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
@@ -1,5677 +1,5709 @@
 (*  Title:      HOL/Analysis/Path_Connected.thy
     Authors:    LC Paulson and Robert Himmelmann (TU Muenchen), based on material from HOL Light
 *)
 
 section \<open>Homotopy of Maps\<close>
 
 theory Homotopy
   imports Path_Connected Continuum_Not_Denumerable Product_Topology
 begin
 
 definition\<^marker>\<open>tag important\<close> homotopic_with
 where
  "homotopic_with P X Y f g \<equiv>
    (\<exists>h. continuous_map (prod_topology (subtopology euclideanreal {0..1}) 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)"
   apply (intro continuous_map_compose [OF _ h] continuous_map_id [unfolded id_def] continuous_intros)
   apply (simp add: t)
   done
 
 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
 
 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
   apply (simp add: homotopic_with_def)
   apply (erule ex_forward)
   apply (force simp: o_def dest: continuous_map_o_Pair intro: Q)
   done
 
 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
     where conth: "continuous_map (prod_topology (subtopology euclideanreal {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 (subtopology euclideanreal {0..1}) X) Y ?h"
   proof (rule continuous_map_eq)
     show "continuous_map (prod_topology (subtopology euclideanreal {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)
 
 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)
 
 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"
   apply (simp add: homotopic_with_def)
   apply (fast elim!: continuous_on_subset ex_forward)
   done
 
 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"
   apply (simp add: homotopic_with_def)
   apply (fast elim!: continuous_on_subset ex_forward)
   done
 
 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)
 
 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"
     apply (intro continuous_intros)
     apply (simp_all add: prod_topology_subtopology continuous_map_from_subtopology [OF continuous_map_fst])
     done
   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)
     apply (rename_tac h)
     apply (rule_tac x="h \<circ> (\<lambda>y. (1 - fst y, snd y))" in exI)
     apply (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 (rule fst continuous_map_compose [OF _ contk1] continuous_intros continuous_map_into_subtopology | simp)+
           apply (intro continuous_intros fst continuous_map_from_subtopology)
          apply (force simp: prod_topology_subtopology)
         using continuous_map_snd continuous_map_from_subtopology by blast
       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 (rule fst continuous_map_compose [OF _ contk2] continuous_intros continuous_map_into_subtopology | simp)+
           apply (rule continuous_intros fst continuous_map_from_subtopology | simp)+
          apply (force simp: prod_topology_subtopology)
         using continuous_map_snd  continuous_map_from_subtopology by blast
       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))"
       using P
       apply (clarsimp simp add: k_def)
       apply (case_tac "t \<le> 1/2", auto)
       done
   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
   apply (rename_tac k)
   apply (rule_tac x="h \<circ> k" in exI)
   apply (rule conjI continuous_map_compose | simp add: o_def)+
   done
 
 lemma homotopic_compose_continuous_map_left:
    "\<lbrakk>homotopic_with (\<lambda>k. True) X1 X2 f g; continuous_map X2 X3 h\<rbrakk>
         \<Longrightarrow> homotopic_with (\<lambda>k. True) X1 X3 (h \<circ> f) (h \<circ> g)"
   by (simp add: homotopic_with_compose_continuous_map_left)
 
 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
 
 lemma homotopic_compose_continuous_map_right:
    "\<lbrakk>homotopic_with (\<lambda>k. True) X2 X3 f g; continuous_map X1 X2 h\<rbrakk>
         \<Longrightarrow> homotopic_with (\<lambda>k. True) X1 X3 (f \<circ> h) (g \<circ> h)"
   by (meson homotopic_with_compose_continuous_map_right)
 
 corollary homotopic_compose:
       shows "\<lbrakk>homotopic_with (\<lambda>x. True) X Y f f'; homotopic_with (\<lambda>x. True) Y Z g g'\<rbrakk>
              \<Longrightarrow> homotopic_with (\<lambda>x. True) X Z (g \<circ> f) (g' \<circ> f')"
   apply (rule homotopic_with_trans [where g = "g \<circ> f'"])
   apply (simp add: homotopic_compose_continuous_map_left homotopic_with_imp_continuous_maps)
   by (simp add: homotopic_compose_continuous_map_right homotopic_with_imp_continuous_maps)
 
 
 
 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)
   apply (rename_tac k)
   apply (rule_tac x="k \<circ> (\<lambda>y. (fst y, h (snd y)))" in exI)
   apply (rule conjI continuous_intros continuous_on_compose [where f=snd and g=h, unfolded o_def] | simp)+
   apply (erule continuous_on_subset)
   apply (fastforce simp: o_def)+
   done
 
 proposition homotopic_compose_continuous_right:
      "\<lbrakk>homotopic_with_canon (\<lambda>f. True) X Y f g; continuous_on W h; h ` W \<subseteq> X\<rbrakk>
       \<Longrightarrow> homotopic_with_canon (\<lambda>f. True) W Y (f \<circ> h) (g \<circ> h)"
   using homotopic_with_compose_continuous_right by fastforce
 
 proposition homotopic_with_compose_continuous_left:
      "\<lbrakk>homotopic_with_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)
   apply (rename_tac k)
   apply (rule_tac x="h \<circ> k" in exI)
   apply (rule conjI continuous_intros continuous_on_compose [where f=snd and g=h, unfolded o_def] | simp)+
   apply (erule continuous_on_subset)
   apply (fastforce simp: o_def)+
   done
 
 proposition homotopic_compose_continuous_left:
    "\<lbrakk>homotopic_with_canon (\<lambda>_. True) X Y f g;
      continuous_on Y h; h ` Y \<subseteq> Z\<rbrakk>
     \<Longrightarrow> homotopic_with_canon (\<lambda>f. True) X Z (h \<circ> f) (h \<circ> g)"
   using homotopic_with_compose_continuous_left by fastforce
 
 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
   apply (erule ex_forward)
   by (simp add: continuous_map_from_subtopology prod_topology_subtopology(2))
 
 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)
 
 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)))"
       apply (rule continuous_map_compose [OF _ conth])
       apply (rule continuous_intros c | simp)+
       done
     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 safe
   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
 
 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
       unfolding continuous_map_pairwise case_prod_unfold
       apply (intro conjI that  continuous_intros continuous_map_compose [OF _ h, unfolded o_def])
       using continuous_map_componentwise continuous_map_snd that apply fastforce
       done
     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
       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"
 
 lemma homotopic_paths:
    "homotopic_paths s p q \<longleftrightarrow>
       (\<exists>h. continuous_on ({0..1} \<times> {0..1}) h \<and>
           h ` ({0..1} \<times> {0..1}) \<subseteq> s \<and>
           (\<forall>x \<in> {0..1}. h(0,x) = p x) \<and>
           (\<forall>x \<in> {0..1}. h(1,x) = q x) \<and>
           (\<forall>t \<in> {0..1::real}. pathstart(h \<circ> Pair t) = pathstart p \<and>
                         pathfinish(h \<circ> Pair t) = pathfinish p))"
   by (auto simp: homotopic_paths_def homotopic_with pathstart_def pathfinish_def)
 
 proposition homotopic_paths_imp_pathstart:
      "homotopic_paths s p q \<Longrightarrow> pathstart p = pathstart q"
   by (metis (mono_tags, lifting) homotopic_paths_def homotopic_with_imp_property)
 
 proposition homotopic_paths_imp_pathfinish:
      "homotopic_paths s p q \<Longrightarrow> pathfinish p = pathfinish q"
   by (metis (mono_tags, lifting) homotopic_paths_def homotopic_with_imp_property)
 
 lemma homotopic_paths_imp_path:
      "homotopic_paths s p q \<Longrightarrow> path p \<and> path q"
   using homotopic_paths_def homotopic_with_imp_continuous_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"
   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"
   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"
   by (metis (mono_tags) homotopic_paths_def homotopic_paths_imp_pathfinish homotopic_paths_imp_pathstart homotopic_with_symD)
 
 proposition homotopic_paths_sym_eq: "homotopic_paths s p q \<longleftrightarrow> homotopic_paths s q p"
   by (metis homotopic_paths_sym)
 
 proposition homotopic_paths_trans [trans]:
   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
 
 proposition homotopic_paths_eq:
      "\<lbrakk>path p; path_image p \<subseteq> s; \<And>t. t \<in> {0..1} \<Longrightarrow> p t = q t\<rbrakk> \<Longrightarrow> homotopic_paths s p q"
   apply (simp add: homotopic_paths_def)
   apply (rule homotopic_with_eq)
   apply (auto simp: path_def pathstart_def pathfinish_def path_image_def elim: continuous_on_eq)
   done
 
 proposition homotopic_paths_reparametrize:
   assumes "path p"
       and pips: "path_image p \<subseteq> s"
       and contf: "continuous_on {0..1} f"
       and f01:"f ` {0..1} \<subseteq> {0..1}"
       and [simp]: "f(0) = 0" "f(1) = 1"
       and q: "\<And>t. t \<in> {0..1} \<Longrightarrow> q(t) = p(f t)"
     shows "homotopic_paths s p q"
 proof -
   have contp: "continuous_on {0..1} p"
     by (metis \<open>path p\<close> path_def)
   then have "continuous_on {0..1} (p \<circ> f)"
     using contf continuous_on_compose continuous_on_subset f01 by blast
   then have "path q"
     by (simp add: path_def) (metis q continuous_on_cong)
   have piqs: "path_image q \<subseteq> s"
     by (metis (no_types, hide_lams) pips f01 image_subset_iff path_image_def q)
   have fb0: "\<And>a b. \<lbrakk>0 \<le> a; a \<le> 1; 0 \<le> b; b \<le> 1\<rbrakk> \<Longrightarrow> 0 \<le> (1 - a) * f b + a * b"
     using f01 by force
   have fb1: "\<lbrakk>0 \<le> a; a \<le> 1; 0 \<le> b; b \<le> 1\<rbrakk> \<Longrightarrow> (1 - a) * f b + a * b \<le> 1" for a b
     using f01 [THEN subsetD, of "f b"] by (simp add: convex_bound_le)
   have "homotopic_paths s q p"
   proof (rule homotopic_paths_trans)
     show "homotopic_paths s q (p \<circ> f)"
       using q by (force intro: homotopic_paths_eq [OF  \<open>path q\<close> piqs])
   next
     show "homotopic_paths s (p \<circ> f) p"
       apply (simp add: homotopic_paths_def homotopic_with_def)
       apply (rule_tac x="p \<circ> (\<lambda>y. (1 - (fst y)) *\<^sub>R ((f \<circ> snd) y) + (fst y) *\<^sub>R snd y)"  in exI)
       apply (rule conjI contf continuous_intros continuous_on_subset [OF contp] | simp)+
       using pips [unfolded path_image_def]
       apply (auto simp: fb0 fb1 pathstart_def pathfinish_def)
       done
   qed
   then show ?thesis
     by (simp add: homotopic_paths_sym)
 qed
 
 lemma homotopic_paths_subset: "\<lbrakk>homotopic_paths s p q; s \<subseteq> t\<rbrakk> \<Longrightarrow> homotopic_paths t p q"
   unfolding homotopic_paths by fast
 
 
 text\<open> A slightly ad-hoc but useful lemma in constructing homotopies.\<close>
 lemma homotopic_join_lemma:
   fixes q :: "[real,real] \<Rightarrow> 'a::topological_space"
   assumes p: "continuous_on ({0..1} \<times> {0..1}) (\<lambda>y. p (fst y) (snd y))"
       and q: "continuous_on ({0..1} \<times> {0..1}) (\<lambda>y. q (fst y) (snd y))"
       and pf: "\<And>t. t \<in> {0..1} \<Longrightarrow> pathfinish(p t) = pathstart(q t)"
     shows "continuous_on ({0..1} \<times> {0..1}) (\<lambda>y. (p(fst y) +++ q(fst y)) (snd y))"
 proof -
   have 1: "(\<lambda>y. p (fst y) (2 * snd y)) = (\<lambda>y. p (fst y) (snd y)) \<circ> (\<lambda>y. (fst y, 2 * snd y))"
     by (rule ext) (simp)
   have 2: "(\<lambda>y. q (fst y) (2 * snd y - 1)) = (\<lambda>y. q (fst y) (snd y)) \<circ> (\<lambda>y. (fst y, 2 * snd y - 1))"
     by (rule ext) (simp)
   show ?thesis
     apply (simp add: joinpaths_def)
     apply (rule continuous_on_cases_le)
     apply (simp_all only: 1 2)
     apply (rule continuous_intros continuous_on_subset [OF p] continuous_on_subset [OF q] | force)+
     using pf
     apply (auto simp: mult.commute pathstart_def pathfinish_def)
     done
 qed
 
 text\<open> Congruence properties of homotopy w.r.t. path-combining operations.\<close>
 
 lemma homotopic_paths_reversepath_D:
       assumes "homotopic_paths s p q"
       shows   "homotopic_paths s (reversepath p) (reversepath q)"
   using assms
   apply (simp add: homotopic_paths_def homotopic_with_def, clarify)
   apply (rule_tac x="h \<circ> (\<lambda>x. (fst x, 1 - snd x))" in exI)
   apply (rule conjI continuous_intros)+
   apply (auto simp: reversepath_def pathstart_def pathfinish_def elim!: continuous_on_subset)
   done
 
 proposition homotopic_paths_reversepath:
      "homotopic_paths s (reversepath p) (reversepath q) \<longleftrightarrow> homotopic_paths s p q"
   using homotopic_paths_reversepath_D by force
 
 
 proposition homotopic_paths_join:
     "\<lbrakk>homotopic_paths s p p'; homotopic_paths s q q'; pathfinish p = pathstart q\<rbrakk> \<Longrightarrow> homotopic_paths s (p +++ q) (p' +++ q')"
   apply (simp add: homotopic_paths_def homotopic_with_def, clarify)
   apply (rename_tac k1 k2)
   apply (rule_tac x="(\<lambda>y. ((k1 \<circ> Pair (fst y)) +++ (k2 \<circ> Pair (fst y))) (snd y))" in exI)
   apply (rule conjI continuous_intros homotopic_join_lemma)+
   apply (auto simp: joinpaths_def pathstart_def pathfinish_def path_image_def)
   done
 
 proposition homotopic_paths_continuous_image:
     "\<lbrakk>homotopic_paths s f g; continuous_on s h; h ` s \<subseteq> t\<rbrakk> \<Longrightarrow> homotopic_paths t (h \<circ> f) (h \<circ> g)"
   unfolding homotopic_paths_def
   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:
     "\<lbrakk>path p; path_image p \<subseteq> s\<rbrakk> \<Longrightarrow> homotopic_paths s (p +++ linepath (pathfinish p) (pathfinish p)) p"
   apply (subst homotopic_paths_sym)
   apply (rule homotopic_paths_reparametrize [where f = "\<lambda>t. if  t \<le> 1 / 2 then 2 *\<^sub>R t else 1"])
   apply (simp_all del: le_divide_eq_numeral1)
   apply (subst split_01)
   apply (rule continuous_on_cases continuous_intros | force simp: pathfinish_def joinpaths_def)+
   done
 
 proposition homotopic_paths_lid:
    "\<lbrakk>path p; path_image p \<subseteq> s\<rbrakk> \<Longrightarrow> homotopic_paths s (linepath (pathstart p) (pathstart p) +++ p) p"
   using homotopic_paths_rid [of "reversepath p" s]
   by (metis homotopic_paths_reversepath path_image_reversepath path_reversepath pathfinish_linepath
         pathfinish_reversepath reversepath_joinpaths reversepath_linepath)
 
 proposition homotopic_paths_assoc:
    "\<lbrakk>path p; path_image p \<subseteq> s; path q; path_image q \<subseteq> s; path r; path_image r \<subseteq> s; pathfinish p = pathstart q;
      pathfinish q = pathstart r\<rbrakk>
     \<Longrightarrow> homotopic_paths s (p +++ (q +++ r)) ((p +++ q) +++ r)"
   apply (subst homotopic_paths_sym)
   apply (rule homotopic_paths_reparametrize
            [where f = "\<lambda>t. if  t \<le> 1 / 2 then inverse 2 *\<^sub>R t
                            else if  t \<le> 3 / 4 then t - (1 / 4)
                            else 2 *\<^sub>R t - 1"])
   apply (simp_all del: le_divide_eq_numeral1)
   apply (simp add: subset_path_image_join)
   apply (rule continuous_on_cases_1 continuous_intros)+
   apply (auto simp: joinpaths_def)
   done
 
 proposition homotopic_paths_rinv:
   assumes "path p" "path_image p \<subseteq> s"
     shows "homotopic_paths s (p +++ reversepath p) (linepath (pathstart p) (pathstart p))"
 proof -
   have "continuous_on ({0..1} \<times> {0..1}) (\<lambda>x. (subpath 0 (fst x) p +++ reversepath (subpath 0 (fst x) p)) (snd x))"
     using assms
     apply (simp add: joinpaths_def subpath_def reversepath_def path_def del: le_divide_eq_numeral1)
     apply (rule continuous_on_cases_le)
     apply (rule_tac [2] continuous_on_compose [of _ _ p, unfolded o_def])
     apply (rule continuous_on_compose [of _ _ p, unfolded o_def])
     apply (auto intro!: continuous_intros simp del: eq_divide_eq_numeral1)
     apply (force elim!: continuous_on_subset simp add: mult_le_one)+
     done
   then show ?thesis
     using assms
     apply (subst homotopic_paths_sym_eq)
     unfolding homotopic_paths_def homotopic_with_def
     apply (rule_tac x="(\<lambda>y. (subpath 0 (fst y) p +++ reversepath(subpath 0 (fst y) p)) (snd y))" in exI)
     apply (simp add: path_defs joinpaths_def subpath_def reversepath_def)
     apply (force simp: mult_le_one)
     done
 qed
 
 proposition homotopic_paths_linv:
   assumes "path p" "path_image p \<subseteq> s"
     shows "homotopic_paths s (reversepath p +++ p) (linepath (pathfinish p) (pathfinish p))"
   using homotopic_paths_rinv [of "reversepath p" s] assms by simp
 
 
 subsection\<open>Homotopy of loops without requiring preservation of endpoints\<close>
 
 definition\<^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"
 
 lemma homotopic_loops:
    "homotopic_loops s p q \<longleftrightarrow>
       (\<exists>h. continuous_on ({0..1::real} \<times> {0..1}) h \<and>
           image h ({0..1} \<times> {0..1}) \<subseteq> s \<and>
           (\<forall>x \<in> {0..1}. h(0,x) = p x) \<and>
           (\<forall>x \<in> {0..1}. h(1,x) = q x) \<and>
           (\<forall>t \<in> {0..1}. pathfinish(h \<circ> Pair t) = pathstart(h \<circ> Pair t)))"
   by (simp add: homotopic_loops_def pathstart_def pathfinish_def homotopic_with)
 
 proposition homotopic_loops_imp_loop:
      "homotopic_loops s p q \<Longrightarrow> pathfinish p = pathstart p \<and> pathfinish q = pathstart q"
 using homotopic_with_imp_property homotopic_loops_def by blast
 
 proposition homotopic_loops_imp_path:
      "homotopic_loops s p q \<Longrightarrow> path p \<and> path q"
   unfolding homotopic_loops_def path_def
   using homotopic_with_imp_continuous_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"
   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"
   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"
   by (simp add: homotopic_loops_def homotopic_with_sym)
 
 proposition homotopic_loops_sym_eq: "homotopic_loops s p q \<longleftrightarrow> homotopic_loops s q p"
   by (metis homotopic_loops_sym)
 
 proposition homotopic_loops_trans:
    "\<lbrakk>homotopic_loops s p q; homotopic_loops s q r\<rbrakk> \<Longrightarrow> homotopic_loops s p r"
   unfolding homotopic_loops_def by (blast intro: homotopic_with_trans)
 
 proposition homotopic_loops_subset:
    "\<lbrakk>homotopic_loops s p q; s \<subseteq> t\<rbrakk> \<Longrightarrow> homotopic_loops t p q"
   by (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"
   unfolding homotopic_loops_def
   apply (rule homotopic_with_eq)
   apply (rule homotopic_with_refl [where f = p, THEN iffD2])
   apply (simp_all add: path_image_def path_def pathstart_def pathfinish_def)
   done
 
 proposition homotopic_loops_continuous_image:
    "\<lbrakk>homotopic_loops s f g; continuous_on s h; h ` s \<subseteq> t\<rbrakk> \<Longrightarrow> homotopic_loops t (h \<circ> f) (h \<circ> g)"
   unfolding homotopic_loops_def
   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"
   by (auto simp: homotopic_with_def homotopic_paths_def homotopic_loops_def)
 
 proposition homotopic_loops_imp_homotopic_paths_null:
   assumes "homotopic_loops s p (linepath a a)"
     shows "homotopic_paths s p (linepath (pathstart p) (pathstart p))"
 proof -
   have "path p" by (metis assms homotopic_loops_imp_path)
   have ploop: "pathfinish p = pathstart p" by (metis assms homotopic_loops_imp_loop)
   have pip: "path_image p \<subseteq> s" by (metis assms homotopic_loops_imp_subset)
   obtain h where conth: "continuous_on ({0..1::real} \<times> {0..1}) h"
              and hs: "h ` ({0..1} \<times> {0..1}) \<subseteq> s"
              and [simp]: "\<And>x. x \<in> {0..1} \<Longrightarrow> h(0,x) = p x"
              and [simp]: "\<And>x. x \<in> {0..1} \<Longrightarrow> h(1,x) = a"
              and ends: "\<And>t. t \<in> {0..1} \<Longrightarrow> pathfinish (h \<circ> Pair t) = pathstart (h \<circ> Pair t)"
     using assms by (auto simp: homotopic_loops homotopic_with)
   have conth0: "path (\<lambda>u. h (u, 0))"
     unfolding path_def
     apply (rule continuous_on_compose [of _ _ h, unfolded o_def])
     apply (force intro: continuous_intros continuous_on_subset [OF conth])+
     done
   have pih0: "path_image (\<lambda>u. h (u, 0)) \<subseteq> s"
     using hs by (force simp: path_image_def)
   have c1: "continuous_on ({0..1} \<times> {0..1}) (\<lambda>x. h (fst x * snd x, 0))"
     apply (rule continuous_on_compose [of _ _ h, unfolded o_def])
     apply (force simp: mult_le_one intro: continuous_intros continuous_on_subset [OF conth])+
     done
   have c2: "continuous_on ({0..1} \<times> {0..1}) (\<lambda>x. h (fst x - fst x * snd x, 0))"
     apply (rule continuous_on_compose [of _ _ h, unfolded o_def])
     apply (force simp: mult_left_le mult_le_one intro: continuous_intros continuous_on_subset [OF conth])+
     apply (rule continuous_on_subset [OF conth])
     apply (auto simp: algebra_simps add_increasing2 mult_left_le)
     done
   have [simp]: "\<And>t. \<lbrakk>0 \<le> t \<and> t \<le> 1\<rbrakk> \<Longrightarrow> h (t, 1) = h (t, 0)"
     using ends by (simp add: pathfinish_def pathstart_def)
   have adhoc_le: "c * 4 \<le> 1 + c * (d * 4)" if "\<not> d * 4 \<le> 3" "0 \<le> c" "c \<le> 1" for c d::real
   proof -
     have "c * 3 \<le> c * (d * 4)" using that less_eq_real_def by auto
     with \<open>c \<le> 1\<close> show ?thesis by fastforce
   qed
   have *: "\<And>p x. (path p \<and> path(reversepath p)) \<and>
                   (path_image p \<subseteq> s \<and> path_image(reversepath p) \<subseteq> s) \<and>
                   (pathfinish p = pathstart(linepath a a +++ reversepath p) \<and>
                    pathstart(reversepath p) = a) \<and> pathstart p = x
                   \<Longrightarrow> homotopic_paths s (p +++ linepath a a +++ reversepath p) (linepath x x)"
     by (metis homotopic_paths_lid homotopic_paths_join
               homotopic_paths_trans homotopic_paths_sym homotopic_paths_rinv)
   have 1: "homotopic_paths s p (p +++ linepath (pathfinish p) (pathfinish p))"
     using \<open>path p\<close> homotopic_paths_rid homotopic_paths_sym pip by blast
   moreover have "homotopic_paths s (p +++ linepath (pathfinish p) (pathfinish p))
                                    (linepath (pathstart p) (pathstart p) +++ p +++ linepath (pathfinish p) (pathfinish p))"
     apply (rule homotopic_paths_sym)
     using homotopic_paths_lid [of "p +++ linepath (pathfinish p) (pathfinish p)" s]
     by (metis 1 homotopic_paths_imp_path homotopic_paths_imp_pathstart homotopic_paths_imp_subset)
   moreover have "homotopic_paths s (linepath (pathstart p) (pathstart p) +++ p +++ linepath (pathfinish p) (pathfinish p))
                                    ((\<lambda>u. h (u, 0)) +++ linepath a a +++ reversepath (\<lambda>u. h (u, 0)))"
     apply (simp add: homotopic_paths_def homotopic_with_def)
     apply (rule_tac x="\<lambda>y. (subpath 0 (fst y) (\<lambda>u. h (u, 0)) +++ (\<lambda>u. h (Pair (fst y) u)) +++ subpath (fst y) 0 (\<lambda>u. h (u, 0))) (snd y)" in exI)
     apply (simp add: subpath_reversepath)
     apply (intro conjI homotopic_join_lemma)
     using ploop
     apply (simp_all add: path_defs joinpaths_def o_def subpath_def conth c1 c2)
     apply (force simp: algebra_simps mult_le_one mult_left_le intro: hs [THEN subsetD] adhoc_le)
     done
   moreover have "homotopic_paths s ((\<lambda>u. h (u, 0)) +++ linepath a a +++ reversepath (\<lambda>u. h (u, 0)))
                                    (linepath (pathstart p) (pathstart p))"
     apply (rule *)
     apply (simp add: pih0 pathstart_def pathfinish_def conth0)
     apply (simp add: reversepath_def joinpaths_def)
     done
   ultimately show ?thesis
     by (blast intro: homotopic_paths_trans)
 qed
 
 proposition homotopic_loops_conjugate:
   fixes s :: "'a::real_normed_vector set"
   assumes "path p" "path q" and pip: "path_image p \<subseteq> s" and piq: "path_image q \<subseteq> s"
       and papp: "pathfinish p = pathstart q" and qloop: "pathfinish q = pathstart q"
     shows "homotopic_loops s (p +++ q +++ reversepath p) q"
 proof -
   have contp: "continuous_on {0..1} p"  using \<open>path p\<close> [unfolded path_def] by blast
   have contq: "continuous_on {0..1} q"  using \<open>path q\<close> [unfolded path_def] by blast
   have c1: "continuous_on ({0..1} \<times> {0..1}) (\<lambda>x. p ((1 - fst x) * snd x + fst x))"
     apply (rule continuous_on_compose [of _ _ p, unfolded o_def])
     apply (force simp: mult_le_one intro!: continuous_intros)
     apply (rule continuous_on_subset [OF contp])
     apply (auto simp: algebra_simps add_increasing2 mult_right_le_one_le sum_le_prod1)
     done
   have c2: "continuous_on ({0..1} \<times> {0..1}) (\<lambda>x. p ((fst x - 1) * snd x + 1))"
     apply (rule continuous_on_compose [of _ _ p, unfolded o_def])
     apply (force simp: mult_le_one intro!: continuous_intros)
     apply (rule continuous_on_subset [OF contp])
     apply (auto simp: algebra_simps add_increasing2 mult_left_le_one_le)
     done
   have ps1: "\<And>a b. \<lbrakk>b * 2 \<le> 1; 0 \<le> b; 0 \<le> a; a \<le> 1\<rbrakk> \<Longrightarrow> p ((1 - a) * (2 * b) + a) \<in> s"
     using sum_le_prod1
     by (force simp: algebra_simps add_increasing2 mult_left_le intro: pip [unfolded path_image_def, THEN subsetD])
   have ps2: "\<And>a b. \<lbrakk>\<not> 4 * b \<le> 3; b \<le> 1; 0 \<le> a; a \<le> 1\<rbrakk> \<Longrightarrow> p ((a - 1) * (4 * b - 3) + 1) \<in> s"
     apply (rule pip [unfolded path_image_def, THEN subsetD])
     apply (rule image_eqI, blast)
     apply (simp add: algebra_simps)
     by (metis add_mono_thms_linordered_semiring(1) affine_ineq linear mult.commute mult.left_neutral mult_right_mono not_le
               add.commute zero_le_numeral)
   have qs: "\<And>a b. \<lbrakk>4 * b \<le> 3; \<not> b * 2 \<le> 1\<rbrakk> \<Longrightarrow> q (4 * b - 2) \<in> s"
     using path_image_def piq by fastforce
   have "homotopic_loops s (p +++ q +++ reversepath p)
                           (linepath (pathstart q) (pathstart q) +++ q +++ linepath (pathstart q) (pathstart q))"
     apply (simp add: homotopic_loops_def homotopic_with_def)
     apply (rule_tac x="(\<lambda>y. (subpath (fst y) 1 p +++ q +++ subpath 1 (fst y) p) (snd y))" in exI)
     apply (simp add: subpath_refl subpath_reversepath)
     apply (intro conjI homotopic_join_lemma)
     using papp qloop
     apply (simp_all add: path_defs joinpaths_def o_def subpath_def c1 c2)
     apply (force simp: contq intro: continuous_on_compose [of _ _ q, unfolded o_def] continuous_on_id continuous_on_snd)
     apply (auto simp: ps1 ps2 qs)
     done
   moreover have "homotopic_loops s (linepath (pathstart q) (pathstart q) +++ q +++ linepath (pathstart q) (pathstart q)) q"
   proof -
     have "homotopic_paths s (linepath (pathfinish q) (pathfinish q) +++ q) q"
       using \<open>path q\<close> homotopic_paths_lid qloop piq by auto
     hence 1: "\<And>f. homotopic_paths s f q \<or> \<not> homotopic_paths s f (linepath (pathfinish q) (pathfinish q) +++ q)"
       using homotopic_paths_trans by blast
     hence "homotopic_paths s (linepath (pathfinish q) (pathfinish q) +++ q +++ linepath (pathfinish q) (pathfinish q)) q"
     proof -
       have "homotopic_paths s (q +++ linepath (pathfinish q) (pathfinish q)) q"
         by (simp add: \<open>path q\<close> homotopic_paths_rid piq)
       thus ?thesis
         by (metis (no_types) 1 \<open>path q\<close> homotopic_paths_join homotopic_paths_rinv homotopic_paths_sym
                   homotopic_paths_trans qloop pathfinish_linepath piq)
     qed
     thus ?thesis
       by (metis (no_types) qloop homotopic_loops_sym homotopic_paths_imp_homotopic_loops homotopic_paths_imp_pathfinish homotopic_paths_sym)
   qed
   ultimately show ?thesis
     by (blast intro: homotopic_loops_trans)
 qed
 
 lemma homotopic_paths_loop_parts:
   assumes loops: "homotopic_loops S (p +++ reversepath q) (linepath a a)" and "path q"
   shows "homotopic_paths S p q"
 proof -
   have paths: "homotopic_paths S (p +++ reversepath q) (linepath (pathstart p) (pathstart p))"
     using homotopic_loops_imp_homotopic_paths_null [OF loops] by simp
   then have "path p"
     using \<open>path q\<close> homotopic_loops_imp_path loops path_join path_join_path_ends path_reversepath by blast
   show ?thesis
   proof (cases "pathfinish p = pathfinish q")
     case True
     have pipq: "path_image p \<subseteq> S" "path_image q \<subseteq> S"
       by (metis Un_subset_iff paths \<open>path p\<close> \<open>path q\<close> homotopic_loops_imp_subset homotopic_paths_imp_path loops
            path_image_join path_image_reversepath path_imp_reversepath path_join_eq)+
     have "homotopic_paths S p (p +++ (linepath (pathfinish p) (pathfinish p)))"
       using \<open>path p\<close> \<open>path_image p \<subseteq> S\<close> homotopic_paths_rid homotopic_paths_sym by blast
     moreover have "homotopic_paths S (p +++ (linepath (pathfinish p) (pathfinish p))) (p +++ (reversepath q +++ q))"
       by (simp add: True \<open>path p\<close> \<open>path q\<close> pipq homotopic_paths_join homotopic_paths_linv homotopic_paths_sym)
     moreover have "homotopic_paths S (p +++ (reversepath q +++ q)) ((p +++ reversepath q) +++ q)"
       by (simp add: True \<open>path p\<close> \<open>path q\<close> homotopic_paths_assoc pipq)
     moreover have "homotopic_paths S ((p +++ reversepath q) +++ q) (linepath (pathstart p) (pathstart p) +++ q)"
       by (simp add: \<open>path q\<close> homotopic_paths_join paths pipq)
     moreover then have "homotopic_paths S (linepath (pathstart p) (pathstart p) +++ q) q"
       by (metis \<open>path q\<close> homotopic_paths_imp_path homotopic_paths_lid linepath_trivial path_join_path_ends pathfinish_def pipq(2))
     ultimately show ?thesis
       using homotopic_paths_trans by metis
   next
     case False
     then show ?thesis
       using \<open>path q\<close> homotopic_loops_imp_path loops path_join_path_ends by fastforce
   qed
 qed
 
 
 subsection\<^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"
   apply (simp add: homotopic_with_def)
   apply (rule_tac x="\<lambda>y. ((1 - (fst y)) *\<^sub>R f(snd y) + (fst y) *\<^sub>R g(snd y))" in exI)
   apply (intro conjI)
   apply (rule subset_refl continuous_intros continuous_on_subset [OF contf] continuous_on_compose2 [where g=f]
                                             continuous_on_subset [OF contg] continuous_on_compose2 [where g=g]| simp)+
   using sub closed_segment_def apply fastforce+
   done
 
 lemma homotopic_paths_linear:
   fixes g h :: "real \<Rightarrow> 'a::real_normed_vector"
   assumes "path g" "path h" "pathstart h = pathstart g" "pathfinish h = pathfinish g"
           "\<And>t. t \<in> {0..1} \<Longrightarrow> closed_segment (g t) (h t) \<subseteq> s"
     shows "homotopic_paths s g h"
   using assms
   unfolding path_def
   apply (simp add: closed_segment_def pathstart_def pathfinish_def homotopic_paths_def homotopic_with_def)
   apply (rule_tac x="\<lambda>y. ((1 - (fst y)) *\<^sub>R (g \<circ> snd) y + (fst y) *\<^sub>R (h \<circ> snd) y)" in exI)
   apply (intro conjI subsetI continuous_intros; force)
   done
 
 lemma homotopic_loops_linear:
   fixes g h :: "real \<Rightarrow> 'a::real_normed_vector"
   assumes "path g" "path h" "pathfinish g = pathstart g" "pathfinish h = pathstart h"
           "\<And>t x. t \<in> {0..1} \<Longrightarrow> closed_segment (g t) (h t) \<subseteq> s"
     shows "homotopic_loops s g h"
   using assms
   unfolding path_def
   apply (simp add: pathstart_def pathfinish_def homotopic_loops_def homotopic_with_def)
   apply (rule_tac x="\<lambda>y. ((1 - (fst y)) *\<^sub>R g(snd y) + (fst y) *\<^sub>R h(snd y))" in exI)
   apply (auto intro!: continuous_intros intro: continuous_on_compose2 [where g=g] continuous_on_compose2 [where g=h])
   apply (force simp: closed_segment_def)
   done
 
 lemma homotopic_paths_nearby_explicit:
   assumes "path g" "path h" "pathstart h = pathstart g" "pathfinish h = pathfinish g"
       and no: "\<And>t x. \<lbrakk>t \<in> {0..1}; x \<notin> s\<rbrakk> \<Longrightarrow> norm(h t - g t) < norm(g t - x)"
     shows "homotopic_paths s g h"
   apply (rule homotopic_paths_linear [OF assms(1-4)])
   by (metis no segment_bound(1) subsetI norm_minus_commute not_le)
 
 lemma homotopic_loops_nearby_explicit:
   assumes "path g" "path h" "pathfinish g = pathstart g" "pathfinish h = pathstart h"
       and no: "\<And>t x. \<lbrakk>t \<in> {0..1}; x \<notin> s\<rbrakk> \<Longrightarrow> norm(h t - g t) < norm(g t - x)"
     shows "homotopic_loops s g h"
   apply (rule homotopic_loops_linear [OF assms(1-4)])
   by (metis no segment_bound(1) subsetI norm_minus_commute not_le)
 
 lemma homotopic_nearby_paths:
   fixes g h :: "real \<Rightarrow> 'a::euclidean_space"
   assumes "path g" "open s" "path_image g \<subseteq> s"
     shows "\<exists>e. 0 < e \<and>
                (\<forall>h. path h \<and>
                     pathstart h = pathstart g \<and> pathfinish h = pathfinish g \<and>
                     (\<forall>t \<in> {0..1}. norm(h t - g t) < e) \<longrightarrow> homotopic_paths s g h)"
 proof -
   obtain e where "e > 0" and e: "\<And>x y. x \<in> path_image g \<Longrightarrow> y \<in> - s \<Longrightarrow> e \<le> dist x y"
     using separate_compact_closed [of "path_image g" "-s"] assms by force
   show ?thesis
     apply (intro exI conjI)
     using e [unfolded dist_norm]
     apply (auto simp: intro!: homotopic_paths_nearby_explicit assms  \<open>e > 0\<close>)
     by (metis atLeastAtMost_iff imageI le_less_trans not_le path_image_def)
 qed
 
 lemma homotopic_nearby_loops:
   fixes g h :: "real \<Rightarrow> 'a::euclidean_space"
   assumes "path g" "open s" "path_image g \<subseteq> s" "pathfinish g = pathstart g"
     shows "\<exists>e. 0 < e \<and>
                (\<forall>h. path h \<and> pathfinish h = pathstart h \<and>
                     (\<forall>t \<in> {0..1}. norm(h t - g t) < e) \<longrightarrow> homotopic_loops s g h)"
 proof -
   obtain e where "e > 0" and e: "\<And>x y. x \<in> path_image g \<Longrightarrow> y \<in> - s \<Longrightarrow> e \<le> dist x y"
     using separate_compact_closed [of "path_image g" "-s"] assms by force
   show ?thesis
     apply (intro exI conjI)
     using e [unfolded dist_norm]
     apply (auto simp: intro!: homotopic_loops_nearby_explicit assms  \<open>e > 0\<close>)
     by (metis atLeastAtMost_iff imageI le_less_trans not_le path_image_def)
 qed
 
 
 subsection\<open> Homotopy and subpaths\<close>
 
 lemma homotopic_join_subpaths1:
   assumes "path g" and pag: "path_image g \<subseteq> s"
       and u: "u \<in> {0..1}" and v: "v \<in> {0..1}" and w: "w \<in> {0..1}" "u \<le> v" "v \<le> w"
     shows "homotopic_paths s (subpath u v g +++ subpath v w g) (subpath u w g)"
 proof -
   have 1: "t * 2 \<le> 1 \<Longrightarrow> u + t * (v * 2) \<le> v + t * (u * 2)" for t
     using affine_ineq \<open>u \<le> v\<close> by fastforce
   have 2: "t * 2 > 1 \<Longrightarrow> u + (2*t - 1) * v \<le> v + (2*t - 1) * w" for t
     by (metis add_mono_thms_linordered_semiring(1) diff_gt_0_iff_gt less_eq_real_def mult.commute mult_right_mono \<open>u \<le> v\<close> \<open>v \<le> w\<close>)
   have t2: "\<And>t::real. t*2 = 1 \<Longrightarrow> t = 1/2" by auto
   show ?thesis
     apply (rule homotopic_paths_subset [OF _ pag])
     using assms
     apply (cases "w = u")
     using homotopic_paths_rinv [of "subpath u v g" "path_image g"]
     apply (force simp: closed_segment_eq_real_ivl image_mono path_image_def subpath_refl)
       apply (rule homotopic_paths_sym)
       apply (rule homotopic_paths_reparametrize
              [where f = "\<lambda>t. if  t \<le> 1 / 2
                              then inverse((w - u)) *\<^sub>R (2 * (v - u)) *\<^sub>R t
                              else inverse((w - u)) *\<^sub>R ((v - u) + (w - v) *\<^sub>R (2 *\<^sub>R t - 1))"])
       using \<open>path g\<close> path_subpath u w apply blast
       using \<open>path g\<close> path_image_subpath_subset u w(1) apply blast
       apply simp_all
       apply (subst split_01)
       apply (rule continuous_on_cases continuous_intros | force simp: pathfinish_def joinpaths_def)+
       apply (simp_all add: field_simps not_le)
       apply (force dest!: t2)
       apply (force simp: algebra_simps mult_left_mono affine_ineq dest!: 1 2)
       apply (simp add: joinpaths_def subpath_def)
       apply (force simp: algebra_simps)
       done
 qed
 
 lemma homotopic_join_subpaths2:
   assumes "homotopic_paths s (subpath u v g +++ subpath v w g) (subpath u w g)"
     shows "homotopic_paths s (subpath w v g +++ subpath v u g) (subpath w u g)"
 by (metis assms homotopic_paths_reversepath_D pathfinish_subpath pathstart_subpath reversepath_joinpaths reversepath_subpath)
 
 lemma homotopic_join_subpaths3:
   assumes hom: "homotopic_paths s (subpath u v g +++ subpath v w g) (subpath u w g)"
       and "path g" and pag: "path_image g \<subseteq> s"
       and u: "u \<in> {0..1}" and v: "v \<in> {0..1}" and w: "w \<in> {0..1}"
     shows "homotopic_paths s (subpath v w g +++ subpath w u g) (subpath v u g)"
 proof -
   have "homotopic_paths s (subpath u w g +++ subpath w v g) ((subpath u v g +++ subpath v w g) +++ subpath w v g)"
     apply (rule homotopic_paths_join)
     using hom homotopic_paths_sym_eq apply blast
     apply (metis \<open>path g\<close> homotopic_paths_eq pag path_image_subpath_subset path_subpath subset_trans v w, simp)
     done
   also have "homotopic_paths s ((subpath u v g +++ subpath v w g) +++ subpath w v g) (subpath u v g +++ subpath v w g +++ subpath w v g)"
     apply (rule homotopic_paths_sym [OF homotopic_paths_assoc])
     using assms by (simp_all add: path_image_subpath_subset [THEN order_trans])
   also have "homotopic_paths s (subpath u v g +++ subpath v w g +++ subpath w v g)
                                (subpath u v g +++ linepath (pathfinish (subpath u v g)) (pathfinish (subpath u v g)))"
     apply (rule homotopic_paths_join)
     apply (metis \<open>path g\<close> homotopic_paths_eq order.trans pag path_image_subpath_subset path_subpath u v)
     apply (metis (no_types, lifting) \<open>path g\<close> homotopic_paths_linv order_trans pag path_image_subpath_subset path_subpath pathfinish_subpath reversepath_subpath v w)
     apply simp
     done
   also have "homotopic_paths s (subpath u v g +++ linepath (pathfinish (subpath u v g)) (pathfinish (subpath u v g))) (subpath u v g)"
     apply (rule homotopic_paths_rid)
     using \<open>path g\<close> path_subpath u v apply blast
     apply (meson \<open>path g\<close> order.trans pag path_image_subpath_subset u v)
     done
   finally have "homotopic_paths s (subpath u w g +++ subpath w v g) (subpath u v g)" .
   then show ?thesis
     using homotopic_join_subpaths2 by blast
 qed
 
 proposition homotopic_join_subpaths:
    "\<lbrakk>path g; path_image g \<subseteq> s; u \<in> {0..1}; v \<in> {0..1}; w \<in> {0..1}\<rbrakk>
     \<Longrightarrow> homotopic_paths s (subpath u v g +++ subpath v w g) (subpath u w g)"
   apply (rule le_cases3 [of u v w])
 using homotopic_join_subpaths1 homotopic_join_subpaths2 homotopic_join_subpaths3 by metis+
 
 text\<open>Relating homotopy of trivial loops to path-connectedness.\<close>
 
 lemma path_component_imp_homotopic_points:
     "path_component S a b \<Longrightarrow> homotopic_loops S (linepath a a) (linepath b b)"
 apply (simp add: path_component_def homotopic_loops_def homotopic_with_def
                  pathstart_def pathfinish_def path_image_def path_def, clarify)
 apply (rule_tac x="g \<circ> fst" in exI)
 apply (intro conjI continuous_intros continuous_on_compose)+
 apply (auto elim!: continuous_on_subset)
 done
 
 lemma homotopic_loops_imp_path_component_value:
    "\<lbrakk>homotopic_loops S p q; 0 \<le> t; t \<le> 1\<rbrakk>
         \<Longrightarrow> path_component S (p t) (q t)"
 apply (simp add: path_component_def homotopic_loops_def homotopic_with_def
                  pathstart_def pathfinish_def path_image_def path_def, clarify)
 apply (rule_tac x="h \<circ> (\<lambda>u. (u, t))" in exI)
 apply (intro conjI continuous_intros continuous_on_compose)+
 apply (auto elim!: continuous_on_subset)
 done
 
 lemma homotopic_points_eq_path_component:
    "homotopic_loops S (linepath a a) (linepath b b) \<longleftrightarrow>
         path_component S a b"
 by (auto simp: path_component_imp_homotopic_points
          dest: homotopic_loops_imp_path_component_value [where t=1])
 
 lemma path_connected_eq_homotopic_points:
     "path_connected S \<longleftrightarrow>
       (\<forall>a b. a \<in> S \<and> b \<in> S \<longrightarrow> homotopic_loops S (linepath a a) (linepath b b))"
 by (auto simp: path_connected_def path_component_def homotopic_points_eq_path_component)
 
 
 subsection\<open>Simply connected sets\<close>
 
 text\<^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))"
 apply (simp add: simply_connected_def)
 apply (rule iffI, force, clarify)
 apply (rule_tac q = "linepath (pathstart p) (pathstart p)" in homotopic_loops_trans)
 apply (fastforce simp add:)
 using homotopic_loops_sym apply blast
 done
 
 lemma simply_connected_eq_contractible_loop_some:
   fixes S :: "_::real_normed_vector set"
   shows "simply_connected S \<longleftrightarrow>
                 path_connected S \<and>
                 (\<forall>p. path p \<and> path_image p \<subseteq> S \<and> pathfinish p = pathstart p
                     \<longrightarrow> (\<exists>a. a \<in> S \<and> homotopic_loops S p (linepath a a)))"
 apply (rule iffI)
  apply (fastforce simp: simply_connected_imp_path_connected simply_connected_eq_contractible_loop_any)
 apply (clarsimp simp add: simply_connected_eq_contractible_loop_any)
 apply (drule_tac x=p in spec)
 using homotopic_loops_trans path_connected_eq_homotopic_points
   apply blast
 done
 
 lemma simply_connected_eq_contractible_loop_all:
   fixes S :: "_::real_normed_vector set"
   shows "simply_connected S \<longleftrightarrow>
          S = {} \<or>
          (\<exists>a \<in> S. \<forall>p. path p \<and> path_image p \<subseteq> S \<and> pathfinish p = pathstart p
                 \<longrightarrow> homotopic_loops S p (linepath a a))"
         (is "?lhs = ?rhs")
 proof (cases "S = {}")
   case True then show ?thesis by force
 next
   case False
   then obtain a where "a \<in> S" by blast
   show ?thesis
   proof
     assume "simply_connected S"
     then show ?rhs
       using \<open>a \<in> S\<close> \<open>simply_connected S\<close> simply_connected_eq_contractible_loop_any
       by blast
   next
     assume ?rhs
     then show "simply_connected S"
       apply (simp add: simply_connected_eq_contractible_loop_any False)
       by (meson homotopic_loops_refl homotopic_loops_sym homotopic_loops_trans
              path_component_imp_homotopic_points path_component_refl)
   qed
 qed
 
 lemma simply_connected_eq_contractible_path:
   fixes S :: "_::real_normed_vector set"
   shows "simply_connected S \<longleftrightarrow>
            path_connected S \<and>
            (\<forall>p. path p \<and> path_image p \<subseteq> S \<and> pathfinish p = pathstart p
             \<longrightarrow> homotopic_paths S p (linepath (pathstart p) (pathstart p)))"
 apply (rule iffI)
  apply (simp add: simply_connected_imp_path_connected)
  apply (metis simply_connected_eq_contractible_loop_some homotopic_loops_imp_homotopic_paths_null)
 by (meson homotopic_paths_imp_homotopic_loops pathfinish_linepath pathstart_in_path_image
          simply_connected_eq_contractible_loop_some subset_iff)
 
 lemma simply_connected_eq_homotopic_paths:
   fixes S :: "_::real_normed_vector set"
   shows "simply_connected S \<longleftrightarrow>
           path_connected S \<and>
           (\<forall>p q. path p \<and> path_image p \<subseteq> S \<and>
                 path q \<and> path_image q \<subseteq> S \<and>
                 pathstart q = pathstart p \<and> pathfinish q = pathfinish p
                 \<longrightarrow> homotopic_paths S p q)"
          (is "?lhs = ?rhs")
 proof
   assume ?lhs
   then have pc: "path_connected S"
         and *:  "\<And>p. \<lbrakk>path p; path_image p \<subseteq> S;
                        pathfinish p = pathstart p\<rbrakk>
                       \<Longrightarrow> homotopic_paths S p (linepath (pathstart p) (pathstart p))"
     by (auto simp: simply_connected_eq_contractible_path)
   have "homotopic_paths S p q"
         if "path p" "path_image p \<subseteq> S" "path q"
            "path_image q \<subseteq> S" "pathstart q = pathstart p"
            "pathfinish q = pathfinish p" for p q
   proof -
     have "homotopic_paths S p (p +++ linepath (pathfinish p) (pathfinish p))"
       by (simp add: homotopic_paths_rid homotopic_paths_sym that)
     also have "homotopic_paths S (p +++ linepath (pathfinish p) (pathfinish p))
                                  (p +++ reversepath q +++ q)"
       using that
       by (metis homotopic_paths_join homotopic_paths_linv homotopic_paths_refl homotopic_paths_sym_eq pathstart_linepath)
     also have "homotopic_paths S (p +++ reversepath q +++ q)
                                  ((p +++ reversepath q) +++ q)"
       by (simp add: that homotopic_paths_assoc)
     also have "homotopic_paths S ((p +++ reversepath q) +++ q)
                                  (linepath (pathstart q) (pathstart q) +++ q)"
       using * [of "p +++ reversepath q"] that
       by (simp add: homotopic_paths_join path_image_join)
     also have "homotopic_paths S (linepath (pathstart q) (pathstart q) +++ q) q"
       using that homotopic_paths_lid by blast
     finally show ?thesis .
   qed
   then show ?rhs
     by (blast intro: pc *)
 next
   assume ?rhs
   then show ?lhs
     by (force simp: simply_connected_eq_contractible_path)
 qed
 
 proposition simply_connected_Times:
   fixes S :: "'a::real_normed_vector set" and T :: "'b::real_normed_vector set"
   assumes S: "simply_connected S" and T: "simply_connected T"
     shows "simply_connected(S \<times> T)"
 proof -
   have "homotopic_loops (S \<times> T) p (linepath (a, b) (a, b))"
        if "path p" "path_image p \<subseteq> S \<times> T" "p 1 = p 0" "a \<in> S" "b \<in> T"
        for p a b
   proof -
     have "path (fst \<circ> p)"
       apply (rule Path_Connected.path_continuous_image [OF \<open>path p\<close>])
       apply (rule continuous_intros)+
       done
     moreover have "path_image (fst \<circ> p) \<subseteq> S"
       using that apply (simp add: path_image_def) by force
     ultimately have p1: "homotopic_loops S (fst \<circ> p) (linepath a a)"
       using S that
       apply (simp add: simply_connected_eq_contractible_loop_any)
       apply (drule_tac x="fst \<circ> p" in spec)
       apply (drule_tac x=a in spec)
       apply (auto simp: pathstart_def pathfinish_def)
       done
     have "path (snd \<circ> p)"
       apply (rule Path_Connected.path_continuous_image [OF \<open>path p\<close>])
       apply (rule continuous_intros)+
       done
     moreover have "path_image (snd \<circ> p) \<subseteq> T"
       using that apply (simp add: path_image_def) by force
     ultimately have p2: "homotopic_loops T (snd \<circ> p) (linepath b b)"
       using T that
       apply (simp add: simply_connected_eq_contractible_loop_any)
       apply (drule_tac x="snd \<circ> p" in spec)
       apply (drule_tac x=b in spec)
       apply (auto simp: pathstart_def pathfinish_def)
       done
     show ?thesis
       using p1 p2
       apply (simp add: homotopic_loops, clarify)
       apply (rename_tac h k)
       apply (rule_tac x="\<lambda>z. Pair (h z) (k z)" in exI)
       apply (intro conjI continuous_intros | assumption)+
       apply (auto simp: pathstart_def pathfinish_def)
       done
   qed
   with assms show ?thesis
     by (simp add: simply_connected_eq_contractible_loop_any pathfinish_def pathstart_def)
 qed
 
 
 subsection\<open>Contractible sets\<close>
 
 definition\<^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)
   show ?thesis
     apply (simp add: simply_connected_eq_contractible_loop_all False)
     apply (rule bexI [OF _ \<open>a \<in> S\<close>])
     using a apply (simp add: homotopic_loops_def homotopic_with_def path_def path_image_def pathfinish_def pathstart_def, clarify)
     apply (rule_tac x="(h \<circ> (\<lambda>y. (fst y, (p \<circ> snd) y)))" in exI)
     apply (intro conjI continuous_on_compose continuous_intros)
     apply (erule continuous_on_subset | force)+
     done
 qed
 
 corollary contractible_imp_connected:
   fixes S :: "_::real_normed_vector set"
   shows "contractible S \<Longrightarrow> connected S"
 by (simp add: contractible_imp_simply_connected simply_connected_imp_connected)
 
 lemma contractible_imp_path_connected:
   fixes S :: "_::real_normed_vector set"
   shows "contractible S \<Longrightarrow> path_connected S"
 by (simp add: contractible_imp_simply_connected simply_connected_imp_path_connected)
 
 lemma nullhomotopic_through_contractible:
   fixes S :: "_::topological_space set"
   assumes f: "continuous_on S f" "f ` S \<subseteq> T"
       and g: "continuous_on T g" "g ` T \<subseteq> U"
       and T: "contractible T"
     obtains c where "homotopic_with_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_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)"
 apply (rule nullhomotopic_through_contractible [OF f, of id T])
 using assms
 apply (auto)
 done
 
 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)"
 apply (rule nullhomotopic_through_contractible [OF continuous_on_id _ f S, of S])
 using assms
 apply (auto simp: comp_def)
 done
 
 lemma homotopic_through_contractible:
   fixes S :: "_::real_normed_vector set"
   assumes "continuous_on S f1" "f1 ` S \<subseteq> T"
           "continuous_on T g1" "g1 ` T \<subseteq> U"
           "continuous_on S f2" "f2 ` S \<subseteq> T"
           "continuous_on T g2" "g2 ` T \<subseteq> U"
           "contractible T" "path_connected U"
    shows "homotopic_with_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)"
     apply (rule nullhomotopic_through_contractible [of S f1 T g1 U])
     using assms apply auto
     done
   obtain c2 where c2: "homotopic_with_canon (\<lambda>h. True) S U (g2 \<circ> f2) (\<lambda>x. c2)"
     apply (rule nullhomotopic_through_contractible [of S f2 T g2 U])
     using assms apply auto
     done
   have *: "S = {} \<or> (\<exists>t. path_connected t \<and> t \<subseteq> U \<and> c2 \<in> t \<and> c1 \<in> t)"
   proof (cases "S = {}")
     case True then show ?thesis by force
   next
     case False
     with c1 c2 have "c1 \<in> U" "c2 \<in> U"
       using homotopic_with_imp_continuous_maps by fastforce+
     with \<open>path_connected U\<close> show ?thesis by blast
   qed
   show ?thesis
     apply (rule homotopic_with_trans [OF c1])
     apply (rule homotopic_with_symD)
     apply (rule homotopic_with_trans [OF c2])
     apply (simp add: path_component homotopic_constant_maps *)
     done
 qed
 
 lemma homotopic_into_contractible:
   fixes S :: "'a::real_normed_vector set" and T:: "'b::real_normed_vector set"
   assumes f: "continuous_on S f" "f ` S \<subseteq> T"
       and g: "continuous_on S g" "g ` S \<subseteq> T"
       and T: "contractible T"
     shows "homotopic_with_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"
+    apply (rule rel_interior_closure_convex_segment [OF \<open>convex S\<close> a])
+    using assms by blast
+  show ?thesis
+    unfolding starlike_def
+    apply (rule bexI [OF _ \<open>a \<in> T\<close>])
+    apply (simp add: closed_segment_eq_open)
+    apply (intro conjI ballI a \<open>a \<in> T\<close> rel_interior_closure_convex_segment [OF \<open>convex S\<close> a])
+    apply (simp add: order_trans [OF * ST])
+    done
+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 "(\<lambda>y. (1 - fst y) *\<^sub>R snd y + fst y *\<^sub>R a) ` ({0..1} \<times> S) \<subseteq> S"
     apply clarify
     apply (erule a [unfolded closed_segment_def, THEN subsetD], simp)
     apply (metis add_diff_cancel_right' diff_ge_0_iff_ge le_add_diff_inverse pth_c(1))
     done
   then show ?thesis
     apply (rule_tac a=a in that)
     using \<open>a \<in> S\<close>
     apply (simp add: homotopic_with_def)
     apply (rule_tac x="\<lambda>y. (1 - (fst y)) *\<^sub>R snd y + (fst y) *\<^sub>R a" in exI)
     apply (intro conjI ballI continuous_on_compose continuous_intros)
     apply (simp_all add: P)
     done
 qed
 
 lemma starlike_imp_contractible:
   fixes S :: "'a::real_normed_vector set"
   shows "starlike S \<Longrightarrow> contractible S"
 using starlike_imp_contractible_gen contractible_def by (fastforce simp: id_def)
 
 lemma contractible_UNIV [simp]: "contractible (UNIV :: 'a::real_normed_vector set)"
   by (simp add: starlike_imp_contractible)
 
 lemma starlike_imp_simply_connected:
   fixes S :: "'a::real_normed_vector set"
   shows "starlike S \<Longrightarrow> simply_connected S"
 by (simp add: contractible_imp_simply_connected starlike_imp_contractible)
 
 lemma convex_imp_simply_connected:
   fixes S :: "'a::real_normed_vector set"
   shows "convex S \<Longrightarrow> simply_connected S"
 using convex_imp_starlike starlike_imp_simply_connected by blast
 
 lemma starlike_imp_path_connected:
   fixes S :: "'a::real_normed_vector set"
   shows "starlike S \<Longrightarrow> path_connected S"
 by (simp add: simply_connected_imp_path_connected starlike_imp_simply_connected)
 
 lemma starlike_imp_connected:
   fixes S :: "'a::real_normed_vector set"
   shows "starlike S \<Longrightarrow> connected S"
 by (simp add: path_connected_imp_connected starlike_imp_path_connected)
 
 lemma is_interval_simply_connected_1:
   fixes S :: "real set"
   shows "is_interval S \<longleftrightarrow> simply_connected S"
 using convex_imp_simply_connected is_interval_convex_1 is_interval_path_connected_1 simply_connected_imp_path_connected by auto
 
 lemma contractible_empty [simp]: "contractible {}"
   by (simp add: contractible_def homotopic_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
     apply (rule starlike_imp_contractible)
     apply (rule starlike_convex_tweak_boundary_points [OF \<open>convex S\<close> False TS])
     done
 qed
 
 lemma convex_imp_contractible:
   fixes S :: "'a::real_normed_vector set"
   shows "convex S \<Longrightarrow> contractible S"
   using contractible_empty convex_imp_starlike starlike_imp_contractible by blast
 
 lemma contractible_sing [simp]:
   fixes a :: "'a::real_normed_vector"
   shows "contractible {a}"
 by (rule convex_imp_contractible [OF convex_singleton])
 
 lemma is_interval_contractible_1:
   fixes S :: "real set"
   shows  "is_interval S \<longleftrightarrow> contractible S"
 using contractible_imp_simply_connected convex_imp_contractible is_interval_convex_1
       is_interval_simply_connected_1 by auto
 
 lemma contractible_Times:
   fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set"
   assumes S: "contractible S" and T: "contractible T"
   shows "contractible (S \<times> T)"
 proof -
   obtain a h where conth: "continuous_on ({0..1} \<times> S) h"
              and hsub: "h ` ({0..1} \<times> S) \<subseteq> S"
              and [simp]: "\<And>x. x \<in> S \<Longrightarrow> h (0, x) = x"
              and [simp]: "\<And>x. x \<in> S \<Longrightarrow>  h (1::real, x) = a"
     using S by (auto simp: contractible_def homotopic_with)
   obtain b k where contk: "continuous_on ({0..1} \<times> T) k"
              and ksub: "k ` ({0..1} \<times> T) \<subseteq> T"
              and [simp]: "\<And>x. x \<in> T \<Longrightarrow> k (0, x) = x"
              and [simp]: "\<And>x. x \<in> T \<Longrightarrow>  k (1::real, x) = b"
     using T by (auto simp: contractible_def homotopic_with)
   show ?thesis
     apply (simp add: contractible_def homotopic_with)
     apply (rule exI [where x=a])
     apply (rule exI [where x=b])
     apply (rule exI [where x = "\<lambda>z. (h (fst z, fst(snd z)), k (fst z, snd(snd z)))"])
     apply (intro conjI ballI continuous_intros continuous_on_compose2 [OF conth] continuous_on_compose2 [OF contk])
     using hsub ksub
     apply auto
     done
 qed
 
