src/HOL/Multivariate_Analysis/Path_Connected.thy
author paulson <lp15@cam.ac.uk>
Mon, 07 Mar 2016 14:34:45 +0000
changeset 62533 bc25f3916a99
parent 62398 a4b68bf18f8d
child 62618 f7f2467ab854
permissions -rw-r--r--
new material to Blochj's theorem, as well as supporting lemmas
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(*  Title:      HOL/Multivariate_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>Continuous paths and path-connected sets\<close>
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theory Path_Connected
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imports Convex_Euclidean_Space
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begin
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subsection \<open>Paths and Arcs\<close>
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definition 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 pathstart :: "(real \<Rightarrow> 'a::topological_space) \<Rightarrow> 'a"
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  where "pathstart g = g 0"
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definition pathfinish :: "(real \<Rightarrow> 'a::topological_space) \<Rightarrow> 'a"
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  where "pathfinish g = g 1"
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definition 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 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 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 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 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\<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 o g)"
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  unfolding path_def path_image_def
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  using continuous_on_compose by blast
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lemma path_translation_eq:
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  fixes g :: "real \<Rightarrow> 'a :: real_normed_vector"
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  shows "path((\<lambda>x. a + x) o g) = path g"
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proof -
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  have g: "g = (\<lambda>x. -a + x) o ((\<lambda>x. a + x) o g)"
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    by (rule ext) simp
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  show ?thesis
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    unfolding path_def
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    apply safe
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    apply (subst g)
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    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 o 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 o (f o 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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qed
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lemma pathstart_translation: "pathstart((\<lambda>x. a + x) o 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 o 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) o 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 o g) = f(pathfinish g)"
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  by (simp add: pathfinish_def)
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lemma path_image_translation: "path_image((\<lambda>x. a + x) o g) = (\<lambda>x. a + x) ` (path_image g)"
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  by (simp add: image_comp path_image_def)
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lemma path_image_linear_image: "linear f \<Longrightarrow> path_image(f o g) = f ` (path_image g)"
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  by (simp add: image_comp path_image_def)
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lemma reversepath_translation: "reversepath((\<lambda>x. a + x) o g) = (\<lambda>x. a + x) o reversepath g"
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  by (rule ext) (simp add: reversepath_def)
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lemma reversepath_linear_image: "linear f \<Longrightarrow> reversepath(f o g) = f o reversepath g"
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  by (rule ext) (simp add: reversepath_def)
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lemma joinpaths_translation:
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    "((\<lambda>x. a + x) o g1) +++ ((\<lambda>x. a + x) o g2) = (\<lambda>x. a + x) o (g1 +++ g2)"
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  by (rule ext) (simp add: joinpaths_def)
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lemma joinpaths_linear_image: "linear f \<Longrightarrow> (f o g1) +++ (f o g2) = f o (g1 +++ g2)"
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  by (rule ext) (simp add: joinpaths_def)
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lemma simple_path_translation_eq:
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  fixes g :: "real \<Rightarrow> 'a::euclidean_space"
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  shows "simple_path((\<lambda>x. a + x) o g) = simple_path g"
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  by (simp add: simple_path_def path_translation_eq)
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lemma simple_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 "simple_path(f o g) = simple_path g"
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  using assms inj_on_eq_iff [of f]
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  by (auto simp: path_linear_image_eq simple_path_def path_translation_eq)
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lemma arc_translation_eq:
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  fixes g :: "real \<Rightarrow> 'a::euclidean_space"
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  shows "arc((\<lambda>x. a + x) o g) = arc g"
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  by (auto simp: arc_def inj_on_def path_translation_eq)
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lemma arc_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  "arc(f o g) = arc g"
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  using assms inj_on_eq_iff [of f]
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  by (auto simp: arc_def inj_on_def path_linear_image_eq)
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subsection\<open>Basic lemmas about paths\<close>
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lemma arc_imp_simple_path: "arc g \<Longrightarrow> simple_path g"
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  by (simp add: arc_def inj_on_def simple_path_def)
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lemma arc_imp_path: "arc g \<Longrightarrow> path g"
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  using arc_def by blast
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   142
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lemma simple_path_imp_path: "simple_path g \<Longrightarrow> path g"
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   144
  using simple_path_def by blast
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lemma simple_path_cases: "simple_path g \<Longrightarrow> arc g \<or> pathfinish g = pathstart g"
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  unfolding simple_path_def arc_def inj_on_def pathfinish_def pathstart_def
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  by (force)
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diff changeset
   150
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
   151
  using simple_path_cases by auto
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   152
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   153
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
   154
  unfolding arc_def inj_on_def pathfinish_def pathstart_def
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   155
  by fastforce
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   156
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   157
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
   158
  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
   159
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   160
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
   161
  by (simp add: arc_simple_path)
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
   162
60974
6a6f15d8fbc4 New material and fixes related to the forthcoming Stone-Weierstrass development
paulson <lp15@cam.ac.uk>
parents: 60809
diff changeset
   163
lemma path_image_nonempty [simp]: "path_image g \<noteq> {}"
56188
0268784f60da use cbox to relax class constraints
immler
parents: 53640
diff changeset
   164
  unfolding path_image_def image_is_empty box_eq_empty
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   165
  by auto
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
   166
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   167
lemma pathstart_in_path_image[intro]: "pathstart g \<in> path_image g"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   168
  unfolding pathstart_def path_image_def
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   169
  by auto
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
   170
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   171
lemma pathfinish_in_path_image[intro]: "pathfinish g \<in> path_image g"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   172
  unfolding pathfinish_def path_image_def
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   173
  by auto
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   174
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   175
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
   176
  unfolding path_def path_image_def
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   177
  using connected_continuous_image connected_Icc by blast
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
   178
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   179
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
   180
  unfolding path_def path_image_def
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   181
  using compact_continuous_image connected_Icc by blast
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
   182
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   183
lemma reversepath_reversepath[simp]: "reversepath (reversepath g) = g"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   184
  unfolding reversepath_def
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   185
  by auto
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
   186
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   187
lemma pathstart_reversepath[simp]: "pathstart (reversepath g) = pathfinish g"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   188
  unfolding pathstart_def reversepath_def pathfinish_def
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   189
  by auto
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
   190
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   191
lemma pathfinish_reversepath[simp]: "pathfinish (reversepath g) = pathstart g"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   192
  unfolding pathstart_def reversepath_def pathfinish_def
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   193
  by auto
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
   194
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   195
lemma pathstart_join[simp]: "pathstart (g1 +++ g2) = pathstart g1"
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   196
  unfolding pathstart_def joinpaths_def pathfinish_def
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   197
  by auto
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
   198
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   199
lemma pathfinish_join[simp]: "pathfinish (g1 +++ g2) = pathfinish g2"
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   200
  unfolding pathstart_def joinpaths_def pathfinish_def
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   201
  by auto
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
   202
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   203
lemma path_image_reversepath[simp]: "path_image (reversepath g) = path_image g"
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   204
proof -
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   205
  have *: "\<And>g. path_image (reversepath g) \<subseteq> path_image g"
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   206
    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
   207
    by force
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   208
  show ?thesis
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   209
    using *[of g] *[of "reversepath g"]
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   210
    unfolding reversepath_reversepath
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   211
    by auto
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   212
qed
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
   213
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   214
lemma path_reversepath [simp]: "path (reversepath g) \<longleftrightarrow> path g"
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   215
proof -
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   216
  have *: "\<And>g. path g \<Longrightarrow> path (reversepath g)"
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   217
    unfolding path_def reversepath_def
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   218
    apply (rule continuous_on_compose[unfolded o_def, of _ "\<lambda>x. 1 - x"])
56371
fb9ae0727548 extend continuous_intros; remove continuous_on_intros and isCont_intros
hoelzl
parents: 56188
diff changeset
   219
    apply (intro continuous_intros)
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   220
    apply (rule continuous_on_subset[of "{0..1}"])
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   221
    apply assumption
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   222
    apply auto
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   223
    done
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   224
  show ?thesis
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   225
    using *[of "reversepath g"] *[of g]
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   226
    unfolding reversepath_reversepath
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   227
    by (rule iffI)
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   228
qed
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   229
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   230
lemma arc_reversepath:
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   231
  assumes "arc g" shows "arc(reversepath g)"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   232
proof -
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   233
  have injg: "inj_on g {0..1}"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   234
    using assms
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   235
    by (simp add: arc_def)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   236
  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
   237
    by simp
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   238
  show ?thesis
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   239
    apply (auto simp: arc_def inj_on_def path_reversepath)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   240
    apply (simp add: arc_imp_path assms)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   241
    apply (rule **)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   242
    apply (rule inj_onD [OF injg])
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   243
    apply (auto simp: reversepath_def)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   244
    done
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   245
qed
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   246
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   247
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
   248
  apply (simp add: simple_path_def)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   249
  apply (force simp: reversepath_def)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   250
  done
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   251
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   252
lemmas reversepath_simps =
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   253
  path_reversepath path_image_reversepath pathstart_reversepath pathfinish_reversepath
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
   254
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   255
lemma path_join[simp]:
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   256
  assumes "pathfinish g1 = pathstart g2"
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   257
  shows "path (g1 +++ g2) \<longleftrightarrow> path g1 \<and> path g2"
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   258
  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
   259
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
   260
  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
   261
  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
   262
    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
   263
  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
   264
    using assms
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   265
    by (intro continuous_on_cong refl) (auto simp: joinpaths_def pathfinish_def pathstart_def)
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   266
  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
   267
    unfolding g1 g2
56371
fb9ae0727548 extend continuous_intros; remove continuous_on_intros and isCont_intros
hoelzl
parents: 56188
diff changeset
   268
    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
   269
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
   270
  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
   271
  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
   272
    by auto
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   273
  {
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   274
    fix x :: real
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   275
    assume "0 \<le> x" and "x \<le> 1"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   276
    then have "x \<in> (\<lambda>x. x * 2) ` {0..1 / 2}"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   277
      by (intro image_eqI[where x="x/2"]) auto
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   278
  }
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
   279
  note 1 = this
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   280
  {
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   281
    fix x :: real
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   282
    assume "0 \<le> x" and "x \<le> 1"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   283
    then have "x \<in> (\<lambda>x. x * 2 - 1) ` {1 / 2..1}"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   284
      by (intro image_eqI[where x="x/2 + 1/2"]) auto
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   285
  }
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
   286
  note 2 = this
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   287
  show "continuous_on {0..1} (g1 +++ g2)"
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   288
    using assms
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   289
    unfolding joinpaths_def 01
56371
fb9ae0727548 extend continuous_intros; remove continuous_on_intros and isCont_intros
hoelzl
parents: 56188
diff changeset
   290
    apply (intro continuous_on_cases closed_atLeastAtMost g1g2[THEN continuous_on_compose2] continuous_intros)
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   291
    apply (auto simp: field_simps pathfinish_def pathstart_def intro!: 1 2)
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   292
    done
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   293
qed
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
   294
60420
884f54e01427 isabelle update_cartouches;
wenzelm
parents: 60303
diff changeset
   295
section \<open>Path Images\<close>
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   296
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   297
lemma bounded_path_image: "path g \<Longrightarrow> bounded(path_image g)"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   298
  by (simp add: compact_imp_bounded compact_path_image)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   299
60809
457abb82fb9e the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents: 60420
diff changeset
   300
lemma closed_path_image:
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   301
  fixes g :: "real \<Rightarrow> 'a::t2_space"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   302
  shows "path g \<Longrightarrow> closed(path_image g)"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   303
  by (metis compact_path_image compact_imp_closed)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   304
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   305
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
   306
  by (metis connected_path_image simple_path_imp_path)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   307
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   308
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
   309
  by (metis compact_path_image simple_path_imp_path)
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 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
   312
  by (metis bounded_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 closed_simple_path_image:
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   315
  fixes g :: "real \<Rightarrow> 'a::t2_space"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   316
  shows "simple_path g \<Longrightarrow> closed(path_image g)"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   317
  by (metis closed_path_image simple_path_imp_path)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   318
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   319
lemma connected_arc_image: "arc g \<Longrightarrow> connected(path_image g)"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   320
  by (metis connected_path_image arc_imp_path)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   321
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   322
lemma compact_arc_image: "arc g \<Longrightarrow> compact(path_image g)"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   323
  by (metis compact_path_image arc_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 bounded_arc_image: "arc g \<Longrightarrow> bounded(path_image g)"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   326
  by (metis bounded_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 closed_arc_image:
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   329
  fixes g :: "real \<Rightarrow> 'a::t2_space"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   330
  shows "arc g \<Longrightarrow> closed(path_image g)"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   331
  by (metis closed_path_image arc_imp_path)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   332
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   333
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
   334
  unfolding path_image_def joinpaths_def
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   335
  by auto
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
   336
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
   337
lemma subset_path_image_join:
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   338
  assumes "path_image g1 \<subseteq> s"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   339
    and "path_image g2 \<subseteq> s"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   340
  shows "path_image (g1 +++ g2) \<subseteq> s"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   341
  using path_image_join_subset[of g1 g2] and assms
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   342
  by auto
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
   343
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
   344
lemma path_image_join:
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   345
    "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
   346
  apply (rule subset_antisym [OF path_image_join_subset])
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   347
  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
   348
  apply (drule sym)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   349
  apply (rule_tac x="xa/2" in bexI, auto)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   350
  apply (rule ccontr)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   351
  apply (drule_tac x="(xa+1)/2" in bspec)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   352
  apply (auto simp: field_simps)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   353
  apply (drule_tac x="1/2" in bspec, auto)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   354
  done
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
   355
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
   356
lemma not_in_path_image_join:
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   357
  assumes "x \<notin> path_image g1"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   358
    and "x \<notin> path_image g2"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   359
  shows "x \<notin> path_image (g1 +++ g2)"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   360
  using assms and path_image_join_subset[of g1 g2]
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   361
  by auto
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
   362
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   363
lemma pathstart_compose: "pathstart(f o p) = f(pathstart p)"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   364
  by (simp add: pathstart_def)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   365
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   366
lemma pathfinish_compose: "pathfinish(f o p) = f(pathfinish p)"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   367
  by (simp add: pathfinish_def)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   368
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   369
lemma path_image_compose: "path_image (f o p) = f ` (path_image p)"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   370
  by (simp add: image_comp path_image_def)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   371
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   372
lemma path_compose_join: "f o (p +++ q) = (f o p) +++ (f o q)"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   373
  by (rule ext) (simp add: joinpaths_def)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   374
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   375
lemma path_compose_reversepath: "f o reversepath p = reversepath(f o p)"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   376
  by (rule ext) (simp add: reversepath_def)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   377
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
   378
lemma joinpaths_eq:
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   379
  "(\<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
   380
   (\<And>t. t \<in> {0..1} \<Longrightarrow> q t = q' t)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   381
   \<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
   382
  by (auto simp: joinpaths_def)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   383
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   384
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
   385
  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
   386
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   387
60420
884f54e01427 isabelle update_cartouches;
wenzelm
parents: 60303
diff changeset
   388
subsection\<open>Simple paths with the endpoints removed\<close>
60303
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_endless:
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   391
    "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
   392
  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
   393
  apply (metis eq_iff le_less_linear)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   394
  apply (metis leD linear)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   395
  using less_eq_real_def zero_le_one apply blast
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   396
  using less_eq_real_def zero_le_one apply blast
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   397
  done
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
   398
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   399
lemma connected_simple_path_endless:
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   400
    "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
   401
apply (simp add: simple_path_endless)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   402
apply (rule connected_continuous_image)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   403
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
   404
by auto
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   405
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   406
lemma nonempty_simple_path_endless:
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   407
    "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
   408
  by (simp add: simple_path_endless)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   409
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   410
60420
884f54e01427 isabelle update_cartouches;
wenzelm
parents: 60303
diff changeset
   411
subsection\<open>The operations on paths\<close>
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   412
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   413
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
   414
  by (auto simp: path_image_def reversepath_def)
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
lemma path_imp_reversepath: "path g \<Longrightarrow> path(reversepath g)"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   417
  apply (auto simp: path_def reversepath_def)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   418
  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
   419
  apply (auto simp: continuous_on_op_minus)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   420
  done
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   421
61204
3e491e34a62e new lemmas and movement of lemmas into place
paulson
parents: 60974
diff changeset
   422
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
   423
  by simp
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   424
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   425
lemma continuous_on_joinpaths:
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   426
  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
   427
    shows "continuous_on {0..1} (g1 +++ g2)"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   428
proof -
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   429
  have *: "{0..1::real} = {0..1/2} \<union> {1/2..1}"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   430
    by auto
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   431
  have gg: "g2 0 = g1 1"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   432
    by (metis assms(3) pathfinish_def pathstart_def)
61204
3e491e34a62e new lemmas and movement of lemmas into place
paulson
parents: 60974
diff changeset
   433
  have 1: "continuous_on {0..1/2} (g1 +++ g2)"
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   434
    apply (rule continuous_on_eq [of _ "g1 o (\<lambda>x. 2*x)"])
61204
3e491e34a62e new lemmas and movement of lemmas into place
paulson
parents: 60974
diff changeset
   435
    apply (rule continuous_intros | simp add: joinpaths_def assms)+
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   436
    done
61204
3e491e34a62e new lemmas and movement of lemmas into place
paulson
parents: 60974
diff changeset
   437
  have "continuous_on {1/2..1} (g2 o (\<lambda>x. 2*x-1))"
3e491e34a62e new lemmas and movement of lemmas into place
paulson
parents: 60974
diff changeset
   438
    apply (rule continuous_on_subset [of "{1/2..1}"])
3e491e34a62e new lemmas and movement of lemmas into place
paulson
parents: 60974
diff changeset
   439
    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
   440
    done
3e491e34a62e new lemmas and movement of lemmas into place
paulson
parents: 60974
diff changeset
   441
  then have 2: "continuous_on {1/2..1} (g1 +++ g2)"
3e491e34a62e new lemmas and movement of lemmas into place
paulson
parents: 60974
diff changeset
   442
    apply (rule continuous_on_eq [of "{1/2..1}" "g2 o (\<lambda>x. 2*x-1)"])
3e491e34a62e new lemmas and movement of lemmas into place
paulson
parents: 60974
diff changeset
   443
    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
   444
    done
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   445
  show ?thesis
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   446
    apply (subst *)
62397
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62381
diff changeset
   447
    apply (rule continuous_on_closed_Un)
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   448
    using 1 2
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   449
    apply auto
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
qed
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   452
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   453
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
   454
  by (simp add: path_join)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   455
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
   456
lemma simple_path_join_loop:
60809
457abb82fb9e the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents: 60420
diff changeset
   457
  assumes "arc g1" "arc g2"
457abb82fb9e the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents: 60420
diff changeset
   458
          "pathfinish g1 = pathstart g2"  "pathfinish g2 = pathstart g1"
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   459
          "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
   460
  shows "simple_path(g1 +++ g2)"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   461
proof -
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   462
  have injg1: "inj_on g1 {0..1}"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   463
    using assms
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   464
    by (simp add: arc_def)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   465
  have injg2: "inj_on g2 {0..1}"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   466
    using assms
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   467
    by (simp add: arc_def)
60809
457abb82fb9e the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents: 60420
diff changeset
   468
  have g12: "g1 1 = g2 0"
457abb82fb9e the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents: 60420
diff changeset
   469
   and g21: "g2 1 = g1 0"
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   470
   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
   471
    using assms
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   472
    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
   473
  { fix x and y::real
60809
457abb82fb9e the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents: 60420
diff changeset
   474
    assume xyI: "x = 1 \<longrightarrow> y \<noteq> 0"
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   475
       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
   476
    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
   477
      using xy
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   478
      apply simp
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   479
      apply (rule_tac x="2 * x - 1" in image_eqI, auto)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   480
      done
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   481
    have False
60809
457abb82fb9e the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents: 60420
diff changeset
   482
      using subsetD [OF sb g1im] xy
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   483
      apply auto
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   484
      apply (drule inj_onD [OF injg1])
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   485
      using g21 [symmetric] xyI
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   486
      apply (auto dest: inj_onD [OF injg2])
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   487
      done
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   488
   } note * = this
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   489
  { fix x and y::real
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   490
    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
   491
    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
   492
      using xy
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   493
      apply simp
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   494
      apply (rule_tac x="2 * x" in image_eqI, auto)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   495
      done
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   496
    have "x = 0 \<and> y = 1"
60809
457abb82fb9e the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents: 60420
diff changeset
   497
      using subsetD [OF sb g1im] xy
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   498
      apply auto
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   499
      apply (force dest: inj_onD [OF injg1])
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   500
      using  g21 [symmetric]
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   501
      apply (auto dest: inj_onD [OF injg2])
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   502
      done
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   503
   } note ** = this
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   504
  show ?thesis
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   505
    using assms
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   506
    apply (simp add: arc_def simple_path_def path_join, clarify)
62390
842917225d56 more canonical names
nipkow
parents: 62087
diff changeset
   507
    apply (simp add: joinpaths_def split: if_split_asm)
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   508
    apply (force dest: inj_onD [OF injg1])
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   509
    apply (metis *)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   510
    apply (metis **)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   511
    apply (force dest: inj_onD [OF injg2])
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   512
    done
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   513
qed
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   514
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   515
lemma arc_join:
60809
457abb82fb9e the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents: 60420
diff changeset
   516
  assumes "arc g1" "arc g2"
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   517
          "pathfinish g1 = pathstart g2"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   518
          "path_image g1 \<inter> path_image g2 \<subseteq> {pathstart g2}"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   519
    shows "arc(g1 +++ g2)"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   520
proof -
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   521
  have injg1: "inj_on g1 {0..1}"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   522
    using assms
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   523
    by (simp add: arc_def)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   524
  have injg2: "inj_on g2 {0..1}"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   525
    using assms
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   526
    by (simp add: arc_def)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   527
  have g11: "g1 1 = g2 0"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   528
   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
   529
    using assms
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   530
    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
   531
  { fix x and y::real
60809
457abb82fb9e the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents: 60420
diff changeset
   532
    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
   533
    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
   534
      using xy
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   535
      apply simp
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   536
      apply (rule_tac x="2 * x - 1" in image_eqI, auto)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   537
      done
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   538
    have False
60809
457abb82fb9e the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents: 60420
diff changeset
   539
      using subsetD [OF sb g1im] xy
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   540
      by (auto dest: inj_onD [OF injg2])
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   541
   } note * = this
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   542
  show ?thesis
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   543
    apply (simp add: arc_def inj_on_def)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   544
    apply (clarsimp simp add: arc_imp_path assms path_join)
62390
842917225d56 more canonical names
nipkow
parents: 62087
diff changeset
   545
    apply (simp add: joinpaths_def split: if_split_asm)
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   546
    apply (force dest: inj_onD [OF injg1])
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   547
    apply (metis *)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   548
    apply (metis *)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   549
    apply (force dest: inj_onD [OF injg2])
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   550
    done
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   551
qed
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   552
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   553
lemma reversepath_joinpaths:
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   554
    "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
   555
  unfolding reversepath_def pathfinish_def pathstart_def joinpaths_def
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   556
  by (rule ext) (auto simp: mult.commute)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   557
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   558
62533
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   559
subsection\<open>Some reversed and "if and only if" versions of joining theorems\<close>
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   560
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   561
lemma path_join_path_ends: 
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   562
  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
   563
  assumes "path(g1 +++ g2)" "path g2" 
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   564
    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
   565
proof (rule ccontr)
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   566
  def e \<equiv> "dist (g1 1) (g2 0)"
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   567
  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
   568
  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
   569
    by auto
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   570
  then have "e > 0"
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   571
    by (metis e_def pathfinish_def pathstart_def) 
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   572
  then obtain d1 where "d1 > 0" 
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   573
       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
   574
    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
   575
    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
   576
    by (metis half_gt_zero_iff norm_conv_dist)
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   577
  obtain d2 where "d2 > 0" 
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   578
       and d2: "\<And>x'. \<lbrakk>x'\<in>{0..1}; dist x' (1/2) < d2\<rbrakk> 
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   579
                      \<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
   580
    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
   581
    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
   582
    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
   583
    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
   584
    done
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   585
  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
   586
    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
   587
  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
   588
    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
   589
  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
   590
    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
   591
  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
   592
    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
   593
