diff --git a/src/HOL/Analysis/Homotopy.thy b/src/HOL/Analysis/Homotopy.thy
--- a/src/HOL/Analysis/Homotopy.thy
+++ b/src/HOL/Analysis/Homotopy.thy
@@ -1,5520 +1,5457 @@
 (*  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 Product_Topology Uncountable_Sets
 begin
 
 definition\<^marker>\<open>tag important\<close> homotopic_with
 where
  "homotopic_with P X Y f g \<equiv>
    (\<exists>h. continuous_map (prod_topology (top_of_set {0..1::real}) X) Y h \<and>
        (\<forall>x. h(0, x) = f x) \<and>
        (\<forall>x. h(1, x) = g 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>
 
 abbreviation homotopic_with_canon ::
   "[('a::topological_space \<Rightarrow> 'b::topological_space) \<Rightarrow> bool, 'a set, 'b set, 'a \<Rightarrow> 'b, 'a \<Rightarrow> 'b] \<Rightarrow> bool"
 where
  "homotopic_with_canon P S T p q \<equiv> homotopic_with P (top_of_set S) (top_of_set T) p q"
 
 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
 
 lemma image_Pair_const: "(\<lambda>x. (x, c)) ` A = A \<times> {c}"
   by auto
 
 lemma fst_o_paired [simp]: "fst \<circ> (\<lambda>(x,y). (f x y, g x y)) = (\<lambda>(x,y). f x y)"
   by auto
 
 lemma snd_o_paired [simp]: "snd \<circ> (\<lambda>(x,y). (f x y, g x y)) = (\<lambda>(x,y). g x y)"
   by auto
 
 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 continuous_map_o_Pair: 
   assumes h: "continuous_map (prod_topology X Y) Z h" and t: "t \<in> topspace X"
   shows "continuous_map Y Z (h \<circ> Pair t)"
   by (intro continuous_map_compose [OF _ h] continuous_intros; simp add: t)
 
 subsection\<^marker>\<open>tag unimportant\<close>\<open>Trivial properties\<close>
 
 text \<open>We often want to just localize the ending function equality or whatever.\<close>
 text\<^marker>\<open>tag important\<close> \<open>%whitespace\<close>
 proposition homotopic_with:
   assumes "\<And>h k. (\<And>x. x \<in> topspace X \<Longrightarrow> h x = k x) \<Longrightarrow> (P h \<longleftrightarrow> P k)"
   shows "homotopic_with P X Y p q \<longleftrightarrow>
            (\<exists>h. continuous_map (prod_topology (subtopology euclideanreal {0..1}) X) Y h \<and>
               (\<forall>x \<in> topspace X. h(0,x) = p x) \<and>
               (\<forall>x \<in> topspace 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> topspace X then h(u,v) else if u = 0 then p v else q v" in exI)
-  apply auto
-  using continuous_map_eq apply fastforce
-  apply (drule_tac x=t in bspec, force)
-  apply (subst assms; simp)
-  done
+  apply simp
+  by (smt (verit, best) SigmaE assms case_prod_conv continuous_map_eq topspace_prod_topology)
 
 lemma homotopic_with_mono:
   assumes hom: "homotopic_with P X Y f g"
     and Q: "\<And>h. \<lbrakk>continuous_map X Y h; P h\<rbrakk> \<Longrightarrow> Q h"
   shows "homotopic_with Q X Y f g"
   using hom unfolding homotopic_with_def
   by (force simp: o_def dest: continuous_map_o_Pair intro: Q)
 
 lemma homotopic_with_imp_continuous_maps:
     assumes "homotopic_with P X Y f g"
     shows "continuous_map X Y f \<and> continuous_map X Y g"
 proof -
   obtain h :: "real \<times> 'a \<Rightarrow> 'b"
     where conth: "continuous_map (prod_topology (top_of_set {0..1}) X) Y 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_map X Y (h \<circ> (\<lambda>x. (t,x)))" for t
     by (rule continuous_map_compose [OF _ conth]) (simp add: o_def continuous_map_pairwise)
   show ?thesis
     using h *[of 0] *[of 1] by (simp add: continuous_map_eq)
 qed
 
 lemma homotopic_with_imp_continuous:
     assumes "homotopic_with_canon P X Y f g"
     shows "continuous_on X f \<and> continuous_on X g"
   by (meson assms continuous_map_subtopology_eu homotopic_with_imp_continuous_maps)
 
 lemma homotopic_with_imp_property:
   assumes "homotopic_with P X Y f g"
   shows "P f \<and> P g"
 proof
   obtain h where h: "\<And>x. h(0, x) = f x" "\<And>x. h(1, x) = g x" and P: "\<And>t. t \<in> {0..1::real} \<Longrightarrow> P(\<lambda>x. h(t,x))"
     using assms by (force simp: homotopic_with_def)
   show "P f" "P g"
     using P [of 0] P [of 1] by (force simp: h)+
 qed
 
 lemma homotopic_with_equal:
   assumes "P f" "P g" and contf: "continuous_map X Y f" and fg: "\<And>x. x \<in> topspace X \<Longrightarrow> f x = g x"
   shows "homotopic_with P X Y f g"
   unfolding homotopic_with_def
 proof (intro exI conjI allI ballI)
   let ?h = "\<lambda>(t::real,x). if t = 1 then g x else f x"
   show "continuous_map (prod_topology (top_of_set {0..1}) X) Y ?h"
   proof (rule continuous_map_eq)
     show "continuous_map (prod_topology (top_of_set {0..1}) X) Y (f \<circ> snd)"
       by (simp add: contf continuous_map_of_snd)
   qed (auto simp: fg)
   show "P (\<lambda>x. ?h (t, x))" if "t \<in> {0..1}" for t
     by (cases "t = 1") (simp_all add: assms)
 qed auto
 
 lemma homotopic_with_imp_subset1:
      "homotopic_with_canon P X Y f g \<Longrightarrow> f ` X \<subseteq> Y"
-  by (simp add: homotopic_with_def image_subset_iff) (metis atLeastAtMost_iff order_refl zero_le_one)
+  by (meson continuous_map_subtopology_eu homotopic_with_imp_continuous_maps)
 
 lemma homotopic_with_imp_subset2:
      "homotopic_with_canon 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)
+  by (meson continuous_map_subtopology_eu homotopic_with_imp_continuous_maps)
 
 lemma homotopic_with_subset_left:
      "\<lbrakk>homotopic_with_canon P X Y f g; Z \<subseteq> X\<rbrakk> \<Longrightarrow> homotopic_with_canon P Z Y f g"
   unfolding homotopic_with_def by (auto elim!: continuous_on_subset ex_forward)
 
 lemma homotopic_with_subset_right:
      "\<lbrakk>homotopic_with_canon P X Y f g; Y \<subseteq> Z\<rbrakk> \<Longrightarrow> homotopic_with_canon P X Z f g"
   unfolding homotopic_with_def by (auto elim!: continuous_on_subset ex_forward)
 
 subsection\<open>Homotopy with P is an equivalence relation\<close>
 
 text \<open>(on continuous functions mapping X into Y that satisfy P, though this only affects reflexivity)\<close>
 
 lemma homotopic_with_refl [simp]: "homotopic_with P X Y f f \<longleftrightarrow> continuous_map X Y f \<and> P f"
-  by (auto simp: homotopic_with_imp_continuous_maps intro: homotopic_with_equal dest: homotopic_with_imp_property)
+  by (metis homotopic_with_equal homotopic_with_imp_continuous_maps homotopic_with_imp_property)
 
 lemma homotopic_with_symD:
     assumes "homotopic_with P X Y f g"
       shows "homotopic_with P X Y g f"
 proof -
   let ?I01 = "subtopology euclideanreal {0..1}"
   let ?j = "\<lambda>y. (1 - fst y, snd y)"
   have 1: "continuous_map (prod_topology ?I01 X) (prod_topology euclideanreal X) ?j"
     by (intro continuous_intros; simp add: continuous_map_subtopology_fst prod_topology_subtopology)
   have *: "continuous_map (prod_topology ?I01 X) (prod_topology ?I01 X) ?j"
   proof -
     have "continuous_map (prod_topology ?I01 X) (subtopology (prod_topology euclideanreal X) ({0..1} \<times> topspace X)) ?j"
       by (simp add: continuous_map_into_subtopology [OF 1] image_subset_iff)
     then show ?thesis
       by (simp add: prod_topology_subtopology(1))
   qed
   show ?thesis
     using assms
     apply (clarsimp simp add: homotopic_with_def)
     subgoal for h
       by (rule_tac x="h \<circ> (\<lambda>y. (1 - fst y, snd y))" in exI) (simp add: continuous_map_compose [OF *])
     done
 qed
 
 lemma homotopic_with_sym:
    "homotopic_with P X Y f g \<longleftrightarrow> homotopic_with P X Y g f"
   by (metis homotopic_with_symD)
 
 proposition homotopic_with_trans:
     assumes "homotopic_with P X Y f g"  "homotopic_with P X Y g h"
     shows "homotopic_with P X Y f h"
 proof -
   let ?X01 = "prod_topology (subtopology euclideanreal {0..1}) X"
   obtain k1 k2
     where contk1: "continuous_map ?X01 Y k1" and contk2: "continuous_map ?X01 Y k2"
       and k12: "\<forall>x. k1 (1, x) = g x" "\<forall>x. k2 (0, x) = g x"
       "\<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))"
     using assms by (auto simp: homotopic_with_def)
   define k where "k \<equiv> \<lambda>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"
   have keq: "k1 (2 * u, v) = k2 (2 * u -1, v)" if "u = 1/2"  for u v
     by (simp add: k12 that)
   show ?thesis
     unfolding homotopic_with_def
   proof (intro exI conjI)
     show "continuous_map ?X01 Y k"
       unfolding k_def
     proof (rule continuous_map_cases_le)
       show fst: "continuous_map ?X01 euclideanreal fst"
         using continuous_map_fst continuous_map_in_subtopology by blast
       show "continuous_map ?X01 euclideanreal (\<lambda>x. 1/2)"
         by simp
       show "continuous_map (subtopology ?X01 {y \<in> topspace ?X01. fst y \<le> 1/2}) Y
                (k1 \<circ> (\<lambda>x. (2 *\<^sub>R fst x, snd x)))"
         apply (intro fst continuous_map_compose [OF _ contk1] continuous_intros continuous_map_into_subtopology continuous_map_from_subtopology | simp)+
         by (force simp: prod_topology_subtopology)
       show "continuous_map (subtopology ?X01 {y \<in> topspace ?X01. 1/2 \<le> fst y}) Y
                (k2 \<circ> (\<lambda>x. (2 *\<^sub>R fst x -1, snd x)))"
         apply (intro fst continuous_map_compose [OF _ contk2] continuous_intros continuous_map_into_subtopology continuous_map_from_subtopology | simp)+
         by (force simp: prod_topology_subtopology)
       show "(k1 \<circ> (\<lambda>x. (2 *\<^sub>R fst x, snd x))) y = (k2 \<circ> (\<lambda>x. (2 *\<^sub>R fst x -1, snd x))) y"
         if "y \<in> topspace ?X01" and "fst y = 1/2" for y
         using that by (simp add: keq)
     qed
     show "\<forall>x. k (0, x) = f x"
       by (simp add: k12 k_def)
     show "\<forall>x. k (1, x) = h x"
       by (simp add: k12 k_def)
     show "\<forall>t\<in>{0..1}. P (\<lambda>x. k (t, x))"
     proof 
       fix t show "t\<in>{0..1} \<Longrightarrow> P (\<lambda>x. k (t, x))"
         by (cases "t \<le> 1/2") (auto simp add: k_def P)
     qed
   qed
 qed
 
 lemma homotopic_with_id2: 
   "(\<And>x. x \<in> topspace X \<Longrightarrow> g (f x) = x) \<Longrightarrow> homotopic_with (\<lambda>x. True) X X (g \<circ> f) id"
   by (metis comp_apply continuous_map_id eq_id_iff homotopic_with_equal homotopic_with_symD)
 
 subsection\<open>Continuity lemmas\<close>
 
 lemma homotopic_with_compose_continuous_map_left:
   "\<lbrakk>homotopic_with p X1 X2 f g; continuous_map X2 X3 h; \<And>j. p j \<Longrightarrow> q(h \<circ> j)\<rbrakk>
    \<Longrightarrow> homotopic_with q X1 X3 (h \<circ> f) (h \<circ> g)"
   unfolding homotopic_with_def
   apply clarify
   subgoal for k
     by (rule_tac x="h \<circ> k" in exI) (rule conjI continuous_map_compose | simp add: o_def)+
   done
 
 lemma homotopic_with_compose_continuous_map_right:
   assumes hom: "homotopic_with p X2 X3 f g" and conth: "continuous_map X1 X2 h"
     and q: "\<And>j. p j \<Longrightarrow> q(j \<circ> h)"
   shows "homotopic_with q X1 X3 (f \<circ> h) (g \<circ> h)"
 proof -
   obtain k
     where contk: "continuous_map (prod_topology (subtopology euclideanreal {0..1}) X2) X3 k"
       and k: "\<forall>x. k (0, x) = f x" "\<forall>x. k (1, x) = g x" and p: "\<And>t. t\<in>{0..1} \<Longrightarrow> p (\<lambda>x. k (t, x))"
     using hom unfolding homotopic_with_def by blast
   have hsnd: "continuous_map (prod_topology (subtopology euclideanreal {0..1}) X1) X2 (h \<circ> snd)"
     by (rule continuous_map_compose [OF continuous_map_snd conth])
   let ?h = "k \<circ> (\<lambda>(t,x). (t,h x))"
   show ?thesis
     unfolding homotopic_with_def
   proof (intro exI conjI allI ballI)
     have "continuous_map (prod_topology (top_of_set {0..1}) X1)
      (prod_topology (top_of_set {0..1::real}) X2) (\<lambda>(t, x). (t, h x))"
       by (metis (mono_tags, lifting) case_prod_beta' comp_def continuous_map_eq continuous_map_fst continuous_map_pairedI hsnd)
     then show "continuous_map (prod_topology (subtopology euclideanreal {0..1}) X1) X3 ?h"
       by (intro conjI continuous_map_compose [OF _ contk])
     show "q (\<lambda>x. ?h (t, x))" if "t \<in> {0..1}" for t
       using q [OF p [OF that]] by (simp add: o_def)
   qed (auto simp: k)
 qed
 
 corollary homotopic_compose:
   assumes "homotopic_with (\<lambda>x. True) X Y f f'" "homotopic_with (\<lambda>x. True) Y Z g g'"
   shows "homotopic_with (\<lambda>x. True) X Z (g \<circ> f) (g' \<circ> f')"
-proof (rule homotopic_with_trans [where g = "g \<circ> f'"])
-  show "homotopic_with (\<lambda>x. True) X Z (g \<circ> f) (g \<circ> f')"
-    using assms by (simp add: homotopic_with_compose_continuous_map_left homotopic_with_imp_continuous_maps)
-  show "homotopic_with (\<lambda>x. True) X Z (g \<circ> f') (g' \<circ> f')"
-    using assms by (simp add: homotopic_with_compose_continuous_map_right homotopic_with_imp_continuous_maps)
-qed
+  by (metis assms homotopic_with_compose_continuous_map_left homotopic_with_compose_continuous_map_right homotopic_with_imp_continuous_maps homotopic_with_trans)
 
 proposition homotopic_with_compose_continuous_right:
     "\<lbrakk>homotopic_with_canon (\<lambda>f. p (f \<circ> h)) X Y f g; continuous_on W h; h ` W \<subseteq> X\<rbrakk>
      \<Longrightarrow> homotopic_with_canon p W Y (f \<circ> h) (g \<circ> h)"
-  apply (clarsimp simp add: homotopic_with_def)
-  subgoal for k
-    apply (rule_tac x="k \<circ> (\<lambda>y. (fst y, h (snd y)))" in exI)
-    by (intro conjI continuous_intros continuous_on_compose2 [where f=snd and g=h]; fastforce simp: o_def elim: continuous_on_subset)
-  done
+  by (simp add: homotopic_with_compose_continuous_map_right)
 
 proposition homotopic_with_compose_continuous_left:
      "\<lbrakk>homotopic_with_canon (\<lambda>f. p (h \<circ> f)) X Y f g; continuous_on Y h; h ` Y \<subseteq> Z\<rbrakk>
       \<Longrightarrow> homotopic_with_canon p X Z (h \<circ> f) (h \<circ> g)"
-  apply (clarsimp simp add: homotopic_with_def)
-  subgoal for k
-  apply (rule_tac x="h \<circ> k" in exI)
-    by (intro conjI continuous_intros continuous_on_compose [where f=snd and g=h, unfolded o_def]; fastforce simp: o_def elim: continuous_on_subset)
-  done
+  by (simp add: homotopic_with_compose_continuous_map_left)
 
 lemma homotopic_from_subtopology:
-   "homotopic_with P X X' f g \<Longrightarrow> homotopic_with P (subtopology X s) X' f g"
-  unfolding homotopic_with_def
-  by (force simp add: continuous_map_from_subtopology prod_topology_subtopology(2) elim!: ex_forward)
+   "homotopic_with P X X' f g \<Longrightarrow> homotopic_with P (subtopology X S) X' f g"
+  by (metis continuous_map_id_subt homotopic_with_compose_continuous_map_right o_id)
 
 lemma homotopic_on_emptyI:
     assumes "topspace X = {}" "P f" "P g"
     shows "homotopic_with P X X' f g"
-  unfolding homotopic_with_def
-proof (intro exI conjI ballI)
-  show "P (\<lambda>x. (\<lambda>(t,x). if t = 0 then f x else g x) (t, x))" if "t \<in> {0..1}" for t::real
-    by (cases "t = 0", auto simp: assms)
-qed (auto simp: continuous_map_atin assms)
+  by (metis assms continuous_map_on_empty empty_iff homotopic_with_equal)
 
 lemma homotopic_on_empty:
    "topspace X = {} \<Longrightarrow> (homotopic_with P X X' f g \<longleftrightarrow> P f \<and> P g)"
   using homotopic_on_emptyI homotopic_with_imp_property by metis
 
 lemma homotopic_with_canon_on_empty [simp]: "homotopic_with_canon (\<lambda>x. True) {} t f g"
   by (auto intro: homotopic_with_equal)
 
 lemma homotopic_constant_maps:
    "homotopic_with (\<lambda>x. True) X X' (\<lambda>x. a) (\<lambda>x. b) \<longleftrightarrow>
     topspace X = {} \<or> path_component_of X' a b" (is "?lhs = ?rhs")
 proof (cases "topspace X = {}")
   case False
   then obtain c where c: "c \<in> topspace X"
     by blast
   have "\<exists>g. continuous_map (top_of_set {0..1::real}) X' g \<and> g 0 = a \<and> g 1 = b"
     if "x \<in> topspace X" and hom: "homotopic_with (\<lambda>x. True) X X' (\<lambda>x. a) (\<lambda>x. b)" for x
   proof -
     obtain h :: "real \<times> 'a \<Rightarrow> 'b"
       where conth: "continuous_map (prod_topology (top_of_set {0..1}) X) X' h"
         and h: "\<And>x. h (0, x) = a" "\<And>x. h (1, x) = b"
       using hom by (auto simp: homotopic_with_def)
     have cont: "continuous_map (top_of_set {0..1}) X' (h \<circ> (\<lambda>t. (t, c)))"
       by (rule continuous_map_compose [OF _ conth] continuous_intros c | simp)+
     then show ?thesis
       by (force simp: h)
   qed
   moreover have "homotopic_with (\<lambda>x. True) X X' (\<lambda>x. g 0) (\<lambda>x. g 1)"
     if "x \<in> topspace X" "a = g 0" "b = g 1" "continuous_map (top_of_set {0..1}) X' g"
     for x and g :: "real \<Rightarrow> 'b"
     unfolding homotopic_with_def
     by (force intro!: continuous_map_compose continuous_intros c that)
   ultimately show ?thesis
     using False by (auto simp: path_component_of_def pathin_def)
 qed (simp add: homotopic_on_empty)
 
