src/HOL/Analysis/Path_Connected.thy
author wenzelm
Fri, 12 Apr 2019 22:09:25 +0200
changeset 70136 f03a01a18c6e
parent 69986 f2d327275065
child 70178 4900351361b0
permissions -rw-r--r--
modernized tags: default scope excludes proof;
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
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(*  Title:      HOL/Analysis/Path_Connected.thy
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    Authors:    LC Paulson and Robert Himmelmann (TU Muenchen), based on material from HOL Light
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*)
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section \<open>Path-Connectedness\<close>
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theory Path_Connected
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  imports Starlike
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begin
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subsection \<open>Paths and Arcs\<close>
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definition\<^marker>\<open>tag important\<close> path :: "(real \<Rightarrow> 'a::topological_space) \<Rightarrow> bool"
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  where "path g \<longleftrightarrow> continuous_on {0..1} g"
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definition\<^marker>\<open>tag important\<close> pathstart :: "(real \<Rightarrow> 'a::topological_space) \<Rightarrow> 'a"
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  where "pathstart g = g 0"
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definition\<^marker>\<open>tag important\<close> pathfinish :: "(real \<Rightarrow> 'a::topological_space) \<Rightarrow> 'a"
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  where "pathfinish g = g 1"
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definition\<^marker>\<open>tag important\<close> path_image :: "(real \<Rightarrow> 'a::topological_space) \<Rightarrow> 'a set"
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  where "path_image g = g ` {0 .. 1}"
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definition\<^marker>\<open>tag important\<close> reversepath :: "(real \<Rightarrow> 'a::topological_space) \<Rightarrow> real \<Rightarrow> 'a"
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  where "reversepath g = (\<lambda>x. g(1 - x))"
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definition\<^marker>\<open>tag important\<close> joinpaths :: "(real \<Rightarrow> 'a::topological_space) \<Rightarrow> (real \<Rightarrow> 'a) \<Rightarrow> real \<Rightarrow> 'a"
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    (infixr "+++" 75)
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  where "g1 +++ g2 = (\<lambda>x. if x \<le> 1/2 then g1 (2 * x) else g2 (2 * x - 1))"
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definition\<^marker>\<open>tag important\<close> simple_path :: "(real \<Rightarrow> 'a::topological_space) \<Rightarrow> bool"
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  where "simple_path g \<longleftrightarrow>
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     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)"
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definition\<^marker>\<open>tag important\<close> arc :: "(real \<Rightarrow> 'a :: topological_space) \<Rightarrow> bool"
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  where "arc g \<longleftrightarrow> path g \<and> inj_on g {0..1}"
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subsection\<^marker>\<open>tag unimportant\<close>\<open>Invariance theorems\<close>
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lemma path_eq: "path p \<Longrightarrow> (\<And>t. t \<in> {0..1} \<Longrightarrow> p t = q t) \<Longrightarrow> path q"
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  using continuous_on_eq path_def by blast
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lemma path_continuous_image: "path g \<Longrightarrow> continuous_on (path_image g) f \<Longrightarrow> path(f \<circ> g)"
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  unfolding path_def path_image_def
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parents: 59557
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    47
  using continuous_on_compose by blast
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    48
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lemma path_translation_eq:
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    50
  fixes g :: "real \<Rightarrow> 'a :: real_normed_vector"
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    51
  shows "path((\<lambda>x. a + x) \<circ> g) = path g"
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proof -
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paulson <lp15@cam.ac.uk>
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    53
  have g: "g = (\<lambda>x. -a + x) \<circ> ((\<lambda>x. a + x) \<circ> g)"
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    by (rule ext) simp
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  show ?thesis
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    56
    unfolding path_def
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    apply safe
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    apply (subst g)
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    59
    apply (rule continuous_on_compose)
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    apply (auto intro: continuous_intros)
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    done
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qed
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lemma path_linear_image_eq:
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  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
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   assumes "linear f" "inj f"
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     shows "path(f \<circ> g) = path g"
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proof -
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  from linear_injective_left_inverse [OF assms]
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  obtain h where h: "linear h" "h \<circ> f = id"
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    by blast
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  then have g: "g = h \<circ> (f \<circ> g)"
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    by (metis comp_assoc id_comp)
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  show ?thesis
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    unfolding path_def
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    using h assms
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    by (metis g continuous_on_compose linear_continuous_on linear_conv_bounded_linear)
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    78
qed
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    79
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lemma pathstart_translation: "pathstart((\<lambda>x. a + x) \<circ> g) = a + pathstart g"
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  by (simp add: pathstart_def)
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lemma pathstart_linear_image_eq: "linear f \<Longrightarrow> pathstart(f \<circ> g) = f(pathstart g)"
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  by (simp add: pathstart_def)
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lemma pathfinish_translation: "pathfinish((\<lambda>x. a + x) \<circ> g) = a + pathfinish g"
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  by (simp add: pathfinish_def)
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lemma pathfinish_linear_image: "linear f \<Longrightarrow> pathfinish(f \<circ> g) = f(pathfinish g)"
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  by (simp add: pathfinish_def)
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    91
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lemma path_image_translation: "path_image((\<lambda>x. a + x) \<circ> g) = (\<lambda>x. a + x) ` (path_image g)"
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    93
  by (simp add: image_comp path_image_def)
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lemma path_image_linear_image: "linear f \<Longrightarrow> path_image(f \<circ> g) = f ` (path_image g)"
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    96
  by (simp add: image_comp path_image_def)
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    97
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    98
lemma reversepath_translation: "reversepath((\<lambda>x. a + x) \<circ> g) = (\<lambda>x. a + x) \<circ> reversepath g"
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    99
  by (rule ext) (simp add: reversepath_def)
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   101
lemma reversepath_linear_image: "linear f \<Longrightarrow> reversepath(f \<circ> g) = f \<circ> reversepath g"
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   102
  by (rule ext) (simp add: reversepath_def)
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   103
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   104
lemma joinpaths_translation:
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   105
    "((\<lambda>x. a + x) \<circ> g1) +++ ((\<lambda>x. a + x) \<circ> g2) = (\<lambda>x. a + x) \<circ> (g1 +++ g2)"
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diff changeset
   106
  by (rule ext) (simp add: joinpaths_def)
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paulson
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diff changeset
   107
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   108
lemma joinpaths_linear_image: "linear f \<Longrightarrow> (f \<circ> g1) +++ (f \<circ> g2) = f \<circ> (g1 +++ g2)"
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   109
  by (rule ext) (simp add: joinpaths_def)
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   110
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   111
lemma simple_path_translation_eq:
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   112
  fixes g :: "real \<Rightarrow> 'a::euclidean_space"
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parents: 68072
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   113
  shows "simple_path((\<lambda>x. a + x) \<circ> g) = simple_path g"
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   114
  by (simp add: simple_path_def path_translation_eq)
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   115
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   116
lemma simple_path_linear_image_eq:
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diff changeset
   117
  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
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paulson
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diff changeset
   118
  assumes "linear f" "inj f"
68096
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paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
   119
    shows "simple_path(f \<circ> g) = simple_path g"
60303
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diff changeset
   120
  using assms inj_on_eq_iff [of f]
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diff changeset
   121
  by (auto simp: path_linear_image_eq simple_path_def path_translation_eq)
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diff changeset
   122
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parents: 59557
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   123
lemma arc_translation_eq:
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parents: 59557
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   124
  fixes g :: "real \<Rightarrow> 'a::euclidean_space"
68096
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paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
   125
  shows "arc((\<lambda>x. a + x) \<circ> g) = arc g"
60303
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parents: 59557
diff changeset
   126
  by (auto simp: arc_def inj_on_def path_translation_eq)
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paulson
parents: 59557
diff changeset
   127
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paulson
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   128
lemma arc_linear_image_eq:
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paulson
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diff changeset
   129
  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
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paulson
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   130
   assumes "linear f" "inj f"
68096
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paulson <lp15@cam.ac.uk>
parents: 68072
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   131
     shows  "arc(f \<circ> g) = arc g"
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   132
  using assms inj_on_eq_iff [of f]
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   133
  by (auto simp: arc_def inj_on_def path_linear_image_eq)
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diff changeset
   134
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parents: 69508
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   135
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parents: 69986
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   136
subsection\<^marker>\<open>tag unimportant\<close>\<open>Basic lemmas about paths\<close>
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parents: 59557
diff changeset
   137
69939
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paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
   138
lemma pathin_iff_path_real [simp]: "pathin euclideanreal g \<longleftrightarrow> path g"
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parents: 69712
diff changeset
   139
  by (simp add: pathin_def path_def)
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
   140
64788
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
   141
lemma continuous_on_path: "path f \<Longrightarrow> t \<subseteq> {0..1} \<Longrightarrow> continuous_on t f"
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
   142
  using continuous_on_subset path_def by blast
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
   143
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   144
lemma arc_imp_simple_path: "arc g \<Longrightarrow> simple_path g"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   145
  by (simp add: arc_def inj_on_def simple_path_def)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   146
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   147
lemma arc_imp_path: "arc g \<Longrightarrow> path g"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   148
  using arc_def by blast
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   149
64788
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
   150
lemma arc_imp_inj_on: "arc g \<Longrightarrow> inj_on g {0..1}"
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
   151
  by (auto simp: arc_def)
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
   152
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   153
lemma simple_path_imp_path: "simple_path g \<Longrightarrow> path g"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   154
  using simple_path_def by blast
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   155
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   156
lemma simple_path_cases: "simple_path g \<Longrightarrow> arc g \<or> pathfinish g = pathstart g"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   157
  unfolding simple_path_def arc_def inj_on_def pathfinish_def pathstart_def
68096
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
   158
  by force
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   159
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   160
lemma simple_path_imp_arc: "simple_path g \<Longrightarrow> pathfinish g \<noteq> pathstart g \<Longrightarrow> arc g"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   161
  using simple_path_cases by auto
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   162
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   163
lemma arc_distinct_ends: "arc g \<Longrightarrow> pathfinish g \<noteq> pathstart g"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   164
  unfolding arc_def inj_on_def pathfinish_def pathstart_def
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   165
  by fastforce
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   166
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   167
lemma arc_simple_path: "arc g \<longleftrightarrow> simple_path g \<and> pathfinish g \<noteq> pathstart g"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   168
  using arc_distinct_ends arc_imp_simple_path simple_path_cases by blast
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   169
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   170
lemma simple_path_eq_arc: "pathfinish g \<noteq> pathstart g \<Longrightarrow> (simple_path g = arc g)"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   171
  by (simp add: arc_simple_path)
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
   172
66793
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66456
diff changeset
   173
lemma path_image_const [simp]: "path_image (\<lambda>t. a) = {a}"
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66456
diff changeset
   174
  by (force simp: path_image_def)
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66456
diff changeset
   175
60974
6a6f15d8fbc4 New material and fixes related to the forthcoming Stone-Weierstrass development
paulson <lp15@cam.ac.uk>
parents: 60809
diff changeset
   176
lemma path_image_nonempty [simp]: "path_image g \<noteq> {}"
56188
0268784f60da use cbox to relax class constraints
immler
parents: 53640
diff changeset
   177
  unfolding path_image_def image_is_empty box_eq_empty
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   178
  by auto
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
   179
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   180
lemma pathstart_in_path_image[intro]: "pathstart g \<in> path_image g"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   181
  unfolding pathstart_def path_image_def
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   182
  by auto
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
   183
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   184
lemma pathfinish_in_path_image[intro]: "pathfinish g \<in> path_image g"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   185
  unfolding pathfinish_def path_image_def
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   186
  by auto
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   187
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   188
lemma connected_path_image[intro]: "path g \<Longrightarrow> connected (path_image g)"
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
   189
  unfolding path_def path_image_def
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   190
  using connected_continuous_image connected_Icc by blast
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
   191
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   192
lemma compact_path_image[intro]: "path g \<Longrightarrow> compact (path_image g)"
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
   193
  unfolding path_def path_image_def
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   194
  using compact_continuous_image connected_Icc by blast
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
   195
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   196
lemma reversepath_reversepath[simp]: "reversepath (reversepath g) = g"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   197
  unfolding reversepath_def
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   198
  by auto
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
   199
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   200
lemma pathstart_reversepath[simp]: "pathstart (reversepath g) = pathfinish g"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   201
  unfolding pathstart_def reversepath_def pathfinish_def
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   202
  by auto
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
   203
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   204
lemma pathfinish_reversepath[simp]: "pathfinish (reversepath g) = pathstart g"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   205
  unfolding pathstart_def reversepath_def pathfinish_def
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   206
  by auto
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
   207
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   208
lemma pathstart_join[simp]: "pathstart (g1 +++ g2) = pathstart g1"
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   209
  unfolding pathstart_def joinpaths_def pathfinish_def
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   210
  by auto
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
   211
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   212
lemma pathfinish_join[simp]: "pathfinish (g1 +++ g2) = pathfinish g2"
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   213
  unfolding pathstart_def joinpaths_def pathfinish_def
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   214
  by auto
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
   215
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   216
lemma path_image_reversepath[simp]: "path_image (reversepath g) = path_image g"
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   217
proof -
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   218
  have *: "\<And>g. path_image (reversepath g) \<subseteq> path_image g"
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   219
    unfolding path_image_def subset_eq reversepath_def Ball_def image_iff
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   220
    by force
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   221
  show ?thesis
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   222
    using *[of g] *[of "reversepath g"]
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   223
    unfolding reversepath_reversepath
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   224
    by auto
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   225
qed
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
   226
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   227
lemma path_reversepath [simp]: "path (reversepath g) \<longleftrightarrow> path g"
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   228
proof -
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   229
  have *: "\<And>g. path g \<Longrightarrow> path (reversepath g)"
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   230
    unfolding path_def reversepath_def
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   231
    apply (rule continuous_on_compose[unfolded o_def, of _ "\<lambda>x. 1 - x"])
68096
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
   232
    apply (auto intro: continuous_intros continuous_on_subset[of "{0..1}"])
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   233
    done
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   234
  show ?thesis
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   235
    using *[of "reversepath g"] *[of g]
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   236
    unfolding reversepath_reversepath
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   237
    by (rule iffI)
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   238
qed
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   239
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   240
lemma arc_reversepath:
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   241
  assumes "arc g" shows "arc(reversepath g)"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   242
proof -
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   243
  have injg: "inj_on g {0..1}"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   244
    using assms
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   245
    by (simp add: arc_def)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   246
  have **: "\<And>x y::real. 1-x = 1-y \<Longrightarrow> x = y"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   247
    by simp
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   248
  show ?thesis
68096
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
   249
    using assms  by (clarsimp simp: arc_def intro!: inj_onI) (simp add: inj_onD reversepath_def **)
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   250
qed
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   251
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   252
lemma simple_path_reversepath: "simple_path g \<Longrightarrow> simple_path (reversepath g)"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   253
  apply (simp add: simple_path_def)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   254
  apply (force simp: reversepath_def)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   255
  done
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   256
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   257
lemmas reversepath_simps =
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   258
  path_reversepath path_image_reversepath pathstart_reversepath pathfinish_reversepath
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
   259
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   260
lemma path_join[simp]:
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   261
  assumes "pathfinish g1 = pathstart g2"
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   262
  shows "path (g1 +++ g2) \<longleftrightarrow> path g1 \<and> path g2"
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   263
  unfolding path_def pathfinish_def pathstart_def
51478
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 50935
diff changeset
   264
proof safe
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 50935
diff changeset
   265
  assume cont: "continuous_on {0..1} (g1 +++ g2)"
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 50935
diff changeset
   266
  have g1: "continuous_on {0..1} g1 \<longleftrightarrow> continuous_on {0..1} ((g1 +++ g2) \<circ> (\<lambda>x. x / 2))"
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 50935
diff changeset
   267
    by (intro continuous_on_cong refl) (auto simp: joinpaths_def)
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 50935
diff changeset
   268
  have g2: "continuous_on {0..1} g2 \<longleftrightarrow> continuous_on {0..1} ((g1 +++ g2) \<circ> (\<lambda>x. x / 2 + 1/2))"
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   269
    using assms
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   270
    by (intro continuous_on_cong refl) (auto simp: joinpaths_def pathfinish_def pathstart_def)
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   271
  show "continuous_on {0..1} g1" and "continuous_on {0..1} g2"
51481
ef949192e5d6 move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents: 51478
diff changeset
   272
    unfolding g1 g2
56371
fb9ae0727548 extend continuous_intros; remove continuous_on_intros and isCont_intros
hoelzl
parents: 56188
diff changeset
   273
    by (auto intro!: continuous_intros continuous_on_subset[OF cont] simp del: o_apply)
51478
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 50935
diff changeset
   274
next
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 50935
diff changeset
   275
  assume g1g2: "continuous_on {0..1} g1" "continuous_on {0..1} g2"
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 50935
diff changeset
   276
  have 01: "{0 .. 1} = {0..1/2} \<union> {1/2 .. 1::real}"
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
   277
    by auto
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   278
  {
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   279
    fix x :: real
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   280
    assume "0 \<le> x" and "x \<le> 1"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   281
    then have "x \<in> (\<lambda>x. x * 2) ` {0..1 / 2}"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   282
      by (intro image_eqI[where x="x/2"]) auto
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   283
  }
51478
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 50935
diff changeset
   284
  note 1 = this
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   285
  {
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   286
    fix x :: real
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   287
    assume "0 \<le> x" and "x \<le> 1"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   288
    then have "x \<in> (\<lambda>x. x * 2 - 1) ` {1 / 2..1}"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   289
      by (intro image_eqI[where x="x/2 + 1/2"]) auto
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   290
  }
51478
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 50935
diff changeset
   291
  note 2 = this
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   292
  show "continuous_on {0..1} (g1 +++ g2)"
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   293
    using assms
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   294
    unfolding joinpaths_def 01
56371
fb9ae0727548 extend continuous_intros; remove continuous_on_intros and isCont_intros
hoelzl
parents: 56188
diff changeset
   295
    apply (intro continuous_on_cases closed_atLeastAtMost g1g2[THEN continuous_on_compose2] continuous_intros)
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   296
    apply (auto simp: field_simps pathfinish_def pathstart_def intro!: 1 2)
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   297
    done
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   298
qed
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
   299
69514
58a77f548bb6 tuned headers
nipkow
parents: 69508
diff changeset
   300
70136
f03a01a18c6e modernized tags: default scope excludes proof;
wenzelm
parents: 69986
diff changeset
   301
subsection\<^marker>\<open>tag unimportant\<close> \<open>Path Images\<close>
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   302
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   303
lemma bounded_path_image: "path g \<Longrightarrow> bounded(path_image g)"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   304
  by (simp add: compact_imp_bounded compact_path_image)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   305
60809
457abb82fb9e the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents: 60420
diff changeset
   306
lemma closed_path_image:
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   307
  fixes g :: "real \<Rightarrow> 'a::t2_space"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   308
  shows "path g \<Longrightarrow> closed(path_image g)"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   309
  by (metis compact_path_image compact_imp_closed)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   310
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   311
lemma connected_simple_path_image: "simple_path g \<Longrightarrow> connected(path_image g)"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   312
  by (metis connected_path_image simple_path_imp_path)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   313
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   314
lemma compact_simple_path_image: "simple_path g \<Longrightarrow> compact(path_image g)"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   315
  by (metis compact_path_image simple_path_imp_path)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   316
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   317
lemma bounded_simple_path_image: "simple_path g \<Longrightarrow> bounded(path_image g)"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   318
  by (metis bounded_path_image simple_path_imp_path)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   319
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   320
lemma closed_simple_path_image:
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   321
  fixes g :: "real \<Rightarrow> 'a::t2_space"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   322
  shows "simple_path g \<Longrightarrow> closed(path_image g)"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   323
  by (metis closed_path_image simple_path_imp_path)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   324
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   325
lemma connected_arc_image: "arc g \<Longrightarrow> connected(path_image g)"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   326
  by (metis connected_path_image arc_imp_path)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   327
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   328
lemma compact_arc_image: "arc g \<Longrightarrow> compact(path_image g)"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   329
  by (metis compact_path_image arc_imp_path)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   330
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   331
lemma bounded_arc_image: "arc g \<Longrightarrow> bounded(path_image g)"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   332
  by (metis bounded_path_image arc_imp_path)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   333
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   334
lemma closed_arc_image:
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   335
  fixes g :: "real \<Rightarrow> 'a::t2_space"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   336
  shows "arc g \<Longrightarrow> closed(path_image g)"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   337
  by (metis closed_path_image arc_imp_path)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   338
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   339
lemma path_image_join_subset: "path_image (g1 +++ g2) \<subseteq> path_image g1 \<union> path_image g2"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   340
  unfolding path_image_def joinpaths_def
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   341
  by auto
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
   342
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
   343
lemma subset_path_image_join:
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   344
  assumes "path_image g1 \<subseteq> s"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   345
    and "path_image g2 \<subseteq> s"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   346
  shows "path_image (g1 +++ g2) \<subseteq> s"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   347
  using path_image_join_subset[of g1 g2] and assms
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   348
  by auto
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
   349
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
   350
lemma path_image_join:
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   351
    "pathfinish g1 = pathstart g2 \<Longrightarrow> path_image(g1 +++ g2) = path_image g1 \<union> path_image g2"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   352
  apply (rule subset_antisym [OF path_image_join_subset])
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   353
  apply (auto simp: pathfinish_def pathstart_def path_image_def joinpaths_def image_def)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   354
  apply (drule sym)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   355
  apply (rule_tac x="xa/2" in bexI, auto)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   356
  apply (rule ccontr)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   357
  apply (drule_tac x="(xa+1)/2" in bspec)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   358
  apply (auto simp: field_simps)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   359
  apply (drule_tac x="1/2" in bspec, auto)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   360
  done
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
   361
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
   362
lemma not_in_path_image_join:
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   363
  assumes "x \<notin> path_image g1"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   364
    and "x \<notin> path_image g2"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   365
  shows "x \<notin> path_image (g1 +++ g2)"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   366
  using assms and path_image_join_subset[of g1 g2]
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   367
  by auto
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
   368
68096
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
   369
lemma pathstart_compose: "pathstart(f \<circ> p) = f(pathstart p)"
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   370
  by (simp add: pathstart_def)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   371
68096
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
   372
lemma pathfinish_compose: "pathfinish(f \<circ> p) = f(pathfinish p)"
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   373
  by (simp add: pathfinish_def)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   374
68096
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
   375
lemma path_image_compose: "path_image (f \<circ> p) = f ` (path_image p)"
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   376
  by (simp add: image_comp path_image_def)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   377
68096
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
   378
lemma path_compose_join: "f \<circ> (p +++ q) = (f \<circ> p) +++ (f \<circ> q)"
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   379
  by (rule ext) (simp add: joinpaths_def)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   380
68096
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
   381
lemma path_compose_reversepath: "f \<circ> reversepath p = reversepath(f \<circ> p)"
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   382
  by (rule ext) (simp add: reversepath_def)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   383
61762
d50b993b4fb9 Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents: 61738
diff changeset
   384
lemma joinpaths_eq:
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   385
  "(\<And>t. t \<in> {0..1} \<Longrightarrow> p t = p' t) \<Longrightarrow>
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   386
   (\<And>t. t \<in> {0..1} \<Longrightarrow> q t = q' t)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   387
   \<Longrightarrow>  t \<in> {0..1} \<Longrightarrow> (p +++ q) t = (p' +++ q') t"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   388
  by (auto simp: joinpaths_def)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   389
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   390
lemma simple_path_inj_on: "simple_path g \<Longrightarrow> inj_on g {0<..<1}"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   391
  by (auto simp: simple_path_def path_image_def inj_on_def less_eq_real_def Ball_def)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   392
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   393
70136
f03a01a18c6e modernized tags: default scope excludes proof;
wenzelm
parents: 69986
diff changeset
   394
subsection\<^marker>\<open>tag unimportant\<close>\<open>Simple paths with the endpoints removed\<close>
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   395
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   396
lemma simple_path_endless:
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   397
    "simple_path c \<Longrightarrow> path_image c - {pathstart c,pathfinish c} = c ` {0<..<1}"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   398
  apply (auto simp: simple_path_def path_image_def pathstart_def pathfinish_def Ball_def Bex_def image_def)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   399
  apply (metis eq_iff le_less_linear)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   400
  apply (metis leD linear)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   401
  using less_eq_real_def zero_le_one apply blast
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   402
  using less_eq_real_def zero_le_one apply blast
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   403
  done
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
   404
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   405
lemma connected_simple_path_endless:
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   406
    "simple_path c \<Longrightarrow> connected(path_image c - {pathstart c,pathfinish c})"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   407
apply (simp add: simple_path_endless)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   408
apply (rule connected_continuous_image)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   409
apply (meson continuous_on_subset greaterThanLessThan_subseteq_atLeastAtMost_iff le_numeral_extra(3) le_numeral_extra(4) path_def simple_path_imp_path)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   410
by auto
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   411
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   412
lemma nonempty_simple_path_endless:
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   413
    "simple_path c \<Longrightarrow> path_image c - {pathstart c,pathfinish c} \<noteq> {}"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   414
  by (simp add: simple_path_endless)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   415
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   416
70136
f03a01a18c6e modernized tags: default scope excludes proof;
wenzelm
parents: 69986
diff changeset
   417
subsection\<^marker>\<open>tag unimportant\<close>\<open>The operations on paths\<close>
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   418
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   419
lemma path_image_subset_reversepath: "path_image(reversepath g) \<le> path_image g"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   420
  by (auto simp: path_image_def reversepath_def)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   421
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   422
lemma path_imp_reversepath: "path g \<Longrightarrow> path(reversepath g)"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   423
  apply (auto simp: path_def reversepath_def)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   424
  using continuous_on_compose [of "{0..1}" "\<lambda>x. 1 - x" g]
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   425
  apply (auto simp: continuous_on_op_minus)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   426
  done
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   427
61204
3e491e34a62e new lemmas and movement of lemmas into place
paulson
parents: 60974
diff changeset
   428
lemma half_bounded_equal: "1 \<le> x * 2 \<Longrightarrow> x * 2 \<le> 1 \<longleftrightarrow> x = (1/2::real)"
3e491e34a62e new lemmas and movement of lemmas into place
paulson
parents: 60974
diff changeset
   429
  by simp
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   430
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   431
lemma continuous_on_joinpaths:
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   432
  assumes "continuous_on {0..1} g1" "continuous_on {0..1} g2" "pathfinish g1 = pathstart g2"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   433
    shows "continuous_on {0..1} (g1 +++ g2)"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   434
proof -
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   435
  have *: "{0..1::real} = {0..1/2} \<union> {1/2..1}"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   436
    by auto
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   437
  have gg: "g2 0 = g1 1"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   438
    by (metis assms(3) pathfinish_def pathstart_def)
61204
3e491e34a62e new lemmas and movement of lemmas into place
paulson
parents: 60974
diff changeset
   439
  have 1: "continuous_on {0..1/2} (g1 +++ g2)"
68096
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
   440
    apply (rule continuous_on_eq [of _ "g1 \<circ> (\<lambda>x. 2*x)"])
61204
3e491e34a62e new lemmas and movement of lemmas into place
paulson
parents: 60974
diff changeset
   441
    apply (rule continuous_intros | simp add: joinpaths_def assms)+
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   442
    done
68096
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
   443
  have "continuous_on {1/2..1} (g2 \<circ> (\<lambda>x. 2*x-1))"
61204
3e491e34a62e new lemmas and movement of lemmas into place
paulson
parents: 60974
diff changeset
   444
    apply (rule continuous_on_subset [of "{1/2..1}"])
3e491e34a62e new lemmas and movement of lemmas into place
paulson
parents: 60974
diff changeset
   445
    apply (rule continuous_intros | simp add: image_affinity_atLeastAtMost_diff assms)+
3e491e34a62e new lemmas and movement of lemmas into place
paulson
parents: 60974
diff changeset
   446
    done
3e491e34a62e new lemmas and movement of lemmas into place
paulson
parents: 60974
diff changeset
   447
  then have 2: "continuous_on {1/2..1} (g1 +++ g2)"
68096
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
   448
    apply (rule continuous_on_eq [of "{1/2..1}" "g2 \<circ> (\<lambda>x. 2*x-1)"])
61204
3e491e34a62e new lemmas and movement of lemmas into place
paulson
parents: 60974
diff changeset
   449
    apply (rule assms continuous_intros | simp add: joinpaths_def mult.commute half_bounded_equal gg)+
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   450
    done
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   451
  show ?thesis
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   452
    apply (subst *)
62397
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62381
diff changeset
   453
    apply (rule continuous_on_closed_Un)
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   454
    using 1 2
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   455
    apply auto
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   456
    done
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   457
qed
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   458
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   459
lemma path_join_imp: "\<lbrakk>path g1; path g2; pathfinish g1 = pathstart g2\<rbrakk> \<Longrightarrow> path(g1 +++ g2)"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   460
  by (simp add: path_join)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   461
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
   462
lemma simple_path_join_loop:
60809
457abb82fb9e the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents: 60420
diff changeset
   463
  assumes "arc g1" "arc g2"
457abb82fb9e the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents: 60420
diff changeset
   464
          "pathfinish g1 = pathstart g2"  "pathfinish g2 = pathstart g1"
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   465
          "path_image g1 \<inter> path_image g2 \<subseteq> {pathstart g1, pathstart g2}"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   466
  shows "simple_path(g1 +++ g2)"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   467
proof -
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   468
  have injg1: "inj_on g1 {0..1}"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   469
    using assms
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   470
    by (simp add: arc_def)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   471
  have injg2: "inj_on g2 {0..1}"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   472
    using assms
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   473
    by (simp add: arc_def)
60809
457abb82fb9e the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents: 60420
diff changeset
   474
  have g12: "g1 1 = g2 0"
457abb82fb9e the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents: 60420
diff changeset
   475
   and g21: "g2 1 = g1 0"
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   476
   and sb:  "g1 ` {0..1} \<inter> g2 ` {0..1} \<subseteq> {g1 0, g2 0}"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   477
    using assms
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   478
    by (simp_all add: arc_def pathfinish_def pathstart_def path_image_def)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   479
  { fix x and y::real
60809
457abb82fb9e the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents: 60420
diff changeset
   480
    assume xyI: "x = 1 \<longrightarrow> y \<noteq> 0"
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   481
       and xy: "x \<le> 1" "0 \<le> y" " y * 2 \<le> 1" "\<not> x * 2 \<le> 1" "g2 (2 * x - 1) = g1 (2 * y)"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   482
    have g1im: "g1 (2 * y) \<in> g1 ` {0..1} \<inter> g2 ` {0..1}"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   483
      using xy
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   484
      apply simp
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   485
      apply (rule_tac x="2 * x - 1" in image_eqI, auto)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   486
      done
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   487
    have False
60809
457abb82fb9e the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents: 60420
diff changeset
   488
      using subsetD [OF sb g1im] xy
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   489
      apply auto
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   490
      apply (drule inj_onD [OF injg1])
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   491
      using g21 [symmetric] xyI
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   492
      apply (auto dest: inj_onD [OF injg2])
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   493
      done
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   494
   } note * = this
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   495
  { fix x and y::real
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   496
    assume xy: "y \<le> 1" "0 \<le> x" "\<not> y * 2 \<le> 1" "x * 2 \<le> 1" "g1 (2 * x) = g2 (2 * y - 1)"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   497
    have g1im: "g1 (2 * x) \<in> g1 ` {0..1} \<inter> g2 ` {0..1}"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   498
      using xy
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   499
      apply simp
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   500
      apply (rule_tac x="2 * x" in image_eqI, auto)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   501
      done
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   502
    have "x = 0 \<and> y = 1"
60809
457abb82fb9e the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents: 60420
diff changeset
   503
      using subsetD [OF sb g1im] xy
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   504
      apply auto
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   505
      apply (force dest: inj_onD [OF injg1])
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   506
      using  g21 [symmetric]
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   507
      apply (auto dest: inj_onD [OF injg2])
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   508
      done
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   509
   } note ** = this
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   510
  show ?thesis
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   511
    using assms
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   512
    apply (simp add: arc_def simple_path_def path_join, clarify)
62390
842917225d56 more canonical names
nipkow
parents: 62087
diff changeset
   513
    apply (simp add: joinpaths_def split: if_split_asm)
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   514
    apply (force dest: inj_onD [OF injg1])
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   515
    apply (metis *)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   516
    apply (metis **)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   517
    apply (force dest: inj_onD [OF injg2])
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   518
    done
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   519
qed
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   520
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   521
lemma arc_join:
60809
457abb82fb9e the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents: 60420
diff changeset
   522
  assumes "arc g1" "arc g2"
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   523
          "pathfinish g1 = pathstart g2"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   524
          "path_image g1 \<inter> path_image g2 \<subseteq> {pathstart g2}"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   525
