collinear insert_commute)
            have "¬ collinear {x1, u, x2}"
            proof
              assume "collinear {x1, u, x2}"
              then have "collinear {?m, c, u}"
                by (metis (full_types) ‹c ≠ ?m› coll_x1 coll_x2 collinear_3_trans insert_commute ncoll ‹x1 ≠ x2›)
              with ncoll show False ..
            qed
            then have "closed_segment x1 u ∩ closed_segment u x2 = {u}"
              by (blast intro!: Int_closed_segment)
            then have "w = u"
              using closed_segment_commute w by auto
            show ?thesis
              using ‹u ∉ S› ‹w = u› that(7) by auto
          qed
          then have disj: "disjoint ((⋃z∈closed_segment ?m c. {closed_segment z u ∩ S}))"
            by (fastforce simp: pairwise_def disjnt_def)
          have cou: "countable ((⋃z ∈ closed_segment ?m c. {closed_segment z u ∩ S}) - {{}})"
            apply (rule pairwise_disjnt_countable_Union [OF _ pairwise_subset [OF disj]])
             apply (rule countable_subset [OF _ ‹countable S›], auto)
            done
          define f where "f ≡ λX. (THE z. z ∈ closed_segment ?m c ∧ X = closed_segment z u ∩ S)"
          show ?thesis
          proof (rule countable_subset [OF _ countable_image [OF cou, where f=f]], clarify)
            fix x
            assume x: "x ∈ closed_segment ?m c" "closed_segment x u ∩ S ≠ {}"
            show "x ∈ f ` ((⋃z∈closed_segment ?m c. {closed_segment z u ∩ S}) - {{}})"
            proof (rule_tac x="closed_segment x u ∩ S" in image_eqI)
              show "x = f (closed_segment x u ∩ S)"
                unfolding f_def 
                by (rule the_equality [symmetric]) (use x in ‹auto dest: **›)
            qed (use x in auto)
          qed
        qed
        have "uncountable (closed_segment ?m c)"
          by (metis ‹c ≠ ?m› uncountable_closed_segment)
        then show False
          using closb * [OF ‹a ∈ U› ‹a ∉ S› ncoll_mca] * [OF ‹b ∈ U› ‹b ∉ S› ncoll_mcb]
          by (simp add: closed_segment_commute countable_subset)
      qed
      then show ?thesis
        by (force intro: that)
    qed
    show ?thesis
    proof (intro exI conjI)
      have "path_image (linepath a z +++ linepath z b) ⊆ U"
        by (metis ‹a ∈ U› ‹b ∈ U› ‹closed_segment ?m c ⊆ U› z ‹convex U› closed_segment_subset contra_subsetD path_image_linepath subset_path_image_join)
      with disjS show "path_image (linepath a z +++ linepath z b) ⊆ U - S"
        by (force simp: path_image_join)
    qed auto
  qed
qed


corollary connected_convex_diff_countable:
  fixes U :: "'a::euclidean_space set"
  assumes "convex U" "¬ collinear U" "countable S"
  shows "connected(U - S)"
  by (simp add: assms path_connected_convex_diff_countable path_connected_imp_connected)

lemma path_connected_punctured_convex:
  assumes "convex S" and aff: "aff_dim S ≠ 1"
    shows "path_connected(S - {a})"
proof -
  consider "aff_dim S = -1" | "aff_dim S = 0" | "aff_dim S ≥ 2"
    using assms aff_dim_geq [of S] by linarith
  then show ?thesis
  proof cases
    assume "aff_dim S = -1"
    then show ?thesis
      by (metis aff_dim_empty empty_Diff path_connected_empty)
  next
    assume "aff_dim S = 0"
    then show ?thesis
      by (metis aff_dim_eq_0 Diff_cancel Diff_empty Diff_insert0 convex_empty convex_imp_path_connected path_connected_singleton singletonD)
  next
    assume ge2: "aff_dim S ≥ 2"
    then have "¬ collinear S"
    proof (clarsimp simp: collinear_affine_hull)
      fix u v
      assume "S ⊆ affine hull {u, v}"
      then have "aff_dim S ≤ aff_dim {u, v}"
        by (metis (no_types) aff_dim_affine_hull aff_dim_subset)
      with ge2 show False
        by (metis (no_types) aff_dim_2 antisym aff not_numeral_le_zero one_le_numeral order_trans)
    qed
    moreover have "countable {a}"
      by simp
    ultimately show ?thesis
      by (metis path_connected_convex_diff_countable [OF ‹convex S›])
  qed
qed

lemma connected_punctured_convex:
  shows "⟦convex S; aff_dim S ≠ 1⟧ ⟹ connected(S - {a})"
  using path_connected_imp_connected path_connected_punctured_convex by blast

lemma path_connected_complement_countable:
  fixes S :: "'a::euclidean_space set"
  assumes "2 ≤ DIM('a)" "countable S"
  shows "path_connected(- S)"
proof -
  have "¬ collinear (UNIV::'a set)"
    using assms by (auto simp: collinear_aff_dim [of "UNIV :: 'a set"])
  then have "path_connected(UNIV - S)"
    by (simp add: ‹countable S› path_connected_convex_diff_countable)
  then show ?thesis
    by (simp add: Compl_eq_Diff_UNIV)
qed

proposition path_connected_openin_diff_countable:
  fixes S :: "'a::euclidean_space set"
  assumes "connected S" and ope: "openin (top_of_set (affine hull S)) S"
      and "¬ collinear S" "countable T"
    shows "path_connected(S - T)"
proof (clarsimp simp: path_connected_component)
  fix x y
  assume xy: "x ∈ S" "x ∉ T" "y ∈ S" "y ∉ T"
  show "path_component (S - T) x y"
  proof (rule connected_equivalence_relation_gen [OF ‹connected S›, where P = "λx. x ∉ T"])
    show "∃z. z ∈ U ∧ z ∉ T" if opeU: "openin (top_of_set S) U" and "x ∈ U" for U x
    proof -
      have "openin (top_of_set (affine hull S)) U"
        using opeU ope openin_trans by blast
      with ‹x ∈ U› obtain r where Usub: "U ⊆ affine hull S" and "r > 0"
                              and subU: "ball x r ∩ affine hull S ⊆ U"
        by (auto simp: openin_contains_ball)
      with ‹x ∈ U› have x: "x ∈ ball x r ∩ affine hull S"
        by auto
      have "¬ S ⊆ {x}"
        using ‹¬ collinear S›  collinear_subset by blast
      then obtain x' where "x' ≠ x" "x' ∈ S"
        by blast
      obtain y where y: "y ≠ x" "y ∈ ball x r ∩ affine hull S"
      proof
        show "x + (r / 2 / norm(x' - x)) *⇩R (x' - x) ≠ x"
          using ‹x' ≠ x› ‹r > 0› by auto
        show "x + (r / 2 / norm (x' - x)) *⇩R (x' - x) ∈ ball x r ∩ affine hull S"
          using ‹x' ≠ x› ‹r > 0› ‹x' ∈ S› x
          by (simp add: dist_norm mem_affine_3_minus hull_inc)
      qed
      have "convex (ball x r ∩ affine hull S)"
        by (simp add: affine_imp_convex convex_Int)
      with x y subU have "uncountable U"
        by (meson countable_subset uncountable_convex)
      then have "¬ U ⊆ T"
        using ‹countable T› countable_subset by blast
      then show ?thesis by blast
    qed
    show "∃U. openin (top_of_set S) U ∧ x ∈ U ∧
              (∀x∈U. ∀y∈U. x ∉ T ∧ y ∉ T ⟶ path_component (S - T) x y)"
          if "x ∈ S" for x
    proof -
      obtain r where Ssub: "S ⊆ affine hull S" and "r > 0"
                 and subS: "ball x r ∩ affine hull S ⊆ S"
        using ope ‹x ∈ S› by (auto simp: openin_contains_ball)
      then have conv: "convex (ball x r ∩ affine hull S)"
        by (simp add: affine_imp_convex convex_Int)
      have "¬ aff_dim (affine hull S) ≤ 1"
        using ‹¬ collinear S› collinear_aff_dim by auto
      then have "¬ aff_dim (ball x r ∩ affine hull S) ≤ 1"
        by (metis (no_types, opaque_lifting) aff_dim_convex_Int_open IntI open_ball ‹0 < r› aff_dim_affine_hull affine_affine_hull affine_imp_convex centre_in_ball empty_iff hull_subset inf_commute subsetCE that)
      then have "¬ collinear (ball x r ∩ affine hull S)"
        by (simp add: collinear_aff_dim)
      then have *: "path_connected ((ball x r ∩ affine hull S) - T)"
        by (rule path_connected_convex_diff_countable [OF conv _ ‹countable T›])
      have ST: "ball x r ∩ affine hull S - T ⊆ S - T"
        using subS by auto
      show ?thesis
      proof (intro exI conjI)
        show "x ∈ ball x r ∩ affine hull S"
          using ‹x ∈ S› ‹r > 0› by (simp add: hull_inc)
        have "openin (top_of_set (affine hull S)) (ball x r ∩ affine hull S)"
          by (subst inf.commute) (simp add: openin_Int_open)
        then show "openin (top_of_set S) (ball x r ∩ affine hull S)"
          by (rule openin_subset_trans [OF _ subS Ssub])
      qed (use * path_component_trans in ‹auto simp: path_connected_component path_component_of_subset [OF ST]›)
    qed
  qed (use xy path_component_trans in auto)
qed

