author | wenzelm |
Mon, 22 May 2017 00:23:25 +0200 | |
changeset 65897 | 94b0da1b242e |
parent 64911 | f0e07600de47 |
child 66112 | 0e640e04fc56 |
permissions | -rw-r--r-- |
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HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
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|
1 |
(* Title: HOL/Analysis/Tagged_Division.thy |
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HOL-Analysis: move gauges and (tagged) divisions to its own theory file
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2 |
Author: John Harrison |
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HOL-Analysis: move gauges and (tagged) divisions to its own theory file
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parents:
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3 |
Author: Robert Himmelmann, TU Muenchen (Translation from HOL light); proofs reworked by LCP |
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HOL-Analysis: move gauges and (tagged) divisions to its own theory file
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parents:
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4 |
*) |
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HOL-Analysis: move gauges and (tagged) divisions to its own theory file
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parents:
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5 |
|
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HOL-Analysis: move gauges and (tagged) divisions to its own theory file
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6 |
section \<open>Tagged divisions used for the Henstock-Kurzweil gauge integration\<close> |
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HOL-Analysis: move gauges and (tagged) divisions to its own theory file
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7 |
|
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HOL-Analysis: move gauges and (tagged) divisions to its own theory file
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parents:
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8 |
theory Tagged_Division |
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HOL-Analysis: move gauges and (tagged) divisions to its own theory file
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9 |
imports |
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HOL-Analysis: move gauges and (tagged) divisions to its own theory file
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diff
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10 |
Topology_Euclidean_Space |
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HOL-Analysis: move gauges and (tagged) divisions to its own theory file
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11 |
begin |
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HOL-Analysis: move gauges and (tagged) divisions to its own theory file
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12 |
|
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HOL-Analysis: move gauges and (tagged) divisions to its own theory file
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13 |
lemma finite_product_dependent: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
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14 |
assumes "finite s" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
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parents:
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15 |
and "\<And>x. x \<in> s \<Longrightarrow> finite (t x)" |
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HOL-Analysis: move gauges and (tagged) divisions to its own theory file
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parents:
diff
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16 |
shows "finite {(i, j) |i j. i \<in> s \<and> j \<in> t i}" |
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HOL-Analysis: move gauges and (tagged) divisions to its own theory file
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parents:
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17 |
using assms |
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HOL-Analysis: move gauges and (tagged) divisions to its own theory file
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parents:
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18 |
proof induct |
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HOL-Analysis: move gauges and (tagged) divisions to its own theory file
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parents:
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19 |
case (insert x s) |
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HOL-Analysis: move gauges and (tagged) divisions to its own theory file
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parents:
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20 |
have *: "{(i, j) |i j. i \<in> insert x s \<and> j \<in> t i} = |
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HOL-Analysis: move gauges and (tagged) divisions to its own theory file
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parents:
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21 |
(\<lambda>y. (x,y)) ` (t x) \<union> {(i, j) |i j. i \<in> s \<and> j \<in> t i}" by auto |
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HOL-Analysis: move gauges and (tagged) divisions to its own theory file
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22 |
show ?case |
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HOL-Analysis: move gauges and (tagged) divisions to its own theory file
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23 |
unfolding * |
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HOL-Analysis: move gauges and (tagged) divisions to its own theory file
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24 |
apply (rule finite_UnI) |
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HOL-Analysis: move gauges and (tagged) divisions to its own theory file
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25 |
using insert |
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HOL-Analysis: move gauges and (tagged) divisions to its own theory file
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diff
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26 |
apply auto |
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HOL-Analysis: move gauges and (tagged) divisions to its own theory file
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27 |
done |
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HOL-Analysis: move gauges and (tagged) divisions to its own theory file
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28 |
qed auto |
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HOL-Analysis: move gauges and (tagged) divisions to its own theory file
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29 |
|
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HOL-Analysis: move gauges and (tagged) divisions to its own theory file
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30 |
lemma sum_sum_product: |
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HOL-Analysis: move gauges and (tagged) divisions to its own theory file
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31 |
assumes "finite s" |
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HOL-Analysis: move gauges and (tagged) divisions to its own theory file
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32 |
and "\<forall>i\<in>s. finite (t i)" |
64267 | 33 |
shows "sum (\<lambda>i. sum (x i) (t i)::real) s = |
34 |
sum (\<lambda>(i,j). x i j) {(i,j) | i j. i \<in> s \<and> j \<in> t i}" |
|
63957
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HOL-Analysis: move gauges and (tagged) divisions to its own theory file
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35 |
using assms |
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HOL-Analysis: move gauges and (tagged) divisions to its own theory file
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36 |
proof induct |
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HOL-Analysis: move gauges and (tagged) divisions to its own theory file
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37 |
case (insert a s) |
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HOL-Analysis: move gauges and (tagged) divisions to its own theory file
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parents:
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38 |
have *: "{(i, j) |i j. i \<in> insert a s \<and> j \<in> t i} = |
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HOL-Analysis: move gauges and (tagged) divisions to its own theory file
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parents:
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39 |
(\<lambda>y. (a,y)) ` (t a) \<union> {(i, j) |i j. i \<in> s \<and> j \<in> t i}" by auto |
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HOL-Analysis: move gauges and (tagged) divisions to its own theory file
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40 |
show ?case |
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HOL-Analysis: move gauges and (tagged) divisions to its own theory file
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41 |
unfolding * |
64267 | 42 |
apply (subst sum.union_disjoint) |
43 |
unfolding sum.insert[OF insert(1-2)] |
|
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
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44 |
prefer 4 |
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HOL-Analysis: move gauges and (tagged) divisions to its own theory file
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45 |
apply (subst insert(3)) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
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parents:
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46 |
unfolding add_right_cancel |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
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47 |
proof - |
64267 | 48 |
show "sum (x a) (t a) = (\<Sum>(xa, y)\<in> Pair a ` t a. x xa y)" |
49 |
apply (subst sum.reindex) |
|
63957
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HOL-Analysis: move gauges and (tagged) divisions to its own theory file
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parents:
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50 |
unfolding inj_on_def |
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HOL-Analysis: move gauges and (tagged) divisions to its own theory file
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parents:
diff
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51 |
apply auto |
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HOL-Analysis: move gauges and (tagged) divisions to its own theory file
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52 |
done |
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HOL-Analysis: move gauges and (tagged) divisions to its own theory file
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parents:
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53 |
show "finite {(i, j) |i j. i \<in> s \<and> j \<in> t i}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
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54 |
apply (rule finite_product_dependent) |
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HOL-Analysis: move gauges and (tagged) divisions to its own theory file
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parents:
diff
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55 |
using insert |
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HOL-Analysis: move gauges and (tagged) divisions to its own theory file
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parents:
diff
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56 |
apply auto |
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HOL-Analysis: move gauges and (tagged) divisions to its own theory file
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parents:
diff
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|
57 |
done |
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HOL-Analysis: move gauges and (tagged) divisions to its own theory file
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parents:
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58 |
qed (insert insert, auto) |
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HOL-Analysis: move gauges and (tagged) divisions to its own theory file
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parents:
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59 |
qed auto |
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HOL-Analysis: move gauges and (tagged) divisions to its own theory file
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parents:
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60 |
|
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HOL-Analysis: move gauges and (tagged) divisions to its own theory file
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61 |
lemmas scaleR_simps = scaleR_zero_left scaleR_minus_left scaleR_left_diff_distrib |
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HOL-Analysis: move gauges and (tagged) divisions to its own theory file
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parents:
diff
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62 |
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
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parents:
diff
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63 |
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
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parents:
diff
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64 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
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parents:
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65 |
|
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HOL-Analysis: move gauges and (tagged) divisions to its own theory file
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66 |
subsection \<open>Sundries\<close> |
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HOL-Analysis: move gauges and (tagged) divisions to its own theory file
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67 |
|
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HOL-Analysis: move gauges and (tagged) divisions to its own theory file
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parents:
diff
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68 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
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69 |
text\<open>A transitive relation is well-founded if all initial segments are finite.\<close> |
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HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
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70 |
lemma wf_finite_segments: |
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HOL-Analysis: move gauges and (tagged) divisions to its own theory file
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parents:
diff
changeset
|
71 |
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
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72 |
shows "wf (r)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
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73 |
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
|
74 |
using acyclic_def assms irrefl_def trans_Restr by fastforce |
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HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
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75 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
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parents:
diff
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|
76 |
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
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parents:
diff
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|
77 |
lemma scaling_mono: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
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parents:
diff
changeset
|
78 |
fixes u::"'a::linordered_field" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
79 |
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
|
80 |
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
|
81 |
proof - |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
82 |
have "r/s \<le> 1" using assms |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
83 |
using divide_le_eq_1 by fastforce |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
84 |
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
|
85 |
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
|
86 |
then show ?thesis |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
87 |
by (simp add: field_simps) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
88 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
89 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
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|
90 |
lemma conjunctD2: assumes "a \<and> b" shows a b using assms by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
91 |
lemma conjunctD3: assumes "a \<and> b \<and> c" shows a b c using assms by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
92 |
lemma conjunctD4: assumes "a \<and> b \<and> c \<and> d" shows a b c d using assms by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
93 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
94 |
lemma cond_cases: "(P \<Longrightarrow> Q x) \<Longrightarrow> (\<not> P \<Longrightarrow> Q y) \<Longrightarrow> Q (if P then x else y)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
95 |
by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
96 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
97 |
declare norm_triangle_ineq4[intro] |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
98 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
99 |
lemma transitive_stepwise_le: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
100 |
assumes "\<And>x. R x x" "\<And>x y z. R x y \<Longrightarrow> R y z \<Longrightarrow> R x z" and "\<And>n. R n (Suc n)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
101 |
shows "\<forall>n\<ge>m. R m n" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
102 |
proof (intro allI impI) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
103 |
show "m \<le> n \<Longrightarrow> R m n" for n |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
104 |
by (induction rule: dec_induct) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
105 |
(use assms in blast)+ |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
106 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
107 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
108 |
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
|
109 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
110 |
lemma interior_subset_union_intervals: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
111 |
assumes "i = cbox a b" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
112 |
and "j = cbox c d" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
113 |
and "interior j \<noteq> {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
114 |
and "i \<subseteq> j \<union> s" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
115 |
and "interior i \<inter> interior j = {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
116 |
shows "interior i \<subseteq> interior s" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
117 |
proof - |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
118 |
have "box a b \<inter> cbox c d = {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
119 |
using inter_interval_mixed_eq_empty[of c d a b] and assms(3,5) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
120 |
unfolding assms(1,2) interior_cbox by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
121 |
moreover |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
122 |
have "box a b \<subseteq> cbox c d \<union> s" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
123 |
apply (rule order_trans,rule box_subset_cbox) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
124 |
using assms(4) unfolding assms(1,2) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
125 |
apply auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
126 |
done |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
127 |
ultimately |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
128 |
show ?thesis |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
129 |
unfolding assms interior_cbox |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
130 |
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
|
131 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
132 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
133 |
lemma interior_Union_subset_cbox: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
134 |
assumes "finite f" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
135 |
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
|
136 |
and t: "closed t" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
137 |
shows "interior (\<Union>f) \<subseteq> t" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
138 |
proof - |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
139 |
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
|
140 |
using f by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
141 |
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
|
142 |
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
|
143 |
using \<open>finite f\<close> by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
144 |
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
|
145 |
proof (induction E rule: finite_subset_induct') |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
146 |
case (insert s f') |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
147 |
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
|
148 |
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
|
149 |
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
|
150 |
using insert.hyps by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
151 |
finally show ?case |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
152 |
by (simp add: insert.IH) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
153 |
qed simp |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
154 |
also have "\<dots> \<subseteq> \<Union>(f - E)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
155 |
by (rule interior_subset) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
156 |
also have "\<dots> \<subseteq> t" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
157 |
proof (rule Union_least) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
158 |
fix s assume "s \<in> f - E" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
159 |
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
|
160 |
by (fastforce simp: E_def) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
161 |
have "closure (interior s) \<subseteq> closure t" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
162 |
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
|
163 |
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
|
164 |
by simp |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
165 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
166 |
finally show ?thesis . |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
167 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
168 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
169 |
lemma inter_interior_unions_intervals: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
170 |
"finite f \<Longrightarrow> open s \<Longrightarrow> \<forall>t\<in>f. \<exists>a b. t = cbox a b \<Longrightarrow> \<forall>t\<in>f. s \<inter> (interior t) = {} \<Longrightarrow> s \<inter> interior (\<Union>f) = {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
171 |
using interior_Union_subset_cbox[of f "UNIV - s"] by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
172 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
173 |
lemma interval_split: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
174 |
fixes a :: "'a::euclidean_space" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
175 |
assumes "k \<in> Basis" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
176 |
shows |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
177 |
"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
|
178 |
"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" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
179 |
apply (rule_tac[!] set_eqI) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
180 |
unfolding Int_iff mem_box mem_Collect_eq |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
181 |
using assms |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
182 |
apply auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
183 |
done |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
184 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
185 |
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
|
186 |
by (simp add: box_ne_empty) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
187 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
188 |
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
|
189 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
190 |
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
|
191 |
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
|
192 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
193 |
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
|
194 |
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
|
195 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
196 |
lemma interval_upperbound[simp]: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
197 |
"\<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
|
198 |
interval_upperbound (cbox a b) = (b::'a::euclidean_space)" |
64267 | 199 |
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
|
200 |
by (safe intro!: cSup_eq) auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
201 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
202 |
lemma interval_lowerbound[simp]: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
203 |
"\<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
|
204 |
interval_lowerbound (cbox a b) = (a::'a::euclidean_space)" |
64267 | 205 |
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
|
206 |
by (safe intro!: cInf_eq) auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
207 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
208 |
lemmas interval_bounds = interval_upperbound interval_lowerbound |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
209 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
210 |
lemma |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
211 |
fixes X::"real set" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
212 |
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
|
213 |
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
|
214 |
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
|
215 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
216 |
lemma interval_bounds'[simp]: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
217 |
assumes "cbox a b \<noteq> {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
218 |
shows "interval_upperbound (cbox a b) = b" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
219 |
and "interval_lowerbound (cbox a b) = a" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
220 |
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
|
221 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
222 |
lemma interval_upperbound_Times: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
223 |
assumes "A \<noteq> {}" and "B \<noteq> {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
224 |
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
|
225 |
proof- |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
226 |
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
|
227 |
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
|
228 |
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
|
229 |
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
|
230 |
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
|
231 |
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
|
232 |
ultimately show ?thesis unfolding interval_upperbound_def |
64267 | 233 |
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
|
234 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
235 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
236 |
lemma interval_lowerbound_Times: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
237 |
assumes "A \<noteq> {}" and "B \<noteq> {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
238 |
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
|
239 |
proof- |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
240 |
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
|
241 |
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
|
242 |
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
|
243 |
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
|
244 |
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
|
245 |
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
|
246 |
ultimately show ?thesis unfolding interval_lowerbound_def |
64267 | 247 |
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
