author | hoelzl |
Thu, 29 Nov 2012 09:59:20 +0100 | |
changeset 50283 | e79a8341dd6b |
parent 50252 | 4aa34bd43228 |
child 50386 | d00e2b0ca069 |
permissions | -rw-r--r-- |
41983 | 1 |
(* Title: HOL/Probability/Sigma_Algebra.thy |
42067 | 2 |
Author: Stefan Richter, Markus Wenzel, TU München |
3 |
Author: Johannes Hölzl, TU München |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41959
diff
changeset
|
4 |
Plus material from the Hurd/Coble measure theory development, |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41959
diff
changeset
|
5 |
translated by Lawrence Paulson. |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
6 |
*) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
7 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
8 |
header {* Sigma Algebras *} |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
9 |
|
41413
64cd30d6b0b8
explicit file specifications -- avoid secondary load path;
wenzelm
parents:
41095
diff
changeset
|
10 |
theory Sigma_Algebra |
64cd30d6b0b8
explicit file specifications -- avoid secondary load path;
wenzelm
parents:
41095
diff
changeset
|
11 |
imports |
42145 | 12 |
Complex_Main |
50245
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset
|
13 |
"~~/src/HOL/Library/Countable_Set" |
41413
64cd30d6b0b8
explicit file specifications -- avoid secondary load path;
wenzelm
parents:
41095
diff
changeset
|
14 |
"~~/src/HOL/Library/FuncSet" |
64cd30d6b0b8
explicit file specifications -- avoid secondary load path;
wenzelm
parents:
41095
diff
changeset
|
15 |
"~~/src/HOL/Library/Indicator_Function" |
47694 | 16 |
"~~/src/HOL/Library/Extended_Real" |
41413
64cd30d6b0b8
explicit file specifications -- avoid secondary load path;
wenzelm
parents:
41095
diff
changeset
|
17 |
begin |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
18 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
19 |
text {* Sigma algebras are an elementary concept in measure |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
20 |
theory. To measure --- that is to integrate --- functions, we first have |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
21 |
to measure sets. Unfortunately, when dealing with a large universe, |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
22 |
it is often not possible to consistently assign a measure to every |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
23 |
subset. Therefore it is necessary to define the set of measurable |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
24 |
subsets of the universe. A sigma algebra is such a set that has |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
25 |
three very natural and desirable properties. *} |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
26 |
|
47762 | 27 |
subsection {* Families of sets *} |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
28 |
|
47694 | 29 |
locale subset_class = |
30 |
fixes \<Omega> :: "'a set" and M :: "'a set set" |
|
31 |
assumes space_closed: "M \<subseteq> Pow \<Omega>" |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
32 |
|
47694 | 33 |
lemma (in subset_class) sets_into_space: "x \<in> M \<Longrightarrow> x \<subseteq> \<Omega>" |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
34 |
by (metis PowD contra_subsetD space_closed) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
35 |
|
47762 | 36 |
subsection {* Semiring of sets *} |
37 |
||
38 |
subsubsection {* Disjoint sets *} |
|
39 |
||
40 |
definition "disjoint A \<longleftrightarrow> (\<forall>a\<in>A. \<forall>b\<in>A. a \<noteq> b \<longrightarrow> a \<inter> b = {})" |
|
41 |
||
42 |
lemma disjointI: |
|
43 |
"(\<And>a b. a \<in> A \<Longrightarrow> b \<in> A \<Longrightarrow> a \<noteq> b \<Longrightarrow> a \<inter> b = {}) \<Longrightarrow> disjoint A" |
|
44 |
unfolding disjoint_def by auto |
|
45 |
||
46 |
lemma disjointD: |
|
47 |
"disjoint A \<Longrightarrow> a \<in> A \<Longrightarrow> b \<in> A \<Longrightarrow> a \<noteq> b \<Longrightarrow> a \<inter> b = {}" |
|
48 |
unfolding disjoint_def by auto |
|
49 |
||
50 |
lemma disjoint_empty[iff]: "disjoint {}" |
|
51 |
by (auto simp: disjoint_def) |
|
42065
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
parents:
41983
diff
changeset
|
52 |
|
47762 | 53 |
lemma disjoint_union: |
54 |
assumes C: "disjoint C" and B: "disjoint B" and disj: "\<Union>C \<inter> \<Union>B = {}" |
|
55 |
shows "disjoint (C \<union> B)" |
|
56 |
proof (rule disjointI) |
|
57 |
fix c d assume sets: "c \<in> C \<union> B" "d \<in> C \<union> B" and "c \<noteq> d" |
|
58 |
show "c \<inter> d = {}" |
|
59 |
proof cases |
|
60 |
assume "(c \<in> C \<and> d \<in> C) \<or> (c \<in> B \<and> d \<in> B)" |
|
61 |
then show ?thesis |
|
62 |
proof |
|
63 |
assume "c \<in> C \<and> d \<in> C" with `c \<noteq> d` C show "c \<inter> d = {}" |
|
64 |
by (auto simp: disjoint_def) |
|
65 |
next |
|
66 |
assume "c \<in> B \<and> d \<in> B" with `c \<noteq> d` B show "c \<inter> d = {}" |
|