corollary connected_openin_diff_countable:
  fixes S :: "'a::euclidean_space set"
  assumes "connected S" and ope: "openin (top_of_set (affine hull S)) S"
      and "¬ collinear S" "countable T"
    shows "connected(S - T)"
  by (metis path_connected_imp_connected path_connected_openin_diff_countable [OF assms])

corollary path_connected_open_diff_countable:
  fixes S :: "'a::euclidean_space set"
  assumes "2 ≤ DIM('a)" "open S" "connected S" "countable T"
  shows "path_connected(S - T)"
proof (cases "S = {}")
  case True
  then show ?thesis
    by (simp)
next
  case False
  show ?thesis
  proof (rule path_connected_openin_diff_countable)
    show "openin (top_of_set (affine hull S)) S"
      by (simp add: assms hull_subset open_subset)
    show "¬ collinear S"
      using assms False by (simp add: collinear_aff_dim aff_dim_open)
  qed (simp_all add: assms)
qed

corollary connected_open_diff_countable:
  fixes S :: "'a::euclidean_space set"
  assumes "2 ≤ DIM('a)" "open S" "connected S" "countable T"
  shows "connected(S - T)"
by (simp add: assms path_connected_imp_connected path_connected_open_diff_countable)



subsection✐‹tag unimportant› ‹Self-homeomorphisms shuffling points about›

subsubsection✐‹tag unimportant›‹The theorem ‹homeomorphism_moving_points_exists››

lemma homeomorphism_moving_point_1:
  fixes a :: "'a::euclidean_space"
  assumes "affine T" "a ∈ T" and u: "u ∈ ball a r ∩ T"
  obtains f g where "homeomorphism (cball a r ∩ T) (cball a r ∩ T) f g"
                    "f a = u" "⋀x. x ∈ sphere a r ⟹ f x = x"
proof -
  have nou: "norm (u - a) < r" and "u ∈ T"
    using u by (auto simp: dist_norm norm_minus_commute)
  then have "0 < r"
    by (metis DiffD1 Diff_Diff_Int ball_eq_empty centre_in_ball not_le u)
  define f where "f ≡ λx. (1 - norm(x - a) / r) *⇩R (u - a) + x"
  have *: "False" if eq: "x + (norm y / r) *⇩R u = y + (norm x / r) *⇩R u"
                  and nou: "norm u < r" and yx: "norm y < norm x" for x y and u::'a
  proof -
    have "x = y + (norm x / r - (norm y / r)) *⇩R u"
      using eq by (simp add: algebra_simps)
    then have "norm x = norm (y + ((norm x - norm y) / r) *⇩R u)"
      by (metis diff_divide_distrib)
    also have "… ≤ norm y + norm(((norm x - norm y) / r) *⇩R u)"
      using norm_triangle_ineq by blast
    also have "… = norm y + (norm x - norm y) * (norm u / r)"
      using yx ‹r > 0›
      by (simp add: field_split_simps)
    also have "… < norm y + (norm x - norm y) * 1"
    proof (subst add_less_cancel_left)
      show "(norm x - norm y) * (norm u / r) < (norm x - norm y) * 1"
      proof (rule mult_strict_left_mono)
        show "norm u / r < 1"
          using ‹0 < r› divide_less_eq_1_pos nou by blast
      qed (simp add: yx)
    qed
    also have "… = norm x"
      by simp
    finally show False by simp
  qed
  have "inj f"
    unfolding f_def
  proof (clarsimp simp: inj_on_def)
    fix x y
    assume "(1 - norm (x - a) / r) *⇩R (u - a) + x =
            (1 - norm (y - a) / r) *⇩R (u - a) + y"
    then have eq: "(x - a) + (norm (y - a) / r) *⇩R (u - a) = (y - a) + (norm (x - a) / r) *⇩R (u - a)"
      by (auto simp: algebra_simps)
    show "x=y"
    proof (cases "norm (x - a) = norm (y - a)")
      case True
      then show ?thesis
        using eq by auto
    next
      case False
      then consider "norm (x - a) < norm (y - a)" | "norm (x - a) > norm (y - a)"
        by linarith
      then have "False"
      proof cases
        case 1 show False
          using * [OF _ nou 1] eq by simp
      next
        case 2 with * [OF eq nou] show False
          by auto
      qed
      then show "x=y" ..
    qed
  qed
  then have inj_onf: "inj_on f (cball a r ∩ T)"
    using inj_on_Int by fastforce
  have contf: "continuous_on (cball a r ∩ T) f"
    unfolding f_def using ‹0 < r›  by (intro continuous_intros) blast
  have fim: "f ` (cball a r ∩ T) = cball a r ∩ T"
  proof
    have *: "norm (y + (1 - norm y / r) *⇩R u) ≤ r" if "norm y ≤ r" "norm u < r" for y u::'a
    proof -
      have "norm (y + (1 - norm y / r) *⇩R u) ≤ norm y + norm((1 - norm y / r) *⇩R u)"
        using norm_triangle_ineq by blast
      also have "… = norm y + abs(1 - norm y / r) * norm u"
        by simp
      also have "… ≤ r"
      proof -
        have "(r - norm u) * (r - norm y) ≥ 0"
          using that by auto
        then have "r * norm u + r * norm y ≤ r * r + norm u * norm y"
          by (simp add: algebra_simps)
        then show ?thesis
        using that ‹0 < r› by (simp add: abs_if field_simps)
      qed
      finally show ?thesis .
    qed
    have "f ` (cball a r) ⊆ cball a r"
      using * nou
      apply (clarsimp simp: dist_norm norm_minus_commute f_def)
      by (metis diff_add_eq diff_diff_add diff_diff_eq2 norm_minus_commute)
    moreover have "f ` T ⊆ T"
      unfolding f_def using ‹affine T› ‹a ∈ T› ‹u ∈ T›
      by (force simp: add.commute mem_affine_3_minus)
    ultimately show "f ` (cball a r ∩ T) ⊆ cball a r ∩ T"
      by blast
  next
    show "cball a r ∩ T ⊆ f ` (cball a r ∩ T)"
    proof (clarsimp simp: dist_norm norm_minus_commute)
      fix x
      assume x: "norm (x - a) ≤ r" and "x ∈ T"
      have "∃v ∈ {0..1}. ((1 - v) * r - norm ((x - a) - v *⇩R (u - a))) ∙ 1 = 0"
        by (rule ivt_decreasing_component_on_1) (auto simp: x continuous_intros)
      then obtain v where "0 ≤ v" "v ≤ 1"
        and v: "(1 - v) * r = norm ((x - a) - v *⇩R (u - a))"
        by auto
      then have n: "norm (a - (x - v *⇩R (u - a))) = r - r * v"
        by (simp add: field_simps norm_minus_commute)
      show "x ∈ f ` (cball a r ∩ T)"
      proof (rule image_eqI)
        show "x = f (x - v *⇩R (u - a))"
          using ‹r > 0› v by (simp add: f_def) (simp add: field_simps)
        have "x - v *⇩R (u - a) ∈ cball a r"
          using ‹r > 0›‹0 ≤ v›
          by (simp add: dist_norm n)
        moreover have "x - v *⇩R (u - a) ∈ T"
          by (simp add: f_def ‹u ∈ T› ‹x ∈ T› assms mem_affine_3_minus2)
        ultimately show "x - v *⇩R (u - a) ∈ cball a r ∩ T"
          by blast
      qed
    qed
  qed
  have "compact (cball a r ∩ T)"
    by (simp add: affine_closed compact_Int_closed ‹affine T›)
  then obtain g where "homeomorphism (cball a r ∩ T) (cball a r ∩ T) f g"
    by (metis homeomorphism_compact [OF _ contf fim inj_onf])
  then show thesis
    apply (rule_tac f=f in that)
    using ‹r > 0› by (simp_all add: f_def dist_norm norm_minus_commute)
qed

corollary✐‹tag unimportant› homeomorphism_moving_point_2:
  fixes a :: "'a::euclidean_space"
  assumes "affine T" "a ∈ T" and u: "u ∈ ball a r ∩ T" and v: "v ∈ ball a r ∩ T"
  obtains f g where "homeomorphism (cball a r ∩ T) (cball a r ∩ T) f g"
                    "f u = v" "⋀x. ⟦x ∈ sphere a r; x ∈ T⟧ ⟹ f x = x"
proof -
  have "0 < r"
    by (metis DiffD1 Diff_Diff_Int ball_eq_empty centre_in_ball not_le u)
  obtain f1 g1 where hom1: "homeomorphism (cball a r ∩ T) (cball a r ∩ T) f1 g1"
                 and "f1 a = u" and f1: "⋀x. x ∈ sphere a r ⟹ f1 x = x"
    using homeomorphism_moving_point_1 [OF ‹affine T› ‹a ∈ T› u] by blast
  obtain f2 g2 where hom2: "homeomorphism (cball a r ∩ T) (cball a r ∩ T) f2 g2"
                 and "f2 a = v" and f2: "⋀x. x ∈ sphere a r ⟹ f2 x = x"
    using homeomorphism_moving_point_1 [OF ‹affine T› ‹a ∈ T› v] by blast
  show ?thesis
  proof
    show "homeomorphism (cball a r ∩ T) (cball a r ∩ T) (f2 ∘ g1) (f1 ∘ g2)"
      by (metis homeomorphism_compose homeomorphism_symD hom1 hom2)
    have "g1 u = a"
      using ‹0 < r› ‹f1 a = u› assms hom1 homeomorphism_apply1 by fastforce
    then show "(f2 ∘ g1) u = v"
      by (simp add: ‹f2 a = v›)
    show "⋀x. ⟦x ∈ sphere a r; x ∈ T⟧ ⟹ (f2 ∘ g1) x = x"
      using f1 f2 hom1 homeomorphism_apply1 by fastforce
  qed
qed


