author | paulson <lp15@cam.ac.uk> |
Mon, 23 May 2016 15:33:24 +0100 | |
changeset 63114 | 27afe7af7379 |
parent 63092 | a949b2a5f51d |
child 63125 | f2d3f8427bc2 |
permissions | -rw-r--r-- |
41959 | 1 |
(* Title: HOL/Multivariate_Analysis/Path_Connected.thy |
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Cauchy's integral formula for circles. Starting to fix eventually_mono.
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parents:
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diff
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Authors: LC Paulson and Robert Himmelmann (TU Muenchen), based on material from HOL Light |
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*) |
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section \<open>Continuous paths and path-connected sets\<close> |
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|
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theory Path_Connected |
|
37674
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convert theorem path_connected_sphere to euclidean_space class
huffman
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imports Convex_Euclidean_Space |
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begin |
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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" |
18 |
||
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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}" |
24 |
||
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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))" |
27 |
||
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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) |
30 |
where "g1 +++ g2 = (\<lambda>x. if x \<le> 1/2 then g1 (2 * x) else g2 (2 * x - 1))" |
|
31 |
||
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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)" |
36583 | 35 |
|
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definition arc :: "(real \<Rightarrow> 'a :: topological_space) \<Rightarrow> bool" |
37 |
where "arc g \<longleftrightarrow> path g \<and> inj_on g {0..1}" |
|
36583 | 38 |
|
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|
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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 |
|
44 |
||
45 |
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 |
|
48 |
||
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lemma path_translation_eq: |
|
50 |
fixes g :: "real \<Rightarrow> 'a :: real_normed_vector" |
|
51 |
shows "path((\<lambda>x. a + x) o g) = path g" |
|
52 |
proof - |
|
53 |
have g: "g = (\<lambda>x. -a + x) o ((\<lambda>x. a + x) o g)" |
|
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by (rule ext) simp |
|
55 |
show ?thesis |
|
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unfolding path_def |
|
57 |
apply safe |
|
58 |
apply (subst g) |
|
59 |
apply (rule continuous_on_compose) |
|
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apply (auto intro: continuous_intros) |
|
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done |
|
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qed |
|
63 |
||
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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] |
|
70 |
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 |
|
79 |
||
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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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85 |
||
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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) |
|
88 |
||
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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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91 |
||
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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) |
|
94 |
||
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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) |
|
36583 | 100 |
|
60303 | 101 |
lemma reversepath_linear_image: "linear f \<Longrightarrow> reversepath(f o g) = f o reversepath g" |
102 |
by (rule ext) (simp add: reversepath_def) |
|
103 |
||
104 |
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) |
|
107 |
||
108 |
lemma joinpaths_linear_image: "linear f \<Longrightarrow> (f o g1) +++ (f o g2) = f o (g1 +++ g2)" |
|
109 |
by (rule ext) (simp add: joinpaths_def) |
|
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||
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the Cauchy integral theorem and related material
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parents:
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diff
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lemma simple_path_translation_eq: |
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fixes g :: "real \<Rightarrow> 'a::euclidean_space" |
113 |
shows "simple_path((\<lambda>x. a + x) o g) = simple_path g" |
|
114 |
by (simp add: simple_path_def path_translation_eq) |
|
115 |
||
116 |
lemma simple_path_linear_image_eq: |
|
117 |
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space" |
|
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assumes "linear f" "inj f" |
|
119 |
shows "simple_path(f o g) = simple_path g" |
|
120 |
using assms inj_on_eq_iff [of f] |
|
121 |
by (auto simp: path_linear_image_eq simple_path_def path_translation_eq) |
|
122 |
||
123 |
lemma arc_translation_eq: |
|
124 |
fixes g :: "real \<Rightarrow> 'a::euclidean_space" |
|
125 |
shows "arc((\<lambda>x. a + x) o g) = arc g" |
|
126 |
by (auto simp: arc_def inj_on_def path_translation_eq) |
|
127 |
||
128 |
lemma arc_linear_image_eq: |
|
129 |
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space" |
|
130 |
assumes "linear f" "inj f" |
|
131 |
shows "arc(f o g) = arc g" |
|
132 |
using assms inj_on_eq_iff [of f] |
|
133 |
by (auto simp: arc_def inj_on_def path_linear_image_eq) |
|
134 |
||
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subsection\<open>Basic lemmas about paths\<close> |
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|
137 |
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) |
|
139 |
||
140 |
lemma arc_imp_path: "arc g \<Longrightarrow> path g" |
|
141 |
using arc_def by blast |
|
142 |
||
143 |
lemma simple_path_imp_path: "simple_path g \<Longrightarrow> path g" |
|
144 |
using simple_path_def by blast |
|
145 |
||
146 |
lemma simple_path_cases: "simple_path g \<Longrightarrow> arc g \<or> pathfinish g = pathstart g" |
|
147 |
unfolding simple_path_def arc_def inj_on_def pathfinish_def pathstart_def |
|
148 |
by (force) |
|
149 |
||
150 |
lemma simple_path_imp_arc: "simple_path g \<Longrightarrow> pathfinish g \<noteq> pathstart g \<Longrightarrow> arc g" |
|
151 |
using simple_path_cases by auto |
|
152 |
||
153 |
lemma arc_distinct_ends: "arc g \<Longrightarrow> pathfinish g \<noteq> pathstart g" |
|
154 |
unfolding arc_def inj_on_def pathfinish_def pathstart_def |
|
155 |
by fastforce |
|
156 |
||
157 |
lemma arc_simple_path: "arc g \<longleftrightarrow> simple_path g \<and> pathfinish g \<noteq> pathstart g" |
|
158 |
using arc_distinct_ends arc_imp_simple_path simple_path_cases by blast |
|
159 |
||
160 |
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) |
|
36583 | 162 |
|
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New material and fixes related to the forthcoming Stone-Weierstrass development
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lemma path_image_nonempty [simp]: "path_image g \<noteq> {}" |
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unfolding path_image_def image_is_empty box_eq_empty |
53640 | 165 |
by auto |
36583 | 166 |
|
53640 | 167 |
lemma pathstart_in_path_image[intro]: "pathstart g \<in> path_image g" |
168 |
unfolding pathstart_def path_image_def |
|
169 |
by auto |
|
36583 | 170 |
|
53640 | 171 |
lemma pathfinish_in_path_image[intro]: "pathfinish g \<in> path_image g" |
172 |
unfolding pathfinish_def path_image_def |
|
173 |
by auto |
|
174 |
||
175 |
lemma connected_path_image[intro]: "path g \<Longrightarrow> connected (path_image g)" |
|
36583 | 176 |
unfolding path_def path_image_def |
60303 | 177 |
using connected_continuous_image connected_Icc by blast |
36583 | 178 |
|
53640 | 179 |
lemma compact_path_image[intro]: "path g \<Longrightarrow> compact (path_image g)" |
36583 | 180 |
unfolding path_def path_image_def |
60303 | 181 |
using compact_continuous_image connected_Icc by blast |
36583 | 182 |
|
53640 | 183 |
lemma reversepath_reversepath[simp]: "reversepath (reversepath g) = g" |
184 |
unfolding reversepath_def |
|
185 |
by auto |
|
36583 | 186 |
|
53640 | 187 |
lemma pathstart_reversepath[simp]: "pathstart (reversepath g) = pathfinish g" |
188 |
unfolding pathstart_def reversepath_def pathfinish_def |
|
189 |
by auto |
|
36583 | 190 |
|
53640 | 191 |
lemma pathfinish_reversepath[simp]: "pathfinish (reversepath g) = pathstart g" |
192 |
unfolding pathstart_def reversepath_def pathfinish_def |
|
193 |
by auto |
|
36583 | 194 |
|
49653 | 195 |
lemma pathstart_join[simp]: "pathstart (g1 +++ g2) = pathstart g1" |
53640 | 196 |
unfolding pathstart_def joinpaths_def pathfinish_def |
197 |
by auto |
|
36583 | 198 |
|
49653 | 199 |
lemma pathfinish_join[simp]: "pathfinish (g1 +++ g2) = pathfinish g2" |
53640 | 200 |
unfolding pathstart_def joinpaths_def pathfinish_def |
201 |
by auto |
|
36583 | 202 |
|
53640 | 203 |
lemma path_image_reversepath[simp]: "path_image (reversepath g) = path_image g" |
49653 | 204 |
proof - |
53640 | 205 |
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 |
60303 | 207 |
by force |
49653 | 208 |
show ?thesis |
209 |
using *[of g] *[of "reversepath g"] |
|
53640 | 210 |
unfolding reversepath_reversepath |
211 |
by auto |
|
49653 | 212 |
qed |
36583 | 213 |
|
53640 | 214 |
lemma path_reversepath [simp]: "path (reversepath g) \<longleftrightarrow> path g" |
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proof - |
216 |
have *: "\<And>g. path g \<Longrightarrow> path (reversepath g)" |
|
217 |
unfolding path_def reversepath_def |
|
218 |
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:
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219 |
apply (intro continuous_intros) |
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apply (rule continuous_on_subset[of "{0..1}"]) |
221 |
apply assumption |
|
49653 | 222 |
apply auto |
223 |
done |
|
224 |
show ?thesis |
|
225 |
using *[of "reversepath g"] *[of g] |
|
226 |
unfolding reversepath_reversepath |
|
227 |
by (rule iffI) |
|
228 |
qed |
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229 |
||
60303 | 230 |
lemma arc_reversepath: |
231 |
assumes "arc g" shows "arc(reversepath g)" |
|
232 |
proof - |
|
233 |
have injg: "inj_on g {0..1}" |
|
234 |
using assms |
|
235 |
by (simp add: arc_def) |
|
236 |
have **: "\<And>x y::real. 1-x = 1-y \<Longrightarrow> x = y" |
|
237 |
by simp |
|
238 |
show ?thesis |
|
239 |
apply (auto simp: arc_def inj_on_def path_reversepath) |
|
240 |
apply (simp add: arc_imp_path assms) |
|
241 |
apply (rule **) |
|
242 |
apply (rule inj_onD [OF injg]) |
|
243 |
apply (auto simp: reversepath_def) |
|
244 |
done |
|
245 |
qed |
|
246 |
||
247 |
lemma simple_path_reversepath: "simple_path g \<Longrightarrow> simple_path (reversepath g)" |
|
248 |
apply (simp add: simple_path_def) |
|
249 |
apply (force simp: reversepath_def) |
|
250 |
done |
|
251 |
||
49653 | 252 |
lemmas reversepath_simps = |
253 |
path_reversepath path_image_reversepath pathstart_reversepath pathfinish_reversepath |
|
36583 | 254 |
|
49653 | 255 |
lemma path_join[simp]: |
256 |
assumes "pathfinish g1 = pathstart g2" |
|
257 |
shows "path (g1 +++ g2) \<longleftrightarrow> path g1 \<and> path g2" |
|
258 |
unfolding path_def pathfinish_def pathstart_def |
|
51478
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
50935
diff
changeset
|
259 |
proof safe |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
50935
diff
changeset
|
260 |
assume cont: "continuous_on {0..1} (g1 +++ g2)" |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
50935
diff
changeset
|
261 |
have g1: "continuous_on {0..1} g1 \<longleftrightarrow> continuous_on {0..1} ((g1 +++ g2) \<circ> (\<lambda>x. x / 2))" |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
50935
diff
changeset
|
262 |
by (intro continuous_on_cong refl) (auto simp: joinpaths_def) |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
50935
diff
changeset
|
263 |
have g2: "continuous_on {0..1} g2 \<longleftrightarrow> continuous_on {0..1} ((g1 +++ g2) \<circ> (\<lambda>x. x / 2 + 1/2))" |
53640 | 264 |
using assms |
265 |
by (intro continuous_on_cong refl) (auto simp: joinpaths_def pathfinish_def pathstart_def) |
|
266 |
show "continuous_on {0..1} g1" and "continuous_on {0..1} g2" |
|
51481
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51478
diff
changeset
|
267 |
unfolding g1 g2 |
56371
fb9ae0727548
extend continuous_intros; remove continuous_on_intros and isCont_intros
hoelzl
parents:
56188
diff
changeset
|
268 |
by (auto intro!: continuous_intros continuous_on_subset[OF cont] simp del: o_apply) |
51478
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
50935
diff
changeset
|
269 |
next |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
50935
diff
changeset
|
270 |
assume g1g2: "continuous_on {0..1} g1" "continuous_on {0..1} g2" |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
50935
diff
changeset
|
271 |
have 01: "{0 .. 1} = {0..1/2} \<union> {1/2 .. 1::real}" |
36583 | 272 |
by auto |
53640 | 273 |
{ |
274 |
fix x :: real |
|
275 |
assume "0 \<le> x" and "x \<le> 1" |
|
276 |
then have "x \<in> (\<lambda>x. x * 2) ` {0..1 / 2}" |
|
277 |
by (intro image_eqI[where x="x/2"]) auto |
|
278 |
} |
|
51478
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
50935
diff
changeset
|
279 |
note 1 = this |
53640 | 280 |
{ |
281 |
fix x :: real |
|
282 |
assume "0 \<le> x" and "x \<le> 1" |
|
283 |
then have "x \<in> (\<lambda>x. x * 2 - 1) ` {1 / 2..1}" |
|
284 |
by (intro image_eqI[where x="x/2 + 1/2"]) auto |
|
285 |
} |
|
51478
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
50935
diff
changeset
|
286 |
note 2 = this |
49653 | 287 |
show "continuous_on {0..1} (g1 +++ g2)" |
53640 | 288 |
using assms |
289 |
unfolding joinpaths_def 01 |
|
56371
fb9ae0727548
extend continuous_intros; remove continuous_on_intros and isCont_intros
hoelzl
parents:
56188
diff
changeset
|
290 |
apply (intro continuous_on_cases closed_atLeastAtMost g1g2[THEN continuous_on_compose2] continuous_intros) |
53640 | 291 |
apply (auto simp: field_simps pathfinish_def pathstart_def intro!: 1 2) |
292 |
done |
|
49653 | 293 |
qed |
36583 | 294 |
|
60420 | 295 |
section \<open>Path Images\<close> |
60303 | 296 |
|
297 |
lemma bounded_path_image: "path g \<Longrightarrow> bounded(path_image g)" |
|
298 |
by (simp add: compact_imp_bounded compact_path_image) |
|
299 |
||
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
60420
diff
changeset
|
300 |
lemma closed_path_image: |
60303 | 301 |
fixes g :: "real \<Rightarrow> 'a::t2_space" |
302 |
shows "path g \<Longrightarrow> closed(path_image g)" |
|
303 |
by (metis compact_path_image compact_imp_closed) |
|
304 |
||
305 |
lemma connected_simple_path_image: "simple_path g \<Longrightarrow> connected(path_image g)" |
|
306 |
by (metis connected_path_image simple_path_imp_path) |
|
307 |
||
308 |
lemma compact_simple_path_image: "simple_path g \<Longrightarrow> compact(path_image g)" |
|
309 |
by (metis compact_path_image simple_path_imp_path) |
|
310 |
||
311 |
lemma bounded_simple_path_image: "simple_path g \<Longrightarrow> bounded(path_image g)" |
|
312 |
by (metis bounded_path_image simple_path_imp_path) |
|
313 |
||
314 |
lemma closed_simple_path_image: |
|
315 |
fixes g :: "real \<Rightarrow> 'a::t2_space" |
|
316 |
shows "simple_path g \<Longrightarrow> closed(path_image g)" |
|
317 |
by (metis closed_path_image simple_path_imp_path) |
|
318 |
||
319 |
lemma connected_arc_image: "arc g \<Longrightarrow> connected(path_image g)" |
|
320 |
by (metis connected_path_image arc_imp_path) |
|
321 |
||
322 |
lemma compact_arc_image: "arc g \<Longrightarrow> compact(path_image g)" |
|
323 |
by (metis compact_path_image arc_imp_path) |
|
324 |
||
325 |
lemma bounded_arc_image: "arc g \<Longrightarrow> bounded(path_image g)" |
|
326 |
by (metis bounded_path_image arc_imp_path) |
|
327 |
||
328 |
lemma closed_arc_image: |
|
329 |
fixes g :: "real \<Rightarrow> 'a::t2_space" |
|
330 |
shows "arc g \<Longrightarrow> closed(path_image g)" |
|
331 |
by (metis closed_path_image arc_imp_path) |
|
332 |
||
53640 | 333 |
lemma path_image_join_subset: "path_image (g1 +++ g2) \<subseteq> path_image g1 \<union> path_image g2" |
334 |
unfolding path_image_def joinpaths_def |
|
335 |
by auto |
|
36583 | 336 |
|
337 |
lemma subset_path_image_join: |
|
53640 | 338 |
assumes "path_image g1 \<subseteq> s" |
339 |
and "path_image g2 \<subseteq> s" |
|
340 |
shows "path_image (g1 +++ g2) \<subseteq> s" |
|
341 |
using path_image_join_subset[of g1 g2] and assms |
|
342 |
by auto |
|
36583 | 343 |
|
344 |
lemma path_image_join: |
|
60303 | 345 |
"pathfinish g1 = pathstart g2 \<Longrightarrow> path_image(g1 +++ g2) = path_image g1 \<union> path_image g2" |
346 |
apply (rule subset_antisym [OF path_image_join_subset]) |
|
347 |
apply (auto simp: pathfinish_def pathstart_def path_image_def joinpaths_def image_def) |
|
348 |
apply (drule sym) |
|
349 |
apply (rule_tac x="xa/2" in bexI, auto) |
|
350 |
apply (rule ccontr) |
|
351 |
apply (drule_tac x="(xa+1)/2" in bspec) |
|
352 |
apply (auto simp: field_simps) |
|
353 |
apply (drule_tac x="1/2" in bspec, auto) |
|
354 |
done |
|
36583 | 355 |
|
356 |
lemma not_in_path_image_join: |
|
53640 | 357 |
assumes "x \<notin> path_image g1" |
358 |
and "x \<notin> path_image g2" |
|
359 |
shows "x \<notin> path_image (g1 +++ g2)" |
|
360 |
using assms and path_image_join_subset[of g1 g2] |
|
361 |
by auto |
|
36583 | 362 |
|
60303 | 363 |
lemma pathstart_compose: "pathstart(f o p) = f(pathstart p)" |
364 |
by (simp add: pathstart_def) |
|
365 |
||
366 |
lemma pathfinish_compose: "pathfinish(f o p) = f(pathfinish p)" |
|
367 |
by (simp add: pathfinish_def) |
|
368 |
||
369 |
lemma path_image_compose: "path_image (f o p) = f ` (path_image p)" |
|
370 |
by (simp add: image_comp path_image_def) |
|
371 |
||
372 |
lemma path_compose_join: "f o (p +++ q) = (f o p) +++ (f o q)" |
|
373 |
by (rule ext) (simp add: joinpaths_def) |
|
374 |
||
375 |
lemma path_compose_reversepath: "f o reversepath p = reversepath(f o p)" |
|
376 |
by (rule ext) (simp add: reversepath_def) |
|
377 |
||
61762
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61738
diff
changeset
|
378 |
lemma joinpaths_eq: |
60303 | 379 |
"(\<And>t. t \<in> {0..1} \<Longrightarrow> p t = p' t) \<Longrightarrow> |
380 |
(\<And>t. t \<in> {0..1} \<Longrightarrow> q t = q' t) |
|
381 |
\<Longrightarrow> t \<in> {0..1} \<Longrightarrow> (p +++ q) t = (p' +++ q') t" |
|
382 |
by (auto simp: joinpaths_def) |
|
383 |
||
384 |
lemma simple_path_inj_on: "simple_path g \<Longrightarrow> inj_on g {0<..<1}" |
|
385 |
by (auto simp: simple_path_def path_image_def inj_on_def less_eq_real_def Ball_def) |
|
386 |
||
387 |
||
60420 | 388 |
subsection\<open>Simple paths with the endpoints removed\<close> |
60303 | 389 |
|
390 |
lemma simple_path_endless: |
|
391 |
"simple_path c \<Longrightarrow> path_image c - {pathstart c,pathfinish c} = c ` {0<..<1}" |
|
392 |
apply (auto simp: simple_path_def path_image_def pathstart_def pathfinish_def Ball_def Bex_def image_def) |
|
393 |
apply (metis eq_iff le_less_linear) |
|
394 |
apply (metis leD linear) |
|
395 |
using less_eq_real_def zero_le_one apply blast |
|
396 |
using less_eq_real_def zero_le_one apply blast |
|
49653 | 397 |
done |
36583 | 398 |
|
60303 | 399 |
lemma connected_simple_path_endless: |
400 |
"simple_path c \<Longrightarrow> connected(path_image c - {pathstart c,pathfinish c})" |
|
401 |
apply (simp add: simple_path_endless) |
|
402 |
apply (rule connected_continuous_image) |
|
403 |
apply (meson continuous_on_subset greaterThanLessThan_subseteq_atLeastAtMost_iff le_numeral_extra(3) le_numeral_extra(4) path_def simple_path_imp_path) |
|
404 |
by auto |
|
405 |
||
406 |
lemma nonempty_simple_path_endless: |
|
407 |
"simple_path c \<Longrightarrow> path_image c - {pathstart c,pathfinish c} \<noteq> {}" |
|
408 |
by (simp add: simple_path_endless) |
|
409 |
||
410 |
||
60420 | 411 |
subsection\<open>The operations on paths\<close> |
60303 | 412 |
|
413 |
lemma path_image_subset_reversepath: "path_image(reversepath g) \<le> path_image g" |
|
414 |
by (auto simp: path_image_def reversepath_def) |
|
415 |
||
416 |
lemma path_imp_reversepath: "path g \<Longrightarrow> path(reversepath g)" |
|
417 |
apply (auto simp: path_def reversepath_def) |
|
418 |
using continuous_on_compose [of "{0..1}" "\<lambda>x. 1 - x" g] |
|
419 |
apply (auto simp: continuous_on_op_minus) |
|
420 |
done |
|
421 |
||
61204 | 422 |
lemma half_bounded_equal: "1 \<le> x * 2 \<Longrightarrow> x * 2 \<le> 1 \<longleftrightarrow> x = (1/2::real)" |
423 |
by simp |
|
60303 | 424 |
|
425 |
lemma continuous_on_joinpaths: |
|
426 |
assumes "continuous_on {0..1} g1" "continuous_on {0..1} g2" "pathfinish g1 = pathstart g2" |
|
427 |
shows "continuous_on {0..1} (g1 +++ g2)" |
|
428 |
proof - |
|
429 |
have *: "{0..1::real} = {0..1/2} \<union> {1/2..1}" |
|
430 |
by auto |
|
431 |
have gg: "g2 0 = g1 1" |
|
432 |
by (metis assms(3) pathfinish_def pathstart_def) |
|
61204 | 433 |
have 1: "continuous_on {0..1/2} (g1 +++ g2)" |
60303 | 434 |
apply (rule continuous_on_eq [of _ "g1 o (\<lambda>x. 2*x)"]) |
61204 | 435 |
apply (rule continuous_intros | simp add: joinpaths_def assms)+ |
60303 | 436 |
done |
61204 | 437 |
have "continuous_on {1/2..1} (g2 o (\<lambda>x. 2*x-1))" |
438 |
apply (rule continuous_on_subset [of "{1/2..1}"]) |
|
439 |
apply (rule continuous_intros | simp add: image_affinity_atLeastAtMost_diff assms)+ |
|
440 |
done |
|
441 |
then have 2: "continuous_on {1/2..1} (g1 +++ g2)" |
|
442 |
apply (rule continuous_on_eq [of "{1/2..1}" "g2 o (\<lambda>x. 2*x-1)"]) |
|
443 |
apply (rule assms continuous_intros | simp add: joinpaths_def mult.commute half_bounded_equal gg)+ |
|
60303 | 444 |
done |
445 |
show ?thesis |
|
446 |
apply (subst *) |
|
62397
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62381
diff
changeset
|
447 |
apply (rule continuous_on_closed_Un) |
60303 | 448 |
using 1 2 |
449 |
apply auto |
|
450 |
done |
|
451 |
qed |
|
452 |
||
453 |
lemma path_join_imp: "\<lbrakk>path g1; path g2; pathfinish g1 = pathstart g2\<rbrakk> \<Longrightarrow> path(g1 +++ g2)" |
|
454 |
by (simp add: path_join) |
|
455 |
||
36583 | 456 |
lemma simple_path_join_loop: |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
60420
diff
changeset
|
457 |
assumes "arc g1" "arc g2" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
60420
diff
changeset
|
458 |
"pathfinish g1 = pathstart g2" "pathfinish g2 = pathstart g1" |
60303 | 459 |
"path_image g1 \<inter> path_image g2 \<subseteq> {pathstart g1, pathstart g2}" |
460 |
shows "simple_path(g1 +++ g2)" |
|
461 |
proof - |
|
462 |
have injg1: "inj_on g1 {0..1}" |
|
463 |
using assms |
|
464 |
by (simp add: arc_def) |
|
465 |
have injg2: "inj_on g2 {0..1}" |
|
466 |
using assms |
|
467 |
by (simp add: arc_def) |
|
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
60420
diff
changeset
|
468 |
have g12: "g1 1 = g2 0" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
60420
diff
changeset
|
469 |
and g21: "g2 1 = g1 0" |
60303 | 470 |
and sb: "g1 ` {0..1} \<inter> g2 ` {0..1} \<subseteq> {g1 0, g2 0}" |
471 |
using assms |
|
472 |
by (simp_all add: arc_def pathfinish_def pathstart_def path_image_def) |
|
473 |
{ fix x and y::real |
|
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
60420
diff
changeset
|
474 |
assume xyI: "x = 1 \<longrightarrow> y \<noteq> 0" |
60303 | 475 |
and xy: "x \<le> 1" "0 \<le> y" " y * 2 \<le> 1" "\<not> x * 2 \<le> 1" "g2 (2 * x - 1) = g1 (2 * y)" |
476 |
have g1im: "g1 (2 * y) \<in> g1 ` {0..1} \<inter> g2 ` {0..1}" |
|
477 |
using xy |
|
478 |
apply simp |
|
479 |
apply (rule_tac x="2 * x - 1" in image_eqI, auto) |
|
480 |
done |
|
481 |
have False |
|
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
60420
diff
changeset
|
482 |
using subsetD [OF sb g1im] xy |
60303 | 483 |
apply auto |
484 |
apply (drule inj_onD [OF injg1]) |
|
485 |
using g21 [symmetric] xyI |
|
486 |
apply (auto dest: inj_onD [OF injg2]) |
|
487 |
done |
|
488 |
} note * = this |
|
489 |
{ fix x and y::real |
|
490 |
assume xy: "y \<le> 1" "0 \<le> x" "\<not> y * 2 \<le> 1" "x * 2 \<le> 1" "g1 (2 * x) = g2 (2 * y - 1)" |
|
491 |
have g1im: "g1 (2 * x) \<in> g1 ` {0..1} \<inter> g2 ` {0..1}" |
|
492 |
using xy |
|
493 |
apply simp |
|
494 |
apply (rule_tac x="2 * x" in image_eqI, auto) |
|
495 |
done |
|
496 |
have "x = 0 \<and> y = 1" |
|
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
60420
diff
changeset
|
497 |
using subsetD [OF sb g1im] xy |
60303 | 498 |
apply auto |
499 |
apply (force dest: inj_onD [OF injg1]) |
|
500 |
using g21 [symmetric] |
|
501 |
apply (auto dest: inj_onD [OF injg2]) |
|
502 |
done |
|
503 |
} note ** = this |
|
504 |
show ?thesis |
|
505 |
using assms |
|
506 |
apply (simp add: arc_def simple_path_def path_join, clarify) |
|
62390 | 507 |
apply (simp add: joinpaths_def split: if_split_asm) |
60303 | 508 |
apply (force dest: inj_onD [OF injg1]) |
509 |
apply (metis *) |
|
510 |
apply (metis **) |
|
511 |
apply (force dest: inj_onD [OF injg2]) |
|
512 |
done |
|
513 |
qed |
|
514 |
||
515 |
lemma arc_join: |
|
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
60420
diff
changeset
|
516 |
assumes "arc g1" "arc g2" |
60303 | 517 |
"pathfinish g1 = pathstart g2" |
518 |
"path_image g1 \<inter> path_image g2 \<subseteq> {pathstart g2}" |
|
519 |
shows "arc(g1 +++ g2)" |
|
520 |
proof - |
|
521 |
have injg1: "inj_on g1 {0..1}" |
|
522 |
using assms |
|
523 |
by (simp add: arc_def) |
|
524 |
have injg2: "inj_on g2 {0..1}" |
|
525 |
using assms |
|
526 |
by (simp add: arc_def) |
|
527 |
have g11: "g1 1 = g2 0" |
|
528 |
and sb: "g1 ` {0..1} \<inter> g2 ` {0..1} \<subseteq> {g2 0}" |
|
529 |
using assms |
|
530 |
by (simp_all add: arc_def pathfinish_def pathstart_def path_image_def) |
|
531 |
{ fix x and y::real |
|
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
60420
diff
changeset
|
532 |
assume xy: "x \<le> 1" "0 \<le> y" " y * 2 \<le> 1" "\<not> x * 2 \<le> 1" "g2 (2 * x - 1) = g1 (2 * y)" |
60303 | 533 |
have g1im: "g1 (2 * y) \<in> g1 ` {0..1} \<inter> g2 ` {0..1}" |
534 |
using xy |
|
535 |
apply simp |
|
536 |
apply (rule_tac x="2 * x - 1" in image_eqI, auto) |
|
537 |
done |
|
538 |
have False |
|
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
60420
diff
changeset
|
539 |
using subsetD [OF sb g1im] xy |
60303 | 540 |
by (auto dest: inj_onD [OF injg2]) |
541 |
} note * = this |
|
542 |
show ?thesis |
|
543 |
apply (simp add: arc_def inj_on_def) |
|
544 |
apply (clarsimp simp add: arc_imp_path assms path_join) |
|
62390 | 545 |
apply (simp add: joinpaths_def split: if_split_asm) |
60303 | 546 |
apply (force dest: inj_onD [OF injg1]) |
547 |
apply (metis *) |
|
548 |
apply (metis *) |
|
549 |
apply (force dest: inj_onD [OF injg2]) |
|
550 |
done |
|
551 |
qed |
|
552 |
||
553 |
lemma reversepath_joinpaths: |
|
554 |
"pathfinish g1 = pathstart g2 \<Longrightarrow> reversepath(g1 +++ g2) = reversepath g2 +++ reversepath g1" |
|
555 |
unfolding reversepath_def pathfinish_def pathstart_def joinpaths_def |
|
556 |
by (rule ext) (auto simp: mult.commute) |
|
557 |
||
558 |
||
62533
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
559 |
subsection\<open>Some reversed and "if and only if" versions of joining theorems\<close> |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
560 |
|
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
561 |
lemma path_join_path_ends: |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
562 |
fixes g1 :: "real \<Rightarrow> 'a::metric_space" |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
563 |
assumes "path(g1 +++ g2)" "path g2" |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
564 |
shows "pathfinish g1 = pathstart g2" |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
565 |
proof (rule ccontr) |
63040 | 566 |
define e where "e = dist (g1 1) (g2 0)" |
62533
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
567 |
assume Neg: "pathfinish g1 \<noteq> pathstart g2" |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
568 |
then have "0 < dist (pathfinish g1) (pathstart g2)" |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
569 |
by auto |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
570 |
then have "e > 0" |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
571 |
by (metis e_def pathfinish_def pathstart_def) |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
572 |
then obtain d1 where "d1 > 0" |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
573 |
and d1: "\<And>x'. \<lbrakk>x'\<in>{0..1}; norm x' < d1\<rbrakk> \<Longrightarrow> dist (g2 x') (g2 0) < e/2" |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
574 |
using assms(2) unfolding path_def continuous_on_iff |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
575 |
apply (drule_tac x=0 in bspec, simp) |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
576 |
by (metis half_gt_zero_iff norm_conv_dist) |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
577 |
obtain d2 where "d2 > 0" |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
578 |
and d2: "\<And>x'. \<lbrakk>x'\<in>{0..1}; dist x' (1/2) < d2\<rbrakk> |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
579 |
\<Longrightarrow> dist ((g1 +++ g2) x') (g1 1) < e/2" |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
580 |
using assms(1) \<open>e > 0\<close> unfolding path_def continuous_on_iff |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
581 |
apply (drule_tac x="1/2" in bspec, simp) |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
582 |
apply (drule_tac x="e/2" in spec) |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
583 |
apply (force simp: joinpaths_def) |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
584 |
done |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
585 |
have int01_1: "min (1/2) (min d1 d2) / 2 \<in> {0..1}" |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
586 |
using \<open>d1 > 0\<close> \<open>d2 > 0\<close> by (simp add: min_def) |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
587 |
have dist1: "norm (min (1 / 2) (min d1 d2) / 2) < d1" |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
588 |
using \<open>d1 > 0\<close> \<open>d2 > 0\<close> by (simp add: min_def dist_norm) |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
589 |
have int01_2: "1/2 + min (1/2) (min d1 d2) / 4 \<in> {0..1}" |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
590 |
using \<open>d1 > 0\<close> \<open>d2 > 0\<close> by (simp add: min_def) |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
591 |
have dist2: "dist (1 / 2 + min (1 / 2) (min d1 d2) / 4) (1 / 2) < d2" |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
592 |
using \<open>d1 > 0\<close> \<open>d2 > 0\<close> by (simp add: min_def dist_norm) |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
593 |
have [simp]: "~ min (1 / 2) (min d1 d2) \<le> 0" |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
594 |
using \<open>d1 > 0\<close> \<open>d2 > 0\<close> by (simp add: min_def) |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
595 |
have "dist (g2 (min (1 / 2) (min d1 d2) / 2)) (g1 1) < e/2" |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
596 |
"dist (g2 (min (1 / 2) (min d1 d2) / 2)) (g2 0) < e/2" |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
597 |
using d1 [OF int01_1 dist1] d2 [OF int01_2 dist2] by (simp_all add: joinpaths_def) |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
598 |
then have "dist (g1 1) (g2 0) < e/2 + e/2" |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
599 |
using dist_triangle_half_r e_def by blast |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
600 |
then show False |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
601 |
by (simp add: e_def [symmetric]) |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
602 |
qed |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
603 |
|
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
604 |
lemma path_join_eq [simp]: |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
605 |
fixes g1 :: "real \<Rightarrow> 'a::metric_space" |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
606 |
assumes "path g1" "path g2" |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
607 |
shows "path(g1 +++ g2) \<longleftrightarrow> pathfinish g1 = pathstart g2" |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
608 |
using assms by (metis path_join_path_ends path_join_imp) |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
609 |
|
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
610 |
lemma simple_path_joinE: |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
611 |
assumes "simple_path(g1 +++ g2)" and "pathfinish g1 = pathstart g2" |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
612 |
obtains "arc g1" "arc g2" |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
613 |
"path_image g1 \<inter> path_image g2 \<subseteq> {pathstart g1, pathstart g2}" |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
614 |
proof - |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
615 |
have *: "\<And>x y. \<lbrakk>0 \<le> x; x \<le> 1; 0 \<le> y; y \<le> 1; (g1 +++ g2) x = (g1 +++ g2) y\<rbrakk> |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
616 |
\<Longrightarrow> x = y \<or> x = 0 \<and> y = 1 \<or> x = 1 \<and> y = 0" |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
617 |
using assms by (simp add: simple_path_def) |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
618 |
have "path g1" |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
619 |
using assms path_join simple_path_imp_path by blast |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
620 |
moreover have "inj_on g1 {0..1}" |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
621 |
proof (clarsimp simp: inj_on_def) |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
622 |
fix x y |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
623 |
assume "g1 x = g1 y" "0 \<le> x" "x \<le> 1" "0 \<le> y" "y \<le> 1" |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
624 |
then show "x = y" |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
625 |
using * [of "x/2" "y/2"] by (simp add: joinpaths_def split_ifs) |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
626 |
qed |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
627 |
ultimately have "arc g1" |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
628 |
using assms by (simp add: arc_def) |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
629 |
have [simp]: "g2 0 = g1 1" |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
630 |
using assms by (metis pathfinish_def pathstart_def) |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
631 |
have "path g2" |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
632 |
using assms path_join simple_path_imp_path by blast |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
633 |
moreover have "inj_on g2 {0..1}" |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
634 |
proof (clarsimp simp: inj_on_def) |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
635 |
fix x y |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
636 |
assume "g2 x = g2 y" "0 \<le> x" "x \<le> 1" "0 \<le> y" "y \<le> 1" |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
637 |
then show "x = y" |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
638 |
using * [of "(x + 1) / 2" "(y + 1) / 2"] |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
639 |
by (force simp: joinpaths_def split_ifs divide_simps) |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
640 |
qed |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
641 |
ultimately have "arc g2" |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
642 |
using assms by (simp add: arc_def) |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
643 |
have "g2 y = g1 0 \<or> g2 y = g1 1" |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
644 |
if "g1 x = g2 y" "0 \<le> x" "x \<le> 1" "0 \<le> y" "y \<le> 1" for x y |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
645 |
using * [of "x / 2" "(y + 1) / 2"] that |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
646 |
by (auto simp: joinpaths_def split_ifs divide_simps) |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
647 |
then have "path_image g1 \<inter> path_image g2 \<subseteq> {pathstart g1, pathstart g2}" |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
648 |
by (fastforce simp: pathstart_def pathfinish_def path_image_def) |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
649 |
with \<open>arc g1\<close> \<open>arc g2\<close> show ?thesis using that by blast |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
650 |
qed |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
651 |
|
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
652 |
lemma simple_path_join_loop_eq: |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
653 |
assumes "pathfinish g2 = pathstart g1" "pathfinish g1 = pathstart g2" |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
654 |
shows "simple_path(g1 +++ g2) \<longleftrightarrow> |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
655 |
arc g1 \<and> arc g2 \<and> path_image g1 \<inter> path_image g2 \<subseteq> {pathstart g1, pathstart g2}" |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
656 |
by (metis assms simple_path_joinE simple_path_join_loop) |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
657 |
|
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
658 |
lemma arc_join_eq: |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
659 |
assumes "pathfinish g1 = pathstart g2" |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
660 |
shows "arc(g1 +++ g2) \<longleftrightarrow> |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
661 |
arc g1 \<and> arc g2 \<and> path_image g1 \<inter> path_image g2 \<subseteq> {pathstart g2}" |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
662 |
(is "?lhs = ?rhs") |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
663 |
proof |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
664 |
assume ?lhs |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
665 |
then have "simple_path(g1 +++ g2)" by (rule arc_imp_simple_path) |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
666 |
then have *: "\<And>x y. \<lbrakk>0 \<le> x; x \<le> 1; 0 \<le> y; y \<le> 1; (g1 +++ g2) x = (g1 +++ g2) y\<rbrakk> |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
667 |
\<Longrightarrow> x = y \<or> x = 0 \<and> y = 1 \<or> x = 1 \<and> y = 0" |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
668 |
using assms by (simp add: simple_path_def) |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
669 |
have False if "g1 0 = g2 u" "0 \<le> u" "u \<le> 1" for u |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
670 |
using * [of 0 "(u + 1) / 2"] that assms arc_distinct_ends [OF \<open>?lhs\<close>] |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
671 |
by (auto simp: joinpaths_def pathstart_def pathfinish_def split_ifs divide_simps) |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
672 |
then have n1: "~ (pathstart g1 \<in> path_image g2)" |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
673 |
unfolding pathstart_def path_image_def |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
674 |
using atLeastAtMost_iff by blast |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
675 |
show ?rhs using \<open>?lhs\<close> |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
676 |
apply (rule simple_path_joinE [OF arc_imp_simple_path assms]) |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
677 |
using n1 by force |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
678 |
next |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
679 |
assume ?rhs then show ?lhs |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
680 |
using assms |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
681 |
by (fastforce simp: pathfinish_def pathstart_def intro!: arc_join) |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
682 |
qed |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
683 |
|
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
684 |
lemma arc_join_eq_alt: |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
685 |
"pathfinish g1 = pathstart g2 |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
686 |
\<Longrightarrow> (arc(g1 +++ g2) \<longleftrightarrow> |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
687 |
arc g1 \<and> arc g2 \<and> |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
688 |
path_image g1 \<inter> path_image g2 = {pathstart g2})" |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
689 |
using pathfinish_in_path_image by (fastforce simp: arc_join_eq) |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
690 |
|
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
691 |
|
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
692 |
subsection\<open>The joining of paths is associative\<close> |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
693 |
|
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
694 |
lemma path_assoc: |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
695 |
"\<lbrakk>pathfinish p = pathstart q; pathfinish q = pathstart r\<rbrakk> |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
696 |
\<Longrightarrow> path(p +++ (q +++ r)) \<longleftrightarrow> path((p +++ q) +++ r)" |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
697 |
by simp |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
698 |
|
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
699 |
lemma simple_path_assoc: |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
700 |
assumes "pathfinish p = pathstart q" "pathfinish q = pathstart r" |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
701 |
shows "simple_path (p +++ (q +++ r)) \<longleftrightarrow> simple_path ((p +++ q) +++ r)" |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
702 |
proof (cases "pathstart p = pathfinish r") |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
703 |
case True show ?thesis |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
704 |
proof |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
705 |
assume "simple_path (p +++ q +++ r)" |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
706 |
with assms True show "simple_path ((p +++ q) +++ r)" |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
707 |
by (fastforce simp add: simple_path_join_loop_eq arc_join_eq path_image_join |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
708 |
dest: arc_distinct_ends [of r]) |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
709 |
next |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
710 |
assume 0: "simple_path ((p +++ q) +++ r)" |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
711 |
with assms True have q: "pathfinish r \<notin> path_image q" |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
712 |
using arc_distinct_ends |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
713 |
by (fastforce simp add: simple_path_join_loop_eq arc_join_eq path_image_join) |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
714 |
have "pathstart r \<notin> path_image p" |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
715 |
using assms |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
716 |
by (metis 0 IntI arc_distinct_ends arc_join_eq_alt empty_iff insert_iff |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
717 |
pathfinish_in_path_image pathfinish_join simple_path_joinE) |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
718 |
with assms 0 q True show "simple_path (p +++ q +++ r)" |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
719 |
by (auto simp: simple_path_join_loop_eq arc_join_eq path_image_join |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
720 |
dest!: subsetD [OF _ IntI]) |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
721 |
qed |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
722 |
next |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
723 |
case False |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
724 |
{ fix x :: 'a |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
725 |
assume a: "path_image p \<inter> path_image q \<subseteq> {pathstart q}" |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
726 |
"(path_image p \<union> path_image q) \<inter> path_image r \<subseteq> {pathstart r}" |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
727 |
"x \<in> path_image p" "x \<in> path_image r" |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
728 |
have "pathstart r \<in> path_image q" |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
729 |
by (metis assms(2) pathfinish_in_path_image) |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
730 |
with a have "x = pathstart q" |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
731 |
by blast |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
732 |
} |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
733 |
with False assms show ?thesis |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
734 |
by (auto simp: simple_path_eq_arc simple_path_join_loop_eq arc_join_eq path_image_join) |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
735 |
qed |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
736 |
|
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
737 |
lemma arc_assoc: |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
738 |
"\<lbrakk>pathfinish p = pathstart q; pathfinish q = pathstart r\<rbrakk> |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
739 |
\<Longrightarrow> arc(p +++ (q +++ r)) \<longleftrightarrow> arc((p +++ q) +++ r)" |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
740 |
by (simp add: arc_simple_path simple_path_assoc) |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
741 |
|
62620
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
742 |
subsubsection\<open>Symmetry and loops\<close> |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
743 |
|
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
744 |
lemma path_sym: |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
745 |
"\<lbrakk>pathfinish p = pathstart q; pathfinish q = pathstart p\<rbrakk> \<Longrightarrow> path(p +++ q) \<longleftrightarrow> path(q +++ p)" |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
746 |
by auto |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
747 |
|
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
748 |
lemma simple_path_sym: |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
749 |
"\<lbrakk>pathfinish p = pathstart q; pathfinish q = pathstart p\<rbrakk> |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
750 |
\<Longrightarrow> simple_path(p +++ q) \<longleftrightarrow> simple_path(q +++ p)" |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
751 |
by (metis (full_types) inf_commute insert_commute simple_path_joinE simple_path_join_loop) |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
752 |
|
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
753 |
lemma path_image_sym: |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
754 |
"\<lbrakk>pathfinish p = pathstart q; pathfinish q = pathstart p\<rbrakk> |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
755 |
\<Longrightarrow> path_image(p +++ q) = path_image(q +++ p)" |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
756 |
by (simp add: path_image_join sup_commute) |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
757 |
|
62533
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
758 |
|
61518
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
759 |
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
|
760 |
|
60303 | 761 |
definition subpath :: "real \<Rightarrow> real \<Rightarrow> (real \<Rightarrow> 'a) \<Rightarrow> real \<Rightarrow> 'a::real_normed_vector" |
762 |
where "subpath a b g \<equiv> \<lambda>x. g((b - a) * x + a)" |
|
763 |
||
61762
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61738
diff
changeset
|
764 |
lemma path_image_subpath_gen: |
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61738
diff
changeset
|
765 |
fixes g :: "_ \<Rightarrow> 'a::real_normed_vector" |
60303 | 766 |
shows "path_image(subpath u v g) = g ` (closed_segment u v)" |
767 |
apply (simp add: closed_segment_real_eq path_image_def subpath_def) |
|
768 |
apply (subst o_def [of g, symmetric]) |
|
769 |
apply (simp add: image_comp [symmetric]) |
|
770 |
done |
|
771 |
||
61762
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61738
diff
changeset
|
772 |
lemma path_image_subpath: |
60303 | 773 |
fixes g :: "real \<Rightarrow> 'a::real_normed_vector" |
774 |
shows "path_image(subpath u v g) = (if u \<le> v then g ` {u..v} else g ` {v..u})" |
|
61762
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61738
diff
changeset
|
775 |
by (simp add: path_image_subpath_gen closed_segment_eq_real_ivl) |
60303 | 776 |
|
777 |
lemma path_subpath [simp]: |
|
778 |
fixes g :: "real \<Rightarrow> 'a::real_normed_vector" |
|
779 |
assumes "path g" "u \<in> {0..1}" "v \<in> {0..1}" |
|
780 |
shows "path(subpath u v g)" |
|
781 |
proof - |
|
782 |
have "continuous_on {0..1} (g o (\<lambda>x. ((v-u) * x+ u)))" |
|
783 |
apply (rule continuous_intros | simp)+ |
|
784 |
apply (simp add: image_affinity_atLeastAtMost [where c=u]) |
|
785 |
using assms |
|
786 |
apply (auto simp: path_def continuous_on_subset) |
|
787 |
done |
|
788 |
then show ?thesis |
|
789 |
by (simp add: path_def subpath_def) |
|
49653 | 790 |
qed |
36583 | 791 |
|
60303 | 792 |
lemma pathstart_subpath [simp]: "pathstart(subpath u v g) = g(u)" |
793 |
by (simp add: pathstart_def subpath_def) |
|
794 |
||
795 |
lemma pathfinish_subpath [simp]: "pathfinish(subpath u v g) = g(v)" |
|
796 |
by (simp add: pathfinish_def subpath_def) |
|
797 |
||
798 |
lemma subpath_trivial [simp]: "subpath 0 1 g = g" |
|
799 |
by (simp add: subpath_def) |
|
800 |
||
801 |
lemma subpath_reversepath: "subpath 1 0 g = reversepath g" |
|
802 |
by (simp add: reversepath_def subpath_def) |
|
803 |
||
804 |
lemma reversepath_subpath: "reversepath(subpath u v g) = subpath v u g" |
|
805 |
by (simp add: reversepath_def subpath_def algebra_simps) |
|
806 |
||
807 |
lemma subpath_translation: "subpath u v ((\<lambda>x. a + x) o g) = (\<lambda>x. a + x) o subpath u v g" |
|
808 |
by (rule ext) (simp add: subpath_def) |
|
809 |
||
810 |
lemma subpath_linear_image: "linear f \<Longrightarrow> subpath u v (f o g) = f o subpath u v g" |
|
811 |
by (rule ext) (simp add: subpath_def) |
|
812 |
||
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
60420
diff
changeset
|
813 |
lemma affine_ineq: |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
60420
diff
changeset
|
814 |
fixes x :: "'a::linordered_idom" |
61762
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61738
diff
changeset
|
815 |
assumes "x \<le> 1" "v \<le> u" |
60303 | 816 |
shows "v + x * u \<le> u + x * v" |
817 |
proof - |
|
818 |
have "(1-x)*(u-v) \<ge> 0" |
|
819 |
using assms by auto |
|
820 |
then show ?thesis |
|
821 |
by (simp add: algebra_simps) |
|
49653 | 822 |
qed |
36583 | 823 |
|
61711
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
824 |
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
|
825 |
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
|
826 |
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
|
827 |
|
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
60420
diff
changeset
|
828 |
lemma simple_path_subpath_eq: |
60303 | 829 |
"simple_path(subpath u v g) \<longleftrightarrow> |
830 |
path(subpath u v g) \<and> u\<noteq>v \<and> |
|
831 |
(\<forall>x y. x \<in> closed_segment u v \<and> y \<in> closed_segment u v \<and> g x = g y |
|
832 |
\<longrightarrow> x = y \<or> x = u \<and> y = v \<or> x = v \<and> y = u)" |
|
833 |
(is "?lhs = ?rhs") |
|
834 |
proof (rule iffI) |
|
835 |
assume ?lhs |
|
836 |
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
|
837 |
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 | 838 |
\<Longrightarrow> x = y \<or> x = 0 \<and> y = 1 \<or> x = 1 \<and> y = 0)" |
839 |
by (auto simp: simple_path_def subpath_def) |
|
840 |
{ fix x y |
|
841 |
assume "x \<in> closed_segment u v" "y \<in> closed_segment u v" "g x = g y" |
|
842 |
then have "x = y \<or> x = u \<and> y = v \<or> x = v \<and> y = u" |
|
843 |
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
|
844 |
by (auto simp: closed_segment_real_eq image_affinity_atLeastAtMost divide_simps |
62390 | 845 |
split: if_split_asm) |
60303 | 846 |
} moreover |
847 |
have "path(subpath u v g) \<and> u\<noteq>v" |
|
848 |
using sim [of "1/3" "2/3"] p |
|
849 |
by (auto simp: subpath_def) |
|
850 |
ultimately show ?rhs |
|
851 |
by metis |
|
852 |
next |
|
853 |
assume ?rhs |
|
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
60420
diff
changeset
|
854 |
then |
60303 | 855 |
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" |
856 |
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" |
|
857 |
and ne: "u < v \<or> v < u" |
|
858 |
and psp: "path (subpath u v g)" |
|
859 |
by (auto simp: closed_segment_real_eq image_affinity_atLeastAtMost) |
|
860 |
have [simp]: "\<And>x. u + x * v = v + x * u \<longleftrightarrow> u=v \<or> x=1" |
|
861 |
by algebra |
|
862 |
show ?lhs using psp ne |
|
863 |
unfolding simple_path_def subpath_def |
|
864 |
by (fastforce simp add: algebra_simps affine_ineq mult_left_mono crossproduct_eq dest: d1 d2) |
|
865 |
qed |
|
866 |
||
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
60420
diff
changeset
|
867 |
lemma arc_subpath_eq: |
60303 | 868 |
"arc(subpath u v g) \<longleftrightarrow> path(subpath u v g) \<and> u\<noteq>v \<and> inj_on g (closed_segment u v)" |
869 |
(is "?lhs = ?rhs") |
|
870 |
proof (rule iffI) |
|
871 |
assume ?lhs |
|
872 |
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
|
873 |
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 | 874 |
\<Longrightarrow> x = y)" |
875 |
by (auto simp: arc_def inj_on_def subpath_def) |
|
876 |
{ fix x y |
|
877 |
assume "x \<in> closed_segment u v" "y \<in> closed_segment u v" "g x = g y" |
|
878 |
then have "x = y" |
|
879 |
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
|
880 |
by (force simp add: inj_on_def closed_segment_real_eq image_affinity_atLeastAtMost divide_simps |
62390 | 881 |
split: if_split_asm) |
60303 | 882 |
} moreover |
883 |
have "path(subpath u v g) \<and> u\<noteq>v" |
|
884 |
using sim [of "1/3" "2/3"] p |
|
885 |
by (auto simp: subpath_def) |
|
886 |
ultimately show ?rhs |
|
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
60420
diff
changeset
|
887 |
unfolding inj_on_def |
60303 | 888 |
by metis |
889 |
next |
|
890 |
assume ?rhs |
|
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
60420
diff
changeset
|
891 |
then |
60303 | 892 |
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" |
893 |
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" |
|
894 |
and ne: "u < v \<or> v < u" |
|
895 |
and psp: "path (subpath u v g)" |
|
896 |
by (auto simp: inj_on_def closed_segment_real_eq image_affinity_atLeastAtMost) |
|
897 |
show ?lhs using psp ne |
|
898 |
unfolding arc_def subpath_def inj_on_def |
|
899 |
by (auto simp: algebra_simps affine_ineq mult_left_mono crossproduct_eq dest: d1 d2) |
|
900 |
qed |
|
901 |
||
902 |
||
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
60420
diff
changeset
|
903 |
lemma simple_path_subpath: |
60303 | 904 |
assumes "simple_path g" "u \<in> {0..1}" "v \<in> {0..1}" "u \<noteq> v" |
905 |
shows "simple_path(subpath u v g)" |
|
906 |
using assms |
|
907 |
apply (simp add: simple_path_subpath_eq simple_path_imp_path) |
|
908 |
apply (simp add: simple_path_def closed_segment_real_eq image_affinity_atLeastAtMost, fastforce) |
|
909 |
done |
|
910 |
||
911 |
lemma arc_simple_path_subpath: |
|
912 |
"\<lbrakk>simple_path g; u \<in> {0..1}; v \<in> {0..1}; g u \<noteq> g v\<rbrakk> \<Longrightarrow> arc(subpath u v g)" |
|
913 |
by (force intro: simple_path_subpath simple_path_imp_arc) |
|
914 |
||
915 |
lemma arc_subpath_arc: |
|
916 |
"\<lbrakk>arc g; u \<in> {0..1}; v \<in> {0..1}; u \<noteq> v\<rbrakk> \<Longrightarrow> arc(subpath u v g)" |
|
917 |
by (meson arc_def arc_imp_simple_path arc_simple_path_subpath inj_onD) |
|
918 |
||
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
60420
diff
changeset
|
919 |
lemma arc_simple_path_subpath_interior: |
60303 | 920 |
"\<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)" |
921 |
apply (rule arc_simple_path_subpath) |
|
922 |
apply (force simp: simple_path_def)+ |
|
923 |
done |
|
924 |
||
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
60420
diff
changeset
|
925 |
lemma path_image_subpath_subset: |
60303 | 926 |
"\<lbrakk>path g; u \<in> {0..1}; v \<in> {0..1}\<rbrakk> \<Longrightarrow> path_image(subpath u v g) \<subseteq> path_image g" |
61762
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61738
diff
changeset
|
927 |
apply (simp add: closed_segment_real_eq image_affinity_atLeastAtMost path_image_subpath) |
60303 | 928 |
apply (auto simp: path_image_def) |
929 |
done |
|
930 |
||
931 |
lemma join_subpaths_middle: "subpath (0) ((1 / 2)) p +++ subpath ((1 / 2)) 1 p = p" |
|
932 |
by (rule ext) (simp add: joinpaths_def subpath_def divide_simps) |
|
53640 | 933 |
|
61518
