author | paulson <lp15@cam.ac.uk> |
Fri, 20 Nov 2015 14:44:53 +0000 | |
changeset 61711 | 21d7910d6816 |
parent 61699 | a81dc5c4d6a9 |
child 61738 | c4f6031f1310 |
permissions | -rw-r--r-- |
41959 | 1 |
(* 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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||
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section \<open>Continuous paths and path-connected sets\<close> |
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|
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theory Path_Connected |
|
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convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
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imports Convex_Euclidean_Space |
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begin |
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||
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(*FIXME move up?*) |
12 |
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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the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
60420
diff
changeset
|
22 |
using le_diff_eq scaleR_le_cancel_left_neg |
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apply fastforce |
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done |
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||
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subsection \<open>Paths and Arcs\<close> |
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|
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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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|
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definition pathstart :: "(real \<Rightarrow> 'a::topological_space) \<Rightarrow> 'a" |
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where "pathstart g = g 0" |
33 |
||
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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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||
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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}" |
39 |
||
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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))" |
42 |
||
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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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||
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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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|
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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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|
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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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||
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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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||
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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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||
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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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||
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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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||
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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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||
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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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||
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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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||
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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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||
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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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||
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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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||
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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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||
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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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||
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457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
60420
diff
changeset
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lemma simple_path_translation_eq: |
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fixes g :: "real \<Rightarrow> 'a::euclidean_space" |
128 |
shows "simple_path((\<lambda>x. a + x) o g) = simple_path g" |
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by (simp add: simple_path_def path_translation_eq) |
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||
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lemma simple_path_linear_image_eq: |
|
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fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space" |
|
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assumes "linear f" "inj f" |
|
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shows "simple_path(f o g) = simple_path g" |
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using assms inj_on_eq_iff [of f] |
|
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by (auto simp: path_linear_image_eq simple_path_def path_translation_eq) |
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137 |
||
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lemma arc_translation_eq: |
|
139 |
fixes g :: "real \<Rightarrow> 'a::euclidean_space" |
|
140 |
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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142 |
||
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lemma arc_linear_image_eq: |
|
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fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space" |
|
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assumes "linear f" "inj f" |
|
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shows "arc(f o g) = arc g" |
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using assms inj_on_eq_iff [of f] |
|
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by (auto simp: arc_def inj_on_def path_linear_image_eq) |
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||
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subsection\<open>Basic lemmas about paths\<close> |
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|
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lemma arc_imp_simple_path: "arc g \<Longrightarrow> simple_path g" |
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by (simp add: arc_def inj_on_def simple_path_def) |
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154 |
||
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lemma arc_imp_path: "arc g \<Longrightarrow> path g" |
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using arc_def by blast |
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||
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lemma simple_path_imp_path: "simple_path g \<Longrightarrow> path g" |
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using simple_path_def by blast |
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160 |
||
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lemma simple_path_cases: "simple_path g \<Longrightarrow> arc g \<or> pathfinish g = pathstart g" |
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unfolding simple_path_def arc_def inj_on_def pathfinish_def pathstart_def |
|
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by (force) |
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164 |
||
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lemma simple_path_imp_arc: "simple_path g \<Longrightarrow> pathfinish g \<noteq> pathstart g \<Longrightarrow> arc g" |
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using simple_path_cases by auto |
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||
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lemma arc_distinct_ends: "arc g \<Longrightarrow> pathfinish g \<noteq> pathstart g" |
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unfolding arc_def inj_on_def pathfinish_def pathstart_def |
|
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by fastforce |
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171 |
||
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lemma arc_simple_path: "arc g \<longleftrightarrow> simple_path g \<and> pathfinish g \<noteq> pathstart g" |
|
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using arc_distinct_ends arc_imp_simple_path simple_path_cases by blast |
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||
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lemma simple_path_eq_arc: "pathfinish g \<noteq> pathstart g \<Longrightarrow> (simple_path g = arc g)" |
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by (simp add: arc_simple_path) |
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New material and fixes related to the forthcoming Stone-Weierstrass development
paulson <lp15@cam.ac.uk>
parents:
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178 |
lemma path_image_nonempty [simp]: "path_image g \<noteq> {}" |
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unfolding path_image_def image_is_empty box_eq_empty |
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by auto |
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|
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lemma pathstart_in_path_image[intro]: "pathstart g \<in> path_image g" |
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unfolding pathstart_def path_image_def |
|
184 |
by auto |
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36583 | 185 |
|
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lemma pathfinish_in_path_image[intro]: "pathfinish g \<in> path_image g" |
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unfolding pathfinish_def path_image_def |
|
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by auto |
|
189 |
||
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lemma connected_path_image[intro]: "path g \<Longrightarrow> connected (path_image g)" |
|
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unfolding path_def path_image_def |
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using connected_continuous_image connected_Icc by blast |
36583 | 193 |
|
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lemma compact_path_image[intro]: "path g \<Longrightarrow> compact (path_image g)" |
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unfolding path_def path_image_def |
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using compact_continuous_image connected_Icc by blast |
36583 | 197 |
|
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lemma reversepath_reversepath[simp]: "reversepath (reversepath g) = g" |
199 |
unfolding reversepath_def |
|
200 |
by auto |
|
36583 | 201 |
|
53640 | 202 |
lemma pathstart_reversepath[simp]: "pathstart (reversepath g) = pathfinish g" |
203 |
unfolding pathstart_def reversepath_def pathfinish_def |
|
204 |
by auto |
|
36583 | 205 |
|
53640 | 206 |
lemma pathfinish_reversepath[simp]: "pathfinish (reversepath g) = pathstart g" |
207 |
unfolding pathstart_def reversepath_def pathfinish_def |
|
208 |
by auto |
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36583 | 209 |
|
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lemma pathstart_join[simp]: "pathstart (g1 +++ g2) = pathstart g1" |
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unfolding pathstart_def joinpaths_def pathfinish_def |
212 |
by auto |
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36583 | 213 |
|
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lemma pathfinish_join[simp]: "pathfinish (g1 +++ g2) = pathfinish g2" |
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unfolding pathstart_def joinpaths_def pathfinish_def |
216 |
by auto |
|
36583 | 217 |
|
53640 | 218 |
lemma path_image_reversepath[simp]: "path_image (reversepath g) = path_image g" |
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proof - |
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have *: "\<And>g. path_image (reversepath g) \<subseteq> path_image g" |
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unfolding path_image_def subset_eq reversepath_def Ball_def image_iff |
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by force |
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show ?thesis |
224 |
using *[of g] *[of "reversepath g"] |
|
53640 | 225 |
unfolding reversepath_reversepath |
226 |
by auto |
|
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qed |
36583 | 228 |
|
53640 | 229 |
lemma path_reversepath [simp]: "path (reversepath g) \<longleftrightarrow> path g" |
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proof - |
231 |
have *: "\<And>g. path g \<Longrightarrow> path (reversepath g)" |
|
232 |
unfolding path_def reversepath_def |
|
233 |
apply (rule continuous_on_compose[unfolded o_def, of _ "\<lambda>x. 1 - x"]) |
|
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fb9ae0727548
extend continuous_intros; remove continuous_on_intros and isCont_intros
hoelzl
parents:
56188
diff
changeset
|
234 |
apply (intro continuous_intros) |
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apply (rule continuous_on_subset[of "{0..1}"]) |
236 |
apply assumption |
|
49653 | 237 |
apply auto |
238 |
done |
|
239 |
show ?thesis |
|
240 |
using *[of "reversepath g"] *[of g] |
|
241 |
unfolding reversepath_reversepath |
|
242 |
by (rule iffI) |
|
243 |
qed |
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244 |
||
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lemma arc_reversepath: |
246 |
assumes "arc g" shows "arc(reversepath g)" |
|
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proof - |
|
248 |
have injg: "inj_on g {0..1}" |
|
249 |
using assms |
|
250 |
by (simp add: arc_def) |
|
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have **: "\<And>x y::real. 1-x = 1-y \<Longrightarrow> x = y" |
|
252 |
by simp |
|
253 |
show ?thesis |
|
254 |
apply (auto simp: arc_def inj_on_def path_reversepath) |
|
255 |
apply (simp add: arc_imp_path assms) |
|
256 |
apply (rule **) |
|
257 |
apply (rule inj_onD [OF injg]) |
|
258 |
apply (auto simp: reversepath_def) |
|
259 |
done |
|
260 |
qed |
|
261 |
||
262 |
lemma simple_path_reversepath: "simple_path g \<Longrightarrow> simple_path (reversepath g)" |
|
263 |
apply (simp add: simple_path_def) |
|
264 |
apply (force simp: reversepath_def) |
|
265 |
done |
|
266 |
||
49653 | 267 |
lemmas reversepath_simps = |
268 |
path_reversepath path_image_reversepath pathstart_reversepath pathfinish_reversepath |
|
36583 | 269 |
|
49653 | 270 |
lemma path_join[simp]: |
271 |
assumes "pathfinish g1 = pathstart g2" |
|
272 |
shows "path (g1 +++ g2) \<longleftrightarrow> path g1 \<and> path g2" |
|
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 | 279 |
using assms |
280 |
by (intro continuous_on_cong refl) (auto simp: joinpaths_def pathfinish_def pathstart_def) |
|
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 | 287 |
by auto |
53640 | 288 |
{ |
289 |
fix x :: real |
|
290 |
assume "0 \<le> x" and "x \<le> 1" |
|
291 |
then have "x \<in> (\<lambda>x. x * 2) ` {0..1 / 2}" |
|
292 |
by (intro image_eqI[where x="x/2"]) auto |
|
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 | 295 |
{ |
296 |
fix x :: real |
|
297 |
assume "0 \<le> x" and "x \<le> 1" |
|
298 |
then have "x \<in> (\<lambda>x. x * 2 - 1) ` {1 / 2..1}" |
|
299 |
by (intro image_eqI[where x="x/2 + 1/2"]) auto |
|
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 | 302 |
show "continuous_on {0..1} (g1 +++ g2)" |
53640 | 303 |
using assms |
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 | 306 |
apply (auto simp: field_simps pathfinish_def pathstart_def intro!: 1 2) |
307 |
done |
|
49653 | 308 |
qed |
36583 | 309 |
|
60420 | 310 |
section \<open>Path Images\<close> |
60303 | 311 |
|
312 |
lemma bounded_path_image: "path g \<Longrightarrow> bounded(path_image g)" |
|
313 |
by (simp add: compact_imp_bounded compact_path_image) |
|
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 | 316 |
fixes g :: "real \<Rightarrow> 'a::t2_space" |
317 |
shows "path g \<Longrightarrow> closed(path_image g)" |
|
318 |
by (metis compact_path_image compact_imp_closed) |
|
319 |
||
320 |
lemma connected_simple_path_image: "simple_path g \<Longrightarrow> connected(path_image g)" |
|
321 |
by (metis connected_path_image simple_path_imp_path) |
|
322 |
||
323 |
lemma compact_simple_path_image: "simple_path g \<Longrightarrow> compact(path_image g)" |
|
324 |
by (metis compact_path_image simple_path_imp_path) |
|
325 |
||
326 |
lemma bounded_simple_path_image: "simple_path g \<Longrightarrow> bounded(path_image g)" |
|
327 |
by (metis bounded_path_image simple_path_imp_path) |
|
328 |
||
329 |
lemma closed_simple_path_image: |
|
330 |
fixes g :: "real \<Rightarrow> 'a::t2_space" |
|
331 |
shows "simple_path g \<Longrightarrow> closed(path_image g)" |
|
332 |
by (metis closed_path_image simple_path_imp_path) |
|
333 |
||
334 |
lemma connected_arc_image: "arc g \<Longrightarrow> connected(path_image g)" |
|
335 |
by (metis connected_path_image arc_imp_path) |
|
336 |
||
337 |
lemma compact_arc_image: "arc g \<Longrightarrow> compact(path_image g)" |
|
338 |
by (metis compact_path_image arc_imp_path) |
|
339 |
||
340 |
lemma bounded_arc_image: "arc g \<Longrightarrow> bounded(path_image g)" |
|
341 |
by (metis bounded_path_image arc_imp_path) |
|
342 |
||
343 |
lemma closed_arc_image: |
|
344 |
fixes g :: "real \<Rightarrow> 'a::t2_space" |
|
345 |
shows "arc g \<Longrightarrow> closed(path_image g)" |
|
346 |
by (metis closed_path_image arc_imp_path) |
|
347 |
||
53640 | 348 |
lemma path_image_join_subset: "path_image (g1 +++ g2) \<subseteq> path_image g1 \<union> path_image g2" |
349 |
unfolding path_image_def joinpaths_def |
|
350 |
by auto |
|
36583 | 351 |
|
352 |
lemma subset_path_image_join: |
|
53640 | 353 |
assumes "path_image g1 \<subseteq> s" |
354 |
and "path_image g2 \<subseteq> s" |
|
355 |
shows "path_image (g1 +++ g2) \<subseteq> s" |
|
356 |
using path_image_join_subset[of g1 g2] and assms |
|
357 |
by auto |
|
36583 | 358 |
|
359 |
lemma path_image_join: |
|
60303 | 360 |
"pathfinish g1 = pathstart g2 \<Longrightarrow> path_image(g1 +++ g2) = path_image g1 \<union> path_image g2" |
361 |
apply (rule subset_antisym [OF path_image_join_subset]) |
|
362 |
apply (auto simp: pathfinish_def pathstart_def path_image_def joinpaths_def image_def) |
|
363 |
apply (drule sym) |
|
364 |
apply (rule_tac x="xa/2" in bexI, auto) |
|
365 |
apply (rule ccontr) |
|
366 |
apply (drule_tac x="(xa+1)/2" in bspec) |
|
367 |
apply (auto simp: field_simps) |
|
368 |
apply (drule_tac x="1/2" in bspec, auto) |
|
369 |
done |
|
36583 | 370 |
|
371 |
lemma not_in_path_image_join: |
|
53640 | 372 |
assumes "x \<notin> path_image g1" |
373 |
and "x \<notin> path_image g2" |
|
374 |
shows "x \<notin> path_image (g1 +++ g2)" |
|
375 |
using assms and path_image_join_subset[of g1 g2] |
|
376 |
by auto |
|
36583 | 377 |
|
60303 | 378 |
lemma pathstart_compose: "pathstart(f o p) = f(pathstart p)" |
379 |
by (simp add: pathstart_def) |
|
380 |
||
381 |
lemma pathfinish_compose: "pathfinish(f o p) = f(pathfinish p)" |
|
382 |
by (simp add: pathfinish_def) |
|
383 |
||
384 |
lemma path_image_compose: "path_image (f o p) = f ` (path_image p)" |
|
385 |
by (simp add: image_comp path_image_def) |
|
386 |
||
387 |
lemma path_compose_join: "f o (p +++ q) = (f o p) +++ (f o q)" |
|
388 |
by (rule ext) (simp add: joinpaths_def) |
|
389 |
||
390 |
lemma path_compose_reversepath: "f o reversepath p = reversepath(f o p)" |
|
391 |
by (rule ext) (simp add: reversepath_def) |
|
392 |
||
393 |
lemma join_paths_eq: |
|
394 |
"(\<And>t. t \<in> {0..1} \<Longrightarrow> p t = p' t) \<Longrightarrow> |
|
395 |
(\<And>t. t \<in> {0..1} \<Longrightarrow> q t = q' t) |
|
396 |
\<Longrightarrow> t \<in> {0..1} \<Longrightarrow> (p +++ q) t = (p' +++ q') t" |
|
397 |
by (auto simp: joinpaths_def) |
|
398 |
||
399 |
lemma simple_path_inj_on: "simple_path g \<Longrightarrow> inj_on g {0<..<1}" |
|
400 |
by (auto simp: simple_path_def path_image_def inj_on_def less_eq_real_def Ball_def) |
|
401 |
||
402 |
||
60420 | 403 |
subsection\<open>Simple paths with the endpoints removed\<close> |
60303 | 404 |
|
405 |
lemma simple_path_endless: |
|
406 |
"simple_path c \<Longrightarrow> path_image c - {pathstart c,pathfinish c} = c ` {0<..<1}" |
|
407 |
apply (auto simp: simple_path_def path_image_def pathstart_def pathfinish_def Ball_def Bex_def image_def) |
|
408 |
apply (metis eq_iff le_less_linear) |
|
409 |
apply (metis leD linear) |
|
410 |
using less_eq_real_def zero_le_one apply blast |
|
411 |
using less_eq_real_def zero_le_one apply blast |
|
49653 | 412 |
done |
36583 | 413 |
|
60303 | 414 |
lemma connected_simple_path_endless: |
415 |
"simple_path c \<Longrightarrow> connected(path_image c - {pathstart c,pathfinish c})" |
|
416 |
apply (simp add: simple_path_endless) |
|
417 |
apply (rule connected_continuous_image) |
|
418 |
apply (meson continuous_on_subset greaterThanLessThan_subseteq_atLeastAtMost_iff le_numeral_extra(3) le_numeral_extra(4) path_def simple_path_imp_path) |
|
419 |
by auto |
|
420 |
||
421 |
lemma nonempty_simple_path_endless: |
|
422 |
"simple_path c \<Longrightarrow> path_image c - {pathstart c,pathfinish c} \<noteq> {}" |
|
423 |
by (simp add: simple_path_endless) |
|
424 |
||
425 |
||
60420 | 426 |
subsection\<open>The operations on paths\<close> |
60303 | 427 |
|
428 |
lemma path_image_subset_reversepath: "path_image(reversepath g) \<le> path_image g" |
|
429 |
by (auto simp: path_image_def reversepath_def) |
|
430 |
||
431 |
lemma path_imp_reversepath: "path g \<Longrightarrow> path(reversepath g)" |
|
432 |
apply (auto simp: path_def reversepath_def) |
|
433 |
using continuous_on_compose [of "{0..1}" "\<lambda>x. 1 - x" g] |
|
434 |
apply (auto simp: continuous_on_op_minus) |
|
435 |
done |
|
436 |
||
61204 | 437 |
lemma half_bounded_equal: "1 \<le> x * 2 \<Longrightarrow> x * 2 \<le> 1 \<longleftrightarrow> x = (1/2::real)" |
438 |
by simp |
|
60303 | 439 |
|
440 |
lemma continuous_on_joinpaths: |
|
441 |
assumes "continuous_on {0..1} g1" "continuous_on {0..1} g2" "pathfinish g1 = pathstart g2" |
|
442 |
shows "continuous_on {0..1} (g1 +++ g2)" |
|
443 |
proof - |
|
444 |
have *: "{0..1::real} = {0..1/2} \<union> {1/2..1}" |
|
445 |
by auto |
|
446 |
have gg: "g2 0 = g1 1" |
|
447 |
by (metis assms(3) pathfinish_def pathstart_def) |
|
61204 | 448 |
have 1: "continuous_on {0..1/2} (g1 +++ g2)" |
60303 | 449 |
apply (rule continuous_on_eq [of _ "g1 o (\<lambda>x. 2*x)"]) |
61204 | 450 |
apply (rule continuous_intros | simp add: joinpaths_def assms)+ |
60303 | 451 |
done |
61204 | 452 |
have "continuous_on {1/2..1} (g2 o (\<lambda>x. 2*x-1))" |
453 |
apply (rule continuous_on_subset [of "{1/2..1}"]) |
|
454 |
apply (rule continuous_intros | simp add: image_affinity_atLeastAtMost_diff assms)+ |
|
455 |
done |
|
456 |
then have 2: "continuous_on {1/2..1} (g1 +++ g2)" |
|
457 |
apply (rule continuous_on_eq [of "{1/2..1}" "g2 o (\<lambda>x. 2*x-1)"]) |
|
458 |
apply (rule assms continuous_intros | simp add: joinpaths_def mult.commute half_bounded_equal gg)+ |
|
60303 | 459 |
done |
460 |
show ?thesis |
|
461 |
apply (subst *) |
|
462 |
apply (rule continuous_on_union) |
|
463 |
using 1 2 |
|
464 |
apply auto |
|
465 |
done |
|
466 |
qed |
|
467 |
||
468 |
lemma path_join_imp: "\<lbrakk>path g1; path g2; pathfinish g1 = pathstart g2\<rbrakk> \<Longrightarrow> path(g1 +++ g2)" |
|
469 |
by (simp add: path_join) |
|
470 |
||
471 |
lemmas join_paths_simps = path_join path_image_join pathstart_join pathfinish_join |
|
472 |
||
36583 | 473 |
lemma simple_path_join_loop: |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
60420
diff
changeset
|
474 |
assumes "arc g1" "arc g2" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
60420
diff
changeset
|
475 |
"pathfinish g1 = pathstart g2" "pathfinish g2 = pathstart g1" |
60303 | 476 |
"path_image g1 \<inter> path_image g2 \<subseteq> {pathstart g1, pathstart g2}" |
477 |
shows "simple_path(g1 +++ g2)" |
|
478 |
proof - |
|
479 |
have injg1: "inj_on g1 {0..1}" |
|
480 |
using assms |
|
481 |
by (simp add: arc_def) |
|
482 |
have injg2: "inj_on g2 {0..1}" |
|
483 |
using assms |
|
484 |
by (simp add: arc_def) |
|
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
60420
diff
changeset
|
485 |
have g12: "g1 1 = g2 0" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
60420
diff
changeset
|
486 |
and g21: "g2 1 = g1 0" |
60303 | 487 |
and sb: "g1 ` {0..1} \<inter> g2 ` {0..1} \<subseteq> {g1 0, g2 0}" |
488 |
using assms |
|
489 |
by (simp_all add: arc_def pathfinish_def pathstart_def path_image_def) |
|
490 |
{ fix x and y::real |
|
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
60420
diff
changeset
|
491 |
assume xyI: "x = 1 \<longrightarrow> y \<noteq> 0" |
60303 | 492 |
and xy: "x \<le> 1" "0 \<le> y" " y * 2 \<le> 1" "\<not> x * 2 \<le> 1" "g2 (2 * x - 1) = g1 (2 * y)" |
493 |
have g1im: "g1 (2 * y) \<in> g1 ` {0..1} \<inter> g2 ` {0..1}" |
|
494 |
using xy |
|
495 |
apply simp |
|
496 |
apply (rule_tac x="2 * x - 1" in image_eqI, auto) |
|
497 |
done |
|
498 |
have False |
|
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
60420
diff
changeset
|
499 |
using subsetD [OF sb g1im] xy |
60303 | 500 |
apply auto |
501 |
apply (drule inj_onD [OF injg1]) |
|
502 |
using g21 [symmetric] xyI |
|
503 |
apply (auto dest: inj_onD [OF injg2]) |
|
504 |
done |
|
505 |
} note * = this |
|
506 |
{ fix x and y::real |
|
507 |
assume xy: "y \<le> 1" "0 \<le> x" "\<not> y * 2 \<le> 1" "x * 2 \<le> 1" "g1 (2 * x) = g2 (2 * y - 1)" |
|
508 |
have g1im: "g1 (2 * x) \<in> g1 ` {0..1} \<inter> g2 ` {0..1}" |
|
509 |
using xy |
|
510 |
apply simp |
|
511 |
apply (rule_tac x="2 * x" in image_eqI, auto) |
|
512 |
done |
|
513 |
have "x = 0 \<and> y = 1" |
|
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
60420
diff
changeset
|
514 |
using subsetD [OF sb g1im] xy |
60303 | 515 |
apply auto |
516 |
apply (force dest: inj_onD [OF injg1]) |
|
517 |
using g21 [symmetric] |
|
518 |
apply (auto dest: inj_onD [OF injg2]) |
|
519 |
done |
|
520 |
} note ** = this |
|
521 |
show ?thesis |
|
522 |
using assms |
|
523 |
apply (simp add: arc_def simple_path_def path_join, clarify) |
|
524 |
apply (simp add: joinpaths_def split: split_if_asm) |
|
525 |
apply (force dest: inj_onD [OF injg1]) |
|
526 |
apply (metis *) |
|
527 |
apply (metis **) |
|
528 |
apply (force dest: inj_onD [OF injg2]) |
|
529 |
done |
|
530 |
qed |
|
531 |
||
532 |
lemma arc_join: |
|
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
60420
diff
changeset
|
533 |
assumes "arc g1" "arc g2" |
60303 | 534 |
"pathfinish g1 = pathstart g2" |
535 |
"path_image g1 \<inter> path_image g2 \<subseteq> {pathstart g2}" |
|
536 |
shows "arc(g1 +++ g2)" |
|
537 |
proof - |
|
538 |
have injg1: "inj_on g1 {0..1}" |
|
539 |
using assms |
|
540 |
by (simp add: arc_def) |
|
541 |
have injg2: "inj_on g2 {0..1}" |
|
542 |
using assms |
|
543 |
by (simp add: arc_def) |
|
544 |
have g11: "g1 1 = g2 0" |
|
545 |
and sb: "g1 ` {0..1} \<inter> g2 ` {0..1} \<subseteq> {g2 0}" |
|
546 |
using assms |
|
547 |
by (simp_all add: arc_def pathfinish_def pathstart_def path_image_def) |
|
548 |
{ fix x and y::real |
|
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
60420
diff
changeset
|
549 |
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 | 550 |
have g1im: "g1 (2 * y) \<in> g1 ` {0..1} \<inter> g2 ` {0..1}" |
551 |
using xy |
|
552 |
apply simp |
|
553 |
apply (rule_tac x="2 * x - 1" in image_eqI, auto) |
|
554 |
done |
|
555 |
have False |
|
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
60420
diff
changeset
|
556 |
using subsetD [OF sb g1im] xy |
60303 | 557 |
by (auto dest: inj_onD [OF injg2]) |
558 |
} note * = this |
|
559 |
show ?thesis |
|
560 |
apply (simp add: arc_def inj_on_def) |
|
561 |
apply (clarsimp simp add: arc_imp_path assms path_join) |
|
562 |
apply (simp add: joinpaths_def split: split_if_asm) |
|
563 |
apply (force dest: inj_onD [OF injg1]) |
|
564 |
apply (metis *) |
|
565 |
apply (metis *) |
|
566 |
apply (force dest: inj_onD [OF injg2]) |
|
567 |
done |
|
568 |
qed |
|
569 |
||
570 |
lemma reversepath_joinpaths: |
|
571 |
"pathfinish g1 = pathstart g2 \<Longrightarrow> reversepath(g1 +++ g2) = reversepath g2 +++ reversepath g1" |
|
572 |
unfolding reversepath_def pathfinish_def pathstart_def joinpaths_def |
|
573 |
by (rule ext) (auto simp: mult.commute) |
|
574 |
||
575 |
||
61518
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
576 |
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
|
577 |
|
60303 | 578 |
definition subpath :: "real \<Rightarrow> real \<Rightarrow> (real \<Rightarrow> 'a) \<Rightarrow> real \<Rightarrow> 'a::real_normed_vector" |
579 |
where "subpath a b g \<equiv> \<lambda>x. g((b - a) * x + a)" |
|
580 |
||
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
60420
diff
changeset
|
581 |
lemma path_image_subpath_gen [simp]: |
60303 | 582 |
fixes g :: "real \<Rightarrow> 'a::real_normed_vector" |
583 |
shows "path_image(subpath u v g) = g ` (closed_segment u v)" |
|
584 |
apply (simp add: closed_segment_real_eq path_image_def subpath_def) |
|
585 |
apply (subst o_def [of g, symmetric]) |
|
586 |
apply (simp add: image_comp [symmetric]) |
|
587 |
done |
|
588 |
||
589 |
lemma path_image_subpath [simp]: |
|
590 |
fixes g :: "real \<Rightarrow> 'a::real_normed_vector" |
|
591 |
shows "path_image(subpath u v g) = (if u \<le> v then g ` {u..v} else g ` {v..u})" |
|
592 |
by (simp add: closed_segment_eq_real_ivl) |
|
593 |
||
594 |
lemma path_subpath [simp]: |
|
595 |
fixes g :: "real \<Rightarrow> 'a::real_normed_vector" |
|
596 |
assumes "path g" "u \<in> {0..1}" "v \<in> {0..1}" |
|
597 |
shows "path(subpath u v g)" |
|
598 |
proof - |
|
599 |
have "continuous_on {0..1} (g o (\<lambda>x. ((v-u) * x+ u)))" |
|
600 |
apply (rule continuous_intros | simp)+ |
|
601 |
apply (simp add: image_affinity_atLeastAtMost [where c=u]) |
|
602 |
using assms |
|
603 |
apply (auto simp: path_def continuous_on_subset) |
|
604 |
done |
|
605 |
then show ?thesis |
|
606 |
by (simp add: path_def subpath_def) |
|
49653 | 607 |
qed |
36583 | 608 |
|
60303 | 609 |
lemma pathstart_subpath [simp]: "pathstart(subpath u v g) = g(u)" |
610 |
by (simp add: pathstart_def subpath_def) |
|
611 |
||
612 |
lemma pathfinish_subpath [simp]: "pathfinish(subpath u v g) = g(v)" |
|
613 |
by (simp add: pathfinish_def subpath_def) |
|
614 |
||
615 |
lemma subpath_trivial [simp]: "subpath 0 1 g = g" |
|
616 |
by (simp add: subpath_def) |
|
617 |
||
618 |
lemma subpath_reversepath: "subpath 1 0 g = reversepath g" |
|
619 |
by (simp add: reversepath_def subpath_def) |
|
620 |
||
621 |
lemma reversepath_subpath: "reversepath(subpath u v g) = subpath v u g" |
|
622 |
by (simp add: reversepath_def subpath_def algebra_simps) |
|
623 |
||
624 |
lemma subpath_translation: "subpath u v ((\<lambda>x. a + x) o g) = (\<lambda>x. a + x) o subpath u v g" |
|
625 |
by (rule ext) (simp add: subpath_def) |
|
626 |
||
627 |
lemma subpath_linear_image: "linear f \<Longrightarrow> subpath u v (f o g) = f o subpath u v g" |
|
628 |
by (rule ext) (simp add: subpath_def) |
|
629 |
||
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
60420
diff
changeset
|
630 |
lemma affine_ineq: |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
60420
diff
changeset
|
631 |
fixes x :: "'a::linordered_idom" |
60303 | 632 |
assumes "x \<le> 1" "v < u" |
633 |
shows "v + x * u \<le> u + x * v" |
