src/HOL/Multivariate_Analysis/Path_Connected.thy

new material to Blochj's theorem, as well as supporting lemmas

(*  Title:      HOL/Multivariate_Analysis/Path_Connected.thy
    Authors:    LC Paulson and Robert Himmelmann (TU Muenchen), based on material from HOL Light
*)

section \<open>Continuous paths and path-connected sets\<close>

theory Path_Connected
imports Convex_Euclidean_Space
begin

subsection \<open>Paths and Arcs\<close>

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>
     path g \<and> (\<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 arc :: "(real \<Rightarrow> 'a :: topological_space) \<Rightarrow> bool"
  where "arc g \<longleftrightarrow> path g \<and> inj_on g {0..1}"


subsection\<open>Invariance theorems\<close>

lemma path_eq: "path p \<Longrightarrow> (\<And>t. t \<in> {0..1} \<Longrightarrow> p t = q t) \<Longrightarrow> path q"
  using continuous_on_eq path_def by blast

lemma path_continuous_image: "path g \<Longrightarrow> continuous_on (path_image g) f \<Longrightarrow> path(f o g)"
  unfolding path_def path_image_def
  using continuous_on_compose by blast

lemma path_translation_eq:
  fixes g :: "real \<Rightarrow> 'a :: real_normed_vector"
  shows "path((\<lambda>x. a + x) o g) = path g"
proof -
  have g: "g = (\<lambda>x. -a + x) o ((\<lambda>x. a + x) o g)"
    by (rule ext) simp
  show ?thesis
    unfolding path_def
    apply safe
    apply (subst g)
    apply (rule continuous_on_compose)
    apply (auto intro: continuous_intros)
    done
qed

lemma path_linear_image_eq:
  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes "linear f" "inj f"
     shows "path(f o g) = path g"
proof -
  from linear_injective_left_inverse [OF assms]
  obtain h where h: "linear h" "h \<circ> f = id"
    by blast
  then have g: "g = h o (f o g)"
    by (metis comp_assoc id_comp)
  show ?thesis
    unfolding path_def
    using h assms
    by (metis g continuous_on_compose linear_continuous_on linear_conv_bounded_linear)
qed

lemma pathstart_translation: "pathstart((\<lambda>x. a + x) o g) = a + pathstart g"
  by (simp add: pathstart_def)

lemma pathstart_linear_image_eq: "linear f \<Longrightarrow> pathstart(f o g) = f(pathstart g)"
  by (simp add: pathstart_def)

lemma pathfinish_translation: "pathfinish((\<lambda>x. a + x) o g) = a + pathfinish g"
  by (simp add: pathfinish_def)

lemma pathfinish_linear_image: "linear f \<Longrightarrow> pathfinish(f o g) = f(pathfinish g)"
  by (simp add: pathfinish_def)

lemma path_image_translation: "path_image((\<lambda>x. a + x) o g) = (\<lambda>x. a + x) ` (path_image g)"
  by (simp add: image_comp path_image_def)

lemma path_image_linear_image: "linear f \<Longrightarrow> path_image(f o g) = f ` (path_image g)"
  by (simp add: image_comp path_image_def)

lemma reversepath_translation: "reversepath((\<lambda>x. a + x) o g) = (\<lambda>x. a + x) o reversepath g"
  by (rule ext) (simp add: reversepath_def)

lemma reversepath_linear_image: "linear f \<Longrightarrow> reversepath(f o g) = f o reversepath g"
  by (rule ext) (simp add: reversepath_def)

lemma joinpaths_translation:
    "((\<lambda>x. a + x) o g1) +++ ((\<lambda>x. a + x) o g2) = (\<lambda>x. a + x) o (g1 +++ g2)"
  by (rule ext) (simp add: joinpaths_def)

lemma joinpaths_linear_image: "linear f \<Longrightarrow> (f o g1) +++ (f o g2) = f o (g1 +++ g2)"
  by (rule ext) (simp add: joinpaths_def)

lemma simple_path_translation_eq:
  fixes g :: "real \<Rightarrow> 'a::euclidean_space"
  shows "simple_path((\<lambda>x. a + x) o g) = simple_path g"
  by (simp add: simple_path_def path_translation_eq)

lemma simple_path_linear_image_eq:
  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
  assumes "linear f" "inj f"
    shows "simple_path(f o g) = simple_path g"
  using assms inj_on_eq_iff [of f]
  by (auto simp: path_linear_image_eq simple_path_def path_translation_eq)

lemma arc_translation_eq:
  fixes g :: "real \<Rightarrow> 'a::euclidean_space"
  shows "arc((\<lambda>x. a + x) o g) = arc g"
  by (auto simp: arc_def inj_on_def path_translation_eq)

lemma arc_linear_image_eq:
  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes "linear f" "inj f"
     shows  "arc(f o g) = arc g"
  using assms inj_on_eq_iff [of f]
  by (auto simp: arc_def inj_on_def path_linear_image_eq)

subsection\<open>Basic lemmas about paths\<close>

lemma arc_imp_simple_path: "arc g \<Longrightarrow> simple_path g"
  by (simp add: arc_def inj_on_def simple_path_def)

lemma arc_imp_path: "arc g \<Longrightarrow> path g"
  using arc_def by blast

lemma simple_path_imp_path: "simple_path g \<Longrightarrow> path g"
  using simple_path_def by blast

lemma simple_path_cases: "simple_path g \<Longrightarrow> arc g \<or> pathfinish g = pathstart g"
  unfolding simple_path_def arc_def inj_on_def pathfinish_def pathstart_def
  by (force)

lemma simple_path_imp_arc: "simple_path g \<Longrightarrow> pathfinish g \<noteq> pathstart g \<Longrightarrow> arc g"
  using simple_path_cases by auto

lemma arc_distinct_ends: "arc g \<Longrightarrow> pathfinish g \<noteq> pathstart g"
  unfolding arc_def inj_on_def pathfinish_def pathstart_def
  by fastforce

lemma arc_simple_path: "arc g \<longleftrightarrow> simple_path g \<and> pathfinish g \<noteq> pathstart g"
  using arc_distinct_ends arc_imp_simple_path simple_path_cases by blast

lemma simple_path_eq_arc: "pathfinish g \<noteq> pathstart g \<Longrightarrow> (simple_path g = arc g)"
  by (simp add: arc_simple_path)

lemma path_image_nonempty [simp]: "path_image g \<noteq> {}"
  unfolding path_image_def image_is_empty box_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
  using connected_continuous_image connected_Icc by blast

lemma compact_path_image[intro]: "path g \<Longrightarrow> compact (path_image g)"
  unfolding path_def path_image_def
  using compact_continuous_image connected_Icc by blast