 proposition homotopic_with_eq:
    assumes h: "homotopic_with P X Y f g"
        and f': "\<And>x. x \<in> topspace X \<Longrightarrow> f' x = f x"
        and g': "\<And>x. x \<in> topspace X \<Longrightarrow> g' x = g x"
        and P:  "(\<And>h k. (\<And>x. x \<in> topspace 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 clarify
-  subgoal for h
-    apply (rule_tac x="\<lambda>(u,v). if v \<in> topspace 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_map_eq)
-    apply (subst P; fastforce)
-    done
-  done
+  by (smt (verit, ccfv_SIG) assms homotopic_with)
 
 lemma homotopic_with_prod_topology:
   assumes "homotopic_with p X1 Y1 f f'" and "homotopic_with q X2 Y2 g g'"
     and r: "\<And>i j. \<lbrakk>p i; q j\<rbrakk> \<Longrightarrow> r(\<lambda>(x,y). (i x, j y))"
   shows "homotopic_with r (prod_topology X1 X2) (prod_topology Y1 Y2)
                           (\<lambda>z. (f(fst z),g(snd z))) (\<lambda>z. (f'(fst z), g'(snd z)))"
 proof -
   obtain h
     where h: "continuous_map (prod_topology (subtopology euclideanreal {0..1}) X1) Y1 h"
       and h0: "\<And>x. h (0, x) = f x"
       and h1: "\<And>x. h (1, x) = f' x"
       and p: "\<And>t. \<lbrakk>0 \<le> t; t \<le> 1\<rbrakk> \<Longrightarrow> p (\<lambda>x. h (t,x))"
     using assms unfolding homotopic_with_def by auto
   obtain k
     where k: "continuous_map (prod_topology (subtopology euclideanreal {0..1}) X2) Y2 k"
       and k0: "\<And>x. k (0, x) = g x"
       and k1: "\<And>x. k (1, x) = g' x"
       and q: "\<And>t. \<lbrakk>0 \<le> t; t \<le> 1\<rbrakk> \<Longrightarrow> q (\<lambda>x. k (t,x))"
     using assms unfolding homotopic_with_def by auto
   let ?hk = "\<lambda>(t,x,y). (h(t,x), k(t,y))"
   show ?thesis
     unfolding homotopic_with_def
   proof (intro conjI allI exI)
     show "continuous_map (prod_topology (subtopology euclideanreal {0..1}) (prod_topology X1 X2))
                          (prod_topology Y1 Y2) ?hk"
       unfolding continuous_map_pairwise case_prod_unfold
       by (rule conjI continuous_map_pairedI continuous_intros continuous_map_id [unfolded id_def]
           continuous_map_fst_of [unfolded o_def] continuous_map_snd_of [unfolded o_def]
           continuous_map_compose [OF _ h, unfolded o_def]
           continuous_map_compose [OF _ k, unfolded o_def])+
   next
     fix x
     show "?hk (0, x) = (f (fst x), g (snd x))" "?hk (1, x) = (f' (fst x), g' (snd x))"
       by (auto simp: case_prod_beta h0 k0 h1 k1)
   qed (auto simp: p q r)
 qed
 
 
 lemma homotopic_with_product_topology:
   assumes ht: "\<And>i. i \<in> I \<Longrightarrow> homotopic_with (p i) (X i) (Y i) (f i) (g i)"
     and pq: "\<And>h. (\<And>i. i \<in> I \<Longrightarrow> p i (h i)) \<Longrightarrow> q(\<lambda>x. (\<lambda>i\<in>I. h i (x i)))"
   shows "homotopic_with q (product_topology X I) (product_topology Y I)
                           (\<lambda>z. (\<lambda>i\<in>I. (f i) (z i))) (\<lambda>z. (\<lambda>i\<in>I. (g i) (z i)))"
 proof -
   obtain h
     where h: "\<And>i. i \<in> I \<Longrightarrow> continuous_map (prod_topology (subtopology euclideanreal {0..1}) (X i)) (Y i) (h i)"
       and h0: "\<And>i x. i \<in> I \<Longrightarrow> h i (0, x) = f i x"
       and h1: "\<And>i x. i \<in> I \<Longrightarrow> h i (1, x) = g i x"
       and p: "\<And>i t. \<lbrakk>i \<in> I; t \<in> {0..1}\<rbrakk> \<Longrightarrow> p i (\<lambda>x. h i (t,x))"
     using ht unfolding homotopic_with_def by metis
   show ?thesis
     unfolding homotopic_with_def
   proof (intro conjI allI exI)
     let ?h = "\<lambda>(t,z). \<lambda>i\<in>I. h i (t,z i)"
     have "continuous_map (prod_topology (subtopology euclideanreal {0..1}) (product_topology X I))
                          (Y i) (\<lambda>x. h i (fst x, snd x i))" if "i \<in> I" for i
     proof -
       have \<section>: "continuous_map (prod_topology (top_of_set {0..1}) (product_topology X I)) (X i) (\<lambda>x. snd x i)"
         using continuous_map_componentwise continuous_map_snd that by fastforce
       show ?thesis
         unfolding continuous_map_pairwise case_prod_unfold
         by (intro conjI that \<section> continuous_intros continuous_map_compose [OF _ h, unfolded o_def])
     qed
     then show "continuous_map (prod_topology (subtopology euclideanreal {0..1}) (product_topology X I))
          (product_topology Y I) ?h"
       by (auto simp: continuous_map_componentwise case_prod_beta)
     show "?h (0, x) = (\<lambda>i\<in>I. f i (x i))" "?h (1, x) = (\<lambda>i\<in>I. g i (x i))" for x
       by (auto simp: case_prod_beta h0 h1)
     show "\<forall>t\<in>{0..1}. q (\<lambda>x. ?h (t, x))"
       by (force intro: p pq)
   qed
 qed
 
 text\<open>Homotopic triviality implicitly incorporates path-connectedness.\<close>
 lemma homotopic_triviality:
   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_canon (\<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_canon (\<lambda>x. True) S T f (\<lambda>x. c)))"
           (is "?lhs = ?rhs")
 proof (cases "S = {} \<or> T = {}")
   case True then show ?thesis
     by (auto simp: homotopic_on_emptyI)
 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_canon (\<lambda>x. True) S T (\<lambda>x. a) (\<lambda>x. b)"
         by (simp add: LHS image_subset_iff that)
       then show ?thesis
         using False homotopic_constant_maps [of "top_of_set S" "top_of_set T" a b] by auto
     qed
     moreover
     have "\<exists>c. homotopic_with_canon (\<lambda>x. True) S T f (\<lambda>x. c)" if "continuous_on S f" "f ` S \<subseteq> T" for f
       using False LHS continuous_on_const that by blast
     ultimately show ?rhs
       by (simp add: path_connected_component)
   next
     assume RHS: ?rhs
     with False have T: "path_connected T"
       by blast
     show ?lhs
     proof clarify
       fix f g
       assume "continuous_on S f" "f ` S \<subseteq> T" "continuous_on S g" "g ` S \<subseteq> T"
       obtain c d where c: "homotopic_with_canon (\<lambda>x. True) S T f (\<lambda>x. c)" and d: "homotopic_with_canon (\<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_continuous_maps by fastforce+
       with T have "path_component T c d"
-        using path_connected_component by blast
+        by (metis False ex_in_conv homotopic_with_imp_subset2 image_subset_iff path_connected_component)
       then have "homotopic_with_canon (\<lambda>x. True) S T (\<lambda>x. c) (\<lambda>x. d)"
         by (simp add: homotopic_constant_maps)
       with c d show "homotopic_with_canon (\<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\<^marker>\<open>tag important\<close> homotopic_paths :: "['a set, real \<Rightarrow> 'a, real \<Rightarrow> 'a::topological_space] \<Rightarrow> bool"
   where
-     "homotopic_paths s p q \<equiv>
-       homotopic_with_canon (\<lambda>r. pathstart r = pathstart p \<and> pathfinish r = pathfinish p) {0..1} s p q"
+     "homotopic_paths S p q \<equiv>
+       homotopic_with_canon (\<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>
+   "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>
+          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"
+     "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"
+     "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"
+     "homotopic_paths S p q \<Longrightarrow> path p \<and> path q"
   using homotopic_paths_def homotopic_with_imp_continuous_maps path_def continuous_map_subtopology_eu by blast
 
 lemma homotopic_paths_imp_subset:
-     "homotopic_paths s p q \<Longrightarrow> path_image p \<subseteq> s \<and> path_image q \<subseteq> s"
+     "homotopic_paths S p q \<Longrightarrow> path_image p \<subseteq> S \<and> path_image q \<subseteq> S"
   by (metis (mono_tags) continuous_map_subtopology_eu homotopic_paths_def homotopic_with_imp_continuous_maps path_image_def)
 
-proposition homotopic_paths_refl [simp]: "homotopic_paths s p p \<longleftrightarrow> path p \<and> path_image p \<subseteq> s"
+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 path_def path_image_def)
 
-proposition homotopic_paths_sym: "homotopic_paths s p q \<Longrightarrow> homotopic_paths s q p"
+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"
+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]:
-  assumes "homotopic_paths s p q" "homotopic_paths s q r"
-  shows "homotopic_paths s p r"
-proof -
-  have "pathstart q = pathstart p" "pathfinish q = pathfinish p"
-    using assms by (simp_all add: homotopic_paths_imp_pathstart homotopic_paths_imp_pathfinish)
-  then have "homotopic_with_canon (\<lambda>f. pathstart f = pathstart p \<and> pathfinish f = pathfinish p) {0..1} s q r"
-    using \<open>homotopic_paths s q r\<close> homotopic_paths_def by force
-  then show ?thesis
-    using assms homotopic_paths_def homotopic_with_trans by blast
-qed
+  assumes "homotopic_paths S p q" "homotopic_paths S q r"
+  shows "homotopic_paths S p r"
+  using assms homotopic_paths_imp_pathfinish homotopic_paths_imp_pathstart unfolding homotopic_paths_def
+  by (smt (verit, ccfv_SIG) homotopic_with_mono homotopic_with_trans)
 
 proposition homotopic_paths_eq:
-     "\<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"
-  unfolding homotopic_paths_def
-  by (rule homotopic_with_eq)
-     (auto simp: path_def pathstart_def pathfinish_def path_image_def elim: continuous_on_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"
+  by (smt (verit, best) homotopic_paths homotopic_paths_refl)
 
 proposition homotopic_paths_reparametrize:
   assumes "path p"
-      and pips: "path_image p \<subseteq> s"
+      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"
+    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"
+  have piqs: "path_image q \<subseteq> S"
     by (metis (no_types, opaque_lifting) 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"
+  have "homotopic_paths S q p"
   proof (rule homotopic_paths_trans)
-    show "homotopic_paths s q (p \<circ> f)"
+    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"
+    show "homotopic_paths S (p \<circ> f) p"
       using pips [unfolded path_image_def]
       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)+
       by (auto simp: fb0 fb1 pathstart_def pathfinish_def)
   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"
+lemma homotopic_paths_subset: "\<lbrakk>homotopic_paths S p q; S \<subseteq> t\<rbrakk> \<Longrightarrow> homotopic_paths t p q"
   unfolding homotopic_paths by fast
 
 
 text\<open> A slightly ad-hoc but useful lemma in constructing homotopies.\<close>
 lemma continuous_on_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))" (is "continuous_on ?A ?p")
       and q: "continuous_on ({0..1} \<times> {0..1}) (\<lambda>y. q (fst y) (snd y))" (is "continuous_on ?A ?q")
       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 \<section>: "(\<lambda>t. p (fst t) (2 * snd t)) = ?p \<circ> (\<lambda>y. (fst y, 2 * snd y))"
           "(\<lambda>t. q (fst t) (2 * snd t - 1)) = ?q \<circ> (\<lambda>y. (fst y, 2 * snd y - 1))"
     by force+
   show ?thesis
     unfolding joinpaths_def
   proof (rule continuous_on_cases_le)
     show "continuous_on {y \<in> ?A. snd y \<le> 1/2} (\<lambda>t. p (fst t) (2 * snd t))" 
          "continuous_on {y \<in> ?A. 1/2 \<le> snd y} (\<lambda>t. q (fst t) (2 * snd t - 1))"
          "continuous_on ?A snd"
       unfolding \<section>
       by (rule continuous_intros continuous_on_subset [OF p] continuous_on_subset [OF q] | force)+
   qed (use pf in \<open>auto simp: mult.commute pathstart_def pathfinish_def\<close>)
 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)"
+      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"
+     "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')"
+    "\<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 (clarsimp simp add: homotopic_paths_def homotopic_with_def)
   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 (intro conjI continuous_intros continuous_on_homotopic_join_lemma; force simp: joinpaths_def pathstart_def pathfinish_def path_image_def)
   done
 
 proposition homotopic_paths_continuous_image:
-    "\<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)"
+    "\<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
   by (simp add: homotopic_with_compose_continuous_map_left pathfinish_compose pathstart_compose)
 
 
 subsection\<open>Group properties for homotopy of paths\<close>
 
 text\<^marker>\<open>tag important\<close>\<open>So taking equivalence classes under homotopy would give the fundamental group\<close>
 
 proposition homotopic_paths_rid:
-  assumes "path p" "path_image p \<subseteq> s"
-  shows "homotopic_paths s (p +++ linepath (pathfinish p) (pathfinish p)) p"
+  assumes "path p" "path_image p \<subseteq> S"
+  shows "homotopic_paths S (p +++ linepath (pathfinish p) (pathfinish p)) p"
 proof -
   have \<section>: "continuous_on {0..1} (\<lambda>t::real. if t \<le> 1/2 then 2 *\<^sub>R t else 1)"
     unfolding split_01
     by (rule continuous_on_cases continuous_intros | force simp: pathfinish_def joinpaths_def)+
   show ?thesis
     apply (rule homotopic_paths_sym)
     using assms unfolding pathfinish_def joinpaths_def
     by (intro \<section> continuous_on_cases continuous_intros homotopic_paths_reparametrize [where f = "\<lambda>t. if t \<le> 1/2 then 2 *\<^sub>R t else 1"]; force)
 qed
 
 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]
+   "\<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;
+   "\<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)"
+    \<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 add: subset_path_image_join)
   apply (rule continuous_on_cases_1 continuous_intros | auto simp: joinpaths_def)+
   done
 
 proposition homotopic_paths_rinv:
-  assumes "path p" "path_image p \<subseteq> s"
-    shows "homotopic_paths s (p +++ reversepath p) (linepath (pathstart p) (pathstart p))"
+  assumes "path p" "path_image p \<subseteq> S"
+    shows "homotopic_paths S (p +++ reversepath p) (linepath (pathstart p) (pathstart p))"
 proof -
   have p: "continuous_on {0..1} p" 
     using assms by (auto simp: path_def)
   let ?A = "{0..1} \<times> {0..1}"
   have "continuous_on ?A (\<lambda>x. (subpath 0 (fst x) p +++ reversepath (subpath 0 (fst x) p)) (snd x))"
     unfolding joinpaths_def subpath_def reversepath_def path_def add_0_right diff_0_right
   proof (rule continuous_on_cases_le)
     show "continuous_on {x \<in> ?A. snd x \<le> 1/2} (\<lambda>t. p (fst t * (2 * snd t)))"
          "continuous_on {x \<in> ?A. 1/2 \<le> snd x} (\<lambda>t. p (fst t * (1 - (2 * snd t - 1))))"
          "continuous_on ?A snd"
       by (intro continuous_on_compose2 [OF p] continuous_intros; auto simp add: mult_le_one)+
   qed (auto simp add: algebra_simps)
   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 (force simp: mult_le_one path_defs joinpaths_def subpath_def reversepath_def)
     done
 qed
 
 proposition homotopic_paths_linv:
-  assumes "path p" "path_image p \<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
+  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\<^marker>\<open>tag important\<close> 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_canon (\<lambda>r. pathfinish r = pathstart r) {0..1} s p q"
+ "homotopic_loops S p q \<equiv>
+     homotopic_with_canon (\<lambda>r. pathfinish r = pathstart r) {0..1} S p q"
 
 lemma homotopic_loops:
-   "homotopic_loops s p q \<longleftrightarrow>
+   "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>
+          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"
+     "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"
+     "homotopic_loops S p q \<Longrightarrow> path p \<and> path q"
   unfolding homotopic_loops_def path_def
   using homotopic_with_imp_continuous_maps continuous_map_subtopology_eu by blast
 
 proposition homotopic_loops_imp_subset:
-     "homotopic_loops s p q \<Longrightarrow> path_image p \<subseteq> s \<and> path_image q \<subseteq> s"
+     "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 (meson continuous_map_subtopology_eu homotopic_with_imp_continuous_maps)
 
 proposition homotopic_loops_refl:
-     "homotopic_loops s p p \<longleftrightarrow>
-      path p \<and> path_image p \<subseteq> s \<and> pathfinish p = pathstart p"
+     "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 path_image_def path_def)
 
-proposition homotopic_loops_sym: "homotopic_loops s p q \<Longrightarrow> homotopic_loops s q p"
+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"
+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"
+   "\<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"
+   "\<lbrakk>homotopic_loops S p q; S \<subseteq> t\<rbrakk> \<Longrightarrow> homotopic_loops t p q"
   by (fastforce simp add: homotopic_loops)
 
