author | Manuel Eberl <eberlm@in.tum.de> |
Mon, 02 Dec 2019 10:31:51 +0100 | |
changeset 71191 | 6695aeae8ec9 |
parent 71189 | 954ee5acaae0 |
parent 71184 | d62fdaafdafc |
child 71200 | 3548d54ce3ee |
permissions | -rw-r--r-- |
63627 | 1 |
(* Title: HOL/Analysis/Path_Connected.thy |
61806
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61762
diff
changeset
|
2 |
Authors: LC Paulson and Robert Himmelmann (TU Muenchen), based on material from HOL Light |
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*) |
4 |
||
69620 | 5 |
section \<open>Path-Connectedness\<close> |
36583 | 6 |
|
7 |
theory Path_Connected |
|
70196
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
8 |
imports Starlike T1_Spaces |
36583 | 9 |
begin |
10 |
||
60420 | 11 |
subsection \<open>Paths and Arcs\<close> |
36583 | 12 |
|
70136 | 13 |
definition\<^marker>\<open>tag important\<close> path :: "(real \<Rightarrow> 'a::topological_space) \<Rightarrow> bool" |
53640 | 14 |
where "path g \<longleftrightarrow> continuous_on {0..1} g" |
36583 | 15 |
|
70136 | 16 |
definition\<^marker>\<open>tag important\<close> pathstart :: "(real \<Rightarrow> 'a::topological_space) \<Rightarrow> 'a" |
36583 | 17 |
where "pathstart g = g 0" |
18 |
||
70136 | 19 |
definition\<^marker>\<open>tag important\<close> pathfinish :: "(real \<Rightarrow> 'a::topological_space) \<Rightarrow> 'a" |
36583 | 20 |
where "pathfinish g = g 1" |
21 |
||
70136 | 22 |
definition\<^marker>\<open>tag important\<close> path_image :: "(real \<Rightarrow> 'a::topological_space) \<Rightarrow> 'a set" |
36583 | 23 |
where "path_image g = g ` {0 .. 1}" |
24 |
||
70136 | 25 |
definition\<^marker>\<open>tag important\<close> reversepath :: "(real \<Rightarrow> 'a::topological_space) \<Rightarrow> real \<Rightarrow> 'a" |
36583 | 26 |
where "reversepath g = (\<lambda>x. g(1 - x))" |
27 |
||
70136 | 28 |
definition\<^marker>\<open>tag important\<close> joinpaths :: "(real \<Rightarrow> 'a::topological_space) \<Rightarrow> (real \<Rightarrow> 'a) \<Rightarrow> real \<Rightarrow> 'a" |
36583 | 29 |
(infixr "+++" 75) |
30 |
where "g1 +++ g2 = (\<lambda>x. if x \<le> 1/2 then g1 (2 * x) else g2 (2 * x - 1))" |
|
31 |
||
70136 | 32 |
definition\<^marker>\<open>tag important\<close> simple_path :: "(real \<Rightarrow> 'a::topological_space) \<Rightarrow> bool" |
36583 | 33 |
where "simple_path g \<longleftrightarrow> |
60303 | 34 |
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 |
|
70136 | 36 |
definition\<^marker>\<open>tag important\<close> arc :: "(real \<Rightarrow> 'a :: topological_space) \<Rightarrow> bool" |
60303 | 37 |
where "arc g \<longleftrightarrow> path g \<and> inj_on g {0..1}" |
36583 | 38 |
|
49653 | 39 |
|
70136 | 40 |
subsection\<^marker>\<open>tag unimportant\<close>\<open>Invariance theorems\<close> |
60303 | 41 |
|
42 |
lemma path_eq: "path p \<Longrightarrow> (\<And>t. t \<in> {0..1} \<Longrightarrow> p t = q t) \<Longrightarrow> path q" |
|
43 |
using continuous_on_eq path_def by blast |
|
44 |
||
68096 | 45 |
lemma path_continuous_image: "path g \<Longrightarrow> continuous_on (path_image g) f \<Longrightarrow> path(f \<circ> g)" |
60303 | 46 |
unfolding path_def path_image_def |
47 |
using continuous_on_compose by blast |
|
48 |
||
49 |
lemma path_translation_eq: |
|
50 |
fixes g :: "real \<Rightarrow> 'a :: real_normed_vector" |
|
68096 | 51 |
shows "path((\<lambda>x. a + x) \<circ> g) = path g" |
60303 | 52 |
proof - |
68096 | 53 |
have g: "g = (\<lambda>x. -a + x) \<circ> ((\<lambda>x. a + x) \<circ> g)" |
60303 | 54 |
by (rule ext) simp |
55 |
show ?thesis |
|
56 |
unfolding path_def |
|
57 |
apply safe |
|
58 |
apply (subst g) |
|
59 |
apply (rule continuous_on_compose) |
|
60 |
apply (auto intro: continuous_intros) |
|
61 |
done |
|
62 |
qed |
|
63 |
||
64 |
lemma path_linear_image_eq: |
|
65 |
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space" |
|
66 |
assumes "linear f" "inj f" |
|
68096 | 67 |
shows "path(f \<circ> g) = path g" |
60303 | 68 |
proof - |
69 |
from linear_injective_left_inverse [OF assms] |
|
70 |
obtain h where h: "linear h" "h \<circ> f = id" |
|
71 |
by blast |
|
68096 | 72 |
then have g: "g = h \<circ> (f \<circ> g)" |
60303 | 73 |
by (metis comp_assoc id_comp) |
74 |
show ?thesis |
|
75 |
unfolding path_def |
|
76 |
using h assms |
|
77 |
by (metis g continuous_on_compose linear_continuous_on linear_conv_bounded_linear) |
|
78 |
qed |
|
79 |
||
68096 | 80 |
lemma pathstart_translation: "pathstart((\<lambda>x. a + x) \<circ> g) = a + pathstart g" |
60303 | 81 |
by (simp add: pathstart_def) |
82 |
||
68096 | 83 |
lemma pathstart_linear_image_eq: "linear f \<Longrightarrow> pathstart(f \<circ> g) = f(pathstart g)" |
60303 | 84 |
by (simp add: pathstart_def) |
85 |
||
68096 | 86 |
lemma pathfinish_translation: "pathfinish((\<lambda>x. a + x) \<circ> g) = a + pathfinish g" |
60303 | 87 |
by (simp add: pathfinish_def) |
88 |
||
68096 | 89 |
lemma pathfinish_linear_image: "linear f \<Longrightarrow> pathfinish(f \<circ> g) = f(pathfinish g)" |
60303 | 90 |
by (simp add: pathfinish_def) |
91 |
||
68096 | 92 |
lemma path_image_translation: "path_image((\<lambda>x. a + x) \<circ> g) = (\<lambda>x. a + x) ` (path_image g)" |
60303 | 93 |
by (simp add: image_comp path_image_def) |
94 |
||
68096 | 95 |
lemma path_image_linear_image: "linear f \<Longrightarrow> path_image(f \<circ> g) = f ` (path_image g)" |
60303 | 96 |
by (simp add: image_comp path_image_def) |
97 |
||
68096 | 98 |
lemma reversepath_translation: "reversepath((\<lambda>x. a + x) \<circ> g) = (\<lambda>x. a + x) \<circ> reversepath g" |
60303 | 99 |
by (rule ext) (simp add: reversepath_def) |
36583 | 100 |
|
68096 | 101 |
lemma reversepath_linear_image: "linear f \<Longrightarrow> reversepath(f \<circ> g) = f \<circ> reversepath g" |
60303 | 102 |
by (rule ext) (simp add: reversepath_def) |
103 |
||
104 |
lemma joinpaths_translation: |
|
68096 | 105 |
"((\<lambda>x. a + x) \<circ> g1) +++ ((\<lambda>x. a + x) \<circ> g2) = (\<lambda>x. a + x) \<circ> (g1 +++ g2)" |
60303 | 106 |
by (rule ext) (simp add: joinpaths_def) |
107 |
||
68096 | 108 |
lemma joinpaths_linear_image: "linear f \<Longrightarrow> (f \<circ> g1) +++ (f \<circ> g2) = f \<circ> (g1 +++ g2)" |
60303 | 109 |
by (rule ext) (simp add: joinpaths_def) |
110 |
||
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
60420
diff
changeset
|
111 |
lemma simple_path_translation_eq: |
60303 | 112 |
fixes g :: "real \<Rightarrow> 'a::euclidean_space" |
68096 | 113 |
shows "simple_path((\<lambda>x. a + x) \<circ> g) = simple_path g" |
60303 | 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" |
|
118 |
assumes "linear f" "inj f" |
|
68096 | 119 |
shows "simple_path(f \<circ> g) = simple_path g" |
60303 | 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" |
|
68096 | 125 |
shows "arc((\<lambda>x. a + x) \<circ> g) = arc g" |
60303 | 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" |
|
68096 | 131 |
shows "arc(f \<circ> g) = arc g" |
60303 | 132 |
using assms inj_on_eq_iff [of f] |
133 |
by (auto simp: arc_def inj_on_def path_linear_image_eq) |
|
134 |
||
69514 | 135 |
|
70136 | 136 |
subsection\<^marker>\<open>tag unimportant\<close>\<open>Basic lemmas about paths\<close> |
60303 | 137 |
|
69939
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
138 |
lemma pathin_iff_path_real [simp]: "pathin euclideanreal g \<longleftrightarrow> path g" |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
139 |
by (simp add: pathin_def path_def) |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
140 |
|
64788
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
141 |
lemma continuous_on_path: "path f \<Longrightarrow> t \<subseteq> {0..1} \<Longrightarrow> continuous_on t f" |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
142 |
using continuous_on_subset path_def by blast |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
143 |
|
60303 | 144 |
lemma arc_imp_simple_path: "arc g \<Longrightarrow> simple_path g" |
145 |
by (simp add: arc_def inj_on_def simple_path_def) |
|
146 |
||
147 |
lemma arc_imp_path: "arc g \<Longrightarrow> path g" |
|
148 |
using arc_def by blast |
|
149 |
||
64788
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
150 |
lemma arc_imp_inj_on: "arc g \<Longrightarrow> inj_on g {0..1}" |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
151 |
by (auto simp: arc_def) |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
152 |
|
60303 | 153 |
lemma simple_path_imp_path: "simple_path g \<Longrightarrow> path g" |
154 |
using simple_path_def by blast |
|
155 |
||
156 |
lemma simple_path_cases: "simple_path g \<Longrightarrow> arc g \<or> pathfinish g = pathstart g" |
|
157 |
unfolding simple_path_def arc_def inj_on_def pathfinish_def pathstart_def |
|
68096 | 158 |
by force |
60303 | 159 |
|
160 |
lemma simple_path_imp_arc: "simple_path g \<Longrightarrow> pathfinish g \<noteq> pathstart g \<Longrightarrow> arc g" |
|
161 |
using simple_path_cases by auto |
|
162 |
||
163 |
lemma arc_distinct_ends: "arc g \<Longrightarrow> pathfinish g \<noteq> pathstart g" |
|
164 |
unfolding arc_def inj_on_def pathfinish_def pathstart_def |
|
165 |
by fastforce |
|
166 |
||
167 |
lemma arc_simple_path: "arc g \<longleftrightarrow> simple_path g \<and> pathfinish g \<noteq> pathstart g" |
|
168 |
using arc_distinct_ends arc_imp_simple_path simple_path_cases by blast |
|
169 |
||
170 |
lemma simple_path_eq_arc: "pathfinish g \<noteq> pathstart g \<Longrightarrow> (simple_path g = arc g)" |
|
171 |
by (simp add: arc_simple_path) |
|
36583 | 172 |
|
66793
deabce3ccf1f
new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
66456
diff
changeset
|
173 |
lemma path_image_const [simp]: "path_image (\<lambda>t. a) = {a}" |
deabce3ccf1f
new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
66456
diff
changeset
|
174 |
by (force simp: path_image_def) |
deabce3ccf1f
new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
66456
diff
changeset
|
175 |
|
60974
6a6f15d8fbc4
New material and fixes related to the forthcoming Stone-Weierstrass development
paulson <lp15@cam.ac.uk>
parents:
60809
diff
changeset
|
176 |
lemma path_image_nonempty [simp]: "path_image g \<noteq> {}" |
56188 | 177 |
unfolding path_image_def image_is_empty box_eq_empty |
53640 | 178 |
by auto |
36583 | 179 |
|
53640 | 180 |
lemma pathstart_in_path_image[intro]: "pathstart g \<in> path_image g" |
181 |
unfolding pathstart_def path_image_def |
|
182 |
by auto |
|
36583 | 183 |
|
53640 | 184 |
lemma pathfinish_in_path_image[intro]: "pathfinish g \<in> path_image g" |
185 |
unfolding pathfinish_def path_image_def |
|
186 |
by auto |
|
187 |
||
188 |
lemma connected_path_image[intro]: "path g \<Longrightarrow> connected (path_image g)" |
|
36583 | 189 |
unfolding path_def path_image_def |
60303 | 190 |
using connected_continuous_image connected_Icc by blast |
36583 | 191 |
|
53640 | 192 |
lemma compact_path_image[intro]: "path g \<Longrightarrow> compact (path_image g)" |
36583 | 193 |
unfolding path_def path_image_def |
60303 | 194 |
using compact_continuous_image connected_Icc by blast |
36583 | 195 |
|
53640 | 196 |
lemma reversepath_reversepath[simp]: "reversepath (reversepath g) = g" |
197 |
unfolding reversepath_def |
|
198 |
by auto |
|
36583 | 199 |
|
53640 | 200 |
lemma pathstart_reversepath[simp]: "pathstart (reversepath g) = pathfinish g" |
201 |
unfolding pathstart_def reversepath_def pathfinish_def |
|
202 |
by auto |
|
36583 | 203 |
|
53640 | 204 |
lemma pathfinish_reversepath[simp]: "pathfinish (reversepath g) = pathstart g" |
205 |
unfolding pathstart_def reversepath_def pathfinish_def |
|
206 |
by auto |
|
36583 | 207 |
|
49653 | 208 |
lemma pathstart_join[simp]: "pathstart (g1 +++ g2) = pathstart g1" |
53640 | 209 |
unfolding pathstart_def joinpaths_def pathfinish_def |
210 |
by auto |
|
36583 | 211 |
|
49653 | 212 |
lemma pathfinish_join[simp]: "pathfinish (g1 +++ g2) = pathfinish g2" |
53640 | 213 |
unfolding pathstart_def joinpaths_def pathfinish_def |
214 |
by auto |
|
36583 | 215 |
|
53640 | 216 |
lemma path_image_reversepath[simp]: "path_image (reversepath g) = path_image g" |
49653 | 217 |
proof - |
53640 | 218 |
have *: "\<And>g. path_image (reversepath g) \<subseteq> path_image g" |
49653 | 219 |
unfolding path_image_def subset_eq reversepath_def Ball_def image_iff |
60303 | 220 |
by force |
49653 | 221 |
show ?thesis |
222 |
using *[of g] *[of "reversepath g"] |
|
53640 | 223 |
unfolding reversepath_reversepath |
224 |
by auto |
|
49653 | 225 |
qed |
36583 | 226 |
|
53640 | 227 |
lemma path_reversepath [simp]: "path (reversepath g) \<longleftrightarrow> path g" |
49653 | 228 |
proof - |
229 |
have *: "\<And>g. path g \<Longrightarrow> path (reversepath g)" |
|
230 |
unfolding path_def reversepath_def |
|
231 |
apply (rule continuous_on_compose[unfolded o_def, of _ "\<lambda>x. 1 - x"]) |
|
68096 | 232 |
apply (auto intro: continuous_intros continuous_on_subset[of "{0..1}"]) |
49653 | 233 |
done |
234 |
show ?thesis |
|
235 |
using *[of "reversepath g"] *[of g] |
|
236 |
unfolding reversepath_reversepath |
|
237 |
by (rule iffI) |
|
238 |
qed |
|
239 |
||
60303 | 240 |
lemma arc_reversepath: |
241 |
assumes "arc g" shows "arc(reversepath g)" |
|
242 |
proof - |
|
243 |
have injg: "inj_on g {0..1}" |
|
244 |
using assms |
|
245 |
by (simp add: arc_def) |
|
246 |
have **: "\<And>x y::real. 1-x = 1-y \<Longrightarrow> x = y" |
|
247 |
by simp |
|
248 |
show ?thesis |
|
68096 | 249 |
using assms by (clarsimp simp: arc_def intro!: inj_onI) (simp add: inj_onD reversepath_def **) |
60303 | 250 |
qed |
251 |
||
252 |
lemma simple_path_reversepath: "simple_path g \<Longrightarrow> simple_path (reversepath g)" |
|
253 |
apply (simp add: simple_path_def) |
|
254 |
apply (force simp: reversepath_def) |
|
255 |
done |
|
256 |
||
49653 | 257 |
lemmas reversepath_simps = |
258 |
path_reversepath path_image_reversepath pathstart_reversepath pathfinish_reversepath |
|
36583 | 259 |
|
49653 | 260 |
lemma path_join[simp]: |
261 |
assumes "pathfinish g1 = pathstart g2" |
|
262 |
shows "path (g1 +++ g2) \<longleftrightarrow> path g1 \<and> path g2" |
|
263 |
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
|
264 |
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
|
265 |
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
|
266 |
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
|
267 |
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
|
268 |
have g2: "continuous_on {0..1} g2 \<longleftrightarrow> continuous_on {0..1} ((g1 +++ g2) \<circ> (\<lambda>x. x / 2 + 1/2))" |
53640 | 269 |
using assms |
270 |
by (intro continuous_on_cong refl) (auto simp: joinpaths_def pathfinish_def pathstart_def) |
|
271 |
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
|
272 |
unfolding g1 g2 |
56371
fb9ae0727548
extend continuous_intros; remove continuous_on_intros and isCont_intros
hoelzl
parents:
56188
diff
changeset
|
273 |
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
|
274 |
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
|
275 |
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
|
276 |
have 01: "{0 .. 1} = {0..1/2} \<union> {1/2 .. 1::real}" |
36583 | 277 |
by auto |
53640 | 278 |
{ |
279 |
fix x :: real |
|
280 |
assume "0 \<le> x" and "x \<le> 1" |
|
281 |
then have "x \<in> (\<lambda>x. x * 2) ` {0..1 / 2}" |
|
282 |
by (intro image_eqI[where x="x/2"]) auto |
|
283 |
} |
|
51478
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
50935
diff
changeset
|
284 |
note 1 = this |
53640 | 285 |
{ |
286 |
fix x :: real |
|
287 |
assume "0 \<le> x" and "x \<le> 1" |
|
288 |
then have "x \<in> (\<lambda>x. x * 2 - 1) ` {1 / 2..1}" |
|
289 |
by (intro image_eqI[where x="x/2 + 1/2"]) auto |
|
290 |
} |
|
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
|
291 |
note 2 = this |
49653 | 292 |
show "continuous_on {0..1} (g1 +++ g2)" |
53640 | 293 |
using assms |
294 |
unfolding joinpaths_def 01 |
|
56371
fb9ae0727548
extend continuous_intros; remove continuous_on_intros and isCont_intros
hoelzl
parents:
56188
diff
changeset
|
295 |
apply (intro continuous_on_cases closed_atLeastAtMost g1g2[THEN continuous_on_compose2] continuous_intros) |
53640 | 296 |
apply (auto simp: field_simps pathfinish_def pathstart_def intro!: 1 2) |
297 |
done |
|
49653 | 298 |
qed |
36583 | 299 |
|
69514 | 300 |
|
70136 | 301 |
subsection\<^marker>\<open>tag unimportant\<close> \<open>Path Images\<close> |
60303 | 302 |
|
303 |
lemma bounded_path_image: "path g \<Longrightarrow> bounded(path_image g)" |
|
304 |
by (simp add: compact_imp_bounded compact_path_image) |
|
305 |
||
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
60420
diff
changeset
|
306 |
lemma closed_path_image: |
60303 | 307 |
fixes g :: "real \<Rightarrow> 'a::t2_space" |
308 |
shows "path g \<Longrightarrow> closed(path_image g)" |
|
309 |
by (metis compact_path_image compact_imp_closed) |
|
310 |
||
311 |
lemma connected_simple_path_image: "simple_path g \<Longrightarrow> connected(path_image g)" |
|
312 |
by (metis connected_path_image simple_path_imp_path) |
|
313 |
||
314 |
lemma compact_simple_path_image: "simple_path g \<Longrightarrow> compact(path_image g)" |
|
315 |
by (metis compact_path_image simple_path_imp_path) |
|
316 |
||
317 |
lemma bounded_simple_path_image: "simple_path g \<Longrightarrow> bounded(path_image g)" |
|
318 |
by (metis bounded_path_image simple_path_imp_path) |
|
319 |
||
320 |
lemma closed_simple_path_image: |
|
321 |
fixes g :: "real \<Rightarrow> 'a::t2_space" |
|
322 |
shows "simple_path g \<Longrightarrow> closed(path_image g)" |
|
323 |
by (metis closed_path_image simple_path_imp_path) |
|
324 |
||
325 |
lemma connected_arc_image: "arc g \<Longrightarrow> connected(path_image g)" |
|
326 |
by (metis connected_path_image arc_imp_path) |
|
327 |
||
328 |
lemma compact_arc_image: "arc g \<Longrightarrow> compact(path_image g)" |
|
329 |
by (metis compact_path_image arc_imp_path) |
|
330 |
||
331 |
lemma bounded_arc_image: "arc g \<Longrightarrow> bounded(path_image g)" |
|
332 |
by (metis bounded_path_image arc_imp_path) |
|
333 |
||
334 |
lemma closed_arc_image: |
|
335 |
fixes g :: "real \<Rightarrow> 'a::t2_space" |
|
336 |
shows "arc g \<Longrightarrow> closed(path_image g)" |
|
337 |
by (metis closed_path_image arc_imp_path) |
|
338 |
||
53640 | 339 |
lemma path_image_join_subset: "path_image (g1 +++ g2) \<subseteq> path_image g1 \<union> path_image g2" |
340 |
unfolding path_image_def joinpaths_def |
|
341 |
by auto |
|
36583 | 342 |
|
343 |
lemma subset_path_image_join: |
|
53640 | 344 |
assumes "path_image g1 \<subseteq> s" |
345 |
and "path_image g2 \<subseteq> s" |
|
346 |
shows "path_image (g1 +++ g2) \<subseteq> s" |
|
347 |
using path_image_join_subset[of g1 g2] and assms |
|
348 |
by auto |
|
36583 | 349 |
|
350 |
lemma path_image_join: |
|
60303 | 351 |
"pathfinish g1 = pathstart g2 \<Longrightarrow> path_image(g1 +++ g2) = path_image g1 \<union> path_image g2" |
352 |
apply (rule subset_antisym [OF path_image_join_subset]) |
|
353 |
apply (auto simp: pathfinish_def pathstart_def path_image_def joinpaths_def image_def) |
|
354 |
apply (drule sym) |
|
355 |
apply (rule_tac x="xa/2" in bexI, auto) |
|
356 |
apply (rule ccontr) |
|
357 |
apply (drule_tac x="(xa+1)/2" in bspec) |
|
358 |
apply (auto simp: field_simps) |
|
359 |
apply (drule_tac x="1/2" in bspec, auto) |
|
360 |
done |
|
36583 | 361 |
|
362 |
lemma not_in_path_image_join: |
|
53640 | 363 |
assumes "x \<notin> path_image g1" |
364 |
and "x \<notin> path_image g2" |
|
365 |
shows "x \<notin> path_image (g1 +++ g2)" |
|
366 |
using assms and path_image_join_subset[of g1 g2] |
|
367 |
by auto |
|
36583 | 368 |
|
68096 | 369 |
lemma pathstart_compose: "pathstart(f \<circ> p) = f(pathstart p)" |
60303 | 370 |
by (simp add: pathstart_def) |
371 |
||
68096 | 372 |
lemma pathfinish_compose: "pathfinish(f \<circ> p) = f(pathfinish p)" |
60303 | 373 |
by (simp add: pathfinish_def) |
374 |
||
68096 | 375 |
lemma path_image_compose: "path_image (f \<circ> p) = f ` (path_image p)" |
60303 | 376 |
by (simp add: image_comp path_image_def) |
377 |
||
68096 | 378 |
lemma path_compose_join: "f \<circ> (p +++ q) = (f \<circ> p) +++ (f \<circ> q)" |
60303 | 379 |
by (rule ext) (simp add: joinpaths_def) |
380 |
||
68096 | 381 |
lemma path_compose_reversepath: "f \<circ> reversepath p = reversepath(f \<circ> p)" |
60303 | 382 |
by (rule ext) (simp add: reversepath_def) |
383 |
||
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
|
384 |
lemma joinpaths_eq: |
60303 | 385 |
"(\<And>t. t \<in> {0..1} \<Longrightarrow> p t = p' t) \<Longrightarrow> |
386 |
(\<And>t. t \<in> {0..1} \<Longrightarrow> q t = q' t) |
|
387 |
\<Longrightarrow> t \<in> {0..1} \<Longrightarrow> (p +++ q) t = (p' +++ q') t" |
|
388 |
by (auto simp: joinpaths_def) |
|
389 |
||
390 |
lemma simple_path_inj_on: "simple_path g \<Longrightarrow> inj_on g {0<..<1}" |
|
391 |
by (auto simp: simple_path_def path_image_def inj_on_def less_eq_real_def Ball_def) |
|
392 |
||
393 |
||
70136 | 394 |
subsection\<^marker>\<open>tag unimportant\<close>\<open>Simple paths with the endpoints removed\<close> |
60303 | 395 |
|
396 |
lemma simple_path_endless: |
|
397 |
"simple_path c \<Longrightarrow> path_image c - {pathstart c,pathfinish c} = c ` {0<..<1}" |
|
398 |
apply (auto simp: simple_path_def path_image_def pathstart_def pathfinish_def Ball_def Bex_def image_def) |
|
399 |
apply (metis eq_iff le_less_linear) |
|
400 |
apply (metis leD linear) |
|
401 |
using less_eq_real_def zero_le_one apply blast |
|
402 |
using less_eq_real_def zero_le_one apply blast |
|
49653 | 403 |
done |
36583 | 404 |
|
60303 | 405 |
lemma connected_simple_path_endless: |
406 |
"simple_path c \<Longrightarrow> connected(path_image c - {pathstart c,pathfinish c})" |
|
407 |
apply (simp add: simple_path_endless) |
|
408 |
apply (rule connected_continuous_image) |
|
409 |
apply (meson continuous_on_subset greaterThanLessThan_subseteq_atLeastAtMost_iff le_numeral_extra(3) le_numeral_extra(4) path_def simple_path_imp_path) |
|
410 |
by auto |
|
411 |
||
412 |
lemma nonempty_simple_path_endless: |
|
413 |
"simple_path c \<Longrightarrow> path_image c - {pathstart c,pathfinish c} \<noteq> {}" |
|
414 |
by (simp add: simple_path_endless) |
|
415 |
||
416 |
||
70136 | 417 |
subsection\<^marker>\<open>tag unimportant\<close>\<open>The operations on paths\<close> |
60303 | 418 |
|
419 |
lemma path_image_subset_reversepath: "path_image(reversepath g) \<le> path_image g" |
|
420 |
by (auto simp: path_image_def reversepath_def) |
|
421 |
||
422 |
lemma path_imp_reversepath: "path g \<Longrightarrow> path(reversepath g)" |
|
423 |
apply (auto simp: path_def reversepath_def) |
|
424 |
using continuous_on_compose [of "{0..1}" "\<lambda>x. 1 - x" g] |
|
425 |
apply (auto simp: continuous_on_op_minus) |
|
426 |
done |
|
427 |
||
61204 | 428 |
lemma half_bounded_equal: "1 \<le> x * 2 \<Longrightarrow> x * 2 \<le> 1 \<longleftrightarrow> x = (1/2::real)" |
429 |
by simp |
|
60303 | 430 |
|
431 |
lemma continuous_on_joinpaths: |
|
432 |
assumes "continuous_on {0..1} g1" "continuous_on {0..1} g2" "pathfinish g1 = pathstart g2" |
|
433 |
shows "continuous_on {0..1} (g1 +++ g2)" |
|
434 |
proof - |
|
435 |
have *: "{0..1::real} = {0..1/2} \<union> {1/2..1}" |
|
436 |
by auto |
|
437 |
have gg: "g2 0 = g1 1" |
|
438 |
by (metis assms(3) pathfinish_def pathstart_def) |
|
61204 | 439 |
have 1: "continuous_on {0..1/2} (g1 +++ g2)" |
68096 | 440 |
apply (rule continuous_on_eq [of _ "g1 \<circ> (\<lambda>x. 2*x)"]) |
61204 | 441 |
apply (rule continuous_intros | simp add: joinpaths_def assms)+ |
60303 | 442 |
done |
68096 | 443 |
have "continuous_on {1/2..1} (g2 \<circ> (\<lambda>x. 2*x-1))" |
61204 | 444 |
apply (rule continuous_on_subset [of "{1/2..1}"]) |
445 |
apply (rule continuous_intros | simp add: image_affinity_atLeastAtMost_diff assms)+ |
|
446 |
done |
|
447 |
then have 2: "continuous_on {1/2..1} (g1 +++ g2)" |
|
68096 | 448 |
apply (rule continuous_on_eq [of "{1/2..1}" "g2 \<circ> (\<lambda>x. 2*x-1)"]) |
61204 | 449 |
apply (rule assms continuous_intros | simp add: joinpaths_def mult.commute half_bounded_equal gg)+ |
60303 | 450 |
done |
451 |
show ?thesis |
|
452 |
apply (subst *) |
|
62397
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62381
diff
changeset
|
453 |
apply (rule continuous_on_closed_Un) |
60303 | 454 |
using 1 2 |
455 |
apply auto |
|
456 |
done |
|
457 |
qed |
|
458 |
||
459 |
lemma path_join_imp: "\<lbrakk>path g1; path g2; pathfinish g1 = pathstart g2\<rbrakk> \<Longrightarrow> path(g1 +++ g2)" |
|
71172 | 460 |
by (simp) |
60303 | 461 |
|
36583 | 462 |
lemma simple_path_join_loop: |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
60420
diff
changeset
|
463 |
assumes "arc g1" "arc g2" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
60420
diff
changeset
|
464 |
"pathfinish g1 = pathstart g2" "pathfinish g2 = pathstart g1" |
60303 | 465 |
"path_image g1 \<inter> path_image g2 \<subseteq> {pathstart g1, pathstart g2}" |
466 |
shows "simple_path(g1 +++ g2)" |
|
467 |
proof - |
|
468 |
have injg1: "inj_on g1 {0..1}" |
|
469 |
using assms |
|
470 |
by (simp add: arc_def) |
|
471 |
have injg2: "inj_on g2 {0..1}" |
|
472 |
using assms |
|
473 |
by (simp add: arc_def) |
|
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
60420
diff
changeset
|
474 |
have g12: "g1 1 = g2 0" |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
60420
diff
changeset
|
475 |
and g21: "g2 1 = g1 0" |
60303 | 476 |
and sb: "g1 ` {0..1} \<inter> g2 ` {0..1} \<subseteq> {g1 0, g2 0}" |
477 |
using assms |
|
478 |
by (simp_all add: arc_def pathfinish_def pathstart_def path_image_def) |
|
479 |
{ fix x and y::real |
|
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
60420
diff
changeset
|
480 |
assume xyI: "x = 1 \<longrightarrow> y \<noteq> 0" |
60303 | 481 |
and xy: "x \<le> 1" "0 \<le> y" " y * 2 \<le> 1" "\<not> x * 2 \<le> 1" "g2 (2 * x - 1) = g1 (2 * y)" |
482 |
have g1im: "g1 (2 * y) \<in> g1 ` {0..1} \<inter> g2 ` {0..1}" |
|
483 |
using xy |
|
484 |
apply simp |
|
485 |
apply (rule_tac x="2 * x - 1" in image_eqI, auto) |
|
486 |
done |
|
487 |
have False |
|
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
60420
diff
changeset
|
488 |
using subsetD [OF sb g1im] xy |
60303 | 489 |
apply auto |
490 |
apply (drule inj_onD [OF injg1]) |
|
491 |
using g21 [symmetric] xyI |
|
492 |
apply (auto dest: inj_onD [OF injg2]) |
|
493 |
done |
|
494 |
} note * = this |
|
495 |
{ fix x and y::real |
|
496 |
assume xy: "y \<le> 1" "0 \<le> x" "\<not> y * 2 \<le> 1" "x * 2 \<le> 1" "g1 (2 * x) = g2 (2 * y - 1)" |
|
497 |
have g1im: "g1 (2 * x) \<in> g1 ` {0..1} \<inter> g2 ` {0..1}" |
|
498 |
using xy |
|
499 |
apply simp |
|
500 |
apply (rule_tac x="2 * x" in image_eqI, auto) |
|
501 |
done |
|
502 |
have "x = 0 \<and> y = 1" |
|
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
60420
diff
changeset
|
503 |
using subsetD [OF sb g1im] xy |
60303 | 504 |
apply auto |
505 |
apply (force dest: inj_onD [OF injg1]) |
|
506 |
using g21 [symmetric] |
|
507 |
apply (auto dest: inj_onD [OF injg2]) |
|
508 |
done |
|
509 |
} note ** = this |
|
510 |
show ?thesis |
|
511 |
using assms |
|
512 |
apply (simp add: arc_def simple_path_def path_join, clarify) |
|
62390 | 513 |
apply (simp add: joinpaths_def split: if_split_asm) |
60303 | 514 |
apply (force dest: inj_onD [OF injg1]) |
515 |
apply (metis *) |
|
516 |
apply (metis **) |
|
517 |
apply (force dest: inj_onD [OF injg2]) |
|
518 |
done |
|
519 |
qed |
|
520 |
||
521 |
lemma arc_join: |
|
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
60420
diff
changeset
|
522 |
assumes "arc g1" "arc g2" |
60303 | 523 |
"pathfinish g1 = pathstart g2" |
524 |
"path_image g1 \<inter> path_image g2 \<subseteq> {pathstart g2}" |
|
525 |
shows "arc(g1 +++ g2)" |
|
526 |
proof - |
|
527 |
have injg1: "inj_on g1 {0..1}" |
|
528 |
using assms |
|
529 |
by (simp add: arc_def) |
|
530 |
have injg2: "inj_on g2 {0..1}" |
|
531 |
using assms |
|
532 |
by (simp add: arc_def) |
|
533 |
have g11: "g1 1 = g2 0" |
|
534 |
and sb: "g1 ` {0..1} \<inter> g2 ` {0..1} \<subseteq> {g2 0}" |
|
535 |
using assms |
|
536 |
by (simp_all add: arc_def pathfinish_def pathstart_def path_image_def) |
|
537 |
{ fix x and y::real |
|
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
60420
diff
changeset
|
538 |
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 | 539 |
have g1im: "g1 (2 * y) \<in> g1 ` {0..1} \<inter> g2 ` {0..1}" |
540 |
using xy |
|
541 |
apply simp |
|
542 |
apply (rule_tac x="2 * x - 1" in image_eqI, auto) |
|
543 |
done |
|
544 |
have False |
|
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
60420
diff
changeset
|
545 |
using subsetD [OF sb g1im] xy |
60303 | 546 |
by (auto dest: inj_onD [OF injg2]) |
547 |
} note * = this |
|
548 |
show ?thesis |
|
549 |
apply (simp add: arc_def inj_on_def) |
|
71172 | 550 |
apply (clarsimp simp add: arc_imp_path assms) |
62390 | 551 |
apply (simp add: joinpaths_def split: if_split_asm) |
60303 | 552 |
apply (force dest: inj_onD [OF injg1]) |
553 |
apply (metis *) |
|
554 |
apply (metis *) |
|
555 |
apply (force dest: inj_onD [OF injg2]) |
|
556 |
done |
|
557 |
qed |
|
558 |
||
559 |
lemma reversepath_joinpaths: |
|
560 |
"pathfinish g1 = pathstart g2 \<Longrightarrow> reversepath(g1 +++ g2) = reversepath g2 +++ reversepath g1" |
|
561 |
unfolding reversepath_def pathfinish_def pathstart_def joinpaths_def |
|
562 |
by (rule ext) (auto simp: mult.commute) |
|
563 |
||
564 |
||
70136 | 565 |
subsection\<^marker>\<open>tag unimportant\<close>\<open>Some reversed and "if and only if" versions of joining theorems\<close> |
62533
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
566 |
|
63594
bd218a9320b5
HOL-Multivariate_Analysis: rename theories for more descriptive names
hoelzl
parents:
63540
diff
changeset
|
567 |
lemma path_join_path_ends: |
62533
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
568 |
fixes g1 :: "real \<Rightarrow> 'a::metric_space" |
63594
bd218a9320b5
HOL-Multivariate_Analysis: rename theories for more descriptive names
hoelzl
parents:
63540
diff
changeset
|
569 |
assumes "path(g1 +++ g2)" "path g2" |
62533
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
570 |
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
|
571 |
proof (rule ccontr) |
63040 | 572 |
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
|
573 |
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
|
574 |
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
|
575 |
by auto |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
576 |
then have "e > 0" |
63594
bd218a9320b5
HOL-Multivariate_Analysis: rename theories for more descriptive names
hoelzl
parents:
63540
diff
changeset
|
577 |
by (metis e_def pathfinish_def pathstart_def) |
bd218a9320b5
HOL-Multivariate_Analysis: rename theories for more descriptive names
hoelzl
parents:
63540
diff
changeset
|
578 |
then obtain d1 where "d1 > 0" |
62533
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
579 |
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
|
580 |
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
|
581 |
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
|
582 |
by (metis half_gt_zero_iff norm_conv_dist) |
63594
bd218a9320b5
HOL-Multivariate_Analysis: rename theories for more descriptive names
hoelzl
parents:
63540
diff
changeset
|
583 |
obtain d2 where "d2 > 0" |
bd218a9320b5
HOL-Multivariate_Analysis: rename theories for more descriptive names
hoelzl
parents:
63540
diff
changeset
|
584 |
and d2: "\<And>x'. \<lbrakk>x'\<in>{0..1}; dist x' (1/2) < d2\<rbrakk> |
62533
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
585 |
\<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
|
586 |
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
|
587 |
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
|
588 |
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
|
589 |
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
|
590 |
done |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
591 |
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
|
592 |
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
|
593 |
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
|
594 |
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
|
595 |
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
|
596 |
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
|
597 |
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
|
598 |
using \<open>d1 > 0\<close> \<open>d2 > 0\<close> by (simp add: min_def dist_norm) |
69508 | 599 |
have [simp]: "\<not> min (1 / 2) (min d1 d2) \<le> 0" |
62533
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
600 |
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
|
601 |
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
|
602 |
"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
|
603 |
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
|
604 |
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
|
605 |
using dist_triangle_half_r e_def by blast |
63594
bd218a9320b5
HOL-Multivariate_Analysis: rename theories for more descriptive names
hoelzl
parents:
63540
diff
changeset
|
606 |
then show False |
62533
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
607 |
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
|
608 |
qed |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
609 |
|
63594
bd218a9320b5
HOL-Multivariate_Analysis: rename theories for more descriptive names
hoelzl
parents:
63540
diff
changeset
|
610 |
lemma path_join_eq [simp]: |
62533
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
611 |
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
|
612 |
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
|
613 |
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
|
614 |
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
|
615 |
|
63594
bd218a9320b5
HOL-Multivariate_Analysis: rename theories for more descriptive names
hoelzl
parents:
63540
diff
changeset
|
616 |
lemma simple_path_joinE: |
62533
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
617 |
assumes "simple_path(g1 +++ g2)" and "pathfinish g1 = pathstart g2" |
63594
bd218a9320b5
HOL-Multivariate_Analysis: rename theories for more descriptive names
hoelzl
parents:
63540
diff
changeset
|
618 |
obtains "arc g1" "arc g2" |
62533
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
619 |
"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
|
620 |
proof - |
63594
bd218a9320b5
HOL-Multivariate_Analysis: rename theories for more descriptive names
hoelzl
parents:
63540
diff
changeset
|
621 |
have *: "\<And>x y. \<lbrakk>0 \<le> x; x \<le> 1; 0 \<le> y; y \<le> 1; (g1 +++ g2) x = (g1 +++ g2) y\<rbrakk> |
62533
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
622 |
\<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
|
623 |
using assms by (simp add: simple_path_def) |
63594
bd218a9320b5
HOL-Multivariate_Analysis: rename theories for more descriptive names
hoelzl
parents:
63540
diff
changeset
|
624 |
have "path g1" |
62533
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
625 |
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
|
626 |
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
|
627 |
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
|
628 |
fix x y |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
629 |
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
|
630 |
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
|
631 |
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
|
632 |
qed |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
633 |
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
|
634 |
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
|
635 |
have [simp]: "g2 0 = g1 1" |
63594
bd218a9320b5
HOL-Multivariate_Analysis: rename theories for more descriptive names
hoelzl
parents:
63540
diff
changeset
|
636 |
using assms by (metis pathfinish_def pathstart_def) |
62533
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
637 |
have "path g2" |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
638 |
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
|
639 |
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
|
640 |
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
|
641 |
fix x y |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
642 |
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
|
643 |
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
|
644 |
using * [of "(x + 1) / 2" "(y + 1) / 2"] |
70817
dd675800469d
dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents:
70802
diff
changeset
|
645 |
by (force simp: joinpaths_def split_ifs field_split_simps) |
62533
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
646 |
qed |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
647 |
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
|
648 |
using assms by (simp add: arc_def) |
63594
bd218a9320b5
HOL-Multivariate_Analysis: rename theories for more descriptive names
hoelzl
parents:
63540
diff
changeset
|
649 |
have "g2 y = g1 0 \<or> g2 y = g1 1" |
62533
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
650 |
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
|
651 |
using * [of "x / 2" "(y + 1) / 2"] that |
70817
dd675800469d
dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents:
70802
diff
changeset
|
652 |
by (auto simp: joinpaths_def split_ifs field_split_simps) |
62533
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
653 |
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
|
654 |
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
|
655 |
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
|
656 |
qed |
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 simple_path_join_loop_eq: |
63594
bd218a9320b5
HOL-Multivariate_Analysis: rename theories for more descriptive names
hoelzl
parents:
63540
diff
changeset
|
659 |
assumes "pathfinish g2 = pathstart g1" "pathfinish g1 = pathstart g2" |
62533
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
660 |
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
|
661 |
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
|
662 |
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
|
663 |
|
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
664 |
lemma arc_join_eq: |
63594
bd218a9320b5
HOL-Multivariate_Analysis: rename theories for more descriptive names
hoelzl
parents:
63540
diff
changeset
|
665 |
assumes "pathfinish g1 = pathstart g2" |
62533
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
666 |
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
|
667 |
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
|
668 |
(is "?lhs = ?rhs") |
63594
bd218a9320b5
HOL-Multivariate_Analysis: rename theories for more descriptive names
hoelzl
parents:
63540
diff
changeset
|
669 |
proof |
62533
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
670 |
assume ?lhs |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
671 |
then have "simple_path(g1 +++ g2)" by (rule arc_imp_simple_path) |
63594
bd218a9320b5
HOL-Multivariate_Analysis: rename theories for more descriptive names
hoelzl
parents:
63540
diff
changeset
|
672 |
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> |
62533
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
673 |
\<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
|
674 |
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
|
675 |
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
|
676 |
using * [of 0 "(u + 1) / 2"] that assms arc_distinct_ends [OF \<open>?lhs\<close>] |
70817
dd675800469d
dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents:
70802
diff
changeset
|
677 |
by (auto simp: joinpaths_def pathstart_def pathfinish_def split_ifs field_split_simps) |
69508 | 678 |
then have n1: "pathstart g1 \<notin> path_image g2" |
62533
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
679 |
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
|
680 |
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
|
681 |
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
|
682 |
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
|
683 |
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
|
684 |
next |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
685 |
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
|
686 |
using assms |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
687 |
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
|
688 |
qed |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
689 |
|
63594
bd218a9320b5
HOL-Multivariate_Analysis: rename theories for more descriptive names
hoelzl
parents:
63540
diff
changeset
|
690 |
lemma arc_join_eq_alt: |
62533
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
691 |
"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
|
692 |
\<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
|
693 |
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
|
694 |
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
|
695 |
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
|
696 |
|
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
697 |
|
70136 | 698 |
subsection\<^marker>\<open>tag unimportant\<close>\<open>The joining of paths is associative\<close> |
62533
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
699 |
|
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
700 |