    shows "arc(g1 +++ g2)"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   526
proof -
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   527
  have injg1: "inj_on g1 {0..1}"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   528
    using assms
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   529
    by (simp add: arc_def)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   530
  have injg2: "inj_on g2 {0..1}"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   531
    using assms
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   532
    by (simp add: arc_def)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   533
  have g11: "g1 1 = g2 0"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   534
   and sb:  "g1 ` {0..1} \<inter> g2 ` {0..1} \<subseteq> {g2 0}"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   535
    using assms
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   536
    by (simp_all add: arc_def pathfinish_def pathstart_def path_image_def)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   537
  { fix x and y::real
60809
457abb82fb9e the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents: 60420
diff changeset
   538
    assume xy: "x \<le> 1" "0 \<le> y" " y * 2 \<le> 1" "\<not> x * 2 \<le> 1" "g2 (2 * x - 1) = g1 (2 * y)"
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   539
    have g1im: "g1 (2 * y) \<in> g1 ` {0..1} \<inter> g2 ` {0..1}"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   540
      using xy
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   541
      apply simp
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   542
      apply (rule_tac x="2 * x - 1" in image_eqI, auto)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   543
      done
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   544
    have False
60809
457abb82fb9e the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents: 60420
diff changeset
   545
      using subsetD [OF sb g1im] xy
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   546
      by (auto dest: inj_onD [OF injg2])
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   547
   } note * = this
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   548
  show ?thesis
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   549
    apply (simp add: arc_def inj_on_def)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   550
    apply (clarsimp simp add: arc_imp_path assms path_join)
62390
842917225d56 more canonical names
nipkow
parents: 62087
diff changeset
   551
    apply (simp add: joinpaths_def split: if_split_asm)
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   552
    apply (force dest: inj_onD [OF injg1])
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   553
    apply (metis *)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   554
    apply (metis *)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   555
    apply (force dest: inj_onD [OF injg2])
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   556
    done
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   557
qed
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   558
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   559
lemma reversepath_joinpaths:
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   560
    "pathfinish g1 = pathstart g2 \<Longrightarrow> reversepath(g1 +++ g2) = reversepath g2 +++ reversepath g1"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   561
  unfolding reversepath_def pathfinish_def pathstart_def joinpaths_def
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   562
  by (rule ext) (auto simp: mult.commute)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   563
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   564
70136
f03a01a18c6e modernized tags: default scope excludes proof;
wenzelm
parents: 69986
diff changeset
   565
subsection\<^marker>\<open>tag unimportant\<close>\<open>Some reversed and "if and only if" versions of joining theorems\<close>
62533
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   566
63594
bd218a9320b5 HOL-Multivariate_Analysis: rename theories for more descriptive names
hoelzl
parents: 63540
diff changeset
   567
lemma path_join_path_ends:
62533
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   568
  fixes g1 :: "real \<Rightarrow> 'a::metric_space"
63594
bd218a9320b5 HOL-Multivariate_Analysis: rename theories for more descriptive names
hoelzl
parents: 63540
diff changeset
   569
  assumes "path(g1 +++ g2)" "path g2"
62533
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   570
    shows "pathfinish g1 = pathstart g2"
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   571
proof (rule ccontr)
63040
eb4ddd18d635 eliminated old 'def';
wenzelm
parents: 63016
diff changeset
   572
  define e where "e = dist (g1 1) (g2 0)"
62533
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   573
  assume Neg: "pathfinish g1 \<noteq> pathstart g2"
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   574
  then have "0 < dist (pathfinish g1) (pathstart g2)"
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   575
    by auto
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   576
  then have "e > 0"
63594
bd218a9320b5 HOL-Multivariate_Analysis: rename theories for more descriptive names
hoelzl
parents: 63540
diff changeset
   577
    by (metis e_def pathfinish_def pathstart_def)
bd218a9320b5 HOL-Multivariate_Analysis: rename theories for more descriptive names
hoelzl
parents: 63540
diff changeset
   578
  then obtain d1 where "d1 > 0"
62533
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   579
       and d1: "\<And>x'. \<lbrakk>x'\<in>{0..1}; norm x' < d1\<rbrakk> \<Longrightarrow> dist (g2 x') (g2 0) < e/2"
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   580
    using assms(2) unfolding path_def continuous_on_iff
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   581
    apply (drule_tac x=0 in bspec, simp)
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   582
    by (metis half_gt_zero_iff norm_conv_dist)
63594
bd218a9320b5 HOL-Multivariate_Analysis: rename theories for more descriptive names
hoelzl
parents: 63540
diff changeset
   583
  obtain d2 where "d2 > 0"
bd218a9320b5 HOL-Multivariate_Analysis: rename theories for more descriptive names
hoelzl
parents: 63540
diff changeset
   584
       and d2: "\<And>x'. \<lbrakk>x'\<in>{0..1}; dist x' (1/2) < d2\<rbrakk>
62533
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   585
                      \<Longrightarrow> dist ((g1 +++ g2) x') (g1 1) < e/2"
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   586
    using assms(1) \<open>e > 0\<close> unfolding path_def continuous_on_iff
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   587
    apply (drule_tac x="1/2" in bspec, simp)
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   588
    apply (drule_tac x="e/2" in spec)
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   589
    apply (force simp: joinpaths_def)
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   590
    done
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   591
  have int01_1: "min (1/2) (min d1 d2) / 2 \<in> {0..1}"
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   592
    using \<open>d1 > 0\<close> \<open>d2 > 0\<close> by (simp add: min_def)
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   593
  have dist1: "norm (min (1 / 2) (min d1 d2) / 2) < d1"
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   594
    using \<open>d1 > 0\<close> \<open>d2 > 0\<close> by (simp add: min_def dist_norm)
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   595
  have int01_2: "1/2 + min (1/2) (min d1 d2) / 4 \<in> {0..1}"
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   596
    using \<open>d1 > 0\<close> \<open>d2 > 0\<close> by (simp add: min_def)
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   597
  have dist2: "dist (1 / 2 + min (1 / 2) (min d1 d2) / 4) (1 / 2) < d2"
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   598
    using \<open>d1 > 0\<close> \<open>d2 > 0\<close> by (simp add: min_def dist_norm)
69508
2a4c8a2a3f8e tuned headers; ~ -> \<not>
nipkow
parents: 69313
diff changeset
   599
  have [simp]: "\<not> min (1 / 2) (min d1 d2) \<le> 0"
62533
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   600
    using \<open>d1 > 0\<close> \<open>d2 > 0\<close> by (simp add: min_def)
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   601
  have "dist (g2 (min (1 / 2) (min d1 d2) / 2)) (g1 1) < e/2"
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   602
       "dist (g2 (min (1 / 2) (min d1 d2) / 2)) (g2 0) < e/2"
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   603
    using d1 [OF int01_1 dist1] d2 [OF int01_2 dist2] by (simp_all add: joinpaths_def)
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   604
  then have "dist (g1 1) (g2 0) < e/2 + e/2"
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   605
    using dist_triangle_half_r e_def by blast
63594
bd218a9320b5 HOL-Multivariate_Analysis: rename theories for more descriptive names
hoelzl
parents: 63540
diff changeset
   606
  then show False
62533
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   607
    by (simp add: e_def [symmetric])
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   608
qed
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   609
63594
bd218a9320b5 HOL-Multivariate_Analysis: rename theories for more descriptive names
hoelzl
parents: 63540
diff changeset
   610
lemma path_join_eq [simp]:
62533
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   611
  fixes g1 :: "real \<Rightarrow> 'a::metric_space"
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   612
  assumes "path g1" "path g2"
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   613
    shows "path(g1 +++ g2) \<longleftrightarrow> pathfinish g1 = pathstart g2"
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   614
  using assms by (metis path_join_path_ends path_join_imp)
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   615
63594
bd218a9320b5 HOL-Multivariate_Analysis: rename theories for more descriptive names
hoelzl
parents: 63540
diff changeset
   616
lemma simple_path_joinE:
62533
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   617
  assumes "simple_path(g1 +++ g2)" and "pathfinish g1 = pathstart g2"
63594
bd218a9320b5 HOL-Multivariate_Analysis: rename theories for more descriptive names
hoelzl
parents: 63540
diff changeset
   618
  obtains "arc g1" "arc g2"
62533
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   619
          "path_image g1 \<inter> path_image g2 \<subseteq> {pathstart g1, pathstart g2}"
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   620
proof -
63594
bd218a9320b5 HOL-Multivariate_Analysis: rename theories for more descriptive names
hoelzl
parents: 63540
diff changeset
   621
  have *: "\<And>x y. \<lbrakk>0 \<le> x; x \<le> 1; 0 \<le> y; y \<le> 1; (g1 +++ g2) x = (g1 +++ g2) y\<rbrakk>
62533
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   622
               \<Longrightarrow> x = y \<or> x = 0 \<and> y = 1 \<or> x = 1 \<and> y = 0"
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   623
    using assms by (simp add: simple_path_def)
63594
bd218a9320b5 HOL-Multivariate_Analysis: rename theories for more descriptive names
hoelzl
parents: 63540
diff changeset
   624
  have "path g1"
62533
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   625
    using assms path_join simple_path_imp_path by blast
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   626
  moreover have "inj_on g1 {0..1}"
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   627
  proof (clarsimp simp: inj_on_def)
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   628
    fix x y
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   629
    assume "g1 x = g1 y" "0 \<le> x" "x \<le> 1" "0 \<le> y" "y \<le> 1"
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   630
    then show "x = y"
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   631
      using * [of "x/2" "y/2"] by (simp add: joinpaths_def split_ifs)
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   632
  qed
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   633
  ultimately have "arc g1"
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   634
    using assms  by (simp add: arc_def)
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   635
  have [simp]: "g2 0 = g1 1"
63594
bd218a9320b5 HOL-Multivariate_Analysis: rename theories for more descriptive names
hoelzl
parents: 63540
diff changeset
   636
    using assms by (metis pathfinish_def pathstart_def)
62533
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   637
  have "path g2"
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   638
    using assms path_join simple_path_imp_path by blast
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   639
  moreover have "inj_on g2 {0..1}"
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   640
  proof (clarsimp simp: inj_on_def)
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   641
    fix x y
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   642
    assume "g2 x = g2 y" "0 \<le> x" "x \<le> 1" "0 \<le> y" "y \<le> 1"
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   643
    then show "x = y"
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   644
      using * [of "(x + 1) / 2" "(y + 1) / 2"]
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   645
      by (force simp: joinpaths_def split_ifs divide_simps)
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   646
  qed
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   647
  ultimately have "arc g2"
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   648
    using assms  by (simp add: arc_def)
63594
bd218a9320b5 HOL-Multivariate_Analysis: rename theories for more descriptive names
hoelzl
parents: 63540
diff changeset
   649
  have "g2 y = g1 0 \<or> g2 y = g1 1"
62533
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   650
       if "g1 x = g2 y" "0 \<le> x" "x \<le> 1" "0 \<le> y" "y \<le> 1" for x y
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   651
      using * [of "x / 2" "(y + 1) / 2"] that
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   652
      by (auto simp: joinpaths_def split_ifs divide_simps)
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   653
  then have "path_image g1 \<inter> path_image g2 \<subseteq> {pathstart g1, pathstart g2}"
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   654
    by (fastforce simp: pathstart_def pathfinish_def path_image_def)
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   655
  with \<open>arc g1\<close> \<open>arc g2\<close> show ?thesis using that by blast
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   656
qed
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   657
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   658
lemma simple_path_join_loop_eq:
63594
bd218a9320b5 HOL-Multivariate_Analysis: rename theories for more descriptive names
hoelzl
parents: 63540
diff changeset
   659
  assumes "pathfinish g2 = pathstart g1" "pathfinish g1 = pathstart g2"
62533
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   660
    shows "simple_path(g1 +++ g2) \<longleftrightarrow>
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   661
             arc g1 \<and> arc g2 \<and> path_image g1 \<inter> path_image g2 \<subseteq> {pathstart g1, pathstart g2}"
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   662
by (metis assms simple_path_joinE simple_path_join_loop)
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   663
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   664
lemma arc_join_eq:
63594
bd218a9320b5 HOL-Multivariate_Analysis: rename theories for more descriptive names
hoelzl
parents: 63540
diff changeset
   665
  assumes "pathfinish g1 = pathstart g2"
62533
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   666
    shows "arc(g1 +++ g2) \<longleftrightarrow>
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   667
           arc g1 \<and> arc g2 \<and> path_image g1 \<inter> path_image g2 \<subseteq> {pathstart g2}"
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   668
           (is "?lhs = ?rhs")
63594
bd218a9320b5 HOL-Multivariate_Analysis: rename theories for more descriptive names
hoelzl
parents: 63540
diff changeset
   669
proof
62533
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   670
  assume ?lhs
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   671
  then have "simple_path(g1 +++ g2)" by (rule arc_imp_simple_path)
63594
bd218a9320b5 HOL-Multivariate_Analysis: rename theories for more descriptive names
hoelzl
parents: 63540
diff changeset
   672
  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>
62533
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   673
               \<Longrightarrow> x = y \<or> x = 0 \<and> y = 1 \<or> x = 1 \<and> y = 0"
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   674
    using assms by (simp add: simple_path_def)
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   675
  have False if "g1 0 = g2 u" "0 \<le> u" "u \<le> 1" for u
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   676
    using * [of 0 "(u + 1) / 2"] that assms arc_distinct_ends [OF \<open>?lhs\<close>]
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   677
    by (auto simp: joinpaths_def pathstart_def pathfinish_def split_ifs divide_simps)
69508
2a4c8a2a3f8e tuned headers; ~ -> \<not>
nipkow
parents: 69313
diff changeset
   678
  then have n1: "pathstart g1 \<notin> path_image g2"
62533
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   679
    unfolding pathstart_def path_image_def
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   680
    using atLeastAtMost_iff by blast
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   681
  show ?rhs using \<open>?lhs\<close>
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   682
    apply (rule simple_path_joinE [OF arc_imp_simple_path assms])
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   683
    using n1 by force
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   684
next
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   685
  assume ?rhs then show ?lhs
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   686
    using assms
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   687
    by (fastforce simp: pathfinish_def pathstart_def intro!: arc_join)
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   688
qed
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   689
63594
bd218a9320b5 HOL-Multivariate_Analysis: rename theories for more descriptive names
hoelzl
parents: 63540
diff changeset
   690
lemma arc_join_eq_alt:
62533
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   691
        "pathfinish g1 = pathstart g2
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   692
        \<Longrightarrow> (arc(g1 +++ g2) \<longleftrightarrow>
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   693
             arc g1 \<and> arc g2 \<and>
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   694
             path_image g1 \<inter> path_image g2 = {pathstart g2})"
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   695
using pathfinish_in_path_image by (fastforce simp: arc_join_eq)
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   696
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   697
70136
f03a01a18c6e modernized tags: default scope excludes proof;
wenzelm
parents: 69986
diff changeset
   698
subsection\<^marker>\<open>tag unimportant\<close>\<open>The joining of paths is associative\<close>
62533
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   699
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   700
lemma path_assoc:
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   701
    "\<lbrakk>pathfinish p = pathstart q; pathfinish q = pathstart r\<rbrakk>
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   702
     \<Longrightarrow> path(p +++ (q +++ r)) \<longleftrightarrow> path((p +++ q) +++ r)"
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   703
by simp
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   704
63594
bd218a9320b5 HOL-Multivariate_Analysis: rename theories for more descriptive names
hoelzl
parents: 63540
diff changeset
   705
lemma simple_path_assoc:
bd218a9320b5 HOL-Multivariate_Analysis: rename theories for more descriptive names
hoelzl
parents: 63540
diff changeset
   706
  assumes "pathfinish p = pathstart q" "pathfinish q = pathstart r"
62533
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   707
    shows "simple_path (p +++ (q +++ r)) \<longleftrightarrow> simple_path ((p +++ q) +++ r)"
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   708
proof (cases "pathstart p = pathfinish r")
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   709
  case True show ?thesis
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   710
  proof
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   711
    assume "simple_path (p +++ q +++ r)"
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   712
    with assms True show "simple_path ((p +++ q) +++ r)"
63594
bd218a9320b5 HOL-Multivariate_Analysis: rename theories for more descriptive names
hoelzl
parents: 63540
diff changeset
   713
      by (fastforce simp add: simple_path_join_loop_eq arc_join_eq path_image_join
62533
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   714
                    dest: arc_distinct_ends [of r])
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   715
  next
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   716
    assume 0: "simple_path ((p +++ q) +++ r)"
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   717
    with assms True have q: "pathfinish r \<notin> path_image q"
63594
bd218a9320b5 HOL-Multivariate_Analysis: rename theories for more descriptive names
hoelzl
parents: 63540
diff changeset
   718
      using arc_distinct_ends
62533
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   719
      by (fastforce simp add: simple_path_join_loop_eq arc_join_eq path_image_join)
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   720
    have "pathstart r \<notin> path_image p"
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   721
      using assms
63594
bd218a9320b5 HOL-Multivariate_Analysis: rename theories for more descriptive names
hoelzl
parents: 63540
diff changeset
   722
      by (metis 0 IntI arc_distinct_ends arc_join_eq_alt empty_iff insert_iff
62533
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   723
              pathfinish_in_path_image pathfinish_join simple_path_joinE)
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   724
    with assms 0 q True show "simple_path (p +++ q +++ r)"
63594
bd218a9320b5 HOL-Multivariate_Analysis: rename theories for more descriptive names
hoelzl
parents: 63540
diff changeset
   725
      by (auto simp: simple_path_join_loop_eq arc_join_eq path_image_join
62533
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   726
               dest!: subsetD [OF _ IntI])
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   727
  qed
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   728
next
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   729
  case False
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   730
  { fix x :: 'a
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   731
    assume a: "path_image p \<inter> path_image q \<subseteq> {pathstart q}"
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   732
              "(path_image p \<union> path_image q) \<inter> path_image r \<subseteq> {pathstart r}"
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   733
              "x \<in> path_image p" "x \<in> path_image r"
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   734
    have "pathstart r \<in> path_image q"
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   735
      by (metis assms(2) pathfinish_in_path_image)
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   736
    with a have "x = pathstart q"
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   737
      by blast
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   738
  }
63594
bd218a9320b5 HOL-Multivariate_Analysis: rename theories for more descriptive names
hoelzl
parents: 63540
diff changeset
   739
  with False assms show ?thesis
62533
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   740
    by (auto simp: simple_path_eq_arc simple_path_join_loop_eq arc_join_eq path_image_join)
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   741
qed
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   742
63594
bd218a9320b5 HOL-Multivariate_Analysis: rename theories for more descriptive names
hoelzl
parents: 63540
diff changeset
   743
lemma arc_assoc:
62533
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   744
     "\<lbrakk>pathfinish p = pathstart q; pathfinish q = pathstart r\<rbrakk>
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   745
      \<Longrightarrow> arc(p +++ (q +++ r)) \<longleftrightarrow> arc((p +++ q) +++ r)"
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   746
by (simp add: arc_simple_path simple_path_assoc)
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   747
70136
f03a01a18c6e modernized tags: default scope excludes proof;
wenzelm
parents: 69986
diff changeset
   748
subsubsection\<^marker>\<open>tag unimportant\<close>\<open>Symmetry and loops\<close>
62620
d21dab28b3f9 New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents: 62618
diff changeset
   749
d21dab28b3f9 New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents: 62618
diff changeset
   750
lemma path_sym:
d21dab28b3f9 New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents: 62618
diff changeset
   751
    "\<lbrakk>pathfinish p = pathstart q; pathfinish q = pathstart p\<rbrakk> \<Longrightarrow> path(p +++ q) \<longleftrightarrow> path(q +++ p)"
d21dab28b3f9 New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents: 62618
diff changeset
   752
  by auto
d21dab28b3f9 New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents: 62618
diff changeset
   753
d21dab28b3f9 New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents: 62618
diff changeset
   754
lemma simple_path_sym:
d21dab28b3f9 New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents: 62618
diff changeset
   755
    "\<lbrakk>pathfinish p = pathstart q; pathfinish q = pathstart p\<rbrakk>
d21dab28b3f9 New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents: 62618
diff changeset
   756
     \<Longrightarrow> simple_path(p +++ q) \<longleftrightarrow> simple_path(q +++ p)"
d21dab28b3f9 New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents: 62618
diff changeset
   757
by (metis (full_types) inf_commute insert_commute simple_path_joinE simple_path_join_loop)
d21dab28b3f9 New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents: 62618
diff changeset
   758
d21dab28b3f9 New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents: 62618
diff changeset
   759
lemma path_image_sym:
d21dab28b3f9 New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents: 62618
diff changeset
   760
    "\<lbrakk>pathfinish p = pathstart q; pathfinish q = pathstart p\<rbrakk>
d21dab28b3f9 New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents: 62618
diff changeset
   761
     \<Longrightarrow> path_image(p +++ q) = path_image(q +++ p)"
d21dab28b3f9 New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents: 62618
diff changeset
   762
by (simp add: path_image_join sup_commute)
d21dab28b3f9 New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents: 62618
diff changeset
   763
62533
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   764
69518
bf88364c9e94 tuned headers etc, added bib-file
nipkow
parents: 69517
diff changeset
   765
subsection\<open>Subpath\<close>
60809
457abb82fb9e the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents: 60420
diff changeset
   766
70136
f03a01a18c6e modernized tags: default scope excludes proof;
wenzelm
parents: 69986
diff changeset
   767
definition\<^marker>\<open>tag important\<close> subpath :: "real \<Rightarrow> real \<Rightarrow> (real \<Rightarrow> 'a) \<Rightarrow> real \<Rightarrow> 'a::real_normed_vector"
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   768
  where "subpath a b g \<equiv> \<lambda>x. g((b - a) * x + a)"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   769
61762
d50b993b4fb9 Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents: 61738
diff changeset
   770
lemma path_image_subpath_gen:
d50b993b4fb9 Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents: 61738
diff changeset
   771
  fixes g :: "_ \<Rightarrow> 'a::real_normed_vector"
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   772
  shows "path_image(subpath u v g) = g ` (closed_segment u v)"
69661
a03a63b81f44 tuned proofs
haftmann
parents: 69620
diff changeset
   773
  by (auto simp add: closed_segment_real_eq path_image_def subpath_def)
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   774
61762
d50b993b4fb9 Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents: 61738
diff changeset
   775
lemma path_image_subpath:
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   776
  fixes g :: "real \<Rightarrow> 'a::real_normed_vector"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   777
  shows "path_image(subpath u v g) = (if u \<le> v then g ` {u..v} else g ` {v..u})"
61762
d50b993b4fb9 Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents: 61738
diff changeset
   778
  by (simp add: path_image_subpath_gen closed_segment_eq_real_ivl)
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   779
65038
9391ea7daa17 new lemmas about segments, etc. Also recast some theorems to use Union rather than general set comprehensions
paulson <lp15@cam.ac.uk>
parents: 64911
diff changeset
   780
lemma path_image_subpath_commute:
9391ea7daa17 new lemmas about segments, etc. Also recast some theorems to use Union rather than general set comprehensions
paulson <lp15@cam.ac.uk>
parents: 64911
diff changeset
   781
  fixes g :: "real \<Rightarrow> 'a::real_normed_vector"
9391ea7daa17 new lemmas about segments, etc. Also recast some theorems to use Union rather than general set comprehensions
paulson <lp15@cam.ac.uk>
parents: 64911
diff changeset
   782
  shows "path_image(subpath u v g) = path_image(subpath v u g)"
9391ea7daa17 new lemmas about segments, etc. Also recast some theorems to use Union rather than general set comprehensions
paulson <lp15@cam.ac.uk>
parents: 64911
diff changeset
   783
  by (simp add: path_image_subpath_gen closed_segment_eq_real_ivl)
9391ea7daa17 new lemmas about segments, etc. Also recast some theorems to use Union rather than general set comprehensions
paulson <lp15@cam.ac.uk>
parents: 64911
diff changeset
   784
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   785
lemma path_subpath [simp]:
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   786
  fixes g :: "real \<Rightarrow> 'a::real_normed_vector"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   787
  assumes "path g" "u \<in> {0..1}" "v \<in> {0..1}"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   788
    shows "path(subpath u v g)"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   789
proof -
68096
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
   790
  have "continuous_on {0..1} (g \<circ> (\<lambda>x. ((v-u) * x+ u)))"
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   791
    apply (rule continuous_intros | simp)+
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   792
    apply (simp add: image_affinity_atLeastAtMost [where c=u])
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   793
    using assms
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   794
    apply (auto simp: path_def continuous_on_subset)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   795
    done
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   796
  then show ?thesis
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   797
    by (simp add: path_def subpath_def)
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   798
qed
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
   799
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   800
lemma pathstart_subpath [simp]: "pathstart(subpath u v g) = g(u)"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   801
  by (simp add: pathstart_def subpath_def)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   802
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   803
lemma pathfinish_subpath [simp]: "pathfinish(subpath u v g) = g(v)"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   804
  by (simp add: pathfinish_def subpath_def)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   805
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   806
lemma subpath_trivial [simp]: "subpath 0 1 g = g"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   807
  by (simp add: subpath_def)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   808
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   809
lemma subpath_reversepath: "subpath 1 0 g = reversepath g"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   810
  by (simp add: reversepath_def subpath_def)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   811
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   812
lemma reversepath_subpath: "reversepath(subpath u v g) = subpath v u g"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   813
  by (simp add: reversepath_def subpath_def algebra_simps)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   814
68096
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
   815
lemma subpath_translation: "subpath u v ((\<lambda>x. a + x) \<circ> g) = (\<lambda>x. a + x) \<circ> subpath u v g"
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   816
  by (rule ext) (simp add: subpath_def)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   817
68096
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
   818
lemma subpath_linear_image: "linear f \<Longrightarrow> subpath u v (f \<circ> g) = f \<circ> subpath u v g"
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   819
  by (rule ext) (simp add: subpath_def)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   820
60809
457abb82fb9e the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents: 60420
diff changeset
   821
lemma affine_ineq:
457abb82fb9e the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents: 60420
diff changeset
   822
  fixes x :: "'a::linordered_idom"
61762
d50b993b4fb9 Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents: 61738
diff changeset
   823
  assumes "x \<le> 1" "v \<le> u"
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   824
    shows "v + x * u \<le> u + x * v"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   825
proof -
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   826
  have "(1-x)*(u-v) \<ge> 0"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   827
    using assms by auto
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   828
  then show ?thesis
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   829
    by (simp add: algebra_simps)
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   830
qed
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
   831
61711
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
   832
lemma sum_le_prod1:
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
   833
  fixes a::real shows "\<lbrakk>a \<le> 1; b \<le> 1\<rbrakk> \<Longrightarrow> a + b \<le> 1 + a * b"
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
   834
by (metis add.commute affine_ineq less_eq_real_def mult.right_neutral)
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
   835
60809
457abb82fb9e the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents: 60420
diff changeset
   836
lemma simple_path_subpath_eq:
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   837
  "simple_path(subpath u v g) \<longleftrightarrow>
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   838
     path(subpath u v g) \<and> u\<noteq>v \<and>
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   839
     (\<forall>x y. x \<in> closed_segment u v \<and> y \<in> closed_segment u v \<and> g x = g y
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   840
                \<longrightarrow> x = y \<or> x = u \<and> y = v \<or> x = v \<and> y = u)"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   841
    (is "?lhs = ?rhs")
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   842
proof (rule iffI)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   843
  assume ?lhs
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   844
  then have p: "path (\<lambda>x. g ((v - u) * x + u))"
60809
457abb82fb9e the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents: 60420
diff changeset
   845
        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>
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   846
                  \<Longrightarrow> x = y \<or> x = 0 \<and> y = 1 \<or> x = 1 \<and> y = 0)"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   847
    by (auto simp: simple_path_def subpath_def)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   848
  { fix x y
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   849
    assume "x \<in> closed_segment u v" "y \<in> closed_segment u v" "g x = g y"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   850
    then have "x = y \<or> x = u \<and> y = v \<or> x = v \<and> y = u"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   851
    using sim [of "(x-u)/(v-u)" "(y-u)/(v-u)"] p
60809
457abb82fb9e the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents: 60420
diff changeset
   852
    by (auto simp: closed_segment_real_eq image_affinity_atLeastAtMost divide_simps
62390
842917225d56 more canonical names
nipkow
parents: 62087
diff changeset
   853
       split: if_split_asm)
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   854
  } moreover
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   855
  have "path(subpath u v g) \<and> u\<noteq>v"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   856
    using sim [of "1/3" "2/3"] p
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   857
    by (auto simp: subpath_def)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   858
  ultimately show ?rhs
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   859
    by metis
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   860
next
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   861
  assume ?rhs
60809
457abb82fb9e the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents: 60420
diff changeset
   862
  then
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   863
  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"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   864
   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"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   865
   and ne: "u < v \<or> v < u"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   866
   and psp: "path (subpath u v g)"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   867
    by (auto simp: closed_segment_real_eq image_affinity_atLeastAtMost)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   868
  have [simp]: "\<And>x. u + x * v = v + x * u \<longleftrightarrow> u=v \<or> x=1"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   869
    by algebra
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   870
  show ?lhs using psp ne
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   871
    unfolding simple_path_def subpath_def
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   872
    by (fastforce simp add: algebra_simps affine_ineq mult_left_mono crossproduct_eq dest: d1 d2)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   873
qed
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   874
60809
457abb82fb9e the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents: 60420
diff changeset
   875
lemma arc_subpath_eq:
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   876
  "arc(subpath u v g) \<longleftrightarrow> path(subpath u v g) \<and> u\<noteq>v \<and> inj_on g (closed_segment u v)"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   877
    (is "?lhs = ?rhs")
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   878
proof (rule iffI)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   879
  assume ?lhs
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   880
  then have p: "path (\<lambda>x. g ((v - u) * x + u))"
60809
457abb82fb9e the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents: 60420
diff changeset
   881
        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>
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   882
                  \<Longrightarrow> x = y)"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   883
    by (auto simp: arc_def inj_on_def subpath_def)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   884
  { fix x y
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   885
    assume "x \<in> closed_segment u v" "y \<in> closed_segment u v" "g x = g y"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   886
    then have "x = y"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   887
    using sim [of "(x-u)/(v-u)" "(y-u)/(v-u)"] p
68096
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
   888
    by (force simp: inj_on_def closed_segment_real_eq image_affinity_atLeastAtMost divide_simps
62390
842917225d56 more canonical names
nipkow
parents: 62087
diff changeset
   889
       split: if_split_asm)
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   890
  } moreover
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   891
  have "path(subpath u v g) \<and> u\<noteq>v"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   892
    using sim [of "1/3" "2/3"] p
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   893
    by (auto simp: subpath_def)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   894
  ultimately show ?rhs
60809
457abb82fb9e the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents: 60420
diff changeset
   895
    unfolding inj_on_def
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   896
    by metis
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   897
next
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   898
  assume ?rhs
60809
457abb82fb9e the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents: 60420
diff changeset
   899
  then
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   900
  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"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   901
   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"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   902
   and ne: "u < v \<or> v < u"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   903
   and psp: "path (subpath u v g)"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   904
    by (auto simp: inj_on_def closed_segment_real_eq image_affinity_atLeastAtMost)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   905
  show ?lhs using psp ne
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   906
    unfolding arc_def subpath_def inj_on_def
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   907
    by (auto simp: algebra_simps affine_ineq mult_left_mono crossproduct_eq dest: d1 d2)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   908
qed
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   909
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   910
60809
457abb82fb9e the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents: 60420
diff changeset
   911
lemma simple_path_subpath:
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   912
  assumes "simple_path g" "u \<in> {0..1}" "v \<in> {0..1}" "u \<noteq> v"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   913
  shows "simple_path(subpath u v g)"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   914
  using assms
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   915
  apply (simp add: simple_path_subpath_eq simple_path_imp_path)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   916
  apply (simp add: simple_path_def closed_segment_real_eq image_affinity_atLeastAtMost, fastforce)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   917
  done
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   918
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   919
lemma arc_simple_path_subpath:
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   920
    "\<lbrakk>simple_path g; u \<in> {0..1}; v \<in> {0..1}; g u \<noteq> g v\<rbrakk> \<Longrightarrow> arc(subpath u v g)"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   921
  by (force intro: simple_path_subpath simple_path_imp_arc)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   922
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   923
lemma arc_subpath_arc:
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   924
    "\<lbrakk>arc g; u \<in> {0..1}; v \<in> {0..1}; u \<noteq> v\<rbrakk> \<Longrightarrow> arc(subpath u v g)"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   925
  by (meson arc_def arc_imp_simple_path arc_simple_path_subpath inj_onD)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   926
60809
457abb82fb9e the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents: 60420
diff changeset
   927
lemma arc_simple_path_subpath_interior:
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   928
    "\<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)"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   929
    apply (rule arc_simple_path_subpath)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   930
    apply (force simp: simple_path_def)+
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   931
    done
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   932
60809
457abb82fb9e the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents: 60420
diff changeset
   933
lemma path_image_subpath_subset:
68532
f8b98d31ad45 Incorporating new/strengthened proofs from Library and AFP entries
paulson <lp15@cam.ac.uk>
parents: 68310
diff changeset
   934
    "\<lbrakk>u \<in> {0..1}; v \<in> {0..1}\<rbrakk> \<Longrightarrow> path_image(subpath u v g) \<subseteq> path_image g"
61762
d50b993b4fb9 Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents: 61738
diff changeset
   935
  apply (simp add: closed_segment_real_eq image_affinity_atLeastAtMost path_image_subpath)
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   936
  apply (auto simp: path_image_def)
68532
f8b98d31ad45 Incorporating new/strengthened proofs from Library and AFP entries
paulson <lp15@cam.ac.uk>
parents: 68310
diff changeset
   937
  done  
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   938
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   939
lemma join_subpaths_middle: "subpath (0) ((1 / 2)) p +++ subpath ((1 / 2)) 1 p = p"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   940
  by (rule ext) (simp add: joinpaths_def subpath_def divide_simps)
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   941
69514
58a77f548bb6 tuned headers
nipkow
parents: 69508
diff changeset
   942
70136
f03a01a18c6e modernized tags: default scope excludes proof;
wenzelm
parents: 69986
diff changeset
   943
subsection\<^marker>\<open>tag unimportant\<close>\<open>There is a subpath to the frontier\<close>
61518
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   944
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   945
lemma subpath_to_frontier_explicit:
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   946
    fixes S :: "'a::metric_space set"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   947
    assumes g: "path g" and "pathfinish g \<notin> S"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   948
    obtains u where "0 \<le> u" "u \<le> 1"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   949
                "\<And>x. 0 \<le> x \<and> x < u \<Longrightarrow> g x \<in> interior S"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   950
                "(g u \<notin> interior S)" "(u = 0 \<or> g u \<in> closure S)"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   951
proof -
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   952
  have gcon: "continuous_on {0..1} g"     using g by (simp add: path_def)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   953
  then have com: "compact ({0..1} \<inter> {u. g u \<in> closure (- S)})"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   954
    apply (simp add: Int_commute [of "{0..1}"] compact_eq_bounded_closed closed_vimage_Int [unfolded vimage_def])
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   955
    using compact_eq_bounded_closed apply fastforce
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   956
    done
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   957
  have "1 \<in> {u. g u \<in> closure (- S)}"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   958
    using assms by (simp add: pathfinish_def closure_def)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   959
  then have dis: "{0..1} \<inter> {u. g u \<in> closure (- S)} \<noteq> {}"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   960
    using atLeastAtMost_iff zero_le_one by blast
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   961
  then obtain u where "0 \<le> u" "u \<le> 1" and gu: "g u \<in> closure (- S)"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   962
                  and umin: "\<And>t. \<lbrakk>0 \<le> t; t \<le> 1; g t \<in> closure (- S)\<rbrakk> \<Longrightarrow> u \<le> t"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   963
    using compact_attains_inf [OF com dis] by fastforce
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   964
  then have umin': "\<And>t. \<lbrakk>0 \<le> t; t \<le> 1; t < u\<rbrakk> \<Longrightarrow>  g t \<in> S"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   965
    using closure_def by fastforce
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   966
  { assume "u \<noteq> 0"
61808
fc1556774cfe isabelle update_cartouches -c -t;
wenzelm
parents: 61806
diff changeset
   967
    then have "u > 0" using \<open>0 \<le> u\<close> by auto
61518
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   968
    { fix e::real assume "e > 0"
62397
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62381
diff changeset
   969
      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"
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62381
diff changeset
   970
        using continuous_onE [OF gcon _ \<open>e > 0\<close>] \<open>0 \<le> _\<close> \<open>_ \<le> 1\<close> atLeastAtMost_iff by auto
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62381
diff changeset
   971
      have *: "dist (max 0 (u - d / 2)) u \<le> d"
61808
fc1556774cfe isabelle update_cartouches -c -t;
wenzelm
parents: 61806
diff changeset
   972
        using \<open>0 \<le> u\<close> \<open>u \<le> 1\<close> \<open>d > 0\<close> by (simp add: dist_real_def)
61518
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   973
      have "\<exists>y\<in>S. dist y (g u) < e"
61808
fc1556774cfe isabelle update_cartouches -c -t;
wenzelm
parents: 61806
diff changeset
   974
        using \<open>0 < u\<close> \<open>u \<le> 1\<close> \<open>d > 0\<close>
61518
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   975
        by (force intro: d [OF _ *] umin')
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   976
    }
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   977
    then have "g u \<in> closure S"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   978
      by (simp add: frontier_def closure_approachable)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   979
  }
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   980
  then show ?thesis
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   981
    apply (rule_tac u=u in that)
61808
fc1556774cfe isabelle update_cartouches -c -t;
wenzelm
parents: 61806
diff changeset
   982
    apply (auto simp: \<open>0 \<le> u\<close> \<open>u \<le> 1\<close> gu interior_closure umin)
fc1556774cfe isabelle update_cartouches -c -t;
wenzelm
parents: 61806
diff changeset
   983
    using \<open>_ \<le> 1\<close> interior_closure umin apply fastforce
61518
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   984
    done
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   985
qed
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   986
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   987
lemma subpath_to_frontier_strong:
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   988
    assumes g: "path g" and "pathfinish g \<notin> S"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   989
    obtains u where "0 \<le> u" "u \<le> 1" "g u \<notin> interior S"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   990
                    "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"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   991
proof -
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   992
  obtain u where "0 \<le> u" "u \<le> 1"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   993
