author | paulson <lp15@cam.ac.uk> |
Thu, 24 Aug 2017 12:45:46 +0100 | |
changeset 66498 | 97fc319d6089 |
parent 66492 | d7206afe2d28 |
parent 66497 | 18a6478a574c |
child 66703 | 61bf958fa1c1 |
permissions | -rw-r--r-- |
63957
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HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1 |
(* Title: HOL/Analysis/Tagged_Division.thy |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2 |
Author: John Harrison |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
3 |
Author: Robert Himmelmann, TU Muenchen (Translation from HOL light); proofs reworked by LCP |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
4 |
*) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
5 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
6 |
section \<open>Tagged divisions used for the Henstock-Kurzweil gauge integration\<close> |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
7 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
8 |
theory Tagged_Division |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
9 |
imports |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
10 |
Topology_Euclidean_Space |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
11 |
begin |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
12 |
|
66492
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
13 |
term comm_monoid |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
14 |
|
66294
0442b3f45556
refactored some HORRIBLE integration proofs
paulson <lp15@cam.ac.uk>
parents:
66193
diff
changeset
|
15 |
lemma sum_Sigma_product: |
0442b3f45556
refactored some HORRIBLE integration proofs
paulson <lp15@cam.ac.uk>
parents:
66193
diff
changeset
|
16 |
assumes "finite S" |
0442b3f45556
refactored some HORRIBLE integration proofs
paulson <lp15@cam.ac.uk>
parents:
66193
diff
changeset
|
17 |
and "\<And>i. i \<in> S \<Longrightarrow> finite (T i)" |
0442b3f45556
refactored some HORRIBLE integration proofs
paulson <lp15@cam.ac.uk>
parents:
66193
diff
changeset
|
18 |
shows "(\<Sum>i\<in>S. sum (x i) (T i)) = (\<Sum>(i, j)\<in>Sigma S T. x i j)" |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
19 |
using assms |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
20 |
proof induct |
66294
0442b3f45556
refactored some HORRIBLE integration proofs
paulson <lp15@cam.ac.uk>
parents:
66193
diff
changeset
|
21 |
case empty |
0442b3f45556
refactored some HORRIBLE integration proofs
paulson <lp15@cam.ac.uk>
parents:
66193
diff
changeset
|
22 |
then show ?case |
0442b3f45556
refactored some HORRIBLE integration proofs
paulson <lp15@cam.ac.uk>
parents:
66193
diff
changeset
|
23 |
by simp |
0442b3f45556
refactored some HORRIBLE integration proofs
paulson <lp15@cam.ac.uk>
parents:
66193
diff
changeset
|
24 |
next |
0442b3f45556
refactored some HORRIBLE integration proofs
paulson <lp15@cam.ac.uk>
parents:
66193
diff
changeset
|
25 |
case (insert a S) |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
26 |
show ?case |
66294
0442b3f45556
refactored some HORRIBLE integration proofs
paulson <lp15@cam.ac.uk>
parents:
66193
diff
changeset
|
27 |
by (simp add: insert.hyps(1) insert.prems sum.Sigma) |
0442b3f45556
refactored some HORRIBLE integration proofs
paulson <lp15@cam.ac.uk>
parents:
66193
diff
changeset
|
28 |
qed |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
29 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
30 |
lemmas scaleR_simps = scaleR_zero_left scaleR_minus_left scaleR_left_diff_distrib |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
31 |
scaleR_zero_right scaleR_minus_right scaleR_right_diff_distrib scaleR_eq_0_iff |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
32 |
scaleR_cancel_left scaleR_cancel_right scaleR_add_right scaleR_add_left real_vector_class.scaleR_one |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
33 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
34 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
35 |
subsection \<open>Sundries\<close> |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
36 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
37 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
38 |
text\<open>A transitive relation is well-founded if all initial segments are finite.\<close> |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
39 |
lemma wf_finite_segments: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
40 |
assumes "irrefl r" and "trans r" and "\<And>x. finite {y. (y, x) \<in> r}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
41 |
shows "wf (r)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
42 |
apply (simp add: trans_wf_iff wf_iff_acyclic_if_finite converse_def assms) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
43 |
using acyclic_def assms irrefl_def trans_Restr by fastforce |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
44 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
45 |
text\<open>For creating values between @{term u} and @{term v}.\<close> |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
46 |
lemma scaling_mono: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
47 |
fixes u::"'a::linordered_field" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
48 |
assumes "u \<le> v" "0 \<le> r" "r \<le> s" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
49 |
shows "u + r * (v - u) / s \<le> v" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
50 |
proof - |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
51 |
have "r/s \<le> 1" using assms |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
52 |
using divide_le_eq_1 by fastforce |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
53 |
then have "(r/s) * (v - u) \<le> 1 * (v - u)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
54 |
by (meson diff_ge_0_iff_ge mult_right_mono \<open>u \<le> v\<close>) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
55 |
then show ?thesis |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
56 |
by (simp add: field_simps) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
57 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
58 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
59 |
subsection \<open>Some useful lemmas about intervals.\<close> |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
60 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
61 |
lemma interior_subset_union_intervals: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
62 |
assumes "i = cbox a b" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
63 |
and "j = cbox c d" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
64 |
and "interior j \<noteq> {}" |
66299
1b4aa3e3e4e6
partial cleanup of the horrible Tagged_Division
paulson <lp15@cam.ac.uk>
parents:
66296
diff
changeset
|
65 |
and "i \<subseteq> j \<union> S" |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
66 |
and "interior i \<inter> interior j = {}" |
66299
1b4aa3e3e4e6
partial cleanup of the horrible Tagged_Division
paulson <lp15@cam.ac.uk>
parents:
66296
diff
changeset
|
67 |
shows "interior i \<subseteq> interior S" |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
68 |
proof - |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
69 |
have "box a b \<inter> cbox c d = {}" |
66299
1b4aa3e3e4e6
partial cleanup of the horrible Tagged_Division
paulson <lp15@cam.ac.uk>
parents:
66296
diff
changeset
|
70 |
using inter_interval_mixed_eq_empty[of c d a b] assms |
1b4aa3e3e4e6
partial cleanup of the horrible Tagged_Division
paulson <lp15@cam.ac.uk>
parents:
66296
diff
changeset
|
71 |
unfolding interior_cbox by auto |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
72 |
moreover |
66299
1b4aa3e3e4e6
partial cleanup of the horrible Tagged_Division
paulson <lp15@cam.ac.uk>
parents:
66296
diff
changeset
|
73 |
have "box a b \<subseteq> cbox c d \<union> S" |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
74 |
apply (rule order_trans,rule box_subset_cbox) |
66299
1b4aa3e3e4e6
partial cleanup of the horrible Tagged_Division
paulson <lp15@cam.ac.uk>
parents:
66296
diff
changeset
|
75 |
using assms by auto |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
76 |
ultimately |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
77 |
show ?thesis |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
78 |
unfolding assms interior_cbox |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
79 |
by auto (metis IntI UnE empty_iff interior_maximal open_box subsetCE subsetI) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
80 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
81 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
82 |
lemma interior_Union_subset_cbox: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
83 |
assumes "finite f" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
84 |
assumes f: "\<And>s. s \<in> f \<Longrightarrow> \<exists>a b. s = cbox a b" "\<And>s. s \<in> f \<Longrightarrow> interior s \<subseteq> t" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
85 |
and t: "closed t" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
86 |
shows "interior (\<Union>f) \<subseteq> t" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
87 |
proof - |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
88 |
have [simp]: "s \<in> f \<Longrightarrow> closed s" for s |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
89 |
using f by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
90 |
define E where "E = {s\<in>f. interior s = {}}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
91 |
then have "finite E" "E \<subseteq> {s\<in>f. interior s = {}}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
92 |
using \<open>finite f\<close> by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
93 |
then have "interior (\<Union>f) = interior (\<Union>(f - E))" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
94 |
proof (induction E rule: finite_subset_induct') |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
95 |
case (insert s f') |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
96 |
have "interior (\<Union>(f - insert s f') \<union> s) = interior (\<Union>(f - insert s f'))" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
97 |
using insert.hyps \<open>finite f\<close> by (intro interior_closed_Un_empty_interior) auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
98 |
also have "\<Union>(f - insert s f') \<union> s = \<Union>(f - f')" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
99 |
using insert.hyps by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
100 |
finally show ?case |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
101 |
by (simp add: insert.IH) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
102 |
qed simp |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
103 |
also have "\<dots> \<subseteq> \<Union>(f - E)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
104 |
by (rule interior_subset) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
105 |
also have "\<dots> \<subseteq> t" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
106 |
proof (rule Union_least) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
107 |
fix s assume "s \<in> f - E" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
108 |
with f[of s] obtain a b where s: "s \<in> f" "s = cbox a b" "box a b \<noteq> {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
109 |
by (fastforce simp: E_def) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
110 |
have "closure (interior s) \<subseteq> closure t" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
111 |
by (intro closure_mono f \<open>s \<in> f\<close>) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
112 |
with s \<open>closed t\<close> show "s \<subseteq> t" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
113 |
by simp |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
114 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
115 |
finally show ?thesis . |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
116 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
117 |
|
66299
1b4aa3e3e4e6
partial cleanup of the horrible Tagged_Division
paulson <lp15@cam.ac.uk>
parents:
66296
diff
changeset
|
118 |
lemma Int_interior_Union_intervals: |
1b4aa3e3e4e6
partial cleanup of the horrible Tagged_Division
paulson <lp15@cam.ac.uk>
parents:
66296
diff
changeset
|
119 |
"\<lbrakk>finite \<F>; open S; \<And>T. T\<in>\<F> \<Longrightarrow> \<exists>a b. T = cbox a b; \<And>T. T\<in>\<F> \<Longrightarrow> S \<inter> (interior T) = {}\<rbrakk> |
1b4aa3e3e4e6
partial cleanup of the horrible Tagged_Division
paulson <lp15@cam.ac.uk>
parents:
66296
diff
changeset
|
120 |
\<Longrightarrow> S \<inter> interior (\<Union>\<F>) = {}" |
1b4aa3e3e4e6
partial cleanup of the horrible Tagged_Division
paulson <lp15@cam.ac.uk>
parents:
66296
diff
changeset
|
121 |
using interior_Union_subset_cbox[of \<F> "UNIV - S"] by auto |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
122 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
123 |
lemma interval_split: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
124 |
fixes a :: "'a::euclidean_space" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
125 |
assumes "k \<in> Basis" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
126 |
shows |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
127 |
"cbox a b \<inter> {x. x\<bullet>k \<le> c} = cbox a (\<Sum>i\<in>Basis. (if i = k then min (b\<bullet>k) c else b\<bullet>i) *\<^sub>R i)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
128 |
"cbox a b \<inter> {x. x\<bullet>k \<ge> c} = cbox (\<Sum>i\<in>Basis. (if i = k then max (a\<bullet>k) c else a\<bullet>i) *\<^sub>R i) b" |
66299
1b4aa3e3e4e6
partial cleanup of the horrible Tagged_Division
paulson <lp15@cam.ac.uk>
parents:
66296
diff
changeset
|
129 |
using assms by (rule_tac set_eqI; auto simp: mem_box)+ |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
130 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
131 |
lemma interval_not_empty: "\<forall>i\<in>Basis. a\<bullet>i \<le> b\<bullet>i \<Longrightarrow> cbox a b \<noteq> {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
132 |
by (simp add: box_ne_empty) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
133 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
134 |
subsection \<open>Bounds on intervals where they exist.\<close> |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
135 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
136 |
definition interval_upperbound :: "('a::euclidean_space) set \<Rightarrow> 'a" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
137 |
where "interval_upperbound s = (\<Sum>i\<in>Basis. (SUP x:s. x\<bullet>i) *\<^sub>R i)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
138 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
139 |
definition interval_lowerbound :: "('a::euclidean_space) set \<Rightarrow> 'a" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
140 |
where "interval_lowerbound s = (\<Sum>i\<in>Basis. (INF x:s. x\<bullet>i) *\<^sub>R i)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
141 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
142 |
lemma interval_upperbound[simp]: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
143 |
"\<forall>i\<in>Basis. a\<bullet>i \<le> b\<bullet>i \<Longrightarrow> |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
144 |
interval_upperbound (cbox a b) = (b::'a::euclidean_space)" |
64267 | 145 |
unfolding interval_upperbound_def euclidean_representation_sum cbox_def |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
146 |
by (safe intro!: cSup_eq) auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
147 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
148 |
lemma interval_lowerbound[simp]: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
149 |
"\<forall>i\<in>Basis. a\<bullet>i \<le> b\<bullet>i \<Longrightarrow> |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
150 |
interval_lowerbound (cbox a b) = (a::'a::euclidean_space)" |
64267 | 151 |
unfolding interval_lowerbound_def euclidean_representation_sum cbox_def |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
152 |
by (safe intro!: cInf_eq) auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
153 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
154 |
lemmas interval_bounds = interval_upperbound interval_lowerbound |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
155 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
156 |
lemma |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
157 |
fixes X::"real set" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
158 |
shows interval_upperbound_real[simp]: "interval_upperbound X = Sup X" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
159 |
and interval_lowerbound_real[simp]: "interval_lowerbound X = Inf X" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
160 |
by (auto simp: interval_upperbound_def interval_lowerbound_def) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
161 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
162 |
lemma interval_bounds'[simp]: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
163 |
assumes "cbox a b \<noteq> {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
164 |
shows "interval_upperbound (cbox a b) = b" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
165 |
and "interval_lowerbound (cbox a b) = a" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
166 |
using assms unfolding box_ne_empty by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
167 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
168 |
lemma interval_upperbound_Times: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
169 |
assumes "A \<noteq> {}" and "B \<noteq> {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
170 |
shows "interval_upperbound (A \<times> B) = (interval_upperbound A, interval_upperbound B)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
171 |
proof- |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
172 |
from assms have fst_image_times': "A = fst ` (A \<times> B)" by simp |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
173 |
have "(\<Sum>i\<in>Basis. (SUP x:A \<times> B. x \<bullet> (i, 0)) *\<^sub>R i) = (\<Sum>i\<in>Basis. (SUP x:A. x \<bullet> i) *\<^sub>R i)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
174 |
by (subst (2) fst_image_times') (simp del: fst_image_times add: o_def inner_Pair_0) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
175 |
moreover from assms have snd_image_times': "B = snd ` (A \<times> B)" by simp |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
176 |
have "(\<Sum>i\<in>Basis. (SUP x:A \<times> B. x \<bullet> (0, i)) *\<^sub>R i) = (\<Sum>i\<in>Basis. (SUP x:B. x \<bullet> i) *\<^sub>R i)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
177 |
by (subst (2) snd_image_times') (simp del: snd_image_times add: o_def inner_Pair_0) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
178 |
ultimately show ?thesis unfolding interval_upperbound_def |
64267 | 179 |
by (subst sum_Basis_prod_eq) (auto simp add: sum_prod) |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
180 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
181 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
182 |
lemma interval_lowerbound_Times: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
183 |
assumes "A \<noteq> {}" and "B \<noteq> {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
184 |
shows "interval_lowerbound (A \<times> B) = (interval_lowerbound A, interval_lowerbound B)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
185 |
proof- |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
186 |
from assms have fst_image_times': "A = fst ` (A \<times> B)" by simp |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
187 |
have "(\<Sum>i\<in>Basis. (INF x:A \<times> B. x \<bullet> (i, 0)) *\<^sub>R i) = (\<Sum>i\<in>Basis. (INF x:A. x \<bullet> i) *\<^sub>R i)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
188 |
by (subst (2) fst_image_times') (simp del: fst_image_times add: o_def inner_Pair_0) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
189 |
moreover from assms have snd_image_times': "B = snd ` (A \<times> B)" by simp |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
190 |
have "(\<Sum>i\<in>Basis. (INF x:A \<times> B. x \<bullet> (0, i)) *\<^sub>R i) = (\<Sum>i\<in>Basis. (INF x:B. x \<bullet> i) *\<^sub>R i)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
191 |
by (subst (2) snd_image_times') (simp del: snd_image_times add: o_def inner_Pair_0) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
192 |
ultimately show ?thesis unfolding interval_lowerbound_def |
64267 | 193 |
by (subst sum_Basis_prod_eq) (auto simp add: sum_prod) |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
194 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
195 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
196 |
subsection \<open>The notion of a gauge --- simply an open set containing the point.\<close> |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
197 |
|
66317 | 198 |
definition "gauge \<gamma> \<longleftrightarrow> (\<forall>x. x \<in> \<gamma> x \<and> open (\<gamma> x))" |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
199 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
200 |
lemma gaugeI: |
66317 | 201 |
assumes "\<And>x. x \<in> \<gamma> x" |
202 |
and "\<And>x. open (\<gamma> x)" |
|
203 |
shows "gauge \<gamma>" |
|
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
204 |
using assms unfolding gauge_def by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
205 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
206 |
lemma gaugeD[dest]: |
66317 | 207 |
assumes "gauge \<gamma>" |
208 |
shows "x \<in> \<gamma> x" |
|
209 |
and "open (\<gamma> x)" |
|
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
210 |
using assms unfolding gauge_def by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
211 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
212 |
lemma gauge_ball_dependent: "\<forall>x. 0 < e x \<Longrightarrow> gauge (\<lambda>x. ball x (e x))" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
213 |
unfolding gauge_def by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
214 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
215 |
lemma gauge_ball[intro]: "0 < e \<Longrightarrow> gauge (\<lambda>x. ball x e)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
216 |
unfolding gauge_def by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
217 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
218 |
lemma gauge_trivial[intro!]: "gauge (\<lambda>x. ball x 1)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
219 |
by (rule gauge_ball) auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
220 |
|
66317 | 221 |
lemma gauge_Int[intro]: "gauge \<gamma>1 \<Longrightarrow> gauge \<gamma>2 \<Longrightarrow> gauge (\<lambda>x. \<gamma>1 x \<inter> \<gamma>2 x)" |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
222 |
unfolding gauge_def by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
223 |
|
66164
2d79288b042c
New theorems and much tidying up of the old ones
paulson <lp15@cam.ac.uk>
parents:
66154
diff
changeset
|
224 |
lemma gauge_reflect: |
66299
1b4aa3e3e4e6
partial cleanup of the horrible Tagged_Division
paulson <lp15@cam.ac.uk>
parents:
66296
diff
changeset
|
225 |
fixes \<gamma> :: "'a::euclidean_space \<Rightarrow> 'a set" |
1b4aa3e3e4e6
partial cleanup of the horrible Tagged_Division
paulson <lp15@cam.ac.uk>
parents:
66296
diff
changeset
|
226 |
shows "gauge \<gamma> \<Longrightarrow> gauge (\<lambda>x. uminus ` \<gamma> (- x))" |
66164
2d79288b042c
New theorems and much tidying up of the old ones
paulson <lp15@cam.ac.uk>
parents:
66154
diff
changeset
|
227 |
using equation_minus_iff |
2d79288b042c
New theorems and much tidying up of the old ones
paulson <lp15@cam.ac.uk>
parents:
66154
diff
changeset
|
228 |
by (auto simp: gauge_def surj_def intro!: open_surjective_linear_image linear_uminus) |
2d79288b042c
New theorems and much tidying up of the old ones
paulson <lp15@cam.ac.uk>
parents:
66154
diff
changeset
|
229 |
|
2d79288b042c
New theorems and much tidying up of the old ones
paulson <lp15@cam.ac.uk>
parents:
66154
diff
changeset
|
230 |
lemma gauge_Inter: |
66299
1b4aa3e3e4e6
partial cleanup of the horrible Tagged_Division
paulson <lp15@cam.ac.uk>
parents:
66296
diff
changeset
|
231 |
assumes "finite S" |
66317 | 232 |
and "\<And>u. u\<in>S \<Longrightarrow> gauge (f u)" |
233 |
shows "gauge (\<lambda>x. \<Inter>{f u x | u. u \<in> S})" |
|
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
234 |
proof - |
66317 | 235 |
have *: "\<And>x. {f u x |u. u \<in> S} = (\<lambda>u. f u x) ` S" |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
236 |
by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
237 |
show ?thesis |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
238 |
unfolding gauge_def unfolding * |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
239 |
using assms unfolding Ball_def Inter_iff mem_Collect_eq gauge_def by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
240 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
241 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
242 |
lemma gauge_existence_lemma: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
243 |
"(\<forall>x. \<exists>d :: real. p x \<longrightarrow> 0 < d \<and> q d x) \<longleftrightarrow> (\<forall>x. \<exists>d>0. p x \<longrightarrow> q d x)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
244 |
by (metis zero_less_one) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
245 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
246 |
subsection \<open>Attempt a systematic general set of "offset" results for components.\<close> |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
247 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
248 |
lemma gauge_modify: |
66299
1b4aa3e3e4e6
partial cleanup of the horrible Tagged_Division
paulson <lp15@cam.ac.uk>
parents:
66296
diff
changeset
|
249 |
assumes "(\<forall>S. open S \<longrightarrow> open {x. f(x) \<in> S})" "gauge d" |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
250 |
shows "gauge (\<lambda>x. {y. f y \<in> d (f x)})" |
66299
1b4aa3e3e4e6
partial cleanup of the horrible Tagged_Division
paulson <lp15@cam.ac.uk>
parents:
66296
diff
changeset
|
251 |
using assms unfolding gauge_def by force |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
252 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
253 |
subsection \<open>Divisions.\<close> |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
254 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
255 |
definition division_of (infixl "division'_of" 40) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
256 |
where |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
257 |
"s division_of i \<longleftrightarrow> |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
258 |
finite s \<and> |
66113
571b698659c0
Repaired an inadvertent reordering of the premises of two theorems
paulson <lp15@cam.ac.uk>
parents:
66112
diff
changeset
|
259 |
(\<forall>K\<in>s. K \<subseteq> i \<and> K \<noteq> {} \<and> (\<exists>a b. K = cbox a b)) \<and> |
571b698659c0
Repaired an inadvertent reordering of the premises of two theorems
paulson <lp15@cam.ac.uk>
parents:
66112
diff
changeset
|
260 |
(\<forall>K1\<in>s. \<forall>K2\<in>s. K1 \<noteq> K2 \<longrightarrow> interior(K1) \<inter> interior(K2) = {}) \<and> |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
261 |
(\<Union>s = i)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
262 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
263 |
lemma division_ofD[dest]: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
264 |
assumes "s division_of i" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
265 |
shows "finite s" |
66113
571b698659c0
Repaired an inadvertent reordering of the premises of two theorems
paulson <lp15@cam.ac.uk>
parents:
66112
diff
changeset
|
266 |
and "\<And>K. K \<in> s \<Longrightarrow> K \<subseteq> i" |
571b698659c0
Repaired an inadvertent reordering of the premises of two theorems
paulson <lp15@cam.ac.uk>
parents:
66112
diff
changeset
|
267 |
and "\<And>K. K \<in> s \<Longrightarrow> K \<noteq> {}" |
571b698659c0
Repaired an inadvertent reordering of the premises of two theorems
paulson <lp15@cam.ac.uk>
parents:
66112
diff
changeset
|
268 |
and "\<And>K. K \<in> s \<Longrightarrow> \<exists>a b. K = cbox a b" |
571b698659c0
Repaired an inadvertent reordering of the premises of two theorems
paulson <lp15@cam.ac.uk>
parents:
66112
diff
changeset
|
269 |
and "\<And>K1 K2. K1 \<in> s \<Longrightarrow> K2 \<in> s \<Longrightarrow> K1 \<noteq> K2 \<Longrightarrow> interior(K1) \<inter> interior(K2) = {}" |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
270 |
and "\<Union>s = i" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
271 |
using assms unfolding division_of_def by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
272 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
273 |
lemma division_ofI: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
274 |
assumes "finite s" |
66113
571b698659c0
Repaired an inadvertent reordering of the premises of two theorems
paulson <lp15@cam.ac.uk>
parents:
66112
diff
changeset
|
275 |
and "\<And>K. K \<in> s \<Longrightarrow> K \<subseteq> i" |
571b698659c0
Repaired an inadvertent reordering of the premises of two theorems
paulson <lp15@cam.ac.uk>
parents:
66112
diff
changeset
|
276 |
and "\<And>K. K \<in> s \<Longrightarrow> K \<noteq> {}" |
571b698659c0
Repaired an inadvertent reordering of the premises of two theorems
paulson <lp15@cam.ac.uk>
parents:
66112
diff
changeset
|
277 |
and "\<And>K. K \<in> s \<Longrightarrow> \<exists>a b. K = cbox a b" |
571b698659c0
Repaired an inadvertent reordering of the premises of two theorems
paulson <lp15@cam.ac.uk>
parents:
66112
diff
changeset
|
278 |
and "\<And>K1 K2. K1 \<in> s \<Longrightarrow> K2 \<in> s \<Longrightarrow> K1 \<noteq> K2 \<Longrightarrow> interior K1 \<inter> interior K2 = {}" |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
279 |
and "\<Union>s = i" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
280 |
shows "s division_of i" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
281 |
using assms unfolding division_of_def by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
282 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
283 |
lemma division_of_finite: "s division_of i \<Longrightarrow> finite s" |
66296
33a47f2d9edc
New theory of Equiintegrability / Continuity of the indefinite integral / improper integration
paulson <lp15@cam.ac.uk>
parents:
66295
diff
changeset
|
284 |
by auto |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
285 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
286 |
lemma division_of_self[intro]: "cbox a b \<noteq> {} \<Longrightarrow> {cbox a b} division_of (cbox a b)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
287 |
unfolding division_of_def by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
288 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
289 |
lemma division_of_trivial[simp]: "s division_of {} \<longleftrightarrow> s = {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
290 |
unfolding division_of_def by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
291 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
292 |
lemma division_of_sing[simp]: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
293 |
"s division_of cbox a (a::'a::euclidean_space) \<longleftrightarrow> s = {cbox a a}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
294 |
(is "?l = ?r") |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
295 |
proof |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
296 |
assume ?r |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
297 |
moreover |
66113
571b698659c0
Repaired an inadvertent reordering of the premises of two theorems
paulson <lp15@cam.ac.uk>
parents:
66112
diff
changeset
|
298 |
{ fix K |
571b698659c0
Repaired an inadvertent reordering of the premises of two theorems
paulson <lp15@cam.ac.uk>
parents:
66112
diff
changeset
|
299 |
assume "s = {{a}}" "K\<in>s" |
571b698659c0
Repaired an inadvertent reordering of the premises of two theorems
paulson <lp15@cam.ac.uk>
parents:
66112
diff
changeset
|
300 |
then have "\<exists>x y. K = cbox x y" |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
301 |
apply (rule_tac x=a in exI)+ |
66317 | 302 |
apply force |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
303 |
done |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
304 |
} |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
305 |
ultimately show ?l |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
306 |
unfolding division_of_def cbox_sing by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
307 |
next |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
308 |
assume ?l |
66317 | 309 |
have "x = {a}" if "x \<in> s" for x |
310 |
by (metis \<open>s division_of cbox a a\<close> cbox_sing division_ofD(2) division_ofD(3) subset_singletonD that) |
|
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
311 |