lemma path_assoc: |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
701 |
"\<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
|
702 |
\<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
|
703 |
by simp |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
704 |
|
63594
bd218a9320b5
HOL-Multivariate_Analysis: rename theories for more descriptive names
hoelzl
parents:
63540
diff
changeset
|
705 |
lemma simple_path_assoc: |
bd218a9320b5
HOL-Multivariate_Analysis: rename theories for more descriptive names
hoelzl
parents:
63540
diff
changeset
|
706 |
assumes "pathfinish p = pathstart q" "pathfinish q = pathstart r" |
62533
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
707 |
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
|
708 |
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
|
709 |
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
|
710 |
proof |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
711 |
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
|
712 |
with assms True show "simple_path ((p +++ q) +++ r)" |
63594
bd218a9320b5
HOL-Multivariate_Analysis: rename theories for more descriptive names
hoelzl
parents:
63540
diff
changeset
|
713 |
by (fastforce simp add: simple_path_join_loop_eq arc_join_eq path_image_join |
62533
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
714 |
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
|
715 |
next |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
716 |
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
|
717 |
with assms True have q: "pathfinish r \<notin> path_image q" |
63594
bd218a9320b5
HOL-Multivariate_Analysis: rename theories for more descriptive names
hoelzl
parents:
63540
diff
changeset
|
718 |
using arc_distinct_ends |
62533
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
719 |
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
|
720 |
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
|
721 |
using assms |
63594
bd218a9320b5
HOL-Multivariate_Analysis: rename theories for more descriptive names
hoelzl
parents:
63540
diff
changeset
|
722 |
by (metis 0 IntI arc_distinct_ends arc_join_eq_alt empty_iff insert_iff |
62533
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
723 |
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
|
724 |
with assms 0 q True show "simple_path (p +++ q +++ r)" |
63594
bd218a9320b5
HOL-Multivariate_Analysis: rename theories for more descriptive names
hoelzl
parents:
63540
diff
changeset
|
725 |
by (auto simp: simple_path_join_loop_eq arc_join_eq path_image_join |
62533
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
726 |
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
|
727 |
qed |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
728 |
next |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
729 |
case False |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
730 |
{ fix x :: 'a |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
731 |
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
|
732 |
"(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
|
733 |
"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
|
734 |
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
|
735 |
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
|
736 |
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
|
737 |
by blast |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
738 |
} |
63594
bd218a9320b5
HOL-Multivariate_Analysis: rename theories for more descriptive names
hoelzl
parents:
63540
diff
changeset
|
739 |
with False assms show ?thesis |
62533
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
740 |
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
|
741 |
qed |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
742 |
|
63594
bd218a9320b5
HOL-Multivariate_Analysis: rename theories for more descriptive names
hoelzl
parents:
63540
diff
changeset
|
743 |
lemma arc_assoc: |
62533
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
744 |
"\<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
|
745 |
\<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
|
746 |
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
|
747 |
|
70136 | 748 |
subsubsection\<^marker>\<open>tag unimportant\<close>\<open>Symmetry and loops\<close> |
62620
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
749 |
|
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
750 |
lemma path_sym: |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
751 |
"\<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
|
752 |
by auto |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
753 |
|
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
754 |
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
|
755 |
"\<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
|
756 |
\<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
|
757 |
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
|
758 |
|
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
759 |
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
|
760 |
"\<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
|
761 |
\<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
|
762 |
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
|
763 |
|
62533
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
764 |
|
69518 | 765 |
subsection\<open>Subpath\<close> |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
60420
diff
changeset
|
766 |
|
70136 | 767 |
definition\<^marker>\<open>tag important\<close> subpath :: "real \<Rightarrow> real \<Rightarrow> (real \<Rightarrow> 'a) \<Rightarrow> real \<Rightarrow> 'a::real_normed_vector" |
60303 | 768 |
where "subpath a b g \<equiv> \<lambda>x. g((b - a) * x + a)" |
769 |
||
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
|
770 |
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
|
771 |
fixes g :: "_ \<Rightarrow> 'a::real_normed_vector" |
60303 | 772 |
shows "path_image(subpath u v g) = g ` (closed_segment u v)" |
69661 | 773 |
by (auto simp add: closed_segment_real_eq path_image_def subpath_def) |
60303 | 774 |
|
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 |
lemma path_image_subpath: |
60303 | 776 |
fixes g :: "real \<Rightarrow> 'a::real_normed_vector" |
777 |
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
|
778 |
by (simp add: path_image_subpath_gen closed_segment_eq_real_ivl) |
60303 | 779 |
|
65038
9391ea7daa17
new lemmas about segments, etc. Also recast some theorems to use Union rather than general set comprehensions
paulson <lp15@cam.ac.uk>
parents:
64911
diff
changeset
|
780 |
lemma path_image_subpath_commute: |
9391ea7daa17
new lemmas about segments, etc. Also recast some theorems to use Union rather than general set comprehensions
paulson <lp15@cam.ac.uk>
parents:
64911
diff
changeset
|
781 |
fixes g :: "real \<Rightarrow> 'a::real_normed_vector" |
9391ea7daa17
new lemmas about segments, etc. Also recast some theorems to use Union rather than general set comprehensions
paulson <lp15@cam.ac.uk>
parents:
64911
diff
changeset
|
782 |
shows "path_image(subpath u v g) = path_image(subpath v u g)" |
9391ea7daa17
new lemmas about segments, etc. Also recast some theorems to use Union rather than general set comprehensions
paulson <lp15@cam.ac.uk>
parents:
64911
diff
changeset
|
783 |
by (simp add: path_image_subpath_gen closed_segment_eq_real_ivl) |
9391ea7daa17
new lemmas about segments, etc. Also recast some theorems to use Union rather than general set comprehensions
paulson <lp15@cam.ac.uk>
parents:
64911
diff
changeset
|
784 |
|
60303 | 785 |
lemma path_subpath [simp]: |
786 |
fixes g :: "real \<Rightarrow> 'a::real_normed_vector" |
|
787 |
assumes "path g" "u \<in> {0..1}" "v \<in> {0..1}" |
|
788 |
shows "path(subpath u v g)" |
|
789 |
proof - |
|
68096 | 790 |
have "continuous_on {0..1} (g \<circ> (\<lambda>x. ((v-u) * x+ u)))" |
60303 | 791 |
apply (rule continuous_intros | simp)+ |
792 |
apply (simp add: image_affinity_atLeastAtMost [where c=u]) |
|
793 |
using assms |
|
794 |
apply (auto simp: path_def continuous_on_subset) |
|
795 |
done |
|
796 |
then show ?thesis |
|
797 |
by (simp add: path_def subpath_def) |
|
49653 | 798 |
qed |
36583 | 799 |
|
60303 | 800 |
lemma pathstart_subpath [simp]: "pathstart(subpath u v g) = g(u)" |
801 |
by (simp add: pathstart_def subpath_def) |
|
802 |
||
803 |
lemma pathfinish_subpath [simp]: "pathfinish(subpath u v g) = g(v)" |
|
804 |
by (simp add: pathfinish_def subpath_def) |
|
805 |
||
806 |
lemma subpath_trivial [simp]: "subpath 0 1 g = g" |
|
807 |
by (simp add: subpath_def) |
|
808 |
||
809 |
lemma subpath_reversepath: "subpath 1 0 g = reversepath g" |
|
810 |
by (simp add: reversepath_def subpath_def) |
|
811 |
||
812 |
lemma reversepath_subpath: "reversepath(subpath u v g) = subpath v u g" |
|
813 |
by (simp add: reversepath_def subpath_def algebra_simps) |
|
814 |
||
68096 | 815 |
lemma subpath_translation: "subpath u v ((\<lambda>x. a + x) \<circ> g) = (\<lambda>x. a + x) \<circ> subpath u v g" |
60303 | 816 |
by (rule ext) (simp add: subpath_def) |
817 |
||
70971 | 818 |
lemma subpath_image: "subpath u v (f \<circ> g) = f \<circ> subpath u v g" |
60303 | 819 |
by (rule ext) (simp add: subpath_def) |
820 |
||
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
60420
diff
changeset
|
821 |
lemma affine_ineq: |
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
60420
diff
changeset
|
822 |
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
|
823 |
assumes "x \<le> 1" "v \<le> u" |
60303 | 824 |
shows "v + x * u \<le> u + x * v" |
825 |
proof - |
|
826 |
have "(1-x)*(u-v) \<ge> 0" |
|
827 |
using assms by auto |
|
828 |
then show ?thesis |
|
829 |
by (simp add: algebra_simps) |
|
49653 | 830 |
qed |
36583 | 831 |
|
61711
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
832 |
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
|
833 |
fixes a::real shows "\<lbrakk>a \<le> 1; b \<le> 1\<rbrakk> \<Longrightarrow> a + b \<le> 1 + a * b" |
71172 | 834 |
by (metis add.commute affine_ineq mult.right_neutral) |
61711
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
835 |
|
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
60420
diff
changeset
|
836 |
lemma simple_path_subpath_eq: |
60303 | 837 |
"simple_path(subpath u v g) \<longleftrightarrow> |
838 |
path(subpath u v g) \<and> u\<noteq>v \<and> |
|
839 |
(\<forall>x y. x \<in> closed_segment u v \<and> y \<in> closed_segment u v \<and> g x = g y |
|
840 |
\<longrightarrow> x = y \<or> x = u \<and> y = v \<or> x = v \<and> y = u)" |
|
841 |
(is "?lhs = ?rhs") |
|
842 |
proof (rule iffI) |
|
843 |
assume ?lhs |
|
844 |
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
|
845 |
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 | 846 |
\<Longrightarrow> x = y \<or> x = 0 \<and> y = 1 \<or> x = 1 \<and> y = 0)" |
847 |
by (auto simp: simple_path_def subpath_def) |
|
848 |
{ fix x y |
|
849 |
assume "x \<in> closed_segment u v" "y \<in> closed_segment u v" "g x = g y" |
|
850 |
then have "x = y \<or> x = u \<and> y = v \<or> x = v \<and> y = u" |
|
70817
dd675800469d
dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents:
70802
diff
changeset
|
851 |
using sim [of "(x-u)/(v-u)" "(y-u)/(v-u)"] p |
dd675800469d
dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents:
70802
diff
changeset
|
852 |
by (auto split: if_split_asm simp add: closed_segment_real_eq image_affinity_atLeastAtMost) |
dd675800469d
dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents:
70802
diff
changeset
|
853 |
(simp_all add: field_split_simps) |
60303 | 854 |
} moreover |
855 |
have "path(subpath u v g) \<and> u\<noteq>v" |
|
856 |
using sim [of "1/3" "2/3"] p |
|
857 |
by (auto simp: subpath_def) |
|
858 |
ultimately show ?rhs |
|
859 |
by metis |
|
860 |
next |
|
861 |
assume ?rhs |
|
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
60420
diff
changeset
|
862 |
then |
60303 | 863 |
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" |
864 |
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" |
|
865 |
and ne: "u < v \<or> v < u" |
|
866 |
and psp: "path (subpath u v g)" |
|
867 |
by (auto simp: closed_segment_real_eq image_affinity_atLeastAtMost) |
|
868 |
have [simp]: "\<And>x. u + x * v = v + x * u \<longleftrightarrow> u=v \<or> x=1" |
|
869 |
by algebra |
|
870 |
show ?lhs using psp ne |
|
871 |
unfolding simple_path_def subpath_def |
|
872 |
by (fastforce simp add: algebra_simps affine_ineq mult_left_mono crossproduct_eq dest: d1 d2) |
|
873 |
qed |
|
874 |
||
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
60420
diff
changeset
|
875 |
lemma arc_subpath_eq: |
60303 | 876 |
"arc(subpath u v g) \<longleftrightarrow> path(subpath u v g) \<and> u\<noteq>v \<and> inj_on g (closed_segment u v)" |
877 |
(is "?lhs = ?rhs") |
|
878 |
proof (rule iffI) |
|
879 |
assume ?lhs |
|
880 |
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
|
881 |
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 | 882 |
\<Longrightarrow> x = y)" |
883 |
by (auto simp: arc_def inj_on_def subpath_def) |
|
884 |
{ fix x y |
|
885 |
assume "x \<in> closed_segment u v" "y \<in> closed_segment u v" "g x = g y" |
|
886 |
then have "x = y" |
|
70817
dd675800469d
dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents:
70802
diff
changeset
|
887 |
using sim [of "(x-u)/(v-u)" "(y-u)/(v-u)"] p |
dd675800469d
dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents:
70802
diff
changeset
|
888 |
by (cases "v = u") |
dd675800469d
dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents:
70802
diff
changeset
|
889 |
(simp_all split: if_split_asm add: inj_on_def closed_segment_real_eq image_affinity_atLeastAtMost, |
dd675800469d
dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents:
70802
diff
changeset
|
890 |
simp add: field_simps) |
60303 | 891 |
} moreover |
892 |
have "path(subpath u v g) \<and> u\<noteq>v" |
|
893 |
using sim [of "1/3" "2/3"] p |
|
894 |
by (auto simp: subpath_def) |
|
895 |
ultimately show ?rhs |
|
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
60420
diff
changeset
|
896 |
unfolding inj_on_def |
60303 | 897 |
by metis |
898 |
next |
|
899 |
assume ?rhs |
|
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
60420
diff
changeset
|
900 |
then |
60303 | 901 |
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" |
902 |
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" |
|
903 |
and ne: "u < v \<or> v < u" |
|
904 |
and psp: "path (subpath u v g)" |
|
905 |
by (auto simp: inj_on_def closed_segment_real_eq image_affinity_atLeastAtMost) |
|
906 |
show ?lhs using psp ne |
|
907 |
unfolding arc_def subpath_def inj_on_def |
|
908 |
by (auto simp: algebra_simps affine_ineq mult_left_mono crossproduct_eq dest: d1 d2) |
|
909 |
qed |
|
910 |
||
911 |
||
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
60420
diff
changeset
|
912 |
lemma simple_path_subpath: |
60303 | 913 |
assumes "simple_path g" "u \<in> {0..1}" "v \<in> {0..1}" "u \<noteq> v" |
914 |
shows "simple_path(subpath u v g)" |
|
915 |
using assms |
|
916 |
apply (simp add: simple_path_subpath_eq simple_path_imp_path) |
|
917 |
apply (simp add: simple_path_def closed_segment_real_eq image_affinity_atLeastAtMost, fastforce) |
|
918 |
done |
|
919 |
||
920 |
lemma arc_simple_path_subpath: |
|
921 |
"\<lbrakk>simple_path g; u \<in> {0..1}; v \<in> {0..1}; g u \<noteq> g v\<rbrakk> \<Longrightarrow> arc(subpath u v g)" |
|
922 |
by (force intro: simple_path_subpath simple_path_imp_arc) |
|
923 |
||
924 |
lemma arc_subpath_arc: |
|
925 |
"\<lbrakk>arc g; u \<in> {0..1}; v \<in> {0..1}; u \<noteq> v\<rbrakk> \<Longrightarrow> arc(subpath u v g)" |
|
926 |
by (meson arc_def arc_imp_simple_path arc_simple_path_subpath inj_onD) |
|
927 |
||
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
60420
diff
changeset
|
928 |
lemma arc_simple_path_subpath_interior: |
60303 | 929 |
"\<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)" |
930 |
apply (rule arc_simple_path_subpath) |
|
931 |
apply (force simp: simple_path_def)+ |
|
932 |
done |
|
933 |
||
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
60420
diff
changeset
|
934 |
lemma path_image_subpath_subset: |
68532
f8b98d31ad45
Incorporating new/strengthened proofs from Library and AFP entries
paulson <lp15@cam.ac.uk>
parents:
68310
diff
changeset
|
935 |
"\<lbrakk>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
|
936 |
apply (simp add: closed_segment_real_eq image_affinity_atLeastAtMost path_image_subpath) |
60303 | 937 |
apply (auto simp: path_image_def) |
68532
f8b98d31ad45
Incorporating new/strengthened proofs from Library and AFP entries
paulson <lp15@cam.ac.uk>
parents:
68310
diff
changeset
|
938 |
done |
60303 | 939 |
|
940 |
lemma join_subpaths_middle: "subpath (0) ((1 / 2)) p +++ subpath ((1 / 2)) 1 p = p" |
|
70817
dd675800469d
dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents:
70802
diff
changeset
|
941 |
by (rule ext) (simp add: joinpaths_def subpath_def field_split_simps) |
53640 | 942 |
|
69514 | 943 |
|
70136 | 944 |
subsection\<^marker>\<open>tag unimportant\<close>\<open>There is a subpath to the frontier\<close> |
61518
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
945 |
|
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
946 |
lemma subpath_to_frontier_explicit: |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
947 |
fixes S :: "'a::metric_space set" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
948 |
assumes g: "path g" and "pathfinish g \<notin> S" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
949 |
obtains u where "0 \<le> u" "u \<le> 1" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
950 |
"\<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
|
951 |
"(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
|
952 |
proof - |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
953 |
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
|
954 |
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
|
955 |
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
|
956 |
using compact_eq_bounded_closed apply fastforce |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
957 |
done |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
958 |
have "1 \<in> {u. g u \<in> closure (- S)}" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
959 |
using assms by (simp add: pathfinish_def closure_def) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
960 |
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
|
961 |
using atLeastAtMost_iff zero_le_one by blast |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
962 |
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
|
963 |
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
|
964 |
using compact_attains_inf [OF com dis] by fastforce |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
965 |
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
|
966 |
using closure_def by fastforce |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
967 |
{ assume "u \<noteq> 0" |
61808 | 968 |
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
|
969 |
{ 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
|
970 |
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
|
971 |
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
|
972 |
have *: "dist (max 0 (u - d / 2)) u \<le> d" |
61808 | 973 |
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
|
974 |
have "\<exists>y\<in>S. dist y (g u) < e" |
61808 | 975 |
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
|
976 |
by (force intro: d [OF _ *] umin') |
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 |
then have "g u \<in> closure S" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
979 |
by (simp add: frontier_def closure_approachable) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
980 |
} |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
981 |
then show ?thesis |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
982 |
apply (rule_tac u=u in that) |
61808 | 983 |
apply (auto simp: \<open>0 \<le> u\<close> \<open>u \<le> 1\<close> gu interior_closure umin) |
984 |
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
|
985 |
done |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
986 |
qed |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
987 |
|
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
988 |
lemma subpath_to_frontier_strong: |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
989 |
assumes g: "path g" and "pathfinish g \<notin> S" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
990 |
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
|
991 |
"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
|
992 |
proof - |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
993 |
obtain u where "0 \<le> u" "u \<le> 1" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
994 |
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
|
995 |
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
|
996 |
using subpath_to_frontier_explicit [OF assms] by blast |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
997 |
show ?thesis |
61808 | 998 |
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
|
999 |
apply (simp add: gunot) |
61808 | 1000 |
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
|
1001 |
qed |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
1002 |
|
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
1003 |
lemma subpath_to_frontier: |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
1004 |
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
|
1005 |
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
|
1006 |
proof - |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
1007 |
obtain u where "0 \<le> u" "u \<le> 1" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
1008 |
and notin: "g u \<notin> interior S" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
1009 |
and disj: "u = 0 \<or> |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
1010 |
(\<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
|
1011 |
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
|
1012 |
show ?thesis |
61808 | 1013 |
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
|
1014 |
apply (metis DiffI disj frontier_def g0 notin pathstart_def) |
61808 | 1015 |
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
|
1016 |
apply (simp add: path_image_subpath_gen) |
61518
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
1017 |
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
|
1018 |
apply (rename_tac y) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
1019 |
apply (drule_tac x="y/u" in spec) |
62390 | 1020 |
apply (auto split: if_split_asm) |
61518
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
1021 |
done |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
1022 |
qed |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
1023 |
|
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
1024 |
lemma exists_path_subpath_to_frontier: |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
1025 |
fixes S :: "'a::real_normed_vector set" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
1026 |
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
|
1027 |
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
|
1028 |
"path_image h - {pathfinish h} \<subseteq> interior S" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
1029 |
"pathfinish h \<in> frontier S" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
1030 |
proof - |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
1031 |
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
|
1032 |
using subpath_to_frontier [OF assms] by blast |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
1033 |
show ?thesis |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
1034 |
apply (rule that [of "subpath 0 u g"]) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
1035 |
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
|
1036 |
apply (simp_all add: path_image_subpath) |
61518
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
1037 |
apply (simp add: pathstart_def) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
1038 |
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
|
1039 |
done |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
1040 |
qed |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
1041 |
|
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
1042 |
lemma exists_path_subpath_to_frontier_closed: |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
1043 |
fixes S :: "'a::real_normed_vector set" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
1044 |
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
|
1045 |
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
|
1046 |
"pathfinish h \<in> frontier S" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
1047 |
proof - |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
1048 |
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
|
1049 |
"path_image h - {pathfinish h} \<subseteq> interior S" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
1050 |
"pathfinish h \<in> frontier S" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
1051 |
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
|
1052 |
show ?thesis |
61808 | 1053 |
apply (rule that [OF \<open>path h\<close>]) |
61518
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
1054 |
using assms h |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
1055 |
apply auto |
62087
44841d07ef1d
revisions to limits and derivatives, plus new lemmas
paulson
parents:
61808
diff
changeset
|
1056 |
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
|
1057 |
done |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
1058 |
qed |
49653 | 1059 |
|
69514 | 1060 |
|
1061 |
subsection \<open>Shift Path to Start at Some Given Point\<close> |
|
36583 | 1062 |
|
70136 | 1063 |
definition\<^marker>\<open>tag important\<close> shiftpath :: "real \<Rightarrow> (real \<Rightarrow> 'a::topological_space) \<Rightarrow> real \<Rightarrow> 'a" |
53640 | 1064 |
where "shiftpath a f = (\<lambda>x. if (a + x) \<le> 1 then f (a + x) else f (a + x - 1))" |
36583 | 1065 |
|
71189
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71172
diff
changeset
|
1066 |
lemma shiftpath_alt_def: "shiftpath a f = (\<lambda>x. if x \<le> 1-a then f (a + x) else f (a + x - 1))" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71172
diff
changeset
|
1067 |
by (auto simp: shiftpath_def) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71172
diff
changeset
|
1068 |
|
53640 | 1069 |
lemma pathstart_shiftpath: "a \<le> 1 \<Longrightarrow> pathstart (shiftpath a g) = g a" |
36583 | 1070 |
unfolding pathstart_def shiftpath_def by auto |
1071 |
||
49653 | 1072 |
lemma pathfinish_shiftpath: |
53640 | 1073 |
assumes "0 \<le> a" |
1074 |
and "pathfinish g = pathstart g" |
|
1075 |
shows "pathfinish (shiftpath a g) = g a" |
|
1076 |
using assms |
|
1077 |
unfolding pathstart_def pathfinish_def shiftpath_def |
|
36583 | 1078 |
by auto |
1079 |
||
1080 |
lemma endpoints_shiftpath: |
|
53640 | 1081 |
assumes "pathfinish g = pathstart g" |
1082 |
and "a \<in> {0 .. 1}" |
|
1083 |
shows "pathfinish (shiftpath a g) = g a" |
|
1084 |
and "pathstart (shiftpath a g) = g a" |
|
1085 |
using assms |
|
1086 |
by (auto intro!: pathfinish_shiftpath pathstart_shiftpath) |
|
36583 | 1087 |
|
1088 |
lemma closed_shiftpath: |
|
53640 | 1089 |
assumes "pathfinish g = pathstart g" |
1090 |
and "a \<in> {0..1}" |
|
1091 |
shows "pathfinish (shiftpath a g) = pathstart (shiftpath a g)" |
|
1092 |
using endpoints_shiftpath[OF assms] |
|
1093 |
by auto |
|
36583 | 1094 |
|
1095 |
lemma path_shiftpath: |
|
53640 | 1096 |
assumes "path g" |
1097 |
and "pathfinish g = pathstart g" |
|
1098 |
and "a \<in> {0..1}" |
|
1099 |
shows "path (shiftpath a g)" |
|
49653 | 1100 |
proof - |
53640 | 1101 |
have *: "{0 .. 1} = {0 .. 1-a} \<union> {1-a .. 1}" |
1102 |
using assms(3) by auto |
|
49653 | 1103 |
have **: "\<And>x. x + a = 1 \<Longrightarrow> g (x + a - 1) = g (x + a)" |
53640 | 1104 |
using assms(2)[unfolded pathfinish_def pathstart_def] |
1105 |
by auto |
|
49653 | 1106 |
show ?thesis |
1107 |
unfolding path_def shiftpath_def * |
|
68096 | 1108 |
proof (rule continuous_on_closed_Un) |
1109 |
have contg: "continuous_on {0..1} g" |
|
1110 |
using \<open>path g\<close> path_def by blast |
|
1111 |
show "continuous_on {0..1-a} (\<lambda>x. if a + x \<le> 1 then g (a + x) else g (a + x - 1))" |
|
1112 |
proof (rule continuous_on_eq) |
|
1113 |
show "continuous_on {0..1-a} (g \<circ> (+) a)" |
|
1114 |
by (intro continuous_intros continuous_on_subset [OF contg]) (use \<open>a \<in> {0..1}\<close> in auto) |
|
1115 |
qed auto |
|
1116 |
show "continuous_on {1-a..1} (\<lambda>x. if a + x \<le> 1 then g (a + x) else g (a + x - 1))" |
|
1117 |
proof (rule continuous_on_eq) |
|
1118 |
show "continuous_on {1-a..1} (g \<circ> (+) (a - 1))" |
|
1119 |
by (intro continuous_intros continuous_on_subset [OF contg]) (use \<open>a \<in> {0..1}\<close> in auto) |
|
1120 |
qed (auto simp: "**" add.commute add_diff_eq) |
|
1121 |
qed auto |
|
49653 | 1122 |
qed |
36583 | 1123 |
|
49653 | 1124 |
lemma shiftpath_shiftpath: |
53640 | 1125 |
assumes "pathfinish g = pathstart g" |
1126 |
and "a \<in> {0..1}" |
|
1127 |
and "x \<in> {0..1}" |
|
36583 | 1128 |
shows "shiftpath (1 - a) (shiftpath a g) x = g x" |
53640 | 1129 |
using assms |
1130 |
unfolding pathfinish_def pathstart_def shiftpath_def |
|
1131 |
by auto |
|
36583 | 1132 |
|
1133 |
lemma path_image_shiftpath: |
|
68096 | 1134 |
assumes a: "a \<in> {0..1}" |
53640 | 1135 |
and "pathfinish g = pathstart g" |
1136 |
shows "path_image (shiftpath a g) = path_image g" |
|
49653 | 1137 |
proof - |
1138 |
{ fix x |
|
68096 | 1139 |
assume g: "g 1 = g 0" "x \<in> {0..1::real}" and gne: "\<And>y. y\<in>{0..1} \<inter> {x. \<not> a + x \<le> 1} \<Longrightarrow> g x \<noteq> g (a + y - 1)" |
49654 | 1140 |
then have "\<exists>y\<in>{0..1} \<inter> {x. a + x \<le> 1}. g x = g (a + y)" |
49653 | 1141 |
proof (cases "a \<le> x") |
1142 |
case False |
|
49654 | 1143 |
then show ?thesis |
49653 | 1144 |
apply (rule_tac x="1 + x - a" in bexI) |
68096 | 1145 |
using g gne[of "1 + x - a"] a |
1146 |
apply (force simp: field_simps)+ |
|
49653 | 1147 |
done |
1148 |
next |
|
1149 |
case True |
|
53640 | 1150 |
then show ?thesis |
68096 | 1151 |
using g a by (rule_tac x="x - a" in bexI) (auto simp: field_simps) |
49653 | 1152 |
qed |
1153 |
} |
|
49654 | 1154 |
then show ?thesis |
53640 | 1155 |
using assms |
1156 |
unfolding shiftpath_def path_image_def pathfinish_def pathstart_def |
|
68096 | 1157 |
by (auto simp: image_iff) |
49653 | 1158 |
qed |
1159 |
||
64788
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
1160 |
lemma simple_path_shiftpath: |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
1161 |
assumes "simple_path g" "pathfinish g = pathstart g" and a: "0 \<le> a" "a \<le> 1" |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
1162 |
shows "simple_path (shiftpath a g)" |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
1163 |
unfolding simple_path_def |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
1164 |
proof (intro conjI impI ballI) |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
1165 |
show "path (shiftpath a g)" |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
1166 |
by (simp add: assms path_shiftpath simple_path_imp_path) |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
1167 |
have *: "\<And>x y. \<lbrakk>g x = g y; x \<in> {0..1}; y \<in> {0..1}\<rbrakk> \<Longrightarrow> x = y \<or> x = 0 \<and> y = 1 \<or> x = 1 \<and> y = 0" |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
1168 |
using assms by (simp add: simple_path_def) |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
1169 |
show "x = y \<or> x = 0 \<and> y = 1 \<or> x = 1 \<and> y = 0" |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
1170 |
if "x \<in> {0..1}" "y \<in> {0..1}" "shiftpath a g x = shiftpath a g y" for x y |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
1171 |
using that a unfolding shiftpath_def |
68096 | 1172 |
by (force split: if_split_asm dest!: *) |
64788
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
1173 |
qed |
36583 | 1174 |
|
69514 | 1175 |
|
1176 |
subsection \<open>Straight-Line Paths\<close> |
|
36583 | 1177 |
|
70136 | 1178 |
definition\<^marker>\<open>tag important\<close> linepath :: "'a::real_normed_vector \<Rightarrow> 'a \<Rightarrow> real \<Rightarrow> 'a" |
49653 | 1179 |
where "linepath a b = (\<lambda>x. (1 - x) *\<^sub>R a + x *\<^sub>R b)" |
36583 | 1180 |
|
53640 | 1181 |
lemma pathstart_linepath[simp]: "pathstart (linepath a b) = a" |
1182 |
unfolding pathstart_def linepath_def |
|
1183 |
by auto |
|
36583 | 1184 |
|
53640 | 1185 |
lemma pathfinish_linepath[simp]: "pathfinish (linepath a b) = b" |
1186 |
unfolding pathfinish_def linepath_def |
|
1187 |
by auto |
|
36583 | 1188 |
|
68721 | 1189 |
lemma linepath_inner: "linepath a b x \<bullet> v = linepath (a \<bullet> v) (b \<bullet> v) x" |
1190 |
by (simp add: linepath_def algebra_simps) |
|
1191 |
||
1192 |
lemma Re_linepath': "Re (linepath a b x) = linepath (Re a) (Re b) x" |
|
1193 |
by (simp add: linepath_def) |
|
1194 |
||
1195 |
lemma Im_linepath': "Im (linepath a b x) = linepath (Im a) (Im b) x" |
|
1196 |
by (simp add: linepath_def) |
|
1197 |
||
1198 |
lemma linepath_0': "linepath a b 0 = a" |
|
1199 |
by (simp add: linepath_def) |
|
1200 |
||
1201 |
lemma linepath_1': "linepath a b 1 = b" |
|
1202 |
by (simp add: linepath_def) |
|
1203 |
||
36583 | 1204 |
lemma continuous_linepath_at[intro]: "continuous (at x) (linepath a b)" |
53640 | 1205 |
unfolding linepath_def |
1206 |
by (intro continuous_intros) |
|
36583 | 1207 |
|
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
|
1208 |
lemma continuous_on_linepath [intro,continuous_intros]: "continuous_on s (linepath a b)" |
53640 | 1209 |
using continuous_linepath_at |
1210 |
by (auto intro!: continuous_at_imp_continuous_on) |
|
36583 | 1211 |
|
62618
f7f2467ab854
Refactoring (moving theorems into better locations), plus a bit of new material
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
1212 |
lemma path_linepath[iff]: "path (linepath a b)" |
53640 | 1213 |
unfolding path_def |
1214 |
by (rule continuous_on_linepath) |
|
36583 | 1215 |
|
53640 | 1216 |
lemma path_image_linepath[simp]: "path_image (linepath a b) = closed_segment a b" |
49653 | 1217 |
unfolding path_image_def segment linepath_def |
60303 | 1218 |
by auto |
49653 | 1219 |
|
53640 | 1220 |
lemma reversepath_linepath[simp]: "reversepath (linepath a b) = linepath b a" |
49653 | 1221 |
unfolding reversepath_def linepath_def |
36583 | 1222 |
by auto |
1223 |
||
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
|
1224 |
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
|
1225 |
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
|
1226 |
|
68721 | 1227 |
lemma linepath_cnj: "cnj (linepath a b x) = linepath (cnj a) (cnj b) x" |
1228 |
by (simp add: linepath_def) |
|
1229 |
||
60303 | 1230 |
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
|
1231 |
assumes "a \<noteq> b" shows [simp]: "arc (linepath a b)" |
36583 | 1232 |
proof - |
53640 | 1233 |
{ |
1234 |
fix x y :: "real" |
|
36583 | 1235 |
assume "x *\<^sub>R b + y *\<^sub>R a = x *\<^sub>R a + y *\<^sub>R b" |
53640 | 1236 |
then have "(x - y) *\<^sub>R a = (x - y) *\<^sub>R b" |
1237 |
by (simp add: algebra_simps) |
|
1238 |
with assms have "x = y" |
|
1239 |
by simp |
|
1240 |
} |
|
49654 | 1241 |
then show ?thesis |
60809
457abb82fb9e
the Cauchy integral theorem and related material
paulson <lp15@cam.ac.uk>
parents:
60420
diff
changeset
|
1242 |
unfolding arc_def inj_on_def |
68096 | 1243 |
by (fastforce simp: algebra_simps linepath_def) |
49653 | 1244 |
qed |
36583 | 1245 |
|
53640 | 1246 |
lemma simple_path_linepath[intro]: "a \<noteq> b \<Longrightarrow> simple_path (linepath a b)" |
68096 | 1247 |
by (simp add: arc_imp_simple_path) |
49653 | 1248 |
|
61711
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
1249 |
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
|
1250 |
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
|
1251 |
|
64394 | 1252 |
lemma linepath_refl: "linepath a a = (\<lambda>x. a)" |
1253 |
by auto |
|
1254 |
||
61711
21d7910d6816
Theory of homotopic paths (from HOL Light), plus comments and minor refinements
paulson <lp15@cam.ac.uk>
parents:
61699
diff
changeset
|
1255 |
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
|
1256 |
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
|
1257 |
|
62618
f7f2467ab854
Refactoring (moving theorems into better locations), plus a bit of new material
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
1258 |
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
|
1259 |
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
|
1260 |
|
f7f2467ab854
Refactoring (moving theorems into better locations), plus a bit of new material
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
1261 |
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
|
1262 |
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
|
1263 |
|
63881
b746b19197bd
lots of new results about topology, affine dimension etc
paulson <lp15@cam.ac.uk>
parents:
63627
diff
changeset
|
1264 |
lemma inj_on_linepath: |
b746b19197bd
lots of new results about topology, affine dimension etc
paulson <lp15@cam.ac.uk>
parents:
63627
diff
changeset
|
1265 |
assumes "a \<noteq> b" shows "inj_on (linepath a b) {0..1}" |
b746b19197bd
lots of new results about topology, affine dimension etc
paulson <lp15@cam.ac.uk>
parents:
63627
diff
changeset
|
1266 |
proof (clarsimp simp: inj_on_def linepath_def) |
b746b19197bd
lots of new results about topology, affine dimension etc
paulson <lp15@cam.ac.uk>
parents:
63627
diff
changeset
|
1267 |
fix x y |
b746b19197bd
lots of new results about topology, affine dimension etc
paulson <lp15@cam.ac.uk>
parents:
63627
diff
changeset
|
1268 |
assume "(1 - x) *\<^sub>R a + x *\<^sub>R b = (1 - y) *\<^sub>R a + y *\<^sub>R b" "0 \<le> x" "x \<le> 1" "0 \<le> y" "y \<le> 1" |
b746b19197bd
lots of new results about topology, affine dimension etc
paulson <lp15@cam.ac.uk>
parents:
63627
diff
changeset
|
1269 |
then have "x *\<^sub>R (a - b) = y *\<^sub>R (a - b)" |
b746b19197bd
lots of new results about topology, affine dimension etc
paulson <lp15@cam.ac.uk>
parents:
63627
diff
changeset
|
1270 |
by (auto simp: algebra_simps) |
b746b19197bd
lots of new results about topology, affine dimension etc
paulson <lp15@cam.ac.uk>
parents:
63627
diff
changeset
|
1271 |
then show "x=y" |
b746b19197bd
lots of new results about topology, affine dimension etc
paulson <lp15@cam.ac.uk>
parents:
63627
diff
changeset
|
1272 |
using assms by auto |
b746b19197bd
lots of new results about topology, affine dimension etc
paulson <lp15@cam.ac.uk>
parents:
63627
diff
changeset
|
1273 |
qed |
b746b19197bd
lots of new results about topology, affine dimension etc
paulson <lp15@cam.ac.uk>
parents:
63627
diff
changeset
|
1274 |
|
69144
f13b82281715
new theory Abstract_Topology with lots of stuff from HOL Light's metric.sml
paulson <lp15@cam.ac.uk>
parents:
69064
diff
changeset
|
1275 |
lemma linepath_le_1: |
f13b82281715
new theory Abstract_Topology with lots of stuff from HOL Light's metric.sml
paulson <lp15@cam.ac.uk>
parents:
69064
diff
changeset
|
1276 |
fixes a::"'a::linordered_idom" shows "\<lbrakk>a \<le> 1; b \<le> 1; 0 \<le> u; u \<le> 1\<rbrakk> \<Longrightarrow> (1 - u) * a + u * b \<le> 1" |
f13b82281715
new theory Abstract_Topology with lots of stuff from HOL Light's metric.sml
paulson <lp15@cam.ac.uk>
parents:
69064
diff
changeset
|
1277 |
using mult_left_le [of a "1-u"] mult_left_le [of b u] by auto |
f13b82281715
new theory Abstract_Topology with lots of stuff from HOL Light's metric.sml
paulson <lp15@cam.ac.uk>
parents:
69064
diff
changeset
|
1278 |
|
71189
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71172
diff
changeset
|
1279 |
lemma linepath_in_path: |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71172
diff
changeset
|
1280 |
shows "x \<in> {0..1} \<Longrightarrow> linepath a b x \<in> closed_segment a b" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71172
diff
changeset
|
1281 |
by (auto simp: segment linepath_def) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71172
diff
changeset
|
1282 |
|
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71172
diff
changeset
|
1283 |
lemma linepath_image_01: "linepath a b ` {0..1} = closed_segment a b" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71172
diff
changeset
|
1284 |
by (auto simp: segment linepath_def) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71172
diff
changeset
|
1285 |
|
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71172
diff
changeset
|
1286 |
lemma linepath_in_convex_hull: |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71172
diff
changeset
|
1287 |
fixes x::real |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71172
diff
changeset
|
1288 |
assumes a: "a \<in> convex hull s" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71172
diff
changeset
|
1289 |
and b: "b \<in> convex hull s" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71172
diff
changeset
|
1290 |
and x: "0\<le>x" "x\<le>1" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71172
diff
changeset
|
1291 |
