(*  Title:      HOL/Multivariate_Analysis/Path_Connected.thy

     Author:     Robert Himmelmann, TU Muenchen

 *)

 header {* Continuous paths and path-connected sets *}

 theory Path_Connected

 imports Convex_Euclidean_Space

 begin

 subsection {* Paths. *}

 definition

   path :: "(real \<Rightarrow> 'a::topological_space) \<Rightarrow> bool"

   where "path g \<longleftrightarrow> continuous_on {0 .. 1} g"

 definition

   pathstart :: "(real \<Rightarrow> 'a::topological_space) \<Rightarrow> 'a"

   where "pathstart g = g 0"

 definition

   pathfinish :: "(real \<Rightarrow> 'a::topological_space) \<Rightarrow> 'a"

   where "pathfinish g = g 1"

 definition

   path_image :: "(real \<Rightarrow> 'a::topological_space) \<Rightarrow> 'a set"

   where "path_image g = g ` {0 .. 1}"

 definition

   reversepath :: "(real \<Rightarrow> 'a::topological_space) \<Rightarrow> (real \<Rightarrow> 'a)"

   where "reversepath g = (\<lambda>x. g(1 - x))"

 definition

   joinpaths :: "(real \<Rightarrow> 'a::topological_space) \<Rightarrow> (real \<Rightarrow> 'a) \<Rightarrow> (real \<Rightarrow> 'a)"

     (infixr "+++" 75)

   where "g1 +++ g2 = (\<lambda>x. if x \<le> 1/2 then g1 (2 * x) else g2 (2 * x - 1))"

 definition

   simple_path :: "(real \<Rightarrow> 'a::topological_space) \<Rightarrow> bool"

   where "simple_path g \<longleftrightarrow>

   (\<forall>x\<in>{0..1}. \<forall>y\<in>{0..1}. g x = g y \<longrightarrow> x = y \<or> x = 0 \<and> y = 1 \<or> x = 1 \<and> y = 0)"

 definition

   injective_path :: "(real \<Rightarrow> 'a::topological_space) \<Rightarrow> bool"

   where "injective_path g \<longleftrightarrow> (\<forall>x\<in>{0..1}. \<forall>y\<in>{0..1}. g x = g y \<longrightarrow> x = y)"

 subsection {* Some lemmas about these concepts. *}

 lemma injective_imp_simple_path:

   "injective_path g \<Longrightarrow> simple_path g"

   unfolding injective_path_def simple_path_def by auto

 lemma path_image_nonempty: "path_image g \<noteq> {}"

   unfolding path_image_def image_is_empty interval_eq_empty by auto 

 lemma pathstart_in_path_image[intro]: "(pathstart g) \<in> path_image g"

   unfolding pathstart_def path_image_def by auto

 lemma pathfinish_in_path_image[intro]: "(pathfinish g) \<in> path_image g"

   unfolding pathfinish_def path_image_def by auto

 lemma connected_path_image[intro]: "path g \<Longrightarrow> connected(path_image g)"

   unfolding path_def path_image_def

   apply (erule connected_continuous_image)

   by(rule convex_connected, rule convex_real_interval)

 lemma compact_path_image[intro]: "path g \<Longrightarrow> compact(path_image g)"

   unfolding path_def path_image_def

   by (erule compact_continuous_image, rule compact_interval)

 lemma reversepath_reversepath[simp]: "reversepath(reversepath g) = g"

   unfolding reversepath_def by auto

 lemma pathstart_reversepath[simp]: "pathstart(reversepath g) = pathfinish g"

   unfolding pathstart_def reversepath_def pathfinish_def by auto

 lemma pathfinish_reversepath[simp]: "pathfinish(reversepath g) = pathstart g"

   unfolding pathstart_def reversepath_def pathfinish_def by auto

 lemma pathstart_join[simp]: "pathstart(g1 +++ g2) = pathstart g1"

   unfolding pathstart_def joinpaths_def pathfinish_def by auto

 lemma pathfinish_join[simp]:"pathfinish(g1 +++ g2) = pathfinish g2"

   unfolding pathstart_def joinpaths_def pathfinish_def by auto

 lemma path_image_reversepath[simp]: "path_image(reversepath g) = path_image g" proof-

   have *:"\<And>g. path_image(reversepath g) \<subseteq> path_image g"

     unfolding path_image_def subset_eq reversepath_def Ball_def image_iff apply(rule,rule,erule bexE)  

     apply(rule_tac x="1 - xa" in bexI) by auto

   show ?thesis using *[of g] *[of "reversepath g"] unfolding reversepath_reversepath by auto qed

 lemma path_reversepath[simp]: "path(reversepath g) \<longleftrightarrow> path g" proof-

   have *:"\<And>g. path g \<Longrightarrow> path(reversepath g)" unfolding path_def reversepath_def

     apply(rule continuous_on_compose[unfolded o_def, of _ "\<lambda>x. 1 - x"])

     apply(intro continuous_on_intros)

     apply(rule continuous_on_subset[of "{0..1}"], assumption) by auto

   show ?thesis using *[of "reversepath g"] *[of g] unfolding reversepath_reversepath by (rule iffI) qed

 lemmas reversepath_simps = path_reversepath path_image_reversepath pathstart_reversepath pathfinish_reversepath

 lemma path_join[simp]: assumes "pathfinish g1 = pathstart g2" shows "path (g1 +++ g2) \<longleftrightarrow>  path g1 \<and> path g2"

   unfolding path_def pathfinish_def pathstart_def apply rule defer apply(erule conjE) proof-

   assume as:"continuous_on {0..1} (g1 +++ g2)"

   have *:"g1 = (\<lambda>x. g1 (2 *\<^sub>R x)) \<circ> (\<lambda>x. (1/2) *\<^sub>R x)" 