67 |
by (auto simp: disjoint_def) |
|
68 |
qed |
|
69 |
next |
|
70 |
assume "\<not> ((c \<in> C \<and> d \<in> C) \<or> (c \<in> B \<and> d \<in> B))" |
|
71 |
with sets have "(c \<subseteq> \<Union>C \<and> d \<subseteq> \<Union>B) \<or> (c \<subseteq> \<Union>B \<and> d \<subseteq> \<Union>C)" |
|
72 |
by auto |
|
73 |
with disj show "c \<inter> d = {}" by auto |
|
74 |
qed |
|
75 |
qed |
|
76 |
||
77 |
locale semiring_of_sets = subset_class + |
|
78 |
assumes empty_sets[iff]: "{} \<in> M" |
|
79 |
assumes Int[intro]: "\<And>a b. a \<in> M \<Longrightarrow> b \<in> M \<Longrightarrow> a \<inter> b \<in> M" |
|
80 |
assumes Diff_cover: |
|
81 |
"\<And>a b. a \<in> M \<Longrightarrow> b \<in> M \<Longrightarrow> \<exists>C\<subseteq>M. finite C \<and> disjoint C \<and> a - b = \<Union>C" |
|
82 |
||
83 |
lemma (in semiring_of_sets) finite_INT[intro]: |
|
84 |
assumes "finite I" "I \<noteq> {}" "\<And>i. i \<in> I \<Longrightarrow> A i \<in> M" |
|
85 |
shows "(\<Inter>i\<in>I. A i) \<in> M" |
|
86 |
using assms by (induct rule: finite_ne_induct) auto |
|
87 |
||
88 |
lemma (in semiring_of_sets) Int_space_eq1 [simp]: "x \<in> M \<Longrightarrow> \<Omega> \<inter> x = x" |
|
89 |
by (metis Int_absorb1 sets_into_space) |
|
90 |
||
91 |
lemma (in semiring_of_sets) Int_space_eq2 [simp]: "x \<in> M \<Longrightarrow> x \<inter> \<Omega> = x" |
|
92 |
by (metis Int_absorb2 sets_into_space) |
|
93 |
||
94 |
lemma (in semiring_of_sets) sets_Collect_conj: |
|
95 |
assumes "{x\<in>\<Omega>. P x} \<in> M" "{x\<in>\<Omega>. Q x} \<in> M" |
|
96 |
shows "{x\<in>\<Omega>. Q x \<and> P x} \<in> M" |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
97 |
proof - |
47762 | 98 |
have "{x\<in>\<Omega>. Q x \<and> P x} = {x\<in>\<Omega>. Q x} \<inter> {x\<in>\<Omega>. P x}" |
42065
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
parents:
41983
diff
changeset
|
99 |
by auto |
47762 | 100 |
with assms show ?thesis by auto |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
101 |
qed |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
102 |
|
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
103 |
lemma (in semiring_of_sets) sets_Collect_finite_All': |
47762 | 104 |
assumes "\<And>i. i \<in> S \<Longrightarrow> {x\<in>\<Omega>. P i x} \<in> M" "finite S" "S \<noteq> {}" |
105 |
shows "{x\<in>\<Omega>. \<forall>i\<in>S. P i x} \<in> M" |
|
106 |
proof - |
|
107 |
have "{x\<in>\<Omega>. \<forall>i\<in>S. P i x} = (\<Inter>i\<in>S. {x\<in>\<Omega>. P i x})" |
|
108 |
using `S \<noteq> {}` by auto |
|
109 |
with assms show ?thesis by auto |
|
110 |
qed |
|
111 |
||
112 |
locale ring_of_sets = semiring_of_sets + |
|
113 |
assumes Un [intro]: "\<And>a b. a \<in> M \<Longrightarrow> b \<in> M \<Longrightarrow> a \<union> b \<in> M" |
|
114 |
||
42065
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
parents:
41983
diff
changeset
|
115 |
lemma (in ring_of_sets) finite_Union [intro]: |
47694 | 116 |
"finite X \<Longrightarrow> X \<subseteq> M \<Longrightarrow> Union X \<in> M" |
38656 | 117 |
by (induct set: finite) (auto simp add: Un) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
118 |
|
42065
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
parents:
41983
diff
changeset
|
119 |
lemma (in ring_of_sets) finite_UN[intro]: |
47694 | 120 |
assumes "finite I" and "\<And>i. i \<in> I \<Longrightarrow> A i \<in> M" |
121 |
shows "(\<Union>i\<in>I. A i) \<in> M" |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41959
diff
changeset
|
122 |
using assms by induct auto |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41959
diff
changeset
|
123 |
|
47762 | 124 |
lemma (in ring_of_sets) Diff [intro]: |
125 |
assumes "a \<in> M" "b \<in> M" shows "a - b \<in> M" |
|
126 |
using Diff_cover[OF assms] by auto |
|
127 |
||
128 |
lemma ring_of_setsI: |
|
129 |
assumes space_closed: "M \<subseteq> Pow \<Omega>" |
|
130 |
assumes empty_sets[iff]: "{} \<in> M" |
|
131 |
assumes Un[intro]: "\<And>a b. a \<in> M \<Longrightarrow> b \<in> M \<Longrightarrow> a \<union> b \<in> M" |
|
132 |
assumes Diff[intro]: "\<And>a b. a \<in> M \<Longrightarrow> b \<in> M \<Longrightarrow> a - b \<in> M" |
|
133 |
shows "ring_of_sets \<Omega> M" |
|
134 |
proof |
|
135 |
fix a b assume ab: "a \<in> M" "b \<in> M" |
|
136 |
from ab show "\<exists>C\<subseteq>M. finite C \<and> disjoint C \<and> a - b = \<Union>C" |
|
137 |
by (intro exI[of _ "{a - b}"]) (auto simp: disjoint_def) |
|
138 |
have "a \<inter> b = a - (a - b)" by auto |
|
139 |
also have "\<dots> \<in> M" using ab by auto |
|
140 |
finally show "a \<inter> b \<in> M" . |
|
141 |
qed fact+ |
|
142 |
||
143 |
lemma ring_of_sets_iff: "ring_of_sets \<Omega> M \<longleftrightarrow> M \<subseteq> Pow \<Omega> \<and> {} \<in> M \<and> (\<forall>a\<in>M. \<forall>b\<in>M. a \<union> b \<in> M) \<and> (\<forall>a\<in>M. \<forall>b\<in>M. a - b \<in> M)" |
|
144 |
proof |
|
145 |
assume "ring_of_sets \<Omega> M" |
|
146 |