  have [simp]: "~ min (1 / 2) (min d1 d2) \<le> 0"
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)
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   595
  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
   596
       "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
   597
    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
   598
  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
   599
    using dist_triangle_half_r e_def by blast
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   600
  then show False 
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   601
    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
   602
qed
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   603
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   604
lemma path_join_eq [simp]:  
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   605
  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
   606
  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
   607
    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
   608
  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
   609
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   610
lemma simple_path_joinE: 
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   611
  assumes "simple_path(g1 +++ g2)" and "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
   612
  obtains "arc g1" "arc g2" 
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   613
          "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
   614
proof -
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   615
  have *: "\<And>x y. \<lbrakk>0 \<le> x; x \<le> 1; 0 \<le> y; y \<le> 1; (g1 +++ g2) x = (g1 +++ g2) y\<rbrakk> 
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   616
               \<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
   617
    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
   618
  have "path g1" 
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   619
    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
   620
  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
   621
  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
   622
    fix x y
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   623
    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
   624
    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
   625
      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
   626
  qed
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   627
  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
   628
    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
   629
  have [simp]: "g2 0 = g1 1"
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   630
    using assms by (metis pathfinish_def pathstart_def) 
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   631
  have "path g2"
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   632
    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
   633
  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
   634
  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
   635
    fix x y
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   636
    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
   637
    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
   638
      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
   639
      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
   640
  qed
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   641
  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
   642
    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
   643
  have "g2 y = g1 0 \<or> g2 y = g1 1" 
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   644
       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
   645
      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
   646
      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
   647
  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
   648
    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
   649
  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
   650
qed
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   651
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   652
lemma simple_path_join_loop_eq:
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   653
  assumes "pathfinish g2 = pathstart g1" "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
   654
    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
   655
             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
   656
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
   657
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   658
lemma arc_join_eq:
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   659
  assumes "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
   660
    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
   661
           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
   662
           (is "?lhs = ?rhs")
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   663
proof 
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   664
  assume ?lhs
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   665
  then have "simple_path(g1 +++ g2)" by (rule arc_imp_simple_path)
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   666
  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> 
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   667
               \<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
   668
    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
   669
  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
   670
    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
   671
    by (auto simp: joinpaths_def pathstart_def pathfinish_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
   672
  then have n1: "~ (pathstart g1 \<in> path_image g2)"
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   673
    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
   674
    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
   675
  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
   676
    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
   677
    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
   678
next
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   679
  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
   680
    using assms
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   681
    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
   682
qed
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   683
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   684
lemma arc_join_eq_alt: 
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   685
        "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
   686
        \<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
   687
             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
   688
             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
   689
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
   690
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   691
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   692
subsection\<open>The joining of paths is associative\<close>
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   693
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   694
lemma path_assoc:
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   695
    "\<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
   696
     \<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
   697
by simp
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   698
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   699
lemma simple_path_assoc: 
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   700
  assumes "pathfinish p = pathstart q" "pathfinish q = pathstart r" 
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   701
    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
   702
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
   703
  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
   704
  proof
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   705
    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
   706
    with assms True show "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
   707
      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
   708
                    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
   709
  next
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   710
    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
   711
    with assms True have q: "pathfinish r \<notin> path_image q"
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   712
      using arc_distinct_ends  
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   713
      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
   714
    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
   715
      using assms
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   716
      by (metis 0 IntI arc_distinct_ends arc_join_eq_alt empty_iff insert_iff 
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   717
              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
   718
    with assms 0 q True show "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
   719
      by (auto simp: 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
               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
   721
  qed
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   722
next
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   723
  case False
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   724
  { fix x :: 'a
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   725
    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
   726
              "(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
   727
              "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
   728
    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
   729
      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
   730
    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
   731
      by blast
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   732
  }
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   733
  with False assms show ?thesis 
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   734
    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
   735
qed
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   736
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   737
lemma arc_assoc: 
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   738
     "\<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
   739
      \<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
   740
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
   741
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
   742
61518
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   743
section\<open>Choosing a subpath of an existing path\<close>
60809
457abb82fb9e the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents: 60420
diff changeset
   744
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   745
definition subpath :: "real \<Rightarrow> real \<Rightarrow> (real \<Rightarrow> 'a) \<Rightarrow> real \<Rightarrow> 'a::real_normed_vector"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   746
  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
   747
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
   748
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
   749
  fixes g :: "_ \<Rightarrow> 'a::real_normed_vector"
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   750
  shows "path_image(subpath u v g) = g ` (closed_segment u v)"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   751
  apply (simp add: closed_segment_real_eq path_image_def subpath_def)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   752
  apply (subst o_def [of g, symmetric])
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   753
  apply (simp add: image_comp [symmetric])
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   754
  done
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   755
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
   756
lemma path_image_subpath:
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   757
  fixes g :: "real \<Rightarrow> 'a::real_normed_vector"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   758
  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
   759
  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
   760
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   761
lemma path_subpath [simp]:
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   762
  fixes g :: "real \<Rightarrow> 'a::real_normed_vector"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   763
  assumes "path g" "u \<in> {0..1}" "v \<in> {0..1}"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   764
    shows "path(subpath u v g)"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   765
proof -
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   766
  have "continuous_on {0..1} (g o (\<lambda>x. ((v-u) * x+ u)))"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   767
    apply (rule continuous_intros | simp)+
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   768
    apply (simp add: image_affinity_atLeastAtMost [where c=u])
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   769
    using assms
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   770
    apply (auto simp: path_def continuous_on_subset)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   771
    done
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   772
  then show ?thesis
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   773
    by (simp add: path_def subpath_def)
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   774
qed
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
   775
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   776
lemma pathstart_subpath [simp]: "pathstart(subpath u v g) = g(u)"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   777
  by (simp add: pathstart_def subpath_def)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   778
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   779
lemma pathfinish_subpath [simp]: "pathfinish(subpath u v g) = g(v)"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   780
  by (simp add: pathfinish_def subpath_def)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   781
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   782
lemma subpath_trivial [simp]: "subpath 0 1 g = g"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   783
  by (simp add: subpath_def)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   784
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   785
lemma subpath_reversepath: "subpath 1 0 g = reversepath g"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   786
  by (simp add: reversepath_def subpath_def)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   787
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   788
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
   789
  by (simp add: reversepath_def subpath_def algebra_simps)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   790
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   791
lemma subpath_translation: "subpath u v ((\<lambda>x. a + x) o g) = (\<lambda>x. a + x) o subpath u v g"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   792
  by (rule ext) (simp add: subpath_def)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   793
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   794
lemma subpath_linear_image: "linear f \<Longrightarrow> subpath u v (f o g) = f o subpath u v g"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   795
  by (rule ext) (simp add: subpath_def)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   796
60809
457abb82fb9e the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents: 60420
diff changeset
   797
lemma affine_ineq:
457abb82fb9e the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents: 60420
diff changeset
   798
  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
   799
  assumes "x \<le> 1" "v \<le> u"
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   800
    shows "v + x * u \<le> u + x * v"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   801
proof -
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   802
  have "(1-x)*(u-v) \<ge> 0"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   803
    using assms by auto
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   804
  then show ?thesis
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   805
    by (simp add: algebra_simps)
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   806
qed
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
   807
61711
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
   808
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
   809
  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
   810
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
   811
60809
457abb82fb9e the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents: 60420
diff changeset
   812
lemma simple_path_subpath_eq:
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   813
  "simple_path(subpath u v g) \<longleftrightarrow>
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   814
     path(subpath u v g) \<and> u\<noteq>v \<and>
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   815
     (\<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
   816
                \<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
   817
    (is "?lhs = ?rhs")
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   818
proof (rule iffI)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   819
  assume ?lhs
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   820
  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
   821
        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
   822
                  \<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
   823
    by (auto simp: simple_path_def subpath_def)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   824
  { fix x y
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   825
    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
   826
    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
   827
    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
   828
    by (auto simp: closed_segment_real_eq image_affinity_atLeastAtMost divide_simps
62390
842917225d56 more canonical names
nipkow
parents: 62087
diff changeset
   829
       split: if_split_asm)
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   830
  } moreover
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   831
  have "path(subpath u v g) \<and> u\<noteq>v"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   832
    using sim [of "1/3" "2/3"] p
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   833
    by (auto simp: subpath_def)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   834
  ultimately show ?rhs
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   835
    by metis
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   836
next
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   837
  assume ?rhs
60809
457abb82fb9e the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents: 60420
diff changeset
   838
  then
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   839
  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
   840
   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
   841
   and ne: "u < v \<or> v < u"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   842
   and psp: "path (subpath u v g)"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   843
    by (auto simp: closed_segment_real_eq image_affinity_atLeastAtMost)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   844
  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
   845
    by algebra
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   846
  show ?lhs using psp ne
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   847
    unfolding simple_path_def subpath_def
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   848
    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
   849
qed
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   850
60809
457abb82fb9e the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents: 60420
diff changeset
   851
lemma arc_subpath_eq:
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   852
  "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
   853
    (is "?lhs = ?rhs")
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   854
proof (rule iffI)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   855
  assume ?lhs
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   856
  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
   857
        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
   858
                  \<Longrightarrow> x = y)"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   859
    by (auto simp: arc_def inj_on_def subpath_def)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   860
  { fix x y
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   861
    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
   862
    then have "x = y"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   863
    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
   864
    by (force simp add: inj_on_def closed_segment_real_eq image_affinity_atLeastAtMost divide_simps
62390
842917225d56 more canonical names
nipkow
parents: 62087
diff changeset
   865
       split: if_split_asm)
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   866
  } moreover
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   867
  have "path(subpath u v g) \<and> u\<noteq>v"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   868
    using sim [of "1/3" "2/3"] p
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   869
    by (auto simp: subpath_def)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   870
  ultimately show ?rhs
60809
457abb82fb9e the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents: 60420
diff changeset
   871
    unfolding inj_on_def
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   872
    by metis
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   873
next
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   874
  assume ?rhs
60809
457abb82fb9e the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents: 60420
diff changeset
   875
  then
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   876
  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
   877
   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
   878
   and ne: "u < v \<or> v < u"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   879
   and psp: "path (subpath u v g)"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   880
    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
   881
  show ?lhs using psp ne
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   882
    unfolding arc_def subpath_def inj_on_def
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   883
    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
   884
qed
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   885
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   886
60809
457abb82fb9e the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents: 60420
diff changeset
   887
lemma simple_path_subpath:
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   888
  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
   889
  shows "simple_path(subpath u v g)"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   890
  using assms
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   891
  apply (simp add: simple_path_subpath_eq simple_path_imp_path)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   892
  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
   893
  done
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   894
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   895
lemma arc_simple_path_subpath:
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   896
    "\<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
   897
  by (force intro: simple_path_subpath simple_path_imp_arc)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   898
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   899
lemma arc_subpath_arc:
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   900
    "\<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
   901
  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
   902
60809
457abb82fb9e the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents: 60420
diff changeset
   903
lemma arc_simple_path_subpath_interior:
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   904
    "\<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
   905
    apply (rule arc_simple_path_subpath)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   906
    apply (force simp: simple_path_def)+
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   907
    done
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   908
60809
457abb82fb9e the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents: 60420
diff changeset
   909
lemma path_image_subpath_subset:
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   910
    "\<lbrakk>path g; 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
   911
  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
   912
  apply (auto simp: path_image_def)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   913
  done
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   914
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   915
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
   916
  by (rule ext) (simp add: joinpaths_def subpath_def divide_simps)
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   917
61518
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   918
subsection\<open>There is a subpath to the frontier\<close>
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   919
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   920
lemma subpath_to_frontier_explicit:
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   921
    fixes S :: "'a::metric_space set"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   922
    assumes g: "path g" and "pathfinish g \<notin> S"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   923
    obtains u where "0 \<le> u" "u \<le> 1"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   924
                "\<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
   925
                "(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
   926
proof -
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   927
  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
   928
  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
   929
    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
   930
    using compact_eq_bounded_closed apply fastforce
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   931
    done
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   932
  have "1 \<in> {u. g u \<in> closure (- S)}"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   933
    using assms by (simp add: pathfinish_def closure_def)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   934
  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
   935
    using atLeastAtMost_iff zero_le_one by blast
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   936
  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
   937
                  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
   938
    using compact_attains_inf [OF com dis] by fastforce
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   939
  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
   940
    using closure_def by fastforce
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   941
  { assume "u \<noteq> 0"
61808
fc1556774cfe isabelle update_cartouches -c -t;
wenzelm
parents: 61806
diff changeset
   942
    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
   943
    { 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
   944
      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
   945
        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
   946
      have *: "dist (max 0 (u - d / 2)) u \<le> d"
61808
fc1556774cfe isabelle update_cartouches -c -t;
wenzelm
parents: 61806
diff changeset
   947
        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
   948
      have "\<exists>y\<in>S. dist y (g u) < e"
61808
fc1556774cfe isabelle update_cartouches -c -t;
wenzelm
parents: 61806
diff changeset
   949
        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
   950
        by (force intro: d [OF _ *] umin')
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   951
    }
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   952
    then have "g u \<in> closure S"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   953
      by (simp add: frontier_def closure_approachable)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   954
  }
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   955
  then show ?thesis
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   956
    apply (rule_tac u=u in that)
61808
fc1556774cfe isabelle update_cartouches -c -t;
wenzelm
parents: 61806
diff changeset
   957
    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
   958
    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
   959
    done
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   960
qed
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   961
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   962
lemma subpath_to_frontier_strong:
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   963
    assumes g: "path g" and "pathfinish g \<notin> S"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   964
    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
   965
                    "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
   966
proof -
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   967
  obtain u where "0 \<le> u" "u \<le> 1"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   968
             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
   969
             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
   970
    using subpath_to_frontier_explicit [OF assms] by blast
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   971
  show ?thesis
61808
fc1556774cfe isabelle update_cartouches -c -t;
wenzelm
parents: 61806
diff changeset
   972
    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
   973
    apply (simp add: gunot)
61808
fc1556774cfe isabelle update_cartouches -c -t;
wenzelm
parents: 61806
diff changeset
   974
    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
   975
qed
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
lemma subpath_to_frontier:
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   978
    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
   979
    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
   980
proof -
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   981
  obtain u where "0 \<le> u" "u \<le> 1"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   982
             and notin: "g u \<notin> interior S"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   983
             and disj: "u = 0 \<or>
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   984
                        (\<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
   985
    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
   986
  show ?thesis
61808
fc1556774cfe isabelle update_cartouches -c -t;
wenzelm
parents: 61806
diff changeset
   987
    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
   988
    apply (metis DiffI disj frontier_def g0 notin pathstart_def)
61808
fc1556774cfe isabelle update_cartouches -c -t;
wenzelm
parents: 61806
diff changeset
   989
    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
   990
    apply (simp add: path_image_subpath_gen)
61518
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   991
    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
   992
    apply (rename_tac y)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   993
    apply (drule_tac x="y/u" in spec)
62390
842917225d56 more canonical names
nipkow
parents: 62087
diff changeset
   994
    apply (auto split: if_split_asm)
61518
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   995
    done
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   996
qed
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   997
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   998
lemma exists_path_subpath_to_frontier:
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   999
    fixes S :: "'a::real_normed_vector set"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  1000
    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
  1001
    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
  1002
                    "path_image h - {pathfinish h} \<subseteq> interior S"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  1003
                    "pathfinish h \<in> frontier S"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  1004
proof -
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  1005
  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
  1006
    using subpath_to_frontier [OF assms] by blast
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  1007
  show ?thesis
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  1008
    apply (rule that [of "subpath 0 u g"])
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  1009
    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
  1010
    apply (simp_all add: path_image_subpath)
61518
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  1011
    apply (simp add: pathstart_def)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  1012
    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
  1013
    done
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  1014
qed
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  1015
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  1016
lemma exists_path_subpath_to_frontier_closed:
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  1017
    fixes S :: "'a::real_normed_vector set"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  1018
    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
  1019
    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
  1020
                    "pathfinish h \<in> frontier S"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  1021
proof -
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  1022
  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
  1023
                    "path_image h - {pathfinish h} \<subseteq> interior S"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  1024
                    "pathfinish h \<in> frontier S"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  1025
    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
  1026
  show ?thesis
61808
fc1556774cfe isabelle update_cartouches -c -t;
wenzelm
parents: 61806
diff changeset
  1027
    apply (rule that [OF \<open>path h\<close>])
61518
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  1028
    using assms h
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  1029
    apply auto
62087
44841d07ef1d revisions to limits and derivatives, plus new lemmas
paulson
parents: 61808
diff changeset
  1030
    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
  1031
    done
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  1032
qed
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1033
60420
884f54e01427 isabelle update_cartouches;
wenzelm
parents: 60303
diff changeset
  1034
subsection \<open>Reparametrizing a closed curve to start at some chosen point\<close>
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1035
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1036
definition shiftpath :: "real \<Rightarrow> (real \<Rightarrow> 'a::topological_space) \<Rightarrow> real \<Rightarrow> 'a"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1037
  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
  1038
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1039
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
  1040
  unfolding pathstart_def shiftpath_def by auto
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1041
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1042
lemma pathfinish_shiftpath:
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1043
  assumes "0 \<le> a"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1044
    and "pathfinish g = pathstart g"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1045
  shows "pathfinish (shiftpath a g) = g a"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1046
  using assms
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1047
  unfolding pathstart_def pathfinish_def shiftpath_def
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1048
  by auto
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1049
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1050
lemma endpoints_shiftpath:
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1051
  assumes "pathfinish g = pathstart g"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1052
    and "a \<in> {0 .. 1}"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1053
  shows "pathfinish (shiftpath a g) = g a"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1054
    and "pathstart (shiftpath a g) = g a"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1055
  using assms
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1056
  by (auto intro!: pathfinish_shiftpath pathstart_shiftpath)
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1057
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1058
lemma closed_shiftpath:
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1059
  assumes "pathfinish g = pathstart g"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1060
    and "a \<in> {0..1}"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1061
  shows "pathfinish (shiftpath a g) = pathstart (shiftpath a g)"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1062
  using endpoints_shiftpath[OF assms]
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1063
  by auto
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1064
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1065
lemma path_shiftpath:
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1066
  assumes "path g"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1067
    and "pathfinish g = pathstart g"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1068
    and "a \<in> {0..1}"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1069
  shows "path (shiftpath a g)"
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1070
proof -
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1071
  have *: "{0 .. 1} = {0 .. 1-a} \<union> {1-a .. 1}"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1072
    using assms(3) by auto
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1073
  have **: "\<And>x. x + a = 1 \<Longrightarrow> g (x + a - 1) = g (x + a)"
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1074
    using assms(2)[unfolded pathfinish_def pathstart_def]
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1075
    by auto
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1076
  show ?thesis
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1077
    unfolding path_def shiftpath_def *
62397
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62381
diff changeset
  1078
    apply (rule continuous_on_closed_Un)
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1079
    apply (rule closed_real_atLeastAtMost)+
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1080
    apply (rule continuous_on_eq[of _ "g \<circ> (\<lambda>x. a + x)"])
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1081
    prefer 3
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1082
    apply (rule continuous_on_eq[of _ "g \<circ> (\<lambda>x. a - 1 + x)"])
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1083
    prefer 3
56371
fb9ae0727548 extend continuous_intros; remove continuous_on_intros and isCont_intros
hoelzl
parents: 56188
diff changeset
  1084
    apply (rule continuous_intros)+
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1085
    prefer 2
56371
fb9ae0727548 extend continuous_intros; remove continuous_on_intros and isCont_intros
hoelzl
parents: 56188
diff changeset
  1086
    apply (rule continuous_intros)+
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1087
    apply (rule_tac[1-2] continuous_on_subset[OF assms(1)[unfolded path_def]])
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1088
    using assms(3) and **
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1089
    apply auto
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1090
    apply (auto simp add: field_simps)
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1091
    done
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1092
qed
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1093
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1094
lemma shiftpath_shiftpath:
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1095
  assumes "pathfinish g = pathstart g"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1096
    and "a \<in> {0..1}"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1097
    and "x \<in> {0..1}"
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1098
  shows "shiftpath (1 - a) (shiftpath a g) x = g x"
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1099
  using assms
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1100
  unfolding pathfinish_def pathstart_def shiftpath_def
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1101
  by auto
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1102
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1103
lemma path_image_shiftpath:
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1104
  assumes "a \<in> {0..1}"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1105
    and "pathfinish g = pathstart g"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1106
  shows "path_image (shiftpath a g) = path_image g"
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1107
proof -
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1108
  { fix x
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1109
    assume as: "g 1 = g 0" "x \<in> {0..1::real}" " \<forall>y\<in>{0..1} \<inter> {x. \<not> a + x \<le> 1}. g x \<noteq> g (a + y - 1)"
49654
366d8b41ca17 tuned proofs;
wenzelm
parents: 49653
diff changeset
  1110
    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
  1111
    proof (cases "a \<le> x")
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1112
      case False
49654
366d8b41ca17 tuned proofs;
wenzelm
parents: 49653
diff changeset
  1113
      then show ?thesis
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1114
        apply (rule_tac x="1 + x - a" in bexI)
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1115
        using as(1,2) and as(3)[THEN bspec[where x="1 + x - a"]] and assms(1)
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1116
        apply (auto simp add: field_simps atomize_not)
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1117
        done
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1118
    next
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1119
      case True
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1120
      then show ?thesis
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1121
        using as(1-2) and assms(1)
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1122
        apply (rule_tac x="x - a" in bexI)
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1123
        apply (auto simp add: field_simps)
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1124
        done
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1125
    qed
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1126
  }
49654
366d8b41ca17 tuned proofs;
wenzelm
parents: 49653
diff changeset
  1127
  then show ?thesis
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1128
    using assms
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1129
    unfolding shiftpath_def path_image_def pathfinish_def pathstart_def
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1130
    by (auto simp add: image_iff)
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1131
qed
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1132
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1133
60420
884f54e01427 isabelle update_cartouches;
wenzelm
parents: 60303
diff changeset
  1134
subsection \<open>Special case of straight-line paths\<close>
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1135
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1136
definition linepath :: "'a::real_normed_vector \<Rightarrow> 'a \<Rightarrow> real \<Rightarrow> 'a"
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1137
  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
  1138
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1139
lemma pathstart_linepath[simp]: "pathstart (linepath a b) = a"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1140
  unfolding pathstart_def linepath_def
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1141
  by auto
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1142
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1143
lemma pathfinish_linepath[simp]: "pathfinish (linepath a b) = b"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1144