          "g2 = (\<lambda>x. g2 (2 *\<^sub>R x - 1)) \<circ> (\<lambda>x. (1/2) *\<^sub>R (x + 1))"

     unfolding o_def by (auto simp add: add_divide_distrib)

   have "op *\<^sub>R (1 / 2) ` {0::real..1} \<subseteq> {0..1}"  "(\<lambda>x. (1 / 2) *\<^sub>R (x + 1)) ` {(0::real)..1} \<subseteq> {0..1}"

     by auto

   thus "continuous_on {0..1} g1 \<and> continuous_on {0..1} g2" apply -apply rule

     apply(subst *) defer apply(subst *) apply (rule_tac[!] continuous_on_compose)

     apply (intro continuous_on_intros) defer

     apply (intro continuous_on_intros)

     apply(rule_tac[!] continuous_on_eq[of _ "g1 +++ g2"]) defer prefer 3

     apply(rule_tac[1-2] continuous_on_subset[of "{0 .. 1}"]) apply(rule as, assumption, rule as, assumption)

     apply(rule) defer apply rule proof-

     fix x assume "x \<in> op *\<^sub>R (1 / 2) ` {0::real..1}"

     hence "x \<le> 1 / 2" unfolding image_iff by auto

     thus "(g1 +++ g2) x = g1 (2 *\<^sub>R x)" unfolding joinpaths_def by auto next

     fix x assume "x \<in> (\<lambda>x. (1 / 2) *\<^sub>R (x + 1)) ` {0::real..1}"

     hence "x \<ge> 1 / 2" unfolding image_iff by auto

     thus "(g1 +++ g2) x = g2 (2 *\<^sub>R x - 1)" proof(cases "x = 1 / 2")

       case True hence "x = (1/2) *\<^sub>R 1" by auto 

       thus ?thesis unfolding joinpaths_def using assms[unfolded pathstart_def pathfinish_def] by (auto simp add: mult_ac)

     qed (auto simp add:le_less joinpaths_def) qed

 next assume as:"continuous_on {0..1} g1" "continuous_on {0..1} g2"

   have *:"{0 .. 1::real} = {0.. (1/2)*\<^sub>R 1} \<union> {(1/2) *\<^sub>R 1 .. 1}" by auto

   have **:"op *\<^sub>R 2 ` {0..(1 / 2) *\<^sub>R 1} = {0..1::real}" apply(rule set_eqI, rule) unfolding image_iff 

     defer apply(rule_tac x="(1/2)*\<^sub>R x" in bexI) by auto

   have ***:"(\<lambda>x. 2 *\<^sub>R x - 1) ` {(1 / 2) *\<^sub>R 1..1} = {0..1::real}"

     apply (auto simp add: image_def)

     apply (rule_tac x="(x + 1) / 2" in bexI)

     apply (auto simp add: add_divide_distrib)

     done

   show "continuous_on {0..1} (g1 +++ g2)" unfolding * apply(rule continuous_on_union) apply (rule closed_real_atLeastAtMost)+ proof-

     show "continuous_on {0..(1 / 2) *\<^sub>R 1} (g1 +++ g2)" apply(rule continuous_on_eq[of _ "\<lambda>x. g1 (2 *\<^sub>R x)"]) defer

       unfolding o_def[THEN sym] apply(rule continuous_on_compose) apply (intro continuous_on_intros)

       unfolding ** apply(rule as(1)) unfolding joinpaths_def by auto next

     show "continuous_on {(1/2)*\<^sub>R1..1} (g1 +++ g2)" apply(rule continuous_on_eq[of _ "g2 \<circ> (\<lambda>x. 2 *\<^sub>R x - 1)"]) defer

       apply(rule continuous_on_compose) apply (intro continuous_on_intros)

       unfolding *** o_def joinpaths_def apply(rule as(2)) using assms[unfolded pathstart_def pathfinish_def]

       by (auto simp add: mult_ac) qed qed

 lemma path_image_join_subset: "path_image(g1 +++ g2) \<subseteq> (path_image g1 \<union> path_image g2)" proof

   fix x assume "x \<in> path_image (g1 +++ g2)"

   then obtain y where y:"y\<in>{0..1}" "x = (if y \<le> 1 / 2 then g1 (2 *\<^sub>R y) else g2 (2 *\<^sub>R y - 1))"

     unfolding path_image_def image_iff joinpaths_def by auto

   thus "x \<in> path_image g1 \<union> path_image g2" apply(cases "y \<le> 1/2")

     apply(rule_tac UnI1) defer apply(rule_tac UnI2) unfolding y(2) path_image_def using y(1)

     by(auto intro!: imageI) qed

 lemma subset_path_image_join:

   assumes "path_image g1 \<subseteq> s" "path_image g2 \<subseteq> s" shows "path_image(g1 +++ g2) \<subseteq> s"

   using path_image_join_subset[of g1 g2] and assms by auto

 lemma path_image_join:

   assumes "path g1" "path g2" "pathfinish g1 = pathstart g2"

   shows "path_image(g1 +++ g2) = (path_image g1) \<union> (path_image g2)"

 apply(rule, rule path_image_join_subset, rule) unfolding Un_iff proof(erule disjE)

   fix x assume "x \<in> path_image g1"

   then obtain y where y:"y\<in>{0..1}" "x = g1 y" unfolding path_image_def image_iff by auto

   thus "x \<in> path_image (g1 +++ g2)" unfolding joinpaths_def path_image_def image_iff

     apply(rule_tac x="(1/2) *\<^sub>R y" in bexI) by auto next

   fix x assume "x \<in> path_image g2"

   then obtain y where y:"y\<in>{0..1}" "x = g2 y" unfolding path_image_def image_iff by auto

   then show "x \<in> path_image (g1 +++ g2)" unfolding joinpaths_def path_image_def image_iff

     apply(rule_tac x="(1/2) *\<^sub>R (y + 1)" in bexI) using assms(3)[unfolded pathfinish_def pathstart_def]

     by (auto simp add: add_divide_distrib) qed

 lemma not_in_path_image_join:

   assumes "x \<notin> path_image g1" "x \<notin> path_image g2" shows "x \<notin> path_image(g1 +++ g2)"

   using assms and path_image_join_subset[of g1 g2] by auto

 lemma simple_path_reversepath: assumes "simple_path g" shows "simple_path (reversepath g)"

   using assms unfolding simple_path_def reversepath_def apply- apply(rule ballI)+

   apply(erule_tac x="1-x" in ballE, erule_tac x="1-y" in ballE)

   by auto

 lemma simple_path_join_loop:

   assumes "injective_path g1" "injective_path g2" "pathfinish g2 = pathstart g1"

   "(path_image g1 \<inter> path_image g2) \<subseteq> {pathstart g1,pathstart g2}"

   shows "simple_path(g1 +++ g2)"

 unfolding simple_path_def proof((rule ballI)+, rule impI) let ?g = "g1 +++ g2"

   note inj = assms(1,2)[unfolded injective_path_def, rule_format]

   fix x y::"real" assume xy:"x \<in> {0..1}" "y \<in> {0..1}" "?g x = ?g y"

   show "x = y \<or> x = 0 \<and> y = 1 \<or> x = 1 \<and> y = 0" proof(case_tac "x \<le> 1/2",case_tac[!] "y \<le> 1/2", unfold not_le)

     assume as:"x \<le> 1 / 2" "y \<le> 1 / 2"

     hence "g1 (2 *\<^sub>R x) = g1 (2 *\<^sub>R y)" using xy(3) unfolding joinpaths_def by auto

     moreover have "2 *\<^sub>R x \<in> {0..1}" "2 *\<^sub>R y \<in> {0..1}" using xy(1,2) as

       by auto

     ultimately show ?thesis using inj(1)[of "2*\<^sub>R x" "2*\<^sub>R y"] by auto

   next assume as:"x > 1 / 2" "y > 1 / 2"

     hence "g2 (2 *\<^sub>R x - 1) = g2 (2 *\<^sub>R y - 1)" using xy(3) unfolding joinpaths_def by auto

     moreover have "2 *\<^sub>R x - 1 \<in> {0..1}" "2 *\<^sub>R y - 1 \<in> {0..1}" using xy(1,2) as by auto

     ultimately show ?thesis using inj(2)[of "2*\<^sub>R x - 1" "2*\<^sub>R y - 1"] by auto

   next assume as:"x \<le> 1 / 2" "y > 1 / 2"

     hence "?g x \<in> path_image g1" "?g y \<in> path_image g2" unfolding path_image_def joinpaths_def

       using xy(1,2) by auto

     moreover have "?g y \<noteq> pathstart g2" using as(2) unfolding pathstart_def joinpaths_def

       using inj(2)[of "2 *\<^sub>R y - 1" 0] and xy(2)

       by (auto simp add: field_simps)

     ultimately have *:"?g x = pathstart g1" using assms(4) unfolding xy(3) by auto

     hence "x = 0" unfolding pathstart_def joinpaths_def using as(1) and xy(1)

       using inj(1)[of "2 *\<^sub>R x" 0] by auto

     moreover have "y = 1" using * unfolding xy(3) assms(3)[THEN sym]

       unfolding joinpaths_def pathfinish_def using as(2) and xy(2)

       using inj(2)[of "2 *\<^sub>R y - 1" 1] by auto

     ultimately show ?thesis by auto

   next assume as:"x > 1 / 2" "y \<le> 1 / 2"

     hence "?g x \<in> path_image g2" "?g y \<in> path_image g1" unfolding path_image_def joinpaths_def

       using xy(1,2) by auto

     moreover have "?g x \<noteq> pathstart g2" using as(1) unfolding pathstart_def joinpaths_def

       using inj(2)[of "2 *\<^sub>R x - 1" 0] and xy(1)

       by (auto simp add: field_simps)

     ultimately have *:"?g y = pathstart g1" using assms(4) unfolding xy(3) by auto

     hence "y = 0" unfolding pathstart_def joinpaths_def using as(2) and xy(2)

       using inj(1)[of "2 *\<^sub>R y" 0] by auto

     moreover have "x = 1" using * unfolding xy(3)[THEN sym] assms(3)[THEN sym]

       unfolding joinpaths_def pathfinish_def using as(1) and xy(1)

       using inj(2)[of "2 *\<^sub>R x - 1" 1] by auto

     ultimately show ?thesis by auto qed qed

 lemma injective_path_join:

   assumes "injective_path g1" "injective_path g2" "pathfinish g1 = pathstart g2"