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
    by force
  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_intros)
    apply (rule continuous_on_subset[of "{0..1}"])
    apply assumption
    apply auto
    done
  show ?thesis
    using *[of "reversepath g"] *[of g]
    unfolding reversepath_reversepath
    by (rule iffI)
qed

lemma arc_reversepath:
  assumes "arc g" shows "arc(reversepath g)"
proof -
  have injg: "inj_on g {0..1}"
    using assms
    by (simp add: arc_def)
  have **: "\<And>x y::real. 1-x = 1-y \<Longrightarrow> x = y"
    by simp
  show ?thesis
    apply (auto simp: arc_def inj_on_def path_reversepath)
    apply (simp add: arc_imp_path assms)
    apply (rule **)
    apply (rule inj_onD [OF injg])
    apply (auto simp: reversepath_def)
    done
qed

lemma simple_path_reversepath: "simple_path g \<Longrightarrow> simple_path (reversepath g)"
  apply (simp add: simple_path_def)
  apply (force simp: reversepath_def)
  done

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
proof safe
  assume cont: "continuous_on {0..1} (g1 +++ g2)"
  have g1: "continuous_on {0..1} g1 \<longleftrightarrow> continuous_on {0..1} ((g1 +++ g2) \<circ> (\<lambda>x. x / 2))"
    by (intro continuous_on_cong refl) (auto simp: joinpaths_def)
  have g2: "continuous_on {0..1} g2 \<longleftrightarrow> continuous_on {0..1} ((g1 +++ g2) \<circ> (\<lambda>x. x / 2 + 1/2))"
    using assms
    by (intro continuous_on_cong refl) (auto simp: joinpaths_def pathfinish_def pathstart_def)
  show "continuous_on {0..1} g1" and "continuous_on {0..1} g2"
    unfolding g1 g2
    by (auto intro!: continuous_intros continuous_on_subset[OF cont] simp del: o_apply)
next
  assume g1g2: "continuous_on {0..1} g1" "continuous_on {0..1} g2"
  have 01: "{0 .. 1} = {0..1/2} \<union> {1/2 .. 1::real}"
    by auto
  {
    fix x :: real
    assume "0 \<le> x" and "x \<le> 1"
    then have "x \<in> (\<lambda>x. x * 2) ` {0..1 / 2}"
      by (intro image_eqI[where x="x/2"]) auto
  }
  note 1 = this
  {
    fix x :: real
    assume "0 \<le> x" and "x \<le> 1"
    then have "x \<in> (\<lambda>x. x * 2 - 1) ` {1 / 2..1}"
      by (intro image_eqI[where x="x/2 + 1/2"]) auto
  }
  note 2 = this
  show "continuous_on {0..1} (g1 +++ g2)"
    using assms
    unfolding joinpaths_def 01
    apply (intro continuous_on_cases closed_atLeastAtMost g1g2[THEN continuous_on_compose2] continuous_intros)
    apply (auto simp: field_simps pathfinish_def pathstart_def intro!: 1 2)
    done
qed

section \<open>Path Images\<close>

lemma bounded_path_image: "path g \<Longrightarrow> bounded(path_image g)"
  by (simp add: compact_imp_bounded compact_path_image)

lemma closed_path_image:
  fixes g :: "real \<Rightarrow> 'a::t2_space"
  shows "path g \<Longrightarrow> closed(path_image g)"
  by (metis compact_path_image compact_imp_closed)

lemma connected_simple_path_image: "simple_path g \<Longrightarrow> connected(path_image g)"
  by (metis connected_path_image simple_path_imp_path)

lemma compact_simple_path_image: "simple_path g \<Longrightarrow> compact(path_image g)"
  by (metis compact_path_image simple_path_imp_path)

lemma bounded_simple_path_image: "simple_path g \<Longrightarrow> bounded(path_image g)"
  by (metis bounded_path_image simple_path_imp_path)

lemma closed_simple_path_image:
  fixes g :: "real \<Rightarrow> 'a::t2_space"
  shows "simple_path g \<Longrightarrow> closed(path_image g)"
  by (metis closed_path_image simple_path_imp_path)

lemma connected_arc_image: "arc g \<Longrightarrow> connected(path_image g)"
  by (metis connected_path_image arc_imp_path)

lemma compact_arc_image: "arc g \<Longrightarrow> compact(path_image g)"
  by (metis compact_path_image arc_imp_path)

lemma bounded_arc_image: "arc g \<Longrightarrow> bounded(path_image g)"
  by (metis bounded_path_image arc_imp_path)

lemma closed_arc_image:
  fixes g :: "real \<Rightarrow> 'a::t2_space"
  shows "arc g \<Longrightarrow> closed(path_image g)"
  by (metis closed_path_image arc_imp_path)

lemma path_image_join_subset: "path_image (g1 +++ g2) \<subseteq> path_image g1 \<union> path_image g2"
  unfolding path_image_def joinpaths_def
  by auto

lemma subset_path_image_join:
  assumes "path_image g1 \<subseteq> s"
    and "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:
    "pathfinish g1 = pathstart g2 \<Longrightarrow> path_image(g1 +++ g2) = path_image g1 \<union> path_image g2"
  apply (rule subset_antisym [OF path_image_join_subset])
  apply (auto simp: pathfinish_def pathstart_def path_image_def joinpaths_def image_def)
  apply (drule sym)
  apply (rule_tac x="xa/2" in bexI, auto)
  apply (rule ccontr)
  apply (drule_tac x="(xa+1)/2" in bspec)
  apply (auto simp: field_simps)
  apply (drule_tac x="1/2" in bspec, auto)
  done

lemma not_in_path_image_join:
  assumes "x \<notin> path_image g1"
    and "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 pathstart_compose: "pathstart(f o p) = f(pathstart p)"
  by (simp add: pathstart_def)

lemma pathfinish_compose: "pathfinish(f o p) = f(pathfinish p)"
  by (simp add: pathfinish_def)

lemma path_image_compose: "path_image (f o p) = f ` (path_image p)"
  by (simp add: image_comp path_image_def)

lemma path_compose_join: "f o (p +++ q) = (f o p) +++ (f o q)"
  by (rule ext) (simp add: joinpaths_def)