             and gxin: "\<And>x. 0 \<le> x \<and> x < u \<Longrightarrow> g x \<in> interior S"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   994
             and gunot: "(g u \<notin> interior S)" and u0: "(u = 0 \<or> g u \<in> closure S)"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   995
    using subpath_to_frontier_explicit [OF assms] by blast
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   996
  show ?thesis
61808
fc1556774cfe isabelle update_cartouches -c -t;
wenzelm
parents: 61806
diff changeset
   997
    apply (rule that [OF \<open>0 \<le> u\<close> \<open>u \<le> 1\<close>])
61518
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   998
    apply (simp add: gunot)
61808
fc1556774cfe isabelle update_cartouches -c -t;
wenzelm
parents: 61806
diff changeset
   999
    using \<open>0 \<le> u\<close> u0 by (force simp: subpath_def gxin)
61518
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  1000
qed
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  1001
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  1002
lemma subpath_to_frontier:
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  1003
    assumes g: "path g" and g0: "pathstart g \<in> closure S" and g1: "pathfinish g \<notin> S"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  1004
    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"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  1005
proof -
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  1006
  obtain u where "0 \<le> u" "u \<le> 1"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  1007
             and notin: "g u \<notin> interior S"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  1008
             and disj: "u = 0 \<or>
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  1009
                        (\<forall>x. 0 \<le> x \<and> x < 1 \<longrightarrow> subpath 0 u g x \<in> interior S) \<and> g u \<in> closure S"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  1010
    using subpath_to_frontier_strong [OF g g1] by blast
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  1011
  show ?thesis
61808
fc1556774cfe isabelle update_cartouches -c -t;
wenzelm
parents: 61806
diff changeset
  1012
    apply (rule that [OF \<open>0 \<le> u\<close> \<open>u \<le> 1\<close>])
61518
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  1013
    apply (metis DiffI disj frontier_def g0 notin pathstart_def)
61808
fc1556774cfe isabelle update_cartouches -c -t;
wenzelm
parents: 61806
diff changeset
  1014
    using \<open>0 \<le> u\<close> g0 disj
61762
d50b993b4fb9 Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents: 61738
diff changeset
  1015
    apply (simp add: path_image_subpath_gen)
61518
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  1016
    apply (auto simp: closed_segment_eq_real_ivl pathstart_def pathfinish_def subpath_def)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  1017
    apply (rename_tac y)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  1018
    apply (drule_tac x="y/u" in spec)
62390
842917225d56 more canonical names
nipkow
parents: 62087
diff changeset
  1019
    apply (auto split: if_split_asm)
61518
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  1020
    done
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  1021
qed
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  1022
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  1023
lemma exists_path_subpath_to_frontier:
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  1024
    fixes S :: "'a::real_normed_vector set"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  1025
    assumes "path g" "pathstart g \<in> closure S" "pathfinish g \<notin> S"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  1026
    obtains h where "path h" "pathstart h = pathstart g" "path_image h \<subseteq> path_image g"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  1027
                    "path_image h - {pathfinish h} \<subseteq> interior S"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  1028
                    "pathfinish h \<in> frontier S"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  1029
proof -
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  1030
  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"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  1031
    using subpath_to_frontier [OF assms] by blast
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  1032
  show ?thesis
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  1033
    apply (rule that [of "subpath 0 u g"])
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  1034
    using assms u
61762
d50b993b4fb9 Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents: 61738
diff changeset
  1035
    apply (simp_all add: path_image_subpath)
61518
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  1036
    apply (simp add: pathstart_def)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  1037
    apply (force simp: closed_segment_eq_real_ivl path_image_def)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  1038
    done
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  1039
qed
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  1040
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  1041
lemma exists_path_subpath_to_frontier_closed:
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  1042
    fixes S :: "'a::real_normed_vector set"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  1043
    assumes S: "closed S" and g: "path g" and g0: "pathstart g \<in> S" and g1: "pathfinish g \<notin> S"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  1044
    obtains h where "path h" "pathstart h = pathstart g" "path_image h \<subseteq> path_image g \<inter> S"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  1045
                    "pathfinish h \<in> frontier S"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  1046
proof -
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  1047
  obtain h where h: "path h" "pathstart h = pathstart g" "path_image h \<subseteq> path_image g"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  1048
                    "path_image h - {pathfinish h} \<subseteq> interior S"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  1049
                    "pathfinish h \<in> frontier S"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  1050
    using exists_path_subpath_to_frontier [OF g _ g1] closure_closed [OF S] g0 by auto
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  1051
  show ?thesis
61808
fc1556774cfe isabelle update_cartouches -c -t;
wenzelm
parents: 61806
diff changeset
  1052
    apply (rule that [OF \<open>path h\<close>])
61518
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  1053
    using assms h
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  1054
    apply auto
62087
44841d07ef1d revisions to limits and derivatives, plus new lemmas
paulson
parents: 61808
diff changeset
  1055
    apply (metis Diff_single_insert frontier_subset_eq insert_iff interior_subset subset_iff)
61518
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  1056
    done
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  1057
qed
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1058
69514
58a77f548bb6 tuned headers
nipkow
parents: 69508
diff changeset
  1059
58a77f548bb6 tuned headers
nipkow
parents: 69508
diff changeset
  1060
subsection \<open>Shift Path to Start at Some Given Point\<close>
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1061
70136
f03a01a18c6e modernized tags: default scope excludes proof;
wenzelm
parents: 69986
diff changeset
  1062
definition\<^marker>\<open>tag important\<close> shiftpath :: "real \<Rightarrow> (real \<Rightarrow> 'a::topological_space) \<Rightarrow> real \<Rightarrow> 'a"
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1063
  where "shiftpath a f = (\<lambda>x. if (a + x) \<le> 1 then f (a + x) else f (a + x - 1))"
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1064
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1065
lemma pathstart_shiftpath: "a \<le> 1 \<Longrightarrow> pathstart (shiftpath a g) = g a"
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1066
  unfolding pathstart_def shiftpath_def by auto
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1067
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1068
lemma pathfinish_shiftpath:
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1069
  assumes "0 \<le> a"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1070
    and "pathfinish g = pathstart g"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1071
  shows "pathfinish (shiftpath a g) = g a"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1072
  using assms
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1073
  unfolding pathstart_def pathfinish_def shiftpath_def
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1074
  by auto
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1075
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1076
lemma endpoints_shiftpath:
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1077
  assumes "pathfinish g = pathstart g"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1078
    and "a \<in> {0 .. 1}"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1079
  shows "pathfinish (shiftpath a g) = g a"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1080
    and "pathstart (shiftpath a g) = g a"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1081
  using assms
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1082
  by (auto intro!: pathfinish_shiftpath pathstart_shiftpath)
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1083
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1084
lemma closed_shiftpath:
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1085
  assumes "pathfinish g = pathstart g"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1086
    and "a \<in> {0..1}"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1087
  shows "pathfinish (shiftpath a g) = pathstart (shiftpath a g)"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1088
  using endpoints_shiftpath[OF assms]
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1089
  by auto
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1090
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1091
lemma path_shiftpath:
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1092
  assumes "path g"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1093
    and "pathfinish g = pathstart g"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1094
    and "a \<in> {0..1}"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1095
  shows "path (shiftpath a g)"
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1096
proof -
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1097
  have *: "{0 .. 1} = {0 .. 1-a} \<union> {1-a .. 1}"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1098
    using assms(3) by auto
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1099
  have **: "\<And>x. x + a = 1 \<Longrightarrow> g (x + a - 1) = g (x + a)"
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1100
    using assms(2)[unfolded pathfinish_def pathstart_def]
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1101
    by auto
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1102
  show ?thesis
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1103
    unfolding path_def shiftpath_def *
68096
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  1104
  proof (rule continuous_on_closed_Un)
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  1105
    have contg: "continuous_on {0..1} g"
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  1106
      using \<open>path g\<close> path_def by blast
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  1107
    show "continuous_on {0..1-a} (\<lambda>x. if a + x \<le> 1 then g (a + x) else g (a + x - 1))"
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  1108
    proof (rule continuous_on_eq)
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  1109
      show "continuous_on {0..1-a} (g \<circ> (+) a)"
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  1110
        by (intro continuous_intros continuous_on_subset [OF contg]) (use \<open>a \<in> {0..1}\<close> in auto)
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  1111
    qed auto
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  1112
    show "continuous_on {1-a..1} (\<lambda>x. if a + x \<le> 1 then g (a + x) else g (a + x - 1))"
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  1113
    proof (rule continuous_on_eq)
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  1114
      show "continuous_on {1-a..1} (g \<circ> (+) (a - 1))"
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  1115
        by (intro continuous_intros continuous_on_subset [OF contg]) (use \<open>a \<in> {0..1}\<close> in auto)
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  1116
    qed (auto simp:  "**" add.commute add_diff_eq)
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  1117
  qed auto
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1118
qed
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1119
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1120
lemma shiftpath_shiftpath:
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1121
  assumes "pathfinish g = pathstart g"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1122
    and "a \<in> {0..1}"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1123
    and "x \<in> {0..1}"
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1124
  shows "shiftpath (1 - a) (shiftpath a g) x = g x"
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1125
  using assms
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1126
  unfolding pathfinish_def pathstart_def shiftpath_def
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1127
  by auto
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1128
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1129
lemma path_image_shiftpath:
68096
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  1130
  assumes a: "a \<in> {0..1}"
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1131
    and "pathfinish g = pathstart g"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1132
  shows "path_image (shiftpath a g) = path_image g"
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1133
proof -
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1134
  { fix x
68096
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  1135
    assume g: "g 1 = g 0" "x \<in> {0..1::real}" and gne: "\<And>y. y\<in>{0..1} \<inter> {x. \<not> a + x \<le> 1} \<Longrightarrow> g x \<noteq> g (a + y - 1)"
49654
366d8b41ca17 tuned proofs;
wenzelm
parents: 49653
diff changeset
  1136
    then have "\<exists>y\<in>{0..1} \<inter> {x. a + x \<le> 1}. g x = g (a + y)"
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1137
    proof (cases "a \<le> x")
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1138
      case False
49654
366d8b41ca17 tuned proofs;
wenzelm
parents: 49653
diff changeset
  1139
      then show ?thesis
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1140
        apply (rule_tac x="1 + x - a" in bexI)
68096
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  1141
        using g gne[of "1 + x - a"] a
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  1142
        apply (force simp: field_simps)+
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1143
        done
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1144
    next
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1145
      case True
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1146
      then show ?thesis
68096
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  1147
        using g a  by (rule_tac x="x - a" in bexI) (auto simp: field_simps)
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1148
    qed
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1149
  }
49654
366d8b41ca17 tuned proofs;
wenzelm
parents: 49653
diff changeset
  1150
  then show ?thesis
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1151
    using assms
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1152
    unfolding shiftpath_def path_image_def pathfinish_def pathstart_def
68096
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  1153
    by (auto simp: image_iff)
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1154
qed
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1155
64788
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  1156
lemma simple_path_shiftpath:
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  1157
  assumes "simple_path g" "pathfinish g = pathstart g" and a: "0 \<le> a" "a \<le> 1"
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  1158
    shows "simple_path (shiftpath a g)"
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  1159
  unfolding simple_path_def
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  1160
proof (intro conjI impI ballI)
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  1161
  show "path (shiftpath a g)"
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  1162
    by (simp add: assms path_shiftpath simple_path_imp_path)
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  1163
  have *: "\<And>x y. \<lbrakk>g x = g y; x \<in> {0..1}; y \<in> {0..1}\<rbrakk> \<Longrightarrow> x = y \<or> x = 0 \<and> y = 1 \<or> x = 1 \<and> y = 0"
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  1164
    using assms by (simp add:  simple_path_def)
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  1165
  show "x = y \<or> x = 0 \<and> y = 1 \<or> x = 1 \<and> y = 0"
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  1166
    if "x \<in> {0..1}" "y \<in> {0..1}" "shiftpath a g x = shiftpath a g y" for x y
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  1167
    using that a unfolding shiftpath_def
68096
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  1168
    by (force split: if_split_asm dest!: *)
64788
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  1169
qed
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1170
69514
58a77f548bb6 tuned headers
nipkow
parents: 69508
diff changeset
  1171
58a77f548bb6 tuned headers
nipkow
parents: 69508
diff changeset
  1172
subsection \<open>Straight-Line Paths\<close>
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1173
70136
f03a01a18c6e modernized tags: default scope excludes proof;
wenzelm
parents: 69986
diff changeset
  1174
definition\<^marker>\<open>tag important\<close> linepath :: "'a::real_normed_vector \<Rightarrow> 'a \<Rightarrow> real \<Rightarrow> 'a"
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1175
  where "linepath a b = (\<lambda>x. (1 - x) *\<^sub>R a + x *\<^sub>R b)"
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1176
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1177
lemma pathstart_linepath[simp]: "pathstart (linepath a b) = a"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1178
  unfolding pathstart_def linepath_def
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1179
  by auto
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1180
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1181
lemma pathfinish_linepath[simp]: "pathfinish (linepath a b) = b"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1182
  unfolding pathfinish_def linepath_def
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1183
  by auto
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1184
68721
53ad5c01be3f Small lemmas about analysis
eberlm <eberlm@in.tum.de>
parents: 68607
diff changeset
  1185
lemma linepath_inner: "linepath a b x \<bullet> v = linepath (a \<bullet> v) (b \<bullet> v) x"
53ad5c01be3f Small lemmas about analysis
eberlm <eberlm@in.tum.de>
parents: 68607
diff changeset
  1186
  by (simp add: linepath_def algebra_simps)
53ad5c01be3f Small lemmas about analysis
eberlm <eberlm@in.tum.de>
parents: 68607
diff changeset
  1187
53ad5c01be3f Small lemmas about analysis
eberlm <eberlm@in.tum.de>
parents: 68607
diff changeset
  1188
lemma Re_linepath': "Re (linepath a b x) = linepath (Re a) (Re b) x"
53ad5c01be3f Small lemmas about analysis
eberlm <eberlm@in.tum.de>
parents: 68607
diff changeset
  1189
  by (simp add: linepath_def)
53ad5c01be3f Small lemmas about analysis
eberlm <eberlm@in.tum.de>
parents: 68607
diff changeset
  1190
53ad5c01be3f Small lemmas about analysis
eberlm <eberlm@in.tum.de>
parents: 68607
diff changeset
  1191
lemma Im_linepath': "Im (linepath a b x) = linepath (Im a) (Im b) x"
53ad5c01be3f Small lemmas about analysis
eberlm <eberlm@in.tum.de>
parents: 68607
diff changeset
  1192
  by (simp add: linepath_def)
53ad5c01be3f Small lemmas about analysis
eberlm <eberlm@in.tum.de>
parents: 68607
diff changeset
  1193
53ad5c01be3f Small lemmas about analysis
eberlm <eberlm@in.tum.de>
parents: 68607
diff changeset
  1194
lemma linepath_0': "linepath a b 0 = a"
53ad5c01be3f Small lemmas about analysis
eberlm <eberlm@in.tum.de>
parents: 68607
diff changeset
  1195
  by (simp add: linepath_def)
53ad5c01be3f Small lemmas about analysis
eberlm <eberlm@in.tum.de>
parents: 68607
diff changeset
  1196
53ad5c01be3f Small lemmas about analysis
eberlm <eberlm@in.tum.de>
parents: 68607
diff changeset
  1197
lemma linepath_1': "linepath a b 1 = b"
53ad5c01be3f Small lemmas about analysis
eberlm <eberlm@in.tum.de>
parents: 68607
diff changeset
  1198
  by (simp add: linepath_def)
53ad5c01be3f Small lemmas about analysis
eberlm <eberlm@in.tum.de>
parents: 68607
diff changeset
  1199
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1200
lemma continuous_linepath_at[intro]: "continuous (at x) (linepath a b)"
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1201
  unfolding linepath_def
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1202
  by (intro continuous_intros)
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1203
61762
d50b993b4fb9 Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents: 61738
diff changeset
  1204
lemma continuous_on_linepath [intro,continuous_intros]: "continuous_on s (linepath a b)"
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1205
  using continuous_linepath_at
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1206
  by (auto intro!: continuous_at_imp_continuous_on)
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1207
62618
f7f2467ab854 Refactoring (moving theorems into better locations), plus a bit of new material
paulson <lp15@cam.ac.uk>
parents: 62533
diff changeset
  1208
lemma path_linepath[iff]: "path (linepath a b)"
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1209
  unfolding path_def
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1210
  by (rule continuous_on_linepath)
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1211
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1212
lemma path_image_linepath[simp]: "path_image (linepath a b) = closed_segment a b"
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1213
  unfolding path_image_def segment linepath_def
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
  1214
  by auto
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1215
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1216
lemma reversepath_linepath[simp]: "reversepath (linepath a b) = linepath b a"
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1217
  unfolding reversepath_def linepath_def
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1218
  by auto
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1219
61762
d50b993b4fb9 Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents: 61738
diff changeset
  1220
lemma linepath_0 [simp]: "linepath 0 b x = x *\<^sub>R b"
d50b993b4fb9 Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents: 61738
diff changeset
  1221
  by (simp add: linepath_def)
d50b993b4fb9 Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents: 61738
diff changeset
  1222
68721
53ad5c01be3f Small lemmas about analysis
eberlm <eberlm@in.tum.de>
parents: 68607
diff changeset
  1223
lemma linepath_cnj: "cnj (linepath a b x) = linepath (cnj a) (cnj b) x"
53ad5c01be3f Small lemmas about analysis
eberlm <eberlm@in.tum.de>
parents: 68607
diff changeset
  1224
  by (simp add: linepath_def)
53ad5c01be3f Small lemmas about analysis
eberlm <eberlm@in.tum.de>
parents: 68607
diff changeset
  1225
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
  1226
lemma arc_linepath:
62618
f7f2467ab854 Refactoring (moving theorems into better locations), plus a bit of new material
paulson <lp15@cam.ac.uk>
parents: 62533
diff changeset
  1227
  assumes "a \<noteq> b" shows [simp]: "arc (linepath a b)"
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1228
proof -
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1229
  {
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1230
    fix x y :: "real"
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1231
    assume "x *\<^sub>R b + y *\<^sub>R a = x *\<^sub>R a + y *\<^sub>R b"
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1232
    then have "(x - y) *\<^sub>R a = (x - y) *\<^sub>R b"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1233
      by (simp add: algebra_simps)
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1234
    with assms have "x = y"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1235
      by simp
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1236
  }
49654
366d8b41ca17 tuned proofs;
wenzelm
parents: 49653
diff changeset
  1237
  then show ?thesis
60809
457abb82fb9e the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents: 60420
diff changeset
  1238
    unfolding arc_def inj_on_def
68096
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  1239
    by (fastforce simp: algebra_simps linepath_def)
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1240
qed
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1241
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1242
lemma simple_path_linepath[intro]: "a \<noteq> b \<Longrightarrow> simple_path (linepath a b)"
68096
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  1243
  by (simp add: arc_imp_simple_path)
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1244
61711
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  1245
lemma linepath_trivial [simp]: "linepath a a x = a"
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  1246
  by (simp add: linepath_def real_vector.scale_left_diff_distrib)
61738
c4f6031f1310 New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents: 61711
diff changeset
  1247
64394
141e1ed8d5a0 more new material
paulson <lp15@cam.ac.uk>
parents: 64267
diff changeset
  1248
lemma linepath_refl: "linepath a a = (\<lambda>x. a)"
141e1ed8d5a0 more new material
paulson <lp15@cam.ac.uk>
parents: 64267
diff changeset
  1249
  by auto
141e1ed8d5a0 more new material
paulson <lp15@cam.ac.uk>
parents: 64267
diff changeset
  1250
61711
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  1251
lemma subpath_refl: "subpath a a g = linepath (g a) (g a)"
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  1252
  by (simp add: subpath_def linepath_def algebra_simps)
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  1253
62618
f7f2467ab854 Refactoring (moving theorems into better locations), plus a bit of new material
paulson <lp15@cam.ac.uk>
parents: 62533
diff changeset
  1254
lemma linepath_of_real: "(linepath (of_real a) (of_real b) x) = of_real ((1 - x)*a + x*b)"
f7f2467ab854 Refactoring (moving theorems into better locations), plus a bit of new material
paulson <lp15@cam.ac.uk>
parents: 62533
diff changeset
  1255
  by (simp add: scaleR_conv_of_real linepath_def)
f7f2467ab854 Refactoring (moving theorems into better locations), plus a bit of new material
paulson <lp15@cam.ac.uk>
parents: 62533
diff changeset
  1256
f7f2467ab854 Refactoring (moving theorems into better locations), plus a bit of new material
paulson <lp15@cam.ac.uk>
parents: 62533
diff changeset
  1257
lemma of_real_linepath: "of_real (linepath a b x) = linepath (of_real a) (of_real b) x"
f7f2467ab854 Refactoring (moving theorems into better locations), plus a bit of new material
paulson <lp15@cam.ac.uk>
parents: 62533
diff changeset
  1258
  by (metis linepath_of_real mult.right_neutral of_real_def real_scaleR_def)
f7f2467ab854 Refactoring (moving theorems into better locations), plus a bit of new material
paulson <lp15@cam.ac.uk>
parents: 62533
diff changeset
  1259
63881
b746b19197bd lots of new results about topology, affine dimension etc
paulson <lp15@cam.ac.uk>
parents: 63627
diff changeset
  1260
lemma inj_on_linepath:
b746b19197bd lots of new results about topology, affine dimension etc
paulson <lp15@cam.ac.uk>
parents: 63627
diff changeset
  1261
  assumes "a \<noteq> b" shows "inj_on (linepath a b) {0..1}"
b746b19197bd lots of new results about topology, affine dimension etc
paulson <lp15@cam.ac.uk>
parents: 63627
diff changeset
  1262
proof (clarsimp simp: inj_on_def linepath_def)
b746b19197bd lots of new results about topology, affine dimension etc
paulson <lp15@cam.ac.uk>
parents: 63627
diff changeset
  1263
  fix x y
b746b19197bd lots of new results about topology, affine dimension etc
paulson <lp15@cam.ac.uk>
parents: 63627
diff changeset
  1264
  assume "(1 - x) *\<^sub>R a + x *\<^sub>R b = (1 - y) *\<^sub>R a + y *\<^sub>R b" "0 \<le> x" "x \<le> 1" "0 \<le> y" "y \<le> 1"
b746b19197bd lots of new results about topology, affine dimension etc
paulson <lp15@cam.ac.uk>
parents: 63627
diff changeset
  1265
  then have "x *\<^sub>R (a - b) = y *\<^sub>R (a - b)"
b746b19197bd lots of new results about topology, affine dimension etc
paulson <lp15@cam.ac.uk>
parents: 63627
diff changeset
  1266
    by (auto simp: algebra_simps)
b746b19197bd lots of new results about topology, affine dimension etc
paulson <lp15@cam.ac.uk>
parents: 63627
diff changeset
  1267
  then show "x=y"
b746b19197bd lots of new results about topology, affine dimension etc
paulson <lp15@cam.ac.uk>
parents: 63627
diff changeset
  1268
    using assms by auto
b746b19197bd lots of new results about topology, affine dimension etc
paulson <lp15@cam.ac.uk>
parents: 63627
diff changeset
  1269
qed
b746b19197bd lots of new results about topology, affine dimension etc
paulson <lp15@cam.ac.uk>
parents: 63627
diff changeset
  1270
69144
f13b82281715 new theory Abstract_Topology with lots of stuff from HOL Light's metric.sml
paulson <lp15@cam.ac.uk>
parents: 69064
diff changeset
  1271
lemma linepath_le_1:
f13b82281715 new theory Abstract_Topology with lots of stuff from HOL Light's metric.sml
paulson <lp15@cam.ac.uk>
parents: 69064
diff changeset
  1272
  fixes a::"'a::linordered_idom" shows "\<lbrakk>a \<le> 1; b \<le> 1; 0 \<le> u; u \<le> 1\<rbrakk> \<Longrightarrow> (1 - u) * a + u * b \<le> 1"
f13b82281715 new theory Abstract_Topology with lots of stuff from HOL Light's metric.sml
paulson <lp15@cam.ac.uk>
parents: 69064
diff changeset
  1273
  using mult_left_le [of a "1-u"] mult_left_le [of b u] by auto
f13b82281715 new theory Abstract_Topology with lots of stuff from HOL Light's metric.sml
paulson <lp15@cam.ac.uk>
parents: 69064
diff changeset
  1274
62618
f7f2467ab854 Refactoring (moving theorems into better locations), plus a bit of new material
paulson <lp15@cam.ac.uk>
parents: 62533
diff changeset
  1275
70136
f03a01a18c6e modernized tags: default scope excludes proof;
wenzelm
parents: 69986
diff changeset
  1276
subsection\<^marker>\<open>tag unimportant\<close>\<open>Segments via convex hulls\<close>
62618
f7f2467ab854 Refactoring (moving theorems into better locations), plus a bit of new material
paulson <lp15@cam.ac.uk>
parents: 62533
diff changeset
  1277
f7f2467ab854 Refactoring (moving theorems into better locations), plus a bit of new material
paulson <lp15@cam.ac.uk>
parents: 62533
diff changeset
  1278
lemma segments_subset_convex_hull:
f7f2467ab854 Refactoring (moving theorems into better locations), plus a bit of new material
paulson <lp15@cam.ac.uk>
parents: 62533
diff changeset
  1279
    "closed_segment a b \<subseteq> (convex hull {a,b,c})"
f7f2467ab854 Refactoring (moving theorems into better locations), plus a bit of new material
paulson <lp15@cam.ac.uk>
parents: 62533
diff changeset
  1280
    "closed_segment a c \<subseteq> (convex hull {a,b,c})"
f7f2467ab854 Refactoring (moving theorems into better locations), plus a bit of new material
paulson <lp15@cam.ac.uk>
parents: 62533
diff changeset
  1281
    "closed_segment b c \<subseteq> (convex hull {a,b,c})"
f7f2467ab854 Refactoring (moving theorems into better locations), plus a bit of new material
paulson <lp15@cam.ac.uk>
parents: 62533
diff changeset
  1282
    "closed_segment b a \<subseteq> (convex hull {a,b,c})"
f7f2467ab854 Refactoring (moving theorems into better locations), plus a bit of new material
paulson <lp15@cam.ac.uk>
parents: 62533
diff changeset
  1283
    "closed_segment c a \<subseteq> (convex hull {a,b,c})"
f7f2467ab854 Refactoring (moving theorems into better locations), plus a bit of new material
paulson <lp15@cam.ac.uk>
parents: 62533
diff changeset
  1284
    "closed_segment c b \<subseteq> (convex hull {a,b,c})"
f7f2467ab854 Refactoring (moving theorems into better locations), plus a bit of new material
paulson <lp15@cam.ac.uk>
parents: 62533
diff changeset
  1285
by (auto simp: segment_convex_hull linepath_of_real  elim!: rev_subsetD [OF _ hull_mono])
f7f2467ab854 Refactoring (moving theorems into better locations), plus a bit of new material
paulson <lp15@cam.ac.uk>
parents: 62533
diff changeset
  1286
f7f2467ab854 Refactoring (moving theorems into better locations), plus a bit of new material
paulson <lp15@cam.ac.uk>
parents: 62533
diff changeset
  1287
lemma midpoints_in_convex_hull:
f7f2467ab854 Refactoring (moving theorems into better locations), plus a bit of new material
paulson <lp15@cam.ac.uk>
parents: 62533
diff changeset
  1288
  assumes "x \<in> convex hull s" "y \<in> convex hull s"
f7f2467ab854 Refactoring (moving theorems into better locations), plus a bit of new material
paulson <lp15@cam.ac.uk>
parents: 62533
diff changeset
  1289
    shows "midpoint x y \<in> convex hull s"
f7f2467ab854 Refactoring (moving theorems into better locations), plus a bit of new material
paulson <lp15@cam.ac.uk>
parents: 62533
diff changeset
  1290
proof -
f7f2467ab854 Refactoring (moving theorems into better locations), plus a bit of new material
paulson <lp15@cam.ac.uk>
parents: 62533
diff changeset
  1291
  have "(1 - inverse(2)) *\<^sub>R x + inverse(2) *\<^sub>R y \<in> convex hull s"
68096
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  1292
    by (rule convexD_alt) (use assms in auto)
62618
f7f2467ab854 Refactoring (moving theorems into better locations), plus a bit of new material
paulson <lp15@cam.ac.uk>
parents: 62533
diff changeset
  1293
  then show ?thesis
f7f2467ab854 Refactoring (moving theorems into better locations), plus a bit of new material
paulson <lp15@cam.ac.uk>
parents: 62533
diff changeset
  1294
    by (simp add: midpoint_def algebra_simps)
f7f2467ab854 Refactoring (moving theorems into better locations), plus a bit of new material
paulson <lp15@cam.ac.uk>
parents: 62533
diff changeset
  1295
qed
f7f2467ab854 Refactoring (moving theorems into better locations), plus a bit of new material
paulson <lp15@cam.ac.uk>
parents: 62533
diff changeset
  1296
f7f2467ab854 Refactoring (moving theorems into better locations), plus a bit of new material
paulson <lp15@cam.ac.uk>
parents: 62533
diff changeset
  1297
lemma not_in_interior_convex_hull_3:
f7f2467ab854 Refactoring (moving theorems into better locations), plus a bit of new material
paulson <lp15@cam.ac.uk>
parents: 62533
diff changeset
  1298
  fixes a :: "complex"
f7f2467ab854 Refactoring (moving theorems into better locations), plus a bit of new material
paulson <lp15@cam.ac.uk>
parents: 62533
diff changeset
  1299
  shows "a \<notin> interior(convex hull {a,b,c})"
f7f2467ab854 Refactoring (moving theorems into better locations), plus a bit of new material
paulson <lp15@cam.ac.uk>
parents: 62533
diff changeset
  1300
        "b \<notin> interior(convex hull {a,b,c})"
f7f2467ab854 Refactoring (moving theorems into better locations), plus a bit of new material
paulson <lp15@cam.ac.uk>
parents: 62533
diff changeset
  1301
        "c \<notin> interior(convex hull {a,b,c})"
f7f2467ab854 Refactoring (moving theorems into better locations), plus a bit of new material
paulson <lp15@cam.ac.uk>
parents: 62533
diff changeset
  1302
  by (auto simp: card_insert_le_m1 not_in_interior_convex_hull)
f7f2467ab854 Refactoring (moving theorems into better locations), plus a bit of new material
paulson <lp15@cam.ac.uk>
parents: 62533
diff changeset
  1303
f7f2467ab854 Refactoring (moving theorems into better locations), plus a bit of new material
paulson <lp15@cam.ac.uk>
parents: 62533
diff changeset
  1304
lemma midpoint_in_closed_segment [simp]: "midpoint a b \<in> closed_segment a b"
f7f2467ab854 Refactoring (moving theorems into better locations), plus a bit of new material
paulson <lp15@cam.ac.uk>
parents: 62533
diff changeset
  1305
  using midpoints_in_convex_hull segment_convex_hull by blast
f7f2467ab854 Refactoring (moving theorems into better locations), plus a bit of new material
paulson <lp15@cam.ac.uk>
parents: 62533
diff changeset
  1306
f7f2467ab854 Refactoring (moving theorems into better locations), plus a bit of new material
paulson <lp15@cam.ac.uk>
parents: 62533
diff changeset
  1307
lemma midpoint_in_open_segment [simp]: "midpoint a b \<in> open_segment a b \<longleftrightarrow> a \<noteq> b"
64122
74fde524799e invariance of domain
paulson <lp15@cam.ac.uk>
parents: 64006
diff changeset
  1308
  by (simp add: open_segment_def)
74fde524799e invariance of domain
paulson <lp15@cam.ac.uk>
parents: 64006
diff changeset
  1309
74fde524799e invariance of domain
paulson <lp15@cam.ac.uk>
parents: 64006
diff changeset
  1310
lemma continuous_IVT_local_extremum:
74fde524799e invariance of domain
paulson <lp15@cam.ac.uk>
parents: 64006
diff changeset
  1311
  fixes f :: "'a::euclidean_space \<Rightarrow> real"
74fde524799e invariance of domain
paulson <lp15@cam.ac.uk>
parents: 64006
diff changeset
  1312
  assumes contf: "continuous_on (closed_segment a b) f"
74fde524799e invariance of domain
paulson <lp15@cam.ac.uk>
parents: 64006
diff changeset
  1313
      and "a \<noteq> b" "f a = f b"
74fde524799e invariance of domain
paulson <lp15@cam.ac.uk>
parents: 64006
diff changeset
  1314
  obtains z where "z \<in> open_segment a b"
74fde524799e invariance of domain
paulson <lp15@cam.ac.uk>
parents: 64006
diff changeset
  1315
                  "(\<forall>w \<in> closed_segment a b. (f w) \<le> (f z)) \<or>
74fde524799e invariance of domain
paulson <lp15@cam.ac.uk>
parents: 64006
diff changeset
  1316
                   (\<forall>w \<in> closed_segment a b. (f z) \<le> (f w))"
74fde524799e invariance of domain
paulson <lp15@cam.ac.uk>
parents: 64006
diff changeset
  1317
proof -
74fde524799e invariance of domain
paulson <lp15@cam.ac.uk>
parents: 64006
diff changeset
  1318
  obtain c where "c \<in> closed_segment a b" and c: "\<And>y. y \<in> closed_segment a b \<Longrightarrow> f y \<le> f c"
74fde524799e invariance of domain
paulson <lp15@cam.ac.uk>
parents: 64006
diff changeset
  1319
    using continuous_attains_sup [of "closed_segment a b" f] contf by auto
74fde524799e invariance of domain
paulson <lp15@cam.ac.uk>
parents: 64006
diff changeset
  1320
  obtain d where "d \<in> closed_segment a b" and d: "\<And>y. y \<in> closed_segment a b \<Longrightarrow> f d \<le> f y"
74fde524799e invariance of domain
paulson <lp15@cam.ac.uk>
parents: 64006
diff changeset
  1321
    using continuous_attains_inf [of "closed_segment a b" f] contf by auto
74fde524799e invariance of domain
paulson <lp15@cam.ac.uk>
parents: 64006
diff changeset
  1322
  show ?thesis
74fde524799e invariance of domain
paulson <lp15@cam.ac.uk>
parents: 64006
diff changeset
  1323
  proof (cases "c \<in> open_segment a b \<or> d \<in> open_segment a b")
74fde524799e invariance of domain
paulson <lp15@cam.ac.uk>
parents: 64006
diff changeset
  1324
    case True
74fde524799e invariance of domain
paulson <lp15@cam.ac.uk>
parents: 64006
diff changeset
  1325
    then show ?thesis
74fde524799e invariance of domain
paulson <lp15@cam.ac.uk>
parents: 64006
diff changeset
  1326
      using c d that by blast
74fde524799e invariance of domain
paulson <lp15@cam.ac.uk>
parents: 64006
diff changeset
  1327
  next
74fde524799e invariance of domain
paulson <lp15@cam.ac.uk>
parents: 64006
diff changeset
  1328
    case False
74fde524799e invariance of domain
paulson <lp15@cam.ac.uk>
parents: 64006
diff changeset
  1329
    then have "(c = a \<or> c = b) \<and> (d = a \<or> d = b)"
74fde524799e invariance of domain
paulson <lp15@cam.ac.uk>
parents: 64006
diff changeset
  1330
      by (simp add: \<open>c \<in> closed_segment a b\<close> \<open>d \<in> closed_segment a b\<close> open_segment_def)
74fde524799e invariance of domain
paulson <lp15@cam.ac.uk>
parents: 64006
diff changeset
  1331
    with \<open>a \<noteq> b\<close> \<open>f a = f b\<close> c d show ?thesis
74fde524799e invariance of domain
paulson <lp15@cam.ac.uk>
parents: 64006
diff changeset
  1332
      by (rule_tac z = "midpoint a b" in that) (fastforce+)
74fde524799e invariance of domain
paulson <lp15@cam.ac.uk>
parents: 64006
diff changeset
  1333
  qed
74fde524799e invariance of domain
paulson <lp15@cam.ac.uk>
parents: 64006
diff changeset
  1334
qed
74fde524799e invariance of domain
paulson <lp15@cam.ac.uk>
parents: 64006
diff changeset
  1335
74fde524799e invariance of domain
paulson <lp15@cam.ac.uk>
parents: 64006
diff changeset
  1336
text\<open>An injective map into R is also an open map w.r.T. the universe, and conversely. \<close>
74fde524799e invariance of domain
paulson <lp15@cam.ac.uk>
parents: 64006
diff changeset
  1337
proposition injective_eq_1d_open_map_UNIV:
74fde524799e invariance of domain
paulson <lp15@cam.ac.uk>
parents: 64006
diff changeset
  1338
  fixes f :: "real \<Rightarrow> real"
74fde524799e invariance of domain
paulson <lp15@cam.ac.uk>
parents: 64006
diff changeset
  1339
  assumes contf: "continuous_on S f" and S: "is_interval S"
74fde524799e invariance of domain
paulson <lp15@cam.ac.uk>
parents: 64006
diff changeset
  1340
    shows "inj_on f S \<longleftrightarrow> (\<forall>T. open T \<and> T \<subseteq> S \<longrightarrow> open(f ` T))"
74fde524799e invariance of domain
paulson <lp15@cam.ac.uk>
parents: 64006
diff changeset
  1341
          (is "?lhs = ?rhs")
74fde524799e invariance of domain
paulson <lp15@cam.ac.uk>
parents: 64006
diff changeset
  1342
proof safe
74fde524799e invariance of domain
paulson <lp15@cam.ac.uk>
parents: 64006
diff changeset
  1343
  fix T
74fde524799e invariance of domain
paulson <lp15@cam.ac.uk>
parents: 64006
diff changeset
  1344
  assume injf: ?lhs and "open T" and "T \<subseteq> S"
74fde524799e invariance of domain
paulson <lp15@cam.ac.uk>
parents: 64006
diff changeset
  1345
  have "\<exists>U. open U \<and> f x \<in> U \<and> U \<subseteq> f ` T" if "x \<in> T" for x
74fde524799e invariance of domain
paulson <lp15@cam.ac.uk>
parents: 64006
diff changeset
  1346
  proof -
74fde524799e invariance of domain
paulson <lp15@cam.ac.uk>
parents: 64006
diff changeset
  1347
    obtain \<delta> where "\<delta> > 0" and \<delta>: "cball x \<delta> \<subseteq> T"
74fde524799e invariance of domain
paulson <lp15@cam.ac.uk>
parents: 64006
diff changeset
  1348
      using \<open>open T\<close> \<open>x \<in> T\<close> open_contains_cball_eq by blast
74fde524799e invariance of domain
paulson <lp15@cam.ac.uk>
parents: 64006
diff changeset
  1349
    show ?thesis
74fde524799e invariance of domain
paulson <lp15@cam.ac.uk>
parents: 64006
diff changeset
  1350
    proof (intro exI conjI)
74fde524799e invariance of domain
paulson <lp15@cam.ac.uk>
parents: 64006
diff changeset
  1351
      have "closed_segment (x-\<delta>) (x+\<delta>) = {x-\<delta>..x+\<delta>}"
74fde524799e invariance of domain
paulson <lp15@cam.ac.uk>
parents: 64006
diff changeset
  1352
        using \<open>0 < \<delta>\<close> by (auto simp: closed_segment_eq_real_ivl)
68096
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  1353
      also have "\<dots> \<subseteq> S"
64122
74fde524799e invariance of domain
paulson <lp15@cam.ac.uk>
parents: 64006
diff changeset
  1354
        using \<delta> \<open>T \<subseteq> S\<close> by (auto simp: dist_norm subset_eq)
74fde524799e invariance of domain
paulson <lp15@cam.ac.uk>
parents: 64006
diff changeset
  1355
      finally have "f ` (open_segment (x-\<delta>) (x+\<delta>)) = open_segment (f (x-\<delta>)) (f (x+\<delta>))"
74fde524799e invariance of domain
paulson <lp15@cam.ac.uk>
parents: 64006
diff changeset
  1356
        using continuous_injective_image_open_segment_1
74fde524799e invariance of domain
paulson <lp15@cam.ac.uk>
parents: 64006
diff changeset
  1357
        by (metis continuous_on_subset [OF contf] inj_on_subset [OF injf])
74fde524799e invariance of domain
paulson <lp15@cam.ac.uk>
parents: 64006
diff changeset
  1358
      then show "open (f ` {x-\<delta><..<x+\<delta>})"
74fde524799e invariance of domain
paulson <lp15@cam.ac.uk>
parents: 64006
diff changeset
  1359
        using \<open>0 < \<delta>\<close> by (simp add: open_segment_eq_real_ivl)
74fde524799e invariance of domain
paulson <lp15@cam.ac.uk>
parents: 64006
diff changeset
  1360
      show "f x \<in> f ` {x - \<delta><..<x + \<delta>}"
74fde524799e invariance of domain
paulson <lp15@cam.ac.uk>
parents: 64006
diff changeset
  1361
        by (auto simp: \<open>\<delta> > 0\<close>)
74fde524799e invariance of domain
paulson <lp15@cam.ac.uk>
parents: 64006
diff changeset
  1362
      show "f ` {x - \<delta><..<x + \<delta>} \<subseteq> f ` T"
74fde524799e invariance of domain
paulson <lp15@cam.ac.uk>
parents: 64006
diff changeset
  1363
        using \<delta> by (auto simp: dist_norm subset_iff)
74fde524799e invariance of domain
paulson <lp15@cam.ac.uk>
parents: 64006
diff changeset
  1364
    qed
74fde524799e invariance of domain
paulson <lp15@cam.ac.uk>
parents: 64006
diff changeset
  1365
  qed
74fde524799e invariance of domain
paulson <lp15@cam.ac.uk>
parents: 64006
diff changeset
  1366
  with open_subopen show "open (f ` T)"
74fde524799e invariance of domain
paulson <lp15@cam.ac.uk>
parents: 64006
diff changeset
  1367
    by blast
74fde524799e invariance of domain
paulson <lp15@cam.ac.uk>
parents: 64006
diff changeset
  1368
next
74fde524799e invariance of domain
paulson <lp15@cam.ac.uk>
parents: 64006
diff changeset
  1369
  assume R: ?rhs
74fde524799e invariance of domain
paulson <lp15@cam.ac.uk>
parents: 64006
diff changeset
  1370
  have False if xy: "x \<in> S" "y \<in> S" and "f x = f y" "x \<noteq> y" for x y
74fde524799e invariance of domain
paulson <lp15@cam.ac.uk>
parents: 64006
diff changeset
  1371
  proof -
74fde524799e invariance of domain
paulson <lp15@cam.ac.uk>
parents: 64006
diff changeset
  1372
    have "open (f ` open_segment x y)"
74fde524799e invariance of domain
paulson <lp15@cam.ac.uk>
parents: 64006
diff changeset
  1373
      using R
74fde524799e invariance of domain
paulson <lp15@cam.ac.uk>
parents: 64006
diff changeset
  1374
      by (metis S convex_contains_open_segment is_interval_convex open_greaterThanLessThan open_segment_eq_real_ivl xy)
74fde524799e invariance of domain
paulson <lp15@cam.ac.uk>
parents: 64006
diff changeset
  1375
    moreover
74fde524799e invariance of domain
paulson <lp15@cam.ac.uk>
parents: 64006
diff changeset
  1376
    have "continuous_on (closed_segment x y) f"
74fde524799e invariance of domain
paulson <lp15@cam.ac.uk>
parents: 64006
diff changeset
  1377
      by (meson S closed_segment_subset contf continuous_on_subset is_interval_convex that)
74fde524799e invariance of domain
paulson <lp15@cam.ac.uk>
parents: 64006
diff changeset
  1378
    then obtain \<xi> where "\<xi> \<in> open_segment x y"
74fde524799e invariance of domain
paulson <lp15@cam.ac.uk>
parents: 64006
diff changeset
  1379
                    and \<xi>: "(\<forall>w \<in> closed_segment x y. (f w) \<le> (f \<xi>)) \<or>
74fde524799e invariance of domain
paulson <lp15@cam.ac.uk>
parents: 64006
diff changeset
  1380
                            (\<forall>w \<in> closed_segment x y. (f \<xi>) \<le> (f w))"
74fde524799e invariance of domain
paulson <lp15@cam.ac.uk>
parents: 64006
diff changeset
  1381
      using continuous_IVT_local_extremum [of x y f] \<open>f x = f y\<close> \<open>x \<noteq> y\<close> by blast
74fde524799e invariance of domain
paulson <lp15@cam.ac.uk>
parents: 64006
diff changeset
  1382
    ultimately obtain e where "e>0" and e: "\<And>u. dist u (f \<xi>) < e \<Longrightarrow> u \<in> f ` open_segment x y"
74fde524799e invariance of domain
paulson <lp15@cam.ac.uk>
parents: 64006
diff changeset
  1383
      using open_dist by (metis image_eqI)
74fde524799e invariance of domain
paulson <lp15@cam.ac.uk>
parents: 64006
diff changeset
  1384
    have fin: "f \<xi> + (e/2) \<in> f ` open_segment x y" "f \<xi> - (e/2) \<in> f ` open_segment x y"
74fde524799e invariance of domain
paulson <lp15@cam.ac.uk>
parents: 64006
diff changeset
  1385
      using e [of "f \<xi> + (e/2)"] e [of "f \<xi> - (e/2)"] \<open>e > 0\<close> by (auto simp: dist_norm)
74fde524799e invariance of domain
paulson <lp15@cam.ac.uk>
parents: 64006
diff changeset
  1386
    show ?thesis
74fde524799e invariance of domain
paulson <lp15@cam.ac.uk>
parents: 64006
diff changeset
  1387
      using \<xi> \<open>0 < e\<close> fin open_closed_segment by fastforce
74fde524799e invariance of domain
paulson <lp15@cam.ac.uk>
parents: 64006
diff changeset
  1388
  qed
74fde524799e invariance of domain
paulson <lp15@cam.ac.uk>
parents: 64006
diff changeset
  1389
  then show ?lhs
74fde524799e invariance of domain
paulson <lp15@cam.ac.uk>
parents: 64006
diff changeset
  1390
    by (force simp: inj_on_def)
74fde524799e invariance of domain
paulson <lp15@cam.ac.uk>
parents: 64006
diff changeset
  1391
qed
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1392
69514
58a77f548bb6 tuned headers
nipkow
parents: 69508
diff changeset
  1393
70136
f03a01a18c6e modernized tags: default scope excludes proof;
wenzelm
parents: 69986
diff changeset
  1394
subsection\<^marker>\<open>tag unimportant\<close> \<open>Bounding a point away from a path\<close>
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1395
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1396
lemma not_on_path_ball:
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1397
  fixes g :: "real \<Rightarrow> 'a::heine_borel"
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1398
  assumes "path g"
68096
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  1399
    and z: "z \<notin> path_image g"
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1400
  shows "\<exists>e > 0. ball z e \<inter> path_image g = {}"
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1401
proof -
68096
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  1402
  have "closed (path_image g)"
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  1403
    by (simp add: \<open>path g\<close> closed_path_image)
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  1404
  then obtain a where "a \<in> path_image g" "\<forall>y \<in> path_image g. dist z a \<le> dist z y"
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  1405
    by (auto intro: distance_attains_inf[OF _ path_image_nonempty, of g z])
49654
366d8b41ca17 tuned proofs;
wenzelm
parents: 49653
diff changeset
  1406
  then show ?thesis
68096
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  1407
    by (rule_tac x="dist z a" in exI) (use dist_commute z in auto)
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1408
qed
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1409
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1410
lemma not_on_path_cball:
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1411
  fixes g :: "real \<Rightarrow> 'a::heine_borel"
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1412
  assumes "path g"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1413
    and "z \<notin> path_image g"
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1414
  shows "\<exists>e>0. cball z e \<inter> (path_image g) = {}"
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1415
proof -
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1416
  obtain e where "ball z e \<inter> path_image g = {}" "e > 0"
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1417
    using not_on_path_ball[OF assms] by auto
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1418
  moreover have "cball z (e/2) \<subseteq> ball z e"
60420
884f54e01427 isabelle update_cartouches;
wenzelm
parents: 60303
diff changeset
  1419
    using \<open>e > 0\<close> by auto
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1420
  ultimately show ?thesis
68096
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  1421
    by (rule_tac x="e/2" in exI) auto
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1422
qed
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1423
69518
bf88364c9e94 tuned headers etc, added bib-file
nipkow
parents: 69517
diff changeset
  1424
subsection \<open>Path component\<close>
bf88364c9e94 tuned headers etc, added bib-file
nipkow
parents: 69517
diff changeset
  1425
