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
p