lemma path_compose_reversepath: "f o reversepath p = reversepath(f o p)"
  by (rule ext) (simp add: reversepath_def)

lemma joinpaths_eq:
  "(\<And>t. t \<in> {0..1} \<Longrightarrow> p t = p' t) \<Longrightarrow>
   (\<And>t. t \<in> {0..1} \<Longrightarrow> q t = q' t)
   \<Longrightarrow>  t \<in> {0..1} \<Longrightarrow> (p +++ q) t = (p' +++ q') t"
  by (auto simp: joinpaths_def)

lemma simple_path_inj_on: "simple_path g \<Longrightarrow> inj_on g {0<..<1}"
  by (auto simp: simple_path_def path_image_def inj_on_def less_eq_real_def Ball_def)


subsection\<open>Simple paths with the endpoints removed\<close>

lemma simple_path_endless:
    "simple_path c \<Longrightarrow> path_image c - {pathstart c,pathfinish c} = c ` {0<..<1}"
  apply (auto simp: simple_path_def path_image_def pathstart_def pathfinish_def Ball_def Bex_def image_def)
  apply (metis eq_iff le_less_linear)
  apply (metis leD linear)
  using less_eq_real_def zero_le_one apply blast
  using less_eq_real_def zero_le_one apply blast
  done

lemma connected_simple_path_endless:
    "simple_path c \<Longrightarrow> connected(path_image c - {pathstart c,pathfinish c})"
apply (simp add: simple_path_endless)
apply (rule connected_continuous_image)
apply (meson continuous_on_subset greaterThanLessThan_subseteq_atLeastAtMost_iff le_numeral_extra(3) le_numeral_extra(4) path_def simple_path_imp_path)
by auto

lemma nonempty_simple_path_endless:
    "simple_path c \<Longrightarrow> path_image c - {pathstart c,pathfinish c} \<noteq> {}"
  by (simp add: simple_path_endless)


subsection\<open>The operations on paths\<close>

lemma path_image_subset_reversepath: "path_image(reversepath g) \<le> path_image g"
  by (auto simp: path_image_def reversepath_def)

lemma path_imp_reversepath: "path g \<Longrightarrow> path(reversepath g)"
  apply (auto simp: path_def reversepath_def)
  using continuous_on_compose [of "{0..1}" "\<lambda>x. 1 - x" g]
  apply (auto simp: continuous_on_op_minus)
  done

lemma half_bounded_equal: "1 \<le> x * 2 \<Longrightarrow> x * 2 \<le> 1 \<longleftrightarrow> x = (1/2::real)"
  by simp

lemma continuous_on_joinpaths:
  assumes "continuous_on {0..1} g1" "continuous_on {0..1} g2" "pathfinish g1 = pathstart g2"
    shows "continuous_on {0..1} (g1 +++ g2)"
proof -
  have *: "{0..1::real} = {0..1/2} \<union> {1/2..1}"
    by auto
  have gg: "g2 0 = g1 1"
    by (metis assms(3) pathfinish_def pathstart_def)
  have 1: "continuous_on {0..1/2} (g1 +++ g2)"
    apply (rule continuous_on_eq [of _ "g1 o (\<lambda>x. 2*x)"])
    apply (rule continuous_intros | simp add: joinpaths_def assms)+
    done
  have "continuous_on {1/2..1} (g2 o (\<lambda>x. 2*x-1))"
    apply (rule continuous_on_subset [of "{1/2..1}"])
    apply (rule continuous_intros | simp add: image_affinity_atLeastAtMost_diff assms)+
    done
  then have 2: "continuous_on {1/2..1} (g1 +++ g2)"
    apply (rule continuous_on_eq [of "{1/2..1}" "g2 o (\<lambda>x. 2*x-1)"])
    apply (rule assms continuous_intros | simp add: joinpaths_def mult.commute half_bounded_equal gg)+
    done
  show ?thesis
    apply (subst *)
    apply (rule continuous_on_closed_Un)
    using 1 2
    apply auto
    done
qed

lemma path_join_imp: "\<lbrakk>path g1; path g2; pathfinish g1 = pathstart g2\<rbrakk> \<Longrightarrow> path(g1 +++ g2)"
  by (simp add: path_join)

lemma simple_path_join_loop:
  assumes "arc g1" "arc g2"
          "pathfinish g1 = pathstart g2"  "pathfinish g2 = pathstart g1"
          "path_image g1 \<inter> path_image g2 \<subseteq> {pathstart g1, pathstart g2}"
  shows "simple_path(g1 +++ g2)"
proof -
  have injg1: "inj_on g1 {0..1}"
    using assms
    by (simp add: arc_def)
  have injg2: "inj_on g2 {0..1}"
    using assms
    by (simp add: arc_def)
  have g12: "g1 1 = g2 0"
   and g21: "g2 1 = g1 0"
   and sb:  "g1 ` {0..1} \<inter> g2 ` {0..1} \<subseteq> {g1 0, g2 0}"
    using assms
    by (simp_all add: arc_def pathfinish_def pathstart_def path_image_def)
  { fix x and y::real
    assume xyI: "x = 1 \<longrightarrow> y \<noteq> 0"
       and xy: "x \<le> 1" "0 \<le> y" " y * 2 \<le> 1" "\<not> x * 2 \<le> 1" "g2 (2 * x - 1) = g1 (2 * y)"
    have g1im: "g1 (2 * y) \<in> g1 ` {0..1} \<inter> g2 ` {0..1}"
      using xy
      apply simp
      apply (rule_tac x="2 * x - 1" in image_eqI, auto)
      done
    have False
      using subsetD [OF sb g1im] xy
      apply auto
      apply (drule inj_onD [OF injg1])
      using g21 [symmetric] xyI
      apply (auto dest: inj_onD [OF injg2])
      done
   } note * = this
  { fix x and y::real
    assume xy: "y \<le> 1" "0 \<le> x" "\<not> y * 2 \<le> 1" "x * 2 \<le> 1" "g1 (2 * x) = g2 (2 * y - 1)"
    have g1im: "g1 (2 * x) \<in> g1 ` {0..1} \<inter> g2 ` {0..1}"
      using xy
      apply simp
      apply (rule_tac x="2 * x" in image_eqI, auto)
      done
    have "x = 0 \<and> y = 1"
      using subsetD [OF sb g1im] xy
      apply auto
      apply (force dest: inj_onD [OF injg1])
      using  g21 [symmetric]
      apply (auto dest: inj_onD [OF injg2])
      done
   } note ** = this
  show ?thesis
    using assms
    apply (simp add: arc_def simple_path_def path_join, clarify)
    apply (simp add: joinpaths_def split: if_split_asm)
    apply (force dest: inj_onD [OF injg1])
    apply (metis *)
    apply (metis **)
    apply (force dest: inj_onD [OF injg2])
    done
qed