  unfolding pathfinish_def linepath_def
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1145
  by auto
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1146
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1147
lemma continuous_linepath_at[intro]: "continuous (at x) (linepath a b)"
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1148
  unfolding linepath_def
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1149
  by (intro continuous_intros)
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1150
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
  1151
lemma continuous_on_linepath [intro,continuous_intros]: "continuous_on s (linepath a b)"
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1152
  using continuous_linepath_at
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1153
  by (auto intro!: continuous_at_imp_continuous_on)
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1154
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1155
lemma path_linepath[intro]: "path (linepath a b)"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1156
  unfolding path_def
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1157
  by (rule continuous_on_linepath)
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1158
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1159
lemma path_image_linepath[simp]: "path_image (linepath a b) = closed_segment a b"
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1160
  unfolding path_image_def segment linepath_def
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
  1161
  by auto
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1162
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1163
lemma reversepath_linepath[simp]: "reversepath (linepath a b) = linepath b a"
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1164
  unfolding reversepath_def linepath_def
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1165
  by auto
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1166
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
  1167
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
  1168
  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
  1169
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
  1170
lemma arc_linepath:
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1171
  assumes "a \<noteq> b"
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
  1172
  shows "arc (linepath a b)"
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1173
proof -
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1174
  {
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1175
    fix x y :: "real"
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1176
    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
  1177
    then have "(x - y) *\<^sub>R a = (x - y) *\<^sub>R b"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1178
      by (simp add: algebra_simps)
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1179
    with assms have "x = y"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1180
      by simp
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1181
  }
49654
366d8b41ca17 tuned proofs;
wenzelm
parents: 49653
diff changeset
  1182
  then show ?thesis
60809
457abb82fb9e the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents: 60420
diff changeset
  1183
    unfolding arc_def inj_on_def
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
  1184
    by (simp add:  path_linepath) (force simp: algebra_simps linepath_def)
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1185
qed
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1186
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1187
lemma simple_path_linepath[intro]: "a \<noteq> b \<Longrightarrow> simple_path (linepath a b)"
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
  1188
  by (simp add: arc_imp_simple_path arc_linepath)
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1189
61711
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  1190
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
  1191
  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
  1192
61711
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  1193
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
  1194
  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
  1195
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1196
60420
884f54e01427 isabelle update_cartouches;
wenzelm
parents: 60303
diff changeset
  1197
subsection \<open>Bounding a point away from a path\<close>
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1198
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1199
lemma not_on_path_ball:
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1200
  fixes g :: "real \<Rightarrow> 'a::heine_borel"
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1201
  assumes "path g"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1202
    and "z \<notin> path_image g"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1203
  shows "\<exists>e > 0. ball z e \<inter> path_image g = {}"
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1204
proof -
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1205
  obtain a where "a \<in> path_image g" "\<forall>y \<in> path_image g. dist z a \<le> dist z y"
62381
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  1206
    apply (rule distance_attains_inf[OF _ path_image_nonempty, of g z]) 
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1207
    using compact_path_image[THEN compact_imp_closed, OF assms(1)] by auto
49654
366d8b41ca17 tuned proofs;
wenzelm
parents: 49653
diff changeset
  1208
  then show ?thesis
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1209
    apply (rule_tac x="dist z a" in exI)
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1210
    using assms(2)
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1211
    apply (auto intro!: dist_pos_lt)
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1212
    done
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1213
qed
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1214
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1215
lemma not_on_path_cball:
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1216
  fixes g :: "real \<Rightarrow> 'a::heine_borel"
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1217
  assumes "path g"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1218
    and "z \<notin> path_image g"
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1219
  shows "\<exists>e>0. cball z e \<inter> (path_image g) = {}"
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1220
proof -
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1221
  obtain e where "ball z e \<inter> path_image g = {}" "e > 0"
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1222
    using not_on_path_ball[OF assms] by auto
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1223
  moreover have "cball z (e/2) \<subseteq> ball z e"
60420
884f54e01427 isabelle update_cartouches;
wenzelm
parents: 60303
diff changeset
  1224
    using \<open>e > 0\<close> by auto
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1225
  ultimately show ?thesis
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1226
    apply (rule_tac x="e/2" in exI)
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1227
    apply auto
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1228
    done
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1229
qed
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1230
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1231
61518
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  1232
section \<open>Path component, considered as a "joinability" relation (from Tom Hales)\<close>
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1233
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1234
definition "path_component s x y \<longleftrightarrow>
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1235
  (\<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
  1236
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1237
abbreviation
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1238
   "path_component_set s x \<equiv> Collect (path_component 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
  1239
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1240
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
  1241
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1242
lemma path_component_mem:
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1243
  assumes "path_component s x y"
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1244
  shows "x \<in> s" and "y \<in> s"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1245
  using assms
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1246
  unfolding path_defs
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1247
  by auto
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1248
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1249
lemma path_component_refl:
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1250
  assumes "x \<in> s"
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1251
  shows "path_component s x x"
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1252
  unfolding path_defs
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1253
  apply (rule_tac x="\<lambda>u. x" in exI)
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1254
  using assms
56371
fb9ae0727548 extend continuous_intros; remove continuous_on_intros and isCont_intros
hoelzl
parents: 56188
diff changeset
  1255
  apply (auto intro!: continuous_intros)
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1256
  done
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1257
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1258
lemma path_component_refl_eq: "path_component s x x \<longleftrightarrow> x \<in> s"
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1259
  by (auto intro!: path_component_mem path_component_refl)
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1260
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1261
lemma path_component_sym: "path_component s x y \<Longrightarrow> path_component s y x"
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1262
  using assms
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1263
  unfolding path_component_def
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1264
  apply (erule exE)
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1265
  apply (rule_tac x="reversepath g" in exI)
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1266
  apply auto
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1267
  done
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1268
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1269
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
  1270
  assumes "path_component s x y" and "path_component s y z"
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1271
  shows "path_component s x z"
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1272
  using assms
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1273
  unfolding path_component_def
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1274
  apply (elim exE)
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1275
  apply (rule_tac x="g +++ ga" in exI)
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1276
  apply (auto simp add: path_image_join)
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1277
  done
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1278
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1279
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
  1280
  unfolding path_component_def by auto
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1281
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1282
lemma path_connected_linepath:
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1283
    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
  1284
    shows "closed_segment a b \<subseteq> s \<Longrightarrow> path_component s 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
  1285
  apply (simp add: path_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
  1286
  apply (rule_tac x="linepath a b" in exI, auto)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1287
  done
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1288
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1289
60420
884f54e01427 isabelle update_cartouches;
wenzelm
parents: 60303
diff changeset
  1290
text \<open>Can also consider it as a set, as the name suggests.\<close>
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1291
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1292
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
  1293
  "path_component_set s x =
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1294
    {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
  1295
  by (auto simp: path_component_def)
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1296
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1297
lemma path_component_subset: "path_component_set s x \<subseteq> s"
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
  1298
  by (auto simp add: path_component_mem(2))
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1299
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1300
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
  1301
  using path_component_mem path_component_refl_eq
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
  1302
    by fastforce
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1303
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1304
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
  1305
     "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
  1306
  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
  1307
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1308
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
  1309
   "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
  1310
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
  1311
60420
884f54e01427 isabelle update_cartouches;
wenzelm
parents: 60303
diff changeset
  1312
subsection \<open>Path connectedness of a space\<close>
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1313
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1314
definition "path_connected s \<longleftrightarrow>
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1315
  (\<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
  1316
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1317
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
  1318
  unfolding path_connected_def path_component_def by auto
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1319
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1320
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
  1321
  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
  1322
  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
  1323
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1324
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
  1325
     "\<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
  1326
  by (metis path_component_mono path_connected_component_set)
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1327
60420
884f54e01427 isabelle update_cartouches;
wenzelm
parents: 60303
diff changeset
  1328
subsection \<open>Some useful lemmas about path-connectedness\<close>
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1329
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1330
lemma convex_imp_path_connected:
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1331
  fixes s :: "'a::real_normed_vector set"
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1332
  assumes "convex s"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1333
  shows "path_connected s"
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1334
  unfolding path_connected_def
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1335
  apply rule
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1336
  apply rule
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1337
  apply (rule_tac x = "linepath x y" in exI)
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1338
  unfolding path_image_linepath
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1339
  using assms [unfolded convex_contains_segment]
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1340
  apply auto
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1341
  done
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1342
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1343
lemma path_connected_imp_connected:
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1344
  assumes "path_connected s"
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1345
  shows "connected s"
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1346
  unfolding connected_def not_ex
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1347
  apply rule
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1348
  apply rule
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1349
  apply (rule ccontr)
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1350
  unfolding not_not
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1351
  apply (elim conjE)
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1352
proof -
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1353
  fix e1 e2
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1354
  assume as: "open e1" "open e2" "s \<subseteq> e1 \<union> e2" "e1 \<inter> e2 \<inter> s = {}" "e1 \<inter> s \<noteq> {}" "e2 \<inter> s \<noteq> {}"
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1355
  then obtain x1 x2 where obt:"x1 \<in> e1 \<inter> s" "x2 \<in> e2 \<inter> s"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1356
    by auto
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1357
  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
  1358
    using assms[unfolded path_connected_def,rule_format,of x1 x2] by auto
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1359
  have *: "connected {0..1::real}"
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1360
    by (auto intro!: convex_connected convex_real_interval)
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1361
  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
  1362
    using as(3) g(2)[unfolded path_defs] by blast
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1363
  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
  1364
    using as(4) g(2)[unfolded path_defs]
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1365
    unfolding subset_eq
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1366
    by auto
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1367
  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
  1368
    using g(3,4)[unfolded path_defs]
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1369
    using obt
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1370
    by (simp add: ex_in_conv [symmetric], metis zero_le_one order_refl)
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1371
  ultimately show False
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1372
    using *[unfolded connected_local not_ex, rule_format,
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1373
      of "{x\<in>{0..1}. g x \<in> e1}" "{x\<in>{0..1}. g x \<in> e2}"]
61518
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  1374
    using continuous_openin_preimage[OF g(1)[unfolded path_def] as(1)]
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  1375
    using continuous_openin_preimage[OF g(1)[unfolded path_def] as(2)]
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1376
    by auto
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1377
qed
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1378
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1379
lemma open_path_component:
53593
a7bcbb5a17d8 removed outdated comments
huffman
parents: 51481
diff changeset
  1380
  fixes s :: "'a::real_normed_vector set"
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1381
  assumes "open 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
  1382
  shows "open (path_component_set s x)"
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1383
  unfolding open_contains_ball
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1384
proof
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1385
  fix 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
  1386
  assume as: "y \<in> path_component_set s x"
49654
366d8b41ca17 tuned proofs;
wenzelm
parents: 49653
diff changeset
  1387
  then have "y \<in> s"
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1388
    apply -
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1389
    apply (rule path_component_mem(2))
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1390
    unfolding mem_Collect_eq
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1391
    apply auto
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1392
    done
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1393
  then obtain e where e: "e > 0" "ball y e \<subseteq> s"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1394
    using assms[unfolded open_contains_ball]
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1395
    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
  1396
  show "\<exists>e > 0. ball y e \<subseteq> path_component_set s x"
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1397
    apply (rule_tac x=e in exI)
60420
884f54e01427 isabelle update_cartouches;
wenzelm
parents: 60303
diff changeset
  1398
    apply (rule,rule \<open>e>0\<close>)
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1399
    apply rule
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1400
    unfolding mem_ball mem_Collect_eq
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1401
  proof -
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1402
    fix z
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1403
    assume "dist y z < e"
49654
366d8b41ca17 tuned proofs;
wenzelm
parents: 49653
diff changeset
  1404
    then show "path_component s x z"
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1405
      apply (rule_tac path_component_trans[of _ _ y])
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1406
      defer
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1407
      apply (rule path_component_of_subset[OF e(2)])
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1408
      apply (rule convex_imp_path_connected[OF convex_ball, unfolded path_connected_component, rule_format])
60420
884f54e01427 isabelle update_cartouches;
wenzelm
parents: 60303
diff changeset
  1409
      using \<open>e > 0\<close> as
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1410
      apply auto
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1411
      done
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1412
  qed
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1413
qed
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1414
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1415
lemma open_non_path_component:
53593
a7bcbb5a17d8 removed outdated comments
huffman
parents: 51481
diff changeset
  1416
  fixes s :: "'a::real_normed_vector set"
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1417
  assumes "open 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
  1418
  shows "open (s - path_component_set s x)"
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1419
  unfolding open_contains_ball
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1420
proof
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1421
  fix 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
  1422
  assume as: "y \<in> s - path_component_set s x"
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1423
  then obtain e where e: "e > 0" "ball y e \<subseteq> s"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1424
    using assms [unfolded open_contains_ball]
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1425
    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
  1426
  show "\<exists>e>0. ball y e \<subseteq> s - path_component_set s x"
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1427
    apply (rule_tac x=e in exI)
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1428
    apply rule
60420
884f54e01427 isabelle update_cartouches;
wenzelm
parents: 60303
diff changeset
  1429
    apply (rule \<open>e>0\<close>)
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1430
    apply rule
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1431
    apply rule
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1432
    defer
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1433
  proof (rule ccontr)
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1434
    fix 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
  1435
    assume "z \<in> ball y e" "\<not> z \<notin> 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
  1436
    then have "y \<in> path_component_set s x"
60420
884f54e01427 isabelle update_cartouches;
wenzelm
parents: 60303
diff changeset
  1437
      unfolding not_not mem_Collect_eq using \<open>e>0\<close>
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1438
      apply -
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1439
      apply (rule path_component_trans, assumption)
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1440
      apply (rule path_component_of_subset[OF e(2)])
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1441
      apply (rule convex_imp_path_connected[OF convex_ball, unfolded path_connected_component, rule_format])
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1442
      apply auto
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1443
      done
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1444
    then show False
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1445
      using as by auto
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1446
  qed (insert e(2), auto)
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1447
qed
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1448
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1449
lemma connected_open_path_connected:
53593
a7bcbb5a17d8 removed outdated comments
huffman
parents: 51481
diff changeset
  1450
  fixes s :: "'a::real_normed_vector set"
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1451
  assumes "open s"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1452
    and "connected s"
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1453
  shows "path_connected s"
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1454
  unfolding path_connected_component_set
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1455
proof (rule, rule, rule path_component_subset, rule)
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1456
  fix x y
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1457
  assume "x \<in> s" and "y \<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
  1458
  show "y \<in> path_component_set s x"
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1459
  proof (rule ccontr)
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1460
    assume "\<not> ?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
  1461
    moreover have "path_component_set s x \<inter> s \<noteq> {}"
60420
884f54e01427 isabelle update_cartouches;
wenzelm
parents: 60303
diff changeset
  1462
      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
  1463
      by auto
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1464
    ultimately
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1465
    show False
60420
884f54e01427 isabelle update_cartouches;
wenzelm
parents: 60303
diff changeset
  1466
      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
  1467
      using assms(2)[unfolded connected_def not_ex, rule_format,
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1468
        of "path_component_set s x" "s - path_component_set s x"]
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1469
      by auto
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1470
  qed
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1471
qed
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1472
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1473
lemma path_connected_continuous_image:
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1474
  assumes "continuous_on s f"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1475
    and "path_connected s"
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1476
  shows "path_connected (f ` s)"
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1477
  unfolding path_connected_def
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1478
proof (rule, rule)
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1479
  fix x' y'
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1480
  assume "x' \<in> f ` s" "y' \<in> f ` s"
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1481
  then obtain x y where x: "x \<in> s" and y: "y \<in> s" and x': "x' = f x" and y': "y' = f y"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1482
    by auto
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1483
  from x y obtain g where "path g \<and> path_image g \<subseteq> s \<and> pathstart g = x \<and> pathfinish g = y"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1484
    using assms(2)[unfolded path_connected_def] by fast
49654
366d8b41ca17 tuned proofs;
wenzelm
parents: 49653
diff changeset
  1485
  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
  1486
    unfolding x' y'
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1487
    apply (rule_tac x="f \<circ> g" in exI)
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1488
    unfolding path_defs
51481
ef949192e5d6 move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents: 51478
diff changeset
  1489
    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
  1490
    apply auto
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1491
    done
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1492
qed
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1493
61518
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  1494
lemma path_connected_segment:
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  1495
    fixes a :: "'a::real_normed_vector"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  1496
    shows "path_connected (closed_segment a b)"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  1497
  by (simp add: convex_imp_path_connected)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  1498
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  1499
lemma path_connected_open_segment:
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  1500
    fixes a :: "'a::real_normed_vector"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  1501
    shows "path_connected (open_segment a b)"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  1502
  by (simp add: convex_imp_path_connected)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  1503
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1504
lemma homeomorphic_path_connectedness:
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1505
  "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
  1506
  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
  1507
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1508
lemma path_connected_empty: "path_connected {}"
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1509
  unfolding path_connected_def by auto
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1510
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1511
lemma path_connected_singleton: "path_connected {a}"
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1512
  unfolding path_connected_def pathstart_def pathfinish_def path_image_def
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1513
  apply clarify
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1514
  apply (rule_tac x="\<lambda>x. a" in exI)
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1515
  apply (simp add: image_constant_conv)
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1516
  apply (simp add: path_def continuous_on_const)
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1517
  done
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1518
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1519
lemma path_connected_Un:
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1520
  assumes "path_connected s"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1521
    and "path_connected t"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1522
    and "s \<inter> t \<noteq> {}"
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1523
  shows "path_connected (s \<union> t)"
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1524
  unfolding path_connected_component
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1525
proof (rule, rule)
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1526
  fix x y
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1527
  assume as: "x \<in> s \<union> t" "y \<in> s \<union> t"
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1528
  from assms(3) obtain z where "z \<in> s \<inter> t"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1529
    by auto
49654
366d8b41ca17 tuned proofs;
wenzelm
parents: 49653
diff changeset
  1530
  then show "path_component (s \<union> t) x y"
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1531
    using as and assms(1-2)[unfolded path_connected_component]
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1532
    apply -
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1533
    apply (erule_tac[!] UnE)+
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1534
    apply (rule_tac[2-3] path_component_trans[of _ _ z])
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1535
    apply (auto simp add:path_component_of_subset [OF Un_upper1] path_component_of_subset[OF Un_upper2])
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1536
    done
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1537
qed
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1538
37674
f86de9c00c47 convert theorem path_connected_sphere to euclidean_space class
huffman
parents: 37489
diff changeset
  1539
lemma path_connected_UNION:
f86de9c00c47 convert theorem path_connected_sphere to euclidean_space class
huffman
parents: 37489
diff changeset
  1540
  assumes "\<And>i. i \<in> A \<Longrightarrow> path_connected (S i)"
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1541
    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
  1542
  shows "path_connected (\<Union>i\<in>A. S i)"
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1543
  unfolding path_connected_component
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1544
proof clarify
37674
f86de9c00c47 convert theorem path_connected_sphere to euclidean_space class
huffman
parents: 37489
diff changeset
  1545
  fix x i y j
f86de9c00c47 convert theorem path_connected_sphere to euclidean_space class
huffman
parents: 37489
diff changeset
  1546
  assume *: "i \<in> A" "x \<in> S i" "j \<in> A" "y \<in> S j"
49654
366d8b41ca17 tuned proofs;
wenzelm
parents: 49653
diff changeset
  1547
  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
  1548
    using assms by (simp_all add: path_connected_component)
49654
366d8b41ca17 tuned proofs;
wenzelm
parents: 49653
diff changeset
  1549
  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
  1550
    using *(1,3) by (auto elim!: path_component_of_subset [rotated])
49654
366d8b41ca17 tuned proofs;
wenzelm
parents: 49653
diff changeset
  1551
  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
  1552
    by (rule path_component_trans)
f86de9c00c47 convert theorem path_connected_sphere to euclidean_space class
huffman
parents: 37489
diff changeset
  1553
qed
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1554
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1555
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
  1556
  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
  1557
  shows "path_component (path_image p) (pathstart p) x"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1558
using x
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1559
proof (clarsimp simp add: path_image_def)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1560
  fix y
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1561
  assume "x = p y" and y: "0 \<le> y" "y \<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
  1562
  show "path_component (p ` {0..1}) (pathstart p) (p y)"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1563
  proof (cases "y=0")
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1564
    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
  1565
      by (simp add: path_component_refl_eq pathstart_def)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1566
  next
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1567
    case False have "continuous_on {0..1} (p o (op*y))"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1568
      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
  1569
      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
  1570
      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
  1571
      done
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1572
    then have "path (\<lambda>u. p (y * u))"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1573
      by (simp add: path_def)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1574
    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
  1575
      apply (simp add: path_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
  1576
      apply (rule_tac x = "\<lambda>u. p (y * u)" 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
  1577
      apply (intro conjI)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1578
      using y False
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1579
      apply (auto simp: mult_le_one pathstart_def pathfinish_def path_image_def)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1580
      done
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1581
  qed
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1582
qed
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1583
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1584
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
  1585
  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
  1586
  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
  1587
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1588
lemma 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
  1589
   "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
  1590
proof -
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1591
  { 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
  1592
    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
  1593
    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
  1594
      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
  1595
    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
  1596
      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
  1597
    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
  1598
      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
  1599
      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
  1600
      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
  1601
  }
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1602
  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
  1603
    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
  1604
qed
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1605
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1606
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)"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1607
  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
  1608
  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
  1609
  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
  1610
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1611
lemma path_component_path_component [simp]:
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1612
   "path_component_set (path_component_set s x) x = 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
  1613
proof (cases "x \<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
  1614
  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
  1615
    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
  1616
    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
  1617
    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
  1618
next
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1619
  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
  1620
    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
  1621
qed
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1622
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1623
lemma path_component_subset_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
  1624
   "(path_component_set s x) \<subseteq> (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
  1625
proof (cases "x \<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
  1626
  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
  1627
    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
  1628
    apply (auto simp: True path_component_subset path_component_refl path_connected_imp_connected 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
  1629
    done
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1630
next
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1631
  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
  1632
    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
  1633
qed
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1634
60420
884f54e01427 isabelle update_cartouches;
wenzelm
parents: 60303
diff changeset
  1635
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
  1636
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1637
lemma path_connected_punctured_universe:
37674
f86de9c00c47 convert theorem path_connected_sphere to euclidean_space class
huffman
parents: 37489
diff changeset
  1638
  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
  1639
  shows "path_connected (- {a::'a})"
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1640
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
  1641
  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
  1642
  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
  1643
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1644
  have A: "path_connected ?A"
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1645
    unfolding Collect_bex_eq
37674
f86de9c00c47 convert theorem path_connected_sphere to euclidean_space class
huffman
parents: 37489
diff changeset
  1646
  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
  1647
    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
  1648
    assume "i \<in> Basis"
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1649
    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
  1650
      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
  1651
    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
  1652
      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
  1653
      by (simp add: inner_commute)
37674
f86de9c00c47 convert theorem path_connected_sphere to euclidean_space class
huffman
parents: 37489
diff changeset
  1654
  qed
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1655
  have B: "path_connected ?B"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1656
    unfolding Collect_bex_eq
37674
f86de9c00c47 convert theorem path_connected_sphere to euclidean_space class
huffman
parents: 37489
diff changeset
  1657
  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
  1658
    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
  1659
    assume "i \<in> Basis"
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1660
    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
  1661
      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
  1662
    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
  1663
      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
  1664
      by (simp add: inner_commute)
37674
f86de9c00c47 convert theorem path_connected_sphere to euclidean_space class
huffman
parents: 37489
diff changeset
  1665
  qed
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1666
  obtain S :: "'a set" where "S \<subseteq> Basis" and "card S = Suc (Suc 0)"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1667
    using ex_card[OF assms]
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1668
    by auto
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1669
  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
  1670
    unfolding card_Suc_eq by auto
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1671
  then have "a + b0 - b1 \<in> ?A \<inter> ?B"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1672
    by (auto simp: inner_simps inner_Basis)
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1673
  then have "?A \<inter> ?B \<noteq> {}"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1674
    by fast
37674
f86de9c00c47 convert theorem path_connected_sphere to euclidean_space class
huffman
parents: 37489
diff changeset
  1675
  with A B have "path_connected (?A \<union> ?B)"
f86de9c00c47 convert theorem path_connected_sphere to euclidean_space class
huffman
parents: 37489
diff changeset
  1676
    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
  1677
  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
  1678
    unfolding neq_iff bex_disj_distrib Collect_disj_eq ..