|
634 |
proof - |
|
635 |
have "(1-x)*(u-v) \<ge> 0" |
|
636 |
using assms by auto |
|
637 |
then show ?thesis |
|
638 |
by (simp add: algebra_simps) |
|
49653 | 639 |
qed |
36583 | 640 |
|
61711
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
641 |
lemma sum_le_prod1: |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
642 |
fixes a::real shows "\<lbrakk>a \<le> 1; b \<le> 1\<rbrakk> \<Longrightarrow> a + b \<le> 1 + a * b" |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
643 |
by (metis add.commute affine_ineq less_eq_real_def mult.right_neutral) |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
644 |
|
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
60420
diff
changeset
|
645 |
lemma simple_path_subpath_eq: |
60303 | 646 |
"simple_path(subpath u v g) \<longleftrightarrow> |
647 |
path(subpath u v g) \<and> u\<noteq>v \<and> |
|
648 |
(\<forall>x y. x \<in> closed_segment u v \<and> y \<in> closed_segment u v \<and> g x = g y |
|
649 |
\<longrightarrow> x = y \<or> x = u \<and> y = v \<or> x = v \<and> y = u)" |
|
650 |
(is "?lhs = ?rhs") |
|
651 |
proof (rule iffI) |
|
652 |
assume ?lhs |
|
653 |
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
|
654 |
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 | 655 |
\<Longrightarrow> x = y \<or> x = 0 \<and> y = 1 \<or> x = 1 \<and> y = 0)" |
656 |
by (auto simp: simple_path_def subpath_def) |
|
657 |
{ fix x y |
|
658 |
assume "x \<in> closed_segment u v" "y \<in> closed_segment u v" "g x = g y" |
|
659 |
then have "x = y \<or> x = u \<and> y = v \<or> x = v \<and> y = u" |
|
660 |
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
|
661 |
by (auto simp: closed_segment_real_eq image_affinity_atLeastAtMost divide_simps |
60303 | 662 |
split: split_if_asm) |
663 |
} moreover |
|
664 |
have "path(subpath u v g) \<and> u\<noteq>v" |
|
665 |
using sim [of "1/3" "2/3"] p |
|
666 |
by (auto simp: subpath_def) |
|
667 |
ultimately show ?rhs |
|
668 |
by metis |
|
669 |
next |
|
670 |
assume ?rhs |
|
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
60420
diff
changeset
|
671 |
then |
60303 | 672 |
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" |
673 |
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" |
|
674 |
and ne: "u < v \<or> v < u" |
|
675 |
and psp: "path (subpath u v g)" |
|
676 |
by (auto simp: closed_segment_real_eq image_affinity_atLeastAtMost) |
|
677 |
have [simp]: "\<And>x. u + x * v = v + x * u \<longleftrightarrow> u=v \<or> x=1" |
|
678 |
by algebra |
|
679 |
show ?lhs using psp ne |
|
680 |
unfolding simple_path_def subpath_def |
|
681 |
by (fastforce simp add: algebra_simps affine_ineq mult_left_mono crossproduct_eq dest: d1 d2) |
|
682 |
qed |
|
683 |
||
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
60420
diff
changeset
|
684 |
lemma arc_subpath_eq: |
60303 | 685 |
"arc(subpath u v g) \<longleftrightarrow> path(subpath u v g) \<and> u\<noteq>v \<and> inj_on g (closed_segment u v)" |
686 |
(is "?lhs = ?rhs") |
|
687 |
proof (rule iffI) |
|
688 |
assume ?lhs |
|
689 |
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
|
690 |
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 | 691 |
\<Longrightarrow> x = y)" |
692 |
by (auto simp: arc_def inj_on_def subpath_def) |
|
693 |
{ fix x y |
|
694 |
assume "x \<in> closed_segment u v" "y \<in> closed_segment u v" "g x = g y" |
|
695 |
then have "x = y" |
|
696 |
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
|
697 |
by (force simp add: inj_on_def closed_segment_real_eq image_affinity_atLeastAtMost divide_simps |
60303 | 698 |
split: split_if_asm) |
699 |
} moreover |
|
700 |
have "path(subpath u v g) \<and> u\<noteq>v" |
|
701 |
using sim [of "1/3" "2/3"] p |
|
702 |
by (auto simp: subpath_def) |
|
703 |
ultimately show ?rhs |
|
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
60420
diff
changeset
|
704 |
unfolding inj_on_def |
60303 | 705 |
by metis |
706 |
next |
|
707 |
assume ?rhs |
|
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
60420
diff
changeset
|
708 |
then |
60303 | 709 |
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" |
710 |
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" |
|
711 |
and ne: "u < v \<or> v < u" |
|
712 |
and psp: "path (subpath u v g)" |
|
713 |
by (auto simp: inj_on_def closed_segment_real_eq image_affinity_atLeastAtMost) |
|
714 |
show ?lhs using psp ne |
|
715 |
unfolding arc_def subpath_def inj_on_def |
|
716 |
by (auto simp: algebra_simps affine_ineq mult_left_mono crossproduct_eq dest: d1 d2) |
|
717 |
qed |
|
718 |
||
719 |
||
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
60420
diff
changeset
|
720 |
lemma simple_path_subpath: |
60303 | 721 |
assumes "simple_path g" "u \<in> {0..1}" "v \<in> {0..1}" "u \<noteq> v" |
722 |
shows "simple_path(subpath u v g)" |
|
723 |
using assms |
|
724 |
apply (simp add: simple_path_subpath_eq simple_path_imp_path) |
|
725 |
apply (simp add: simple_path_def closed_segment_real_eq image_affinity_atLeastAtMost, fastforce) |
|
726 |
done |
|
727 |
||
728 |
lemma arc_simple_path_subpath: |
|
729 |
"\<lbrakk>simple_path g; u \<in> {0..1}; v \<in> {0..1}; g u \<noteq> g v\<rbrakk> \<Longrightarrow> arc(subpath u v g)" |
|
730 |
by (force intro: simple_path_subpath simple_path_imp_arc) |
|
731 |
||
732 |
lemma arc_subpath_arc: |
|
733 |
"\<lbrakk>arc g; u \<in> {0..1}; v \<in> {0..1}; u \<noteq> v\<rbrakk> \<Longrightarrow> arc(subpath u v g)" |
|
734 |
by (meson arc_def arc_imp_simple_path arc_simple_path_subpath inj_onD) |
|
735 |
||
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
60420
diff
changeset
|
736 |
lemma arc_simple_path_subpath_interior: |
60303 | 737 |
"\<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)" |
738 |
apply (rule arc_simple_path_subpath) |
|
739 |
apply (force simp: simple_path_def)+ |
|
740 |
done |
|
741 |
||
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
60420
diff
changeset
|
742 |
lemma path_image_subpath_subset: |
60303 | 743 |
"\<lbrakk>path g; u \<in> {0..1}; v \<in> {0..1}\<rbrakk> \<Longrightarrow> path_image(subpath u v g) \<subseteq> path_image g" |
744 |
apply (simp add: closed_segment_real_eq image_affinity_atLeastAtMost) |
|
745 |
apply (auto simp: path_image_def) |
|
746 |
done |
|
747 |
||
748 |
lemma join_subpaths_middle: "subpath (0) ((1 / 2)) p +++ subpath ((1 / 2)) 1 p = p" |
|
749 |
by (rule ext) (simp add: joinpaths_def subpath_def divide_simps) |
|
53640 | 750 |
|
61518
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
751 |
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
|
752 |
|
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
753 |
lemma subpath_to_frontier_explicit: |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
754 |
fixes S :: "'a::metric_space set" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
755 |
assumes g: "path g" and "pathfinish g \<notin> S" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
756 |
obtains u where "0 \<le> u" "u \<le> 1" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
757 |
"\<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
|
758 |
"(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
|
759 |
proof - |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
760 |
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
|
761 |
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
|
762 |
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
|
763 |
using compact_eq_bounded_closed apply fastforce |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
764 |
done |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
765 |
have "1 \<in> {u. g u \<in> closure (- S)}" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
766 |
using assms by (simp add: pathfinish_def closure_def) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
767 |
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
|
768 |
using atLeastAtMost_iff zero_le_one by blast |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
769 |
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
|
770 |
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
|
771 |
using compact_attains_inf [OF com dis] by fastforce |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
772 |
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
|
773 |
using closure_def by fastforce |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
774 |
{ assume "u \<noteq> 0" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
775 |
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
|
776 |
{ fix e::real assume "e > 0" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
777 |
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
|
778 |
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
|
779 |
have *: "dist (max 0 (u - d / 2)) u < d" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
780 |
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
|
781 |
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
|
782 |
using `0 < u` `u \<le> 1` `d > 0` |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
783 |
by (force intro: d [OF _ *] umin') |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
784 |
} |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
785 |
then have "g u \<in> closure S" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
786 |
by (simp add: frontier_def closure_approachable) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
787 |
} |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
788 |
then show ?thesis |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
789 |
apply (rule_tac u=u in that) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
790 |
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
|
791 |
using `_ \<le> 1` interior_closure umin apply fastforce |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
792 |
done |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
793 |
qed |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
794 |
|
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
795 |
lemma subpath_to_frontier_strong: |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
796 |
assumes g: "path g" and "pathfinish g \<notin> S" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
797 |
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
|
798 |
"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
|
799 |
proof - |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
800 |
obtain u where "0 \<le> u" "u \<le> 1" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
801 |
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
|
802 |
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
|
803 |
using subpath_to_frontier_explicit [OF assms] by blast |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
804 |
show ?thesis |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
805 |
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
|
806 |
apply (simp add: gunot) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
807 |
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
|
808 |
qed |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
809 |
|
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
810 |
lemma subpath_to_frontier: |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
811 |
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
|
812 |
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
|
813 |
proof - |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
814 |
obtain u where "0 \<le> u" "u \<le> 1" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
815 |
and notin: "g u \<notin> interior S" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
816 |
and disj: "u = 0 \<or> |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
817 |
(\<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
|
818 |
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
|
819 |
show ?thesis |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
820 |
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
|
821 |
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
|
822 |
using `0 \<le> u` g0 disj |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
823 |
apply (simp add:) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
824 |
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
|
825 |
apply (rename_tac y) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
826 |
apply (drule_tac x="y/u" in spec) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
827 |
apply (auto split: split_if_asm) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
828 |
done |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
829 |
qed |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
830 |
|
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
831 |
lemma exists_path_subpath_to_frontier: |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
832 |
fixes S :: "'a::real_normed_vector set" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
833 |
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
|
834 |
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
|
835 |
"path_image h - {pathfinish h} \<subseteq> interior S" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
836 |
"pathfinish h \<in> frontier S" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
837 |
proof - |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
838 |
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
|
839 |
using subpath_to_frontier [OF assms] by blast |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
840 |
show ?thesis |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
841 |
apply (rule that [of "subpath 0 u g"]) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
842 |
using assms u |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
843 |
apply simp_all |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
844 |
apply (simp add: pathstart_def) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
845 |
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
|
846 |
done |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
847 |
qed |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
848 |
|
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
849 |
lemma exists_path_subpath_to_frontier_closed: |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
850 |
fixes S :: "'a::real_normed_vector set" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
851 |
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
|
852 |
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
|
853 |
"pathfinish h \<in> frontier S" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
854 |
proof - |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
855 |
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
|
856 |
"path_image h - {pathfinish h} \<subseteq> interior S" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
857 |
"pathfinish h \<in> frontier S" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
858 |
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
|
859 |
show ?thesis |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
860 |
apply (rule that [OF `path h`]) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
861 |
using assms h |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
862 |
apply auto |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
863 |
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
|
864 |
done |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
865 |
qed |
49653 | 866 |
|
60420 | 867 |
subsection \<open>Reparametrizing a closed curve to start at some chosen point\<close> |
36583 | 868 |
|
53640 | 869 |
definition shiftpath :: "real \<Rightarrow> (real \<Rightarrow> 'a::topological_space) \<Rightarrow> real \<Rightarrow> 'a" |
870 |
where "shiftpath a f = (\<lambda>x. if (a + x) \<le> 1 then f (a + x) else f (a + x - 1))" |
|
36583 | 871 |
|
53640 | 872 |
lemma pathstart_shiftpath: "a \<le> 1 \<Longrightarrow> pathstart (shiftpath a g) = g a" |
36583 | 873 |
unfolding pathstart_def shiftpath_def by auto |
874 |
||
49653 | 875 |
lemma pathfinish_shiftpath: |
53640 | 876 |
assumes "0 \<le> a" |
877 |
and "pathfinish g = pathstart g" |
|
878 |
shows "pathfinish (shiftpath a g) = g a" |
|
879 |
using assms |
|
880 |
unfolding pathstart_def pathfinish_def shiftpath_def |
|
36583 | 881 |
by auto |
882 |
||
883 |
lemma endpoints_shiftpath: |
|
53640 | 884 |
assumes "pathfinish g = pathstart g" |
885 |
and "a \<in> {0 .. 1}" |
|
886 |
shows "pathfinish (shiftpath a g) = g a" |
|
887 |
and "pathstart (shiftpath a g) = g a" |
|
888 |
using assms |
|
889 |
by (auto intro!: pathfinish_shiftpath pathstart_shiftpath) |
|
36583 | 890 |
|
891 |
lemma closed_shiftpath: |
|
53640 | 892 |
assumes "pathfinish g = pathstart g" |
893 |
and "a \<in> {0..1}" |
|
894 |
shows "pathfinish (shiftpath a g) = pathstart (shiftpath a g)" |
|
895 |
using endpoints_shiftpath[OF assms] |
|
896 |
by auto |
|
36583 | 897 |
|
898 |
lemma path_shiftpath: |
|
53640 | 899 |
assumes "path g" |
900 |
and "pathfinish g = pathstart g" |
|
901 |
and "a \<in> {0..1}" |
|
902 |
shows "path (shiftpath a g)" |
|
49653 | 903 |
proof - |
53640 | 904 |
have *: "{0 .. 1} = {0 .. 1-a} \<union> {1-a .. 1}" |
905 |
using assms(3) by auto |
|
49653 | 906 |
have **: "\<And>x. x + a = 1 \<Longrightarrow> g (x + a - 1) = g (x + a)" |
53640 | 907 |
using assms(2)[unfolded pathfinish_def pathstart_def] |
908 |
by auto |
|
49653 | 909 |
show ?thesis |
910 |
unfolding path_def shiftpath_def * |
|
911 |
apply (rule continuous_on_union) |
|
912 |
apply (rule closed_real_atLeastAtMost)+ |
|
53640 | 913 |
apply (rule continuous_on_eq[of _ "g \<circ> (\<lambda>x. a + x)"]) |
914 |
prefer 3 |
|
915 |
apply (rule continuous_on_eq[of _ "g \<circ> (\<lambda>x. a - 1 + x)"]) |
|
916 |
prefer 3 |
|
56371
fb9ae0727548
extend continuous_intros; remove continuous_on_intros and isCont_intros
hoelzl
parents:
56188
diff
changeset
|
917 |
apply (rule continuous_intros)+ |
53640 | 918 |
prefer 2 |
56371
fb9ae0727548
extend continuous_intros; remove continuous_on_intros and isCont_intros
hoelzl
parents:
56188
diff
changeset
|
919 |
apply (rule continuous_intros)+ |
49653 | 920 |
apply (rule_tac[1-2] continuous_on_subset[OF assms(1)[unfolded path_def]]) |
921 |
using assms(3) and ** |
|
53640 | 922 |
apply auto |
923 |
apply (auto simp add: field_simps) |
|
49653 | 924 |
done |
925 |
qed |
|
36583 | 926 |
|
49653 | 927 |
lemma shiftpath_shiftpath: |
53640 | 928 |
assumes "pathfinish g = pathstart g" |
929 |
and "a \<in> {0..1}" |
|
930 |
and "x \<in> {0..1}" |
|
36583 | 931 |
shows "shiftpath (1 - a) (shiftpath a g) x = g x" |
53640 | 932 |
using assms |
933 |
unfolding pathfinish_def pathstart_def shiftpath_def |
|
934 |
by auto |
|
36583 | 935 |
|
936 |
lemma path_image_shiftpath: |
|
53640 | 937 |
assumes "a \<in> {0..1}" |
938 |
and "pathfinish g = pathstart g" |
|
939 |
shows "path_image (shiftpath a g) = path_image g" |
|
49653 | 940 |
proof - |
941 |
{ fix x |
|
53640 | 942 |
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 | 943 |
then have "\<exists>y\<in>{0..1} \<inter> {x. a + x \<le> 1}. g x = g (a + y)" |
49653 | 944 |
proof (cases "a \<le> x") |
945 |
case False |
|
49654 | 946 |
then show ?thesis |
49653 | 947 |
apply (rule_tac x="1 + x - a" in bexI) |
36583 | 948 |
using as(1,2) and as(3)[THEN bspec[where x="1 + x - a"]] and assms(1) |
49653 | 949 |
apply (auto simp add: field_simps atomize_not) |
950 |
done |
|
951 |
next |
|
952 |
case True |
|
53640 | 953 |
then show ?thesis |
954 |
using as(1-2) and assms(1) |
|
955 |
apply (rule_tac x="x - a" in bexI) |
|
956 |
apply (auto simp add: field_simps) |
|
957 |
done |
|
49653 | 958 |
qed |
959 |
} |
|
49654 | 960 |
then show ?thesis |
53640 | 961 |
using assms |
962 |
unfolding shiftpath_def path_image_def pathfinish_def pathstart_def |
|
963 |
by (auto simp add: image_iff) |
|
49653 | 964 |
qed |
965 |
||
36583 | 966 |
|
60420 | 967 |
subsection \<open>Special case of straight-line paths\<close> |
36583 | 968 |
|
49653 | 969 |
definition linepath :: "'a::real_normed_vector \<Rightarrow> 'a \<Rightarrow> real \<Rightarrow> 'a" |
970 |
where "linepath a b = (\<lambda>x. (1 - x) *\<^sub>R a + x *\<^sub>R b)" |
|
36583 | 971 |
|
53640 | 972 |
lemma pathstart_linepath[simp]: "pathstart (linepath a b) = a" |
973 |
unfolding pathstart_def linepath_def |
|
974 |
by auto |
|
36583 | 975 |
|
53640 | 976 |
lemma pathfinish_linepath[simp]: "pathfinish (linepath a b) = b" |
977 |
unfolding pathfinish_def linepath_def |
|
978 |
by auto |
|
36583 | 979 |
|
980 |
lemma continuous_linepath_at[intro]: "continuous (at x) (linepath a b)" |
|
53640 | 981 |
unfolding linepath_def |
982 |
by (intro continuous_intros) |
|
36583 | 983 |
|
984 |
lemma continuous_on_linepath[intro]: "continuous_on s (linepath a b)" |
|
53640 | 985 |
using continuous_linepath_at |
986 |
by (auto intro!: continuous_at_imp_continuous_on) |
|
36583 | 987 |
|
53640 | 988 |
lemma path_linepath[intro]: "path (linepath a b)" |
989 |
unfolding path_def |
|
990 |
by (rule continuous_on_linepath) |
|
36583 | 991 |
|
53640 | 992 |
lemma path_image_linepath[simp]: "path_image (linepath a b) = closed_segment a b" |
49653 | 993 |
unfolding path_image_def segment linepath_def |
60303 | 994 |
by auto |
49653 | 995 |
|
53640 | 996 |
lemma reversepath_linepath[simp]: "reversepath (linepath a b) = linepath b a" |
49653 | 997 |
unfolding reversepath_def linepath_def |
36583 | 998 |
by auto |
999 |
||
60303 | 1000 |
lemma arc_linepath: |
49653 | 1001 |
assumes "a \<noteq> b" |
60303 | 1002 |
shows "arc (linepath a b)" |
36583 | 1003 |
proof - |
53640 | 1004 |
{ |
1005 |
fix x y :: "real" |
|
36583 | 1006 |
assume "x *\<^sub>R b + y *\<^sub>R a = x *\<^sub>R a + y *\<^sub>R b" |
53640 | 1007 |
then have "(x - y) *\<^sub>R a = (x - y) *\<^sub>R b" |
1008 |
by (simp add: algebra_simps) |
|
1009 |
with assms have "x = y" |
|
1010 |
by simp |
|
1011 |
} |
|
49654 | 1012 |
then show ?thesis |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
60420
diff
changeset
|
1013 |
unfolding arc_def inj_on_def |
60303 | 1014 |
by (simp add: path_linepath) (force simp: algebra_simps linepath_def) |
49653 | 1015 |
qed |
36583 | 1016 |
|
53640 | 1017 |
lemma simple_path_linepath[intro]: "a \<noteq> b \<Longrightarrow> simple_path (linepath a b)" |
60303 | 1018 |
by (simp add: arc_imp_simple_path arc_linepath) |
49653 | 1019 |
|
61711
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
1020 |
lemma linepath_trivial [simp]: "linepath a a x = a" |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
1021 |
by (simp add: linepath_def real_vector.scale_left_diff_distrib) |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
1022 |
|
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
1023 |
lemma subpath_refl: "subpath a a g = linepath (g a) (g a)" |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
1024 |
by (simp add: subpath_def linepath_def algebra_simps) |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
1025 |
|
36583 | 1026 |
|
60420 | 1027 |
subsection \<open>Bounding a point away from a path\<close> |
36583 | 1028 |
|
1029 |
lemma not_on_path_ball: |
|
1030 |
fixes g :: "real \<Rightarrow> 'a::heine_borel" |
|
53640 | 1031 |
assumes "path g" |
1032 |
and "z \<notin> path_image g" |
|
1033 |
shows "\<exists>e > 0. ball z e \<inter> path_image g = {}" |
|
49653 | 1034 |
proof - |
1035 |
obtain a where "a \<in> path_image g" "\<forall>y \<in> path_image g. dist z a \<le> dist z y" |
|
36583 | 1036 |
using distance_attains_inf[OF _ path_image_nonempty, of g z] |
1037 |
using compact_path_image[THEN compact_imp_closed, OF assms(1)] by auto |
|
49654 | 1038 |
then show ?thesis |
49653 | 1039 |
apply (rule_tac x="dist z a" in exI) |
1040 |
using assms(2) |
|
1041 |
apply (auto intro!: dist_pos_lt) |
|
1042 |
done |
|
1043 |
qed |
|
36583 | 1044 |
|
1045 |
lemma not_on_path_cball: |
|
1046 |
fixes g :: "real \<Rightarrow> 'a::heine_borel" |
|
53640 | 1047 |
assumes "path g" |
1048 |
and "z \<notin> path_image g" |
|
49653 | 1049 |
shows "\<exists>e>0. cball z e \<inter> (path_image g) = {}" |
1050 |
proof - |
|
53640 | 1051 |
obtain e where "ball z e \<inter> path_image g = {}" "e > 0" |
49653 | 1052 |
using not_on_path_ball[OF assms] by auto |
53640 | 1053 |
moreover have "cball z (e/2) \<subseteq> ball z e" |
60420 | 1054 |
using \<open>e > 0\<close> by auto |
53640 | 1055 |
ultimately show ?thesis |
1056 |
apply (rule_tac x="e/2" in exI) |
|
1057 |
apply auto |
|
1058 |
done |
|
49653 | 1059 |
qed |
1060 |
||
36583 | 1061 |
|
61518
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
1062 |
section \<open>Path component, considered as a "joinability" relation (from Tom Hales)\<close> |
36583 | 1063 |
|
49653 | 1064 |
definition "path_component s x y \<longleftrightarrow> |
1065 |
(\<exists>g. path g \<and> path_image g \<subseteq> s \<and> pathstart g = x \<and> pathfinish g = y)" |
|
36583 | 1066 |
|
61426
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1067 |
abbreviation |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1068 |
"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
|
1069 |
|
53640 | 1070 |
lemmas path_defs = path_def pathstart_def pathfinish_def path_image_def path_component_def |
36583 | 1071 |
|
49653 | 1072 |
lemma path_component_mem: |
1073 |
assumes "path_component s x y" |
|
53640 | 1074 |
shows "x \<in> s" and "y \<in> s" |
1075 |
using assms |
|
1076 |
unfolding path_defs |
|
1077 |
by auto |
|
36583 | 1078 |
|
49653 | 1079 |
lemma path_component_refl: |
1080 |
assumes "x \<in> s" |
|
1081 |
shows "path_component s x x" |
|
1082 |
unfolding path_defs |
|
1083 |
apply (rule_tac x="\<lambda>u. x" in exI) |
|
53640 | 1084 |
using assms |
56371
fb9ae0727548
extend continuous_intros; remove continuous_on_intros and isCont_intros
hoelzl
parents:
56188
diff
changeset
|
1085 |
apply (auto intro!: continuous_intros) |
53640 | 1086 |
done |
36583 | 1087 |
|
1088 |
lemma path_component_refl_eq: "path_component s x x \<longleftrightarrow> x \<in> s" |
|
49653 | 1089 |
by (auto intro!: path_component_mem path_component_refl) |
36583 | 1090 |
|
1091 |
lemma path_component_sym: "path_component s x y \<Longrightarrow> path_component s y x" |
|
49653 | 1092 |
using assms |
1093 |
unfolding path_component_def |
|
1094 |
apply (erule exE) |
|
1095 |
apply (rule_tac x="reversepath g" in exI) |
|
1096 |
apply auto |
|
1097 |
done |
|
36583 | 1098 |
|
49653 | 1099 |
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
|
1100 |
assumes "path_component s x y" and "path_component s y z" |
49653 | 1101 |
shows "path_component s x z" |
1102 |
using assms |
|
1103 |
unfolding path_component_def |
|
53640 | 1104 |
apply (elim exE) |
49653 | 1105 |
apply (rule_tac x="g +++ ga" in exI) |
1106 |
apply (auto simp add: path_image_join) |
|
1107 |
done |
|
36583 | 1108 |
|
53640 | 1109 |
lemma path_component_of_subset: "s \<subseteq> t \<Longrightarrow> path_component s x y \<Longrightarrow> path_component t x y" |
36583 | 1110 |
unfolding path_component_def by auto |
1111 |
||
61426
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1112 |
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
|
1113 |
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
|
1114 |
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
|
1115 |
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
|
1116 |
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
|
1117 |
done |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1118 |
|
49653 | 1119 |
|
60420 | 1120 |
text \<open>Can also consider it as a set, as the name suggests.\<close> |
36583 | 1121 |
|
49653 | 1122 |
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
|
1123 |
"path_component_set s x = |
49653 | 1124 |
{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
|
1125 |
by (auto simp: path_component_def) |
36583 | 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_subset: "path_component_set s x \<subseteq> s" |
60303 | 1128 |
by (auto simp add: path_component_mem(2)) |
36583 | 1129 |
|
61426
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1130 |
lemma path_component_eq_empty: "path_component_set s x = {} \<longleftrightarrow> x \<notin> s" |
60303 | 1131 |
using path_component_mem path_component_refl_eq |
1132 |
by fastforce |
|
36583 | 1133 |
|
61426
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1134 |
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
|
1135 |
"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
|
1136 |
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
|
1137 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1138 |
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
|
1139 |
"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
|
1140 |
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
|
1141 |
|
60420 | 1142 |
subsection \<open>Path connectedness of a space\<close> |
36583 | 1143 |
|
49653 | 1144 |
definition "path_connected s \<longleftrightarrow> |
53640 | 1145 |
(\<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 | 1146 |
|
1147 |
lemma path_connected_component: "path_connected s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>y\<in>s. path_component s x y)" |
|
1148 |
unfolding path_connected_def path_component_def by auto |
|
1149 |
||
61426
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1150 |
lemma path_connected_component_set: "path_connected s \<longleftrightarrow> (\<forall>x\<in>s. path_component_set s x = s)" |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
1151 |
unfolding path_connected_component path_component_subset |
61426
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1152 |
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
|
1153 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1154 |
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
|
1155 |
"\<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
|
1156 |
by (metis path_component_mono path_connected_component_set) |
36583 | 1157 |
|
60420 | 1158 |
subsection \<open>Some useful lemmas about path-connectedness\<close> |
36583 | 1159 |
|
1160 |
lemma convex_imp_path_connected: |
|
1161 |
fixes s :: "'a::real_normed_vector set" |
|
53640 | 1162 |
assumes "convex s" |
1163 |
shows "path_connected s" |
|
49653 | 1164 |
unfolding path_connected_def |
53640 | 1165 |
apply rule |
1166 |
apply rule |
|
1167 |
apply (rule_tac x = "linepath x y" in exI) |
|
49653 | 1168 |
unfolding path_image_linepath |
1169 |
using assms [unfolded convex_contains_segment] |
|
1170 |
apply auto |
|
1171 |
done |
|
36583 | 1172 |
|
49653 | 1173 |
lemma path_connected_imp_connected: |
1174 |
assumes "path_connected s" |
|
1175 |
shows "connected s" |
|
1176 |
unfolding connected_def not_ex |
|
53640 | 1177 |
apply rule |
1178 |
apply rule |
|
1179 |
apply (rule ccontr) |
|
49653 | 1180 |
unfolding not_not |
53640 | 1181 |
apply (elim conjE) |
49653 | 1182 |
proof - |
1183 |
fix e1 e2 |
|
1184 |
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 | 1185 |
then obtain x1 x2 where obt:"x1 \<in> e1 \<inter> s" "x2 \<in> e2 \<inter> s" |
1186 |
by auto |
|
1187 |
then obtain g where g: "path g" "path_image g \<subseteq> s" "pathstart g = x1" "pathfinish g = x2" |
|
36583 | 1188 |
using assms[unfolded path_connected_def,rule_format,of x1 x2] by auto |
49653 | 1189 |
have *: "connected {0..1::real}" |
1190 |
by (auto intro!: convex_connected convex_real_interval) |
|
1191 |
have "{0..1} \<subseteq> {x \<in> {0..1}. g x \<in> e1} \<union> {x \<in> {0..1}. g x \<in> e2}" |
|
1192 |
using as(3) g(2)[unfolded path_defs] by blast |
|
1193 |
moreover have "{x \<in> {0..1}. g x \<in> e1} \<inter> {x \<in> {0..1}. g x \<in> e2} = {}" |
|
53640 | 1194 |
using as(4) g(2)[unfolded path_defs] |
1195 |
unfolding subset_eq |
|
1196 |
by auto |
|
49653 | 1197 |
moreover have "{x \<in> {0..1}. g x \<in> e1} \<noteq> {} \<and> {x \<in> {0..1}. g x \<in> e2} \<noteq> {}" |
53640 | 1198 |
using g(3,4)[unfolded path_defs] |
1199 |
using obt |
|
36583 | 1200 |
by (simp add: ex_in_conv [symmetric], metis zero_le_one order_refl) |
49653 | 1201 |
ultimately show False |
53640 | 1202 |
using *[unfolded connected_local not_ex, rule_format, |
1203 |
of "{x\<in>{0..1}. g x \<in> e1}" "{x\<in>{0..1}. g x \<in> e2}"] |
|
61518
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
1204 |
using continuous_openin_preimage[OF g(1)[unfolded path_def] as(1)] |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
1205 |
using continuous_openin_preimage[OF g(1)[unfolded path_def] as(2)] |
49653 | 1206 |
by auto |
1207 |
qed |
|
36583 | 1208 |
|
1209 |
lemma open_path_component: |
|