lemma arc_join:
  assumes "arc g1" "arc g2"
          "pathfinish g1 = pathstart g2"
          "path_image g1 \<inter> path_image g2 \<subseteq> {pathstart g2}"
    shows "arc(g1 +++ g2)"
proof -
  have injg1: "inj_on g1 {0..1}"
    using assms
    by (simp add: arc_def)
  have injg2: "inj_on g2 {0..1}"
    using assms
    by (simp add: arc_def)
  have g11: "g1 1 = g2 0"
   and sb:  "g1 ` {0..1} \<inter> g2 ` {0..1} \<subseteq> {g2 0}"
    using assms
    by (simp_all add: arc_def pathfinish_def pathstart_def path_image_def)
  { fix x and y::real
    assume xy: "x \<le> 1" "0 \<le> y" " y * 2 \<le> 1" "\<not> x * 2 \<le> 1" "g2 (2 * x - 1) = g1 (2 * y)"
    have g1im: "g1 (2 * y) \<in> g1 ` {0..1} \<inter> g2 ` {0..1}"
      using xy
      apply simp
      apply (rule_tac x="2 * x - 1" in image_eqI, auto)
      done
    have False
      using subsetD [OF sb g1im] xy
      by (auto dest: inj_onD [OF injg2])
   } note * = this
  show ?thesis
    apply (simp add: arc_def inj_on_def)
    apply (clarsimp simp add: arc_imp_path assms path_join)
    apply (simp add: joinpaths_def split: if_split_asm)
    apply (force dest: inj_onD [OF injg1])
    apply (metis *)
    apply (metis *)
    apply (force dest: inj_onD [OF injg2])
    done
qed

lemma reversepath_joinpaths:
    "pathfinish g1 = pathstart g2 \<Longrightarrow> reversepath(g1 +++ g2) = reversepath g2 +++ reversepath g1"
  unfolding reversepath_def pathfinish_def pathstart_def joinpaths_def
  by (rule ext) (auto simp: mult.commute)


subsection\<open>Some reversed and "if and only if" versions of joining theorems\<close>

lemma path_join_path_ends: 
  fixes g1 :: "real \<Rightarrow> 'a::metric_space"
  assumes "path(g1 +++ g2)" "path g2" 
    shows "pathfinish g1 = pathstart g2"
proof (rule ccontr)
  def e \<equiv> "dist (g1 1) (g2 0)"
  assume Neg: "pathfinish g1 \<noteq> pathstart g2"
  then have "0 < dist (pathfinish g1) (pathstart g2)"
    by auto
  then have "e > 0"
    by (metis e_def pathfinish_def pathstart_def) 
  then obtain d1 where "d1 > 0" 
       and d1: "\<And>x'. \<lbrakk>x'\<in>{0..1}; norm x' < d1\<rbrakk> \<Longrightarrow> dist (g2 x') (g2 0) < e/2"
    using assms(2) unfolding path_def continuous_on_iff
    apply (drule_tac x=0 in bspec, simp)
    by (metis half_gt_zero_iff norm_conv_dist)
  obtain d2 where "d2 > 0" 
       and d2: "\<And>x'. \<lbrakk>x'\<in>{0..1}; dist x' (1/2) < d2\<rbrakk> 
                      \<Longrightarrow> dist ((g1 +++ g2) x') (g1 1) < e/2"
    using assms(1) \<open>e > 0\<close> unfolding path_def continuous_on_iff
    apply (drule_tac x="1/2" in bspec, simp)
    apply (drule_tac x="e/2" in spec)
    apply (force simp: joinpaths_def)
    done
  have int01_1: "min (1/2) (min d1 d2) / 2 \<in> {0..1}"
    using \<open>d1 > 0\<close> \<open>d2 > 0\<close> by (simp add: min_def)
  have dist1: "norm (min (1 / 2) (min d1 d2) / 2) < d1"
    using \<open>d1 > 0\<close> \<open>d2 > 0\<close> by (simp add: min_def dist_norm)
  have int01_2: "1/2 + min (1/2) (min d1 d2) / 4 \<in> {0..1}"
    using \<open>d1 > 0\<close> \<open>d2 > 0\<close> by (simp add: min_def)
  have dist2: "dist (1 / 2 + min (1 / 2) (min d1 d2) / 4) (1 / 2) < d2"
    using \<open>d1 > 0\<close> \<open>d2 > 0\<close> by (simp add: min_def dist_norm)
  have [simp]: "~ min (1 / 2) (min d1 d2) \<le> 0"
    using \<open>d1 > 0\<close> \<open>d2 > 0\<close> by (simp add: min_def)
  have "dist (g2 (min (1 / 2) (min d1 d2) / 2)) (g1 1) < e/2"
       "dist (g2 (min (1 / 2) (min d1 d2) / 2)) (g2 0) < e/2"
    using d1 [OF int01_1 dist1] d2 [OF int01_2 dist2] by (simp_all add: joinpaths_def)
  then have "dist (g1 1) (g2 0) < e/2 + e/2"
    using dist_triangle_half_r e_def by blast
  then show False 
    by (simp add: e_def [symmetric])
qed

lemma path_join_eq [simp]:  
  fixes g1 :: "real \<Rightarrow> 'a::metric_space"
  assumes "path g1" "path g2"
    shows "path(g1 +++ g2) \<longleftrightarrow> pathfinish g1 = pathstart g2"
  using assms by (metis path_join_path_ends path_join_imp)