f86de9c00c47 convert theorem path_connected_sphere to euclidean_space class
huffman
parents: 37489
diff changeset
  1679
  also have "\<dots> = {x. x \<noteq> a}"
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1680
    unfolding euclidean_eq_iff [where 'a='a]
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1681
    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
  1682
  also have "\<dots> = - {a}"
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1683
    by auto
37674
f86de9c00c47 convert theorem path_connected_sphere to euclidean_space class
huffman
parents: 37489
diff changeset
  1684
  finally show ?thesis .
f86de9c00c47 convert theorem path_connected_sphere to euclidean_space class
huffman
parents: 37489
diff changeset
  1685
qed
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1686
37674
f86de9c00c47 convert theorem path_connected_sphere to euclidean_space class
huffman
parents: 37489
diff changeset
  1687
lemma path_connected_sphere:
f86de9c00c47 convert theorem path_connected_sphere to euclidean_space class
huffman
parents: 37489
diff changeset
  1688
  assumes "2 \<le> DIM('a::euclidean_space)"
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1689
  shows "path_connected {x::'a. norm (x - a) = r}"
37674
f86de9c00c47 convert theorem path_connected_sphere to euclidean_space class
huffman
parents: 37489
diff changeset
  1690
proof (rule linorder_cases [of r 0])
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1691
  assume "r < 0"
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1692
  then have "{x::'a. norm(x - a) = r} = {}"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1693
    by auto
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1694
  then show ?thesis
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1695
    using path_connected_empty by simp
37674
f86de9c00c47 convert theorem path_connected_sphere to euclidean_space class
huffman
parents: 37489
diff changeset
  1696
next
f86de9c00c47 convert theorem path_connected_sphere to euclidean_space class
huffman
parents: 37489
diff changeset
  1697
  assume "r = 0"
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1698
  then show ?thesis
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1699
    using path_connected_singleton by simp
37674
f86de9c00c47 convert theorem path_connected_sphere to euclidean_space class
huffman
parents: 37489
diff changeset
  1700
next
f86de9c00c47 convert theorem path_connected_sphere to euclidean_space class
huffman
parents: 37489
diff changeset
  1701
  assume r: "0 < r"
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1702
  have *: "{x::'a. norm(x - a) = r} = (\<lambda>x. a + r *\<^sub>R x) ` {x. norm x = 1}"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1703
    apply (rule set_eqI)
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1704
    apply rule
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1705
    unfolding image_iff
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1706
    apply (rule_tac x="(1/r) *\<^sub>R (x - a)" in bexI)
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1707
    unfolding mem_Collect_eq norm_scaleR
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1708
    using r
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1709
    apply (auto simp add: scaleR_right_diff_distrib)
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1710
    done
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1711
  have **: "{x::'a. norm x = 1} = (\<lambda>x. (1/norm x) *\<^sub>R x) ` (- {0})"
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1712
    apply (rule set_eqI)
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1713
    apply rule
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1714
    unfolding image_iff
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1715
    apply (rule_tac x=x in bexI)
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1716
    unfolding mem_Collect_eq
62390
842917225d56 more canonical names
nipkow
parents: 62087
diff changeset
  1717
    apply (auto split: if_split_asm)
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1718
    done
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1719
  have "continuous_on (- {0}) (\<lambda>x::'a. 1 / norm x)"
59557
ebd8ecacfba6 establish unique preferred fact names
haftmann
parents: 58877
diff changeset
  1720
    by (auto intro!: continuous_intros)
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1721
  then show ?thesis
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1722
    unfolding * **
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1723
    using path_connected_punctured_universe[OF assms]
56371
fb9ae0727548 extend continuous_intros; remove continuous_on_intros and isCont_intros
hoelzl
parents: 56188
diff changeset
  1724
    by (auto intro!: path_connected_continuous_image continuous_intros)
37674
f86de9c00c47 convert theorem path_connected_sphere to euclidean_space class
huffman
parents: 37489
diff changeset
  1725
qed
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1726
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1727
corollary connected_sphere: "2 \<le> DIM('a::euclidean_space) \<Longrightarrow> connected {x::'a. norm (x - a) = r}"
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1728
  using path_connected_sphere path_connected_imp_connected
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1729
  by auto
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1730
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1731
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
  1732
    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
  1733
    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
  1734
    shows "path_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
  1735
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
  1736
  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
  1737
    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
  1738
next
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1739
  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
  1740
  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
  1741
  { fix x y assume "x \<notin> s" "y \<notin> s"
61808
fc1556774cfe isabelle update_cartouches -c -t;
wenzelm
parents: 61806
diff changeset
  1742
    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
  1743
    then have bxy: "bounded(insert x (insert y s))"
61808
fc1556774cfe isabelle update_cartouches -c -t;
wenzelm
parents: 61806
diff changeset
  1744
      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
  1745
    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
  1746
                          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
  1747
      using bounded_subset_ballD [OF bxy, of a] by (auto simp: 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
  1748
    def C == "B / norm(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
  1749
    { 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
  1750
      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
  1751
      have CC: "1 \<le> 1 + (C - 1) * u"
61808
fc1556774cfe isabelle update_cartouches -c -t;
wenzelm
parents: 61806
diff changeset
  1752
        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
  1753
        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
  1754
        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
  1755
      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
  1756
        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
  1757
      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
  1758
            (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
  1759
        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
  1760
      also have "... = (1 + (u / (1 + C * u - u))) *\<^sub>R a + (1 + (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
  1761
        using CC by (simp add: 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
  1762
      also have "... = x + (1 + (u / (1 + C * u - u))) *\<^sub>R a + (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
  1763
        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
  1764
      also have "... = x + ((1 / (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
  1765
              ((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
  1766
        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
  1767
      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
  1768
        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
  1769
      have False
61808
fc1556774cfe isabelle update_cartouches -c -t;
wenzelm
parents: 61806
diff changeset
  1770
        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
  1771
        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
  1772
        apply (drule_tac x=a in bspec)
61808
fc1556774cfe isabelle update_cartouches -c -t;
wenzelm
parents: 61806
diff changeset
  1773
         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
  1774
        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
  1775
         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
  1776
        apply (drule_tac x="1 / (1 + (C - 1) * u)" in spec)
61808
fc1556774cfe isabelle update_cartouches -c -t;
wenzelm
parents: 61806
diff changeset
  1777
        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
  1778
        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
  1779
        done
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1780
    }
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1781
    then have pcx: "path_component (- s) x (a + C *\<^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
  1782
      by (force simp: closed_segment_def intro!: path_connected_linepath)
61808
fc1556774cfe isabelle update_cartouches -c -t;
wenzelm
parents: 61806
diff changeset
  1783
    def 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
  1784
    { 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
  1785
      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
  1786
      have DD: "1 \<le> 1 + (D - 1) * u"
61808
fc1556774cfe isabelle update_cartouches -c -t;
wenzelm
parents: 61806
diff changeset
  1787
        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
  1788
        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
  1789
        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
  1790
      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
  1791
        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
  1792
      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
  1793
            (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
  1794
        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
  1795
      also have "... = (1 + (u / (1 + D * u - u))) *\<^sub>R a + (1 + (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
  1796
        using DD by (simp add: 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
  1797
      also have "... = y + (1 + (u / (1 + D * u - u))) *\<^sub>R a + (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
  1798
        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
  1799
      also have "... = y + ((1 / (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
  1800
              ((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
  1801
        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
  1802
      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
  1803
        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
  1804
      have False
61808
fc1556774cfe isabelle update_cartouches -c -t;
wenzelm
parents: 61806
diff changeset
  1805
        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
  1806
        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
  1807
        apply (drule_tac x=a in bspec)
61808
fc1556774cfe isabelle update_cartouches -c -t;
wenzelm
parents: 61806
diff changeset
  1808
         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
  1809
        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
  1810
         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
  1811
        apply (drule_tac x="1 / (1 + (D - 1) * u)" in spec)
61808
fc1556774cfe isabelle update_cartouches -c -t;
wenzelm
parents: 61806
diff changeset
  1812
        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
  1813
        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
  1814
        done
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1815
    }
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1816
    then have pdy: "path_component (- s) y (a + D *\<^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
  1817
      by (force simp: closed_segment_def intro!: path_connected_linepath)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1818
    have pyx: "path_component (- s) (a + D *\<^sub>R (y - a)) (a + C *\<^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
  1819
      apply (rule path_component_of_subset [of "{x. norm(x - a) = B}"])
61808
fc1556774cfe isabelle update_cartouches -c -t;
wenzelm
parents: 61806
diff changeset
  1820
       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
  1821
       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
  1822
      apply (rule path_connected_sphere [OF 2, of a B, simplified path_connected_component, rule_format])
61808
fc1556774cfe isabelle update_cartouches -c -t;
wenzelm
parents: 61806
diff changeset
  1823
      using \<open>x \<noteq> a\<close>  using \<open>y \<noteq> a\<close>  B apply (auto simp: 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
  1824
      done
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1825
    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
  1826
      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
  1827
  }
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1828
  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
  1829
    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
  1830
qed
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1831
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1832
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1833
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
  1834
    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
  1835
    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
  1836
      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
  1837
  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
  1838
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1839
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
  1840
    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
  1841
    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
  1842
      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
  1843
  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
  1844
  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
  1845
  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
  1846
  done
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1847
62381
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  1848
proposition connected_open_delete:
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  1849
  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
  1850
    shows "connected(S - {a::'N})"
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  1851
proof (cases "a \<in> S")
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  1852
  case True
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  1853
  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
  1854
    using open_contains_cball_eq by blast
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  1855
  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
  1856
    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
  1857
  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
  1858
    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
  1859
    by auto
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  1860
  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
  1861
    by auto
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  1862
  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
  1863
    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
  1864
  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
  1865
    apply (rule UnionI [of "S - ball a (min \<epsilon> (dist a x) / 2)"])
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  1866
     using that \<open>0 < \<epsilon>\<close> apply (simp_all add:)
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  1867
    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
  1868
    apply auto
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  1869
    done
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  1870
  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
  1871
    by auto
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  1872
  then show ?thesis
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  1873
    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
  1874
next
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  1875
  case False then show ?thesis
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  1876
    by (simp add: \<open>connected S\<close>)
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  1877
qed
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  1878
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  1879
corollary path_connected_open_delete:
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  1880
  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
  1881
    shows "path_connected(S - {a::'N})"
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  1882
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
  1883
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  1884
corollary path_connected_punctured_ball:
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  1885
   "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
  1886
by (simp add: path_connected_open_delete)
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  1887
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  1888
lemma connected_punctured_ball:
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  1889
   "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
  1890
by (simp add: connected_open_delete)
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  1891
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1892
subsection\<open>Relations between components and path components\<close>
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1893
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1894
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
  1895
  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
  1896
  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
  1897
    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
  1898
    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
  1899
    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
  1900
     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
  1901
    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
  1902
    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
  1903
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1904
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
  1905
    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
  1906
    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
  1907
  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
  1908
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1909
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
  1910
  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
  1911
  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
  1912
  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
  1913
proof -
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1914
  { 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
  1915
    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
  1916
      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
  1917
    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
  1918
      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
  1919
  }
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1920
  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
  1921
    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
  1922
qed
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1923
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1924
subsection\<open>Existence of unbounded components\<close>
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1925
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1926
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
  1927
    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
  1928
    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
  1929
      shows "\<exists>x. x \<in> s \<and> ~ bounded (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
  1930
proof -
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1931
  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
  1932
    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
  1933
  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
  1934
    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
  1935
  then have *: "\<And>x. B \<le> norm x \<Longrightarrow> x \<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
  1936
    by (force simp add: ball_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
  1937
  have unbounded_inner: "~ bounded {x. inner i x \<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
  1938
    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
  1939
    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
  1940
    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
  1941
    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
  1942
    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
  1943
    done
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1944
  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
  1945
    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
  1946
    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
  1947
    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
  1948
    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
  1949
    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
  1950
  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
  1951
    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
  1952
  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
  1953
    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
  1954
    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
  1955
    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
  1956
    done
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1957
qed
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1958
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1959
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
  1960
    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
  1961
    assumes bs: "bounded (-s)" and "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
  1962
        and bo: "~ bounded(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
  1963
                "~ bounded(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
  1964
      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
  1965
proof -
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1966
  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
  1967
    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
  1968
  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
  1969
    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
  1970
  then have *: "\<And>x. B \<le> norm x \<Longrightarrow> x \<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
  1971
    by (force simp add: ball_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
  1972
  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
  1973
    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
  1974
  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
  1975
    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
  1976
    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
  1977
  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
  1978
    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
  1979
    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
  1980
  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
  1981
    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
  1982
    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
  1983
    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
  1984
    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
  1985
    done
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1986
  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
  1987
    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
  1988
    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
  1989
    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
  1990
qed
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1991
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1992
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
  1993
    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
  1994
    shows "bounded (-s) \<Longrightarrow> \<exists>c. c \<in> components s \<and> ~bounded c"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1995
  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
  1996
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1997
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
  1998
    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
  1999
    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
  2000
  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
  2001
  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
  2002
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2003
lemma cobounded_has_bounded_component:
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2004
    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
  2005
    shows "\<lbrakk>bounded (- s); ~connected s; 2 \<le> DIM('a)\<rbrakk> \<Longrightarrow> \<exists>c. c \<in> components s \<and> bounded c"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2006
  by (meson cobounded_unique_unbounded_components connected_eq_connected_components_eq)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2007
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2008
61518
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2009
section\<open>The "inside" and "outside" 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
  2010
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2011
text\<open>The inside comprises the points in a bounded connected component of the set's complement.