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
934 |
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
|
935 |
|
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
936 |
lemma subpath_to_frontier_explicit: |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
937 |
fixes S :: "'a::metric_space set" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
938 |
assumes g: "path g" and "pathfinish g \<notin> S" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
939 |
obtains u where "0 \<le> u" "u \<le> 1" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
940 |
"\<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
|
941 |
"(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
|
942 |
proof - |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
943 |
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
|
944 |
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
|
945 |
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
|
946 |
using compact_eq_bounded_closed apply fastforce |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
947 |
done |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
948 |
have "1 \<in> {u. g u \<in> closure (- S)}" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
949 |
using assms by (simp add: pathfinish_def closure_def) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
950 |
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
|
951 |
using atLeastAtMost_iff zero_le_one by blast |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
952 |
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
|
953 |
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
|
954 |
using compact_attains_inf [OF com dis] by fastforce |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
955 |
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
|
956 |
using closure_def by fastforce |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
957 |
{ assume "u \<noteq> 0" |
61808 | 958 |
then have "u > 0" using \<open>0 \<le> u\<close> by auto |
61518
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
959 |
{ fix e::real assume "e > 0" |
62397
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62381
diff
changeset
|
960 |
obtain d where "d>0" and d: "\<And>x'. \<lbrakk>x' \<in> {0..1}; dist x' u \<le> d\<rbrakk> \<Longrightarrow> dist (g x') (g u) < e" |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62381
diff
changeset
|
961 |
using continuous_onE [OF gcon _ \<open>e > 0\<close>] \<open>0 \<le> _\<close> \<open>_ \<le> 1\<close> atLeastAtMost_iff by auto |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62381
diff
changeset
|
962 |
have *: "dist (max 0 (u - d / 2)) u \<le> d" |
61808 | 963 |
using \<open>0 \<le> u\<close> \<open>u \<le> 1\<close> \<open>d > 0\<close> by (simp add: dist_real_def) |
61518
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
964 |
have "\<exists>y\<in>S. dist y (g u) < e" |
61808 | 965 |
using \<open>0 < u\<close> \<open>u \<le> 1\<close> \<open>d > 0\<close> |
61518
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
966 |
by (force intro: d [OF _ *] umin') |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
967 |
} |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
968 |
then have "g u \<in> closure S" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
969 |
by (simp add: frontier_def closure_approachable) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
970 |
} |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
971 |
then show ?thesis |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
972 |
apply (rule_tac u=u in that) |
61808 | 973 |
apply (auto simp: \<open>0 \<le> u\<close> \<open>u \<le> 1\<close> gu interior_closure umin) |
974 |
using \<open>_ \<le> 1\<close> interior_closure umin apply fastforce |
|
61518
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
975 |
done |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
976 |
qed |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
977 |
|
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
978 |
lemma subpath_to_frontier_strong: |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
979 |
assumes g: "path g" and "pathfinish g \<notin> S" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
980 |
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
|
981 |
"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
|
982 |
proof - |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
983 |
obtain u where "0 \<le> u" "u \<le> 1" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
984 |
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
|
985 |
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
|
986 |
using subpath_to_frontier_explicit [OF assms] by blast |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
987 |
show ?thesis |
61808 | 988 |
apply (rule that [OF \<open>0 \<le> u\<close> \<open>u \<le> 1\<close>]) |
61518
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
989 |
apply (simp add: gunot) |
61808 | 990 |
using \<open>0 \<le> u\<close> u0 by (force simp: subpath_def gxin) |
61518
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
991 |
qed |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
992 |
|
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
993 |
lemma subpath_to_frontier: |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
994 |
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
|
995 |
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
|
996 |
proof - |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
997 |
obtain u where "0 \<le> u" "u \<le> 1" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
998 |
and notin: "g u \<notin> interior S" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
999 |
and disj: "u = 0 \<or> |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
1000 |
(\<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
|
1001 |
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
|
1002 |
show ?thesis |
61808 | 1003 |
apply (rule that [OF \<open>0 \<le> u\<close> \<open>u \<le> 1\<close>]) |
61518
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
1004 |
apply (metis DiffI disj frontier_def g0 notin pathstart_def) |
61808 | 1005 |
using \<open>0 \<le> u\<close> g0 disj |
61762
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61738
diff
changeset
|
1006 |
apply (simp add: path_image_subpath_gen) |
61518
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
1007 |
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
|
1008 |
apply (rename_tac y) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
1009 |
apply (drule_tac x="y/u" in spec) |
62390 | 1010 |
apply (auto split: if_split_asm) |
61518
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
1011 |
done |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
1012 |
qed |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
1013 |
|
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
1014 |
lemma exists_path_subpath_to_frontier: |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
1015 |
fixes S :: "'a::real_normed_vector set" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
1016 |
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
|
1017 |
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
|
1018 |
"path_image h - {pathfinish h} \<subseteq> interior S" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
1019 |
"pathfinish h \<in> frontier S" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
1020 |
proof - |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
1021 |
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
|
1022 |
using subpath_to_frontier [OF assms] by blast |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
1023 |
show ?thesis |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
1024 |
apply (rule that [of "subpath 0 u g"]) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
1025 |
using assms u |
61762
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61738
diff
changeset
|
1026 |
apply (simp_all add: path_image_subpath) |
61518
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
1027 |
apply (simp add: pathstart_def) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
1028 |
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
|
1029 |
done |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
1030 |
qed |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
1031 |
|
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
1032 |
lemma exists_path_subpath_to_frontier_closed: |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
1033 |
fixes S :: "'a::real_normed_vector set" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
1034 |
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
|
1035 |
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
|
1036 |
"pathfinish h \<in> frontier S" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
1037 |
proof - |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
1038 |
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
|
1039 |
"path_image h - {pathfinish h} \<subseteq> interior S" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
1040 |
"pathfinish h \<in> frontier S" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
1041 |
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
|
1042 |
show ?thesis |
61808 | 1043 |
apply (rule that [OF \<open>path h\<close>]) |
61518
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
1044 |
using assms h |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
1045 |
apply auto |
62087
44841d07ef1d
revisions to limits and derivatives, plus new lemmas
paulson
parents:
61808
diff
changeset
|
1046 |
apply (metis Diff_single_insert frontier_subset_eq insert_iff interior_subset subset_iff) |
61518
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
1047 |
done |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
1048 |
qed |
49653 | 1049 |
|
60420 | 1050 |
subsection \<open>Reparametrizing a closed curve to start at some chosen point\<close> |
36583 | 1051 |
|
53640 | 1052 |
definition shiftpath :: "real \<Rightarrow> (real \<Rightarrow> 'a::topological_space) \<Rightarrow> real \<Rightarrow> 'a" |
1053 |
where "shiftpath a f = (\<lambda>x. if (a + x) \<le> 1 then f (a + x) else f (a + x - 1))" |
|
36583 | 1054 |
|
53640 | 1055 |
lemma pathstart_shiftpath: "a \<le> 1 \<Longrightarrow> pathstart (shiftpath a g) = g a" |
36583 | 1056 |
unfolding pathstart_def shiftpath_def by auto |
1057 |
||
49653 | 1058 |
lemma pathfinish_shiftpath: |
53640 | 1059 |
assumes "0 \<le> a" |
1060 |
and "pathfinish g = pathstart g" |
|
1061 |
shows "pathfinish (shiftpath a g) = g a" |
|
1062 |
using assms |
|
1063 |
unfolding pathstart_def pathfinish_def shiftpath_def |
|
36583 | 1064 |
by auto |
1065 |
||
1066 |
lemma endpoints_shiftpath: |
|
53640 | 1067 |
assumes "pathfinish g = pathstart g" |
1068 |
and "a \<in> {0 .. 1}" |
|
1069 |
shows "pathfinish (shiftpath a g) = g a" |
|
1070 |
and "pathstart (shiftpath a g) = g a" |
|
1071 |
using assms |
|
1072 |
by (auto intro!: pathfinish_shiftpath pathstart_shiftpath) |
|
36583 | 1073 |
|
1074 |
lemma closed_shiftpath: |
|
53640 | 1075 |
assumes "pathfinish g = pathstart g" |
1076 |
and "a \<in> {0..1}" |
|
1077 |
shows "pathfinish (shiftpath a g) = pathstart (shiftpath a g)" |
|
1078 |
using endpoints_shiftpath[OF assms] |
|
1079 |
by auto |
|
36583 | 1080 |
|
1081 |
lemma path_shiftpath: |
|
53640 | 1082 |
assumes "path g" |
1083 |
and "pathfinish g = pathstart g" |
|
1084 |
and "a \<in> {0..1}" |
|
1085 |
shows "path (shiftpath a g)" |
|
49653 | 1086 |
proof - |
53640 | 1087 |
have *: "{0 .. 1} = {0 .. 1-a} \<union> {1-a .. 1}" |
1088 |
using assms(3) by auto |
|
49653 | 1089 |
have **: "\<And>x. x + a = 1 \<Longrightarrow> g (x + a - 1) = g (x + a)" |
53640 | 1090 |
using assms(2)[unfolded pathfinish_def pathstart_def] |
1091 |
by auto |
|
49653 | 1092 |
show ?thesis |
1093 |
unfolding path_def shiftpath_def * |
|
62397
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62381
diff
changeset
|
1094 |
apply (rule continuous_on_closed_Un) |
49653 | 1095 |
apply (rule closed_real_atLeastAtMost)+ |
53640 | 1096 |
apply (rule continuous_on_eq[of _ "g \<circ> (\<lambda>x. a + x)"]) |
1097 |
prefer 3 |
|
1098 |
apply (rule continuous_on_eq[of _ "g \<circ> (\<lambda>x. a - 1 + x)"]) |
|
1099 |
prefer 3 |
|
56371
fb9ae0727548
extend continuous_intros; remove continuous_on_intros and isCont_intros
hoelzl
parents:
56188
diff
changeset
|
1100 |
apply (rule continuous_intros)+ |
53640 | 1101 |
prefer 2 |
56371
fb9ae0727548
extend continuous_intros; remove continuous_on_intros and isCont_intros
hoelzl
parents:
56188
diff
changeset
|
1102 |
apply (rule continuous_intros)+ |
49653 | 1103 |
apply (rule_tac[1-2] continuous_on_subset[OF assms(1)[unfolded path_def]]) |
1104 |
using assms(3) and ** |
|
53640 | 1105 |
apply auto |
1106 |
apply (auto simp add: field_simps) |
|
49653 | 1107 |
done |
1108 |
qed |
|
36583 | 1109 |
|
49653 | 1110 |
lemma shiftpath_shiftpath: |
53640 | 1111 |
assumes "pathfinish g = pathstart g" |
1112 |
and "a \<in> {0..1}" |
|
1113 |
and "x \<in> {0..1}" |
|
36583 | 1114 |
shows "shiftpath (1 - a) (shiftpath a g) x = g x" |
53640 | 1115 |
using assms |
1116 |
unfolding pathfinish_def pathstart_def shiftpath_def |
|
1117 |
by auto |
|
36583 | 1118 |
|
1119 |
lemma path_image_shiftpath: |
|
53640 | 1120 |
assumes "a \<in> {0..1}" |
1121 |
and "pathfinish g = pathstart g" |
|
1122 |
shows "path_image (shiftpath a g) = path_image g" |
|
49653 | 1123 |
proof - |
1124 |
{ fix x |
|
53640 | 1125 |
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 | 1126 |
then have "\<exists>y\<in>{0..1} \<inter> {x. a + x \<le> 1}. g x = g (a + y)" |
49653 | 1127 |
proof (cases "a \<le> x") |
1128 |
case False |
|
49654 | 1129 |
then show ?thesis |
49653 | 1130 |
apply (rule_tac x="1 + x - a" in bexI) |
36583 | 1131 |
using as(1,2) and as(3)[THEN bspec[where x="1 + x - a"]] and assms(1) |
49653 | 1132 |
apply (auto simp add: field_simps atomize_not) |
1133 |
done |
|
1134 |
next |
|
1135 |
case True |
|
53640 | 1136 |
then show ?thesis |
1137 |
using as(1-2) and assms(1) |
|
1138 |
apply (rule_tac x="x - a" in bexI) |
|
1139 |
apply (auto simp add: field_simps) |
|
1140 |
done |
|
49653 | 1141 |
qed |
1142 |
} |
|
49654 | 1143 |
then show ?thesis |
53640 | 1144 |
using assms |
1145 |
unfolding shiftpath_def path_image_def pathfinish_def pathstart_def |
|
1146 |
by (auto simp add: image_iff) |
|
49653 | 1147 |
qed |
1148 |
||
36583 | 1149 |
|
60420 | 1150 |
subsection \<open>Special case of straight-line paths\<close> |
36583 | 1151 |
|
49653 | 1152 |
definition linepath :: "'a::real_normed_vector \<Rightarrow> 'a \<Rightarrow> real \<Rightarrow> 'a" |
1153 |
where "linepath a b = (\<lambda>x. (1 - x) *\<^sub>R a + x *\<^sub>R b)" |
|
36583 | 1154 |
|
53640 | 1155 |
lemma pathstart_linepath[simp]: "pathstart (linepath a b) = a" |
1156 |
unfolding pathstart_def linepath_def |
|
1157 |
by auto |
|
36583 | 1158 |
|
53640 | 1159 |
lemma pathfinish_linepath[simp]: "pathfinish (linepath a b) = b" |
1160 |
unfolding pathfinish_def linepath_def |
|
1161 |
by auto |
|
36583 | 1162 |
|
1163 |
lemma continuous_linepath_at[intro]: "continuous (at x) (linepath a b)" |
|
53640 | 1164 |
unfolding linepath_def |
1165 |
by (intro continuous_intros) |
|
36583 | 1166 |
|
61762
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61738
diff
changeset
|
1167 |
lemma continuous_on_linepath [intro,continuous_intros]: "continuous_on s (linepath a b)" |
53640 | 1168 |
using continuous_linepath_at |
1169 |
by (auto intro!: continuous_at_imp_continuous_on) |
|
36583 | 1170 |
|
62618
f7f2467ab854
Refactoring (moving theorems into better locations), plus a bit of new material
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
1171 |
lemma path_linepath[iff]: "path (linepath a b)" |
53640 | 1172 |
unfolding path_def |
1173 |
by (rule continuous_on_linepath) |
|
36583 | 1174 |
|
53640 | 1175 |
lemma path_image_linepath[simp]: "path_image (linepath a b) = closed_segment a b" |
49653 | 1176 |
unfolding path_image_def segment linepath_def |
60303 | 1177 |
by auto |
49653 | 1178 |
|
53640 | 1179 |
lemma reversepath_linepath[simp]: "reversepath (linepath a b) = linepath b a" |
49653 | 1180 |
unfolding reversepath_def linepath_def |
36583 | 1181 |
by auto |
1182 |
||
61762
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61738
diff
changeset
|
1183 |
lemma linepath_0 [simp]: "linepath 0 b x = x *\<^sub>R b" |
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61738
diff
changeset
|
1184 |
by (simp add: linepath_def) |
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61738
diff
changeset
|
1185 |
|
60303 | 1186 |
lemma arc_linepath: |
62618
f7f2467ab854
Refactoring (moving theorems into better locations), plus a bit of new material
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
1187 |
assumes "a \<noteq> b" shows [simp]: "arc (linepath a b)" |
36583 | 1188 |
proof - |
53640 | 1189 |
{ |
1190 |
fix x y :: "real" |
|
36583 | 1191 |
assume "x *\<^sub>R b + y *\<^sub>R a = x *\<^sub>R a + y *\<^sub>R b" |
53640 | 1192 |
then have "(x - y) *\<^sub>R a = (x - y) *\<^sub>R b" |
1193 |
by (simp add: algebra_simps) |
|
1194 |
with assms have "x = y" |
|
1195 |
by simp |
|
1196 |
} |
|
49654 | 1197 |
then show ?thesis |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
60420
diff
changeset
|
1198 |
unfolding arc_def inj_on_def |
60303 | 1199 |
by (simp add: path_linepath) (force simp: algebra_simps linepath_def) |
49653 | 1200 |
qed |
36583 | 1201 |
|
53640 | 1202 |
lemma simple_path_linepath[intro]: "a \<noteq> b \<Longrightarrow> simple_path (linepath a b)" |
60303 | 1203 |
by (simp add: arc_imp_simple_path arc_linepath) |
49653 | 1204 |
|
61711
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
1205 |
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
|
1206 |
by (simp add: linepath_def real_vector.scale_left_diff_distrib) |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1207 |
|
61711
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
1208 |
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
|
1209 |
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
|
1210 |
|
62618
f7f2467ab854
Refactoring (moving theorems into better locations), plus a bit of new material
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
1211 |
lemma linepath_of_real: "(linepath (of_real a) (of_real b) x) = of_real ((1 - x)*a + x*b)" |
f7f2467ab854
Refactoring (moving theorems into better locations), plus a bit of new material
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
1212 |
by (simp add: scaleR_conv_of_real linepath_def) |
f7f2467ab854
Refactoring (moving theorems into better locations), plus a bit of new material
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
1213 |
|
f7f2467ab854
Refactoring (moving theorems into better locations), plus a bit of new material
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
1214 |
lemma of_real_linepath: "of_real (linepath a b x) = linepath (of_real a) (of_real b) x" |
f7f2467ab854
Refactoring (moving theorems into better locations), plus a bit of new material
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
1215 |
by (metis linepath_of_real mult.right_neutral of_real_def real_scaleR_def) |
f7f2467ab854
Refactoring (moving theorems into better locations), plus a bit of new material
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
1216 |
|
f7f2467ab854
Refactoring (moving theorems into better locations), plus a bit of new material
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
1217 |
|
f7f2467ab854
Refactoring (moving theorems into better locations), plus a bit of new material
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
1218 |
subsection\<open>Segments via convex hulls\<close> |
f7f2467ab854
Refactoring (moving theorems into better locations), plus a bit of new material
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
1219 |
|
f7f2467ab854
Refactoring (moving theorems into better locations), plus a bit of new material
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
1220 |
lemma segments_subset_convex_hull: |
f7f2467ab854
Refactoring (moving theorems into better locations), plus a bit of new material
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
1221 |
"closed_segment a b \<subseteq> (convex hull {a,b,c})" |
f7f2467ab854
Refactoring (moving theorems into better locations), plus a bit of new material
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
1222 |
"closed_segment a c \<subseteq> (convex hull {a,b,c})" |
f7f2467ab854
Refactoring (moving theorems into better locations), plus a bit of new material
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
1223 |
"closed_segment b c \<subseteq> (convex hull {a,b,c})" |
f7f2467ab854
Refactoring (moving theorems into better locations), plus a bit of new material
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
1224 |
"closed_segment b a \<subseteq> (convex hull {a,b,c})" |
f7f2467ab854
Refactoring (moving theorems into better locations), plus a bit of new material
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
1225 |
"closed_segment c a \<subseteq> (convex hull {a,b,c})" |
f7f2467ab854
Refactoring (moving theorems into better locations), plus a bit of new material
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
1226 |
"closed_segment c b \<subseteq> (convex hull {a,b,c})" |
f7f2467ab854
Refactoring (moving theorems into better locations), plus a bit of new material
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
1227 |
by (auto simp: segment_convex_hull linepath_of_real elim!: rev_subsetD [OF _ hull_mono]) |
f7f2467ab854
Refactoring (moving theorems into better locations), plus a bit of new material
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
1228 |
|
f7f2467ab854
Refactoring (moving theorems into better locations), plus a bit of new material
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
1229 |
lemma midpoints_in_convex_hull: |
f7f2467ab854
Refactoring (moving theorems into better locations), plus a bit of new material
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
1230 |
assumes "x \<in> convex hull s" "y \<in> convex hull s" |
f7f2467ab854
Refactoring (moving theorems into better locations), plus a bit of new material
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
1231 |
shows "midpoint x y \<in> convex hull s" |
f7f2467ab854
Refactoring (moving theorems into better locations), plus a bit of new material
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
1232 |
proof - |
f7f2467ab854
Refactoring (moving theorems into better locations), plus a bit of new material
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
1233 |
have "(1 - inverse(2)) *\<^sub>R x + inverse(2) *\<^sub>R y \<in> convex hull s" |
f7f2467ab854
Refactoring (moving theorems into better locations), plus a bit of new material
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
1234 |
apply (rule convexD_alt) |
f7f2467ab854
Refactoring (moving theorems into better locations), plus a bit of new material
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
1235 |
using assms |
f7f2467ab854
Refactoring (moving theorems into better locations), plus a bit of new material
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
1236 |
apply (auto simp: convex_convex_hull) |
f7f2467ab854
Refactoring (moving theorems into better locations), plus a bit of new material
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
1237 |
done |
f7f2467ab854
Refactoring (moving theorems into better locations), plus a bit of new material
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
1238 |
then show ?thesis |
f7f2467ab854
Refactoring (moving theorems into better locations), plus a bit of new material
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
1239 |
by (simp add: midpoint_def algebra_simps) |
f7f2467ab854
Refactoring (moving theorems into better locations), plus a bit of new material
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
1240 |
qed |
f7f2467ab854
Refactoring (moving theorems into better locations), plus a bit of new material
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
1241 |
|
f7f2467ab854
Refactoring (moving theorems into better locations), plus a bit of new material
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
1242 |
lemma not_in_interior_convex_hull_3: |
f7f2467ab854
Refactoring (moving theorems into better locations), plus a bit of new material
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
1243 |
fixes a :: "complex" |
f7f2467ab854
Refactoring (moving theorems into better locations), plus a bit of new material
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
1244 |
shows "a \<notin> interior(convex hull {a,b,c})" |
f7f2467ab854
Refactoring (moving theorems into better locations), plus a bit of new material
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
1245 |
"b \<notin> interior(convex hull {a,b,c})" |
f7f2467ab854
Refactoring (moving theorems into better locations), plus a bit of new material
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
1246 |
"c \<notin> interior(convex hull {a,b,c})" |
f7f2467ab854
Refactoring (moving theorems into better locations), plus a bit of new material
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
1247 |
by (auto simp: card_insert_le_m1 not_in_interior_convex_hull) |
f7f2467ab854
Refactoring (moving theorems into better locations), plus a bit of new material
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
1248 |
|
f7f2467ab854
Refactoring (moving theorems into better locations), plus a bit of new material
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
1249 |
lemma midpoint_in_closed_segment [simp]: "midpoint a b \<in> closed_segment a b" |
f7f2467ab854
Refactoring (moving theorems into better locations), plus a bit of new material
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
1250 |
using midpoints_in_convex_hull segment_convex_hull by blast |
f7f2467ab854
Refactoring (moving theorems into better locations), plus a bit of new material
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
1251 |
|
f7f2467ab854
Refactoring (moving theorems into better locations), plus a bit of new material
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
1252 |
lemma midpoint_in_open_segment [simp]: "midpoint a b \<in> open_segment a b \<longleftrightarrow> a \<noteq> b" |
f7f2467ab854
Refactoring (moving theorems into better locations), plus a bit of new material
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
1253 |
by (simp add: midpoint_eq_endpoint(1) midpoint_eq_endpoint(2) open_segment_def) |
f7f2467ab854
Refactoring (moving theorems into better locations), plus a bit of new material
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
1254 |
|
36583 | 1255 |
|
60420 | 1256 |
subsection \<open>Bounding a point away from a path\<close> |
36583 | 1257 |
|
1258 |
lemma not_on_path_ball: |
|
1259 |
fixes g :: "real \<Rightarrow> 'a::heine_borel" |
|
53640 | 1260 |
assumes "path g" |
1261 |
and "z \<notin> path_image g" |
|
1262 |
shows "\<exists>e > 0. ball z e \<inter> path_image g = {}" |
|
49653 | 1263 |
proof - |
1264 |
obtain a where "a \<in> path_image g" "\<forall>y \<in> path_image g. dist z a \<le> dist z y" |
|
62381
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
1265 |
apply (rule distance_attains_inf[OF _ path_image_nonempty, of g z]) |
36583 | 1266 |
using compact_path_image[THEN compact_imp_closed, OF assms(1)] by auto |
49654 | 1267 |
then show ?thesis |
49653 | 1268 |
apply (rule_tac x="dist z a" in exI) |
1269 |
using assms(2) |
|
1270 |
apply (auto intro!: dist_pos_lt) |
|
1271 |
done |
|
1272 |
qed |
|
36583 | 1273 |
|
1274 |
lemma not_on_path_cball: |
|
1275 |
fixes g :: "real \<Rightarrow> 'a::heine_borel" |
|
53640 | 1276 |
assumes "path g" |
1277 |
and "z \<notin> path_image g" |
|
49653 | 1278 |
shows "\<exists>e>0. cball z e \<inter> (path_image g) = {}" |
1279 |
proof - |
|
53640 | 1280 |
obtain e where "ball z e \<inter> path_image g = {}" "e > 0" |
49653 | 1281 |
using not_on_path_ball[OF assms] by auto |
53640 | 1282 |
moreover have "cball z (e/2) \<subseteq> ball z e" |
60420 | 1283 |
using \<open>e > 0\<close> by auto |
53640 | 1284 |
ultimately show ?thesis |
1285 |
apply (rule_tac x="e/2" in exI) |
|
1286 |
apply auto |
|
1287 |
done |
|
49653 | 1288 |
qed |
1289 |
||
36583 | 1290 |
|
61518
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
1291 |
section \<open>Path component, considered as a "joinability" relation (from Tom Hales)\<close> |
36583 | 1292 |
|
49653 | 1293 |
definition "path_component s x y \<longleftrightarrow> |
1294 |
(\<exists>g. path g \<and> path_image g \<subseteq> s \<and> pathstart g = x \<and> pathfinish g = y)" |
|
36583 | 1295 |
|
61426
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1296 |
abbreviation |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1297 |
"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
|
1298 |
|
53640 | 1299 |
lemmas path_defs = path_def pathstart_def pathfinish_def path_image_def path_component_def |
36583 | 1300 |
|
49653 | 1301 |
lemma path_component_mem: |
1302 |
assumes "path_component s x y" |
|
53640 | 1303 |
shows "x \<in> s" and "y \<in> s" |
1304 |
using assms |
|
1305 |
unfolding path_defs |
|
1306 |
by auto |
|
36583 | 1307 |
|
49653 | 1308 |
lemma path_component_refl: |
1309 |
assumes "x \<in> s" |
|
1310 |
shows "path_component s x x" |
|
1311 |
unfolding path_defs |
|
1312 |
apply (rule_tac x="\<lambda>u. x" in exI) |
|
53640 | 1313 |
using assms |
56371
fb9ae0727548
extend continuous_intros; remove continuous_on_intros and isCont_intros
hoelzl
parents:
56188
diff
changeset
|
1314 |
apply (auto intro!: continuous_intros) |
53640 | 1315 |
done |
36583 | 1316 |
|
1317 |
lemma path_component_refl_eq: "path_component s x x \<longleftrightarrow> x \<in> s" |
|
49653 | 1318 |
by (auto intro!: path_component_mem path_component_refl) |
36583 | 1319 |
|
1320 |
lemma path_component_sym: "path_component s x y \<Longrightarrow> path_component s y x" |
|
49653 | 1321 |
unfolding path_component_def |
1322 |
apply (erule exE) |
|
1323 |
apply (rule_tac x="reversepath g" in exI) |
|
1324 |
apply auto |
|
1325 |
done |
|
36583 | 1326 |
|
49653 | 1327 |
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
|
1328 |
assumes "path_component s x y" and "path_component s y z" |
49653 | 1329 |
shows "path_component s x z" |
1330 |
using assms |
|
1331 |
unfolding path_component_def |
|
53640 | 1332 |
apply (elim exE) |
49653 | 1333 |
apply (rule_tac x="g +++ ga" in exI) |
1334 |
apply (auto simp add: path_image_join) |
|
1335 |
done |
|
36583 | 1336 |
|
53640 | 1337 |
lemma path_component_of_subset: "s \<subseteq> t \<Longrightarrow> path_component s x y \<Longrightarrow> path_component t x y" |
36583 | 1338 |
unfolding path_component_def by auto |
1339 |
||
61426
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1340 |
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
|
1341 |
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
|
1342 |
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
|
1343 |
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
|
1344 |
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
|
1345 |
done |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1346 |
|
49653 | 1347 |
|
62620
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
1348 |
subsubsection \<open>Path components as sets\<close> |
36583 | 1349 |
|
49653 | 1350 |
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
|
1351 |
"path_component_set s x = |
49653 | 1352 |
{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
|
1353 |
by (auto simp: path_component_def) |
36583 | 1354 |
|
61426
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1355 |
lemma path_component_subset: "path_component_set s x \<subseteq> s" |
60303 | 1356 |
by (auto simp add: path_component_mem(2)) |
36583 | 1357 |
|
61426
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1358 |
lemma path_component_eq_empty: "path_component_set s x = {} \<longleftrightarrow> x \<notin> s" |
60303 | 1359 |
using path_component_mem path_component_refl_eq |
1360 |
by fastforce |
|
36583 | 1361 |
|
61426
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1362 |
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
|
1363 |
"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
|
1364 |
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
|
1365 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1366 |
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
|
1367 |
"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
|
1368 |
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
|
1369 |
|
60420 | 1370 |
subsection \<open>Path connectedness of a space\<close> |
36583 | 1371 |
|
49653 | 1372 |
definition "path_connected s \<longleftrightarrow> |
53640 | 1373 |
(\<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 | 1374 |
|
1375 |
lemma path_connected_component: "path_connected s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>y\<in>s. path_component s x y)" |
|
1376 |
unfolding path_connected_def path_component_def by auto |
|
1377 |
||
61426
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1378 |
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
|
1379 |
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
|
1380 |
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
|
1381 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1382 |
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
|
1383 |
"\<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
|
1384 |
by (metis path_component_mono path_connected_component_set) |
36583 | 1385 |
|
1386 |
lemma convex_imp_path_connected: |
|
1387 |
fixes s :: "'a::real_normed_vector set" |
|
53640 | 1388 |
assumes "convex s" |
1389 |
shows "path_connected s" |
|
49653 | 1390 |
unfolding path_connected_def |
53640 | 1391 |
apply rule |
1392 |
apply rule |
|
1393 |
apply (rule_tac x = "linepath x y" in exI) |
|
49653 | 1394 |
unfolding path_image_linepath |
1395 |
using assms [unfolded convex_contains_segment] |
|
1396 |
apply auto |
|
1397 |
done |
|
36583 | 1398 |
|
62620
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
1399 |
lemma path_connected_UNIV [iff]: "path_connected (UNIV :: 'a::real_normed_vector set)" |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
1400 |
by (simp add: convex_imp_path_connected) |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
1401 |
|
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
1402 |
lemma path_component_UNIV: "path_component_set UNIV x = (UNIV :: 'a::real_normed_vector set)" |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
1403 |
using path_connected_component_set by auto |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
1404 |
|
49653 | 1405 |
lemma path_connected_imp_connected: |
1406 |
assumes "path_connected s" |
|
1407 |
shows "connected s" |
|
1408 |
unfolding connected_def not_ex |
|
53640 | 1409 |
apply rule |
1410 |
apply rule |
|
1411 |
apply (rule ccontr) |
|
49653 | 1412 |
unfolding not_not |
53640 | 1413 |
apply (elim conjE) |
49653 | 1414 |
proof - |
1415 |
fix e1 e2 |
|
1416 |
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 | 1417 |
then obtain x1 x2 where obt:"x1 \<in> e1 \<inter> s" "x2 \<in> e2 \<inter> s" |
1418 |
by auto |
|
1419 |
then obtain g where g: "path g" "path_image g \<subseteq> s" "pathstart g = x1" "pathfinish g = x2" |
|
36583 | 1420 |
using assms[unfolded path_connected_def,rule_format,of x1 x2] by auto |
49653 | 1421 |
have *: "connected {0..1::real}" |
1422 |
by (auto intro!: convex_connected convex_real_interval) |
|
1423 |
have "{0..1} \<subseteq> {x \<in> {0..1}. g x \<in> e1} \<union> {x \<in> {0..1}. g x \<in> e2}" |
|
1424 |
using as(3) g(2)[unfolded path_defs] by blast |
|
1425 |
moreover have "{x \<in> {0..1}. g x \<in> e1} \<inter> {x \<in> {0..1}. g x \<in> e2} = {}" |
|
53640 | 1426 |
using as(4) g(2)[unfolded path_defs] |
1427 |
unfolding subset_eq |
|
1428 |
by auto |
|
49653 | 1429 |
moreover have "{x \<in> {0..1}. g x \<in> e1} \<noteq> {} \<and> {x \<in> {0..1}. g x \<in> e2} \<noteq> {}" |
53640 | 1430 |
using g(3,4)[unfolded path_defs] |
1431 |
using obt |
|
36583 | 1432 |
by (simp add: ex_in_conv [symmetric], metis zero_le_one order_refl) |
49653 | 1433 |
ultimately show False |
53640 | 1434 |
using *[unfolded connected_local not_ex, rule_format, |
1435 |
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
|
1436 |
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
|
1437 |
using continuous_openin_preimage[OF g(1)[unfolded path_def] as(2)] |
49653 | 1438 |
by auto |
1439 |
qed |
|
36583 | 1440 |
|
1441 |
lemma open_path_component: |
|
53593 | 1442 |
fixes s :: "'a::real_normed_vector set" |
49653 | 1443 |
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
|
1444 |
shows "open (path_component_set s x)" |
49653 | 1445 |
unfolding open_contains_ball |
1446 |
proof |
|
1447 |
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
|
1448 |
assume as: "y \<in> path_component_set s x" |
49654 | 1449 |
then have "y \<in> s" |
49653 | 1450 |
apply - |
1451 |
apply (rule path_component_mem(2)) |
|
1452 |
unfolding mem_Collect_eq |
|
1453 |
apply auto |
|
1454 |
done |
|
53640 | 1455 |
then obtain e where e: "e > 0" "ball y e \<subseteq> s" |
1456 |
using assms[unfolded open_contains_ball] |
|
1457 |
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
|
1458 |
show "\<exists>e > 0. ball y e \<subseteq> path_component_set s x" |
49653 | 1459 |
apply (rule_tac x=e in exI) |
60420 | 1460 |
apply (rule,rule \<open>e>0\<close>) |
53640 | 1461 |
apply rule |
49653 | 1462 |
unfolding mem_ball mem_Collect_eq |
1463 |
proof - |
|
1464 |
fix z |
|
1465 |
assume "dist y z < e" |
|
49654 | 1466 |
then show "path_component s x z" |
53640 | 1467 |
apply (rule_tac path_component_trans[of _ _ y]) |
1468 |
defer |
|
49653 | 1469 |
apply (rule path_component_of_subset[OF e(2)]) |
1470 |
apply (rule convex_imp_path_connected[OF convex_ball, unfolded path_connected_component, rule_format]) |
|
60420 | 1471 |
using \<open>e > 0\<close> as |
49653 | 1472 |
apply auto |
1473 |
done |
|
1474 |
qed |
|
1475 |
qed |
|
36583 | 1476 |
|
1477 |
lemma open_non_path_component: |
|
53593 | 1478 |
fixes s :: "'a::real_normed_vector set" |
49653 | 1479 |
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
|
1480 |
shows "open (s - path_component_set s x)" |
49653 | 1481 |
unfolding open_contains_ball |
1482 |
proof |
|
1483 |
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
|
1484 |
assume as: "y \<in> s - path_component_set s x" |
53640 | 1485 |
then obtain e where e: "e > 0" "ball y e \<subseteq> s" |
1486 |
using assms [unfolded open_contains_ball] |
|
1487 |
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
|
1488 |
show "\<exists>e>0. ball y e \<subseteq> s - path_component_set s x" |
49653 | 1489 |
apply (rule_tac x=e in exI) |
53640 | 1490 |
apply rule |
60420 | 1491 |
apply (rule \<open>e>0\<close>) |
53640 | 1492 |
apply rule |
1493 |
apply rule |
|
1494 |
defer |
|
49653 | 1495 |
proof (rule ccontr) |
1496 |
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
|
1497 |
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
|
1498 |
then have "y \<in> path_component_set s x" |
60420 | 1499 |
unfolding not_not mem_Collect_eq using \<open>e>0\<close> |
49653 | 1500 |
apply - |
1501 |
apply (rule path_component_trans, assumption) |
|
1502 |
apply (rule path_component_of_subset[OF e(2)]) |
|
1503 |
apply (rule convex_imp_path_connected[OF convex_ball, unfolded path_connected_component, rule_format]) |
|
1504 |
apply auto |
|
1505 |
done |
|
53640 | 1506 |
then show False |
1507 |
using as by auto |
|
49653 | 1508 |
qed (insert e(2), auto) |
1509 |
qed |
|
36583 | 1510 |
|
1511 |
lemma connected_open_path_connected: |
|
53593 | 1512 |
fixes s :: "'a::real_normed_vector set" |
53640 | 1513 |
assumes "open s" |
1514 |
and "connected s" |
|
49653 | 1515 |
shows "path_connected s" |
1516 |
unfolding path_connected_component_set |
|
1517 |
proof (rule, rule, rule path_component_subset, rule) |
|
1518 |
fix x y |
|
53640 | 1519 |
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
|
1520 |
show "y \<in> path_component_set s x" |
49653 | 1521 |
proof (rule ccontr) |
53640 | 1522 |
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
|
1523 |
moreover have "path_component_set s x \<inter> s \<noteq> {}" |
60420 | 1524 |
using \<open>x \<in> s\<close> path_component_eq_empty path_component_subset[of s x] |
53640 | 1525 |
by auto |
49653 | 1526 |
ultimately |
1527 |
show False |
|
60420 | 1528 |
using \<open>y \<in> s\<close> open_non_path_component[OF assms(1)] open_path_component[OF assms(1)] |
53640 | 1529 |
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
|
1530 |
of "path_component_set s x" "s - path_component_set s x"] |
49653 | 1531 |
by auto |
1532 |
qed |
|
1533 |
qed |
|
36583 | 1534 |
|
1535 |
lemma path_connected_continuous_image: |
|
53640 | 1536 |
assumes "continuous_on s f" |
1537 |
and "path_connected s" |
|
49653 | 1538 |
shows "path_connected (f ` s)" |
1539 |
unfolding path_connected_def |
|
1540 |
proof (rule, rule) |
|
1541 |
fix x' y' |
|
1542 |
assume "x' \<in> f ` s" "y' \<in> f ` s" |
|
53640 | 1543 |
then obtain x y where x: "x \<in> s" and y: "y \<in> s" and x': "x' = f x" and y': "y' = f y" |
1544 |
by auto |
|
1545 |
from x y obtain g where "path g \<and> path_image g \<subseteq> s \<and> pathstart g = x \<and> pathfinish g = y" |
|
1546 |
using assms(2)[unfolded path_connected_def] by fast |
|
49654 | 1547 |
then show "\<exists>g. path g \<and> path_image g \<subseteq> f ` s \<and> pathstart g = x' \<and> pathfinish g = y'" |
53640 | 1548 |
unfolding x' y' |
49653 | 1549 |
apply (rule_tac x="f \<circ> g" in exI) |
1550 |
unfolding path_defs |
|
51481
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51478
diff
changeset
|
1551 |
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
|
1552 |
apply auto |
49653 | 1553 |
done |
1554 |
qed |
|
36583 | 1555 |
|
61518
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
1556 |
lemma path_connected_segment: |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
1557 |
fixes a :: "'a::real_normed_vector" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
1558 |
shows "path_connected (closed_segment a b)" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
1559 |
by (simp add: convex_imp_path_connected) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
1560 |
|
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
1561 |
lemma path_connected_open_segment: |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
1562 |
fixes a :: "'a::real_normed_vector" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
1563 |
shows "path_connected (open_segment a b)" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
1564 |
by (simp add: convex_imp_path_connected) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
1565 |
|
36583 | 1566 |
lemma homeomorphic_path_connectedness: |
53640 | 1567 |
"s homeomorphic t \<Longrightarrow> path_connected s \<longleftrightarrow> path_connected t" |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
1568 |
unfolding homeomorphic_def homeomorphism_def by (metis path_connected_continuous_image) |
36583 | 1569 |
|
1570 |
lemma path_connected_empty: "path_connected {}" |
|
1571 |
unfolding path_connected_def by auto |
|
1572 |
||
1573 |
lemma path_connected_singleton: "path_connected {a}" |
|
1574 |
unfolding path_connected_def pathstart_def pathfinish_def path_image_def |
|
53640 | 1575 |
apply clarify |
1576 |
apply (rule_tac x="\<lambda>x. a" in exI) |
|
1577 |
apply (simp add: image_constant_conv) |
|
36583 | 1578 |
apply (simp add: path_def continuous_on_const) |
1579 |
done |
|
1580 |
||
49653 | 1581 |
lemma path_connected_Un: |
53640 | 1582 |
assumes "path_connected s" |
1583 |
and "path_connected t" |
|
1584 |
and "s \<inter> t \<noteq> {}" |
|
49653 | 1585 |
shows "path_connected (s \<union> t)" |
1586 |
unfolding path_connected_component |
|
1587 |
proof (rule, rule) |
|
1588 |
fix x y |
|
1589 |
assume as: "x \<in> s \<union> t" "y \<in> s \<union> t" |
|
53640 | 1590 |
from assms(3) obtain z where "z \<in> s \<inter> t" |
1591 |
by auto |
|
49654 | 1592 |
then show "path_component (s \<union> t) x y" |
49653 | 1593 |
using as and assms(1-2)[unfolded path_connected_component] |
53640 | 1594 |
apply - |
49653 | 1595 |
apply (erule_tac[!] UnE)+ |
1596 |
apply (rule_tac[2-3] path_component_trans[of _ _ z]) |
|
1597 |
apply (auto simp add:path_component_of_subset [OF Un_upper1] path_component_of_subset[OF Un_upper2]) |
|
1598 |
done |
|
1599 |
qed |
|
36583 | 1600 |
|
37674
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
1601 |
lemma path_connected_UNION: |
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
1602 |
assumes "\<And>i. i \<in> A \<Longrightarrow> path_connected (S i)" |
49653 | 1603 |
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
|
1604 |
shows "path_connected (\<Union>i\<in>A. S i)" |
49653 | 1605 |
unfolding path_connected_component |
1606 |
proof clarify |
|
37674
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
1607 |
fix x i y j |
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
1608 |
assume *: "i \<in> A" "x \<in> S i" "j \<in> A" "y \<in> S j" |
49654 | 1609 |
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
|
1610 |
using assms by (simp_all add: path_connected_component) |
49654 | 1611 |
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
|
1612 |
using *(1,3) by (auto elim!: path_component_of_subset [rotated]) |
49654 | 1613 |
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
|
1614 |
by (rule path_component_trans) |
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
1615 |
qed |
36583 | 1616 |
|
61426
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1617 |
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
|
1618 |
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
|
1619 |
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
|
1620 |
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
|
1621 |
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
|
1622 |
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
|
1623 |
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
|
1624 |
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
|
1625 |
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
|
1626 |
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
|
1627 |
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
|
1628 |
next |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1629 |
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
|
1630 |
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
|
1631 |
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
|
1632 |
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
|
1633 |
done |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1634 |
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
|
1635 |
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
|
1636 |
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
|
1637 |
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
|
1638 |
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
|
1639 |
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
|
1640 |
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
|
1641 |
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
|
1642 |
done |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1643 |
qed |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1644 |
qed |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1645 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1646 |
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
|
1647 |
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
|
1648 |
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
|
1649 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1650 |
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
|
1651 |
"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
|
1652 |
proof - |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1653 |
{ 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
|
1654 |
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
|
1655 |
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
|
1656 |
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
|
1657 |
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
|
1658 |
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
|
1659 |
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
|
1660 |
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
|
1661 |
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
|
1662 |
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
|
1663 |
} |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1664 |
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
|
1665 |
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
|
1666 |
qed |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1667 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1668 |
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
|
1669 |
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
|
1670 |
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
|
1671 |
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
|
1672 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1673 |
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
|
1674 |
"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
|
1675 |
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
|
1676 |
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
|
1677 |
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
|
1678 |
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
|
1679 |
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
|
1680 |
next |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1681 |
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
|
1682 |
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
|
1683 |
qed |
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 |
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
|
1686 |
"(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
|
1687 |
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
|
1688 |
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
|
1689 |
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
|
1690 |
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
|
1691 |
done |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1692 |
next |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1693 |
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
|
1694 |
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
|
1695 |
qed |
49653 | 1696 |
|
62620
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
1697 |
subsection\<open>Lemmas about path-connectedness\<close> |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
1698 |
|
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
1699 |
lemma path_connected_linear_image: |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
1700 |
fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector" |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
1701 |
assumes "path_connected s" "bounded_linear f" |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
1702 |
shows "path_connected(f ` s)" |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
1703 |
by (auto simp: linear_continuous_on assms path_connected_continuous_image) |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
1704 |
|
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
1705 |
lemma is_interval_path_connected: "is_interval s \<Longrightarrow> path_connected s" |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
1706 |
by (simp add: convex_imp_path_connected is_interval_convex) |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
1707 |
|
62843
313d3b697c9a
Mostly renaming (from HOL Light to Isabelle conventions), with a couple of new results
paulson <lp15@cam.ac.uk>
parents:
62626
diff
changeset
|
1708 |
lemma linear_homeomorphism_image: |
313d3b697c9a
Mostly renaming (from HOL Light to Isabelle conventions), with a couple of new results
paulson <lp15@cam.ac.uk>
parents:
62626
diff
changeset
|
1709 |
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space" |
62620
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
1710 |
assumes "linear f" "inj f" |
62843
313d3b697c9a
Mostly renaming (from HOL Light to Isabelle conventions), with a couple of new results
paulson <lp15@cam.ac.uk>
parents:
62626
diff
changeset
|
1711 |
obtains g where "homeomorphism (f ` S) S g f" |
313d3b697c9a
Mostly renaming (from HOL Light to Isabelle conventions), with a couple of new results
paulson <lp15@cam.ac.uk>
parents:
62626
diff
changeset
|
1712 |
using linear_injective_left_inverse [OF assms] |
313d3b697c9a
Mostly renaming (from HOL Light to Isabelle conventions), with a couple of new results
paulson <lp15@cam.ac.uk>
parents:
62626
diff
changeset
|
1713 |
apply clarify |
313d3b697c9a
Mostly renaming (from HOL Light to Isabelle conventions), with a couple of new results
paulson <lp15@cam.ac.uk>
parents:
62626
diff
changeset
|
1714 |
apply (rule_tac g=g in that) |
313d3b697c9a
Mostly renaming (from HOL Light to Isabelle conventions), with a couple of new results
paulson <lp15@cam.ac.uk>
parents:
62626
diff
changeset
|
1715 |
using assms |
313d3b697c9a
Mostly renaming (from HOL Light to Isabelle conventions), with a couple of new results
paulson <lp15@cam.ac.uk>
parents:
62626
diff
changeset
|
1716 |
apply (auto simp: homeomorphism_def eq_id_iff [symmetric] image_comp comp_def linear_conv_bounded_linear linear_continuous_on) |
313d3b697c9a
Mostly renaming (from HOL Light to Isabelle conventions), with a couple of new results
paulson <lp15@cam.ac.uk>
parents:
62626
diff
changeset
|
1717 |
done |
313d3b697c9a
Mostly renaming (from HOL Light to Isabelle conventions), with a couple of new results
paulson <lp15@cam.ac.uk>
parents:
62626
diff
changeset
|
1718 |
|
313d3b697c9a
Mostly renaming (from HOL Light to Isabelle conventions), with a couple of new results
paulson <lp15@cam.ac.uk>
parents:
62626
diff
changeset
|
1719 |
lemma linear_homeomorphic_image: |
313d3b697c9a
Mostly renaming (from HOL Light to Isabelle conventions), with a couple of new results
paulson <lp15@cam.ac.uk>
parents:
62626
diff
changeset
|
1720 |
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space" |
313d3b697c9a
Mostly renaming (from HOL Light to Isabelle conventions), with a couple of new results
paulson <lp15@cam.ac.uk>
parents:
62626
diff
changeset
|
1721 |
assumes "linear f" "inj f" |
313d3b697c9a
Mostly renaming (from HOL Light to Isabelle conventions), with a couple of new results
paulson <lp15@cam.ac.uk>
parents:
62626
diff
changeset
|
1722 |
shows "S homeomorphic f ` S" |
313d3b697c9a
Mostly renaming (from HOL Light to Isabelle conventions), with a couple of new results
paulson <lp15@cam.ac.uk>
parents:
62626
diff
changeset
|
1723 |
by (meson homeomorphic_def homeomorphic_sym linear_homeomorphism_image [OF assms]) |
62620
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
1724 |
|
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
1725 |
lemma path_connected_Times: |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
1726 |
assumes "path_connected s" "path_connected t" |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
1727 |
shows "path_connected (s \<times> t)" |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
1728 |
proof (simp add: path_connected_def Sigma_def, clarify) |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
1729 |
fix x1 y1 x2 y2 |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
1730 |
assume "x1 \<in> s" "y1 \<in> t" "x2 \<in> s" "y2 \<in> t" |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
1731 |
obtain g where "path g" and g: "path_image g \<subseteq> s" and gs: "pathstart g = x1" and gf: "pathfinish g = x2" |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
1732 |
using \<open>x1 \<in> s\<close> \<open>x2 \<in> s\<close> assms by (force simp: path_connected_def) |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
1733 |
obtain h where "path h" and h: "path_image h \<subseteq> t" and hs: "pathstart h = y1" and hf: "pathfinish h = y2" |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
1734 |
using \<open>y1 \<in> t\<close> \<open>y2 \<in> t\<close> assms by (force simp: path_connected_def) |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
1735 |
have "path (\<lambda>z. (x1, h z))" |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
1736 |
using \<open>path h\<close> |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
1737 |
apply (simp add: path_def) |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
1738 |
apply (rule continuous_on_compose2 [where f = h]) |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
1739 |
apply (rule continuous_intros | force)+ |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
1740 |
done |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
1741 |
moreover have "path (\<lambda>z. (g z, y2))" |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
1742 |
using \<open>path g\<close> |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
1743 |
apply (simp add: path_def) |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
1744 |
apply (rule continuous_on_compose2 [where f = g]) |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
1745 |
apply (rule continuous_intros | force)+ |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
1746 |
done |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
1747 |
ultimately have 1: "path ((\<lambda>z. (x1, h z)) +++ (\<lambda>z. (g z, y2)))" |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
1748 |
by (metis hf gs path_join_imp pathstart_def pathfinish_def) |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
1749 |
have "path_image ((\<lambda>z. (x1, h z)) +++ (\<lambda>z. (g z, y2))) \<subseteq> path_image (\<lambda>z. (x1, h z)) \<union> path_image (\<lambda>z. (g z, y2))" |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