lemma simple_path_joinE: 
  assumes "simple_path(g1 +++ g2)" and "pathfinish g1 = pathstart g2"
  obtains "arc g1" "arc g2" 
          "path_image g1 \<inter> path_image g2 \<subseteq> {pathstart g1, pathstart g2}"
proof -
  have *: "\<And>x y. \<lbrakk>0 \<le> x; x \<le> 1; 0 \<le> y; y \<le> 1; (g1 +++ g2) x = (g1 +++ g2) y\<rbrakk> 
               \<Longrightarrow> x = y \<or> x = 0 \<and> y = 1 \<or> x = 1 \<and> y = 0"
    using assms by (simp add: simple_path_def)
  have "path g1" 
    using assms path_join simple_path_imp_path by blast
  moreover have "inj_on g1 {0..1}"
  proof (clarsimp simp: inj_on_def)
    fix x y
    assume "g1 x = g1 y" "0 \<le> x" "x \<le> 1" "0 \<le> y" "y \<le> 1"
    then show "x = y"
      using * [of "x/2" "y/2"] by (simp add: joinpaths_def split_ifs)
  qed
  ultimately have "arc g1"
    using assms  by (simp add: arc_def)
  have [simp]: "g2 0 = g1 1"
    using assms by (metis pathfinish_def pathstart_def) 
  have "path g2"
    using assms path_join simple_path_imp_path by blast
  moreover have "inj_on g2 {0..1}"
  proof (clarsimp simp: inj_on_def)
    fix x y
    assume "g2 x = g2 y" "0 \<le> x" "x \<le> 1" "0 \<le> y" "y \<le> 1"
    then show "x = y"
      using * [of "(x + 1) / 2" "(y + 1) / 2"]
      by (force simp: joinpaths_def split_ifs divide_simps)
  qed
  ultimately have "arc g2"
    using assms  by (simp add: arc_def)
  have "g2 y = g1 0 \<or> g2 y = g1 1" 
       if "g1 x = g2 y" "0 \<le> x" "x \<le> 1" "0 \<le> y" "y \<le> 1" for x y
      using * [of "x / 2" "(y + 1) / 2"] that
      by (auto simp: joinpaths_def split_ifs divide_simps)
  then have "path_image g1 \<inter> path_image g2 \<subseteq> {pathstart g1, pathstart g2}"
    by (fastforce simp: pathstart_def pathfinish_def path_image_def)
  with \<open>arc g1\<close> \<open>arc g2\<close> show ?thesis using that by blast
qed

lemma simple_path_join_loop_eq:
  assumes "pathfinish g2 = pathstart g1" "pathfinish g1 = pathstart g2" 
    shows "simple_path(g1 +++ g2) \<longleftrightarrow>
             arc g1 \<and> arc g2 \<and> path_image g1 \<inter> path_image g2 \<subseteq> {pathstart g1, pathstart g2}"
by (metis assms simple_path_joinE simple_path_join_loop)

lemma arc_join_eq:
  assumes "pathfinish g1 = pathstart g2" 
    shows "arc(g1 +++ g2) \<longleftrightarrow>
           arc g1 \<and> arc g2 \<and> path_image g1 \<inter> path_image g2 \<subseteq> {pathstart g2}"
           (is "?lhs = ?rhs")
proof 
  assume ?lhs
  then have "simple_path(g1 +++ g2)" by (rule arc_imp_simple_path)
  then have *: "\<And>x y. \<lbrakk>0 \<le> x; x \<le> 1; 0 \<le> y; y \<le> 1; (g1 +++ g2) x = (g1 +++ g2) y\<rbrakk> 
               \<Longrightarrow> x = y \<or> x = 0 \<and> y = 1 \<or> x = 1 \<and> y = 0"
    using assms by (simp add: simple_path_def)
  have False if "g1 0 = g2 u" "0 \<le> u" "u \<le> 1" for u
    using * [of 0 "(u + 1) / 2"] that assms arc_distinct_ends [OF \<open>?lhs\<close>]
    by (auto simp: joinpaths_def pathstart_def pathfinish_def split_ifs divide_simps)
  then have n1: "~ (pathstart g1 \<in> path_image g2)"
    unfolding pathstart_def path_image_def
    using atLeastAtMost_iff by blast
  show ?rhs using \<open>?lhs\<close>
    apply (rule simple_path_joinE [OF arc_imp_simple_path assms])
    using n1 by force
next
  assume ?rhs then show ?lhs
    using assms
    by (fastforce simp: pathfinish_def pathstart_def intro!: arc_join)
qed

lemma arc_join_eq_alt: 
        "pathfinish g1 = pathstart g2
        \<Longrightarrow> (arc(g1 +++ g2) \<longleftrightarrow>
             arc g1 \<and> arc g2 \<and>
             path_image g1 \<inter> path_image g2 = {pathstart g2})"
using pathfinish_in_path_image by (fastforce simp: arc_join_eq)


subsection\<open>The joining of paths is associative\<close>

lemma path_assoc:
    "\<lbrakk>pathfinish p = pathstart q; pathfinish q = pathstart r\<rbrakk>
     \<Longrightarrow> path(p +++ (q +++ r)) \<longleftrightarrow> path((p +++ q) +++ r)"
by simp

lemma simple_path_assoc: 
  assumes "pathfinish p = pathstart q" "pathfinish q = pathstart r" 
    shows "simple_path (p +++ (q +++ r)) \<longleftrightarrow> simple_path ((p +++ q) +++ r)"
proof (cases "pathstart p = pathfinish r")
  case True show ?thesis
  proof
    assume "simple_path (p +++ q +++ r)"
    with assms True show "simple_path ((p +++ q) +++ r)"
      by (fastforce simp add: simple_path_join_loop_eq arc_join_eq path_image_join 
                    dest: arc_distinct_ends [of r])
  next
    assume 0: "simple_path ((p +++ q) +++ r)"
    with assms True have q: "pathfinish r \<notin> path_image q"
      using arc_distinct_ends  
      by (fastforce simp add: simple_path_join_loop_eq arc_join_eq path_image_join)
    have "pathstart r \<notin> path_image p"
      using assms
      by (metis 0 IntI arc_distinct_ends arc_join_eq_alt empty_iff insert_iff 
              pathfinish_in_path_image pathfinish_join simple_path_joinE)
    with assms 0 q True show "simple_path (p +++ q +++ r)"
      by (auto simp: simple_path_join_loop_eq arc_join_eq path_image_join 
               dest!: subsetD [OF _ IntI])
  qed
next
  case False
  { fix x :: 'a
    assume a: "path_image p \<inter> path_image q \<subseteq> {pathstart q}"
              "(path_image p \<union> path_image q) \<inter> path_image r \<subseteq> {pathstart r}"
              "x \<in> path_image p" "x \<in> path_image r"
    have "pathstart r \<in> path_image q"
      by (metis assms(2) pathfinish_in_path_image)
    with a have "x = pathstart q"
      by blast
  }
  with False assms show ?thesis 
    by (auto simp: simple_path_eq_arc simple_path_join_loop_eq arc_join_eq path_image_join)
qed