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2012
  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
  2013
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2014
definition inside where
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2015
  "inside s \<equiv> {x. (x \<notin> s) \<and> bounded(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
  2016
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2017
definition outside where
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2018
  "outside s \<equiv> -s \<inter> {x. ~ bounded(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
  2019
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2020
lemma outside: "outside s = {x. ~ bounded(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
  2021
  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
  2022
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2023
lemma inside_no_overlap [simp]: "inside s \<inter> s = {}"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2024
  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
  2025
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2026
lemma outside_no_overlap [simp]:
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2027
   "outside s \<inter> s = {}"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2028
  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
  2029
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2030
lemma inside_inter_outside [simp]: "inside s \<inter> 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
  2031
  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
  2032
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2033
lemma inside_union_outside [simp]: "inside s \<union> outside s = (- s)"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2034
  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
  2035
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2036
lemma inside_eq_outside:
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2037
   "inside s = outside s \<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
  2038
  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
  2039
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2040
lemma inside_outside: "inside s = (- (s \<union> 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
  2041
  by (force simp add: inside_def outside)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2042
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2043
lemma outside_inside: "outside s = (- (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
  2044
  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
  2045
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2046
lemma union_with_inside: "s \<union> inside s = - 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
  2047
  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
  2048
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2049
lemma union_with_outside: "s \<union> outside s = - 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
  2050
  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
  2051
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2052
lemma outside_mono: "s \<subseteq> t \<Longrightarrow> outside 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
  2053
  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
  2054
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2055
lemma inside_mono: "s \<subseteq> t \<Longrightarrow> inside s - t \<subseteq> inside t"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2056
  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
  2057
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2058
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
  2059
  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
  2060
  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
  2061
  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
  2062
proof -
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2063
  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
  2064
    using assms by auto (metis add.commute diff_add_cancel)
61808
fc1556774cfe isabelle update_cartouches -c -t;
wenzelm
parents: 61806
diff changeset
  2065
  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
  2066
    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
  2067
qed
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2068
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2069
lemma cobounded_outside:
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2070
  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
  2071
  assumes "bounded s" shows "bounded (- 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
  2072
proof -
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2073
  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
  2074
    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
  2075
  { 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
  2076
    assume Bno: "B \<le> norm x" and C: "0 < C"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2077
    have "\<exists>y. connected_component (- s) x y \<and> norm y > C"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2078
    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
  2079
      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
  2080
    next
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2081
      case False with B C 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
  2082
        apply (rule_tac x="((B+C)/norm x) *\<^sub>R x" 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
  2083
        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
  2084
        apply (rule_tac x="closed_segment x (((B+C)/norm x) *\<^sub>R x)" 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
  2085
        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
  2086
        apply (rule_tac y="- ball 0 B" in 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
  2087
         prefer 2 apply force
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2088
        apply (simp add: closed_segment_def ball_def dist_norm, clarify)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2089
        apply (simp add: real_vector_class.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
  2090
        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
  2091
        by (meson le_add_same_cancel1 less_eq_real_def 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
  2092
    qed
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2093
  }
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2094
  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
  2095
    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
  2096
    apply (rule bounded_subset [OF bounded_ball [of 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
  2097
    apply (force simp add: dist_norm not_less 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
  2098
    done
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2099
qed
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2100
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2101
lemma unbounded_outside:
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2102
    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
  2103
    shows "bounded s \<Longrightarrow> ~ bounded(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
  2104
  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
  2105
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2106
lemma bounded_inside:
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2107
    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
  2108
    shows "bounded s \<Longrightarrow> bounded(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
  2109
  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
  2110
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2111
lemma connected_outside:
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2112
    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
  2113
    assumes "bounded 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
  2114
      shows "connected(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
  2115
  apply (simp add: connected_iff_connected_component, clarify)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2116
  apply (simp add: outside)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2117
  apply (rule_tac s="connected_component_set (- s) x" in 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
  2118
  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
  2119
  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
  2120
  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
  2121
  done
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2122
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2123
lemma 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
  2124
    "outside s = {x. \<forall>B. \<exists>y. B < norm(y) \<and> 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
  2125
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
  2126
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
  2127
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
  2128
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2129
lemma outside_connected_component_le:
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2130
   "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
  2131
            {x. \<forall>B. \<exists>y. B \<le> norm(y) \<and>
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2132
                         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
  2133
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
  2134
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
  2135
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
  2136
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2137
lemma not_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
  2138
    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
  2139
    assumes s: "bounded s" and "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
  2140
      shows "- (outside s) = {x. \<forall>B. \<exists>y. B < norm(y) \<and> ~ (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
  2141
proof -
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2142
  obtain B::real where B: "0 < B" and Bno: "\<And>x. x \<in> s \<Longrightarrow> norm x \<le> B"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2143
    using s [simplified bounded_pos] 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
  2144
  { 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
  2145
    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
  2146
    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
  2147
      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
  2148
      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
  2149
      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
  2150
      by (auto simp: 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
  2151
    then have "connected_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
  2152
      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
  2153
      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
  2154
      done
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2155
  } 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
  2156
  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
  2157
    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
  2158
    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
  2159
    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
  2160
    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
  2161
qed
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2162
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2163
lemma not_outside_connected_component_le:
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2164
    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
  2165
    assumes s: "bounded 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
  2166
      shows "- (outside s) = {x. \<forall>B. \<exists>y. B \<le> norm(y) \<and> ~ (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
  2167
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
  2168
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
  2169
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2170
lemma inside_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
  2171
    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
  2172
    assumes s: "bounded 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
  2173
      shows "inside s = {x. (x \<notin> s) \<and> (\<forall>B. \<exists>y. B < norm(y) \<and> ~(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
  2174
  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
  2175
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2176
lemma inside_connected_component_le:
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2177
    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
  2178
    assumes s: "bounded 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
  2179
      shows "inside s = {x. (x \<notin> s) \<and> (\<forall>B. \<exists>y. B \<le> norm(y) \<and> ~(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
  2180
  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
  2181
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2182
lemma 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
  2183
  assumes "connected u" and "~bounded u" and "t \<union> u = - s"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2184
  shows "inside s \<subseteq> t"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2185
apply (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
  2186
by (metis bounded_subset [of "connected_component_set (- s) _"] 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
  2187
       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
  2188
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2189
lemma frontier_interiors: "frontier s = - interior(s) - interior(-s)"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2190
  by (simp add: Int_commute frontier_def interior_closure)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2191
62381
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  2192
lemma frontier_interior_subset: "frontier(interior S) \<subseteq> frontier S"
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  2193
  by (simp add: Diff_mono frontier_interiors interior_mono interior_subset)
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  2194
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  2195
lemma connected_Int_frontier:
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2196
     "\<lbrakk>connected s; s \<inter> t \<noteq> {}; s - t \<noteq> {}\<rbrakk> \<Longrightarrow> (s \<inter> frontier t \<noteq> {})"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2197
  apply (simp add: frontier_interiors connected_open_in, safe)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2198
  apply (drule_tac x="s \<inter> interior t" in spec, safe)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2199
   apply (drule_tac [2] x="s \<inter> interior (-t)" in spec)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2200
   apply (auto simp: disjoint_eq_subset_Compl dest: interior_subset [THEN subsetD])
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2201
  done
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2202
62381
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  2203
lemma frontier_not_empty:
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  2204
  fixes S :: "'a :: real_normed_vector set"
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  2205
  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
  2206
    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
  2207
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  2208
lemma frontier_eq_empty:
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  2209
  fixes S :: "'a :: real_normed_vector set"
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  2210
  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
  2211
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
  2212
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  2213
lemma frontier_of_connected_component_subset:
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  2214
  fixes S :: "'a::real_normed_vector set"
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  2215
  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
  2216
proof -
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  2217
  { fix y
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  2218
    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
  2219
       and y2: "y \<notin> interior (connected_component_set S x)"
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  2220
    have 1: "y \<in> closure S"
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  2221
      using y1 closure_mono connected_component_subset by blast
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  2222
    have "z \<in> interior (connected_component_set S x)"
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  2223
          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
  2224
    proof -
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  2225
      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
  2226
        apply (rule connected_component_maximal)
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  2227
        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
  2228
        done
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  2229
      then show ?thesis
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  2230
        using y1 apply (simp add: closure_approachable open_contains_ball_eq [OF open_interior])
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  2231
        by (metis (no_types, hide_lams) connected_component_eq_eq connected_component_in subsetD
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  2232
                       dist_commute mem_Collect_eq mem_ball mem_interior \<open>0 < e\<close> y2)
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  2233
    qed
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  2234
    then have 2: "y \<notin> interior S"
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  2235
      using y2 by (force simp: open_contains_ball_eq [OF open_interior])
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  2236
    note 1 2
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  2237
  }
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  2238
  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
  2239
qed
a6479cb85944 New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents: 62087
diff changeset
  2240
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2241
lemma 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
  2242
    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
  2243
    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
  2244
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
  2245
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
  2246
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2247
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
  2248
    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
  2249
    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
  2250
  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
  2251
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2252
lemma components_univ [simp]: "components UNIV = {UNIV :: '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
  2253
  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
  2254
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2255
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
  2256
    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
  2257
    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
  2258
      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
  2259
proof -
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2260
  { 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
  2261
    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
  2262
       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
  2263
    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
  2264
      apply (rule connected_Int_frontier, 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
  2265
      apply (metis IntI cc connected_component_in connected_component_refl empty_iff interiorE mem_Collect_eq set_rev_mp x)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2266
      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
  2267
      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
  2268
    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
  2269
      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
  2270
  }
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2271
  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
  2272
    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
  2273
    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
  2274
    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
  2275
    done
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2276
qed
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2277
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2278
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
  2279
  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
  2280
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2281
lemma outside_empty [simp]: "outside {} = (UNIV :: '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
  2282
using inside_empty inside_union_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
  2283
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2284
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
  2285
   "\<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
  2286
  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
  2287
  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
  2288
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2289
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
  2290
   "\<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
  2291
  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
  2292
  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
  2293
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2294
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
  2295
  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
  2296
  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
  2297
    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
  2298
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
  2299
  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
  2300
next
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2301
  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
  2302
  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
  2303
  { 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
  2304
    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
  2305
      by (metis 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
  2306
    def C \<equiv> "((B + 1 + norm z) / norm (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
  2307
    have "C > 0"
61808
fc1556774cfe isabelle update_cartouches -c -t;
wenzelm
parents: 61806
diff changeset
  2308
      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
  2309
    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
  2310
      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
  2311
    moreover have "norm (C *\<^sub>R (z-a)) > norm z + B"
61808
fc1556774cfe isabelle update_cartouches -c -t;
wenzelm
parents: 61806
diff changeset
  2312
      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
  2313
    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
  2314
    { 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
  2315
      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
  2316
      then have Cpos: "1 + u * C > 0"
61808
fc1556774cfe isabelle update_cartouches -c -t;
wenzelm
parents: 61806
diff changeset
  2317
        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
  2318
      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
  2319
        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
  2320
      then have False
61808
fc1556774cfe isabelle update_cartouches -c -t;
wenzelm
parents: 61806
diff changeset
  2321
        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
  2322
        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
  2323
    } 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
  2324
    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
  2325
      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
  2326
      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
  2327
      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
  2328
      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
  2329
      done
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2330
    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
  2331
      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
  2332
      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
  2333
  }
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2334
  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
  2335
    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
  2336
qed
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2337
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2338
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
  2339
  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
  2340
  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
  2341
    shows "outside s = - s"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2342
  by (metis ComplD assms convex_in_outside equalityI inside_union_outside subsetI sup.cobounded2)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2343
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2344
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
  2345
  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
  2346
  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
  2347
  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
  2348
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2349
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
  2350
  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
  2351
  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
  2352
  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
  2353
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2354
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
  2355
    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
  2356
    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
  2357
    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
  2358
  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
  2359
proof -
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2360
  { 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
  2361
    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
  2362
      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
  2363
    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
  2364
      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
  2365
  }
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2366
  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
  2367
    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
  2368
qed
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2369
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2370
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
  2371
  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
  2372
    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
  2373
      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
  2374
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
  2375
          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
  2376
61518
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2377
lemma inside_frontier_eq_interior:
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2378
     fixes s :: "'a :: {real_normed_vector, perfect_space} set"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2379
     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
  2380
  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
  2381
  using closure_subset interior_subset
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2382
  apply (auto simp add: frontier_def)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2383
  done
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2384
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2385
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
  2386
    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
  2387
    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
  2388
      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
  2389
proof -
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2390
  { 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
  2391
    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
  2392
      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
  2393
    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
  2394
      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
  2395
      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
  2396
      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
  2397
    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
  2398
      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
  2399
  }
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2400
  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
  2401
    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
  2402
qed
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2403
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2404
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
  2405
    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
  2406
    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
  2407
      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
  2408
proof -
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2409
  { 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
  2410
    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
  2411
      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
  2412
    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
  2413
      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
  2414
      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
  2415
      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
  2416
    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
  2417
      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
  2418
  }
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2419
  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
  2420
    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
  2421
qed
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2422
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2423
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
  2424
    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
  2425
    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
  2426
      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
  2427
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
  2428
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2429
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
  2430
    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
  2431
    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
  2432
      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
  2433
proof -
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2434
  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
  2435
    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
  2436
  moreover have "- inside s \<inter> - outside s = s"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2437
    by (metis (no_types) compl_sup double_compl inside_union_outside)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2438
  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
  2439
    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
  2440
  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
  2441
    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
  2442
  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
  2443
    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
  2444
qed
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  2445
61518
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2446
lemma closure_outside_subset:
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2447
    fixes s :: "'a::real_normed_vector set"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2448
    assumes "closed s"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2449
      shows "closure(outside s) \<subseteq> s \<union> outside s"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2450
  apply (rule closure_minimal, simp)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2451
  by (metis assms closed_open inside_outside open_inside)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2452
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2453
lemma frontier_outside_subset:
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2454
    fixes s :: "'a::real_normed_vector set"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2455
    assumes "closed s"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2456
      shows "frontier(outside s) \<subseteq> s"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2457
  apply (simp add: frontier_def open_outside interior_open)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2458
  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
  2459
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2460
lemma inside_complement_unbounded_connected_empty:
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2461
     "\<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
  2462
  apply (simp add: inside_def)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2463
  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
  2464
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2465
lemma inside_bounded_complement_connected_empty:
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2466
    fixes s :: "'a::{real_normed_vector, perfect_space} set"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2467
    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
  2468
  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
  2469
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2470
lemma inside_inside:
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2471
    assumes "s \<subseteq> inside t"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2472
    shows "inside s - t \<subseteq> inside t"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2473
unfolding inside_def
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2474
proof clarify
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2475
  fix x
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2476
  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
  2477
  show "bounded (connected_component_set (- t) x)"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2478
  proof (cases "s \<inter> connected_component_set (- t) x = {}")
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2479
    case True show ?thesis
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2480
      apply (rule bounded_subset [OF bo])
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2481
      apply (rule connected_component_maximal)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2482
      using x True apply auto
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2483
      done
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2484
  next
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2485
    case False then show ?thesis
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2486
      using assms [unfolded inside_def] x
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2487
      apply (simp add: disjoint_iff_not_equal, clarify)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2488
      apply (drule subsetD, assumption, auto)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2489
      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
  2490
  qed
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2491
qed
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2492
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2493
lemma inside_inside_subset: "inside(inside s) \<subseteq> s"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2494
  using inside_inside union_with_outside by fastforce
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2495
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2496
lemma inside_outside_intersect_connected:
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2497
      "\<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
  2498
  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
  2499
  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
  2500
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2501
lemma outside_bounded_nonempty:
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2502
  fixes s :: "'a :: {real_normed_vector, perfect_space} set"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2503
    assumes "bounded s" shows "outside s \<noteq> {}"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2504
  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
  2505
                   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
  2506
                   double_complement order_refl outside_convex outside_def)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2507
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2508
lemma outside_compact_in_open:
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2509
    fixes s :: "'a :: {real_normed_vector,perfect_space} set"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2510
    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
  2511
      shows "outside s \<inter> t \<noteq> {}"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2512
proof -
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2513
  have "outside s \<noteq> {}"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2514
    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
  2515
  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
  2516
  show ?thesis
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2517
  proof (cases "a \<in> t")
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2518
    case True with a show ?thesis by blast
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2519
  next
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2520
    case False
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2521
      have front: "frontier t \<subseteq> - s"
61808
fc1556774cfe isabelle update_cartouches -c -t;
wenzelm
parents: 61806
diff changeset
  2522
        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
  2523
      { fix \<gamma>
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2524
        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
  2525
           and pf: "pathfinish \<gamma> \<in> frontier t" and ps: "pathstart \<gamma> = a"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2526
        def c \<equiv> "pathfinish \<gamma>"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2527
        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
  2528
        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
  2529
        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
  2530
          using open_contains_cball[of "-s"] s by blast
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2531
        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
  2532
          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
  2533
          by (metis Diff_iff frontier_def)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2534
        then have "d \<in> -s" using \<epsilon>
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2535
          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
  2536
        have pimg_sbs_cos: "path_image \<gamma> \<subseteq> -s"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2537
          using pimg_sbs apply (auto simp: path_image_def)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2538
          apply (drule subsetD)
61808
fc1556774cfe isabelle update_cartouches -c -t;
wenzelm
parents: 61806
diff changeset
  2539
          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
  2540
          done
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2541
        have "closed_segment c d \<le> cball c \<epsilon>"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2542
          apply (simp add: segment_convex_hull)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2543
          apply (rule hull_minimal)
61808
fc1556774cfe isabelle update_cartouches -c -t;
wenzelm
parents: 61806
diff changeset
  2544
          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
  2545
          done
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2546
        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
  2547
        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
  2548
          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
  2549
        ultimately have "connected_component (- s) a d"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2550
          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
  2551
        then have "outside s \<inter> t \<noteq> {}"
61808
fc1556774cfe isabelle update_cartouches -c -t;
wenzelm
parents: 61806
diff changeset
  2552
          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
  2553
      } note * = this
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2554
      have pal: "pathstart (linepath a b) \<in> closure (- t)"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2555
        by (auto simp: False closure_def)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2556
      show ?thesis
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2557
        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
  2558
  qed
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2559
qed
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2560
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2561
lemma inside_inside_compact_connected:
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2562
    fixes s :: "'a :: euclidean_space set"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2563
    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
  2564
      shows "inside s \<subseteq> inside t"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2565
proof (cases "inside t = {}")
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2566
  case True with assms show ?thesis by auto
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2567
next
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2568
  case False
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2569
  consider "DIM('a) = 1" | "DIM('a) \<ge> 2"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2570
    using antisym not_less_eq_eq by fastforce
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2571
  then show ?thesis
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2572
  proof cases
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2573
    case 1 then show ?thesis
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2574
             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
  2575
  next
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2576
    case 2
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2577
    have coms: "compact s"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2578
      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
  2579
       by (meson bounded_inside bounded_subset compact_imp_bounded)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2580
    then have bst: "bounded (s \<union> t)"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2581
      by (simp add: compact_imp_bounded t)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2582
    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
  2583
      using bounded_subset_ballD by blast
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2584
    have outst: "outside s \<inter> outside t \<noteq> {}"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2585
    proof -
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2586
      have "- ball 0 r \<subseteq> outside s"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2587
        apply (rule outside_subset_convex)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2588
        using r by auto
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2589
      moreover have "- ball 0 r \<subseteq> outside t"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2590
        apply (rule outside_subset_convex)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2591
        using r by auto
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2592
      ultimately show ?thesis
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2593
        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
  2594
    qed
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2595
    have "s \<inter> t = {}" using assms
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2596
      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
  2597
    moreover have "outside s \<inter> inside t \<noteq> {}"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2598
      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
  2599
    ultimately have "inside s \<inter> t = {}"
61808
fc1556774cfe isabelle update_cartouches -c -t;
wenzelm
parents: 61806
diff changeset
  2600
      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
  2601
      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
  2602
    then show ?thesis
61808
fc1556774cfe isabelle update_cartouches -c -t;
wenzelm
parents: 61806
diff changeset
  2603
      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
  2604
  qed
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2605
qed
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2606
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2607
lemma connected_with_inside:
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2608
    fixes s :: "'a :: real_normed_vector set"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2609
    assumes s: "closed s" and cons: "connected s"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2610
      shows "connected(s \<union> inside s)"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2611
proof (cases "s \<union> inside s = UNIV")
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2612
  case True with assms show ?thesis by auto
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2613
next
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2614
  case False
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2615
  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
  2616
  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
  2617
  using that proof
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2618
    assume "a \<in> s" then show ?thesis
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2619
      apply (rule_tac x=a in exI)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2620
      apply (rule_tac x="{a}" in exI)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2621
      apply (simp add:)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2622
      done
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2623
  next
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2624
    assume a: "a \<in> inside s"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2625
    show ?thesis
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2626
      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
  2627
      using a apply (simp add: closure_def)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2628
      apply (simp add: b)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2629
      apply (rule_tac x="pathfinish h" in exI)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2630
      apply (rule_tac x="path_image h" in exI)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2631
      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
  2632
      using frontier_inside_subset s apply fastforce
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2633
      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
  2634
  qed
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2635
  show ?thesis
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2636
    apply (simp add: connected_iff_connected_component)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2637
    apply (simp add: connected_component_def)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2638
    apply (clarify dest!: *)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2639
    apply (rename_tac u u' t t')
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2640
    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
  2641
    apply (auto simp: intro!: connected_Un cons)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2642
    done
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2643
qed
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2644
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2645
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
  2646
lemma connected_with_outside:
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2647
    fixes s :: "'a :: real_normed_vector set"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2648
    assumes s: "closed s" and cons: "connected s"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2649
      shows "connected(s \<union> outside s)"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2650
proof (cases "s \<union> outside s = UNIV")
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2651
  case True with assms show ?thesis by auto
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2652
next
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2653
  case False
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2654
  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
  2655
  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
  2656
  using that proof
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2657
    assume "a \<in> s" then show ?thesis
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2658
      apply (rule_tac x=a in exI)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2659
      apply (rule_tac x="{a}" in exI)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2660
      apply (simp add:)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2661
      done
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2662
  next
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2663
    assume a: "a \<in> outside s"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2664
    show ?thesis
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2665
      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
  2666
      using a apply (simp add: closure_def)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2667
      apply (simp add: b)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2668
      apply (rule_tac x="pathfinish h" in exI)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2669
      apply (rule_tac x="path_image h" in exI)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2670
      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
  2671
      using frontier_outside_subset s apply fastforce
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2672
      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
  2673
  qed
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2674
  show ?thesis
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2675
    apply (simp add: connected_iff_connected_component)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2676
    apply (simp add: connected_component_def)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2677
    apply (clarify dest!: *)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2678
    apply (rename_tac u u' t t')
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2679
    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
  2680
    apply (auto simp: intro!: connected_Un cons)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2681
    done
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2682
qed
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2683
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2684
lemma inside_inside_eq_empty [simp]:
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2685
    fixes s :: "'a :: {real_normed_vector, perfect_space} set"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2686
    assumes s: "closed s" and cons: "connected s"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2687
      shows "inside (inside s) = {}"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2688
  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
  2689
           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
  2690
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2691
lemma inside_in_components:
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2692
     "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
  2693
  apply (simp add: in_components_maximal)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2694
  apply (auto intro: inside_same_component connected_componentI)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2695
  apply (metis IntI empty_iff inside_no_overlap)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2696
  done
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2697
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2698
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
  2699
lemma outside_in_components:
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2700
     "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
  2701
  apply (simp add: in_components_maximal)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2702
  apply (auto intro: outside_same_component connected_componentI)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2703
  apply (metis IntI empty_iff outside_no_overlap)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2704
  done
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2705
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2706
lemma bounded_unique_outside:
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2707
    fixes s :: "'a :: euclidean_space set"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2708
    shows "\<lbrakk>bounded s; DIM('a) \<ge> 2\<rbrakk> \<Longrightarrow> (c \<in> components (- s) \<and> ~bounded c \<longleftrightarrow> c = outside s)"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2709
  apply (rule iffI)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  2710
  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
  2711
  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
  2712
62533
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
  2713
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
  2714
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
  2715
lemma ball_subset_open_map_image:
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
  2716
  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
  2717
  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
  2718
      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
  2719
      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
  2720
      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
  2721
    shows "ball (f a) r \<subseteq> f ` S"
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
  2722
proof (cases "f ` S = UNIV")
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
  2723
  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
  2724
next
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
  2725
  case False
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
  2726
    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
  2727
               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
  2728
      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
  2729
      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
  2730
    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
  2731
      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
  2732
    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
  2733
    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
  2734
      by metis
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
  2735
    then obtain y K where y: "y \<in> closure S" and "subseq K" and Klim: "(z \<circ> K) \<longlonglongrightarrow> y"
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
  2736
      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
  2737
      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
  2738
      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
  2739
      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
  2740
      done
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
  2741
    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
  2742
      by (metis LIMSEQ_subseq_LIMSEQ fun.map_cong0 fz tendsw)
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
  2743
    have zKs: "\<And>n. (z o K) n \<in> S" by (simp add: zs)
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
  2744
    have "f \<circ> z = \<xi>"  "(\<lambda>n. f (z n)) = \<xi>"
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
  2745
      using fz by auto
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
  2746
    moreover then have "(\<xi> \<circ> K) \<longlonglongrightarrow> f y"
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
  2747
      by (metis (no_types) Klim zKs y contf comp_assoc continuous_on_closure_sequentially)
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
  2748
    ultimately have wy: "w = f y" using fz LIMSEQ_unique ftendsw by auto
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
  2749
    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
  2750
      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
  2751
      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
  2752
      by (force simp: imageI image_mono interiorI interior_subset frontier_def y)
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
  2753
    have **: "(~(b \<inter> (- S) = {}) \<and> ~(b - (- S) = {}) \<Longrightarrow> (b \<inter> f \<noteq> {}))
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
  2754
              \<Longrightarrow> (b \<inter> S \<noteq> {}) \<Longrightarrow> b \<inter> f = {} \<Longrightarrow>
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
  2755
              b \<subseteq> S" for b f and S :: "'b set" 
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
  2756
      by blast
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
  2757
    show ?thesis
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
  2758
      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
  2759
      apply (rule connected_Int_frontier [where t = "f`S", OF connected_ball])
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
  2760
      using \<open>a \<in> S\<close> \<open>0 < r\<close> 
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
  2761
      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
  2762
      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
  2763
qed
bc25f3916a99 new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents: 62398
diff changeset
  2764
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
  2765
section\<open> Homotopy of maps p,q : X=>Y with property P of all intermediate maps.\<close>
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2766
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2767
text\<open> We often just want to require that it fixes some subset, but to take in
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2768
  the case of a loop homotopy, it's convenient to have a general property P.\<close>
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2769
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2770
definition homotopic_with ::
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2771
  "[('a::topological_space \<Rightarrow> 'b::topological_space) \<Rightarrow> bool, 'a set, 'b set, 'a \<Rightarrow> 'b, 'a \<Rightarrow> 'b] \<Rightarrow> bool"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2772
where
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2773
 "homotopic_with P X Y p q \<equiv>
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2774
   (\<exists>h:: real \<times> 'a \<Rightarrow> 'b.