 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"
+   "\<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 path_image_def path_def pathstart_def pathfinish_def
   by (auto intro: homotopic_with_eq [OF homotopic_with_refl [where f = p, THEN iffD2]])
 
 proposition homotopic_loops_continuous_image:
-   "\<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)"
+   "\<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
   by (simp add: homotopic_with_compose_continuous_map_left 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"
+    "\<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_with_def homotopic_paths_def homotopic_loops_def)
 
 proposition homotopic_loops_imp_homotopic_paths_null:
-  assumes "homotopic_loops s p (linepath a a)"
-    shows "homotopic_paths s p (linepath (pathstart p) (pathstart p))"
+  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)
+  have pip: "path_image p \<subseteq> S" by (metis assms homotopic_loops_imp_subset)
   let ?A = "{0..1::real} \<times> {0..1::real}"
   obtain h where conth: "continuous_on ?A h"
-             and hs: "h ` ?A \<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 hs: "h ` ?A \<subseteq> S"
+             and h0[simp]: "\<And>x. x \<in> {0..1} \<Longrightarrow> h(0,x) = p x"
+             and h1[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
   proof (rule continuous_on_compose [of _ _ h, unfolded o_def])
     show "continuous_on ((\<lambda>x. (x, 0)) ` {0..1}) h"
       by (force intro: continuous_on_subset [OF conth])
   qed (force intro: continuous_intros)
-  have pih0: "path_image (\<lambda>u. h (u, 0)) \<subseteq> s"
+  have pih0: "path_image (\<lambda>u. h (u, 0)) \<subseteq> S"
     using hs by (force simp: path_image_def)
   have c1: "continuous_on ?A (\<lambda>x. h (fst x * snd x, 0))"
   proof (rule continuous_on_compose [of _ _ h, unfolded o_def])
     show "continuous_on ((\<lambda>x. (fst x * snd x, 0)) ` ?A) h"
       by (force simp: mult_le_one intro: continuous_on_subset [OF conth])
   qed (force intro: continuous_intros)+
   have c2: "continuous_on ?A (\<lambda>x. h (fst x - fst x * snd x, 0))"
   proof (rule continuous_on_compose [of _ _ h, unfolded o_def])
     show "continuous_on ((\<lambda>x. (fst x - fst x * snd x, 0)) ` ?A) h"
       by (auto simp: algebra_simps add_increasing2 mult_left_le intro: continuous_on_subset [OF conth])
   qed (force intro: continuous_intros)
   have [simp]: "\<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. \<lbrakk>path p \<and> path(reversepath p);
-                  path_image p \<subseteq> s \<and> path_image(reversepath p) \<subseteq> s;
+                  path_image p \<subseteq> S \<and> path_image(reversepath p) \<subseteq> S;
                   pathfinish p = pathstart(linepath a a +++ reversepath p) \<and>
                    pathstart(reversepath p) = a \<and> pathstart p = x\<rbrakk>
-                  \<Longrightarrow> homotopic_paths s (p +++ linepath a a +++ reversepath p) (linepath x 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))"
+  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))
+  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)
+    using homotopic_paths_lid [of "p +++ linepath (pathfinish p) (pathfinish p)" S]
+    by (metis 1 homotopic_paths_imp_path homotopic_paths_imp_subset homotopic_paths_sym pathstart_join)
   moreover 
-  have "homotopic_paths s (linepath (pathstart p) (pathstart p) +++ p +++ linepath (pathfinish p) (pathfinish p))
+  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)))"
     unfolding homotopic_paths_def homotopic_with_def
   proof (intro exI strip conjI)
     let ?h = "\<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)" 
     have "continuous_on ?A ?h"
       by (intro continuous_on_homotopic_join_lemma; simp add: path_defs joinpaths_def subpath_def conth c1 c2)
-    moreover have "?h ` ?A \<subseteq> s"
+    moreover have "?h ` ?A \<subseteq> S"
       unfolding joinpaths_def subpath_def
       by (force simp: algebra_simps mult_le_one mult_left_le intro: hs [THEN subsetD] adhoc_le)
   ultimately show "continuous_map (prod_topology (top_of_set {0..1}) (top_of_set {0..1}))
-                         (top_of_set s) ?h"
+                         (top_of_set S) ?h"
     by (simp add: subpath_reversepath)
   qed (use ploop in \<open>simp_all add: reversepath_def path_defs joinpaths_def o_def subpath_def conth c1 c2\<close>)
-  moreover have "homotopic_paths s ((\<lambda>u. h (u, 0)) +++ linepath a a +++ reversepath (\<lambda>u. h (u, 0)))
+  moreover have "homotopic_paths S ((\<lambda>u. h (u, 0)) +++ linepath a a +++ reversepath (\<lambda>u. h (u, 0)))
                                    (linepath (pathstart p) (pathstart p))"
-  proof (rule *; simp add: pih0 pathstart_def pathfinish_def conth0)
-    show "a = (linepath a a +++ reversepath (\<lambda>u. h (u, 0))) 0 \<and> reversepath (\<lambda>u. h (u, 0)) 0 = a"
-      by (simp_all add: reversepath_def joinpaths_def)
-  qed
+    by (rule *; simp add: pih0 pathstart_def pathfinish_def conth0; simp add: reversepath_def joinpaths_def)
   ultimately show ?thesis
     by (blast intro: homotopic_paths_trans)
 qed
 
 proposition homotopic_loops_conjugate:
-  fixes s :: "'a::real_normed_vector set"
-  assumes "path p" "path q" and pip: "path_image p \<subseteq> s" and piq: "path_image q \<subseteq> s"
+  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 pq: "pathfinish p = pathstart q" and qloop: "pathfinish q = pathstart q"
-    shows "homotopic_loops s (p +++ q +++ reversepath p) 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
   let ?A = "{0..1::real} \<times> {0..1::real}"
   have c1: "continuous_on ?A (\<lambda>x. p ((1 - fst x) * snd x + fst x))"
   proof (rule continuous_on_compose [of _ _ p, unfolded o_def])
     show "continuous_on ((\<lambda>x. (1 - fst x) * snd x + fst x) ` ?A) p"
       by (auto intro: continuous_on_subset [OF contp] simp: algebra_simps add_increasing2 mult_right_le_one_le sum_le_prod1)
   qed (force intro: continuous_intros)
   have c2: "continuous_on ?A (\<lambda>x. p ((fst x - 1) * snd x + 1))"
   proof (rule continuous_on_compose [of _ _ p, unfolded o_def])
     show "continuous_on ((\<lambda>x. (fst x - 1) * snd x + 1) ` ?A) p"
       by (auto intro: continuous_on_subset [OF contp] simp: algebra_simps add_increasing2 mult_left_le_one_le)
   qed (force intro: continuous_intros)
 
-  have ps1: "\<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"
+  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"
+  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
+    by (metis add_mono affine_ineq linear mult.commute mult.left_neutral mult_right_mono
               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"
+  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)
+  have "homotopic_loops S (p +++ q +++ reversepath p)
                           (linepath (pathstart q) (pathstart q) +++ q +++ linepath (pathstart q) (pathstart q))"
     unfolding homotopic_loops_def homotopic_with_def
   proof (intro exI strip conjI)
     let ?h = "(\<lambda>y. (subpath (fst y) 1 p +++ q +++ subpath 1 (fst y) p) (snd y))" 
     have "continuous_on ?A (\<lambda>y. q (snd y))"
       by (force simp: contq intro: continuous_on_compose [of _ _ q, unfolded o_def] continuous_on_id continuous_on_snd)
     then have "continuous_on ?A ?h"
       using pq qloop
       by (intro continuous_on_homotopic_join_lemma) (auto simp: path_defs joinpaths_def subpath_def c1 c2)
-    then show "continuous_map (prod_topology (top_of_set {0..1}) (top_of_set {0..1})) (top_of_set s) ?h"
+    then show "continuous_map (prod_topology (top_of_set {0..1}) (top_of_set {0..1})) (top_of_set S) ?h"
       by (auto simp: joinpaths_def subpath_def  ps1 ps2 qs)
     show "?h (1,x) = (linepath (pathstart q) (pathstart q) +++ q +++ linepath (pathstart q) (pathstart q)) x"  for x
       using pq by (simp add: pathfinish_def subpath_refl)
   qed (auto simp: subpath_reversepath)
-  moreover have "homotopic_loops s (linepath (pathstart q) (pathstart q) +++ q +++ linepath (pathstart q) (pathstart q)) q"
+  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"
+    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)"
+    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
+    hence "homotopic_paths S (linepath (pathfinish q) (pathfinish q) +++ q +++ linepath (pathfinish q) (pathfinish q)) q"
+      by (smt (verit, best) \<open>path q\<close> homotopic_paths_imp_path homotopic_paths_imp_subset homotopic_paths_lid 
+          homotopic_paths_rid homotopic_paths_trans pathstart_join piq qloop)
     thus ?thesis
       by (metis (no_types) qloop homotopic_loops_sym homotopic_paths_imp_homotopic_loops homotopic_paths_imp_pathfinish homotopic_paths_sym)
   qed
   ultimately show ?thesis
     by (blast intro: homotopic_loops_trans)
 qed
 
 lemma homotopic_paths_loop_parts:
   assumes loops: "homotopic_loops S (p +++ reversepath q) (linepath a a)" and "path q"
   shows "homotopic_paths S p q"
 proof -
   have paths: "homotopic_paths S (p +++ reversepath q) (linepath (pathstart p) (pathstart p))"
     using homotopic_loops_imp_homotopic_paths_null [OF loops] by simp
   then have "path p"
     using \<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"
+    obtain 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)+
+           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
+      by (metis \<open>path q\<close> homotopic_paths_imp_path homotopic_paths_lid homotopic_paths_trans path_join_path_ends pathfinish_linepath pipq(2))
   next
     case False
     then show ?thesis
       using \<open>path q\<close> homotopic_loops_imp_path loops path_join_path_ends by fastforce
   qed
 qed
 
 
 subsection\<^marker>\<open>tag unimportant\<close>\<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_canon (\<lambda>z. True) S t f g"
   unfolding 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)
   using sub closed_segment_def
      by (fastforce intro: continuous_intros continuous_on_subset [OF contf] continuous_on_compose2 [where g=f]
             continuous_on_subset [OF contg] continuous_on_compose2 [where g=g])
 
 lemma homotopic_paths_linear:
   fixes g h :: "real \<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_defs 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)
   by (force simp: closed_segment_def intro!: continuous_intros intro: continuous_on_compose2 [where g=g] continuous_on_compose2 [where g=h])
 
 lemma homotopic_paths_nearby_explicit:
   assumes \<section>: "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"
-proof (rule homotopic_paths_linear [OF \<section>])
-  show "\<And>t. t \<in> {0..1} \<Longrightarrow> closed_segment (g t) (h t) \<subseteq> S"
-  by (metis no segment_bound(1) subsetI norm_minus_commute not_le)
-qed
+  using homotopic_paths_linear [OF \<section>] by (metis linorder_not_le no norm_minus_commute segment_bound1 subsetI)
 
 lemma homotopic_loops_nearby_explicit:
   assumes \<section>: "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"
-proof (rule homotopic_loops_linear [OF \<section>])
-  show "\<And>t. t \<in> {0..1} \<Longrightarrow> closed_segment (g t) (h t) \<subseteq> S"
-  by (metis no segment_bound(1) subsetI norm_minus_commute not_le)
-qed
+  using homotopic_loops_linear [OF \<section>] by (metis linorder_not_le no norm_minus_commute segment_bound1 subsetI)
 
 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
     using e [unfolded dist_norm] \<open>e > 0\<close>
     by (fastforce simp: path_image_def intro!: homotopic_paths_nearby_explicit assms exI)
 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
     using e [unfolded dist_norm] \<open>e > 0\<close>
     by (fastforce simp: path_image_def intro!: homotopic_loops_nearby_explicit assms exI)
 qed
 
 
 subsection\<open> Homotopy and subpaths\<close>
 
 lemma homotopic_join_subpaths1:
-  assumes "path g" and pag: "path_image g \<subseteq> s"
+  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)"
+    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
   have "homotopic_paths (path_image g) (subpath u v g +++ subpath v w g) (subpath u w g)"
   proof (cases "w = u")
     case True
     then show ?thesis
       by (metis \<open>path g\<close> homotopic_paths_rinv path_image_subpath_subset path_subpath pathstart_subpath reversepath_subpath subpath_refl u v)
   next
     case False
     let ?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))"
     show ?thesis
     proof (rule homotopic_paths_sym [OF homotopic_paths_reparametrize [where f = ?f]])
       show "path (subpath u w g)"
         using assms(1) path_subpath u w(1) by blast
       show "path_image (subpath u w g) \<subseteq> path_image g"
         by (meson path_image_subpath_subset u w(1))
       show "continuous_on {0..1} ?f"
         unfolding split_01
         by (rule continuous_on_cases continuous_intros | force simp: pathfinish_def joinpaths_def dest!: t2)+
       show "?f ` {0..1} \<subseteq> {0..1}"
         using False assms
         by (force simp: field_simps not_le mult_left_mono affine_ineq dest!: 1 2)
       show "(subpath u v g +++ subpath v w g) t = subpath u w g (?f t)" if "t \<in> {0..1}" for t 
         using assms
         unfolding joinpaths_def subpath_def by (auto simp add: divide_simps add.commute mult.commute mult.left_commute)
     qed (use False in auto)
   qed
   then show ?thesis
     by (rule homotopic_paths_subset [OF _ pag])
 qed
 
 lemma homotopic_join_subpaths2:
-  assumes "homotopic_paths s (subpath u v g +++ subpath v w g) (subpath u w g)"
-    shows "homotopic_paths s (subpath w v g +++ subpath v u g) (subpath w u g)"
-by (metis assms homotopic_paths_reversepath_D pathfinish_subpath pathstart_subpath reversepath_joinpaths reversepath_subpath)
+  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"
+  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)"
+    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)"
-  proof (rule homotopic_paths_join)
-    show "homotopic_paths s (subpath u w g) (subpath u v g +++ subpath v w g)"
-      using hom homotopic_paths_sym_eq by blast
-    show "homotopic_paths s (subpath w v g) (subpath w v g)"
-      by (metis \<open>path g\<close> homotopic_paths_eq pag path_image_subpath_subset path_subpath subset_trans v w)
-  qed auto
-  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)"
-    by (rule homotopic_paths_sym [OF homotopic_paths_assoc])
-       (use assms in \<open>simp_all add: path_image_subpath_subset [THEN order_trans]\<close>)
-  also have "homotopic_paths s (subpath u v g +++ subpath v w g +++ subpath w v g)
+  obtain wvg: "path (subpath w v g)" "path_image (subpath w v g) \<subseteq> S" 
+     and wug: "path (subpath w u g)" "path_image (subpath w u g) \<subseteq> S"
+     and vug: "path (subpath v u g)" "path_image (subpath v u g) \<subseteq> S"
+    by (meson \<open>path g\<close> pag path_image_subpath_subset path_subpath subset_trans u v w)
+  have "homotopic_paths S (subpath u w g +++ subpath w v g) 
+                          ((subpath u v g +++ subpath v w g) +++ subpath w v g)"
+    by (simp add: hom homotopic_paths_join homotopic_paths_sym wvg)
+  also have "homotopic_paths S ((subpath u v g +++ subpath v w g) +++ subpath w v g) 
+                               (subpath u v g +++ subpath v w g +++ subpath w v g)"
+    using wvg vug \<open>path g\<close>
+    by (metis homotopic_paths_assoc homotopic_paths_sym path_image_subpath_commute path_subpath
+        pathfinish_subpath pathstart_subpath u v w)
+  also have "homotopic_paths S (subpath u v g +++ subpath v w g +++ subpath w v g)
                                (subpath u v g +++ linepath (pathfinish (subpath u v g)) (pathfinish (subpath u v g)))"
-  proof (rule homotopic_paths_join; simp)
-    show "path (subpath u v g) \<and> path_image (subpath u v g) \<subseteq> s"
-      by (metis \<open>path g\<close> order.trans pag path_image_subpath_subset path_subpath u v)
-    show "homotopic_paths s (subpath v w g +++ subpath w v g) (linepath (g v) (g v))"
-      by (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)
-  qed 
-  also have "homotopic_paths s (subpath u v g +++ linepath (pathfinish (subpath u v g)) (pathfinish (subpath u v g))) (subpath u v g)"
-  proof (rule homotopic_paths_rid)
-    show "path (subpath u v g)"
-      using \<open>path g\<close> path_subpath u v by blast
-    show "path_image (subpath u v g) \<subseteq> s"
-      by (meson \<open>path g\<close> order.trans pag path_image_subpath_subset u v)
-  qed
-  finally have "homotopic_paths s (subpath u w g +++ subpath w v g) (subpath u v g)" .
+    using wvg vug \<open>path g\<close>
+    by (metis homotopic_paths_join homotopic_paths_linv homotopic_paths_refl path_image_subpath_commute
+        path_subpath pathfinish_subpath pathstart_join pathstart_subpath reversepath_subpath u v)
+  also have "homotopic_paths S (subpath u v g +++ linepath (pathfinish (subpath u v g)) (pathfinish (subpath u v g))) (subpath u v g)"
+    using vug \<open>path g\<close> by (metis homotopic_paths_rid path_image_subpath_commute path_subpath u v)
+  finally have "homotopic_paths S (subpath u w g +++ subpath w v g) (subpath u v g)" .
   then show ?thesis
     using homotopic_join_subpaths2 by blast
 qed
 
 proposition homotopic_join_subpaths:
-   "\<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)"
-  using le_cases3 [of u v w] homotopic_join_subpaths1 homotopic_join_subpaths2 homotopic_join_subpaths3 
-  by metis
+   "\<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)"
+  by (smt (verit, del_insts) homotopic_join_subpaths1 homotopic_join_subpaths2 homotopic_join_subpaths3)
 
 text\<open>Relating homotopy of trivial loops to path-connectedness.\<close>
 
 lemma path_component_imp_homotopic_points:
   assumes "path_component S a b"
   shows "homotopic_loops S (linepath a a) (linepath b b)"
 proof -
   obtain g :: "real \<Rightarrow> 'a" where g: "continuous_on {0..1} g" "g ` {0..1} \<subseteq> S" "g 0 = a" "g 1 = b"
     using assms by (auto simp: path_defs)
   then have "continuous_on ({0..1} \<times> {0..1}) (g \<circ> fst)"
     by (fastforce intro!: continuous_intros)+
   with g show ?thesis
     by (auto simp add: homotopic_loops_def homotopic_with_def path_defs image_subset_iff)
 qed
 
 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 (clarsimp simp add: homotopic_loops_def homotopic_with_def path_defs)
 apply (rule_tac x="h \<circ> (\<lambda>u. (u, t))" in exI)
 apply (fastforce elim!: continuous_on_subset intro!: continuous_intros)
 done
 
 lemma homotopic_points_eq_path_component:
    "homotopic_loops S (linepath a a) (linepath b b) \<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\<^marker>\<open>tag important\<close>\<open>defined as "all loops are homotopic (as loops)\<close>
 
 definition\<^marker>\<open>tag important\<close> 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))"
         (is "?lhs = ?rhs")
 proof
   assume ?lhs then show ?rhs
     unfolding simply_connected_def by force
 next
   assume ?rhs then show ?lhs
     unfolding simply_connected_def
     by (metis pathfinish_in_path_image subsetD  homotopic_loops_trans homotopic_loops_sym)
 qed
 
 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)))"
      (is "?lhs = ?rhs")
 proof
   assume ?lhs
   then show ?rhs
   using simply_connected_eq_contractible_loop_any by (blast intro: simply_connected_imp_path_connected)
 next
   assume r: ?rhs
   note pa = r [THEN conjunct2, rule_format]
   show ?lhs
   proof (clarsimp simp add: simply_connected_eq_contractible_loop_any)
     fix p a
     assume "path p" and "path_image p \<subseteq> S" "pathfinish p = pathstart p"
       and "a \<in> S"
     with pa [of p] show "homotopic_loops S p (linepath a a)"
       using homotopic_loops_trans path_connected_eq_homotopic_points r by blast
   qed
 qed
 