lemma arc_assoc: 
     "\<lbrakk>pathfinish p = pathstart q; pathfinish q = pathstart r\<rbrakk>
      \<Longrightarrow> arc(p +++ (q +++ r)) \<longleftrightarrow> arc((p +++ q) +++ r)"
by (simp add: arc_simple_path simple_path_assoc)


section\<open>Choosing a subpath of an existing path\<close>

definition subpath :: "real \<Rightarrow> real \<Rightarrow> (real \<Rightarrow> 'a) \<Rightarrow> real \<Rightarrow> 'a::real_normed_vector"
  where "subpath a b g \<equiv> \<lambda>x. g((b - a) * x + a)"

lemma path_image_subpath_gen:
  fixes g :: "_ \<Rightarrow> 'a::real_normed_vector"
  shows "path_image(subpath u v g) = g ` (closed_segment u v)"
  apply (simp add: closed_segment_real_eq path_image_def subpath_def)
  apply (subst o_def [of g, symmetric])
  apply (simp add: image_comp [symmetric])
  done

lemma path_image_subpath:
  fixes g :: "real \<Rightarrow> 'a::real_normed_vector"
  shows "path_image(subpath u v g) = (if u \<le> v then g ` {u..v} else g ` {v..u})"
  by (simp add: path_image_subpath_gen closed_segment_eq_real_ivl)

lemma path_subpath [simp]:
  fixes g :: "real \<Rightarrow> 'a::real_normed_vector"
  assumes "path g" "u \<in> {0..1}" "v \<in> {0..1}"
    shows "path(subpath u v g)"
proof -
  have "continuous_on {0..1} (g o (\<lambda>x. ((v-u) * x+ u)))"
    apply (rule continuous_intros | simp)+
    apply (simp add: image_affinity_atLeastAtMost [where c=u])
    using assms
    apply (auto simp: path_def continuous_on_subset)
    done
  then show ?thesis
    by (simp add: path_def subpath_def)
qed

lemma pathstart_subpath [simp]: "pathstart(subpath u v g) = g(u)"
  by (simp add: pathstart_def subpath_def)

lemma pathfinish_subpath [simp]: "pathfinish(subpath u v g) = g(v)"
  by (simp add: pathfinish_def subpath_def)

lemma subpath_trivial [simp]: "subpath 0 1 g = g"
  by (simp add: subpath_def)

lemma subpath_reversepath: "subpath 1 0 g = reversepath g"
  by (simp add: reversepath_def subpath_def)

lemma reversepath_subpath: "reversepath(subpath u v g) = subpath v u g"
  by (simp add: reversepath_def subpath_def algebra_simps)

lemma subpath_translation: "subpath u v ((\<lambda>x. a + x) o g) = (\<lambda>x. a + x) o subpath u v g"
  by (rule ext) (simp add: subpath_def)

lemma subpath_linear_image: "linear f \<Longrightarrow> subpath u v (f o g) = f o subpath u v g"
  by (rule ext) (simp add: subpath_def)

lemma affine_ineq:
  fixes x :: "'a::linordered_idom"
  assumes "x \<le> 1" "v \<le> u"
    shows "v + x * u \<le> u + x * v"
proof -
  have "(1-x)*(u-v) \<ge> 0"
    using assms by auto
  then show ?thesis
    by (simp add: algebra_simps)
qed

lemma sum_le_prod1:
  fixes a::real shows "\<lbrakk>a \<le> 1; b \<le> 1\<rbrakk> \<Longrightarrow> a + b \<le> 1 + a * b"
by (metis add.commute affine_ineq less_eq_real_def mult.right_neutral)

lemma simple_path_subpath_eq:
  "simple_path(subpath u v g) \<longleftrightarrow>
     path(subpath u v g) \<and> u\<noteq>v \<and>
     (\<forall>x y. x \<in> closed_segment u v \<and> y \<in> closed_segment u v \<and> g x = g y
                \<longrightarrow> x = y \<or> x = u \<and> y = v \<or> x = v \<and> y = u)"
    (is "?lhs = ?rhs")
proof (rule iffI)
  assume ?lhs
  then have p: "path (\<lambda>x. g ((v - u) * x + u))"
        and sim: "(\<And>x y. \<lbrakk>x\<in>{0..1}; y\<in>{0..1}; g ((v - u) * x + u) = g ((v - u) * y + u)\<rbrakk>
                  \<Longrightarrow> x = y \<or> x = 0 \<and> y = 1 \<or> x = 1 \<and> y = 0)"
    by (auto simp: simple_path_def subpath_def)
  { fix x y
    assume "x \<in> closed_segment u v" "y \<in> closed_segment u v" "g x = g y"
    then have "x = y \<or> x = u \<and> y = v \<or> x = v \<and> y = u"
    using sim [of "(x-u)/(v-u)" "(y-u)/(v-u)"] p
    by (auto simp: closed_segment_real_eq image_affinity_atLeastAtMost divide_simps
       split: if_split_asm)
  } moreover
  have "path(subpath u v g) \<and> u\<noteq>v"
    using sim [of "1/3" "2/3"] p
    by (auto simp: subpath_def)
  ultimately show ?rhs
    by metis
next
  assume ?rhs
  then
  have d1: "\<And>x y. \<lbrakk>g x = g y; u \<le> x; x \<le> v; u \<le> y; y \<le> v\<rbrakk> \<Longrightarrow> x = y \<or> x = u \<and> y = v \<or> x = v \<and> y = u"
   and d2: "\<And>x y. \<lbrakk>g x = g y; v \<le> x; x \<le> u; v \<le> y; y \<le> u\<rbrakk> \<Longrightarrow> x = y \<or> x = u \<and> y = v \<or> x = v \<and> y = u"
   and ne: "u < v \<or> v < u"
   and psp: "path (subpath u v g)"
    by (auto simp: closed_segment_real_eq image_affinity_atLeastAtMost)
  have [simp]: "\<And>x. u + x * v = v + x * u \<longleftrightarrow> u=v \<or> x=1"
    by algebra
  show ?lhs using psp ne
    unfolding simple_path_def subpath_def
    by (fastforce simp add: algebra_simps affine_ineq mult_left_mono crossproduct_eq dest: d1 d2)
qed

