src/HOL/Multivariate_Analysis/Path_Connected.thy
author paulson
Mon, 26 Oct 2015 23:41:27 +0000
changeset 61518 ff12606337e9
parent 61426 d53db136e8fd
child 61694 6571c78c9667
permissions -rw-r--r--
new lemmas about topology, etc., for Cauchy integral formula
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(*  Title:      HOL/Multivariate_Analysis/Path_Connected.thy
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    Author:     Robert Himmelmann, TU Muenchen, and LCP with 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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(*FIXME move up?*)
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lemma image_affinity_interval:
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  fixes c :: "'a::ordered_real_vector"
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  shows "((\<lambda>x. m *\<^sub>R x + c) ` {a..b}) = (if {a..b}={} then {}
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            else if 0 <= m then {m *\<^sub>R a + c .. m  *\<^sub>R b + c}
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            else {m *\<^sub>R b + c .. m *\<^sub>R a + c})"
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  apply (case_tac "m=0", force)
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  apply (auto simp: scaleR_left_mono)
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  apply (rule_tac x="inverse m *\<^sub>R (x-c)" in rev_image_eqI, auto simp: pos_le_divideR_eq le_diff_eq scaleR_left_mono_neg)
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  apply (metis diff_le_eq inverse_inverse_eq order.not_eq_order_implies_strict pos_le_divideR_eq positive_imp_inverse_positive)
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  apply (rule_tac x="inverse m *\<^sub>R (x-c)" in rev_image_eqI, auto simp: not_le neg_le_divideR_eq diff_le_eq)
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  using le_diff_eq scaleR_le_cancel_left_neg
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  apply fastforce
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  done
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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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   129
  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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   135
  using assms inj_on_eq_iff [of f]
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   136
  by (auto simp: path_linear_image_eq simple_path_def path_translation_eq)
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   137
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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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   144
  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
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   145
   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]
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   148
  by (auto simp: arc_def inj_on_def path_linear_image_eq)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   149
60420
884f54e01427 isabelle update_cartouches;
wenzelm
parents: 60303
diff changeset
   150
subsection\<open>Basic lemmas about paths\<close>
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   151
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   152
lemma arc_imp_simple_path: "arc g \<Longrightarrow> simple_path g"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   153
  by (simp add: arc_def inj_on_def simple_path_def)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   154
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   155
lemma arc_imp_path: "arc g \<Longrightarrow> path g"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   156
  using arc_def by blast
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   157
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   158
lemma simple_path_imp_path: "simple_path g \<Longrightarrow> path g"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   159
  using simple_path_def by blast
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   160
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   161
lemma simple_path_cases: "simple_path g \<Longrightarrow> arc g \<or> pathfinish g = pathstart g"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   162
  unfolding simple_path_def arc_def inj_on_def pathfinish_def pathstart_def
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   163
  by (force)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   164
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   165
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
   166
  using simple_path_cases by auto
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   167
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   168
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
   169
  unfolding arc_def inj_on_def pathfinish_def pathstart_def
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   170
  by fastforce
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   171
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   172
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
   173
  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
   174
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   175
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
   176
  by (simp add: arc_simple_path)
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
   177
60974
6a6f15d8fbc4 New material and fixes related to the forthcoming Stone-Weierstrass development
paulson <lp15@cam.ac.uk>
parents: 60809
diff changeset
   178
lemma path_image_nonempty [simp]: "path_image g \<noteq> {}"
56188
0268784f60da use cbox to relax class constraints
immler
parents: 53640
diff changeset
   179
  unfolding path_image_def image_is_empty box_eq_empty
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   180
  by auto
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
   181
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   182
lemma pathstart_in_path_image[intro]: "pathstart g \<in> path_image g"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   183
  unfolding pathstart_def path_image_def
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   184
  by auto
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
   185
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   186
lemma pathfinish_in_path_image[intro]: "pathfinish g \<in> path_image g"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   187
  unfolding pathfinish_def path_image_def
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   188
  by auto
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   189
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   190
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
   191
  unfolding path_def path_image_def
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   192
  using connected_continuous_image connected_Icc by blast
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
   193
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   194
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
   195
  unfolding path_def path_image_def
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   196
  using compact_continuous_image connected_Icc by blast
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
   197
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   198
lemma reversepath_reversepath[simp]: "reversepath (reversepath g) = g"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   199
  unfolding reversepath_def
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   200
  by auto
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
   201
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   202
lemma pathstart_reversepath[simp]: "pathstart (reversepath g) = pathfinish g"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   203
  unfolding pathstart_def reversepath_def pathfinish_def
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   204
  by auto
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
   205
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   206
lemma pathfinish_reversepath[simp]: "pathfinish (reversepath g) = pathstart g"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   207
  unfolding pathstart_def reversepath_def pathfinish_def
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   208
  by auto
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
   209
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   210
lemma pathstart_join[simp]: "pathstart (g1 +++ g2) = pathstart g1"
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   211
  unfolding pathstart_def joinpaths_def pathfinish_def
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   212
  by auto
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
   213
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   214
lemma pathfinish_join[simp]: "pathfinish (g1 +++ g2) = pathfinish g2"
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   215
  unfolding pathstart_def joinpaths_def pathfinish_def
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   216
  by auto
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
   217
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   218
lemma path_image_reversepath[simp]: "path_image (reversepath g) = path_image g"
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   219
proof -
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   220
  have *: "\<And>g. path_image (reversepath g) \<subseteq> path_image g"
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   221
    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
   222
    by force
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   223
  show ?thesis
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   224
    using *[of g] *[of "reversepath g"]
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   225
    unfolding reversepath_reversepath
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   226
    by auto
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   227
qed
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
   228
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   229
lemma path_reversepath [simp]: "path (reversepath g) \<longleftrightarrow> path g"
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   230
proof -
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   231
  have *: "\<And>g. path g \<Longrightarrow> path (reversepath g)"
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   232
    unfolding path_def reversepath_def
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   233
    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
   234
    apply (intro continuous_intros)
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   235
    apply (rule continuous_on_subset[of "{0..1}"])
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   236
    apply assumption
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   237
    apply auto
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   238
    done
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   239
  show ?thesis
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   240
    using *[of "reversepath g"] *[of g]
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   241
    unfolding reversepath_reversepath
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   242
    by (rule iffI)
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   243
qed
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   244
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   245
lemma arc_reversepath:
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   246
  assumes "arc g" shows "arc(reversepath g)"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   247
proof -