1750 |
by (rule Path_Connected.path_image_join_subset) |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
1751 |
also have "... \<subseteq> (\<Union>x\<in>s. \<Union>x1\<in>t. {(x, x1)})" |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
1752 |
using g h \<open>x1 \<in> s\<close> \<open>y2 \<in> t\<close> by (force simp: path_image_def) |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
1753 |
finally have 2: "path_image ((\<lambda>z. (x1, h z)) +++ (\<lambda>z. (g z, y2))) \<subseteq> (\<Union>x\<in>s. \<Union>x1\<in>t. {(x, x1)})" . |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
1754 |
show "\<exists>g. path g \<and> path_image g \<subseteq> (\<Union>x\<in>s. \<Union>x1\<in>t. {(x, x1)}) \<and> |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
1755 |
pathstart g = (x1, y1) \<and> pathfinish g = (x2, y2)" |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
1756 |
apply (intro exI conjI) |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
1757 |
apply (rule 1) |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
1758 |
apply (rule 2) |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
1759 |
apply (metis hs pathstart_def pathstart_join) |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
1760 |
by (metis gf pathfinish_def pathfinish_join) |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
1761 |
qed |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
1762 |
|
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
1763 |
lemma is_interval_path_connected_1: |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
1764 |
fixes s :: "real set" |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
1765 |
shows "is_interval s \<longleftrightarrow> path_connected s" |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
1766 |
using is_interval_connected_1 is_interval_path_connected path_connected_imp_connected by blast |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
1767 |
|
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
1768 |
|
62948
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
1769 |
lemma Union_path_component [simp]: |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
1770 |
"Union {path_component_set S x |x. x \<in> S} = S" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
1771 |
apply (rule subset_antisym) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
1772 |
using path_component_subset apply force |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
1773 |
using path_component_refl by auto |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
1774 |
|
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
1775 |
lemma path_component_disjoint: |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
1776 |
"disjnt (path_component_set S a) (path_component_set S b) \<longleftrightarrow> |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
1777 |
(a \<notin> path_component_set S b)" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
1778 |
apply (auto simp: disjnt_def) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
1779 |
using path_component_eq apply fastforce |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
1780 |
using path_component_sym path_component_trans by blast |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
1781 |
|
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
1782 |
lemma path_component_eq_eq: |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
1783 |
"path_component S x = path_component S y \<longleftrightarrow> |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
1784 |
(x \<notin> S) \<and> (y \<notin> S) \<or> x \<in> S \<and> y \<in> S \<and> path_component S x y" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
1785 |
apply (rule iffI, metis (no_types) path_component_mem(1) path_component_refl) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
1786 |
apply (erule disjE, metis Collect_empty_eq_bot path_component_eq_empty) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
1787 |
apply (rule ext) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
1788 |
apply (metis path_component_trans path_component_sym) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
1789 |
done |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
1790 |
|
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
1791 |
lemma path_component_unique: |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
1792 |
assumes "x \<in> c" "c \<subseteq> S" "path_connected c" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
1793 |
"\<And>c'. \<lbrakk>x \<in> c'; c' \<subseteq> S; path_connected c'\<rbrakk> \<Longrightarrow> c' \<subseteq> c" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
1794 |
shows "path_component_set S x = c" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
1795 |
apply (rule subset_antisym) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
1796 |
using assms |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
1797 |
apply (metis mem_Collect_eq subsetCE path_component_eq_eq path_component_subset path_connected_path_component) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
1798 |
by (simp add: assms path_component_maximal) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
1799 |
|
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
1800 |
lemma path_component_intermediate_subset: |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
1801 |
"path_component_set u a \<subseteq> t \<and> t \<subseteq> u |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
1802 |
\<Longrightarrow> path_component_set t a = path_component_set u a" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
1803 |
by (metis (no_types) path_component_mono path_component_path_component subset_antisym) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
1804 |
|
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
1805 |
lemma complement_path_component_Union: |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
1806 |
fixes x :: "'a :: topological_space" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
1807 |
shows "S - path_component_set S x = |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
1808 |
\<Union>({path_component_set S y| y. y \<in> S} - {path_component_set S x})" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
1809 |
proof - |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
1810 |
have *: "(\<And>x. x \<in> S - {a} \<Longrightarrow> disjnt a x) \<Longrightarrow> \<Union>S - a = \<Union>(S - {a})" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
1811 |
for a::"'a set" and S |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
1812 |
by (auto simp: disjnt_def) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
1813 |
have "\<And>y. y \<in> {path_component_set S x |x. x \<in> S} - {path_component_set S x} |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
1814 |
\<Longrightarrow> disjnt (path_component_set S x) y" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
1815 |
using path_component_disjoint path_component_eq by fastforce |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
1816 |
then have "\<Union>{path_component_set S x |x. x \<in> S} - path_component_set S x = |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
1817 |
\<Union>({path_component_set S y |y. y \<in> S} - {path_component_set S x})" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
1818 |
by (meson *) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
1819 |
then show ?thesis by simp |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
1820 |
qed |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
1821 |
|
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
1822 |
|
60420 | 1823 |
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
|
1824 |
|
36583 | 1825 |
lemma path_connected_punctured_universe: |
37674
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
1826 |
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
|
1827 |
shows "path_connected (- {a::'a})" |
49653 | 1828 |
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
|
1829 |
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
|
1830 |
let ?B = "{x::'a. \<exists>i\<in>Basis. a \<bullet> i < x \<bullet> i}" |
36583 | 1831 |
|
49653 | 1832 |
have A: "path_connected ?A" |
1833 |
unfolding Collect_bex_eq |
|
37674
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
1834 |
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
|
1835 |
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
|
1836 |
assume "i \<in> Basis" |
53640 | 1837 |
then show "(\<Sum>i\<in>Basis. (a \<bullet> i - 1)*\<^sub>R i) \<in> {x::'a. x \<bullet> i < a \<bullet> i}" |
1838 |
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
|
1839 |
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
|
1840 |
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
|
1841 |
by (simp add: inner_commute) |
37674
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
1842 |
qed |
53640 | 1843 |
have B: "path_connected ?B" |
1844 |
unfolding Collect_bex_eq |
|
37674
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
1845 |
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
|
1846 |
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
|
1847 |
assume "i \<in> Basis" |
53640 | 1848 |
then show "(\<Sum>i\<in>Basis. (a \<bullet> i + 1) *\<^sub>R i) \<in> {x::'a. a \<bullet> i < x \<bullet> i}" |
1849 |
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
|
1850 |
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
|
1851 |
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
|
1852 |
by (simp add: inner_commute) |
37674
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
1853 |
qed |
53640 | 1854 |
obtain S :: "'a set" where "S \<subseteq> Basis" and "card S = Suc (Suc 0)" |
1855 |
using ex_card[OF assms] |
|
1856 |
by auto |
|
1857 |
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
|
1858 |
unfolding card_Suc_eq by auto |
53640 | 1859 |
then have "a + b0 - b1 \<in> ?A \<inter> ?B" |
1860 |
by (auto simp: inner_simps inner_Basis) |
|
1861 |
then have "?A \<inter> ?B \<noteq> {}" |
|
1862 |
by fast |
|
37674
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
1863 |
with A B have "path_connected (?A \<union> ?B)" |
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
1864 |
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
|
1865 |
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
|
1866 |
unfolding neq_iff bex_disj_distrib Collect_disj_eq .. |
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
1867 |
also have "\<dots> = {x. x \<noteq> a}" |
53640 | 1868 |
unfolding euclidean_eq_iff [where 'a='a] |
1869 |
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
|
1870 |
also have "\<dots> = - {a}" |
53640 | 1871 |
by auto |
37674
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
1872 |
finally show ?thesis . |
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
1873 |
qed |
36583 | 1874 |
|
37674
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
1875 |
lemma path_connected_sphere: |
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
1876 |
assumes "2 \<le> DIM('a::euclidean_space)" |
53640 | 1877 |
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
|
1878 |
proof (rule linorder_cases [of r 0]) |
49653 | 1879 |
assume "r < 0" |
53640 | 1880 |
then have "{x::'a. norm(x - a) = r} = {}" |
1881 |
by auto |
|
1882 |
then show ?thesis |
|
1883 |
using path_connected_empty by simp |
|
37674
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
1884 |
next |
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
1885 |
assume "r = 0" |
53640 | 1886 |
then show ?thesis |
1887 |
using path_connected_singleton by simp |
|
37674
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
1888 |
next |
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
1889 |
assume r: "0 < r" |
53640 | 1890 |
have *: "{x::'a. norm(x - a) = r} = (\<lambda>x. a + r *\<^sub>R x) ` {x. norm x = 1}" |
1891 |
apply (rule set_eqI) |
|
1892 |
apply rule |
|
49653 | 1893 |
unfolding image_iff |
1894 |
apply (rule_tac x="(1/r) *\<^sub>R (x - a)" in bexI) |
|
1895 |
unfolding mem_Collect_eq norm_scaleR |
|
53640 | 1896 |
using r |
49653 | 1897 |
apply (auto simp add: scaleR_right_diff_distrib) |
1898 |
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
|
1899 |
have **: "{x::'a. norm x = 1} = (\<lambda>x. (1/norm x) *\<^sub>R x) ` (- {0})" |
53640 | 1900 |
apply (rule set_eqI) |
1901 |
apply rule |
|
49653 | 1902 |
unfolding image_iff |
1903 |
apply (rule_tac x=x in bexI) |
|
1904 |
unfolding mem_Collect_eq |
|
62390 | 1905 |
apply (auto split: if_split_asm) |
49653 | 1906 |
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
|
1907 |
have "continuous_on (- {0}) (\<lambda>x::'a. 1 / norm x)" |
59557 | 1908 |
by (auto intro!: continuous_intros) |
53640 | 1909 |
then show ?thesis |
1910 |
unfolding * ** |
|
1911 |
using path_connected_punctured_universe[OF assms] |
|
56371
fb9ae0727548
extend continuous_intros; remove continuous_on_intros and isCont_intros
hoelzl
parents:
56188
diff
changeset
|
1912 |
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
|
1913 |
qed |
36583 | 1914 |
|
61426
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1915 |
corollary connected_sphere: "2 \<le> DIM('a::euclidean_space) \<Longrightarrow> connected {x::'a. norm (x - a) = r}" |
53640 | 1916 |
using path_connected_sphere path_connected_imp_connected |
1917 |
by auto |
|
36583 | 1918 |
|
61426
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1919 |
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
|
1920 |
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
|
1921 |
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
|
1922 |
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
|
1923 |
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
|
1924 |
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
|
1925 |
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
|
1926 |
next |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1927 |
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
|
1928 |
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
|
1929 |
{ fix x y assume "x \<notin> s" "y \<notin> s" |
61808 | 1930 |
then have "x \<noteq> a" "y \<noteq> a" using \<open>a \<in> s\<close> by auto |
61426
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1931 |
then have bxy: "bounded(insert x (insert y s))" |
61808 | 1932 |
by (simp add: \<open>bounded s\<close>) |
61426
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1933 |
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
|
1934 |
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
|
1935 |
using bounded_subset_ballD [OF bxy, of a] by (auto simp: dist_norm) |
63040 | 1936 |
define C where "C = B / norm(x - a)" |
61426
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1937 |
{ 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
|
1938 |
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
|
1939 |
have CC: "1 \<le> 1 + (C - 1) * u" |
61808 | 1940 |
using \<open>x \<noteq> a\<close> \<open>0 \<le> u\<close> |
61426
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1941 |
apply (simp add: C_def divide_simps norm_minus_commute) |
61762
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61738
diff
changeset
|
1942 |
using Bx by auto |
61426
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1943 |
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
|
1944 |
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
|
1945 |
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
|
1946 |
(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
|
1947 |
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
|
1948 |
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
|
1949 |
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
|
1950 |
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
|
1951 |
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
|
1952 |
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
|
1953 |
((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
|
1954 |
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
|
1955 |
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
|
1956 |
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
|
1957 |
have False |
61808 | 1958 |
using \<open>convex s\<close> |
61426
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1959 |
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
|
1960 |
apply (drule_tac x=a in bspec) |
61808 | 1961 |
apply (rule \<open>a \<in> s\<close>) |
61426
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1962 |
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
|
1963 |
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
|
1964 |
apply (drule_tac x="1 / (1 + (C - 1) * u)" in spec) |
61808 | 1965 |
using \<open>x \<noteq> a\<close> \<open>x \<notin> s\<close> \<open>0 \<le> u\<close> CC |
61426
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1966 |
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
|
1967 |
done |
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 |
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
|
1970 |
by (force simp: closed_segment_def intro!: path_connected_linepath) |
63040 | 1971 |
define D where "D = B / norm(y - a)" \<comment>\<open>massive duplication with the proof above\<close> |
61426
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1972 |
{ 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
|
1973 |
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
|
1974 |
have DD: "1 \<le> 1 + (D - 1) * u" |
61808 | 1975 |
using \<open>y \<noteq> a\<close> \<open>0 \<le> u\<close> |
61426
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1976 |
apply (simp add: D_def divide_simps norm_minus_commute) |
61762
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61738
diff
changeset
|
1977 |
using By by auto |
61426
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1978 |
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
|
1979 |
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
|
1980 |
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
|
1981 |
(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
|
1982 |
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
|
1983 |
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
|
1984 |
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
|
1985 |
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
|
1986 |
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
|
1987 |
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
|
1988 |
((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
|
1989 |
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
|
1990 |
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
|
1991 |
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
|
1992 |
have False |
61808 | 1993 |
using \<open>convex s\<close> |
61426
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1994 |
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
|
1995 |
apply (drule_tac x=a in bspec) |
61808 | 1996 |
apply (rule \<open>a \<in> s\<close>) |
61426
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1997 |
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
|
1998 |
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
|
1999 |
apply (drule_tac x="1 / (1 + (D - 1) * u)" in spec) |
61808 | 2000 |
using \<open>y \<noteq> a\<close> \<open>y \<notin> s\<close> \<open>0 \<le> u\<close> DD |
61426
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2001 |
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
|
2002 |
done |
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 |
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
|
2005 |
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
|
2006 |
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
|
2007 |
apply (rule path_component_of_subset [of "{x. norm(x - a) = B}"]) |
61808 | 2008 |
using \<open>s \<subseteq> ball a B\<close> |
61426
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2009 |
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
|
2010 |
apply (rule path_connected_sphere [OF 2, of a B, simplified path_connected_component, rule_format]) |
61808 | 2011 |
using \<open>x \<noteq> a\<close> using \<open>y \<noteq> a\<close> B apply (auto simp: C_def D_def) |
61426
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2012 |
done |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2013 |
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
|
2014 |
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
|
2015 |
} |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2016 |
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
|
2017 |
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
|
2018 |
qed |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2019 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2020 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2021 |
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
|
2022 |
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
|
2023 |
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
|
2024 |
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
|
2025 |
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
|
2026 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2027 |
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
|
2028 |
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
|
2029 |
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
|
2030 |
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
|
2031 |
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
|
2032 |
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
|
2033 |
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
|
2034 |
done |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2035 |
|
62381
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
2036 |
proposition connected_open_delete: |
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
2037 |
assumes "open S" "connected S" and 2: "2 \<le> DIM('N::euclidean_space)" |
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
2038 |
shows "connected(S - {a::'N})" |
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
2039 |
proof (cases "a \<in> S") |
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
2040 |
case True |
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
2041 |
with \<open>open S\<close> obtain \<epsilon> where "\<epsilon> > 0" and \<epsilon>: "cball a \<epsilon> \<subseteq> S" |
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
2042 |
using open_contains_cball_eq by blast |
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
2043 |
have "dist a (a + \<epsilon> *\<^sub>R (SOME i. i \<in> Basis)) = \<epsilon>" |
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
2044 |
by (simp add: dist_norm SOME_Basis \<open>0 < \<epsilon>\<close> less_imp_le) |
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
2045 |
with \<epsilon> have "\<Inter>{S - ball a r |r. 0 < r \<and> r < \<epsilon>} \<subseteq> {} \<Longrightarrow> False" |
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
2046 |
apply (drule_tac c="a + scaleR (\<epsilon>) ((SOME i. i \<in> Basis))" in subsetD) |
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
2047 |
by auto |
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
2048 |
then have nonemp: "(\<Inter>{S - ball a r |r. 0 < r \<and> r < \<epsilon>}) = {} \<Longrightarrow> False" |
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
2049 |
by auto |
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
2050 |
have con: "\<And>r. r < \<epsilon> \<Longrightarrow> connected (S - ball a r)" |
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
2051 |
using \<epsilon> by (force intro: connected_diff_ball [OF \<open>connected S\<close> _ 2]) |
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
2052 |
have "x \<in> \<Union>{S - ball a r |r. 0 < r \<and> r < \<epsilon>}" if "x \<in> S - {a}" for x |
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
2053 |
apply (rule UnionI [of "S - ball a (min \<epsilon> (dist a x) / 2)"]) |
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
2054 |
using that \<open>0 < \<epsilon>\<close> apply (simp_all add:) |
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
2055 |
apply (rule_tac x="min \<epsilon> (dist a x) / 2" in exI) |
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
2056 |
apply auto |
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
2057 |
done |
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
2058 |
then have "S - {a} = \<Union>{S - ball a r | r. 0 < r \<and> r < \<epsilon>}" |
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
2059 |
by auto |
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
2060 |
then show ?thesis |
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
2061 |
by (auto intro: connected_Union con dest!: nonemp) |
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
2062 |
next |
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
2063 |
case False then show ?thesis |
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
2064 |
by (simp add: \<open>connected S\<close>) |
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
2065 |
qed |
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
2066 |
|
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
2067 |
corollary path_connected_open_delete: |
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
2068 |
assumes "open S" "connected S" and 2: "2 \<le> DIM('N::euclidean_space)" |
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
2069 |
shows "path_connected(S - {a::'N})" |
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
2070 |
by (simp add: assms connected_open_delete connected_open_path_connected open_delete) |
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
2071 |
|
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
2072 |
corollary path_connected_punctured_ball: |
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
2073 |
"2 \<le> DIM('N::euclidean_space) \<Longrightarrow> path_connected(ball a r - {a::'N})" |
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
2074 |
by (simp add: path_connected_open_delete) |
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
2075 |
|
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
2076 |
lemma connected_punctured_ball: |
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
2077 |
"2 \<le> DIM('N::euclidean_space) \<Longrightarrow> connected(ball a r - {a::'N})" |
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
2078 |
by (simp add: connected_open_delete) |
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
2079 |
|
61426
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2080 |
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
|
2081 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2082 |
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
|
2083 |
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
|
2084 |
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
|
2085 |
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
|
2086 |
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
|
2087 |
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
|
2088 |
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
|
2089 |
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
|
2090 |
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
|
2091 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2092 |
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
|
2093 |
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
|
2094 |
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
|
2095 |
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
|
2096 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2097 |
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
|
2098 |
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
|
2099 |
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
|
2100 |
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
|
2101 |
proof - |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2102 |
{ 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
|
2103 |
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
|
2104 |
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
|
2105 |
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
|
2106 |
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
|
2107 |
} |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2108 |
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
|
2109 |
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
|
2110 |
qed |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2111 |
|
63114
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
2112 |
lemma connected_disjoint_Union_open_pick: |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
2113 |
assumes "pairwise disjnt B" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
2114 |
"\<And>S. S \<in> A \<Longrightarrow> connected S \<and> S \<noteq> {}" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
2115 |
"\<And>S. S \<in> B \<Longrightarrow> open S" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
2116 |
"\<Union>A \<subseteq> \<Union>B" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
2117 |
"S \<in> A" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
2118 |
obtains T where "T \<in> B" "S \<subseteq> T" "S \<inter> \<Union>(B - {T}) = {}" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
2119 |
proof - |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
2120 |
have "S \<subseteq> \<Union>B" "connected S" "S \<noteq> {}" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
2121 |
using assms \<open>S \<in> A\<close> by blast+ |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
2122 |
then obtain T where "T \<in> B" "S \<inter> T \<noteq> {}" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
2123 |
by (metis Sup_inf_eq_bot_iff inf.absorb_iff2 inf_commute) |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
2124 |
have 1: "open T" by (simp add: \<open>T \<in> B\<close> assms) |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
2125 |
have 2: "open (\<Union>(B-{T}))" using assms by blast |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
2126 |
have 3: "S \<subseteq> T \<union> \<Union>(B - {T})" using \<open>S \<subseteq> \<Union>B\<close> by blast |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
2127 |
have "T \<inter> \<Union>(B - {T}) = {}" using \<open>T \<in> B\<close> \<open>pairwise disjnt B\<close> |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
2128 |
by (auto simp: pairwise_def disjnt_def) |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
2129 |
then have 4: "T \<inter> \<Union>(B - {T}) \<inter> S = {}" by auto |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
2130 |
from connectedD [OF \<open>connected S\<close> 1 2 3 4] |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
2131 |
have "S \<inter> \<Union>(B-{T}) = {}" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
2132 |
by (auto simp: Int_commute \<open>S \<inter> T \<noteq> {}\<close>) |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
2133 |
with \<open>T \<in> B\<close> have "S \<subseteq> T" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
2134 |
using "3" by auto |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
2135 |
show ?thesis |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
2136 |
using \<open>S \<inter> \<Union>(B - {T}) = {}\<close> \<open>S \<subseteq> T\<close> \<open>T \<in> B\<close> that by auto |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
2137 |
qed |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
2138 |
|
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
2139 |
lemma connected_disjoint_Union_open_subset: |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
2140 |
assumes A: "pairwise disjnt A" and B: "pairwise disjnt B" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
2141 |
and SA: "\<And>S. S \<in> A \<Longrightarrow> open S \<and> connected S \<and> S \<noteq> {}" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
2142 |
and SB: "\<And>S. S \<in> B \<Longrightarrow> open S \<and> connected S \<and> S \<noteq> {}" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
2143 |
and eq [simp]: "\<Union>A = \<Union>B" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
2144 |
shows "A \<subseteq> B" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
2145 |
proof |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
2146 |
fix S |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
2147 |
assume "S \<in> A" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
2148 |
obtain T where "T \<in> B" "S \<subseteq> T" "S \<inter> \<Union>(B - {T}) = {}" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
2149 |
apply (rule connected_disjoint_Union_open_pick [OF B, of A]) |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
2150 |
using SA SB \<open>S \<in> A\<close> by auto |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
2151 |
moreover obtain S' where "S' \<in> A" "T \<subseteq> S'" "T \<inter> \<Union>(A - {S'}) = {}" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
2152 |
apply (rule connected_disjoint_Union_open_pick [OF A, of B]) |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
2153 |
using SA SB \<open>T \<in> B\<close> by auto |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
2154 |
ultimately have "S' = S" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
2155 |
by (metis A Int_subset_iff SA \<open>S \<in> A\<close> disjnt_def inf.orderE pairwise_def) |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
2156 |
with \<open>T \<subseteq> S'\<close> have "T \<subseteq> S" by simp |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
2157 |
with \<open>S \<subseteq> T\<close> have "S = T" by blast |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
2158 |
with \<open>T \<in> B\<close> show "S \<in> B" by simp |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
2159 |
qed |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
2160 |
|
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
2161 |
lemma connected_disjoint_Union_open_unique: |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
2162 |
assumes A: "pairwise disjnt A" and B: "pairwise disjnt B" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
2163 |
and SA: "\<And>S. S \<in> A \<Longrightarrow> open S \<and> connected S \<and> S \<noteq> {}" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
2164 |
and SB: "\<And>S. S \<in> B \<Longrightarrow> open S \<and> connected S \<and> S \<noteq> {}" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
2165 |
and eq [simp]: "\<Union>A = \<Union>B" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
2166 |
shows "A = B" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
2167 |
by (rule subset_antisym; metis connected_disjoint_Union_open_subset assms) |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
2168 |
|
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
2169 |
proposition components_open_unique: |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
2170 |
fixes S :: "'a::real_normed_vector set" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
2171 |
assumes "pairwise disjnt A" "\<Union>A = S" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
2172 |
"\<And>X. X \<in> A \<Longrightarrow> open X \<and> connected X \<and> X \<noteq> {}" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
2173 |
shows "components S = A" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
2174 |
proof - |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
2175 |
have "open S" using assms by blast |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
2176 |
show ?thesis |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
2177 |
apply (rule connected_disjoint_Union_open_unique) |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
2178 |
apply (simp add: components_eq disjnt_def pairwise_def) |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
2179 |
using \<open>open S\<close> |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
2180 |
apply (simp_all add: assms open_components in_components_connected in_components_nonempty) |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
2181 |
done |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
2182 |
qed |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
2183 |
|
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
2184 |
|
61426
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2185 |
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
|
2186 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2187 |
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
|
2188 |
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
|
2189 |
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
|
2190 |
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
|
2191 |
proof - |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2192 |
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
|
2193 |
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
|
2194 |
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
|
2195 |
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
|
2196 |
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
|
2197 |
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
|
2198 |
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
|
2199 |
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
|
2200 |
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
|
2201 |
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
|
2202 |
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
|
2203 |
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
|
2204 |
done |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2205 |
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
|
2206 |
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
|
2207 |
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
|
2208 |
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
|
2209 |
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
|
2210 |
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
|
2211 |
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
|
2212 |
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
|
2213 |
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
|
2214 |
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
|
2215 |
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
|
2216 |
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
|
2217 |
done |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2218 |
qed |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2219 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2220 |
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
|
2221 |
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
|
2222 |
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
|
2223 |
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
|
2224 |
"~ 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
|
2225 |
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
|
2226 |
proof - |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2227 |
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
|
2228 |
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
|
2229 |
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
|
2230 |
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
|
2231 |
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
|
2232 |
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
|
2233 |
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
|
2234 |
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
|
2235 |
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
|
2236 |
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
|
2237 |
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
|
2238 |
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
|
2239 |
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
|
2240 |
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
|
2241 |
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
|
2242 |
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
|
2243 |
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
|
2244 |
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
|
2245 |
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
|
2246 |
done |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2247 |
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
|
2248 |
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
|
2249 |
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
|
2250 |
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
|
2251 |
qed |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2252 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2253 |
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
|
2254 |
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
|
2255 |
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
|
2256 |
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
|
2257 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2258 |
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
|
2259 |
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
|
2260 |
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
|
2261 |
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
|
2262 |
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
|
2263 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2264 |
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
|
2265 |
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
|
2266 |
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
|
2267 |
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
|
2268 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2269 |
|
61518
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2270 |
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
|
2271 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2272 |
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
|
2273 |
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
|
2274 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2275 |
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
|
2276 |
"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
|
2277 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2278 |
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
|
2279 |
"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
|
2280 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2281 |
lemma outside: "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
|
2282 |
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
|
2283 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2284 |
lemma inside_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
|
2285 |
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
|
2286 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2287 |
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
|
2288 |
"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
|
2289 |
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
|
2290 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2291 |
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
|
2292 |
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
|
2293 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2294 |
lemma 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
|
2295 |
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
|
2296 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2297 |
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
|
2298 |
"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
|
2299 |
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
|
2300 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2301 |
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
|
2302 |
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
|
2303 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2304 |
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
|
2305 |
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
|
2306 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2307 |
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
|
2308 |
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
|
2309 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2310 |
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
|
2311 |
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
|
2312 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2313 |
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
|
2314 |
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
|
2315 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2316 |
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
|
2317 |
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
|
2318 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2319 |
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
|
2320 |
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
|
2321 |
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
|
2322 |
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
|
2323 |
proof - |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2324 |
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
|
2325 |
using assms by auto (metis add.commute diff_add_cancel) |
61808 | 2326 |
with \<open>0 \<le> u\<close> \<open>u \<le> 1\<close> show ?thesis |
61426
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2327 |
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
|
2328 |
qed |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2329 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2330 |
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
|
2331 |
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
|
2332 |
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
|
2333 |
proof - |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2334 |
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
|
2335 |
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
|
2336 |
{ 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
|
2337 |
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
|
2338 |
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
|
2339 |
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
|
2340 |
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
|
2341 |
next |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2342 |
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
|
2343 |
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
|
2344 |
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
|
2345 |
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
|
2346 |
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
|
2347 |
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
|
2348 |
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
|
2349 |
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
|
2350 |
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
|
2351 |
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
|
2352 |
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
|
2353 |
qed |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2354 |
} |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2355 |
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
|
2356 |
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
|
2357 |
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
|
2358 |
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
|
2359 |
done |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2360 |
qed |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2361 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2362 |
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
|
2363 |
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
|
2364 |
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
|
2365 |
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
|
2366 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2367 |
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
|
2368 |
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
|
2369 |
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
|
2370 |
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
|
2371 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2372 |
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
|
2373 |
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
|
2374 |
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
|
2375 |
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
|
2376 |
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
|
2377 |
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
|
2378 |
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
|
2379 |
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
|
2380 |
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
|
2381 |
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
|
2382 |
done |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2383 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2384 |
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
|
2385 |
"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
|
2386 |
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
|
2387 |
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
|
2388 |
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
|
2389 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2390 |
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
|
2391 |
"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
|
2392 |
{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
|
2393 |
connected_component (- s) x y}" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2394 |
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
|
2395 |
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
|
2396 |
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
|
2397 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2398 |
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
|
2399 |
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
|
2400 |
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
|
2401 |
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
|
2402 |
proof - |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2403 |
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
|
2404 |
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
|
2405 |
{ 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
|
2406 |
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
|
2407 |
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
|
2408 |
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
|
2409 |
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
|
2410 |
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
|
2411 |
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
|
2412 |
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
|
2413 |
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
|
2414 |
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
|
2415 |
done |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2416 |
} 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
|
2417 |
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
|
2418 |
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
|
2419 |
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
|
2420 |
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
|
2421 |
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
|
2422 |
qed |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2423 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2424 |
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
|
2425 |
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
|
2426 |
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
|
2427 |
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
|
2428 |
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
|
2429 |
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
|
2430 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2431 |
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
|
2432 |
fixes s :: "'a::euclidean_space set" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2433 |
assumes 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
|
2434 |
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
|
2435 |
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
|
2436 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2437 |
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
|
2438 |
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
|
2439 |
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
|
2440 |
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
|
2441 |
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
|
2442 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2443 |
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
|
2444 |
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
|
2445 |
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
|
2446 |
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
|
2447 |
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
|
2448 |
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
|
2449 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2450 |
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
|
2451 |
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
|
2452 |
|
62381
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
2453 |
lemma frontier_interior_subset: "frontier(interior S) \<subseteq> frontier S" |
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
2454 |
by (simp add: Diff_mono frontier_interiors interior_mono interior_subset) |
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
2455 |
|
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
2456 |
lemma connected_Int_frontier: |
61426
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2457 |
"\<lbrakk>connected s; s \<inter> t \<noteq> {}; s - t \<noteq> {}\<rbrakk> \<Longrightarrow> (s \<inter> frontier t \<noteq> {})" |
62843
313d3b697c9a
Mostly renaming (from HOL Light to Isabelle conventions), with a couple of new results
paulson <lp15@cam.ac.uk>
parents:
62626
diff
changeset
|
2458 |
apply (simp add: frontier_interiors connected_openin, safe) |
61426
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2459 |
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
|
2460 |
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
|
2461 |
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
|
2462 |
done |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2463 |
|
62381
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
2464 |
lemma frontier_not_empty: |
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
2465 |
fixes S :: "'a :: real_normed_vector set" |
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
2466 |
shows "\<lbrakk>S \<noteq> {}; S \<noteq> UNIV\<rbrakk> \<Longrightarrow> frontier S \<noteq> {}" |
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
2467 |
using connected_Int_frontier [of UNIV S] by auto |
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
2468 |
|
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
2469 |
lemma frontier_eq_empty: |
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
2470 |
fixes S :: "'a :: real_normed_vector set" |
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
2471 |
shows "frontier S = {} \<longleftrightarrow> S = {} \<or> S = UNIV" |
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
2472 |
using frontier_UNIV frontier_empty frontier_not_empty by blast |
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
2473 |
|
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
2474 |
lemma frontier_of_connected_component_subset: |
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
2475 |
fixes S :: "'a::real_normed_vector set" |
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
2476 |
shows "frontier(connected_component_set S x) \<subseteq> frontier S" |
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
2477 |
proof - |
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
2478 |
{ fix y |
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
2479 |
assume y1: "y \<in> closure (connected_component_set S x)" |
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
2480 |
and y2: "y \<notin> interior (connected_component_set S x)" |
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
2481 |
have 1: "y \<in> closure S" |
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
2482 |
using y1 closure_mono connected_component_subset by blast |
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
2483 |
have "z \<in> interior (connected_component_set S x)" |
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
2484 |
if "0 < e" "ball y e \<subseteq> interior S" "dist y z < e" for e z |
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
2485 |
proof - |
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
2486 |
have "ball y e \<subseteq> connected_component_set S y" |
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
2487 |
apply (rule connected_component_maximal) |
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
2488 |
using that interior_subset mem_ball apply auto |
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
2489 |
done |
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
2490 |
then show ?thesis |
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
2491 |
using y1 apply (simp add: closure_approachable open_contains_ball_eq [OF open_interior]) |
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
2492 |
by (metis (no_types, hide_lams) connected_component_eq_eq connected_component_in subsetD |
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
2493 |
dist_commute mem_Collect_eq mem_ball mem_interior \<open>0 < e\<close> y2) |
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
2494 |
qed |