   "(path_image g1 \<inter> path_image g2) \<subseteq> {pathstart g2}"

   shows "injective_path(g1 +++ g2)"

   unfolding injective_path_def proof(rule,rule,rule) let ?g = "g1 +++ g2"

   note inj = assms(1,2)[unfolded injective_path_def, rule_format]

   fix x y assume xy:"x \<in> {0..1}" "y \<in> {0..1}" "(g1 +++ g2) x = (g1 +++ g2) y"

   show "x = y" proof(cases "x \<le> 1/2", case_tac[!] "y \<le> 1/2", unfold not_le)

     assume "x \<le> 1 / 2" "y \<le> 1 / 2" thus ?thesis using inj(1)[of "2*\<^sub>R x" "2*\<^sub>R y"] and xy

       unfolding joinpaths_def by auto

   next assume "x > 1 / 2" "y > 1 / 2" thus ?thesis using inj(2)[of "2*\<^sub>R x - 1" "2*\<^sub>R y - 1"] and xy

       unfolding joinpaths_def by auto

   next assume as:"x \<le> 1 / 2" "y > 1 / 2" 

     hence "?g x \<in> path_image g1" "?g y \<in> path_image g2" unfolding path_image_def joinpaths_def

       using xy(1,2) by auto

     hence "?g x = pathfinish g1" "?g y = pathstart g2" using assms(4) unfolding assms(3) xy(3) by auto

     thus ?thesis using as and inj(1)[of "2 *\<^sub>R x" 1] inj(2)[of "2 *\<^sub>R y - 1" 0] and xy(1,2)

       unfolding pathstart_def pathfinish_def joinpaths_def

       by auto

   next assume as:"x > 1 / 2" "y \<le> 1 / 2" 

     hence "?g x \<in> path_image g2" "?g y \<in> path_image g1" unfolding path_image_def joinpaths_def

       using xy(1,2) by auto

     hence "?g x = pathstart g2" "?g y = pathfinish g1" using assms(4) unfolding assms(3) xy(3) by auto

     thus ?thesis using as and inj(2)[of "2 *\<^sub>R x - 1" 0] inj(1)[of "2 *\<^sub>R y" 1] and xy(1,2)

       unfolding pathstart_def pathfinish_def joinpaths_def

       by auto qed qed

 lemmas join_paths_simps = path_join path_image_join pathstart_join pathfinish_join

 subsection {* Reparametrizing a closed curve to start at some chosen point. *}

 definition "shiftpath a (f::real \<Rightarrow> 'a::topological_space) =

   (\<lambda>x. if (a + x) \<le> 1 then f(a + x) else f(a + x - 1))"

 lemma pathstart_shiftpath: "a \<le> 1 \<Longrightarrow> pathstart(shiftpath a g) = g a"

   unfolding pathstart_def shiftpath_def by auto

 lemma pathfinish_shiftpath: assumes "0 \<le> a" "pathfinish g = pathstart g"

   shows "pathfinish(shiftpath a g) = g a"

   using assms unfolding pathstart_def pathfinish_def shiftpath_def

   by auto

 lemma endpoints_shiftpath:

   assumes "pathfinish g = pathstart g" "a \<in> {0 .. 1}" 

   shows "pathfinish(shiftpath a g) = g a" "pathstart(shiftpath a g) = g a"

   using assms by(auto intro!:pathfinish_shiftpath pathstart_shiftpath)

 lemma closed_shiftpath:

   assumes "pathfinish g = pathstart g" "a \<in> {0..1}"

   shows "pathfinish(shiftpath a g) = pathstart(shiftpath a g)"

   using endpoints_shiftpath[OF assms] by auto

 lemma path_shiftpath:

   assumes "path g" "pathfinish g = pathstart g" "a \<in> {0..1}"

   shows "path(shiftpath a g)" proof-

   have *:"{0 .. 1} = {0 .. 1-a} \<union> {1-a .. 1}" using assms(3) by auto

   have **:"\<And>x. x + a = 1 \<Longrightarrow> g (x + a - 1) = g (x + a)"

     using assms(2)[unfolded pathfinish_def pathstart_def] by auto

   show ?thesis unfolding path_def shiftpath_def * apply(rule continuous_on_union)

     apply(rule closed_real_atLeastAtMost)+ apply(rule continuous_on_eq[of _ "g \<circ> (\<lambda>x. a + x)"]) prefer 3

     apply(rule continuous_on_eq[of _ "g \<circ> (\<lambda>x. a - 1 + x)"]) defer prefer 3