bf88364c9e94 tuned headers etc, added bib-file
nipkow
parents: 69517
diff changeset
  1426
text \<open>Original formalization by Tom Hales\<close>
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1427
70136
f03a01a18c6e modernized tags: default scope excludes proof;
wenzelm
parents: 69986
diff changeset
  1428
definition\<^marker>\<open>tag important\<close> "path_component s x y \<longleftrightarrow>
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1429
  (\<exists>g. path g \<and> path_image g \<subseteq> s \<and> pathstart g = x \<and> pathfinish g = y)"
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1430
70136
f03a01a18c6e modernized tags: default scope excludes proof;
wenzelm
parents: 69986
diff changeset
  1431
abbreviation\<^marker>\<open>tag important\<close>
69518
bf88364c9e94 tuned headers etc, added bib-file
nipkow
parents: 69517
diff changeset
  1432
  "path_component_set s x \<equiv> Collect (path_component s x)"
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1433
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1434
lemmas path_defs = path_def pathstart_def pathfinish_def path_image_def path_component_def
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1435
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1436
lemma path_component_mem:
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1437
  assumes "path_component s x y"
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1438
  shows "x \<in> s" and "y \<in> s"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1439
  using assms
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1440
  unfolding path_defs
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1441
  by auto
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1442
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1443
lemma path_component_refl:
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1444
  assumes "x \<in> s"
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1445
  shows "path_component s x x"
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1446
  unfolding path_defs
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1447
  apply (rule_tac x="\<lambda>u. x" in exI)
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1448
  using assms
56371
fb9ae0727548 extend continuous_intros; remove continuous_on_intros and isCont_intros
hoelzl
parents: 56188
diff changeset
  1449
  apply (auto intro!: continuous_intros)
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1450
  done
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1451
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1452
lemma path_component_refl_eq: "path_component s x x \<longleftrightarrow> x \<in> s"
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1453
  by (auto intro!: path_component_mem path_component_refl)
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1454
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1455
lemma path_component_sym: "path_component s x y \<Longrightarrow> path_component s y x"
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1456
  unfolding path_component_def
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1457
  apply (erule exE)
68096
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  1458
  apply (rule_tac x="reversepath g" in exI, auto)
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1459
  done
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1460
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1461
lemma path_component_trans:
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1462
  assumes "path_component s x y" and "path_component s y z"
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1463
  shows "path_component s x z"
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1464
  using assms
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1465
  unfolding path_component_def
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1466
  apply (elim exE)
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1467
  apply (rule_tac x="g +++ ga" in exI)
68096
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  1468
  apply (auto simp: path_image_join)
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1469
  done
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1470
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1471
lemma path_component_of_subset: "s \<subseteq> t \<Longrightarrow> path_component s x y \<Longrightarrow> path_component t x y"
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1472
  unfolding path_component_def by auto
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1473
69939
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  1474
lemma path_component_linepath:
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1475
    fixes s :: "'a::real_normed_vector set"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1476
    shows "closed_segment a b \<subseteq> s \<Longrightarrow> path_component s a b"
68096
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  1477
  unfolding path_component_def
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  1478
  by (rule_tac x="linepath a b" in exI, auto)
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1479
70136
f03a01a18c6e modernized tags: default scope excludes proof;
wenzelm
parents: 69986
diff changeset
  1480
subsubsection\<^marker>\<open>tag unimportant\<close> \<open>Path components as sets\<close>
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1481
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1482
lemma path_component_set:
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1483
  "path_component_set s x =
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1484
    {y. (\<exists>g. path g \<and> path_image g \<subseteq> s \<and> pathstart g = x \<and> pathfinish g = y)}"
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1485
  by (auto simp: path_component_def)
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1486
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1487
lemma path_component_subset: "path_component_set s x \<subseteq> s"
68096
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  1488
  by (auto simp: path_component_mem(2))
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1489
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1490
lemma path_component_eq_empty: "path_component_set s x = {} \<longleftrightarrow> x \<notin> s"
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
  1491
  using path_component_mem path_component_refl_eq
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
  1492
    by fastforce
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1493
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1494
lemma path_component_mono:
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1495
     "s \<subseteq> t \<Longrightarrow> (path_component_set s x) \<subseteq> (path_component_set t x)"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1496
  by (simp add: Collect_mono path_component_of_subset)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1497
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1498
lemma path_component_eq:
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1499
   "y \<in> path_component_set s x \<Longrightarrow> path_component_set s y = path_component_set s x"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1500
by (metis (no_types, lifting) Collect_cong mem_Collect_eq path_component_sym path_component_trans)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1501
69514
58a77f548bb6 tuned headers
nipkow
parents: 69508
diff changeset
  1502
60420
884f54e01427 isabelle update_cartouches;
wenzelm
parents: 60303
diff changeset
  1503
subsection \<open>Path connectedness of a space\<close>
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1504
70136
f03a01a18c6e modernized tags: default scope excludes proof;
wenzelm
parents: 69986
diff changeset
  1505
definition\<^marker>\<open>tag important\<close> "path_connected s \<longleftrightarrow>
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1506
  (\<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)"
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1507
69939
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  1508
lemma path_connectedin_iff_path_connected_real [simp]:
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  1509
     "path_connectedin euclideanreal S \<longleftrightarrow> path_connected S"
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  1510
  by (simp add: path_connectedin path_connected_def path_defs)
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  1511
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1512
lemma path_connected_component: "path_connected s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>y\<in>s. path_component s x y)"
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1513
  unfolding path_connected_def path_component_def by auto
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1514
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1515
lemma path_connected_component_set: "path_connected s \<longleftrightarrow> (\<forall>x\<in>s. path_component_set s x = s)"
61694
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  1516
  unfolding path_connected_component path_component_subset
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1517
  using path_component_mem by blast
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1518
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1519
lemma path_component_maximal:
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1520
     "\<lbrakk>x \<in> t; path_connected t; t \<subseteq> s\<rbrakk> \<Longrightarrow> t \<subseteq> (path_component_set s x)"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1521
  by (metis path_component_mono path_connected_component_set)
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1522
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1523
lemma convex_imp_path_connected:
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1524
  fixes s :: "'a::real_normed_vector set"
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1525
  assumes "convex s"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1526
  shows "path_connected s"
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1527
  unfolding path_connected_def
66793
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66456
diff changeset
  1528
  using assms convex_contains_segment by fastforce
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1529
62620
d21dab28b3f9 New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents: 62618
diff changeset
  1530
lemma path_connected_UNIV [iff]: "path_connected (UNIV :: 'a::real_normed_vector set)"
d21dab28b3f9 New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents: 62618
diff changeset
  1531
  by (simp add: convex_imp_path_connected)
d21dab28b3f9 New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents: 62618
diff changeset
  1532
d21dab28b3f9 New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents: 62618
diff changeset
  1533
lemma path_component_UNIV: "path_component_set UNIV x = (UNIV :: 'a::real_normed_vector set)"
d21dab28b3f9 New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents: 62618
diff changeset
  1534
  using path_connected_component_set by auto
d21dab28b3f9 New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents: 62618
diff changeset
  1535
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1536
lemma path_connected_imp_connected:
64788
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  1537
  assumes "path_connected S"
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  1538
  shows "connected S"
66793
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66456
diff changeset
  1539
proof (rule connectedI)
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1540
  fix e1 e2
64788
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  1541
  assume as: "open e1" "open e2" "S \<subseteq> e1 \<union> e2" "e1 \<inter> e2 \<inter> S = {}" "e1 \<inter> S \<noteq> {}" "e2 \<inter> S \<noteq> {}"
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  1542
  then obtain x1 x2 where obt:"x1 \<in> e1 \<inter> S" "x2 \<in> e2 \<inter> S"
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1543
    by auto
64788
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  1544
  then obtain g where g: "path g" "path_image g \<subseteq> S" "pathstart g = x1" "pathfinish g = x2"
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1545
    using assms[unfolded path_connected_def,rule_format,of x1 x2] by auto
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1546
  have *: "connected {0..1::real}"
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1547
    by (auto intro!: convex_connected convex_real_interval)
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1548
  have "{0..1} \<subseteq> {x \<in> {0..1}. g x \<in> e1} \<union> {x \<in> {0..1}. g x \<in> e2}"
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1549
    using as(3) g(2)[unfolded path_defs] by blast
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1550
  moreover have "{x \<in> {0..1}. g x \<in> e1} \<inter> {x \<in> {0..1}. g x \<in> e2} = {}"
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1551
    using as(4) g(2)[unfolded path_defs]
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1552
    unfolding subset_eq
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1553
    by auto
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1554
  moreover have "{x \<in> {0..1}. g x \<in> e1} \<noteq> {} \<and> {x \<in> {0..1}. g x \<in> e2} \<noteq> {}"
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1555
    using g(3,4)[unfolded path_defs]
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1556
    using obt
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1557
    by (simp add: ex_in_conv [symmetric], metis zero_le_one order_refl)
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1558
  ultimately show False
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1559
    using *[unfolded connected_local not_ex, rule_format,
66884
c2128ab11f61 Switching to inverse image and constant_on, plus some new material
paulson <lp15@cam.ac.uk>
parents: 66827
diff changeset
  1560
      of "{0..1} \<inter> g -` e1" "{0..1} \<inter> g -` e2"]
63301
d3c87eb0bad2 new results about topology
paulson <lp15@cam.ac.uk>
parents: 63151
diff changeset
  1561
    using continuous_openin_preimage_gen[OF g(1)[unfolded path_def] as(1)]
d3c87eb0bad2 new results about topology
paulson <lp15@cam.ac.uk>
parents: 63151
diff changeset
  1562
    using continuous_openin_preimage_gen[OF g(1)[unfolded path_def] as(2)]
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1563
    by auto
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1564
qed
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1565
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1566
lemma open_path_component:
64788
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  1567
  fixes S :: "'a::real_normed_vector set"
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  1568
  assumes "open S"
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  1569
  shows "open (path_component_set S x)"
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1570
  unfolding open_contains_ball
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1571
proof
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1572
  fix y
64788
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  1573
  assume as: "y \<in> path_component_set S x"
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  1574
  then have "y \<in> S"
66793
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66456
diff changeset
  1575
    by (simp add: path_component_mem(2))
64788
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  1576
  then obtain e where e: "e > 0" "ball y e \<subseteq> S"
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1577
    using assms[unfolded open_contains_ball]
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1578
    by auto
66793
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66456
diff changeset
  1579
have "\<And>u. dist y u < e \<Longrightarrow> path_component S x u"
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66456
diff changeset
  1580
      by (metis (full_types) as centre_in_ball convex_ball convex_imp_path_connected e mem_Collect_eq mem_ball path_component_eq path_component_of_subset path_connected_component)
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66456
diff changeset
  1581
  then show "\<exists>e > 0. ball y e \<subseteq> path_component_set S x"
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66456
diff changeset
  1582
    using \<open>e>0\<close> by auto
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1583
qed
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1584
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1585
lemma open_non_path_component:
64788
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  1586
  fixes S :: "'a::real_normed_vector set"
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  1587
  assumes "open S"
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  1588
  shows "open (S - path_component_set S x)"
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1589
  unfolding open_contains_ball
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1590
proof
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1591
  fix y
68096
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  1592
  assume y: "y \<in> S - path_component_set S x"
64788
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  1593
  then obtain e where e: "e > 0" "ball y e \<subseteq> S"
68096
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  1594
    using assms openE by auto
64788
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  1595
  show "\<exists>e>0. ball y e \<subseteq> S - path_component_set S x"
68096
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  1596
  proof (intro exI conjI subsetI DiffI notI)
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  1597
    show "\<And>x. x \<in> ball y e \<Longrightarrow> x \<in> S"
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  1598
      using e by blast
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  1599
    show False if "z \<in> ball y e" "z \<in> path_component_set S x" for z
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  1600
    proof -
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  1601
      have "y \<in> path_component_set S z"
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  1602
        by (meson assms convex_ball convex_imp_path_connected e open_contains_ball_eq open_path_component path_component_maximal that(1))
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  1603
      then have "y \<in> path_component_set S x"
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  1604
        using path_component_eq that(2) by blast
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  1605
      then show False
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  1606
        using y by blast
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  1607
    qed
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  1608
  qed (use e in auto)
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1609
qed
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1610
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1611
lemma connected_open_path_connected:
64788
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  1612
  fixes S :: "'a::real_normed_vector set"
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  1613
  assumes "open S"
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  1614
    and "connected S"
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  1615
  shows "path_connected S"
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1616
  unfolding path_connected_component_set
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1617
proof (rule, rule, rule path_component_subset, rule)
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1618
  fix x y
64788
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  1619
  assume "x \<in> S" and "y \<in> S"
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  1620
  show "y \<in> path_component_set S x"
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1621
  proof (rule ccontr)
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1622
    assume "\<not> ?thesis"
64788
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  1623
    moreover have "path_component_set S x \<inter> S \<noteq> {}"
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  1624
      using \<open>x \<in> S\<close> path_component_eq_empty path_component_subset[of S x]
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1625
      by auto
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1626
    ultimately
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1627
    show False
64788
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  1628
      using \<open>y \<in> S\<close> open_non_path_component[OF assms(1)] open_path_component[OF assms(1)]
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1629
      using assms(2)[unfolded connected_def not_ex, rule_format,
64788
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  1630
        of "path_component_set S x" "S - path_component_set S x"]
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1631
      by auto
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1632
  qed
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1633
qed
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1634
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1635
lemma path_connected_continuous_image:
64788
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  1636
  assumes "continuous_on S f"
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  1637
    and "path_connected S"
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  1638
  shows "path_connected (f ` S)"
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1639
  unfolding path_connected_def
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1640
proof (rule, rule)
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1641
  fix x' y'
64788
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  1642
  assume "x' \<in> f ` S" "y' \<in> f ` S"
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  1643
  then obtain x y where x: "x \<in> S" and y: "y \<in> S" and x': "x' = f x" and y': "y' = f y"
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1644
    by auto
64788
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  1645
  from x y obtain g where "path g \<and> path_image g \<subseteq> S \<and> pathstart g = x \<and> pathfinish g = y"
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1646
    using assms(2)[unfolded path_connected_def] by fast
64788
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  1647
  then show "\<exists>g. path g \<and> path_image g \<subseteq> f ` S \<and> pathstart g = x' \<and> pathfinish g = y'"
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1648
    unfolding x' y'
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1649
    apply (rule_tac x="f \<circ> g" in exI)
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1650
    unfolding path_defs
51481
ef949192e5d6 move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents: 51478
diff changeset
  1651
    apply (intro conjI continuous_on_compose continuous_on_subset[OF assms(1)])
ef949192e5d6 move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents: 51478
diff changeset
  1652
    apply auto
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1653
    done
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1654
qed
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1655
64788
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  1656
lemma path_connected_translationI:
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  1657
  fixes a :: "'a :: topological_group_add"
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  1658
  assumes "path_connected S" shows "path_connected ((\<lambda>x. a + x) ` S)"
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  1659
  by (intro path_connected_continuous_image assms continuous_intros)
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  1660
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  1661
lemma path_connected_translation:
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  1662
  fixes a :: "'a :: topological_group_add"
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  1663
  shows "path_connected ((\<lambda>x. a + x) ` S) = path_connected S"
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  1664
proof -
67399
eab6ce8368fa ran isabelle update_op on all sources
nipkow
parents: 67240
diff changeset
  1665
  have "\<forall>x y. (+) (x::'a) ` (+) (0 - x) ` y = y"
64788
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  1666
    by (simp add: image_image)
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  1667
  then show ?thesis
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  1668
    by (metis (no_types) path_connected_translationI)
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  1669
qed
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  1670
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  1671
lemma path_connected_segment [simp]:
61518
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  1672
    fixes a :: "'a::real_normed_vector"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  1673
    shows "path_connected (closed_segment a b)"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  1674
  by (simp add: convex_imp_path_connected)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  1675
64788
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  1676
lemma path_connected_open_segment [simp]:
61518
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  1677
    fixes a :: "'a::real_normed_vector"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  1678
    shows "path_connected (open_segment a b)"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  1679
  by (simp add: convex_imp_path_connected)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  1680
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1681
lemma homeomorphic_path_connectedness:
68096
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  1682
  "S homeomorphic T \<Longrightarrow> path_connected S \<longleftrightarrow> path_connected T"
61738
c4f6031f1310 New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents: 61711
diff changeset
  1683
  unfolding homeomorphic_def homeomorphism_def by (metis path_connected_continuous_image)
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1684
64788
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  1685
lemma path_connected_empty [simp]: "path_connected {}"
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1686
  unfolding path_connected_def by auto
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1687
64788
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  1688
lemma path_connected_singleton [simp]: "path_connected {a}"
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1689
  unfolding path_connected_def pathstart_def pathfinish_def path_image_def
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1690
  apply clarify
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1691
  apply (rule_tac x="\<lambda>x. a" in exI)
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1692
  apply (simp add: image_constant_conv)
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1693
  apply (simp add: path_def continuous_on_const)
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1694
  done
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1695
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1696
lemma path_connected_Un:
68096
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  1697
  assumes "path_connected S"
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  1698
    and "path_connected T"
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  1699
    and "S \<inter> T \<noteq> {}"
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  1700
  shows "path_connected (S \<union> T)"
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1701
  unfolding path_connected_component
68096
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  1702
proof (intro ballI)
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1703
  fix x y
68096
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  1704
  assume x: "x \<in> S \<union> T" and y: "y \<in> S \<union> T"
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  1705
  from assms obtain z where z: "z \<in> S" "z \<in> T"
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1706
    by auto
68096
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  1707
  show "path_component (S \<union> T) x y"
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  1708
    using x y
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  1709
  proof safe
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  1710
    assume "x \<in> S" "y \<in> S"
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  1711
    then show "path_component (S \<union> T) x y"
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  1712
      by (meson Un_upper1 \<open>path_connected S\<close> path_component_of_subset path_connected_component)
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  1713
  next
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  1714
    assume "x \<in> S" "y \<in> T"
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  1715
    then show "path_component (S \<union> T) x y"
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  1716
      by (metis z assms(1-2) le_sup_iff order_refl path_component_of_subset path_component_trans path_connected_component)
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  1717
  next
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  1718
  assume "x \<in> T" "y \<in> S"
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  1719
    then show "path_component (S \<union> T) x y"
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  1720
      by (metis z assms(1-2) le_sup_iff order_refl path_component_of_subset path_component_trans path_connected_component)
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  1721
  next
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  1722
    assume "x \<in> T" "y \<in> T"
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  1723
    then show "path_component (S \<union> T) x y"
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  1724
      by (metis Un_upper1 assms(2) path_component_of_subset path_connected_component sup_commute)
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  1725
  qed
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1726
qed
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1727
37674
f86de9c00c47 convert theorem path_connected_sphere to euclidean_space class
huffman
parents: 37489
diff changeset
  1728
lemma path_connected_UNION:
f86de9c00c47 convert theorem path_connected_sphere to euclidean_space class
huffman
parents: 37489
diff changeset
  1729
  assumes "\<And>i. i \<in> A \<Longrightarrow> path_connected (S i)"
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1730
    and "\<And>i. i \<in> A \<Longrightarrow> z \<in> S i"
37674
f86de9c00c47 convert theorem path_connected_sphere to euclidean_space class
huffman
parents: 37489
diff changeset
  1731
  shows "path_connected (\<Union>i\<in>A. S i)"
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1732
  unfolding path_connected_component
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1733
proof clarify
37674
f86de9c00c47 convert theorem path_connected_sphere to euclidean_space class
huffman
parents: 37489
diff changeset
  1734
  fix x i y j
f86de9c00c47 convert theorem path_connected_sphere to euclidean_space class
huffman
parents: 37489
diff changeset
  1735
  assume *: "i \<in> A" "x \<in> S i" "j \<in> A" "y \<in> S j"
49654
366d8b41ca17 tuned proofs;
wenzelm
parents: 49653
diff changeset
  1736
  then have "path_component (S i) x z" and "path_component (S j) z y"
37674
f86de9c00c47 convert theorem path_connected_sphere to euclidean_space class
huffman
parents: 37489
diff changeset
  1737
    using assms by (simp_all add: path_connected_component)
49654
366d8b41ca17 tuned proofs;
wenzelm
parents: 49653
diff changeset
  1738
  then have "path_component (\<Union>i\<in>A. S i) x z" and "path_component (\<Union>i\<in>A. S i) z y"
48125
602dc0215954 tuned proofs -- prefer direct "rotated" instead of old-style COMP;
wenzelm
parents: 44647
diff changeset
  1739
    using *(1,3) by (auto elim!: path_component_of_subset [rotated])
49654
366d8b41ca17 tuned proofs;
wenzelm
parents: 49653
diff changeset
  1740
  then show "path_component (\<Union>i\<in>A. S i) x y"
37674
f86de9c00c47 convert theorem path_connected_sphere to euclidean_space class
huffman
parents: 37489
diff changeset
  1741
    by (rule path_component_trans)
f86de9c00c47 convert theorem path_connected_sphere to euclidean_space class
huffman
parents: 37489
diff changeset
  1742
qed
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1743
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1744
lemma path_component_path_image_pathstart:
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1745
  assumes p: "path p" and x: "x \<in> path_image p"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1746
  shows "path_component (path_image p) (pathstart p) x"
68096
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  1747
proof -
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  1748
  obtain y where x: "x = p y" and y: "0 \<le> y" "y \<le> 1"
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  1749
    using x by (auto simp: path_image_def)
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  1750
  show ?thesis
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  1751
    unfolding path_component_def 
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  1752
  proof (intro exI conjI)
69064
5840724b1d71 Prefix form of infix with * on either side no longer needs special treatment
nipkow
parents: 68913
diff changeset
  1753
    have "continuous_on {0..1} (p \<circ> ((*) y))"
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1754
      apply (rule continuous_intros)+
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1755
      using p [unfolded path_def] y
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1756
      apply (auto simp: mult_le_one intro: continuous_on_subset [of _ p])
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1757
      done
68096
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  1758
    then show "path (\<lambda>u. p (y * u))"
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1759
      by (simp add: path_def)
68096
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  1760
    show "path_image (\<lambda>u. p (y * u)) \<subseteq> path_image p"
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  1761
      using y mult_le_one by (fastforce simp: path_image_def image_iff)
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  1762
  qed (auto simp: pathstart_def pathfinish_def x)
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1763
qed
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1764
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1765
lemma path_connected_path_image: "path p \<Longrightarrow> path_connected(path_image p)"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1766
  unfolding path_connected_component
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1767
  by (meson path_component_path_image_pathstart path_component_sym path_component_trans)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1768
64788
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  1769
lemma path_connected_path_component [simp]:
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1770
   "path_connected (path_component_set s x)"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1771
proof -
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1772
  { fix y z
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1773
    assume pa: "path_component s x y" "path_component s x z"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1774
    then have pae: "path_component_set s x = path_component_set s y"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1775
      using path_component_eq by auto
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1776
    have yz: "path_component s y z"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1777
      using pa path_component_sym path_component_trans by blast
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1778
    then have "\<exists>g. path g \<and> path_image g \<subseteq> path_component_set s x \<and> pathstart g = y \<and> pathfinish g = z"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1779
      apply (simp add: path_component_def, clarify)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1780
      apply (rule_tac x=g in exI)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1781
      by (simp add: pae path_component_maximal path_connected_path_image pathstart_in_path_image)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1782
  }
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1783
  then show ?thesis
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1784
    by (simp add: path_connected_def)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1785
qed
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1786
68532
f8b98d31ad45 Incorporating new/strengthened proofs from Library and AFP entries
paulson <lp15@cam.ac.uk>
parents: 68310
diff changeset
  1787
lemma path_component: "path_component S x y \<longleftrightarrow> (\<exists>t. path_connected t \<and> t \<subseteq> S \<and> x \<in> t \<and> y \<in> t)"
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1788
  apply (intro iffI)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1789
  apply (metis path_connected_path_image path_defs(5) pathfinish_in_path_image pathstart_in_path_image)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1790
  using path_component_of_subset path_connected_component by blast
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1791
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1792
lemma path_component_path_component [simp]:
68532
f8b98d31ad45 Incorporating new/strengthened proofs from Library and AFP entries
paulson <lp15@cam.ac.uk>
parents: 68310
diff changeset
  1793
   "path_component_set (path_component_set S x) x = path_component_set S x"
f8b98d31ad45 Incorporating new/strengthened proofs from Library and AFP entries
paulson <lp15@cam.ac.uk>
parents: 68310
diff changeset
  1794
proof (cases "x \<in> S")
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1795
  case True show ?thesis
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1796
    apply (rule subset_antisym)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1797
    apply (simp add: path_component_subset)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1798
    by (simp add: True path_component_maximal path_component_refl path_connected_path_component)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1799
next
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1800
  case False then show ?thesis
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1801
    by (metis False empty_iff path_component_eq_empty)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1802
qed
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1803
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1804
lemma path_component_subset_connected_component:
68532
f8b98d31ad45 Incorporating new/strengthened proofs from Library and AFP entries
paulson <lp15@cam.ac.uk>
parents: 68310
diff changeset
  1805
   "(path_component_set S x) \<subseteq> (connected_component_set S x)"
f8b98d31ad45 Incorporating new/strengthened proofs from Library and AFP entries
paulson <lp15@cam.ac.uk>
parents: 68310
diff changeset
  1806
proof (cases "x \<in> S")
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1807
  case True show ?thesis
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1808
    apply (rule connected_component_maximal)
68532
f8b98d31ad45 Incorporating new/strengthened proofs from Library and AFP entries
paulson <lp15@cam.ac.uk>
parents: 68310
diff changeset
  1809
    apply (auto simp: True path_component_subset path_component_refl path_connected_imp_connected)
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1810
    done
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1811
next
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1812
  case False then show ?thesis
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1813
    using path_component_eq_empty by auto
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1814
qed
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1815
69514
58a77f548bb6 tuned headers
nipkow
parents: 69508
diff changeset
  1816
70136
f03a01a18c6e modernized tags: default scope excludes proof;
wenzelm
parents: 69986
diff changeset
  1817
subsection\<^marker>\<open>tag unimportant\<close>\<open>Lemmas about path-connectedness\<close>
62620
d21dab28b3f9 New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents: 62618
diff changeset
  1818
d21dab28b3f9 New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents: 62618
diff changeset
  1819
lemma path_connected_linear_image:
d21dab28b3f9 New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents: 62618
diff changeset
  1820
  fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
68532
f8b98d31ad45 Incorporating new/strengthened proofs from Library and AFP entries
paulson <lp15@cam.ac.uk>
parents: 68310
diff changeset
  1821
  assumes "path_connected S" "bounded_linear f"
f8b98d31ad45 Incorporating new/strengthened proofs from Library and AFP entries
paulson <lp15@cam.ac.uk>
parents: 68310
diff changeset
  1822
    shows "path_connected(f ` S)"
62620
d21dab28b3f9 New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents: 62618
diff changeset
  1823
by (auto simp: linear_continuous_on assms path_connected_continuous_image)
d21dab28b3f9 New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents: 62618
diff changeset
  1824
68532
f8b98d31ad45 Incorporating new/strengthened proofs from Library and AFP entries
paulson <lp15@cam.ac.uk>
parents: 68310
diff changeset
  1825
lemma is_interval_path_connected: "is_interval S \<Longrightarrow> path_connected S"
62620
d21dab28b3f9 New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents: 62618
diff changeset
  1826
  by (simp add: convex_imp_path_connected is_interval_convex)
d21dab28b3f9 New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents: 62618
diff changeset
  1827
69939
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  1828
lemma path_connectedin_path_image:
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  1829
  assumes "pathin X g" shows "path_connectedin X (g ` ({0..1}))"
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  1830
  unfolding pathin_def
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  1831
proof (rule path_connectedin_continuous_map_image)
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  1832
  show "continuous_map (subtopology euclideanreal {0..1}) X g"
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  1833
    using assms pathin_def by blast
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  1834
qed (auto simp: is_interval_1 is_interval_path_connected)
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  1835
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  1836
lemma path_connected_space_subconnected:
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  1837
     "path_connected_space X \<longleftrightarrow>
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  1838
      (\<forall>x \<in> topspace X. \<forall>y \<in> topspace X. \<exists>S. path_connectedin X S \<and> x \<in> S \<and> y \<in> S)"
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  1839
  unfolding path_connected_space_def Ball_def
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  1840
  apply (intro all_cong1 imp_cong refl, safe)
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  1841
  using path_connectedin_path_image apply fastforce
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  1842
  by (meson path_connectedin)
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  1843
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  1844
lemma connectedin_path_image: "pathin X g \<Longrightarrow> connectedin X (g ` ({0..1}))"
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  1845
  by (simp add: path_connectedin_imp_connectedin path_connectedin_path_image)
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  1846
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  1847
lemma compactin_path_image: "pathin X g \<Longrightarrow> compactin X (g ` ({0..1}))"
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  1848
  unfolding pathin_def
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  1849
  by (rule image_compactin [of "top_of_set {0..1}"]) auto
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  1850
62843
313d3b697c9a Mostly renaming (from HOL Light to Isabelle conventions), with a couple of new results
paulson <lp15@cam.ac.uk>
parents: 62626
diff changeset
  1851
lemma linear_homeomorphism_image:
313d3b697c9a Mostly renaming (from HOL Light to Isabelle conventions), with a couple of new results
paulson <lp15@cam.ac.uk>
parents: 62626
diff changeset
  1852
  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
62620
d21dab28b3f9 New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents: 62618
diff changeset
  1853
  assumes "linear f" "inj f"
62843
313d3b697c9a Mostly renaming (from HOL Light to Isabelle conventions), with a couple of new results
paulson <lp15@cam.ac.uk>
parents: 62626
diff changeset
  1854
    obtains g where "homeomorphism (f ` S) S g f"
313d3b697c9a Mostly renaming (from HOL Light to Isabelle conventions), with a couple of new results
paulson <lp15@cam.ac.uk>
parents: 62626
diff changeset
  1855
using linear_injective_left_inverse [OF assms]
313d3b697c9a Mostly renaming (from HOL Light to Isabelle conventions), with a couple of new results
paulson <lp15@cam.ac.uk>
parents: 62626
diff changeset
  1856
apply clarify
313d3b697c9a Mostly renaming (from HOL Light to Isabelle conventions), with a couple of new results
paulson <lp15@cam.ac.uk>
parents: 62626
diff changeset
  1857
apply (rule_tac g=g in that)
313d3b697c9a Mostly renaming (from HOL Light to Isabelle conventions), with a couple of new results
paulson <lp15@cam.ac.uk>
parents: 62626
diff changeset
  1858
using assms
313d3b697c9a Mostly renaming (from HOL Light to Isabelle conventions), with a couple of new results
paulson <lp15@cam.ac.uk>
parents: 62626
diff changeset
  1859
apply (auto simp: homeomorphism_def eq_id_iff [symmetric] image_comp comp_def linear_conv_bounded_linear linear_continuous_on)
313d3b697c9a Mostly renaming (from HOL Light to Isabelle conventions), with a couple of new results
paulson <lp15@cam.ac.uk>
parents: 62626
diff changeset
  1860
done
313d3b697c9a Mostly renaming (from HOL Light to Isabelle conventions), with a couple of new results
paulson <lp15@cam.ac.uk>
parents: 62626
diff changeset
  1861
313d3b697c9a Mostly renaming (from HOL Light to Isabelle conventions), with a couple of new results
paulson <lp15@cam.ac.uk>
parents: 62626
diff changeset
  1862
lemma linear_homeomorphic_image:
313d3b697c9a Mostly renaming (from HOL Light to Isabelle conventions), with a couple of new results
paulson <lp15@cam.ac.uk>
parents: 62626
diff changeset
  1863
  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
313d3b697c9a Mostly renaming (from HOL Light to Isabelle conventions), with a couple of new results
paulson <lp15@cam.ac.uk>
parents: 62626
diff changeset
  1864
  assumes "linear f" "inj f"
313d3b697c9a Mostly renaming (from HOL Light to Isabelle conventions), with a couple of new results
paulson <lp15@cam.ac.uk>
parents: 62626
diff changeset
  1865
    shows "S homeomorphic f ` S"
313d3b697c9a Mostly renaming (from HOL Light to Isabelle conventions), with a couple of new results
paulson <lp15@cam.ac.uk>
parents: 62626
diff changeset
  1866
by (meson homeomorphic_def homeomorphic_sym linear_homeomorphism_image [OF assms])
62620
d21dab28b3f9 New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents: 62618
diff changeset
  1867
d21dab28b3f9 New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents: 62618
diff changeset
  1868
lemma path_connected_Times:
d21dab28b3f9 New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents: 62618
diff changeset
  1869
  assumes "path_connected s" "path_connected t"
d21dab28b3f9 New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents: 62618
diff changeset
  1870
    shows "path_connected (s \<times> t)"
d21dab28b3f9 New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents: 62618
diff changeset
  1871
proof (simp add: path_connected_def Sigma_def, clarify)
d21dab28b3f9 New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents: 62618
diff changeset
  1872
  fix x1 y1 x2 y2
d21dab28b3f9 New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents: 62618
diff changeset
  1873
  assume "x1 \<in> s" "y1 \<in> t" "x2 \<in> s" "y2 \<in> t"
d21dab28b3f9 New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents: 62618
diff changeset
  1874
  obtain g where "path g" and g: "path_image g \<subseteq> s" and gs: "pathstart g = x1" and gf: "pathfinish g = x2"
d21dab28b3f9 New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents: 62618
diff changeset
  1875
    using \<open>x1 \<in> s\<close> \<open>x2 \<in> s\<close> assms by (force simp: path_connected_def)
d21dab28b3f9 New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents: 62618
diff changeset
  1876
  obtain h where "path h" and h: "path_image h \<subseteq> t" and hs: "pathstart h = y1" and hf: "pathfinish h = y2"
d21dab28b3f9 New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents: 62618
diff changeset
  1877
    using \<open>y1 \<in> t\<close> \<open>y2 \<in> t\<close> assms by (force simp: path_connected_def)
d21dab28b3f9 New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents: 62618
diff changeset
  1878
  have "path (\<lambda>z. (x1, h z))"
d21dab28b3f9 New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents: 62618
diff changeset
  1879
    using \<open>path h\<close>
d21dab28b3f9 New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents: 62618
diff changeset
  1880
    apply (simp add: path_def)
d21dab28b3f9 New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents: 62618
diff changeset
  1881
    apply (rule continuous_on_compose2 [where f = h])
d21dab28b3f9 New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents: 62618
diff changeset
  1882
    apply (rule continuous_intros | force)+
d21dab28b3f9 New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents: 62618
diff changeset
  1883
    done
d21dab28b3f9 New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents: 62618
diff changeset
  1884
  moreover have "path (\<lambda>z. (g z, y2))"
d21dab28b3f9 New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents: 62618
diff changeset
  1885
    using \<open>path g\<close>
d21dab28b3f9 New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents: 62618
diff changeset
  1886
    apply (simp add: path_def)
d21dab28b3f9 New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents: 62618
diff changeset
  1887
    apply (rule continuous_on_compose2 [where f = g])
d21dab28b3f9 New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents: 62618
diff changeset
  1888
    apply (rule continuous_intros | force)+
d21dab28b3f9 New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents: 62618
diff changeset
  1889
    done
d21dab28b3f9 New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents: 62618
diff changeset
  1890
  ultimately have 1: "path ((\<lambda>z. (x1, h z)) +++ (\<lambda>z. (g z, y2)))"
d21dab28b3f9 New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents: 62618
diff changeset
  1891
    by (metis hf gs path_join_imp pathstart_def pathfinish_def)
d21dab28b3f9 New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents: 62618
diff changeset
  1892
  have "path_image ((\<lambda>z. (x1, h z)) +++ (\<lambda>z. (g z, y2))) \<subseteq> path_image (\<lambda>z. (x1, h z)) \<union> path_image (\<lambda>z. (g z, y2))"
d21dab28b3f9 New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents: 62618
diff changeset
  1893
    by (rule Path_Connected.path_image_join_subset)
68096
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  1894
  also have "\<dots> \<subseteq> (\<Union>x\<in>s. \<Union>x1\<in>t. {(x, x1)})"
62620
d21dab28b3f9 New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents: 62618
diff changeset
  1895
    using g h \<open>x1 \<in> s\<close> \<open>y2 \<in> t\<close> by (force simp: path_image_def)
d21dab28b3f9 New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents: 62618
diff changeset
  1896
  finally have 2: "path_image ((\<lambda>z. (x1, h z)) +++ (\<lambda>z. (g z, y2))) \<subseteq> (\<Union>x\<in>s. \<Union>x1\<in>t. {(x, x1)})" .