|
248 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
249 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
250 |
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
|
251 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
252 |
definition "gauge d \<longleftrightarrow> (\<forall>x. x \<in> d x \<and> open (d x))" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
253 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
254 |
lemma gaugeI: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
255 |
assumes "\<And>x. x \<in> g x" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
256 |
and "\<And>x. open (g x)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
257 |
shows "gauge g" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
258 |
using assms unfolding gauge_def by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
259 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
260 |
lemma gaugeD[dest]: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
261 |
assumes "gauge d" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
262 |
shows "x \<in> d x" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
263 |
and "open (d x)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
264 |
using assms unfolding gauge_def by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
265 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
266 |
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
|
267 |
unfolding gauge_def by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
268 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
269 |
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
|
270 |
unfolding gauge_def by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
271 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
272 |
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
|
273 |
by (rule gauge_ball) auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
274 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
275 |
lemma gauge_inter[intro]: "gauge d1 \<Longrightarrow> gauge d2 \<Longrightarrow> gauge (\<lambda>x. d1 x \<inter> d2 x)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
276 |
unfolding gauge_def by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
277 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
278 |
lemma gauge_inters: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
279 |
assumes "finite s" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
280 |
and "\<forall>d\<in>s. gauge (f d)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
281 |
shows "gauge (\<lambda>x. \<Inter>{f d x | d. d \<in> s})" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
282 |
proof - |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
283 |
have *: "\<And>x. {f d x |d. d \<in> s} = (\<lambda>d. f d x) ` s" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
284 |
by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
285 |
show ?thesis |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
286 |
unfolding gauge_def unfolding * |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
287 |
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
|
288 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
289 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
290 |
lemma gauge_existence_lemma: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
291 |
"(\<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
|
292 |
by (metis zero_less_one) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
293 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
294 |
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
|
295 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
296 |
lemma gauge_modify: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
297 |
assumes "(\<forall>s. open s \<longrightarrow> open {x. f(x) \<in> s})" "gauge d" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
298 |
shows "gauge (\<lambda>x. {y. f y \<in> d (f x)})" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
299 |
using assms |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
300 |
unfolding gauge_def |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
301 |
apply safe |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
302 |
defer |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
303 |
apply (erule_tac x="f x" in allE) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
304 |
apply (erule_tac x="d (f x)" in allE) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
305 |
apply auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
306 |
done |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
307 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
308 |
subsection \<open>Divisions.\<close> |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
309 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
310 |
definition division_of (infixl "division'_of" 40) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
311 |
where |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
312 |
"s division_of i \<longleftrightarrow> |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
313 |
finite s \<and> |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
314 |
(\<forall>k\<in>s. k \<subseteq> i \<and> k \<noteq> {} \<and> (\<exists>a b. k = cbox a b)) \<and> |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
315 |
(\<forall>k1\<in>s. \<forall>k2\<in>s. k1 \<noteq> k2 \<longrightarrow> interior(k1) \<inter> interior(k2) = {}) \<and> |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
316 |
(\<Union>s = i)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
317 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
318 |
lemma division_ofD[dest]: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
319 |
assumes "s division_of i" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
320 |
shows "finite s" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
321 |
and "\<And>k. k \<in> s \<Longrightarrow> k \<subseteq> i" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
322 |
and "\<And>k. k \<in> s \<Longrightarrow> k \<noteq> {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
323 |
and "\<And>k. k \<in> s \<Longrightarrow> \<exists>a b. k = cbox a b" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
324 |
and "\<And>k1 k2. k1 \<in> s \<Longrightarrow> k2 \<in> s \<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
|
325 |
and "\<Union>s = i" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
326 |
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
|
327 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
328 |
lemma division_ofI: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
329 |
assumes "finite s" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
330 |
and "\<And>k. k \<in> s \<Longrightarrow> k \<subseteq> i" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
331 |
and "\<And>k. k \<in> s \<Longrightarrow> k \<noteq> {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
332 |
and "\<And>k. k \<in> s \<Longrightarrow> \<exists>a b. k = cbox a b" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
333 |
and "\<And>k1 k2. k1 \<in> s \<Longrightarrow> k2 \<in> s \<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
|
334 |
and "\<Union>s = i" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
335 |
shows "s division_of i" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
336 |
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
|
337 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
338 |
lemma division_of_finite: "s division_of i \<Longrightarrow> finite s" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
339 |
unfolding division_of_def by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
340 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
341 |
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
|
342 |
unfolding division_of_def 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 |
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
|
345 |
unfolding division_of_def by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
346 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
347 |
lemma division_of_sing[simp]: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
348 |
"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
|
349 |
(is "?l = ?r") |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
350 |
proof |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
351 |
assume ?r |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
352 |
moreover |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
353 |
{ fix k |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
354 |
assume "s = {{a}}" "k\<in>s" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
355 |
then have "\<exists>x y. k = cbox x y" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
356 |
apply (rule_tac x=a in exI)+ |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
357 |
apply (force simp: cbox_sing) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
358 |
done |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
359 |
} |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
360 |
ultimately show ?l |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
361 |
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
|
362 |
next |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
363 |
assume ?l |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
364 |
note * = conjunctD4[OF this[unfolded division_of_def cbox_sing]] |
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 |
fix x |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
367 |
assume x: "x \<in> s" have "x = {a}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
368 |
using *(2)[rule_format,OF x] by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
369 |
} |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
370 |
moreover have "s \<noteq> {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
371 |
using *(4) by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
372 |
ultimately show ?r |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
373 |
unfolding cbox_sing by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
374 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
375 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
376 |
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
|
377 |
unfolding division_of_trivial by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
378 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
379 |
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
|
380 |
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
|
381 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
382 |
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
|
383 |
unfolding division_of_def by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
384 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
385 |
lemma forall_in_division: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
386 |
"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
|
387 |
unfolding division_of_def by fastforce |
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 |
lemma division_of_subset: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
390 |
assumes "p division_of (\<Union>p)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
391 |
and "q \<subseteq> p" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
392 |
shows "q division_of (\<Union>q)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
393 |
proof (rule division_ofI) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
394 |
note * = division_ofD[OF assms(1)] |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
395 |
show "finite q" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
396 |
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
|
397 |
{ |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
398 |
fix k |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
399 |
assume "k \<in> q" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
400 |
then have kp: "k \<in> p" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
401 |
using assms(2) by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
402 |
show "k \<subseteq> \<Union>q" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
403 |
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
|
404 |
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
|
405 |
using *(4)[OF kp] by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
406 |
show "k \<noteq> {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
407 |
using *(3)[OF kp] by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
408 |
} |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
409 |
fix k1 k2 |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
410 |
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
|
411 |
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
|
412 |
using assms(2) by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
413 |
show "interior k1 \<inter> interior k2 = {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
414 |
using *(5)[OF **] by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
415 |
qed auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
416 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
417 |
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
|
418 |
unfolding division_of_def by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
419 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
420 |
lemma division_inter: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
421 |
fixes s1 s2 :: "'a::euclidean_space set" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
422 |
assumes "p1 division_of s1" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
423 |
and "p2 division_of s2" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
424 |
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
|
425 |
(is "?A' division_of _") |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
426 |
proof - |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
427 |
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
|
428 |
have *: "?A' = ?A" by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
429 |
show ?thesis |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
430 |
unfolding * |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
431 |
proof (rule division_ofI) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
432 |
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
|
433 |
by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
434 |
moreover have "finite (p1 \<times> p2)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
435 |
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
|
436 |
ultimately show "finite ?A" by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
437 |
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
|
438 |
by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
439 |
show "\<Union>?A = s1 \<inter> s2" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
440 |
apply (rule set_eqI) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
441 |
unfolding * and UN_iff |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
442 |
using division_ofD(6)[OF assms(1)] and division_ofD(6)[OF assms(2)] |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
443 |
apply auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
444 |
done |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
445 |
{ |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
446 |
fix k |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
447 |
assume "k \<in> ?A" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
448 |
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
|
449 |
by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
450 |
then show "k \<noteq> {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
451 |
by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
452 |
show "k \<subseteq> s1 \<inter> s2" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
453 |
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
|
454 |
unfolding k by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
455 |
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
|
456 |
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
|
457 |
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
|
458 |
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
|
459 |
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
|
460 |
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
|
461 |
} |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
462 |
fix k1 k2 |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
463 |
assume "k1 \<in> ?A" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
464 |
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
|
465 |
by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
466 |
assume "k2 \<in> ?A" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
467 |
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
|
468 |
by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
469 |
assume "k1 \<noteq> k2" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
470 |
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
|
471 |
unfolding k1 k2 by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
472 |
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
|
473 |
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
|
474 |
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
|
475 |
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
|
476 |
show "interior k1 \<inter> interior k2 = {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
477 |
unfolding k1 k2 |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
478 |
apply (rule *) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
479 |
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
|
480 |
done |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
481 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
482 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
483 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
484 |
lemma division_inter_1: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
485 |
assumes "d division_of i" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
486 |
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
|
487 |
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
|
488 |
proof (cases "cbox a b = {}") |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
489 |
case True |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
490 |
show ?thesis |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
491 |
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
|
492 |
next |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
493 |
case False |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
494 |
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
|
495 |
show ?thesis |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
496 |
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
|
497 |
unfolding * by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
498 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
499 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
500 |
lemma elementary_inter: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
501 |
fixes s t :: "'a::euclidean_space set" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
502 |
assumes "p1 division_of s" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
503 |
and "p2 division_of t" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
504 |
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
|
505 |
using assms division_inter by blast |
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 |
lemma elementary_inters: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
508 |
assumes "finite f" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
509 |
and "f \<noteq> {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
510 |
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
|
511 |
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
|
512 |
using assms |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
513 |
proof (induct f rule: finite_induct) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
514 |
case (insert x f) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
515 |
show ?case |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
516 |
proof (cases "f = {}") |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
517 |
case True |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
518 |
then show ?thesis |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
519 |
unfolding True using insert by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
520 |
next |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
521 |
case False |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
522 |
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
|
523 |
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
|
524 |
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
|
525 |
using insert(5)[rule_format,OF insertI1] .. |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
526 |
ultimately show ?thesis |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
527 |
by (simp add: elementary_inter Inter_insert) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
528 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
529 |
qed auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
530 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
531 |
lemma division_disjoint_union: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
532 |
assumes "p1 division_of s1" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
533 |
and "p2 division_of s2" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
534 |
and "interior s1 \<inter> interior s2 = {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
535 |
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
|
536 |
proof (rule division_ofI) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
537 |
note d1 = division_ofD[OF assms(1)] |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
538 |
note d2 = division_ofD[OF assms(2)] |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
539 |
show "finite (p1 \<union> p2)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
540 |
using d1(1) d2(1) by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
541 |
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
|
542 |
using d1(6) d2(6) by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
543 |
{ |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
544 |
fix k1 k2 |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
545 |
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
|
546 |
moreover |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
547 |
let ?g="interior k1 \<inter> interior k2 = {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
548 |
{ |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
549 |
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
|
550 |
have ?g |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
551 |
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
|
552 |
using assms(3) by blast |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
553 |
} |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
554 |
moreover |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
555 |
{ |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
556 |
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
|
557 |
have ?g |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
558 |
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
|
559 |
using assms(3) by blast |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
560 |
} |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
561 |
ultimately show ?g |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
562 |
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
|
563 |
} |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
564 |
fix k |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
565 |
assume k: "k \<in> p1 \<union> p2" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
566 |
show "k \<subseteq> s1 \<union> s2" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
567 |
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
|
568 |
show "k \<noteq> {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
569 |
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
|
570 |
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
|
571 |
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
|
572 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
573 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
574 |
lemma partial_division_extend_1: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
575 |
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
|
576 |
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
|
577 |
and nonempty: "cbox c d \<noteq> {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
578 |
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
|
579 |
proof |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
580 |
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
|
581 |
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
|
582 |
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
|
583 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
584 |
show "cbox c d \<in> p" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
585 |
unfolding p_def |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
586 |
by (auto simp add: box_eq_empty cbox_def intro!: image_eqI[where x="\<lambda>(i::'a)\<in>Basis. (c, d)"]) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
587 |
{ |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
588 |
fix i :: 'a |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
589 |
assume "i \<in> Basis" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
590 |
with incl nonempty have "a \<bullet> i \<le> c \<bullet> i" "c \<bullet> i \<le> d \<bullet> i" "d \<bullet> i \<le> b \<bullet> i" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
591 |
unfolding box_eq_empty subset_box by (auto simp: not_le) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
592 |
} |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
593 |
note ord = this |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
594 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
595 |
show "p division_of (cbox a b)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
596 |
proof (rule division_ofI) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
597 |
show "finite p" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
598 |
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
|
599 |
{ |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
600 |
fix k |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
601 |
assume "k \<in> p" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
602 |
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
|
603 |
by (auto simp: p_def) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
604 |
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
|
605 |
by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
606 |
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
|
607 |
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
|
608 |
fix i :: 'a |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
609 |
assume i: "i \<in> Basis" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
610 |
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
|
611 |
by (auto simp: PiE_iff) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
612 |
with i ord[of i] |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
613 |
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
|
614 |
by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
615 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
616 |
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
|
617 |
by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
618 |
{ |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
619 |
fix l |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
620 |
assume "l \<in> p" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
621 |
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
|
622 |
by (auto simp: p_def) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
623 |
assume "l \<noteq> k" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
624 |
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
|
625 |
proof (rule ccontr) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
626 |
assume "\<not> ?thesis" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
627 |
with f g have "f = g" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
628 |
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
|
629 |
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
|
630 |
by (simp add: l k) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
631 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
632 |
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
|
633 |
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
|
634 |
"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
|
635 |
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
|
636 |
with * ord[of i] show "interior l \<inter> interior k = {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
637 |
by (auto simp add: l k interior_cbox disjoint_interval intro!: bexI[of _ i]) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
638 |
} |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
639 |
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
|
640 |
} |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