     apply(rule continuous_on_intros)+ prefer 2 apply(rule continuous_on_intros)+

     apply(rule_tac[1-2] continuous_on_subset[OF assms(1)[unfolded path_def]])

     using assms(3) and ** by(auto, auto simp add: field_simps) qed

 lemma shiftpath_shiftpath: assumes "pathfinish g = pathstart g" "a \<in> {0..1}" "x \<in> {0..1}" 

   shows "shiftpath (1 - a) (shiftpath a g) x = g x"

   using assms unfolding pathfinish_def pathstart_def shiftpath_def by auto

 lemma path_image_shiftpath:

   assumes "a \<in> {0..1}" "pathfinish g = pathstart g"

   shows "path_image(shiftpath a g) = path_image g" proof-

   { fix x assume as:"g 1 = g 0" "x \<in> {0..1::real}" " \<forall>y\<in>{0..1} \<inter> {x. \<not> a + x \<le> 1}. g x \<noteq> g (a + y - 1)" 

     hence "\<exists>y\<in>{0..1} \<inter> {x. a + x \<le> 1}. g x = g (a + y)" proof(cases "a \<le> x")

       case False thus ?thesis apply(rule_tac x="1 + x - a" in bexI)

         using as(1,2) and as(3)[THEN bspec[where x="1 + x - a"]] and assms(1)

         by(auto simp add: field_simps atomize_not) next

       case True thus ?thesis using as(1-2) and assms(1) apply(rule_tac x="x - a" in bexI)

         by(auto simp add: field_simps) qed }

   thus ?thesis using assms unfolding shiftpath_def path_image_def pathfinish_def pathstart_def

     by(auto simp add: image_iff) qed

 subsection {* Special case of straight-line paths. *}

 definition

   linepath :: "'a::real_normed_vector \<Rightarrow> 'a \<Rightarrow> real \<Rightarrow> 'a" where

   "linepath a b = (\<lambda>x. (1 - x) *\<^sub>R a + x *\<^sub>R b)"

 lemma pathstart_linepath[simp]: "pathstart(linepath a b) = a"

   unfolding pathstart_def linepath_def by auto

 lemma pathfinish_linepath[simp]: "pathfinish(linepath a b) = b"

   unfolding pathfinish_def linepath_def by auto

 lemma continuous_linepath_at[intro]: "continuous (at x) (linepath a b)"

   unfolding linepath_def by (intro continuous_intros)

 lemma continuous_on_linepath[intro]: "continuous_on s (linepath a b)"

   using continuous_linepath_at by(auto intro!: continuous_at_imp_continuous_on)

 lemma path_linepath[intro]: "path(linepath a b)"

   unfolding path_def by(rule continuous_on_linepath)

 lemma path_image_linepath[simp]: "path_image(linepath a b) = (closed_segment a b)"

   unfolding path_image_def segment linepath_def apply (rule set_eqI, rule) defer

   unfolding mem_Collect_eq image_iff apply(erule exE) apply(rule_tac x="u *\<^sub>R 1" in bexI)

   by auto

 lemma reversepath_linepath[simp]:  "reversepath(linepath a b) = linepath b a"

   unfolding reversepath_def linepath_def by(rule ext, auto)

 lemma injective_path_linepath:

   assumes "a \<noteq> b" shows "injective_path(linepath a b)"

 proof -

   { fix x y :: "real"

     assume "x *\<^sub>R b + y *\<^sub>R a = x *\<^sub>R a + y *\<^sub>R b"

     hence "(x - y) *\<^sub>R a = (x - y) *\<^sub>R b" by (simp add: algebra_simps)

     with assms have "x = y" by simp }

   thus ?thesis unfolding injective_path_def linepath_def by(auto simp add: algebra_simps) qed

 lemma simple_path_linepath[intro]: "a \<noteq> b \<Longrightarrow> simple_path(linepath a b)" by(auto intro!: injective_imp_simple_path injective_path_linepath)

 subsection {* Bounding a point away from a path. *}

 lemma not_on_path_ball:

   fixes g :: "real \<Rightarrow> 'a::heine_borel"

   assumes "path g" "z \<notin> path_image g"

   shows "\<exists>e>0. ball z e \<inter> (path_image g) = {}" proof-

   obtain a where "a\<in>path_image g" "\<forall>y\<in>path_image g. dist z a \<le> dist z y"

     using distance_attains_inf[OF _ path_image_nonempty, of g z]

     using compact_path_image[THEN compact_imp_closed, OF assms(1)] by auto

   thus ?thesis apply(rule_tac x="dist z a" in exI) using assms(2) by(auto intro!: dist_pos_lt) qed

 lemma not_on_path_cball:

   fixes g :: "real \<Rightarrow> 'a::heine_borel"

   assumes "path g" "z \<notin> path_image g"

   shows "\<exists>e>0. cball z e \<inter> (path_image g) = {}" proof-

   obtain e where "ball z e \<inter> path_image g = {}" "e>0" using not_on_path_ball[OF assms] by auto

   moreover have "cball z (e/2) \<subseteq> ball z e" using `e>0` by auto

   ultimately show ?thesis apply(rule_tac x="e/2" in exI) by auto qed

 subsection {* Path component, considered as a "joinability" relation (from Tom Hales). *}