53593 | 1210 |
fixes s :: "'a::real_normed_vector set" |
49653 | 1211 |
assumes "open s" |
61426
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1212 |
shows "open (path_component_set s x)" |
49653 | 1213 |
unfolding open_contains_ball |
1214 |
proof |
|
1215 |
fix y |
|
61426
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1216 |
assume as: "y \<in> path_component_set s x" |
49654 | 1217 |
then have "y \<in> s" |
49653 | 1218 |
apply - |
1219 |
apply (rule path_component_mem(2)) |
|
1220 |
unfolding mem_Collect_eq |
|
1221 |
apply auto |
|
1222 |
done |
|
53640 | 1223 |
then obtain e where e: "e > 0" "ball y e \<subseteq> s" |
1224 |
using assms[unfolded open_contains_ball] |
|
1225 |
by auto |
|
61426
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1226 |
show "\<exists>e > 0. ball y e \<subseteq> path_component_set s x" |
49653 | 1227 |
apply (rule_tac x=e in exI) |
60420 | 1228 |
apply (rule,rule \<open>e>0\<close>) |
53640 | 1229 |
apply rule |
49653 | 1230 |
unfolding mem_ball mem_Collect_eq |
1231 |
proof - |
|
1232 |
fix z |
|
1233 |
assume "dist y z < e" |
|
49654 | 1234 |
then show "path_component s x z" |
53640 | 1235 |
apply (rule_tac path_component_trans[of _ _ y]) |
1236 |
defer |
|
49653 | 1237 |
apply (rule path_component_of_subset[OF e(2)]) |
1238 |
apply (rule convex_imp_path_connected[OF convex_ball, unfolded path_connected_component, rule_format]) |
|
60420 | 1239 |
using \<open>e > 0\<close> as |
49653 | 1240 |
apply auto |
1241 |
done |
|
1242 |
qed |
|
1243 |
qed |
|
36583 | 1244 |
|
1245 |
lemma open_non_path_component: |
|
53593 | 1246 |
fixes s :: "'a::real_normed_vector set" |
49653 | 1247 |
assumes "open s" |
61426
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1248 |
shows "open (s - path_component_set s x)" |
49653 | 1249 |
unfolding open_contains_ball |
1250 |
proof |
|
1251 |
fix y |
|
61426
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1252 |
assume as: "y \<in> s - path_component_set s x" |
53640 | 1253 |
then obtain e where e: "e > 0" "ball y e \<subseteq> s" |
1254 |
using assms [unfolded open_contains_ball] |
|
1255 |
by auto |
|
61426
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1256 |
show "\<exists>e>0. ball y e \<subseteq> s - path_component_set s x" |
49653 | 1257 |
apply (rule_tac x=e in exI) |
53640 | 1258 |
apply rule |
60420 | 1259 |
apply (rule \<open>e>0\<close>) |
53640 | 1260 |
apply rule |
1261 |
apply rule |
|
1262 |
defer |
|
49653 | 1263 |
proof (rule ccontr) |
1264 |
fix z |
|
61426
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1265 |
assume "z \<in> ball y e" "\<not> z \<notin> path_component_set s x" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1266 |
then have "y \<in> path_component_set s x" |
60420 | 1267 |
unfolding not_not mem_Collect_eq using \<open>e>0\<close> |
49653 | 1268 |
apply - |
1269 |
apply (rule path_component_trans, assumption) |
|
1270 |
apply (rule path_component_of_subset[OF e(2)]) |
|
1271 |
apply (rule convex_imp_path_connected[OF convex_ball, unfolded path_connected_component, rule_format]) |
|
1272 |
apply auto |
|
1273 |
done |
|
53640 | 1274 |
then show False |
1275 |
using as by auto |
|
49653 | 1276 |
qed (insert e(2), auto) |
1277 |
qed |
|
36583 | 1278 |
|
1279 |
lemma connected_open_path_connected: |
|
53593 | 1280 |
fixes s :: "'a::real_normed_vector set" |
53640 | 1281 |
assumes "open s" |
1282 |
and "connected s" |
|
49653 | 1283 |
shows "path_connected s" |
1284 |
unfolding path_connected_component_set |
|
1285 |
proof (rule, rule, rule path_component_subset, rule) |
|
1286 |
fix x y |
|
53640 | 1287 |
assume "x \<in> s" and "y \<in> s" |
61426
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1288 |
show "y \<in> path_component_set s x" |
49653 | 1289 |
proof (rule ccontr) |
53640 | 1290 |
assume "\<not> ?thesis" |
61426
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1291 |
moreover have "path_component_set s x \<inter> s \<noteq> {}" |
60420 | 1292 |
using \<open>x \<in> s\<close> path_component_eq_empty path_component_subset[of s x] |
53640 | 1293 |
by auto |
49653 | 1294 |
ultimately |
1295 |
show False |
|
60420 | 1296 |
using \<open>y \<in> s\<close> open_non_path_component[OF assms(1)] open_path_component[OF assms(1)] |
53640 | 1297 |
using assms(2)[unfolded connected_def not_ex, rule_format, |
61426
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1298 |
of "path_component_set s x" "s - path_component_set s x"] |
49653 | 1299 |
by auto |
1300 |
qed |
|
1301 |
qed |
|
36583 | 1302 |
|
1303 |
lemma path_connected_continuous_image: |
|
53640 | 1304 |
assumes "continuous_on s f" |
1305 |
and "path_connected s" |
|
49653 | 1306 |
shows "path_connected (f ` s)" |
1307 |
unfolding path_connected_def |
|
1308 |
proof (rule, rule) |
|
1309 |
fix x' y' |
|
1310 |
assume "x' \<in> f ` s" "y' \<in> f ` s" |
|
53640 | 1311 |
then obtain x y where x: "x \<in> s" and y: "y \<in> s" and x': "x' = f x" and y': "y' = f y" |
1312 |
by auto |
|
1313 |
from x y obtain g where "path g \<and> path_image g \<subseteq> s \<and> pathstart g = x \<and> pathfinish g = y" |
|
1314 |
using assms(2)[unfolded path_connected_def] by fast |
|
49654 | 1315 |
then show "\<exists>g. path g \<and> path_image g \<subseteq> f ` s \<and> pathstart g = x' \<and> pathfinish g = y'" |
53640 | 1316 |
unfolding x' y' |
49653 | 1317 |
apply (rule_tac x="f \<circ> g" in exI) |
1318 |
unfolding path_defs |
|
51481
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51478
diff
changeset
|
1319 |
apply (intro conjI continuous_on_compose continuous_on_subset[OF assms(1)]) |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51478
diff
changeset
|
1320 |
apply auto |
49653 | 1321 |
done |
1322 |
qed |
|
36583 | 1323 |
|
61518
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
1324 |
lemma path_connected_segment: |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
1325 |
fixes a :: "'a::real_normed_vector" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
1326 |
shows "path_connected (closed_segment a b)" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
1327 |
by (simp add: convex_imp_path_connected) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
1328 |
|
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
1329 |
lemma path_connected_open_segment: |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
1330 |
fixes a :: "'a::real_normed_vector" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
1331 |
shows "path_connected (open_segment a b)" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
1332 |
by (simp add: convex_imp_path_connected) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
1333 |
|
36583 | 1334 |
lemma homeomorphic_path_connectedness: |
53640 | 1335 |
"s homeomorphic t \<Longrightarrow> path_connected s \<longleftrightarrow> path_connected t" |
49653 | 1336 |
unfolding homeomorphic_def homeomorphism_def |
53640 | 1337 |
apply (erule exE|erule conjE)+ |
49653 | 1338 |
apply rule |
53640 | 1339 |
apply (drule_tac f=f in path_connected_continuous_image) |
1340 |
prefer 3 |
|
49653 | 1341 |
apply (drule_tac f=g in path_connected_continuous_image) |
1342 |
apply auto |
|
1343 |
done |
|
36583 | 1344 |
|
1345 |
lemma path_connected_empty: "path_connected {}" |
|
1346 |
unfolding path_connected_def by auto |
|
1347 |
||
1348 |
lemma path_connected_singleton: "path_connected {a}" |
|
1349 |
unfolding path_connected_def pathstart_def pathfinish_def path_image_def |
|
53640 | 1350 |
apply clarify |
1351 |
apply (rule_tac x="\<lambda>x. a" in exI) |
|
1352 |
apply (simp add: image_constant_conv) |
|
36583 | 1353 |
apply (simp add: path_def continuous_on_const) |
1354 |
done |
|
1355 |
||
49653 | 1356 |
lemma path_connected_Un: |
53640 | 1357 |
assumes "path_connected s" |
1358 |
and "path_connected t" |
|
1359 |
and "s \<inter> t \<noteq> {}" |
|
49653 | 1360 |
shows "path_connected (s \<union> t)" |
1361 |
unfolding path_connected_component |
|
1362 |
proof (rule, rule) |
|
1363 |
fix x y |
|
1364 |
assume as: "x \<in> s \<union> t" "y \<in> s \<union> t" |
|
53640 | 1365 |
from assms(3) obtain z where "z \<in> s \<inter> t" |
1366 |
by auto |
|
49654 | 1367 |
then show "path_component (s \<union> t) x y" |
49653 | 1368 |
using as and assms(1-2)[unfolded path_connected_component] |
53640 | 1369 |
apply - |
49653 | 1370 |
apply (erule_tac[!] UnE)+ |
1371 |
apply (rule_tac[2-3] path_component_trans[of _ _ z]) |
|
1372 |
apply (auto simp add:path_component_of_subset [OF Un_upper1] path_component_of_subset[OF Un_upper2]) |
|
1373 |
done |
|
1374 |
qed |
|
36583 | 1375 |
|
37674
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
1376 |
lemma path_connected_UNION: |
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
1377 |
assumes "\<And>i. i \<in> A \<Longrightarrow> path_connected (S i)" |
49653 | 1378 |
and "\<And>i. i \<in> A \<Longrightarrow> z \<in> S i" |
37674
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
1379 |
shows "path_connected (\<Union>i\<in>A. S i)" |
49653 | 1380 |
unfolding path_connected_component |
1381 |
proof clarify |
|
37674
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
1382 |
fix x i y j |
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
1383 |
assume *: "i \<in> A" "x \<in> S i" "j \<in> A" "y \<in> S j" |
49654 | 1384 |
then have "path_component (S i) x z" and "path_component (S j) z y" |
37674
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
1385 |
using assms by (simp_all add: path_connected_component) |
49654 | 1386 |
then have "path_component (\<Union>i\<in>A. S i) x z" and "path_component (\<Union>i\<in>A. S i) z y" |
48125
602dc0215954
tuned proofs -- prefer direct "rotated" instead of old-style COMP;
wenzelm
parents:
44647
diff
changeset
|
1387 |
using *(1,3) by (auto elim!: path_component_of_subset [rotated]) |
49654 | 1388 |
then show "path_component (\<Union>i\<in>A. S i) x y" |
37674
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
1389 |
by (rule path_component_trans) |
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
1390 |
qed |
36583 | 1391 |
|
61426
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1392 |
lemma path_component_path_image_pathstart: |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1393 |
assumes p: "path p" and x: "x \<in> path_image p" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1394 |
shows "path_component (path_image p) (pathstart p) x" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1395 |
using x |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1396 |
proof (clarsimp simp add: path_image_def) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1397 |
fix y |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1398 |
assume "x = p y" and y: "0 \<le> y" "y \<le> 1" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1399 |
show "path_component (p ` {0..1}) (pathstart p) (p y)" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1400 |
proof (cases "y=0") |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1401 |
case True then show ?thesis |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1402 |
by (simp add: path_component_refl_eq pathstart_def) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1403 |
next |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1404 |
case False have "continuous_on {0..1} (p o (op*y))" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1405 |
apply (rule continuous_intros)+ |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1406 |
using p [unfolded path_def] y |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1407 |
apply (auto simp: mult_le_one intro: continuous_on_subset [of _ p]) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1408 |
done |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1409 |
then have "path (\<lambda>u. p (y * u))" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1410 |
by (simp add: path_def) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1411 |
then show ?thesis |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1412 |
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
|
1413 |
apply (rule_tac x = "\<lambda>u. p (y * u)" in exI) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1414 |
apply (intro conjI) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1415 |
using y False |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1416 |
apply (auto simp: mult_le_one pathstart_def pathfinish_def path_image_def) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1417 |
done |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1418 |
qed |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1419 |
qed |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1420 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1421 |
lemma path_connected_path_image: "path p \<Longrightarrow> path_connected(path_image p)" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1422 |
unfolding path_connected_component |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1423 |
by (meson path_component_path_image_pathstart path_component_sym path_component_trans) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1424 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1425 |
lemma path_connected_path_component: |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1426 |
"path_connected (path_component_set s x)" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1427 |
proof - |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1428 |
{ fix y z |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1429 |
assume pa: "path_component s x y" "path_component s x z" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1430 |
then have pae: "path_component_set s x = path_component_set s y" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1431 |
using path_component_eq by auto |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1432 |
have yz: "path_component s y z" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1433 |
using pa path_component_sym path_component_trans by blast |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1434 |
then have "\<exists>g. path g \<and> path_image g \<subseteq> path_component_set s x \<and> pathstart g = y \<and> pathfinish g = z" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1435 |
apply (simp add: path_component_def, clarify) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1436 |
apply (rule_tac x=g in exI) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1437 |
by (simp add: pae path_component_maximal path_connected_path_image pathstart_in_path_image) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1438 |
} |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1439 |
then show ?thesis |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1440 |
by (simp add: path_connected_def) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1441 |
qed |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1442 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1443 |
lemma path_component: "path_component s x y \<longleftrightarrow> (\<exists>t. path_connected t \<and> t \<subseteq> s \<and> x \<in> t \<and> y \<in> t)" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1444 |
apply (intro iffI) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1445 |
apply (metis path_connected_path_image path_defs(5) pathfinish_in_path_image pathstart_in_path_image) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1446 |
using path_component_of_subset path_connected_component by blast |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1447 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1448 |
lemma path_component_path_component [simp]: |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1449 |
"path_component_set (path_component_set s x) x = path_component_set s x" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1450 |
proof (cases "x \<in> s") |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1451 |
case True show ?thesis |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1452 |
apply (rule subset_antisym) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1453 |
apply (simp add: path_component_subset) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1454 |
by (simp add: True path_component_maximal path_component_refl path_connected_path_component) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1455 |
next |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1456 |
case False then show ?thesis |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1457 |
by (metis False empty_iff path_component_eq_empty) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1458 |
qed |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1459 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1460 |
lemma path_component_subset_connected_component: |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1461 |
"(path_component_set s x) \<subseteq> (connected_component_set s x)" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1462 |
proof (cases "x \<in> s") |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1463 |
case True show ?thesis |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1464 |
apply (rule connected_component_maximal) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1465 |
apply (auto simp: True path_component_subset path_component_refl path_connected_imp_connected path_connected_path_component) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1466 |
done |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1467 |
next |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1468 |
case False then show ?thesis |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1469 |
using path_component_eq_empty by auto |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1470 |
qed |
49653 | 1471 |
|
60420 | 1472 |
subsection \<open>Sphere is path-connected\<close> |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
36583
diff
changeset
|
1473 |
|
36583 | 1474 |
lemma path_connected_punctured_universe: |
37674
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
1475 |
assumes "2 \<le> DIM('a::euclidean_space)" |
61426
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1476 |
shows "path_connected (- {a::'a})" |
49653 | 1477 |
proof - |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
49654
diff
changeset
|
1478 |
let ?A = "{x::'a. \<exists>i\<in>Basis. x \<bullet> i < a \<bullet> i}" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
49654
diff
changeset
|
1479 |
let ?B = "{x::'a. \<exists>i\<in>Basis. a \<bullet> i < x \<bullet> i}" |
36583 | 1480 |
|
49653 | 1481 |
have A: "path_connected ?A" |
1482 |
unfolding Collect_bex_eq |
|
37674
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
1483 |
proof (rule path_connected_UNION) |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
49654
diff
changeset
|
1484 |
fix i :: 'a |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
49654
diff
changeset
|
1485 |
assume "i \<in> Basis" |
53640 | 1486 |
then show "(\<Sum>i\<in>Basis. (a \<bullet> i - 1)*\<^sub>R i) \<in> {x::'a. x \<bullet> i < a \<bullet> i}" |
1487 |
by simp |
|
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
49654
diff
changeset
|
1488 |
show "path_connected {x. x \<bullet> i < a \<bullet> i}" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
49654
diff
changeset
|
1489 |
using convex_imp_path_connected [OF convex_halfspace_lt, of i "a \<bullet> i"] |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
49654
diff
changeset
|
1490 |
by (simp add: inner_commute) |
37674
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
1491 |
qed |
53640 | 1492 |
have B: "path_connected ?B" |
1493 |
unfolding Collect_bex_eq |
|
37674
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
1494 |
proof (rule path_connected_UNION) |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
49654
diff
changeset
|
1495 |
fix i :: 'a |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
49654
diff
changeset
|
1496 |
assume "i \<in> Basis" |
53640 | 1497 |
then show "(\<Sum>i\<in>Basis. (a \<bullet> i + 1) *\<^sub>R i) \<in> {x::'a. a \<bullet> i < x \<bullet> i}" |
1498 |
by simp |
|
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
49654
diff
changeset
|
1499 |
show "path_connected {x. a \<bullet> i < x \<bullet> i}" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
49654
diff
changeset
|
1500 |
using convex_imp_path_connected [OF convex_halfspace_gt, of "a \<bullet> i" i] |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
49654
diff
changeset
|
1501 |
by (simp add: inner_commute) |
37674
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
1502 |
qed |
53640 | 1503 |
obtain S :: "'a set" where "S \<subseteq> Basis" and "card S = Suc (Suc 0)" |
1504 |
using ex_card[OF assms] |
|
1505 |
by auto |
|
1506 |
then obtain b0 b1 :: 'a where "b0 \<in> Basis" and "b1 \<in> Basis" and "b0 \<noteq> b1" |
|
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
49654
diff
changeset
|
1507 |
unfolding card_Suc_eq by auto |
53640 | 1508 |
then have "a + b0 - b1 \<in> ?A \<inter> ?B" |
1509 |
by (auto simp: inner_simps inner_Basis) |
|
1510 |
then have "?A \<inter> ?B \<noteq> {}" |
|
1511 |
by fast |
|
37674
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
1512 |
with A B have "path_connected (?A \<union> ?B)" |
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
1513 |
by (rule path_connected_Un) |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
49654
diff
changeset
|
1514 |
also have "?A \<union> ?B = {x. \<exists>i\<in>Basis. x \<bullet> i \<noteq> a \<bullet> i}" |
37674
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
1515 |
unfolding neq_iff bex_disj_distrib Collect_disj_eq .. |
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
1516 |
also have "\<dots> = {x. x \<noteq> a}" |
53640 | 1517 |
unfolding euclidean_eq_iff [where 'a='a] |
1518 |
by (simp add: Bex_def) |
|
61426
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1519 |
also have "\<dots> = - {a}" |
53640 | 1520 |
by auto |
37674
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
1521 |
finally show ?thesis . |
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
1522 |
qed |
36583 | 1523 |
|
37674
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
1524 |
lemma path_connected_sphere: |
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
1525 |
assumes "2 \<le> DIM('a::euclidean_space)" |
53640 | 1526 |
shows "path_connected {x::'a. norm (x - a) = r}" |
37674
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
1527 |
proof (rule linorder_cases [of r 0]) |
49653 | 1528 |
assume "r < 0" |
53640 | 1529 |
then have "{x::'a. norm(x - a) = r} = {}" |
1530 |
by auto |
|
1531 |
then show ?thesis |
|
1532 |
using path_connected_empty by simp |
|
37674
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
1533 |
next |
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
1534 |
assume "r = 0" |
53640 | 1535 |
then show ?thesis |
1536 |
using path_connected_singleton by simp |
|
37674
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
1537 |
next |
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
1538 |
assume r: "0 < r" |
53640 | 1539 |
have *: "{x::'a. norm(x - a) = r} = (\<lambda>x. a + r *\<^sub>R x) ` {x. norm x = 1}" |
1540 |
apply (rule set_eqI) |
|
1541 |
apply rule |
|
49653 | 1542 |
unfolding image_iff |
1543 |
apply (rule_tac x="(1/r) *\<^sub>R (x - a)" in bexI) |
|
1544 |
unfolding mem_Collect_eq norm_scaleR |
|
53640 | 1545 |
using r |
49653 | 1546 |
apply (auto simp add: scaleR_right_diff_distrib) |
1547 |
done |
|
61426
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1548 |
have **: "{x::'a. norm x = 1} = (\<lambda>x. (1/norm x) *\<^sub>R x) ` (- {0})" |
53640 | 1549 |
apply (rule set_eqI) |
1550 |
apply rule |
|
49653 | 1551 |
unfolding image_iff |
1552 |
apply (rule_tac x=x in bexI) |
|
1553 |
unfolding mem_Collect_eq |
|
53640 | 1554 |
apply (auto split: split_if_asm) |
49653 | 1555 |
done |
61426
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1556 |
have "continuous_on (- {0}) (\<lambda>x::'a. 1 / norm x)" |
59557 | 1557 |
by (auto intro!: continuous_intros) |
53640 | 1558 |
then show ?thesis |
1559 |
unfolding * ** |
|
1560 |
using path_connected_punctured_universe[OF assms] |
|
56371
fb9ae0727548
extend continuous_intros; remove continuous_on_intros and isCont_intros
hoelzl
parents:
56188
diff
changeset
|
1561 |
by (auto intro!: path_connected_continuous_image continuous_intros) |
37674
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
1562 |
qed |
36583 | 1563 |
|
61426
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1564 |
corollary connected_sphere: "2 \<le> DIM('a::euclidean_space) \<Longrightarrow> connected {x::'a. norm (x - a) = r}" |
53640 | 1565 |
using path_connected_sphere path_connected_imp_connected |
1566 |
by auto |
|
36583 | 1567 |
|
61426
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1568 |
corollary path_connected_complement_bounded_convex: |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1569 |
fixes s :: "'a :: euclidean_space set" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1570 |
assumes "bounded s" "convex s" and 2: "2 \<le> DIM('a)" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1571 |
shows "path_connected (- s)" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1572 |
proof (cases "s={}") |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1573 |
case True then show ?thesis |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1574 |
using convex_imp_path_connected by auto |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1575 |
next |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1576 |
case False |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1577 |
then obtain a where "a \<in> s" by auto |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1578 |
{ fix x y assume "x \<notin> s" "y \<notin> s" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1579 |
then have "x \<noteq> a" "y \<noteq> a" using `a \<in> s` by auto |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1580 |
then have bxy: "bounded(insert x (insert y s))" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1581 |
by (simp add: `bounded s`) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1582 |
then obtain B::real where B: "0 < B" and Bx: "norm (a - x) < B" and By: "norm (a - y) < B" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1583 |
and "s \<subseteq> ball a B" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1584 |
using bounded_subset_ballD [OF bxy, of a] by (auto simp: dist_norm) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1585 |
def C == "B / norm(x - a)" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1586 |
{ fix u |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1587 |
assume u: "(1 - u) *\<^sub>R x + u *\<^sub>R (a + C *\<^sub>R (x - a)) \<in> s" and "0 \<le> u" "u \<le> 1" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1588 |
have CC: "1 \<le> 1 + (C - 1) * u" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1589 |
using `x \<noteq> a` `0 \<le> u` |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1590 |
apply (simp add: C_def divide_simps norm_minus_commute) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1591 |
by (metis Bx diff_le_iff(1) less_eq_real_def mult_nonneg_nonneg) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1592 |
have *: "\<And>v. (1 - u) *\<^sub>R x + u *\<^sub>R (a + v *\<^sub>R (x - a)) = a + (1 + (v - 1) * u) *\<^sub>R (x - a)" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1593 |
by (simp add: algebra_simps) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1594 |
have "a + ((1 / (1 + C * u - u)) *\<^sub>R x + ((u / (1 + C * u - u)) *\<^sub>R a + (C * u / (1 + C * u - u)) *\<^sub>R x)) = |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1595 |
(1 + (u / (1 + C * u - u))) *\<^sub>R a + ((1 / (1 + C * u - u)) + (C * u / (1 + C * u - u))) *\<^sub>R x" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1596 |
by (simp add: algebra_simps) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1597 |
also have "... = (1 + (u / (1 + C * u - u))) *\<^sub>R a + (1 + (u / (1 + C * u - u))) *\<^sub>R x" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1598 |
using CC by (simp add: field_simps) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1599 |
also have "... = x + (1 + (u / (1 + C * u - u))) *\<^sub>R a + (u / (1 + C * u - u)) *\<^sub>R x" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1600 |
by (simp add: algebra_simps) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1601 |
also have "... = x + ((1 / (1 + C * u - u)) *\<^sub>R a + |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1602 |
((u / (1 + C * u - u)) *\<^sub>R x + (C * u / (1 + C * u - u)) *\<^sub>R a))" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1603 |
using CC by (simp add: field_simps) (simp add: add_divide_distrib scaleR_add_left) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1604 |
finally have xeq: "(1 - 1 / (1 + (C - 1) * u)) *\<^sub>R a + (1 / (1 + (C - 1) * u)) *\<^sub>R (a + (1 + (C - 1) * u) *\<^sub>R (x - a)) = x" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1605 |
by (simp add: algebra_simps) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1606 |
have False |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1607 |
using `convex s` |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1608 |
apply (simp add: convex_alt) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1609 |
apply (drule_tac x=a in bspec) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1610 |
apply (rule `a \<in> s`) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1611 |
apply (drule_tac x="a + (1 + (C - 1) * u) *\<^sub>R (x - a)" in bspec) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1612 |
using u apply (simp add: *) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1613 |
apply (drule_tac x="1 / (1 + (C - 1) * u)" in spec) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1614 |
using `x \<noteq> a` `x \<notin> s` `0 \<le> u` CC |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1615 |
apply (auto simp: xeq) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1616 |
done |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1617 |
} |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1618 |
then have pcx: "path_component (- s) x (a + C *\<^sub>R (x - a))" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1619 |
by (force simp: closed_segment_def intro!: path_connected_linepath) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1620 |
def D == "B / norm(y - a)" --{*massive duplication with the proof above*} |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1621 |
{ fix u |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1622 |
assume u: "(1 - u) *\<^sub>R y + u *\<^sub>R (a + D *\<^sub>R (y - a)) \<in> s" and "0 \<le> u" "u \<le> 1" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1623 |
have DD: "1 \<le> 1 + (D - 1) * u" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1624 |
using `y \<noteq> a` `0 \<le> u` |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1625 |
apply (simp add: D_def divide_simps norm_minus_commute) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1626 |
by (metis By diff_le_iff(1) less_eq_real_def mult_nonneg_nonneg) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1627 |
have *: "\<And>v. (1 - u) *\<^sub>R y + u *\<^sub>R (a + v *\<^sub>R (y - a)) = a + (1 + (v - 1) * u) *\<^sub>R (y - a)" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1628 |
by (simp add: algebra_simps) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1629 |
have "a + ((1 / (1 + D * u - u)) *\<^sub>R y + ((u / (1 + D * u - u)) *\<^sub>R a + (D * u / (1 + D * u - u)) *\<^sub>R y)) = |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1630 |
(1 + (u / (1 + D * u - u))) *\<^sub>R a + ((1 / (1 + D * u - u)) + (D * u / (1 + D * u - u))) *\<^sub>R y" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1631 |
by (simp add: algebra_simps) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1632 |
also have "... = (1 + (u / (1 + D * u - u))) *\<^sub>R a + (1 + (u / (1 + D * u - u))) *\<^sub>R y" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1633 |
using DD by (simp add: field_simps) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1634 |
also have "... = y + (1 + (u / (1 + D * u - u))) *\<^sub>R a + (u / (1 + D * u - u)) *\<^sub>R y" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1635 |
by (simp add: algebra_simps) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1636 |