641 |
moreover |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
642 |
{ |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
643 |
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
|
644 |
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
|
645 |
proof |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
646 |
fix i :: 'a |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
647 |
assume "i \<in> Basis" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
648 |
with x ord[of i] |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
649 |
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
|
650 |
(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
|
651 |
by (auto simp: cbox_def) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
652 |
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
|
653 |
by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
654 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
655 |
then obtain f where |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
656 |
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
|
657 |
unfolding bchoice_iff .. |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
658 |
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
|
659 |
by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
660 |
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
|
661 |
by (auto simp: mem_box) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
662 |
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
|
663 |
unfolding p_def by blast |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
664 |
} |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
665 |
ultimately show "\<Union>p = cbox a b" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
666 |
by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
667 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
668 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
669 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
670 |
lemma partial_division_extend_interval: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
671 |
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
|
672 |
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
|
673 |
proof (cases "p = {}") |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
674 |
case True |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
675 |
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
|
676 |
by (rule elementary_interval) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
677 |
then show ?thesis |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
678 |
using True that by blast |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
679 |
next |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
680 |
case False |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
681 |
note p = division_ofD[OF assms(1)] |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
682 |
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
|
683 |
proof |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
684 |
fix k |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
685 |
assume kp: "k \<in> p" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
686 |
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
|
687 |
using p(4)[OF kp] by blast |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
688 |
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
|
689 |
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
|
690 |
by (blast intro: order.trans)+ |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
691 |
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
|
692 |
by (rule partial_division_extend_1[OF *]) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
693 |
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
|
694 |
unfolding k by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
695 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
696 |
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
|
697 |
using bchoice[OF div_cbox] by blast |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
698 |
{ fix x |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
699 |
assume x: "x \<in> p" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
700 |
have "q x division_of \<Union>q x" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
701 |
apply (rule division_ofI) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
702 |
using division_ofD[OF q(1)[OF x]] |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
703 |
apply auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
704 |
done } |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
705 |
then have "\<And>x. x \<in> p \<Longrightarrow> \<exists>d. d division_of \<Union>(q x - {x})" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
706 |
by (meson Diff_subset division_of_subset) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
707 |
then have "\<exists>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
|
708 |
apply - |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
709 |
apply (rule elementary_inters [OF finite_imageI[OF p(1)]]) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
710 |
apply (auto simp: False elementary_inters [OF finite_imageI[OF p(1)]]) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
711 |
done |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
712 |
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
|
713 |
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
|
714 |
proof - |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
715 |
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 |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
716 |
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
|
717 |
proof (rule te[OF False], clarify) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
718 |
fix i |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
719 |
assume i: "i \<in> p" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
720 |
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
|
721 |
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
|
722 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
723 |
{ fix k |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
724 |
assume k: "k \<in> p" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
725 |
have *: "\<And>u t s. t \<inter> s = {} \<Longrightarrow> u \<subseteq> s \<Longrightarrow> u \<inter> t = {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
726 |
by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
727 |
have "interior (\<Inter>i\<in>p. \<Union>(q i - {i})) \<inter> interior k = {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
728 |
proof (rule *[OF inter_interior_unions_intervals]) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
729 |
note qk=division_ofD[OF q(1)[OF k]] |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
730 |
show "finite (q k - {k})" "open (interior k)" "\<forall>t\<in>q k - {k}. \<exists>a b. t = cbox a b" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
731 |
using qk by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
732 |
show "\<forall>t\<in>q k - {k}. interior k \<inter> interior t = {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
733 |
using qk(5) using q(2)[OF k] by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
734 |
show "interior (\<Inter>i\<in>p. \<Union>(q i - {i})) \<subseteq> interior (\<Union>(q k - {k}))" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
735 |
apply (rule interior_mono)+ |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
736 |
using k |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
737 |
apply auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
738 |
done |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
739 |
qed } note [simp] = this |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
740 |
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
|
741 |
unfolding cbox_eq |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
742 |
apply (rule division_disjoint_union[OF d assms(1)]) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
743 |
apply (rule inter_interior_unions_intervals) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
744 |
apply (rule p open_interior ballI)+ |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
745 |
apply simp_all |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
746 |
done |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
747 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
748 |
then show ?thesis |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
749 |
by (meson Un_upper2 that) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
750 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
751 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
752 |
lemma elementary_bounded[dest]: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
753 |
fixes s :: "'a::euclidean_space set" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
754 |
shows "p division_of s \<Longrightarrow> bounded s" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
755 |
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
|
756 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
757 |
lemma elementary_subset_cbox: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
758 |
"p division_of s \<Longrightarrow> \<exists>a b. s \<subseteq> cbox a (b::'a::euclidean_space)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
759 |
by (meson elementary_bounded bounded_subset_cbox) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
760 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
761 |
lemma division_union_intervals_exists: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
762 |
fixes a b :: "'a::euclidean_space" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
763 |
assumes "cbox a b \<noteq> {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
764 |
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
|
765 |
proof (cases "cbox c d = {}") |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
766 |
case True |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
767 |
show ?thesis |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
768 |
apply (rule that[of "{}"]) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
769 |
unfolding True |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
770 |
using assms |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
771 |
apply auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
772 |
done |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
773 |
next |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
774 |
case False |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
775 |
show ?thesis |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
776 |
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
|
777 |
case True |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
778 |
then show ?thesis |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
779 |
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
|
780 |
next |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
781 |
case False |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
782 |
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
|
783 |
unfolding Int_interval by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
784 |
have uv_sub: "cbox u v \<subseteq> cbox c d" using uv by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
785 |
obtain p where "p division_of cbox c d" "cbox u v \<in> p" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
786 |
by (rule partial_division_extend_1[OF uv_sub False[unfolded uv]]) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
787 |
note p = this division_ofD[OF this(1)] |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
788 |
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
|
789 |
apply (rule arg_cong[of _ _ interior]) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
790 |
using p(8) uv by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
791 |
also have "\<dots> = {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
792 |
unfolding interior_Int |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
793 |
apply (rule inter_interior_unions_intervals) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
794 |
using p(6) p(7)[OF p(2)] p(3) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
795 |
apply auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
796 |
done |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
797 |
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
|
798 |
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
|
799 |
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
|
800 |
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
|
801 |
proof - |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
802 |
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
|
803 |
by (simp add: assms division_of_self) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
804 |
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
|
805 |
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
|
806 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
807 |
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
|
808 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
809 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
810 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
811 |
lemma division_of_unions: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
812 |
assumes "finite f" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
813 |
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
|
814 |
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
|
815 |
shows "\<Union>f division_of \<Union>\<Union>f" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
816 |
using assms |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
817 |
by (auto intro!: division_ofI) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
818 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
819 |
lemma elementary_union_interval: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
820 |
fixes a b :: "'a::euclidean_space" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
821 |
assumes "p division_of \<Union>p" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
822 |
obtains q where "q division_of (cbox a b \<union> \<Union>p)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
823 |
proof - |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
824 |
note assm = division_ofD[OF assms] |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
825 |
have lem1: "\<And>f s. \<Union>\<Union>(f ` s) = \<Union>((\<lambda>x. \<Union>(f x)) ` s)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
826 |
by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
827 |
have lem2: "\<And>f s. f \<noteq> {} \<Longrightarrow> \<Union>{s \<union> t |t. t \<in> f} = s \<union> \<Union>f" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
828 |
by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
829 |
{ |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
830 |
presume "p = {} \<Longrightarrow> thesis" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
831 |
"cbox a b = {} \<Longrightarrow> thesis" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
832 |
"cbox a b \<noteq> {} \<Longrightarrow> interior (cbox a b) = {} \<Longrightarrow> thesis" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
833 |
"p \<noteq> {} \<Longrightarrow> interior (cbox a b)\<noteq>{} \<Longrightarrow> cbox a b \<noteq> {} \<Longrightarrow> thesis" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
834 |
then show thesis by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
835 |
next |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
836 |
assume as: "p = {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
837 |
obtain 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
|
838 |
by (rule elementary_interval) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
839 |
then show thesis |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
840 |
using as that by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
841 |
next |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
842 |
assume as: "cbox a b = {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
843 |
show thesis |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
844 |
using as assms that by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
845 |
next |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
846 |
assume as: "interior (cbox a b) = {}" "cbox a b \<noteq> {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
847 |
show thesis |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
848 |
apply (rule that[of "insert (cbox a b) p"],rule division_ofI) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
849 |
unfolding finite_insert |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
850 |
apply (rule assm(1)) unfolding Union_insert |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
851 |
using assm(2-4) as |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
852 |
apply - |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
853 |
apply (fast dest: assm(5))+ |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
854 |
done |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
855 |
next |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
856 |
assume as: "p \<noteq> {}" "interior (cbox a b) \<noteq> {}" "cbox a b \<noteq> {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
857 |
have "\<forall>k\<in>p. \<exists>q. (insert (cbox a b) q) division_of (cbox a b \<union> k)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
858 |
proof |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
859 |
fix k |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
860 |
assume kp: "k \<in> p" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
861 |
from assm(4)[OF kp] obtain c d where "k = cbox c d" by blast |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
862 |
then show "\<exists>q. (insert (cbox a b) q) division_of (cbox a b \<union> k)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
863 |
by (meson as(3) division_union_intervals_exists) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
864 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
865 |
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
|
866 |
note q = division_ofD[OF this[rule_format]] |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
867 |
let ?D = "\<Union>{insert (cbox a b) (q k) | k. k \<in> p}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
868 |
show thesis |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
869 |
proof (rule that[OF division_ofI]) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
870 |
have *: "{insert (cbox a b) (q k) |k. k \<in> p} = (\<lambda>k. insert (cbox a b) (q k)) ` p" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
871 |
by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
872 |
show "finite ?D" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
873 |
using "*" assm(1) q(1) by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
874 |
show "\<Union>?D = cbox a b \<union> \<Union>p" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
875 |
unfolding * lem1 |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
876 |
unfolding lem2[OF as(1), of "cbox a b", symmetric] |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
877 |
using q(6) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
878 |
by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
879 |
fix k |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
880 |
assume k: "k \<in> ?D" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
881 |
then show "k \<subseteq> cbox a b \<union> \<Union>p" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
882 |
using q(2) by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
883 |
show "k \<noteq> {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
884 |
using q(3) k by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
885 |
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
|
886 |
using q(4) k by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
887 |
fix k' |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
888 |
assume k': "k' \<in> ?D" "k \<noteq> k'" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
889 |
obtain x where x: "k \<in> insert (cbox a b) (q x)" "x\<in>p" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
890 |
using k by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
891 |
obtain x' where x': "k'\<in>insert (cbox a b) (q x')" "x'\<in>p" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
892 |
using k' by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
893 |
show "interior k \<inter> interior k' = {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
894 |
proof (cases "x = x'") |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
895 |
case True |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
896 |
show ?thesis |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
897 |
using True k' q(5) x' x by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
898 |
next |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
899 |
case False |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
900 |
{ |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
901 |
presume "k = cbox a b \<Longrightarrow> ?thesis" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
902 |
and "k' = cbox a b \<Longrightarrow> ?thesis" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
903 |
and "k \<noteq> cbox a b \<Longrightarrow> k' \<noteq> cbox a b \<Longrightarrow> ?thesis" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
904 |
then show ?thesis by linarith |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
905 |
next |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
906 |
assume as': "k = cbox a b" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
907 |
show ?thesis |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
908 |
using as' k' q(5) x' by blast |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
909 |
next |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
910 |
assume as': "k' = cbox a b" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
911 |
show ?thesis |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
912 |
using as' k'(2) q(5) x by blast |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
913 |
} |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
914 |
assume as': "k \<noteq> cbox a b" "k' \<noteq> cbox a b" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
915 |
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
|
916 |
using q(4)[OF x(2,1)] by blast |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
917 |
have "interior k \<inter> interior (cbox a b) = {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
918 |
using as' k'(2) q(5) x by blast |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
919 |
then have "interior k \<subseteq> interior x" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
920 |
using interior_subset_union_intervals |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
921 |
by (metis as(2) k q(2) x interior_subset_union_intervals) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
922 |
moreover |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
923 |
obtain c d where c_d: "k' = cbox c d" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
924 |
using q(4)[OF x'(2,1)] by blast |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
925 |
have "interior k' \<inter> interior (cbox a b) = {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
926 |
using as'(2) q(5) x' by blast |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
927 |
then have "interior k' \<subseteq> interior x'" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
928 |
by (metis as(2) c_d interior_subset_union_intervals q(2) x'(1) x'(2)) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
929 |
ultimately show ?thesis |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
930 |
using assm(5)[OF x(2) x'(2) False] by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
931 |
qed |
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 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
935 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
936 |
lemma elementary_unions_intervals: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
937 |
assumes fin: "finite f" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
938 |
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
|
939 |
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
|
940 |
proof - |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
941 |
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
|
942 |
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
|
943 |
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
|
944 |
next |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
945 |
fix x F |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
946 |
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
|
947 |
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
|
948 |
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
|
949 |
have *: "\<Union>F = \<Union>p" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
950 |
using division_ofD[OF p] by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
951 |
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
|
952 |
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
|
953 |
unfolding Union_insert x * by metis |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
954 |
qed (insert assms, auto) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
955 |
then show ?thesis |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
956 |
using that by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
957 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
958 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
959 |
lemma elementary_union: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
960 |
fixes s t :: "'a::euclidean_space set" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
961 |
assumes "ps division_of s" "pt division_of t" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
962 |
obtains p where "p division_of (s \<union> t)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
963 |
proof - |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
964 |
have *: "s \<union> t = \<Union>ps \<union> \<Union>pt" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
965 |
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
|
966 |
show ?thesis |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
967 |
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
|
968 |
using assms apply auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
969 |
by (simp add: * that) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
970 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
971 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
972 |
lemma partial_division_extend: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
973 |
fixes t :: "'a::euclidean_space set" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
974 |
assumes "p division_of s" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
975 |
and "q division_of t" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
976 |
and "s \<subseteq> t" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
977 |
obtains r where "p \<subseteq> r" and "r division_of t" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
978 |
proof - |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
979 |
note divp = division_ofD[OF assms(1)] and divq = division_ofD[OF assms(2)] |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
980 |
obtain a b where ab: "t \<subseteq> cbox a b" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
981 |
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
|
982 |
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
|
983 |
using assms |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
984 |
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
|
985 |
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
|
986 |
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
|
987 |
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
|
988 |
using r1(3,6) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
989 |
apply auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
990 |
done |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
991 |
then obtain r2 where r2: "r2 division_of (\<Union>(r1 - p)) \<inter> (\<Union>q)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
992 |
by (metis assms(2) divq(6) elementary_inter) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
993 |
{ |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
994 |
fix x |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
995 |
assume x: "x \<in> t" "x \<notin> s" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
996 |
then have "x\<in>\<Union>r1" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
997 |
unfolding r1 using ab by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
998 |
then obtain r where r: "r \<in> r1" "x \<in> r" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
999 |
unfolding Union_iff .. |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1000 |
moreover |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1001 |
have "r \<notin> p" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1002 |
proof |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1003 |
assume "r \<in> p" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1004 |