 definition "path_component s x y \<longleftrightarrow> (\<exists>g. path g \<and> path_image g \<subseteq> s \<and> pathstart g = x \<and> pathfinish g = y)"

 lemmas path_defs = path_def pathstart_def pathfinish_def path_image_def path_component_def 

 lemma path_component_mem: assumes "path_component s x y" shows "x \<in> s" "y \<in> s"

   using assms unfolding path_defs by auto

 lemma path_component_refl: assumes "x \<in> s" shows "path_component s x x"

   unfolding path_defs apply(rule_tac x="\<lambda>u. x" in exI) using assms 

   by(auto intro!:continuous_on_intros)

 lemma path_component_refl_eq: "path_component s x x \<longleftrightarrow> x \<in> s"

   by(auto intro!: path_component_mem path_component_refl)

 lemma path_component_sym: "path_component s x y \<Longrightarrow> path_component s y x"

   using assms unfolding path_component_def apply(erule exE) apply(rule_tac x="reversepath g" in exI)

   by auto

 lemma path_component_trans: assumes "path_component s x y" "path_component s y z" shows "path_component s x z"

   using assms unfolding path_component_def apply- apply(erule exE)+ apply(rule_tac x="g +++ ga" in exI) by(auto simp add: path_image_join)

 lemma path_component_of_subset: "s \<subseteq> t \<Longrightarrow>  path_component s x y \<Longrightarrow> path_component t x y"

   unfolding path_component_def by auto

 subsection {* Can also consider it as a set, as the name suggests. *}

 lemma path_component_set: "{y. path_component s x y} = { y. (\<exists>g. path g \<and> path_image g \<subseteq> s \<and> pathstart g = x \<and> pathfinish g = y )}"

   apply(rule set_eqI) unfolding mem_Collect_eq unfolding path_component_def by auto

 lemma path_component_subset: "{y. path_component s x y} \<subseteq> s"

   apply(rule, rule path_component_mem(2)) by auto

 lemma path_component_eq_empty: "{y. path_component s x y} = {} \<longleftrightarrow> x \<notin> s"

   apply rule apply(drule equals0D[of _ x]) defer apply(rule equals0I) unfolding mem_Collect_eq

   apply(drule path_component_mem(1)) using path_component_refl by auto

 subsection {* Path connectedness of a space. *}

 definition "path_connected s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>y\<in>s. \<exists>g. path g \<and> (path_image g) \<subseteq> s \<and> pathstart g = x \<and> pathfinish g = y)"

 lemma path_connected_component: "path_connected s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>y\<in>s. path_component s x y)"

   unfolding path_connected_def path_component_def by auto

 lemma path_connected_component_set: "path_connected s \<longleftrightarrow> (\<forall>x\<in>s. {y. path_component s x y} = s)" 

   unfolding path_connected_component apply(rule, rule, rule, rule path_component_subset) 

   unfolding subset_eq mem_Collect_eq Ball_def by auto

 subsection {* Some useful lemmas about path-connectedness. *}

 lemma convex_imp_path_connected:

   fixes s :: "'a::real_normed_vector set"

   assumes "convex s" shows "path_connected s"

   unfolding path_connected_def apply(rule,rule,rule_tac x="linepath x y" in exI)

   unfolding path_image_linepath using assms[unfolded convex_contains_segment] by auto

 lemma path_connected_imp_connected: assumes "path_connected s" shows "connected s"

   unfolding connected_def not_ex apply(rule,rule,rule ccontr) unfolding not_not apply(erule conjE)+ proof-

   fix e1 e2 assume as:"open e1" "open e2" "s \<subseteq> e1 \<union> e2" "e1 \<inter> e2 \<inter> s = {}" "e1 \<inter> s \<noteq> {}" "e2 \<inter> s \<noteq> {}"

   then obtain x1 x2 where obt:"x1\<in>e1\<inter>s" "x2\<in>e2\<inter>s" by auto

   then obtain g where g:"path g" "path_image g \<subseteq> s" "pathstart g = x1" "pathfinish g = x2"

     using assms[unfolded path_connected_def,rule_format,of x1 x2] by auto

   have *:"connected {0..1::real}" by(auto intro!: convex_connected convex_real_interval)

   have "{0..1} \<subseteq> {x \<in> {0..1}. g x \<in> e1} \<union> {x \<in> {0..1}. g x \<in> e2}" using as(3) g(2)[unfolded path_defs] by blast

   moreover have "{x \<in> {0..1}. g x \<in> e1} \<inter> {x \<in> {0..1}. g x \<in> e2} = {}" using as(4) g(2)[unfolded path_defs] unfolding subset_eq by auto 

   moreover have "{x \<in> {0..1}. g x \<in> e1} \<noteq> {} \<and> {x \<in> {0..1}. g x \<in> e2} \<noteq> {}" using g(3,4)[unfolded path_defs] using obt

     by (simp add: ex_in_conv [symmetric], metis zero_le_one order_refl)

   ultimately show False using *[unfolded connected_local not_ex,rule_format, of "{x\<in>{0..1}. g x \<in> e1}" "{x\<in>{0..1}. g x \<in> e2}"]

     using continuous_open_in_preimage[OF g(1)[unfolded path_def] as(1)]