then interpret ring_of_sets \<Omega> M . |
|
147 |
show "M \<subseteq> Pow \<Omega> \<and> {} \<in> M \<and> (\<forall>a\<in>M. \<forall>b\<in>M. a \<union> b \<in> M) \<and> (\<forall>a\<in>M. \<forall>b\<in>M. a - b \<in> M)" |
|
148 |
using space_closed by auto |
|
149 |
qed (auto intro!: ring_of_setsI) |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41959
diff
changeset
|
150 |
|
42065
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
parents:
41983
diff
changeset
|
151 |
lemma (in ring_of_sets) insert_in_sets: |
47694 | 152 |
assumes "{x} \<in> M" "A \<in> M" shows "insert x A \<in> M" |
38656 | 153 |
proof - |
47694 | 154 |
have "{x} \<union> A \<in> M" using assms by (rule Un) |
38656 | 155 |
thus ?thesis by auto |
156 |
qed |
|
157 |
||
42867 | 158 |
lemma (in ring_of_sets) sets_Collect_disj: |
47694 | 159 |
assumes "{x\<in>\<Omega>. P x} \<in> M" "{x\<in>\<Omega>. Q x} \<in> M" |
160 |
shows "{x\<in>\<Omega>. Q x \<or> P x} \<in> M" |
|
42867 | 161 |
proof - |
47694 | 162 |
have "{x\<in>\<Omega>. Q x \<or> P x} = {x\<in>\<Omega>. Q x} \<union> {x\<in>\<Omega>. P x}" |
42867 | 163 |
by auto |
164 |
with assms show ?thesis by auto |
|
165 |
qed |
|
166 |
||
167 |
lemma (in ring_of_sets) sets_Collect_finite_Ex: |
|
47694 | 168 |
assumes "\<And>i. i \<in> S \<Longrightarrow> {x\<in>\<Omega>. P i x} \<in> M" "finite S" |
169 |
shows "{x\<in>\<Omega>. \<exists>i\<in>S. P i x} \<in> M" |
|
42867 | 170 |
proof - |
47694 | 171 |
have "{x\<in>\<Omega>. \<exists>i\<in>S. P i x} = (\<Union>i\<in>S. {x\<in>\<Omega>. P i x})" |
42867 | 172 |
by auto |
173 |
with assms show ?thesis by auto |
|
174 |
qed |
|
175 |
||
42065
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
parents:
41983
diff
changeset
|
176 |
locale algebra = ring_of_sets + |
47694 | 177 |
assumes top [iff]: "\<Omega> \<in> M" |
42065
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
parents:
41983
diff
changeset
|
178 |
|
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
parents:
41983
diff
changeset
|
179 |
lemma (in algebra) compl_sets [intro]: |
47694 | 180 |
"a \<in> M \<Longrightarrow> \<Omega> - a \<in> M" |
42065
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
parents:
41983
diff
changeset
|
181 |
by auto |
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
parents:
41983
diff
changeset
|
182 |
|
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
parents:
41983
diff
changeset
|
183 |
lemma algebra_iff_Un: |
47694 | 184 |
"algebra \<Omega> M \<longleftrightarrow> |
185 |
M \<subseteq> Pow \<Omega> \<and> |
|
186 |
{} \<in> M \<and> |
|
187 |
(\<forall>a \<in> M. \<Omega> - a \<in> M) \<and> |
|
188 |
(\<forall>a \<in> M. \<forall> b \<in> M. a \<union> b \<in> M)" (is "_ \<longleftrightarrow> ?Un") |
|
42065
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
parents:
41983
diff
changeset
|
189 |
proof |
47694 | 190 |
assume "algebra \<Omega> M" |
191 |
then interpret algebra \<Omega> M . |
|
42065
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
parents:
41983
diff
changeset
|
192 |
show ?Un using sets_into_space by auto |
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
parents:
41983
diff
changeset
|
193 |
next |
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
parents:
41983
diff
changeset
|
194 |
assume ?Un |
47762 | 195 |
then have "\<Omega> \<in> M" by auto |
196 |
interpret ring_of_sets \<Omega> M |
|
197 |
proof (rule ring_of_setsI) |
|
198 |
show \<Omega>: "M \<subseteq> Pow \<Omega>" "{} \<in> M" |
|
42065
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
parents:
41983
diff
changeset
|
199 |
using `?Un` by auto |
47694 | 200 |
fix a b assume a: "a \<in> M" and b: "b \<in> M" |
201 |
then show "a \<union> b \<in> M" using `?Un` by auto |
|
202 |
have "a - b = \<Omega> - ((\<Omega> - a) \<union> b)" |
|
203 |
using \<Omega> a b by auto |
|
204 |
then show "a - b \<in> M" |
|
42065
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
parents:
41983
diff
changeset
|
205 |
using a b `?Un` by auto |
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
parents:
41983
diff
changeset
|
206 |
qed |
47762 | 207 |
show "algebra \<Omega> M" proof qed fact |
42065
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
parents:
41983
diff
changeset
|
208 |
qed |
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
parents:
41983
diff
changeset
|
209 |
|
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
parents:
41983
diff
changeset
|
210 |
lemma algebra_iff_Int: |
47694 | 211 |
"algebra \<Omega> M \<longleftrightarrow> |
212 |
M \<subseteq> Pow \<Omega> & {} \<in> M & |
|
213 |
(\<forall>a \<in> M. \<Omega> - a \<in> M) & |
|
214 |
(\<forall>a \<in> M. \<forall> b \<in> M. a \<inter> b \<in> M)" (is "_ \<longleftrightarrow> ?Int") |
|
42065
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
parents:
41983
diff
changeset
|
215 |
proof |
47694 | 216 |
assume "algebra \<Omega> M" |
217 |
then interpret algebra \<Omega> M . |
|
42065