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
2495 |
then have 2: "y \<notin> interior S" |
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
2496 |
using y2 by (force simp: open_contains_ball_eq [OF open_interior]) |
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
2497 |
note 1 2 |
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
2498 |
} |
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
2499 |
then show ?thesis by (auto simp: frontier_def) |
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
2500 |
qed |
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
2501 |
|
62620
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
2502 |
lemma frontier_Union_subset_closure: |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
2503 |
fixes F :: "'a::real_normed_vector set set" |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
2504 |
shows "frontier(\<Union>F) \<subseteq> closure(\<Union>t \<in> F. frontier t)" |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
2505 |
proof - |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
2506 |
have "\<exists>y\<in>F. \<exists>y\<in>frontier y. dist y x < e" |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
2507 |
if "T \<in> F" "y \<in> T" "dist y x < e" |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
2508 |
"x \<notin> interior (\<Union>F)" "0 < e" for x y e T |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
2509 |
proof (cases "x \<in> T") |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
2510 |
case True with that show ?thesis |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
2511 |
by (metis Diff_iff Sup_upper closure_subset contra_subsetD dist_self frontier_def interior_mono) |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
2512 |
next |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
2513 |
case False |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
2514 |
have 1: "closed_segment x y \<inter> T \<noteq> {}" using \<open>y \<in> T\<close> by blast |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
2515 |
have 2: "closed_segment x y - T \<noteq> {}" |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
2516 |
using False by blast |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
2517 |
obtain c where "c \<in> closed_segment x y" "c \<in> frontier T" |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
2518 |
using False connected_Int_frontier [OF connected_segment 1 2] by auto |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
2519 |
then show ?thesis |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
2520 |
proof - |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
2521 |
have "norm (y - x) < e" |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
2522 |
by (metis dist_norm \<open>dist y x < e\<close>) |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
2523 |
moreover have "norm (c - x) \<le> norm (y - x)" |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
2524 |
by (simp add: \<open>c \<in> closed_segment x y\<close> segment_bound(1)) |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
2525 |
ultimately have "norm (c - x) < e" |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
2526 |
by linarith |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
2527 |
then show ?thesis |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
2528 |
by (metis (no_types) \<open>c \<in> frontier T\<close> dist_norm that(1)) |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
2529 |
qed |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
2530 |
qed |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
2531 |
then show ?thesis |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
2532 |
by (fastforce simp add: frontier_def closure_approachable) |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
2533 |
qed |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
2534 |
|
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
2535 |
lemma frontier_Union_subset: |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
2536 |
fixes F :: "'a::real_normed_vector set set" |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
2537 |
shows "finite F \<Longrightarrow> frontier(\<Union>F) \<subseteq> (\<Union>t \<in> F. frontier t)" |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
2538 |
by (rule order_trans [OF frontier_Union_subset_closure]) |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
2539 |
(auto simp: closure_subset_eq) |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
2540 |
|
62843
313d3b697c9a
Mostly renaming (from HOL Light to Isabelle conventions), with a couple of new results
paulson <lp15@cam.ac.uk>
parents:
62626
diff
changeset
|
2541 |
lemma connected_component_UNIV [simp]: |
61426
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2542 |
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
|
2543 |
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
|
2544 |
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
|
2545 |
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
|
2546 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2547 |
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
|
2548 |
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
|
2549 |
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
|
2550 |
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
|
2551 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2552 |
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
|
2553 |
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
|
2554 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2555 |
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
|
2556 |
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
|
2557 |
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
|
2558 |
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
|
2559 |
proof - |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2560 |
{ 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
|
2561 |
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
|
2562 |
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
|
2563 |
have "connected_component_set (- frontier s) x \<inter> frontier s \<noteq> {}" |
62381
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
2564 |
apply (rule connected_Int_frontier, simp) |
61426
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2565 |
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
|
2566 |
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
|
2567 |
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
|
2568 |
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
|
2569 |
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
|
2570 |
} |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2571 |
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
|
2572 |
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
|
2573 |
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
|
2574 |
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
|
2575 |
done |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2576 |
qed |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2577 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2578 |
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
|
2579 |
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
|
2580 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2581 |
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
|
2582 |
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
|
2583 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2584 |
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
|
2585 |
"\<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
|
2586 |
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
|
2587 |
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
|
2588 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2589 |
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
|
2590 |
"\<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
|
2591 |
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
|
2592 |
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
|
2593 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2594 |
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
|
2595 |
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
|
2596 |
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
|
2597 |
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
|
2598 |
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
|
2599 |
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
|
2600 |
next |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2601 |
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
|
2602 |
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
|
2603 |
{ 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
|
2604 |
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
|
2605 |
by (metis mem_Collect_eq) |
63040 | 2606 |
define C where "C = (B + 1 + norm z) / norm (z-a)" |
61426
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2607 |
have "C > 0" |
61808 | 2608 |
using \<open>0 < B\<close> zna by (simp add: C_def divide_simps add_strict_increasing) |
61426
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2609 |
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
|
2610 |
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
|
2611 |
moreover have "norm (C *\<^sub>R (z-a)) > norm z + B" |
61808 | 2612 |
using zna \<open>B>0\<close> by (simp add: C_def le_max_iff_disj field_simps) |
61426
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2613 |
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
|
2614 |
{ 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
|
2615 |
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
|
2616 |
then have Cpos: "1 + u * C > 0" |
61808 | 2617 |
by (meson \<open>0 < C\<close> add_pos_nonneg less_eq_real_def zero_le_mult_iff zero_less_one) |
61426
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2618 |
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
|
2619 |
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
|
2620 |
then have False |
61808 | 2621 |
using convexD_alt [OF s \<open>a \<in> s\<close> ins, of "1/(u*C + 1)"] \<open>C>0\<close> \<open>z \<notin> s\<close> Cpos u |
61426
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2622 |
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
|
2623 |
} 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
|
2624 |
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
|
2625 |
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
|
2626 |
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
|
2627 |
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
|
2628 |
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
|
2629 |
done |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2630 |
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
|
2631 |
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
|
2632 |
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
|
2633 |
} |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2634 |
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
|
2635 |
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
|
2636 |
qed |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2637 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2638 |
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
|
2639 |
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
|
2640 |
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
|
2641 |
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
|
2642 |
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
|
2643 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2644 |
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
|
2645 |
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
|
2646 |
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
|
2647 |
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
|
2648 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2649 |
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
|
2650 |
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
|
2651 |
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
|
2652 |
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
|
2653 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2654 |
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
|
2655 |
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
|
2656 |
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
|
2657 |
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
|
2658 |
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
|
2659 |
proof - |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2660 |
{ 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
|
2661 |
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
|
2662 |
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
|
2663 |
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
|
2664 |
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
|
2665 |
} |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2666 |
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
|
2667 |
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
|
2668 |
qed |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2669 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2670 |
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
|
2671 |
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
|
2672 |
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
|
2673 |
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
|
2674 |
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
|
2675 |
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
|
2676 |
|
61518
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2677 |
lemma inside_frontier_eq_interior: |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2678 |
fixes s :: "'a :: {real_normed_vector, perfect_space} set" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2679 |
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
|
2680 |
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
|
2681 |
using closure_subset interior_subset |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2682 |
apply (auto simp add: frontier_def) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2683 |
done |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2684 |
|
61426
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2685 |
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
|
2686 |
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
|
2687 |
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
|
2688 |
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
|
2689 |
proof - |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2690 |
{ 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
|
2691 |
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
|
2692 |
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
|
2693 |
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
|
2694 |
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
|
2695 |
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
|
2696 |
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
|
2697 |
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
|
2698 |
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
|
2699 |
} |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2700 |
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
|
2701 |
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
|
2702 |
qed |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2703 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2704 |
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
|
2705 |
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
|
2706 |
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
|
2707 |
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
|
2708 |
proof - |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2709 |
{ 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
|
2710 |
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
|
2711 |
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
|
2712 |
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
|
2713 |
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
|
2714 |
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
|
2715 |
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
|
2716 |
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
|
2717 |
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
|
2718 |
} |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2719 |
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
|
2720 |
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
|
2721 |
qed |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2722 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2723 |
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
|
2724 |
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
|
2725 |
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
|
2726 |
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
|
2727 |
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
|
2728 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2729 |
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
|
2730 |
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
|
2731 |
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
|
2732 |
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
|
2733 |
proof - |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2734 |
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
|
2735 |
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
|
2736 |
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
|
2737 |
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
|
2738 |
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
|
2739 |
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
|
2740 |
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
|
2741 |
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
|
2742 |
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
|
2743 |
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
|
2744 |
qed |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2745 |
|
61518
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2746 |
lemma closure_outside_subset: |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2747 |
fixes s :: "'a::real_normed_vector set" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2748 |
assumes "closed s" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2749 |
shows "closure(outside s) \<subseteq> s \<union> outside s" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2750 |
apply (rule closure_minimal, simp) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2751 |
by (metis assms closed_open inside_outside open_inside) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2752 |
|
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2753 |
lemma frontier_outside_subset: |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2754 |
fixes s :: "'a::real_normed_vector set" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2755 |
assumes "closed s" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2756 |
shows "frontier(outside s) \<subseteq> s" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2757 |
apply (simp add: frontier_def open_outside interior_open) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2758 |
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
|
2759 |
|
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2760 |
lemma inside_complement_unbounded_connected_empty: |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2761 |
"\<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
|
2762 |
apply (simp add: inside_def) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2763 |
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
|
2764 |
|
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2765 |
lemma inside_bounded_complement_connected_empty: |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2766 |
fixes s :: "'a::{real_normed_vector, perfect_space} set" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2767 |
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
|
2768 |
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
|
2769 |
|
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2770 |
lemma inside_inside: |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2771 |
assumes "s \<subseteq> inside t" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2772 |
shows "inside s - t \<subseteq> inside t" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2773 |
unfolding inside_def |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2774 |
proof clarify |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2775 |
fix x |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2776 |
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
|
2777 |
show "bounded (connected_component_set (- t) x)" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2778 |
proof (cases "s \<inter> connected_component_set (- t) x = {}") |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2779 |
case True show ?thesis |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2780 |
apply (rule bounded_subset [OF bo]) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2781 |
apply (rule connected_component_maximal) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2782 |
using x True apply auto |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2783 |
done |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2784 |
next |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2785 |
case False then show ?thesis |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2786 |
using assms [unfolded inside_def] x |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2787 |
apply (simp add: disjoint_iff_not_equal, clarify) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2788 |
apply (drule subsetD, assumption, auto) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2789 |
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
|
2790 |
qed |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2791 |
qed |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2792 |
|
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2793 |
lemma inside_inside_subset: "inside(inside s) \<subseteq> s" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2794 |
using inside_inside union_with_outside by fastforce |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2795 |
|
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2796 |
lemma inside_outside_intersect_connected: |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2797 |
"\<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
|
2798 |
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
|
2799 |
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
|
2800 |
|
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2801 |
lemma outside_bounded_nonempty: |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2802 |
fixes s :: "'a :: {real_normed_vector, perfect_space} set" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2803 |
assumes "bounded s" shows "outside s \<noteq> {}" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2804 |
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
|
2805 |
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
|
2806 |
double_complement order_refl outside_convex outside_def) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2807 |
|
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2808 |
lemma outside_compact_in_open: |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2809 |
fixes s :: "'a :: {real_normed_vector,perfect_space} set" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2810 |
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
|
2811 |
shows "outside s \<inter> t \<noteq> {}" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2812 |
proof - |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2813 |
have "outside s \<noteq> {}" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2814 |
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
|
2815 |
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
|
2816 |
show ?thesis |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2817 |
proof (cases "a \<in> t") |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2818 |
case True with a show ?thesis by blast |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2819 |
next |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2820 |
case False |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2821 |
have front: "frontier t \<subseteq> - s" |
61808 | 2822 |
using \<open>s \<subseteq> t\<close> frontier_disjoint_eq t by auto |
61518
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2823 |
{ fix \<gamma> |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2824 |
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
|
2825 |
and pf: "pathfinish \<gamma> \<in> frontier t" and ps: "pathstart \<gamma> = a" |
63040 | 2826 |
define c where "c = pathfinish \<gamma>" |
61518
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2827 |
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
|
2828 |
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
|
2829 |
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
|
2830 |
using open_contains_cball[of "-s"] s by blast |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2831 |
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
|
2832 |
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
|
2833 |
by (metis Diff_iff frontier_def) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2834 |
then have "d \<in> -s" using \<epsilon> |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2835 |
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
|
2836 |
have pimg_sbs_cos: "path_image \<gamma> \<subseteq> -s" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2837 |
using pimg_sbs apply (auto simp: path_image_def) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2838 |
apply (drule subsetD) |
61808 | 2839 |
using \<open>c \<in> - s\<close> \<open>s \<subseteq> t\<close> interior_subset apply (auto simp: c_def) |
61518
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2840 |
done |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2841 |
have "closed_segment c d \<le> cball c \<epsilon>" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2842 |
apply (simp add: segment_convex_hull) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2843 |
apply (rule hull_minimal) |
61808 | 2844 |
using \<open>\<epsilon> > 0\<close> d apply (auto simp: dist_commute) |
61518
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2845 |
done |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2846 |
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
|
2847 |
moreover have con_gcd: "connected (path_image \<gamma> \<union> closed_segment c d)" |
61808 | 2848 |
by (rule connected_Un) (auto simp: c_def \<open>path \<gamma>\<close> connected_path_image) |
61518
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2849 |
ultimately have "connected_component (- s) a d" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2850 |
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
|
2851 |
then have "outside s \<inter> t \<noteq> {}" |
61808 | 2852 |
using outside_same_component [OF _ a] by (metis IntI \<open>d \<in> t\<close> empty_iff) |
61518
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2853 |
} note * = this |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2854 |
have pal: "pathstart (linepath a b) \<in> closure (- t)" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2855 |
by (auto simp: False closure_def) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2856 |
show ?thesis |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2857 |
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
|
2858 |
qed |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2859 |
qed |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2860 |
|
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2861 |
lemma inside_inside_compact_connected: |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2862 |
fixes s :: "'a :: euclidean_space set" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2863 |
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
|
2864 |
shows "inside s \<subseteq> inside t" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2865 |
proof (cases "inside t = {}") |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2866 |
case True with assms show ?thesis by auto |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2867 |
next |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2868 |
case False |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2869 |
consider "DIM('a) = 1" | "DIM('a) \<ge> 2" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2870 |
using antisym not_less_eq_eq by fastforce |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2871 |
then show ?thesis |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2872 |
proof cases |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2873 |
case 1 then show ?thesis |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2874 |
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
|
2875 |
next |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2876 |
case 2 |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2877 |
have coms: "compact s" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2878 |
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
|
2879 |
by (meson bounded_inside bounded_subset compact_imp_bounded) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2880 |
then have bst: "bounded (s \<union> t)" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2881 |
by (simp add: compact_imp_bounded t) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2882 |
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
|
2883 |
using bounded_subset_ballD by blast |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2884 |
have outst: "outside s \<inter> outside t \<noteq> {}" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2885 |
proof - |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2886 |
have "- ball 0 r \<subseteq> outside s" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2887 |
apply (rule outside_subset_convex) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2888 |
using r by auto |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2889 |
moreover have "- ball 0 r \<subseteq> outside t" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2890 |
apply (rule outside_subset_convex) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2891 |
using r by auto |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2892 |
ultimately show ?thesis |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2893 |
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
|
2894 |
qed |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2895 |
have "s \<inter> t = {}" using assms |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2896 |
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
|
2897 |
moreover have "outside s \<inter> inside t \<noteq> {}" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2898 |
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
|
2899 |
ultimately have "inside s \<inter> t = {}" |
61808 | 2900 |
using inside_outside_intersect_connected [OF \<open>connected t\<close>, of s] |
61518
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2901 |
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
|
2902 |
then show ?thesis |
61808 | 2903 |
using inside_inside [OF \<open>s \<subseteq> inside t\<close>] by blast |
61518
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2904 |
qed |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2905 |
qed |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2906 |
|
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2907 |
lemma connected_with_inside: |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2908 |
fixes s :: "'a :: real_normed_vector set" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2909 |
assumes s: "closed s" and cons: "connected s" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2910 |
shows "connected(s \<union> inside s)" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2911 |
proof (cases "s \<union> inside s = UNIV") |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2912 |
case True with assms show ?thesis by auto |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2913 |
next |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2914 |
case False |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2915 |
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
|
2916 |
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
|
2917 |
using that proof |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2918 |
assume "a \<in> s" then show ?thesis |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2919 |
apply (rule_tac x=a in exI) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2920 |
apply (rule_tac x="{a}" in exI) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2921 |
apply (simp add:) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2922 |
done |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2923 |
next |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2924 |
assume a: "a \<in> inside s" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2925 |
show ?thesis |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2926 |
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
|
2927 |
using a apply (simp add: closure_def) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2928 |
apply (simp add: b) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2929 |
apply (rule_tac x="pathfinish h" in exI) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2930 |
apply (rule_tac x="path_image h" in exI) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2931 |
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
|
2932 |
using frontier_inside_subset s apply fastforce |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2933 |
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
|
2934 |
qed |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2935 |
show ?thesis |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2936 |
apply (simp add: connected_iff_connected_component) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2937 |
apply (simp add: connected_component_def) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2938 |
apply (clarify dest!: *) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2939 |
apply (rename_tac u u' t t') |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2940 |
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
|
2941 |
apply (auto simp: intro!: connected_Un cons) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2942 |
done |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2943 |
qed |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2944 |
|
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2945 |
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
|
2946 |
lemma connected_with_outside: |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2947 |
fixes s :: "'a :: real_normed_vector set" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2948 |
assumes s: "closed s" and cons: "connected s" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2949 |
shows "connected(s \<union> outside s)" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2950 |
proof (cases "s \<union> outside s = UNIV") |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2951 |
case True with assms show ?thesis by auto |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2952 |
next |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2953 |
case False |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2954 |
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
|
2955 |
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
|
2956 |
using that proof |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2957 |
assume "a \<in> s" then show ?thesis |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2958 |
apply (rule_tac x=a in exI) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2959 |
apply (rule_tac x="{a}" in exI) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2960 |
apply (simp add:) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2961 |
done |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2962 |
next |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2963 |
assume a: "a \<in> outside s" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2964 |
show ?thesis |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2965 |
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
|
2966 |
using a apply (simp add: closure_def) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2967 |
apply (simp add: b) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2968 |
apply (rule_tac x="pathfinish h" in exI) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2969 |
apply (rule_tac x="path_image h" in exI) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2970 |
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
|
2971 |
using frontier_outside_subset s apply fastforce |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2972 |
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
|
2973 |
qed |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2974 |
show ?thesis |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2975 |
apply (simp add: connected_iff_connected_component) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2976 |
apply (simp add: connected_component_def) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2977 |
apply (clarify dest!: *) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2978 |
apply (rename_tac u u' t t') |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2979 |
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
|
2980 |
apply (auto simp: intro!: connected_Un cons) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2981 |
done |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2982 |
qed |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2983 |
|
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2984 |
lemma inside_inside_eq_empty [simp]: |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2985 |
fixes s :: "'a :: {real_normed_vector, perfect_space} set" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2986 |
assumes s: "closed s" and cons: "connected s" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2987 |
shows "inside (inside s) = {}" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2988 |
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
|
2989 |
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
|
2990 |
|
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2991 |
lemma inside_in_components: |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2992 |
"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
|
2993 |
apply (simp add: in_components_maximal) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2994 |
apply (auto intro: inside_same_component connected_componentI) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2995 |
apply (metis IntI empty_iff inside_no_overlap) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2996 |
done |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2997 |
|
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
2998 |
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
|
2999 |
lemma outside_in_components: |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3000 |
"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
|
3001 |
apply (simp add: in_components_maximal) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3002 |
apply (auto intro: outside_same_component connected_componentI) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3003 |
apply (metis IntI empty_iff outside_no_overlap) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3004 |
done |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3005 |
|
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3006 |
lemma bounded_unique_outside: |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3007 |
fixes s :: "'a :: euclidean_space set" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3008 |
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
|
3009 |
apply (rule iffI) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3010 |
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
|
3011 |
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
|
3012 |
|
62533
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
3013 |
subsection\<open>Condition for an open map's image to contain a ball\<close> |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
3014 |
|
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
3015 |
lemma ball_subset_open_map_image: |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
3016 |
fixes f :: "'a::heine_borel \<Rightarrow> 'b :: {real_normed_vector,heine_borel}" |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
3017 |
assumes contf: "continuous_on (closure S) f" |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
3018 |
and oint: "open (f ` interior S)" |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
3019 |
and le_no: "\<And>z. z \<in> frontier S \<Longrightarrow> r \<le> norm(f z - f a)" |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
3020 |
and "bounded S" "a \<in> S" "0 < r" |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
3021 |
shows "ball (f a) r \<subseteq> f ` S" |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
3022 |
proof (cases "f ` S = UNIV") |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
3023 |
case True then show ?thesis by simp |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
3024 |
next |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
3025 |
case False |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
3026 |
obtain w where w: "w \<in> frontier (f ` S)" |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
3027 |
and dw_le: "\<And>y. y \<in> frontier (f ` S) \<Longrightarrow> norm (f a - w) \<le> norm (f a - y)" |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
3028 |
apply (rule distance_attains_inf [of "frontier(f ` S)" "f a"]) |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
3029 |
using \<open>a \<in> S\<close> by (auto simp: frontier_eq_empty dist_norm False) |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
3030 |
then obtain \<xi> where \<xi>: "\<And>n. \<xi> n \<in> f ` S" and tendsw: "\<xi> \<longlonglongrightarrow> w" |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
3031 |
by (metis Diff_iff frontier_def closure_sequential) |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
3032 |
then have "\<And>n. \<exists>x \<in> S. \<xi> n = f x" by force |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
3033 |
then obtain z where zs: "\<And>n. z n \<in> S" and fz: "\<And>n. \<xi> n = f (z n)" |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
3034 |
by metis |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
3035 |
then obtain y K where y: "y \<in> closure S" and "subseq K" and Klim: "(z \<circ> K) \<longlonglongrightarrow> y" |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
3036 |
using \<open>bounded S\<close> |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
3037 |
apply (simp add: compact_closure [symmetric] compact_def) |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
3038 |
apply (drule_tac x=z in spec) |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
3039 |
using closure_subset apply force |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
3040 |
done |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
3041 |
then have ftendsw: "((\<lambda>n. f (z n)) \<circ> K) \<longlonglongrightarrow> w" |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
3042 |
by (metis LIMSEQ_subseq_LIMSEQ fun.map_cong0 fz tendsw) |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
3043 |
have zKs: "\<And>n. (z o K) n \<in> S" by (simp add: zs) |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
3044 |
have "f \<circ> z = \<xi>" "(\<lambda>n. f (z n)) = \<xi>" |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
3045 |
using fz by auto |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
3046 |
moreover then have "(\<xi> \<circ> K) \<longlonglongrightarrow> f y" |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
3047 |
by (metis (no_types) Klim zKs y contf comp_assoc continuous_on_closure_sequentially) |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
3048 |
ultimately have wy: "w = f y" using fz LIMSEQ_unique ftendsw by auto |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
3049 |
have rle: "r \<le> norm (f y - f a)" |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
3050 |
apply (rule le_no) |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
3051 |
using w wy oint |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
3052 |
by (force simp: imageI image_mono interiorI interior_subset frontier_def y) |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
3053 |
have **: "(~(b \<inter> (- S) = {}) \<and> ~(b - (- S) = {}) \<Longrightarrow> (b \<inter> f \<noteq> {})) |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
3054 |
\<Longrightarrow> (b \<inter> S \<noteq> {}) \<Longrightarrow> b \<inter> f = {} \<Longrightarrow> |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
3055 |
b \<subseteq> S" for b f and S :: "'b set" |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
3056 |
by blast |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
3057 |
show ?thesis |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
3058 |
apply (rule **) (*such a horrible mess*) |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
3059 |
apply (rule connected_Int_frontier [where t = "f`S", OF connected_ball]) |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
3060 |
using \<open>a \<in> S\<close> \<open>0 < r\<close> |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
3061 |
apply (auto simp: disjoint_iff_not_equal dist_norm) |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
3062 |
by (metis dw_le norm_minus_commute not_less order_trans rle wy) |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
3063 |
qed |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
3064 |
|
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
|
3065 |
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
|
3066 |
|
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3067 |
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
|
3068 |
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
|
3069 |
|
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3070 |
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
|
3071 |
"[('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
|
3072 |
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
|
3073 |
"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
|
3074 |
(\<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
|
3075 |
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
|
3076 |
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
|
3077 |
(\<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
|
3078 |
(\<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
|
3079 |
(\<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
|
3080 |
|
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3081 |
|
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3082 |
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
|
3083 |
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
|
3084 |
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
|
3085 |
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
|
3086 |
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
|
3087 |
(\<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
|
3088 |
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
|
3089 |
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
|
3090 |
(\<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
|
3091 |
(\<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
|
3092 |
(\<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
|
3093 |
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
|
3094 |
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
|
3095 |
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
|
3096 |
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
|
3097 |
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
|
3098 |
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
|
3099 |
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
|
3100 |
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
|
3101 |
|
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3102 |
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
|
3103 |
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
|
3104 |
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
|
3105 |
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
|
3106 |
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
|
3107 |
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
|
3108 |
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
|
3109 |
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
|
3110 |
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
|
3111 |
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
|
3112 |
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
|
3113 |
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
|
3114 |
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
|
3115 |
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
|
3116 |
|
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3117 |
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
|
3118 |
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
|
3119 |
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
|
3120 |
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
|
3121 |
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
|
3122 |
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
|
3123 |
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
|
3124 |
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
|
3125 |
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
|
3126 |
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
|
3127 |
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
|
3128 |
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
|
3129 |
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
|
3130 |
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
|
3131 |
|
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3132 |
|
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3133 |
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
|
3134 |
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
|
3135 |
|
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3136 |
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
|
3137 |
"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
|
3138 |
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
|
3139 |
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
|
3140 |
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
|
3141 |
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
|
3142 |
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
|
3143 |
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
|
3144 |
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
|
3145 |
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
|
3146 |
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
|
3147 |
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
|
3148 |
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
|
3149 |
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
|
3150 |
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
|
3151 |
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
|
3152 |
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
|
3153 |
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
|
3154 |
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
|
3155 |
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
|
3156 |
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
|
3157 |
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
|
3158 |
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
|
3159 |
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
|
3160 |
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
|
3161 |
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
|
3162 |
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
|
3163 |
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
|
3164 |
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
|
3165 |
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
|
3166 |
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
|
3167 |
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
|
3168 |
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
|
3169 |
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
|
3170 |
|
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3171 |
|
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3172 |
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
|
3173 |
|
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3174 |
lemma homotopic_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
|
3175 |
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
|
3176 |
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
|
3177 |
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
|
3178 |
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
|
3179 |
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
|
3180 |
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
|
3181 |
|
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3182 |
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
|
3183 |
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
|
3184 |
|
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3185 |
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
|
3186 |
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
|
3187 |
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
|
3188 |
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
|
3189 |
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
|
3190 |
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
|
3191 |
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
|
3192 |
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
|
3193 |
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
|
3194 |
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
|
3195 |
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
|
3196 |
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
|
3197 |
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
|
3198 |
|
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3199 |
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
|
3200 |
"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
|
3201 |
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
|
3202 |
|
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3203 |
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
|
3204 |
"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
|
3205 |
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
|
3206 |
|
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3207 |
proposition homotopic_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
|
3208 |
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
|
3209 |
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
|
3210 |
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
|
3211 |
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
|
3212 |
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
|
3213 |
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
|
3214 |
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
|
3215 |
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
|
3216 |
|
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3217 |
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
|
3218 |
"\<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
|
3219 |
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
|
3220 |
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
|
3221 |
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
|
3222 |
|
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3223 |
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
|
3224 |
"\<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
|
3225 |
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
|
3226 |
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
|
3227 |
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
|
3228 |
|
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3229 |
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
|
3230 |
"\<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
|
3231 |
\<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
|
3232 |
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
|
3233 |
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
|
3234 |
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
|
3235 |
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
|
3236 |
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
|
3237 |
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
|
3238 |
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
|
3239 |
|
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3240 |
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
|
3241 |
"\<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
|
3242 |
\<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
|
3243 |
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
|
3244 |
|
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3245 |
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