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   248
  have injg: "inj_on g {0..1}"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   249
    using assms
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   250
    by (simp add: arc_def)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   251
  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
   252
    by simp
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   253
  show ?thesis
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   254
    apply (auto simp: arc_def inj_on_def path_reversepath)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   255
    apply (simp add: arc_imp_path assms)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   256
    apply (rule **)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   257
    apply (rule inj_onD [OF injg])
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   258
    apply (auto simp: reversepath_def)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   259
    done
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   260
qed
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   261
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   262
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
   263
  apply (simp add: simple_path_def)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   264
  apply (force simp: reversepath_def)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   265
  done
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   266
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   267
lemmas reversepath_simps =
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   268
  path_reversepath path_image_reversepath pathstart_reversepath pathfinish_reversepath
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
   269
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   270
lemma path_join[simp]:
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   271
  assumes "pathfinish g1 = pathstart g2"
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   272
  shows "path (g1 +++ g2) \<longleftrightarrow> path g1 \<and> path g2"
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   273
  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
   274
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
   275
  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
   276
  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
   277
    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
   278
  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
   279
    using assms
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   280
    by (intro continuous_on_cong refl) (auto simp: joinpaths_def pathfinish_def pathstart_def)
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   281
  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
   282
    unfolding g1 g2
56371
fb9ae0727548 extend continuous_intros; remove continuous_on_intros and isCont_intros
hoelzl
parents: 56188
diff changeset
   283
    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
   284
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
   285
  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
   286
  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
   287
    by auto
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   288
  {
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   289
    fix x :: real
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   290
    assume "0 \<le> x" and "x \<le> 1"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   291
    then have "x \<in> (\<lambda>x. x * 2) ` {0..1 / 2}"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   292
      by (intro image_eqI[where x="x/2"]) auto
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   293
  }
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
   294
  note 1 = this
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   295
  {
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   296
    fix x :: real
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   297
    assume "0 \<le> x" and "x \<le> 1"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   298
    then have "x \<in> (\<lambda>x. x * 2 - 1) ` {1 / 2..1}"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   299
      by (intro image_eqI[where x="x/2 + 1/2"]) auto
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   300
  }
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
   301
  note 2 = this
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   302
  show "continuous_on {0..1} (g1 +++ g2)"
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   303
    using assms
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   304
    unfolding joinpaths_def 01
56371
fb9ae0727548 extend continuous_intros; remove continuous_on_intros and isCont_intros
hoelzl
parents: 56188
diff changeset
   305
    apply (intro continuous_on_cases closed_atLeastAtMost g1g2[THEN continuous_on_compose2] continuous_intros)
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   306
    apply (auto simp: field_simps pathfinish_def pathstart_def intro!: 1 2)
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   307
    done
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   308
qed
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
   309
60420
884f54e01427 isabelle update_cartouches;
wenzelm
parents: 60303
diff changeset
   310
section \<open>Path Images\<close>
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   311
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   312
lemma bounded_path_image: "path g \<Longrightarrow> bounded(path_image g)"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   313
  by (simp add: compact_imp_bounded compact_path_image)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   314
60809
457abb82fb9e the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents: 60420
diff changeset
   315
lemma closed_path_image:
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   316
  fixes g :: "real \<Rightarrow> 'a::t2_space"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   317
  shows "path g \<Longrightarrow> closed(path_image g)"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   318
  by (metis compact_path_image compact_imp_closed)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   319
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   320
lemma 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
   321
  by (metis connected_path_image simple_path_imp_path)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   322
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   323
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
   324
  by (metis compact_path_image simple_path_imp_path)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   325
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   326
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
   327
  by (metis bounded_path_image simple_path_imp_path)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   328
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   329
lemma closed_simple_path_image:
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   330
  fixes g :: "real \<Rightarrow> 'a::t2_space"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   331
  shows "simple_path g \<Longrightarrow> closed(path_image g)"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   332
  by (metis closed_path_image simple_path_imp_path)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   333
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   334
lemma connected_arc_image: "arc g \<Longrightarrow> connected(path_image g)"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   335
  by (metis connected_path_image arc_imp_path)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   336
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   337
lemma compact_arc_image: "arc g \<Longrightarrow> compact(path_image g)"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   338
  by (metis compact_path_image arc_imp_path)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   339
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   340
lemma bounded_arc_image: "arc g \<Longrightarrow> bounded(path_image g)"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   341
  by (metis bounded_path_image arc_imp_path)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   342
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   343
lemma closed_arc_image:
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   344
  fixes g :: "real \<Rightarrow> 'a::t2_space"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   345
  shows "arc g \<Longrightarrow> closed(path_image g)"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   346
  by (metis closed_path_image arc_imp_path)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   347
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   348
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
   349
  unfolding path_image_def joinpaths_def
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   350
  by auto
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
   351
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
   352
lemma subset_path_image_join:
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   353
  assumes "path_image g1 \<subseteq> s"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   354
    and "path_image g2 \<subseteq> s"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   355
  shows "path_image (g1 +++ g2) \<subseteq> s"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   356
  using path_image_join_subset[of g1 g2] and assms
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   357
  by auto
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
   358
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
   359
lemma path_image_join:
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   360
    "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
   361
  apply (rule subset_antisym [OF path_image_join_subset])
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   362
  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
   363
  apply (drule sym)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   364
  apply (rule_tac x="xa/2" in bexI, auto)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   365
  apply (rule ccontr)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   366
  apply (drule_tac x="(xa+1)/2" in bspec)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   367
  apply (auto simp: field_simps)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   368
  apply (drule_tac x="1/2" in bspec, auto)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   369
  done
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
   370
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
   371
lemma not_in_path_image_join:
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   372
  assumes "x \<notin> path_image g1"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   373
    and "x \<notin> path_image g2"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   374
  shows "x \<notin> path_image (g1 +++ g2)"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   375
  using assms and path_image_join_subset[of g1 g2]
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   376
  by auto
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
   377
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   378
lemma pathstart_compose: "pathstart(f o p) = f(pathstart p)"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   379
  by (simp add: pathstart_def)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   380
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   381
lemma pathfinish_compose: "pathfinish(f o p) = f(pathfinish p)"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   382