lemma arc_subpath_eq:
  "arc(subpath u v g) \<longleftrightarrow> path(subpath u v g) \<and> u\<noteq>v \<and> inj_on g (closed_segment u v)"
    (is "?lhs = ?rhs")
proof (rule iffI)
  assume ?lhs
  then have p: "path (\<lambda>x. g ((v - u) * x + u))"
        and sim: "(\<And>x y. \<lbrakk>x\<in>{0..1}; y\<in>{0..1}; g ((v - u) * x + u) = g ((v - u) * y + u)\<rbrakk>
                  \<Longrightarrow> x = y)"
    by (auto simp: arc_def inj_on_def subpath_def)
  { fix x y
    assume "x \<in> closed_segment u v" "y \<in> closed_segment u v" "g x = g y"
    then have "x = y"
    using sim [of "(x-u)/(v-u)" "(y-u)/(v-u)"] p
    by (force simp add: inj_on_def closed_segment_real_eq image_affinity_atLeastAtMost divide_simps
       split: if_split_asm)
  } moreover
  have "path(subpath u v g) \<and> u\<noteq>v"
    using sim [of "1/3" "2/3"] p
    by (auto simp: subpath_def)
  ultimately show ?rhs
    unfolding inj_on_def
    by metis
next
  assume ?rhs
  then
  have d1: "\<And>x y. \<lbrakk>g x = g y; u \<le> x; x \<le> v; u \<le> y; y \<le> v\<rbrakk> \<Longrightarrow> x = y"
   and d2: "\<And>x y. \<lbrakk>g x = g y; v \<le> x; x \<le> u; v \<le> y; y \<le> u\<rbrakk> \<Longrightarrow> x = y"
   and ne: "u < v \<or> v < u"
   and psp: "path (subpath u v g)"
    by (auto simp: inj_on_def closed_segment_real_eq image_affinity_atLeastAtMost)
  show ?lhs using psp ne
    unfolding arc_def subpath_def inj_on_def
    by (auto simp: algebra_simps affine_ineq mult_left_mono crossproduct_eq dest: d1 d2)
qed


lemma simple_path_subpath:
  assumes "simple_path g" "u \<in> {0..1}" "v \<in> {0..1}" "u \<noteq> v"
  shows "simple_path(subpath u v g)"
  using assms
  apply (simp add: simple_path_subpath_eq simple_path_imp_path)
  apply (simp add: simple_path_def closed_segment_real_eq image_affinity_atLeastAtMost, fastforce)
  done

lemma arc_simple_path_subpath:
    "\<lbrakk>simple_path g; u \<in> {0..1}; v \<in> {0..1}; g u \<noteq> g v\<rbrakk> \<Longrightarrow> arc(subpath u v g)"
  by (force intro: simple_path_subpath simple_path_imp_arc)

lemma arc_subpath_arc:
    "\<lbrakk>arc g; u \<in> {0..1}; v \<in> {0..1}; u \<noteq> v\<rbrakk> \<Longrightarrow> arc(subpath u v g)"
  by (meson arc_def arc_imp_simple_path arc_simple_path_subpath inj_onD)

lemma arc_simple_path_subpath_interior:
    "\<lbrakk>simple_path g; u \<in> {0..1}; v \<in> {0..1}; u \<noteq> v; \<bar>u-v\<bar> < 1\<rbrakk> \<Longrightarrow> arc(subpath u v g)"
    apply (rule arc_simple_path_subpath)
    apply (force simp: simple_path_def)+
    done

lemma path_image_subpath_subset:
    "\<lbrakk>path g; u \<in> {0..1}; v \<in> {0..1}\<rbrakk> \<Longrightarrow> path_image(subpath u v g) \<subseteq> path_image g"
  apply (simp add: closed_segment_real_eq image_affinity_atLeastAtMost path_image_subpath)
  apply (auto simp: path_image_def)
  done

lemma join_subpaths_middle: "subpath (0) ((1 / 2)) p +++ subpath ((1 / 2)) 1 p = p"
  by (rule ext) (simp add: joinpaths_def subpath_def divide_simps)

subsection\<open>There is a subpath to the frontier\<close>

lemma subpath_to_frontier_explicit:
    fixes S :: "'a::metric_space set"
    assumes g: "path g" and "pathfinish g \<notin> S"
    obtains u where "0 \<le> u" "u \<le> 1"
                "\<And>x. 0 \<le> x \<and> x < u \<Longrightarrow> g x \<in> interior S"
                "(g u \<notin> interior S)" "(u = 0 \<or> g u \<in> closure S)"
proof -
  have gcon: "continuous_on {0..1} g"     using g by (simp add: path_def)
  then have com: "compact ({0..1} \<inter> {u. g u \<in> closure (- S)})"
    apply (simp add: Int_commute [of "{0..1}"] compact_eq_bounded_closed closed_vimage_Int [unfolded vimage_def])
    using compact_eq_bounded_closed apply fastforce
    done
  have "1 \<in> {u. g u \<in> closure (- S)}"
    using assms by (simp add: pathfinish_def closure_def)
  then have dis: "{0..1} \<inter> {u. g u \<in> closure (- S)} \<noteq> {}"
    using atLeastAtMost_iff zero_le_one by blast
  then obtain u where "0 \<le> u" "u \<le> 1" and gu: "g u \<in> closure (- S)"
                  and umin: "\<And>t. \<lbrakk>0 \<le> t; t \<le> 1; g t \<in> closure (- S)\<rbrakk> \<Longrightarrow> u \<le> t"
    using compact_attains_inf [OF com dis] by fastforce
  then have umin': "\<And>t. \<lbrakk>0 \<le> t; t \<le> 1; t < u\<rbrakk> \<Longrightarrow>  g t \<in> S"
    using closure_def by fastforce
  { assume "u \<noteq> 0"
    then have "u > 0" using \<open>0 \<le> u\<close> by auto
    { fix e::real assume "e > 0"
      obtain d where "d>0" and d: "\<And>x'. \<lbrakk>x' \<in> {0..1}; dist x' u \<le> d\<rbrakk> \<Longrightarrow> dist (g x') (g u) < e"
        using continuous_onE [OF gcon _ \<open>e > 0\<close>] \<open>0 \<le> _\<close> \<open>_ \<le> 1\<close> atLeastAtMost_iff by auto
      have *: "dist (max 0 (u - d / 2)) u \<le> d"
        using \<open>0 \<le> u\<close> \<open>u \<le> 1\<close> \<open>d > 0\<close> by (simp add: dist_real_def)
      have "\<exists>y\<in>S. dist y (g u) < e"
        using \<open>0 < u\<close> \<open>u \<le> 1\<close> \<open>d > 0\<close>
        by (force intro: d [OF _ *] umin')
    }
    then have "g u \<in> closure S"
      by (simp add: frontier_def closure_approachable)
  }
  then show ?thesis
    apply (rule_tac u=u in that)
    apply (auto simp: \<open>0 \<le> u\<close> \<open>u \<le> 1\<close> gu interior_closure umin)
    using \<open>_ \<le> 1\<close> interior_closure umin apply fastforce
    done
qed