corollary✐‹tag unimportant› homeomorphism_moving_point_3:
  fixes a :: "'a::euclidean_space"
  assumes "affine T" "a ∈ T" and ST: "ball a r ∩ T ⊆ S" "S ⊆ T"
      and u: "u ∈ ball a r ∩ T" and v: "v ∈ ball a r ∩ T"
  obtains f g where "homeomorphism S S f g"
                    "f u = v" "{x. ¬ (f x = x ∧ g x = x)} ⊆ ball a r ∩ T"
proof -
  obtain f g where hom: "homeomorphism (cball a r ∩ T) (cball a r ∩ T) f g"
               and "f u = v" and fid: "⋀x. ⟦x ∈ sphere a r; x ∈ T⟧ ⟹ f x = x"
    using homeomorphism_moving_point_2 [OF ‹affine T› ‹a ∈ T› u v] by blast
  have gid: "⋀x. ⟦x ∈ sphere a r; x ∈ T⟧ ⟹ g x = x"
    using fid hom homeomorphism_apply1 by fastforce
  define ff where "ff ≡ λx. if x ∈ ball a r ∩ T then f x else x"
  define gg where "gg ≡ λx. if x ∈ ball a r ∩ T then g x else x"
  show ?thesis
  proof
    show "homeomorphism S S ff gg"
    proof (rule homeomorphismI)
      have "continuous_on ((cball a r ∩ T) ∪ (T - ball a r)) ff"
        unfolding ff_def
        using homeomorphism_cont1 [OF hom] 
        by (intro continuous_on_cases) (auto simp: affine_closed ‹affine T› fid)
      then show "continuous_on S ff"
        by (rule continuous_on_subset) (use ST in auto)
      have "continuous_on ((cball a r ∩ T) ∪ (T - ball a r)) gg"
        unfolding gg_def
        using homeomorphism_cont2 [OF hom] 
        by (intro continuous_on_cases) (auto simp: affine_closed ‹affine T› gid)
      then show "continuous_on S gg"
        by (rule continuous_on_subset) (use ST in auto)
      show "ff ` S ⊆ S"
      proof (clarsimp simp: ff_def)
        fix x
        assume "x ∈ S" and x: "dist a x < r" and "x ∈ T"
        then have "f x ∈ cball a r ∩ T"
          using homeomorphism_image1 [OF hom] by force
        then show "f x ∈ S"
          using ST(1) ‹x ∈ T› gid hom homeomorphism_def x by fastforce
      qed
      show "gg ` S ⊆ S"
      proof (clarsimp simp: gg_def)
        fix x
        assume "x ∈ S" and x: "dist a x < r" and "x ∈ T"
        then have "g x ∈ cball a r ∩ T"
          using homeomorphism_image2 [OF hom] by force
        then have "g x ∈ ball a r"
          using homeomorphism_apply2 [OF hom]
            by (metis Diff_Diff_Int Diff_iff  ‹x ∈ T› cball_def fid le_less mem_Collect_eq mem_ball mem_sphere x)
        then show "g x ∈ S"
          using ST(1) ‹g x ∈ cball a r ∩ T› by force
        qed
      show "⋀x. x ∈ S ⟹ gg (ff x) = x"
        unfolding ff_def gg_def
        using homeomorphism_apply1 [OF hom] homeomorphism_image1 [OF hom]
        by simp (metis Int_iff homeomorphism_apply1 [OF hom] fid image_eqI less_eq_real_def mem_cball mem_sphere)
      show "⋀x. x ∈ S ⟹ ff (gg x) = x"
        unfolding ff_def gg_def
        using homeomorphism_apply2 [OF hom] homeomorphism_image2 [OF hom]
        by simp (metis Int_iff fid image_eqI less_eq_real_def mem_cball mem_sphere)
    qed
    show "ff u = v"
      using u by (auto simp: ff_def ‹f u = v›)
    show "{x. ¬ (ff x = x ∧ gg x = x)} ⊆ ball a r ∩ T"
      by (auto simp: ff_def gg_def)
  qed
qed


proposition✐‹tag unimportant› homeomorphism_moving_point:
  fixes a :: "'a::euclidean_space"
  assumes ope: "openin (top_of_set (affine hull S)) S"
      and "S ⊆ T"
      and TS: "T ⊆ affine hull S"
      and S: "connected S" "a ∈ S" "b ∈ S"
  obtains f g where "homeomorphism T T f g" "f a = b"
                    "{x. ¬ (f x = x ∧ g x = x)} ⊆ S"
                    "bounded {x. ¬ (f x = x ∧ g x = x)}"
proof -
  have 1: "∃h k. homeomorphism T T h k ∧ h (f d) = d ∧
              {x. ¬ (h x = x ∧ k x = x)} ⊆ S ∧ bounded {x. ¬ (h x = x ∧ k x = x)}"
        if "d ∈ S" "f d ∈ S" and homfg: "homeomorphism T T f g"
        and S: "{x. ¬ (f x = x ∧ g x = x)} ⊆ S"
        and bo: "bounded {x. ¬ (f x = x ∧ g x = x)}" for d f g
  proof (intro exI conjI)
    show homgf: "homeomorphism T T g f"
      by (metis homeomorphism_symD homfg)
    then show "g (f d) = d"
      by (meson ‹S ⊆ T› homeomorphism_def subsetD ‹d ∈ S›)
    show "{x. ¬ (g x = x ∧ f x = x)} ⊆ S"
      using S by blast
    show "bounded {x. ¬ (g x = x ∧ f x = x)}"
      using bo by (simp add: conj_commute)
  qed
  have 2: "∃f g. homeomorphism T T f g ∧ f x = f2 (f1 x) ∧
                 {x. ¬ (f x = x ∧ g x = x)} ⊆ S ∧ bounded {x. ¬ (f x = x ∧ g x = x)}"
             if "x ∈ S" "f1 x ∈ S" "f2 (f1 x) ∈ S"
                and hom: "homeomorphism T T f1 g1" "homeomorphism T T f2 g2"
                and sub: "{x. ¬ (f1 x = x ∧ g1 x = x)} ⊆ S"   "{x. ¬ (f2 x = x ∧ g2 x = x)} ⊆ S"
                and bo: "bounded {x. ¬ (f1 x = x ∧ g1 x = x)}"  "bounded {x. ¬ (f2 x = x ∧ g2 x = x)}"
             for x f1 f2 g1 g2
  proof (intro exI conjI)
    show homgf: "homeomorphism T T (f2 ∘ f1) (g1 ∘ g2)"
      by (metis homeomorphism_compose hom)
    then show "(f2 ∘ f1) x = f2 (f1 x)"
      by force
    show "{x. ¬ ((f2 ∘ f1) x = x ∧ (g1 ∘ g2) x = x)} ⊆ S"
      using sub by force
    have "bounded ({x. ¬(f1 x = x ∧ g1 x = x)} ∪ {x. ¬(f2 x = x ∧ g2 x = x)})"
      using bo by simp
    then show "bounded {x. ¬ ((f2 ∘ f1) x = x ∧ (g1 ∘ g2) x = x)}"
      by (rule bounded_subset) auto
  qed
  have 3: "∃U. openin (top_of_set S) U ∧
              d ∈ U ∧
              (∀x∈U.
                  ∃f g. homeomorphism T T f g ∧ f d = x ∧
                        {x. ¬ (f x = x ∧ g x = x)} ⊆ S ∧
                        bounded {x. ¬ (f x = x ∧ g x = x)})"
           if "d ∈ S" for d
  proof -
    obtain r where "r > 0" and r: "ball d r ∩ affine hull S ⊆ S"
      by (metis ‹d ∈ S› ope openin_contains_ball)
    have *: "∃f g. homeomorphism T T f g ∧ f d = e ∧
                   {x. ¬ (f x = x ∧ g x = x)} ⊆ S ∧
                   bounded {x. ¬ (f x = x ∧ g x = x)}" if "e ∈ S" "e ∈ ball d r" for e
      apply (rule homeomorphism_moving_point_3 [of "affine hull S" d r T d e])
      using r ‹S ⊆ T› TS that 
            apply (auto simp: ‹d ∈ S› ‹0 < r› hull_inc)
      using bounded_subset by blast
    show ?thesis
      by (rule_tac x="S ∩ ball d r" in exI) (fastforce simp: openin_open_Int ‹0 < r› that intro: *)
  qed
  have "∃f g. homeomorphism T T f g ∧ f a = b ∧
              {x. ¬ (f x = x ∧ g x = x)} ⊆ S ∧ bounded {x. ¬ (f x = x ∧ g x = x)}"
    by (rule connected_equivalence_relation [OF S]; blast intro: 1 2 3)
  then show ?thesis
    using that by auto
qed