shows "linepath a b x \<in> convex hull s" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71172
diff
changeset
|
1292 |
apply (rule closed_segment_subset_convex_hull [OF a b, THEN subsetD]) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71172
diff
changeset
|
1293 |
using x |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71172
diff
changeset
|
1294 |
apply (auto simp: linepath_image_01 [symmetric]) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71172
diff
changeset
|
1295 |
done |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71172
diff
changeset
|
1296 |
|
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71172
diff
changeset
|
1297 |
lemma Re_linepath: "Re(linepath (of_real a) (of_real b) x) = (1 - x)*a + x*b" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71172
diff
changeset
|
1298 |
by (simp add: linepath_def) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71172
diff
changeset
|
1299 |
|
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71172
diff
changeset
|
1300 |
lemma Im_linepath: "Im(linepath (of_real a) (of_real b) x) = 0" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71172
diff
changeset
|
1301 |
by (simp add: linepath_def) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71172
diff
changeset
|
1302 |
|
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71172
diff
changeset
|
1303 |
lemma bounded_linear_linepath: |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71172
diff
changeset
|
1304 |
assumes "bounded_linear f" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71172
diff
changeset
|
1305 |
shows "f (linepath a b x) = linepath (f a) (f b) x" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71172
diff
changeset
|
1306 |
proof - |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71172
diff
changeset
|
1307 |
interpret f: bounded_linear f by fact |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71172
diff
changeset
|
1308 |
show ?thesis by (simp add: linepath_def f.add f.scale) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71172
diff
changeset
|
1309 |
qed |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71172
diff
changeset
|
1310 |
|
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71172
diff
changeset
|
1311 |
lemma bounded_linear_linepath': |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71172
diff
changeset
|
1312 |
assumes "bounded_linear f" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71172
diff
changeset
|
1313 |
shows "f \<circ> linepath a b = linepath (f a) (f b)" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71172
diff
changeset
|
1314 |
using bounded_linear_linepath[OF assms] by (simp add: fun_eq_iff) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71172
diff
changeset
|
1315 |
|
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71172
diff
changeset
|
1316 |
lemma linepath_cnj': "cnj \<circ> linepath a b = linepath (cnj a) (cnj b)" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71172
diff
changeset
|
1317 |
by (simp add: linepath_def fun_eq_iff) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71172
diff
changeset
|
1318 |
|
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71172
diff
changeset
|
1319 |
lemma differentiable_linepath [intro]: "linepath a b differentiable at x within A" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71172
diff
changeset
|
1320 |
by (auto simp: linepath_def) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71172
diff
changeset
|
1321 |
|
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71172
diff
changeset
|
1322 |
lemma has_vector_derivative_linepath_within: |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71172
diff
changeset
|
1323 |
"(linepath a b has_vector_derivative (b - a)) (at x within s)" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71172
diff
changeset
|
1324 |
apply (simp add: linepath_def has_vector_derivative_def algebra_simps) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71172
diff
changeset
|
1325 |
apply (rule derivative_eq_intros | simp)+ |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71172
diff
changeset
|
1326 |
done |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71172
diff
changeset
|
1327 |
|
62618
f7f2467ab854
Refactoring (moving theorems into better locations), plus a bit of new material
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
1328 |
|
70136 | 1329 |
subsection\<^marker>\<open>tag unimportant\<close>\<open>Segments via convex hulls\<close> |
62618
f7f2467ab854
Refactoring (moving theorems into better locations), plus a bit of new material
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
1330 |
|
f7f2467ab854
Refactoring (moving theorems into better locations), plus a bit of new material
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
1331 |
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
|
1332 |
"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
|
1333 |
"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
|
1334 |
"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
|
1335 |
"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
|
1336 |
"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
|
1337 |
"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
|
1338 |
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
|
1339 |
|
f7f2467ab854
Refactoring (moving theorems into better locations), plus a bit of new material
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
1340 |
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
|
1341 |
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
|
1342 |
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
|
1343 |
proof - |
f7f2467ab854
Refactoring (moving theorems into better locations), plus a bit of new material
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
1344 |
have "(1 - inverse(2)) *\<^sub>R x + inverse(2) *\<^sub>R y \<in> convex hull s" |
68096 | 1345 |
by (rule convexD_alt) (use assms in auto) |
62618
f7f2467ab854
Refactoring (moving theorems into better locations), plus a bit of new material
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
1346 |
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
|
1347 |
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
|
1348 |
qed |
f7f2467ab854
Refactoring (moving theorems into better locations), plus a bit of new material
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
1349 |
|
f7f2467ab854
Refactoring (moving theorems into better locations), plus a bit of new material
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
1350 |
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
|
1351 |
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
|
1352 |
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
|
1353 |
"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
|
1354 |
"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
|
1355 |
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
|
1356 |
|
f7f2467ab854
Refactoring (moving theorems into better locations), plus a bit of new material
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
1357 |
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
|
1358 |
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
|
1359 |
|
f7f2467ab854
Refactoring (moving theorems into better locations), plus a bit of new material
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
1360 |
lemma midpoint_in_open_segment [simp]: "midpoint a b \<in> open_segment a b \<longleftrightarrow> a \<noteq> b" |
64122 | 1361 |
by (simp add: open_segment_def) |
1362 |
||
1363 |
lemma continuous_IVT_local_extremum: |
|
1364 |
fixes f :: "'a::euclidean_space \<Rightarrow> real" |
|
1365 |
assumes contf: "continuous_on (closed_segment a b) f" |
|
1366 |
and "a \<noteq> b" "f a = f b" |
|
1367 |
obtains z where "z \<in> open_segment a b" |
|
1368 |
"(\<forall>w \<in> closed_segment a b. (f w) \<le> (f z)) \<or> |
|
1369 |
(\<forall>w \<in> closed_segment a b. (f z) \<le> (f w))" |
|
1370 |
proof - |
|
1371 |
obtain c where "c \<in> closed_segment a b" and c: "\<And>y. y \<in> closed_segment a b \<Longrightarrow> f y \<le> f c" |
|
1372 |
using continuous_attains_sup [of "closed_segment a b" f] contf by auto |
|
1373 |
obtain d where "d \<in> closed_segment a b" and d: "\<And>y. y \<in> closed_segment a b \<Longrightarrow> f d \<le> f y" |
|
1374 |
using continuous_attains_inf [of "closed_segment a b" f] contf by auto |
|
1375 |
show ?thesis |
|
1376 |
proof (cases "c \<in> open_segment a b \<or> d \<in> open_segment a b") |
|
1377 |
case True |
|
1378 |
then show ?thesis |
|
1379 |
using c d that by blast |
|
1380 |
next |
|
1381 |
case False |
|
1382 |
then have "(c = a \<or> c = b) \<and> (d = a \<or> d = b)" |
|
1383 |
by (simp add: \<open>c \<in> closed_segment a b\<close> \<open>d \<in> closed_segment a b\<close> open_segment_def) |
|
1384 |
with \<open>a \<noteq> b\<close> \<open>f a = f b\<close> c d show ?thesis |
|
1385 |
by (rule_tac z = "midpoint a b" in that) (fastforce+) |
|
1386 |
qed |
|
1387 |
qed |
|
1388 |
||
1389 |
text\<open>An injective map into R is also an open map w.r.T. the universe, and conversely. \<close> |
|
1390 |
proposition injective_eq_1d_open_map_UNIV: |
|
1391 |
fixes f :: "real \<Rightarrow> real" |
|
1392 |
assumes contf: "continuous_on S f" and S: "is_interval S" |
|
1393 |
shows "inj_on f S \<longleftrightarrow> (\<forall>T. open T \<and> T \<subseteq> S \<longrightarrow> open(f ` T))" |
|
1394 |
(is "?lhs = ?rhs") |
|
1395 |
proof safe |
|
1396 |
fix T |
|
1397 |
assume injf: ?lhs and "open T" and "T \<subseteq> S" |
|
1398 |
have "\<exists>U. open U \<and> f x \<in> U \<and> U \<subseteq> f ` T" if "x \<in> T" for x |
|
1399 |
proof - |
|
1400 |
obtain \<delta> where "\<delta> > 0" and \<delta>: "cball x \<delta> \<subseteq> T" |
|
1401 |
using \<open>open T\<close> \<open>x \<in> T\<close> open_contains_cball_eq by blast |
|
1402 |
show ?thesis |
|
1403 |
proof (intro exI conjI) |
|
1404 |
have "closed_segment (x-\<delta>) (x+\<delta>) = {x-\<delta>..x+\<delta>}" |
|
1405 |
using \<open>0 < \<delta>\<close> by (auto simp: closed_segment_eq_real_ivl) |
|
68096 | 1406 |
also have "\<dots> \<subseteq> S" |
64122 | 1407 |
using \<delta> \<open>T \<subseteq> S\<close> by (auto simp: dist_norm subset_eq) |
1408 |
finally have "f ` (open_segment (x-\<delta>) (x+\<delta>)) = open_segment (f (x-\<delta>)) (f (x+\<delta>))" |
|
1409 |
using continuous_injective_image_open_segment_1 |
|
1410 |
by (metis continuous_on_subset [OF contf] inj_on_subset [OF injf]) |
|
1411 |
then show "open (f ` {x-\<delta><..<x+\<delta>})" |
|
1412 |
using \<open>0 < \<delta>\<close> by (simp add: open_segment_eq_real_ivl) |
|
1413 |
show "f x \<in> f ` {x - \<delta><..<x + \<delta>}" |
|
1414 |
by (auto simp: \<open>\<delta> > 0\<close>) |
|
1415 |
show "f ` {x - \<delta><..<x + \<delta>} \<subseteq> f ` T" |
|
1416 |
using \<delta> by (auto simp: dist_norm subset_iff) |
|
1417 |
qed |
|
1418 |
qed |
|
1419 |
with open_subopen show "open (f ` T)" |
|
1420 |
by blast |
|
1421 |
next |
|
1422 |
assume R: ?rhs |
|
1423 |
have False if xy: "x \<in> S" "y \<in> S" and "f x = f y" "x \<noteq> y" for x y |
|
1424 |
proof - |
|
1425 |
have "open (f ` open_segment x y)" |
|
1426 |
using R |
|
1427 |
by (metis S convex_contains_open_segment is_interval_convex open_greaterThanLessThan open_segment_eq_real_ivl xy) |
|
1428 |
moreover |
|
1429 |
have "continuous_on (closed_segment x y) f" |
|
1430 |
by (meson S closed_segment_subset contf continuous_on_subset is_interval_convex that) |
|
1431 |
then obtain \<xi> where "\<xi> \<in> open_segment x y" |
|
1432 |
and \<xi>: "(\<forall>w \<in> closed_segment x y. (f w) \<le> (f \<xi>)) \<or> |
|
1433 |
(\<forall>w \<in> closed_segment x y. (f \<xi>) \<le> (f w))" |
|
1434 |
using continuous_IVT_local_extremum [of x y f] \<open>f x = f y\<close> \<open>x \<noteq> y\<close> by blast |
|
1435 |
ultimately obtain e where "e>0" and e: "\<And>u. dist u (f \<xi>) < e \<Longrightarrow> u \<in> f ` open_segment x y" |
|
1436 |
using open_dist by (metis image_eqI) |
|
1437 |
have fin: "f \<xi> + (e/2) \<in> f ` open_segment x y" "f \<xi> - (e/2) \<in> f ` open_segment x y" |
|
1438 |
using e [of "f \<xi> + (e/2)"] e [of "f \<xi> - (e/2)"] \<open>e > 0\<close> by (auto simp: dist_norm) |
|
1439 |
show ?thesis |
|
1440 |
using \<xi> \<open>0 < e\<close> fin open_closed_segment by fastforce |
|
1441 |
qed |
|
1442 |
then show ?lhs |
|
1443 |
by (force simp: inj_on_def) |
|
1444 |
qed |
|
36583 | 1445 |
|
69514 | 1446 |
|
70136 | 1447 |
subsection\<^marker>\<open>tag unimportant\<close> \<open>Bounding a point away from a path\<close> |
36583 | 1448 |
|
1449 |
lemma not_on_path_ball: |
|
1450 |
fixes g :: "real \<Rightarrow> 'a::heine_borel" |
|
53640 | 1451 |
assumes "path g" |
68096 | 1452 |
and z: "z \<notin> path_image g" |
53640 | 1453 |
shows "\<exists>e > 0. ball z e \<inter> path_image g = {}" |
49653 | 1454 |
proof - |
68096 | 1455 |
have "closed (path_image g)" |
1456 |
by (simp add: \<open>path g\<close> closed_path_image) |
|
1457 |
then obtain a where "a \<in> path_image g" "\<forall>y \<in> path_image g. dist z a \<le> dist z y" |
|
1458 |
by (auto intro: distance_attains_inf[OF _ path_image_nonempty, of g z]) |
|
49654 | 1459 |
then show ?thesis |
68096 | 1460 |
by (rule_tac x="dist z a" in exI) (use dist_commute z in auto) |
49653 | 1461 |
qed |
36583 | 1462 |
|
1463 |
lemma not_on_path_cball: |
|
1464 |
fixes g :: "real \<Rightarrow> 'a::heine_borel" |
|
53640 | 1465 |
assumes "path g" |
1466 |
and "z \<notin> path_image g" |
|
49653 | 1467 |
shows "\<exists>e>0. cball z e \<inter> (path_image g) = {}" |
1468 |
proof - |
|
53640 | 1469 |
obtain e where "ball z e \<inter> path_image g = {}" "e > 0" |
49653 | 1470 |
using not_on_path_ball[OF assms] by auto |
53640 | 1471 |
moreover have "cball z (e/2) \<subseteq> ball z e" |
60420 | 1472 |
using \<open>e > 0\<close> by auto |
53640 | 1473 |
ultimately show ?thesis |
68096 | 1474 |
by (rule_tac x="e/2" in exI) auto |
49653 | 1475 |
qed |
1476 |
||
69518 | 1477 |
subsection \<open>Path component\<close> |
1478 |
||
1479 |
text \<open>Original formalization by Tom Hales\<close> |
|
36583 | 1480 |
|
70136 | 1481 |
definition\<^marker>\<open>tag important\<close> "path_component s x y \<longleftrightarrow> |
49653 | 1482 |
(\<exists>g. path g \<and> path_image g \<subseteq> s \<and> pathstart g = x \<and> pathfinish g = y)" |
36583 | 1483 |
|
70136 | 1484 |
abbreviation\<^marker>\<open>tag important\<close> |
69518 | 1485 |
"path_component_set s x \<equiv> Collect (path_component s x)" |
61426
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1486 |
|
53640 | 1487 |
lemmas path_defs = path_def pathstart_def pathfinish_def path_image_def path_component_def |
36583 | 1488 |
|
49653 | 1489 |
lemma path_component_mem: |
1490 |
assumes "path_component s x y" |
|
53640 | 1491 |
shows "x \<in> s" and "y \<in> s" |
1492 |
using assms |
|
1493 |
unfolding path_defs |
|
1494 |
by auto |
|
36583 | 1495 |
|
49653 | 1496 |
lemma path_component_refl: |
1497 |
assumes "x \<in> s" |
|
1498 |
shows "path_component s x x" |
|
1499 |
unfolding path_defs |
|
1500 |
apply (rule_tac x="\<lambda>u. x" in exI) |
|
53640 | 1501 |
using assms |
56371
fb9ae0727548
extend continuous_intros; remove continuous_on_intros and isCont_intros
hoelzl
parents:
56188
diff
changeset
|
1502 |
apply (auto intro!: continuous_intros) |
53640 | 1503 |
done |
36583 | 1504 |
|
1505 |
lemma path_component_refl_eq: "path_component s x x \<longleftrightarrow> x \<in> s" |
|
49653 | 1506 |
by (auto intro!: path_component_mem path_component_refl) |
36583 | 1507 |
|
1508 |
lemma path_component_sym: "path_component s x y \<Longrightarrow> path_component s y x" |
|
49653 | 1509 |
unfolding path_component_def |
1510 |
apply (erule exE) |
|
68096 | 1511 |
apply (rule_tac x="reversepath g" in exI, auto) |
49653 | 1512 |
done |
36583 | 1513 |
|
49653 | 1514 |
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
|
1515 |
assumes "path_component s x y" and "path_component s y z" |
49653 | 1516 |
shows "path_component s x z" |
1517 |
using assms |
|
1518 |
unfolding path_component_def |
|
53640 | 1519 |
apply (elim exE) |
49653 | 1520 |
apply (rule_tac x="g +++ ga" in exI) |
68096 | 1521 |
apply (auto simp: path_image_join) |
49653 | 1522 |
done |
36583 | 1523 |
|
53640 | 1524 |
lemma path_component_of_subset: "s \<subseteq> t \<Longrightarrow> path_component s x y \<Longrightarrow> path_component t x y" |
36583 | 1525 |
unfolding path_component_def by auto |
1526 |
||
69939
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
1527 |
lemma path_component_linepath: |
61426
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1528 |
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
|
1529 |
shows "closed_segment a b \<subseteq> s \<Longrightarrow> path_component s a b" |
68096 | 1530 |
unfolding path_component_def |
1531 |
by (rule_tac x="linepath a b" in exI, 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
|
1532 |
|
70136 | 1533 |
subsubsection\<^marker>\<open>tag unimportant\<close> \<open>Path components as sets\<close> |
36583 | 1534 |
|
49653 | 1535 |
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
|
1536 |
"path_component_set s x = |
49653 | 1537 |
{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
|
1538 |
by (auto simp: path_component_def) |
36583 | 1539 |
|
61426
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1540 |
lemma path_component_subset: "path_component_set s x \<subseteq> s" |
68096 | 1541 |
by (auto simp: path_component_mem(2)) |
36583 | 1542 |
|
61426
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1543 |
lemma path_component_eq_empty: "path_component_set s x = {} \<longleftrightarrow> x \<notin> s" |
60303 | 1544 |
using path_component_mem path_component_refl_eq |
1545 |
by fastforce |
|
36583 | 1546 |
|
61426
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1547 |
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
|
1548 |
"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
|
1549 |
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
|
1550 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1551 |
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
|
1552 |
"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
|
1553 |
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
|
1554 |
|
69514 | 1555 |
|
60420 | 1556 |
subsection \<open>Path connectedness of a space\<close> |
36583 | 1557 |
|
70136 | 1558 |
definition\<^marker>\<open>tag important\<close> "path_connected s \<longleftrightarrow> |
53640 | 1559 |
(\<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 | 1560 |
|
69939
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
1561 |
lemma path_connectedin_iff_path_connected_real [simp]: |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
1562 |
"path_connectedin euclideanreal S \<longleftrightarrow> path_connected S" |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
1563 |
by (simp add: path_connectedin path_connected_def path_defs) |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
1564 |
|
36583 | 1565 |
lemma path_connected_component: "path_connected s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>y\<in>s. path_component s x y)" |
1566 |
unfolding path_connected_def path_component_def by auto |
|
1567 |
||
61426
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1568 |
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
|
1569 |
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
|
1570 |
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
|
1571 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1572 |
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
|
1573 |
"\<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
|
1574 |
by (metis path_component_mono path_connected_component_set) |
36583 | 1575 |
|
1576 |
lemma convex_imp_path_connected: |
|
1577 |
fixes s :: "'a::real_normed_vector set" |
|
53640 | 1578 |
assumes "convex s" |
1579 |
shows "path_connected s" |
|
49653 | 1580 |
unfolding path_connected_def |
66793
deabce3ccf1f
new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
66456
diff
changeset
|
1581 |
using assms convex_contains_segment by fastforce |
36583 | 1582 |
|
62620
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
1583 |
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
|
1584 |
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
|
1585 |
|
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
1586 |
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
|
1587 |
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
|
1588 |
|
49653 | 1589 |
lemma path_connected_imp_connected: |
64788
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
1590 |
assumes "path_connected S" |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
1591 |
shows "connected S" |
66793
deabce3ccf1f
new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
66456
diff
changeset
|
1592 |
proof (rule connectedI) |
49653 | 1593 |
fix e1 e2 |
64788
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
1594 |
assume as: "open e1" "open e2" "S \<subseteq> e1 \<union> e2" "e1 \<inter> e2 \<inter> S = {}" "e1 \<inter> S \<noteq> {}" "e2 \<inter> S \<noteq> {}" |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
1595 |
then obtain x1 x2 where obt:"x1 \<in> e1 \<inter> S" "x2 \<in> e2 \<inter> S" |
53640 | 1596 |
by auto |
64788
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
1597 |
then obtain g where g: "path g" "path_image g \<subseteq> S" "pathstart g = x1" "pathfinish g = x2" |
36583 | 1598 |
using assms[unfolded path_connected_def,rule_format,of x1 x2] by auto |
49653 | 1599 |
have *: "connected {0..1::real}" |
71172 | 1600 |
by (auto intro!: convex_connected) |
49653 | 1601 |
have "{0..1} \<subseteq> {x \<in> {0..1}. g x \<in> e1} \<union> {x \<in> {0..1}. g x \<in> e2}" |
1602 |
using as(3) g(2)[unfolded path_defs] by blast |
|
1603 |
moreover have "{x \<in> {0..1}. g x \<in> e1} \<inter> {x \<in> {0..1}. g x \<in> e2} = {}" |
|
53640 | 1604 |
using as(4) g(2)[unfolded path_defs] |
1605 |
unfolding subset_eq |
|
1606 |
by auto |
|
49653 | 1607 |
moreover have "{x \<in> {0..1}. g x \<in> e1} \<noteq> {} \<and> {x \<in> {0..1}. g x \<in> e2} \<noteq> {}" |
53640 | 1608 |
using g(3,4)[unfolded path_defs] |
1609 |
using obt |
|
36583 | 1610 |
by (simp add: ex_in_conv [symmetric], metis zero_le_one order_refl) |
49653 | 1611 |
ultimately show False |
53640 | 1612 |
using *[unfolded connected_local not_ex, rule_format, |
66884
c2128ab11f61
Switching to inverse image and constant_on, plus some new material
paulson <lp15@cam.ac.uk>
parents:
66827
diff
changeset
|
1613 |
of "{0..1} \<inter> g -` e1" "{0..1} \<inter> g -` e2"] |
63301 | 1614 |
using continuous_openin_preimage_gen[OF g(1)[unfolded path_def] as(1)] |
1615 |
using continuous_openin_preimage_gen[OF g(1)[unfolded path_def] as(2)] |
|
49653 | 1616 |
by auto |
1617 |
qed |
|
36583 | 1618 |
|
1619 |
lemma open_path_component: |
|
64788
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
1620 |
fixes S :: "'a::real_normed_vector set" |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
1621 |
assumes "open S" |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
1622 |
shows "open (path_component_set S x)" |
49653 | 1623 |
unfolding open_contains_ball |
1624 |
proof |
|
1625 |
fix y |
|
64788
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
1626 |
assume as: "y \<in> path_component_set S x" |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
1627 |
then have "y \<in> S" |
66793
deabce3ccf1f
new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
66456
diff
changeset
|
1628 |
by (simp add: path_component_mem(2)) |
64788
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
1629 |
then obtain e where e: "e > 0" "ball y e \<subseteq> S" |
53640 | 1630 |
using assms[unfolded open_contains_ball] |
1631 |
by auto |
|
66793
deabce3ccf1f
new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
66456
diff
changeset
|
1632 |
have "\<And>u. dist y u < e \<Longrightarrow> path_component S x u" |
deabce3ccf1f
new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
66456
diff
changeset
|
1633 |
by (metis (full_types) as centre_in_ball convex_ball convex_imp_path_connected e mem_Collect_eq mem_ball path_component_eq path_component_of_subset path_connected_component) |
deabce3ccf1f
new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
66456
diff
changeset
|
1634 |
then show "\<exists>e > 0. ball y e \<subseteq> path_component_set S x" |
deabce3ccf1f
new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
66456
diff
changeset
|
1635 |
using \<open>e>0\<close> by auto |
49653 | 1636 |
qed |
36583 | 1637 |
|
1638 |
lemma open_non_path_component: |
|
64788
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
1639 |
fixes S :: "'a::real_normed_vector set" |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
1640 |
assumes "open S" |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
1641 |
shows "open (S - path_component_set S x)" |
49653 | 1642 |
unfolding open_contains_ball |
1643 |
proof |
|
1644 |
fix y |
|
68096 | 1645 |
assume y: "y \<in> S - path_component_set S x" |
64788
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
1646 |
then obtain e where e: "e > 0" "ball y e \<subseteq> S" |
68096 | 1647 |
using assms openE by auto |
64788
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
1648 |
show "\<exists>e>0. ball y e \<subseteq> S - path_component_set S x" |
68096 | 1649 |
proof (intro exI conjI subsetI DiffI notI) |
1650 |
show "\<And>x. x \<in> ball y e \<Longrightarrow> x \<in> S" |
|
1651 |
using e by blast |
|
1652 |
show False if "z \<in> ball y e" "z \<in> path_component_set S x" for z |
|
1653 |
proof - |
|
1654 |
have "y \<in> path_component_set S z" |
|
1655 |
by (meson assms convex_ball convex_imp_path_connected e open_contains_ball_eq open_path_component path_component_maximal that(1)) |
|
1656 |
then have "y \<in> path_component_set S x" |
|
1657 |
using path_component_eq that(2) by blast |
|
1658 |
then show False |
|
1659 |
using y by blast |
|
1660 |
qed |
|
1661 |
qed (use e in auto) |
|
49653 | 1662 |
qed |
36583 | 1663 |
|
1664 |
lemma connected_open_path_connected: |
|
64788
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
1665 |
fixes S :: "'a::real_normed_vector set" |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
1666 |
assumes "open S" |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
1667 |
and "connected S" |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
1668 |
shows "path_connected S" |
49653 | 1669 |
unfolding path_connected_component_set |
1670 |
proof (rule, rule, rule path_component_subset, rule) |
|
1671 |
fix x y |
|
64788
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
1672 |
assume "x \<in> S" and "y \<in> S" |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
1673 |
show "y \<in> path_component_set S x" |
49653 | 1674 |
proof (rule ccontr) |
53640 | 1675 |
assume "\<not> ?thesis" |
64788
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
1676 |
moreover have "path_component_set S x \<inter> S \<noteq> {}" |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
1677 |
using \<open>x \<in> S\<close> path_component_eq_empty path_component_subset[of S x] |
53640 | 1678 |
by auto |
49653 | 1679 |
ultimately |
1680 |
show False |
|
64788
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
1681 |
using \<open>y \<in> S\<close> open_non_path_component[OF assms(1)] open_path_component[OF assms(1)] |
53640 | 1682 |
using assms(2)[unfolded connected_def not_ex, rule_format, |
64788
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
1683 |
of "path_component_set S x" "S - path_component_set S x"] |
49653 | 1684 |
by auto |
1685 |
qed |
|
1686 |
qed |
|
36583 | 1687 |
|
1688 |
lemma path_connected_continuous_image: |
|
64788
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
1689 |
assumes "continuous_on S f" |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
1690 |
and "path_connected S" |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
1691 |
shows "path_connected (f ` S)" |
49653 | 1692 |
unfolding path_connected_def |
1693 |
proof (rule, rule) |
|
1694 |
fix x' y' |
|
64788
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
1695 |
assume "x' \<in> f ` S" "y' \<in> f ` S" |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
1696 |
then obtain x y where x: "x \<in> S" and y: "y \<in> S" and x': "x' = f x" and y': "y' = f y" |
53640 | 1697 |
by auto |
64788
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
1698 |
from x y obtain g where "path g \<and> path_image g \<subseteq> S \<and> pathstart g = x \<and> pathfinish g = y" |
53640 | 1699 |
using assms(2)[unfolded path_connected_def] by fast |
64788
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
1700 |
then show "\<exists>g. path g \<and> path_image g \<subseteq> f ` S \<and> pathstart g = x' \<and> pathfinish g = y'" |
53640 | 1701 |
unfolding x' y' |
49653 | 1702 |
apply (rule_tac x="f \<circ> g" in exI) |
1703 |
unfolding path_defs |
|
51481
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51478
diff
changeset
|
1704 |
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
|
1705 |
apply auto |
49653 | 1706 |
done |
1707 |
qed |
|
36583 | 1708 |
|
64788
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
1709 |
lemma path_connected_translationI: |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
1710 |
fixes a :: "'a :: topological_group_add" |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
1711 |
assumes "path_connected S" shows "path_connected ((\<lambda>x. a + x) ` S)" |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
1712 |
by (intro path_connected_continuous_image assms continuous_intros) |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
1713 |
|
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
1714 |
lemma path_connected_translation: |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
1715 |
fixes a :: "'a :: topological_group_add" |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
1716 |
shows "path_connected ((\<lambda>x. a + x) ` S) = path_connected S" |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
1717 |
proof - |
67399 | 1718 |
have "\<forall>x y. (+) (x::'a) ` (+) (0 - x) ` y = y" |
64788
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
1719 |
by (simp add: image_image) |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
1720 |
then show ?thesis |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
1721 |
by (metis (no_types) path_connected_translationI) |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
1722 |
qed |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
1723 |
|
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
1724 |
lemma path_connected_segment [simp]: |
61518
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
1725 |
fixes a :: "'a::real_normed_vector" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
1726 |
shows "path_connected (closed_segment a b)" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
1727 |
by (simp add: convex_imp_path_connected) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
1728 |
|
64788
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
1729 |
lemma path_connected_open_segment [simp]: |
61518
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
1730 |
fixes a :: "'a::real_normed_vector" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
1731 |
shows "path_connected (open_segment a b)" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
1732 |
by (simp add: convex_imp_path_connected) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
1733 |
|
36583 | 1734 |
lemma homeomorphic_path_connectedness: |
68096 | 1735 |
"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
|
1736 |
unfolding homeomorphic_def homeomorphism_def by (metis path_connected_continuous_image) |
36583 | 1737 |
|
64788
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
1738 |
lemma path_connected_empty [simp]: "path_connected {}" |
36583 | 1739 |
unfolding path_connected_def by auto |
1740 |
||
64788
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
1741 |
lemma path_connected_singleton [simp]: "path_connected {a}" |
36583 | 1742 |
unfolding path_connected_def pathstart_def pathfinish_def path_image_def |
53640 | 1743 |
apply clarify |
1744 |
apply (rule_tac x="\<lambda>x. a" in exI) |
|
1745 |
apply (simp add: image_constant_conv) |
|
71172 | 1746 |
apply (simp add: path_def) |
36583 | 1747 |
done |
1748 |
||
49653 | 1749 |
lemma path_connected_Un: |
68096 | 1750 |
assumes "path_connected S" |
1751 |
and "path_connected T" |
|
1752 |
and "S \<inter> T \<noteq> {}" |
|
1753 |
shows "path_connected (S \<union> T)" |
|
49653 | 1754 |
unfolding path_connected_component |
68096 | 1755 |
proof (intro ballI) |
49653 | 1756 |
fix x y |
68096 | 1757 |
assume x: "x \<in> S \<union> T" and y: "y \<in> S \<union> T" |
1758 |
from assms obtain z where z: "z \<in> S" "z \<in> T" |
|
53640 | 1759 |
by auto |
68096 | 1760 |
show "path_component (S \<union> T) x y" |
1761 |
using x y |
|
1762 |
proof safe |
|
1763 |
assume "x \<in> S" "y \<in> S" |
|
1764 |
then show "path_component (S \<union> T) x y" |
|
1765 |
by (meson Un_upper1 \<open>path_connected S\<close> path_component_of_subset path_connected_component) |
|
1766 |
next |
|
1767 |
assume "x \<in> S" "y \<in> T" |
|
1768 |
then show "path_component (S \<union> T) x y" |
|
1769 |
by (metis z assms(1-2) le_sup_iff order_refl path_component_of_subset path_component_trans path_connected_component) |
|
1770 |
next |
|
1771 |
assume "x \<in> T" "y \<in> S" |
|
1772 |
then show "path_component (S \<union> T) x y" |
|
1773 |
by (metis z assms(1-2) le_sup_iff order_refl path_component_of_subset path_component_trans path_connected_component) |
|
1774 |
next |
|
1775 |
assume "x \<in> T" "y \<in> T" |
|
1776 |
then show "path_component (S \<union> T) x y" |
|
1777 |
by (metis Un_upper1 assms(2) path_component_of_subset path_connected_component sup_commute) |
|
1778 |
qed |
|
49653 | 1779 |
qed |
36583 | 1780 |
|
37674
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
1781 |
lemma path_connected_UNION: |
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
1782 |
assumes "\<And>i. i \<in> A \<Longrightarrow> path_connected (S i)" |
49653 | 1783 |
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
|
1784 |
shows "path_connected (\<Union>i\<in>A. S i)" |
49653 | 1785 |
unfolding path_connected_component |
1786 |
proof clarify |
|
37674
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
1787 |
fix x i y j |
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
1788 |
assume *: "i \<in> A" "x \<in> S i" "j \<in> A" "y \<in> S j" |
49654 | 1789 |
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
|
1790 |
using assms by (simp_all add: path_connected_component) |
49654 | 1791 |
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
|
1792 |
using *(1,3) by (auto elim!: path_component_of_subset [rotated]) |
49654 | 1793 |
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
|
1794 |
by (rule path_component_trans) |
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
1795 |
qed |
36583 | 1796 |
|
61426
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1797 |
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
|
1798 |
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
|
1799 |
shows "path_component (path_image p) (pathstart p) x" |
68096 | 1800 |
proof - |
1801 |
obtain y where x: "x = p y" and y: "0 \<le> y" "y \<le> 1" |
|
1802 |
using x by (auto simp: path_image_def) |
|
1803 |
show ?thesis |
|
1804 |
unfolding path_component_def |
|
1805 |
proof (intro exI conjI) |
|
69064
5840724b1d71
Prefix form of infix with * on either side no longer needs special treatment
nipkow
parents:
68913
diff
changeset
|
1806 |
have "continuous_on {0..1} (p \<circ> ((*) 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
|
1807 |
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
|
1808 |
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
|
1809 |
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
|
1810 |
done |
68096 | 1811 |
then show "path (\<lambda>u. p (y * 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
|
1812 |
by (simp add: path_def) |
68096 | 1813 |
show "path_image (\<lambda>u. p (y * u)) \<subseteq> path_image p" |
1814 |
using y mult_le_one by (fastforce simp: path_image_def image_iff) |
|
1815 |
qed (auto simp: pathstart_def pathfinish_def x) |
|
61426
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1816 |
qed |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1817 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1818 |
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
|
1819 |
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
|
1820 |
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
|
1821 |
|
64788
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
1822 |
lemma path_connected_path_component [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
|
1823 |
"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
|
1824 |
proof - |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1825 |
{ 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
|
1826 |
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
|
1827 |
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
|
1828 |
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
|
1829 |
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
|
1830 |
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
|
1831 |
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
|
1832 |
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
|
1833 |
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
|
1834 |
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
|
1835 |
} |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1836 |
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
|
1837 |
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
|
1838 |
qed |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1839 |
|
68532
f8b98d31ad45
Incorporating new/strengthened proofs from Library and AFP entries
paulson <lp15@cam.ac.uk>
parents:
68310
diff
changeset
|
1840 |
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)" |
61426
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1841 |
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
|
1842 |
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
|
1843 |
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
|
1844 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1845 |
lemma path_component_path_component [simp]: |
68532
f8b98d31ad45
Incorporating new/strengthened proofs from Library and AFP entries
paulson <lp15@cam.ac.uk>
parents:
68310
diff
changeset
|
1846 |
"path_component_set (path_component_set S x) x = path_component_set S x" |
f8b98d31ad45
Incorporating new/strengthened proofs from Library and AFP entries
paulson <lp15@cam.ac.uk>
parents:
68310
diff
changeset
|
1847 |
proof (cases "x \<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
|
1848 |
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
|
1849 |
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
|
1850 |
apply (simp add: path_component_subset) |
71172 | 1851 |
by (simp add: True path_component_maximal path_component_refl) |
61426
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1852 |
next |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1853 |
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
|
1854 |
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
|
1855 |
qed |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1856 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1857 |
lemma path_component_subset_connected_component: |
68532
f8b98d31ad45
Incorporating new/strengthened proofs from Library and AFP entries
paulson <lp15@cam.ac.uk>
parents:
68310
diff
changeset
|
1858 |
"(path_component_set S x) \<subseteq> (connected_component_set S x)" |
f8b98d31ad45
Incorporating new/strengthened proofs from Library and AFP entries
paulson <lp15@cam.ac.uk>
parents:
68310
diff
changeset
|
1859 |
proof (cases "x \<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
|
1860 |
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
|
1861 |
apply (rule connected_component_maximal) |
68532
f8b98d31ad45
Incorporating new/strengthened proofs from Library and AFP entries
paulson <lp15@cam.ac.uk>
parents:
68310
diff
changeset
|
1862 |
apply (auto simp: True path_component_subset path_component_refl path_connected_imp_connected) |
61426
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1863 |
done |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1864 |
next |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1865 |
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
|
1866 |
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
|
1867 |
qed |
49653 | 1868 |
|
69514 | 1869 |
|
70136 | 1870 |
subsection\<^marker>\<open>tag unimportant\<close>\<open>Lemmas about path-connectedness\<close> |
62620
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
1871 |
|
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
1872 |
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
|
1873 |
fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector" |
68532
f8b98d31ad45
Incorporating new/strengthened proofs from Library and AFP entries
paulson <lp15@cam.ac.uk>
parents:
68310