then have "x \<in> s" using divp(2) r by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1005 |
then show False using x by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1006 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1007 |
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
|
1008 |
} |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1009 |
then have *: "t = \<Union>p \<union> (\<Union>(r1 - p) \<inter> \<Union>q)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1010 |
unfolding divp divq using assms(3) by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1011 |
show ?thesis |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1012 |
apply (rule that[of "p \<union> r2"]) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1013 |
unfolding * |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1014 |
defer |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1015 |
apply (rule division_disjoint_union) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1016 |
unfolding divp(6) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1017 |
apply(rule assms r2)+ |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1018 |
proof - |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1019 |
have "interior s \<inter> interior (\<Union>(r1-p)) = {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1020 |
proof (rule inter_interior_unions_intervals) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1021 |
show "finite (r1 - p)" and "open (interior s)" and "\<forall>t\<in>r1-p. \<exists>a b. t = cbox a b" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1022 |
using r1 by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1023 |
have *: "\<And>s. (\<And>x. x \<in> s \<Longrightarrow> False) \<Longrightarrow> s = {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1024 |
by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1025 |
show "\<forall>t\<in>r1-p. interior s \<inter> interior t = {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1026 |
proof |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1027 |
fix m x |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1028 |
assume as: "m \<in> r1 - p" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1029 |
have "interior m \<inter> interior (\<Union>p) = {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1030 |
proof (rule inter_interior_unions_intervals) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1031 |
show "finite p" and "open (interior m)" and "\<forall>t\<in>p. \<exists>a b. t = cbox a b" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1032 |
using divp by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1033 |
show "\<forall>t\<in>p. interior m \<inter> interior t = {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1034 |
by (metis DiffD1 DiffD2 as r1(1) r1(7) set_rev_mp) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1035 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1036 |
then show "interior s \<inter> interior m = {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1037 |
unfolding divp by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1038 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1039 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1040 |
then show "interior s \<inter> interior (\<Union>(r1-p) \<inter> (\<Union>q)) = {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1041 |
using interior_subset by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1042 |
qed auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1043 |
qed |
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 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1046 |
lemma division_split: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1047 |
fixes a :: "'a::euclidean_space" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1048 |
assumes "p division_of (cbox a b)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1049 |
and k: "k\<in>Basis" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1050 |
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
|
1051 |
(is "?p1 division_of ?I1") |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1052 |
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
|
1053 |
(is "?p2 division_of ?I2") |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1054 |
proof (rule_tac[!] division_ofI) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1055 |
note p = division_ofD[OF assms(1)] |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1056 |
show "finite ?p1" "finite ?p2" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1057 |
using p(1) by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1058 |
show "\<Union>?p1 = ?I1" "\<Union>?p2 = ?I2" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1059 |
unfolding p(6)[symmetric] by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1060 |
{ |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1061 |
fix k |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1062 |
assume "k \<in> ?p1" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1063 |
then guess l unfolding mem_Collect_eq by (elim exE conjE) note l=this |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1064 |
guess u v using p(4)[OF l(2)] by (elim exE) note uv=this |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1065 |
show "k \<subseteq> ?I1" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1066 |
using l p(2) uv by force |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1067 |
show "k \<noteq> {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1068 |
by (simp add: l) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1069 |
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
|
1070 |
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
|
1071 |
apply (subst interval_split[OF k]) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1072 |
apply (auto intro: order.trans) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1073 |
done |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1074 |
fix k' |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1075 |
assume "k' \<in> ?p1" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1076 |
then guess l' unfolding mem_Collect_eq by (elim exE conjE) note l'=this |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1077 |
assume "k \<noteq> k'" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1078 |
then show "interior k \<inter> interior k' = {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1079 |
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
|
1080 |
} |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1081 |
{ |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1082 |
fix k |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1083 |
assume "k \<in> ?p2" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1084 |
then guess l unfolding mem_Collect_eq by (elim exE conjE) note l=this |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1085 |
guess u v using p(4)[OF l(2)] by (elim exE) note uv=this |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1086 |
show "k \<subseteq> ?I2" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1087 |
using l p(2) uv by force |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1088 |
show "k \<noteq> {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1089 |
by (simp add: l) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1090 |
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
|
1091 |
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
|
1092 |
apply (subst interval_split[OF k]) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1093 |
apply (auto intro: order.trans) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1094 |
done |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1095 |
fix k' |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1096 |
assume "k' \<in> ?p2" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1097 |
then guess l' unfolding mem_Collect_eq by (elim exE conjE) note l'=this |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1098 |
assume "k \<noteq> k'" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1099 |
then show "interior k \<inter> interior k' = {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1100 |
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
|
1101 |
} |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1102 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1103 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1104 |
subsection \<open>Tagged (partial) divisions.\<close> |
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 |
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
|
1107 |
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
|
1108 |
finite s \<and> |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1109 |
(\<forall>x k. (x, k) \<in> s \<longrightarrow> x \<in> k \<and> k \<subseteq> i \<and> (\<exists>a b. k = cbox a b)) \<and> |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1110 |
(\<forall>x1 k1 x2 k2. (x1, k1) \<in> s \<and> (x2, k2) \<in> s \<and> (x1, k1) \<noteq> (x2, k2) \<longrightarrow> |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1111 |
interior k1 \<inter> interior k2 = {})" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1112 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1113 |
lemma tagged_partial_division_ofD[dest]: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1114 |
assumes "s tagged_partial_division_of i" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1115 |
shows "finite s" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1116 |
and "\<And>x k. (x,k) \<in> s \<Longrightarrow> x \<in> k" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1117 |
and "\<And>x k. (x,k) \<in> s \<Longrightarrow> k \<subseteq> i" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1118 |
and "\<And>x k. (x,k) \<in> s \<Longrightarrow> \<exists>a b. k = cbox a b" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1119 |
and "\<And>x1 k1 x2 k2. (x1,k1) \<in> s \<Longrightarrow> |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1120 |
(x2, k2) \<in> s \<Longrightarrow> (x1, k1) \<noteq> (x2, k2) \<Longrightarrow> interior k1 \<inter> interior k2 = {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1121 |
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
|
1122 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1123 |
definition tagged_division_of (infixr "tagged'_division'_of" 40) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1124 |
where "s tagged_division_of i \<longleftrightarrow> s tagged_partial_division_of i \<and> (\<Union>{k. \<exists>x. (x,k) \<in> s} = i)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1125 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1126 |
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
|
1127 |
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
|
1128 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1129 |
lemma tagged_division_of: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1130 |
"s tagged_division_of i \<longleftrightarrow> |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1131 |
finite s \<and> |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1132 |
(\<forall>x k. (x, k) \<in> s \<longrightarrow> x \<in> k \<and> k \<subseteq> i \<and> (\<exists>a b. k = cbox a b)) \<and> |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1133 |
(\<forall>x1 k1 x2 k2. (x1, k1) \<in> s \<and> (x2, k2) \<in> s \<and> (x1, k1) \<noteq> (x2, k2) \<longrightarrow> |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1134 |
interior k1 \<inter> interior k2 = {}) \<and> |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1135 |
(\<Union>{k. \<exists>x. (x,k) \<in> s} = i)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1136 |
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
|
1137 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1138 |
lemma tagged_division_ofI: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1139 |
assumes "finite s" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1140 |
and "\<And>x k. (x,k) \<in> s \<Longrightarrow> x \<in> k" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1141 |
and "\<And>x k. (x,k) \<in> s \<Longrightarrow> k \<subseteq> i" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1142 |
and "\<And>x k. (x,k) \<in> s \<Longrightarrow> \<exists>a b. k = cbox a b" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1143 |
and "\<And>x1 k1 x2 k2. (x1,k1) \<in> s \<Longrightarrow> (x2, k2) \<in> s \<Longrightarrow> (x1, k1) \<noteq> (x2, k2) \<Longrightarrow> |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1144 |
interior k1 \<inter> interior k2 = {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1145 |
and "(\<Union>{k. \<exists>x. (x,k) \<in> s} = i)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1146 |
shows "s tagged_division_of i" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1147 |
unfolding tagged_division_of |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1148 |
using assms |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1149 |
apply auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1150 |
apply fastforce+ |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1151 |
done |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1152 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1153 |
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
|
1154 |
assumes "s tagged_division_of i" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1155 |
shows "finite s" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1156 |
and "\<And>x k. (x,k) \<in> s \<Longrightarrow> x \<in> k" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1157 |
and "\<And>x k. (x,k) \<in> s \<Longrightarrow> k \<subseteq> i" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1158 |
and "\<And>x k. (x,k) \<in> s \<Longrightarrow> \<exists>a b. k = cbox a b" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1159 |
and "\<And>x1 k1 x2 k2. (x1, k1) \<in> s \<Longrightarrow> (x2, k2) \<in> s \<Longrightarrow> (x1, k1) \<noteq> (x2, k2) \<Longrightarrow> |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1160 |
interior k1 \<inter> interior k2 = {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1161 |
and "(\<Union>{k. \<exists>x. (x,k) \<in> s} = i)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1162 |
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
|
1163 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1164 |
lemma division_of_tagged_division: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1165 |
assumes "s tagged_division_of i" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1166 |
shows "(snd ` s) division_of i" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1167 |
proof (rule division_ofI) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1168 |
note assm = tagged_division_ofD[OF assms] |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1169 |
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
|
1170 |
using assm by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1171 |
fix k |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1172 |
assume k: "k \<in> snd ` s" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1173 |
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
|
1174 |
by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1175 |
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
|
1176 |
using assm by fastforce+ |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1177 |
fix k' |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1178 |
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
|
1179 |
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
|
1180 |
by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1181 |
then show "interior k \<inter> interior k' = {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1182 |
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
|
1183 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1184 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1185 |
lemma partial_division_of_tagged_division: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1186 |
assumes "s tagged_partial_division_of i" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1187 |
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
|
1188 |
proof (rule division_ofI) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1189 |
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
|
1190 |
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
|
1191 |
using assm by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1192 |
fix k |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1193 |
assume k: "k \<in> snd ` s" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1194 |
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
|
1195 |
by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1196 |
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
|
1197 |
using assm by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1198 |
fix k' |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1199 |
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
|
1200 |
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
|
1201 |
by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1202 |
then show "interior k \<inter> interior k' = {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1203 |
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
|
1204 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1205 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1206 |
lemma tagged_partial_division_subset: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1207 |
assumes "s tagged_partial_division_of i" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1208 |
and "t \<subseteq> s" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1209 |
shows "t tagged_partial_division_of i" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1210 |
using assms |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1211 |
unfolding tagged_partial_division_of_def |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1212 |
using finite_subset[OF assms(2)] |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1213 |
by blast |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1214 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1215 |
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
|
1216 |
by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1217 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1218 |
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
|
1219 |
unfolding tagged_division_of by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1220 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1221 |
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
|
1222 |
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
|
1223 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1224 |
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
|
1225 |
unfolding tagged_division_of by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1226 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1227 |
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
|
1228 |
by (rule tagged_division_ofI) auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1229 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1230 |
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
|
1231 |
unfolding box_real[symmetric] |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1232 |
by (rule tagged_division_of_self) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1233 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1234 |
lemma tagged_division_union: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1235 |
assumes "p1 tagged_division_of s1" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1236 |
and "p2 tagged_division_of s2" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1237 |
and "interior s1 \<inter> interior s2 = {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1238 |
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
|
1239 |
proof (rule tagged_division_ofI) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1240 |
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
|
1241 |
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
|
1242 |
show "finite (p1 \<union> p2)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1243 |
using p1(1) p2(1) by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1244 |
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
|
1245 |
using p1(6) p2(6) by blast |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1246 |
fix x k |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1247 |
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
|
1248 |
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
|
1249 |
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
|
1250 |
show "k \<subseteq> s1 \<union> s2" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1251 |
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
|
1252 |
fix x' k' |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1253 |
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
|
1254 |
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
|
1255 |
using assms(3) interior_mono by blast |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1256 |
show "interior k \<inter> interior k' = {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1257 |
apply (cases "(x, k) \<in> p1") |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1258 |
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
|
1259 |
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
|
1260 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1261 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1262 |
lemma tagged_division_unions: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1263 |
assumes "finite iset" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1264 |
and "\<forall>i\<in>iset. pfn i tagged_division_of i" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1265 |
and "\<forall>i1\<in>iset. \<forall>i2\<in>iset. i1 \<noteq> i2 \<longrightarrow> interior(i1) \<inter> interior(i2) = {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1266 |
shows "\<Union>(pfn ` iset) tagged_division_of (\<Union>iset)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1267 |
proof (rule tagged_division_ofI) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1268 |
note assm = tagged_division_ofD[OF assms(2)[rule_format]] |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1269 |
show "finite (\<Union>(pfn ` iset))" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1270 |
using assms by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1271 |
have "\<Union>{k. \<exists>x. (x, k) \<in> \<Union>(pfn ` iset)} = \<Union>((\<lambda>i. \<Union>{k. \<exists>x. (x, k) \<in> pfn i}) ` iset)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1272 |
by blast |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1273 |
also have "\<dots> = \<Union>iset" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1274 |
using assm(6) by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1275 |
finally show "\<Union>{k. \<exists>x. (x, k) \<in> \<Union>(pfn ` iset)} = \<Union>iset" . |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1276 |
fix x k |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1277 |
assume xk: "(x, k) \<in> \<Union>(pfn ` iset)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1278 |
then obtain i where i: "i \<in> iset" "(x, k) \<in> pfn i" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1279 |
by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1280 |
show "x \<in> k" "\<exists>a b. k = cbox a b" "k \<subseteq> \<Union>iset" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1281 |
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
|
1282 |
fix x' k' |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1283 |
assume xk': "(x', k') \<in> \<Union>(pfn ` iset)" "(x, k) \<noteq> (x', k')" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1284 |
then obtain i' where i': "i' \<in> iset" "(x', k') \<in> pfn i'" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1285 |
by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1286 |
have *: "\<And>a b. i \<noteq> i' \<Longrightarrow> a \<subseteq> i \<Longrightarrow> b \<subseteq> i' \<Longrightarrow> interior a \<inter> interior b = {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1287 |
using i(1) i'(1) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1288 |
using assms(3)[rule_format] interior_mono |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1289 |
by blast |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1290 |
show "interior k \<inter> interior k' = {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1291 |
apply (cases "i = i'") |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1292 |
using assm(5) i' i(2) xk'(2) apply blast |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1293 |
using "*" assm(3) i' i by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1294 |
qed |
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 |
lemma tagged_partial_division_of_union_self: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1297 |
assumes "p tagged_partial_division_of s" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1298 |
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
|
1299 |
apply (rule tagged_division_ofI) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1300 |
using tagged_partial_division_ofD[OF assms] |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1301 |
apply auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1302 |
done |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1303 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1304 |
lemma tagged_division_of_union_self: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1305 |
assumes "p tagged_division_of s" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1306 |
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
|
1307 |
apply (rule tagged_division_ofI) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1308 |
using tagged_division_ofD[OF assms] |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1309 |
apply auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1310 |
done |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1311 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1312 |
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
|
1313 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1314 |
text \<open>This auxiliary structure is used to sum up over the elements of a division. Main theorem is |
64911 | 1315 |
\<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
|
1316 |
@{typ bool}.\<close> |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1317 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1318 |
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
|
1319 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1320 |
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
|
1321 |
where |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1322 |
"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
|
1323 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1324 |
lemma lift_option_simps[simp]: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1325 |
"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
|
1326 |
"lift_option f None b' = None" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1327 |
"lift_option f a' None = None" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1328 |
by (auto simp: lift_option_def) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1329 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1330 |
lemma comm_monoid_lift_option: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1331 |
assumes "comm_monoid f z" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1332 |
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
|
1333 |
proof - |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1334 |
from assms interpret comm_monoid f z . |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1335 |
show ?thesis |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1336 |
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
|
1337 |
qed |
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 |
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
|
1340 |
by standard auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1341 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1342 |
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
|
1343 |
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
|
1344 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1345 |
paragraph \<open>Operative\<close> |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1346 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1347 |
definition (in comm_monoid) operative :: "('b::euclidean_space set \<Rightarrow> 'a) \<Rightarrow> bool" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1348 |
where "operative g \<longleftrightarrow> |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1349 |
(\<forall>a b. box a b = {} \<longrightarrow> g (cbox a b) = \<^bold>1) \<and> |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1350 |
(\<forall>a b c. \<forall>k\<in>Basis. 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}))" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1351 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1352 |
lemma (in comm_monoid) operativeD[dest]: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1353 |
