split off Homotopy.thy
authorimmler
Mon, 07 Jan 2019 14:57:45 +0100
changeset 69620 19d8a59481db
parent 69619 3f7d8e05e0f2
child 69621 9c22ff18125b
split off Homotopy.thy
src/HOL/Analysis/Brouwer_Fixpoint.thy
src/HOL/Analysis/Homeomorphism.thy
src/HOL/Analysis/Homotopy.thy
src/HOL/Analysis/Path_Connected.thy
--- a/src/HOL/Analysis/Brouwer_Fixpoint.thy	Mon Jan 07 14:06:54 2019 +0100
+++ b/src/HOL/Analysis/Brouwer_Fixpoint.thy	Mon Jan 07 14:57:45 2019 +0100
@@ -15,7 +15,10 @@
 section \<open>Brouwer's Fixed Point Theorem\<close>
 
 theory Brouwer_Fixpoint
-imports Path_Connected Homeomorphism
+  imports
+    Path_Connected
+    Homeomorphism
+    Continuous_Extension
 begin
 
 (* FIXME mv topology euclidean space *)
--- a/src/HOL/Analysis/Homeomorphism.thy	Mon Jan 07 14:06:54 2019 +0100
+++ b/src/HOL/Analysis/Homeomorphism.thy	Mon Jan 07 14:57:45 2019 +0100
@@ -5,7 +5,7 @@
 section%important \<open>Homeomorphism Theorems\<close>
 
 theory Homeomorphism
-imports Path_Connected
+imports Homotopy
 begin
 
 lemma%unimportant homeomorphic_spheres':
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Analysis/Homotopy.thy	Mon Jan 07 14:57:45 2019 +0100
@@ -0,0 +1,5159 @@
+(*  Title:      HOL/Analysis/Path_Connected.thy
+    Authors:    LC Paulson and Robert Himmelmann (TU Muenchen), based on material from HOL Light
+*)
+
+section \<open>Homotopy of Maps\<close>
+
+theory Homotopy
+  imports Path_Connected Continuum_Not_Denumerable
+begin
+
+definition%important homotopic_with ::
+  "[('a::topological_space \<Rightarrow> 'b::topological_space) \<Rightarrow> bool, 'a set, 'b set, 'a \<Rightarrow> 'b, 'a \<Rightarrow> 'b] \<Rightarrow> bool"
+where
+ "homotopic_with P X Y p q \<equiv>
+   (\<exists>h:: real \<times> 'a \<Rightarrow> 'b.
+       continuous_on ({0..1} \<times> X) h \<and>
+       h ` ({0..1} \<times> X) \<subseteq> Y \<and>
+       (\<forall>x. h(0, x) = p x) \<and>
+       (\<forall>x. h(1, x) = q x) \<and>
+       (\<forall>t \<in> {0..1}. P(\<lambda>x. h(t, x))))"
+
+text\<open>\<open>p\<close>, \<open>q\<close> are functions \<open>X \<rightarrow> Y\<close>, and the property \<open>P\<close> restricts all intermediate maps.
+We often just want to require that \<open>P\<close> fixes some subset, but to include the case of a loop homotopy,
+it is convenient to have a general property \<open>P\<close>.\<close>
+
+text \<open>We often want to just localize the ending function equality or whatever.\<close>
+text%important \<open>%whitespace\<close>
+proposition homotopic_with:
+  fixes X :: "'a::topological_space set" and Y :: "'b::topological_space set"
+  assumes "\<And>h k. (\<And>x. x \<in> X \<Longrightarrow> h x = k x) \<Longrightarrow> (P h \<longleftrightarrow> P k)"
+  shows "homotopic_with P X Y p q \<longleftrightarrow>
+           (\<exists>h :: real \<times> 'a \<Rightarrow> 'b.
+              continuous_on ({0..1} \<times> X) h \<and>
+              h ` ({0..1} \<times> X) \<subseteq> Y \<and>
+              (\<forall>x \<in> X. h(0,x) = p x) \<and>
+              (\<forall>x \<in> X. h(1,x) = q x) \<and>
+              (\<forall>t \<in> {0..1}. P(\<lambda>x. h(t, x))))"
+  unfolding homotopic_with_def
+  apply (rule iffI, blast, clarify)
+  apply (rule_tac x="\<lambda>(u,v). if v \<in> X then h(u,v) else if u = 0 then p v else q v" in exI)
+  apply auto
+  apply (force elim: continuous_on_eq)
+  apply (drule_tac x=t in bspec, force)
+  apply (subst assms; simp)
+  done
+
+proposition homotopic_with_eq:
+   assumes h: "homotopic_with P X Y f g"
+       and f': "\<And>x. x \<in> X \<Longrightarrow> f' x = f x"
+       and g': "\<And>x. x \<in> X \<Longrightarrow> g' x = g x"
+       and P:  "(\<And>h k. (\<And>x. x \<in> X \<Longrightarrow> h x = k x) \<Longrightarrow> (P h \<longleftrightarrow> P k))"
+   shows "homotopic_with P X Y f' g'"
+  using h unfolding homotopic_with_def
+  apply safe
+  apply (rule_tac x="\<lambda>(u,v). if v \<in> X then h(u,v) else if u = 0 then f' v else g' v" in exI)
+  apply (simp add: f' g', safe)
+  apply (fastforce intro: continuous_on_eq, fastforce)
+  apply (subst P; fastforce)
+  done
+
+proposition homotopic_with_equal:
+   assumes contf: "continuous_on X f" and fXY: "f ` X \<subseteq> Y"
+       and gf: "\<And>x. x \<in> X \<Longrightarrow> g x = f x"
+       and P:  "P f" "P g"
+   shows "homotopic_with P X Y f g"
+  unfolding homotopic_with_def
+  apply (rule_tac x="\<lambda>(u,v). if u = 1 then g v else f v" in exI)
+  using assms
+  apply (intro conjI)
+  apply (rule continuous_on_eq [where f = "f \<circ> snd"])
+  apply (rule continuous_intros | force)+
+  apply clarify
+  apply (case_tac "t=1"; force)
+  done
+
+
+lemma image_Pair_const: "(\<lambda>x. (x, c)) ` A = A \<times> {c}"
+  by auto
+
+lemma homotopic_constant_maps:
+   "homotopic_with (\<lambda>x. True) s t (\<lambda>x. a) (\<lambda>x. b) \<longleftrightarrow> s = {} \<or> path_component t a b"
+proof (cases "s = {} \<or> t = {}")
+  case True with continuous_on_const show ?thesis
+    by (auto simp: homotopic_with path_component_def)
+next
+  case False
+  then obtain c where "c \<in> s" by blast
+  show ?thesis
+  proof
+    assume "homotopic_with (\<lambda>x. True) s t (\<lambda>x. a) (\<lambda>x. b)"
+    then obtain h :: "real \<times> 'a \<Rightarrow> 'b"
+        where conth: "continuous_on ({0..1} \<times> s) h"
+          and h: "h ` ({0..1} \<times> s) \<subseteq> t" "(\<forall>x\<in>s. h (0, x) = a)" "(\<forall>x\<in>s. h (1, x) = b)"
+      by (auto simp: homotopic_with)
+    have "continuous_on {0..1} (h \<circ> (\<lambda>t. (t, c)))"
+      apply (rule continuous_intros conth | simp add: image_Pair_const)+
+      apply (blast intro:  \<open>c \<in> s\<close> continuous_on_subset [OF conth])
+      done
+    with \<open>c \<in> s\<close> h show "s = {} \<or> path_component t a b"
+      apply (simp_all add: homotopic_with path_component_def, auto)
+      apply (drule_tac x="h \<circ> (\<lambda>t. (t, c))" in spec)
+      apply (auto simp: pathstart_def pathfinish_def path_image_def path_def)
+      done
+  next
+    assume "s = {} \<or> path_component t a b"
+    with False show "homotopic_with (\<lambda>x. True) s t (\<lambda>x. a) (\<lambda>x. b)"
+      apply (clarsimp simp: homotopic_with path_component_def pathstart_def pathfinish_def path_image_def path_def)
+      apply (rule_tac x="g \<circ> fst" in exI)
+      apply (rule conjI continuous_intros | force)+
+      done
+  qed
+qed
+
+
+subsection%unimportant\<open>Trivial properties\<close>
+
+lemma homotopic_with_imp_property: "homotopic_with P X Y f g \<Longrightarrow> P f \<and> P g"
+  unfolding homotopic_with_def Ball_def
+  apply clarify
+  apply (frule_tac x=0 in spec)
+  apply (drule_tac x=1 in spec, auto)
+  done
+
+lemma continuous_on_o_Pair: "\<lbrakk>continuous_on (T \<times> X) h; t \<in> T\<rbrakk> \<Longrightarrow> continuous_on X (h \<circ> Pair t)"
+  by (fast intro: continuous_intros elim!: continuous_on_subset)
+
+lemma homotopic_with_imp_continuous:
+    assumes "homotopic_with P X Y f g"
+    shows "continuous_on X f \<and> continuous_on X g"
+proof -
+  obtain h :: "real \<times> 'a \<Rightarrow> 'b"
+    where conth: "continuous_on ({0..1} \<times> X) h"
+      and h: "\<forall>x. h (0, x) = f x" "\<forall>x. h (1, x) = g x"
+    using assms by (auto simp: homotopic_with_def)
+  have *: "t \<in> {0..1} \<Longrightarrow> continuous_on X (h \<circ> (\<lambda>x. (t,x)))" for t
+    by (rule continuous_intros continuous_on_subset [OF conth] | force)+
+  show ?thesis
+    using h *[of 0] *[of 1] by auto
+qed
+
+proposition homotopic_with_imp_subset1:
+     "homotopic_with P X Y f g \<Longrightarrow> f ` X \<subseteq> Y"
+  by (simp add: homotopic_with_def image_subset_iff) (metis atLeastAtMost_iff order_refl zero_le_one)
+
+proposition homotopic_with_imp_subset2:
+     "homotopic_with P X Y f g \<Longrightarrow> g ` X \<subseteq> Y"
+  by (simp add: homotopic_with_def image_subset_iff) (metis atLeastAtMost_iff order_refl zero_le_one)
+
+proposition homotopic_with_mono:
+    assumes hom: "homotopic_with P X Y f g"
+        and Q: "\<And>h. \<lbrakk>continuous_on X h; image h X \<subseteq> Y \<and> P h\<rbrakk> \<Longrightarrow> Q h"
+      shows "homotopic_with Q X Y f g"
+  using hom
+  apply (simp add: homotopic_with_def)
+  apply (erule ex_forward)
+  apply (force simp: intro!: Q dest: continuous_on_o_Pair)
+  done
+
+proposition homotopic_with_subset_left:
+     "\<lbrakk>homotopic_with P X Y f g; Z \<subseteq> X\<rbrakk> \<Longrightarrow> homotopic_with P Z Y f g"
+  apply (simp add: homotopic_with_def)
+  apply (fast elim!: continuous_on_subset ex_forward)
+  done
+
+proposition homotopic_with_subset_right:
+     "\<lbrakk>homotopic_with P X Y f g; Y \<subseteq> Z\<rbrakk> \<Longrightarrow> homotopic_with P X Z f g"
+  apply (simp add: homotopic_with_def)
+  apply (fast elim!: continuous_on_subset ex_forward)
+  done
+
+proposition homotopic_with_compose_continuous_right:
+    "\<lbrakk>homotopic_with (\<lambda>f. p (f \<circ> h)) X Y f g; continuous_on W h; h ` W \<subseteq> X\<rbrakk>
+     \<Longrightarrow> homotopic_with p W Y (f \<circ> h) (g \<circ> h)"
+  apply (clarsimp simp add: homotopic_with_def)
+  apply (rename_tac k)
+  apply (rule_tac x="k \<circ> (\<lambda>y. (fst y, h (snd y)))" in exI)
+  apply (rule conjI continuous_intros continuous_on_compose [where f=snd and g=h, unfolded o_def] | simp)+
+  apply (erule continuous_on_subset)
+  apply (fastforce simp: o_def)+
+  done
+
+proposition homotopic_compose_continuous_right:
+     "\<lbrakk>homotopic_with (\<lambda>f. True) X Y f g; continuous_on W h; h ` W \<subseteq> X\<rbrakk>
+      \<Longrightarrow> homotopic_with (\<lambda>f. True) W Y (f \<circ> h) (g \<circ> h)"
+  using homotopic_with_compose_continuous_right by fastforce
+
+proposition homotopic_with_compose_continuous_left:
+     "\<lbrakk>homotopic_with (\<lambda>f. p (h \<circ> f)) X Y f g; continuous_on Y h; h ` Y \<subseteq> Z\<rbrakk>
+      \<Longrightarrow> homotopic_with p X Z (h \<circ> f) (h \<circ> g)"
+  apply (clarsimp simp add: homotopic_with_def)
+  apply (rename_tac k)
+  apply (rule_tac x="h \<circ> k" in exI)
+  apply (rule conjI continuous_intros continuous_on_compose [where f=snd and g=h, unfolded o_def] | simp)+
+  apply (erule continuous_on_subset)
+  apply (fastforce simp: o_def)+
+  done
+
+proposition homotopic_compose_continuous_left:
+   "\<lbrakk>homotopic_with (\<lambda>_. True) X Y f g;
+     continuous_on Y h; h ` Y \<subseteq> Z\<rbrakk>
+    \<Longrightarrow> homotopic_with (\<lambda>f. True) X Z (h \<circ> f) (h \<circ> g)"
+  using homotopic_with_compose_continuous_left by fastforce
+
+proposition homotopic_with_Pair:
+   assumes hom: "homotopic_with p s t f g" "homotopic_with p' s' t' f' g'"
+       and q: "\<And>f g. \<lbrakk>p f; p' g\<rbrakk> \<Longrightarrow> q(\<lambda>(x,y). (f x, g y))"
+     shows "homotopic_with q (s \<times> s') (t \<times> t')
+                  (\<lambda>(x,y). (f x, f' y)) (\<lambda>(x,y). (g x, g' y))"
+  using hom
+  apply (clarsimp simp add: homotopic_with_def)
+  apply (rename_tac k k')
+  apply (rule_tac x="\<lambda>z. ((k \<circ> (\<lambda>x. (fst x, fst (snd x)))) z, (k' \<circ> (\<lambda>x. (fst x, snd (snd x)))) z)" in exI)
+  apply (rule conjI continuous_intros | erule continuous_on_subset | clarsimp)+
+  apply (auto intro!: q [unfolded case_prod_unfold])
+  done
+
+lemma homotopic_on_empty [simp]: "homotopic_with (\<lambda>x. True) {} t f g"
+  by (metis continuous_on_def empty_iff homotopic_with_equal image_subset_iff)
+
+
+text\<open>Homotopy with P is an equivalence relation (on continuous functions mapping X into Y that satisfy P,
+     though this only affects reflexivity.\<close>
+
+
+proposition homotopic_with_refl:
+   "homotopic_with P X Y f f \<longleftrightarrow> continuous_on X f \<and> image f X \<subseteq> Y \<and> P f"
+  apply (rule iffI)
+  using homotopic_with_imp_continuous homotopic_with_imp_property homotopic_with_imp_subset2 apply blast
+  apply (simp add: homotopic_with_def)
+  apply (rule_tac x="f \<circ> snd" in exI)
+  apply (rule conjI continuous_intros | force)+
+  done
+
+lemma homotopic_with_symD:
+  fixes X :: "'a::real_normed_vector set"
+    assumes "homotopic_with P X Y f g"
+      shows "homotopic_with P X Y g f"
+  using assms
+  apply (clarsimp simp add: homotopic_with_def)
+  apply (rename_tac h)
+  apply (rule_tac x="h \<circ> (\<lambda>y. (1 - fst y, snd y))" in exI)
+  apply (rule conjI continuous_intros | erule continuous_on_subset | force simp: image_subset_iff)+
+  done
+
+proposition homotopic_with_sym:
+    fixes X :: "'a::real_normed_vector set"
+    shows "homotopic_with P X Y f g \<longleftrightarrow> homotopic_with P X Y g f"
+  using homotopic_with_symD by blast
+
+lemma split_01: "{0..1::real} = {0..1/2} \<union> {1/2..1}"
+  by force
+
+lemma split_01_prod: "{0..1::real} \<times> X = ({0..1/2} \<times> X) \<union> ({1/2..1} \<times> X)"
+  by force
+
+proposition homotopic_with_trans:
+    fixes X :: "'a::real_normed_vector set"
+    assumes "homotopic_with P X Y f g" and "homotopic_with P X Y g h"
+      shows "homotopic_with P X Y f h"
+proof -
+  have clo1: "closedin (subtopology euclidean ({0..1/2} \<times> X \<union> {1/2..1} \<times> X)) ({0..1/2::real} \<times> X)"
+    apply (simp add: closedin_closed split_01_prod [symmetric])
+    apply (rule_tac x="{0..1/2} \<times> UNIV" in exI)
+    apply (force simp: closed_Times)
+    done
+  have clo2: "closedin (subtopology euclidean ({0..1/2} \<times> X \<union> {1/2..1} \<times> X)) ({1/2..1::real} \<times> X)"
+    apply (simp add: closedin_closed split_01_prod [symmetric])
+    apply (rule_tac x="{1/2..1} \<times> UNIV" in exI)
+    apply (force simp: closed_Times)
+    done
+  { fix k1 k2:: "real \<times> 'a \<Rightarrow> 'b"
+    assume cont: "continuous_on ({0..1} \<times> X) k1" "continuous_on ({0..1} \<times> X) k2"
+       and Y: "k1 ` ({0..1} \<times> X) \<subseteq> Y" "k2 ` ({0..1} \<times> X) \<subseteq> Y"
+       and geq: "\<forall>x. k1 (1, x) = g x" "\<forall>x. k2 (0, x) = g x"
+       and k12: "\<forall>x. k1 (0, x) = f x" "\<forall>x. k2 (1, x) = h x"
+       and P:   "\<forall>t\<in>{0..1}. P (\<lambda>x. k1 (t, x))" "\<forall>t\<in>{0..1}. P (\<lambda>x. k2 (t, x))"
+    define k where "k y =
+      (if fst y \<le> 1 / 2
+       then (k1 \<circ> (\<lambda>x. (2 *\<^sub>R fst x, snd x))) y
+       else (k2 \<circ> (\<lambda>x. (2 *\<^sub>R fst x -1, snd x))) y)" for y
+    have keq: "k1 (2 * u, v) = k2 (2 * u - 1, v)" if "u = 1/2"  for u v
+      by (simp add: geq that)
+    have "continuous_on ({0..1} \<times> X) k"
+      using cont
+      apply (simp add: split_01_prod k_def)
+      apply (rule clo1 clo2 continuous_on_cases_local continuous_intros | erule continuous_on_subset | simp add: linear image_subset_iff)+
+      apply (force simp: keq)
+      done
+    moreover have "k ` ({0..1} \<times> X) \<subseteq> Y"
+      using Y by (force simp: k_def)
+    moreover have "\<forall>x. k (0, x) = f x"
+      by (simp add: k_def k12)
+    moreover have "(\<forall>x. k (1, x) = h x)"
+      by (simp add: k_def k12)
+    moreover have "\<forall>t\<in>{0..1}. P (\<lambda>x. k (t, x))"
+      using P
+      apply (clarsimp simp add: k_def)
+      apply (case_tac "t \<le> 1/2", auto)
+      done
+    ultimately have *: "\<exists>k :: real \<times> 'a \<Rightarrow> 'b.
+                       continuous_on ({0..1} \<times> X) k \<and> k ` ({0..1} \<times> X) \<subseteq> Y \<and>
+                       (\<forall>x. k (0, x) = f x) \<and> (\<forall>x. k (1, x) = h x) \<and> (\<forall>t\<in>{0..1}. P (\<lambda>x. k (t, x)))"
+      by blast
+  } note * = this
+  show ?thesis
+    using assms by (auto intro: * simp add: homotopic_with_def)
+qed
+
+proposition homotopic_compose:
+      fixes s :: "'a::real_normed_vector set"
+      shows "\<lbrakk>homotopic_with (\<lambda>x. True) s t f f'; homotopic_with (\<lambda>x. True) t u g g'\<rbrakk>
+             \<Longrightarrow> homotopic_with (\<lambda>x. True) s u (g \<circ> f) (g' \<circ> f')"
+  apply (rule homotopic_with_trans [where g = "g \<circ> f'"])
+  apply (metis homotopic_compose_continuous_left homotopic_with_imp_continuous homotopic_with_imp_subset1)
+  by (metis homotopic_compose_continuous_right homotopic_with_imp_continuous homotopic_with_imp_subset2)
+
+
+text\<open>Homotopic triviality implicitly incorporates path-connectedness.\<close>
+lemma homotopic_triviality:
+  fixes S :: "'a::real_normed_vector set"
+  shows  "(\<forall>f g. continuous_on S f \<and> f ` S \<subseteq> T \<and>
+                 continuous_on S g \<and> g ` S \<subseteq> T
+                 \<longrightarrow> homotopic_with (\<lambda>x. True) S T f g) \<longleftrightarrow>
+          (S = {} \<or> path_connected T) \<and>
+          (\<forall>f. continuous_on S f \<and> f ` S \<subseteq> T \<longrightarrow> (\<exists>c. homotopic_with (\<lambda>x. True) S T f (\<lambda>x. c)))"
+          (is "?lhs = ?rhs")
+proof (cases "S = {} \<or> T = {}")
+  case True then show ?thesis by auto
+next
+  case False show ?thesis
+  proof
+    assume LHS [rule_format]: ?lhs
+    have pab: "path_component T a b" if "a \<in> T" "b \<in> T" for a b
+    proof -
+      have "homotopic_with (\<lambda>x. True) S T (\<lambda>x. a) (\<lambda>x. b)"
+        by (simp add: LHS continuous_on_const image_subset_iff that)
+      then show ?thesis
+        using False homotopic_constant_maps by blast
+    qed
+      moreover
+    have "\<exists>c. homotopic_with (\<lambda>x. True) S T f (\<lambda>x. c)" if "continuous_on S f" "f ` S \<subseteq> T" for f
+      by (metis (full_types) False LHS equals0I homotopic_constant_maps homotopic_with_imp_continuous homotopic_with_imp_subset2 pab that)
+    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 \<subseteq> T" "continuous_on S g" "g ` S \<subseteq> T"
+      obtain c d where c: "homotopic_with (\<lambda>x. True) S T f (\<lambda>x. c)" and d: "homotopic_with (\<lambda>x. True) S T g (\<lambda>x. d)"
+        using False \<open>continuous_on S f\<close> \<open>f ` S \<subseteq> T\<close>  RHS \<open>continuous_on S g\<close> \<open>g ` S \<subseteq> T\<close> by blast
+      then have "c \<in> T" "d \<in> T"
+        using False homotopic_with_imp_subset2 by fastforce+
+      with T have "path_component T c d"
+        using path_connected_component by blast
+      then have "homotopic_with (\<lambda>x. True) S T (\<lambda>x. c) (\<lambda>x. d)"
+        by (simp add: homotopic_constant_maps)
+      with c d show "homotopic_with (\<lambda>x. True) S T f g"
+        by (meson homotopic_with_symD homotopic_with_trans)
+    qed
+  qed
+qed
+
+
+subsection\<open>Homotopy of paths, maintaining the same endpoints\<close>
+
+
+definition%important homotopic_paths :: "['a set, real \<Rightarrow> 'a, real \<Rightarrow> 'a::topological_space] \<Rightarrow> bool"
+  where
+     "homotopic_paths s p q \<equiv>
+       homotopic_with (\<lambda>r. pathstart r = pathstart p \<and> pathfinish r = pathfinish p) {0..1} s p q"
+
+lemma homotopic_paths:
+   "homotopic_paths s p q \<longleftrightarrow>
+      (\<exists>h. continuous_on ({0..1} \<times> {0..1}) h \<and>
+          h ` ({0..1} \<times> {0..1}) \<subseteq> s \<and>
+          (\<forall>x \<in> {0..1}. h(0,x) = p x) \<and>
+          (\<forall>x \<in> {0..1}. h(1,x) = q x) \<and>
+          (\<forall>t \<in> {0..1::real}. pathstart(h \<circ> Pair t) = pathstart p \<and>
+                        pathfinish(h \<circ> 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 \<Longrightarrow> 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 \<Longrightarrow> 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 \<Longrightarrow> path p \<and> path q"
+  using homotopic_paths_def homotopic_with_imp_continuous path_def by blast
+
+lemma homotopic_paths_imp_subset:
+     "homotopic_paths s p q \<Longrightarrow> path_image p \<subseteq> s \<and> path_image q \<subseteq> s"
+  by (simp add: homotopic_paths_def homotopic_with_imp_subset1 homotopic_with_imp_subset2 path_image_def)
+
+proposition homotopic_paths_refl [simp]: "homotopic_paths s p p \<longleftrightarrow> path p \<and> path_image p \<subseteq> s"
+by (simp add: homotopic_paths_def homotopic_with_refl path_def path_image_def)
+
+proposition homotopic_paths_sym: "homotopic_paths s p q \<Longrightarrow> 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 \<longleftrightarrow> homotopic_paths s q p"
+  by (metis homotopic_paths_sym)
+
+proposition homotopic_paths_trans [trans]:
+     "\<lbrakk>homotopic_paths s p q; homotopic_paths s q r\<rbrakk> \<Longrightarrow> homotopic_paths s p r"
+  apply (simp add: homotopic_paths_def)
+  apply (rule homotopic_with_trans, assumption)
+  by (metis (mono_tags, lifting) homotopic_with_imp_property homotopic_with_mono)
+
+proposition homotopic_paths_eq:
+     "\<lbrakk>path p; path_image p \<subseteq> s; \<And>t. t \<in> {0..1} \<Longrightarrow> p t = q t\<rbrakk> \<Longrightarrow> homotopic_paths s p q"
+  apply (simp add: homotopic_paths_def)
+  apply (rule homotopic_with_eq)
+  apply (auto simp: path_def homotopic_with_refl pathstart_def pathfinish_def path_image_def elim: continuous_on_eq)
+  done
+
+proposition homotopic_paths_reparametrize:
+  assumes "path p"
+      and pips: "path_image p \<subseteq> s"
+      and contf: "continuous_on {0..1} f"
+      and f01:"f ` {0..1} \<subseteq> {0..1}"
+      and [simp]: "f(0) = 0" "f(1) = 1"
+      and q: "\<And>t. t \<in> {0..1} \<Longrightarrow> q(t) = p(f t)"
+    shows "homotopic_paths s p q"
+proof -
+  have contp: "continuous_on {0..1} p"
+    by (metis \<open>path p\<close> path_def)
+  then have "continuous_on {0..1} (p \<circ> f)"
+    using contf continuous_on_compose continuous_on_subset f01 by blast
+  then have "path q"
+    by (simp add: path_def) (metis q continuous_on_cong)
+  have piqs: "path_image q \<subseteq> s"
+    by (metis (no_types, hide_lams) pips f01 image_subset_iff path_image_def q)
+  have fb0: "\<And>a b. \<lbrakk>0 \<le> a; a \<le> 1; 0 \<le> b; b \<le> 1\<rbrakk> \<Longrightarrow> 0 \<le> (1 - a) * f b + a * b"
+    using f01 by force
+  have fb1: "\<lbrakk>0 \<le> a; a \<le> 1; 0 \<le> b; b \<le> 1\<rbrakk> \<Longrightarrow> (1 - a) * f b + a * b \<le> 1" for a b
+    using f01 [THEN subsetD, of "f b"] by (simp add: convex_bound_le)
+  have "homotopic_paths s q p"
+  proof (rule homotopic_paths_trans)
+    show "homotopic_paths s q (p \<circ> f)"
+      using q by (force intro: homotopic_paths_eq [OF  \<open>path q\<close> piqs])
+  next
+    show "homotopic_paths s (p \<circ> f) p"
+      apply (simp add: homotopic_paths_def homotopic_with_def)
+      apply (rule_tac x="p \<circ> (\<lambda>y. (1 - (fst y)) *\<^sub>R ((f \<circ> snd) y) + (fst y) *\<^sub>R snd y)"  in exI)
+      apply (rule conjI contf continuous_intros continuous_on_subset [OF contp] | simp)+
+      using pips [unfolded path_image_def]
+      apply (auto simp: fb0 fb1 pathstart_def pathfinish_def)
+      done
+  qed
+  then show ?thesis
+    by (simp add: homotopic_paths_sym)
+qed
+
+lemma homotopic_paths_subset: "\<lbrakk>homotopic_paths s p q; s \<subseteq> t\<rbrakk> \<Longrightarrow> homotopic_paths t p q"
+  using homotopic_paths_def homotopic_with_subset_right by blast
+
+
+text\<open> A slightly ad-hoc but useful lemma in constructing homotopies.\<close>
+lemma homotopic_join_lemma:
+  fixes q :: "[real,real] \<Rightarrow> 'a::topological_space"
+  assumes p: "continuous_on ({0..1} \<times> {0..1}) (\<lambda>y. p (fst y) (snd y))"
+      and q: "continuous_on ({0..1} \<times> {0..1}) (\<lambda>y. q (fst y) (snd y))"
+      and pf: "\<And>t. t \<in> {0..1} \<Longrightarrow> pathfinish(p t) = pathstart(q t)"
+    shows "continuous_on ({0..1} \<times> {0..1}) (\<lambda>y. (p(fst y) +++ q(fst y)) (snd y))"
+proof -
+  have 1: "(\<lambda>y. p (fst y) (2 * snd y)) = (\<lambda>y. p (fst y) (snd y)) \<circ> (\<lambda>y. (fst y, 2 * snd y))"
+    by (rule ext) (simp)
+  have 2: "(\<lambda>y. q (fst y) (2 * snd y - 1)) = (\<lambda>y. q (fst y) (snd y)) \<circ> (\<lambda>y. (fst y, 2 * snd y - 1))"
+    by (rule ext) (simp)
+  show ?thesis
+    apply (simp add: joinpaths_def)
+    apply (rule continuous_on_cases_le)
+    apply (simp_all only: 1 2)
+    apply (rule continuous_intros continuous_on_subset [OF p] continuous_on_subset [OF q] | force)+
+    using pf
+    apply (auto simp: mult.commute pathstart_def pathfinish_def)
+    done
+qed
+
+text\<open> Congruence properties of homotopy w.r.t. path-combining operations.\<close>
+
+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 \<circ> (\<lambda>x. (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) \<longleftrightarrow> homotopic_paths s p q"
+  using homotopic_paths_reversepath_D by force
+
+
+proposition homotopic_paths_join:
+    "\<lbrakk>homotopic_paths s p p'; homotopic_paths s q q'; pathfinish p = pathstart q\<rbrakk> \<Longrightarrow> homotopic_paths s (p +++ q) (p' +++ q')"
+  apply (simp add: homotopic_paths_def homotopic_with_def, clarify)
+  apply (rename_tac k1 k2)
+  apply (rule_tac x="(\<lambda>y. ((k1 \<circ> Pair (fst y)) +++ (k2 \<circ> Pair (fst y))) (snd y))" in exI)
+  apply (rule conjI continuous_intros homotopic_join_lemma)+
+  apply (auto simp: joinpaths_def pathstart_def pathfinish_def path_image_def)
+  done
+
+proposition homotopic_paths_continuous_image:
+    "\<lbrakk>homotopic_paths s f g; continuous_on s h; h ` s \<subseteq> t\<rbrakk> \<Longrightarrow> homotopic_paths t (h \<circ> f) (h \<circ> g)"
+  unfolding homotopic_paths_def
+  apply (rule homotopic_with_compose_continuous_left [of _ _ _ s])
+  apply (auto simp: pathstart_def pathfinish_def elim!: homotopic_with_mono)
+  done
+
+
+subsection\<open>Group properties for homotopy of paths\<close>
+
+text%important\<open>So taking equivalence classes under homotopy would give the fundamental group\<close>
+
+proposition homotopic_paths_rid:
+    "\<lbrakk>path p; path_image p \<subseteq> s\<rbrakk> \<Longrightarrow> homotopic_paths s (p +++ linepath (pathfinish p) (pathfinish p)) p"
+  apply (subst homotopic_paths_sym)
+  apply (rule homotopic_paths_reparametrize [where f = "\<lambda>t. if  t \<le> 1 / 2 then 2 *\<^sub>R t else 1"])
+  apply (simp_all del: le_divide_eq_numeral1)
+  apply (subst split_01)
+  apply (rule continuous_on_cases continuous_intros | force simp: pathfinish_def joinpaths_def)+
+  done
+
+proposition homotopic_paths_lid:
+   "\<lbrakk>path p; path_image p \<subseteq> s\<rbrakk> \<Longrightarrow> 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)
+
+proposition homotopic_paths_assoc:
+   "\<lbrakk>path p; path_image p \<subseteq> s; path q; path_image q \<subseteq> s; path r; path_image r \<subseteq> s; pathfinish p = pathstart q;
+     pathfinish q = pathstart r\<rbrakk>
+    \<Longrightarrow> homotopic_paths s (p +++ (q +++ r)) ((p +++ q) +++ r)"
+  apply (subst homotopic_paths_sym)
+  apply (rule homotopic_paths_reparametrize
+           [where f = "\<lambda>t. if  t \<le> 1 / 2 then inverse 2 *\<^sub>R t
+                           else if  t \<le> 3 / 4 then t - (1 / 4)
+                           else 2 *\<^sub>R t - 1"])
+  apply (simp_all del: le_divide_eq_numeral1)
+  apply (simp add: subset_path_image_join)
+  apply (rule continuous_on_cases_1 continuous_intros)+
+  apply (auto simp: joinpaths_def)
+  done
+
+proposition homotopic_paths_rinv:
+  assumes "path p" "path_image p \<subseteq> s"
+    shows "homotopic_paths s (p +++ reversepath p) (linepath (pathstart p) (pathstart p))"
+proof -
+  have "continuous_on ({0..1} \<times> {0..1}) (\<lambda>x. (subpath 0 (fst x) p +++ reversepath (subpath 0 (fst x) p)) (snd x))"
+    using assms
+    apply (simp add: joinpaths_def subpath_def reversepath_def path_def del: le_divide_eq_numeral1)
+    apply (rule continuous_on_cases_le)
+    apply (rule_tac [2] continuous_on_compose [of _ _ p, unfolded o_def])
+    apply (rule continuous_on_compose [of _ _ p, unfolded o_def])
+    apply (auto intro!: continuous_intros simp del: eq_divide_eq_numeral1)
+    apply (force elim!: continuous_on_subset simp add: mult_le_one)+
+    done
+  then show ?thesis
+    using assms
+    apply (subst homotopic_paths_sym_eq)
+    unfolding homotopic_paths_def homotopic_with_def
+    apply (rule_tac x="(\<lambda>y. (subpath 0 (fst y) p +++ reversepath(subpath 0 (fst y) p)) (snd y))" in exI)
+    apply (simp add: path_defs joinpaths_def subpath_def reversepath_def)
+    apply (force simp: mult_le_one)
+    done
+qed
+
+proposition homotopic_paths_linv:
+  assumes "path p" "path_image p \<subseteq> 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\<open>Homotopy of loops without requiring preservation of endpoints\<close>
+
+definition%important homotopic_loops :: "'a::topological_space set \<Rightarrow> (real \<Rightarrow> 'a) \<Rightarrow> (real \<Rightarrow> 'a) \<Rightarrow> bool"  where
+ "homotopic_loops s p q \<equiv>
+     homotopic_with (\<lambda>r. pathfinish r = pathstart r) {0..1} s p q"
+
+lemma homotopic_loops:
+   "homotopic_loops s p q \<longleftrightarrow>
+      (\<exists>h. continuous_on ({0..1::real} \<times> {0..1}) h \<and>
+          image h ({0..1} \<times> {0..1}) \<subseteq> s \<and>
+          (\<forall>x \<in> {0..1}. h(0,x) = p x) \<and>
+          (\<forall>x \<in> {0..1}. h(1,x) = q x) \<and>
+          (\<forall>t \<in> {0..1}. pathfinish(h \<circ> Pair t) = pathstart(h \<circ> Pair t)))"
+  by (simp add: homotopic_loops_def pathstart_def pathfinish_def homotopic_with)
+
+proposition homotopic_loops_imp_loop:
+     "homotopic_loops s p q \<Longrightarrow> pathfinish p = pathstart p \<and> pathfinish q = pathstart q"
+using homotopic_with_imp_property homotopic_loops_def by blast
+
+proposition homotopic_loops_imp_path:
+     "homotopic_loops s p q \<Longrightarrow> path p \<and> path q"
+  unfolding homotopic_loops_def path_def
+  using homotopic_with_imp_continuous by blast
+
+proposition homotopic_loops_imp_subset:
+     "homotopic_loops s p q \<Longrightarrow> path_image p \<subseteq> s \<and> path_image q \<subseteq> s"
+  unfolding homotopic_loops_def path_image_def
+  by (metis homotopic_with_imp_subset1 homotopic_with_imp_subset2)
+
+proposition homotopic_loops_refl:
+     "homotopic_loops s p p \<longleftrightarrow>
+      path p \<and> path_image p \<subseteq> s \<and> pathfinish p = pathstart p"
+  by (simp add: homotopic_loops_def homotopic_with_refl path_image_def path_def)
+
+proposition homotopic_loops_sym: "homotopic_loops s p q \<Longrightarrow> homotopic_loops s q p"
+  by (simp add: homotopic_loops_def homotopic_with_sym)
+
+proposition homotopic_loops_sym_eq: "homotopic_loops s p q \<longleftrightarrow> homotopic_loops s q p"
+  by (metis homotopic_loops_sym)
+
+proposition homotopic_loops_trans:
+   "\<lbrakk>homotopic_loops s p q; homotopic_loops s q r\<rbrakk> \<Longrightarrow> homotopic_loops s p r"
+  unfolding homotopic_loops_def by (blast intro: homotopic_with_trans)
+
+proposition homotopic_loops_subset:
+   "\<lbrakk>homotopic_loops s p q; s \<subseteq> t\<rbrakk> \<Longrightarrow> homotopic_loops t p q"
+  by (simp add: homotopic_loops_def homotopic_with_subset_right)
+
+proposition homotopic_loops_eq:
+   "\<lbrakk>path p; path_image p \<subseteq> s; pathfinish p = pathstart p; \<And>t. t \<in> {0..1} \<Longrightarrow> p(t) = q(t)\<rbrakk>
+          \<Longrightarrow> homotopic_loops s p q"
+  unfolding homotopic_loops_def
+  apply (rule homotopic_with_eq)
+  apply (rule homotopic_with_refl [where f = p, THEN iffD2])
+  apply (simp_all add: path_image_def path_def pathstart_def pathfinish_def)
+  done
+
+proposition homotopic_loops_continuous_image:
+   "\<lbrakk>homotopic_loops s f g; continuous_on s h; h ` s \<subseteq> t\<rbrakk> \<Longrightarrow> homotopic_loops t (h \<circ> f) (h \<circ> g)"
+  unfolding homotopic_loops_def
+  apply (rule homotopic_with_compose_continuous_left)
+  apply (erule homotopic_with_mono)
+  by (simp add: pathfinish_def pathstart_def)
+
+
+subsection\<open>Relations between the two variants of homotopy\<close>
+
+proposition homotopic_paths_imp_homotopic_loops:
+    "\<lbrakk>homotopic_paths s p q; pathfinish p = pathstart p; pathfinish q = pathstart p\<rbrakk> \<Longrightarrow> homotopic_loops s p q"
+  by (auto simp: homotopic_paths_def homotopic_loops_def intro: homotopic_with_mono)
+
+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 \<subseteq> s" by (metis assms homotopic_loops_imp_subset)
+  obtain h where conth: "continuous_on ({0..1::real} \<times> {0..1}) h"
+             and hs: "h ` ({0..1} \<times> {0..1}) \<subseteq> s"
+             and [simp]: "\<And>x. x \<in> {0..1} \<Longrightarrow> h(0,x) = p x"
+             and [simp]: "\<And>x. x \<in> {0..1} \<Longrightarrow> h(1,x) = a"
+             and ends: "\<And>t. t \<in> {0..1} \<Longrightarrow> pathfinish (h \<circ> Pair t) = pathstart (h \<circ> Pair t)"
+    using assms by (auto simp: homotopic_loops homotopic_with)
+  have conth0: "path (\<lambda>u. h (u, 0))"
+    unfolding path_def
+    apply (rule continuous_on_compose [of _ _ h, unfolded o_def])
+    apply (force intro: continuous_intros continuous_on_subset [OF conth])+
+    done
+  have pih0: "path_image (\<lambda>u. h (u, 0)) \<subseteq> s"
+    using hs by (force simp: path_image_def)
+  have c1: "continuous_on ({0..1} \<times> {0..1}) (\<lambda>x. h (fst x * snd x, 0))"
+    apply (rule continuous_on_compose [of _ _ h, unfolded o_def])
+    apply (force simp: mult_le_one intro: continuous_intros continuous_on_subset [OF conth])+
+    done
+  have c2: "continuous_on ({0..1} \<times> {0..1}) (\<lambda>x. h (fst x - fst x * snd x, 0))"
+    apply (rule continuous_on_compose [of _ _ h, unfolded o_def])
+    apply (force simp: mult_left_le mult_le_one intro: continuous_intros continuous_on_subset [OF conth])+
+    apply (rule continuous_on_subset [OF conth])
+    apply (auto simp: algebra_simps add_increasing2 mult_left_le)
+    done
+  have [simp]: "\<And>t. \<lbrakk>0 \<le> t \<and> t \<le> 1\<rbrakk> \<Longrightarrow> h (t, 1) = h (t, 0)"
+    using ends by (simp add: pathfinish_def pathstart_def)
+  have adhoc_le: "c * 4 \<le> 1 + c * (d * 4)" if "\<not> d * 4 \<le> 3" "0 \<le> c" "c \<le> 1" for c d::real
+  proof -
+    have "c * 3 \<le> c * (d * 4)" using that less_eq_real_def by auto
+    with \<open>c \<le> 1\<close> show ?thesis by fastforce
+  qed
+  have *: "\<And>p x. (path p \<and> path(reversepath p)) \<and>
+                  (path_image p \<subseteq> s \<and> path_image(reversepath p) \<subseteq> s) \<and>
+                  (pathfinish p = pathstart(linepath a a +++ reversepath p) \<and>
+                   pathstart(reversepath p) = a) \<and> pathstart p = x
+                  \<Longrightarrow> 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 \<open>path p\<close> 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))"
+    apply (rule homotopic_paths_sym)
+    using homotopic_paths_lid [of "p +++ linepath (pathfinish p) (pathfinish p)" s]
+    by (metis 1 homotopic_paths_imp_path homotopic_paths_imp_pathstart homotopic_paths_imp_subset)
+  moreover have "homotopic_paths s (linepath (pathstart p) (pathstart p) +++ p +++ linepath (pathfinish p) (pathfinish p))
+                                   ((\<lambda>u. h (u, 0)) +++ linepath a a +++ reversepath (\<lambda>u. h (u, 0)))"
+    apply (simp add: homotopic_paths_def homotopic_with_def)
+    apply (rule_tac x="\<lambda>y. (subpath 0 (fst y) (\<lambda>u. h (u, 0)) +++ (\<lambda>u. h (Pair (fst y) u)) +++ subpath (fst y) 0 (\<lambda>u. h (u, 0))) (snd y)" in exI)
+    apply (simp add: subpath_reversepath)
+    apply (intro conjI homotopic_join_lemma)
+    using ploop
+    apply (simp_all add: path_defs joinpaths_def o_def subpath_def conth c1 c2)
+    apply (force simp: algebra_simps mult_le_one mult_left_le intro: hs [THEN subsetD] adhoc_le)
+    done
+  moreover have "homotopic_paths s ((\<lambda>u. h (u, 0)) +++ linepath a a +++ reversepath (\<lambda>u. h (u, 0)))
+                                   (linepath (pathstart p) (pathstart p))"
+    apply (rule *)
+    apply (simp add: pih0 pathstart_def pathfinish_def conth0)
+    apply (simp add: reversepath_def joinpaths_def)
+    done
+  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 \<subseteq> s" and piq: "path_image q \<subseteq> s"
+      and papp: "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 \<open>path p\<close> [unfolded path_def] by blast
+  have contq: "continuous_on {0..1} q"  using \<open>path q\<close> [unfolded path_def] by blast
+  have c1: "continuous_on ({0..1} \<times> {0..1}) (\<lambda>x. p ((1 - fst x) * snd x + fst x))"
+    apply (rule continuous_on_compose [of _ _ p, unfolded o_def])
+    apply (force simp: mult_le_one intro!: continuous_intros)
+    apply (rule continuous_on_subset [OF contp])
+    apply (auto simp: algebra_simps add_increasing2 mult_right_le_one_le sum_le_prod1)
+    done
+  have c2: "continuous_on ({0..1} \<times> {0..1}) (\<lambda>x. p ((fst x - 1) * snd x + 1))"
+    apply (rule continuous_on_compose [of _ _ p, unfolded o_def])
+    apply (force simp: mult_le_one intro!: continuous_intros)
+    apply (rule continuous_on_subset [OF contp])
+    apply (auto simp: algebra_simps add_increasing2 mult_left_le_one_le)
+    done
+  have ps1: "\<And>a b. \<lbrakk>b * 2 \<le> 1; 0 \<le> b; 0 \<le> a; a \<le> 1\<rbrakk> \<Longrightarrow> p ((1 - a) * (2 * b) + a) \<in> 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: "\<And>a b. \<lbrakk>\<not> 4 * b \<le> 3; b \<le> 1; 0 \<le> a; a \<le> 1\<rbrakk> \<Longrightarrow> p ((a - 1) * (4 * b - 3) + 1) \<in> s"
+    apply (rule pip [unfolded path_image_def, THEN subsetD])
+    apply (rule image_eqI, blast)
+    apply (simp add: algebra_simps)
+    by (metis add_mono_thms_linordered_semiring(1) affine_ineq linear mult.commute mult.left_neutral mult_right_mono not_le
+              add.commute zero_le_numeral)
+  have qs: "\<And>a b. \<lbrakk>4 * b \<le> 3; \<not> b * 2 \<le> 1\<rbrakk> \<Longrightarrow> q (4 * b - 2) \<in> 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))"
+    apply (simp add: homotopic_loops_def homotopic_with_def)
+    apply (rule_tac x="(\<lambda>y. (subpath (fst y) 1 p +++ q +++ subpath 1 (fst y) p) (snd y))" in exI)
+    apply (simp add: subpath_refl subpath_reversepath)
+    apply (intro conjI homotopic_join_lemma)
+    using papp qloop
+    apply (simp_all add: path_defs joinpaths_def o_def subpath_def c1 c2)
+    apply (force simp: contq intro: continuous_on_compose [of _ _ q, unfolded o_def] continuous_on_id continuous_on_snd)
+    apply (auto simp: ps1 ps2 qs)
+    done
+  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 \<open>path q\<close> homotopic_paths_lid qloop piq by auto
+    hence 1: "\<And>f. homotopic_paths s f q \<or> \<not> 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"
+    proof -
+      have "homotopic_paths s (q +++ linepath (pathfinish q) (pathfinish q)) q"
+        by (simp add: \<open>path q\<close> homotopic_paths_rid piq)
+      thus ?thesis
+        by (metis (no_types) 1 \<open>path q\<close> homotopic_paths_join homotopic_paths_rinv homotopic_paths_sym
+                  homotopic_paths_trans qloop pathfinish_linepath piq)
+    qed
+    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 \<open>path q\<close> 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
+    have pipq: "path_image p \<subseteq> S" "path_image q \<subseteq> S"
+      by (metis Un_subset_iff paths \<open>path p\<close> \<open>path q\<close> 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 \<open>path p\<close> \<open>path_image p \<subseteq> S\<close> 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 \<open>path p\<close> \<open>path q\<close> 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 \<open>path p\<close> \<open>path q\<close> homotopic_paths_assoc pipq)
+    moreover have "homotopic_paths S ((p +++ reversepath q) +++ q) (linepath (pathstart p) (pathstart p) +++ q)"
+      by (simp add: \<open>path q\<close> homotopic_paths_join paths pipq)
+    moreover then have "homotopic_paths S (linepath (pathstart p) (pathstart p) +++ q) q"
+      by (metis \<open>path q\<close> homotopic_paths_imp_path homotopic_paths_lid linepath_trivial path_join_path_ends pathfinish_def pipq(2))
+    ultimately show ?thesis
+      using homotopic_paths_trans by metis
+  next
+    case False
+    then show ?thesis
+      using \<open>path q\<close> homotopic_loops_imp_path loops path_join_path_ends by fastforce
+  qed
+qed
+
+
+subsection%unimportant\<open>Homotopy of "nearby" function, paths and loops\<close>
+
+lemma homotopic_with_linear:
+  fixes f g :: "_ \<Rightarrow> 'b::real_normed_vector"
+  assumes contf: "continuous_on s f"
+      and contg:"continuous_on s g"
+      and sub: "\<And>x. x \<in> s \<Longrightarrow> closed_segment (f x) (g x) \<subseteq> t"
+    shows "homotopic_with (\<lambda>z. True) s t f g"
+  apply (simp add: homotopic_with_def)
+  apply (rule_tac x="\<lambda>y. ((1 - (fst y)) *\<^sub>R f(snd y) + (fst y) *\<^sub>R g(snd y))" in exI)
+  apply (intro conjI)
+  apply (rule subset_refl continuous_intros continuous_on_subset [OF contf] continuous_on_compose2 [where g=f]
+                                            continuous_on_subset [OF contg] continuous_on_compose2 [where g=g]| simp)+
+  using sub closed_segment_def apply fastforce+
+  done
+
+lemma homotopic_paths_linear:
+  fixes g h :: "real \<Rightarrow> 'a::real_normed_vector"
+  assumes "path g" "path h" "pathstart h = pathstart g" "pathfinish h = pathfinish g"