     using continuous_open_in_preimage[OF g(1)[unfolded path_def] as(2)] by auto qed

 lemma open_path_component:

   fixes s :: "'a::real_normed_vector set" (*TODO: generalize to metric_space*)

   assumes "open s" shows "open {y. path_component s x y}"

   unfolding open_contains_ball proof

   fix y assume as:"y \<in> {y. path_component s x y}"

   hence "y\<in>s" apply- apply(rule path_component_mem(2)) unfolding mem_Collect_eq by auto

   then obtain e where e:"e>0" "ball y e \<subseteq> s" using assms[unfolded open_contains_ball] by auto

   show "\<exists>e>0. ball y e \<subseteq> {y. path_component s x y}" apply(rule_tac x=e in exI) apply(rule,rule `e>0`,rule) unfolding mem_ball mem_Collect_eq proof-

     fix z assume "dist y z < e" thus "path_component s x z" apply(rule_tac path_component_trans[of _ _ y]) defer 

       apply(rule path_component_of_subset[OF e(2)]) apply(rule convex_imp_path_connected[OF convex_ball, unfolded path_connected_component, rule_format]) using `e>0`

       using as by auto qed qed

 lemma open_non_path_component:

   fixes s :: "'a::real_normed_vector set" (*TODO: generalize to metric_space*)

   assumes "open s" shows "open(s - {y. path_component s x y})"

   unfolding open_contains_ball proof

   fix y assume as:"y\<in>s - {y. path_component s x y}"

   then obtain e where e:"e>0" "ball y e \<subseteq> s" using assms[unfolded open_contains_ball] by auto

   show "\<exists>e>0. ball y e \<subseteq> s - {y. path_component s x y}" apply(rule_tac x=e in exI) apply(rule,rule `e>0`,rule,rule) defer proof(rule ccontr)

     fix z assume "z\<in>ball y e" "\<not> z \<notin> {y. path_component s x y}"

     hence "y \<in> {y. path_component s x y}" unfolding not_not mem_Collect_eq using `e>0`

       apply- apply(rule path_component_trans,assumption) apply(rule path_component_of_subset[OF e(2)])

       apply(rule convex_imp_path_connected[OF convex_ball, unfolded path_connected_component, rule_format]) by auto

     thus False using as by auto qed(insert e(2), auto) qed

 lemma connected_open_path_connected:

   fixes s :: "'a::real_normed_vector set" (*TODO: generalize to metric_space*)

   assumes "open s" "connected s" shows "path_connected s"

   unfolding path_connected_component_set proof(rule,rule,rule path_component_subset, rule)

   fix x y assume "x \<in> s" "y \<in> s" show "y \<in> {y. path_component s x y}" proof(rule ccontr)

     assume "y \<notin> {y. path_component s x y}" moreover

     have "{y. path_component s x y} \<inter> s \<noteq> {}" using `x\<in>s` path_component_eq_empty path_component_subset[of s x] by auto

     ultimately show False using `y\<in>s` open_non_path_component[OF assms(1)] open_path_component[OF assms(1)]

     using assms(2)[unfolded connected_def not_ex, rule_format, of"{y. path_component s x y}" "s - {y. path_component s x y}"] by auto

 qed qed

 lemma path_connected_continuous_image:

   assumes "continuous_on s f" "path_connected s" shows "path_connected (f ` s)"

   unfolding path_connected_def proof(rule,rule)

   fix x' y' assume "x' \<in> f ` s" "y' \<in> f ` s"

   then obtain x y where xy:"x\<in>s" "y\<in>s" "x' = f x" "y' = f y" by auto

   guess g using assms(2)[unfolded path_connected_def,rule_format,OF xy(1,2)] ..

   thus "\<exists>g. path g \<and> path_image g \<subseteq> f ` s \<and> pathstart g = x' \<and> pathfinish g = y'"

     unfolding xy apply(rule_tac x="f \<circ> g" in exI) unfolding path_defs

     using assms(1) by(auto intro!: continuous_on_compose continuous_on_subset[of _ _ "g ` {0..1}"]) qed

 lemma homeomorphic_path_connectedness:

   "s homeomorphic t \<Longrightarrow> (path_connected s \<longleftrightarrow> path_connected t)"

   unfolding homeomorphic_def homeomorphism_def apply(erule exE|erule conjE)+ apply rule

   apply(drule_tac f=f in path_connected_continuous_image) prefer 3

   apply(drule_tac f=g in path_connected_continuous_image) by auto

 lemma path_connected_empty: "path_connected {}"

   unfolding path_connected_def by auto

 lemma path_connected_singleton: "path_connected {a}"

   unfolding path_connected_def pathstart_def pathfinish_def path_image_def

   apply (clarify, rule_tac x="\<lambda>x. a" in exI, simp add: image_constant_conv)

   apply (simp add: path_def continuous_on_const)

   done

 lemma path_connected_Un: assumes "path_connected s" "path_connected t" "s \<inter> t \<noteq> {}"

   shows "path_connected (s \<union> t)" unfolding path_connected_component proof(rule,rule)

   fix x y assume as:"x \<in> s \<union> t" "y \<in> s \<union> t" 