also have "... = y + ((1 / (1 + D * u - u)) *\<^sub>R a + |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1637 |
((u / (1 + D * u - u)) *\<^sub>R y + (D * u / (1 + D * u - u)) *\<^sub>R a))" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1638 |
using DD by (simp add: field_simps) (simp add: add_divide_distrib scaleR_add_left) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1639 |
finally have xeq: "(1 - 1 / (1 + (D - 1) * u)) *\<^sub>R a + (1 / (1 + (D - 1) * u)) *\<^sub>R (a + (1 + (D - 1) * u) *\<^sub>R (y - a)) = y" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1640 |
by (simp add: algebra_simps) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1641 |
have False |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1642 |
using `convex s` |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1643 |
apply (simp add: convex_alt) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1644 |
apply (drule_tac x=a in bspec) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1645 |
apply (rule `a \<in> s`) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1646 |
apply (drule_tac x="a + (1 + (D - 1) * u) *\<^sub>R (y - a)" in bspec) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1647 |
using u apply (simp add: *) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1648 |
apply (drule_tac x="1 / (1 + (D - 1) * u)" in spec) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1649 |
using `y \<noteq> a` `y \<notin> s` `0 \<le> u` DD |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1650 |
apply (auto simp: xeq) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1651 |
done |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1652 |
} |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1653 |
then have pdy: "path_component (- s) y (a + D *\<^sub>R (y - a))" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1654 |
by (force simp: closed_segment_def intro!: path_connected_linepath) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1655 |
have pyx: "path_component (- s) (a + D *\<^sub>R (y - a)) (a + C *\<^sub>R (x - a))" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1656 |
apply (rule path_component_of_subset [of "{x. norm(x - 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
|
1657 |
using `s \<subseteq> ball a B` |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1658 |
apply (force simp: ball_def dist_norm norm_minus_commute) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1659 |
apply (rule path_connected_sphere [OF 2, of a B, simplified path_connected_component, rule_format]) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1660 |
using `x \<noteq> a` using `y \<noteq> a` B apply (auto simp: C_def D_def) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1661 |
done |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1662 |
have "path_component (- s) x y" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1663 |
by (metis path_component_trans path_component_sym pcx pdy pyx) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1664 |
} |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1665 |
then show ?thesis |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1666 |
by (auto simp: path_connected_component) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1667 |
qed |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1668 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1669 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1670 |
lemma connected_complement_bounded_convex: |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1671 |
fixes s :: "'a :: euclidean_space set" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1672 |
assumes "bounded s" "convex s" "2 \<le> DIM('a)" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1673 |
shows "connected (- s)" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1674 |
using path_connected_complement_bounded_convex [OF assms] path_connected_imp_connected by blast |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1675 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1676 |
lemma connected_diff_ball: |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1677 |
fixes s :: "'a :: euclidean_space set" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1678 |
assumes "connected s" "cball a r \<subseteq> s" "2 \<le> DIM('a)" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1679 |
shows "connected (s - ball a r)" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1680 |
apply (rule connected_diff_open_from_closed [OF ball_subset_cball]) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1681 |
using assms connected_sphere |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1682 |
apply (auto simp: cball_diff_eq_sphere dist_norm) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1683 |
done |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1684 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1685 |
subsection\<open>Relations between components and path components\<close> |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1686 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1687 |
lemma open_connected_component: |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1688 |
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
|
1689 |
shows "open s \<Longrightarrow> open (connected_component_set s x)" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1690 |
apply (simp add: open_contains_ball, clarify) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1691 |
apply (rename_tac y) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1692 |
apply (drule_tac x=y in bspec) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1693 |
apply (simp add: connected_component_in, clarify) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1694 |
apply (rule_tac x=e in exI) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1695 |
by (metis mem_Collect_eq connected_component_eq connected_component_maximal centre_in_ball connected_ball) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1696 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1697 |
corollary open_components: |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1698 |
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
|
1699 |
shows "\<lbrakk>open u; s \<in> components u\<rbrakk> \<Longrightarrow> open s" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1700 |
by (simp add: components_iff) (metis open_connected_component) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1701 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1702 |
lemma in_closure_connected_component: |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1703 |
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
|
1704 |
assumes x: "x \<in> s" and s: "open s" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1705 |
shows "x \<in> closure (connected_component_set s y) \<longleftrightarrow> x \<in> connected_component_set s y" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1706 |
proof - |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1707 |
{ assume "x \<in> closure (connected_component_set s y)" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1708 |
moreover have "x \<in> connected_component_set s x" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1709 |
using x by simp |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1710 |
ultimately have "x \<in> connected_component_set s y" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1711 |
using s by (meson Compl_disjoint closure_iff_nhds_not_empty connected_component_disjoint disjoint_eq_subset_Compl open_connected_component) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1712 |
} |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1713 |
then show ?thesis |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1714 |
by (auto simp: closure_def) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1715 |
qed |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1716 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1717 |
subsection\<open>Existence of unbounded components\<close> |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1718 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1719 |
lemma cobounded_unbounded_component: |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1720 |
fixes s :: "'a :: euclidean_space set" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1721 |
assumes "bounded (-s)" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1722 |
shows "\<exists>x. x \<in> s \<and> ~ bounded (connected_component_set s x)" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1723 |
proof - |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1724 |
obtain i::'a where i: "i \<in> Basis" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1725 |
using nonempty_Basis by blast |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1726 |
obtain B where B: "B>0" "-s \<subseteq> ball 0 B" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1727 |
using bounded_subset_ballD [OF assms, of 0] by auto |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1728 |
then have *: "\<And>x. B \<le> norm x \<Longrightarrow> x \<in> s" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1729 |
by (force simp add: ball_def dist_norm) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1730 |
have unbounded_inner: "~ bounded {x. inner i x \<ge> B}" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1731 |
apply (auto simp: bounded_def dist_norm) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1732 |
apply (rule_tac x="x + (max B e + 1 + \<bar>i \<bullet> x\<bar>) *\<^sub>R i" in exI) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1733 |
apply simp |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1734 |
using i |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1735 |
apply (auto simp: algebra_simps) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1736 |
done |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1737 |
have **: "{x. B \<le> i \<bullet> x} \<subseteq> connected_component_set s (B *\<^sub>R i)" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1738 |
apply (rule connected_component_maximal) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1739 |
apply (auto simp: i intro: convex_connected convex_halfspace_ge [of B]) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1740 |
apply (rule *) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1741 |
apply (rule order_trans [OF _ Basis_le_norm [OF i]]) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1742 |
by (simp add: inner_commute) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1743 |
have "B *\<^sub>R i \<in> s" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1744 |
by (rule *) (simp add: norm_Basis [OF i]) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1745 |
then show ?thesis |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1746 |
apply (rule_tac x="B *\<^sub>R i" in exI, clarify) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1747 |
apply (frule bounded_subset [of _ "{x. B \<le> i \<bullet> x}", OF _ **]) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1748 |
using unbounded_inner apply blast |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1749 |
done |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1750 |
qed |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1751 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1752 |
lemma cobounded_unique_unbounded_component: |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1753 |
fixes s :: "'a :: euclidean_space set" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1754 |
assumes bs: "bounded (-s)" and "2 \<le> DIM('a)" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1755 |
and bo: "~ bounded(connected_component_set s x)" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1756 |
"~ bounded(connected_component_set s y)" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1757 |
shows "connected_component_set s x = connected_component_set s y" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1758 |
proof - |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1759 |
obtain i::'a where i: "i \<in> Basis" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1760 |
using nonempty_Basis by blast |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1761 |
obtain B where B: "B>0" "-s \<subseteq> ball 0 B" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1762 |
using bounded_subset_ballD [OF bs, of 0] by auto |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1763 |
then have *: "\<And>x. B \<le> norm x \<Longrightarrow> x \<in> s" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1764 |
by (force simp add: ball_def dist_norm) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1765 |
have ccb: "connected (- ball 0 B :: 'a set)" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1766 |
using assms by (auto intro: connected_complement_bounded_convex) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1767 |
obtain x' where x': "connected_component s x x'" "norm x' > B" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1768 |
using bo [unfolded bounded_def dist_norm, simplified, rule_format] |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1769 |
by (metis diff_zero norm_minus_commute not_less) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1770 |
obtain y' where y': "connected_component s y y'" "norm y' > B" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1771 |
using bo [unfolded bounded_def dist_norm, simplified, rule_format] |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1772 |
by (metis diff_zero norm_minus_commute not_less) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1773 |
have x'y': "connected_component s x' y'" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1774 |
apply (simp add: connected_component_def) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1775 |
apply (rule_tac x="- ball 0 B" in exI) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1776 |
using x' y' |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1777 |
apply (auto simp: ccb dist_norm *) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1778 |
done |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1779 |
show ?thesis |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1780 |
apply (rule connected_component_eq) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1781 |
using x' y' x'y' |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1782 |
by (metis (no_types, lifting) connected_component_eq_empty connected_component_eq_eq connected_component_idemp connected_component_in) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1783 |
qed |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1784 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1785 |
lemma cobounded_unbounded_components: |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1786 |
fixes s :: "'a :: euclidean_space set" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1787 |
shows "bounded (-s) \<Longrightarrow> \<exists>c. c \<in> components s \<and> ~bounded c" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1788 |
by (metis cobounded_unbounded_component components_def imageI) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1789 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1790 |
lemma cobounded_unique_unbounded_components: |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1791 |
fixes s :: "'a :: euclidean_space set" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1792 |
shows "\<lbrakk>bounded (- s); c \<in> components s; \<not> bounded c; c' \<in> components s; \<not> bounded c'; 2 \<le> DIM('a)\<rbrakk> \<Longrightarrow> c' = c" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1793 |
unfolding components_iff |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1794 |
by (metis cobounded_unique_unbounded_component) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1795 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1796 |
lemma cobounded_has_bounded_component: |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1797 |
fixes s :: "'a :: euclidean_space set" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1798 |
shows "\<lbrakk>bounded (- s); ~connected s; 2 \<le> DIM('a)\<rbrakk> \<Longrightarrow> \<exists>c. c \<in> components s \<and> bounded c" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1799 |
by (meson cobounded_unique_unbounded_components connected_eq_connected_components_eq) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1800 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1801 |
|
61518
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
1802 |
section\<open>The "inside" and "outside" of a set\<close> |
61426
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1803 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1804 |
text\<open>The inside comprises the points in a bounded connected component of the set's complement. |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1805 |
The outside comprises the points in unbounded connected component of the complement.\<close> |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1806 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1807 |
definition inside where |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1808 |
"inside s \<equiv> {x. (x \<notin> s) \<and> bounded(connected_component_set ( - s) x)}" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1809 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1810 |
definition outside where |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1811 |
"outside s \<equiv> -s \<inter> {x. ~ bounded(connected_component_set (- s) x)}" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1812 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1813 |
lemma outside: "outside s = {x. ~ bounded(connected_component_set (- s) x)}" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1814 |
by (auto simp: outside_def) (metis Compl_iff bounded_empty connected_component_eq_empty) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1815 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1816 |
lemma inside_no_overlap [simp]: "inside s \<inter> s = {}" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1817 |
by (auto simp: inside_def) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1818 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1819 |
lemma outside_no_overlap [simp]: |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1820 |
"outside s \<inter> s = {}" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1821 |
by (auto simp: outside_def) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1822 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1823 |
lemma inside_inter_outside [simp]: "inside s \<inter> outside s = {}" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1824 |
by (auto simp: inside_def outside_def) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1825 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1826 |
lemma inside_union_outside [simp]: "inside s \<union> outside s = (- s)" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1827 |
by (auto simp: inside_def outside_def) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1828 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1829 |
lemma inside_eq_outside: |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1830 |
"inside s = outside s \<longleftrightarrow> s = UNIV" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1831 |
by (auto simp: inside_def outside_def) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1832 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1833 |
lemma inside_outside: "inside s = (- (s \<union> outside s))" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1834 |
by (force simp add: inside_def outside) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1835 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1836 |
lemma outside_inside: "outside s = (- (s \<union> inside s))" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1837 |
by (auto simp: inside_outside) (metis IntI equals0D outside_no_overlap) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1838 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1839 |
lemma union_with_inside: "s \<union> inside s = - outside s" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1840 |
by (auto simp: inside_outside) (simp add: outside_inside) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1841 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1842 |
lemma union_with_outside: "s \<union> outside s = - inside s" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1843 |
by (simp add: inside_outside) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1844 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1845 |
lemma outside_mono: "s \<subseteq> t \<Longrightarrow> outside t \<subseteq> outside s" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1846 |
by (auto simp: outside bounded_subset connected_component_mono) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1847 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1848 |
lemma inside_mono: "s \<subseteq> t \<Longrightarrow> inside s - t \<subseteq> inside t" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1849 |
by (auto simp: inside_def bounded_subset connected_component_mono) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1850 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1851 |
lemma segment_bound_lemma: |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1852 |
fixes u::real |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1853 |
assumes "x \<ge> B" "y \<ge> B" "0 \<le> u" "u \<le> 1" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1854 |
shows "(1 - u) * x + u * y \<ge> B" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1855 |
proof - |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1856 |
obtain dx dy where "dx \<ge> 0" "dy \<ge> 0" "x = B + dx" "y = B + dy" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1857 |
using assms by auto (metis add.commute diff_add_cancel) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1858 |
with `0 \<le> u` `u \<le> 1` show ?thesis |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1859 |
by (simp add: add_increasing2 mult_left_le field_simps) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1860 |
qed |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1861 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1862 |
lemma cobounded_outside: |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1863 |
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
|
1864 |
assumes "bounded s" shows "bounded (- outside s)" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1865 |
proof - |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1866 |
obtain B where B: "B>0" "s \<subseteq> ball 0 B" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1867 |
using bounded_subset_ballD [OF assms, of 0] by auto |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1868 |
{ fix x::'a and C::real |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1869 |
assume Bno: "B \<le> norm x" and C: "0 < C" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1870 |
have "\<exists>y. connected_component (- s) x y \<and> norm y > C" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1871 |
proof (cases "x = 0") |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1872 |
case True with B Bno show ?thesis by force |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1873 |
next |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1874 |
case False with B C show ?thesis |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1875 |
apply (rule_tac x="((B+C)/norm x) *\<^sub>R x" in exI) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1876 |
apply (simp add: connected_component_def) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1877 |
apply (rule_tac x="closed_segment x (((B+C)/norm x) *\<^sub>R x)" in exI) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1878 |
apply simp |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1879 |
apply (rule_tac y="- ball 0 B" in order_trans) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1880 |
prefer 2 apply force |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1881 |
apply (simp add: closed_segment_def ball_def dist_norm, clarify) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1882 |
apply (simp add: real_vector_class.scaleR_add_left [symmetric] divide_simps) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1883 |
using segment_bound_lemma [of B "norm x" "B+C" ] Bno |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1884 |
by (meson le_add_same_cancel1 less_eq_real_def not_le) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1885 |
qed |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1886 |
} |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1887 |
then show ?thesis |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1888 |
apply (simp add: outside_def assms) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1889 |
apply (rule bounded_subset [OF bounded_ball [of 0 B]]) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1890 |
apply (force simp add: dist_norm not_less bounded_pos) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1891 |
done |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1892 |
qed |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1893 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1894 |
lemma unbounded_outside: |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1895 |
fixes s :: "'a::{real_normed_vector, perfect_space} set" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1896 |
shows "bounded s \<Longrightarrow> ~ bounded(outside s)" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1897 |
using cobounded_imp_unbounded cobounded_outside by blast |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1898 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1899 |
lemma bounded_inside: |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1900 |
fixes s :: "'a::{real_normed_vector, perfect_space} set" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1901 |
shows "bounded s \<Longrightarrow> bounded(inside s)" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1902 |
by (simp add: bounded_Int cobounded_outside inside_outside) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1903 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1904 |
lemma connected_outside: |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1905 |
fixes s :: "'a::euclidean_space set" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1906 |
assumes "bounded s" "2 \<le> DIM('a)" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1907 |
shows "connected(outside s)" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1908 |
apply (simp add: connected_iff_connected_component, clarify) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1909 |
apply (simp add: outside) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1910 |
apply (rule_tac s="connected_component_set (- s) x" in connected_component_of_subset) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1911 |
apply (metis (no_types) assms cobounded_unbounded_component cobounded_unique_unbounded_component connected_component_eq_eq connected_component_idemp double_complement mem_Collect_eq) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1912 |
apply clarify |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1913 |
apply (metis connected_component_eq_eq connected_component_in) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1914 |
done |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1915 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1916 |
lemma outside_connected_component_lt: |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1917 |
"outside s = {x. \<forall>B. \<exists>y. B < norm(y) \<and> connected_component (- s) x y}" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1918 |
apply (auto simp: outside bounded_def dist_norm) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1919 |
apply (metis diff_0 norm_minus_cancel not_less) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1920 |
by (metis less_diff_eq norm_minus_commute norm_triangle_ineq2 order.trans pinf(6)) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1921 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1922 |
lemma outside_connected_component_le: |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1923 |
"outside s = |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1924 |
{x. \<forall>B. \<exists>y. B \<le> norm(y) \<and> |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1925 |
connected_component (- s) x y}" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1926 |
apply (simp add: outside_connected_component_lt) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1927 |
apply (simp add: Set.set_eq_iff) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1928 |
by (meson gt_ex leD le_less_linear less_imp_le order.trans) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1929 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1930 |
lemma not_outside_connected_component_lt: |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1931 |
fixes s :: "'a::euclidean_space set" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1932 |
assumes s: "bounded s" and "2 \<le> DIM('a)" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1933 |
shows "- (outside s) = {x. \<forall>B. \<exists>y. B < norm(y) \<and> ~ (connected_component (- s) x y)}" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1934 |
proof - |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1935 |
obtain B::real where B: "0 < B" and Bno: "\<And>x. x \<in> s \<Longrightarrow> norm x \<le> B" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1936 |
using s [simplified bounded_pos] by auto |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1937 |
{ fix y::'a and z::'a |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1938 |
assume yz: "B < norm z" "B < norm y" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1939 |
have "connected_component (- cball 0 B) y z" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1940 |
apply (rule connected_componentI [OF _ subset_refl]) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1941 |
apply (rule connected_complement_bounded_convex) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1942 |
using assms yz |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1943 |
by (auto simp: dist_norm) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1944 |
then have "connected_component (- s) y z" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1945 |
apply (rule connected_component_of_subset) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1946 |
apply (metis Bno Compl_anti_mono mem_cball_0 subset_iff) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1947 |
done |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1948 |
} note cyz = this |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1949 |
show ?thesis |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1950 |
apply (auto simp: outside) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1951 |
apply (metis Compl_iff bounded_iff cobounded_imp_unbounded mem_Collect_eq not_le) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1952 |
apply (simp add: bounded_pos) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1953 |
by (metis B connected_component_trans cyz not_le) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1954 |
qed |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1955 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1956 |
lemma not_outside_connected_component_le: |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1957 |
fixes s :: "'a::euclidean_space set" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1958 |
assumes s: "bounded s" "2 \<le> DIM('a)" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1959 |
shows "- (outside s) = {x. \<forall>B. \<exists>y. B \<le> norm(y) \<and> ~ (connected_component (- s) x y)}" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1960 |
apply (auto intro: less_imp_le simp: not_outside_connected_component_lt [OF assms]) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1961 |
by (meson gt_ex less_le_trans) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1962 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1963 |