 
 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
 apply (simp add: locally_def)
 apply (erule all_forward)+
 apply (rule impI)
 apply (erule impCE)
  using openin_trans apply blast
 apply (erule ex_forward)
 by (metis (no_types, hide_lams) Int_absorb1 Int_lower1 Int_subset_iff openin_open openin_subtopology_Int_subset)
 
 lemma locally_diff_closed:
     "\<lbrakk>locally P S; closedin (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}"
 apply (simp add: locally_def openin_euclidean_subtopology_iff subset_singleton_iff conj_disj_distribR cong: conj_cong)
 using zero_less_one by blast
 
 lemma locally_iff:
     "locally P S \<longleftrightarrow>
      (\<forall>T x. open T \<and> x \<in> S \<inter> T \<longrightarrow> (\<exists>U. open U \<and> (\<exists>v. P v \<and> x \<in> S \<inter> U \<and> S \<inter> U \<subseteq> v \<and> v \<subseteq> S \<inter> T)))"
 apply (simp add: le_inf_iff locally_def openin_open, safe)
 apply (metis IntE IntI le_inf_iff)
 apply (metis IntI Int_subset_iff)
 done
 
 lemma locally_Int:
   assumes S: "locally P S" and t: "locally P t"
       and P: "\<And>S t. P S \<and> P t \<Longrightarrow> P(S \<inter> t)"
     shows "locally P (S \<inter> t)"
 using S t unfolding locally_iff
 apply clarify
 apply (drule_tac x=T in spec)+
 apply (drule_tac x=x in spec)+
 apply clarsimp
 apply (rename_tac U1 U2 V1 V2)
 apply (rule_tac x="U1 \<inter> U2" in exI)
 apply (simp add: open_Int)
 apply (rule_tac x="V1 \<inter> V2" in exI)
 apply (auto intro: P)
   done
 
 lemma locally_Times:
   fixes S :: "('a::metric_space) set" and T :: "('b::metric_space) set"
   assumes PS: "locally P S" and QT: "locally Q T" and R: "\<And>S T. P S \<and> Q T \<Longrightarrow> R(S \<times> T)"
   shows "locally R (S \<times> T)"
     unfolding locally_def
 proof (clarify)
   fix W x y
   assume W: "openin (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)
   with \<open>U \<times> V \<subseteq> W\<close> 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"
     apply (rule_tac x="U1 \<times> V1" in exI)
     apply (rule_tac x="U2 \<times> V2" in exI)
     apply (auto simp: openin_Times R openin_Times_eq)
     done
 qed
 
 
 proposition homeomorphism_locally_imp:
   fixes S :: "'a::metric_space set" and t :: "'b::t2_space set"
   assumes S: "locally P S" and hom: "homeomorphism S t f g"
       and Q: "\<And>S S'. \<lbrakk>P S; homeomorphism S S' f g\<rbrakk> \<Longrightarrow> Q S'"
     shows "locally Q t"
 proof (clarsimp simp: locally_def)
   fix W y
   assume "y \<in> W" and "openin (top_of_set t) W"
   then obtain T where T: "open T" "W = t \<inter> T"
     by (force simp: openin_open)
   then have "W \<subseteq> t" by auto
   have f: "\<And>x. x \<in> S \<Longrightarrow> g(f x) = x" "f ` S = t" "continuous_on S f"
    and g: "\<And>y. y \<in> t \<Longrightarrow> f(g y) = y" "g ` t = S" "continuous_on t g"
     using hom by (auto simp: homeomorphism_def)
   have gw: "g ` W = S \<inter> f -` W"
     using \<open>W \<subseteq> t\<close>
     apply auto
     using \<open>g ` t = S\<close> \<open>W \<subseteq> t\<close> apply blast
     using g \<open>W \<subseteq> t\<close> apply auto[1]
     by (simp add: f rev_image_eqI)
   have \<circ>: "openin (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 homv: "homeomorphism v (f ` v) f g"
     using \<open>v \<subseteq> S\<close> \<open>W \<subseteq> t\<close> f
     apply (simp add: homeomorphism_def contvf contvg, auto)
     by (metis f(1) rev_image_eqI rev_subsetD)
   have 1: "openin (top_of_set t) (t \<inter> g -` u)"
     apply (rule continuous_on_open [THEN iffD1, rule_format])
     apply (rule \<open>continuous_on t g\<close>)
     using \<open>g ` t = S\<close> apply (simp add: osu)
     done
   have 2: "\<exists>V. Q V \<and> y \<in> (t \<inter> g -` u) \<and> (t \<inter> g -` u) \<subseteq> V \<and> V \<subseteq> W"
     apply (rule_tac x="f ` v" in exI)
     apply (intro conjI Q [OF \<open>P v\<close> homv])
     using \<open>W \<subseteq> t\<close> \<open>y \<in> W\<close>  \<open>f ` v \<subseteq> W\<close>  uv  apply (auto simp: fv)
     done
   show "\<exists>U. openin (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"
 apply (rule iffI)
 apply (erule homeomorphism_locally_imp [OF _ hom])
 apply (simp add: eq)
 apply (erule homeomorphism_locally_imp)
 using eq homeomorphism_sym homeomorphism_symD [OF hom] apply blast+
 done
 
 lemma homeomorphic_locally:
   fixes S:: "'a::metric_space set" and T:: "'b::metric_space set"
   assumes hom: "S homeomorphic T"
           and iff: "\<And>X Y. X homeomorphic Y \<Longrightarrow> (P X \<longleftrightarrow> Q Y)"
     shows "locally P S \<longleftrightarrow> locally Q T"
 proof -
   obtain f g where hom: "homeomorphism S T f g"
     using assms by (force simp: homeomorphic_def)
   then show ?thesis
     using homeomorphic_def local.iff
     by (blast intro!: homeomorphism_locally)
 qed
 
 lemma homeomorphic_local_compactness:
   fixes S:: "'a::metric_space set" and T:: "'b::metric_space set"
   shows "S homeomorphic T \<Longrightarrow> locally compact S \<longleftrightarrow> locally compact T"
 by (simp add: homeomorphic_compactness homeomorphic_locally)
 
 lemma locally_translation:
   fixes P :: "'a :: real_normed_vector set \<Rightarrow> bool"
   shows
    "(\<And>S. P (image (\<lambda>x. a + x) S) \<longleftrightarrow> P S)
         \<Longrightarrow> locally P (image (\<lambda>x. a + x) S) \<longleftrightarrow> locally P S"
 apply (rule homeomorphism_locally [OF homeomorphism_translation])
 apply (simp add: homeomorphism_def)
 by metis
 
 lemma locally_injective_linear_image:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes f: "linear f" "inj f" and iff: "\<And>S. P (f ` S) \<longleftrightarrow> Q S"
     shows "locally P (f ` S) \<longleftrightarrow> locally Q S"
 apply (rule linear_homeomorphism_image [OF f])
 apply (rule_tac f=g and g = f in homeomorphism_locally, assumption)
 by (metis iff homeomorphism_def)
 
 lemma locally_open_map_image:
   fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
   assumes P: "locally P S"
       and f: "continuous_on S f"
       and oo: "\<And>t. openin (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 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)"
     apply (rule_tac x="f ` U" in exI)
     apply (rule conjI, blast intro!: oo)
     apply (rule_tac x="f ` V" in exI)
     apply (force simp: \<open>f x = y\<close> rev_image_eqI intro: Q)
     done
 qed
 
 
 subsection\<open>An induction principle for connected sets\<close>
 
 proposition connected_induction:
   assumes "connected S"
       and opD: "\<And>T a. \<lbrakk>openin (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 -
   have 1: "openin (top_of_set S)
              {b. \<exists>T. openin (top_of_set S) T \<and>
                      b \<in> T \<and> (\<forall>x\<in>T. P x \<longrightarrow> Q x)}"
     apply (subst openin_subopen, clarify)
     apply (rule_tac x=T in exI, auto)
     done
   have 2: "openin (top_of_set S)
              {b. \<exists>T. openin (top_of_set S) T \<and>
                      b \<in> T \<and> (\<forall>x\<in>T. P x \<longrightarrow> \<not> Q x)}"
     apply (subst openin_subopen, clarify)
     apply (rule_tac x=T in exI, auto)
     done
   show ?thesis
     using \<open>connected S\<close>
     apply (simp only: connected_openin HOL.not_ex HOL.de_Morgan_conj)
     apply (elim disjE allE)
          apply (blast intro: 1)
         apply (blast intro: 2, simp_all)
        apply clarify apply (metis opI)
       using opD apply (blast intro: etc elim: dest:)
      using opI etc apply meson+
     done
 qed
 
 lemma connected_equivalence_relation_gen:
   assumes "connected S"
       and etc: "a \<in> S" "b \<in> S" "P a" "P b"
       and trans: "\<And>x y z. \<lbrakk>R x y; R y z\<rbrakk> \<Longrightarrow> R x z"
       and opD: "\<And>T a. \<lbrakk>openin (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"
 apply (rule connected_induction [OF \<open>connected S\<close> _, where P = "\<lambda>x. True"], blast)
 apply (frule opI)
 using etc apply simp_all
 done
 
 lemma connected_equivalence_relation:
   assumes "connected S"
       and etc: "a \<in> S" "b \<in> S"
       and sym: "\<And>x y. \<lbrakk>R x y; x \<in> S; y \<in> S\<rbrakk> \<Longrightarrow> R y x"
       and trans: "\<And>x y z. \<lbrakk>R x y; R y z; x \<in> S; y \<in> S; z \<in> S\<rbrakk> \<Longrightarrow> R x z"
       and opI: "\<And>a. a \<in> S \<Longrightarrow> \<exists>T. openin (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:
      "connected S \<Longrightarrow> locally (\<lambda>U. f constant_on U) S \<longleftrightarrow> f constant_on S"
 apply (simp add: locally_def)
 apply (rule iffI)
  apply (rule locally_constant_imp_constant, assumption)
  apply (metis (mono_tags, hide_lams) constant_on_def constant_on_subset openin_subtopology_self)
 by (meson constant_on_subset openin_imp_subset order_refl)
 
 
 subsection\<open>Basic properties of local compactness\<close>
 
 proposition locally_compact:
   fixes s :: "'a :: metric_space set"
   shows
     "locally compact s \<longleftrightarrow>
      (\<forall>x \<in> s. \<exists>u v. x \<in> u \<and> u \<subseteq> v \<and> v \<subseteq> s \<and>
                     openin (top_of_set s) u \<and> compact v)"
      (is "?lhs = ?rhs")
 proof
   assume ?lhs
   then show ?rhs
     apply clarify
     apply (erule_tac w = "s \<inter> ball x 1" in locallyE)
     by auto
 next
   assume r [rule_format]: ?rhs
   have *: "\<exists>u v.
               openin (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 e where "e>0" and e: "cball x e \<subseteq> T"
       using open_contains_cball \<open>open T\<close> \<open>x \<in> T\<close> by blast
     show ?thesis
       apply (rule_tac x="(s \<inter> ball x e) \<inter> u" in exI)
       apply (rule_tac x="cball x e \<inter> v" in exI)
       using that \<open>e > 0\<close> e uv
       apply auto
       done
   qed
   show ?lhs
     apply (rule locallyI)
     apply (subst (asm) openin_open)
     apply (blast intro: *)
     done
 qed
 
 lemma locally_compactE:
   fixes s :: "'a :: metric_space set"
   assumes "locally compact s"
   obtains u v where "\<And>x. x \<in> s \<Longrightarrow> x \<in> u x \<and> u x \<subseteq> v x \<and> v x \<subseteq> s \<and>
                              openin (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)"
 apply (simp add: locally_compact)
 apply (intro ball_cong ex_cong refl iffI)
 apply (metis bounded_subset closure_eq closure_mono compact_eq_bounded_closed dual_order.trans)
 by (meson closure_subset compact_closure)
 
 lemma locally_compact_Int_cball:
   fixes s :: "'a :: heine_borel set"
   shows "locally compact s \<longleftrightarrow> (\<forall>x \<in> s. \<exists>e. 0 < e \<and> closed(cball x e \<inter> s))"
         (is "?lhs = ?rhs")
 proof
   assume ?lhs
   then show ?rhs
     apply (simp add: locally_compact openin_contains_cball)
     apply (clarify | assumption | drule bspec)+
     by (metis (no_types, lifting)  compact_cball compact_imp_closed compact_Int inf.absorb_iff2 inf.orderE inf_sup_aci(2))
 next
   assume ?rhs
   then show ?lhs
     apply (simp add: locally_compact openin_contains_cball)
     apply (clarify | assumption | drule bspec)+
     apply (rule_tac x="ball x e \<inter> s" in exI, simp)
     apply (rule_tac x="cball x e \<inter> s" in exI)
     using compact_eq_bounded_closed
     apply auto
     apply (metis open_ball le_infI1 mem_ball open_contains_cball_eq)
     done
 qed
 
 lemma locally_compact_compact:
   fixes s :: "'a :: heine_borel set"
   shows "locally compact s \<longleftrightarrow>
          (\<forall>k. k \<subseteq> s \<and> compact k
               \<longrightarrow> (\<exists>u v. k \<subseteq> u \<and> u \<subseteq> v \<and> v \<subseteq> s \<and>
                          openin (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 simp: dest!: uv)
     show ?thesis
       apply (rule_tac x="\<Union>(u ` T)" in exI)
       apply (rule_tac x="\<Union>(v ` T)" in exI)
       apply (simp add: Tuv)
       using T that
       apply (auto simp: dest!: uv)
       done
   qed
   show ?rhs
     by (blast intro: *)
 next
   assume ?rhs
   then show ?lhs
     apply (clarsimp simp add: locally_compact)
     apply (drule_tac x="{x}" in spec, simp)
     done
 qed
 
 lemma open_imp_locally_compact:
   fixes s :: "'a :: heine_borel set"
   assumes "open s"
     shows "locally compact s"
 proof -
   have *: "\<exists>u v. x \<in> u \<and> u \<subseteq> v \<and> v \<subseteq> s \<and> openin (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
       apply (rule_tac x="ball x e" in exI)
       apply (rule_tac x="cball x e" in exI)
       using \<open>e > 0\<close> e apply (auto simp: ope)
       done
   qed
   show ?thesis
     unfolding locally_compact
     by (blast intro: *)
 qed
 
 lemma closed_imp_locally_compact:
   fixes s :: "'a :: heine_borel set"
   assumes "closed s"
     shows "locally compact s"
 proof -
   have *: "\<exists>u v. x \<in> u \<and> u \<subseteq> v \<and> v \<subseteq> s \<and>
                  openin (top_of_set s) u \<and> compact v"
           if "x \<in> s" for x
   proof -
     show ?thesis
       apply (rule_tac x = "s \<inter> ball x 1" in exI)
       apply (rule_tac x = "s \<inter> cball x 1" in exI)
       using \<open>x \<in> s\<close> assms apply auto
       done
   qed
   show ?thesis
     unfolding locally_compact
     by (blast intro: *)
 qed
 
 lemma locally_compact_UNIV: "locally compact (UNIV :: 'a :: heine_borel set)"
   by (simp add: closed_imp_locally_compact)
 
 lemma locally_compact_Int:
   fixes s :: "'a :: t2_space set"
   shows "\<lbrakk>locally compact s; locally compact t\<rbrakk> \<Longrightarrow> locally compact (s \<inter> t)"
 by (simp add: compact_Int locally_Int)
 
 lemma locally_compact_closedin:
   fixes s :: "'a :: heine_borel set"
   shows "\<lbrakk>closedin (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
     apply (simp only: locally_def)
     apply (erule all_forward imp_forward asm_rl exE)+
     apply (rule_tac x = "u \<inter> ball x 1" in exI)
     apply (rule_tac x = "v \<inter> cball x 1" in exI)
     apply (force intro: openin_trans)
     done
 next
   assume ?rhs then show ?lhs
     using compact_eq_bounded_closed locally_mono by blast
 qed
 
 lemma locally_compact_openin_Un:
   fixes S :: "'a::euclidean_space set"
   assumes LCS: "locally compact S" and LCT:"locally compact T"
       and opS: "openin (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 show ?thesis
       apply (rule_tac x="min e1 e2" in exI)
       apply (auto simp: cball_min_Int \<open>e2 > 0\<close> inf_assoc closed_Int)
       by (metis closed_Int closed_cball inf_left_commute)
   qed
   moreover have "\<exists>e>0. closed (cball x e \<inter> (S \<union> T))" if "x \<in> T" for x
   proof -
     obtain e1 where "e1 > 0" and e1: "closed (cball x e1 \<inter> T)"
       using LCT \<open>x \<in> T\<close> unfolding locally_compact_Int_cball by blast
     moreover obtain e2 where "e2 > 0" and e2: "cball x e2 \<inter> (S \<union> T) \<subseteq> T"
       by (meson \<open>x \<in> T\<close> opT openin_contains_cball)
     then have "cball x e2 \<inter> (S \<union> T) = cball x e2 \<inter> T"
       by force
     ultimately show ?thesis
       apply (rule_tac x="min e1 e2" in exI)
       apply (auto simp: cball_min_Int \<open>e2 > 0\<close> inf_assoc closed_Int)
       by (metis closed_Int closed_cball inf_left_commute)
   qed
   ultimately show ?thesis
     by (force simp: locally_compact_Int_cball)
 qed
 
 lemma locally_compact_closedin_Un:
   fixes S :: "'a::euclidean_space set"
   assumes LCS: "locally compact S" and LCT:"locally compact T"
       and clS: "closedin (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
     ultimately show ?thesis
       apply (rule_tac x="min e1 e2" in exI)
       apply (auto simp: cball_min_Int \<open>e2 > 0\<close> inf_assoc closed_Int Int_Un_distrib)
       by (metis closed_Int closed_Un closed_cball inf_left_commute)
   qed
   moreover
   have "\<exists>e>0. closed (cball x e \<inter> (S \<union> T))" if x: "x \<in> S" "x \<notin> T" for x
   proof -
     obtain e1 where "e1 > 0" and e1: "closed (cball x e1 \<inter> S)"
       using LCS \<open>x \<in> S\<close> unfolding locally_compact_Int_cball by blast
     moreover
     obtain e2 where "e2>0" and "cball x e2 \<inter> (S \<union> T) \<subseteq> S - T"
       using clT x by (fastforce simp: openin_contains_cball closedin_def)
     then have "closed (cball x e2 \<inter> T)"
     proof -
       have "{} = T - (T - cball x e2)"
         using Diff_subset Int_Diff \<open>cball x e2 \<inter> (S \<union> T) \<subseteq> S - T\<close> by auto
       then show ?thesis
         by (simp add: Diff_Diff_Int inf_commute)
     qed
     ultimately show ?thesis
       apply (rule_tac x="min e1 e2" in exI)
       apply (auto simp: cball_min_Int \<open>e2 > 0\<close> inf_assoc closed_Int Int_Un_distrib)
       by (metis closed_Int closed_Un closed_cball inf_left_commute)
   qed
   moreover
   have "\<exists>e>0. closed (cball x e \<inter> (S \<union> T))" if x: "x \<notin> S" "x \<in> T" for x
   proof -
     obtain e1 where "e1 > 0" and e1: "closed (cball x e1 \<inter> T)"
       using LCT \<open>x \<in> T\<close> unfolding locally_compact_Int_cball by blast
     moreover
     obtain e2 where "e2>0" and "cball x e2 \<inter> (S \<union> T) \<subseteq> S \<union> T - S"
       using clS x by (fastforce simp: openin_contains_cball closedin_def)
     then have "closed (cball x e2 \<inter> S)"
       by (metis Diff_disjoint Int_empty_right closed_empty inf.left_commute inf.orderE inf_sup_absorb)
     ultimately show ?thesis
       apply (rule_tac x="min e1 e2" in exI)
       apply (auto simp: cball_min_Int \<open>e2 > 0\<close> inf_assoc closed_Int Int_Un_distrib)
       by (metis closed_Int closed_Un closed_cball inf_left_commute)
   qed
   ultimately show ?thesis
     by (auto simp: locally_compact_Int_cball)
 qed
 
 lemma locally_compact_Times:
   fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set"
   shows "\<lbrakk>locally compact S; locally compact T\<rbrakk> \<Longrightarrow> locally compact (S \<times> T)"
   by (auto simp: compact_Times locally_Times)
 
 lemma locally_compact_compact_subopen:
   fixes S :: "'a :: heine_borel set"
   shows
    "locally compact S \<longleftrightarrow>
     (\<forall>K T. K \<subseteq> S \<and> compact K \<and> open T \<and> K \<subseteq> T
           \<longrightarrow> (\<exists>U V. K \<subseteq> U \<and> U \<subseteq> V \<and> U \<subseteq> T \<and> V \<subseteq> S \<and>
                      openin (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"
             apply safe
             using \<open>D \<subseteq> V1 \<union> V2\<close> \<open>open V1\<close> \<open>open V2\<close> V12
                apply (simp_all add: closure_subset [THEN subsetD] closure_iff_nhds_not_empty, blast+)
             done
           ultimately have cloDV1: "closedin (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 "\<Inter>?\<T> \<subseteq> D \<inter> V2"
               apply (rule Inter_lower)
               using * apply simp
               by (meson cloDV2 \<open>open V2\<close> cloD closedin_trans le_inf_iff opeD openin_Int_open)
             then show False
               using K1 V12 \<open>K1 \<noteq> {}\<close> \<open>K1 \<subseteq> V1\<close> closedin_imp_subset by blast
           qed
           moreover have "D \<inter> U2 \<inter> C \<noteq> {}"
           proof
             assume "D \<inter> U2 \<inter> C = {}"
             then have *: "C \<subseteq> D \<inter> V1"
               using D2 DV12 \<open>C \<subseteq> D\<close> by auto
             have "\<Inter>?\<T> \<subseteq> D \<inter> V1"
               apply (rule Inter_lower)
               using * apply simp
               by (meson cloDV1 \<open>open V1\<close> cloD closedin_trans le_inf_iff opeD openin_Int_open)
             then show False
               using K2 V12 \<open>K2 \<noteq> {}\<close> \<open>K2 \<subseteq> V2\<close> closedin_imp_subset by blast
           qed
           ultimately show False
             using \<open>connected C\<close> unfolding connected_closed
             apply (simp only: not_ex)
             apply (drule_tac x="D \<inter> U1" in spec)
             apply (drule_tac x="D \<inter> U2" in spec)
             using \<open>C \<subseteq> D\<close> D1 D2 V12 DV12 \<open>closed U1\<close> \<open>closed U2\<close> \<open>closed D\<close>
             by blast
         qed
       qed
       show False
         by (metis (full_types) DiffE UnE Un_upper2 SV12_ne \<open>K1 \<subseteq> V1\<close> \<open>K2 \<subseteq> V2\<close> disjoint_iff_not_equal subsetCE sup_ge1 K12_Un)
     qed
     then show "connected (\<Inter>?\<T>)"
       by (auto simp: connected_closedin_eq)
     show "\<Inter>?\<T> \<subseteq> S"
       by (fastforce simp: C in_components_subset)
   qed
   with x show "\<Inter>?\<T> \<subseteq> C" by simp
 qed auto
 
 
 corollary Sura_Bura_clopen_subset:
   fixes S :: "'a::euclidean_space set"
   assumes S: "locally compact S" and C: "C \<in> components S" and "compact C"
       and U: "open U" "C \<subseteq> U"
   obtains K where "openin (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>
     apply (simp add: locally_compact_compact_subopen)
     by (meson C in_components_subset)
   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"
 apply (clarsimp simp add: locally_def)
 apply (drule assms; blast)
 done
 
 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"
 apply (clarsimp simp add: locally_def)
 apply (drule assms; blast)
 done
 
 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(1) assms(2) assms(3) mem_Collect_eq path_component_subset path_connected_path_component)
 qed
 
 proposition locally_path_connected:
   "locally path_connected S \<longleftrightarrow>
    (\<forall>v x. openin (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
     apply (clarsimp simp add: locally_connected_open_component)
     apply (subst openin_subopen)
     apply (blast intro: *)
     done
 qed
 
 proposition locally_path_connected_im_kleinen:
   "locally path_connected S \<longleftrightarrow>
    (\<forall>v x. openin (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
       apply (rule_tac x=U in exI)
       apply (auto simp: U)
       apply (metis U c path_component_trans path_component_def)
       done
   qed
   show ?lhs
     apply (clarsimp simp add: locally_path_connected_open_path_component)
     apply (subst openin_subopen)
     apply (blast intro: *)
     done
 qed
 
 lemma locally_path_connected_imp_locally_connected:
   "locally path_connected S \<Longrightarrow> locally connected S"
 using locally_mono path_connected_imp_connected by blast
 
 lemma locally_connected_components:
   "\<lbrakk>locally connected S; c \<in> components S\<rbrakk> \<Longrightarrow> locally connected c"
 by (meson locally_connected_open_component locally_open_subset openin_subtopology_self)
 
 lemma locally_path_connected_components:
   "\<lbrakk>locally path_connected S; c \<in> components S\<rbrakk> \<Longrightarrow> locally path_connected c"
 by (meson locally_connected_open_component locally_open_subset locally_path_connected_imp_locally_connected openin_subtopology_self)
 
 lemma locally_path_connected_connected_component:
   "locally path_connected S \<Longrightarrow> locally path_connected (connected_component_set S x)"
 by (metis components_iff connected_component_eq_empty locally_empty locally_path_connected_components)
 
 lemma open_imp_locally_path_connected:
   fixes S :: "'a :: real_normed_vector set"
   shows "open S \<Longrightarrow> locally path_connected S"
 apply (rule locally_mono [of convex])
 apply (simp_all add: locally_def openin_open_eq convex_imp_path_connected)
 apply (meson open_ball centre_in_ball convex_ball openE order_trans)
 done
 
 lemma open_imp_locally_connected:
   fixes S :: "'a :: real_normed_vector set"
   shows "open S \<Longrightarrow> locally connected S"
 by (simp add: locally_path_connected_imp_locally_connected open_imp_locally_path_connected)
 
 lemma locally_path_connected_UNIV: "locally path_connected (UNIV::'a :: real_normed_vector set)"
   by (simp add: open_imp_locally_path_connected)
 
 lemma locally_connected_UNIV: "locally connected (UNIV::'a :: real_normed_vector set)"
   by (simp add: open_imp_locally_connected)
 
 lemma openin_connected_component_locally_connected:
     "locally connected S
      \<Longrightarrow> openin (top_of_set S) (connected_component_set S x)"
 apply (simp add: locally_connected_open_connected_component)
 by (metis connected_component_eq_empty connected_component_subset open_empty open_subset openin_subtopology_self)
 
 lemma openin_components_locally_connected:
     "\<lbrakk>locally connected S; c \<in> components S\<rbrakk> \<Longrightarrow> openin (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:
     "locally path_connected S
         \<Longrightarrow> closedin (top_of_set S) (path_component_set S x)"
 apply  (simp add: closedin_def path_component_subset complement_path_component_Union)
 apply (rule openin_Union)
 using openin_path_component_locally_path_connected by auto
 
 lemma convex_imp_locally_path_connected:
   fixes S :: "'a:: real_normed_vector set"
   shows "convex S \<Longrightarrow> locally path_connected S"
 apply (clarsimp simp add: locally_path_connected)
 apply (subst (asm) openin_open)
 apply clarify
 apply (erule (1) openE)
 apply (rule_tac x = "S \<inter> ball x e" in exI)
 apply (force simp: convex_Int convex_imp_path_connected)
 done
 
 lemma convex_imp_locally_connected:
   fixes S :: "'a:: real_normed_vector set"
   shows "convex S \<Longrightarrow> locally connected S"
   by (simp add: locally_path_connected_imp_locally_connected convex_imp_locally_path_connected)
 
 
 subsection\<open>Relations between components and path components\<close>
 
 lemma path_component_eq_connected_component:
   assumes "locally path_connected S"
     shows "(path_component S x = connected_component S x)"
 proof (cases "x \<in> S")
   case True
   have "openin (top_of_set (connected_component_set S x)) (path_component_set S x)"
     apply (rule openin_subset_trans [of S])
     apply (intro conjI openin_path_component_locally_path_connected [OF assms])
     using path_component_subset_connected_component   apply (auto simp: connected_component_subset)
     done
   moreover have "closedin (top_of_set (connected_component_set S x)) (path_component_set S x)"
     apply (rule closedin_subset_trans [of S])
     apply (intro conjI closedin_path_component_locally_path_connected [OF assms])
     using path_component_subset_connected_component   apply (auto simp: connected_component_subset)
     done
   ultimately have *: "path_component_set S x = connected_component_set S x"
     by (metis connected_connected_component connected_clopen True path_component_eq_empty)
   then show ?thesis
     by blast
 next
   case False then show ?thesis
     by (metis Collect_empty_eq_bot connected_component_eq_empty path_component_eq_empty)
 qed
 
 lemma path_component_eq_connected_component_set:
      "locally path_connected S \<Longrightarrow> (path_component_set S x = connected_component_set S x)"
 by (simp add: path_component_eq_connected_component)
 
 lemma locally_path_connected_path_component:
      "locally path_connected S \<Longrightarrow> locally path_connected (path_component_set S x)"
 using locally_path_connected_connected_component path_component_eq_connected_component by fastforce
 
 lemma open_path_connected_component:
   fixes S :: "'a :: real_normed_vector set"
   shows "open S \<Longrightarrow> path_component S x = connected_component S x"
 by (simp add: path_component_eq_connected_component open_imp_locally_path_connected)
 
 lemma open_path_connected_component_set:
   fixes S :: "'a :: real_normed_vector set"
   shows "open S \<Longrightarrow> path_component_set S x = connected_component_set S x"
 by (simp add: open_path_connected_component)
 
 proposition locally_connected_quotient_image:
   assumes lcS: "locally connected S"
       and oo: "\<And>T. T \<subseteq> f ` S
                 \<Longrightarrow> openin (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"
         apply (rule continuous_on_subset)
         using connected_component_subset apply blast
         done
       then have "connected (f ` connected_component_set (S \<inter> f -` U) x)"
         by (rule connected_continuous_image [OF _ connected_connected_component])
       moreover have "f ` connected_component_set (S \<inter> f -` U) x \<subseteq> U"
         using connected_component_in by blast
       moreover have "C \<inter> f ` connected_component_set (S \<inter> f -` U) x \<noteq> {}"
         using \<open>C \<subseteq> U\<close> \<open>f x \<in> C\<close> \<open>x \<in> S\<close> by fastforce
       ultimately have fC: "f ` (connected_component_set (S \<inter> f -` U) x) \<subseteq> C"
         by (rule components_maximal [OF \<open>C \<in> components U\<close>])
       have cUC: "connected_component_set (S \<inter> f -` U) x \<subseteq> (S \<inter> f -` C)"
         using connected_component_subset fC by blast
       have "connected_component_set (S \<inter> f -` U) x \<subseteq> connected_component_set (S \<inter> f -` C) x"
       proof -
         { assume "x \<in> connected_component_set (S \<inter> f -` U) x"
           then have ?thesis
             using cUC connected_component_idemp connected_component_mono by blast }
         then show ?thesis
           using connected_component_eq_empty by auto
       qed
       also have "\<dots> \<subseteq> (S \<inter> f -` C)"
         by (rule connected_component_subset)
       finally show "connected_component_set (S \<inter> f -` U) x \<subseteq> (S \<inter> f -` C)" .
     qed
   qed
   ultimately show "openin (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"
         apply (rule continuous_on_subset)
         using path_component_subset apply blast
         done
       then have "path_connected (f ` path_component_set (S \<inter> f -` U) x)"
         by (simp add: path_connected_continuous_image)
       moreover have "f ` path_component_set (S \<inter> f -` U) x \<subseteq> U"
         using path_component_mem by fastforce
       moreover have "f x \<in> f ` path_component_set (S \<inter> f -` U) x"
         by (force simp: \<open>x \<in> S\<close> \<open>f x \<in> U\<close> path_component_refl_eq)
       ultimately have "f ` (path_component_set (S \<inter> f -` U) x) \<subseteq> path_component_set U (f x)"
         by (meson path_component_maximal)
        also have  "\<dots> \<subseteq> path_component_set U y"
         by (simp add: Uyfx path_component_maximal path_component_subset path_component_sym)
       finally have fC: "f ` (path_component_set (S \<inter> f -` U) x) \<subseteq> path_component_set U y" .
       have cUC: "path_component_set (S \<inter> f -` U) x \<subseteq> (S \<inter> f -` path_component_set U y)"
         using path_component_subset fC by blast
       have "path_component_set (S \<inter> f -` U) x \<subseteq> path_component_set (S \<inter> f -` path_component_set U y) x"
       proof -
         have "\<And>a. path_component_set (path_component_set (S \<inter> f -` U) x) a \<subseteq> path_component_set (S \<inter> f -` path_component_set U y) a"
           using cUC path_component_mono by blast
         then show ?thesis
           using path_component_path_component by blast
       qed
       also have "\<dots> \<subseteq> (S \<inter> f -` path_component_set U y)"
         by (rule path_component_subset)
       finally show "path_component_set (S \<inter> f -` U) x \<subseteq> (S \<inter> f -` path_component_set U y)" .
     qed
   qed
   ultimately show "openin (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"
 apply (rule continuous_on_components_gen)
 apply (auto simp: assms intro: openin_components_locally_connected)
 done
 
 lemma continuous_on_components_eq:
     "locally connected S
      \<Longrightarrow> (continuous_on S f \<longleftrightarrow> (\<forall>c \<in> components S. continuous_on c f))"
 by (meson continuous_on_components continuous_on_subset in_components_subset)
 
 lemma continuous_on_components_open:
  fixes S :: "'a::real_normed_vector set"
   assumes "open S "
           "\<And>c. c \<in> components S \<Longrightarrow> continuous_on c f"
     shows "continuous_on S f"
 using continuous_on_components open_imp_locally_connected assms by blast
 
 lemma continuous_on_components_open_eq:
   fixes S :: "'a::real_normed_vector set"
   shows "open S \<Longrightarrow> (continuous_on S f \<longleftrightarrow> (\<forall>c \<in> components S. continuous_on c f))"
 using continuous_on_subset in_components_subset
 by (blast intro: continuous_on_components_open)
 
 lemma closedin_union_complement_components:
   assumes u: "locally connected u"
       and S: "closedin (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 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))"
     apply (rule openin_Union)
     apply (rule openin_trans [of "u - S"])
     apply (simp add: u S locally_diff_closed openin_components_locally_connected)
     apply (simp add: openin_diff S)
     done
   have "openin (top_of_set u) (u - (u - \<Union>(components (u - S) - c)))"
     apply (rule openin_diff, simp)
     apply (metis closedin_diff closedin_topspace topspace_euclidean_subtopology *)
     done
   then show ?thesis
     by (force simp: eq closedin_def)
 qed
 
 lemma closed_union_complement_components:
   fixes S :: "'a::real_normed_vector set"
   assumes S: "closed S" and c: "c \<subseteq> components(- S)"
     shows "closed(S \<union> \<Union> c)"
 proof -
   have "closedin (top_of_set UNIV) (S \<union> \<Union>c)"
     apply (rule closedin_union_complement_components [OF locally_connected_UNIV])
     using S c apply (simp_all add: Compl_eq_Diff_UNIV)
     done
   then show ?thesis by simp
 qed
 
 lemma closedin_Un_complement_component:
   fixes S :: "'a::real_normed_vector set"
   assumes u: "locally connected u"
       and S: "closedin (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
     apply (auto simp: pairwise_def orthogonal_clauses)
     by (meson subsetD image_eqI inj_on_def)
   obtain f where "linear f" and ffb: "\<And>x. x \<in> B \<Longrightarrow> f x = fb x"
     using linear_independent_extend \<open>independent B\<close> by fastforce
   have "span (f ` B) \<subseteq> span C"
     by (metis \<open>fb ` B \<subseteq> C\<close> ffb image_cong span_mono)
   then have "f ` S \<subseteq> T"
     unfolding \<open>span B = S\<close> \<open>span C = T\<close> span_linear_image[OF \<open>linear f\<close>] .
   have [simp]: "\<And>x. x \<in> B \<Longrightarrow> norm (fb x) = norm x"
     using B1 C1 \<open>fb ` B \<subseteq> C\<close> by auto
   have "norm (f x) = norm x" if "x \<in> S" for x
   proof -
     interpret linear f by fact
     obtain a where x: "x = (\<Sum>v \<in> B. a v *\<^sub>R v)"
       using \<open>finite B\<close> \<open>span B = S\<close> \<open>x \<in> S\<close> span_finite by fastforce
     have "norm (f x)^2 = norm (\<Sum>v\<in>B. a v *\<^sub>R fb v)^2" by (simp add: sum scale ffb x)
     also have "\<dots> = (\<Sum>v\<in>B. norm ((a v *\<^sub>R fb v))^2)"
       apply (rule norm_sum_Pythagorean [OF \<open>finite B\<close>])
       apply (rule pairwise_ortho_scaleR [OF pairwise_orth_fb])
       done
     also have "\<dots> = norm x ^2"
       by (simp add: x pairwise_ortho_scaleR Borth norm_sum_Pythagorean [OF \<open>finite B\<close>])
     finally show ?thesis
       by (simp add: norm_eq_sqrt_inner)
   qed
   then show ?thesis
     by (rule that [OF \<open>linear f\<close> \<open>f ` S \<subseteq> T\<close>])
 qed
 
 proposition isometries_subspaces:
   fixes S :: "'a::euclidean_space set"
     and T :: "'b::euclidean_space set"
   assumes S: "subspace S"
       and T: "subspace T"
       and d: "dim S = dim T"
   obtains f g where "linear f" "linear g" "f ` S = T" "g ` T = S"
                     "\<And>x. x \<in> S \<Longrightarrow> norm(f x) = norm x"
                     "\<And>x. x \<in> T \<Longrightarrow> norm(g x) = norm x"
                     "\<And>x. x \<in> S \<Longrightarrow> g(f x) = x"
                     "\<And>x. x \<in> T \<Longrightarrow> f(g x) = x"
 proof -
   obtain B where "B \<subseteq> S" and Borth: "pairwise orthogonal B"
              and B1: "\<And>x. x \<in> B \<Longrightarrow> norm x = 1"
              and "independent B" "finite B" "card B = dim S" "span B = S"
     by (metis orthonormal_basis_subspace [OF S] independent_finite)
   obtain C where "C \<subseteq> T" and Corth: "pairwise orthogonal C"
              and C1:"\<And>x. x \<in> C \<Longrightarrow> norm x = 1"
              and "independent C" "finite C" "card C = dim T" "span C = T"
     by (metis orthonormal_basis_subspace [OF T] independent_finite)
   obtain fb where "bij_betw fb B C"
     by (metis \<open>finite B\<close> \<open>finite C\<close> bij_betw_iff_card \<open>card B = dim S\<close> \<open>card C = dim T\<close> d)
   then have pairwise_orth_fb: "pairwise (\<lambda>v j. orthogonal (fb v) (fb j)) B"
     using Corth
     apply (auto simp: pairwise_def orthogonal_clauses bij_betw_def)
     by (meson subsetD image_eqI inj_on_def)
   obtain f where "linear f" and ffb: "\<And>x. x \<in> B \<Longrightarrow> f x = fb x"
     using linear_independent_extend \<open>independent B\<close> by fastforce
   interpret f: linear f by fact
   define gb where "gb \<equiv> inv_into B fb"
   then have pairwise_orth_gb: "pairwise (\<lambda>v j. orthogonal (gb v) (gb j)) C"
     using Borth
     apply (auto simp: pairwise_def orthogonal_clauses bij_betw_def)
     by (metis \<open>bij_betw fb B C\<close> bij_betw_imp_surj_on bij_betw_inv_into_right inv_into_into)
   obtain g where "linear g" and ggb: "\<And>x. x \<in> C \<Longrightarrow> g x = gb x"
     using linear_independent_extend \<open>independent C\<close> by fastforce
   interpret g: linear g by fact
   have "span (f ` B) \<subseteq> span C"
     by (metis \<open>bij_betw fb B C\<close> bij_betw_imp_surj_on eq_iff ffb image_cong)
   then have "f ` S \<subseteq> T"
     unfolding \<open>span B = S\<close> \<open>span C = T\<close>
       span_linear_image[OF \<open>linear f\<close>] .
   have [simp]: "\<And>x. x \<in> B \<Longrightarrow> norm (fb x) = norm x"
     using B1 C1 \<open>bij_betw fb B C\<close> bij_betw_imp_surj_on by fastforce
   have f [simp]: "norm (f x) = norm x" "g (f x) = x" if "x \<in> S" for x
   proof -
     obtain a where x: "x = (\<Sum>v \<in> B. a v *\<^sub>R v)"
       using \<open>finite B\<close> \<open>span B = S\<close> \<open>x \<in> S\<close> span_finite by fastforce
     have "f x = (\<Sum>v \<in> B. f (a v *\<^sub>R v))"
       using linear_sum [OF \<open>linear f\<close>] x by auto
     also have "\<dots> = (\<Sum>v \<in> B. a v *\<^sub>R f v)"
       by (simp add: f.sum f.scale)
     also have "\<dots> = (\<Sum>v \<in> B. a v *\<^sub>R fb v)"
       by (simp add: ffb cong: sum.cong)
     finally have *: "f x = (\<Sum>v\<in>B. a v *\<^sub>R fb v)" .
     then have "(norm (f x))\<^sup>2 = (norm (\<Sum>v\<in>B. a v *\<^sub>R fb v))\<^sup>2" by simp
     also have "\<dots> = (\<Sum>v\<in>B. norm ((a v *\<^sub>R fb v))^2)"
       apply (rule norm_sum_Pythagorean [OF \<open>finite B\<close>])
       apply (rule pairwise_ortho_scaleR [OF pairwise_orth_fb])
       done
     also have "\<dots> = (norm x)\<^sup>2"
       by (simp add: x pairwise_ortho_scaleR Borth norm_sum_Pythagorean [OF \<open>finite B\<close>])
     finally show "norm (f x) = norm x"
       by (simp add: norm_eq_sqrt_inner)
     have "g (f x) = g (\<Sum>v\<in>B. a v *\<^sub>R fb v)" by (simp add: *)
     also have "\<dots> = (\<Sum>v\<in>B. g (a v *\<^sub>R fb v))"
       by (simp add: g.sum g.scale)
     also have "\<dots> = (\<Sum>v\<in>B. a v *\<^sub>R g (fb v))"
       by (simp add: g.scale)
     also have "\<dots> = (\<Sum>v\<in>B. a v *\<^sub>R v)"
       apply (rule sum.cong [OF refl])
       using \<open>bij_betw fb B C\<close> gb_def bij_betwE bij_betw_inv_into_left gb_def ggb by fastforce
     also have "\<dots> = x"
       using x by blast
     finally show "g (f x) = x" .
   qed
   have [simp]: "\<And>x. x \<in> C \<Longrightarrow> norm (gb x) = norm x"
     by (metis B1 C1 \<open>bij_betw fb B C\<close> bij_betw_imp_surj_on gb_def inv_into_into)
   have g [simp]: "f (g x) = x" if "x \<in> T" for x
   proof -
     obtain a where x: "x = (\<Sum>v \<in> C. a v *\<^sub>R v)"
       using \<open>finite C\<close> \<open>span C = T\<close> \<open>x \<in> T\<close> span_finite by fastforce
     have "g x = (\<Sum>v \<in> C. g (a v *\<^sub>R v))"
       by (simp add: x g.sum)
     also have "\<dots> = (\<Sum>v \<in> C. a v *\<^sub>R g v)"
       by (simp add: g.scale)
     also have "\<dots> = (\<Sum>v \<in> C. a v *\<^sub>R gb v)"
       by (simp add: ggb cong: sum.cong)
     finally have "f (g x) = f (\<Sum>v\<in>C. a v *\<^sub>R gb v)" by simp
     also have "\<dots> = (\<Sum>v\<in>C. f (a v *\<^sub>R gb v))"
       by (simp add: f.scale f.sum)
     also have "\<dots> = (\<Sum>v\<in>C. a v *\<^sub>R f (gb v))"
       by (simp add: f.scale f.sum)
     also have "\<dots> = (\<Sum>v\<in>C. a v *\<^sub>R v)"
       using \<open>bij_betw fb B C\<close>
       by (simp add: bij_betw_def gb_def bij_betw_inv_into_right ffb inv_into_into)
     also have "\<dots> = x"
       using x by blast
     finally show "f (g x) = x" .
   qed
   have gim: "g ` T = S"
     by (metis (full_types) S T \<open>f ` S \<subseteq> T\<close> d dim_eq_span dim_image_le f(2) g.linear_axioms
         image_iff linear_subspace_image span_eq_iff subset_iff)
   have fim: "f ` S = T"
     using \<open>g ` T = S\<close> image_iff by fastforce
   have [simp]: "norm (g x) = norm x" if "x \<in> T" for x
     using fim that by auto
   show ?thesis
     apply (rule that [OF \<open>linear f\<close> \<open>linear g\<close>])
     apply (simp_all add: fim gim)
     done
 qed
 
 corollary isometry_subspaces:
   fixes S :: "'a::euclidean_space set"
     and T :: "'b::euclidean_space set"
   assumes S: "subspace S"
       and T: "subspace T"
       and d: "dim S = dim T"
   obtains f where "linear f" "f ` S = T" "\<And>x. x \<in> S \<Longrightarrow> norm(f x) = norm x"
 using isometries_subspaces [OF assms]
 by metis
 
 corollary isomorphisms_UNIV_UNIV:
   assumes "DIM('M) = DIM('N)"
   obtains f::"'M::euclidean_space \<Rightarrow>'N::euclidean_space" and g
   where "linear f" "linear g"
                     "\<And>x. norm(f x) = norm x" "\<And>y. norm(g y) = norm y"
                     "\<And>x. g (f x) = x" "\<And>y. f(g y) = y"
   using assms by (auto intro: isometries_subspaces [of "UNIV::'M set" "UNIV::'N set"])
 
 lemma homeomorphic_subspaces:
   fixes S :: "'a::euclidean_space set"
     and T :: "'b::euclidean_space set"
   assumes S: "subspace S"
       and T: "subspace T"
       and d: "dim S = dim T"
     shows "S homeomorphic T"
 proof -
   obtain f g where "linear f" "linear g" "f ` S = T" "g ` T = S"
                    "\<And>x. x \<in> S \<Longrightarrow> g(f x) = x" "\<And>x. x \<in> T \<Longrightarrow> f(g x) = x"
     by (blast intro: isometries_subspaces [OF assms])
   then show ?thesis
     apply (simp add: homeomorphic_def homeomorphism_def)
     apply (rule_tac x=f in exI)
     apply (rule_tac x=g in exI)
     apply (auto simp: linear_continuous_on linear_conv_bounded_linear)
     done
 qed
 
 lemma homeomorphic_affine_sets:
   assumes "affine S" "affine T" "aff_dim S = aff_dim T"
     shows "S homeomorphic T"
 proof (cases "S = {} \<or> T = {}")
   case True  with assms aff_dim_empty homeomorphic_empty show ?thesis
     by metis
 next
   case False
   then obtain a b where ab: "a \<in> S" "b \<in> T" by auto
   then have ss: "subspace ((+) (- a) ` S)" "subspace ((+) (- b) ` T)"
     using affine_diffs_subspace assms by blast+
   have dd: "dim ((+) (- a) ` S) = dim ((+) (- b) ` T)"
     using assms ab  by (simp add: aff_dim_eq_dim  [OF hull_inc] image_def)
   have "S homeomorphic ((+) (- a) ` S)"
     by (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"
       and contk: "continuous_on t k"
       and imk: "k ` t \<subseteq> s"
       and idhk: "\<And>y. y \<in> t \<Longrightarrow> h(k y) = y"
 
 begin
 
 lemma homotopically_trivial_retraction_gen:
   assumes P: "\<And>f. \<lbrakk>continuous_on u f; f ` u \<subseteq> t; Q f\<rbrakk> \<Longrightarrow> P(k \<circ> f)"
       and Q: "\<And>f. \<lbrakk>continuous_on u f; f ` u \<subseteq> s; P f\<rbrakk> \<Longrightarrow> Q(h \<circ> f)"
       and Qeq: "\<And>h k. (\<And>x. x \<in> u \<Longrightarrow> h x = k x) \<Longrightarrow> Q h = Q k"
       and hom: "\<And>f g. \<lbrakk>continuous_on u f; f ` u \<subseteq> s; P f;
                        continuous_on u g; g ` u \<subseteq> s; P g\<rbrakk>
                        \<Longrightarrow> homotopic_with_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 feq: "\<And>x. x \<in> u \<Longrightarrow> (h \<circ> (k \<circ> f)) x = f x" using idhk imf by auto
   have geq: "\<And>x. x \<in> u \<Longrightarrow> (h \<circ> (k \<circ> g)) x = g x" using idhk img by auto
   have "continuous_on u (k \<circ> f)"
     using contf continuous_on_compose continuous_on_subset contk imf by blast
   moreover have "(k \<circ> f) ` u \<subseteq> s"
     using imf imk by fastforce
   moreover have "P (k \<circ> f)"
     by (simp add: P Qf contf imf)
   moreover have "continuous_on u (k \<circ> g)"
     using contg continuous_on_compose continuous_on_subset contk img by blast
   moreover have "(k \<circ> g) ` u \<subseteq> s"
     using img imk by fastforce
   moreover have "P (k \<circ> g)"
     by (simp add: P Qg contg img)
   ultimately have "homotopic_with_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
     apply (rule homotopic_with_eq)
     using feq geq apply fastforce+
     using Qeq topspace_euclidean_subtopology by blast
 qed
 
 lemma homotopically_trivial_retraction_null_gen:
   assumes P: "\<And>f. \<lbrakk>continuous_on u f; f ` u \<subseteq> t; Q f\<rbrakk> \<Longrightarrow> P(k \<circ> f)"
       and Q: "\<And>f. \<lbrakk>continuous_on u f; f ` u \<subseteq> s; P f\<rbrakk> \<Longrightarrow> Q(h \<circ> f)"
       and Qeq: "\<And>h k. (\<And>x. x \<in> u \<Longrightarrow> h x = k x) \<Longrightarrow> Q h = Q k"
       and hom: "\<And>f. \<lbrakk>continuous_on u f; f ` u \<subseteq> s; P f\<rbrakk>
                      \<Longrightarrow> \<exists>c. homotopic_with_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"
     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)"
     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 show ?thesis
     apply (rule_tac c = "h c" in that)
     apply (erule homotopic_with_eq)
     using feq apply auto[1]
     apply simp
     using Qeq topspace_euclidean_subtopology 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 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 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)"
     using contf conth continuous_on_compose imh by blast
   moreover have "(f \<circ> h) ` s \<subseteq> u"
     using imf imh by fastforce
   moreover have "P (f \<circ> h)"
     by (simp add: P Qf contf imf)
   moreover have "continuous_on s (g \<circ> h)"
     using contg continuous_on_compose continuous_on_subset conth imh by blast
   moreover have "(g \<circ> h) ` s \<subseteq> u"
     using img imh by fastforce
   moreover have "P (g \<circ> h)"
     by (simp add: P Qg contg img)
   ultimately have "homotopic_with_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
     apply (rule homotopic_with_eq)
     using feq apply auto[1]
     using geq apply auto[1]
     using Qeq topspace_euclidean_subtopology by 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 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 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)"
     using contf conth continuous_on_compose imh by blast
   moreover have "(f \<circ> h) ` s \<subseteq> u"
     using imf imh by fastforce
   moreover have "P (f \<circ> h)"
     by (simp add: P Qf contf imf)
   ultimately obtain c where "homotopic_with_canon P s u (f \<circ> h) (\<lambda>x. c)"
     by (metis hom)
   then have "homotopic_with_canon Q t u (f \<circ> h \<circ> k) ((\<lambda>x. c) \<circ> k)"
     apply (rule homotopic_with_compose_continuous_right [OF homotopic_with_mono])
     using Q by (auto simp: contk imk)
   then show ?thesis
     apply (rule_tac c = c in that)
     apply (erule homotopic_with_eq)
     using feq apply auto[1]
     apply simp
     using Qeq topspace_euclidean_subtopology 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_compose_continuous_map_right by blast
   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_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"
     apply (simp add: o_assoc)
     apply (blast intro: homotopic_with_trans)
     done
   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"
     apply (simp add: o_assoc)
     apply (blast intro: homotopic_with_trans)
     done
   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>
     \<Longrightarrow> X homotopy_equivalent_space Y"
   unfolding homotopy_equivalent_space_def retraction_maps_def
   apply (rule_tac x=r in exI)
   apply (rule_tac x=s in exI)
   apply (auto simp: homotopic_with_equal continuous_map_compose)
   done
 
 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)"
             using homotopic_with_symD continuous_map_into_fulltopology f homotopic_compose_continuous_map_left r by blast
           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"
   apply (simp add: 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"
   apply (simp add: 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
     apply (auto simp: contractible_space_def)
     using homotopic_with_imp_continuous_maps by fastforce
 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
       using that unfolding path_component_of_def
       apply (rule_tac x="h \<circ> (\<lambda>x. (x,b))" in exI)
       apply (simp add: h pathin_def)
       apply (rule continuous_map_compose [OF _ conth])
       apply (auto simp: continuous_map_pairwise intro!: continuous_intros continuous_map_compose continuous_map_id [unfolded id_def])
       done
   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
       apply (rule homotopic_with_trans [OF a])
       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
 next
   assume R: ?rhs
   then show ?lhs
     apply (simp add: 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_compose_continuous_map_right
            [OF homotopic_compose_continuous_map_left [OF b g] f]
     by (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_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 "X homotopy_equivalent_space subtopology X {a}"
       unfolding homotopy_equivalent_space_def
       apply (rule_tac x="\<lambda>x. a" in exI)
       apply (rule_tac x=id in exI)
       apply (auto simp: homotopic_with_sym topspace_subtopology_subset a)
       using connectedin_absolute connectedin_sing contractible_space_alt contractible_space_subtopology_singleton by fastforce
     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_compose_continuous_map_left homotopic_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)
       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)"
         apply (auto simp: case_prod_unfold)
          apply (intro continuous_intros)
         using  \<open>z \<in> S\<close> apply (auto intro: convexD [OF \<open>convex S\<close>, simplified])
         done
     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
   apply (erule ex_forward)+
   apply auto
    apply (metis homotopic_with_id2 topspace_euclidean_subtopology)
   by (metis (full_types) homotopic_with_id2 imageE 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"
 apply (rule homeomorphic_imp_homotopy_eqv)
 using assms homeomorphic_sym linear_homeomorphic_image by auto
 
 lemma homotopy_eqv_translation:
     fixes S :: "'a::real_normed_vector set"
     shows "(+) a ` S homotopy_eqv S"
   apply (rule homeomorphic_imp_homotopy_eqv)
   using homeomorphic_translation homeomorphic_sym by blast
 
 lemma homotopy_eqv_homotopic_triviality_imp:
   fixes S :: "'a::real_normed_vector set"
     and T :: "'b::real_normed_vector set"
     and U :: "'c::real_normed_vector set"
   assumes "S homotopy_eqv T"
       and f: "continuous_on U f" "f ` U \<subseteq> T"
       and g: "continuous_on U g" "g ` U \<subseteq> T"
       and homUS: "\<And>f g. \<lbrakk>continuous_on U f; f ` U \<subseteq> S;
                          continuous_on U g; g ` U \<subseteq> S\<rbrakk>
                          \<Longrightarrow> homotopic_with_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)"
     apply (rule homUS)
     using f g k
     apply (safe intro!: continuous_on_compose h k f elim!: continuous_on_subset)
     apply (force simp: o_def)+
     done
   then have "homotopic_with_canon (\<lambda>x. True) U T (h \<circ> (k \<circ> f)) (h \<circ> (k \<circ> g))"
     apply (rule homotopic_with_compose_continuous_left)
     apply (simp_all add: h)
     done
   moreover have "homotopic_with_canon (\<lambda>x. True) U T (h \<circ> k \<circ> f) (id \<circ> f)"
     apply (rule homotopic_with_compose_continuous_right [where X=T and Y=T])
     apply (auto simp: hom f)
     done
   moreover have "homotopic_with_canon (\<lambda>x. True) U T (h \<circ> k \<circ> g) (id \<circ> g)"
     apply (rule homotopic_with_compose_continuous_right [where X=T and Y=T])
     apply (auto simp: hom g)
     done
   ultimately show "homotopic_with_canon (\<lambda>x. True) U T f g"
     apply (simp add: o_assoc)
     using homotopic_with_trans homotopic_with_sym by blast
 qed
 
 lemma homotopy_eqv_homotopic_triviality:
   fixes S :: "'a::real_normed_vector set"
     and T :: "'b::real_normed_vector set"
     and U :: "'c::real_normed_vector set"
   assumes "S homotopy_eqv T"
     shows "(\<forall>f g. continuous_on U f \<and> f ` U \<subseteq> S \<and>
                    continuous_on U g \<and> g ` U \<subseteq> S
                    \<longrightarrow> homotopic_with_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)"
     apply (rule exE [OF homSU [of "f \<circ> h"]])
     apply (intro continuous_on_compose h)
     using h f  apply (force elim!: continuous_on_subset)+
     done
   then have "homotopic_with_canon (\<lambda>x. True) T U ((f \<circ> h) \<circ> k) ((\<lambda>x. c) \<circ> k)"
     apply (rule homotopic_with_compose_continuous_right [where X=S])
     using k by auto
   moreover have "homotopic_with_canon (\<lambda>x. True) T U (f \<circ> id) (f \<circ> (h \<circ> k))"
     apply (rule homotopic_with_compose_continuous_left [where Y=T])
       apply (simp add: hom homotopic_with_symD)
      using f apply auto
     done
   ultimately show ?thesis
     apply (rule_tac c=c in that)
     apply (simp add: o_def)
     using homotopic_with_trans by blast
 qed
 
 lemma homotopy_eqv_cohomotopic_triviality_null:
   fixes S :: "'a::real_normed_vector set"
     and T :: "'b::real_normed_vector set"
     and U :: "'c::real_normed_vector set"
   assumes "S homotopy_eqv T"
     shows "(\<forall>f. continuous_on S f \<and> f ` S \<subseteq> U
                 \<longrightarrow> (\<exists>c. homotopic_with_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)))"
 apply (rule iffI)
 apply (metis assms homotopy_eqv_cohomotopic_triviality_null_imp)
 by (metis assms homotopy_eqv_cohomotopic_triviality_null_imp homotopy_equivalent_space_sym)
 
 lemma homotopy_eqv_homotopic_triviality_null_imp:
   fixes S :: "'a::real_normed_vector set"
     and T :: "'b::real_normed_vector set"
     and U :: "'c::real_normed_vector set"
   assumes "S homotopy_eqv T"
       and f: "continuous_on U f" "f ` U \<subseteq> T"
       and homSU: "\<And>f. \<lbrakk>continuous_on U f; f ` U \<subseteq> S\<rbrakk>
                       \<Longrightarrow> \<exists>c. homotopic_with_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)"
     apply (rule exE [OF homSU [of "k \<circ> f"]])
     apply (intro continuous_on_compose h)
     using k f  apply (force elim!: continuous_on_subset)+
     done
   then have "homotopic_with_canon (\<lambda>x. True) U T (h \<circ> (k \<circ> f)) (h \<circ> (\<lambda>x. c))"
     apply (rule homotopic_with_compose_continuous_left [where Y=S])
     using h by auto
   moreover have "homotopic_with_canon (\<lambda>x. True) U T (id \<circ> f) ((h \<circ> k) \<circ> f)"
     apply (rule homotopic_with_compose_continuous_right [where X=T])
       apply (simp add: hom homotopic_with_symD)
      using f apply auto
     done
   ultimately show ?thesis
     using homotopic_with_trans by (fastforce simp add: o_def)
 qed
 
 lemma homotopy_eqv_homotopic_triviality_null:
   fixes S :: "'a::real_normed_vector set"
     and T :: "'b::real_normed_vector set"
     and U :: "'c::real_normed_vector set"
   assumes "S homotopy_eqv T"
     shows "(\<forall>f. continuous_on U f \<and> f ` U \<subseteq> S
                   \<longrightarrow> (\<exists>c. homotopic_with_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)))"
 apply (rule iffI)
 apply (metis assms homotopy_eqv_homotopic_triviality_null_imp)
 by (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)
     apply (rule_tac x="\<lambda>x. a" in exI)
     apply (intro assms conjI continuous_on_id' homotopic_into_contractible)
     apply (auto simp: o_def)
     done
 qed
 
 lemma homotopy_eqv_empty1 [simp]:
   fixes S :: "'a::real_normed_vector set"
   shows "S homotopy_eqv ({}::'b::real_normed_vector set) \<longleftrightarrow> S = {}"
   apply (rule iffI)
    apply (metis Abstract_Topology.continuous_map_subtopology_eu emptyE equals0I homotopy_equivalent_space_def image_subset_iff)
   by (simp add: homotopy_eqv_contractible_sets)
 
 lemma homotopy_eqv_empty2 [simp]:
   fixes S :: "'a::real_normed_vector set"
   shows "({}::'b::real_normed_vector set) homotopy_eqv S \<longleftrightarrow> S = {}"
   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 True then show ?thesis
     by simp
 next
   case False then show ?thesis
     by (metis contractible_sing empty_not_insert homotopy_eqv_contractibility homotopy_eqv_contractible_sets)
 qed
 
 lemma homeomorphic_contractible_eq:
   fixes S :: "'a::real_normed_vector set" and T :: "'b::real_normed_vector set"
   shows "S homeomorphic T \<Longrightarrow> (contractible S \<longleftrightarrow> contractible T)"
 by (simp add: homeomorphic_imp_homotopy_eqv homotopy_eqv_contractibility)
 
 lemma homeomorphic_contractible:
   fixes S :: "'a::real_normed_vector set" and T :: "'b::real_normed_vector set"
   shows "\<lbrakk>contractible S; S homeomorphic T\<rbrakk> \<Longrightarrow> contractible T"
   by (metis homeomorphic_contractible_eq)
 
 
 subsection\<^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>x. g(f x) = x" "\<And>y. f(g y) = y"
     by (rule isomorphisms_UNIV_UNIV) blast
   with \<open>bounded S\<close> have "bounded (f ` S)"
     using bounded_linear_image linear_linear by blast
   have "connected (f ` (-S))"
     using connected_linear_image assms \<open>linear f\<close> by blast
   moreover have "f ` (-S) = - (f ` S)"
     apply (rule bij_image_Compl_eq)
     apply (auto simp: bij_def)
      apply (metis \<open>\<And>x. g (f x) = x\<close> injI)
     by (metis UNIV_I \<open>\<And>y. f (g y) = y\<close> image_iff)
   finally have "connected (- (f ` S))"
     by simp
   then have "f ` S = {}"
     using \<open>bounded (f ` S)\<close> bounded_connected_Compl_real by blast
   then show ?thesis
     by blast
 qed
 
 
 subsection\<^marker>\<open>tag unimportant\<close>\<open>Some Uncountable Sets\<close>
 
 lemma uncountable_closed_segment:
   fixes a :: "'a::real_normed_vector"
   assumes "a \<noteq> b" shows "uncountable (closed_segment a b)"
 unfolding path_image_linepath [symmetric] path_image_def
   using inj_on_linepath [OF assms] uncountable_closed_interval [of 0 1]
         countable_image_inj_on by auto
 
 lemma uncountable_open_segment:
   fixes a :: "'a::real_normed_vector"
   assumes "a \<noteq> b" shows "uncountable (open_segment a b)"
   by (simp add: assms open_segment_def uncountable_closed_segment uncountable_minus_countable)
 
 lemma uncountable_convex:
   fixes a :: "'a::real_normed_vector"
   assumes "convex S" "a \<in> S" "b \<in> S" "a \<noteq> b"
     shows "uncountable S"
 proof -
   have "uncountable (closed_segment a b)"
     by (simp add: uncountable_closed_segment assms)
   then show ?thesis
     by (meson assms convex_contains_segment countable_subset)
 qed
 
 lemma uncountable_ball:
   fixes a :: "'a::euclidean_space"
   assumes "r > 0"
     shows "uncountable (ball a r)"
 proof -
   have "uncountable (open_segment a (a + r *\<^sub>R (SOME i. i \<in> Basis)))"
     by (metis Basis_zero SOME_Basis add_cancel_right_right assms less_le scale_eq_0_iff uncountable_open_segment)
   moreover have "open_segment a (a + r *\<^sub>R (SOME i. i \<in> Basis)) \<subseteq> ball a r"
     using assms by (auto simp: in_segment algebra_simps dist_norm SOME_Basis)
   ultimately show ?thesis
     by (metis countable_subset)
 qed
 
 lemma ball_minus_countable_nonempty:
   assumes "countable (A :: 'a :: euclidean_space set)" "r > 0"
   shows   "ball z r - A \<noteq> {}"
 proof
   assume *: "ball z r - A = {}"
   have "uncountable (ball z r - A)"
     by (intro uncountable_minus_countable assms uncountable_ball)
   thus False by (subst (asm) *) auto
 qed
 
 lemma uncountable_cball:
   fixes a :: "'a::euclidean_space"
   assumes "r > 0"
   shows "uncountable (cball a r)"
   using assms countable_subset uncountable_ball by auto
 
 lemma pairwise_disjnt_countable:
   fixes \<N> :: "nat set set"
   assumes "pairwise disjnt \<N>"
     shows "countable \<N>"
 proof -
   have "inj_on (\<lambda>X. SOME n. n \<in> X) (\<N> - {{}})"
     apply (clarsimp simp add: inj_on_def)
     by (metis assms disjnt_insert2 insert_absorb pairwise_def subsetI subset_empty tfl_some)
   then show ?thesis
     by (metis countable_Diff_eq countable_def)
 qed
 
 lemma pairwise_disjnt_countable_Union:
     assumes "countable (\<Union>\<N>)" and pwd: "pairwise disjnt \<N>"
     shows "countable \<N>"
 proof -
   obtain f :: "_ \<Rightarrow> nat" where f: "inj_on f (\<Union>\<N>)"
     using assms by blast
   then have "pairwise disjnt (\<Union> X \<in> \<N>. {f ` X})"
     using assms by (force simp: pairwise_def disjnt_inj_on_iff [OF f])
   then have "countable (\<Union> X \<in> \<N>. {f ` X})"
     using pairwise_disjnt_countable by blast
   then show ?thesis
     by (meson pwd countable_image_inj_on disjoint_image f inj_on_image pairwise_disjnt_countable)
 qed
 
 lemma connected_uncountable:
   fixes S :: "'a::metric_space set"
   assumes "connected S" "a \<in> S" "b \<in> S" "a \<noteq> b" shows "uncountable S"
 proof -
   have "continuous_on S (dist a)"
     by (intro continuous_intros)
   then have "connected (dist a ` S)"
     by (metis connected_continuous_image \<open>connected S\<close>)
   then have "closed_segment 0 (dist a b) \<subseteq> (dist a ` S)"
     by (simp add: assms closed_segment_subset is_interval_connected_1 is_interval_convex)
   then have "uncountable (dist a ` S)"
     by (metis \<open>a \<noteq> b\<close> countable_subset dist_eq_0_iff uncountable_closed_segment)
   then show ?thesis
     by blast
 qed
 