  by (simp add: pathfinish_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 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
   385
  by (simp add: image_comp path_image_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
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
   388
  by (rule ext) (simp add: joinpaths_def)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   389
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   390
lemma path_compose_reversepath: "f o reversepath p = reversepath(f o p)"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   391
  by (rule ext) (simp add: reversepath_def)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   392
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   393
lemma join_paths_eq:
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   394
  "(\<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
   395
   (\<And>t. t \<in> {0..1} \<Longrightarrow> q t = q' t)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   396
   \<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
   397
  by (auto simp: joinpaths_def)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   398
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   399
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
   400
  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
   401
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   402
60420
884f54e01427 isabelle update_cartouches;
wenzelm
parents: 60303
diff changeset
   403
subsection\<open>Simple paths with the endpoints removed\<close>
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   404
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   405
lemma simple_path_endless:
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   406
    "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
   407
  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
   408
  apply (metis eq_iff le_less_linear)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   409
  apply (metis leD linear)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   410
  using less_eq_real_def zero_le_one apply blast
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   411
  using less_eq_real_def zero_le_one apply blast
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   412
  done
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
   413
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   414
lemma connected_simple_path_endless:
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   415
    "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
   416
apply (simp add: simple_path_endless)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   417
apply (rule connected_continuous_image)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   418
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
   419
by auto
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   420
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   421
lemma nonempty_simple_path_endless:
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   422
    "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
   423
  by (simp add: simple_path_endless)
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
60420
884f54e01427 isabelle update_cartouches;
wenzelm
parents: 60303
diff changeset
   426
subsection\<open>The operations on paths\<close>
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   427
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   428
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
   429
  by (auto simp: path_image_def reversepath_def)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   430
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   431
lemma continuous_on_op_minus: "continuous_on (s::real set) (op - x)"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   432
  by (rule continuous_intros | simp)+
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   433
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   434
lemma path_imp_reversepath: "path g \<Longrightarrow> path(reversepath g)"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   435
  apply (auto simp: path_def reversepath_def)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   436
  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
   437
  apply (auto simp: continuous_on_op_minus)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   438
  done
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   439
61204
3e491e34a62e new lemmas and movement of lemmas into place
paulson
parents: 60974
diff changeset
   440
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
   441
  by simp
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   442
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   443
lemma continuous_on_joinpaths:
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   444
  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
   445
    shows "continuous_on {0..1} (g1 +++ g2)"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   446
proof -
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   447
  have *: "{0..1::real} = {0..1/2} \<union> {1/2..1}"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   448
    by auto
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   449
  have gg: "g2 0 = g1 1"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   450
    by (metis assms(3) pathfinish_def pathstart_def)
61204
3e491e34a62e new lemmas and movement of lemmas into place
paulson
parents: 60974
diff changeset
   451
  have 1: "continuous_on {0..1/2} (g1 +++ g2)"
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   452
    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
   453
    apply (rule continuous_intros | simp add: joinpaths_def assms)+
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   454
    done
61204
3e491e34a62e new lemmas and movement of lemmas into place
paulson
parents: 60974
diff changeset
   455
  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
   456
    apply (rule continuous_on_subset [of "{1/2..1}"])
3e491e34a62e new lemmas and movement of lemmas into place
paulson
parents: 60974
diff changeset
   457
    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
   458
    done
3e491e34a62e new lemmas and movement of lemmas into place
paulson
parents: 60974
diff changeset
   459
  then have 2: "continuous_on {1/2..1} (g1 +++ g2)"
3e491e34a62e new lemmas and movement of lemmas into place
paulson
parents: 60974
diff changeset
   460
    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
   461
    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
   462
    done
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   463
  show ?thesis
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   464
    apply (subst *)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   465
    apply (rule continuous_on_union)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   466
    using 1 2
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   467
    apply auto
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   468
    done
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   469
qed
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   470
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   471
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
   472
  by (simp add: path_join)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   473
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   474
lemmas join_paths_simps = path_join path_image_join pathstart_join pathfinish_join
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   475
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
   476
lemma simple_path_join_loop:
60809
457abb82fb9e the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents: 60420
diff changeset
   477
  assumes "arc g1" "arc g2"
457abb82fb9e the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents: 60420
diff changeset
   478
          "pathfinish g1 = pathstart g2"  "pathfinish g2 = pathstart g1"
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   479
          "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
   480
  shows "simple_path(g1 +++ g2)"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   481
proof -
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   482
  have injg1: "inj_on g1 {0..1}"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   483
    using assms
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   484
    by (simp add: arc_def)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   485
  have injg2: "inj_on g2 {0..1}"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   486
    using assms
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   487
    by (simp add: arc_def)
60809
457abb82fb9e the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents: 60420
diff changeset
   488
  have g12: "g1 1 = g2 0"
457abb82fb9e the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents: 60420
diff changeset
   489
   and g21: "g2 1 = g1 0"
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   490
   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
   491
    using assms
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   492
    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
   493
  { fix x and y::real
60809
457abb82fb9e the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents: 60420
diff changeset
   494
    assume xyI: "x = 1 \<longrightarrow> y \<noteq> 0"
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   495
       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
   496
    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
   497
      using xy
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   498
      apply simp
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   499
      apply (rule_tac x="2 * x - 1" in image_eqI, auto)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   500
      done
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   501
    have False
60809
457abb82fb9e the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents: 60420
diff changeset
   502
      using subsetD [OF sb g1im] xy
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   503
      apply auto
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   504
      apply (drule inj_onD [OF injg1])
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   505
      using g21 [symmetric] xyI
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   506
      apply (auto dest: inj_onD [OF injg2])
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   507
      done
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   508
   } note * = this
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   509
  { fix x and y::real
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   510
    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
   511
    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
   512
      using xy
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   513
      apply simp
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   514
      apply (rule_tac x="2 * x" in image_eqI, auto)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   515
      done
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   516
    have "x = 0 \<and> y = 1"
60809
457abb82fb9e the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents: 60420
diff changeset
   517
      using subsetD [OF sb g1im] xy
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   518
      apply auto
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   519
      apply (force dest: inj_onD [OF injg1])
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   520
      using  g21 [symmetric]
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   521
      apply (auto dest: inj_onD [OF injg2])
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   522