diff
changeset
|
1874 |
assumes "path_connected S" "bounded_linear f" |
f8b98d31ad45
Incorporating new/strengthened proofs from Library and AFP entries
paulson <lp15@cam.ac.uk>
parents:
68310
diff
changeset
|
1875 |
shows "path_connected(f ` S)" |
62620
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
1876 |
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
|
1877 |
|
68532
f8b98d31ad45
Incorporating new/strengthened proofs from Library and AFP entries
paulson <lp15@cam.ac.uk>
parents:
68310
diff
changeset
|
1878 |
lemma is_interval_path_connected: "is_interval S \<Longrightarrow> path_connected S" |
62620
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
1879 |
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
|
1880 |
|
71025
be8cec1abcbb
reduce dependencies of Ordered_Euclidean_Space; move more general material from Cartesian_Euclidean_Space
immler
parents:
70971
diff
changeset
|
1881 |
lemma path_connected_Ioi[simp]: "path_connected {a<..}" for a :: real |
be8cec1abcbb
reduce dependencies of Ordered_Euclidean_Space; move more general material from Cartesian_Euclidean_Space
immler
parents:
70971
diff
changeset
|
1882 |
by (simp add: convex_imp_path_connected) |
be8cec1abcbb
reduce dependencies of Ordered_Euclidean_Space; move more general material from Cartesian_Euclidean_Space
immler
parents:
70971
diff
changeset
|
1883 |
|
be8cec1abcbb
reduce dependencies of Ordered_Euclidean_Space; move more general material from Cartesian_Euclidean_Space
immler
parents:
70971
diff
changeset
|
1884 |
lemma path_connected_Ici[simp]: "path_connected {a..}" for a :: real |
be8cec1abcbb
reduce dependencies of Ordered_Euclidean_Space; move more general material from Cartesian_Euclidean_Space
immler
parents:
70971
diff
changeset
|
1885 |
by (simp add: convex_imp_path_connected) |
be8cec1abcbb
reduce dependencies of Ordered_Euclidean_Space; move more general material from Cartesian_Euclidean_Space
immler
parents:
70971
diff
changeset
|
1886 |
|
be8cec1abcbb
reduce dependencies of Ordered_Euclidean_Space; move more general material from Cartesian_Euclidean_Space
immler
parents:
70971
diff
changeset
|
1887 |
lemma path_connected_Iio[simp]: "path_connected {..<a}" for a :: real |
be8cec1abcbb
reduce dependencies of Ordered_Euclidean_Space; move more general material from Cartesian_Euclidean_Space
immler
parents:
70971
diff
changeset
|
1888 |
by (simp add: convex_imp_path_connected) |
be8cec1abcbb
reduce dependencies of Ordered_Euclidean_Space; move more general material from Cartesian_Euclidean_Space
immler
parents:
70971
diff
changeset
|
1889 |
|
be8cec1abcbb
reduce dependencies of Ordered_Euclidean_Space; move more general material from Cartesian_Euclidean_Space
immler
parents:
70971
diff
changeset
|
1890 |
lemma path_connected_Iic[simp]: "path_connected {..a}" for a :: real |
be8cec1abcbb
reduce dependencies of Ordered_Euclidean_Space; move more general material from Cartesian_Euclidean_Space
immler
parents:
70971
diff
changeset
|
1891 |
by (simp add: convex_imp_path_connected) |
be8cec1abcbb
reduce dependencies of Ordered_Euclidean_Space; move more general material from Cartesian_Euclidean_Space
immler
parents:
70971
diff
changeset
|
1892 |
|
be8cec1abcbb
reduce dependencies of Ordered_Euclidean_Space; move more general material from Cartesian_Euclidean_Space
immler
parents:
70971
diff
changeset
|
1893 |
lemma path_connected_Ioo[simp]: "path_connected {a<..<b}" for a b :: real |
be8cec1abcbb
reduce dependencies of Ordered_Euclidean_Space; move more general material from Cartesian_Euclidean_Space
immler
parents:
70971
diff
changeset
|
1894 |
by (simp add: convex_imp_path_connected) |
be8cec1abcbb
reduce dependencies of Ordered_Euclidean_Space; move more general material from Cartesian_Euclidean_Space
immler
parents:
70971
diff
changeset
|
1895 |
|
be8cec1abcbb
reduce dependencies of Ordered_Euclidean_Space; move more general material from Cartesian_Euclidean_Space
immler
parents:
70971
diff
changeset
|
1896 |
lemma path_connected_Ioc[simp]: "path_connected {a<..b}" for a b :: real |
be8cec1abcbb
reduce dependencies of Ordered_Euclidean_Space; move more general material from Cartesian_Euclidean_Space
immler
parents:
70971
diff
changeset
|
1897 |
by (simp add: convex_imp_path_connected) |
be8cec1abcbb
reduce dependencies of Ordered_Euclidean_Space; move more general material from Cartesian_Euclidean_Space
immler
parents:
70971
diff
changeset
|
1898 |
|
be8cec1abcbb
reduce dependencies of Ordered_Euclidean_Space; move more general material from Cartesian_Euclidean_Space
immler
parents:
70971
diff
changeset
|
1899 |
lemma path_connected_Ico[simp]: "path_connected {a..<b}" for a b :: real |
be8cec1abcbb
reduce dependencies of Ordered_Euclidean_Space; move more general material from Cartesian_Euclidean_Space
immler
parents:
70971
diff
changeset
|
1900 |
by (simp add: convex_imp_path_connected) |
be8cec1abcbb
reduce dependencies of Ordered_Euclidean_Space; move more general material from Cartesian_Euclidean_Space
immler
parents:
70971
diff
changeset
|
1901 |
|
69939
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
1902 |
lemma path_connectedin_path_image: |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
1903 |
assumes "pathin X g" shows "path_connectedin X (g ` ({0..1}))" |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
1904 |
unfolding pathin_def |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
1905 |
proof (rule path_connectedin_continuous_map_image) |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
1906 |
show "continuous_map (subtopology euclideanreal {0..1}) X g" |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
1907 |
using assms pathin_def by blast |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
1908 |
qed (auto simp: is_interval_1 is_interval_path_connected) |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
1909 |
|
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
1910 |
lemma path_connected_space_subconnected: |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
1911 |
"path_connected_space X \<longleftrightarrow> |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
1912 |
(\<forall>x \<in> topspace X. \<forall>y \<in> topspace X. \<exists>S. path_connectedin X S \<and> x \<in> S \<and> y \<in> S)" |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
1913 |
unfolding path_connected_space_def Ball_def |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
1914 |
apply (intro all_cong1 imp_cong refl, safe) |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
1915 |
using path_connectedin_path_image apply fastforce |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
1916 |
by (meson path_connectedin) |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
1917 |
|
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
1918 |
lemma connectedin_path_image: "pathin X g \<Longrightarrow> connectedin X (g ` ({0..1}))" |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
1919 |
by (simp add: path_connectedin_imp_connectedin path_connectedin_path_image) |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
1920 |
|
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
1921 |
lemma compactin_path_image: "pathin X g \<Longrightarrow> compactin X (g ` ({0..1}))" |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
1922 |
unfolding pathin_def |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
1923 |
by (rule image_compactin [of "top_of_set {0..1}"]) auto |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
1924 |
|
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
|
1925 |
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
|
1926 |
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
|
1927 |
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
|
1928 |
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
|
1929 |
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
|
1930 |
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
|
1931 |
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
|
1932 |
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
|
1933 |
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
|
1934 |
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
|
1935 |
|
313d3b697c9a
Mostly renaming (from HOL Light to Isabelle conventions), with a couple of new results
paulson <lp15@cam.ac.uk>
parents:
62626
diff
changeset
|
1936 |
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
|
1937 |
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
|
1938 |
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
|
1939 |
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
|
1940 |
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
|
1941 |
|
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
1942 |
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
|
1943 |
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
|
1944 |
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
|
1945 |
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
|
1946 |
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
|
1947 |
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
|
1948 |
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
|
1949 |
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
|
1950 |
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
|
1951 |
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
|
1952 |
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
|
1953 |
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
|
1954 |
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
|
1955 |
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
|
1956 |
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
|
1957 |
done |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
1958 |
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
|
1959 |
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
|
1960 |
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
|
1961 |
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
|
1962 |
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
|
1963 |
done |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
1964 |
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
|
1965 |
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
|
1966 |
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
|
1967 |
by (rule Path_Connected.path_image_join_subset) |
68096 | 1968 |
also have "\<dots> \<subseteq> (\<Union>x\<in>s. \<Union>x1\<in>t. {(x, x1)})" |
62620
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
1969 |
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
|
1970 |
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
|
1971 |
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
|
1972 |
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
|
1973 |
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
|
1974 |
apply (rule 1) |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
1975 |
apply (rule 2) |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
1976 |
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
|
1977 |
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
|
1978 |
qed |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
1979 |
|
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
1980 |
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
|
1981 |
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
|
1982 |
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
|
1983 |
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
|
1984 |
|
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
1985 |
|
70136 | 1986 |
subsection\<^marker>\<open>tag unimportant\<close>\<open>Path components\<close> |
66793
deabce3ccf1f
new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
66456
diff
changeset
|
1987 |
|
62948
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
1988 |
lemma Union_path_component [simp]: |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
1989 |
"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
|
1990 |
apply (rule subset_antisym) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
1991 |
using path_component_subset apply force |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
1992 |
using path_component_refl by auto |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
1993 |
|
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
1994 |
lemma path_component_disjoint: |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
1995 |
"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
|
1996 |
(a \<notin> path_component_set S b)" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
1997 |
apply (auto simp: disjnt_def) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
1998 |
using path_component_eq apply fastforce |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
1999 |
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
|
2000 |
|
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
2001 |
lemma path_component_eq_eq: |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
2002 |
"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
|
2003 |
(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
|
2004 |
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
|
2005 |
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
|
2006 |
apply (rule ext) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
2007 |
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
|
2008 |
done |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
2009 |
|
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
2010 |
lemma path_component_unique: |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
2011 |
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
|
2012 |
"\<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
|
2013 |
shows "path_component_set S x = c" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
2014 |
apply (rule subset_antisym) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
2015 |
using assms |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
2016 |
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
|
2017 |
by (simp add: assms path_component_maximal) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
2018 |
|
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
2019 |
lemma path_component_intermediate_subset: |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
2020 |
"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
|
2021 |
\<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
|
2022 |
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
|
2023 |
|
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
2024 |
lemma complement_path_component_Union: |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
2025 |
fixes x :: "'a :: topological_space" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
2026 |
shows "S - path_component_set S x = |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
2027 |
\<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
|
2028 |
proof - |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
2029 |
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
|
2030 |
for a::"'a set" and S |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
2031 |
by (auto simp: disjnt_def) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
2032 |
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
|
2033 |
\<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
|
2034 |
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
|
2035 |
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
|
2036 |
\<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
|
2037 |
by (meson *) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
2038 |
then show ?thesis by simp |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
2039 |
qed |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
2040 |
|
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62843
diff
changeset
|
2041 |
|
69939
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2042 |
subsection\<open>Path components\<close> |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2043 |
|
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2044 |
definition path_component_of |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2045 |
where "path_component_of X x y \<equiv> \<exists>g. pathin X g \<and> g 0 = x \<and> g 1 = y" |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2046 |
|
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2047 |
abbreviation path_component_of_set |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2048 |
where "path_component_of_set X x \<equiv> Collect (path_component_of X x)" |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2049 |
|
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2050 |
definition path_components_of :: "'a topology \<Rightarrow> 'a set set" |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2051 |
where "path_components_of X \<equiv> path_component_of_set X ` topspace X" |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2052 |
|
69986
f2d327275065
generalised homotopic_with to topologies; homotopic_with_canon is the old version
paulson <lp15@cam.ac.uk>
parents:
69939
diff
changeset
|
2053 |
lemma pathin_canon_iff: "pathin (top_of_set T) g \<longleftrightarrow> path g \<and> g ` {0..1} \<subseteq> T" |
f2d327275065
generalised homotopic_with to topologies; homotopic_with_canon is the old version
paulson <lp15@cam.ac.uk>
parents:
69939
diff
changeset
|
2054 |
by (simp add: path_def pathin_def) |
f2d327275065
generalised homotopic_with to topologies; homotopic_with_canon is the old version
paulson <lp15@cam.ac.uk>
parents:
69939
diff
changeset
|
2055 |
|
f2d327275065
generalised homotopic_with to topologies; homotopic_with_canon is the old version
paulson <lp15@cam.ac.uk>
parents:
69939
diff
changeset
|
2056 |
lemma path_component_of_canon_iff [simp]: |
f2d327275065
generalised homotopic_with to topologies; homotopic_with_canon is the old version
paulson <lp15@cam.ac.uk>
parents:
69939
diff
changeset
|
2057 |
"path_component_of (top_of_set T) a b \<longleftrightarrow> path_component T a b" |
f2d327275065
generalised homotopic_with to topologies; homotopic_with_canon is the old version
paulson <lp15@cam.ac.uk>
parents:
69939
diff
changeset
|
2058 |
by (simp add: path_component_of_def pathin_canon_iff path_defs) |
f2d327275065
generalised homotopic_with to topologies; homotopic_with_canon is the old version
paulson <lp15@cam.ac.uk>
parents:
69939
diff
changeset
|
2059 |
|
69939
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2060 |
lemma path_component_in_topspace: |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2061 |
"path_component_of X x y \<Longrightarrow> x \<in> topspace X \<and> y \<in> topspace X" |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2062 |
by (auto simp: path_component_of_def pathin_def continuous_map_def) |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2063 |
|
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2064 |
lemma path_component_of_refl: |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2065 |
"path_component_of X x x \<longleftrightarrow> x \<in> topspace X" |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2066 |
apply (auto simp: path_component_in_topspace) |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2067 |
apply (force simp: path_component_of_def pathin_const) |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2068 |
done |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2069 |
|
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2070 |
lemma path_component_of_sym: |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2071 |
assumes "path_component_of X x y" |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2072 |
shows "path_component_of X y x" |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2073 |
using assms |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2074 |
apply (clarsimp simp: path_component_of_def pathin_def) |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2075 |
apply (rule_tac x="g \<circ> (\<lambda>t. 1 - t)" in exI) |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2076 |
apply (auto intro!: continuous_map_compose) |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2077 |
apply (force simp: continuous_map_in_subtopology continuous_on_op_minus) |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2078 |
done |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2079 |
|
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2080 |
lemma path_component_of_sym_iff: |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2081 |
"path_component_of X x y \<longleftrightarrow> path_component_of X y x" |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2082 |
by (metis path_component_of_sym) |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2083 |
|
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2084 |
lemma path_component_of_trans: |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2085 |
assumes "path_component_of X x y" and "path_component_of X y z" |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2086 |
shows "path_component_of X x z" |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2087 |
unfolding path_component_of_def pathin_def |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2088 |
proof - |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2089 |
let ?T01 = "top_of_set {0..1::real}" |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2090 |
obtain g1 g2 where g1: "continuous_map ?T01 X g1" "x = g1 0" "y = g1 1" |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2091 |
and g2: "continuous_map ?T01 X g2" "g2 0 = g1 1" "z = g2 1" |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2092 |
using assms unfolding path_component_of_def pathin_def by blast |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2093 |
let ?g = "\<lambda>x. if x \<le> 1/2 then (g1 \<circ> (\<lambda>t. 2 * t)) x else (g2 \<circ> (\<lambda>t. 2 * t -1)) x" |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2094 |
show "\<exists>g. continuous_map ?T01 X g \<and> g 0 = x \<and> g 1 = z" |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2095 |
proof (intro exI conjI) |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2096 |
show "continuous_map (subtopology euclideanreal {0..1}) X ?g" |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2097 |
proof (intro continuous_map_cases_le continuous_map_compose, force, force) |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2098 |
show "continuous_map (subtopology ?T01 {x \<in> topspace ?T01. x \<le> 1/2}) ?T01 ((*) 2)" |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2099 |
by (auto simp: continuous_map_in_subtopology continuous_map_from_subtopology) |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2100 |
have "continuous_map |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2101 |
(subtopology (top_of_set {0..1}) {x. 0 \<le> x \<and> x \<le> 1 \<and> 1 \<le> x * 2}) |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2102 |
euclideanreal (\<lambda>t. 2 * t - 1)" |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2103 |
by (intro continuous_intros) (force intro: continuous_map_from_subtopology) |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2104 |
then show "continuous_map (subtopology ?T01 {x \<in> topspace ?T01. 1/2 \<le> x}) ?T01 (\<lambda>t. 2 * t - 1)" |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2105 |
by (force simp: continuous_map_in_subtopology) |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2106 |
show "(g1 \<circ> (*) 2) x = (g2 \<circ> (\<lambda>t. 2 * t - 1)) x" if "x \<in> topspace ?T01" "x = 1/2" for x |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2107 |
using that by (simp add: g2(2) mult.commute continuous_map_from_subtopology) |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2108 |
qed (auto simp: g1 g2) |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2109 |
qed (auto simp: g1 g2) |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2110 |
qed |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2111 |
|
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2112 |
lemma path_component_of_mono: |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2113 |
"\<lbrakk>path_component_of (subtopology X S) x y; S \<subseteq> T\<rbrakk> \<Longrightarrow> path_component_of (subtopology X T) x y" |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2114 |
unfolding path_component_of_def |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2115 |
by (metis subsetD pathin_subtopology) |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2116 |
|
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2117 |
lemma path_component_of: |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2118 |
"path_component_of X x y \<longleftrightarrow> (\<exists>T. path_connectedin X T \<and> x \<in> T \<and> y \<in> T)" |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2119 |
apply (auto simp: path_component_of_def) |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2120 |
using path_connectedin_path_image apply fastforce |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2121 |
apply (metis path_connectedin) |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2122 |
done |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2123 |
|
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2124 |
lemma path_component_of_set: |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2125 |
"path_component_of X x y \<longleftrightarrow> (\<exists>g. pathin X g \<and> g 0 = x \<and> g 1 = y)" |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2126 |
by (auto simp: path_component_of_def) |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2127 |
|
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2128 |
lemma path_component_of_subset_topspace: |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2129 |
"Collect(path_component_of X x) \<subseteq> topspace X" |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2130 |
using path_component_in_topspace by fastforce |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2131 |
|
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2132 |
lemma path_component_of_eq_empty: |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2133 |
"Collect(path_component_of X x) = {} \<longleftrightarrow> (x \<notin> topspace X)" |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2134 |
using path_component_in_topspace path_component_of_refl by fastforce |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2135 |
|
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2136 |
lemma path_connected_space_iff_path_component: |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2137 |
"path_connected_space X \<longleftrightarrow> (\<forall>x \<in> topspace X. \<forall>y \<in> topspace X. path_component_of X x y)" |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2138 |
by (simp add: path_component_of path_connected_space_subconnected) |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2139 |
|
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2140 |
lemma path_connected_space_imp_path_component_of: |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2141 |
"\<lbrakk>path_connected_space X; a \<in> topspace X; b \<in> topspace X\<rbrakk> |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2142 |
\<Longrightarrow> path_component_of X a b" |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2143 |
by (simp add: path_connected_space_iff_path_component) |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2144 |
|
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2145 |
lemma path_connected_space_path_component_set: |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2146 |
"path_connected_space X \<longleftrightarrow> (\<forall>x \<in> topspace X. Collect(path_component_of X x) = topspace X)" |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2147 |
using path_component_of_subset_topspace path_connected_space_iff_path_component by fastforce |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2148 |
|
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2149 |
lemma path_component_of_maximal: |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2150 |
"\<lbrakk>path_connectedin X s; x \<in> s\<rbrakk> \<Longrightarrow> s \<subseteq> Collect(path_component_of X x)" |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2151 |
using path_component_of by fastforce |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2152 |
|
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2153 |
lemma path_component_of_equiv: |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2154 |
"path_component_of X x y \<longleftrightarrow> x \<in> topspace X \<and> y \<in> topspace X \<and> path_component_of X x = path_component_of X y" |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2155 |
(is "?lhs = ?rhs") |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2156 |
proof |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2157 |
assume ?lhs |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2158 |
then show ?rhs |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2159 |
apply (simp add: fun_eq_iff path_component_in_topspace) |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2160 |
apply (meson path_component_of_sym path_component_of_trans) |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2161 |
done |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2162 |
qed (simp add: path_component_of_refl) |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2163 |
|
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2164 |
lemma path_component_of_disjoint: |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2165 |
"disjnt (Collect (path_component_of X x)) (Collect (path_component_of X y)) \<longleftrightarrow> |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2166 |
~(path_component_of X x y)" |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2167 |
by (force simp: disjnt_def path_component_of_eq_empty path_component_of_equiv) |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2168 |
|
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2169 |
lemma path_component_of_eq: |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2170 |
"path_component_of X x = path_component_of X y \<longleftrightarrow> |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2171 |
(x \<notin> topspace X) \<and> (y \<notin> topspace X) \<or> |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2172 |
x \<in> topspace X \<and> y \<in> topspace X \<and> path_component_of X x y" |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2173 |
by (metis Collect_empty_eq_bot path_component_of_eq_empty path_component_of_equiv) |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2174 |
|
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2175 |
lemma path_connectedin_path_component_of: |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2176 |
"path_connectedin X (Collect (path_component_of X x))" |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2177 |
proof - |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2178 |
have "\<And>y. path_component_of X x y |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2179 |
\<Longrightarrow> path_component_of (subtopology X (Collect (path_component_of X x))) x y" |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2180 |
by (meson path_component_of path_component_of_maximal path_connectedin_subtopology) |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2181 |
then show ?thesis |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2182 |
apply (simp add: path_connectedin_def path_component_of_subset_topspace path_connected_space_iff_path_component) |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2183 |
by (metis Int_absorb1 mem_Collect_eq path_component_of_equiv path_component_of_subset_topspace topspace_subtopology) |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2184 |
qed |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2185 |
|
70178
4900351361b0
Lindelöf spaces and supporting material
paulson <lp15@cam.ac.uk>
parents:
70136
diff
changeset
|
2186 |
lemma path_connectedin_euclidean [simp]: |
4900351361b0
Lindelöf spaces and supporting material
paulson <lp15@cam.ac.uk>
parents:
70136
diff
changeset
|
2187 |
"path_connectedin euclidean S \<longleftrightarrow> path_connected S" |
4900351361b0
Lindelöf spaces and supporting material
paulson <lp15@cam.ac.uk>
parents:
70136
diff
changeset
|
2188 |
by (auto simp: path_connectedin_def path_connected_space_iff_path_component path_connected_component) |
4900351361b0
Lindelöf spaces and supporting material
paulson <lp15@cam.ac.uk>
parents:
70136
diff
changeset
|
2189 |
|
4900351361b0
Lindelöf spaces and supporting material
paulson <lp15@cam.ac.uk>
parents:
70136
diff
changeset
|
2190 |
lemma path_connected_space_euclidean_subtopology [simp]: |
4900351361b0
Lindelöf spaces and supporting material
paulson <lp15@cam.ac.uk>
parents:
70136
diff
changeset
|
2191 |
"path_connected_space(subtopology euclidean S) \<longleftrightarrow> path_connected S" |
4900351361b0
Lindelöf spaces and supporting material
paulson <lp15@cam.ac.uk>
parents:
70136
diff
changeset
|
2192 |
using path_connectedin_topspace by force |
4900351361b0
Lindelöf spaces and supporting material
paulson <lp15@cam.ac.uk>
parents:
70136
diff
changeset
|
2193 |
|
69939
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2194 |
lemma Union_path_components_of: |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2195 |
"\<Union>(path_components_of X) = topspace X" |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2196 |
by (auto simp: path_components_of_def path_component_of_equiv) |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2197 |
|
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2198 |
lemma path_components_of_maximal: |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2199 |
"\<lbrakk>C \<in> path_components_of X; path_connectedin X S; ~disjnt C S\<rbrakk> \<Longrightarrow> S \<subseteq> C" |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2200 |
apply (auto simp: path_components_of_def path_component_of_equiv) |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2201 |
using path_component_of_maximal path_connectedin_def apply fastforce |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2202 |
by (meson disjnt_subset2 path_component_of_disjoint path_component_of_equiv path_component_of_maximal) |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2203 |
|
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2204 |
lemma pairwise_disjoint_path_components_of: |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2205 |
"pairwise disjnt (path_components_of X)" |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2206 |
by (auto simp: path_components_of_def pairwise_def path_component_of_disjoint path_component_of_equiv) |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2207 |
|
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2208 |
lemma complement_path_components_of_Union: |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2209 |
"C \<in> path_components_of X |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2210 |
\<Longrightarrow> topspace X - C = \<Union>(path_components_of X - {C})" |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2211 |
by (metis Diff_cancel Diff_subset Union_path_components_of cSup_singleton diff_Union_pairwise_disjoint insert_subset pairwise_disjoint_path_components_of) |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2212 |
|
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2213 |
lemma nonempty_path_components_of: |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2214 |
"C \<in> path_components_of X \<Longrightarrow> (C \<noteq> {})" |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2215 |
apply (clarsimp simp: path_components_of_def path_component_of_eq_empty) |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2216 |
by (meson path_component_of_refl) |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2217 |
|
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2218 |
lemma path_components_of_subset: "C \<in> path_components_of X \<Longrightarrow> C \<subseteq> topspace X" |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2219 |
by (auto simp: path_components_of_def path_component_of_equiv) |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2220 |
|
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2221 |
lemma path_connectedin_path_components_of: |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2222 |
"C \<in> path_components_of X \<Longrightarrow> path_connectedin X C" |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2223 |
by (auto simp: path_components_of_def path_connectedin_path_component_of) |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2224 |
|
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2225 |
lemma path_component_in_path_components_of: |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2226 |
"Collect (path_component_of X a) \<in> path_components_of X \<longleftrightarrow> a \<in> topspace X" |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2227 |
apply (rule iffI) |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2228 |
using nonempty_path_components_of path_component_of_eq_empty apply fastforce |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2229 |
by (simp add: path_components_of_def) |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2230 |
|
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2231 |
lemma path_connectedin_Union: |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2232 |
assumes \<A>: "\<And>S. S \<in> \<A> \<Longrightarrow> path_connectedin X S" "\<Inter>\<A> \<noteq> {}" |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2233 |
shows "path_connectedin X (\<Union>\<A>)" |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2234 |
proof - |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2235 |
obtain a where "\<And>S. S \<in> \<A> \<Longrightarrow> a \<in> S" |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2236 |
using assms by blast |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2237 |
then have "\<And>x. x \<in> topspace (subtopology X (\<Union>\<A>)) \<Longrightarrow> path_component_of (subtopology X (\<Union>\<A>)) a x" |
71172 | 2238 |
apply (simp) |
69939
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2239 |
by (meson Union_upper \<A> path_component_of path_connectedin_subtopology) |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2240 |
then show ?thesis |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2241 |
using \<A> unfolding path_connectedin_def |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2242 |
by (metis Sup_le_iff path_component_of_equiv path_connected_space_iff_path_component) |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2243 |
qed |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2244 |
|
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2245 |
lemma path_connectedin_Un: |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2246 |
"\<lbrakk>path_connectedin X S; path_connectedin X T; S \<inter> T \<noteq> {}\<rbrakk> |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2247 |
\<Longrightarrow> path_connectedin X (S \<union> T)" |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2248 |
by (blast intro: path_connectedin_Union [of "{S,T}", simplified]) |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2249 |
|
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2250 |
lemma path_connected_space_iff_components_eq: |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2251 |
"path_connected_space X \<longleftrightarrow> |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2252 |
(\<forall>C \<in> path_components_of X. \<forall>C' \<in> path_components_of X. C = C')" |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2253 |
unfolding path_components_of_def |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2254 |
proof (intro iffI ballI) |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2255 |
assume "\<forall>C \<in> path_component_of_set X ` topspace X. |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2256 |
\<forall>C' \<in> path_component_of_set X ` topspace X. C = C'" |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2257 |
then show "path_connected_space X" |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2258 |
using path_component_of_refl path_connected_space_iff_path_component by fastforce |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2259 |
qed (auto simp: path_connected_space_path_component_set) |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2260 |
|
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2261 |
lemma path_components_of_eq_empty: |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2262 |
"path_components_of X = {} \<longleftrightarrow> topspace X = {}" |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2263 |
using Union_path_components_of nonempty_path_components_of by fastforce |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2264 |
|
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2265 |
lemma path_components_of_empty_space: |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2266 |
"topspace X = {} \<Longrightarrow> path_components_of X = {}" |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2267 |