assumes "operative g" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1354 |
shows "\<And>a b. box a b = {} \<Longrightarrow> g (cbox a b) = \<^bold>1" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1355 |
and "\<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})" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1356 |
using assms unfolding operative_def by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1357 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1358 |
lemma (in comm_monoid) operative_empty: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1359 |
assumes g: "operative g" shows "g {} = \<^bold>1" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1360 |
proof - |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1361 |
have *: "cbox One (-One) = ({}::'b set)" |
64267 | 1362 |
by (auto simp: box_eq_empty inner_sum_left inner_Basis sum.If_cases ex_in_conv) |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1363 |
moreover have "box One (-One) = ({}::'b set)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1364 |
using box_subset_cbox[of One "-One"] by (auto simp: *) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1365 |
ultimately show ?thesis |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1366 |
using operativeD(1)[OF g, of One "-One"] by simp |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1367 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1368 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1369 |
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
|
1370 |
{(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
|
1371 |
(\<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
|
1372 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1373 |
lemma division_points_finite: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1374 |
fixes i :: "'a::euclidean_space set" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1375 |
assumes "d division_of i" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1376 |
shows "finite (division_points i d)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1377 |
proof - |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1378 |
note assm = division_ofD[OF assms] |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1379 |
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
|
1380 |
(\<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
|
1381 |
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
|
1382 |
unfolding division_points_def by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1383 |
show ?thesis |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1384 |
unfolding * using assm by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1385 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1386 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1387 |
lemma division_points_subset: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1388 |
fixes a :: "'a::euclidean_space" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1389 |
assumes "d division_of (cbox a b)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1390 |
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
|
1391 |
and k: "k \<in> Basis" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1392 |
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
|
1393 |
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
|
1394 |
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
|
1395 |
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
|
1396 |
proof - |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1397 |
note assm = division_ofD[OF assms(1)] |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1398 |
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
|
1399 |
"\<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
|
1400 |
"\<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
|
1401 |
"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
|
1402 |
using assms using less_imp_le by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1403 |
show ?t1 (*FIXME a horrible mess*) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1404 |
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
|
1405 |
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
|
1406 |
unfolding * |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1407 |
apply (rule subsetI) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1408 |
unfolding mem_Collect_eq split_beta |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1409 |
apply (erule bexE conjE)+ |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1410 |
apply (simp add: ) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1411 |
apply (erule exE conjE)+ |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1412 |
proof |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1413 |
fix i l x |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1414 |
assume as: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1415 |
"a \<bullet> fst x < snd x" "snd x < (if fst x = k then c else b \<bullet> fst x)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1416 |
"interval_lowerbound i \<bullet> fst x = snd x \<or> interval_upperbound i \<bullet> fst x = snd x" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1417 |
"i = l \<inter> {x. x \<bullet> k \<le> c}" "l \<in> d" "l \<inter> {x. x \<bullet> k \<le> c} \<noteq> {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1418 |
and fstx: "fst x \<in> Basis" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1419 |
from assm(4)[OF this(5)] guess u v apply-by(erule exE)+ note l=this |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1420 |
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" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1421 |
using as(6) unfolding l interval_split[OF k] box_ne_empty as . |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1422 |
have **: "\<forall>i\<in>Basis. u\<bullet>i \<le> v\<bullet>i" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1423 |
using l using as(6) unfolding box_ne_empty[symmetric] by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1424 |
show "\<exists>i\<in>d. interval_lowerbound i \<bullet> fst x = snd x \<or> interval_upperbound i \<bullet> fst x = snd x" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1425 |
apply (rule bexI[OF _ \<open>l \<in> d\<close>]) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1426 |
using as(1-3,5) fstx |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1427 |
unfolding l interval_bounds[OF **] interval_bounds[OF *] interval_split[OF k] as |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1428 |
apply (auto split: if_split_asm) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1429 |
done |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1430 |
show "snd x < b \<bullet> fst x" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1431 |
using as(2) \<open>c < b\<bullet>k\<close> by (auto split: if_split_asm) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1432 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1433 |
show ?t2 |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1434 |
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
|
1435 |
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
|
1436 |
unfolding * |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1437 |
unfolding subset_eq |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1438 |
apply rule |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1439 |
unfolding mem_Collect_eq split_beta |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1440 |
apply (erule bexE conjE)+ |
64267 | 1441 |
apply (simp only: mem_Collect_eq inner_sum_left_Basis simp_thms) |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1442 |
apply (erule exE conjE)+ |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1443 |
proof |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1444 |
fix i l x |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1445 |
assume as: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1446 |
"(if fst x = k then c else a \<bullet> fst x) < snd x" "snd x < b \<bullet> fst x" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1447 |
"interval_lowerbound i \<bullet> fst x = snd x \<or> interval_upperbound i \<bullet> fst x = snd x" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1448 |
"i = l \<inter> {x. c \<le> x \<bullet> k}" "l \<in> d" "l \<inter> {x. c \<le> x \<bullet> k} \<noteq> {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1449 |
and fstx: "fst x \<in> Basis" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1450 |
from assm(4)[OF this(5)] guess u v by (elim exE) note l=this |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1451 |
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" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1452 |
using as(6) unfolding l interval_split[OF k] box_ne_empty as . |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1453 |
have **: "\<forall>i\<in>Basis. u\<bullet>i \<le> v\<bullet>i" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1454 |
using l using as(6) unfolding box_ne_empty[symmetric] by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1455 |
show "\<exists>i\<in>d. interval_lowerbound i \<bullet> fst x = snd x \<or> interval_upperbound i \<bullet> fst x = snd x" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1456 |
apply (rule bexI[OF _ \<open>l \<in> d\<close>]) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1457 |
using as(1-3,5) fstx |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1458 |
unfolding l interval_bounds[OF **] interval_bounds[OF *] interval_split[OF k] as |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1459 |
apply (auto split: if_split_asm) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1460 |
done |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1461 |
show "a \<bullet> fst x < snd x" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1462 |
using as(1) \<open>a\<bullet>k < c\<close> by (auto split: if_split_asm) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1463 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1464 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1465 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1466 |
lemma division_points_psubset: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1467 |
fixes a :: "'a::euclidean_space" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1468 |
assumes "d division_of (cbox a b)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1469 |
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
|
1470 |
and "l \<in> d" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1471 |
and "interval_lowerbound l\<bullet>k = c \<or> interval_upperbound l\<bullet>k = c" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1472 |
and k: "k \<in> Basis" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1473 |
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
|
1474 |
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
|
1475 |
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
|
1476 |
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
|
1477 |
proof - |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1478 |
have 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
|
1479 |
using assms(2) by (auto intro!:less_imp_le) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1480 |
guess u v using division_ofD(4)[OF assms(1,5)] by (elim exE) note l=this |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1481 |
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" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1482 |
using division_ofD(2,2,3)[OF assms(1,5)] unfolding l box_ne_empty |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1483 |
using subset_box(1) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1484 |
apply auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1485 |
apply blast+ |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1486 |
done |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1487 |
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
|
1488 |
"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
|
1489 |
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
|
1490 |
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
|
1491 |
by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1492 |
have "\<exists>x. x \<in> ?D - ?D1" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1493 |
using assms(3-) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1494 |
unfolding division_points_def interval_bounds[OF ab] |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1495 |
apply - |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1496 |
apply (erule disjE) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1497 |
apply (rule_tac x="(k,(interval_lowerbound l)\<bullet>k)" in exI, force simp add: *) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1498 |
apply (rule_tac x="(k,(interval_upperbound l)\<bullet>k)" in exI, force simp add: *) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1499 |
done |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1500 |
moreover have "?D1 \<subseteq> ?D" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1501 |
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
|
1502 |
ultimately show "?D1 \<subset> ?D" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1503 |
by blast |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1504 |
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
|
1505 |
"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
|
1506 |
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
|
1507 |
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
|
1508 |
by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1509 |
have "\<exists>x. x \<in> ?D - ?D2" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1510 |
using assms(3-) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1511 |
unfolding division_points_def interval_bounds[OF ab] |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1512 |
apply - |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1513 |
apply (erule disjE) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1514 |
apply (rule_tac x="(k,(interval_lowerbound l)\<bullet>k)" in exI, force simp add: *) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1515 |
apply (rule_tac x="(k,(interval_upperbound l)\<bullet>k)" in exI, force simp add: *) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1516 |
done |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1517 |
moreover have "?D2 \<subseteq> ?D" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1518 |
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
|
1519 |
ultimately show "?D2 \<subset> ?D" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1520 |
by blast |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1521 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1522 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1523 |
lemma division_split_left_inj: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1524 |
fixes type :: "'a::euclidean_space" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1525 |
assumes "d division_of i" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1526 |
and "k1 \<in> d" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1527 |
and "k2 \<in> d" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1528 |
and "k1 \<noteq> k2" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1529 |
and "k1 \<inter> {x::'a. x\<bullet>k \<le> c} = k2 \<inter> {x. x\<bullet>k \<le> c}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1530 |
and k: "k\<in>Basis" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1531 |
shows "interior (k1 \<inter> {x. x\<bullet>k \<le> c}) = {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1532 |
proof - |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1533 |
note d=division_ofD[OF assms(1)] |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1534 |
guess u1 v1 using d(4)[OF assms(2)] by (elim exE) note uv1=this |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1535 |
guess u2 v2 using d(4)[OF assms(3)] by (elim exE) note uv2=this |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1536 |
have **: "\<And>s t u. s \<inter> t = {} \<Longrightarrow> u \<subseteq> s \<Longrightarrow> u \<subseteq> t \<Longrightarrow> u = {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1537 |
by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1538 |
show ?thesis |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1539 |
unfolding uv1 uv2 |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1540 |
apply (rule **[OF d(5)[OF assms(2-4)]]) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1541 |
apply (simp add: uv1) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1542 |
using assms(5) uv1 by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1543 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1544 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1545 |
lemma division_split_right_inj: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1546 |
fixes type :: "'a::euclidean_space" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1547 |
assumes "d division_of i" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1548 |
and "k1 \<in> d" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1549 |
and "k2 \<in> d" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1550 |
and "k1 \<noteq> k2" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1551 |
and "k1 \<inter> {x::'a. x\<bullet>k \<ge> c} = k2 \<inter> {x. x\<bullet>k \<ge> c}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1552 |
and k: "k \<in> Basis" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1553 |
shows "interior (k1 \<inter> {x. x\<bullet>k \<ge> c}) = {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1554 |
proof - |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1555 |
note d=division_ofD[OF assms(1)] |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1556 |
guess u1 v1 using d(4)[OF assms(2)] by (elim exE) note uv1=this |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1557 |
guess u2 v2 using d(4)[OF assms(3)] by (elim exE) note uv2=this |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1558 |
have **: "\<And>s t u. s \<inter> t = {} \<Longrightarrow> u \<subseteq> s \<Longrightarrow> u \<subseteq> t \<Longrightarrow> u = {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1559 |
by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1560 |
show ?thesis |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1561 |
unfolding uv1 uv2 |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1562 |
apply (rule **[OF d(5)[OF assms(2-4)]]) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1563 |
apply (simp add: uv1) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1564 |
using assms(5) uv1 by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1565 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1566 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1567 |
lemma interval_doublesplit: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1568 |
fixes a :: "'a::euclidean_space" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1569 |
assumes "k \<in> Basis" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1570 |
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
|
1571 |
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
|
1572 |
(\<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
|
1573 |
proof - |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1574 |
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
|
1575 |
by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1576 |
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
|
1577 |
by blast |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1578 |
show ?thesis |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1579 |
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
|
1580 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1581 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1582 |
lemma division_doublesplit: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1583 |
fixes a :: "'a::euclidean_space" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1584 |
assumes "p division_of (cbox a b)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1585 |
and k: "k \<in> Basis" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1586 |
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
|
1587 |
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
|
1588 |
proof - |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1589 |
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
|
1590 |
by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1591 |
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
|
1592 |
by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1593 |
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
|
1594 |
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
|
1595 |
then show ?thesis |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1596 |
apply (rule **) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1597 |
subgoal |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1598 |
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
|
1599 |
apply (rule equalityI) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1600 |
apply blast |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1601 |
apply clarsimp |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1602 |
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
|
1603 |
apply auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1604 |
done |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1605 |
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
|
1606 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1607 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1608 |
lemma (in comm_monoid_set) operative_division: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1609 |
fixes g :: "'b::euclidean_space set \<Rightarrow> 'a" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1610 |
assumes g: "operative g" and d: "d division_of (cbox a b)" shows "F g d = g (cbox a b)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1611 |
proof - |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1612 |
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
|
1613 |
then show ?thesis |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1614 |
using d |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1615 |
proof (induction C arbitrary: a b d rule: less_induct) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1616 |
case (less a b d) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1617 |
show ?case |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1618 |
proof cases |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1619 |
assume "box a b = {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1620 |
{ fix k assume "k\<in>d" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1621 |
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
|
1622 |
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
|
1623 |
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
|
1624 |
by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1625 |
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
|
1626 |
unfolding subset_box by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1627 |
then have "g k = \<^bold>1" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1628 |
using operativeD(1)[OF g, of a' b'] k by (simp add: \<open>box a b = {}\<close>) } |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1629 |
then show "box a b = {} \<Longrightarrow> F g d = g (cbox a b)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1630 |
by (auto intro!: neutral simp: operativeD(1)[OF g]) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1631 |
next |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1632 |
assume "box a b \<noteq> {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1633 |
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
|
1634 |
by (auto simp: box_ne_empty) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1635 |
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
|
1636 |
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
|
1637 |
case True |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1638 |
{ fix u v and j :: 'b |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1639 |
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
|
1640 |
then have "cbox u v \<noteq> {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1641 |
using less.prems by blast |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1642 |
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
|
1643 |
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
|
1644 |
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
|
1645 |
using as j by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1646 |
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
|
1647 |
"(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
|
1648 |
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
|
1649 |
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
|
1650 |
moreover |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1651 |
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
|
1652 |
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
|
1653 |
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
|
1654 |
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
|
1655 |
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
|
1656 |
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
|
1657 |
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
|
1658 |
(\<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
|
1659 |
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
|
1660 |
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
|
1661 |
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
|
1662 |
note this[unfolded division_ofD(6)[OF \<open>d division_of cbox a b\<close>,symmetric] Union_iff] |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1663 |
then guess i .. note i=this |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1664 |
guess u v using d'[rule_format,OF i(1)] by (elim exE conjE) note uv=this |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1665 |
have "cbox a b \<in> d" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1666 |
proof - |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1667 |
have "u = a" "v = b" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1668 |
unfolding euclidean_eq_iff[where 'a='b] |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1669 |
proof safe |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1670 |
fix j :: 'b |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1671 |
assume j: "j \<in> Basis" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1672 |
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
|
1673 |
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
|
1674 |
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
|
1675 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1676 |
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
|
1677 |
then show ?thesis using i by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1678 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1679 |
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
|
1680 |
by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1681 |
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
|
1682 |
proof (intro neutral ballI) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1683 |
fix x |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1684 |
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
|
1685 |
then have "x\<in>d" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1686 |
by auto note d'[rule_format,OF this] |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1687 |
then guess u v by (elim exE conjE) note uv=this |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1688 |
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
|
1689 |
using x[unfolded uv] by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1690 |
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
|
1691 |
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
|
1692 |
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
|
1693 |
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
|
1694 |
then have "box u v = {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1695 |
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
|
1696 |
then show "g x = \<^bold>1" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1697 |
unfolding uv(1) by (rule operativeD(1)[OF g]) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1698 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1699 |
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
|
1700 |
using division_ofD[OF less.prems] |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1701 |
apply (subst deq) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1702 |
apply (subst insert) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1703 |
apply auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1704 |
done |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1705 |
next |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1706 |
case False |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1707 |
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
|
1708 |
by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1709 |
then guess k c |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1710 |