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
parents:
41983
diff
changeset
|
218 |
show ?Int using sets_into_space by auto |
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
parents:
41983
diff
changeset
|
219 |
next |
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
parents:
41983
diff
changeset
|
220 |
assume ?Int |
47694 | 221 |
show "algebra \<Omega> M" |
42065
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
parents:
41983
diff
changeset
|
222 |
proof (unfold algebra_iff_Un, intro conjI ballI) |
47694 | 223 |
show \<Omega>: "M \<subseteq> Pow \<Omega>" "{} \<in> M" |
42065
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
parents:
41983
diff
changeset
|
224 |
using `?Int` by auto |
47694 | 225 |
from `?Int` show "\<And>a. a \<in> M \<Longrightarrow> \<Omega> - a \<in> M" by auto |
226 |
fix a b assume M: "a \<in> M" "b \<in> M" |
|
227 |
hence "a \<union> b = \<Omega> - ((\<Omega> - a) \<inter> (\<Omega> - b))" |
|
228 |
using \<Omega> by blast |
|
229 |
also have "... \<in> M" |
|
230 |
using M `?Int` by auto |
|
231 |
finally show "a \<union> b \<in> M" . |
|
42065
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
parents:
41983
diff
changeset
|
232 |
qed |
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
parents:
41983
diff
changeset
|
233 |
qed |
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
parents:
41983
diff
changeset
|
234 |
|
42867 | 235 |
lemma (in algebra) sets_Collect_neg: |
47694 | 236 |
assumes "{x\<in>\<Omega>. P x} \<in> M" |
237 |
shows "{x\<in>\<Omega>. \<not> P x} \<in> M" |
|
42867 | 238 |
proof - |
47694 | 239 |
have "{x\<in>\<Omega>. \<not> P x} = \<Omega> - {x\<in>\<Omega>. P x}" by auto |
42867 | 240 |
with assms show ?thesis by auto |
241 |
qed |
|
242 |
||
243 |
lemma (in algebra) sets_Collect_imp: |
|
47694 | 244 |
"{x\<in>\<Omega>. P x} \<in> M \<Longrightarrow> {x\<in>\<Omega>. Q x} \<in> M \<Longrightarrow> {x\<in>\<Omega>. Q x \<longrightarrow> P x} \<in> M" |
42867 | 245 |
unfolding imp_conv_disj by (intro sets_Collect_disj sets_Collect_neg) |
246 |
||
247 |
lemma (in algebra) sets_Collect_const: |
|
47694 | 248 |
"{x\<in>\<Omega>. P} \<in> M" |
42867 | 249 |
by (cases P) auto |
250 |
||
42984 | 251 |
lemma algebra_single_set: |
47762 | 252 |
"X \<subseteq> S \<Longrightarrow> algebra S { {}, X, S - X, S }" |
253 |
by (auto simp: algebra_iff_Int) |
|
42984 | 254 |
|
39092 | 255 |
section {* Restricted algebras *} |
256 |
||
257 |
abbreviation (in algebra) |
|
47694 | 258 |
"restricted_space A \<equiv> (op \<inter> A) ` M" |
39092 | 259 |
|
38656 | 260 |
lemma (in algebra) restricted_algebra: |
47694 | 261 |
assumes "A \<in> M" shows "algebra A (restricted_space A)" |
47762 | 262 |
using assms by (auto simp: algebra_iff_Int) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
263 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
264 |
subsection {* Sigma Algebras *} |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
265 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
266 |
locale sigma_algebra = algebra + |
47694 | 267 |
assumes countable_nat_UN [intro]: "\<And>A. range A \<subseteq> M \<Longrightarrow> (\<Union>i::nat. A i) \<in> M" |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
268 |
|
42984 | 269 |
lemma (in algebra) is_sigma_algebra: |
47694 | 270 |
assumes "finite M" |
271 |
shows "sigma_algebra \<Omega> M" |
|
42984 | 272 |
proof |
47694 | 273 |
fix A :: "nat \<Rightarrow> 'a set" assume "range A \<subseteq> M" |
274 |
then have "(\<Union>i. A i) = (\<Union>s\<in>M \<inter> range A. s)" |
|
42984 | 275 |
by auto |
47694 | 276 |
also have "(\<Union>s\<in>M \<inter> range A. s) \<in> M" |
277 |
using `finite M` by auto |
|
278 |
finally show "(\<Union>i. A i) \<in> M" . |
|
42984 | 279 |
qed |
280 |
||
38656 | 281 |
lemma countable_UN_eq: |
282 |
fixes A :: "'i::countable \<Rightarrow> 'a set" |
|
47694 | 283 |
shows "(range A \<subseteq> M \<longrightarrow> (\<Union>i. A i) \<in> M) \<longleftrightarrow> |
284 |
(range (A \<circ> from_nat) \<subseteq> M \<longrightarrow> (\<Union>i. (A \<circ> from_nat) i) \<in> M)" |
|
38656 | 285 |
proof - |
286 |
let ?A' = "A \<circ> from_nat" |
|
287 |
have *: "(\<Union>i. ?A' i) = (\<Union>i. A i)" (is "?l = ?r") |
|
288 |
proof safe |
|
289 |
fix x i assume "x \<in> A i" thus "x \<in> ?l" |
|
290 |
by (auto intro!: exI[of _ "to_nat i"]) |
|
291 |
next |
|
292 |
fix x i assume "x \<in> ?A' i" thus "x \<in> ?r" |
|
293 |
by (auto intro!: exI[of _ "from_nat i"]) |
|
294 |
qed |
|
295 |
have **: "range ?A' = range A" |
|
40702 | 296 |
using surj_from_nat |
38656 | 297 |
by (auto simp: image_compose intro!: imageI) |
298 |
show ?thesis unfolding * ** .. |
|
299 |
qed |
|
300 |
||
50245
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset
|
301 |
lemma (in sigma_algebra) countable_Union [intro]: |
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset
|
302 |
assumes "countable X" "X \<subseteq> M" shows "Union X \<in> M" |