moreover have "s \<noteq> {}" |
66193 | 312 |
using \<open>s division_of cbox a a\<close> by auto |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
313 |
ultimately show ?r |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
314 |
unfolding cbox_sing by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
315 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
316 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
317 |
lemma elementary_empty: obtains p where "p division_of {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
318 |
unfolding division_of_trivial by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
319 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
320 |
lemma elementary_interval: obtains p where "p division_of (cbox a b)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
321 |
by (metis division_of_trivial division_of_self) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
322 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
323 |
lemma division_contains: "s division_of i \<Longrightarrow> \<forall>x\<in>i. \<exists>k\<in>s. x \<in> k" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
324 |
unfolding division_of_def by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
325 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
326 |
lemma forall_in_division: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
327 |
"d division_of i \<Longrightarrow> (\<forall>x\<in>d. P x) \<longleftrightarrow> (\<forall>a b. cbox a b \<in> d \<longrightarrow> P (cbox a b))" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
328 |
unfolding division_of_def by fastforce |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
329 |
|
66296
33a47f2d9edc
New theory of Equiintegrability / Continuity of the indefinite integral / improper integration
paulson <lp15@cam.ac.uk>
parents:
66295
diff
changeset
|
330 |
lemma cbox_division_memE: |
33a47f2d9edc
New theory of Equiintegrability / Continuity of the indefinite integral / improper integration
paulson <lp15@cam.ac.uk>
parents:
66295
diff
changeset
|
331 |
assumes \<D>: "K \<in> \<D>" "\<D> division_of S" |
33a47f2d9edc
New theory of Equiintegrability / Continuity of the indefinite integral / improper integration
paulson <lp15@cam.ac.uk>
parents:
66295
diff
changeset
|
332 |
obtains a b where "K = cbox a b" "K \<noteq> {}" "K \<subseteq> S" |
33a47f2d9edc
New theory of Equiintegrability / Continuity of the indefinite integral / improper integration
paulson <lp15@cam.ac.uk>
parents:
66295
diff
changeset
|
333 |
using assms unfolding division_of_def by metis |
33a47f2d9edc
New theory of Equiintegrability / Continuity of the indefinite integral / improper integration
paulson <lp15@cam.ac.uk>
parents:
66295
diff
changeset
|
334 |
|
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
335 |
lemma division_of_subset: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
336 |
assumes "p division_of (\<Union>p)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
337 |
and "q \<subseteq> p" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
338 |
shows "q division_of (\<Union>q)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
339 |
proof (rule division_ofI) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
340 |
note * = division_ofD[OF assms(1)] |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
341 |
show "finite q" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
342 |
using "*"(1) assms(2) infinite_super by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
343 |
{ |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
344 |
fix k |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
345 |
assume "k \<in> q" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
346 |
then have kp: "k \<in> p" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
347 |
using assms(2) by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
348 |
show "k \<subseteq> \<Union>q" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
349 |
using \<open>k \<in> q\<close> by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
350 |
show "\<exists>a b. k = cbox a b" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
351 |
using *(4)[OF kp] by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
352 |
show "k \<noteq> {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
353 |
using *(3)[OF kp] by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
354 |
} |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
355 |
fix k1 k2 |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
356 |
assume "k1 \<in> q" "k2 \<in> q" "k1 \<noteq> k2" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
357 |
then have **: "k1 \<in> p" "k2 \<in> p" "k1 \<noteq> k2" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
358 |
using assms(2) by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
359 |
show "interior k1 \<inter> interior k2 = {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
360 |
using *(5)[OF **] by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
361 |
qed auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
362 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
363 |
lemma division_of_union_self[intro]: "p division_of s \<Longrightarrow> p division_of (\<Union>p)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
364 |
unfolding division_of_def by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
365 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
366 |
lemma division_inter: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
367 |
fixes s1 s2 :: "'a::euclidean_space set" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
368 |
assumes "p1 division_of s1" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
369 |
and "p2 division_of s2" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
370 |
shows "{k1 \<inter> k2 | k1 k2. k1 \<in> p1 \<and> k2 \<in> p2 \<and> k1 \<inter> k2 \<noteq> {}} division_of (s1 \<inter> s2)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
371 |
(is "?A' division_of _") |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
372 |
proof - |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
373 |
let ?A = "{s. s \<in> (\<lambda>(k1,k2). k1 \<inter> k2) ` (p1 \<times> p2) \<and> s \<noteq> {}}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
374 |
have *: "?A' = ?A" by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
375 |
show ?thesis |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
376 |
unfolding * |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
377 |
proof (rule division_ofI) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
378 |
have "?A \<subseteq> (\<lambda>(x, y). x \<inter> y) ` (p1 \<times> p2)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
379 |
by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
380 |
moreover have "finite (p1 \<times> p2)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
381 |
using assms unfolding division_of_def by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
382 |
ultimately show "finite ?A" by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
383 |
have *: "\<And>s. \<Union>{x\<in>s. x \<noteq> {}} = \<Union>s" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
384 |
by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
385 |
show "\<Union>?A = s1 \<inter> s2" |
66317 | 386 |
unfolding * |
387 |
using division_ofD(6)[OF assms(1)] and division_ofD(6)[OF assms(2)] by auto |
|
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
388 |
{ |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
389 |
fix k |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
390 |
assume "k \<in> ?A" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
391 |
then obtain k1 k2 where k: "k = k1 \<inter> k2" "k1 \<in> p1" "k2 \<in> p2" "k \<noteq> {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
392 |
by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
393 |
then show "k \<noteq> {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
394 |
by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
395 |
show "k \<subseteq> s1 \<inter> s2" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
396 |
using division_ofD(2)[OF assms(1) k(2)] and division_ofD(2)[OF assms(2) k(3)] |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
397 |
unfolding k by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
398 |
obtain a1 b1 where k1: "k1 = cbox a1 b1" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
399 |
using division_ofD(4)[OF assms(1) k(2)] by blast |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
400 |
obtain a2 b2 where k2: "k2 = cbox a2 b2" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
401 |
using division_ofD(4)[OF assms(2) k(3)] by blast |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
402 |
show "\<exists>a b. k = cbox a b" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
403 |
unfolding k k1 k2 unfolding Int_interval by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
404 |
} |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
405 |
fix k1 k2 |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
406 |
assume "k1 \<in> ?A" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
407 |
then obtain x1 y1 where k1: "k1 = x1 \<inter> y1" "x1 \<in> p1" "y1 \<in> p2" "k1 \<noteq> {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
408 |
by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
409 |
assume "k2 \<in> ?A" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
410 |
then obtain x2 y2 where k2: "k2 = x2 \<inter> y2" "x2 \<in> p1" "y2 \<in> p2" "k2 \<noteq> {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
411 |
by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
412 |
assume "k1 \<noteq> k2" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
413 |
then have th: "x1 \<noteq> x2 \<or> y1 \<noteq> y2" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
414 |
unfolding k1 k2 by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
415 |
have *: "interior x1 \<inter> interior x2 = {} \<or> interior y1 \<inter> interior y2 = {} \<Longrightarrow> |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
416 |
interior (x1 \<inter> y1) \<subseteq> interior x1 \<Longrightarrow> interior (x1 \<inter> y1) \<subseteq> interior y1 \<Longrightarrow> |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
417 |
interior (x2 \<inter> y2) \<subseteq> interior x2 \<Longrightarrow> interior (x2 \<inter> y2) \<subseteq> interior y2 \<Longrightarrow> |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
418 |
interior (x1 \<inter> y1) \<inter> interior (x2 \<inter> y2) = {}" by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
419 |
show "interior k1 \<inter> interior k2 = {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
420 |
unfolding k1 k2 |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
421 |
apply (rule *) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
422 |
using assms division_ofD(5) k1 k2(2) k2(3) th apply auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
423 |
done |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
424 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
425 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
426 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
427 |
lemma division_inter_1: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
428 |
assumes "d division_of i" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
429 |
and "cbox a (b::'a::euclidean_space) \<subseteq> i" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
430 |
shows "{cbox a b \<inter> k | k. k \<in> d \<and> cbox a b \<inter> k \<noteq> {}} division_of (cbox a b)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
431 |
proof (cases "cbox a b = {}") |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
432 |
case True |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
433 |
show ?thesis |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
434 |
unfolding True and division_of_trivial by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
435 |
next |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
436 |
case False |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
437 |
have *: "cbox a b \<inter> i = cbox a b" using assms(2) by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
438 |
show ?thesis |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
439 |
using division_inter[OF division_of_self[OF False] assms(1)] |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
440 |
unfolding * by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
441 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
442 |
|
66317 | 443 |
lemma elementary_Int: |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
444 |
fixes s t :: "'a::euclidean_space set" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
445 |
assumes "p1 division_of s" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
446 |
and "p2 division_of t" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
447 |
shows "\<exists>p. p division_of (s \<inter> t)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
448 |
using assms division_inter by blast |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
449 |
|
66317 | 450 |
lemma elementary_Inter: |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
451 |
assumes "finite f" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
452 |
and "f \<noteq> {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
453 |
and "\<forall>s\<in>f. \<exists>p. p division_of (s::('a::euclidean_space) set)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
454 |
shows "\<exists>p. p division_of (\<Inter>f)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
455 |
using assms |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
456 |
proof (induct f rule: finite_induct) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
457 |
case (insert x f) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
458 |
show ?case |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
459 |
proof (cases "f = {}") |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
460 |
case True |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
461 |
then show ?thesis |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
462 |
unfolding True using insert by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
463 |
next |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
464 |
case False |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
465 |
obtain p where "p division_of \<Inter>f" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
466 |
using insert(3)[OF False insert(5)[unfolded ball_simps,THEN conjunct2]] .. |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
467 |
moreover obtain px where "px division_of x" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
468 |
using insert(5)[rule_format,OF insertI1] .. |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
469 |
ultimately show ?thesis |
66317 | 470 |
by (simp add: elementary_Int Inter_insert) |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
471 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
472 |
qed auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
473 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
474 |
lemma division_disjoint_union: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
475 |
assumes "p1 division_of s1" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
476 |
and "p2 division_of s2" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
477 |
and "interior s1 \<inter> interior s2 = {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
478 |
shows "(p1 \<union> p2) division_of (s1 \<union> s2)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
479 |
proof (rule division_ofI) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
480 |
note d1 = division_ofD[OF assms(1)] |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
481 |
note d2 = division_ofD[OF assms(2)] |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
482 |
show "finite (p1 \<union> p2)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
483 |
using d1(1) d2(1) by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
484 |
show "\<Union>(p1 \<union> p2) = s1 \<union> s2" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
485 |
using d1(6) d2(6) by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
486 |
{ |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
487 |
fix k1 k2 |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
488 |
assume as: "k1 \<in> p1 \<union> p2" "k2 \<in> p1 \<union> p2" "k1 \<noteq> k2" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
489 |
moreover |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
490 |
let ?g="interior k1 \<inter> interior k2 = {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
491 |
{ |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
492 |
assume as: "k1\<in>p1" "k2\<in>p2" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
493 |
have ?g |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
494 |
using interior_mono[OF d1(2)[OF as(1)]] interior_mono[OF d2(2)[OF as(2)]] |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
495 |
using assms(3) by blast |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
496 |
} |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
497 |
moreover |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
498 |
{ |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
499 |
assume as: "k1\<in>p2" "k2\<in>p1" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
500 |
have ?g |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
501 |
using interior_mono[OF d1(2)[OF as(2)]] interior_mono[OF d2(2)[OF as(1)]] |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
502 |
using assms(3) by blast |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
503 |
} |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
504 |
ultimately show ?g |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
505 |
using d1(5)[OF _ _ as(3)] and d2(5)[OF _ _ as(3)] by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
506 |
} |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
507 |
fix k |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
508 |
assume k: "k \<in> p1 \<union> p2" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
509 |
show "k \<subseteq> s1 \<union> s2" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
510 |
using k d1(2) d2(2) by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
511 |
show "k \<noteq> {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
512 |
using k d1(3) d2(3) by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
513 |
show "\<exists>a b. k = cbox a b" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
514 |
using k d1(4) d2(4) by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
515 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
516 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
517 |
lemma partial_division_extend_1: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
518 |
fixes a b c d :: "'a::euclidean_space" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
519 |
assumes incl: "cbox c d \<subseteq> cbox a b" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
520 |
and nonempty: "cbox c d \<noteq> {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
521 |
obtains p where "p division_of (cbox a b)" "cbox c d \<in> p" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
522 |
proof |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
523 |
let ?B = "\<lambda>f::'a\<Rightarrow>'a \<times> 'a. |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
524 |
cbox (\<Sum>i\<in>Basis. (fst (f i) \<bullet> i) *\<^sub>R i) (\<Sum>i\<in>Basis. (snd (f i) \<bullet> i) *\<^sub>R i)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
525 |
define p where "p = ?B ` (Basis \<rightarrow>\<^sub>E {(a, c), (c, d), (d, b)})" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
526 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
527 |
show "cbox c d \<in> p" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
528 |
unfolding p_def |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
529 |
by (auto simp add: box_eq_empty cbox_def intro!: image_eqI[where x="\<lambda>(i::'a)\<in>Basis. (c, d)"]) |
66318 | 530 |
have ord: "a \<bullet> i \<le> c \<bullet> i" "c \<bullet> i \<le> d \<bullet> i" "d \<bullet> i \<le> b \<bullet> i" if "i \<in> Basis" for i |
531 |
using incl nonempty that |
|
532 |
unfolding box_eq_empty subset_box by (auto simp: not_le) |
|
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
533 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
534 |
show "p division_of (cbox a b)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
535 |
proof (rule division_ofI) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
536 |
show "finite p" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
537 |
unfolding p_def by (auto intro!: finite_PiE) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
538 |
{ |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
539 |
fix k |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
540 |
assume "k \<in> p" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
541 |
then obtain f where f: "f \<in> Basis \<rightarrow>\<^sub>E {(a, c), (c, d), (d, b)}" and k: "k = ?B f" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
542 |
by (auto simp: p_def) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
543 |
then show "\<exists>a b. k = cbox a b" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
544 |
by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
545 |
have "k \<subseteq> cbox a b \<and> k \<noteq> {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
546 |
proof (simp add: k box_eq_empty subset_box not_less, safe) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
547 |
fix i :: 'a |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
548 |
assume i: "i \<in> Basis" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
549 |
with f have "f i = (a, c) \<or> f i = (c, d) \<or> f i = (d, b)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
550 |
by (auto simp: PiE_iff) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
551 |
with i ord[of i] |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
552 |
show "a \<bullet> i \<le> fst (f i) \<bullet> i" "snd (f i) \<bullet> i \<le> b \<bullet> i" "fst (f i) \<bullet> i \<le> snd (f i) \<bullet> i" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
553 |
by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
554 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
555 |
then show "k \<noteq> {}" "k \<subseteq> cbox a b" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
556 |
by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
557 |
{ |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
558 |
fix l |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
559 |
assume "l \<in> p" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
560 |
then obtain g where g: "g \<in> Basis \<rightarrow>\<^sub>E {(a, c), (c, d), (d, b)}" and l: "l = ?B g" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
561 |
by (auto simp: p_def) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
562 |
assume "l \<noteq> k" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
563 |
have "\<exists>i\<in>Basis. f i \<noteq> g i" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
564 |
proof (rule ccontr) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
565 |
assume "\<not> ?thesis" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
566 |
with f g have "f = g" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
567 |
by (auto simp: PiE_iff extensional_def fun_eq_iff) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
568 |
with \<open>l \<noteq> k\<close> show False |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
569 |
by (simp add: l k) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
570 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
571 |
then obtain i where *: "i \<in> Basis" "f i \<noteq> g i" .. |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
572 |
then have "f i = (a, c) \<or> f i = (c, d) \<or> f i = (d, b)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
573 |
"g i = (a, c) \<or> g i = (c, d) \<or> g i = (d, b)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
574 |
using f g by (auto simp: PiE_iff) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
575 |
with * ord[of i] show "interior l \<inter> interior k = {}" |
66492
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
576 |
by (auto simp add: l k disjoint_interval intro!: bexI[of _ i]) |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
577 |
} |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
578 |
note \<open>k \<subseteq> cbox a b\<close> |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
579 |
} |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
580 |
moreover |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
581 |
{ |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
582 |
fix x assume x: "x \<in> cbox a b" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
583 |
have "\<forall>i\<in>Basis. \<exists>l. x \<bullet> i \<in> {fst l \<bullet> i .. snd l \<bullet> i} \<and> l \<in> {(a, c), (c, d), (d, b)}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
584 |
proof |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
585 |
fix i :: 'a |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
586 |
assume "i \<in> Basis" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
587 |
with x ord[of i] |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
588 |
have "(a \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> c \<bullet> i) \<or> (c \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> d \<bullet> i) \<or> |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
589 |
(d \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> b \<bullet> i)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
590 |
by (auto simp: cbox_def) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
591 |
then show "\<exists>l. x \<bullet> i \<in> {fst l \<bullet> i .. snd l \<bullet> i} \<and> l \<in> {(a, c), (c, d), (d, b)}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
592 |
by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
593 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
594 |
then obtain f where |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
595 |
f: "\<forall>i\<in>Basis. x \<bullet> i \<in> {fst (f i) \<bullet> i..snd (f i) \<bullet> i} \<and> f i \<in> {(a, c), (c, d), (d, b)}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
596 |
unfolding bchoice_iff .. |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
597 |
moreover from f have "restrict f Basis \<in> Basis \<rightarrow>\<^sub>E {(a, c), (c, d), (d, b)}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
598 |
by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
599 |
moreover from f have "x \<in> ?B (restrict f Basis)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
600 |
by (auto simp: mem_box) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
601 |
ultimately have "\<exists>k\<in>p. x \<in> k" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
602 |
unfolding p_def by blast |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
603 |
} |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
604 |
ultimately show "\<Union>p = cbox a b" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
605 |
by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
606 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
607 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
608 |
|
66154
bc5e6461f759
Tidying up integration theory and some new theorems
paulson <lp15@cam.ac.uk>
parents:
66113
diff
changeset
|
609 |
proposition partial_division_extend_interval: |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
610 |
assumes "p division_of (\<Union>p)" "(\<Union>p) \<subseteq> cbox a b" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
611 |
obtains q where "p \<subseteq> q" "q division_of cbox a (b::'a::euclidean_space)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
612 |
proof (cases "p = {}") |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
613 |
case True |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
614 |
obtain q where "q division_of (cbox a b)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
615 |
by (rule elementary_interval) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
616 |
then show ?thesis |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
617 |
using True that by blast |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
618 |
next |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
619 |
case False |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
620 |
note p = division_ofD[OF assms(1)] |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
621 |
have div_cbox: "\<forall>k\<in>p. \<exists>q. q division_of cbox a b \<and> k \<in> q" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
622 |
proof |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
623 |
fix k |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
624 |
assume kp: "k \<in> p" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
625 |
obtain c d where k: "k = cbox c d" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
626 |
using p(4)[OF kp] by blast |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
627 |
have *: "cbox c d \<subseteq> cbox a b" "cbox c d \<noteq> {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
628 |
using p(2,3)[OF kp, unfolded k] using assms(2) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
629 |
by (blast intro: order.trans)+ |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
630 |
obtain q where "q division_of cbox a b" "cbox c d \<in> q" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
631 |
by (rule partial_division_extend_1[OF *]) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
632 |
then show "\<exists>q. q division_of cbox a b \<and> k \<in> q" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
633 |
unfolding k by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
634 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
635 |
obtain q where q: "\<And>x. x \<in> p \<Longrightarrow> q x division_of cbox a b" "\<And>x. x \<in> p \<Longrightarrow> x \<in> q x" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
636 |
using bchoice[OF div_cbox] by blast |
66299
1b4aa3e3e4e6
partial cleanup of the horrible Tagged_Division
paulson <lp15@cam.ac.uk>
parents:
66296
diff
changeset
|
637 |
have "q x division_of \<Union>q x" if x: "x \<in> p" for x |
1b4aa3e3e4e6
partial cleanup of the horrible Tagged_Division
paulson <lp15@cam.ac.uk>
parents:
66296
diff
changeset
|
638 |
apply (rule division_ofI) |
1b4aa3e3e4e6
partial cleanup of the horrible Tagged_Division
paulson <lp15@cam.ac.uk>
parents:
66296
diff
changeset
|
639 |
using division_ofD[OF q(1)[OF x]] by auto |
1b4aa3e3e4e6
partial cleanup of the horrible Tagged_Division
paulson <lp15@cam.ac.uk>
parents:
66296
diff
changeset
|
640 |
then have di: "\<And>x. x \<in> p \<Longrightarrow> \<exists>d. d division_of \<Union>(q x - {x})" |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
641 |
by (meson Diff_subset division_of_subset) |
66299
1b4aa3e3e4e6
partial cleanup of the horrible Tagged_Division
paulson <lp15@cam.ac.uk>
parents:
66296
diff
changeset
|
642 |
have "\<exists>d. d division_of \<Inter>((\<lambda>i. \<Union>(q i - {i})) ` p)" |
66317 | 643 |
apply (rule elementary_Inter [OF finite_imageI[OF p(1)]]) |
644 |
apply (auto dest: di simp: False elementary_Inter [OF finite_imageI[OF p(1)]]) |
|
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
645 |
done |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
646 |
then obtain d where d: "d division_of \<Inter>((\<lambda>i. \<Union>(q i - {i})) ` p)" .. |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
647 |
have "d \<union> p division_of cbox a b" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
648 |
proof - |
66299
1b4aa3e3e4e6
partial cleanup of the horrible Tagged_Division
paulson <lp15@cam.ac.uk>
parents:
66296
diff
changeset
|
649 |
have te: "\<And>S f T. S \<noteq> {} \<Longrightarrow> \<forall>i\<in>S. f i \<union> i = T \<Longrightarrow> T = \<Inter>(f ` S) \<union> \<Union>S" by auto |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
650 |
have cbox_eq: "cbox a b = \<Inter>((\<lambda>i. \<Union>(q i - {i})) ` p) \<union> \<Union>p" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
651 |
proof (rule te[OF False], clarify) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
652 |
fix i |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
653 |
assume i: "i \<in> p" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
654 |
show "\<Union>(q i - {i}) \<union> i = cbox a b" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
655 |
using division_ofD(6)[OF q(1)[OF i]] using q(2)[OF i] by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
656 |
qed |
66299
1b4aa3e3e4e6
partial cleanup of the horrible Tagged_Division
paulson <lp15@cam.ac.uk>
parents:
66296
diff
changeset
|
657 |
{ fix K |
1b4aa3e3e4e6
partial cleanup of the horrible Tagged_Division
paulson <lp15@cam.ac.uk>
parents:
66296
diff
changeset