+          "\<And>t. t \<in> {0..1} \<Longrightarrow> closed_segment (g t) (h t) \<subseteq> 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="\<lambda>y. ((1 - (fst y)) *\<^sub>R (g \<circ> snd) y + (fst y) *\<^sub>R (h \<circ> snd) y)" in exI)
+  apply (intro conjI subsetI continuous_intros; force)
+  done
+
+lemma homotopic_loops_linear:
+  fixes g h :: "real \<Rightarrow> 'a::real_normed_vector"
+  assumes "path g" "path h" "pathfinish g = pathstart g" "pathfinish h = pathstart h"
+          "\<And>t x. t \<in> {0..1} \<Longrightarrow> closed_segment (g t) (h t) \<subseteq> s"
+    shows "homotopic_loops s g h"
+  using assms
+  unfolding path_def
+  apply (simp add: pathstart_def pathfinish_def homotopic_loops_def homotopic_with_def)
+  apply (rule_tac x="\<lambda>y. ((1 - (fst y)) *\<^sub>R g(snd y) + (fst y) *\<^sub>R h(snd y))" in exI)
+  apply (auto intro!: continuous_intros intro: continuous_on_compose2 [where g=g] continuous_on_compose2 [where g=h])
+  apply (force simp: closed_segment_def)
+  done
+
+lemma homotopic_paths_nearby_explicit:
+  assumes "path g" "path h" "pathstart h = pathstart g" "pathfinish h = pathfinish g"
+      and no: "\<And>t x. \<lbrakk>t \<in> {0..1}; x \<notin> s\<rbrakk> \<Longrightarrow> norm(h t - g t) < norm(g t - x)"
+    shows "homotopic_paths s g h"
+  apply (rule homotopic_paths_linear [OF assms(1-4)])
+  by (metis no segment_bound(1) subsetI norm_minus_commute not_le)
+
+lemma homotopic_loops_nearby_explicit:
+  assumes "path g" "path h" "pathfinish g = pathstart g" "pathfinish h = pathstart h"
+      and no: "\<And>t x. \<lbrakk>t \<in> {0..1}; x \<notin> s\<rbrakk> \<Longrightarrow> norm(h t - g t) < norm(g t - x)"
+    shows "homotopic_loops s g h"
+  apply (rule homotopic_loops_linear [OF assms(1-4)])
+  by (metis no segment_bound(1) subsetI norm_minus_commute not_le)
+
+lemma homotopic_nearby_paths:
+  fixes g h :: "real \<Rightarrow> 'a::euclidean_space"
+  assumes "path g" "open s" "path_image g \<subseteq> s"
+    shows "\<exists>e. 0 < e \<and>
+               (\<forall>h. path h \<and>
+                    pathstart h = pathstart g \<and> pathfinish h = pathfinish g \<and>
+                    (\<forall>t \<in> {0..1}. norm(h t - g t) < e) \<longrightarrow> homotopic_paths s g h)"
+proof -
+  obtain e where "e > 0" and e: "\<And>x y. x \<in> path_image g \<Longrightarrow> y \<in> - s \<Longrightarrow> e \<le> dist x y"
+    using separate_compact_closed [of "path_image g" "-s"] assms by force
+  show ?thesis
+    apply (intro exI conjI)
+    using e [unfolded dist_norm]
+    apply (auto simp: intro!: homotopic_paths_nearby_explicit assms  \<open>e > 0\<close>)
+    by (metis atLeastAtMost_iff imageI le_less_trans not_le path_image_def)
+qed
+
+lemma homotopic_nearby_loops:
+  fixes g h :: "real \<Rightarrow> 'a::euclidean_space"
+  assumes "path g" "open s" "path_image g \<subseteq> s" "pathfinish g = pathstart g"
+    shows "\<exists>e. 0 < e \<and>
+               (\<forall>h. path h \<and> pathfinish h = pathstart h \<and>
+                    (\<forall>t \<in> {0..1}. norm(h t - g t) < e) \<longrightarrow> homotopic_loops s g h)"
+proof -
+  obtain e where "e > 0" and e: "\<And>x y. x \<in> path_image g \<Longrightarrow> y \<in> - s \<Longrightarrow> e \<le> dist x y"
+    using separate_compact_closed [of "path_image g" "-s"] assms by force
+  show ?thesis
+    apply (intro exI conjI)
+    using e [unfolded dist_norm]
+    apply (auto simp: intro!: homotopic_loops_nearby_explicit assms  \<open>e > 0\<close>)
+    by (metis atLeastAtMost_iff imageI le_less_trans not_le path_image_def)
+qed
+
+
+subsection\<open> Homotopy and subpaths\<close>
+
+lemma homotopic_join_subpaths1:
+  assumes "path g" and pag: "path_image g \<subseteq> s"
+      and u: "u \<in> {0..1}" and v: "v \<in> {0..1}" and w: "w \<in> {0..1}" "u \<le> v" "v \<le> w"
+    shows "homotopic_paths s (subpath u v g +++ subpath v w g) (subpath u w g)"
+proof -
+  have 1: "t * 2 \<le> 1 \<Longrightarrow> u + t * (v * 2) \<le> v + t * (u * 2)" for t
+    using affine_ineq \<open>u \<le> v\<close> by fastforce
+  have 2: "t * 2 > 1 \<Longrightarrow> u + (2*t - 1) * v \<le> 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 \<open>u \<le> v\<close> \<open>v \<le> w\<close>)
+  have t2: "\<And>t::real. t*2 = 1 \<Longrightarrow> t = 1/2" by auto
+  show ?thesis
+    apply (rule homotopic_paths_subset [OF _ pag])
+    using assms
+    apply (cases "w = u")
+    using homotopic_paths_rinv [of "subpath u v g" "path_image g"]
+    apply (force simp: closed_segment_eq_real_ivl image_mono path_image_def subpath_refl)
+      apply (rule homotopic_paths_sym)
+      apply (rule homotopic_paths_reparametrize
+             [where f = "\<lambda>t. if  t \<le> 1 / 2
+                             then inverse((w - u)) *\<^sub>R (2 * (v - u)) *\<^sub>R t
+                             else inverse((w - u)) *\<^sub>R ((v - u) + (w - v) *\<^sub>R (2 *\<^sub>R t - 1))"])
+      using \<open>path g\<close> path_subpath u w apply blast
+      using \<open>path g\<close> path_image_subpath_subset u w(1) apply blast
+      apply simp_all
+      apply (subst split_01)
+      apply (rule continuous_on_cases continuous_intros | force simp: pathfinish_def joinpaths_def)+
+      apply (simp_all add: field_simps not_le)
+      apply (force dest!: t2)
+      apply (force simp: algebra_simps mult_left_mono affine_ineq dest!: 1 2)
+      apply (simp add: joinpaths_def subpath_def)
+      apply (force simp: algebra_simps)
+      done
+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 \<subseteq> s"
+      and u: "u \<in> {0..1}" and v: "v \<in> {0..1}" and w: "w \<in> {0..1}"
+    shows "homotopic_paths s (subpath v w g +++ subpath w u g) (subpath v u g)"
+proof -
+  have "homotopic_paths s (subpath u w g +++ subpath w v g) ((subpath u v g +++ subpath v w g) +++ subpath w v g)"
+    apply (rule homotopic_paths_join)
+    using hom homotopic_paths_sym_eq apply blast
+    apply (metis \<open>path g\<close> homotopic_paths_eq pag path_image_subpath_subset path_subpath subset_trans v w, simp)
+    done
+  also have "homotopic_paths s ((subpath u v g +++ subpath v w g) +++ subpath w v g) (subpath u v g +++ subpath v w g +++ subpath w v g)"
+    apply (rule homotopic_paths_sym [OF homotopic_paths_assoc])
+    using assms by (simp_all add: path_image_subpath_subset [THEN order_trans])
+  also have "homotopic_paths s (subpath u v g +++ subpath v w g +++ subpath w v g)
+                               (subpath u v g +++ linepath (pathfinish (subpath u v g)) (pathfinish (subpath u v g)))"
+    apply (rule homotopic_paths_join)
+    apply (metis \<open>path g\<close> homotopic_paths_eq order.trans pag path_image_subpath_subset path_subpath u v)
+    apply (metis (no_types, lifting) \<open>path g\<close> homotopic_paths_linv order_trans pag path_image_subpath_subset path_subpath pathfinish_subpath reversepath_subpath v w)
+    apply simp
+    done
+  also have "homotopic_paths s (subpath u v g +++ linepath (pathfinish (subpath u v g)) (pathfinish (subpath u v g))) (subpath u v g)"
+    apply (rule homotopic_paths_rid)
+    using \<open>path g\<close> path_subpath u v apply blast
+    apply (meson \<open>path g\<close> order.trans pag path_image_subpath_subset u v)
+    done
+  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:
+   "\<lbrakk>path g; path_image g \<subseteq> s; u \<in> {0..1}; v \<in> {0..1}; w \<in> {0..1}\<rbrakk>
+    \<Longrightarrow> homotopic_paths s (subpath u v g +++ subpath v w g) (subpath u w g)"
+  apply (rule le_cases3 [of u v w])
+using homotopic_join_subpaths1 homotopic_join_subpaths2 homotopic_join_subpaths3 by metis+
+
+text\<open>Relating homotopy of trivial loops to path-connectedness.\<close>
+
+lemma path_component_imp_homotopic_points:
+    "path_component S a b \<Longrightarrow> homotopic_loops S (linepath a a) (linepath b b)"
+apply (simp add: path_component_def homotopic_loops_def homotopic_with_def
+                 pathstart_def pathfinish_def path_image_def path_def, clarify)
+apply (rule_tac x="g \<circ> fst" in exI)
+apply (intro conjI continuous_intros continuous_on_compose)+
+apply (auto elim!: continuous_on_subset)
+done
+
+lemma homotopic_loops_imp_path_component_value:
+   "\<lbrakk>homotopic_loops S p q; 0 \<le> t; t \<le> 1\<rbrakk>
+        \<Longrightarrow> path_component S (p t) (q t)"
+apply (simp add: path_component_def homotopic_loops_def homotopic_with_def
+                 pathstart_def pathfinish_def path_image_def path_def, clarify)
+apply (rule_tac x="h \<circ> (\<lambda>u. (u, t))" in exI)
+apply (intro conjI continuous_intros continuous_on_compose)+
+apply (auto elim!: continuous_on_subset)
+done
+
+lemma homotopic_points_eq_path_component:
+   "homotopic_loops S (linepath a a) (linepath b b) \<longleftrightarrow>
+        path_component S a b"
+by (auto simp: path_component_imp_homotopic_points
+         dest: homotopic_loops_imp_path_component_value [where t=1])
+
+lemma path_connected_eq_homotopic_points:
+    "path_connected S \<longleftrightarrow>
+      (\<forall>a b. a \<in> S \<and> b \<in> S \<longrightarrow> homotopic_loops S (linepath a a) (linepath b b))"
+by (auto simp: path_connected_def path_component_def homotopic_points_eq_path_component)
+
+
+subsection\<open>Simply connected sets\<close>
+
+text%important\<open>defined as "all loops are homotopic (as loops)\<close>
+
+definition%important simply_connected where
+  "simply_connected S \<equiv>
+        \<forall>p q. path p \<and> pathfinish p = pathstart p \<and> path_image p \<subseteq> S \<and>
+              path q \<and> pathfinish q = pathstart q \<and> path_image q \<subseteq> S
+              \<longrightarrow> 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 \<Longrightarrow> 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 \<Longrightarrow> 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 \<longleftrightarrow>
+            (\<forall>p a. path p \<and> path_image p \<subseteq> S \<and>
+                  pathfinish p = pathstart p \<and> a \<in> S
+                  \<longrightarrow> homotopic_loops S p (linepath a a))"
+apply (simp add: simply_connected_def)
+apply (rule iffI, force, clarify)
+apply (rule_tac q = "linepath (pathstart p) (pathstart p)" in homotopic_loops_trans)
+apply (fastforce simp add:)
+using homotopic_loops_sym apply blast
+done
+
+lemma simply_connected_eq_contractible_loop_some:
+  fixes S :: "_::real_normed_vector set"
+  shows "simply_connected S \<longleftrightarrow>
+                path_connected S \<and>
+                (\<forall>p. path p \<and> path_image p \<subseteq> S \<and> pathfinish p = pathstart p
+                    \<longrightarrow> (\<exists>a. a \<in> S \<and> homotopic_loops S p (linepath a a)))"
+apply (rule iffI)
+ apply (fastforce simp: simply_connected_imp_path_connected simply_connected_eq_contractible_loop_any)
+apply (clarsimp simp add: simply_connected_eq_contractible_loop_any)
+apply (drule_tac x=p in spec)
+using homotopic_loops_trans path_connected_eq_homotopic_points
+  apply blast
+done
+
+lemma simply_connected_eq_contractible_loop_all:
+  fixes S :: "_::real_normed_vector set"
+  shows "simply_connected S \<longleftrightarrow>
+         S = {} \<or>
+         (\<exists>a \<in> S. \<forall>p. path p \<and> path_image p \<subseteq> S \<and> pathfinish p = pathstart p
+                \<longrightarrow> homotopic_loops S p (linepath a a))"
+        (is "?lhs = ?rhs")
+proof (cases "S = {}")
+  case True then show ?thesis by force
+next
+  case False
+  then obtain a where "a \<in> S" by blast
+  show ?thesis
+  proof
+    assume "simply_connected S"
+    then show ?rhs
+      using \<open>a \<in> S\<close> \<open>simply_connected S\<close> simply_connected_eq_contractible_loop_any
+      by blast
+  next
+    assume ?rhs
+    then show "simply_connected S"
+      apply (simp add: simply_connected_eq_contractible_loop_any False)
+      by (meson homotopic_loops_refl homotopic_loops_sym homotopic_loops_trans
+             path_component_imp_homotopic_points path_component_refl)
+  qed
+qed
+
+lemma simply_connected_eq_contractible_path:
+  fixes S :: "_::real_normed_vector set"
+  shows "simply_connected S \<longleftrightarrow>
+           path_connected S \<and>
+           (\<forall>p. path p \<and> path_image p \<subseteq> S \<and> pathfinish p = pathstart p
+            \<longrightarrow> homotopic_paths S p (linepath (pathstart p) (pathstart p)))"
+apply (rule iffI)
+ apply (simp add: simply_connected_imp_path_connected)
+ apply (metis simply_connected_eq_contractible_loop_some homotopic_loops_imp_homotopic_paths_null)
+by (meson homotopic_paths_imp_homotopic_loops pathfinish_linepath pathstart_in_path_image
+         simply_connected_eq_contractible_loop_some subset_iff)
+
+lemma simply_connected_eq_homotopic_paths:
+  fixes S :: "_::real_normed_vector set"
+  shows "simply_connected S \<longleftrightarrow>
+          path_connected S \<and>
+          (\<forall>p q. path p \<and> path_image p \<subseteq> S \<and>
+                path q \<and> path_image q \<subseteq> S \<and>
+                pathstart q = pathstart p \<and> pathfinish q = pathfinish p
+                \<longrightarrow> homotopic_paths S p q)"
+         (is "?lhs = ?rhs")
+proof
+  assume ?lhs
+  then have pc: "path_connected S"
+        and *:  "\<And>p. \<lbrakk>path p; path_image p \<subseteq> S;
+                       pathfinish p = pathstart p\<rbrakk>
+                      \<Longrightarrow> 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 \<subseteq> S" "path q"
+           "path_image q \<subseteq> S" "pathstart q = pathstart p"
+           "pathfinish q = pathfinish p" for p q
+  proof -
+    have "homotopic_paths S p (p +++ linepath (pathfinish p) (pathfinish p))"
+      by (simp add: homotopic_paths_rid homotopic_paths_sym that)
+    also have "homotopic_paths S (p +++ linepath (pathfinish p) (pathfinish p))
+                                 (p +++ reversepath q +++ q)"
+      using that
+      by (metis homotopic_paths_join homotopic_paths_linv homotopic_paths_refl homotopic_paths_sym_eq pathstart_linepath)
+    also have "homotopic_paths S (p +++ reversepath q +++ q)
+                                 ((p +++ reversepath q) +++ q)"
+      by (simp add: that homotopic_paths_assoc)
+    also have "homotopic_paths S ((p +++ reversepath q) +++ q)
+                                 (linepath (pathstart q) (pathstart q) +++ q)"
+      using * [of "p +++ reversepath q"] that
+      by (simp add: homotopic_paths_join path_image_join)
+    also have "homotopic_paths S (linepath (pathstart q) (pathstart q) +++ q) 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 \<times> T)"
+proof -
+  have "homotopic_loops (S \<times> T) p (linepath (a, b) (a, b))"
+       if "path p" "path_image p \<subseteq> S \<times> T" "p 1 = p 0" "a \<in> S" "b \<in> T"
+       for p a b
+  proof -
+    have "path (fst \<circ> p)"
+      apply (rule Path_Connected.path_continuous_image [OF \<open>path p\<close>])
+      apply (rule continuous_intros)+
+      done
+    moreover have "path_image (fst \<circ> p) \<subseteq> S"
+      using that apply (simp add: path_image_def) by force
+    ultimately have p1: "homotopic_loops S (fst \<circ> p) (linepath a a)"
+      using S that
+      apply (simp add: simply_connected_eq_contractible_loop_any)
+      apply (drule_tac x="fst \<circ> p" in spec)
+      apply (drule_tac x=a in spec)
+      apply (auto simp: pathstart_def pathfinish_def)
+      done
+    have "path (snd \<circ> p)"
+      apply (rule Path_Connected.path_continuous_image [OF \<open>path p\<close>])
+      apply (rule continuous_intros)+
+      done
+    moreover have "path_image (snd \<circ> p) \<subseteq> T"
+      using that apply (simp add: path_image_def) by force
+    ultimately have p2: "homotopic_loops T (snd \<circ> p) (linepath b b)"
+      using T that
+      apply (simp add: simply_connected_eq_contractible_loop_any)
+      apply (drule_tac x="snd \<circ> p" in spec)
+      apply (drule_tac x=b in spec)
+      apply (auto simp: pathstart_def pathfinish_def)
+      done
+    show ?thesis
+      using p1 p2
+      apply (simp add: homotopic_loops, clarify)
+      apply (rename_tac h k)
+      apply (rule_tac x="\<lambda>z. Pair (h z) (k z)" in exI)
+      apply (intro conjI continuous_intros | assumption)+
+      apply (auto simp: pathstart_def pathfinish_def)
+      done
+  qed
+  with assms show ?thesis
+    by (simp add: simply_connected_eq_contractible_loop_any pathfinish_def pathstart_def)
+qed
+
+
+subsection\<open>Contractible sets\<close>
+
+definition%important contractible where
+ "contractible S \<equiv> \<exists>a. homotopic_with (\<lambda>x. True) S S id (\<lambda>x. 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 (\<lambda>x. True) S S id (\<lambda>x. a)"
+    using assms by (force simp: contractible_def)
+  then have "a \<in> S"
+    by (metis False homotopic_constant_maps homotopic_with_symD homotopic_with_trans path_component_mem(2))
+  show ?thesis
+    apply (simp add: simply_connected_eq_contractible_loop_all False)
+    apply (rule bexI [OF _ \<open>a \<in> S\<close>])
+    using a apply (simp add: homotopic_loops_def homotopic_with_def path_def path_image_def pathfinish_def pathstart_def, clarify)
+    apply (rule_tac x="(h \<circ> (\<lambda>y. (fst y, (p \<circ> snd) y)))" in exI)
+    apply (intro conjI continuous_on_compose continuous_intros)
+    apply (erule continuous_on_subset | force)+
+    done
+qed
+
+corollary contractible_imp_connected:
+  fixes S :: "_::real_normed_vector set"
+  shows "contractible S \<Longrightarrow> 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 \<Longrightarrow> 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 \<subseteq> T"
+      and g: "continuous_on T g" "g ` T \<subseteq> U"
+      and T: "contractible T"
+    obtains c where "homotopic_with (\<lambda>h. True) S U (g \<circ> f) (\<lambda>x. c)"
+proof -
+  obtain b where b: "homotopic_with (\<lambda>x. True) T T id (\<lambda>x. b)"
+    using assms by (force simp: contractible_def)
+  have "homotopic_with (\<lambda>f. True) T U (g \<circ> id) (g \<circ> (\<lambda>x. b))"
+    by (rule homotopic_compose_continuous_left [OF b g])
+  then have "homotopic_with (\<lambda>f. True) S U (g \<circ> id \<circ> f) (g \<circ> (\<lambda>x. b) \<circ> f)"
+    by (rule homotopic_compose_continuous_right [OF _ f])
+  then show ?thesis
+    by (simp add: comp_def that)
+qed
+
+lemma nullhomotopic_into_contractible:
+  assumes f: "continuous_on S f" "f ` S \<subseteq> T"
+      and T: "contractible T"
+    obtains c where "homotopic_with (\<lambda>h. True) S T f (\<lambda>x. c)"
+apply (rule nullhomotopic_through_contractible [OF f, of id T])
+using assms
+apply (auto simp: continuous_on_id)
+done
+
+lemma nullhomotopic_from_contractible:
+  assumes f: "continuous_on S f" "f ` S \<subseteq> T"
+      and S: "contractible S"
+    obtains c where "homotopic_with (\<lambda>h. True) S T f (\<lambda>x. c)"
+apply (rule nullhomotopic_through_contractible [OF continuous_on_id _ f S, of S])
+using assms
+apply (auto simp: comp_def)
+done
+
+lemma homotopic_through_contractible:
+  fixes S :: "_::real_normed_vector set"
+  assumes "continuous_on S f1" "f1 ` S \<subseteq> T"
+          "continuous_on T g1" "g1 ` T \<subseteq> U"
+          "continuous_on S f2" "f2 ` S \<subseteq> T"
+          "continuous_on T g2" "g2 ` T \<subseteq> U"
+          "contractible T" "path_connected U"
+   shows "homotopic_with (\<lambda>h. True) S U (g1 \<circ> f1) (g2 \<circ> f2)"
+proof -
+  obtain c1 where c1: "homotopic_with (\<lambda>h. True) S U (g1 \<circ> f1) (\<lambda>x. c1)"
+    apply (rule nullhomotopic_through_contractible [of S f1 T g1 U])
+    using assms apply auto
+    done
+  obtain c2 where c2: "homotopic_with (\<lambda>h. True) S U (g2 \<circ> f2) (\<lambda>x. c2)"
+    apply (rule nullhomotopic_through_contractible [of S f2 T g2 U])
+    using assms apply auto
+    done
+  have *: "S = {} \<or> (\<exists>t. path_connected t \<and> t \<subseteq> U \<and> c2 \<in> t \<and> c1 \<in> t)"
+  proof (cases "S = {}")
+    case True then show ?thesis by force
+  next
+    case False
+    with c1 c2 have "c1 \<in> U" "c2 \<in> U"
+      using homotopic_with_imp_subset2 all_not_in_conv image_subset_iff by blast+
+    with \<open>path_connected U\<close> show ?thesis by blast
+  qed
+  show ?thesis
+    apply (rule homotopic_with_trans [OF c1])
+    apply (rule homotopic_with_symD)
+    apply (rule homotopic_with_trans [OF c2])
+    apply (simp add: path_component homotopic_constant_maps *)
+    done
+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 \<subseteq> T"
+      and g: "continuous_on S g" "g ` S \<subseteq> T"
+      and T: "contractible T"
+    shows "homotopic_with (\<lambda>h. 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 continuous_on_id)
+
+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 \<subseteq> T"
+      and g: "continuous_on S g" "g ` S \<subseteq> T"
+      and "contractible S" "path_connected T"
+    shows "homotopic_with (\<lambda>h. 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 continuous_on_id)
+
+lemma starlike_imp_contractible_gen:
+  fixes S :: "'a::real_normed_vector set"
+  assumes S: "starlike S"
+      and P: "\<And>a T. \<lbrakk>a \<in> S; 0 \<le> T; T \<le> 1\<rbrakk> \<Longrightarrow> P(\<lambda>x. (1 - T) *\<^sub>R x + T *\<^sub>R a)"
+    obtains a where "homotopic_with P S S (\<lambda>x. x) (\<lambda>x. a)"
+proof -
+  obtain a where "a \<in> S" and a: "\<And>x. x \<in> S \<Longrightarrow> closed_segment a x \<subseteq> S"
+    using S by (auto simp: starlike_def)
+  have "(\<lambda>y. (1 - fst y) *\<^sub>R snd y + fst y *\<^sub>R a) ` ({0..1} \<times> S) \<subseteq> S"
+    apply clarify
+    apply (erule a [unfolded closed_segment_def, THEN subsetD], simp)
+    apply (metis add_diff_cancel_right' diff_ge_0_iff_ge le_add_diff_inverse pth_c(1))
+    done
+  then show ?thesis
+    apply (rule_tac a=a in that)
+    using \<open>a \<in> S\<close>
+    apply (simp add: homotopic_with_def)
+    apply (rule_tac x="\<lambda>y. (1 - (fst y)) *\<^sub>R snd y + (fst y) *\<^sub>R a" in exI)
+    apply (intro conjI ballI continuous_on_compose continuous_intros)
+    apply (simp_all add: P)
+    done
+qed
+
+lemma starlike_imp_contractible:
+  fixes S :: "'a::real_normed_vector set"
+  shows "starlike S \<Longrightarrow> 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 \<Longrightarrow> 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 \<Longrightarrow> 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 \<Longrightarrow> 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 \<Longrightarrow> 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 \<longleftrightarrow> simply_connected S"
+using convex_imp_simply_connected is_interval_convex_1 is_interval_path_connected_1 simply_connected_imp_path_connected by auto
+
+lemma contractible_empty [simp]: "contractible {}"
+  by (simp add: contractible_def homotopic_with)
+
+lemma contractible_convex_tweak_boundary_points:
+  fixes S :: "'a::euclidean_space set"
+  assumes "convex S" and TS: "rel_interior S \<subseteq> T" "T \<subseteq> closure S"
+  shows "contractible T"
+proof (cases "S = {}")
+  case True
+  with assms show ?thesis
+    by (simp add: subsetCE)
+next
+  case False
+  show ?thesis
+    apply (rule starlike_imp_contractible)
+    apply (rule starlike_convex_tweak_boundary_points [OF \<open>convex S\<close> False TS])
+    done
+qed
+
+lemma convex_imp_contractible:
+  fixes S :: "'a::real_normed_vector set"
+  shows "convex S \<Longrightarrow> 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 \<longleftrightarrow> 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 \<times> T)"
+proof -
+  obtain a h where conth: "continuous_on ({0..1} \<times> S) h"
+             and hsub: "h ` ({0..1} \<times> S) \<subseteq> S"
+             and [simp]: "\<And>x. x \<in> S \<Longrightarrow> h (0, x) = x"
+             and [simp]: "\<And>x. x \<in> S \<Longrightarrow>  h (1::real, x) = a"
+    using S by (auto simp: contractible_def homotopic_with)
+  obtain b k where contk: "continuous_on ({0..1} \<times> T) k"
+             and ksub: "k ` ({0..1} \<times> T) \<subseteq> T"
+             and [simp]: "\<And>x. x \<in> T \<Longrightarrow> k (0, x) = x"
+             and [simp]: "\<And>x. x \<in> T \<Longrightarrow>  k (1::real, x) = b"
+    using T by (auto 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 = "\<lambda>z. (h (fst z, fst(snd z)), k (fst z, snd(snd z)))"])
+    apply (intro conjI ballI continuous_intros continuous_on_compose2 [OF conth] continuous_on_compose2 [OF contk])
+    using hsub ksub
+    apply auto
+    done
+qed
+
+lemma homotopy_dominated_contractibility:
+  fixes S :: "'a::real_normed_vector set" and T :: "'b::real_normed_vector set"
+  assumes S: "contractible S"
+      and f: "continuous_on S f" "image f S \<subseteq> T"
+      and g: "continuous_on T g" "image g T \<subseteq> S"
+      and hom: "homotopic_with (\<lambda>x. True) T T (f \<circ> g) id"
+    shows "contractible T"
+proof -
+  obtain b where "homotopic_with (\<lambda>h. True) S T f (\<lambda>x. b)"
+    using nullhomotopic_from_contractible [OF f S] .
+  then have homg: "homotopic_with (\<lambda>x. True) T T ((\<lambda>x. b) \<circ> g) (f \<circ> g)"
+    by (rule homotopic_with_compose_continuous_right [OF homotopic_with_symD g])
+  show ?thesis
+    apply (simp add: contractible_def)
+    apply (rule exI [where x = b])
+    apply (rule homotopic_with_symD)
+    apply (rule homotopic_with_trans [OF _ hom])
+    using homg apply (simp add: o_def)
+    done
+qed
+
+
+subsection\<open>Local versions of topological properties in general\<close>
+
+definition%important locally :: "('a::topological_space set \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> bool"
+where
+ "locally P S \<equiv>
+        \<forall>w x. openin (subtopology euclidean S) w \<and> x \<in> w
+              \<longrightarrow> (\<exists>u v. openin (subtopology euclidean S) u \<and> P v \<and>
+                        x \<in> u \<and> u \<subseteq> v \<and> v \<subseteq> w)"
+
+lemma locallyI:
+  assumes "\<And>w x. \<lbrakk>openin (subtopology euclidean S) w; x \<in> w\<rbrakk>
+                  \<Longrightarrow> \<exists>u v. openin (subtopology euclidean S) u \<and> P v \<and>
+                        x \<in> u \<and> u \<subseteq> v \<and> v \<subseteq> w"
+    shows "locally P S"
+using assms by (force simp: locally_def)
+
+lemma locallyE:
+  assumes "locally P S" "openin (subtopology euclidean S) w" "x \<in> w"
+  obtains u v where "openin (subtopology euclidean S) u"
+                    "P v" "x \<in> u" "u \<subseteq> v" "v \<subseteq> w"
+  using assms unfolding locally_def by meson
+
+lemma locally_mono:
+  assumes "locally P S" "\<And>t. P t \<Longrightarrow> Q t"
+    shows "locally Q S"
+by (metis assms locally_def)
+
+lemma locally_open_subset:
+  assumes "locally P S" "openin (subtopology euclidean S) t"
+    shows "locally P t"
+using assms
+apply (simp add: locally_def)
+apply (erule all_forward)+
+apply (rule impI)
+apply (erule impCE)
+ using openin_trans apply blast
+apply (erule ex_forward)
+by (metis (no_types, hide_lams) Int_absorb1 Int_lower1 Int_subset_iff openin_open openin_subtopology_Int_subset)
+
+lemma locally_diff_closed:
+    "\<lbrakk>locally P S; closedin (subtopology euclidean S) t\<rbrakk> \<Longrightarrow> 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} \<longleftrightarrow> P {a}"
+apply (simp add: locally_def openin_euclidean_subtopology_iff subset_singleton_iff conj_disj_distribR cong: conj_cong)
+using zero_less_one by blast
+
+lemma locally_iff:
+    "locally P S \<longleftrightarrow>
+     (\<forall>T x. open T \<and> x \<in> S \<inter> T \<longrightarrow> (\<exists>U. open U \<and> (\<exists>v. P v \<and> x \<in> S \<inter> U \<and> S \<inter> U \<subseteq> v \<and> v \<subseteq> S \<inter> T)))"
+apply (simp add: le_inf_iff locally_def openin_open, safe)
+apply (metis IntE IntI le_inf_iff)
+apply (metis IntI Int_subset_iff)
+done
+
+lemma locally_Int:
+  assumes S: "locally P S" and t: "locally P t"
+      and P: "\<And>S t. P S \<and> P t \<Longrightarrow> P(S \<inter> t)"
+    shows "locally P (S \<inter> t)"
+using S t unfolding locally_iff
+apply clarify
+apply (drule_tac x=T in spec)+
+apply (drule_tac x=x in spec)+
+apply clarsimp
+apply (rename_tac U1 U2 V1 V2)
+apply (rule_tac x="U1 \<inter> U2" in exI)
+apply (simp add: open_Int)
+apply (rule_tac x="V1 \<inter> V2" in exI)
+apply (auto intro: P)
+done
+
+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: "\<And>S T. P S \<and> Q T \<Longrightarrow> R(S \<times> T)"
+  shows "locally R (S \<times> T)"
+    unfolding locally_def
+proof (clarify)
+  fix W x y
+  assume W: "openin (subtopology euclidean (S \<times> T)) W" and xy: "(x, y) \<in> W"
+  then obtain U V where "openin (subtopology euclidean S) U" "x \<in> U"
+                        "openin (subtopology euclidean T) V" "y \<in> V" "U \<times> V \<subseteq> W"
+    using Times_in_interior_subtopology by metis
+  then obtain U1 U2 V1 V2
+         where opeS: "openin (subtopology euclidean S) U1 \<and> P U2 \<and> x \<in> U1 \<and> U1 \<subseteq> U2 \<and> U2 \<subseteq> U"
+           and opeT: "openin (subtopology euclidean T) V1 \<and> Q V2 \<and> y \<in> V1 \<and> V1 \<subseteq> V2 \<and> V2 \<subseteq> V"
+    by (meson PS QT locallyE)
+  with \<open>U \<times> V \<subseteq> W\<close> show "\<exists>u v. openin (subtopology euclidean (S \<times> T)) u \<and> R v \<and> (x,y) \<in> u \<and> u \<subseteq> v \<and> v \<subseteq> W"
+    apply (rule_tac x="U1 \<times> V1" in exI)
+    apply (rule_tac x="U2 \<times> V2" in exI)
+    apply (auto simp: openin_Times R)
+    done
+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: "\<And>S S'. \<lbrakk>P S; homeomorphism S S' f g\<rbrakk> \<Longrightarrow> Q S'"
+    shows "locally Q t"
+proof (clarsimp simp: locally_def)
+  fix W y
+  assume "y \<in> W" and "openin (subtopology euclidean t) W"
+  then obtain T where T: "open T" "W = t \<inter> T"
+    by (force simp: openin_open)
+  then have "W \<subseteq> t" by auto
+  have f: "\<And>x. x \<in> S \<Longrightarrow> g(f x) = x" "f ` S = t" "continuous_on S f"
+   and g: "\<And>y. y \<in> t \<Longrightarrow> f(g y) = y" "g ` t = S" "continuous_on t g"
+    using hom by (auto simp: homeomorphism_def)
+  have gw: "g ` W = S \<inter> f -` W"
+    using \<open>W \<subseteq> t\<close>
+    apply auto
+    using \<open>g ` t = S\<close> \<open>W \<subseteq> t\<close> apply blast
+    using g \<open>W \<subseteq> t\<close> apply auto[1]
+    by (simp add: f rev_image_eqI)
+  have \<circ>: "openin (subtopology euclidean S) (g ` W)"
+  proof -
+    have "continuous_on S f"
+      using f(3) by blast
+    then show "openin (subtopology euclidean S) (g ` W)"
+      by (simp add: gw Collect_conj_eq \<open>openin (subtopology euclidean t) W\<close> continuous_on_open f(2))
+  qed
+  then obtain u v
+    where osu: "openin (subtopology euclidean S) u" and uv: "P v" "g y \<in> u" "u \<subseteq> v" "v \<subseteq> g ` W"
+    using S [unfolded locally_def, rule_format, of "g ` W" "g y"] \<open>y \<in> W\<close> by force
+  have "v \<subseteq> S" using uv by (simp add: gw)
+  have fv: "f ` v = t \<inter> {x. g x \<in> v}"
+    using \<open>f ` S = t\<close> f \<open>v \<subseteq> S\<close> by auto
+  have "f ` v \<subseteq> W"
+    using uv using Int_lower2 gw image_subsetI mem_Collect_eq subset_iff by auto
+  have contvf: "continuous_on v f"
+    using \<open>v \<subseteq> S\<close> continuous_on_subset f(3) by blast
+  have contvg: "continuous_on (f ` v) g"
+    using \<open>f ` v \<subseteq> W\<close> \<open>W \<subseteq> t\<close> continuous_on_subset [OF g(3)] by blast
+  have homv: "homeomorphism v (f ` v) f g"
+    using \<open>v \<subseteq> S\<close> \<open>W \<subseteq> t\<close> f
+    apply (simp add: homeomorphism_def contvf contvg, auto)
+    by (metis f(1) rev_image_eqI rev_subsetD)
+  have 1: "openin (subtopology euclidean t) (t \<inter> g -` u)"
+    apply (rule continuous_on_open [THEN iffD1, rule_format])
+    apply (rule \<open>continuous_on t g\<close>)
+    using \<open>g ` t = S\<close> apply (simp add: osu)
+    done
+  have 2: "\<exists>V. Q V \<and> y \<in> (t \<inter> g -` u) \<and> (t \<inter> g -` u) \<subseteq> V \<and> V \<subseteq> W"
+    apply (rule_tac x="f ` v" in exI)
+    apply (intro conjI Q [OF \<open>P v\<close> homv])
+    using \<open>W \<subseteq> t\<close> \<open>y \<in> W\<close>  \<open>f ` v \<subseteq> W\<close>  uv  apply (auto simp: fv)
+    done
+  show "\<exists>U. openin (subtopology euclidean t) U \<and> (\<exists>v. Q v \<and> y \<in> U \<and> U \<subseteq> v \<and> v \<subseteq> W)"
+    by (meson 1 2)
+qed
+
+lemma homeomorphism_locally:
+  fixes f:: "'a::metric_space \<Rightarrow> 'b::metric_space"
+  assumes hom: "homeomorphism S t f g"
+      and eq: "\<And>S t. homeomorphism S t f g \<Longrightarrow> (P S \<longleftrightarrow> Q t)"
+    shows "locally P S \<longleftrightarrow> locally Q t"
+apply (rule iffI)
+apply (erule homeomorphism_locally_imp [OF _ hom])
+apply (simp add: eq)
+apply (erule homeomorphism_locally_imp)
+using eq homeomorphism_sym homeomorphism_symD [OF hom] apply blast+
+done
+
+lemma homeomorphic_locally:
+  fixes S:: "'a::metric_space set" and T:: "'b::metric_space set"
+  assumes hom: "S homeomorphic T"
+          and iff: "\<And>X Y. X homeomorphic Y \<Longrightarrow> (P X \<longleftrightarrow> Q Y)"
+    shows "locally P S \<longleftrightarrow> locally Q T"
+proof -
+  obtain f g where hom: "homeomorphism S T f g"
+    using assms by (force simp: homeomorphic_def)
+  then show ?thesis
+    using homeomorphic_def local.iff
+    by (blast intro!: homeomorphism_locally)
+qed
+
+lemma homeomorphic_local_compactness:
+  fixes S:: "'a::metric_space set" and T:: "'b::metric_space set"
+  shows "S homeomorphic T \<Longrightarrow> locally compact S \<longleftrightarrow> locally compact T"
+by (simp add: homeomorphic_compactness homeomorphic_locally)
+
+lemma locally_translation:
+  fixes P :: "'a :: real_normed_vector set \<Rightarrow> bool"
+  shows
+   "(\<And>S. P (image (\<lambda>x. a + x) S) \<longleftrightarrow> P S)
+        \<Longrightarrow> locally P (image (\<lambda>x. a + x) S) \<longleftrightarrow> locally P S"
+apply (rule homeomorphism_locally [OF homeomorphism_translation])
+apply (simp add: homeomorphism_def)
+by metis
+
+lemma locally_injective_linear_image:
+  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
+  assumes f: "linear f" "inj f" and iff: "\<And>S. P (f ` S) \<longleftrightarrow> Q S"
+    shows "locally P (f ` S) \<longleftrightarrow> locally Q S"
+apply (rule linear_homeomorphism_image [OF f])
+apply (rule_tac f=g and g = f in homeomorphism_locally, assumption)
+by (metis iff homeomorphism_def)
+
+lemma locally_open_map_image:
+  fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
+  assumes P: "locally P S"
+      and f: "continuous_on S f"
+      and oo: "\<And>t. openin (subtopology euclidean S) t
+                   \<Longrightarrow> openin (subtopology euclidean (f ` S)) (f ` t)"
+      and Q: "\<And>t. \<lbrakk>t \<subseteq> S; P t\<rbrakk> \<Longrightarrow> Q(f ` t)"
+    shows "locally Q (f ` S)"
+proof (clarsimp simp add: locally_def)
+  fix W y
+  assume oiw: "openin (subtopology euclidean (f ` S)) W" and "y \<in> W"
+  then have "W \<subseteq> f ` S" by (simp add: openin_euclidean_subtopology_iff)
+  have oivf: "openin (subtopology euclidean S) (S \<inter> f -` W)"
+    by (rule continuous_on_open [THEN iffD1, rule_format, OF f oiw])
+  then obtain x where "x \<in> S" "f x = y"
+    using \<open>W \<subseteq> f ` S\<close> \<open>y \<in> W\<close> by blast
+  then obtain U V
+    where "openin (subtopology euclidean S) U" "P V" "x \<in> U" "U \<subseteq> V" "V \<subseteq> S \<inter> f -` W"
+    using P [unfolded locally_def, rule_format, of "(S \<inter> f -` W)" x] oivf \<open>y \<in> W\<close>
+    by auto
+  then show "\<exists>X. openin (subtopology euclidean (f ` S)) X \<and> (\<exists>Y. Q Y \<and> y \<in> X \<and> X \<subseteq> Y \<and> Y \<subseteq> W)"
+    apply (rule_tac x="f ` U" in exI)
+    apply (rule conjI, blast intro!: oo)
+    apply (rule_tac x="f ` V" in exI)
+    apply (force simp: \<open>f x = y\<close> rev_image_eqI intro: Q)
+    done
+qed
+
+
+subsection\<open>An induction principle for connected sets\<close>
+
+proposition connected_induction:
+  assumes "connected S"
+      and opD: "\<And>T a. \<lbrakk>openin (subtopology euclidean S) T; a \<in> T\<rbrakk> \<Longrightarrow> \<exists>z. z \<in> T \<and> P z"
+      and opI: "\<And>a. a \<in> S
+             \<Longrightarrow> \<exists>T. openin (subtopology euclidean S) T \<and> a \<in> T \<and>
+                     (\<forall>x \<in> T. \<forall>y \<in> T. P x \<and> P y \<and> Q x \<longrightarrow> Q y)"
+      and etc: "a \<in> S" "b \<in> S" "P a" "P b" "Q a"
+    shows "Q b"
+proof -
+  have 1: "openin (subtopology euclidean S)
+             {b. \<exists>T. openin (subtopology euclidean S) T \<and>
+                     b \<in> T \<and> (\<forall>x\<in>T. P x \<longrightarrow> Q x)}"
+    apply (subst openin_subopen, clarify)
+    apply (rule_tac x=T in exI, auto)
+    done
+  have 2: "openin (subtopology euclidean S)
+             {b. \<exists>T. openin (subtopology euclidean S) T \<and>
+                     b \<in> T \<and> (\<forall>x\<in>T. P x \<longrightarrow> \<not> Q x)}"
+    apply (subst openin_subopen, clarify)
+    apply (rule_tac x=T in exI, auto)
+    done
+  show ?thesis
+    using \<open>connected S\<close>
+    apply (simp only: connected_openin HOL.not_ex HOL.de_Morgan_conj)
+    apply (elim disjE allE)
+         apply (blast intro: 1)
+        apply (blast intro: 2, simp_all)
+       apply clarify apply (metis opI)
+      using opD apply (blast intro: etc elim: dest:)
+     using opI etc apply meson+
+    done
+qed
+
+lemma connected_equivalence_relation_gen:
+  assumes "connected S"
+      and etc: "a \<in> S" "b \<in> S" "P a" "P b"
+      and trans: "\<And>x y z. \<lbrakk>R x y; R y z\<rbrakk> \<Longrightarrow> R x z"
+      and opD: "\<And>T a. \<lbrakk>openin (subtopology euclidean S) T; a \<in> T\<rbrakk> \<Longrightarrow> \<exists>z. z \<in> T \<and> P z"
+      and opI: "\<And>a. a \<in> S
+             \<Longrightarrow> \<exists>T. openin (subtopology euclidean S) T \<and> a \<in> T \<and>
+                     (\<forall>x \<in> T. \<forall>y \<in> T. P x \<and> P y \<longrightarrow> R x y)"
+    shows "R a b"
+proof -
+  have "\<And>a b c. \<lbrakk>a \<in> S; P a; b \<in> S; c \<in> S; P b; P c; R a b\<rbrakk> \<Longrightarrow> R a c"
+    apply (rule connected_induction [OF \<open>connected S\<close> 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 \<in> S" "b \<in> S" "P a"
+      and opI: "\<And>a. a \<in> S
+             \<Longrightarrow> \<exists>T. openin (subtopology euclidean S) T \<and> a \<in> T \<and>
+                     (\<forall>x \<in> T. \<forall>y \<in> T. P x \<longrightarrow> P y)"
+    shows "P b"
+apply (rule connected_induction [OF \<open>connected S\<close> _, where P = "\<lambda>x. True"], blast)
+apply (frule opI)
+using etc apply simp_all
+done
+
+lemma connected_equivalence_relation:
+  assumes "connected S"
+      and etc: "a \<in> S" "b \<in> S"
+      and sym: "\<And>x y. \<lbrakk>R x y; x \<in> S; y \<in> S\<rbrakk> \<Longrightarrow> R y x"
+      and trans: "\<And>x y z. \<lbrakk>R x y; R y z; x \<in> S; y \<in> S; z \<in> S\<rbrakk> \<Longrightarrow> R x z"
+      and opI: "\<And>a. a \<in> S \<Longrightarrow> \<exists>T. openin (subtopology euclidean S) T \<and> a \<in> T \<and> (\<forall>x \<in> T. R a x)"
+    shows "R a b"
+proof -
+  have "\<And>a b c. \<lbrakk>a \<in> S; b \<in> S; c \<in> S; R a b\<rbrakk> \<Longrightarrow> R a c"
+    apply (rule connected_induction_simple [OF \<open>connected S\<close>], simp_all)
+    by (meson local.sym local.trans opI openin_imp_subset subsetCE)
+  then show ?thesis by (metis etc opI)
+qed
+
+lemma locally_constant_imp_constant:
+  assumes "connected S"
+      and opI: "\<And>a. a \<in> S
+             \<Longrightarrow> \<exists>T. openin (subtopology euclidean S) T \<and> a \<in> T \<and> (\<forall>x \<in> T. f x = f a)"
+    shows "f constant_on S"
+proof -
+  have "\<And>x y. x \<in> S \<Longrightarrow> y \<in> S \<Longrightarrow> f x = f y"
+    apply (rule connected_equivalence_relation [OF \<open>connected S\<close>], simp_all)
+    by (metis opI)
+  then show ?thesis
+    by (metis constant_on_def)
+qed
+
+lemma locally_constant:
+     "connected S \<Longrightarrow> locally (\<lambda>U. f constant_on U) S \<longleftrightarrow> f constant_on S"
+apply (simp add: locally_def)
+apply (rule iffI)
+ apply (rule locally_constant_imp_constant, assumption)
+ apply (metis (mono_tags, hide_lams) constant_on_def constant_on_subset openin_subtopology_self)
+by (meson constant_on_subset openin_imp_subset order_refl)
+
+
+subsection\<open>Basic properties of local compactness\<close>
+
+proposition locally_compact:
+  fixes s :: "'a :: metric_space set"
+  shows
+    "locally compact s \<longleftrightarrow>
+     (\<forall>x \<in> s. \<exists>u v. x \<in> u \<and> u \<subseteq> v \<and> v \<subseteq> s \<and>
+                    openin (subtopology euclidean s) u \<and> compact v)"
+     (is "?lhs = ?rhs")
+proof
+  assume ?lhs
+  then show ?rhs
+    apply clarify
+    apply (erule_tac w = "s \<inter> ball x 1" in locallyE)
+    by auto
+next
+  assume r [rule_format]: ?rhs
+  have *: "\<exists>u v.