lemma inside_connected_component_lt: |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1964 |
fixes s :: "'a::euclidean_space set" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1965 |
assumes s: "bounded s" "2 \<le> DIM('a)" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1966 |
shows "inside s = {x. (x \<notin> s) \<and> (\<forall>B. \<exists>y. B < norm(y) \<and> ~(connected_component (- s) x y))}" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1967 |
by (auto simp: inside_outside not_outside_connected_component_lt [OF assms]) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1968 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1969 |
lemma inside_connected_component_le: |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1970 |
fixes s :: "'a::euclidean_space set" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1971 |
assumes s: "bounded s" "2 \<le> DIM('a)" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1972 |
shows "inside s = {x. (x \<notin> s) \<and> (\<forall>B. \<exists>y. B \<le> norm(y) \<and> ~(connected_component (- s) x y))}" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1973 |
by (auto simp: inside_outside not_outside_connected_component_le [OF assms]) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1974 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1975 |
lemma inside_subset: |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1976 |
assumes "connected u" and "~bounded u" and "t \<union> u = - s" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1977 |
shows "inside s \<subseteq> t" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1978 |
apply (auto simp: inside_def) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1979 |
by (metis bounded_subset [of "connected_component_set (- s) _"] connected_component_maximal |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1980 |
Compl_iff Un_iff assms subsetI) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1981 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1982 |
lemma frontier_interiors: "frontier s = - interior(s) - interior(-s)" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1983 |
by (simp add: Int_commute frontier_def interior_closure) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1984 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1985 |
lemma connected_inter_frontier: |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1986 |
"\<lbrakk>connected s; s \<inter> t \<noteq> {}; s - t \<noteq> {}\<rbrakk> \<Longrightarrow> (s \<inter> frontier t \<noteq> {})" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1987 |
apply (simp add: frontier_interiors connected_open_in, safe) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1988 |
apply (drule_tac x="s \<inter> interior t" in spec, safe) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1989 |
apply (drule_tac [2] x="s \<inter> interior (-t)" in spec) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1990 |
apply (auto simp: disjoint_eq_subset_Compl dest: interior_subset [THEN subsetD]) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1991 |
done |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1992 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1993 |
lemma connected_component_UNIV: |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1994 |
fixes x :: "'a::real_normed_vector" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1995 |
shows "connected_component_set UNIV x = UNIV" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1996 |
using connected_iff_eq_connected_component_set [of "UNIV::'a set"] connected_UNIV |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1997 |
by auto |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1998 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1999 |
lemma connected_component_eq_UNIV: |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2000 |
fixes x :: "'a::real_normed_vector" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2001 |
shows "connected_component_set s x = UNIV \<longleftrightarrow> s = UNIV" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2002 |
using connected_component_in connected_component_UNIV by blast |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2003 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2004 |
lemma components_univ [simp]: "components UNIV = {UNIV :: 'a::real_normed_vector set}" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2005 |
by (auto simp: components_eq_sing_iff) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2006 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2007 |
lemma interior_inside_frontier: |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2008 |
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
|
2009 |
assumes "bounded s" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2010 |
shows "interior s \<subseteq> inside (frontier s)" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2011 |
proof - |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2012 |
{ fix x y |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2013 |
assume x: "x \<in> interior s" and y: "y \<notin> s" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2014 |
and cc: "connected_component (- frontier s) x y" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2015 |
have "connected_component_set (- frontier s) x \<inter> frontier s \<noteq> {}" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2016 |
apply (rule connected_inter_frontier, simp) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2017 |
apply (metis IntI cc connected_component_in connected_component_refl empty_iff interiorE mem_Collect_eq set_rev_mp x) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2018 |
using y cc |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2019 |
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
|
2020 |
then have "bounded (connected_component_set (- frontier s) x)" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2021 |
using connected_component_in by auto |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2022 |
} |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2023 |
then show ?thesis |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2024 |
apply (auto simp: inside_def frontier_def) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2025 |
apply (rule classical) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2026 |
apply (rule bounded_subset [OF assms], blast) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2027 |
done |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2028 |
qed |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2029 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2030 |
lemma inside_empty [simp]: "inside {} = ({} :: 'a :: {real_normed_vector, perfect_space} set)" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2031 |
by (simp add: inside_def connected_component_UNIV) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2032 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2033 |
lemma outside_empty [simp]: "outside {} = (UNIV :: 'a :: {real_normed_vector, perfect_space} set)" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2034 |
using inside_empty inside_union_outside by blast |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2035 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2036 |
lemma inside_same_component: |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2037 |
"\<lbrakk>connected_component (- s) x y; x \<in> inside s\<rbrakk> \<Longrightarrow> y \<in> inside s" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2038 |
using connected_component_eq connected_component_in |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2039 |
by (fastforce simp add: inside_def) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2040 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2041 |
lemma outside_same_component: |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2042 |
"\<lbrakk>connected_component (- s) x y; x \<in> outside s\<rbrakk> \<Longrightarrow> y \<in> outside s" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2043 |
using connected_component_eq connected_component_in |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2044 |
by (fastforce simp add: outside_def) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2045 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2046 |
lemma convex_in_outside: |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2047 |
fixes s :: "'a :: {real_normed_vector, perfect_space} set" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2048 |
assumes s: "convex s" and z: "z \<notin> s" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2049 |
shows "z \<in> outside s" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2050 |
proof (cases "s={}") |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2051 |
case True then show ?thesis by simp |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2052 |
next |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2053 |
case False then obtain a where "a \<in> s" by blast |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2054 |
with z have zna: "z \<noteq> a" by auto |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2055 |
{ assume "bounded (connected_component_set (- s) z)" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2056 |
with bounded_pos_less obtain B where "B>0" and B: "\<And>x. connected_component (- s) z x \<Longrightarrow> norm x < B" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2057 |
by (metis mem_Collect_eq) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2058 |
def C \<equiv> "((B + 1 + norm z) / norm (z-a))" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2059 |
have "C > 0" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2060 |
using `0 < B` zna by (simp add: C_def divide_simps add_strict_increasing) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2061 |
have "\<bar>norm (z + C *\<^sub>R (z-a)) - norm (C *\<^sub>R (z-a))\<bar> \<le> norm z" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2062 |
by (metis add_diff_cancel norm_triangle_ineq3) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2063 |
moreover have "norm (C *\<^sub>R (z-a)) > norm z + B" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2064 |
using zna `B>0` by (simp add: C_def le_max_iff_disj field_simps) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2065 |
ultimately have C: "norm (z + C *\<^sub>R (z-a)) > B" by linarith |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2066 |
{ fix u::real |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2067 |
assume u: "0\<le>u" "u\<le>1" and ins: "(1 - u) *\<^sub>R z + u *\<^sub>R (z + C *\<^sub>R (z - a)) \<in> s" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2068 |
then have Cpos: "1 + u * C > 0" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2069 |
by (meson `0 < C` add_pos_nonneg less_eq_real_def zero_le_mult_iff zero_less_one) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2070 |
then have *: "(1 / (1 + u * C)) *\<^sub>R z + (u * C / (1 + u * C)) *\<^sub>R z = z" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2071 |
by (simp add: scaleR_add_left [symmetric] divide_simps) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2072 |
then have False |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2073 |
using convexD_alt [OF s `a \<in> s` ins, of "1/(u*C + 1)"] `C>0` `z \<notin> s` Cpos u |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2074 |
by (simp add: * divide_simps algebra_simps) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2075 |
} note contra = this |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2076 |
have "connected_component (- s) z (z + C *\<^sub>R (z-a))" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2077 |
apply (rule connected_componentI [OF connected_segment [of z "z + C *\<^sub>R (z-a)"]]) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2078 |
apply (simp add: closed_segment_def) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2079 |
using contra |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2080 |
apply auto |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2081 |
done |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2082 |
then have False |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2083 |
using zna B [of "z + C *\<^sub>R (z-a)"] C |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2084 |
by (auto simp: divide_simps max_mult_distrib_right) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2085 |
} |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2086 |
then show ?thesis |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2087 |
by (auto simp: outside_def z) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2088 |
qed |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2089 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2090 |
lemma outside_convex: |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2091 |
fixes s :: "'a :: {real_normed_vector, perfect_space} set" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2092 |
assumes "convex s" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2093 |
shows "outside s = - s" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2094 |
by (metis ComplD assms convex_in_outside equalityI inside_union_outside subsetI sup.cobounded2) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2095 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2096 |
lemma inside_convex: |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2097 |
fixes s :: "'a :: {real_normed_vector, perfect_space} set" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2098 |
shows "convex s \<Longrightarrow> inside s = {}" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2099 |
by (simp add: inside_outside outside_convex) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2100 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2101 |
lemma outside_subset_convex: |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2102 |
fixes s :: "'a :: {real_normed_vector, perfect_space} set" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2103 |
shows "\<lbrakk>convex t; s \<subseteq> t\<rbrakk> \<Longrightarrow> - t \<subseteq> outside s" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2104 |
using outside_convex outside_mono by blast |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2105 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2106 |
lemma outside_frontier_misses_closure: |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2107 |
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
|
2108 |
assumes "bounded s" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2109 |
shows "outside(frontier s) \<subseteq> - closure s" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2110 |
unfolding outside_inside Lattices.boolean_algebra_class.compl_le_compl_iff |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2111 |
proof - |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2112 |
{ assume "interior s \<subseteq> inside (frontier s)" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2113 |
hence "interior s \<union> inside (frontier s) = inside (frontier s)" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2114 |
by (simp add: subset_Un_eq) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2115 |
then have "closure s \<subseteq> frontier s \<union> inside (frontier s)" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2116 |
using frontier_def by auto |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2117 |
} |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2118 |
then show "closure s \<subseteq> frontier s \<union> inside (frontier s)" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2119 |
using interior_inside_frontier [OF assms] by blast |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2120 |
qed |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2121 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2122 |
lemma outside_frontier_eq_complement_closure: |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2123 |
fixes s :: "'a :: {real_normed_vector, perfect_space} set" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2124 |
assumes "bounded s" "convex s" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2125 |
shows "outside(frontier s) = - closure s" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2126 |
by (metis Diff_subset assms convex_closure frontier_def outside_frontier_misses_closure |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2127 |
outside_subset_convex subset_antisym) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2128 |
|
61518
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2129 |
lemma inside_frontier_eq_interior: |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2130 |
fixes s :: "'a :: {real_normed_vector, perfect_space} set" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2131 |
shows "\<lbrakk>bounded s; convex s\<rbrakk> \<Longrightarrow> inside(frontier s) = interior s" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2132 |
apply (simp add: inside_outside outside_frontier_eq_complement_closure) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2133 |
using closure_subset interior_subset |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2134 |
apply (auto simp add: frontier_def) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2135 |
done |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2136 |
|
61426
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2137 |
lemma open_inside: |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2138 |
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
|
2139 |
assumes "closed s" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2140 |
shows "open (inside s)" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2141 |
proof - |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2142 |
{ fix x assume x: "x \<in> inside s" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2143 |
have "open (connected_component_set (- s) x)" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2144 |
using assms open_connected_component by blast |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2145 |
then obtain e where e: "e>0" and e: "\<And>y. dist y x < e \<longrightarrow> connected_component (- s) x y" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2146 |
using dist_not_less_zero |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2147 |
apply (simp add: open_dist) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2148 |
by (metis (no_types, lifting) Compl_iff connected_component_refl_eq inside_def mem_Collect_eq x) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2149 |
then have "\<exists>e>0. ball x e \<subseteq> inside s" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2150 |
by (metis e dist_commute inside_same_component mem_ball subsetI x) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2151 |
} |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2152 |
then show ?thesis |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2153 |
by (simp add: open_contains_ball) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2154 |
qed |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2155 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2156 |
lemma open_outside: |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2157 |
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
|
2158 |
assumes "closed s" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2159 |
shows "open (outside s)" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2160 |
proof - |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2161 |
{ fix x assume x: "x \<in> outside s" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2162 |
have "open (connected_component_set (- s) x)" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2163 |
using assms open_connected_component by blast |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2164 |
then obtain e where e: "e>0" and e: "\<And>y. dist y x < e \<longrightarrow> connected_component (- s) x y" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2165 |
using dist_not_less_zero |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2166 |
apply (simp add: open_dist) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2167 |
by (metis Int_iff outside_def connected_component_refl_eq x) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2168 |
then have "\<exists>e>0. ball x e \<subseteq> outside s" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2169 |
by (metis e dist_commute outside_same_component mem_ball subsetI x) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2170 |
} |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2171 |
then show ?thesis |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2172 |
by (simp add: open_contains_ball) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2173 |
qed |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2174 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2175 |
lemma closure_inside_subset: |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2176 |
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
|
2177 |
assumes "closed s" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2178 |
shows "closure(inside s) \<subseteq> s \<union> inside s" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2179 |
by (metis assms closure_minimal open_closed open_outside sup.cobounded2 union_with_inside) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2180 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2181 |
lemma frontier_inside_subset: |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2182 |
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
|
2183 |
assumes "closed s" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2184 |
shows "frontier(inside s) \<subseteq> s" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2185 |
proof - |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2186 |
have "closure (inside s) \<inter> - inside s = closure (inside s) - interior (inside s)" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2187 |
by (metis (no_types) Diff_Compl assms closure_closed interior_closure open_closed open_inside) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2188 |
moreover have "- inside s \<inter> - outside s = s" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2189 |
by (metis (no_types) compl_sup double_compl inside_union_outside) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2190 |
moreover have "closure (inside s) \<subseteq> - outside s" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2191 |
by (metis (no_types) assms closure_inside_subset union_with_inside) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2192 |
ultimately have "closure (inside s) - interior (inside s) \<subseteq> s" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2193 |
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
|
2194 |
then show ?thesis |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2195 |
by (simp add: frontier_def open_inside interior_open) |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2196 |
qed |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2197 |
|
61518
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2198 |
lemma closure_outside_subset: |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2199 |
fixes s :: "'a::real_normed_vector set" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2200 |
assumes "closed s" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2201 |
shows "closure(outside s) \<subseteq> s \<union> outside s" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2202 |
apply (rule closure_minimal, simp) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2203 |
by (metis assms closed_open inside_outside open_inside) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2204 |
|
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2205 |
lemma frontier_outside_subset: |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2206 |
fixes s :: "'a::real_normed_vector set" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2207 |
assumes "closed s" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2208 |
shows "frontier(outside s) \<subseteq> s" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2209 |
apply (simp add: frontier_def open_outside interior_open) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2210 |
by (metis Diff_subset_conv assms closure_outside_subset interior_eq open_outside sup.commute) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2211 |
|
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2212 |
lemma inside_complement_unbounded_connected_empty: |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2213 |
"\<lbrakk>connected (- s); \<not> bounded (- s)\<rbrakk> \<Longrightarrow> inside s = {}" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2214 |
apply (simp add: inside_def) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2215 |
by (meson Compl_iff bounded_subset connected_component_maximal order_refl) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2216 |
|
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2217 |
lemma inside_bounded_complement_connected_empty: |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2218 |
fixes s :: "'a::{real_normed_vector, perfect_space} set" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2219 |
shows "\<lbrakk>connected (- s); bounded s\<rbrakk> \<Longrightarrow> inside s = {}" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2220 |
by (metis inside_complement_unbounded_connected_empty cobounded_imp_unbounded) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2221 |
|
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2222 |
lemma inside_inside: |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2223 |
assumes "s \<subseteq> inside t" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2224 |
shows "inside s - t \<subseteq> inside t" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2225 |
unfolding inside_def |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2226 |
proof clarify |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2227 |
fix x |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2228 |
assume x: "x \<notin> t" "x \<notin> s" and bo: "bounded (connected_component_set (- s) x)" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2229 |
show "bounded (connected_component_set (- t) x)" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2230 |
proof (cases "s \<inter> connected_component_set (- t) x = {}") |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2231 |
case True show ?thesis |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2232 |
apply (rule bounded_subset [OF bo]) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2233 |
apply (rule connected_component_maximal) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2234 |
using x True apply auto |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2235 |
done |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2236 |
next |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2237 |
case False then show ?thesis |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2238 |
using assms [unfolded inside_def] x |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2239 |
apply (simp add: disjoint_iff_not_equal, clarify) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2240 |
apply (drule subsetD, assumption, auto) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2241 |
by (metis (no_types, hide_lams) ComplI connected_component_eq_eq) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2242 |
qed |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2243 |
qed |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2244 |
|
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2245 |
lemma inside_inside_subset: "inside(inside s) \<subseteq> s" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2246 |
using inside_inside union_with_outside by fastforce |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2247 |
|
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2248 |
lemma inside_outside_intersect_connected: |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2249 |
"\<lbrakk>connected t; inside s \<inter> t \<noteq> {}; outside s \<inter> t \<noteq> {}\<rbrakk> \<Longrightarrow> s \<inter> t \<noteq> {}" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2250 |
apply (simp add: inside_def outside_def ex_in_conv [symmetric] disjoint_eq_subset_Compl, clarify) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2251 |
by (metis (no_types, hide_lams) Compl_anti_mono connected_component_eq connected_component_maximal contra_subsetD double_compl) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2252 |
|
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2253 |
lemma outside_bounded_nonempty: |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2254 |
fixes s :: "'a :: {real_normed_vector, perfect_space} set" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2255 |
assumes "bounded s" shows "outside s \<noteq> {}" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2256 |
by (metis (no_types, lifting) Collect_empty_eq Collect_mem_eq Compl_eq_Diff_UNIV Diff_cancel |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2257 |
Diff_disjoint UNIV_I assms ball_eq_empty bounded_diff cobounded_outside convex_ball |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2258 |
double_complement order_refl outside_convex outside_def) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2259 |
|
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2260 |
lemma outside_compact_in_open: |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2261 |
fixes s :: "'a :: {real_normed_vector,perfect_space} set" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2262 |
assumes s: "compact s" and t: "open t" and "s \<subseteq> t" "t \<noteq> {}" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2263 |
shows "outside s \<inter> t \<noteq> {}" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2264 |
proof - |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2265 |
have "outside s \<noteq> {}" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2266 |
by (simp add: compact_imp_bounded outside_bounded_nonempty s) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2267 |
with assms obtain a b where a: "a \<in> outside s" and b: "b \<in> t" by auto |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2268 |
show ?thesis |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2269 |
proof (cases "a \<in> t") |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2270 |
case True with a show ?thesis by blast |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2271 |
next |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2272 |
case False |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2273 |
have front: "frontier t \<subseteq> - s" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2274 |
using `s \<subseteq> t` frontier_disjoint_eq t by auto |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2275 |
{ fix \<gamma> |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2276 |
assume "path \<gamma>" and pimg_sbs: "path_image \<gamma> - {pathfinish \<gamma>} \<subseteq> interior (- t)" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2277 |
and pf: "pathfinish \<gamma> \<in> frontier t" and ps: "pathstart \<gamma> = a" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2278 |
def c \<equiv> "pathfinish \<gamma>" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2279 |
have "c \<in> -s" unfolding c_def using front pf by blast |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2280 |
moreover have "open (-s)" using s compact_imp_closed by blast |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2281 |
ultimately obtain \<epsilon>::real where "\<epsilon> > 0" and \<epsilon>: "cball c \<epsilon> \<subseteq> -s" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2282 |
using open_contains_cball[of "-s"] s by blast |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2283 |
then obtain d where "d \<in> t" and d: "dist d c < \<epsilon>" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2284 |
using closure_approachable [of c t] pf unfolding c_def |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2285 |
by (metis Diff_iff frontier_def) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2286 |