lemma homeomorphism_moving_points_exists_gen:
  assumes K: "finite K" "⋀i. i ∈ K ⟹ x i ∈ S ∧ y i ∈ S"
             "pairwise (λi j. (x i ≠ x j) ∧ (y i ≠ y j)) K"
      and "2 ≤ aff_dim S"
      and ope: "openin (top_of_set (affine hull S)) S"
      and "S ⊆ T" "T ⊆ affine hull S" "connected S"
  shows "∃f g. homeomorphism T T f g ∧ (∀i ∈ K. f(x i) = y i) ∧
               {x. ¬ (f x = x ∧ g x = x)} ⊆ S ∧ bounded {x. ¬ (f x = x ∧ g x = x)}"
  using assms
proof (induction K)
  case empty
  then show ?case
    by (force simp: homeomorphism_ident)
next
  case (insert i K)
  then have xney: "⋀j. ⟦j ∈ K; j ≠ i⟧ ⟹ x i ≠ x j ∧ y i ≠ y j"
       and pw: "pairwise (λi j. x i ≠ x j ∧ y i ≠ y j) K"
       and "x i ∈ S" "y i ∈ S"
       and xyS: "⋀i. i ∈ K ⟹ x i ∈ S ∧ y i ∈ S"
    by (simp_all add: pairwise_insert)
  obtain f g where homfg: "homeomorphism T T f g" and feq: "⋀i. i ∈ K ⟹ f(x i) = y i"
               and fg_sub: "{x. ¬ (f x = x ∧ g x = x)} ⊆ S"
               and bo_fg: "bounded {x. ¬ (f x = x ∧ g x = x)}"
    using insert.IH [OF xyS pw] insert.prems by (blast intro: that)
  then have "∃f g. homeomorphism T T f g ∧ (∀i ∈ K. f(x i) = y i) ∧
                   {x. ¬ (f x = x ∧ g x = x)} ⊆ S ∧ bounded {x. ¬ (f x = x ∧ g x = x)}"
    using insert by blast
  have aff_eq: "affine hull (S - y ` K) = affine hull S"
  proof (rule affine_hull_Diff [OF ope])
    show "finite (y ` K)"
      by (simp add: insert.hyps(1))
    show "y ` K ⊂ S"
      using ‹y i ∈ S› insert.hyps(2) xney xyS by fastforce
  qed
  have f_in_S: "f x ∈ S" if "x ∈ S" for x
    using homfg fg_sub homeomorphism_apply1 ‹S ⊆ T›
  proof -
    have "(f (f x) ≠ f x ∨ g (f x) ≠ f x) ∨ f x ∈ S"
      by (metis ‹S ⊆ T› homfg subsetD homeomorphism_apply1 that)
    then show ?thesis
      using fg_sub by force
  qed
  obtain h k where homhk: "homeomorphism T T h k" and heq: "h (f (x i)) = y i"
               and hk_sub: "{x. ¬ (h x = x ∧ k x = x)} ⊆ S - y ` K"
               and bo_hk:  "bounded {x. ¬ (h x = x ∧ k x = x)}"
  proof (rule homeomorphism_moving_point [of "S - y`K" T "f(x i)" "y i"])
    show "openin (top_of_set (affine hull (S - y ` K))) (S - y ` K)"
      by (simp add: aff_eq openin_diff finite_imp_closedin image_subset_iff hull_inc insert xyS)
    show "S - y ` K ⊆ T"
      using ‹S ⊆ T› by auto
    show "T ⊆ affine hull (S - y ` K)"
      using insert by (simp add: aff_eq)
    show "connected (S - y ` K)"
    proof (rule connected_openin_diff_countable [OF ‹connected S› ope])
      show "¬ collinear S"
        using collinear_aff_dim ‹2 ≤ aff_dim S› by force
      show "countable (y ` K)"
        using countable_finite insert.hyps(1) by blast
    qed
    have "⋀k. ⟦f (x i) = y k; k ∈ K⟧ ⟹ False"
        by (metis feq homfg ‹x i ∈ S› homeomorphism_def ‹S ⊆ T› ‹i ∉ K› subsetCE xney xyS)
    then show "f (x i) ∈ S - y ` K"
      by (auto simp: f_in_S ‹x i ∈ S›)
    show "y i ∈ S - y ` K"
      using insert.hyps xney by (auto simp: ‹y i ∈ S›)
  qed blast
  show ?case
  proof (intro exI conjI)
    show "homeomorphism T T (h ∘ f) (g ∘ k)"
      using homfg homhk homeomorphism_compose by blast
    show "∀i ∈ insert i K. (h ∘ f) (x i) = y i"
      using feq hk_sub by (auto simp: heq)
    show "{x. ¬ ((h ∘ f) x = x ∧ (g ∘ k) x = x)} ⊆ S"
      using fg_sub hk_sub by force
    have "bounded ({x. ¬(f x = x ∧ g x = x)} ∪ {x. ¬(h x = x ∧ k x = x)})"
      using bo_fg bo_hk bounded_Un by blast
    then show "bounded {x. ¬ ((h ∘ f) x = x ∧ (g ∘ k) x = x)}"
      by (rule bounded_subset) auto
  qed
qed

proposition✐‹tag unimportant› homeomorphism_moving_points_exists:
  fixes S :: "'a::euclidean_space set"
  assumes 2: "2 ≤ DIM('a)" "open S" "connected S" "S ⊆ T" "finite K"
      and KS: "⋀i. i ∈ K ⟹ x i ∈ S ∧ y i ∈ S"
      and pw: "pairwise (λi j. (x i ≠ x j) ∧ (y i ≠ y j)) K"
      and S: "S ⊆ T" "T ⊆ affine hull S" "connected S"
  obtains f g where "homeomorphism T T f g" "⋀i. i ∈ K ⟹ f(x i) = y i"
                    "{x. ¬ (f x = x ∧ g x = x)} ⊆ S" "bounded {x. (¬ (f x = x ∧ g x = x))}"
proof (cases "S = {}")
  case True
  then show ?thesis
    using KS homeomorphism_ident that by fastforce
next
  case False
  then have affS: "affine hull S = UNIV"
    by (simp add: affine_hull_open ‹open S›)
  then have ope: "openin (top_of_set (affine hull S)) S"
    using ‹open S› open_openin by auto
  have "2 ≤ DIM('a)" by (rule 2)
  also have "… = aff_dim (UNIV :: 'a set)"
    by simp
  also have "… ≤ aff_dim S"
    by (metis aff_dim_UNIV aff_dim_affine_hull aff_dim_le_DIM affS)
  finally have "2 ≤ aff_dim S"
    by linarith
  then show ?thesis
    using homeomorphism_moving_points_exists_gen [OF ‹finite K› KS pw _ ope S] that by fastforce
qed

subsubsection✐‹tag unimportant›‹The theorem ‹homeomorphism_grouping_points_exists››

lemma homeomorphism_grouping_point_1:
  fixes a::real and c::real
  assumes "a < b" "c < d"
  obtains f g where "homeomorphism (cbox a b) (cbox c d) f g" "f a = c" "f b = d"
proof -
  define f where "f ≡ λx. ((d - c) / (b - a)) * x + (c - a * ((d - c) / (b - a)))"
  have "∃g. homeomorphism (cbox a b) (cbox c d) f g"
  proof (rule homeomorphism_compact)
    show "continuous_on (cbox a b) f"
      unfolding f_def by (intro continuous_intros)
    have "f ` {a..b} = {c..d}"
      unfolding f_def image_affinity_atLeastAtMost
      using assms sum_sqs_eq by (auto simp: field_split_simps)
    then show "f ` cbox a b = cbox c d"
      by auto
    show "inj_on f (cbox a b)"
      unfolding f_def inj_on_def using assms by auto
  qed auto
  then obtain g where "homeomorphism (cbox a b) (cbox c d) f g" ..
  then show ?thesis
  proof
    show "f a = c"
      by (simp add: f_def)
    show "f b = d"
      using assms sum_sqs_eq [of a b] by (auto simp: f_def field_split_simps)
  qed
qed