 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"
       unfolding simply_connected_eq_contractible_loop_any 
       by (meson False 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)))"
      (is "?lhs = ?rhs")
 proof
   assume ?lhs
   then show ?rhs
     unfolding simply_connected_imp_path_connected
     by (metis simply_connected_eq_contractible_loop_some homotopic_loops_imp_homotopic_paths_null)
 next
   assume  ?rhs
   then show ?lhs
     using homotopic_paths_imp_homotopic_loops simply_connected_eq_contractible_loop_some by fastforce
 qed
 
 lemma simply_connected_eq_homotopic_paths:
   fixes S :: "_::real_normed_vector set"
   shows "simply_connected S \<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)"
       by (simp add: continuous_on_fst Path_Connected.path_continuous_image [OF \<open>path p\<close>])
     moreover have "path_image (fst \<circ> p) \<subseteq> S"
       using that by (force simp add: path_image_def)
     ultimately have p1: "homotopic_loops S (fst \<circ> p) (linepath a a)"
       using S that
       by (simp add: simply_connected_eq_contractible_loop_any pathfinish_def pathstart_def)
     have "path (snd \<circ> p)"
       by (simp add: continuous_on_snd Path_Connected.path_continuous_image [OF \<open>path p\<close>])
     moreover have "path_image (snd \<circ> p) \<subseteq> T"
       using that by (force simp: path_image_def)
     ultimately have p2: "homotopic_loops T (snd \<circ> p) (linepath b b)"
       using T that
       by (simp add: simply_connected_eq_contractible_loop_any pathfinish_def pathstart_def)
     show ?thesis
       using p1 p2 unfolding homotopic_loops
       apply clarify
       subgoal for h k
         by (rule_tac x="\<lambda>z. (h z, k z)" in exI) (force intro: continuous_intros simp: path_defs)
       done
   qed
   with assms show ?thesis
     by (simp add: simply_connected_eq_contractible_loop_any pathfinish_def pathstart_def)
 qed
 
 
 subsection\<open>Contractible sets\<close>
 
 definition\<^marker>\<open>tag important\<close> contractible where
  "contractible S \<equiv> \<exists>a. homotopic_with_canon (\<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_canon (\<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_in_topspace topspace_euclidean_subtopology)
   have "\<forall>p. path p \<and>
             path_image p \<subseteq> S \<and> pathfinish p = pathstart p \<longrightarrow>
             homotopic_loops S p (linepath a a)"
     using a apply (clarsimp simp add: homotopic_loops_def homotopic_with_def path_defs)
     apply (rule_tac x="(h \<circ> (\<lambda>y. (fst y, (p \<circ> snd) y)))" in exI)
     apply (intro conjI continuous_on_compose continuous_intros; force elim: continuous_on_subset)
     done
   with \<open>a \<in> S\<close> show ?thesis
     by (auto simp add: simply_connected_eq_contractible_loop_all False)
 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_canon (\<lambda>h. True) S U (g \<circ> f) (\<lambda>x. c)"
 proof -
   obtain b where b: "homotopic_with_canon (\<lambda>x. True) T T id (\<lambda>x. b)"
     using assms by (force simp: contractible_def)
   have "homotopic_with_canon (\<lambda>f. True) T U (g \<circ> id) (g \<circ> (\<lambda>x. b))"
     by (metis Abstract_Topology.continuous_map_subtopology_eu b g homotopic_with_compose_continuous_map_left)
   then have "homotopic_with_canon (\<lambda>f. True) S U (g \<circ> id \<circ> f) (g \<circ> (\<lambda>x. b) \<circ> f)"
     by (simp add: f homotopic_with_compose_continuous_map_right)
   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_canon (\<lambda>h. True) S T f (\<lambda>x. c)"
   by (rule nullhomotopic_through_contractible [OF f, of id T]) (use assms in auto)
 
 lemma nullhomotopic_from_contractible:
   assumes f: "continuous_on S f" "f ` S \<subseteq> T"
       and S: "contractible S"
     obtains c where "homotopic_with_canon (\<lambda>h. True) S T f (\<lambda>x. c)"
   by (auto simp: comp_def intro: nullhomotopic_through_contractible [OF continuous_on_id _ f S])
 
 lemma homotopic_through_contractible:
   fixes S :: "_::real_normed_vector set"
   assumes "continuous_on S f1" "f1 ` S \<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_canon (\<lambda>h. True) S U (g1 \<circ> f1) (g2 \<circ> f2)"
 proof -
   obtain c1 where c1: "homotopic_with_canon (\<lambda>h. True) S U (g1 \<circ> f1) (\<lambda>x. c1)"
     by (rule nullhomotopic_through_contractible [of S f1 T g1 U]) (use assms in auto)
   obtain c2 where c2: "homotopic_with_canon (\<lambda>h. True) S U (g2 \<circ> f2) (\<lambda>x. c2)"
     by (rule nullhomotopic_through_contractible [of S f2 T g2 U]) (use assms in auto)
   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_continuous_maps by fastforce+
     with \<open>path_connected U\<close> show ?thesis by blast
   qed
   then have "homotopic_with_canon (\<lambda>h. True) S U (\<lambda>x. c2) (\<lambda>x. c1)"
     by (simp add: path_component homotopic_constant_maps)
   then show ?thesis
     using c1 c2 homotopic_with_symD homotopic_with_trans by blast
 qed
 
 lemma homotopic_into_contractible:
   fixes S :: "'a::real_normed_vector set" and T:: "'b::real_normed_vector set"
   assumes f: "continuous_on S f" "f ` S \<subseteq> T"
       and g: "continuous_on S g" "g ` S \<subseteq> T"
       and T: "contractible T"
     shows "homotopic_with_canon (\<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)
 
 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_canon (\<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)
 
 subsection\<open>Starlike sets\<close>
 
 definition\<^marker>\<open>tag important\<close> "starlike S \<longleftrightarrow> (\<exists>a\<in>S. \<forall>x\<in>S. closed_segment a x \<subseteq> S)"
 
 lemma starlike_UNIV [simp]: "starlike UNIV"
   by (simp add: starlike_def)
 
 lemma convex_imp_starlike:
   "convex S \<Longrightarrow> S \<noteq> {} \<Longrightarrow> starlike S"
   unfolding convex_contains_segment starlike_def by auto
 
 lemma starlike_convex_tweak_boundary_points:
   fixes S :: "'a::euclidean_space set"
   assumes "convex S" "S \<noteq> {}" and ST: "rel_interior S \<subseteq> T" and TS: "T \<subseteq> closure S"
   shows "starlike T"
 proof -
   have "rel_interior S \<noteq> {}"
     by (simp add: assms rel_interior_eq_empty)
   then obtain a where a: "a \<in> rel_interior S"  by blast
   with ST have "a \<in> T"  by blast
   have "\<And>x. x \<in> T \<Longrightarrow> open_segment a x \<subseteq> rel_interior S"
     by (rule rel_interior_closure_convex_segment [OF \<open>convex S\<close> a]) (use assms in auto)
   then have "\<forall>x\<in>T. a \<in> T \<and> open_segment a x \<subseteq> T"
     using ST by (blast intro: a \<open>a \<in> T\<close> rel_interior_closure_convex_segment [OF \<open>convex S\<close> a])
   then show ?thesis
     unfolding starlike_def using bexI [OF _ \<open>a \<in> T\<close>]
     by (simp add: closed_segment_eq_open)
 qed
 
 lemma starlike_imp_contractible_gen:
   fixes S :: "'a::real_normed_vector set"
   assumes S: "starlike S"
       and P: "\<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_canon 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 "\<And>t b. 0 \<le> t \<and> t \<le> 1 \<Longrightarrow>
               \<exists>u. (1 - t) *\<^sub>R b + t *\<^sub>R a = (1 - u) *\<^sub>R a + u *\<^sub>R b \<and> 0 \<le> u \<and> u \<le> 1"
     by (metis add_diff_cancel_right' diff_ge_0_iff_ge le_add_diff_inverse pth_c(1))
   then have "(\<lambda>y. (1 - fst y) *\<^sub>R snd y + fst y *\<^sub>R a) ` ({0..1} \<times> S) \<subseteq> S"
     using a [unfolded closed_segment_def] by force
   then have "homotopic_with_canon P S S (\<lambda>x. x) (\<lambda>x. a)"
     using \<open>a \<in> S\<close>
     unfolding homotopic_with_def
     apply (rule_tac x="\<lambda>y. (1 - (fst y)) *\<^sub>R snd y + (fst y) *\<^sub>R a" in exI)
     apply (force simp add: P intro: continuous_intros)
     done
   then show ?thesis
     using that by blast
 qed
 
 lemma starlike_imp_contractible:
   fixes S :: "'a::real_normed_vector set"
   shows "starlike S \<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_on_emptyI)
 
 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
     by (meson False assms starlike_convex_tweak_boundary_points starlike_imp_contractible)
 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)))"])
     using hsub ksub
     apply (fastforce intro!: continuous_intros continuous_on_compose2 [OF conth] continuous_on_compose2 [OF contk])
     done
 qed
 
 
 subsection\<open>Local versions of topological properties in general\<close>
 
 definition\<^marker>\<open>tag important\<close> locally :: "('a::topological_space set \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> bool"
 where
  "locally P S \<equiv>
         \<forall>w x. openin (top_of_set S) w \<and> x \<in> w
               \<longrightarrow> (\<exists>u v. openin (top_of_set 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 (top_of_set S) w; x \<in> w\<rbrakk>
                   \<Longrightarrow> \<exists>u v. openin (top_of_set 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 (top_of_set S) w" "x \<in> w"
   obtains u v where "openin (top_of_set 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 (top_of_set S) t"
     shows "locally P t"
   using assms
   unfolding locally_def
   by (elim all_forward) (meson dual_order.trans openin_imp_subset openin_subset_trans openin_trans)
 
 lemma locally_diff_closed:
     "\<lbrakk>locally P S; closedin (top_of_set 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}"
 proof -
   have "\<forall>x::real. \<not> 0 < x \<Longrightarrow> P {a}"
     using zero_less_one by blast
   then show ?thesis
     unfolding locally_def
     by (auto simp add: openin_euclidean_subtopology_iff subset_singleton_iff conj_disj_distribR)
 qed
 
 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)"
   unfolding locally_iff
 proof clarify
   fix A x
   assume "open A" "x \<in> A" "x \<in> S" "x \<in> T"
   then obtain U1 V1 U2 V2 
     where "open U1" "P V1" "x \<in> S \<inter> U1" "S \<inter> U1 \<subseteq> V1 \<and> V1 \<subseteq> S \<inter> A" 
           "open U2" "P V2" "x \<in> T \<inter> U2" "T \<inter> U2 \<subseteq> V2 \<and> V2 \<subseteq> T \<inter> A"
     using S T unfolding locally_iff by (meson IntI)
   then have "S \<inter> T \<inter> (U1 \<inter> U2) \<subseteq> V1 \<inter> V2" "V1 \<inter> V2 \<subseteq> S \<inter> T \<inter> A" "x \<in> S \<inter> T \<inter> (U1 \<inter> U2)"
     by blast+
   moreover have "P (V1 \<inter> V2)"
     by (simp add: P \<open>P V1\<close> \<open>P V2\<close>)
   ultimately show "\<exists>U. open U \<and> (\<exists>V. P V \<and> x \<in> S \<inter> T \<inter> U \<and> S \<inter> T \<inter> U \<subseteq> V \<and> V \<subseteq> S \<inter> T \<inter> A)"
     using \<open>open U1\<close> \<open>open U2\<close> by blast
 qed
 
 
 lemma locally_Times:
   fixes S :: "('a::metric_space) set" and T :: "('b::metric_space) set"
   assumes PS: "locally P S" and QT: "locally Q T" and R: "\<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 (top_of_set (S \<times> T)) W" and xy: "(x, y) \<in> W"
   then obtain U V where "openin (top_of_set S) U" "x \<in> U"
                         "openin (top_of_set 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 (top_of_set S) U1 \<and> P U2 \<and> x \<in> U1 \<and> U1 \<subseteq> U2 \<and> U2 \<subseteq> U"
            and opeT: "openin (top_of_set T) V1 \<and> Q V2 \<and> y \<in> V1 \<and> V1 \<subseteq> V2 \<and> V2 \<subseteq> V"
     by (meson PS QT locallyE)
   then have "openin (top_of_set (S \<times> T)) (U1 \<times> V1)"
     by (simp add: openin_Times)
   moreover have "R (U2 \<times> V2)"
     by (simp add: R opeS opeT)
   moreover have "U1 \<times> V1 \<subseteq> U2 \<times> V2 \<and> U2 \<times> V2 \<subseteq> W"
     using opeS opeT \<open>U \<times> V \<subseteq> W\<close> by auto 
   ultimately show "\<exists>U V. openin (top_of_set (S \<times> T)) U \<and> R V \<and> (x,y) \<in> U \<and> U \<subseteq> V \<and> V \<subseteq> W"
     using opeS opeT by auto 
 qed
 
 
 proposition homeomorphism_locally_imp:
   fixes S :: "'a::metric_space set" and T :: "'b::t2_space set"
   assumes S: "locally P S" and hom: "homeomorphism S T f g"
       and Q: "\<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 (top_of_set T) W"
   then obtain A where T: "open A" "W = T \<inter> A"
     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> g by force
   have \<circ>: "openin (top_of_set S) (g ` W)"
   proof -
     have "continuous_on S f"
       using f(3) by blast
     then show "openin (top_of_set S) (g ` W)"
       by (simp add: gw Collect_conj_eq \<open>openin (top_of_set T) W\<close> continuous_on_open f(2))
   qed
   then obtain U V
     where osu: "openin (top_of_set 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 "V \<subseteq> g ` f ` V"
     by (metis \<open>V \<subseteq> S\<close> hom homeomorphism_def homeomorphism_of_subsets order_refl)
   then have homv: "homeomorphism V (f ` V) f g"
     using \<open>V \<subseteq> S\<close> f by (auto simp add: homeomorphism_def contvf contvg)
   have "openin (top_of_set (g ` T)) U"
     using \<open>g ` T = S\<close> by (simp add: osu)
   then have 1: "openin (top_of_set T) (T \<inter> g -` U)"
     using \<open>continuous_on T g\<close> continuous_on_open [THEN iffD1] by blast
   have 2: "\<exists>V. Q V \<and> y \<in> (T \<inter> g -` U) \<and> (T \<inter> g -` U) \<subseteq> V \<and> V \<subseteq> W"
   proof (intro exI conjI)
     show "Q (f ` V)"
       using Q homv \<open>P V\<close> by blast
     show "y \<in> T \<inter> g -` U"
       using T(2) \<open>y \<in> W\<close> \<open>g y \<in> U\<close> by blast
     show "T \<inter> g -` U \<subseteq> f ` V"
       using g(1) image_iff uv(3) by fastforce
     show "f ` V \<subseteq> W"
       using \<open>f ` V \<subseteq> W\<close> by blast
   qed
   show "\<exists>U. openin (top_of_set 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"
      (is "?lhs = ?rhs")
 proof
   assume ?lhs
   then show ?rhs
     using eq hom homeomorphism_locally_imp by blast
 next
   assume ?rhs
   then show ?lhs
     using eq homeomorphism_sym homeomorphism_symD [OF hom] 
     by (blast intro: homeomorphism_locally_imp) 
 qed
 
 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 ((+) a ` S) = P S) \<Longrightarrow> locally P ((+) a ` S) = locally P S"
   using homeomorphism_locally [OF homeomorphism_translation]
   by (metis (full_types) homeomorphism_image2)
 
 lemma locally_injective_linear_image:
   fixes f :: "'a::euclidean_space \<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"
   using homeomorphism_locally [of "f`S" _ _ f] linear_homeomorphism_image [OF f]
   by (metis (no_types, lifting) homeomorphism_image2 iff)
 
 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 (top_of_set S) T \<Longrightarrow> openin (top_of_set (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 (top_of_set (f ` S)) W" and "y \<in> W"
   then have "W \<subseteq> f ` S" by (simp add: openin_euclidean_subtopology_iff)
   have oivf: "openin (top_of_set 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 (top_of_set 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 have "openin (top_of_set (f ` S)) (f ` U)"
     by (simp add: oo)
   then show "\<exists>X. openin (top_of_set (f ` S)) X \<and> (\<exists>Y. Q Y \<and> y \<in> X \<and> X \<subseteq> Y \<and> Y \<subseteq> W)"
     using Q \<open>P V\<close> \<open>U \<subseteq> V\<close> \<open>V \<subseteq> S \<inter> f -` W\<close> \<open>f x = y\<close> \<open>x \<in> U\<close> by blast
 qed
 
 
 subsection\<open>An induction principle for connected sets\<close>
 
 proposition connected_induction:
   assumes "connected S"
       and opD: "\<And>T a. \<lbrakk>openin (top_of_set 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 (top_of_set 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 -
   let ?A = "{b. \<exists>T. openin (top_of_set S) T \<and> b \<in> T \<and> (\<forall>x\<in>T. P x \<longrightarrow> Q x)}"
   let ?B = "{b. \<exists>T. openin (top_of_set S) T \<and> b \<in> T \<and> (\<forall>x\<in>T. P x \<longrightarrow> \<not> Q x)}"
   have 1: "openin (top_of_set S) ?A"
     by (subst openin_subopen, blast)
   have 2: "openin (top_of_set S) ?B"
     by (subst openin_subopen, blast)
   have \<section>: "?A \<inter> ?B = {}"
     by (clarsimp simp: set_eq_iff) (metis (no_types, opaque_lifting) Int_iff opD openin_Int)
   have *: "S \<subseteq> ?A \<union> ?B"
     by clarsimp (meson opI)
   have "?A = {} \<or> ?B = {}"
     using \<open>connected S\<close> [unfolded connected_openin, simplified, rule_format, OF 1 \<section> * 2] 
     by blast
   then show ?thesis
     by clarsimp (meson opI etc)
 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 (top_of_set 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 (top_of_set 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 (top_of_set S) T \<and> a \<in> T \<and>
                      (\<forall>x \<in> T. \<forall>y \<in> T. P x \<longrightarrow> P y)"
     shows "P b"
   by (rule connected_induction [OF \<open>connected S\<close> _, where P = "\<lambda>x. True"])
      (use opI etc in auto)
 
 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 (top_of_set 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 (top_of_set 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:
   assumes "connected S"
   shows "locally (\<lambda>U. f constant_on U) S \<longleftrightarrow> f constant_on S" (is "?lhs = ?rhs")
 proof
   assume ?lhs
   then have "\<And>a. a \<in> S \<Longrightarrow> \<exists>T. openin (top_of_set S) T \<and> a \<in> T \<and> (\<forall>x\<in>T. f x = f a)"
     unfolding locally_def
     by (metis (mono_tags, opaque_lifting) constant_on_def constant_on_subset openin_subtopology_self)
   then show ?rhs
     using assms
     by (simp add: locally_constant_imp_constant)
 next
   assume ?rhs then show ?lhs
     using assms by (metis constant_on_subset locallyI openin_imp_subset order_refl)
 qed
 
 
 subsection\<open>Basic properties of local compactness\<close>
 
 proposition locally_compact:
-  fixes s :: "'a :: metric_space set"
+  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 (top_of_set s) u \<and> compact v)"
+    "locally compact S \<longleftrightarrow>
+     (\<forall>x \<in> S. \<exists>u v. x \<in> u \<and> u \<subseteq> v \<and> v \<subseteq> S \<and>
+                    openin (top_of_set S) u \<and> compact v)"
      (is "?lhs = ?rhs")
 proof
   assume ?lhs
   then show ?rhs
     by (meson locallyE openin_subtopology_self)
 next
   assume r [rule_format]: ?rhs
   have *: "\<exists>u v.
-              openin (top_of_set 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
+              openin (top_of_set 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 (top_of_set s) u"
-      using r [OF \<open>x \<in> s\<close>] by auto
+    obtain u v where uv: "x \<in> u" "u \<subseteq> v" "v \<subseteq> S" "compact v" "openin (top_of_set 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="(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
     by (rule locallyI) (metis "*" Int_iff openin_open)
 qed
 
 lemma locally_compactE:
   fixes S :: "'a :: metric_space set"
   assumes "locally compact S"
   obtains u v where "\<And>x. x \<in> S \<Longrightarrow> x \<in> u x \<and> u x \<subseteq> v x \<and> v x \<subseteq> S \<and>
                              openin (top_of_set 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 (top_of_set S) U \<and> compact(closure U) \<and> closure U \<subseteq> S)"
         (is "?lhs = ?rhs")
 proof
   assume ?lhs
   then show ?rhs
     by (meson bounded_subset closure_minimal compact_closure compact_imp_bounded 
               compact_imp_closed dual_order.trans locally_compactE)
 next
   assume ?rhs then show ?lhs
     by (meson closure_subset locally_compact)
 qed
 