then have "d \<in> -s" using \<epsilon> |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2287 |
using dist_commute by (metis contra_subsetD mem_cball not_le not_less_iff_gr_or_eq) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2288 |
have pimg_sbs_cos: "path_image \<gamma> \<subseteq> -s" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2289 |
using pimg_sbs apply (auto simp: path_image_def) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2290 |
apply (drule subsetD) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2291 |
using `c \<in> - s` `s \<subseteq> t` interior_subset apply (auto simp: c_def) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2292 |
done |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2293 |
have "closed_segment c d \<le> cball c \<epsilon>" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2294 |
apply (simp add: segment_convex_hull) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2295 |
apply (rule hull_minimal) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2296 |
using `\<epsilon> > 0` d apply (auto simp: dist_commute) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2297 |
done |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2298 |
with \<epsilon> have "closed_segment c d \<subseteq> -s" by blast |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2299 |
moreover have con_gcd: "connected (path_image \<gamma> \<union> closed_segment c d)" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2300 |
by (rule connected_Un) (auto simp: c_def `path \<gamma>` connected_path_image) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2301 |
ultimately have "connected_component (- s) a d" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2302 |
unfolding connected_component_def using pimg_sbs_cos ps by blast |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2303 |
then have "outside s \<inter> t \<noteq> {}" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2304 |
using outside_same_component [OF _ a] by (metis IntI `d \<in> t` empty_iff) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2305 |
} note * = this |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2306 |
have pal: "pathstart (linepath a b) \<in> closure (- t)" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2307 |
by (auto simp: False closure_def) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2308 |
show ?thesis |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2309 |
by (rule exists_path_subpath_to_frontier [OF path_linepath pal _ *]) (auto simp: b) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2310 |
qed |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2311 |
qed |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2312 |
|
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2313 |
lemma inside_inside_compact_connected: |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2314 |
fixes s :: "'a :: euclidean_space set" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2315 |
assumes s: "closed s" and t: "compact t" and "connected t" "s \<subseteq> inside t" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2316 |
shows "inside s \<subseteq> inside t" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2317 |
proof (cases "inside t = {}") |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2318 |
case True with assms show ?thesis by auto |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2319 |
next |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2320 |
case False |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2321 |
consider "DIM('a) = 1" | "DIM('a) \<ge> 2" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2322 |
using antisym not_less_eq_eq by fastforce |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2323 |
then show ?thesis |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2324 |
proof cases |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2325 |
case 1 then show ?thesis |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2326 |
using connected_convex_1_gen assms False inside_convex by blast |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2327 |
next |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2328 |
case 2 |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2329 |
have coms: "compact s" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2330 |
using assms apply (simp add: s compact_eq_bounded_closed) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2331 |
by (meson bounded_inside bounded_subset compact_imp_bounded) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2332 |
then have bst: "bounded (s \<union> t)" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2333 |
by (simp add: compact_imp_bounded t) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2334 |
then obtain r where "0 < r" and r: "s \<union> t \<subseteq> ball 0 r" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2335 |
using bounded_subset_ballD by blast |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2336 |
have outst: "outside s \<inter> outside t \<noteq> {}" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2337 |
proof - |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2338 |
have "- ball 0 r \<subseteq> outside s" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2339 |
apply (rule outside_subset_convex) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2340 |
using r by auto |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2341 |
moreover have "- ball 0 r \<subseteq> outside t" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2342 |
apply (rule outside_subset_convex) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2343 |
using r by auto |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2344 |
ultimately show ?thesis |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2345 |
by (metis Compl_subset_Compl_iff Int_subset_iff bounded_ball inf.orderE outside_bounded_nonempty outside_no_overlap) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2346 |
qed |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2347 |
have "s \<inter> t = {}" using assms |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2348 |
by (metis disjoint_iff_not_equal inside_no_overlap subsetCE) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2349 |
moreover have "outside s \<inter> inside t \<noteq> {}" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2350 |
by (meson False assms(4) compact_eq_bounded_closed coms open_inside outside_compact_in_open t) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2351 |
ultimately have "inside s \<inter> t = {}" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2352 |
using inside_outside_intersect_connected [OF `connected t`, of s] |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2353 |
by (metis "2" compact_eq_bounded_closed coms connected_outside inf.commute inside_outside_intersect_connected outst) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2354 |
then show ?thesis |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2355 |
using inside_inside [OF `s \<subseteq> inside t`] by blast |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2356 |
qed |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2357 |
qed |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2358 |
|
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2359 |
lemma connected_with_inside: |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2360 |
fixes s :: "'a :: real_normed_vector set" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2361 |
assumes s: "closed s" and cons: "connected s" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2362 |
shows "connected(s \<union> inside s)" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2363 |
proof (cases "s \<union> inside s = UNIV") |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2364 |
case True with assms show ?thesis by auto |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2365 |
next |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2366 |
case False |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2367 |
then obtain b where b: "b \<notin> s" "b \<notin> inside s" by blast |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2368 |
have *: "\<exists>y t. y \<in> s \<and> connected t \<and> a \<in> t \<and> y \<in> t \<and> t \<subseteq> (s \<union> inside s)" if "a \<in> (s \<union> inside s)" for a |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2369 |
using that proof |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2370 |
assume "a \<in> s" then show ?thesis |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2371 |
apply (rule_tac x=a in exI) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2372 |
apply (rule_tac x="{a}" in exI) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2373 |
apply (simp add:) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2374 |
done |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2375 |
next |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2376 |
assume a: "a \<in> inside s" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2377 |
show ?thesis |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2378 |
apply (rule exists_path_subpath_to_frontier [OF path_linepath [of a b], of "inside s"]) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2379 |
using a apply (simp add: closure_def) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2380 |
apply (simp add: b) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2381 |
apply (rule_tac x="pathfinish h" in exI) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2382 |
apply (rule_tac x="path_image h" in exI) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2383 |
apply (simp add: pathfinish_in_path_image connected_path_image, auto) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2384 |
using frontier_inside_subset s apply fastforce |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2385 |
by (metis (no_types, lifting) frontier_inside_subset insertE insert_Diff interior_eq open_inside pathfinish_in_path_image s subsetCE) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2386 |
qed |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2387 |
show ?thesis |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2388 |
apply (simp add: connected_iff_connected_component) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2389 |
apply (simp add: connected_component_def) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2390 |
apply (clarify dest!: *) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2391 |
apply (rename_tac u u' t t') |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2392 |
apply (rule_tac x="(s \<union> t \<union> t')" in exI) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2393 |
apply (auto simp: intro!: connected_Un cons) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2394 |
done |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2395 |
qed |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2396 |
|
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2397 |
text\<open>The proof is virtually the same as that above.\<close> |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2398 |
lemma connected_with_outside: |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2399 |
fixes s :: "'a :: real_normed_vector set" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2400 |
assumes s: "closed s" and cons: "connected s" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2401 |
shows "connected(s \<union> outside s)" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2402 |
proof (cases "s \<union> outside s = UNIV") |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2403 |
case True with assms show ?thesis by auto |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2404 |
next |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2405 |
case False |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2406 |
then obtain b where b: "b \<notin> s" "b \<notin> outside s" by blast |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2407 |
have *: "\<exists>y t. y \<in> s \<and> connected t \<and> a \<in> t \<and> y \<in> t \<and> t \<subseteq> (s \<union> outside s)" if "a \<in> (s \<union> outside s)" for a |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2408 |
using that proof |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2409 |
assume "a \<in> s" then show ?thesis |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2410 |
apply (rule_tac x=a in exI) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2411 |
apply (rule_tac x="{a}" in exI) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2412 |
apply (simp add:) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2413 |
done |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2414 |
next |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2415 |
assume a: "a \<in> outside s" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2416 |
show ?thesis |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2417 |
apply (rule exists_path_subpath_to_frontier [OF path_linepath [of a b], of "outside s"]) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2418 |
using a apply (simp add: closure_def) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2419 |
apply (simp add: b) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2420 |
apply (rule_tac x="pathfinish h" in exI) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2421 |
apply (rule_tac x="path_image h" in exI) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2422 |
apply (simp add: pathfinish_in_path_image connected_path_image, auto) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2423 |
using frontier_outside_subset s apply fastforce |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2424 |
by (metis (no_types, lifting) frontier_outside_subset insertE insert_Diff interior_eq open_outside pathfinish_in_path_image s subsetCE) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2425 |
qed |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2426 |
show ?thesis |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2427 |
apply (simp add: connected_iff_connected_component) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2428 |
apply (simp add: connected_component_def) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2429 |
apply (clarify dest!: *) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2430 |
apply (rename_tac u u' t t') |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2431 |
apply (rule_tac x="(s \<union> t \<union> t')" in exI) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2432 |
apply (auto simp: intro!: connected_Un cons) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2433 |
done |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2434 |
qed |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2435 |
|
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2436 |
lemma inside_inside_eq_empty [simp]: |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2437 |
fixes s :: "'a :: {real_normed_vector, perfect_space} set" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2438 |
assumes s: "closed s" and cons: "connected s" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2439 |
shows "inside (inside s) = {}" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2440 |
by (metis (no_types) unbounded_outside connected_with_outside [OF assms] bounded_Un |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2441 |
inside_complement_unbounded_connected_empty unbounded_outside union_with_outside) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2442 |
|
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2443 |
lemma inside_in_components: |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2444 |
"inside s \<in> components (- s) \<longleftrightarrow> connected(inside s) \<and> inside s \<noteq> {}" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2445 |
apply (simp add: in_components_maximal) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2446 |
apply (auto intro: inside_same_component connected_componentI) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2447 |
apply (metis IntI empty_iff inside_no_overlap) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2448 |
done |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2449 |
|
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2450 |
text\<open>The proof is virtually the same as that above.\<close> |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2451 |
lemma outside_in_components: |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2452 |
"outside s \<in> components (- s) \<longleftrightarrow> connected(outside s) \<and> outside s \<noteq> {}" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2453 |
apply (simp add: in_components_maximal) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2454 |
apply (auto intro: outside_same_component connected_componentI) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2455 |
apply (metis IntI empty_iff outside_no_overlap) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2456 |
done |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2457 |
|
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2458 |
lemma bounded_unique_outside: |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2459 |
fixes s :: "'a :: euclidean_space set" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2460 |
shows "\<lbrakk>bounded s; DIM('a) \<ge> 2\<rbrakk> \<Longrightarrow> (c \<in> components (- s) \<and> ~bounded c \<longleftrightarrow> c = outside s)" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2461 |
apply (rule iffI) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2462 |
apply (metis cobounded_unique_unbounded_components connected_outside double_compl outside_bounded_nonempty outside_in_components unbounded_outside) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2463 |
by (simp add: connected_outside outside_bounded_nonempty outside_in_components unbounded_outside) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2464 |
|
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2465 |
section\<open> Homotopy of maps p,q : X=>Y with property P of all intermediate maps.\<close> |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2466 |
|
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2467 |
text\<open> We often just want to require that it fixes some subset, but to take in |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2468 |
the case of a loop homotopy, it's convenient to have a general property P.\<close> |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2469 |
|
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2470 |
definition homotopic_with :: |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2471 |
"[('a::topological_space \<Rightarrow> 'b::topological_space) \<Rightarrow> bool, 'a set, 'b set, 'a \<Rightarrow> 'b, 'a \<Rightarrow> 'b] \<Rightarrow> bool" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2472 |
where |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2473 |
"homotopic_with P X Y p q \<equiv> |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2474 |
(\<exists>h:: real \<times> 'a \<Rightarrow> 'b. |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2475 |
continuous_on ({0..1} \<times> X) h \<and> |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2476 |
h ` ({0..1} \<times> X) \<subseteq> Y \<and> |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2477 |
(\<forall>x. h(0, x) = p x) \<and> |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2478 |
(\<forall>x. h(1, x) = q x) \<and> |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2479 |
(\<forall>t \<in> {0..1}. P(\<lambda>x. h(t, x))))" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2480 |
|
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2481 |
|
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2482 |
text\<open> We often want to just localize the ending function equality or whatever.\<close> |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2483 |
proposition homotopic_with: |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2484 |
fixes X :: "'a::topological_space set" and Y :: "'b::topological_space set" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2485 |
assumes "\<And>h k. (\<And>x. x \<in> X \<Longrightarrow> h x = k x) \<Longrightarrow> (P h \<longleftrightarrow> P k)" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2486 |
shows "homotopic_with P X Y p q \<longleftrightarrow> |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2487 |
(\<exists>h :: real \<times> 'a \<Rightarrow> 'b. |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2488 |
continuous_on ({0..1} \<times> X) h \<and> |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2489 |
h ` ({0..1} \<times> X) \<subseteq> Y \<and> |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2490 |
(\<forall>x \<in> X. h(0,x) = p x) \<and> |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2491 |
(\<forall>x \<in> X. h(1,x) = q x) \<and> |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2492 |
(\<forall>t \<in> {0..1}. P(\<lambda>x. h(t, x))))" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2493 |
unfolding homotopic_with_def |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2494 |
apply (rule iffI, blast, clarify) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2495 |
apply (rule_tac x="\<lambda>(u,v). if v \<in> X then h(u,v) else if u = 0 then p v else q v" in exI) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2496 |
apply (auto simp:) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2497 |
apply (force elim: continuous_on_eq) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2498 |
apply (drule_tac x=t in bspec, force) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2499 |
apply (subst assms; simp) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2500 |
done |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2501 |
|
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2502 |
proposition homotopic_with_eq: |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2503 |
assumes h: "homotopic_with P X Y f g" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2504 |
and f': "\<And>x. x \<in> X \<Longrightarrow> f' x = f x" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2505 |
and g': "\<And>x. x \<in> X \<Longrightarrow> g' x = g x" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2506 |
and P: "(\<And>h k. (\<And>x. x \<in> X \<Longrightarrow> h x = k x) \<Longrightarrow> (P h \<longleftrightarrow> P k))" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2507 |
shows "homotopic_with P X Y f' g'" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2508 |
using h unfolding homotopic_with_def |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2509 |
apply safe |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2510 |
apply (rule_tac x="\<lambda>(u,v). if v \<in> X then h(u,v) else if u = 0 then f' v else g' v" in exI) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2511 |
apply (simp add: f' g', safe) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2512 |
apply (fastforce intro: continuous_on_eq) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2513 |
apply fastforce |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2514 |
apply (subst P; fastforce) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2515 |
done |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2516 |
|
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2517 |
proposition homotopic_with_equal: |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2518 |
assumes contf: "continuous_on X f" and fXY: "f ` X \<subseteq> Y" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2519 |
and gf: "\<And>x. x \<in> X \<Longrightarrow> g x = f x" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2520 |
and P: "P f" "P g" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2521 |
shows "homotopic_with P X Y f g" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2522 |
unfolding homotopic_with_def |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2523 |
apply (rule_tac x="\<lambda>(u,v). if u = 1 then g v else f v" in exI) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2524 |
using assms |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2525 |
apply (intro conjI) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2526 |
apply (rule continuous_on_eq [where f = "f o snd"]) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2527 |
apply (rule continuous_intros | force)+ |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2528 |
apply clarify |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2529 |
apply (case_tac "t=1"; force) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2530 |
done |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2531 |
|
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2532 |
|
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2533 |
lemma image_Pair_const: "(\<lambda>x. (x, c)) ` A = A \<times> {c}" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2534 |
by (auto simp:) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2535 |
|
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2536 |
lemma homotopic_constant_maps: |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2537 |
"homotopic_with (\<lambda>x. True) s t (\<lambda>x. a) (\<lambda>x. b) \<longleftrightarrow> s = {} \<or> path_component t a b" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2538 |
proof (cases "s = {} \<or> t = {}") |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2539 |
case True with continuous_on_const show ?thesis |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2540 |
by (auto simp: homotopic_with path_component_def) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2541 |
next |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2542 |
case False |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2543 |
then obtain c where "c \<in> s" by blast |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2544 |
show ?thesis |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2545 |
proof |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2546 |
assume "homotopic_with (\<lambda>x. True) s t (\<lambda>x. a) (\<lambda>x. b)" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2547 |
then obtain h :: "real \<times> 'a \<Rightarrow> 'b" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2548 |
where conth: "continuous_on ({0..1} \<times> s) h" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2549 |
and h: "h ` ({0..1} \<times> s) \<subseteq> t" "(\<forall>x\<in>s. h (0, x) = a)" "(\<forall>x\<in>s. h (1, x) = b)" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2550 |
by (auto simp: homotopic_with) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2551 |
have "continuous_on {0..1} (h \<circ> (\<lambda>t. (t, c)))" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2552 |
apply (rule continuous_intros conth | simp add: image_Pair_const)+ |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2553 |
apply (blast intro: \<open>c \<in> s\<close> continuous_on_subset [OF conth] ) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2554 |
done |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2555 |
with \<open>c \<in> s\<close> h show "s = {} \<or> path_component t a b" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2556 |
apply (simp_all add: homotopic_with path_component_def) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2557 |
apply (auto simp:) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2558 |
apply (drule_tac x="h o (\<lambda>t. (t, c))" in spec) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2559 |
apply (auto simp: pathstart_def pathfinish_def path_image_def path_def) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2560 |
done |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2561 |
next |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2562 |
assume "s = {} \<or> path_component t a b" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2563 |
with False show "homotopic_with (\<lambda>x. True) s t (\<lambda>x. a) (\<lambda>x. b)" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2564 |
apply (clarsimp simp: homotopic_with path_component_def pathstart_def pathfinish_def path_image_def path_def) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2565 |
apply (rule_tac x="g o fst" in exI) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2566 |
apply (rule conjI continuous_intros | force)+ |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2567 |
done |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2568 |
qed |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2569 |
qed |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2570 |
|
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2571 |
|
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2572 |
subsection\<open> Trivial properties.\<close> |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2573 |
|
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2574 |
lemma homotopic_with_imp_property: "homotopic_with P X Y f g \<Longrightarrow> P f \<and> P g" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2575 |
unfolding homotopic_with_def Ball_def |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2576 |
apply clarify |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2577 |
apply (frule_tac x=0 in spec) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2578 |
apply (drule_tac x=1 in spec) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2579 |
apply (auto simp:) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2580 |
done |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2581 |
|
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2582 |
lemma continuous_on_o_Pair: "\<lbrakk>continuous_on (T \<times> X) h; t \<in> T\<rbrakk> \<Longrightarrow> continuous_on X (h o Pair t)" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2583 |
by (fast intro: continuous_intros elim!: continuous_on_subset) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2584 |
|
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2585 |
lemma homotopic_with_imp_continuous: |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2586 |
assumes "homotopic_with P X Y f g" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2587 |
shows "continuous_on X f \<and> continuous_on X g" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2588 |
proof - |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2589 |
obtain h :: "real \<times> 'a \<Rightarrow> 'b" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2590 |
where conth: "continuous_on ({0..1} \<times> X) h" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2591 |
and h: "\<forall>x. h (0, x) = f x" "\<forall>x. h (1, x) = g x" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2592 |
using assms by (auto simp: homotopic_with_def) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2593 |
have *: "t \<in> {0..1} \<Longrightarrow> continuous_on X (h o (\<lambda>x. (t,x)))" for t |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2594 |
by (rule continuous_intros continuous_on_subset [OF conth] | force)+ |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2595 |
show ?thesis |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2596 |
using h *[of 0] *[of 1] by auto |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2597 |
qed |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2598 |
|
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2599 |
proposition homotopic_with_imp_subset1: |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2600 |
"homotopic_with P X Y f g \<Longrightarrow> f ` X \<subseteq> Y" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2601 |
by (simp add: homotopic_with_def image_subset_iff) (metis atLeastAtMost_iff order_refl zero_le_one) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2602 |
|
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2603 |
proposition homotopic_with_imp_subset2: |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2604 |
"homotopic_with P X Y f g \<Longrightarrow> g ` X \<subseteq> Y" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2605 |
by (simp add: homotopic_with_def image_subset_iff) (metis atLeastAtMost_iff order_refl zero_le_one) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2606 |
|
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2607 |