|
658 |
assume K: "K \<in> p" |
1b4aa3e3e4e6
partial cleanup of the horrible Tagged_Division
paulson <lp15@cam.ac.uk>
parents:
66296
diff
changeset
|
659 |
note qk=division_ofD[OF q(1)[OF K]] |
1b4aa3e3e4e6
partial cleanup of the horrible Tagged_Division
paulson <lp15@cam.ac.uk>
parents:
66296
diff
changeset
|
660 |
have *: "\<And>u T S. T \<inter> S = {} \<Longrightarrow> u \<subseteq> S \<Longrightarrow> u \<inter> T = {}" |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
661 |
by auto |
66299
1b4aa3e3e4e6
partial cleanup of the horrible Tagged_Division
paulson <lp15@cam.ac.uk>
parents:
66296
diff
changeset
|
662 |
have "interior (\<Inter>i\<in>p. \<Union>(q i - {i})) \<inter> interior K = {}" |
1b4aa3e3e4e6
partial cleanup of the horrible Tagged_Division
paulson <lp15@cam.ac.uk>
parents:
66296
diff
changeset
|
663 |
proof (rule *[OF Int_interior_Union_intervals]) |
1b4aa3e3e4e6
partial cleanup of the horrible Tagged_Division
paulson <lp15@cam.ac.uk>
parents:
66296
diff
changeset
|
664 |
show "\<And>T. T\<in>q K - {K} \<Longrightarrow> interior K \<inter> interior T = {}" |
1b4aa3e3e4e6
partial cleanup of the horrible Tagged_Division
paulson <lp15@cam.ac.uk>
parents:
66296
diff
changeset
|
665 |
using qk(5) using q(2)[OF K] by auto |
1b4aa3e3e4e6
partial cleanup of the horrible Tagged_Division
paulson <lp15@cam.ac.uk>
parents:
66296
diff
changeset
|
666 |
show "interior (\<Inter>i\<in>p. \<Union>(q i - {i})) \<subseteq> interior (\<Union>(q K - {K}))" |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
667 |
apply (rule interior_mono)+ |
66299
1b4aa3e3e4e6
partial cleanup of the horrible Tagged_Division
paulson <lp15@cam.ac.uk>
parents:
66296
diff
changeset
|
668 |
using K by auto |
1b4aa3e3e4e6
partial cleanup of the horrible Tagged_Division
paulson <lp15@cam.ac.uk>
parents:
66296
diff
changeset
|
669 |
qed (use qk in auto)} note [simp] = this |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
670 |
show "d \<union> p division_of (cbox a b)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
671 |
unfolding cbox_eq |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
672 |
apply (rule division_disjoint_union[OF d assms(1)]) |
66299
1b4aa3e3e4e6
partial cleanup of the horrible Tagged_Division
paulson <lp15@cam.ac.uk>
parents:
66296
diff
changeset
|
673 |
apply (rule Int_interior_Union_intervals) |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
674 |
apply (rule p open_interior ballI)+ |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
675 |
apply simp_all |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
676 |
done |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
677 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
678 |
then show ?thesis |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
679 |
by (meson Un_upper2 that) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
680 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
681 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
682 |
lemma elementary_bounded[dest]: |
66299
1b4aa3e3e4e6
partial cleanup of the horrible Tagged_Division
paulson <lp15@cam.ac.uk>
parents:
66296
diff
changeset
|
683 |
fixes S :: "'a::euclidean_space set" |
1b4aa3e3e4e6
partial cleanup of the horrible Tagged_Division
paulson <lp15@cam.ac.uk>
parents:
66296
diff
changeset
|
684 |
shows "p division_of S \<Longrightarrow> bounded S" |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
685 |
unfolding division_of_def by (metis bounded_Union bounded_cbox) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
686 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
687 |
lemma elementary_subset_cbox: |
66299
1b4aa3e3e4e6
partial cleanup of the horrible Tagged_Division
paulson <lp15@cam.ac.uk>
parents:
66296
diff
changeset
|
688 |
"p division_of S \<Longrightarrow> \<exists>a b. S \<subseteq> cbox a (b::'a::euclidean_space)" |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
689 |
by (meson elementary_bounded bounded_subset_cbox) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
690 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
691 |
lemma division_union_intervals_exists: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
692 |
fixes a b :: "'a::euclidean_space" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
693 |
assumes "cbox a b \<noteq> {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
694 |
obtains p where "(insert (cbox a b) p) division_of (cbox a b \<union> cbox c d)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
695 |
proof (cases "cbox c d = {}") |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
696 |
case True |
66318 | 697 |
with assms that show ?thesis by force |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
698 |
next |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
699 |
case False |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
700 |
show ?thesis |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
701 |
proof (cases "cbox a b \<inter> cbox c d = {}") |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
702 |
case True |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
703 |
then show ?thesis |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
704 |
by (metis that False assms division_disjoint_union division_of_self insert_is_Un interior_Int interior_empty) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
705 |
next |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
706 |
case False |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
707 |
obtain u v where uv: "cbox a b \<inter> cbox c d = cbox u v" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
708 |
unfolding Int_interval by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
709 |
have uv_sub: "cbox u v \<subseteq> cbox c d" using uv by auto |
66299
1b4aa3e3e4e6
partial cleanup of the horrible Tagged_Division
paulson <lp15@cam.ac.uk>
parents:
66296
diff
changeset
|
710 |
obtain p where pd: "p division_of cbox c d" and "cbox u v \<in> p" |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
711 |
by (rule partial_division_extend_1[OF uv_sub False[unfolded uv]]) |
66299
1b4aa3e3e4e6
partial cleanup of the horrible Tagged_Division
paulson <lp15@cam.ac.uk>
parents:
66296
diff
changeset
|
712 |
note p = this division_ofD[OF pd] |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
713 |
have "interior (cbox a b \<inter> \<Union>(p - {cbox u v})) = interior(cbox u v \<inter> \<Union>(p - {cbox u v}))" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
714 |
apply (rule arg_cong[of _ _ interior]) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
715 |
using p(8) uv by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
716 |
also have "\<dots> = {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
717 |
unfolding interior_Int |
66299
1b4aa3e3e4e6
partial cleanup of the horrible Tagged_Division
paulson <lp15@cam.ac.uk>
parents:
66296
diff
changeset
|
718 |
apply (rule Int_interior_Union_intervals) |
1b4aa3e3e4e6
partial cleanup of the horrible Tagged_Division
paulson <lp15@cam.ac.uk>
parents:
66296
diff
changeset
|
719 |
using p(6) p(7)[OF p(2)] \<open>finite p\<close> |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
720 |
apply auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
721 |
done |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
722 |
finally have [simp]: "interior (cbox a b) \<inter> interior (\<Union>(p - {cbox u v})) = {}" by simp |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
723 |
have cbe: "cbox a b \<union> cbox c d = cbox a b \<union> \<Union>(p - {cbox u v})" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
724 |
using p(8) unfolding uv[symmetric] by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
725 |
have "insert (cbox a b) (p - {cbox u v}) division_of cbox a b \<union> \<Union>(p - {cbox u v})" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
726 |
proof - |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
727 |
have "{cbox a b} division_of cbox a b" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
728 |
by (simp add: assms division_of_self) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
729 |
then show "insert (cbox a b) (p - {cbox u v}) division_of cbox a b \<union> \<Union>(p - {cbox u v})" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
730 |
by (metis (no_types) Diff_subset \<open>interior (cbox a b) \<inter> interior (\<Union>(p - {cbox u v})) = {}\<close> division_disjoint_union division_of_subset insert_is_Un p(1) p(8)) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
731 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
732 |
with that[of "p - {cbox u v}"] show ?thesis by (simp add: cbe) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
733 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
734 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
735 |
|
66299
1b4aa3e3e4e6
partial cleanup of the horrible Tagged_Division
paulson <lp15@cam.ac.uk>
parents:
66296
diff
changeset
|
736 |
lemma division_of_Union: |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
737 |
assumes "finite f" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
738 |
and "\<And>p. p \<in> f \<Longrightarrow> p division_of (\<Union>p)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
739 |
and "\<And>k1 k2. k1 \<in> \<Union>f \<Longrightarrow> k2 \<in> \<Union>f \<Longrightarrow> k1 \<noteq> k2 \<Longrightarrow> interior k1 \<inter> interior k2 = {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
740 |
shows "\<Union>f division_of \<Union>\<Union>f" |
66299
1b4aa3e3e4e6
partial cleanup of the horrible Tagged_Division
paulson <lp15@cam.ac.uk>
parents:
66296
diff
changeset
|
741 |
using assms by (auto intro!: division_ofI) |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
742 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
743 |
lemma elementary_union_interval: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
744 |
fixes a b :: "'a::euclidean_space" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
745 |
assumes "p division_of \<Union>p" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
746 |
obtains q where "q division_of (cbox a b \<union> \<Union>p)" |
66299
1b4aa3e3e4e6
partial cleanup of the horrible Tagged_Division
paulson <lp15@cam.ac.uk>
parents:
66296
diff
changeset
|
747 |
proof (cases "p = {} \<or> cbox a b = {}") |
1b4aa3e3e4e6
partial cleanup of the horrible Tagged_Division
paulson <lp15@cam.ac.uk>
parents:
66296
diff
changeset
|
748 |
case True |
1b4aa3e3e4e6
partial cleanup of the horrible Tagged_Division
paulson <lp15@cam.ac.uk>
parents:
66296
diff
changeset
|
749 |
obtain p where "p division_of (cbox a b)" |
1b4aa3e3e4e6
partial cleanup of the horrible Tagged_Division
paulson <lp15@cam.ac.uk>
parents:
66296
diff
changeset
|
750 |
by (rule elementary_interval) |
1b4aa3e3e4e6
partial cleanup of the horrible Tagged_Division
paulson <lp15@cam.ac.uk>
parents:
66296
diff
changeset
|
751 |
then show thesis |
1b4aa3e3e4e6
partial cleanup of the horrible Tagged_Division
paulson <lp15@cam.ac.uk>
parents:
66296
diff
changeset
|
752 |
using True assms that by auto |
1b4aa3e3e4e6
partial cleanup of the horrible Tagged_Division
paulson <lp15@cam.ac.uk>
parents:
66296
diff
changeset
|
753 |
next |
1b4aa3e3e4e6
partial cleanup of the horrible Tagged_Division
paulson <lp15@cam.ac.uk>
parents:
66296
diff
changeset
|
754 |
case False |
1b4aa3e3e4e6
partial cleanup of the horrible Tagged_Division
paulson <lp15@cam.ac.uk>
parents:
66296
diff
changeset
|
755 |
then have "p \<noteq> {}" "cbox a b \<noteq> {}" |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
756 |
by auto |
66299
1b4aa3e3e4e6
partial cleanup of the horrible Tagged_Division
paulson <lp15@cam.ac.uk>
parents:
66296
diff
changeset
|
757 |
note pdiv = division_ofD[OF assms] |
1b4aa3e3e4e6
partial cleanup of the horrible Tagged_Division
paulson <lp15@cam.ac.uk>
parents:
66296
diff
changeset
|
758 |
show ?thesis |
1b4aa3e3e4e6
partial cleanup of the horrible Tagged_Division
paulson <lp15@cam.ac.uk>
parents:
66296
diff
changeset
|
759 |
proof (cases "interior (cbox a b) = {}") |
1b4aa3e3e4e6
partial cleanup of the horrible Tagged_Division
paulson <lp15@cam.ac.uk>
parents:
66296
diff
changeset
|
760 |
case True |
1b4aa3e3e4e6
partial cleanup of the horrible Tagged_Division
paulson <lp15@cam.ac.uk>
parents:
66296
diff
changeset
|
761 |
show ?thesis |
1b4aa3e3e4e6
partial cleanup of the horrible Tagged_Division
paulson <lp15@cam.ac.uk>
parents:
66296
diff
changeset
|
762 |
apply (rule that [of "insert (cbox a b) p", OF division_ofI]) |
1b4aa3e3e4e6
partial cleanup of the horrible Tagged_Division
paulson <lp15@cam.ac.uk>
parents:
66296
diff
changeset
|
763 |
using pdiv(1-4) True False apply auto |
1b4aa3e3e4e6
partial cleanup of the horrible Tagged_Division
paulson <lp15@cam.ac.uk>
parents:
66296
diff
changeset
|
764 |
apply (metis IntI equals0D pdiv(5)) |
1b4aa3e3e4e6
partial cleanup of the horrible Tagged_Division
paulson <lp15@cam.ac.uk>
parents:
66296
diff
changeset
|
765 |
by (metis disjoint_iff_not_equal pdiv(5)) |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
766 |
next |
66299
1b4aa3e3e4e6
partial cleanup of the horrible Tagged_Division
paulson <lp15@cam.ac.uk>
parents:
66296
diff
changeset
|
767 |
case False |
66300 | 768 |
have "\<forall>K\<in>p. \<exists>q. (insert (cbox a b) q) division_of (cbox a b \<union> K)" |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
769 |
proof |
66300 | 770 |
fix K |
771 |
assume kp: "K \<in> p" |
|
772 |
from pdiv(4)[OF kp] obtain c d where "K = cbox c d" by blast |
|
773 |
then show "\<exists>q. (insert (cbox a b) q) division_of (cbox a b \<union> K)" |
|
66299
1b4aa3e3e4e6
partial cleanup of the horrible Tagged_Division
paulson <lp15@cam.ac.uk>
parents:
66296
diff
changeset
|
774 |
by (meson \<open>cbox a b \<noteq> {}\<close> division_union_intervals_exists) |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
775 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
776 |
from bchoice[OF this] obtain q where "\<forall>x\<in>p. insert (cbox a b) (q x) division_of (cbox a b) \<union> x" .. |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
777 |
note q = division_ofD[OF this[rule_format]] |
66300 | 778 |
let ?D = "\<Union>{insert (cbox a b) (q K) | K. K \<in> p}" |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
779 |
show thesis |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
780 |
proof (rule that[OF division_ofI]) |
66300 | 781 |
have *: "{insert (cbox a b) (q K) |K. K \<in> p} = (\<lambda>K. insert (cbox a b) (q K)) ` p" |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
782 |
by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
783 |
show "finite ?D" |
66299
1b4aa3e3e4e6
partial cleanup of the horrible Tagged_Division
paulson <lp15@cam.ac.uk>
parents:
66296
diff
changeset
|
784 |
using "*" pdiv(1) q(1) by auto |
66318 | 785 |
have "\<Union>?D = (\<Union>x\<in>p. \<Union>insert (cbox a b) (q x))" |
66299
1b4aa3e3e4e6
partial cleanup of the horrible Tagged_Division
paulson <lp15@cam.ac.uk>
parents:
66296
diff
changeset
|
786 |
by auto |
66318 | 787 |
also have "... = \<Union>{cbox a b \<union> t |t. t \<in> p}" |
788 |
using q(6) by auto |
|
789 |
also have "... = cbox a b \<union> \<Union>p" |
|
790 |
using \<open>p \<noteq> {}\<close> by auto |
|
791 |
finally show "\<Union>?D = cbox a b \<union> \<Union>p" . |
|
66300 | 792 |
show "K \<subseteq> cbox a b \<union> \<Union>p" "K \<noteq> {}" if "K \<in> ?D" for K |
66299
1b4aa3e3e4e6
partial cleanup of the horrible Tagged_Division
paulson <lp15@cam.ac.uk>
parents:
66296
diff
changeset
|
793 |
using q that by blast+ |
66300 | 794 |
show "\<exists>a b. K = cbox a b" if "K \<in> ?D" for K |
66299
1b4aa3e3e4e6
partial cleanup of the horrible Tagged_Division
paulson <lp15@cam.ac.uk>
parents:
66296
diff
changeset
|
795 |
using q(4) that by auto |
1b4aa3e3e4e6
partial cleanup of the horrible Tagged_Division
paulson <lp15@cam.ac.uk>
parents:
66296
diff
changeset
|
796 |
next |
66300 | 797 |
fix K' K |
798 |
assume K: "K \<in> ?D" and K': "K' \<in> ?D" "K \<noteq> K'" |
|
799 |
obtain x where x: "K \<in> insert (cbox a b) (q x)" "x\<in>p" |
|
800 |
using K by auto |
|
801 |
obtain x' where x': "K'\<in>insert (cbox a b) (q x')" "x'\<in>p" |
|
802 |
using K' by auto |
|
803 |
show "interior K \<inter> interior K' = {}" |
|
804 |
proof (cases "x = x' \<or> K = cbox a b \<or> K' = cbox a b") |
|
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
805 |
case True |
66299
1b4aa3e3e4e6
partial cleanup of the horrible Tagged_Division
paulson <lp15@cam.ac.uk>
parents:
66296
diff
changeset
|
806 |
then show ?thesis |
66300 | 807 |
using True K' q(5) x' x by auto |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
808 |
next |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
809 |
case False |
66300 | 810 |
then have as': "K \<noteq> cbox a b" "K' \<noteq> cbox a b" |
66299
1b4aa3e3e4e6
partial cleanup of the horrible Tagged_Division
paulson <lp15@cam.ac.uk>
parents:
66296
diff
changeset
|
811 |
by auto |
66300 | 812 |
obtain c d where K: "K = cbox c d" |
813 |
using q(4) x by blast |
|
814 |
have "interior K \<inter> interior (cbox a b) = {}" |
|
815 |
using as' K'(2) q(5) x by blast |
|
816 |
then have "interior K \<subseteq> interior x" |
|
817 |
by (metis \<open>interior (cbox a b) \<noteq> {}\<close> K q(2) x interior_subset_union_intervals) |
|
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
818 |
moreover |
66300 | 819 |
obtain c d where c_d: "K' = cbox c d" |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
820 |
using q(4)[OF x'(2,1)] by blast |
66300 | 821 |
have "interior K' \<inter> interior (cbox a b) = {}" |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
822 |
using as'(2) q(5) x' by blast |
66300 | 823 |
then have "interior K' \<subseteq> interior x'" |
66299
1b4aa3e3e4e6
partial cleanup of the horrible Tagged_Division
paulson <lp15@cam.ac.uk>
parents:
66296
diff
changeset
|
824 |
by (metis \<open>interior (cbox a b) \<noteq> {}\<close> c_d interior_subset_union_intervals q(2) x'(1) x'(2)) |
1b4aa3e3e4e6
partial cleanup of the horrible Tagged_Division
paulson <lp15@cam.ac.uk>
parents:
66296
diff
changeset
|
825 |
moreover have "interior x \<inter> interior x' = {}" |
1b4aa3e3e4e6
partial cleanup of the horrible Tagged_Division
paulson <lp15@cam.ac.uk>
parents:
66296
diff
changeset
|
826 |
by (meson False assms division_ofD(5) x'(2) x(2)) |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
827 |
ultimately show ?thesis |
66300 | 828 |
using \<open>interior K \<subseteq> interior x\<close> \<open>interior K' \<subseteq> interior x'\<close> by auto |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
829 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
830 |
qed |
66299
1b4aa3e3e4e6
partial cleanup of the horrible Tagged_Division
paulson <lp15@cam.ac.uk>
parents:
66296
diff
changeset
|
831 |
qed |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
832 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
833 |
|
66299
1b4aa3e3e4e6
partial cleanup of the horrible Tagged_Division
paulson <lp15@cam.ac.uk>
parents:
66296
diff
changeset
|
834 |
|
1b4aa3e3e4e6
partial cleanup of the horrible Tagged_Division
paulson <lp15@cam.ac.uk>
parents:
66296
diff
changeset
|
835 |
|
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
836 |
lemma elementary_unions_intervals: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
837 |
assumes fin: "finite f" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
838 |
and "\<And>s. s \<in> f \<Longrightarrow> \<exists>a b. s = cbox a (b::'a::euclidean_space)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
839 |
obtains p where "p division_of (\<Union>f)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
840 |
proof - |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
841 |
have "\<exists>p. p division_of (\<Union>f)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
842 |
proof (induct_tac f rule:finite_subset_induct) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
843 |
show "\<exists>p. p division_of \<Union>{}" using elementary_empty by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
844 |
next |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
845 |
fix x F |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
846 |
assume as: "finite F" "x \<notin> F" "\<exists>p. p division_of \<Union>F" "x\<in>f" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
847 |
from this(3) obtain p where p: "p division_of \<Union>F" .. |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
848 |
from assms(2)[OF as(4)] obtain a b where x: "x = cbox a b" by blast |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
849 |
have *: "\<Union>F = \<Union>p" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
850 |
using division_ofD[OF p] by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
851 |
show "\<exists>p. p division_of \<Union>insert x F" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
852 |
using elementary_union_interval[OF p[unfolded *], of a b] |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
853 |
unfolding Union_insert x * by metis |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
854 |
qed (insert assms, auto) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
855 |
then show ?thesis |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
856 |
using that by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
857 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
858 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
859 |
lemma elementary_union: |
66300 | 860 |
fixes S T :: "'a::euclidean_space set" |
861 |
assumes "ps division_of S" "pt division_of T" |
|
862 |
obtains p where "p division_of (S \<union> T)" |
|
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
863 |
proof - |
66300 | 864 |
have *: "S \<union> T = \<Union>ps \<union> \<Union>pt" |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
865 |
using assms unfolding division_of_def by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
866 |
show ?thesis |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
867 |
apply (rule elementary_unions_intervals[of "ps \<union> pt"]) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
868 |
using assms apply auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
869 |
by (simp add: * that) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
870 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
871 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
872 |
lemma partial_division_extend: |
66300 | 873 |
fixes T :: "'a::euclidean_space set" |
874 |
assumes "p division_of S" |
|
875 |
and "q division_of T" |
|
876 |
and "S \<subseteq> T" |
|
877 |
obtains r where "p \<subseteq> r" and "r division_of T" |
|
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
878 |
proof - |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
879 |
note divp = division_ofD[OF assms(1)] and divq = division_ofD[OF assms(2)] |
66300 | 880 |
obtain a b where ab: "T \<subseteq> cbox a b" |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
881 |
using elementary_subset_cbox[OF assms(2)] by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
882 |
obtain r1 where "p \<subseteq> r1" "r1 division_of (cbox a b)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
883 |
using assms |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
884 |
by (metis ab dual_order.trans partial_division_extend_interval divp(6)) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
885 |
note r1 = this division_ofD[OF this(2)] |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
886 |
obtain p' where "p' division_of \<Union>(r1 - p)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
887 |
apply (rule elementary_unions_intervals[of "r1 - p"]) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
888 |
using r1(3,6) |
66300 | 889 |
apply auto |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
890 |
done |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
891 |
then obtain r2 where r2: "r2 division_of (\<Union>(r1 - p)) \<inter> (\<Union>q)" |
66317 | 892 |
by (metis assms(2) divq(6) elementary_Int) |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
893 |
{ |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
894 |
fix x |
66300 | 895 |
assume x: "x \<in> T" "x \<notin> S" |
896 |
then obtain R where r: "R \<in> r1" "x \<in> R" |
|
897 |
unfolding r1 using ab |
|
898 |
by (meson division_contains r1(2) subsetCE) |
|
899 |
moreover have "R \<notin> p" |
|
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
900 |
proof |
66300 | 901 |
assume "R \<in> p" |
902 |
then have "x \<in> S" using divp(2) r by auto |
|
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
903 |
then show False using x by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
904 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
905 |
ultimately have "x\<in>\<Union>(r1 - p)" by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
906 |
} |
66300 | 907 |
then have Teq: "T = \<Union>p \<union> (\<Union>(r1 - p) \<inter> \<Union>q)" |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
908 |
unfolding divp divq using assms(3) by auto |
66300 | 909 |
have "interior S \<inter> interior (\<Union>(r1-p)) = {}" |
910 |
proof (rule Int_interior_Union_intervals) |
|
911 |
have *: "\<And>S. (\<And>x. x \<in> S \<Longrightarrow> False) \<Longrightarrow> S = {}" |
|
912 |
by auto |
|
913 |
show "interior S \<inter> interior m = {}" if "m \<in> r1 - p" for m |
|
914 |
proof - |
|
915 |
have "interior m \<inter> interior (\<Union>p) = {}" |
|
916 |
proof (rule Int_interior_Union_intervals) |
|
917 |
show "\<And>T. T\<in>p \<Longrightarrow> interior m \<inter> interior T = {}" |
|
918 |
by (metis DiffD1 DiffD2 that r1(1) r1(7) set_rev_mp) |
|
919 |
qed (use divp in auto) |
|
920 |
then show "interior S \<inter> interior m = {}" |
|
921 |
unfolding divp by auto |
|
922 |
qed |
|
923 |
qed (use r1 in auto) |
|
924 |
then have "interior S \<inter> interior (\<Union>(r1-p) \<inter> (\<Union>q)) = {}" |
|
925 |
using interior_subset by auto |
|
926 |
then have div: "p \<union> r2 division_of \<Union>p \<union> \<Union>(r1 - p) \<inter> \<Union>q" |
|
927 |
by (simp add: assms(1) division_disjoint_union divp(6) r2) |
|
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
928 |
show ?thesis |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
929 |
apply (rule that[of "p \<union> r2"]) |
66300 | 930 |
apply (auto simp: div Teq) |
931 |
done |
|
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
932 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
933 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
934 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
935 |
lemma division_split: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
936 |
fixes a :: "'a::euclidean_space" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
937 |
assumes "p division_of (cbox a b)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
938 |
and k: "k\<in>Basis" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
939 |
shows "{l \<inter> {x. x\<bullet>k \<le> c} | l. l \<in> p \<and> l \<inter> {x. x\<bullet>k \<le> c} \<noteq> {}} division_of(cbox a b \<inter> {x. x\<bullet>k \<le> c})" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
940 |
(is "?p1 division_of ?I1") |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
941 |
and "{l \<inter> {x. x\<bullet>k \<ge> c} | l. l \<in> p \<and> l \<inter> {x. x\<bullet>k \<ge> c} \<noteq> {}} division_of (cbox a b \<inter> {x. x\<bullet>k \<ge> c})" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
942 |
(is "?p2 division_of ?I2") |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
943 |
proof (rule_tac[!] division_ofI) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
944 |
note p = division_ofD[OF assms(1)] |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
945 |
show "finite ?p1" "finite ?p2" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
946 |
using p(1) by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
947 |
show "\<Union>?p1 = ?I1" "\<Union>?p2 = ?I2" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
948 |
unfolding p(6)[symmetric] by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
949 |
{ |
66300 | 950 |
fix K |
951 |
assume "K \<in> ?p1" |
|
952 |
then obtain l where l: "K = l \<inter> {x. x \<bullet> k \<le> c}" "l \<in> p" "l \<inter> {x. x \<bullet> k \<le> c} \<noteq> {}" |
|
953 |
by blast |
|
954 |
obtain u v where uv: "l = cbox u v" |
|
955 |
using assms(1) l(2) by blast |
|
956 |
show "K \<subseteq> ?I1" |
|
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
957 |
using l p(2) uv by force |
66300 | 958 |
show "K \<noteq> {}" |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
959 |
by (simp add: l) |
66300 | 960 |
show "\<exists>a b. K = cbox a b" |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
961 |
apply (simp add: l uv p(2-3)[OF l(2)]) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
962 |
apply (subst interval_split[OF k]) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
963 |
apply (auto intro: order.trans) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
964 |
done |
66300 | 965 |
fix K' |
966 |
assume "K' \<in> ?p1" |
|
967 |
then obtain l' where l': "K' = l' \<inter> {x. x \<bullet> k \<le> c}" "l' \<in> p" "l' \<inter> {x. x \<bullet> k \<le> c} \<noteq> {}" |
|
968 |
by blast |
|
969 |
assume "K \<noteq> K'" |
|
970 |
then show "interior K \<inter> interior K' = {}" |
|
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
971 |
unfolding l l' using p(5)[OF l(2) l'(2)] by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
972 |
} |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
973 |
{ |
66300 | 974 |
fix K |
975 |
assume "K \<in> ?p2" |
|
976 |
then obtain l where l: "K = l \<inter> {x. c \<le> x \<bullet> k}" "l \<in> p" "l \<inter> {x. c \<le> x \<bullet> k} \<noteq> {}" |
|
977 |
by blast |
|
978 |
obtain u v where uv: "l = cbox u v" |
|
979 |
using l(2) p(4) by blast |
|
980 |
show "K \<subseteq> ?I2" |
|
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
981 |
using l p(2) uv by force |
66300 | 982 |
show "K \<noteq> {}" |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
983 |
by (simp add: l) |
66300 | 984 |
show "\<exists>a b. K = cbox a b" |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
985 |
apply (simp add: l uv p(2-3)[OF l(2)]) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
986 |
apply (subst interval_split[OF k]) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
987 |
apply (auto intro: order.trans) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
988 |
done |
66300 | 989 |
fix K' |
990 |
assume "K' \<in> ?p2" |
|
991 |
then obtain l' where l': "K' = l' \<inter> {x. c \<le> x \<bullet> k}" "l' \<in> p" "l' \<inter> {x. c \<le> x \<bullet> k} \<noteq> {}" |
|
992 |
by blast |
|
993 |
assume "K \<noteq> K'" |
|
994 |
then show "interior K \<inter> interior K' = {}" |
|
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
995 |
unfolding l l' using p(5)[OF l(2) l'(2)] by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
996 |
} |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
997 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
998 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
999 |
subsection \<open>Tagged (partial) divisions.\<close> |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1000 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1001 |
definition tagged_partial_division_of (infixr "tagged'_partial'_division'_of" 40) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1002 |
where "s tagged_partial_division_of i \<longleftrightarrow> |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1003 |
finite s \<and> |
66113
571b698659c0
Repaired an inadvertent reordering of the premises of two theorems
paulson <lp15@cam.ac.uk>
parents:
66112
diff
changeset
|
1004 |
(\<forall>x K. (x, K) \<in> s \<longrightarrow> x \<in> K \<and> K \<subseteq> i \<and> (\<exists>a b. K = cbox a b)) \<and> |
571b698659c0
Repaired an inadvertent reordering of the premises of two theorems
paulson <lp15@cam.ac.uk>
parents:
66112
diff
changeset
|
1005 |
(\<forall>x1 K1 x2 K2. (x1, K1) \<in> s \<and> (x2, K2) \<in> s \<and> (x1, K1) \<noteq> (x2, K2) \<longrightarrow> |
571b698659c0
Repaired an inadvertent reordering of the premises of two theorems
paulson <lp15@cam.ac.uk>
parents:
66112
diff
changeset
|
1006 |
interior K1 \<inter> interior K2 = {})" |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1007 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1008 |
lemma tagged_partial_division_ofD[dest]: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1009 |
assumes "s tagged_partial_division_of i" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1010 |
shows "finite s" |
66113
571b698659c0
Repaired an inadvertent reordering of the premises of two theorems
paulson <lp15@cam.ac.uk>
parents:
66112
diff
changeset
|
1011 |
and "\<And>x K. (x,K) \<in> s \<Longrightarrow> x \<in> K" |
571b698659c0
Repaired an inadvertent reordering of the premises of two theorems
paulson <lp15@cam.ac.uk>
parents:
66112
diff
changeset
|
1012 |
and "\<And>x K. (x,K) \<in> s \<Longrightarrow> K \<subseteq> i" |
571b698659c0
Repaired an inadvertent reordering of the premises of two theorems