 lemma path_connected_uncountable:
   fixes S :: "'a::metric_space set"
   assumes "path_connected S" "a \<in> S" "b \<in> S" "a \<noteq> b" shows "uncountable S"
   using path_connected_imp_connected assms connected_uncountable by metis
 
 lemma connected_finite_iff_sing:
   fixes S :: "'a::metric_space set"
   assumes "connected S"
   shows "finite S \<longleftrightarrow> S = {} \<or> (\<exists>a. S = {a})"  (is "_ = ?rhs")
 proof -
   have "uncountable S" if "\<not> ?rhs"
     using connected_uncountable assms that by blast
   then show ?thesis
     using uncountable_infinite by auto
 qed
 
 lemma connected_card_eq_iff_nontrivial:
   fixes S :: "'a::metric_space set"
   shows "connected S \<Longrightarrow> uncountable S \<longleftrightarrow> \<not>(\<exists>a. S \<subseteq> {a})"
   apply (auto simp: countable_finite finite_subset)
   by (metis connected_uncountable is_singletonI' is_singleton_the_elem subset_singleton_iff)
 
 lemma simple_path_image_uncountable:
   fixes g :: "real \<Rightarrow> 'a::metric_space"
   assumes "simple_path g"
   shows "uncountable (path_image g)"
 proof -
   have "g 0 \<in> path_image g" "g (1/2) \<in> path_image g"
     by (simp_all add: path_defs)
   moreover have "g 0 \<noteq> g (1/2)"
     using assms by (fastforce simp add: simple_path_def)
   ultimately show ?thesis
     apply (simp add: assms connected_card_eq_iff_nontrivial connected_simple_path_image)
     by blast
 qed
 
 lemma arc_image_uncountable:
   fixes g :: "real \<Rightarrow> 'a::metric_space"
   assumes "arc g"
   shows "uncountable (path_image g)"
   by (simp add: arc_imp_simple_path assms simple_path_image_uncountable)
 
 
 subsection\<^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
                 apply (rule the_equality [symmetric])
                 using x  apply (auto simp: dest: **)
                 done
             qed (use x in auto)
           qed
         qed
         have "uncountable (closed_segment ?m c)"
           by (metis \<open>c \<noteq> ?m\<close> uncountable_closed_segment)
         then show False
           using closb * [OF \<open>a \<in> U\<close> \<open>a \<notin> S\<close> ncoll_mca] * [OF \<open>b \<in> U\<close> \<open>b \<notin> S\<close> ncoll_mcb]
           apply (simp add: closed_segment_commute)
           by (simp add: countable_subset)
       qed
       then show ?thesis
         by (force intro: that)
     qed
     show ?thesis
     proof (intro exI conjI)
       have "path_image (linepath a z +++ linepath z b) \<subseteq> U"
         by (metis \<open>a \<in> U\<close> \<open>b \<in> U\<close> \<open>closed_segment ?m c \<subseteq> U\<close> z \<open>convex U\<close> closed_segment_subset contra_subsetD path_image_linepath subset_path_image_join)
       with disjS show "path_image (linepath a z +++ linepath z b) \<subseteq> U - S"
         by (force simp: path_image_join)
     qed auto
   qed
 qed
 
 
 corollary connected_convex_diff_countable:
   fixes U :: "'a::euclidean_space set"
   assumes "convex U" "\<not> collinear U" "countable S"
   shows "connected(U - S)"
   by (simp add: assms path_connected_convex_diff_countable path_connected_imp_connected)
 
 lemma path_connected_punctured_convex:
   assumes "convex S" and aff: "aff_dim S \<noteq> 1"
     shows "path_connected(S - {a})"
 proof -
   consider "aff_dim S = -1" | "aff_dim S = 0" | "aff_dim S \<ge> 2"
     using assms aff_dim_geq [of S] by linarith
   then show ?thesis
   proof cases
     assume "aff_dim S = -1"
     then show ?thesis
       by (metis aff_dim_empty empty_Diff path_connected_empty)
   next
     assume "aff_dim S = 0"
     then show ?thesis
       by (metis aff_dim_eq_0 Diff_cancel Diff_empty Diff_insert0 convex_empty convex_imp_path_connected path_connected_singleton singletonD)
   next
     assume ge2: "aff_dim S \<ge> 2"
     then have "\<not> collinear S"
     proof (clarsimp simp add: collinear_affine_hull)
       fix u v
       assume "S \<subseteq> affine hull {u, v}"
       then have "aff_dim S \<le> aff_dim {u, v}"
         by (metis (no_types) aff_dim_affine_hull aff_dim_subset)
       with ge2 show False
         by (metis (no_types) aff_dim_2 antisym aff not_numeral_le_zero one_le_numeral order_trans)
     qed
     then show ?thesis
       apply (rule path_connected_convex_diff_countable [OF \<open>convex S\<close>])
       by simp
   qed
 qed
 
 lemma connected_punctured_convex:
   shows "\<lbrakk>convex S; aff_dim S \<noteq> 1\<rbrakk> \<Longrightarrow> connected(S - {a})"
   using path_connected_imp_connected path_connected_punctured_convex by blast
 
 lemma path_connected_complement_countable:
   fixes S :: "'a::euclidean_space set"
   assumes "2 \<le> DIM('a)" "countable S"
   shows "path_connected(- S)"
 proof -
   have "path_connected(UNIV - S)"
     apply (rule path_connected_convex_diff_countable)
     using assms by (auto simp: collinear_aff_dim [of "UNIV :: 'a set"])
   then show ?thesis
     by (simp add: Compl_eq_Diff_UNIV)
 qed
 
 proposition path_connected_openin_diff_countable:
   fixes S :: "'a::euclidean_space set"
   assumes "connected S" and ope: "openin (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> collinear (ball x r \<inter> affine hull S)"
         apply (simp add: collinear_aff_dim)
         by (metis (no_types, hide_lams) aff_dim_convex_Int_open IntI open_ball \<open>0 < r\<close> aff_dim_affine_hull affine_affine_hull affine_imp_convex centre_in_ball empty_iff hull_subset inf_commute subsetCE that)
       then have *: "path_connected ((ball x r \<inter> affine hull S) - T)"
         by (rule path_connected_convex_diff_countable [OF conv _ \<open>countable T\<close>])
       have ST: "ball x r \<inter> affine hull S - T \<subseteq> S - T"
         using subS by auto
       show ?thesis
       proof (intro exI conjI)
         show "x \<in> ball x r \<inter> affine hull S"
           using \<open>x \<in> S\<close> \<open>r > 0\<close> by (simp add: hull_inc)
         have "openin (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"
       apply (subst add_less_cancel_left)
       apply (rule mult_strict_left_mono)
       using nou \<open>0 < r\<close> yx
        apply (simp_all add: field_simps)
       done
     also have "\<dots> = norm x"
       by simp
     finally show False by simp
   qed
   have "inj f"
     unfolding f_def
   proof (clarsimp simp: inj_on_def)
     fix x y
     assume "(1 - norm (x - a) / r) *\<^sub>R (u - a) + x =
             (1 - norm (y - a) / r) *\<^sub>R (u - a) + y"
     then have eq: "(x - a) + (norm (y - a) / r) *\<^sub>R (u - a) = (y - a) + (norm (x - a) / r) *\<^sub>R (u - a)"
       by (auto simp: algebra_simps)
     show "x=y"
     proof (cases "norm (x - a) = norm (y - a)")
       case True
       then show ?thesis
         using eq by auto
     next
       case False
       then consider "norm (x - a) < norm (y - a)" | "norm (x - a) > norm (y - a)"
         by linarith
       then have "False"
       proof cases
         case 1 show False
           using * [OF _ nou 1] eq by simp
       next
         case 2 with * [OF eq nou] show False
           by auto
       qed
       then show "x=y" ..
     qed
   qed
   then have inj_onf: "inj_on f (cball a r \<inter> T)"
     using inj_on_Int by fastforce
   have contf: "continuous_on (cball a r \<inter> T) f"
     unfolding f_def using \<open>0 < r\<close>  by (intro continuous_intros) blast
   have fim: "f ` (cball a r \<inter> T) = cball a r \<inter> T"
   proof
     have *: "norm (y + (1 - norm y / r) *\<^sub>R u) \<le> r" if "norm y \<le> r" "norm u < r" for y u::'a
     proof -
       have "norm (y + (1 - norm y / r) *\<^sub>R u) \<le> norm y + norm((1 - norm y / r) *\<^sub>R u)"
         using norm_triangle_ineq by blast
       also have "\<dots> = norm y + abs(1 - norm y / r) * norm u"
         by simp
       also have "\<dots> \<le> r"
       proof -
         have "(r - norm u) * (r - norm y) \<ge> 0"
           using that by auto
         then have "r * norm u + r * norm y \<le> r * r + norm u * norm y"
           by (simp add: algebra_simps)
         then show ?thesis
         using that \<open>0 < r\<close> by (simp add: abs_if field_simps)
       qed
       finally show ?thesis .
     qed
     have "f ` (cball a r) \<subseteq> cball a r"
       apply (clarsimp simp add: dist_norm norm_minus_commute f_def)
       using * by (metis diff_add_eq diff_diff_add diff_diff_eq2 norm_minus_commute nou)
     moreover have "f ` T \<subseteq> T"
       unfolding f_def using \<open>affine T\<close> \<open>a \<in> T\<close> \<open>u \<in> T\<close>
       by (force simp: add.commute mem_affine_3_minus)
     ultimately show "f ` (cball a r \<inter> T) \<subseteq> cball a r \<inter> T"
       by blast
   next
     show "cball a r \<inter> T \<subseteq> f ` (cball a r \<inter> T)"
     proof (clarsimp simp add: dist_norm norm_minus_commute)
       fix x
       assume x: "norm (x - a) \<le> r" and "x \<in> T"
       have "\<exists>v \<in> {0..1}. ((1 - v) * r - norm ((x - a) - v *\<^sub>R (u - a))) \<bullet> 1 = 0"
         by (rule ivt_decreasing_component_on_1) (auto simp: x continuous_intros)
       then obtain v where "0 \<le> v" "v \<le> 1"
         and v: "(1 - v) * r = norm ((x - a) - v *\<^sub>R (u - a))"
         by auto
       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 "\<exists>g. homeomorphism (cball a r \<inter> T) (cball a r \<inter> T) f g"
     apply (rule homeomorphism_compact [OF _ contf fim inj_onf])
     apply (simp add: affine_closed compact_Int_closed \<open>affine T\<close>)
     done
   then show ?thesis
     apply (rule exE)
     apply (erule_tac f=f in that)
     using \<open>r > 0\<close>
      apply (simp_all add: f_def dist_norm norm_minus_commute)
     done
 qed
 
 corollary\<^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"
         apply (simp add: ff_def)
         apply (rule continuous_on_cases)
         using homeomorphism_cont1 [OF hom]
             apply (auto simp: affine_closed \<open>affine T\<close> fid)
         done
       then show "continuous_on S ff"
         apply (rule continuous_on_subset)
         using ST by auto
       have "continuous_on ((cball a r \<inter> T) \<union> (T - ball a r)) gg"
         apply (simp add: gg_def)
         apply (rule continuous_on_cases)
         using homeomorphism_cont2 [OF hom]
             apply (auto simp: affine_closed \<open>affine T\<close> gid)
         done
       then show "continuous_on S gg"
         apply (rule continuous_on_subset)
         using ST by auto
       show "ff ` S \<subseteq> S"
       proof (clarsimp simp add: ff_def)
         fix x
         assume "x \<in> S" and x: "dist a x < r" and "x \<in> T"
         then have "f x \<in> cball a r \<inter> T"
           using homeomorphism_image1 [OF hom] by force
         then show "f x \<in> S"
           using ST(1) \<open>x \<in> T\<close> gid hom homeomorphism_def x by fastforce
       qed
       show "gg ` S \<subseteq> S"
       proof (clarsimp simp add: gg_def)
         fix x
         assume "x \<in> S" and x: "dist a x < r" and "x \<in> T"
         then have "g x \<in> cball a r \<inter> T"
           using homeomorphism_image2 [OF hom] by force
         then have "g x \<in> ball a r"
           using homeomorphism_apply2 [OF hom]
             by (metis Diff_Diff_Int Diff_iff  \<open>x \<in> T\<close> cball_def fid le_less mem_Collect_eq mem_ball mem_sphere x)
         then show "g x \<in> S"
           using ST(1) \<open>g x \<in> cball a r \<inter> T\<close> by force
         qed
       show "\<And>x. x \<in> S \<Longrightarrow> gg (ff x) = x"
         apply (simp add: ff_def gg_def)
         using homeomorphism_apply1 [OF hom] homeomorphism_image1 [OF hom]
         apply auto
         apply (metis Int_iff homeomorphism_apply1 [OF hom] fid image_eqI less_eq_real_def mem_cball mem_sphere)
         done
       show "\<And>x. x \<in> S \<Longrightarrow> ff (gg x) = x"
         apply (simp add: ff_def gg_def)
         using homeomorphism_apply2 [OF hom] homeomorphism_image2 [OF hom]
         apply auto
         apply (metis Int_iff fid image_eqI less_eq_real_def mem_cball mem_sphere)
         done
     qed
     show "ff u = v"
       using u by (auto simp: ff_def \<open>f u = v\<close>)
     show "{x. \<not> (ff x = x \<and> gg x = x)} \<subseteq> ball a r \<inter> T"
       by (auto simp: ff_def gg_def)
   qed
 qed
 
 
 proposition\<^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
       apply (rule_tac x="S \<inter> ball d r" in exI)
       apply (intro conjI)
         apply (simp add: openin_open_Int)
        apply (simp add: \<open>0 < r\<close> that)
       apply (blast intro: *)
       done
   qed
   have "\<exists>f g. homeomorphism T T f g \<and> f a = b \<and>
               {x. \<not> (f x = x \<and> g x = x)} \<subseteq> S \<and> bounded {x. \<not> (f x = x \<and> g x = x)}"
     apply (rule connected_equivalence_relation [OF S], safe)
       apply (blast intro: 1 2 3)+
     done
   then show ?thesis
     using that by auto
 qed
 
 
 lemma homeomorphism_moving_points_exists_gen:
   assumes K: "finite K" "\<And>i. i \<in> K \<Longrightarrow> x i \<in> S \<and> y i \<in> S"
              "pairwise (\<lambda>i j. (x i \<noteq> x j) \<and> (y i \<noteq> y j)) K"
       and "2 \<le> aff_dim S"
       and ope: "openin (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"
     apply (rule affine_hull_Diff)
     apply (auto simp: insert)
     using \<open>y i \<in> S\<close> insert.hyps(2) xney xyS by fastforce
   have f_in_S: "f x \<in> S" if "x \<in> S" for x
     using homfg fg_sub homeomorphism_apply1 \<open>S \<subseteq> T\<close>
   proof -
     have "(f (f x) \<noteq> f x \<or> g (f x) \<noteq> f x) \<or> f x \<in> S"
       by (metis \<open>S \<subseteq> T\<close> homfg subsetD homeomorphism_apply1 that)
     then show ?thesis
       using fg_sub by force
   qed
   obtain h k where homhk: "homeomorphism T T h k" and heq: "h (f (x i)) = y i"
                and hk_sub: "{x. \<not> (h x = x \<and> k x = x)} \<subseteq> S - y ` K"
                and bo_hk:  "bounded {x. \<not> (h x = x \<and> k x = x)}"
   proof (rule homeomorphism_moving_point [of "S - y`K" T "f(x i)" "y i"])
     show "openin (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
     show "f (x i) \<in> S - y ` K"
       apply (auto simp: f_in_S \<open>x i \<in> S\<close>)
         by (metis feq homfg \<open>x i \<in> S\<close> homeomorphism_def \<open>S \<subseteq> T\<close> \<open>i \<notin> K\<close> subsetCE xney xyS)
     show "y i \<in> S - y ` K"
       using insert.hyps xney by (auto simp: \<open>y i \<in> S\<close>)
   qed blast
   show ?case
   proof (intro exI conjI)
     show "homeomorphism T T (h \<circ> f) (g \<circ> k)"
       using homfg homhk homeomorphism_compose by blast
     show "\<forall>i \<in> insert i K. (h \<circ> f) (x i) = y i"
       using feq hk_sub by (auto simp: heq)
     show "{x. \<not> ((h \<circ> f) x = x \<and> (g \<circ> k) x = x)} \<subseteq> S"
       using fg_sub hk_sub by force
     have "bounded ({x. \<not>(f x = x \<and> g x = x)} \<union> {x. \<not>(h x = x \<and> k x = x)})"
       using bo_fg bo_hk bounded_Un by blast
     then show "bounded {x. \<not> ((h \<circ> f) x = x \<and> (g \<circ> k) x = x)}"
       by (rule bounded_subset) auto
   qed
 qed
 
 proposition\<^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"
       apply (simp add: f_def)
       apply (intro continuous_intros)
       using assms by auto
     have "f ` {a..b} = {c..d}"
       unfolding f_def image_affinity_atLeastAtMost
       using assms sum_sqs_eq by (auto simp: 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"
       apply (simp add: f_def)
       apply (rule continuous_on_cases_le [OF continuous_on_subset [OF cf1] continuous_on_subset [OF cf2]])
       using le eq apply (force)+
       done
     have "f ` cbox a b = f1 ` cbox a b" "f ` cbox b c = f2 ` cbox b c"
       unfolding f_def using eq by force+
     then show "f ` cbox a c = cbox u w"
       apply (simp only: ac uw image_Un)
       by (metis hom_ab hom_bc homeomorphism_def)
     have neq12: "f1 x \<noteq> f2 y" if x: "a \<le> x" "x \<le> b" and y: "b < y" "y \<le> c" for x y
     proof -
       have "f1 x \<in> cbox u v"
         by (metis hom_ab homeomorphism_def image_eqI mem_box_real(2) x)
       moreover have "f2 y \<in> cbox v w"
         by (metis (full_types) hom_bc homeomorphism_def image_subset_iff mem_box_real(2) not_le not_less_iff_gr_or_eq order_refl y)
       moreover have "f2 y \<noteq> f2 b"
         by (metis cancel_comm_monoid_add_class.diff_cancel diff_gt_0_iff_gt hom_bc homeomorphism_def le(2) less_imp_le less_numeral_extra(3) mem_box_real(2) order_refl y)
       ultimately show ?thesis
         using le eq by simp
     qed
     have "inj_on f1 (cbox a b)"
       by (metis (full_types) hom_ab homeomorphism_def inj_onI)
     moreover have "inj_on f2 (cbox b c)"
       by (metis (full_types) hom_bc homeomorphism_def inj_onI)
     ultimately show "inj_on f (cbox a c)"
       apply (simp (no_asm) add: inj_on_def)
       apply (simp add: f_def inj_on_eq_iff)
       using neq12  apply force
       done
   qed auto
   then obtain g where "homeomorphism (cbox a c) (cbox u w) f g" ..
   then show ?thesis
     apply (rule that)
     using eq le by (auto simp: f_def)
 qed
 
 lemma homeomorphism_grouping_point_3:
   fixes a::real
   assumes cbox_sub: "cbox c d \<subseteq> box a b" "cbox u v \<subseteq> box a b"
       and box_ne: "box c d \<noteq> {}" "box u v \<noteq> {}"
   obtains f g where "homeomorphism (cbox a b) (cbox a b) f g" "f a = a" "f b = b"
                     "\<And>x. x \<in> cbox c d \<Longrightarrow> f x \<in> cbox u v"
 proof -
   have less: "a < c" "a < u" "d < b" "v < b" "c < d" "u < v" "cbox c d \<noteq> {}"
     using assms
     by (simp_all add: cbox_sub subset_eq)
   obtain f1 g1 where 1: "homeomorphism (cbox a c) (cbox a u) f1 g1"
                    and f1_eq: "f1 a = a" "f1 c = u"
     using homeomorphism_grouping_point_1 [OF \<open>a < c\<close> \<open>a < u\<close>] .
   obtain f2 g2 where 2: "homeomorphism (cbox c d) (cbox u v) f2 g2"
                    and f2_eq: "f2 c = u" "f2 d = v"
     using homeomorphism_grouping_point_1 [OF \<open>c < d\<close> \<open>u < v\<close>] .
   obtain f3 g3 where 3: "homeomorphism (cbox d b) (cbox v b) f3 g3"
                    and f3_eq: "f3 d = v" "f3 b = b"
     using homeomorphism_grouping_point_1 [OF \<open>d < b\<close> \<open>v < b\<close>] .
   obtain f4 g4 where 4: "homeomorphism (cbox a d) (cbox a v) f4 g4" and "f4 a = a" "f4 d = v"
                  and f4_eq: "\<And>x. x \<in> cbox a c \<Longrightarrow> f4 x = f1 x" "\<And>x. x \<in> cbox c d \<Longrightarrow> f4 x = f2 x"
     using homeomorphism_grouping_point_2 [OF 1 2] less  by (auto simp: f1_eq f2_eq)
   obtain f g where fg: "homeomorphism (cbox a b) (cbox a b) f g" "f a = a" "f b = b"
                and f_eq: "\<And>x. x \<in> cbox a d \<Longrightarrow> f x = f4 x" "\<And>x. x \<in> cbox d b \<Longrightarrow> f x = f3 x"
     using homeomorphism_grouping_point_2 [OF 4 3] less by (auto simp: f4_eq f3_eq f2_eq f1_eq)
   show ?thesis
     apply (rule that [OF fg])
     using f4_eq f_eq homeomorphism_image1 [OF 2]
     apply simp
     by (metis atLeastAtMost_iff box_real(1) box_real(2) cbox_sub(1) greaterThanLessThan_iff imageI less_eq_real_def subset_eq)
 qed
 
 
 lemma homeomorphism_grouping_point_4:
   fixes T :: "real set"
   assumes "open U" "open S" "connected S" "U \<noteq> {}" "finite K" "K \<subseteq> S" "U \<subseteq> S" "S \<subseteq> T"
   obtains f g where "homeomorphism T T f g"
                     "\<And>x. x \<in> K \<Longrightarrow> f x \<in> U" "{x. (\<not> (f x = x \<and> g x = x))} \<subseteq> S"
                     "bounded {x. (\<not> (f x = x \<and> g x = x))}"
 proof -
   obtain c d where "box c d \<noteq> {}" "cbox c d \<subseteq> U"
   proof -
     obtain u where "u \<in> U"
       using \<open>U \<noteq> {}\<close> by blast
     then obtain e where "e > 0" "cball u e \<subseteq> U"
       using \<open>open U\<close> open_contains_cball by blast
     then show ?thesis
       by (rule_tac c=u and d="u+e" in that) (auto simp: dist_norm subset_iff)
   qed
   have "compact K"
     by (simp add: \<open>finite K\<close> finite_imp_compact)
   obtain a b where "box a b \<noteq> {}" "K \<subseteq> cbox a b" "cbox a b \<subseteq> S"
   proof (cases "K = {}")
     case True then show ?thesis
       using \<open>box c d \<noteq> {}\<close> \<open>cbox c d \<subseteq> U\<close> \<open>U \<subseteq> S\<close> that by blast
   next
     case False
     then obtain a b where "a \<in> K" "b \<in> K"
             and a: "\<And>x. x \<in> K \<Longrightarrow> a \<le> x" and b: "\<And>x. x \<in> K \<Longrightarrow> x \<le> b"
       using compact_attains_inf compact_attains_sup by (metis \<open>compact K\<close>)+
     obtain e where "e > 0" "cball b e \<subseteq> S"
       using \<open>open S\<close> open_contains_cball
       by (metis \<open>b \<in> K\<close> \<open>K \<subseteq> S\<close> subsetD)
     show ?thesis
     proof
       show "box a (b + e) \<noteq> {}"
         using \<open>0 < e\<close> \<open>b \<in> K\<close> a by force
       show "K \<subseteq> cbox a (b + e)"
         using \<open>0 < e\<close> a b by fastforce
       have "a \<in> S"
         using \<open>a \<in> K\<close> assms(6) by blast
       have "b + e \<in> S"
         using \<open>0 < e\<close> \<open>cball b e \<subseteq> S\<close>  by (force simp: dist_norm)
       show "cbox a (b + e) \<subseteq> S"
         using \<open>a \<in> S\<close> \<open>b + e \<in> S\<close> \<open>connected S\<close> connected_contains_Icc by auto
     qed
   qed
   obtain w z where "cbox w z \<subseteq> S" and sub_wz: "cbox a b \<union> cbox c d \<subseteq> box w z"
   proof -
     have "a \<in> S" "b \<in> S"
       using \<open>box a b \<noteq> {}\<close> \<open>cbox a b \<subseteq> S\<close> by auto
     moreover have "c \<in> S" "d \<in> S"
       using \<open>box c d \<noteq> {}\<close> \<open>cbox c d \<subseteq> U\<close> \<open>U \<subseteq> S\<close> by force+
     ultimately have "min a c \<in> S" "max b d \<in> S"
       by linarith+
     then obtain e1 e2 where "e1 > 0" "cball (min a c) e1 \<subseteq> S" "e2 > 0" "cball (max b d) e2 \<subseteq> S"
       using \<open>open S\<close> open_contains_cball by metis
     then have *: "min a c - e1 \<in> S" "max b d + e2 \<in> S"
       by (auto simp: dist_norm)
     show ?thesis
     proof
       show "cbox (min a c - e1) (max b d+ e2) \<subseteq> S"
         using * \<open>connected S\<close> connected_contains_Icc by auto
       show "cbox a b \<union> cbox c d \<subseteq> box (min a c - e1) (max b d + e2)"
         using \<open>0 < e1\<close> \<open>0 < e2\<close> by auto
     qed
   qed
   then
   obtain f g where hom: "homeomorphism (cbox w z) (cbox w z) f g"
                and "f w = w" "f z = z"
                and fin: "\<And>x. x \<in> cbox a b \<Longrightarrow> f x \<in> cbox c d"
     using homeomorphism_grouping_point_3 [of a b w z c d]
     using \<open>box a b \<noteq> {}\<close> \<open>box c d \<noteq> {}\<close> by blast
   have contfg: "continuous_on (cbox w z) f" "continuous_on (cbox w z) g"
     using hom homeomorphism_def by blast+
   define f' where "f' \<equiv> \<lambda>x. if x \<in> cbox w z then f x else x"
   define g' where "g' \<equiv> \<lambda>x. if x \<in> cbox w z then g x else x"
   show ?thesis
   proof
     have T: "cbox w z \<union> (T - box w z) = T"
       using \<open>cbox w z \<subseteq> S\<close> \<open>S \<subseteq> T\<close> by auto
     show "homeomorphism T T f' g'"
     proof
       have clo: "closedin (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 "continuous_on (cbox w z \<union> (T - box w z)) f'" "continuous_on (cbox w z \<union> (T - box w z)) g'"
         unfolding f'_def g'_def
          apply (safe intro!: continuous_on_cases_local contfg continuous_on_id clo)
          apply (simp_all add: closed_subset)
         using \<open>f w = w\<close> \<open>f z = z\<close> apply force
         by (metis \<open>f w = w\<close> \<open>f z = z\<close> hom homeomorphism_def less_eq_real_def mem_box_real(2))
       then show "continuous_on T f'" "continuous_on T g'"
         by (simp_all only: T)
       show "f' ` T \<subseteq> T"
         unfolding f'_def
         by clarsimp (metis \<open>cbox w z \<subseteq> S\<close> \<open>S \<subseteq> T\<close> subsetD hom homeomorphism_def imageI mem_box_real(2))
       show "g' ` T \<subseteq> T"
         unfolding g'_def
         by clarsimp (metis \<open>cbox w z \<subseteq> S\<close> \<open>S \<subseteq> T\<close> subsetD hom homeomorphism_def imageI mem_box_real(2))
       show "\<And>x. x \<in> T \<Longrightarrow> g' (f' x) = x"
         unfolding f'_def g'_def
         using homeomorphism_apply1 [OF hom]  homeomorphism_image1 [OF hom] by fastforce
       show "\<And>y. y \<in> T \<Longrightarrow> f' (g' y) = y"
         unfolding f'_def g'_def
         using homeomorphism_apply2 [OF hom]  homeomorphism_image2 [OF hom] by fastforce
     qed
     show "\<And>x. x \<in> K \<Longrightarrow> f' x \<in> U"
       using fin sub_wz \<open>K \<subseteq> cbox a b\<close> \<open>cbox c d \<subseteq> U\<close> by (force simp: f'_def)
     show "{x. \<not> (f' x = x \<and> g' x = x)} \<subseteq> S"
       using \<open>cbox w z \<subseteq> S\<close> by (auto simp: f'_def g'_def)
     show "bounded {x. \<not> (f' x = x \<and> g' x = x)}"
       apply (rule bounded_subset [of "cbox w z"])
       using bounded_cbox apply blast
       apply (auto simp: f'_def g'_def)
       done
   qed
 qed
 
 proposition\<^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, hide_lams) DIM_real UNIV_eq_I range_eqI)
   obtain f g where hom: "homeomorphism (h ` T) (h ` T) f g"
                and f: "\<And>x. x \<in> h ` K \<Longrightarrow> f x \<in> h ` U"
                and sub: "{x. \<not> (f x = x \<and> g x = x)} \<subseteq> h ` S"
                and bou: "bounded {x. \<not> (f x = x \<and> g x = x)}"
     apply (rule homeomorphism_grouping_point_4 [of "h ` U" "h ` S" "h ` K" "h ` T"])
     by (simp_all add: assms image_mono  \<open>linear h\<close> open_surjective_linear_image connected_linear_image ranh)
   have jf: "j (f (h x)) = x \<longleftrightarrow> f (h x) = h x" for x
     by (metis hj)
   have jg: "j (g (h x)) = x \<longleftrightarrow> g (h x) = h x" for x
     by (metis hj)
   have cont_hj: "continuous_on X h"  "continuous_on Y j" for X Y
     by (simp_all add: \<open>linear h\<close> \<open>linear j\<close> linear_linear linear_continuous_on)
   show ?thesis
   proof
     show "homeomorphism T T (j \<circ> f \<circ> h) (j \<circ> g \<circ> h)"
     proof
       show "continuous_on T (j \<circ> f \<circ> h)" "continuous_on T (j \<circ> g \<circ> h)"
         using hom homeomorphism_def
         by (blast intro: continuous_on_compose cont_hj)+
       show "(j \<circ> f \<circ> h) ` T \<subseteq> T" "(j \<circ> g \<circ> h) ` T \<subseteq> T"
         by auto (metis (mono_tags, hide_lams) hj(1) hom homeomorphism_def imageE imageI)+
       show "\<And>x. x \<in> T \<Longrightarrow> (j \<circ> g \<circ> h) ((j \<circ> f \<circ> h) x) = x"
         using hj hom homeomorphism_apply1 by fastforce
       show "\<And>y. y \<in> T \<Longrightarrow> (j \<circ> f \<circ> h) ((j \<circ> g \<circ> h) y) = y"
         using hj hom homeomorphism_apply2 by fastforce
     qed
     show "{x. \<not> ((j \<circ> f \<circ> h) x = x \<and> (j \<circ> g \<circ> h) x = x)} \<subseteq> S"
       apply (clarsimp simp: jf jg hj)
       using sub hj
       apply (drule_tac c="h x" in subsetD, force)
       by (metis imageE)
     have "bounded (j ` {x. (\<not> (f x = x \<and> g x = x))})"
       by (rule bounded_linear_image [OF bou]) (use \<open>linear j\<close> linear_conv_bounded_linear in auto)
     moreover
     have *: "{x. \<not>((j \<circ> f \<circ> h) x = x \<and> (j \<circ> g \<circ> h) x = x)} = j ` {x. (\<not> (f x = x \<and> g x = x))}"
       using hj by (auto simp: jf jg image_iff, metis+)
     ultimately show "bounded {x. \<not> ((j \<circ> f \<circ> h) x = x \<and> (j \<circ> g \<circ> h) x = x)}"
       by metis
     show "\<And>x. x \<in> K \<Longrightarrow> (j \<circ> f \<circ> h) x \<in> U"
       using f hj by fastforce
   qed
 qed
 
 
 proposition\<^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"
     apply (rule infinite_openin [OF opeU \<open>u \<in> U\<close>])
     apply (rule connected_imp_perfect_aff_dim [OF \<open>connected S\<close> _ \<open>u \<in> S\<close>])
     using True apply simp
     done
   then obtain P where "P \<subseteq> U" "finite P" "card K = card P"
     using infinite_arbitrarily_large by metis
   then obtain \<gamma> where \<gamma>: "bij_betw \<gamma> K P"
     using \<open>finite K\<close> finite_same_card_bij by blast
   have "\<exists>f g. homeomorphism T T f g \<and> (\<forall>i \<in> K. f(id i) = \<gamma> i) \<and>
                {x. \<not> (f x = x \<and> g x = x)} \<subseteq> S \<and> bounded {x. \<not> (f x = x \<and> g x = x)}"
   proof (rule homeomorphism_moving_points_exists_gen [OF \<open>finite K\<close> _ _ True opeS S])
     show "\<And>i. i \<in> K \<Longrightarrow> id i \<in> S \<and> \<gamma> i \<in> S"
       by (metis id_apply opeU openin_contains_cball subsetCE \<open>P \<subseteq> U\<close> \<open>bij_betw \<gamma> K P\<close> \<open>K \<subseteq> S\<close> bij_betwE)
     show "pairwise (\<lambda>i j. id i \<noteq> id j \<and> \<gamma> i \<noteq> \<gamma> j) K"
       using \<gamma> by (auto simp: pairwise_def bij_betw_def inj_on_def)
   qed
   then show ?thesis
     using \<gamma> \<open>P \<subseteq> U\<close> bij_betwE by (fastforce simp add: intro!: that)
 next
   case False
   with aff_dim_geq [of S] consider "aff_dim S = -1" | "aff_dim S = 0" | "aff_dim S = 1" by linarith
   then show ?thesis
   proof cases
     assume "aff_dim S = -1"
     then have "S = {}"
       using aff_dim_empty by blast
     then have "False"
       using \<open>U \<noteq> {}\<close> \<open>K \<subseteq> S\<close> openin_imp_subset [OF opeU] by blast
     then show ?thesis ..
   next
     assume "aff_dim S = 0"
     then obtain a where "S = {a}"
       using aff_dim_eq_0 by blast
     then have "K \<subseteq> U"
       using \<open>U \<noteq> {}\<close> \<open>K \<subseteq> S\<close> openin_imp_subset [OF opeU] by blast
     show ?thesis
       apply (rule that [of id id])
       using \<open>K \<subseteq> U\<close> by (auto 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
       apply (rule continuous_on_subset [OF _ \<open>T \<subseteq> affine hull S\<close>])
       using homeomorphism_def homhj apply blast
       by (meson continuous_on_subset homeomorphism_def homhj top_greatest)
     define f' where "f' \<equiv> \<lambda>x. if x \<in> affine hull S then (j \<circ> f \<circ> h) x else x"
     define g' where "g' \<equiv> \<lambda>x. if x \<in> affine hull S then (j \<circ> g \<circ> h) x else x"
     show ?thesis
     proof
       show "homeomorphism T T f' g'"
       proof
         have "continuous_on T (j \<circ> f \<circ> h)"
           apply (intro continuous_on_compose cont_hj)
           using hom homeomorphism_def by blast
         then show "continuous_on T f'"
           apply (rule continuous_on_eq)
           using \<open>T \<subseteq> affine hull S\<close> f'_def by auto
         have "continuous_on T (j \<circ> g \<circ> h)"
           apply (intro continuous_on_compose cont_hj)
           using hom homeomorphism_def by blast
         then show "continuous_on T g'"
           apply (rule continuous_on_eq)
           using \<open>T \<subseteq> affine hull S\<close> g'_def by auto
         show "f' ` T \<subseteq> T"
         proof (clarsimp simp: f'_def)
           fix x assume "x \<in> T"
           then have "f (h x) \<in> h ` T"
             by (metis (no_types) hom homeomorphism_def image_subset_iff subset_refl)
           then show "j (f (h x)) \<in> T"
             using \<open>T \<subseteq> affine hull S\<close> h by auto
         qed
         show "g' ` T \<subseteq> T"
         proof (clarsimp simp: g'_def)
           fix x assume "x \<in> T"
           then have "g (h x) \<in> h ` T"
             by (metis (no_types) hom homeomorphism_def image_subset_iff subset_refl)
           then show "j (g (h x)) \<in> T"
             using \<open>T \<subseteq> affine hull S\<close> h by auto
         qed
         show "\<And>x. x \<in> T \<Longrightarrow> g' (f' x) = x"
           using h j hom homeomorphism_apply1 by (fastforce simp add: f'_def g'_def)
         show "\<And>y. y \<in> T \<Longrightarrow> f' (g' y) = y"
           using h j hom homeomorphism_apply2 by (fastforce simp add: f'_def g'_def)
       qed
     next
       show "{x. \<not> (f' x = x \<and> g' x = x)} \<subseteq> S"
         apply (clarsimp simp: f'_def g'_def jf jg)
         apply (rule imageE [OF subsetD [OF sub]], force)
         by (metis h hull_inc)
     next
       have "compact (j ` closure {x. \<not> (f x = x \<and> g x = x)})"
         using bou by (auto simp: compact_continuous_image cont_hj)
       then have "bounded (j ` {x. \<not> (f x = x \<and> g x = x)})"
         by (rule bounded_closure_image [OF compact_imp_bounded])
       moreover
       have *: "{x \<in> affine hull S. j (f (h x)) \<noteq> x \<or> j (g (h x)) \<noteq> x} = j ` {x. (\<not> (f x = x \<and> g x = x))}"
         using h j by (auto simp: image_iff; metis)
       ultimately have "bounded {x \<in> affine hull S. j (f (h x)) \<noteq> x \<or> j (g (h x)) \<noteq> x}"
         by metis
       then show "bounded {x. \<not> (f' x = x \<and> g' x = x)}"
         by (simp add: f'_def g'_def Collect_mono bounded_subset)
     next
       show "f' x \<in> U" if "x \<in> K" for x
       proof -
         have "U \<subseteq> S"
           using opeU openin_imp_subset by blast
         then have "j (f (h x)) \<in> U"
           using f h hull_subset that by fastforce
         then show "f' x \<in> U"
           using \<open>K \<subseteq> S\<close> S f'_def that by auto
       qed
     qed
   qed
 qed
 
 
 subsection\<open>Nullhomotopic mappings\<close>
 
 text\<open> A mapping out of a sphere is nullhomotopic iff it extends to the ball.
 This even works out in the degenerate cases when the radius is \<open>\<le>\<close> 0, and
 we also don't need to explicitly assume continuity since it's already implicit
 in both sides of the equivalence.\<close>
 
 lemma nullhomotopic_from_lemma:
   assumes contg: "continuous_on (cball a r - {a}) g"
       and fa: "\<And>e. 0 < e
                \<Longrightarrow> \<exists>d. 0 < d \<and> (\<forall>x. x \<noteq> a \<and> norm(x - a) < d \<longrightarrow> norm(g x - f a) < e)"
       and r: "\<And>x. x \<in> cball a r \<and> x \<noteq> a \<Longrightarrow> f x = g x"
     shows "continuous_on (cball a r) f"
 proof (clarsimp simp: continuous_on_eq_continuous_within Ball_def)
   fix x
   assume x: "dist a x \<le> r"
   show "continuous (at x within cball a r) f"
   proof (cases "x=a")
     case True
     then show ?thesis
       by (metis continuous_within_eps_delta fa dist_norm dist_self r)
   next
     case False
     show ?thesis
     proof (rule continuous_transform_within [where f=g and d = "norm(x-a)"])
       have "\<exists>d>0. \<forall>x'\<in>cball a r.
                       dist x' x < d \<longrightarrow> dist (g x') (g x) < e" if "e>0" for e
       proof -
         obtain d where "d > 0"
            and d: "\<And>x'. \<lbrakk>dist x' a \<le> r; x' \<noteq> a; dist x' x < d\<rbrakk> \<Longrightarrow>
                                  dist (g x') (g x) < e"
           using contg False x \<open>e>0\<close>
           unfolding continuous_on_iff by (fastforce simp add: dist_commute intro: that)
         show ?thesis
           using \<open>d > 0\<close> \<open>x \<noteq> a\<close>
           by (rule_tac x="min d (norm(x - a))" in exI)
              (auto simp: dist_commute dist_norm [symmetric]  intro!: d)
       qed
       then show "continuous (at x within cball a r) g"
         using contg False by (auto simp: continuous_within_eps_delta)
       show "0 < norm (x - a)"
         using False by force
       show "x \<in> cball a r"
         by (simp add: x)
       show "\<And>x'. \<lbrakk>x' \<in> cball a r; dist x' x < norm (x - a)\<rbrakk>
         \<Longrightarrow> g x' = f x'"
         by (metis dist_commute dist_norm less_le r)
     qed
   qed
 qed
 
 proposition nullhomotopic_from_sphere_extension:
   fixes f :: "'M::euclidean_space \<Rightarrow> 'a::real_normed_vector"
   shows  "(\<exists>c. homotopic_with_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
   then show ?thesis
     apply (auto simp: homotopic_with)
     apply (rule_tac x="\<lambda>x. h (0, a)" in exI)
      apply (fastforce simp add:)
     using continuous_on_const by blast
 next
   case greater
   let ?P = "continuous_on {x. norm(x - a) = r} f \<and> f ` {x. norm(x - a) = r} \<subseteq> S"
   have ?P if ?lhs using that
   proof
     fix c
     assume c: "homotopic_with_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
     show ?P
       apply (safe elim!: continuous_on_eq [OF continuous_on_subset])
       apply (auto simp: dist_norm norm_minus_commute)
       by (metis dist_norm image_subset_iff mem_sphere norm_minus_commute sphere_cball subsetCE)
   qed
   moreover have ?thesis if ?P
   proof
     assume ?lhs
     then obtain c where "homotopic_with_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> S"
       apply (rule him [THEN subsetD])
       apply (rule_tac x = "(0, a + r *\<^sub>R b1)" in image_eqI)
       using h greater \<open>b1 \<in> Basis\<close>
        apply (auto simp: dist_norm)
       done
     have uconth: "uniformly_continuous_on ({0..1::real} \<times> (sphere a r)) h"
       by (force intro: compact_Times conth compact_uniformly_continuous)
     let ?g = "\<lambda>x. h (norm (x - a)/r,
                      a + (if x = a then r *\<^sub>R b1 else (r / norm(x - a)) *\<^sub>R (x - a)))"
     let ?g' = "\<lambda>x. h (norm (x - a)/r, a + (r / norm(x - a)) *\<^sub>R (x - a))"
     show ?rhs
     proof (intro exI conjI)
       have "continuous_on (cball a r - {a}) ?g'"
         apply (rule continuous_on_compose2 [OF conth])
          apply (intro continuous_intros)
         using greater apply (auto simp: dist_norm norm_minus_commute)
         done
       then show "continuous_on (cball a r) ?g"
       proof (rule nullhomotopic_from_lemma)
         show "\<exists>d>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < d \<longrightarrow> norm (?g' x - ?g a) < e" if "0 < e" for e
         proof -
           obtain d where "0 < d"
              and d: "\<And>x x'. \<lbrakk>x \<in> {0..1} \<times> sphere a r; x' \<in> {0..1} \<times> sphere a r; dist x' x < d\<rbrakk>
                         \<Longrightarrow> dist (h x') (h x) < e"
             using uniformly_continuous_onE [OF uconth \<open>0 < e\<close>] by auto
           have *: "norm (h (norm (x - a) / r,
                          a + (r / norm (x - a)) *\<^sub>R (x - a)) - h (0, a + r *\<^sub>R b1)) < e"
                    if "x \<noteq> a" "norm (x - a) < r" "norm (x - a) < d * r" for x
           proof -
             have "norm (h (norm (x - a) / r, a + (r / norm (x - a)) *\<^sub>R (x - a)) - h (0, a + r *\<^sub>R b1)) =
                   norm (h (norm (x - a) / r, a + (r / norm (x - a)) *\<^sub>R (x - a)) - h (0, a + (r / norm (x - a)) *\<^sub>R (x - a)))"
               by (simp add: h)
             also have "\<dots> < e"
               apply (rule d [unfolded dist_norm])
               using greater \<open>0 < d\<close> \<open>b1 \<in> Basis\<close> that
                 by (simp_all add: dist_norm) (simp add: field_simps)
             finally show ?thesis .
           qed
           show ?thesis
             apply (rule_tac x = "min r (d * r)" in exI)
             using greater \<open>0 < d\<close> by (auto simp: *)
         qed
         show "\<And>x. x \<in> cball a r \<and> x \<noteq> a \<Longrightarrow> ?g x = ?g' x"
           by auto
       qed
     next
       show "?g ` cball a r \<subseteq> S"
         using greater him \<open>c \<in> S\<close>
         by (force simp: h dist_norm norm_minus_commute)
     next
       show "\<forall>x\<in>sphere a r. ?g x = f x"
         using greater by (auto simp: h dist_norm norm_minus_commute)
     qed
   next
     assume ?rhs
     then obtain g where contg: "continuous_on (cball a r) g"
                     and gim: "g ` cball a r \<subseteq> S"
                     and gf: "\<forall>x \<in> sphere a r. g x = f x"
       by auto
     let ?h = "\<lambda>y. g (a + (fst y) *\<^sub>R (snd y - a))"
     have "continuous_on ({0..1} \<times> sphere a r) ?h"
     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
diff --git a/src/HOL/Analysis/Starlike.thy b/src/HOL/Analysis/Starlike.thy
--- a/src/HOL/Analysis/Starlike.thy
+++ b/src/HOL/Analysis/Starlike.thy
@@ -1,6664 +1,6631 @@
 (* Title:      HOL/Analysis/Starlike.thy
    Author:     L C Paulson, University of Cambridge
    Author:     Robert Himmelmann, TU Muenchen
    Author:     Bogdan Grechuk, University of Edinburgh
    Author:     Armin Heller, TU Muenchen
    Author:     Johannes Hoelzl, TU Muenchen
 *)
 chapter \<open>Unsorted\<close>
 
 theory Starlike
   imports
     Convex_Euclidean_Space
     Abstract_Limits
     Line_Segment
 begin
 
-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 affine_hull_closed_segment [simp]:
      "affine hull (closed_segment a b) = affine hull {a,b}"
   by (simp add: segment_convex_hull)
 
 lemma affine_hull_open_segment [simp]:
     fixes a :: "'a::euclidean_space"
     shows "affine hull (open_segment a b) = (if a = b then {} else affine hull {a,b})"
 by (metis affine_hull_convex_hull affine_hull_empty closure_open_segment closure_same_affine_hull segment_convex_hull)
 
 lemma rel_interior_closure_convex_segment:
   fixes S :: "_::euclidean_space set"
   assumes "convex S" "a \<in> rel_interior S" "b \<in> closure S"
     shows "open_segment a b \<subseteq> rel_interior S"
 proof
   fix x
   have [simp]: "(1 - u) *\<^sub>R a + u *\<^sub>R b = b - (1 - u) *\<^sub>R (b - a)" for u
     by (simp add: algebra_simps)
   assume "x \<in> open_segment a b"
   then show "x \<in> rel_interior S"
     unfolding closed_segment_def open_segment_def  using assms
     by (auto intro: rel_interior_closure_convex_shrink)
 qed
 
 lemma convex_hull_insert_segments:
    "convex hull (insert a S) =
     (if S = {} then {a} else  \<Union>x \<in> convex hull S. closed_segment a x)"
   by (force simp add: convex_hull_insert_alt in_segment)
 
 lemma Int_convex_hull_insert_rel_exterior:
   fixes z :: "'a::euclidean_space"
   assumes "convex C" "T \<subseteq> C" and z: "z \<in> rel_interior C" and dis: "disjnt S (rel_interior C)"
   shows "S \<inter> (convex hull (insert z T)) = S \<inter> (convex hull T)" (is "?lhs = ?rhs")
 proof
   have "T = {} \<Longrightarrow> z \<notin> S"
     using dis z by (auto simp add: disjnt_def)
   then show "?lhs \<subseteq> ?rhs"
   proof (clarsimp simp add: convex_hull_insert_segments)
     fix x y
     assume "x \<in> S" and y: "y \<in> convex hull T" and "x \<in> closed_segment z y"
     have "y \<in> closure C"
       by (metis y \<open>convex C\<close> \<open>T \<subseteq> C\<close> closure_subset contra_subsetD convex_hull_eq hull_mono)
     moreover have "x \<notin> rel_interior C"
       by (meson \<open>x \<in> S\<close> dis disjnt_iff)
     moreover have "x \<in> open_segment z y \<union> {z, y}"
       using \<open>x \<in> closed_segment z y\<close> closed_segment_eq_open by blast
     ultimately show "x \<in> convex hull T"
       using rel_interior_closure_convex_segment [OF \<open>convex C\<close> z]
       using y z by blast
   qed
   show "?rhs \<subseteq> ?lhs"
     by (meson hull_mono inf_mono subset_insertI subset_refl)
 qed
 
 subsection\<^marker>\<open>tag unimportant\<close> \<open>Shrinking towards the interior of a convex set\<close>
 
 lemma mem_interior_convex_shrink:
   fixes S :: "'a::euclidean_space set"
   assumes "convex S"
     and "c \<in> interior S"
     and "x \<in> S"
     and "0 < e"
     and "e \<le> 1"
   shows "x - e *\<^sub>R (x - c) \<in> interior S"
 proof -
   obtain d where "d > 0" and d: "ball c d \<subseteq> S"
     using assms(2) unfolding mem_interior by auto
   show ?thesis
     unfolding mem_interior
   proof (intro exI subsetI conjI)
     fix y
     assume "y \<in> ball (x - e *\<^sub>R (x - c)) (e*d)"
     then have as: "dist (x - e *\<^sub>R (x - c)) y < e * d"
       by simp
     have *: "y = (1 - (1 - e)) *\<^sub>R ((1 / e) *\<^sub>R y - ((1 - e) / e) *\<^sub>R x) + (1 - e) *\<^sub>R x"
       using \<open>e > 0\<close> by (auto simp add: scaleR_left_diff_distrib scaleR_right_diff_distrib)
     have "dist c ((1 / e) *\<^sub>R y - ((1 - e) / e) *\<^sub>R x) = \<bar>1/e\<bar> * norm (e *\<^sub>R c - y + (1 - e) *\<^sub>R x)"
       unfolding dist_norm
       unfolding norm_scaleR[symmetric]
       apply (rule arg_cong[where f=norm])
       using \<open>e > 0\<close>
       by (auto simp add: euclidean_eq_iff[where 'a='a] field_simps inner_simps)
     also have "\<dots> = \<bar>1/e\<bar> * norm (x - e *\<^sub>R (x - c) - y)"
       by (auto intro!:arg_cong[where f=norm] simp add: algebra_simps)
     also have "\<dots> < d"
       using as[unfolded dist_norm] and \<open>e > 0\<close>
       by (auto simp add:pos_divide_less_eq[OF \<open>e > 0\<close>] mult.commute)
     finally show "y \<in> S"
       apply (subst *)
       apply (rule assms(1)[unfolded convex_alt,rule_format])
       apply (rule d[unfolded subset_eq,rule_format])
       unfolding mem_ball
       using assms(3-5)
       apply auto
       done
   qed (insert \<open>e>0\<close> \<open>d>0\<close>, auto)
 qed
 
 lemma mem_interior_closure_convex_shrink:
   fixes S :: "'a::euclidean_space set"
   assumes "convex S"
     and "c \<in> interior S"
     and "x \<in> closure S"
     and "0 < e"
     and "e \<le> 1"
   shows "x - e *\<^sub>R (x - c) \<in> interior S"
 proof -
   obtain d where "d > 0" and d: "ball c d \<subseteq> S"
     using assms(2) unfolding mem_interior by auto
   have "\<exists>y\<in>S. norm (y - x) * (1 - e) < e * d"
   proof (cases "x \<in> S")
     case True
     then show ?thesis
       using \<open>e > 0\<close> \<open>d > 0\<close>
       apply (rule_tac bexI[where x=x])
       apply (auto)
       done
   next
     case False
     then have x: "x islimpt S"
       using assms(3)[unfolded closure_def] by auto
     show ?thesis
     proof (cases "e = 1")
       case True
       obtain y where "y \<in> S" "y \<noteq> x" "dist y x < 1"
         using x[unfolded islimpt_approachable,THEN spec[where x=1]] by auto
       then show ?thesis
         apply (rule_tac x=y in bexI)
         unfolding True
         using \<open>d > 0\<close>
         apply auto
         done
     next
       case False
       then have "0 < e * d / (1 - e)" and *: "1 - e > 0"
         using \<open>e \<le> 1\<close> \<open>e > 0\<close> \<open>d > 0\<close> by auto
       then obtain y where "y \<in> S" "y \<noteq> x" "dist y x < e * d / (1 - e)"
         using x[unfolded islimpt_approachable,THEN spec[where x="e*d / (1 - e)"]] by auto
       then show ?thesis
         apply (rule_tac x=y in bexI)
         unfolding dist_norm
         using pos_less_divide_eq[OF *]
         apply auto
         done
     qed
   qed
   then obtain y where "y \<in> S" and y: "norm (y - x) * (1 - e) < e * d"
     by auto
   define z where "z = c + ((1 - e) / e) *\<^sub>R (x - y)"
   have *: "x - e *\<^sub>R (x - c) = y - e *\<^sub>R (y - z)"
     unfolding z_def using \<open>e > 0\<close>
     by (auto simp add: scaleR_right_diff_distrib scaleR_right_distrib scaleR_left_diff_distrib)
   have "z \<in> interior S"
     apply (rule interior_mono[OF d,unfolded subset_eq,rule_format])
     unfolding interior_open[OF open_ball] mem_ball z_def dist_norm using y and assms(4,5)
     by simp (simp add: field_simps norm_minus_commute)
   then show ?thesis
     unfolding *
     using mem_interior_convex_shrink \<open>y \<in> S\<close> assms by blast
 qed
 
 lemma in_interior_closure_convex_segment:
   fixes S :: "'a::euclidean_space set"
   assumes "convex S" and a: "a \<in> interior S" and b: "b \<in> closure S"
     shows "open_segment a b \<subseteq> interior S"
 proof (clarsimp simp: in_segment)
   fix u::real
   assume u: "0 < u" "u < 1"
   have "(1 - u) *\<^sub>R a + u *\<^sub>R b = b - (1 - u) *\<^sub>R (b - a)"
     by (simp add: algebra_simps)
   also have "... \<in> interior S" using mem_interior_closure_convex_shrink [OF assms] u
     by simp
   finally show "(1 - u) *\<^sub>R a + u *\<^sub>R b \<in> interior S" .
 qed
 
 lemma convex_closure_interior:
   fixes S :: "'a::euclidean_space set"
   assumes "convex S" and int: "interior S \<noteq> {}"
   shows "closure(interior S) = closure S"
 proof -
   obtain a where a: "a \<in> interior S"
     using int by auto
   have "closure S \<subseteq> closure(interior S)"
   proof
     fix x
     assume x: "x \<in> closure S"
     show "x \<in> closure (interior S)"
     proof (cases "x=a")
       case True
       then show ?thesis
         using \<open>a \<in> interior S\<close> closure_subset by blast
     next
       case False
       show ?thesis
       proof (clarsimp simp add: closure_def islimpt_approachable)
         fix e::real
         assume xnotS: "x \<notin> interior S" and "0 < e"
         show "\<exists>x'\<in>interior S. x' \<noteq> x \<and> dist x' x < e"
         proof (intro bexI conjI)
           show "x - min (e/2 / norm (x - a)) 1 *\<^sub>R (x - a) \<noteq> x"
             using False \<open>0 < e\<close> by (auto simp: algebra_simps min_def)
           show "dist (x - min (e/2 / norm (x - a)) 1 *\<^sub>R (x - a)) x < e"
             using \<open>0 < e\<close> by (auto simp: dist_norm min_def)
           show "x - min (e/2 / norm (x - a)) 1 *\<^sub>R (x - a) \<in> interior S"
             apply (clarsimp simp add: min_def a)
             apply (rule mem_interior_closure_convex_shrink [OF \<open>convex S\<close> a x])
             using \<open>0 < e\<close> False apply (auto simp: field_split_simps)
             done
         qed
       qed
     qed
   qed
   then show ?thesis
     by (simp add: closure_mono interior_subset subset_antisym)
 qed
 
 lemma closure_convex_Int_superset:
   fixes S :: "'a::euclidean_space set"
   assumes "convex S" "interior S \<noteq> {}" "interior S \<subseteq> closure T"
   shows "closure(S \<inter> T) = closure S"
 proof -
   have "closure S \<subseteq> closure(interior S)"
     by (simp add: convex_closure_interior assms)
   also have "... \<subseteq> closure (S \<inter> T)"
     using interior_subset [of S] assms
     by (metis (no_types, lifting) Int_assoc Int_lower2 closure_mono closure_open_Int_superset inf.orderE open_interior)
   finally show ?thesis
     by (simp add: closure_mono dual_order.antisym)
 qed
 
 
 subsection\<^marker>\<open>tag unimportant\<close> \<open>Some obvious but surprisingly hard simplex lemmas\<close>
 
 lemma simplex:
   assumes "finite S"
     and "0 \<notin> S"
   shows "convex hull (insert 0 S) = {y. \<exists>u. (\<forall>x\<in>S. 0 \<le> u x) \<and> sum u S \<le> 1 \<and> sum (\<lambda>x. u x *\<^sub>R x) S = y}"
 proof (simp add: convex_hull_finite set_eq_iff assms, safe)
   fix x and u :: "'a \<Rightarrow> real"
   assume "0 \<le> u 0" "\<forall>x\<in>S. 0 \<le> u x" "u 0 + sum u S = 1"
   then show "\<exists>v. (\<forall>x\<in>S. 0 \<le> v x) \<and> sum v S \<le> 1 \<and> (\<Sum>x\<in>S. v x *\<^sub>R x) = (\<Sum>x\<in>S. u x *\<^sub>R x)"
     by force
 next
   fix x and u :: "'a \<Rightarrow> real"
   assume "\<forall>x\<in>S. 0 \<le> u x" "sum u S \<le> 1"
   then show "\<exists>v. 0 \<le> v 0 \<and> (\<forall>x\<in>S. 0 \<le> v x) \<and> v 0 + sum v S = 1 \<and> (\<Sum>x\<in>S. v x *\<^sub>R x) = (\<Sum>x\<in>S. u x *\<^sub>R x)"
     by (rule_tac x="\<lambda>x. if x = 0 then 1 - sum u S else u x" in exI) (auto simp: sum_delta_notmem assms if_smult)
 qed
 
 lemma substd_simplex:
   assumes d: "d \<subseteq> Basis"
   shows "convex hull (insert 0 d) =
     {x. (\<forall>i\<in>Basis. 0 \<le> x\<bullet>i) \<and> (\<Sum>i\<in>d. x\<bullet>i) \<le> 1 \<and> (\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x\<bullet>i = 0)}"
   (is "convex hull (insert 0 ?p) = ?s")
 proof -
   let ?D = d
   have "0 \<notin> ?p"
     using assms by (auto simp: image_def)
   from d have "finite d"
     by (blast intro: finite_subset finite_Basis)
   show ?thesis
     unfolding simplex[OF \<open>finite d\<close> \<open>0 \<notin> ?p\<close>]
   proof (intro set_eqI; safe)
     fix u :: "'a \<Rightarrow> real"
     assume as: "\<forall>x\<in>?D. 0 \<le> u x" "sum u ?D \<le> 1" 
     let ?x = "(\<Sum>x\<in>?D. u x *\<^sub>R x)"
     have ind: "\<forall>i\<in>Basis. i \<in> d \<longrightarrow> u i = ?x \<bullet> i"
       and notind: "(\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> ?x \<bullet> i = 0)"
       using substdbasis_expansion_unique[OF assms] by blast+
     then have **: "sum u ?D = sum ((\<bullet>) ?x) ?D"
       using assms by (auto intro!: sum.cong)
     show "0 \<le> ?x \<bullet> i" if "i \<in> Basis" for i
       using as(1) ind notind that by fastforce
     show "sum ((\<bullet>) ?x) ?D \<le> 1"
       using "**" as(2) by linarith
     show "?x \<bullet> i = 0" if "i \<in> Basis" "i \<notin> d" for i
       using notind that by blast
   next
     fix x 
     assume "\<forall>i\<in>Basis. 0 \<le> x \<bullet> i" "sum ((\<bullet>) x) ?D \<le> 1" "(\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = 0)"
     with d show "\<exists>u. (\<forall>x\<in>?D. 0 \<le> u x) \<and> sum u ?D \<le> 1 \<and> (\<Sum>x\<in>?D. u x *\<^sub>R x) = x"
       unfolding substdbasis_expansion_unique[OF assms] 
       by (rule_tac x="inner x" in exI) auto
   qed
 qed
 
 lemma std_simplex:
   "convex hull (insert 0 Basis) =
     {x::'a::euclidean_space. (\<forall>i\<in>Basis. 0 \<le> x\<bullet>i) \<and> sum (\<lambda>i. x\<bullet>i) Basis \<le> 1}"
   using substd_simplex[of Basis] by auto
 
 lemma interior_std_simplex:
   "interior (convex hull (insert 0 Basis)) =
     {x::'a::euclidean_space. (\<forall>i\<in>Basis. 0 < x\<bullet>i) \<and> sum (\<lambda>i. x\<bullet>i) Basis < 1}"
   unfolding set_eq_iff mem_interior std_simplex
 proof (intro allI iffI CollectI; clarify)
   fix x :: 'a
   fix e
   assume "e > 0" and as: "ball x e \<subseteq> {x. (\<forall>i\<in>Basis. 0 \<le> x \<bullet> i) \<and> sum ((\<bullet>) x) Basis \<le> 1}"
   show "(\<forall>i\<in>Basis. 0 < x \<bullet> i) \<and> sum ((\<bullet>) x) Basis < 1"
   proof safe
     fix i :: 'a
     assume i: "i \<in> Basis"
     then show "0 < x \<bullet> i"
       using as[THEN subsetD[where c="x - (e / 2) *\<^sub>R i"]] and \<open>e > 0\<close> 
       by (force simp add: inner_simps)
   next
     have **: "dist x (x + (e / 2) *\<^sub>R (SOME i. i\<in>Basis)) < e" using \<open>e > 0\<close>
       unfolding dist_norm
       by (auto intro!: mult_strict_left_mono simp: SOME_Basis)
     have "\<And>i. i \<in> Basis \<Longrightarrow> (x + (e / 2) *\<^sub>R (SOME i. i\<in>Basis)) \<bullet> i =
       x\<bullet>i + (if i = (SOME i. i\<in>Basis) then e/2 else 0)"
       by (auto simp: SOME_Basis inner_Basis inner_simps)
     then have *: "sum ((\<bullet>) (x + (e / 2) *\<^sub>R (SOME i. i\<in>Basis))) Basis =
       sum (\<lambda>i. x\<bullet>i + (if (SOME i. i\<in>Basis) = i then e/2 else 0)) Basis"
       by (auto simp: intro!: sum.cong)
     have "sum ((\<bullet>) x) Basis < sum ((\<bullet>) (x + (e / 2) *\<^sub>R (SOME i. i\<in>Basis))) Basis"
       using \<open>e > 0\<close> DIM_positive by (auto simp: SOME_Basis sum.distrib *)
     also have "\<dots> \<le> 1"
       using ** as by force
     finally show "sum ((\<bullet>) x) Basis < 1" by auto
   qed 
 next
   fix x :: 'a
   assume as: "\<forall>i\<in>Basis. 0 < x \<bullet> i" "sum ((\<bullet>) x) Basis < 1"
   obtain a :: 'b where "a \<in> UNIV" using UNIV_witness ..
   let ?d = "(1 - sum ((\<bullet>) x) Basis) / real (DIM('a))"
   show "\<exists>e>0. ball x e \<subseteq> {x. (\<forall>i\<in>Basis. 0 \<le> x \<bullet> i) \<and> sum ((\<bullet>) x) Basis \<le> 1}"
   proof (rule_tac x="min (Min (((\<bullet>) x) ` Basis)) D" for D in exI, intro conjI subsetI CollectI)
     fix y
     assume y: "y \<in> ball x (min (Min ((\<bullet>) x ` Basis)) ?d)"
     have "sum ((\<bullet>) y) Basis \<le> sum (\<lambda>i. x\<bullet>i + ?d) Basis"
     proof (rule sum_mono)
       fix i :: 'a
       assume i: "i \<in> Basis"
       have "\<bar>y\<bullet>i - x\<bullet>i\<bar> \<le> norm (y - x)"
         by (metis Basis_le_norm i inner_commute inner_diff_right)
       also have "... < ?d"
         using y by (simp add: dist_norm norm_minus_commute)
       finally have "\<bar>y\<bullet>i - x\<bullet>i\<bar> < ?d" .
       then show "y \<bullet> i \<le> x \<bullet> i + ?d" by auto
     qed
     also have "\<dots> \<le> 1"
       unfolding sum.distrib sum_constant
       by (auto simp add: Suc_le_eq)
     finally show "sum ((\<bullet>) y) Basis \<le> 1" .
     show "(\<forall>i\<in>Basis. 0 \<le> y \<bullet> i)"
     proof safe
       fix i :: 'a
       assume i: "i \<in> Basis"
       have "norm (x - y) < Min (((\<bullet>) x) ` Basis)"
         using y by (auto simp: dist_norm less_eq_real_def)
       also have "... \<le> x\<bullet>i"
         using i by auto
       finally have "norm (x - y) < x\<bullet>i" .
       then show "0 \<le> y\<bullet>i"
         using Basis_le_norm[OF i, of "x - y"] and as(1)[rule_format, OF i]
         by (auto simp: inner_simps)
     qed
   next
     have "Min (((\<bullet>) x) ` Basis) > 0"
       using as by simp
     moreover have "?d > 0"
       using as by (auto simp: Suc_le_eq)
     ultimately show "0 < min (Min ((\<bullet>) x ` Basis)) ((1 - sum ((\<bullet>) x) Basis) / real DIM('a))"
       by linarith
   qed 
 qed
 