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2775
       continuous_on ({0..1} \<times> X) h \<and>
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2776
       h ` ({0..1} \<times> X) \<subseteq> Y \<and>
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2777
       (\<forall>x. h(0, x) = p x) \<and>
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2778
       (\<forall>x. h(1, x) = q x) \<and>
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2779
       (\<forall>t \<in> {0..1}. P(\<lambda>x. h(t, x))))"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2780
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2781
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2782
text\<open> We often want to just localize the ending function equality or whatever.\<close>
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2783
proposition homotopic_with:
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2784
  fixes X :: "'a::topological_space set" and Y :: "'b::topological_space set"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2785
  assumes "\<And>h k. (\<And>x. x \<in> X \<Longrightarrow> h x = k x) \<Longrightarrow> (P h \<longleftrightarrow> P k)"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2786
  shows "homotopic_with P X Y p q \<longleftrightarrow>
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2787
           (\<exists>h :: real \<times> 'a \<Rightarrow> 'b.
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2788
              continuous_on ({0..1} \<times> X) h \<and>
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2789
              h ` ({0..1} \<times> X) \<subseteq> Y \<and>
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2790
              (\<forall>x \<in> X. h(0,x) = p x) \<and>
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2791
              (\<forall>x \<in> X. h(1,x) = q x) \<and>
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2792
              (\<forall>t \<in> {0..1}. P(\<lambda>x. h(t, x))))"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2793
  unfolding homotopic_with_def
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2794
  apply (rule iffI, blast, clarify)
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2795
  apply (rule_tac x="\<lambda>(u,v). if v \<in> X then h(u,v) else if u = 0 then p v else q v" in exI)
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2796
  apply (auto simp:)
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2797
  apply (force elim: continuous_on_eq)
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2798
  apply (drule_tac x=t in bspec, force)
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2799
  apply (subst assms; simp)
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2800
  done
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2801
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2802
proposition homotopic_with_eq:
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2803
   assumes h: "homotopic_with P X Y f g"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2804
       and f': "\<And>x. x \<in> X \<Longrightarrow> f' x = f x"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2805
       and g': "\<And>x. x \<in> X \<Longrightarrow> g' x = g x"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2806
       and P:  "(\<And>h k. (\<And>x. x \<in> X \<Longrightarrow> h x = k x) \<Longrightarrow> (P h \<longleftrightarrow> P k))"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2807
   shows "homotopic_with P X Y f' g'"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2808
  using h unfolding homotopic_with_def
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2809
  apply safe
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2810
  apply (rule_tac x="\<lambda>(u,v). if v \<in> X then h(u,v) else if u = 0 then f' v else g' v" in exI)
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2811
  apply (simp add: f' g', safe)
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2812
  apply (fastforce intro: continuous_on_eq)
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2813
  apply fastforce
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2814
  apply (subst P; fastforce)
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2815
  done
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2816
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2817
proposition homotopic_with_equal:
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2818
   assumes contf: "continuous_on X f" and fXY: "f ` X \<subseteq> Y"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2819
       and gf: "\<And>x. x \<in> X \<Longrightarrow> g x = f x"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2820
       and P:  "P f" "P g"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2821
   shows "homotopic_with P X Y f g"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2822
  unfolding homotopic_with_def
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2823
  apply (rule_tac x="\<lambda>(u,v). if u = 1 then g v else f v" in exI)
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2824
  using assms
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2825
  apply (intro conjI)
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2826
  apply (rule continuous_on_eq [where f = "f o snd"])
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2827
  apply (rule continuous_intros | force)+
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2828
  apply clarify
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2829
  apply (case_tac "t=1"; force)
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2830
  done
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2831
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2832
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2833
lemma image_Pair_const: "(\<lambda>x. (x, c)) ` A = A \<times> {c}"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2834
  by (auto simp:)
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2835
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2836
lemma homotopic_constant_maps:
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2837
   "homotopic_with (\<lambda>x. True) s t (\<lambda>x. a) (\<lambda>x. b) \<longleftrightarrow> s = {} \<or> path_component t a b"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2838
proof (cases "s = {} \<or> t = {}")
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2839
  case True with continuous_on_const show ?thesis
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2840
    by (auto simp: homotopic_with path_component_def)
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2841
next
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2842
  case False
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2843
  then obtain c where "c \<in> s" by blast
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2844
  show ?thesis
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2845
  proof
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2846
    assume "homotopic_with (\<lambda>x. True) s t (\<lambda>x. a) (\<lambda>x. b)"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2847
    then obtain h :: "real \<times> 'a \<Rightarrow> 'b"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2848
        where conth: "continuous_on ({0..1} \<times> s) h"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2849
          and h: "h ` ({0..1} \<times> s) \<subseteq> t" "(\<forall>x\<in>s. h (0, x) = a)" "(\<forall>x\<in>s. h (1, x) = b)"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2850
      by (auto simp: homotopic_with)
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2851
    have "continuous_on {0..1} (h \<circ> (\<lambda>t. (t, c)))"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2852
      apply (rule continuous_intros conth | simp add: image_Pair_const)+
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2853
      apply (blast intro:  \<open>c \<in> s\<close> continuous_on_subset [OF conth] )
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2854
      done
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2855
    with \<open>c \<in> s\<close> h show "s = {} \<or> path_component t a b"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2856
      apply (simp_all add: homotopic_with path_component_def)
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2857
      apply (auto simp:)
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2858
      apply (drule_tac x="h o (\<lambda>t. (t, c))" in spec)
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2859
      apply (auto simp: pathstart_def pathfinish_def path_image_def path_def)
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2860
      done
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2861
  next
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2862
    assume "s = {} \<or> path_component t a b"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2863
    with False show "homotopic_with (\<lambda>x. True) s t (\<lambda>x. a) (\<lambda>x. b)"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2864
      apply (clarsimp simp: homotopic_with path_component_def pathstart_def pathfinish_def path_image_def path_def)
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2865
      apply (rule_tac x="g o fst" in exI)
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2866
      apply (rule conjI continuous_intros | force)+
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2867
      done
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2868
  qed
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2869
qed
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2870
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2871
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2872
subsection\<open> Trivial properties.\<close>
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2873
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2874
lemma homotopic_with_imp_property: "homotopic_with P X Y f g \<Longrightarrow> P f \<and> P g"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2875
  unfolding homotopic_with_def Ball_def
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2876
  apply clarify
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2877
  apply (frule_tac x=0 in spec)
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2878
  apply (drule_tac x=1 in spec)
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2879
  apply (auto simp:)
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2880
  done
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2881
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2882
lemma continuous_on_o_Pair: "\<lbrakk>continuous_on (T \<times> X) h; t \<in> T\<rbrakk> \<Longrightarrow> continuous_on X (h o Pair t)"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2883
  by (fast intro: continuous_intros elim!: continuous_on_subset)
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2884
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2885
lemma homotopic_with_imp_continuous:
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2886
    assumes "homotopic_with P X Y f g"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2887
    shows "continuous_on X f \<and> continuous_on X g"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2888
proof -
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2889
  obtain h :: "real \<times> 'a \<Rightarrow> 'b"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2890
    where conth: "continuous_on ({0..1} \<times> X) h"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2891
      and h: "\<forall>x. h (0, x) = f x" "\<forall>x. h (1, x) = g x"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2892
    using assms by (auto simp: homotopic_with_def)
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2893
  have *: "t \<in> {0..1} \<Longrightarrow> continuous_on X (h o (\<lambda>x. (t,x)))" for t
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2894
    by (rule continuous_intros continuous_on_subset [OF conth] | force)+
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2895
  show ?thesis
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2896
    using h *[of 0] *[of 1] by auto
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2897
qed
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2898
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2899
proposition homotopic_with_imp_subset1:
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2900
     "homotopic_with P X Y f g \<Longrightarrow> f ` X \<subseteq> Y"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2901
  by (simp add: homotopic_with_def image_subset_iff) (metis atLeastAtMost_iff order_refl zero_le_one)
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2902
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2903
proposition homotopic_with_imp_subset2:
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2904
     "homotopic_with P X Y f g \<Longrightarrow> g ` X \<subseteq> Y"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2905
  by (simp add: homotopic_with_def image_subset_iff) (metis atLeastAtMost_iff order_refl zero_le_one)
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2906
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2907
proposition homotopic_with_mono:
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2908
    assumes hom: "homotopic_with P X Y f g"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2909
        and Q: "\<And>h. \<lbrakk>continuous_on X h; image h X \<subseteq> Y \<and> P h\<rbrakk> \<Longrightarrow> Q h"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2910
      shows "homotopic_with Q X Y f g"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2911
  using hom
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2912
  apply (simp add: homotopic_with_def)
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2913
  apply (erule ex_forward)
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2914
  apply (force simp: intro!: Q dest: continuous_on_o_Pair)
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2915
  done
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2916
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2917
proposition homotopic_with_subset_left:
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2918
     "\<lbrakk>homotopic_with P X Y f g; Z \<subseteq> X\<rbrakk> \<Longrightarrow> homotopic_with P Z Y f g"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2919
  apply (simp add: homotopic_with_def)
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2920
  apply (fast elim!: continuous_on_subset ex_forward)
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2921
  done
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2922
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2923
proposition homotopic_with_subset_right:
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2924
     "\<lbrakk>homotopic_with P X Y f g; Y \<subseteq> Z\<rbrakk> \<Longrightarrow> homotopic_with P X Z f g"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2925
  apply (simp add: homotopic_with_def)
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2926
  apply (fast elim!: continuous_on_subset ex_forward)
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2927
  done
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2928
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2929
proposition homotopic_with_compose_continuous_right:
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2930
    "\<lbrakk>homotopic_with (\<lambda>f. p (f \<circ> h)) X Y f g; continuous_on W h; h ` W \<subseteq> X\<rbrakk>
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2931
     \<Longrightarrow> homotopic_with p W Y (f o h) (g o h)"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2932
  apply (clarsimp simp add: homotopic_with_def)
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2933
  apply (rename_tac k)
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2934
  apply (rule_tac x="k o (\<lambda>y. (fst y, h (snd y)))" in exI)
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2935
  apply (rule conjI continuous_intros continuous_on_compose [where f=snd and g=h, unfolded o_def] | simp)+
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2936
  apply (erule continuous_on_subset)
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2937
  apply (fastforce simp: o_def)+
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2938
  done
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2939
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2940
proposition homotopic_compose_continuous_right:
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2941
     "\<lbrakk>homotopic_with (\<lambda>f. True) X Y f g; continuous_on W h; h ` W \<subseteq> X\<rbrakk>
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2942
      \<Longrightarrow> homotopic_with (\<lambda>f. True) W Y (f o h) (g o h)"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2943
  using homotopic_with_compose_continuous_right by fastforce
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2944
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2945
proposition homotopic_with_compose_continuous_left:
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2946
     "\<lbrakk>homotopic_with (\<lambda>f. p (h \<circ> f)) X Y f g; continuous_on Y h; h ` Y \<subseteq> Z\<rbrakk>
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2947
      \<Longrightarrow> homotopic_with p X Z (h o f) (h o g)"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2948
  apply (clarsimp simp add: homotopic_with_def)
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2949
  apply (rename_tac k)
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2950
  apply (rule_tac x="h o k" in exI)
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2951
  apply (rule conjI continuous_intros continuous_on_compose [where f=snd and g=h, unfolded o_def] | simp)+
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2952
  apply (erule continuous_on_subset)
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2953
  apply (fastforce simp: o_def)+
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2954
  done
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2955
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2956
proposition homotopic_compose_continuous_left:
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2957
   "homotopic_with (\<lambda>f. True) X Y f g \<and>
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2958
        continuous_on Y h \<and> image h Y \<subseteq> Z
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2959
        \<Longrightarrow> homotopic_with (\<lambda>f. True) X Z (h o f) (h o g)"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2960
  using homotopic_with_compose_continuous_left by fastforce
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2961
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2962
proposition homotopic_with_Pair:
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2963
   assumes hom: "homotopic_with p s t f g" "homotopic_with p' s' t' f' g'"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2964
       and q: "\<And>f g. \<lbrakk>p f; p' g\<rbrakk> \<Longrightarrow> q(\<lambda>(x,y). (f x, g y))"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2965
     shows "homotopic_with q (s \<times> s') (t \<times> t')
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2966
                  (\<lambda>(x,y). (f x, f' y)) (\<lambda>(x,y). (g x, g' y))"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2967
  using hom
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2968
  apply (clarsimp simp add: homotopic_with_def)
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2969
  apply (rename_tac k k')
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2970
  apply (rule_tac x="\<lambda>z. ((k o (\<lambda>x. (fst x, fst (snd x)))) z, (k' o (\<lambda>x. (fst x, snd (snd x)))) z)" in exI)
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2971
  apply (rule conjI continuous_intros | erule continuous_on_subset | clarsimp)+
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2972
  apply (auto intro!: q [unfolded case_prod_unfold])
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2973
  done
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2974
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2975
lemma homotopic_on_empty: "homotopic_with (\<lambda>x. True) {} t f g"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2976
  by (metis continuous_on_def empty_iff homotopic_with_equal image_subset_iff)
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2977
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2978
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2979
text\<open>Homotopy with P is an equivalence relation (on continuous functions mapping X into Y that satisfy P,
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2980
     though this only affects reflexivity.\<close>
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2981
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2982
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2983
proposition homotopic_with_refl:
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2984
   "homotopic_with P X Y f f \<longleftrightarrow> continuous_on X f \<and> image f X \<subseteq> Y \<and> P f"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2985
  apply (rule iffI)
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2986
  using homotopic_with_imp_continuous homotopic_with_imp_property homotopic_with_imp_subset2 apply blast
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2987
  apply (simp add: homotopic_with_def)
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2988
  apply (rule_tac x="f o snd" in exI)
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2989
  apply (rule conjI continuous_intros | force)+
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2990
  done
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2991
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2992
lemma homotopic_with_symD:
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2993
  fixes X :: "'a::real_normed_vector set"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2994
    assumes "homotopic_with P X Y f g"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2995
      shows "homotopic_with P X Y g f"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2996
  using assms
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2997
  apply (clarsimp simp add: homotopic_with_def)
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2998
  apply (rename_tac h)
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  2999
  apply (rule_tac x="h o (\<lambda>y. (1 - fst y, snd y))" in exI)
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3000
  apply (rule conjI continuous_intros | erule continuous_on_subset | force simp add: image_subset_iff)+
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3001
  done
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3002
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3003
proposition homotopic_with_sym:
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3004
    fixes X :: "'a::real_normed_vector set"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3005
    shows "homotopic_with P X Y f g \<longleftrightarrow> homotopic_with P X Y g f"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3006
  using homotopic_with_symD by blast
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3007
61699
a81dc5c4d6a9 New theorems mostly from Peter Gammie
paulson <lp15@cam.ac.uk>
parents: 61694
diff changeset
  3008
lemma split_01: "{0..1::real} = {0..1/2} \<union> {1/2..1}"
a81dc5c4d6a9 New theorems mostly from Peter Gammie
paulson <lp15@cam.ac.uk>
parents: 61694
diff changeset
  3009
  by force
a81dc5c4d6a9 New theorems mostly from Peter Gammie
paulson <lp15@cam.ac.uk>
parents: 61694
diff changeset
  3010
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
  3011
lemma split_01_prod: "{0..1::real} \<times> X = ({0..1/2} \<times> X) \<union> ({1/2..1} \<times> X)"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3012
  by force
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3013
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3014
proposition homotopic_with_trans:
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3015
    fixes X :: "'a::real_normed_vector set"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3016
    assumes "homotopic_with P X Y f g" and "homotopic_with P X Y g h"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3017
      shows "homotopic_with P X Y f h"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3018
proof -
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3019
  have clo1: "closedin (subtopology euclidean ({0..1/2} \<times> X \<union> {1/2..1} \<times> X)) ({0..1/2::real} \<times> X)"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3020
    apply (simp add: closedin_closed split_01_prod [symmetric])
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3021
    apply (rule_tac x="{0..1/2} \<times> UNIV" in exI)
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3022
    apply (force simp add: closed_Times)
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3023
    done
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3024
  have clo2: "closedin (subtopology euclidean ({0..1/2} \<times> X \<union> {1/2..1} \<times> X)) ({1/2..1::real} \<times> X)"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3025
    apply (simp add: closedin_closed split_01_prod [symmetric])
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3026
    apply (rule_tac x="{1/2..1} \<times> UNIV" in exI)
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3027
    apply (force simp add: closed_Times)
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3028
    done
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3029
  { fix k1 k2:: "real \<times> 'a \<Rightarrow> 'b"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3030
    assume cont: "continuous_on ({0..1} \<times> X) k1" "continuous_on ({0..1} \<times> X) k2"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3031
       and Y: "k1 ` ({0..1} \<times> X) \<subseteq> Y" "k2 ` ({0..1} \<times> X) \<subseteq> Y"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3032
       and geq: "\<forall>x. k1 (1, x) = g x" "\<forall>x. k2 (0, x) = g x"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3033
       and k12: "\<forall>x. k1 (0, x) = f x" "\<forall>x. k2 (1, x) = h x"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3034
       and P:   "\<forall>t\<in>{0..1}. P (\<lambda>x. k1 (t, x))" "\<forall>t\<in>{0..1}. P (\<lambda>x. k2 (t, x))"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3035
    def k \<equiv> "\<lambda>y. if fst y \<le> 1 / 2 then (k1 o (\<lambda>x. (2 *\<^sub>R fst x, snd x))) y
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3036
                                   else (k2 o (\<lambda>x. (2 *\<^sub>R fst x -1, snd x))) y"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3037
    have keq: "k1 (2 * u, v) = k2 (2 * u - 1, v)" if "u = 1/2"  for u v
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3038
      by (simp add: geq that)
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3039
    have "continuous_on ({0..1} \<times> X) k"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3040
      using cont
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3041
      apply (simp add: split_01_prod k_def)
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3042
      apply (rule clo1 clo2 continuous_on_cases_local continuous_intros | erule continuous_on_subset | simp add: linear image_subset_iff)+
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3043
      apply (force simp add: keq)
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3044
      done
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3045
    moreover have "k ` ({0..1} \<times> X) \<subseteq> Y"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3046
      using Y by (force simp add: k_def)
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3047
    moreover have "\<forall>x. k (0, x) = f x"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3048
      by (simp add: k_def k12)
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3049
    moreover have "(\<forall>x. k (1, x) = h x)"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3050
      by (simp add: k_def k12)
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3051
    moreover have "\<forall>t\<in>{0..1}. P (\<lambda>x. k (t, x))"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3052
      using P
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3053
      apply (clarsimp simp add: k_def)
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3054
      apply (case_tac "t \<le> 1/2")
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3055
      apply (auto simp:)
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3056
      done
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3057
    ultimately have *: "\<exists>k :: real \<times> 'a \<Rightarrow> 'b.