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset
|
303 |
proof cases |
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset
|
304 |
assume "X \<noteq> {}" |
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset
|
305 |
hence "\<Union>X = (\<Union>n. from_nat_into X n)" |
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset
|
306 |
using assms by (auto intro: from_nat_into) (metis from_nat_into_surj) |
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset
|
307 |
also have "\<dots> \<in> M" using assms |
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset
|
308 |
by (auto intro!: countable_nat_UN) (metis `X \<noteq> {}` from_nat_into set_mp) |
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset
|
309 |
finally show ?thesis . |
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset
|
310 |
qed simp |
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset
|
311 |
|
38656 | 312 |
lemma (in sigma_algebra) countable_UN[intro]: |
313 |
fixes A :: "'i::countable \<Rightarrow> 'a set" |
|
47694 | 314 |
assumes "A`X \<subseteq> M" |
315 |
shows "(\<Union>x\<in>X. A x) \<in> M" |
|
38656 | 316 |
proof - |
46731 | 317 |
let ?A = "\<lambda>i. if i \<in> X then A i else {}" |
47694 | 318 |
from assms have "range ?A \<subseteq> M" by auto |
38656 | 319 |
with countable_nat_UN[of "?A \<circ> from_nat"] countable_UN_eq[of ?A M] |
47694 | 320 |
have "(\<Union>x. ?A x) \<in> M" by auto |
38656 | 321 |
moreover have "(\<Union>x. ?A x) = (\<Union>x\<in>X. A x)" by (auto split: split_if_asm) |
322 |
ultimately show ?thesis by simp |
|
323 |
qed |
|
324 |
||
33533
40b44cb20c8c
New theory Probability/Borel.thy, and some associated lemmas
paulson
parents:
33271
diff
changeset
|
325 |
lemma (in sigma_algebra) countable_INT [intro]: |
38656 | 326 |
fixes A :: "'i::countable \<Rightarrow> 'a set" |
47694 | 327 |
assumes A: "A`X \<subseteq> M" "X \<noteq> {}" |
328 |
shows "(\<Inter>i\<in>X. A i) \<in> M" |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
329 |
proof - |
47694 | 330 |
from A have "\<forall>i\<in>X. A i \<in> M" by fast |
331 |
hence "\<Omega> - (\<Union>i\<in>X. \<Omega> - A i) \<in> M" by blast |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
332 |
moreover |
47694 | 333 |
have "(\<Inter>i\<in>X. A i) = \<Omega> - (\<Union>i\<in>X. \<Omega> - A i)" using space_closed A |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
334 |
by blast |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
335 |
ultimately show ?thesis by metis |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
336 |
qed |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
337 |
|
47694 | 338 |
lemma ring_of_sets_Pow: "ring_of_sets sp (Pow sp)" |
47762 | 339 |
by (auto simp: ring_of_sets_iff) |
42145 | 340 |
|
47694 | 341 |
lemma algebra_Pow: "algebra sp (Pow sp)" |
47762 | 342 |
by (auto simp: algebra_iff_Un) |
38656 | 343 |
|
344 |
lemma sigma_algebra_iff: |
|
47694 | 345 |
"sigma_algebra \<Omega> M \<longleftrightarrow> |
346 |
algebra \<Omega> M \<and> (\<forall>A. range A \<subseteq> M \<longrightarrow> (\<Union>i::nat. A i) \<in> M)" |
|
38656 | 347 |
by (simp add: sigma_algebra_def sigma_algebra_axioms_def) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
348 |
|
47762 | 349 |
lemma sigma_algebra_Pow: "sigma_algebra sp (Pow sp)" |
350 |
by (auto simp: sigma_algebra_iff algebra_iff_Int) |
|
351 |
||
42867 | 352 |
lemma (in sigma_algebra) sets_Collect_countable_All: |
47694 | 353 |
assumes "\<And>i. {x\<in>\<Omega>. P i x} \<in> M" |
354 |
shows "{x\<in>\<Omega>. \<forall>i::'i::countable. P i x} \<in> M" |
|
42867 | 355 |
proof - |
47694 | 356 |
have "{x\<in>\<Omega>. \<forall>i::'i::countable. P i x} = (\<Inter>i. {x\<in>\<Omega>. P i x})" by auto |
42867 | 357 |
with assms show ?thesis by auto |
358 |
qed |
|
359 |
||
360 |
lemma (in sigma_algebra) sets_Collect_countable_Ex: |
|
47694 | 361 |
assumes "\<And>i. {x\<in>\<Omega>. P i x} \<in> M" |
362 |
shows "{x\<in>\<Omega>. \<exists>i::'i::countable. P i x} \<in> M" |
|
42867 | 363 |
proof - |
47694 | 364 |
have "{x\<in>\<Omega>. \<exists>i::'i::countable. P i x} = (\<Union>i. {x\<in>\<Omega>. P i x})" by auto |
42867 | 365 |
with assms show ?thesis by auto |
366 |
qed |
|
367 |
||
368 |
lemmas (in sigma_algebra) sets_Collect = |
|
369 |
sets_Collect_imp sets_Collect_disj sets_Collect_conj sets_Collect_neg sets_Collect_const |
|
370 |
sets_Collect_countable_All sets_Collect_countable_Ex sets_Collect_countable_All |
|
371 |
||
47694 | 372 |
lemma (in sigma_algebra) sets_Collect_countable_Ball: |
373 |
assumes "\<And>i. {x\<in>\<Omega>. P i x} \<in> M" |
|
374 |
shows "{x\<in>\<Omega>. \<forall>i::'i::countable\<in>X. P i x} \<in> M" |
|
375 |
unfolding Ball_def by (intro sets_Collect assms) |
|
376 |
||
377 |
lemma (in sigma_algebra) sets_Collect_countable_Bex: |
|
378 |
assumes "\<And>i. {x\<in>\<Omega>. P i x} \<in> M" |
|
379 |
shows "{x\<in>\<Omega>. \<exists>i::'i::countable\<in>X. P i x} \<in> M" |