proposition homotopic_with_mono: |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2608 |
assumes hom: "homotopic_with P X Y f g" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2609 |
and Q: "\<And>h. \<lbrakk>continuous_on X h; image h X \<subseteq> Y \<and> P h\<rbrakk> \<Longrightarrow> Q h" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2610 |
shows "homotopic_with Q X Y f g" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2611 |
using hom |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2612 |
apply (simp add: homotopic_with_def) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2613 |
apply (erule ex_forward) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2614 |
apply (force simp: intro!: Q dest: continuous_on_o_Pair) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2615 |
done |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2616 |
|
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2617 |
proposition homotopic_with_subset_left: |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2618 |
"\<lbrakk>homotopic_with P X Y f g; Z \<subseteq> X\<rbrakk> \<Longrightarrow> homotopic_with P Z Y f g" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2619 |
apply (simp add: homotopic_with_def) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2620 |
apply (fast elim!: continuous_on_subset ex_forward) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2621 |
done |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2622 |
|
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2623 |
proposition homotopic_with_subset_right: |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2624 |
"\<lbrakk>homotopic_with P X Y f g; Y \<subseteq> Z\<rbrakk> \<Longrightarrow> homotopic_with P X Z f g" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2625 |
apply (simp add: homotopic_with_def) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2626 |
apply (fast elim!: continuous_on_subset ex_forward) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2627 |
done |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2628 |
|
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2629 |
proposition homotopic_with_compose_continuous_right: |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2630 |
"\<lbrakk>homotopic_with (\<lambda>f. p (f \<circ> h)) X Y f g; continuous_on W h; h ` W \<subseteq> X\<rbrakk> |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2631 |
\<Longrightarrow> homotopic_with p W Y (f o h) (g o h)" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2632 |
apply (clarsimp simp add: homotopic_with_def) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2633 |
apply (rename_tac k) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2634 |
apply (rule_tac x="k o (\<lambda>y. (fst y, h (snd y)))" in exI) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2635 |
apply (rule conjI continuous_intros continuous_on_compose [where f=snd and g=h, unfolded o_def] | simp)+ |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2636 |
apply (erule continuous_on_subset) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2637 |
apply (fastforce simp: o_def)+ |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2638 |
done |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2639 |
|
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2640 |
proposition homotopic_compose_continuous_right: |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2641 |
"\<lbrakk>homotopic_with (\<lambda>f. True) X Y f g; continuous_on W h; h ` W \<subseteq> X\<rbrakk> |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2642 |
\<Longrightarrow> homotopic_with (\<lambda>f. True) W Y (f o h) (g o h)" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2643 |
using homotopic_with_compose_continuous_right by fastforce |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2644 |
|
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2645 |
proposition homotopic_with_compose_continuous_left: |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2646 |
"\<lbrakk>homotopic_with (\<lambda>f. p (h \<circ> f)) X Y f g; continuous_on Y h; h ` Y \<subseteq> Z\<rbrakk> |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2647 |
\<Longrightarrow> homotopic_with p X Z (h o f) (h o g)" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2648 |
apply (clarsimp simp add: homotopic_with_def) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2649 |
apply (rename_tac k) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2650 |
apply (rule_tac x="h o k" in exI) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2651 |
apply (rule conjI continuous_intros continuous_on_compose [where f=snd and g=h, unfolded o_def] | simp)+ |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2652 |
apply (erule continuous_on_subset) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2653 |
apply (fastforce simp: o_def)+ |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2654 |
done |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2655 |
|
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2656 |
proposition homotopic_compose_continuous_left: |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2657 |
"homotopic_with (\<lambda>f. True) X Y f g \<and> |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2658 |
continuous_on Y h \<and> image h Y \<subseteq> Z |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2659 |
\<Longrightarrow> homotopic_with (\<lambda>f. True) X Z (h o f) (h o g)" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2660 |
using homotopic_with_compose_continuous_left by fastforce |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2661 |
|
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2662 |
proposition homotopic_with_Pair: |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2663 |
assumes hom: "homotopic_with p s t f g" "homotopic_with p' s' t' f' g'" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2664 |
and q: "\<And>f g. \<lbrakk>p f; p' g\<rbrakk> \<Longrightarrow> q(\<lambda>(x,y). (f x, g y))" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2665 |
shows "homotopic_with q (s \<times> s') (t \<times> t') |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2666 |
(\<lambda>(x,y). (f x, f' y)) (\<lambda>(x,y). (g x, g' y))" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2667 |
using hom |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2668 |
apply (clarsimp simp add: homotopic_with_def) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2669 |
apply (rename_tac k k') |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2670 |
apply (rule_tac x="\<lambda>z. ((k o (\<lambda>x. (fst x, fst (snd x)))) z, (k' o (\<lambda>x. (fst x, snd (snd x)))) z)" in exI) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2671 |
apply (rule conjI continuous_intros | erule continuous_on_subset | clarsimp)+ |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2672 |
apply (auto intro!: q [unfolded case_prod_unfold]) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2673 |
done |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2674 |
|
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2675 |
lemma homotopic_on_empty: "homotopic_with (\<lambda>x. True) {} t f g" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2676 |
by (metis continuous_on_def empty_iff homotopic_with_equal image_subset_iff) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2677 |
|
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2678 |
|
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2679 |
text\<open>Homotopy with P is an equivalence relation (on continuous functions mapping X into Y that satisfy P, |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2680 |
though this only affects reflexivity.\<close> |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2681 |
|
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2682 |
|
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2683 |
proposition homotopic_with_refl: |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2684 |
"homotopic_with P X Y f f \<longleftrightarrow> continuous_on X f \<and> image f X \<subseteq> Y \<and> P f" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2685 |
apply (rule iffI) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2686 |
using homotopic_with_imp_continuous homotopic_with_imp_property homotopic_with_imp_subset2 apply blast |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2687 |
apply (simp add: homotopic_with_def) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2688 |
apply (rule_tac x="f o snd" in exI) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2689 |
apply (rule conjI continuous_intros | force)+ |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2690 |
done |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2691 |
|
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2692 |
lemma homotopic_with_symD: |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2693 |
fixes X :: "'a::real_normed_vector set" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2694 |
assumes "homotopic_with P X Y f g" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2695 |
shows "homotopic_with P X Y g f" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2696 |
using assms |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2697 |
apply (clarsimp simp add: homotopic_with_def) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2698 |
apply (rename_tac h) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2699 |
apply (rule_tac x="h o (\<lambda>y. (1 - fst y, snd y))" in exI) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2700 |
apply (rule conjI continuous_intros | erule continuous_on_subset | force simp add: image_subset_iff)+ |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2701 |
done |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2702 |
|
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2703 |
proposition homotopic_with_sym: |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2704 |
fixes X :: "'a::real_normed_vector set" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2705 |
shows "homotopic_with P X Y f g \<longleftrightarrow> homotopic_with P X Y g f" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2706 |
using homotopic_with_symD by blast |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2707 |
|
61699
a81dc5c4d6a9
New theorems mostly from Peter Gammie
paulson <lp15@cam.ac.uk>
parents:
61694
diff
changeset
|
2708 |
lemma split_01: "{0..1::real} = {0..1/2} \<union> {1/2..1}" |
a81dc5c4d6a9
New theorems mostly from Peter Gammie
paulson <lp15@cam.ac.uk>
parents:
61694
diff
changeset
|
2709 |
by force |
a81dc5c4d6a9
New theorems mostly from Peter Gammie
paulson <lp15@cam.ac.uk>
parents:
61694
diff
changeset
|
2710 |
|
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2711 |
lemma split_01_prod: "{0..1::real} \<times> X = ({0..1/2} \<times> X) \<union> ({1/2..1} \<times> X)" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2712 |
by force |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2713 |
|
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2714 |
proposition homotopic_with_trans: |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2715 |
fixes X :: "'a::real_normed_vector set" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2716 |
assumes "homotopic_with P X Y f g" and "homotopic_with P X Y g h" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2717 |
shows "homotopic_with P X Y f h" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2718 |
proof - |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2719 |
have clo1: "closedin (subtopology euclidean ({0..1/2} \<times> X \<union> {1/2..1} \<times> X)) ({0..1/2::real} \<times> X)" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2720 |
apply (simp add: closedin_closed split_01_prod [symmetric]) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2721 |
apply (rule_tac x="{0..1/2} \<times> UNIV" in exI) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2722 |
apply (force simp add: closed_Times) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2723 |
done |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2724 |
have clo2: "closedin (subtopology euclidean ({0..1/2} \<times> X \<union> {1/2..1} \<times> X)) ({1/2..1::real} \<times> X)" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2725 |
apply (simp add: closedin_closed split_01_prod [symmetric]) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2726 |
apply (rule_tac x="{1/2..1} \<times> UNIV" in exI) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2727 |
apply (force simp add: closed_Times) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2728 |
done |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2729 |
{ fix k1 k2:: "real \<times> 'a \<Rightarrow> 'b" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2730 |
assume cont: "continuous_on ({0..1} \<times> X) k1" "continuous_on ({0..1} \<times> X) k2" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2731 |
and Y: "k1 ` ({0..1} \<times> X) \<subseteq> Y" "k2 ` ({0..1} \<times> X) \<subseteq> Y" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2732 |
and geq: "\<forall>x. k1 (1, x) = g x" "\<forall>x. k2 (0, x) = g x" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2733 |
and k12: "\<forall>x. k1 (0, x) = f x" "\<forall>x. k2 (1, x) = h x" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2734 |
and P: "\<forall>t\<in>{0..1}. P (\<lambda>x. k1 (t, x))" "\<forall>t\<in>{0..1}. P (\<lambda>x. k2 (t, x))" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2735 |
def k \<equiv> "\<lambda>y. if fst y \<le> 1 / 2 then (k1 o (\<lambda>x. (2 *\<^sub>R fst x, snd x))) y |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2736 |
else (k2 o (\<lambda>x. (2 *\<^sub>R fst x -1, snd x))) y" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2737 |
have keq: "k1 (2 * u, v) = k2 (2 * u - 1, v)" if "u = 1/2" for u v |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2738 |
by (simp add: geq that) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2739 |
have "continuous_on ({0..1} \<times> X) k" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2740 |
using cont |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2741 |
apply (simp add: split_01_prod k_def) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2742 |
apply (rule clo1 clo2 continuous_on_cases_local continuous_intros | erule continuous_on_subset | simp add: linear image_subset_iff)+ |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2743 |
apply (force simp add: keq) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2744 |
done |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2745 |
moreover have "k ` ({0..1} \<times> X) \<subseteq> Y" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2746 |
using Y by (force simp add: k_def) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2747 |
moreover have "\<forall>x. k (0, x) = f x" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2748 |
by (simp add: k_def k12) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2749 |
moreover have "(\<forall>x. k (1, x) = h x)" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2750 |
by (simp add: k_def k12) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2751 |
moreover have "\<forall>t\<in>{0..1}. P (\<lambda>x. k (t, x))" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2752 |
using P |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2753 |
apply (clarsimp simp add: k_def) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2754 |
apply (case_tac "t \<le> 1/2") |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2755 |
apply (auto simp:) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2756 |
done |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2757 |
ultimately have *: "\<exists>k :: real \<times> 'a \<Rightarrow> 'b. |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2758 |
continuous_on ({0..1} \<times> X) k \<and> k ` ({0..1} \<times> X) \<subseteq> Y \<and> |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2759 |
(\<forall>x. k (0, x) = f x) \<and> (\<forall>x. k (1, x) = h x) \<and> (\<forall>t\<in>{0..1}. P (\<lambda>x. k (t, x)))" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2760 |
by blast |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2761 |
} note * = this |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2762 |
show ?thesis |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2763 |
using assms by (auto intro: * simp add: homotopic_with_def) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2764 |
qed |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2765 |
|
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2766 |
proposition homotopic_compose: |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2767 |
fixes s :: "'a::real_normed_vector set" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2768 |
shows "\<lbrakk>homotopic_with (\<lambda>x. True) s t f f'; homotopic_with (\<lambda>x. True) t u g g'\<rbrakk> |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2769 |
\<Longrightarrow> homotopic_with (\<lambda>x. True) s u (g o f) (g' o f')" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2770 |
apply (rule homotopic_with_trans [where g = "g o f'"]) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2771 |
apply (metis homotopic_compose_continuous_left homotopic_with_imp_continuous homotopic_with_imp_subset1) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2772 |
by (metis homotopic_compose_continuous_right homotopic_with_imp_continuous homotopic_with_imp_subset2) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2773 |
|
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2774 |
|
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2775 |
subsection\<open>Homotopy of paths, maintaining the same endpoints.\<close> |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2776 |
|
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2777 |
|
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2778 |
definition homotopic_paths :: "['a set, real \<Rightarrow> 'a, real \<Rightarrow> 'a::topological_space] \<Rightarrow> bool" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2779 |
where |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2780 |
"homotopic_paths s p q \<equiv> |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2781 |
homotopic_with (\<lambda>r. pathstart r = pathstart p \<and> pathfinish r = pathfinish p) {0..1} s p q" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2782 |
|
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2783 |
lemma homotopic_paths: |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2784 |
"homotopic_paths s p q \<longleftrightarrow> |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2785 |
(\<exists>h. continuous_on ({0..1} \<times> {0..1}) h \<and> |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2786 |
h ` ({0..1} \<times> {0..1}) \<subseteq> s \<and> |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2787 |
(\<forall>x \<in> {0..1}. h(0,x) = p x) \<and> |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2788 |
(\<forall>x \<in> {0..1}. h(1,x) = q x) \<and> |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2789 |
(\<forall>t \<in> {0..1::real}. pathstart(h o Pair t) = pathstart p \<and> |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2790 |
pathfinish(h o Pair t) = pathfinish p))" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2791 |
by (auto simp: homotopic_paths_def homotopic_with pathstart_def pathfinish_def) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2792 |
|
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2793 |
proposition homotopic_paths_imp_pathstart: |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2794 |
"homotopic_paths s p q \<Longrightarrow> pathstart p = pathstart q" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2795 |
by (metis (mono_tags, lifting) homotopic_paths_def homotopic_with_imp_property) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2796 |
|
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2797 |
proposition homotopic_paths_imp_pathfinish: |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2798 |
"homotopic_paths s p q \<Longrightarrow> pathfinish p = pathfinish q" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2799 |
by (metis (mono_tags, lifting) homotopic_paths_def homotopic_with_imp_property) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2800 |
|
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2801 |
lemma homotopic_paths_imp_path: |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2802 |
"homotopic_paths s p q \<Longrightarrow> path p \<and> path q" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2803 |
using homotopic_paths_def homotopic_with_imp_continuous path_def by blast |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2804 |
|
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2805 |
lemma homotopic_paths_imp_subset: |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2806 |
"homotopic_paths s p q \<Longrightarrow> path_image p \<subseteq> s \<and> path_image q \<subseteq> s" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2807 |
by (simp add: homotopic_paths_def homotopic_with_imp_subset1 homotopic_with_imp_subset2 path_image_def) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2808 |
|
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2809 |
proposition homotopic_paths_refl [simp]: "homotopic_paths s p p \<longleftrightarrow> path p \<and> path_image p \<subseteq> s" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2810 |
by (simp add: homotopic_paths_def homotopic_with_refl path_def path_image_def) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2811 |
|
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2812 |
proposition homotopic_paths_sym: "homotopic_paths s p q \<longleftrightarrow> homotopic_paths s q p" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2813 |
by (metis (mono_tags) homotopic_paths_def homotopic_paths_imp_pathfinish homotopic_paths_imp_pathstart homotopic_with_symD)+ |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2814 |
|
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2815 |
proposition homotopic_paths_trans: |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2816 |
"\<lbrakk>homotopic_paths s p q; homotopic_paths s q r\<rbrakk> \<Longrightarrow> homotopic_paths s p r" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2817 |
apply (simp add: homotopic_paths_def) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2818 |
apply (rule homotopic_with_trans, assumption) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2819 |
by (metis (mono_tags, lifting) homotopic_with_imp_property homotopic_with_mono) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2820 |
|
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2821 |
proposition homotopic_paths_eq: |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2822 |
"\<lbrakk>path p; path_image p \<subseteq> s; \<And>t. t \<in> {0..1} \<Longrightarrow> p t = q t\<rbrakk> \<Longrightarrow> homotopic_paths s p q" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2823 |
apply (simp add: homotopic_paths_def) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2824 |
apply (rule homotopic_with_eq) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2825 |
apply (auto simp: path_def homotopic_with_refl pathstart_def pathfinish_def path_image_def elim: continuous_on_eq) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2826 |
done |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2827 |
|
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2828 |
proposition homotopic_paths_reparametrize: |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2829 |
assumes "path p" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2830 |
and pips: "path_image p \<subseteq> s" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2831 |
and contf: "continuous_on {0..1} f" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2832 |
and f01:"f ` {0..1} \<subseteq> {0..1}" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2833 |
and [simp]: "f(0) = 0" "f(1) = 1" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2834 |
and q: "\<And>t. t \<in> {0..1} \<Longrightarrow> q(t) = p(f t)" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2835 |
shows "homotopic_paths s p q" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2836 |
proof - |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2837 |
have contp: "continuous_on {0..1} p" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2838 |
by (metis \<open>path p\<close> path_def) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2839 |
then have "continuous_on {0..1} (p o f)" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2840 |
using contf continuous_on_compose continuous_on_subset f01 by blast |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2841 |
then have "path q" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2842 |
by (simp add: path_def) (metis q continuous_on_cong) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2843 |
have piqs: "path_image q \<subseteq> s" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2844 |
by (metis (no_types, hide_lams) pips f01 image_subset_iff path_image_def q) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2845 |
have fb0: "\<And>a b. \<lbrakk>0 \<le> a; a \<le> 1; 0 \<le> b; b \<le> 1\<rbrakk> \<Longrightarrow> 0 \<le> (1 - a) * f b + a * b" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2846 |
using f01 by force |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2847 |
have fb1: "\<lbrakk>0 \<le> a; a \<le> 1; 0 \<le> b; b \<le> 1\<rbrakk> \<Longrightarrow> (1 - a) * f b + a * b \<le> 1" for a b |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2848 |
using f01 [THEN subsetD, of "f b"] by (simp add: convex_bound_le) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2849 |
have "homotopic_paths s q p" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2850 |
proof (rule homotopic_paths_trans) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2851 |
show "homotopic_paths s q (p \<circ> f)" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2852 |
using q by (force intro: homotopic_paths_eq [OF \<open>path q\<close> piqs]) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2853 |
next |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2854 |
show "homotopic_paths s (p \<circ> f) p" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2855 |
apply (simp add: homotopic_paths_def homotopic_with_def) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2856 |
apply (rule_tac x="p o (\<lambda>y. (1 - (fst y)) *\<^sub>R ((f o snd) y) + (fst y) *\<^sub>R snd y)" in exI) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2857 |
apply (rule conjI contf continuous_intros continuous_on_subset [OF contp] | simp)+ |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2858 |
using pips [unfolded path_image_def] |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2859 |
apply (auto simp: fb0 fb1 pathstart_def pathfinish_def) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2860 |
done |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2861 |
qed |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2862 |
then show ?thesis |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2863 |
by (simp add: homotopic_paths_sym) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2864 |
qed |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2865 |
|
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2866 |
lemma homotopic_paths_subset: "\<lbrakk>homotopic_paths s p q; s \<subseteq> t\<rbrakk> \<Longrightarrow> homotopic_paths t p q" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2867 |
using homotopic_paths_def homotopic_with_subset_right by blast |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2868 |
|
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2869 |
|
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2870 |
text\<open> A slightly ad-hoc but useful lemma in constructing homotopies.\<close> |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2871 |
lemma homotopic_join_lemma: |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2872 |
fixes q :: "[real,real] \<Rightarrow> 'a::topological_space" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2873 |
assumes p: "continuous_on ({0..1} \<times> {0..1}) (\<lambda>y. p (fst y) (snd y))" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2874 |
and q: "continuous_on ({0..1} \<times> {0..1}) (\<lambda>y. q (fst y) (snd y))" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2875 |
and pf: "\<And>t. t \<in> {0..1} \<Longrightarrow> pathfinish(p t) = pathstart(q t)" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2876 |
shows "continuous_on ({0..1} \<times> {0..1}) (\<lambda>y. (p(fst y) +++ q(fst y)) (snd y))" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2877 |
proof - |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2878 |
have 1: "(\<lambda>y. p (fst y) (2 * snd y)) = (\<lambda>y. p (fst y) (snd y)) o (\<lambda>y. (fst y, 2 * snd y))" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2879 |
by (rule ext) (simp ) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2880 |
have 2: "(\<lambda>y. q (fst y) (2 * snd y - 1)) = (\<lambda>y. q (fst y) (snd y)) o (\<lambda>y. (fst y, 2 * snd y - 1))" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2881 |
by (rule ext) (simp ) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2882 |
show ?thesis |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2883 |
apply (simp add: joinpaths_def) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2884 |
apply (rule continuous_on_cases_le) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2885 |
apply (simp_all only: 1 2) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2886 |
apply (rule continuous_intros continuous_on_subset [OF p] continuous_on_subset [OF q] | force)+ |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2887 |
using pf |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2888 |
apply (auto simp: mult.commute pathstart_def pathfinish_def) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2889 |
done |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2890 |
qed |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2891 |
|
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2892 |
text\<open> Congruence properties of homotopy w.r.t. path-combining operations.\<close> |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2893 |
|
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2894 |
lemma homotopic_paths_reversepath_D: |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2895 |
assumes "homotopic_paths s p q" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2896 |
shows "homotopic_paths s (reversepath p) (reversepath q)" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2897 |
using assms |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2898 |
apply (simp add: homotopic_paths_def homotopic_with_def, clarify) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2899 |
apply (rule_tac x="h o (\<lambda>x. (fst x, 1 - snd x))" in exI) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2900 |
apply (rule conjI continuous_intros)+ |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2901 |
apply (auto simp: reversepath_def pathstart_def pathfinish_def elim!: continuous_on_subset) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2902 |
done |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2903 |
|
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2904 |
proposition homotopic_paths_reversepath: |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2905 |
"homotopic_paths s (reversepath p) (reversepath q) \<longleftrightarrow> homotopic_paths s p q" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2906 |
using homotopic_paths_reversepath_D by force |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2907 |
|
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2908 |
|
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2909 |
proposition homotopic_paths_join: |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2910 |
"\<lbrakk>homotopic_paths s p p'; homotopic_paths s q q'; pathfinish p = pathstart q\<rbrakk> \<Longrightarrow> homotopic_paths s (p +++ q) (p' +++ q')" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2911 |