by (simp add: path_components_of_eq_empty) |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2268 |
|
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2269 |
lemma path_components_of_subset_singleton: |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2270 |
"path_components_of X \<subseteq> {S} \<longleftrightarrow> |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2271 |
path_connected_space X \<and> (topspace X = {} \<or> topspace X = S)" |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2272 |
proof (cases "topspace X = {}") |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2273 |
case True |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2274 |
then show ?thesis |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2275 |
by (auto simp: path_components_of_empty_space path_connected_space_topspace_empty) |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2276 |
next |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2277 |
case False |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2278 |
have "(path_components_of X = {S}) \<longleftrightarrow> (path_connected_space X \<and> topspace X = S)" |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2279 |
proof (intro iffI conjI) |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2280 |
assume L: "path_components_of X = {S}" |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2281 |
then show "path_connected_space X" |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2282 |
by (simp add: path_connected_space_iff_components_eq) |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2283 |
show "topspace X = S" |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2284 |
by (metis L ccpo_Sup_singleton [of S] Union_path_components_of) |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2285 |
next |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2286 |
assume R: "path_connected_space X \<and> topspace X = S" |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2287 |
then show "path_components_of X = {S}" |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2288 |
using ccpo_Sup_singleton [of S] |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2289 |
by (metis False all_not_in_conv insert_iff mk_disjoint_insert path_component_in_path_components_of path_connected_space_iff_components_eq path_connected_space_path_component_set) |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2290 |
qed |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2291 |
with False show ?thesis |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2292 |
by (simp add: path_components_of_eq_empty subset_singleton_iff) |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2293 |
qed |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2294 |
|
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2295 |
lemma path_connected_space_iff_components_subset_singleton: |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2296 |
"path_connected_space X \<longleftrightarrow> (\<exists>a. path_components_of X \<subseteq> {a})" |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2297 |
by (simp add: path_components_of_subset_singleton) |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2298 |
|
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2299 |
lemma path_components_of_eq_singleton: |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2300 |
"path_components_of X = {S} \<longleftrightarrow> path_connected_space X \<and> topspace X \<noteq> {} \<and> S = topspace X" |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2301 |
by (metis cSup_singleton insert_not_empty path_components_of_subset_singleton subset_singleton_iff) |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2302 |
|
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2303 |
lemma path_components_of_path_connected_space: |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2304 |
"path_connected_space X \<Longrightarrow> path_components_of X = (if topspace X = {} then {} else {topspace X})" |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2305 |
by (simp add: path_components_of_eq_empty path_components_of_eq_singleton) |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2306 |
|
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2307 |
lemma path_component_subset_connected_component_of: |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2308 |
"path_component_of_set X x \<subseteq> connected_component_of_set X x" |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2309 |
proof (cases "x \<in> topspace X") |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2310 |
case True |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2311 |
then show ?thesis |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2312 |
by (simp add: connected_component_of_maximal path_component_of_refl path_connectedin_imp_connectedin path_connectedin_path_component_of) |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2313 |
next |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2314 |
case False |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2315 |
then show ?thesis |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2316 |
using path_component_of_eq_empty by fastforce |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2317 |
qed |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2318 |
|
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2319 |
lemma exists_path_component_of_superset: |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2320 |
assumes S: "path_connectedin X S" and ne: "topspace X \<noteq> {}" |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2321 |
obtains C where "C \<in> path_components_of X" "S \<subseteq> C" |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2322 |
proof (cases "S = {}") |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2323 |
case True |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2324 |
then show ?thesis |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2325 |
using ne path_components_of_eq_empty that by fastforce |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2326 |
next |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2327 |
case False |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2328 |
then obtain a where "a \<in> S" |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2329 |
by blast |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2330 |
show ?thesis |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2331 |
proof |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2332 |
show "Collect (path_component_of X a) \<in> path_components_of X" |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2333 |
by (meson \<open>a \<in> S\<close> S subsetD path_component_in_path_components_of path_connectedin_subset_topspace) |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2334 |
show "S \<subseteq> Collect (path_component_of X a)" |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2335 |
by (simp add: S \<open>a \<in> S\<close> path_component_of_maximal) |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2336 |
qed |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2337 |
qed |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2338 |
|
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2339 |
lemma path_component_of_eq_overlap: |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2340 |
"path_component_of X x = path_component_of X y \<longleftrightarrow> |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2341 |
(x \<notin> topspace X) \<and> (y \<notin> topspace X) \<or> |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2342 |
Collect (path_component_of X x) \<inter> Collect (path_component_of X y) \<noteq> {}" |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2343 |
by (metis disjnt_def empty_iff inf_bot_right mem_Collect_eq path_component_of_disjoint path_component_of_eq path_component_of_eq_empty) |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2344 |
|
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2345 |
lemma path_component_of_nonoverlap: |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2346 |
"Collect (path_component_of X x) \<inter> Collect (path_component_of X y) = {} \<longleftrightarrow> |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2347 |
(x \<notin> topspace X) \<or> (y \<notin> topspace X) \<or> |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2348 |
path_component_of X x \<noteq> path_component_of X y" |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2349 |
by (metis inf.idem path_component_of_eq_empty path_component_of_eq_overlap) |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2350 |
|
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2351 |
lemma path_component_of_overlap: |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2352 |
"Collect (path_component_of X x) \<inter> Collect (path_component_of X y) \<noteq> {} \<longleftrightarrow> |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2353 |
x \<in> topspace X \<and> y \<in> topspace X \<and> path_component_of X x = path_component_of X y" |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2354 |
by (meson path_component_of_nonoverlap) |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2355 |
|
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2356 |
lemma path_components_of_disjoint: |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2357 |
"\<lbrakk>C \<in> path_components_of X; C' \<in> path_components_of X\<rbrakk> \<Longrightarrow> disjnt C C' \<longleftrightarrow> C \<noteq> C'" |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2358 |
by (auto simp: path_components_of_def path_component_of_disjoint path_component_of_equiv) |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2359 |
|
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2360 |
lemma path_components_of_overlap: |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2361 |
"\<lbrakk>C \<in> path_components_of X; C' \<in> path_components_of X\<rbrakk> \<Longrightarrow> C \<inter> C' \<noteq> {} \<longleftrightarrow> C = C'" |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2362 |
by (auto simp: path_components_of_def path_component_of_equiv) |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2363 |
|
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2364 |
lemma path_component_of_unique: |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2365 |
"\<lbrakk>x \<in> C; path_connectedin X C; \<And>C'. \<lbrakk>x \<in> C'; path_connectedin X C'\<rbrakk> \<Longrightarrow> C' \<subseteq> C\<rbrakk> |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2366 |
\<Longrightarrow> Collect (path_component_of X x) = C" |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2367 |
by (meson subsetD eq_iff path_component_of_maximal path_connectedin_path_component_of) |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2368 |
|
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2369 |
lemma path_component_of_discrete_topology [simp]: |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2370 |
"Collect (path_component_of (discrete_topology U) x) = (if x \<in> U then {x} else {})" |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2371 |
proof - |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2372 |
have "\<And>C'. \<lbrakk>x \<in> C'; path_connectedin (discrete_topology U) C'\<rbrakk> \<Longrightarrow> C' \<subseteq> {x}" |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2373 |
by (metis path_connectedin_discrete_topology subsetD singletonD) |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2374 |
then have "x \<in> U \<Longrightarrow> Collect (path_component_of (discrete_topology U) x) = {x}" |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2375 |
by (simp add: path_component_of_unique) |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2376 |
then show ?thesis |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2377 |
using path_component_in_topspace by fastforce |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2378 |
qed |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2379 |
|
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2380 |
lemma path_component_of_discrete_topology_iff [simp]: |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2381 |
"path_component_of (discrete_topology U) x y \<longleftrightarrow> x \<in> U \<and> y=x" |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2382 |
by (metis empty_iff insertI1 mem_Collect_eq path_component_of_discrete_topology singletonD) |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2383 |
|
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2384 |
lemma path_components_of_discrete_topology [simp]: |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2385 |
"path_components_of (discrete_topology U) = (\<lambda>x. {x}) ` U" |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2386 |
by (auto simp: path_components_of_def image_def fun_eq_iff) |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2387 |
|
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2388 |
lemma homeomorphic_map_path_component_of: |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2389 |
assumes f: "homeomorphic_map X Y f" and x: "x \<in> topspace X" |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2390 |
shows "Collect (path_component_of Y (f x)) = f ` Collect(path_component_of X x)" |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2391 |
proof - |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2392 |
obtain g where g: "homeomorphic_maps X Y f g" |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2393 |
using f homeomorphic_map_maps by blast |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2394 |
show ?thesis |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2395 |
proof |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2396 |
have "Collect (path_component_of Y (f x)) \<subseteq> topspace Y" |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2397 |
by (simp add: path_component_of_subset_topspace) |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2398 |
moreover have "g ` Collect(path_component_of Y (f x)) \<subseteq> Collect (path_component_of X (g (f x)))" |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2399 |
using g x unfolding homeomorphic_maps_def |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2400 |
by (metis f homeomorphic_imp_surjective_map imageI mem_Collect_eq path_component_of_maximal path_component_of_refl path_connectedin_continuous_map_image path_connectedin_path_component_of) |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2401 |
ultimately show "Collect (path_component_of Y (f x)) \<subseteq> f ` Collect (path_component_of X x)" |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2402 |
using g x unfolding homeomorphic_maps_def continuous_map_def image_iff subset_iff |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2403 |
by metis |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2404 |
show "f ` Collect (path_component_of X x) \<subseteq> Collect (path_component_of Y (f x))" |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2405 |
proof (rule path_component_of_maximal) |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2406 |
show "path_connectedin Y (f ` Collect (path_component_of X x))" |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2407 |
by (meson f homeomorphic_map_path_connectedness_eq path_connectedin_path_component_of) |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2408 |
qed (simp add: path_component_of_refl x) |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2409 |
qed |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2410 |
qed |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2411 |
|
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2412 |
lemma homeomorphic_map_path_components_of: |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2413 |
assumes "homeomorphic_map X Y f" |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2414 |
shows "path_components_of Y = (image f) ` (path_components_of X)" |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2415 |
unfolding path_components_of_def homeomorphic_imp_surjective_map [OF assms, symmetric] |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2416 |
apply safe |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2417 |
using assms homeomorphic_map_path_component_of apply fastforce |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2418 |
by (metis assms homeomorphic_map_path_component_of imageI) |
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2419 |
|
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2420 |
|
60420 | 2421 |
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
|
2422 |
|
36583 | 2423 |
lemma path_connected_punctured_universe: |
37674
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
2424 |
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
|
2425 |
shows "path_connected (- {a::'a})" |
49653 | 2426 |
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
|
2427 |
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
|
2428 |
let ?B = "{x::'a. \<exists>i\<in>Basis. a \<bullet> i < x \<bullet> i}" |
36583 | 2429 |
|
49653 | 2430 |
have A: "path_connected ?A" |
2431 |
unfolding Collect_bex_eq |
|
37674
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
2432 |
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
|
2433 |
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
|
2434 |
assume "i \<in> Basis" |
53640 | 2435 |
then show "(\<Sum>i\<in>Basis. (a \<bullet> i - 1)*\<^sub>R i) \<in> {x::'a. x \<bullet> i < a \<bullet> i}" |
2436 |
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
|
2437 |
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
|
2438 |
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
|
2439 |
by (simp add: inner_commute) |
37674
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
2440 |
qed |
53640 | 2441 |
have B: "path_connected ?B" |
2442 |
unfolding Collect_bex_eq |
|
37674
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
2443 |
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
|
2444 |
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
|
2445 |
assume "i \<in> Basis" |
53640 | 2446 |
then show "(\<Sum>i\<in>Basis. (a \<bullet> i + 1) *\<^sub>R i) \<in> {x::'a. a \<bullet> i < x \<bullet> i}" |
2447 |
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
|
2448 |
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
|
2449 |
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
|
2450 |
by (simp add: inner_commute) |
37674
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
2451 |
qed |
53640 | 2452 |
obtain S :: "'a set" where "S \<subseteq> Basis" and "card S = Suc (Suc 0)" |
2453 |
using ex_card[OF assms] |
|
2454 |
by auto |
|
2455 |
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
|
2456 |
unfolding card_Suc_eq by auto |
53640 | 2457 |
then have "a + b0 - b1 \<in> ?A \<inter> ?B" |
2458 |
by (auto simp: inner_simps inner_Basis) |
|
2459 |
then have "?A \<inter> ?B \<noteq> {}" |
|
2460 |
by fast |
|
37674
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
2461 |
with A B have "path_connected (?A \<union> ?B)" |
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
2462 |
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
|
2463 |
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
|
2464 |
unfolding neq_iff bex_disj_distrib Collect_disj_eq .. |
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
2465 |
also have "\<dots> = {x. x \<noteq> a}" |
53640 | 2466 |
unfolding euclidean_eq_iff [where 'a='a] |
2467 |
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
|
2468 |
also have "\<dots> = - {a}" |
53640 | 2469 |
by auto |
37674
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
2470 |
finally show ?thesis . |
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
2471 |
qed |
36583 | 2472 |
|
64006
0de4736dad8b
new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
63978
diff
changeset
|
2473 |
corollary connected_punctured_universe: |
0de4736dad8b
new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
63978
diff
changeset
|
2474 |
"2 \<le> DIM('N::euclidean_space) \<Longrightarrow> connected(- {a::'N})" |
0de4736dad8b
new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
63978
diff
changeset
|
2475 |
by (simp add: path_connected_punctured_universe path_connected_imp_connected) |
0de4736dad8b
new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
63978
diff
changeset
|
2476 |
|
68607
67bb59e49834
make theorem, corollary, and proposition %important for HOL-Analysis manual
immler
parents:
68532
diff
changeset
|
2477 |
proposition path_connected_sphere: |
64788
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
2478 |
fixes a :: "'a :: euclidean_space" |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
2479 |
assumes "2 \<le> DIM('a)" |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
2480 |
shows "path_connected(sphere a r)" |
68607
67bb59e49834
make theorem, corollary, and proposition %important for HOL-Analysis manual
immler
parents:
68532
diff
changeset
|
2481 |
proof (cases r "0::real" rule: linorder_cases) |
64788
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
2482 |
case less |
53640 | 2483 |
then show ?thesis |
71172 | 2484 |
by (simp) |
37674
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
2485 |
next |
64788
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
2486 |
case equal |
53640 | 2487 |
then show ?thesis |
71172 | 2488 |
by (simp) |
37674
f86de9c00c47
convert theorem path_connected_sphere to euclidean_space class
huffman
parents:
37489
diff
changeset
|
2489 |
next |
64788
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
2490 |
case greater |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
2491 |
then have eq: "(sphere (0::'a) r) = (\<lambda>x. (r / norm x) *\<^sub>R x) ` (- {0::'a})" |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
2492 |
by (force simp: image_iff split: if_split_asm) |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
2493 |
have "continuous_on (- {0::'a}) (\<lambda>x. (r / norm x) *\<^sub>R x)" |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
2494 |
by (intro continuous_intros) auto |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
2495 |
then have "path_connected ((\<lambda>x. (r / norm x) *\<^sub>R x) ` (- {0::'a}))" |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
2496 |
by (intro path_connected_continuous_image path_connected_punctured_universe assms) |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
2497 |
with eq have "path_connected (sphere (0::'a) r)" |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
2498 |
by auto |
67399 | 2499 |
then have "path_connected((+) a ` (sphere (0::'a) r))" |
64788
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
2500 |
by (simp add: path_connected_translation) |
53640 | 2501 |
then show ?thesis |
64788
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
2502 |
by (metis add.right_neutral sphere_translation) |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
2503 |
qed |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
2504 |
|
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
2505 |
lemma connected_sphere: |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
2506 |
fixes a :: "'a :: euclidean_space" |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
2507 |
assumes "2 \<le> DIM('a)" |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
2508 |
shows "connected(sphere a r)" |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
2509 |
using path_connected_sphere [OF assms] |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
2510 |
by (simp add: path_connected_imp_connected) |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
2511 |
|
36583 | 2512 |
|
61426
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2513 |
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
|
2514 |
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
|
2515 |
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
|
2516 |
shows "path_connected (- s)" |
64788
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
2517 |
proof (cases "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
|
2518 |
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
|
2519 |
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
|
2520 |
next |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2521 |
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
|
2522 |
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
|
2523 |
{ fix x y assume "x \<notin> s" "y \<notin> s" |
61808 | 2524 |
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
|
2525 |
then have bxy: "bounded(insert x (insert y s))" |
61808 | 2526 |
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
|
2527 |
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
|
2528 |
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
|
2529 |
using bounded_subset_ballD [OF bxy, of a] by (auto simp: dist_norm) |
63040 | 2530 |
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
|
2531 |
{ 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
|
2532 |
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
|
2533 |
have CC: "1 \<le> 1 + (C - 1) * u" |
70817
dd675800469d
dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents:
70802
diff
changeset
|
2534 |
using \<open>x \<noteq> a\<close> \<open>0 \<le> u\<close> Bx |
dd675800469d
dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents:
70802
diff
changeset
|
2535 |
by (auto simp add: C_def norm_minus_commute) |
61426
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2536 |
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
|
2537 |
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
|
2538 |
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
|
2539 |
(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
|
2540 |
by (simp add: algebra_simps) |
68096 | 2541 |
also have "\<dots> = (1 + (u / (1 + C * u - u))) *\<^sub>R a + (1 + (u / (1 + C * u - u))) *\<^sub>R x" |
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 |
using CC by (simp add: field_simps) |
68096 | 2543 |
also have "\<dots> = x + (1 + (u / (1 + C * u - u))) *\<^sub>R a + (u / (1 + C * u - u)) *\<^sub>R x" |
61426
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2544 |
by (simp add: algebra_simps) |
68096 | 2545 |
also have "\<dots> = x + ((1 / (1 + C * u - u)) *\<^sub>R 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
|
2546 |
((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
|
2547 |
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
|
2548 |
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
|
2549 |
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
|
2550 |
have False |
61808 | 2551 |
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
|
2552 |
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
|
2553 |
apply (drule_tac x=a in bspec) |
61808 | 2554 |
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
|
2555 |
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
|
2556 |
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
|
2557 |
apply (drule_tac x="1 / (1 + (C - 1) * u)" in spec) |
61808 | 2558 |
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
|
2559 |
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
|
2560 |
done |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2561 |
} |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2562 |
then have pcx: "path_component (- s) x (a + C *\<^sub>R (x - a))" |
69939
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2563 |
by (force simp: closed_segment_def intro!: path_component_linepath) |
67443
3abf6a722518
standardized towards new-style formal comments: isabelle update_comments;
wenzelm
parents:
67399
diff
changeset
|
2564 |
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
|
2565 |
{ 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
|
2566 |
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
|
2567 |
have DD: "1 \<le> 1 + (D - 1) * u" |
70817
dd675800469d
dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents:
70802
diff
changeset
|
2568 |
using \<open>y \<noteq> a\<close> \<open>0 \<le> u\<close> By |
dd675800469d
dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents:
70802
diff
changeset
|
2569 |
by (auto simp add: D_def norm_minus_commute) |
61426
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2570 |
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
|
2571 |
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
|
2572 |
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
|
2573 |
(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
|
2574 |
by (simp add: algebra_simps) |
68096 | 2575 |
also have "\<dots> = (1 + (u / (1 + D * u - u))) *\<^sub>R a + (1 + (u / (1 + D * u - u))) *\<^sub>R 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
|
2576 |
using DD by (simp add: field_simps) |
68096 | 2577 |
also have "\<dots> = y + (1 + (u / (1 + D * u - u))) *\<^sub>R a + (u / (1 + D * u - u)) *\<^sub>R 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
|
2578 |
by (simp add: algebra_simps) |
68096 | 2579 |
also have "\<dots> = y + ((1 / (1 + D * u - u)) *\<^sub>R 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
|
2580 |
((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
|
2581 |
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
|
2582 |
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
|
2583 |
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
|
2584 |
have False |
61808 | 2585 |
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
|
2586 |
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
|
2587 |
apply (drule_tac x=a in bspec) |
61808 | 2588 |
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
|
2589 |
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
|
2590 |
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
|
2591 |
apply (drule_tac x="1 / (1 + (D - 1) * u)" in spec) |
61808 | 2592 |
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
|
2593 |
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
|
2594 |
done |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2595 |
} |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2596 |
then have pdy: "path_component (- s) y (a + D *\<^sub>R (y - a))" |
69939
812ce526da33
new material on topology: products, etc. Some renamings, esp continuous_on_topo -> continuous_map
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
2597 |
by (force simp: closed_segment_def intro!: path_component_linepath) |
61426
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2598 |
have pyx: "path_component (- s) (a + D *\<^sub>R (y - a)) (a + C *\<^sub>R (x - a))" |
64788
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
2599 |
apply (rule path_component_of_subset [of "sphere a B"]) |
61808 | 2600 |
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
|
2601 |
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
|
2602 |
apply (rule path_connected_sphere [OF 2, of a B, simplified path_connected_component, rule_format]) |
64788
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
2603 |
using \<open>x \<noteq> a\<close> using \<open>y \<noteq> a\<close> B apply (auto simp: dist_norm 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
|
2604 |
done |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2605 |
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
|
2606 |
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
|
2607 |
} |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2608 |
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
|
2609 |
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
|
2610 |
qed |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2611 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2612 |
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
|
2613 |
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
|
2614 |
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
|
2615 |
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
|
2616 |
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
|
2617 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2618 |
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
|
2619 |
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
|
2620 |
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
|
2621 |
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
|
2622 |
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
|
2623 |
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
|
2624 |
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
|
2625 |
done |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2626 |
|
62381
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
2627 |
proposition connected_open_delete: |
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
2628 |
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
|
2629 |
shows "connected(S - {a::'N})" |
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
2630 |
proof (cases "a \<in> S") |
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
2631 |
case True |
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
2632 |
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
|
2633 |
using open_contains_cball_eq by blast |
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
2634 |
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
|
2635 |
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
|
2636 |
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
|
2637 |
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
|
2638 |
by auto |
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
2639 |
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
|
2640 |
by auto |
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
2641 |
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
|
2642 |
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
|
2643 |
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
|
2644 |
apply (rule UnionI [of "S - ball a (min \<epsilon> (dist a x) / 2)"]) |
68096 | 2645 |
using that \<open>0 < \<epsilon>\<close> apply simp_all |
62381
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
2646 |
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
|
2647 |
apply auto |
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
2648 |
done |
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
2649 |
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
|
2650 |
by auto |
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
2651 |
then show ?thesis |
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
2652 |
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
|
2653 |
next |
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
2654 |
case False then show ?thesis |
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
2655 |
by (simp add: \<open>connected S\<close>) |
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
2656 |
qed |
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
2657 |
|
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
2658 |
corollary path_connected_open_delete: |
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
2659 |
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
|
2660 |
shows "path_connected(S - {a::'N})" |
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
2661 |
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
|
2662 |
|
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
2663 |
corollary path_connected_punctured_ball: |
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
2664 |
"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
|
2665 |
by (simp add: path_connected_open_delete) |
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
2666 |
|
63151
82df5181d699
updated proof of Residue Theorem (form Wenda Li)
paulson <lp15@cam.ac.uk>
parents:
63126
diff
changeset
|
2667 |
corollary connected_punctured_ball: |
62381
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
2668 |
"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
|
2669 |
by (simp add: connected_open_delete) |
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
2670 |
|
63151
82df5181d699
updated proof of Residue Theorem (form Wenda Li)
paulson <lp15@cam.ac.uk>
parents:
63126
diff
changeset
|
2671 |
corollary connected_open_delete_finite: |
82df5181d699
updated proof of Residue Theorem (form Wenda Li)
paulson <lp15@cam.ac.uk>
parents:
63126
diff
changeset
|
2672 |
fixes S T::"'a::euclidean_space set" |
82df5181d699
updated proof of Residue Theorem (form Wenda Li)
paulson <lp15@cam.ac.uk>
parents:
63126
diff
changeset
|
2673 |
assumes S: "open S" "connected S" and 2: "2 \<le> DIM('a)" and "finite T" |
63594
bd218a9320b5
HOL-Multivariate_Analysis: rename theories for more descriptive names
hoelzl
parents:
63540
diff
changeset
|
2674 |
shows "connected(S - T)" |
bd218a9320b5
HOL-Multivariate_Analysis: rename theories for more descriptive names
hoelzl
parents:
63540
diff
changeset
|
2675 |
using \<open>finite T\<close> S |
63151
82df5181d699
updated proof of Residue Theorem (form Wenda Li)
paulson <lp15@cam.ac.uk>
parents:
63126
diff
changeset
|
2676 |
proof (induct T) |
82df5181d699
updated proof of Residue Theorem (form Wenda Li)
paulson <lp15@cam.ac.uk>
parents:
63126
diff
changeset
|
2677 |
case empty |
82df5181d699
updated proof of Residue Theorem (form Wenda Li)
paulson <lp15@cam.ac.uk>
parents:
63126
diff
changeset
|
2678 |
show ?case using \<open>connected S\<close> by simp |
82df5181d699
updated proof of Residue Theorem (form Wenda Li)
paulson <lp15@cam.ac.uk>
parents:
63126
diff
changeset
|
2679 |
next |
82df5181d699
updated proof of Residue Theorem (form Wenda Li)
paulson <lp15@cam.ac.uk>
parents:
63126
diff
changeset
|
2680 |
case (insert x F) |
82df5181d699
updated proof of Residue Theorem (form Wenda Li)
paulson <lp15@cam.ac.uk>
parents:
63126
diff
changeset
|
2681 |
then have "connected (S-F)" by auto |
82df5181d699
updated proof of Residue Theorem (form Wenda Li)
paulson <lp15@cam.ac.uk>
parents:
63126
diff
changeset
|
2682 |
moreover have "open (S - F)" using finite_imp_closed[OF \<open>finite F\<close>] \<open>open S\<close> by auto |
82df5181d699
updated proof of Residue Theorem (form Wenda Li)
paulson <lp15@cam.ac.uk>
parents:
63126
diff
changeset
|
2683 |
ultimately have "connected (S - F - {x})" using connected_open_delete[OF _ _ 2] by auto |
82df5181d699
updated proof of Residue Theorem (form Wenda Li)
paulson <lp15@cam.ac.uk>
parents:
63126
diff
changeset
|
2684 |
thus ?case by (metis Diff_insert) |
82df5181d699
updated proof of Residue Theorem (form Wenda Li)
paulson <lp15@cam.ac.uk>
parents:
63126
diff
changeset
|
2685 |
qed |
82df5181d699
updated proof of Residue Theorem (form Wenda Li)
paulson <lp15@cam.ac.uk>
parents:
63126
diff
changeset
|
2686 |
|
64788
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
2687 |
lemma sphere_1D_doubleton_zero: |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
2688 |
assumes 1: "DIM('a) = 1" and "r > 0" |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
2689 |
obtains x y::"'a::euclidean_space" |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
2690 |
where "sphere 0 r = {x,y} \<and> dist x y = 2*r" |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
2691 |
proof - |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
2692 |
obtain b::'a where b: "Basis = {b}" |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
2693 |
using 1 card_1_singletonE by blast |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
2694 |
show ?thesis |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
2695 |
proof (intro that conjI) |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
2696 |
have "x = norm x *\<^sub>R b \<or> x = - norm x *\<^sub>R b" if "r = norm x" for x |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
2697 |
proof - |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
2698 |
have xb: "(x \<bullet> b) *\<^sub>R b = x" |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
2699 |
using euclidean_representation [of x, unfolded b] by force |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
2700 |
then have "norm ((x \<bullet> b) *\<^sub>R b) = norm x" |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
2701 |
by simp |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
2702 |
with b have "\<bar>x \<bullet> b\<bar> = norm x" |
68310 | 2703 |
using norm_Basis by (simp add: b) |
64788
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
2704 |
with xb show ?thesis |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
2705 |
apply (simp add: abs_if split: if_split_asm) |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
2706 |
apply (metis add.inverse_inverse real_vector.scale_minus_left xb) |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
2707 |
done |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
2708 |
qed |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
2709 |
with \<open>r > 0\<close> b show "sphere 0 r = {r *\<^sub>R b, - r *\<^sub>R b}" |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
2710 |
by (force simp: sphere_def dist_norm) |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
2711 |
have "dist (r *\<^sub>R b) (- r *\<^sub>R b) = norm (r *\<^sub>R b + r *\<^sub>R b)" |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
2712 |