 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 L: ?lhs
   then have "\<And>x U V e. \<lbrakk>U \<subseteq> V; V \<subseteq> S; compact V; 0 < e; cball x e \<inter> S \<subseteq> U\<rbrakk>
        \<Longrightarrow> closed (cball x e \<inter> S)"
     by (metis compact_Int compact_cball compact_imp_closed inf.absorb_iff2 inf.assoc inf.orderE)
   with L show ?rhs
     by (meson locally_compactE openin_contains_cball)
 next
   assume R: ?rhs
   show ?lhs unfolding locally_compact 
   proof
     fix x
     assume "x \<in> S"
     then obtain e where "e>0" and e: "closed (cball x e \<inter> S)"
       using R by blast
     then have "compact (cball x e \<inter> S)"
       by (simp add: bounded_Int compact_eq_bounded_closed)
     moreover have "\<forall>y\<in>ball x e \<inter> S. \<exists>\<epsilon>>0. cball y \<epsilon> \<inter> S \<subseteq> ball x e"
       by (meson Elementary_Metric_Spaces.open_ball IntD1 le_infI1 open_contains_cball_eq)
     moreover have "openin (top_of_set S) (ball x e \<inter> S)"
       by (simp add: inf_commute openin_open_Int)
     ultimately show "\<exists>U V. x \<in> U \<and> U \<subseteq> V \<and> V \<subseteq> S \<and> openin (top_of_set S) U \<and> compact V"
       by (metis Int_iff \<open>0 < e\<close> \<open>x \<in> S\<close> ball_subset_cball centre_in_ball inf_commute inf_le1 inf_mono order_refl)
   qed
 qed
 
 lemma locally_compact_compact:
   fixes S :: "'a :: heine_borel set"
   shows "locally compact S \<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 (top_of_set 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 (top_of_set 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 (top_of_set S) U \<and> compact V"
           if "K \<subseteq> S" "compact K" for K
   proof -
     have "\<And>C. (\<forall>c\<in>C. openin (top_of_set 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 (top_of_set K) c"
       using that by clarify (metis subsetD inf.absorb_iff2 openin_subset openin_subtopology_Int_subset topspace_euclidean_subtopology uv)
     moreover have "K \<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 dest!: uv)
     moreover
     have "openin (top_of_set S) (\<Union> (u ` T))"
       using T that uv by fastforce
     moreover
     have "compact (\<Union> (v ` T))"
       by (meson T compact_UN subset_eq that(1) uv)
     moreover have "\<Union> (v ` T) \<subseteq> S"
       by (metis SUP_least T(1) subset_eq that(1) uv)
     ultimately show ?thesis
       using T by auto 
   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 (top_of_set 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 (top_of_set S) (ball x e)"
       by (meson e open_ball ball_subset_cball dual_order.trans open_subset)
     show ?thesis
     proof (intro exI conjI)
       let ?U = "ball x e"
       let ?V = "cball x e"
       show "x \<in> ?U" "?U \<subseteq> ?V" "?V \<subseteq> S" "compact ?V"
         using \<open>e > 0\<close> e by auto
       show "openin (top_of_set S) ?U"
         using ope by blast
     qed
   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 (top_of_set S) U \<and> compact V"
           if "x \<in> S" for x
     apply (rule_tac x = "S \<inter> ball x 1" in exI, rule_tac x = "S \<inter> cball x 1" in exI)
     using \<open>x \<in> S\<close> assms by auto
   show ?thesis
     unfolding locally_compact by (blast intro: *)
 qed
 
 lemma locally_compact_UNIV: "locally compact (UNIV :: 'a :: heine_borel set)"
   by (simp add: closed_imp_locally_compact)
 
 lemma locally_compact_Int:
   fixes S :: "'a :: t2_space set"
   shows "\<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 (top_of_set 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
     unfolding locally_def
     apply (elim 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 (top_of_set (S \<union> T)) S"
       and opT: "openin (top_of_set (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 have "closed (cball x (min e1 e2) \<inter> (S \<union> T))"
       by (metis (no_types, lifting) cball_min_Int closed_Int closed_cball inf_assoc inf_commute)
     then show ?thesis
       by (metis \<open>0 < e1\<close> \<open>0 < e2\<close> min_def)
   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
     moreover have "closed (cball x e1 \<inter> (cball x e2 \<inter> T))"
       by (metis closed_Int closed_cball e1 inf_left_commute)
     ultimately show ?thesis
       by (rule_tac x="min e1 e2" in exI) (simp add: \<open>0 < e2\<close> cball_min_Int inf_assoc)
   qed
   ultimately show ?thesis
     by (force simp: locally_compact_Int_cball)
 qed
 
 lemma locally_compact_closedin_Un:
   fixes S :: "'a::euclidean_space set"
   assumes LCS: "locally compact S" and LCT:"locally compact T"
       and clS: "closedin (top_of_set (S \<union> T)) S"
       and clT: "closedin (top_of_set (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
     moreover have "closed (cball x (min e1 e2) \<inter> (S \<union> T))"
     proof -
       have "closed (cball x e1 \<inter> (cball x e2 \<inter> S))"
         by (metis closed_Int closed_cball e1 inf_left_commute)
       then show ?thesis
         by (simp add: Int_Un_distrib cball_min_Int closed_Int closed_Un e2 inf_assoc)
     qed
     ultimately show ?thesis
       by (rule_tac x="min e1 e2" in exI) linarith
   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
     with e1 have "closed ((cball x e1 \<inter> cball x e2) \<inter> (S \<union> T))"
       apply (simp add: inf_commute inf_sup_distrib2)
       by (metis closed_Int closed_Un closed_cball inf_assoc inf_left_commute)
     then have "closed (cball x (min e1 e2) \<inter> (S \<union> T))"
       by (simp add: cball_min_Int inf_commute)
     ultimately show ?thesis
       using \<open>0 < e2\<close> by (rule_tac x="min e1 e2" in exI) linarith 
   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)
     with e1 have "closed ((cball x e1 \<inter> cball x e2) \<inter> (S \<union> T))"
       apply (simp add: inf_commute inf_sup_distrib2)
       by (metis closed_Int closed_Un closed_cball inf_assoc inf_left_commute)
     then have "closed (cball x (min e1 e2) \<inter> (S \<union> T))"
       by (auto simp: cball_min_Int)
     ultimately show ?thesis
       using \<open>0 < e2\<close> by (rule_tac x="min e1 e2" in exI) linarith
   qed
   ultimately show ?thesis
     by (auto simp: locally_compact_Int_cball)
 qed
 
 lemma locally_compact_Times:
   fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set"
   shows "\<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 (top_of_set 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 (top_of_set 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 (top_of_set 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 (top_of_set 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 (top_of_set S) T \<and>
                            closedin (top_of_set 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 (top_of_set (\<Inter>?\<T>)) K1" and
          K2: "closedin (top_of_set (\<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 (top_of_set S) D"
                    and cloD: "closedin (top_of_set 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 (top_of_set S) (\<Inter>\<F>)"
                 using openin_Inter \<open>finite \<F>\<close> False \<F> by blast
               then show "closedin (top_of_set 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 (top_of_set D) (D \<inter> closure V1)"
             and cloV2: "closedin (top_of_set 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"
             using \<open>D \<subseteq> V1 \<union> V2\<close> \<open>open V1\<close> \<open>open V2\<close> V12
             by (auto simp add: closure_subset [THEN subsetD] closure_iff_nhds_not_empty, blast+)
           ultimately have cloDV1: "closedin (top_of_set D) (D \<inter> V1)"
                       and cloDV2:  "closedin (top_of_set 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 1: "openin (top_of_set S) (D \<inter> V2)"
               by (simp add: \<open>open V2\<close> opeD openin_Int_open)
             have 2: "closedin (top_of_set S) (D \<inter> V2)"
               using cloD cloDV2 closedin_trans by blast
             have "\<Inter> ?\<T> \<subseteq> D \<inter> V2"
               by (rule Inter_lower) (use * 1 2 in simp)
             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 1: "openin (top_of_set S) (D \<inter> V1)"
               by (simp add: \<open>open V1\<close> opeD openin_Int_open)
             have 2: "closedin (top_of_set S) (D \<inter> V1)"
               using cloD cloDV1 closedin_trans by blast
             have "\<Inter>?\<T> \<subseteq> D \<inter> V1"
               by (rule Inter_lower) (use * 1 2 in simp)
             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> [unfolded connected_closed, simplified, rule_format, of concl: "D \<inter> U1" "D \<inter> U2"]
             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 (top_of_set S) K" "compact K" "C \<subseteq> K" "K \<subseteq> U"
 proof (rule ccontr)
   assume "\<not> thesis"
   with that have neg: "\<nexists>K. openin (top_of_set 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 (top_of_set S) V"
     using S U \<open>compact C\<close> by (meson C in_components_subset locally_compact_compact_subopen)
   let ?\<T> = "{T. C \<subseteq> T \<and> openin (top_of_set 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 (top_of_set K) T \<and> closedin (top_of_set 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 (top_of_set K) (\<Inter>\<F>)"
           using False \<F> \<open>finite \<F>\<close> by blast
         then have opeVF: "openin (top_of_set 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 (top_of_set 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 (top_of_set S) U" and "C \<subseteq> U"
   obtains K where "openin (top_of_set 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 (top_of_set 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 (top_of_set 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 (top_of_set S) V; x \<in> V\<rbrakk> \<Longrightarrow> \<exists>U. openin (top_of_set S) U \<and> connected U \<and> x \<in> U \<and> U \<subseteq> V"
    shows "locally connected S"
   by (metis assms locally_def)
 
 lemma locally_connected_2:
   assumes "locally connected S"
           "openin (top_of_set S) t"
           "x \<in> t"
    shows "openin (top_of_set S) (connected_component_set t x)"
 proof -
   { fix y :: 'a
     let ?SS = "top_of_set S"
     assume 1: "openin ?SS t"
               "\<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 (top_of_set S) t; x \<in> t\<rbrakk>
               \<Longrightarrow> openin (top_of_set S)
                           (connected_component_set t x)"
           "openin (top_of_set S) v" "x \<in> v"
    shows  "\<exists>u. openin (top_of_set 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 (top_of_set S) v \<and> x \<in> v
           \<longrightarrow> (\<exists>u. openin (top_of_set 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 (top_of_set S) t \<and> x \<in> t
           \<longrightarrow> openin (top_of_set S) (connected_component_set t x))"
 by (metis locally_connected_1 locally_connected_2 locally_connected_3)
 
 lemma locally_path_connected_1:
   assumes
     "\<And>v x. \<lbrakk>openin (top_of_set S) v; x \<in> v\<rbrakk>
               \<Longrightarrow> \<exists>u. openin (top_of_set S) u \<and> path_connected u \<and> x \<in> u \<and> u \<subseteq> v"
    shows "locally path_connected S"
   by (force simp add: locally_def dest: assms)
 
 lemma locally_path_connected_2:
   assumes "locally path_connected S"
           "openin (top_of_set S) t"
           "x \<in> t"
    shows "openin (top_of_set S) (path_component_set t x)"
 proof -
   { fix y :: 'a
     let ?SS = "top_of_set S"
     assume 1: "openin ?SS t"
               "\<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 rev_subsetD 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 (top_of_set S) t; x \<in> t\<rbrakk>
               \<Longrightarrow> openin (top_of_set S) (path_component_set t x)"
           "openin (top_of_set S) v" "x \<in> v"
    shows  "\<exists>u. openin (top_of_set 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 mem_Collect_eq path_component_subset path_connected_path_component)
 qed
 
 proposition locally_path_connected:
   "locally path_connected S \<longleftrightarrow>
    (\<forall>V x. openin (top_of_set S) V \<and> x \<in> V
           \<longrightarrow> (\<exists>U. openin (top_of_set 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 (top_of_set S) t \<and> x \<in> t
           \<longrightarrow> openin (top_of_set S) (path_component_set t x))"
   by (metis locally_path_connected_1 locally_path_connected_2 locally_path_connected_3)
 
 lemma locally_connected_open_component:
   "locally connected S \<longleftrightarrow>
    (\<forall>t c. openin (top_of_set S) t \<and> c \<in> components t
           \<longrightarrow> openin (top_of_set S) c)"
 by (metis components_iff locally_connected_open_connected_component)
 
 proposition locally_connected_im_kleinen:
   "locally connected S \<longleftrightarrow>
    (\<forall>v x. openin (top_of_set S) v \<and> x \<in> v
        \<longrightarrow> (\<exists>u. openin (top_of_set 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 (top_of_set S) T \<and> x \<in> T \<and> T \<subseteq> c"
        if "openin (top_of_set 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 (top_of_set 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
     unfolding locally_connected_open_component by (meson "*" openin_subopen)
 qed
 
 proposition locally_path_connected_im_kleinen:
   "locally path_connected S \<longleftrightarrow>
    (\<forall>v x. openin (top_of_set S) v \<and> x \<in> v
        \<longrightarrow> (\<exists>u. openin (top_of_set 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 (top_of_set S) T \<and>
                x \<in> T \<and> T \<subseteq> path_component_set u z"
        if "openin (top_of_set 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 (top_of_set 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
       by (metis U c mem_Collect_eq path_component_def path_component_eq subsetI)
   qed
   show ?lhs
     unfolding locally_path_connected_open_path_component
     using "*" openin_subopen by fastforce
 qed
 
 lemma locally_path_connected_imp_locally_connected:
   "locally path_connected S \<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"
   assumes "open S"
   shows "locally path_connected S"
 proof (rule locally_mono)
   show "locally convex S"
     using assms unfolding locally_def
     by (meson open_ball centre_in_ball convex_ball openE open_subset openin_imp_subset openin_open_trans subset_trans)
   show "\<And>T::'a set. convex T \<Longrightarrow> path_connected T"
     using convex_imp_path_connected by blast
 qed
 
 lemma open_imp_locally_connected:
   fixes S :: "'a :: real_normed_vector set"
   shows "open S \<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 (top_of_set S) (connected_component_set S x)"
   by (metis connected_component_eq_empty locally_connected_2 openin_empty openin_subtopology_self)
 
 lemma openin_components_locally_connected:
     "\<lbrakk>locally connected S; c \<in> components S\<rbrakk> \<Longrightarrow> openin (top_of_set S) c"
   using locally_connected_open_component openin_subtopology_self by blast
 
 lemma openin_path_component_locally_path_connected:
   "locally path_connected S
         \<Longrightarrow> openin (top_of_set S) (path_component_set S x)"
 by (metis (no_types) empty_iff locally_path_connected_2 openin_subopen openin_subtopology_self path_component_eq_empty)
 
 lemma closedin_path_component_locally_path_connected:
   assumes "locally path_connected S"
   shows "closedin (top_of_set S) (path_component_set S x)"
 proof -
   have "openin (top_of_set S) (\<Union> ({path_component_set S y |y. y \<in> S} - {path_component_set S x}))"
     using locally_path_connected_2 assms by fastforce
   then show ?thesis
     by  (simp add: closedin_def path_component_subset complement_path_component_Union)
 qed
 
 lemma convex_imp_locally_path_connected:
   fixes S :: "'a:: real_normed_vector set"
   assumes "convex S"
   shows "locally path_connected S"
 proof (clarsimp simp add: locally_path_connected)
   fix V x
   assume "openin (top_of_set S) V" and "x \<in> V"
   then obtain T e where  "V = S \<inter> T" "x \<in> S" "0 < e" "ball x e \<subseteq> T"
     by (metis Int_iff openE openin_open)
   then have "openin (top_of_set S) (S \<inter> ball x e)" "path_connected (S \<inter> ball x e)"
     by (simp_all add: assms convex_Int convex_imp_path_connected openin_open_Int)
   then show "\<exists>U. openin (top_of_set S) U \<and> path_connected U \<and> x \<in> U \<and> U \<subseteq> V"
     using \<open>0 < e\<close> \<open>V = S \<inter> T\<close> \<open>ball x e \<subseteq> T\<close> \<open>x \<in> S\<close> by auto
 qed
 
 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 (top_of_set (connected_component_set S x)) (path_component_set S x)"
   proof (rule openin_subset_trans)
     show "openin (top_of_set S) (path_component_set S x)"
       by (simp add: True assms locally_path_connected_2)
     show "connected_component_set S x \<subseteq> S"
       by (simp add: connected_component_subset)
   qed (simp add: path_component_subset_connected_component)
   moreover have "closedin (top_of_set (connected_component_set S x)) (path_component_set S x)"
     proof (rule closedin_subset_trans [of S])
   show "closedin (top_of_set S) (path_component_set S x)"
     by (simp add: assms closedin_path_component_locally_path_connected)
   show "connected_component_set S x \<subseteq> S"
     by (simp add: connected_component_subset)
   qed (simp add: path_component_subset_connected_component)
   ultimately have *: "path_component_set S x = connected_component_set S x"
     by (metis connected_connected_component connected_clopen True path_component_eq_empty)
   then show ?thesis
     by blast
 next
   case False then show ?thesis
     by (metis Collect_empty_eq_bot connected_component_eq_empty path_component_eq_empty)
 qed
 
 lemma path_component_eq_connected_component_set:
      "locally path_connected S \<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 (top_of_set S) (S \<inter> f -` T) \<longleftrightarrow>
                     openin (top_of_set (f ` S)) T"
     shows "locally connected (f ` S)"
 proof (clarsimp simp: locally_connected_open_component)
   fix U C
   assume opefSU: "openin (top_of_set (f ` S)) U" and "C \<in> components U"
   then have "C \<subseteq> U" "U \<subseteq> f ` S"
     by (meson in_components_subset openin_imp_subset)+
   then have "openin (top_of_set (f ` S)) C \<longleftrightarrow>
              openin (top_of_set S) (S \<inter> f -` C)"
     by (auto simp: oo)
   moreover have "openin (top_of_set S) (S \<inter> f -` C)"
   proof (subst openin_subopen, clarify)
     fix x
     assume "x \<in> S" "f x \<in> C"
     show "\<exists>T. openin (top_of_set S) T \<and> x \<in> T \<and> T \<subseteq> (S \<inter> f -` C)"
     proof (intro conjI exI)
       show "openin (top_of_set S) (connected_component_set (S \<inter> f -` U) x)"
       proof (rule ccontr)
         assume **: "\<not> openin (top_of_set 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"
         by (meson connected_component_subset continuous_on_subset inf.boundedE)
       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 (top_of_set (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 (top_of_set S) (S \<inter> f -` T) \<longleftrightarrow> openin (top_of_set (f ` S)) T"
     shows "locally path_connected (f ` S)"
 proof (clarsimp simp: locally_path_connected_open_path_component)
   fix U y
   assume opefSU: "openin (top_of_set (f ` S)) U" and "y \<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 (top_of_set (f ` S)) (path_component_set U y) \<longleftrightarrow>
              openin (top_of_set 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 (top_of_set 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 (top_of_set S) T \<and> x \<in> T \<and> T \<subseteq> (S \<inter> f -` path_component_set U y)"
     proof (intro conjI exI)
       show "openin (top_of_set S) (path_component_set (S \<inter> f -` U) x)"
       proof (rule ccontr)
         assume **: "\<not> openin (top_of_set 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"
         by (meson Int_lower1 continuous_on_subset path_component_subset)
       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 (top_of_set (f ` S)) (path_component_set U y)"
     by metis
 qed
 
 subsection\<^marker>\<open>tag unimportant\<close>\<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 (top_of_set 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 (top_of_set 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"
 proof (rule continuous_on_components_gen)
   fix C
   assume "C \<in> components S"
   then show "openin (top_of_set S) C \<and> continuous_on C f"
     by (simp add: assms openin_components_locally_connected)
 qed
 