      done
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   523
   } note ** = this
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   524
  show ?thesis
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
    apply (simp add: arc_def simple_path_def path_join, clarify)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   527
    apply (simp add: joinpaths_def split: split_if_asm)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   528
    apply (force dest: inj_onD [OF injg1])
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   529
    apply (metis *)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   530
    apply (metis **)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   531
    apply (force dest: inj_onD [OF injg2])
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   532
    done
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   533
qed
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   534
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   535
lemma arc_join:
60809
457abb82fb9e the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents: 60420
diff changeset
   536
  assumes "arc g1" "arc g2"
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   537
          "pathfinish g1 = pathstart g2"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   538
          "path_image g1 \<inter> path_image g2 \<subseteq> {pathstart g2}"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   539
    shows "arc(g1 +++ g2)"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   540
proof -
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   541
  have injg1: "inj_on g1 {0..1}"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   542
    using assms
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   543
    by (simp add: arc_def)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   544
  have injg2: "inj_on g2 {0..1}"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   545
    using assms
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   546
    by (simp add: arc_def)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   547
  have g11: "g1 1 = g2 0"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   548
   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
   549
    using assms
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   550
    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
   551
  { fix x and y::real
60809
457abb82fb9e the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents: 60420
diff changeset
   552
    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
   553
    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
   554
      using xy
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   555
      apply simp
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   556
      apply (rule_tac x="2 * x - 1" in image_eqI, auto)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   557
      done
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   558
    have False
60809
457abb82fb9e the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents: 60420
diff changeset
   559
      using subsetD [OF sb g1im] xy
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   560
      by (auto dest: inj_onD [OF injg2])
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   561
   } note * = this
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   562
  show ?thesis
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   563
    apply (simp add: arc_def inj_on_def)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   564
    apply (clarsimp simp add: arc_imp_path assms path_join)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   565
    apply (simp add: joinpaths_def split: split_if_asm)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   566
    apply (force dest: inj_onD [OF injg1])
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   567
    apply (metis *)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   568
    apply (metis *)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   569
    apply (force dest: inj_onD [OF injg2])
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   570
    done
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   571
qed
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   572
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   573
lemma reversepath_joinpaths:
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   574
    "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
   575
  unfolding reversepath_def pathfinish_def pathstart_def joinpaths_def
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   576
  by (rule ext) (auto simp: mult.commute)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   577
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   578
61518
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   579
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
   580
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   581
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
   582
  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
   583
60809
457abb82fb9e the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents: 60420
diff changeset
   584
lemma path_image_subpath_gen [simp]:
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   585
  fixes g :: "real \<Rightarrow> 'a::real_normed_vector"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   586
  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
   587
  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
   588
  apply (subst o_def [of g, symmetric])
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   589
  apply (simp add: image_comp [symmetric])
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   590
  done
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   591
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   592
lemma path_image_subpath [simp]:
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   593
  fixes g :: "real \<Rightarrow> 'a::real_normed_vector"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   594
  shows "path_image(subpath u v g) = (if u \<le> v then g ` {u..v} else g ` {v..u})"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   595
  by (simp add: closed_segment_eq_real_ivl)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   596
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   597
lemma path_subpath [simp]:
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   598
  fixes g :: "real \<Rightarrow> 'a::real_normed_vector"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   599
  assumes "path g" "u \<in> {0..1}" "v \<in> {0..1}"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   600
    shows "path(subpath u v g)"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   601
proof -
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   602
  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
   603
    apply (rule continuous_intros | simp)+
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   604
    apply (simp add: image_affinity_atLeastAtMost [where c=u])
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   605
    using assms
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   606
    apply (auto simp: path_def continuous_on_subset)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   607
    done
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   608
  then show ?thesis
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   609
    by (simp add: path_def subpath_def)
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   610
qed
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
   611
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   612
lemma pathstart_subpath [simp]: "pathstart(subpath u v g) = g(u)"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   613
  by (simp add: pathstart_def subpath_def)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   614
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   615
lemma pathfinish_subpath [simp]: "pathfinish(subpath u v g) = g(v)"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   616
  by (simp add: pathfinish_def subpath_def)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   617
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   618
lemma subpath_trivial [simp]: "subpath 0 1 g = g"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   619
  by (simp add: subpath_def)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   620
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   621
lemma subpath_reversepath: "subpath 1 0 g = reversepath g"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   622
  by (simp add: reversepath_def subpath_def)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   623
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   624
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
   625
  by (simp add: reversepath_def subpath_def algebra_simps)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   626
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   627
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
   628
  by (rule ext) (simp add: subpath_def)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   629
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   630
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
   631
  by (rule ext) (simp add: subpath_def)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   632
60809
457abb82fb9e the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents: 60420
diff changeset
   633
lemma affine_ineq:
457abb82fb9e the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents: 60420
diff changeset
   634
  fixes x :: "'a::linordered_idom"
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   635
  assumes "x \<le> 1" "v < u"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   636
    shows "v + x * u \<le> u + x * v"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   637
proof -
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   638
  have "(1-x)*(u-v) \<ge> 0"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   639
    using assms by auto
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   640
  then show ?thesis
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   641
    by (simp add: algebra_simps)
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   642
qed
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
   643
60809
457abb82fb9e the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents: 60420
diff changeset
   644
lemma simple_path_subpath_eq:
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   645
  "simple_path(subpath u v g) \<longleftrightarrow>
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   646
     path(subpath u v g) \<and> u\<noteq>v \<and>
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   647
     (\<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
   648
                \<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
   649
    (is "?lhs = ?rhs")
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   650
proof (rule iffI)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   651
  assume ?lhs
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   652
  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
   653
        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
   654
                  \<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
   655
    by (auto simp: simple_path_def subpath_def)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   656
  { fix x y
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   657
    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
   658
    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
   659
    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
   660