lemma homeomorphism_grouping_point_2:
  fixes a::real and w::real
  assumes hom_ab: "homeomorphism (cbox a b) (cbox u v) f1 g1"
      and hom_bc: "homeomorphism (cbox b c) (cbox v w) f2 g2"
      and "b ∈ cbox a c" "v ∈ cbox u w"
      and eq: "f1 a = u" "f1 b = v" "f2 b = v" "f2 c = w"
 obtains f g where "homeomorphism (cbox a c) (cbox u w) f g" "f a = u" "f c = w"
                   "⋀x. x ∈ cbox a b ⟹ f x = f1 x" "⋀x. x ∈ cbox b c ⟹ f x = f2 x"
proof -
  have le: "a ≤ b" "b ≤ c" "u ≤ v" "v ≤ w"
    using assms by simp_all
  then have ac: "cbox a c = cbox a b ∪ cbox b c" and uw: "cbox u w = cbox u v ∪ cbox v w"
    by auto
  define f where "f ≡ λx. if x ≤ b then f1 x else f2 x"
  have "∃g. homeomorphism (cbox a c) (cbox u w) f g"
  proof (rule homeomorphism_compact)
    have cf1: "continuous_on (cbox a b) f1"
      using hom_ab homeomorphism_cont1 by blast
    have cf2: "continuous_on (cbox b c) f2"
      using hom_bc homeomorphism_cont1 by blast
    show "continuous_on (cbox a c) f"
      unfolding f_def using le eq
      by (force intro: continuous_on_cases_le [OF continuous_on_subset [OF cf1] continuous_on_subset [OF cf2]])
    have "f ` cbox a b = f1 ` cbox a b" "f ` cbox b c = f2 ` cbox b c"
      unfolding f_def using eq by force+
    then show "f ` cbox a c = cbox u w"
      unfolding ac uw image_Un by (metis hom_ab hom_bc homeomorphism_def)
    have neq12: "f1 x ≠ f2 y" if x: "a ≤ x" "x ≤ b" and y: "b < y" "y ≤ c" for x y
    proof -
      have "f1 x ∈ cbox u v"
        by (metis hom_ab homeomorphism_def image_eqI mem_box_real(2) x)
      moreover have "f2 y ∈ cbox v w"
        by (metis (full_types) hom_bc homeomorphism_def image_subset_iff mem_box_real(2) not_le not_less_iff_gr_or_eq order_refl y)
      moreover have "f2 y ≠ f2 b"
        by (metis cancel_comm_monoid_add_class.diff_cancel diff_gt_0_iff_gt hom_bc homeomorphism_def le(2) less_imp_le less_numeral_extra(3) mem_box_real(2) order_refl y)
      ultimately show ?thesis
        using le eq by simp
    qed
    have "inj_on f1 (cbox a b)"
      by (metis (full_types) hom_ab homeomorphism_def inj_onI)
    moreover have "inj_on f2 (cbox b c)"
      by (metis (full_types) hom_bc homeomorphism_def inj_onI)
    ultimately show "inj_on f (cbox a c)"
      apply (simp (no_asm) add: inj_on_def)
      apply (simp add: f_def inj_on_eq_iff)
      using neq12 by force
  qed auto
  then obtain g where "homeomorphism (cbox a c) (cbox u w) f g" ..
  then show ?thesis
    using eq f_def le that by force
qed

lemma homeomorphism_grouping_point_3:
  fixes a::real
  assumes cbox_sub: "cbox c d ⊆ box a b" "cbox u v ⊆ box a b"
      and box_ne: "box c d ≠ {}" "box u v ≠ {}"
  obtains f g where "homeomorphism (cbox a b) (cbox a b) f g" "f a = a" "f b = b"
                    "⋀x. x ∈ cbox c d ⟹ f x ∈ cbox u v"
proof -
  have less: "a < c" "a < u" "d < b" "v < b" "c < d" "u < v" "cbox c d ≠ {}"
    using assms
    by (simp_all add: cbox_sub subset_eq)
  obtain f1 g1 where 1: "homeomorphism (cbox a c) (cbox a u) f1 g1"
                   and f1_eq: "f1 a = a" "f1 c = u"
    using homeomorphism_grouping_point_1 [OF ‹a < c› ‹a < u›] .
  obtain f2 g2 where 2: "homeomorphism (cbox c d) (cbox u v) f2 g2"
                   and f2_eq: "f2 c = u" "f2 d = v"
    using homeomorphism_grouping_point_1 [OF ‹c < d› ‹u < v›] .
  obtain f3 g3 where 3: "homeomorphism (cbox d b) (cbox v b) f3 g3"
                   and f3_eq: "f3 d = v" "f3 b = b"
    using homeomorphism_grouping_point_1 [OF ‹d < b› ‹v < b›] .
  obtain f4 g4 where 4: "homeomorphism (cbox a d) (cbox a v) f4 g4" and "f4 a = a" "f4 d = v"
                 and f4_eq: "⋀x. x ∈ cbox a c ⟹ f4 x = f1 x" "⋀x. x ∈ cbox c d ⟹ f4 x = f2 x"
    using homeomorphism_grouping_point_2 [OF 1 2] less  by (auto simp: f1_eq f2_eq)
  obtain f g where fg: "homeomorphism (cbox a b) (cbox a b) f g" "f a = a" "f b = b"
               and f_eq: "⋀x. x ∈ cbox a d ⟹ f x = f4 x" "⋀x. x ∈ cbox d b ⟹ f x = f3 x"
    using homeomorphism_grouping_point_2 [OF 4 3] less by (auto simp: f4_eq f3_eq f2_eq f1_eq)
  show ?thesis
  proof (rule that [OF fg])
    show "f x ∈ cbox u v" if "x ∈ cbox c d" for x 
      using that f4_eq f_eq homeomorphism_image1 [OF 2]
      by (metis atLeastAtMost_iff box_real(2) image_eqI less(1) less_eq_real_def order_trans)
  qed
qed


lemma homeomorphism_grouping_point_4:
  fixes T :: "real set"
  assumes "open U" "open S" "connected S" "U ≠ {}" "finite K" "K ⊆ S" "U ⊆ S" "S ⊆ T"
  obtains f g where "homeomorphism T T f g"
                    "⋀x. x ∈ K ⟹ f x ∈ U" "{x. (¬ (f x = x ∧ g x = x))} ⊆ S"
                    "bounded {x. (¬ (f x = x ∧ g x = x))}"
proof -
  obtain c d where "box c d ≠ {}" "cbox c d ⊆ U"
  proof -
    obtain u where "u ∈ U"
      using ‹U ≠ {}› by blast
    then obtain e where "e > 0" "cball u e ⊆ U"
      using ‹open U› open_contains_cball by blast
    then show ?thesis
      by (rule_tac c=u and d="u+e" in that) (auto simp: dist_norm subset_iff)
  qed
  have "compact K"
    by (simp add: ‹finite K› finite_imp_compact)
  obtain a b where "box a b ≠ {}" "K ⊆ cbox a b" "cbox a b ⊆ S"
  proof (cases "K = {}")
    case True then show ?thesis
      using ‹box c d ≠ {}› ‹cbox c d ⊆ U› ‹U ⊆ S› that by blast
  next
    case False
    then obtain a b where "a ∈ K" "b ∈ K"
            and a: "⋀x. x ∈ K ⟹ a ≤ x" and b: "⋀x. x ∈ K ⟹ x ≤ b"
      using compact_attains_inf compact_attains_sup by (metis ‹compact K›)+
    obtain e where "e > 0" "cball b e ⊆ S"
      using ‹open S› open_contains_cball
      by (metis ‹b ∈ K› ‹K ⊆ S› subsetD)
    show ?thesis
    proof
      show "box a (b + e) ≠ {}"
        using ‹0 < e› ‹b ∈ K› a by force
      show "K ⊆ cbox a (b + e)"
        using ‹0 < e› a b by fastforce
      have "a ∈ S"
        using ‹a ∈ K› assms(6) by blast
      have "b + e ∈ S"
        using ‹0 < e› ‹cball b e ⊆ S›  by (force simp: dist_norm)
      show "cbox a (b + e) ⊆ S"
        using ‹a ∈ S› ‹b + e ∈ S› ‹connected S› connected_contains_Icc by auto
    qed
  qed
  obtain w z where "cbox w z ⊆ S" and sub_wz: "cbox a b ∪ cbox c d ⊆ box w z"
  proof -
    have "a ∈ S" "b ∈ S"
      using ‹box a b ≠ {}› ‹cbox a b ⊆ S› by auto
    moreover have "c ∈ S" "d ∈ S"
      using ‹box c d ≠ {}› ‹cbox c d ⊆ U› ‹U ⊆ S› by force+
    ultimately have "min a c ∈ S" "max b d ∈ S"
      by linarith+
    then obtain e1 e2 where "e1 > 0" "cball (min a c) e1 ⊆ S" "e2 > 0" "cball (max b d) e2 ⊆ S"
      using ‹open S› open_contains_cball by metis
    then have *: "min a c - e1 ∈ S" "max b d + e2 ∈ S"
      by (auto simp: dist_norm)
    show ?thesis
    proof
      show "cbox (min a c - e1) (max b d+ e2) ⊆ S"
        using * ‹connected S› connected_contains_Icc by auto
      show "cbox a b ∪ cbox c d ⊆ box (min a c - e1) (max b d + e2)"
        using ‹0 < e1› ‹0 < e2› by auto
    qed
  qed
  then
  obtain f g where hom: "homeomorphism (cbox w z) (cbox w z) f g"
               and "f w = w" "f z = z"
               and fin: "⋀x. x ∈ cbox a b ⟹ f x ∈ cbox c d"
    using homeomorphism_grouping_point_3 [of a b w z c d]
    using ‹box a b ≠ {}› ‹box c d ≠ {}› by blast
  have contfg: "continuous_on (cbox w z) f" "continuous_on (cbox w z) g"
    using hom homeomorphism_def by blast+
  define f' where "f' ≡ λx. if x ∈ cbox w z then f x else x"
  define g' where "g' ≡ λx. if x ∈ cbox w z then g x else x"
  show ?thesis
  proof
    have T: "cbox w z ∪ (T - box w z) = T"
      using ‹cbox w z ⊆ S› ‹S ⊆ T› by auto
    show "homeomorphism T T f' g'"
    proof
      have clo: "closedin (top_of_set (cbox w z ∪ (T - box w z))) (T - box w z)"
        by (metis Diff_Diff_Int Diff_subset T closedin_def open_box openin_open_Int topspace_euclidean_subtopology)
      have "⋀x. ⟦w ≤ x ∧ x ≤ z; w < x ⟶ ¬ x < z⟧ ⟹ f x = x"
        using ‹f w = w› ‹f z = z› by auto
      moreover have "⋀x. ⟦w ≤ x ∧ x ≤ z; w < x ⟶ ¬ x < z⟧ ⟹ g x = x"
        using ‹f w = w› ‹f z = z› hom homeomorphism_apply1 by fastforce
      ultimately
      have "continuous_on (cbox w z ∪ (T - box w z)) f'" "continuous_on (cbox w z ∪ (T - box w z)) g'"
        unfolding f'_def g'_def
        by (intro continuous_on_cases_local contfg continuous_on_id clo; auto simp: closed_subset)+
      then show "continuous_on T f'" "continuous_on T g'"
        by (simp_all only: T)
      show "f' ` T ⊆ T"
        unfolding f'_def
        by clarsimp (metis ‹cbox w z ⊆ S› ‹S ⊆ T› subsetD hom homeomorphism_def imageI mem_box_real(2))
      show "g' ` T ⊆ T"
        unfolding g'_def
        by clarsimp (metis ‹cbox w z ⊆ S› ‹S ⊆ T› subsetD hom homeomorphism_def imageI mem_box_real(2))
      show "⋀x. x ∈ T ⟹ g' (f' x) = x"
        unfolding f'_def g'_def
        using homeomorphism_apply1 [OF hom]  homeomorphism_image1 [OF hom] by fastforce
      show "⋀y. y ∈ T ⟹ f' (g' y) = y"
        unfolding f'_def g'_def
        using homeomorphism_apply2 [OF hom]  homeomorphism_image2 [OF hom] by fastforce
    qed
    show "⋀x. x ∈ K ⟹ f' x ∈ U"
      using fin sub_wz ‹K ⊆ cbox a b› ‹cbox c d ⊆ U› by (force simp: f'_def)
    show "{x. ¬ (f' x = x ∧ g' x = x)} ⊆ S"
      using ‹cbox w z ⊆ S› by (auto simp: f'_def g'_def)
    show "bounded {x. ¬ (f' x = x ∧ g' x = x)}"
    proof (rule bounded_subset [of "cbox w z"])
      show "bounded (cbox w z)"
        using bounded_cbox by blast
      show "{x. ¬ (f' x = x ∧ g' x = x)} ⊆ cbox w z"
        by (auto simp: f'_def g'_def)
    qed
  qed
qed