by (simp add: dist_norm) |
68096 | 2713 |
also have "\<dots> = norm ((2*r) *\<^sub>R b)" |
64788
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
2714 |
by (metis mult_2 scaleR_add_left) |
68096 | 2715 |
also have "\<dots> = 2*r" |
64788
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
2716 |
using \<open>r > 0\<close> b norm_Basis by fastforce |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
2717 |
finally show "dist (r *\<^sub>R b) (- r *\<^sub>R b) = 2*r" . |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
2718 |
qed |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
2719 |
qed |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
2720 |
|
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
2721 |
lemma sphere_1D_doubleton: |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
2722 |
fixes a :: "'a :: euclidean_space" |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
2723 |
assumes "DIM('a) = 1" and "r > 0" |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
2724 |
obtains x y where "sphere a r = {x,y} \<and> dist x y = 2*r" |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
2725 |
proof - |
67399 | 2726 |
have "sphere a r = (+) a ` sphere 0 r" |
64788
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
2727 |
by (metis add.right_neutral sphere_translation) |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
2728 |
then show ?thesis |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
2729 |
using sphere_1D_doubleton_zero [OF assms] |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
2730 |
by (metis (mono_tags, lifting) dist_add_cancel image_empty image_insert that) |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
2731 |
qed |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
2732 |
|
64006
0de4736dad8b
new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
63978
diff
changeset
|
2733 |
lemma psubset_sphere_Compl_connected: |
0de4736dad8b
new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
63978
diff
changeset
|
2734 |
fixes S :: "'a::euclidean_space set" |
0de4736dad8b
new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
63978
diff
changeset
|
2735 |
assumes S: "S \<subset> sphere a r" and "0 < r" and 2: "2 \<le> DIM('a)" |
0de4736dad8b
new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
63978
diff
changeset
|
2736 |
shows "connected(- S)" |
0de4736dad8b
new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
63978
diff
changeset
|
2737 |
proof - |
0de4736dad8b
new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
63978
diff
changeset
|
2738 |
have "S \<subseteq> sphere a r" |
0de4736dad8b
new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
63978
diff
changeset
|
2739 |
using S by blast |
0de4736dad8b
new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
63978
diff
changeset
|
2740 |
obtain b where "dist a b = r" and "b \<notin> S" |
0de4736dad8b
new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
63978
diff
changeset
|
2741 |
using S mem_sphere by blast |
0de4736dad8b
new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
63978
diff
changeset
|
2742 |
have CS: "- S = {x. dist a x \<le> r \<and> (x \<notin> S)} \<union> {x. r \<le> dist a x \<and> (x \<notin> S)}" |
68096 | 2743 |
by auto |
64006
0de4736dad8b
new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
63978
diff
changeset
|
2744 |
have "{x. dist a x \<le> r \<and> x \<notin> S} \<inter> {x. r \<le> dist a x \<and> x \<notin> S} \<noteq> {}" |
0de4736dad8b
new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
63978
diff
changeset
|
2745 |
using \<open>b \<notin> S\<close> \<open>dist a b = r\<close> by blast |
0de4736dad8b
new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
63978
diff
changeset
|
2746 |
moreover have "connected {x. dist a x \<le> r \<and> x \<notin> S}" |
0de4736dad8b
new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
63978
diff
changeset
|
2747 |
apply (rule connected_intermediate_closure [of "ball a r"]) |
0de4736dad8b
new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
63978
diff
changeset
|
2748 |
using assms by auto |
0de4736dad8b
new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
63978
diff
changeset
|
2749 |
moreover |
0de4736dad8b
new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
63978
diff
changeset
|
2750 |
have "connected {x. r \<le> dist a x \<and> x \<notin> S}" |
0de4736dad8b
new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
63978
diff
changeset
|
2751 |
apply (rule connected_intermediate_closure [of "- cball a r"]) |
0de4736dad8b
new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
63978
diff
changeset
|
2752 |
using assms apply (auto intro: connected_complement_bounded_convex) |
0de4736dad8b
new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
63978
diff
changeset
|
2753 |
apply (metis ComplI interior_cball interior_closure mem_ball not_less) |
0de4736dad8b
new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
63978
diff
changeset
|
2754 |
done |
0de4736dad8b
new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
63978
diff
changeset
|
2755 |
ultimately show ?thesis |
0de4736dad8b
new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
63978
diff
changeset
|
2756 |
by (simp add: CS connected_Un) |
0de4736dad8b
new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
63978
diff
changeset
|
2757 |
qed |
0de4736dad8b
new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
63978
diff
changeset
|
2758 |
|
64788
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
2759 |
|
66793
deabce3ccf1f
new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
66456
diff
changeset
|
2760 |
subsection\<open>Every annulus is a connected set\<close> |
deabce3ccf1f
new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
66456
diff
changeset
|
2761 |
|
deabce3ccf1f
new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
66456
diff
changeset
|
2762 |
lemma path_connected_2DIM_I: |
deabce3ccf1f
new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
66456
diff
changeset
|
2763 |
fixes a :: "'N::euclidean_space" |
deabce3ccf1f
new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
66456
diff
changeset
|
2764 |
assumes 2: "2 \<le> DIM('N)" and pc: "path_connected {r. 0 \<le> r \<and> P r}" |
deabce3ccf1f
new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
66456
diff
changeset
|
2765 |
shows "path_connected {x. P(norm(x - a))}" |
deabce3ccf1f
new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
66456
diff
changeset
|
2766 |
proof - |
67399 | 2767 |
have "{x. P(norm(x - a))} = (+) a ` {x. P(norm x)}" |
66793
deabce3ccf1f
new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
66456
diff
changeset
|
2768 |
by force |
deabce3ccf1f
new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
66456
diff
changeset
|
2769 |
moreover have "path_connected {x::'N. P(norm x)}" |
deabce3ccf1f
new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
66456
diff
changeset
|
2770 |
proof - |
deabce3ccf1f
new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
66456
diff
changeset
|
2771 |
let ?D = "{x. 0 \<le> x \<and> P x} \<times> sphere (0::'N) 1" |
deabce3ccf1f
new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
66456
diff
changeset
|
2772 |
have "x \<in> (\<lambda>z. fst z *\<^sub>R snd z) ` ?D" |
deabce3ccf1f
new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
66456
diff
changeset
|
2773 |
if "P (norm x)" for x::'N |
deabce3ccf1f
new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
66456
diff
changeset
|
2774 |
proof (cases "x=0") |
deabce3ccf1f
new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
66456
diff
changeset
|
2775 |
case True |
deabce3ccf1f
new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
66456
diff
changeset
|
2776 |
with that show ?thesis |
deabce3ccf1f
new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
66456
diff
changeset
|
2777 |
apply (simp add: image_iff) |
deabce3ccf1f
new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
66456
diff
changeset
|
2778 |
apply (rule_tac x=0 in exI, simp) |
deabce3ccf1f
new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
66456
diff
changeset
|
2779 |
using vector_choose_size zero_le_one by blast |
deabce3ccf1f
new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
66456
diff
changeset
|
2780 |
next |
deabce3ccf1f
new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
66456
diff
changeset
|
2781 |
case False |
deabce3ccf1f
new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
66456
diff
changeset
|
2782 |
with that show ?thesis |
deabce3ccf1f
new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
66456
diff
changeset
|
2783 |
by (rule_tac x="(norm x, x /\<^sub>R norm x)" in image_eqI) auto |
deabce3ccf1f
new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
66456
diff
changeset
|
2784 |
qed |
deabce3ccf1f
new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
66456
diff
changeset
|
2785 |
then have *: "{x::'N. P(norm x)} = (\<lambda>z. fst z *\<^sub>R snd z) ` ?D" |
deabce3ccf1f
new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
66456
diff
changeset
|
2786 |
by auto |
deabce3ccf1f
new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
66456
diff
changeset
|
2787 |
have "continuous_on ?D (\<lambda>z:: real\<times>'N. fst z *\<^sub>R snd z)" |
deabce3ccf1f
new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
66456
diff
changeset
|
2788 |
by (intro continuous_intros) |
deabce3ccf1f
new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
66456
diff
changeset
|
2789 |
moreover have "path_connected ?D" |
deabce3ccf1f
new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
66456
diff
changeset
|
2790 |
by (metis path_connected_Times [OF pc] path_connected_sphere 2) |
deabce3ccf1f
new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
66456
diff
changeset
|
2791 |
ultimately show ?thesis |
deabce3ccf1f
new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
66456
diff
changeset
|
2792 |
apply (subst *) |
deabce3ccf1f
new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
66456
diff
changeset
|
2793 |
apply (rule path_connected_continuous_image, auto) |
deabce3ccf1f
new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
66456
diff
changeset
|
2794 |
done |
deabce3ccf1f
new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
66456
diff
changeset
|
2795 |
qed |
deabce3ccf1f
new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
66456
diff
changeset
|
2796 |
ultimately show ?thesis |
deabce3ccf1f
new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
66456
diff
changeset
|
2797 |
using path_connected_translation by metis |
deabce3ccf1f
new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
66456
diff
changeset
|
2798 |
qed |
deabce3ccf1f
new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
66456
diff
changeset
|
2799 |
|
68607
67bb59e49834
make theorem, corollary, and proposition %important for HOL-Analysis manual
immler
parents:
68532
diff
changeset
|
2800 |
proposition path_connected_annulus: |
66793
deabce3ccf1f
new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
66456
diff
changeset
|
2801 |
fixes a :: "'N::euclidean_space" |
deabce3ccf1f
new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
66456
diff
changeset
|
2802 |
assumes "2 \<le> DIM('N)" |
deabce3ccf1f
new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
66456
diff
changeset
|
2803 |
shows "path_connected {x. r1 < norm(x - a) \<and> norm(x - a) < r2}" |
deabce3ccf1f
new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
66456
diff
changeset
|
2804 |
"path_connected {x. r1 < norm(x - a) \<and> norm(x - a) \<le> r2}" |
deabce3ccf1f
new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
66456
diff
changeset
|
2805 |
"path_connected {x. r1 \<le> norm(x - a) \<and> norm(x - a) < r2}" |
deabce3ccf1f
new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
66456
diff
changeset
|
2806 |
"path_connected {x. r1 \<le> norm(x - a) \<and> norm(x - a) \<le> r2}" |
68607
67bb59e49834
make theorem, corollary, and proposition %important for HOL-Analysis manual
immler
parents:
68532
diff
changeset
|
2807 |
by (auto simp: is_interval_def intro!: is_interval_convex convex_imp_path_connected path_connected_2DIM_I [OF assms]) |
67bb59e49834
make theorem, corollary, and proposition %important for HOL-Analysis manual
immler
parents:
68532
diff
changeset
|
2808 |
|
67bb59e49834
make theorem, corollary, and proposition %important for HOL-Analysis manual
immler
parents:
68532
diff
changeset
|
2809 |
proposition connected_annulus: |
66793
deabce3ccf1f
new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
66456
diff
changeset
|
2810 |
fixes a :: "'N::euclidean_space" |
deabce3ccf1f
new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
66456
diff
changeset
|
2811 |
assumes "2 \<le> DIM('N::euclidean_space)" |
deabce3ccf1f
new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
66456
diff
changeset
|
2812 |
shows "connected {x. r1 < norm(x - a) \<and> norm(x - a) < r2}" |
deabce3ccf1f
new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
66456
diff
changeset
|
2813 |
"connected {x. r1 < norm(x - a) \<and> norm(x - a) \<le> r2}" |
deabce3ccf1f
new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
66456
diff
changeset
|
2814 |
"connected {x. r1 \<le> norm(x - a) \<and> norm(x - a) < r2}" |
deabce3ccf1f
new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
66456
diff
changeset
|
2815 |
"connected {x. r1 \<le> norm(x - a) \<and> norm(x - a) \<le> r2}" |
68607
67bb59e49834
make theorem, corollary, and proposition %important for HOL-Analysis manual
immler
parents:
68532
diff
changeset
|
2816 |
by (auto simp: path_connected_annulus [OF assms] path_connected_imp_connected) |
67962 | 2817 |
|
2818 |
||
70136 | 2819 |
subsection\<^marker>\<open>tag unimportant\<close>\<open>Relations between components and path components\<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
|
2820 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2821 |
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
|
2822 |
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
|
2823 |
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
|
2824 |
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
|
2825 |
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
|
2826 |
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
|
2827 |
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
|
2828 |
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
|
2829 |
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
|
2830 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2831 |
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
|
2832 |
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
|
2833 |
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
|
2834 |
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
|
2835 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2836 |
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
|
2837 |
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
|
2838 |
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
|
2839 |
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
|
2840 |
proof - |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2841 |
{ 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
|
2842 |
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
|
2843 |
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
|
2844 |
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
|
2845 |
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
|
2846 |
} |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2847 |
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
|
2848 |
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
|
2849 |
qed |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2850 |
|
63114
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
2851 |
lemma connected_disjoint_Union_open_pick: |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
2852 |
assumes "pairwise disjnt B" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
2853 |
"\<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
|
2854 |
"\<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
|
2855 |
"\<Union>A \<subseteq> \<Union>B" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
2856 |
"S \<in> A" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
2857 |
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
|
2858 |
proof - |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
2859 |
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
|
2860 |
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
|
2861 |
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
|
2862 |
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
|
2863 |
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
|
2864 |
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
|
2865 |
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
|
2866 |
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
|
2867 |
by (auto simp: pairwise_def disjnt_def) |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
2868 |
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
|
2869 |
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
|
2870 |
have "S \<inter> \<Union>(B-{T}) = {}" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
2871 |
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
|
2872 |
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
|
2873 |
using "3" by auto |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
2874 |
show ?thesis |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
2875 |
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
|
2876 |
qed |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
2877 |
|
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
2878 |
lemma connected_disjoint_Union_open_subset: |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
2879 |
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
|
2880 |
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
|
2881 |
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
|
2882 |
and eq [simp]: "\<Union>A = \<Union>B" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
2883 |
shows "A \<subseteq> B" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
2884 |
proof |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
2885 |
fix S |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
2886 |
assume "S \<in> A" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
2887 |
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
|
2888 |
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
|
2889 |
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
|
2890 |
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
|
2891 |
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
|
2892 |
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
|
2893 |
ultimately have "S' = S" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
2894 |
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
|
2895 |
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
|
2896 |
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
|
2897 |
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
|
2898 |
qed |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
2899 |
|
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
2900 |
lemma connected_disjoint_Union_open_unique: |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
2901 |
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
|
2902 |
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
|
2903 |
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
|
2904 |
and eq [simp]: "\<Union>A = \<Union>B" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
2905 |
shows "A = B" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
2906 |
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
|
2907 |
|
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
2908 |
proposition components_open_unique: |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
2909 |
fixes S :: "'a::real_normed_vector set" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
2910 |
assumes "pairwise disjnt A" "\<Union>A = S" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
2911 |
"\<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
|
2912 |
shows "components S = A" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
2913 |
proof - |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
2914 |
have "open S" using assms by blast |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
2915 |
show ?thesis |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
2916 |
apply (rule connected_disjoint_Union_open_unique) |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
2917 |
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
|
2918 |
using \<open>open S\<close> |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
2919 |
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
|
2920 |
done |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
2921 |
qed |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
2922 |
|
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63092
diff
changeset
|
2923 |
|
70136 | 2924 |
subsection\<^marker>\<open>tag unimportant\<close>\<open>Existence of unbounded components\<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
|
2925 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2926 |
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
|
2927 |
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
|
2928 |
assumes "bounded (-s)" |
69508 | 2929 |
shows "\<exists>x. x \<in> s \<and> \<not> bounded (connected_component_set s x)" |
61426
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2930 |
proof - |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2931 |
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
|
2932 |
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
|
2933 |
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
|
2934 |
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
|
2935 |
then have *: "\<And>x. B \<le> norm x \<Longrightarrow> x \<in> s" |
68096 | 2936 |
by (force simp: ball_def dist_norm) |
69508 | 2937 |
have unbounded_inner: "\<not> bounded {x. inner i x \<ge> B}" |
61426
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2938 |
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
|
2939 |
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
|
2940 |
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
|
2941 |
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
|
2942 |
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
|
2943 |
done |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2944 |
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
|
2945 |
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
|
2946 |
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
|
2947 |
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
|
2948 |
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
|
2949 |
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
|
2950 |
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
|
2951 |
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
|
2952 |
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
|
2953 |
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
|
2954 |
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
|
2955 |
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
|
2956 |
done |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2957 |
qed |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2958 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2959 |
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
|
2960 |
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
|
2961 |
assumes bs: "bounded (-s)" and "2 \<le> DIM('a)" |
69508 | 2962 |
and bo: "\<not> bounded(connected_component_set s x)" |
2963 |
"\<not> bounded(connected_component_set s 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
|
2964 |
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
|
2965 |
proof - |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2966 |
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
|
2967 |
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
|
2968 |
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
|
2969 |
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
|
2970 |
then have *: "\<And>x. B \<le> norm x \<Longrightarrow> x \<in> s" |
68096 | 2971 |
by (force simp: ball_def dist_norm) |
61426
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2972 |
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
|
2973 |
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
|
2974 |
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
|
2975 |
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
|
2976 |
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
|
2977 |
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
|
2978 |
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
|
2979 |
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
|
2980 |
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
|
2981 |
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
|
2982 |
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
|
2983 |
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
|
2984 |
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
|
2985 |
done |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2986 |
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
|
2987 |
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
|
2988 |
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
|
2989 |
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
|
2990 |
qed |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2991 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2992 |
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
|
2993 |
fixes s :: "'a :: euclidean_space set" |
69508 | 2994 |
shows "bounded (-s) \<Longrightarrow> \<exists>c. c \<in> components s \<and> \<not>bounded c" |
61426
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2995 |
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
|
2996 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
2997 |
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
|
2998 |
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
|
2999 |
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
|
3000 |
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
|
3001 |
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
|
3002 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
3003 |
lemma cobounded_has_bounded_component: |
64122 | 3004 |
fixes S :: "'a :: euclidean_space set" |
3005 |
assumes "bounded (- S)" "\<not> connected S" "2 \<le> DIM('a)" |
|
3006 |
obtains C where "C \<in> components S" "bounded C" |
|
3007 |
by (meson cobounded_unique_unbounded_components connected_eq_connected_components_eq assms) |
|
61426
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
3008 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
3009 |
|
69620 | 3010 |
subsection\<open>The \<open>inside\<close> and \<open>outside\<close> 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
|
3011 |
|
70136 | 3012 |
text\<^marker>\<open>tag important\<close>\<open>The inside comprises the points in a bounded connected component of the set's complement. |
61426
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
3013 |
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
|
3014 |
|
70136 | 3015 |
definition\<^marker>\<open>tag important\<close> inside where |
68096 | 3016 |
"inside S \<equiv> {x. (x \<notin> S) \<and> bounded(connected_component_set ( - S) x)}" |
61426
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
3017 |
|
70136 | 3018 |
definition\<^marker>\<open>tag important\<close> outside where |
69508 | 3019 |
"outside S \<equiv> -S \<inter> {x. \<not> bounded(connected_component_set (- S) x)}" |
3020 |
||
3021 |
lemma outside: "outside S = {x. \<not> bounded(connected_component_set (- S) x)}" |
|
61426
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
3022 |
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
|
3023 |
|
68096 | 3024 |
lemma inside_no_overlap [simp]: "inside S \<inter> 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
|
3025 |
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
|
3026 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
3027 |
lemma outside_no_overlap [simp]: |
68096 | 3028 |
"outside S \<inter> 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
|
3029 |
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
|
3030 |
|
68096 | 3031 |
lemma inside_Int_outside [simp]: "inside S \<inter> outside 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
|
3032 |
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
|
3033 |
|
68096 | 3034 |
lemma inside_Un_outside [simp]: "inside S \<union> outside S = (- 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
|
3035 |
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
|
3036 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
3037 |
lemma inside_eq_outside: |
68096 | 3038 |
"inside S = outside S \<longleftrightarrow> S = UNIV" |
61426
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
3039 |
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
|
3040 |
|
68096 | 3041 |
lemma inside_outside: "inside S = (- (S \<union> outside S))" |
3042 |
by (force simp: inside_def outside) |
|
3043 |
||
3044 |
lemma outside_inside: "outside S = (- (S \<union> inside 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
|
3045 |
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
|
3046 |
|
68096 | 3047 |
lemma union_with_inside: "S \<union> inside S = - outside 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
|
3048 |
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
|
3049 |
|
68096 | 3050 |
lemma union_with_outside: "S \<union> outside S = - inside 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
|
3051 |
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
|
3052 |
|
68096 | 3053 |
lemma outside_mono: "S \<subseteq> T \<Longrightarrow> outside T \<subseteq> outside 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
|
3054 |
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
|
3055 |
|
68096 | 3056 |
lemma inside_mono: "S \<subseteq> T \<Longrightarrow> inside S - T \<subseteq> inside T" |
61426
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
3057 |
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
|
3058 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
3059 |
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
|
3060 |
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
|
3061 |
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
|
3062 |
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
|
3063 |
proof - |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
3064 |
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
|
3065 |
using assms by auto (metis add.commute diff_add_cancel) |
61808 | 3066 |
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
|
3067 |
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
|
3068 |
qed |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
3069 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
3070 |
lemma cobounded_outside: |
68096 | 3071 |
fixes S :: "'a :: real_normed_vector set" |
3072 |
assumes "bounded S" shows "bounded (- outside 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
|
3073 |
proof - |
68096 | 3074 |
obtain B where B: "B>0" "S \<subseteq> ball 0 B" |
61426
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
3075 |
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
|
3076 |
{ 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
|
3077 |
assume Bno: "B \<le> norm x" and C: "0 < C" |
68096 | 3078 |
have "\<exists>y. connected_component (- S) x y \<and> norm y > C" |
61426
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
3079 |
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
|
3080 |
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
|
3081 |
next |
68096 | 3082 |
case False |
3083 |
have "closed_segment x (((B + C) / norm x) *\<^sub>R x) \<subseteq> - ball 0 B" |
|
70817
dd675800469d
dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents:
70802
diff
changeset
|
3084 |
proof |
dd675800469d
dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents:
70802
diff
changeset
|
3085 |
fix w |
dd675800469d
dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents:
70802
diff
changeset
|
3086 |
assume "w \<in> closed_segment x (((B + C) / norm x) *\<^sub>R x)" |
dd675800469d
dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents:
70802
diff
changeset
|
3087 |
then obtain u where |
dd675800469d
dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents:
70802
diff
changeset
|
3088 |
w: "w = (1 - u + u * (B + C) / norm x) *\<^sub>R x" "0 \<le> u" "u \<le> 1" |
dd675800469d
dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents:
70802
diff
changeset
|
3089 |
by (auto simp add: closed_segment_def real_vector_class.scaleR_add_left [symmetric]) |
dd675800469d
dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents:
70802
diff
changeset
|
3090 |
with False B C have "B \<le> (1 - u) * norm x + u * (B + C)" |
dd675800469d
dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents:
70802
diff
changeset
|
3091 |
using segment_bound_lemma [of B "norm x" "B + C" u] Bno |
dd675800469d
dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents:
70802
diff
changeset
|
3092 |
by simp |
dd675800469d
dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents:
70802
diff
changeset
|
3093 |
with False B C show "w \<in> - ball 0 B" |
dd675800469d
dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents:
70802
diff
changeset
|
3094 |
using distrib_right [of _ _ "norm x"] |
dd675800469d
dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents:
70802
diff
changeset
|
3095 |
by (simp add: ball_def w not_less) |
dd675800469d
dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents:
70802
diff
changeset
|
3096 |
qed |
68096 | 3097 |
also have "... \<subseteq> -S" |
3098 |
by (simp add: B) |
|
3099 |
finally have "\<exists>T. connected T \<and> T \<subseteq> - S \<and> x \<in> T \<and> ((B + C) / norm x) *\<^sub>R x \<in> T" |
|
3100 |
by (rule_tac x="closed_segment x (((B+C)/norm x) *\<^sub>R x)" in exI) simp |
|
3101 |
with False B |
|
3102 |
show ?thesis |
|
3103 |
by (rule_tac x="((B+C)/norm x) *\<^sub>R x" in exI) (simp add: connected_component_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
|
3104 |
qed |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
3105 |
} |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
3106 |
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
|
3107 |
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
|
3108 |
apply (rule bounded_subset [OF bounded_ball [of 0 B]]) |
68096 | 3109 |
apply (force simp: dist_norm not_less bounded_pos) |
61426
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
3110 |
done |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
3111 |
qed |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
3112 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
3113 |
lemma unbounded_outside: |
68096 | 3114 |
fixes S :: "'a::{real_normed_vector, perfect_space} set" |
69508 | 3115 |
shows "bounded S \<Longrightarrow> \<not> bounded(outside 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
|
3116 |
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
|
3117 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
3118 |
lemma bounded_inside: |
68096 | 3119 |
fixes S :: "'a::{real_normed_vector, perfect_space} set" |
3120 |
shows "bounded S \<Longrightarrow> bounded(inside 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
|
3121 |
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
|
3122 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
3123 |
lemma connected_outside: |
68096 | 3124 |
fixes S :: "'a::euclidean_space set" |
3125 |
assumes "bounded S" "2 \<le> DIM('a)" |
|
3126 |
shows "connected(outside S)" |
|
3127 |
apply (clarsimp simp add: connected_iff_connected_component outside) |
|
3128 |
apply (rule_tac s="connected_component_set (- S) x" in connected_component_of_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
|
3129 |
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
|
3130 |
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
|
3131 |
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
|
3132 |
done |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
3133 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
3134 |
lemma outside_connected_component_lt: |
68096 | 3135 |
"outside S = {x. \<forall>B. \<exists>y. B < norm(y) \<and> connected_component (- S) x 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
|
3136 |
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
|
3137 |
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
|
3138 |
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
|
3139 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
3140 |
lemma outside_connected_component_le: |
68096 | 3141 |
"outside 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
|
3142 |
{x. \<forall>B. \<exists>y. B \<le> norm(y) \<and> |
68096 | 3143 |
connected_component (- S) x 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
|
3144 |
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
|
3145 |
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
|
3146 |
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
|
3147 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
3148 |
lemma not_outside_connected_component_lt: |
68096 | 3149 |
fixes S :: "'a::euclidean_space set" |
3150 |
assumes S: "bounded S" and "2 \<le> DIM('a)" |
|
69508 | 3151 |
shows "- (outside S) = {x. \<forall>B. \<exists>y. B < norm(y) \<and> \<not> connected_component (- S) x 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
|
3152 |
proof - |
68096 | 3153 |
obtain B::real where B: "0 < B" and Bno: "\<And>x. x \<in> S \<Longrightarrow> norm x \<le> B" |
3154 |
using S [simplified bounded_pos] 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
|
3155 |
{ 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
|
3156 |
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
|
3157 |
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
|
3158 |
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
|
3159 |
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
|
3160 |
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
|
3161 |
by (auto simp: dist_norm) |
68096 | 3162 |
then have "connected_component (- S) y 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
|
3163 |
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
|
3164 |
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
|
3165 |
done |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
3166 |
} 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
|
3167 |
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
|
3168 |
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
|
3169 |
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
|
3170 |
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
|
3171 |
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
|
3172 |
qed |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
3173 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
3174 |
lemma not_outside_connected_component_le: |
68096 | 3175 |
fixes S :: "'a::euclidean_space set" |
3176 |
assumes S: "bounded S" "2 \<le> DIM('a)" |
|
69508 | 3177 |
shows "- (outside S) = {x. \<forall>B. \<exists>y. B \<le> norm(y) \<and> \<not> connected_component (- S) x 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
|
3178 |
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
|
3179 |
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
|
3180 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
3181 |
lemma inside_connected_component_lt: |
68096 | 3182 |
fixes S :: "'a::euclidean_space set" |
3183 |
assumes S: "bounded S" "2 \<le> DIM('a)" |
|
69508 | 3184 |
shows "inside S = {x. (x \<notin> S) \<and> (\<forall>B. \<exists>y. B < norm(y) \<and> \<not> connected_component (- S) x 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
|
3185 |
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
|
3186 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
3187 |
lemma inside_connected_component_le: |
68096 | 3188 |
fixes S :: "'a::euclidean_space set" |
3189 |
assumes S: "bounded S" "2 \<le> DIM('a)" |
|
69508 | 3190 |
shows "inside S = {x. (x \<notin> S) \<and> (\<forall>B. \<exists>y. B \<le> norm(y) \<and> \<not> connected_component (- S) x 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
|
3191 |
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
|
3192 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
3193 |
lemma inside_subset: |
69508 | 3194 |
assumes "connected U" and "\<not> bounded U" and "T \<union> U = - S" |
68096 | 3195 |
shows "inside S \<subseteq> T" |
61426
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
3196 |
apply (auto simp: inside_def) |
68096 | 3197 |
by (metis bounded_subset [of "connected_component_set (- S) _"] connected_component_maximal |
61426
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
3198 |
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
|
3199 |
|
62381
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
3200 |
lemma frontier_not_empty: |
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