|
3246 |
"\<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
|
3247 |
\<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
|
3248 |
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
|
3249 |
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
|
3250 |
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
|
3251 |
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
|
3252 |
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
|
3253 |
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
|
3254 |
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
|
3255 |
|
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3256 |
proposition homotopic_compose_continuous_left: |
62626
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
3257 |
"\<lbrakk>homotopic_with (\<lambda>_. True) X Y f g; |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
3258 |
continuous_on Y h; h ` Y \<subseteq> Z\<rbrakk> |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
3259 |
\<Longrightarrow> homotopic_with (\<lambda>f. True) X Z (h o f) (h o g)" |
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
|
3260 |
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
|
3261 |
|
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3262 |
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
|
3263 |
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
|
3264 |
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
|
3265 |
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
|
3266 |
(\<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
|
3267 |
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
|
3268 |
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
|
3269 |
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
|
3270 |
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
|
3271 |
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
|
3272 |
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
|
3273 |
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
|
3274 |
|
62843
313d3b697c9a
Mostly renaming (from HOL Light to Isabelle conventions), with a couple of new results
paulson <lp15@cam.ac.uk>
parents:
62626
diff
changeset
|
3275 |
lemma homotopic_on_empty [simp]: "homotopic_with (\<lambda>x. True) {} t f g" |
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
|
3276 |
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
|
3277 |
|
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3278 |
|
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3279 |
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
|
3280 |
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
|
3281 |
|
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3282 |
|
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3283 |
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
|
3284 |
"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
|
3285 |
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
|
3286 |
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
|
3287 |
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
|
3288 |
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
|
3289 |
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
|
3290 |
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
|
3291 |
|
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3292 |
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
|
3293 |
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
|
3294 |
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
|
3295 |
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
|
3296 |
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
|
3297 |
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
|
3298 |
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
|
3299 |
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
|
3300 |
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
|
3301 |
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
|
3302 |
|
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3303 |
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
|
3304 |
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
|
3305 |
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
|
3306 |
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
|
3307 |
|
61699
a81dc5c4d6a9
New theorems mostly from Peter Gammie
paulson <lp15@cam.ac.uk>
parents:
61694
diff
changeset
|
3308 |
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
|
3309 |
by force |
a81dc5c4d6a9
New theorems mostly from Peter Gammie
paulson <lp15@cam.ac.uk>
parents:
61694
diff
changeset
|
3310 |
|
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
|
3311 |
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
|
3312 |
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
|
3313 |
|
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3314 |
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
|
3315 |
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
|
3316 |
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
|
3317 |
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
|
3318 |
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
|
3319 |
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
|
3320 |
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
|
3321 |
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
|
3322 |
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
|
3323 |
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
|
3324 |
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
|
3325 |
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
|
3326 |
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
|
3327 |
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
|
3328 |
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
|
3329 |
{ 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
|
3330 |
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
|
3331 |
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
|
3332 |
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
|
3333 |
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
|
3334 |
and P: "\<forall>t\<in>{0..1}. P (\<lambda>x. k1 (t, x))" "\<forall>t\<in>{0..1}. P (\<lambda>x. k2 (t, x))" |
63040 | 3335 |
define k where "k y = |
3336 |
(if fst y \<le> 1 / 2 |
|
3337 |
then (k1 o (\<lambda>x. (2 *\<^sub>R fst x, snd x))) y |
|
3338 |
else (k2 o (\<lambda>x. (2 *\<^sub>R fst x -1, snd x))) y)" for y |
|
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
|
3339 |
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
|
3340 |
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
|
3341 |
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
|
3342 |
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
|
3343 |
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
|
3344 |
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
|
3345 |
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
|
3346 |
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
|
3347 |
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
|
3348 |
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
|
3349 |
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
|
3350 |
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
|
3351 |
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
|
3352 |
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
|
3353 |
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
|
3354 |
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
|
3355 |
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
|
3356 |
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
|
3357 |
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
|
3358 |
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
|
3359 |
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
|
3360 |
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
|
3361 |
(\<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
|
3362 |
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
|
3363 |
} 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
|
3364 |
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
|
3365 |
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
|
3366 |
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
|
3367 |
|
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3368 |
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
|
3369 |
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
|
3370 |
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
|
3371 |
\<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
|
3372 |
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
|
3373 |
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
|
3374 |
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
|
3375 |
|
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3376 |
|
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3377 |
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
|
3378 |
|
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3379 |
|
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3380 |
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
|
3381 |
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
|
3382 |
"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
|
3383 |
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
|
3384 |
|
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3385 |
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
|
3386 |
"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
|
3387 |
(\<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
|
3388 |
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
|
3389 |
(\<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
|
3390 |
(\<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
|
3391 |
(\<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
|
3392 |
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
|
3393 |
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
|
3394 |
|
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3395 |
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
|
3396 |
"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
|
3397 |
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
|
3398 |
|
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3399 |
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
|
3400 |
"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
|
3401 |
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
|
3402 |
|
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3403 |
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
|
3404 |
"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
|
3405 |
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
|
3406 |
|
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3407 |
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
|
3408 |
"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
|
3409 |
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
|
3410 |
|
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3411 |
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
|
3412 |
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
|
3413 |
|
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3414 |
proposition homotopic_paths_sym: "homotopic_paths s p q \<Longrightarrow> homotopic_paths s q p" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3415 |
by (metis (mono_tags) homotopic_paths_def homotopic_paths_imp_pathfinish homotopic_paths_imp_pathstart homotopic_with_symD) |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3416 |
|
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3417 |
proposition homotopic_paths_sym_eq: "homotopic_paths s p q \<longleftrightarrow> homotopic_paths s q p" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3418 |
by (metis homotopic_paths_sym) |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3419 |
|
61762
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61738
diff
changeset
|
3420 |
proposition homotopic_paths_trans [trans]: |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3421 |
"\<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
|
3422 |
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
|
3423 |
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
|
3424 |
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
|
3425 |
|
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3426 |
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
|
3427 |
"\<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
|
3428 |
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
|
3429 |
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
|
3430 |
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
|
3431 |
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
|
3432 |
|
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3433 |
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
|
3434 |
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
|
3435 |
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
|
3436 |
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
|
3437 |
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
|
3438 |
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
|
3439 |
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
|
3440 |
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
|
3441 |
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
|
3442 |
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
|
3443 |
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
|
3444 |
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
|
3445 |
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
|
3446 |
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
|
3447 |
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
|
3448 |
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
|
3449 |
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
|
3450 |
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
|
3451 |
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
|
3452 |
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
|
3453 |
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
|
3454 |
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
|
3455 |
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
|
3456 |
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
|
3457 |
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
|
3458 |
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
|
3459 |
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
|
3460 |
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
|
3461 |
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
|
3462 |
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
|
3463 |
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
|
3464 |
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
|
3465 |
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
|
3466 |
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
|
3467 |
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
|
3468 |
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
|
3469 |
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
|
3470 |
|
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3471 |
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
|
3472 |
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
|
3473 |
|
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3474 |
|
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3475 |
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
|
3476 |
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
|
3477 |
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
|
3478 |
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
|
3479 |
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
|
3480 |
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
|
3481 |
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
|
3482 |
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
|
3483 |
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
|
3484 |
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
|
3485 |
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
|
3486 |
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
|
3487 |
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
|
3488 |
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
|
3489 |
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
|
3490 |
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
|
3491 |
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
|
3492 |
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
|
3493 |
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
|
3494 |
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
|
3495 |
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
|
3496 |
|
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3497 |
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
|
3498 |
|
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3499 |
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
|
3500 |
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
|
3501 |
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
|
3502 |
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
|
3503 |
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
|
3504 |
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
|
3505 |
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
|
3506 |
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
|
3507 |
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
|
3508 |
|
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3509 |
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
|
3510 |
"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
|
3511 |
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
|
3512 |
|
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3513 |
|
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3514 |
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
|
3515 |
"\<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
|
3516 |
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
|
3517 |
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
|
3518 |
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
|
3519 |
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
|
3520 |
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
|
3521 |
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
|
3522 |
|
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
3523 |
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
|
3524 |
"\<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
|
3525 |
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
|
3526 |
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
|
3527 |
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
|
3528 |
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
|
3529 |
|
61711
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3530 |
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
|
3531 |
|
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3532 |
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
|
3533 |
|
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3534 |
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
|
3535 |
"\<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
|
3536 |
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
|
3537 |
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
|
3538 |
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
|
3539 |
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
|
3540 |
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
|
3541 |
done |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3542 |
|
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3543 |
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
|
3544 |
"\<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
|
3545 |
using homotopic_paths_rid [of "reversepath p" s] |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3546 |
by (metis homotopic_paths_reversepath path_image_reversepath path_reversepath pathfinish_linepath |
61711
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3547 |
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
|
3548 |
|
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3549 |
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
|
3550 |
"\<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
|
3551 |
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
|
3552 |
\<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
|
3553 |
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
|
3554 |
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
|
3555 |
[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
|
3556 |
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
|
3557 |
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
|
3558 |
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
|
3559 |
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
|
3560 |
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
|
3561 |
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
|
3562 |
done |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3563 |
|
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3564 |
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
|
3565 |
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
|
3566 |
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
|
3567 |
proof - |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3568 |
have "continuous_on ({0..1} \<times> {0..1}) (\<lambda>x. (subpath 0 (fst x) p +++ reversepath (subpath 0 (fst x) p)) (snd x))" |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3569 |
using assms |
61711
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3570 |
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
|
3571 |
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
|
3572 |
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
|
3573 |
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
|
3574 |
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
|
3575 |
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
|
3576 |
done |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3577 |
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
|
3578 |
using assms |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3579 |
apply (subst homotopic_paths_sym_eq) |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3580 |
unfolding homotopic_paths_def homotopic_with_def |
61711
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3581 |
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
|
3582 |
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
|
3583 |
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
|
3584 |
done |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3585 |
qed |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3586 |
|
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3587 |
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
|
3588 |
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
|
3589 |
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
|
3590 |
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
|
3591 |
|
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3592 |
|
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3593 |
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
|
3594 |
|
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3595 |
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
|
3596 |
"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
|
3597 |
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
|
3598 |
|
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3599 |
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
|
3600 |
"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
|
3601 |
(\<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
|
3602 |
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
|
3603 |
(\<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
|
3604 |
(\<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
|
3605 |
(\<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
|
3606 |
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
|
3607 |
|
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3608 |
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
|
3609 |
"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
|
3610 |
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
|
3611 |
|
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3612 |
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
|
3613 |
"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
|
3614 |
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
|
3615 |
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
|
3616 |
|
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3617 |
proposition homotopic_loops_imp_subset: |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3618 |
"homotopic_loops s p q \<Longrightarrow> path_image p \<subseteq> s \<and> path_image q \<subseteq> s" |
61711
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3619 |
unfolding homotopic_loops_def path_image_def |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3620 |
by (metis homotopic_with_imp_subset1 homotopic_with_imp_subset2) |
61711
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3621 |
|
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3622 |
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
|
3623 |
"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
|
3624 |
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
|
3625 |
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
|
3626 |
|
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3627 |
proposition homotopic_loops_sym: "homotopic_loops s p q \<Longrightarrow> homotopic_loops s q p" |
61711
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3628 |
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
|
3629 |
|
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3630 |
proposition homotopic_loops_sym_eq: "homotopic_loops s p q \<longleftrightarrow> homotopic_loops s q p" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3631 |
by (metis homotopic_loops_sym) |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3632 |
|
61711
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3633 |
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
|
3634 |
"\<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
|
3635 |
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
|
3636 |
|
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3637 |
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
|
3638 |
"\<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
|
3639 |
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
|
3640 |
|
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3641 |
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
|
3642 |
"\<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
|
3643 |
\<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
|
3644 |
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
|
3645 |
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
|
3646 |
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
|
3647 |
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
|
3648 |
done |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3649 |
|
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3650 |
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
|
3651 |
"\<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
|
3652 |
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
|
3653 |
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
|
3654 |
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
|
3655 |
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
|
3656 |
|
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3657 |
|
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3658 |
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
|
3659 |
|
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3660 |
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
|
3661 |
"\<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
|
3662 |
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
|
3663 |
|
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3664 |
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
|
3665 |
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
|
3666 |
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
|
3667 |
proof - |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3668 |
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
|
3669 |
have ploop: "pathfinish p = pathstart p" by (metis assms homotopic_loops_imp_loop) |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3670 |
have pip: "path_image p \<subseteq> s" by (metis assms homotopic_loops_imp_subset) |
61711
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3671 |
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
|
3672 |
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
|
3673 |
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
|
3674 |
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
|
3675 |
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
|
3676 |
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
|
3677 |
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
|
3678 |
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
|
3679 |
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
|
3680 |
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
|
3681 |
done |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3682 |
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
|
3683 |
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
|
3684 |
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
|
3685 |
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
|
3686 |
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
|
3687 |
done |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3688 |
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
|
3689 |
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
|
3690 |
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
|
3691 |
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
|
3692 |
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
|
3693 |
done |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3694 |
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
|
3695 |
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
|
3696 |
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
|
3697 |
proof - |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3698 |
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
|
3699 |
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
|
3700 |
qed |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3701 |
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
|
3702 |
(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
|
3703 |
(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
|
3704 |
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
|
3705 |
\<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
|
3706 |
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
|
3707 |
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
|
3708 |
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
|
3709 |
using \<open>path p\<close> homotopic_paths_rid homotopic_paths_sym pip by blast |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3710 |
moreover have "homotopic_paths s (p +++ linepath (pathfinish p) (pathfinish p)) |
61711
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3711 |
(linepath (pathstart p) (pathstart p) +++ p +++ linepath (pathfinish p) (pathfinish p))" |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3712 |
apply (rule homotopic_paths_sym) |
61711
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3713 |
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
|
3714 |
by (metis 1 homotopic_paths_imp_path homotopic_paths_imp_pathstart homotopic_paths_imp_subset) |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3715 |
moreover have "homotopic_paths s (linepath (pathstart p) (pathstart p) +++ p +++ linepath (pathfinish p) (pathfinish p)) |
61711
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3716 |
((\<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
|
3717 |
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
|
3718 |
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
|
3719 |
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
|
3720 |
apply (intro conjI homotopic_join_lemma) |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3721 |
using ploop |
61711
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3722 |
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
|
3723 |
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
|
3724 |
done |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3725 |
moreover have "homotopic_paths s ((\<lambda>u. h (u, 0)) +++ linepath a a +++ reversepath (\<lambda>u. h (u, 0))) |
61711
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3726 |
(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
|
3727 |
apply (rule *) |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3728 |
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
|
3729 |
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
|
3730 |
done |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3731 |
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
|
3732 |
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
|
3733 |
qed |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3734 |
|
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3735 |
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
|
3736 |
fixes s :: "'a::real_normed_vector set" |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3737 |
assumes "path p" "path q" and pip: "path_image p \<subseteq> s" and piq: "path_image q \<subseteq> s" |
61711
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3738 |
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
|
3739 |
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
|
3740 |
proof - |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3741 |
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
|
3742 |
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
|
3743 |
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
|
3744 |
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
|
3745 |
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
|
3746 |
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
|
3747 |
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
|
3748 |
done |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3749 |
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
|
3750 |
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
|
3751 |
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
|
3752 |
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
|
3753 |
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
|
3754 |
done |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3755 |
have ps1: "\<And>a b. \<lbrakk>b * 2 \<le> 1; 0 \<le> b; 0 \<le> a; a \<le> 1\<rbrakk> \<Longrightarrow> p ((1 - a) * (2 * b) + a) \<in> s" |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3756 |
using sum_le_prod1 |
61711
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3757 |
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
|
3758 |
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
|
3759 |
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
|
3760 |
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
|
3761 |
apply (simp add: algebra_simps) |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3762 |
by (metis add_mono_thms_linordered_semiring(1) affine_ineq linear mult.commute mult.left_neutral mult_right_mono not_le |
61711
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3763 |
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
|
3764 |
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
|
3765 |
using path_image_def piq by fastforce |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3766 |
have "homotopic_loops s (p +++ q +++ reversepath p) |
61711
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3767 |
(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
|
3768 |
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
|
3769 |
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
|
3770 |
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
|
3771 |
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
|
3772 |
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
|
3773 |
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
|
3774 |
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
|
3775 |
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
|
3776 |
done |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3777 |
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
|
3778 |
proof - |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3779 |
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
|
3780 |
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
|
3781 |
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
|
3782 |
using homotopic_paths_trans by blast |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3783 |
hence "homotopic_paths s (linepath (pathfinish q) (pathfinish q) +++ q +++ linepath (pathfinish q) (pathfinish q)) q" |
61711
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3784 |
proof - |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3785 |
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
|
3786 |
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
|
3787 |
thus ?thesis |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3788 |
by (metis (no_types) 1 \<open>path q\<close> homotopic_paths_join homotopic_paths_rinv homotopic_paths_sym |
61711
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3789 |
homotopic_paths_trans qloop pathfinish_linepath piq) |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3790 |
qed |
61711
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3791 |
thus ?thesis |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3792 |
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
|
3793 |
qed |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3794 |
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
|
3795 |
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
|
3796 |
qed |
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
3797 |
|
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3798 |
|
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3799 |
subsection\<open> Homotopy of "nearby" function, paths and loops.\<close> |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3800 |
|
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3801 |
lemma homotopic_with_linear: |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3802 |
fixes f g :: "_ \<Rightarrow> 'b::real_normed_vector" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3803 |
assumes contf: "continuous_on s f" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3804 |
and contg:"continuous_on s g" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3805 |
and sub: "\<And>x. x \<in> s \<Longrightarrow> closed_segment (f x) (g x) \<subseteq> t" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3806 |
shows "homotopic_with (\<lambda>z. True) s t f g" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3807 |
apply (simp add: homotopic_with_def) |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3808 |
apply (rule_tac x="\<lambda>y. ((1 - (fst y)) *\<^sub>R f(snd y) + (fst y) *\<^sub>R g(snd y))" in exI) |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3809 |
apply (intro conjI) |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3810 |
apply (rule subset_refl continuous_intros continuous_on_subset [OF contf] continuous_on_compose2 [where g=f] |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3811 |
continuous_on_subset [OF contg] continuous_on_compose2 [where g=g]| simp)+ |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3812 |
using sub closed_segment_def apply fastforce+ |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3813 |
done |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3814 |
|
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3815 |
lemma homotopic_paths_linear: |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3816 |
fixes g h :: "real \<Rightarrow> 'a::real_normed_vector" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3817 |
assumes "path g" "path h" "pathstart h = pathstart g" "pathfinish h = pathfinish g" |
63114
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
3818 |
"\<And>t. t \<in> {0..1} \<Longrightarrow> closed_segment (g t) (h t) \<subseteq> s" |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3819 |
shows "homotopic_paths s g h" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3820 |
using assms |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3821 |
unfolding path_def |
62948
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
3822 |
apply (simp add: closed_segment_def pathstart_def pathfinish_def homotopic_paths_def homotopic_with_def) |
63114
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
3823 |
apply (rule_tac x="\<lambda>y. ((1 - (fst y)) *\<^sub>R (g o snd) y + (fst y) *\<^sub>R (h o snd) y)" in exI) |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
3824 |
apply (intro conjI subsetI continuous_intros; force) |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3825 |
done |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3826 |
|
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3827 |
lemma homotopic_loops_linear: |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3828 |
fixes g h :: "real \<Rightarrow> 'a::real_normed_vector" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3829 |
assumes "path g" "path h" "pathfinish g = pathstart g" "pathfinish h = pathstart h" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3830 |
"\<And>t x. t \<in> {0..1} \<Longrightarrow> closed_segment (g t) (h t) \<subseteq> s" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3831 |
shows "homotopic_loops s g h" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3832 |
using assms |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3833 |
unfolding path_def |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3834 |
apply (simp add: pathstart_def pathfinish_def homotopic_loops_def homotopic_with_def) |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3835 |
apply (rule_tac x="\<lambda>y. ((1 - (fst y)) *\<^sub>R g(snd y) + (fst y) *\<^sub>R h(snd y))" in exI) |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3836 |
apply (auto intro!: continuous_intros intro: continuous_on_compose2 [where g=g] continuous_on_compose2 [where g=h]) |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3837 |
apply (force simp: closed_segment_def) |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3838 |
done |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3839 |
|
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3840 |
lemma homotopic_paths_nearby_explicit: |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3841 |
assumes "path g" "path h" "pathstart h = pathstart g" "pathfinish h = pathfinish g" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3842 |
and no: "\<And>t x. \<lbrakk>t \<in> {0..1}; x \<notin> s\<rbrakk> \<Longrightarrow> norm(h t - g t) < norm(g t - x)" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3843 |
shows "homotopic_paths s g h" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3844 |
apply (rule homotopic_paths_linear [OF assms(1-4)]) |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3845 |
by (metis no segment_bound(1) subsetI norm_minus_commute not_le) |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3846 |
|
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3847 |
lemma homotopic_loops_nearby_explicit: |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3848 |
assumes "path g" "path h" "pathfinish g = pathstart g" "pathfinish h = pathstart h" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3849 |
and no: "\<And>t x. \<lbrakk>t \<in> {0..1}; x \<notin> s\<rbrakk> \<Longrightarrow> norm(h t - g t) < norm(g t - x)" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3850 |
shows "homotopic_loops s g h" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3851 |
apply (rule homotopic_loops_linear [OF assms(1-4)]) |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3852 |
by (metis no segment_bound(1) subsetI norm_minus_commute not_le) |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3853 |
|
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3854 |
lemma homotopic_nearby_paths: |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3855 |
fixes g h :: "real \<Rightarrow> 'a::euclidean_space" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3856 |
assumes "path g" "open s" "path_image g \<subseteq> s" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3857 |
shows "\<exists>e. 0 < e \<and> |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3858 |
(\<forall>h. path h \<and> |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3859 |
pathstart h = pathstart g \<and> pathfinish h = pathfinish g \<and> |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3860 |
(\<forall>t \<in> {0..1}. norm(h t - g t) < e) \<longrightarrow> homotopic_paths s g h)" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3861 |
proof - |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3862 |
obtain e where "e > 0" and e: "\<And>x y. x \<in> path_image g \<Longrightarrow> y \<in> - s \<Longrightarrow> e \<le> dist x y" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3863 |
using separate_compact_closed [of "path_image g" "-s"] assms by force |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3864 |
show ?thesis |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3865 |
apply (intro exI conjI) |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3866 |
using e [unfolded dist_norm] |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3867 |
apply (auto simp: intro!: homotopic_paths_nearby_explicit assms \<open>e > 0\<close>) |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3868 |
by (metis atLeastAtMost_iff imageI le_less_trans not_le path_image_def) |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3869 |
qed |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3870 |
|
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3871 |
lemma homotopic_nearby_loops: |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3872 |
fixes g h :: "real \<Rightarrow> 'a::euclidean_space" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3873 |
assumes "path g" "open s" "path_image g \<subseteq> s" "pathfinish g = pathstart g" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3874 |
shows "\<exists>e. 0 < e \<and> |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3875 |
(\<forall>h. path h \<and> pathfinish h = pathstart h \<and> |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3876 |
(\<forall>t \<in> {0..1}. norm(h t - g t) < e) \<longrightarrow> homotopic_loops s g h)" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3877 |
proof - |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3878 |
obtain e where "e > 0" and e: "\<And>x y. x \<in> path_image g \<Longrightarrow> y \<in> - s \<Longrightarrow> e \<le> dist x y" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3879 |
using separate_compact_closed [of "path_image g" "-s"] assms by force |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3880 |
show ?thesis |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3881 |
apply (intro exI conjI) |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3882 |
using e [unfolded dist_norm] |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3883 |
apply (auto simp: intro!: homotopic_loops_nearby_explicit assms \<open>e > 0\<close>) |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3884 |
by (metis atLeastAtMost_iff imageI le_less_trans not_le path_image_def) |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3885 |
qed |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61711
diff
changeset
|
3886 |
|
61762
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61738
diff
changeset
|
3887 |
subsection\<open> Homotopy and subpaths\<close> |
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61738
diff
changeset
|
3888 |
|
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61738
diff
changeset
|
3889 |
lemma homotopic_join_subpaths1: |
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61738
diff
changeset
|
3890 |
assumes "path g" and pag: "path_image g \<subseteq> s" |
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61738
diff
changeset
|
3891 |
and u: "u \<in> {0..1}" and v: "v \<in> {0..1}" and w: "w \<in> {0..1}" "u \<le> v" "v \<le> w" |
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61738
diff
changeset
|
3892 |
shows "homotopic_paths s (subpath u v g +++ subpath v w g) (subpath u w g)" |
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61738
diff
changeset
|
3893 |
proof - |
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61738
diff
changeset
|
3894 |
have 1: "t * 2 \<le> 1 \<Longrightarrow> u + t * (v * 2) \<le> v + t * (u * 2)" for t |
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61738
diff
changeset
|
3895 |
using affine_ineq \<open>u \<le> v\<close> by fastforce |
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61738
diff
changeset
|
3896 |
have 2: "t * 2 > 1 \<Longrightarrow> u + (2*t - 1) * v \<le> v + (2*t - 1) * w" for t |
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61738
diff
changeset
|
3897 |
by (metis add_mono_thms_linordered_semiring(1) diff_gt_0_iff_gt less_eq_real_def mult.commute mult_right_mono \<open>u \<le> v\<close> \<open>v \<le> w\<close>) |
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61738
diff
changeset
|
3898 |
have t2: "\<And>t::real. t*2 = 1 \<Longrightarrow> t = 1/2" by auto |
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61738
diff
changeset
|
3899 |
show ?thesis |
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61738
diff
changeset
|
3900 |
apply (rule homotopic_paths_subset [OF _ pag]) |
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61738
diff
changeset
|
3901 |
using assms |
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61738
diff
changeset
|
3902 |
apply (cases "w = u") |
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61738
diff
changeset
|
3903 |
using homotopic_paths_rinv [of "subpath u v g" "path_image g"] |
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61738
diff
changeset
|
3904 |
apply (force simp: closed_segment_eq_real_ivl image_mono path_image_def subpath_refl) |
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61738
diff
changeset
|
3905 |
apply (rule homotopic_paths_sym) |
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61738
diff
changeset
|
3906 |
apply (rule homotopic_paths_reparametrize |
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61738
diff
changeset
|
3907 |
[where f = "\<lambda>t. if t \<le> 1 / 2 |
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61738
diff
changeset
|
3908 |
then inverse((w - u)) *\<^sub>R (2 * (v - u)) *\<^sub>R t |
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61738
diff
changeset
|
3909 |
else inverse((w - u)) *\<^sub>R ((v - u) + (w - v) *\<^sub>R (2 *\<^sub>R t - 1))"]) |
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61738
diff
changeset
|
3910 |
using \<open>path g\<close> path_subpath u w apply blast |
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61738
diff
changeset
|
3911 |
using \<open>path g\<close> path_image_subpath_subset u w(1) apply blast |
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61738
diff
changeset
|
3912 |
apply simp_all |
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61738
diff
changeset
|
3913 |
apply (subst split_01) |
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61738
diff
changeset
|
3914 |
apply (rule continuous_on_cases continuous_intros | force simp: pathfinish_def joinpaths_def)+ |
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61738
diff
changeset
|
3915 |
apply (simp_all add: field_simps not_le) |
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61738
diff
changeset
|
3916 |
apply (force dest!: t2) |
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61738
diff
changeset
|
3917 |
apply (force simp: algebra_simps mult_left_mono affine_ineq dest!: 1 2) |
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61738
diff
changeset
|
3918 |
apply (simp add: joinpaths_def subpath_def) |
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61738
diff
changeset
|
3919 |
apply (force simp: algebra_simps) |
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61738
diff
changeset
|
3920 |
done |
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61738
diff
changeset
|
3921 |
qed |
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61738
diff
changeset
|
3922 |
|
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61738
diff
changeset
|
3923 |
lemma homotopic_join_subpaths2: |
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61738
diff
changeset
|
3924 |
assumes "homotopic_paths s (subpath u v g +++ subpath v w g) (subpath u w g)" |
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61738
diff
changeset
|
3925 |
shows "homotopic_paths s (subpath w v g +++ subpath v u g) (subpath w u g)" |
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61738
diff
changeset
|
3926 |
by (metis assms homotopic_paths_reversepath_D pathfinish_subpath pathstart_subpath reversepath_joinpaths reversepath_subpath) |
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61738
diff
changeset
|
3927 |
|
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61738
diff
changeset
|
3928 |
lemma homotopic_join_subpaths3: |
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61738
diff
changeset
|
3929 |
assumes hom: "homotopic_paths s (subpath u v g +++ subpath v w g) (subpath u w g)" |
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61738
diff
changeset
|
3930 |
and "path g" and pag: "path_image g \<subseteq> s" |
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61738
diff
changeset
|
3931 |
and u: "u \<in> {0..1}" and v: "v \<in> {0..1}" and w: "w \<in> {0..1}" |
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61738
diff
changeset
|
3932 |
shows "homotopic_paths s (subpath v w g +++ subpath w u g) (subpath v u g)" |
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61738
diff
changeset
|
3933 |
proof - |
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61738
diff
changeset
|
3934 |
have "homotopic_paths s (subpath u w g +++ subpath w v g) ((subpath u v g +++ subpath v w g) +++ subpath w v g)" |
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61738
diff
changeset
|
3935 |
apply (rule homotopic_paths_join) |
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61738
diff
changeset
|
3936 |
using hom homotopic_paths_sym_eq apply blast |
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61738
diff
changeset
|
3937 |
apply (metis \<open>path g\<close> homotopic_paths_eq pag path_image_subpath_subset path_subpath subset_trans v w) |
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61738
diff
changeset
|
3938 |
apply (simp add:) |
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61738
diff
changeset
|
3939 |
done |
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61738
diff
changeset
|
3940 |
also have "homotopic_paths s ((subpath u v g +++ subpath v w g) +++ subpath w v g) (subpath u v g +++ subpath v w g +++ subpath w v g)" |
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61738
diff
changeset
|
3941 |
apply (rule homotopic_paths_sym [OF homotopic_paths_assoc]) |
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61738
diff
changeset
|
3942 |
using assms by (simp_all add: path_image_subpath_subset [THEN order_trans]) |
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61738
diff
changeset
|
3943 |
also have "homotopic_paths s (subpath u v g +++ subpath v w g +++ subpath w v g) |
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61738
diff
changeset
|
3944 |
(subpath u v g +++ linepath (pathfinish (subpath u v g)) (pathfinish (subpath u v g)))" |
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61738
diff
changeset
|
3945 |
apply (rule homotopic_paths_join) |
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61738
diff
changeset
|
3946 |
apply (metis \<open>path g\<close> homotopic_paths_eq order.trans pag path_image_subpath_subset path_subpath u v) |
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61738
diff
changeset
|
3947 |
apply (metis (no_types, lifting) \<open>path g\<close> homotopic_paths_linv order_trans pag path_image_subpath_subset path_subpath pathfinish_subpath reversepath_subpath v w) |
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61738
diff
changeset
|
3948 |
apply (simp add:) |
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61738
diff
changeset
|
3949 |
done |
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61738
diff
changeset
|
3950 |
also have "homotopic_paths s (subpath u v g +++ linepath (pathfinish (subpath u v g)) (pathfinish (subpath u v g))) (subpath u v g)" |
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61738
diff
changeset
|
3951 |
apply (rule homotopic_paths_rid) |
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61738
diff
changeset
|
3952 |
using \<open>path g\<close> path_subpath u v apply blast |
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61738
diff
changeset
|
3953 |
apply (meson \<open>path g\<close> order.trans pag path_image_subpath_subset u v) |
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61738
diff
changeset
|
3954 |
done |
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61738
diff
changeset
|
3955 |