+              openin (subtopology euclidean s) u \<and>
+              compact v \<and> x \<in> u \<and> u \<subseteq> v \<and> v \<subseteq> s \<inter> T"
+          if "open T" "x \<in> s" "x \<in> T" for x T
+  proof -
+    obtain u v where uv: "x \<in> u" "u \<subseteq> v" "v \<subseteq> s" "compact v" "openin (subtopology euclidean s) u"
+      using r [OF \<open>x \<in> s\<close>] by auto
+    obtain e where "e>0" and e: "cball x e \<subseteq> T"
+      using open_contains_cball \<open>open T\<close> \<open>x \<in> T\<close> by blast
+    show ?thesis
+      apply (rule_tac x="(s \<inter> ball x e) \<inter> u" in exI)
+      apply (rule_tac x="cball x e \<inter> v" in exI)
+      using that \<open>e > 0\<close> e uv
+      apply auto
+      done
+  qed
+  show ?lhs
+    apply (rule locallyI)
+    apply (subst (asm) openin_open)
+    apply (blast intro: *)
+    done
+qed
+
+lemma locally_compactE:
+  fixes s :: "'a :: metric_space set"
+  assumes "locally compact s"
+  obtains u v where "\<And>x. x \<in> s \<Longrightarrow> x \<in> u x \<and> u x \<subseteq> v x \<and> v x \<subseteq> s \<and>
+                             openin (subtopology euclidean s) (u x) \<and> compact (v x)"
+using assms
+unfolding locally_compact by metis
+
+lemma locally_compact_alt:
+  fixes s :: "'a :: heine_borel set"
+  shows "locally compact s \<longleftrightarrow>
+         (\<forall>x \<in> s. \<exists>u. x \<in> u \<and>
+                    openin (subtopology euclidean s) u \<and> compact(closure u) \<and> closure u \<subseteq> s)"
+apply (simp add: locally_compact)
+apply (intro ball_cong ex_cong refl iffI)
+apply (metis bounded_subset closure_eq closure_mono compact_eq_bounded_closed dual_order.trans)
+by (meson closure_subset compact_closure)
+
+lemma locally_compact_Int_cball:
+  fixes s :: "'a :: heine_borel set"
+  shows "locally compact s \<longleftrightarrow> (\<forall>x \<in> s. \<exists>e. 0 < e \<and> closed(cball x e \<inter> s))"
+        (is "?lhs = ?rhs")
+proof
+  assume ?lhs
+  then show ?rhs
+    apply (simp add: locally_compact openin_contains_cball)
+    apply (clarify | assumption | drule bspec)+
+    by (metis (no_types, lifting)  compact_cball compact_imp_closed compact_Int inf.absorb_iff2 inf.orderE inf_sup_aci(2))
+next
+  assume ?rhs
+  then show ?lhs
+    apply (simp add: locally_compact openin_contains_cball)
+    apply (clarify | assumption | drule bspec)+
+    apply (rule_tac x="ball x e \<inter> s" in exI, simp)
+    apply (rule_tac x="cball x e \<inter> s" in exI)
+    using compact_eq_bounded_closed
+    apply auto
+    apply (metis open_ball le_infI1 mem_ball open_contains_cball_eq)
+    done
+qed
+
+lemma locally_compact_compact:
+  fixes s :: "'a :: heine_borel set"
+  shows "locally compact s \<longleftrightarrow>
+         (\<forall>k. k \<subseteq> s \<and> compact k
+              \<longrightarrow> (\<exists>u v. k \<subseteq> u \<and> u \<subseteq> v \<and> v \<subseteq> s \<and>
+                         openin (subtopology euclidean s) u \<and> compact v))"
+        (is "?lhs = ?rhs")
+proof
+  assume ?lhs
+  then obtain u v where
+    uv: "\<And>x. x \<in> s \<Longrightarrow> x \<in> u x \<and> u x \<subseteq> v x \<and> v x \<subseteq> s \<and>
+                             openin (subtopology euclidean s) (u x) \<and> compact (v x)"
+    by (metis locally_compactE)
+  have *: "\<exists>u v. k \<subseteq> u \<and> u \<subseteq> v \<and> v \<subseteq> s \<and> openin (subtopology euclidean s) u \<and> compact v"
+          if "k \<subseteq> s" "compact k" for k
+  proof -
+    have "\<And>C. (\<forall>c\<in>C. openin (subtopology euclidean k) c) \<and> k \<subseteq> \<Union>C \<Longrightarrow>
+                    \<exists>D\<subseteq>C. finite D \<and> k \<subseteq> \<Union>D"
+      using that by (simp add: compact_eq_openin_cover)
+    moreover have "\<forall>c \<in> (\<lambda>x. k \<inter> u x) ` k. openin (subtopology euclidean k) c"
+      using that by clarify (metis subsetD inf.absorb_iff2 openin_subset openin_subtopology_Int_subset topspace_euclidean_subtopology uv)
+    moreover have "k \<subseteq> \<Union>((\<lambda>x. k \<inter> u x) ` k)"
+      using that by clarsimp (meson subsetCE uv)
+    ultimately obtain D where "D \<subseteq> (\<lambda>x. k \<inter> u x) ` k" "finite D" "k \<subseteq> \<Union>D"
+      by metis
+    then obtain T where T: "T \<subseteq> k" "finite T" "k \<subseteq> \<Union>((\<lambda>x. k \<inter> u x) ` T)"
+      by (metis finite_subset_image)
+    have Tuv: "\<Union>(u ` T) \<subseteq> \<Union>(v ` T)"
+      using T that by (force simp: dest!: uv)
+    show ?thesis
+      apply (rule_tac x="\<Union>(u ` T)" in exI)
+      apply (rule_tac x="\<Union>(v ` T)" in exI)
+      apply (simp add: Tuv)
+      using T that
+      apply (auto simp: dest!: uv)
+      done
+  qed
+  show ?rhs
+    by (blast intro: *)
+next
+  assume ?rhs
+  then show ?lhs
+    apply (clarsimp simp add: 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 *: "\<exists>u v. x \<in> u \<and> u \<subseteq> v \<and> v \<subseteq> s \<and> openin (subtopology euclidean s) u \<and> compact v"
+          if "x \<in> s" for x
+  proof -
+    obtain e where "e>0" and e: "cball x e \<subseteq> s"
+      using open_contains_cball assms \<open>x \<in> s\<close> by blast
+    have ope: "openin (subtopology euclidean s) (ball x e)"
+      by (meson e open_ball ball_subset_cball dual_order.trans open_subset)
+    show ?thesis
+      apply (rule_tac x="ball x e" in exI)
+      apply (rule_tac x="cball x e" in exI)
+      using \<open>e > 0\<close> e apply (auto simp: ope)
+      done
+  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 *: "\<exists>u v. x \<in> u \<and> u \<subseteq> v \<and> v \<subseteq> s \<and>
+                 openin (subtopology euclidean s) u \<and> compact v"
+          if "x \<in> s" for x
+  proof -
+    show ?thesis
+      apply (rule_tac x = "s \<inter> ball x 1" in exI)
+      apply (rule_tac x = "s \<inter> cball x 1" in exI)
+      using \<open>x \<in> s\<close> assms apply auto
+      done
+  qed
+  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 "\<lbrakk>locally compact s; locally compact t\<rbrakk> \<Longrightarrow> locally compact (s \<inter> t)"
+by (simp add: compact_Int locally_Int)
+
+lemma locally_compact_closedin:
+  fixes s :: "'a :: heine_borel set"
+  shows "\<lbrakk>closedin (subtopology euclidean s) t; locally compact s\<rbrakk>
+        \<Longrightarrow> 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 \<Longrightarrow> 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 \<longleftrightarrow> locally compact s"
+        (is "?lhs = ?rhs")
+proof
+  assume ?lhs
+  then show ?rhs
+    apply (simp only: locally_def)
+    apply (erule all_forward imp_forward asm_rl exE)+
+    apply (rule_tac x = "u \<inter> ball x 1" in exI)
+    apply (rule_tac x = "v \<inter> 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 (subtopology euclidean (S \<union> T)) S"
+      and opT: "openin (subtopology euclidean (S \<union> T)) T"
+    shows "locally compact (S \<union> T)"
+proof -
+  have "\<exists>e>0. closed (cball x e \<inter> (S \<union> T))" if "x \<in> S" for x
+  proof -
+    obtain e1 where "e1 > 0" and e1: "closed (cball x e1 \<inter> S)"
+      using LCS \<open>x \<in> S\<close> unfolding locally_compact_Int_cball by blast
+    moreover obtain e2 where "e2 > 0" and e2: "cball x e2 \<inter> (S \<union> T) \<subseteq> S"
+      by (meson \<open>x \<in> S\<close> opS openin_contains_cball)
+    then have "cball x e2 \<inter> (S \<union> T) = cball x e2 \<inter> S"
+      by force
+    ultimately show ?thesis
+      apply (rule_tac x="min e1 e2" in exI)
+      apply (auto simp: cball_min_Int \<open>e2 > 0\<close> inf_assoc closed_Int)
+      by (metis closed_Int closed_cball inf_left_commute)
+  qed
+  moreover have "\<exists>e>0. closed (cball x e \<inter> (S \<union> T))" if "x \<in> T" for x
+  proof -
+    obtain e1 where "e1 > 0" and e1: "closed (cball x e1 \<inter> T)"
+      using LCT \<open>x \<in> T\<close> unfolding locally_compact_Int_cball by blast
+    moreover obtain e2 where "e2 > 0" and e2: "cball x e2 \<inter> (S \<union> T) \<subseteq> T"
+      by (meson \<open>x \<in> T\<close> opT openin_contains_cball)
+    then have "cball x e2 \<inter> (S \<union> T) = cball x e2 \<inter> T"
+      by force
+    ultimately show ?thesis
+      apply (rule_tac x="min e1 e2" in exI)
+      apply (auto simp: cball_min_Int \<open>e2 > 0\<close> inf_assoc closed_Int)
+      by (metis closed_Int closed_cball inf_left_commute)
+  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 (subtopology euclidean (S \<union> T)) S"
+      and clT: "closedin (subtopology euclidean (S \<union> T)) T"
+    shows "locally compact (S \<union> T)"
+proof -
+  have "\<exists>e>0. closed (cball x e \<inter> (S \<union> T))" if "x \<in> S" "x \<in> T" for x
+  proof -
+    obtain e1 where "e1 > 0" and e1: "closed (cball x e1 \<inter> S)"
+      using LCS \<open>x \<in> S\<close> unfolding locally_compact_Int_cball by blast
+    moreover
+    obtain e2 where "e2 > 0" and e2: "closed (cball x e2 \<inter> T)"
+      using LCT \<open>x \<in> T\<close> unfolding locally_compact_Int_cball by blast
+    ultimately show ?thesis
+      apply (rule_tac x="min e1 e2" in exI)
+      apply (auto simp: cball_min_Int \<open>e2 > 0\<close> inf_assoc closed_Int Int_Un_distrib)
+      by (metis closed_Int closed_Un closed_cball inf_left_commute)
+  qed
+  moreover
+  have "\<exists>e>0. closed (cball x e \<inter> (S \<union> T))" if x: "x \<in> S" "x \<notin> T" for x
+  proof -
+    obtain e1 where "e1 > 0" and e1: "closed (cball x e1 \<inter> S)"
+      using LCS \<open>x \<in> S\<close> unfolding locally_compact_Int_cball by blast
+    moreover
+    obtain e2 where "e2>0" and "cball x e2 \<inter> (S \<union> T) \<subseteq> S - T"
+      using clT x by (fastforce simp: openin_contains_cball closedin_def)
+    then have "closed (cball x e2 \<inter> T)"
+    proof -
+      have "{} = T - (T - cball x e2)"
+        using Diff_subset Int_Diff \<open>cball x e2 \<inter> (S \<union> T) \<subseteq> S - T\<close> by auto
+      then show ?thesis
+        by (simp add: Diff_Diff_Int inf_commute)
+    qed
+    ultimately show ?thesis
+      apply (rule_tac x="min e1 e2" in exI)
+      apply (auto simp: cball_min_Int \<open>e2 > 0\<close> inf_assoc closed_Int Int_Un_distrib)
+      by (metis closed_Int closed_Un closed_cball inf_left_commute)
+  qed
+  moreover
+  have "\<exists>e>0. closed (cball x e \<inter> (S \<union> T))" if x: "x \<notin> S" "x \<in> T" for x
+  proof -
+    obtain e1 where "e1 > 0" and e1: "closed (cball x e1 \<inter> T)"
+      using LCT \<open>x \<in> T\<close> unfolding locally_compact_Int_cball by blast
+    moreover
+    obtain e2 where "e2>0" and "cball x e2 \<inter> (S \<union> T) \<subseteq> S \<union> T - S"
+      using clS x by (fastforce simp: openin_contains_cball closedin_def)
+    then have "closed (cball x e2 \<inter> S)"
+      by (metis Diff_disjoint Int_empty_right closed_empty inf.left_commute inf.orderE inf_sup_absorb)
+    ultimately show ?thesis
+      apply (rule_tac x="min e1 e2" in exI)
+      apply (auto simp: cball_min_Int \<open>e2 > 0\<close> inf_assoc closed_Int Int_Un_distrib)
+      by (metis closed_Int closed_Un closed_cball inf_left_commute)
+  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 "\<lbrakk>locally compact S; locally compact T\<rbrakk> \<Longrightarrow> locally compact (S \<times> T)"
+  by (auto simp: compact_Times locally_Times)
+
+lemma locally_compact_compact_subopen:
+  fixes S :: "'a :: heine_borel set"
+  shows
+   "locally compact S \<longleftrightarrow>
+    (\<forall>K T. K \<subseteq> S \<and> compact K \<and> open T \<and> K \<subseteq> T
+          \<longrightarrow> (\<exists>U V. K \<subseteq> U \<and> U \<subseteq> V \<and> U \<subseteq> T \<and> V \<subseteq> S \<and>
+                     openin (subtopology euclidean S) U \<and> compact V))"
+   (is "?lhs = ?rhs")
+proof
+  assume L: ?lhs
+  show ?rhs
+  proof clarify
+    fix K :: "'a set" and T :: "'a set"
+    assume "K \<subseteq> S" and "compact K" and "open T" and "K \<subseteq> T"
+    obtain U V where "K \<subseteq> U" "U \<subseteq> V" "V \<subseteq> S" "compact V"
+                 and ope: "openin (subtopology euclidean S) U"
+      using L unfolding locally_compact_compact by (meson \<open>K \<subseteq> S\<close> \<open>compact K\<close>)
+    show "\<exists>U V. K \<subseteq> U \<and> U \<subseteq> V \<and> U \<subseteq> T \<and> V \<subseteq> S \<and>
+                openin (subtopology euclidean S) U \<and> compact V"
+    proof (intro exI conjI)
+      show "K \<subseteq> U \<inter> T"
+        by (simp add: \<open>K \<subseteq> T\<close> \<open>K \<subseteq> U\<close>)
+      show "U \<inter> T \<subseteq> closure(U \<inter> T)"
+        by (rule closure_subset)
+      show "closure (U \<inter> T) \<subseteq> S"
+        by (metis \<open>U \<subseteq> V\<close> \<open>V \<subseteq> S\<close> \<open>compact V\<close> closure_closed closure_mono compact_imp_closed inf.cobounded1 subset_trans)
+      show "openin (subtopology euclidean S) (U \<inter> T)"
+        by (simp add: \<open>open T\<close> ope openin_Int_open)
+      show "compact (closure (U \<inter> T))"
+        by (meson Int_lower1 \<open>U \<subseteq> V\<close> \<open>compact V\<close> 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\<open>Sura-Bura's results about compact components of sets\<close>
+
+proposition Sura_Bura_compact:
+  fixes S :: "'a::euclidean_space set"
+  assumes "compact S" and C: "C \<in> components S"
+  shows "C = \<Inter>{T. C \<subseteq> T \<and> openin (subtopology euclidean S) T \<and>
+                           closedin (subtopology euclidean S) T}"
+         (is "C = \<Inter>?\<T>")
+proof
+  obtain x where x: "C = connected_component_set S x" and "x \<in> S"
+    using C by (auto simp: components_def)
+  have "C \<subseteq> S"
+    by (simp add: C in_components_subset)
+  have "\<Inter>?\<T> \<subseteq> connected_component_set S x"
+  proof (rule connected_component_maximal)
+    have "x \<in> C"
+      by (simp add: \<open>x \<in> S\<close> x)
+    then show "x \<in> \<Inter>?\<T>"
+      by blast
+    have clo: "closed (\<Inter>?\<T>)"
+      by (simp add: \<open>compact S\<close> closed_Inter closedin_compact_eq compact_imp_closed)
+    have False
+      if K1: "closedin (subtopology euclidean (\<Inter>?\<T>)) K1" and
+         K2: "closedin (subtopology euclidean (\<Inter>?\<T>)) K2" and
+         K12_Int: "K1 \<inter> K2 = {}" and K12_Un: "K1 \<union> K2 = \<Inter>?\<T>" and "K1 \<noteq> {}" "K2 \<noteq> {}"
+       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 \<subseteq> V1" "K2 \<subseteq> V2" and V12: "V1 \<inter> V2 = {}"
+        using separation_normal \<open>K1 \<inter> K2 = {}\<close> by metis
+      have SV12_ne: "(S - (V1 \<union> V2)) \<inter> (\<Inter>?\<T>) \<noteq> {}"
+      proof (rule compact_imp_fip)
+        show "compact (S - (V1 \<union> V2))"
+          by (simp add: \<open>open V1\<close> \<open>open V2\<close> \<open>compact S\<close> compact_diff open_Un)
+        show clo\<T>: "closed T" if "T \<in> ?\<T>" for T
+          using that \<open>compact S\<close>
+          by (force intro: closedin_closed_trans simp add: compact_imp_closed)
+        show "(S - (V1 \<union> V2)) \<inter> \<Inter>\<F> \<noteq> {}" if "finite \<F>" and \<F>: "\<F> \<subseteq> ?\<T>" for \<F>
+        proof
+          assume djo: "(S - (V1 \<union> V2)) \<inter> \<Inter>\<F> = {}"
+          obtain D where opeD: "openin (subtopology euclidean S) D"
+                   and cloD: "closedin (subtopology euclidean S) D"
+                   and "C \<subseteq> D" and DV12: "D \<subseteq> V1 \<union> V2"
+          proof (cases "\<F> = {}")
+            case True
+            with \<open>C \<subseteq> S\<close> djo that show ?thesis
+              by force
+          next
+            case False show ?thesis
+            proof
+              show ope: "openin (subtopology euclidean S) (\<Inter>\<F>)"
+                using openin_Inter \<open>finite \<F>\<close> False \<F> by blast
+              then show "closedin (subtopology euclidean S) (\<Inter>\<F>)"
+                by (meson clo\<T> \<F> closed_Inter closed_subset openin_imp_subset subset_eq)
+              show "C \<subseteq> \<Inter>\<F>"
+                using \<F> by auto
+              show "\<Inter>\<F> \<subseteq> V1 \<union> V2"
+                using ope djo openin_imp_subset by fastforce
+            qed
+          qed
+          have "connected C"
+            by (simp add: x)
+          have "closed D"
+            using \<open>compact S\<close> cloD closedin_closed_trans compact_imp_closed by blast
+          have cloV1: "closedin (subtopology euclidean D) (D \<inter> closure V1)"
+            and cloV2: "closedin (subtopology euclidean D) (D \<inter> closure V2)"
+            by (simp_all add: closedin_closed_Int)
+          moreover have "D \<inter> closure V1 = D \<inter> V1" "D \<inter> closure V2 = D \<inter> V2"
+            apply safe
+            using \<open>D \<subseteq> V1 \<union> V2\<close> \<open>open V1\<close> \<open>open V2\<close> V12
+               apply (simp_all add: closure_subset [THEN subsetD] closure_iff_nhds_not_empty, blast+)
+            done
+          ultimately have cloDV1: "closedin (subtopology euclidean D) (D \<inter> V1)"
+                      and cloDV2:  "closedin (subtopology euclidean D) (D \<inter> V2)"
+            by metis+
+          then obtain U1 U2 where "closed U1" "closed U2"
+               and D1: "D \<inter> V1 = D \<inter> U1" and D2: "D \<inter> V2 = D \<inter> U2"
+            by (auto simp: closedin_closed)
+          have "D \<inter> U1 \<inter> C \<noteq> {}"
+          proof
+            assume "D \<inter> U1 \<inter> C = {}"
+            then have *: "C \<subseteq> D \<inter> V2"
+              using D1 DV12 \<open>C \<subseteq> D\<close> by auto
+            have "\<Inter>?\<T> \<subseteq> D \<inter> V2"
+              apply (rule Inter_lower)
+              using * apply simp
+              by (meson cloDV2 \<open>open V2\<close> cloD closedin_trans le_inf_iff opeD openin_Int_open)
+            then show False
+              using K1 V12 \<open>K1 \<noteq> {}\<close> \<open>K1 \<subseteq> V1\<close> closedin_imp_subset by blast
+          qed
+          moreover have "D \<inter> U2 \<inter> C \<noteq> {}"
+          proof
+            assume "D \<inter> U2 \<inter> C = {}"
+            then have *: "C \<subseteq> D \<inter> V1"
+              using D2 DV12 \<open>C \<subseteq> D\<close> by auto
+            have "\<Inter>?\<T> \<subseteq> D \<inter> V1"
+              apply (rule Inter_lower)
+              using * apply simp
+              by (meson cloDV1 \<open>open V1\<close> cloD closedin_trans le_inf_iff opeD openin_Int_open)
+            then show False
+              using K2 V12 \<open>K2 \<noteq> {}\<close> \<open>K2 \<subseteq> V2\<close> closedin_imp_subset by blast
+          qed
+          ultimately show False
+            using \<open>connected C\<close> unfolding connected_closed
+            apply (simp only: not_ex)
+            apply (drule_tac x="D \<inter> U1" in spec)
+            apply (drule_tac x="D \<inter> U2" in spec)
+            using \<open>C \<subseteq> D\<close> D1 D2 V12 DV12 \<open>closed U1\<close> \<open>closed U2\<close> \<open>closed D\<close>
+            by blast
+        qed
+      qed
+      show False
+        by (metis (full_types) DiffE UnE Un_upper2 SV12_ne \<open>K1 \<subseteq> V1\<close> \<open>K2 \<subseteq> V2\<close> disjoint_iff_not_equal subsetCE sup_ge1 K12_Un)
+    qed
+    then show "connected (\<Inter>?\<T>)"
+      by (auto simp: connected_closedin_eq)
+    show "\<Inter>?\<T> \<subseteq> S"
+      by (fastforce simp: C in_components_subset)
+  qed
+  with x show "\<Inter>?\<T> \<subseteq> C" by simp
+qed auto
+
+
+corollary Sura_Bura_clopen_subset:
+  fixes S :: "'a::euclidean_space set"
+  assumes S: "locally compact S" and C: "C \<in> components S" and "compact C"
+      and U: "open U" "C \<subseteq> U"
+  obtains K where "openin (subtopology euclidean S) K" "compact K" "C \<subseteq> K" "K \<subseteq> U"
+proof (rule ccontr)
+  assume "\<not> thesis"
+  with that have neg: "\<nexists>K. openin (subtopology euclidean S) K \<and> compact K \<and> C \<subseteq> K \<and> K \<subseteq> U"
+    by metis
+  obtain V K where "C \<subseteq> V" "V \<subseteq> U" "V \<subseteq> K" "K \<subseteq> S" "compact K"
+               and opeSV: "openin (subtopology euclidean S) V"
+    using S U \<open>compact C\<close>
+    apply (simp add: locally_compact_compact_subopen)
+    by (meson C in_components_subset)
+  let ?\<T> = "{T. C \<subseteq> T \<and> openin (subtopology euclidean K) T \<and> compact T \<and> T \<subseteq> K}"
+  have CK: "C \<in> components K"
+    by (meson C \<open>C \<subseteq> V\<close> \<open>K \<subseteq> S\<close> \<open>V \<subseteq> K\<close> components_intermediate_subset subset_trans)
+  with \<open>compact K\<close>
+  have "C = \<Inter>{T. C \<subseteq> T \<and> openin (subtopology euclidean K) T \<and> closedin (subtopology euclidean K) T}"
+    by (simp add: Sura_Bura_compact)
+  then have Ceq: "C = \<Inter>?\<T>"
+    by (simp add: closedin_compact_eq \<open>compact K\<close>)
+  obtain W where "open W" and W: "V = S \<inter> W"
+    using opeSV by (auto simp: openin_open)
+  have "-(U \<inter> W) \<inter> \<Inter>?\<T> \<noteq> {}"
+  proof (rule closed_imp_fip_compact)
+    show "- (U \<inter> W) \<inter> \<Inter>\<F> \<noteq> {}"
+      if "finite \<F>" and \<F>: "\<F> \<subseteq> ?\<T>" for \<F>
+    proof (cases "\<F> = {}")
+      case True
+      have False if "U = UNIV" "W = UNIV"
+      proof -
+        have "V = S"
+          by (simp add: W \<open>W = UNIV\<close>)
+        with neg show False
+          using \<open>C \<subseteq> V\<close> \<open>K \<subseteq> S\<close> \<open>V \<subseteq> K\<close> \<open>V \<subseteq> U\<close> \<open>compact K\<close> by auto
+      qed
+      with True show ?thesis
+        by auto
+    next
+      case False
+      show ?thesis
+      proof
+        assume "- (U \<inter> W) \<inter> \<Inter>\<F> = {}"
+        then have FUW: "\<Inter>\<F> \<subseteq> U \<inter> W"
+          by blast
+        have "C \<subseteq> \<Inter>\<F>"
+          using \<F> by auto
+        moreover have "compact (\<Inter>\<F>)"
+          by (metis (no_types, lifting) compact_Inter False mem_Collect_eq subsetCE \<F>)
+        moreover have "\<Inter>\<F> \<subseteq> K"
+          using False that(2) by fastforce
+        moreover have opeKF: "openin (subtopology euclidean K) (\<Inter>\<F>)"
+          using False \<F> \<open>finite \<F>\<close> by blast
+        then have opeVF: "openin (subtopology euclidean V) (\<Inter>\<F>)"
+          using W \<open>K \<subseteq> S\<close> \<open>V \<subseteq> K\<close> opeKF \<open>\<Inter>\<F> \<subseteq> K\<close> FUW openin_subset_trans by fastforce
+        then have "openin (subtopology euclidean S) (\<Inter>\<F>)"
+          by (metis opeSV openin_trans)
+        moreover have "\<Inter>\<F> \<subseteq> U"
+          by (meson \<open>V \<subseteq> U\<close> opeVF dual_order.trans openin_imp_subset)
+        ultimately show False
+          using neg by blast
+      qed
+    qed
+  qed (use \<open>open W\<close> \<open>open U\<close> in auto)
+  with W Ceq \<open>C \<subseteq> V\<close> \<open>C \<subseteq> U\<close> 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 \<in> components S" and "compact C"
+      and opeSU: "openin (subtopology euclidean S) U" and "C \<subseteq> U"
+  obtains K where "openin (subtopology euclidean S) K" "compact K" "C \<subseteq> K" "K \<subseteq> U"
+proof -
+  obtain V where "open V" "U = S \<inter> V"
+    using opeSU by (auto simp: openin_open)
+  with \<open>C \<subseteq> U\<close> have "C \<subseteq> V"
+    by auto
+  then show ?thesis
+    using Sura_Bura_clopen_subset [OF S C \<open>compact C\<close> \<open>open V\<close>]
+    by (metis \<open>U = S \<inter> V\<close> inf.bounded_iff openin_imp_subset that)
+qed
+
+corollary Sura_Bura:
+  fixes S :: "'a::euclidean_space set"
+  assumes "locally compact S" "C \<in> components S" "compact C"
+  shows "C = \<Inter> {K. C \<subseteq> K \<and> compact K \<and> openin (subtopology euclidean S) K}"
+         (is "C = ?rhs")
+proof
+  show "?rhs \<subseteq> C"
+  proof (clarsimp, rule ccontr)
+    fix x
+    assume *: "\<forall>X. C \<subseteq> X \<and> compact X \<and> openin (subtopology euclidean S) X \<longrightarrow> x \<in> X"
+      and "x \<notin> C"
+    obtain U V where "open U" "open V" "{x} \<subseteq> U" "C \<subseteq> V" "U \<inter> V = {}"
+      using separation_normal [of "{x}" C]
+      by (metis Int_empty_left \<open>x \<notin> C\<close> \<open>compact C\<close> closed_empty closed_insert compact_imp_closed insert_disjoint(1))
+    have "x \<notin> V"
+      using \<open>U \<inter> V = {}\<close> \<open>{x} \<subseteq> U\<close> by blast
+    then show False
+      by (meson "*" Sura_Bura_clopen_subset \<open>C \<subseteq> V\<close> \<open>open V\<close> assms(1) assms(2) assms(3) subsetCE)
+  qed
+qed blast
+
+
+subsection\<open>Special cases of local connectedness and path connectedness\<close>
+
+lemma locally_connected_1:
+  assumes
+    "\<And>v x. \<lbrakk>openin (subtopology euclidean S) v; x \<in> v\<rbrakk>
+              \<Longrightarrow> \<exists>u. openin (subtopology euclidean S) u \<and>
+                      connected u \<and> x \<in> u \<and> u \<subseteq> v"
+   shows "locally connected S"
+apply (clarsimp simp add: locally_def)
+apply (drule assms; blast)
+done
+
+lemma locally_connected_2:
+  assumes "locally connected S"
+          "openin (subtopology euclidean S) t"
+          "x \<in> t"
+   shows "openin (subtopology euclidean S) (connected_component_set t x)"
+proof -
+  { fix y :: 'a
+    let ?SS = "subtopology euclidean S"
+    assume 1: "openin ?SS t"
+              "\<forall>w x. openin ?SS w \<and> x \<in> w \<longrightarrow> (\<exists>u. openin ?SS u \<and> (\<exists>v. connected v \<and> x \<in> u \<and> u \<subseteq> v \<and> v \<subseteq> w))"
+    and "connected_component t x y"
+    then have "y \<in> t" and y: "y \<in> connected_component_set t x"
+      using connected_component_subset by blast+
+    obtain F where
+      "\<forall>x y. (\<exists>w. openin ?SS w \<and> (\<exists>u. connected u \<and> x \<in> w \<and> w \<subseteq> u \<and> u \<subseteq> y)) = (openin ?SS (F x y) \<and> (\<exists>u. connected u \<and> x \<in> F x y \<and> F x y \<subseteq> u \<and> u \<subseteq> y))"
+      by moura
+    then obtain G where
+       "\<forall>a A. (\<exists>U. openin ?SS U \<and> (\<exists>V. connected V \<and> a \<in> U \<and> U \<subseteq> V \<and> V \<subseteq> A)) = (openin ?SS (F a A) \<and> connected (G a A) \<and> a \<in> F a A \<and> F a A \<subseteq> G a A \<and> G a A \<subseteq> A)"
+      by moura
+    then have *: "openin ?SS (F y t) \<and> connected (G y t) \<and> y \<in> F y t \<and> F y t \<subseteq> G y t \<and> G y t \<subseteq> t"
+      using 1 \<open>y \<in> t\<close> by presburger
+    have "G y t \<subseteq> connected_component_set t y"
+      by (metis (no_types) * connected_component_eq_self connected_component_mono contra_subsetD)
+    then have "\<exists>A. openin ?SS A \<and> y \<in> A \<and> A \<subseteq> 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 "\<And>t x. \<lbrakk>openin (subtopology euclidean S) t; x \<in> t\<rbrakk>
+              \<Longrightarrow> openin (subtopology euclidean S)
+                          (connected_component_set t x)"
+          "openin (subtopology euclidean S) v" "x \<in> v"
+   shows  "\<exists>u. openin (subtopology euclidean S) u \<and> connected u \<and> x \<in> u \<and> u \<subseteq> v"
+using assms connected_component_subset by fastforce
+
+lemma locally_connected:
+  "locally connected S \<longleftrightarrow>
+   (\<forall>v x. openin (subtopology euclidean S) v \<and> x \<in> v
+          \<longrightarrow> (\<exists>u. openin (subtopology euclidean S) u \<and> connected u \<and> x \<in> u \<and> u \<subseteq> v))"
+by (metis locally_connected_1 locally_connected_2 locally_connected_3)
+
+lemma locally_connected_open_connected_component:
+  "locally connected S \<longleftrightarrow>
+   (\<forall>t x. openin (subtopology euclidean S) t \<and> x \<in> t
+          \<longrightarrow> openin (subtopology euclidean S) (connected_component_set t x))"
+by (metis locally_connected_1 locally_connected_2 locally_connected_3)
+
+lemma locally_path_connected_1:
+  assumes
+    "\<And>v x. \<lbrakk>openin (subtopology euclidean S) v; x \<in> v\<rbrakk>
+              \<Longrightarrow> \<exists>u. openin (subtopology euclidean S) u \<and> path_connected u \<and> x \<in> u \<and> u \<subseteq> v"
+   shows "locally path_connected S"
+apply (clarsimp simp add: locally_def)
+apply (drule assms; blast)
+done
+
+lemma locally_path_connected_2:
+  assumes "locally path_connected S"
+          "openin (subtopology euclidean S) t"
+          "x \<in> t"
+   shows "openin (subtopology euclidean S) (path_component_set t x)"
+proof -
+  { fix y :: 'a
+    let ?SS = "subtopology euclidean S"
+    assume 1: "openin ?SS t"
+              "\<forall>w x. openin ?SS w \<and> x \<in> w \<longrightarrow> (\<exists>u. openin ?SS u \<and> (\<exists>v. path_connected v \<and> x \<in> u \<and> u \<subseteq> v \<and> v \<subseteq> w))"
+    and "path_component t x y"
+    then have "y \<in> t" and y: "y \<in> path_component_set t x"
+      using path_component_mem(2) by blast+
+    obtain F where
+      "\<forall>x y. (\<exists>w. openin ?SS w \<and> (\<exists>u. path_connected u \<and> x \<in> w \<and> w \<subseteq> u \<and> u \<subseteq> y)) = (openin ?SS (F x y) \<and> (\<exists>u. path_connected u \<and> x \<in> F x y \<and> F x y \<subseteq> u \<and> u \<subseteq> y))"
+      by moura
+    then obtain G where
+       "\<forall>a A. (\<exists>U. openin ?SS U \<and> (\<exists>V. path_connected V \<and> a \<in> U \<and> U \<subseteq> V \<and> V \<subseteq> A)) = (openin ?SS (F a A) \<and> path_connected (G a A) \<and> a \<in> F a A \<and> F a A \<subseteq> G a A \<and> G a A \<subseteq> A)"
+      by moura
+    then have *: "openin ?SS (F y t) \<and> path_connected (G y t) \<and> y \<in> F y t \<and> F y t \<subseteq> G y t \<and> G y t \<subseteq> t"
+      using 1 \<open>y \<in> t\<close> by presburger
+    have "G y t \<subseteq> path_component_set t y"
+      using * path_component_maximal set_rev_mp by blast
+    then have "\<exists>A. openin ?SS A \<and> y \<in> A \<and> A \<subseteq> path_component_set t x"
+      by (metis "*" \<open>G y t \<subseteq> path_component_set t y\<close> 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 "\<And>t x. \<lbrakk>openin (subtopology euclidean S) t; x \<in> t\<rbrakk>
+              \<Longrightarrow> openin (subtopology euclidean S) (path_component_set t x)"
+          "openin (subtopology euclidean S) v" "x \<in> v"
+   shows  "\<exists>u. openin (subtopology euclidean S) u \<and> path_connected u \<and> x \<in> u \<and> u \<subseteq> v"
+proof -
+  have "path_component v x x"
+    by (meson assms(3) path_component_refl)
+  then show ?thesis
+    by (metis assms(1) assms(2) assms(3) mem_Collect_eq path_component_subset path_connected_path_component)
+qed
+
+proposition locally_path_connected:
+  "locally path_connected S \<longleftrightarrow>
+   (\<forall>v x. openin (subtopology euclidean S) v \<and> x \<in> v
+          \<longrightarrow> (\<exists>u. openin (subtopology euclidean S) u \<and> path_connected u \<and> x \<in> u \<and> u \<subseteq> 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 \<longleftrightarrow>
+   (\<forall>t x. openin (subtopology euclidean S) t \<and> x \<in> t
+          \<longrightarrow> openin (subtopology euclidean 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 \<longleftrightarrow>
+   (\<forall>t c. openin (subtopology euclidean S) t \<and> c \<in> components t
+          \<longrightarrow> openin (subtopology euclidean S) c)"
+by (metis components_iff locally_connected_open_connected_component)
+
+proposition locally_connected_im_kleinen:
+  "locally connected S \<longleftrightarrow>
+   (\<forall>v x. openin (subtopology euclidean S) v \<and> x \<in> v
+       \<longrightarrow> (\<exists>u. openin (subtopology euclidean S) u \<and>
+                x \<in> u \<and> u \<subseteq> v \<and>
+                (\<forall>y. y \<in> u \<longrightarrow> (\<exists>c. connected c \<and> c \<subseteq> v \<and> x \<in> c \<and> y \<in> c))))"
+   (is "?lhs = ?rhs")
+proof
+  assume ?lhs
+  then show ?rhs
+    by (fastforce simp add: locally_connected)
+next
+  assume ?rhs
+  have *: "\<exists>T. openin (subtopology euclidean S) T \<and> x \<in> T \<and> T \<subseteq> c"
+       if "openin (subtopology euclidean S) t" and c: "c \<in> components t" and "x \<in> c" for t c x
+  proof -
+    from that \<open>?rhs\<close> [rule_format, of t x]
+    obtain u where u:
+      "openin (subtopology euclidean S) u \<and> x \<in> u \<and> u \<subseteq> t \<and>
+       (\<forall>y. y \<in> u \<longrightarrow> (\<exists>c. connected c \<and> c \<subseteq> t \<and> x \<in> c \<and> y \<in> c))"
+      using in_components_subset by auto
+    obtain F :: "'a set \<Rightarrow> 'a set \<Rightarrow> 'a" where
+      "\<forall>x y. (\<exists>z. z \<in> x \<and> y = connected_component_set x z) = (F x y \<in> x \<and> y = connected_component_set x (F x y))"
+      by moura
+    then have F: "F t c \<in> t \<and> c = connected_component_set t (F t c)"
+      by (meson components_iff c)
+    obtain G :: "'a set \<Rightarrow> 'a set \<Rightarrow> 'a" where
+        G: "\<forall>x y. (\<exists>z. z \<in> y \<and> z \<notin> x) = (G x y \<in> y \<and> G x y \<notin> x)"
+      by moura
+     have "G c u \<notin> u \<or> G c u \<in> 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
+    apply (clarsimp simp add: locally_connected_open_component)
+    apply (subst openin_subopen)
+    apply (blast intro: *)
+    done
+qed
+
+proposition locally_path_connected_im_kleinen:
+  "locally path_connected S \<longleftrightarrow>
+   (\<forall>v x. openin (subtopology euclidean S) v \<and> x \<in> v
+       \<longrightarrow> (\<exists>u. openin (subtopology euclidean S) u \<and>
+                x \<in> u \<and> u \<subseteq> v \<and>
+                (\<forall>y. y \<in> u \<longrightarrow> (\<exists>p. path p \<and> path_image p \<subseteq> v \<and>
+                                pathstart p = x \<and> 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 *: "\<exists>T. openin (subtopology euclidean S) T \<and>
+               x \<in> T \<and> T \<subseteq> path_component_set u z"
+       if "openin (subtopology euclidean S) u" and "z \<in> u" and c: "path_component u z x" for u z x
+  proof -
+    have "x \<in> u"
+      by (meson c path_component_mem(2))
+    with that \<open>?rhs\<close> [rule_format, of u x]
+    obtain U where U:
+      "openin (subtopology euclidean S) U \<and> x \<in> U \<and> U \<subseteq> u \<and>
+       (\<forall>y. y \<in> U \<longrightarrow> (\<exists>p. path p \<and> path_image p \<subseteq> u \<and> pathstart p = x \<and> pathfinish p = y))"
+       by blast
+    show ?thesis
+      apply (rule_tac x=U in exI)
+      apply (auto simp: U)
+      apply (metis U c path_component_trans path_component_def)
+      done
+  qed
+  show ?lhs
+    apply (clarsimp simp add: locally_path_connected_open_path_component)
+    apply (subst openin_subopen)
+    apply (blast intro: *)
+    done
+qed
+
+lemma locally_path_connected_imp_locally_connected:
+  "locally path_connected S \<Longrightarrow> locally connected S"
+using locally_mono path_connected_imp_connected by blast
+
+lemma locally_connected_components:
+  "\<lbrakk>locally connected S; c \<in> components S\<rbrakk> \<Longrightarrow> locally connected c"
+by (meson locally_connected_open_component locally_open_subset openin_subtopology_self)
+
+lemma locally_path_connected_components:
+  "\<lbrakk>locally path_connected S; c \<in> components S\<rbrakk> \<Longrightarrow> 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 \<Longrightarrow> 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"
+  shows "open S \<Longrightarrow> locally path_connected S"
+apply (rule locally_mono [of convex])
+apply (simp_all add: locally_def openin_open_eq convex_imp_path_connected)
+apply (meson open_ball centre_in_ball convex_ball openE order_trans)
+done
+
+lemma open_imp_locally_connected:
+  fixes S :: "'a :: real_normed_vector set"
+  shows "open S \<Longrightarrow> 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
+     \<Longrightarrow> openin (subtopology euclidean S) (connected_component_set S x)"
+apply (simp add: locally_connected_open_connected_component)
+by (metis connected_component_eq_empty connected_component_subset open_empty open_subset openin_subtopology_self)
+
+lemma openin_components_locally_connected:
+    "\<lbrakk>locally connected S; c \<in> components S\<rbrakk> \<Longrightarrow> openin (subtopology euclidean S) c"
+  using locally_connected_open_component openin_subtopology_self by blast
+
+lemma openin_path_component_locally_path_connected:
+  "locally path_connected S
+        \<Longrightarrow> openin (subtopology euclidean 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:
+    "locally path_connected S
+        \<Longrightarrow> closedin (subtopology euclidean S) (path_component_set S x)"
+apply  (simp add: closedin_def path_component_subset complement_path_component_Union)
+apply (rule openin_Union)
+using openin_path_component_locally_path_connected by auto
+
+lemma convex_imp_locally_path_connected:
+  fixes S :: "'a:: real_normed_vector set"
+  shows "convex S \<Longrightarrow> locally path_connected S"
+apply (clarsimp simp add: locally_path_connected)
+apply (subst (asm) openin_open)
+apply clarify
+apply (erule (1) openE)
+apply (rule_tac x = "S \<inter> ball x e" in exI)
+apply (force simp: convex_Int convex_imp_path_connected)
+done
+
+lemma convex_imp_locally_connected:
+  fixes S :: "'a:: real_normed_vector set"
+  shows "convex S \<Longrightarrow> locally connected S"
+  by (simp add: locally_path_connected_imp_locally_connected convex_imp_locally_path_connected)
+
+
+subsection\<open>Relations between components and path components\<close>
+
+lemma path_component_eq_connected_component:
+  assumes "locally path_connected S"
+    shows "(path_component S x = connected_component S x)"
+proof (cases "x \<in> S")
+  case True
+  have "openin (subtopology euclidean (connected_component_set S x)) (path_component_set S x)"
+    apply (rule openin_subset_trans [of S])
+    apply (intro conjI openin_path_component_locally_path_connected [OF assms])
+    using path_component_subset_connected_component   apply (auto simp: connected_component_subset)
+    done
+  moreover have "closedin (subtopology euclidean (connected_component_set S x)) (path_component_set S x)"
+    apply (rule closedin_subset_trans [of S])
+    apply (intro conjI closedin_path_component_locally_path_connected [OF assms])
+    using path_component_subset_connected_component   apply (auto simp: connected_component_subset)
+    done
+  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 \<Longrightarrow> (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 \<Longrightarrow> 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 \<Longrightarrow> 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 \<Longrightarrow> 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: "\<And>T. T \<subseteq> f ` S
+                \<Longrightarrow> openin (subtopology euclidean S) (S \<inter> f -` T) \<longleftrightarrow>
+                    openin (subtopology euclidean (f ` S)) T"
+    shows "locally connected (f ` S)"
+proof (clarsimp simp: locally_connected_open_component)
+  fix U C
+  assume opefSU: "openin (subtopology euclidean (f ` S)) U" and "C \<in> components U"
+  then have "C \<subseteq> U" "U \<subseteq> f ` S"
+    by (meson in_components_subset openin_imp_subset)+
+  then have "openin (subtopology euclidean (f ` S)) C \<longleftrightarrow>
+             openin (subtopology euclidean S) (S \<inter> f -` C)"
+    by (auto simp: oo)
+  moreover have "openin (subtopology euclidean S) (S \<inter> f -` C)"
+  proof (subst openin_subopen, clarify)
+    fix x
+    assume "x \<in> S" "f x \<in> C"
+    show "\<exists>T. openin (subtopology euclidean S) T \<and> x \<in> T \<and> T \<subseteq> (S \<inter> f -` C)"
+    proof (intro conjI exI)
+      show "openin (subtopology euclidean S) (connected_component_set (S \<inter> f -` U) x)"
+      proof (rule ccontr)
+        assume **: "\<not> openin (subtopology euclidean S) (connected_component_set (S \<inter> f -` U) x)"
+        then have "x \<notin> (S \<inter> f -` U)"
+          using \<open>U \<subseteq> f ` S\<close> 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 \<in> connected_component_set (S \<inter> f -` U) x"
+        using \<open>C \<subseteq> U\<close> \<open>f x \<in> C\<close> \<open>x \<in> S\<close> 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 \<inter> f -` U) x) f"
+        apply (rule continuous_on_subset)
+        using connected_component_subset apply blast
+        done
+      then have "connected (f ` connected_component_set (S \<inter> f -` U) x)"
+        by (rule connected_continuous_image [OF _ connected_connected_component])
+      moreover have "f ` connected_component_set (S \<inter> f -` U) x \<subseteq> U"
+        using connected_component_in by blast
+      moreover have "C \<inter> f ` connected_component_set (S \<inter> f -` U) x \<noteq> {}"
+        using \<open>C \<subseteq> U\<close> \<open>f x \<in> C\<close> \<open>x \<in> S\<close> by fastforce
+      ultimately have fC: "f ` (connected_component_set (S \<inter> f -` U) x) \<subseteq> C"
+        by (rule components_maximal [OF \<open>C \<in> components U\<close>])
+      have cUC: "connected_component_set (S \<inter> f -` U) x \<subseteq> (S \<inter> f -` C)"
+        using connected_component_subset fC by blast
+      have "connected_component_set (S \<inter> f -` U) x \<subseteq> connected_component_set (S \<inter> f -` C) x"
+      proof -
+        { assume "x \<in> connected_component_set (S \<inter> 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 "\<dots> \<subseteq> (S \<inter> f -` C)"
+        by (rule connected_component_subset)
+      finally show "connected_component_set (S \<inter> f -` U) x \<subseteq> (S \<inter> f -` C)" .
+    qed
+  qed
+  ultimately show "openin (subtopology euclidean (f ` S)) C"
+    by metis
+qed
+
+text\<open>The proof resembles that above but is not identical!\<close>
+proposition locally_path_connected_quotient_image:
+  assumes lcS: "locally path_connected S"
+      and oo: "\<And>T. T \<subseteq> f ` S
+                \<Longrightarrow> openin (subtopology euclidean S) (S \<inter> f -` T) \<longleftrightarrow> openin (subtopology euclidean (f ` S)) T"
+    shows "locally path_connected (f ` S)"
+proof (clarsimp simp: locally_path_connected_open_path_component)
+  fix U y
+  assume opefSU: "openin (subtopology euclidean (f ` S)) U" and "y \<in> U"
+  then have "path_component_set U y \<subseteq> U" "U \<subseteq> f ` S"
+    by (meson path_component_subset openin_imp_subset)+
+  then have "openin (subtopology euclidean (f ` S)) (path_component_set U y) \<longleftrightarrow>
+             openin (subtopology euclidean S) (S \<inter> f -` path_component_set U y)"
+  proof -
+    have "path_component_set U y \<subseteq> f ` S"
+      using \<open>U \<subseteq> f ` S\<close> \<open>path_component_set U y \<subseteq> U\<close> by blast
+    then show ?thesis
+      using oo by blast
+  qed
+  moreover have "openin (subtopology euclidean S) (S \<inter> f -` path_component_set U y)"
+  proof (subst openin_subopen, clarify)
+    fix x
+    assume "x \<in> S" and Uyfx: "path_component U y (f x)"
+    then have "f x \<in> U"
+      using path_component_mem by blast
+    show "\<exists>T. openin (subtopology euclidean S) T \<and> x \<in> T \<and> T \<subseteq> (S \<inter> f -` path_component_set U y)"
+    proof (intro conjI exI)
+      show "openin (subtopology euclidean S) (path_component_set (S \<inter> f -` U) x)"
+      proof (rule ccontr)
+        assume **: "\<not> openin (subtopology euclidean S) (path_component_set (S \<inter> f -` U) x)"
+        then have "x \<notin> (S \<inter> f -` U)"
+          by (metis (no_types, lifting) \<open>U \<subseteq> f ` S\<close> opefSU lcS oo locally_path_connected_open_path_component)
+        then show False
+          using ** \<open>path_component_set U y \<subseteq> U\<close>  \<open>x \<in> S\<close> \<open>path_component U y (f x)\<close> by blast
+      qed
+    next
+      show "x \<in> path_component_set (S \<inter> f -` U) x"
+        by (simp add: \<open>f x \<in> U\<close> \<open>x \<in> S\<close> 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 \<inter> f -` U) x) f"
+        apply (rule continuous_on_subset)
+        using path_component_subset apply blast
+        done
+      then have "path_connected (f ` path_component_set (S \<inter> f -` U) x)"
+        by (simp add: path_connected_continuous_image)
+      moreover have "f ` path_component_set (S \<inter> f -` U) x \<subseteq> U"
+        using path_component_mem by fastforce
+      moreover have "f x \<in> f ` path_component_set (S \<inter> f -` U) x"
+        by (force simp: \<open>x \<in> S\<close> \<open>f x \<in> U\<close> path_component_refl_eq)
+      ultimately have "f ` (path_component_set (S \<inter> f -` U) x) \<subseteq> path_component_set U (f x)"
+        by (meson path_component_maximal)
+       also have  "\<dots> \<subseteq> 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 \<inter> f -` U) x) \<subseteq> path_component_set U y" .
+      have cUC: "path_component_set (S \<inter> f -` U) x \<subseteq> (S \<inter> f -` path_component_set U y)"
+        using path_component_subset fC by blast
+      have "path_component_set (S \<inter> f -` U) x \<subseteq> path_component_set (S \<inter> f -` path_component_set U y) x"
+      proof -
+        have "\<And>a. path_component_set (path_component_set (S \<inter> f -` U) x) a \<subseteq> path_component_set (S \<inter> 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 "\<dots> \<subseteq> (S \<inter> f -` path_component_set U y)"
+        by (rule path_component_subset)
+      finally show "path_component_set (S \<inter> f -` U) x \<subseteq> (S \<inter> f -` path_component_set U y)" .
+    qed
+  qed
+  ultimately show "openin (subtopology euclidean (f ` S)) (path_component_set U y)"
+    by metis
+qed
+
+subsection%unimportant\<open>Components, continuity, openin, closedin\<close>
+
+lemma continuous_on_components_gen:
+ fixes f :: "'a::topological_space \<Rightarrow> 'b::topological_space"
+  assumes "\<And>c. c \<in> components S \<Longrightarrow>
+              openin (subtopology euclidean S) c \<and> 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 \<inter> f -` t = (\<Union>c \<in> components S. c \<inter> f -` t)"
+    by auto
+  show "openin (subtopology euclidean S) (S \<inter> f -` t)"
+    unfolding * using \<open>open t\<close> assms continuous_openin_preimage_gen openin_trans openin_Union by blast
+qed
+
+lemma continuous_on_components:
+ fixes f :: "'a::topological_space \<Rightarrow> 'b::topological_space"
+  assumes "locally connected S "
+          "\<And>c. c \<in> components S \<Longrightarrow> continuous_on c f"
+    shows "continuous_on S f"
+apply (rule continuous_on_components_gen)
+apply (auto simp: assms intro: openin_components_locally_connected)
+done
+
+lemma continuous_on_components_eq:
+    "locally connected S
+     \<Longrightarrow> (continuous_on S f \<longleftrightarrow> (\<forall>c \<in> 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 "
+          "\<And>c. c \<in> components S \<Longrightarrow> 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 \<Longrightarrow> (continuous_on S f \<longleftrightarrow> (\<forall>c \<in> 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 (subtopology euclidean u) S"
+      and cuS: "c \<subseteq> components(u - S)"
+    shows "closedin (subtopology euclidean u) (S \<union> \<Union>c)"
+proof -
+  have di: "(\<And>S t. S \<in> c \<and> t \<in> c' \<Longrightarrow> disjnt S t) \<Longrightarrow> disjnt (\<Union> c) (\<Union> c')" for c'
+    by (simp add: disjnt_def) blast
+  have "S \<subseteq> u"
+    using S closedin_imp_subset by blast
+  moreover have "u - S = \<Union>c \<union> \<Union>(components (u - S) - c)"
+    by (metis Diff_partition Union_components Union_Un_distrib assms(3))
+  moreover have "disjnt (\<Union>c) (\<Union>(components (u - S) - c))"
+    apply (rule di)
+    by (metis DiffD1 DiffD2 assms(3) components_nonoverlap disjnt_def subsetCE)
+  ultimately have eq: "S \<union> \<Union>c = u - (\<Union>(components(u - S) - c))"
+    by (auto simp: disjnt_def)
+  have *: "openin (subtopology euclidean u) (\<Union>(components (u - S) - c))"
+    apply (rule openin_Union)
+    apply (rule openin_trans [of "u - S"])
+    apply (simp add: u S locally_diff_closed openin_components_locally_connected)
+    apply (simp add: openin_diff S)
+    done
+  have "openin (subtopology euclidean u) (u - (u - \<Union>(components (u - S) - c)))"
+    apply (rule openin_diff, simp)
+    apply (metis closedin_diff closedin_topspace topspace_euclidean_subtopology *)
+    done
+  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 \<subseteq> components(- S)"
+    shows "closed(S \<union> \<Union> c)"
+proof -
+  have "closedin (subtopology euclidean UNIV) (S \<union> \<Union>c)"
+    apply (rule closedin_union_complement_components [OF locally_connected_UNIV])
+    using S c apply (simp_all add: Compl_eq_Diff_UNIV)
+    done
+  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 (subtopology euclidean u) S"
+      and c: " c \<in> components(u - S)"
+    shows "closedin (subtopology euclidean u) (S \<union> c)"
+proof -
+  have "closedin (subtopology euclidean u) (S \<union> \<Union>{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 \<in> components(-S)"
+    shows "closed (S \<union> c)"
+  by (metis Compl_eq_Diff_UNIV S c closed_closedin closedin_Un_complement_component
+      locally_connected_UNIV subtopology_UNIV)
+
+
+subsection\<open>Existence of isometry between subspaces of same dimension\<close>
+
+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 \<le> dim T"
+  obtains f where "linear f" "f ` S \<subseteq> T" "\<And>x. x \<in> S \<Longrightarrow> norm(f x) = norm x"
+proof -
+  obtain B where "B \<subseteq> S" and Borth: "pairwise orthogonal B"
+             and B1: "\<And>x. x \<in> B \<Longrightarrow> norm x = 1"
+             and "independent B" "finite B" "card B = dim S" "span B = S"
+    by (metis orthonormal_basis_subspace [OF S] independent_finite)
+  obtain C where "C \<subseteq> T" and Corth: "pairwise orthogonal C"
+             and C1:"\<And>x. x \<in> C \<Longrightarrow> norm x = 1"
+             and "independent C" "finite C" "card C = dim T" "span C = T"
+    by (metis orthonormal_basis_subspace [OF T] independent_finite)
+  obtain fb where "fb ` B \<subseteq> C" "inj_on fb B"
+    by (metis \<open>card B = dim S\<close> \<open>card C = dim T\<close> \<open>finite B\<close> \<open>finite C\<close> card_le_inj d)
+  then have pairwise_orth_fb: "pairwise (\<lambda>v j. orthogonal (fb v) (fb j)) B"
+    using Corth
+    apply (auto simp: pairwise_def orthogonal_clauses)
+    by (meson subsetD image_eqI inj_on_def)
+  obtain f where "linear f" and ffb: "\<And>x. x \<in> B \<Longrightarrow> f x = fb x"
+    using linear_independent_extend \<open>independent B\<close> by fastforce
+  have "span (f ` B) \<subseteq> span C"
+    by (metis \<open>fb ` B \<subseteq> C\<close> ffb image_cong span_mono)
+  then have "f ` S \<subseteq> T"
+    unfolding \<open>span B = S\<close> \<open>span C = T\<close> span_linear_image[OF \<open>linear f\<close>] .