   from assms(3) obtain z where "z \<in> s \<inter> t" by auto

   thus "path_component (s \<union> t) x y" using as using assms(1-2)[unfolded path_connected_component] apply- 

     apply(erule_tac[!] UnE)+ apply(rule_tac[2-3] path_component_trans[of _ _ z])

     by(auto simp add:path_component_of_subset [OF Un_upper1] path_component_of_subset[OF Un_upper2]) qed

 lemma path_connected_UNION:

   assumes "\<And>i. i \<in> A \<Longrightarrow> path_connected (S i)"

   assumes "\<And>i. i \<in> A \<Longrightarrow> z \<in> S i"

   shows "path_connected (\<Union>i\<in>A. S i)"

 unfolding path_connected_component proof(clarify)

   fix x i y j

   assume *: "i \<in> A" "x \<in> S i" "j \<in> A" "y \<in> S j"

   hence "path_component (S i) x z" and "path_component (S j) z y"

     using assms by (simp_all add: path_connected_component)

   hence "path_component (\<Union>i\<in>A. S i) x z" and "path_component (\<Union>i\<in>A. S i) z y"

     using *(1,3) by (auto elim!: path_component_of_subset [rotated])

   thus "path_component (\<Union>i\<in>A. S i) x y"

     by (rule path_component_trans)

 qed

 subsection {* sphere is path-connected. *}

 lemma path_connected_punctured_universe:

   assumes "2 \<le> DIM('a::euclidean_space)"

   shows "path_connected((UNIV::'a::euclidean_space set) - {a})"

 proof-

   let ?A = "{x::'a. \<exists>i\<in>{..<DIM('a)}. x $$ i < a $$ i}"

   let ?B = "{x::'a. \<exists>i\<in>{..<DIM('a)}. a $$ i < x $$ i}"

   have A: "path_connected ?A" unfolding Collect_bex_eq

   proof (rule path_connected_UNION)

     fix i assume "i \<in> {..<DIM('a)}"

     thus "(\<chi>\<chi> i. a $$ i - 1) \<in> {x::'a. x $$ i < a $$ i}" by simp

     show "path_connected {x. x $$ i < a $$ i}" unfolding euclidean_component_def

       by (rule convex_imp_path_connected [OF convex_halfspace_lt])

   qed

   have B: "path_connected ?B" unfolding Collect_bex_eq

   proof (rule path_connected_UNION)

     fix i assume "i \<in> {..<DIM('a)}"

     thus "(\<chi>\<chi> i. a $$ i + 1) \<in> {x::'a. a $$ i < x $$ i}" by simp

     show "path_connected {x. a $$ i < x $$ i}" unfolding euclidean_component_def

       by (rule convex_imp_path_connected [OF convex_halfspace_gt])

   qed

   from assms have "1 < DIM('a)" by auto

   hence "a + basis 0 - basis 1 \<in> ?A \<inter> ?B" by auto

   hence "?A \<inter> ?B \<noteq> {}" by fast

   with A B have "path_connected (?A \<union> ?B)"

     by (rule path_connected_Un)

   also have "?A \<union> ?B = {x. \<exists>i\<in>{..<DIM('a)}. x $$ i \<noteq> a $$ i}"

     unfolding neq_iff bex_disj_distrib Collect_disj_eq ..

   also have "\<dots> = {x. x \<noteq> a}"

     unfolding Bex_def euclidean_eq [where 'a='a] by simp

   also have "\<dots> = UNIV - {a}" by auto

   finally show ?thesis .

 qed

 lemma path_connected_sphere:

   assumes "2 \<le> DIM('a::euclidean_space)"

   shows "path_connected {x::'a::euclidean_space. norm(x - a) = r}"

 proof (rule linorder_cases [of r 0])

   assume "r < 0" hence "{x::'a. norm(x - a) = r} = {}" by auto

   thus ?thesis using path_connected_empty by simp

 next

   assume "r = 0"

   thus ?thesis using path_connected_singleton by simp

 next

   assume r: "0 < r"

   hence *:"{x::'a. norm(x - a) = r} = (\<lambda>x. a + r *\<^sub>R x) ` {x. norm x = 1}" apply -apply(rule set_eqI,rule)

     unfolding image_iff apply(rule_tac x="(1/r) *\<^sub>R (x - a)" in bexI) unfolding mem_Collect_eq norm_scaleR by (auto simp add: scaleR_right_diff_distrib)

   have **:"{x::'a. norm x = 1} = (\<lambda>x. (1/norm x) *\<^sub>R x) ` (UNIV - {0})" apply(rule set_eqI,rule)

     unfolding image_iff apply(rule_tac x=x in bexI) unfolding mem_Collect_eq by(auto split:split_if_asm)

   have "continuous_on (UNIV - {0}) (\<lambda>x::'a. 1 / norm x)"

     unfolding field_divide_inverse by (simp add: continuous_on_intros)

   thus ?thesis unfolding * ** using path_connected_punctured_universe[OF assms]

     by(auto intro!: path_connected_continuous_image continuous_on_intros)

 qed

 lemma connected_sphere: "2 \<le> DIM('a::euclidean_space) \<Longrightarrow> connected {x::'a. norm(x - a) = r}"

   using path_connected_sphere path_connected_imp_connected by auto

 end