finally have "homotopic_paths s (subpath u w g +++ subpath w v g) (subpath u v g)" . |
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61738
diff
changeset
|
3956 |
then show ?thesis |
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61738
diff
changeset
|
3957 |
using homotopic_join_subpaths2 by blast |
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61738
diff
changeset
|
3958 |
qed |
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61738
diff
changeset
|
3959 |
|
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61738
diff
changeset
|
3960 |
proposition homotopic_join_subpaths: |
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61738
diff
changeset
|
3961 |
"\<lbrakk>path g; path_image g \<subseteq> s; u \<in> {0..1}; v \<in> {0..1}; w \<in> {0..1}\<rbrakk> |
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61738
diff
changeset
|
3962 |
\<Longrightarrow> homotopic_paths s (subpath u v g +++ subpath v w g) (subpath u w g)" |
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61738
diff
changeset
|
3963 |
apply (rule le_cases3 [of u v w]) |
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61738
diff
changeset
|
3964 |
using homotopic_join_subpaths1 homotopic_join_subpaths2 homotopic_join_subpaths3 by metis+ |
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61738
diff
changeset
|
3965 |
|
62620
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
3966 |
text\<open>Relating homotopy of trivial loops to path-connectedness.\<close> |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
3967 |
|
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
3968 |
lemma path_component_imp_homotopic_points: |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
3969 |
"path_component S a b \<Longrightarrow> homotopic_loops S (linepath a a) (linepath b b)" |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
3970 |
apply (simp add: path_component_def homotopic_loops_def homotopic_with_def |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
3971 |
pathstart_def pathfinish_def path_image_def path_def, clarify) |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
3972 |
apply (rule_tac x="g o fst" in exI) |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
3973 |
apply (intro conjI continuous_intros continuous_on_compose)+ |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
3974 |
apply (auto elim!: continuous_on_subset) |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
3975 |
done |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
3976 |
|
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
3977 |
lemma homotopic_loops_imp_path_component_value: |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
3978 |
"\<lbrakk>homotopic_loops S p q; 0 \<le> t; t \<le> 1\<rbrakk> |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
3979 |
\<Longrightarrow> path_component S (p t) (q t)" |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
3980 |
apply (simp add: path_component_def homotopic_loops_def homotopic_with_def |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
3981 |
pathstart_def pathfinish_def path_image_def path_def, clarify) |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
3982 |
apply (rule_tac x="h o (\<lambda>u. (u, t))" in exI) |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
3983 |
apply (intro conjI continuous_intros continuous_on_compose)+ |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
3984 |
apply (auto elim!: continuous_on_subset) |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
3985 |
done |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
3986 |
|
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
3987 |
lemma homotopic_points_eq_path_component: |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
3988 |
"homotopic_loops S (linepath a a) (linepath b b) \<longleftrightarrow> |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
3989 |
path_component S a b" |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
3990 |
by (auto simp: path_component_imp_homotopic_points |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
3991 |
dest: homotopic_loops_imp_path_component_value [where t=1]) |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
3992 |
|
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
3993 |
lemma path_connected_eq_homotopic_points: |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
3994 |
"path_connected S \<longleftrightarrow> |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
3995 |
(\<forall>a b. a \<in> S \<and> b \<in> S \<longrightarrow> homotopic_loops S (linepath a a) (linepath b b))" |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
3996 |
by (auto simp: path_connected_def path_component_def homotopic_points_eq_path_component) |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
3997 |
|
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
3998 |
|
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
3999 |
subsection\<open>Simply connected sets\<close> |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4000 |
|
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4001 |
text\<open>defined as "all loops are homotopic (as loops)\<close> |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4002 |
|
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4003 |
definition simply_connected where |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4004 |
"simply_connected S \<equiv> |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4005 |
\<forall>p q. path p \<and> pathfinish p = pathstart p \<and> path_image p \<subseteq> S \<and> |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4006 |
path q \<and> pathfinish q = pathstart q \<and> path_image q \<subseteq> S |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4007 |
\<longrightarrow> homotopic_loops S p q" |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4008 |
|
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4009 |
lemma simply_connected_empty [iff]: "simply_connected {}" |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4010 |
by (simp add: simply_connected_def) |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4011 |
|
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4012 |
lemma simply_connected_imp_path_connected: |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4013 |
fixes S :: "_::real_normed_vector set" |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4014 |
shows "simply_connected S \<Longrightarrow> path_connected S" |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4015 |
by (simp add: simply_connected_def path_connected_eq_homotopic_points) |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4016 |
|
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4017 |
lemma simply_connected_imp_connected: |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4018 |
fixes S :: "_::real_normed_vector set" |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4019 |
shows "simply_connected S \<Longrightarrow> connected S" |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4020 |
by (simp add: path_connected_imp_connected simply_connected_imp_path_connected) |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4021 |
|
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4022 |
lemma simply_connected_eq_contractible_loop_any: |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4023 |
fixes S :: "_::real_normed_vector set" |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4024 |
shows "simply_connected S \<longleftrightarrow> |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4025 |
(\<forall>p a. path p \<and> path_image p \<subseteq> S \<and> |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4026 |
pathfinish p = pathstart p \<and> a \<in> S |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4027 |
\<longrightarrow> homotopic_loops S p (linepath a a))" |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4028 |
apply (simp add: simply_connected_def) |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4029 |
apply (rule iffI, force, clarify) |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4030 |
apply (rule_tac q = "linepath (pathstart p) (pathstart p)" in homotopic_loops_trans) |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4031 |
apply (fastforce simp add:) |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4032 |
using homotopic_loops_sym apply blast |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4033 |
done |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4034 |
|
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4035 |
lemma simply_connected_eq_contractible_loop_some: |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4036 |
fixes S :: "_::real_normed_vector set" |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4037 |
shows "simply_connected S \<longleftrightarrow> |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4038 |
path_connected S \<and> |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4039 |
(\<forall>p. path p \<and> path_image p \<subseteq> S \<and> pathfinish p = pathstart p |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4040 |
\<longrightarrow> (\<exists>a. a \<in> S \<and> homotopic_loops S p (linepath a a)))" |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4041 |
apply (rule iffI) |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4042 |
apply (fastforce simp: simply_connected_imp_path_connected simply_connected_eq_contractible_loop_any) |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4043 |
apply (clarsimp simp add: simply_connected_eq_contractible_loop_any) |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4044 |
apply (drule_tac x=p in spec) |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4045 |
using homotopic_loops_trans path_connected_eq_homotopic_points |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4046 |
apply blast |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4047 |
done |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4048 |
|
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4049 |
lemma simply_connected_eq_contractible_loop_all: |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4050 |
fixes S :: "_::real_normed_vector set" |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4051 |
shows "simply_connected S \<longleftrightarrow> |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4052 |
S = {} \<or> |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4053 |
(\<exists>a \<in> S. \<forall>p. path p \<and> path_image p \<subseteq> S \<and> pathfinish p = pathstart p |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4054 |
\<longrightarrow> homotopic_loops S p (linepath a a))" |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4055 |
(is "?lhs = ?rhs") |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4056 |
proof (cases "S = {}") |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4057 |
case True then show ?thesis by force |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4058 |
next |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4059 |
case False |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4060 |
then obtain a where "a \<in> S" by blast |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4061 |
show ?thesis |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4062 |
proof |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4063 |
assume "simply_connected S" |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4064 |
then show ?rhs |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4065 |
using \<open>a \<in> S\<close> \<open>simply_connected S\<close> simply_connected_eq_contractible_loop_any |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4066 |
by blast |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4067 |
next |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4068 |
assume ?rhs |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4069 |
then show "simply_connected S" |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4070 |
apply (simp add: simply_connected_eq_contractible_loop_any False) |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4071 |
by (meson homotopic_loops_refl homotopic_loops_sym homotopic_loops_trans |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4072 |
path_component_imp_homotopic_points path_component_refl) |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4073 |
qed |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4074 |
qed |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4075 |
|
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4076 |
lemma simply_connected_eq_contractible_path: |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4077 |
fixes S :: "_::real_normed_vector set" |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4078 |
shows "simply_connected S \<longleftrightarrow> |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4079 |
path_connected S \<and> |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4080 |
(\<forall>p. path p \<and> path_image p \<subseteq> S \<and> pathfinish p = pathstart p |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4081 |
\<longrightarrow> homotopic_paths S p (linepath (pathstart p) (pathstart p)))" |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4082 |
apply (rule iffI) |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4083 |
apply (simp add: simply_connected_imp_path_connected) |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4084 |
apply (metis simply_connected_eq_contractible_loop_some homotopic_loops_imp_homotopic_paths_null) |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4085 |
by (meson homotopic_paths_imp_homotopic_loops pathfinish_linepath pathstart_in_path_image |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4086 |
simply_connected_eq_contractible_loop_some subset_iff) |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4087 |
|
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4088 |
lemma simply_connected_eq_homotopic_paths: |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4089 |
fixes S :: "_::real_normed_vector set" |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4090 |
shows "simply_connected S \<longleftrightarrow> |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4091 |
path_connected S \<and> |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4092 |
(\<forall>p q. path p \<and> path_image p \<subseteq> S \<and> |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4093 |
path q \<and> path_image q \<subseteq> S \<and> |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4094 |
pathstart q = pathstart p \<and> pathfinish q = pathfinish p |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4095 |
\<longrightarrow> homotopic_paths S p q)" |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4096 |
(is "?lhs = ?rhs") |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4097 |
proof |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4098 |
assume ?lhs |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4099 |
then have pc: "path_connected S" |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4100 |
and *: "\<And>p. \<lbrakk>path p; path_image p \<subseteq> S; |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4101 |
pathfinish p = pathstart p\<rbrakk> |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4102 |
\<Longrightarrow> homotopic_paths S p (linepath (pathstart p) (pathstart p))" |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4103 |
by (auto simp: simply_connected_eq_contractible_path) |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4104 |
have "homotopic_paths S p q" |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4105 |
if "path p" "path_image p \<subseteq> S" "path q" |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4106 |
"path_image q \<subseteq> S" "pathstart q = pathstart p" |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4107 |
"pathfinish q = pathfinish p" for p q |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4108 |
proof - |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4109 |
have "homotopic_paths S p (p +++ linepath (pathfinish p) (pathfinish p))" |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4110 |
by (simp add: homotopic_paths_rid homotopic_paths_sym that) |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4111 |
also have "homotopic_paths S (p +++ linepath (pathfinish p) (pathfinish p)) |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4112 |
(p +++ reversepath q +++ q)" |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4113 |
using that |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4114 |
by (metis homotopic_paths_join homotopic_paths_linv homotopic_paths_refl homotopic_paths_sym_eq pathstart_linepath) |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4115 |
also have "homotopic_paths S (p +++ reversepath q +++ q) |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4116 |
((p +++ reversepath q) +++ q)" |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4117 |
by (simp add: that homotopic_paths_assoc) |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4118 |
also have "homotopic_paths S ((p +++ reversepath q) +++ q) |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4119 |
(linepath (pathstart q) (pathstart q) +++ q)" |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4120 |
using * [of "p +++ reversepath q"] that |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4121 |
by (simp add: homotopic_paths_join path_image_join) |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4122 |
also have "homotopic_paths S (linepath (pathstart q) (pathstart q) +++ q) q" |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4123 |
using that homotopic_paths_lid by blast |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4124 |
finally show ?thesis . |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4125 |
qed |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4126 |
then show ?rhs |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4127 |
by (blast intro: pc *) |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4128 |
next |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4129 |
assume ?rhs |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4130 |
then show ?lhs |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4131 |
by (force simp: simply_connected_eq_contractible_path) |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4132 |
qed |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4133 |
|
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4134 |
proposition simply_connected_Times: |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4135 |
fixes S :: "'a::real_normed_vector set" and T :: "'b::real_normed_vector set" |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4136 |
assumes S: "simply_connected S" and T: "simply_connected T" |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4137 |
shows "simply_connected(S \<times> T)" |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4138 |
proof - |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4139 |
have "homotopic_loops (S \<times> T) p (linepath (a, b) (a, b))" |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4140 |
if "path p" "path_image p \<subseteq> S \<times> T" "p 1 = p 0" "a \<in> S" "b \<in> T" |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4141 |
for p a b |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4142 |
proof - |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4143 |
have "path (fst \<circ> p)" |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4144 |
apply (rule Path_Connected.path_continuous_image [OF \<open>path p\<close>]) |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4145 |
apply (rule continuous_intros)+ |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4146 |
done |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4147 |
moreover have "path_image (fst \<circ> p) \<subseteq> S" |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4148 |
using that apply (simp add: path_image_def) by force |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4149 |
ultimately have p1: "homotopic_loops S (fst o p) (linepath a a)" |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4150 |
using S that |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4151 |
apply (simp add: simply_connected_eq_contractible_loop_any) |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4152 |
apply (drule_tac x="fst o p" in spec) |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4153 |
apply (drule_tac x=a in spec) |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4154 |
apply (auto simp: pathstart_def pathfinish_def) |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4155 |
done |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4156 |
have "path (snd \<circ> p)" |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4157 |
apply (rule Path_Connected.path_continuous_image [OF \<open>path p\<close>]) |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4158 |
apply (rule continuous_intros)+ |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4159 |
done |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4160 |
moreover have "path_image (snd \<circ> p) \<subseteq> T" |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4161 |
using that apply (simp add: path_image_def) by force |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4162 |
ultimately have p2: "homotopic_loops T (snd o p) (linepath b b)" |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4163 |
using T that |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4164 |
apply (simp add: simply_connected_eq_contractible_loop_any) |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4165 |
apply (drule_tac x="snd o p" in spec) |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4166 |
apply (drule_tac x=b in spec) |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4167 |
apply (auto simp: pathstart_def pathfinish_def) |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4168 |
done |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4169 |
show ?thesis |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4170 |
using p1 p2 |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4171 |
apply (simp add: homotopic_loops, clarify) |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4172 |
apply (rename_tac h k) |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4173 |
apply (rule_tac x="\<lambda>z. Pair (h z) (k z)" in exI) |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4174 |
apply (intro conjI continuous_intros | assumption)+ |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4175 |
apply (auto simp: pathstart_def pathfinish_def) |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4176 |
done |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4177 |
qed |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4178 |
with assms show ?thesis |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4179 |
by (simp add: simply_connected_eq_contractible_loop_any pathfinish_def pathstart_def) |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4180 |
qed |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
4181 |
|
62626
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4182 |
subsection\<open>Contractible sets\<close> |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4183 |
|
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4184 |
definition contractible where |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4185 |
"contractible S \<equiv> \<exists>a. homotopic_with (\<lambda>x. True) S S id (\<lambda>x. a)" |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4186 |
|
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4187 |
proposition contractible_imp_simply_connected: |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4188 |
fixes S :: "_::real_normed_vector set" |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4189 |
assumes "contractible S" shows "simply_connected S" |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4190 |
proof (cases "S = {}") |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4191 |
case True then show ?thesis by force |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4192 |
next |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4193 |
case False |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4194 |
obtain a where a: "homotopic_with (\<lambda>x. True) S S id (\<lambda>x. a)" |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4195 |
using assms by (force simp add: contractible_def) |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4196 |
then have "a \<in> S" |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4197 |
by (metis False homotopic_constant_maps homotopic_with_symD homotopic_with_trans path_component_mem(2)) |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4198 |
show ?thesis |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4199 |
apply (simp add: simply_connected_eq_contractible_loop_all False) |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4200 |
apply (rule bexI [OF _ \<open>a \<in> S\<close>]) |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4201 |
using a apply (simp add: homotopic_loops_def homotopic_with_def path_def path_image_def pathfinish_def pathstart_def) |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4202 |
apply clarify |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4203 |
apply (rule_tac x="(h o (\<lambda>y. (fst y, (p \<circ> snd) y)))" in exI) |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4204 |
apply (intro conjI continuous_on_compose continuous_intros) |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4205 |
apply (erule continuous_on_subset | force)+ |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4206 |
done |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4207 |
qed |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4208 |
|
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4209 |
corollary contractible_imp_connected: |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4210 |
fixes S :: "_::real_normed_vector set" |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4211 |
shows "contractible S \<Longrightarrow> connected S" |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4212 |
by (simp add: contractible_imp_simply_connected simply_connected_imp_connected) |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4213 |
|
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4214 |
lemma contractible_imp_path_connected: |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4215 |
fixes S :: "_::real_normed_vector set" |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4216 |
shows "contractible S \<Longrightarrow> path_connected S" |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4217 |
by (simp add: contractible_imp_simply_connected simply_connected_imp_path_connected) |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4218 |
|
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4219 |
lemma nullhomotopic_through_contractible: |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4220 |
fixes S :: "_::topological_space set" |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4221 |
assumes f: "continuous_on S f" "f ` S \<subseteq> T" |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4222 |
and g: "continuous_on T g" "g ` T \<subseteq> U" |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4223 |
and T: "contractible T" |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4224 |
obtains c where "homotopic_with (\<lambda>h. True) S U (g o f) (\<lambda>x. c)" |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4225 |
proof - |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4226 |
obtain b where b: "homotopic_with (\<lambda>x. True) T T id (\<lambda>x. b)" |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4227 |
using assms by (force simp add: contractible_def) |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4228 |
have "homotopic_with (\<lambda>f. True) T U (g \<circ> id) (g \<circ> (\<lambda>x. b))" |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4229 |
by (rule homotopic_compose_continuous_left [OF b g]) |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4230 |
then have "homotopic_with (\<lambda>f. True) S U (g \<circ> id \<circ> f) (g \<circ> (\<lambda>x. b) \<circ> f)" |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4231 |
by (rule homotopic_compose_continuous_right [OF _ f]) |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4232 |
then show ?thesis |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4233 |
by (simp add: comp_def that) |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4234 |
qed |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4235 |
|
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4236 |
lemma nullhomotopic_into_contractible: |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4237 |
assumes f: "continuous_on S f" "f ` S \<subseteq> T" |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4238 |
and T: "contractible T" |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4239 |
obtains c where "homotopic_with (\<lambda>h. True) S T f (\<lambda>x. c)" |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4240 |
apply (rule nullhomotopic_through_contractible [OF f, of id T]) |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4241 |
using assms |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4242 |
apply (auto simp: continuous_on_id) |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4243 |
done |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4244 |
|
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4245 |
lemma nullhomotopic_from_contractible: |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4246 |
assumes f: "continuous_on S f" "f ` S \<subseteq> T" |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4247 |
and S: "contractible S" |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4248 |
obtains c where "homotopic_with (\<lambda>h. True) S T f (\<lambda>x. c)" |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4249 |
apply (rule nullhomotopic_through_contractible [OF continuous_on_id _ f S, of S]) |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4250 |
using assms |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4251 |
apply (auto simp: comp_def) |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4252 |
done |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4253 |
|
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4254 |
lemma homotopic_through_contractible: |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4255 |
fixes S :: "_::real_normed_vector set" |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4256 |
assumes "continuous_on S f1" "f1 ` S \<subseteq> T" |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4257 |
"continuous_on T g1" "g1 ` T \<subseteq> U" |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4258 |
"continuous_on S f2" "f2 ` S \<subseteq> T" |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4259 |
"continuous_on T g2" "g2 ` T \<subseteq> U" |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4260 |
"contractible T" "path_connected U" |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4261 |
shows "homotopic_with (\<lambda>h. True) S U (g1 o f1) (g2 o f2)" |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4262 |
proof - |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4263 |
obtain c1 where c1: "homotopic_with (\<lambda>h. True) S U (g1 o f1) (\<lambda>x. c1)" |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4264 |
apply (rule nullhomotopic_through_contractible [of S f1 T g1 U]) |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4265 |
using assms apply (auto simp: ) |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4266 |
done |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4267 |
obtain c2 where c2: "homotopic_with (\<lambda>h. True) S U (g2 o f2) (\<lambda>x. c2)" |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4268 |
apply (rule nullhomotopic_through_contractible [of S f2 T g2 U]) |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4269 |
using assms apply (auto simp: ) |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4270 |
done |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4271 |
have *: "S = {} \<or> (\<exists>t. path_connected t \<and> t \<subseteq> U \<and> c2 \<in> t \<and> c1 \<in> t)" |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4272 |
proof (cases "S = {}") |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4273 |
case True then show ?thesis by force |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4274 |
next |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4275 |
case False |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4276 |
with c1 c2 have "c1 \<in> U" "c2 \<in> U" |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4277 |
using homotopic_with_imp_subset2 all_not_in_conv image_subset_iff by blast+ |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4278 |
with \<open>path_connected U\<close> show ?thesis by blast |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4279 |
qed |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4280 |
show ?thesis |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4281 |
apply (rule homotopic_with_trans [OF c1]) |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4282 |
apply (rule homotopic_with_symD) |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4283 |
apply (rule homotopic_with_trans [OF c2]) |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4284 |
apply (simp add: path_component homotopic_constant_maps *) |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4285 |
done |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4286 |
qed |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4287 |
|
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4288 |
lemma homotopic_into_contractible: |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4289 |
fixes S :: "'a::real_normed_vector set" and T:: "'b::real_normed_vector set" |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4290 |
assumes f: "continuous_on S f" "f ` S \<subseteq> T" |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4291 |
and g: "continuous_on S g" "g ` S \<subseteq> T" |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4292 |
and T: "contractible T" |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4293 |
shows "homotopic_with (\<lambda>h. True) S T f g" |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4294 |
using homotopic_through_contractible [of S f T id T g id] |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4295 |
by (simp add: assms contractible_imp_path_connected continuous_on_id) |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4296 |
|
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4297 |
lemma homotopic_from_contractible: |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4298 |
fixes S :: "'a::real_normed_vector set" and T:: "'b::real_normed_vector set" |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4299 |
assumes f: "continuous_on S f" "f ` S \<subseteq> T" |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4300 |
and g: "continuous_on S g" "g ` S \<subseteq> T" |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4301 |
and "contractible S" "path_connected T" |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4302 |
shows "homotopic_with (\<lambda>h. True) S T f g" |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4303 |
using homotopic_through_contractible [of S id S f T id g] |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4304 |
by (simp add: assms contractible_imp_path_connected continuous_on_id) |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4305 |
|
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4306 |
lemma starlike_imp_contractible_gen: |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4307 |
fixes S :: "'a::real_normed_vector set" |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4308 |
assumes S: "starlike S" |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4309 |
and P: "\<And>a T. \<lbrakk>a \<in> S; 0 \<le> T; T \<le> 1\<rbrakk> \<Longrightarrow> P(\<lambda>x. (1 - T) *\<^sub>R x + T *\<^sub>R a)" |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4310 |
obtains a where "homotopic_with P S S (\<lambda>x. x) (\<lambda>x. a)" |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4311 |
proof - |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4312 |
obtain a where "a \<in> S" and a: "\<And>x. x \<in> S \<Longrightarrow> closed_segment a x \<subseteq> S" |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4313 |
using S by (auto simp add: starlike_def) |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4314 |
have "(\<lambda>y. (1 - fst y) *\<^sub>R snd y + fst y *\<^sub>R a) ` ({0..1} \<times> S) \<subseteq> S" |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4315 |
apply clarify |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4316 |
apply (erule a [unfolded closed_segment_def, THEN subsetD]) |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4317 |
apply (simp add: ) |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4318 |
apply (metis add_diff_cancel_right' diff_ge_0_iff_ge le_add_diff_inverse pth_c(1)) |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4319 |
done |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4320 |
then show ?thesis |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4321 |
apply (rule_tac a="a" in that) |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4322 |
using \<open>a \<in> S\<close> |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4323 |
apply (simp add: homotopic_with_def) |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4324 |
apply (rule_tac x="\<lambda>y. (1 - (fst y)) *\<^sub>R snd y + (fst y) *\<^sub>R a" in exI) |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4325 |
apply (intro conjI ballI continuous_on_compose continuous_intros) |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4326 |
apply (simp_all add: P) |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4327 |
done |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4328 |
qed |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4329 |
|
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4330 |
lemma starlike_imp_contractible: |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4331 |
fixes S :: "'a::real_normed_vector set" |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4332 |
shows "starlike S \<Longrightarrow> contractible S" |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4333 |
using starlike_imp_contractible_gen contractible_def by (fastforce simp: id_def) |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4334 |
|
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4335 |
lemma contractible_UNIV: "contractible (UNIV :: 'a::real_normed_vector set)" |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4336 |
by (simp add: starlike_imp_contractible) |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4337 |
|
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4338 |
lemma starlike_imp_simply_connected: |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4339 |
fixes S :: "'a::real_normed_vector set" |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4340 |
shows "starlike S \<Longrightarrow> simply_connected S" |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4341 |
by (simp add: contractible_imp_simply_connected starlike_imp_contractible) |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4342 |
|
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4343 |
lemma convex_imp_simply_connected: |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4344 |
fixes S :: "'a::real_normed_vector set" |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4345 |
shows "convex S \<Longrightarrow> simply_connected S" |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4346 |
using convex_imp_starlike starlike_imp_simply_connected by blast |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4347 |
|
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4348 |
lemma starlike_imp_path_connected: |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4349 |
fixes S :: "'a::real_normed_vector set" |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4350 |
shows "starlike S \<Longrightarrow> path_connected S" |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4351 |
by (simp add: simply_connected_imp_path_connected starlike_imp_simply_connected) |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4352 |
|
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4353 |
lemma starlike_imp_connected: |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4354 |
fixes S :: "'a::real_normed_vector set" |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4355 |
shows "starlike S \<Longrightarrow> connected S" |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4356 |
by (simp add: path_connected_imp_connected starlike_imp_path_connected) |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4357 |
|
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4358 |
lemma is_interval_simply_connected_1: |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4359 |
fixes S :: "real set" |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4360 |
shows "is_interval S \<longleftrightarrow> simply_connected S" |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4361 |
using convex_imp_simply_connected is_interval_convex_1 is_interval_path_connected_1 simply_connected_imp_path_connected by auto |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4362 |
|
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4363 |
lemma contractible_empty: "contractible {}" |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4364 |
by (simp add: continuous_on_empty contractible_def homotopic_with) |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4365 |
|
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4366 |
lemma contractible_convex_tweak_boundary_points: |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4367 |
fixes S :: "'a::euclidean_space set" |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4368 |
assumes "convex S" and TS: "rel_interior S \<subseteq> T" "T \<subseteq> closure S" |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4369 |
shows "contractible T" |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4370 |
proof (cases "S = {}") |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4371 |
case True |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4372 |
with assms show ?thesis |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4373 |
by (simp add: contractible_empty subsetCE) |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4374 |
next |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4375 |
case False |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4376 |
show ?thesis |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4377 |
apply (rule starlike_imp_contractible) |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4378 |
apply (rule starlike_convex_tweak_boundary_points [OF \<open>convex S\<close> False TS]) |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4379 |
done |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4380 |
qed |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4381 |
|
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4382 |
lemma convex_imp_contractible: |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4383 |
fixes S :: "'a::real_normed_vector set" |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4384 |
shows "convex S \<Longrightarrow> contractible S" |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4385 |
using contractible_empty convex_imp_starlike starlike_imp_contractible by auto |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4386 |
|
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4387 |
lemma contractible_sing: |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4388 |
fixes a :: "'a::real_normed_vector" |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4389 |
shows "contractible {a}" |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4390 |
by (rule convex_imp_contractible [OF convex_singleton]) |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4391 |
|
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4392 |
lemma is_interval_contractible_1: |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4393 |
fixes S :: "real set" |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4394 |
shows "is_interval S \<longleftrightarrow> contractible S" |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4395 |
using contractible_imp_simply_connected convex_imp_contractible is_interval_convex_1 |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4396 |
is_interval_simply_connected_1 by auto |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4397 |
|
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4398 |
lemma contractible_times: |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4399 |
fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set" |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4400 |
assumes S: "contractible S" and T: "contractible T" |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4401 |
shows "contractible (S \<times> T)" |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4402 |
proof - |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4403 |
obtain a h where conth: "continuous_on ({0..1} \<times> S) h" |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4404 |
and hsub: "h ` ({0..1} \<times> S) \<subseteq> S" |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4405 |
and [simp]: "\<And>x. x \<in> S \<Longrightarrow> h (0, x) = x" |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4406 |
and [simp]: "\<And>x. x \<in> S \<Longrightarrow> h (1::real, x) = a" |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4407 |
using S by (auto simp add: contractible_def homotopic_with) |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4408 |
obtain b k where contk: "continuous_on ({0..1} \<times> T) k" |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4409 |
and ksub: "k ` ({0..1} \<times> T) \<subseteq> T" |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4410 |
and [simp]: "\<And>x. x \<in> T \<Longrightarrow> k (0, x) = x" |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4411 |
and [simp]: "\<And>x. x \<in> T \<Longrightarrow> k (1::real, x) = b" |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4412 |
using T by (auto simp add: contractible_def homotopic_with) |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4413 |
show ?thesis |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4414 |
apply (simp add: contractible_def homotopic_with) |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4415 |
apply (rule exI [where x=a]) |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4416 |
apply (rule exI [where x=b]) |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4417 |
apply (rule exI [where x = "\<lambda>z. (h (fst z, fst(snd z)), k (fst z, snd(snd z)))"]) |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4418 |
apply (intro conjI ballI continuous_intros continuous_on_compose2 [OF conth] continuous_on_compose2 [OF contk]) |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4419 |
using hsub ksub |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4420 |
apply (auto simp: ) |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4421 |
done |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4422 |
qed |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4423 |
|
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4424 |
lemma homotopy_dominated_contractibility: |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4425 |
fixes S :: "'a::real_normed_vector set" and T :: "'b::real_normed_vector set" |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4426 |
assumes S: "contractible S" |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4427 |
and f: "continuous_on S f" "image f S \<subseteq> T" |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4428 |
and g: "continuous_on T g" "image g T \<subseteq> S" |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4429 |
and hom: "homotopic_with (\<lambda>x. True) T T (f o g) id" |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4430 |
shows "contractible T" |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4431 |
proof - |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4432 |
obtain b where "homotopic_with (\<lambda>h. True) S T f (\<lambda>x. b)" |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4433 |
using nullhomotopic_from_contractible [OF f S] . |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4434 |
then have homg: "homotopic_with (\<lambda>x. True) T T ((\<lambda>x. b) \<circ> g) (f \<circ> g)" |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4435 |
by (rule homotopic_with_compose_continuous_right [OF homotopic_with_symD g]) |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4436 |
show ?thesis |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4437 |
apply (simp add: contractible_def) |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4438 |
apply (rule exI [where x = b]) |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4439 |
apply (rule homotopic_with_symD) |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4440 |
apply (rule homotopic_with_trans [OF _ hom]) |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4441 |
using homg apply (simp add: o_def) |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4442 |
done |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4443 |
qed |
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62620
diff
changeset
|
4444 |
|
62948
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4445 |
subsection\<open>Local versions of topological properties in general\<close> |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4446 |
|
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4447 |
definition locally :: "('a::topological_space set \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> bool" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4448 |
where |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4449 |
"locally P S \<equiv> |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4450 |
\<forall>w x. openin (subtopology euclidean S) w \<and> x \<in> w |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4451 |
\<longrightarrow> (\<exists>u v. openin (subtopology euclidean S) u \<and> P v \<and> |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4452 |
x \<in> u \<and> u \<subseteq> v \<and> v \<subseteq> w)" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4453 |
|
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4454 |
lemma locallyI: |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4455 |
assumes "\<And>w x. \<lbrakk>openin (subtopology euclidean S) w; x \<in> w\<rbrakk> |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4456 |
\<Longrightarrow> \<exists>u v. openin (subtopology euclidean S) u \<and> P v \<and> |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4457 |
x \<in> u \<and> u \<subseteq> v \<and> v \<subseteq> w" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4458 |
shows "locally P S" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4459 |
using assms by (force simp: locally_def) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4460 |
|
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4461 |
lemma locallyE: |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4462 |
assumes "locally P S" "openin (subtopology euclidean S) w" "x \<in> w" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4463 |
obtains u v where "openin (subtopology euclidean S) u" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4464 |
"P v" "x \<in> u" "u \<subseteq> v" "v \<subseteq> w" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4465 |
using assms by (force simp: locally_def) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4466 |
|
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4467 |
lemma locally_mono: |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4468 |
assumes "locally P S" "\<And>t. P t \<Longrightarrow> Q t" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4469 |
shows "locally Q S" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4470 |
by (metis assms locally_def) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4471 |
|
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4472 |
lemma locally_open_subset: |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4473 |
assumes "locally P S" "openin (subtopology euclidean S) t" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4474 |
shows "locally P t" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4475 |
using assms |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4476 |
apply (simp add: locally_def) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4477 |
apply (erule all_forward)+ |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4478 |
apply (rule impI) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4479 |
apply (erule impCE) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4480 |
using openin_trans apply blast |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4481 |
apply (erule ex_forward) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4482 |
by (metis (no_types, hide_lams) Int_absorb1 Int_lower1 Int_subset_iff openin_open openin_subtopology_Int_subset) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4483 |
|
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4484 |
lemma locally_diff_closed: |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4485 |
"\<lbrakk>locally P S; closedin (subtopology euclidean S) t\<rbrakk> \<Longrightarrow> locally P (S - t)" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4486 |
using locally_open_subset closedin_def by fastforce |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4487 |
|
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4488 |
lemma locally_empty [iff]: "locally P {}" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4489 |
by (simp add: locally_def openin_subtopology) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4490 |
|
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4491 |
lemma locally_singleton [iff]: |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4492 |
fixes a :: "'a::metric_space" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4493 |
shows "locally P {a} \<longleftrightarrow> P {a}" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4494 |
apply (simp add: locally_def openin_euclidean_subtopology_iff subset_singleton_iff conj_disj_distribR cong: conj_cong) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4495 |
using zero_less_one by blast |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4496 |
|
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4497 |
lemma locally_iff: |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4498 |
"locally P S \<longleftrightarrow> |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4499 |
(\<forall>T x. open T \<and> x \<in> S \<inter> T \<longrightarrow> (\<exists>U. open U \<and> (\<exists>v. P v \<and> x \<in> S \<inter> U \<and> S \<inter> U \<subseteq> v \<and> v \<subseteq> S \<inter> T)))" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4500 |
apply (simp add: le_inf_iff locally_def openin_open, safe) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4501 |
apply (metis IntE IntI le_inf_iff) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4502 |
apply (metis IntI Int_subset_iff) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4503 |
done |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4504 |
|
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4505 |
lemma locally_Int: |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4506 |
assumes S: "locally P S" and t: "locally P t" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4507 |
and P: "\<And>S t. P S \<and> P t \<Longrightarrow> P(S \<inter> t)" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4508 |
shows "locally P (S \<inter> t)" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4509 |
using S t unfolding locally_iff |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4510 |
apply clarify |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4511 |
apply (drule_tac x=T in spec)+ |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4512 |
apply (drule_tac x=x in spec)+ |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4513 |
apply clarsimp |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4514 |
apply (rename_tac U1 U2 V1 V2) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4515 |
apply (rule_tac x="U1 \<inter> U2" in exI) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4516 |
apply (simp add: open_Int) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4517 |
apply (rule_tac x="V1 \<inter> V2" in exI) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4518 |
apply (auto intro: P) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4519 |
done |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4520 |
|
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4521 |
|
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4522 |
proposition homeomorphism_locally_imp: |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4523 |
fixes S :: "'a::metric_space set" and t :: "'b::t2_space set" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4524 |
assumes S: "locally P S" and hom: "homeomorphism S t f g" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4525 |
and Q: "\<And>S t. \<lbrakk>P S; homeomorphism S t f g\<rbrakk> \<Longrightarrow> Q t" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4526 |