+  have [simp]: "\<And>x. x \<in> B \<Longrightarrow> norm (fb x) = norm x"
+    using B1 C1 \<open>fb ` B \<subseteq> C\<close> by auto
+  have "norm (f x) = norm x" if "x \<in> S" for x
+  proof -
+    interpret linear f by fact
+    obtain a where x: "x = (\<Sum>v \<in> B. a v *\<^sub>R v)"
+      using \<open>finite B\<close> \<open>span B = S\<close> \<open>x \<in> S\<close> span_finite by fastforce
+    have "norm (f x)^2 = norm (\<Sum>v\<in>B. a v *\<^sub>R fb v)^2" by (simp add: sum scale ffb x)
+    also have "\<dots> = (\<Sum>v\<in>B. norm ((a v *\<^sub>R fb v))^2)"
+      apply (rule norm_sum_Pythagorean [OF \<open>finite B\<close>])
+      apply (rule pairwise_ortho_scaleR [OF pairwise_orth_fb])
+      done
+    also have "\<dots> = norm x ^2"
+      by (simp add: x pairwise_ortho_scaleR Borth norm_sum_Pythagorean [OF \<open>finite B\<close>])
+    finally show ?thesis
+      by (simp add: norm_eq_sqrt_inner)
+  qed
+  then show ?thesis
+    by (rule that [OF \<open>linear f\<close> \<open>f ` S \<subseteq> T\<close>])
+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"
+                    "\<And>x. x \<in> S \<Longrightarrow> norm(f x) = norm x"
+                    "\<And>x. x \<in> T \<Longrightarrow> norm(g x) = norm x"
+                    "\<And>x. x \<in> S \<Longrightarrow> g(f x) = x"
+                    "\<And>x. x \<in> T \<Longrightarrow> f(g x) = x"
+proof -
+  obtain B where "B \<subseteq> S" and Borth: "pairwise orthogonal B"
+             and B1: "\<And>x. x \<in> B \<Longrightarrow> norm x = 1"
+             and "independent B" "finite B" "card B = dim S" "span B = S"
+    by (metis orthonormal_basis_subspace [OF S] independent_finite)
+  obtain C where "C \<subseteq> T" and Corth: "pairwise orthogonal C"
+             and C1:"\<And>x. x \<in> C \<Longrightarrow> norm x = 1"
+             and "independent C" "finite C" "card C = dim T" "span C = T"
+    by (metis orthonormal_basis_subspace [OF T] independent_finite)
+  obtain fb where "bij_betw fb B C"
+    by (metis \<open>finite B\<close> \<open>finite C\<close> bij_betw_iff_card \<open>card B = dim S\<close> \<open>card C = dim T\<close> d)
+  then have pairwise_orth_fb: "pairwise (\<lambda>v j. orthogonal (fb v) (fb j)) B"
+    using Corth
+    apply (auto simp: pairwise_def orthogonal_clauses bij_betw_def)
+    by (meson subsetD image_eqI inj_on_def)
+  obtain f where "linear f" and ffb: "\<And>x. x \<in> B \<Longrightarrow> f x = fb x"
+    using linear_independent_extend \<open>independent B\<close> by fastforce
+  interpret f: linear f by fact
+  define gb where "gb \<equiv> inv_into B fb"
+  then have pairwise_orth_gb: "pairwise (\<lambda>v j. orthogonal (gb v) (gb j)) C"
+    using Borth
+    apply (auto simp: pairwise_def orthogonal_clauses bij_betw_def)
+    by (metis \<open>bij_betw fb B C\<close> bij_betw_imp_surj_on bij_betw_inv_into_right inv_into_into)
+  obtain g where "linear g" and ggb: "\<And>x. x \<in> C \<Longrightarrow> g x = gb x"
+    using linear_independent_extend \<open>independent C\<close> by fastforce
+  interpret g: linear g by fact
+  have "span (f ` B) \<subseteq> span C"
+    by (metis \<open>bij_betw fb B C\<close> bij_betw_imp_surj_on eq_iff ffb image_cong)
+  then have "f ` S \<subseteq> T"
+    unfolding \<open>span B = S\<close> \<open>span C = T\<close>
+      span_linear_image[OF \<open>linear f\<close>] .
+  have [simp]: "\<And>x. x \<in> B \<Longrightarrow> norm (fb x) = norm x"
+    using B1 C1 \<open>bij_betw fb B C\<close> bij_betw_imp_surj_on by fastforce
+  have f [simp]: "norm (f x) = norm x" "g (f x) = x" if "x \<in> S" for x
+  proof -
+    obtain a where x: "x = (\<Sum>v \<in> B. a v *\<^sub>R v)"
+      using \<open>finite B\<close> \<open>span B = S\<close> \<open>x \<in> S\<close> span_finite by fastforce
+    have "f x = (\<Sum>v \<in> B. f (a v *\<^sub>R v))"
+      using linear_sum [OF \<open>linear f\<close>] x by auto
+    also have "\<dots> = (\<Sum>v \<in> B. a v *\<^sub>R f v)"
+      by (simp add: f.sum f.scale)
+    also have "\<dots> = (\<Sum>v \<in> B. a v *\<^sub>R fb v)"
+      by (simp add: ffb cong: sum.cong)
+    finally have *: "f x = (\<Sum>v\<in>B. a v *\<^sub>R fb v)" .
+    then have "(norm (f x))\<^sup>2 = (norm (\<Sum>v\<in>B. a v *\<^sub>R fb v))\<^sup>2" by simp
+    also have "\<dots> = (\<Sum>v\<in>B. norm ((a v *\<^sub>R fb v))^2)"
+      apply (rule norm_sum_Pythagorean [OF \<open>finite B\<close>])
+      apply (rule pairwise_ortho_scaleR [OF pairwise_orth_fb])
+      done
+    also have "\<dots> = (norm x)\<^sup>2"
+      by (simp add: x pairwise_ortho_scaleR Borth norm_sum_Pythagorean [OF \<open>finite B\<close>])
+    finally show "norm (f x) = norm x"
+      by (simp add: norm_eq_sqrt_inner)
+    have "g (f x) = g (\<Sum>v\<in>B. a v *\<^sub>R fb v)" by (simp add: *)
+    also have "\<dots> = (\<Sum>v\<in>B. g (a v *\<^sub>R fb v))"
+      by (simp add: g.sum g.scale)
+    also have "\<dots> = (\<Sum>v\<in>B. a v *\<^sub>R g (fb v))"
+      by (simp add: g.scale)
+    also have "\<dots> = (\<Sum>v\<in>B. a v *\<^sub>R v)"
+      apply (rule sum.cong [OF refl])
+      using \<open>bij_betw fb B C\<close> gb_def bij_betwE bij_betw_inv_into_left gb_def ggb by fastforce
+    also have "\<dots> = x"
+      using x by blast
+    finally show "g (f x) = x" .
+  qed
+  have [simp]: "\<And>x. x \<in> C \<Longrightarrow> norm (gb x) = norm x"
+    by (metis B1 C1 \<open>bij_betw fb B C\<close> bij_betw_imp_surj_on gb_def inv_into_into)
+  have g [simp]: "f (g x) = x" if "x \<in> T" for x
+  proof -
+    obtain a where x: "x = (\<Sum>v \<in> C. a v *\<^sub>R v)"
+      using \<open>finite C\<close> \<open>span C = T\<close> \<open>x \<in> T\<close> span_finite by fastforce
+    have "g x = (\<Sum>v \<in> C. g (a v *\<^sub>R v))"
+      by (simp add: x g.sum)
+    also have "\<dots> = (\<Sum>v \<in> C. a v *\<^sub>R g v)"
+      by (simp add: g.scale)
+    also have "\<dots> = (\<Sum>v \<in> C. a v *\<^sub>R gb v)"
+      by (simp add: ggb cong: sum.cong)
+    finally have "f (g x) = f (\<Sum>v\<in>C. a v *\<^sub>R gb v)" by simp
+    also have "\<dots> = (\<Sum>v\<in>C. f (a v *\<^sub>R gb v))"
+      by (simp add: f.scale f.sum)
+    also have "\<dots> = (\<Sum>v\<in>C. a v *\<^sub>R f (gb v))"
+      by (simp add: f.scale f.sum)
+    also have "\<dots> = (\<Sum>v\<in>C. a v *\<^sub>R v)"
+      using \<open>bij_betw fb B C\<close>
+      by (simp add: bij_betw_def gb_def bij_betw_inv_into_right ffb inv_into_into)
+    also have "\<dots> = x"
+      using x by blast
+    finally show "f (g x) = x" .
+  qed
+  have gim: "g ` T = S"
+    by (metis (full_types) S T \<open>f ` S \<subseteq> T\<close> 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 \<open>g ` T = S\<close> image_iff by fastforce
+  have [simp]: "norm (g x) = norm x" if "x \<in> T" for x
+    using fim that by auto
+  show ?thesis
+    apply (rule that [OF \<open>linear f\<close> \<open>linear g\<close>])
+    apply (simp_all add: fim gim)
+    done
+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" "\<And>x. x \<in> S \<Longrightarrow> 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 \<Rightarrow>'N::euclidean_space" and g
+  where "linear f" "linear g"
+                    "\<And>x. norm(f x) = norm x" "\<And>y. norm(g y) = norm y"
+                    "\<And>x. g (f x) = x" "\<And>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"
+                   "\<And>x. x \<in> S \<Longrightarrow> g(f x) = x" "\<And>x. x \<in> T \<Longrightarrow> f(g x) = x"
+    by (blast intro: isometries_subspaces [OF assms])
+  then show ?thesis
+    apply (simp add: homeomorphic_def homeomorphism_def)
+    apply (rule_tac x=f in exI)
+    apply (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 = {} \<or> T = {}")
+  case True  with assms aff_dim_empty homeomorphic_empty show ?thesis
+    by metis
+next
+  case False
+  then obtain a b where ab: "a \<in> S" "b \<in> 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 (simp add: homeomorphic_translation)
+  also have "\<dots> homeomorphic ((+) (- b) ` T)"
+    by (rule homeomorphic_subspaces [OF ss dd])
+  also have "\<dots> homeomorphic T"
+    using homeomorphic_sym homeomorphic_translation by auto
+  finally show ?thesis .
+qed
+
+
+subsection\<open>Retracts, in a general sense, preserve (co)homotopic triviality)\<close>
+
+locale%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 \<subseteq> s"
+      and idhk: "\<And>y. y \<in> t \<Longrightarrow> h(k y) = y"
+
+begin
+
+lemma homotopically_trivial_retraction_gen:
+  assumes P: "\<And>f. \<lbrakk>continuous_on u f; f ` u \<subseteq> t; Q f\<rbrakk> \<Longrightarrow> P(k \<circ> f)"
+      and Q: "\<And>f. \<lbrakk>continuous_on u f; f ` u \<subseteq> s; P f\<rbrakk> \<Longrightarrow> Q(h \<circ> f)"
+      and Qeq: "\<And>h k. (\<And>x. x \<in> u \<Longrightarrow> h x = k x) \<Longrightarrow> Q h = Q k"
+      and hom: "\<And>f g. \<lbrakk>continuous_on u f; f ` u \<subseteq> s; P f;
+                       continuous_on u g; g ` u \<subseteq> s; P g\<rbrakk>
+                       \<Longrightarrow> homotopic_with P u s f g"
+      and contf: "continuous_on u f" and imf: "f ` u \<subseteq> t" and Qf: "Q f"
+      and contg: "continuous_on u g" and img: "g ` u \<subseteq> t" and Qg: "Q g"
+    shows "homotopic_with Q u t f g"
+proof -
+  have feq: "\<And>x. x \<in> u \<Longrightarrow> (h \<circ> (k \<circ> f)) x = f x" using idhk imf by auto
+  have geq: "\<And>x. x \<in> u \<Longrightarrow> (h \<circ> (k \<circ> g)) x = g x" using idhk img by auto
+  have "continuous_on u (k \<circ> f)"
+    using contf continuous_on_compose continuous_on_subset contk imf by blast
+  moreover have "(k \<circ> f) ` u \<subseteq> s"
+    using imf imk by fastforce
+  moreover have "P (k \<circ> f)"
+    by (simp add: P Qf contf imf)
+  moreover have "continuous_on u (k \<circ> g)"
+    using contg continuous_on_compose continuous_on_subset contk img by blast
+  moreover have "(k \<circ> g) ` u \<subseteq> s"
+    using img imk by fastforce
+  moreover have "P (k \<circ> g)"
+    by (simp add: P Qg contg img)
+  ultimately have "homotopic_with P u s (k \<circ> f) (k \<circ> g)"
+    by (rule hom)
+  then have "homotopic_with Q u t (h \<circ> (k \<circ> f)) (h \<circ> (k \<circ> g))"
+    apply (rule homotopic_with_compose_continuous_left [OF homotopic_with_mono])
+    using Q by (auto simp: conth imh)
+  then show ?thesis
+    apply (rule homotopic_with_eq)
+    apply (metis feq)
+    apply (metis geq)
+    apply (metis Qeq)
+    done
+qed
+
+lemma homotopically_trivial_retraction_null_gen:
+  assumes P: "\<And>f. \<lbrakk>continuous_on u f; f ` u \<subseteq> t; Q f\<rbrakk> \<Longrightarrow> P(k \<circ> f)"
+      and Q: "\<And>f. \<lbrakk>continuous_on u f; f ` u \<subseteq> s; P f\<rbrakk> \<Longrightarrow> Q(h \<circ> f)"
+      and Qeq: "\<And>h k. (\<And>x. x \<in> u \<Longrightarrow> h x = k x) \<Longrightarrow> Q h = Q k"
+      and hom: "\<And>f. \<lbrakk>continuous_on u f; f ` u \<subseteq> s; P f\<rbrakk>
+                     \<Longrightarrow> \<exists>c. homotopic_with P u s f (\<lambda>x. c)"
+      and contf: "continuous_on u f" and imf:"f ` u \<subseteq> t" and Qf: "Q f"
+  obtains c where "homotopic_with Q u t f (\<lambda>x. c)"
+proof -
+  have feq: "\<And>x. x \<in> u \<Longrightarrow> (h \<circ> (k \<circ> f)) x = f x" using idhk imf by auto
+  have "continuous_on u (k \<circ> f)"
+    using contf continuous_on_compose continuous_on_subset contk imf by blast
+  moreover have "(k \<circ> f) ` u \<subseteq> s"
+    using imf imk by fastforce
+  moreover have "P (k \<circ> f)"
+    by (simp add: P Qf contf imf)
+  ultimately obtain c where "homotopic_with P u s (k \<circ> f) (\<lambda>x. c)"
+    by (metis hom)
+  then have "homotopic_with Q u t (h \<circ> (k \<circ> f)) (h \<circ> (\<lambda>x. c))"
+    apply (rule homotopic_with_compose_continuous_left [OF homotopic_with_mono])
+    using Q by (auto simp: conth imh)
+  then show ?thesis
+    apply (rule_tac c = "h c" in that)
+    apply (erule homotopic_with_eq)
+    apply (metis feq, simp)
+    apply (metis Qeq)
+    done
+qed
+
+lemma cohomotopically_trivial_retraction_gen:
+  assumes P: "\<And>f. \<lbrakk>continuous_on t f; f ` t \<subseteq> u; Q f\<rbrakk> \<Longrightarrow> P(f \<circ> h)"
+      and Q: "\<And>f. \<lbrakk>continuous_on s f; f ` s \<subseteq> u; P f\<rbrakk> \<Longrightarrow> Q(f \<circ> k)"
+      and Qeq: "\<And>h k. (\<And>x. x \<in> t \<Longrightarrow> h x = k x) \<Longrightarrow> Q h = Q k"
+      and hom: "\<And>f g. \<lbrakk>continuous_on s f; f ` s \<subseteq> u; P f;
+                       continuous_on s g; g ` s \<subseteq> u; P g\<rbrakk>
+                       \<Longrightarrow> homotopic_with P s u f g"
+      and contf: "continuous_on t f" and imf: "f ` t \<subseteq> u" and Qf: "Q f"
+      and contg: "continuous_on t g" and img: "g ` t \<subseteq> u" and Qg: "Q g"
+    shows "homotopic_with Q t u f g"
+proof -
+  have feq: "\<And>x. x \<in> t \<Longrightarrow> (f \<circ> h \<circ> k) x = f x" using idhk imf by auto
+  have geq: "\<And>x. x \<in> t \<Longrightarrow> (g \<circ> h \<circ> k) x = g x" using idhk img by auto
+  have "continuous_on s (f \<circ> h)"
+    using contf conth continuous_on_compose imh by blast
+  moreover have "(f \<circ> h) ` s \<subseteq> u"
+    using imf imh by fastforce
+  moreover have "P (f \<circ> h)"
+    by (simp add: P Qf contf imf)
+  moreover have "continuous_on s (g \<circ> h)"
+    using contg continuous_on_compose continuous_on_subset conth imh by blast
+  moreover have "(g \<circ> h) ` s \<subseteq> u"
+    using img imh by fastforce
+  moreover have "P (g \<circ> h)"
+    by (simp add: P Qg contg img)
+  ultimately have "homotopic_with P s u (f \<circ> h) (g \<circ> h)"
+    by (rule hom)
+  then have "homotopic_with Q t u (f \<circ> h \<circ> k) (g \<circ> h \<circ> k)"
+    apply (rule homotopic_with_compose_continuous_right [OF homotopic_with_mono])
+    using Q by (auto simp: contk imk)
+  then show ?thesis
+    apply (rule homotopic_with_eq)
+    apply (metis feq)
+    apply (metis geq)
+    apply (metis Qeq)
+    done
+qed
+
+lemma cohomotopically_trivial_retraction_null_gen:
+  assumes P: "\<And>f. \<lbrakk>continuous_on t f; f ` t \<subseteq> u; Q f\<rbrakk> \<Longrightarrow> P(f \<circ> h)"
+      and Q: "\<And>f. \<lbrakk>continuous_on s f; f ` s \<subseteq> u; P f\<rbrakk> \<Longrightarrow> Q(f \<circ> k)"
+      and Qeq: "\<And>h k. (\<And>x. x \<in> t \<Longrightarrow> h x = k x) \<Longrightarrow> Q h = Q k"
+      and hom: "\<And>f g. \<lbrakk>continuous_on s f; f ` s \<subseteq> u; P f\<rbrakk>
+                       \<Longrightarrow> \<exists>c. homotopic_with P s u f (\<lambda>x. c)"
+      and contf: "continuous_on t f" and imf: "f ` t \<subseteq> u" and Qf: "Q f"
+  obtains c where "homotopic_with Q t u f (\<lambda>x. c)"
+proof -
+  have feq: "\<And>x. x \<in> t \<Longrightarrow> (f \<circ> h \<circ> k) x = f x" using idhk imf by auto
+  have "continuous_on s (f \<circ> h)"
+    using contf conth continuous_on_compose imh by blast
+  moreover have "(f \<circ> h) ` s \<subseteq> u"
+    using imf imh by fastforce
+  moreover have "P (f \<circ> h)"
+    by (simp add: P Qf contf imf)
+  ultimately obtain c where "homotopic_with P s u (f \<circ> h) (\<lambda>x. c)"
+    by (metis hom)
+  then have "homotopic_with Q t u (f \<circ> h \<circ> k) ((\<lambda>x. c) \<circ> k)"
+    apply (rule homotopic_with_compose_continuous_right [OF homotopic_with_mono])
+    using Q by (auto simp: contk imk)
+  then show ?thesis
+    apply (rule_tac c = c in that)
+    apply (erule homotopic_with_eq)
+    apply (metis feq, simp)
+    apply (metis Qeq)
+    done
+qed
+
+end
+
+lemma simply_connected_retraction_gen:
+  shows "\<lbrakk>simply_connected S; continuous_on S h; h ` S = T;
+          continuous_on T k; k ` T \<subseteq> S; \<And>y. y \<in> T \<Longrightarrow> h(k y) = y\<rbrakk>
+        \<Longrightarrow> 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 _ "\<lambda>p. pathfinish p = pathstart p"  "\<lambda>p. pathfinish p = pathstart p"])
+apply (simp_all add: Retracts_def pathfinish_def pathstart_def)
+done
+
+lemma homeomorphic_simply_connected:
+    "\<lbrakk>S homeomorphic T; simply_connected S\<rbrakk> \<Longrightarrow> simply_connected T"
+  by (auto simp: homeomorphic_def homeomorphism_def intro: simply_connected_retraction_gen)
+
+lemma homeomorphic_simply_connected_eq:
+    "S homeomorphic T \<Longrightarrow> (simply_connected S \<longleftrightarrow> simply_connected T)"
+  by (metis homeomorphic_simply_connected homeomorphic_sym)
+
+
+subsection\<open>Homotopy equivalence\<close>
+
+definition%important homotopy_eqv :: "'a::topological_space set \<Rightarrow> 'b::topological_space set \<Rightarrow> bool"
+             (infix "homotopy'_eqv" 50)
+  where "S homotopy_eqv T \<equiv>
+        \<exists>f g. continuous_on S f \<and> f ` S \<subseteq> T \<and>
+              continuous_on T g \<and> g ` T \<subseteq> S \<and>
+              homotopic_with (\<lambda>x. True) S S (g \<circ> f) id \<and>
+              homotopic_with (\<lambda>x. True) T T (f \<circ> g) id"
+
+lemma homeomorphic_imp_homotopy_eqv: "S homeomorphic T \<Longrightarrow> S homotopy_eqv T"
+  unfolding homeomorphic_def homotopy_eqv_def homeomorphism_def
+  by (fastforce intro!: homotopic_with_equal continuous_on_compose)
+
+lemma homotopy_eqv_refl: "S homotopy_eqv S"
+  by (rule homeomorphic_imp_homotopy_eqv homeomorphic_refl)+
+
+lemma homotopy_eqv_sym: "S homotopy_eqv T \<longleftrightarrow> T homotopy_eqv S"
+  by (auto simp: homotopy_eqv_def)
+
+lemma homotopy_eqv_trans [trans]:
+    fixes S :: "'a::real_normed_vector set" and U :: "'c::real_normed_vector set"
+  assumes ST: "S homotopy_eqv T" and TU: "T homotopy_eqv U"
+    shows "S homotopy_eqv U"
+proof -
+  obtain f1 g1 where f1: "continuous_on S f1" "f1 ` S \<subseteq> T"
+                 and g1: "continuous_on T g1" "g1 ` T \<subseteq> S"
+                 and hom1: "homotopic_with (\<lambda>x. True) S S (g1 \<circ> f1) id"
+                           "homotopic_with (\<lambda>x. True) T T (f1 \<circ> g1) id"
+    using ST by (auto simp: homotopy_eqv_def)
+  obtain f2 g2 where f2: "continuous_on T f2" "f2 ` T \<subseteq> U"
+                 and g2: "continuous_on U g2" "g2 ` U \<subseteq> T"
+                 and hom2: "homotopic_with (\<lambda>x. True) T T (g2 \<circ> f2) id"
+                           "homotopic_with (\<lambda>x. True) U U (f2 \<circ> g2) id"
+    using TU by (auto simp: homotopy_eqv_def)
+  have "homotopic_with (\<lambda>f. True) S T (g2 \<circ> f2 \<circ> f1) (id \<circ> f1)"
+    by (rule homotopic_with_compose_continuous_right hom2 f1)+
+  then have "homotopic_with (\<lambda>f. True) S T (g2 \<circ> (f2 \<circ> f1)) (id \<circ> f1)"
+    by (simp add: o_assoc)
+  then have "homotopic_with (\<lambda>x. True) S S
+         (g1 \<circ> (g2 \<circ> (f2 \<circ> f1))) (g1 \<circ> (id \<circ> f1))"
+    by (simp add: g1 homotopic_with_compose_continuous_left)
+  moreover have "homotopic_with (\<lambda>x. True) S S (g1 \<circ> id \<circ> f1) id"
+    using hom1 by simp
+  ultimately have SS: "homotopic_with (\<lambda>x. True) S S (g1 \<circ> g2 \<circ> (f2 \<circ> f1)) id"
+    apply (simp add: o_assoc)
+    apply (blast intro: homotopic_with_trans)
+    done
+  have "homotopic_with (\<lambda>f. True) U T (f1 \<circ> g1 \<circ> g2) (id \<circ> g2)"
+    by (rule homotopic_with_compose_continuous_right hom1 g2)+
+  then have "homotopic_with (\<lambda>f. True) U T (f1 \<circ> (g1 \<circ> g2)) (id \<circ> g2)"
+    by (simp add: o_assoc)
+  then have "homotopic_with (\<lambda>x. True) U U
+         (f2 \<circ> (f1 \<circ> (g1 \<circ> g2))) (f2 \<circ> (id \<circ> g2))"
+    by (simp add: f2 homotopic_with_compose_continuous_left)
+  moreover have "homotopic_with (\<lambda>x. True) U U (f2 \<circ> id \<circ> g2) id"
+    using hom2 by simp
+  ultimately have UU: "homotopic_with (\<lambda>x. True) U U (f2 \<circ> f1 \<circ> (g1 \<circ> g2)) id"
+    apply (simp add: o_assoc)
+    apply (blast intro: homotopic_with_trans)
+    done
+  show ?thesis
+    unfolding homotopy_eqv_def
+    apply (rule_tac x = "f2 \<circ> f1" in exI)
+    apply (rule_tac x = "g1 \<circ> g2" in exI)
+    apply (intro conjI continuous_on_compose SS UU)
+    using f1 f2 g1 g2  apply (force simp: elim!: continuous_on_subset)+
+    done
+qed
+
+lemma homotopy_eqv_inj_linear_image:
+  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
+  assumes "linear f" "inj f"
+    shows "(f ` S) homotopy_eqv S"
+apply (rule homeomorphic_imp_homotopy_eqv)
+using assms homeomorphic_sym linear_homeomorphic_image by auto
+
+lemma homotopy_eqv_translation:
+    fixes S :: "'a::real_normed_vector set"
+    shows "(+) a ` S homotopy_eqv S"
+  apply (rule homeomorphic_imp_homotopy_eqv)
+  using 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 \<subseteq> T"
+      and g: "continuous_on U g" "g ` U \<subseteq> T"
+      and homUS: "\<And>f g. \<lbrakk>continuous_on U f; f ` U \<subseteq> S;
+                         continuous_on U g; g ` U \<subseteq> S\<rbrakk>
+                         \<Longrightarrow> homotopic_with (\<lambda>x. True) U S f g"
+    shows "homotopic_with (\<lambda>x. True) U T f g"
+proof -
+  obtain h k where h: "continuous_on S h" "h ` S \<subseteq> T"
+               and k: "continuous_on T k" "k ` T \<subseteq> S"
+               and hom: "homotopic_with (\<lambda>x. True) S S (k \<circ> h) id"
+                        "homotopic_with (\<lambda>x. True) T T (h \<circ> k) id"
+    using assms by (auto simp: homotopy_eqv_def)
+  have "homotopic_with (\<lambda>f. True) U S (k \<circ> f) (k \<circ> g)"
+    apply (rule homUS)
+    using f g k
+    apply (safe intro!: continuous_on_compose h k f elim!: continuous_on_subset)
+    apply (force simp: o_def)+
+    done
+  then have "homotopic_with (\<lambda>x. True) U T (h \<circ> (k \<circ> f)) (h \<circ> (k \<circ> g))"
+    apply (rule homotopic_with_compose_continuous_left)
+    apply (simp_all add: h)
+    done
+  moreover have "homotopic_with (\<lambda>x. True) U T (h \<circ> k \<circ> f) (id \<circ> f)"
+    apply (rule homotopic_with_compose_continuous_right [where X=T and Y=T])
+    apply (auto simp: hom f)
+    done
+  moreover have "homotopic_with (\<lambda>x. True) U T (h \<circ> k \<circ> g) (id \<circ> g)"
+    apply (rule homotopic_with_compose_continuous_right [where X=T and Y=T])
+    apply (auto simp: hom g)
+    done
+  ultimately show "homotopic_with (\<lambda>x. True) U T f g"
+    apply (simp add: o_assoc)
+    using homotopic_with_trans homotopic_with_sym by blast
+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 "(\<forall>f g. continuous_on U f \<and> f ` U \<subseteq> S \<and>
+                   continuous_on U g \<and> g ` U \<subseteq> S
+                   \<longrightarrow> homotopic_with (\<lambda>x. True) U S f g) \<longleftrightarrow>
+           (\<forall>f g. continuous_on U f \<and> f ` U \<subseteq> T \<and>
+                  continuous_on U g \<and> g ` U \<subseteq> T
+                  \<longrightarrow> homotopic_with (\<lambda>x. True) U T f g)"
+apply (rule iffI)
+apply (metis assms homotopy_eqv_homotopic_triviality_imp)
+by (metis (no_types) assms homotopy_eqv_homotopic_triviality_imp homotopy_eqv_sym)
+
+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 \<subseteq> U"
+      and homSU: "\<And>f. \<lbrakk>continuous_on S f; f ` S \<subseteq> U\<rbrakk>
+                      \<Longrightarrow> \<exists>c. homotopic_with (\<lambda>x. True) S U f (\<lambda>x. c)"
+  obtains c where "homotopic_with (\<lambda>x. True) T U f (\<lambda>x. c)"
+proof -
+  obtain h k where h: "continuous_on S h" "h ` S \<subseteq> T"
+               and k: "continuous_on T k" "k ` T \<subseteq> S"
+               and hom: "homotopic_with (\<lambda>x. True) S S (k \<circ> h) id"
+                        "homotopic_with (\<lambda>x. True) T T (h \<circ> k) id"
+    using assms by (auto simp: homotopy_eqv_def)
+  obtain c where "homotopic_with (\<lambda>x. True) S U (f \<circ> h) (\<lambda>x. c)"
+    apply (rule exE [OF homSU [of "f \<circ> h"]])
+    apply (intro continuous_on_compose h)
+    using h f  apply (force elim!: continuous_on_subset)+
+    done
+  then have "homotopic_with (\<lambda>x. True) T U ((f \<circ> h) \<circ> k) ((\<lambda>x. c) \<circ> k)"
+    apply (rule homotopic_with_compose_continuous_right [where X=S])
+    using k by auto
+  moreover have "homotopic_with (\<lambda>x. True) T U (f \<circ> id) (f \<circ> (h \<circ> k))"
+    apply (rule homotopic_with_compose_continuous_left [where Y=T])
+      apply (simp add: hom homotopic_with_symD)
+     using f apply auto
+    done
+  ultimately show ?thesis
+    apply (rule_tac c=c in that)
+    apply (simp add: o_def)
+    using homotopic_with_trans by blast
+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 "(\<forall>f. continuous_on S f \<and> f ` S \<subseteq> U
+                \<longrightarrow> (\<exists>c. homotopic_with (\<lambda>x. True) S U f (\<lambda>x. c))) \<longleftrightarrow>
+           (\<forall>f. continuous_on T f \<and> f ` T \<subseteq> U
+                \<longrightarrow> (\<exists>c. homotopic_with (\<lambda>x. True) T U f (\<lambda>x. c)))"
+apply (rule iffI)
+apply (metis assms homotopy_eqv_cohomotopic_triviality_null_imp)
+by (metis assms homotopy_eqv_cohomotopic_triviality_null_imp homotopy_eqv_sym)
+
+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 \<subseteq> T"
+      and homSU: "\<And>f. \<lbrakk>continuous_on U f; f ` U \<subseteq> S\<rbrakk>
+                      \<Longrightarrow> \<exists>c. homotopic_with (\<lambda>x. True) U S f (\<lambda>x. c)"
+    shows "\<exists>c. homotopic_with (\<lambda>x. True) U T f (\<lambda>x. c)"
+proof -
+  obtain h k where h: "continuous_on S h" "h ` S \<subseteq> T"
+               and k: "continuous_on T k" "k ` T \<subseteq> S"
+               and hom: "homotopic_with (\<lambda>x. True) S S (k \<circ> h) id"
+                        "homotopic_with (\<lambda>x. True) T T (h \<circ> k) id"
+    using assms by (auto simp: homotopy_eqv_def)
+  obtain c::'a where "homotopic_with (\<lambda>x. True) U S (k \<circ> f) (\<lambda>x. c)"
+    apply (rule exE [OF homSU [of "k \<circ> f"]])
+    apply (intro continuous_on_compose h)
+    using k f  apply (force elim!: continuous_on_subset)+
+    done
+  then have "homotopic_with (\<lambda>x. True) U T (h \<circ> (k \<circ> f)) (h \<circ> (\<lambda>x. c))"
+    apply (rule homotopic_with_compose_continuous_left [where Y=S])
+    using h by auto
+  moreover have "homotopic_with (\<lambda>x. True) U T (id \<circ> f) ((h \<circ> k) \<circ> f)"
+    apply (rule homotopic_with_compose_continuous_right [where X=T])
+      apply (simp add: hom homotopic_with_symD)
+     using f apply auto
+    done
+  ultimately show ?thesis
+    using homotopic_with_trans by (fastforce simp add: 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 "(\<forall>f. continuous_on U f \<and> f ` U \<subseteq> S
+                  \<longrightarrow> (\<exists>c. homotopic_with (\<lambda>x. True) U S f (\<lambda>x. c))) \<longleftrightarrow>
+           (\<forall>f. continuous_on U f \<and> f ` U \<subseteq> T
+                  \<longrightarrow> (\<exists>c. homotopic_with (\<lambda>x. True) U T f (\<lambda>x. c)))"
+apply (rule iffI)
+apply (metis assms homotopy_eqv_homotopic_triviality_null_imp)
+by (metis assms homotopy_eqv_homotopic_triviality_null_imp homotopy_eqv_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 = {} \<longleftrightarrow> T = {}"
+    shows "S homotopy_eqv T"
+proof (cases "S = {}")
+  case True with assms show ?thesis
+    by (simp add: homeomorphic_imp_homotopy_eqv)
+next
+  case False
+  with assms obtain a b where "a \<in> S" "b \<in> T"
+    by auto
+  then show ?thesis
+    unfolding homotopy_eqv_def
+    apply (rule_tac x="\<lambda>x. b" in exI)
+    apply (rule_tac x="\<lambda>x. a" in exI)
+    apply (intro assms conjI continuous_on_id' homotopic_into_contractible)
+    apply (auto simp: o_def continuous_on_const)
+    done
+qed
+
+lemma homotopy_eqv_empty1 [simp]:
+  fixes S :: "'a::real_normed_vector set"
+  shows "S homotopy_eqv ({}::'b::real_normed_vector set) \<longleftrightarrow> S = {}"
+apply (rule iffI)
+using homotopy_eqv_def apply fastforce
+by (simp add: homotopy_eqv_contractible_sets)
+
+lemma homotopy_eqv_empty2 [simp]:
+  fixes S :: "'a::real_normed_vector set"
+  shows "({}::'b::real_normed_vector set) homotopy_eqv S \<longleftrightarrow> S = {}"
+by (metis homotopy_eqv_empty1 homotopy_eqv_sym)
+
+lemma homotopy_eqv_contractibility:
+  fixes S :: "'a::real_normed_vector set" and T :: "'b::real_normed_vector set"
+  shows "S homotopy_eqv T \<Longrightarrow> (contractible S \<longleftrightarrow> contractible T)"
+unfolding homotopy_eqv_def
+by (blast intro: homotopy_dominated_contractibility)
+
+lemma homotopy_eqv_sing:
+  fixes S :: "'a::real_normed_vector set" and a :: "'b::real_normed_vector"
+  shows "S homotopy_eqv {a} \<longleftrightarrow> S \<noteq> {} \<and> contractible S"
+proof (cases "S = {}")
+  case True then show ?thesis
+    by simp
+next
+  case False then show ?thesis
+    by (metis contractible_sing empty_not_insert homotopy_eqv_contractibility homotopy_eqv_contractible_sets)
+qed
+
+lemma homeomorphic_contractible_eq:
+  fixes S :: "'a::real_normed_vector set" and T :: "'b::real_normed_vector set"
+  shows "S homeomorphic T \<Longrightarrow> (contractible S \<longleftrightarrow> 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 "\<lbrakk>contractible S; S homeomorphic T\<rbrakk> \<Longrightarrow> contractible T"
+  by (metis homeomorphic_contractible_eq)
+
+
+subsection%unimportant\<open>Misc other results\<close>
+
+lemma bounded_connected_Compl_real:
+  fixes S :: "real set"
+  assumes "bounded S" and conn: "connected(- S)"
+    shows "S = {}"
+proof -
+  obtain a b where "S \<subseteq> box a b"
+    by (meson assms bounded_subset_box_symmetric)
+  then have "a \<notin> S" "b \<notin> S"
+    by auto
+  then have "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> x \<in> - S"
+    by (meson Compl_iff conn connected_iff_interval)
+  then show ?thesis
+    using \<open>S \<subseteq> box a b\<close> by auto
+qed
+
+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 \<Rightarrow> real" and g
+  where "linear f" "\<And>x. norm(f x) = norm x" "\<And>x. g(f x) = x" "\<And>y. f(g y) = y"
+    by (rule isomorphisms_UNIV_UNIV) blast
+  with \<open>bounded S\<close> have "bounded (f ` S)"
+    using bounded_linear_image linear_linear by blast
+  have "connected (f ` (-S))"
+    using connected_linear_image assms \<open>linear f\<close> by blast
+  moreover have "f ` (-S) = - (f ` S)"
+    apply (rule bij_image_Compl_eq)
+    apply (auto simp: bij_def)
+     apply (metis \<open>\<And>x. g (f x) = x\<close> injI)
+    by (metis UNIV_I \<open>\<And>y. f (g y) = y\<close> image_iff)
+  finally have "connected (- (f ` S))"
+    by simp
+  then have "f ` S = {}"
+    using \<open>bounded (f ` S)\<close> bounded_connected_Compl_real by blast
+  then show ?thesis
+    by blast
+qed
+
+
+subsection%unimportant\<open>Some Uncountable Sets\<close>
+
+lemma uncountable_closed_segment:
+  fixes a :: "'a::real_normed_vector"
+  assumes "a \<noteq> b" shows "uncountable (closed_segment a b)"
+unfolding path_image_linepath [symmetric] path_image_def
+  using inj_on_linepath [OF assms] uncountable_closed_interval [of 0 1]
+        countable_image_inj_on by auto
+
+lemma uncountable_open_segment:
+  fixes a :: "'a::real_normed_vector"
+  assumes "a \<noteq> b" shows "uncountable (open_segment a b)"
+  by (simp add: assms open_segment_def uncountable_closed_segment uncountable_minus_countable)
+
+lemma uncountable_convex:
+  fixes a :: "'a::real_normed_vector"
+  assumes "convex S" "a \<in> S" "b \<in> S" "a \<noteq> b"
+    shows "uncountable S"
+proof -
+  have "uncountable (closed_segment a b)"
+    by (simp add: uncountable_closed_segment assms)
+  then show ?thesis
+    by (meson assms convex_contains_segment countable_subset)
+qed
+
+lemma uncountable_ball:
+  fixes a :: "'a::euclidean_space"
+  assumes "r > 0"
+    shows "uncountable (ball a r)"
+proof -
+  have "uncountable (open_segment a (a + r *\<^sub>R (SOME i. i \<in> Basis)))"
+    by (metis Basis_zero SOME_Basis add_cancel_right_right assms less_le scale_eq_0_iff uncountable_open_segment)
+  moreover have "open_segment a (a + r *\<^sub>R (SOME i. i \<in> Basis)) \<subseteq> ball a r"
+    using assms by (auto simp: in_segment algebra_simps dist_norm SOME_Basis)
+  ultimately show ?thesis
+    by (metis countable_subset)
+qed
+
+lemma ball_minus_countable_nonempty:
+  assumes "countable (A :: 'a :: euclidean_space set)" "r > 0"
+  shows   "ball z r - A \<noteq> {}"
+proof
+  assume *: "ball z r - A = {}"
+  have "uncountable (ball z r - A)"
+    by (intro uncountable_minus_countable assms uncountable_ball)
+  thus False by (subst (asm) *) auto
+qed
+
+lemma uncountable_cball:
+  fixes a :: "'a::euclidean_space"
+  assumes "r > 0"
+  shows "uncountable (cball a r)"
+  using assms countable_subset uncountable_ball by auto
+
+lemma pairwise_disjnt_countable:
+  fixes \<N> :: "nat set set"
+  assumes "pairwise disjnt \<N>"
+    shows "countable \<N>"
+proof -
+  have "inj_on (\<lambda>X. SOME n. n \<in> X) (\<N> - {{}})"
+    apply (clarsimp simp add: inj_on_def)
+    by (metis assms disjnt_insert2 insert_absorb pairwise_def subsetI subset_empty tfl_some)
+  then show ?thesis
+    by (metis countable_Diff_eq countable_def)
+qed
+
+lemma pairwise_disjnt_countable_Union:
+    assumes "countable (\<Union>\<N>)" and pwd: "pairwise disjnt \<N>"
+    shows "countable \<N>"
+proof -
+  obtain f :: "_ \<Rightarrow> nat" where f: "inj_on f (\<Union>\<N>)"
+    using assms by blast
+  then have "pairwise disjnt (\<Union> X \<in> \<N>. {f ` X})"
+    using assms by (force simp: pairwise_def disjnt_inj_on_iff [OF f])
+  then have "countable (\<Union> X \<in> \<N>. {f ` X})"
+    using pairwise_disjnt_countable by blast
+  then show ?thesis
+    by (meson pwd countable_image_inj_on disjoint_image f inj_on_image pairwise_disjnt_countable)
+qed
+
+lemma connected_uncountable:
+  fixes S :: "'a::metric_space set"
+  assumes "connected S" "a \<in> S" "b \<in> S" "a \<noteq> b" shows "uncountable S"
+proof -
+  have "continuous_on S (dist a)"
+    by (intro continuous_intros)
+  then have "connected (dist a ` S)"
+    by (metis connected_continuous_image \<open>connected S\<close>)
+  then have "closed_segment 0 (dist a b) \<subseteq> (dist a ` S)"
+    by (simp add: assms closed_segment_subset is_interval_connected_1 is_interval_convex)
+  then have "uncountable (dist a ` S)"
+    by (metis \<open>a \<noteq> b\<close> countable_subset dist_eq_0_iff uncountable_closed_segment)
+  then show ?thesis
+    by blast
+qed
+
+lemma path_connected_uncountable:
+  fixes S :: "'a::metric_space set"
+  assumes "path_connected S" "a \<in> S" "b \<in> S" "a \<noteq> b" shows "uncountable S"
+  using path_connected_imp_connected assms connected_uncountable by metis
+
+lemma connected_finite_iff_sing:
+  fixes S :: "'a::metric_space set"
+  assumes "connected S"
+  shows "finite S \<longleftrightarrow> S = {} \<or> (\<exists>a. S = {a})"  (is "_ = ?rhs")
+proof -
+  have "uncountable S" if "\<not> ?rhs"
+    using connected_uncountable assms that by blast
+  then show ?thesis
+    using uncountable_infinite by auto
+qed
+
+lemma connected_card_eq_iff_nontrivial:
+  fixes S :: "'a::metric_space set"
+  shows "connected S \<Longrightarrow> uncountable S \<longleftrightarrow> \<not>(\<exists>a. S \<subseteq> {a})"
+  apply (auto simp: countable_finite finite_subset)
+  by (metis connected_uncountable is_singletonI' is_singleton_the_elem subset_singleton_iff)
+
+lemma simple_path_image_uncountable:
+  fixes g :: "real \<Rightarrow> 'a::metric_space"
+  assumes "simple_path g"
+  shows "uncountable (path_image g)"
+proof -
+  have "g 0 \<in> path_image g" "g (1/2) \<in> path_image g"
+    by (simp_all add: path_defs)
+  moreover have "g 0 \<noteq> g (1/2)"
+    using assms by (fastforce simp add: simple_path_def)
+  ultimately show ?thesis
+    apply (simp add: assms connected_card_eq_iff_nontrivial connected_simple_path_image)
+    by blast
+qed
+
+lemma arc_image_uncountable:
+  fixes g :: "real \<Rightarrow> 'a::metric_space"
+  assumes "arc g"
+  shows "uncountable (path_image g)"
+  by (simp add: arc_imp_simple_path assms simple_path_image_uncountable)
+
+
+subsection%unimportant\<open> Some simple positive connection theorems\<close>
+
+proposition path_connected_convex_diff_countable:
+  fixes U :: "'a::euclidean_space set"
+  assumes "convex U" "\<not> collinear U" "countable S"
+    shows "path_connected(U - S)"
+proof (clarsimp simp add: path_connected_def)
+  fix a b
+  assume "a \<in> U" "a \<notin> S" "b \<in> U" "b \<notin> S"
+  let ?m = "midpoint a b"
+  show "\<exists>g. path g \<and> path_image g \<subseteq> U - S \<and> pathstart g = a \<and> pathfinish g = b"
+  proof (cases "a = b")
+    case True
+    then show ?thesis
+      by (metis DiffI \<open>a \<in> U\<close> \<open>a \<notin> S\<close> path_component_def path_component_refl)
+  next
+    case False
+    then have "a \<noteq> ?m" "b \<noteq> ?m"
+      using midpoint_eq_endpoint by fastforce+
+    have "?m \<in> U"
+      using \<open>a \<in> U\<close> \<open>b \<in> U\<close> \<open>convex U\<close> convex_contains_segment by force
+    obtain c where "c \<in> U" and nc_abc: "\<not> collinear {a,b,c}"
+      by (metis False \<open>a \<in> U\<close> \<open>b \<in> U\<close> \<open>\<not> collinear U\<close> collinear_triples insert_absorb)
+    have ncoll_mca: "\<not> collinear {?m,c,a}"
+      by (metis (full_types) \<open>a \<noteq> ?m\<close> collinear_3_trans collinear_midpoint insert_commute nc_abc)
+    have ncoll_mcb: "\<not> collinear {?m,c,b}"
+      by (metis (full_types) \<open>b \<noteq> ?m\<close> collinear_3_trans collinear_midpoint insert_commute nc_abc)
+    have "c \<noteq> ?m"
+      by (metis collinear_midpoint insert_commute nc_abc)
+    then have "closed_segment ?m c \<subseteq> U"
+      by (simp add: \<open>c \<in> U\<close> \<open>?m \<in> U\<close> \<open>convex U\<close> closed_segment_subset)
+    then obtain z where z: "z \<in> closed_segment ?m c"
+                    and disjS: "(closed_segment a z \<union> closed_segment z b) \<inter> S = {}"
+    proof -
+      have False if "closed_segment ?m c \<subseteq> {z. (closed_segment a z \<union> closed_segment z b) \<inter> S \<noteq> {}}"
+      proof -
+        have closb: "closed_segment ?m c \<subseteq>
+                 {z \<in> closed_segment ?m c. closed_segment a z \<inter> S \<noteq> {}} \<union> {z \<in> closed_segment ?m c. closed_segment z b \<inter> S \<noteq> {}}"
+          using that by blast
+        have *: "countable {z \<in> closed_segment ?m c. closed_segment z u \<inter> S \<noteq> {}}"
+          if "u \<in> U" "u \<notin> S" and ncoll: "\<not> collinear {?m, c, u}" for u
+        proof -
+          have **: False if x1: "x1 \<in> closed_segment ?m c" and x2: "x2 \<in> closed_segment ?m c"
+                            and "x1 \<noteq> x2" "x1 \<noteq> u"
+                            and w: "w \<in> closed_segment x1 u" "w \<in> closed_segment x2 u"
+                            and "w \<in> S" for x1 x2 w
+          proof -
+            have "x1 \<in> affine hull {?m,c}" "x2 \<in> 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 "\<not> collinear {x1, u, x2}"
+            proof
+              assume "collinear {x1, u, x2}"
+              then have "collinear {?m, c, u}"
+                by (metis (full_types) \<open>c \<noteq> ?m\<close> coll_x1 coll_x2 collinear_3_trans insert_commute ncoll \<open>x1 \<noteq> x2\<close>)
+              with ncoll show False ..