d21dab28b3f9 New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents: 62618
diff changeset
  1897
  show "\<exists>g. path g \<and> path_image g \<subseteq> (\<Union>x\<in>s. \<Union>x1\<in>t. {(x, x1)}) \<and>
d21dab28b3f9 New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents: 62618
diff changeset
  1898
            pathstart g = (x1, y1) \<and> pathfinish g = (x2, y2)"
d21dab28b3f9 New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents: 62618
diff changeset
  1899
    apply (intro exI conjI)
d21dab28b3f9 New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents: 62618
diff changeset
  1900
       apply (rule 1)
d21dab28b3f9 New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents: 62618
diff changeset
  1901
      apply (rule 2)
d21dab28b3f9 New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents: 62618
diff changeset
  1902
     apply (metis hs pathstart_def pathstart_join)
d21dab28b3f9 New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents: 62618
diff changeset
  1903
    by (metis gf pathfinish_def pathfinish_join)
d21dab28b3f9 New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents: 62618
diff changeset
  1904
qed
d21dab28b3f9 New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents: 62618
diff changeset
  1905
d21dab28b3f9 New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents: 62618
diff changeset
  1906
lemma is_interval_path_connected_1:
d21dab28b3f9 New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents: 62618
diff changeset
  1907
  fixes s :: "real set"
d21dab28b3f9 New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents: 62618
diff changeset
  1908
  shows "is_interval s \<longleftrightarrow> path_connected s"
d21dab28b3f9 New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents: 62618
diff changeset
  1909
using is_interval_connected_1 is_interval_path_connected path_connected_imp_connected by blast
d21dab28b3f9 New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents: 62618
diff changeset
  1910
d21dab28b3f9 New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents: 62618
diff changeset
  1911
70136
f03a01a18c6e modernized tags: default scope excludes proof;
wenzelm
parents: 69986
diff changeset
  1912
subsection\<^marker>\<open>tag unimportant\<close>\<open>Path components\<close>
66793
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66456
diff changeset
  1913
62948
7700f467892b lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 62843
diff changeset
  1914
lemma Union_path_component [simp]:
7700f467892b lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 62843
diff changeset
  1915
   "Union {path_component_set S x |x. x \<in> S} = S"
7700f467892b lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 62843
diff changeset
  1916
apply (rule subset_antisym)
7700f467892b lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 62843
diff changeset
  1917
using path_component_subset apply force
7700f467892b lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 62843
diff changeset
  1918
using path_component_refl by auto
7700f467892b lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 62843
diff changeset
  1919
7700f467892b lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 62843
diff changeset
  1920
lemma path_component_disjoint:
7700f467892b lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 62843
diff changeset
  1921
   "disjnt (path_component_set S a) (path_component_set S b) \<longleftrightarrow>
7700f467892b lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 62843
diff changeset
  1922
    (a \<notin> path_component_set S b)"
7700f467892b lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 62843
diff changeset
  1923
apply (auto simp: disjnt_def)
7700f467892b lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 62843
diff changeset
  1924
using path_component_eq apply fastforce
7700f467892b lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 62843
diff changeset
  1925
using path_component_sym path_component_trans by blast
7700f467892b lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 62843
diff changeset
  1926
7700f467892b lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 62843
diff changeset
  1927
lemma path_component_eq_eq:
7700f467892b lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 62843
diff changeset
  1928
   "path_component S x = path_component S y \<longleftrightarrow>
7700f467892b lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 62843
diff changeset
  1929
        (x \<notin> S) \<and> (y \<notin> S) \<or> x \<in> S \<and> y \<in> S \<and> path_component S x y"
7700f467892b lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 62843
diff changeset
  1930
apply (rule iffI, metis (no_types) path_component_mem(1) path_component_refl)
7700f467892b lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 62843
diff changeset
  1931
apply (erule disjE, metis Collect_empty_eq_bot path_component_eq_empty)
7700f467892b lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 62843
diff changeset
  1932
apply (rule ext)
7700f467892b lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 62843
diff changeset
  1933
apply (metis path_component_trans path_component_sym)
7700f467892b lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 62843
diff changeset
  1934
done
7700f467892b lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 62843
diff changeset
  1935
7700f467892b lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 62843
diff changeset
  1936
lemma path_component_unique:
7700f467892b lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 62843
diff changeset
  1937
  assumes "x \<in> c" "c \<subseteq> S" "path_connected c"
7700f467892b lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 62843
diff changeset
  1938
          "\<And>c'. \<lbrakk>x \<in> c'; c' \<subseteq> S; path_connected c'\<rbrakk> \<Longrightarrow> c' \<subseteq> c"
7700f467892b lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 62843
diff changeset
  1939
   shows "path_component_set S x = c"
7700f467892b lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 62843
diff changeset
  1940
apply (rule subset_antisym)
7700f467892b lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 62843
diff changeset
  1941
using assms
7700f467892b lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 62843
diff changeset
  1942
apply (metis mem_Collect_eq subsetCE path_component_eq_eq path_component_subset path_connected_path_component)
7700f467892b lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 62843
diff changeset
  1943
by (simp add: assms path_component_maximal)
7700f467892b lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 62843
diff changeset
  1944
7700f467892b lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 62843
diff changeset
  1945
lemma path_component_intermediate_subset:
7700f467892b lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 62843
diff changeset
  1946
   "path_component_set u a \<subseteq> t \<and> t \<subseteq> u
7700f467892b lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 62843
diff changeset
  1947
        \<Longrightarrow> path_component_set t a = path_component_set u a"
7700f467892b lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 62843
diff changeset
  1948
by (metis (no_types) path_component_mono path_component_path_component subset_antisym)
7700f467892b lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 62843
diff changeset
  1949
7700f467892b lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 62843
diff changeset
  1950
lemma complement_path_component_Union:
7700f467892b lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 62843
diff changeset
  1951
  fixes x :: "'a :: topological_space"
7700f467892b lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 62843
diff changeset
  1952
  shows "S - path_component_set S x =
7700f467892b lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 62843
diff changeset
  1953
         \<Union>({path_component_set S y| y. y \<in> S} - {path_component_set S x})"
7700f467892b lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 62843
diff changeset
  1954
proof -
7700f467892b lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 62843
diff changeset
  1955
  have *: "(\<And>x. x \<in> S - {a} \<Longrightarrow> disjnt a x) \<Longrightarrow> \<Union>S - a = \<Union>(S - {a})"
7700f467892b lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 62843
diff changeset
  1956
    for a::"'a set" and S
7700f467892b lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 62843
diff changeset
  1957
    by (auto simp: disjnt_def)
7700f467892b lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 62843
diff changeset
  1958
  have "\<And>y. y \<in> {path_component_set S x |x. x \<in> S} - {path_component_set S x}
7700f467892b lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 62843
diff changeset
  1959
            \<Longrightarrow> disjnt (path_component_set S x) y"
7700f467892b lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 62843
diff changeset
  1960
    using path_component_disjoint path_component_eq by fastforce
7700f467892b lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 62843
diff changeset
  1961
  then have "\<Union>{path_component_set S x |x. x \<in> S} - path_component_set S x =
7700f467892b lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 62843
diff changeset
  1962
             \<Union>({path_component_set S y |y. y \<in> S} - {path_component_set S x})"
7700f467892b lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 62843
diff changeset
  1963
    by (meson *)
7700f467892b lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 62843
diff changeset
  1964
  then show ?thesis by simp
7700f467892b lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 62843
diff changeset
  1965
qed
7700f467892b lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 62843
diff changeset
  1966
7700f467892b lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 62843
diff changeset
  1967
69939
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  1968
subsection\<open>Path components\<close>
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  1969
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  1970
definition path_component_of
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  1971
  where "path_component_of X x y \<equiv> \<exists>g. pathin X g \<and> g 0 = x \<and> g 1 = y"
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  1972
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  1973
abbreviation path_component_of_set
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  1974
  where "path_component_of_set X x \<equiv> Collect (path_component_of X x)"
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  1975
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  1976
definition path_components_of :: "'a topology \<Rightarrow> 'a set set"
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  1977
  where "path_components_of X \<equiv> path_component_of_set X ` topspace X"
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  1978
69986
f2d327275065 generalised homotopic_with to topologies; homotopic_with_canon is the old version
paulson <lp15@cam.ac.uk>
parents: 69939
diff changeset
  1979
lemma pathin_canon_iff: "pathin (top_of_set T) g \<longleftrightarrow> path g \<and> g ` {0..1} \<subseteq> T"
f2d327275065 generalised homotopic_with to topologies; homotopic_with_canon is the old version
paulson <lp15@cam.ac.uk>
parents: 69939
diff changeset
  1980
  by (simp add: path_def pathin_def)
f2d327275065 generalised homotopic_with to topologies; homotopic_with_canon is the old version
paulson <lp15@cam.ac.uk>
parents: 69939
diff changeset
  1981
f2d327275065 generalised homotopic_with to topologies; homotopic_with_canon is the old version
paulson <lp15@cam.ac.uk>
parents: 69939
diff changeset
  1982
lemma path_component_of_canon_iff [simp]:
f2d327275065 generalised homotopic_with to topologies; homotopic_with_canon is the old version
paulson <lp15@cam.ac.uk>
parents: 69939
diff changeset
  1983
  "path_component_of (top_of_set T) a b \<longleftrightarrow> path_component T a b"
f2d327275065 generalised homotopic_with to topologies; homotopic_with_canon is the old version
paulson <lp15@cam.ac.uk>
parents: 69939
diff changeset
  1984
  by (simp add: path_component_of_def pathin_canon_iff path_defs)
f2d327275065 generalised homotopic_with to topologies; homotopic_with_canon is the old version
paulson <lp15@cam.ac.uk>
parents: 69939
diff changeset
  1985
69939
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  1986
lemma path_component_in_topspace:
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  1987
   "path_component_of X x y \<Longrightarrow> x \<in> topspace X \<and> y \<in> topspace X"
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  1988
  by (auto simp: path_component_of_def pathin_def continuous_map_def)
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  1989
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  1990
lemma path_component_of_refl:
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  1991
   "path_component_of X x x \<longleftrightarrow> x \<in> topspace X"
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  1992
  apply (auto simp: path_component_in_topspace)
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  1993
  apply (force simp: path_component_of_def pathin_const)
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  1994
  done
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  1995
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  1996
lemma path_component_of_sym:
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  1997
  assumes "path_component_of X x y"
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  1998
  shows "path_component_of X y x"
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  1999
  using assms
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2000
  apply (clarsimp simp: path_component_of_def pathin_def)
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2001
  apply (rule_tac x="g \<circ> (\<lambda>t. 1 - t)" in exI)
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2002
  apply (auto intro!: continuous_map_compose)
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2003
  apply (force simp: continuous_map_in_subtopology continuous_on_op_minus)
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2004
  done
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2005
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2006
lemma path_component_of_sym_iff:
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2007
   "path_component_of X x y \<longleftrightarrow> path_component_of X y x"
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2008
  by (metis path_component_of_sym)
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2009
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2010
lemma path_component_of_trans:
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2011
  assumes "path_component_of X x y" and "path_component_of X y z"
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2012
  shows "path_component_of X x z"
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2013
  unfolding path_component_of_def pathin_def
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2014
proof -
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2015
  let ?T01 = "top_of_set {0..1::real}"
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2016
  obtain g1 g2 where g1: "continuous_map ?T01 X g1" "x = g1 0" "y = g1 1"
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2017
    and g2: "continuous_map ?T01 X g2" "g2 0 = g1 1" "z = g2 1"
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2018
    using assms unfolding path_component_of_def pathin_def by blast
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2019
  let ?g = "\<lambda>x. if x \<le> 1/2 then (g1 \<circ> (\<lambda>t. 2 * t)) x else (g2 \<circ> (\<lambda>t. 2 * t -1)) x"
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2020
  show "\<exists>g. continuous_map ?T01 X g \<and> g 0 = x \<and> g 1 = z"
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2021
  proof (intro exI conjI)
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2022
    show "continuous_map (subtopology euclideanreal {0..1}) X ?g"
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2023
    proof (intro continuous_map_cases_le continuous_map_compose, force, force)
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2024
      show "continuous_map (subtopology ?T01 {x \<in> topspace ?T01. x \<le> 1/2}) ?T01 ((*) 2)"
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2025
        by (auto simp: continuous_map_in_subtopology continuous_map_from_subtopology)
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2026
      have "continuous_map
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2027
             (subtopology (top_of_set {0..1}) {x. 0 \<le> x \<and> x \<le> 1 \<and> 1 \<le> x * 2})
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2028
             euclideanreal (\<lambda>t. 2 * t - 1)"
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2029
        by (intro continuous_intros) (force intro: continuous_map_from_subtopology)
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2030
      then show "continuous_map (subtopology ?T01 {x \<in> topspace ?T01. 1/2 \<le> x}) ?T01 (\<lambda>t. 2 * t - 1)"
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2031
        by (force simp: continuous_map_in_subtopology)
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2032
      show "(g1 \<circ> (*) 2) x = (g2 \<circ> (\<lambda>t. 2 * t - 1)) x" if "x \<in> topspace ?T01" "x = 1/2" for x
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2033
        using that by (simp add: g2(2) mult.commute continuous_map_from_subtopology)
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2034
    qed (auto simp: g1 g2)
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2035
  qed (auto simp: g1 g2)
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2036
qed
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2037
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2038
lemma path_component_of_mono:
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2039
   "\<lbrakk>path_component_of (subtopology X S) x y; S \<subseteq> T\<rbrakk> \<Longrightarrow> path_component_of (subtopology X T) x y"
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2040
  unfolding path_component_of_def
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2041
  by (metis subsetD pathin_subtopology)
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2042
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2043
lemma path_component_of:
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2044
  "path_component_of X x y \<longleftrightarrow> (\<exists>T. path_connectedin X T \<and> x \<in> T \<and> y \<in> T)"
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2045
  apply (auto simp: path_component_of_def)
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2046
  using path_connectedin_path_image apply fastforce
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2047
  apply (metis path_connectedin)
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2048
  done
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2049
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2050
lemma path_component_of_set:
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2051
   "path_component_of X x y \<longleftrightarrow> (\<exists>g. pathin X g \<and> g 0 = x \<and> g 1 = y)"
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2052
  by (auto simp: path_component_of_def)
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2053
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2054
lemma path_component_of_subset_topspace:
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2055
   "Collect(path_component_of X x) \<subseteq> topspace X"
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2056
  using path_component_in_topspace by fastforce
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2057
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2058
lemma path_component_of_eq_empty:
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2059
   "Collect(path_component_of X x) = {} \<longleftrightarrow> (x \<notin> topspace X)"
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2060
  using path_component_in_topspace path_component_of_refl by fastforce
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2061
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2062
lemma path_connected_space_iff_path_component:
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2063
   "path_connected_space X \<longleftrightarrow> (\<forall>x \<in> topspace X. \<forall>y \<in> topspace X. path_component_of X x y)"
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2064
  by (simp add: path_component_of path_connected_space_subconnected)
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2065
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2066
lemma path_connected_space_imp_path_component_of:
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2067
   "\<lbrakk>path_connected_space X; a \<in> topspace X; b \<in> topspace X\<rbrakk>
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2068
        \<Longrightarrow> path_component_of X a b"
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2069
  by (simp add: path_connected_space_iff_path_component)
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2070
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2071
lemma path_connected_space_path_component_set:
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2072
   "path_connected_space X \<longleftrightarrow> (\<forall>x \<in> topspace X. Collect(path_component_of X x) = topspace X)"
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2073
  using path_component_of_subset_topspace path_connected_space_iff_path_component by fastforce
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2074
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2075
lemma path_component_of_maximal:
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2076
   "\<lbrakk>path_connectedin X s; x \<in> s\<rbrakk> \<Longrightarrow> s \<subseteq> Collect(path_component_of X x)"
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2077
  using path_component_of by fastforce
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2078
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2079
lemma path_component_of_equiv:
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2080
   "path_component_of X x y \<longleftrightarrow> x \<in> topspace X \<and> y \<in> topspace X \<and> path_component_of X x = path_component_of X y"
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2081
    (is "?lhs = ?rhs")
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2082
proof
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2083
  assume ?lhs
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2084
  then show ?rhs
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2085
    apply (simp add: fun_eq_iff path_component_in_topspace)
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2086
    apply (meson path_component_of_sym path_component_of_trans)
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2087
    done
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2088
qed (simp add: path_component_of_refl)
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2089
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2090
lemma path_component_of_disjoint:
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2091
     "disjnt (Collect (path_component_of X x)) (Collect (path_component_of X y)) \<longleftrightarrow>
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2092
      ~(path_component_of X x y)"
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2093
  by (force simp: disjnt_def path_component_of_eq_empty path_component_of_equiv)
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2094
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2095
lemma path_component_of_eq:
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2096
   "path_component_of X x = path_component_of X y \<longleftrightarrow>
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2097
        (x \<notin> topspace X) \<and> (y \<notin> topspace X) \<or>
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2098
        x \<in> topspace X \<and> y \<in> topspace X \<and> path_component_of X x y"
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2099
  by (metis Collect_empty_eq_bot path_component_of_eq_empty path_component_of_equiv)
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2100
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2101
lemma path_connectedin_path_component_of:
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2102
  "path_connectedin X (Collect (path_component_of X x))"
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2103
proof -
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2104
  have "\<And>y. path_component_of X x y
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2105
        \<Longrightarrow> path_component_of (subtopology X (Collect (path_component_of X x))) x y"
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2106
    by (meson path_component_of path_component_of_maximal path_connectedin_subtopology)
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2107
  then show ?thesis
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2108
    apply (simp add: path_connectedin_def path_component_of_subset_topspace path_connected_space_iff_path_component)
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2109
    by (metis Int_absorb1 mem_Collect_eq path_component_of_equiv path_component_of_subset_topspace topspace_subtopology)
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2110
qed
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2111
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2112
lemma Union_path_components_of:
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2113
     "\<Union>(path_components_of X) = topspace X"
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2114
  by (auto simp: path_components_of_def path_component_of_equiv)
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2115
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2116
lemma path_components_of_maximal:
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2117
   "\<lbrakk>C \<in> path_components_of X; path_connectedin X S; ~disjnt C S\<rbrakk> \<Longrightarrow> S \<subseteq> C"
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2118
  apply (auto simp: path_components_of_def path_component_of_equiv)
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2119
  using path_component_of_maximal path_connectedin_def apply fastforce
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2120
  by (meson disjnt_subset2 path_component_of_disjoint path_component_of_equiv path_component_of_maximal)
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2121
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2122
lemma pairwise_disjoint_path_components_of:
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2123
     "pairwise disjnt (path_components_of X)"
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2124
  by (auto simp: path_components_of_def pairwise_def path_component_of_disjoint path_component_of_equiv)
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2125
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2126
lemma complement_path_components_of_Union:
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2127
   "C \<in> path_components_of X
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2128
        \<Longrightarrow> topspace X - C = \<Union>(path_components_of X - {C})"
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2129
  by (metis Diff_cancel Diff_subset Union_path_components_of cSup_singleton diff_Union_pairwise_disjoint insert_subset pairwise_disjoint_path_components_of)
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2130
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2131
lemma nonempty_path_components_of:
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2132
  "C \<in> path_components_of X \<Longrightarrow> (C \<noteq> {})"
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2133
  apply (clarsimp simp: path_components_of_def path_component_of_eq_empty)
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2134
  by (meson path_component_of_refl)
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2135
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2136
lemma path_components_of_subset: "C \<in> path_components_of X \<Longrightarrow> C \<subseteq> topspace X"
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2137
  by (auto simp: path_components_of_def path_component_of_equiv)
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2138
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2139
lemma path_connectedin_path_components_of:
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2140
   "C \<in> path_components_of X \<Longrightarrow> path_connectedin X C"
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2141
  by (auto simp: path_components_of_def path_connectedin_path_component_of)
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2142
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2143
lemma path_component_in_path_components_of:
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2144
  "Collect (path_component_of X a) \<in> path_components_of X \<longleftrightarrow> a \<in> topspace X"
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2145
  apply (rule iffI)
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2146
  using nonempty_path_components_of path_component_of_eq_empty apply fastforce
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2147
  by (simp add: path_components_of_def)
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2148
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2149
lemma path_connectedin_Union:
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2150
  assumes \<A>: "\<And>S. S \<in> \<A> \<Longrightarrow> path_connectedin X S" "\<Inter>\<A> \<noteq> {}"
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2151
  shows "path_connectedin X (\<Union>\<A>)"
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2152
proof -
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2153
  obtain a where "\<And>S. S \<in> \<A> \<Longrightarrow> a \<in> S"
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2154
    using assms by blast
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2155
  then have "\<And>x. x \<in> topspace (subtopology X (\<Union>\<A>)) \<Longrightarrow> path_component_of (subtopology X (\<Union>\<A>)) a x"
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2156
    apply (simp add: topspace_subtopology)
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2157
    by (meson Union_upper \<A> path_component_of path_connectedin_subtopology)
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2158
  then show ?thesis
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2159
    using \<A> unfolding path_connectedin_def
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2160
    by (metis Sup_le_iff path_component_of_equiv path_connected_space_iff_path_component)
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2161
qed
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2162
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2163
lemma path_connectedin_Un:
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2164
   "\<lbrakk>path_connectedin X S; path_connectedin X T; S \<inter> T \<noteq> {}\<rbrakk>
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2165
    \<Longrightarrow> path_connectedin X (S \<union> T)"
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2166
  by (blast intro: path_connectedin_Union [of "{S,T}", simplified])
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2167
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2168
lemma path_connected_space_iff_components_eq:
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2169
  "path_connected_space X \<longleftrightarrow>
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2170
    (\<forall>C \<in> path_components_of X. \<forall>C' \<in> path_components_of X. C = C')"
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2171
  unfolding path_components_of_def
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2172
proof (intro iffI ballI)
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2173
  assume "\<forall>C \<in> path_component_of_set X ` topspace X.
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2174
             \<forall>C' \<in> path_component_of_set X ` topspace X. C = C'"
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2175
  then show "path_connected_space X"
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2176
    using path_component_of_refl path_connected_space_iff_path_component by fastforce
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2177
qed (auto simp: path_connected_space_path_component_set)
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2178
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2179
lemma path_components_of_eq_empty:
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2180
   "path_components_of X = {} \<longleftrightarrow> topspace X = {}"
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2181
  using Union_path_components_of nonempty_path_components_of by fastforce
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2182
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2183
lemma path_components_of_empty_space:
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2184
   "topspace X = {} \<Longrightarrow> path_components_of X = {}"
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2185
  by (simp add: path_components_of_eq_empty)
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2186
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2187
lemma path_components_of_subset_singleton:
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2188
  "path_components_of X \<subseteq> {S} \<longleftrightarrow>
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2189
        path_connected_space X \<and> (topspace X = {} \<or> topspace X = S)"
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2190
proof (cases "topspace X = {}")
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2191
  case True
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2192
  then show ?thesis
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2193
    by (auto simp: path_components_of_empty_space path_connected_space_topspace_empty)
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2194
next
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2195
  case False
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2196
  have "(path_components_of X = {S}) \<longleftrightarrow> (path_connected_space X \<and> topspace X = S)"
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2197
  proof (intro iffI conjI)
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2198
    assume L: "path_components_of X = {S}"
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2199
    then show "path_connected_space X"
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2200
      by (simp add: path_connected_space_iff_components_eq)
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2201
    show "topspace X = S"
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2202
      by (metis L ccpo_Sup_singleton [of S] Union_path_components_of)
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2203
  next
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2204
    assume R: "path_connected_space X \<and> topspace X = S"
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2205
    then show "path_components_of X = {S}"
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2206
      using ccpo_Sup_singleton [of S]
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2207
      by (metis False all_not_in_conv insert_iff mk_disjoint_insert path_component_in_path_components_of path_connected_space_iff_components_eq path_connected_space_path_component_set)
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2208
  qed
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2209
  with False show ?thesis
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2210
    by (simp add: path_components_of_eq_empty subset_singleton_iff)
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2211
qed
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2212
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2213
lemma path_connected_space_iff_components_subset_singleton:
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2214
   "path_connected_space X \<longleftrightarrow> (\<exists>a. path_components_of X \<subseteq> {a})"
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2215
  by (simp add: path_components_of_subset_singleton)
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2216
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2217
lemma path_components_of_eq_singleton:
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2218
   "path_components_of X = {S} \<longleftrightarrow> path_connected_space X \<and> topspace X \<noteq> {} \<and> S = topspace X"
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2219
  by (metis cSup_singleton insert_not_empty path_components_of_subset_singleton subset_singleton_iff)
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2220
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2221
lemma path_components_of_path_connected_space:
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2222
   "path_connected_space X \<Longrightarrow> path_components_of X = (if topspace X = {} then {} else {topspace X})"
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2223
  by (simp add: path_components_of_eq_empty path_components_of_eq_singleton)
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2224
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2225
lemma path_component_subset_connected_component_of:
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2226
   "path_component_of_set X x \<subseteq> connected_component_of_set X x"
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2227
proof (cases "x \<in> topspace X")
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2228
  case True
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2229
  then show ?thesis
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2230
    by (simp add: connected_component_of_maximal path_component_of_refl path_connectedin_imp_connectedin path_connectedin_path_component_of)
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2231
next
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2232
  case False
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2233
  then show ?thesis
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2234
    using path_component_of_eq_empty by fastforce
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2235
qed
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2236
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2237
lemma exists_path_component_of_superset:
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2238
  assumes S: "path_connectedin X S" and ne: "topspace X \<noteq> {}"
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2239
  obtains C where "C \<in> path_components_of X" "S \<subseteq> C"
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2240
proof (cases "S = {}")
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2241
  case True
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2242
  then show ?thesis
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2243
    using ne path_components_of_eq_empty that by fastforce
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2244
next
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2245
  case False
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2246
  then obtain a where "a \<in> S"
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2247
    by blast
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2248
  show ?thesis
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2249
  proof
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2250
    show "Collect (path_component_of X a) \<in> path_components_of X"
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2251
      by (meson \<open>a \<in> S\<close> S subsetD path_component_in_path_components_of path_connectedin_subset_topspace)
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2252
    show "S \<subseteq> Collect (path_component_of X a)"
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2253
      by (simp add: S \<open>a \<in> S\<close> path_component_of_maximal)
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2254
  qed
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2255
qed
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2256
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2257
lemma path_component_of_eq_overlap:
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2258
   "path_component_of X x = path_component_of X y \<longleftrightarrow>
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2259
      (x \<notin> topspace X) \<and> (y \<notin> topspace X) \<or>
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2260
      Collect (path_component_of X x) \<inter> Collect (path_component_of X y) \<noteq> {}"
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2261
  by (metis disjnt_def empty_iff inf_bot_right mem_Collect_eq path_component_of_disjoint path_component_of_eq path_component_of_eq_empty)
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2262
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2263
lemma path_component_of_nonoverlap:
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2264
   "Collect (path_component_of X x) \<inter> Collect (path_component_of X y) = {} \<longleftrightarrow>
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2265
    (x \<notin> topspace X) \<or> (y \<notin> topspace X) \<or>
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2266
    path_component_of X x \<noteq> path_component_of X y"
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2267
  by (metis inf.idem path_component_of_eq_empty path_component_of_eq_overlap)
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2268
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2269
lemma path_component_of_overlap:
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2270
   "Collect (path_component_of X x) \<inter> Collect (path_component_of X y) \<noteq> {} \<longleftrightarrow>
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2271
    x \<in> topspace X \<and> y \<in> topspace X \<and> path_component_of X x = path_component_of X y"
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2272
  by (meson path_component_of_nonoverlap)
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2273
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2274
lemma path_components_of_disjoint:
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2275
     "\<lbrakk>C \<in> path_components_of X; C' \<in> path_components_of X\<rbrakk> \<Longrightarrow> disjnt C C' \<longleftrightarrow> C \<noteq> C'"
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2276
  by (auto simp: path_components_of_def path_component_of_disjoint path_component_of_equiv)
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2277
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2278
lemma path_components_of_overlap:
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2279
    "\<lbrakk>C \<in> path_components_of X; C' \<in> path_components_of X\<rbrakk> \<Longrightarrow> C \<inter> C' \<noteq> {} \<longleftrightarrow> C = C'"
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2280
  by (auto simp: path_components_of_def path_component_of_equiv)
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2281
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2282
lemma path_component_of_unique:
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2283
   "\<lbrakk>x \<in> C; path_connectedin X C; \<And>C'. \<lbrakk>x \<in> C'; path_connectedin X C'\<rbrakk> \<Longrightarrow> C' \<subseteq> C\<rbrakk>
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2284
        \<Longrightarrow> Collect (path_component_of X x) = C"
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2285
  by (meson subsetD eq_iff path_component_of_maximal path_connectedin_path_component_of)
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2286
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2287
lemma path_component_of_discrete_topology [simp]:
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2288
  "Collect (path_component_of (discrete_topology U) x) = (if x \<in> U then {x} else {})"
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2289
proof -
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2290
  have "\<And>C'. \<lbrakk>x \<in> C'; path_connectedin (discrete_topology U) C'\<rbrakk> \<Longrightarrow> C' \<subseteq> {x}"
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2291
    by (metis path_connectedin_discrete_topology subsetD singletonD)
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2292
  then have "x \<in> U \<Longrightarrow> Collect (path_component_of (discrete_topology U) x) = {x}"
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2293
    by (simp add: path_component_of_unique)
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2294
  then show ?thesis
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2295
    using path_component_in_topspace by fastforce
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2296
qed
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2297
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2298
lemma path_component_of_discrete_topology_iff [simp]:
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2299
  "path_component_of (discrete_topology U) x y \<longleftrightarrow> x \<in> U \<and> y=x"
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2300
  by (metis empty_iff insertI1 mem_Collect_eq path_component_of_discrete_topology singletonD)
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2301
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2302
lemma path_components_of_discrete_topology [simp]:
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2303
   "path_components_of (discrete_topology U) = (\<lambda>x. {x}) ` U"
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2304
  by (auto simp: path_components_of_def image_def fun_eq_iff)
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2305
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2306
lemma homeomorphic_map_path_component_of:
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2307
  assumes f: "homeomorphic_map X Y f" and x: "x \<in> topspace X"
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2308
  shows "Collect (path_component_of Y (f x)) = f ` Collect(path_component_of X x)"
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2309
proof -
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2310
  obtain g where g: "homeomorphic_maps X Y f g"
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2311
    using f homeomorphic_map_maps by blast
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2312
  show ?thesis
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2313
  proof
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2314
    have "Collect (path_component_of Y (f x)) \<subseteq> topspace Y"
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2315
      by (simp add: path_component_of_subset_topspace)
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2316
    moreover have "g ` Collect(path_component_of Y (f x)) \<subseteq> Collect (path_component_of X (g (f x)))"
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2317
      using g x unfolding homeomorphic_maps_def
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2318
      by (metis f homeomorphic_imp_surjective_map imageI mem_Collect_eq path_component_of_maximal path_component_of_refl path_connectedin_continuous_map_image path_connectedin_path_component_of)
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2319
    ultimately show "Collect (path_component_of Y (f x)) \<subseteq> f ` Collect (path_component_of X x)"
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2320
      using g x unfolding homeomorphic_maps_def continuous_map_def image_iff subset_iff
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2321
      by metis
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2322
    show "f ` Collect (path_component_of X x) \<subseteq> Collect (path_component_of Y (f x))"
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2323
    proof (rule path_component_of_maximal)
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2324
      show "path_connectedin Y (f ` Collect (path_component_of X x))"
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2325
        by (meson f homeomorphic_map_path_connectedness_eq path_connectedin_path_component_of)
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2326
    qed (simp add: path_component_of_refl x)
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2327
  qed
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2328
qed
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2329
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2330
lemma homeomorphic_map_path_components_of:
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2331
  assumes "homeomorphic_map X Y f"
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2332
  shows "path_components_of Y = (image f) ` (path_components_of X)"
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2333
  unfolding path_components_of_def homeomorphic_imp_surjective_map [OF assms, symmetric]
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2334
  apply safe
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2335
  using assms homeomorphic_map_path_component_of apply fastforce
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2336
  by (metis assms homeomorphic_map_path_component_of imageI)
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2337
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2338
60420
884f54e01427 isabelle update_cartouches;
wenzelm
parents: 60303
diff changeset
  2339
subsection \<open>Sphere is path-connected\<close>
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36583
diff changeset
  2340
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  2341
lemma path_connected_punctured_universe:
37674
f86de9c00c47 convert theorem path_connected_sphere to euclidean_space class
huffman
parents: 37489
diff changeset
  2342
  assumes "2 \<le> DIM('a::euclidean_space)"
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2343
  shows "path_connected (- {a::'a})"
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  2344
proof -
50526
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 49654
diff changeset
  2345
  let ?A = "{x::'a. \<exists>i\<in>Basis. x \<bullet> i < a \<bullet> i}"
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 49654
diff changeset
  2346
  let ?B = "{x::'a. \<exists>i\<in>Basis. a \<bullet> i < x \<bullet> i}"
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  2347
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  2348
  have A: "path_connected ?A"
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  2349
    unfolding Collect_bex_eq
37674
f86de9c00c47 convert theorem path_connected_sphere to euclidean_space class
huffman
parents: 37489
diff changeset
  2350
  proof (rule path_connected_UNION)
50526
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 49654
diff changeset
  2351
    fix i :: 'a
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 49654
diff changeset
  2352
    assume "i \<in> Basis"
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  2353
    then show "(\<Sum>i\<in>Basis. (a \<bullet> i - 1)*\<^sub>R i) \<in> {x::'a. x \<bullet> i < a \<bullet> i}"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  2354
      by simp
50526
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 49654
diff changeset
  2355
    show "path_connected {x. x \<bullet> i < a \<bullet> i}"
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 49654
diff changeset
  2356
      using convex_imp_path_connected [OF convex_halfspace_lt, of i "a \<bullet> i"]
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 49654
diff changeset
  2357
      by (simp add: inner_commute)
37674
f86de9c00c47 convert theorem path_connected_sphere to euclidean_space class
huffman
parents: 37489
diff changeset
  2358
  qed
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  2359
  have B: "path_connected ?B"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  2360
    unfolding Collect_bex_eq
37674
f86de9c00c47 convert theorem path_connected_sphere to euclidean_space class
huffman
parents: 37489
diff changeset
  2361
  proof (rule path_connected_UNION)
50526
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 49654
diff changeset
  2362
    fix i :: 'a
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 49654
diff changeset
  2363
    assume "i \<in> Basis"
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  2364
    then show "(\<Sum>i\<in>Basis. (a \<bullet> i + 1) *\<^sub>R i) \<in> {x::'a. a \<bullet> i < x \<bullet> i}"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  2365
      by simp
50526
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 49654
diff changeset
  2366
    show "path_connected {x. a \<bullet> i < x \<bullet> i}"
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 49654
diff changeset
  2367
      using convex_imp_path_connected [OF convex_halfspace_gt, of "a \<bullet> i" i]
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 49654
diff changeset
  2368
      by (simp add: inner_commute)
37674
f86de9c00c47 convert theorem path_connected_sphere to euclidean_space class
huffman
parents: 37489
diff changeset
  2369
  qed
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  2370
  obtain S :: "'a set" where "S \<subseteq> Basis" and "card S = Suc (Suc 0)"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  2371
    using ex_card[OF assms]
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  2372
    by auto
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  2373
  then obtain b0 b1 :: 'a where "b0 \<in> Basis" and "b1 \<in> Basis" and "b0 \<noteq> b1"
50526
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 49654
diff changeset
  2374
    unfolding card_Suc_eq by auto
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  2375
  then have "a + b0 - b1 \<in> ?A \<inter> ?B"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  2376
    by (auto simp: inner_simps inner_Basis)
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  2377
  then have "?A \<inter> ?B \<noteq> {}"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  2378
    by fast
37674
f86de9c00c47 convert theorem path_connected_sphere to euclidean_space class
huffman
parents: 37489
diff changeset
  2379
  with A B have "path_connected (?A \<union> ?B)"
f86de9c00c47 convert theorem path_connected_sphere to euclidean_space class
huffman
parents: 37489
diff changeset
  2380
    by (rule path_connected_Un)
50526
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 49654
diff changeset
  2381
  also have "?A \<union> ?B = {x. \<exists>i\<in>Basis. x \<bullet> i \<noteq> a \<bullet> i}"
37674
f86de9c00c47 convert theorem path_connected_sphere to euclidean_space class
huffman
parents: 37489
diff changeset
  2382
    unfolding neq_iff bex_disj_distrib Collect_disj_eq ..