paulson <lp15@cam.ac.uk>
parents:
66112
diff
changeset
|
1013 |
and "\<And>x K. (x,K) \<in> s \<Longrightarrow> \<exists>a b. K = cbox a b" |
571b698659c0
Repaired an inadvertent reordering of the premises of two theorems
paulson <lp15@cam.ac.uk>
parents:
66112
diff
changeset
|
1014 |
and "\<And>x1 K1 x2 K2. (x1,K1) \<in> s \<Longrightarrow> |
571b698659c0
Repaired an inadvertent reordering of the premises of two theorems
paulson <lp15@cam.ac.uk>
parents:
66112
diff
changeset
|
1015 |
(x2, K2) \<in> s \<Longrightarrow> (x1, K1) \<noteq> (x2, K2) \<Longrightarrow> interior K1 \<inter> interior K2 = {}" |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1016 |
using assms unfolding tagged_partial_division_of_def by blast+ |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1017 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1018 |
definition tagged_division_of (infixr "tagged'_division'_of" 40) |
66113
571b698659c0
Repaired an inadvertent reordering of the premises of two theorems
paulson <lp15@cam.ac.uk>
parents:
66112
diff
changeset
|
1019 |
where "s tagged_division_of i \<longleftrightarrow> s tagged_partial_division_of i \<and> (\<Union>{K. \<exists>x. (x,K) \<in> s} = i)" |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1020 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1021 |
lemma tagged_division_of_finite: "s tagged_division_of i \<Longrightarrow> finite s" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1022 |
unfolding tagged_division_of_def tagged_partial_division_of_def by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1023 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1024 |
lemma tagged_division_of: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1025 |
"s tagged_division_of i \<longleftrightarrow> |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1026 |
finite s \<and> |
66113
571b698659c0
Repaired an inadvertent reordering of the premises of two theorems
paulson <lp15@cam.ac.uk>
parents:
66112
diff
changeset
|
1027 |
(\<forall>x K. (x, K) \<in> s \<longrightarrow> x \<in> K \<and> K \<subseteq> i \<and> (\<exists>a b. K = cbox a b)) \<and> |
571b698659c0
Repaired an inadvertent reordering of the premises of two theorems
paulson <lp15@cam.ac.uk>
parents:
66112
diff
changeset
|
1028 |
(\<forall>x1 K1 x2 K2. (x1, K1) \<in> s \<and> (x2, K2) \<in> s \<and> (x1, K1) \<noteq> (x2, K2) \<longrightarrow> |
571b698659c0
Repaired an inadvertent reordering of the premises of two theorems
paulson <lp15@cam.ac.uk>
parents:
66112
diff
changeset
|
1029 |
interior K1 \<inter> interior K2 = {}) \<and> |
571b698659c0
Repaired an inadvertent reordering of the premises of two theorems
paulson <lp15@cam.ac.uk>
parents:
66112
diff
changeset
|
1030 |
(\<Union>{K. \<exists>x. (x,K) \<in> s} = i)" |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1031 |
unfolding tagged_division_of_def tagged_partial_division_of_def by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1032 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1033 |
lemma tagged_division_ofI: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1034 |
assumes "finite s" |
66113
571b698659c0
Repaired an inadvertent reordering of the premises of two theorems
paulson <lp15@cam.ac.uk>
parents:
66112
diff
changeset
|
1035 |
and "\<And>x K. (x,K) \<in> s \<Longrightarrow> x \<in> K" |
571b698659c0
Repaired an inadvertent reordering of the premises of two theorems
paulson <lp15@cam.ac.uk>
parents:
66112
diff
changeset
|
1036 |
and "\<And>x K. (x,K) \<in> s \<Longrightarrow> K \<subseteq> i" |
571b698659c0
Repaired an inadvertent reordering of the premises of two theorems
paulson <lp15@cam.ac.uk>
parents:
66112
diff
changeset
|
1037 |
and "\<And>x K. (x,K) \<in> s \<Longrightarrow> \<exists>a b. K = cbox a b" |
571b698659c0
Repaired an inadvertent reordering of the premises of two theorems
paulson <lp15@cam.ac.uk>
parents:
66112
diff
changeset
|
1038 |
and "\<And>x1 K1 x2 K2. (x1,K1) \<in> s \<Longrightarrow> (x2, K2) \<in> s \<Longrightarrow> (x1, K1) \<noteq> (x2, K2) \<Longrightarrow> |
571b698659c0
Repaired an inadvertent reordering of the premises of two theorems
paulson <lp15@cam.ac.uk>
parents:
66112
diff
changeset
|
1039 |
interior K1 \<inter> interior K2 = {}" |
571b698659c0
Repaired an inadvertent reordering of the premises of two theorems
paulson <lp15@cam.ac.uk>
parents:
66112
diff
changeset
|
1040 |
and "(\<Union>{K. \<exists>x. (x,K) \<in> s} = i)" |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1041 |
shows "s tagged_division_of i" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1042 |
unfolding tagged_division_of |
66300 | 1043 |
using assms by fastforce |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1044 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1045 |
lemma tagged_division_ofD[dest]: (*FIXME USE A LOCALE*) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1046 |
assumes "s tagged_division_of i" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1047 |
shows "finite s" |
66113
571b698659c0
Repaired an inadvertent reordering of the premises of two theorems
paulson <lp15@cam.ac.uk>
parents:
66112
diff
changeset
|
1048 |
and "\<And>x K. (x,K) \<in> s \<Longrightarrow> x \<in> K" |
571b698659c0
Repaired an inadvertent reordering of the premises of two theorems
paulson <lp15@cam.ac.uk>
parents:
66112
diff
changeset
|
1049 |
and "\<And>x K. (x,K) \<in> s \<Longrightarrow> K \<subseteq> i" |
571b698659c0
Repaired an inadvertent reordering of the premises of two theorems
paulson <lp15@cam.ac.uk>
parents:
66112
diff
changeset
|
1050 |
and "\<And>x K. (x,K) \<in> s \<Longrightarrow> \<exists>a b. K = cbox a b" |
571b698659c0
Repaired an inadvertent reordering of the premises of two theorems
paulson <lp15@cam.ac.uk>
parents:
66112
diff
changeset
|
1051 |
and "\<And>x1 K1 x2 K2. (x1, K1) \<in> s \<Longrightarrow> (x2, K2) \<in> s \<Longrightarrow> (x1, K1) \<noteq> (x2, K2) \<Longrightarrow> |
571b698659c0
Repaired an inadvertent reordering of the premises of two theorems
paulson <lp15@cam.ac.uk>
parents:
66112
diff
changeset
|
1052 |
interior K1 \<inter> interior K2 = {}" |
571b698659c0
Repaired an inadvertent reordering of the premises of two theorems
paulson <lp15@cam.ac.uk>
parents:
66112
diff
changeset
|
1053 |
and "(\<Union>{K. \<exists>x. (x,K) \<in> s} = i)" |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1054 |
using assms unfolding tagged_division_of by blast+ |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1055 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1056 |
lemma division_of_tagged_division: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1057 |
assumes "s tagged_division_of i" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1058 |
shows "(snd ` s) division_of i" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1059 |
proof (rule division_ofI) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1060 |
note assm = tagged_division_ofD[OF assms] |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1061 |
show "\<Union>(snd ` s) = i" "finite (snd ` s)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1062 |
using assm by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1063 |
fix k |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1064 |
assume k: "k \<in> snd ` s" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1065 |
then obtain xk where xk: "(xk, k) \<in> s" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1066 |
by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1067 |
then show "k \<subseteq> i" "k \<noteq> {}" "\<exists>a b. k = cbox a b" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1068 |
using assm by fastforce+ |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1069 |
fix k' |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1070 |
assume k': "k' \<in> snd ` s" "k \<noteq> k'" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1071 |
from this(1) obtain xk' where xk': "(xk', k') \<in> s" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1072 |
by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1073 |
then show "interior k \<inter> interior k' = {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1074 |
using assm(5) k'(2) xk by blast |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1075 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1076 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1077 |
lemma partial_division_of_tagged_division: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1078 |
assumes "s tagged_partial_division_of i" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1079 |
shows "(snd ` s) division_of \<Union>(snd ` s)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1080 |
proof (rule division_ofI) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1081 |
note assm = tagged_partial_division_ofD[OF assms] |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1082 |
show "finite (snd ` s)" "\<Union>(snd ` s) = \<Union>(snd ` s)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1083 |
using assm by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1084 |
fix k |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1085 |
assume k: "k \<in> snd ` s" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1086 |
then obtain xk where xk: "(xk, k) \<in> s" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1087 |
by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1088 |
then show "k \<noteq> {}" "\<exists>a b. k = cbox a b" "k \<subseteq> \<Union>(snd ` s)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1089 |
using assm by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1090 |
fix k' |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1091 |
assume k': "k' \<in> snd ` s" "k \<noteq> k'" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1092 |
from this(1) obtain xk' where xk': "(xk', k') \<in> s" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1093 |
by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1094 |
then show "interior k \<inter> interior k' = {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1095 |
using assm(5) k'(2) xk by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1096 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1097 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1098 |
lemma tagged_partial_division_subset: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1099 |
assumes "s tagged_partial_division_of i" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1100 |
and "t \<subseteq> s" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1101 |
shows "t tagged_partial_division_of i" |
66318 | 1102 |
using assms finite_subset[OF assms(2)] |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1103 |
unfolding tagged_partial_division_of_def |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1104 |
by blast |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1105 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1106 |
lemma tag_in_interval: "p tagged_division_of i \<Longrightarrow> (x, k) \<in> p \<Longrightarrow> x \<in> i" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1107 |
by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1108 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1109 |
lemma tagged_division_of_empty: "{} tagged_division_of {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1110 |
unfolding tagged_division_of by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1111 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1112 |
lemma tagged_partial_division_of_trivial[simp]: "p tagged_partial_division_of {} \<longleftrightarrow> p = {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1113 |
unfolding tagged_partial_division_of_def by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1114 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1115 |
lemma tagged_division_of_trivial[simp]: "p tagged_division_of {} \<longleftrightarrow> p = {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1116 |
unfolding tagged_division_of by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1117 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1118 |
lemma tagged_division_of_self: "x \<in> cbox a b \<Longrightarrow> {(x,cbox a b)} tagged_division_of (cbox a b)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1119 |
by (rule tagged_division_ofI) auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1120 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1121 |
lemma tagged_division_of_self_real: "x \<in> {a .. b::real} \<Longrightarrow> {(x,{a .. b})} tagged_division_of {a .. b}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1122 |
unfolding box_real[symmetric] |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1123 |
by (rule tagged_division_of_self) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1124 |
|
66497
18a6478a574c
More tidying, and renaming of theorems
paulson <lp15@cam.ac.uk>
parents:
66365
diff
changeset
|
1125 |
lemma tagged_division_Un: |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1126 |
assumes "p1 tagged_division_of s1" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1127 |
and "p2 tagged_division_of s2" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1128 |
and "interior s1 \<inter> interior s2 = {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1129 |
shows "(p1 \<union> p2) tagged_division_of (s1 \<union> s2)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1130 |
proof (rule tagged_division_ofI) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1131 |
note p1 = tagged_division_ofD[OF assms(1)] |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1132 |
note p2 = tagged_division_ofD[OF assms(2)] |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1133 |
show "finite (p1 \<union> p2)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1134 |
using p1(1) p2(1) by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1135 |
show "\<Union>{k. \<exists>x. (x, k) \<in> p1 \<union> p2} = s1 \<union> s2" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1136 |
using p1(6) p2(6) by blast |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1137 |
fix x k |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1138 |
assume xk: "(x, k) \<in> p1 \<union> p2" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1139 |
show "x \<in> k" "\<exists>a b. k = cbox a b" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1140 |
using xk p1(2,4) p2(2,4) by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1141 |
show "k \<subseteq> s1 \<union> s2" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1142 |
using xk p1(3) p2(3) by blast |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1143 |
fix x' k' |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1144 |
assume xk': "(x', k') \<in> p1 \<union> p2" "(x, k) \<noteq> (x', k')" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1145 |
have *: "\<And>a b. a \<subseteq> s1 \<Longrightarrow> b \<subseteq> s2 \<Longrightarrow> interior a \<inter> interior b = {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1146 |
using assms(3) interior_mono by blast |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1147 |
show "interior k \<inter> interior k' = {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1148 |
apply (cases "(x, k) \<in> p1") |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1149 |
apply (meson "*" UnE assms(1) assms(2) p1(5) tagged_division_ofD(3) xk'(1) xk'(2)) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1150 |
by (metis "*" UnE assms(1) assms(2) inf_sup_aci(1) p2(5) tagged_division_ofD(3) xk xk'(1) xk'(2)) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1151 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1152 |
|
66497
18a6478a574c
More tidying, and renaming of theorems
paulson <lp15@cam.ac.uk>
parents:
66365
diff
changeset
|
1153 |
lemma tagged_division_Union: |
66318 | 1154 |
assumes "finite I" |
66497
18a6478a574c
More tidying, and renaming of theorems
paulson <lp15@cam.ac.uk>
parents:
66365
diff
changeset
|
1155 |
and tag: "\<And>i. i\<in>I \<Longrightarrow> pfn i tagged_division_of i" |
18a6478a574c
More tidying, and renaming of theorems
paulson <lp15@cam.ac.uk>
parents:
66365
diff
changeset
|
1156 |
and disj: "\<And>i1 i2. \<lbrakk>i1 \<in> I; i2 \<in> I; i1 \<noteq> i2\<rbrakk> \<Longrightarrow> interior(i1) \<inter> interior(i2) = {}" |
66318 | 1157 |
shows "\<Union>(pfn ` I) tagged_division_of (\<Union>I)" |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1158 |
proof (rule tagged_division_ofI) |
66497
18a6478a574c
More tidying, and renaming of theorems
paulson <lp15@cam.ac.uk>
parents:
66365
diff
changeset
|
1159 |
note assm = tagged_division_ofD[OF tag] |
66318 | 1160 |
show "finite (\<Union>(pfn ` I))" |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1161 |
using assms by auto |
66318 | 1162 |
have "\<Union>{k. \<exists>x. (x, k) \<in> \<Union>(pfn ` I)} = \<Union>((\<lambda>i. \<Union>{k. \<exists>x. (x, k) \<in> pfn i}) ` I)" |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1163 |
by blast |
66318 | 1164 |
also have "\<dots> = \<Union>I" |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1165 |
using assm(6) by auto |
66318 | 1166 |
finally show "\<Union>{k. \<exists>x. (x, k) \<in> \<Union>(pfn ` I)} = \<Union>I" . |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1167 |
fix x k |
66318 | 1168 |
assume xk: "(x, k) \<in> \<Union>(pfn ` I)" |
1169 |
then obtain i where i: "i \<in> I" "(x, k) \<in> pfn i" |
|
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1170 |
by auto |
66318 | 1171 |
show "x \<in> k" "\<exists>a b. k = cbox a b" "k \<subseteq> \<Union>I" |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1172 |
using assm(2-4)[OF i] using i(1) by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1173 |
fix x' k' |
66318 | 1174 |
assume xk': "(x', k') \<in> \<Union>(pfn ` I)" "(x, k) \<noteq> (x', k')" |
1175 |
then obtain i' where i': "i' \<in> I" "(x', k') \<in> pfn i'" |
|
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1176 |
by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1177 |
have *: "\<And>a b. i \<noteq> i' \<Longrightarrow> a \<subseteq> i \<Longrightarrow> b \<subseteq> i' \<Longrightarrow> interior a \<inter> interior b = {}" |
66497
18a6478a574c
More tidying, and renaming of theorems
paulson <lp15@cam.ac.uk>
parents:
66365
diff
changeset
|
1178 |
using i(1) i'(1) disj interior_mono by blast |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1179 |
show "interior k \<inter> interior k' = {}" |
66497
18a6478a574c
More tidying, and renaming of theorems
paulson <lp15@cam.ac.uk>
parents:
66365
diff
changeset
|
1180 |
proof (cases "i = i'") |
18a6478a574c
More tidying, and renaming of theorems
paulson <lp15@cam.ac.uk>
parents:
66365
diff
changeset
|
1181 |
case True then show ?thesis |
18a6478a574c
More tidying, and renaming of theorems
paulson <lp15@cam.ac.uk>
parents:
66365
diff
changeset
|
1182 |
using assm(5) i' i xk'(2) by blast |
18a6478a574c
More tidying, and renaming of theorems
paulson <lp15@cam.ac.uk>
parents:
66365
diff
changeset
|
1183 |
next |
18a6478a574c
More tidying, and renaming of theorems
paulson <lp15@cam.ac.uk>
parents:
66365
diff
changeset
|
1184 |
case False then show ?thesis |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1185 |
using "*" assm(3) i' i by auto |
66497
18a6478a574c
More tidying, and renaming of theorems
paulson <lp15@cam.ac.uk>
parents:
66365
diff
changeset
|
1186 |
qed |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1187 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1188 |
|
66497
18a6478a574c
More tidying, and renaming of theorems
paulson <lp15@cam.ac.uk>
parents:
66365
diff
changeset
|
1189 |
lemma tagged_partial_division_of_Union_self: |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1190 |
assumes "p tagged_partial_division_of s" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1191 |
shows "p tagged_division_of (\<Union>(snd ` p))" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1192 |
apply (rule tagged_division_ofI) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1193 |
using tagged_partial_division_ofD[OF assms] |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1194 |
apply auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1195 |
done |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1196 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1197 |
lemma tagged_division_of_union_self: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1198 |
assumes "p tagged_division_of s" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1199 |
shows "p tagged_division_of (\<Union>(snd ` p))" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1200 |
apply (rule tagged_division_ofI) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1201 |
using tagged_division_ofD[OF assms] |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1202 |
apply auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1203 |
done |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1204 |
|
66498 | 1205 |
lemma tagged_division_Un_interval: |
66492
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1206 |
fixes a :: "'a::euclidean_space" |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1207 |
assumes "p1 tagged_division_of (cbox a b \<inter> {x. x\<bullet>k \<le> (c::real)})" |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1208 |
and "p2 tagged_division_of (cbox a b \<inter> {x. x\<bullet>k \<ge> c})" |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1209 |
and k: "k \<in> Basis" |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1210 |
shows "(p1 \<union> p2) tagged_division_of (cbox a b)" |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1211 |
proof - |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1212 |
have *: "cbox a b = (cbox a b \<inter> {x. x\<bullet>k \<le> c}) \<union> (cbox a b \<inter> {x. x\<bullet>k \<ge> c})" |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1213 |
by auto |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1214 |
show ?thesis |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1215 |
apply (subst *) |
66498 | 1216 |
apply (rule tagged_division_Un[OF assms(1-2)]) |
66492
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1217 |
unfolding interval_split[OF k] interior_cbox |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1218 |
using k |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1219 |
apply (auto simp add: box_def elim!: ballE[where x=k]) |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1220 |
done |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1221 |
qed |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1222 |
|
66498 | 1223 |
lemma tagged_division_Un_interval_real: |
66492
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1224 |
fixes a :: real |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1225 |
assumes "p1 tagged_division_of ({a .. b} \<inter> {x. x\<bullet>k \<le> (c::real)})" |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1226 |
and "p2 tagged_division_of ({a .. b} \<inter> {x. x\<bullet>k \<ge> c})" |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1227 |
and k: "k \<in> Basis" |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1228 |
shows "(p1 \<union> p2) tagged_division_of {a .. b}" |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1229 |
using assms |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1230 |
unfolding box_real[symmetric] |
66498 | 1231 |
by (rule tagged_division_Un_interval) |
66492
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1232 |
|
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1233 |
lemma tagged_division_split_left_inj: |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1234 |
assumes d: "d tagged_division_of i" |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1235 |
and tags: "(x1, K1) \<in> d" "(x2, K2) \<in> d" |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1236 |
and "K1 \<noteq> K2" |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1237 |
and eq: "K1 \<inter> {x. x \<bullet> k \<le> c} = K2 \<inter> {x. x \<bullet> k \<le> c}" |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1238 |
shows "interior (K1 \<inter> {x. x\<bullet>k \<le> c}) = {}" |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1239 |
proof - |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1240 |
have "interior (K1 \<inter> K2) = {} \<or> (x2, K2) = (x1, K1)" |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1241 |
using tags d by (metis (no_types) interior_Int tagged_division_ofD(5)) |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1242 |
then show ?thesis |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1243 |
using eq \<open>K1 \<noteq> K2\<close> by (metis (no_types) inf_assoc inf_bot_left inf_left_idem interior_Int old.prod.inject) |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1244 |
qed |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1245 |
|
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1246 |
lemma tagged_division_split_right_inj: |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1247 |
assumes d: "d tagged_division_of i" |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1248 |
and tags: "(x1, K1) \<in> d" "(x2, K2) \<in> d" |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1249 |
and "K1 \<noteq> K2" |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1250 |
and eq: "K1 \<inter> {x. x\<bullet>k \<ge> c} = K2 \<inter> {x. x\<bullet>k \<ge> c}" |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1251 |
shows "interior (K1 \<inter> {x. x\<bullet>k \<ge> c}) = {}" |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1252 |
proof - |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1253 |
have "interior (K1 \<inter> K2) = {} \<or> (x2, K2) = (x1, K1)" |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1254 |
using tags d by (metis (no_types) interior_Int tagged_division_ofD(5)) |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1255 |
then show ?thesis |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1256 |
using eq \<open>K1 \<noteq> K2\<close> by (metis (no_types) inf_assoc inf_bot_left inf_left_idem interior_Int old.prod.inject) |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1257 |
qed |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1258 |
|
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1259 |
lemma (in comm_monoid_set) over_tagged_division_lemma: |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1260 |
assumes "p tagged_division_of i" |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1261 |
and "\<And>u v. box u v = {} \<Longrightarrow> d (cbox u v) = \<^bold>1" |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1262 |
shows "F (\<lambda>(_, k). d k) p = F d (snd ` p)" |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1263 |
proof - |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1264 |
have *: "(\<lambda>(_ ,k). d k) = d \<circ> snd" |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1265 |
by (simp add: fun_eq_iff) |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1266 |
note assm = tagged_division_ofD[OF assms(1)] |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1267 |
show ?thesis |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1268 |
unfolding * |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1269 |
proof (rule reindex_nontrivial[symmetric]) |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1270 |
show "finite p" |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1271 |
using assm by auto |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1272 |
fix x y |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1273 |
assume "x\<in>p" "y\<in>p" "x\<noteq>y" "snd x = snd y" |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1274 |
obtain a b where ab: "snd x = cbox a b" |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1275 |
using assm(4)[of "fst x" "snd x"] \<open>x\<in>p\<close> by auto |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1276 |
have "(fst x, snd y) \<in> p" "(fst x, snd y) \<noteq> y" |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1277 |
using \<open>x \<in> p\<close> \<open>x \<noteq> y\<close> \<open>snd x = snd y\<close> [symmetric] by auto |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1278 |
with \<open>x\<in>p\<close> \<open>y\<in>p\<close> have "interior (snd x) \<inter> interior (snd y) = {}" |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1279 |
by (intro assm(5)[of "fst x" _ "fst y"]) auto |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1280 |
then have "box a b = {}" |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1281 |
unfolding \<open>snd x = snd y\<close>[symmetric] ab by auto |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1282 |
then have "d (cbox a b) = \<^bold>1" |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1283 |
using assm(2)[of "fst x" "snd x"] \<open>x\<in>p\<close> ab[symmetric] by (intro assms(2)) auto |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1284 |
then show "d (snd x) = \<^bold>1" |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1285 |
unfolding ab by auto |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1286 |
qed |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1287 |
qed |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1288 |
|
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1289 |
|
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1290 |
subsection \<open>Functions closed on boxes: morphisms from boxes to monoids\<close> |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1291 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1292 |
text \<open>This auxiliary structure is used to sum up over the elements of a division. Main theorem is |
64911 | 1293 |
\<open>operative_division\<close>. Instances for the monoid are @{typ "'a option"}, @{typ real}, and |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1294 |
@{typ bool}.\<close> |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1295 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1296 |
paragraph \<open>Using additivity of lifted function to encode definedness.\<close> |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1297 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1298 |
definition lift_option :: "('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> 'a option \<Rightarrow> 'b option \<Rightarrow> 'c option" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1299 |
where |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1300 |
"lift_option f a' b' = Option.bind a' (\<lambda>a. Option.bind b' (\<lambda>b. Some (f a b)))" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1301 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1302 |
lemma lift_option_simps[simp]: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1303 |
"lift_option f (Some a) (Some b) = Some (f a b)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1304 |
"lift_option f None b' = None" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1305 |
"lift_option f a' None = None" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1306 |
by (auto simp: lift_option_def) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1307 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1308 |
lemma comm_monoid_lift_option: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1309 |
assumes "comm_monoid f z" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1310 |
shows "comm_monoid (lift_option f) (Some z)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1311 |
proof - |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1312 |
from assms interpret comm_monoid f z . |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1313 |
show ?thesis |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1314 |
by standard (auto simp: lift_option_def ac_simps split: bind_split) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1315 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1316 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1317 |
lemma comm_monoid_and: "comm_monoid HOL.conj True" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1318 |
by standard auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1319 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1320 |
lemma comm_monoid_set_and: "comm_monoid_set HOL.conj True" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1321 |
by (rule comm_monoid_set.intro) (fact comm_monoid_and) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1322 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1323 |
|
66492
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1324 |
paragraph \<open>Misc\<close> |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1325 |
|
66492
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1326 |
lemma interval_real_split: |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1327 |
"{a .. b::real} \<inter> {x. x \<le> c} = {a .. min b c}" |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1328 |
"{a .. b} \<inter> {x. c \<le> x} = {max a c .. b}" |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1329 |
apply (metis Int_atLeastAtMostL1 atMost_def) |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1330 |
apply (metis Int_atLeastAtMostL2 atLeast_def) |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1331 |