+            qed
+            then have "closed_segment x1 u \<inter> 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 \<open>u \<notin> S\<close> \<open>w = u\<close> that(7) by auto
+          qed
+          then have disj: "disjoint ((\<Union>z\<in>closed_segment ?m c. {closed_segment z u \<inter> S}))"
+            by (fastforce simp: pairwise_def disjnt_def)
+          have cou: "countable ((\<Union>z \<in> closed_segment ?m c. {closed_segment z u \<inter> S}) - {{}})"
+            apply (rule pairwise_disjnt_countable_Union [OF _ pairwise_subset [OF disj]])
+             apply (rule countable_subset [OF _ \<open>countable S\<close>], auto)
+            done
+          define f where "f \<equiv> \<lambda>X. (THE z. z \<in> closed_segment ?m c \<and> X = closed_segment z u \<inter> S)"
+          show ?thesis
+          proof (rule countable_subset [OF _ countable_image [OF cou, where f=f]], clarify)
+            fix x
+            assume x: "x \<in> closed_segment ?m c" "closed_segment x u \<inter> S \<noteq> {}"
+            show "x \<in> f ` ((\<Union>z\<in>closed_segment ?m c. {closed_segment z u \<inter> S}) - {{}})"
+            proof (rule_tac x="closed_segment x u \<inter> S" in image_eqI)
+              show "x = f (closed_segment x u \<inter> S)"
+                unfolding f_def
+                apply (rule the_equality [symmetric])
+                using x  apply (auto simp: dest: **)
+                done
+            qed (use x in auto)
+          qed
+        qed
+        have "uncountable (closed_segment ?m c)"
+          by (metis \<open>c \<noteq> ?m\<close> uncountable_closed_segment)
+        then show False
+          using closb * [OF \<open>a \<in> U\<close> \<open>a \<notin> S\<close> ncoll_mca] * [OF \<open>b \<in> U\<close> \<open>b \<notin> S\<close> ncoll_mcb]
+          apply (simp add: closed_segment_commute)
+          by (simp add: 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) \<subseteq> U"
+        by (metis \<open>a \<in> U\<close> \<open>b \<in> U\<close> \<open>closed_segment ?m c \<subseteq> U\<close> z \<open>convex U\<close> closed_segment_subset contra_subsetD path_image_linepath subset_path_image_join)
+      with disjS show "path_image (linepath a z +++ linepath z b) \<subseteq> 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" "\<not> 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 \<noteq> 1"
+    shows "path_connected(S - {a})"
+proof -
+  consider "aff_dim S = -1" | "aff_dim S = 0" | "aff_dim S \<ge> 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 \<ge> 2"
+    then have "\<not> collinear S"
+    proof (clarsimp simp add: collinear_affine_hull)
+      fix u v
+      assume "S \<subseteq> affine hull {u, v}"
+      then have "aff_dim S \<le> 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
+    then show ?thesis
+      apply (rule path_connected_convex_diff_countable [OF \<open>convex S\<close>])
+      by simp
+  qed
+qed
+
+lemma connected_punctured_convex:
+  shows "\<lbrakk>convex S; aff_dim S \<noteq> 1\<rbrakk> \<Longrightarrow> 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 \<le> DIM('a)" "countable S"
+  shows "path_connected(- S)"
+proof -
+  have "path_connected(UNIV - S)"
+    apply (rule path_connected_convex_diff_countable)
+    using assms by (auto simp: collinear_aff_dim [of "UNIV :: 'a set"])
+  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 (subtopology euclidean (affine hull S)) S"
+      and "\<not> collinear S" "countable T"
+    shows "path_connected(S - T)"
+proof (clarsimp simp add: path_connected_component)
+  fix x y
+  assume xy: "x \<in> S" "x \<notin> T" "y \<in> S" "y \<notin> T"
+  show "path_component (S - T) x y"
+  proof (rule connected_equivalence_relation_gen [OF \<open>connected S\<close>, where P = "\<lambda>x. x \<notin> T"])
+    show "\<exists>z. z \<in> U \<and> z \<notin> T" if opeU: "openin (subtopology euclidean S) U" and "x \<in> U" for U x
+    proof -
+      have "openin (subtopology euclidean (affine hull S)) U"
+        using opeU ope openin_trans by blast
+      with \<open>x \<in> U\<close> obtain r where Usub: "U \<subseteq> affine hull S" and "r > 0"
+                              and subU: "ball x r \<inter> affine hull S \<subseteq> U"
+        by (auto simp: openin_contains_ball)
+      with \<open>x \<in> U\<close> have x: "x \<in> ball x r \<inter> affine hull S"
+        by auto
+      have "\<not> S \<subseteq> {x}"
+        using \<open>\<not> collinear S\<close>  collinear_subset by blast
+      then obtain x' where "x' \<noteq> x" "x' \<in> S"
+        by blast
+      obtain y where y: "y \<noteq> x" "y \<in> ball x r \<inter> affine hull S"
+      proof
+        show "x + (r / 2 / norm(x' - x)) *\<^sub>R (x' - x) \<noteq> x"
+          using \<open>x' \<noteq> x\<close> \<open>r > 0\<close> by auto
+        show "x + (r / 2 / norm (x' - x)) *\<^sub>R (x' - x) \<in> ball x r \<inter> affine hull S"
+          using \<open>x' \<noteq> x\<close> \<open>r > 0\<close> \<open>x' \<in> S\<close> x
+          by (simp add: dist_norm mem_affine_3_minus hull_inc)
+      qed
+      have "convex (ball x r \<inter> 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 "\<not> U \<subseteq> T"
+        using \<open>countable T\<close> countable_subset by blast
+      then show ?thesis by blast
+    qed
+    show "\<exists>U. openin (subtopology euclidean S) U \<and> x \<in> U \<and>
+              (\<forall>x\<in>U. \<forall>y\<in>U. x \<notin> T \<and> y \<notin> T \<longrightarrow> path_component (S - T) x y)"
+          if "x \<in> S" for x
+    proof -
+      obtain r where Ssub: "S \<subseteq> affine hull S" and "r > 0"
+                 and subS: "ball x r \<inter> affine hull S \<subseteq> S"
+        using ope \<open>x \<in> S\<close> by (auto simp: openin_contains_ball)
+      then have conv: "convex (ball x r \<inter> affine hull S)"
+        by (simp add: affine_imp_convex convex_Int)
+      have "\<not> aff_dim (affine hull S) \<le> 1"
+        using \<open>\<not> collinear S\<close> collinear_aff_dim by auto
+      then have "\<not> collinear (ball x r \<inter> affine hull S)"
+        apply (simp add: collinear_aff_dim)
+        by (metis (no_types, hide_lams) aff_dim_convex_Int_open IntI open_ball \<open>0 < r\<close> aff_dim_affine_hull affine_affine_hull affine_imp_convex centre_in_ball empty_iff hull_subset inf_commute subsetCE that)
+      then have *: "path_connected ((ball x r \<inter> affine hull S) - T)"
+        by (rule path_connected_convex_diff_countable [OF conv _ \<open>countable T\<close>])
+      have ST: "ball x r \<inter> affine hull S - T \<subseteq> S - T"
+        using subS by auto
+      show ?thesis
+      proof (intro exI conjI)
+        show "x \<in> ball x r \<inter> affine hull S"
+          using \<open>x \<in> S\<close> \<open>r > 0\<close> by (simp add: hull_inc)
+        have "openin (subtopology euclidean (affine hull S)) (ball x r \<inter> affine hull S)"
+          by (subst inf.commute) (simp add: openin_Int_open)
+        then show "openin (subtopology euclidean S) (ball x r \<inter> affine hull S)"
+          by (rule openin_subset_trans [OF _ subS Ssub])
+      qed (use * path_component_trans in \<open>auto simp: path_connected_component path_component_of_subset [OF ST]\<close>)
+    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 (subtopology euclidean (affine hull S)) S"
+      and "\<not> 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 \<le> DIM('a)" "open S" "connected S" "countable T"
+  shows "path_connected(S - T)"
+proof (cases "S = {}")
+  case True
+  then show ?thesis
+    by (simp add: path_connected_empty)
+next
+  case False
+  show ?thesis
+  proof (rule path_connected_openin_diff_countable)
+    show "openin (subtopology euclidean (affine hull S)) S"
+      by (simp add: assms hull_subset open_subset)
+    show "\<not> 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 \<le> DIM('a)" "open S" "connected S" "countable T"
+  shows "connected(S - T)"
+by (simp add: assms path_connected_imp_connected path_connected_open_diff_countable)
+
+
+
+subsection%unimportant \<open>Self-homeomorphisms shuffling points about\<close>
+
+subsubsection%unimportant\<open>The theorem \<open>homeomorphism_moving_points_exists\<close>\<close>
+
+lemma homeomorphism_moving_point_1:
+  fixes a :: "'a::euclidean_space"
+  assumes "affine T" "a \<in> T" and u: "u \<in> ball a r \<inter> T"
+  obtains f g where "homeomorphism (cball a r \<inter> T) (cball a r \<inter> T) f g"
+                    "f a = u" "\<And>x. x \<in> sphere a r \<Longrightarrow> f x = x"
+proof -
+  have nou: "norm (u - a) < r" and "u \<in> 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 \<equiv> \<lambda>x. (1 - norm(x - a) / r) *\<^sub>R (u - a) + x"
+  have *: "False" if eq: "x + (norm y / r) *\<^sub>R u = y + (norm x / r) *\<^sub>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)) *\<^sub>R u"
+      using eq by (simp add: algebra_simps)
+    then have "norm x = norm (y + ((norm x - norm y) / r) *\<^sub>R u)"
+      by (metis diff_divide_distrib)
+    also have "\<dots> \<le> norm y + norm(((norm x - norm y) / r) *\<^sub>R u)"
+      using norm_triangle_ineq by blast
+    also have "\<dots> = norm y + (norm x - norm y) * (norm u / r)"
+      using yx \<open>r > 0\<close>
+      by (simp add: divide_simps)
+    also have "\<dots> < norm y + (norm x - norm y) * 1"
+      apply (subst add_less_cancel_left)
+      apply (rule mult_strict_left_mono)
+      using nou \<open>0 < r\<close> yx
+       apply (simp_all add: field_simps)
+      done
+    also have "\<dots> = 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) *\<^sub>R (u - a) + x =
+            (1 - norm (y - a) / r) *\<^sub>R (u - a) + y"
+    then have eq: "(x - a) + (norm (y - a) / r) *\<^sub>R (u - a) = (y - a) + (norm (x - a) / r) *\<^sub>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 \<inter> T)"
+    using inj_on_Int by fastforce
+  have contf: "continuous_on (cball a r \<inter> T) f"
+    unfolding f_def using \<open>0 < r\<close>  by (intro continuous_intros) blast
+  have fim: "f ` (cball a r \<inter> T) = cball a r \<inter> T"
+  proof
+    have *: "norm (y + (1 - norm y / r) *\<^sub>R u) \<le> r" if "norm y \<le> r" "norm u < r" for y u::'a
+    proof -
+      have "norm (y + (1 - norm y / r) *\<^sub>R u) \<le> norm y + norm((1 - norm y / r) *\<^sub>R u)"
+        using norm_triangle_ineq by blast
+      also have "\<dots> = norm y + abs(1 - norm y / r) * norm u"
+        by simp
+      also have "\<dots> \<le> r"
+      proof -
+        have "(r - norm u) * (r - norm y) \<ge> 0"
+          using that by auto
+        then have "r * norm u + r * norm y \<le> r * r + norm u * norm y"
+          by (simp add: algebra_simps)
+        then show ?thesis
+        using that \<open>0 < r\<close> by (simp add: abs_if field_simps)
+      qed
+      finally show ?thesis .
+    qed
+    have "f ` (cball a r) \<subseteq> cball a r"
+      apply (clarsimp simp add: dist_norm norm_minus_commute f_def)
+      using * by (metis diff_add_eq diff_diff_add diff_diff_eq2 norm_minus_commute nou)
+    moreover have "f ` T \<subseteq> T"
+      unfolding f_def using \<open>affine T\<close> \<open>a \<in> T\<close> \<open>u \<in> T\<close>
+      by (force simp: add.commute mem_affine_3_minus)
+    ultimately show "f ` (cball a r \<inter> T) \<subseteq> cball a r \<inter> T"
+      by blast
+  next
+    show "cball a r \<inter> T \<subseteq> f ` (cball a r \<inter> T)"
+    proof (clarsimp simp add: dist_norm norm_minus_commute)
+      fix x
+      assume x: "norm (x - a) \<le> r" and "x \<in> T"
+      have "\<exists>v \<in> {0..1}. ((1 - v) * r - norm ((x - a) - v *\<^sub>R (u - a))) \<bullet> 1 = 0"
+        by (rule ivt_decreasing_component_on_1) (auto simp: x continuous_intros)
+      then obtain v where "0\<le>v" "v\<le>1" and v: "(1 - v) * r = norm ((x - a) - v *\<^sub>R (u - a))"
+        by auto
+      show "x \<in> f ` (cball a r \<inter> T)"
+      proof (rule image_eqI)
+        show "x = f (x - v *\<^sub>R (u - a))"
+          using \<open>r > 0\<close> v by (simp add: f_def field_simps)
+        have "x - v *\<^sub>R (u - a) \<in> cball a r"
+          using \<open>r > 0\<close> v \<open>0 \<le> v\<close>
+          apply (simp add: field_simps dist_norm norm_minus_commute)
+          by (metis le_add_same_cancel2 order.order_iff_strict zero_le_mult_iff)
+        moreover have "x - v *\<^sub>R (u - a) \<in> T"
+          by (simp add: f_def \<open>affine T\<close> \<open>u \<in> T\<close> \<open>x \<in> T\<close> assms mem_affine_3_minus2)
+        ultimately show "x - v *\<^sub>R (u - a) \<in> cball a r \<inter> T"
+          by blast
+      qed
+    qed
+  qed
+  have "\<exists>g. homeomorphism (cball a r \<inter> T) (cball a r \<inter> T) f g"
+    apply (rule homeomorphism_compact [OF _ contf fim inj_onf])
+    apply (simp add: affine_closed compact_Int_closed \<open>affine T\<close>)
+    done
+  then show ?thesis
+    apply (rule exE)
+    apply (erule_tac f=f in that)
+    using \<open>r > 0\<close>
+     apply (simp_all add: f_def dist_norm norm_minus_commute)
+    done
+qed
+
+corollary%unimportant homeomorphism_moving_point_2:
+  fixes a :: "'a::euclidean_space"
+  assumes "affine T" "a \<in> T" and u: "u \<in> ball a r \<inter> T" and v: "v \<in> ball a r \<inter> T"
+  obtains f g where "homeomorphism (cball a r \<inter> T) (cball a r \<inter> T) f g"
+                    "f u = v" "\<And>x. \<lbrakk>x \<in> sphere a r; x \<in> T\<rbrakk> \<Longrightarrow> 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 \<inter> T) (cball a r \<inter> T) f1 g1"
+                 and "f1 a = u" and f1: "\<And>x. x \<in> sphere a r \<Longrightarrow> f1 x = x"
+    using homeomorphism_moving_point_1 [OF \<open>affine T\<close> \<open>a \<in> T\<close> u] by blast
+  obtain f2 g2 where hom2: "homeomorphism (cball a r \<inter> T) (cball a r \<inter> T) f2 g2"
+                 and "f2 a = v" and f2: "\<And>x. x \<in> sphere a r \<Longrightarrow> f2 x = x"
+    using homeomorphism_moving_point_1 [OF \<open>affine T\<close> \<open>a \<in> T\<close> v] by blast
+  show ?thesis
+  proof
+    show "homeomorphism (cball a r \<inter> T) (cball a r \<inter> T) (f2 \<circ> g1) (f1 \<circ> g2)"
+      by (metis homeomorphism_compose homeomorphism_symD hom1 hom2)
+    have "g1 u = a"
+      using \<open>0 < r\<close> \<open>f1 a = u\<close> assms hom1 homeomorphism_apply1 by fastforce
+    then show "(f2 \<circ> g1) u = v"
+      by (simp add: \<open>f2 a = v\<close>)
+    show "\<And>x. \<lbrakk>x \<in> sphere a r; x \<in> T\<rbrakk> \<Longrightarrow> (f2 \<circ> g1) x = x"
+      using f1 f2 hom1 homeomorphism_apply1 by fastforce
+  qed
+qed
+
+
+corollary%unimportant homeomorphism_moving_point_3:
+  fixes a :: "'a::euclidean_space"
+  assumes "affine T" "a \<in> T" and ST: "ball a r \<inter> T \<subseteq> S" "S \<subseteq> T"
+      and u: "u \<in> ball a r \<inter> T" and v: "v \<in> ball a r \<inter> T"
+  obtains f g where "homeomorphism S S f g"
+                    "f u = v" "{x. \<not> (f x = x \<and> g x = x)} \<subseteq> ball a r \<inter> T"
+proof -
+  obtain f g where hom: "homeomorphism (cball a r \<inter> T) (cball a r \<inter> T) f g"
+               and "f u = v" and fid: "\<And>x. \<lbrakk>x \<in> sphere a r; x \<in> T\<rbrakk> \<Longrightarrow> f x = x"
+    using homeomorphism_moving_point_2 [OF \<open>affine T\<close> \<open>a \<in> T\<close> u v] by blast
+  have gid: "\<And>x. \<lbrakk>x \<in> sphere a r; x \<in> T\<rbrakk> \<Longrightarrow> g x = x"
+    using fid hom homeomorphism_apply1 by fastforce
+  define ff where "ff \<equiv> \<lambda>x. if x \<in> ball a r \<inter> T then f x else x"
+  define gg where "gg \<equiv> \<lambda>x. if x \<in> ball a r \<inter> T then g x else x"
+  show ?thesis
+  proof
+    show "homeomorphism S S ff gg"
+    proof (rule homeomorphismI)
+      have "continuous_on ((cball a r \<inter> T) \<union> (T - ball a r)) ff"
+        apply (simp add: ff_def)
+        apply (rule continuous_on_cases)
+        using homeomorphism_cont1 [OF hom]
+            apply (auto simp: affine_closed \<open>affine T\<close> continuous_on_id fid)
+        done
+      then show "continuous_on S ff"
+        apply (rule continuous_on_subset)
+        using ST by auto
+      have "continuous_on ((cball a r \<inter> T) \<union> (T - ball a r)) gg"
+        apply (simp add: gg_def)
+        apply (rule continuous_on_cases)
+        using homeomorphism_cont2 [OF hom]
+            apply (auto simp: affine_closed \<open>affine T\<close> continuous_on_id gid)
+        done
+      then show "continuous_on S gg"
+        apply (rule continuous_on_subset)
+        using ST by auto
+      show "ff ` S \<subseteq> S"
+      proof (clarsimp simp add: ff_def)
+        fix x
+        assume "x \<in> S" and x: "dist a x < r" and "x \<in> T"
+        then have "f x \<in> cball a r \<inter> T"
+          using homeomorphism_image1 [OF hom] by force
+        then show "f x \<in> S"
+          using ST(1) \<open>x \<in> T\<close> gid hom homeomorphism_def x by fastforce
+      qed
+      show "gg ` S \<subseteq> S"
+      proof (clarsimp simp add: gg_def)
+        fix x
+        assume "x \<in> S" and x: "dist a x < r" and "x \<in> T"
+        then have "g x \<in> cball a r \<inter> T"
+          using homeomorphism_image2 [OF hom] by force
+        then have "g x \<in> ball a r"
+          using homeomorphism_apply2 [OF hom]
+            by (metis Diff_Diff_Int Diff_iff  \<open>x \<in> T\<close> cball_def fid le_less mem_Collect_eq mem_ball mem_sphere x)
+        then show "g x \<in> S"
+          using ST(1) \<open>g x \<in> cball a r \<inter> T\<close> by force
+        qed
+      show "\<And>x. x \<in> S \<Longrightarrow> gg (ff x) = x"
+        apply (simp add: ff_def gg_def)
+        using homeomorphism_apply1 [OF hom] homeomorphism_image1 [OF hom]
+        apply auto
+        apply (metis Int_iff homeomorphism_apply1 [OF hom] fid image_eqI less_eq_real_def mem_cball mem_sphere)
+        done
+      show "\<And>x. x \<in> S \<Longrightarrow> ff (gg x) = x"
+        apply (simp add: ff_def gg_def)
+        using homeomorphism_apply2 [OF hom] homeomorphism_image2 [OF hom]
+        apply auto
+        apply (metis Int_iff fid image_eqI less_eq_real_def mem_cball mem_sphere)
+        done
+    qed
+    show "ff u = v"
+      using u by (auto simp: ff_def \<open>f u = v\<close>)
+    show "{x. \<not> (ff x = x \<and> gg x = x)} \<subseteq> ball a r \<inter> T"
+      by (auto simp: ff_def gg_def)
+  qed
+qed
+
+
+proposition%unimportant homeomorphism_moving_point:
+  fixes a :: "'a::euclidean_space"
+  assumes ope: "openin (subtopology euclidean (affine hull S)) S"
+      and "S \<subseteq> T"
+      and TS: "T \<subseteq> affine hull S"
+      and S: "connected S" "a \<in> S" "b \<in> S"
+  obtains f g where "homeomorphism T T f g" "f a = b"
+                    "{x. \<not> (f x = x \<and> g x = x)} \<subseteq> S"
+                    "bounded {x. \<not> (f x = x \<and> g x = x)}"
+proof -
+  have 1: "\<exists>h k. homeomorphism T T h k \<and> h (f d) = d \<and>
+              {x. \<not> (h x = x \<and> k x = x)} \<subseteq> S \<and> bounded {x. \<not> (h x = x \<and> k x = x)}"
+        if "d \<in> S" "f d \<in> S" and homfg: "homeomorphism T T f g"
+        and S: "{x. \<not> (f x = x \<and> g x = x)} \<subseteq> S"
+        and bo: "bounded {x. \<not> (f x = x \<and> 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 \<open>S \<subseteq> T\<close> homeomorphism_def subsetD \<open>d \<in> S\<close>)
+    show "{x. \<not> (g x = x \<and> f x = x)} \<subseteq> S"
+      using S by blast
+    show "bounded {x. \<not> (g x = x \<and> f x = x)}"
+      using bo by (simp add: conj_commute)
+  qed
+  have 2: "\<exists>f g. homeomorphism T T f g \<and> f x = f2 (f1 x) \<and>
+                 {x. \<not> (f x = x \<and> g x = x)} \<subseteq> S \<and> bounded {x. \<not> (f x = x \<and> g x = x)}"
+             if "x \<in> S" "f1 x \<in> S" "f2 (f1 x) \<in> S"
+                and hom: "homeomorphism T T f1 g1" "homeomorphism T T f2 g2"
+                and sub: "{x. \<not> (f1 x = x \<and> g1 x = x)} \<subseteq> S"   "{x. \<not> (f2 x = x \<and> g2 x = x)} \<subseteq> S"
+                and bo: "bounded {x. \<not> (f1 x = x \<and> g1 x = x)}"  "bounded {x. \<not> (f2 x = x \<and> g2 x = x)}"
+             for x f1 f2 g1 g2
+  proof (intro exI conjI)
+    show homgf: "homeomorphism T T (f2 \<circ> f1) (g1 \<circ> g2)"
+      by (metis homeomorphism_compose hom)
+    then show "(f2 \<circ> f1) x = f2 (f1 x)"
+      by force
+    show "{x. \<not> ((f2 \<circ> f1) x = x \<and> (g1 \<circ> g2) x = x)} \<subseteq> S"
+      using sub by force
+    have "bounded ({x. \<not>(f1 x = x \<and> g1 x = x)} \<union> {x. \<not>(f2 x = x \<and> g2 x = x)})"
+      using bo by simp
+    then show "bounded {x. \<not> ((f2 \<circ> f1) x = x \<and> (g1 \<circ> g2) x = x)}"
+      by (rule bounded_subset) auto
+  qed
+  have 3: "\<exists>U. openin (subtopology euclidean S) U \<and>
+              d \<in> U \<and>
+              (\<forall>x\<in>U.
+                  \<exists>f g. homeomorphism T T f g \<and> f d = x \<and>
+                        {x. \<not> (f x = x \<and> g x = x)} \<subseteq> S \<and>
+                        bounded {x. \<not> (f x = x \<and> g x = x)})"
+           if "d \<in> S" for d
+  proof -
+    obtain r where "r > 0" and r: "ball d r \<inter> affine hull S \<subseteq> S"
+      by (metis \<open>d \<in> S\<close> ope openin_contains_ball)
+    have *: "\<exists>f g. homeomorphism T T f g \<and> f d = e \<and>
+                   {x. \<not> (f x = x \<and> g x = x)} \<subseteq> S \<and>
+                   bounded {x. \<not> (f x = x \<and> g x = x)}" if "e \<in> S" "e \<in> ball d r" for e
+      apply (rule homeomorphism_moving_point_3 [of "affine hull S" d r T d e])
+      using r \<open>S \<subseteq> T\<close> TS that
+            apply (auto simp: \<open>d \<in> S\<close> \<open>0 < r\<close> hull_inc)
+      using bounded_subset by blast
+    show ?thesis
+      apply (rule_tac x="S \<inter> ball d r" in exI)
+      apply (intro conjI)
+        apply (simp add: openin_open_Int)
+       apply (simp add: \<open>0 < r\<close> that)
+      apply (blast intro: *)
+      done
+  qed
+  have "\<exists>f g. homeomorphism T T f g \<and> f a = b \<and>
+              {x. \<not> (f x = x \<and> g x = x)} \<subseteq> S \<and> bounded {x. \<not> (f x = x \<and> g x = x)}"
+    apply (rule connected_equivalence_relation [OF S], safe)
+      apply (blast intro: 1 2 3)+
+    done
+  then show ?thesis
+    using that by auto
+qed
+
+
+lemma homeomorphism_moving_points_exists_gen:
+  assumes K: "finite K" "\<And>i. i \<in> K \<Longrightarrow> x i \<in> S \<and> y i \<in> S"
+             "pairwise (\<lambda>i j. (x i \<noteq> x j) \<and> (y i \<noteq> y j)) K"
+      and "2 \<le> aff_dim S"
+      and ope: "openin (subtopology euclidean (affine hull S)) S"
+      and "S \<subseteq> T" "T \<subseteq> affine hull S" "connected S"
+  shows "\<exists>f g. homeomorphism T T f g \<and> (\<forall>i \<in> K. f(x i) = y i) \<and>
+               {x. \<not> (f x = x \<and> g x = x)} \<subseteq> S \<and> bounded {x. \<not> (f x = x \<and> 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: "\<And>j. \<lbrakk>j \<in> K; j \<noteq> i\<rbrakk> \<Longrightarrow> x i \<noteq> x j \<and> y i \<noteq> y j"
+       and pw: "pairwise (\<lambda>i j. x i \<noteq> x j \<and> y i \<noteq> y j) K"
+       and "x i \<in> S" "y i \<in> S"
+       and xyS: "\<And>i. i \<in> K \<Longrightarrow> x i \<in> S \<and> y i \<in> S"
+    by (simp_all add: pairwise_insert)
+  obtain f g where homfg: "homeomorphism T T f g" and feq: "\<And>i. i \<in> K \<Longrightarrow> f(x i) = y i"
+               and fg_sub: "{x. \<not> (f x = x \<and> g x = x)} \<subseteq> S"
+               and bo_fg: "bounded {x. \<not> (f x = x \<and> g x = x)}"
+    using insert.IH [OF xyS pw] insert.prems by (blast intro: that)
+  then have "\<exists>f g. homeomorphism T T f g \<and> (\<forall>i \<in> K. f(x i) = y i) \<and>
+                   {x. \<not> (f x = x \<and> g x = x)} \<subseteq> S \<and> bounded {x. \<not> (f x = x \<and> g x = x)}"
+    using insert by blast
+  have aff_eq: "affine hull (S - y ` K) = affine hull S"
+    apply (rule affine_hull_Diff)
+    apply (auto simp: insert)
+    using \<open>y i \<in> S\<close> insert.hyps(2) xney xyS by fastforce
+  have f_in_S: "f x \<in> S" if "x \<in> S" for x
+    using homfg fg_sub homeomorphism_apply1 \<open>S \<subseteq> T\<close>
+  proof -
+    have "(f (f x) \<noteq> f x \<or> g (f x) \<noteq> f x) \<or> f x \<in> S"
+      by (metis \<open>S \<subseteq> T\<close> 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. \<not> (h x = x \<and> k x = x)} \<subseteq> S - y ` K"
+               and bo_hk:  "bounded {x. \<not> (h x = x \<and> k x = x)}"
+  proof (rule homeomorphism_moving_point [of "S - y`K" T "f(x i)" "y i"])
+    show "openin (subtopology euclidean (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 \<subseteq> T"
+      using \<open>S \<subseteq> T\<close> by auto
+    show "T \<subseteq> affine hull (S - y ` K)"
+      using insert by (simp add: aff_eq)
+    show "connected (S - y ` K)"
+    proof (rule connected_openin_diff_countable [OF \<open>connected S\<close> ope])
+      show "\<not> collinear S"
+        using collinear_aff_dim \<open>2 \<le> aff_dim S\<close> by force
+      show "countable (y ` K)"
+        using countable_finite insert.hyps(1) by blast
+    qed
+    show "f (x i) \<in> S - y ` K"
+      apply (auto simp: f_in_S \<open>x i \<in> S\<close>)
+        by (metis feq homfg \<open>x i \<in> S\<close> homeomorphism_def \<open>S \<subseteq> T\<close> \<open>i \<notin> K\<close> subsetCE xney xyS)
+    show "y i \<in> S - y ` K"
+      using insert.hyps xney by (auto simp: \<open>y i \<in> S\<close>)
+  qed blast
+  show ?case
+  proof (intro exI conjI)
+    show "homeomorphism T T (h \<circ> f) (g \<circ> k)"
+      using homfg homhk homeomorphism_compose by blast
+    show "\<forall>i \<in> insert i K. (h \<circ> f) (x i) = y i"
+      using feq hk_sub by (auto simp: heq)
+    show "{x. \<not> ((h \<circ> f) x = x \<and> (g \<circ> k) x = x)} \<subseteq> S"
+      using fg_sub hk_sub by force
+    have "bounded ({x. \<not>(f x = x \<and> g x = x)} \<union> {x. \<not>(h x = x \<and> k x = x)})"
+      using bo_fg bo_hk bounded_Un by blast
+    then show "bounded {x. \<not> ((h \<circ> f) x = x \<and> (g \<circ> k) x = x)}"
+      by (rule bounded_subset) auto
+  qed
+qed
+
+proposition%unimportant homeomorphism_moving_points_exists:
+  fixes S :: "'a::euclidean_space set"
+  assumes 2: "2 \<le> DIM('a)" "open S" "connected S" "S \<subseteq> T" "finite K"
+      and KS: "\<And>i. i \<in> K \<Longrightarrow> x i \<in> S \<and> y i \<in> S"
+      and pw: "pairwise (\<lambda>i j. (x i \<noteq> x j) \<and> (y i \<noteq> y j)) K"
+      and S: "S \<subseteq> T" "T \<subseteq> affine hull S" "connected S"
+  obtains f g where "homeomorphism T T f g" "\<And>i. i \<in> K \<Longrightarrow> f(x i) = y i"
+                    "{x. \<not> (f x = x \<and> g x = x)} \<subseteq> S" "bounded {x. (\<not> (f x = x \<and> 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>open S\<close>)
+  then have ope: "openin (subtopology euclidean (affine hull S)) S"
+    using \<open>open S\<close> open_openin by auto
+  have "2 \<le> DIM('a)" by (rule 2)
+  also have "\<dots> = aff_dim (UNIV :: 'a set)"
+    by simp
+  also have "\<dots> \<le> aff_dim S"
+    by (metis aff_dim_UNIV aff_dim_affine_hull aff_dim_le_DIM affS)
+  finally have "2 \<le> aff_dim S"
+    by linarith
+  then show ?thesis
+    using homeomorphism_moving_points_exists_gen [OF \<open>finite K\<close> KS pw _ ope S] that by fastforce
+qed
+
+subsubsection%unimportant\<open>The theorem \<open>homeomorphism_grouping_points_exists\<close>\<close>
+
+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 \<equiv> \<lambda>x. ((d - c) / (b - a)) * x + (c - a * ((d - c) / (b - a)))"
+  have "\<exists>g. homeomorphism (cbox a b) (cbox c d) f g"
+  proof (rule homeomorphism_compact)
+    show "continuous_on (cbox a b) f"
+      apply (simp add: f_def)
+      apply (intro continuous_intros)
+      using assms by auto
+    have "f ` {a..b} = {c..d}"
+      unfolding f_def image_affinity_atLeastAtMost
+      using assms sum_sqs_eq by (auto simp: divide_simps algebra_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 divide_simps algebra_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 \<in> cbox a c" "v \<in> 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"
+                   "\<And>x. x \<in> cbox a b \<Longrightarrow> f x = f1 x" "\<And>x. x \<in> cbox b c \<Longrightarrow> f x = f2 x"
+proof -
+  have le: "a \<le> b" "b \<le> c" "u \<le> v" "v \<le> w"
+    using assms by simp_all
+  then have ac: "cbox a c = cbox a b \<union> cbox b c" and uw: "cbox u w = cbox u v \<union> cbox v w"
+    by auto
+  define f where "f \<equiv> \<lambda>x. if x \<le> b then f1 x else f2 x"
+  have "\<exists>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"
+      apply (simp add: f_def)
+      apply (rule continuous_on_cases_le [OF continuous_on_subset [OF cf1] continuous_on_subset [OF cf2]])
+      using le eq apply (force simp: continuous_on_id)+
+      done
+    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"
+      apply (simp only: ac uw image_Un)
+      by (metis hom_ab hom_bc homeomorphism_def)
+    have neq12: "f1 x \<noteq> f2 y" if x: "a \<le> x" "x \<le> b" and y: "b < y" "y \<le> c" for x y
+    proof -
+      have "f1 x \<in> cbox u v"
+        by (metis hom_ab homeomorphism_def image_eqI mem_box_real(2) x)
+      moreover have "f2 y \<in> 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 \<noteq> 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  apply force
+      done
+  qed auto
+  then obtain g where "homeomorphism (cbox a c) (cbox u w) f g" ..
+  then show ?thesis
+    apply (rule that)
+    using eq le by (auto simp: f_def)
+qed
+
+lemma homeomorphism_grouping_point_3:
+  fixes a::real
+  assumes cbox_sub: "cbox c d \<subseteq> box a b" "cbox u v \<subseteq> box a b"
+      and box_ne: "box c d \<noteq> {}" "box u v \<noteq> {}"
+  obtains f g where "homeomorphism (cbox a b) (cbox a b) f g" "f a = a" "f b = b"
+                    "\<And>x. x \<in> cbox c d \<Longrightarrow> f x \<in> cbox u v"
+proof -
+  have less: "a < c" "a < u" "d < b" "v < b" "c < d" "u < v" "cbox c d \<noteq> {}"
+    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 \<open>a < c\<close> \<open>a < u\<close>] .
+  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 \<open>c < d\<close> \<open>u < v\<close>] .
+  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 \<open>d < b\<close> \<open>v < b\<close>] .
+  obtain f4 g4 where 4: "homeomorphism (cbox a d) (cbox a v) f4 g4" and "f4 a = a" "f4 d = v"
+                 and f4_eq: "\<And>x. x \<in> cbox a c \<Longrightarrow> f4 x = f1 x" "\<And>x. x \<in> cbox c d \<Longrightarrow> 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: "\<And>x. x \<in> cbox a d \<Longrightarrow> f x = f4 x" "\<And>x. x \<in> cbox d b \<Longrightarrow> 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
+    apply (rule that [OF fg])
+    using f4_eq f_eq homeomorphism_image1 [OF 2]
+    apply simp
+    by (metis atLeastAtMost_iff box_real(1) box_real(2) cbox_sub(1) greaterThanLessThan_iff imageI less_eq_real_def subset_eq)
+qed
+
+
+lemma homeomorphism_grouping_point_4:
+  fixes T :: "real set"
+  assumes "open U" "open S" "connected S" "U \<noteq> {}" "finite K" "K \<subseteq> S" "U \<subseteq> S" "S \<subseteq> T"
+  obtains f g where "homeomorphism T T f g"
+                    "\<And>x. x \<in> K \<Longrightarrow> f x \<in> U" "{x. (\<not> (f x = x \<and> g x = x))} \<subseteq> S"
+                    "bounded {x. (\<not> (f x = x \<and> g x = x))}"
+proof -
+  obtain c d where "box c d \<noteq> {}" "cbox c d \<subseteq> U"
+  proof -
+    obtain u where "u \<in> U"
+      using \<open>U \<noteq> {}\<close> by blast
+    then obtain e where "e > 0" "cball u e \<subseteq> U"
+      using \<open>open U\<close> 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: \<open>finite K\<close> finite_imp_compact)
+  obtain a b where "box a b \<noteq> {}" "K \<subseteq> cbox a b" "cbox a b \<subseteq> S"
+  proof (cases "K = {}")
+    case True then show ?thesis
+      using \<open>box c d \<noteq> {}\<close> \<open>cbox c d \<subseteq> U\<close> \<open>U \<subseteq> S\<close> that by blast
+  next
+    case False
+    then obtain a b where "a \<in> K" "b \<in> K"
+            and a: "\<And>x. x \<in> K \<Longrightarrow> a \<le> x" and b: "\<And>x. x \<in> K \<Longrightarrow> x \<le> b"
+      using compact_attains_inf compact_attains_sup by (metis \<open>compact K\<close>)+
+    obtain e where "e > 0" "cball b e \<subseteq> S"
+      using \<open>open S\<close> open_contains_cball
+      by (metis \<open>b \<in> K\<close> \<open>K \<subseteq> S\<close> subsetD)
+    show ?thesis
+    proof
+      show "box a (b + e) \<noteq> {}"
+        using \<open>0 < e\<close> \<open>b \<in> K\<close> a by force
+      show "K \<subseteq> cbox a (b + e)"
+        using \<open>0 < e\<close> a b by fastforce
+      have "a \<in> S"
+        using \<open>a \<in> K\<close> assms(6) by blast
+      have "b + e \<in> S"
+        using \<open>0 < e\<close> \<open>cball b e \<subseteq> S\<close>  by (force simp: dist_norm)
+      show "cbox a (b + e) \<subseteq> S"
+        using \<open>a \<in> S\<close> \<open>b + e \<in> S\<close> \<open>connected S\<close> connected_contains_Icc by auto
+    qed
+  qed
+  obtain w z where "cbox w z \<subseteq> S" and sub_wz: "cbox a b \<union> cbox c d \<subseteq> box w z"
+  proof -
+    have "a \<in> S" "b \<in> S"
+      using \<open>box a b \<noteq> {}\<close> \<open>cbox a b \<subseteq> S\<close> by auto
+    moreover have "c \<in> S" "d \<in> S"
+      using \<open>box c d \<noteq> {}\<close> \<open>cbox c d \<subseteq> U\<close> \<open>U \<subseteq> S\<close> by force+
+    ultimately have "min a c \<in> S" "max b d \<in> S"
+      by linarith+
+    then obtain e1 e2 where "e1 > 0" "cball (min a c) e1 \<subseteq> S" "e2 > 0" "cball (max b d) e2 \<subseteq> S"
+      using \<open>open S\<close> open_contains_cball by metis
+    then have *: "min a c - e1 \<in> S" "max b d + e2 \<in> S"
+      by (auto simp: dist_norm)
+    show ?thesis
+    proof
+      show "cbox (min a c - e1) (max b d+ e2) \<subseteq> S"
+        using * \<open>connected S\<close> connected_contains_Icc by auto
+      show "cbox a b \<union> cbox c d \<subseteq> box (min a c - e1) (max b d + e2)"
+        using \<open>0 < e1\<close> \<open>0 < e2\<close> 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: "\<And>x. x \<in> cbox a b \<Longrightarrow> f x \<in> cbox c d"
+    using homeomorphism_grouping_point_3 [of a b w z c d]
+    using \<open>box a b \<noteq> {}\<close> \<open>box c d \<noteq> {}\<close> 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' \<equiv> \<lambda>x. if x \<in> cbox w z then f x else x"
+  define g' where "g' \<equiv> \<lambda>x. if x \<in> cbox w z then g x else x"
+  show ?thesis
+  proof
+    have T: "cbox w z \<union> (T - box w z) = T"
+      using \<open>cbox w z \<subseteq> S\<close> \<open>S \<subseteq> T\<close> by auto
+    show "homeomorphism T T f' g'"
+    proof
+      have clo: "closedin (subtopology euclidean (cbox w z \<union> (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 "continuous_on (cbox w z \<union> (T - box w z)) f'" "continuous_on (cbox w z \<union> (T - box w z)) g'"
+        unfolding f'_def g'_def
+         apply (safe intro!: continuous_on_cases_local contfg continuous_on_id clo)
+         apply (simp_all add: closed_subset)
+        using \<open>f w = w\<close> \<open>f z = z\<close> apply force
+        by (metis \<open>f w = w\<close> \<open>f z = z\<close> hom homeomorphism_def less_eq_real_def mem_box_real(2))
+      then show "continuous_on T f'" "continuous_on T g'"
+        by (simp_all only: T)
+      show "f' ` T \<subseteq> T"
+        unfolding f'_def
+        by clarsimp (metis \<open>cbox w z \<subseteq> S\<close> \<open>S \<subseteq> T\<close> subsetD hom homeomorphism_def imageI mem_box_real(2))
+      show "g' ` T \<subseteq> T"
+        unfolding g'_def
+        by clarsimp (metis \<open>cbox w z \<subseteq> S\<close> \<open>S \<subseteq> T\<close> subsetD hom homeomorphism_def imageI mem_box_real(2))
+      show "\<And>x. x \<in> T \<Longrightarrow> g' (f' x) = x"
+        unfolding f'_def g'_def
+        using homeomorphism_apply1 [OF hom]  homeomorphism_image1 [OF hom] by fastforce
+      show "\<And>y. y \<in> T \<Longrightarrow> f' (g' y) = y"
+        unfolding f'_def g'_def
+        using homeomorphism_apply2 [OF hom]  homeomorphism_image2 [OF hom] by fastforce
+    qed
+    show "\<And>x. x \<in> K \<Longrightarrow> f' x \<in> U"
+      using fin sub_wz \<open>K \<subseteq> cbox a b\<close> \<open>cbox c d \<subseteq> U\<close> by (force simp: f'_def)
+    show "{x. \<not> (f' x = x \<and> g' x = x)} \<subseteq> S"
+      using \<open>cbox w z \<subseteq> S\<close> by (auto simp: f'_def g'_def)
+    show "bounded {x. \<not> (f' x = x \<and> g' x = x)}"
+      apply (rule bounded_subset [of "cbox w z"])
+      using bounded_cbox apply blast
+      apply (auto simp: f'_def g'_def)
+      done
+  qed
+qed
+
+proposition%unimportant homeomorphism_grouping_points_exists:
+  fixes S :: "'a::euclidean_space set"
+  assumes "open U" "open S" "connected S" "U \<noteq> {}" "finite K" "K \<subseteq> S" "U \<subseteq> S" "S \<subseteq> T"
+  obtains f g where "homeomorphism T T f g" "{x. (\<not> (f x = x \<and> g x = x))} \<subseteq> S"
+                    "bounded {x. (\<not> (f x = x \<and> g x = x))}" "\<And>x. x \<in> K \<Longrightarrow> f x \<in> U"
+proof (cases "2 \<le> DIM('a)")
+  case True
+  have TS: "T \<subseteq> affine hull S"
+    using affine_hull_open assms by blast
+  have "infinite U"
+    using \<open>open U\<close> \<open>U \<noteq> {}\<close> finite_imp_not_open by blast
+  then obtain P where "P \<subseteq> U" "finite P" "card K = card P"
+    using infinite_arbitrarily_large by metis
+  then obtain \<gamma> where \<gamma>: "bij_betw \<gamma> K P"
+    using \<open>finite K\<close> finite_same_card_bij by blast
+  obtain f g where "homeomorphism T T f g" "\<And>i. i \<in> K \<Longrightarrow> f (id i) = \<gamma> i" "{x. \<not> (f x = x \<and> g x = x)} \<subseteq> S" "bounded {x. \<not> (f x = x \<and> g x = x)}"
+  proof (rule homeomorphism_moving_points_exists [OF True \<open>open S\<close> \<open>connected S\<close> \<open>S \<subseteq> T\<close> \<open>finite K\<close>])
+    show "\<And>i. i \<in> K \<Longrightarrow> id i \<in> S \<and> \<gamma> i \<in> S"
+      using \<open>P \<subseteq> U\<close> \<open>bij_betw \<gamma> K P\<close> \<open>K \<subseteq> S\<close> \<open>U \<subseteq> S\<close> bij_betwE by blast
+    show "pairwise (\<lambda>i j. id i \<noteq> id j \<and> \<gamma> i \<noteq> \<gamma> j) K"
+      using \<gamma> by (auto simp: pairwise_def bij_betw_def inj_on_def)
+  qed (use affine_hull_open assms that in auto)
+  then show ?thesis
+    using \<gamma> \<open>P \<subseteq> U\<close> bij_betwE by (fastforce simp add: intro!: that)
+next
+  case False
+  with DIM_positive have "DIM('a) = 1"
+    by (simp add: dual_order.antisym)
+  then obtain h::"'a \<Rightarrow>real" and j
+  where "linear h" "linear j"
+    and noh: "\<And>x. norm(h x) = norm x" and noj: "\<And>y. norm(j y) = norm y"
+    and hj:  "\<And>x. j(h x) = x" "\<And>y. h(j y) = y"
+    and ranh: "surj h"
+    using isomorphisms_UNIV_UNIV
+    by (metis (mono_tags, hide_lams) DIM_real UNIV_eq_I range_eqI)
+  obtain f g where hom: "homeomorphism (h ` T) (h ` T) f g"
+               and f: "\<And>x. x \<in> h ` K \<Longrightarrow> f x \<in> h ` U"
+               and sub: "{x. \<not> (f x = x \<and> g x = x)} \<subseteq> h ` S"
+               and bou: "bounded {x. \<not> (f x = x \<and> 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  \<open>linear h\<close> open_surjective_linear_image connected_linear_image ranh)
+  have jf: "j (f (h x)) = x \<longleftrightarrow> f (h x) = h x" for x
+    by (metis hj)
+  have jg: "j (g (h x)) = x \<longleftrightarrow> 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: \<open>linear h\<close> \<open>linear j\<close> linear_linear linear_continuous_on)
+  show ?thesis
+  proof
+    show "homeomorphism T T (j \<circ> f \<circ> h) (j \<circ> g \<circ> h)"
+    proof
+      show "continuous_on T (j \<circ> f \<circ> h)" "continuous_on T (j \<circ> g \<circ> h)"
+        using hom homeomorphism_def
+        by (blast intro: continuous_on_compose cont_hj)+
+      show "(j \<circ> f \<circ> h) ` T \<subseteq> T" "(j \<circ> g \<circ> h) ` T \<subseteq> T"
+        by auto (metis (mono_tags, hide_lams) hj(1) hom homeomorphism_def imageE imageI)+
+      show "\<And>x. x \<in> T \<Longrightarrow> (j \<circ> g \<circ> h) ((j \<circ> f \<circ> h) x) = x"
+        using hj hom homeomorphism_apply1 by fastforce
+      show "\<And>y. y \<in> T \<Longrightarrow> (j \<circ> f \<circ> h) ((j \<circ> g \<circ> h) y) = y"
+        using hj hom homeomorphism_apply2 by fastforce
+    qed
+    show "{x. \<not> ((j \<circ> f \<circ> h) x = x \<and> (j \<circ> g \<circ> h) x = x)} \<subseteq> S"
+      apply (clarsimp simp: jf jg hj)
+      using sub hj
+      apply (drule_tac c="h x" in subsetD, force)
+      by (metis imageE)
+    have "bounded (j ` {x. (\<not> (f x = x \<and> g x = x))})"
+      by (rule bounded_linear_image [OF bou]) (use \<open>linear j\<close> linear_conv_bounded_linear in auto)
+    moreover
+    have *: "{x. \<not>((j \<circ> f \<circ> h) x = x \<and> (j \<circ> g \<circ> h) x = x)} = j ` {x. (\<not> (f x = x \<and> g x = x))}"
+      using hj by (auto simp: jf jg image_iff, metis+)
+    ultimately show "bounded {x. \<not> ((j \<circ> f \<circ> h) x = x \<and> (j \<circ> g \<circ> h) x = x)}"
+      by metis
+    show "\<And>x. x \<in> K \<Longrightarrow> (j \<circ> f \<circ> h) x \<in> U"
+      using f hj by fastforce
+  qed
+qed
+
+
+proposition%unimportant homeomorphism_grouping_points_exists_gen:
+  fixes S :: "'a::euclidean_space set"
+  assumes opeU: "openin (subtopology euclidean S) U"
+      and opeS: "openin (subtopology euclidean (affine hull S)) S"
+      and "U \<noteq> {}" "finite K" "K \<subseteq> S" and S: "S \<subseteq> T" "T \<subseteq> affine hull S" "connected S"
+  obtains f g where "homeomorphism T T f g" "{x. (\<not> (f x = x \<and> g x = x))} \<subseteq> S"
+                    "bounded {x. (\<not> (f x = x \<and> g x = x))}" "\<And>x. x \<in> K \<Longrightarrow> f x \<in> U"
+proof (cases "2 \<le> aff_dim S")
+  case True
+  have opeU': "openin (subtopology euclidean (affine hull S)) U"
+    using opeS opeU openin_trans by blast
+  obtain u where "u \<in> U" "u \<in> S"
+    using \<open>U \<noteq> {}\<close> opeU openin_imp_subset by fastforce+
+  have "infinite U"
+    apply (rule infinite_openin [OF opeU \<open>u \<in> U\<close>])
+    apply (rule connected_imp_perfect_aff_dim [OF \<open>connected S\<close> _ \<open>u \<in> S\<close>])
+    using True apply simp
+    done
+  then obtain P where "P \<subseteq> U" "finite P" "card K = card P"
+    using infinite_arbitrarily_large by metis
+  then obtain \<gamma> where \<gamma>: "bij_betw \<gamma> K P"
+    using \<open>finite K\<close> finite_same_card_bij by blast
+  have "\<exists>f g. homeomorphism T T f g \<and> (\<forall>i \<in> K. f(id i) = \<gamma> i) \<and>
+               {x. \<not> (f x = x \<and> g x = x)} \<subseteq> S \<and> bounded {x. \<not> (f x = x \<and> g x = x)}"
+  proof (rule homeomorphism_moving_points_exists_gen [OF \<open>finite K\<close> _ _ True opeS S])
+    show "\<And>i. i \<in> K \<Longrightarrow> id i \<in> S \<and> \<gamma> i \<in> S"
+      by (metis id_apply opeU openin_contains_cball subsetCE \<open>P \<subseteq> U\<close> \<open>bij_betw \<gamma> K P\<close> \<open>K \<subseteq> S\<close> bij_betwE)
+    show "pairwise (\<lambda>i j. id i \<noteq> id j \<and> \<gamma> i \<noteq> \<gamma> j) K"
+      using \<gamma> by (auto simp: pairwise_def bij_betw_def inj_on_def)
+  qed
+  then show ?thesis
+    using \<gamma> \<open>P \<subseteq> U\<close> bij_betwE by (fastforce simp add: 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 \<open>U \<noteq> {}\<close> \<open>K \<subseteq> S\<close> 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 \<subseteq> U"
+      using \<open>U \<noteq> {}\<close> \<open>K \<subseteq> S\<close> openin_imp_subset [OF opeU] by blast
+    show ?thesis
+      apply (rule that [of id id])
+      using \<open>K \<subseteq> U\<close> by (auto simp: continuous_on_id intro: homeomorphismI)
+  next
+    assume "aff_dim S = 1"
+    then have "affine hull S homeomorphic (UNIV :: real set)"
+      by (auto simp: homeomorphic_affine_sets)
+    then obtain h::"'a\<Rightarrow>real" and j where homhj: "homeomorphism (affine hull S) UNIV h j"
+      using homeomorphic_def by blast
+    then have h: "\<And>x. x \<in> affine hull S \<Longrightarrow> j(h(x)) = x" and j: "\<And>y. j y \<in> affine hull S \<and> h(j y) = y"
+      by (auto simp: homeomorphism_def)
+    have connh: "connected (h ` S)"
+      by (meson Topological_Spaces.connected_continuous_image \<open>connected S\<close> homeomorphism_cont1 homeomorphism_of_subsets homhj hull_subset top_greatest)
+    have hUS: "h ` U \<subseteq> h ` S"
+      by (meson homeomorphism_imp_open_map homeomorphism_of_subsets homhj hull_subset opeS opeU open_UNIV openin_open_eq)
+    have opn: "openin (subtopology euclidean (affine hull S)) U \<Longrightarrow> 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: "\<And>x. x \<in> h ` K \<Longrightarrow> f x \<in> h ` U"
+                 and sub: "{x. \<not> (f x = x \<and> g x = x)} \<subseteq> h ` S"
+                 and bou: "bounded {x. \<not> (f x = x \<and> 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: "\<And>x. x \<in> affine hull S \<Longrightarrow> j (f (h x)) = x \<longleftrightarrow> f (h x) = h x"
+      by (metis h j)
+    have jg: "\<And>x. x \<in> affine hull S \<Longrightarrow> j (g (h x)) = x \<longleftrightarrow> g (h x) = h x"
+      by (metis h j)
+    have cont_hj: "continuous_on T h"  "continuous_on Y j" for Y
+      apply (rule continuous_on_subset [OF _ \<open>T \<subseteq> affine hull S\<close>])
+      using homeomorphism_def homhj apply blast
+      by (meson continuous_on_subset homeomorphism_def homhj top_greatest)
+    define f' where "f' \<equiv> \<lambda>x. if x \<in> affine hull S then (j \<circ> f \<circ> h) x else x"
+    define g' where "g' \<equiv> \<lambda>x. if x \<in> affine hull S then (j \<circ> g \<circ> h) x else x"
+    show ?thesis
+    proof
+      show "homeomorphism T T f' g'"
+      proof
+        have "continuous_on T (j \<circ> f \<circ> h)"
+          apply (intro continuous_on_compose cont_hj)
+          using hom homeomorphism_def by blast
+        then show "continuous_on T f'"
+          apply (rule continuous_on_eq)
+          using \<open>T \<subseteq> affine hull S\<close> f'_def by auto
+        have "continuous_on T (j \<circ> g \<circ> h)"
+          apply (intro continuous_on_compose cont_hj)
+          using hom homeomorphism_def by blast
+        then show "continuous_on T g'"
+          apply (rule continuous_on_eq)
+          using \<open>T \<subseteq> affine hull S\<close> g'_def by auto
+        show "f' ` T \<subseteq> T"
+        proof (clarsimp simp: f'_def)
+          fix x assume "x \<in> T"
+          then have "f (h x) \<in> h ` T"
+            by (metis (no_types) hom homeomorphism_def image_subset_iff subset_refl)
+          then show "j (f (h x)) \<in> T"
+            using \<open>T \<subseteq> affine hull S\<close> h by auto
+        qed
+        show "g' ` T \<subseteq> T"
+        proof (clarsimp simp: g'_def)
+          fix x assume "x \<in> T"
+          then have "g (h x) \<in> h ` T"
+            by (metis (no_types) hom homeomorphism_def image_subset_iff subset_refl)
+          then show "j (g (h x)) \<in> T"
+            using \<open>T \<subseteq> affine hull S\<close> h by auto
+        qed
+        show "\<And>x. x \<in> T \<Longrightarrow> g' (f' x) = x"
+          using h j hom homeomorphism_apply1 by (fastforce simp add: f'_def g'_def)
+        show "\<And>y. y \<in> T \<Longrightarrow> f' (g' y) = y"
+          using h j hom homeomorphism_apply2 by (fastforce simp add: f'_def g'_def)
+      qed
+    next
+      show "{x. \<not> (f' x = x \<and> g' x = x)} \<subseteq> S"
+        apply (clarsimp simp: f'_def g'_def jf jg)
+        apply (rule imageE [OF subsetD [OF sub]], force)
+        by (metis h hull_inc)
+    next
+      have "compact (j ` closure {x. \<not> (f x = x \<and> g x = x)})"
+        using bou by (auto simp: compact_continuous_image cont_hj)
+      then have "bounded (j ` {x. \<not> (f x = x \<and> g x = x)})"
+        by (rule bounded_closure_image [OF compact_imp_bounded])
+      moreover
+      have *: "{x \<in> affine hull S. j (f (h x)) \<noteq> x \<or> j (g (h x)) \<noteq> x} = j ` {x. (\<not> (f x = x \<and> g x = x))}"
+        using h j by (auto simp: image_iff; metis)
+      ultimately have "bounded {x \<in> affine hull S. j (f (h x)) \<noteq> x \<or> j (g (h x)) \<noteq> x}"
+        by metis
+      then show "bounded {x. \<not> (f' x = x \<and> g' x = x)}"
+        by (simp add: f'_def g'_def Collect_mono bounded_subset)
+    next
+      show "f' x \<in> U" if "x \<in> K" for x
+      proof -
+        have "U \<subseteq> S"
+          using opeU openin_imp_subset by blast
+        then have "j (f (h x)) \<in> U"
+          using f h hull_subset that by fastforce
+        then show "f' x \<in> U"
+          using \<open>K \<subseteq> S\<close> S f'_def that by auto
+      qed
+    qed
+  qed
+qed
+
+
+subsection\<open>Nullhomotopic mappings\<close>
+
+text\<open> 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 \<open>\<le>\<close> 0, and
+we also don't need to explicitly assume continuity since it's already implicit
+in both sides of the equivalence.\<close>
+
+lemma nullhomotopic_from_lemma:
+  assumes contg: "continuous_on (cball a r - {a}) g"
+      and fa: "\<And>e. 0 < e
+               \<Longrightarrow> \<exists>d. 0 < d \<and> (\<forall>x. x \<noteq> a \<and> norm(x - a) < d \<longrightarrow> norm(g x - f a) < e)"
+      and r: "\<And>x. x \<in> cball a r \<and> x \<noteq> a \<Longrightarrow> 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 \<le> 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 "\<exists>d>0. \<forall>x'\<in>cball a r.