3201 |
fixes S :: "'a :: real_normed_vector set" |
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
3202 |
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
|
3203 |
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
|
3204 |
|
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
3205 |
lemma frontier_eq_empty: |
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
3206 |
fixes S :: "'a :: real_normed_vector set" |
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
3207 |
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
|
3208 |
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
|
3209 |
|
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
3210 |
lemma frontier_of_connected_component_subset: |
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
3211 |
fixes S :: "'a::real_normed_vector set" |
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
3212 |
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
|
3213 |
proof - |
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
3214 |
{ fix y |
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
3215 |
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
|
3216 |
and y2: "y \<notin> interior (connected_component_set S x)" |
64006
0de4736dad8b
new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
63978
diff
changeset
|
3217 |
have "y \<in> closure S" |
62381
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
3218 |
using y1 closure_mono connected_component_subset by blast |
64006
0de4736dad8b
new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
63978
diff
changeset
|
3219 |
moreover have "z \<in> interior (connected_component_set S x)" |
62381
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
3220 |
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
|
3221 |
proof - |
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
3222 |
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
|
3223 |
apply (rule connected_component_maximal) |
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
3224 |
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
|
3225 |
done |
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
3226 |
then show ?thesis |
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
3227 |
using y1 apply (simp add: closure_approachable open_contains_ball_eq [OF open_interior]) |
64006
0de4736dad8b
new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
63978
diff
changeset
|
3228 |
by (metis connected_component_eq dist_commute mem_Collect_eq mem_ball mem_interior subsetD \<open>0 < e\<close> y2) |
62381
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
3229 |
qed |
64006
0de4736dad8b
new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
63978
diff
changeset
|
3230 |
then have "y \<notin> interior S" |
62381
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
3231 |
using y2 by (force simp: open_contains_ball_eq [OF open_interior]) |
64006
0de4736dad8b
new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
63978
diff
changeset
|
3232 |
ultimately have "y \<in> frontier S" |
0de4736dad8b
new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
63978
diff
changeset
|
3233 |
by (auto simp: frontier_def) |
62381
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
3234 |
} |
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
3235 |
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
|
3236 |
qed |
a6479cb85944
New and revised material for (multivariate) analysis
paulson <lp15@cam.ac.uk>
parents:
62087
diff
changeset
|
3237 |
|
62620
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
3238 |
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
|
3239 |
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
|
3240 |
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
|
3241 |
proof - |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
3242 |
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
|
3243 |
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
|
3244 |
"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
|
3245 |
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
|
3246 |
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
|
3247 |
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
|
3248 |
next |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
3249 |
case False |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
3250 |
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
|
3251 |
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
|
3252 |
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
|
3253 |
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
|
3254 |
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
|
3255 |
then show ?thesis |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
3256 |
proof - |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
3257 |
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
|
3258 |
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
|
3259 |
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
|
3260 |
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
|
3261 |
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
|
3262 |
by linarith |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
3263 |
then show ?thesis |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
3264 |
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
|
3265 |
qed |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
3266 |
qed |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
3267 |
then show ?thesis |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
3268 |
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
|
3269 |
qed |
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
3270 |
|
d21dab28b3f9
New results about paths, segments, etc. The notion of simply_connected.
paulson <lp15@cam.ac.uk>
parents:
62618
diff
changeset
|
3271 |
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
|
3272 |
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
|
3273 |
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
|
3274 |
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
|
3275 |
(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
|
3276 |
|
64006
0de4736dad8b
new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
63978
diff
changeset
|
3277 |
lemma frontier_of_components_subset: |
0de4736dad8b
new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
63978
diff
changeset
|
3278 |
fixes S :: "'a::real_normed_vector set" |
0de4736dad8b
new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
63978
diff
changeset
|
3279 |
shows "C \<in> components S \<Longrightarrow> frontier C \<subseteq> frontier S" |
0de4736dad8b
new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
63978
diff
changeset
|
3280 |
by (metis Path_Connected.frontier_of_connected_component_subset components_iff) |
0de4736dad8b
new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
63978
diff
changeset
|
3281 |
|
0de4736dad8b
new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
63978
diff
changeset
|
3282 |
lemma frontier_of_components_closed_complement: |
0de4736dad8b
new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
63978
diff
changeset
|
3283 |
fixes S :: "'a::real_normed_vector set" |
0de4736dad8b
new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
63978
diff
changeset
|
3284 |
shows "\<lbrakk>closed S; C \<in> components (- S)\<rbrakk> \<Longrightarrow> frontier C \<subseteq> S" |
0de4736dad8b
new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
63978
diff
changeset
|
3285 |
using frontier_complement frontier_of_components_subset frontier_subset_eq by blast |
0de4736dad8b
new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
63978
diff
changeset
|
3286 |
|
0de4736dad8b
new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
63978
diff
changeset
|
3287 |
lemma frontier_minimal_separating_closed: |
0de4736dad8b
new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
63978
diff
changeset
|
3288 |
fixes S :: "'a::real_normed_vector set" |
0de4736dad8b
new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
63978
diff
changeset
|
3289 |
assumes "closed S" |
69508 | 3290 |
and nconn: "\<not> connected(- S)" |
64006
0de4736dad8b
new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
63978
diff
changeset
|
3291 |
and C: "C \<in> components (- S)" |
0de4736dad8b
new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
63978
diff
changeset
|
3292 |
and conn: "\<And>T. \<lbrakk>closed T; T \<subset> S\<rbrakk> \<Longrightarrow> connected(- T)" |
0de4736dad8b
new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
63978
diff
changeset
|
3293 |
shows "frontier C = S" |
0de4736dad8b
new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
63978
diff
changeset
|
3294 |
proof (rule ccontr) |
0de4736dad8b
new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
63978
diff
changeset
|
3295 |
assume "frontier C \<noteq> S" |
0de4736dad8b
new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
63978
diff
changeset
|
3296 |
then have "frontier C \<subset> S" |
0de4736dad8b
new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
63978
diff
changeset
|
3297 |
using frontier_of_components_closed_complement [OF \<open>closed S\<close> C] by blast |
0de4736dad8b
new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
63978
diff
changeset
|
3298 |
then have "connected(- (frontier C))" |
0de4736dad8b
new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
63978
diff
changeset
|
3299 |
by (simp add: conn) |
0de4736dad8b
new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
63978
diff
changeset
|
3300 |
have "\<not> connected(- (frontier C))" |
0de4736dad8b
new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
63978
diff
changeset
|
3301 |
unfolding connected_def not_not |
0de4736dad8b
new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
63978
diff
changeset
|
3302 |
proof (intro exI conjI) |
0de4736dad8b
new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
63978
diff
changeset
|
3303 |
show "open C" |
0de4736dad8b
new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
63978
diff
changeset
|
3304 |
using C \<open>closed S\<close> open_components by blast |
0de4736dad8b
new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
63978
diff
changeset
|
3305 |
show "open (- closure C)" |
0de4736dad8b
new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
63978
diff
changeset
|
3306 |
by blast |
0de4736dad8b
new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
63978
diff
changeset
|
3307 |
show "C \<inter> - closure C \<inter> - frontier C = {}" |
0de4736dad8b
new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
63978
diff
changeset
|
3308 |
using closure_subset by blast |
0de4736dad8b
new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
63978
diff
changeset
|
3309 |
show "C \<inter> - frontier C \<noteq> {}" |
0de4736dad8b
new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
63978
diff
changeset
|
3310 |
using C \<open>open C\<close> components_eq frontier_disjoint_eq by fastforce |
0de4736dad8b
new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
63978
diff
changeset
|
3311 |
show "- frontier C \<subseteq> C \<union> - closure C" |
0de4736dad8b
new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
63978
diff
changeset
|
3312 |
by (simp add: \<open>open C\<close> closed_Compl frontier_closures) |
0de4736dad8b
new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
63978
diff
changeset
|
3313 |
then show "- closure C \<inter> - frontier C \<noteq> {}" |
0de4736dad8b
new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
63978
diff
changeset
|
3314 |
by (metis (no_types, lifting) C Compl_subset_Compl_iff \<open>frontier C \<subset> S\<close> compl_sup frontier_closures in_components_subset psubsetE sup.absorb_iff2 sup.boundedE sup_bot.right_neutral sup_inf_absorb) |
0de4736dad8b
new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
63978
diff
changeset
|
3315 |
qed |
0de4736dad8b
new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
63978
diff
changeset
|
3316 |
then show False |
0de4736dad8b
new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
63978
diff
changeset
|
3317 |
using \<open>connected (- frontier C)\<close> by blast |
0de4736dad8b
new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
63978
diff
changeset
|
3318 |
qed |
0de4736dad8b
new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
63978
diff
changeset
|
3319 |
|
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
|
3320 |
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
|
3321 |
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
|
3322 |
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
|
3323 |
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
|
3324 |
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
|
3325 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
3326 |
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
|
3327 |
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
|
3328 |
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
|
3329 |
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
|
3330 |
|
64788
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
3331 |
lemma components_UNIV [simp]: "components UNIV = {UNIV :: 'a::real_normed_vector 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
|
3332 |
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
|
3333 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
3334 |
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
|
3335 |
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
|
3336 |
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
|
3337 |
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
|
3338 |
proof - |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
3339 |
{ 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
|
3340 |
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
|
3341 |
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
|
3342 |
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
|
3343 |
apply (rule connected_Int_frontier, simp) |
69712 | 3344 |
apply (metis IntI cc connected_component_in connected_component_refl empty_iff interiorE mem_Collect_eq rev_subsetD x) |
61426
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
3345 |
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
|
3346 |
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
|
3347 |
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
|
3348 |
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
|
3349 |
} |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
3350 |
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
|
3351 |
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
|
3352 |
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
|
3353 |
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
|
3354 |
done |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
3355 |
qed |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
3356 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
3357 |
lemma inside_empty [simp]: "inside {} = ({} :: 'a :: {real_normed_vector, perfect_space} set)" |
71172 | 3358 |
by (simp add: inside_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
|
3359 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
3360 |
lemma outside_empty [simp]: "outside {} = (UNIV :: 'a :: {real_normed_vector, perfect_space} set)" |
63955 | 3361 |
using inside_empty inside_Un_outside by blast |
61426
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
3362 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
3363 |
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
|
3364 |
"\<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
|
3365 |
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
|
3366 |
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
|
3367 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
3368 |
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
|
3369 |
"\<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
|
3370 |
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
|
3371 |
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
|
3372 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
3373 |
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
|
3374 |
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
|
3375 |
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
|
3376 |
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
|
3377 |
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
|
3378 |
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
|
3379 |
next |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
3380 |
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
|
3381 |
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
|
3382 |
{ 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
|
3383 |
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
|
3384 |
by (metis mem_Collect_eq) |
63040 | 3385 |
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
|
3386 |
have "C > 0" |
70817
dd675800469d
dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents:
70802
diff
changeset
|
3387 |
using \<open>0 < B\<close> zna by (simp add: C_def field_split_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
|
3388 |
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
|
3389 |
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
|
3390 |
moreover have "norm (C *\<^sub>R (z-a)) > norm z + B" |
70802
160eaf566bcb
formally augmented corresponding rules for field_simps
haftmann
parents:
70196
diff
changeset
|
3391 |
using zna \<open>B>0\<close> by (simp add: C_def le_max_iff_disj) |
61426
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
3392 |
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
|
3393 |
{ 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
|
3394 |
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
|
3395 |
then have Cpos: "1 + u * C > 0" |
61808 | 3396 |
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
|
3397 |
then have *: "(1 / (1 + u * C)) *\<^sub>R z + (u * C / (1 + u * C)) *\<^sub>R z = z" |
70817
dd675800469d
dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents:
70802
diff
changeset
|
3398 |
by (simp add: scaleR_add_left [symmetric] field_split_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
|
3399 |
then have False |
61808 | 3400 |
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 |
71172 | 3401 |
by (simp add: * field_split_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
|
3402 |
} 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
|
3403 |
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
|
3404 |
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
|
3405 |
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
|
3406 |
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
|
3407 |
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
|
3408 |
done |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
3409 |
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
|
3410 |
using zna B [of "z + C *\<^sub>R (z-a)"] C |
70817
dd675800469d
dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents:
70802
diff
changeset
|
3411 |
by (auto simp: field_split_simps max_mult_distrib_right) |
61426
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
3412 |
} |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
3413 |
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
|
3414 |
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
|
3415 |
qed |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
3416 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
3417 |
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
|
3418 |
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
|
3419 |
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
|
3420 |
shows "outside s = - s" |
63955 | 3421 |
by (metis ComplD assms convex_in_outside equalityI inside_Un_outside subsetI sup.cobounded2) |
61426
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
3422 |
|
66793
deabce3ccf1f
new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
66456
diff
changeset
|
3423 |
lemma outside_singleton [simp]: |
deabce3ccf1f
new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
66456
diff
changeset
|
3424 |
fixes x :: "'a :: {real_normed_vector, perfect_space}" |
deabce3ccf1f
new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
66456
diff
changeset
|
3425 |
shows "outside {x} = -{x}" |
deabce3ccf1f
new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
66456
diff
changeset
|
3426 |
by (auto simp: outside_convex) |
deabce3ccf1f
new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
66456
diff
changeset
|
3427 |
|
61426
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
3428 |
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
|
3429 |
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
|
3430 |
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
|
3431 |
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
|
3432 |
|
66793
deabce3ccf1f
new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
66456
diff
changeset
|
3433 |
lemma inside_singleton [simp]: |
deabce3ccf1f
new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
66456
diff
changeset
|
3434 |
fixes x :: "'a :: {real_normed_vector, perfect_space}" |
deabce3ccf1f
new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
66456
diff
changeset
|
3435 |
shows "inside {x} = {}" |
deabce3ccf1f
new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
66456
diff
changeset
|
3436 |
by (auto simp: inside_convex) |
deabce3ccf1f
new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
66456
diff
changeset
|
3437 |
|
61426
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
3438 |
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
|
3439 |
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
|
3440 |
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
|
3441 |
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
|
3442 |
|
64788
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
3443 |
lemma outside_Un_outside_Un: |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
3444 |
fixes S :: "'a::real_normed_vector set" |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
3445 |
assumes "S \<inter> outside(T \<union> U) = {}" |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
3446 |
shows "outside(T \<union> U) \<subseteq> outside(T \<union> S)" |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
3447 |
proof |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
3448 |
fix x |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
3449 |
assume x: "x \<in> outside (T \<union> U)" |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
3450 |
have "Y \<subseteq> - S" if "connected Y" "Y \<subseteq> - T" "Y \<subseteq> - U" "x \<in> Y" "u \<in> Y" for u Y |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
3451 |
proof - |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
3452 |
have "Y \<subseteq> connected_component_set (- (T \<union> U)) x" |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
3453 |
by (simp add: connected_component_maximal that) |
68096 | 3454 |
also have "\<dots> \<subseteq> outside(T \<union> U)" |
64788
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
3455 |
by (metis (mono_tags, lifting) Collect_mono mem_Collect_eq outside outside_same_component x) |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
3456 |
finally have "Y \<subseteq> outside(T \<union> U)" . |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
3457 |
with assms show ?thesis by auto |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
3458 |
qed |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
3459 |
with x show "x \<in> outside (T \<union> S)" |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
3460 |
by (simp add: outside_connected_component_lt connected_component_def) meson |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
3461 |
qed |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64394
diff
changeset
|
3462 |
|
61426
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
3463 |
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
|
3464 |
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
|
3465 |
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
|
3466 |
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
|
3467 |
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
|
3468 |
proof - |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
3469 |
{ 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
|
3470 |
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
|
3471 |
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
|
3472 |
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
|
3473 |
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
|
3474 |
} |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
3475 |
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
|
3476 |
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
|
3477 |
qed |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
3478 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
3479 |
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
|
3480 |
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
|
3481 |
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
|
3482 |
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
|
3483 |
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
|
3484 |
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
|
3485 |
|
61518
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3486 |
lemma inside_frontier_eq_interior: |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3487 |
fixes s :: "'a :: {real_normed_vector, perfect_space} set" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3488 |
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
|
3489 |
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
|
3490 |
using closure_subset interior_subset |
68096 | 3491 |
apply (auto simp: frontier_def) |
61518
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3492 |
done |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3493 |
|
61426
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
3494 |
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
|
3495 |
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
|
3496 |
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
|
3497 |
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
|
3498 |
proof - |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
3499 |
{ 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
|
3500 |
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
|
3501 |
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
|
3502 |
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
|
3503 |
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
|
3504 |
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
|
3505 |
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
|
3506 |
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
|
3507 |
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
|
3508 |
} |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
3509 |
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
|
3510 |
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
|
3511 |
qed |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
3512 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
3513 |
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
|
3514 |
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
|
3515 |
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
|
3516 |
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
|
3517 |
proof - |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
3518 |
{ 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
|
3519 |
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
|
3520 |
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
|
3521 |
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
|
3522 |
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
|
3523 |
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
|
3524 |
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
|
3525 |
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
|
3526 |
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
|
3527 |
} |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
3528 |
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
|
3529 |
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
|
3530 |
qed |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
3531 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
3532 |
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
|
3533 |
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
|
3534 |
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
|
3535 |
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
|
3536 |
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
|
3537 |
|
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
3538 |
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
|
3539 |
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
|
3540 |
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
|
3541 |
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
|
3542 |
proof - |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
3543 |
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
|
3544 |
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
|
3545 |
moreover have "- inside s \<inter> - outside s = s" |
63955 | 3546 |
by (metis (no_types) compl_sup double_compl inside_Un_outside) |
61426
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
3547 |
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
|
3548 |
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
|
3549 |
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
|
3550 |
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
|
3551 |
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
|
3552 |
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
|
3553 |
qed |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
3554 |
|
61518
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3555 |
lemma closure_outside_subset: |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3556 |
fixes s :: "'a::real_normed_vector set" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3557 |
assumes "closed s" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3558 |
shows "closure(outside s) \<subseteq> s \<union> outside s" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3559 |
apply (rule closure_minimal, simp) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3560 |
by (metis assms closed_open inside_outside open_inside) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3561 |
|
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3562 |
lemma frontier_outside_subset: |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3563 |
fixes s :: "'a::real_normed_vector set" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3564 |
assumes "closed s" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3565 |
shows "frontier(outside s) \<subseteq> s" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3566 |
apply (simp add: frontier_def open_outside interior_open) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3567 |
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
|
3568 |
|
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3569 |
lemma inside_complement_unbounded_connected_empty: |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3570 |
"\<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
|
3571 |
apply (simp add: inside_def) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3572 |
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
|
3573 |
|
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3574 |
lemma inside_bounded_complement_connected_empty: |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3575 |
fixes s :: "'a::{real_normed_vector, perfect_space} set" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3576 |
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
|
3577 |
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
|
3578 |
|
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3579 |
lemma inside_inside: |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3580 |
assumes "s \<subseteq> inside t" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3581 |
shows "inside s - t \<subseteq> inside t" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3582 |
unfolding inside_def |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3583 |
proof clarify |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3584 |
fix x |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3585 |
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
|
3586 |
show "bounded (connected_component_set (- t) x)" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3587 |
proof (cases "s \<inter> connected_component_set (- t) x = {}") |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3588 |
case True show ?thesis |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3589 |
apply (rule bounded_subset [OF bo]) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3590 |
apply (rule connected_component_maximal) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3591 |
using x True apply auto |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3592 |
done |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3593 |
next |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3594 |
case False then show ?thesis |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3595 |
using assms [unfolded inside_def] x |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3596 |
apply (simp add: disjoint_iff_not_equal, clarify) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3597 |
apply (drule subsetD, assumption, auto) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3598 |
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
|
3599 |
qed |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3600 |
qed |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3601 |
|
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3602 |
lemma inside_inside_subset: "inside(inside s) \<subseteq> s" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3603 |
using inside_inside union_with_outside by fastforce |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3604 |
|
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3605 |
lemma inside_outside_intersect_connected: |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3606 |
"\<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
|
3607 |
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
|
3608 |
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
|
3609 |
|
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3610 |
lemma outside_bounded_nonempty: |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3611 |
fixes s :: "'a :: {real_normed_vector, perfect_space} set" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3612 |
assumes "bounded s" shows "outside s \<noteq> {}" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3613 |
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
|
3614 |
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
|
3615 |
double_complement order_refl outside_convex outside_def) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3616 |
|
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3617 |
lemma outside_compact_in_open: |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3618 |
fixes s :: "'a :: {real_normed_vector,perfect_space} set" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3619 |
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
|
3620 |
shows "outside s \<inter> t \<noteq> {}" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3621 |
proof - |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3622 |
have "outside s \<noteq> {}" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3623 |
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
|
3624 |
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
|
3625 |
show ?thesis |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3626 |
proof (cases "a \<in> t") |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3627 |
case True with a show ?thesis by blast |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3628 |
next |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3629 |
case False |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3630 |
have front: "frontier t \<subseteq> - s" |
61808 | 3631 |
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
|
3632 |
{ fix \<gamma> |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3633 |
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
|
3634 |
and pf: "pathfinish \<gamma> \<in> frontier t" and ps: "pathstart \<gamma> = a" |
63040 | 3635 |
define c where "c = pathfinish \<gamma>" |
61518
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3636 |
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
|
3637 |
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
|
3638 |
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
|
3639 |
using open_contains_cball[of "-s"] s by blast |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3640 |
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
|
3641 |
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
|
3642 |
by (metis Diff_iff frontier_def) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3643 |
then have "d \<in> -s" using \<epsilon> |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3644 |
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
|
3645 |
have pimg_sbs_cos: "path_image \<gamma> \<subseteq> -s" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3646 |
using pimg_sbs apply (auto simp: path_image_def) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3647 |
apply (drule subsetD) |
61808 | 3648 |
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
|
3649 |
done |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3650 |
have "closed_segment c d \<le> cball c \<epsilon>" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3651 |
apply (simp add: segment_convex_hull) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3652 |
apply (rule hull_minimal) |
61808 | 3653 |
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
|
3654 |
done |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3655 |
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
|
3656 |
moreover have con_gcd: "connected (path_image \<gamma> \<union> closed_segment c d)" |
61808 | 3657 |
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
|
3658 |
ultimately have "connected_component (- s) a d" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3659 |
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
|
3660 |
then have "outside s \<inter> t \<noteq> {}" |
61808 | 3661 |
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
|
3662 |
} note * = this |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3663 |
have pal: "pathstart (linepath a b) \<in> closure (- t)" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3664 |
by (auto simp: False closure_def) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3665 |
show ?thesis |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3666 |
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
|
3667 |
qed |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3668 |
qed |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3669 |
|
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3670 |
lemma inside_inside_compact_connected: |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3671 |
fixes s :: "'a :: euclidean_space set" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3672 |
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
|
3673 |
shows "inside s \<subseteq> inside t" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3674 |
proof (cases "inside t = {}") |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3675 |
case True with assms show ?thesis by auto |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3676 |
next |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3677 |
case False |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3678 |
consider "DIM('a) = 1" | "DIM('a) \<ge> 2" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3679 |
using antisym not_less_eq_eq by fastforce |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3680 |
then show ?thesis |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3681 |
proof cases |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3682 |
case 1 then show ?thesis |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3683 |
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
|
3684 |
next |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3685 |
case 2 |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3686 |
have coms: "compact s" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3687 |
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
|
3688 |
by (meson bounded_inside bounded_subset compact_imp_bounded) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3689 |
then have bst: "bounded (s \<union> t)" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3690 |
by (simp add: compact_imp_bounded t) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3691 |
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
|
3692 |