 lemma interior_std_simplex_nonempty:
   obtains a :: "'a::euclidean_space" where
     "a \<in> interior(convex hull (insert 0 Basis))"
 proof -
   let ?D = "Basis :: 'a set"
   let ?a = "sum (\<lambda>b::'a. inverse (2 * real DIM('a)) *\<^sub>R b) Basis"
   {
     fix i :: 'a
     assume i: "i \<in> Basis"
     have "?a \<bullet> i = inverse (2 * real DIM('a))"
       by (rule trans[of _ "sum (\<lambda>j. if i = j then inverse (2 * real DIM('a)) else 0) ?D"])
          (simp_all add: sum.If_cases i) }
   note ** = this
   show ?thesis
     apply (rule that[of ?a])
     unfolding interior_std_simplex mem_Collect_eq
   proof safe
     fix i :: 'a
     assume i: "i \<in> Basis"
     show "0 < ?a \<bullet> i"
       unfolding **[OF i] by (auto simp add: Suc_le_eq)
   next
     have "sum ((\<bullet>) ?a) ?D = sum (\<lambda>i. inverse (2 * real DIM('a))) ?D"
       apply (rule sum.cong)
       apply rule
       apply auto
       done
     also have "\<dots> < 1"
       unfolding sum_constant divide_inverse[symmetric]
       by (auto simp add: field_simps)
     finally show "sum ((\<bullet>) ?a) ?D < 1" by auto
   qed
 qed
 
 lemma rel_interior_substd_simplex:
   assumes D: "D \<subseteq> Basis"
   shows "rel_interior (convex hull (insert 0 D)) =
     {x::'a::euclidean_space. (\<forall>i\<in>D. 0 < x\<bullet>i) \<and> (\<Sum>i\<in>D. x\<bullet>i) < 1 \<and> (\<forall>i\<in>Basis. i \<notin> D \<longrightarrow> x\<bullet>i = 0)}"
   (is "rel_interior (convex hull (insert 0 ?p)) = ?s")
 proof -
   have "finite D"
     using D finite_Basis finite_subset by blast
   show ?thesis
   proof (cases "D = {}")
     case True
     then show ?thesis
       using rel_interior_sing using euclidean_eq_iff[of _ 0] by auto
   next
     case False
     have h0: "affine hull (convex hull (insert 0 ?p)) =
       {x::'a::euclidean_space. (\<forall>i\<in>Basis. i \<notin> D \<longrightarrow> x\<bullet>i = 0)}"
       using affine_hull_convex_hull affine_hull_substd_basis assms by auto
     have aux: "\<And>x::'a. \<forall>i\<in>Basis. (\<forall>i\<in>D. 0 \<le> x\<bullet>i) \<and> (\<forall>i\<in>Basis. i \<notin> D \<longrightarrow> x\<bullet>i = 0) \<longrightarrow> 0 \<le> x\<bullet>i"
       by auto
     {
       fix x :: "'a::euclidean_space"
       assume x: "x \<in> rel_interior (convex hull (insert 0 ?p))"
       then obtain e where "e > 0" and
         "ball x e \<inter> {xa. (\<forall>i\<in>Basis. i \<notin> D \<longrightarrow> xa\<bullet>i = 0)} \<subseteq> convex hull (insert 0 ?p)"
         using mem_rel_interior_ball[of x "convex hull (insert 0 ?p)"] h0 by auto
       then have as [rule_format]: "\<And>y. dist x y < e \<and> (\<forall>i\<in>Basis. i \<notin> D \<longrightarrow> y\<bullet>i = 0) \<longrightarrow>
         (\<forall>i\<in>D. 0 \<le> y \<bullet> i) \<and> sum ((\<bullet>) y) D \<le> 1"
         unfolding ball_def unfolding substd_simplex[OF assms] using assms by auto
       have x0: "(\<forall>i\<in>Basis. i \<notin> D \<longrightarrow> x\<bullet>i = 0)"
         using x rel_interior_subset  substd_simplex[OF assms] by auto
       have "(\<forall>i\<in>D. 0 < x \<bullet> i) \<and> sum ((\<bullet>) x) D < 1 \<and> (\<forall>i\<in>Basis. i \<notin> D \<longrightarrow> x\<bullet>i = 0)"
       proof (intro conjI ballI)
         fix i :: 'a
         assume "i \<in> D"
         then have "\<forall>j\<in>D. 0 \<le> (x - (e / 2) *\<^sub>R i) \<bullet> j"
           apply -
           apply (rule as[THEN conjunct1])
           using D \<open>e > 0\<close> x0
           apply (auto simp: dist_norm inner_simps inner_Basis)
           done
         then show "0 < x \<bullet> i"
           using \<open>e > 0\<close> \<open>i \<in> D\<close> D  by (force simp: inner_simps inner_Basis)
       next
         obtain a where a: "a \<in> D"
           using \<open>D \<noteq> {}\<close> by auto
         then have **: "dist x (x + (e / 2) *\<^sub>R a) < e"
           using \<open>e > 0\<close> norm_Basis[of a] D
           unfolding dist_norm
           by auto
         have "\<And>i. i \<in> Basis \<Longrightarrow> (x + (e / 2) *\<^sub>R a) \<bullet> i = x\<bullet>i + (if i = a then e/2 else 0)"
           using a D by (auto simp: inner_simps inner_Basis)
         then have *: "sum ((\<bullet>) (x + (e / 2) *\<^sub>R a)) D =
           sum (\<lambda>i. x\<bullet>i + (if a = i then e/2 else 0)) D"
           using D by (intro sum.cong) auto
         have "a \<in> Basis"
           using \<open>a \<in> D\<close> D by auto
         then have h1: "(\<forall>i\<in>Basis. i \<notin> D \<longrightarrow> (x + (e / 2) *\<^sub>R a) \<bullet> i = 0)"
           using x0 D \<open>a\<in>D\<close> by (auto simp add: inner_add_left inner_Basis)
         have "sum ((\<bullet>) x) D < sum ((\<bullet>) (x + (e / 2) *\<^sub>R a)) D"
           using \<open>e > 0\<close> \<open>a \<in> D\<close> \<open>finite D\<close> by (auto simp add: * sum.distrib)
         also have "\<dots> \<le> 1"
           using ** h1 as[rule_format, of "x + (e / 2) *\<^sub>R a"]
           by auto
         finally show "sum ((\<bullet>) x) D < 1" "\<And>i. i\<in>Basis \<Longrightarrow> i \<notin> D \<longrightarrow> x\<bullet>i = 0"
           using x0 by auto
       qed
     }
     moreover
     {
       fix x :: "'a::euclidean_space"
       assume as: "x \<in> ?s"
       have "\<forall>i. 0 < x\<bullet>i \<or> 0 = x\<bullet>i \<longrightarrow> 0 \<le> x\<bullet>i"
         by auto
       moreover have "\<forall>i. i \<in> D \<or> i \<notin> D" by auto
       ultimately
       have "\<forall>i. (\<forall>i\<in>D. 0 < x\<bullet>i) \<and> (\<forall>i. i \<notin> D \<longrightarrow> x\<bullet>i = 0) \<longrightarrow> 0 \<le> x\<bullet>i"
         by metis
       then have h2: "x \<in> convex hull (insert 0 ?p)"
         using as assms
         unfolding substd_simplex[OF assms] by fastforce
       obtain a where a: "a \<in> D"
         using \<open>D \<noteq> {}\<close> by auto
       let ?d = "(1 - sum ((\<bullet>) x) D) / real (card D)"
       have "0 < card D" using \<open>D \<noteq> {}\<close> \<open>finite D\<close>
         by (simp add: card_gt_0_iff)
       have "Min (((\<bullet>) x) ` D) > 0"
         using as \<open>D \<noteq> {}\<close> \<open>finite D\<close> by (simp)
       moreover have "?d > 0" using as using \<open>0 < card D\<close> by auto
       ultimately have h3: "min (Min (((\<bullet>) x) ` D)) ?d > 0"
         by auto
 
       have "x \<in> rel_interior (convex hull (insert 0 ?p))"
         unfolding rel_interior_ball mem_Collect_eq h0
         apply (rule,rule h2)
         unfolding substd_simplex[OF assms]
         apply (rule_tac x="min (Min (((\<bullet>) x) ` D)) ?d" in exI)
         apply (rule, rule h3)
         apply safe
         unfolding mem_ball
       proof -
         fix y :: 'a
         assume y: "dist x y < min (Min ((\<bullet>) x ` D)) ?d"
         assume y2: "\<forall>i\<in>Basis. i \<notin> D \<longrightarrow> y\<bullet>i = 0"
         have "sum ((\<bullet>) y) D \<le> sum (\<lambda>i. x\<bullet>i + ?d) D"
         proof (rule sum_mono)
           fix i
           assume "i \<in> D"
           with D have i: "i \<in> Basis"
             by auto
           have "\<bar>y\<bullet>i - x\<bullet>i\<bar> \<le> norm (y - x)"
             by (metis i inner_commute inner_diff_right norm_bound_Basis_le order_refl)
           also have "... < ?d"
             by (metis dist_norm min_less_iff_conj norm_minus_commute y)
           finally have "\<bar>y\<bullet>i - x\<bullet>i\<bar> < ?d" .
           then show "y \<bullet> i \<le> x \<bullet> i + ?d" by auto
         qed
         also have "\<dots> \<le> 1"
           unfolding sum.distrib sum_constant  using \<open>0 < card D\<close>
           by auto
         finally show "sum ((\<bullet>) y) D \<le> 1" .
 
         fix i :: 'a
         assume i: "i \<in> Basis"
         then show "0 \<le> y\<bullet>i"
         proof (cases "i\<in>D")
           case True
           have "norm (x - y) < x\<bullet>i"
             using y[unfolded min_less_iff_conj dist_norm, THEN conjunct1]
             using Min_gr_iff[of "(\<bullet>) x ` D" "norm (x - y)"] \<open>0 < card D\<close> \<open>i \<in> D\<close>
             by (simp add: card_gt_0_iff)
           then show "0 \<le> y\<bullet>i"
             using Basis_le_norm[OF i, of "x - y"] and as(1)[rule_format]
             by (auto simp: inner_simps)
         qed (insert y2, auto)
       qed
     }
     ultimately have
       "\<And>x. x \<in> rel_interior (convex hull insert 0 D) \<longleftrightarrow>
         x \<in> {x. (\<forall>i\<in>D. 0 < x \<bullet> i) \<and> sum ((\<bullet>) x) D < 1 \<and> (\<forall>i\<in>Basis. i \<notin> D \<longrightarrow> x \<bullet> i = 0)}"
       by blast
     then show ?thesis by (rule set_eqI)
   qed
 qed
 
 lemma rel_interior_substd_simplex_nonempty:
   assumes "D \<noteq> {}"
     and "D \<subseteq> Basis"
   obtains a :: "'a::euclidean_space"
     where "a \<in> rel_interior (convex hull (insert 0 D))"
 proof -
   let ?D = D
   let ?a = "sum (\<lambda>b::'a::euclidean_space. inverse (2 * real (card D)) *\<^sub>R b) ?D"
   have "finite D"
     apply (rule finite_subset)
     using assms(2)
     apply auto
     done
   then have d1: "0 < real (card D)"
     using \<open>D \<noteq> {}\<close> by auto
   {
     fix i
     assume "i \<in> D"
     have "?a \<bullet> i = inverse (2 * real (card D))"
       apply (rule trans[of _ "sum (\<lambda>j. if i = j then inverse (2 * real (card D)) else 0) ?D"])
       unfolding inner_sum_left
       apply (rule sum.cong)
       using \<open>i \<in> D\<close> \<open>finite D\<close> sum.delta'[of D i "(\<lambda>k. inverse (2 * real (card D)))"]
         d1 assms(2)
       by (auto simp: inner_Basis rev_subsetD[OF _ assms(2)])
   }
   note ** = this
   show ?thesis
     apply (rule that[of ?a])
     unfolding rel_interior_substd_simplex[OF assms(2)] mem_Collect_eq
   proof safe
     fix i
     assume "i \<in> D"
     have "0 < inverse (2 * real (card D))"
       using d1 by auto
     also have "\<dots> = ?a \<bullet> i" using **[of i] \<open>i \<in> D\<close>
       by auto
     finally show "0 < ?a \<bullet> i" by auto
   next
     have "sum ((\<bullet>) ?a) ?D = sum (\<lambda>i. inverse (2 * real (card D))) ?D"
       by (rule sum.cong) (rule refl, rule **)
     also have "\<dots> < 1"
       unfolding sum_constant divide_real_def[symmetric]
       by (auto simp add: field_simps)
     finally show "sum ((\<bullet>) ?a) ?D < 1" by auto
   next
     fix i
     assume "i \<in> Basis" and "i \<notin> D"
     have "?a \<in> span D"
     proof (rule span_sum[of D "(\<lambda>b. b /\<^sub>R (2 * real (card D)))" D])
       {
         fix x :: "'a::euclidean_space"
         assume "x \<in> D"
         then have "x \<in> span D"
           using span_base[of _ "D"] by auto
         then have "x /\<^sub>R (2 * real (card D)) \<in> span D"
           using span_mul[of x "D" "(inverse (real (card D)) / 2)"] by auto
       }
       then show "\<And>x. x\<in>D \<Longrightarrow> x /\<^sub>R (2 * real (card D)) \<in> span D"
         by auto
     qed
     then show "?a \<bullet> i = 0 "
       using \<open>i \<notin> D\<close> unfolding span_substd_basis[OF assms(2)] using \<open>i \<in> Basis\<close> by auto
   qed
 qed
 
 
 subsection\<^marker>\<open>tag unimportant\<close> \<open>Relative interior of convex set\<close>
 
 lemma rel_interior_convex_nonempty_aux:
   fixes S :: "'n::euclidean_space set"
   assumes "convex S"
     and "0 \<in> S"
   shows "rel_interior S \<noteq> {}"
 proof (cases "S = {0}")
   case True
   then show ?thesis using rel_interior_sing by auto
 next
   case False
   obtain B where B: "independent B \<and> B \<le> S \<and> S \<le> span B \<and> card B = dim S"
     using basis_exists[of S] by metis
   then have "B \<noteq> {}"
     using B assms \<open>S \<noteq> {0}\<close> span_empty by auto
   have "insert 0 B \<le> span B"
     using subspace_span[of B] subspace_0[of "span B"]
       span_superset by auto
   then have "span (insert 0 B) \<le> span B"
     using span_span[of B] span_mono[of "insert 0 B" "span B"] by blast
   then have "convex hull insert 0 B \<le> span B"
     using convex_hull_subset_span[of "insert 0 B"] by auto
   then have "span (convex hull insert 0 B) \<le> span B"
     using span_span[of B]
       span_mono[of "convex hull insert 0 B" "span B"] by blast
   then have *: "span (convex hull insert 0 B) = span B"
     using span_mono[of B "convex hull insert 0 B"] hull_subset[of "insert 0 B"] by auto
   then have "span (convex hull insert 0 B) = span S"
     using B span_mono[of B S] span_mono[of S "span B"]
       span_span[of B] by auto
   moreover have "0 \<in> affine hull (convex hull insert 0 B)"
     using hull_subset[of "convex hull insert 0 B"] hull_subset[of "insert 0 B"] by auto
   ultimately have **: "affine hull (convex hull insert 0 B) = affine hull S"
     using affine_hull_span_0[of "convex hull insert 0 B"] affine_hull_span_0[of "S"]
       assms hull_subset[of S]
     by auto
   obtain d and f :: "'n \<Rightarrow> 'n" where
     fd: "card d = card B" "linear f" "f ` B = d"
       "f ` span B = {x. \<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = (0::real)} \<and> inj_on f (span B)"
     and d: "d \<subseteq> Basis"
     using basis_to_substdbasis_subspace_isomorphism[of B,OF _ ] B by auto
   then have "bounded_linear f"
     using linear_conv_bounded_linear by auto
   have "d \<noteq> {}"
     using fd B \<open>B \<noteq> {}\<close> by auto
   have "insert 0 d = f ` (insert 0 B)"
     using fd linear_0 by auto
   then have "(convex hull (insert 0 d)) = f ` (convex hull (insert 0 B))"
     using convex_hull_linear_image[of f "(insert 0 d)"]
       convex_hull_linear_image[of f "(insert 0 B)"] \<open>linear f\<close>
     by auto
   moreover have "rel_interior (f ` (convex hull insert 0 B)) =
     f ` rel_interior (convex hull insert 0 B)"
     apply (rule  rel_interior_injective_on_span_linear_image[of f "(convex hull insert 0 B)"])
     using \<open>bounded_linear f\<close> fd *
     apply auto
     done
   ultimately have "rel_interior (convex hull insert 0 B) \<noteq> {}"
     using rel_interior_substd_simplex_nonempty[OF \<open>d \<noteq> {}\<close> d]
     apply auto
     apply blast
     done
   moreover have "convex hull (insert 0 B) \<subseteq> S"
     using B assms hull_mono[of "insert 0 B" "S" "convex"] convex_hull_eq
     by auto
   ultimately show ?thesis
     using subset_rel_interior[of "convex hull insert 0 B" S] ** by auto
 qed
 
 lemma rel_interior_eq_empty:
   fixes S :: "'n::euclidean_space set"
   assumes "convex S"
   shows "rel_interior S = {} \<longleftrightarrow> S = {}"
 proof -
   {
     assume "S \<noteq> {}"
     then obtain a where "a \<in> S" by auto
     then have "0 \<in> (+) (-a) ` S"
       using assms exI[of "(\<lambda>x. x \<in> S \<and> - a + x = 0)" a] by auto
     then have "rel_interior ((+) (-a) ` S) \<noteq> {}"
       using rel_interior_convex_nonempty_aux[of "(+) (-a) ` S"]
         convex_translation[of S "-a"] assms
       by auto
     then have "rel_interior S \<noteq> {}"
       using rel_interior_translation [of "- a"] by simp
   }
   then show ?thesis by auto
 qed
 
 lemma interior_simplex_nonempty:
   fixes S :: "'N :: euclidean_space set"
   assumes "independent S" "finite S" "card S = DIM('N)"
   obtains a where "a \<in> interior (convex hull (insert 0 S))"
 proof -
   have "affine hull (insert 0 S) = UNIV"
     by (simp add: hull_inc affine_hull_span_0 dim_eq_full[symmetric]
          assms(1) assms(3) dim_eq_card_independent)
   moreover have "rel_interior (convex hull insert 0 S) \<noteq> {}"
     using rel_interior_eq_empty [of "convex hull (insert 0 S)"] by auto
   ultimately have "interior (convex hull insert 0 S) \<noteq> {}"
     by (simp add: rel_interior_interior)
   with that show ?thesis
     by auto
 qed
 
 lemma convex_rel_interior:
   fixes S :: "'n::euclidean_space set"
   assumes "convex S"
   shows "convex (rel_interior S)"
 proof -
   {
     fix x y and u :: real
     assume assm: "x \<in> rel_interior S" "y \<in> rel_interior S" "0 \<le> u" "u \<le> 1"
     then have "x \<in> S"
       using rel_interior_subset by auto
     have "x - u *\<^sub>R (x-y) \<in> rel_interior S"
     proof (cases "0 = u")
       case False
       then have "0 < u" using assm by auto
       then show ?thesis
         using assm rel_interior_convex_shrink[of S y x u] assms \<open>x \<in> S\<close> by auto
     next
       case True
       then show ?thesis using assm by auto
     qed
     then have "(1 - u) *\<^sub>R x + u *\<^sub>R y \<in> rel_interior S"
       by (simp add: algebra_simps)
   }
   then show ?thesis
     unfolding convex_alt by auto
 qed
 
 lemma convex_closure_rel_interior:
   fixes S :: "'n::euclidean_space set"
   assumes "convex S"
   shows "closure (rel_interior S) = closure S"
 proof -
   have h1: "closure (rel_interior S) \<le> closure S"
     using closure_mono[of "rel_interior S" S] rel_interior_subset[of S] by auto
   show ?thesis
   proof (cases "S = {}")
     case False
     then obtain a where a: "a \<in> rel_interior S"
       using rel_interior_eq_empty assms by auto
     { fix x
       assume x: "x \<in> closure S"
       {
         assume "x = a"
         then have "x \<in> closure (rel_interior S)"
           using a unfolding closure_def by auto
       }
       moreover
       {
         assume "x \<noteq> a"
          {
            fix e :: real
            assume "e > 0"
            define e1 where "e1 = min 1 (e/norm (x - a))"
            then have e1: "e1 > 0" "e1 \<le> 1" "e1 * norm (x - a) \<le> e"
              using \<open>x \<noteq> a\<close> \<open>e > 0\<close> le_divide_eq[of e1 e "norm (x - a)"]
              by simp_all
            then have *: "x - e1 *\<^sub>R (x - a) \<in> rel_interior S"
              using rel_interior_closure_convex_shrink[of S a x e1] assms x a e1_def
              by auto
            have "\<exists>y. y \<in> rel_interior S \<and> y \<noteq> x \<and> dist y x \<le> e"
               apply (rule_tac x="x - e1 *\<^sub>R (x - a)" in exI)
               using * e1 dist_norm[of "x - e1 *\<^sub>R (x - a)" x] \<open>x \<noteq> a\<close>
               apply simp
               done
         }
         then have "x islimpt rel_interior S"
           unfolding islimpt_approachable_le by auto
         then have "x \<in> closure(rel_interior S)"
           unfolding closure_def by auto
       }
       ultimately have "x \<in> closure(rel_interior S)" by auto
     }
     then show ?thesis using h1 by auto
   next
     case True
     then have "rel_interior S = {}" by auto
     then have "closure (rel_interior S) = {}"
       using closure_empty by auto
     with True show ?thesis by auto
   qed
 qed
 
 lemma rel_interior_same_affine_hull:
   fixes S :: "'n::euclidean_space set"
   assumes "convex S"
   shows "affine hull (rel_interior S) = affine hull S"
   by (metis assms closure_same_affine_hull convex_closure_rel_interior)
 
 lemma rel_interior_aff_dim:
   fixes S :: "'n::euclidean_space set"
   assumes "convex S"
   shows "aff_dim (rel_interior S) = aff_dim S"
   by (metis aff_dim_affine_hull2 assms rel_interior_same_affine_hull)
 
 lemma rel_interior_rel_interior:
   fixes S :: "'n::euclidean_space set"
   assumes "convex S"
   shows "rel_interior (rel_interior S) = rel_interior S"
 proof -
   have "openin (top_of_set (affine hull (rel_interior S))) (rel_interior S)"
     using openin_rel_interior[of S] rel_interior_same_affine_hull[of S] assms by auto
   then show ?thesis
     using rel_interior_def by auto
 qed
 
 lemma rel_interior_rel_open:
   fixes S :: "'n::euclidean_space set"
   assumes "convex S"
   shows "rel_open (rel_interior S)"
   unfolding rel_open_def using rel_interior_rel_interior assms by auto
 
 lemma convex_rel_interior_closure_aux:
   fixes x y z :: "'n::euclidean_space"
   assumes "0 < a" "0 < b" "(a + b) *\<^sub>R z = a *\<^sub>R x + b *\<^sub>R y"
   obtains e where "0 < e" "e \<le> 1" "z = y - e *\<^sub>R (y - x)"
 proof -
   define e where "e = a / (a + b)"
   have "z = (1 / (a + b)) *\<^sub>R ((a + b) *\<^sub>R z)"
     using assms  by (simp add: eq_vector_fraction_iff)
   also have "\<dots> = (1 / (a + b)) *\<^sub>R (a *\<^sub>R x + b *\<^sub>R y)"
     using assms scaleR_cancel_left[of "1/(a+b)" "(a + b) *\<^sub>R z" "a *\<^sub>R x + b *\<^sub>R y"]
     by auto
   also have "\<dots> = y - e *\<^sub>R (y-x)"
     using e_def
     apply (simp add: algebra_simps)
     using scaleR_left_distrib[of "a/(a+b)" "b/(a+b)" y] assms add_divide_distrib[of a b "a+b"]
     apply auto
     done
   finally have "z = y - e *\<^sub>R (y-x)"
     by auto
   moreover have "e > 0" using e_def assms by auto
   moreover have "e \<le> 1" using e_def assms by auto
   ultimately show ?thesis using that[of e] by auto
 qed
 
 lemma convex_rel_interior_closure:
   fixes S :: "'n::euclidean_space set"
   assumes "convex S"
   shows "rel_interior (closure S) = rel_interior S"
 proof (cases "S = {}")
   case True
   then show ?thesis
     using assms rel_interior_eq_empty by auto
 next
   case False
   have "rel_interior (closure S) \<supseteq> rel_interior S"
     using subset_rel_interior[of S "closure S"] closure_same_affine_hull closure_subset
     by auto
   moreover
   {
     fix z
     assume z: "z \<in> rel_interior (closure S)"
     obtain x where x: "x \<in> rel_interior S"
       using \<open>S \<noteq> {}\<close> assms rel_interior_eq_empty by auto
     have "z \<in> rel_interior S"
     proof (cases "x = z")
       case True
       then show ?thesis using x by auto
     next
       case False
       obtain e where e: "e > 0" "cball z e \<inter> affine hull closure S \<le> closure S"
         using z rel_interior_cball[of "closure S"] by auto
       hence *: "0 < e/norm(z-x)" using e False by auto
       define y where "y = z + (e/norm(z-x)) *\<^sub>R (z-x)"
       have yball: "y \<in> cball z e"
         using y_def dist_norm[of z y] e by auto
       have "x \<in> affine hull closure S"
         using x rel_interior_subset_closure hull_inc[of x "closure S"] by blast
       moreover have "z \<in> affine hull closure S"
         using z rel_interior_subset hull_subset[of "closure S"] by blast
       ultimately have "y \<in> affine hull closure S"
         using y_def affine_affine_hull[of "closure S"]
           mem_affine_3_minus [of "affine hull closure S" z z x "e/norm(z-x)"] by auto
       then have "y \<in> closure S" using e yball by auto
       have "(1 + (e/norm(z-x))) *\<^sub>R z = (e/norm(z-x)) *\<^sub>R x + y"
         using y_def by (simp add: algebra_simps)
       then obtain e1 where "0 < e1" "e1 \<le> 1" "z = y - e1 *\<^sub>R (y - x)"
         using * convex_rel_interior_closure_aux[of "e / norm (z - x)" 1 z x y]
         by (auto simp add: algebra_simps)
       then show ?thesis
         using rel_interior_closure_convex_shrink assms x \<open>y \<in> closure S\<close>
         by auto
     qed
   }
   ultimately show ?thesis by auto
 qed
 
 lemma convex_interior_closure:
   fixes S :: "'n::euclidean_space set"
   assumes "convex S"
   shows "interior (closure S) = interior S"
   using closure_aff_dim[of S] interior_rel_interior_gen[of S]
     interior_rel_interior_gen[of "closure S"]
     convex_rel_interior_closure[of S] assms
   by auto
 
 lemma closure_eq_rel_interior_eq:
   fixes S1 S2 :: "'n::euclidean_space set"
   assumes "convex S1"
     and "convex S2"
   shows "closure S1 = closure S2 \<longleftrightarrow> rel_interior S1 = rel_interior S2"
   by (metis convex_rel_interior_closure convex_closure_rel_interior assms)
 
 lemma closure_eq_between:
   fixes S1 S2 :: "'n::euclidean_space set"
   assumes "convex S1"
     and "convex S2"
   shows "closure S1 = closure S2 \<longleftrightarrow> rel_interior S1 \<le> S2 \<and> S2 \<subseteq> closure S1"
   (is "?A \<longleftrightarrow> ?B")
 proof
   assume ?A
   then show ?B
     by (metis assms closure_subset convex_rel_interior_closure rel_interior_subset)
 next
   assume ?B
   then have "closure S1 \<subseteq> closure S2"
     by (metis assms(1) convex_closure_rel_interior closure_mono)
   moreover from \<open>?B\<close> have "closure S1 \<supseteq> closure S2"
     by (metis closed_closure closure_minimal)
   ultimately show ?A ..
 qed
 
 lemma open_inter_closure_rel_interior:
   fixes S A :: "'n::euclidean_space set"
   assumes "convex S"
     and "open A"
   shows "A \<inter> closure S = {} \<longleftrightarrow> A \<inter> rel_interior S = {}"
   by (metis assms convex_closure_rel_interior open_Int_closure_eq_empty)
 
 lemma rel_interior_open_segment:
   fixes a :: "'a :: euclidean_space"
   shows "rel_interior(open_segment a b) = open_segment a b"
 proof (cases "a = b")
   case True then show ?thesis by auto
 next
   case False then show ?thesis
     apply (simp add: rel_interior_eq openin_open)
     apply (rule_tac x="ball (inverse 2 *\<^sub>R (a + b)) (norm(b - a) / 2)" in exI)
     apply (simp add: open_segment_as_ball)
     done
 qed
 
 lemma rel_interior_closed_segment:
   fixes a :: "'a :: euclidean_space"
   shows "rel_interior(closed_segment a b) =
          (if a = b then {a} else open_segment a b)"
 proof (cases "a = b")
   case True then show ?thesis by auto
 next
   case False then show ?thesis
     by simp
        (metis closure_open_segment convex_open_segment convex_rel_interior_closure
               rel_interior_open_segment)
 qed
 
 lemmas rel_interior_segment = rel_interior_closed_segment rel_interior_open_segment
 
-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"
-    apply (rule rel_interior_closure_convex_segment [OF \<open>convex S\<close> a])
-    using assms by blast
-  show ?thesis
-    unfolding starlike_def
-    apply (rule bexI [OF _ \<open>a \<in> T\<close>])
-    apply (simp add: closed_segment_eq_open)
-    apply (intro conjI ballI a \<open>a \<in> T\<close> rel_interior_closure_convex_segment [OF \<open>convex S\<close> a])
-    apply (simp add: order_trans [OF * ST])
-    done
-qed
-
 subsection\<open>The relative frontier of a set\<close>
 
 definition\<^marker>\<open>tag important\<close> "rel_frontier S = closure S - rel_interior S"
 
 lemma rel_frontier_empty [simp]: "rel_frontier {} = {}"
   by (simp add: rel_frontier_def)
 
 lemma rel_frontier_eq_empty:
     fixes S :: "'n::euclidean_space set"
     shows "rel_frontier S = {} \<longleftrightarrow> affine S"
   unfolding rel_frontier_def
   using rel_interior_subset_closure  by (auto simp add: rel_interior_eq_closure [symmetric])
 
 lemma rel_frontier_sing [simp]:
     fixes a :: "'n::euclidean_space"
     shows "rel_frontier {a} = {}"
   by (simp add: rel_frontier_def)
 
 lemma rel_frontier_affine_hull:
   fixes S :: "'a::euclidean_space set"
   shows "rel_frontier S \<subseteq> affine hull S"
 using closure_affine_hull rel_frontier_def by fastforce
 
 lemma rel_frontier_cball [simp]:
     fixes a :: "'n::euclidean_space"
     shows "rel_frontier(cball a r) = (if r = 0 then {} else sphere a r)"
 proof (cases rule: linorder_cases [of r 0])
   case less then show ?thesis
     by (force simp: sphere_def)
 next
   case equal then show ?thesis by simp
 next
   case greater then show ?thesis
     apply simp
     by (metis centre_in_ball empty_iff frontier_cball frontier_def interior_cball interior_rel_interior_gen rel_frontier_def)
 qed
 
 lemma rel_frontier_translation:
   fixes a :: "'a::euclidean_space"
   shows "rel_frontier((\<lambda>x. a + x) ` S) = (\<lambda>x. a + x) ` (rel_frontier S)"
 by (simp add: rel_frontier_def translation_diff rel_interior_translation closure_translation)
 
 lemma rel_frontier_nonempty_interior:
   fixes S :: "'n::euclidean_space set"
   shows "interior S \<noteq> {} \<Longrightarrow> rel_frontier S = frontier S"
 by (metis frontier_def interior_rel_interior_gen rel_frontier_def)
 
 lemma rel_frontier_frontier:
   fixes S :: "'n::euclidean_space set"
   shows "affine hull S = UNIV \<Longrightarrow> rel_frontier S = frontier S"
 by (simp add: frontier_def rel_frontier_def rel_interior_interior)
 
 lemma closest_point_in_rel_frontier:
    "\<lbrakk>closed S; S \<noteq> {}; x \<in> affine hull S - rel_interior S\<rbrakk>
    \<Longrightarrow> closest_point S x \<in> rel_frontier S"
   by (simp add: closest_point_in_rel_interior closest_point_in_set rel_frontier_def)
 
 lemma closed_rel_frontier [iff]:
   fixes S :: "'n::euclidean_space set"
   shows "closed (rel_frontier S)"
 proof -
   have *: "closedin (top_of_set (affine hull S)) (closure S - rel_interior S)"
     by (simp add: closed_subset closedin_diff closure_affine_hull openin_rel_interior)
   show ?thesis
     apply (rule closedin_closed_trans[of "affine hull S" "rel_frontier S"])
     unfolding rel_frontier_def
     using * closed_affine_hull
     apply auto
     done
 qed
 
 lemma closed_rel_boundary:
   fixes S :: "'n::euclidean_space set"
   shows "closed S \<Longrightarrow> closed(S - rel_interior S)"
 by (metis closed_rel_frontier closure_closed rel_frontier_def)
 
 lemma compact_rel_boundary:
   fixes S :: "'n::euclidean_space set"
   shows "compact S \<Longrightarrow> compact(S - rel_interior S)"
 by (metis bounded_diff closed_rel_boundary closure_eq compact_closure compact_imp_closed)
 
 lemma bounded_rel_frontier:
   fixes S :: "'n::euclidean_space set"
   shows "bounded S \<Longrightarrow> bounded(rel_frontier S)"
 by (simp add: bounded_closure bounded_diff rel_frontier_def)
 
 lemma compact_rel_frontier_bounded:
   fixes S :: "'n::euclidean_space set"
   shows "bounded S \<Longrightarrow> compact(rel_frontier S)"
 using bounded_rel_frontier closed_rel_frontier compact_eq_bounded_closed by blast
 
 lemma compact_rel_frontier:
   fixes S :: "'n::euclidean_space set"
   shows "compact S \<Longrightarrow> compact(rel_frontier S)"
 by (meson compact_eq_bounded_closed compact_rel_frontier_bounded)
 
 lemma convex_same_rel_interior_closure:
   fixes S :: "'n::euclidean_space set"
   shows "\<lbrakk>convex S; convex T\<rbrakk>
          \<Longrightarrow> rel_interior S = rel_interior T \<longleftrightarrow> closure S = closure T"
 by (simp add: closure_eq_rel_interior_eq)
 
 lemma convex_same_rel_interior_closure_straddle:
   fixes S :: "'n::euclidean_space set"
   shows "\<lbrakk>convex S; convex T\<rbrakk>
          \<Longrightarrow> rel_interior S = rel_interior T \<longleftrightarrow>
              rel_interior S \<subseteq> T \<and> T \<subseteq> closure S"
 by (simp add: closure_eq_between convex_same_rel_interior_closure)
 
 lemma convex_rel_frontier_aff_dim:
   fixes S1 S2 :: "'n::euclidean_space set"
   assumes "convex S1"
     and "convex S2"
     and "S2 \<noteq> {}"
     and "S1 \<le> rel_frontier S2"
   shows "aff_dim S1 < aff_dim S2"
 proof -
   have "S1 \<subseteq> closure S2"
     using assms unfolding rel_frontier_def by auto
   then have *: "affine hull S1 \<subseteq> affine hull S2"
     using hull_mono[of "S1" "closure S2"] closure_same_affine_hull[of S2] by blast
   then have "aff_dim S1 \<le> aff_dim S2"
     using * aff_dim_affine_hull[of S1] aff_dim_affine_hull[of S2]
       aff_dim_subset[of "affine hull S1" "affine hull S2"]
     by auto
   moreover
   {
     assume eq: "aff_dim S1 = aff_dim S2"
     then have "S1 \<noteq> {}"
       using aff_dim_empty[of S1] aff_dim_empty[of S2] \<open>S2 \<noteq> {}\<close> by auto
     have **: "affine hull S1 = affine hull S2"
        apply (rule affine_dim_equal)
        using * affine_affine_hull
        apply auto
        using \<open>S1 \<noteq> {}\<close> hull_subset[of S1]
        apply auto
        using eq aff_dim_affine_hull[of S1] aff_dim_affine_hull[of S2]
        apply auto
        done
     obtain a where a: "a \<in> rel_interior S1"
       using \<open>S1 \<noteq> {}\<close> rel_interior_eq_empty assms by auto
     obtain T where T: "open T" "a \<in> T \<inter> S1" "T \<inter> affine hull S1 \<subseteq> S1"
        using mem_rel_interior[of a S1] a by auto
     then have "a \<in> T \<inter> closure S2"
       using a assms unfolding rel_frontier_def by auto
     then obtain b where b: "b \<in> T \<inter> rel_interior S2"
       using open_inter_closure_rel_interior[of S2 T] assms T by auto
     then have "b \<in> affine hull S1"
       using rel_interior_subset hull_subset[of S2] ** by auto
     then have "b \<in> S1"
       using T b by auto
     then have False
       using b assms unfolding rel_frontier_def by auto
   }
   ultimately show ?thesis
     using less_le by auto
 qed
 
 lemma convex_rel_interior_if:
   fixes S ::  "'n::euclidean_space set"
   assumes "convex S"
     and "z \<in> rel_interior S"
   shows "\<forall>x\<in>affine hull S. \<exists>m. m > 1 \<and> (\<forall>e. e > 1 \<and> e \<le> m \<longrightarrow> (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> S)"
 proof -
   obtain e1 where e1: "e1 > 0 \<and> cball z e1 \<inter> affine hull S \<subseteq> S"
     using mem_rel_interior_cball[of z S] assms by auto
   {
     fix x
     assume x: "x \<in> affine hull S"
     {
       assume "x \<noteq> z"
       define m where "m = 1 + e1/norm(x-z)"
       hence "m > 1" using e1 \<open>x \<noteq> z\<close> by auto
       {
         fix e
         assume e: "e > 1 \<and> e \<le> m"
         have "z \<in> affine hull S"
           using assms rel_interior_subset hull_subset[of S] by auto
         then have *: "(1 - e)*\<^sub>R x + e *\<^sub>R z \<in> affine hull S"
           using mem_affine[of "affine hull S" x z "(1-e)" e] affine_affine_hull[of S] x
           by auto
         have "norm (z + e *\<^sub>R x - (x + e *\<^sub>R z)) = norm ((e - 1) *\<^sub>R (x - z))"
           by (simp add: algebra_simps)
         also have "\<dots> = (e - 1) * norm (x-z)"
           using norm_scaleR e by auto
         also have "\<dots> \<le> (m - 1) * norm (x - z)"
           using e mult_right_mono[of _ _ "norm(x-z)"] by auto
         also have "\<dots> = (e1 / norm (x - z)) * norm (x - z)"
           using m_def by auto
         also have "\<dots> = e1"
           using \<open>x \<noteq> z\<close> e1 by simp
         finally have **: "norm (z + e *\<^sub>R x - (x + e *\<^sub>R z)) \<le> e1"
           by auto
         have "(1 - e)*\<^sub>R x+ e *\<^sub>R z \<in> cball z e1"
           using m_def **
           unfolding cball_def dist_norm
           by (auto simp add: algebra_simps)
         then have "(1 - e) *\<^sub>R x+ e *\<^sub>R z \<in> S"
           using e * e1 by auto
       }
       then have "\<exists>m. m > 1 \<and> (\<forall>e. e > 1 \<and> e \<le> m \<longrightarrow> (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> S )"
         using \<open>m> 1 \<close> by auto
     }
     moreover
     {
       assume "x = z"
       define m where "m = 1 + e1"
       then have "m > 1"
         using e1 by auto
       {
         fix e
         assume e: "e > 1 \<and> e \<le> m"
         then have "(1 - e) *\<^sub>R x + e *\<^sub>R z \<in> S"
           using e1 x \<open>x = z\<close> by (auto simp add: algebra_simps)
         then have "(1 - e) *\<^sub>R x + e *\<^sub>R z \<in> S"
           using e by auto
       }
       then have "\<exists>m. m > 1 \<and> (\<forall>e. e > 1 \<and> e \<le> m \<longrightarrow> (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> S)"
         using \<open>m > 1\<close> by auto
     }
     ultimately have "\<exists>m. m > 1 \<and> (\<forall>e. e > 1 \<and> e \<le> m \<longrightarrow> (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> S )"
       by blast
   }
   then show ?thesis by auto
 qed
 
 lemma convex_rel_interior_if2:
   fixes S :: "'n::euclidean_space set"
   assumes "convex S"
   assumes "z \<in> rel_interior S"
   shows "\<forall>x\<in>affine hull S. \<exists>e. e > 1 \<and> (1 - e)*\<^sub>R x + e *\<^sub>R z \<in> S"
   using convex_rel_interior_if[of S z] assms by auto
 
 lemma convex_rel_interior_only_if:
   fixes S :: "'n::euclidean_space set"
   assumes "convex S"
     and "S \<noteq> {}"
   assumes "\<forall>x\<in>S. \<exists>e. e > 1 \<and> (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> S"
   shows "z \<in> rel_interior S"
 proof -
   obtain x where x: "x \<in> rel_interior S"
     using rel_interior_eq_empty assms by auto
   then have "x \<in> S"
     using rel_interior_subset by auto
   then obtain e where e: "e > 1 \<and> (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> S"
     using assms by auto
   define y where [abs_def]: "y = (1 - e) *\<^sub>R x + e *\<^sub>R z"
   then have "y \<in> S" using e by auto
   define e1 where "e1 = 1/e"
   then have "0 < e1 \<and> e1 < 1" using e by auto
   then have "z  =y - (1 - e1) *\<^sub>R (y - x)"
     using e1_def y_def by (auto simp add: algebra_simps)
   then show ?thesis
     using rel_interior_convex_shrink[of S x y "1-e1"] \<open>0 < e1 \<and> e1 < 1\<close> \<open>y \<in> S\<close> x assms
     by auto
 qed
 
 lemma convex_rel_interior_iff:
   fixes S :: "'n::euclidean_space set"
   assumes "convex S"
     and "S \<noteq> {}"
   shows "z \<in> rel_interior S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e. e > 1 \<and> (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> S)"
   using assms hull_subset[of S "affine"]
     convex_rel_interior_if[of S z] convex_rel_interior_only_if[of S z]
   by auto
 
 lemma convex_rel_interior_iff2:
   fixes S :: "'n::euclidean_space set"
   assumes "convex S"
     and "S \<noteq> {}"
   shows "z \<in> rel_interior S \<longleftrightarrow> (\<forall>x\<in>affine hull S. \<exists>e. e > 1 \<and> (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> S)"
   using assms hull_subset[of S] convex_rel_interior_if2[of S z] convex_rel_interior_only_if[of S z]
   by auto
 
 lemma convex_interior_iff:
   fixes S :: "'n::euclidean_space set"
   assumes "convex S"
   shows "z \<in> interior S \<longleftrightarrow> (\<forall>x. \<exists>e. e > 0 \<and> z + e *\<^sub>R x \<in> S)"
 proof (cases "aff_dim S = int DIM('n)")
   case False
   { assume "z \<in> interior S"
     then have False
       using False interior_rel_interior_gen[of S] by auto }
   moreover
   { assume r: "\<forall>x. \<exists>e. e > 0 \<and> z + e *\<^sub>R x \<in> S"
     { fix x
       obtain e1 where e1: "e1 > 0 \<and> z + e1 *\<^sub>R (x - z) \<in> S"
         using r by auto
       obtain e2 where e2: "e2 > 0 \<and> z + e2 *\<^sub>R (z - x) \<in> S"
         using r by auto
       define x1 where [abs_def]: "x1 = z + e1 *\<^sub>R (x - z)"
       then have x1: "x1 \<in> affine hull S"
         using e1 hull_subset[of S] by auto
       define x2 where [abs_def]: "x2 = z + e2 *\<^sub>R (z - x)"
       then have x2: "x2 \<in> affine hull S"
         using e2 hull_subset[of S] by auto
       have *: "e1/(e1+e2) + e2/(e1+e2) = 1"
         using add_divide_distrib[of e1 e2 "e1+e2"] e1 e2 by simp
       then have "z = (e2/(e1+e2)) *\<^sub>R x1 + (e1/(e1+e2)) *\<^sub>R x2"
         using x1_def x2_def
         apply (auto simp add: algebra_simps)
         using scaleR_left_distrib[of "e1/(e1+e2)" "e2/(e1+e2)" z]
         apply auto
         done
       then have z: "z \<in> affine hull S"
         using mem_affine[of "affine hull S" x1 x2 "e2/(e1+e2)" "e1/(e1+e2)"]
           x1 x2 affine_affine_hull[of S] *
         by auto
       have "x1 - x2 = (e1 + e2) *\<^sub>R (x - z)"
         using x1_def x2_def by (auto simp add: algebra_simps)
       then have "x = z+(1/(e1+e2)) *\<^sub>R (x1-x2)"
         using e1 e2 by simp
       then have "x \<in> affine hull S"
         using mem_affine_3_minus[of "affine hull S" z x1 x2 "1/(e1+e2)"]
           x1 x2 z affine_affine_hull[of S]
         by auto
     }
     then have "affine hull S = UNIV"
       by auto
     then have "aff_dim S = int DIM('n)"
       using aff_dim_affine_hull[of S] by (simp)
     then have False
       using False by auto
   }
   ultimately show ?thesis by auto
 next
   case True
   then have "S \<noteq> {}"
     using aff_dim_empty[of S] by auto
   have *: "affine hull S = UNIV"
     using True affine_hull_UNIV by auto
   {
     assume "z \<in> interior S"
     then have "z \<in> rel_interior S"
       using True interior_rel_interior_gen[of S] by auto
     then have **: "\<forall>x. \<exists>e. e > 1 \<and> (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> S"
       using convex_rel_interior_iff2[of S z] assms \<open>S \<noteq> {}\<close> * by auto
     fix x
     obtain e1 where e1: "e1 > 1" "(1 - e1) *\<^sub>R (z - x) + e1 *\<^sub>R z \<in> S"
       using **[rule_format, of "z-x"] by auto
     define e where [abs_def]: "e = e1 - 1"
     then have "(1 - e1) *\<^sub>R (z - x) + e1 *\<^sub>R z = z + e *\<^sub>R x"
       by (simp add: algebra_simps)
     then have "e > 0" "z + e *\<^sub>R x \<in> S"
       using e1 e_def by auto
     then have "\<exists>e. e > 0 \<and> z + e *\<^sub>R x \<in> S"
       by auto
   }
   moreover
   {
     assume r: "\<forall>x. \<exists>e. e > 0 \<and> z + e *\<^sub>R x \<in> S"
     {
       fix x
       obtain e1 where e1: "e1 > 0" "z + e1 *\<^sub>R (z - x) \<in> S"
         using r[rule_format, of "z-x"] by auto
       define e where "e = e1 + 1"
       then have "z + e1 *\<^sub>R (z - x) = (1 - e) *\<^sub>R x + e *\<^sub>R z"
         by (simp add: algebra_simps)
       then have "e > 1" "(1 - e)*\<^sub>R x + e *\<^sub>R z \<in> S"
         using e1 e_def by auto
       then have "\<exists>e. e > 1 \<and> (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> S" by auto
     }
     then have "z \<in> rel_interior S"
       using convex_rel_interior_iff2[of S z] assms \<open>S \<noteq> {}\<close> by auto
     then have "z \<in> interior S"
       using True interior_rel_interior_gen[of S] by auto
   }
   ultimately show ?thesis by auto
 qed
 
 
 subsubsection\<^marker>\<open>tag unimportant\<close> \<open>Relative interior and closure under common operations\<close>
 
 lemma rel_interior_inter_aux: "\<Inter>{rel_interior S |S. S \<in> I} \<subseteq> \<Inter>I"
 proof -
   {
     fix y
     assume "y \<in> \<Inter>{rel_interior S |S. S \<in> I}"
     then have y: "\<forall>S \<in> I. y \<in> rel_interior S"
       by auto
     {
       fix S
       assume "S \<in> I"
       then have "y \<in> S"
         using rel_interior_subset y by auto
     }
     then have "y \<in> \<Inter>I" by auto
   }
   then show ?thesis by auto
 qed
 
 lemma convex_closure_rel_interior_inter:
   assumes "\<forall>S\<in>I. convex (S :: 'n::euclidean_space set)"
     and "\<Inter>{rel_interior S |S. S \<in> I} \<noteq> {}"
   shows "\<Inter>{closure S |S. S \<in> I} \<le> closure (\<Inter>{rel_interior S |S. S \<in> I})"
 proof -
   obtain x where x: "\<forall>S\<in>I. x \<in> rel_interior S"
     using assms by auto
   {
     fix y
     assume "y \<in> \<Inter>{closure S |S. S \<in> I}"
     then have y: "\<forall>S \<in> I. y \<in> closure S"
       by auto
     {
       assume "y = x"
       then have "y \<in> closure (\<Inter>{rel_interior S |S. S \<in> I})"
         using x closure_subset[of "\<Inter>{rel_interior S |S. S \<in> I}"] by auto
     }
     moreover
     {
       assume "y \<noteq> x"
       { fix e :: real
         assume e: "e > 0"
         define e1 where "e1 = min 1 (e/norm (y - x))"
         then have e1: "e1 > 0" "e1 \<le> 1" "e1 * norm (y - x) \<le> e"
           using \<open>y \<noteq> x\<close> \<open>e > 0\<close> le_divide_eq[of e1 e "norm (y - x)"]
           by simp_all
         define z where "z = y - e1 *\<^sub>R (y - x)"
         {
           fix S
           assume "S \<in> I"
           then have "z \<in> rel_interior S"
             using rel_interior_closure_convex_shrink[of S x y e1] assms x y e1 z_def
             by auto
         }
         then have *: "z \<in> \<Inter>{rel_interior S |S. S \<in> I}"
           by auto
         have "\<exists>z. z \<in> \<Inter>{rel_interior S |S. S \<in> I} \<and> z \<noteq> y \<and> dist z y \<le> e"
           apply (rule_tac x="z" in exI)
           using \<open>y \<noteq> x\<close> z_def * e1 e dist_norm[of z y]
           apply simp
           done
       }
       then have "y islimpt \<Inter>{rel_interior S |S. S \<in> I}"
         unfolding islimpt_approachable_le by blast
       then have "y \<in> closure (\<Inter>{rel_interior S |S. S \<in> I})"
         unfolding closure_def by auto
     }
     ultimately have "y \<in> closure (\<Inter>{rel_interior S |S. S \<in> I})"
       by auto
   }
   then show ?thesis by auto
 qed
 
 lemma convex_closure_inter:
   assumes "\<forall>S\<in>I. convex (S :: 'n::euclidean_space set)"
     and "\<Inter>{rel_interior S |S. S \<in> I} \<noteq> {}"
   shows "closure (\<Inter>I) = \<Inter>{closure S |S. S \<in> I}"
 proof -
   have "\<Inter>{closure S |S. S \<in> I} \<le> closure (\<Inter>{rel_interior S |S. S \<in> I})"
     using convex_closure_rel_interior_inter assms by auto
   moreover
   have "closure (\<Inter>{rel_interior S |S. S \<in> I}) \<le> closure (\<Inter>I)"
     using rel_interior_inter_aux closure_mono[of "\<Inter>{rel_interior S |S. S \<in> I}" "\<Inter>I"]
     by auto
   ultimately show ?thesis
     using closure_Int[of I] by auto
 qed
 
 lemma convex_inter_rel_interior_same_closure:
   assumes "\<forall>S\<in>I. convex (S :: 'n::euclidean_space set)"
     and "\<Inter>{rel_interior S |S. S \<in> I} \<noteq> {}"
   shows "closure (\<Inter>{rel_interior S |S. S \<in> I}) = closure (\<Inter>I)"
 proof -
   have "\<Inter>{closure S |S. S \<in> I} \<le> closure (\<Inter>{rel_interior S |S. S \<in> I})"
     using convex_closure_rel_interior_inter assms by auto
   moreover
   have "closure (\<Inter>{rel_interior S |S. S \<in> I}) \<le> closure (\<Inter>I)"
     using rel_interior_inter_aux closure_mono[of "\<Inter>{rel_interior S |S. S \<in> I}" "\<Inter>I"]
     by auto
   ultimately show ?thesis
     using closure_Int[of I] by auto
 qed
 
 lemma convex_rel_interior_inter:
   assumes "\<forall>S\<in>I. convex (S :: 'n::euclidean_space set)"
     and "\<Inter>{rel_interior S |S. S \<in> I} \<noteq> {}"
   shows "rel_interior (\<Inter>I) \<subseteq> \<Inter>{rel_interior S |S. S \<in> I}"
 proof -
   have "convex (\<Inter>I)"
     using assms convex_Inter by auto
   moreover
   have "convex (\<Inter>{rel_interior S |S. S \<in> I})"
     apply (rule convex_Inter)
     using assms convex_rel_interior
     apply auto
     done
   ultimately
   have "rel_interior (\<Inter>{rel_interior S |S. S \<in> I}) = rel_interior (\<Inter>I)"
     using convex_inter_rel_interior_same_closure assms
       closure_eq_rel_interior_eq[of "\<Inter>{rel_interior S |S. S \<in> I}" "\<Inter>I"]
     by blast
   then show ?thesis
     using rel_interior_subset[of "\<Inter>{rel_interior S |S. S \<in> I}"] by auto
 qed
 
 lemma convex_rel_interior_finite_inter:
   assumes "\<forall>S\<in>I. convex (S :: 'n::euclidean_space set)"
     and "\<Inter>{rel_interior S |S. S \<in> I} \<noteq> {}"
     and "finite I"
   shows "rel_interior (\<Inter>I) = \<Inter>{rel_interior S |S. S \<in> I}"
 proof -
   have "\<Inter>I \<noteq> {}"
     using assms rel_interior_inter_aux[of I] by auto
   have "convex (\<Inter>I)"
     using convex_Inter assms by auto
   show ?thesis
   proof (cases "I = {}")
     case True
     then show ?thesis
       using Inter_empty rel_interior_UNIV by auto
   next
     case False
     {
       fix z
       assume z: "z \<in> \<Inter>{rel_interior S |S. S \<in> I}"
       {
         fix x
         assume x: "x \<in> \<Inter>I"
         {
           fix S
           assume S: "S \<in> I"
           then have "z \<in> rel_interior S" "x \<in> S"
             using z x by auto
           then have "\<exists>m. m > 1 \<and> (\<forall>e. e > 1 \<and> e \<le> m \<longrightarrow> (1 - e)*\<^sub>R x + e *\<^sub>R z \<in> S)"
             using convex_rel_interior_if[of S z] S assms hull_subset[of S] by auto
         }
         then obtain mS where
           mS: "\<forall>S\<in>I. mS S > 1 \<and> (\<forall>e. e > 1 \<and> e \<le> mS S \<longrightarrow> (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> S)" by metis
         define e where "e = Min (mS ` I)"
         then have "e \<in> mS ` I" using assms \<open>I \<noteq> {}\<close> by simp
         then have "e > 1" using mS by auto
         moreover have "\<forall>S\<in>I. e \<le> mS S"
           using e_def assms by auto
         ultimately have "\<exists>e > 1. (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> \<Inter>I"
           using mS by auto
       }
       then have "z \<in> rel_interior (\<Inter>I)"
         using convex_rel_interior_iff[of "\<Inter>I" z] \<open>\<Inter>I \<noteq> {}\<close> \<open>convex (\<Inter>I)\<close> by auto
     }
     then show ?thesis
       using convex_rel_interior_inter[of I] assms by auto
   qed
 qed
 
 lemma convex_closure_inter_two:
   fixes S T :: "'n::euclidean_space set"
   assumes "convex S"
     and "convex T"
   assumes "rel_interior S \<inter> rel_interior T \<noteq> {}"
   shows "closure (S \<inter> T) = closure S \<inter> closure T"
   using convex_closure_inter[of "{S,T}"] assms by auto
 
 lemma convex_rel_interior_inter_two:
   fixes S T :: "'n::euclidean_space set"
   assumes "convex S"
     and "convex T"
     and "rel_interior S \<inter> rel_interior T \<noteq> {}"
   shows "rel_interior (S \<inter> T) = rel_interior S \<inter> rel_interior T"
   using convex_rel_interior_finite_inter[of "{S,T}"] assms by auto
 
 lemma convex_affine_closure_Int:
   fixes S T :: "'n::euclidean_space set"
   assumes "convex S"
     and "affine T"
     and "rel_interior S \<inter> T \<noteq> {}"
   shows "closure (S \<inter> T) = closure S \<inter> T"
 proof -
   have "affine hull T = T"
     using assms by auto
   then have "rel_interior T = T"
     using rel_interior_affine_hull[of T] by metis
   moreover have "closure T = T"
     using assms affine_closed[of T] by auto
   ultimately show ?thesis
     using convex_closure_inter_two[of S T] assms affine_imp_convex by auto
 qed
 
 lemma connected_component_1_gen:
   fixes S :: "'a :: euclidean_space set"
   assumes "DIM('a) = 1"
   shows "connected_component S a b \<longleftrightarrow> closed_segment a b \<subseteq> S"
 unfolding connected_component_def
 by (metis (no_types, lifting) assms subsetD subsetI convex_contains_segment convex_segment(1)
             ends_in_segment connected_convex_1_gen)
 
 lemma connected_component_1:
   fixes S :: "real set"
   shows "connected_component S a b \<longleftrightarrow> closed_segment a b \<subseteq> S"
 by (simp add: connected_component_1_gen)
 
 lemma convex_affine_rel_interior_Int:
   fixes S T :: "'n::euclidean_space set"
   assumes "convex S"
     and "affine T"
     and "rel_interior S \<inter> T \<noteq> {}"
   shows "rel_interior (S \<inter> T) = rel_interior S \<inter> T"
 proof -
   have "affine hull T = T"
     using assms by auto
   then have "rel_interior T = T"
     using rel_interior_affine_hull[of T] by metis
   moreover have "closure T = T"
     using assms affine_closed[of T] by auto
   ultimately show ?thesis
     using convex_rel_interior_inter_two[of S T] assms affine_imp_convex by auto
 qed
 
 lemma convex_affine_rel_frontier_Int:
    fixes S T :: "'n::euclidean_space set"
   assumes "convex S"
     and "affine T"
     and "interior S \<inter> T \<noteq> {}"
   shows "rel_frontier(S \<inter> T) = frontier S \<inter> T"
 using assms
 apply (simp add: rel_frontier_def convex_affine_closure_Int frontier_def)
 by (metis Diff_Int_distrib2 Int_emptyI convex_affine_closure_Int convex_affine_rel_interior_Int empty_iff interior_rel_interior_gen)
 
 lemma rel_interior_convex_Int_affine:
   fixes S :: "'a::euclidean_space set"
   assumes "convex S" "affine T" "interior S \<inter> T \<noteq> {}"
     shows "rel_interior(S \<inter> T) = interior S \<inter> T"
 proof -
   obtain a where aS: "a \<in> interior S" and aT:"a \<in> T"
     using assms by force
   have "rel_interior S = interior S"
     by (metis (no_types) aS affine_hull_nonempty_interior equals0D rel_interior_interior)
   then show ?thesis
     by (metis (no_types) affine_imp_convex assms convex_rel_interior_inter_two hull_same rel_interior_affine_hull)
 qed
 
 lemma closure_convex_Int_affine:
   fixes S :: "'a::euclidean_space set"
   assumes "convex S" "affine T" "rel_interior S \<inter> T \<noteq> {}"
   shows "closure(S \<inter> T) = closure S \<inter> T"
 proof
   have "closure (S \<inter> T) \<subseteq> closure T"
     by (simp add: closure_mono)
   also have "... \<subseteq> T"
     by (simp add: affine_closed assms)
   finally show "closure(S \<inter> T) \<subseteq> closure S \<inter> T"
     by (simp add: closure_mono)
 next
   obtain a where "a \<in> rel_interior S" "a \<in> T"
     using assms by auto
   then have ssT: "subspace ((\<lambda>x. (-a)+x) ` T)" and "a \<in> S"
     using affine_diffs_subspace rel_interior_subset assms by blast+
   show "closure S \<inter> T \<subseteq> closure (S \<inter> T)"
   proof
     fix x  assume "x \<in> closure S \<inter> T"
     show "x \<in> closure (S \<inter> T)"
     proof (cases "x = a")
       case True
       then show ?thesis
         using \<open>a \<in> S\<close> \<open>a \<in> T\<close> closure_subset by fastforce
     next
       case False
       then have "x \<in> closure(open_segment a x)"
         by auto
       then show ?thesis
         using \<open>x \<in> closure S \<inter> T\<close> assms convex_affine_closure_Int by blast
     qed
   qed
 qed
 
 lemma subset_rel_interior_convex:
   fixes S T :: "'n::euclidean_space set"
   assumes "convex S"
     and "convex T"
     and "S \<le> closure T"
     and "\<not> S \<subseteq> rel_frontier T"
   shows "rel_interior S \<subseteq> rel_interior T"
 proof -
   have *: "S \<inter> closure T = S"
     using assms by auto
   have "\<not> rel_interior S \<subseteq> rel_frontier T"
     using closure_mono[of "rel_interior S" "rel_frontier T"] closed_rel_frontier[of T]
       closure_closed[of S] convex_closure_rel_interior[of S] closure_subset[of S] assms
     by auto
   then have "rel_interior S \<inter> rel_interior (closure T) \<noteq> {}"
     using assms rel_frontier_def[of T] rel_interior_subset convex_rel_interior_closure[of T]
     by auto
   then have "rel_interior S \<inter> rel_interior T = rel_interior (S \<inter> closure T)"
     using assms convex_closure convex_rel_interior_inter_two[of S "closure T"]
       convex_rel_interior_closure[of T]
     by auto
   also have "\<dots> = rel_interior S"
     using * by auto
   finally show ?thesis
     by auto
 qed
 