unfolding split_paired_Ex division_points_def mem_Collect_eq split_conv |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1711 |
apply (elim exE conjE) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1712 |
done |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1713 |
note this(2-4,1) note kc=this[unfolded interval_bounds[OF ab']] |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1714 |
from this(3) guess j .. note j=this |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1715 |
define d1 where "d1 = {l \<inter> {x. x\<bullet>k \<le> c} | l. l \<in> d \<and> l \<inter> {x. x\<bullet>k \<le> c} \<noteq> {}}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1716 |
define d2 where "d2 = {l \<inter> {x. x\<bullet>k \<ge> c} | l. l \<in> d \<and> l \<inter> {x. x\<bullet>k \<ge> c} \<noteq> {}}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1717 |
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
|
1718 |
define ca where "ca = (\<Sum>i\<in>Basis. (if i = k then c else a\<bullet>i) *\<^sub>R i)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1719 |
note division_points_psubset[OF \<open>d division_of cbox a b\<close> ab kc(1-2) j] |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1720 |
note psubset_card_mono[OF _ this(1)] psubset_card_mono[OF _ this(2)] |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1721 |
then have *: "F g d1 = g (cbox a b \<inter> {x. x\<bullet>k \<le> c})" "F g d2 = g (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
|
1722 |
unfolding interval_split[OF kc(4)] |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1723 |
apply (rule_tac[!] "less.hyps"[rule_format]) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1724 |
using division_split[OF \<open>d division_of cbox a b\<close>, where k=k and c=c] |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1725 |
apply (simp_all add: interval_split kc d1_def d2_def division_points_finite[OF \<open>d division_of cbox a b\<close>]) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1726 |
done |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1727 |
{ fix l y |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1728 |
assume as: "l \<in> d" "y \<in> d" "l \<inter> {x. x \<bullet> k \<le> c} = y \<inter> {x. x \<bullet> k \<le> c}" "l \<noteq> y" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1729 |
from division_ofD(4)[OF \<open>d division_of cbox a b\<close> this(1)] guess u v by (elim exE) note leq=this |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1730 |
have "g (l \<inter> {x. x \<bullet> k \<le> c}) = \<^bold>1" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1731 |
unfolding leq interval_split[OF kc(4)] |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1732 |
apply (rule operativeD[OF g]) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1733 |
unfolding interior_cbox[symmetric] interval_split[symmetric, OF kc(4)] |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1734 |
using division_split_left_inj less as kc leq by blast |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1735 |
} note fxk_le = this |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1736 |
{ fix l y |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1737 |
assume as: "l \<in> d" "y \<in> d" "l \<inter> {x. c \<le> x \<bullet> k} = y \<inter> {x. c \<le> x \<bullet> k}" "l \<noteq> y" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1738 |
from division_ofD(4)[OF \<open>d division_of cbox a b\<close> this(1)] guess u v by (elim exE) note leq=this |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1739 |
have "g (l \<inter> {x. x \<bullet> k \<ge> c}) = \<^bold>1" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1740 |
unfolding leq interval_split[OF kc(4)] |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1741 |
apply (rule operativeD(1)[OF g]) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1742 |
unfolding interior_cbox[symmetric] interval_split[symmetric,OF kc(4)] |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1743 |
using division_split_right_inj less leq as kc by blast |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1744 |
} note fxk_ge = this |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1745 |
have d1_alt: "d1 = (\<lambda>l. l \<inter> {x. x\<bullet>k \<le> c}) ` {l \<in> d. l \<inter> {x. x\<bullet>k \<le> c} \<noteq> {}}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1746 |
using d1_def by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1747 |
have d2_alt: "d2 = (\<lambda>l. l \<inter> {x. x\<bullet>k \<ge> c}) ` {l \<in> d. l \<inter> {x. x\<bullet>k \<ge> c} \<noteq> {}}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1748 |
using d2_def by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1749 |
have "g (cbox a b) = F g d1 \<^bold>* F g d2" (is "_ = ?prev") |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1750 |
unfolding * using g kc(4) by blast |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1751 |
also have "F g d1 = F (\<lambda>l. g (l \<inter> {x. x\<bullet>k \<le> c})) d" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1752 |
unfolding d1_alt using division_of_finite[OF less.prems] fxk_le |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1753 |
by (subst reindex_nontrivial) (auto intro!: mono_neutral_cong_left simp: operative_empty[OF g]) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1754 |
also have "F g d2 = F (\<lambda>l. g (l \<inter> {x. x\<bullet>k \<ge> c})) d" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1755 |
unfolding d2_alt using division_of_finite[OF less.prems] fxk_ge |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1756 |
by (subst reindex_nontrivial) (auto intro!: mono_neutral_cong_left simp: operative_empty[OF g]) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1757 |
also have *: "\<forall>x\<in>d. g x = g (x \<inter> {x. x \<bullet> k \<le> c}) \<^bold>* g (x \<inter> {x. c \<le> x \<bullet> k})" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1758 |
unfolding forall_in_division[OF \<open>d division_of cbox a b\<close>] |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1759 |
using g kc(4) by blast |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1760 |
have "F (\<lambda>l. g (l \<inter> {x. x\<bullet>k \<le> c})) d \<^bold>* F (\<lambda>l. g (l \<inter> {x. x\<bullet>k \<ge> c})) d = F g d" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1761 |
using * by (simp add: distrib) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1762 |
finally show ?thesis by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1763 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1764 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1765 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1766 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1767 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1768 |
lemma (in comm_monoid_set) over_tagged_division_lemma: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1769 |
assumes "p tagged_division_of i" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1770 |
and "\<And>u v. cbox u v \<noteq> {} \<Longrightarrow> box u v = {} \<Longrightarrow> d (cbox u v) = \<^bold>1" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1771 |
shows "F (\<lambda>(x,k). d k) p = F d (snd ` p)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1772 |
proof - |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1773 |
have *: "(\<lambda>(x,k). d k) = d \<circ> snd" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1774 |
unfolding o_def by (rule ext) auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1775 |
note assm = tagged_division_ofD[OF assms(1)] |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1776 |
show ?thesis |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1777 |
unfolding * |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1778 |
proof (rule reindex_nontrivial[symmetric]) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1779 |
show "finite p" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1780 |
using assm by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1781 |
fix x y |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1782 |
assume "x\<in>p" "y\<in>p" "x\<noteq>y" "snd x = snd y" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1783 |
obtain a b where ab: "snd x = cbox a b" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1784 |
using assm(4)[of "fst x" "snd x"] \<open>x\<in>p\<close> by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1785 |
have "(fst x, snd y) \<in> p" "(fst x, snd y) \<noteq> y" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1786 |
by (metis prod.collapse \<open>x\<in>p\<close> \<open>snd x = snd y\<close> \<open>x \<noteq> y\<close>)+ |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1787 |
with \<open>x\<in>p\<close> \<open>y\<in>p\<close> have "interior (snd x) \<inter> interior (snd y) = {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1788 |
by (intro assm(5)[of "fst x" _ "fst y"]) auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1789 |
then have "box a b = {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1790 |
unfolding \<open>snd x = snd y\<close>[symmetric] ab by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1791 |
then have "d (cbox a b) = \<^bold>1" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1792 |
using assm(2)[of "fst x" "snd x"] \<open>x\<in>p\<close> ab[symmetric] by (intro assms(2)) auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1793 |
then show "d (snd x) = \<^bold>1" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1794 |
unfolding ab by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1795 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1796 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1797 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1798 |
lemma (in comm_monoid_set) operative_tagged_division: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1799 |
assumes f: "operative g" and d: "d tagged_division_of (cbox a b)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1800 |
shows "F (\<lambda>(x, l). g l) d = g (cbox a b)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1801 |
unfolding d[THEN division_of_tagged_division, THEN operative_division[OF f], symmetric] |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1802 |
by (simp add: f[THEN operativeD(1)] over_tagged_division_lemma[OF d]) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1803 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1804 |
lemma interval_real_split: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1805 |
"{a .. b::real} \<inter> {x. x \<le> c} = {a .. min b c}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1806 |
"{a .. b} \<inter> {x. c \<le> x} = {max a c .. b}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1807 |
apply (metis Int_atLeastAtMostL1 atMost_def) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1808 |
apply (metis Int_atLeastAtMostL2 atLeast_def) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1809 |
done |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1810 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1811 |
lemma (in comm_monoid) operative_1_lt: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1812 |
"operative (g :: real set \<Rightarrow> 'a) \<longleftrightarrow> |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1813 |
((\<forall>a b. b \<le> a \<longrightarrow> g {a .. b} = \<^bold>1) \<and> (\<forall>a b c. a < c \<and> 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
|
1814 |
apply (simp add: operative_def atMost_def[symmetric] atLeast_def[symmetric]) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1815 |
proof safe |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1816 |
fix a b c :: real |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1817 |
assume *: "\<forall>a b c. g {a..b} = g {a..min b c} \<^bold>* g {max a c..b}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1818 |
assume "a < c" "c < b" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1819 |
with *[rule_format, of a b c] show "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
|
1820 |
by (simp add: less_imp_le min.absorb2 max.absorb2) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1821 |
next |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1822 |
fix a b c :: real |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1823 |
assume as: "\<forall>a b. b \<le> a \<longrightarrow> g {a..b} = \<^bold>1" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1824 |
"\<forall>a b c. a < c \<and> 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
|
1825 |
from as(1)[rule_format, of 0 1] as(1)[rule_format, of a a for a] as(2) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1826 |
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
|
1827 |
"\<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
|
1828 |
by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1829 |
show "g {a..b} = g {a..min b c} \<^bold>* g {max a c..b}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1830 |
by (auto simp: min_def max_def le_less) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1831 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1832 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1833 |
lemma (in comm_monoid) operative_1_le: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1834 |
"operative (g :: real set \<Rightarrow> 'a) \<longleftrightarrow> |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1835 |
((\<forall>a b. b \<le> a \<longrightarrow> g {a..b} = \<^bold>1) \<and> (\<forall>a b c. a \<le> c \<and> c \<le> 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
|
1836 |
unfolding operative_1_lt |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1837 |
proof safe |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1838 |
fix a b c :: real |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1839 |
assume as: "\<forall>a b c. a \<le> c \<and> c \<le> b \<longrightarrow> g {a..c} \<^bold>* g {c..b} = g {a..b}" "a < c" "c < b" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1840 |
show "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
|
1841 |
apply (rule as(1)[rule_format]) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1842 |
using as(2-) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1843 |
apply auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1844 |
done |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1845 |
next |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1846 |
fix a b c :: real |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1847 |
assume "\<forall>a b. b \<le> a \<longrightarrow> g {a .. b} = \<^bold>1" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1848 |
and "\<forall>a b c. a < c \<and> 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
|
1849 |
and "a \<le> c" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1850 |
and "c \<le> b" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1851 |
note as = this[rule_format] |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1852 |
show "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
|
1853 |
proof (cases "c = a \<or> c = b") |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1854 |
case False |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1855 |
then show ?thesis |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1856 |
apply - |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1857 |
apply (subst as(2)) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1858 |
using as(3-) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1859 |
apply auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1860 |
done |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1861 |
next |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1862 |
case True |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1863 |
then show ?thesis |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1864 |
proof |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1865 |
assume *: "c = a" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1866 |
then have "g {a .. c} = \<^bold>1" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1867 |
apply - |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1868 |
apply (rule as(1)[rule_format]) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1869 |
apply auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1870 |
done |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1871 |
then show ?thesis |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1872 |
unfolding * by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1873 |
next |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1874 |
assume *: "c = b" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1875 |
then have "g {c .. b} = \<^bold>1" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1876 |
apply - |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1877 |
apply (rule as(1)[rule_format]) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1878 |
apply auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1879 |
done |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1880 |
then show ?thesis |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1881 |
unfolding * by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1882 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1883 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1884 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1885 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1886 |
lemma tagged_division_union_interval: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1887 |
fixes a :: "'a::euclidean_space" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1888 |
assumes "p1 tagged_division_of (cbox a b \<inter> {x. x\<bullet>k \<le> (c::real)})" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1889 |
and "p2 tagged_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
|
1890 |
and k: "k \<in> Basis" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1891 |
shows "(p1 \<union> p2) tagged_division_of (cbox a b)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1892 |
proof - |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1893 |
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})" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1894 |
by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1895 |
show ?thesis |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1896 |
apply (subst *) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1897 |
apply (rule tagged_division_union[OF assms(1-2)]) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1898 |
unfolding interval_split[OF k] interior_cbox |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1899 |
using k |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1900 |
apply (auto simp add: box_def elim!: ballE[where x=k]) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1901 |
done |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1902 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1903 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1904 |
lemma tagged_division_union_interval_real: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1905 |
fixes a :: real |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1906 |
assumes "p1 tagged_division_of ({a .. b} \<inter> {x. x\<bullet>k \<le> (c::real)})" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1907 |
and "p2 tagged_division_of ({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
|
1908 |
and k: "k \<in> Basis" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1909 |
shows "(p1 \<union> p2) tagged_division_of {a .. b}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1910 |
using assms |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1911 |
unfolding box_real[symmetric] |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1912 |
by (rule tagged_division_union_interval) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1913 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1914 |
lemma tagged_division_split_left_inj: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1915 |
"d tagged_division_of i \<Longrightarrow> (x1, k1) \<in> d \<Longrightarrow> (x2, k2) \<in> d \<Longrightarrow> k1 \<noteq> k2 \<Longrightarrow> |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1916 |
k1 \<inter> {x. x\<bullet>k \<le> c} = k2 \<inter> {x. x\<bullet>k \<le> c} \<Longrightarrow> k \<in> Basis \<Longrightarrow> |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1917 |
interior (k1 \<inter> {x. x\<bullet>k \<le> c}) = {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1918 |
by (intro division_split_left_inj[of "snd`d" i k1 k2, OF division_of_tagged_division]) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1919 |
(auto simp add: snd_def[abs_def] image_iff split: prod.split ) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1920 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1921 |
lemma tagged_division_split_right_inj: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1922 |
"d tagged_division_of i \<Longrightarrow> (x1, k1) \<in> d \<Longrightarrow> (x2, k2) \<in> d \<Longrightarrow> k1 \<noteq> k2 \<Longrightarrow> |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1923 |
k1 \<inter> {x. x\<bullet>k \<ge> c} = k2 \<inter> {x. x\<bullet>k \<ge> c} \<Longrightarrow> k \<in> Basis \<Longrightarrow> |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1924 |
interior (k1 \<inter> {x. x\<bullet>k \<ge> c}) = {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1925 |
by (intro division_split_right_inj[of "snd`d" i k1 k2, OF division_of_tagged_division]) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1926 |
(auto simp add: snd_def[abs_def] image_iff split: prod.split ) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1927 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1928 |
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
|
1929 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1930 |
lemma additive_tagged_division_1: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1931 |
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
|
1932 |
assumes "a \<le> b" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1933 |
and "p tagged_division_of {a..b}" |
64267 | 1934 |
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
|
1935 |
proof - |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1936 |
let ?f = "(\<lambda>k::(real) set. if k = {} then 0 else f(interval_upperbound k) - f(interval_lowerbound k))" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1937 |
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
|
1938 |
using assms by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1939 |
have *: "add.operative ?f" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1940 |
unfolding add.operative_1_lt box_eq_empty |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1941 |
by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1942 |
have **: "cbox a b \<noteq> {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1943 |
using assms(1) by auto |
64267 | 1944 |
note sum.operative_tagged_division[OF * assms(2)[simplified box_real[symmetric]]] |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1945 |
note * = this[unfolded if_not_P[OF **] interval_bounds[OF ***],symmetric] |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1946 |
show ?thesis |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1947 |
unfolding * |
64267 | 1948 |
apply (rule sum.cong) |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1949 |
unfolding split_paired_all split_conv |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1950 |
using assms(2) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1951 |
apply auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1952 |
done |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1953 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1954 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1955 |
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)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1956 |
by (meson zero_less_one) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1957 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1958 |
lemma additive_tagged_division_1': |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1959 |
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
|
1960 |
assumes "a \<le> b" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1961 |
and "p tagged_division_of {a..b}" |
64267 | 1962 |
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
|
1963 |
using additive_tagged_division_1[OF _ assms(2), of f] |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1964 |
using assms(1) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1965 |
by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1966 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1967 |
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
|
1968 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1969 |
definition fine (infixr "fine" 46) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1970 |
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
|
1971 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1972 |
lemma fineI: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1973 |
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
|
1974 |
shows "d fine s" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1975 |
using assms unfolding fine_def by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1976 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1977 |
lemma fineD[dest]: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1978 |
assumes "d fine s" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1979 |
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
|
1980 |
using assms unfolding fine_def by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1981 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1982 |
lemma fine_inter: "(\<lambda>x. d1 x \<inter> d2 x) fine p \<longleftrightarrow> d1 fine p \<and> d2 fine p" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1983 |
unfolding fine_def by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1984 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1985 |
lemma fine_inters: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1986 |
"(\<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
|
1987 |
unfolding fine_def by blast |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1988 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1989 |
lemma fine_union: "d fine p1 \<Longrightarrow> d fine p2 \<Longrightarrow> d fine (p1 \<union> p2)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1990 |
unfolding fine_def by blast |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1991 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1992 |
lemma fine_unions: "(\<And>p. p \<in> ps \<Longrightarrow> d fine p) \<Longrightarrow> d fine (\<Union>ps)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1993 |
unfolding fine_def by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1994 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1995 |
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
|
1996 |
unfolding fine_def by blast |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1997 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1998 |
subsection \<open>Some basic combining lemmas.\<close> |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
1999 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2000 |
lemma tagged_division_unions_exists: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2001 |
assumes "finite iset" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2002 |
and "\<forall>i\<in>iset. \<exists>p. 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
|
2003 |
and "\<forall>i1\<in>iset. \<forall>i2\<in>iset. i1 \<noteq> i2 \<longrightarrow> interior i1 \<inter> interior i2 = {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2004 |
and "\<Union>iset = i" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2005 |
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
|
2006 |
proof - |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2007 |
obtain pfn where pfn: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2008 |
"\<And>x. x \<in> iset \<Longrightarrow> pfn x tagged_division_of x" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2009 |
"\<And>x. x \<in> iset \<Longrightarrow> d fine pfn x" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2010 |
using bchoice[OF assms(2)] by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2011 |
show thesis |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2012 |
apply (rule_tac p="\<Union>(pfn ` iset)" in that) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2013 |
using assms(1) assms(3) assms(4) pfn(1) tagged_division_unions apply force |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2014 |
by (metis (mono_tags, lifting) fine_unions imageE pfn(2)) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2015 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2016 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2017 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2018 |
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
|
2019 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2020 |
lemma division_of_closed: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2021 |
fixes i :: "'n::euclidean_space set" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2022 |