 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 (top_of_set U) S"
       and cuS: "c \<subseteq> components(U - S)"
     shows "closedin (top_of_set 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 di 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 (top_of_set U) (\<Union>(components (U - S) - c))"
   proof (rule openin_Union [OF openin_trans [of "U - S"]])
     show "openin (top_of_set (U - S)) T" if "T \<in> components (U - S) - c" for T
       using that by (simp add: U S locally_diff_closed openin_components_locally_connected)
     show "openin (top_of_set U) (U - S)" if "T \<in> components (U - S) - c" for T
       using that by (simp add: openin_diff S)
   qed
   have "closedin (top_of_set U) (U - \<Union> (components (U - S) - c))"
     by (metis closedin_diff closedin_topspace topspace_euclidean_subtopology *)
   then have "openin (top_of_set U) (U - (U - \<Union>(components (U - S) - c)))"
     by (simp add: openin_diff)
   then show ?thesis
     by (force simp: eq closedin_def)
 qed
 
 lemma closed_union_complement_components:
   fixes S :: "'a::real_normed_vector set"
   assumes S: "closed S" and c: "c \<subseteq> components(- S)"
     shows "closed(S \<union> \<Union> c)"
 proof -
   have "closedin (top_of_set UNIV) (S \<union> \<Union>c)"
     by (metis Compl_eq_Diff_UNIV S c closed_closedin closedin_union_complement_components locally_connected_UNIV subtopology_UNIV)
   then show ?thesis by simp
 qed
 
 lemma closedin_Un_complement_component:
   fixes S :: "'a::real_normed_vector set"
   assumes u: "locally connected u"
       and S: "closedin (top_of_set u) S"
       and c: " c \<in> components(u - S)"
     shows "closedin (top_of_set u) (S \<union> c)"
 proof -
   have "closedin (top_of_set 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 unfolding pairwise_def inj_on_def
     by (blast intro: orthogonal_clauses)
   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)"
     proof (rule norm_sum_Pythagorean [OF \<open>finite B\<close>])
       show "pairwise (\<lambda>v j. orthogonal (a v *\<^sub>R fb v) (a j *\<^sub>R fb j)) B"
         by (rule pairwise_ortho_scaleR [OF pairwise_orth_fb])
     qed
     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 unfolding pairwise_def inj_on_def bij_betw_def
     by (blast intro: orthogonal_clauses)
   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 \<open>bij_betw fb B C\<close> unfolding pairwise_def bij_betw_def by force
   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)"
     proof (rule norm_sum_Pythagorean [OF \<open>finite B\<close>])
       show "pairwise (\<lambda>v j. orthogonal (a v *\<^sub>R fb v) (a j *\<^sub>R fb j)) B"
         by (rule pairwise_ortho_scaleR [OF pairwise_orth_fb])
     qed
     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)"
     proof (rule sum.cong [OF refl])
       show "a x *\<^sub>R g (fb x) = a x *\<^sub>R x" if "x \<in> B" for x
         using that \<open>bij_betw fb B C\<close> bij_betwE bij_betw_inv_into_left gb_def ggb by fastforce
     qed
     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
     by (rule that [OF \<open>linear f\<close> \<open>linear g\<close>]) (simp_all add: fim gim)
 qed
 
 corollary isometry_subspaces:
   fixes S :: "'a::euclidean_space set"
     and T :: "'b::euclidean_space set"
   assumes S: "subspace S"
       and T: "subspace T"
       and d: "dim S = dim T"
   obtains f where "linear f" "f ` S = T" "\<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
     unfolding homeomorphic_def homeomorphism_def
     apply (rule_tac x=f in exI, rule_tac x=g in exI)
     apply (auto simp: linear_continuous_on linear_conv_bounded_linear)
     done
 qed
 
 lemma homeomorphic_affine_sets:
   assumes "affine S" "affine T" "aff_dim S = aff_dim T"
     shows "S homeomorphic T"
 proof (cases "S = {} \<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 (fact homeomorphic_translation)
   also have "\<dots> homeomorphic ((+) (- b) ` T)"
     by (rule homeomorphic_subspaces [OF ss dd])
   also have "\<dots> homeomorphic T"
     using homeomorphic_translation [of T "- b"] by (simp add: homeomorphic_sym [of T])
   finally show ?thesis .
 qed
 
 
 subsection\<open>Retracts, in a general sense, preserve (co)homotopic triviality)\<close>
 
 locale\<^marker>\<open>tag important\<close> Retracts =
-  fixes s h t k
-  assumes conth: "continuous_on s h"
-      and imh: "h ` s = t"
+  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 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 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_canon P U s f g"
+      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_canon 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_canon Q U t f g"
 proof -
   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"
+  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"
+  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_canon P U s (k \<circ> f) (k \<circ> g)"
+  ultimately have "homotopic_with_canon P U S (k \<circ> f) (k \<circ> g)"
     by (rule hom)
   then have "homotopic_with_canon 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
   proof (rule homotopic_with_eq; simp)
     show "\<And>h k. (\<And>x. x \<in> U \<Longrightarrow> h x = k x) \<Longrightarrow> Q h = Q k"
       using Qeq topspace_euclidean_subtopology by blast
     show "\<And>x. x \<in> U \<Longrightarrow> f x = h (k (f x))" "\<And>x. x \<in> U \<Longrightarrow> g x = h (k (g x))"
       using idhk imf img by auto
   qed 
 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 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_canon P U s f (\<lambda>x. c)"
+      and hom: "\<And>f. \<lbrakk>continuous_on U f; f ` U \<subseteq> S; P f\<rbrakk>
+                     \<Longrightarrow> \<exists>c. homotopic_with_canon 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_canon 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"
+  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_canon P U s (k \<circ> f) (\<lambda>x. c)"
+  ultimately obtain c where "homotopic_with_canon P U S (k \<circ> f) (\<lambda>x. c)"
     by (metis hom)
   then have "homotopic_with_canon 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 have "homotopic_with_canon Q U t f (\<lambda>x. h c)"
   proof (rule homotopic_with_eq)
     show "\<And>x. x \<in> topspace (top_of_set U) \<Longrightarrow> f x = (h \<circ> (k \<circ> f)) x"
       using feq by auto
     show "\<And>h k. (\<And>x. x \<in> topspace (top_of_set U) \<Longrightarrow> h x = k x) \<Longrightarrow> Q h = Q k"
       using Qeq topspace_euclidean_subtopology by blast
   qed auto
   then show ?thesis
     using that by blast
 qed
 
 lemma cohomotopically_trivial_retraction_gen:
   assumes P: "\<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 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_canon P s U f g"
+      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_canon 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_canon 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)"
+  have "continuous_on S (f \<circ> h)"
     using contf conth continuous_on_compose imh by blast
-  moreover have "(f \<circ> h) ` s \<subseteq> U"
+  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)"
+  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"
+  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_canon P s U (f \<circ> h) (g \<circ> h)"
+  ultimately have "homotopic_with_canon P S U (f \<circ> h) (g \<circ> h)"
     by (rule hom)
   then have "homotopic_with_canon 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
   proof (rule homotopic_with_eq)
     show "f x = (f \<circ> h \<circ> k) x" "g x = (g \<circ> h \<circ> k) x" 
       if "x \<in> topspace (top_of_set t)" for x
       using feq geq that by force+
   qed (use Qeq topspace_euclidean_subtopology in blast)
 qed
 
 lemma cohomotopically_trivial_retraction_null_gen:
   assumes P: "\<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 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_canon P s U f (\<lambda>x. c)"
+      and hom: "\<And>f g. \<lbrakk>continuous_on S f; f ` S \<subseteq> U; P f\<rbrakk>
+                       \<Longrightarrow> \<exists>c. homotopic_with_canon 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_canon 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)"
+  have "continuous_on S (f \<circ> h)"
     using contf conth continuous_on_compose imh by blast
-  moreover have "(f \<circ> h) ` s \<subseteq> U"
+  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_canon P s U (f \<circ> h) (\<lambda>x. c)"
+  ultimately obtain c where "homotopic_with_canon P S U (f \<circ> h) (\<lambda>x. c)"
     by (metis hom)
   then have \<section>: "homotopic_with_canon Q t U (f \<circ> h \<circ> k) ((\<lambda>x. c) \<circ> k)"
   proof (rule homotopic_with_compose_continuous_right [OF homotopic_with_mono])
-    show "\<And>h. \<lbrakk>continuous_map (top_of_set s) (top_of_set U) h; P h\<rbrakk> \<Longrightarrow> Q (h \<circ> k)"
+    show "\<And>h. \<lbrakk>continuous_map (top_of_set S) (top_of_set U) h; P h\<rbrakk> \<Longrightarrow> Q (h \<circ> k)"
       using Q by auto
   qed (auto simp: contk imk)
   moreover have "homotopic_with_canon Q t U f (\<lambda>x. c)"
     using homotopic_with_eq [OF \<section>] feq Qeq by fastforce
   ultimately show ?thesis 
     using that by blast
 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>
 
 subsection\<open>Homotopy equivalence of topological spaces.\<close>
 
 definition\<^marker>\<open>tag important\<close> homotopy_equivalent_space
              (infix "homotopy'_equivalent'_space" 50)
   where "X homotopy_equivalent_space Y \<equiv>
         (\<exists>f g. continuous_map X Y f \<and>
               continuous_map Y X g \<and>
               homotopic_with (\<lambda>x. True) X X (g \<circ> f) id \<and>
               homotopic_with (\<lambda>x. True) Y Y (f \<circ> g) id)"
 
 lemma homeomorphic_imp_homotopy_equivalent_space:
   "X homeomorphic_space Y \<Longrightarrow> X homotopy_equivalent_space Y"
   unfolding homeomorphic_space_def homotopy_equivalent_space_def
   apply (erule ex_forward)+
   by (simp add: homotopic_with_equal homotopic_with_sym homeomorphic_maps_def)
 
 lemma homotopy_equivalent_space_refl:
    "X homotopy_equivalent_space X"
   by (simp add: homeomorphic_imp_homotopy_equivalent_space homeomorphic_space_refl)
 
 lemma homotopy_equivalent_space_sym:
    "X homotopy_equivalent_space Y \<longleftrightarrow> Y homotopy_equivalent_space X"
   by (meson homotopy_equivalent_space_def)
 
 lemma homotopy_eqv_trans [trans]:
   assumes 1: "X homotopy_equivalent_space Y" and 2: "Y homotopy_equivalent_space U"
     shows "X homotopy_equivalent_space U"
 proof -
   obtain f1 g1 where f1: "continuous_map X Y f1"
                  and g1: "continuous_map Y X g1"
                  and hom1: "homotopic_with (\<lambda>x. True) X X (g1 \<circ> f1) id"
                            "homotopic_with (\<lambda>x. True) Y Y (f1 \<circ> g1) id"
     using 1 by (auto simp: homotopy_equivalent_space_def)
   obtain f2 g2 where f2: "continuous_map Y U f2"
                  and g2: "continuous_map U Y g2"
                  and hom2: "homotopic_with (\<lambda>x. True) Y Y (g2 \<circ> f2) id"
                            "homotopic_with (\<lambda>x. True) U U (f2 \<circ> g2) id"
     using 2 by (auto simp: homotopy_equivalent_space_def)
   have "homotopic_with (\<lambda>f. True) X Y (g2 \<circ> f2 \<circ> f1) (id \<circ> f1)"
     using f1 hom2(1) homotopic_with_compose_continuous_map_right by metis
   then have "homotopic_with (\<lambda>f. True) X Y (g2 \<circ> (f2 \<circ> f1)) (id \<circ> f1)"
     by (simp add: o_assoc)
   then have "homotopic_with (\<lambda>x. True) X X
          (g1 \<circ> (g2 \<circ> (f2 \<circ> f1))) (g1 \<circ> (id \<circ> f1))"
     by (simp add: g1 homotopic_with_compose_continuous_map_left)
   moreover have "homotopic_with (\<lambda>x. True) X X (g1 \<circ> id \<circ> f1) id"
     using hom1 by simp
   ultimately have SS: "homotopic_with (\<lambda>x. True) X X (g1 \<circ> g2 \<circ> (f2 \<circ> f1)) id"
     by (metis comp_assoc homotopic_with_trans id_comp)
   have "homotopic_with (\<lambda>f. True) U Y (f1 \<circ> g1 \<circ> g2) (id \<circ> g2)"
     using g2 hom1(2) homotopic_with_compose_continuous_map_right by fastforce
   then have "homotopic_with (\<lambda>f. True) U Y (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_map_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"
     by (simp add: fun.map_comp hom2(2) homotopic_with_trans)
   show ?thesis
     unfolding homotopy_equivalent_space_def
     by (blast intro: f1 f2 g1 g2 continuous_map_compose SS UU)
 qed
 
 lemma deformation_retraction_imp_homotopy_equivalent_space:
-  "\<lbrakk>homotopic_with (\<lambda>x. True) X X (s \<circ> r) id; retraction_maps X Y r s\<rbrakk>
+  "\<lbrakk>homotopic_with (\<lambda>x. True) X X (S \<circ> r) id; retraction_maps X Y r S\<rbrakk>
     \<Longrightarrow> X homotopy_equivalent_space Y"
   unfolding homotopy_equivalent_space_def retraction_maps_def
   using homotopic_with_id2 by fastforce
 
 lemma deformation_retract_imp_homotopy_equivalent_space:
    "\<lbrakk>homotopic_with (\<lambda>x. True) X X r id; retraction_maps X Y r id\<rbrakk>
     \<Longrightarrow> X homotopy_equivalent_space Y"
   using deformation_retraction_imp_homotopy_equivalent_space by force
 
 lemma deformation_retract_of_space:
   "S \<subseteq> topspace X \<and>
    (\<exists>r. homotopic_with (\<lambda>x. True) X X id r \<and> retraction_maps X (subtopology X S) r id) \<longleftrightarrow>
    S retract_of_space X \<and> (\<exists>f. homotopic_with (\<lambda>x. True) X X id f \<and> f ` (topspace X) \<subseteq> S)"
 proof (cases "S \<subseteq> topspace X")
   case True
   moreover have "(\<exists>r. homotopic_with (\<lambda>x. True) X X id r \<and> retraction_maps X (subtopology X S) r id)
              \<longleftrightarrow> (S retract_of_space X \<and> (\<exists>f. homotopic_with (\<lambda>x. True) X X id f \<and> f ` topspace X \<subseteq> S))"
     unfolding retract_of_space_def
   proof safe
     fix f r
     assume f: "homotopic_with (\<lambda>x. True) X X id f"
       and fS: "f ` topspace X \<subseteq> S"
       and r: "continuous_map X (subtopology X S) r"
       and req: "\<forall>x\<in>S. r x = x"
     show "\<exists>r. homotopic_with (\<lambda>x. True) X X id r \<and> retraction_maps X (subtopology X S) r id"
     proof (intro exI conjI)
       have "homotopic_with (\<lambda>x. True) X X f r"
         proof (rule homotopic_with_eq)
           show "homotopic_with (\<lambda>x. True) X X (r \<circ> f) (r \<circ> id)"
             by (metis continuous_map_into_fulltopology f homotopic_with_compose_continuous_map_left homotopic_with_symD r)
           show "f x = (r \<circ> f) x" if "x \<in> topspace X" for x
             using that fS req by auto
         qed auto
       then show "homotopic_with (\<lambda>x. True) X X id r"
         by (rule homotopic_with_trans [OF f])
     next
       show "retraction_maps X (subtopology X S) r id"
         by (simp add: r req retraction_maps_def)
     qed
   qed (use True in \<open>auto simp: retraction_maps_def topspace_subtopology_subset continuous_map_in_subtopology\<close>)
   ultimately show ?thesis by simp
 qed (auto simp: retract_of_space_def retraction_maps_def)
 
 
 
 subsection\<open>Contractible spaces\<close>
 
 text\<open>The definition (which agrees with "contractible" on subsets of Euclidean space)
 is a little cryptic because we don't in fact assume that the constant "a" is in the space.
 This forces the convention that the empty space / set is contractible, avoiding some special cases. \<close>
 
 definition contractible_space where
   "contractible_space X \<equiv> \<exists>a. homotopic_with (\<lambda>x. True) X X id (\<lambda>x. a)"
 
 lemma contractible_space_top_of_set [simp]:"contractible_space (top_of_set S) \<longleftrightarrow> contractible S"
   by (auto simp: contractible_space_def contractible_def)
 
 lemma contractible_space_empty:
    "topspace X = {} \<Longrightarrow> contractible_space X"
   unfolding contractible_space_def homotopic_with_def
   apply (rule_tac x=undefined in exI)
   apply (rule_tac x="\<lambda>(t,x). if t = 0 then x else undefined" in exI)
   apply (auto simp: continuous_map_on_empty)
   done
 
 lemma contractible_space_singleton:
   "topspace X = {a} \<Longrightarrow> contractible_space X"
   unfolding contractible_space_def homotopic_with_def
   apply (rule_tac x=a in exI)
   apply (rule_tac x="\<lambda>(t,x). if t = 0 then x else a" in exI)
   apply (auto intro: continuous_map_eq [where f = "\<lambda>z. a"])
   done
 
 lemma contractible_space_subset_singleton:
    "topspace X \<subseteq> {a} \<Longrightarrow> contractible_space X"
   by (meson contractible_space_empty contractible_space_singleton subset_singletonD)
 
 lemma contractible_space_subtopology_singleton:
    "contractible_space(subtopology X {a})"
   by (meson contractible_space_subset_singleton insert_subset path_connectedin_singleton path_connectedin_subtopology subsetI)
 
 lemma contractible_space:
    "contractible_space X \<longleftrightarrow>
         topspace X = {} \<or>
         (\<exists>a \<in> topspace X. homotopic_with (\<lambda>x. True) X X id (\<lambda>x. a))"
 proof (cases "topspace X = {}")
   case False
   then show ?thesis
     using homotopic_with_imp_continuous_maps  by (fastforce simp: contractible_space_def)
 qed (simp add: contractible_space_empty)
 
 lemma contractible_imp_path_connected_space:
   assumes "contractible_space X" shows "path_connected_space X"
 proof (cases "topspace X = {}")
   case False
   have *: "path_connected_space X"
     if "a \<in> topspace X" and conth: "continuous_map (prod_topology (top_of_set {0..1}) X) X h"
       and h: "\<forall>x. h (0, x) = x" "\<forall>x. h (1, x) = a"
     for a and h :: "real \<times> 'a \<Rightarrow> 'a"
   proof -
     have "path_component_of X b a" if "b \<in> topspace X" for b
       unfolding path_component_of_def
     proof (intro exI conjI)
       let ?g = "h \<circ> (\<lambda>x. (x,b))"
       show "pathin X ?g"
         unfolding pathin_def
       proof (rule continuous_map_compose [OF _ conth])
         show "continuous_map (top_of_set {0..1}) (prod_topology (top_of_set {0..1}) X) (\<lambda>x. (x, b))"
           using that by (auto intro!: continuous_intros)
       qed
     qed (use h in auto)
   then show ?thesis
     by (metis path_component_of_equiv path_connected_space_iff_path_component)
   qed
   show ?thesis
     using assms False by (auto simp: contractible_space homotopic_with_def *)
 qed (simp add: path_connected_space_topspace_empty)
 