proposition✐‹tag unimportant› homeomorphism_grouping_points_exists:
  fixes S :: "'a::euclidean_space set"
  assumes "open U" "open S" "connected S" "U ≠ {}" "finite K" "K ⊆ S" "U ⊆ S" "S ⊆ T"
  obtains f g where "homeomorphism T T f g" "{x. (¬ (f x = x ∧ g x = x))} ⊆ S"
                    "bounded {x. (¬ (f x = x ∧ g x = x))}" "⋀x. x ∈ K ⟹ f x ∈ U"
proof (cases "2 ≤ DIM('a)")
  case True
  have TS: "T ⊆ affine hull S"
    using affine_hull_open assms by blast
  have "infinite U"
    using ‹open U› ‹U ≠ {}› finite_imp_not_open by blast
  then obtain P where "P ⊆ U" "finite P" "card K = card P"
    using infinite_arbitrarily_large by metis
  then obtain γ where γ: "bij_betw γ K P"
    using ‹finite K› finite_same_card_bij by blast
  obtain f g where "homeomorphism T T f g" "⋀i. i ∈ K ⟹ f (id i) = γ i" "{x. ¬ (f x = x ∧ g x = x)} ⊆ S" "bounded {x. ¬ (f x = x ∧ g x = x)}"
  proof (rule homeomorphism_moving_points_exists [OF True ‹open S› ‹connected S› ‹S ⊆ T› ‹finite K›])
    show "⋀i. i ∈ K ⟹ id i ∈ S ∧ γ i ∈ S"
      using ‹P ⊆ U› ‹bij_betw γ K P› ‹K ⊆ S› ‹U ⊆ S› bij_betwE by blast
    show "pairwise (λi j. id i ≠ id j ∧ γ i ≠ γ j) K"
      using γ by (auto simp: pairwise_def bij_betw_def inj_on_def)
  qed (use affine_hull_open assms that in auto)
  then show ?thesis
    using γ ‹P ⊆ U› bij_betwE by (fastforce simp: intro!: that)
next
  case False
  with DIM_positive have "DIM('a) = 1"
    by (simp add: dual_order.antisym)
  then obtain h::"'a ⇒real" and j
  where "linear h" "linear j"
    and noh: "⋀x. norm(h x) = norm x" and noj: "⋀y. norm(j y) = norm y"
    and hj:  "⋀x. j(h x) = x" "⋀y. h(j y) = y"
    and ranh: "surj h"
    using isomorphisms_UNIV_UNIV
    by (metis (mono_tags, opaque_lifting) DIM_real UNIV_eq_I range_eqI)
  obtain f g where hom: "homeomorphism (h ` T) (h ` T) f g"
               and f: "⋀x. x ∈ h ` K ⟹ f x ∈ h ` U"
               and sub: "{x. ¬ (f x = x ∧ g x = x)} ⊆ h ` S"
               and bou: "bounded {x. ¬ (f x = x ∧ g x = x)}"
    apply (rule homeomorphism_grouping_point_4 [of "h ` U" "h ` S" "h ` K" "h ` T"])
    by (simp_all add: assms image_mono  ‹linear h› open_surjective_linear_image connected_linear_image ranh)
  have jf: "j (f (h x)) = x ⟷ f (h x) = h x" for x
    by (metis hj)
  have jg: "j (g (h x)) = x ⟷ g (h x) = h x" for x
    by (metis hj)
  have cont_hj: "continuous_on X h"  "continuous_on Y j" for X Y
    by (simp_all add: ‹linear h› ‹linear j› linear_linear linear_continuous_on)
  show ?thesis
  proof
    show "homeomorphism T T (j ∘ f ∘ h) (j ∘ g ∘ h)"
    proof
      show "continuous_on T (j ∘ f ∘ h)" "continuous_on T (j ∘ g ∘ h)"
        using hom homeomorphism_def
        by (blast intro: continuous_on_compose cont_hj)+
      show "(j ∘ f ∘ h) ` T ⊆ T" "(j ∘ g ∘ h) ` T ⊆ T"
        by auto (metis (mono_tags, opaque_lifting) hj(1) hom homeomorphism_def imageE imageI)+
      show "⋀x. x ∈ T ⟹ (j ∘ g ∘ h) ((j ∘ f ∘ h) x) = x"
        using hj hom homeomorphism_apply1 by fastforce
      show "⋀y. y ∈ T ⟹ (j ∘ f ∘ h) ((j ∘ g ∘ h) y) = y"
        using hj hom homeomorphism_apply2 by fastforce
    qed
    show "{x. ¬ ((j ∘ f ∘ h) x = x ∧ (j ∘ g ∘ h) x = x)} ⊆ S"
    proof (clarsimp simp: jf jg hj)
      show "f (h x) = h x ⟶ g (h x) ≠ h x ⟹ x ∈ S" for x
        using sub [THEN subsetD, of "h x"] hj by simp (metis imageE)
    qed
    have "bounded (j ` {x. (¬ (f x = x ∧ g x = x))})"
      by (rule bounded_linear_image [OF bou]) (use ‹linear j› linear_conv_bounded_linear in auto)
    moreover
    have *: "{x. ¬((j ∘ f ∘ h) x = x ∧ (j ∘ g ∘ h) x = x)} = j ` {x. (¬ (f x = x ∧ g x = x))}"
      using hj by (auto simp: jf jg image_iff, metis+)
    ultimately show "bounded {x. ¬ ((j ∘ f ∘ h) x = x ∧ (j ∘ g ∘ h) x = x)}"
      by metis
    show "⋀x. x ∈ K ⟹ (j ∘ f ∘ h) x ∈ U"
      using f hj by fastforce
  qed
qed