    by (auto simp: closed_segment_real_eq image_affinity_atLeastAtMost divide_simps
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   661
       split: split_if_asm)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   662
  } moreover
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   663
  have "path(subpath u v g) \<and> u\<noteq>v"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   664
    using sim [of "1/3" "2/3"] p
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   665
    by (auto simp: subpath_def)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   666
  ultimately show ?rhs
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   667
    by metis
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   668
next
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   669
  assume ?rhs
60809
457abb82fb9e the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents: 60420
diff changeset
   670
  then
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   671
  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
   672
   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
   673
   and ne: "u < v \<or> v < u"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   674
   and psp: "path (subpath u v g)"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   675
    by (auto simp: closed_segment_real_eq image_affinity_atLeastAtMost)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   676
  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
   677
    by algebra
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   678
  show ?lhs using psp ne
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   679
    unfolding simple_path_def subpath_def
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   680
    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
   681
qed
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   682
60809
457abb82fb9e the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents: 60420
diff changeset
   683
lemma arc_subpath_eq:
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   684
  "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
   685
    (is "?lhs = ?rhs")
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   686
proof (rule iffI)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   687
  assume ?lhs
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   688
  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
   689
        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
   690
                  \<Longrightarrow> x = y)"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   691
    by (auto simp: arc_def inj_on_def subpath_def)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   692
  { fix x y
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   693
    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
   694
    then have "x = y"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   695
    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
   696
    by (force simp add: inj_on_def closed_segment_real_eq image_affinity_atLeastAtMost divide_simps
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   697
       split: split_if_asm)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   698
  } moreover
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   699
  have "path(subpath u v g) \<and> u\<noteq>v"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   700
    using sim [of "1/3" "2/3"] p
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   701
    by (auto simp: subpath_def)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   702
  ultimately show ?rhs
60809
457abb82fb9e the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents: 60420
diff changeset
   703
    unfolding inj_on_def
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   704
    by metis
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   705
next
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   706
  assume ?rhs
60809
457abb82fb9e the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents: 60420
diff changeset
   707
  then
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   708
  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
   709
   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
   710
   and ne: "u < v \<or> v < u"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   711
   and psp: "path (subpath u v g)"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   712
    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
   713
  show ?lhs using psp ne
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   714
    unfolding arc_def subpath_def inj_on_def
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   715
    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
   716
qed
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   717
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   718
60809
457abb82fb9e the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents: 60420
diff changeset
   719
lemma simple_path_subpath:
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   720
  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
   721
  shows "simple_path(subpath u v g)"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   722
  using assms
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   723
  apply (simp add: simple_path_subpath_eq simple_path_imp_path)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   724
  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
   725
  done
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   726
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   727
lemma arc_simple_path_subpath:
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   728
    "\<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
   729
  by (force intro: simple_path_subpath simple_path_imp_arc)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   730
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   731
lemma arc_subpath_arc:
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   732
    "\<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
   733
  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
   734
60809
457abb82fb9e the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents: 60420
diff changeset
   735
lemma arc_simple_path_subpath_interior:
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   736
    "\<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
   737
    apply (rule arc_simple_path_subpath)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   738
    apply (force simp: simple_path_def)+
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   739
    done
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   740
60809
457abb82fb9e the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents: 60420
diff changeset
   741
lemma path_image_subpath_subset:
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   742
    "\<lbrakk>path g; u \<in> {0..1}; v \<in> {0..1}\<rbrakk> \<Longrightarrow> path_image(subpath u v g) \<subseteq> path_image g"
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   743
  apply (simp add: closed_segment_real_eq image_affinity_atLeastAtMost)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   744
  apply (auto simp: path_image_def)
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   745
  done
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   746
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   747
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
   748
  by (rule ext) (simp add: joinpaths_def subpath_def divide_simps)
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   749
61518
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   750
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
   751
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   752
lemma subpath_to_frontier_explicit:
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   753
    fixes S :: "'a::metric_space set"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   754
    assumes g: "path g" and "pathfinish g \<notin> S"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   755
    obtains u where "0 \<le> u" "u \<le> 1"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   756
                "\<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
   757
                "(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
   758
proof -
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   759
  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
   760
  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
   761
    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
   762
    using compact_eq_bounded_closed apply fastforce
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   763
    done
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   764
  have "1 \<in> {u. g u \<in> closure (- S)}"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   765
    using assms by (simp add: pathfinish_def closure_def)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   766
  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
   767
    using atLeastAtMost_iff zero_le_one by blast
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   768
  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
   769
                  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
   770
    using compact_attains_inf [OF com dis] by fastforce
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   771
  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
   772
    using closure_def by fastforce
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   773
  { assume "u \<noteq> 0"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   774
    then have "u > 0" using `0 \<le> u` by auto
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   775
    { fix e::real assume "e > 0"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   776
      obtain d where "d>0" and d: "\<And>x'. \<lbrakk>x' \<in> {0..1}; dist x' u < d\<rbrakk> \<Longrightarrow> dist (g x') (g u) < e"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   777
        using continuous_onD [OF gcon _ `e > 0`] `0 \<le> _` `_ \<le> 1` atLeastAtMost_iff by auto
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   778
      have *: "dist (max 0 (u - d / 2)) u < d"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   779
        using `0 \<le> u` `u \<le> 1` `d > 0` by (simp add: dist_real_def)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   780
      have "\<exists>y\<in>S. dist y (g u) < e"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   781
        using `0 < u` `u \<le> 1` `d > 0`
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   782
        by (force intro: d [OF _ *] umin')
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   783
    }
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   784
    then have "g u \<in> closure S"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   785
      by (simp add: frontier_def closure_approachable)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   786
  }
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   787
  then show ?thesis
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   788
    apply (rule_tac u=u in that)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   789
    apply (auto simp: `0 \<le> u` `u \<le> 1` gu interior_closure umin)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   790
    using `_ \<le> 1` interior_closure umin apply fastforce
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   791
    done
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   792
qed
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   793
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   794
lemma subpath_to_frontier_strong:
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   795