f86de9c00c47 convert theorem path_connected_sphere to euclidean_space class
huffman
parents: 37489
diff changeset
  2383
  also have "\<dots> = {x. x \<noteq> a}"
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  2384
    unfolding euclidean_eq_iff [where 'a='a]
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  2385
    by (simp add: Bex_def)
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2386
  also have "\<dots> = - {a}"
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  2387
    by auto
37674
f86de9c00c47 convert theorem path_connected_sphere to euclidean_space class
huffman
parents: 37489
diff changeset
  2388
  finally show ?thesis .
f86de9c00c47 convert theorem path_connected_sphere to euclidean_space class
huffman
parents: 37489
diff changeset
  2389
qed
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  2390
64006
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents: 63978
diff changeset
  2391
corollary connected_punctured_universe:
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents: 63978
diff changeset
  2392
  "2 \<le> DIM('N::euclidean_space) \<Longrightarrow> connected(- {a::'N})"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents: 63978
diff changeset
  2393
  by (simp add: path_connected_punctured_universe path_connected_imp_connected)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents: 63978
diff changeset
  2394
68607
67bb59e49834 make theorem, corollary, and proposition %important for HOL-Analysis manual
immler
parents: 68532
diff changeset
  2395
proposition path_connected_sphere:
64788
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  2396
  fixes a :: "'a :: euclidean_space"
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  2397
  assumes "2 \<le> DIM('a)"
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  2398
  shows "path_connected(sphere a r)"
68607
67bb59e49834 make theorem, corollary, and proposition %important for HOL-Analysis manual
immler
parents: 68532
diff changeset
  2399
proof (cases r "0::real" rule: linorder_cases)
64788
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  2400
  case less
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  2401
  then show ?thesis
64788
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  2402
    by (simp add: path_connected_empty)
37674
f86de9c00c47 convert theorem path_connected_sphere to euclidean_space class
huffman
parents: 37489
diff changeset
  2403
next
64788
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  2404
  case equal
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  2405
  then show ?thesis
64788
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  2406
    by (simp add: path_connected_singleton)
37674
f86de9c00c47 convert theorem path_connected_sphere to euclidean_space class
huffman
parents: 37489
diff changeset
  2407
next
64788
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  2408
  case greater
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  2409
  then have eq: "(sphere (0::'a) r) = (\<lambda>x. (r / norm x) *\<^sub>R x) ` (- {0::'a})"
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  2410
    by (force simp: image_iff split: if_split_asm)
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  2411
  have "continuous_on (- {0::'a}) (\<lambda>x. (r / norm x) *\<^sub>R x)"
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  2412
    by (intro continuous_intros) auto
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  2413
  then have "path_connected ((\<lambda>x. (r / norm x) *\<^sub>R x) ` (- {0::'a}))"
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  2414
    by (intro path_connected_continuous_image path_connected_punctured_universe assms)
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  2415
  with eq have "path_connected (sphere (0::'a) r)"
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  2416
    by auto
67399
eab6ce8368fa ran isabelle update_op on all sources
nipkow
parents: 67240
diff changeset
  2417
  then have "path_connected((+) a ` (sphere (0::'a) r))"
64788
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  2418
    by (simp add: path_connected_translation)
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  2419
  then show ?thesis
64788
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  2420
    by (metis add.right_neutral sphere_translation)
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  2421
qed
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  2422
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  2423
lemma connected_sphere:
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  2424
    fixes a :: "'a :: euclidean_space"
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  2425
    assumes "2 \<le> DIM('a)"
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  2426
      shows "connected(sphere a r)"
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  2427
  using path_connected_sphere [OF assms]
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  2428
  by (simp add: path_connected_imp_connected)
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  2429
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  2430
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2431
corollary path_connected_complement_bounded_convex:
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2432
    fixes s :: "'a :: euclidean_space set"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2433
    assumes "bounded s" "convex s" and 2: "2 \<le> DIM('a)"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2434
    shows "path_connected (- s)"
64788
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  2435
proof (cases "s = {}")
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2436
  case True then show ?thesis
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2437
    using convex_imp_path_connected by auto
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2438
next
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2439
  case False
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2440
  then obtain a where "a \<in> s" by auto
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2441
  { fix x y assume "x \<notin> s" "y \<notin> s"
61808
fc1556774cfe isabelle update_cartouches -c -t;
wenzelm
parents: 61806
diff changeset
  2442
    then have "x \<noteq> a" "y \<noteq> a" using \<open>a \<in> s\<close> by auto
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2443
    then have bxy: "bounded(insert x (insert y s))"
61808
fc1556774cfe isabelle update_cartouches -c -t;
wenzelm
parents: 61806
diff changeset
  2444
      by (simp add: \<open>bounded s\<close>)
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2445
    then obtain B::real where B: "0 < B" and Bx: "norm (a - x) < B" and By: "norm (a - y) < B"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2446
                          and "s \<subseteq> ball a B"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2447
      using bounded_subset_ballD [OF bxy, of a] by (auto simp: dist_norm)
63040
eb4ddd18d635 eliminated old 'def';
wenzelm
parents: 63016
diff changeset
  2448
    define C where "C = B / norm(x - a)"
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2449
    { fix u
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2450
      assume u: "(1 - u) *\<^sub>R x + u *\<^sub>R (a + C *\<^sub>R (x - a)) \<in> s" and "0 \<le> u" "u \<le> 1"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2451
      have CC: "1 \<le> 1 + (C - 1) * u"
61808
fc1556774cfe isabelle update_cartouches -c -t;
wenzelm
parents: 61806
diff changeset
  2452
        using \<open>x \<noteq> a\<close> \<open>0 \<le> u\<close>
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2453
        apply (simp add: C_def divide_simps norm_minus_commute)
61762
d50b993b4fb9 Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents: 61738
diff changeset
  2454
        using Bx by auto
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2455
      have *: "\<And>v. (1 - u) *\<^sub>R x + u *\<^sub>R (a + v *\<^sub>R (x - a)) = a + (1 + (v - 1) * u) *\<^sub>R (x - a)"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2456
        by (simp add: algebra_simps)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2457
      have "a + ((1 / (1 + C * u - u)) *\<^sub>R x + ((u / (1 + C * u - u)) *\<^sub>R a + (C * u / (1 + C * u - u)) *\<^sub>R x)) =
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2458
            (1 + (u / (1 + C * u - u))) *\<^sub>R a + ((1 / (1 + C * u - u)) + (C * u / (1 + C * u - u))) *\<^sub>R x"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2459
        by (simp add: algebra_simps)
68096
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  2460
      also have "\<dots> = (1 + (u / (1 + C * u - u))) *\<^sub>R a + (1 + (u / (1 + C * u - u))) *\<^sub>R x"
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2461
        using CC by (simp add: field_simps)
68096
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  2462
      also have "\<dots> = x + (1 + (u / (1 + C * u - u))) *\<^sub>R a + (u / (1 + C * u - u)) *\<^sub>R x"
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2463
        by (simp add: algebra_simps)
68096
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  2464
      also have "\<dots> = x + ((1 / (1 + C * u - u)) *\<^sub>R a +
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2465
              ((u / (1 + C * u - u)) *\<^sub>R x + (C * u / (1 + C * u - u)) *\<^sub>R a))"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2466
        using CC by (simp add: field_simps) (simp add: add_divide_distrib scaleR_add_left)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2467
      finally have xeq: "(1 - 1 / (1 + (C - 1) * u)) *\<^sub>R a + (1 / (1 + (C - 1) * u)) *\<^sub>R (a + (1 + (C - 1) * u) *\<^sub>R (x - a)) = x"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2468
        by (simp add: algebra_simps)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2469
      have False
61808
fc1556774cfe isabelle update_cartouches -c -t;
wenzelm
parents: 61806
diff changeset
  2470
        using \<open>convex s\<close>
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2471
        apply (simp add: convex_alt)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2472
        apply (drule_tac x=a in bspec)
61808
fc1556774cfe isabelle update_cartouches -c -t;
wenzelm
parents: 61806
diff changeset
  2473
         apply (rule  \<open>a \<in> s\<close>)
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2474
        apply (drule_tac x="a + (1 + (C - 1) * u) *\<^sub>R (x - a)" in bspec)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2475
         using u apply (simp add: *)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2476
        apply (drule_tac x="1 / (1 + (C - 1) * u)" in spec)
61808
fc1556774cfe isabelle update_cartouches -c -t;
wenzelm
parents: 61806
diff changeset
  2477
        using \<open>x \<noteq> a\<close> \<open>x \<notin> s\<close> \<open>0 \<le> u\<close> CC
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2478
        apply (auto simp: xeq)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2479
        done
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2480
    }
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2481
    then have pcx: "path_component (- s) x (a + C *\<^sub>R (x - a))"
69939
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2482
      by (force simp: closed_segment_def intro!: path_component_linepath)
67443
3abf6a722518 standardized towards new-style formal comments: isabelle update_comments;
wenzelm
parents: 67399
diff changeset
  2483
    define D where "D = B / norm(y - a)"  \<comment> \<open>massive duplication with the proof above\<close>
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2484
    { fix u
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2485
      assume u: "(1 - u) *\<^sub>R y + u *\<^sub>R (a + D *\<^sub>R (y - a)) \<in> s" and "0 \<le> u" "u \<le> 1"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2486
      have DD: "1 \<le> 1 + (D - 1) * u"
61808
fc1556774cfe isabelle update_cartouches -c -t;
wenzelm
parents: 61806
diff changeset
  2487
        using \<open>y \<noteq> a\<close> \<open>0 \<le> u\<close>
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2488
        apply (simp add: D_def divide_simps norm_minus_commute)
61762
d50b993b4fb9 Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents: 61738
diff changeset
  2489
        using By by auto
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2490
      have *: "\<And>v. (1 - u) *\<^sub>R y + u *\<^sub>R (a + v *\<^sub>R (y - a)) = a + (1 + (v - 1) * u) *\<^sub>R (y - a)"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2491
        by (simp add: algebra_simps)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2492
      have "a + ((1 / (1 + D * u - u)) *\<^sub>R y + ((u / (1 + D * u - u)) *\<^sub>R a + (D * u / (1 + D * u - u)) *\<^sub>R y)) =
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2493
            (1 + (u / (1 + D * u - u))) *\<^sub>R a + ((1 / (1 + D * u - u)) + (D * u / (1 + D * u - u))) *\<^sub>R y"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2494
        by (simp add: algebra_simps)
68096
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  2495
      also have "\<dots> = (1 + (u / (1 + D * u - u))) *\<^sub>R a + (1 + (u / (1 + D * u - u))) *\<^sub>R y"
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2496
        using DD by (simp add: field_simps)
68096
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  2497
      also have "\<dots> = y + (1 + (u / (1 + D * u - u))) *\<^sub>R a + (u / (1 + D * u - u)) *\<^sub>R y"
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2498
        by (simp add: algebra_simps)
68096
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  2499
      also have "\<dots> = y + ((1 / (1 + D * u - u)) *\<^sub>R a +
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2500
              ((u / (1 + D * u - u)) *\<^sub>R y + (D * u / (1 + D * u - u)) *\<^sub>R a))"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2501
        using DD by (simp add: field_simps) (simp add: add_divide_distrib scaleR_add_left)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2502
      finally have xeq: "(1 - 1 / (1 + (D - 1) * u)) *\<^sub>R a + (1 / (1 + (D - 1) * u)) *\<^sub>R (a + (1 + (D - 1) * u) *\<^sub>R (y - a)) = y"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2503
        by (simp add: algebra_simps)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2504
      have False
61808
fc1556774cfe isabelle update_cartouches -c -t;
wenzelm
parents: 61806
diff changeset
  2505
        using \<open>convex s\<close>
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2506
        apply (simp add: convex_alt)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2507
        apply (drule_tac x=a in bspec)
61808
fc1556774cfe isabelle update_cartouches -c -t;
wenzelm
parents: 61806
diff changeset
  2508
         apply (rule  \<open>a \<in> s\<close>)
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2509
        apply (drule_tac x="a + (1 + (D - 1) * u) *\<^sub>R (y - a)" in bspec)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2510
         using u apply (simp add: *)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2511
        apply (drule_tac x="1 / (1 + (D - 1) * u)" in spec)
61808
fc1556774cfe isabelle update_cartouches -c -t;
wenzelm
parents: 61806
diff changeset
  2512
        using \<open>y \<noteq> a\<close> \<open>y \<notin> s\<close> \<open>0 \<le> u\<close> DD
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2513
        apply (auto simp: xeq)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2514
        done
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2515
    }
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2516
    then have pdy: "path_component (- s) y (a + D *\<^sub>R (y - a))"
69939
812ce526da33 new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents: 69712
diff changeset
  2517
      by (force simp: closed_segment_def intro!: path_component_linepath)
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2518
    have pyx: "path_component (- s) (a + D *\<^sub>R (y - a)) (a + C *\<^sub>R (x - a))"
64788
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  2519
      apply (rule path_component_of_subset [of "sphere a B"])
61808
fc1556774cfe isabelle update_cartouches -c -t;
wenzelm
parents: 61806
diff changeset
  2520
       using \<open>s \<subseteq> ball a B\<close>
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2521
       apply (force simp: ball_def dist_norm norm_minus_commute)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2522
      apply (rule path_connected_sphere [OF 2, of a B, simplified path_connected_component, rule_format])
64788
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  2523
       using \<open>x \<noteq> a\<close>  using \<open>y \<noteq> a\<close>  B apply (auto simp: dist_norm C_def D_def)
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2524
      done
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2525
    have "path_component (- s) x y"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2526
      by (metis path_component_trans path_component_sym pcx pdy pyx)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2527
  }
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2528
  then show ?thesis
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2529
    by (auto simp: path_connected_component)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2530
qed
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2531
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2532
lemma connected_complement_bounded_convex:
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2533
    fixes s :: "'a :: euclidean_space set"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2534
    assumes "bounded s" "convex s" "2 \<le> DIM('a)"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2535
      shows  "connected (- s)"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2536
  using path_connected_complement_bounded_convex [OF assms] path_connected_imp_connected by blast
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2537
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2538
lemma connected_diff_ball:
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2539
    fixes s :: "'a :: euclidean_space set"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2540
    assumes "connected s" "cball a r \<subseteq> s" "2 \<le> DIM('a)"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2541
      shows "connected (s - ball a r)"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2542
  apply (rule connected_diff_open_from_closed [OF ball_subset_cball])
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2543
  using assms connected_sphere
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2544
  apply (auto simp: cball_diff_eq_sphere dist_norm)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2545
  done
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2546
62381
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  2547
proposition connected_open_delete:
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  2548
  assumes "open S" "connected S" and 2: "2 \<le> DIM('N::euclidean_space)"
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  2549
    shows "connected(S - {a::'N})"
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  2550
proof (cases "a \<in> S")
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  2551
  case True
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  2552
  with \<open>open S\<close> obtain \<epsilon> where "\<epsilon> > 0" and \<epsilon>: "cball a \<epsilon> \<subseteq> S"
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  2553
    using open_contains_cball_eq by blast
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  2554
  have "dist a (a + \<epsilon> *\<^sub>R (SOME i. i \<in> Basis)) = \<epsilon>"
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  2555
    by (simp add: dist_norm SOME_Basis \<open>0 < \<epsilon>\<close> less_imp_le)
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  2556
  with \<epsilon> have "\<Inter>{S - ball a r |r. 0 < r \<and> r < \<epsilon>} \<subseteq> {} \<Longrightarrow> False"
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  2557
    apply (drule_tac c="a + scaleR (\<epsilon>) ((SOME i. i \<in> Basis))" in subsetD)
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  2558
    by auto
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  2559
  then have nonemp: "(\<Inter>{S - ball a r |r. 0 < r \<and> r < \<epsilon>}) = {} \<Longrightarrow> False"
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  2560
    by auto
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  2561
  have con: "\<And>r. r < \<epsilon> \<Longrightarrow> connected (S - ball a r)"
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  2562
    using \<epsilon> by (force intro: connected_diff_ball [OF \<open>connected S\<close> _ 2])
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  2563
  have "x \<in> \<Union>{S - ball a r |r. 0 < r \<and> r < \<epsilon>}" if "x \<in> S - {a}" for x
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  2564
    apply (rule UnionI [of "S - ball a (min \<epsilon> (dist a x) / 2)"])
68096
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  2565
     using that \<open>0 < \<epsilon>\<close> apply simp_all
62381
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  2566
    apply (rule_tac x="min \<epsilon> (dist a x) / 2" in exI)
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  2567
    apply auto
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  2568
    done
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  2569
  then have "S - {a} = \<Union>{S - ball a r | r. 0 < r \<and> r < \<epsilon>}"
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  2570
    by auto
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  2571
  then show ?thesis
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  2572
    by (auto intro: connected_Union con dest!: nonemp)
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  2573
next
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  2574
  case False then show ?thesis
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  2575
    by (simp add: \<open>connected S\<close>)
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  2576
qed
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  2577
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  2578
corollary path_connected_open_delete:
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  2579
  assumes "open S" "connected S" and 2: "2 \<le> DIM('N::euclidean_space)"
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  2580
    shows "path_connected(S - {a::'N})"
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  2581
by (simp add: assms connected_open_delete connected_open_path_connected open_delete)
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  2582
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  2583
corollary path_connected_punctured_ball:
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  2584
   "2 \<le> DIM('N::euclidean_space) \<Longrightarrow> path_connected(ball a r - {a::'N})"
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  2585
by (simp add: path_connected_open_delete)
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  2586
63151
82df5181d699 updated proof of Residue Theorem (form Wenda Li)
paulson <lp15@cam.ac.uk>
parents: 63126
diff changeset
  2587
corollary connected_punctured_ball:
62381
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  2588
   "2 \<le> DIM('N::euclidean_space) \<Longrightarrow> connected(ball a r - {a::'N})"
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  2589
by (simp add: connected_open_delete)
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  2590
63151
82df5181d699 updated proof of Residue Theorem (form Wenda Li)
paulson <lp15@cam.ac.uk>
parents: 63126
diff changeset
  2591
corollary connected_open_delete_finite:
82df5181d699 updated proof of Residue Theorem (form Wenda Li)
paulson <lp15@cam.ac.uk>
parents: 63126
diff changeset
  2592
  fixes S T::"'a::euclidean_space set"
82df5181d699 updated proof of Residue Theorem (form Wenda Li)
paulson <lp15@cam.ac.uk>
parents: 63126
diff changeset
  2593
  assumes S: "open S" "connected S" and 2: "2 \<le> DIM('a)" and "finite T"
63594
bd218a9320b5 HOL-Multivariate_Analysis: rename theories for more descriptive names
hoelzl
parents: 63540
diff changeset
  2594
  shows "connected(S - T)"
bd218a9320b5 HOL-Multivariate_Analysis: rename theories for more descriptive names
hoelzl
parents: 63540
diff changeset
  2595
  using \<open>finite T\<close> S
63151
82df5181d699 updated proof of Residue Theorem (form Wenda Li)
paulson <lp15@cam.ac.uk>
parents: 63126
diff changeset
  2596
proof (induct T)
82df5181d699 updated proof of Residue Theorem (form Wenda Li)
paulson <lp15@cam.ac.uk>
parents: 63126
diff changeset
  2597
  case empty
82df5181d699 updated proof of Residue Theorem (form Wenda Li)
paulson <lp15@cam.ac.uk>
parents: 63126
diff changeset
  2598
  show ?case using \<open>connected S\<close> by simp
82df5181d699 updated proof of Residue Theorem (form Wenda Li)
paulson <lp15@cam.ac.uk>
parents: 63126
diff changeset
  2599
next
82df5181d699 updated proof of Residue Theorem (form Wenda Li)
paulson <lp15@cam.ac.uk>
parents: 63126
diff changeset
  2600
  case (insert x F)
82df5181d699 updated proof of Residue Theorem (form Wenda Li)
paulson <lp15@cam.ac.uk>
parents: 63126
diff changeset
  2601
  then have "connected (S-F)" by auto
82df5181d699 updated proof of Residue Theorem (form Wenda Li)
paulson <lp15@cam.ac.uk>
parents: 63126
diff changeset
  2602
  moreover have "open (S - F)" using finite_imp_closed[OF \<open>finite F\<close>] \<open>open S\<close> by auto
82df5181d699 updated proof of Residue Theorem (form Wenda Li)
paulson <lp15@cam.ac.uk>
parents: 63126
diff changeset
  2603
  ultimately have "connected (S - F - {x})" using connected_open_delete[OF _ _ 2] by auto
82df5181d699 updated proof of Residue Theorem (form Wenda Li)
paulson <lp15@cam.ac.uk>
parents: 63126
diff changeset
  2604
  thus ?case by (metis Diff_insert)
82df5181d699 updated proof of Residue Theorem (form Wenda Li)
paulson <lp15@cam.ac.uk>
parents: 63126
diff changeset
  2605
qed
82df5181d699 updated proof of Residue Theorem (form Wenda Li)
paulson <lp15@cam.ac.uk>
parents: 63126
diff changeset
  2606
64788
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  2607
lemma sphere_1D_doubleton_zero:
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  2608
  assumes 1: "DIM('a) = 1" and "r > 0"
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  2609
  obtains x y::"'a::euclidean_space"
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  2610
    where "sphere 0 r = {x,y} \<and> dist x y = 2*r"
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  2611
proof -
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  2612
  obtain b::'a where b: "Basis = {b}"
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  2613
    using 1 card_1_singletonE by blast
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  2614
  show ?thesis
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  2615
  proof (intro that conjI)
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  2616
    have "x = norm x *\<^sub>R b \<or> x = - norm x *\<^sub>R b" if "r = norm x" for x
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  2617
    proof -
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  2618
      have xb: "(x \<bullet> b) *\<^sub>R b = x"
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  2619
        using euclidean_representation [of x, unfolded b] by force
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  2620
      then have "norm ((x \<bullet> b) *\<^sub>R b) = norm x"
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  2621
        by simp
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  2622
      with b have "\<bar>x \<bullet> b\<bar> = norm x"
68310
d0a7ddf5450e more general tidying
paulson <lp15@cam.ac.uk>
parents: 68296
diff changeset
  2623
        using norm_Basis by (simp add: b)
64788
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  2624
      with xb show ?thesis
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  2625
        apply (simp add: abs_if split: if_split_asm)
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  2626
        apply (metis add.inverse_inverse real_vector.scale_minus_left xb)
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  2627
        done
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  2628
    qed
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  2629
    with \<open>r > 0\<close> b show "sphere 0 r = {r *\<^sub>R b, - r *\<^sub>R b}"
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  2630
      by (force simp: sphere_def dist_norm)
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  2631
    have "dist (r *\<^sub>R b) (- r *\<^sub>R b) = norm (r *\<^sub>R b + r *\<^sub>R b)"
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  2632
      by (simp add: dist_norm)
68096
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  2633
    also have "\<dots> = norm ((2*r) *\<^sub>R b)"
64788
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  2634
      by (metis mult_2 scaleR_add_left)
68096
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  2635
    also have "\<dots> = 2*r"
64788
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  2636
      using \<open>r > 0\<close> b norm_Basis by fastforce
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  2637
    finally show "dist (r *\<^sub>R b) (- r *\<^sub>R b) = 2*r" .