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3058
                       continuous_on ({0..1} \<times> X) k \<and> k ` ({0..1} \<times> X) \<subseteq> Y \<and>
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3059
                       (\<forall>x. k (0, x) = f x) \<and> (\<forall>x. k (1, x) = h x) \<and> (\<forall>t\<in>{0..1}. P (\<lambda>x. k (t, x)))"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3060
      by blast
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3061
  } note * = this
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3062
  show ?thesis
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3063
    using assms by (auto intro: * simp add: homotopic_with_def)
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3064
qed
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3065
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3066
proposition homotopic_compose:
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3067
      fixes s :: "'a::real_normed_vector set"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3068
      shows "\<lbrakk>homotopic_with (\<lambda>x. True) s t f f'; homotopic_with (\<lambda>x. True) t u g g'\<rbrakk>
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3069
             \<Longrightarrow> homotopic_with (\<lambda>x. True) s u (g o f) (g' o f')"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3070
  apply (rule homotopic_with_trans [where g = "g o f'"])
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3071
  apply (metis homotopic_compose_continuous_left homotopic_with_imp_continuous homotopic_with_imp_subset1)
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3072
  by (metis homotopic_compose_continuous_right homotopic_with_imp_continuous homotopic_with_imp_subset2)
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3073
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3074
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3075
subsection\<open>Homotopy of paths, maintaining the same endpoints.\<close>
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3076
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3077
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3078
definition homotopic_paths :: "['a set, real \<Rightarrow> 'a, real \<Rightarrow> 'a::topological_space] \<Rightarrow> bool"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3079
  where
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3080
     "homotopic_paths s p q \<equiv>
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3081
       homotopic_with (\<lambda>r. pathstart r = pathstart p \<and> pathfinish r = pathfinish p) {0..1} s p q"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3082
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3083
lemma homotopic_paths:
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3084
   "homotopic_paths s p q \<longleftrightarrow>
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3085
      (\<exists>h. continuous_on ({0..1} \<times> {0..1}) h \<and>
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3086
          h ` ({0..1} \<times> {0..1}) \<subseteq> s \<and>
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3087
          (\<forall>x \<in> {0..1}. h(0,x) = p x) \<and>
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3088
          (\<forall>x \<in> {0..1}. h(1,x) = q x) \<and>
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3089
          (\<forall>t \<in> {0..1::real}. pathstart(h o Pair t) = pathstart p \<and>
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3090
                        pathfinish(h o Pair t) = pathfinish p))"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3091
  by (auto simp: homotopic_paths_def homotopic_with pathstart_def pathfinish_def)
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3092
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3093
proposition homotopic_paths_imp_pathstart:
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3094
     "homotopic_paths s p q \<Longrightarrow> pathstart p = pathstart q"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3095
  by (metis (mono_tags, lifting) homotopic_paths_def homotopic_with_imp_property)
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3096
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3097
proposition homotopic_paths_imp_pathfinish:
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3098
     "homotopic_paths s p q \<Longrightarrow> pathfinish p = pathfinish q"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3099
  by (metis (mono_tags, lifting) homotopic_paths_def homotopic_with_imp_property)
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3100
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3101
lemma homotopic_paths_imp_path:
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3102
     "homotopic_paths s p q \<Longrightarrow> path p \<and> path q"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3103
  using homotopic_paths_def homotopic_with_imp_continuous path_def by blast
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3104
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3105
lemma homotopic_paths_imp_subset:
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3106
     "homotopic_paths s p q \<Longrightarrow> path_image p \<subseteq> s \<and> path_image q \<subseteq> s"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3107
  by (simp add: homotopic_paths_def homotopic_with_imp_subset1 homotopic_with_imp_subset2 path_image_def)
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3108
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3109
proposition homotopic_paths_refl [simp]: "homotopic_paths s p p \<longleftrightarrow> path p \<and> path_image p \<subseteq> s"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3110
by (simp add: homotopic_paths_def homotopic_with_refl path_def path_image_def)
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3111
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
  3112
proposition homotopic_paths_sym: "homotopic_paths s p q \<Longrightarrow> homotopic_paths s q p"
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
  3113
  by (metis (mono_tags) homotopic_paths_def homotopic_paths_imp_pathfinish homotopic_paths_imp_pathstart homotopic_with_symD)
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
  3114
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
  3115
proposition homotopic_paths_sym_eq: "homotopic_paths s p q \<longleftrightarrow> homotopic_paths s q p"
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
  3116
  by (metis homotopic_paths_sym)
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
  3117
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
  3118
proposition homotopic_paths_trans [trans]:
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
  3119
     "\<lbrakk>homotopic_paths s p q; homotopic_paths s q r\<rbrakk> \<Longrightarrow> homotopic_paths s p r"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3120
  apply (simp add: homotopic_paths_def)
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3121
  apply (rule homotopic_with_trans, assumption)
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3122
  by (metis (mono_tags, lifting) homotopic_with_imp_property homotopic_with_mono)
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3123
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3124
proposition homotopic_paths_eq:
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3125
     "\<lbrakk>path p; path_image p \<subseteq> s; \<And>t. t \<in> {0..1} \<Longrightarrow> p t = q t\<rbrakk> \<Longrightarrow> homotopic_paths s p q"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3126
  apply (simp add: homotopic_paths_def)
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3127
  apply (rule homotopic_with_eq)
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3128
  apply (auto simp: path_def homotopic_with_refl pathstart_def pathfinish_def path_image_def elim: continuous_on_eq)
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3129
  done
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3130
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3131
proposition homotopic_paths_reparametrize:
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3132
  assumes "path p"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3133
      and pips: "path_image p \<subseteq> s"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3134
      and contf: "continuous_on {0..1} f"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3135
      and f01:"f ` {0..1} \<subseteq> {0..1}"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3136
      and [simp]: "f(0) = 0" "f(1) = 1"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3137
      and q: "\<And>t. t \<in> {0..1} \<Longrightarrow> q(t) = p(f t)"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3138
    shows "homotopic_paths s p q"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3139
proof -
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3140
  have contp: "continuous_on {0..1} p"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3141
    by (metis \<open>path p\<close> path_def)
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3142
  then have "continuous_on {0..1} (p o f)"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3143
    using contf continuous_on_compose continuous_on_subset f01 by blast
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3144
  then have "path q"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3145
    by (simp add: path_def) (metis q continuous_on_cong)
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3146
  have piqs: "path_image q \<subseteq> s"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3147
    by (metis (no_types, hide_lams) pips f01 image_subset_iff path_image_def q)
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3148
  have fb0: "\<And>a b. \<lbrakk>0 \<le> a; a \<le> 1; 0 \<le> b; b \<le> 1\<rbrakk> \<Longrightarrow> 0 \<le> (1 - a) * f b + a * b"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3149
    using f01 by force
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3150
  have fb1: "\<lbrakk>0 \<le> a; a \<le> 1; 0 \<le> b; b \<le> 1\<rbrakk> \<Longrightarrow> (1 - a) * f b + a * b \<le> 1" for a b
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3151
    using f01 [THEN subsetD, of "f b"] by (simp add: convex_bound_le)
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3152
  have "homotopic_paths s q p"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3153
  proof (rule homotopic_paths_trans)
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3154
    show "homotopic_paths s q (p \<circ> f)"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3155
      using q by (force intro: homotopic_paths_eq [OF  \<open>path q\<close> piqs])
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3156
  next
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3157
    show "homotopic_paths s (p \<circ> f) p"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3158
      apply (simp add: homotopic_paths_def homotopic_with_def)
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3159
      apply (rule_tac x="p o (\<lambda>y. (1 - (fst y)) *\<^sub>R ((f o snd) y) + (fst y) *\<^sub>R snd y)"  in exI)
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3160
      apply (rule conjI contf continuous_intros continuous_on_subset [OF contp] | simp)+
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3161
      using pips [unfolded path_image_def]
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3162
      apply (auto simp: fb0 fb1 pathstart_def pathfinish_def)
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3163
      done
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3164
  qed
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3165
  then show ?thesis
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3166
    by (simp add: homotopic_paths_sym)
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3167
qed
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3168
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3169
lemma homotopic_paths_subset: "\<lbrakk>homotopic_paths s p q; s \<subseteq> t\<rbrakk> \<Longrightarrow> homotopic_paths t p q"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3170
  using homotopic_paths_def homotopic_with_subset_right by blast
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3171
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3172
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3173
text\<open> A slightly ad-hoc but useful lemma in constructing homotopies.\<close>
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3174
lemma homotopic_join_lemma:
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3175
  fixes q :: "[real,real] \<Rightarrow> 'a::topological_space"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3176
  assumes p: "continuous_on ({0..1} \<times> {0..1}) (\<lambda>y. p (fst y) (snd y))"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3177
      and q: "continuous_on ({0..1} \<times> {0..1}) (\<lambda>y. q (fst y) (snd y))"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3178
      and pf: "\<And>t. t \<in> {0..1} \<Longrightarrow> pathfinish(p t) = pathstart(q t)"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3179
    shows "continuous_on ({0..1} \<times> {0..1}) (\<lambda>y. (p(fst y) +++ q(fst y)) (snd y))"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3180
proof -
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3181
  have 1: "(\<lambda>y. p (fst y) (2 * snd y)) = (\<lambda>y. p (fst y) (snd y)) o (\<lambda>y. (fst y, 2 * snd y))"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3182
    by (rule ext) (simp )
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3183
  have 2: "(\<lambda>y. q (fst y) (2 * snd y - 1)) = (\<lambda>y. q (fst y) (snd y)) o (\<lambda>y. (fst y, 2 * snd y - 1))"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3184
    by (rule ext) (simp )
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3185
  show ?thesis
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3186
    apply (simp add: joinpaths_def)
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3187
    apply (rule continuous_on_cases_le)
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3188
    apply (simp_all only: 1 2)
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3189
    apply (rule continuous_intros continuous_on_subset [OF p] continuous_on_subset [OF q] | force)+
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3190
    using pf
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3191
    apply (auto simp: mult.commute pathstart_def pathfinish_def)
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3192
    done
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3193
qed
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3194
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3195
text\<open> Congruence properties of homotopy w.r.t. path-combining operations.\<close>
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3196
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3197
lemma homotopic_paths_reversepath_D:
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3198
      assumes "homotopic_paths s p q"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3199
      shows   "homotopic_paths s (reversepath p) (reversepath q)"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3200
  using assms
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3201
  apply (simp add: homotopic_paths_def homotopic_with_def, clarify)
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3202
  apply (rule_tac x="h o (\<lambda>x. (fst x, 1 - snd x))" in exI)
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3203
  apply (rule conjI continuous_intros)+
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3204
  apply (auto simp: reversepath_def pathstart_def pathfinish_def elim!: continuous_on_subset)
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3205
  done
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3206
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3207
proposition homotopic_paths_reversepath:
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3208
     "homotopic_paths s (reversepath p) (reversepath q) \<longleftrightarrow> homotopic_paths s p q"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3209
  using homotopic_paths_reversepath_D by force
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3210
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3211
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3212
proposition homotopic_paths_join:
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3213
    "\<lbrakk>homotopic_paths s p p'; homotopic_paths s q q'; pathfinish p = pathstart q\<rbrakk> \<Longrightarrow> homotopic_paths s (p +++ q) (p' +++ q')"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3214
  apply (simp add: homotopic_paths_def homotopic_with_def, clarify)
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3215
  apply (rename_tac k1 k2)
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3216
  apply (rule_tac x="(\<lambda>y. ((k1 o Pair (fst y)) +++ (k2 o Pair (fst y))) (snd y))" in exI)
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3217
  apply (rule conjI continuous_intros homotopic_join_lemma)+
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3218
  apply (auto simp: joinpaths_def pathstart_def pathfinish_def path_image_def)
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3219
  done
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3220
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3221
proposition homotopic_paths_continuous_image:
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3222
    "\<lbrakk>homotopic_paths s f g; continuous_on s h; h ` s \<subseteq> t\<rbrakk> \<Longrightarrow> homotopic_paths t (h o f) (h o g)"
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3223
  unfolding homotopic_paths_def
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3224
  apply (rule homotopic_with_compose_continuous_left [of _ _ _ s])
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3225
  apply (auto simp: pathstart_def pathfinish_def elim!: homotopic_with_mono)
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3226
  done
6571c78c9667 Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents: 61518
diff changeset
  3227
61711
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3228
subsection\<open>Group properties for homotopy of paths\<close>
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3229
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3230
text\<open>So taking equivalence classes under homotopy would give the fundamental group\<close>
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3231
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3232
proposition homotopic_paths_rid:
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3233
    "\<lbrakk>path p; path_image p \<subseteq> s\<rbrakk> \<Longrightarrow> homotopic_paths s (p +++ linepath (pathfinish p) (pathfinish p)) p"
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3234
  apply (subst homotopic_paths_sym)
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3235
  apply (rule homotopic_paths_reparametrize [where f = "\<lambda>t. if  t \<le> 1 / 2 then 2 *\<^sub>R t else 1"])
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3236
  apply (simp_all del: le_divide_eq_numeral1)
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3237
  apply (subst split_01)
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3238
  apply (rule continuous_on_cases continuous_intros | force simp: pathfinish_def joinpaths_def)+
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3239
  done
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3240
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3241
proposition homotopic_paths_lid:
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3242
   "\<lbrakk>path p; path_image p \<subseteq> s\<rbrakk> \<Longrightarrow> homotopic_paths s (linepath (pathstart p) (pathstart p) +++ p) p"
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3243
using homotopic_paths_rid [of "reversepath p" s]
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
  3244
  by (metis homotopic_paths_reversepath path_image_reversepath path_reversepath pathfinish_linepath
61711
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3245
        pathfinish_reversepath reversepath_joinpaths reversepath_linepath)
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3246
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3247
proposition homotopic_paths_assoc:
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3248
   "\<lbrakk>path p; path_image p \<subseteq> s; path q; path_image q \<subseteq> s; path r; path_image r \<subseteq> s; pathfinish p = pathstart q;
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3249
     pathfinish q = pathstart r\<rbrakk>
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3250
    \<Longrightarrow> homotopic_paths s (p +++ (q +++ r)) ((p +++ q) +++ r)"
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3251
  apply (subst homotopic_paths_sym)
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3252
  apply (rule homotopic_paths_reparametrize
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3253
           [where f = "\<lambda>t. if  t \<le> 1 / 2 then inverse 2 *\<^sub>R t
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3254
                           else if  t \<le> 3 / 4 then t - (1 / 4)
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3255
                           else 2 *\<^sub>R t - 1"])
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3256
  apply (simp_all del: le_divide_eq_numeral1)
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3257
  apply (simp add: subset_path_image_join)
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3258
  apply (rule continuous_on_cases_1 continuous_intros)+
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3259
  apply (auto simp: joinpaths_def)
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3260
  done
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3261
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3262
proposition homotopic_paths_rinv:
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3263
  assumes "path p" "path_image p \<subseteq> s"
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3264
    shows "homotopic_paths s (p +++ reversepath p) (linepath (pathstart p) (pathstart p))"
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3265
proof -
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3266
  have "continuous_on ({0..1} \<times> {0..1}) (\<lambda>x. (subpath 0 (fst x) p +++ reversepath (subpath 0 (fst x) p)) (snd x))"
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
  3267
    using assms
61711
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3268
    apply (simp add: joinpaths_def subpath_def reversepath_def path_def del: le_divide_eq_numeral1)
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3269
    apply (rule continuous_on_cases_le)
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3270
    apply (rule_tac [2] continuous_on_compose [of _ _ p, unfolded o_def])
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3271
    apply (rule continuous_on_compose [of _ _ p, unfolded o_def])
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3272
    apply (auto intro!: continuous_intros simp del: eq_divide_eq_numeral1)
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3273
    apply (force elim!: continuous_on_subset simp add: mult_le_one)+
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3274
    done
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3275
  then show ?thesis
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3276
    using assms
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
  3277
    apply (subst homotopic_paths_sym_eq)
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
  3278
    unfolding homotopic_paths_def homotopic_with_def
61711
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3279
    apply (rule_tac x="(\<lambda>y. (subpath 0 (fst y) p +++ reversepath(subpath 0 (fst y) p)) (snd y))" in exI)
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3280
    apply (simp add: path_defs joinpaths_def subpath_def reversepath_def)
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3281
    apply (force simp: mult_le_one)
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3282
    done
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3283
qed
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3284
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3285
proposition homotopic_paths_linv:
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3286
  assumes "path p" "path_image p \<subseteq> s"
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3287
    shows "homotopic_paths s (reversepath p +++ p) (linepath (pathfinish p) (pathfinish p))"
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3288
using homotopic_paths_rinv [of "reversepath p" s] assms by simp
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3289
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3290
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3291
subsection\<open> Homotopy of loops without requiring preservation of endpoints.\<close>
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3292
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3293
definition homotopic_loops :: "'a::topological_space set \<Rightarrow> (real \<Rightarrow> 'a) \<Rightarrow> (real \<Rightarrow> 'a) \<Rightarrow> bool"  where
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3294
 "homotopic_loops s p q \<equiv>
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3295
     homotopic_with (\<lambda>r. pathfinish r = pathstart r) {0..1} s p q"
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3296
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3297
lemma homotopic_loops:
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3298
   "homotopic_loops s p q \<longleftrightarrow>
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3299
      (\<exists>h. continuous_on ({0..1::real} \<times> {0..1}) h \<and>
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3300
          image h ({0..1} \<times> {0..1}) \<subseteq> s \<and>
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3301
          (\<forall>x \<in> {0..1}. h(0,x) = p x) \<and>
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3302
          (\<forall>x \<in> {0..1}. h(1,x) = q x) \<and>
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3303
          (\<forall>t \<in> {0..1}. pathfinish(h o Pair t) = pathstart(h o Pair t)))"
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3304
  by (simp add: homotopic_loops_def pathstart_def pathfinish_def homotopic_with)
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3305
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3306
proposition homotopic_loops_imp_loop:
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3307
     "homotopic_loops s p q \<Longrightarrow> pathfinish p = pathstart p \<and> pathfinish q = pathstart q"
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3308
using homotopic_with_imp_property homotopic_loops_def by blast
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3309
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3310
proposition homotopic_loops_imp_path:
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3311
     "homotopic_loops s p q \<Longrightarrow> path p \<and> path q"
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3312
  unfolding homotopic_loops_def path_def
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3313
  using homotopic_with_imp_continuous by blast
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3314
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
  3315
proposition homotopic_loops_imp_subset:
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
  3316
     "homotopic_loops s p q \<Longrightarrow> path_image p \<subseteq> s \<and> path_image q \<subseteq> s"
61711
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3317
  unfolding homotopic_loops_def path_image_def
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
  3318
  by (metis homotopic_with_imp_subset1 homotopic_with_imp_subset2)
61711
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3319
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3320
proposition homotopic_loops_refl:
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3321
     "homotopic_loops s p p \<longleftrightarrow>
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3322
      path p \<and> path_image p \<subseteq> s \<and> pathfinish p = pathstart p"
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3323
  by (simp add: homotopic_loops_def homotopic_with_refl path_image_def path_def)
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3324
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
  3325
proposition homotopic_loops_sym: "homotopic_loops s p q \<Longrightarrow> homotopic_loops s q p"
61711
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3326
  by (simp add: homotopic_loops_def homotopic_with_sym)
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3327
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
  3328
proposition homotopic_loops_sym_eq: "homotopic_loops s p q \<longleftrightarrow> homotopic_loops s q p"
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
  3329
  by (metis homotopic_loops_sym)
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
  3330
61711
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3331
proposition homotopic_loops_trans:
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3332
   "\<lbrakk>homotopic_loops s p q; homotopic_loops s q r\<rbrakk> \<Longrightarrow> homotopic_loops s p r"
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3333
  unfolding homotopic_loops_def by (blast intro: homotopic_with_trans)
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3334
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3335
proposition homotopic_loops_subset:
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3336
   "\<lbrakk>homotopic_loops s p q; s \<subseteq> t\<rbrakk> \<Longrightarrow> homotopic_loops t p q"
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3337
  by (simp add: homotopic_loops_def homotopic_with_subset_right)
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3338
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3339
proposition homotopic_loops_eq:
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3340
   "\<lbrakk>path p; path_image p \<subseteq> s; pathfinish p = pathstart p; \<And>t. t \<in> {0..1} \<Longrightarrow> p(t) = q(t)\<rbrakk>
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3341
          \<Longrightarrow> homotopic_loops s p q"
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3342
  unfolding homotopic_loops_def
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3343
  apply (rule homotopic_with_eq)
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3344
  apply (rule homotopic_with_refl [where f = p, THEN iffD2])
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3345
  apply (simp_all add: path_image_def path_def pathstart_def pathfinish_def)
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3346
  done
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3347
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3348
proposition homotopic_loops_continuous_image:
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3349
   "\<lbrakk>homotopic_loops s f g; continuous_on s h; h ` s \<subseteq> t\<rbrakk> \<Longrightarrow> homotopic_loops t (h \<circ> f) (h \<circ> g)"
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3350
  unfolding homotopic_loops_def
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3351
  apply (rule homotopic_with_compose_continuous_left)
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3352
  apply (erule homotopic_with_mono)
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3353
  by (simp add: pathfinish_def pathstart_def)
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3354
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3355
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3356
subsection\<open>Relations between the two variants of homotopy\<close>
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3357
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3358
proposition homotopic_paths_imp_homotopic_loops:
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3359
    "\<lbrakk>homotopic_paths s p q; pathfinish p = pathstart p; pathfinish q = pathstart p\<rbrakk> \<Longrightarrow> homotopic_loops s p q"
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3360
  by (auto simp: homotopic_paths_def homotopic_loops_def intro: homotopic_with_mono)
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3361
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3362
proposition homotopic_loops_imp_homotopic_paths_null:
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3363
  assumes "homotopic_loops s p (linepath a a)"
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3364
    shows "homotopic_paths s p (linepath (pathstart p) (pathstart p))"
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3365
proof -
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3366
  have "path p" by (metis assms homotopic_loops_imp_path)
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3367
  have ploop: "pathfinish p = pathstart p" by (metis assms homotopic_loops_imp_loop)
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
  3368
  have pip: "path_image p \<subseteq> s" by (metis assms homotopic_loops_imp_subset)
61711
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3369
  obtain h where conth: "continuous_on ({0..1::real} \<times> {0..1}) h"
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3370
             and hs: "h ` ({0..1} \<times> {0..1}) \<subseteq> s"
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3371
             and [simp]: "\<And>x. x \<in> {0..1} \<Longrightarrow> h(0,x) = p x"
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3372
             and [simp]: "\<And>x. x \<in> {0..1} \<Longrightarrow> h(1,x) = a"
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3373
             and ends: "\<And>t. t \<in> {0..1} \<Longrightarrow> pathfinish (h \<circ> Pair t) = pathstart (h \<circ> Pair t)"
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3374