 lemma rel_interior_convex_linear_image:
   fixes f :: "'m::euclidean_space \<Rightarrow> 'n::euclidean_space"
   assumes "linear f"
     and "convex S"
   shows "f ` (rel_interior S) = rel_interior (f ` S)"
 proof (cases "S = {}")
   case True
   then show ?thesis
     using assms by auto
 next
   case False
   interpret linear f by fact
   have *: "f ` (rel_interior S) \<subseteq> f ` S"
     unfolding image_mono using rel_interior_subset by auto
   have "f ` S \<subseteq> f ` (closure S)"
     unfolding image_mono using closure_subset by auto
   also have "\<dots> = f ` (closure (rel_interior S))"
     using convex_closure_rel_interior assms by auto
   also have "\<dots> \<subseteq> closure (f ` (rel_interior S))"
     using closure_linear_image_subset assms by auto
   finally have "closure (f ` S) = closure (f ` rel_interior S)"
     using closure_mono[of "f ` S" "closure (f ` rel_interior S)"] closure_closure
       closure_mono[of "f ` rel_interior S" "f ` S"] *
     by auto
   then have "rel_interior (f ` S) = rel_interior (f ` rel_interior S)"
     using assms convex_rel_interior
       linear_conv_bounded_linear[of f] convex_linear_image[of _ S]
       convex_linear_image[of _ "rel_interior S"]
       closure_eq_rel_interior_eq[of "f ` S" "f ` rel_interior S"]
     by auto
   then have "rel_interior (f ` S) \<subseteq> f ` rel_interior S"
     using rel_interior_subset by auto
   moreover
   {
     fix z
     assume "z \<in> f ` rel_interior S"
     then obtain z1 where z1: "z1 \<in> rel_interior S" "f z1 = z" by auto
     {
       fix x
       assume "x \<in> f ` S"
       then obtain x1 where x1: "x1 \<in> S" "f x1 = x" by auto
       then obtain e where e: "e > 1" "(1 - e) *\<^sub>R x1 + e *\<^sub>R z1 \<in> S"
         using convex_rel_interior_iff[of S z1] \<open>convex S\<close> x1 z1 by auto
       moreover have "f ((1 - e) *\<^sub>R x1 + e *\<^sub>R z1) = (1 - e) *\<^sub>R x + e *\<^sub>R z"
         using x1 z1 by (simp add: linear_add linear_scale \<open>linear f\<close>)
       ultimately have "(1 - e) *\<^sub>R x + e *\<^sub>R z \<in> f ` S"
         using imageI[of "(1 - e) *\<^sub>R x1 + e *\<^sub>R z1" S f] by auto
       then have "\<exists>e. e > 1 \<and> (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> f ` S"
         using e by auto
     }
     then have "z \<in> rel_interior (f ` S)"
       using convex_rel_interior_iff[of "f ` S" z] \<open>convex S\<close> \<open>linear f\<close>
         \<open>S \<noteq> {}\<close> convex_linear_image[of f S]  linear_conv_bounded_linear[of f]
       by auto
   }
   ultimately show ?thesis by auto
 qed
 
 lemma rel_interior_convex_linear_preimage:
   fixes f :: "'m::euclidean_space \<Rightarrow> 'n::euclidean_space"
   assumes "linear f"
     and "convex S"
     and "f -` (rel_interior S) \<noteq> {}"
   shows "rel_interior (f -` S) = f -` (rel_interior S)"
 proof -
   interpret linear f by fact
   have "S \<noteq> {}"
     using assms by auto
   have nonemp: "f -` S \<noteq> {}"
     by (metis assms(3) rel_interior_subset subset_empty vimage_mono)
   then have "S \<inter> (range f) \<noteq> {}"
     by auto
   have conv: "convex (f -` S)"
     using convex_linear_vimage assms by auto
   then have "convex (S \<inter> range f)"
     by (simp add: assms(2) convex_Int convex_linear_image linear_axioms)
   {
     fix z
     assume "z \<in> f -` (rel_interior S)"
     then have z: "f z \<in> rel_interior S"
       by auto
     {
       fix x
       assume "x \<in> f -` S"
       then have "f x \<in> S" by auto
       then obtain e where e: "e > 1" "(1 - e) *\<^sub>R f x + e *\<^sub>R f z \<in> S"
         using convex_rel_interior_iff[of S "f z"] z assms \<open>S \<noteq> {}\<close> by auto
       moreover have "(1 - e) *\<^sub>R f x + e *\<^sub>R f z = f ((1 - e) *\<^sub>R x + e *\<^sub>R z)"
         using \<open>linear f\<close> by (simp add: linear_iff)
       ultimately have "\<exists>e. e > 1 \<and> (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> f -` S"
         using e by auto
     }
     then have "z \<in> rel_interior (f -` S)"
       using convex_rel_interior_iff[of "f -` S" z] conv nonemp by auto
   }
   moreover
   {
     fix z
     assume z: "z \<in> rel_interior (f -` S)"
     {
       fix x
       assume "x \<in> S \<inter> range f"
       then obtain y where y: "f y = x" "y \<in> f -` S" by auto
       then obtain e where e: "e > 1" "(1 - e) *\<^sub>R y + e *\<^sub>R z \<in> f -` S"
         using convex_rel_interior_iff[of "f -` S" z] z conv by auto
       moreover have "(1 - e) *\<^sub>R x + e *\<^sub>R f z = f ((1 - e) *\<^sub>R y + e *\<^sub>R z)"
         using \<open>linear f\<close> y by (simp add: linear_iff)
       ultimately have "\<exists>e. e > 1 \<and> (1 - e) *\<^sub>R x + e *\<^sub>R f z \<in> S \<inter> range f"
         using e by auto
     }
     then have "f z \<in> rel_interior (S \<inter> range f)"
       using \<open>convex (S \<inter> (range f))\<close> \<open>S \<inter> range f \<noteq> {}\<close>
         convex_rel_interior_iff[of "S \<inter> (range f)" "f z"]
       by auto
     moreover have "affine (range f)"
       by (simp add: linear_axioms linear_subspace_image subspace_imp_affine)
     ultimately have "f z \<in> rel_interior S"
       using convex_affine_rel_interior_Int[of S "range f"] assms by auto
     then have "z \<in> f -` (rel_interior S)"
       by auto
   }
   ultimately show ?thesis by auto
 qed
 
 lemma rel_interior_Times:
   fixes S :: "'n::euclidean_space set"
     and T :: "'m::euclidean_space set"
   assumes "convex S"
     and "convex T"
   shows "rel_interior (S \<times> T) = rel_interior S \<times> rel_interior T"
 proof -
   { assume "S = {}"
     then have ?thesis
       by auto
   }
   moreover
   { assume "T = {}"
     then have ?thesis
        by auto
   }
   moreover
   {
     assume "S \<noteq> {}" "T \<noteq> {}"
     then have ri: "rel_interior S \<noteq> {}" "rel_interior T \<noteq> {}"
       using rel_interior_eq_empty assms by auto
     then have "fst -` rel_interior S \<noteq> {}"
       using fst_vimage_eq_Times[of "rel_interior S"] by auto
     then have "rel_interior ((fst :: 'n * 'm \<Rightarrow> 'n) -` S) = fst -` rel_interior S"
       using fst_linear \<open>convex S\<close> rel_interior_convex_linear_preimage[of fst S] by auto
     then have s: "rel_interior (S \<times> (UNIV :: 'm set)) = rel_interior S \<times> UNIV"
       by (simp add: fst_vimage_eq_Times)
     from ri have "snd -` rel_interior T \<noteq> {}"
       using snd_vimage_eq_Times[of "rel_interior T"] by auto
     then have "rel_interior ((snd :: 'n * 'm \<Rightarrow> 'm) -` T) = snd -` rel_interior T"
       using snd_linear \<open>convex T\<close> rel_interior_convex_linear_preimage[of snd T] by auto
     then have t: "rel_interior ((UNIV :: 'n set) \<times> T) = UNIV \<times> rel_interior T"
       by (simp add: snd_vimage_eq_Times)
     from s t have *: "rel_interior (S \<times> (UNIV :: 'm set)) \<inter> rel_interior ((UNIV :: 'n set) \<times> T) =
       rel_interior S \<times> rel_interior T" by auto
     have "S \<times> T = S \<times> (UNIV :: 'm set) \<inter> (UNIV :: 'n set) \<times> T"
       by auto
     then have "rel_interior (S \<times> T) = rel_interior ((S \<times> (UNIV :: 'm set)) \<inter> ((UNIV :: 'n set) \<times> T))"
       by auto
     also have "\<dots> = rel_interior (S \<times> (UNIV :: 'm set)) \<inter> rel_interior ((UNIV :: 'n set) \<times> T)"
        apply (subst convex_rel_interior_inter_two[of "S \<times> (UNIV :: 'm set)" "(UNIV :: 'n set) \<times> T"])
        using * ri assms convex_Times
        apply auto
        done
     finally have ?thesis using * by auto
   }
   ultimately show ?thesis by blast
 qed
 
 lemma rel_interior_scaleR:
   fixes S :: "'n::euclidean_space set"
   assumes "c \<noteq> 0"
   shows "((*\<^sub>R) c) ` (rel_interior S) = rel_interior (((*\<^sub>R) c) ` S)"
   using rel_interior_injective_linear_image[of "((*\<^sub>R) c)" S]
     linear_conv_bounded_linear[of "(*\<^sub>R) c"] linear_scaleR injective_scaleR[of c] assms
   by auto
 
 lemma rel_interior_convex_scaleR:
   fixes S :: "'n::euclidean_space set"
   assumes "convex S"
   shows "((*\<^sub>R) c) ` (rel_interior S) = rel_interior (((*\<^sub>R) c) ` S)"
   by (metis assms linear_scaleR rel_interior_convex_linear_image)
 
 lemma convex_rel_open_scaleR:
   fixes S :: "'n::euclidean_space set"
   assumes "convex S"
     and "rel_open S"
   shows "convex (((*\<^sub>R) c) ` S) \<and> rel_open (((*\<^sub>R) c) ` S)"
   by (metis assms convex_scaling rel_interior_convex_scaleR rel_open_def)
 
 lemma convex_rel_open_finite_inter:
   assumes "\<forall>S\<in>I. convex (S :: 'n::euclidean_space set) \<and> rel_open S"
     and "finite I"
   shows "convex (\<Inter>I) \<and> rel_open (\<Inter>I)"
 proof (cases "\<Inter>{rel_interior S |S. S \<in> I} = {}")
   case True
   then have "\<Inter>I = {}"
     using assms unfolding rel_open_def by auto
   then show ?thesis
     unfolding rel_open_def by auto
 next
   case False
   then have "rel_open (\<Inter>I)"
     using assms unfolding rel_open_def
     using convex_rel_interior_finite_inter[of I]
     by auto
   then show ?thesis
     using convex_Inter assms by auto
 qed
 
 lemma convex_rel_open_linear_image:
   fixes f :: "'m::euclidean_space \<Rightarrow> 'n::euclidean_space"
   assumes "linear f"
     and "convex S"
     and "rel_open S"
   shows "convex (f ` S) \<and> rel_open (f ` S)"
   by (metis assms convex_linear_image rel_interior_convex_linear_image rel_open_def)
 
 lemma convex_rel_open_linear_preimage:
   fixes f :: "'m::euclidean_space \<Rightarrow> 'n::euclidean_space"
   assumes "linear f"
     and "convex S"
     and "rel_open S"
   shows "convex (f -` S) \<and> rel_open (f -` S)"
 proof (cases "f -` (rel_interior S) = {}")
   case True
   then have "f -` S = {}"
     using assms unfolding rel_open_def by auto
   then show ?thesis
     unfolding rel_open_def by auto
 next
   case False
   then have "rel_open (f -` S)"
     using assms unfolding rel_open_def
     using rel_interior_convex_linear_preimage[of f S]
     by auto
   then show ?thesis
     using convex_linear_vimage assms
     by auto
 qed
 
 lemma rel_interior_projection:
   fixes S :: "('m::euclidean_space \<times> 'n::euclidean_space) set"
     and f :: "'m::euclidean_space \<Rightarrow> 'n::euclidean_space set"
   assumes "convex S"
     and "f = (\<lambda>y. {z. (y, z) \<in> S})"
   shows "(y, z) \<in> rel_interior S \<longleftrightarrow> (y \<in> rel_interior {y. (f y \<noteq> {})} \<and> z \<in> rel_interior (f y))"
 proof -
   {
     fix y
     assume "y \<in> {y. f y \<noteq> {}}"
     then obtain z where "(y, z) \<in> S"
       using assms by auto
     then have "\<exists>x. x \<in> S \<and> y = fst x"
       apply (rule_tac x="(y, z)" in exI)
       apply auto
       done
     then obtain x where "x \<in> S" "y = fst x"
       by blast
     then have "y \<in> fst ` S"
       unfolding image_def by auto
   }
   then have "fst ` S = {y. f y \<noteq> {}}"
     unfolding fst_def using assms by auto
   then have h1: "fst ` rel_interior S = rel_interior {y. f y \<noteq> {}}"
     using rel_interior_convex_linear_image[of fst S] assms fst_linear by auto
   {
     fix y
     assume "y \<in> rel_interior {y. f y \<noteq> {}}"
     then have "y \<in> fst ` rel_interior S"
       using h1 by auto
     then have *: "rel_interior S \<inter> fst -` {y} \<noteq> {}"
       by auto
     moreover have aff: "affine (fst -` {y})"
       unfolding affine_alt by (simp add: algebra_simps)
     ultimately have **: "rel_interior (S \<inter> fst -` {y}) = rel_interior S \<inter> fst -` {y}"
       using convex_affine_rel_interior_Int[of S "fst -` {y}"] assms by auto
     have conv: "convex (S \<inter> fst -` {y})"
       using convex_Int assms aff affine_imp_convex by auto
     {
       fix x
       assume "x \<in> f y"
       then have "(y, x) \<in> S \<inter> (fst -` {y})"
         using assms by auto
       moreover have "x = snd (y, x)" by auto
       ultimately have "x \<in> snd ` (S \<inter> fst -` {y})"
         by blast
     }
     then have "snd ` (S \<inter> fst -` {y}) = f y"
       using assms by auto
     then have ***: "rel_interior (f y) = snd ` rel_interior (S \<inter> fst -` {y})"
       using rel_interior_convex_linear_image[of snd "S \<inter> fst -` {y}"] snd_linear conv
       by auto
     {
       fix z
       assume "z \<in> rel_interior (f y)"
       then have "z \<in> snd ` rel_interior (S \<inter> fst -` {y})"
         using *** by auto
       moreover have "{y} = fst ` rel_interior (S \<inter> fst -` {y})"
         using * ** rel_interior_subset by auto
       ultimately have "(y, z) \<in> rel_interior (S \<inter> fst -` {y})"
         by force
       then have "(y,z) \<in> rel_interior S"
         using ** by auto
     }
     moreover
     {
       fix z
       assume "(y, z) \<in> rel_interior S"
       then have "(y, z) \<in> rel_interior (S \<inter> fst -` {y})"
         using ** by auto
       then have "z \<in> snd ` rel_interior (S \<inter> fst -` {y})"
         by (metis Range_iff snd_eq_Range)
       then have "z \<in> rel_interior (f y)"
         using *** by auto
     }
     ultimately have "\<And>z. (y, z) \<in> rel_interior S \<longleftrightarrow> z \<in> rel_interior (f y)"
       by auto
   }
   then have h2: "\<And>y z. y \<in> rel_interior {t. f t \<noteq> {}} \<Longrightarrow>
     (y, z) \<in> rel_interior S \<longleftrightarrow> z \<in> rel_interior (f y)"
     by auto
   {
     fix y z
     assume asm: "(y, z) \<in> rel_interior S"
     then have "y \<in> fst ` rel_interior S"
       by (metis Domain_iff fst_eq_Domain)
     then have "y \<in> rel_interior {t. f t \<noteq> {}}"
       using h1 by auto
     then have "y \<in> rel_interior {t. f t \<noteq> {}}" and "(z \<in> rel_interior (f y))"
       using h2 asm by auto
   }
   then show ?thesis using h2 by blast
 qed
 
 lemma rel_frontier_Times:
   fixes S :: "'n::euclidean_space set"
     and T :: "'m::euclidean_space set"
   assumes "convex S"
     and "convex T"
   shows "rel_frontier S \<times> rel_frontier T \<subseteq> rel_frontier (S \<times> T)"
     by (force simp: rel_frontier_def rel_interior_Times assms closure_Times)
 
 
 subsubsection\<^marker>\<open>tag unimportant\<close> \<open>Relative interior of convex cone\<close>
 
 lemma cone_rel_interior:
   fixes S :: "'m::euclidean_space set"
   assumes "cone S"
   shows "cone ({0} \<union> rel_interior S)"
 proof (cases "S = {}")
   case True
   then show ?thesis
     by (simp add: cone_0)
 next
   case False
   then have *: "0 \<in> S \<and> (\<forall>c. c > 0 \<longrightarrow> (*\<^sub>R) c ` S = S)"
     using cone_iff[of S] assms by auto
   then have *: "0 \<in> ({0} \<union> rel_interior S)"
     and "\<forall>c. c > 0 \<longrightarrow> (*\<^sub>R) c ` ({0} \<union> rel_interior S) = ({0} \<union> rel_interior S)"
     by (auto simp add: rel_interior_scaleR)
   then show ?thesis
     using cone_iff[of "{0} \<union> rel_interior S"] by auto
 qed
 
 lemma rel_interior_convex_cone_aux:
   fixes S :: "'m::euclidean_space set"
   assumes "convex S"
   shows "(c, x) \<in> rel_interior (cone hull ({(1 :: real)} \<times> S)) \<longleftrightarrow>
     c > 0 \<and> x \<in> (((*\<^sub>R) c) ` (rel_interior S))"
 proof (cases "S = {}")
   case True
   then show ?thesis
     by (simp add: cone_hull_empty)
 next
   case False
   then obtain s where "s \<in> S" by auto
   have conv: "convex ({(1 :: real)} \<times> S)"
     using convex_Times[of "{(1 :: real)}" S] assms convex_singleton[of "1 :: real"]
     by auto
   define f where "f y = {z. (y, z) \<in> cone hull ({1 :: real} \<times> S)}" for y
   then have *: "(c, x) \<in> rel_interior (cone hull ({(1 :: real)} \<times> S)) =
     (c \<in> rel_interior {y. f y \<noteq> {}} \<and> x \<in> rel_interior (f c))"
     apply (subst rel_interior_projection[of "cone hull ({(1 :: real)} \<times> S)" f c x])
     using convex_cone_hull[of "{(1 :: real)} \<times> S"] conv
     apply auto
     done
   {
     fix y :: real
     assume "y \<ge> 0"
     then have "y *\<^sub>R (1,s) \<in> cone hull ({1 :: real} \<times> S)"
       using cone_hull_expl[of "{(1 :: real)} \<times> S"] \<open>s \<in> S\<close> by auto
     then have "f y \<noteq> {}"
       using f_def by auto
   }
   then have "{y. f y \<noteq> {}} = {0..}"
     using f_def cone_hull_expl[of "{1 :: real} \<times> S"] by auto
   then have **: "rel_interior {y. f y \<noteq> {}} = {0<..}"
     using rel_interior_real_semiline by auto
   {
     fix c :: real
     assume "c > 0"
     then have "f c = ((*\<^sub>R) c ` S)"
       using f_def cone_hull_expl[of "{1 :: real} \<times> S"] by auto
     then have "rel_interior (f c) = (*\<^sub>R) c ` rel_interior S"
       using rel_interior_convex_scaleR[of S c] assms by auto
   }
   then show ?thesis using * ** by auto
 qed
 
 lemma rel_interior_convex_cone:
   fixes S :: "'m::euclidean_space set"
   assumes "convex S"
   shows "rel_interior (cone hull ({1 :: real} \<times> S)) =
     {(c, c *\<^sub>R x) | c x. c > 0 \<and> x \<in> rel_interior S}"
   (is "?lhs = ?rhs")
 proof -
   {
     fix z
     assume "z \<in> ?lhs"
     have *: "z = (fst z, snd z)"
       by auto
     then have "z \<in> ?rhs"
       using rel_interior_convex_cone_aux[of S "fst z" "snd z"] assms \<open>z \<in> ?lhs\<close> by fastforce
   }
   moreover
   {
     fix z
     assume "z \<in> ?rhs"
     then have "z \<in> ?lhs"
       using rel_interior_convex_cone_aux[of S "fst z" "snd z"] assms
       by auto
   }
   ultimately show ?thesis by blast
 qed
 
 lemma convex_hull_finite_union:
   assumes "finite I"
   assumes "\<forall>i\<in>I. convex (S i) \<and> (S i) \<noteq> {}"
   shows "convex hull (\<Union>(S ` I)) =
     {sum (\<lambda>i. c i *\<^sub>R s i) I | c s. (\<forall>i\<in>I. c i \<ge> 0) \<and> sum c I = 1 \<and> (\<forall>i\<in>I. s i \<in> S i)}"
   (is "?lhs = ?rhs")
 proof -
   have "?lhs \<supseteq> ?rhs"
   proof
     fix x
     assume "x \<in> ?rhs"
     then obtain c s where *: "sum (\<lambda>i. c i *\<^sub>R s i) I = x" "sum c I = 1"
       "(\<forall>i\<in>I. c i \<ge> 0) \<and> (\<forall>i\<in>I. s i \<in> S i)" by auto
     then have "\<forall>i\<in>I. s i \<in> convex hull (\<Union>(S ` I))"
       using hull_subset[of "\<Union>(S ` I)" convex] by auto
     then show "x \<in> ?lhs"
       unfolding *(1)[symmetric]
       apply (subst convex_sum[of I "convex hull \<Union>(S ` I)" c s])
       using * assms convex_convex_hull
       apply auto
       done
   qed
 
   {
     fix i
     assume "i \<in> I"
     with assms have "\<exists>p. p \<in> S i" by auto
   }
   then obtain p where p: "\<forall>i\<in>I. p i \<in> S i" by metis
 
   {
     fix i
     assume "i \<in> I"
     {
       fix x
       assume "x \<in> S i"
       define c where "c j = (if j = i then 1::real else 0)" for j
       then have *: "sum c I = 1"
         using \<open>finite I\<close> \<open>i \<in> I\<close> sum.delta[of I i "\<lambda>j::'a. 1::real"]
         by auto
       define s where "s j = (if j = i then x else p j)" for j
       then have "\<forall>j. c j *\<^sub>R s j = (if j = i then x else 0)"
         using c_def by (auto simp add: algebra_simps)
       then have "x = sum (\<lambda>i. c i *\<^sub>R s i) I"
         using s_def c_def \<open>finite I\<close> \<open>i \<in> I\<close> sum.delta[of I i "\<lambda>j::'a. x"]
         by auto
       then have "x \<in> ?rhs"
         apply auto
         apply (rule_tac x = c in exI)
         apply (rule_tac x = s in exI)
         using * c_def s_def p \<open>x \<in> S i\<close>
         apply auto
         done
     }
     then have "?rhs \<supseteq> S i" by auto
   }
   then have *: "?rhs \<supseteq> \<Union>(S ` I)" by auto
 
   {
     fix u v :: real
     assume uv: "u \<ge> 0 \<and> v \<ge> 0 \<and> u + v = 1"
     fix x y
     assume xy: "x \<in> ?rhs \<and> y \<in> ?rhs"
     from xy obtain c s where
       xc: "x = sum (\<lambda>i. c i *\<^sub>R s i) I \<and> (\<forall>i\<in>I. c i \<ge> 0) \<and> sum c I = 1 \<and> (\<forall>i\<in>I. s i \<in> S i)"
       by auto
     from xy obtain d t where
       yc: "y = sum (\<lambda>i. d i *\<^sub>R t i) I \<and> (\<forall>i\<in>I. d i \<ge> 0) \<and> sum d I = 1 \<and> (\<forall>i\<in>I. t i \<in> S i)"
       by auto
     define e where "e i = u * c i + v * d i" for i
     have ge0: "\<forall>i\<in>I. e i \<ge> 0"
       using e_def xc yc uv by simp
     have "sum (\<lambda>i. u * c i) I = u * sum c I"
       by (simp add: sum_distrib_left)
     moreover have "sum (\<lambda>i. v * d i) I = v * sum d I"
       by (simp add: sum_distrib_left)
     ultimately have sum1: "sum e I = 1"
       using e_def xc yc uv by (simp add: sum.distrib)
     define q where "q i = (if e i = 0 then p i else (u * c i / e i) *\<^sub>R s i + (v * d i / e i) *\<^sub>R t i)"
       for i
     {
       fix i
       assume i: "i \<in> I"
       have "q i \<in> S i"
       proof (cases "e i = 0")
         case True
         then show ?thesis using i p q_def by auto
       next
         case False
         then show ?thesis
           using mem_convex_alt[of "S i" "s i" "t i" "u * (c i)" "v * (d i)"]
             mult_nonneg_nonneg[of u "c i"] mult_nonneg_nonneg[of v "d i"]
             assms q_def e_def i False xc yc uv
           by (auto simp del: mult_nonneg_nonneg)
       qed
     }
     then have qs: "\<forall>i\<in>I. q i \<in> S i" by auto
     {
       fix i
       assume i: "i \<in> I"
       have "(u * c i) *\<^sub>R s i + (v * d i) *\<^sub>R t i = e i *\<^sub>R q i"
       proof (cases "e i = 0")
         case True
         have ge: "u * (c i) \<ge> 0 \<and> v * d i \<ge> 0"
           using xc yc uv i by simp
         moreover from ge have "u * c i \<le> 0 \<and> v * d i \<le> 0"
           using True e_def i by simp
         ultimately have "u * c i = 0 \<and> v * d i = 0" by auto
         with True show ?thesis by auto
       next
         case False
         then have "(u * (c i)/(e i))*\<^sub>R (s i)+(v * (d i)/(e i))*\<^sub>R (t i) = q i"
           using q_def by auto
         then have "e i *\<^sub>R ((u * (c i)/(e i))*\<^sub>R (s i)+(v * (d i)/(e i))*\<^sub>R (t i))
                = (e i) *\<^sub>R (q i)" by auto
         with False show ?thesis by (simp add: algebra_simps)
       qed
     }
     then have *: "\<forall>i\<in>I. (u * c i) *\<^sub>R s i + (v * d i) *\<^sub>R t i = e i *\<^sub>R q i"
       by auto
     have "u *\<^sub>R x + v *\<^sub>R y = sum (\<lambda>i. (u * c i) *\<^sub>R s i + (v * d i) *\<^sub>R t i) I"
       using xc yc by (simp add: algebra_simps scaleR_right.sum sum.distrib)
     also have "\<dots> = sum (\<lambda>i. e i *\<^sub>R q i) I"
       using * by auto
     finally have "u *\<^sub>R x + v *\<^sub>R y = sum (\<lambda>i. (e i) *\<^sub>R (q i)) I"
       by auto
     then have "u *\<^sub>R x + v *\<^sub>R y \<in> ?rhs"
       using ge0 sum1 qs by auto
   }
   then have "convex ?rhs" unfolding convex_def by auto
   then show ?thesis
     using \<open>?lhs \<supseteq> ?rhs\<close> * hull_minimal[of "\<Union>(S ` I)" ?rhs convex]
     by blast
 qed
 
 lemma convex_hull_union_two:
   fixes S T :: "'m::euclidean_space set"
   assumes "convex S"
     and "S \<noteq> {}"
     and "convex T"
     and "T \<noteq> {}"
   shows "convex hull (S \<union> T) =
     {u *\<^sub>R s + v *\<^sub>R t | u v s t. u \<ge> 0 \<and> v \<ge> 0 \<and> u + v = 1 \<and> s \<in> S \<and> t \<in> T}"
   (is "?lhs = ?rhs")
 proof
   define I :: "nat set" where "I = {1, 2}"
   define s where "s i = (if i = (1::nat) then S else T)" for i
   have "\<Union>(s ` I) = S \<union> T"
     using s_def I_def by auto
   then have "convex hull (\<Union>(s ` I)) = convex hull (S \<union> T)"
     by auto
   moreover have "convex hull \<Union>(s ` I) =
     {\<Sum> i\<in>I. c i *\<^sub>R sa i | c sa. (\<forall>i\<in>I. 0 \<le> c i) \<and> sum c I = 1 \<and> (\<forall>i\<in>I. sa i \<in> s i)}"
       apply (subst convex_hull_finite_union[of I s])
       using assms s_def I_def
       apply auto
       done
   moreover have
     "{\<Sum>i\<in>I. c i *\<^sub>R sa i | c sa. (\<forall>i\<in>I. 0 \<le> c i) \<and> sum c I = 1 \<and> (\<forall>i\<in>I. sa i \<in> s i)} \<le> ?rhs"
     using s_def I_def by auto
   ultimately show "?lhs \<subseteq> ?rhs" by auto
   {
     fix x
     assume "x \<in> ?rhs"
     then obtain u v s t where *: "x = u *\<^sub>R s + v *\<^sub>R t \<and> u \<ge> 0 \<and> v \<ge> 0 \<and> u + v = 1 \<and> s \<in> S \<and> t \<in> T"
       by auto
     then have "x \<in> convex hull {s, t}"
       using convex_hull_2[of s t] by auto
     then have "x \<in> convex hull (S \<union> T)"
       using * hull_mono[of "{s, t}" "S \<union> T"] by auto
   }
   then show "?lhs \<supseteq> ?rhs" by blast
 qed
 
 proposition ray_to_rel_frontier:
   fixes a :: "'a::real_inner"
   assumes "bounded S"
       and a: "a \<in> rel_interior S"
       and aff: "(a + l) \<in> affine hull S"
       and "l \<noteq> 0"
   obtains d where "0 < d" "(a + d *\<^sub>R l) \<in> rel_frontier S"
            "\<And>e. \<lbrakk>0 \<le> e; e < d\<rbrakk> \<Longrightarrow> (a + e *\<^sub>R l) \<in> rel_interior S"
 proof -
   have aaff: "a \<in> affine hull S"
     by (meson a hull_subset rel_interior_subset rev_subsetD)
   let ?D = "{d. 0 < d \<and> a + d *\<^sub>R l \<notin> rel_interior S}"
   obtain B where "B > 0" and B: "S \<subseteq> ball a B"
     using bounded_subset_ballD [OF \<open>bounded S\<close>] by blast
   have "a + (B / norm l) *\<^sub>R l \<notin> ball a B"
     by (simp add: dist_norm \<open>l \<noteq> 0\<close>)
   with B have "a + (B / norm l) *\<^sub>R l \<notin> rel_interior S"
     using rel_interior_subset subsetCE by blast
   with \<open>B > 0\<close> \<open>l \<noteq> 0\<close> have nonMT: "?D \<noteq> {}"
     using divide_pos_pos zero_less_norm_iff by fastforce
   have bdd: "bdd_below ?D"
     by (metis (no_types, lifting) bdd_belowI le_less mem_Collect_eq)
   have relin_Ex: "\<And>x. x \<in> rel_interior S \<Longrightarrow>
                     \<exists>e>0. \<forall>x'\<in>affine hull S. dist x' x < e \<longrightarrow> x' \<in> rel_interior S"
     using openin_rel_interior [of S] by (simp add: openin_euclidean_subtopology_iff)
   define d where "d = Inf ?D"
   obtain \<epsilon> where "0 < \<epsilon>" and \<epsilon>: "\<And>\<eta>. \<lbrakk>0 \<le> \<eta>; \<eta> < \<epsilon>\<rbrakk> \<Longrightarrow> (a + \<eta> *\<^sub>R l) \<in> rel_interior S"
   proof -
     obtain e where "e>0"
             and e: "\<And>x'. x' \<in> affine hull S \<Longrightarrow> dist x' a < e \<Longrightarrow> x' \<in> rel_interior S"
       using relin_Ex a by blast
     show thesis
     proof (rule_tac \<epsilon> = "e / norm l" in that)
       show "0 < e / norm l" by (simp add: \<open>0 < e\<close> \<open>l \<noteq> 0\<close>)
     next
       show "a + \<eta> *\<^sub>R l \<in> rel_interior S" if "0 \<le> \<eta>" "\<eta> < e / norm l" for \<eta>
       proof (rule e)
         show "a + \<eta> *\<^sub>R l \<in> affine hull S"
           by (metis (no_types) add_diff_cancel_left' aff affine_affine_hull mem_affine_3_minus aaff)
         show "dist (a + \<eta> *\<^sub>R l) a < e"
           using that by (simp add: \<open>l \<noteq> 0\<close> dist_norm pos_less_divide_eq)
       qed
     qed
   qed
   have inint: "\<And>e. \<lbrakk>0 \<le> e; e < d\<rbrakk> \<Longrightarrow> a + e *\<^sub>R l \<in> rel_interior S"
     unfolding d_def using cInf_lower [OF _ bdd]
     by (metis (no_types, lifting) a add.right_neutral le_less mem_Collect_eq not_less real_vector.scale_zero_left)
   have "\<epsilon> \<le> d"
     unfolding d_def
     apply (rule cInf_greatest [OF nonMT])
     using \<epsilon> dual_order.strict_implies_order le_less_linear by blast
   with \<open>0 < \<epsilon>\<close> have "0 < d" by simp
   have "a + d *\<^sub>R l \<notin> rel_interior S"
   proof
     assume adl: "a + d *\<^sub>R l \<in> rel_interior S"
     obtain e where "e > 0"
              and e: "\<And>x'. x' \<in> affine hull S \<Longrightarrow> dist x' (a + d *\<^sub>R l) < e \<Longrightarrow> x' \<in> rel_interior S"
       using relin_Ex adl by blast
     have "d + e / norm l \<le> Inf {d. 0 < d \<and> a + d *\<^sub>R l \<notin> rel_interior S}"
     proof (rule cInf_greatest [OF nonMT], clarsimp)
       fix x::real
       assume "0 < x" and nonrel: "a + x *\<^sub>R l \<notin> rel_interior S"
       show "d + e / norm l \<le> x"
       proof (cases "x < d")
         case True with inint nonrel \<open>0 < x\<close>
           show ?thesis by auto
       next
         case False
           then have dle: "x < d + e / norm l \<Longrightarrow> dist (a + x *\<^sub>R l) (a + d *\<^sub>R l) < e"
             by (simp add: field_simps \<open>l \<noteq> 0\<close>)
           have ain: "a + x *\<^sub>R l \<in> affine hull S"
             by (metis add_diff_cancel_left' aff affine_affine_hull mem_affine_3_minus aaff)
           show ?thesis
             using e [OF ain] nonrel dle by force
       qed
     qed
     then show False
       using \<open>0 < e\<close> \<open>l \<noteq> 0\<close> by (simp add: d_def [symmetric] field_simps)
   qed
   moreover have "a + d *\<^sub>R l \<in> closure S"
   proof (clarsimp simp: closure_approachable)
     fix \<eta>::real assume "0 < \<eta>"
     have 1: "a + (d - min d (\<eta> / 2 / norm l)) *\<^sub>R l \<in> S"
       apply (rule subsetD [OF rel_interior_subset inint])
       using \<open>l \<noteq> 0\<close> \<open>0 < d\<close> \<open>0 < \<eta>\<close> by auto
     have "norm l * min d (\<eta> / (norm l * 2)) \<le> norm l * (\<eta> / (norm l * 2))"
       by (metis min_def mult_left_mono norm_ge_zero order_refl)
     also have "... < \<eta>"
       using \<open>l \<noteq> 0\<close> \<open>0 < \<eta>\<close> by (simp add: field_simps)
     finally have 2: "norm l * min d (\<eta> / (norm l * 2)) < \<eta>" .
     show "\<exists>y\<in>S. dist y (a + d *\<^sub>R l) < \<eta>"
       apply (rule_tac x="a + (d - min d (\<eta> / 2 / norm l)) *\<^sub>R l" in bexI)
       using 1 2 \<open>0 < d\<close> \<open>0 < \<eta>\<close> apply (auto simp: algebra_simps)
       done
   qed
   ultimately have infront: "a + d *\<^sub>R l \<in> rel_frontier S"
     by (simp add: rel_frontier_def)
   show ?thesis
     by (rule that [OF \<open>0 < d\<close> infront inint])
 qed
 
 corollary ray_to_frontier:
   fixes a :: "'a::euclidean_space"
   assumes "bounded S"
       and a: "a \<in> interior S"
       and "l \<noteq> 0"
   obtains d where "0 < d" "(a + d *\<^sub>R l) \<in> frontier S"
            "\<And>e. \<lbrakk>0 \<le> e; e < d\<rbrakk> \<Longrightarrow> (a + e *\<^sub>R l) \<in> interior S"
 proof -
   have "interior S = rel_interior S"
     using a rel_interior_nonempty_interior by auto
   then have "a \<in> rel_interior S"
     using a by simp
   then show ?thesis
     apply (rule ray_to_rel_frontier [OF \<open>bounded S\<close> _ _ \<open>l \<noteq> 0\<close>])
      using a affine_hull_nonempty_interior apply blast
     by (simp add: \<open>interior S = rel_interior S\<close> frontier_def rel_frontier_def that)
 qed
 
 
 lemma segment_to_rel_frontier_aux:
   fixes x :: "'a::euclidean_space"
   assumes "convex S" "bounded S" and x: "x \<in> rel_interior S" and y: "y \<in> S" and xy: "x \<noteq> y"
   obtains z where "z \<in> rel_frontier S" "y \<in> closed_segment x z"
                    "open_segment x z \<subseteq> rel_interior S"
 proof -
   have "x + (y - x) \<in> affine hull S"
     using hull_inc [OF y] by auto
   then obtain d where "0 < d" and df: "(x + d *\<^sub>R (y-x)) \<in> rel_frontier S"
                   and di: "\<And>e. \<lbrakk>0 \<le> e; e < d\<rbrakk> \<Longrightarrow> (x + e *\<^sub>R (y-x)) \<in> rel_interior S"
     by (rule ray_to_rel_frontier [OF \<open>bounded S\<close> x]) (use xy in auto)
   show ?thesis
   proof
     show "x + d *\<^sub>R (y - x) \<in> rel_frontier S"
       by (simp add: df)
   next
     have "open_segment x y \<subseteq> rel_interior S"
       using rel_interior_closure_convex_segment [OF \<open>convex S\<close> x] closure_subset y by blast
     moreover have "x + d *\<^sub>R (y - x) \<in> open_segment x y" if "d < 1"
       using xy
       apply (auto simp: in_segment)
       apply (rule_tac x="d" in exI)
       using \<open>0 < d\<close> that apply (auto simp: algebra_simps)
       done
     ultimately have "1 \<le> d"
       using df rel_frontier_def by fastforce
     moreover have "x = (1 / d) *\<^sub>R x + ((d - 1) / d) *\<^sub>R x"
       by (metis \<open>0 < d\<close> add.commute add_divide_distrib diff_add_cancel divide_self_if less_irrefl scaleR_add_left scaleR_one)
     ultimately show "y \<in> closed_segment x (x + d *\<^sub>R (y - x))"
       apply (auto simp: in_segment)
       apply (rule_tac x="1/d" in exI)
       apply (auto simp: algebra_simps)
       done
   next
     show "open_segment x (x + d *\<^sub>R (y - x)) \<subseteq> rel_interior S"
       apply (rule rel_interior_closure_convex_segment [OF \<open>convex S\<close> x])
       using df rel_frontier_def by auto
   qed
 qed
 
 lemma segment_to_rel_frontier:
   fixes x :: "'a::euclidean_space"
   assumes S: "convex S" "bounded S" and x: "x \<in> rel_interior S"
       and y: "y \<in> S" and xy: "\<not>(x = y \<and> S = {x})"
   obtains z where "z \<in> rel_frontier S" "y \<in> closed_segment x z"
                   "open_segment x z \<subseteq> rel_interior S"
 proof (cases "x=y")
   case True
   with xy have "S \<noteq> {x}"
     by blast
   with True show ?thesis
     by (metis Set.set_insert all_not_in_conv ends_in_segment(1) insert_iff segment_to_rel_frontier_aux[OF S x] that y)
 next
   case False
   then show ?thesis
     using segment_to_rel_frontier_aux [OF S x y] that by blast
 qed
 
 proposition rel_frontier_not_sing:
   fixes a :: "'a::euclidean_space"
   assumes "bounded S"
     shows "rel_frontier S \<noteq> {a}"
 proof (cases "S = {}")
   case True  then show ?thesis  by simp
 next
   case False
   then obtain z where "z \<in> S"
     by blast
   then show ?thesis
   proof (cases "S = {z}")
     case True then show ?thesis  by simp
   next
     case False
     then obtain w where "w \<in> S" "w \<noteq> z"
       using \<open>z \<in> S\<close> by blast
     show ?thesis
     proof
       assume "rel_frontier S = {a}"
       then consider "w \<notin> rel_frontier S" | "z \<notin> rel_frontier S"
         using \<open>w \<noteq> z\<close> by auto
       then show False
       proof cases
         case 1
         then have w: "w \<in> rel_interior S"
           using \<open>w \<in> S\<close> closure_subset rel_frontier_def by fastforce
         have "w + (w - z) \<in> affine hull S"
           by (metis \<open>w \<in> S\<close> \<open>z \<in> S\<close> affine_affine_hull hull_inc mem_affine_3_minus scaleR_one)
         then obtain e where "0 < e" "(w + e *\<^sub>R (w - z)) \<in> rel_frontier S"
           using \<open>w \<noteq> z\<close>  \<open>z \<in> S\<close> by (metis assms ray_to_rel_frontier right_minus_eq w)
         moreover obtain d where "0 < d" "(w + d *\<^sub>R (z - w)) \<in> rel_frontier S"
           using ray_to_rel_frontier [OF \<open>bounded S\<close> w, of "1 *\<^sub>R (z - w)"]  \<open>w \<noteq> z\<close>  \<open>z \<in> S\<close>
           by (metis add.commute add.right_neutral diff_add_cancel hull_inc scaleR_one)
         ultimately have "d *\<^sub>R (z - w) = e *\<^sub>R (w - z)"
           using \<open>rel_frontier S = {a}\<close> by force
         moreover have "e \<noteq> -d "
           using \<open>0 < e\<close> \<open>0 < d\<close> by force
         ultimately show False
           by (metis (no_types, lifting) \<open>w \<noteq> z\<close> eq_iff_diff_eq_0 minus_diff_eq real_vector.scale_cancel_right real_vector.scale_minus_right scaleR_left.minus)
       next
         case 2
         then have z: "z \<in> rel_interior S"
           using \<open>z \<in> S\<close> closure_subset rel_frontier_def by fastforce
         have "z + (z - w) \<in> affine hull S"
           by (metis \<open>z \<in> S\<close> \<open>w \<in> S\<close> affine_affine_hull hull_inc mem_affine_3_minus scaleR_one)
         then obtain e where "0 < e" "(z + e *\<^sub>R (z - w)) \<in> rel_frontier S"
           using \<open>w \<noteq> z\<close>  \<open>w \<in> S\<close> by (metis assms ray_to_rel_frontier right_minus_eq z)
         moreover obtain d where "0 < d" "(z + d *\<^sub>R (w - z)) \<in> rel_frontier S"
           using ray_to_rel_frontier [OF \<open>bounded S\<close> z, of "1 *\<^sub>R (w - z)"]  \<open>w \<noteq> z\<close>  \<open>w \<in> S\<close>
           by (metis add.commute add.right_neutral diff_add_cancel hull_inc scaleR_one)
         ultimately have "d *\<^sub>R (w - z) = e *\<^sub>R (z - w)"
           using \<open>rel_frontier S = {a}\<close> by force
         moreover have "e \<noteq> -d "
           using \<open>0 < e\<close> \<open>0 < d\<close> by force
         ultimately show False
           by (metis (no_types, lifting) \<open>w \<noteq> z\<close> eq_iff_diff_eq_0 minus_diff_eq real_vector.scale_cancel_right real_vector.scale_minus_right scaleR_left.minus)
       qed
     qed
   qed
 qed
 
 
 subsection\<^marker>\<open>tag unimportant\<close> \<open>Convexity on direct sums\<close>
 
 lemma closure_sum:
   fixes S T :: "'a::real_normed_vector set"
   shows "closure S + closure T \<subseteq> closure (S + T)"
   unfolding set_plus_image closure_Times [symmetric] split_def
   by (intro closure_bounded_linear_image_subset bounded_linear_add
     bounded_linear_fst bounded_linear_snd)
 
 lemma rel_interior_sum:
   fixes S T :: "'n::euclidean_space set"
   assumes "convex S"
     and "convex T"
   shows "rel_interior (S + T) = rel_interior S + rel_interior T"
 proof -
   have "rel_interior S + rel_interior T = (\<lambda>(x,y). x + y) ` (rel_interior S \<times> rel_interior T)"
     by (simp add: set_plus_image)
   also have "\<dots> = (\<lambda>(x,y). x + y) ` rel_interior (S \<times> T)"
     using rel_interior_Times assms by auto
   also have "\<dots> = rel_interior (S + T)"
     using fst_snd_linear convex_Times assms
       rel_interior_convex_linear_image[of "(\<lambda>(x,y). x + y)" "S \<times> T"]
     by (auto simp add: set_plus_image)
   finally show ?thesis ..
 qed
 
 lemma rel_interior_sum_gen:
   fixes S :: "'a \<Rightarrow> 'n::euclidean_space set"
   assumes "\<forall>i\<in>I. convex (S i)"
   shows "rel_interior (sum S I) = sum (\<lambda>i. rel_interior (S i)) I"
   apply (subst sum_set_cond_linear[of convex])
   using rel_interior_sum rel_interior_sing[of "0"] assms
   apply (auto simp add: convex_set_plus)
   done
 
 lemma convex_rel_open_direct_sum:
   fixes S T :: "'n::euclidean_space set"
   assumes "convex S"
     and "rel_open S"
     and "convex T"
     and "rel_open T"
   shows "convex (S \<times> T) \<and> rel_open (S \<times> T)"
   by (metis assms convex_Times rel_interior_Times rel_open_def)
 
 lemma convex_rel_open_sum:
   fixes S T :: "'n::euclidean_space set"
   assumes "convex S"
     and "rel_open S"
     and "convex T"
     and "rel_open T"
   shows "convex (S + T) \<and> rel_open (S + T)"
   by (metis assms convex_set_plus rel_interior_sum rel_open_def)
 
 lemma convex_hull_finite_union_cones:
   assumes "finite I"
     and "I \<noteq> {}"
   assumes "\<forall>i\<in>I. convex (S i) \<and> cone (S i) \<and> S i \<noteq> {}"
   shows "convex hull (\<Union>(S ` I)) = sum S I"
   (is "?lhs = ?rhs")
 proof -
   {
     fix x
     assume "x \<in> ?lhs"
     then obtain c xs where
       x: "x = sum (\<lambda>i. c i *\<^sub>R xs i) I \<and> (\<forall>i\<in>I. c i \<ge> 0) \<and> sum c I = 1 \<and> (\<forall>i\<in>I. xs i \<in> S i)"
       using convex_hull_finite_union[of I S] assms by auto
     define s where "s i = c i *\<^sub>R xs i" for i
     {
       fix i
       assume "i \<in> I"
       then have "s i \<in> S i"
         using s_def x assms mem_cone[of "S i" "xs i" "c i"] by auto
     }
     then have "\<forall>i\<in>I. s i \<in> S i" by auto
     moreover have "x = sum s I" using x s_def by auto
     ultimately have "x \<in> ?rhs"
       using set_sum_alt[of I S] assms by auto
   }
   moreover
   {
     fix x
     assume "x \<in> ?rhs"
     then obtain s where x: "x = sum s I \<and> (\<forall>i\<in>I. s i \<in> S i)"
       using set_sum_alt[of I S] assms by auto
     define xs where "xs i = of_nat(card I) *\<^sub>R s i" for i
     then have "x = sum (\<lambda>i. ((1 :: real) / of_nat(card I)) *\<^sub>R xs i) I"
       using x assms by auto
     moreover have "\<forall>i\<in>I. xs i \<in> S i"
       using x xs_def assms by (simp add: cone_def)
     moreover have "\<forall>i\<in>I. (1 :: real) / of_nat (card I) \<ge> 0"
       by auto
     moreover have "sum (\<lambda>i. (1 :: real) / of_nat (card I)) I = 1"
       using assms by auto
     ultimately have "x \<in> ?lhs"
       apply (subst convex_hull_finite_union[of I S])
       using assms
       apply blast
       using assms
       apply blast
       apply rule
       apply (rule_tac x = "(\<lambda>i. (1 :: real) / of_nat (card I))" in exI)
       apply auto
       done
   }
   ultimately show ?thesis by auto
 qed
 
 lemma convex_hull_union_cones_two:
   fixes S T :: "'m::euclidean_space set"
   assumes "convex S"
     and "cone S"
     and "S \<noteq> {}"
   assumes "convex T"
     and "cone T"
     and "T \<noteq> {}"
   shows "convex hull (S \<union> T) = S + T"
 proof -
   define I :: "nat set" where "I = {1, 2}"
   define A where "A i = (if i = (1::nat) then S else T)" for i
   have "\<Union>(A ` I) = S \<union> T"
     using A_def I_def by auto
   then have "convex hull (\<Union>(A ` I)) = convex hull (S \<union> T)"
     by auto
   moreover have "convex hull \<Union>(A ` I) = sum A I"
     apply (subst convex_hull_finite_union_cones[of I A])
     using assms A_def I_def
     apply auto
     done
   moreover have "sum A I = S + T"
     using A_def I_def
     unfolding set_plus_def
     apply auto
     unfolding set_plus_def
     apply auto
     done
   ultimately show ?thesis by auto
 qed
 
 lemma rel_interior_convex_hull_union:
   fixes S :: "'a \<Rightarrow> 'n::euclidean_space set"
   assumes "finite I"
     and "\<forall>i\<in>I. convex (S i) \<and> S i \<noteq> {}"
   shows "rel_interior (convex hull (\<Union>(S ` I))) =
     {sum (\<lambda>i. c i *\<^sub>R s i) I | c s. (\<forall>i\<in>I. c i > 0) \<and> sum c I = 1 \<and>
       (\<forall>i\<in>I. s i \<in> rel_interior(S i))}"
   (is "?lhs = ?rhs")
 proof (cases "I = {}")
   case True
   then show ?thesis
     using convex_hull_empty by auto
 next
   case False
   define C0 where "C0 = convex hull (\<Union>(S ` I))"
   have "\<forall>i\<in>I. C0 \<ge> S i"
     unfolding C0_def using hull_subset[of "\<Union>(S ` I)"] by auto
   define K0 where "K0 = cone hull ({1 :: real} \<times> C0)"
   define K where "K i = cone hull ({1 :: real} \<times> S i)" for i
   have "\<forall>i\<in>I. K i \<noteq> {}"
     unfolding K_def using assms
     by (simp add: cone_hull_empty_iff[symmetric])
   {
     fix i
     assume "i \<in> I"
     then have "convex (K i)"
       unfolding K_def
       apply (subst convex_cone_hull)
       apply (subst convex_Times)
       using assms
       apply auto
       done
   }
   then have convK: "\<forall>i\<in>I. convex (K i)"
     by auto
   {
     fix i
     assume "i \<in> I"
     then have "K0 \<supseteq> K i"
       unfolding K0_def K_def
       apply (subst hull_mono)
       using \<open>\<forall>i\<in>I. C0 \<ge> S i\<close>
       apply auto
       done
   }
   then have "K0 \<supseteq> \<Union>(K ` I)" by auto
   moreover have "convex K0"
     unfolding K0_def
     apply (subst convex_cone_hull)
     apply (subst convex_Times)
     unfolding C0_def
     using convex_convex_hull
     apply auto
     done
   ultimately have geq: "K0 \<supseteq> convex hull (\<Union>(K ` I))"
     using hull_minimal[of _ "K0" "convex"] by blast
   have "\<forall>i\<in>I. K i \<supseteq> {1 :: real} \<times> S i"
     using K_def by (simp add: hull_subset)
   then have "\<Union>(K ` I) \<supseteq> {1 :: real} \<times> \<Union>(S ` I)"
     by auto
   then have "convex hull \<Union>(K ` I) \<supseteq> convex hull ({1 :: real} \<times> \<Union>(S ` I))"
     by (simp add: hull_mono)
   then have "convex hull \<Union>(K ` I) \<supseteq> {1 :: real} \<times> C0"
     unfolding C0_def
     using convex_hull_Times[of "{(1 :: real)}" "\<Union>(S ` I)"] convex_hull_singleton
     by auto
   moreover have "cone (convex hull (\<Union>(K ` I)))"
     apply (subst cone_convex_hull)
     using cone_Union[of "K ` I"]
     apply auto
     unfolding K_def
     using cone_cone_hull
     apply auto
     done
   ultimately have "convex hull (\<Union>(K ` I)) \<supseteq> K0"
     unfolding K0_def
     using hull_minimal[of _ "convex hull (\<Union>(K ` I))" "cone"]
     by blast
   then have "K0 = convex hull (\<Union>(K ` I))"
     using geq by auto
   also have "\<dots> = sum K I"
     apply (subst convex_hull_finite_union_cones[of I K])
     using assms
     apply blast
     using False
     apply blast
     unfolding K_def
     apply rule
     apply (subst convex_cone_hull)
     apply (subst convex_Times)
     using assms cone_cone_hull \<open>\<forall>i\<in>I. K i \<noteq> {}\<close> K_def
     apply auto
     done
   finally have "K0 = sum K I" by auto
   then have *: "rel_interior K0 = sum (\<lambda>i. (rel_interior (K i))) I"
     using rel_interior_sum_gen[of I K] convK by auto
   {
     fix x
     assume "x \<in> ?lhs"
     then have "(1::real, x) \<in> rel_interior K0"
       using K0_def C0_def rel_interior_convex_cone_aux[of C0 "1::real" x] convex_convex_hull
       by auto
     then obtain k where k: "(1::real, x) = sum k I \<and> (\<forall>i\<in>I. k i \<in> rel_interior (K i))"
       using \<open>finite I\<close> * set_sum_alt[of I "\<lambda>i. rel_interior (K i)"] by auto
     {
       fix i
       assume "i \<in> I"
       then have "convex (S i) \<and> k i \<in> rel_interior (cone hull {1} \<times> S i)"
         using k K_def assms by auto
       then have "\<exists>ci si. k i = (ci, ci *\<^sub>R si) \<and> 0 < ci \<and> si \<in> rel_interior (S i)"
         using rel_interior_convex_cone[of "S i"] by auto
     }
     then obtain c s where
       cs: "\<forall>i\<in>I. k i = (c i, c i *\<^sub>R s i) \<and> 0 < c i \<and> s i \<in> rel_interior (S i)"
       by metis
     then have "x = (\<Sum>i\<in>I. c i *\<^sub>R s i) \<and> sum c I = 1"
       using k by (simp add: sum_prod)
     then have "x \<in> ?rhs"
       using k cs by auto
   }
   moreover
   {
     fix x
     assume "x \<in> ?rhs"
     then obtain c s where cs: "x = sum (\<lambda>i. c i *\<^sub>R s i) I \<and>
         (\<forall>i\<in>I. c i > 0) \<and> sum c I = 1 \<and> (\<forall>i\<in>I. s i \<in> rel_interior (S i))"
       by auto
     define k where "k i = (c i, c i *\<^sub>R s i)" for i
     {
       fix i assume "i \<in> I"
       then have "k i \<in> rel_interior (K i)"
         using k_def K_def assms cs rel_interior_convex_cone[of "S i"]
         by auto
     }
     then have "(1::real, x) \<in> rel_interior K0"
       using K0_def * set_sum_alt[of I "(\<lambda>i. rel_interior (K i))"] assms k_def cs
       apply auto
       apply (rule_tac x = k in exI)
       apply (simp add: sum_prod)
       done
     then have "x \<in> ?lhs"
       using K0_def C0_def rel_interior_convex_cone_aux[of C0 1 x]
       by auto
   }
   ultimately show ?thesis by blast
 qed
 
 
 lemma convex_le_Inf_differential:
   fixes f :: "real \<Rightarrow> real"
   assumes "convex_on I f"
     and "x \<in> interior I"
     and "y \<in> I"
   shows "f y \<ge> f x + Inf ((\<lambda>t. (f x - f t) / (x - t)) ` ({x<..} \<inter> I)) * (y - x)"
   (is "_ \<ge> _ + Inf (?F x) * (y - x)")
 proof (cases rule: linorder_cases)
   assume "x < y"
   moreover
   have "open (interior I)" by auto
   from openE[OF this \<open>x \<in> interior I\<close>]
   obtain e where e: "0 < e" "ball x e \<subseteq> interior I" .
   moreover define t where "t = min (x + e / 2) ((x + y) / 2)"
   ultimately have "x < t" "t < y" "t \<in> ball x e"
     by (auto simp: dist_real_def field_simps split: split_min)
   with \<open>x \<in> interior I\<close> e interior_subset[of I] have "t \<in> I" "x \<in> I" by auto
 
   have "open (interior I)" by auto
   from openE[OF this \<open>x \<in> interior I\<close>]
   obtain e where "0 < e" "ball x e \<subseteq> interior I" .
   moreover define K where "K = x - e / 2"
   with \<open>0 < e\<close> have "K \<in> ball x e" "K < x"
     by (auto simp: dist_real_def)
   ultimately have "K \<in> I" "K < x" "x \<in> I"
     using interior_subset[of I] \<open>x \<in> interior I\<close> by auto
 
   have "Inf (?F x) \<le> (f x - f y) / (x - y)"
   proof (intro bdd_belowI cInf_lower2)
     show "(f x - f t) / (x - t) \<in> ?F x"
       using \<open>t \<in> I\<close> \<open>x < t\<close> by auto
     show "(f x - f t) / (x - t) \<le> (f x - f y) / (x - y)"
       using \<open>convex_on I f\<close> \<open>x \<in> I\<close> \<open>y \<in> I\<close> \<open>x < t\<close> \<open>t < y\<close>
       by (rule convex_on_diff)
   next
     fix y
     assume "y \<in> ?F x"
     with order_trans[OF convex_on_diff[OF \<open>convex_on I f\<close> \<open>K \<in> I\<close> _ \<open>K < x\<close> _]]
     show "(f K - f x) / (K - x) \<le> y" by auto
   qed
   then show ?thesis
     using \<open>x < y\<close> by (simp add: field_simps)
 next
   assume "y < x"
   moreover
   have "open (interior I)" by auto
   from openE[OF this \<open>x \<in> interior I\<close>]
   obtain e where e: "0 < e" "ball x e \<subseteq> interior I" .
   moreover define t where "t = x + e / 2"
   ultimately have "x < t" "t \<in> ball x e"
     by (auto simp: dist_real_def field_simps)
   with \<open>x \<in> interior I\<close> e interior_subset[of I] have "t \<in> I" "x \<in> I" by auto
 
   have "(f x - f y) / (x - y) \<le> Inf (?F x)"
   proof (rule cInf_greatest)
     have "(f x - f y) / (x - y) = (f y - f x) / (y - x)"
       using \<open>y < x\<close> by (auto simp: field_simps)
     also
     fix z
     assume "z \<in> ?F x"
     with order_trans[OF convex_on_diff[OF \<open>convex_on I f\<close> \<open>y \<in> I\<close> _ \<open>y < x\<close>]]
     have "(f y - f x) / (y - x) \<le> z"
       by auto
     finally show "(f x - f y) / (x - y) \<le> z" .
   next
     have "open (interior I)" by auto
     from openE[OF this \<open>x \<in> interior I\<close>]
     obtain e where e: "0 < e" "ball x e \<subseteq> interior I" .
     then have "x + e / 2 \<in> ball x e"
       by (auto simp: dist_real_def)
     with e interior_subset[of I] have "x + e / 2 \<in> {x<..} \<inter> I"
       by auto
     then show "?F x \<noteq> {}"
       by blast
   qed
   then show ?thesis
     using \<open>y < x\<close> by (simp add: field_simps)
 qed simp
 
 subsection\<^marker>\<open>tag unimportant\<close>\<open>Explicit formulas for interior and relative interior of convex hull\<close>
 
 lemma at_within_cbox_finite:
   assumes "x \<in> box a b" "x \<notin> S" "finite S"
   shows "(at x within cbox a b - S) = at x"
 proof -
   have "interior (cbox a b - S) = box a b - S"
     using \<open>finite S\<close> by (simp add: interior_diff finite_imp_closed)
   then show ?thesis
     using at_within_interior assms by fastforce
 qed
 
 lemma affine_independent_convex_affine_hull:
   fixes s :: "'a::euclidean_space set"
   assumes "\<not> affine_dependent s" "t \<subseteq> s"
     shows "convex hull t = affine hull t \<inter> convex hull s"
 proof -
   have fin: "finite s" "finite t" using assms aff_independent_finite finite_subset by auto
     { fix u v x
       assume uv: "sum u t = 1" "\<forall>x\<in>s. 0 \<le> v x" "sum v s = 1"
                  "(\<Sum>x\<in>s. v x *\<^sub>R x) = (\<Sum>v\<in>t. u v *\<^sub>R v)" "x \<in> t"
       then have s: "s = (s - t) \<union> t" \<comment> \<open>split into separate cases\<close>
         using assms by auto
       have [simp]: "(\<Sum>x\<in>t. v x *\<^sub>R x) + (\<Sum>x\<in>s - t. v x *\<^sub>R x) = (\<Sum>x\<in>t. u x *\<^sub>R x)"
                    "sum v t + sum v (s - t) = 1"
         using uv fin s
         by (auto simp: sum.union_disjoint [symmetric] Un_commute)
       have "(\<Sum>x\<in>s. if x \<in> t then v x - u x else v x) = 0"
            "(\<Sum>x\<in>s. (if x \<in> t then v x - u x else v x) *\<^sub>R x) = 0"
         using uv fin
         by (subst s, subst sum.union_disjoint, auto simp: algebra_simps sum_subtractf)+
     } note [simp] = this
   have "convex hull t \<subseteq> affine hull t"
     using convex_hull_subset_affine_hull by blast
   moreover have "convex hull t \<subseteq> convex hull s"
     using assms hull_mono by blast
   moreover have "affine hull t \<inter> convex hull s \<subseteq> convex hull t"
     using assms
     apply (simp add: convex_hull_finite affine_hull_finite fin affine_dependent_explicit)
     apply (drule_tac x=s in spec)
     apply (auto simp: fin)
     apply (rule_tac x=u in exI)
     apply (rename_tac v)
     apply (drule_tac x="\<lambda>x. if x \<in> t then v x - u x else v x" in spec)
     apply (force)+
     done
   ultimately show ?thesis
     by blast
 qed
 
 lemma affine_independent_span_eq:
   fixes s :: "'a::euclidean_space set"
   assumes "\<not> affine_dependent s" "card s = Suc (DIM ('a))"
     shows "affine hull s = UNIV"
 proof (cases "s = {}")
   case True then show ?thesis
     using assms by simp
 next
   case False
     then obtain a t where t: "a \<notin> t" "s = insert a t"
       by blast
     then have fin: "finite t" using assms
       by (metis finite_insert aff_independent_finite)
     show ?thesis
     using assms t fin
       apply (simp add: affine_dependent_iff_dependent affine_hull_insert_span_gen)
       apply (rule subset_antisym)
       apply force
       apply (rule Fun.vimage_subsetD)
       apply (metis add.commute diff_add_cancel surj_def)
       apply (rule card_ge_dim_independent)
       apply (auto simp: card_image inj_on_def dim_subset_UNIV)
       done
 qed
 
 lemma affine_independent_span_gt:
   fixes s :: "'a::euclidean_space set"
   assumes ind: "\<not> affine_dependent s" and dim: "DIM ('a) < card s"
     shows "affine hull s = UNIV"
   apply (rule affine_independent_span_eq [OF ind])
   apply (rule antisym)
   using assms
   apply auto
   apply (metis add_2_eq_Suc' not_less_eq_eq affine_dependent_biggerset aff_independent_finite)
   done
 
 lemma empty_interior_affine_hull:
   fixes s :: "'a::euclidean_space set"
   assumes "finite s" and dim: "card s \<le> DIM ('a)"
     shows "interior(affine hull s) = {}"
   using assms
   apply (induct s rule: finite_induct)
   apply (simp_all add:  affine_dependent_iff_dependent affine_hull_insert_span_gen interior_translation)
   apply (rule empty_interior_lowdim)
   by (auto simp: Suc_le_lessD card_image_le dual_order.trans intro!: dim_le_card'[THEN le_less_trans])
 
 lemma empty_interior_convex_hull:
   fixes s :: "'a::euclidean_space set"
   assumes "finite s" and dim: "card s \<le> DIM ('a)"
     shows "interior(convex hull s) = {}"
   by (metis Diff_empty Diff_eq_empty_iff convex_hull_subset_affine_hull
             interior_mono empty_interior_affine_hull [OF assms])
 
 lemma explicit_subset_rel_interior_convex_hull:
   fixes s :: "'a::euclidean_space set"
   shows "finite s
          \<Longrightarrow> {y. \<exists>u. (\<forall>x \<in> s. 0 < u x \<and> u x < 1) \<and> sum u s = 1 \<and> sum (\<lambda>x. u x *\<^sub>R x) s = y}
              \<subseteq> rel_interior (convex hull s)"
   by (force simp add:  rel_interior_convex_hull_union [where S="\<lambda>x. {x}" and I=s, simplified])
 