    assumes g: "path g" and "pathfinish g \<notin> S"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   796
    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
   797
                    "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
   798
proof -
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   799
  obtain u where "0 \<le> u" "u \<le> 1"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   800
             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
   801
             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
   802
    using subpath_to_frontier_explicit [OF assms] by blast
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   803
  show ?thesis
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   804
    apply (rule that [OF `0 \<le> u` `u \<le> 1`])
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   805
    apply (simp add: gunot)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   806
    using `0 \<le> u` u0 by (force simp: subpath_def gxin)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   807
qed
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   808
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   809
lemma subpath_to_frontier:
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   810
    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
   811
    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
   812
proof -
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   813
  obtain u where "0 \<le> u" "u \<le> 1"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   814
             and notin: "g u \<notin> interior S"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   815
             and disj: "u = 0 \<or>
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   816
                        (\<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
   817
    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
   818
  show ?thesis
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   819
    apply (rule that [OF `0 \<le> u` `u \<le> 1`])
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   820
    apply (metis DiffI disj frontier_def g0 notin pathstart_def)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   821
    using `0 \<le> u` g0 disj
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   822
    apply (simp add:)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   823
    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
   824
    apply (rename_tac y)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   825
    apply (drule_tac x="y/u" in spec)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   826
    apply (auto split: split_if_asm)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   827
    done
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   828
qed
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   829
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   830
lemma exists_path_subpath_to_frontier:
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   831
    fixes S :: "'a::real_normed_vector set"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   832
    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
   833
    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
   834
                    "path_image h - {pathfinish h} \<subseteq> interior S"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   835
                    "pathfinish h \<in> frontier S"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   836
proof -
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   837
  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
   838
    using subpath_to_frontier [OF assms] by blast
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   839
  show ?thesis
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   840
    apply (rule that [of "subpath 0 u g"])
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   841
    using assms u
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   842
    apply simp_all
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   843
    apply (simp add: pathstart_def)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   844
    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
   845
    done
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   846
qed
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   847
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   848
lemma exists_path_subpath_to_frontier_closed:
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   849
    fixes S :: "'a::real_normed_vector set"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   850
    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
   851
    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
   852
                    "pathfinish h \<in> frontier S"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   853
proof -
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   854
  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
   855
                    "path_image h - {pathfinish h} \<subseteq> interior S"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   856
                    "pathfinish h \<in> frontier S"
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   857
    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
   858
  show ?thesis
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   859
    apply (rule that [OF `path h`])
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   860
    using assms h
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   861
    apply auto
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   862
    apply (metis diff_single_insert frontier_subset_eq insert_iff interior_subset subset_iff)
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   863
    done
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
   864
qed
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   865
60420
884f54e01427 isabelle update_cartouches;
wenzelm
parents: 60303
diff changeset
   866
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
   867
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   868
definition shiftpath :: "real \<Rightarrow> (real \<Rightarrow> 'a::topological_space) \<Rightarrow> real \<Rightarrow> 'a"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   869
  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
   870
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   871
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
   872
  unfolding pathstart_def shiftpath_def by auto
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
   873
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   874
lemma pathfinish_shiftpath:
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   875
  assumes "0 \<le> a"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   876
    and "pathfinish g = pathstart g"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   877
  shows "pathfinish (shiftpath a g) = g a"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   878
  using assms
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   879
  unfolding pathstart_def pathfinish_def shiftpath_def
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
   880
  by auto
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
   881
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
   882
lemma endpoints_shiftpath:
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   883
  assumes "pathfinish g = pathstart g"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   884
    and "a \<in> {0 .. 1}"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   885
  shows "pathfinish (shiftpath a g) = g a"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   886
    and "pathstart (shiftpath a g) = g a"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   887
  using assms
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   888
  by (auto intro!: pathfinish_shiftpath pathstart_shiftpath)
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
   889
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
   890
lemma closed_shiftpath:
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   891
  assumes "pathfinish g = pathstart g"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   892
    and "a \<in> {0..1}"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   893
  shows "pathfinish (shiftpath a g) = pathstart (shiftpath a g)"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   894
  using endpoints_shiftpath[OF assms]
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   895
  by auto
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
   896
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
   897
lemma path_shiftpath:
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   898
  assumes "path g"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   899
    and "pathfinish g = pathstart g"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   900
    and "a \<in> {0..1}"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   901
  shows "path (shiftpath a g)"
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   902
proof -
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   903
  have *: "{0 .. 1} = {0 .. 1-a} \<union> {1-a .. 1}"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   904
    using assms(3) by auto
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   905
  have **: "\<And>x. x + a = 1 \<Longrightarrow> g (x + a - 1) = g (x + a)"
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   906
    using assms(2)[unfolded pathfinish_def pathstart_def]
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   907
    by auto
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   908
  show ?thesis
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   909
    unfolding path_def shiftpath_def *
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   910
    apply (rule continuous_on_union)
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   911
    apply (rule closed_real_atLeastAtMost)+
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   912
    apply (rule continuous_on_eq[of _ "g \<circ> (\<lambda>x. a + x)"])
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   913
    prefer 3
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   914
    apply (rule continuous_on_eq[of _ "g \<circ> (\<lambda>x. a - 1 + x)"])
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   915
    prefer 3
56371
fb9ae0727548 extend continuous_intros; remove continuous_on_intros and isCont_intros
hoelzl
parents: 56188
diff changeset
   916
    apply (rule continuous_intros)+
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   917
    prefer 2
56371
fb9ae0727548 extend continuous_intros; remove continuous_on_intros and isCont_intros
hoelzl
parents: 56188
diff changeset
   918
    apply (rule continuous_intros)+
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   919
    apply (rule_tac[1-2] continuous_on_subset[OF assms(1)[unfolded path_def]])
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   920
    using assms(3) and **
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   921
    apply auto
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   922
    apply (auto simp add: field_simps)
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   923
    done
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   924
qed
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
   925
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   926
lemma shiftpath_shiftpath:
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   927
  assumes "pathfinish g = pathstart g"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   928
    and "a \<in> {0..1}"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   929
    and "x \<in> {0..1}"
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