done |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1332 |
|
66492
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1333 |
lemma bgauge_existence_lemma: "(\<forall>x\<in>s. \<exists>d::real. 0 < d \<and> q d x) \<longleftrightarrow> (\<forall>x. \<exists>d>0. x\<in>s \<longrightarrow> q d x)" |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1334 |
by (meson zero_less_one) |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1335 |
|
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1336 |
|
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1337 |
paragraph \<open>Division points\<close> |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1338 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1339 |
definition "division_points (k::('a::euclidean_space) set) d = |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1340 |
{(j,x). j \<in> Basis \<and> (interval_lowerbound k)\<bullet>j < x \<and> x < (interval_upperbound k)\<bullet>j \<and> |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1341 |
(\<exists>i\<in>d. (interval_lowerbound i)\<bullet>j = x \<or> (interval_upperbound i)\<bullet>j = x)}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1342 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1343 |
lemma division_points_finite: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1344 |
fixes i :: "'a::euclidean_space set" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1345 |
assumes "d division_of i" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1346 |
shows "finite (division_points i d)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1347 |
proof - |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1348 |
note assm = division_ofD[OF assms] |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1349 |
let ?M = "\<lambda>j. {(j,x)|x. (interval_lowerbound i)\<bullet>j < x \<and> x < (interval_upperbound i)\<bullet>j \<and> |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1350 |
(\<exists>i\<in>d. (interval_lowerbound i)\<bullet>j = x \<or> (interval_upperbound i)\<bullet>j = x)}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1351 |
have *: "division_points i d = \<Union>(?M ` Basis)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1352 |
unfolding division_points_def by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1353 |
show ?thesis |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1354 |
unfolding * using assm by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1355 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1356 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1357 |
lemma division_points_subset: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1358 |
fixes a :: "'a::euclidean_space" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1359 |
assumes "d division_of (cbox a b)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1360 |
and "\<forall>i\<in>Basis. a\<bullet>i < b\<bullet>i" "a\<bullet>k < c" "c < b\<bullet>k" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1361 |
and k: "k \<in> Basis" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1362 |
shows "division_points (cbox a b \<inter> {x. x\<bullet>k \<le> c}) {l \<inter> {x. x\<bullet>k \<le> c} | l . l \<in> d \<and> l \<inter> {x. x\<bullet>k \<le> c} \<noteq> {}} \<subseteq> |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1363 |
division_points (cbox a b) d" (is ?t1) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1364 |
and "division_points (cbox a b \<inter> {x. x\<bullet>k \<ge> c}) {l \<inter> {x. x\<bullet>k \<ge> c} | l . l \<in> d \<and> ~(l \<inter> {x. x\<bullet>k \<ge> c} = {})} \<subseteq> |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1365 |
division_points (cbox a b) d" (is ?t2) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1366 |
proof - |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1367 |
note assm = division_ofD[OF assms(1)] |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1368 |
have *: "\<forall>i\<in>Basis. a\<bullet>i \<le> b\<bullet>i" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1369 |
"\<forall>i\<in>Basis. a\<bullet>i \<le> (\<Sum>i\<in>Basis. (if i = k then min (b \<bullet> k) c else b \<bullet> i) *\<^sub>R i) \<bullet> i" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1370 |
"\<forall>i\<in>Basis. (\<Sum>i\<in>Basis. (if i = k then max (a \<bullet> k) c else a \<bullet> i) *\<^sub>R i) \<bullet> i \<le> b\<bullet>i" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1371 |
"min (b \<bullet> k) c = c" "max (a \<bullet> k) c = c" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1372 |
using assms using less_imp_le by auto |
66300 | 1373 |
have "\<exists>i\<in>d. interval_lowerbound i \<bullet> x = y \<or> interval_upperbound i \<bullet> x = y" |
1374 |
if "a \<bullet> x < y" "y < (if x = k then c else b \<bullet> x)" |
|
1375 |
"interval_lowerbound i \<bullet> x = y \<or> interval_upperbound i \<bullet> x = y" |
|
1376 |
"i = l \<inter> {x. x \<bullet> k \<le> c}" "l \<in> d" "l \<inter> {x. x \<bullet> k \<le> c} \<noteq> {}" |
|
1377 |
"x \<in> Basis" for i l x y |
|
1378 |
proof - |
|
1379 |
obtain u v where l: "l = cbox u v" |
|
1380 |
using \<open>l \<in> d\<close> assms(1) by blast |
|
1381 |
have *: "\<forall>i\<in>Basis. u \<bullet> i \<le> (\<Sum>i\<in>Basis. (if i = k then min (v \<bullet> k) c else v \<bullet> i) *\<^sub>R i) \<bullet> i" |
|
1382 |
using that(6) unfolding l interval_split[OF k] box_ne_empty that . |
|
1383 |
have **: "\<forall>i\<in>Basis. u\<bullet>i \<le> v\<bullet>i" |
|
1384 |
using l using that(6) unfolding box_ne_empty[symmetric] by auto |
|
1385 |
show ?thesis |
|
1386 |
apply (rule bexI[OF _ \<open>l \<in> d\<close>]) |
|
1387 |
using that(1-3,5) \<open>x \<in> Basis\<close> |
|
1388 |
unfolding l interval_bounds[OF **] interval_bounds[OF *] interval_split[OF k] that |
|
1389 |
apply (auto split: if_split_asm) |
|
1390 |
done |
|
1391 |
qed |
|
1392 |
moreover have "\<And>x y. \<lbrakk>y < (if x = k then c else b \<bullet> x)\<rbrakk> \<Longrightarrow> y < b \<bullet> x" |
|
1393 |
using \<open>c < b\<bullet>k\<close> by (auto split: if_split_asm) |
|
1394 |
ultimately show ?t1 |
|
1395 |
unfolding division_points_def interval_split[OF k, of a b] |
|
1396 |
unfolding interval_bounds[OF *(1)] interval_bounds[OF *(2)] interval_bounds[OF *(3)] |
|
1397 |
unfolding * by force |
|
1398 |
have "\<And>x y i l. (if x = k then c else a \<bullet> x) < y \<Longrightarrow> a \<bullet> x < y" |
|
1399 |
using \<open>a\<bullet>k < c\<close> by (auto split: if_split_asm) |
|
1400 |
moreover have "\<exists>i\<in>d. interval_lowerbound i \<bullet> x = y \<or> |
|
1401 |
interval_upperbound i \<bullet> x = y" |
|
1402 |
if "(if x = k then c else a \<bullet> x) < y" "y < b \<bullet> x" |
|
1403 |
"interval_lowerbound i \<bullet> x = y \<or> interval_upperbound i \<bullet> x = y" |
|
1404 |
"i = l \<inter> {x. c \<le> x \<bullet> k}" "l \<in> d" "l \<inter> {x. c \<le> x \<bullet> k} \<noteq> {}" |
|
1405 |
"x \<in> Basis" for x y i l |
|
1406 |
proof - |
|
1407 |
obtain u v where l: "l = cbox u v" |
|
1408 |
using \<open>l \<in> d\<close> assm(4) by blast |
|
1409 |
have *: "\<forall>i\<in>Basis. (\<Sum>i\<in>Basis. (if i = k then max (u \<bullet> k) c else u \<bullet> i) *\<^sub>R i) \<bullet> i \<le> v \<bullet> i" |
|
1410 |
using that(6) unfolding l interval_split[OF k] box_ne_empty that . |
|
1411 |
have **: "\<forall>i\<in>Basis. u\<bullet>i \<le> v\<bullet>i" |
|
1412 |
using l using that(6) unfolding box_ne_empty[symmetric] by auto |
|
1413 |
show "\<exists>i\<in>d. interval_lowerbound i \<bullet> x = y \<or> interval_upperbound i \<bullet> x = y" |
|
1414 |
apply (rule bexI[OF _ \<open>l \<in> d\<close>]) |
|
1415 |
using that(1-3,5) \<open>x \<in> Basis\<close> |
|
1416 |
unfolding l interval_bounds[OF **] interval_bounds[OF *] interval_split[OF k] that |
|
1417 |
apply (auto split: if_split_asm) |
|
1418 |
done |
|
1419 |
qed |
|
1420 |
ultimately show ?t2 |
|
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1421 |
unfolding division_points_def interval_split[OF k, of a b] |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1422 |
unfolding interval_bounds[OF *(1)] interval_bounds[OF *(2)] interval_bounds[OF *(3)] |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1423 |
unfolding * |
66300 | 1424 |
by force |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1425 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1426 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1427 |
lemma division_points_psubset: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1428 |
fixes a :: "'a::euclidean_space" |
66314 | 1429 |
assumes d: "d division_of (cbox a b)" |
1430 |
and altb: "\<forall>i\<in>Basis. a\<bullet>i < b\<bullet>i" "a\<bullet>k < c" "c < b\<bullet>k" |
|
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1431 |
and "l \<in> d" |
66300 | 1432 |
and disj: "interval_lowerbound l\<bullet>k = c \<or> interval_upperbound l\<bullet>k = c" |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1433 |
and k: "k \<in> Basis" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1434 |
shows "division_points (cbox a b \<inter> {x. x\<bullet>k \<le> c}) {l \<inter> {x. x\<bullet>k \<le> c} | l. l\<in>d \<and> l \<inter> {x. x\<bullet>k \<le> c} \<noteq> {}} \<subset> |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1435 |
division_points (cbox a b) d" (is "?D1 \<subset> ?D") |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1436 |
and "division_points (cbox a b \<inter> {x. x\<bullet>k \<ge> c}) {l \<inter> {x. x\<bullet>k \<ge> c} | l. l\<in>d \<and> l \<inter> {x. x\<bullet>k \<ge> c} \<noteq> {}} \<subset> |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1437 |
division_points (cbox a b) d" (is "?D2 \<subset> ?D") |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1438 |
proof - |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1439 |
have ab: "\<forall>i\<in>Basis. a\<bullet>i \<le> b\<bullet>i" |
66314 | 1440 |
using altb by (auto intro!:less_imp_le) |
1441 |
obtain u v where l: "l = cbox u v" |
|
1442 |
using d \<open>l \<in> d\<close> by blast |
|
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1443 |
have uv: "\<forall>i\<in>Basis. u\<bullet>i \<le> v\<bullet>i" "\<forall>i\<in>Basis. a\<bullet>i \<le> u\<bullet>i \<and> v\<bullet>i \<le> b\<bullet>i" |
66314 | 1444 |
apply (metis assms(5) box_ne_empty(1) cbox_division_memE d l) |
1445 |
by (metis assms(5) box_ne_empty(1) cbox_division_memE d l subset_box(1)) |
|
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1446 |
have *: "interval_upperbound (cbox a b \<inter> {x. x \<bullet> k \<le> interval_upperbound l \<bullet> k}) \<bullet> k = interval_upperbound l \<bullet> k" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1447 |
"interval_upperbound (cbox a b \<inter> {x. x \<bullet> k \<le> interval_lowerbound l \<bullet> k}) \<bullet> k = interval_lowerbound l \<bullet> k" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1448 |
unfolding l interval_split[OF k] interval_bounds[OF uv(1)] |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1449 |
using uv[rule_format, of k] ab k |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1450 |
by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1451 |
have "\<exists>x. x \<in> ?D - ?D1" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1452 |
using assms(3-) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1453 |
unfolding division_points_def interval_bounds[OF ab] |
66318 | 1454 |
by (force simp add: *) |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1455 |
moreover have "?D1 \<subseteq> ?D" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1456 |
by (auto simp add: assms division_points_subset) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1457 |
ultimately show "?D1 \<subset> ?D" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1458 |
by blast |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1459 |
have *: "interval_lowerbound (cbox a b \<inter> {x. x \<bullet> k \<ge> interval_lowerbound l \<bullet> k}) \<bullet> k = interval_lowerbound l \<bullet> k" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1460 |
"interval_lowerbound (cbox a b \<inter> {x. x \<bullet> k \<ge> interval_upperbound l \<bullet> k}) \<bullet> k = interval_upperbound l \<bullet> k" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1461 |
unfolding l interval_split[OF k] interval_bounds[OF uv(1)] |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1462 |
using uv[rule_format, of k] ab k |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1463 |
by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1464 |
have "\<exists>x. x \<in> ?D - ?D2" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1465 |
using assms(3-) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1466 |
unfolding division_points_def interval_bounds[OF ab] |
66318 | 1467 |
by (force simp add: *) |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1468 |
moreover have "?D2 \<subseteq> ?D" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1469 |
by (auto simp add: assms division_points_subset) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1470 |
ultimately show "?D2 \<subset> ?D" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1471 |
by blast |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1472 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1473 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1474 |
lemma division_split_left_inj: |
66112
0e640e04fc56
New theorems; stronger theorems; tidier theorems. Also some renaming
paulson <lp15@cam.ac.uk>
parents:
64911
diff
changeset
|
1475 |
fixes S :: "'a::euclidean_space set" |
0e640e04fc56
New theorems; stronger theorems; tidier theorems. Also some renaming
paulson <lp15@cam.ac.uk>
parents:
64911
diff
changeset
|
1476 |
assumes div: "\<D> division_of S" |
0e640e04fc56
New theorems; stronger theorems; tidier theorems. Also some renaming
paulson <lp15@cam.ac.uk>
parents:
64911
diff
changeset
|
1477 |
and eq: "K1 \<inter> {x::'a. x\<bullet>k \<le> c} = K2 \<inter> {x. x\<bullet>k \<le> c}" |
0e640e04fc56
New theorems; stronger theorems; tidier theorems. Also some renaming
paulson <lp15@cam.ac.uk>
parents:
64911
diff
changeset
|
1478 |
and "K1 \<in> \<D>" "K2 \<in> \<D>" "K1 \<noteq> K2" |
0e640e04fc56
New theorems; stronger theorems; tidier theorems. Also some renaming
paulson <lp15@cam.ac.uk>
parents:
64911
diff
changeset
|
1479 |
shows "interior (K1 \<inter> {x. x\<bullet>k \<le> c}) = {}" |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1480 |
proof - |
66112
0e640e04fc56
New theorems; stronger theorems; tidier theorems. Also some renaming
paulson <lp15@cam.ac.uk>
parents:
64911
diff
changeset
|
1481 |
have "interior K2 \<inter> interior {a. a \<bullet> k \<le> c} = interior K1 \<inter> interior {a. a \<bullet> k \<le> c}" |
0e640e04fc56
New theorems; stronger theorems; tidier theorems. Also some renaming
paulson <lp15@cam.ac.uk>
parents:
64911
diff
changeset
|
1482 |
by (metis (no_types) eq interior_Int) |
0e640e04fc56
New theorems; stronger theorems; tidier theorems. Also some renaming
paulson <lp15@cam.ac.uk>
parents:
64911
diff
changeset
|
1483 |
moreover have "\<And>A. interior A \<inter> interior K2 = {} \<or> A = K2 \<or> A \<notin> \<D>" |
0e640e04fc56
New theorems; stronger theorems; tidier theorems. Also some renaming
paulson <lp15@cam.ac.uk>
parents:
64911
diff
changeset
|
1484 |
by (meson div \<open>K2 \<in> \<D>\<close> division_of_def) |
0e640e04fc56
New theorems; stronger theorems; tidier theorems. Also some renaming
paulson <lp15@cam.ac.uk>
parents:
64911
diff
changeset
|
1485 |
ultimately show ?thesis |
0e640e04fc56
New theorems; stronger theorems; tidier theorems. Also some renaming
paulson <lp15@cam.ac.uk>
parents:
64911
diff
changeset
|
1486 |
using \<open>K1 \<in> \<D>\<close> \<open>K1 \<noteq> K2\<close> by auto |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1487 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1488 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1489 |
lemma division_split_right_inj: |
66112
0e640e04fc56
New theorems; stronger theorems; tidier theorems. Also some renaming
paulson <lp15@cam.ac.uk>
parents:
64911
diff
changeset
|
1490 |
fixes S :: "'a::euclidean_space set" |
0e640e04fc56
New theorems; stronger theorems; tidier theorems. Also some renaming
paulson <lp15@cam.ac.uk>
parents:
64911
diff
changeset
|
1491 |
assumes div: "\<D> division_of S" |
0e640e04fc56
New theorems; stronger theorems; tidier theorems. Also some renaming
paulson <lp15@cam.ac.uk>
parents:
64911
diff
changeset
|
1492 |
and eq: "K1 \<inter> {x::'a. x\<bullet>k \<ge> c} = K2 \<inter> {x. x\<bullet>k \<ge> c}" |
0e640e04fc56
New theorems; stronger theorems; tidier theorems. Also some renaming
paulson <lp15@cam.ac.uk>
parents:
64911
diff
changeset
|
1493 |
and "K1 \<in> \<D>" "K2 \<in> \<D>" "K1 \<noteq> K2" |
0e640e04fc56
New theorems; stronger theorems; tidier theorems. Also some renaming
paulson <lp15@cam.ac.uk>
parents:
64911
diff
changeset
|
1494 |
shows "interior (K1 \<inter> {x. x\<bullet>k \<ge> c}) = {}" |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1495 |
proof - |
66112
0e640e04fc56
New theorems; stronger theorems; tidier theorems. Also some renaming
paulson <lp15@cam.ac.uk>
parents:
64911
diff
changeset
|
1496 |
have "interior K2 \<inter> interior {a. a \<bullet> k \<ge> c} = interior K1 \<inter> interior {a. a \<bullet> k \<ge> c}" |
0e640e04fc56
New theorems; stronger theorems; tidier theorems. Also some renaming
paulson <lp15@cam.ac.uk>
parents:
64911
diff
changeset
|
1497 |
by (metis (no_types) eq interior_Int) |
0e640e04fc56
New theorems; stronger theorems; tidier theorems. Also some renaming
paulson <lp15@cam.ac.uk>
parents:
64911
diff
changeset
|
1498 |
moreover have "\<And>A. interior A \<inter> interior K2 = {} \<or> A = K2 \<or> A \<notin> \<D>" |
0e640e04fc56
New theorems; stronger theorems; tidier theorems. Also some renaming
paulson <lp15@cam.ac.uk>
parents:
64911
diff
changeset
|
1499 |
by (meson div \<open>K2 \<in> \<D>\<close> division_of_def) |
0e640e04fc56
New theorems; stronger theorems; tidier theorems. Also some renaming
paulson <lp15@cam.ac.uk>
parents:
64911
diff
changeset
|
1500 |
ultimately show ?thesis |
0e640e04fc56
New theorems; stronger theorems; tidier theorems. Also some renaming
paulson <lp15@cam.ac.uk>
parents:
64911
diff
changeset
|
1501 |
using \<open>K1 \<in> \<D>\<close> \<open>K1 \<noteq> K2\<close> by auto |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1502 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1503 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1504 |
lemma interval_doublesplit: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1505 |
fixes a :: "'a::euclidean_space" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1506 |
assumes "k \<in> Basis" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1507 |
shows "cbox a b \<inter> {x . \<bar>x\<bullet>k - c\<bar> \<le> (e::real)} = |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1508 |
cbox (\<Sum>i\<in>Basis. (if i = k then max (a\<bullet>k) (c - e) else a\<bullet>i) *\<^sub>R i) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1509 |
(\<Sum>i\<in>Basis. (if i = k then min (b\<bullet>k) (c + e) else b\<bullet>i) *\<^sub>R i)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1510 |
proof - |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1511 |
have *: "\<And>x c e::real. \<bar>x - c\<bar> \<le> e \<longleftrightarrow> x \<ge> c - e \<and> x \<le> c + e" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1512 |
by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1513 |
have **: "\<And>s P Q. s \<inter> {x. P x \<and> Q x} = (s \<inter> {x. Q x}) \<inter> {x. P x}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1514 |
by blast |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1515 |
show ?thesis |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1516 |
unfolding * ** interval_split[OF assms] by (rule refl) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1517 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1518 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1519 |
lemma division_doublesplit: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1520 |
fixes a :: "'a::euclidean_space" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1521 |
assumes "p division_of (cbox a b)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1522 |
and k: "k \<in> Basis" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1523 |
shows "(\<lambda>l. l \<inter> {x. \<bar>x\<bullet>k - c\<bar> \<le> e}) ` {l\<in>p. l \<inter> {x. \<bar>x\<bullet>k - c\<bar> \<le> e} \<noteq> {}} |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1524 |
division_of (cbox a b \<inter> {x. \<bar>x\<bullet>k - c\<bar> \<le> e})" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1525 |
proof - |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1526 |
have *: "\<And>x c. \<bar>x - c\<bar> \<le> e \<longleftrightarrow> x \<ge> c - e \<and> x \<le> c + e" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1527 |
by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1528 |
have **: "\<And>p q p' q'. p division_of q \<Longrightarrow> p = p' \<Longrightarrow> q = q' \<Longrightarrow> p' division_of q'" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1529 |
by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1530 |
note division_split(1)[OF assms, where c="c+e",unfolded interval_split[OF k]] |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1531 |
note division_split(2)[OF this, where c="c-e" and k=k,OF k] |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1532 |
then show ?thesis |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1533 |
apply (rule **) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1534 |
subgoal |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1535 |
apply (simp add: abs_diff_le_iff field_simps Collect_conj_eq setcompr_eq_image[symmetric]) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1536 |
apply (rule equalityI) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1537 |
apply blast |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1538 |
apply clarsimp |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1539 |
apply (rule_tac x="l \<inter> {x. c + e \<ge> x \<bullet> k}" in exI) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1540 |
apply auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1541 |
done |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1542 |
by (simp add: interval_split k interval_doublesplit) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1543 |
qed |
66306 | 1544 |
|
66492
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1545 |
paragraph \<open>Operative\<close> |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1546 |
|
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1547 |
locale operative = comm_monoid_set + |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1548 |
fixes g :: "'b::euclidean_space set \<Rightarrow> 'a" |
66492
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1549 |
assumes box_empty_imp: "\<And>a b. box a b = {} \<Longrightarrow> g (cbox a b) = \<^bold>1" |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1550 |
and Basis_imp: "\<And>a b c k. k \<in> Basis \<Longrightarrow> g (cbox a b) = g (cbox a b \<inter> {x. x\<bullet>k \<le> c}) \<^bold>* g (cbox a b \<inter> {x. x\<bullet>k \<ge> c})" |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1551 |
begin |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1552 |
|
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1553 |
lemma empty [simp]: |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1554 |
"g {} = \<^bold>1" |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1555 |
proof - |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1556 |
have *: "cbox One (-One) = ({}::'b set)" |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1557 |
by (auto simp: box_eq_empty inner_sum_left inner_Basis sum.If_cases ex_in_conv) |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1558 |
moreover have "box One (-One) = ({}::'b set)" |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1559 |
using box_subset_cbox[of One "-One"] by (auto simp: *) |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1560 |
ultimately show ?thesis |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1561 |
using box_empty_imp [of One "-One"] by simp |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1562 |
qed |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1563 |
|
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1564 |
lemma division: |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1565 |
"F g d = g (cbox a b)" if "d division_of (cbox a b)" |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1566 |
proof - |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1567 |
define C where [abs_def]: "C = card (division_points (cbox a b) d)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1568 |
then show ?thesis |
66492
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1569 |
using that proof (induction C arbitrary: a b d rule: less_induct) |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1570 |
case (less a b d) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1571 |
show ?case |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1572 |
proof cases |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1573 |
assume "box a b = {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1574 |
{ fix k assume "k\<in>d" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1575 |
then obtain a' b' where k: "k = cbox a' b'" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1576 |
using division_ofD(4)[OF less.prems] by blast |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1577 |
with \<open>k\<in>d\<close> division_ofD(2)[OF less.prems] have "cbox a' b' \<subseteq> cbox a b" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1578 |
by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1579 |
then have "box a' b' \<subseteq> box a b" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1580 |
unfolding subset_box by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1581 |
then have "g k = \<^bold>1" |
66492
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1582 |
using box_empty_imp [of a' b'] k by (simp add: \<open>box a b = {}\<close>) } |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1583 |
then show "box a b = {} \<Longrightarrow> F g d = g (cbox a b)" |
66492
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1584 |
by (auto intro!: neutral simp: box_empty_imp) |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1585 |
next |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1586 |
assume "box a b \<noteq> {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1587 |
then have ab: "\<forall>i\<in>Basis. a\<bullet>i < b\<bullet>i" and ab': "\<forall>i\<in>Basis. a\<bullet>i \<le> b\<bullet>i" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1588 |
by (auto simp: box_ne_empty) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1589 |
show "F g d = g (cbox a b)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1590 |
proof (cases "division_points (cbox a b) d = {}") |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1591 |
case True |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1592 |
{ fix u v and j :: 'b |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1593 |
assume j: "j \<in> Basis" and as: "cbox u v \<in> d" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1594 |
then have "cbox u v \<noteq> {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1595 |
using less.prems by blast |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1596 |
then have uv: "\<forall>i\<in>Basis. u\<bullet>i \<le> v\<bullet>i" "u\<bullet>j \<le> v\<bullet>j" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1597 |
using j unfolding box_ne_empty by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1598 |
have *: "\<And>p r Q. \<not> j\<in>Basis \<or> p \<or> r \<or> (\<forall>x\<in>d. Q x) \<Longrightarrow> p \<or> r \<or> Q (cbox u v)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1599 |
using as j by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1600 |
have "(j, u\<bullet>j) \<notin> division_points (cbox a b) d" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1601 |
"(j, v\<bullet>j) \<notin> division_points (cbox a b) d" using True by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1602 |
note this[unfolded de_Morgan_conj division_points_def mem_Collect_eq split_conv interval_bounds[OF ab'] bex_simps] |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1603 |
note *[OF this(1)] *[OF this(2)] note this[unfolded interval_bounds[OF uv(1)]] |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1604 |
moreover |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1605 |
have "a\<bullet>j \<le> u\<bullet>j" "v\<bullet>j \<le> b\<bullet>j" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1606 |
using division_ofD(2,2,3)[OF \<open>d division_of cbox a b\<close> as] |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1607 |
apply (metis j subset_box(1) uv(1)) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1608 |
by (metis \<open>cbox u v \<subseteq> cbox a b\<close> j subset_box(1) uv(1)) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1609 |
ultimately have "u\<bullet>j = a\<bullet>j \<and> v\<bullet>j = a\<bullet>j \<or> u\<bullet>j = b\<bullet>j \<and> v\<bullet>j = b\<bullet>j \<or> u\<bullet>j = a\<bullet>j \<and> v\<bullet>j = b\<bullet>j" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1610 |
unfolding not_less de_Morgan_disj using ab[rule_format,of j] uv(2) j by force } |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1611 |
then have d': "\<forall>i\<in>d. \<exists>u v. i = cbox u v \<and> |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1612 |
(\<forall>j\<in>Basis. u\<bullet>j = a\<bullet>j \<and> v\<bullet>j = a\<bullet>j \<or> u\<bullet>j = b\<bullet>j \<and> v\<bullet>j = b\<bullet>j \<or> u\<bullet>j = a\<bullet>j \<and> v\<bullet>j = b\<bullet>j)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1613 |
unfolding forall_in_division[OF less.prems] by blast |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1614 |
have "(1/2) *\<^sub>R (a+b) \<in> cbox a b" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1615 |
unfolding mem_box using ab by (auto simp: inner_simps) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1616 |
note this[unfolded division_ofD(6)[OF \<open>d division_of cbox a b\<close>,symmetric] Union_iff] |
66300 | 1617 |
then obtain i where i: "i \<in> d" "(1 / 2) *\<^sub>R (a + b) \<in> i" .. |
1618 |
obtain u v where uv: "i = cbox u v" |
|
1619 |
"\<forall>j\<in>Basis. u \<bullet> j = a \<bullet> j \<and> v \<bullet> j = a \<bullet> j \<or> |
|
1620 |
u \<bullet> j = b \<bullet> j \<and> v \<bullet> j = b \<bullet> j \<or> |
|
1621 |
u \<bullet> j = a \<bullet> j \<and> v \<bullet> j = b \<bullet> j" |
|
1622 |
using d' i(1) by auto |
|
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1623 |
have "cbox a b \<in> d" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1624 |
proof - |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1625 |
have "u = a" "v = b" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1626 |
unfolding euclidean_eq_iff[where 'a='b] |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1627 |
proof safe |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1628 |
fix j :: 'b |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1629 |
assume j: "j \<in> Basis" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1630 |
note i(2)[unfolded uv mem_box,rule_format,of j] |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1631 |
then show "u \<bullet> j = a \<bullet> j" and "v \<bullet> j = b \<bullet> j" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1632 |
using uv(2)[rule_format,of j] j by (auto simp: inner_simps) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1633 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1634 |
then have "i = cbox a b" using uv by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1635 |
then show ?thesis using i by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1636 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1637 |
then have deq: "d = insert (cbox a b) (d - {cbox a b})" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1638 |
by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1639 |
have "F g (d - {cbox a b}) = \<^bold>1" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1640 |
proof (intro neutral ballI) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1641 |
fix x |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1642 |
assume x: "x \<in> d - {cbox a b}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1643 |
then have "x\<in>d" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1644 |
by auto note d'[rule_format,OF this] |
66300 | 1645 |
then obtain u v where uv: "x = cbox u v" |
1646 |
"\<forall>j\<in>Basis. u \<bullet> j = a \<bullet> j \<and> v \<bullet> j = a \<bullet> j \<or> |
|
1647 |
u \<bullet> j = b \<bullet> j \<and> v \<bullet> j = b \<bullet> j \<or> |