 lemma explicit_subset_rel_interior_convex_hull_minimal:
   fixes s :: "'a::euclidean_space set"
   shows "finite s
          \<Longrightarrow> {y. \<exists>u. (\<forall>x \<in> s. 0 < u x) \<and> sum u s = 1 \<and> sum (\<lambda>x. u x *\<^sub>R x) s = y}
              \<subseteq> rel_interior (convex hull s)"
   by (force simp add:  rel_interior_convex_hull_union [where S="\<lambda>x. {x}" and I=s, simplified])
 
 lemma rel_interior_convex_hull_explicit:
   fixes s :: "'a::euclidean_space set"
   assumes "\<not> affine_dependent s"
   shows "rel_interior(convex hull s) =
          {y. \<exists>u. (\<forall>x \<in> s. 0 < u x) \<and> sum u s = 1 \<and> sum (\<lambda>x. u x *\<^sub>R x) s = y}"
          (is "?lhs = ?rhs")
 proof
   show "?rhs \<le> ?lhs"
     by (simp add: aff_independent_finite explicit_subset_rel_interior_convex_hull_minimal assms)
 next
   show "?lhs \<le> ?rhs"
   proof (cases "\<exists>a. s = {a}")
     case True then show "?lhs \<le> ?rhs"
       by force
   next
     case False
     have fs: "finite s"
       using assms by (simp add: aff_independent_finite)
     { fix a b and d::real
       assume ab: "a \<in> s" "b \<in> s" "a \<noteq> b"
       then have s: "s = (s - {a,b}) \<union> {a,b}" \<comment> \<open>split into separate cases\<close>
         by auto
       have "(\<Sum>x\<in>s. if x = a then - d else if x = b then d else 0) = 0"
            "(\<Sum>x\<in>s. (if x = a then - d else if x = b then d else 0) *\<^sub>R x) = d *\<^sub>R b - d *\<^sub>R a"
         using ab fs
         by (subst s, subst sum.union_disjoint, auto)+
     } note [simp] = this
     { fix y
       assume y: "y \<in> convex hull s" "y \<notin> ?rhs"
       { fix u T a
         assume ua: "\<forall>x\<in>s. 0 \<le> u x" "sum u s = 1" "\<not> 0 < u a" "a \<in> s"
            and yT: "y = (\<Sum>x\<in>s. u x *\<^sub>R x)" "y \<in> T" "open T"
            and sb: "T \<inter> affine hull s \<subseteq> {w. \<exists>u. (\<forall>x\<in>s. 0 \<le> u x) \<and> sum u s = 1 \<and> (\<Sum>x\<in>s. u x *\<^sub>R x) = w}"
         have ua0: "u a = 0"
           using ua by auto
         obtain b where b: "b\<in>s" "a \<noteq> b"
           using ua False by auto
         obtain e where e: "0 < e" "ball (\<Sum>x\<in>s. u x *\<^sub>R x) e \<subseteq> T"
           using yT by (auto elim: openE)
         with b obtain d where d: "0 < d" "norm(d *\<^sub>R (a-b)) < e"
           by (auto intro: that [of "e / 2 / norm(a-b)"])
         have "(\<Sum>x\<in>s. u x *\<^sub>R x) \<in> affine hull s"
           using yT y by (metis affine_hull_convex_hull hull_redundant_eq)
         then have "(\<Sum>x\<in>s. u x *\<^sub>R x) - d *\<^sub>R (a - b) \<in> affine hull s"
           using ua b by (auto simp: hull_inc intro: mem_affine_3_minus2)
         then have "y - d *\<^sub>R (a - b) \<in> T \<inter> affine hull s"
           using d e yT by auto
         then obtain v where "\<forall>x\<in>s. 0 \<le> v x"
                             "sum v s = 1"
                             "(\<Sum>x\<in>s. v x *\<^sub>R x) = (\<Sum>x\<in>s. u x *\<^sub>R x) - d *\<^sub>R (a - b)"
           using subsetD [OF sb] yT
           by auto
         then have False
           using assms
           apply (simp add: affine_dependent_explicit_finite fs)
           apply (drule_tac x="\<lambda>x. (v x - u x) - (if x = a then -d else if x = b then d else 0)" in spec)
           using ua b d
           apply (auto simp: algebra_simps sum_subtractf sum.distrib)
           done
       } note * = this
       have "y \<notin> rel_interior (convex hull s)"
         using y
         apply (simp add: mem_rel_interior)
         apply (auto simp: convex_hull_finite [OF fs])
         apply (drule_tac x=u in spec)
         apply (auto intro: *)
         done
     } with rel_interior_subset show "?lhs \<le> ?rhs"
       by blast
   qed
 qed
 
 lemma interior_convex_hull_explicit_minimal:
   fixes s :: "'a::euclidean_space set"
   shows
    "\<not> affine_dependent s
         ==> interior(convex hull s) =
              (if card(s) \<le> DIM('a) then {}
               else {y. \<exists>u. (\<forall>x \<in> s. 0 < u x) \<and> sum u s = 1 \<and> (\<Sum>x\<in>s. u x *\<^sub>R x) = y})"
   apply (simp add: aff_independent_finite empty_interior_convex_hull, clarify)
   apply (rule trans [of _ "rel_interior(convex hull s)"])
   apply (simp add: affine_independent_span_gt rel_interior_interior)
   by (simp add: rel_interior_convex_hull_explicit)
 
 lemma interior_convex_hull_explicit:
   fixes s :: "'a::euclidean_space set"
   assumes "\<not> affine_dependent s"
   shows
    "interior(convex hull s) =
              (if card(s) \<le> DIM('a) then {}
               else {y. \<exists>u. (\<forall>x \<in> s. 0 < u x \<and> u x < 1) \<and> sum u s = 1 \<and> (\<Sum>x\<in>s. u x *\<^sub>R x) = y})"
 proof -
   { fix u :: "'a \<Rightarrow> real" and a
     assume "card Basis < card s" and u: "\<And>x. x\<in>s \<Longrightarrow> 0 < u x" "sum u s = 1" and a: "a \<in> s"
     then have cs: "Suc 0 < card s"
       by (metis DIM_positive less_trans_Suc)
     obtain b where b: "b \<in> s" "a \<noteq> b"
     proof (cases "s \<le> {a}")
       case True
       then show thesis
         using cs subset_singletonD by fastforce
     next
       case False
       then show thesis
       by (blast intro: that)
     qed
     have "u a + u b \<le> sum u {a,b}"
       using a b by simp
     also have "... \<le> sum u s"
       apply (rule Groups_Big.sum_mono2)
       using a b u
       apply (auto simp: less_imp_le aff_independent_finite assms)
       done
     finally have "u a < 1"
       using \<open>b \<in> s\<close> u by fastforce
   } note [simp] = this
   show ?thesis
     using assms
     apply (auto simp: interior_convex_hull_explicit_minimal)
     apply (rule_tac x=u in exI)
     apply (auto simp: not_le)
     done
 qed
 
 lemma interior_closed_segment_ge2:
   fixes a :: "'a::euclidean_space"
   assumes "2 \<le> DIM('a)"
     shows  "interior(closed_segment a b) = {}"
 using assms unfolding segment_convex_hull
 proof -
   have "card {a, b} \<le> DIM('a)"
     using assms
     by (simp add: card_insert_if linear not_less_eq_eq numeral_2_eq_2)
   then show "interior (convex hull {a, b}) = {}"
     by (metis empty_interior_convex_hull finite.insertI finite.emptyI)
 qed
 
 lemma interior_open_segment:
   fixes a :: "'a::euclidean_space"
   shows  "interior(open_segment a b) =
                  (if 2 \<le> DIM('a) then {} else open_segment a b)"
 proof (simp add: not_le, intro conjI impI)
   assume "2 \<le> DIM('a)"
   then show "interior (open_segment a b) = {}"
     apply (simp add: segment_convex_hull open_segment_def)
     apply (metis Diff_subset interior_mono segment_convex_hull subset_empty interior_closed_segment_ge2)
     done
 next
   assume le2: "DIM('a) < 2"
   show "interior (open_segment a b) = open_segment a b"
   proof (cases "a = b")
     case True then show ?thesis by auto
   next
     case False
     with le2 have "affine hull (open_segment a b) = UNIV"
       apply simp
       apply (rule affine_independent_span_gt)
       apply (simp_all add: affine_dependent_def insert_Diff_if)
       done
     then show "interior (open_segment a b) = open_segment a b"
       using rel_interior_interior rel_interior_open_segment by blast
   qed
 qed
 
 lemma interior_closed_segment:
   fixes a :: "'a::euclidean_space"
   shows "interior(closed_segment a b) =
                  (if 2 \<le> DIM('a) then {} else open_segment a b)"
 proof (cases "a = b")
   case True then show ?thesis by simp
 next
   case False
   then have "closure (open_segment a b) = closed_segment a b"
     by simp
   then show ?thesis
     by (metis (no_types) convex_interior_closure convex_open_segment interior_open_segment)
 qed
 
 lemmas interior_segment = interior_closed_segment interior_open_segment
 
 lemma closed_segment_eq [simp]:
   fixes a :: "'a::euclidean_space"
   shows "closed_segment a b = closed_segment c d \<longleftrightarrow> {a,b} = {c,d}"
 proof
   assume abcd: "closed_segment a b = closed_segment c d"
   show "{a,b} = {c,d}"
   proof (cases "a=b \<or> c=d")
     case True with abcd show ?thesis by force
   next
     case False
     then have neq: "a \<noteq> b \<and> c \<noteq> d" by force
     have *: "closed_segment c d - {a, b} = rel_interior (closed_segment c d)"
       using neq abcd by (metis (no_types) open_segment_def rel_interior_closed_segment)
     have "b \<in> {c, d}"
     proof -
       have "insert b (closed_segment c d) = closed_segment c d"
         using abcd by blast
       then show ?thesis
         by (metis DiffD2 Diff_insert2 False * insertI1 insert_Diff_if open_segment_def rel_interior_closed_segment)
     qed
     moreover have "a \<in> {c, d}"
       by (metis Diff_iff False * abcd ends_in_segment(1) insertI1 open_segment_def rel_interior_closed_segment)
     ultimately show "{a, b} = {c, d}"
       using neq by fastforce
   qed
 next
   assume "{a,b} = {c,d}"
   then show "closed_segment a b = closed_segment c d"
     by (simp add: segment_convex_hull)
 qed
 
 lemma closed_open_segment_eq [simp]:
   fixes a :: "'a::euclidean_space"
   shows "closed_segment a b \<noteq> open_segment c d"
 by (metis DiffE closed_segment_neq_empty closure_closed_segment closure_open_segment ends_in_segment(1) insertI1 open_segment_def)
 
 lemma open_closed_segment_eq [simp]:
   fixes a :: "'a::euclidean_space"
   shows "open_segment a b \<noteq> closed_segment c d"
 using closed_open_segment_eq by blast
 
 lemma open_segment_eq [simp]:
   fixes a :: "'a::euclidean_space"
   shows "open_segment a b = open_segment c d \<longleftrightarrow> a = b \<and> c = d \<or> {a,b} = {c,d}"
         (is "?lhs = ?rhs")
 proof
   assume abcd: ?lhs
   show ?rhs
   proof (cases "a=b \<or> c=d")
     case True with abcd show ?thesis
       using finite_open_segment by fastforce
   next
     case False
     then have a2: "a \<noteq> b \<and> c \<noteq> d" by force
     with abcd show ?rhs
       unfolding open_segment_def
       by (metis (no_types) abcd closed_segment_eq closure_open_segment)
   qed
 next
   assume ?rhs
   then show ?lhs
     by (metis Diff_cancel convex_hull_singleton insert_absorb2 open_segment_def segment_convex_hull)
 qed
 
 subsection\<^marker>\<open>tag unimportant\<close>\<open>Similar results for closure and (relative or absolute) frontier\<close>
 
 lemma closure_convex_hull [simp]:
   fixes s :: "'a::euclidean_space set"
   shows "compact s ==> closure(convex hull s) = convex hull s"
   by (simp add: compact_imp_closed compact_convex_hull)
 
 lemma rel_frontier_convex_hull_explicit:
   fixes s :: "'a::euclidean_space set"
   assumes "\<not> affine_dependent s"
   shows "rel_frontier(convex hull s) =
          {y. \<exists>u. (\<forall>x \<in> s. 0 \<le> u x) \<and> (\<exists>x \<in> s. u x = 0) \<and> sum u s = 1 \<and> sum (\<lambda>x. u x *\<^sub>R x) s = y}"
 proof -
   have fs: "finite s"
     using assms by (simp add: aff_independent_finite)
   show ?thesis
     apply (simp add: rel_frontier_def finite_imp_compact rel_interior_convex_hull_explicit assms fs)
     apply (auto simp: convex_hull_finite fs)
     apply (drule_tac x=u in spec)
     apply (rule_tac x=u in exI)
     apply force
     apply (rename_tac v)
     apply (rule notE [OF assms])
     apply (simp add: affine_dependent_explicit)
     apply (rule_tac x=s in exI)
     apply (auto simp: fs)
     apply (rule_tac x = "\<lambda>x. u x - v x" in exI)
     apply (force simp: sum_subtractf scaleR_diff_left)
     done
 qed
 
 lemma frontier_convex_hull_explicit:
   fixes s :: "'a::euclidean_space set"
   assumes "\<not> affine_dependent s"
   shows "frontier(convex hull s) =
          {y. \<exists>u. (\<forall>x \<in> s. 0 \<le> u x) \<and> (DIM ('a) < card s \<longrightarrow> (\<exists>x \<in> s. u x = 0)) \<and>
              sum u s = 1 \<and> sum (\<lambda>x. u x *\<^sub>R x) s = y}"
 proof -
   have fs: "finite s"
     using assms by (simp add: aff_independent_finite)
   show ?thesis
   proof (cases "DIM ('a) < card s")
     case True
     with assms fs show ?thesis
       by (simp add: rel_frontier_def frontier_def rel_frontier_convex_hull_explicit [symmetric]
                     interior_convex_hull_explicit_minimal rel_interior_convex_hull_explicit)
   next
     case False
     then have "card s \<le> DIM ('a)"
       by linarith
     then show ?thesis
       using assms fs
       apply (simp add: frontier_def interior_convex_hull_explicit finite_imp_compact)
       apply (simp add: convex_hull_finite)
       done
   qed
 qed
 
 lemma rel_frontier_convex_hull_cases:
   fixes s :: "'a::euclidean_space set"
   assumes "\<not> affine_dependent s"
   shows "rel_frontier(convex hull s) = \<Union>{convex hull (s - {x}) |x. x \<in> s}"
 proof -
   have fs: "finite s"
     using assms by (simp add: aff_independent_finite)
   { fix u a
   have "\<forall>x\<in>s. 0 \<le> u x \<Longrightarrow> a \<in> s \<Longrightarrow> u a = 0 \<Longrightarrow> sum u s = 1 \<Longrightarrow>
             \<exists>x v. x \<in> s \<and>
                   (\<forall>x\<in>s - {x}. 0 \<le> v x) \<and>
                       sum v (s - {x}) = 1 \<and> (\<Sum>x\<in>s - {x}. v x *\<^sub>R x) = (\<Sum>x\<in>s. u x *\<^sub>R x)"
     apply (rule_tac x=a in exI)
     apply (rule_tac x=u in exI)
     apply (simp add: Groups_Big.sum_diff1 fs)
     done }
   moreover
   { fix a u
     have "a \<in> s \<Longrightarrow> \<forall>x\<in>s - {a}. 0 \<le> u x \<Longrightarrow> sum u (s - {a}) = 1 \<Longrightarrow>
             \<exists>v. (\<forall>x\<in>s. 0 \<le> v x) \<and>
                  (\<exists>x\<in>s. v x = 0) \<and> sum v s = 1 \<and> (\<Sum>x\<in>s. v x *\<^sub>R x) = (\<Sum>x\<in>s - {a}. u x *\<^sub>R x)"
     apply (rule_tac x="\<lambda>x. if x = a then 0 else u x" in exI)
     apply (auto simp: sum.If_cases Diff_eq if_smult fs)
     done }
   ultimately show ?thesis
     using assms
     apply (simp add: rel_frontier_convex_hull_explicit)
     apply (simp add: convex_hull_finite fs Union_SetCompr_eq, auto)
     done
 qed
 
 lemma frontier_convex_hull_eq_rel_frontier:
   fixes s :: "'a::euclidean_space set"
   assumes "\<not> affine_dependent s"
   shows "frontier(convex hull s) =
            (if card s \<le> DIM ('a) then convex hull s else rel_frontier(convex hull s))"
   using assms
   unfolding rel_frontier_def frontier_def
   by (simp add: affine_independent_span_gt rel_interior_interior
                 finite_imp_compact empty_interior_convex_hull aff_independent_finite)
 
 lemma frontier_convex_hull_cases:
   fixes s :: "'a::euclidean_space set"
   assumes "\<not> affine_dependent s"
   shows "frontier(convex hull s) =
            (if card s \<le> DIM ('a) then convex hull s else \<Union>{convex hull (s - {x}) |x. x \<in> s})"
 by (simp add: assms frontier_convex_hull_eq_rel_frontier rel_frontier_convex_hull_cases)
 
 lemma in_frontier_convex_hull:
   fixes s :: "'a::euclidean_space set"
   assumes "finite s" "card s \<le> Suc (DIM ('a))" "x \<in> s"
   shows   "x \<in> frontier(convex hull s)"
 proof (cases "affine_dependent s")
   case True
   with assms show ?thesis
     apply (auto simp: affine_dependent_def frontier_def finite_imp_compact hull_inc)
     by (metis card.insert_remove convex_hull_subset_affine_hull empty_interior_affine_hull finite_Diff hull_redundant insert_Diff insert_Diff_single insert_not_empty interior_mono not_less_eq_eq subset_empty)
 next
   case False
   { assume "card s = Suc (card Basis)"
     then have cs: "Suc 0 < card s"
       by (simp)
     with subset_singletonD have "\<exists>y \<in> s. y \<noteq> x"
       by (cases "s \<le> {x}") fastforce+
   } note [dest!] = this
   show ?thesis using assms
     unfolding frontier_convex_hull_cases [OF False] Union_SetCompr_eq
     by (auto simp: le_Suc_eq hull_inc)
 qed
 
 lemma not_in_interior_convex_hull:
   fixes s :: "'a::euclidean_space set"
   assumes "finite s" "card s \<le> Suc (DIM ('a))" "x \<in> s"
   shows   "x \<notin> interior(convex hull s)"
 using in_frontier_convex_hull [OF assms]
 by (metis Diff_iff frontier_def)
 
 lemma interior_convex_hull_eq_empty:
   fixes s :: "'a::euclidean_space set"
   assumes "card s = Suc (DIM ('a))"
   shows   "interior(convex hull s) = {} \<longleftrightarrow> affine_dependent s"
 proof -
   { fix a b
     assume ab: "a \<in> interior (convex hull s)" "b \<in> s" "b \<in> affine hull (s - {b})"
     then have "interior(affine hull s) = {}" using assms
       by (metis DIM_positive One_nat_def Suc_mono card.remove card_infinite empty_interior_affine_hull eq_iff hull_redundant insert_Diff not_less zero_le_one)
     then have False using ab
       by (metis convex_hull_subset_affine_hull equals0D interior_mono subset_eq)
   } then
   show ?thesis
     using assms
     apply auto
     apply (metis UNIV_I affine_hull_convex_hull affine_hull_empty affine_independent_span_eq convex_convex_hull empty_iff rel_interior_interior rel_interior_same_affine_hull)
     apply (auto simp: affine_dependent_def)
     done
 qed
 
 
 subsection \<open>Coplanarity, and collinearity in terms of affine hull\<close>
 
 definition\<^marker>\<open>tag important\<close> coplanar  where
    "coplanar s \<equiv> \<exists>u v w. s \<subseteq> affine hull {u,v,w}"
 
 lemma collinear_affine_hull:
   "collinear s \<longleftrightarrow> (\<exists>u v. s \<subseteq> affine hull {u,v})"
 proof (cases "s={}")
   case True then show ?thesis
     by simp
 next
   case False
   then obtain x where x: "x \<in> s" by auto
   { fix u
     assume *: "\<And>x y. \<lbrakk>x\<in>s; y\<in>s\<rbrakk> \<Longrightarrow> \<exists>c. x - y = c *\<^sub>R u"
     have "\<exists>u v. s \<subseteq> {a *\<^sub>R u + b *\<^sub>R v |a b. a + b = 1}"
       apply (rule_tac x=x in exI)
       apply (rule_tac x="x+u" in exI, clarify)
       apply (erule exE [OF * [OF x]])
       apply (rename_tac c)
       apply (rule_tac x="1+c" in exI)
       apply (rule_tac x="-c" in exI)
       apply (simp add: algebra_simps)
       done
   } moreover
   { fix u v x y
     assume *: "s \<subseteq> {a *\<^sub>R u + b *\<^sub>R v |a b. a + b = 1}"
     have "x\<in>s \<Longrightarrow> y\<in>s \<Longrightarrow> \<exists>c. x - y = c *\<^sub>R (v-u)"
       apply (drule subsetD [OF *])+
       apply simp
       apply clarify
       apply (rename_tac r1 r2)
       apply (rule_tac x="r1-r2" in exI)
       apply (simp add: algebra_simps)
       apply (metis scaleR_left.add)
       done
   } ultimately
   show ?thesis
   unfolding collinear_def affine_hull_2
     by blast
 qed
 
 lemma collinear_closed_segment [simp]: "collinear (closed_segment a b)"
 by (metis affine_hull_convex_hull collinear_affine_hull hull_subset segment_convex_hull)
 
 lemma collinear_open_segment [simp]: "collinear (open_segment a b)"
   unfolding open_segment_def
   by (metis convex_hull_subset_affine_hull segment_convex_hull dual_order.trans
     convex_hull_subset_affine_hull Diff_subset collinear_affine_hull)
 
 lemma collinear_between_cases:
   fixes c :: "'a::euclidean_space"
   shows "collinear {a,b,c} \<longleftrightarrow> between (b,c) a \<or> between (c,a) b \<or> between (a,b) c"
          (is "?lhs = ?rhs")
 proof
   assume ?lhs
   then obtain u v where uv: "\<And>x. x \<in> {a, b, c} \<Longrightarrow> \<exists>c. x = u + c *\<^sub>R v"
     by (auto simp: collinear_alt)
   show ?rhs
     using uv [of a] uv [of b] uv [of c] by (auto simp: between_1)
 next
   assume ?rhs
   then show ?lhs
     unfolding between_mem_convex_hull
     by (metis (no_types, hide_lams) collinear_closed_segment collinear_subset hull_redundant hull_subset insert_commute segment_convex_hull)
 qed
 
 
 lemma subset_continuous_image_segment_1:
   fixes f :: "'a::euclidean_space \<Rightarrow> real"
   assumes "continuous_on (closed_segment a b) f"
   shows "closed_segment (f a) (f b) \<subseteq> image f (closed_segment a b)"
 by (metis connected_segment convex_contains_segment ends_in_segment imageI
            is_interval_connected_1 is_interval_convex connected_continuous_image [OF assms])
 
 lemma continuous_injective_image_segment_1:
   fixes f :: "'a::euclidean_space \<Rightarrow> real"
   assumes contf: "continuous_on (closed_segment a b) f"
       and injf: "inj_on f (closed_segment a b)"
   shows "f ` (closed_segment a b) = closed_segment (f a) (f b)"
 proof
   show "closed_segment (f a) (f b) \<subseteq> f ` closed_segment a b"
     by (metis subset_continuous_image_segment_1 contf)
   show "f ` closed_segment a b \<subseteq> closed_segment (f a) (f b)"
   proof (cases "a = b")
     case True
     then show ?thesis by auto
   next
     case False
     then have fnot: "f a \<noteq> f b"
       using inj_onD injf by fastforce
     moreover
     have "f a \<notin> open_segment (f c) (f b)" if c: "c \<in> closed_segment a b" for c
     proof (clarsimp simp add: open_segment_def)
       assume fa: "f a \<in> closed_segment (f c) (f b)"
       moreover have "closed_segment (f c) (f b) \<subseteq> f ` closed_segment c b"
         by (meson closed_segment_subset contf continuous_on_subset convex_closed_segment ends_in_segment(2) subset_continuous_image_segment_1 that)
       ultimately have "f a \<in> f ` closed_segment c b"
         by blast
       then have a: "a \<in> closed_segment c b"
         by (meson ends_in_segment inj_on_image_mem_iff_alt injf subset_closed_segment that)
       have cb: "closed_segment c b \<subseteq> closed_segment a b"
         by (simp add: closed_segment_subset that)
       show "f a = f c"
       proof (rule between_antisym)
         show "between (f c, f b) (f a)"
           by (simp add: between_mem_segment fa)
         show "between (f a, f b) (f c)"
           by (metis a cb between_antisym between_mem_segment between_triv1 subset_iff)
       qed
     qed
     moreover
     have "f b \<notin> open_segment (f a) (f c)" if c: "c \<in> closed_segment a b" for c
     proof (clarsimp simp add: open_segment_def fnot eq_commute)
       assume fb: "f b \<in> closed_segment (f a) (f c)"
       moreover have "closed_segment (f a) (f c) \<subseteq> f ` closed_segment a c"
         by (meson contf continuous_on_subset ends_in_segment(1) subset_closed_segment subset_continuous_image_segment_1 that)
       ultimately have "f b \<in> f ` closed_segment a c"
         by blast
       then have b: "b \<in> closed_segment a c"
         by (meson ends_in_segment inj_on_image_mem_iff_alt injf subset_closed_segment that)
       have ca: "closed_segment a c \<subseteq> closed_segment a b"
         by (simp add: closed_segment_subset that)
       show "f b = f c"
       proof (rule between_antisym)
         show "between (f c, f a) (f b)"
           by (simp add: between_commute between_mem_segment fb)
         show "between (f b, f a) (f c)"
           by (metis b between_antisym between_commute between_mem_segment between_triv2 that)
       qed
     qed
     ultimately show ?thesis
       by (force simp: closed_segment_eq_real_ivl open_segment_eq_real_ivl split: if_split_asm)
   qed
 qed
 
 lemma continuous_injective_image_open_segment_1:
   fixes f :: "'a::euclidean_space \<Rightarrow> real"
   assumes contf: "continuous_on (closed_segment a b) f"
       and injf: "inj_on f (closed_segment a b)"
     shows "f ` (open_segment a b) = open_segment (f a) (f b)"
 proof -
   have "f ` (open_segment a b) = f ` (closed_segment a b) - {f a, f b}"
     by (metis (no_types, hide_lams) empty_subsetI ends_in_segment image_insert image_is_empty inj_on_image_set_diff injf insert_subset open_segment_def segment_open_subset_closed)
   also have "... = open_segment (f a) (f b)"
     using continuous_injective_image_segment_1 [OF assms]
     by (simp add: open_segment_def inj_on_image_set_diff [OF injf])
   finally show ?thesis .
 qed
 
 lemma collinear_imp_coplanar:
   "collinear s ==> coplanar s"
 by (metis collinear_affine_hull coplanar_def insert_absorb2)
 
 lemma collinear_small:
   assumes "finite s" "card s \<le> 2"
     shows "collinear s"
 proof -
   have "card s = 0 \<or> card s = 1 \<or> card s = 2"
     using assms by linarith
   then show ?thesis using assms
     using card_eq_SucD
     by auto (metis collinear_2 numeral_2_eq_2)
 qed
 
 lemma coplanar_small:
   assumes "finite s" "card s \<le> 3"
     shows "coplanar s"
 proof -
   have "card s \<le> 2 \<or> card s = Suc (Suc (Suc 0))"
     using assms by linarith
   then show ?thesis using assms
     apply safe
     apply (simp add: collinear_small collinear_imp_coplanar)
     apply (safe dest!: card_eq_SucD)
     apply (auto simp: coplanar_def)
     apply (metis hull_subset insert_subset)
     done
 qed
 
 lemma coplanar_empty: "coplanar {}"
   by (simp add: coplanar_small)
 
 lemma coplanar_sing: "coplanar {a}"
   by (simp add: coplanar_small)
 
 lemma coplanar_2: "coplanar {a,b}"
   by (auto simp: card_insert_if coplanar_small)
 
 lemma coplanar_3: "coplanar {a,b,c}"
   by (auto simp: card_insert_if coplanar_small)
 
 lemma collinear_affine_hull_collinear: "collinear(affine hull s) \<longleftrightarrow> collinear s"
   unfolding collinear_affine_hull
   by (metis affine_affine_hull subset_hull hull_hull hull_mono)
 
 lemma coplanar_affine_hull_coplanar: "coplanar(affine hull s) \<longleftrightarrow> coplanar s"
   unfolding coplanar_def
   by (metis affine_affine_hull subset_hull hull_hull hull_mono)
 
 lemma coplanar_linear_image:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
   assumes "coplanar s" "linear f" shows "coplanar(f ` s)"
 proof -
   { fix u v w
     assume "s \<subseteq> affine hull {u, v, w}"
     then have "f ` s \<subseteq> f ` (affine hull {u, v, w})"
       by (simp add: image_mono)
     then have "f ` s \<subseteq> affine hull (f ` {u, v, w})"
       by (metis assms(2) linear_conv_bounded_linear affine_hull_linear_image)
   } then
   show ?thesis
     by auto (meson assms(1) coplanar_def)
 qed
 
 lemma coplanar_translation_imp: "coplanar s \<Longrightarrow> coplanar ((\<lambda>x. a + x) ` s)"
   unfolding coplanar_def
   apply clarify
   apply (rule_tac x="u+a" in exI)
   apply (rule_tac x="v+a" in exI)
   apply (rule_tac x="w+a" in exI)
   using affine_hull_translation [of a "{u,v,w}" for u v w]
   apply (force simp: add.commute)
   done
 
 lemma coplanar_translation_eq: "coplanar((\<lambda>x. a + x) ` s) \<longleftrightarrow> coplanar s"
     by (metis (no_types) coplanar_translation_imp translation_galois)
 
 lemma coplanar_linear_image_eq:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes "linear f" "inj f" shows "coplanar(f ` s) = coplanar s"
 proof
   assume "coplanar s"
   then show "coplanar (f ` s)"
     unfolding coplanar_def
     using affine_hull_linear_image [of f "{u,v,w}" for u v w]  assms
     by (meson coplanar_def coplanar_linear_image)
 next
   obtain g where g: "linear g" "g \<circ> f = id"
     using linear_injective_left_inverse [OF assms]
     by blast
   assume "coplanar (f ` s)"
   then obtain u v w where "f ` s \<subseteq> affine hull {u, v, w}"
     by (auto simp: coplanar_def)
   then have "g ` f ` s \<subseteq> g ` (affine hull {u, v, w})"
     by blast
   then have "s \<subseteq> g ` (affine hull {u, v, w})"
     using g by (simp add: Fun.image_comp)
   then show "coplanar s"
     unfolding coplanar_def
     using affine_hull_linear_image [of g "{u,v,w}" for u v w]  \<open>linear g\<close> linear_conv_bounded_linear
     by fastforce
 qed
 (*The HOL Light proof is simply
     MATCH_ACCEPT_TAC(LINEAR_INVARIANT_RULE COPLANAR_LINEAR_IMAGE));;
 *)
 
 lemma coplanar_subset: "\<lbrakk>coplanar t; s \<subseteq> t\<rbrakk> \<Longrightarrow> coplanar s"
   by (meson coplanar_def order_trans)
 
 lemma affine_hull_3_imp_collinear: "c \<in> affine hull {a,b} \<Longrightarrow> collinear {a,b,c}"
   by (metis collinear_2 collinear_affine_hull_collinear hull_redundant insert_commute)
 
 lemma collinear_3_imp_in_affine_hull: "\<lbrakk>collinear {a,b,c}; a \<noteq> b\<rbrakk> \<Longrightarrow> c \<in> affine hull {a,b}"
   unfolding collinear_def
   apply clarify
   apply (frule_tac x=b in bspec, blast, drule_tac x=a in bspec, blast, erule exE)
   apply (drule_tac x=c in bspec, blast, drule_tac x=a in bspec, blast, erule exE)
   apply (rename_tac y x)
   apply (simp add: affine_hull_2)
   apply (rule_tac x="1 - x/y" in exI)
   apply (simp add: algebra_simps)
   done
 
 lemma collinear_3_affine_hull:
   assumes "a \<noteq> b"
     shows "collinear {a,b,c} \<longleftrightarrow> c \<in> affine hull {a,b}"
 using affine_hull_3_imp_collinear assms collinear_3_imp_in_affine_hull by blast
 
 lemma collinear_3_eq_affine_dependent:
   "collinear{a,b,c} \<longleftrightarrow> a = b \<or> a = c \<or> b = c \<or> affine_dependent {a,b,c}"
 apply (case_tac "a=b", simp)
 apply (case_tac "a=c")
 apply (simp add: insert_commute)
 apply (case_tac "b=c")
 apply (simp add: insert_commute)
 apply (auto simp: affine_dependent_def collinear_3_affine_hull insert_Diff_if)
 apply (metis collinear_3_affine_hull insert_commute)+
 done
 
 lemma affine_dependent_imp_collinear_3:
   "affine_dependent {a,b,c} \<Longrightarrow> collinear{a,b,c}"
 by (simp add: collinear_3_eq_affine_dependent)
 
 lemma collinear_3: "NO_MATCH 0 x \<Longrightarrow> collinear {x,y,z} \<longleftrightarrow> collinear {0, x-y, z-y}"
   by (auto simp add: collinear_def)
 
 lemma collinear_3_expand:
    "collinear{a,b,c} \<longleftrightarrow> a = c \<or> (\<exists>u. b = u *\<^sub>R a + (1 - u) *\<^sub>R c)"
 proof -
   have "collinear{a,b,c} = collinear{a,c,b}"
     by (simp add: insert_commute)
   also have "... = collinear {0, a - c, b - c}"
     by (simp add: collinear_3)
   also have "... \<longleftrightarrow> (a = c \<or> b = c \<or> (\<exists>ca. b - c = ca *\<^sub>R (a - c)))"
     by (simp add: collinear_lemma)
   also have "... \<longleftrightarrow> a = c \<or> (\<exists>u. b = u *\<^sub>R a + (1 - u) *\<^sub>R c)"
     by (cases "a = c \<or> b = c") (auto simp: algebra_simps)
   finally show ?thesis .
 qed
 
 lemma collinear_aff_dim: "collinear S \<longleftrightarrow> aff_dim S \<le> 1"
 proof
   assume "collinear S"
   then obtain u and v :: "'a" where "aff_dim S \<le> aff_dim {u,v}"
     by (metis \<open>collinear S\<close> aff_dim_affine_hull aff_dim_subset collinear_affine_hull)
   then show "aff_dim S \<le> 1"
     using order_trans by fastforce
 next
   assume "aff_dim S \<le> 1"
   then have le1: "aff_dim (affine hull S) \<le> 1"
     by simp
   obtain B where "B \<subseteq> S" and B: "\<not> affine_dependent B" "affine hull S = affine hull B"
     using affine_basis_exists [of S] by auto
   then have "finite B" "card B \<le> 2"
     using B le1 by (auto simp: affine_independent_iff_card)
   then have "collinear B"
     by (rule collinear_small)
   then show "collinear S"
     by (metis \<open>affine hull S = affine hull B\<close> collinear_affine_hull_collinear)
 qed
 
 lemma collinear_midpoint: "collinear{a,midpoint a b,b}"
   apply (auto simp: collinear_3 collinear_lemma)
   apply (drule_tac x="-1" in spec)
   apply (simp add: algebra_simps)
   done
 
 lemma midpoint_collinear:
   fixes a b c :: "'a::real_normed_vector"
   assumes "a \<noteq> c"
     shows "b = midpoint a c \<longleftrightarrow> collinear{a,b,c} \<and> dist a b = dist b c"
 proof -
   have *: "a - (u *\<^sub>R a + (1 - u) *\<^sub>R c) = (1 - u) *\<^sub>R (a - c)"
           "u *\<^sub>R a + (1 - u) *\<^sub>R c - c = u *\<^sub>R (a - c)"
           "\<bar>1 - u\<bar> = \<bar>u\<bar> \<longleftrightarrow> u = 1/2" for u::real
     by (auto simp: algebra_simps)
   have "b = midpoint a c \<Longrightarrow> collinear{a,b,c} "
     using collinear_midpoint by blast
   moreover have "collinear{a,b,c} \<Longrightarrow> b = midpoint a c \<longleftrightarrow> dist a b = dist b c"
     apply (auto simp: collinear_3_expand assms dist_midpoint)
     apply (simp add: dist_norm * assms midpoint_def del: divide_const_simps)
     apply (simp add: algebra_simps)
     done
   ultimately show ?thesis by blast
 qed
 
 lemma between_imp_collinear:
   fixes x :: "'a :: euclidean_space"
   assumes "between (a,b) x"
     shows "collinear {a,x,b}"
 proof (cases "x = a \<or> x = b \<or> a = b")
   case True with assms show ?thesis
     by (auto simp: dist_commute)
 next
   case False with assms show ?thesis
     apply (auto simp: collinear_3 collinear_lemma between_norm)
     apply (drule_tac x="-(norm(b - x) / norm(x - a))" in spec)
     apply (simp add: vector_add_divide_simps real_vector.scale_minus_right [symmetric])
     done
 qed
 
 lemma midpoint_between:
   fixes a b :: "'a::euclidean_space"
   shows "b = midpoint a c \<longleftrightarrow> between (a,c) b \<and> dist a b = dist b c"
 proof (cases "a = c")
   case True then show ?thesis
     by (auto simp: dist_commute)
 next
   case False
   show ?thesis
     apply (rule iffI)
     apply (simp add: between_midpoint(1) dist_midpoint)
     using False between_imp_collinear midpoint_collinear by blast
 qed
 
 lemma collinear_triples:
   assumes "a \<noteq> b"
     shows "collinear(insert a (insert b S)) \<longleftrightarrow> (\<forall>x \<in> S. collinear{a,b,x})"
           (is "?lhs = ?rhs")
 proof safe
   fix x
   assume ?lhs and "x \<in> S"
   then show "collinear {a, b, x}"
     using collinear_subset by force
 next
   assume ?rhs
   then have "\<forall>x \<in> S. collinear{a,x,b}"
     by (simp add: insert_commute)
   then have *: "\<exists>u. x = u *\<^sub>R a + (1 - u) *\<^sub>R b" if "x \<in> (insert a (insert b S))" for x
     using that assms collinear_3_expand by fastforce+
   show ?lhs
     unfolding collinear_def
     apply (rule_tac x="b-a" in exI)
     apply (clarify dest!: *)
     by (metis (no_types, hide_lams) add.commute diff_add_cancel diff_diff_eq2 real_vector.scale_right_diff_distrib scaleR_left.diff)
 qed
 
 lemma collinear_4_3:
   assumes "a \<noteq> b"
     shows "collinear {a,b,c,d} \<longleftrightarrow> collinear{a,b,c} \<and> collinear{a,b,d}"
   using collinear_triples [OF assms, of "{c,d}"] by (force simp:)
 
 lemma collinear_3_trans:
   assumes "collinear{a,b,c}" "collinear{b,c,d}" "b \<noteq> c"
     shows "collinear{a,b,d}"
 proof -
   have "collinear{b,c,a,d}"
     by (metis (full_types) assms collinear_4_3 insert_commute)
   then show ?thesis
     by (simp add: collinear_subset)
 qed
 
 lemma affine_hull_eq_empty [simp]: "affine hull S = {} \<longleftrightarrow> S = {}"
   using affine_hull_nonempty by blast
 
 lemma affine_hull_2_alt:
   fixes a b :: "'a::real_vector"
   shows "affine hull {a,b} = range (\<lambda>u. a + u *\<^sub>R (b - a))"
 apply (simp add: affine_hull_2, safe)
 apply (rule_tac x=v in image_eqI)
 apply (simp add: algebra_simps)
 apply (metis scaleR_add_left scaleR_one, simp)
 apply (rule_tac x="1-u" in exI)
 apply (simp add: algebra_simps)
 done
 
 lemma interior_convex_hull_3_minimal:
   fixes a :: "'a::euclidean_space"
   shows "\<lbrakk>\<not> collinear{a,b,c}; DIM('a) = 2\<rbrakk>
          \<Longrightarrow> interior(convex hull {a,b,c}) =
                 {v. \<exists>x y z. 0 < x \<and> 0 < y \<and> 0 < z \<and> x + y + z = 1 \<and>
                             x *\<^sub>R a + y *\<^sub>R b + z *\<^sub>R c = v}"
 apply (simp add: collinear_3_eq_affine_dependent interior_convex_hull_explicit_minimal, safe)
 apply (rule_tac x="u a" in exI, simp)
 apply (rule_tac x="u b" in exI, simp)
 apply (rule_tac x="u c" in exI, simp)
 apply (rename_tac uu x y z)
 apply (rule_tac x="\<lambda>r. (if r=a then x else if r=b then y else if r=c then z else 0)" in exI)
 apply simp
 done
 
 
 subsection\<^marker>\<open>tag unimportant\<close>\<open>Basic lemmas about hyperplanes and halfspaces\<close>
 
 lemma halfspace_Int_eq:
      "{x. a \<bullet> x \<le> b} \<inter> {x. b \<le> a \<bullet> x} = {x. a \<bullet> x = b}"
      "{x. b \<le> a \<bullet> x} \<inter> {x. a \<bullet> x \<le> b} = {x. a \<bullet> x = b}"
   by auto
 
 lemma hyperplane_eq_Ex:
   assumes "a \<noteq> 0" obtains x where "a \<bullet> x = b"
   by (rule_tac x = "(b / (a \<bullet> a)) *\<^sub>R a" in that) (simp add: assms)
 
 lemma hyperplane_eq_empty:
      "{x. a \<bullet> x = b} = {} \<longleftrightarrow> a = 0 \<and> b \<noteq> 0"
   using hyperplane_eq_Ex apply auto[1]
   using inner_zero_right by blast
 
 lemma hyperplane_eq_UNIV:
    "{x. a \<bullet> x = b} = UNIV \<longleftrightarrow> a = 0 \<and> b = 0"
 proof -
   have "UNIV \<subseteq> {x. a \<bullet> x = b} \<Longrightarrow> a = 0 \<and> b = 0"
     apply (drule_tac c = "((b+1) / (a \<bullet> a)) *\<^sub>R a" in subsetD)
     apply simp_all
     by (metis add_cancel_right_right zero_neq_one)
   then show ?thesis by force
 qed
 
 lemma halfspace_eq_empty_lt:
    "{x. a \<bullet> x < b} = {} \<longleftrightarrow> a = 0 \<and> b \<le> 0"
 proof -
   have "{x. a \<bullet> x < b} \<subseteq> {} \<Longrightarrow> a = 0 \<and> b \<le> 0"
     apply (rule ccontr)
     apply (drule_tac c = "((b-1) / (a \<bullet> a)) *\<^sub>R a" in subsetD)
     apply force+
     done
   then show ?thesis by force
 qed
 
 lemma halfspace_eq_empty_gt:
    "{x. a \<bullet> x > b} = {} \<longleftrightarrow> a = 0 \<and> b \<ge> 0"
 using halfspace_eq_empty_lt [of "-a" "-b"]
 by simp
 
 lemma halfspace_eq_empty_le:
    "{x. a \<bullet> x \<le> b} = {} \<longleftrightarrow> a = 0 \<and> b < 0"
 proof -
   have "{x. a \<bullet> x \<le> b} \<subseteq> {} \<Longrightarrow> a = 0 \<and> b < 0"
     apply (rule ccontr)
     apply (drule_tac c = "((b-1) / (a \<bullet> a)) *\<^sub>R a" in subsetD)
     apply force+
     done
   then show ?thesis by force
 qed
 
 lemma halfspace_eq_empty_ge:
    "{x. a \<bullet> x \<ge> b} = {} \<longleftrightarrow> a = 0 \<and> b > 0"
 using halfspace_eq_empty_le [of "-a" "-b"]
 by simp
 
 subsection\<^marker>\<open>tag unimportant\<close>\<open>Use set distance for an easy proof of separation properties\<close>
 
 proposition\<^marker>\<open>tag unimportant\<close> separation_closures:
   fixes S :: "'a::euclidean_space set"
   assumes "S \<inter> closure T = {}" "T \<inter> closure S = {}"
   obtains U V where "U \<inter> V = {}" "open U" "open V" "S \<subseteq> U" "T \<subseteq> V"
 proof (cases "S = {} \<or> T = {}")
   case True with that show ?thesis by auto
 next
   case False
   define f where "f \<equiv> \<lambda>x. setdist {x} T - setdist {x} S"
   have contf: "continuous_on UNIV f"
     unfolding f_def by (intro continuous_intros continuous_on_setdist)
   show ?thesis
   proof (rule_tac U = "{x. f x > 0}" and V = "{x. f x < 0}" in that)
     show "{x. 0 < f x} \<inter> {x. f x < 0} = {}"
       by auto
     show "open {x. 0 < f x}"
       by (simp add: open_Collect_less contf)
     show "open {x. f x < 0}"
       by (simp add: open_Collect_less contf)
     show "S \<subseteq> {x. 0 < f x}"
       apply (clarsimp simp add: f_def setdist_sing_in_set)
       using assms
       by (metis False IntI empty_iff le_less setdist_eq_0_sing_2 setdist_pos_le setdist_sym)
     show "T \<subseteq> {x. f x < 0}"
       apply (clarsimp simp add: f_def setdist_sing_in_set)
       using assms
       by (metis False IntI empty_iff le_less setdist_eq_0_sing_2 setdist_pos_le setdist_sym)
   qed
 qed
 
 lemma separation_normal:
   fixes S :: "'a::euclidean_space set"
   assumes "closed S" "closed T" "S \<inter> T = {}"
   obtains U V where "open U" "open V" "S \<subseteq> U" "T \<subseteq> V" "U \<inter> V = {}"
 using separation_closures [of S T]
 by (metis assms closure_closed disjnt_def inf_commute)
 
 lemma separation_normal_local:
   fixes S :: "'a::euclidean_space set"
   assumes US: "closedin (top_of_set U) S"
       and UT: "closedin (top_of_set U) T"
       and "S \<inter> T = {}"
   obtains S' T' where "openin (top_of_set U) S'"
                       "openin (top_of_set U) T'"
                       "S \<subseteq> S'"  "T \<subseteq> T'"  "S' \<inter> T' = {}"
 proof (cases "S = {} \<or> T = {}")
   case True with that show ?thesis
     using UT US by (blast dest: closedin_subset)
 next
   case False
   define f where "f \<equiv> \<lambda>x. setdist {x} T - setdist {x} S"
   have contf: "continuous_on U f"
     unfolding f_def by (intro continuous_intros)
   show ?thesis
   proof (rule_tac S' = "(U \<inter> f -` {0<..})" and T' = "(U \<inter> f -` {..<0})" in that)
     show "(U \<inter> f -` {0<..}) \<inter> (U \<inter> f -` {..<0}) = {}"
       by auto
     show "openin (top_of_set U) (U \<inter> f -` {0<..})"
       by (rule continuous_openin_preimage [where T=UNIV]) (simp_all add: contf)
   next
     show "openin (top_of_set U) (U \<inter> f -` {..<0})"
       by (rule continuous_openin_preimage [where T=UNIV]) (simp_all add: contf)
   next
     have "S \<subseteq> U" "T \<subseteq> U"
       using closedin_imp_subset assms by blast+
     then show "S \<subseteq> U \<inter> f -` {0<..}" "T \<subseteq> U \<inter> f -` {..<0}"
       using assms False by (force simp add: f_def setdist_sing_in_set intro!: setdist_gt_0_closedin)+
   qed
 qed
 
 lemma separation_normal_compact:
   fixes S :: "'a::euclidean_space set"
   assumes "compact S" "closed T" "S \<inter> T = {}"
   obtains U V where "open U" "compact(closure U)" "open V" "S \<subseteq> U" "T \<subseteq> V" "U \<inter> V = {}"
 proof -
   have "closed S" "bounded S"
     using assms by (auto simp: compact_eq_bounded_closed)
   then obtain r where "r>0" and r: "S \<subseteq> ball 0 r"
     by (auto dest!: bounded_subset_ballD)
   have **: "closed (T \<union> - ball 0 r)" "S \<inter> (T \<union> - ball 0 r) = {}"
     using assms r by blast+
   then show ?thesis
     apply (rule separation_normal [OF \<open>closed S\<close>])
     apply (rule_tac U=U and V=V in that)
     by auto (meson bounded_ball bounded_subset compl_le_swap2 disjoint_eq_subset_Compl)
 qed
 
 subsection\<open>Connectedness of the intersection of a chain\<close>
 
 proposition connected_chain:
   fixes \<F> :: "'a :: euclidean_space set set"
   assumes cc: "\<And>S. S \<in> \<F> \<Longrightarrow> compact S \<and> connected S"
       and linear: "\<And>S T. S \<in> \<F> \<and> T \<in> \<F> \<Longrightarrow> S \<subseteq> T \<or> T \<subseteq> S"
   shows "connected(\<Inter>\<F>)"
 proof (cases "\<F> = {}")
   case True then show ?thesis
     by auto
 next
   case False
   then have cf: "compact(\<Inter>\<F>)"
     by (simp add: cc compact_Inter)
   have False if AB: "closed A" "closed B" "A \<inter> B = {}"
                 and ABeq: "A \<union> B = \<Inter>\<F>" and "A \<noteq> {}" "B \<noteq> {}" for A B
   proof -
     obtain U V where "open U" "open V" "A \<subseteq> U" "B \<subseteq> V" "U \<inter> V = {}"
       using separation_normal [OF AB] by metis
     obtain K where "K \<in> \<F>" "compact K"
       using cc False by blast
     then obtain N where "open N" and "K \<subseteq> N"
       by blast
     let ?\<C> = "insert (U \<union> V) ((\<lambda>S. N - S) ` \<F>)"
     obtain \<D> where "\<D> \<subseteq> ?\<C>" "finite \<D>" "K \<subseteq> \<Union>\<D>"
     proof (rule compactE [OF \<open>compact K\<close>])
       show "K \<subseteq> \<Union>(insert (U \<union> V) ((-) N ` \<F>))"
         using \<open>K \<subseteq> N\<close> ABeq \<open>A \<subseteq> U\<close> \<open>B \<subseteq> V\<close> by auto
       show "\<And>B. B \<in> insert (U \<union> V) ((-) N ` \<F>) \<Longrightarrow> open B"
         by (auto simp:  \<open>open U\<close> \<open>open V\<close> open_Un \<open>open N\<close> cc compact_imp_closed open_Diff)
     qed
     then have "finite(\<D> - {U \<union> V})"
       by blast
     moreover have "\<D> - {U \<union> V} \<subseteq> (\<lambda>S. N - S) ` \<F>"
       using \<open>\<D> \<subseteq> ?\<C>\<close> by blast
     ultimately obtain \<G> where "\<G> \<subseteq> \<F>" "finite \<G>" and Deq: "\<D> - {U \<union> V} = (\<lambda>S. N-S) ` \<G>"
       using finite_subset_image by metis
     obtain J where "J \<in> \<F>" and J: "(\<Union>S\<in>\<G>. N - S) \<subseteq> N - J"
     proof (cases "\<G> = {}")
       case True
       with \<open>\<F> \<noteq> {}\<close> that show ?thesis
         by auto
     next
       case False
       have "\<And>S T. \<lbrakk>S \<in> \<G>; T \<in> \<G>\<rbrakk> \<Longrightarrow> S \<subseteq> T \<or> T \<subseteq> S"
         by (meson \<open>\<G> \<subseteq> \<F>\<close> in_mono local.linear)
       with \<open>finite \<G>\<close> \<open>\<G> \<noteq> {}\<close>
       have "\<exists>J \<in> \<G>. (\<Union>S\<in>\<G>. N - S) \<subseteq> N - J"
       proof induction
         case (insert X \<H>)
         show ?case
         proof (cases "\<H> = {}")
           case True then show ?thesis by auto
         next
           case False
           then have "\<And>S T. \<lbrakk>S \<in> \<H>; T \<in> \<H>\<rbrakk> \<Longrightarrow> S \<subseteq> T \<or> T \<subseteq> S"
             by (simp add: insert.prems)
           with insert.IH False obtain J where "J \<in> \<H>" and J: "(\<Union>Y\<in>\<H>. N - Y) \<subseteq> N - J"
             by metis
           have "N - J \<subseteq> N - X \<or> N - X \<subseteq> N - J"
             by (meson Diff_mono \<open>J \<in> \<H>\<close> insert.prems(2) insert_iff order_refl)
           then show ?thesis
           proof
             assume "N - J \<subseteq> N - X" with J show ?thesis
               by auto
           next
             assume "N - X \<subseteq> N - J"
             with J have "N - X \<union> \<Union> ((-) N ` \<H>) \<subseteq> N - J"
               by auto
             with \<open>J \<in> \<H>\<close> show ?thesis
               by blast
           qed
         qed
       qed simp
       with \<open>\<G> \<subseteq> \<F>\<close> show ?thesis by (blast intro: that)
     qed
     have "K \<subseteq> \<Union>(insert (U \<union> V) (\<D> - {U \<union> V}))"
       using \<open>K \<subseteq> \<Union>\<D>\<close> by auto
     also have "... \<subseteq> (U \<union> V) \<union> (N - J)"
       by (metis (no_types, hide_lams) Deq Un_subset_iff Un_upper2 J Union_insert order_trans sup_ge1)
     finally have "J \<inter> K \<subseteq> U \<union> V"
       by blast
     moreover have "connected(J \<inter> K)"
       by (metis Int_absorb1 \<open>J \<in> \<F>\<close> \<open>K \<in> \<F>\<close> cc inf.orderE local.linear)
     moreover have "U \<inter> (J \<inter> K) \<noteq> {}"
       using ABeq \<open>J \<in> \<F>\<close> \<open>K \<in> \<F>\<close> \<open>A \<noteq> {}\<close> \<open>A \<subseteq> U\<close> by blast
     moreover have "V \<inter> (J \<inter> K) \<noteq> {}"
       using ABeq \<open>J \<in> \<F>\<close> \<open>K \<in> \<F>\<close> \<open>B \<noteq> {}\<close> \<open>B \<subseteq> V\<close> by blast
     ultimately show False
         using connectedD [of "J \<inter> K" U V] \<open>open U\<close> \<open>open V\<close> \<open>U \<inter> V = {}\<close>  by auto
   qed
   with cf show ?thesis
     by (auto simp: connected_closed_set compact_imp_closed)
 qed
 
 lemma connected_chain_gen:
   fixes \<F> :: "'a :: euclidean_space set set"
   assumes X: "X \<in> \<F>" "compact X"
       and cc: "\<And>T. T \<in> \<F> \<Longrightarrow> closed T \<and> connected T"
       and linear: "\<And>S T. S \<in> \<F> \<and> T \<in> \<F> \<Longrightarrow> S \<subseteq> T \<or> T \<subseteq> S"
   shows "connected(\<Inter>\<F>)"
 proof -
   have "\<Inter>\<F> = (\<Inter>T\<in>\<F>. X \<inter> T)"
     using X by blast
   moreover have "connected (\<Inter>T\<in>\<F>. X \<inter> T)"
   proof (rule connected_chain)
     show "\<And>T. T \<in> (\<inter>) X ` \<F> \<Longrightarrow> compact T \<and> connected T"
       using cc X by auto (metis inf.absorb2 inf.orderE local.linear)
     show "\<And>S T. S \<in> (\<inter>) X ` \<F> \<and> T \<in> (\<inter>) X ` \<F> \<Longrightarrow> S \<subseteq> T \<or> T \<subseteq> S"
       using local.linear by blast
   qed
   ultimately show ?thesis
     by metis
 qed
 
 lemma connected_nest:
   fixes S :: "'a::linorder \<Rightarrow> 'b::euclidean_space set"
   assumes S: "\<And>n. compact(S n)" "\<And>n. connected(S n)"
     and nest: "\<And>m n. m \<le> n \<Longrightarrow> S n \<subseteq> S m"
   shows "connected(\<Inter> (range S))"
   apply (rule connected_chain)
   using S apply blast
   by (metis image_iff le_cases nest)
 
 lemma connected_nest_gen:
   fixes S :: "'a::linorder \<Rightarrow> 'b::euclidean_space set"
   assumes S: "\<And>n. closed(S n)" "\<And>n. connected(S n)" "compact(S k)"
     and nest: "\<And>m n. m \<le> n \<Longrightarrow> S n \<subseteq> S m"
   shows "connected(\<Inter> (range S))"
   apply (rule connected_chain_gen [of "S k"])
   using S apply auto
   by (meson le_cases nest subsetCE)
 
 subsection\<open>Proper maps, including projections out of compact sets\<close>
 
 lemma finite_indexed_bound:
   assumes A: "finite A" "\<And>x. x \<in> A \<Longrightarrow> \<exists>n::'a::linorder. P x n"
     shows "\<exists>m. \<forall>x \<in> A. \<exists>k\<le>m. P x k"
 using A
 proof (induction A)
   case empty then show ?case by force
 next
   case (insert a A)
     then obtain m n where "\<forall>x \<in> A. \<exists>k\<le>m. P x k" "P a n"
       by force
     then show ?case
       apply (rule_tac x="max m n" in exI, safe)
       using max.cobounded2 apply blast
       by (meson le_max_iff_disj)
 qed
 
 proposition proper_map:
   fixes f :: "'a::heine_borel \<Rightarrow> 'b::heine_borel"
   assumes "closedin (top_of_set S) K"
       and com: "\<And>U. \<lbrakk>U \<subseteq> T; compact U\<rbrakk> \<Longrightarrow> compact (S \<inter> f -` U)"
       and "f ` S \<subseteq> T"
     shows "closedin (top_of_set T) (f ` K)"
 proof -
   have "K \<subseteq> S"
     using assms closedin_imp_subset by metis
   obtain C where "closed C" and Keq: "K = S \<inter> C"
     using assms by (auto simp: closedin_closed)
   have *: "y \<in> f ` K" if "y \<in> T" and y: "y islimpt f ` K" for y
   proof -
     obtain h where "\<forall>n. (\<exists>x\<in>K. h n = f x) \<and> h n \<noteq> y" "inj h" and hlim: "(h \<longlongrightarrow> y) sequentially"
       using \<open>y \<in> T\<close> y by (force simp: limpt_sequential_inj)
     then obtain X where X: "\<And>n. X n \<in> K \<and> h n = f (X n) \<and> h n \<noteq> y"
       by metis
     then have fX: "\<And>n. f (X n) = h n"
       by metis
     have "compact (C \<inter> (S \<inter> f -` insert y (range (\<lambda>i. f(X(n + i))))))" for n
       apply (rule closed_Int_compact [OF \<open>closed C\<close>])
       apply (rule com)
        using X \<open>K \<subseteq> S\<close> \<open>f ` S \<subseteq> T\<close> \<open>y \<in> T\<close> apply blast
       apply (rule compact_sequence_with_limit)
       apply (simp add: fX add.commute [of n] LIMSEQ_ignore_initial_segment [OF hlim])
       done
     then have comf: "compact {a \<in> K. f a \<in> insert y (range (\<lambda>i. f(X(n + i))))}" for n
       by (simp add: Keq Int_def conj_commute)
     have ne: "\<Inter>\<F> \<noteq> {}"
              if "finite \<F>"
                 and \<F>: "\<And>t. t \<in> \<F> \<Longrightarrow>
                            (\<exists>n. t = {a \<in> K. f a \<in> insert y (range (\<lambda>i. f (X (n + i))))})"
              for \<F>
     proof -
       obtain m where m: "\<And>t. t \<in> \<F> \<Longrightarrow> \<exists>k\<le>m. t = {a \<in> K. f a \<in> insert y (range (\<lambda>i. f (X (k + i))))}"
         apply (rule exE)
         apply (rule finite_indexed_bound [OF \<open>finite \<F>\<close> \<F>], assumption, force)
         done
       have "X m \<in> \<Inter>\<F>"
         using X le_Suc_ex by (fastforce dest: m)
       then show ?thesis by blast
     qed
     have "\<Inter>{{a. a \<in> K \<and> f a \<in> insert y (range (\<lambda>i. f(X(n + i))))} |n. n \<in> UNIV}
                \<noteq> {}"
       apply (rule compact_fip_Heine_Borel)
        using comf apply force
       using ne  apply (simp add: subset_iff del: insert_iff)
       done
     then have "\<exists>x. x \<in> (\<Inter>n. {a \<in> K. f a \<in> insert y (range (\<lambda>i. f (X (n + i))))})"
       by blast
     then show ?thesis
       apply (simp add: image_iff fX)
       by (metis \<open>inj h\<close> le_add1 not_less_eq_eq rangeI range_ex1_eq)
   qed
   with assms closedin_subset show ?thesis
     by (force simp: closedin_limpt)
 qed
 
 
 lemma compact_continuous_image_eq:
   fixes f :: "'a::heine_borel \<Rightarrow> 'b::heine_borel"
   assumes f: "inj_on f S"
   shows "continuous_on S f \<longleftrightarrow> (\<forall>T. compact T \<and> T \<subseteq> S \<longrightarrow> compact(f ` T))"
            (is "?lhs = ?rhs")
 proof
   assume ?lhs then show ?rhs
     by (metis continuous_on_subset compact_continuous_image)
 next
   assume RHS: ?rhs
   obtain g where gf: "\<And>x. x \<in> S \<Longrightarrow> g (f x) = x"
     by (metis inv_into_f_f f)
   then have *: "(S \<inter> f -` U) = g ` U" if "U \<subseteq> f ` S" for U
     using that by fastforce
   have gfim: "g ` f ` S \<subseteq> S" using gf by auto
   have **: "compact (f ` S \<inter> g -` C)" if C: "C \<subseteq> S" "compact C" for C
   proof -
     obtain h where "h C \<in> C \<and> h C \<notin> S \<or> compact (f ` C)"
       by (force simp: C RHS)
     moreover have "f ` C = (f ` S \<inter> g -` C)"
       using C gf by auto
     ultimately show ?thesis
       using C by auto
   qed
   show ?lhs
     using proper_map [OF _ _ gfim] **
     by (simp add: continuous_on_closed * closedin_imp_subset)
 qed
 
 subsection\<^marker>\<open>tag unimportant\<close>\<open>Trivial fact: convexity equals connectedness for collinear sets\<close>
 
 lemma convex_connected_collinear:
   fixes S :: "'a::euclidean_space set"
   assumes "collinear S"
     shows "convex S \<longleftrightarrow> connected S"
 proof
   assume "convex S"
   then show "connected S"
     using convex_connected by blast
 next
   assume S: "connected S"
   show "convex S"
   proof (cases "S = {}")
     case True
     then show ?thesis by simp
   next
     case False
     then obtain a where "a \<in> S" by auto
     have "collinear (affine hull S)"
       by (simp add: assms collinear_affine_hull_collinear)
     then obtain z where "z \<noteq> 0" "\<And>x. x \<in> affine hull S \<Longrightarrow> \<exists>c. x - a = c *\<^sub>R z"
       by (meson \<open>a \<in> S\<close> collinear hull_inc)
     then obtain f where f: "\<And>x. x \<in> affine hull S \<Longrightarrow> x - a = f x *\<^sub>R z"
       by metis
     then have inj_f: "inj_on f (affine hull S)"
       by (metis diff_add_cancel inj_onI)
     have diff: "x - y = (f x - f y) *\<^sub>R z" if x: "x \<in> affine hull S" and y: "y \<in> affine hull S" for x y
     proof -
       have "f x *\<^sub>R z = x - a"
         by (simp add: f hull_inc x)
       moreover have "f y *\<^sub>R z = y - a"
         by (simp add: f hull_inc y)
       ultimately show ?thesis
         by (simp add: scaleR_left.diff)
     qed
     have cont_f: "continuous_on (affine hull S) f"
       apply (clarsimp simp: dist_norm continuous_on_iff diff)
       by (metis \<open>z \<noteq> 0\<close> mult.commute mult_less_cancel_left_pos norm_minus_commute real_norm_def zero_less_mult_iff zero_less_norm_iff)
     then have conn_fS: "connected (f ` S)"
       by (meson S connected_continuous_image continuous_on_subset hull_subset)
     show ?thesis
     proof (clarsimp simp: convex_contains_segment)
       fix x y z
       assume "x \<in> S" "y \<in> S" "z \<in> closed_segment x y"
       have False if "z \<notin> S"
       proof -
         have "f ` (closed_segment x y) = closed_segment (f x) (f y)"
           apply (rule continuous_injective_image_segment_1)
           apply (meson \<open>x \<in> S\<close> \<open>y \<in> S\<close> convex_affine_hull convex_contains_segment hull_inc continuous_on_subset [OF cont_f])
           by (meson \<open>x \<in> S\<close> \<open>y \<in> S\<close> convex_affine_hull convex_contains_segment hull_inc inj_on_subset [OF inj_f])
         then have fz: "f z \<in> closed_segment (f x) (f y)"
           using \<open>z \<in> closed_segment x y\<close> by blast
         have "z \<in> affine hull S"
           by (meson \<open>x \<in> S\<close> \<open>y \<in> S\<close> \<open>z \<in> closed_segment x y\<close> convex_affine_hull convex_contains_segment hull_inc subset_eq)
         then have fz_notin: "f z \<notin> f ` S"
           using hull_subset inj_f inj_onD that by fastforce
         moreover have "{..<f z} \<inter> f ` S \<noteq> {}" "{f z<..} \<inter> f ` S \<noteq> {}"
         proof -
           have "{..<f z} \<inter> f ` {x,y} \<noteq> {}"  "{f z<..} \<inter> f ` {x,y} \<noteq> {}"
             using fz fz_notin \<open>x \<in> S\<close> \<open>y \<in> S\<close>
              apply (auto simp: closed_segment_eq_real_ivl split: if_split_asm)
              apply (metis image_eqI less_eq_real_def)+
             done
           then show "{..<f z} \<inter> f ` S \<noteq> {}" "{f z<..} \<inter> f ` S \<noteq> {}"
             using \<open>x \<in> S\<close> \<open>y \<in> S\<close> by blast+
         qed
         ultimately show False
           using connectedD [OF conn_fS, of "{..<f z}" "{f z<..}"] by force
       qed
       then show "z \<in> S" by meson
     qed
   qed
 qed
 