+                      dist x' x < d \<longrightarrow> dist (g x') (g x) < e" if "e>0" for e
+      proof -
+        obtain d where "d > 0"
+           and d: "\<And>x'. \<lbrakk>dist x' a \<le> r; x' \<noteq> a; dist x' x < d\<rbrakk> \<Longrightarrow>
+                                 dist (g x') (g x) < e"
+          using contg False x \<open>e>0\<close>
+          unfolding continuous_on_iff by (fastforce simp add: dist_commute intro: that)
+        show ?thesis
+          using \<open>d > 0\<close> \<open>x \<noteq> a\<close>
+          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 \<in> cball a r"
+        by (simp add: x)
+      show "\<And>x'. \<lbrakk>x' \<in> cball a r; dist x' x < norm (x - a)\<rbrakk>
+        \<Longrightarrow> 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 \<Rightarrow> 'a::real_normed_vector"
+  shows  "(\<exists>c. homotopic_with (\<lambda>x. True) (sphere a r) S f (\<lambda>x. c)) \<longleftrightarrow>
+          (\<exists>g. continuous_on (cball a r) g \<and> g ` (cball a r) \<subseteq> S \<and>
+               (\<forall>x \<in> sphere a r. g x = f x))"
+         (is "?lhs = ?rhs")
+proof (cases r "0::real" rule: linorder_cases)
+  case equal
+  then show ?thesis
+    apply (auto simp: homotopic_with)
+    apply (rule_tac x="\<lambda>x. h (0, a)" in exI)
+     apply (fastforce simp add:)
+    using continuous_on_const by blast
+next
+  case greater
+  let ?P = "continuous_on {x. norm(x - a) = r} f \<and> f ` {x. norm(x - a) = r} \<subseteq> S"
+  have ?P if ?lhs using that
+  proof
+    fix c
+    assume c: "homotopic_with (\<lambda>x. True) (sphere a r) S f (\<lambda>x. c)"
+    then have contf: "continuous_on (sphere a r) f" and fim: "f ` sphere a r \<subseteq> S"
+      by (auto simp: homotopic_with_imp_subset1 homotopic_with_imp_continuous)
+    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 \<and> g ` cball a r \<subseteq> S \<and> (\<forall>xa\<in>sphere a r. g xa = f xa)"
+    then
+    show ?P
+      apply (safe elim!: continuous_on_eq [OF continuous_on_subset])
+      apply (auto simp: dist_norm norm_minus_commute)
+      by (metis dist_norm image_subset_iff mem_sphere norm_minus_commute sphere_cball subsetCE)
+  qed
+  moreover have ?thesis if ?P
+  proof
+    assume ?lhs
+    then obtain c where "homotopic_with (\<lambda>x. True) (sphere a r) S (\<lambda>x. c) f"
+      using homotopic_with_sym by blast
+    then obtain h where conth: "continuous_on ({0..1::real} \<times> sphere a r) h"
+                    and him: "h ` ({0..1} \<times> sphere a r) \<subseteq> S"
+                    and h: "\<And>x. h(0, x) = c" "\<And>x. h(1, x) = f x"
+      by (auto simp: homotopic_with_def)
+    obtain b1::'M where "b1 \<in> Basis"
+      using SOME_Basis by auto
+    have "c \<in> S"
+      apply (rule him [THEN subsetD])
+      apply (rule_tac x = "(0, a + r *\<^sub>R b1)" in image_eqI)
+      using h greater \<open>b1 \<in> Basis\<close>
+       apply (auto simp: dist_norm)
+      done
+    have uconth: "uniformly_continuous_on ({0..1::real} \<times> (sphere a r)) h"
+      by (force intro: compact_Times conth compact_uniformly_continuous)
+    let ?g = "\<lambda>x. h (norm (x - a)/r,
+                     a + (if x = a then r *\<^sub>R b1 else (r / norm(x - a)) *\<^sub>R (x - a)))"
+    let ?g' = "\<lambda>x. h (norm (x - a)/r, a + (r / norm(x - a)) *\<^sub>R (x - a))"
+    show ?rhs
+    proof (intro exI conjI)
+      have "continuous_on (cball a r - {a}) ?g'"
+        apply (rule continuous_on_compose2 [OF conth])
+         apply (intro continuous_intros)
+        using greater apply (auto simp: dist_norm norm_minus_commute)
+        done
+      then show "continuous_on (cball a r) ?g"
+      proof (rule nullhomotopic_from_lemma)
+        show "\<exists>d>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < d \<longrightarrow> norm (?g' x - ?g a) < e" if "0 < e" for e
+        proof -
+          obtain d where "0 < d"
+             and d: "\<And>x x'. \<lbrakk>x \<in> {0..1} \<times> sphere a r; x' \<in> {0..1} \<times> sphere a r; dist x' x < d\<rbrakk>
+                        \<Longrightarrow> dist (h x') (h x) < e"
+            using uniformly_continuous_onE [OF uconth \<open>0 < e\<close>] by auto
+          have *: "norm (h (norm (x - a) / r,
+                         a + (r / norm (x - a)) *\<^sub>R (x - a)) - h (0, a + r *\<^sub>R b1)) < e"
+                   if "x \<noteq> a" "norm (x - a) < r" "norm (x - a) < d * r" for x
+          proof -
+            have "norm (h (norm (x - a) / r, a + (r / norm (x - a)) *\<^sub>R (x - a)) - h (0, a + r *\<^sub>R b1)) =
+                  norm (h (norm (x - a) / r, a + (r / norm (x - a)) *\<^sub>R (x - a)) - h (0, a + (r / norm (x - a)) *\<^sub>R (x - a)))"
+              by (simp add: h)
+            also have "\<dots> < e"
+              apply (rule d [unfolded dist_norm])
+              using greater \<open>0 < d\<close> \<open>b1 \<in> Basis\<close> that
+                by (auto simp: dist_norm divide_simps)
+            finally show ?thesis .
+          qed
+          show ?thesis
+            apply (rule_tac x = "min r (d * r)" in exI)
+            using greater \<open>0 < d\<close> by (auto simp: *)
+        qed
+        show "\<And>x. x \<in> cball a r \<and> x \<noteq> a \<Longrightarrow> ?g x = ?g' x"
+          by auto
+      qed
+    next
+      show "?g ` cball a r \<subseteq> S"
+        using greater him \<open>c \<in> S\<close>
+        by (force simp: h dist_norm norm_minus_commute)
+    next
+      show "\<forall>x\<in>sphere a r. ?g x = f x"
+        using greater by (auto simp: h dist_norm norm_minus_commute)
+    qed
+  next
+    assume ?rhs
+    then obtain g where contg: "continuous_on (cball a r) g"
+                    and gim: "g ` cball a r \<subseteq> S"
+                    and gf: "\<forall>x \<in> sphere a r. g x = f x"
+      by auto
+    let ?h = "\<lambda>y. g (a + (fst y) *\<^sub>R (snd y - a))"
+    have "continuous_on ({0..1} \<times> sphere a r) ?h"
+      apply (rule continuous_on_compose2 [OF contg])
+       apply (intro continuous_intros)
+      apply (auto simp: dist_norm norm_minus_commute mult_left_le_one_le)
+      done
+    moreover
+    have "?h ` ({0..1} \<times> sphere a r) \<subseteq> S"
+      by (auto simp: dist_norm norm_minus_commute mult_left_le_one_le gim [THEN subsetD])
+    moreover
+    have "\<forall>x\<in>sphere a r. ?h (0, x) = g a" "\<forall>x\<in>sphere a r. ?h (1, x) = f x"
+      by (auto simp: dist_norm norm_minus_commute mult_left_le_one_le gf)
+    ultimately
+    show ?lhs
+      apply (subst homotopic_with_sym)
+      apply (rule_tac x="g a" in exI)
+      apply (auto simp: homotopic_with)
+      done
+  qed
+  ultimately
+  show ?thesis by meson
+qed simp
+
+end
\ No newline at end of file
--- a/src/HOL/Analysis/Path_Connected.thy	Mon Jan 07 14:06:54 2019 +0100
+++ b/src/HOL/Analysis/Path_Connected.thy	Mon Jan 07 14:57:45 2019 +0100
@@ -2,10 +2,10 @@
     Authors:    LC Paulson and Robert Himmelmann (TU Muenchen), based on material from HOL Light
 *)
 
-section \<open>Continuous Paths\<close>
+section \<open>Path-Connectedness\<close>
 
 theory Path_Connected
-  imports Continuous_Extension Continuum_Not_Denumerable
+  imports Starlike
 begin
 
 subsection \<open>Paths and Arcs\<close>
@@ -295,7 +295,7 @@
 qed
 
 
-section%unimportant \<open>Path Images\<close>
+subsection%unimportant \<open>Path Images\<close>
 
 lemma bounded_path_image: "path g \<Longrightarrow> bounded(path_image g)"
   by (simp add: compact_imp_bounded compact_path_image)
@@ -1421,8 +1421,6 @@
     by (rule_tac x="e/2" in exI) auto
 qed
 
-section "Path-Connectedness" (* TODO: separate theory? *)
-
 subsection \<open>Path component\<close>
 
 text \<open>Original formalization by Tom Hales\<close>
@@ -2531,7 +2529,7 @@
   by (meson cobounded_unique_unbounded_components connected_eq_connected_components_eq assms)
 
 
-section\<open>The \<open>inside\<close> and \<open>outside\<close> of a Set\<close>
+subsection\<open>The \<open>inside\<close> and \<open>outside\<close> of a Set\<close>
 
 text%important\<open>The inside comprises the points in a bounded connected component of the set's complement.
   The outside comprises the points in unbounded connected component of the complement.\<close>
@@ -3386,5156 +3384,4 @@
       by (metis dw_le norm_minus_commute not_less order_trans rle wy)
 qed
 
-
-section \<open>Homotopy of Maps\<close> (* TODO separate theory? *)
-
-
-definition%important homotopic_with ::
-  "[('a::topological_space \<Rightarrow> 'b::topological_space) \<Rightarrow> bool, 'a set, 'b set, 'a \<Rightarrow> 'b, 'a \<Rightarrow> 'b] \<Rightarrow> bool"
-where
- "homotopic_with P X Y p q \<equiv>
-   (\<exists>h:: real \<times> 'a \<Rightarrow> 'b.
-       continuous_on ({0..1} \<times> X) h \<and>
-       h ` ({0..1} \<times> X) \<subseteq> Y \<and>
-       (\<forall>x. h(0, x) = p x) \<and>
-       (\<forall>x. h(1, x) = q x) \<and>
-       (\<forall>t \<in> {0..1}. P(\<lambda>x. h(t, x))))"
-
-text\<open>\<open>p\<close>, \<open>q\<close> are functions \<open>X \<rightarrow> Y\<close>, and the property \<open>P\<close> restricts all intermediate maps.
-We often just want to require that \<open>P\<close> fixes some subset, but to include the case of a loop homotopy,
-it is convenient to have a general property \<open>P\<close>.\<close>
-
-text \<open>We often want to just localize the ending function equality or whatever.\<close>
-text%important \<open>%whitespace\<close>
-proposition homotopic_with:
-  fixes X :: "'a::topological_space set" and Y :: "'b::topological_space set"
-  assumes "\<And>h k. (\<And>x. x \<in> X \<Longrightarrow> h x = k x) \<Longrightarrow> (P h \<longleftrightarrow> P k)"
-  shows "homotopic_with P X Y p q \<longleftrightarrow>
-           (\<exists>h :: real \<times> 'a \<Rightarrow> 'b.
-              continuous_on ({0..1} \<times> X) h \<and>
-              h ` ({0..1} \<times> X) \<subseteq> Y \<and>
-              (\<forall>x \<in> X. h(0,x) = p x) \<and>
-              (\<forall>x \<in> X. h(1,x) = q x) \<and>
-              (\<forall>t \<in> {0..1}. P(\<lambda>x. h(t, x))))"
-  unfolding homotopic_with_def
-  apply (rule iffI, blast, clarify)
-  apply (rule_tac x="\<lambda>(u,v). if v \<in> X then h(u,v) else if u = 0 then p v else q v" in exI)
-  apply auto
-  apply (force elim: continuous_on_eq)
-  apply (drule_tac x=t in bspec, force)
-  apply (subst assms; simp)
-  done
-
-proposition homotopic_with_eq:
-   assumes h: "homotopic_with P X Y f g"
-       and f': "\<And>x. x \<in> X \<Longrightarrow> f' x = f x"
-       and g': "\<And>x. x \<in> X \<Longrightarrow> g' x = g x"
-       and P:  "(\<And>h k. (\<And>x. x \<in> X \<Longrightarrow> h x = k x) \<Longrightarrow> (P h \<longleftrightarrow> P k))"
-   shows "homotopic_with P X Y f' g'"
-  using h unfolding homotopic_with_def
-  apply safe
-  apply (rule_tac x="\<lambda>(u,v). if v \<in> X then h(u,v) else if u = 0 then f' v else g' v" in exI)
-  apply (simp add: f' g', safe)
-  apply (fastforce intro: continuous_on_eq, fastforce)
-  apply (subst P; fastforce)
-  done
-
-proposition homotopic_with_equal:
-   assumes contf: "continuous_on X f" and fXY: "f ` X \<subseteq> Y"
-       and gf: "\<And>x. x \<in> X \<Longrightarrow> g x = f x"
-       and P:  "P f" "P g"
-   shows "homotopic_with P X Y f g"
-  unfolding homotopic_with_def
-  apply (rule_tac x="\<lambda>(u,v). if u = 1 then g v else f v" in exI)
-  using assms
-  apply (intro conjI)
-  apply (rule continuous_on_eq [where f = "f \<circ> snd"])
-  apply (rule continuous_intros | force)+
-  apply clarify
-  apply (case_tac "t=1"; force)
-  done
-
-
-lemma image_Pair_const: "(\<lambda>x. (x, c)) ` A = A \<times> {c}"
-  by auto
-
-lemma homotopic_constant_maps:
-   "homotopic_with (\<lambda>x. True) s t (\<lambda>x. a) (\<lambda>x. b) \<longleftrightarrow> s = {} \<or> path_component t a b"
-proof (cases "s = {} \<or> t = {}")
-  case True with continuous_on_const show ?thesis
-    by (auto simp: homotopic_with path_component_def)
-next
-  case False
-  then obtain c where "c \<in> s" by blast
-  show ?thesis
-  proof
-    assume "homotopic_with (\<lambda>x. True) s t (\<lambda>x. a) (\<lambda>x. b)"
-    then obtain h :: "real \<times> 'a \<Rightarrow> 'b"
-        where conth: "continuous_on ({0..1} \<times> s) h"
-          and h: "h ` ({0..1} \<times> s) \<subseteq> t" "(\<forall>x\<in>s. h (0, x) = a)" "(\<forall>x\<in>s. h (1, x) = b)"
-      by (auto simp: homotopic_with)
-    have "continuous_on {0..1} (h \<circ> (\<lambda>t. (t, c)))"
-      apply (rule continuous_intros conth | simp add: image_Pair_const)+
-      apply (blast intro:  \<open>c \<in> s\<close> continuous_on_subset [OF conth])
-      done
-    with \<open>c \<in> s\<close> h show "s = {} \<or> path_component t a b"
-      apply (simp_all add: homotopic_with path_component_def, auto)
-      apply (drule_tac x="h \<circ> (\<lambda>t. (t, c))" in spec)
-      apply (auto simp: pathstart_def pathfinish_def path_image_def path_def)
-      done
-  next
-    assume "s = {} \<or> path_component t a b"
-    with False show "homotopic_with (\<lambda>x. True) s t (\<lambda>x. a) (\<lambda>x. b)"
-      apply (clarsimp simp: homotopic_with path_component_def pathstart_def pathfinish_def path_image_def path_def)
-      apply (rule_tac x="g \<circ> fst" in exI)
-      apply (rule conjI continuous_intros | force)+
-      done
-  qed
-qed
-
-
-subsection%unimportant\<open>Trivial properties\<close>
-
-lemma homotopic_with_imp_property: "homotopic_with P X Y f g \<Longrightarrow> P f \<and> P g"
-  unfolding homotopic_with_def Ball_def
-  apply clarify
-  apply (frule_tac x=0 in spec)
-  apply (drule_tac x=1 in spec, auto)
-  done
-
-lemma continuous_on_o_Pair: "\<lbrakk>continuous_on (T \<times> X) h; t \<in> T\<rbrakk> \<Longrightarrow> continuous_on X (h \<circ> Pair t)"
-  by (fast intro: continuous_intros elim!: continuous_on_subset)
-
-lemma homotopic_with_imp_continuous:
-    assumes "homotopic_with P X Y f g"
-    shows "continuous_on X f \<and> continuous_on X g"
-proof -
-  obtain h :: "real \<times> 'a \<Rightarrow> 'b"
-    where conth: "continuous_on ({0..1} \<times> X) h"
-      and h: "\<forall>x. h (0, x) = f x" "\<forall>x. h (1, x) = g x"
-    using assms by (auto simp: homotopic_with_def)
-  have *: "t \<in> {0..1} \<Longrightarrow> continuous_on X (h \<circ> (\<lambda>x. (t,x)))" for t
-    by (rule continuous_intros continuous_on_subset [OF conth] | force)+
-  show ?thesis
-    using h *[of 0] *[of 1] by auto
-qed
-
-proposition homotopic_with_imp_subset1:
-     "homotopic_with P X Y f g \<Longrightarrow> f ` X \<subseteq> Y"
-  by (simp add: homotopic_with_def image_subset_iff) (metis atLeastAtMost_iff order_refl zero_le_one)
-
-proposition homotopic_with_imp_subset2:
-     "homotopic_with P X Y f g \<Longrightarrow> g ` X \<subseteq> Y"
-  by (simp add: homotopic_with_def image_subset_iff) (metis atLeastAtMost_iff order_refl zero_le_one)
-
-proposition homotopic_with_mono:
-    assumes hom: "homotopic_with P X Y f g"
-        and Q: "\<And>h. \<lbrakk>continuous_on X h; image h X \<subseteq> Y \<and> P h\<rbrakk> \<Longrightarrow> Q h"
-      shows "homotopic_with Q X Y f g"
-  using hom
-  apply (simp add: homotopic_with_def)
-  apply (erule ex_forward)
-  apply (force simp: intro!: Q dest: continuous_on_o_Pair)
-  done
-
-proposition homotopic_with_subset_left:
-     "\<lbrakk>homotopic_with P X Y f g; Z \<subseteq> X\<rbrakk> \<Longrightarrow> homotopic_with P Z Y f g"
-  apply (simp add: homotopic_with_def)
-  apply (fast elim!: continuous_on_subset ex_forward)
-  done
-
-proposition homotopic_with_subset_right:
-     "\<lbrakk>homotopic_with P X Y f g; Y \<subseteq> Z\<rbrakk> \<Longrightarrow> homotopic_with P X Z f g"
-  apply (simp add: homotopic_with_def)
-  apply (fast elim!: continuous_on_subset ex_forward)
-  done
-
-proposition homotopic_with_compose_continuous_right:
-    "\<lbrakk>homotopic_with (\<lambda>f. p (f \<circ> h)) X Y f g; continuous_on W h; h ` W \<subseteq> X\<rbrakk>
-     \<Longrightarrow> homotopic_with p W Y (f \<circ> h) (g \<circ> h)"
-  apply (clarsimp simp add: homotopic_with_def)
-  apply (rename_tac k)
-  apply (rule_tac x="k \<circ> (\<lambda>y. (fst y, h (snd y)))" in exI)
-  apply (rule conjI continuous_intros continuous_on_compose [where f=snd and g=h, unfolded o_def] | simp)+
-  apply (erule continuous_on_subset)
-  apply (fastforce simp: o_def)+
-  done
-
-proposition homotopic_compose_continuous_right:
-     "\<lbrakk>homotopic_with (\<lambda>f. True) X Y f g; continuous_on W h; h ` W \<subseteq> X\<rbrakk>
-      \<Longrightarrow> homotopic_with (\<lambda>f. True) W Y (f \<circ> h) (g \<circ> h)"
-  using homotopic_with_compose_continuous_right by fastforce
-
-proposition homotopic_with_compose_continuous_left:
-     "\<lbrakk>homotopic_with (\<lambda>f. p (h \<circ> f)) X Y f g; continuous_on Y h; h ` Y \<subseteq> Z\<rbrakk>
-      \<Longrightarrow> homotopic_with p X Z (h \<circ> f) (h \<circ> g)"
-  apply (clarsimp simp add: homotopic_with_def)
-  apply (rename_tac k)
-  apply (rule_tac x="h \<circ> k" in exI)
-  apply (rule conjI continuous_intros continuous_on_compose [where f=snd and g=h, unfolded o_def] | simp)+
-  apply (erule continuous_on_subset)
-  apply (fastforce simp: o_def)+
-  done
-
-proposition homotopic_compose_continuous_left:
-   "\<lbrakk>homotopic_with (\<lambda>_. True) X Y f g;
-     continuous_on Y h; h ` Y \<subseteq> Z\<rbrakk>
-    \<Longrightarrow> homotopic_with (\<lambda>f. True) X Z (h \<circ> f) (h \<circ> g)"
-  using homotopic_with_compose_continuous_left by fastforce
-
-proposition homotopic_with_Pair:
-   assumes hom: "homotopic_with p s t f g" "homotopic_with p' s' t' f' g'"
-       and q: "\<And>f g. \<lbrakk>p f; p' g\<rbrakk> \<Longrightarrow> q(\<lambda>(x,y). (f x, g y))"
-     shows "homotopic_with q (s \<times> s') (t \<times> t')
-                  (\<lambda>(x,y). (f x, f' y)) (\<lambda>(x,y). (g x, g' y))"
-  using hom
-  apply (clarsimp simp add: homotopic_with_def)
-  apply (rename_tac k k')
-  apply (rule_tac x="\<lambda>z. ((k \<circ> (\<lambda>x. (fst x, fst (snd x)))) z, (k' \<circ> (\<lambda>x. (fst x, snd (snd x)))) z)" in exI)
-  apply (rule conjI continuous_intros | erule continuous_on_subset | clarsimp)+
-  apply (auto intro!: q [unfolded case_prod_unfold])
-  done
-
-lemma homotopic_on_empty [simp]: "homotopic_with (\<lambda>x. True) {} t f g"
-  by (metis continuous_on_def empty_iff homotopic_with_equal image_subset_iff)
-
-
-text\<open>Homotopy with P is an equivalence relation (on continuous functions mapping X into Y that satisfy P,
-     though this only affects reflexivity.\<close>
-
-
-proposition homotopic_with_refl:
-   "homotopic_with P X Y f f \<longleftrightarrow> continuous_on X f \<and> image f X \<subseteq> Y \<and> P f"
-  apply (rule iffI)
-  using homotopic_with_imp_continuous homotopic_with_imp_property homotopic_with_imp_subset2 apply blast
-  apply (simp add: homotopic_with_def)
-  apply (rule_tac x="f \<circ> snd" in exI)
-  apply (rule conjI continuous_intros | force)+
-  done
-
-lemma homotopic_with_symD:
-  fixes X :: "'a::real_normed_vector set"
-    assumes "homotopic_with P X Y f g"
-      shows "homotopic_with P X Y g f"
-  using assms
-  apply (clarsimp simp add: homotopic_with_def)
-  apply (rename_tac h)
-  apply (rule_tac x="h \<circ> (\<lambda>y. (1 - fst y, snd y))" in exI)
-  apply (rule conjI continuous_intros | erule continuous_on_subset | force simp: image_subset_iff)+
-  done
-
-proposition homotopic_with_sym:
-    fixes X :: "'a::real_normed_vector set"
-    shows "homotopic_with P X Y f g \<longleftrightarrow> homotopic_with P X Y g f"
-  using homotopic_with_symD by blast
-
-lemma split_01: "{0..1::real} = {0..1/2} \<union> {1/2..1}"
-  by force
-
-lemma split_01_prod: "{0..1::real} \<times> X = ({0..1/2} \<times> X) \<union> ({1/2..1} \<times> X)"
-  by force
-
-proposition homotopic_with_trans:
-    fixes X :: "'a::real_normed_vector set"
-    assumes "homotopic_with P X Y f g" and "homotopic_with P X Y g h"
-      shows "homotopic_with P X Y f h"
-proof -
-  have clo1: "closedin (subtopology euclidean ({0..1/2} \<times> X \<union> {1/2..1} \<times> X)) ({0..1/2::real} \<times> X)"
-    apply (simp add: closedin_closed split_01_prod [symmetric])
-    apply (rule_tac x="{0..1/2} \<times> UNIV" in exI)
-    apply (force simp: closed_Times)
-    done
-  have clo2: "closedin (subtopology euclidean ({0..1/2} \<times> X \<union> {1/2..1} \<times> X)) ({1/2..1::real} \<times> X)"
-    apply (simp add: closedin_closed split_01_prod [symmetric])
-    apply (rule_tac x="{1/2..1} \<times> UNIV" in exI)
-    apply (force simp: closed_Times)
-    done
-  { fix k1 k2:: "real \<times> 'a \<Rightarrow> 'b"
-    assume cont: "continuous_on ({0..1} \<times> X) k1" "continuous_on ({0..1} \<times> X) k2"
-       and Y: "k1 ` ({0..1} \<times> X) \<subseteq> Y" "k2 ` ({0..1} \<times> X) \<subseteq> Y"
-       and geq: "\<forall>x. k1 (1, x) = g x" "\<forall>x. k2 (0, x) = g x"
-       and k12: "\<forall>x. k1 (0, x) = f x" "\<forall>x. k2 (1, x) = h x"
-       and P:   "\<forall>t\<in>{0..1}. P (\<lambda>x. k1 (t, x))" "\<forall>t\<in>{0..1}. P (\<lambda>x. k2 (t, x))"
-    define k where "k y =
-      (if fst y \<le> 1 / 2
-       then (k1 \<circ> (\<lambda>x. (2 *\<^sub>R fst x, snd x))) y
-       else (k2 \<circ> (\<lambda>x. (2 *\<^sub>R fst x -1, snd x))) y)" for y
-    have keq: "k1 (2 * u, v) = k2 (2 * u - 1, v)" if "u = 1/2"  for u v
-      by (simp add: geq that)
-    have "continuous_on ({0..1} \<times> X) k"
-      using cont
-      apply (simp add: split_01_prod k_def)
-      apply (rule clo1 clo2 continuous_on_cases_local continuous_intros | erule continuous_on_subset | simp add: linear image_subset_iff)+
-      apply (force simp: keq)
-      done
-    moreover have "k ` ({0..1} \<times> X) \<subseteq> Y"
-      using Y by (force simp: k_def)
-    moreover have "\<forall>x. k (0, x) = f x"
-      by (simp add: k_def k12)
-    moreover have "(\<forall>x. k (1, x) = h x)"
-      by (simp add: k_def k12)
-    moreover have "\<forall>t\<in>{0..1}. P (\<lambda>x. k (t, x))"
-      using P
-      apply (clarsimp simp add: k_def)
-      apply (case_tac "t \<le> 1/2", auto)
-      done
-    ultimately have *: "\<exists>k :: real \<times> 'a \<Rightarrow> 'b.
-                       continuous_on ({0..1} \<times> X) k \<and> k ` ({0..1} \<times> X) \<subseteq> Y \<and>
-                       (\<forall>x. k (0, x) = f x) \<and> (\<forall>x. k (1, x) = h x) \<and> (\<forall>t\<in>{0..1}. P (\<lambda>x. k (t, x)))"
-      by blast
-  } note * = this
-  show ?thesis
-    using assms by (auto intro: * simp add: homotopic_with_def)
-qed
-
-proposition homotopic_compose:
-      fixes s :: "'a::real_normed_vector set"
-      shows "\<lbrakk>homotopic_with (\<lambda>x. True) s t f f'; homotopic_with (\<lambda>x. True) t u g g'\<rbrakk>
-             \<Longrightarrow> homotopic_with (\<lambda>x. True) s u (g \<circ> f) (g' \<circ> f')"
-  apply (rule homotopic_with_trans [where g = "g \<circ> f'"])
-  apply (metis homotopic_compose_continuous_left homotopic_with_imp_continuous homotopic_with_imp_subset1)
-  by (metis homotopic_compose_continuous_right homotopic_with_imp_continuous homotopic_with_imp_subset2)
-
-
-text\<open>Homotopic triviality implicitly incorporates path-connectedness.\<close>
-lemma homotopic_triviality:
-  fixes S :: "'a::real_normed_vector set"
-  shows  "(\<forall>f g. continuous_on S f \<and> f ` S \<subseteq> T \<and>
-                 continuous_on S g \<and> g ` S \<subseteq> T
-                 \<longrightarrow> homotopic_with (\<lambda>x. True) S T f g) \<longleftrightarrow>
-          (S = {} \<or> path_connected T) \<and>
-          (\<forall>f. continuous_on S f \<and> f ` S \<subseteq> T \<longrightarrow> (\<exists>c. homotopic_with (\<lambda>x. True) S T f (\<lambda>x. c)))"
-          (is "?lhs = ?rhs")
-proof (cases "S = {} \<or> T = {}")
-  case True then show ?thesis by auto
-next
-  case False show ?thesis
-  proof
-    assume LHS [rule_format]: ?lhs
-    have pab: "path_component T a b" if "a \<in> T" "b \<in> T" for a b
-    proof -
-      have "homotopic_with (\<lambda>x. True) S T (\<lambda>x. a) (\<lambda>x. b)"
-        by (simp add: LHS continuous_on_const image_subset_iff that)
-      then show ?thesis
-        using False homotopic_constant_maps by blast
-    qed
-      moreover
-    have "\<exists>c. homotopic_with (\<lambda>x. True) S T f (\<lambda>x. c)" if "continuous_on S f" "f ` S \<subseteq> T" for f
-      by (metis (full_types) False LHS equals0I homotopic_constant_maps homotopic_with_imp_continuous homotopic_with_imp_subset2 pab that)
-    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 \<subseteq> T" "continuous_on S g" "g ` S \<subseteq> T"
-      obtain c d where c: "homotopic_with (\<lambda>x. True) S T f (\<lambda>x. c)" and d: "homotopic_with (\<lambda>x. True) S T g (\<lambda>x. d)"
-        using False \<open>continuous_on S f\<close> \<open>f ` S \<subseteq> T\<close>  RHS \<open>continuous_on S g\<close> \<open>g ` S \<subseteq> T\<close> by blast
-      then have "c \<in> T" "d \<in> T"
-        using False homotopic_with_imp_subset2 by fastforce+
-      with T have "path_component T c d"
-        using path_connected_component by blast
-      then have "homotopic_with (\<lambda>x. True) S T (\<lambda>x. c) (\<lambda>x. d)"
-        by (simp add: homotopic_constant_maps)
-      with c d show "homotopic_with (\<lambda>x. True) S T f g"
-        by (meson homotopic_with_symD homotopic_with_trans)
-    qed
-  qed
-qed
-
-
-subsection\<open>Homotopy of paths, maintaining the same endpoints\<close>
-
-
-definition%important homotopic_paths :: "['a set, real \<Rightarrow> 'a, real \<Rightarrow> 'a::topological_space] \<Rightarrow> bool"
-  where
-     "homotopic_paths s p q \<equiv>
-       homotopic_with (\<lambda>r. pathstart r = pathstart p \<and> pathfinish r = pathfinish p) {0..1} s p q"
-
-lemma homotopic_paths:
-   "homotopic_paths s p q \<longleftrightarrow>
-      (\<exists>h. continuous_on ({0..1} \<times> {0..1}) h \<and>
-          h ` ({0..1} \<times> {0..1}) \<subseteq> s \<and>
-          (\<forall>x \<in> {0..1}. h(0,x) = p x) \<and>
-          (\<forall>x \<in> {0..1}. h(1,x) = q x) \<and>
-          (\<forall>t \<in> {0..1::real}. pathstart(h \<circ> Pair t) = pathstart p \<and>
-                        pathfinish(h \<circ> 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 \<Longrightarrow> 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 \<Longrightarrow> 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 \<Longrightarrow> path p \<and> path q"
-  using homotopic_paths_def homotopic_with_imp_continuous path_def by blast
-
-lemma homotopic_paths_imp_subset:
-     "homotopic_paths s p q \<Longrightarrow> path_image p \<subseteq> s \<and> path_image q \<subseteq> s"
-  by (simp add: homotopic_paths_def homotopic_with_imp_subset1 homotopic_with_imp_subset2 path_image_def)
-
-proposition homotopic_paths_refl [simp]: "homotopic_paths s p p \<longleftrightarrow> path p \<and> path_image p \<subseteq> s"
-by (simp add: homotopic_paths_def homotopic_with_refl path_def path_image_def)
-
-proposition homotopic_paths_sym: "homotopic_paths s p q \<Longrightarrow> 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 \<longleftrightarrow> homotopic_paths s q p"
-  by (metis homotopic_paths_sym)
-
-proposition homotopic_paths_trans [trans]:
-     "\<lbrakk>homotopic_paths s p q; homotopic_paths s q r\<rbrakk> \<Longrightarrow> homotopic_paths s p r"
-  apply (simp add: homotopic_paths_def)
-  apply (rule homotopic_with_trans, assumption)
-  by (metis (mono_tags, lifting) homotopic_with_imp_property homotopic_with_mono)
-
-proposition homotopic_paths_eq:
-     "\<lbrakk>path p; path_image p \<subseteq> s; \<And>t. t \<in> {0..1} \<Longrightarrow> p t = q t\<rbrakk> \<Longrightarrow> homotopic_paths s p q"
-  apply (simp add: homotopic_paths_def)
-  apply (rule homotopic_with_eq)
-  apply (auto simp: path_def homotopic_with_refl pathstart_def pathfinish_def path_image_def elim: continuous_on_eq)
-  done
-
-proposition homotopic_paths_reparametrize:
-  assumes "path p"
-      and pips: "path_image p \<subseteq> s"
-      and contf: "continuous_on {0..1} f"
-      and f01:"f ` {0..1} \<subseteq> {0..1}"
-      and [simp]: "f(0) = 0" "f(1) = 1"
-      and q: "\<And>t. t \<in> {0..1} \<Longrightarrow> q(t) = p(f t)"
-    shows "homotopic_paths s p q"
-proof -
-  have contp: "continuous_on {0..1} p"
-    by (metis \<open>path p\<close> path_def)
-  then have "continuous_on {0..1} (p \<circ> f)"
-    using contf continuous_on_compose continuous_on_subset f01 by blast
-  then have "path q"
-    by (simp add: path_def) (metis q continuous_on_cong)
-  have piqs: "path_image q \<subseteq> s"
-    by (metis (no_types, hide_lams) pips f01 image_subset_iff path_image_def q)
-  have fb0: "\<And>a b. \<lbrakk>0 \<le> a; a \<le> 1; 0 \<le> b; b \<le> 1\<rbrakk> \<Longrightarrow> 0 \<le> (1 - a) * f b + a * b"
-    using f01 by force
-  have fb1: "\<lbrakk>0 \<le> a; a \<le> 1; 0 \<le> b; b \<le> 1\<rbrakk> \<Longrightarrow> (1 - a) * f b + a * b \<le> 1" for a b
-    using f01 [THEN subsetD, of "f b"] by (simp add: convex_bound_le)
-  have "homotopic_paths s q p"
-  proof (rule homotopic_paths_trans)
-    show "homotopic_paths s q (p \<circ> f)"
-      using q by (force intro: homotopic_paths_eq [OF  \<open>path q\<close> piqs])
-  next
-    show "homotopic_paths s (p \<circ> f) p"
-      apply (simp add: homotopic_paths_def homotopic_with_def)
-      apply (rule_tac x="p \<circ> (\<lambda>y. (1 - (fst y)) *\<^sub>R ((f \<circ> snd) y) + (fst y) *\<^sub>R snd y)"  in exI)
-      apply (rule conjI contf continuous_intros continuous_on_subset [OF contp] | simp)+
-      using pips [unfolded path_image_def]
-      apply (auto simp: fb0 fb1 pathstart_def pathfinish_def)
-      done
-  qed
-  then show ?thesis
-    by (simp add: homotopic_paths_sym)
-qed
-
-lemma homotopic_paths_subset: "\<lbrakk>homotopic_paths s p q; s \<subseteq> t\<rbrakk> \<Longrightarrow> homotopic_paths t p q"
-  using homotopic_paths_def homotopic_with_subset_right by blast
-
-
-text\<open> A slightly ad-hoc but useful lemma in constructing homotopies.\<close>
-lemma homotopic_join_lemma:
-  fixes q :: "[real,real] \<Rightarrow> 'a::topological_space"
-  assumes p: "continuous_on ({0..1} \<times> {0..1}) (\<lambda>y. p (fst y) (snd y))"
-      and q: "continuous_on ({0..1} \<times> {0..1}) (\<lambda>y. q (fst y) (snd y))"
-      and pf: "\<And>t. t \<in> {0..1} \<Longrightarrow> pathfinish(p t) = pathstart(q t)"
-    shows "continuous_on ({0..1} \<times> {0..1}) (\<lambda>y. (p(fst y) +++ q(fst y)) (snd y))"
-proof -
-  have 1: "(\<lambda>y. p (fst y) (2 * snd y)) = (\<lambda>y. p (fst y) (snd y)) \<circ> (\<lambda>y. (fst y, 2 * snd y))"
-    by (rule ext) (simp)
-  have 2: "(\<lambda>y. q (fst y) (2 * snd y - 1)) = (\<lambda>y. q (fst y) (snd y)) \<circ> (\<lambda>y. (fst y, 2 * snd y - 1))"
-    by (rule ext) (simp)
-  show ?thesis
-    apply (simp add: joinpaths_def)
-    apply (rule continuous_on_cases_le)
-    apply (simp_all only: 1 2)
-    apply (rule continuous_intros continuous_on_subset [OF p] continuous_on_subset [OF q] | force)+
-    using pf
-    apply (auto simp: mult.commute pathstart_def pathfinish_def)
-    done
-qed
-
-text\<open> Congruence properties of homotopy w.r.t. path-combining operations.\<close>
-
-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 \<circ> (\<lambda>x. (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) \<longleftrightarrow> homotopic_paths s p q"
-  using homotopic_paths_reversepath_D by force
-
-
-proposition homotopic_paths_join:
-    "\<lbrakk>homotopic_paths s p p'; homotopic_paths s q q'; pathfinish p = pathstart q\<rbrakk> \<Longrightarrow> homotopic_paths s (p +++ q) (p' +++ q')"
-  apply (simp add: homotopic_paths_def homotopic_with_def, clarify)
-  apply (rename_tac k1 k2)
-  apply (rule_tac x="(\<lambda>y. ((k1 \<circ> Pair (fst y)) +++ (k2 \<circ> Pair (fst y))) (snd y))" in exI)
-  apply (rule conjI continuous_intros homotopic_join_lemma)+
-  apply (auto simp: joinpaths_def pathstart_def pathfinish_def path_image_def)
-  done
-
-proposition homotopic_paths_continuous_image:
-    "\<lbrakk>homotopic_paths s f g; continuous_on s h; h ` s \<subseteq> t\<rbrakk> \<Longrightarrow> homotopic_paths t (h \<circ> f) (h \<circ> g)"
-  unfolding homotopic_paths_def
-  apply (rule homotopic_with_compose_continuous_left [of _ _ _ s])
-  apply (auto simp: pathstart_def pathfinish_def elim!: homotopic_with_mono)
-  done
-
-
-subsection\<open>Group properties for homotopy of paths\<close>
-
-text%important\<open>So taking equivalence classes under homotopy would give the fundamental group\<close>
-
-proposition homotopic_paths_rid:
-    "\<lbrakk>path p; path_image p \<subseteq> s\<rbrakk> \<Longrightarrow> homotopic_paths s (p +++ linepath (pathfinish p) (pathfinish p)) p"
-  apply (subst homotopic_paths_sym)
-  apply (rule homotopic_paths_reparametrize [where f = "\<lambda>t. if  t \<le> 1 / 2 then 2 *\<^sub>R t else 1"])
-  apply (simp_all del: le_divide_eq_numeral1)
-  apply (subst split_01)
-  apply (rule continuous_on_cases continuous_intros | force simp: pathfinish_def joinpaths_def)+
-  done
-
-proposition homotopic_paths_lid:
-   "\<lbrakk>path p; path_image p \<subseteq> s\<rbrakk> \<Longrightarrow> 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)
-
-proposition homotopic_paths_assoc:
-   "\<lbrakk>path p; path_image p \<subseteq> s; path q; path_image q \<subseteq> s; path r; path_image r \<subseteq> s; pathfinish p = pathstart q;
-     pathfinish q = pathstart r\<rbrakk>
-    \<Longrightarrow> homotopic_paths s (p +++ (q +++ r)) ((p +++ q) +++ r)"
-  apply (subst homotopic_paths_sym)
-  apply (rule homotopic_paths_reparametrize
-           [where f = "\<lambda>t. if  t \<le> 1 / 2 then inverse 2 *\<^sub>R t
-                           else if  t \<le> 3 / 4 then t - (1 / 4)
-                           else 2 *\<^sub>R t - 1"])
-  apply (simp_all del: le_divide_eq_numeral1)
-  apply (simp add: subset_path_image_join)
-  apply (rule continuous_on_cases_1 continuous_intros)+
-  apply (auto simp: joinpaths_def)
-  done
-
-proposition homotopic_paths_rinv:
-  assumes "path p" "path_image p \<subseteq> s"
-    shows "homotopic_paths s (p +++ reversepath p) (linepath (pathstart p) (pathstart p))"
-proof -
-  have "continuous_on ({0..1} \<times> {0..1}) (\<lambda>x. (subpath 0 (fst x) p +++ reversepath (subpath 0 (fst x) p)) (snd x))"
-    using assms
-    apply (simp add: joinpaths_def subpath_def reversepath_def path_def del: le_divide_eq_numeral1)
-    apply (rule continuous_on_cases_le)
-    apply (rule_tac [2] continuous_on_compose [of _ _ p, unfolded o_def])
-    apply (rule continuous_on_compose [of _ _ p, unfolded o_def])
-    apply (auto intro!: continuous_intros simp del: eq_divide_eq_numeral1)
-    apply (force elim!: continuous_on_subset simp add: mult_le_one)+
-    done
-  then show ?thesis
-    using assms
-    apply (subst homotopic_paths_sym_eq)
-    unfolding homotopic_paths_def homotopic_with_def
-    apply (rule_tac x="(\<lambda>y. (subpath 0 (fst y) p +++ reversepath(subpath 0 (fst y) p)) (snd y))" in exI)
-    apply (simp add: path_defs joinpaths_def subpath_def reversepath_def)
-    apply (force simp: mult_le_one)
-    done
-qed
-
-proposition homotopic_paths_linv:
-  assumes "path p" "path_image p \<subseteq> 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\<open>Homotopy of loops without requiring preservation of endpoints\<close>
-
-definition%important homotopic_loops :: "'a::topological_space set \<Rightarrow> (real \<Rightarrow> 'a) \<Rightarrow> (real \<Rightarrow> 'a) \<Rightarrow> bool"  where
- "homotopic_loops s p q \<equiv>
-     homotopic_with (\<lambda>r. pathfinish r = pathstart r) {0..1} s p q"
-
-lemma homotopic_loops:
-   "homotopic_loops s p q \<longleftrightarrow>
-      (\<exists>h. continuous_on ({0..1::real} \<times> {0..1}) h \<and>
-          image h ({0..1} \<times> {0..1}) \<subseteq> s \<and>
-          (\<forall>x \<in> {0..1}. h(0,x) = p x) \<and>
-          (\<forall>x \<in> {0..1}. h(1,x) = q x) \<and>
-          (\<forall>t \<in> {0..1}. pathfinish(h \<circ> Pair t) = pathstart(h \<circ> Pair t)))"
-  by (simp add: homotopic_loops_def pathstart_def pathfinish_def homotopic_with)
-
-proposition homotopic_loops_imp_loop:
-     "homotopic_loops s p q \<Longrightarrow> pathfinish p = pathstart p \<and> pathfinish q = pathstart q"
-using homotopic_with_imp_property homotopic_loops_def by blast
-
-proposition homotopic_loops_imp_path:
-     "homotopic_loops s p q \<Longrightarrow> path p \<and> path q"
-  unfolding homotopic_loops_def path_def
-  using homotopic_with_imp_continuous by blast
-
-proposition homotopic_loops_imp_subset:
-     "homotopic_loops s p q \<Longrightarrow> path_image p \<subseteq> s \<and> path_image q \<subseteq> s"
-  unfolding homotopic_loops_def path_image_def
-  by (metis homotopic_with_imp_subset1 homotopic_with_imp_subset2)
-
-proposition homotopic_loops_refl:
-     "homotopic_loops s p p \<longleftrightarrow>
-      path p \<and> path_image p \<subseteq> s \<and> pathfinish p = pathstart p"
-  by (simp add: homotopic_loops_def homotopic_with_refl path_image_def path_def)
-
-proposition homotopic_loops_sym: "homotopic_loops s p q \<Longrightarrow> homotopic_loops s q p"
-  by (simp add: homotopic_loops_def homotopic_with_sym)
-
-proposition homotopic_loops_sym_eq: "homotopic_loops s p q \<longleftrightarrow> homotopic_loops s q p"
-  by (metis homotopic_loops_sym)
-
-proposition homotopic_loops_trans:
-   "\<lbrakk>homotopic_loops s p q; homotopic_loops s q r\<rbrakk> \<Longrightarrow> homotopic_loops s p r"
-  unfolding homotopic_loops_def by (blast intro: homotopic_with_trans)
-
-proposition homotopic_loops_subset:
-   "\<lbrakk>homotopic_loops s p q; s \<subseteq> t\<rbrakk> \<Longrightarrow> homotopic_loops t p q"
-  by (simp add: homotopic_loops_def homotopic_with_subset_right)
-
-proposition homotopic_loops_eq:
-   "\<lbrakk>path p; path_image p \<subseteq> s; pathfinish p = pathstart p; \<And>t. t \<in> {0..1} \<Longrightarrow> p(t) = q(t)\<rbrakk>
-          \<Longrightarrow> homotopic_loops s p q"
-  unfolding homotopic_loops_def
-  apply (rule homotopic_with_eq)
-  apply (rule homotopic_with_refl [where f = p, THEN iffD2])
-  apply (simp_all add: path_image_def path_def pathstart_def pathfinish_def)
-  done
-
-proposition homotopic_loops_continuous_image:
-   "\<lbrakk>homotopic_loops s f g; continuous_on s h; h ` s \<subseteq> t\<rbrakk> \<Longrightarrow> homotopic_loops t (h \<circ> f) (h \<circ> g)"
-  unfolding homotopic_loops_def
-  apply (rule homotopic_with_compose_continuous_left)
-  apply (erule homotopic_with_mono)
-  by (simp add: pathfinish_def pathstart_def)
-
-
-subsection\<open>Relations between the two variants of homotopy\<close>
-
-proposition homotopic_paths_imp_homotopic_loops:
-    "\<lbrakk>homotopic_paths s p q; pathfinish p = pathstart p; pathfinish q = pathstart p\<rbrakk> \<Longrightarrow> homotopic_loops s p q"
-  by (auto simp: homotopic_paths_def homotopic_loops_def intro: homotopic_with_mono)
-
-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 \<subseteq> s" by (metis assms homotopic_loops_imp_subset)
-  obtain h where conth: "continuous_on ({0..1::real} \<times> {0..1}) h"
-             and hs: "h ` ({0..1} \<times> {0..1}) \<subseteq> s"
-             and [simp]: "\<And>x. x \<in> {0..1} \<Longrightarrow> h(0,x) = p x"
-             and [simp]: "\<And>x. x \<in> {0..1} \<Longrightarrow> h(1,x) = a"
-             and ends: "\<And>t. t \<in> {0..1} \<Longrightarrow> pathfinish (h \<circ> Pair t) = pathstart (h \<circ> Pair t)"
-    using assms by (auto simp: homotopic_loops homotopic_with)
-  have conth0: "path (\<lambda>u. h (u, 0))"
-    unfolding path_def
-    apply (rule continuous_on_compose [of _ _ h, unfolded o_def])
-    apply (force intro: continuous_intros continuous_on_subset [OF conth])+
-    done
-  have pih0: "path_image (\<lambda>u. h (u, 0)) \<subseteq> s"
-    using hs by (force simp: path_image_def)
-  have c1: "continuous_on ({0..1} \<times> {0..1}) (\<lambda>x. h (fst x * snd x, 0))"
-    apply (rule continuous_on_compose [of _ _ h, unfolded o_def])
-    apply (force simp: mult_le_one intro: continuous_intros continuous_on_subset [OF conth])+
-    done
-  have c2: "continuous_on ({0..1} \<times> {0..1}) (\<lambda>x. h (fst x - fst x * snd x, 0))"
-    apply (rule continuous_on_compose [of _ _ h, unfolded o_def])
-    apply (force simp: mult_left_le mult_le_one intro: continuous_intros continuous_on_subset [OF conth])+
-    apply (rule continuous_on_subset [OF conth])
-    apply (auto simp: algebra_simps add_increasing2 mult_left_le)
-    done
-  have [simp]: "\<And>t. \<lbrakk>0 \<le> t \<and> t \<le> 1\<rbrakk> \<Longrightarrow> h (t, 1) = h (t, 0)"
-    using ends by (simp add: pathfinish_def pathstart_def)
-  have adhoc_le: "c * 4 \<le> 1 + c * (d * 4)" if "\<not> d * 4 \<le> 3" "0 \<le> c" "c \<le> 1" for c d::real
-  proof -
-    have "c * 3 \<le> c * (d * 4)" using that less_eq_real_def by auto
-    with \<open>c \<le> 1\<close> show ?thesis by fastforce
-  qed
-  have *: "\<And>p x. (path p \<and> path(reversepath p)) \<and>
-                  (path_image p \<subseteq> s \<and> path_image(reversepath p) \<subseteq> s) \<and>
-                  (pathfinish p = pathstart(linepath a a +++ reversepath p) \<and>
-                   pathstart(reversepath p) = a) \<and> pathstart p = x
-                  \<Longrightarrow> 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 \<open>path p\<close> 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))"
-    apply (rule homotopic_paths_sym)
-    using homotopic_paths_lid [of "p +++ linepath (pathfinish p) (pathfinish p)" s]
-    by (metis 1 homotopic_paths_imp_path homotopic_paths_imp_pathstart homotopic_paths_imp_subset)
-  moreover have "homotopic_paths s (linepath (pathstart p) (pathstart p) +++ p +++ linepath (pathfinish p) (pathfinish p))
-                                   ((\<lambda>u. h (u, 0)) +++ linepath a a +++ reversepath (\<lambda>u. h (u, 0)))"
-    apply (simp add: homotopic_paths_def homotopic_with_def)
-    apply (rule_tac x="\<lambda>y. (subpath 0 (fst y) (\<lambda>u. h (u, 0)) +++ (\<lambda>u. h (Pair (fst y) u)) +++ subpath (fst y) 0 (\<lambda>u. h (u, 0))) (snd y)" in exI)
-    apply (simp add: subpath_reversepath)
-    apply (intro conjI homotopic_join_lemma)
-    using ploop
-    apply (simp_all add: path_defs joinpaths_def o_def subpath_def conth c1 c2)
-    apply (force simp: algebra_simps mult_le_one mult_left_le intro: hs [THEN subsetD] adhoc_le)
-    done
-  moreover have "homotopic_paths s ((\<lambda>u. h (u, 0)) +++ linepath a a +++ reversepath (\<lambda>u. h (u, 0)))
-                                   (linepath (pathstart p) (pathstart p))"
-    apply (rule *)
-    apply (simp add: pih0 pathstart_def pathfinish_def conth0)
-    apply (simp add: reversepath_def joinpaths_def)
-    done
-  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 \<subseteq> s" and piq: "path_image q \<subseteq> s"
-      and papp: "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 \<open>path p\<close> [unfolded path_def] by blast
-  have contq: "continuous_on {0..1} q"  using \<open>path q\<close> [unfolded path_def] by blast
-  have c1: "continuous_on ({0..1} \<times> {0..1}) (\<lambda>x. p ((1 - fst x) * snd x + fst x))"
-    apply (rule continuous_on_compose [of _ _ p, unfolded o_def])
-    apply (force simp: mult_le_one intro!: continuous_intros)
-    apply (rule continuous_on_subset [OF contp])
-    apply (auto simp: algebra_simps add_increasing2 mult_right_le_one_le sum_le_prod1)
-    done
-  have c2: "continuous_on ({0..1} \<times> {0..1}) (\<lambda>x. p ((fst x - 1) * snd x + 1))"
-    apply (rule continuous_on_compose [of _ _ p, unfolded o_def])
-    apply (force simp: mult_le_one intro!: continuous_intros)
-    apply (rule continuous_on_subset [OF contp])
-    apply (auto simp: algebra_simps add_increasing2 mult_left_le_one_le)
-    done
-  have ps1: "\<And>a b. \<lbrakk>b * 2 \<le> 1; 0 \<le> b; 0 \<le> a; a \<le> 1\<rbrakk> \<Longrightarrow> p ((1 - a) * (2 * b) + a) \<in> 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: "\<And>a b. \<lbrakk>\<not> 4 * b \<le> 3; b \<le> 1; 0 \<le> a; a \<le> 1\<rbrakk> \<Longrightarrow> p ((a - 1) * (4 * b - 3) + 1) \<in> s"
-    apply (rule pip [unfolded path_image_def, THEN subsetD])
-    apply (rule image_eqI, blast)
-    apply (simp add: algebra_simps)
-    by (metis add_mono_thms_linordered_semiring(1) affine_ineq linear mult.commute mult.left_neutral mult_right_mono not_le
-              add.commute zero_le_numeral)
-  have qs: "\<And>a b. \<lbrakk>4 * b \<le> 3; \<not> b * 2 \<le> 1\<rbrakk> \<Longrightarrow> q (4 * b - 2) \<in> 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))"