|
1648 |
u \<bullet> j = a \<bullet> j \<and> v \<bullet> j = b \<bullet> j" |
|
1649 |
by blast |
|
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1650 |
have "u \<noteq> a \<or> v \<noteq> b" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1651 |
using x[unfolded uv] by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1652 |
then obtain j where "u\<bullet>j \<noteq> a\<bullet>j \<or> v\<bullet>j \<noteq> b\<bullet>j" and j: "j \<in> Basis" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1653 |
unfolding euclidean_eq_iff[where 'a='b] by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1654 |
then have "u\<bullet>j = v\<bullet>j" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1655 |
using uv(2)[rule_format,OF j] by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1656 |
then have "box u v = {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1657 |
using j unfolding box_eq_empty by (auto intro!: bexI[of _ j]) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1658 |
then show "g x = \<^bold>1" |
66492
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1659 |
unfolding uv(1) by (rule box_empty_imp) |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1660 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1661 |
then show "F g d = g (cbox a b)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1662 |
using division_ofD[OF less.prems] |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1663 |
apply (subst deq) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1664 |
apply (subst insert) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1665 |
apply auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1666 |
done |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1667 |
next |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1668 |
case False |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1669 |
then have "\<exists>x. x \<in> division_points (cbox a b) d" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1670 |
by auto |
66306 | 1671 |
then obtain k c |
1672 |
where "k \<in> Basis" "interval_lowerbound (cbox a b) \<bullet> k < c" |
|
1673 |
"c < interval_upperbound (cbox a b) \<bullet> k" |
|
1674 |
"\<exists>i\<in>d. interval_lowerbound i \<bullet> k = c \<or> interval_upperbound i \<bullet> k = c" |
|
1675 |
unfolding division_points_def by auto |
|
1676 |
then obtain j where "a \<bullet> k < c" "c < b \<bullet> k" |
|
1677 |
and "j \<in> d" and j: "interval_lowerbound j \<bullet> k = c \<or> interval_upperbound j \<bullet> k = c" |
|
1678 |
by (metis division_of_trivial empty_iff interval_bounds' less.prems) |
|
1679 |
let ?lec = "{x. x\<bullet>k \<le> c}" let ?gec = "{x. x\<bullet>k \<ge> c}" |
|
1680 |
define d1 where "d1 = {l \<inter> ?lec | l. l \<in> d \<and> l \<inter> ?lec \<noteq> {}}" |
|
1681 |
define d2 where "d2 = {l \<inter> ?gec | l. l \<in> d \<and> l \<inter> ?gec \<noteq> {}}" |
|
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1682 |
define cb where "cb = (\<Sum>i\<in>Basis. (if i = k then c else b\<bullet>i) *\<^sub>R i)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1683 |
define ca where "ca = (\<Sum>i\<in>Basis. (if i = k then c else a\<bullet>i) *\<^sub>R i)" |
66306 | 1684 |
have "division_points (cbox a b \<inter> ?lec) {l \<inter> ?lec |l. l \<in> d \<and> l \<inter> ?lec \<noteq> {}} |
1685 |
\<subset> division_points (cbox a b) d" |
|
1686 |
by (rule division_points_psubset[OF \<open>d division_of cbox a b\<close> ab \<open>a \<bullet> k < c\<close> \<open>c < b \<bullet> k\<close> \<open>j \<in> d\<close> j \<open>k \<in> Basis\<close>]) |
|
1687 |
with division_points_finite[OF \<open>d division_of cbox a b\<close>] |
|
1688 |
have "card |
|
1689 |
(division_points (cbox a b \<inter> ?lec) {l \<inter> ?lec |l. l \<in> d \<and> l \<inter> ?lec \<noteq> {}}) |
|
1690 |
< card (division_points (cbox a b) d)" |
|
1691 |
by (rule psubset_card_mono) |
|
1692 |
moreover have "division_points (cbox a b \<inter> {x. c \<le> x \<bullet> k}) {l \<inter> {x. c \<le> x \<bullet> k} |l. l \<in> d \<and> l \<inter> {x. c \<le> x \<bullet> k} \<noteq> {}} |
|
1693 |
\<subset> division_points (cbox a b) d" |
|
1694 |
by (rule division_points_psubset[OF \<open>d division_of cbox a b\<close> ab \<open>a \<bullet> k < c\<close> \<open>c < b \<bullet> k\<close> \<open>j \<in> d\<close> j \<open>k \<in> Basis\<close>]) |
|
1695 |
with division_points_finite[OF \<open>d division_of cbox a b\<close>] |
|
1696 |
have "card (division_points (cbox a b \<inter> ?gec) {l \<inter> ?gec |l. l \<in> d \<and> l \<inter> ?gec \<noteq> {}}) |
|
1697 |
< card (division_points (cbox a b) d)" |
|
1698 |
by (rule psubset_card_mono) |
|
1699 |
ultimately have *: "F g d1 = g (cbox a b \<inter> ?lec)" "F g d2 = g (cbox a b \<inter> ?gec)" |
|
1700 |
unfolding interval_split[OF \<open>k \<in> Basis\<close>] |
|
1701 |
apply (rule_tac[!] less.hyps) |
|
1702 |
using division_split[OF \<open>d division_of cbox a b\<close>, where k=k and c=c] \<open>k \<in> Basis\<close> |
|
1703 |
by (simp_all add: interval_split d1_def d2_def division_points_finite[OF \<open>d division_of cbox a b\<close>]) |
|
1704 |
have fxk_le: "g (l \<inter> ?lec) = \<^bold>1" |
|
1705 |
if "l \<in> d" "y \<in> d" "l \<inter> ?lec = y \<inter> ?lec" "l \<noteq> y" for l y |
|
1706 |
proof - |
|
1707 |
obtain u v where leq: "l = cbox u v" |
|
1708 |
using \<open>l \<in> d\<close> less.prems by auto |
|
1709 |
have "interior (cbox u v \<inter> ?lec) = {}" |
|
1710 |
using that division_split_left_inj leq less.prems by blast |
|
1711 |
then show ?thesis |
|
66492
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1712 |
unfolding leq interval_split [OF \<open>k \<in> Basis\<close>] |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1713 |
by (auto intro: box_empty_imp) |
66306 | 1714 |
qed |
1715 |
have fxk_ge: "g (l \<inter> {x. x \<bullet> k \<ge> c}) = \<^bold>1" |
|
1716 |
if "l \<in> d" "y \<in> d" "l \<inter> ?gec = y \<inter> ?gec" "l \<noteq> y" for l y |
|
1717 |
proof - |
|
1718 |
obtain u v where leq: "l = cbox u v" |
|
1719 |
using \<open>l \<in> d\<close> less.prems by auto |
|
1720 |
have "interior (cbox u v \<inter> ?gec) = {}" |
|
1721 |
using that division_split_right_inj leq less.prems by blast |
|
1722 |
then show ?thesis |
|
66492
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1723 |
unfolding leq interval_split[OF \<open>k \<in> Basis\<close>] |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1724 |
by (auto intro: box_empty_imp) |
66306 | 1725 |
qed |
1726 |
have d1_alt: "d1 = (\<lambda>l. l \<inter> ?lec) ` {l \<in> d. l \<inter> ?lec \<noteq> {}}" |
|
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1727 |
using d1_def by auto |
66306 | 1728 |
have d2_alt: "d2 = (\<lambda>l. l \<inter> ?gec) ` {l \<in> d. l \<inter> ?gec \<noteq> {}}" |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1729 |
using d2_def by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1730 |
have "g (cbox a b) = F g d1 \<^bold>* F g d2" (is "_ = ?prev") |
66492
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1731 |
unfolding * using \<open>k \<in> Basis\<close> |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1732 |
by (auto dest: Basis_imp) |
66306 | 1733 |
also have "F g d1 = F (\<lambda>l. g (l \<inter> ?lec)) d" |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1734 |
unfolding d1_alt using division_of_finite[OF less.prems] fxk_le |
66492
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1735 |
by (subst reindex_nontrivial) (auto intro!: mono_neutral_cong_left) |
66306 | 1736 |
also have "F g d2 = F (\<lambda>l. g (l \<inter> ?gec)) d" |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1737 |
unfolding d2_alt using division_of_finite[OF less.prems] fxk_ge |
66492
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1738 |
by (subst reindex_nontrivial) (auto intro!: mono_neutral_cong_left) |
66306 | 1739 |
also have *: "\<forall>x\<in>d. g x = g (x \<inter> ?lec) \<^bold>* g (x \<inter> ?gec)" |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1740 |
unfolding forall_in_division[OF \<open>d division_of cbox a b\<close>] |
66492
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1741 |
using \<open>k \<in> Basis\<close> |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1742 |
by (auto dest: Basis_imp) |
66306 | 1743 |
have "F (\<lambda>l. g (l \<inter> ?lec)) d \<^bold>* F (\<lambda>l. g (l \<inter> ?gec)) d = F g d" |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1744 |
using * by (simp add: distrib) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1745 |
finally show ?thesis by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1746 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1747 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1748 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1749 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1750 |
|
66492
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1751 |
lemma tagged_division: |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1752 |
assumes "d tagged_division_of (cbox a b)" |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1753 |
shows "F (\<lambda>(_, l). g l) d = g (cbox a b)" |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1754 |
proof - |
66492
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1755 |
have "F (\<lambda>(_, k). g k) d = F g (snd ` d)" |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1756 |
using assms box_empty_imp by (rule over_tagged_division_lemma) |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1757 |
then show ?thesis |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1758 |
unfolding assms [THEN division_of_tagged_division, THEN division] . |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1759 |
qed |
66492
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1760 |
|
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1761 |
end |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1762 |
|
66492
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1763 |
locale operative_real = comm_monoid_set + |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1764 |
fixes g :: "real set \<Rightarrow> 'a" |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1765 |
assumes neutral: "b \<le> a \<Longrightarrow> g {a..b} = \<^bold>1" |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1766 |
assumes coalesce_less: "a < c \<Longrightarrow> c < b \<Longrightarrow> g {a..c} \<^bold>* g {c..b} = g {a..b}" |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1767 |
begin |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1768 |
|
66492
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1769 |
sublocale operative where g = g |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1770 |
rewrites "box = (greaterThanLessThan :: real \<Rightarrow> _)" |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1771 |
and "cbox = (atLeastAtMost :: real \<Rightarrow> _)" |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1772 |
and "\<And>x::real. x \<in> Basis \<longleftrightarrow> x = 1" |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1773 |
proof - |
66492
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1774 |
show "operative f z g" |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1775 |
proof |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1776 |
show "g (cbox a b) = \<^bold>1" if "box a b = {}" for a b |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1777 |
using that by (simp add: neutral) |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1778 |
show "g (cbox a b) = g (cbox a b \<inter> {x. x \<bullet> k \<le> c}) \<^bold>* g (cbox a b \<inter> {x. c \<le> x \<bullet> k})" |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1779 |
if "k \<in> Basis" for a b c k |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1780 |
proof - |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1781 |
from that have [simp]: "k = 1" |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1782 |
by simp |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1783 |
from neutral [of 0 1] neutral [of a a for a] coalesce_less |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1784 |
have [simp]: "g {} = \<^bold>1" "\<And>a. g {a} = \<^bold>1" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1785 |
"\<And>a b c. a < c \<Longrightarrow> c < b \<Longrightarrow> g {a..c} \<^bold>* g {c..b} = g {a..b}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1786 |
by auto |
66492
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1787 |
have "g {a..b} = g {a..min b c} \<^bold>* g {max a c..b}" |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1788 |
by (auto simp: min_def max_def le_less) |
66492
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1789 |
then show "g (cbox a b) = g (cbox a b \<inter> {x. x \<bullet> k \<le> c}) \<^bold>* g (cbox a b \<inter> {x. c \<le> x \<bullet> k})" |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1790 |
by (simp add: atMost_def [symmetric] atLeast_def [symmetric]) |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1791 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1792 |
qed |
66492
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1793 |
show "box = (greaterThanLessThan :: real \<Rightarrow> _)" |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1794 |
and "cbox = (atLeastAtMost :: real \<Rightarrow> _)" |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1795 |
and "\<And>x::real. x \<in> Basis \<longleftrightarrow> x = 1" |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1796 |
by (simp_all add: fun_eq_iff) |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1797 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1798 |
|
66492
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1799 |
lemma coalesce_less_eq: |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1800 |
"g {a..c} \<^bold>* g {c..b} = g {a..b}" if "a \<le> c" "c \<le> b" |
66498 | 1801 |
proof (cases "c = a \<or> c = b") |
1802 |
case False |
|
66492
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1803 |
with that have "a < c" "c < b" |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1804 |
by auto |
66498 | 1805 |
then show ?thesis |
66492
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1806 |
by (rule coalesce_less) |
66498 | 1807 |
next |
1808 |
case True |
|
66492
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1809 |
with that box_empty_imp [of a a] box_empty_imp [of b b] show ?thesis |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1810 |
by safe simp_all |
66498 | 1811 |
qed |
66492
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1812 |
|
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1813 |
end |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1814 |
|
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1815 |
lemma operative_realI: |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1816 |
"operative_real f z g" if "operative f z g" |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1817 |
proof - |
66492
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1818 |
interpret operative f z g |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1819 |
using that . |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1820 |
show ?thesis |
66492
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1821 |
proof |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1822 |
show "g {a..b} = z" if "b \<le> a" for a b |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1823 |
using that box_empty_imp by simp |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1824 |
show "f (g {a..c}) (g {c..b}) = g {a..b}" if "a < c" "c < b" for a b c |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1825 |
using that |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1826 |
using Basis_imp [of 1 a b c] |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1827 |
by (simp_all add: atMost_def [symmetric] atLeast_def [symmetric] max_def min_def) |
66498 | 1828 |
qed |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1829 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1830 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1831 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1832 |
subsection \<open>Special case of additivity we need for the FTC.\<close> |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1833 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1834 |
lemma additive_tagged_division_1: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1835 |
fixes f :: "real \<Rightarrow> 'a::real_normed_vector" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1836 |
assumes "a \<le> b" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1837 |
and "p tagged_division_of {a..b}" |
64267 | 1838 |
shows "sum (\<lambda>(x,k). f(Sup k) - f(Inf k)) p = f b - f a" |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1839 |
proof - |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1840 |
let ?f = "(\<lambda>k::(real) set. if k = {} then 0 else f(interval_upperbound k) - f(interval_lowerbound k))" |
66492
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1841 |
interpret operative_real plus 0 ?f |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1842 |
rewrites "comm_monoid_set.F op + 0 = sum" |
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1843 |
by standard[1] (auto simp add: sum_def) |
66112
0e640e04fc56
New theorems; stronger theorems; tidier theorems. Also some renaming
paulson <lp15@cam.ac.uk>
parents:
64911
diff
changeset
|
1844 |
have p_td: "p tagged_division_of cbox a b" |
0e640e04fc56
New theorems; stronger theorems; tidier theorems. Also some renaming
paulson <lp15@cam.ac.uk>
parents:
64911
diff
changeset
|
1845 |
using assms(2) box_real(2) by presburger |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1846 |
have **: "cbox a b \<noteq> {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1847 |
using assms(1) by auto |
66112
0e640e04fc56
New theorems; stronger theorems; tidier theorems. Also some renaming
paulson <lp15@cam.ac.uk>
parents:
64911
diff
changeset
|
1848 |
then have "f b - f a = (\<Sum>(x, l)\<in>p. if l = {} then 0 else f (interval_upperbound l) - f (interval_lowerbound l))" |
0e640e04fc56
New theorems; stronger theorems; tidier theorems. Also some renaming
paulson <lp15@cam.ac.uk>
parents:
64911
diff
changeset
|
1849 |
proof - |
0e640e04fc56
New theorems; stronger theorems; tidier theorems. Also some renaming
paulson <lp15@cam.ac.uk>
parents:
64911
diff
changeset
|
1850 |
have "(if cbox a b = {} then 0 else f (interval_upperbound (cbox a b)) - f (interval_lowerbound (cbox a b))) = f b - f a" |
0e640e04fc56
New theorems; stronger theorems; tidier theorems. Also some renaming
paulson <lp15@cam.ac.uk>
parents:
64911
diff
changeset
|
1851 |
using assms by auto |
0e640e04fc56
New theorems; stronger theorems; tidier theorems. Also some renaming
paulson <lp15@cam.ac.uk>
parents:
64911
diff
changeset
|
1852 |
then show ?thesis |
66492
d7206afe2d28
dedicated local for "operative" avoids namespace pollution
haftmann
parents:
66365
diff
changeset
|
1853 |
using p_td assms by (simp add: tagged_division) |
66112
0e640e04fc56
New theorems; stronger theorems; tidier theorems. Also some renaming
paulson <lp15@cam.ac.uk>
parents:
64911
diff
changeset
|
1854 |
qed |
0e640e04fc56
New theorems; stronger theorems; tidier theorems. Also some renaming
paulson <lp15@cam.ac.uk>
parents:
64911
diff
changeset
|
1855 |
then show ?thesis |
0e640e04fc56
New theorems; stronger theorems; tidier theorems. Also some renaming
paulson <lp15@cam.ac.uk>
parents:
64911
diff
changeset
|
1856 |
using assms by (auto intro!: sum.cong) |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1857 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1858 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1859 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1860 |
subsection \<open>Fine-ness of a partition w.r.t. a gauge.\<close> |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1861 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1862 |
definition fine (infixr "fine" 46) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1863 |
where "d fine s \<longleftrightarrow> (\<forall>(x,k) \<in> s. k \<subseteq> d x)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1864 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1865 |
lemma fineI: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1866 |
assumes "\<And>x k. (x, k) \<in> s \<Longrightarrow> k \<subseteq> d x" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1867 |
shows "d fine s" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1868 |
using assms unfolding fine_def by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1869 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1870 |
lemma fineD[dest]: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1871 |
assumes "d fine s" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1872 |
shows "\<And>x k. (x,k) \<in> s \<Longrightarrow> k \<subseteq> d x" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1873 |
using assms unfolding fine_def by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1874 |
|
66192
e5b84854baa4
A few renamings and several tidied-up proofs
paulson <lp15@cam.ac.uk>
parents:
66164
diff
changeset
|
1875 |
lemma fine_Int: "(\<lambda>x. d1 x \<inter> d2 x) fine p \<longleftrightarrow> d1 fine p \<and> d2 fine p" |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1876 |
unfolding fine_def by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1877 |
|
66192
e5b84854baa4
A few renamings and several tidied-up proofs
paulson <lp15@cam.ac.uk>
parents:
66164
diff
changeset
|
1878 |
lemma fine_Inter: |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1879 |
"(\<lambda>x. \<Inter>{f d x | d. d \<in> s}) fine p \<longleftrightarrow> (\<forall>d\<in>s. (f d) fine p)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1880 |
unfolding fine_def by blast |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1881 |
|
66192
e5b84854baa4
A few renamings and several tidied-up proofs
paulson <lp15@cam.ac.uk>
parents:
66164
diff
changeset
|
1882 |
lemma fine_Un: "d fine p1 \<Longrightarrow> d fine p2 \<Longrightarrow> d fine (p1 \<union> p2)" |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1883 |
unfolding fine_def by blast |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1884 |
|
66192
e5b84854baa4
A few renamings and several tidied-up proofs
paulson <lp15@cam.ac.uk>
parents:
66164
diff
changeset
|
1885 |
lemma fine_Union: "(\<And>p. p \<in> ps \<Longrightarrow> d fine p) \<Longrightarrow> d fine (\<Union>ps)" |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1886 |
unfolding fine_def by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1887 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1888 |
lemma fine_subset: "p \<subseteq> q \<Longrightarrow> d fine q \<Longrightarrow> d fine p" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1889 |
unfolding fine_def by blast |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1890 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1891 |
subsection \<open>Some basic combining lemmas.\<close> |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1892 |
|
66192
e5b84854baa4
A few renamings and several tidied-up proofs
paulson <lp15@cam.ac.uk>
parents:
66164
diff
changeset
|
1893 |
lemma tagged_division_Union_exists: |
66318 | 1894 |
assumes "finite I" |
1895 |
and "\<forall>i\<in>I. \<exists>p. p tagged_division_of i \<and> d fine p" |
|
1896 |
and "\<forall>i1\<in>I. \<forall>i2\<in>I. i1 \<noteq> i2 \<longrightarrow> interior i1 \<inter> interior i2 = {}" |
|
1897 |
and "\<Union>I = i" |
|
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1898 |
obtains p where "p tagged_division_of i" and "d fine p" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1899 |
proof - |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1900 |
obtain pfn where pfn: |
66318 | 1901 |
"\<And>x. x \<in> I \<Longrightarrow> pfn x tagged_division_of x" |
1902 |
"\<And>x. x \<in> I \<Longrightarrow> d fine pfn x" |
|
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1903 |
using bchoice[OF assms(2)] by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1904 |
show thesis |
66318 | 1905 |
apply (rule_tac p="\<Union>(pfn ` I)" in that) |
66497
18a6478a574c
More tidying, and renaming of theorems
paulson <lp15@cam.ac.uk>
parents:
66365
diff
changeset
|
1906 |
using assms(1) assms(3) assms(4) pfn(1) tagged_division_Union apply force |
66192
e5b84854baa4
A few renamings and several tidied-up proofs
paulson <lp15@cam.ac.uk>
parents:
66164
diff
changeset
|
1907 |
by (metis (mono_tags, lifting) fine_Union imageE pfn(2)) |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1908 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1909 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1910 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1911 |
subsection \<open>The set we're concerned with must be closed.\<close> |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1912 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1913 |
lemma division_of_closed: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1914 |
fixes i :: "'n::euclidean_space set" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1915 |
shows "s division_of i \<Longrightarrow> closed i" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1916 |
unfolding division_of_def by fastforce |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1917 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1918 |
subsection \<open>General bisection principle for intervals; might be useful elsewhere.\<close> |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1919 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1920 |
lemma interval_bisection_step: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1921 |
fixes type :: "'a::euclidean_space" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1922 |
assumes "P {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1923 |
and "\<forall>s t. P s \<and> P t \<and> interior(s) \<inter> interior(t) = {} \<longrightarrow> P (s \<union> t)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1924 |
and "\<not> P (cbox a (b::'a))" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1925 |
obtains c d where "\<not> P (cbox c d)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1926 |
and "\<forall>i\<in>Basis. a\<bullet>i \<le> c\<bullet>i \<and> c\<bullet>i \<le> d\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i \<and> 2 * (d\<bullet>i - c\<bullet>i) \<le> b\<bullet>i - a\<bullet>i" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1927 |
proof - |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1928 |
have "cbox a b \<noteq> {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1929 |
using assms(1,3) by metis |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1930 |
then have ab: "\<And>i. i\<in>Basis \<Longrightarrow> a \<bullet> i \<le> b \<bullet> i" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1931 |
by (force simp: mem_box) |
66318 | 1932 |
have UN_cases: "\<lbrakk>finite f; |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1933 |
\<And>s. s\<in>f \<Longrightarrow> P s; |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1934 |
\<And>s. s\<in>f \<Longrightarrow> \<exists>a b. s = cbox a b; |
66318 | 1935 |
\<And>s t. s\<in>f \<Longrightarrow> t\<in>f \<Longrightarrow> s \<noteq> t \<Longrightarrow> interior s \<inter> interior t = {}\<rbrakk> \<Longrightarrow> P (\<Union>f)" for f |
1936 |
proof (induct f rule: finite_induct) |
|
1937 |
case empty |
|
1938 |
show ?case |
|
1939 |
using assms(1) by auto |
|
1940 |
next |
|
1941 |
case (insert x f) |
|
1942 |
show ?case |
|
1943 |
unfolding Union_insert |
|
1944 |
apply (rule assms(2)[rule_format]) |
|
1945 |
using Int_interior_Union_intervals [of f "interior x"] |
|
1946 |
by (metis (no_types, lifting) insert insert_iff open_interior) |
|
1947 |
qed |
|
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1948 |
let ?A = "{cbox c d | c d::'a. \<forall>i\<in>Basis. (c\<bullet>i = a\<bullet>i) \<and> (d\<bullet>i = (a\<bullet>i + b\<bullet>i) / 2) \<or> |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1949 |
(c\<bullet>i = (a\<bullet>i + b\<bullet>i) / 2) \<and> (d\<bullet>i = b\<bullet>i)}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1950 |
let ?PP = "\<lambda>c d. \<forall>i\<in>Basis. a\<bullet>i \<le> c\<bullet>i \<and> c\<bullet>i \<le> d\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i \<and> 2 * (d\<bullet>i - c\<bullet>i) \<le> b\<bullet>i - a\<bullet>i" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1951 |
{ |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1952 |
presume "\<forall>c d. ?PP c d \<longrightarrow> P (cbox c d) \<Longrightarrow> False" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1953 |
then show thesis |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1954 |
unfolding atomize_not not_all |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1955 |
by (blast intro: that) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1956 |
} |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1957 |
assume as: "\<forall>c d. ?PP c d \<longrightarrow> P (cbox c d)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1958 |
have "P (\<Union>?A)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1959 |
proof (rule UN_cases) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1960 |
let ?B = "(\<lambda>s. cbox (\<Sum>i\<in>Basis. (if i \<in> s then a\<bullet>i else (a\<bullet>i + b\<bullet>i) / 2) *\<^sub>R i::'a) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1961 |
(\<Sum>i\<in>Basis. (if i \<in> s then (a\<bullet>i + b\<bullet>i) / 2 else b\<bullet>i) *\<^sub>R i)) ` {s. s \<subseteq> Basis}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1962 |
have "?A \<subseteq> ?B" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1963 |
proof |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1964 |
fix x |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1965 |
assume "x \<in> ?A" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1966 |
then obtain c d |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1967 |
where x: "x = cbox c d" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1968 |
"\<And>i. i \<in> Basis \<Longrightarrow> |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1969 |
c \<bullet> i = a \<bullet> i \<and> d \<bullet> i = (a \<bullet> i + b \<bullet> i) / 2 \<or> |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1970 |
c \<bullet> i = (a \<bullet> i + b \<bullet> i) / 2 \<and> d \<bullet> i = b \<bullet> i" by blast |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1971 |
show "x \<in> ?B" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1972 |
unfolding image_iff x |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1973 |
apply (rule_tac x="{i. i\<in>Basis \<and> c\<bullet>i = a\<bullet>i}" in bexI) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1974 |
apply (rule arg_cong2 [where f = cbox]) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1975 |
using x(2) ab |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1976 |
apply (auto simp add: euclidean_eq_iff[where 'a='a]) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1977 |
by fastforce |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1978 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1979 |
then show "finite ?A" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1980 |
by (rule finite_subset) auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1981 |
next |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1982 |
fix s |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1983 |
assume "s \<in> ?A" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1984 |
then obtain c d |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1985 |
where s: "s = cbox c d" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1986 |
"\<And>i. i \<in> Basis \<Longrightarrow> |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1987 |
c \<bullet> i = a \<bullet> i \<and> d \<bullet> i = (a \<bullet> i + b \<bullet> i) / 2 \<or> |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1988 |
c \<bullet> i = (a \<bullet> i + b \<bullet> i) / 2 \<and> d \<bullet> i = b \<bullet> i" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1989 |
by blast |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1990 |
show "P s" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1991 |
unfolding s |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1992 |
apply (rule as[rule_format]) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1993 |
using ab s(2) by force |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1994 |
show "\<exists>a b. s = cbox a b" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1995 |
unfolding s by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1996 |
fix t |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1997 |
assume "t \<in> ?A" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1998 |
then obtain e f where t: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1999 |
"t = cbox e f" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2000 |