|
380 |
unfolding Bex_def by (intro sets_Collect assms) |
|
381 |
||
42984 | 382 |
lemma sigma_algebra_single_set: |
383 |
assumes "X \<subseteq> S" |
|
47694 | 384 |
shows "sigma_algebra S { {}, X, S - X, S }" |
42984 | 385 |
using algebra.is_sigma_algebra[OF algebra_single_set[OF `X \<subseteq> S`]] by simp |
386 |
||
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
387 |
subsection {* Binary Unions *} |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
388 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
389 |
definition binary :: "'a \<Rightarrow> 'a \<Rightarrow> nat \<Rightarrow> 'a" |
50252 | 390 |
where "binary a b = (\<lambda>x. b)(0 := a)" |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
391 |
|
38656 | 392 |
lemma range_binary_eq: "range(binary a b) = {a,b}" |
393 |
by (auto simp add: binary_def) |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
394 |
|
38656 | 395 |
lemma Un_range_binary: "a \<union> b = (\<Union>i::nat. binary a b i)" |
44106 | 396 |
by (simp add: SUP_def range_binary_eq) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
397 |
|
38656 | 398 |
lemma Int_range_binary: "a \<inter> b = (\<Inter>i::nat. binary a b i)" |
44106 | 399 |
by (simp add: INF_def range_binary_eq) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
400 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
401 |
lemma sigma_algebra_iff2: |
47694 | 402 |
"sigma_algebra \<Omega> M \<longleftrightarrow> |
403 |
M \<subseteq> Pow \<Omega> \<and> |
|
404 |
{} \<in> M \<and> (\<forall>s \<in> M. \<Omega> - s \<in> M) \<and> |
|
405 |
(\<forall>A. range A \<subseteq> M \<longrightarrow> (\<Union>i::nat. A i) \<in> M)" |
|
38656 | 406 |
by (auto simp add: range_binary_eq sigma_algebra_def sigma_algebra_axioms_def |
42065
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
parents:
41983
diff
changeset
|
407 |
algebra_iff_Un Un_range_binary) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
408 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
409 |
subsection {* Initial Sigma Algebra *} |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
410 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
411 |
text {*Sigma algebras can naturally be created as the closure of any set of |
47694 | 412 |
M with regard to the properties just postulated. *} |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
413 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
414 |
inductive_set |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
415 |
sigma_sets :: "'a set \<Rightarrow> 'a set set \<Rightarrow> 'a set set" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
416 |
for sp :: "'a set" and A :: "'a set set" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
417 |
where |
47694 | 418 |
Basic[intro, simp]: "a \<in> A \<Longrightarrow> a \<in> sigma_sets sp A" |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
419 |
| Empty: "{} \<in> sigma_sets sp A" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
420 |
| Compl: "a \<in> sigma_sets sp A \<Longrightarrow> sp - a \<in> sigma_sets sp A" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
421 |
| Union: "(\<And>i::nat. a i \<in> sigma_sets sp A) \<Longrightarrow> (\<Union>i. a i) \<in> sigma_sets sp A" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
422 |
|
41543 | 423 |
lemma (in sigma_algebra) sigma_sets_subset: |
47694 | 424 |
assumes a: "a \<subseteq> M" |
425 |
shows "sigma_sets \<Omega> a \<subseteq> M" |
|
41543 | 426 |
proof |
427 |
fix x |
|
47694 | 428 |
assume "x \<in> sigma_sets \<Omega> a" |
429 |
from this show "x \<in> M" |
|
41543 | 430 |
by (induct rule: sigma_sets.induct, auto) (metis a subsetD) |
431 |
qed |
|
432 |
||
433 |
lemma sigma_sets_into_sp: "A \<subseteq> Pow sp \<Longrightarrow> x \<in> sigma_sets sp A \<Longrightarrow> x \<subseteq> sp" |
|
434 |
by (erule sigma_sets.induct, auto) |
|
435 |
||
436 |
lemma sigma_algebra_sigma_sets: |
|
47694 | 437 |
"a \<subseteq> Pow \<Omega> \<Longrightarrow> sigma_algebra \<Omega> (sigma_sets \<Omega> a)" |
41543 | 438 |
by (auto simp add: sigma_algebra_iff2 dest: sigma_sets_into_sp |
439 |
intro!: sigma_sets.Union sigma_sets.Empty sigma_sets.Compl) |
|
440 |
||
441 |
lemma sigma_sets_least_sigma_algebra: |
|
442 |
assumes "A \<subseteq> Pow S" |
|
47694 | 443 |
shows "sigma_sets S A = \<Inter>{B. A \<subseteq> B \<and> sigma_algebra S B}" |
41543 | 444 |
proof safe |
47694 | 445 |
fix B X assume "A \<subseteq> B" and sa: "sigma_algebra S B" |
41543 | 446 |
and X: "X \<in> sigma_sets S A" |
447 |
from sigma_algebra.sigma_sets_subset[OF sa, simplified, OF `A \<subseteq> B`] X |
|
448 |
show "X \<in> B" by auto |
|
449 |
next |
|
47694 | 450 |
fix X assume "X \<in> \<Inter>{B. A \<subseteq> B \<and> sigma_algebra S B}" |
451 |
then have [intro!]: "\<And>B. A \<subseteq> B \<Longrightarrow> sigma_algebra S B \<Longrightarrow> X \<in> B" |
|
41543 | 452 |
by simp |
47694 | 453 |