-    apply (simp add: homotopic_loops_def homotopic_with_def)
-    apply (rule_tac x="(\<lambda>y. (subpath (fst y) 1 p +++ q +++ subpath 1 (fst y) p) (snd y))" in exI)
-    apply (simp add: subpath_refl subpath_reversepath)
-    apply (intro conjI homotopic_join_lemma)
-    using papp qloop
-    apply (simp_all add: path_defs joinpaths_def o_def subpath_def c1 c2)
-    apply (force simp: contq intro: continuous_on_compose [of _ _ q, unfolded o_def] continuous_on_id continuous_on_snd)
-    apply (auto simp: ps1 ps2 qs)
-    done
-  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 \<open>path q\<close> homotopic_paths_lid qloop piq by auto
-    hence 1: "\<And>f. homotopic_paths s f q \<or> \<not> 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"
-    proof -
-      have "homotopic_paths s (q +++ linepath (pathfinish q) (pathfinish q)) q"
-        by (simp add: \<open>path q\<close> homotopic_paths_rid piq)
-      thus ?thesis
-        by (metis (no_types) 1 \<open>path q\<close> homotopic_paths_join homotopic_paths_rinv homotopic_paths_sym
-                  homotopic_paths_trans qloop pathfinish_linepath piq)
-    qed
-    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 \<open>path q\<close> 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
-    have pipq: "path_image p \<subseteq> S" "path_image q \<subseteq> S"
-      by (metis Un_subset_iff paths \<open>path p\<close> \<open>path q\<close> 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 \<open>path p\<close> \<open>path_image p \<subseteq> S\<close> 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 \<open>path p\<close> \<open>path q\<close> 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 \<open>path p\<close> \<open>path q\<close> homotopic_paths_assoc pipq)
-    moreover have "homotopic_paths S ((p +++ reversepath q) +++ q) (linepath (pathstart p) (pathstart p) +++ q)"
-      by (simp add: \<open>path q\<close> homotopic_paths_join paths pipq)
-    moreover then have "homotopic_paths S (linepath (pathstart p) (pathstart p) +++ q) q"
-      by (metis \<open>path q\<close> homotopic_paths_imp_path homotopic_paths_lid linepath_trivial path_join_path_ends pathfinish_def pipq(2))
-    ultimately show ?thesis
-      using homotopic_paths_trans by metis
-  next
-    case False
-    then show ?thesis
-      using \<open>path q\<close> homotopic_loops_imp_path loops path_join_path_ends by fastforce
-  qed
-qed
-
-
-subsection%unimportant\<open>Homotopy of "nearby" function, paths and loops\<close>
-
-lemma homotopic_with_linear:
-  fixes f g :: "_ \<Rightarrow> 'b::real_normed_vector"
-  assumes contf: "continuous_on s f"
-      and contg:"continuous_on s g"
-      and sub: "\<And>x. x \<in> s \<Longrightarrow> closed_segment (f x) (g x) \<subseteq> t"
-    shows "homotopic_with (\<lambda>z. True) s t f g"
-  apply (simp add: homotopic_with_def)
-  apply (rule_tac x="\<lambda>y. ((1 - (fst y)) *\<^sub>R f(snd y) + (fst y) *\<^sub>R g(snd y))" in exI)
-  apply (intro conjI)
-  apply (rule subset_refl continuous_intros continuous_on_subset [OF contf] continuous_on_compose2 [where g=f]
-                                            continuous_on_subset [OF contg] continuous_on_compose2 [where g=g]| simp)+
-  using sub closed_segment_def apply fastforce+
-  done
-
-lemma homotopic_paths_linear:
-  fixes g h :: "real \<Rightarrow> 'a::real_normed_vector"
-  assumes "path g" "path h" "pathstart h = pathstart g" "pathfinish h = pathfinish g"
-          "\<And>t. t \<in> {0..1} \<Longrightarrow> closed_segment (g t) (h t) \<subseteq> 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="\<lambda>y. ((1 - (fst y)) *\<^sub>R (g \<circ> snd) y + (fst y) *\<^sub>R (h \<circ> snd) y)" in exI)
-  apply (intro conjI subsetI continuous_intros; force)
-  done
-
-lemma homotopic_loops_linear:
-  fixes g h :: "real \<Rightarrow> 'a::real_normed_vector"
-  assumes "path g" "path h" "pathfinish g = pathstart g" "pathfinish h = pathstart h"
-          "\<And>t x. t \<in> {0..1} \<Longrightarrow> closed_segment (g t) (h t) \<subseteq> s"
-    shows "homotopic_loops s g h"
-  using assms
-  unfolding path_def
-  apply (simp add: pathstart_def pathfinish_def homotopic_loops_def homotopic_with_def)
-  apply (rule_tac x="\<lambda>y. ((1 - (fst y)) *\<^sub>R g(snd y) + (fst y) *\<^sub>R h(snd y))" in exI)
-  apply (auto intro!: continuous_intros intro: continuous_on_compose2 [where g=g] continuous_on_compose2 [where g=h])
-  apply (force simp: closed_segment_def)
-  done
-
-lemma homotopic_paths_nearby_explicit:
-  assumes "path g" "path h" "pathstart h = pathstart g" "pathfinish h = pathfinish g"
-      and no: "\<And>t x. \<lbrakk>t \<in> {0..1}; x \<notin> s\<rbrakk> \<Longrightarrow> norm(h t - g t) < norm(g t - x)"
-    shows "homotopic_paths s g h"
-  apply (rule homotopic_paths_linear [OF assms(1-4)])
-  by (metis no segment_bound(1) subsetI norm_minus_commute not_le)
-
-lemma homotopic_loops_nearby_explicit:
-  assumes "path g" "path h" "pathfinish g = pathstart g" "pathfinish h = pathstart h"
-      and no: "\<And>t x. \<lbrakk>t \<in> {0..1}; x \<notin> s\<rbrakk> \<Longrightarrow> norm(h t - g t) < norm(g t - x)"
-    shows "homotopic_loops s g h"
-  apply (rule homotopic_loops_linear [OF assms(1-4)])
-  by (metis no segment_bound(1) subsetI norm_minus_commute not_le)
-
-lemma homotopic_nearby_paths:
-  fixes g h :: "real \<Rightarrow> 'a::euclidean_space"
-  assumes "path g" "open s" "path_image g \<subseteq> s"
-    shows "\<exists>e. 0 < e \<and>
-               (\<forall>h. path h \<and>
-                    pathstart h = pathstart g \<and> pathfinish h = pathfinish g \<and>
-                    (\<forall>t \<in> {0..1}. norm(h t - g t) < e) \<longrightarrow> homotopic_paths s g h)"
-proof -
-  obtain e where "e > 0" and e: "\<And>x y. x \<in> path_image g \<Longrightarrow> y \<in> - s \<Longrightarrow> e \<le> dist x y"
-    using separate_compact_closed [of "path_image g" "-s"] assms by force
-  show ?thesis
-    apply (intro exI conjI)
-    using e [unfolded dist_norm]
-    apply (auto simp: intro!: homotopic_paths_nearby_explicit assms  \<open>e > 0\<close>)
-    by (metis atLeastAtMost_iff imageI le_less_trans not_le path_image_def)
-qed
-
-lemma homotopic_nearby_loops:
-  fixes g h :: "real \<Rightarrow> 'a::euclidean_space"
-  assumes "path g" "open s" "path_image g \<subseteq> s" "pathfinish g = pathstart g"
-    shows "\<exists>e. 0 < e \<and>
-               (\<forall>h. path h \<and> pathfinish h = pathstart h \<and>
-                    (\<forall>t \<in> {0..1}. norm(h t - g t) < e) \<longrightarrow> homotopic_loops s g h)"
-proof -
-  obtain e where "e > 0" and e: "\<And>x y. x \<in> path_image g \<Longrightarrow> y \<in> - s \<Longrightarrow> e \<le> dist x y"
-    using separate_compact_closed [of "path_image g" "-s"] assms by force
-  show ?thesis
-    apply (intro exI conjI)
-    using e [unfolded dist_norm]
-    apply (auto simp: intro!: homotopic_loops_nearby_explicit assms  \<open>e > 0\<close>)
-    by (metis atLeastAtMost_iff imageI le_less_trans not_le path_image_def)
-qed
-
-
-subsection\<open> Homotopy and subpaths\<close>
-
-lemma homotopic_join_subpaths1:
-  assumes "path g" and pag: "path_image g \<subseteq> s"
-      and u: "u \<in> {0..1}" and v: "v \<in> {0..1}" and w: "w \<in> {0..1}" "u \<le> v" "v \<le> w"
-    shows "homotopic_paths s (subpath u v g +++ subpath v w g) (subpath u w g)"
-proof -
-  have 1: "t * 2 \<le> 1 \<Longrightarrow> u + t * (v * 2) \<le> v + t * (u * 2)" for t
-    using affine_ineq \<open>u \<le> v\<close> by fastforce
-  have 2: "t * 2 > 1 \<Longrightarrow> u + (2*t - 1) * v \<le> 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 \<open>u \<le> v\<close> \<open>v \<le> w\<close>)
-  have t2: "\<And>t::real. t*2 = 1 \<Longrightarrow> t = 1/2" by auto
-  show ?thesis
-    apply (rule homotopic_paths_subset [OF _ pag])
-    using assms
-    apply (cases "w = u")
-    using homotopic_paths_rinv [of "subpath u v g" "path_image g"]
-    apply (force simp: closed_segment_eq_real_ivl image_mono path_image_def subpath_refl)
-      apply (rule homotopic_paths_sym)
-      apply (rule homotopic_paths_reparametrize
-             [where f = "\<lambda>t. if  t \<le> 1 / 2
-                             then inverse((w - u)) *\<^sub>R (2 * (v - u)) *\<^sub>R t
-                             else inverse((w - u)) *\<^sub>R ((v - u) + (w - v) *\<^sub>R (2 *\<^sub>R t - 1))"])
-      using \<open>path g\<close> path_subpath u w apply blast
-      using \<open>path g\<close> path_image_subpath_subset u w(1) apply blast
-      apply simp_all
-      apply (subst split_01)
-      apply (rule continuous_on_cases continuous_intros | force simp: pathfinish_def joinpaths_def)+
-      apply (simp_all add: field_simps not_le)
-      apply (force dest!: t2)
-      apply (force simp: algebra_simps mult_left_mono affine_ineq dest!: 1 2)
-      apply (simp add: joinpaths_def subpath_def)
-      apply (force simp: algebra_simps)
-      done
-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 \<subseteq> s"
-      and u: "u \<in> {0..1}" and v: "v \<in> {0..1}" and w: "w \<in> {0..1}"
-    shows "homotopic_paths s (subpath v w g +++ subpath w u g) (subpath v u g)"
-proof -
-  have "homotopic_paths s (subpath u w g +++ subpath w v g) ((subpath u v g +++ subpath v w g) +++ subpath w v g)"
-    apply (rule homotopic_paths_join)
-    using hom homotopic_paths_sym_eq apply blast
-    apply (metis \<open>path g\<close> homotopic_paths_eq pag path_image_subpath_subset path_subpath subset_trans v w, simp)
-    done
-  also have "homotopic_paths s ((subpath u v g +++ subpath v w g) +++ subpath w v g) (subpath u v g +++ subpath v w g +++ subpath w v g)"
-    apply (rule homotopic_paths_sym [OF homotopic_paths_assoc])
-    using assms by (simp_all add: path_image_subpath_subset [THEN order_trans])
-  also have "homotopic_paths s (subpath u v g +++ subpath v w g +++ subpath w v g)
-                               (subpath u v g +++ linepath (pathfinish (subpath u v g)) (pathfinish (subpath u v g)))"
-    apply (rule homotopic_paths_join)
-    apply (metis \<open>path g\<close> homotopic_paths_eq order.trans pag path_image_subpath_subset path_subpath u v)
-    apply (metis (no_types, lifting) \<open>path g\<close> homotopic_paths_linv order_trans pag path_image_subpath_subset path_subpath pathfinish_subpath reversepath_subpath v w)
-    apply simp
-    done
-  also have "homotopic_paths s (subpath u v g +++ linepath (pathfinish (subpath u v g)) (pathfinish (subpath u v g))) (subpath u v g)"
-    apply (rule homotopic_paths_rid)
-    using \<open>path g\<close> path_subpath u v apply blast
-    apply (meson \<open>path g\<close> order.trans pag path_image_subpath_subset u v)
-    done
-  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:
-   "\<lbrakk>path g; path_image g \<subseteq> s; u \<in> {0..1}; v \<in> {0..1}; w \<in> {0..1}\<rbrakk>
-    \<Longrightarrow> homotopic_paths s (subpath u v g +++ subpath v w g) (subpath u w g)"
-  apply (rule le_cases3 [of u v w])
-using homotopic_join_subpaths1 homotopic_join_subpaths2 homotopic_join_subpaths3 by metis+
-
-text\<open>Relating homotopy of trivial loops to path-connectedness.\<close>
-
-lemma path_component_imp_homotopic_points:
-    "path_component S a b \<Longrightarrow> homotopic_loops S (linepath a a) (linepath b b)"
-apply (simp add: path_component_def homotopic_loops_def homotopic_with_def
-                 pathstart_def pathfinish_def path_image_def path_def, clarify)
-apply (rule_tac x="g \<circ> fst" in exI)
-apply (intro conjI continuous_intros continuous_on_compose)+
-apply (auto elim!: continuous_on_subset)
-done
-
-lemma homotopic_loops_imp_path_component_value:
-   "\<lbrakk>homotopic_loops S p q; 0 \<le> t; t \<le> 1\<rbrakk>
-        \<Longrightarrow> path_component S (p t) (q t)"
-apply (simp add: path_component_def homotopic_loops_def homotopic_with_def
-                 pathstart_def pathfinish_def path_image_def path_def, clarify)
-apply (rule_tac x="h \<circ> (\<lambda>u. (u, t))" in exI)
-apply (intro conjI continuous_intros continuous_on_compose)+
-apply (auto elim!: continuous_on_subset)
-done
-
-lemma homotopic_points_eq_path_component:
-   "homotopic_loops S (linepath a a) (linepath b b) \<longleftrightarrow>
-        path_component S a b"
-by (auto simp: path_component_imp_homotopic_points
-         dest: homotopic_loops_imp_path_component_value [where t=1])
-
-lemma path_connected_eq_homotopic_points:
-    "path_connected S \<longleftrightarrow>
-      (\<forall>a b. a \<in> S \<and> b \<in> S \<longrightarrow> homotopic_loops S (linepath a a) (linepath b b))"
-by (auto simp: path_connected_def path_component_def homotopic_points_eq_path_component)
-
-
-subsection\<open>Simply connected sets\<close>
-
-text%important\<open>defined as "all loops are homotopic (as loops)\<close>
-
-definition%important simply_connected where
-  "simply_connected S \<equiv>
-        \<forall>p q. path p \<and> pathfinish p = pathstart p \<and> path_image p \<subseteq> S \<and>
-              path q \<and> pathfinish q = pathstart q \<and> path_image q \<subseteq> S
-              \<longrightarrow> 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 \<Longrightarrow> 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 \<Longrightarrow> 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 \<longleftrightarrow>
-            (\<forall>p a. path p \<and> path_image p \<subseteq> S \<and>
-                  pathfinish p = pathstart p \<and> a \<in> S
-                  \<longrightarrow> homotopic_loops S p (linepath a a))"
-apply (simp add: simply_connected_def)
-apply (rule iffI, force, clarify)
-apply (rule_tac q = "linepath (pathstart p) (pathstart p)" in homotopic_loops_trans)
-apply (fastforce simp add:)
-using homotopic_loops_sym apply blast
-done
-
-lemma simply_connected_eq_contractible_loop_some:
-  fixes S :: "_::real_normed_vector set"
-  shows "simply_connected S \<longleftrightarrow>
-                path_connected S \<and>
-                (\<forall>p. path p \<and> path_image p \<subseteq> S \<and> pathfinish p = pathstart p
-                    \<longrightarrow> (\<exists>a. a \<in> S \<and> homotopic_loops S p (linepath a a)))"
-apply (rule iffI)
- apply (fastforce simp: simply_connected_imp_path_connected simply_connected_eq_contractible_loop_any)
-apply (clarsimp simp add: simply_connected_eq_contractible_loop_any)
-apply (drule_tac x=p in spec)
-using homotopic_loops_trans path_connected_eq_homotopic_points
-  apply blast
-done
-
-lemma simply_connected_eq_contractible_loop_all:
-  fixes S :: "_::real_normed_vector set"
-  shows "simply_connected S \<longleftrightarrow>
-         S = {} \<or>
-         (\<exists>a \<in> S. \<forall>p. path p \<and> path_image p \<subseteq> S \<and> pathfinish p = pathstart p
-                \<longrightarrow> homotopic_loops S p (linepath a a))"
-        (is "?lhs = ?rhs")
-proof (cases "S = {}")
-  case True then show ?thesis by force
-next
-  case False
-  then obtain a where "a \<in> S" by blast
-  show ?thesis
-  proof
-    assume "simply_connected S"
-    then show ?rhs
-      using \<open>a \<in> S\<close> \<open>simply_connected S\<close> simply_connected_eq_contractible_loop_any
-      by blast
-  next
-    assume ?rhs
-    then show "simply_connected S"
-      apply (simp add: simply_connected_eq_contractible_loop_any False)
-      by (meson homotopic_loops_refl homotopic_loops_sym homotopic_loops_trans
-             path_component_imp_homotopic_points path_component_refl)
-  qed
-qed
-
-lemma simply_connected_eq_contractible_path:
-  fixes S :: "_::real_normed_vector set"
-  shows "simply_connected S \<longleftrightarrow>
-           path_connected S \<and>
-           (\<forall>p. path p \<and> path_image p \<subseteq> S \<and> pathfinish p = pathstart p
-            \<longrightarrow> homotopic_paths S p (linepath (pathstart p) (pathstart p)))"
-apply (rule iffI)
- apply (simp add: simply_connected_imp_path_connected)
- apply (metis simply_connected_eq_contractible_loop_some homotopic_loops_imp_homotopic_paths_null)
-by (meson homotopic_paths_imp_homotopic_loops pathfinish_linepath pathstart_in_path_image
-         simply_connected_eq_contractible_loop_some subset_iff)
-
-lemma simply_connected_eq_homotopic_paths:
-  fixes S :: "_::real_normed_vector set"
-  shows "simply_connected S \<longleftrightarrow>
-          path_connected S \<and>
-          (\<forall>p q. path p \<and> path_image p \<subseteq> S \<and>
-                path q \<and> path_image q \<subseteq> S \<and>
-                pathstart q = pathstart p \<and> pathfinish q = pathfinish p
-                \<longrightarrow> homotopic_paths S p q)"
-         (is "?lhs = ?rhs")
-proof
-  assume ?lhs
-  then have pc: "path_connected S"
-        and *:  "\<And>p. \<lbrakk>path p; path_image p \<subseteq> S;
-                       pathfinish p = pathstart p\<rbrakk>
-                      \<Longrightarrow> 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 \<subseteq> S" "path q"
-           "path_image q \<subseteq> S" "pathstart q = pathstart p"
-           "pathfinish q = pathfinish p" for p q
-  proof -
-    have "homotopic_paths S p (p +++ linepath (pathfinish p) (pathfinish p))"
-      by (simp add: homotopic_paths_rid homotopic_paths_sym that)
-    also have "homotopic_paths S (p +++ linepath (pathfinish p) (pathfinish p))
-                                 (p +++ reversepath q +++ q)"
-      using that
-      by (metis homotopic_paths_join homotopic_paths_linv homotopic_paths_refl homotopic_paths_sym_eq pathstart_linepath)
-    also have "homotopic_paths S (p +++ reversepath q +++ q)
-                                 ((p +++ reversepath q) +++ q)"
-      by (simp add: that homotopic_paths_assoc)
-    also have "homotopic_paths S ((p +++ reversepath q) +++ q)
-                                 (linepath (pathstart q) (pathstart q) +++ q)"
-      using * [of "p +++ reversepath q"] that
-      by (simp add: homotopic_paths_join path_image_join)
-    also have "homotopic_paths S (linepath (pathstart q) (pathstart q) +++ q) 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 \<times> T)"
-proof -
-  have "homotopic_loops (S \<times> T) p (linepath (a, b) (a, b))"
-       if "path p" "path_image p \<subseteq> S \<times> T" "p 1 = p 0" "a \<in> S" "b \<in> T"
-       for p a b
-  proof -
-    have "path (fst \<circ> p)"
-      apply (rule Path_Connected.path_continuous_image [OF \<open>path p\<close>])
-      apply (rule continuous_intros)+
-      done
-    moreover have "path_image (fst \<circ> p) \<subseteq> S"
-      using that apply (simp add: path_image_def) by force
-    ultimately have p1: "homotopic_loops S (fst \<circ> p) (linepath a a)"
-      using S that
-      apply (simp add: simply_connected_eq_contractible_loop_any)
-      apply (drule_tac x="fst \<circ> p" in spec)
-      apply (drule_tac x=a in spec)
-      apply (auto simp: pathstart_def pathfinish_def)
-      done
-    have "path (snd \<circ> p)"
-      apply (rule Path_Connected.path_continuous_image [OF \<open>path p\<close>])
-      apply (rule continuous_intros)+
-      done
-    moreover have "path_image (snd \<circ> p) \<subseteq> T"
-      using that apply (simp add: path_image_def) by force
-    ultimately have p2: "homotopic_loops T (snd \<circ> p) (linepath b b)"
-      using T that
-      apply (simp add: simply_connected_eq_contractible_loop_any)
-      apply (drule_tac x="snd \<circ> p" in spec)
-      apply (drule_tac x=b in spec)
-      apply (auto simp: pathstart_def pathfinish_def)
-      done
-    show ?thesis
-      using p1 p2
-      apply (simp add: homotopic_loops, clarify)
-      apply (rename_tac h k)
-      apply (rule_tac x="\<lambda>z. Pair (h z) (k z)" in exI)
-      apply (intro conjI continuous_intros | assumption)+
-      apply (auto simp: pathstart_def pathfinish_def)
-      done
-  qed
-  with assms show ?thesis
-    by (simp add: simply_connected_eq_contractible_loop_any pathfinish_def pathstart_def)
-qed
-
-
-subsection\<open>Contractible sets\<close>
-
-definition%important contractible where
- "contractible S \<equiv> \<exists>a. homotopic_with (\<lambda>x. True) S S id (\<lambda>x. 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 (\<lambda>x. True) S S id (\<lambda>x. a)"
-    using assms by (force simp: contractible_def)
-  then have "a \<in> S"
-    by (metis False homotopic_constant_maps homotopic_with_symD homotopic_with_trans path_component_mem(2))
-  show ?thesis
-    apply (simp add: simply_connected_eq_contractible_loop_all False)
-    apply (rule bexI [OF _ \<open>a \<in> S\<close>])
-    using a apply (simp add: homotopic_loops_def homotopic_with_def path_def path_image_def pathfinish_def pathstart_def, clarify)
-    apply (rule_tac x="(h \<circ> (\<lambda>y. (fst y, (p \<circ> snd) y)))" in exI)
-    apply (intro conjI continuous_on_compose continuous_intros)
-    apply (erule continuous_on_subset | force)+
-    done
-qed
-
-corollary contractible_imp_connected:
-  fixes S :: "_::real_normed_vector set"
-  shows "contractible S \<Longrightarrow> 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 \<Longrightarrow> 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 \<subseteq> T"
-      and g: "continuous_on T g" "g ` T \<subseteq> U"
-      and T: "contractible T"
-    obtains c where "homotopic_with (\<lambda>h. True) S U (g \<circ> f) (\<lambda>x. c)"
-proof -
-  obtain b where b: "homotopic_with (\<lambda>x. True) T T id (\<lambda>x. b)"
-    using assms by (force simp: contractible_def)
-  have "homotopic_with (\<lambda>f. True) T U (g \<circ> id) (g \<circ> (\<lambda>x. b))"
-    by (rule homotopic_compose_continuous_left [OF b g])
-  then have "homotopic_with (\<lambda>f. True) S U (g \<circ> id \<circ> f) (g \<circ> (\<lambda>x. b) \<circ> f)"
-    by (rule homotopic_compose_continuous_right [OF _ f])
-  then show ?thesis
-    by (simp add: comp_def that)
-qed
-
-lemma nullhomotopic_into_contractible:
-  assumes f: "continuous_on S f" "f ` S \<subseteq> T"
-      and T: "contractible T"
-    obtains c where "homotopic_with (\<lambda>h. True) S T f (\<lambda>x. c)"
-apply (rule nullhomotopic_through_contractible [OF f, of id T])
-using assms
-apply (auto simp: continuous_on_id)
-done
-
-lemma nullhomotopic_from_contractible:
-  assumes f: "continuous_on S f" "f ` S \<subseteq> T"
-      and S: "contractible S"
-    obtains c where "homotopic_with (\<lambda>h. True) S T f (\<lambda>x. c)"
-apply (rule nullhomotopic_through_contractible [OF continuous_on_id _ f S, of S])
-using assms
-apply (auto simp: comp_def)
-done
-
-lemma homotopic_through_contractible:
-  fixes S :: "_::real_normed_vector set"
-  assumes "continuous_on S f1" "f1 ` S \<subseteq> T"
-          "continuous_on T g1" "g1 ` T \<subseteq> U"
-          "continuous_on S f2" "f2 ` S \<subseteq> T"
-          "continuous_on T g2" "g2 ` T \<subseteq> U"
-          "contractible T" "path_connected U"
-   shows "homotopic_with (\<lambda>h. True) S U (g1 \<circ> f1) (g2 \<circ> f2)"
-proof -
-  obtain c1 where c1: "homotopic_with (\<lambda>h. True) S U (g1 \<circ> f1) (\<lambda>x. c1)"
-    apply (rule nullhomotopic_through_contractible [of S f1 T g1 U])
-    using assms apply auto
-    done
-  obtain c2 where c2: "homotopic_with (\<lambda>h. True) S U (g2 \<circ> f2) (\<lambda>x. c2)"
-    apply (rule nullhomotopic_through_contractible [of S f2 T g2 U])
-    using assms apply auto
-    done
-  have *: "S = {} \<or> (\<exists>t. path_connected t \<and> t \<subseteq> U \<and> c2 \<in> t \<and> c1 \<in> t)"
-  proof (cases "S = {}")
-    case True then show ?thesis by force
-  next
-    case False
-    with c1 c2 have "c1 \<in> U" "c2 \<in> U"
-      using homotopic_with_imp_subset2 all_not_in_conv image_subset_iff by blast+
-    with \<open>path_connected U\<close> show ?thesis by blast
-  qed
-  show ?thesis
-    apply (rule homotopic_with_trans [OF c1])
-    apply (rule homotopic_with_symD)
-    apply (rule homotopic_with_trans [OF c2])
-    apply (simp add: path_component homotopic_constant_maps *)
-    done
-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 \<subseteq> T"
-      and g: "continuous_on S g" "g ` S \<subseteq> T"
-      and T: "contractible T"
-    shows "homotopic_with (\<lambda>h. 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 continuous_on_id)
-
-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 \<subseteq> T"
-      and g: "continuous_on S g" "g ` S \<subseteq> T"
-      and "contractible S" "path_connected T"
-    shows "homotopic_with (\<lambda>h. 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 continuous_on_id)
-
-lemma starlike_imp_contractible_gen:
-  fixes S :: "'a::real_normed_vector set"
-  assumes S: "starlike S"
-      and P: "\<And>a T. \<lbrakk>a \<in> S; 0 \<le> T; T \<le> 1\<rbrakk> \<Longrightarrow> P(\<lambda>x. (1 - T) *\<^sub>R x + T *\<^sub>R a)"
-    obtains a where "homotopic_with P S S (\<lambda>x. x) (\<lambda>x. a)"
-proof -
-  obtain a where "a \<in> S" and a: "\<And>x. x \<in> S \<Longrightarrow> closed_segment a x \<subseteq> S"
-    using S by (auto simp: starlike_def)
-  have "(\<lambda>y. (1 - fst y) *\<^sub>R snd y + fst y *\<^sub>R a) ` ({0..1} \<times> S) \<subseteq> S"
-    apply clarify
-    apply (erule a [unfolded closed_segment_def, THEN subsetD], simp)
-    apply (metis add_diff_cancel_right' diff_ge_0_iff_ge le_add_diff_inverse pth_c(1))
-    done
-  then show ?thesis
-    apply (rule_tac a=a in that)
-    using \<open>a \<in> S\<close>
-    apply (simp add: homotopic_with_def)
-    apply (rule_tac x="\<lambda>y. (1 - (fst y)) *\<^sub>R snd y + (fst y) *\<^sub>R a" in exI)
-    apply (intro conjI ballI continuous_on_compose continuous_intros)
-    apply (simp_all add: P)
-    done
-qed
-
-lemma starlike_imp_contractible:
-  fixes S :: "'a::real_normed_vector set"
-  shows "starlike S \<Longrightarrow> 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 \<Longrightarrow> 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 \<Longrightarrow> 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 \<Longrightarrow> 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 \<Longrightarrow> 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 \<longleftrightarrow> simply_connected S"
-using convex_imp_simply_connected is_interval_convex_1 is_interval_path_connected_1 simply_connected_imp_path_connected by auto
-
-lemma contractible_empty [simp]: "contractible {}"
-  by (simp add: contractible_def homotopic_with)
-
-lemma contractible_convex_tweak_boundary_points:
-  fixes S :: "'a::euclidean_space set"
-  assumes "convex S" and TS: "rel_interior S \<subseteq> T" "T \<subseteq> closure S"
-  shows "contractible T"
-proof (cases "S = {}")
-  case True
-  with assms show ?thesis
-    by (simp add: subsetCE)
-next
-  case False
-  show ?thesis
-    apply (rule starlike_imp_contractible)
-    apply (rule starlike_convex_tweak_boundary_points [OF \<open>convex S\<close> False TS])
-    done
-qed
-
-lemma convex_imp_contractible:
-  fixes S :: "'a::real_normed_vector set"
-  shows "convex S \<Longrightarrow> 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 \<longleftrightarrow> 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 \<times> T)"
-proof -
-  obtain a h where conth: "continuous_on ({0..1} \<times> S) h"
-             and hsub: "h ` ({0..1} \<times> S) \<subseteq> S"
-             and [simp]: "\<And>x. x \<in> S \<Longrightarrow> h (0, x) = x"
-             and [simp]: "\<And>x. x \<in> S \<Longrightarrow>  h (1::real, x) = a"
-    using S by (auto simp: contractible_def homotopic_with)
-  obtain b k where contk: "continuous_on ({0..1} \<times> T) k"
-             and ksub: "k ` ({0..1} \<times> T) \<subseteq> T"
-             and [simp]: "\<And>x. x \<in> T \<Longrightarrow> k (0, x) = x"
-             and [simp]: "\<And>x. x \<in> T \<Longrightarrow>  k (1::real, x) = b"
-    using T by (auto 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 = "\<lambda>z. (h (fst z, fst(snd z)), k (fst z, snd(snd z)))"])
-    apply (intro conjI ballI continuous_intros continuous_on_compose2 [OF conth] continuous_on_compose2 [OF contk])
-    using hsub ksub
-    apply auto
-    done
-qed
-
-lemma homotopy_dominated_contractibility:
-  fixes S :: "'a::real_normed_vector set" and T :: "'b::real_normed_vector set"
-  assumes S: "contractible S"
-      and f: "continuous_on S f" "image f S \<subseteq> T"
-      and g: "continuous_on T g" "image g T \<subseteq> S"
-      and hom: "homotopic_with (\<lambda>x. True) T T (f \<circ> g) id"
-    shows "contractible T"
-proof -
-  obtain b where "homotopic_with (\<lambda>h. True) S T f (\<lambda>x. b)"
-    using nullhomotopic_from_contractible [OF f S] .
-  then have homg: "homotopic_with (\<lambda>x. True) T T ((\<lambda>x. b) \<circ> g) (f \<circ> g)"
-    by (rule homotopic_with_compose_continuous_right [OF homotopic_with_symD g])
-  show ?thesis
-    apply (simp add: contractible_def)
-    apply (rule exI [where x = b])
-    apply (rule homotopic_with_symD)
-    apply (rule homotopic_with_trans [OF _ hom])
-    using homg apply (simp add: o_def)
-    done
-qed
-
-
-subsection\<open>Local versions of topological properties in general\<close>
-
-definition%important locally :: "('a::topological_space set \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> bool"
-where
- "locally P S \<equiv>
-        \<forall>w x. openin (subtopology euclidean S) w \<and> x \<in> w
-              \<longrightarrow> (\<exists>u v. openin (subtopology euclidean S) u \<and> P v \<and>
-                        x \<in> u \<and> u \<subseteq> v \<and> v \<subseteq> w)"
-
-lemma locallyI:
-  assumes "\<And>w x. \<lbrakk>openin (subtopology euclidean S) w; x \<in> w\<rbrakk>
-                  \<Longrightarrow> \<exists>u v. openin (subtopology euclidean S) u \<and> P v \<and>
-                        x \<in> u \<and> u \<subseteq> v \<and> v \<subseteq> w"
-    shows "locally P S"
-using assms by (force simp: locally_def)
-
-lemma locallyE:
-  assumes "locally P S" "openin (subtopology euclidean S) w" "x \<in> w"
-  obtains u v where "openin (subtopology euclidean S) u"
-                    "P v" "x \<in> u" "u \<subseteq> v" "v \<subseteq> w"
-  using assms unfolding locally_def by meson
-
-lemma locally_mono:
-  assumes "locally P S" "\<And>t. P t \<Longrightarrow> Q t"
-    shows "locally Q S"
-by (metis assms locally_def)
-
-lemma locally_open_subset:
-  assumes "locally P S" "openin (subtopology euclidean S) t"
-    shows "locally P t"
-using assms
-apply (simp add: locally_def)
-apply (erule all_forward)+
-apply (rule impI)
-apply (erule impCE)
- using openin_trans apply blast
-apply (erule ex_forward)
-by (metis (no_types, hide_lams) Int_absorb1 Int_lower1 Int_subset_iff openin_open openin_subtopology_Int_subset)
-
-lemma locally_diff_closed:
-    "\<lbrakk>locally P S; closedin (subtopology euclidean S) t\<rbrakk> \<Longrightarrow> 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} \<longleftrightarrow> P {a}"
-apply (simp add: locally_def openin_euclidean_subtopology_iff subset_singleton_iff conj_disj_distribR cong: conj_cong)
-using zero_less_one by blast
-
-lemma locally_iff:
-    "locally P S \<longleftrightarrow>
-     (\<forall>T x. open T \<and> x \<in> S \<inter> T \<longrightarrow> (\<exists>U. open U \<and> (\<exists>v. P v \<and> x \<in> S \<inter> U \<and> S \<inter> U \<subseteq> v \<and> v \<subseteq> S \<inter> T)))"
-apply (simp add: le_inf_iff locally_def openin_open, safe)
-apply (metis IntE IntI le_inf_iff)
-apply (metis IntI Int_subset_iff)
-done
-
-lemma locally_Int:
-  assumes S: "locally P S" and t: "locally P t"
-      and P: "\<And>S t. P S \<and> P t \<Longrightarrow> P(S \<inter> t)"
-    shows "locally P (S \<inter> t)"
-using S t unfolding locally_iff
-apply clarify
-apply (drule_tac x=T in spec)+
-apply (drule_tac x=x in spec)+
-apply clarsimp
-apply (rename_tac U1 U2 V1 V2)
-apply (rule_tac x="U1 \<inter> U2" in exI)
-apply (simp add: open_Int)
-apply (rule_tac x="V1 \<inter> V2" in exI)
-apply (auto intro: P)
-done
-
-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: "\<And>S T. P S \<and> Q T \<Longrightarrow> R(S \<times> T)"
-  shows "locally R (S \<times> T)"
-    unfolding locally_def
-proof (clarify)
-  fix W x y
-  assume W: "openin (subtopology euclidean (S \<times> T)) W" and xy: "(x, y) \<in> W"
-  then obtain U V where "openin (subtopology euclidean S) U" "x \<in> U"
-                        "openin (subtopology euclidean T) V" "y \<in> V" "U \<times> V \<subseteq> W"
-    using Times_in_interior_subtopology by metis
-  then obtain U1 U2 V1 V2
-         where opeS: "openin (subtopology euclidean S) U1 \<and> P U2 \<and> x \<in> U1 \<and> U1 \<subseteq> U2 \<and> U2 \<subseteq> U"
-           and opeT: "openin (subtopology euclidean T) V1 \<and> Q V2 \<and> y \<in> V1 \<and> V1 \<subseteq> V2 \<and> V2 \<subseteq> V"
-    by (meson PS QT locallyE)
-  with \<open>U \<times> V \<subseteq> W\<close> show "\<exists>u v. openin (subtopology euclidean (S \<times> T)) u \<and> R v \<and> (x,y) \<in> u \<and> u \<subseteq> v \<and> v \<subseteq> W"
-    apply (rule_tac x="U1 \<times> V1" in exI)
-    apply (rule_tac x="U2 \<times> V2" in exI)
-    apply (auto simp: openin_Times R)
-    done
-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: "\<And>S S'. \<lbrakk>P S; homeomorphism S S' f g\<rbrakk> \<Longrightarrow> Q S'"
-    shows "locally Q t"
-proof (clarsimp simp: locally_def)
-  fix W y
-  assume "y \<in> W" and "openin (subtopology euclidean t) W"
-  then obtain T where T: "open T" "W = t \<inter> T"
-    by (force simp: openin_open)
-  then have "W \<subseteq> t" by auto
-  have f: "\<And>x. x \<in> S \<Longrightarrow> g(f x) = x" "f ` S = t" "continuous_on S f"
-   and g: "\<And>y. y \<in> t \<Longrightarrow> f(g y) = y" "g ` t = S" "continuous_on t g"
-    using hom by (auto simp: homeomorphism_def)
-  have gw: "g ` W = S \<inter> f -` W"
-    using \<open>W \<subseteq> t\<close>
-    apply auto
-    using \<open>g ` t = S\<close> \<open>W \<subseteq> t\<close> apply blast
-    using g \<open>W \<subseteq> t\<close> apply auto[1]
-    by (simp add: f rev_image_eqI)
-  have \<circ>: "openin (subtopology euclidean S) (g ` W)"
-  proof -
-    have "continuous_on S f"
-      using f(3) by blast
-    then show "openin (subtopology euclidean S) (g ` W)"
-      by (simp add: gw Collect_conj_eq \<open>openin (subtopology euclidean t) W\<close> continuous_on_open f(2))
-  qed
-  then obtain u v
-    where osu: "openin (subtopology euclidean S) u" and uv: "P v" "g y \<in> u" "u \<subseteq> v" "v \<subseteq> g ` W"
-    using S [unfolded locally_def, rule_format, of "g ` W" "g y"] \<open>y \<in> W\<close> by force
-  have "v \<subseteq> S" using uv by (simp add: gw)
-  have fv: "f ` v = t \<inter> {x. g x \<in> v}"
-    using \<open>f ` S = t\<close> f \<open>v \<subseteq> S\<close> by auto
-  have "f ` v \<subseteq> W"
-    using uv using Int_lower2 gw image_subsetI mem_Collect_eq subset_iff by auto
-  have contvf: "continuous_on v f"
-    using \<open>v \<subseteq> S\<close> continuous_on_subset f(3) by blast
-  have contvg: "continuous_on (f ` v) g"
-    using \<open>f ` v \<subseteq> W\<close> \<open>W \<subseteq> t\<close> continuous_on_subset [OF g(3)] by blast
-  have homv: "homeomorphism v (f ` v) f g"
-    using \<open>v \<subseteq> S\<close> \<open>W \<subseteq> t\<close> f
-    apply (simp add: homeomorphism_def contvf contvg, auto)
-    by (metis f(1) rev_image_eqI rev_subsetD)
-  have 1: "openin (subtopology euclidean t) (t \<inter> g -` u)"
-    apply (rule continuous_on_open [THEN iffD1, rule_format])
-    apply (rule \<open>continuous_on t g\<close>)
-    using \<open>g ` t = S\<close> apply (simp add: osu)
-    done
-  have 2: "\<exists>V. Q V \<and> y \<in> (t \<inter> g -` u) \<and> (t \<inter> g -` u) \<subseteq> V \<and> V \<subseteq> W"
-    apply (rule_tac x="f ` v" in exI)
-    apply (intro conjI Q [OF \<open>P v\<close> homv])
-    using \<open>W \<subseteq> t\<close> \<open>y \<in> W\<close>  \<open>f ` v \<subseteq> W\<close>  uv  apply (auto simp: fv)
-    done
-  show "\<exists>U. openin (subtopology euclidean t) U \<and> (\<exists>v. Q v \<and> y \<in> U \<and> U \<subseteq> v \<and> v \<subseteq> W)"
-    by (meson 1 2)
-qed
-
-lemma homeomorphism_locally:
-  fixes f:: "'a::metric_space \<Rightarrow> 'b::metric_space"
-  assumes hom: "homeomorphism S t f g"
-      and eq: "\<And>S t. homeomorphism S t f g \<Longrightarrow> (P S \<longleftrightarrow> Q t)"
-    shows "locally P S \<longleftrightarrow> locally Q t"
-apply (rule iffI)
-apply (erule homeomorphism_locally_imp [OF _ hom])
-apply (simp add: eq)
-apply (erule homeomorphism_locally_imp)
-using eq homeomorphism_sym homeomorphism_symD [OF hom] apply blast+
-done
-
-lemma homeomorphic_locally:
-  fixes S:: "'a::metric_space set" and T:: "'b::metric_space set"
-  assumes hom: "S homeomorphic T"
-          and iff: "\<And>X Y. X homeomorphic Y \<Longrightarrow> (P X \<longleftrightarrow> Q Y)"
-    shows "locally P S \<longleftrightarrow> locally Q T"
-proof -
-  obtain f g where hom: "homeomorphism S T f g"
-    using assms by (force simp: homeomorphic_def)
-  then show ?thesis
-    using homeomorphic_def local.iff
-    by (blast intro!: homeomorphism_locally)
-qed
-
-lemma homeomorphic_local_compactness:
-  fixes S:: "'a::metric_space set" and T:: "'b::metric_space set"
-  shows "S homeomorphic T \<Longrightarrow> locally compact S \<longleftrightarrow> locally compact T"
-by (simp add: homeomorphic_compactness homeomorphic_locally)
-
-lemma locally_translation:
-  fixes P :: "'a :: real_normed_vector set \<Rightarrow> bool"
-  shows
-   "(\<And>S. P (image (\<lambda>x. a + x) S) \<longleftrightarrow> P S)
-        \<Longrightarrow> locally P (image (\<lambda>x. a + x) S) \<longleftrightarrow> locally P S"
-apply (rule homeomorphism_locally [OF homeomorphism_translation])
-apply (simp add: homeomorphism_def)
-by metis
-
-lemma locally_injective_linear_image:
-  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
-  assumes f: "linear f" "inj f" and iff: "\<And>S. P (f ` S) \<longleftrightarrow> Q S"
-    shows "locally P (f ` S) \<longleftrightarrow> locally Q S"
-apply (rule linear_homeomorphism_image [OF f])
-apply (rule_tac f=g and g = f in homeomorphism_locally, assumption)
-by (metis iff homeomorphism_def)
-
-lemma locally_open_map_image:
-  fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
-  assumes P: "locally P S"
-      and f: "continuous_on S f"
-      and oo: "\<And>t. openin (subtopology euclidean S) t
-                   \<Longrightarrow> openin (subtopology euclidean (f ` S)) (f ` t)"
-      and Q: "\<And>t. \<lbrakk>t \<subseteq> S; P t\<rbrakk> \<Longrightarrow> Q(f ` t)"
-    shows "locally Q (f ` S)"
-proof (clarsimp simp add: locally_def)
-  fix W y
-  assume oiw: "openin (subtopology euclidean (f ` S)) W" and "y \<in> W"
-  then have "W \<subseteq> f ` S" by (simp add: openin_euclidean_subtopology_iff)
-  have oivf: "openin (subtopology euclidean S) (S \<inter> f -` W)"
-    by (rule continuous_on_open [THEN iffD1, rule_format, OF f oiw])
-  then obtain x where "x \<in> S" "f x = y"
-    using \<open>W \<subseteq> f ` S\<close> \<open>y \<in> W\<close> by blast
-  then obtain U V
-    where "openin (subtopology euclidean S) U" "P V" "x \<in> U" "U \<subseteq> V" "V \<subseteq> S \<inter> f -` W"
-    using P [unfolded locally_def, rule_format, of "(S \<inter> f -` W)" x] oivf \<open>y \<in> W\<close>
-    by auto
-  then show "\<exists>X. openin (subtopology euclidean (f ` S)) X \<and> (\<exists>Y. Q Y \<and> y \<in> X \<and> X \<subseteq> Y \<and> Y \<subseteq> W)"
-    apply (rule_tac x="f ` U" in exI)
-    apply (rule conjI, blast intro!: oo)
-    apply (rule_tac x="f ` V" in exI)
-    apply (force simp: \<open>f x = y\<close> rev_image_eqI intro: Q)
-    done
-qed
-
-
-subsection\<open>An induction principle for connected sets\<close>
-
-proposition connected_induction:
-  assumes "connected S"
-      and opD: "\<And>T a. \<lbrakk>openin (subtopology euclidean S) T; a \<in> T\<rbrakk> \<Longrightarrow> \<exists>z. z \<in> T \<and> P z"
-      and opI: "\<And>a. a \<in> S
-             \<Longrightarrow> \<exists>T. openin (subtopology euclidean S) T \<and> a \<in> T \<and>
-                     (\<forall>x \<in> T. \<forall>y \<in> T. P x \<and> P y \<and> Q x \<longrightarrow> Q y)"
-      and etc: "a \<in> S" "b \<in> S" "P a" "P b" "Q a"
-    shows "Q b"
-proof -
-  have 1: "openin (subtopology euclidean S)
-             {b. \<exists>T. openin (subtopology euclidean S) T \<and>
-                     b \<in> T \<and> (\<forall>x\<in>T. P x \<longrightarrow> Q x)}"
-    apply (subst openin_subopen, clarify)
-    apply (rule_tac x=T in exI, auto)
-    done
-  have 2: "openin (subtopology euclidean S)
-             {b. \<exists>T. openin (subtopology euclidean S) T \<and>
-                     b \<in> T \<and> (\<forall>x\<in>T. P x \<longrightarrow> \<not> Q x)}"
-    apply (subst openin_subopen, clarify)
-    apply (rule_tac x=T in exI, auto)
-    done
-  show ?thesis
-    using \<open>connected S\<close>
-    apply (simp only: connected_openin HOL.not_ex HOL.de_Morgan_conj)
-    apply (elim disjE allE)
-         apply (blast intro: 1)
-        apply (blast intro: 2, simp_all)
-       apply clarify apply (metis opI)
-      using opD apply (blast intro: etc elim: dest:)
-     using opI etc apply meson+
-    done
-qed
-
-lemma connected_equivalence_relation_gen:
-  assumes "connected S"
-      and etc: "a \<in> S" "b \<in> S" "P a" "P b"
-      and trans: "\<And>x y z. \<lbrakk>R x y; R y z\<rbrakk> \<Longrightarrow> R x z"
-      and opD: "\<And>T a. \<lbrakk>openin (subtopology euclidean S) T; a \<in> T\<rbrakk> \<Longrightarrow> \<exists>z. z \<in> T \<and> P z"
-      and opI: "\<And>a. a \<in> S
-             \<Longrightarrow> \<exists>T. openin (subtopology euclidean S) T \<and> a \<in> T \<and>
-                     (\<forall>x \<in> T. \<forall>y \<in> T. P x \<and> P y \<longrightarrow> R x y)"
-    shows "R a b"
-proof -
-  have "\<And>a b c. \<lbrakk>a \<in> S; P a; b \<in> S; c \<in> S; P b; P c; R a b\<rbrakk> \<Longrightarrow> R a c"
-    apply (rule connected_induction [OF \<open>connected S\<close> 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 \<in> S" "b \<in> S" "P a"
-      and opI: "\<And>a. a \<in> S
-             \<Longrightarrow> \<exists>T. openin (subtopology euclidean S) T \<and> a \<in> T \<and>
-                     (\<forall>x \<in> T. \<forall>y \<in> T. P x \<longrightarrow> P y)"
-    shows "P b"
-apply (rule connected_induction [OF \<open>connected S\<close> _, where P = "\<lambda>x. True"], blast)
-apply (frule opI)
-using etc apply simp_all
-done
-
-lemma connected_equivalence_relation:
-  assumes "connected S"
-      and etc: "a \<in> S" "b \<in> S"
-      and sym: "\<And>x y. \<lbrakk>R x y; x \<in> S; y \<in> S\<rbrakk> \<Longrightarrow> R y x"
-      and trans: "\<And>x y z. \<lbrakk>R x y; R y z; x \<in> S; y \<in> S; z \<in> S\<rbrakk> \<Longrightarrow> R x z"
-      and opI: "\<And>a. a \<in> S \<Longrightarrow> \<exists>T. openin (subtopology euclidean S) T \<and> a \<in> T \<and> (\<forall>x \<in> T. R a x)"
-    shows "R a b"
-proof -
-  have "\<And>a b c. \<lbrakk>a \<in> S; b \<in> S; c \<in> S; R a b\<rbrakk> \<Longrightarrow> R a c"
-    apply (rule connected_induction_simple [OF \<open>connected S\<close>], simp_all)
-    by (meson local.sym local.trans opI openin_imp_subset subsetCE)
-  then show ?thesis by (metis etc opI)
-qed
-
-lemma locally_constant_imp_constant:
-  assumes "connected S"
-      and opI: "\<And>a. a \<in> S
-             \<Longrightarrow> \<exists>T. openin (subtopology euclidean S) T \<and> a \<in> T \<and> (\<forall>x \<in> T. f x = f a)"
-    shows "f constant_on S"
-proof -
-  have "\<And>x y. x \<in> S \<Longrightarrow> y \<in> S \<Longrightarrow> f x = f y"
-    apply (rule connected_equivalence_relation [OF \<open>connected S\<close>], simp_all)
-    by (metis opI)
-  then show ?thesis
-    by (metis constant_on_def)
-qed
-
-lemma locally_constant:
-     "connected S \<Longrightarrow> locally (\<lambda>U. f constant_on U) S \<longleftrightarrow> f constant_on S"
-apply (simp add: locally_def)
-apply (rule iffI)
- apply (rule locally_constant_imp_constant, assumption)
- apply (metis (mono_tags, hide_lams) constant_on_def constant_on_subset openin_subtopology_self)
-by (meson constant_on_subset openin_imp_subset order_refl)
-
-
-subsection\<open>Basic properties of local compactness\<close>
-
-proposition locally_compact:
-  fixes s :: "'a :: metric_space set"
-  shows
-    "locally compact s \<longleftrightarrow>
-     (\<forall>x \<in> s. \<exists>u v. x \<in> u \<and> u \<subseteq> v \<and> v \<subseteq> s \<and>
-                    openin (subtopology euclidean s) u \<and> compact v)"
-     (is "?lhs = ?rhs")
-proof
-  assume ?lhs
-  then show ?rhs
-    apply clarify
-    apply (erule_tac w = "s \<inter> ball x 1" in locallyE)
-    by auto
-next
-  assume r [rule_format]: ?rhs
-  have *: "\<exists>u v.