proposition✐‹tag unimportant› homeomorphism_grouping_points_exists_gen:
  fixes S :: "'a::euclidean_space set"
  assumes opeU: "openin (top_of_set S) U"
      and opeS: "openin (top_of_set (affine hull S)) S"
      and "U ≠ {}" "finite K" "K ⊆ S" and S: "S ⊆ T" "T ⊆ affine hull S" "connected S"
  obtains f g where "homeomorphism T T f g" "{x. (¬ (f x = x ∧ g x = x))} ⊆ S"
                    "bounded {x. (¬ (f x = x ∧ g x = x))}" "⋀x. x ∈ K ⟹ f x ∈ U"
proof (cases "2 ≤ aff_dim S")
  case True
  have opeU': "openin (top_of_set (affine hull S)) U"
    using opeS opeU openin_trans by blast
  obtain u where "u ∈ U" "u ∈ S"
    using ‹U ≠ {}› opeU openin_imp_subset by fastforce+
  have "infinite U"
  proof (rule infinite_openin [OF opeU ‹u ∈ U›])
    show "u islimpt S"
      using True ‹u ∈ S› assms(8) connected_imp_perfect_aff_dim by fastforce
  qed
  then obtain P where "P ⊆ U" "finite P" "card K = card P"
    using infinite_arbitrarily_large by metis
  then obtain γ where γ: "bij_betw γ K P"
    using ‹finite K› finite_same_card_bij by blast
  have "∃f g. homeomorphism T T f g ∧ (∀i ∈ K. f(id i) = γ i) ∧
               {x. ¬ (f x = x ∧ g x = x)} ⊆ S ∧ bounded {x. ¬ (f x = x ∧ g x = x)}"
  proof (rule homeomorphism_moving_points_exists_gen [OF ‹finite K› _ _ True opeS S])
    show "⋀i. i ∈ K ⟹ id i ∈ S ∧ γ i ∈ S"
      by (metis id_apply opeU openin_contains_cball subsetCE ‹P ⊆ U› ‹bij_betw γ K P› ‹K ⊆ S› bij_betwE)
    show "pairwise (λi j. id i ≠ id j ∧ γ i ≠ γ j) K"
      using γ by (auto simp: pairwise_def bij_betw_def inj_on_def)
  qed
  then show ?thesis
    using γ ‹P ⊆ U› bij_betwE by (fastforce simp: intro!: that)
next
  case False
  with aff_dim_geq [of S] consider "aff_dim S = -1" | "aff_dim S = 0" | "aff_dim S = 1" by linarith
  then show ?thesis
  proof cases
    assume "aff_dim S = -1"
    then have "S = {}"
      using aff_dim_empty by blast
    then have "False"
      using ‹U ≠ {}› ‹K ⊆ S› openin_imp_subset [OF opeU] by blast
    then show ?thesis ..
  next
    assume "aff_dim S = 0"
    then obtain a where "S = {a}"
      using aff_dim_eq_0 by blast
    then have "K ⊆ U"
      using ‹U ≠ {}› ‹K ⊆ S› openin_imp_subset [OF opeU] by blast
    show ?thesis
      using ‹K ⊆ U› by (intro that [of id id]) (auto intro: homeomorphismI)
  next
    assume "aff_dim S = 1"
    then have "affine hull S homeomorphic (UNIV :: real set)"
      by (auto simp: homeomorphic_affine_sets)
    then obtain h::"'a⇒real" and j where homhj: "homeomorphism (affine hull S) UNIV h j"
      using homeomorphic_def by blast
    then have h: "⋀x. x ∈ affine hull S ⟹ j(h(x)) = x" and j: "⋀y. j y ∈ affine hull S ∧ h(j y) = y"
      by (auto simp: homeomorphism_def)
    have connh: "connected (h ` S)"
      by (meson Topological_Spaces.connected_continuous_image ‹connected S› homeomorphism_cont1 homeomorphism_of_subsets homhj hull_subset top_greatest)
    have hUS: "h ` U ⊆ h ` S"
      by (meson homeomorphism_imp_open_map homeomorphism_of_subsets homhj hull_subset opeS opeU open_UNIV openin_open_eq)
    have opn: "openin (top_of_set (affine hull S)) U ⟹ open (h ` U)" for U
      using homeomorphism_imp_open_map [OF homhj]  by simp
    have "open (h ` U)" "open (h ` S)"
      by (auto intro: opeS opeU openin_trans opn)
    then obtain f g where hom: "homeomorphism (h ` T) (h ` T) f g"
                 and f: "⋀x. x ∈ h ` K ⟹ f x ∈ h ` U"
                 and sub: "{x. ¬ (f x = x ∧ g x = x)} ⊆ h ` S"
                 and bou: "bounded {x. ¬ (f x = x ∧ g x = x)}"
      apply (rule homeomorphism_grouping_points_exists [of "h ` U" "h ` S" "h ` K" "h ` T"])
      using assms by (auto simp: connh hUS)
    have jf: "⋀x. x ∈ affine hull S ⟹ j (f (h x)) = x ⟷ f (h x) = h x"
      by (metis h j)
    have jg: "⋀x. x ∈ affine hull S ⟹ j (g (h x)) = x ⟷ g (h x) = h x"
      by (metis h j)
    have cont_hj: "continuous_on T h"  "continuous_on Y j" for Y
    proof (rule continuous_on_subset [OF _ ‹T ⊆ affine hull S›])
      show "continuous_on (affine hull S) h"
        using homeomorphism_def homhj by blast
    qed (meson continuous_on_subset homeomorphism_def homhj top_greatest)
    define f' where "f' ≡ λx. if x ∈ affine hull S then (j ∘ f ∘ h) x else x"
    define g' where "g' ≡ λx. if x ∈ affine hull S then (j ∘ g ∘ h) x else x"
    show ?thesis
    proof
      show "homeomorphism T T f' g'"
      proof
        have "continuous_on T (j ∘ f ∘ h)"
          using hom homeomorphism_def by (intro continuous_on_compose cont_hj) blast
        then show "continuous_on T f'"
          apply (rule continuous_on_eq)
          using ‹T ⊆ affine hull S› f'_def by auto
        have "continuous_on T (j ∘ g ∘ h)"
          using hom homeomorphism_def by (intro continuous_on_compose cont_hj) blast
        then show "continuous_on T g'"
          apply (rule continuous_on_eq)
          using ‹T ⊆ affine hull S› g'_def by auto
        show "f' ` T ⊆ T"
        proof (clarsimp simp: f'_def)
          fix x assume "x ∈ T"
          then have "f (h x) ∈ h ` T"
            by (metis (no_types) hom homeomorphism_def image_subset_iff subset_refl)
          then show "j (f (h x)) ∈ T"
            using ‹T ⊆ affine hull S› h by auto
        qed
        show "g' ` T ⊆ T"
        proof (clarsimp simp: g'_def)
          fix x assume "x ∈ T"
          then have "g (h x) ∈ h ` T"
            by (metis (no_types) hom homeomorphism_def image_subset_iff subset_refl)
          then show "j (g (h x)) ∈ T"
            using ‹T ⊆ affine hull S› h by auto
        qed
        show "⋀x. x ∈ T ⟹ g' (f' x) = x"
          using h j hom homeomorphism_apply1 by (fastforce simp: f'_def g'_def)
        show "⋀y. y ∈ T ⟹ f' (g' y) = y"
          using h j hom homeomorphism_apply2 by (fastforce simp: f'_def g'_def)
      qed
    next
      have §: "⋀x y. ⟦x ∈ affine hull S; h x = h y; y ∈ S⟧ ⟹ x ∈ S"
        by (metis h hull_inc)
      show "{x. ¬ (f' x = x ∧ g' x = x)} ⊆ S"
        using sub by (simp add: f'_def g'_def jf jg) (force elim: §)
    next
      have "compact (j ` closure {x. ¬ (f x = x ∧ g x = x)})"
        using bou by (auto simp: compact_continuous_image cont_hj)
      then have "bounded (j ` {x. ¬ (f x = x ∧ g x = x)})"
        by (rule bounded_closure_image [OF compact_imp_bounded])
      moreover
      have *: "{x ∈ affine hull S. j (f (h x)) ≠ x ∨ j (g (h x)) ≠ x} = j ` {x. (¬ (f x = x ∧ g x = x))}"
        using h j by (auto simp: image_iff; metis)
      ultimately have "bounded {x ∈ affine hull S. j (f (h x)) ≠ x ∨ j (g (h x)) ≠ x}"
        by metis
      then show "bounded {x. ¬ (f' x = x ∧ g' x = x)}"
        by (simp add: f'_def g'_def Collect_mono bounded_subset)
    next
      show "f' x ∈ U" if "x ∈ K" for x
      proof -
        have "U ⊆ S"
          using opeU openin_imp_subset by blast
        then have "j (f (h x)) ∈ U"
          using f h hull_subset that by fastforce
        then show "f' x ∈ U"
          using ‹K ⊆ S› S f'_def that by auto
      qed
    qed
  qed
qed


subsection‹Nullhomotopic mappings›

text‹ A mapping out of a sphere is nullhomotopic iff it extends to the ball.
This even works out in the degenerate cases when the radius is ‹≤› 0, and
we also don't need to explicitly assume continuity since it's already implicit
in both sides of the equivalence.›