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  2638
  qed
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  2639
qed
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  2640
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  2641
lemma sphere_1D_doubleton:
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  2642
  fixes a :: "'a :: euclidean_space"
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  2643
  assumes "DIM('a) = 1" and "r > 0"
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  2644
  obtains x y where "sphere a r = {x,y} \<and> dist x y = 2*r"
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  2645
proof -
67399
eab6ce8368fa ran isabelle update_op on all sources
nipkow
parents: 67240
diff changeset
  2646
  have "sphere a r = (+) a ` sphere 0 r"
64788
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  2647
    by (metis add.right_neutral sphere_translation)
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  2648
  then show ?thesis
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  2649
    using sphere_1D_doubleton_zero [OF assms]
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  2650
    by (metis (mono_tags, lifting) dist_add_cancel image_empty image_insert that)
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  2651
qed
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  2652
64006
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents: 63978
diff changeset
  2653
lemma psubset_sphere_Compl_connected:
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents: 63978
diff changeset
  2654
  fixes S :: "'a::euclidean_space set"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents: 63978
diff changeset
  2655
  assumes S: "S \<subset> sphere a r" and "0 < r" and 2: "2 \<le> DIM('a)"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents: 63978
diff changeset
  2656
  shows "connected(- S)"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents: 63978
diff changeset
  2657
proof -
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents: 63978
diff changeset
  2658
  have "S \<subseteq> sphere a r"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents: 63978
diff changeset
  2659
    using S by blast
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents: 63978
diff changeset
  2660
  obtain b where "dist a b = r" and "b \<notin> S"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents: 63978
diff changeset
  2661
    using S mem_sphere by blast
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents: 63978
diff changeset
  2662
  have CS: "- S = {x. dist a x \<le> r \<and> (x \<notin> S)} \<union> {x. r \<le> dist a x \<and> (x \<notin> S)}"
68096
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  2663
    by auto
64006
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents: 63978
diff changeset
  2664
  have "{x. dist a x \<le> r \<and> x \<notin> S} \<inter> {x. r \<le> dist a x \<and> x \<notin> S} \<noteq> {}"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents: 63978
diff changeset
  2665
    using \<open>b \<notin> S\<close> \<open>dist a b = r\<close> by blast
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents: 63978
diff changeset
  2666
  moreover have "connected {x. dist a x \<le> r \<and> x \<notin> S}"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents: 63978
diff changeset
  2667
    apply (rule connected_intermediate_closure [of "ball a r"])
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents: 63978
diff changeset
  2668
    using assms by auto
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents: 63978
diff changeset
  2669
  moreover
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents: 63978
diff changeset
  2670
  have "connected {x. r \<le> dist a x \<and> x \<notin> S}"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents: 63978
diff changeset
  2671
    apply (rule connected_intermediate_closure [of "- cball a r"])
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents: 63978
diff changeset
  2672
    using assms apply (auto intro: connected_complement_bounded_convex)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents: 63978
diff changeset
  2673
    apply (metis ComplI interior_cball interior_closure mem_ball not_less)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents: 63978
diff changeset
  2674
    done
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents: 63978
diff changeset
  2675
  ultimately show ?thesis
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents: 63978
diff changeset
  2676
    by (simp add: CS connected_Un)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents: 63978
diff changeset
  2677
qed
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents: 63978
diff changeset
  2678
64788
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  2679
66793
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66456
diff changeset
  2680
subsection\<open>Every annulus is a connected set\<close>
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66456
diff changeset
  2681
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66456
diff changeset
  2682
lemma path_connected_2DIM_I:
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66456
diff changeset
  2683
  fixes a :: "'N::euclidean_space"
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66456
diff changeset
  2684
  assumes 2: "2 \<le> DIM('N)" and pc: "path_connected {r. 0 \<le> r \<and> P r}"
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66456
diff changeset
  2685
  shows "path_connected {x. P(norm(x - a))}"
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66456
diff changeset
  2686
proof -
67399
eab6ce8368fa ran isabelle update_op on all sources
nipkow
parents: 67240
diff changeset
  2687
  have "{x. P(norm(x - a))} = (+) a ` {x. P(norm x)}"
66793
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66456
diff changeset
  2688
    by force
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66456
diff changeset
  2689
  moreover have "path_connected {x::'N. P(norm x)}"
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66456
diff changeset
  2690
  proof -
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66456
diff changeset
  2691
    let ?D = "{x. 0 \<le> x \<and> P x} \<times> sphere (0::'N) 1"
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66456
diff changeset
  2692
    have "x \<in> (\<lambda>z. fst z *\<^sub>R snd z) ` ?D"
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66456
diff changeset
  2693
      if "P (norm x)" for x::'N
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66456
diff changeset
  2694
    proof (cases "x=0")
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66456
diff changeset
  2695
      case True
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66456
diff changeset
  2696
      with that show ?thesis
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66456
diff changeset
  2697
        apply (simp add: image_iff)
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66456
diff changeset
  2698
        apply (rule_tac x=0 in exI, simp)
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66456
diff changeset
  2699
        using vector_choose_size zero_le_one by blast
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66456
diff changeset
  2700
    next
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66456
diff changeset
  2701
      case False
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66456
diff changeset
  2702
      with that show ?thesis
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66456
diff changeset
  2703
        by (rule_tac x="(norm x, x /\<^sub>R norm x)" in image_eqI) auto
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66456
diff changeset
  2704
    qed
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66456
diff changeset
  2705
    then have *: "{x::'N. P(norm x)} =  (\<lambda>z. fst z *\<^sub>R snd z) ` ?D"
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66456
diff changeset
  2706
      by auto
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66456
diff changeset
  2707
    have "continuous_on ?D (\<lambda>z:: real\<times>'N. fst z *\<^sub>R snd z)"
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66456
diff changeset
  2708
      by (intro continuous_intros)
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66456
diff changeset
  2709
    moreover have "path_connected ?D"
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66456
diff changeset
  2710
      by (metis path_connected_Times [OF pc] path_connected_sphere 2)
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66456
diff changeset
  2711
    ultimately show ?thesis
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66456
diff changeset
  2712
      apply (subst *)
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66456
diff changeset
  2713
      apply (rule path_connected_continuous_image, auto)
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66456
diff changeset
  2714
      done
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66456
diff changeset
  2715
  qed
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66456
diff changeset
  2716
  ultimately show ?thesis
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66456
diff changeset
  2717
    using path_connected_translation by metis
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66456
diff changeset
  2718
qed
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66456
diff changeset
  2719
68607
67bb59e49834 make theorem, corollary, and proposition %important for HOL-Analysis manual
immler
parents: 68532
diff changeset
  2720
proposition path_connected_annulus:
66793
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66456
diff changeset
  2721
  fixes a :: "'N::euclidean_space"
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66456
diff changeset
  2722
  assumes "2 \<le> DIM('N)"
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66456
diff changeset
  2723
  shows "path_connected {x. r1 < norm(x - a) \<and> norm(x - a) < r2}"
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66456
diff changeset
  2724
        "path_connected {x. r1 < norm(x - a) \<and> norm(x - a) \<le> r2}"
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66456
diff changeset
  2725
        "path_connected {x. r1 \<le> norm(x - a) \<and> norm(x - a) < r2}"
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66456
diff changeset
  2726
        "path_connected {x. r1 \<le> norm(x - a) \<and> norm(x - a) \<le> r2}"
68607
67bb59e49834 make theorem, corollary, and proposition %important for HOL-Analysis manual
immler
parents: 68532
diff changeset
  2727
  by (auto simp: is_interval_def intro!: is_interval_convex convex_imp_path_connected path_connected_2DIM_I [OF assms])
67bb59e49834 make theorem, corollary, and proposition %important for HOL-Analysis manual
immler
parents: 68532
diff changeset
  2728
67bb59e49834 make theorem, corollary, and proposition %important for HOL-Analysis manual
immler
parents: 68532
diff changeset
  2729
proposition connected_annulus:
66793
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66456
diff changeset
  2730
  fixes a :: "'N::euclidean_space"
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66456
diff changeset
  2731
  assumes "2 \<le> DIM('N::euclidean_space)"
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66456
diff changeset
  2732
  shows "connected {x. r1 < norm(x - a) \<and> norm(x - a) < r2}"
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66456
diff changeset
  2733
        "connected {x. r1 < norm(x - a) \<and> norm(x - a) \<le> r2}"
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66456
diff changeset
  2734
        "connected {x. r1 \<le> norm(x - a) \<and> norm(x - a) < r2}"
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66456
diff changeset
  2735
        "connected {x. r1 \<le> norm(x - a) \<and> norm(x - a) \<le> r2}"
68607
67bb59e49834 make theorem, corollary, and proposition %important for HOL-Analysis manual
immler
parents: 68532
diff changeset
  2736
  by (auto simp: path_connected_annulus [OF assms] path_connected_imp_connected)
67962
0acdcd8f4ba1 a first shot at tagging for HOL-Analysis manual
immler
parents: 67443
diff changeset
  2737
0acdcd8f4ba1 a first shot at tagging for HOL-Analysis manual
immler
parents: 67443
diff changeset
  2738
70136
f03a01a18c6e modernized tags: default scope excludes proof;
wenzelm
parents: 69986
diff changeset
  2739
subsection\<^marker>\<open>tag unimportant\<close>\<open>Relations between components and path components\<close>
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2740
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2741
lemma open_connected_component:
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2742
  fixes s :: "'a::real_normed_vector set"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2743
  shows "open s \<Longrightarrow> open (connected_component_set s x)"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2744
    apply (simp add: open_contains_ball, clarify)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2745
    apply (rename_tac y)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2746
    apply (drule_tac x=y in bspec)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2747
     apply (simp add: connected_component_in, clarify)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2748
    apply (rule_tac x=e in exI)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2749
    by (metis mem_Collect_eq connected_component_eq connected_component_maximal centre_in_ball connected_ball)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2750
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2751
corollary open_components:
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2752
    fixes s :: "'a::real_normed_vector set"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2753
    shows "\<lbrakk>open u; s \<in> components u\<rbrakk> \<Longrightarrow> open s"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2754
  by (simp add: components_iff) (metis open_connected_component)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2755
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2756
lemma in_closure_connected_component:
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2757
  fixes s :: "'a::real_normed_vector set"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2758
  assumes x: "x \<in> s" and s: "open s"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2759
  shows "x \<in> closure (connected_component_set s y) \<longleftrightarrow>  x \<in> connected_component_set s y"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2760
proof -
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2761
  { assume "x \<in> closure (connected_component_set s y)"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2762
    moreover have "x \<in> connected_component_set s x"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2763
      using x by simp
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2764
    ultimately have "x \<in> connected_component_set s y"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2765
      using s by (meson Compl_disjoint closure_iff_nhds_not_empty connected_component_disjoint disjoint_eq_subset_Compl open_connected_component)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2766
  }
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2767
  then show ?thesis
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2768
    by (auto simp: closure_def)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2769
qed
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2770
63114
27afe7af7379 Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 63092
diff changeset
  2771
lemma connected_disjoint_Union_open_pick:
27afe7af7379 Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 63092
diff changeset
  2772
  assumes "pairwise disjnt B"
27afe7af7379 Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 63092
diff changeset
  2773
          "\<And>S. S \<in> A \<Longrightarrow> connected S \<and> S \<noteq> {}"
27afe7af7379 Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 63092
diff changeset
  2774
          "\<And>S. S \<in> B \<Longrightarrow> open S"
27afe7af7379 Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 63092
diff changeset
  2775
          "\<Union>A \<subseteq> \<Union>B"
27afe7af7379 Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 63092
diff changeset
  2776
          "S \<in> A"
27afe7af7379 Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 63092
diff changeset
  2777
  obtains T where "T \<in> B" "S \<subseteq> T" "S \<inter> \<Union>(B - {T}) = {}"
27afe7af7379 Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 63092
diff changeset
  2778
proof -
27afe7af7379 Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 63092
diff changeset
  2779
  have "S \<subseteq> \<Union>B" "connected S" "S \<noteq> {}"
27afe7af7379 Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 63092
diff changeset
  2780
    using assms \<open>S \<in> A\<close> by blast+
27afe7af7379 Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 63092
diff changeset
  2781
  then obtain T where "T \<in> B" "S \<inter> T \<noteq> {}"
27afe7af7379 Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 63092
diff changeset
  2782
    by (metis Sup_inf_eq_bot_iff inf.absorb_iff2 inf_commute)
27afe7af7379 Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 63092
diff changeset
  2783
  have 1: "open T" by (simp add: \<open>T \<in> B\<close> assms)
27afe7af7379 Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 63092
diff changeset
  2784
  have 2: "open (\<Union>(B-{T}))" using assms by blast
27afe7af7379 Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 63092
diff changeset
  2785
  have 3: "S \<subseteq> T \<union> \<Union>(B - {T})" using \<open>S \<subseteq> \<Union>B\<close> by blast
27afe7af7379 Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 63092
diff changeset
  2786
  have "T \<inter> \<Union>(B - {T}) = {}" using \<open>T \<in> B\<close> \<open>pairwise disjnt B\<close>
27afe7af7379 Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 63092
diff changeset
  2787
    by (auto simp: pairwise_def disjnt_def)
27afe7af7379 Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 63092
diff changeset
  2788
  then have 4: "T \<inter> \<Union>(B - {T}) \<inter> S = {}" by auto
27afe7af7379 Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 63092
diff changeset
  2789
  from connectedD [OF \<open>connected S\<close> 1 2 3 4]
27afe7af7379 Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 63092
diff changeset
  2790
  have "S \<inter> \<Union>(B-{T}) = {}"
27afe7af7379 Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 63092
diff changeset
  2791
    by (auto simp: Int_commute \<open>S \<inter> T \<noteq> {}\<close>)
27afe7af7379 Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 63092
diff changeset
  2792
  with \<open>T \<in> B\<close> have "S \<subseteq> T"
27afe7af7379 Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 63092
diff changeset
  2793
    using "3" by auto
27afe7af7379 Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 63092
diff changeset
  2794
  show ?thesis
27afe7af7379 Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 63092
diff changeset
  2795
    using \<open>S \<inter> \<Union>(B - {T}) = {}\<close> \<open>S \<subseteq> T\<close> \<open>T \<in> B\<close> that by auto
27afe7af7379 Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 63092
diff changeset
  2796
qed
27afe7af7379 Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 63092
diff changeset
  2797
27afe7af7379 Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 63092
diff changeset
  2798
lemma connected_disjoint_Union_open_subset:
27afe7af7379 Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 63092
diff changeset
  2799
  assumes A: "pairwise disjnt A" and B: "pairwise disjnt B"
27afe7af7379 Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 63092
diff changeset
  2800
      and SA: "\<And>S. S \<in> A \<Longrightarrow> open S \<and> connected S \<and> S \<noteq> {}"
27afe7af7379 Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 63092
diff changeset
  2801
      and SB: "\<And>S. S \<in> B \<Longrightarrow> open S \<and> connected S \<and> S \<noteq> {}"
27afe7af7379 Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 63092
diff changeset
  2802
      and eq [simp]: "\<Union>A = \<Union>B"
27afe7af7379 Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 63092
diff changeset
  2803
    shows "A \<subseteq> B"
27afe7af7379 Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 63092
diff changeset
  2804
proof
27afe7af7379 Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 63092
diff changeset
  2805
  fix S
27afe7af7379 Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 63092
diff changeset
  2806
  assume "S \<in> A"
27afe7af7379 Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 63092
diff changeset
  2807
  obtain T where "T \<in> B" "S \<subseteq> T" "S \<inter> \<Union>(B - {T}) = {}"
27afe7af7379 Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 63092
diff changeset
  2808
      apply (rule connected_disjoint_Union_open_pick [OF B, of A])
27afe7af7379 Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 63092
diff changeset
  2809
      using SA SB \<open>S \<in> A\<close> by auto
27afe7af7379 Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 63092
diff changeset
  2810
  moreover obtain S' where "S' \<in> A" "T \<subseteq> S'" "T \<inter> \<Union>(A - {S'}) = {}"
27afe7af7379 Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 63092
diff changeset
  2811
      apply (rule connected_disjoint_Union_open_pick [OF A, of B])
27afe7af7379 Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 63092
diff changeset
  2812
      using SA SB \<open>T \<in> B\<close> by auto
27afe7af7379 Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 63092
diff changeset
  2813
  ultimately have "S' = S"
27afe7af7379 Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 63092
diff changeset
  2814
    by (metis A Int_subset_iff SA \<open>S \<in> A\<close> disjnt_def inf.orderE pairwise_def)
27afe7af7379 Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 63092
diff changeset
  2815
  with \<open>T \<subseteq> S'\<close> have "T \<subseteq> S" by simp
27afe7af7379 Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 63092
diff changeset
  2816
  with \<open>S \<subseteq> T\<close> have "S = T" by blast
27afe7af7379 Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 63092
diff changeset
  2817
  with \<open>T \<in> B\<close> show "S \<in> B" by simp
27afe7af7379 Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 63092
diff changeset
  2818
qed
27afe7af7379 Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 63092
diff changeset
  2819
27afe7af7379 Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 63092
diff changeset
  2820
lemma connected_disjoint_Union_open_unique:
27afe7af7379 Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 63092
diff changeset
  2821
  assumes A: "pairwise disjnt A" and B: "pairwise disjnt B"
27afe7af7379 Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 63092
diff changeset
  2822
      and SA: "\<And>S. S \<in> A \<Longrightarrow> open S \<and> connected S \<and> S \<noteq> {}"
27afe7af7379 Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 63092
diff changeset
  2823
      and SB: "\<And>S. S \<in> B \<Longrightarrow> open S \<and> connected S \<and> S \<noteq> {}"
27afe7af7379 Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 63092
diff changeset
  2824
      and eq [simp]: "\<Union>A = \<Union>B"
27afe7af7379 Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 63092
diff changeset
  2825
    shows "A = B"
27afe7af7379 Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 63092
diff changeset
  2826
by (rule subset_antisym; metis connected_disjoint_Union_open_subset assms)
27afe7af7379 Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 63092
diff changeset
  2827
27afe7af7379 Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 63092
diff changeset
  2828
proposition components_open_unique:
27afe7af7379 Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 63092
diff changeset
  2829
 fixes S :: "'a::real_normed_vector set"
27afe7af7379 Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 63092
diff changeset
  2830
  assumes "pairwise disjnt A" "\<Union>A = S"
27afe7af7379 Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 63092
diff changeset
  2831
          "\<And>X. X \<in> A \<Longrightarrow> open X \<and> connected X \<and> X \<noteq> {}"
27afe7af7379 Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 63092
diff changeset
  2832
    shows "components S = A"
27afe7af7379 Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 63092
diff changeset
  2833
proof -
27afe7af7379 Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 63092
diff changeset
  2834
  have "open S" using assms by blast
27afe7af7379 Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 63092
diff changeset
  2835
  show ?thesis
27afe7af7379 Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 63092
diff changeset
  2836
    apply (rule connected_disjoint_Union_open_unique)
27afe7af7379 Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 63092
diff changeset
  2837
    apply (simp add: components_eq disjnt_def pairwise_def)
27afe7af7379 Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 63092
diff changeset
  2838
    using \<open>open S\<close>
27afe7af7379 Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 63092
diff changeset
  2839
    apply (simp_all add: assms open_components in_components_connected in_components_nonempty)
27afe7af7379 Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 63092
diff changeset
  2840
    done
27afe7af7379 Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 63092
diff changeset
  2841
qed
27afe7af7379 Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 63092
diff changeset
  2842
27afe7af7379 Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 63092
diff changeset
  2843
70136
f03a01a18c6e modernized tags: default scope excludes proof;
wenzelm
parents: 69986
diff changeset
  2844
subsection\<^marker>\<open>tag unimportant\<close>\<open>Existence of unbounded components\<close>
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2845
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2846
lemma cobounded_unbounded_component:
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2847
    fixes s :: "'a :: euclidean_space set"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2848
    assumes "bounded (-s)"
69508
2a4c8a2a3f8e tuned headers; ~ -> \<not>
nipkow
parents: 69313
diff changeset
  2849
      shows "\<exists>x. x \<in> s \<and> \<not> bounded (connected_component_set s x)"
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2850
proof -
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2851
  obtain i::'a where i: "i \<in> Basis"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2852
    using nonempty_Basis by blast
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2853
  obtain B where B: "B>0" "-s \<subseteq> ball 0 B"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2854
    using bounded_subset_ballD [OF assms, of 0] by auto
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2855
  then have *: "\<And>x. B \<le> norm x \<Longrightarrow> x \<in> s"
68096
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  2856
    by (force simp: ball_def dist_norm)
69508
2a4c8a2a3f8e tuned headers; ~ -> \<not>
nipkow
parents: 69313
diff changeset
  2857
  have unbounded_inner: "\<not> bounded {x. inner i x \<ge> B}"
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2858
    apply (auto simp: bounded_def dist_norm)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2859
    apply (rule_tac x="x + (max B e + 1 + \<bar>i \<bullet> x\<bar>) *\<^sub>R i" in exI)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2860
    apply simp
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2861
    using i
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2862
    apply (auto simp: algebra_simps)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2863
    done
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2864
  have **: "{x. B \<le> i \<bullet> x} \<subseteq> connected_component_set s (B *\<^sub>R i)"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2865
    apply (rule connected_component_maximal)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2866
    apply (auto simp: i intro: convex_connected convex_halfspace_ge [of B])
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2867
    apply (rule *)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2868
    apply (rule order_trans [OF _ Basis_le_norm [OF i]])
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2869
    by (simp add: inner_commute)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2870
  have "B *\<^sub>R i \<in> s"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2871
    by (rule *) (simp add: norm_Basis [OF i])
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2872
  then show ?thesis
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2873
    apply (rule_tac x="B *\<^sub>R i" in exI, clarify)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2874
    apply (frule bounded_subset [of _ "{x. B \<le> i \<bullet> x}", OF _ **])
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2875
    using unbounded_inner apply blast
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2876
    done
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2877
qed
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2878
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2879
lemma cobounded_unique_unbounded_component:
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2880
    fixes s :: "'a :: euclidean_space set"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2881
    assumes bs: "bounded (-s)" and "2 \<le> DIM('a)"
69508
2a4c8a2a3f8e tuned headers; ~ -> \<not>
nipkow
parents: 69313
diff changeset
  2882
        and bo: "\<not> bounded(connected_component_set s x)"
2a4c8a2a3f8e tuned headers; ~ -> \<not>
nipkow
parents: 69313
diff changeset
  2883
                "\<not> bounded(connected_component_set s y)"
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2884
      shows "connected_component_set s x = connected_component_set s y"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2885
proof -
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2886
  obtain i::'a where i: "i \<in> Basis"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2887
    using nonempty_Basis by blast
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2888
  obtain B where B: "B>0" "-s \<subseteq> ball 0 B"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2889
    using bounded_subset_ballD [OF bs, of 0] by auto
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2890
  then have *: "\<And>x. B \<le> norm x \<Longrightarrow> x \<in> s"
68096
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  2891
    by (force simp: ball_def dist_norm)
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2892
  have ccb: "connected (- ball 0 B :: 'a set)"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2893
    using assms by (auto intro: connected_complement_bounded_convex)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2894
  obtain x' where x': "connected_component s x x'" "norm x' > B"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2895
    using bo [unfolded bounded_def dist_norm, simplified, rule_format]
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2896
    by (metis diff_zero norm_minus_commute not_less)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2897
  obtain y' where y': "connected_component s y y'" "norm y' > B"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2898
    using bo [unfolded bounded_def dist_norm, simplified, rule_format]
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2899
    by (metis diff_zero norm_minus_commute not_less)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2900
  have x'y': "connected_component s x' y'"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2901
    apply (simp add: connected_component_def)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2902
    apply (rule_tac x="- ball 0 B" in exI)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2903
    using x' y'
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2904
    apply (auto simp: ccb dist_norm *)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2905
    done
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2906
  show ?thesis
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2907
    apply (rule connected_component_eq)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2908
    using x' y' x'y'
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2909
    by (metis (no_types, lifting) connected_component_eq_empty connected_component_eq_eq connected_component_idemp connected_component_in)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2910
qed
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2911
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2912
lemma cobounded_unbounded_components:
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2913
    fixes s :: "'a :: euclidean_space set"
69508
2a4c8a2a3f8e tuned headers; ~ -> \<not>
nipkow
parents: 69313
diff changeset
  2914
    shows "bounded (-s) \<Longrightarrow> \<exists>c. c \<in> components s \<and> \<not>bounded c"
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2915
  by (metis cobounded_unbounded_component components_def imageI)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2916
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2917
lemma cobounded_unique_unbounded_components:
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2918
    fixes s :: "'a :: euclidean_space set"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2919
    shows  "\<lbrakk>bounded (- s); c \<in> components s; \<not> bounded c; c' \<in> components s; \<not> bounded c'; 2 \<le> DIM('a)\<rbrakk> \<Longrightarrow> c' = c"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2920
  unfolding components_iff
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2921
  by (metis cobounded_unique_unbounded_component)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2922
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2923
lemma cobounded_has_bounded_component:
64122
74fde524799e invariance of domain
paulson <lp15@cam.ac.uk>
parents: 64006
diff changeset
  2924
  fixes S :: "'a :: euclidean_space set"
74fde524799e invariance of domain
paulson <lp15@cam.ac.uk>
parents: 64006
diff changeset
  2925
  assumes "bounded (- S)" "\<not> connected S" "2 \<le> DIM('a)"
74fde524799e invariance of domain
paulson <lp15@cam.ac.uk>
parents: 64006
diff changeset
  2926
  obtains C where "C \<in> components S" "bounded C"
74fde524799e invariance of domain
paulson <lp15@cam.ac.uk>
parents: 64006
diff changeset
  2927
  by (meson cobounded_unique_unbounded_components connected_eq_connected_components_eq assms)
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2928
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2929
69620
19d8a59481db split off Homotopy.thy
immler
parents: 69566
diff changeset
  2930
subsection\<open>The \<open>inside\<close> and \<open>outside\<close> of a Set\<close>
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2931
70136
f03a01a18c6e modernized tags: default scope excludes proof;
wenzelm
parents: 69986
diff changeset
  2932
text\<^marker>\<open>tag important\<close>\<open>The inside comprises the points in a bounded connected component of the set's complement.
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2933
  The outside comprises the points in unbounded connected component of the complement.\<close>
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2934
70136
f03a01a18c6e modernized tags: default scope excludes proof;
wenzelm
parents: 69986
diff changeset
  2935
definition\<^marker>\<open>tag important\<close> inside where
68096
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  2936
  "inside S \<equiv> {x. (x \<notin> S) \<and> bounded(connected_component_set ( - S) x)}"
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2937
70136
f03a01a18c6e modernized tags: default scope excludes proof;
wenzelm
parents: 69986
diff changeset
  2938
definition\<^marker>\<open>tag important\<close> outside where
69508
2a4c8a2a3f8e tuned headers; ~ -> \<not>
nipkow
parents: 69313
diff changeset
  2939
  "outside S \<equiv> -S \<inter> {x. \<not> bounded(connected_component_set (- S) x)}"
2a4c8a2a3f8e tuned headers; ~ -> \<not>
nipkow
parents: 69313
diff changeset
  2940
2a4c8a2a3f8e tuned headers; ~ -> \<not>
nipkow
parents: 69313
diff changeset
  2941
lemma outside: "outside S = {x. \<not> bounded(connected_component_set (- S) x)}"
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2942
  by (auto simp: outside_def) (metis Compl_iff bounded_empty connected_component_eq_empty)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2943
68096
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  2944
lemma inside_no_overlap [simp]: "inside S \<inter> S = {}"
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2945
  by (auto simp: inside_def)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2946
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2947
lemma outside_no_overlap [simp]:
68096
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  2948
   "outside S \<inter> S = {}"
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2949
  by (auto simp: outside_def)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2950
68096
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  2951
lemma inside_Int_outside [simp]: "inside S \<inter> outside S = {}"
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2952
  by (auto simp: inside_def outside_def)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2953
68096
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  2954
lemma inside_Un_outside [simp]: "inside S \<union> outside S = (- S)"
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2955
  by (auto simp: inside_def outside_def)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2956
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2957
lemma inside_eq_outside:
68096
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  2958
   "inside S = outside S \<longleftrightarrow> S = UNIV"
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2959
  by (auto simp: inside_def outside_def)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2960
68096
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  2961
lemma inside_outside: "inside S = (- (S \<union> outside S))"
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  2962
  by (force simp: inside_def outside)
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  2963
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  2964
lemma outside_inside: "outside S = (- (S \<union> inside S))"
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2965
  by (auto simp: inside_outside) (metis IntI equals0D outside_no_overlap)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2966
68096
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  2967
lemma union_with_inside: "S \<union> inside S = - outside S"
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2968
  by (auto simp: inside_outside) (simp add: outside_inside)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2969
68096
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  2970
lemma union_with_outside: "S \<union> outside S = - inside S"
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2971
  by (simp add: inside_outside)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2972
68096
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  2973
lemma outside_mono: "S \<subseteq> T \<Longrightarrow> outside T \<subseteq> outside S"
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2974
  by (auto simp: outside bounded_subset connected_component_mono)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2975
68096
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  2976
lemma inside_mono: "S \<subseteq> T \<Longrightarrow> inside S - T \<subseteq> inside T"
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2977
  by (auto simp: inside_def bounded_subset connected_component_mono)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2978
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2979
lemma segment_bound_lemma:
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2980
  fixes u::real
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2981
  assumes "x \<ge> B" "y \<ge> B" "0 \<le> u" "u \<le> 1"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2982
  shows "(1 - u) * x + u * y \<ge> B"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2983
proof -
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2984
  obtain dx dy where "dx \<ge> 0" "dy \<ge> 0" "x = B + dx" "y = B + dy"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2985
    using assms by auto (metis add.commute diff_add_cancel)
61808
fc1556774cfe isabelle update_cartouches -c -t;
wenzelm
parents: 61806
diff changeset
  2986
  with \<open>0 \<le> u\<close> \<open>u \<le> 1\<close> show ?thesis
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2987
    by (simp add: add_increasing2 mult_left_le field_simps)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2988
qed
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2989
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2990
lemma cobounded_outside:
68096
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  2991
  fixes S :: "'a :: real_normed_vector set"
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  2992
  assumes "bounded S" shows "bounded (- outside S)"
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2993
proof -
68096
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  2994
  obtain B where B: "B>0" "S \<subseteq> ball 0 B"
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2995
    using bounded_subset_ballD [OF assms, of 0] by auto
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2996
  { fix x::'a and C::real
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2997
    assume Bno: "B \<le> norm x" and C: "0 < C"
68096
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  2998
    have "\<exists>y. connected_component (- S) x y \<and> norm y > C"
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2999
    proof (cases "x = 0")
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3000
      case True with B Bno show ?thesis by force
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3001
    next
68096
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  3002
      case False 
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  3003
      with B C
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  3004
      have "closed_segment x (((B + C) / norm x) *\<^sub>R x) \<subseteq> - ball 0 B"
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  3005
        apply (clarsimp simp add: closed_segment_def ball_def dist_norm real_vector_class.scaleR_add_left [symmetric] divide_simps)
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3006
        using segment_bound_lemma [of B "norm x" "B+C" ] Bno
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3007
        by (meson le_add_same_cancel1 less_eq_real_def not_le)
68096
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  3008
      also have "... \<subseteq> -S"
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  3009
        by (simp add: B)
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  3010
      finally have "\<exists>T. connected T \<and> T \<subseteq> - S \<and> x \<in> T \<and> ((B + C) / norm x) *\<^sub>R x \<in> T"
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  3011
        by (rule_tac x="closed_segment x (((B+C)/norm x) *\<^sub>R x)" in exI) simp
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  3012
      with False B
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  3013
      show ?thesis
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  3014
        by (rule_tac x="((B+C)/norm x) *\<^sub>R x" in exI) (simp add: connected_component_def)
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3015
    qed
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3016
  }
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3017
  then show ?thesis
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3018
    apply (simp add: outside_def assms)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3019
    apply (rule bounded_subset [OF bounded_ball [of 0 B]])
68096
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  3020
    apply (force simp: dist_norm not_less bounded_pos)
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3021
    done
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3022
qed
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3023
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3024
lemma unbounded_outside:
68096
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  3025
    fixes S :: "'a::{real_normed_vector, perfect_space} set"
69508
2a4c8a2a3f8e tuned headers; ~ -> \<not>
nipkow
parents: 69313
diff changeset
  3026
    shows "bounded S \<Longrightarrow> \<not> bounded(outside S)"
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3027
  using cobounded_imp_unbounded cobounded_outside by blast
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3028
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3029
lemma bounded_inside:
68096
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  3030
    fixes S :: "'a::{real_normed_vector, perfect_space} set"
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  3031
    shows "bounded S \<Longrightarrow> bounded(inside S)"
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3032
  by (simp add: bounded_Int cobounded_outside inside_outside)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3033
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3034
lemma connected_outside:
68096
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  3035
    fixes S :: "'a::euclidean_space set"
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  3036
    assumes "bounded S" "2 \<le> DIM('a)"
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  3037
      shows "connected(outside S)"
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  3038
  apply (clarsimp simp add: connected_iff_connected_component outside)
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  3039
  apply (rule_tac s="connected_component_set (- S) x" in connected_component_of_subset)
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3040
  apply (metis (no_types) assms cobounded_unbounded_component cobounded_unique_unbounded_component connected_component_eq_eq connected_component_idemp double_complement mem_Collect_eq)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3041
  apply clarify
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3042
  apply (metis connected_component_eq_eq connected_component_in)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3043
  done
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3044
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3045
lemma outside_connected_component_lt:
68096
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  3046
    "outside S = {x. \<forall>B. \<exists>y. B < norm(y) \<and> connected_component (- S) x y}"
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3047
apply (auto simp: outside bounded_def dist_norm)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3048
apply (metis diff_0 norm_minus_cancel not_less)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3049
by (metis less_diff_eq norm_minus_commute norm_triangle_ineq2 order.trans pinf(6))
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3050
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3051
lemma outside_connected_component_le:
68096
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  3052
   "outside S =
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3053
            {x. \<forall>B. \<exists>y. B \<le> norm(y) \<and>
68096
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  3054
                         connected_component (- S) x y}"
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3055
apply (simp add: outside_connected_component_lt)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3056
apply (simp add: Set.set_eq_iff)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3057
by (meson gt_ex leD le_less_linear less_imp_le order.trans)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3058
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3059
lemma not_outside_connected_component_lt:
68096
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  3060
    fixes S :: "'a::euclidean_space set"
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  3061
    assumes S: "bounded S" and "2 \<le> DIM('a)"
69508
2a4c8a2a3f8e tuned headers; ~ -> \<not>
nipkow
parents: 69313
diff changeset
  3062
      shows "- (outside S) = {x. \<forall>B. \<exists>y. B < norm(y) \<and> \<not> connected_component (- S) x y}"
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3063
proof -
68096
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  3064
  obtain B::real where B: "0 < B" and Bno: "\<And>x. x \<in> S \<Longrightarrow> norm x \<le> B"
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  3065
    using S [simplified bounded_pos] by auto
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3066
  { fix y::'a and z::'a
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3067
    assume yz: "B < norm z" "B < norm y"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3068
    have "connected_component (- cball 0 B) y z"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3069
      apply (rule connected_componentI [OF _ subset_refl])
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3070
      apply (rule connected_complement_bounded_convex)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3071
      using assms yz
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3072
      by (auto simp: dist_norm)
68096
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  3073
    then have "connected_component (- S) y z"
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3074
      apply (rule connected_component_of_subset)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3075
      apply (metis Bno Compl_anti_mono mem_cball_0 subset_iff)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3076
      done
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3077
  } note cyz = this
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3078
  show ?thesis
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3079
    apply (auto simp: outside)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3080
    apply (metis Compl_iff bounded_iff cobounded_imp_unbounded mem_Collect_eq not_le)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3081
    apply (simp add: bounded_pos)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3082
    by (metis B connected_component_trans cyz not_le)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3083
qed
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3084
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3085
lemma not_outside_connected_component_le:
68096
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  3086
    fixes S :: "'a::euclidean_space set"
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  3087
    assumes S: "bounded S"  "2 \<le> DIM('a)"
69508
2a4c8a2a3f8e tuned headers; ~ -> \<not>
nipkow
parents: 69313
diff changeset
  3088
      shows "- (outside S) = {x. \<forall>B. \<exists>y. B \<le> norm(y) \<and> \<not> connected_component (- S) x y}"
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3089
apply (auto intro: less_imp_le simp: not_outside_connected_component_lt [OF assms])
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3090
by (meson gt_ex less_le_trans)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3091
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3092
lemma inside_connected_component_lt:
68096
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  3093
    fixes S :: "'a::euclidean_space set"
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  3094
    assumes S: "bounded S"  "2 \<le> DIM('a)"
69508
2a4c8a2a3f8e tuned headers; ~ -> \<not>
nipkow
parents: 69313
diff changeset
  3095
      shows "inside S = {x. (x \<notin> S) \<and> (\<forall>B. \<exists>y. B < norm(y) \<and> \<not> connected_component (- S) x y)}"
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3096
  by (auto simp: inside_outside not_outside_connected_component_lt [OF assms])
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3097
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3098
lemma inside_connected_component_le:
68096
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  3099
    fixes S :: "'a::euclidean_space set"
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  3100
    assumes S: "bounded S"  "2 \<le> DIM('a)"
69508
2a4c8a2a3f8e tuned headers; ~ -> \<not>
nipkow
parents: 69313
diff changeset
  3101
      shows "inside S = {x. (x \<notin> S) \<and> (\<forall>B. \<exists>y. B \<le> norm(y) \<and> \<not> connected_component (- S) x y)}"
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3102
  by (auto simp: inside_outside not_outside_connected_component_le [OF assms])
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3103
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3104
lemma inside_subset:
69508
2a4c8a2a3f8e tuned headers; ~ -> \<not>
nipkow
parents: 69313
diff changeset
  3105
  assumes "connected U" and "\<not> bounded U" and "T \<union> U = - S"
68096
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  3106
  shows "inside S \<subseteq> T"
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3107
apply (auto simp: inside_def)
68096
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  3108
by (metis bounded_subset [of "connected_component_set (- S) _"] connected_component_maximal
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3109
       Compl_iff Un_iff assms subsetI)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3110
62381
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  3111
lemma frontier_not_empty:
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  3112
  fixes S :: "'a :: real_normed_vector set"
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  3113
  shows "\<lbrakk>S \<noteq> {}; S \<noteq> UNIV\<rbrakk> \<Longrightarrow> frontier S \<noteq> {}"
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  3114
    using connected_Int_frontier [of UNIV S] by auto
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  3115
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  3116
lemma frontier_eq_empty:
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  3117
  fixes S :: "'a :: real_normed_vector set"
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  3118
  shows "frontier S = {} \<longleftrightarrow> S = {} \<or> S = UNIV"
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  3119
using frontier_UNIV frontier_empty frontier_not_empty by blast
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  3120
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  3121
lemma frontier_of_connected_component_subset:
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  3122
  fixes S :: "'a::real_normed_vector set"
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  3123
  shows "frontier(connected_component_set S x) \<subseteq> frontier S"
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  3124
proof -
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  3125
  { fix y
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  3126
    assume y1: "y \<in> closure (connected_component_set S x)"
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  3127
       and y2: "y \<notin> interior (connected_component_set S x)"
64006
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents: 63978
diff changeset
  3128
    have "y \<in> closure S"
62381
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  3129
      using y1 closure_mono connected_component_subset by blast
64006
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents: 63978
diff changeset
  3130
    moreover have "z \<in> interior (connected_component_set S x)"
62381
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  3131
          if "0 < e" "ball y e \<subseteq> interior S" "dist y z < e" for e z
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  3132
    proof -
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  3133
      have "ball y e \<subseteq> connected_component_set S y"
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  3134
        apply (rule connected_component_maximal)
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  3135
        using that interior_subset mem_ball apply auto
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  3136
        done
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  3137
      then show ?thesis
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  3138
        using y1 apply (simp add: closure_approachable open_contains_ball_eq [OF open_interior])
64006
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents: 63978
diff changeset
  3139
        by (metis connected_component_eq dist_commute mem_Collect_eq mem_ball mem_interior subsetD \<open>0 < e\<close> y2)
62381
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  3140
    qed
64006
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents: 63978
diff changeset
  3141
    then have "y \<notin> interior S"
62381
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  3142
      using y2 by (force simp: open_contains_ball_eq [OF open_interior])
64006
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents: 63978
diff changeset
  3143
    ultimately have "y \<in> frontier S"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents: 63978
diff changeset
  3144
      by (auto simp: frontier_def)
62381
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  3145
  }
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  3146
  then show ?thesis by (auto simp: frontier_def)
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  3147
qed
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  3148
62620
d21dab28b3f9 New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents: 62618
diff changeset
  3149
lemma frontier_Union_subset_closure:
d21dab28b3f9 New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents: 62618
diff changeset
  3150
  fixes F :: "'a::real_normed_vector set set"
d21dab28b3f9 New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents: 62618
diff changeset
  3151
  shows "frontier(\<Union>F) \<subseteq> closure(\<Union>t \<in> F. frontier t)"
d21dab28b3f9 New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents: 62618
diff changeset
  3152
proof -
d21dab28b3f9 New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents: 62618
diff changeset
  3153
  have "\<exists>y\<in>F. \<exists>y\<in>frontier y. dist y x < e"
d21dab28b3f9 New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents: 62618
diff changeset
  3154
       if "T \<in> F" "y \<in> T" "dist y x < e"
d21dab28b3f9 New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents: 62618
diff changeset
  3155
          "x \<notin> interior (\<Union>F)" "0 < e" for x y e T
d21dab28b3f9 New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents: 62618
diff changeset
  3156
  proof (cases "x \<in> T")
d21dab28b3f9 New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents: 62618
diff changeset
  3157
    case True with that show ?thesis
d21dab28b3f9 New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents: 62618
diff changeset
  3158
      by (metis Diff_iff Sup_upper closure_subset contra_subsetD dist_self frontier_def interior_mono)
d21dab28b3f9 New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents: 62618
diff changeset
  3159
  next
d21dab28b3f9 New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents: 62618
diff changeset
  3160
    case False
d21dab28b3f9 New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents: 62618
diff changeset
  3161
    have 1: "closed_segment x y \<inter> T \<noteq> {}" using \<open>y \<in> T\<close> by blast
d21dab28b3f9 New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents: 62618
diff changeset
  3162
    have 2: "closed_segment x y - T \<noteq> {}"
d21dab28b3f9 New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents: 62618
diff changeset
  3163
      using False by blast
d21dab28b3f9 New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents: 62618
diff changeset
  3164
    obtain c where "c \<in> closed_segment x y" "c \<in> frontier T"
d21dab28b3f9 New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents: 62618
diff changeset
  3165
       using False connected_Int_frontier [OF connected_segment 1 2] by auto
d21dab28b3f9 New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents: 62618
diff changeset
  3166
    then show ?thesis
d21dab28b3f9 New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents: 62618
diff changeset
  3167
    proof -
d21dab28b3f9 New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents: 62618
diff changeset
  3168
      have "norm (y - x) < e"
d21dab28b3f9 New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents: 62618
diff changeset
  3169
        by (metis dist_norm \<open>dist y x < e\<close>)
d21dab28b3f9 New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents: 62618
diff changeset
  3170
      moreover have "norm (c - x) \<le> norm (y - x)"
d21dab28b3f9 New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents: 62618
diff changeset
  3171
        by (simp add: \<open>c \<in> closed_segment x y\<close> segment_bound(1))
d21dab28b3f9 New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents: 62618
diff changeset
  3172
      ultimately have "norm (c - x) < e"
d21dab28b3f9 New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents: 62618
diff changeset
  3173
        by linarith
d21dab28b3f9 New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents: 62618
diff changeset
  3174
      then show ?thesis
d21dab28b3f9 New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents: 62618
diff changeset
  3175
        by (metis (no_types) \<open>c \<in> frontier T\<close> dist_norm that(1))
d21dab28b3f9 New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents: 62618
diff changeset
  3176
    qed
d21dab28b3f9 New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents: 62618
diff changeset
  3177
  qed
d21dab28b3f9 New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents: 62618
diff changeset
  3178
  then show ?thesis
d21dab28b3f9 New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents: 62618
diff changeset
  3179
    by (fastforce simp add: frontier_def closure_approachable)
d21dab28b3f9 New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents: 62618
diff changeset
  3180
qed
d21dab28b3f9 New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents: 62618
diff changeset
  3181
d21dab28b3f9 New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents: 62618
diff changeset
  3182
lemma frontier_Union_subset:
d21dab28b3f9 New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents: 62618
diff changeset
  3183
  fixes F :: "'a::real_normed_vector set set"
d21dab28b3f9 New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents: 62618
diff changeset
  3184
  shows "finite F \<Longrightarrow> frontier(\<Union>F) \<subseteq> (\<Union>t \<in> F. frontier t)"
d21dab28b3f9 New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents: 62618
diff changeset
  3185
by (rule order_trans [OF frontier_Union_subset_closure])
d21dab28b3f9 New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents: 62618
diff changeset
  3186
   (auto simp: closure_subset_eq)
d21dab28b3f9 New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents: 62618
diff changeset
  3187
64006
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents: 63978
diff changeset
  3188
lemma frontier_of_components_subset:
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents: 63978
diff changeset
  3189
  fixes S :: "'a::real_normed_vector set"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents: 63978
diff changeset
  3190
  shows "C \<in> components S \<Longrightarrow> frontier C \<subseteq> frontier S"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents: 63978
diff changeset
  3191
  by (metis Path_Connected.frontier_of_connected_component_subset components_iff)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents: 63978
diff changeset
  3192
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents: 63978
diff changeset
  3193
lemma frontier_of_components_closed_complement:
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents: 63978
diff changeset
  3194
  fixes S :: "'a::real_normed_vector set"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents: 63978
diff changeset
  3195
  shows "\<lbrakk>closed S; C \<in> components (- S)\<rbrakk> \<Longrightarrow> frontier C \<subseteq> S"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents: 63978
diff changeset
  3196
  using frontier_complement frontier_of_components_subset frontier_subset_eq by blast
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents: 63978
diff changeset
  3197
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents: 63978
diff changeset
  3198
lemma frontier_minimal_separating_closed:
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents: 63978
diff changeset
  3199
  fixes S :: "'a::real_normed_vector set"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents: 63978
diff changeset
  3200
  assumes "closed S"
69508
2a4c8a2a3f8e tuned headers; ~ -> \<not>
nipkow
parents: 69313
diff changeset
  3201
      and nconn: "\<not> connected(- S)"
64006
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents: 63978
diff changeset
  3202
      and C: "C \<in> components (- S)"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents: 63978
diff changeset
  3203
      and conn: "\<And>T. \<lbrakk>closed T; T \<subset> S\<rbrakk> \<Longrightarrow> connected(- T)"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents: 63978
diff changeset
  3204
    shows "frontier C = S"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents: 63978
diff changeset
  3205
proof (rule ccontr)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents: 63978
diff changeset
  3206
  assume "frontier C \<noteq> S"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents: 63978
diff changeset
  3207
  then have "frontier C \<subset> S"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents: 63978
diff changeset
  3208
    using frontier_of_components_closed_complement [OF \<open>closed S\<close> C] by blast
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents: 63978
diff changeset
  3209
  then have "connected(- (frontier C))"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents: 63978
diff changeset
  3210
    by (simp add: conn)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents: 63978
diff changeset
  3211
  have "\<not> connected(- (frontier C))"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents: 63978
diff changeset
  3212
    unfolding connected_def not_not
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents: 63978
diff changeset
  3213
  proof (intro exI conjI)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents: 63978
diff changeset
  3214
    show "open C"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents: 63978
diff changeset
  3215
      using C \<open>closed S\<close> open_components by blast
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents: 63978
diff changeset
  3216
    show "open (- closure C)"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents: 63978
diff changeset
  3217
      by blast
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents: 63978
diff changeset
  3218
    show "C \<inter> - closure C \<inter> - frontier C = {}"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents: 63978
diff changeset
  3219
      using closure_subset by blast
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents: 63978
diff changeset
  3220
    show "C \<inter> - frontier C \<noteq> {}"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents: 63978
diff changeset
  3221
      using C \<open>open C\<close> components_eq frontier_disjoint_eq by fastforce
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents: 63978
diff changeset
  3222
    show "- frontier C \<subseteq> C \<union> - closure C"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents: 63978
diff changeset
  3223
      by (simp add: \<open>open C\<close> closed_Compl frontier_closures)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents: 63978
diff changeset
  3224
    then show "- closure C \<inter> - frontier C \<noteq> {}"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents: 63978
diff changeset
  3225
      by (metis (no_types, lifting) C Compl_subset_Compl_iff \<open>frontier C \<subset> S\<close> compl_sup frontier_closures in_components_subset psubsetE sup.absorb_iff2 sup.boundedE sup_bot.right_neutral sup_inf_absorb)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents: 63978
diff changeset
  3226
  qed
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents: 63978
diff changeset
  3227
  then show False
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents: 63978
diff changeset
  3228
    using \<open>connected (- frontier C)\<close> by blast
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents: 63978
diff changeset
  3229
qed
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents: 63978
diff changeset
  3230
62843
313d3b697c9a Mostly renaming (from HOL Light to Isabelle conventions), with a couple of new results
paulson <lp15@cam.ac.uk>
parents: 62626
diff changeset
  3231
lemma connected_component_UNIV [simp]:
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3232
    fixes x :: "'a::real_normed_vector"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3233
    shows "connected_component_set UNIV x = UNIV"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3234
using connected_iff_eq_connected_component_set [of "UNIV::'a set"] connected_UNIV
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3235
by auto
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3236
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3237
lemma connected_component_eq_UNIV:
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3238
    fixes x :: "'a::real_normed_vector"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3239
    shows "connected_component_set s x = UNIV \<longleftrightarrow> s = UNIV"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3240
  using connected_component_in connected_component_UNIV by blast
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3241
64788
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  3242
lemma components_UNIV [simp]: "components UNIV = {UNIV :: 'a::real_normed_vector set}"
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3243
  by (auto simp: components_eq_sing_iff)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3244
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3245
lemma interior_inside_frontier:
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3246
    fixes s :: "'a::real_normed_vector set"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3247
    assumes "bounded s"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3248
      shows "interior s \<subseteq> inside (frontier s)"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3249
proof -
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3250
  { fix x y
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3251
    assume x: "x \<in> interior s" and y: "y \<notin> s"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3252
       and cc: "connected_component (- frontier s) x y"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3253
    have "connected_component_set (- frontier s) x \<inter> frontier s \<noteq> {}"
62381
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  3254
      apply (rule connected_Int_frontier, simp)
69712
dc85b5b3a532 renamings and new material
paulson <lp15@cam.ac.uk>
parents: 69661
diff changeset
  3255
      apply (metis IntI cc connected_component_in connected_component_refl empty_iff interiorE mem_Collect_eq rev_subsetD x)
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3256
      using  y cc
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3257
      by blast
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3258
    then have "bounded (connected_component_set (- frontier s) x)"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3259
      using connected_component_in by auto
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3260
  }
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3261
  then show ?thesis
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3262
    apply (auto simp: inside_def frontier_def)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3263
    apply (rule classical)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3264
    apply (rule bounded_subset [OF assms], blast)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3265
    done
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3266
qed
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3267
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3268
lemma inside_empty [simp]: "inside {} = ({} :: 'a :: {real_normed_vector, perfect_space} set)"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3269
  by (simp add: inside_def connected_component_UNIV)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3270
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3271
lemma outside_empty [simp]: "outside {} = (UNIV :: 'a :: {real_normed_vector, perfect_space} set)"
63955
51a3d38d2281 more new material
paulson <lp15@cam.ac.uk>
parents: 63928
diff changeset
  3272
using inside_empty inside_Un_outside by blast
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3273
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3274
lemma inside_same_component:
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3275
   "\<lbrakk>connected_component (- s) x y; x \<in> inside s\<rbrakk> \<Longrightarrow> y \<in> inside s"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3276
  using connected_component_eq connected_component_in
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3277
  by (fastforce simp add: inside_def)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3278
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3279
lemma outside_same_component:
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3280
   "\<lbrakk>connected_component (- s) x y; x \<in> outside s\<rbrakk> \<Longrightarrow> y \<in> outside s"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3281
  using connected_component_eq connected_component_in
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3282
  by (fastforce simp add: outside_def)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3283
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3284
lemma convex_in_outside:
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3285
  fixes s :: "'a :: {real_normed_vector, perfect_space} set"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3286
  assumes s: "convex s" and z: "z \<notin> s"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3287
    shows "z \<in> outside s"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3288
proof (cases "s={}")
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3289
  case True then show ?thesis by simp
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3290
next
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3291
  case False then obtain a where "a \<in> s" by blast
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3292
  with z have zna: "z \<noteq> a" by auto
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3293
  { assume "bounded (connected_component_set (- s) z)"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3294
    with bounded_pos_less obtain B where "B>0" and B: "\<And>x. connected_component (- s) z x \<Longrightarrow> norm x < B"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3295
      by (metis mem_Collect_eq)
63040
eb4ddd18d635 eliminated old 'def';
wenzelm
parents: 63016
diff changeset
  3296
    define C where "C = (B + 1 + norm z) / norm (z-a)"
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3297
    have "C > 0"
61808
fc1556774cfe isabelle update_cartouches -c -t;
wenzelm
parents: 61806
diff changeset
  3298
      using \<open>0 < B\<close> zna by (simp add: C_def divide_simps add_strict_increasing)
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3299
    have "\<bar>norm (z + C *\<^sub>R (z-a)) - norm (C *\<^sub>R (z-a))\<bar> \<le> norm z"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3300
      by (metis add_diff_cancel norm_triangle_ineq3)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3301
    moreover have "norm (C *\<^sub>R (z-a)) > norm z + B"
61808
fc1556774cfe isabelle update_cartouches -c -t;
wenzelm
parents: 61806
diff changeset
  3302
      using zna \<open>B>0\<close> by (simp add: C_def le_max_iff_disj field_simps)
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3303
    ultimately have C: "norm (z + C *\<^sub>R (z-a)) > B" by linarith
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3304
    { fix u::real
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3305
      assume u: "0\<le>u" "u\<le>1" and ins: "(1 - u) *\<^sub>R z + u *\<^sub>R (z + C *\<^sub>R (z - a)) \<in> s"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3306
      then have Cpos: "1 + u * C > 0"
61808
fc1556774cfe isabelle update_cartouches -c -t;
wenzelm
parents: 61806
diff changeset
  3307
        by (meson \<open>0 < C\<close> add_pos_nonneg less_eq_real_def zero_le_mult_iff zero_less_one)
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3308
      then have *: "(1 / (1 + u * C)) *\<^sub>R z + (u * C / (1 + u * C)) *\<^sub>R z = z"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3309
        by (simp add: scaleR_add_left [symmetric] divide_simps)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3310
      then have False
61808
fc1556774cfe isabelle update_cartouches -c -t;
wenzelm
parents: 61806
diff changeset
  3311
        using convexD_alt [OF s \<open>a \<in> s\<close> ins, of "1/(u*C + 1)"] \<open>C>0\<close> \<open>z \<notin> s\<close> Cpos u
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3312
        by (simp add: * divide_simps algebra_simps)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3313
    } note contra = this
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3314
    have "connected_component (- s) z (z + C *\<^sub>R (z-a))"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3315
      apply (rule connected_componentI [OF connected_segment [of z "z + C *\<^sub>R (z-a)"]])
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3316
      apply (simp add: closed_segment_def)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3317
      using contra
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3318
      apply auto
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3319
      done
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3320
    then have False
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3321
      using zna B [of "z + C *\<^sub>R (z-a)"] C
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3322
      by (auto simp: divide_simps max_mult_distrib_right)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3323
  }
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3324
  then show ?thesis
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3325
    by (auto simp: outside_def z)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3326
qed
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3327
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3328
lemma outside_convex:
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3329
  fixes s :: "'a :: {real_normed_vector, perfect_space} set"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3330
  assumes "convex s"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3331
    shows "outside s = - s"
63955
51a3d38d2281 more new material
paulson <lp15@cam.ac.uk>
parents: 63928
diff changeset
  3332
  by (metis ComplD assms convex_in_outside equalityI inside_Un_outside subsetI sup.cobounded2)
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3333
66793
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66456
diff changeset
  3334
lemma outside_singleton [simp]:
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66456
diff changeset
  3335
  fixes x :: "'a :: {real_normed_vector, perfect_space}"
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66456
diff changeset
  3336
  shows "outside {x} = -{x}"
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66456
diff changeset
  3337
  by (auto simp: outside_convex)
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66456
diff changeset
  3338
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3339
lemma inside_convex:
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3340
  fixes s :: "'a :: {real_normed_vector, perfect_space} set"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3341
  shows "convex s \<Longrightarrow> inside s = {}"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3342
  by (simp add: inside_outside outside_convex)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3343
66793
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66456
diff changeset
  3344
lemma inside_singleton [simp]:
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66456
diff changeset
  3345
  fixes x :: "'a :: {real_normed_vector, perfect_space}"
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66456
diff changeset
  3346
  shows "inside {x} = {}"
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66456
diff changeset
  3347
  by (auto simp: inside_convex)
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66456
diff changeset
  3348
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3349
lemma outside_subset_convex:
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3350
  fixes s :: "'a :: {real_normed_vector, perfect_space} set"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3351
  shows "\<lbrakk>convex t; s \<subseteq> t\<rbrakk> \<Longrightarrow> - t \<subseteq> outside s"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3352
  using outside_convex outside_mono by blast
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3353
64788
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  3354
lemma outside_Un_outside_Un:
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  3355
  fixes S :: "'a::real_normed_vector set"
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  3356
  assumes "S \<inter> outside(T \<union> U) = {}"
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  3357
  shows "outside(T \<union> U) \<subseteq> outside(T \<union> S)"
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  3358
proof
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  3359
  fix x
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  3360
  assume x: "x \<in> outside (T \<union> U)"
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  3361
  have "Y \<subseteq> - S" if "connected Y" "Y \<subseteq> - T" "Y \<subseteq> - U" "x \<in> Y" "u \<in> Y" for u Y
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  3362
  proof -
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  3363
    have "Y \<subseteq> connected_component_set (- (T \<union> U)) x"
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  3364
      by (simp add: connected_component_maximal that)
68096
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  3365
    also have "\<dots> \<subseteq> outside(T \<union> U)"
64788
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  3366
      by (metis (mono_tags, lifting) Collect_mono mem_Collect_eq outside outside_same_component x)
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  3367
    finally have "Y \<subseteq> outside(T \<union> U)" .