 lemma contractible_imp_connected_space:
    "contractible_space X \<Longrightarrow> connected_space X"
   by (simp add: contractible_imp_path_connected_space path_connected_imp_connected_space)
 
 lemma contractible_space_alt:
    "contractible_space X \<longleftrightarrow> (\<forall>a \<in> topspace X. homotopic_with (\<lambda>x. True) X X id (\<lambda>x. a))" (is "?lhs = ?rhs")
 proof
   assume X: ?lhs
   then obtain a where a: "homotopic_with (\<lambda>x. True) X X id (\<lambda>x. a)"
     by (auto simp: contractible_space_def)
   show ?rhs
   proof
     show "homotopic_with (\<lambda>x. True) X X id (\<lambda>x. b)" if "b \<in> topspace X" for b
     proof (rule homotopic_with_trans [OF a])
       show "homotopic_with (\<lambda>x. True) X X (\<lambda>x. a) (\<lambda>x. b)"
         using homotopic_constant_maps path_connected_space_imp_path_component_of
         by (metis (full_types) X a continuous_map_const contractible_imp_path_connected_space homotopic_with_imp_continuous_maps that)
     qed
   qed
 next
   assume R: ?rhs
   then show ?lhs
     unfolding contractible_space_def by (metis equals0I homotopic_on_emptyI)
 qed
 
 
 lemma compose_const [simp]: "f \<circ> (\<lambda>x. a) = (\<lambda>x. f a)" "(\<lambda>x. a) \<circ> g = (\<lambda>x. a)"
   by (simp_all add: o_def)
 
 lemma nullhomotopic_through_contractible_space:
   assumes f: "continuous_map X Y f" and g: "continuous_map Y Z g" and Y: "contractible_space Y"
   obtains c where "homotopic_with (\<lambda>h. True) X Z (g \<circ> f) (\<lambda>x. c)"
 proof -
   obtain b where b: "homotopic_with (\<lambda>x. True) Y Y id (\<lambda>x. b)"
     using Y by (auto simp: contractible_space_def)
   show thesis
     using homotopic_with_compose_continuous_map_right
            [OF homotopic_with_compose_continuous_map_left [OF b g] f]
     by (force simp add: that)
 qed
 
 lemma nullhomotopic_into_contractible_space:
   assumes f: "continuous_map X Y f" and Y: "contractible_space Y"
   obtains c where "homotopic_with (\<lambda>h. True) X Y f (\<lambda>x. c)"
   using nullhomotopic_through_contractible_space [OF f _ Y]
   by (metis continuous_map_id id_comp)
 
 lemma nullhomotopic_from_contractible_space:
   assumes f: "continuous_map X Y f" and X: "contractible_space X"
   obtains c where "homotopic_with (\<lambda>h. True) X Y f (\<lambda>x. c)"
   using nullhomotopic_through_contractible_space [OF _ f X]
   by (metis comp_id continuous_map_id)
 
 lemma homotopy_dominated_contractibility:
   assumes f: "continuous_map X Y f" and g: "continuous_map Y X g"
     and hom: "homotopic_with (\<lambda>x. True) Y Y (f \<circ> g) id" and X: "contractible_space X"
   shows "contractible_space Y"
 proof -
   obtain c where c: "homotopic_with (\<lambda>h. True) X Y f (\<lambda>x. c)"
     using nullhomotopic_from_contractible_space [OF f X] .
   have "homotopic_with (\<lambda>x. True) Y Y (f \<circ> g) (\<lambda>x. c)"
     using homotopic_with_compose_continuous_map_right [OF c g] by fastforce
   then have "homotopic_with (\<lambda>x. True) Y Y id (\<lambda>x. c)"
     using homotopic_with_trans [OF _ hom] homotopic_with_symD by blast
   then show ?thesis
     unfolding contractible_space_def ..
 qed
 
 lemma homotopy_equivalent_space_contractibility:
    "X homotopy_equivalent_space Y \<Longrightarrow> (contractible_space X \<longleftrightarrow> contractible_space Y)"
   unfolding homotopy_equivalent_space_def
   by (blast intro: homotopy_dominated_contractibility)
 
 lemma homeomorphic_space_contractibility:
    "X homeomorphic_space Y
         \<Longrightarrow> (contractible_space X \<longleftrightarrow> contractible_space Y)"
   by (simp add: homeomorphic_imp_homotopy_equivalent_space homotopy_equivalent_space_contractibility)
 
 lemma contractible_eq_homotopy_equivalent_singleton_subtopology:
    "contractible_space X \<longleftrightarrow>
         topspace X = {} \<or> (\<exists>a \<in> topspace X. X homotopy_equivalent_space (subtopology X {a}))"(is "?lhs = ?rhs")
 proof (cases "topspace X = {}")
   case False
   show ?thesis
   proof
     assume ?lhs
     then obtain a where a: "homotopic_with (\<lambda>x. True) X X id (\<lambda>x. a)"
       by (auto simp: contractible_space_def)
     then have "a \<in> topspace X"
       by (metis False continuous_map_const homotopic_with_imp_continuous_maps)
     then have "homotopic_with (\<lambda>x. True) (subtopology X {a}) (subtopology X {a}) id (\<lambda>x. a)"
       using connectedin_absolute connectedin_sing contractible_space_alt contractible_space_subtopology_singleton by fastforce
     then have "X homotopy_equivalent_space subtopology X {a}"
       unfolding homotopy_equivalent_space_def using \<open>a \<in> topspace X\<close>
       by (metis (full_types) a comp_id continuous_map_const continuous_map_id_subt empty_subsetI homotopic_with_symD
            id_comp insertI1 insert_subset topspace_subtopology_subset)
     with \<open>a \<in> topspace X\<close> show ?rhs
       by blast
   next
     assume ?rhs
     then show ?lhs
       by (meson False contractible_space_subtopology_singleton homotopy_equivalent_space_contractibility)
   qed
 qed (simp add: contractible_space_empty)
 
 lemma contractible_space_retraction_map_image:
   assumes "retraction_map X Y f" and X: "contractible_space X"
   shows "contractible_space Y"
 proof -
   obtain g where f: "continuous_map X Y f" and g: "continuous_map Y X g" and fg: "\<forall>y \<in> topspace Y. f(g y) = y"
     using assms by (auto simp: retraction_map_def retraction_maps_def)
   obtain a where a: "homotopic_with (\<lambda>x. True) X X id (\<lambda>x. a)"
     using X by (auto simp: contractible_space_def)
   have "homotopic_with (\<lambda>x. True) Y Y id (\<lambda>x. f a)"
   proof (rule homotopic_with_eq)
     show "homotopic_with (\<lambda>x. True) Y Y (f \<circ> id \<circ> g) (f \<circ> (\<lambda>x. a) \<circ> g)"
       using f g a homotopic_with_compose_continuous_map_left homotopic_with_compose_continuous_map_right by metis
   qed (use fg in auto)
   then show ?thesis
     unfolding contractible_space_def by blast
 qed
 
 lemma contractible_space_prod_topology:
    "contractible_space(prod_topology X Y) \<longleftrightarrow>
     topspace X = {} \<or> topspace Y = {} \<or> contractible_space X \<and> contractible_space Y"
 proof (cases "topspace X = {} \<or> topspace Y = {}")
   case True
   then have "topspace (prod_topology X Y) = {}"
     by simp
   then show ?thesis
     by (auto simp: contractible_space_empty)
 next
   case False
   have "contractible_space(prod_topology X Y) \<longleftrightarrow> contractible_space X \<and> contractible_space Y"
   proof safe
     assume XY: "contractible_space (prod_topology X Y)"
     with False have "retraction_map (prod_topology X Y) X fst"
       by (auto simp: contractible_space False retraction_map_fst)
     then show "contractible_space X"
       by (rule contractible_space_retraction_map_image [OF _ XY])
     have "retraction_map (prod_topology X Y) Y snd"
       using False XY  by (auto simp: contractible_space False retraction_map_snd)
     then show "contractible_space Y"
       by (rule contractible_space_retraction_map_image [OF _ XY])
   next
     assume "contractible_space X" and "contractible_space Y"
     with False obtain a b
       where "a \<in> topspace X" and a: "homotopic_with (\<lambda>x. True) X X id (\<lambda>x. a)"
         and "b \<in> topspace Y" and b: "homotopic_with (\<lambda>x. True) Y Y id (\<lambda>x. b)"
       by (auto simp: contractible_space)
     with False show "contractible_space (prod_topology X Y)"
       apply (simp add: contractible_space)
       apply (rule_tac x=a in bexI)
        apply (rule_tac x=b in bexI)
       using homotopic_with_prod_topology [OF a b]
         apply (metis (no_types, lifting) case_prod_Pair case_prod_beta' eq_id_iff)
        apply auto
       done
   qed
   with False show ?thesis
     by auto
 qed
 
 
 lemma contractible_space_product_topology:
   "contractible_space(product_topology X I) \<longleftrightarrow>
     topspace (product_topology X I) = {} \<or> (\<forall>i \<in> I. contractible_space(X i))"
 proof (cases "topspace (product_topology X I) = {}")
   case False
   have 1: "contractible_space (X i)"
     if XI: "contractible_space (product_topology X I)" and "i \<in> I"
     for i
   proof (rule contractible_space_retraction_map_image [OF _ XI])
     show "retraction_map (product_topology X I) (X i) (\<lambda>x. x i)"
       using False by (simp add: retraction_map_product_projection \<open>i \<in> I\<close>)
   qed
   have 2: "contractible_space (product_topology X I)"
     if "x \<in> topspace (product_topology X I)" and cs: "\<forall>i\<in>I. contractible_space (X i)"
     for x :: "'a \<Rightarrow> 'b"
   proof -
     obtain f where f: "\<And>i. i\<in>I \<Longrightarrow> homotopic_with (\<lambda>x. True) (X i) (X i) id (\<lambda>x. f i)"
       using cs unfolding contractible_space_def by metis
     have "homotopic_with (\<lambda>x. True)
                          (product_topology X I) (product_topology X I) id (\<lambda>x. restrict f I)"
       by (rule homotopic_with_eq [OF homotopic_with_product_topology [OF f]]) (auto)
     then show ?thesis
       by (auto simp: contractible_space_def)
   qed
   show ?thesis
     using False 1 2 by blast
 qed (simp add: contractible_space_empty)
 
 
 lemma contractible_space_subtopology_euclideanreal [simp]:
   "contractible_space(subtopology euclideanreal S) \<longleftrightarrow> is_interval S"
   (is "?lhs = ?rhs")
 proof
   assume ?lhs
   then have "path_connectedin (subtopology euclideanreal S) S"
     using contractible_imp_path_connected_space path_connectedin_topspace path_connectedin_absolute
     by (simp add: contractible_imp_path_connected) 
   then show ?rhs
     by (simp add: is_interval_path_connected_1)
 next
   assume ?rhs
   then have "convex S"
     by (simp add: is_interval_convex_1)
   show ?lhs
   proof (cases "S = {}")
     case False
     then obtain z where "z \<in> S"
       by blast
     show ?thesis
       unfolding contractible_space_def homotopic_with_def
     proof (intro exI conjI allI)
       note \<section> = convexD [OF \<open>convex S\<close>, simplified]
       show "continuous_map (prod_topology (top_of_set {0..1}) (top_of_set S)) (top_of_set S)
                            (\<lambda>(t,x). (1 - t) * x + t * z)"
         using  \<open>z \<in> S\<close> 
         by (auto simp add: case_prod_unfold intro!: continuous_intros \<section>)
     qed auto
   qed (simp add: contractible_space_empty)
 qed
 
 
 corollary contractible_space_euclideanreal: "contractible_space euclideanreal"
 proof -
   have "contractible_space (subtopology euclideanreal UNIV)"
     using contractible_space_subtopology_euclideanreal by blast
   then show ?thesis
     by simp
 qed
 
 
 abbreviation\<^marker>\<open>tag important\<close> homotopy_eqv :: "'a::topological_space set \<Rightarrow> 'b::topological_space set \<Rightarrow> bool"
              (infix "homotopy'_eqv" 50)
   where "S homotopy_eqv T \<equiv> top_of_set S homotopy_equivalent_space top_of_set T"
 
 
 
 
 
 lemma homeomorphic_imp_homotopy_eqv: "S homeomorphic T \<Longrightarrow> S homotopy_eqv T"
   unfolding homeomorphic_def homeomorphism_def homotopy_equivalent_space_def
   by (metis continuous_map_subtopology_eu homotopic_with_id2 openin_imp_subset openin_subtopology_self topspace_euclidean_subtopology)
 
 
 lemma homotopy_eqv_inj_linear_image:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes "linear f" "inj f"
     shows "(f ` S) homotopy_eqv S"
   by (metis assms homeomorphic_sym homeomorphic_imp_homotopy_eqv linear_homeomorphic_image)
 
 lemma homotopy_eqv_translation:
     fixes S :: "'a::real_normed_vector set"
     shows "(+) a ` S homotopy_eqv S"
   using homeomorphic_imp_homotopy_eqv homeomorphic_translation homeomorphic_sym by blast
 
 lemma homotopy_eqv_homotopic_triviality_imp:
   fixes S :: "'a::real_normed_vector set"
     and T :: "'b::real_normed_vector set"
     and U :: "'c::real_normed_vector set"
   assumes "S homotopy_eqv T"
       and f: "continuous_on U f" "f ` U \<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_canon (\<lambda>x. True) U S f g"
     shows "homotopic_with_canon (\<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_canon (\<lambda>x. True) S S (k \<circ> h) id"
                         "homotopic_with_canon (\<lambda>x. True) T T (h \<circ> k) id"
     using assms by (auto simp: homotopy_equivalent_space_def)
   have "homotopic_with_canon (\<lambda>f. True) U S (k \<circ> f) (k \<circ> g)"
   proof (rule homUS)
     show "continuous_on U (k \<circ> f)" "continuous_on U (k \<circ> g)"
       using continuous_on_compose continuous_on_subset f g k by blast+
   qed (use f g k in \<open>(force simp: o_def)+\<close> )
   then have "homotopic_with_canon (\<lambda>x. True) U T (h \<circ> (k \<circ> f)) (h \<circ> (k \<circ> g))"
     by (rule homotopic_with_compose_continuous_left; simp add: h)
   moreover have "homotopic_with_canon (\<lambda>x. True) U T (h \<circ> k \<circ> f) (id \<circ> f)"
     by (rule homotopic_with_compose_continuous_right [where X=T and Y=T]; simp add: hom f)
   moreover have "homotopic_with_canon (\<lambda>x. True) U T (h \<circ> k \<circ> g) (id \<circ> g)"
     by (rule homotopic_with_compose_continuous_right [where X=T and Y=T]; simp add: hom g)
   ultimately show "homotopic_with_canon (\<lambda>x. True) U T f g"
     unfolding o_assoc
     by (metis homotopic_with_trans homotopic_with_sym id_comp) 
 qed
 
 lemma homotopy_eqv_homotopic_triviality:
   fixes S :: "'a::real_normed_vector set"
     and T :: "'b::real_normed_vector set"
     and U :: "'c::real_normed_vector set"
   assumes "S homotopy_eqv T"
     shows "(\<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_canon (\<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_canon (\<lambda>x. True) U T f g)"
       (is "?lhs = ?rhs")
 proof
   assume ?lhs
   then show ?rhs
     by (metis assms homotopy_eqv_homotopic_triviality_imp)
 next
   assume ?rhs
   moreover
   have "T homotopy_eqv S"
     using assms homotopy_equivalent_space_sym by blast
   ultimately show ?lhs
     by (blast intro: homotopy_eqv_homotopic_triviality_imp)
 qed
 
 
 lemma homotopy_eqv_cohomotopic_triviality_null_imp:
   fixes S :: "'a::real_normed_vector set"
     and T :: "'b::real_normed_vector set"
     and U :: "'c::real_normed_vector set"
   assumes "S homotopy_eqv T"
       and f: "continuous_on T f" "f ` T \<subseteq> U"
       and homSU: "\<And>f. \<lbrakk>continuous_on S f; f ` S \<subseteq> U\<rbrakk>
                       \<Longrightarrow> \<exists>c. homotopic_with_canon (\<lambda>x. True) S U f (\<lambda>x. c)"
   obtains c where "homotopic_with_canon (\<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_canon (\<lambda>x. True) S S (k \<circ> h) id"
                         "homotopic_with_canon (\<lambda>x. True) T T (h \<circ> k) id"
     using assms by (auto simp: homotopy_equivalent_space_def)
   obtain c where "homotopic_with_canon (\<lambda>x. True) S U (f \<circ> h) (\<lambda>x. c)"
   proof (rule exE [OF homSU])
     show "continuous_on S (f \<circ> h)"
       using continuous_on_compose continuous_on_subset f h by blast
   qed (use f h in force)
   then have "homotopic_with_canon (\<lambda>x. True) T U ((f \<circ> h) \<circ> k) ((\<lambda>x. c) \<circ> k)"
     by (rule homotopic_with_compose_continuous_right [where X=S]) (use k in auto)
   moreover have "homotopic_with_canon (\<lambda>x. True) T U (f \<circ> id) (f \<circ> (h \<circ> k))"
     by (rule homotopic_with_compose_continuous_left [where Y=T])
        (use f in \<open>auto simp add: hom homotopic_with_symD\<close>)
   ultimately show ?thesis
     using that homotopic_with_trans by (fastforce simp add: o_def)
 qed
 
 lemma homotopy_eqv_cohomotopic_triviality_null:
   fixes S :: "'a::real_normed_vector set"
     and T :: "'b::real_normed_vector set"
     and U :: "'c::real_normed_vector set"
   assumes "S homotopy_eqv T"
     shows "(\<forall>f. continuous_on S f \<and> f ` S \<subseteq> U
                 \<longrightarrow> (\<exists>c. homotopic_with_canon (\<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_canon (\<lambda>x. True) T U f (\<lambda>x. c)))"
 by (rule iffI; metis assms homotopy_eqv_cohomotopic_triviality_null_imp homotopy_equivalent_space_sym)
 
 text \<open>Similar to the proof above\<close>
 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_canon (\<lambda>x. True) U S f (\<lambda>x. c)"
     shows "\<exists>c. homotopic_with_canon (\<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_canon (\<lambda>x. True) S S (k \<circ> h) id"
                         "homotopic_with_canon (\<lambda>x. True) T T (h \<circ> k) id"
     using assms by (auto simp: homotopy_equivalent_space_def)
   obtain c::'a where "homotopic_with_canon (\<lambda>x. True) U S (k \<circ> f) (\<lambda>x. c)"
   proof (rule exE [OF homSU [of "k \<circ> f"]])
     show "continuous_on U (k \<circ> f)"
       using continuous_on_compose continuous_on_subset f k by blast
   qed (use f k in force)
   then have "homotopic_with_canon (\<lambda>x. True) U T (h \<circ> (k \<circ> f)) (h \<circ> (\<lambda>x. c))"
     by (rule homotopic_with_compose_continuous_left [where Y=S]) (use h in auto)
   moreover have "homotopic_with_canon (\<lambda>x. True) U T (id \<circ> f) ((h \<circ> k) \<circ> f)"
     by (rule homotopic_with_compose_continuous_right [where X=T])
        (use f in \<open>auto simp add: hom homotopic_with_symD\<close>)
   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_canon (\<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_canon (\<lambda>x. True) U T f (\<lambda>x. c)))"
 by (rule iffI; metis assms homotopy_eqv_homotopic_triviality_null_imp homotopy_equivalent_space_sym)
 
 lemma homotopy_eqv_contractible_sets:
   fixes S :: "'a::real_normed_vector set"
     and T :: "'b::real_normed_vector set"
   assumes "contractible S" "contractible T" "S = {} \<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_equivalent_space_def
     apply (rule_tac x="\<lambda>x. b" in exI, rule_tac x="\<lambda>x. a" in exI)
     apply (intro assms conjI continuous_on_id' homotopic_into_contractible; force)
     done
 qed
 
 lemma homotopy_eqv_empty1 [simp]:
   fixes S :: "'a::real_normed_vector set"
   shows "S homotopy_eqv ({}::'b::real_normed_vector set) \<longleftrightarrow> S = {}" (is "?lhs = ?rhs")
 proof
   assume ?lhs then show ?rhs
     by (metis continuous_map_subtopology_eu empty_iff equalityI homotopy_equivalent_space_def image_subset_iff subsetI)
 qed (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 = {}"
   using homotopy_equivalent_space_sym homotopy_eqv_empty1 by blast
 
 lemma homotopy_eqv_contractibility:
   fixes S :: "'a::real_normed_vector set" and T :: "'b::real_normed_vector set"
   shows "S homotopy_eqv T \<Longrightarrow> (contractible S \<longleftrightarrow> contractible T)"
   by (meson contractible_space_top_of_set homotopy_equivalent_space_contractibility)
 
 lemma homotopy_eqv_sing:
   fixes S :: "'a::real_normed_vector set" and a :: "'b::real_normed_vector"
   shows "S homotopy_eqv {a} \<longleftrightarrow> S \<noteq> {} \<and> contractible S"
 proof (cases "S = {}")
   case False then show ?thesis
     by (metis contractible_sing empty_not_insert homotopy_eqv_contractibility homotopy_eqv_contractible_sets)
 qed simp
 