shows "locally Q t" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4527 |
proof (clarsimp simp: locally_def) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4528 |
fix w y |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4529 |
assume "y \<in> w" and "openin (subtopology euclidean t) w" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4530 |
then obtain T where T: "open T" "w = t \<inter> T" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4531 |
by (force simp: openin_open) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4532 |
then have "w \<subseteq> t" by auto |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4533 |
have f: "\<And>x. x \<in> S \<Longrightarrow> g(f x) = x" "f ` S = t" "continuous_on S f" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4534 |
and g: "\<And>y. y \<in> t \<Longrightarrow> f(g y) = y" "g ` t = S" "continuous_on t g" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4535 |
using hom by (auto simp: homeomorphism_def) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4536 |
have gw: "g ` w = S \<inter> {x. f x \<in> w}" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4537 |
using \<open>w \<subseteq> t\<close> |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4538 |
apply auto |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4539 |
using \<open>g ` t = S\<close> \<open>w \<subseteq> t\<close> apply blast |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4540 |
using g \<open>w \<subseteq> t\<close> apply auto[1] |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4541 |
by (simp add: f rev_image_eqI) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4542 |
have o: "openin (subtopology euclidean S) (g ` w)" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4543 |
proof - |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4544 |
have "continuous_on S f" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4545 |
using f(3) by blast |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4546 |
then show "openin (subtopology euclidean S) (g ` w)" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4547 |
by (simp add: gw Collect_conj_eq \<open>openin (subtopology euclidean t) w\<close> continuous_on_open f(2)) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4548 |
qed |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4549 |
then obtain u v |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4550 |
where osu: "openin (subtopology euclidean S) u" and uv: "P v" "g y \<in> u" "u \<subseteq> v" "v \<subseteq> g ` w" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4551 |
using S [unfolded locally_def, rule_format, of "g ` w" "g y"] \<open>y \<in> w\<close> by force |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4552 |
have "v \<subseteq> S" using uv by (simp add: gw) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4553 |
have fv: "f ` v = t \<inter> {x. g x \<in> v}" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4554 |
using \<open>f ` S = t\<close> f \<open>v \<subseteq> S\<close> by auto |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4555 |
have "f ` v \<subseteq> w" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4556 |
using uv using Int_lower2 gw image_subsetI mem_Collect_eq subset_iff by auto |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4557 |
have contvf: "continuous_on v f" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4558 |
using \<open>v \<subseteq> S\<close> continuous_on_subset f(3) by blast |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4559 |
have contvg: "continuous_on (f ` v) g" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4560 |
using \<open>f ` v \<subseteq> w\<close> \<open>w \<subseteq> t\<close> continuous_on_subset g(3) by blast |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4561 |
have homv: "homeomorphism v (f ` v) f g" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4562 |
using \<open>v \<subseteq> S\<close> \<open>w \<subseteq> t\<close> f |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4563 |
apply (simp add: homeomorphism_def contvf contvg, auto) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4564 |
by (metis f(1) rev_image_eqI rev_subsetD) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4565 |
have 1: "openin (subtopology euclidean t) {x \<in> t. g x \<in> u}" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4566 |
apply (rule continuous_on_open [THEN iffD1, rule_format]) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4567 |
apply (rule \<open>continuous_on t g\<close>) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4568 |
using \<open>g ` t = S\<close> apply (simp add: osu) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4569 |
done |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4570 |
have 2: "\<exists>v. Q v \<and> y \<in> {x \<in> t. g x \<in> u} \<and> {x \<in> t. g x \<in> u} \<subseteq> v \<and> v \<subseteq> w" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4571 |
apply (rule_tac x="f ` v" in exI) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4572 |
apply (intro conjI Q [OF \<open>P v\<close> homv]) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4573 |
using \<open>w \<subseteq> t\<close> \<open>y \<in> w\<close> \<open>f ` v \<subseteq> w\<close> uv apply (auto simp: fv) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4574 |
done |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4575 |
show "\<exists>u. openin (subtopology euclidean t) u \<and> |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4576 |
(\<exists>v. Q v \<and> y \<in> u \<and> u \<subseteq> v \<and> v \<subseteq> w)" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4577 |
by (meson 1 2) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4578 |
qed |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4579 |
|
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4580 |
lemma homeomorphism_locally: |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4581 |
fixes f:: "'a::metric_space \<Rightarrow> 'b::metric_space" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4582 |
assumes hom: "homeomorphism S t f g" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4583 |
and eq: "\<And>S t. homeomorphism S t f g \<Longrightarrow> (P S \<longleftrightarrow> Q t)" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4584 |
shows "locally P S \<longleftrightarrow> locally Q t" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4585 |
apply (rule iffI) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4586 |
apply (erule homeomorphism_locally_imp [OF _ hom]) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4587 |
apply (simp add: eq) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4588 |
apply (erule homeomorphism_locally_imp) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4589 |
using eq homeomorphism_sym homeomorphism_symD [OF hom] apply blast+ |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4590 |
done |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4591 |
|
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4592 |
lemma locally_translation: |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4593 |
fixes P :: "'a :: real_normed_vector set \<Rightarrow> bool" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4594 |
shows |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4595 |
"(\<And>S. P (image (\<lambda>x. a + x) S) \<longleftrightarrow> P S) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4596 |
\<Longrightarrow> locally P (image (\<lambda>x. a + x) S) \<longleftrightarrow> locally P S" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4597 |
apply (rule homeomorphism_locally [OF homeomorphism_translation]) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4598 |
apply (simp add: homeomorphism_def) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4599 |
by metis |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4600 |
|
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4601 |
lemma locally_injective_linear_image: |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4602 |
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4603 |
assumes f: "linear f" "inj f" and iff: "\<And>S. P (f ` S) \<longleftrightarrow> Q S" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4604 |
shows "locally P (f ` S) \<longleftrightarrow> locally Q S" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4605 |
apply (rule linear_homeomorphism_image [OF f]) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4606 |
apply (rule_tac f=g and g = f in homeomorphism_locally, assumption) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4607 |
by (metis iff homeomorphism_def) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4608 |
|
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4609 |
lemma locally_open_map_image: |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4610 |
fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4611 |
assumes P: "locally P S" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4612 |
and f: "continuous_on S f" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4613 |
and oo: "\<And>t. openin (subtopology euclidean S) t |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4614 |
\<Longrightarrow> openin (subtopology euclidean (f ` S)) (f ` t)" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4615 |
and Q: "\<And>t. \<lbrakk>t \<subseteq> S; P t\<rbrakk> \<Longrightarrow> Q(f ` t)" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4616 |
shows "locally Q (f ` S)" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4617 |
proof (clarsimp simp add: locally_def) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4618 |
fix w y |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4619 |
assume oiw: "openin (subtopology euclidean (f ` S)) w" and "y \<in> w" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4620 |
then have "w \<subseteq> f ` S" by (simp add: openin_euclidean_subtopology_iff) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4621 |
have oivf: "openin (subtopology euclidean S) {x \<in> S. f x \<in> w}" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4622 |
by (rule continuous_on_open [THEN iffD1, rule_format, OF f oiw]) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4623 |
then obtain x where "x \<in> S" "f x = y" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4624 |
using \<open>w \<subseteq> f ` S\<close> \<open>y \<in> w\<close> by blast |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4625 |
then obtain u v |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4626 |
where "openin (subtopology euclidean S) u" "P v" "x \<in> u" "u \<subseteq> v" "v \<subseteq> {x \<in> S. f x \<in> w}" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4627 |
using P [unfolded locally_def, rule_format, of "{x. x \<in> S \<and> f x \<in> w}" x] oivf \<open>y \<in> w\<close> |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4628 |
by auto |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4629 |
then show "\<exists>u. openin (subtopology euclidean (f ` S)) u \<and> |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4630 |
(\<exists>v. Q v \<and> y \<in> u \<and> u \<subseteq> v \<and> v \<subseteq> w)" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4631 |
apply (rule_tac x="f ` u" in exI) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4632 |
apply (rule conjI, blast intro!: oo) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4633 |
apply (rule_tac x="f ` v" in exI) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4634 |
apply (force simp: \<open>f x = y\<close> rev_image_eqI intro: Q) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4635 |
done |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4636 |
qed |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4637 |
|
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4638 |
subsection\<open>Basic properties of local compactness\<close> |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4639 |
|
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4640 |
lemma locally_compact: |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4641 |
fixes s :: "'a :: metric_space set" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4642 |
shows |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4643 |
"locally compact s \<longleftrightarrow> |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4644 |
(\<forall>x \<in> s. \<exists>u v. x \<in> u \<and> u \<subseteq> v \<and> v \<subseteq> s \<and> |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4645 |
openin (subtopology euclidean s) u \<and> compact v)" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4646 |
(is "?lhs = ?rhs") |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4647 |
proof |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4648 |
assume ?lhs |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4649 |
then show ?rhs |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4650 |
apply clarify |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4651 |
apply (erule_tac w = "s \<inter> ball x 1" in locallyE) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4652 |
by auto |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4653 |
next |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4654 |
assume r [rule_format]: ?rhs |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4655 |
have *: "\<exists>u v. |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4656 |
openin (subtopology euclidean s) u \<and> |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4657 |
compact v \<and> x \<in> u \<and> u \<subseteq> v \<and> v \<subseteq> s \<inter> T" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4658 |
if "open T" "x \<in> s" "x \<in> T" for x T |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4659 |
proof - |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4660 |
obtain u v where uv: "x \<in> u" "u \<subseteq> v" "v \<subseteq> s" "compact v" "openin (subtopology euclidean s) u" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4661 |
using r [OF \<open>x \<in> s\<close>] by auto |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4662 |
obtain e where "e>0" and e: "cball x e \<subseteq> T" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4663 |
using open_contains_cball \<open>open T\<close> \<open>x \<in> T\<close> by blast |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4664 |
show ?thesis |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4665 |
apply (rule_tac x="(s \<inter> ball x e) \<inter> u" in exI) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4666 |
apply (rule_tac x="cball x e \<inter> v" in exI) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4667 |
using that \<open>e > 0\<close> e uv |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4668 |
apply auto |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4669 |
done |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4670 |
qed |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4671 |
show ?lhs |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4672 |
apply (rule locallyI) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4673 |
apply (subst (asm) openin_open) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4674 |
apply (blast intro: *) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4675 |
done |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4676 |
qed |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4677 |
|
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4678 |
lemma locally_compactE: |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4679 |
fixes s :: "'a :: metric_space set" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4680 |
assumes "locally compact s" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4681 |
obtains u v where "\<And>x. x \<in> s \<Longrightarrow> x \<in> u x \<and> u x \<subseteq> v x \<and> v x \<subseteq> s \<and> |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4682 |
openin (subtopology euclidean s) (u x) \<and> compact (v x)" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4683 |
using assms |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4684 |
unfolding locally_compact by metis |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4685 |
|
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4686 |
lemma locally_compact_alt: |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4687 |
fixes s :: "'a :: heine_borel set" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4688 |
shows "locally compact s \<longleftrightarrow> |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4689 |
(\<forall>x \<in> s. \<exists>u. x \<in> u \<and> |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4690 |
openin (subtopology euclidean s) u \<and> compact(closure u) \<and> closure u \<subseteq> s)" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4691 |
apply (simp add: locally_compact) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4692 |
apply (intro ball_cong ex_cong refl iffI) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4693 |
apply (metis bounded_subset closure_eq closure_mono compact_eq_bounded_closed dual_order.trans) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4694 |
by (meson closure_subset compact_closure) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4695 |
|
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4696 |
lemma locally_compact_Int_cball: |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4697 |
fixes s :: "'a :: heine_borel set" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4698 |
shows "locally compact s \<longleftrightarrow> (\<forall>x \<in> s. \<exists>e. 0 < e \<and> closed(cball x e \<inter> s))" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4699 |
(is "?lhs = ?rhs") |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4700 |
proof |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4701 |
assume ?lhs |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4702 |
then show ?rhs |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4703 |
apply (simp add: locally_compact openin_contains_cball) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4704 |
apply (clarify | assumption | drule bspec)+ |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4705 |
by (metis (no_types, lifting) compact_cball compact_imp_closed compact_Int inf.absorb_iff2 inf.orderE inf_sup_aci(2)) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4706 |
next |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4707 |
assume ?rhs |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4708 |
then show ?lhs |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4709 |
apply (simp add: locally_compact openin_contains_cball) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4710 |
apply (clarify | assumption | drule bspec)+ |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4711 |
apply (rule_tac x="ball x e \<inter> s" in exI, simp) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4712 |
apply (rule_tac x="cball x e \<inter> s" in exI) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4713 |
using compact_eq_bounded_closed |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4714 |
apply auto |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4715 |
apply (metis open_ball le_infI1 mem_ball open_contains_cball_eq) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4716 |
done |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4717 |
qed |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4718 |
|
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4719 |
lemma locally_compact_compact: |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4720 |
fixes s :: "'a :: heine_borel set" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4721 |
shows "locally compact s \<longleftrightarrow> |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4722 |
(\<forall>k. k \<subseteq> s \<and> compact k |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4723 |
\<longrightarrow> (\<exists>u v. k \<subseteq> u \<and> u \<subseteq> v \<and> v \<subseteq> s \<and> |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4724 |
openin (subtopology euclidean s) u \<and> compact v))" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4725 |
(is "?lhs = ?rhs") |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4726 |
proof |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4727 |
assume ?lhs |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4728 |
then obtain u v where |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4729 |
uv: "\<And>x. x \<in> s \<Longrightarrow> x \<in> u x \<and> u x \<subseteq> v x \<and> v x \<subseteq> s \<and> |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4730 |
openin (subtopology euclidean s) (u x) \<and> compact (v x)" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4731 |
by (metis locally_compactE) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4732 |
have *: "\<exists>u v. k \<subseteq> u \<and> u \<subseteq> v \<and> v \<subseteq> s \<and> openin (subtopology euclidean s) u \<and> compact v" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4733 |
if "k \<subseteq> s" "compact k" for k |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4734 |
proof - |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4735 |
have "\<And>C. (\<forall>c\<in>C. openin (subtopology euclidean k) c) \<and> k \<subseteq> \<Union>C \<Longrightarrow> |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4736 |
\<exists>D\<subseteq>C. finite D \<and> k \<subseteq> \<Union>D" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4737 |
using that by (simp add: compact_eq_openin_cover) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4738 |
moreover have "\<forall>c \<in> (\<lambda>x. k \<inter> u x) ` k. openin (subtopology euclidean k) c" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4739 |
using that by clarify (metis subsetD inf.absorb_iff2 openin_subset openin_subtopology_Int_subset topspace_euclidean_subtopology uv) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4740 |
moreover have "k \<subseteq> \<Union>((\<lambda>x. k \<inter> u x) ` k)" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4741 |
using that by clarsimp (meson subsetCE uv) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4742 |
ultimately obtain D where "D \<subseteq> (\<lambda>x. k \<inter> u x) ` k" "finite D" "k \<subseteq> \<Union>D" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4743 |
by metis |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4744 |
then obtain T where T: "T \<subseteq> k" "finite T" "k \<subseteq> \<Union>((\<lambda>x. k \<inter> u x) ` T)" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4745 |
by (metis finite_subset_image) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4746 |
have Tuv: "UNION T u \<subseteq> UNION T v" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4747 |
using T that by (force simp: dest!: uv) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4748 |
show ?thesis |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4749 |
apply (rule_tac x="\<Union>(u ` T)" in exI) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4750 |
apply (rule_tac x="\<Union>(v ` T)" in exI) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4751 |
apply (simp add: Tuv) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4752 |
using T that |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4753 |
apply (auto simp: dest!: uv) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4754 |
done |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4755 |
qed |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4756 |
show ?rhs |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4757 |
by (blast intro: *) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4758 |
next |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4759 |
assume ?rhs |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4760 |
then show ?lhs |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4761 |
apply (clarsimp simp add: locally_compact) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4762 |
apply (drule_tac x="{x}" in spec, simp) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4763 |
done |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4764 |
qed |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4765 |
|
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4766 |
lemma open_imp_locally_compact: |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4767 |
fixes s :: "'a :: heine_borel set" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4768 |
assumes "open s" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4769 |
shows "locally compact s" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4770 |
proof - |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4771 |
have *: "\<exists>u v. x \<in> u \<and> u \<subseteq> v \<and> v \<subseteq> s \<and> openin (subtopology euclidean s) u \<and> compact v" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4772 |
if "x \<in> s" for x |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4773 |
proof - |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4774 |
obtain e where "e>0" and e: "cball x e \<subseteq> s" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4775 |
using open_contains_cball assms \<open>x \<in> s\<close> by blast |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4776 |
have ope: "openin (subtopology euclidean s) (ball x e)" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4777 |
by (meson e open_ball ball_subset_cball dual_order.trans open_subset) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4778 |
show ?thesis |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4779 |
apply (rule_tac x="ball x e" in exI) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4780 |
apply (rule_tac x="cball x e" in exI) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4781 |
using \<open>e > 0\<close> e apply (auto simp: ope) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4782 |
done |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4783 |
qed |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4784 |
show ?thesis |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4785 |
unfolding locally_compact |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4786 |
by (blast intro: *) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4787 |
qed |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4788 |
|
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4789 |
lemma closed_imp_locally_compact: |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4790 |
fixes s :: "'a :: heine_borel set" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4791 |
assumes "closed s" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4792 |
shows "locally compact s" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4793 |
proof - |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4794 |
have *: "\<exists>u v. x \<in> u \<and> u \<subseteq> v \<and> v \<subseteq> s \<and> |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4795 |
openin (subtopology euclidean s) u \<and> compact v" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4796 |
if "x \<in> s" for x |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4797 |
proof - |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4798 |
show ?thesis |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4799 |
apply (rule_tac x = "s \<inter> ball x 1" in exI) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4800 |
apply (rule_tac x = "s \<inter> cball x 1" in exI) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4801 |
using \<open>x \<in> s\<close> assms apply auto |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4802 |
done |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4803 |
qed |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4804 |
show ?thesis |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4805 |
unfolding locally_compact |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4806 |
by (blast intro: *) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4807 |
qed |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4808 |
|
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4809 |
lemma locally_compact_UNIV: "locally compact (UNIV :: 'a :: heine_borel set)" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4810 |
by (simp add: closed_imp_locally_compact) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4811 |
|
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4812 |
lemma locally_compact_Int: |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4813 |
fixes s :: "'a :: t2_space set" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4814 |
shows "\<lbrakk>locally compact s; locally compact t\<rbrakk> \<Longrightarrow> locally compact (s \<inter> t)" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4815 |
by (simp add: compact_Int locally_Int) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4816 |
|
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4817 |
lemma locally_compact_closedin: |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4818 |
fixes s :: "'a :: heine_borel set" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4819 |
shows "\<lbrakk>closedin (subtopology euclidean s) t; locally compact s\<rbrakk> |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4820 |
\<Longrightarrow> locally compact t" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4821 |
unfolding closedin_closed |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4822 |
using closed_imp_locally_compact locally_compact_Int by blast |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4823 |
|
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4824 |
lemma locally_compact_delete: |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4825 |
fixes s :: "'a :: t1_space set" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4826 |
shows "locally compact s \<Longrightarrow> locally compact (s - {a})" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4827 |
by (auto simp: openin_delete locally_open_subset) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4828 |
|
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4829 |
lemma locally_closed: |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4830 |
fixes s :: "'a :: heine_borel set" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4831 |
shows "locally closed s \<longleftrightarrow> locally compact s" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4832 |
(is "?lhs = ?rhs") |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4833 |
proof |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4834 |
assume ?lhs |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4835 |
then show ?rhs |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4836 |
apply (simp only: locally_def) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4837 |
apply (erule all_forward imp_forward asm_rl exE)+ |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4838 |
apply (rule_tac x = "u \<inter> ball x 1" in exI) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4839 |
apply (rule_tac x = "v \<inter> cball x 1" in exI) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4840 |
apply (force intro: openin_trans) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4841 |
done |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4842 |
next |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4843 |
assume ?rhs then show ?lhs |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4844 |
using compact_eq_bounded_closed locally_mono by blast |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4845 |
qed |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4846 |
|
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4847 |
subsection\<open>Important special cases of local connectedness and path connectedness\<close> |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4848 |
|
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4849 |
lemma locally_connected_1: |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4850 |
assumes |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4851 |
"\<And>v x. \<lbrakk>openin (subtopology euclidean S) v; x \<in> v\<rbrakk> |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4852 |
\<Longrightarrow> \<exists>u. openin (subtopology euclidean S) u \<and> |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4853 |
connected u \<and> x \<in> u \<and> u \<subseteq> v" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4854 |
shows "locally connected S" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4855 |
apply (clarsimp simp add: locally_def) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4856 |
apply (drule assms; blast) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4857 |
done |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4858 |
|
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4859 |
lemma locally_connected_2: |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4860 |
assumes "locally connected S" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4861 |
"openin (subtopology euclidean S) t" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4862 |
"x \<in> t" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4863 |
shows "openin (subtopology euclidean S) (connected_component_set t x)" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4864 |
proof - |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4865 |
{ fix y :: 'a |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4866 |
let ?SS = "subtopology euclidean S" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4867 |
assume 1: "openin ?SS t" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4868 |
"\<forall>w x. openin ?SS w \<and> x \<in> w \<longrightarrow> (\<exists>u. openin ?SS u \<and> (\<exists>v. connected v \<and> x \<in> u \<and> u \<subseteq> v \<and> v \<subseteq> w))" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4869 |
and "connected_component t x y" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4870 |
then have "y \<in> t" and y: "y \<in> connected_component_set t x" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4871 |
using connected_component_subset by blast+ |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4872 |
obtain F where |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4873 |
"\<forall>x y. (\<exists>w. openin ?SS w \<and> (\<exists>u. connected u \<and> x \<in> w \<and> w \<subseteq> u \<and> u \<subseteq> y)) = (openin ?SS (F x y) \<and> (\<exists>u. connected u \<and> x \<in> F x y \<and> F x y \<subseteq> u \<and> u \<subseteq> y))" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4874 |
by moura |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4875 |
then obtain G where |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4876 |
"\<forall>a A. (\<exists>U. openin ?SS U \<and> (\<exists>V. connected V \<and> a \<in> U \<and> U \<subseteq> V \<and> V \<subseteq> A)) = (openin ?SS (F a A) \<and> connected (G a A) \<and> a \<in> F a A \<and> F a A \<subseteq> G a A \<and> G a A \<subseteq> A)" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4877 |
by moura |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4878 |
then have *: "openin ?SS (F y t) \<and> connected (G y t) \<and> y \<in> F y t \<and> F y t \<subseteq> G y t \<and> G y t \<subseteq> t" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4879 |
using 1 \<open>y \<in> t\<close> by presburger |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4880 |
have "G y t \<subseteq> connected_component_set t y" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4881 |
by (metis (no_types) * connected_component_eq_self connected_component_mono contra_subsetD) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4882 |
then have "\<exists>A. openin ?SS A \<and> y \<in> A \<and> A \<subseteq> connected_component_set t x" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4883 |
by (metis (no_types) * connected_component_eq dual_order.trans y) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4884 |
} |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4885 |
then show ?thesis |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4886 |
using assms openin_subopen by (force simp: locally_def) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4887 |
qed |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4888 |
|
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4889 |
lemma locally_connected_3: |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4890 |
assumes "\<And>t x. \<lbrakk>openin (subtopology euclidean S) t; x \<in> t\<rbrakk> |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4891 |
\<Longrightarrow> openin (subtopology euclidean S) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4892 |
(connected_component_set t x)" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4893 |
"openin (subtopology euclidean S) v" "x \<in> v" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4894 |
shows "\<exists>u. openin (subtopology euclidean S) u \<and> connected u \<and> x \<in> u \<and> u \<subseteq> v" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4895 |
using assms connected_component_subset by fastforce |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4896 |
|
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4897 |
lemma locally_connected: |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4898 |
"locally connected S \<longleftrightarrow> |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4899 |
(\<forall>v x. openin (subtopology euclidean S) v \<and> x \<in> v |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4900 |
\<longrightarrow> (\<exists>u. openin (subtopology euclidean S) u \<and> connected u \<and> x \<in> u \<and> u \<subseteq> v))" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4901 |
by (metis locally_connected_1 locally_connected_2 locally_connected_3) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4902 |
|
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4903 |
lemma locally_connected_open_connected_component: |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4904 |
"locally connected S \<longleftrightarrow> |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4905 |
(\<forall>t x. openin (subtopology euclidean S) t \<and> x \<in> t |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4906 |
\<longrightarrow> openin (subtopology euclidean S) (connected_component_set t x))" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4907 |
by (metis locally_connected_1 locally_connected_2 locally_connected_3) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4908 |
|
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4909 |
lemma locally_path_connected_1: |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4910 |
assumes |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4911 |
"\<And>v x. \<lbrakk>openin (subtopology euclidean S) v; x \<in> v\<rbrakk> |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4912 |
\<Longrightarrow> \<exists>u. openin (subtopology euclidean S) u \<and> path_connected u \<and> x \<in> u \<and> u \<subseteq> v" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4913 |
shows "locally path_connected S" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4914 |
apply (clarsimp simp add: locally_def) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4915 |
apply (drule assms; blast) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4916 |
done |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4917 |
|
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4918 |
lemma locally_path_connected_2: |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4919 |
assumes "locally path_connected S" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4920 |
"openin (subtopology euclidean S) t" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4921 |
"x \<in> t" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4922 |
shows "openin (subtopology euclidean S) (path_component_set t x)" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4923 |
proof - |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4924 |
{ fix y :: 'a |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4925 |
let ?SS = "subtopology euclidean S" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4926 |
assume 1: "openin ?SS t" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4927 |
"\<forall>w x. openin ?SS w \<and> x \<in> w \<longrightarrow> (\<exists>u. openin ?SS u \<and> (\<exists>v. path_connected v \<and> x \<in> u \<and> u \<subseteq> v \<and> v \<subseteq> w))" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4928 |
and "path_component t x y" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4929 |
then have "y \<in> t" and y: "y \<in> path_component_set t x" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4930 |
using path_component_mem(2) by blast+ |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4931 |
obtain F where |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4932 |
"\<forall>x y. (\<exists>w. openin ?SS w \<and> (\<exists>u. path_connected u \<and> x \<in> w \<and> w \<subseteq> u \<and> u \<subseteq> y)) = (openin ?SS (F x y) \<and> (\<exists>u. path_connected u \<and> x \<in> F x y \<and> F x y \<subseteq> u \<and> u \<subseteq> y))" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4933 |
by moura |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4934 |
then obtain G where |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4935 |
"\<forall>a A. (\<exists>U. openin ?SS U \<and> (\<exists>V. path_connected V \<and> a \<in> U \<and> U \<subseteq> V \<and> V \<subseteq> A)) = (openin ?SS (F a A) \<and> path_connected (G a A) \<and> a \<in> F a A \<and> F a A \<subseteq> G a A \<and> G a A \<subseteq> A)" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4936 |
by moura |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4937 |
then have *: "openin ?SS (F y t) \<and> path_connected (G y t) \<and> y \<in> F y t \<and> F y t \<subseteq> G y t \<and> G y t \<subseteq> t" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4938 |
using 1 \<open>y \<in> t\<close> by presburger |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4939 |
have "G y t \<subseteq> path_component_set t y" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4940 |
using * path_component_maximal set_rev_mp by blast |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4941 |
then have "\<exists>A. openin ?SS A \<and> y \<in> A \<and> A \<subseteq> path_component_set t x" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4942 |
by (metis "*" \<open>G y t \<subseteq> path_component_set t y\<close> dual_order.trans path_component_eq y) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4943 |
} |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4944 |
then show ?thesis |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4945 |
using assms openin_subopen by (force simp: locally_def) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4946 |
qed |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4947 |
|
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4948 |
lemma locally_path_connected_3: |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4949 |
assumes "\<And>t x. \<lbrakk>openin (subtopology euclidean S) t; x \<in> t\<rbrakk> |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4950 |
\<Longrightarrow> openin (subtopology euclidean S) (path_component_set t x)" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4951 |
"openin (subtopology euclidean S) v" "x \<in> v" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4952 |
shows "\<exists>u. openin (subtopology euclidean S) u \<and> path_connected u \<and> x \<in> u \<and> u \<subseteq> v" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4953 |
proof - |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4954 |
have "path_component v x x" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4955 |
by (meson assms(3) path_component_refl) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4956 |
then show ?thesis |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4957 |
by (metis assms(1) assms(2) assms(3) mem_Collect_eq path_component_subset path_connected_path_component) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4958 |
qed |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4959 |
|
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4960 |
proposition locally_path_connected: |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4961 |
"locally path_connected S \<longleftrightarrow> |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4962 |
(\<forall>v x. openin (subtopology euclidean S) v \<and> x \<in> v |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4963 |
\<longrightarrow> (\<exists>u. openin (subtopology euclidean S) u \<and> path_connected u \<and> x \<in> u \<and> u \<subseteq> v))" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4964 |
by (metis locally_path_connected_1 locally_path_connected_2 locally_path_connected_3) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4965 |
|
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4966 |
proposition locally_path_connected_open_path_connected_component: |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4967 |
"locally path_connected S \<longleftrightarrow> |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4968 |
(\<forall>t x. openin (subtopology euclidean S) t \<and> x \<in> t |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4969 |
\<longrightarrow> openin (subtopology euclidean S) (path_component_set t x))" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4970 |
by (metis locally_path_connected_1 locally_path_connected_2 locally_path_connected_3) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4971 |
|
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4972 |
lemma locally_connected_open_component: |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4973 |
"locally connected S \<longleftrightarrow> |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4974 |
(\<forall>t c. openin (subtopology euclidean S) t \<and> c \<in> components t |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4975 |
\<longrightarrow> openin (subtopology euclidean S) c)" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4976 |
by (metis components_iff locally_connected_open_connected_component) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4977 |
|
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4978 |
proposition locally_connected_im_kleinen: |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4979 |
"locally connected S \<longleftrightarrow> |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4980 |
(\<forall>v x. openin (subtopology euclidean S) v \<and> x \<in> v |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4981 |
\<longrightarrow> (\<exists>u. openin (subtopology euclidean S) u \<and> |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4982 |
x \<in> u \<and> u \<subseteq> v \<and> |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4983 |
(\<forall>y. y \<in> u \<longrightarrow> (\<exists>c. connected c \<and> c \<subseteq> v \<and> x \<in> c \<and> y \<in> c))))" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4984 |
(is "?lhs = ?rhs") |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4985 |
proof |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4986 |
assume ?lhs |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4987 |
then show ?rhs |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4988 |
by (fastforce simp add: locally_connected) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4989 |
next |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4990 |
assume ?rhs |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4991 |
have *: "\<exists>T. openin (subtopology euclidean S) T \<and> x \<in> T \<and> T \<subseteq> c" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4992 |
if "openin (subtopology euclidean S) t" and c: "c \<in> components t" and "x \<in> c" for t c x |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4993 |
proof - |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4994 |
from that \<open>?rhs\<close> [rule_format, of t x] |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4995 |
obtain u where u: |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4996 |
"openin (subtopology euclidean S) u \<and> x \<in> u \<and> u \<subseteq> t \<and> |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4997 |
(\<forall>y. y \<in> u \<longrightarrow> (\<exists>c. connected c \<and> c \<subseteq> t \<and> x \<in> c \<and> y \<in> c))" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4998 |
by auto (meson subsetD in_components_subset) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
4999 |
obtain F :: "'a set \<Rightarrow> 'a set \<Rightarrow> 'a" where |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5000 |
"\<forall>x y. (\<exists>z. z \<in> x \<and> y = connected_component_set x z) = (F x y \<in> x \<and> y = connected_component_set x (F x y))" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5001 |
by moura |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5002 |
then have F: "F t c \<in> t \<and> c = connected_component_set t (F t c)" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5003 |
by (meson components_iff c) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5004 |
obtain G :: "'a set \<Rightarrow> 'a set \<Rightarrow> 'a" where |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5005 |
G: "\<forall>x y. (\<exists>z. z \<in> y \<and> z \<notin> x) = (G x y \<in> y \<and> G x y \<notin> x)" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5006 |
by moura |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5007 |
have "G c u \<notin> u \<or> G c u \<in> c" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5008 |
using F by (metis (full_types) u connected_componentI connected_component_eq mem_Collect_eq that(3)) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5009 |
then show ?thesis |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5010 |
using G u by auto |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5011 |
qed |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5012 |
show ?lhs |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5013 |
apply (clarsimp simp add: locally_connected_open_component) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5014 |
apply (subst openin_subopen) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5015 |
apply (blast intro: *) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5016 |
done |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5017 |
qed |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5018 |
|
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5019 |
proposition locally_path_connected_im_kleinen: |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5020 |
"locally path_connected S \<longleftrightarrow> |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5021 |
(\<forall>v x. openin (subtopology euclidean S) v \<and> x \<in> v |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5022 |
\<longrightarrow> (\<exists>u. openin (subtopology euclidean S) u \<and> |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5023 |
x \<in> u \<and> u \<subseteq> v \<and> |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5024 |
(\<forall>y. y \<in> u \<longrightarrow> (\<exists>p. path p \<and> path_image p \<subseteq> v \<and> |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5025 |
pathstart p = x \<and> pathfinish p = y))))" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5026 |
(is "?lhs = ?rhs") |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5027 |
proof |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5028 |
assume ?lhs |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5029 |
then show ?rhs |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5030 |
apply (simp add: locally_path_connected path_connected_def) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5031 |
apply (erule all_forward ex_forward imp_forward conjE | simp)+ |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5032 |
by (meson dual_order.trans) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5033 |
next |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5034 |
assume ?rhs |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5035 |
have *: "\<exists>T. openin (subtopology euclidean S) T \<and> |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5036 |
x \<in> T \<and> T \<subseteq> path_component_set u z" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5037 |
if "openin (subtopology euclidean S) u" and "z \<in> u" and c: "path_component u z x" for u z x |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5038 |
proof - |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5039 |
have "x \<in> u" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5040 |
by (meson c path_component_mem(2)) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5041 |
with that \<open>?rhs\<close> [rule_format, of u x] |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5042 |
obtain U where U: |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5043 |
"openin (subtopology euclidean S) U \<and> x \<in> U \<and> U \<subseteq> u \<and> |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5044 |
(\<forall>y. y \<in> U \<longrightarrow> (\<exists>p. path p \<and> path_image p \<subseteq> u \<and> pathstart p = x \<and> pathfinish p = y))" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5045 |
by blast |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5046 |
show ?thesis |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5047 |
apply (rule_tac x=U in exI) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5048 |
apply (auto simp: U) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5049 |
apply (metis U c path_component_trans path_component_def) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5050 |
done |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5051 |
qed |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5052 |
show ?lhs |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5053 |
apply (clarsimp simp add: locally_path_connected_open_path_connected_component) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5054 |
apply (subst openin_subopen) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5055 |
apply (blast intro: *) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5056 |
done |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5057 |
qed |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5058 |
|
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5059 |
lemma locally_path_connected_imp_locally_connected: |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5060 |
"locally path_connected S \<Longrightarrow> locally connected S" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5061 |
using locally_mono path_connected_imp_connected by blast |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5062 |
|
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5063 |
lemma locally_connected_components: |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5064 |
"\<lbrakk>locally connected S; c \<in> components S\<rbrakk> \<Longrightarrow> locally connected c" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5065 |
by (meson locally_connected_open_component locally_open_subset openin_subtopology_self) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5066 |
|
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5067 |
lemma locally_path_connected_components: |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5068 |
"\<lbrakk>locally path_connected S; c \<in> components S\<rbrakk> \<Longrightarrow> locally path_connected c" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5069 |
by (meson locally_connected_open_component locally_open_subset locally_path_connected_imp_locally_connected openin_subtopology_self) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5070 |
|
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5071 |
lemma locally_path_connected_connected_component: |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5072 |
"locally path_connected S \<Longrightarrow> locally path_connected (connected_component_set S x)" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5073 |
by (metis components_iff connected_component_eq_empty locally_empty locally_path_connected_components) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5074 |
|
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5075 |
lemma open_imp_locally_path_connected: |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5076 |
fixes S :: "'a :: real_normed_vector set" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5077 |
shows "open S \<Longrightarrow> locally path_connected S" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5078 |
apply (rule locally_mono [of convex]) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5079 |
apply (simp_all add: locally_def openin_open_eq convex_imp_path_connected) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5080 |
apply (meson Topology_Euclidean_Space.open_ball centre_in_ball convex_ball openE order_trans) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5081 |
done |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5082 |
|
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5083 |
lemma open_imp_locally_connected: |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5084 |
fixes S :: "'a :: real_normed_vector set" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5085 |
shows "open S \<Longrightarrow> locally connected S" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5086 |
by (simp add: locally_path_connected_imp_locally_connected open_imp_locally_path_connected) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5087 |