    using assms by (auto simp: homotopic_loops homotopic_with)
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3375
  have conth0: "path (\<lambda>u. h (u, 0))"
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3376
    unfolding path_def
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3377
    apply (rule continuous_on_compose [of _ _ h, unfolded o_def])
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3378
    apply (force intro: continuous_intros continuous_on_subset [OF conth])+
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3379
    done
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3380
  have pih0: "path_image (\<lambda>u. h (u, 0)) \<subseteq> s"
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3381
    using hs by (force simp: path_image_def)
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3382
  have c1: "continuous_on ({0..1} \<times> {0..1}) (\<lambda>x. h (fst x * snd x, 0))"
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3383
    apply (rule continuous_on_compose [of _ _ h, unfolded o_def])
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3384
    apply (force simp: mult_le_one intro: continuous_intros continuous_on_subset [OF conth])+
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3385
    done
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3386
  have c2: "continuous_on ({0..1} \<times> {0..1}) (\<lambda>x. h (fst x - fst x * snd x, 0))"
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3387
    apply (rule continuous_on_compose [of _ _ h, unfolded o_def])
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3388
    apply (force simp: mult_left_le mult_le_one intro: continuous_intros continuous_on_subset [OF conth])+
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3389
    apply (rule continuous_on_subset [OF conth])
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3390
    apply (auto simp: algebra_simps add_increasing2 mult_left_le)
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3391
    done
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3392
  have [simp]: "\<And>t. \<lbrakk>0 \<le> t \<and> t \<le> 1\<rbrakk> \<Longrightarrow> h (t, 1) = h (t, 0)"
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3393
    using ends by (simp add: pathfinish_def pathstart_def)
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3394
  have adhoc_le: "c * 4 \<le> 1 + c * (d * 4)" if "\<not> d * 4 \<le> 3" "0 \<le> c" "c \<le> 1" for c d::real
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3395
  proof -
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3396
    have "c * 3 \<le> c * (d * 4)" using that less_eq_real_def by auto
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3397
    with \<open>c \<le> 1\<close> show ?thesis by fastforce
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3398
  qed
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3399
  have *: "\<And>p x. (path p \<and> path(reversepath p)) \<and>
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3400
                  (path_image p \<subseteq> s \<and> path_image(reversepath p) \<subseteq> s) \<and>
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3401
                  (pathfinish p = pathstart(linepath a a +++ reversepath p) \<and>
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3402
                   pathstart(reversepath p) = a) \<and> pathstart p = x
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3403
                  \<Longrightarrow> homotopic_paths s (p +++ linepath a a +++ reversepath p) (linepath x x)"
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3404
    by (metis homotopic_paths_lid homotopic_paths_join
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3405
              homotopic_paths_trans homotopic_paths_sym homotopic_paths_rinv)
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3406
  have 1: "homotopic_paths s p (p +++ linepath (pathfinish p) (pathfinish p))"
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3407
    using \<open>path p\<close> homotopic_paths_rid homotopic_paths_sym pip by blast
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
  3408
  moreover have "homotopic_paths s (p +++ linepath (pathfinish p) (pathfinish p))
61711
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3409
                                   (linepath (pathstart p) (pathstart p) +++ p +++ linepath (pathfinish p) (pathfinish p))"
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
  3410
    apply (rule homotopic_paths_sym)
61711
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3411
    using homotopic_paths_lid [of "p +++ linepath (pathfinish p) (pathfinish p)" s]
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3412
    by (metis 1 homotopic_paths_imp_path homotopic_paths_imp_pathstart homotopic_paths_imp_subset)
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
  3413
  moreover have "homotopic_paths s (linepath (pathstart p) (pathstart p) +++ p +++ linepath (pathfinish p) (pathfinish p))
61711
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3414
                                   ((\<lambda>u. h (u, 0)) +++ linepath a a +++ reversepath (\<lambda>u. h (u, 0)))"
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3415
    apply (simp add: homotopic_paths_def homotopic_with_def)
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3416
    apply (rule_tac x="\<lambda>y. (subpath 0 (fst y) (\<lambda>u. h (u, 0)) +++ (\<lambda>u. h (Pair (fst y) u)) +++ subpath (fst y) 0 (\<lambda>u. h (u, 0))) (snd y)" in exI)
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3417
    apply (simp add: subpath_reversepath)
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3418
    apply (intro conjI homotopic_join_lemma)
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
  3419
    using ploop
61711
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3420
    apply (simp_all add: path_defs joinpaths_def o_def subpath_def conth c1 c2)
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3421
    apply (force simp: algebra_simps mult_le_one mult_left_le intro: hs [THEN subsetD] adhoc_le)
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3422
    done
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
  3423
  moreover have "homotopic_paths s ((\<lambda>u. h (u, 0)) +++ linepath a a +++ reversepath (\<lambda>u. h (u, 0)))
61711
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3424
                                   (linepath (pathstart p) (pathstart p))"
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3425
    apply (rule *)
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3426
    apply (simp add: pih0 pathstart_def pathfinish_def conth0)
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3427
    apply (simp add: reversepath_def joinpaths_def)
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3428
    done
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3429
  ultimately show ?thesis
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3430
    by (blast intro: homotopic_paths_trans)
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3431
qed
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3432
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3433
proposition homotopic_loops_conjugate:
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3434
  fixes s :: "'a::real_normed_vector set"
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
  3435
  assumes "path p" "path q" and pip: "path_image p \<subseteq> s" and piq: "path_image q \<subseteq> s"
61711
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3436
      and papp: "pathfinish p = pathstart q" and qloop: "pathfinish q = pathstart q"
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3437
    shows "homotopic_loops s (p +++ q +++ reversepath p) q"
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3438
proof -
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3439
  have contp: "continuous_on {0..1} p"  using \<open>path p\<close> [unfolded path_def] by blast
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3440
  have contq: "continuous_on {0..1} q"  using \<open>path q\<close> [unfolded path_def] by blast
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3441
  have c1: "continuous_on ({0..1} \<times> {0..1}) (\<lambda>x. p ((1 - fst x) * snd x + fst x))"
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3442
    apply (rule continuous_on_compose [of _ _ p, unfolded o_def])
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3443
    apply (force simp: mult_le_one intro!: continuous_intros)
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3444
    apply (rule continuous_on_subset [OF contp])
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3445
    apply (auto simp: algebra_simps add_increasing2 mult_right_le_one_le 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
  3446
    done
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3447
  have c2: "continuous_on ({0..1} \<times> {0..1}) (\<lambda>x. p ((fst x - 1) * snd x + 1))"
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3448
    apply (rule continuous_on_compose [of _ _ p, unfolded o_def])
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3449
    apply (force simp: mult_le_one intro!: continuous_intros)
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3450
    apply (rule continuous_on_subset [OF contp])
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3451
    apply (auto simp: algebra_simps add_increasing2 mult_left_le_one_le)
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3452
    done
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3453
  have ps1: "\<And>a b. \<lbrakk>b * 2 \<le> 1; 0 \<le> b; 0 \<le> a; a \<le> 1\<rbrakk> \<Longrightarrow> p ((1 - a) * (2 * b) + a) \<in> s"
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
  3454
    using sum_le_prod1
61711
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3455
    by (force simp: algebra_simps add_increasing2 mult_left_le intro: pip [unfolded path_image_def, THEN subsetD])
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3456
  have ps2: "\<And>a b. \<lbrakk>\<not> 4 * b \<le> 3; b \<le> 1; 0 \<le> a; a \<le> 1\<rbrakk> \<Longrightarrow> p ((a - 1) * (4 * b - 3) + 1) \<in> s"
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3457
    apply (rule pip [unfolded path_image_def, THEN subsetD])
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3458
    apply (rule image_eqI, blast)
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3459
    apply (simp add: algebra_simps)
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
  3460
    by (metis add_mono_thms_linordered_semiring(1) affine_ineq linear mult.commute mult.left_neutral mult_right_mono not_le
61711
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3461
              add.commute zero_le_numeral)
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3462
  have qs: "\<And>a b. \<lbrakk>4 * b \<le> 3; \<not> b * 2 \<le> 1\<rbrakk> \<Longrightarrow> q (4 * b - 2) \<in> s"
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3463
    using path_image_def piq by fastforce
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
  3464
  have "homotopic_loops s (p +++ q +++ reversepath p)
61711
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3465
                          (linepath (pathstart q) (pathstart q) +++ q +++ linepath (pathstart q) (pathstart q))"
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3466
    apply (simp add: homotopic_loops_def homotopic_with_def)
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3467
    apply (rule_tac x="(\<lambda>y. (subpath (fst y) 1 p +++ q +++ subpath 1 (fst y) p) (snd y))" in exI)
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3468
    apply (simp add: subpath_refl subpath_reversepath)
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3469
    apply (intro conjI homotopic_join_lemma)
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3470
    using papp qloop
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3471
    apply (simp_all add: path_defs joinpaths_def o_def subpath_def c1 c2)
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3472
    apply (force simp: contq intro: continuous_on_compose [of _ _ q, unfolded o_def] continuous_on_id continuous_on_snd)
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3473
    apply (auto simp: ps1 ps2 qs)
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3474
    done
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3475
  moreover have "homotopic_loops s (linepath (pathstart q) (pathstart q) +++ q +++ linepath (pathstart q) (pathstart q)) q"
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3476
  proof -
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3477
    have "homotopic_paths s (linepath (pathfinish q) (pathfinish q) +++ q) q"
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3478
      using \<open>path q\<close> homotopic_paths_lid qloop piq by auto
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3479
    hence 1: "\<And>f. homotopic_paths s f q \<or> \<not> homotopic_paths s f (linepath (pathfinish q) (pathfinish q) +++ q)"
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3480
      using homotopic_paths_trans by blast
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
  3481
    hence "homotopic_paths s (linepath (pathfinish q) (pathfinish q) +++ q +++ linepath (pathfinish q) (pathfinish q)) q"
61711
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3482
    proof -
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3483
      have "homotopic_paths s (q +++ linepath (pathfinish q) (pathfinish q)) q"
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3484
        by (simp add: \<open>path q\<close> homotopic_paths_rid piq)
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3485
      thus ?thesis
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
  3486
        by (metis (no_types) 1 \<open>path q\<close> homotopic_paths_join homotopic_paths_rinv homotopic_paths_sym
61711
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3487
                  homotopic_paths_trans qloop pathfinish_linepath piq)
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
  3488
    qed
61711
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3489
    thus ?thesis
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3490
      by (metis (no_types) qloop homotopic_loops_sym homotopic_paths_imp_homotopic_loops homotopic_paths_imp_pathfinish homotopic_paths_sym)
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3491
  qed
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3492
  ultimately show ?thesis
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3493
    by (blast intro: homotopic_loops_trans)
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3494
qed
21d7910d6816 Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents: 61699
diff changeset
  3495
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
  3496
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
  3497
subsection\<open> Homotopy of "nearby" function, paths and loops.\<close>
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
  3498
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
  3499
lemma homotopic_with_linear:
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
  3500
  fixes f g :: "_ \<Rightarrow> 'b::real_normed_vector"
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
  3501
  assumes contf: "continuous_on s f"
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
  3502
      and contg:"continuous_on s g"
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
  3503
      and sub: "\<And>x. x \<in> s \<Longrightarrow> closed_segment (f x) (g x) \<subseteq> t"
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
  3504
    shows "homotopic_with (\<lambda>z. True) s t f g"
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
  3505
  apply (simp add: homotopic_with_def)
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
  3506
  apply (rule_tac x="\<lambda>y. ((1 - (fst y)) *\<^sub>R f(snd y) + (fst y) *\<^sub>R g(snd y))" in exI)
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
  3507
  apply (intro conjI)
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
  3508
  apply (rule subset_refl continuous_intros continuous_on_subset [OF contf] continuous_on_compose2 [where g=f]
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
  3509
                                            continuous_on_subset [OF contg] continuous_on_compose2 [where g=g]| simp)+
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
  3510
  using sub closed_segment_def apply fastforce+
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
  3511
  done
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
  3512
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
  3513
lemma homotopic_paths_linear:
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
  3514
  fixes g h :: "real \<Rightarrow> 'a::real_normed_vector"
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
  3515
  assumes "path g" "path h" "pathstart h = pathstart g" "pathfinish h = pathfinish g"
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
  3516
          "\<And>t x. t \<in> {0..1} \<Longrightarrow> closed_segment (g t) (h t) \<subseteq> s"
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
  3517
    shows "homotopic_paths s g h"
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
  3518
  using assms
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
  3519
  unfolding path_def
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
  3520
  apply (simp add: pathstart_def pathfinish_def homotopic_paths_def homotopic_with_def)
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
  3521
  apply (rule_tac x="\<lambda>y. ((1 - (fst y)) *\<^sub>R g(snd y) + (fst y) *\<^sub>R h(snd y))" in exI)
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
  3522
  apply (auto intro!: continuous_intros intro: continuous_on_compose2 [where g=g] continuous_on_compose2 [where g=h] )
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
  3523
  apply (force simp: closed_segment_def)
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
  3524
  apply (simp_all add: algebra_simps)
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
  3525
  done
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
  3526
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
  3527
lemma homotopic_loops_linear:
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
  3528
  fixes g h :: "real \<Rightarrow> 'a::real_normed_vector"
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
  3529
  assumes "path g" "path h" "pathfinish g = pathstart g" "pathfinish h = pathstart h"
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
  3530
          "\<And>t x. t \<in> {0..1} \<Longrightarrow> closed_segment (g t) (h t) \<subseteq> s"
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
  3531
    shows "homotopic_loops s g h"
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
  3532
  using assms
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
  3533
  unfolding path_def
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
  3534
  apply (simp add: pathstart_def pathfinish_def homotopic_loops_def homotopic_with_def)
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
  3535
  apply (rule_tac x="\<lambda>y. ((1 - (fst y)) *\<^sub>R g(snd y) + (fst y) *\<^sub>R h(snd y))" in exI)
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
  3536
  apply (auto intro!: continuous_intros intro: continuous_on_compose2 [where g=g] continuous_on_compose2 [where g=h])
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
  3537
  apply (force simp: closed_segment_def)
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
  3538
  done
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
  3539
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
  3540
lemma homotopic_paths_nearby_explicit:
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
  3541
  assumes "path g" "path h" "pathstart h = pathstart g" "pathfinish h = pathfinish g"
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
  3542
      and no: "\<And>t x. \<lbrakk>t \<in> {0..1}; x \<notin> s\<rbrakk> \<Longrightarrow> norm(h t - g t) < norm(g t - x)"
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
  3543
    shows "homotopic_paths s g h"
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
  3544
  apply (rule homotopic_paths_linear [OF assms(1-4)])
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
  3545
  by (metis no segment_bound(1) subsetI norm_minus_commute not_le)
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
  3546
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
  3547
lemma homotopic_loops_nearby_explicit:
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
  3548
  assumes "path g" "path h" "pathfinish g = pathstart g" "pathfinish h = pathstart h"
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
  3549
      and no: "\<And>t x. \<lbrakk>t \<in> {0..1}; x \<notin> s\<rbrakk> \<Longrightarrow> norm(h t - g t) < norm(g t - x)"
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
  3550
    shows "homotopic_loops s g h"
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
  3551
  apply (rule homotopic_loops_linear [OF assms(1-4)])
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
  3552
  by (metis no segment_bound(1) subsetI norm_minus_commute not_le)
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
  3553
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
  3554
lemma homotopic_nearby_paths:
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
  3555
  fixes g h :: "real \<Rightarrow> 'a::euclidean_space"
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
  3556
  assumes "path g" "open s" "path_image g \<subseteq> s"
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
  3557
    shows "\<exists>e. 0 < e \<and>
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
  3558
               (\<forall>h. path h \<and>
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
  3559
                    pathstart h = pathstart g \<and> pathfinish h = pathfinish g \<and>
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
  3560
                    (\<forall>t \<in> {0..1}. norm(h t - g t) < e) \<longrightarrow> homotopic_paths s g h)"
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
  3561
proof -
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
  3562
  obtain e where "e > 0" and e: "\<And>x y. x \<in> path_image g \<Longrightarrow> y \<in> - s \<Longrightarrow> e \<le> dist x y"
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
  3563
    using separate_compact_closed [of "path_image g" "-s"] assms by force
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
  3564
  show ?thesis
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
  3565
    apply (intro exI conjI)
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
  3566
    using e [unfolded dist_norm]
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
  3567
    apply (auto simp: intro!: homotopic_paths_nearby_explicit assms  \<open>e > 0\<close>)
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
  3568
    by (metis atLeastAtMost_iff imageI le_less_trans not_le path_image_def)
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
  3569
qed
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
  3570
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
  3571
lemma homotopic_nearby_loops:
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
  3572
  fixes g h :: "real \<Rightarrow> 'a::euclidean_space"
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
  3573
  assumes "path g" "open s" "path_image g \<subseteq> s" "pathfinish g = pathstart g"
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
  3574
    shows "\<exists>e. 0 < e \<and>
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
  3575
               (\<forall>h. path h \<and> pathfinish h = pathstart h \<and>
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
  3576
                    (\<forall>t \<in> {0..1}. norm(h t - g t) < e) \<longrightarrow> homotopic_loops s g h)"
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
  3577
proof -
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
  3578
  obtain e where "e > 0" and e: "\<And>x y. x \<in> path_image g \<Longrightarrow> y \<in> - s \<Longrightarrow> e \<le> dist x y"
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
  3579
    using separate_compact_closed [of "path_image g" "-s"] assms by force
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
  3580
  show ?thesis
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
  3581
    apply (intro exI conjI)
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
  3582
    using e [unfolded dist_norm]
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
  3583
    apply (auto simp: intro!: homotopic_loops_nearby_explicit assms  \<open>e > 0\<close>)
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
  3584
    by (metis atLeastAtMost_iff imageI le_less_trans not_le path_image_def)
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
  3585
qed
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
  3586
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
  3587
subsection\<open> Homotopy and subpaths\<close>
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
  3588
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
  3589
lemma homotopic_join_subpaths1:
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
  3590
  assumes "path g" and pag: "path_image g \<subseteq> s"
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
  3591
      and u: "u \<in> {0..1}" and v: "v \<in> {0..1}" and w: "w \<in> {0..1}" "u \<le> v" "v \<le> w"
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
  3592
    shows "homotopic_paths s (subpath u v g +++ subpath v w g) (subpath u w g)"
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
  3593
proof -
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
  3594
  have 1: "t * 2 \<le> 1 \<Longrightarrow> u + t * (v * 2) \<le> v + t * (u * 2)" for t
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
  3595
    using affine_ineq \<open>u \<le> v\<close> by fastforce
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
  3596
  have 2: "t * 2 > 1 \<Longrightarrow> u + (2*t - 1) * v \<le> v + (2*t - 1) * w" for t
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
  3597
    by (metis add_mono_thms_linordered_semiring(1) diff_gt_0_iff_gt less_eq_real_def mult.commute mult_right_mono \<open>u \<le> v\<close> \<open>v \<le> w\<close>)
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
  3598
  have t2: "\<And>t::real. t*2 = 1 \<Longrightarrow> t = 1/2" by auto
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
  3599
  show ?thesis
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
  3600
    apply (rule homotopic_paths_subset [OF _ pag])
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
  3601
    using assms
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
  3602
    apply (cases "w = u")
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
  3603
    using homotopic_paths_rinv [of "subpath u v g" "path_image g"]
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
  3604
    apply (force simp: closed_segment_eq_real_ivl image_mono path_image_def subpath_refl)
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
  3605
      apply (rule homotopic_paths_sym)
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
  3606
      apply (rule homotopic_paths_reparametrize
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
  3607
             [where f = "\<lambda>t. if  t \<le> 1 / 2
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
  3608
                             then inverse((w - u)) *\<^sub>R (2 * (v - u)) *\<^sub>R t
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
  3609
                             else inverse((w - u)) *\<^sub>R ((v - u) + (w - v) *\<^sub>R (2 *\<^sub>R t - 1))"])
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
  3610
      using \<open>path g\<close> path_subpath u w apply blast
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
  3611
      using \<open>path g\<close> path_image_subpath_subset u w(1) apply blast
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
  3612
      apply simp_all
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
  3613
      apply (subst split_01)
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
  3614
      apply (rule continuous_on_cases continuous_intros | force simp: pathfinish_def joinpaths_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
  3615
      apply (simp_all add: field_simps not_le)
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
  3616
      apply (force dest!: t2)
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
  3617
      apply (force simp: algebra_simps mult_left_mono affine_ineq dest!: 1 2)
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
  3618
      apply (simp add: joinpaths_def subpath_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
  3619
      apply (force simp: algebra_simps)
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
  3620
      done
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
  3621
qed
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
  3622
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
  3623
lemma homotopic_join_subpaths2:
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
  3624
  assumes "homotopic_paths s (subpath u v g +++ subpath v w g) (subpath u w g)"
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
  3625
    shows "homotopic_paths s (subpath w v g +++ subpath v u g) (subpath w u g)"
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
  3626
by (metis assms homotopic_paths_reversepath_D pathfinish_subpath pathstart_subpath reversepath_joinpaths reversepath_subpath)
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
  3627
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
  3628
lemma homotopic_join_subpaths3:
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
  3629
  assumes hom: "homotopic_paths s (subpath u v g +++ subpath v w g) (subpath u w g)"
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
  3630
      and "path g" and pag: "path_image g \<subseteq> s"
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
  3631
      and u: "u \<in> {0..1}" and v: "v \<in> {0..1}" and w: "w \<in> {0..1}"
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
  3632
    shows "homotopic_paths s (subpath v w g +++ subpath w u g) (subpath v u g)"
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
  3633
proof -
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
  3634
  have "homotopic_paths s (subpath u w g +++ subpath w v g) ((subpath u v g +++ subpath v w g) +++ subpath w v g)"
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
  3635
    apply (rule homotopic_paths_join)
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
  3636
    using hom homotopic_paths_sym_eq apply blast
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
  3637
    apply (metis \<open>path g\<close> homotopic_paths_eq pag path_image_subpath_subset path_subpath subset_trans v w)
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
  3638
    apply (simp add:)
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
  3639
    done
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
  3640
  also have "homotopic_paths s ((subpath u v g +++ subpath v w g) +++ subpath w v g) (subpath u v g +++ subpath v w g +++ subpath w v g)"
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
  3641
    apply (rule homotopic_paths_sym [OF homotopic_paths_assoc])
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
  3642
    using assms by (simp_all add: path_image_subpath_subset [THEN order_trans])
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
  3643
  also have "homotopic_paths s (subpath u v g +++ subpath v w g +++ subpath w v g)
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
  3644
                               (subpath u v g +++ linepath (pathfinish (subpath u v g)) (pathfinish (subpath u v g)))"
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
  3645
    apply (rule homotopic_paths_join)
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
  3646
    apply (metis \<open>path g\<close> homotopic_paths_eq order.trans pag path_image_subpath_subset path_subpath u v)
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
  3647
    apply (metis (no_types, lifting) \<open>path g\<close> homotopic_paths_linv order_trans pag path_image_subpath_subset path_subpath pathfinish_subpath reversepath_subpath v w)
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
  3648
    apply (simp add:)
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
  3649
    done
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
  3650
  also have "homotopic_paths s (subpath u v g +++ linepath (pathfinish (subpath u v g)) (pathfinish (subpath u v g))) (subpath u v g)"
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
  3651
    apply (rule homotopic_paths_rid)
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
  3652
    using \<open>path g\<close> path_subpath u v apply blast
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
  3653
    apply (meson \<open>path g\<close> order.trans pag path_image_subpath_subset u v)
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
  3654
    done
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
  3655
  finally have "homotopic_paths s (subpath u w g +++ subpath w v g) (subpath u v g)" .
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
  3656
  then show ?thesis
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
  3657
    using homotopic_join_subpaths2 by blast
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
  3658
qed
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
  3659
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
  3660
proposition homotopic_join_subpaths:
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
  3661
   "\<lbrakk>path g; path_image g \<subseteq> s; u \<in> {0..1}; v \<in> {0..1}; w \<in> {0..1}\<rbrakk>
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
  3662
    \<Longrightarrow> homotopic_paths s (subpath u v g +++ subpath v w g) (subpath u w g)"
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
  3663
apply (rule le_cases3 [of u v w])
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
  3664
using homotopic_join_subpaths1 homotopic_join_subpaths2 homotopic_join_subpaths3 by metis+
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
  3665
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  3666
end