-              openin (subtopology euclidean s) u \<and>
-              compact v \<and> x \<in> u \<and> u \<subseteq> v \<and> v \<subseteq> s \<inter> T"
-          if "open T" "x \<in> s" "x \<in> T" for x T
-  proof -
-    obtain u v where uv: "x \<in> u" "u \<subseteq> v" "v \<subseteq> s" "compact v" "openin (subtopology euclidean s) u"
-      using r [OF \<open>x \<in> s\<close>] by auto
-    obtain e where "e>0" and e: "cball x e \<subseteq> T"
-      using open_contains_cball \<open>open T\<close> \<open>x \<in> T\<close> by blast
-    show ?thesis
-      apply (rule_tac x="(s \<inter> ball x e) \<inter> u" in exI)
-      apply (rule_tac x="cball x e \<inter> v" in exI)
-      using that \<open>e > 0\<close> e uv
-      apply auto
-      done
-  qed
-  show ?lhs
-    apply (rule locallyI)
-    apply (subst (asm) openin_open)
-    apply (blast intro: *)
-    done
-qed
-
-lemma locally_compactE:
-  fixes s :: "'a :: metric_space set"
-  assumes "locally compact s"
-  obtains u v where "\<And>x. x \<in> s \<Longrightarrow> x \<in> u x \<and> u x \<subseteq> v x \<and> v x \<subseteq> s \<and>
-                             openin (subtopology euclidean s) (u x) \<and> compact (v x)"
-using assms
-unfolding locally_compact by metis
-
-lemma locally_compact_alt:
-  fixes s :: "'a :: heine_borel set"
-  shows "locally compact s \<longleftrightarrow>
-         (\<forall>x \<in> s. \<exists>u. x \<in> u \<and>
-                    openin (subtopology euclidean s) u \<and> compact(closure u) \<and> closure u \<subseteq> s)"
-apply (simp add: locally_compact)
-apply (intro ball_cong ex_cong refl iffI)
-apply (metis bounded_subset closure_eq closure_mono compact_eq_bounded_closed dual_order.trans)
-by (meson closure_subset compact_closure)
-
-lemma locally_compact_Int_cball:
-  fixes s :: "'a :: heine_borel set"
-  shows "locally compact s \<longleftrightarrow> (\<forall>x \<in> s. \<exists>e. 0 < e \<and> closed(cball x e \<inter> s))"
-        (is "?lhs = ?rhs")
-proof
-  assume ?lhs
-  then show ?rhs
-    apply (simp add: locally_compact openin_contains_cball)
-    apply (clarify | assumption | drule bspec)+
-    by (metis (no_types, lifting)  compact_cball compact_imp_closed compact_Int inf.absorb_iff2 inf.orderE inf_sup_aci(2))
-next
-  assume ?rhs
-  then show ?lhs
-    apply (simp add: locally_compact openin_contains_cball)
-    apply (clarify | assumption | drule bspec)+
-    apply (rule_tac x="ball x e \<inter> s" in exI, simp)
-    apply (rule_tac x="cball x e \<inter> s" in exI)
-    using compact_eq_bounded_closed
-    apply auto
-    apply (metis open_ball le_infI1 mem_ball open_contains_cball_eq)
-    done
-qed
-
-lemma locally_compact_compact:
-  fixes s :: "'a :: heine_borel set"
-  shows "locally compact s \<longleftrightarrow>
-         (\<forall>k. k \<subseteq> s \<and> compact k
-              \<longrightarrow> (\<exists>u v. k \<subseteq> u \<and> u \<subseteq> v \<and> v \<subseteq> s \<and>
-                         openin (subtopology euclidean s) u \<and> compact v))"
-        (is "?lhs = ?rhs")
-proof
-  assume ?lhs
-  then obtain u v where
-    uv: "\<And>x. x \<in> s \<Longrightarrow> x \<in> u x \<and> u x \<subseteq> v x \<and> v x \<subseteq> s \<and>
-                             openin (subtopology euclidean s) (u x) \<and> compact (v x)"
-    by (metis locally_compactE)
-  have *: "\<exists>u v. k \<subseteq> u \<and> u \<subseteq> v \<and> v \<subseteq> s \<and> openin (subtopology euclidean s) u \<and> compact v"
-          if "k \<subseteq> s" "compact k" for k
-  proof -
-    have "\<And>C. (\<forall>c\<in>C. openin (subtopology euclidean k) c) \<and> k \<subseteq> \<Union>C \<Longrightarrow>
-                    \<exists>D\<subseteq>C. finite D \<and> k \<subseteq> \<Union>D"
-      using that by (simp add: compact_eq_openin_cover)
-    moreover have "\<forall>c \<in> (\<lambda>x. k \<inter> u x) ` k. openin (subtopology euclidean k) c"
-      using that by clarify (metis subsetD inf.absorb_iff2 openin_subset openin_subtopology_Int_subset topspace_euclidean_subtopology uv)
-    moreover have "k \<subseteq> \<Union>((\<lambda>x. k \<inter> u x) ` k)"
-      using that by clarsimp (meson subsetCE uv)
-    ultimately obtain D where "D \<subseteq> (\<lambda>x. k \<inter> u x) ` k" "finite D" "k \<subseteq> \<Union>D"
-      by metis
-    then obtain T where T: "T \<subseteq> k" "finite T" "k \<subseteq> \<Union>((\<lambda>x. k \<inter> u x) ` T)"
-      by (metis finite_subset_image)
-    have Tuv: "\<Union>(u ` T) \<subseteq> \<Union>(v ` T)"
-      using T that by (force simp: dest!: uv)
-    show ?thesis
-      apply (rule_tac x="\<Union>(u ` T)" in exI)
-      apply (rule_tac x="\<Union>(v ` T)" in exI)
-      apply (simp add: Tuv)
-      using T that
-      apply (auto simp: dest!: uv)
-      done
-  qed
-  show ?rhs
-    by (blast intro: *)
-next
-  assume ?rhs
-  then show ?lhs
-    apply (clarsimp simp add: 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 *: "\<exists>u v. x \<in> u \<and> u \<subseteq> v \<and> v \<subseteq> s \<and> openin (subtopology euclidean s) u \<and> compact v"
-          if "x \<in> s" for x
-  proof -
-    obtain e where "e>0" and e: "cball x e \<subseteq> s"
-      using open_contains_cball assms \<open>x \<in> s\<close> by blast
-    have ope: "openin (subtopology euclidean s) (ball x e)"
-      by (meson e open_ball ball_subset_cball dual_order.trans open_subset)
-    show ?thesis
-      apply (rule_tac x="ball x e" in exI)
-      apply (rule_tac x="cball x e" in exI)
-      using \<open>e > 0\<close> e apply (auto simp: ope)
-      done
-  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 *: "\<exists>u v. x \<in> u \<and> u \<subseteq> v \<and> v \<subseteq> s \<and>
-                 openin (subtopology euclidean s) u \<and> compact v"
-          if "x \<in> s" for x
-  proof -
-    show ?thesis
-      apply (rule_tac x = "s \<inter> ball x 1" in exI)
-      apply (rule_tac x = "s \<inter> cball x 1" in exI)
-      using \<open>x \<in> s\<close> assms apply auto
-      done
-  qed
-  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 "\<lbrakk>locally compact s; locally compact t\<rbrakk> \<Longrightarrow> locally compact (s \<inter> t)"
-by (simp add: compact_Int locally_Int)
-
-lemma locally_compact_closedin:
-  fixes s :: "'a :: heine_borel set"
-  shows "\<lbrakk>closedin (subtopology euclidean s) t; locally compact s\<rbrakk>
-        \<Longrightarrow> 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 \<Longrightarrow> 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 \<longleftrightarrow> locally compact s"
-        (is "?lhs = ?rhs")
-proof
-  assume ?lhs
-  then show ?rhs
-    apply (simp only: locally_def)
-    apply (erule all_forward imp_forward asm_rl exE)+
-    apply (rule_tac x = "u \<inter> ball x 1" in exI)
-    apply (rule_tac x = "v \<inter> 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 (subtopology euclidean (S \<union> T)) S"
-      and opT: "openin (subtopology euclidean (S \<union> T)) T"
-    shows "locally compact (S \<union> T)"
-proof -
-  have "\<exists>e>0. closed (cball x e \<inter> (S \<union> T))" if "x \<in> S" for x
-  proof -
-    obtain e1 where "e1 > 0" and e1: "closed (cball x e1 \<inter> S)"
-      using LCS \<open>x \<in> S\<close> unfolding locally_compact_Int_cball by blast
-    moreover obtain e2 where "e2 > 0" and e2: "cball x e2 \<inter> (S \<union> T) \<subseteq> S"
-      by (meson \<open>x \<in> S\<close> opS openin_contains_cball)
-    then have "cball x e2 \<inter> (S \<union> T) = cball x e2 \<inter> S"
-      by force
-    ultimately show ?thesis
-      apply (rule_tac x="min e1 e2" in exI)
-      apply (auto simp: cball_min_Int \<open>e2 > 0\<close> inf_assoc closed_Int)
-      by (metis closed_Int closed_cball inf_left_commute)
-  qed
-  moreover have "\<exists>e>0. closed (cball x e \<inter> (S \<union> T))" if "x \<in> T" for x
-  proof -
-    obtain e1 where "e1 > 0" and e1: "closed (cball x e1 \<inter> T)"
-      using LCT \<open>x \<in> T\<close> unfolding locally_compact_Int_cball by blast
-    moreover obtain e2 where "e2 > 0" and e2: "cball x e2 \<inter> (S \<union> T) \<subseteq> T"
-      by (meson \<open>x \<in> T\<close> opT openin_contains_cball)
-    then have "cball x e2 \<inter> (S \<union> T) = cball x e2 \<inter> T"
-      by force
-    ultimately show ?thesis
-      apply (rule_tac x="min e1 e2" in exI)
-      apply (auto simp: cball_min_Int \<open>e2 > 0\<close> inf_assoc closed_Int)
-      by (metis closed_Int closed_cball inf_left_commute)
-  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 (subtopology euclidean (S \<union> T)) S"
-      and clT: "closedin (subtopology euclidean (S \<union> T)) T"
-    shows "locally compact (S \<union> T)"
-proof -
-  have "\<exists>e>0. closed (cball x e \<inter> (S \<union> T))" if "x \<in> S" "x \<in> T" for x
-  proof -
-    obtain e1 where "e1 > 0" and e1: "closed (cball x e1 \<inter> S)"
-      using LCS \<open>x \<in> S\<close> unfolding locally_compact_Int_cball by blast
-    moreover
-    obtain e2 where "e2 > 0" and e2: "closed (cball x e2 \<inter> T)"
-      using LCT \<open>x \<in> T\<close> unfolding locally_compact_Int_cball by blast
-    ultimately show ?thesis
-      apply (rule_tac x="min e1 e2" in exI)
-      apply (auto simp: cball_min_Int \<open>e2 > 0\<close> inf_assoc closed_Int Int_Un_distrib)
-      by (metis closed_Int closed_Un closed_cball inf_left_commute)
-  qed
-  moreover
-  have "\<exists>e>0. closed (cball x e \<inter> (S \<union> T))" if x: "x \<in> S" "x \<notin> T" for x
-  proof -
-    obtain e1 where "e1 > 0" and e1: "closed (cball x e1 \<inter> S)"
-      using LCS \<open>x \<in> S\<close> unfolding locally_compact_Int_cball by blast
-    moreover
-    obtain e2 where "e2>0" and "cball x e2 \<inter> (S \<union> T) \<subseteq> S - T"
-      using clT x by (fastforce simp: openin_contains_cball closedin_def)
-    then have "closed (cball x e2 \<inter> T)"
-    proof -
-      have "{} = T - (T - cball x e2)"
-        using Diff_subset Int_Diff \<open>cball x e2 \<inter> (S \<union> T) \<subseteq> S - T\<close> by auto
-      then show ?thesis
-        by (simp add: Diff_Diff_Int inf_commute)
-    qed
-    ultimately show ?thesis
-      apply (rule_tac x="min e1 e2" in exI)
-      apply (auto simp: cball_min_Int \<open>e2 > 0\<close> inf_assoc closed_Int Int_Un_distrib)
-      by (metis closed_Int closed_Un closed_cball inf_left_commute)
-  qed
-  moreover
-  have "\<exists>e>0. closed (cball x e \<inter> (S \<union> T))" if x: "x \<notin> S" "x \<in> T" for x
-  proof -
-    obtain e1 where "e1 > 0" and e1: "closed (cball x e1 \<inter> T)"
-      using LCT \<open>x \<in> T\<close> unfolding locally_compact_Int_cball by blast
-    moreover
-    obtain e2 where "e2>0" and "cball x e2 \<inter> (S \<union> T) \<subseteq> S \<union> T - S"
-      using clS x by (fastforce simp: openin_contains_cball closedin_def)
-    then have "closed (cball x e2 \<inter> S)"
-      by (metis Diff_disjoint Int_empty_right closed_empty inf.left_commute inf.orderE inf_sup_absorb)
-    ultimately show ?thesis
-      apply (rule_tac x="min e1 e2" in exI)
-      apply (auto simp: cball_min_Int \<open>e2 > 0\<close> inf_assoc closed_Int Int_Un_distrib)
-      by (metis closed_Int closed_Un closed_cball inf_left_commute)
-  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 "\<lbrakk>locally compact S; locally compact T\<rbrakk> \<Longrightarrow> locally compact (S \<times> T)"
-  by (auto simp: compact_Times locally_Times)
-
-lemma locally_compact_compact_subopen:
-  fixes S :: "'a :: heine_borel set"
-  shows
-   "locally compact S \<longleftrightarrow>
-    (\<forall>K T. K \<subseteq> S \<and> compact K \<and> open T \<and> K \<subseteq> T
-          \<longrightarrow> (\<exists>U V. K \<subseteq> U \<and> U \<subseteq> V \<and> U \<subseteq> T \<and> V \<subseteq> S \<and>
-                     openin (subtopology euclidean S) U \<and> compact V))"
-   (is "?lhs = ?rhs")
-proof
-  assume L: ?lhs
-  show ?rhs
-  proof clarify
-    fix K :: "'a set" and T :: "'a set"
-    assume "K \<subseteq> S" and "compact K" and "open T" and "K \<subseteq> T"
-    obtain U V where "K \<subseteq> U" "U \<subseteq> V" "V \<subseteq> S" "compact V"
-                 and ope: "openin (subtopology euclidean S) U"
-      using L unfolding locally_compact_compact by (meson \<open>K \<subseteq> S\<close> \<open>compact K\<close>)
-    show "\<exists>U V. K \<subseteq> U \<and> U \<subseteq> V \<and> U \<subseteq> T \<and> V \<subseteq> S \<and>
-                openin (subtopology euclidean S) U \<and> compact V"
-    proof (intro exI conjI)
-      show "K \<subseteq> U \<inter> T"
-        by (simp add: \<open>K \<subseteq> T\<close> \<open>K \<subseteq> U\<close>)
-      show "U \<inter> T \<subseteq> closure(U \<inter> T)"
-        by (rule closure_subset)
-      show "closure (U \<inter> T) \<subseteq> S"
-        by (metis \<open>U \<subseteq> V\<close> \<open>V \<subseteq> S\<close> \<open>compact V\<close> closure_closed closure_mono compact_imp_closed inf.cobounded1 subset_trans)
-      show "openin (subtopology euclidean S) (U \<inter> T)"
-        by (simp add: \<open>open T\<close> ope openin_Int_open)
-      show "compact (closure (U \<inter> T))"
-        by (meson Int_lower1 \<open>U \<subseteq> V\<close> \<open>compact V\<close> 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\<open>Sura-Bura's results about compact components of sets\<close>
-
-proposition Sura_Bura_compact:
-  fixes S :: "'a::euclidean_space set"
-  assumes "compact S" and C: "C \<in> components S"
-  shows "C = \<Inter>{T. C \<subseteq> T \<and> openin (subtopology euclidean S) T \<and>
-                           closedin (subtopology euclidean S) T}"
-         (is "C = \<Inter>?\<T>")
-proof
-  obtain x where x: "C = connected_component_set S x" and "x \<in> S"
-    using C by (auto simp: components_def)
-  have "C \<subseteq> S"
-    by (simp add: C in_components_subset)
-  have "\<Inter>?\<T> \<subseteq> connected_component_set S x"
-  proof (rule connected_component_maximal)
-    have "x \<in> C"
-      by (simp add: \<open>x \<in> S\<close> x)
-    then show "x \<in> \<Inter>?\<T>"
-      by blast
-    have clo: "closed (\<Inter>?\<T>)"
-      by (simp add: \<open>compact S\<close> closed_Inter closedin_compact_eq compact_imp_closed)
-    have False
-      if K1: "closedin (subtopology euclidean (\<Inter>?\<T>)) K1" and
-         K2: "closedin (subtopology euclidean (\<Inter>?\<T>)) K2" and
-         K12_Int: "K1 \<inter> K2 = {}" and K12_Un: "K1 \<union> K2 = \<Inter>?\<T>" and "K1 \<noteq> {}" "K2 \<noteq> {}"
-       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 \<subseteq> V1" "K2 \<subseteq> V2" and V12: "V1 \<inter> V2 = {}"
-        using separation_normal \<open>K1 \<inter> K2 = {}\<close> by metis
-      have SV12_ne: "(S - (V1 \<union> V2)) \<inter> (\<Inter>?\<T>) \<noteq> {}"
-      proof (rule compact_imp_fip)
-        show "compact (S - (V1 \<union> V2))"
-          by (simp add: \<open>open V1\<close> \<open>open V2\<close> \<open>compact S\<close> compact_diff open_Un)
-        show clo\<T>: "closed T" if "T \<in> ?\<T>" for T
-          using that \<open>compact S\<close>
-          by (force intro: closedin_closed_trans simp add: compact_imp_closed)
-        show "(S - (V1 \<union> V2)) \<inter> \<Inter>\<F> \<noteq> {}" if "finite \<F>" and \<F>: "\<F> \<subseteq> ?\<T>" for \<F>
-        proof
-          assume djo: "(S - (V1 \<union> V2)) \<inter> \<Inter>\<F> = {}"
-          obtain D where opeD: "openin (subtopology euclidean S) D"
-                   and cloD: "closedin (subtopology euclidean S) D"
-                   and "C \<subseteq> D" and DV12: "D \<subseteq> V1 \<union> V2"
-          proof (cases "\<F> = {}")
-            case True
-            with \<open>C \<subseteq> S\<close> djo that show ?thesis
-              by force
-          next
-            case False show ?thesis
-            proof
-              show ope: "openin (subtopology euclidean S) (\<Inter>\<F>)"
-                using openin_Inter \<open>finite \<F>\<close> False \<F> by blast
-              then show "closedin (subtopology euclidean S) (\<Inter>\<F>)"
-                by (meson clo\<T> \<F> closed_Inter closed_subset openin_imp_subset subset_eq)
-              show "C \<subseteq> \<Inter>\<F>"
-                using \<F> by auto
-              show "\<Inter>\<F> \<subseteq> V1 \<union> V2"
-                using ope djo openin_imp_subset by fastforce
-            qed
-          qed
-          have "connected C"
-            by (simp add: x)
-          have "closed D"
-            using \<open>compact S\<close> cloD closedin_closed_trans compact_imp_closed by blast
-          have cloV1: "closedin (subtopology euclidean D) (D \<inter> closure V1)"
-            and cloV2: "closedin (subtopology euclidean D) (D \<inter> closure V2)"
-            by (simp_all add: closedin_closed_Int)
-          moreover have "D \<inter> closure V1 = D \<inter> V1" "D \<inter> closure V2 = D \<inter> V2"
-            apply safe
-            using \<open>D \<subseteq> V1 \<union> V2\<close> \<open>open V1\<close> \<open>open V2\<close> V12
-               apply (simp_all add: closure_subset [THEN subsetD] closure_iff_nhds_not_empty, blast+)
-            done
-          ultimately have cloDV1: "closedin (subtopology euclidean D) (D \<inter> V1)"
-                      and cloDV2:  "closedin (subtopology euclidean D) (D \<inter> V2)"
-            by metis+
-          then obtain U1 U2 where "closed U1" "closed U2"
-               and D1: "D \<inter> V1 = D \<inter> U1" and D2: "D \<inter> V2 = D \<inter> U2"
-            by (auto simp: closedin_closed)
-          have "D \<inter> U1 \<inter> C \<noteq> {}"
-          proof
-            assume "D \<inter> U1 \<inter> C = {}"
-            then have *: "C \<subseteq> D \<inter> V2"
-              using D1 DV12 \<open>C \<subseteq> D\<close> by auto
-            have "\<Inter>?\<T> \<subseteq> D \<inter> V2"
-              apply (rule Inter_lower)
-              using * apply simp
-              by (meson cloDV2 \<open>open V2\<close> cloD closedin_trans le_inf_iff opeD openin_Int_open)
-            then show False
-              using K1 V12 \<open>K1 \<noteq> {}\<close> \<open>K1 \<subseteq> V1\<close> closedin_imp_subset by blast
-          qed
-          moreover have "D \<inter> U2 \<inter> C \<noteq> {}"
-          proof
-            assume "D \<inter> U2 \<inter> C = {}"
-            then have *: "C \<subseteq> D \<inter> V1"
-              using D2 DV12 \<open>C \<subseteq> D\<close> by auto
-            have "\<Inter>?\<T> \<subseteq> D \<inter> V1"
-              apply (rule Inter_lower)
-              using * apply simp
-              by (meson cloDV1 \<open>open V1\<close> cloD closedin_trans le_inf_iff opeD openin_Int_open)
-            then show False
-              using K2 V12 \<open>K2 \<noteq> {}\<close> \<open>K2 \<subseteq> V2\<close> closedin_imp_subset by blast
-          qed
-          ultimately show False
-            using \<open>connected C\<close> unfolding connected_closed
-            apply (simp only: not_ex)
-            apply (drule_tac x="D \<inter> U1" in spec)
-            apply (drule_tac x="D \<inter> U2" in spec)
-            using \<open>C \<subseteq> D\<close> D1 D2 V12 DV12 \<open>closed U1\<close> \<open>closed U2\<close> \<open>closed D\<close>
-            by blast
-        qed
-      qed
-      show False
-        by (metis (full_types) DiffE UnE Un_upper2 SV12_ne \<open>K1 \<subseteq> V1\<close> \<open>K2 \<subseteq> V2\<close> disjoint_iff_not_equal subsetCE sup_ge1 K12_Un)
-    qed
-    then show "connected (\<Inter>?\<T>)"
-      by (auto simp: connected_closedin_eq)
-    show "\<Inter>?\<T> \<subseteq> S"
-      by (fastforce simp: C in_components_subset)
-  qed
-  with x show "\<Inter>?\<T> \<subseteq> C" by simp
-qed auto
-
-
-corollary Sura_Bura_clopen_subset:
-  fixes S :: "'a::euclidean_space set"
-  assumes S: "locally compact S" and C: "C \<in> components S" and "compact C"
-      and U: "open U" "C \<subseteq> U"
-  obtains K where "openin (subtopology euclidean S) K" "compact K" "C \<subseteq> K" "K \<subseteq> U"
-proof (rule ccontr)
-  assume "\<not> thesis"
-  with that have neg: "\<nexists>K. openin (subtopology euclidean S) K \<and> compact K \<and> C \<subseteq> K \<and> K \<subseteq> U"
-    by metis
-  obtain V K where "C \<subseteq> V" "V \<subseteq> U" "V \<subseteq> K" "K \<subseteq> S" "compact K"
-               and opeSV: "openin (subtopology euclidean S) V"
-    using S U \<open>compact C\<close>
-    apply (simp add: locally_compact_compact_subopen)
-    by (meson C in_components_subset)
-  let ?\<T> = "{T. C \<subseteq> T \<and> openin (subtopology euclidean K) T \<and> compact T \<and> T \<subseteq> K}"
-  have CK: "C \<in> components K"
-    by (meson C \<open>C \<subseteq> V\<close> \<open>K \<subseteq> S\<close> \<open>V \<subseteq> K\<close> components_intermediate_subset subset_trans)
-  with \<open>compact K\<close>
-  have "C = \<Inter>{T. C \<subseteq> T \<and> openin (subtopology euclidean K) T \<and> closedin (subtopology euclidean K) T}"
-    by (simp add: Sura_Bura_compact)
-  then have Ceq: "C = \<Inter>?\<T>"
-    by (simp add: closedin_compact_eq \<open>compact K\<close>)
-  obtain W where "open W" and W: "V = S \<inter> W"
-    using opeSV by (auto simp: openin_open)
-  have "-(U \<inter> W) \<inter> \<Inter>?\<T> \<noteq> {}"
-  proof (rule closed_imp_fip_compact)
-    show "- (U \<inter> W) \<inter> \<Inter>\<F> \<noteq> {}"
-      if "finite \<F>" and \<F>: "\<F> \<subseteq> ?\<T>" for \<F>
-    proof (cases "\<F> = {}")
-      case True
-      have False if "U = UNIV" "W = UNIV"
-      proof -
-        have "V = S"
-          by (simp add: W \<open>W = UNIV\<close>)
-        with neg show False
-          using \<open>C \<subseteq> V\<close> \<open>K \<subseteq> S\<close> \<open>V \<subseteq> K\<close> \<open>V \<subseteq> U\<close> \<open>compact K\<close> by auto
-      qed
-      with True show ?thesis
-        by auto
-    next
-      case False
-      show ?thesis
-      proof
-        assume "- (U \<inter> W) \<inter> \<Inter>\<F> = {}"
-        then have FUW: "\<Inter>\<F> \<subseteq> U \<inter> W"
-          by blast
-        have "C \<subseteq> \<Inter>\<F>"
-          using \<F> by auto
-        moreover have "compact (\<Inter>\<F>)"
-          by (metis (no_types, lifting) compact_Inter False mem_Collect_eq subsetCE \<F>)
-        moreover have "\<Inter>\<F> \<subseteq> K"
-          using False that(2) by fastforce
-        moreover have opeKF: "openin (subtopology euclidean K) (\<Inter>\<F>)"
-          using False \<F> \<open>finite \<F>\<close> by blast
-        then have opeVF: "openin (subtopology euclidean V) (\<Inter>\<F>)"
-          using W \<open>K \<subseteq> S\<close> \<open>V \<subseteq> K\<close> opeKF \<open>\<Inter>\<F> \<subseteq> K\<close> FUW openin_subset_trans by fastforce
-        then have "openin (subtopology euclidean S) (\<Inter>\<F>)"
-          by (metis opeSV openin_trans)
-        moreover have "\<Inter>\<F> \<subseteq> U"
-          by (meson \<open>V \<subseteq> U\<close> opeVF dual_order.trans openin_imp_subset)
-        ultimately show False
-          using neg by blast
-      qed
-    qed
-  qed (use \<open>open W\<close> \<open>open U\<close> in auto)
-  with W Ceq \<open>C \<subseteq> V\<close> \<open>C \<subseteq> U\<close> 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 \<in> components S" and "compact C"
-      and opeSU: "openin (subtopology euclidean S) U" and "C \<subseteq> U"
-  obtains K where "openin (subtopology euclidean S) K" "compact K" "C \<subseteq> K" "K \<subseteq> U"
-proof -
-  obtain V where "open V" "U = S \<inter> V"
-    using opeSU by (auto simp: openin_open)
-  with \<open>C \<subseteq> U\<close> have "C \<subseteq> V"
-    by auto
-  then show ?thesis
-    using Sura_Bura_clopen_subset [OF S C \<open>compact C\<close> \<open>open V\<close>]
-    by (metis \<open>U = S \<inter> V\<close> inf.bounded_iff openin_imp_subset that)
-qed
-
-corollary Sura_Bura:
-  fixes S :: "'a::euclidean_space set"
-  assumes "locally compact S" "C \<in> components S" "compact C"
-  shows "C = \<Inter> {K. C \<subseteq> K \<and> compact K \<and> openin (subtopology euclidean S) K}"
-         (is "C = ?rhs")
-proof
-  show "?rhs \<subseteq> C"
-  proof (clarsimp, rule ccontr)
-    fix x
-    assume *: "\<forall>X. C \<subseteq> X \<and> compact X \<and> openin (subtopology euclidean S) X \<longrightarrow> x \<in> X"
-      and "x \<notin> C"
-    obtain U V where "open U" "open V" "{x} \<subseteq> U" "C \<subseteq> V" "U \<inter> V = {}"
-      using separation_normal [of "{x}" C]
-      by (metis Int_empty_left \<open>x \<notin> C\<close> \<open>compact C\<close> closed_empty closed_insert compact_imp_closed insert_disjoint(1))
-    have "x \<notin> V"
-      using \<open>U \<inter> V = {}\<close> \<open>{x} \<subseteq> U\<close> by blast
-    then show False
-      by (meson "*" Sura_Bura_clopen_subset \<open>C \<subseteq> V\<close> \<open>open V\<close> assms(1) assms(2) assms(3) subsetCE)
-  qed
-qed blast
-
-
-subsection\<open>Special cases of local connectedness and path connectedness\<close>
-
-lemma locally_connected_1:
-  assumes
-    "\<And>v x. \<lbrakk>openin (subtopology euclidean S) v; x \<in> v\<rbrakk>
-              \<Longrightarrow> \<exists>u. openin (subtopology euclidean S) u \<and>
-                      connected u \<and> x \<in> u \<and> u \<subseteq> v"
-   shows "locally connected S"
-apply (clarsimp simp add: locally_def)
-apply (drule assms; blast)
-done
-
-lemma locally_connected_2:
-  assumes "locally connected S"
-          "openin (subtopology euclidean S) t"
-          "x \<in> t"
-   shows "openin (subtopology euclidean S) (connected_component_set t x)"
-proof -
-  { fix y :: 'a
-    let ?SS = "subtopology euclidean S"
-    assume 1: "openin ?SS t"
-              "\<forall>w x. openin ?SS w \<and> x \<in> w \<longrightarrow> (\<exists>u. openin ?SS u \<and> (\<exists>v. connected v \<and> x \<in> u \<and> u \<subseteq> v \<and> v \<subseteq> w))"
-    and "connected_component t x y"
-    then have "y \<in> t" and y: "y \<in> connected_component_set t x"
-      using connected_component_subset by blast+
-    obtain F where
-      "\<forall>x y. (\<exists>w. openin ?SS w \<and> (\<exists>u. connected u \<and> x \<in> w \<and> w \<subseteq> u \<and> u \<subseteq> y)) = (openin ?SS (F x y) \<and> (\<exists>u. connected u \<and> x \<in> F x y \<and> F x y \<subseteq> u \<and> u \<subseteq> y))"
-      by moura
-    then obtain G where
-       "\<forall>a A. (\<exists>U. openin ?SS U \<and> (\<exists>V. connected V \<and> a \<in> U \<and> U \<subseteq> V \<and> V \<subseteq> A)) = (openin ?SS (F a A) \<and> connected (G a A) \<and> a \<in> F a A \<and> F a A \<subseteq> G a A \<and> G a A \<subseteq> A)"
-      by moura
-    then have *: "openin ?SS (F y t) \<and> connected (G y t) \<and> y \<in> F y t \<and> F y t \<subseteq> G y t \<and> G y t \<subseteq> t"
-      using 1 \<open>y \<in> t\<close> by presburger
-    have "G y t \<subseteq> connected_component_set t y"
-      by (metis (no_types) * connected_component_eq_self connected_component_mono contra_subsetD)
-    then have "\<exists>A. openin ?SS A \<and> y \<in> A \<and> A \<subseteq> 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 "\<And>t x. \<lbrakk>openin (subtopology euclidean S) t; x \<in> t\<rbrakk>
-              \<Longrightarrow> openin (subtopology euclidean S)
-                          (connected_component_set t x)"
-          "openin (subtopology euclidean S) v" "x \<in> v"
-   shows  "\<exists>u. openin (subtopology euclidean S) u \<and> connected u \<and> x \<in> u \<and> u \<subseteq> v"
-using assms connected_component_subset by fastforce
-
-lemma locally_connected:
-  "locally connected S \<longleftrightarrow>
-   (\<forall>v x. openin (subtopology euclidean S) v \<and> x \<in> v
-          \<longrightarrow> (\<exists>u. openin (subtopology euclidean S) u \<and> connected u \<and> x \<in> u \<and> u \<subseteq> v))"
-by (metis locally_connected_1 locally_connected_2 locally_connected_3)
-
-lemma locally_connected_open_connected_component:
-  "locally connected S \<longleftrightarrow>
-   (\<forall>t x. openin (subtopology euclidean S) t \<and> x \<in> t
-          \<longrightarrow> openin (subtopology euclidean S) (connected_component_set t x))"
-by (metis locally_connected_1 locally_connected_2 locally_connected_3)
-
-lemma locally_path_connected_1:
-  assumes
-    "\<And>v x. \<lbrakk>openin (subtopology euclidean S) v; x \<in> v\<rbrakk>
-              \<Longrightarrow> \<exists>u. openin (subtopology euclidean S) u \<and> path_connected u \<and> x \<in> u \<and> u \<subseteq> v"
-   shows "locally path_connected S"
-apply (clarsimp simp add: locally_def)
-apply (drule assms; blast)
-done
-
-lemma locally_path_connected_2:
-  assumes "locally path_connected S"
-          "openin (subtopology euclidean S) t"
-          "x \<in> t"
-   shows "openin (subtopology euclidean S) (path_component_set t x)"
-proof -
-  { fix y :: 'a
-    let ?SS = "subtopology euclidean S"
-    assume 1: "openin ?SS t"
-              "\<forall>w x. openin ?SS w \<and> x \<in> w \<longrightarrow> (\<exists>u. openin ?SS u \<and> (\<exists>v. path_connected v \<and> x \<in> u \<and> u \<subseteq> v \<and> v \<subseteq> w))"
-    and "path_component t x y"
-    then have "y \<in> t" and y: "y \<in> path_component_set t x"
-      using path_component_mem(2) by blast+
-    obtain F where
-      "\<forall>x y. (\<exists>w. openin ?SS w \<and> (\<exists>u. path_connected u \<and> x \<in> w \<and> w \<subseteq> u \<and> u \<subseteq> y)) = (openin ?SS (F x y) \<and> (\<exists>u. path_connected u \<and> x \<in> F x y \<and> F x y \<subseteq> u \<and> u \<subseteq> y))"
-      by moura
-    then obtain G where
-       "\<forall>a A. (\<exists>U. openin ?SS U \<and> (\<exists>V. path_connected V \<and> a \<in> U \<and> U \<subseteq> V \<and> V \<subseteq> A)) = (openin ?SS (F a A) \<and> path_connected (G a A) \<and> a \<in> F a A \<and> F a A \<subseteq> G a A \<and> G a A \<subseteq> A)"
-      by moura
-    then have *: "openin ?SS (F y t) \<and> path_connected (G y t) \<and> y \<in> F y t \<and> F y t \<subseteq> G y t \<and> G y t \<subseteq> t"
-      using 1 \<open>y \<in> t\<close> by presburger
-    have "G y t \<subseteq> path_component_set t y"
-      using * path_component_maximal set_rev_mp by blast
-    then have "\<exists>A. openin ?SS A \<and> y \<in> A \<and> A \<subseteq> path_component_set t x"
-      by (metis "*" \<open>G y t \<subseteq> path_component_set t y\<close> 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 "\<And>t x. \<lbrakk>openin (subtopology euclidean S) t; x \<in> t\<rbrakk>
-              \<Longrightarrow> openin (subtopology euclidean S) (path_component_set t x)"
-          "openin (subtopology euclidean S) v" "x \<in> v"
-   shows  "\<exists>u. openin (subtopology euclidean S) u \<and> path_connected u \<and> x \<in> u \<and> u \<subseteq> v"
-proof -
-  have "path_component v x x"
-    by (meson assms(3) path_component_refl)
-  then show ?thesis
-    by (metis assms(1) assms(2) assms(3) mem_Collect_eq path_component_subset path_connected_path_component)
-qed
-
-proposition locally_path_connected:
-  "locally path_connected S \<longleftrightarrow>
-   (\<forall>v x. openin (subtopology euclidean S) v \<and> x \<in> v
-          \<longrightarrow> (\<exists>u. openin (subtopology euclidean S) u \<and> path_connected u \<and> x \<in> u \<and> u \<subseteq> 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 \<longleftrightarrow>
-   (\<forall>t x. openin (subtopology euclidean S) t \<and> x \<in> t
-          \<longrightarrow> openin (subtopology euclidean 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 \<longleftrightarrow>
-   (\<forall>t c. openin (subtopology euclidean S) t \<and> c \<in> components t
-          \<longrightarrow> openin (subtopology euclidean S) c)"
-by (metis components_iff locally_connected_open_connected_component)
-
-proposition locally_connected_im_kleinen:
-  "locally connected S \<longleftrightarrow>
-   (\<forall>v x. openin (subtopology euclidean S) v \<and> x \<in> v
-       \<longrightarrow> (\<exists>u. openin (subtopology euclidean S) u \<and>
-                x \<in> u \<and> u \<subseteq> v \<and>
-                (\<forall>y. y \<in> u \<longrightarrow> (\<exists>c. connected c \<and> c \<subseteq> v \<and> x \<in> c \<and> y \<in> c))))"
-   (is "?lhs = ?rhs")
-proof
-  assume ?lhs
-  then show ?rhs
-    by (fastforce simp add: locally_connected)
-next
-  assume ?rhs
-  have *: "\<exists>T. openin (subtopology euclidean S) T \<and> x \<in> T \<and> T \<subseteq> c"
-       if "openin (subtopology euclidean S) t" and c: "c \<in> components t" and "x \<in> c" for t c x
-  proof -
-    from that \<open>?rhs\<close> [rule_format, of t x]
-    obtain u where u:
-      "openin (subtopology euclidean S) u \<and> x \<in> u \<and> u \<subseteq> t \<and>
-       (\<forall>y. y \<in> u \<longrightarrow> (\<exists>c. connected c \<and> c \<subseteq> t \<and> x \<in> c \<and> y \<in> c))"
-      using in_components_subset by auto
-    obtain F :: "'a set \<Rightarrow> 'a set \<Rightarrow> 'a" where
-      "\<forall>x y. (\<exists>z. z \<in> x \<and> y = connected_component_set x z) = (F x y \<in> x \<and> y = connected_component_set x (F x y))"
-      by moura
-    then have F: "F t c \<in> t \<and> c = connected_component_set t (F t c)"
-      by (meson components_iff c)
-    obtain G :: "'a set \<Rightarrow> 'a set \<Rightarrow> 'a" where
-        G: "\<forall>x y. (\<exists>z. z \<in> y \<and> z \<notin> x) = (G x y \<in> y \<and> G x y \<notin> x)"
-      by moura
-     have "G c u \<notin> u \<or> G c u \<in> 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
-    apply (clarsimp simp add: locally_connected_open_component)
-    apply (subst openin_subopen)
-    apply (blast intro: *)
-    done
-qed
-
-proposition locally_path_connected_im_kleinen:
-  "locally path_connected S \<longleftrightarrow>
-   (\<forall>v x. openin (subtopology euclidean S) v \<and> x \<in> v
-       \<longrightarrow> (\<exists>u. openin (subtopology euclidean S) u \<and>
-                x \<in> u \<and> u \<subseteq> v \<and>
-                (\<forall>y. y \<in> u \<longrightarrow> (\<exists>p. path p \<and> path_image p \<subseteq> v \<and>
-                                pathstart p = x \<and> 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 *: "\<exists>T. openin (subtopology euclidean S) T \<and>
-               x \<in> T \<and> T \<subseteq> path_component_set u z"
-       if "openin (subtopology euclidean S) u" and "z \<in> u" and c: "path_component u z x" for u z x
-  proof -
-    have "x \<in> u"
-      by (meson c path_component_mem(2))
-    with that \<open>?rhs\<close> [rule_format, of u x]
-    obtain U where U:
-      "openin (subtopology euclidean S) U \<and> x \<in> U \<and> U \<subseteq> u \<and>
-       (\<forall>y. y \<in> U \<longrightarrow> (\<exists>p. path p \<and> path_image p \<subseteq> u \<and> pathstart p = x \<and> pathfinish p = y))"
-       by blast
-    show ?thesis
-      apply (rule_tac x=U in exI)
-      apply (auto simp: U)
-      apply (metis U c path_component_trans path_component_def)
-      done
-  qed
-  show ?lhs
-    apply (clarsimp simp add: locally_path_connected_open_path_component)
-    apply (subst openin_subopen)
-    apply (blast intro: *)
-    done
-qed
-
-lemma locally_path_connected_imp_locally_connected:
-  "locally path_connected S \<Longrightarrow> locally connected S"
-using locally_mono path_connected_imp_connected by blast
-
-lemma locally_connected_components:
-  "\<lbrakk>locally connected S; c \<in> components S\<rbrakk> \<Longrightarrow> locally connected c"
-by (meson locally_connected_open_component locally_open_subset openin_subtopology_self)
-
-lemma locally_path_connected_components:
-  "\<lbrakk>locally path_connected S; c \<in> components S\<rbrakk> \<Longrightarrow> 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 \<Longrightarrow> 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"
-  shows "open S \<Longrightarrow> locally path_connected S"
-apply (rule locally_mono [of convex])
-apply (simp_all add: locally_def openin_open_eq convex_imp_path_connected)
-apply (meson open_ball centre_in_ball convex_ball openE order_trans)
-done
-
-lemma open_imp_locally_connected:
-  fixes S :: "'a :: real_normed_vector set"
-  shows "open S \<Longrightarrow> 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
-     \<Longrightarrow> openin (subtopology euclidean S) (connected_component_set S x)"
-apply (simp add: locally_connected_open_connected_component)
-by (metis connected_component_eq_empty connected_component_subset open_empty open_subset openin_subtopology_self)
-
-lemma openin_components_locally_connected:
-    "\<lbrakk>locally connected S; c \<in> components S\<rbrakk> \<Longrightarrow> openin (subtopology euclidean S) c"
-  using locally_connected_open_component openin_subtopology_self by blast
-
-lemma openin_path_component_locally_path_connected:
-  "locally path_connected S
-        \<Longrightarrow> openin (subtopology euclidean 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:
-    "locally path_connected S
-        \<Longrightarrow> closedin (subtopology euclidean S) (path_component_set S x)"
-apply  (simp add: closedin_def path_component_subset complement_path_component_Union)
-apply (rule openin_Union)
-using openin_path_component_locally_path_connected by auto
-
-lemma convex_imp_locally_path_connected:
-  fixes S :: "'a:: real_normed_vector set"
-  shows "convex S \<Longrightarrow> locally path_connected S"
-apply (clarsimp simp add: locally_path_connected)
-apply (subst (asm) openin_open)
-apply clarify
-apply (erule (1) openE)
-apply (rule_tac x = "S \<inter> ball x e" in exI)
-apply (force simp: convex_Int convex_imp_path_connected)
-done
-
-lemma convex_imp_locally_connected:
-  fixes S :: "'a:: real_normed_vector set"
-  shows "convex S \<Longrightarrow> locally connected S"
-  by (simp add: locally_path_connected_imp_locally_connected convex_imp_locally_path_connected)
-
-
-subsection\<open>Relations between components and path components\<close>
-
-lemma path_component_eq_connected_component:
-  assumes "locally path_connected S"
-    shows "(path_component S x = connected_component S x)"
-proof (cases "x \<in> S")
-  case True
-  have "openin (subtopology euclidean (connected_component_set S x)) (path_component_set S x)"
-    apply (rule openin_subset_trans [of S])
-    apply (intro conjI openin_path_component_locally_path_connected [OF assms])
-    using path_component_subset_connected_component   apply (auto simp: connected_component_subset)
-    done
-  moreover have "closedin (subtopology euclidean (connected_component_set S x)) (path_component_set S x)"
-    apply (rule closedin_subset_trans [of S])
-    apply (intro conjI closedin_path_component_locally_path_connected [OF assms])
-    using path_component_subset_connected_component   apply (auto simp: connected_component_subset)
-    done
-  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 \<Longrightarrow> (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 \<Longrightarrow> 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 \<Longrightarrow> 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 \<Longrightarrow> 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: "\<And>T. T \<subseteq> f ` S
-                \<Longrightarrow> openin (subtopology euclidean S) (S \<inter> f -` T) \<longleftrightarrow>
-                    openin (subtopology euclidean (f ` S)) T"
-    shows "locally connected (f ` S)"
-proof (clarsimp simp: locally_connected_open_component)
-  fix U C
-  assume opefSU: "openin (subtopology euclidean (f ` S)) U" and "C \<in> components U"
-  then have "C \<subseteq> U" "U \<subseteq> f ` S"
-    by (meson in_components_subset openin_imp_subset)+
-  then have "openin (subtopology euclidean (f ` S)) C \<longleftrightarrow>
-             openin (subtopology euclidean S) (S \<inter> f -` C)"
-    by (auto simp: oo)
-  moreover have "openin (subtopology euclidean S) (S \<inter> f -` C)"
-  proof (subst openin_subopen, clarify)
-    fix x
-    assume "x \<in> S" "f x \<in> C"
-    show "\<exists>T. openin (subtopology euclidean S) T \<and> x \<in> T \<and> T \<subseteq> (S \<inter> f -` C)"
-    proof (intro conjI exI)
-      show "openin (subtopology euclidean S) (connected_component_set (S \<inter> f -` U) x)"
-      proof (rule ccontr)
-        assume **: "\<not> openin (subtopology euclidean S) (connected_component_set (S \<inter> f -` U) x)"
-        then have "x \<notin> (S \<inter> f -` U)"
-          using \<open>U \<subseteq> f ` S\<close> 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 \<in> connected_component_set (S \<inter> f -` U) x"
-        using \<open>C \<subseteq> U\<close> \<open>f x \<in> C\<close> \<open>x \<in> S\<close> 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 \<inter> f -` U) x) f"
-        apply (rule continuous_on_subset)
-        using connected_component_subset apply blast
-        done
-      then have "connected (f ` connected_component_set (S \<inter> f -` U) x)"
-        by (rule connected_continuous_image [OF _ connected_connected_component])
-      moreover have "f ` connected_component_set (S \<inter> f -` U) x \<subseteq> U"
-        using connected_component_in by blast
-      moreover have "C \<inter> f ` connected_component_set (S \<inter> f -` U) x \<noteq> {}"
-        using \<open>C \<subseteq> U\<close> \<open>f x \<in> C\<close> \<open>x \<in> S\<close> by fastforce
-      ultimately have fC: "f ` (connected_component_set (S \<inter> f -` U) x) \<subseteq> C"
-        by (rule components_maximal [OF \<open>C \<in> components U\<close>])
-      have cUC: "connected_component_set (S \<inter> f -` U) x \<subseteq> (S \<inter> f -` C)"
-        using connected_component_subset fC by blast
-      have "connected_component_set (S \<inter> f -` U) x \<subseteq> connected_component_set (S \<inter> f -` C) x"
-      proof -
-        { assume "x \<in> connected_component_set (S \<inter> 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 "\<dots> \<subseteq> (S \<inter> f -` C)"
-        by (rule connected_component_subset)
-      finally show "connected_component_set (S \<inter> f -` U) x \<subseteq> (S \<inter> f -` C)" .
-    qed
-  qed
-  ultimately show "openin (subtopology euclidean (f ` S)) C"
-    by metis
-qed
-
-text\<open>The proof resembles that above but is not identical!\<close>
-proposition locally_path_connected_quotient_image:
-  assumes lcS: "locally path_connected S"
-      and oo: "\<And>T. T \<subseteq> f ` S
-                \<Longrightarrow> openin (subtopology euclidean S) (S \<inter> f -` T) \<longleftrightarrow> openin (subtopology euclidean (f ` S)) T"
-    shows "locally path_c