have "A \<subseteq> sigma_sets S A" using assms by auto |
454 |
moreover have "sigma_algebra S (sigma_sets S A)" |
|
41543 | 455 |
using assms by (intro sigma_algebra_sigma_sets[of A]) auto |
456 |
ultimately show "X \<in> sigma_sets S A" by auto |
|
457 |
qed |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
458 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
459 |
lemma sigma_sets_top: "sp \<in> sigma_sets sp A" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
460 |
by (metis Diff_empty sigma_sets.Compl sigma_sets.Empty) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
461 |
|
38656 | 462 |
lemma sigma_sets_Un: |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
463 |
"a \<in> sigma_sets sp A \<Longrightarrow> b \<in> sigma_sets sp A \<Longrightarrow> a \<union> b \<in> sigma_sets sp A" |
38656 | 464 |
apply (simp add: Un_range_binary range_binary_eq) |
40859 | 465 |
apply (rule Union, simp add: binary_def) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
466 |
done |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
467 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
468 |
lemma sigma_sets_Inter: |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
469 |
assumes Asb: "A \<subseteq> Pow sp" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
470 |
shows "(\<And>i::nat. a i \<in> sigma_sets sp A) \<Longrightarrow> (\<Inter>i. a i) \<in> sigma_sets sp A" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
471 |
proof - |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
472 |
assume ai: "\<And>i::nat. a i \<in> sigma_sets sp A" |
38656 | 473 |
hence "\<And>i::nat. sp-(a i) \<in> sigma_sets sp A" |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
474 |
by (rule sigma_sets.Compl) |
38656 | 475 |
hence "(\<Union>i. sp-(a i)) \<in> sigma_sets sp A" |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
476 |
by (rule sigma_sets.Union) |
38656 | 477 |
hence "sp-(\<Union>i. sp-(a i)) \<in> sigma_sets sp A" |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
478 |
by (rule sigma_sets.Compl) |
38656 | 479 |
also have "sp-(\<Union>i. sp-(a i)) = sp Int (\<Inter>i. a i)" |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
480 |
by auto |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
481 |
also have "... = (\<Inter>i. a i)" using ai |
38656 | 482 |
by (blast dest: sigma_sets_into_sp [OF Asb]) |
483 |
finally show ?thesis . |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
484 |
qed |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
485 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
486 |
lemma sigma_sets_INTER: |
38656 | 487 |
assumes Asb: "A \<subseteq> Pow sp" |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
488 |
and ai: "\<And>i::nat. i \<in> S \<Longrightarrow> a i \<in> sigma_sets sp A" and non: "S \<noteq> {}" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
489 |
shows "(\<Inter>i\<in>S. a i) \<in> sigma_sets sp A" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
490 |
proof - |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
491 |
from ai have "\<And>i. (if i\<in>S then a i else sp) \<in> sigma_sets sp A" |
47694 | 492 |
by (simp add: sigma_sets.intros(2-) sigma_sets_top) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
493 |
hence "(\<Inter>i. (if i\<in>S then a i else sp)) \<in> sigma_sets sp A" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
494 |
by (rule sigma_sets_Inter [OF Asb]) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
495 |
also have "(\<Inter>i. (if i\<in>S then a i else sp)) = (\<Inter>i\<in>S. a i)" |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
496 |
by auto (metis ai non sigma_sets_into_sp subset_empty subset_iff Asb)+ |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
497 |
finally show ?thesis . |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
498 |
qed |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
499 |
|
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
500 |
lemma (in sigma_algebra) sigma_sets_eq: |
47694 | 501 |
"sigma_sets \<Omega> M = M" |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
502 |
proof |
47694 | 503 |
show "M \<subseteq> sigma_sets \<Omega> M" |
37032 | 504 |
by (metis Set.subsetI sigma_sets.Basic) |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
505 |
next |
47694 | 506 |
show "sigma_sets \<Omega> M \<subseteq> M" |
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
507 |
by (metis sigma_sets_subset subset_refl) |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
508 |
qed |
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
509 |
|
42981 | 510 |
lemma sigma_sets_eqI: |
511 |
assumes A: "\<And>a. a \<in> A \<Longrightarrow> a \<in> sigma_sets M B" |
|
512 |
assumes B: "\<And>b. b \<in> B \<Longrightarrow> b \<in> sigma_sets M A" |
|
513 |
shows "sigma_sets M A = sigma_sets M B" |
|
514 |
proof (intro set_eqI iffI) |
|
515 |
fix a assume "a \<in> sigma_sets M A" |