 lemma compact_convex_collinear_segment_alt:
   fixes S :: "'a::euclidean_space set"
   assumes "S \<noteq> {}" "compact S" "connected S" "collinear S"
   obtains a b where "S = closed_segment a b"
 proof -
   obtain \<xi> where "\<xi> \<in> S" using \<open>S \<noteq> {}\<close> by auto
   have "collinear (affine hull S)"
     by (simp add: assms collinear_affine_hull_collinear)
   then obtain z where "z \<noteq> 0" "\<And>x. x \<in> affine hull S \<Longrightarrow> \<exists>c. x - \<xi> = c *\<^sub>R z"
     by (meson \<open>\<xi> \<in> S\<close> collinear hull_inc)
   then obtain f where f: "\<And>x. x \<in> affine hull S \<Longrightarrow> x - \<xi> = f x *\<^sub>R z"
     by metis
   let ?g = "\<lambda>r. r *\<^sub>R z + \<xi>"
   have gf: "?g (f x) = x" if "x \<in> affine hull S" for x
     by (metis diff_add_cancel f that)
   then have inj_f: "inj_on f (affine hull S)"
     by (metis inj_onI)
   have diff: "x - y = (f x - f y) *\<^sub>R z" if x: "x \<in> affine hull S" and y: "y \<in> affine hull S" for x y
   proof -
     have "f x *\<^sub>R z = x - \<xi>"
       by (simp add: f hull_inc x)
     moreover have "f y *\<^sub>R z = y - \<xi>"
       by (simp add: f hull_inc y)
     ultimately show ?thesis
       by (simp add: scaleR_left.diff)
   qed
   have cont_f: "continuous_on (affine hull S) f"
     apply (clarsimp simp: dist_norm continuous_on_iff diff)
     by (metis \<open>z \<noteq> 0\<close> mult.commute mult_less_cancel_left_pos norm_minus_commute real_norm_def zero_less_mult_iff zero_less_norm_iff)
   then have "connected (f ` S)"
     by (meson \<open>connected S\<close> connected_continuous_image continuous_on_subset hull_subset)
   moreover have "compact (f ` S)"
     by (meson \<open>compact S\<close> compact_continuous_image_eq cont_f hull_subset inj_f)
   ultimately obtain x y where "f ` S = {x..y}"
     by (meson connected_compact_interval_1)
   then have fS_eq: "f ` S = closed_segment x y"
     using \<open>S \<noteq> {}\<close> closed_segment_eq_real_ivl by auto
   obtain a b where "a \<in> S" "f a = x" "b \<in> S" "f b = y"
     by (metis (full_types) ends_in_segment fS_eq imageE)
   have "f ` (closed_segment a b) = closed_segment (f a) (f b)"
     apply (rule continuous_injective_image_segment_1)
      apply (meson \<open>a \<in> S\<close> \<open>b \<in> S\<close> convex_affine_hull convex_contains_segment hull_inc continuous_on_subset [OF cont_f])
     by (meson \<open>a \<in> S\<close> \<open>b \<in> S\<close> convex_affine_hull convex_contains_segment hull_inc inj_on_subset [OF inj_f])
   then have "f ` (closed_segment a b) = f ` S"
     by (simp add: \<open>f a = x\<close> \<open>f b = y\<close> fS_eq)
   then have "?g ` f ` (closed_segment a b) = ?g ` f ` S"
     by simp
   moreover have "(\<lambda>x. f x *\<^sub>R z + \<xi>) ` closed_segment a b = closed_segment a b"
     apply safe
      apply (metis (mono_tags, hide_lams) \<open>a \<in> S\<close> \<open>b \<in> S\<close> convex_affine_hull convex_contains_segment gf hull_inc subsetCE)
     by (metis (mono_tags, lifting) \<open>a \<in> S\<close> \<open>b \<in> S\<close> convex_affine_hull convex_contains_segment gf hull_subset image_iff subsetCE)
   ultimately have "closed_segment a b = S"
     using gf by (simp add: image_comp o_def hull_inc cong: image_cong)
   then show ?thesis
     using that by blast
 qed
 
 lemma compact_convex_collinear_segment:
   fixes S :: "'a::euclidean_space set"
   assumes "S \<noteq> {}" "compact S" "convex S" "collinear S"
   obtains a b where "S = closed_segment a b"
   using assms convex_connected_collinear compact_convex_collinear_segment_alt by blast
 
 
 lemma proper_map_from_compact:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes contf: "continuous_on S f" and imf: "f ` S \<subseteq> T" and "compact S"
           "closedin (top_of_set T) K"
   shows "compact (S \<inter> f -` K)"
 by (rule closedin_compact [OF \<open>compact S\<close>] continuous_closedin_preimage_gen assms)+
 
 lemma proper_map_fst:
   assumes "compact T" "K \<subseteq> S" "compact K"
     shows "compact (S \<times> T \<inter> fst -` K)"
 proof -
   have "(S \<times> T \<inter> fst -` K) = K \<times> T"
     using assms by auto
   then show ?thesis by (simp add: assms compact_Times)
 qed
 
 lemma closed_map_fst:
   fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set"
   assumes "compact T" "closedin (top_of_set (S \<times> T)) c"
    shows "closedin (top_of_set S) (fst ` c)"
 proof -
   have *: "fst ` (S \<times> T) \<subseteq> S"
     by auto
   show ?thesis
     using proper_map [OF _ _ *] by (simp add: proper_map_fst assms)
 qed
 
 lemma proper_map_snd:
   assumes "compact S" "K \<subseteq> T" "compact K"
     shows "compact (S \<times> T \<inter> snd -` K)"
 proof -
   have "(S \<times> T \<inter> snd -` K) = S \<times> K"
     using assms by auto
   then show ?thesis by (simp add: assms compact_Times)
 qed
 
 lemma closed_map_snd:
   fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set"
   assumes "compact S" "closedin (top_of_set (S \<times> T)) c"
    shows "closedin (top_of_set T) (snd ` c)"
 proof -
   have *: "snd ` (S \<times> T) \<subseteq> T"
     by auto
   show ?thesis
     using proper_map [OF _ _ *] by (simp add: proper_map_snd assms)
 qed
 
 lemma closedin_compact_projection:
   fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set"
   assumes "compact S" and clo: "closedin (top_of_set (S \<times> T)) U"
     shows "closedin (top_of_set T) {y. \<exists>x. x \<in> S \<and> (x, y) \<in> U}"
 proof -
   have "U \<subseteq> S \<times> T"
     by (metis clo closedin_imp_subset)
   then have "{y. \<exists>x. x \<in> S \<and> (x, y) \<in> U} = snd ` U"
     by force
   moreover have "closedin (top_of_set T) (snd ` U)"
     by (rule closed_map_snd [OF assms])
   ultimately show ?thesis
     by simp
 qed
 
 
 lemma closed_compact_projection:
   fixes S :: "'a::euclidean_space set"
     and T :: "('a * 'b::euclidean_space) set"
   assumes "compact S" and clo: "closed T"
     shows "closed {y. \<exists>x. x \<in> S \<and> (x, y) \<in> T}"
 proof -
   have *: "{y. \<exists>x. x \<in> S \<and> Pair x y \<in> T} =
         {y. \<exists>x. x \<in> S \<and> Pair x y \<in> ((S \<times> UNIV) \<inter> T)}"
     by auto
   show ?thesis
     apply (subst *)
     apply (rule closedin_closed_trans [OF _ closed_UNIV])
     apply (rule closedin_compact_projection [OF \<open>compact S\<close>])
     by (simp add: clo closedin_closed_Int)
 qed
 
 subsubsection\<^marker>\<open>tag unimportant\<close>\<open>Representing affine hull as a finite intersection of hyperplanes\<close>
 
 proposition\<^marker>\<open>tag unimportant\<close> affine_hull_convex_Int_nonempty_interior:
   fixes S :: "'a::real_normed_vector set"
   assumes "convex S" "S \<inter> interior T \<noteq> {}"
     shows "affine hull (S \<inter> T) = affine hull S"
 proof
   show "affine hull (S \<inter> T) \<subseteq> affine hull S"
     by (simp add: hull_mono)
 next
   obtain a where "a \<in> S" "a \<in> T" and at: "a \<in> interior T"
     using assms interior_subset by blast
   then obtain e where "e > 0" and e: "cball a e \<subseteq> T"
     using mem_interior_cball by blast
   have *: "x \<in> (+) a ` span ((\<lambda>x. x - a) ` (S \<inter> T))" if "x \<in> S" for x
   proof (cases "x = a")
     case True with that span_0 eq_add_iff image_def mem_Collect_eq show ?thesis
       by blast
   next
     case False
     define k where "k = min (1/2) (e / norm (x-a))"
     have k: "0 < k" "k < 1"
       using \<open>e > 0\<close> False by (auto simp: k_def)
     then have xa: "(x-a) = inverse k *\<^sub>R k *\<^sub>R (x-a)"
       by simp
     have "e / norm (x - a) \<ge> k"
       using k_def by linarith
     then have "a + k *\<^sub>R (x - a) \<in> cball a e"
       using \<open>0 < k\<close> False
       by (simp add: dist_norm) (simp add: field_simps)
     then have T: "a + k *\<^sub>R (x - a) \<in> T"
       using e by blast
     have S: "a + k *\<^sub>R (x - a) \<in> S"
       using k \<open>a \<in> S\<close> convexD [OF \<open>convex S\<close> \<open>a \<in> S\<close> \<open>x \<in> S\<close>, of "1-k" k]
       by (simp add: algebra_simps)
     have "inverse k *\<^sub>R k *\<^sub>R (x-a) \<in> span ((\<lambda>x. x - a) ` (S \<inter> T))"
       apply (rule span_mul)
       apply (rule span_base)
       apply (rule image_eqI [where x = "a + k *\<^sub>R (x - a)"])
       apply (auto simp: S T)
       done
     with xa image_iff show ?thesis  by fastforce
   qed
   show "affine hull S \<subseteq> affine hull (S \<inter> T)"
     apply (simp add: subset_hull)
     apply (simp add: \<open>a \<in> S\<close> \<open>a \<in> T\<close> hull_inc affine_hull_span_gen [of a])
     apply (force simp: *)
     done
 qed
 
 corollary affine_hull_convex_Int_open:
   fixes S :: "'a::real_normed_vector set"
   assumes "convex S" "open T" "S \<inter> T \<noteq> {}"
     shows "affine hull (S \<inter> T) = affine hull S"
 using affine_hull_convex_Int_nonempty_interior assms interior_eq by blast
 
 corollary affine_hull_affine_Int_nonempty_interior:
   fixes S :: "'a::real_normed_vector set"
   assumes "affine S" "S \<inter> interior T \<noteq> {}"
     shows "affine hull (S \<inter> T) = affine hull S"
 by (simp add: affine_hull_convex_Int_nonempty_interior affine_imp_convex assms)
 
 corollary affine_hull_affine_Int_open:
   fixes S :: "'a::real_normed_vector set"
   assumes "affine S" "open T" "S \<inter> T \<noteq> {}"
     shows "affine hull (S \<inter> T) = affine hull S"
 by (simp add: affine_hull_convex_Int_open affine_imp_convex assms)
 
 corollary affine_hull_convex_Int_openin:
   fixes S :: "'a::real_normed_vector set"
   assumes "convex S" "openin (top_of_set (affine hull S)) T" "S \<inter> T \<noteq> {}"
     shows "affine hull (S \<inter> T) = affine hull S"
 using assms unfolding openin_open
 by (metis affine_hull_convex_Int_open hull_subset inf.orderE inf_assoc)
 
 corollary affine_hull_openin:
   fixes S :: "'a::real_normed_vector set"
   assumes "openin (top_of_set (affine hull T)) S" "S \<noteq> {}"
     shows "affine hull S = affine hull T"
 using assms unfolding openin_open
 by (metis affine_affine_hull affine_hull_affine_Int_open hull_hull)
 
 corollary affine_hull_open:
   fixes S :: "'a::real_normed_vector set"
   assumes "open S" "S \<noteq> {}"
     shows "affine hull S = UNIV"
 by (metis affine_hull_convex_Int_nonempty_interior assms convex_UNIV hull_UNIV inf_top.left_neutral interior_open)
 
 lemma aff_dim_convex_Int_nonempty_interior:
   fixes S :: "'a::euclidean_space set"
   shows "\<lbrakk>convex S; S \<inter> interior T \<noteq> {}\<rbrakk> \<Longrightarrow> aff_dim(S \<inter> T) = aff_dim S"
 using aff_dim_affine_hull2 affine_hull_convex_Int_nonempty_interior by blast
 
 lemma aff_dim_convex_Int_open:
   fixes S :: "'a::euclidean_space set"
   shows "\<lbrakk>convex S; open T; S \<inter> T \<noteq> {}\<rbrakk> \<Longrightarrow>  aff_dim(S \<inter> T) = aff_dim S"
 using aff_dim_convex_Int_nonempty_interior interior_eq by blast
 
 lemma affine_hull_Diff:
   fixes S:: "'a::real_normed_vector set"
   assumes ope: "openin (top_of_set (affine hull S)) S" and "finite F" "F \<subset> S"
     shows "affine hull (S - F) = affine hull S"
 proof -
   have clo: "closedin (top_of_set S) F"
     using assms finite_imp_closedin by auto
   moreover have "S - F \<noteq> {}"
     using assms by auto
   ultimately show ?thesis
     by (metis ope closedin_def topspace_euclidean_subtopology affine_hull_openin openin_trans)
 qed
 
 lemma affine_hull_halfspace_lt:
   fixes a :: "'a::euclidean_space"
   shows "affine hull {x. a \<bullet> x < r} = (if a = 0 \<and> r \<le> 0 then {} else UNIV)"
 using halfspace_eq_empty_lt [of a r]
 by (simp add: open_halfspace_lt affine_hull_open)
 
 lemma affine_hull_halfspace_le:
   fixes a :: "'a::euclidean_space"
   shows "affine hull {x. a \<bullet> x \<le> r} = (if a = 0 \<and> r < 0 then {} else UNIV)"
 proof (cases "a = 0")
   case True then show ?thesis by simp
 next
   case False
   then have "affine hull closure {x. a \<bullet> x < r} = UNIV"
     using affine_hull_halfspace_lt closure_same_affine_hull by fastforce
   moreover have "{x. a \<bullet> x < r} \<subseteq> {x. a \<bullet> x \<le> r}"
     by (simp add: Collect_mono)
   ultimately show ?thesis using False antisym_conv hull_mono top_greatest
     by (metis affine_hull_halfspace_lt)
 qed
 
 lemma affine_hull_halfspace_gt:
   fixes a :: "'a::euclidean_space"
   shows "affine hull {x. a \<bullet> x > r} = (if a = 0 \<and> r \<ge> 0 then {} else UNIV)"
 using halfspace_eq_empty_gt [of r a]
 by (simp add: open_halfspace_gt affine_hull_open)
 
 lemma affine_hull_halfspace_ge:
   fixes a :: "'a::euclidean_space"
   shows "affine hull {x. a \<bullet> x \<ge> r} = (if a = 0 \<and> r > 0 then {} else UNIV)"
 using affine_hull_halfspace_le [of "-a" "-r"] by simp
 
 lemma aff_dim_halfspace_lt:
   fixes a :: "'a::euclidean_space"
   shows "aff_dim {x. a \<bullet> x < r} =
         (if a = 0 \<and> r \<le> 0 then -1 else DIM('a))"
 by simp (metis aff_dim_open halfspace_eq_empty_lt open_halfspace_lt)
 
 lemma aff_dim_halfspace_le:
   fixes a :: "'a::euclidean_space"
   shows "aff_dim {x. a \<bullet> x \<le> r} =
         (if a = 0 \<and> r < 0 then -1 else DIM('a))"
 proof -
   have "int (DIM('a)) = aff_dim (UNIV::'a set)"
     by (simp)
   then have "aff_dim (affine hull {x. a \<bullet> x \<le> r}) = DIM('a)" if "(a = 0 \<longrightarrow> r \<ge> 0)"
     using that by (simp add: affine_hull_halfspace_le not_less)
   then show ?thesis
     by (force)
 qed
 
 lemma aff_dim_halfspace_gt:
   fixes a :: "'a::euclidean_space"
   shows "aff_dim {x. a \<bullet> x > r} =
         (if a = 0 \<and> r \<ge> 0 then -1 else DIM('a))"
 by simp (metis aff_dim_open halfspace_eq_empty_gt open_halfspace_gt)
 
 lemma aff_dim_halfspace_ge:
   fixes a :: "'a::euclidean_space"
   shows "aff_dim {x. a \<bullet> x \<ge> r} =
         (if a = 0 \<and> r > 0 then -1 else DIM('a))"
 using aff_dim_halfspace_le [of "-a" "-r"] by simp
 
 proposition aff_dim_eq_hyperplane:
   fixes S :: "'a::euclidean_space set"
   shows "aff_dim S = DIM('a) - 1 \<longleftrightarrow> (\<exists>a b. a \<noteq> 0 \<and> affine hull S = {x. a \<bullet> x = b})"
 proof (cases "S = {}")
   case True then show ?thesis
     by (auto simp: dest: hyperplane_eq_Ex)
 next
   case False
   then obtain c where "c \<in> S" by blast
   show ?thesis
   proof (cases "c = 0")
     case True show ?thesis
       using span_zero [of S]
         apply (simp add: aff_dim_eq_dim [of c] affine_hull_span_gen [of c] \<open>c \<in> S\<close> hull_inc dim_eq_hyperplane
           del: One_nat_def)
         apply (auto simp add: \<open>c = 0\<close>)
         done
   next
     case False
     have xc_im: "x \<in> (+) c ` {y. a \<bullet> y = 0}" if "a \<bullet> x = a \<bullet> c" for a x
     proof -
       have "\<exists>y. a \<bullet> y = 0 \<and> c + y = x"
         by (metis that add.commute diff_add_cancel inner_commute inner_diff_left right_minus_eq)
       then show "x \<in> (+) c ` {y. a \<bullet> y = 0}"
         by blast
     qed
     have 2: "span ((\<lambda>x. x - c) ` S) = {x. a \<bullet> x = 0}"
          if "(+) c ` span ((\<lambda>x. x - c) ` S) = {x. a \<bullet> x = b}" for a b
     proof -
       have "b = a \<bullet> c"
         using span_0 that by fastforce
       with that have "(+) c ` span ((\<lambda>x. x - c) ` S) = {x. a \<bullet> x = a \<bullet> c}"
         by simp
       then have "span ((\<lambda>x. x - c) ` S) = (\<lambda>x. x - c) ` {x. a \<bullet> x = a \<bullet> c}"
         by (metis (no_types) image_cong translation_galois uminus_add_conv_diff)
       also have "... = {x. a \<bullet> x = 0}"
         by (force simp: inner_distrib inner_diff_right
              intro: image_eqI [where x="x+c" for x])
       finally show ?thesis .
     qed
     show ?thesis
       apply (simp add: aff_dim_eq_dim [of c] affine_hull_span_gen [of c] \<open>c \<in> S\<close> hull_inc dim_eq_hyperplane
                   del: One_nat_def cong: image_cong_simp, safe)
       apply (fastforce simp add: inner_distrib intro: xc_im)
       apply (force simp: intro!: 2)
       done
   qed
 qed
 
 corollary aff_dim_hyperplane [simp]:
   fixes a :: "'a::euclidean_space"
   shows "a \<noteq> 0 \<Longrightarrow> aff_dim {x. a \<bullet> x = r} = DIM('a) - 1"
 by (metis aff_dim_eq_hyperplane affine_hull_eq affine_hyperplane)
 
 subsection\<^marker>\<open>tag unimportant\<close>\<open>Some stepping theorems\<close>
 
 lemma aff_dim_insert:
   fixes a :: "'a::euclidean_space"
   shows "aff_dim (insert a S) = (if a \<in> affine hull S then aff_dim S else aff_dim S + 1)"
 proof (cases "S = {}")
   case True then show ?thesis
     by simp
 next
   case False
   then obtain x s' where S: "S = insert x s'" "x \<notin> s'"
     by (meson Set.set_insert all_not_in_conv)
   show ?thesis using S
     apply (simp add: hull_redundant cong: aff_dim_affine_hull2)
     apply (simp add: affine_hull_insert_span_gen hull_inc)
     by (force simp add: span_zero insert_commute [of a] hull_inc aff_dim_eq_dim [of x] dim_insert
       cong: image_cong_simp)
 qed
 
 lemma affine_dependent_choose:
   fixes a :: "'a :: euclidean_space"
   assumes "\<not>(affine_dependent S)"
   shows "affine_dependent(insert a S) \<longleftrightarrow> a \<notin> S \<and> a \<in> affine hull S"
         (is "?lhs = ?rhs")
 proof safe
   assume "affine_dependent (insert a S)" and "a \<in> S"
   then show "False"
     using \<open>a \<in> S\<close> assms insert_absorb by fastforce
 next
   assume lhs: "affine_dependent (insert a S)"
   then have "a \<notin> S"
     by (metis (no_types) assms insert_absorb)
   moreover have "finite S"
     using affine_independent_iff_card assms by blast
   moreover have "aff_dim (insert a S) \<noteq> int (card S)"
     using \<open>finite S\<close> affine_independent_iff_card \<open>a \<notin> S\<close> lhs by fastforce
   ultimately show "a \<in> affine hull S"
     by (metis aff_dim_affine_independent aff_dim_insert assms)
 next
   assume "a \<notin> S" and "a \<in> affine hull S"
   show "affine_dependent (insert a S)"
     by (simp add: \<open>a \<in> affine hull S\<close> \<open>a \<notin> S\<close> affine_dependent_def)
 qed
 
 lemma affine_independent_insert:
   fixes a :: "'a :: euclidean_space"
   shows "\<lbrakk>\<not> affine_dependent S; a \<notin> affine hull S\<rbrakk> \<Longrightarrow> \<not> affine_dependent(insert a S)"
   by (simp add: affine_dependent_choose)
 
 lemma subspace_bounded_eq_trivial:
   fixes S :: "'a::real_normed_vector set"
   assumes "subspace S"
     shows "bounded S \<longleftrightarrow> S = {0}"
 proof -
   have "False" if "bounded S" "x \<in> S" "x \<noteq> 0" for x
   proof -
     obtain B where B: "\<And>y. y \<in> S \<Longrightarrow> norm y < B" "B > 0"
       using \<open>bounded S\<close> by (force simp: bounded_pos_less)
     have "(B / norm x) *\<^sub>R x \<in> S"
       using assms subspace_mul \<open>x \<in> S\<close> by auto
     moreover have "norm ((B / norm x) *\<^sub>R x) = B"
       using that B by (simp add: algebra_simps)
     ultimately show False using B by force
   qed
   then have "bounded S \<Longrightarrow> S = {0}"
     using assms subspace_0 by fastforce
   then show ?thesis
     by blast
 qed
 
 lemma affine_bounded_eq_trivial:
   fixes S :: "'a::real_normed_vector set"
   assumes "affine S"
     shows "bounded S \<longleftrightarrow> S = {} \<or> (\<exists>a. S = {a})"
 proof (cases "S = {}")
   case True then show ?thesis
     by simp
 next
   case False
   then obtain b where "b \<in> S" by blast
   with False assms show ?thesis
     apply safe
     using affine_diffs_subspace [OF assms \<open>b \<in> S\<close>]
     apply (metis (no_types, lifting) subspace_bounded_eq_trivial ab_left_minus bounded_translation
                 image_empty image_insert translation_invert)
     apply force
     done
 qed
 
 lemma affine_bounded_eq_lowdim:
   fixes S :: "'a::euclidean_space set"
   assumes "affine S"
     shows "bounded S \<longleftrightarrow> aff_dim S \<le> 0"
 apply safe
 using affine_bounded_eq_trivial assms apply fastforce
 by (metis aff_dim_sing aff_dim_subset affine_dim_equal affine_sing all_not_in_conv assms bounded_empty bounded_insert dual_order.antisym empty_subsetI insert_subset)
 
 
 lemma bounded_hyperplane_eq_trivial_0:
   fixes a :: "'a::euclidean_space"
   assumes "a \<noteq> 0"
   shows "bounded {x. a \<bullet> x = 0} \<longleftrightarrow> DIM('a) = 1"
 proof
   assume "bounded {x. a \<bullet> x = 0}"
   then have "aff_dim {x. a \<bullet> x = 0} \<le> 0"
     by (simp add: affine_bounded_eq_lowdim affine_hyperplane)
   with assms show "DIM('a) = 1"
     by (simp add: le_Suc_eq)
 next
   assume "DIM('a) = 1"
   then show "bounded {x. a \<bullet> x = 0}"
     by (simp add: affine_bounded_eq_lowdim affine_hyperplane assms)
 qed
 
 lemma bounded_hyperplane_eq_trivial:
   fixes a :: "'a::euclidean_space"
   shows "bounded {x. a \<bullet> x = r} \<longleftrightarrow> (if a = 0 then r \<noteq> 0 else DIM('a) = 1)"
 proof (simp add: bounded_hyperplane_eq_trivial_0, clarify)
   assume "r \<noteq> 0" "a \<noteq> 0"
   have "aff_dim {x. y \<bullet> x = 0} = aff_dim {x. a \<bullet> x = r}" if "y \<noteq> 0" for y::'a
     by (metis that \<open>a \<noteq> 0\<close> aff_dim_hyperplane)
   then show "bounded {x. a \<bullet> x = r} = (DIM('a) = Suc 0)"
     by (metis One_nat_def \<open>a \<noteq> 0\<close> affine_bounded_eq_lowdim affine_hyperplane bounded_hyperplane_eq_trivial_0)
 qed
 
 subsection\<^marker>\<open>tag unimportant\<close>\<open>General case without assuming closure and getting non-strict separation\<close>
 
 proposition\<^marker>\<open>tag unimportant\<close> separating_hyperplane_closed_point_inset:
   fixes S :: "'a::euclidean_space set"
   assumes "convex S" "closed S" "S \<noteq> {}" "z \<notin> S"
   obtains a b where "a \<in> S" "(a - z) \<bullet> z < b" "\<And>x. x \<in> S \<Longrightarrow> b < (a - z) \<bullet> x"
 proof -
   obtain y where "y \<in> S" and y: "\<And>u. u \<in> S \<Longrightarrow> dist z y \<le> dist z u"
     using distance_attains_inf [of S z] assms by auto
   then have *: "(y - z) \<bullet> z < (y - z) \<bullet> z + (norm (y - z))\<^sup>2 / 2"
     using \<open>y \<in> S\<close> \<open>z \<notin> S\<close> by auto
   show ?thesis
   proof (rule that [OF \<open>y \<in> S\<close> *])
     fix x
     assume "x \<in> S"
     have yz: "0 < (y - z) \<bullet> (y - z)"
       using \<open>y \<in> S\<close> \<open>z \<notin> S\<close> by auto
     { assume 0: "0 < ((z - y) \<bullet> (x - y))"
       with any_closest_point_dot [OF \<open>convex S\<close> \<open>closed S\<close>]
       have False
         using y \<open>x \<in> S\<close> \<open>y \<in> S\<close> not_less by blast
     }
     then have "0 \<le> ((y - z) \<bullet> (x - y))"
       by (force simp: not_less inner_diff_left)
     with yz have "0 < 2 * ((y - z) \<bullet> (x - y)) + (y - z) \<bullet> (y - z)"
       by (simp add: algebra_simps)
     then show "(y - z) \<bullet> z + (norm (y - z))\<^sup>2 / 2 < (y - z) \<bullet> x"
       by (simp add: field_simps inner_diff_left inner_diff_right dot_square_norm [symmetric])
   qed
 qed
 
 lemma separating_hyperplane_closed_0_inset:
   fixes S :: "'a::euclidean_space set"
   assumes "convex S" "closed S" "S \<noteq> {}" "0 \<notin> S"
   obtains a b where "a \<in> S" "a \<noteq> 0" "0 < b" "\<And>x. x \<in> S \<Longrightarrow> a \<bullet> x > b"
 using separating_hyperplane_closed_point_inset [OF assms]
 by simp (metis \<open>0 \<notin> S\<close>)
 
 
 proposition\<^marker>\<open>tag unimportant\<close> separating_hyperplane_set_0_inspan:
   fixes S :: "'a::euclidean_space set"
   assumes "convex S" "S \<noteq> {}" "0 \<notin> S"
   obtains a where "a \<in> span S" "a \<noteq> 0" "\<And>x. x \<in> S \<Longrightarrow> 0 \<le> a \<bullet> x"
 proof -
   define k where [abs_def]: "k c = {x. 0 \<le> c \<bullet> x}" for c :: 'a
   have *: "span S \<inter> frontier (cball 0 1) \<inter> \<Inter>f' \<noteq> {}"
           if f': "finite f'" "f' \<subseteq> k ` S" for f'
   proof -
     obtain C where "C \<subseteq> S" "finite C" and C: "f' = k ` C"
       using finite_subset_image [OF f'] by blast
     obtain a where "a \<in> S" "a \<noteq> 0"
       using \<open>S \<noteq> {}\<close> \<open>0 \<notin> S\<close> ex_in_conv by blast
     then have "norm (a /\<^sub>R (norm a)) = 1"
       by simp
     moreover have "a /\<^sub>R (norm a) \<in> span S"
       by (simp add: \<open>a \<in> S\<close> span_scale span_base)
     ultimately have ass: "a /\<^sub>R (norm a) \<in> span S \<inter> sphere 0 1"
       by simp
     show ?thesis
     proof (cases "C = {}")
       case True with C ass show ?thesis
         by auto
     next
       case False
       have "closed (convex hull C)"
         using \<open>finite C\<close> compact_eq_bounded_closed finite_imp_compact_convex_hull by auto
       moreover have "convex hull C \<noteq> {}"
         by (simp add: False)
       moreover have "0 \<notin> convex hull C"
         by (metis \<open>C \<subseteq> S\<close> \<open>convex S\<close> \<open>0 \<notin> S\<close> convex_hull_subset hull_same insert_absorb insert_subset)
       ultimately obtain a b
             where "a \<in> convex hull C" "a \<noteq> 0" "0 < b"
                   and ab: "\<And>x. x \<in> convex hull C \<Longrightarrow> a \<bullet> x > b"
         using separating_hyperplane_closed_0_inset by blast
       then have "a \<in> S"
         by (metis \<open>C \<subseteq> S\<close> assms(1) subsetCE subset_hull)
       moreover have "norm (a /\<^sub>R (norm a)) = 1"
         using \<open>a \<noteq> 0\<close> by simp
       moreover have "a /\<^sub>R (norm a) \<in> span S"
         by (simp add: \<open>a \<in> S\<close> span_scale span_base)
       ultimately have ass: "a /\<^sub>R (norm a) \<in> span S \<inter> sphere 0 1"
         by simp
       have aa: "a /\<^sub>R (norm a) \<in> (\<Inter>c\<in>C. {x. 0 \<le> c \<bullet> x})"
         apply (clarsimp simp add: field_split_simps)
         using ab \<open>0 < b\<close>
         by (metis hull_inc inner_commute less_eq_real_def less_trans)
       show ?thesis
         apply (simp add: C k_def)
         using ass aa Int_iff empty_iff by blast
     qed
   qed
   have "(span S \<inter> frontier(cball 0 1)) \<inter> (\<Inter> (k ` S)) \<noteq> {}"
     apply (rule compact_imp_fip)
     apply (blast intro: compact_cball)
     using closed_halfspace_ge k_def apply blast
     apply (metis *)
     done
   then show ?thesis
     unfolding set_eq_iff k_def
     by simp (metis inner_commute norm_eq_zero that zero_neq_one)
 qed
 
 
 lemma separating_hyperplane_set_point_inaff:
   fixes S :: "'a::euclidean_space set"
   assumes "convex S" "S \<noteq> {}" and zno: "z \<notin> S"
   obtains a b where "(z + a) \<in> affine hull (insert z S)"
                 and "a \<noteq> 0" and "a \<bullet> z \<le> b"
                 and "\<And>x. x \<in> S \<Longrightarrow> a \<bullet> x \<ge> b"
 proof -
   from separating_hyperplane_set_0_inspan [of "image (\<lambda>x. -z + x) S"]
   have "convex ((+) (- z) ` S)"
     using \<open>convex S\<close> by simp
   moreover have "(+) (- z) ` S \<noteq> {}"
     by (simp add: \<open>S \<noteq> {}\<close>)
   moreover have "0 \<notin> (+) (- z) ` S"
     using zno by auto
   ultimately obtain a where "a \<in> span ((+) (- z) ` S)" "a \<noteq> 0"
                   and a:  "\<And>x. x \<in> ((+) (- z) ` S) \<Longrightarrow> 0 \<le> a \<bullet> x"
     using separating_hyperplane_set_0_inspan [of "image (\<lambda>x. -z + x) S"]
     by blast
   then have szx: "\<And>x. x \<in> S \<Longrightarrow> a \<bullet> z \<le> a \<bullet> x"
     by (metis (no_types, lifting) imageI inner_minus_right inner_right_distrib minus_add neg_le_0_iff_le neg_le_iff_le real_add_le_0_iff)
   show ?thesis
     apply (rule_tac a=a and b = "a  \<bullet> z" in that, simp_all)
     using \<open>a \<in> span ((+) (- z) ` S)\<close> affine_hull_insert_span_gen apply blast
     apply (simp_all add: \<open>a \<noteq> 0\<close> szx)
     done
 qed
 
 proposition\<^marker>\<open>tag unimportant\<close> supporting_hyperplane_rel_boundary:
   fixes S :: "'a::euclidean_space set"
   assumes "convex S" "x \<in> S" and xno: "x \<notin> rel_interior S"
   obtains a where "a \<noteq> 0"
               and "\<And>y. y \<in> S \<Longrightarrow> a \<bullet> x \<le> a \<bullet> y"
               and "\<And>y. y \<in> rel_interior S \<Longrightarrow> a \<bullet> x < a \<bullet> y"
 proof -
   obtain a b where aff: "(x + a) \<in> affine hull (insert x (rel_interior S))"
                   and "a \<noteq> 0" and "a \<bullet> x \<le> b"
                   and ageb: "\<And>u. u \<in> (rel_interior S) \<Longrightarrow> a \<bullet> u \<ge> b"
     using separating_hyperplane_set_point_inaff [of "rel_interior S" x] assms
     by (auto simp: rel_interior_eq_empty convex_rel_interior)
   have le_ay: "a \<bullet> x \<le> a \<bullet> y" if "y \<in> S" for y
   proof -
     have con: "continuous_on (closure (rel_interior S)) ((\<bullet>) a)"
       by (rule continuous_intros continuous_on_subset | blast)+
     have y: "y \<in> closure (rel_interior S)"
       using \<open>convex S\<close> closure_def convex_closure_rel_interior \<open>y \<in> S\<close>
       by fastforce
     show ?thesis
       using continuous_ge_on_closure [OF con y] ageb \<open>a \<bullet> x \<le> b\<close>
       by fastforce
   qed
   have 3: "a \<bullet> x < a \<bullet> y" if "y \<in> rel_interior S" for y
   proof -
     obtain e where "0 < e" "y \<in> S" and e: "cball y e \<inter> affine hull S \<subseteq> S"
       using \<open>y \<in> rel_interior S\<close> by (force simp: rel_interior_cball)
     define y' where "y' = y - (e / norm a) *\<^sub>R ((x + a) - x)"
     have "y' \<in> cball y e"
       unfolding y'_def using \<open>0 < e\<close> by force
     moreover have "y' \<in> affine hull S"
       unfolding y'_def
       by (metis \<open>x \<in> S\<close> \<open>y \<in> S\<close> \<open>convex S\<close> aff affine_affine_hull hull_redundant
                 rel_interior_same_affine_hull hull_inc mem_affine_3_minus2)
     ultimately have "y' \<in> S"
       using e by auto
     have "a \<bullet> x \<le> a \<bullet> y"
       using le_ay \<open>a \<noteq> 0\<close> \<open>y \<in> S\<close> by blast
     moreover have "a \<bullet> x \<noteq> a \<bullet> y"
       using le_ay [OF \<open>y' \<in> S\<close>] \<open>a \<noteq> 0\<close>
       apply (simp add: y'_def inner_diff dot_square_norm power2_eq_square)
       by (metis \<open>0 < e\<close> add_le_same_cancel1 inner_commute inner_real_def inner_zero_left le_diff_eq norm_le_zero_iff real_mult_le_cancel_iff2)
     ultimately show ?thesis by force
   qed
   show ?thesis
     by (rule that [OF \<open>a \<noteq> 0\<close> le_ay 3])
 qed
 
 lemma supporting_hyperplane_relative_frontier:
   fixes S :: "'a::euclidean_space set"
   assumes "convex S" "x \<in> closure S" "x \<notin> rel_interior S"
   obtains a where "a \<noteq> 0"
               and "\<And>y. y \<in> closure S \<Longrightarrow> a \<bullet> x \<le> a \<bullet> y"
               and "\<And>y. y \<in> rel_interior S \<Longrightarrow> a \<bullet> x < a \<bullet> y"
 using supporting_hyperplane_rel_boundary [of "closure S" x]
 by (metis assms convex_closure convex_rel_interior_closure)
 
 
 subsection\<^marker>\<open>tag unimportant\<close>\<open> Some results on decomposing convex hulls: intersections, simplicial subdivision\<close>
 
 lemma
   fixes s :: "'a::euclidean_space set"
   assumes "\<not> affine_dependent(s \<union> t)"
     shows convex_hull_Int_subset: "convex hull s \<inter> convex hull t \<subseteq> convex hull (s \<inter> t)" (is ?C)
       and affine_hull_Int_subset: "affine hull s \<inter> affine hull t \<subseteq> affine hull (s \<inter> t)" (is ?A)
 proof -
   have [simp]: "finite s" "finite t"
     using aff_independent_finite assms by blast+
     have "sum u (s \<inter> t) = 1 \<and>
           (\<Sum>v\<in>s \<inter> t. u v *\<^sub>R v) = (\<Sum>v\<in>s. u v *\<^sub>R v)"
       if [simp]:  "sum u s = 1"
                  "sum v t = 1"
          and eq: "(\<Sum>x\<in>t. v x *\<^sub>R x) = (\<Sum>x\<in>s. u x *\<^sub>R x)" for u v
     proof -
     define f where "f x = (if x \<in> s then u x else 0) - (if x \<in> t then v x else 0)" for x
     have "sum f (s \<union> t) = 0"
       apply (simp add: f_def sum_Un sum_subtractf)
       apply (simp add: sum.inter_restrict [symmetric] Int_commute)
       done
     moreover have "(\<Sum>x\<in>(s \<union> t). f x *\<^sub>R x) = 0"
       apply (simp add: f_def sum_Un scaleR_left_diff_distrib sum_subtractf)
       apply (simp add: if_smult sum.inter_restrict [symmetric] Int_commute eq
           cong del: if_weak_cong)
       done
     ultimately have "\<And>v. v \<in> s \<union> t \<Longrightarrow> f v = 0"
       using aff_independent_finite assms unfolding affine_dependent_explicit
       by blast
     then have u [simp]: "\<And>x. x \<in> s \<Longrightarrow> u x = (if x \<in> t then v x else 0)"
       by (simp add: f_def) presburger
     have "sum u (s \<inter> t) = sum u s"
       by (simp add: sum.inter_restrict)
     then have "sum u (s \<inter> t) = 1"
       using that by linarith
     moreover have "(\<Sum>v\<in>s \<inter> t. u v *\<^sub>R v) = (\<Sum>v\<in>s. u v *\<^sub>R v)"
       by (auto simp: if_smult sum.inter_restrict intro: sum.cong)
     ultimately show ?thesis
       by force
     qed
     then show ?A ?C
       by (auto simp: convex_hull_finite affine_hull_finite)
 qed
 
 
 proposition\<^marker>\<open>tag unimportant\<close> affine_hull_Int:
   fixes s :: "'a::euclidean_space set"
   assumes "\<not> affine_dependent(s \<union> t)"
     shows "affine hull (s \<inter> t) = affine hull s \<inter> affine hull t"
 apply (rule subset_antisym)
 apply (simp add: hull_mono)
 by (simp add: affine_hull_Int_subset assms)
 
 proposition\<^marker>\<open>tag unimportant\<close> convex_hull_Int:
   fixes s :: "'a::euclidean_space set"
   assumes "\<not> affine_dependent(s \<union> t)"
     shows "convex hull (s \<inter> t) = convex hull s \<inter> convex hull t"
 apply (rule subset_antisym)
 apply (simp add: hull_mono)
 by (simp add: convex_hull_Int_subset assms)
 
 proposition\<^marker>\<open>tag unimportant\<close>
   fixes s :: "'a::euclidean_space set set"
   assumes "\<not> affine_dependent (\<Union>s)"
     shows affine_hull_Inter: "affine hull (\<Inter>s) = (\<Inter>t\<in>s. affine hull t)" (is "?A")
       and convex_hull_Inter: "convex hull (\<Inter>s) = (\<Inter>t\<in>s. convex hull t)" (is "?C")
 proof -
   have "finite s"
     using aff_independent_finite assms finite_UnionD by blast
   then have "?A \<and> ?C" using assms
   proof (induction s rule: finite_induct)
     case empty then show ?case by auto
   next
     case (insert t F)
     then show ?case
     proof (cases "F={}")
       case True then show ?thesis by simp
     next
       case False
       with "insert.prems" have [simp]: "\<not> affine_dependent (t \<union> \<Inter>F)"
         by (auto intro: affine_dependent_subset)
       have [simp]: "\<not> affine_dependent (\<Union>F)"
         using affine_independent_subset insert.prems by fastforce
       show ?thesis
         by (simp add: affine_hull_Int convex_hull_Int insert.IH)
     qed
   qed
   then show "?A" "?C"
     by auto
 qed
 
 proposition\<^marker>\<open>tag unimportant\<close> in_convex_hull_exchange_unique:
   fixes S :: "'a::euclidean_space set"
   assumes naff: "\<not> affine_dependent S" and a: "a \<in> convex hull S"
       and S: "T \<subseteq> S" "T' \<subseteq> S"
       and x:  "x \<in> convex hull (insert a T)"
       and x': "x \<in> convex hull (insert a T')"
     shows "x \<in> convex hull (insert a (T \<inter> T'))"
 proof (cases "a \<in> S")
   case True
   then have "\<not> affine_dependent (insert a T \<union> insert a T')"
     using affine_dependent_subset assms by auto
   then have "x \<in> convex hull (insert a T \<inter> insert a T')"
     by (metis IntI convex_hull_Int x x')
   then show ?thesis
     by simp
 next
   case False
   then have anot: "a \<notin> T" "a \<notin> T'"
     using assms by auto
   have [simp]: "finite S"
     by (simp add: aff_independent_finite assms)
   then obtain b where b0: "\<And>s. s \<in> S \<Longrightarrow> 0 \<le> b s"
                   and b1: "sum b S = 1" and aeq: "a = (\<Sum>s\<in>S. b s *\<^sub>R s)"
     using a by (auto simp: convex_hull_finite)
   have fin [simp]: "finite T" "finite T'"
     using assms infinite_super \<open>finite S\<close> by blast+
   then obtain c c' where c0: "\<And>t. t \<in> insert a T \<Longrightarrow> 0 \<le> c t"
                      and c1: "sum c (insert a T) = 1"
                      and xeq: "x = (\<Sum>t \<in> insert a T. c t *\<^sub>R t)"
                      and c'0: "\<And>t. t \<in> insert a T' \<Longrightarrow> 0 \<le> c' t"
                      and c'1: "sum c' (insert a T') = 1"
                      and x'eq: "x = (\<Sum>t \<in> insert a T'. c' t *\<^sub>R t)"
     using x x' by (auto simp: convex_hull_finite)
   with fin anot
   have sumTT': "sum c T = 1 - c a" "sum c' T' = 1 - c' a"
    and wsumT: "(\<Sum>t \<in> T. c t *\<^sub>R t) = x - c a *\<^sub>R a"
     by simp_all
   have wsumT': "(\<Sum>t \<in> T'. c' t *\<^sub>R t) = x - c' a *\<^sub>R a"
     using x'eq fin anot by simp
   define cc  where "cc \<equiv> \<lambda>x. if x \<in> T then c x else 0"
   define cc' where "cc' \<equiv> \<lambda>x. if x \<in> T' then c' x else 0"
   define dd  where "dd \<equiv> \<lambda>x. cc x - cc' x + (c a - c' a) * b x"
   have sumSS': "sum cc S = 1 - c a" "sum cc' S = 1 - c' a"
     unfolding cc_def cc'_def  using S
     by (simp_all add: Int_absorb1 Int_absorb2 sum_subtractf sum.inter_restrict [symmetric] sumTT')
   have wsumSS: "(\<Sum>t \<in> S. cc t *\<^sub>R t) = x - c a *\<^sub>R a" "(\<Sum>t \<in> S. cc' t *\<^sub>R t) = x - c' a *\<^sub>R a"
     unfolding cc_def cc'_def  using S
     by (simp_all add: Int_absorb1 Int_absorb2 if_smult sum.inter_restrict [symmetric] wsumT wsumT' cong: if_cong)
   have sum_dd0: "sum dd S = 0"
     unfolding dd_def  using S
     by (simp add: sumSS' comm_monoid_add_class.sum.distrib sum_subtractf
                   algebra_simps sum_distrib_right [symmetric] b1)
   have "(\<Sum>v\<in>S. (b v * x) *\<^sub>R v) = x *\<^sub>R (\<Sum>v\<in>S. b v *\<^sub>R v)" for x
     by (simp add: pth_5 real_vector.scale_sum_right mult.commute)
   then have *: "(\<Sum>v\<in>S. (b v * x) *\<^sub>R v) = x *\<^sub>R a" for x
     using aeq by blast
   have "(\<Sum>v \<in> S. dd v *\<^sub>R v) = 0"
     unfolding dd_def using S
     by (simp add: * wsumSS sum.distrib sum_subtractf algebra_simps)
   then have dd0: "dd v = 0" if "v \<in> S" for v
     using naff that \<open>finite S\<close> sum_dd0 unfolding affine_dependent_explicit
     apply (simp only: not_ex)
     apply (drule_tac x=S in spec)
     apply (drule_tac x=dd in spec, simp)
     done
   consider "c' a \<le> c a" | "c a \<le> c' a" by linarith
   then show ?thesis
   proof cases
     case 1
     then have "sum cc S \<le> sum cc' S"
       by (simp add: sumSS')
     then have le: "cc x \<le> cc' x" if "x \<in> S" for x
       using dd0 [OF that] 1 b0 mult_left_mono that
       by (fastforce simp add: dd_def algebra_simps)
     have cc0: "cc x = 0" if "x \<in> S" "x \<notin> T \<inter> T'" for x
       using le [OF \<open>x \<in> S\<close>] that c0
       by (force simp: cc_def cc'_def split: if_split_asm)
     show ?thesis
     proof (simp add: convex_hull_finite, intro exI conjI)
       show "\<forall>x\<in>T \<inter> T'. 0 \<le> (cc(a := c a)) x"
         by (simp add: c0 cc_def)
       show "0 \<le> (cc(a := c a)) a"
         by (simp add: c0)
       have "sum (cc(a := c a)) (insert a (T \<inter> T')) = c a + sum (cc(a := c a)) (T \<inter> T')"
         by (simp add: anot)
       also have "... = c a + sum (cc(a := c a)) S"
         apply simp
         apply (rule sum.mono_neutral_left)
         using \<open>T \<subseteq> S\<close> apply (auto simp: \<open>a \<notin> S\<close> cc0)
         done
       also have "... = c a + (1 - c a)"
         by (metis \<open>a \<notin> S\<close> fun_upd_other sum.cong sumSS')
       finally show "sum (cc(a := c a)) (insert a (T \<inter> T')) = 1"
         by simp
       have "(\<Sum>x\<in>insert a (T \<inter> T'). (cc(a := c a)) x *\<^sub>R x) = c a *\<^sub>R a + (\<Sum>x \<in> T \<inter> T'. (cc(a := c a)) x *\<^sub>R x)"
         by (simp add: anot)
       also have "... = c a *\<^sub>R a + (\<Sum>x \<in> S. (cc(a := c a)) x *\<^sub>R x)"
         apply simp
         apply (rule sum.mono_neutral_left)
         using \<open>T \<subseteq> S\<close> apply (auto simp: \<open>a \<notin> S\<close> cc0)
         done
       also have "... = c a *\<^sub>R a + x - c a *\<^sub>R a"
         by (simp add: wsumSS \<open>a \<notin> S\<close> if_smult sum_delta_notmem)
       finally show "(\<Sum>x\<in>insert a (T \<inter> T'). (cc(a := c a)) x *\<^sub>R x) = x"
         by simp
     qed
   next
     case 2
     then have "sum cc' S \<le> sum cc S"
       by (simp add: sumSS')
     then have le: "cc' x \<le> cc x" if "x \<in> S" for x
       using dd0 [OF that] 2 b0 mult_left_mono that
       by (fastforce simp add: dd_def algebra_simps)
     have cc0: "cc' x = 0" if "x \<in> S" "x \<notin> T \<inter> T'" for x
       using le [OF \<open>x \<in> S\<close>] that c'0
       by (force simp: cc_def cc'_def split: if_split_asm)
     show ?thesis
     proof (simp add: convex_hull_finite, intro exI conjI)
       show "\<forall>x\<in>T \<inter> T'. 0 \<le> (cc'(a := c' a)) x"
         by (simp add: c'0 cc'_def)
       show "0 \<le> (cc'(a := c' a)) a"
         by (simp add: c'0)
       have "sum (cc'(a := c' a)) (insert a (T \<inter> T')) = c' a + sum (cc'(a := c' a)) (T \<inter> T')"
         by (simp add: anot)
       also have "... = c' a + sum (cc'(a := c' a)) S"
         apply simp
         apply (rule sum.mono_neutral_left)
         using \<open>T \<subseteq> S\<close> apply (auto simp: \<open>a \<notin> S\<close> cc0)
         done
       also have "... = c' a + (1 - c' a)"
         by (metis \<open>a \<notin> S\<close> fun_upd_other sum.cong sumSS')
       finally show "sum (cc'(a := c' a)) (insert a (T \<inter> T')) = 1"
         by simp
       have "(\<Sum>x\<in>insert a (T \<inter> T'). (cc'(a := c' a)) x *\<^sub>R x) = c' a *\<^sub>R a + (\<Sum>x \<in> T \<inter> T'. (cc'(a := c' a)) x *\<^sub>R x)"
         by (simp add: anot)
       also have "... = c' a *\<^sub>R a + (\<Sum>x \<in> S. (cc'(a := c' a)) x *\<^sub>R x)"
         apply simp
         apply (rule sum.mono_neutral_left)
         using \<open>T \<subseteq> S\<close> apply (auto simp: \<open>a \<notin> S\<close> cc0)
         done
       also have "... = c a *\<^sub>R a + x - c a *\<^sub>R a"
         by (simp add: wsumSS \<open>a \<notin> S\<close> if_smult sum_delta_notmem)
       finally show "(\<Sum>x\<in>insert a (T \<inter> T'). (cc'(a := c' a)) x *\<^sub>R x) = x"
         by simp
     qed
   qed
 qed
 
 corollary\<^marker>\<open>tag unimportant\<close> convex_hull_exchange_Int:
   fixes a  :: "'a::euclidean_space"
   assumes "\<not> affine_dependent S" "a \<in> convex hull S" "T \<subseteq> S" "T' \<subseteq> S"
   shows "(convex hull (insert a T)) \<inter> (convex hull (insert a T')) =
          convex hull (insert a (T \<inter> T'))"
 apply (rule subset_antisym)
   using in_convex_hull_exchange_unique assms apply blast
   by (metis hull_mono inf_le1 inf_le2 insert_inter_insert le_inf_iff)
 
 lemma Int_closed_segment:
   fixes b :: "'a::euclidean_space"
   assumes "b \<in> closed_segment a c \<or> \<not> collinear{a,b,c}"
     shows "closed_segment a b \<inter> closed_segment b c = {b}"
 proof (cases "c = a")
   case True
   then show ?thesis
     using assms collinear_3_eq_affine_dependent by fastforce
 next
   case False
   from assms show ?thesis
   proof
     assume "b \<in> closed_segment a c"
     moreover have "\<not> affine_dependent {a, c}"
       by (simp)
     ultimately show ?thesis
       using False convex_hull_exchange_Int [of "{a,c}" b "{a}" "{c}"]
       by (simp add: segment_convex_hull insert_commute)
   next
     assume ncoll: "\<not> collinear {a, b, c}"
     have False if "closed_segment a b \<inter> closed_segment b c \<noteq> {b}"
     proof -
       have "b \<in> closed_segment a b" and "b \<in> closed_segment b c"
         by auto
       with that obtain d where "b \<noteq> d" "d \<in> closed_segment a b" "d \<in> closed_segment b c"
         by force
       then have d: "collinear {a, d, b}"  "collinear {b, d, c}"
         by (auto simp:  between_mem_segment between_imp_collinear)
       have "collinear {a, b, c}"
         apply (rule collinear_3_trans [OF _ _ \<open>b \<noteq> d\<close>])
         using d  by (auto simp: insert_commute)
       with ncoll show False ..
     qed
     then show ?thesis
       by blast
   qed
 qed
 
 lemma affine_hull_finite_intersection_hyperplanes:
   fixes s :: "'a::euclidean_space set"
   obtains f where
      "finite f"
      "of_nat (card f) + aff_dim s = DIM('a)"
      "affine hull s = \<Inter>f"
      "\<And>h. h \<in> f \<Longrightarrow> \<exists>a b. a \<noteq> 0 \<and> h = {x. a \<bullet> x = b}"
 proof -
   obtain b where "b \<subseteq> s"
              and indb: "\<not> affine_dependent b"
              and eq: "affine hull s = affine hull b"
     using affine_basis_exists by blast
   obtain c where indc: "\<not> affine_dependent c" and "b \<subseteq> c"
              and affc: "affine hull c = UNIV"
     by (metis extend_to_affine_basis affine_UNIV hull_same indb subset_UNIV)
   then have "finite c"
     by (simp add: aff_independent_finite)
   then have fbc: "finite b" "card b \<le> card c"
     using \<open>b \<subseteq> c\<close> infinite_super by (auto simp: card_mono)
   have imeq: "(\<lambda>x. affine hull x) ` ((\<lambda>a. c - {a}) ` (c - b)) = ((\<lambda>a. affine hull (c - {a})) ` (c - b))"
     by blast
   have card1: "card ((\<lambda>a. affine hull (c - {a})) ` (c - b)) = card (c - b)"
     apply (rule card_image [OF inj_onI])
     by (metis Diff_eq_empty_iff Diff_iff indc affine_dependent_def hull_subset insert_iff)
   have card2: "(card (c - b)) + aff_dim s = DIM('a)"
   proof -
     have aff: "aff_dim (UNIV::'a set) = aff_dim c"
       by (metis aff_dim_affine_hull affc)
     have "aff_dim b = aff_dim s"
       by (metis (no_types) aff_dim_affine_hull eq)
     then have "int (card b) = 1 + aff_dim s"
       by (simp add: aff_dim_affine_independent indb)
     then show ?thesis
       using fbc aff
       by (simp add: \<open>\<not> affine_dependent c\<close> \<open>b \<subseteq> c\<close> aff_dim_affine_independent card_Diff_subset of_nat_diff)
   qed
   show ?thesis
   proof (cases "c = b")
     case True show ?thesis
       apply (rule_tac f="{}" in that)
       using True affc
       apply (simp_all add: eq [symmetric])
       by (metis aff_dim_UNIV aff_dim_affine_hull)
   next
     case False
     have ind: "\<not> affine_dependent (\<Union>a\<in>c - b. c - {a})"
       by (rule affine_independent_subset [OF indc]) auto
     have affeq: "affine hull s = (\<Inter>x\<in>(\<lambda>a. c - {a}) ` (c - b). affine hull x)"
       using \<open>b \<subseteq> c\<close> False
       apply (subst affine_hull_Inter [OF ind, symmetric])
       apply (simp add: eq double_diff)
       done
     have *: "1 + aff_dim (c - {t}) = int (DIM('a))"
             if t: "t \<in> c" for t
     proof -
       have "insert t c = c"
         using t by blast
       then show ?thesis
         by (metis (full_types) add.commute aff_dim_affine_hull aff_dim_insert aff_dim_UNIV affc affine_dependent_def indc insert_Diff_single t)
     qed
     show ?thesis
       apply (rule_tac f = "(\<lambda>x. affine hull x) ` ((\<lambda>a. c - {a}) ` (c - b))" in that)
          using \<open>finite c\<close> apply blast
         apply (simp add: imeq card1 card2)
       apply (simp add: affeq, clarify)
       apply (metis DIM_positive One_nat_def Suc_leI add_diff_cancel_left' of_nat_1 aff_dim_eq_hyperplane of_nat_diff *)
       done
   qed
 qed
 
 lemma affine_hyperplane_sums_eq_UNIV_0:
   fixes S :: "'a :: euclidean_space set"
   assumes "affine S"
      and "0 \<in> S" and "w \<in> S"
      and "a \<bullet> w \<noteq> 0"
    shows "{x + y| x y. x \<in> S \<and> a \<bullet> y = 0} = UNIV"
 proof -
   have "subspace S"
     by (simp add: assms subspace_affine)
   have span1: "span {y. a \<bullet> y = 0} \<subseteq> span {x + y |x y. x \<in> S \<and> a \<bullet> y = 0}"
     apply (rule span_mono)
     using \<open>0 \<in> S\<close> add.left_neutral by force
   have "w \<notin> span {y. a \<bullet> y = 0}"
     using \<open>a \<bullet> w \<noteq> 0\<close> span_induct subspace_hyperplane by auto
   moreover have "w \<in> span {x + y |x y. x \<in> S \<and> a \<bullet> y = 0}"
     using \<open>w \<in> S\<close>
     by (metis (mono_tags, lifting) inner_zero_right mem_Collect_eq pth_d span_base)
   ultimately have span2: "span {y. a \<bullet> y = 0} \<noteq> span {x + y |x y. x \<in> S \<and> a \<bullet> y = 0}"
     by blast
   have "a \<noteq> 0" using assms inner_zero_left by blast
   then have "DIM('a) - 1 = dim {y. a \<bullet> y = 0}"
     by (simp add: dim_hyperplane)
   also have "... < dim {x + y |x y. x \<in> S \<and> a \<bullet> y = 0}"
     using span1 span2 by (blast intro: dim_psubset)
   finally have DIM_lt: "DIM('a) - 1 < dim {x + y |x y. x \<in> S \<and> a \<bullet> y = 0}" .
   have subs: "subspace {x + y| x y. x \<in> S \<and> a \<bullet> y = 0}"
     using subspace_sums [OF \<open>subspace S\<close> subspace_hyperplane] by simp
   moreover have "span {x + y| x y. x \<in> S \<and> a \<bullet> y = 0} = UNIV"
     apply (rule dim_eq_full [THEN iffD1])
     apply (rule antisym [OF dim_subset_UNIV])
     using DIM_lt apply simp
     done
   ultimately show ?thesis
     by (simp add: subs) (metis (lifting) span_eq_iff subs)
 qed
 
 proposition\<^marker>\<open>tag unimportant\<close> affine_hyperplane_sums_eq_UNIV:
   fixes S :: "'a :: euclidean_space set"
   assumes "affine S"
       and "S \<inter> {v. a \<bullet> v = b} \<noteq> {}"
       and "S - {v. a \<bullet> v = b} \<noteq> {}"
     shows "{x + y| x y. x \<in> S \<and> a \<bullet> y = b} = UNIV"
 proof (cases "a = 0")
   case True with assms show ?thesis
     by (auto simp: if_splits)
 next
   case False
   obtain c where "c \<in> S" and c: "a \<bullet> c = b"
     using assms by force
   with affine_diffs_subspace [OF \<open>affine S\<close>]
   have "subspace ((+) (- c) ` S)" by blast
   then have aff: "affine ((+) (- c) ` S)"
     by (simp add: subspace_imp_affine)
   have 0: "0 \<in> (+) (- c) ` S"
     by (simp add: \<open>c \<in> S\<close>)
   obtain d where "d \<in> S" and "a \<bullet> d \<noteq> b" and dc: "d-c \<in> (+) (- c) ` S"
     using assms by auto
   then have adc: "a \<bullet> (d - c) \<noteq> 0"
     by (simp add: c inner_diff_right)
   let ?U = "(+) (c+c) ` {x + y |x y. x \<in> (+) (- c) ` S \<and> a \<bullet> y = 0}"
   have "u + v \<in> (+) (c + c) ` {x + v |x v. x \<in> (+) (- c) ` S \<and> a \<bullet> v = 0}"
               if "u \<in> S" "b = a \<bullet> v" for u v
     apply (rule_tac x="u+v-c-c" in image_eqI)
     apply (simp_all add: algebra_simps)
     apply (rule_tac x="u-c" in exI)
     apply (rule_tac x="v-c" in exI)
     apply (simp add: algebra_simps that c)
     done
   moreover have "\<lbrakk>a \<bullet> v = 0; u \<in> S\<rbrakk>
        \<Longrightarrow> \<exists>x ya. v + (u + c) = x + ya \<and> x \<in> S \<and> a \<bullet> ya = b" for v u
     by (metis add.left_commute c inner_right_distrib pth_d)
   ultimately have "{x + y |x y. x \<in> S \<and> a \<bullet> y = b} = ?U"
     by (fastforce simp: algebra_simps)
   also have "... = range ((+) (c + c))"
     by (simp only: affine_hyperplane_sums_eq_UNIV_0 [OF aff 0 dc adc])
   also have "... = UNIV"
     by simp
   finally show ?thesis .
 qed
 
 lemma aff_dim_sums_Int_0:
   assumes "affine S"
       and "affine T"
       and "0 \<in> S" "0 \<in> T"
     shows "aff_dim {x + y| x y. x \<in> S \<and> y \<in> T} = (aff_dim S + aff_dim T) - aff_dim(S \<inter> T)"
 proof -
   have "0 \<in> {x + y |x y. x \<in> S \<and> y \<in> T}"
     using assms by force
   then have 0: "0 \<in> affine hull {x + y |x y. x \<in> S \<and> y \<in> T}"
     by (metis (lifting) hull_inc)
   have sub: "subspace S"  "subspace T"
     using assms by (auto simp: subspace_affine)
   show ?thesis
     using dim_sums_Int [OF sub] by (simp add: aff_dim_zero assms 0 hull_inc)
 qed
 