lemma subpath_to_frontier_strong:
    assumes g: "path g" and "pathfinish g \<notin> S"
    obtains u where "0 \<le> u" "u \<le> 1" "g u \<notin> interior S"
                    "u = 0 \<or> (\<forall>x. 0 \<le> x \<and> x < 1 \<longrightarrow> subpath 0 u g x \<in> interior S)  \<and>  g u \<in> closure S"
proof -
  obtain u where "0 \<le> u" "u \<le> 1"
             and gxin: "\<And>x. 0 \<le> x \<and> x < u \<Longrightarrow> g x \<in> interior S"
             and gunot: "(g u \<notin> interior S)" and u0: "(u = 0 \<or> g u \<in> closure S)"
    using subpath_to_frontier_explicit [OF assms] by blast
  show ?thesis
    apply (rule that [OF \<open>0 \<le> u\<close> \<open>u \<le> 1\<close>])
    apply (simp add: gunot)
    using \<open>0 \<le> u\<close> u0 by (force simp: subpath_def gxin)
qed

lemma subpath_to_frontier:
    assumes g: "path g" and g0: "pathstart g \<in> closure S" and g1: "pathfinish g \<notin> S"
    obtains u where "0 \<le> u" "u \<le> 1" "g u \<in> frontier S" "(path_image(subpath 0 u g) - {g u}) \<subseteq> interior S"
proof -
  obtain u where "0 \<le> u" "u \<le> 1"
             and notin: "g u \<notin> interior S"
             and disj: "u = 0 \<or>
                        (\<forall>x. 0 \<le> x \<and> x < 1 \<longrightarrow> subpath 0 u g x \<in> interior S) \<and> g u \<in> closure S"
    using subpath_to_frontier_strong [OF g g1] by blast
  show ?thesis
    apply (rule that [OF \<open>0 \<le> u\<close> \<open>u \<le> 1\<close>])
    apply (metis DiffI disj frontier_def g0 notin pathstart_def)
    using \<open>0 \<le> u\<close> g0 disj
    apply (simp add: path_image_subpath_gen)
    apply (auto simp: closed_segment_eq_real_ivl pathstart_def pathfinish_def subpath_def)
    apply (rename_tac y)
    apply (drule_tac x="y/u" in spec)
    apply (auto split: if_split_asm)
    done
qed

lemma exists_path_subpath_to_frontier:
    fixes S :: "'a::real_normed_vector set"
    assumes "path g" "pathstart g \<in> closure S" "pathfinish g \<notin> S"
    obtains h where "path h" "pathstart h = pathstart g" "path_image h \<subseteq> path_image g"
                    "path_image h - {pathfinish h} \<subseteq> interior S"
                    "pathfinish h \<in> frontier S"
proof -
  obtain u where u: "0 \<le> u" "u \<le> 1" "g u \<in> frontier S" "(path_image(subpath 0 u g) - {g u}) \<subseteq> interior S"
    using subpath_to_frontier [OF assms] by blast
  show ?thesis
    apply (rule that [of "subpath 0 u g"])
    using assms u
    apply (simp_all add: path_image_subpath)
    apply (simp add: pathstart_def)
    apply (force simp: closed_segment_eq_real_ivl path_image_def)
    done
qed

lemma exists_path_subpath_to_frontier_closed:
    fixes S :: "'a::real_normed_vector set"
    assumes S: "closed S" and g: "path g" and g0: "pathstart g \<in> S" and g1: "pathfinish g \<notin> S"
    obtains h where "path h" "pathstart h = pathstart g" "path_image h \<subseteq> path_image g \<inter> S"
                    "pathfinish h \<in> frontier S"
proof -
  obtain h where h: "path h" "pathstart h = pathstart g" "path_image h \<subseteq> path_image g"
                    "path_image h - {pathfinish h} \<subseteq> interior S"
                    "pathfinish h \<in> frontier S"
    using exists_path_subpath_to_frontier [OF g _ g1] closure_closed [OF S] g0 by auto
  show ?thesis
    apply (rule that [OF \<open>path h\<close>])
    using assms h
    apply auto
    apply (metis Diff_single_insert frontier_subset_eq insert_iff interior_subset subset_iff)
    done
qed

subsection \<open>Reparametrizing a closed curve to start at some chosen point\<close>

definition shiftpath :: "real \<Rightarrow> (real \<Rightarrow> 'a::topological_space) \<Rightarrow> real \<Rightarrow> 'a"
  where "shiftpath a f = (\<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"
    and "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"
    and "a \<in> {0 .. 1}"
  shows "pathfinish (shiftpath a g) = g a"
    and "pathstart (shiftpath a g) = g a"
  using assms
  by (auto intro!: pathfinish_shiftpath pathstart_shiftpath)

lemma closed_shiftpath:
  assumes "pathfinish g = pathstart g"
    and "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"
    and "pathfinish g = pathstart g"
    and "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_closed_Un)
    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)"])
    prefer 3
    apply (rule continuous_intros)+
    prefer 2
    apply (rule continuous_intros)+
    apply (rule_tac[1-2] continuous_on_subset[OF assms(1)[unfolded path_def]])
    using assms(3) and **
    apply auto
    apply (auto simp add: field_simps)
    done
qed

lemma shiftpath_shiftpath:
  assumes "pathfinish g = pathstart g"
    and "a \<in> {0..1}"
    and "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}"
    and "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)"
    then have "\<exists>y\<in>{0..1} \<inter> {x. a + x \<le> 1}. g x = g (a + y)"
    proof (cases "a \<le> x")
      case False
      then show ?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)
        apply (auto simp add: field_simps atomize_not)
        done
    next
      case True
      then show ?thesis
        using as(1-2) and assms(1)
        apply (rule_tac x="x - a" in bexI)
        apply (auto simp add: field_simps)
        done
    qed
  }
  then show ?thesis
    using assms
    unfolding shiftpath_def path_image_def pathfinish_def pathstart_def
    by (auto simp add: image_iff)
qed


subsection \<open>Special case of straight-line paths\<close>

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_intros]: "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