|
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5088 |
lemma locally_path_connected_UNIV: "locally path_connected (UNIV::'a :: real_normed_vector set)" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5089 |
by (simp add: open_imp_locally_path_connected) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5090 |
|
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5091 |
lemma locally_connected_UNIV: "locally connected (UNIV::'a :: real_normed_vector set)" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5092 |
by (simp add: open_imp_locally_connected) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5093 |
|
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5094 |
lemma openin_connected_component_locally_connected: |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5095 |
"locally connected S |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5096 |
\<Longrightarrow> openin (subtopology euclidean S) (connected_component_set S x)" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5097 |
apply (simp add: locally_connected_open_connected_component) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5098 |
by (metis connected_component_eq_empty connected_component_subset open_empty open_subset openin_subtopology_self) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5099 |
|
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5100 |
lemma openin_components_locally_connected: |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5101 |
"\<lbrakk>locally connected S; c \<in> components S\<rbrakk> \<Longrightarrow> openin (subtopology euclidean S) c" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5102 |
using locally_connected_open_component openin_subtopology_self by blast |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5103 |
|
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5104 |
lemma openin_path_component_locally_path_connected: |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5105 |
"locally path_connected S |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5106 |
\<Longrightarrow> openin (subtopology euclidean S) (path_component_set S x)" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5107 |
by (metis (no_types) empty_iff locally_path_connected_2 openin_subopen openin_subtopology_self path_component_eq_empty) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5108 |
|
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5109 |
lemma closedin_path_component_locally_path_connected: |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5110 |
"locally path_connected S |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5111 |
\<Longrightarrow> closedin (subtopology euclidean S) (path_component_set S x)" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5112 |
apply (simp add: closedin_def path_component_subset complement_path_component_Union) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5113 |
apply (rule openin_Union) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5114 |
using openin_path_component_locally_path_connected by auto |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5115 |
|
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5116 |
lemma convex_imp_locally_path_connected: |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5117 |
fixes S :: "'a:: real_normed_vector set" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5118 |
shows "convex S \<Longrightarrow> locally path_connected S" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5119 |
apply (clarsimp simp add: locally_path_connected) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5120 |
apply (subst (asm) openin_open) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5121 |
apply clarify |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5122 |
apply (erule (1) Topology_Euclidean_Space.openE) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5123 |
apply (rule_tac x = "S \<inter> ball x e" in exI) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5124 |
apply (force simp: convex_Int convex_imp_path_connected) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5125 |
done |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5126 |
|
63114
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5127 |
subsection\<open>Components, continuity, openin, closedin\<close> |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5128 |
|
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5129 |
lemma continuous_openin_preimage_eq: |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5130 |
"continuous_on S f \<longleftrightarrow> |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5131 |
(\<forall>t. open t \<longrightarrow> openin (subtopology euclidean S) {x. x \<in> S \<and> f x \<in> t})" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5132 |
apply (auto simp: continuous_openin_preimage) |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5133 |
apply (fastforce simp add: continuous_on_open openin_open) |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5134 |
done |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5135 |
|
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5136 |
lemma continuous_closedin_preimage_eq: |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5137 |
"continuous_on S f \<longleftrightarrow> |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5138 |
(\<forall>t. closed t \<longrightarrow> closedin (subtopology euclidean S) {x. x \<in> S \<and> f x \<in> t})" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5139 |
apply safe |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5140 |
apply (simp add: continuous_closedin_preimage) |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5141 |
apply (fastforce simp add: continuous_on_closed closedin_closed) |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5142 |
done |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5143 |
|
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5144 |
lemma continuous_on_components_gen: |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5145 |
fixes f :: "'a::topological_space \<Rightarrow> 'b::topological_space" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5146 |
assumes "\<And>c. c \<in> components S \<Longrightarrow> |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5147 |
openin (subtopology euclidean S) c \<and> continuous_on c f" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5148 |
shows "continuous_on S f" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5149 |
proof (clarsimp simp: continuous_openin_preimage_eq) |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5150 |
fix t :: "'b set" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5151 |
assume "open t" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5152 |
have "{x. x \<in> S \<and> f x \<in> t} = \<Union>{{x. x \<in> c \<and> f x \<in> t} |c. c \<in> components S}" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5153 |
apply auto |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5154 |
apply (metis (lifting) components_iff connected_component_refl_eq mem_Collect_eq) |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5155 |
using Union_components by blast |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5156 |
then show "openin (subtopology euclidean S) {x \<in> S. f x \<in> t}" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5157 |
using \<open>open t\<close> assms |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5158 |
by (fastforce intro: openin_trans continuous_openin_preimage) |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5159 |
qed |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5160 |
|
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5161 |
lemma continuous_on_components: |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5162 |
fixes f :: "'a::topological_space \<Rightarrow> 'b::topological_space" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5163 |
assumes "locally connected S " |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5164 |
"\<And>c. c \<in> components S \<Longrightarrow> continuous_on c f" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5165 |
shows "continuous_on S f" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5166 |
apply (rule continuous_on_components_gen) |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5167 |
apply (auto simp: assms intro: openin_components_locally_connected) |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5168 |
done |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5169 |
|
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5170 |
lemma continuous_on_components_eq: |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5171 |
"locally connected S |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5172 |
\<Longrightarrow> (continuous_on S f \<longleftrightarrow> (\<forall>c \<in> components S. continuous_on c f))" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5173 |
by (meson continuous_on_components continuous_on_subset in_components_subset) |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5174 |
|
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5175 |
lemma continuous_on_components_open: |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5176 |
fixes S :: "'a::real_normed_vector set" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5177 |
assumes "open S " |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5178 |
"\<And>c. c \<in> components S \<Longrightarrow> continuous_on c f" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5179 |
shows "continuous_on S f" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5180 |
using continuous_on_components open_imp_locally_connected assms by blast |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5181 |
|
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5182 |
lemma continuous_on_components_open_eq: |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5183 |
fixes S :: "'a::real_normed_vector set" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5184 |
shows "open S \<Longrightarrow> (continuous_on S f \<longleftrightarrow> (\<forall>c \<in> components S. continuous_on c f))" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5185 |
using continuous_on_subset in_components_subset |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5186 |
by (blast intro: continuous_on_components_open) |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5187 |
|
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5188 |
lemma closedin_union_complement_components: |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5189 |
assumes u: "locally connected u" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5190 |
and S: "closedin (subtopology euclidean u) S" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5191 |
and cuS: "c \<subseteq> components(u - S)" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5192 |
shows "closedin (subtopology euclidean u) (S \<union> \<Union>c)" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5193 |
proof - |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5194 |
have di: "(\<And>S t. S \<in> c \<and> t \<in> c' \<Longrightarrow> disjnt S t) \<Longrightarrow> disjnt (\<Union> c) (\<Union> c')" for c' |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5195 |
by (simp add: disjnt_def) blast |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5196 |
have "S \<subseteq> u" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5197 |
using S closedin_imp_subset by blast |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5198 |
moreover have "u - S = \<Union>c \<union> \<Union>(components (u - S) - c)" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5199 |
by (metis Diff_partition Topology_Euclidean_Space.Union_components Union_Un_distrib assms(3)) |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5200 |
moreover have "disjnt (\<Union>c) (\<Union>(components (u - S) - c))" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5201 |
apply (rule di) |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5202 |
by (metis DiffD1 DiffD2 assms(3) components_nonoverlap disjnt_def subsetCE) |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5203 |
ultimately have eq: "S \<union> \<Union>c = u - (\<Union>(components(u - S) - c))" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5204 |
by (auto simp: disjnt_def) |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5205 |
have *: "openin (subtopology euclidean u) (\<Union>(components (u - S) - c))" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5206 |
apply (rule openin_Union) |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5207 |
apply (rule openin_trans [of "u - S"]) |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5208 |
apply (simp add: u S locally_diff_closed openin_components_locally_connected) |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5209 |
apply (simp add: openin_diff S) |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5210 |
done |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5211 |
have "openin (subtopology euclidean u) (u - (u - \<Union>(components (u - S) - c)))" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5212 |
apply (rule openin_diff, simp) |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5213 |
apply (metis closedin_diff closedin_topspace topspace_euclidean_subtopology *) |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5214 |
done |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5215 |
then show ?thesis |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5216 |
by (force simp: eq closedin_def) |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5217 |
qed |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5218 |
|
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5219 |
lemma closed_union_complement_components: |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5220 |
fixes S :: "'a::real_normed_vector set" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5221 |
assumes S: "closed S" and c: "c \<subseteq> components(- S)" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5222 |
shows "closed(S \<union> \<Union> c)" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5223 |
proof - |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5224 |
have "closedin (subtopology euclidean UNIV) (S \<union> \<Union>c)" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5225 |
apply (rule closedin_union_complement_components [OF locally_connected_UNIV]) |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5226 |
using S apply (simp add: closed_closedin) |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5227 |
using c apply (simp add: Compl_eq_Diff_UNIV) |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5228 |
done |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5229 |
then show ?thesis |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5230 |
by (simp add: closed_closedin) |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5231 |
qed |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5232 |
|
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5233 |
lemma closedin_Un_complement_component: |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5234 |
fixes S :: "'a::real_normed_vector set" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5235 |
assumes u: "locally connected u" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5236 |
and S: "closedin (subtopology euclidean u) S" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5237 |
and c: " c \<in> components(u - S)" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5238 |
shows "closedin (subtopology euclidean u) (S \<union> c)" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5239 |
proof - |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5240 |
have "closedin (subtopology euclidean u) (S \<union> \<Union>{c})" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5241 |
using c by (blast intro: closedin_union_complement_components [OF u S]) |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5242 |
then show ?thesis |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5243 |
by simp |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5244 |
qed |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5245 |
|
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5246 |
lemma closed_Un_complement_component: |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5247 |
fixes S :: "'a::real_normed_vector set" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5248 |
assumes S: "closed S" and c: " c \<in> components(-S)" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5249 |
shows "closed (S \<union> c)" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5250 |
by (metis Compl_eq_Diff_UNIV S c closed_closedin closedin_Un_complement_component locally_connected_UNIV subtopology_UNIV) |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5251 |
|
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5252 |
|
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5253 |
subsection\<open>Existence of isometry between subspaces of same dimension\<close> |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5254 |
|
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5255 |
thm subspace_isomorphism |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5256 |
lemma isometry_subset_subspace: |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5257 |
fixes S :: "'a::euclidean_space set" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5258 |
and T :: "'b::euclidean_space set" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5259 |
assumes S: "subspace S" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5260 |
and T: "subspace T" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5261 |
and d: "dim S \<le> dim T" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5262 |
obtains f where "linear f" "f ` S \<subseteq> T" "\<And>x. x \<in> S \<Longrightarrow> norm(f x) = norm x" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5263 |
proof - |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5264 |
obtain B where "B \<subseteq> S" and Borth: "pairwise orthogonal B" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5265 |
and B1: "\<And>x. x \<in> B \<Longrightarrow> norm x = 1" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5266 |
and "independent B" "finite B" "card B = dim S" "span B = S" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5267 |
by (metis orthonormal_basis_subspace [OF S] independent_finite) |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5268 |
obtain C where "C \<subseteq> T" and Corth: "pairwise orthogonal C" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5269 |
and C1:"\<And>x. x \<in> C \<Longrightarrow> norm x = 1" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5270 |
and "independent C" "finite C" "card C = dim T" "span C = T" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5271 |
by (metis orthonormal_basis_subspace [OF T] independent_finite) |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5272 |
obtain fb where "fb ` B \<subseteq> C" "inj_on fb B" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5273 |
by (metis \<open>card B = dim S\<close> \<open>card C = dim T\<close> \<open>finite B\<close> \<open>finite C\<close> card_le_inj d) |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5274 |
then have pairwise_orth_fb: "pairwise (\<lambda>v j. orthogonal (fb v) (fb j)) B" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5275 |
using Corth |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5276 |
apply (auto simp: pairwise_def orthogonal_clauses) |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5277 |
by (meson subsetD image_eqI inj_on_def) |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5278 |
obtain f where "linear f" and ffb: "\<And>x. x \<in> B \<Longrightarrow> f x = fb x" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5279 |
using linear_independent_extend \<open>independent B\<close> by fastforce |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5280 |
have "f ` S \<subseteq> T" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5281 |
by (metis ffb \<open>fb ` B \<subseteq> C\<close> \<open>linear f\<close> \<open>span B = S\<close> \<open>span C = T\<close> image_cong span_linear_image span_mono) |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5282 |
have [simp]: "\<And>x. x \<in> B \<Longrightarrow> norm (fb x) = norm x" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5283 |
using B1 C1 \<open>fb ` B \<subseteq> C\<close> by auto |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5284 |
have "norm (f x) = norm x" if "x \<in> S" for x |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5285 |
proof - |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5286 |
obtain a where x: "x = (\<Sum>v \<in> B. a v *\<^sub>R v)" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5287 |
using \<open>finite B\<close> \<open>span B = S\<close> \<open>x \<in> S\<close> span_finite by fastforce |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5288 |
have "f x = (\<Sum>v \<in> B. f (a v *\<^sub>R v))" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5289 |
using linear_setsum [OF \<open>linear f\<close>] x by auto |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5290 |
also have "... = (\<Sum>v \<in> B. a v *\<^sub>R f v)" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5291 |
using \<open>linear f\<close> by (simp add: linear_setsum linear.scaleR) |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5292 |
also have "... = (\<Sum>v \<in> B. a v *\<^sub>R fb v)" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5293 |
by (simp add: ffb cong: setsum.cong) |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5294 |
finally have "norm (f x)^2 = norm (\<Sum>v\<in>B. a v *\<^sub>R fb v)^2" by simp |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5295 |
also have "... = (\<Sum>v\<in>B. norm ((a v *\<^sub>R fb v))^2)" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5296 |
apply (rule norm_setsum_Pythagorean [OF \<open>finite B\<close>]) |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5297 |
apply (rule pairwise_ortho_scaleR [OF pairwise_orth_fb]) |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5298 |
done |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5299 |
also have "... = norm x ^2" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5300 |
by (simp add: x pairwise_ortho_scaleR Borth norm_setsum_Pythagorean [OF \<open>finite B\<close>]) |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5301 |
finally show ?thesis |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5302 |
by (simp add: norm_eq_sqrt_inner) |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5303 |
qed |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5304 |
then show ?thesis |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5305 |
by (rule that [OF \<open>linear f\<close> \<open>f ` S \<subseteq> T\<close>]) |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5306 |
qed |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5307 |
|
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5308 |
proposition isometries_subspaces: |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5309 |
fixes S :: "'a::euclidean_space set" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5310 |
and T :: "'b::euclidean_space set" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5311 |
assumes S: "subspace S" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5312 |
and T: "subspace T" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5313 |
and d: "dim S = dim T" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5314 |
obtains f g where "linear f" "linear g" "f ` S = T" "g ` T = S" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5315 |
"\<And>x. x \<in> S \<Longrightarrow> norm(f x) = norm x" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5316 |
"\<And>x. x \<in> T \<Longrightarrow> norm(g x) = norm x" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5317 |
"\<And>x. x \<in> S \<Longrightarrow> g(f x) = x" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5318 |
"\<And>x. x \<in> T \<Longrightarrow> f(g x) = x" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5319 |
proof - |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5320 |
obtain B where "B \<subseteq> S" and Borth: "pairwise orthogonal B" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5321 |
and B1: "\<And>x. x \<in> B \<Longrightarrow> norm x = 1" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5322 |
and "independent B" "finite B" "card B = dim S" "span B = S" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5323 |
by (metis orthonormal_basis_subspace [OF S] independent_finite) |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5324 |
obtain C where "C \<subseteq> T" and Corth: "pairwise orthogonal C" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5325 |
and C1:"\<And>x. x \<in> C \<Longrightarrow> norm x = 1" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5326 |
and "independent C" "finite C" "card C = dim T" "span C = T" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5327 |
by (metis orthonormal_basis_subspace [OF T] independent_finite) |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5328 |
obtain fb where "bij_betw fb B C" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5329 |
by (metis \<open>finite B\<close> \<open>finite C\<close> bij_betw_iff_card \<open>card B = dim S\<close> \<open>card C = dim T\<close> d) |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5330 |
then have pairwise_orth_fb: "pairwise (\<lambda>v j. orthogonal (fb v) (fb j)) B" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5331 |
using Corth |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5332 |
apply (auto simp: pairwise_def orthogonal_clauses bij_betw_def) |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5333 |
by (meson subsetD image_eqI inj_on_def) |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5334 |
obtain f where "linear f" and ffb: "\<And>x. x \<in> B \<Longrightarrow> f x = fb x" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5335 |
using linear_independent_extend \<open>independent B\<close> by fastforce |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5336 |
define gb where "gb \<equiv> inv_into B fb" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5337 |
then have pairwise_orth_gb: "pairwise (\<lambda>v j. orthogonal (gb v) (gb j)) C" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5338 |
using Borth |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5339 |
apply (auto simp: pairwise_def orthogonal_clauses bij_betw_def) |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5340 |
by (metis \<open>bij_betw fb B C\<close> bij_betw_imp_surj_on bij_betw_inv_into_right inv_into_into) |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5341 |
obtain g where "linear g" and ggb: "\<And>x. x \<in> C \<Longrightarrow> g x = gb x" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5342 |
using linear_independent_extend \<open>independent C\<close> by fastforce |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5343 |
have "f ` S \<subseteq> T" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5344 |
by (metis \<open>bij_betw fb B C\<close> bij_betw_imp_surj_on eq_iff ffb \<open>linear f\<close> \<open>span B = S\<close> \<open>span C = T\<close> image_cong span_linear_image) |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5345 |
have [simp]: "\<And>x. x \<in> B \<Longrightarrow> norm (fb x) = norm x" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5346 |
using B1 C1 \<open>bij_betw fb B C\<close> bij_betw_imp_surj_on by fastforce |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5347 |
have f [simp]: "norm (f x) = norm x" "g (f x) = x" if "x \<in> S" for x |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5348 |
proof - |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5349 |
obtain a where x: "x = (\<Sum>v \<in> B. a v *\<^sub>R v)" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5350 |
using \<open>finite B\<close> \<open>span B = S\<close> \<open>x \<in> S\<close> span_finite by fastforce |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5351 |
have "f x = (\<Sum>v \<in> B. f (a v *\<^sub>R v))" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5352 |
using linear_setsum [OF \<open>linear f\<close>] x by auto |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5353 |
also have "... = (\<Sum>v \<in> B. a v *\<^sub>R f v)" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5354 |
using \<open>linear f\<close> by (simp add: linear_setsum linear.scaleR) |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5355 |
also have "... = (\<Sum>v \<in> B. a v *\<^sub>R fb v)" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5356 |
by (simp add: ffb cong: setsum.cong) |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5357 |
finally have *: "f x = (\<Sum>v\<in>B. a v *\<^sub>R fb v)" . |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5358 |
then have "(norm (f x))\<^sup>2 = (norm (\<Sum>v\<in>B. a v *\<^sub>R fb v))\<^sup>2" by simp |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5359 |
also have "... = (\<Sum>v\<in>B. norm ((a v *\<^sub>R fb v))^2)" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5360 |
apply (rule norm_setsum_Pythagorean [OF \<open>finite B\<close>]) |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5361 |
apply (rule pairwise_ortho_scaleR [OF pairwise_orth_fb]) |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5362 |
done |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5363 |
also have "... = (norm x)\<^sup>2" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5364 |
by (simp add: x pairwise_ortho_scaleR Borth norm_setsum_Pythagorean [OF \<open>finite B\<close>]) |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5365 |
finally show "norm (f x) = norm x" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5366 |
by (simp add: norm_eq_sqrt_inner) |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5367 |
have "g (f x) = g (\<Sum>v\<in>B. a v *\<^sub>R fb v)" by (simp add: *) |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5368 |
also have "... = (\<Sum>v\<in>B. g (a v *\<^sub>R fb v))" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5369 |
using \<open>linear g\<close> by (simp add: linear_setsum linear.scaleR) |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5370 |
also have "... = (\<Sum>v\<in>B. a v *\<^sub>R g (fb v))" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5371 |
by (simp add: \<open>linear g\<close> linear.scaleR) |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5372 |
also have "... = (\<Sum>v\<in>B. a v *\<^sub>R v)" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5373 |
apply (rule setsum.cong [OF refl]) |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5374 |
using \<open>bij_betw fb B C\<close> gb_def bij_betwE bij_betw_inv_into_left gb_def ggb by fastforce |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5375 |
also have "... = x" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5376 |
using x by blast |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5377 |
finally show "g (f x) = x" . |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5378 |
qed |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5379 |
have [simp]: "\<And>x. x \<in> C \<Longrightarrow> norm (gb x) = norm x" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5380 |
by (metis B1 C1 \<open>bij_betw fb B C\<close> bij_betw_imp_surj_on gb_def inv_into_into) |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5381 |
have g [simp]: "f (g x) = x" if "x \<in> T" for x |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5382 |
proof - |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5383 |
obtain a where x: "x = (\<Sum>v \<in> C. a v *\<^sub>R v)" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5384 |
using \<open>finite C\<close> \<open>span C = T\<close> \<open>x \<in> T\<close> span_finite by fastforce |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5385 |
have "g x = (\<Sum>v \<in> C. g (a v *\<^sub>R v))" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5386 |
using linear_setsum [OF \<open>linear g\<close>] x by auto |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5387 |
also have "... = (\<Sum>v \<in> C. a v *\<^sub>R g v)" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5388 |
using \<open>linear g\<close> by (simp add: linear_setsum linear.scaleR) |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5389 |
also have "... = (\<Sum>v \<in> C. a v *\<^sub>R gb v)" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5390 |
by (simp add: ggb cong: setsum.cong) |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5391 |
finally have "f (g x) = f (\<Sum>v\<in>C. a v *\<^sub>R gb v)" by simp |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5392 |
also have "... = (\<Sum>v\<in>C. f (a v *\<^sub>R gb v))" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5393 |
using \<open>linear f\<close> by (simp add: linear_setsum linear.scaleR) |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5394 |
also have "... = (\<Sum>v\<in>C. a v *\<^sub>R f (gb v))" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5395 |
by (simp add: \<open>linear f\<close> linear.scaleR) |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5396 |
also have "... = (\<Sum>v\<in>C. a v *\<^sub>R v)" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5397 |
using \<open>bij_betw fb B C\<close> |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5398 |
by (simp add: bij_betw_def gb_def bij_betw_inv_into_right ffb inv_into_into) |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5399 |
also have "... = x" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5400 |
using x by blast |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5401 |
finally show "f (g x) = x" . |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5402 |
qed |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5403 |
have gim: "g ` T = S" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5404 |
by (metis (no_types, lifting) \<open>f ` S \<subseteq> T\<close> \<open>linear g\<close> \<open>span B = S\<close> \<open>span C = T\<close> d dim_eq_span dim_image_le f(2) image_subset_iff span_linear_image span_span subsetI) |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5405 |
have fim: "f ` S = T" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5406 |
using \<open>g ` T = S\<close> image_iff by fastforce |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5407 |
have [simp]: "norm (g x) = norm x" if "x \<in> T" for x |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5408 |
using fim that by auto |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5409 |
show ?thesis |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5410 |
apply (rule that [OF \<open>linear f\<close> \<open>linear g\<close>]) |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5411 |
apply (simp_all add: fim gim) |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5412 |
done |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5413 |
qed |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5414 |
|
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5415 |
(*REPLACE*) |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5416 |
lemma isometry_subspaces: |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5417 |
fixes S :: "'a::euclidean_space set" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5418 |
and T :: "'b::euclidean_space set" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5419 |
assumes S: "subspace S" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5420 |
and T: "subspace T" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5421 |
and d: "dim S = dim T" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5422 |
obtains f where "linear f" "f ` S = T" "\<And>x. x \<in> S \<Longrightarrow> norm(f x) = norm x" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5423 |
using isometries_subspaces [OF assms] |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5424 |
by metis |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5425 |
|
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5426 |
lemma homeomorphic_subspaces: |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5427 |
fixes S :: "'a::euclidean_space set" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5428 |
and T :: "'b::euclidean_space set" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5429 |
assumes S: "subspace S" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5430 |
and T: "subspace T" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5431 |
and d: "dim S = dim T" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5432 |
shows "S homeomorphic T" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5433 |
proof - |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5434 |
obtain f g where "linear f" "linear g" "f ` S = T" "g ` T = S" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5435 |
"\<And>x. x \<in> S \<Longrightarrow> g(f x) = x" "\<And>x. x \<in> T \<Longrightarrow> f(g x) = x" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5436 |
by (blast intro: isometries_subspaces [OF assms]) |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5437 |
then show ?thesis |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5438 |
apply (simp add: homeomorphic_def homeomorphism_def) |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5439 |
apply (rule_tac x=f in exI) |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5440 |
apply (rule_tac x=g in exI) |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5441 |
apply (auto simp: linear_continuous_on linear_conv_bounded_linear) |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5442 |
done |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5443 |
qed |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5444 |
|
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5445 |
lemma homeomorphic_affine_sets: |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5446 |
assumes "affine S" "affine T" "aff_dim S = aff_dim T" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5447 |
shows "S homeomorphic T" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5448 |
proof (cases "S = {} \<or> T = {}") |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5449 |
case True with assms aff_dim_empty homeomorphic_empty show ?thesis |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5450 |
by metis |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5451 |
next |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5452 |
case False |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5453 |
then obtain a b where ab: "a \<in> S" "b \<in> T" by auto |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5454 |
then have ss: "subspace (op + (- a) ` S)" "subspace (op + (- b) ` T)" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5455 |
using affine_diffs_subspace assms by blast+ |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5456 |
have dd: "dim (op + (- a) ` S) = dim (op + (- b) ` T)" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5457 |
using assms ab by (simp add: aff_dim_eq_dim [OF hull_inc] image_def) |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5458 |
have "S homeomorphic (op + (- a) ` S)" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5459 |
by (simp add: homeomorphic_translation) |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5460 |
also have "... homeomorphic (op + (- b) ` T)" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5461 |
by (rule homeomorphic_subspaces [OF ss dd]) |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5462 |
also have "... homeomorphic T" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5463 |
using homeomorphic_sym homeomorphic_translation by auto |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5464 |
finally show ?thesis . |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5465 |
qed |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
5466 |
|
62948
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5467 |
subsection\<open>Retracts, in a general sense, preserve (co)homotopic triviality)\<close> |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5468 |
|
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5469 |
locale Retracts = |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5470 |
fixes s h t k |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5471 |
assumes conth: "continuous_on s h" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5472 |
and imh: "h ` s = t" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5473 |
and contk: "continuous_on t k" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5474 |
and imk: "k ` t \<subseteq> s" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5475 |
and idhk: "\<And>y. y \<in> t \<Longrightarrow> h(k y) = y" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5476 |
|
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5477 |
begin |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5478 |
|
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5479 |
lemma homotopically_trivial_retraction_gen: |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5480 |
assumes P: "\<And>f. \<lbrakk>continuous_on u f; f ` u \<subseteq> t; Q f\<rbrakk> \<Longrightarrow> P(k o f)" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5481 |
and Q: "\<And>f. \<lbrakk>continuous_on u f; f ` u \<subseteq> s; P f\<rbrakk> \<Longrightarrow> Q(h o f)" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5482 |
and Qeq: "\<And>h k. (\<And>x. x \<in> u \<Longrightarrow> h x = k x) \<Longrightarrow> Q h = Q k" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5483 |
and hom: "\<And>f g. \<lbrakk>continuous_on u f; f ` u \<subseteq> s; P f; |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5484 |
continuous_on u g; g ` u \<subseteq> s; P g\<rbrakk> |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5485 |
\<Longrightarrow> homotopic_with P u s f g" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5486 |
and contf: "continuous_on u f" and imf: "f ` u \<subseteq> t" and Qf: "Q f" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5487 |
and contg: "continuous_on u g" and img: "g ` u \<subseteq> t" and Qg: "Q g" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5488 |
shows "homotopic_with Q u t f g" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5489 |
proof - |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5490 |
have feq: "\<And>x. x \<in> u \<Longrightarrow> (h \<circ> (k \<circ> f)) x = f x" using idhk imf by auto |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5491 |
have geq: "\<And>x. x \<in> u \<Longrightarrow> (h \<circ> (k \<circ> g)) x = g x" using idhk img by auto |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5492 |
have "continuous_on u (k \<circ> f)" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5493 |
using contf continuous_on_compose continuous_on_subset contk imf by blast |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5494 |
moreover have "(k \<circ> f) ` u \<subseteq> s" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5495 |
using imf imk by fastforce |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5496 |
moreover have "P (k \<circ> f)" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5497 |
by (simp add: P Qf contf imf) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5498 |
moreover have "continuous_on u (k \<circ> g)" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5499 |
using contg continuous_on_compose continuous_on_subset contk img by blast |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5500 |
moreover have "(k \<circ> g) ` u \<subseteq> s" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5501 |
using img imk by fastforce |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5502 |
moreover have "P (k \<circ> g)" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5503 |
by (simp add: P Qg contg img) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5504 |
ultimately have "homotopic_with P u s (k \<circ> f) (k \<circ> g)" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5505 |
by (rule hom) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5506 |
then have "homotopic_with Q u t (h \<circ> (k \<circ> f)) (h \<circ> (k \<circ> g))" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5507 |
apply (rule homotopic_with_compose_continuous_left [OF homotopic_with_mono]) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5508 |
using Q by (auto simp: conth imh) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5509 |
then show ?thesis |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5510 |
apply (rule homotopic_with_eq) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5511 |
apply (metis feq) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5512 |
apply (metis geq) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5513 |
apply (metis Qeq) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5514 |
done |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5515 |
qed |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5516 |
|
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5517 |
lemma homotopically_trivial_retraction_null_gen: |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5518 |
assumes P: "\<And>f. \<lbrakk>continuous_on u f; f ` u \<subseteq> t; Q f\<rbrakk> \<Longrightarrow> P(k o f)" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5519 |
and Q: "\<And>f. \<lbrakk>continuous_on u f; f ` u \<subseteq> s; P f\<rbrakk> \<Longrightarrow> Q(h o f)" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5520 |
and Qeq: "\<And>h k. (\<And>x. x \<in> u \<Longrightarrow> h x = k x) \<Longrightarrow> Q h = Q k" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5521 |
and hom: "\<And>f. \<lbrakk>continuous_on u f; f ` u \<subseteq> s; P f\<rbrakk> |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5522 |
\<Longrightarrow> \<exists>c. homotopic_with P u s f (\<lambda>x. c)" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5523 |
and contf: "continuous_on u f" and imf:"f ` u \<subseteq> t" and Qf: "Q f" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5524 |
obtains c where "homotopic_with Q u t f (\<lambda>x. c)" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5525 |
proof - |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5526 |
have feq: "\<And>x. x \<in> u \<Longrightarrow> (h \<circ> (k \<circ> f)) x = f x" using idhk imf by auto |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5527 |
have "continuous_on u (k \<circ> f)" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5528 |
using contf continuous_on_compose continuous_on_subset contk imf by blast |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5529 |
moreover have "(k \<circ> f) ` u \<subseteq> s" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5530 |
using imf imk by fastforce |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5531 |
moreover have "P (k \<circ> f)" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5532 |
by (simp add: P Qf contf imf) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5533 |
ultimately obtain c where "homotopic_with P u s (k \<circ> f) (\<lambda>x. c)" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5534 |
by (metis hom) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5535 |
then have "homotopic_with Q u t (h \<circ> (k \<circ> f)) (h o (\<lambda>x. c))" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5536 |
apply (rule homotopic_with_compose_continuous_left [OF homotopic_with_mono]) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5537 |
using Q by (auto simp: conth imh) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5538 |
then show ?thesis |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5539 |
apply (rule_tac c = "h c" in that) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5540 |
apply (erule homotopic_with_eq) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5541 |
apply (metis feq, simp) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5542 |
apply (metis Qeq) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5543 |
done |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5544 |
qed |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5545 |
|
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5546 |
lemma cohomotopically_trivial_retraction_gen: |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5547 |
assumes P: "\<And>f. \<lbrakk>continuous_on t f; f ` t \<subseteq> u; Q f\<rbrakk> \<Longrightarrow> P(f o h)" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5548 |
and Q: "\<And>f. \<lbrakk>continuous_on s f; f ` s \<subseteq> u; P f\<rbrakk> \<Longrightarrow> Q(f o k)" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5549 |
and Qeq: "\<And>h k. (\<And>x. x \<in> t \<Longrightarrow> h x = k x) \<Longrightarrow> Q h = Q k" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5550 |
and hom: "\<And>f g. \<lbrakk>continuous_on s f; f ` s \<subseteq> u; P f; |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5551 |
continuous_on s g; g ` s \<subseteq> u; P g\<rbrakk> |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5552 |
\<Longrightarrow> homotopic_with P s u f g" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5553 |
and contf: "continuous_on t f" and imf: "f ` t \<subseteq> u" and Qf: "Q f" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5554 |
and contg: "continuous_on t g" and img: "g ` t \<subseteq> u" and Qg: "Q g" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5555 |
shows "homotopic_with Q t u f g" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5556 |
proof - |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5557 |
have feq: "\<And>x. x \<in> t \<Longrightarrow> (f \<circ> h \<circ> k) x = f x" using idhk imf by auto |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5558 |
have geq: "\<And>x. x \<in> t \<Longrightarrow> (g \<circ> h \<circ> k) x = g x" using idhk img by auto |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5559 |
have "continuous_on s (f \<circ> h)" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5560 |
using contf conth continuous_on_compose imh by blast |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5561 |
moreover have "(f \<circ> h) ` s \<subseteq> u" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5562 |
using imf imh by fastforce |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5563 |
moreover have "P (f \<circ> h)" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5564 |
by (simp add: P Qf contf imf) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5565 |
moreover have "continuous_on s (g o h)" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5566 |
using contg continuous_on_compose continuous_on_subset conth imh by blast |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5567 |
moreover have "(g \<circ> h) ` s \<subseteq> u" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5568 |
using img imh by fastforce |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5569 |
moreover have "P (g \<circ> h)" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5570 |
by (simp add: P Qg contg img) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5571 |
ultimately have "homotopic_with P s u (f o h) (g \<circ> h)" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5572 |
by (rule hom) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5573 |
then have "homotopic_with Q t u (f o h o k) (g \<circ> h o k)" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5574 |
apply (rule homotopic_with_compose_continuous_right [OF homotopic_with_mono]) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5575 |
using Q by (auto simp: contk imk) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5576 |
then show ?thesis |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5577 |
apply (rule homotopic_with_eq) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5578 |
apply (metis feq) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5579 |
apply (metis geq) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5580 |
apply (metis Qeq) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5581 |
done |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5582 |
qed |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5583 |
|
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5584 |
lemma cohomotopically_trivial_retraction_null_gen: |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5585 |
assumes P: "\<And>f. \<lbrakk>continuous_on t f; f ` t \<subseteq> u; Q f\<rbrakk> \<Longrightarrow> P(f o h)" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5586 |
and Q: "\<And>f. \<lbrakk>continuous_on s f; f ` s \<subseteq> u; P f\<rbrakk> \<Longrightarrow> Q(f o k)" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5587 |
and Qeq: "\<And>h k. (\<And>x. x \<in> t \<Longrightarrow> h x = k x) \<Longrightarrow> Q h = Q k" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5588 |
and hom: "\<And>f g. \<lbrakk>continuous_on s f; f ` s \<subseteq> u; P f\<rbrakk> |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5589 |
\<Longrightarrow> \<exists>c. homotopic_with P s u f (\<lambda>x. c)" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5590 |
and contf: "continuous_on t f" and imf: "f ` t \<subseteq> u" and Qf: "Q f" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5591 |
obtains c where "homotopic_with Q t u f (\<lambda>x. c)" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5592 |
proof - |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5593 |
have feq: "\<And>x. x \<in> t \<Longrightarrow> (f \<circ> h \<circ> k) x = f x" using idhk imf by auto |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5594 |
have "continuous_on s (f \<circ> h)" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5595 |
using contf conth continuous_on_compose imh by blast |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5596 |
moreover have "(f \<circ> h) ` s \<subseteq> u" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5597 |
using imf imh by fastforce |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5598 |
moreover have "P (f \<circ> h)" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5599 |
by (simp add: P Qf contf imf) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5600 |
ultimately obtain c where "homotopic_with P s u (f o h) (\<lambda>x. c)" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5601 |
by (metis hom) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5602 |
then have "homotopic_with Q t u (f o h o k) ((\<lambda>x. c) o k)" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5603 |
apply (rule homotopic_with_compose_continuous_right [OF homotopic_with_mono]) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5604 |
using Q by (auto simp: contk imk) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5605 |
then show ?thesis |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5606 |
apply (rule_tac c = c in that) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5607 |
apply (erule homotopic_with_eq) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5608 |
apply (metis feq, simp) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5609 |
apply (metis Qeq) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5610 |
done |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5611 |
qed |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5612 |
|
36583 | 5613 |
end |
62948
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5614 |
|
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5615 |
lemma simply_connected_retraction_gen: |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5616 |
shows "\<lbrakk>simply_connected S; continuous_on S h; h ` S = T; |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5617 |
continuous_on T k; k ` T \<subseteq> S; \<And>y. y \<in> T \<Longrightarrow> h(k y) = y\<rbrakk> |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5618 |
\<Longrightarrow> simply_connected T" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5619 |
apply (simp add: simply_connected_def path_def path_image_def homotopic_loops_def, clarify) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5620 |
apply (rule Retracts.homotopically_trivial_retraction_gen |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5621 |
[of S h _ k _ "\<lambda>p. pathfinish p = pathstart p" "\<lambda>p. pathfinish p = pathstart p"]) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5622 |
apply (simp_all add: Retracts_def pathfinish_def pathstart_def) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5623 |
done |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5624 |
|
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5625 |
lemma homeomorphic_simply_connected: |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5626 |
"\<lbrakk>S homeomorphic T; simply_connected S\<rbrakk> \<Longrightarrow> simply_connected T" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5627 |
by (auto simp: homeomorphic_def homeomorphism_def intro: simply_connected_retraction_gen) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5628 |
|
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5629 |
lemma homeomorphic_simply_connected_eq: |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5630 |
"S homeomorphic T \<Longrightarrow> (simply_connected S \<longleftrightarrow> simply_connected T)" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5631 |
by (metis homeomorphic_simply_connected homeomorphic_sym) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5632 |
|
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
5633 |
end |