 proposition aff_dim_sums_Int:
   assumes "affine S"
       and "affine T"
       and "S \<inter> T \<noteq> {}"
     shows "aff_dim {x + y| x y. x \<in> S \<and> y \<in> T} = (aff_dim S + aff_dim T) - aff_dim(S \<inter> T)"
 proof -
   obtain a where a: "a \<in> S" "a \<in> T" using assms by force
   have aff: "affine ((+) (-a) ` S)"  "affine ((+) (-a) ` T)"
     using affine_translation [symmetric, of "- a"] assms by (simp_all cong: image_cong_simp)
   have zero: "0 \<in> ((+) (-a) ` S)"  "0 \<in> ((+) (-a) ` T)"
     using a assms by auto
   have "{x + y |x y. x \<in> (+) (- a) ` S \<and> y \<in> (+) (- a) ` T} =
       (+) (- 2 *\<^sub>R a) ` {x + y| x y. x \<in> S \<and> y \<in> T}"
     by (force simp: algebra_simps scaleR_2)
   moreover have "(+) (- a) ` S \<inter> (+) (- a) ` T = (+) (- a) ` (S \<inter> T)"
     by auto
   ultimately show ?thesis
     using aff_dim_sums_Int_0 [OF aff zero] aff_dim_translation_eq
     by (metis (lifting))
 qed
 
 lemma aff_dim_affine_Int_hyperplane:
   fixes a :: "'a::euclidean_space"
   assumes "affine S"
     shows "aff_dim(S \<inter> {x. a \<bullet> x = b}) =
              (if S \<inter> {v. a \<bullet> v = b} = {} then - 1
               else if S \<subseteq> {v. a \<bullet> v = b} then aff_dim S
               else aff_dim S - 1)"
 proof (cases "a = 0")
   case True with assms show ?thesis
     by auto
 next
   case False
   then have "aff_dim (S \<inter> {x. a \<bullet> x = b}) = aff_dim S - 1"
             if "x \<in> S" "a \<bullet> x \<noteq> b" and non: "S \<inter> {v. a \<bullet> v = b} \<noteq> {}" for x
   proof -
     have [simp]: "{x + y| x y. x \<in> S \<and> a \<bullet> y = b} = UNIV"
       using affine_hyperplane_sums_eq_UNIV [OF assms non] that  by blast
     show ?thesis
       using aff_dim_sums_Int [OF assms affine_hyperplane non]
       by (simp add: of_nat_diff False)
   qed
   then show ?thesis
     by (metis (mono_tags, lifting) inf.orderE aff_dim_empty_eq mem_Collect_eq subsetI)
 qed
 
 lemma aff_dim_lt_full:
   fixes S :: "'a::euclidean_space set"
   shows "aff_dim S < DIM('a) \<longleftrightarrow> (affine hull S \<noteq> UNIV)"
 by (metis (no_types) aff_dim_affine_hull aff_dim_le_DIM aff_dim_UNIV affine_hull_UNIV less_le)
 
 lemma aff_dim_openin:
   fixes S :: "'a::euclidean_space set"
   assumes ope: "openin (top_of_set T) S" and "affine T" "S \<noteq> {}"
   shows "aff_dim S = aff_dim T"
 proof -
   show ?thesis
   proof (rule order_antisym)
     show "aff_dim S \<le> aff_dim T"
       by (blast intro: aff_dim_subset [OF openin_imp_subset] ope)
   next
     obtain a where "a \<in> S"
       using \<open>S \<noteq> {}\<close> by blast
     have "S \<subseteq> T"
       using ope openin_imp_subset by auto
     then have "a \<in> T"
       using \<open>a \<in> S\<close> by auto
     then have subT': "subspace ((\<lambda>x. - a + x) ` T)"
       using affine_diffs_subspace \<open>affine T\<close> by auto
     then obtain B where Bsub: "B \<subseteq> ((\<lambda>x. - a + x) ` T)" and po: "pairwise orthogonal B"
                     and eq1: "\<And>x. x \<in> B \<Longrightarrow> norm x = 1" and "independent B"
                     and cardB: "card B = dim ((\<lambda>x. - a + x) ` T)"
                     and spanB: "span B = ((\<lambda>x. - a + x) ` T)"
       by (rule orthonormal_basis_subspace) auto
     obtain e where "0 < e" and e: "cball a e \<inter> T \<subseteq> S"
       by (meson \<open>a \<in> S\<close> openin_contains_cball ope)
     have "aff_dim T = aff_dim ((\<lambda>x. - a + x) ` T)"
       by (metis aff_dim_translation_eq)
     also have "... = dim ((\<lambda>x. - a + x) ` T)"
       using aff_dim_subspace subT' by blast
     also have "... = card B"
       by (simp add: cardB)
     also have "... = card ((\<lambda>x. e *\<^sub>R x) ` B)"
       using \<open>0 < e\<close>  by (force simp: inj_on_def card_image)
     also have "... \<le> dim ((\<lambda>x. - a + x) ` S)"
     proof (simp, rule independent_card_le_dim)
       have e': "cball 0 e \<inter> (\<lambda>x. x - a) ` T \<subseteq> (\<lambda>x. x - a) ` S"
         using e by (auto simp: dist_norm norm_minus_commute subset_eq)
       have "(\<lambda>x. e *\<^sub>R x) ` B \<subseteq> cball 0 e \<inter> (\<lambda>x. x - a) ` T"
         using Bsub \<open>0 < e\<close> eq1 subT' \<open>a \<in> T\<close> by (auto simp: subspace_def)
       then show "(\<lambda>x. e *\<^sub>R x) ` B \<subseteq> (\<lambda>x. x - a) ` S"
         using e' by blast
       show "independent ((\<lambda>x. e *\<^sub>R x) ` B)"
         using linear_scale_self \<open>independent B\<close>
         apply (rule linear_independent_injective_image)
         using \<open>0 < e\<close> inj_on_def by fastforce
     qed
     also have "... = aff_dim S"
       using \<open>a \<in> S\<close> aff_dim_eq_dim hull_inc by (force cong: image_cong_simp)
     finally show "aff_dim T \<le> aff_dim S" .
   qed
 qed
 
 lemma dim_openin:
   fixes S :: "'a::euclidean_space set"
   assumes ope: "openin (top_of_set T) S" and "subspace T" "S \<noteq> {}"
   shows "dim S = dim T"
 proof (rule order_antisym)
   show "dim S \<le> dim T"
     by (metis ope dim_subset openin_subset topspace_euclidean_subtopology)
 next
   have "dim T = aff_dim S"
     using aff_dim_openin
     by (metis aff_dim_subspace \<open>subspace T\<close> \<open>S \<noteq> {}\<close> ope subspace_affine)
   also have "... \<le> dim S"
     by (metis aff_dim_subset aff_dim_subspace dim_span span_superset
         subspace_span)
   finally show "dim T \<le> dim S" by simp
 qed
 
 subsection\<open>Lower-dimensional affine subsets are nowhere dense\<close>
 
 proposition dense_complement_subspace:
   fixes S :: "'a :: euclidean_space set"
   assumes dim_less: "dim T < dim S" and "subspace S" shows "closure(S - T) = S"
 proof -
   have "closure(S - U) = S" if "dim U < dim S" "U \<subseteq> S" for U
   proof -
     have "span U \<subset> span S"
       by (metis neq_iff psubsetI span_eq_dim span_mono that)
     then obtain a where "a \<noteq> 0" "a \<in> span S" and a: "\<And>y. y \<in> span U \<Longrightarrow> orthogonal a y"
       using orthogonal_to_subspace_exists_gen by metis
     show ?thesis
     proof
       have "closed S"
         by (simp add: \<open>subspace S\<close> closed_subspace)
       then show "closure (S - U) \<subseteq> S"
         by (simp add: closure_minimal)
       show "S \<subseteq> closure (S - U)"
       proof (clarsimp simp: closure_approachable)
         fix x and e::real
         assume "x \<in> S" "0 < e"
         show "\<exists>y\<in>S - U. dist y x < e"
         proof (cases "x \<in> U")
           case True
           let ?y = "x + (e/2 / norm a) *\<^sub>R a"
           show ?thesis
           proof
             show "dist ?y x < e"
               using \<open>0 < e\<close> by (simp add: dist_norm)
           next
             have "?y \<in> S"
               by (metis \<open>a \<in> span S\<close> \<open>x \<in> S\<close> assms(2) span_eq_iff subspace_add subspace_scale)
             moreover have "?y \<notin> U"
             proof -
               have "e/2 / norm a \<noteq> 0"
                 using \<open>0 < e\<close> \<open>a \<noteq> 0\<close> by auto
               then show ?thesis
                 by (metis True \<open>a \<noteq> 0\<close> a orthogonal_scaleR orthogonal_self real_vector.scale_eq_0_iff span_add_eq span_base)
             qed
             ultimately show "?y \<in> S - U" by blast
           qed
         next
           case False
           with \<open>0 < e\<close> \<open>x \<in> S\<close> show ?thesis by force
         qed
       qed
     qed
   qed
   moreover have "S - S \<inter> T = S-T"
     by blast
   moreover have "dim (S \<inter> T) < dim S"
     by (metis dim_less dim_subset inf.cobounded2 inf.orderE inf.strict_boundedE not_le)
   ultimately show ?thesis
     by force
 qed
 
 corollary\<^marker>\<open>tag unimportant\<close> dense_complement_affine:
   fixes S :: "'a :: euclidean_space set"
   assumes less: "aff_dim T < aff_dim S" and "affine S" shows "closure(S - T) = S"
 proof (cases "S \<inter> T = {}")
   case True
   then show ?thesis
     by (metis Diff_triv affine_hull_eq \<open>affine S\<close> closure_same_affine_hull closure_subset hull_subset subset_antisym)
 next
   case False
   then obtain z where z: "z \<in> S \<inter> T" by blast
   then have "subspace ((+) (- z) ` S)"
     by (meson IntD1 affine_diffs_subspace \<open>affine S\<close>)
   moreover have "int (dim ((+) (- z) ` T)) < int (dim ((+) (- z) ` S))"
 thm aff_dim_eq_dim
     using z less by (simp add: aff_dim_eq_dim_subtract [of z] hull_inc cong: image_cong_simp)
   ultimately have "closure(((+) (- z) ` S) - ((+) (- z) ` T)) = ((+) (- z) ` S)"
     by (simp add: dense_complement_subspace)
   then show ?thesis
     by (metis closure_translation translation_diff translation_invert)
 qed
 
 corollary\<^marker>\<open>tag unimportant\<close> dense_complement_openin_affine_hull:
   fixes S :: "'a :: euclidean_space set"
   assumes less: "aff_dim T < aff_dim S"
       and ope: "openin (top_of_set (affine hull S)) S"
     shows "closure(S - T) = closure S"
 proof -
   have "affine hull S - T \<subseteq> affine hull S"
     by blast
   then have "closure (S \<inter> closure (affine hull S - T)) = closure (S \<inter> (affine hull S - T))"
     by (rule closure_openin_Int_closure [OF ope])
   then show ?thesis
     by (metis Int_Diff aff_dim_affine_hull affine_affine_hull dense_complement_affine hull_subset inf.orderE less)
 qed
 
 corollary\<^marker>\<open>tag unimportant\<close> dense_complement_convex:
   fixes S :: "'a :: euclidean_space set"
   assumes "aff_dim T < aff_dim S" "convex S"
     shows "closure(S - T) = closure S"
 proof
   show "closure (S - T) \<subseteq> closure S"
     by (simp add: closure_mono)
   have "closure (rel_interior S - T) = closure (rel_interior S)"
     apply (rule dense_complement_openin_affine_hull)
     apply (simp add: assms rel_interior_aff_dim)
     using \<open>convex S\<close> rel_interior_rel_open rel_open by blast
   then show "closure S \<subseteq> closure (S - T)"
     by (metis Diff_mono \<open>convex S\<close> closure_mono convex_closure_rel_interior order_refl rel_interior_subset)
 qed
 
 corollary\<^marker>\<open>tag unimportant\<close> dense_complement_convex_closed:
   fixes S :: "'a :: euclidean_space set"
   assumes "aff_dim T < aff_dim S" "convex S" "closed S"
     shows "closure(S - T) = S"
   by (simp add: assms dense_complement_convex)
 
 
 subsection\<^marker>\<open>tag unimportant\<close>\<open>Parallel slices, etc\<close>
 
 text\<open> If we take a slice out of a set, we can do it perpendicularly,
   with the normal vector to the slice parallel to the affine hull.\<close>
 
 proposition\<^marker>\<open>tag unimportant\<close> affine_parallel_slice:
   fixes S :: "'a :: euclidean_space set"
   assumes "affine S"
       and "S \<inter> {x. a \<bullet> x \<le> b} \<noteq> {}"
       and "\<not> (S \<subseteq> {x. a \<bullet> x \<le> b})"
   obtains a' b' where "a' \<noteq> 0"
                    "S \<inter> {x. a' \<bullet> x \<le> b'} = S \<inter> {x. a \<bullet> x \<le> b}"
                    "S \<inter> {x. a' \<bullet> x = b'} = S \<inter> {x. a \<bullet> x = b}"
                    "\<And>w. w \<in> S \<Longrightarrow> (w + a') \<in> S"
 proof (cases "S \<inter> {x. a \<bullet> x = b} = {}")
   case True
   then obtain u v where "u \<in> S" "v \<in> S" "a \<bullet> u \<le> b" "a \<bullet> v > b"
     using assms by (auto simp: not_le)
   define \<eta> where "\<eta> = u + ((b - a \<bullet> u) / (a \<bullet> v - a \<bullet> u)) *\<^sub>R (v - u)"
   have "\<eta> \<in> S"
     by (simp add: \<eta>_def \<open>u \<in> S\<close> \<open>v \<in> S\<close> \<open>affine S\<close> mem_affine_3_minus)
   moreover have "a \<bullet> \<eta> = b"
     using \<open>a \<bullet> u \<le> b\<close> \<open>b < a \<bullet> v\<close>
     by (simp add: \<eta>_def algebra_simps) (simp add: field_simps)
   ultimately have False
     using True by force
   then show ?thesis ..
 next
   case False
   then obtain z where "z \<in> S" and z: "a \<bullet> z = b"
     using assms by auto
   with affine_diffs_subspace [OF \<open>affine S\<close>]
   have sub: "subspace ((+) (- z) ` S)" by blast
   then have aff: "affine ((+) (- z) ` S)" and span: "span ((+) (- z) ` S) = ((+) (- z) ` S)"
     by (auto simp: subspace_imp_affine)
   obtain a' a'' where a': "a' \<in> span ((+) (- z) ` S)" and a: "a = a' + a''"
                   and "\<And>w. w \<in> span ((+) (- z) ` S) \<Longrightarrow> orthogonal a'' w"
     using orthogonal_subspace_decomp_exists [of "(+) (- z) ` S" "a"] by metis
   then have "\<And>w. w \<in> S \<Longrightarrow> a'' \<bullet> (w-z) = 0"
     by (simp add: span_base orthogonal_def)
   then have a'': "\<And>w. w \<in> S \<Longrightarrow> a'' \<bullet> w = (a - a') \<bullet> z"
     by (simp add: a inner_diff_right)
   then have ba'': "\<And>w. w \<in> S \<Longrightarrow> a'' \<bullet> w = b - a' \<bullet> z"
     by (simp add: inner_diff_left z)
   have "\<And>w. w \<in> (+) (- z) ` S \<Longrightarrow> (w + a') \<in> (+) (- z) ` S"
     by (metis subspace_add a' span_eq_iff sub)
   then have Sclo: "\<And>w. w \<in> S \<Longrightarrow> (w + a') \<in> S"
     by fastforce
   show ?thesis
   proof (cases "a' = 0")
     case True
     with a assms True a'' diff_zero less_irrefl show ?thesis
       by auto
   next
     case False
     show ?thesis
       apply (rule_tac a' = "a'" and b' = "a' \<bullet> z" in that)
       apply (auto simp: a ba'' inner_left_distrib False Sclo)
       done
   qed
 qed
 
 lemma diffs_affine_hull_span:
   assumes "a \<in> S"
     shows "{x - a |x. x \<in> affine hull S} = span {x - a |x. x \<in> S}"
 proof -
   have *: "((\<lambda>x. x - a) ` (S - {a})) = {x. x + a \<in> S} - {0}"
     by (auto simp: algebra_simps)
   show ?thesis
     apply (simp add: affine_hull_span2 [OF assms] *)
     apply (auto simp: algebra_simps)
     done
 qed
 
 lemma aff_dim_dim_affine_diffs:
   fixes S :: "'a :: euclidean_space set"
   assumes "affine S" "a \<in> S"
     shows "aff_dim S = dim {x - a |x. x \<in> S}"
 proof -
   obtain B where aff: "affine hull B = affine hull S"
              and ind: "\<not> affine_dependent B"
              and card: "of_nat (card B) = aff_dim S + 1"
     using aff_dim_basis_exists by blast
   then have "B \<noteq> {}" using assms
     by (metis affine_hull_eq_empty ex_in_conv)
   then obtain c where "c \<in> B" by auto
   then have "c \<in> S"
     by (metis aff affine_hull_eq \<open>affine S\<close> hull_inc)
   have xy: "x - c = y - a \<longleftrightarrow> y = x + 1 *\<^sub>R (a - c)" for x y c and a::'a
     by (auto simp: algebra_simps)
   have *: "{x - c |x. x \<in> S} = {x - a |x. x \<in> S}"
     apply safe
     apply (simp_all only: xy)
     using mem_affine_3_minus [OF \<open>affine S\<close>] \<open>a \<in> S\<close> \<open>c \<in> S\<close> apply blast+
     done
   have affS: "affine hull S = S"
     by (simp add: \<open>affine S\<close>)
   have "aff_dim S = of_nat (card B) - 1"
     using card by simp
   also have "... = dim {x - c |x. x \<in> B}"
     by (simp add: affine_independent_card_dim_diffs [OF ind \<open>c \<in> B\<close>])
   also have "... = dim {x - c | x. x \<in> affine hull B}"
      by (simp add: diffs_affine_hull_span \<open>c \<in> B\<close>)
   also have "... = dim {x - a |x. x \<in> S}"
      by (simp add: affS aff *)
    finally show ?thesis .
 qed
 
 lemma aff_dim_linear_image_le:
   assumes "linear f"
     shows "aff_dim(f ` S) \<le> aff_dim S"
 proof -
   have "aff_dim (f ` T) \<le> aff_dim T" if "affine T" for T
   proof (cases "T = {}")
     case True then show ?thesis by (simp add: aff_dim_geq)
   next
     case False
     then obtain a where "a \<in> T" by auto
     have 1: "((\<lambda>x. x - f a) ` f ` T) = {x - f a |x. x \<in> f ` T}"
       by auto
     have 2: "{x - f a| x. x \<in> f ` T} = f ` {x - a| x. x \<in> T}"
       by (force simp: linear_diff [OF assms])
     have "aff_dim (f ` T) = int (dim {x - f a |x. x \<in> f ` T})"
       by (simp add: \<open>a \<in> T\<close> hull_inc aff_dim_eq_dim [of "f a"] 1 cong: image_cong_simp)
     also have "... = int (dim (f ` {x - a| x. x \<in> T}))"
       by (force simp: linear_diff [OF assms] 2)
     also have "... \<le> int (dim {x - a| x. x \<in> T})"
       by (simp add: dim_image_le [OF assms])
     also have "... \<le> aff_dim T"
       by (simp add: aff_dim_dim_affine_diffs [symmetric] \<open>a \<in> T\<close> \<open>affine T\<close>)
     finally show ?thesis .
   qed
   then
   have "aff_dim (f ` (affine hull S)) \<le> aff_dim (affine hull S)"
     using affine_affine_hull [of S] by blast
   then show ?thesis
     using affine_hull_linear_image assms linear_conv_bounded_linear by fastforce
 qed
 
 lemma aff_dim_injective_linear_image [simp]:
   assumes "linear f" "inj f"
     shows "aff_dim (f ` S) = aff_dim S"
 proof (rule antisym)
   show "aff_dim (f ` S) \<le> aff_dim S"
     by (simp add: aff_dim_linear_image_le assms(1))
 next
   obtain g where "linear g" "g \<circ> f = id"
     using assms(1) assms(2) linear_injective_left_inverse by blast
   then have "aff_dim S \<le> aff_dim(g ` f ` S)"
     by (simp add: image_comp)
   also have "... \<le> aff_dim (f ` S)"
     by (simp add: \<open>linear g\<close> aff_dim_linear_image_le)
   finally show "aff_dim S \<le> aff_dim (f ` S)" .
 qed
 
 
 lemma choose_affine_subset:
   assumes "affine S" "-1 \<le> d" and dle: "d \<le> aff_dim S"
   obtains T where "affine T" "T \<subseteq> S" "aff_dim T = d"
 proof (cases "d = -1 \<or> S={}")
   case True with assms show ?thesis
     by (metis aff_dim_empty affine_empty bot.extremum that eq_iff)
 next
   case False
   with assms obtain a where "a \<in> S" "0 \<le> d" by auto
   with assms have ss: "subspace ((+) (- a) ` S)"
     by (simp add: affine_diffs_subspace_subtract cong: image_cong_simp)
   have "nat d \<le> dim ((+) (- a) ` S)"
     by (metis aff_dim_subspace aff_dim_translation_eq dle nat_int nat_mono ss)
   then obtain T where "subspace T" and Tsb: "T \<subseteq> span ((+) (- a) ` S)"
                   and Tdim: "dim T = nat d"
     using choose_subspace_of_subspace [of "nat d" "(+) (- a) ` S"] by blast
   then have "affine T"
     using subspace_affine by blast
   then have "affine ((+) a ` T)"
     by (metis affine_hull_eq affine_hull_translation)
   moreover have "(+) a ` T \<subseteq> S"
   proof -
     have "T \<subseteq> (+) (- a) ` S"
       by (metis (no_types) span_eq_iff Tsb ss)
     then show "(+) a ` T \<subseteq> S"
       using add_ac by auto
   qed
   moreover have "aff_dim ((+) a ` T) = d"
     by (simp add: aff_dim_subspace Tdim \<open>0 \<le> d\<close> \<open>subspace T\<close> aff_dim_translation_eq)
   ultimately show ?thesis
     by (rule that)
 qed
 
 subsection\<open>Paracompactness\<close>
 
 proposition paracompact:
   fixes S :: "'a :: {metric_space,second_countable_topology} set"
   assumes "S \<subseteq> \<Union>\<C>" and opC: "\<And>T. T \<in> \<C> \<Longrightarrow> open T"
   obtains \<C>' where "S \<subseteq> \<Union> \<C>'"
                and "\<And>U. U \<in> \<C>' \<Longrightarrow> open U \<and> (\<exists>T. T \<in> \<C> \<and> U \<subseteq> T)"
                and "\<And>x. x \<in> S
                        \<Longrightarrow> \<exists>V. open V \<and> x \<in> V \<and> finite {U. U \<in> \<C>' \<and> (U \<inter> V \<noteq> {})}"
 proof (cases "S = {}")
   case True with that show ?thesis by blast
 next
   case False
   have "\<exists>T U. x \<in> U \<and> open U \<and> closure U \<subseteq> T \<and> T \<in> \<C>" if "x \<in> S" for x
   proof -
     obtain T where "x \<in> T" "T \<in> \<C>" "open T"
       using assms \<open>x \<in> S\<close> by blast
     then obtain e where "e > 0" "cball x e \<subseteq> T"
       by (force simp: open_contains_cball)
     then show ?thesis
       apply (rule_tac x = T in exI)
       apply (rule_tac x = "ball x e" in exI)
       using  \<open>T \<in> \<C>\<close>
       apply (simp add: closure_minimal)
       using closed_cball closure_minimal by blast
   qed
   then obtain F G where Gin: "x \<in> G x" and oG: "open (G x)"
                     and clos: "closure (G x) \<subseteq> F x" and Fin: "F x \<in> \<C>"
          if "x \<in> S" for x
     by metis
   then obtain \<F> where "\<F> \<subseteq> G ` S" "countable \<F>" "\<Union>\<F> = \<Union>(G ` S)"
     using Lindelof [of "G ` S"] by (metis image_iff)
   then obtain K where K: "K \<subseteq> S" "countable K" and eq: "\<Union>(G ` K) = \<Union>(G ` S)"
     by (metis countable_subset_image)
   with False Gin have "K \<noteq> {}" by force
   then obtain a :: "nat \<Rightarrow> 'a" where "range a = K"
     by (metis range_from_nat_into \<open>countable K\<close>)
   then have odif: "\<And>n. open (F (a n) - \<Union>{closure (G (a m)) |m. m < n})"
     using \<open>K \<subseteq> S\<close> Fin opC by (fastforce simp add:)
   let ?C = "range (\<lambda>n. F(a n) - \<Union>{closure(G(a m)) |m. m < n})"
   have enum_S: "\<exists>n. x \<in> F(a n) \<and> x \<in> G(a n)" if "x \<in> S" for x
   proof -
     have "\<exists>y \<in> K. x \<in> G y" using eq that Gin by fastforce
     then show ?thesis
       using clos K \<open>range a = K\<close> closure_subset by blast
   qed
   have 1: "S \<subseteq> Union ?C"
   proof
     fix x assume "x \<in> S"
     define n where "n \<equiv> LEAST n. x \<in> F(a n)"
     have n: "x \<in> F(a n)"
       using enum_S [OF \<open>x \<in> S\<close>] by (force simp: n_def intro: LeastI)
     have notn: "x \<notin> F(a m)" if "m < n" for m
       using that not_less_Least by (force simp: n_def)
     then have "x \<notin> \<Union>{closure (G (a m)) |m. m < n}"
       using n \<open>K \<subseteq> S\<close> \<open>range a = K\<close> clos notn by fastforce
     with n show "x \<in> Union ?C"
       by blast
   qed
   have 3: "\<exists>V. open V \<and> x \<in> V \<and> finite {U. U \<in> ?C \<and> (U \<inter> V \<noteq> {})}" if "x \<in> S" for x
   proof -
     obtain n where n: "x \<in> F(a n)" "x \<in> G(a n)"
       using \<open>x \<in> S\<close> enum_S by auto
     have "{U \<in> ?C. U \<inter> G (a n) \<noteq> {}} \<subseteq> (\<lambda>n. F(a n) - \<Union>{closure(G(a m)) |m. m < n}) ` atMost n"
     proof clarsimp
       fix k  assume "(F (a k) - \<Union>{closure (G (a m)) |m. m < k}) \<inter> G (a n) \<noteq> {}"
       then have "k \<le> n"
         by auto (metis closure_subset not_le subsetCE)
       then show "F (a k) - \<Union>{closure (G (a m)) |m. m < k}
                  \<in> (\<lambda>n. F (a n) - \<Union>{closure (G (a m)) |m. m < n}) ` {..n}"
         by force
     qed
     moreover have "finite ((\<lambda>n. F(a n) - \<Union>{closure(G(a m)) |m. m < n}) ` atMost n)"
       by force
     ultimately have *: "finite {U \<in> ?C. U \<inter> G (a n) \<noteq> {}}"
       using finite_subset by blast
     show ?thesis
       apply (rule_tac x="G (a n)" in exI)
       apply (intro conjI oG n *)
       using \<open>K \<subseteq> S\<close> \<open>range a = K\<close> apply blast
       done
   qed
   show ?thesis
     apply (rule that [OF 1 _ 3])
     using Fin \<open>K \<subseteq> S\<close> \<open>range a = K\<close>  apply (auto simp: odif)
     done
 qed
 
 corollary paracompact_closedin:
   fixes S :: "'a :: {metric_space,second_countable_topology} set"
   assumes cin: "closedin (top_of_set U) S"
       and oin: "\<And>T. T \<in> \<C> \<Longrightarrow> openin (top_of_set U) T"
       and "S \<subseteq> \<Union>\<C>"
   obtains \<C>' where "S \<subseteq> \<Union> \<C>'"
                and "\<And>V. V \<in> \<C>' \<Longrightarrow> openin (top_of_set U) V \<and> (\<exists>T. T \<in> \<C> \<and> V \<subseteq> T)"
                and "\<And>x. x \<in> U
                        \<Longrightarrow> \<exists>V. openin (top_of_set U) V \<and> x \<in> V \<and>
                                finite {X. X \<in> \<C>' \<and> (X \<inter> V \<noteq> {})}"
 proof -
   have "\<exists>Z. open Z \<and> (T = U \<inter> Z)" if "T \<in> \<C>" for T
     using oin [OF that] by (auto simp: openin_open)
   then obtain F where opF: "open (F T)" and intF: "U \<inter> F T = T" if "T \<in> \<C>" for T
     by metis
   obtain K where K: "closed K" "U \<inter> K = S"
     using cin by (auto simp: closedin_closed)
   have 1: "U \<subseteq> \<Union>(insert (- K) (F ` \<C>))"
     by clarsimp (metis Int_iff Union_iff \<open>U \<inter> K = S\<close> \<open>S \<subseteq> \<Union>\<C>\<close> subsetD intF)
   have 2: "\<And>T. T \<in> insert (- K) (F ` \<C>) \<Longrightarrow> open T"
     using \<open>closed K\<close> by (auto simp: opF)
   obtain \<D> where "U \<subseteq> \<Union>\<D>"
              and D1: "\<And>U. U \<in> \<D> \<Longrightarrow> open U \<and> (\<exists>T. T \<in> insert (- K) (F ` \<C>) \<and> U \<subseteq> T)"
              and D2: "\<And>x. x \<in> U \<Longrightarrow> \<exists>V. open V \<and> x \<in> V \<and> finite {U \<in> \<D>. U \<inter> V \<noteq> {}}"
     by (blast intro: paracompact [OF 1 2])
   let ?C = "{U \<inter> V |V. V \<in> \<D> \<and> (V \<inter> K \<noteq> {})}"
   show ?thesis
   proof (rule_tac \<C>' = "{U \<inter> V |V. V \<in> \<D> \<and> (V \<inter> K \<noteq> {})}" in that)
     show "S \<subseteq> \<Union>?C"
       using \<open>U \<inter> K = S\<close> \<open>U \<subseteq> \<Union>\<D>\<close> K by (blast dest!: subsetD)
     show "\<And>V. V \<in> ?C \<Longrightarrow> openin (top_of_set U) V \<and> (\<exists>T. T \<in> \<C> \<and> V \<subseteq> T)"
       using D1 intF by fastforce
     have *: "{X. (\<exists>V. X = U \<inter> V \<and> V \<in> \<D> \<and> V \<inter> K \<noteq> {}) \<and> X \<inter> (U \<inter> V) \<noteq> {}} \<subseteq>
              (\<lambda>x. U \<inter> x) ` {U \<in> \<D>. U \<inter> V \<noteq> {}}" for V
       by blast
     show "\<exists>V. openin (top_of_set U) V \<and> x \<in> V \<and> finite {X \<in> ?C. X \<inter> V \<noteq> {}}"
          if "x \<in> U" for x
       using D2 [OF that]
       apply clarify
       apply (rule_tac x="U \<inter> V" in exI)
       apply (auto intro: that finite_subset [OF *])
       done
     qed
 qed
 
 corollary\<^marker>\<open>tag unimportant\<close> paracompact_closed:
   fixes S :: "'a :: {metric_space,second_countable_topology} set"
   assumes "closed S"
       and opC: "\<And>T. T \<in> \<C> \<Longrightarrow> open T"
       and "S \<subseteq> \<Union>\<C>"
   obtains \<C>' where "S \<subseteq> \<Union>\<C>'"
                and "\<And>U. U \<in> \<C>' \<Longrightarrow> open U \<and> (\<exists>T. T \<in> \<C> \<and> U \<subseteq> T)"
                and "\<And>x. \<exists>V. open V \<and> x \<in> V \<and>
                                finite {U. U \<in> \<C>' \<and> (U \<inter> V \<noteq> {})}"
   by (rule paracompact_closedin [of UNIV S \<C>]) (auto simp: assms)
 
   
 subsection\<^marker>\<open>tag unimportant\<close>\<open>Closed-graph characterization of continuity\<close>
 
 lemma continuous_closed_graph_gen:
   fixes T :: "'b::real_normed_vector set"
   assumes contf: "continuous_on S f" and fim: "f ` S \<subseteq> T"
     shows "closedin (top_of_set (S \<times> T)) ((\<lambda>x. Pair x (f x)) ` S)"
 proof -
   have eq: "((\<lambda>x. Pair x (f x)) ` S) =(S \<times> T \<inter> (\<lambda>z. (f \<circ> fst)z - snd z) -` {0})"
     using fim by auto
   show ?thesis
     apply (subst eq)
     apply (intro continuous_intros continuous_closedin_preimage continuous_on_subset [OF contf])
     by auto
 qed
 
 lemma continuous_closed_graph_eq:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes "compact T" and fim: "f ` S \<subseteq> T"
   shows "continuous_on S f \<longleftrightarrow>
          closedin (top_of_set (S \<times> T)) ((\<lambda>x. Pair x (f x)) ` S)"
          (is "?lhs = ?rhs")
 proof -
   have "?lhs" if ?rhs
   proof (clarsimp simp add: continuous_on_closed_gen [OF fim])
     fix U
     assume U: "closedin (top_of_set T) U"
     have eq: "(S \<inter> f -` U) = fst ` (((\<lambda>x. Pair x (f x)) ` S) \<inter> (S \<times> U))"
       by (force simp: image_iff)
     show "closedin (top_of_set S) (S \<inter> f -` U)"
       by (simp add: U closedin_Int closedin_Times closed_map_fst [OF \<open>compact T\<close>] that eq)
   qed
   with continuous_closed_graph_gen assms show ?thesis by blast
 qed
 
 lemma continuous_closed_graph:
   fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
   assumes "closed S" and contf: "continuous_on S f"
   shows "closed ((\<lambda>x. Pair x (f x)) ` S)"
   apply (rule closedin_closed_trans)
    apply (rule continuous_closed_graph_gen [OF contf subset_UNIV])
   by (simp add: \<open>closed S\<close> closed_Times)
 
 lemma continuous_from_closed_graph:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes "compact T" and fim: "f ` S \<subseteq> T" and clo: "closed ((\<lambda>x. Pair x (f x)) ` S)"
   shows "continuous_on S f"
     using fim clo
     by (auto intro: closed_subset simp: continuous_closed_graph_eq [OF \<open>compact T\<close> fim])
 
 lemma continuous_on_Un_local_open:
   assumes opS: "openin (top_of_set (S \<union> T)) S"
       and opT: "openin (top_of_set (S \<union> T)) T"
       and contf: "continuous_on S f" and contg: "continuous_on T f"
     shows "continuous_on (S \<union> T) f"
   using pasting_lemma [of "{S,T}" "top_of_set (S \<union> T)" id euclidean "\<lambda>i. f" f] contf contg opS opT
   by (simp add: subtopology_subtopology) (metis inf.absorb2 openin_imp_subset)  
 
 lemma continuous_on_cases_local_open:
   assumes opS: "openin (top_of_set (S \<union> T)) S"
       and opT: "openin (top_of_set (S \<union> T)) T"
       and contf: "continuous_on S f" and contg: "continuous_on T g"
       and fg: "\<And>x. x \<in> S \<and> \<not>P x \<or> x \<in> T \<and> P x \<Longrightarrow> f x = g x"
     shows "continuous_on (S \<union> T) (\<lambda>x. if P x then f x else g x)"
 proof -
   have "\<And>x. x \<in> S \<Longrightarrow> (if P x then f x else g x) = f x"  "\<And>x. x \<in> T \<Longrightarrow> (if P x then f x else g x) = g x"
     by (simp_all add: fg)
   then have "continuous_on S (\<lambda>x. if P x then f x else g x)" "continuous_on T (\<lambda>x. if P x then f x else g x)"
     by (simp_all add: contf contg cong: continuous_on_cong)
   then show ?thesis
     by (rule continuous_on_Un_local_open [OF opS opT])
 qed
 
 lemma continuous_map_cases_le:
   assumes contp: "continuous_map X euclideanreal p"
     and contq: "continuous_map X euclideanreal q"
     and contf: "continuous_map (subtopology X {x. x \<in> topspace X \<and> p x \<le> q x}) Y f"
     and contg: "continuous_map (subtopology X {x. x \<in> topspace X \<and> q x \<le> p x}) Y g"
     and fg: "\<And>x. \<lbrakk>x \<in> topspace X; p x = q x\<rbrakk> \<Longrightarrow> f x = g x"
   shows "continuous_map X Y (\<lambda>x. if p x \<le> q x then f x else g x)"
 proof -
   have "continuous_map X Y (\<lambda>x. if q x - p x \<in> {0..} then f x else g x)"
   proof (rule continuous_map_cases_function)
     show "continuous_map X euclideanreal (\<lambda>x. q x - p x)"
       by (intro contp contq continuous_intros)
     show "continuous_map (subtopology X {x \<in> topspace X. q x - p x \<in> euclideanreal closure_of {0..}}) Y f"
       by (simp add: contf)
     show "continuous_map (subtopology X {x \<in> topspace X. q x - p x \<in> euclideanreal closure_of (topspace euclideanreal - {0..})}) Y g"
       by (simp add: contg flip: Compl_eq_Diff_UNIV)
   qed (auto simp: fg)
   then show ?thesis
     by simp
 qed
 
 lemma continuous_map_cases_lt:
   assumes contp: "continuous_map X euclideanreal p"
     and contq: "continuous_map X euclideanreal q"
     and contf: "continuous_map (subtopology X {x. x \<in> topspace X \<and> p x \<le> q x}) Y f"
     and contg: "continuous_map (subtopology X {x. x \<in> topspace X \<and> q x \<le> p x}) Y g"
     and fg: "\<And>x. \<lbrakk>x \<in> topspace X; p x = q x\<rbrakk> \<Longrightarrow> f x = g x"
   shows "continuous_map X Y (\<lambda>x. if p x < q x then f x else g x)"
 proof -
   have "continuous_map X Y (\<lambda>x. if q x - p x \<in> {0<..} then f x else g x)"
   proof (rule continuous_map_cases_function)
     show "continuous_map X euclideanreal (\<lambda>x. q x - p x)"
       by (intro contp contq continuous_intros)
     show "continuous_map (subtopology X {x \<in> topspace X. q x - p x \<in> euclideanreal closure_of {0<..}}) Y f"
       by (simp add: contf)
     show "continuous_map (subtopology X {x \<in> topspace X. q x - p x \<in> euclideanreal closure_of (topspace euclideanreal - {0<..})}) Y g"
       by (simp add: contg flip: Compl_eq_Diff_UNIV)
   qed (auto simp: fg)
   then show ?thesis
     by simp
 qed
 
 subsection\<^marker>\<open>tag unimportant\<close>\<open>The union of two collinear segments is another segment\<close>
 
 proposition\<^marker>\<open>tag unimportant\<close> in_convex_hull_exchange:
   fixes a :: "'a::euclidean_space"
   assumes a: "a \<in> convex hull S" and xS: "x \<in> convex hull S"
   obtains b where "b \<in> S" "x \<in> convex hull (insert a (S - {b}))"
 proof (cases "a \<in> S")
   case True
   with xS insert_Diff that  show ?thesis by fastforce
 next
   case False
   show ?thesis
   proof (cases "finite S \<and> card S \<le> Suc (DIM('a))")
     case True
     then obtain u where u0: "\<And>i. i \<in> S \<Longrightarrow> 0 \<le> u i" and u1: "sum u S = 1"
                     and ua: "(\<Sum>i\<in>S. u i *\<^sub>R i) = a"
         using a by (auto simp: convex_hull_finite)
     obtain v where v0: "\<And>i. i \<in> S \<Longrightarrow> 0 \<le> v i" and v1: "sum v S = 1"
                and vx: "(\<Sum>i\<in>S. v i *\<^sub>R i) = x"
       using True xS by (auto simp: convex_hull_finite)
     show ?thesis
     proof (cases "\<exists>b. b \<in> S \<and> v b = 0")
       case True
       then obtain b where b: "b \<in> S" "v b = 0"
         by blast
       show ?thesis
       proof
         have fin: "finite (insert a (S - {b}))"
           using sum.infinite v1 by fastforce
         show "x \<in> convex hull insert a (S - {b})"
           unfolding convex_hull_finite [OF fin] mem_Collect_eq
         proof (intro conjI exI ballI)
           have "(\<Sum>x \<in> insert a (S - {b}). if x = a then 0 else v x) =
                 (\<Sum>x \<in> S - {b}. if x = a then 0 else v x)"
             apply (rule sum.mono_neutral_right)
             using fin by auto
           also have "... = (\<Sum>x \<in> S - {b}. v x)"
             using b False by (auto intro!: sum.cong split: if_split_asm)
           also have "... = (\<Sum>x\<in>S. v x)"
             by (metis \<open>v b = 0\<close> diff_zero sum.infinite sum_diff1 u1 zero_neq_one)
           finally show "(\<Sum>x\<in>insert a (S - {b}). if x = a then 0 else v x) = 1"
             by (simp add: v1)
           show "\<And>x. x \<in> insert a (S - {b}) \<Longrightarrow> 0 \<le> (if x = a then 0 else v x)"
             by (auto simp: v0)
           have "(\<Sum>x \<in> insert a (S - {b}). (if x = a then 0 else v x) *\<^sub>R x) =
                 (\<Sum>x \<in> S - {b}. (if x = a then 0 else v x) *\<^sub>R x)"
             apply (rule sum.mono_neutral_right)
             using fin by auto
           also have "... = (\<Sum>x \<in> S - {b}. v x *\<^sub>R x)"
             using b False by (auto intro!: sum.cong split: if_split_asm)
           also have "... = (\<Sum>x\<in>S. v x *\<^sub>R x)"
             by (metis (no_types, lifting) b(2) diff_zero fin finite.emptyI finite_Diff2 finite_insert scale_eq_0_iff sum_diff1)
           finally show "(\<Sum>x\<in>insert a (S - {b}). (if x = a then 0 else v x) *\<^sub>R x) = x"
             by (simp add: vx)
         qed
       qed (rule \<open>b \<in> S\<close>)
     next
       case False
       have le_Max: "u i / v i \<le> Max ((\<lambda>i. u i / v i) ` S)" if "i \<in> S" for i
         by (simp add: True that)
       have "Max ((\<lambda>i. u i / v i) ` S) \<in> (\<lambda>i. u i / v i) ` S"
         using True v1 by (auto intro: Max_in)
       then obtain b where "b \<in> S" and beq: "Max ((\<lambda>b. u b / v b) ` S) = u b / v b"
         by blast
       then have "0 \<noteq> u b / v b"
         using le_Max beq divide_le_0_iff le_numeral_extra(2) sum_nonpos u1
         by (metis False eq_iff v0)
       then have  "0 < u b" "0 < v b"
         using False \<open>b \<in> S\<close> u0 v0 by force+
       have fin: "finite (insert a (S - {b}))"
         using sum.infinite v1 by fastforce
       show ?thesis
       proof
         show "x \<in> convex hull insert a (S - {b})"
           unfolding convex_hull_finite [OF fin] mem_Collect_eq
         proof (intro conjI exI ballI)
           have "(\<Sum>x \<in> insert a (S - {b}). if x=a then v b / u b else v x - (v b / u b) * u x) =
                 v b / u b + (\<Sum>x \<in> S - {b}. v x - (v b / u b) * u x)"
             using \<open>a \<notin> S\<close> \<open>b \<in> S\<close> True  apply simp
             apply (rule sum.cong, auto)
             done
           also have "... = v b / u b + (\<Sum>x \<in> S - {b}. v x) - (v b / u b) * (\<Sum>x \<in> S - {b}. u x)"
             by (simp add: Groups_Big.sum_subtractf sum_distrib_left)
           also have "... = (\<Sum>x\<in>S. v x)"
             using \<open>0 < u b\<close> True  by (simp add: Groups_Big.sum_diff1 u1 field_simps)
           finally show "sum (\<lambda>x. if x=a then v b / u b else v x - (v b / u b) * u x) (insert a (S - {b})) = 1"
             by (simp add: v1)
           show "0 \<le> (if i = a then v b / u b else v i - v b / u b * u i)"
             if "i \<in> insert a (S - {b})" for i
             using \<open>0 < u b\<close> \<open>0 < v b\<close> v0 [of i] le_Max [of i] beq that False
             by (auto simp: field_simps split: if_split_asm)
           have "(\<Sum>x\<in>insert a (S - {b}). (if x=a then v b / u b else v x - v b / u b * u x) *\<^sub>R x) =
                 (v b / u b) *\<^sub>R a + (\<Sum>x\<in>S - {b}. (v x - v b / u b * u x) *\<^sub>R x)"
             using \<open>a \<notin> S\<close> \<open>b \<in> S\<close> True  apply simp
             apply (rule sum.cong, auto)
             done
           also have "... = (v b / u b) *\<^sub>R a + (\<Sum>x \<in> S - {b}. v x *\<^sub>R x) - (v b / u b) *\<^sub>R (\<Sum>x \<in> S - {b}. u x *\<^sub>R x)"
             by (simp add: Groups_Big.sum_subtractf scaleR_left_diff_distrib sum_distrib_left scale_sum_right)
           also have "... = (\<Sum>x\<in>S. v x *\<^sub>R x)"
             using \<open>0 < u b\<close> True  by (simp add: ua vx Groups_Big.sum_diff1 algebra_simps)
           finally
           show "(\<Sum>x\<in>insert a (S - {b}). (if x=a then v b / u b else v x - v b / u b * u x) *\<^sub>R x) = x"
             by (simp add: vx)
         qed
       qed (rule \<open>b \<in> S\<close>)
     qed
   next
     case False
     obtain T where "finite T" "T \<subseteq> S" and caT: "card T \<le> Suc (DIM('a))" and xT: "x \<in> convex hull T"
       using xS by (auto simp: caratheodory [of S])
     with False obtain b where b: "b \<in> S" "b \<notin> T"
       by (metis antisym subsetI)
     show ?thesis
     proof
       show "x \<in> convex hull insert a (S - {b})"
         using  \<open>T \<subseteq> S\<close> b by (blast intro: subsetD [OF hull_mono xT])
     qed (rule \<open>b \<in> S\<close>)
   qed
 qed
 
 lemma convex_hull_exchange_Union:
   fixes a :: "'a::euclidean_space"
   assumes "a \<in> convex hull S"
   shows "convex hull S = (\<Union>b \<in> S. convex hull (insert a (S - {b})))" (is "?lhs = ?rhs")
 proof
   show "?lhs \<subseteq> ?rhs"
     by (blast intro: in_convex_hull_exchange [OF assms])
   show "?rhs \<subseteq> ?lhs"
   proof clarify
     fix x b
     assume"b \<in> S" "x \<in> convex hull insert a (S - {b})"
     then show "x \<in> convex hull S" if "b \<in> S"
       by (metis (no_types) that assms order_refl hull_mono hull_redundant insert_Diff_single insert_subset subsetCE)
   qed
 qed
 
 lemma Un_closed_segment:
   fixes a :: "'a::euclidean_space"
   assumes "b \<in> closed_segment a c"
     shows "closed_segment a b \<union> closed_segment b c = closed_segment a c"
 proof (cases "c = a")
   case True
   with assms show ?thesis by simp
 next
   case False
   with assms have "convex hull {a, b} \<union> convex hull {b, c} = (\<Union>ba\<in>{a, c}. convex hull insert b ({a, c} - {ba}))"
     by (auto simp: insert_Diff_if insert_commute)
   then show ?thesis
     using convex_hull_exchange_Union
     by (metis assms segment_convex_hull)
 qed
 
 lemma Un_open_segment:
   fixes a :: "'a::euclidean_space"
   assumes "b \<in> open_segment a c"
   shows "open_segment a b \<union> {b} \<union> open_segment b c = open_segment a c"
 proof -
   have b: "b \<in> closed_segment a c"
     by (simp add: assms open_closed_segment)
   have *: "open_segment a c \<subseteq> insert b (open_segment a b \<union> open_segment b c)"
           if "{b,c,a} \<union> open_segment a b \<union> open_segment b c = {c,a} \<union> open_segment a c"
   proof -
     have "insert a (insert c (insert b (open_segment a b \<union> open_segment b c))) = insert a (insert c (open_segment a c))"
       using that by (simp add: insert_commute)
     then show ?thesis
       by (metis (no_types) Diff_cancel Diff_eq_empty_iff Diff_insert2 open_segment_def)
   qed
   show ?thesis
     using Un_closed_segment [OF b]
     apply (simp add: closed_segment_eq_open)
       apply (rule equalityI)
     using assms
      apply (simp add: b subset_open_segment)
       using * by (simp add: insert_commute)
 qed
 
 subsection\<open>Covering an open set by a countable chain of compact sets\<close>
   
 proposition open_Union_compact_subsets:
   fixes S :: "'a::euclidean_space set"
   assumes "open S"
   obtains C where "\<And>n. compact(C n)" "\<And>n. C n \<subseteq> S"
                   "\<And>n. C n \<subseteq> interior(C(Suc n))"
                   "\<Union>(range C) = S"
                   "\<And>K. \<lbrakk>compact K; K \<subseteq> S\<rbrakk> \<Longrightarrow> \<exists>N. \<forall>n\<ge>N. K \<subseteq> (C n)"
 proof (cases "S = {}")
   case True
   then show ?thesis
     by (rule_tac C = "\<lambda>n. {}" in that) auto
 next
   case False
   then obtain a where "a \<in> S"
     by auto
   let ?C = "\<lambda>n. cball a (real n) - (\<Union>x \<in> -S. \<Union>e \<in> ball 0 (1 / real(Suc n)). {x + e})"
   have "\<exists>N. \<forall>n\<ge>N. K \<subseteq> (f n)"
         if "\<And>n. compact(f n)" and sub_int: "\<And>n. f n \<subseteq> interior (f(Suc n))"
             and eq: "\<Union>(range f) = S" and "compact K" "K \<subseteq> S" for f K
   proof -
     have *: "\<forall>n. f n \<subseteq> (\<Union>n. interior (f n))"
       by (meson Sup_upper2 UNIV_I \<open>\<And>n. f n \<subseteq> interior (f (Suc n))\<close> image_iff)
     have mono: "\<And>m n. m \<le> n \<Longrightarrow>f m \<subseteq> f n"
       by (meson dual_order.trans interior_subset lift_Suc_mono_le sub_int)
     obtain I where "finite I" and I: "K \<subseteq> (\<Union>i\<in>I. interior (f i))"
     proof (rule compactE_image [OF \<open>compact K\<close>])
       show "K \<subseteq> (\<Union>n. interior (f n))"
         using \<open>K \<subseteq> S\<close> \<open>\<Union>(f ` UNIV) = S\<close> * by blast
     qed auto
     { fix n
       assume n: "Max I \<le> n"
       have "(\<Union>i\<in>I. interior (f i)) \<subseteq> f n"
         by (rule UN_least) (meson dual_order.trans interior_subset mono I Max_ge [OF \<open>finite I\<close>] n)
       then have "K \<subseteq> f n"
         using I by auto
     }
     then show ?thesis
       by blast
   qed
   moreover have "\<exists>f. (\<forall>n. compact(f n)) \<and> (\<forall>n. (f n) \<subseteq> S) \<and> (\<forall>n. (f n) \<subseteq> interior(f(Suc n))) \<and>
                      ((\<Union>(range f) = S))"
   proof (intro exI conjI allI)
     show "\<And>n. compact (?C n)"
       by (auto simp: compact_diff open_sums)
     show "\<And>n. ?C n \<subseteq> S"
       by auto
     show "?C n \<subseteq> interior (?C (Suc n))" for n
     proof (simp add: interior_diff, rule Diff_mono)
       show "cball a (real n) \<subseteq> ball a (1 + real n)"
         by (simp add: cball_subset_ball_iff)
       have cl: "closed (\<Union>x\<in>- S. \<Union>e\<in>cball 0 (1 / (2 + real n)). {x + e})"
         using assms by (auto intro: closed_compact_sums)
       have "closure (\<Union>x\<in>- S. \<Union>y\<in>ball 0 (1 / (2 + real n)). {x + y})
             \<subseteq> (\<Union>x \<in> -S. \<Union>e \<in> cball 0 (1 / (2 + real n)). {x + e})"
         by (intro closure_minimal UN_mono ball_subset_cball order_refl cl)
       also have "... \<subseteq> (\<Union>x \<in> -S. \<Union>y\<in>ball 0 (1 / (1 + real n)). {x + y})"
         apply (intro UN_mono order_refl)
         apply (simp add: cball_subset_ball_iff field_split_simps)
         done
       finally show "closure (\<Union>x\<in>- S. \<Union>y\<in>ball 0 (1 / (2 + real n)). {x + y})
                     \<subseteq> (\<Union>x \<in> -S. \<Union>y\<in>ball 0 (1 / (1 + real n)). {x + y})" .
     qed
     have "S \<subseteq> \<Union> (range ?C)"
     proof
       fix x
       assume x: "x \<in> S"
       then obtain e where "e > 0" and e: "ball x e \<subseteq> S"
         using assms open_contains_ball by blast
       then obtain N1 where "N1 > 0" and N1: "real N1 > 1/e"
         using reals_Archimedean2
         by (metis divide_less_0_iff less_eq_real_def neq0_conv not_le of_nat_0 of_nat_1 of_nat_less_0_iff)
       obtain N2 where N2: "norm(x - a) \<le> real N2"
         by (meson real_arch_simple)
       have N12: "inverse((N1 + N2) + 1) \<le> inverse(N1)"
         using \<open>N1 > 0\<close> by (auto simp: field_split_simps)
       have "x \<noteq> y + z" if "y \<notin> S" "norm z < 1 / (1 + (real N1 + real N2))" for y z
       proof -
         have "e * real N1 < e * (1 + (real N1 + real N2))"
           by (simp add: \<open>0 < e\<close>)
         then have "1 / (1 + (real N1 + real N2)) < e"
           using N1 \<open>e > 0\<close>
           by (metis divide_less_eq less_trans mult.commute of_nat_add of_nat_less_0_iff of_nat_Suc)
         then have "x - z \<in> ball x e"
           using that by simp
         then have "x - z \<in> S"
           using e by blast
         with that show ?thesis
           by auto
       qed
       with N2 show "x \<in> \<Union> (range ?C)"
         by (rule_tac a = "N1+N2" in UN_I) (auto simp: dist_norm norm_minus_commute)
     qed
     then show "\<Union> (range ?C) = S" by auto
   qed
   ultimately show ?thesis
     using that by metis
 qed
 
 
 subsection\<open>Orthogonal complement\<close>
 
 definition\<^marker>\<open>tag important\<close> orthogonal_comp ("_\<^sup>\<bottom>" [80] 80)
   where "orthogonal_comp W \<equiv> {x. \<forall>y \<in> W. orthogonal y x}"
 
 proposition subspace_orthogonal_comp: "subspace (W\<^sup>\<bottom>)"
   unfolding subspace_def orthogonal_comp_def orthogonal_def
   by (auto simp: inner_right_distrib)
 
 lemma orthogonal_comp_anti_mono:
   assumes "A \<subseteq> B"
   shows "B\<^sup>\<bottom> \<subseteq> A\<^sup>\<bottom>"
 proof
   fix x assume x: "x \<in> B\<^sup>\<bottom>"
   show "x \<in> orthogonal_comp A" using x unfolding orthogonal_comp_def
     by (simp add: orthogonal_def, metis assms in_mono)
 qed
 
 lemma orthogonal_comp_null [simp]: "{0}\<^sup>\<bottom> = UNIV"
   by (auto simp: orthogonal_comp_def orthogonal_def)
 
 lemma orthogonal_comp_UNIV [simp]: "UNIV\<^sup>\<bottom> = {0}"
   unfolding orthogonal_comp_def orthogonal_def
   by auto (use inner_eq_zero_iff in blast)
 
 lemma orthogonal_comp_subset: "U \<subseteq> U\<^sup>\<bottom>\<^sup>\<bottom>"
   by (auto simp: orthogonal_comp_def orthogonal_def inner_commute)
 
 lemma subspace_sum_minimal:
   assumes "S \<subseteq> U" "T \<subseteq> U" "subspace U"
   shows "S + T \<subseteq> U"
 proof
   fix x
   assume "x \<in> S + T"
   then obtain xs xt where "xs \<in> S" "xt \<in> T" "x = xs+xt"
     by (meson set_plus_elim)
   then show "x \<in> U"
     by (meson assms subsetCE subspace_add)
 qed
 
 proposition subspace_sum_orthogonal_comp:
   fixes U :: "'a :: euclidean_space set"
   assumes "subspace U"
   shows "U + U\<^sup>\<bottom> = UNIV"
 proof -
   obtain B where "B \<subseteq> U"
     and ortho: "pairwise orthogonal B" "\<And>x. x \<in> B \<Longrightarrow> norm x = 1"
     and "independent B" "card B = dim U" "span B = U"
     using orthonormal_basis_subspace [OF assms] by metis
   then have "finite B"
     by (simp add: indep_card_eq_dim_span)
   have *: "\<forall>x\<in>B. \<forall>y\<in>B. x \<bullet> y = (if x=y then 1 else 0)"
     using ortho norm_eq_1 by (auto simp: orthogonal_def pairwise_def)
   { fix v
     let ?u = "\<Sum>b\<in>B. (v \<bullet> b) *\<^sub>R b"
     have "v = ?u + (v - ?u)"
       by simp
     moreover have "?u \<in> U"
       by (metis (no_types, lifting) \<open>span B = U\<close> assms subspace_sum span_base span_mul)
     moreover have "(v - ?u) \<in> U\<^sup>\<bottom>"
     proof (clarsimp simp: orthogonal_comp_def orthogonal_def)
       fix y
       assume "y \<in> U"
       with \<open>span B = U\<close> span_finite [OF \<open>finite B\<close>]
       obtain u where u: "y = (\<Sum>b\<in>B. u b *\<^sub>R b)"
         by auto
       have "b \<bullet> (v - ?u) = 0" if "b \<in> B" for b
         using that \<open>finite B\<close>
         by (simp add: * algebra_simps inner_sum_right if_distrib [of "(*)v" for v] inner_commute cong: if_cong)
       then show "y \<bullet> (v - ?u) = 0"
         by (simp add: u inner_sum_left)
     qed
     ultimately have "v \<in> U + U\<^sup>\<bottom>"
       using set_plus_intro by fastforce
   } then show ?thesis
     by auto
 qed
 
 lemma orthogonal_Int_0:
   assumes "subspace U"
   shows "U \<inter> U\<^sup>\<bottom> = {0}"
   using orthogonal_comp_def orthogonal_self
   by (force simp: assms subspace_0 subspace_orthogonal_comp)
 
 lemma orthogonal_comp_self:
   fixes U :: "'a :: euclidean_space set"
   assumes "subspace U"
   shows "U\<^sup>\<bottom>\<^sup>\<bottom> = U"
 proof
   have ssU': "subspace (U\<^sup>\<bottom>)"
     by (simp add: subspace_orthogonal_comp)
   have "u \<in> U" if "u \<in> U\<^sup>\<bottom>\<^sup>\<bottom>" for u
   proof -
     obtain v w where "u = v+w" "v \<in> U" "w \<in> U\<^sup>\<bottom>"
       using subspace_sum_orthogonal_comp [OF assms] set_plus_elim by blast
     then have "u-v \<in> U\<^sup>\<bottom>"
       by simp
     moreover have "v \<in> U\<^sup>\<bottom>\<^sup>\<bottom>"
       using \<open>v \<in> U\<close> orthogonal_comp_subset by blast
     then have "u-v \<in> U\<^sup>\<bottom>\<^sup>\<bottom>"
       by (simp add: subspace_diff subspace_orthogonal_comp that)
     ultimately have "u-v = 0"
       using orthogonal_Int_0 ssU' by blast
     with \<open>v \<in> U\<close> show ?thesis
       by auto
   qed
   then show "U\<^sup>\<bottom>\<^sup>\<bottom> \<subseteq> U"
     by auto
 qed (use orthogonal_comp_subset in auto)
 
 lemma ker_orthogonal_comp_adjoint:
   fixes f :: "'m::euclidean_space \<Rightarrow> 'n::euclidean_space"
   assumes "linear f"
   shows "f -` {0} =  (range (adjoint f))\<^sup>\<bottom>"
   apply (auto simp: orthogonal_comp_def orthogonal_def)
   apply (simp add: adjoint_works assms(1) inner_commute)
   by (metis adjoint_works all_zero_iff assms(1) inner_commute)
 
 subsection\<^marker>\<open>tag unimportant\<close> \<open>A non-injective linear function maps into a hyperplane.\<close>
 
 lemma linear_surj_adj_imp_inj:
   fixes f :: "'m::euclidean_space \<Rightarrow> 'n::euclidean_space"
   assumes "linear f" "surj (adjoint f)"
   shows "inj f"
 proof -
   have "\<exists>x. y = adjoint f x" for y
     using assms by (simp add: surjD)
   then show "inj f"
     using assms unfolding inj_on_def image_def
     by (metis (no_types) adjoint_works euclidean_eqI)
 qed
 
 \<comment> \<open>\<^url>\<open>https://mathonline.wikidot.com/injectivity-and-surjectivity-of-the-adjoint-of-a-linear-map\<close>\<close>
 lemma surj_adjoint_iff_inj [simp]:
   fixes f :: "'m::euclidean_space \<Rightarrow> 'n::euclidean_space"
   assumes "linear f"
   shows  "surj (adjoint f) \<longleftrightarrow> inj f"
 proof
   assume "surj (adjoint f)"
   then show "inj f"
     by (simp add: assms linear_surj_adj_imp_inj)
 next
   assume "inj f"
   have "f -` {0} = {0}"
     using assms \<open>inj f\<close> linear_0 linear_injective_0 by fastforce
   moreover have "f -` {0} = range (adjoint f)\<^sup>\<bottom>"
     by (intro ker_orthogonal_comp_adjoint assms)
   ultimately have "range (adjoint f)\<^sup>\<bottom>\<^sup>\<bottom> = UNIV"
     by (metis orthogonal_comp_null)
   then show "surj (adjoint f)"
     using adjoint_linear \<open>linear f\<close>
     by (subst (asm) orthogonal_comp_self)
       (simp add: adjoint_linear linear_subspace_image)
 qed
 
 lemma inj_adjoint_iff_surj [simp]:
   fixes f :: "'m::euclidean_space \<Rightarrow> 'n::euclidean_space"
   assumes "linear f"
   shows  "inj (adjoint f) \<longleftrightarrow> surj f"
 proof
   assume "inj (adjoint f)"
   have "(adjoint f) -` {0} = {0}"
     by (metis \<open>inj (adjoint f)\<close> adjoint_linear assms surj_adjoint_iff_inj ker_orthogonal_comp_adjoint orthogonal_comp_UNIV)
   then have "(range(f))\<^sup>\<bottom> = {0}"
     by (metis (no_types, hide_lams) adjoint_adjoint adjoint_linear assms ker_orthogonal_comp_adjoint set_zero)
   then show "surj f"
     by (metis \<open>inj (adjoint f)\<close> adjoint_adjoint adjoint_linear assms surj_adjoint_iff_inj)
 next
   assume "surj f"
   then have "range f = (adjoint f -` {0})\<^sup>\<bottom>"
     by (simp add: adjoint_adjoint adjoint_linear assms ker_orthogonal_comp_adjoint)
   then have "{0} = adjoint f -` {0}"
     using \<open>surj f\<close> adjoint_adjoint adjoint_linear assms ker_orthogonal_comp_adjoint by force
   then show "inj (adjoint f)"
     by (simp add: \<open>surj f\<close> adjoint_adjoint adjoint_linear assms linear_surj_adj_imp_inj)
 qed
 
 lemma linear_singular_into_hyperplane:
   fixes f :: "'n::euclidean_space \<Rightarrow> 'n"
   assumes "linear f"
   shows "\<not> inj f \<longleftrightarrow> (\<exists>a. a \<noteq> 0 \<and> (\<forall>x. a \<bullet> f x = 0))" (is "_ = ?rhs")
 proof
   assume "\<not>inj f"
   then show ?rhs
     using all_zero_iff
     by (metis (no_types, hide_lams) adjoint_clauses(2) adjoint_linear assms
         linear_injective_0 linear_injective_imp_surjective linear_surj_adj_imp_inj)
 next
   assume ?rhs
   then show "\<not>inj f"
     by (metis assms linear_injective_isomorphism all_zero_iff)
 qed
 
 lemma linear_singular_image_hyperplane:
   fixes f :: "'n::euclidean_space \<Rightarrow> 'n"
   assumes "linear f" "\<not>inj f"
   obtains a where "a \<noteq> 0" "\<And>S. f ` S \<subseteq> {x. a \<bullet> x = 0}"
   using assms by (fastforce simp add: linear_singular_into_hyperplane)
 
 end