"\<And>i. i \<in> Basis \<Longrightarrow> |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2001 |
e \<bullet> i = a \<bullet> i \<and> f \<bullet> i = (a \<bullet> i + b \<bullet> i) / 2 \<or> |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2002 |
e \<bullet> i = (a \<bullet> i + b \<bullet> i) / 2 \<and> f \<bullet> i = b \<bullet> i" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2003 |
by blast |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2004 |
assume "s \<noteq> t" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2005 |
then have "\<not> (c = e \<and> d = f)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2006 |
unfolding s t by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2007 |
then obtain i where "c\<bullet>i \<noteq> e\<bullet>i \<or> d\<bullet>i \<noteq> f\<bullet>i" and i': "i \<in> Basis" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2008 |
unfolding euclidean_eq_iff[where 'a='a] by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2009 |
then have i: "c\<bullet>i \<noteq> e\<bullet>i" "d\<bullet>i \<noteq> f\<bullet>i" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2010 |
using s(2) t(2) apply fastforce |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2011 |
using t(2)[OF i'] \<open>c \<bullet> i \<noteq> e \<bullet> i \<or> d \<bullet> i \<noteq> f \<bullet> i\<close> i' s(2) t(2) by fastforce |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2012 |
have *: "\<And>s t. (\<And>a. a \<in> s \<Longrightarrow> a \<in> t \<Longrightarrow> False) \<Longrightarrow> s \<inter> t = {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2013 |
by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2014 |
show "interior s \<inter> interior t = {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2015 |
unfolding s t interior_cbox |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2016 |
proof (rule *) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2017 |
fix x |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2018 |
assume "x \<in> box c d" "x \<in> box e f" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2019 |
then have x: "c\<bullet>i < d\<bullet>i" "e\<bullet>i < f\<bullet>i" "c\<bullet>i < f\<bullet>i" "e\<bullet>i < d\<bullet>i" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2020 |
unfolding mem_box using i' |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2021 |
by force+ |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2022 |
show False using s(2)[OF i'] |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2023 |
proof safe |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2024 |
assume as: "c \<bullet> i = a \<bullet> i" "d \<bullet> i = (a \<bullet> i + b \<bullet> i) / 2" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2025 |
show False |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2026 |
using t(2)[OF i'] and i x unfolding as by (fastforce simp add:field_simps) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2027 |
next |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2028 |
assume as: "c \<bullet> i = (a \<bullet> i + b \<bullet> i) / 2" "d \<bullet> i = b \<bullet> i" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2029 |
show False |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2030 |
using t(2)[OF i'] and i x unfolding as by(fastforce simp add:field_simps) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2031 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2032 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2033 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2034 |
also have "\<Union>?A = cbox a b" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2035 |
proof (rule set_eqI,rule) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2036 |
fix x |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2037 |
assume "x \<in> \<Union>?A" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2038 |
then obtain c d where x: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2039 |
"x \<in> cbox c d" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2040 |
"\<And>i. i \<in> Basis \<Longrightarrow> |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2041 |
c \<bullet> i = a \<bullet> i \<and> d \<bullet> i = (a \<bullet> i + b \<bullet> i) / 2 \<or> |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2042 |
c \<bullet> i = (a \<bullet> i + b \<bullet> i) / 2 \<and> d \<bullet> i = b \<bullet> i" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2043 |
by blast |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2044 |
show "x\<in>cbox a b" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2045 |
unfolding mem_box |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2046 |
proof safe |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2047 |
fix i :: 'a |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2048 |
assume i: "i \<in> Basis" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2049 |
then show "a \<bullet> i \<le> x \<bullet> i" "x \<bullet> i \<le> b \<bullet> i" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2050 |
using x(2)[OF i] x(1)[unfolded mem_box,THEN bspec, OF i] by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2051 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2052 |
next |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2053 |
fix x |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2054 |
assume x: "x \<in> cbox a b" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2055 |
have "\<forall>i\<in>Basis. |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2056 |
\<exists>c d. (c = a\<bullet>i \<and> d = (a\<bullet>i + b\<bullet>i) / 2 \<or> c = (a\<bullet>i + b\<bullet>i) / 2 \<and> d = b\<bullet>i) \<and> c\<le>x\<bullet>i \<and> x\<bullet>i \<le> d" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2057 |
(is "\<forall>i\<in>Basis. \<exists>c d. ?P i c d") |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2058 |
unfolding mem_box |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2059 |
proof |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2060 |
fix i :: 'a |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2061 |
assume i: "i \<in> Basis" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2062 |
have "?P i (a\<bullet>i) ((a \<bullet> i + b \<bullet> i) / 2) \<or> ?P i ((a \<bullet> i + b \<bullet> i) / 2) (b\<bullet>i)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2063 |
using x[unfolded mem_box,THEN bspec, OF i] by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2064 |
then show "\<exists>c d. ?P i c d" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2065 |
by blast |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2066 |
qed |
66318 | 2067 |
then obtain \<alpha> \<beta> where |
2068 |
"\<forall>i\<in>Basis. (\<alpha> \<bullet> i = a \<bullet> i \<and> \<beta> \<bullet> i = (a \<bullet> i + b \<bullet> i) / 2 \<or> |
|
2069 |
\<alpha> \<bullet> i = (a \<bullet> i + b \<bullet> i) / 2 \<and> \<beta> \<bullet> i = b \<bullet> i) \<and> \<alpha> \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> \<beta> \<bullet> i" |
|
2070 |
by (auto simp: choice_Basis_iff) |
|
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2071 |
then show "x\<in>\<Union>?A" |
66318 | 2072 |
by (force simp add: mem_box) |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2073 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2074 |
finally show False |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2075 |
using assms by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2076 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2077 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2078 |
lemma interval_bisection: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2079 |
fixes type :: "'a::euclidean_space" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2080 |
assumes "P {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2081 |
and "(\<forall>s t. P s \<and> P t \<and> interior(s) \<inter> interior(t) = {} \<longrightarrow> P(s \<union> t))" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2082 |
and "\<not> P (cbox a (b::'a))" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2083 |
obtains x where "x \<in> cbox a b" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2084 |
and "\<forall>e>0. \<exists>c d. x \<in> cbox c d \<and> cbox c d \<subseteq> ball x e \<and> cbox c d \<subseteq> cbox a b \<and> \<not> P (cbox c d)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2085 |
proof - |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2086 |
have "\<forall>x. \<exists>y. \<not> P (cbox (fst x) (snd x)) \<longrightarrow> (\<not> P (cbox (fst y) (snd y)) \<and> |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2087 |
(\<forall>i\<in>Basis. fst x\<bullet>i \<le> fst y\<bullet>i \<and> fst y\<bullet>i \<le> snd y\<bullet>i \<and> snd y\<bullet>i \<le> snd x\<bullet>i \<and> |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2088 |
2 * (snd y\<bullet>i - fst y\<bullet>i) \<le> snd x\<bullet>i - fst x\<bullet>i))" (is "\<forall>x. ?P x") |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2089 |
proof |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2090 |
show "?P x" for x |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2091 |
proof (cases "P (cbox (fst x) (snd x))") |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2092 |
case True |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2093 |
then show ?thesis by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2094 |
next |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2095 |
case as: False |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2096 |
obtain c d where "\<not> P (cbox c d)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2097 |
"\<forall>i\<in>Basis. |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2098 |
fst x \<bullet> i \<le> c \<bullet> i \<and> |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2099 |
c \<bullet> i \<le> d \<bullet> i \<and> |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2100 |
d \<bullet> i \<le> snd x \<bullet> i \<and> |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2101 |
2 * (d \<bullet> i - c \<bullet> i) \<le> snd x \<bullet> i - fst x \<bullet> i" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2102 |
by (rule interval_bisection_step[of P, OF assms(1-2) as]) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2103 |
then show ?thesis |
66318 | 2104 |
by (rule_tac x="(c,d)" in exI) auto |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2105 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2106 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2107 |
then obtain f where f: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2108 |
"\<forall>x. |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2109 |
\<not> P (cbox (fst x) (snd x)) \<longrightarrow> |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2110 |
\<not> P (cbox (fst (f x)) (snd (f x))) \<and> |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2111 |
(\<forall>i\<in>Basis. |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2112 |
fst x \<bullet> i \<le> fst (f x) \<bullet> i \<and> |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2113 |
fst (f x) \<bullet> i \<le> snd (f x) \<bullet> i \<and> |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2114 |
snd (f x) \<bullet> i \<le> snd x \<bullet> i \<and> |
66318 | 2115 |
2 * (snd (f x) \<bullet> i - fst (f x) \<bullet> i) \<le> snd x \<bullet> i - fst x \<bullet> i)" by metis |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2116 |
define AB A B where ab_def: "AB n = (f ^^ n) (a,b)" "A n = fst(AB n)" "B n = snd(AB n)" for n |
66193 | 2117 |
have [simp]: "A 0 = a" "B 0 = b" and ABRAW: "\<And>n. \<not> P (cbox (A(Suc n)) (B(Suc n))) \<and> |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2118 |
(\<forall>i\<in>Basis. A(n)\<bullet>i \<le> A(Suc n)\<bullet>i \<and> A(Suc n)\<bullet>i \<le> B(Suc n)\<bullet>i \<and> B(Suc n)\<bullet>i \<le> B(n)\<bullet>i \<and> |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2119 |
2 * (B(Suc n)\<bullet>i - A(Suc n)\<bullet>i) \<le> B(n)\<bullet>i - A(n)\<bullet>i)" (is "\<And>n. ?P n") |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2120 |
proof - |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2121 |
show "A 0 = a" "B 0 = b" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2122 |
unfolding ab_def by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2123 |
note S = ab_def funpow.simps o_def id_apply |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2124 |
show "?P n" for n |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2125 |
proof (induct n) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2126 |
case 0 |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2127 |
then show ?case |
66318 | 2128 |
unfolding S using \<open>\<not> P (cbox a b)\<close> f by auto |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2129 |
next |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2130 |
case (Suc n) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2131 |
show ?case |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2132 |
unfolding S |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2133 |
apply (rule f[rule_format]) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2134 |
using Suc |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2135 |
unfolding S |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2136 |
apply auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2137 |
done |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2138 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2139 |
qed |
66193 | 2140 |
then have AB: "A(n)\<bullet>i \<le> A(Suc n)\<bullet>i" "A(Suc n)\<bullet>i \<le> B(Suc n)\<bullet>i" |
2141 |
"B(Suc n)\<bullet>i \<le> B(n)\<bullet>i" "2 * (B(Suc n)\<bullet>i - A(Suc n)\<bullet>i) \<le> B(n)\<bullet>i - A(n)\<bullet>i" |
|
2142 |
if "i\<in>Basis" for i n |
|
2143 |
using that by blast+ |
|
2144 |
have notPAB: "\<not> P (cbox (A(Suc n)) (B(Suc n)))" for n |
|
2145 |
using ABRAW by blast |
|
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2146 |
have interv: "\<exists>n. \<forall>x\<in>cbox (A n) (B n). \<forall>y\<in>cbox (A n) (B n). dist x y < e" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2147 |
if e: "0 < e" for e |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2148 |
proof - |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2149 |
obtain n where n: "(\<Sum>i\<in>Basis. b \<bullet> i - a \<bullet> i) / e < 2 ^ n" |
64267 | 2150 |
using real_arch_pow[of 2 "(sum (\<lambda>i. b\<bullet>i - a\<bullet>i) Basis) / e"] by auto |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2151 |
show ?thesis |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2152 |
proof (rule exI [where x=n], clarify) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2153 |
fix x y |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2154 |
assume xy: "x\<in>cbox (A n) (B n)" "y\<in>cbox (A n) (B n)" |
64267 | 2155 |
have "dist x y \<le> sum (\<lambda>i. \<bar>(x - y)\<bullet>i\<bar>) Basis" |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2156 |
unfolding dist_norm by(rule norm_le_l1) |
64267 | 2157 |
also have "\<dots> \<le> sum (\<lambda>i. B n\<bullet>i - A n\<bullet>i) Basis" |
2158 |
proof (rule sum_mono) |
|
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2159 |
fix i :: 'a |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2160 |
assume i: "i \<in> Basis" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2161 |
show "\<bar>(x - y) \<bullet> i\<bar> \<le> B n \<bullet> i - A n \<bullet> i" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2162 |
using xy[unfolded mem_box,THEN bspec, OF i] |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2163 |
by (auto simp: inner_diff_left) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2164 |
qed |
64267 | 2165 |
also have "\<dots> \<le> sum (\<lambda>i. b\<bullet>i - a\<bullet>i) Basis / 2^n" |
2166 |
unfolding sum_divide_distrib |
|
2167 |
proof (rule sum_mono) |
|
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2168 |
show "B n \<bullet> i - A n \<bullet> i \<le> (b \<bullet> i - a \<bullet> i) / 2 ^ n" if i: "i \<in> Basis" for i |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2169 |
proof (induct n) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2170 |
case 0 |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2171 |
then show ?case |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2172 |
unfolding AB by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2173 |
next |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2174 |
case (Suc n) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2175 |
have "B (Suc n) \<bullet> i - A (Suc n) \<bullet> i \<le> (B n \<bullet> i - A n \<bullet> i) / 2" |
66318 | 2176 |
using AB(3) that AB(4)[of i n] using i by auto |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2177 |
also have "\<dots> \<le> (b \<bullet> i - a \<bullet> i) / 2 ^ Suc n" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2178 |
using Suc by (auto simp add: field_simps) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2179 |
finally show ?case . |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2180 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2181 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2182 |
also have "\<dots> < e" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2183 |
using n using e by (auto simp add: field_simps) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2184 |
finally show "dist x y < e" . |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2185 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2186 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2187 |
{ |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2188 |
fix n m :: nat |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2189 |
assume "m \<le> n" then have "cbox (A n) (B n) \<subseteq> cbox (A m) (B m)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2190 |
proof (induction rule: inc_induct) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2191 |
case (step i) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2192 |
show ?case |
66193 | 2193 |
using AB by (intro order_trans[OF step.IH] subset_box_imp) auto |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2194 |
qed simp |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2195 |
} note ABsubset = this |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2196 |
have "\<exists>a. \<forall>n. a\<in> cbox (A n) (B n)" |
66193 | 2197 |
proof (rule decreasing_closed_nest) |
2198 |
show "\<forall>n. closed (cbox (A n) (B n))" |
|
2199 |
by (simp add: closed_cbox) |
|
2200 |
show "\<forall>n. cbox (A n) (B n) \<noteq> {}" |
|
2201 |
by (meson AB dual_order.trans interval_not_empty) |
|
2202 |
qed (use ABsubset interv in auto) |
|
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2203 |
then obtain x0 where x0: "\<And>n. x0 \<in> cbox (A n) (B n)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2204 |
by blast |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2205 |
show thesis |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2206 |
proof (rule that[rule_format, of x0]) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2207 |
show "x0\<in>cbox a b" |
66193 | 2208 |
using \<open>A 0 = a\<close> \<open>B 0 = b\<close> x0 by blast |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2209 |
fix e :: real |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2210 |
assume "e > 0" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2211 |
from interv[OF this] obtain n |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2212 |
where n: "\<forall>x\<in>cbox (A n) (B n). \<forall>y\<in>cbox (A n) (B n). dist x y < e" .. |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2213 |
have "\<not> P (cbox (A n) (B n))" |
66193 | 2214 |
proof (cases "0 < n") |
2215 |
case True then show ?thesis |
|
2216 |
by (metis Suc_pred' notPAB) |
|
2217 |
next |
|
2218 |
case False then show ?thesis |
|
2219 |
using \<open>A 0 = a\<close> \<open>B 0 = b\<close> \<open>\<not> P (cbox a b)\<close> by blast |
|
2220 |
qed |
|
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2221 |
moreover have "cbox (A n) (B n) \<subseteq> ball x0 e" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2222 |
using n using x0[of n] by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2223 |
moreover have "cbox (A n) (B n) \<subseteq> cbox a b" |
66193 | 2224 |
using ABsubset \<open>A 0 = a\<close> \<open>B 0 = b\<close> by blast |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2225 |
ultimately show "\<exists>c d. x0 \<in> cbox c d \<and> cbox c d \<subseteq> ball x0 e \<and> cbox c d \<subseteq> cbox a b \<and> \<not> P (cbox c d)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2226 |
apply (rule_tac x="A n" in exI) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2227 |
apply (rule_tac x="B n" in exI) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2228 |
apply (auto simp: x0) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2229 |
done |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2230 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2231 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2232 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2233 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2234 |
subsection \<open>Cousin's lemma.\<close> |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2235 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2236 |
lemma fine_division_exists: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2237 |
fixes a b :: "'a::euclidean_space" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2238 |
assumes "gauge g" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2239 |
obtains p where "p tagged_division_of (cbox a b)" "g fine p" |
66318 | 2240 |
proof (cases "\<exists>p. p tagged_division_of (cbox a b) \<and> g fine p") |
2241 |
case True |
|
2242 |
then show ?thesis |
|
2243 |
using that by auto |
|
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2244 |
next |
66318 | 2245 |
case False |
2246 |
assume "\<not> (\<exists>p. p tagged_division_of (cbox a b) \<and> g fine p)" |
|
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2247 |
obtain x where x: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2248 |
"x \<in> (cbox a b)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2249 |
"\<And>e. 0 < e \<Longrightarrow> |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2250 |
\<exists>c d. |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2251 |
x \<in> cbox c d \<and> |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2252 |
cbox c d \<subseteq> ball x e \<and> |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2253 |
cbox c d \<subseteq> (cbox a b) \<and> |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2254 |
\<not> (\<exists>p. p tagged_division_of cbox c d \<and> g fine p)" |
66318 | 2255 |
apply (rule interval_bisection[of "\<lambda>s. \<exists>p. p tagged_division_of s \<and> g fine p", OF _ _ False]) |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2256 |
apply (simp add: fine_def) |
66497
18a6478a574c
More tidying, and renaming of theorems
paulson <lp15@cam.ac.uk>
parents:
66365
diff
changeset
|
2257 |
apply (metis tagged_division_Un fine_Un) |
66318 | 2258 |
apply auto |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2259 |
done |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2260 |
obtain e where e: "e > 0" "ball x e \<subseteq> g x" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2261 |
using gaugeD[OF assms, of x] unfolding open_contains_ball by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2262 |
from x(2)[OF e(1)] |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2263 |
obtain c d where c_d: "x \<in> cbox c d" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2264 |
"cbox c d \<subseteq> ball x e" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2265 |
"cbox c d \<subseteq> cbox a b" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2266 |
"\<not> (\<exists>p. p tagged_division_of cbox c d \<and> g fine p)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2267 |
by blast |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2268 |
have "g fine {(x, cbox c d)}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2269 |
unfolding fine_def using e using c_d(2) by auto |
66318 | 2270 |
then show ?thesis |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2271 |
using tagged_division_of_self[OF c_d(1)] using c_d by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2272 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2273 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2274 |
lemma fine_division_exists_real: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2275 |
fixes a b :: real |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2276 |
assumes "gauge g" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2277 |
obtains p where "p tagged_division_of {a .. b}" "g fine p" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2278 |
by (metis assms box_real(2) fine_division_exists) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2279 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2280 |
subsection \<open>A technical lemma about "refinement" of division.\<close> |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2281 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2282 |
lemma tagged_division_finer: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2283 |
fixes p :: "('a::euclidean_space \<times> ('a::euclidean_space set)) set" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2284 |
assumes "p tagged_division_of (cbox a b)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2285 |
and "gauge d" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2286 |
obtains q where "q tagged_division_of (cbox a b)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2287 |
and "d fine q" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2288 |
and "\<forall>(x,k) \<in> p. k \<subseteq> d(x) \<longrightarrow> (x,k) \<in> q" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2289 |
proof - |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2290 |
let ?P = "\<lambda>p. p tagged_partial_division_of (cbox a b) \<longrightarrow> gauge d \<longrightarrow> |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2291 |
(\<exists>q. q tagged_division_of (\<Union>{k. \<exists>x. (x,k) \<in> p}) \<and> d fine q \<and> |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2292 |
(\<forall>(x,k) \<in> p. k \<subseteq> d(x) \<longrightarrow> (x,k) \<in> q))" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2293 |
{ |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2294 |
have *: "finite p" "p tagged_partial_division_of (cbox a b)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2295 |
using assms(1) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2296 |
unfolding tagged_division_of_def |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2297 |
by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2298 |
presume "\<And>p. finite p \<Longrightarrow> ?P p" |
66300 | 2299 |
from this[rule_format,OF * assms(2)] |
2300 |
obtain q where q: "q tagged_division_of \<Union>{k. \<exists>x. (x, k) \<in> p}" "d fine q" "(\<forall>(x, k)\<in>p. k \<subseteq> d x \<longrightarrow> (x, k) \<in> q)" |
|
2301 |
by auto |
|
2302 |
with that[of q] show ?thesis |
|
2303 |
using assms(1) by auto |
|
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2304 |
} |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2305 |
fix p :: "('a::euclidean_space \<times> ('a::euclidean_space set)) set" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2306 |
assume as: "finite p" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2307 |
show "?P p" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2308 |
apply rule |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2309 |
apply rule |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2310 |
using as |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2311 |
proof (induct p) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2312 |
case empty |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2313 |
show ?case |
66318 | 2314 |
by (force simp add: fine_def) |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2315 |
next |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2316 |
case (insert xk p) |
66314 | 2317 |
obtain x k where xk: "xk = (x, k)" |
2318 |
using surj_pair[of xk] by metis |
|
66193 | 2319 |
obtain q1 where q1: "q1 tagged_division_of \<Union>{k. \<exists>x. (x, k) \<in> p}" |
2320 |
and "d fine q1" |
|
2321 |
and q1I: "\<And>x k. \<lbrakk>(x, k)\<in>p; k \<subseteq> d x\<rbrakk> \<Longrightarrow> (x, k) \<in> q1" |
|
66314 | 2322 |
using case_prodD tagged_partial_division_subset[OF insert(4) subset_insertI] |
2323 |
by (metis (mono_tags, lifting) insert.hyps(3) insert.prems(2)) |
|
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2324 |
have *: "\<Union>{l. \<exists>y. (y,l) \<in> insert xk p} = k \<union> \<Union>{l. \<exists>y. (y,l) \<in> p}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2325 |
unfolding xk by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2326 |
note p = tagged_partial_division_ofD[OF insert(4)] |
66314 | 2327 |
obtain u v where uv: "k = cbox u v" |
2328 |
using p(4)[unfolded xk, OF insertI1] by blast |
|
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2329 |
have "finite {k. \<exists>x. (x, k) \<in> p}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2330 |
apply (rule finite_subset[of _ "snd ` p"]) |
66314 | 2331 |
using image_iff apply fastforce |
2332 |
using insert.hyps(1) by blast |
|
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2333 |
then have int: "interior (cbox u v) \<inter> interior (\<Union>{k. \<exists>x. (x, k) \<in> p}) = {}" |
66314 | 2334 |
proof (rule Int_interior_Union_intervals) |
2335 |
show "open (interior (cbox u v))" |
|
2336 |
by auto |
|
2337 |
show "\<And>T. T \<in> {k. \<exists>x. (x, k) \<in> p} \<Longrightarrow> \<exists>a b. T = cbox a b" |
|
2338 |
using p(4) by auto |
|
2339 |
show "\<And>T. T \<in> {k. \<exists>x. (x, k) \<in> p} \<Longrightarrow> interior (cbox u v) \<inter> interior T = {}" |
|
2340 |
by clarify (metis insert.hyps(2) insert_iff interior_cbox p(5) uv xk) |
|
2341 |
qed |
|
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2342 |
show ?case |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2343 |
proof (cases "cbox u v \<subseteq> d x") |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2344 |
case True |
66314 | 2345 |
have "{(x, cbox u v)} tagged_division_of cbox u v" |
2346 |
by (simp add: p(2) uv xk tagged_division_of_self) |
|
2347 |
then have "{(x, cbox u v)} \<union> q1 tagged_division_of \<Union>{k. \<exists>x. (x, k) \<in> insert xk p}" |
|
66497
18a6478a574c
More tidying, and renaming of theorems
paulson <lp15@cam.ac.uk>
parents:
66365
diff
changeset
|
2348 |
unfolding * uv by (metis (no_types, lifting) int q1 tagged_division_Un) |
66314 | 2349 |
with True show ?thesis |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2350 |
apply (rule_tac x="{(x,cbox u v)} \<union> q1" in exI) |
66314 | 2351 |
using \<open>d fine q1\<close> fine_def q1I uv xk apply fastforce |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2352 |
done |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2353 |
next |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2354 |
case False |
66314 | 2355 |
obtain q2 where q2: "q2 tagged_division_of cbox u v" "d fine q2" |
2356 |
using fine_division_exists[OF assms(2)] by blast |
|
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2357 |
show ?thesis |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2358 |
apply (rule_tac x="q2 \<union> q1" in exI) |
66314 | 2359 |
apply (intro conjI) |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2360 |
unfolding * uv |
66497
18a6478a574c
More tidying, and renaming of theorems
paulson <lp15@cam.ac.uk>
parents:
66365
diff
changeset
|
2361 |
apply (rule tagged_division_Un q2 q1 int fine_Un)+ |
66193 | 2362 |
apply (auto intro: q1 q2 fine_Un \<open>d fine q1\<close> simp add: False q1I uv xk) |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2363 |
done |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2364 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2365 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2366 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2367 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2368 |
subsubsection \<open>Covering lemma\<close> |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2369 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2370 |
text\<open> Some technical lemmas used in the approximation results that follow. Proof of the covering |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2371 |
lemma is an obvious multidimensional generalization of Lemma 3, p65 of Swartz's |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2372 |
"Introduction to Gauge Integrals". \<close> |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2373 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2374 |
