Theory 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

definitiontag 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))))"

textp›, 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)

subsectiontag unimportant›‹Trivial properties›

text ‹We often want to just localize the ending function equality or whatever.›
texttag 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. (λiI. h i (x i)))"
  shows "homotopic_with q (product_topology X I) (product_topology Y I)
                          (λz. (λiI. (f i) (z i))) (λz. (λiI. (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). λiI. 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) = (λiI. f i (x i))" "?h (1, x) = (λiI. 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›


definitiontag 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›

texttag 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›

definitiontag 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


subsectiontag 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›

texttag important›‹defined as "all loops are homotopic (as loops)›

definitiontag 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›

definitiontag 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›

definitiontag important› "starlike S  (aS. xS. 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 "xT. 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›

definitiontag 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  (xT. P x  Q x)}"
  let ?B = "{b. T. openin (top_of_set S) T  b  T  (xT. 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 "yball 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. (cC. openin (top_of_set K) c)  K  C 
                    DC. 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))    {}" if "finite " and : "  ?𝒯" for 
        proof
          assume djo: "(S - (V1  V2))   = {}"
          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 " = {}")
            case True
            with C  S djo that show ?thesis
              by force
          next
            case False show ?thesis
            proof
              show ope: "openin (top_of_set S) ()"
                using openin_Inter finite  False  by blast
              then show "closedin (top_of_set S) ()"
                by (meson clo𝒯  closed_Inter closed_subset openin_imp_subset subset_eq)
              show "C  "
                using  by auto
              show "  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)    {}"
      if "finite " and : "  ?𝒯" for 
    proof (cases " = {}")
      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)   = {}"
        then have FUW: "  U  W"
          by blast
        have "C  "
          using  by auto
        moreover have "compact ()"
          by (metis (no_types, lifting) compact_Inter False mem_Collect_eq subsetCE )
        moreover have "  K"
          using False that(2) by fastforce
        moreover have opeKF: "openin (top_of_set K) ()"
          using False  finite  by blast
        then have opeVF: "openin (top_of_set V) ()"
          using W K  S V  K opeKF   K FUW openin_subset_trans by fastforce
        then have "openin (top_of_set S) ()"
          by (metis opeSV openin_trans)
        moreover have "  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

subsectiontag 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 (vB. a v *R fb v)^2" by (simp add: sum scale ffb x)
    also have " = (vB. 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 = (vB. a v *R fb v)" .
    then have "(norm (f x))2 = (norm (vB. a v *R fb v))2" by simp
    also have " = (vB. 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 (vB. a v *R fb v)" by (simp add: *)
    also have " = (vB. g (a v *R fb v))"
      by (simp add: g.sum g.scale)
    also have " = (vB. a v *R g (fb v))"
      by (simp add: g.scale)
    also have " = (vB. 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 (vC. a v *R gb v)" by simp
    also have " = (vC. f (a v *R gb v))"
      by (simp add: f.scale f.sum)
    also have " = (vC. a v *R f (gb v))"
      by (simp add: f.scale f.sum)
    also have " = (vC. 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)›

localetag 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.›

definitiontag 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: "xS. 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: "iI. contractible_space (X i)"
    for x :: "'a  'b"
  proof -
    obtain f where f: "i. iI  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


abbreviationtag 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)


subsectiontag 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

subsectiontag 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 ((zclosed_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 ` ((zclosed_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 
              (xU. yU. 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)



subsectiontag unimportant› ‹Self-homeomorphisms shuffling points about›

subsubsectiontag 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 > 00  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

corollarytag 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


corollarytag 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


propositiontag 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 
              (xU.
                  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

propositiontag 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

subsubsectiontag 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

propositiontag 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


propositiontag 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::"'areal" 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  (xasphere 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 "xsphere 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 "xsphere a r. ?h (0, x) = g a" "xsphere 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