 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\<^marker>\<open>tag unimportant\<close>\<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
 
 corollary bounded_path_connected_Compl_real:
   fixes S :: "real set"
   assumes "bounded S" "path_connected(- S)" shows "S = {}"
   by (simp add: assms bounded_connected_Compl_real path_connected_imp_connected)
 
 lemma bounded_connected_Compl_1:
   fixes S :: "'a::{euclidean_space} set"
   assumes "bounded S" and conn: "connected(- S)" and 1: "DIM('a) = 1"
     shows "S = {}"
 proof -
   have "DIM('a) = DIM(real)"
     by (simp add: "1")
   then obtain f::"'a \<Rightarrow> real" and g
   where "linear f" "\<And>x. norm(f x) = norm x" and fg: "\<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 "bij f" by (metis fg bijI')
   have "connected (f ` (-S))"
     using connected_linear_image assms \<open>linear f\<close> by blast
   moreover have "f ` (-S) = - (f ` S)"
     by (simp add: \<open>bij f\<close> bij_image_Compl_eq)
   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
 
 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})"
   by (metis connected_uncountable finite.emptyI finite.insertI rev_finite_subset singleton_iff subsetI uncountable_infinite)
 
 lemma connected_finite_iff_sing:
   fixes S :: "'a::metric_space set"
   assumes "connected S"
   shows "finite S \<longleftrightarrow> S = {} \<or> (\<exists>a. S = {a})" 
   using assms connected_uncountable countable_finite by blast
 
 subsection\<^marker>\<open>tag unimportant\<close>\<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 
                 by (rule the_equality [symmetric]) (use x in \<open>auto dest: **\<close>)
             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]
           by (simp add: closed_segment_commute countable_subset)
       qed
       then show ?thesis
         by (force intro: that)
     qed
     show ?thesis
     proof (intro exI conjI)
       have "path_image (linepath a z +++ linepath z b) \<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
     moreover have "countable {a}"
       by simp
     ultimately show ?thesis
       by (metis path_connected_convex_diff_countable [OF \<open>convex S\<close>])
   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 "\<not> collinear (UNIV::'a set)"
     using assms by (auto simp: collinear_aff_dim [of "UNIV :: 'a set"])
   then have "path_connected(UNIV - S)"
     by (simp add: \<open>countable S\<close> path_connected_convex_diff_countable)
   then show ?thesis
     by (simp add: Compl_eq_Diff_UNIV)
 qed
 
 proposition path_connected_openin_diff_countable:
   fixes S :: "'a::euclidean_space set"
   assumes "connected S" and ope: "openin (top_of_set (affine hull S)) S"
       and "\<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 (top_of_set S) U" and "x \<in> U" for U x
     proof -
       have "openin (top_of_set (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 (top_of_set 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> aff_dim (ball x r \<inter> affine hull S) \<le> 1"
         by (metis (no_types, opaque_lifting) 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 "\<not> collinear (ball x r \<inter> affine hull S)"
         by (simp add: collinear_aff_dim)
       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 (top_of_set (affine hull S)) (ball x r \<inter> affine hull S)"
           by (subst inf.commute) (simp add: openin_Int_open)
         then show "openin (top_of_set 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 (top_of_set (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)
 next
   case False
   show ?thesis
   proof (rule path_connected_openin_diff_countable)
     show "openin (top_of_set (affine hull S)) S"
       by (simp add: assms hull_subset open_subset)
     show "\<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\<^marker>\<open>tag unimportant\<close> \<open>Self-homeomorphisms shuffling points about\<close>
 
 subsubsection\<^marker>\<open>tag unimportant\<close>\<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: field_split_simps)
     also have "\<dots> < norm y + (norm x - norm y) * 1"
     proof (subst add_less_cancel_left)
       show "(norm x - norm y) * (norm u / r) < (norm x - norm y) * 1"
       proof (rule mult_strict_left_mono)
         show "norm u / r < 1"
           using \<open>0 < r\<close> divide_less_eq_1_pos nou by blast
       qed (simp add: yx)
     qed
     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"
       using * nou
       apply (clarsimp simp: dist_norm norm_minus_commute f_def)
       by (metis diff_add_eq diff_diff_add diff_diff_eq2 norm_minus_commute)
     moreover have "f ` T \<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
       then have n: "norm (a - (x - v *\<^sub>R (u - a))) = r - r * v"
         by (simp add: field_simps norm_minus_commute)
       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) (simp add: field_simps)
         have "x - v *\<^sub>R (u - a) \<in> cball a r"
           using \<open>r > 0\<close>\<open>0 \<le> v\<close>
           by (simp add: dist_norm n)
         moreover have "x - v *\<^sub>R (u - a) \<in> T"
           by (simp add: f_def \<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 "compact (cball a r \<inter> T)"
     by (simp add: affine_closed compact_Int_closed \<open>affine T\<close>)
   then obtain g where "homeomorphism (cball a r \<inter> T) (cball a r \<inter> T) f g"
     by (metis homeomorphism_compact [OF _ contf fim inj_onf])
   then show thesis
     apply (rule_tac f=f in that)
     using \<open>r > 0\<close> by (simp_all add: f_def dist_norm norm_minus_commute)
 qed
 
 corollary\<^marker>\<open>tag unimportant\<close> 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\<^marker>\<open>tag unimportant\<close> 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"
         unfolding ff_def
         using homeomorphism_cont1 [OF hom] 
         by (intro continuous_on_cases) (auto simp: affine_closed \<open>affine T\<close> fid)
       then show "continuous_on S ff"
         by (rule continuous_on_subset) (use ST in auto)
       have "continuous_on ((cball a r \<inter> T) \<union> (T - ball a r)) gg"
         unfolding gg_def
         using homeomorphism_cont2 [OF hom] 
         by (intro continuous_on_cases) (auto simp: affine_closed \<open>affine T\<close> gid)
       then show "continuous_on S gg"
         by (rule continuous_on_subset) (use ST in 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"
         unfolding ff_def gg_def
         using homeomorphism_apply1 [OF hom] homeomorphism_image1 [OF hom]
         by simp (metis Int_iff homeomorphism_apply1 [OF hom] fid image_eqI less_eq_real_def mem_cball mem_sphere)
       show "\<And>x. x \<in> S \<Longrightarrow> ff (gg x) = x"
         unfolding ff_def gg_def
         using homeomorphism_apply2 [OF hom] homeomorphism_image2 [OF hom]
         by simp (metis Int_iff fid image_eqI less_eq_real_def mem_cball mem_sphere)
     qed
     show "ff u = v"
       using u by (auto simp: ff_def \<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\<^marker>\<open>tag unimportant\<close> homeomorphism_moving_point:
   fixes a :: "'a::euclidean_space"
   assumes ope: "openin (top_of_set (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 (top_of_set 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
       by (rule_tac x="S \<inter> ball d r" in exI) (fastforce simp: openin_open_Int \<open>0 < r\<close> that intro: *)
   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)}"
     by (rule connected_equivalence_relation [OF S]; blast intro: 1 2 3)
   then show ?thesis
     using that by auto
 qed
 
 
 lemma homeomorphism_moving_points_exists_gen:
   assumes K: "finite K" "\<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 (top_of_set (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"
   proof (rule affine_hull_Diff [OF ope])
     show "finite (y ` K)"
       by (simp add: insert.hyps(1))
     show "y ` K \<subset> S"
       using \<open>y i \<in> S\<close> insert.hyps(2) xney xyS by fastforce
   qed
   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 (top_of_set (affine hull (S - y ` K))) (S - y ` K)"
       by (simp add: aff_eq openin_diff finite_imp_closedin image_subset_iff hull_inc insert xyS)
     show "S - y ` K \<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
     have "\<And>k. \<lbrakk>f (x i) = y k; k \<in> K\<rbrakk> \<Longrightarrow> False"
         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)
     then show "f (x i) \<in> S - y ` K"
       by (auto simp: f_in_S \<open>x i \<in> S\<close>)
     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\<^marker>\<open>tag unimportant\<close> 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 (top_of_set (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\<^marker>\<open>tag unimportant\<close>\<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"
       unfolding f_def by (intro continuous_intros)
     have "f ` {a..b} = {c..d}"
       unfolding f_def image_affinity_atLeastAtMost
       using assms sum_sqs_eq by (auto simp: field_split_simps)
     then show "f ` cbox a b = cbox c d"
       by auto
     show "inj_on f (cbox a b)"
       unfolding f_def inj_on_def using assms by auto
   qed auto
   then obtain g where "homeomorphism (cbox a b) (cbox c d) f g" ..
   then show ?thesis
   proof
     show "f a = c"
       by (simp add: f_def)
     show "f b = d"
       using assms sum_sqs_eq [of a b] by (auto simp: f_def field_split_simps)
   qed
 qed
 
 lemma homeomorphism_grouping_point_2:
   fixes a::real and w::real
   assumes hom_ab: "homeomorphism (cbox a b) (cbox u v) f1 g1"
       and hom_bc: "homeomorphism (cbox b c) (cbox v w) f2 g2"
       and "b \<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"
       unfolding f_def using le eq
       by (force intro: continuous_on_cases_le [OF continuous_on_subset [OF cf1] continuous_on_subset [OF cf2]])
     have "f ` cbox a b = f1 ` cbox a b" "f ` cbox b c = f2 ` cbox b c"
       unfolding f_def using eq by force+
     then show "f ` cbox a c = cbox u w"
       unfolding ac uw image_Un by (metis hom_ab hom_bc homeomorphism_def)
     have neq12: "f1 x \<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 by force
   qed auto
   then obtain g where "homeomorphism (cbox a c) (cbox u w) f g" ..
   then show ?thesis
     using eq f_def le that by force
 qed
 
 lemma homeomorphism_grouping_point_3:
   fixes a::real
   assumes cbox_sub: "cbox c d \<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
   proof (rule that [OF fg])
     show "f x \<in> cbox u v" if "x \<in> cbox c d" for x 
       using that f4_eq f_eq homeomorphism_image1 [OF 2]
       by (metis atLeastAtMost_iff box_real(2) image_eqI less(1) less_eq_real_def order_trans)
   qed
 qed
 
 
 lemma homeomorphism_grouping_point_4:
   fixes T :: "real set"
   assumes "open U" "open S" "connected S" "U \<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 (top_of_set (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 "\<And>x. \<lbrakk>w \<le> x \<and> x \<le> z; w < x \<longrightarrow> \<not> x < z\<rbrakk> \<Longrightarrow> f x = x"
         using \<open>f w = w\<close> \<open>f z = z\<close> by auto
       moreover have "\<And>x. \<lbrakk>w \<le> x \<and> x \<le> z; w < x \<longrightarrow> \<not> x < z\<rbrakk> \<Longrightarrow> g x = x"
         using \<open>f w = w\<close> \<open>f z = z\<close> hom homeomorphism_apply1 by fastforce
       ultimately
       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
         by (intro continuous_on_cases_local contfg continuous_on_id clo; auto simp: closed_subset)+
       then show "continuous_on T f'" "continuous_on T g'"
         by (simp_all only: T)
       show "f' ` T \<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)}"
     proof (rule bounded_subset [of "cbox w z"])
       show "bounded (cbox w z)"
         using bounded_cbox by blast
       show "{x. \<not> (f' x = x \<and> g' x = x)} \<subseteq> cbox w z"
         by (auto simp: f'_def g'_def)
     qed
   qed
 qed
 
 proposition\<^marker>\<open>tag unimportant\<close> 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, opaque_lifting) 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, opaque_lifting) 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"
     proof (clarsimp simp: jf jg hj)
       show "f (h x) = h x \<longrightarrow> g (h x) \<noteq> h x \<Longrightarrow> x \<in> S" for x
         using sub [THEN subsetD, of "h x"] hj by simp (metis imageE)
     qed
     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\<^marker>\<open>tag unimportant\<close> homeomorphism_grouping_points_exists_gen:
   fixes S :: "'a::euclidean_space set"
   assumes opeU: "openin (top_of_set S) U"
       and opeS: "openin (top_of_set (affine hull S)) S"
       and "U \<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 (top_of_set (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"
   proof (rule infinite_openin [OF opeU \<open>u \<in> U\<close>])
     show "u islimpt S"
       using True \<open>u \<in> S\<close> assms(8) connected_imp_perfect_aff_dim by fastforce
   qed
   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
       using \<open>K \<subseteq> U\<close> by (intro that [of id id]) (auto intro: homeomorphismI)
   next
     assume "aff_dim S = 1"
     then have "affine hull S homeomorphic (UNIV :: real set)"
       by (auto simp: homeomorphic_affine_sets)
     then obtain h::"'a\<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 (top_of_set (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
     proof (rule continuous_on_subset [OF _ \<open>T \<subseteq> affine hull S\<close>])
       show "continuous_on (affine hull S) h"
         using homeomorphism_def homhj by blast
     qed (meson continuous_on_subset homeomorphism_def homhj top_greatest)
     define f' where "f' \<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)"
           using hom homeomorphism_def by (intro continuous_on_compose cont_hj) 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)"
           using hom homeomorphism_def by (intro continuous_on_compose cont_hj) 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
       have \<section>: "\<And>x y. \<lbrakk>x \<in> affine hull S; h x = h y; y \<in> S\<rbrakk> \<Longrightarrow> x \<in> S"
         by (metis h hull_inc)
       show "{x. \<not> (f' x = x \<and> g' x = x)} \<subseteq> S"
         using sub by (simp add: f'_def g'_def jf jg) (force elim: \<section>)
     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_canon (\<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 less
   then show ?thesis
     by (simp add: homotopic_on_emptyI)
 next
   case equal
   show ?thesis
   proof
     assume L: ?lhs
     with equal have [simp]: "f a \<in> S"
       using homotopic_with_imp_subset1 by fastforce
     obtain h:: "real \<times> 'M \<Rightarrow> 'a" 
       where h: "continuous_on ({0..1} \<times> {a}) h" "h ` ({0..1} \<times> {a}) \<subseteq> S" "h (0, a) = f a"    
       using L equal by (auto simp: homotopic_with)
     then have "continuous_on (cball a r) (\<lambda>x. h (0, a))" "(\<lambda>x. h (0, a)) ` cball a r \<subseteq> S"
       by (auto simp: equal)
     then show ?rhs
       using h(3) local.equal by force
   next
     assume ?rhs
     then show ?lhs
       using equal continuous_on_const by (force simp add: homotopic_with)
   qed
 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_canon (\<lambda>x. True) (sphere a r) S f (\<lambda>x. c)"
     then have contf: "continuous_on (sphere a r) f" 
       by (metis homotopic_with_imp_continuous)
     moreover have fim: "f ` sphere a r \<subseteq> S"
       by (meson continuous_map_subtopology_eu c homotopic_with_imp_continuous_maps)
     show ?P
       using contf fim by (auto simp: sphere_def dist_norm norm_minus_commute)
   qed
   moreover have ?P if ?rhs using that
   proof
     fix g
     assume g: "continuous_on (cball a r) g \<and> g ` cball a r \<subseteq> S \<and> (\<forall>xa\<in>sphere a r. g xa = f xa)"
     then have "f ` {x. norm (x - a) = r} \<subseteq> S"
       using sphere_cball [of a r] unfolding image_subset_iff sphere_def
       by (metis dist_commute dist_norm mem_Collect_eq subset_eq)
     with g show ?P
       by (auto simp: dist_norm norm_minus_commute elim!: continuous_on_eq [OF continuous_on_subset])
   qed
   moreover have ?thesis if ?P
   proof
     assume ?lhs
     then obtain c where "homotopic_with_canon (\<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> h ` ({0..1} \<times> sphere a r)"
     proof
       show "c = h (0, a + r *\<^sub>R b1)"
         by (simp add: h)
       show "(0, a + r *\<^sub>R b1) \<in> {0..1::real} \<times> sphere a r"
         using greater \<open>b1 \<in> Basis\<close> by (auto simp: dist_norm)
     qed
     then have "c \<in> S"
       using him by blast
     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'"
         using greater
         by (force simp: dist_norm norm_minus_commute intro: continuous_on_compose2 [OF conth] continuous_intros)
       then show "continuous_on (cball a r) ?g"
       proof (rule nullhomotopic_from_lemma)
         show "\<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; norm ( x' - x) < d\<rbrakk>
                         \<Longrightarrow> norm (h x' - h x) < e"
             using uniformly_continuous_onE [OF uconth \<open>0 < e\<close>] by (auto simp: dist_norm)
           have *: "norm (h (norm (x - a) / r,
                          a + (r / norm (x - a)) *\<^sub>R (x - a)) - h (0, a + r *\<^sub>R b1)) < e" (is  "norm (?ha - ?hb) < e")
                    if "x \<noteq> a" "norm (x - a) < r" "norm (x - a) < d * r" for x
           proof -
             have "norm (?ha - ?hb) = norm (?ha - h (0, a + (r / norm (x - a)) *\<^sub>R (x - a)))"
               by (simp add: h)
             also have "\<dots> < e"
               using greater \<open>0 < d\<close> \<open>b1 \<in> Basis\<close> that
               by (intro d) (simp_all add: dist_norm, simp add: field_simps) 
             finally show ?thesis .
           qed
           show ?thesis
             using greater \<open>0 < d\<close> 
             by (rule_tac x = "min r (d * r)" in exI) (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"
     proof (rule continuous_on_compose2 [OF contg])
       show "continuous_on ({0..1} \<times> sphere a r) (\<lambda>x. a + fst x *\<^sub>R (snd x - a))"
         by (intro continuous_intros)
       qed (auto simp: dist_norm norm_minus_commute mult_left_le_one_le)
     moreover
     have "?h ` ({0..1} \<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 have "homotopic_with_canon (\<lambda>x. True) (sphere a r) S (\<lambda>x. g a) f"
       by (auto simp: homotopic_with)
     then show ?lhs
       using homotopic_with_symD by blast
   qed
   ultimately
   show ?thesis by meson
 qed 
 
 end
\ No newline at end of file