proposition covering_lemma: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2375 |
assumes "S \<subseteq> cbox a b" "box a b \<noteq> {}" "gauge g" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2376 |
obtains \<D> where |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2377 |
"countable \<D>" "\<Union>\<D> \<subseteq> cbox a b" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2378 |
"\<And>K. K \<in> \<D> \<Longrightarrow> interior K \<noteq> {} \<and> (\<exists>c d. K = cbox c d)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2379 |
"pairwise (\<lambda>A B. interior A \<inter> interior B = {}) \<D>" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2380 |
"\<And>K. K \<in> \<D> \<Longrightarrow> \<exists>x \<in> S \<inter> K. K \<subseteq> g x" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2381 |
"\<And>u v. cbox u v \<in> \<D> \<Longrightarrow> \<exists>n. \<forall>i \<in> Basis. v \<bullet> i - u \<bullet> i = (b \<bullet> i - a \<bullet> i) / 2^n" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2382 |
"S \<subseteq> \<Union>\<D>" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2383 |
proof - |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2384 |
have aibi: "\<And>i. i \<in> Basis \<Longrightarrow> a \<bullet> i \<le> b \<bullet> i" and normab: "0 < norm(b - a)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2385 |
using \<open>box a b \<noteq> {}\<close> box_eq_empty box_sing by fastforce+ |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2386 |
let ?K0 = "\<lambda>(n, f::'a\<Rightarrow>nat). |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2387 |
cbox (\<Sum>i \<in> Basis. (a \<bullet> i + (f i / 2^n) * (b \<bullet> i - a \<bullet> i)) *\<^sub>R i) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2388 |
(\<Sum>i \<in> Basis. (a \<bullet> i + ((f i + 1) / 2^n) * (b \<bullet> i - a \<bullet> i)) *\<^sub>R i)" |
64910 | 2389 |
let ?D0 = "?K0 ` (SIGMA n:UNIV. Pi\<^sub>E Basis (\<lambda>i::'a. lessThan (2^n)))" |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2390 |
obtain \<D>0 where count: "countable \<D>0" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2391 |
and sub: "\<Union>\<D>0 \<subseteq> cbox a b" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2392 |
and int: "\<And>K. K \<in> \<D>0 \<Longrightarrow> (interior K \<noteq> {}) \<and> (\<exists>c d. K = cbox c d)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2393 |
and intdj: "\<And>A B. \<lbrakk>A \<in> \<D>0; B \<in> \<D>0\<rbrakk> \<Longrightarrow> A \<subseteq> B \<or> B \<subseteq> A \<or> interior A \<inter> interior B = {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2394 |
and SK: "\<And>x. x \<in> S \<Longrightarrow> \<exists>K \<in> \<D>0. x \<in> K \<and> K \<subseteq> g x" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2395 |
and cbox: "\<And>u v. cbox u v \<in> \<D>0 \<Longrightarrow> \<exists>n. \<forall>i \<in> Basis. v \<bullet> i - u \<bullet> i = (b \<bullet> i - a \<bullet> i) / 2^n" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2396 |
and fin: "\<And>K. K \<in> \<D>0 \<Longrightarrow> finite {L \<in> \<D>0. K \<subseteq> L}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2397 |
proof |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2398 |
show "countable ?D0" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2399 |
by (simp add: countable_PiE) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2400 |
next |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2401 |
show "\<Union>?D0 \<subseteq> cbox a b" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2402 |
apply (simp add: UN_subset_iff) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2403 |
apply (intro conjI allI ballI subset_box_imp) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2404 |
apply (simp add: divide_simps zero_le_mult_iff aibi) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2405 |
apply (force simp: aibi scaling_mono nat_less_real_le dest: PiE_mem) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2406 |
done |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2407 |
next |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2408 |
show "\<And>K. K \<in> ?D0 \<Longrightarrow> interior K \<noteq> {} \<and> (\<exists>c d. K = cbox c d)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2409 |
using \<open>box a b \<noteq> {}\<close> |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2410 |
by (clarsimp simp: box_eq_empty) (fastforce simp add: divide_simps dest: PiE_mem) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2411 |
next |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2412 |
have realff: "(real w) * 2^m < (real v) * 2^n \<longleftrightarrow> w * 2^m < v * 2^n" for m n v w |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2413 |
using of_nat_less_iff less_imp_of_nat_less by fastforce |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2414 |
have *: "\<forall>v w. ?K0(m,v) \<subseteq> ?K0(n,w) \<or> ?K0(n,w) \<subseteq> ?K0(m,v) \<or> interior(?K0(m,v)) \<inter> interior(?K0(n,w)) = {}" |
64911 | 2415 |
for m n \<comment>\<open>The symmetry argument requires a single HOL formula\<close> |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2416 |
proof (rule linorder_wlog [where a=m and b=n], intro allI impI) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2417 |
fix v w m and n::nat |
64911 | 2418 |
assume "m \<le> n" \<comment>\<open>WLOG we can assume @{term"m \<le> n"}, when the first disjunct becomes impossible\<close> |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2419 |
have "?K0(n,w) \<subseteq> ?K0(m,v) \<or> interior(?K0(m,v)) \<inter> interior(?K0(n,w)) = {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2420 |
apply (simp add: subset_box disjoint_interval) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2421 |
apply (rule ccontr) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2422 |
apply (clarsimp simp add: aibi mult_le_cancel_right divide_le_cancel not_less not_le) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2423 |
apply (drule_tac x=i in bspec, assumption) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2424 |
using \<open>m\<le>n\<close> realff [of _ _ "1+_"] realff [of "1+_"_ "1+_"] |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2425 |
apply (auto simp: divide_simps add.commute not_le nat_le_iff_add realff) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2426 |
apply (simp add: power_add, metis (no_types, hide_lams) mult_Suc mult_less_cancel2 not_less_eq mult.assoc)+ |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2427 |
done |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2428 |
then show "?K0(m,v) \<subseteq> ?K0(n,w) \<or> ?K0(n,w) \<subseteq> ?K0(m,v) \<or> interior(?K0(m,v)) \<inter> interior(?K0(n,w)) = {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2429 |
by meson |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2430 |
qed auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2431 |
show "\<And>A B. \<lbrakk>A \<in> ?D0; B \<in> ?D0\<rbrakk> \<Longrightarrow> A \<subseteq> B \<or> B \<subseteq> A \<or> interior A \<inter> interior B = {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2432 |
apply (erule imageE SigmaE)+ |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2433 |
using * by simp |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2434 |
next |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2435 |
show "\<exists>K \<in> ?D0. x \<in> K \<and> K \<subseteq> g x" if "x \<in> S" for x |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2436 |
proof (simp only: bex_simps split_paired_Bex_Sigma) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2437 |
show "\<exists>n. \<exists>f \<in> Basis \<rightarrow>\<^sub>E {..<2 ^ n}. x \<in> ?K0(n,f) \<and> ?K0(n,f) \<subseteq> g x" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2438 |
proof - |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2439 |
obtain e where "0 < e" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2440 |
and e: "\<And>y. (\<And>i. i \<in> Basis \<Longrightarrow> \<bar>x \<bullet> i - y \<bullet> i\<bar> \<le> e) \<Longrightarrow> y \<in> g x" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2441 |
proof - |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2442 |
have "x \<in> g x" "open (g x)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2443 |
using \<open>gauge g\<close> by (auto simp: gauge_def) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2444 |
then obtain \<epsilon> where "0 < \<epsilon>" and \<epsilon>: "ball x \<epsilon> \<subseteq> g x" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2445 |
using openE by blast |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2446 |
have "norm (x - y) < \<epsilon>" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2447 |
if "(\<And>i. i \<in> Basis \<Longrightarrow> \<bar>x \<bullet> i - y \<bullet> i\<bar> \<le> \<epsilon> / (2 * real DIM('a)))" for y |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2448 |
proof - |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2449 |
have "norm (x - y) \<le> (\<Sum>i\<in>Basis. \<bar>x \<bullet> i - y \<bullet> i\<bar>)" |
64267 | 2450 |
by (metis (no_types, lifting) inner_diff_left norm_le_l1 sum.cong) |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2451 |
also have "... \<le> DIM('a) * (\<epsilon> / (2 * real DIM('a)))" |
64267 | 2452 |
by (meson sum_bounded_above that) |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2453 |
also have "... = \<epsilon> / 2" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2454 |
by (simp add: divide_simps) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2455 |
also have "... < \<epsilon>" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2456 |
by (simp add: \<open>0 < \<epsilon>\<close>) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2457 |
finally show ?thesis . |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2458 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2459 |
then show ?thesis |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2460 |
by (rule_tac e = "\<epsilon> / 2 / DIM('a)" in that) (simp_all add: \<open>0 < \<epsilon>\<close> dist_norm subsetD [OF \<epsilon>]) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2461 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2462 |
have xab: "x \<in> cbox a b" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2463 |
using \<open>x \<in> S\<close> \<open>S \<subseteq> cbox a b\<close> by blast |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2464 |
obtain n where n: "norm (b - a) / 2^n < e" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2465 |
using real_arch_pow_inv [of "e / norm(b - a)" "1/2"] normab \<open>0 < e\<close> |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2466 |
by (auto simp: divide_simps) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2467 |
then have "norm (b - a) < e * 2^n" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2468 |
by (auto simp: divide_simps) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2469 |
then have bai: "b \<bullet> i - a \<bullet> i < e * 2 ^ n" if "i \<in> Basis" for i |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2470 |
proof - |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2471 |
have "b \<bullet> i - a \<bullet> i \<le> norm (b - a)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2472 |
by (metis abs_of_nonneg dual_order.trans inner_diff_left linear norm_ge_zero Basis_le_norm that) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2473 |
also have "... < e * 2 ^ n" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2474 |
using \<open>norm (b - a) < e * 2 ^ n\<close> by blast |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2475 |
finally show ?thesis . |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2476 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2477 |
have D: "(a + n \<le> x \<and> x \<le> a + m) \<Longrightarrow> (a + n \<le> y \<and> y \<le> a + m) \<Longrightarrow> abs(x - y) \<le> m - n" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2478 |
for a m n x and y::real |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2479 |
by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2480 |
have "\<forall>i\<in>Basis. \<exists>k<2 ^ n. (a \<bullet> i + real k * (b \<bullet> i - a \<bullet> i) / 2 ^ n \<le> x \<bullet> i \<and> |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2481 |
x \<bullet> i \<le> a \<bullet> i + (real k + 1) * (b \<bullet> i - a \<bullet> i) / 2 ^ n)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2482 |
proof |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2483 |
fix i::'a assume "i \<in> Basis" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2484 |
consider "x \<bullet> i = b \<bullet> i" | "x \<bullet> i < b \<bullet> i" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2485 |
using \<open>i \<in> Basis\<close> mem_box(2) xab by force |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2486 |
then show "\<exists>k<2 ^ n. (a \<bullet> i + real k * (b \<bullet> i - a \<bullet> i) / 2 ^ n \<le> x \<bullet> i \<and> |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2487 |
x \<bullet> i \<le> a \<bullet> i + (real k + 1) * (b \<bullet> i - a \<bullet> i) / 2 ^ n)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2488 |
proof cases |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2489 |
case 1 then show ?thesis |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2490 |
by (rule_tac x = "2^n - 1" in exI) (auto simp: algebra_simps divide_simps of_nat_diff \<open>i \<in> Basis\<close> aibi) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2491 |
next |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2492 |
case 2 |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2493 |
then have abi_less: "a \<bullet> i < b \<bullet> i" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2494 |
using \<open>i \<in> Basis\<close> xab by (auto simp: mem_box) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2495 |
let ?k = "nat \<lfloor>2 ^ n * (x \<bullet> i - a \<bullet> i) / (b \<bullet> i - a \<bullet> i)\<rfloor>" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2496 |
show ?thesis |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2497 |
proof (intro exI conjI) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2498 |
show "?k < 2 ^ n" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2499 |
using aibi xab \<open>i \<in> Basis\<close> |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2500 |
by (force simp: nat_less_iff floor_less_iff divide_simps 2 mem_box) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2501 |
next |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2502 |
have "a \<bullet> i + real ?k * (b \<bullet> i - a \<bullet> i) / 2 ^ n \<le> |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2503 |
a \<bullet> i + (2 ^ n * (x \<bullet> i - a \<bullet> i) / (b \<bullet> i - a \<bullet> i)) * (b \<bullet> i - a \<bullet> i) / 2 ^ n" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2504 |
apply (intro add_left_mono mult_right_mono divide_right_mono of_nat_floor) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2505 |
using aibi [OF \<open>i \<in> Basis\<close>] xab 2 |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2506 |
apply (simp_all add: \<open>i \<in> Basis\<close> mem_box divide_simps) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2507 |
done |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2508 |
also have "... = x \<bullet> i" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2509 |
using abi_less by (simp add: divide_simps) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2510 |
finally show "a \<bullet> i + real ?k * (b \<bullet> i - a \<bullet> i) / 2 ^ n \<le> x \<bullet> i" . |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2511 |
next |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2512 |
have "x \<bullet> i \<le> a \<bullet> i + (2 ^ n * (x \<bullet> i - a \<bullet> i) / (b \<bullet> i - a \<bullet> i)) * (b \<bullet> i - a \<bullet> i) / 2 ^ n" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2513 |
using abi_less by (simp add: divide_simps algebra_simps) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2514 |
also have "... \<le> a \<bullet> i + (real ?k + 1) * (b \<bullet> i - a \<bullet> i) / 2 ^ n" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2515 |
apply (intro add_left_mono mult_right_mono divide_right_mono of_nat_floor) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2516 |
using aibi [OF \<open>i \<in> Basis\<close>] xab |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2517 |
apply (auto simp: \<open>i \<in> Basis\<close> mem_box divide_simps) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2518 |
done |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2519 |
finally show "x \<bullet> i \<le> a \<bullet> i + (real ?k + 1) * (b \<bullet> i - a \<bullet> i) / 2 ^ n" . |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2520 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2521 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2522 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2523 |
then have "\<exists>f\<in>Basis \<rightarrow>\<^sub>E {..<2 ^ n}. x \<in> ?K0(n,f)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2524 |
apply (simp add: mem_box Bex_def) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2525 |
apply (clarify dest!: bchoice) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2526 |
apply (rule_tac x="restrict f Basis" in exI, simp) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2527 |
done |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2528 |
moreover have "\<And>f. x \<in> ?K0(n,f) \<Longrightarrow> ?K0(n,f) \<subseteq> g x" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2529 |
apply (clarsimp simp add: mem_box) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2530 |
apply (rule e) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2531 |
apply (drule bspec D, assumption)+ |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2532 |
apply (erule order_trans) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2533 |
apply (simp add: divide_simps) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2534 |
using bai by (force simp: algebra_simps) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2535 |
ultimately show ?thesis by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2536 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2537 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2538 |
next |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2539 |
show "\<And>u v. cbox u v \<in> ?D0 \<Longrightarrow> \<exists>n. \<forall>i \<in> Basis. v \<bullet> i - u \<bullet> i = (b \<bullet> i - a \<bullet> i) / 2^n" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2540 |
by (force simp: eq_cbox box_eq_empty field_simps dest!: aibi) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2541 |
next |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2542 |
obtain j::'a where "j \<in> Basis" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2543 |
using nonempty_Basis by blast |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2544 |
have "finite {L \<in> ?D0. ?K0(n,f) \<subseteq> L}" if "f \<in> Basis \<rightarrow>\<^sub>E {..<2 ^ n}" for n f |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2545 |
proof (rule finite_subset) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2546 |
let ?B = "(\<lambda>(n, f::'a\<Rightarrow>nat). cbox (\<Sum>i\<in>Basis. (a \<bullet> i + (f i) / 2^n * (b \<bullet> i - a \<bullet> i)) *\<^sub>R i) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2547 |
(\<Sum>i\<in>Basis. (a \<bullet> i + ((f i) + 1) / 2^n * (b \<bullet> i - a \<bullet> i)) *\<^sub>R i)) |
64910 | 2548 |
` (SIGMA m:atMost n. Pi\<^sub>E Basis (\<lambda>i::'a. lessThan (2^m)))" |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2549 |
have "?K0(m,g) \<in> ?B" if "g \<in> Basis \<rightarrow>\<^sub>E {..<2 ^ m}" "?K0(n,f) \<subseteq> ?K0(m,g)" for m g |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2550 |
proof - |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2551 |
have dd: "w / m \<le> v / n \<and> (v+1) / n \<le> (w+1) / m |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2552 |
\<Longrightarrow> inverse n \<le> inverse m" for w m v n::real |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2553 |
by (auto simp: divide_simps algebra_simps) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2554 |
have bjaj: "b \<bullet> j - a \<bullet> j > 0" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2555 |
using \<open>j \<in> Basis\<close> \<open>box a b \<noteq> {}\<close> box_eq_empty(1) by fastforce |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2556 |
have "((g j) / 2 ^ m) * (b \<bullet> j - a \<bullet> j) \<le> ((f j) / 2 ^ n) * (b \<bullet> j - a \<bullet> j) \<and> |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2557 |
(((f j) + 1) / 2 ^ n) * (b \<bullet> j - a \<bullet> j) \<le> (((g j) + 1) / 2 ^ m) * (b \<bullet> j - a \<bullet> j)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2558 |
using that \<open>j \<in> Basis\<close> by (simp add: subset_box algebra_simps divide_simps aibi) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2559 |
then have "((g j) / 2 ^ m) \<le> ((f j) / 2 ^ n) \<and> |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2560 |
((real(f j) + 1) / 2 ^ n) \<le> ((real(g j) + 1) / 2 ^ m)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2561 |
by (metis bjaj mult.commute of_nat_1 of_nat_add real_mult_le_cancel_iff2) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2562 |
then have "inverse (2^n) \<le> (inverse (2^m) :: real)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2563 |
by (rule dd) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2564 |
then have "m \<le> n" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2565 |
by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2566 |
show ?thesis |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2567 |
by (rule imageI) (simp add: \<open>m \<le> n\<close> that) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2568 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2569 |
then show "{L \<in> ?D0. ?K0(n,f) \<subseteq> L} \<subseteq> ?B" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2570 |
by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2571 |
show "finite ?B" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2572 |
by (intro finite_imageI finite_SigmaI finite_atMost finite_lessThan finite_PiE finite_Basis) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2573 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2574 |
then show "finite {L \<in> ?D0. K \<subseteq> L}" if "K \<in> ?D0" for K |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2575 |
using that by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2576 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2577 |
let ?D1 = "{K \<in> \<D>0. \<exists>x \<in> S \<inter> K. K \<subseteq> g x}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2578 |
obtain \<D> where count: "countable \<D>" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2579 |
and sub: "\<Union>\<D> \<subseteq> cbox a b" "S \<subseteq> \<Union>\<D>" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2580 |
and int: "\<And>K. K \<in> \<D> \<Longrightarrow> (interior K \<noteq> {}) \<and> (\<exists>c d. K = cbox c d)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2581 |
and intdj: "\<And>A B. \<lbrakk>A \<in> \<D>; B \<in> \<D>\<rbrakk> \<Longrightarrow> A \<subseteq> B \<or> B \<subseteq> A \<or> interior A \<inter> interior B = {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2582 |
and SK: "\<And>K. K \<in> \<D> \<Longrightarrow> \<exists>x. x \<in> S \<inter> K \<and> K \<subseteq> g x" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2583 |
and cbox: "\<And>u v. cbox u v \<in> \<D> \<Longrightarrow> \<exists>n. \<forall>i \<in> Basis. v \<bullet> i - u \<bullet> i = (b \<bullet> i - a \<bullet> i) / 2^n" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2584 |
and fin: "\<And>K. K \<in> \<D> \<Longrightarrow> finite {L. L \<in> \<D> \<and> K \<subseteq> L}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2585 |
proof |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2586 |
show "countable ?D1" using count countable_subset |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2587 |
by (simp add: count countable_subset) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2588 |
show "\<Union>?D1 \<subseteq> cbox a b" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2589 |
using sub by blast |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2590 |
show "S \<subseteq> \<Union>?D1" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2591 |
using SK by (force simp:) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2592 |
show "\<And>K. K \<in> ?D1 \<Longrightarrow> (interior K \<noteq> {}) \<and> (\<exists>c d. K = cbox c d)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2593 |
using int by blast |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2594 |
show "\<And>A B. \<lbrakk>A \<in> ?D1; B \<in> ?D1\<rbrakk> \<Longrightarrow> A \<subseteq> B \<or> B \<subseteq> A \<or> interior A \<inter> interior B = {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2595 |
using intdj by blast |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2596 |
show "\<And>K. K \<in> ?D1 \<Longrightarrow> \<exists>x. x \<in> S \<inter> K \<and> K \<subseteq> g x" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2597 |
by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2598 |
show "\<And>u v. cbox u v \<in> ?D1 \<Longrightarrow> \<exists>n. \<forall>i \<in> Basis. v \<bullet> i - u \<bullet> i = (b \<bullet> i - a \<bullet> i) / 2^n" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2599 |
using cbox by blast |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2600 |
show "\<And>K. K \<in> ?D1 \<Longrightarrow> finite {L. L \<in> ?D1 \<and> K \<subseteq> L}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2601 |
using fin by simp (metis (mono_tags, lifting) Collect_mono rev_finite_subset) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2602 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2603 |
let ?\<D> = "{K \<in> \<D>. \<forall>K'. K' \<in> \<D> \<and> K \<noteq> K' \<longrightarrow> ~(K \<subseteq> K')}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2604 |
show ?thesis |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2605 |
proof (rule that) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2606 |
show "countable ?\<D>" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2607 |
by (blast intro: countable_subset [OF _ count]) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2608 |
show "\<Union>?\<D> \<subseteq> cbox a b" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2609 |
using sub by blast |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2610 |
show "S \<subseteq> \<Union>?\<D>" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2611 |
proof clarsimp |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2612 |
fix x |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2613 |
assume "x \<in> S" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2614 |
then obtain X where "x \<in> X" "X \<in> \<D>" using \<open>S \<subseteq> \<Union>\<D>\<close> by blast |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2615 |
let ?R = "{(K,L). K \<in> \<D> \<and> L \<in> \<D> \<and> L \<subset> K}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2616 |
have irrR: "irrefl ?R" by (force simp: irrefl_def) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2617 |
have traR: "trans ?R" by (force simp: trans_def) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2618 |
have finR: "\<And>x. finite {y. (y, x) \<in> ?R}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2619 |
by simp (metis (mono_tags, lifting) fin \<open>X \<in> \<D>\<close> finite_subset mem_Collect_eq psubset_imp_subset subsetI) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2620 |
have "{X \<in> \<D>. x \<in> X} \<noteq> {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2621 |
using \<open>X \<in> \<D>\<close> \<open>x \<in> X\<close> by blast |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2622 |
then obtain Y where "Y \<in> {X \<in> \<D>. x \<in> X}" "\<And>Y'. (Y', Y) \<in> ?R \<Longrightarrow> Y' \<notin> {X \<in> \<D>. x \<in> X}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2623 |
by (rule wfE_min' [OF wf_finite_segments [OF irrR traR finR]]) blast |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2624 |
then show "\<exists>Y. Y \<in> \<D> \<and> (\<forall>K'. K' \<in> \<D> \<and> Y \<noteq> K' \<longrightarrow> \<not> Y \<subseteq> K') \<and> x \<in> Y" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2625 |
by blast |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2626 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2627 |
show "\<And>K. K \<in> ?\<D> \<Longrightarrow> interior K \<noteq> {} \<and> (\<exists>c d. K = cbox c d)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2628 |
by (blast intro: dest: int) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2629 |
show "pairwise (\<lambda>A B. interior A \<inter> interior B = {}) ?\<D>" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2630 |
using intdj by (simp add: pairwise_def) metis |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2631 |
show "\<And>K. K \<in> ?\<D> \<Longrightarrow> \<exists>x \<in> S \<inter> K. K \<subseteq> g x" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2632 |
using SK by force |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2633 |
show "\<And>u v. cbox u v \<in> ?\<D> \<Longrightarrow> \<exists>n. \<forall>i\<in>Basis. v \<bullet> i - u \<bullet> i = (b \<bullet> i - a \<bullet> i) / 2^n" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2634 |
using cbox by force |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2635 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2636 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2637 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2638 |
subsection \<open>Division filter\<close> |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2639 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2640 |
text \<open>Divisions over all gauges towards finer divisions.\<close> |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2641 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2642 |
definition division_filter :: "'a::euclidean_space set \<Rightarrow> ('a \<times> 'a set) set filter" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2643 |
where "division_filter s = (INF g:{g. gauge g}. principal {p. p tagged_division_of s \<and> g fine p})" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2644 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2645 |
lemma eventually_division_filter: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2646 |
"(\<forall>\<^sub>F p in division_filter s. P p) \<longleftrightarrow> |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2647 |
(\<exists>g. gauge g \<and> (\<forall>p. p tagged_division_of s \<and> g fine p \<longrightarrow> P p))" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2648 |
unfolding division_filter_def |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2649 |
proof (subst eventually_INF_base; clarsimp) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2650 |
fix g1 g2 :: "'a \<Rightarrow> 'a set" show "gauge g1 \<Longrightarrow> gauge g2 \<Longrightarrow> \<exists>x. gauge x \<and> |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2651 |
{p. p tagged_division_of s \<and> x fine p} \<subseteq> {p. p tagged_division_of s \<and> g1 fine p} \<and> |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2652 |
{p. p tagged_division_of s \<and> x fine p} \<subseteq> {p. p tagged_division_of s \<and> g2 fine p}" |
66192
e5b84854baa4
A few renamings and several tidied-up proofs
paulson <lp15@cam.ac.uk>
parents:
66164
diff
changeset
|
2653 |
by (intro exI[of _ "\<lambda>x. g1 x \<inter> g2 x"]) (auto simp: fine_Int) |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2654 |
qed (auto simp: eventually_principal) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2655 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2656 |
lemma division_filter_not_empty: "division_filter (cbox a b) \<noteq> bot" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2657 |
unfolding trivial_limit_def eventually_division_filter |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2658 |
by (auto elim: fine_division_exists) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2659 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2660 |
lemma eventually_division_filter_tagged_division: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2661 |
"eventually (\<lambda>p. p tagged_division_of s) (division_filter s)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2662 |
unfolding eventually_division_filter by (intro exI[of _ "\<lambda>x. ball x 1"]) auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2663 |
|
64267 | 2664 |
end |