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
|
2023 |
unfolding division_of_def by fastforce |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2024 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2025 |
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
|
2026 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2027 |
lemma interval_bisection_step: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2028 |
fixes type :: "'a::euclidean_space" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2029 |
assumes "P {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2030 |
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
|
2031 |
and "\<not> P (cbox a (b::'a))" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2032 |
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
|
2033 |
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
|
2034 |
proof - |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2035 |
have "cbox a b \<noteq> {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2036 |
using assms(1,3) by metis |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2037 |
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
|
2038 |
by (force simp: mem_box) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2039 |
{ fix f |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2040 |
have "\<lbrakk>finite f; |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2041 |
\<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
|
2042 |
\<And>s. s\<in>f \<Longrightarrow> \<exists>a b. s = cbox a b; |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2043 |
\<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)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2044 |
proof (induct f rule: finite_induct) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2045 |
case empty |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2046 |
show ?case |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2047 |
using assms(1) by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2048 |
next |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2049 |
case (insert x f) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2050 |
show ?case |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2051 |
unfolding Union_insert |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2052 |
apply (rule assms(2)[rule_format]) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2053 |
using inter_interior_unions_intervals [of f "interior x"] |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2054 |
apply (auto simp: insert) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2055 |
by (metis IntI empty_iff insert.hyps(2) insert.prems(3) insert_iff) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2056 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2057 |
} note UN_cases = this |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2058 |
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
|
2059 |
(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
|
2060 |
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
|
2061 |
{ |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2062 |
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
|
2063 |
then show thesis |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2064 |
unfolding atomize_not not_all |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2065 |
by (blast intro: that) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2066 |
} |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2067 |
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
|
2068 |
have "P (\<Union>?A)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2069 |
proof (rule UN_cases) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2070 |
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
|
2071 |
(\<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
|
2072 |
have "?A \<subseteq> ?B" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2073 |
proof |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2074 |
fix x |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2075 |
assume "x \<in> ?A" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2076 |
then obtain c d |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2077 |
where x: "x = cbox c d" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2078 |
"\<And>i. i \<in> Basis \<Longrightarrow> |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2079 |
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
|
2080 |
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
|
2081 |
show "x \<in> ?B" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2082 |
unfolding image_iff x |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2083 |
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
|
2084 |
apply (rule arg_cong2 [where f = cbox]) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2085 |
using x(2) ab |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2086 |
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
|
2087 |
by fastforce |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2088 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2089 |
then show "finite ?A" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2090 |
by (rule finite_subset) auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2091 |
next |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2092 |
fix s |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2093 |
assume "s \<in> ?A" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2094 |
then obtain c d |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2095 |
where s: "s = cbox c d" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2096 |
"\<And>i. i \<in> Basis \<Longrightarrow> |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2097 |
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
|
2098 |
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
|
2099 |
by blast |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2100 |
show "P s" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2101 |
unfolding s |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2102 |
apply (rule as[rule_format]) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2103 |
using ab s(2) by force |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2104 |
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
|
2105 |
unfolding s by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2106 |
fix t |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2107 |
assume "t \<in> ?A" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2108 |
then obtain e f where t: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2109 |
"t = cbox e f" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2110 |
"\<And>i. i \<in> Basis \<Longrightarrow> |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2111 |
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
|
2112 |
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
|
2113 |
by blast |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2114 |
assume "s \<noteq> t" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2115 |
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
|
2116 |
unfolding s t by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2117 |
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
|
2118 |
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
|
2119 |
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
|
2120 |
using s(2) t(2) apply fastforce |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2121 |
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
|
2122 |
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
|
2123 |
by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2124 |
show "interior s \<inter> interior t = {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2125 |
unfolding s t interior_cbox |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2126 |
proof (rule *) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2127 |
fix x |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2128 |
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
|
2129 |
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
|
2130 |
unfolding mem_box using i' |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2131 |
by force+ |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2132 |
show False using s(2)[OF i'] |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2133 |
proof safe |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2134 |
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
|
2135 |
show False |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2136 |
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
|
2137 |
next |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2138 |
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
|
2139 |
show False |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2140 |
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
|
2141 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2142 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2143 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2144 |
also have "\<Union>?A = cbox a b" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2145 |
proof (rule set_eqI,rule) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2146 |
fix x |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2147 |
assume "x \<in> \<Union>?A" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2148 |
then obtain c d where x: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2149 |
"x \<in> cbox c d" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2150 |
"\<And>i. i \<in> Basis \<Longrightarrow> |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2151 |
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
|
2152 |
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
|
2153 |
by blast |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2154 |
show "x\<in>cbox a b" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2155 |
unfolding mem_box |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2156 |
proof safe |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2157 |
fix i :: 'a |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2158 |
assume i: "i \<in> Basis" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2159 |
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
|
2160 |
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
|
2161 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2162 |
next |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2163 |
fix x |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2164 |
assume x: "x \<in> cbox a b" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2165 |
have "\<forall>i\<in>Basis. |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2166 |
\<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
|
2167 |
(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
|
2168 |
unfolding mem_box |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2169 |
proof |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2170 |
fix i :: 'a |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2171 |
assume i: "i \<in> Basis" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2172 |
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
|
2173 |
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
|
2174 |
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
|
2175 |
by blast |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2176 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2177 |
then show "x\<in>\<Union>?A" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2178 |
unfolding Union_iff Bex_def mem_Collect_eq choice_Basis_iff |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2179 |
apply auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2180 |
apply (rule_tac x="cbox xa xaa" in exI) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2181 |
unfolding mem_box |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2182 |
apply auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2183 |
done |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2184 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2185 |
finally show False |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2186 |
using assms by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2187 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2188 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2189 |
lemma interval_bisection: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2190 |
fixes type :: "'a::euclidean_space" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2191 |
assumes "P {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2192 |
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
|
2193 |
and "\<not> P (cbox a (b::'a))" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2194 |
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
|
2195 |
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
|
2196 |
proof - |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2197 |
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
|
2198 |
(\<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
|
2199 |
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
|
2200 |
proof |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2201 |
show "?P x" for x |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2202 |
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
|
2203 |
case True |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2204 |
then show ?thesis by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2205 |
next |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2206 |
case as: False |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2207 |
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
|
2208 |
"\<forall>i\<in>Basis. |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2209 |
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
|
2210 |
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
|
2211 |
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
|
2212 |
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
|
2213 |
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
|
2214 |
then show ?thesis |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2215 |
apply - |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2216 |
apply (rule_tac x="(c,d)" in exI) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2217 |
apply auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2218 |
done |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2219 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2220 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2221 |
then obtain f where f: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2222 |
"\<forall>x. |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2223 |
\<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
|
2224 |
\<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
|
2225 |
(\<forall>i\<in>Basis. |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2226 |
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
|
2227 |
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
|
2228 |
snd (f x) \<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
|
2229 |
2 * (snd (f x) \<bullet> i - fst (f x) \<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
|
2230 |
apply - |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2231 |
apply (drule choice) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2232 |
apply blast |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2233 |
done |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2234 |
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 |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2235 |
have "A 0 = a" "B 0 = b" "\<And>n. \<not> P (cbox (A(Suc n)) (B(Suc n))) \<and> |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2236 |
(\<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
|
2237 |
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
|
2238 |
proof - |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2239 |
show "A 0 = a" "B 0 = b" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2240 |
unfolding ab_def by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2241 |
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
|
2242 |
show "?P n" for n |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2243 |
proof (induct n) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2244 |
case 0 |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2245 |
then show ?case |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2246 |
unfolding S |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2247 |
apply (rule f[rule_format]) using assms(3) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2248 |
apply auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2249 |
done |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2250 |
next |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2251 |
case (Suc n) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2252 |
show ?case |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2253 |
unfolding S |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2254 |
apply (rule f[rule_format]) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2255 |
using Suc |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2256 |
unfolding S |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2257 |
apply auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2258 |
done |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2259 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2260 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2261 |
note AB = this(1-2) conjunctD2[OF this(3),rule_format] |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2262 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2263 |
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
|
2264 |
if e: "0 < e" for e |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2265 |
proof - |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2266 |
obtain n where n: "(\<Sum>i\<in>Basis. b \<bullet> i - a \<bullet> i) / e < 2 ^ n" |
64267 | 2267 |
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
|
2268 |
show ?thesis |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2269 |
proof (rule exI [where x=n], clarify) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2270 |
fix x y |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2271 |
assume xy: "x\<in>cbox (A n) (B n)" "y\<in>cbox (A n) (B n)" |
64267 | 2272 |
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
|
2273 |
unfolding dist_norm by(rule norm_le_l1) |
64267 | 2274 |
also have "\<dots> \<le> sum (\<lambda>i. B n\<bullet>i - A n\<bullet>i) Basis" |
2275 |
proof (rule sum_mono) |
|
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2276 |
fix i :: 'a |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2277 |
assume i: "i \<in> Basis" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2278 |
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
|
2279 |
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
|
2280 |
by (auto simp: inner_diff_left) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2281 |
qed |
64267 | 2282 |
also have "\<dots> \<le> sum (\<lambda>i. b\<bullet>i - a\<bullet>i) Basis / 2^n" |
2283 |
unfolding sum_divide_distrib |
|
2284 |
proof (rule sum_mono) |
|
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2285 |
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
|
2286 |
proof (induct n) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2287 |
case 0 |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2288 |
then show ?case |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2289 |
unfolding AB by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2290 |
next |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2291 |
case (Suc n) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2292 |
have "B (Suc n) \<bullet> i - A (Suc n) \<bullet> i \<le> (B n \<bullet> i - A n \<bullet> i) / 2" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2293 |
using AB(4)[of i n] using i by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2294 |
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
|
2295 |
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
|
2296 |
finally show ?case . |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2297 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2298 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2299 |
also have "\<dots> < e" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2300 |
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
|
2301 |
finally show "dist x y < e" . |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2302 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2303 |
qed |
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 n m :: nat |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2306 |
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
|
2307 |
proof (induction rule: inc_induct) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2308 |
case (step i) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2309 |
show ?case |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2310 |
using AB(4) by (intro order_trans[OF step.IH] subset_box_imp) auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2311 |
qed simp |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2312 |
} note ABsubset = this |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2313 |
have "\<exists>a. \<forall>n. a\<in> cbox (A n) (B n)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2314 |
by (rule decreasing_closed_nest[rule_format,OF closed_cbox _ ABsubset interv]) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2315 |
(metis nat.exhaust AB(1-3) assms(1,3)) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2316 |
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
|
2317 |
by blast |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2318 |
show thesis |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2319 |
proof (rule that[rule_format, of x0]) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2320 |
show "x0\<in>cbox a b" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2321 |
using x0[of 0] unfolding AB . |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2322 |
fix e :: real |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2323 |
assume "e > 0" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2324 |
from interv[OF this] obtain n |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2325 |
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
|
2326 |
have "\<not> P (cbox (A n) (B n))" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2327 |
apply (cases "0 < n") |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2328 |
using AB(3)[of "n - 1"] assms(3) AB(1-2) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2329 |
apply auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2330 |
done |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2331 |
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
|
2332 |
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
|
2333 |
moreover have "cbox (A n) (B n) \<subseteq> cbox a b" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2334 |
unfolding AB(1-2)[symmetric] by (rule ABsubset) auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2335 |
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
|
2336 |
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
|
2337 |
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
|
2338 |
apply (auto simp: x0) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2339 |
done |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2340 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2341 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2342 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2343 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2344 |
subsection \<open>Cousin's lemma.\<close> |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2345 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2346 |
lemma fine_division_exists: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2347 |
fixes a b :: "'a::euclidean_space" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2348 |
assumes "gauge g" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2349 |
obtains p where "p tagged_division_of (cbox a b)" "g fine p" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2350 |
proof - |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2351 |
presume "\<not> (\<exists>p. p tagged_division_of (cbox a b) \<and> g fine p) \<Longrightarrow> False" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2352 |
then obtain p where "p tagged_division_of (cbox a b)" "g fine p" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2353 |
by blast |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2354 |
then show thesis .. |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2355 |
next |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2356 |
assume as: "\<not> (\<exists>p. p tagged_division_of (cbox a b) \<and> g fine p)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2357 |
obtain x where x: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2358 |
"x \<in> (cbox a b)" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2359 |
"\<And>e. 0 < e \<Longrightarrow> |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2360 |
\<exists>c d. |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2361 |
x \<in> cbox c d \<and> |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2362 |
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
|
2363 |
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
|
2364 |
\<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
|
2365 |
apply (rule interval_bisection[of "\<lambda>s. \<exists>p. p tagged_division_of s \<and> g fine p", OF _ _ as]) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2366 |
apply (simp add: fine_def) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2367 |
apply (metis tagged_division_union fine_union) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2368 |
apply (auto simp: ) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2369 |
done |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2370 |
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
|
2371 |
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
|
2372 |
from x(2)[OF e(1)] |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2373 |
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
|
2374 |
"cbox c d \<subseteq> ball x e" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2375 |
"cbox c d \<subseteq> cbox a b" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2376 |
"\<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
|
2377 |
by blast |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2378 |
have "g fine {(x, cbox c d)}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2379 |
unfolding fine_def using e using c_d(2) by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2380 |
then show False |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2381 |
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
|
2382 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2383 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2384 |
lemma fine_division_exists_real: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2385 |
fixes a b :: real |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2386 |
assumes "gauge g" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2387 |
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
|
2388 |
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
|
2389 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2390 |
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
|
2391 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2392 |
lemma tagged_division_finer: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2393 |
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
|
2394 |
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
|
2395 |
and "gauge d" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2396 |
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
|
2397 |
and "d fine q" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2398 |
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
|
2399 |
proof - |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2400 |
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
|
2401 |
(\<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
|
2402 |
(\<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
|
2403 |
{ |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2404 |
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
|
2405 |
using assms(1) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2406 |
unfolding tagged_division_of_def |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2407 |
by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2408 |
presume "\<And>p. finite p \<Longrightarrow> ?P p" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2409 |
from this[rule_format,OF * assms(2)] guess q .. note q=this |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2410 |
then show ?thesis |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2411 |
apply - |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2412 |
apply (rule that[of q]) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2413 |
unfolding tagged_division_ofD[OF assms(1)] |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2414 |