using bounded_subset_ballD by blast |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3693 |
have outst: "outside s \<inter> outside t \<noteq> {}" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3694 |
proof - |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3695 |
have "- ball 0 r \<subseteq> outside s" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3696 |
apply (rule outside_subset_convex) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3697 |
using r by auto |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3698 |
moreover have "- ball 0 r \<subseteq> outside t" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3699 |
apply (rule outside_subset_convex) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3700 |
using r by auto |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3701 |
ultimately show ?thesis |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3702 |
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
|
3703 |
qed |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3704 |
have "s \<inter> t = {}" using assms |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3705 |
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
|
3706 |
moreover have "outside s \<inter> inside t \<noteq> {}" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3707 |
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
|
3708 |
ultimately have "inside s \<inter> t = {}" |
61808 | 3709 |
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
|
3710 |
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
|
3711 |
then show ?thesis |
61808 | 3712 |
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
|
3713 |
qed |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3714 |
qed |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3715 |
|
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3716 |
lemma connected_with_inside: |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3717 |
fixes s :: "'a :: real_normed_vector set" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3718 |
assumes s: "closed s" and cons: "connected s" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3719 |
shows "connected(s \<union> inside s)" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3720 |
proof (cases "s \<union> inside s = UNIV") |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3721 |
case True with assms show ?thesis by auto |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3722 |
next |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3723 |
case False |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3724 |
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
|
3725 |
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
|
3726 |
using that proof |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3727 |
assume "a \<in> s" then show ?thesis |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3728 |
apply (rule_tac x=a in exI) |
68096 | 3729 |
apply (rule_tac x="{a}" in exI, simp) |
61518
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3730 |
done |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3731 |
next |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3732 |
assume a: "a \<in> inside s" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3733 |
show ?thesis |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3734 |
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
|
3735 |
using a apply (simp add: closure_def) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3736 |
apply (simp add: b) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3737 |
apply (rule_tac x="pathfinish h" in exI) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3738 |
apply (rule_tac x="path_image h" in exI) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3739 |
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
|
3740 |
using frontier_inside_subset s apply fastforce |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3741 |
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
|
3742 |
qed |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3743 |
show ?thesis |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3744 |
apply (simp add: connected_iff_connected_component) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3745 |
apply (simp add: connected_component_def) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3746 |
apply (clarify dest!: *) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3747 |
apply (rename_tac u u' t t') |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3748 |
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
|
3749 |
apply (auto simp: intro!: connected_Un cons) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3750 |
done |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3751 |
qed |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3752 |
|
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3753 |
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
|
3754 |
lemma connected_with_outside: |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3755 |
fixes s :: "'a :: real_normed_vector set" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3756 |
assumes s: "closed s" and cons: "connected s" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3757 |
shows "connected(s \<union> outside s)" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3758 |
proof (cases "s \<union> outside s = UNIV") |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3759 |
case True with assms show ?thesis by auto |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3760 |
next |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3761 |
case False |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3762 |
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
|
3763 |
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
|
3764 |
using that proof |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3765 |
assume "a \<in> s" then show ?thesis |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3766 |
apply (rule_tac x=a in exI) |
68096 | 3767 |
apply (rule_tac x="{a}" in exI, simp) |
61518
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3768 |
done |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3769 |
next |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3770 |
assume a: "a \<in> outside s" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3771 |
show ?thesis |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3772 |
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
|
3773 |
using a apply (simp add: closure_def) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3774 |
apply (simp add: b) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3775 |
apply (rule_tac x="pathfinish h" in exI) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3776 |
apply (rule_tac x="path_image h" in exI) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3777 |
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
|
3778 |
using frontier_outside_subset s apply fastforce |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3779 |
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
|
3780 |
qed |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3781 |
show ?thesis |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3782 |
apply (simp add: connected_iff_connected_component) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3783 |
apply (simp add: connected_component_def) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3784 |
apply (clarify dest!: *) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3785 |
apply (rename_tac u u' t t') |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3786 |
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
|
3787 |
apply (auto simp: intro!: connected_Un cons) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3788 |
done |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3789 |
qed |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3790 |
|
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3791 |
lemma inside_inside_eq_empty [simp]: |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3792 |
fixes s :: "'a :: {real_normed_vector, perfect_space} set" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3793 |
assumes s: "closed s" and cons: "connected s" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3794 |
shows "inside (inside s) = {}" |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3795 |
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
|
3796 |
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
|
3797 |
|
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3798 |
lemma inside_in_components: |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3799 |
"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
|
3800 |
apply (simp add: in_components_maximal) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3801 |
apply (auto intro: inside_same_component connected_componentI) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3802 |
apply (metis IntI empty_iff inside_no_overlap) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3803 |
done |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3804 |
|
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3805 |
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
|
3806 |
lemma outside_in_components: |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3807 |
"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
|
3808 |
apply (simp add: in_components_maximal) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3809 |
apply (auto intro: outside_same_component connected_componentI) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3810 |
apply (metis IntI empty_iff outside_no_overlap) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3811 |
done |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3812 |
|
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3813 |
lemma bounded_unique_outside: |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3814 |
fixes s :: "'a :: euclidean_space set" |
69508 | 3815 |
shows "\<lbrakk>bounded s; DIM('a) \<ge> 2\<rbrakk> \<Longrightarrow> (c \<in> components (- s) \<and> \<not> bounded c \<longleftrightarrow> c = outside s)" |
61518
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3816 |
apply (rule iffI) |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61426
diff
changeset
|
3817 |
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
|
3818 |
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
|
3819 |
|
69514 | 3820 |
|
62533
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
3821 |
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
|
3822 |
|
68607
67bb59e49834
make theorem, corollary, and proposition %important for HOL-Analysis manual
immler
parents:
68532
diff
changeset
|
3823 |
proposition ball_subset_open_map_image: |
62533
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
3824 |
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
|
3825 |
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
|
3826 |
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
|
3827 |
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
|
3828 |
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
|
3829 |
shows "ball (f a) r \<subseteq> f ` S" |
68607
67bb59e49834
make theorem, corollary, and proposition %important for HOL-Analysis manual
immler
parents:
68532
diff
changeset
|
3830 |
proof (cases "f ` S = UNIV") |
62533
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
3831 |
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
|
3832 |
next |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
3833 |
case False |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
3834 |
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
|
3835 |
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
|
3836 |
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
|
3837 |
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
|
3838 |
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
|
3839 |
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
|
3840 |
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
|
3841 |
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
|
3842 |
by metis |
66447
a1f5c5c26fa6
Replaced subseq with strict_mono
eberlm <eberlm@in.tum.de>
parents:
65038
diff
changeset
|
3843 |
then obtain y K where y: "y \<in> closure S" and "strict_mono (K :: nat \<Rightarrow> nat)" |
a1f5c5c26fa6
Replaced subseq with strict_mono
eberlm <eberlm@in.tum.de>
parents:
65038
diff
changeset
|
3844 |
and Klim: "(z \<circ> K) \<longlonglongrightarrow> y" |
62533
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
3845 |
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
|
3846 |
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
|
3847 |
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
|
3848 |
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
|
3849 |
done |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
3850 |
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
|
3851 |
by (metis LIMSEQ_subseq_LIMSEQ fun.map_cong0 fz tendsw) |
68096 | 3852 |
have zKs: "\<And>n. (z \<circ> K) n \<in> S" by (simp add: zs) |
63540 | 3853 |
have fz: "f \<circ> z = \<xi>" "(\<lambda>n. f (z n)) = \<xi>" |
62533
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
3854 |
using fz by auto |
63540 | 3855 |
then have "(\<xi> \<circ> K) \<longlonglongrightarrow> f y" |
62533
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
3856 |
by (metis (no_types) Klim zKs y contf comp_assoc continuous_on_closure_sequentially) |
63540 | 3857 |
with fz have wy: "w = f y" using fz LIMSEQ_unique ftendsw by auto |
62533
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
3858 |
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
|
3859 |
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
|
3860 |
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
|
3861 |
by (force simp: imageI image_mono interiorI interior_subset frontier_def y) |
69508 | 3862 |
have **: "(b \<inter> (- S) \<noteq> {} \<and> b - (- S) \<noteq> {} \<Longrightarrow> b \<inter> f \<noteq> {}) |
62533
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
3863 |
\<Longrightarrow> (b \<inter> S \<noteq> {}) \<Longrightarrow> b \<inter> f = {} \<Longrightarrow> |
63594
bd218a9320b5
HOL-Multivariate_Analysis: rename theories for more descriptive names
hoelzl
parents:
63540
diff
changeset
|
3864 |
b \<subseteq> S" for b f and S :: "'b set" |
62533
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
3865 |
by blast |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
3866 |
show ?thesis |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
3867 |
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
|
3868 |
apply (rule connected_Int_frontier [where t = "f`S", OF connected_ball]) |
63594
bd218a9320b5
HOL-Multivariate_Analysis: rename theories for more descriptive names
hoelzl
parents:
63540
diff
changeset
|
3869 |
using \<open>a \<in> S\<close> \<open>0 < r\<close> |
70196
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3870 |
apply (auto simp: disjoint_iff_not_equal dist_norm) |
62533
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
3871 |
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
|
3872 |
qed |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
3873 |
|
70196
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3874 |
|
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3875 |
subsubsection\<open>Special characterizations of classes of functions into and out of R.\<close> |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3876 |
|
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3877 |
proposition embedding_map_into_euclideanreal: |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3878 |
assumes "path_connected_space X" |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3879 |
shows "embedding_map X euclideanreal f \<longleftrightarrow> |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3880 |
continuous_map X euclideanreal f \<and> inj_on f (topspace X)" |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3881 |
proof safe |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3882 |
show "continuous_map X euclideanreal f" |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3883 |
if "embedding_map X euclideanreal f" |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3884 |
using continuous_map_in_subtopology homeomorphic_imp_continuous_map that |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3885 |
unfolding embedding_map_def by blast |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3886 |
show "inj_on f (topspace X)" |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3887 |
if "embedding_map X euclideanreal f" |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3888 |
using that homeomorphic_imp_injective_map |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3889 |
unfolding embedding_map_def by blast |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3890 |
show "embedding_map X euclideanreal f" |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3891 |
if cont: "continuous_map X euclideanreal f" and inj: "inj_on f (topspace X)" |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3892 |
proof - |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3893 |
obtain g where gf: "\<And>x. x \<in> topspace X \<Longrightarrow> g (f x) = x" |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3894 |
using inv_into_f_f [OF inj] by auto |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3895 |
show ?thesis |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3896 |
unfolding embedding_map_def homeomorphic_map_maps homeomorphic_maps_def |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3897 |
proof (intro exI conjI) |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3898 |
show "continuous_map X (top_of_set (f ` topspace X)) f" |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3899 |
by (simp add: cont continuous_map_in_subtopology) |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3900 |
let ?S = "f ` topspace X" |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3901 |
have eq: "{x \<in> ?S. g x \<in> U} = f ` U" if "openin X U" for U |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3902 |
using openin_subset [OF that] by (auto simp: gf) |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3903 |
have 1: "g ` ?S \<subseteq> topspace X" |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3904 |
using eq by blast |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3905 |
have "openin (top_of_set ?S) {x \<in> ?S. g x \<in> T}" |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3906 |
if "openin X T" for T |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3907 |
proof - |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3908 |
have "T \<subseteq> topspace X" |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3909 |
by (simp add: openin_subset that) |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3910 |
have RR: "\<forall>x \<in> ?S \<inter> g -` T. \<exists>d>0. \<forall>x' \<in> ?S \<inter> ball x d. g x' \<in> T" |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3911 |
proof (clarsimp simp add: gf) |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3912 |
have pcS: "path_connectedin euclidean ?S" |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3913 |
using assms cont path_connectedin_continuous_map_image path_connectedin_topspace by blast |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3914 |
show "\<exists>d>0. \<forall>x'\<in>f ` topspace X \<inter> ball (f x) d. g x' \<in> T" |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3915 |
if "x \<in> T" for x |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3916 |
proof - |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3917 |
have x: "x \<in> topspace X" |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3918 |
using \<open>T \<subseteq> topspace X\<close> \<open>x \<in> T\<close> by blast |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3919 |
obtain u v d where "0 < d" "u \<in> topspace X" "v \<in> topspace X" |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3920 |
and sub_fuv: "?S \<inter> {f x - d .. f x + d} \<subseteq> {f u..f v}" |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3921 |
proof (cases "\<exists>u \<in> topspace X. f u < f x") |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3922 |
case True |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3923 |
then obtain u where u: "u \<in> topspace X" "f u < f x" .. |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3924 |
show ?thesis |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3925 |
proof (cases "\<exists>v \<in> topspace X. f x < f v") |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3926 |
case True |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3927 |
then obtain v where v: "v \<in> topspace X" "f x < f v" .. |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3928 |
show ?thesis |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3929 |
proof |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3930 |
let ?d = "min (f x - f u) (f v - f x)" |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3931 |
show "0 < ?d" |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3932 |
by (simp add: \<open>f u < f x\<close> \<open>f x < f v\<close>) |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3933 |
show "f ` topspace X \<inter> {f x - ?d..f x + ?d} \<subseteq> {f u..f v}" |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3934 |
by fastforce |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3935 |
qed (auto simp: u v) |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3936 |
next |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3937 |
case False |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3938 |
show ?thesis |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3939 |
proof |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3940 |
let ?d = "f x - f u" |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3941 |
show "0 < ?d" |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3942 |
by (simp add: u) |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3943 |
show "f ` topspace X \<inter> {f x - ?d..f x + ?d} \<subseteq> {f u..f x}" |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3944 |
using x u False by auto |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3945 |
qed (auto simp: x u) |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3946 |
qed |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3947 |
next |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3948 |
case False |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3949 |
note no_u = False |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3950 |
show ?thesis |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3951 |
proof (cases "\<exists>v \<in> topspace X. f x < f v") |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3952 |
case True |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3953 |
then obtain v where v: "v \<in> topspace X" "f x < f v" .. |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3954 |
show ?thesis |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3955 |
proof |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3956 |
let ?d = "f v - f x" |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3957 |
show "0 < ?d" |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3958 |
by (simp add: v) |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3959 |
show "f ` topspace X \<inter> {f x - ?d..f x + ?d} \<subseteq> {f x..f v}" |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3960 |
using False by auto |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3961 |
qed (auto simp: x v) |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3962 |
next |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3963 |
case False |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3964 |
show ?thesis |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3965 |
proof |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3966 |
show "f ` topspace X \<inter> {f x - 1..f x + 1} \<subseteq> {f x..f x}" |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3967 |
using False no_u by fastforce |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3968 |
qed (auto simp: x) |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3969 |
qed |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3970 |
qed |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3971 |
then obtain h where "pathin X h" "h 0 = u" "h 1 = v" |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3972 |
using assms unfolding path_connected_space_def by blast |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3973 |
obtain C where "compactin X C" "connectedin X C" "u \<in> C" "v \<in> C" |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3974 |
proof |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3975 |
show "compactin X (h ` {0..1})" |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3976 |
using that by (simp add: \<open>pathin X h\<close> compactin_path_image) |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3977 |
show "connectedin X (h ` {0..1})" |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3978 |
using \<open>pathin X h\<close> connectedin_path_image by blast |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3979 |
qed (use \<open>h 0 = u\<close> \<open>h 1 = v\<close> in auto) |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3980 |
have "continuous_map (subtopology euclideanreal (?S \<inter> {f x - d .. f x + d})) (subtopology X C) g" |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3981 |
proof (rule continuous_inverse_map) |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3982 |
show "compact_space (subtopology X C)" |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3983 |
using \<open>compactin X C\<close> compactin_subspace by blast |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3984 |
show "continuous_map (subtopology X C) euclideanreal f" |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3985 |
by (simp add: cont continuous_map_from_subtopology) |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3986 |
have "{f u .. f v} \<subseteq> f ` topspace (subtopology X C)" |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3987 |
proof (rule connected_contains_Icc) |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3988 |
show "connected (f ` topspace (subtopology X C))" |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3989 |
using connectedin_continuous_map_image [OF cont] |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3990 |
by (simp add: \<open>compactin X C\<close> \<open>connectedin X C\<close> compactin_subset_topspace inf_absorb2) |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3991 |
show "f u \<in> f ` topspace (subtopology X C)" |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3992 |
by (simp add: \<open>u \<in> C\<close> \<open>u \<in> topspace X\<close>) |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3993 |
show "f v \<in> f ` topspace (subtopology X C)" |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3994 |
by (simp add: \<open>v \<in> C\<close> \<open>v \<in> topspace X\<close>) |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3995 |
qed |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3996 |
then show "f ` topspace X \<inter> {f x - d..f x + d} \<subseteq> f ` topspace (subtopology X C)" |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3997 |
using sub_fuv by blast |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3998 |
qed (auto simp: gf) |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
3999 |
then have contg: "continuous_map (subtopology euclideanreal (?S \<inter> {f x - d .. f x + d})) X g" |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
4000 |
using continuous_map_in_subtopology by blast |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
4001 |
have "\<exists>e>0. \<forall>x \<in> ?S \<inter> {f x - d .. f x + d} \<inter> ball (f x) e. g x \<in> T" |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
4002 |
using openin_continuous_map_preimage [OF contg \<open>openin X T\<close>] x \<open>x \<in> T\<close> \<open>0 < d\<close> |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
4003 |
unfolding openin_euclidean_subtopology_iff |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
4004 |
by (force simp: gf dist_commute) |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
4005 |
then obtain e where "e > 0 \<and> (\<forall>x\<in>f ` topspace X \<inter> {f x - d..f x + d} \<inter> ball (f x) e. g x \<in> T)" |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
4006 |
by metis |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
4007 |
with \<open>0 < d\<close> have "min d e > 0" "\<forall>u. u \<in> topspace X \<longrightarrow> \<bar>f x - f u\<bar> < min d e \<longrightarrow> u \<in> T" |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
4008 |
using dist_real_def gf by force+ |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
4009 |
then show ?thesis |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
4010 |
by (metis (full_types) Int_iff dist_real_def image_iff mem_ball gf) |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
4011 |
qed |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
4012 |
qed |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
4013 |
then obtain d where d: "\<And>r. r \<in> ?S \<inter> g -` T \<Longrightarrow> |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
4014 |
d r > 0 \<and> (\<forall>x \<in> ?S \<inter> ball r (d r). g x \<in> T)" |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
4015 |
by metis |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
4016 |
show ?thesis |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
4017 |
unfolding openin_subtopology |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
4018 |
proof (intro exI conjI) |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
4019 |
show "{x \<in> ?S. g x \<in> T} = (\<Union>r \<in> ?S \<inter> g -` T. ball r (d r)) \<inter> f ` topspace X" |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
4020 |
using d by (auto simp: gf) |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
4021 |
qed auto |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
4022 |
qed |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
4023 |
then show "continuous_map (top_of_set ?S) X g" |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
4024 |
by (simp add: continuous_map_def gf) |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
4025 |
qed (auto simp: gf) |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
4026 |
qed |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
4027 |
qed |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
4028 |
|
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
4029 |
subsubsection \<open>An injective function into R is a homeomorphism and so an open map.\<close> |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
4030 |
|
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
4031 |
lemma injective_into_1d_eq_homeomorphism: |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
4032 |
fixes f :: "'a::topological_space \<Rightarrow> real" |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
4033 |
assumes f: "continuous_on S f" and S: "path_connected S" |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
4034 |
shows "inj_on f S \<longleftrightarrow> (\<exists>g. homeomorphism S (f ` S) f g)" |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
4035 |
proof |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
4036 |
show "\<exists>g. homeomorphism S (f ` S) f g" |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
4037 |
if "inj_on f S" |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
4038 |
proof - |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
4039 |
have "embedding_map (top_of_set S) euclideanreal f" |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
4040 |
using that embedding_map_into_euclideanreal [of "top_of_set S" f] assms by auto |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
4041 |
then show ?thesis |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
4042 |
by (simp add: embedding_map_def) (metis all_closedin_homeomorphic_image f homeomorphism_injective_closed_map that) |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
4043 |
qed |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
4044 |
qed (metis homeomorphism_def inj_onI) |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
4045 |
|
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
4046 |
lemma injective_into_1d_imp_open_map: |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
4047 |
fixes f :: "'a::topological_space \<Rightarrow> real" |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
4048 |
assumes "continuous_on S f" "path_connected S" "inj_on f S" "openin (subtopology euclidean S) T" |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
4049 |
shows "openin (subtopology euclidean (f ` S)) (f ` T)" |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
4050 |
using assms homeomorphism_imp_open_map injective_into_1d_eq_homeomorphism by blast |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
4051 |
|
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
4052 |
lemma homeomorphism_into_1d: |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
4053 |
fixes f :: "'a::topological_space \<Rightarrow> real" |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
4054 |
assumes "path_connected S" "continuous_on S f" "f ` S = T" "inj_on f S" |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
4055 |
shows "\<exists>g. homeomorphism S T f g" |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
4056 |
using assms injective_into_1d_eq_homeomorphism by blast |
b7ef9090feed
Added embedding_map_into_euclideanreal; reduced dependence on Equivalence_Lebesgue_Henstock_Integration in Analysis theories by moving a few lemmas
paulson <lp15@cam.ac.uk>
parents:
70178
diff
changeset
|
4057 |
|
71189
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71172
diff
changeset
|
4058 |
subsection\<^marker>\<open>tag unimportant\<close> \<open>Rectangular paths\<close> |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71172
diff
changeset
|
4059 |
|
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71172
diff
changeset
|
4060 |
definition\<^marker>\<open>tag unimportant\<close> rectpath where |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71172
diff
changeset
|
4061 |
"rectpath a1 a3 = (let a2 = Complex (Re a3) (Im a1); a4 = Complex (Re a1) (Im a3) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71172
diff
changeset
|
4062 |
in linepath a1 a2 +++ linepath a2 a3 +++ linepath a3 a4 +++ linepath a4 a1)" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71172
diff
changeset
|
4063 |
|
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71172
diff
changeset
|
4064 |
lemma path_rectpath [simp, intro]: "path (rectpath a b)" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71172
diff
changeset
|
4065 |
by (simp add: Let_def rectpath_def) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71172
diff
changeset
|
4066 |
|
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71172
diff
changeset
|
4067 |
lemma pathstart_rectpath [simp]: "pathstart (rectpath a1 a3) = a1" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71172
diff
changeset
|
4068 |
by (simp add: rectpath_def Let_def) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71172
diff
changeset
|
4069 |
|
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71172
diff
changeset
|
4070 |
lemma pathfinish_rectpath [simp]: "pathfinish (rectpath a1 a3) = a1" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71172
diff
changeset
|
4071 |
by (simp add: rectpath_def Let_def) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71172
diff
changeset
|
4072 |
|
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71172
diff
changeset
|
4073 |
lemma simple_path_rectpath [simp, intro]: |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71172
diff
changeset
|
4074 |
assumes "Re a1 \<noteq> Re a3" "Im a1 \<noteq> Im a3" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71172
diff
changeset
|
4075 |
shows "simple_path (rectpath a1 a3)" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71172
diff
changeset
|
4076 |
unfolding rectpath_def Let_def using assms |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71172
diff
changeset
|
4077 |
by (intro simple_path_join_loop arc_join arc_linepath) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71172
diff
changeset
|
4078 |
(auto simp: complex_eq_iff path_image_join closed_segment_same_Re closed_segment_same_Im) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71172
diff
changeset
|
4079 |
|
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71172
diff
changeset
|
4080 |
lemma path_image_rectpath: |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71172
diff
changeset
|
4081 |
assumes "Re a1 \<le> Re a3" "Im a1 \<le> Im a3" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71172
diff
changeset
|
4082 |
shows "path_image (rectpath a1 a3) = |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71172
diff
changeset
|
4083 |
{z. Re z \<in> {Re a1, Re a3} \<and> Im z \<in> {Im a1..Im a3}} \<union> |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71172
diff
changeset
|
4084 |
{z. Im z \<in> {Im a1, Im a3} \<and> Re z \<in> {Re a1..Re a3}}" (is "?lhs = ?rhs") |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71172
diff
changeset
|
4085 |
proof - |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71172
diff
changeset
|
4086 |
define a2 a4 where "a2 = Complex (Re a3) (Im a1)" and "a4 = Complex (Re a1) (Im a3)" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71172
diff
changeset
|
4087 |
have "?lhs = closed_segment a1 a2 \<union> closed_segment a2 a3 \<union> |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71172
diff
changeset
|
4088 |
closed_segment a4 a3 \<union> closed_segment a1 a4" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71172
diff
changeset
|
4089 |
by (simp_all add: rectpath_def Let_def path_image_join closed_segment_commute |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71172
diff
changeset
|
4090 |
a2_def a4_def Un_assoc) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71172
diff
changeset
|
4091 |
also have "\<dots> = ?rhs" using assms |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71172
diff
changeset
|
4092 |
by (auto simp: rectpath_def Let_def path_image_join a2_def a4_def |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71172
diff
changeset
|
4093 |
closed_segment_same_Re closed_segment_same_Im closed_segment_eq_real_ivl) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71172
diff
changeset
|
4094 |
finally show ?thesis . |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71172
diff
changeset
|
4095 |
qed |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71172
diff
changeset
|
4096 |
|
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71172
diff
changeset
|
4097 |
lemma path_image_rectpath_subset_cbox: |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71172
diff
changeset
|
4098 |
assumes "Re a \<le> Re b" "Im a \<le> Im b" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71172
diff
changeset
|
4099 |
shows "path_image (rectpath a b) \<subseteq> cbox a b" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71172
diff
changeset
|
4100 |
using assms by (auto simp: path_image_rectpath in_cbox_complex_iff) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71172
diff
changeset
|
4101 |
|
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71172
diff
changeset
|
4102 |
lemma path_image_rectpath_inter_box: |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71172
diff
changeset
|
4103 |
assumes "Re a \<le> Re b" "Im a \<le> Im b" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71172
diff
changeset
|
4104 |
shows "path_image (rectpath a b) \<inter> box a b = {}" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71172
diff
changeset
|
4105 |
using assms by (auto simp: path_image_rectpath in_box_complex_iff) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71172
diff
changeset
|
4106 |
|
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71172
diff
changeset
|
4107 |
lemma path_image_rectpath_cbox_minus_box: |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71172
diff
changeset
|
4108 |
assumes "Re a \<le> Re b" "Im a \<le> Im b" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71172
diff
changeset
|
4109 |
shows "path_image (rectpath a b) = cbox a b - box a b" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71172
diff
changeset
|
4110 |
using assms by (auto simp: path_image_rectpath in_cbox_complex_iff |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71172
diff
changeset
|
4111 |
in_box_complex_iff) |
71184
d62fdaafdafc
renamed Analysis/Winding_Numbers to Winding_Numbers_2; reorganised Analysis/Cauchy_Integral_Theorem by splitting it into Contour_Integration, Winding_Numbers,Cauchy_Integral_Theorem and Cauchy_Integral_Formula.
Wenda Li <wl302@cam.ac.uk>
parents:
71172
diff
changeset
|
4112 |
|
36583 | 4113 |
end |