lemma nullhomotopic_from_lemma:
  assumes contg: "continuous_on (cball a r - {a}) g"
      and fa: "⋀e. 0 < e
               ⟹ ∃d. 0 < d ∧ (∀x. x ≠ a ∧ norm(x - a) < d ⟶ norm(g x - f a) < e)"
      and r: "⋀x. x ∈ cball a r ∧ x ≠ a ⟹ f x = g x"
    shows "continuous_on (cball a r) f"
proof (clarsimp simp: continuous_on_eq_continuous_within Ball_def)
  fix x
  assume x: "dist a x ≤ r"
  show "continuous (at x within cball a r) f"
  proof (cases "x=a")
    case True
    then show ?thesis
      by (metis continuous_within_eps_delta fa dist_norm dist_self r)
  next
    case False
    show ?thesis
    proof (rule continuous_transform_within [where f=g and d = "norm(x-a)"])
      have "∃d>0. ∀x'∈cball a r.
                      dist x' x < d ⟶ dist (g x') (g x) < e" if "e>0" for e
      proof -
        obtain d where "d > 0"
           and d: "⋀x'. ⟦dist x' a ≤ r; x' ≠ a; dist x' x < d⟧ ⟹
                                 dist (g x') (g x) < e"
          using contg False x ‹e>0›
          unfolding continuous_on_iff by (fastforce simp: dist_commute intro: that)
        show ?thesis
          using ‹d > 0› ‹x ≠ a›
          by (rule_tac x="min d (norm(x - a))" in exI)
             (auto simp: dist_commute dist_norm [symmetric]  intro!: d)
      qed
      then show "continuous (at x within cball a r) g"
        using contg False by (auto simp: continuous_within_eps_delta)
      show "0 < norm (x - a)"
        using False by force
      show "x ∈ cball a r"
        by (simp add: x)
      show "⋀x'. ⟦x' ∈ cball a r; dist x' x < norm (x - a)⟧
        ⟹ g x' = f x'"
        by (metis dist_commute dist_norm less_le r)
    qed
  qed
qed

proposition nullhomotopic_from_sphere_extension:
  fixes f :: "'M::euclidean_space ⇒ 'a::real_normed_vector"
  shows  "(∃c. homotopic_with_canon (λx. True) (sphere a r) S f (λx. c)) ⟷
          (∃g. continuous_on (cball a r) g ∧ g ` (cball a r) ⊆ S ∧
               (∀x ∈ sphere a r. g x = f x))"
         (is "?lhs = ?rhs")
proof (cases r "0::real" rule: linorder_cases)
  case less
  then show ?thesis
    by (simp add: homotopic_on_emptyI)
next
  case equal
  show ?thesis
  proof
    assume L: ?lhs
    with equal have [simp]: "f a ∈ S"
      using homotopic_with_imp_subset1 by fastforce
    obtain h:: "real × 'M ⇒ 'a" 
      where h: "continuous_on ({0..1} × {a}) h" "h ` ({0..1} × {a}) ⊆ S" "h (0, a) = f a"    
      using L equal by (auto simp: homotopic_with)
    then have "continuous_on (cball a r) (λx. h (0, a))" "(λx. h (0, a)) ` cball a r ⊆ S"
      by (auto simp: equal)
    then show ?rhs
      using h(3) local.equal by force
  next
    assume ?rhs
    then show ?lhs
      using equal continuous_on_const by (force simp: homotopic_with)
  qed
next
  case greater
  let ?P = "continuous_on {x. norm(x - a) = r} f ∧ f ` {x. norm(x - a) = r} ⊆ S"
  have ?P if ?lhs using that
  proof
    fix c
    assume c: "homotopic_with_canon (λx. True) (sphere a r) S f (λx. c)"
    then have contf: "continuous_on (sphere a r) f" 
      by (metis homotopic_with_imp_continuous)
    moreover have fim: "f ` sphere a r ⊆ S"
      by (meson continuous_map_subtopology_eu c homotopic_with_imp_continuous_maps)
    show ?P
      using contf fim by (auto simp: sphere_def dist_norm norm_minus_commute)
  qed
  moreover have ?P if ?rhs using that
  proof
    fix g
    assume g: "continuous_on (cball a r) g ∧ g ` cball a r ⊆ S ∧ (∀xa∈sphere a r. g xa = f xa)"
    then have "f ` {x. norm (x - a) = r} ⊆ S"
      using sphere_cball [of a r] unfolding image_subset_iff sphere_def
      by (metis dist_commute dist_norm mem_Collect_eq subset_eq)
    with g show ?P
      by (auto simp: dist_norm norm_minus_commute elim!: continuous_on_eq [OF continuous_on_subset])
  qed
  moreover have ?thesis if ?P
  proof
    assume ?lhs
    then obtain c where "homotopic_with_canon (λx. True) (sphere a r) S (λx. c) f"
      using homotopic_with_sym by blast
    then obtain h where conth: "continuous_on ({0..1::real} × sphere a r) h"
                    and him: "h ` ({0..1} × sphere a r) ⊆ S"
                    and h: "⋀x. h(0, x) = c" "⋀x. h(1, x) = f x"
      by (auto simp: homotopic_with_def)
    obtain b1::'M where "b1 ∈ Basis"
      using SOME_Basis by auto
    have "c ∈ h ` ({0..1} × sphere a r)"
    proof
      show "c = h (0, a + r *⇩R b1)"
        by (simp add: h)
      show "(0, a + r *⇩R b1) ∈ {0..1::real} × sphere a r"
        using greater ‹b1 ∈ Basis› by (auto simp: dist_norm)
    qed
    then have "c ∈ S"
      using him by blast
    have uconth: "uniformly_continuous_on ({0..1::real} × (sphere a r)) h"
      by (force intro: compact_Times conth compact_uniformly_continuous)
    let ?g = "λx. h (norm (x - a)/r,
                     a + (if x = a then r *⇩R b1 else (r / norm(x - a)) *⇩R (x - a)))"
    let ?g' = "λx. h (norm (x - a)/r, a + (r / norm(x - a)) *⇩R (x - a))"
    show ?rhs
    proof (intro exI conjI)
      have "continuous_on (cball a r - {a}) ?g'"
        using greater
        by (force simp: dist_norm norm_minus_commute intro: continuous_on_compose2 [OF conth] continuous_intros)
      then show "continuous_on (cball a r) ?g"
      proof (rule nullhomotopic_from_lemma)
        show "∃d>0. ∀x. x ≠ a ∧ norm (x - a) < d ⟶ norm (?g' x - ?g a) < e" if "0 < e" for e
        proof -
          obtain d where "0 < d"
             and d: "⋀x x'. ⟦x ∈ {0..1} × sphere a r; x' ∈ {0..1} × sphere a r; norm ( x' - x) < d⟧
                        ⟹ norm (h x' - h x) < e"
            using uniformly_continuous_onE [OF uconth ‹0 < e›] by (auto simp: dist_norm)
          have *: "norm (h (norm (x - a) / r,
                         a + (r / norm (x - a)) *⇩R (x - a)) - h (0, a + r *⇩R b1)) < e" (is  "norm (?ha - ?hb) < e")
                   if "x ≠ a" "norm (x - a) < r" "norm (x - a) < d * r" for x
          proof -
            have "norm (?ha - ?hb) = norm (?ha - h (0, a + (r / norm (x - a)) *⇩R (x - a)))"
              by (simp add: h)
            also have "… < e"
              using greater ‹0 < d› ‹b1 ∈ Basis› that
              by (intro d) (simp_all add: dist_norm, simp add: field_simps) 
            finally show ?thesis .
          qed
          show ?thesis
            using greater ‹0 < d› 
            by (rule_tac x = "min r (d * r)" in exI) (auto simp: *)
        qed
        show "⋀x. x ∈ cball a r ∧ x ≠ a ⟹ ?g x = ?g' x"
          by auto
      qed
    next
      show "?g ` cball a r ⊆ S"
        using greater him ‹c ∈ S›
        by (force simp: h dist_norm norm_minus_commute)
    next
      show "∀x∈sphere a r. ?g x = f x"
        using greater by (auto simp: h dist_norm norm_minus_commute)
    qed
  next
    assume ?rhs
    then obtain g where contg: "continuous_on (cball a r) g"
                    and gim: "g ` cball a r ⊆ S"
                    and gf: "∀x ∈ sphere a r. g x = f x"
      by auto
    let ?h = "λy. g (a + (fst y) *⇩R (snd y - a))"
    have "continuous_on ({0..1} × sphere a r) ?h"
    proof (rule continuous_on_compose2 [OF contg])
      show "continuous_on ({0..1} × sphere a r) (λx. a + fst x *⇩R (snd x - a))"
        by (intro continuous_intros)
      qed (auto simp: dist_norm norm_minus_commute mult_left_le_one_le)
    moreover
    have "?h ` ({0..1} × sphere a r) ⊆ S"
      by (auto simp: dist_norm norm_minus_commute mult_left_le_one_le gim [THEN subsetD])
    moreover
    have "∀x∈sphere a r. ?h (0, x) = g a" "∀x∈sphere a r. ?h (1, x) = f x"
      by (auto simp: dist_norm norm_minus_commute mult_left_le_one_le gf)
    ultimately have "homotopic_with_canon (λx. True) (sphere a r) S (λx. g a) f"
      by (auto simp: homotopic_with)
    then show ?lhs
      using homotopic_with_symD by blast
  qed
  ultimately
  show ?thesis by meson
qed 

end