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  3368
    with assms show ?thesis by auto
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  3369
  qed
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  3370
  with x show "x \<in> outside (T \<union> S)"
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  3371
    by (simp add: outside_connected_component_lt connected_component_def) meson
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  3372
qed
19f3d4af7a7d New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 64394
diff changeset
  3373
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3374
lemma outside_frontier_misses_closure:
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3375
    fixes s :: "'a::real_normed_vector set"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3376
    assumes "bounded s"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3377
    shows  "outside(frontier s) \<subseteq> - closure s"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3378
  unfolding outside_inside Lattices.boolean_algebra_class.compl_le_compl_iff
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3379
proof -
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3380
  { assume "interior s \<subseteq> inside (frontier s)"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3381
    hence "interior s \<union> inside (frontier s) = inside (frontier s)"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3382
      by (simp add: subset_Un_eq)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3383
    then have "closure s \<subseteq> frontier s \<union> inside (frontier s)"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3384
      using frontier_def by auto
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3385
  }
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3386
  then show "closure s \<subseteq> frontier s \<union> inside (frontier s)"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3387
    using interior_inside_frontier [OF assms] by blast
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3388
qed
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3389
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3390
lemma outside_frontier_eq_complement_closure:
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3391
  fixes s :: "'a :: {real_normed_vector, perfect_space} set"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3392
    assumes "bounded s" "convex s"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3393
      shows "outside(frontier s) = - closure s"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3394
by (metis Diff_subset assms convex_closure frontier_def outside_frontier_misses_closure
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3395
          outside_subset_convex subset_antisym)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3396
61518
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3397
lemma inside_frontier_eq_interior:
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3398
     fixes s :: "'a :: {real_normed_vector, perfect_space} set"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3399
     shows "\<lbrakk>bounded s; convex s\<rbrakk> \<Longrightarrow> inside(frontier s) = interior s"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3400
  apply (simp add: inside_outside outside_frontier_eq_complement_closure)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3401
  using closure_subset interior_subset
68096
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  3402
  apply (auto simp: frontier_def)
61518
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3403
  done
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3404
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3405
lemma open_inside:
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3406
    fixes s :: "'a::real_normed_vector set"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3407
    assumes "closed s"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3408
      shows "open (inside s)"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3409
proof -
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3410
  { fix x assume x: "x \<in> inside s"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3411
    have "open (connected_component_set (- s) x)"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3412
      using assms open_connected_component by blast
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3413
    then obtain e where e: "e>0" and e: "\<And>y. dist y x < e \<longrightarrow> connected_component (- s) x y"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3414
      using dist_not_less_zero
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3415
      apply (simp add: open_dist)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3416
      by (metis (no_types, lifting) Compl_iff connected_component_refl_eq inside_def mem_Collect_eq x)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3417
    then have "\<exists>e>0. ball x e \<subseteq> inside s"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3418
      by (metis e dist_commute inside_same_component mem_ball subsetI x)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3419
  }
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3420
  then show ?thesis
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3421
    by (simp add: open_contains_ball)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3422
qed
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3423
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3424
lemma open_outside:
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3425
    fixes s :: "'a::real_normed_vector set"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3426
    assumes "closed s"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3427
      shows "open (outside s)"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3428
proof -
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3429
  { fix x assume x: "x \<in> outside s"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3430
    have "open (connected_component_set (- s) x)"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3431
      using assms open_connected_component by blast
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3432
    then obtain e where e: "e>0" and e: "\<And>y. dist y x < e \<longrightarrow> connected_component (- s) x y"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3433
      using dist_not_less_zero
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3434
      apply (simp add: open_dist)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3435
      by (metis Int_iff outside_def connected_component_refl_eq  x)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3436
    then have "\<exists>e>0. ball x e \<subseteq> outside s"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3437
      by (metis e dist_commute outside_same_component mem_ball subsetI x)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3438
  }
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3439
  then show ?thesis
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3440
    by (simp add: open_contains_ball)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3441
qed
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3442
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3443
lemma closure_inside_subset:
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3444
    fixes s :: "'a::real_normed_vector set"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3445
    assumes "closed s"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3446
      shows "closure(inside s) \<subseteq> s \<union> inside s"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3447
by (metis assms closure_minimal open_closed open_outside sup.cobounded2 union_with_inside)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3448
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3449
lemma frontier_inside_subset:
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3450
    fixes s :: "'a::real_normed_vector set"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3451
    assumes "closed s"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3452
      shows "frontier(inside s) \<subseteq> s"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3453
proof -
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3454
  have "closure (inside s) \<inter> - inside s = closure (inside s) - interior (inside s)"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3455
    by (metis (no_types) Diff_Compl assms closure_closed interior_closure open_closed open_inside)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3456
  moreover have "- inside s \<inter> - outside s = s"
63955
51a3d38d2281 more new material
paulson <lp15@cam.ac.uk>
parents: 63928
diff changeset
  3457
    by (metis (no_types) compl_sup double_compl inside_Un_outside)
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3458
  moreover have "closure (inside s) \<subseteq> - outside s"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3459
    by (metis (no_types) assms closure_inside_subset union_with_inside)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3460
  ultimately have "closure (inside s) - interior (inside s) \<subseteq> s"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3461
    by blast
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3462
  then show ?thesis
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3463
    by (simp add: frontier_def open_inside interior_open)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3464
qed
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  3465
61518
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3466
lemma closure_outside_subset:
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3467
    fixes s :: "'a::real_normed_vector set"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3468
    assumes "closed s"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3469
      shows "closure(outside s) \<subseteq> s \<union> outside s"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3470
  apply (rule closure_minimal, simp)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3471
  by (metis assms closed_open inside_outside open_inside)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3472
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3473
lemma frontier_outside_subset:
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3474
    fixes s :: "'a::real_normed_vector set"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3475
    assumes "closed s"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3476
      shows "frontier(outside s) \<subseteq> s"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3477
  apply (simp add: frontier_def open_outside interior_open)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3478
  by (metis Diff_subset_conv assms closure_outside_subset interior_eq open_outside sup.commute)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3479
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3480
lemma inside_complement_unbounded_connected_empty:
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3481
     "\<lbrakk>connected (- s); \<not> bounded (- s)\<rbrakk> \<Longrightarrow> inside s = {}"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3482
  apply (simp add: inside_def)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3483
  by (meson Compl_iff bounded_subset connected_component_maximal order_refl)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3484
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3485
lemma inside_bounded_complement_connected_empty:
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3486
    fixes s :: "'a::{real_normed_vector, perfect_space} set"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3487
    shows "\<lbrakk>connected (- s); bounded s\<rbrakk> \<Longrightarrow> inside s = {}"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3488
  by (metis inside_complement_unbounded_connected_empty cobounded_imp_unbounded)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3489
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3490
lemma inside_inside:
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3491
    assumes "s \<subseteq> inside t"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3492
    shows "inside s - t \<subseteq> inside t"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3493
unfolding inside_def
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3494
proof clarify
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3495
  fix x
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3496
  assume x: "x \<notin> t" "x \<notin> s" and bo: "bounded (connected_component_set (- s) x)"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3497
  show "bounded (connected_component_set (- t) x)"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3498
  proof (cases "s \<inter> connected_component_set (- t) x = {}")
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3499
    case True show ?thesis
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3500
      apply (rule bounded_subset [OF bo])
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3501
      apply (rule connected_component_maximal)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3502
      using x True apply auto
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3503
      done
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3504
  next
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3505
    case False then show ?thesis
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3506
      using assms [unfolded inside_def] x
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3507
      apply (simp add: disjoint_iff_not_equal, clarify)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3508
      apply (drule subsetD, assumption, auto)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3509
      by (metis (no_types, hide_lams) ComplI connected_component_eq_eq)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3510
  qed
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3511
qed
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3512
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3513
lemma inside_inside_subset: "inside(inside s) \<subseteq> s"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3514
  using inside_inside union_with_outside by fastforce
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3515
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3516
lemma inside_outside_intersect_connected:
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3517
      "\<lbrakk>connected t; inside s \<inter> t \<noteq> {}; outside s \<inter> t \<noteq> {}\<rbrakk> \<Longrightarrow> s \<inter> t \<noteq> {}"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3518
  apply (simp add: inside_def outside_def ex_in_conv [symmetric] disjoint_eq_subset_Compl, clarify)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3519
  by (metis (no_types, hide_lams) Compl_anti_mono connected_component_eq connected_component_maximal contra_subsetD double_compl)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3520
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3521
lemma outside_bounded_nonempty:
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3522
  fixes s :: "'a :: {real_normed_vector, perfect_space} set"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3523
    assumes "bounded s" shows "outside s \<noteq> {}"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3524
  by (metis (no_types, lifting) Collect_empty_eq Collect_mem_eq Compl_eq_Diff_UNIV Diff_cancel
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3525
                   Diff_disjoint UNIV_I assms ball_eq_empty bounded_diff cobounded_outside convex_ball
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3526
                   double_complement order_refl outside_convex outside_def)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3527
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3528
lemma outside_compact_in_open:
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3529
    fixes s :: "'a :: {real_normed_vector,perfect_space} set"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3530
    assumes s: "compact s" and t: "open t" and "s \<subseteq> t" "t \<noteq> {}"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3531
      shows "outside s \<inter> t \<noteq> {}"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3532
proof -
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3533
  have "outside s \<noteq> {}"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3534
    by (simp add: compact_imp_bounded outside_bounded_nonempty s)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3535
  with assms obtain a b where a: "a \<in> outside s" and b: "b \<in> t" by auto
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3536
  show ?thesis
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3537
  proof (cases "a \<in> t")
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3538
    case True with a show ?thesis by blast
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3539
  next
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3540
    case False
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3541
      have front: "frontier t \<subseteq> - s"
61808
fc1556774cfe isabelle update_cartouches -c -t;
wenzelm
parents: 61806
diff changeset
  3542
        using \<open>s \<subseteq> t\<close> frontier_disjoint_eq t by auto
61518
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3543
      { fix \<gamma>
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3544
        assume "path \<gamma>" and pimg_sbs: "path_image \<gamma> - {pathfinish \<gamma>} \<subseteq> interior (- t)"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3545
           and pf: "pathfinish \<gamma> \<in> frontier t" and ps: "pathstart \<gamma> = a"
63040
eb4ddd18d635 eliminated old 'def';
wenzelm
parents: 63016
diff changeset
  3546
        define c where "c = pathfinish \<gamma>"
61518
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3547
        have "c \<in> -s" unfolding c_def using front pf by blast
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3548
        moreover have "open (-s)" using s compact_imp_closed by blast
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3549
        ultimately obtain \<epsilon>::real where "\<epsilon> > 0" and \<epsilon>: "cball c \<epsilon> \<subseteq> -s"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3550
          using open_contains_cball[of "-s"] s by blast
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3551
        then obtain d where "d \<in> t" and d: "dist d c < \<epsilon>"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3552
          using closure_approachable [of c t] pf unfolding c_def
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3553
          by (metis Diff_iff frontier_def)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3554
        then have "d \<in> -s" using \<epsilon>
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3555
          using dist_commute by (metis contra_subsetD mem_cball not_le not_less_iff_gr_or_eq)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3556
        have pimg_sbs_cos: "path_image \<gamma> \<subseteq> -s"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3557
          using pimg_sbs apply (auto simp: path_image_def)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3558
          apply (drule subsetD)
61808
fc1556774cfe isabelle update_cartouches -c -t;
wenzelm
parents: 61806
diff changeset
  3559
          using \<open>c \<in> - s\<close> \<open>s \<subseteq> t\<close> interior_subset apply (auto simp: c_def)
61518
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3560
          done
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3561
        have "closed_segment c d \<le> cball c \<epsilon>"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3562
          apply (simp add: segment_convex_hull)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3563
          apply (rule hull_minimal)
61808
fc1556774cfe isabelle update_cartouches -c -t;
wenzelm
parents: 61806
diff changeset
  3564
          using  \<open>\<epsilon> > 0\<close> d apply (auto simp: dist_commute)
61518
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3565
          done
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3566
        with \<epsilon> have "closed_segment c d \<subseteq> -s" by blast
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3567
        moreover have con_gcd: "connected (path_image \<gamma> \<union> closed_segment c d)"
61808
fc1556774cfe isabelle update_cartouches -c -t;
wenzelm
parents: 61806
diff changeset
  3568
          by (rule connected_Un) (auto simp: c_def \<open>path \<gamma>\<close> connected_path_image)
61518
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3569
        ultimately have "connected_component (- s) a d"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3570
          unfolding connected_component_def using pimg_sbs_cos ps by blast
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3571
        then have "outside s \<inter> t \<noteq> {}"
61808
fc1556774cfe isabelle update_cartouches -c -t;
wenzelm
parents: 61806
diff changeset
  3572
          using outside_same_component [OF _ a]  by (metis IntI \<open>d \<in> t\<close> empty_iff)
61518
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3573
      } note * = this
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3574
      have pal: "pathstart (linepath a b) \<in> closure (- t)"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3575
        by (auto simp: False closure_def)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3576
      show ?thesis
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3577
        by (rule exists_path_subpath_to_frontier [OF path_linepath pal _ *]) (auto simp: b)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3578
  qed
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3579
qed
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3580
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3581
lemma inside_inside_compact_connected:
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3582
    fixes s :: "'a :: euclidean_space set"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3583
    assumes s: "closed s" and t: "compact t" and "connected t" "s \<subseteq> inside t"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3584
      shows "inside s \<subseteq> inside t"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3585
proof (cases "inside t = {}")
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3586
  case True with assms show ?thesis by auto
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3587
next
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3588
  case False
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3589
  consider "DIM('a) = 1" | "DIM('a) \<ge> 2"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3590
    using antisym not_less_eq_eq by fastforce
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3591
  then show ?thesis
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3592
  proof cases
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3593
    case 1 then show ?thesis
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3594
             using connected_convex_1_gen assms False inside_convex by blast
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3595
  next
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3596
    case 2
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3597
    have coms: "compact s"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3598
      using assms apply (simp add: s compact_eq_bounded_closed)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3599
       by (meson bounded_inside bounded_subset compact_imp_bounded)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3600
    then have bst: "bounded (s \<union> t)"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3601
      by (simp add: compact_imp_bounded t)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3602
    then obtain r where "0 < r" and r: "s \<union> t \<subseteq> ball 0 r"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3603
      using bounded_subset_ballD by blast
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3604
    have outst: "outside s \<inter> outside t \<noteq> {}"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3605
    proof -
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3606
      have "- ball 0 r \<subseteq> outside s"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3607
        apply (rule outside_subset_convex)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3608
        using r by auto
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3609
      moreover have "- ball 0 r \<subseteq> outside t"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3610
        apply (rule outside_subset_convex)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3611
        using r by auto
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3612
      ultimately show ?thesis
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3613
        by (metis Compl_subset_Compl_iff Int_subset_iff bounded_ball inf.orderE outside_bounded_nonempty outside_no_overlap)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3614
    qed
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3615
    have "s \<inter> t = {}" using assms
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3616
      by (metis disjoint_iff_not_equal inside_no_overlap subsetCE)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3617
    moreover have "outside s \<inter> inside t \<noteq> {}"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3618
      by (meson False assms(4) compact_eq_bounded_closed coms open_inside outside_compact_in_open t)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3619
    ultimately have "inside s \<inter> t = {}"
61808
fc1556774cfe isabelle update_cartouches -c -t;
wenzelm
parents: 61806
diff changeset
  3620
      using inside_outside_intersect_connected [OF \<open>connected t\<close>, of s]
61518
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3621
      by (metis "2" compact_eq_bounded_closed coms connected_outside inf.commute inside_outside_intersect_connected outst)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3622
    then show ?thesis
61808
fc1556774cfe isabelle update_cartouches -c -t;
wenzelm
parents: 61806
diff changeset
  3623
      using inside_inside [OF \<open>s \<subseteq> inside t\<close>] by blast
61518
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3624
  qed
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3625
qed
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3626
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3627
lemma connected_with_inside:
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3628
    fixes s :: "'a :: real_normed_vector set"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3629
    assumes s: "closed s" and cons: "connected s"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3630
      shows "connected(s \<union> inside s)"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3631
proof (cases "s \<union> inside s = UNIV")
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3632
  case True with assms show ?thesis by auto
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3633
next
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3634
  case False
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3635
  then obtain b where b: "b \<notin> s" "b \<notin> inside s" by blast
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3636
  have *: "\<exists>y t. y \<in> s \<and> connected t \<and> a \<in> t \<and> y \<in> t \<and> t \<subseteq> (s \<union> inside s)" if "a \<in> (s \<union> inside s)" for a
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3637
  using that proof
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3638
    assume "a \<in> s" then show ?thesis
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3639
      apply (rule_tac x=a in exI)
68096
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  3640
      apply (rule_tac x="{a}" in exI, simp)
61518
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3641
      done
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3642
  next
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3643
    assume a: "a \<in> inside s"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3644
    show ?thesis
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3645
      apply (rule exists_path_subpath_to_frontier [OF path_linepath [of a b], of "inside s"])
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3646
      using a apply (simp add: closure_def)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3647
      apply (simp add: b)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3648
      apply (rule_tac x="pathfinish h" in exI)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3649
      apply (rule_tac x="path_image h" in exI)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3650
      apply (simp add: pathfinish_in_path_image connected_path_image, auto)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3651
      using frontier_inside_subset s apply fastforce
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3652
      by (metis (no_types, lifting) frontier_inside_subset insertE insert_Diff interior_eq open_inside pathfinish_in_path_image s subsetCE)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3653
  qed
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3654
  show ?thesis
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3655
    apply (simp add: connected_iff_connected_component)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3656
    apply (simp add: connected_component_def)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3657
    apply (clarify dest!: *)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3658
    apply (rename_tac u u' t t')
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3659
    apply (rule_tac x="(s \<union> t \<union> t')" in exI)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3660
    apply (auto simp: intro!: connected_Un cons)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3661
    done
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3662
qed
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3663
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3664
text\<open>The proof is virtually the same as that above.\<close>
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3665
lemma connected_with_outside:
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3666
    fixes s :: "'a :: real_normed_vector set"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3667
    assumes s: "closed s" and cons: "connected s"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3668
      shows "connected(s \<union> outside s)"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3669
proof (cases "s \<union> outside s = UNIV")
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3670
  case True with assms show ?thesis by auto
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3671
next
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3672
  case False
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3673
  then obtain b where b: "b \<notin> s" "b \<notin> outside s" by blast
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3674
  have *: "\<exists>y t. y \<in> s \<and> connected t \<and> a \<in> t \<and> y \<in> t \<and> t \<subseteq> (s \<union> outside s)" if "a \<in> (s \<union> outside s)" for a
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3675
  using that proof
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3676
    assume "a \<in> s" then show ?thesis
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3677
      apply (rule_tac x=a in exI)
68096
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  3678
      apply (rule_tac x="{a}" in exI, simp)
61518
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3679
      done
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3680
  next
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3681
    assume a: "a \<in> outside s"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3682
    show ?thesis
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3683
      apply (rule exists_path_subpath_to_frontier [OF path_linepath [of a b], of "outside s"])
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3684
      using a apply (simp add: closure_def)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3685
      apply (simp add: b)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3686
      apply (rule_tac x="pathfinish h" in exI)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3687
      apply (rule_tac x="path_image h" in exI)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3688
      apply (simp add: pathfinish_in_path_image connected_path_image, auto)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3689
      using frontier_outside_subset s apply fastforce
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3690
      by (metis (no_types, lifting) frontier_outside_subset insertE insert_Diff interior_eq open_outside pathfinish_in_path_image s subsetCE)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3691
  qed
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3692
  show ?thesis
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3693
    apply (simp add: connected_iff_connected_component)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3694
    apply (simp add: connected_component_def)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3695
    apply (clarify dest!: *)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3696
    apply (rename_tac u u' t t')
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3697
    apply (rule_tac x="(s \<union> t \<union> t')" in exI)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3698
    apply (auto simp: intro!: connected_Un cons)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3699
    done
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3700
qed
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3701
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3702
lemma inside_inside_eq_empty [simp]:
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3703
    fixes s :: "'a :: {real_normed_vector, perfect_space} set"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3704
    assumes s: "closed s" and cons: "connected s"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3705
      shows "inside (inside s) = {}"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3706
  by (metis (no_types) unbounded_outside connected_with_outside [OF assms] bounded_Un
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3707
           inside_complement_unbounded_connected_empty unbounded_outside union_with_outside)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3708
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3709
lemma inside_in_components:
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3710
     "inside s \<in> components (- s) \<longleftrightarrow> connected(inside s) \<and> inside s \<noteq> {}"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3711
  apply (simp add: in_components_maximal)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3712
  apply (auto intro: inside_same_component connected_componentI)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3713
  apply (metis IntI empty_iff inside_no_overlap)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3714
  done
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3715
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3716
text\<open>The proof is virtually the same as that above.\<close>
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3717
lemma outside_in_components:
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3718
     "outside s \<in> components (- s) \<longleftrightarrow> connected(outside s) \<and> outside s \<noteq> {}"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3719
  apply (simp add: in_components_maximal)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3720
  apply (auto intro: outside_same_component connected_componentI)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3721
  apply (metis IntI empty_iff outside_no_overlap)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3722
  done
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3723
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3724
lemma bounded_unique_outside:
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3725
    fixes s :: "'a :: euclidean_space set"
69508
2a4c8a2a3f8e tuned headers; ~ -> \<not>
nipkow
parents: 69313
diff changeset
  3726
    shows "\<lbrakk>bounded s; DIM('a) \<ge> 2\<rbrakk> \<Longrightarrow> (c \<in> components (- s) \<and> \<not> bounded c \<longleftrightarrow> c = outside s)"
61518
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3727
  apply (rule iffI)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3728
  apply (metis cobounded_unique_unbounded_components connected_outside double_compl outside_bounded_nonempty outside_in_components unbounded_outside)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3729
  by (simp add: connected_outside outside_bounded_nonempty outside_in_components unbounded_outside)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  3730
69514
58a77f548bb6 tuned headers
nipkow
parents: 69508
diff changeset
  3731
62533
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
  3732
subsection\<open>Condition for an open map's image to contain a ball\<close>
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
  3733
68607
67bb59e49834 make theorem, corollary, and proposition %important for HOL-Analysis manual
immler
parents: 68532
diff changeset
  3734
proposition ball_subset_open_map_image:
62533
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
  3735
  fixes f :: "'a::heine_borel \<Rightarrow> 'b :: {real_normed_vector,heine_borel}"
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
  3736
  assumes contf: "continuous_on (closure S) f"
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
  3737
      and oint: "open (f ` interior S)"
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
  3738
      and le_no: "\<And>z. z \<in> frontier S \<Longrightarrow> r \<le> norm(f z - f a)"
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
  3739
      and "bounded S" "a \<in> S" "0 < r"
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
  3740
    shows "ball (f a) r \<subseteq> f ` S"
68607
67bb59e49834 make theorem, corollary, and proposition %important for HOL-Analysis manual
immler
parents: 68532
diff changeset
  3741
proof (cases "f ` S = UNIV")
62533
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
  3742
  case True then show ?thesis by simp
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
  3743
next
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
  3744
  case False
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
  3745
    obtain w where w: "w \<in> frontier (f ` S)"
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
  3746
               and dw_le: "\<And>y. y \<in> frontier (f ` S) \<Longrightarrow> norm (f a - w) \<le> norm (f a - y)"
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
  3747
      apply (rule distance_attains_inf [of "frontier(f ` S)" "f a"])
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
  3748
      using \<open>a \<in> S\<close> by (auto simp: frontier_eq_empty dist_norm False)
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
  3749
    then obtain \<xi> where \<xi>: "\<And>n. \<xi> n \<in> f ` S" and tendsw: "\<xi> \<longlonglongrightarrow> w"
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
  3750
      by (metis Diff_iff frontier_def closure_sequential)
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
  3751
    then have "\<And>n. \<exists>x \<in> S. \<xi> n = f x" by force
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
  3752
    then obtain z where zs: "\<And>n. z n \<in> S" and fz: "\<And>n. \<xi> n = f (z n)"
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
  3753
      by metis
66447
a1f5c5c26fa6 Replaced subseq with strict_mono
eberlm <eberlm@in.tum.de>
parents: 65038
diff changeset
  3754
    then obtain y K where y: "y \<in> closure S" and "strict_mono (K :: nat \<Rightarrow> nat)" 
a1f5c5c26fa6 Replaced subseq with strict_mono
eberlm <eberlm@in.tum.de>
parents: 65038
diff changeset
  3755
                      and Klim: "(z \<circ> K) \<longlonglongrightarrow> y"
62533
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
  3756
      using \<open>bounded S\<close>
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
  3757
      apply (simp add: compact_closure [symmetric] compact_def)
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
  3758
      apply (drule_tac x=z in spec)
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
  3759
      using closure_subset apply force
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
  3760
      done
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
  3761
    then have ftendsw: "((\<lambda>n. f (z n)) \<circ> K) \<longlonglongrightarrow> w"
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
  3762
      by (metis LIMSEQ_subseq_LIMSEQ fun.map_cong0 fz tendsw)
68096
e58c9ac761cb more tidying
paulson <lp15@cam.ac.uk>
parents: 68072
diff changeset
  3763
    have zKs: "\<And>n. (z \<circ> K) n \<in> S" by (simp add: zs)
63540
f8652d0534fa tuned proofs -- avoid unstructured calculation;
wenzelm
parents: 63469
diff changeset
  3764
    have fz: "f \<circ> z = \<xi>"  "(\<lambda>n. f (z n)) = \<xi>"
62533
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
  3765
      using fz by auto
63540
f8652d0534fa tuned proofs -- avoid unstructured calculation;
wenzelm
parents: 63469
diff changeset
  3766
    then have "(\<xi> \<circ> K) \<longlonglongrightarrow> f y"
62533
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
  3767
      by (metis (no_types) Klim zKs y contf comp_assoc continuous_on_closure_sequentially)
63540
f8652d0534fa tuned proofs -- avoid unstructured calculation;
wenzelm
parents: 63469
diff changeset
  3768
    with fz have wy: "w = f y" using fz LIMSEQ_unique ftendsw by auto
62533
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
  3769
    have rle: "r \<le> norm (f y - f a)"
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
  3770
      apply (rule le_no)
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
  3771
      using w wy oint
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
  3772
      by (force simp: imageI image_mono interiorI interior_subset frontier_def y)
69508
2a4c8a2a3f8e tuned headers; ~ -> \<not>
nipkow
parents: 69313
diff changeset
  3773
    have **: "(b \<inter> (- S) \<noteq> {} \<and> b - (- S) \<noteq> {} \<Longrightarrow> b \<inter> f \<noteq> {})
62533
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
  3774
              \<Longrightarrow> (b \<inter> S \<noteq> {}) \<Longrightarrow> b \<inter> f = {} \<Longrightarrow>
63594
bd218a9320b5 HOL-Multivariate_Analysis: rename theories for more descriptive names
hoelzl
parents: 63540
diff changeset
  3775
              b \<subseteq> S" for b f and S :: "'b set"
62533
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
  3776
      by blast
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
  3777
    show ?thesis
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
  3778
      apply (rule **)   (*such a horrible mess*)
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
  3779
      apply (rule connected_Int_frontier [where t = "f`S", OF connected_ball])
63594
bd218a9320b5 HOL-Multivariate_Analysis: rename theories for more descriptive names
hoelzl
parents: 63540
diff changeset
  3780
      using \<open>a \<in> S\<close> \<open>0 < r\<close>
62533
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
  3781
      apply (auto simp: disjoint_iff_not_equal  dist_norm)
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
  3782
      by (metis dw_le norm_minus_commute not_less order_trans rle wy)
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
  3783
qed
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
  3784
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  3785
end