   930
  shows "shiftpath (1 - a) (shiftpath a g) x = g x"
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   931
  using assms
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   932
  unfolding pathfinish_def pathstart_def shiftpath_def
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   933
  by auto
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
   934
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
   935
lemma path_image_shiftpath:
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   936
  assumes "a \<in> {0..1}"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   937
    and "pathfinish g = pathstart g"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   938
  shows "path_image (shiftpath a g) = path_image g"
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   939
proof -
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   940
  { fix x
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   941
    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
   942
    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
   943
    proof (cases "a \<le> x")
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   944
      case False
49654
366d8b41ca17 tuned proofs;
wenzelm
parents: 49653
diff changeset
   945
      then show ?thesis
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   946
        apply (rule_tac x="1 + x - a" in bexI)
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
   947
        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
   948
        apply (auto simp add: field_simps atomize_not)
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   949
        done
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   950
    next
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   951
      case True
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   952
      then show ?thesis
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   953
        using as(1-2) and assms(1)
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   954
        apply (rule_tac x="x - a" in bexI)
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   955
        apply (auto simp add: field_simps)
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   956
        done
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   957
    qed
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   958
  }
49654
366d8b41ca17 tuned proofs;
wenzelm
parents: 49653
diff changeset
   959
  then show ?thesis
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   960
    using assms
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   961
    unfolding shiftpath_def path_image_def pathfinish_def pathstart_def
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   962
    by (auto simp add: image_iff)
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   963
qed
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   964
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
   965
60420
884f54e01427 isabelle update_cartouches;
wenzelm
parents: 60303
diff changeset
   966
subsection \<open>Special case of straight-line paths\<close>
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
   967
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   968
definition linepath :: "'a::real_normed_vector \<Rightarrow> 'a \<Rightarrow> real \<Rightarrow> 'a"
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   969
  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
   970
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   971
lemma pathstart_linepath[simp]: "pathstart (linepath a b) = a"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   972
  unfolding pathstart_def linepath_def
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   973
  by auto
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
   974
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   975
lemma pathfinish_linepath[simp]: "pathfinish (linepath a b) = b"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   976
  unfolding pathfinish_def linepath_def
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   977
  by auto
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
   978
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
   979
lemma continuous_linepath_at[intro]: "continuous (at x) (linepath a b)"
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   980
  unfolding linepath_def
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   981
  by (intro continuous_intros)
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
   982
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
   983
lemma continuous_on_linepath[intro]: "continuous_on s (linepath a b)"
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   984
  using continuous_linepath_at
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   985
  by (auto intro!: continuous_at_imp_continuous_on)
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
   986
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   987
lemma path_linepath[intro]: "path (linepath a b)"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   988
  unfolding path_def
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   989
  by (rule continuous_on_linepath)
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
   990
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   991
lemma path_image_linepath[simp]: "path_image (linepath a b) = closed_segment a b"
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   992
  unfolding path_image_def segment linepath_def
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   993
  by auto
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   994
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
   995
lemma reversepath_linepath[simp]: "reversepath (linepath a b) = linepath b a"
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
   996
  unfolding reversepath_def linepath_def
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
   997
  by auto
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
   998
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
   999
lemma arc_linepath:
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1000
  assumes "a \<noteq> b"
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
  1001
  shows "arc (linepath a b)"
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1002
proof -
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1003
  {
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1004
    fix x y :: "real"
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1005
    assume "x *\<^sub>R b + y *\<^sub>R a = x *\<^sub>R a + y *\<^sub>R b"
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1006
    then have "(x - y) *\<^sub>R a = (x - y) *\<^sub>R b"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1007
      by (simp add: algebra_simps)
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1008
    with assms have "x = y"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1009
      by simp
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1010
  }
49654
366d8b41ca17 tuned proofs;
wenzelm
parents: 49653
diff changeset
  1011
  then show ?thesis
60809
457abb82fb9e the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents: 60420
diff changeset
  1012
    unfolding arc_def inj_on_def
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
  1013
    by (simp add:  path_linepath) (force simp: algebra_simps linepath_def)
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1014
qed
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1015
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1016
lemma simple_path_linepath[intro]: "a \<noteq> b \<Longrightarrow> simple_path (linepath a b)"
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
  1017
  by (simp add: arc_imp_simple_path arc_linepath)
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1018
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1019
60420
884f54e01427 isabelle update_cartouches;
wenzelm
parents: 60303
diff changeset
  1020
subsection \<open>Bounding a point away from a path\<close>
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1021
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1022
lemma not_on_path_ball:
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1023
  fixes g :: "real \<Rightarrow> 'a::heine_borel"
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1024
  assumes "path g"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1025
    and "z \<notin> path_image g"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1026
  shows "\<exists>e > 0. ball z e \<inter> path_image g = {}"
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1027
proof -
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1028
  obtain a where "a \<in> path_image g" "\<forall>y \<in> path_image g. dist z a \<le> dist z y"
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1029
    using distance_attains_inf[OF _ path_image_nonempty, of g z]
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1030
    using compact_path_image[THEN compact_imp_closed, OF assms(1)] by auto
49654
366d8b41ca17 tuned proofs;
wenzelm
parents: 49653
diff changeset
  1031
  then show ?thesis
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1032
    apply (rule_tac x="dist z a" in exI)
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1033
    using assms(2)
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1034
    apply (auto intro!: dist_pos_lt)
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1035
    done
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1036
qed
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1037
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1038
lemma not_on_path_cball:
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1039
  fixes g :: "real \<Rightarrow> 'a::heine_borel"
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1040
  assumes "path g"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1041
    and "z \<notin> path_image g"
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1042
  shows "\<exists>e>0. cball z e \<inter> (path_image g) = {}"
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1043
proof -
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1044
  obtain e where "ball z e \<inter> path_image g = {}" "e > 0"
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1045
    using not_on_path_ball[OF assms] by auto
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1046
  moreover have "cball z (e/2) \<subseteq> ball z e"
60420
884f54e01427 isabelle update_cartouches;
wenzelm
parents: 60303
diff changeset
  1047
    using \<open>e > 0\<close> by auto
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1048
  ultimately show ?thesis
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1049
    apply (rule_tac x="e/2" in exI)
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1050
    apply auto
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1051
    done
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1052
qed
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1053
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1054
61518
ff12606337e9 new lemmas about topology, etc., for Cauchy integral formula
paulson
parents: 61426
diff changeset
  1055
section \<open>Path component, considered as a "joinability" relation (from Tom Hales)\<close>
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1056
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1057
definition "path_component s x y \<longleftrightarrow>
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1058
  (\<exists>g. path g \<and> path_image g \<subseteq> s \<and> pathstart g = x \<and> pathfinish g = y)"
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1059
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1060
abbreviation
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1061