apply (simp add: homotopic_paths_def homotopic_with_def, clarify) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2912 |
apply (rename_tac k1 k2) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2913 |
apply (rule_tac x="(\<lambda>y. ((k1 o Pair (fst y)) +++ (k2 o Pair (fst y))) (snd y))" in exI) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2914 |
apply (rule conjI continuous_intros homotopic_join_lemma)+ |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2915 |
apply (auto simp: joinpaths_def pathstart_def pathfinish_def path_image_def) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2916 |
done |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2917 |
|
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2918 |
proposition homotopic_paths_continuous_image: |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2919 |
"\<lbrakk>homotopic_paths s f g; continuous_on s h; h ` s \<subseteq> t\<rbrakk> \<Longrightarrow> homotopic_paths t (h o f) (h o g)" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2920 |
unfolding homotopic_paths_def |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2921 |
apply (rule homotopic_with_compose_continuous_left [of _ _ _ s]) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2922 |
apply (auto simp: pathstart_def pathfinish_def elim!: homotopic_with_mono) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2923 |
done |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
2924 |
|
61711
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
2925 |
subsection\<open>Group properties for homotopy of paths\<close> |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
2926 |
|
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
2927 |
text\<open>So taking equivalence classes under homotopy would give the fundamental group\<close> |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
2928 |
|
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
2929 |
proposition homotopic_paths_rid: |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
2930 |
"\<lbrakk>path p; path_image p \<subseteq> s\<rbrakk> \<Longrightarrow> homotopic_paths s (p +++ linepath (pathfinish p) (pathfinish p)) p" |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
2931 |
apply (subst homotopic_paths_sym) |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
2932 |
apply (rule homotopic_paths_reparametrize [where f = "\<lambda>t. if t \<le> 1 / 2 then 2 *\<^sub>R t else 1"]) |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
2933 |
apply (simp_all del: le_divide_eq_numeral1) |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
2934 |
apply (subst split_01) |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
2935 |
apply (rule continuous_on_cases continuous_intros | force simp: pathfinish_def joinpaths_def)+ |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
2936 |
done |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
2937 |
|
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
2938 |
proposition homotopic_paths_lid: |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
2939 |
"\<lbrakk>path p; path_image p \<subseteq> s\<rbrakk> \<Longrightarrow> homotopic_paths s (linepath (pathstart p) (pathstart p) +++ p) p" |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
2940 |
using homotopic_paths_rid [of "reversepath p" s] |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
2941 |
by (metis homotopic_paths_reversepath path_image_reversepath path_reversepath pathfinish_linepath |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
2942 |
pathfinish_reversepath reversepath_joinpaths reversepath_linepath) |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
2943 |
|
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
2944 |
proposition homotopic_paths_assoc: |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
2945 |
"\<lbrakk>path p; path_image p \<subseteq> s; path q; path_image q \<subseteq> s; path r; path_image r \<subseteq> s; pathfinish p = pathstart q; |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
2946 |
pathfinish q = pathstart r\<rbrakk> |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
2947 |
\<Longrightarrow> homotopic_paths s (p +++ (q +++ r)) ((p +++ q) +++ r)" |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
2948 |
apply (subst homotopic_paths_sym) |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
2949 |
apply (rule homotopic_paths_reparametrize |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
2950 |
[where f = "\<lambda>t. if t \<le> 1 / 2 then inverse 2 *\<^sub>R t |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
2951 |
else if t \<le> 3 / 4 then t - (1 / 4) |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
2952 |
else 2 *\<^sub>R t - 1"]) |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
2953 |
apply (simp_all del: le_divide_eq_numeral1) |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
2954 |
apply (simp add: subset_path_image_join) |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
2955 |
apply (rule continuous_on_cases_1 continuous_intros)+ |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
2956 |
apply (auto simp: joinpaths_def) |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
2957 |
done |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
2958 |
|
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
2959 |
proposition homotopic_paths_rinv: |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
2960 |
assumes "path p" "path_image p \<subseteq> s" |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
2961 |
shows "homotopic_paths s (p +++ reversepath p) (linepath (pathstart p) (pathstart p))" |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
2962 |
proof - |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
2963 |
have "continuous_on ({0..1} \<times> {0..1}) (\<lambda>x. (subpath 0 (fst x) p +++ reversepath (subpath 0 (fst x) p)) (snd x))" |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
2964 |
using assms |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
2965 |
apply (simp add: joinpaths_def subpath_def reversepath_def path_def del: le_divide_eq_numeral1) |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
2966 |
apply (rule continuous_on_cases_le) |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
2967 |
apply (rule_tac [2] continuous_on_compose [of _ _ p, unfolded o_def]) |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
2968 |
apply (rule continuous_on_compose [of _ _ p, unfolded o_def]) |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
2969 |
apply (auto intro!: continuous_intros simp del: eq_divide_eq_numeral1) |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
2970 |
apply (force elim!: continuous_on_subset simp add: mult_le_one)+ |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
2971 |
done |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
2972 |
then show ?thesis |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
2973 |
using assms |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
2974 |
apply (subst homotopic_paths_sym) |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
2975 |
apply (simp add: homotopic_paths_def homotopic_with_def) |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
2976 |
apply (rule_tac x="(\<lambda>y. (subpath 0 (fst y) p +++ reversepath(subpath 0 (fst y) p)) (snd y))" in exI) |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
2977 |
apply (simp add: path_defs joinpaths_def subpath_def reversepath_def) |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
2978 |
apply (force simp: mult_le_one) |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
2979 |
done |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
2980 |
qed |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
2981 |
|
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
2982 |
proposition homotopic_paths_linv: |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
2983 |
assumes "path p" "path_image p \<subseteq> s" |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
2984 |
shows "homotopic_paths s (reversepath p +++ p) (linepath (pathfinish p) (pathfinish p))" |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
2985 |
using homotopic_paths_rinv [of "reversepath p" s] assms by simp |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
2986 |
|
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
2987 |
|
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
2988 |
subsection\<open> Homotopy of loops without requiring preservation of endpoints.\<close> |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
2989 |
|
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
2990 |
definition homotopic_loops :: "'a::topological_space set \<Rightarrow> (real \<Rightarrow> 'a) \<Rightarrow> (real \<Rightarrow> 'a) \<Rightarrow> bool" where |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
2991 |
"homotopic_loops s p q \<equiv> |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
2992 |
homotopic_with (\<lambda>r. pathfinish r = pathstart r) {0..1} s p q" |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
2993 |
|
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
2994 |
lemma homotopic_loops: |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
2995 |
"homotopic_loops s p q \<longleftrightarrow> |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
2996 |
(\<exists>h. continuous_on ({0..1::real} \<times> {0..1}) h \<and> |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
2997 |
image h ({0..1} \<times> {0..1}) \<subseteq> s \<and> |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
2998 |
(\<forall>x \<in> {0..1}. h(0,x) = p x) \<and> |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
2999 |
(\<forall>x \<in> {0..1}. h(1,x) = q x) \<and> |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3000 |
(\<forall>t \<in> {0..1}. pathfinish(h o Pair t) = pathstart(h o Pair t)))" |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3001 |
by (simp add: homotopic_loops_def pathstart_def pathfinish_def homotopic_with) |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3002 |
|
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3003 |
proposition homotopic_loops_imp_loop: |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3004 |
"homotopic_loops s p q \<Longrightarrow> pathfinish p = pathstart p \<and> pathfinish q = pathstart q" |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3005 |
using homotopic_with_imp_property homotopic_loops_def by blast |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3006 |
|
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3007 |
proposition homotopic_loops_imp_path: |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3008 |
"homotopic_loops s p q \<Longrightarrow> path p \<and> path q" |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3009 |
unfolding homotopic_loops_def path_def |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3010 |
using homotopic_with_imp_continuous by blast |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3011 |
|
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3012 |
proposition homotopic_loops_imp_subset1: |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3013 |
"homotopic_loops s p q \<Longrightarrow> path_image p \<subseteq> s" |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3014 |
unfolding homotopic_loops_def path_image_def |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3015 |
using homotopic_with_imp_subset1 by blast |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3016 |
|
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3017 |
proposition homotopic_loops_imp_subset2: |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3018 |
"homotopic_loops s p q \<Longrightarrow> path_image q \<subseteq> s" |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3019 |
unfolding homotopic_loops_def path_image_def |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3020 |
using homotopic_with_imp_subset2 by blast |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3021 |
|
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3022 |
proposition homotopic_loops_refl: |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3023 |
"homotopic_loops s p p \<longleftrightarrow> |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3024 |
path p \<and> path_image p \<subseteq> s \<and> pathfinish p = pathstart p" |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3025 |
by (simp add: homotopic_loops_def homotopic_with_refl path_image_def path_def) |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3026 |
|
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3027 |
proposition homotopic_loops_sym: |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3028 |
"homotopic_loops s p q \<longleftrightarrow> homotopic_loops s q p" |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3029 |
by (simp add: homotopic_loops_def homotopic_with_sym) |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3030 |
|
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3031 |
proposition homotopic_loops_trans: |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3032 |
"\<lbrakk>homotopic_loops s p q; homotopic_loops s q r\<rbrakk> \<Longrightarrow> homotopic_loops s p r" |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3033 |
unfolding homotopic_loops_def by (blast intro: homotopic_with_trans) |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3034 |
|
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3035 |
proposition homotopic_loops_subset: |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3036 |
"\<lbrakk>homotopic_loops s p q; s \<subseteq> t\<rbrakk> \<Longrightarrow> homotopic_loops t p q" |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3037 |
by (simp add: homotopic_loops_def homotopic_with_subset_right) |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3038 |
|
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3039 |
proposition homotopic_loops_eq: |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3040 |
"\<lbrakk>path p; path_image p \<subseteq> s; pathfinish p = pathstart p; \<And>t. t \<in> {0..1} \<Longrightarrow> p(t) = q(t)\<rbrakk> |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3041 |
\<Longrightarrow> homotopic_loops s p q" |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3042 |
unfolding homotopic_loops_def |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3043 |
apply (rule homotopic_with_eq) |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3044 |
apply (rule homotopic_with_refl [where f = p, THEN iffD2]) |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3045 |
apply (simp_all add: path_image_def path_def pathstart_def pathfinish_def) |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3046 |
done |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3047 |
|
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3048 |
proposition homotopic_loops_continuous_image: |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3049 |
"\<lbrakk>homotopic_loops s f g; continuous_on s h; h ` s \<subseteq> t\<rbrakk> \<Longrightarrow> homotopic_loops t (h \<circ> f) (h \<circ> g)" |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3050 |
unfolding homotopic_loops_def |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3051 |
apply (rule homotopic_with_compose_continuous_left) |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3052 |
apply (erule homotopic_with_mono) |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3053 |
by (simp add: pathfinish_def pathstart_def) |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3054 |
|
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3055 |
|
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3056 |
subsection\<open>Relations between the two variants of homotopy\<close> |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3057 |
|
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3058 |
proposition homotopic_paths_imp_homotopic_loops: |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3059 |
"\<lbrakk>homotopic_paths s p q; pathfinish p = pathstart p; pathfinish q = pathstart p\<rbrakk> \<Longrightarrow> homotopic_loops s p q" |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3060 |
by (auto simp: homotopic_paths_def homotopic_loops_def intro: homotopic_with_mono) |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3061 |
|
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3062 |
proposition homotopic_loops_imp_homotopic_paths_null: |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3063 |
assumes "homotopic_loops s p (linepath a a)" |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3064 |
shows "homotopic_paths s p (linepath (pathstart p) (pathstart p))" |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3065 |
proof - |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3066 |
have "path p" by (metis assms homotopic_loops_imp_path) |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3067 |
have ploop: "pathfinish p = pathstart p" by (metis assms homotopic_loops_imp_loop) |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3068 |
have pip: "path_image p \<subseteq> s" by (metis assms homotopic_loops_imp_subset1) |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3069 |
obtain h where conth: "continuous_on ({0..1::real} \<times> {0..1}) h" |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3070 |
and hs: "h ` ({0..1} \<times> {0..1}) \<subseteq> s" |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3071 |
and [simp]: "\<And>x. x \<in> {0..1} \<Longrightarrow> h(0,x) = p x" |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3072 |
and [simp]: "\<And>x. x \<in> {0..1} \<Longrightarrow> h(1,x) = a" |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3073 |
and ends: "\<And>t. t \<in> {0..1} \<Longrightarrow> pathfinish (h \<circ> Pair t) = pathstart (h \<circ> Pair t)" |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3074 |
using assms by (auto simp: homotopic_loops homotopic_with) |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3075 |
have conth0: "path (\<lambda>u. h (u, 0))" |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3076 |
unfolding path_def |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3077 |
apply (rule continuous_on_compose [of _ _ h, unfolded o_def]) |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3078 |
apply (force intro: continuous_intros continuous_on_subset [OF conth])+ |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3079 |
done |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3080 |
have pih0: "path_image (\<lambda>u. h (u, 0)) \<subseteq> s" |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3081 |
using hs by (force simp: path_image_def) |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3082 |
have c1: "continuous_on ({0..1} \<times> {0..1}) (\<lambda>x. h (fst x * snd x, 0))" |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3083 |
apply (rule continuous_on_compose [of _ _ h, unfolded o_def]) |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3084 |
apply (force simp: mult_le_one intro: continuous_intros continuous_on_subset [OF conth])+ |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3085 |
done |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3086 |
have c2: "continuous_on ({0..1} \<times> {0..1}) (\<lambda>x. h (fst x - fst x * snd x, 0))" |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3087 |
apply (rule continuous_on_compose [of _ _ h, unfolded o_def]) |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3088 |
apply (force simp: mult_left_le mult_le_one intro: continuous_intros continuous_on_subset [OF conth])+ |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3089 |
apply (rule continuous_on_subset [OF conth]) |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3090 |
apply (auto simp: algebra_simps add_increasing2 mult_left_le) |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3091 |
done |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3092 |
have [simp]: "\<And>t. \<lbrakk>0 \<le> t \<and> t \<le> 1\<rbrakk> \<Longrightarrow> h (t, 1) = h (t, 0)" |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3093 |
using ends by (simp add: pathfinish_def pathstart_def) |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3094 |
have adhoc_le: "c * 4 \<le> 1 + c * (d * 4)" if "\<not> d * 4 \<le> 3" "0 \<le> c" "c \<le> 1" for c d::real |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3095 |
proof - |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3096 |
have "c * 3 \<le> c * (d * 4)" using that less_eq_real_def by auto |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3097 |
with \<open>c \<le> 1\<close> show ?thesis by fastforce |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3098 |
qed |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3099 |
have *: "\<And>p x. (path p \<and> path(reversepath p)) \<and> |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3100 |
(path_image p \<subseteq> s \<and> path_image(reversepath p) \<subseteq> s) \<and> |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3101 |
(pathfinish p = pathstart(linepath a a +++ reversepath p) \<and> |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3102 |
pathstart(reversepath p) = a) \<and> pathstart p = x |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3103 |
\<Longrightarrow> homotopic_paths s (p +++ linepath a a +++ reversepath p) (linepath x x)" |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3104 |
by (metis homotopic_paths_lid homotopic_paths_join |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3105 |
homotopic_paths_trans homotopic_paths_sym homotopic_paths_rinv) |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3106 |
have 1: "homotopic_paths s p (p +++ linepath (pathfinish p) (pathfinish p))" |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3107 |
using \<open>path p\<close> homotopic_paths_rid homotopic_paths_sym pip by blast |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3108 |
moreover have "homotopic_paths s (p +++ linepath (pathfinish p) (pathfinish p)) |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3109 |
(linepath (pathstart p) (pathstart p) +++ p +++ linepath (pathfinish p) (pathfinish p))" |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3110 |
apply (subst homotopic_paths_sym) |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3111 |
using homotopic_paths_lid [of "p +++ linepath (pathfinish p) (pathfinish p)" s] |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3112 |
by (metis 1 homotopic_paths_imp_path homotopic_paths_imp_pathstart homotopic_paths_imp_subset) |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3113 |
moreover have "homotopic_paths s (linepath (pathstart p) (pathstart p) +++ p +++ linepath (pathfinish p) (pathfinish p)) |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3114 |
((\<lambda>u. h (u, 0)) +++ linepath a a +++ reversepath (\<lambda>u. h (u, 0)))" |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3115 |
apply (simp add: homotopic_paths_def homotopic_with_def) |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3116 |
apply (rule_tac x="\<lambda>y. (subpath 0 (fst y) (\<lambda>u. h (u, 0)) +++ (\<lambda>u. h (Pair (fst y) u)) +++ subpath (fst y) 0 (\<lambda>u. h (u, 0))) (snd y)" in exI) |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3117 |
apply (simp add: subpath_reversepath) |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3118 |
apply (intro conjI homotopic_join_lemma) |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3119 |
using ploop |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3120 |
apply (simp_all add: path_defs joinpaths_def o_def subpath_def conth c1 c2) |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3121 |
apply (force simp: algebra_simps mult_le_one mult_left_le intro: hs [THEN subsetD] adhoc_le) |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3122 |
done |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3123 |
moreover have "homotopic_paths s ((\<lambda>u. h (u, 0)) +++ linepath a a +++ reversepath (\<lambda>u. h (u, 0))) |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3124 |
(linepath (pathstart p) (pathstart p))" |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3125 |
apply (rule *) |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3126 |
apply (simp add: pih0 pathstart_def pathfinish_def conth0) |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3127 |
apply (simp add: reversepath_def joinpaths_def) |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3128 |
done |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3129 |
ultimately show ?thesis |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3130 |
by (blast intro: homotopic_paths_trans) |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3131 |
qed |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3132 |
|
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3133 |
proposition homotopic_loops_conjugate: |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3134 |
fixes s :: "'a::real_normed_vector set" |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3135 |
assumes "path p" "path q" and pip: "path_image p \<subseteq> s" and piq: "path_image q \<subseteq> s" |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3136 |
and papp: "pathfinish p = pathstart q" and qloop: "pathfinish q = pathstart q" |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3137 |
shows "homotopic_loops s (p +++ q +++ reversepath p) q" |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3138 |
proof - |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3139 |
have contp: "continuous_on {0..1} p" using \<open>path p\<close> [unfolded path_def] by blast |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3140 |
have contq: "continuous_on {0..1} q" using \<open>path q\<close> [unfolded path_def] by blast |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3141 |
have c1: "continuous_on ({0..1} \<times> {0..1}) (\<lambda>x. p ((1 - fst x) * snd x + fst x))" |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3142 |
apply (rule continuous_on_compose [of _ _ p, unfolded o_def]) |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3143 |
apply (force simp: mult_le_one intro!: continuous_intros) |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3144 |
apply (rule continuous_on_subset [OF contp]) |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3145 |
apply (auto simp: algebra_simps add_increasing2 mult_right_le_one_le sum_le_prod1) |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3146 |
done |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3147 |
have c2: "continuous_on ({0..1} \<times> {0..1}) (\<lambda>x. p ((fst x - 1) * snd x + 1))" |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3148 |
apply (rule continuous_on_compose [of _ _ p, unfolded o_def]) |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3149 |
apply (force simp: mult_le_one intro!: continuous_intros) |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3150 |
apply (rule continuous_on_subset [OF contp]) |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3151 |
apply (auto simp: algebra_simps add_increasing2 mult_left_le_one_le) |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3152 |
done |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3153 |
have ps1: "\<And>a b. \<lbrakk>b * 2 \<le> 1; 0 \<le> b; 0 \<le> a; a \<le> 1\<rbrakk> \<Longrightarrow> p ((1 - a) * (2 * b) + a) \<in> s" |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3154 |
using sum_le_prod1 |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3155 |
by (force simp: algebra_simps add_increasing2 mult_left_le intro: pip [unfolded path_image_def, THEN subsetD]) |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3156 |
have ps2: "\<And>a b. \<lbrakk>\<not> 4 * b \<le> 3; b \<le> 1; 0 \<le> a; a \<le> 1\<rbrakk> \<Longrightarrow> p ((a - 1) * (4 * b - 3) + 1) \<in> s" |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3157 |
apply (rule pip [unfolded path_image_def, THEN subsetD]) |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3158 |
apply (rule image_eqI, blast) |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3159 |
apply (simp add: algebra_simps) |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3160 |
by (metis add_mono_thms_linordered_semiring(1) affine_ineq linear mult.commute mult.left_neutral mult_right_mono not_le |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3161 |
add.commute zero_le_numeral) |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3162 |
have qs: "\<And>a b. \<lbrakk>4 * b \<le> 3; \<not> b * 2 \<le> 1\<rbrakk> \<Longrightarrow> q (4 * b - 2) \<in> s" |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3163 |
using path_image_def piq by fastforce |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3164 |
have "homotopic_loops s (p +++ q +++ reversepath p) |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3165 |
(linepath (pathstart q) (pathstart q) +++ q +++ linepath (pathstart q) (pathstart q))" |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3166 |
apply (simp add: homotopic_loops_def homotopic_with_def) |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3167 |
apply (rule_tac x="(\<lambda>y. (subpath (fst y) 1 p +++ q +++ subpath 1 (fst y) p) (snd y))" in exI) |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3168 |
apply (simp add: subpath_refl subpath_reversepath) |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3169 |
apply (intro conjI homotopic_join_lemma) |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3170 |
using papp qloop |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3171 |
apply (simp_all add: path_defs joinpaths_def o_def subpath_def c1 c2) |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3172 |
apply (force simp: contq intro: continuous_on_compose [of _ _ q, unfolded o_def] continuous_on_id continuous_on_snd) |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3173 |
apply (auto simp: ps1 ps2 qs) |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3174 |
done |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3175 |
moreover have "homotopic_loops s (linepath (pathstart q) (pathstart q) +++ q +++ linepath (pathstart q) (pathstart q)) q" |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3176 |
proof - |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3177 |
have "homotopic_paths s (linepath (pathfinish q) (pathfinish q) +++ q) q" |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3178 |
using \<open>path q\<close> homotopic_paths_lid qloop piq by auto |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3179 |
hence 1: "\<And>f. homotopic_paths s f q \<or> \<not> homotopic_paths s f (linepath (pathfinish q) (pathfinish q) +++ q)" |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3180 |
using homotopic_paths_trans by blast |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3181 |
hence "homotopic_paths s (linepath (pathfinish q) (pathfinish q) +++ q +++ linepath (pathfinish q) (pathfinish q)) q" |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3182 |
proof - |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3183 |
have "homotopic_paths s (q +++ linepath (pathfinish q) (pathfinish q)) q" |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3184 |
by (simp add: \<open>path q\<close> homotopic_paths_rid piq) |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3185 |
thus ?thesis |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3186 |
by (metis (no_types) 1 \<open>path q\<close> homotopic_paths_join homotopic_paths_rinv homotopic_paths_sym |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3187 |
homotopic_paths_trans qloop pathfinish_linepath piq) |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3188 |
qed |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3189 |
thus ?thesis |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3190 |
by (metis (no_types) qloop homotopic_loops_sym homotopic_paths_imp_homotopic_loops homotopic_paths_imp_pathfinish homotopic_paths_sym) |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3191 |
qed |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3192 |
ultimately show ?thesis |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3193 |
by (blast intro: homotopic_loops_trans) |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3194 |
qed |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3195 |
|
36583 | 3196 |
end |