apply auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2415 |
done |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2416 |
} |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2417 |
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
|
2418 |
assume as: "finite p" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2419 |
show "?P p" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2420 |
apply rule |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2421 |
apply rule |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2422 |
using as |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2423 |
proof (induct p) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2424 |
case empty |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2425 |
show ?case |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2426 |
apply (rule_tac x="{}" in exI) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2427 |
unfolding fine_def |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2428 |
apply auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2429 |
done |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2430 |
next |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2431 |
case (insert xk p) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2432 |
guess x k using surj_pair[of xk] by (elim exE) note xk=this |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2433 |
note tagged_partial_division_subset[OF insert(4) subset_insertI] |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2434 |
from insert(3)[OF this insert(5)] guess q1 .. note q1 = conjunctD3[OF this] |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2435 |
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
|
2436 |
unfolding xk by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2437 |
note p = tagged_partial_division_ofD[OF insert(4)] |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2438 |
from p(4)[unfolded xk, OF insertI1] guess u v by (elim exE) note uv=this |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2439 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2440 |
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
|
2441 |
apply (rule finite_subset[of _ "snd ` p"]) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2442 |
using p |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2443 |
apply safe |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2444 |
apply (metis image_iff snd_conv) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2445 |
apply auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2446 |
done |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2447 |
then have int: "interior (cbox u v) \<inter> interior (\<Union>{k. \<exists>x. (x, k) \<in> p}) = {}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2448 |
apply (rule inter_interior_unions_intervals) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2449 |
apply (rule open_interior) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2450 |
apply (rule_tac[!] ballI) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2451 |
unfolding mem_Collect_eq |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2452 |
apply (erule_tac[!] exE) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2453 |
apply (drule p(4)[OF insertI2]) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2454 |
apply assumption |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2455 |
apply (rule p(5)) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2456 |
unfolding uv xk |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2457 |
apply (rule insertI1) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2458 |
apply (rule insertI2) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2459 |
apply assumption |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2460 |
using insert(2) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2461 |
unfolding uv xk |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2462 |
apply auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2463 |
done |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2464 |
show ?case |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2465 |
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
|
2466 |
case True |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2467 |
then show ?thesis |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2468 |
apply (rule_tac x="{(x,cbox u v)} \<union> q1" in exI) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2469 |
apply rule |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2470 |
unfolding * uv |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2471 |
apply (rule tagged_division_union) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2472 |
apply (rule tagged_division_of_self) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2473 |
apply (rule p[unfolded xk uv] insertI1)+ |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2474 |
apply (rule q1) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2475 |
apply (rule int) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2476 |
apply rule |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2477 |
apply (rule fine_union) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2478 |
apply (subst fine_def) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2479 |
defer |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2480 |
apply (rule q1) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2481 |
unfolding Ball_def split_paired_All split_conv |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2482 |
apply rule |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2483 |
apply rule |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2484 |
apply rule |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2485 |
apply rule |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2486 |
apply (erule insertE) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2487 |
apply (simp add: uv xk) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2488 |
apply (rule UnI2) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2489 |
apply (drule q1(3)[rule_format]) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2490 |
unfolding xk uv |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2491 |
apply auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2492 |
done |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2493 |
next |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2494 |
case False |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2495 |
from fine_division_exists[OF assms(2), of u v] guess q2 . note q2=this |
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 |
apply (rule_tac x="q2 \<union> q1" in exI) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2498 |
apply rule |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2499 |
unfolding * uv |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2500 |
apply (rule tagged_division_union q2 q1 int fine_union)+ |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2501 |
unfolding Ball_def split_paired_All split_conv |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2502 |
apply rule |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2503 |
apply (rule fine_union) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2504 |
apply (rule q1 q2)+ |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2505 |
apply rule |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2506 |
apply rule |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2507 |
apply rule |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2508 |
apply rule |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2509 |
apply (erule insertE) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2510 |
apply (rule UnI2) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2511 |
apply (simp add: False uv xk) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2512 |
apply (drule q1(3)[rule_format]) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2513 |
using False |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2514 |
unfolding xk uv |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2515 |
apply auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2516 |
done |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2517 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2518 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2519 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2520 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2521 |
subsubsection \<open>Covering lemma\<close> |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2522 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2523 |
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
|
2524 |
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
|
2525 |
"Introduction to Gauge Integrals". \<close> |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2526 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2527 |
proposition covering_lemma: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2528 |
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
|
2529 |
obtains \<D> where |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2530 |
"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
|
2531 |
"\<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
|
2532 |
"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
|
2533 |
"\<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
|
2534 |
"\<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
|
2535 |
"S \<subseteq> \<Union>\<D>" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2536 |
proof - |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2537 |
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
|
2538 |
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
|
2539 |
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
|
2540 |
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
|
2541 |
(\<Sum>i \<in> Basis. (a \<bullet> i + ((f i + 1) / 2^n) * (b \<bullet> i - a \<bullet> i)) *\<^sub>R i)" |
64910 | 2542 |
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
|
2543 |
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
|
2544 |
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
|
2545 |
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
|
2546 |
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
|
2547 |
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
|
2548 |
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
|
2549 |
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
|
2550 |
proof |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2551 |
show "countable ?D0" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2552 |
by (simp add: countable_PiE) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2553 |
next |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2554 |
show "\<Union>?D0 \<subseteq> cbox a b" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2555 |
apply (simp add: UN_subset_iff) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2556 |
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
|
2557 |
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
|
2558 |
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
|
2559 |
done |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2560 |
next |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2561 |
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
|
2562 |
using \<open>box a b \<noteq> {}\<close> |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2563 |
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
|
2564 |
next |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2565 |
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
|
2566 |
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
|
2567 |
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 | 2568 |
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
|
2569 |
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
|
2570 |
fix v w m and n::nat |
64911 | 2571 |
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
|
2572 |
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
|
2573 |
apply (simp add: subset_box disjoint_interval) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2574 |
apply (rule ccontr) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2575 |
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
|
2576 |
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
|
2577 |
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
|
2578 |
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
|
2579 |
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
|
2580 |
done |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2581 |
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
|
2582 |
by meson |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2583 |
qed auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2584 |
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
|
2585 |
apply (erule imageE SigmaE)+ |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2586 |
using * by simp |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2587 |
next |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2588 |
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
|
2589 |
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
|
2590 |
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
|
2591 |
proof - |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2592 |
obtain e where "0 < e" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2593 |
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
|
2594 |
proof - |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2595 |
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
|
2596 |
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
|
2597 |
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
|
2598 |
using openE by blast |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2599 |
have "norm (x - y) < \<epsilon>" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2600 |
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
|
2601 |
proof - |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2602 |
have "norm (x - y) \<le> (\<Sum>i\<in>Basis. \<bar>x \<bullet> i - y \<bullet> i\<bar>)" |
64267 | 2603 |
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
|
2604 |
also have "... \<le> DIM('a) * (\<epsilon> / (2 * real DIM('a)))" |
64267 | 2605 |
by (meson sum_bounded_above that) |
63957
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2606 |
also have "... = \<epsilon> / 2" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2607 |
by (simp add: divide_simps) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2608 |
also have "... < \<epsilon>" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2609 |
by (simp add: \<open>0 < \<epsilon>\<close>) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2610 |
finally show ?thesis . |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2611 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2612 |
then show ?thesis |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2613 |
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
|
2614 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2615 |
have xab: "x \<in> cbox a b" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2616 |
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
|
2617 |
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
|
2618 |
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
|
2619 |
by (auto simp: divide_simps) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2620 |
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
|
2621 |
by (auto simp: divide_simps) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2622 |
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
|
2623 |
proof - |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2624 |
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
|
2625 |
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
|
2626 |
also have "... < e * 2 ^ n" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2627 |
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
|
2628 |
finally show ?thesis . |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2629 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2630 |
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
|
2631 |
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
|
2632 |
by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2633 |
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
|
2634 |
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
|
2635 |
proof |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2636 |
fix i::'a assume "i \<in> Basis" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2637 |
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
|
2638 |
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
|
2639 |
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
|
2640 |
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
|
2641 |
proof cases |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2642 |
case 1 then show ?thesis |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2643 |
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
|
2644 |
next |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2645 |
case 2 |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2646 |
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
|
2647 |
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
|
2648 |
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
|
2649 |
show ?thesis |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2650 |
proof (intro exI conjI) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2651 |
show "?k < 2 ^ n" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2652 |
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
|
2653 |
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
|
2654 |
next |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2655 |
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
|
2656 |
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
|
2657 |
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
|
2658 |
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
|
2659 |
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
|
2660 |
done |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2661 |
also have "... = x \<bullet> i" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2662 |
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
|
2663 |
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
|
2664 |
next |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2665 |
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
|
2666 |
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
|
2667 |
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
|
2668 |
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
|
2669 |
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
|
2670 |
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
|
2671 |
done |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2672 |
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
|
2673 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2674 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2675 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2676 |
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
|
2677 |
apply (simp add: mem_box Bex_def) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2678 |
apply (clarify dest!: bchoice) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2679 |
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
|
2680 |
done |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2681 |
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
|
2682 |
apply (clarsimp simp add: mem_box) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2683 |
apply (rule e) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2684 |
apply (drule bspec D, assumption)+ |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2685 |
apply (erule order_trans) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2686 |
apply (simp add: divide_simps) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2687 |
using bai by (force simp: algebra_simps) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2688 |
ultimately show ?thesis by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2689 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2690 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2691 |
next |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2692 |
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
|
2693 |
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
|
2694 |
next |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2695 |
obtain j::'a where "j \<in> Basis" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2696 |
using nonempty_Basis by blast |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2697 |
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
|
2698 |
proof (rule finite_subset) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2699 |
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
|
2700 |
(\<Sum>i\<in>Basis. (a \<bullet> i + ((f i) + 1) / 2^n * (b \<bullet> i - a \<bullet> i)) *\<^sub>R i)) |
64910 | 2701 |
` (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
|
2702 |
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
|
2703 |
proof - |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2704 |
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
|
2705 |
\<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
|
2706 |
by (auto simp: divide_simps algebra_simps) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2707 |
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
|
2708 |
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
|
2709 |
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
|
2710 |
(((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
|
2711 |
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
|
2712 |
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
|
2713 |
((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
|
2714 |
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
|
2715 |
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
|
2716 |
by (rule dd) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2717 |
then have "m \<le> n" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2718 |
by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2719 |
show ?thesis |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2720 |
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
|
2721 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2722 |
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
|
2723 |
by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2724 |
show "finite ?B" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2725 |
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
|
2726 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2727 |
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
|
2728 |
using that by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2729 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2730 |
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
|
2731 |
obtain \<D> where count: "countable \<D>" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2732 |
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
|
2733 |
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
|
2734 |
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
|
2735 |
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
|
2736 |
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
|
2737 |
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
|
2738 |
proof |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2739 |
show "countable ?D1" using count countable_subset |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2740 |
by (simp add: count countable_subset) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2741 |
show "\<Union>?D1 \<subseteq> cbox a b" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2742 |
using sub by blast |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2743 |
show "S \<subseteq> \<Union>?D1" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2744 |
using SK by (force simp:) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2745 |
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
|
2746 |
using int by blast |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2747 |
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
|
2748 |
using intdj by blast |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2749 |
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
|
2750 |
by auto |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2751 |
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
|
2752 |
using cbox by blast |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2753 |
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
|
2754 |
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
|
2755 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2756 |
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
|
2757 |
show ?thesis |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2758 |
proof (rule that) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2759 |
show "countable ?\<D>" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2760 |
by (blast intro: countable_subset [OF _ count]) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2761 |
show "\<Union>?\<D> \<subseteq> cbox a b" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2762 |
using sub by blast |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2763 |
show "S \<subseteq> \<Union>?\<D>" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2764 |
proof clarsimp |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2765 |
fix x |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2766 |
assume "x \<in> S" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2767 |
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
|
2768 |
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
|
2769 |
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
|
2770 |
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
|
2771 |
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
|
2772 |
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
|
2773 |
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
|
2774 |
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
|
2775 |
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
|
2776 |
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
|
2777 |
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
|
2778 |
by blast |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2779 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2780 |
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
|
2781 |
by (blast intro: dest: int) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2782 |
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
|
2783 |
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
|
2784 |
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
|
2785 |
using SK by force |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2786 |
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
|
2787 |
using cbox by force |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2788 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2789 |
qed |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2790 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2791 |
subsection \<open>Division filter\<close> |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2792 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2793 |
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
|
2794 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2795 |
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
|
2796 |
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
|
2797 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2798 |
lemma eventually_division_filter: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2799 |
"(\<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
|
2800 |
(\<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
|
2801 |
unfolding division_filter_def |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2802 |
proof (subst eventually_INF_base; clarsimp) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2803 |
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
|
2804 |
{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
|
2805 |
{p. p tagged_division_of s \<and> x fine p} \<subseteq> {p. p tagged_division_of s \<and> g2 fine p}" |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2806 |
by (intro exI[of _ "\<lambda>x. g1 x \<inter> g2 x"]) (auto simp: fine_inter) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2807 |
qed (auto simp: eventually_principal) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2808 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2809 |
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
|
2810 |
unfolding trivial_limit_def eventually_division_filter |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2811 |
by (auto elim: fine_division_exists) |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2812 |
|
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2813 |
lemma eventually_division_filter_tagged_division: |
c3da799b1b45
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
hoelzl
parents:
diff
changeset
|
2814 |
"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
|
2815 |
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
|
2816 |
|
64267 | 2817 |
end |