   "path_component_set s x \<equiv> Collect (path_component s x)"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1062
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1063
lemmas path_defs = path_def pathstart_def pathfinish_def path_image_def path_component_def
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1064
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1065
lemma path_component_mem:
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1066
  assumes "path_component s x y"
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1067
  shows "x \<in> s" and "y \<in> s"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1068
  using assms
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1069
  unfolding path_defs
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1070
  by auto
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1071
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1072
lemma path_component_refl:
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1073
  assumes "x \<in> s"
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1074
  shows "path_component s x x"
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1075
  unfolding path_defs
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1076
  apply (rule_tac x="\<lambda>u. x" in exI)
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1077
  using assms
56371
fb9ae0727548 extend continuous_intros; remove continuous_on_intros and isCont_intros
hoelzl
parents: 56188
diff changeset
  1078
  apply (auto intro!: continuous_intros)
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1079
  done
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1080
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1081
lemma path_component_refl_eq: "path_component s x x \<longleftrightarrow> x \<in> s"
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1082
  by (auto intro!: path_component_mem path_component_refl)
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1083
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1084
lemma path_component_sym: "path_component s x y \<Longrightarrow> path_component s y x"
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1085
  using assms
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1086
  unfolding path_component_def
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1087
  apply (erule exE)
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1088
  apply (rule_tac x="reversepath g" in exI)
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1089
  apply auto
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1090
  done
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1091
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1092
lemma path_component_trans:
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1093
  assumes "path_component s x y" and "path_component s y z"
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1094
  shows "path_component s x z"
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1095
  using assms
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1096
  unfolding path_component_def
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1097
  apply (elim exE)
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1098
  apply (rule_tac x="g +++ ga" in exI)
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1099
  apply (auto simp add: path_image_join)
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1100
  done
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1101
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1102
lemma path_component_of_subset: "s \<subseteq> t \<Longrightarrow> path_component s x y \<Longrightarrow> path_component t x y"
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1103
  unfolding path_component_def by auto
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1104
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1105
lemma path_connected_linepath:
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1106
    fixes s :: "'a::real_normed_vector set"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1107
    shows "closed_segment a b \<subseteq> s \<Longrightarrow> path_component s a b"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1108
  apply (simp add: path_component_def)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1109
  apply (rule_tac x="linepath a b" in exI, auto)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1110
  done
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1111
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1112
60420
884f54e01427 isabelle update_cartouches;
wenzelm
parents: 60303
diff changeset
  1113
text \<open>Can also consider it as a set, as the name suggests.\<close>
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1114
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1115
lemma path_component_set:
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1116
  "path_component_set s x =
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1117
    {y. (\<exists>g. path g \<and> path_image g \<subseteq> s \<and> pathstart g = x \<and> pathfinish g = y)}"
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1118
  by (auto simp: path_component_def)
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1119
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1120
lemma path_component_subset: "path_component_set s x \<subseteq> s"
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
  1121
  by (auto simp add: path_component_mem(2))
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1122
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1123
lemma path_component_eq_empty: "path_component_set s x = {} \<longleftrightarrow> x \<notin> s"
60303
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
  1124
  using path_component_mem path_component_refl_eq
00c06f1315d0 New material about paths, and some lemmas
paulson
parents: 59557
diff changeset
  1125
    by fastforce
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1126
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1127
lemma path_component_mono:
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1128
     "s \<subseteq> t \<Longrightarrow> (path_component_set s x) \<subseteq> (path_component_set t x)"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1129
  by (simp add: Collect_mono path_component_of_subset)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1130
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1131
lemma path_component_eq:
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1132
   "y \<in> path_component_set s x \<Longrightarrow> path_component_set s y = path_component_set s x"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1133
by (metis (no_types, lifting) Collect_cong mem_Collect_eq path_component_sym path_component_trans)
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1134
60420
884f54e01427 isabelle update_cartouches;
wenzelm
parents: 60303
diff changeset
  1135
subsection \<open>Path connectedness of a space\<close>
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1136
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1137
definition "path_connected s \<longleftrightarrow>
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1138
  (\<forall>x\<in>s. \<forall>y\<in>s. \<exists>g. path g \<and> path_image g \<subseteq> s \<and> pathstart g = x \<and> pathfinish g = y)"
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1139
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1140
lemma path_connected_component: "path_connected s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>y\<in>s. path_component s x y)"
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1141
  unfolding path_connected_def path_component_def by auto
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1142
61426
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1143
lemma path_connected_component_set: "path_connected s \<longleftrightarrow> (\<forall>x\<in>s. path_component_set s x = s)"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1144
  unfolding path_connected_component path_component_subset 
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1145
  using path_component_mem by blast
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1146
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1147
lemma path_component_maximal:
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1148
     "\<lbrakk>x \<in> t; path_connected t; t \<subseteq> s\<rbrakk> \<Longrightarrow> t \<subseteq> (path_component_set s x)"
d53db136e8fd new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents: 61204
diff changeset
  1149
  by (metis path_component_mono path_connected_component_set)
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1150
60420
884f54e01427 isabelle update_cartouches;
wenzelm
parents: 60303
diff changeset
  1151
subsection \<open>Some useful lemmas about path-connectedness\<close>
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1152
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1153
lemma convex_imp_path_connected:
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1154
  fixes s :: "'a::real_normed_vector set"
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1155
  assumes "convex s"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1156
  shows "path_connected s"
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1157
  unfolding path_connected_def
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1158
  apply rule
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1159
  apply rule
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1160
  apply (rule_tac x = "linepath x y" in exI)
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1161
  unfolding path_image_linepath
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1162
  using assms [unfolded convex_contains_segment]
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1163
  apply auto
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1164
  done
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1165
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1166
lemma path_connected_imp_connected:
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1167
  assumes "path_connected s"
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1168
  shows "connected s"
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1169
  unfolding connected_def not_ex
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1170
  apply rule
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1171
  apply rule
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1172
  apply (rule ccontr)
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1173
  unfolding not_not
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1174
  apply (elim conjE)
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1175
proof -
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1176
  fix e1 e2
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1177
  assume as: "open e1" "open e2" "s \<subseteq> e1 \<union> e2" "e1 \<inter> e2 \<inter> s = {}" "e1 \<inter> s \<noteq> {}" "e2 \<inter> s \<noteq> {}"
53640
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1178
  then obtain x1 x2 where obt:"x1 \<in> e1 \<inter> s" "x2 \<in> e2 \<inter> s"
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1179
    by auto
3170b5eb9f5a tuned proofs;
wenzelm
parents: 53593
diff changeset
  1180
  then obtain g where g: "path g" "path_image g \<subseteq> s" "pathstart g = x1" "pathfinish g = x2"
36583
68ce5760c585 move path-related stuff into new theory file
huffman
parents:
diff changeset
  1181
    using assms[unfolded path_connected_def,rule_format,of x1 x2] by auto
49653
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1182
  have *: "connected {0..1::real}"
03bc7afe8814 tuned proofs;
wenzelm
parents: 48125
diff changeset
  1183