|
516 |
from this A show "a \<in> sigma_sets M B" |
|
47694 | 517 |
by induct (auto intro!: sigma_sets.intros(2-) del: sigma_sets.Basic) |
42981 | 518 |
next |
519 |
fix b assume "b \<in> sigma_sets M B" |
|
520 |
from this B show "b \<in> sigma_sets M A" |
|
47694 | 521 |
by induct (auto intro!: sigma_sets.intros(2-) del: sigma_sets.Basic) |
42981 | 522 |
qed |
523 |
||
42984 | 524 |
lemma sigma_sets_subseteq: assumes "A \<subseteq> B" shows "sigma_sets X A \<subseteq> sigma_sets X B" |
525 |
proof |
|
526 |
fix x assume "x \<in> sigma_sets X A" then show "x \<in> sigma_sets X B" |
|
47694 | 527 |
by induct (insert `A \<subseteq> B`, auto intro: sigma_sets.intros(2-)) |
42984 | 528 |
qed |
529 |
||
47762 | 530 |
lemma sigma_sets_mono: assumes "A \<subseteq> sigma_sets X B" shows "sigma_sets X A \<subseteq> sigma_sets X B" |
531 |
proof |
|
532 |
fix x assume "x \<in> sigma_sets X A" then show "x \<in> sigma_sets X B" |
|
533 |
by induct (insert `A \<subseteq> sigma_sets X B`, auto intro: sigma_sets.intros(2-)) |
|
534 |
qed |
|
535 |
||
536 |
lemma sigma_sets_mono': assumes "A \<subseteq> B" shows "sigma_sets X A \<subseteq> sigma_sets X B" |
|
537 |
proof |
|
538 |
fix x assume "x \<in> sigma_sets X A" then show "x \<in> sigma_sets X B" |
|
539 |
by induct (insert `A \<subseteq> B`, auto intro: sigma_sets.intros(2-)) |
|
540 |
qed |
|
541 |
||
542 |
lemma sigma_sets_superset_generator: "A \<subseteq> sigma_sets X A" |
|
543 |
by (auto intro: sigma_sets.Basic) |
|
544 |
||
38656 | 545 |
lemma (in sigma_algebra) restriction_in_sets: |
546 |
fixes A :: "nat \<Rightarrow> 'a set" |
|
47694 | 547 |
assumes "S \<in> M" |
548 |
and *: "range A \<subseteq> (\<lambda>A. S \<inter> A) ` M" (is "_ \<subseteq> ?r") |
|
549 |
shows "range A \<subseteq> M" "(\<Union>i. A i) \<in> (\<lambda>A. S \<inter> A) ` M" |
|
38656 | 550 |
proof - |
551 |
{ fix i have "A i \<in> ?r" using * by auto |
|
47694 | 552 |
hence "\<exists>B. A i = B \<inter> S \<and> B \<in> M" by auto |
553 |
hence "A i \<subseteq> S" "A i \<in> M" using `S \<in> M` by auto } |
|
554 |
thus "range A \<subseteq> M" "(\<Union>i. A i) \<in> (\<lambda>A. S \<inter> A) ` M" |
|
38656 | 555 |
by (auto intro!: image_eqI[of _ _ "(\<Union>i. A i)"]) |
556 |
qed |
|
557 |
||
558 |
lemma (in sigma_algebra) restricted_sigma_algebra: |
|
47694 | 559 |
assumes "S \<in> M" |
560 |
shows "sigma_algebra S (restricted_space S)" |
|
38656 | 561 |
unfolding sigma_algebra_def sigma_algebra_axioms_def |
562 |
proof safe |
|
47694 | 563 |
show "algebra S (restricted_space S)" using restricted_algebra[OF assms] . |
38656 | 564 |
next |
47694 | 565 |
fix A :: "nat \<Rightarrow> 'a set" assume "range A \<subseteq> restricted_space S" |
38656 | 566 |
from restriction_in_sets[OF assms this[simplified]] |
47694 | 567 |
show "(\<Union>i. A i) \<in> restricted_space S" by simp |
38656 | 568 |
qed |
569 |
||
40859 | 570 |
lemma sigma_sets_Int: |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41543
diff
changeset
|
571 |
assumes "A \<in> sigma_sets sp st" "A \<subseteq> sp" |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41543
diff
changeset
|
572 |
shows "op \<inter> A ` sigma_sets sp st = sigma_sets A (op \<inter> A ` st)" |
40859 | 573 |
proof (intro equalityI subsetI) |
574 |
fix x assume "x \<in> op \<inter> A ` sigma_sets sp st" |
|
575 |
then obtain y where "y \<in> sigma_sets sp st" "x = y \<inter> A" by auto |
|
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41543
diff
changeset
|
576 |
then have "x \<in> sigma_sets (A \<inter> sp) (op \<inter> A ` st)" |
40859 | 577 |
proof (induct arbitrary: x) |
578 |
case (Compl a) |
|
579 |
then show ?case |
|
580 |
by (force intro!: sigma_sets.Compl simp: Diff_Int_distrib ac_simps) |
|
581 |
next |
|
582 |
case (Union a) |
|
583 |
then show ?case |
|
584 |
by (auto intro!: sigma_sets.Union |
|
585 |
simp add: UN_extend_simps simp del: UN_simps) |
|
47694 | 586 |
qed (auto intro!: sigma_sets.intros(2-)) |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41543
diff
changeset
|
587 |
then show "x \<in> sigma_sets A (op \<inter> A ` st)" |
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41543
diff
changeset
|
588 |
using `A \<subseteq> sp` by (simp add: Int_absorb2) |
40859 | 589 |
next |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41543
diff
changeset
|
590 |
fix x assume "x \<in> sigma_sets A (op \<inter> A ` st)" |
40859 | 591 |
then show "x \<in> op \<inter> A ` sigma_sets sp st" |
592 |
proof induct |
|
593 |
case (Compl a) |
|
594 |
then obtain x where "a = A \<inter> x" "x \<in> sigma_sets sp st" by auto |
|
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41543
diff
changeset
|
595 |
then show ?case using `A \<subseteq> sp` |
40859 | 596 |
by (force simp add: image_iff intro!: bexI[of _ "sp - x"] sigma_sets.Compl) |
597 |
next |
|
598 |
case (Union a) |
|
599 |
then have "\<forall>i. \<exists>x. x \<in> sigma_sets sp st \<and> a i = A \<inter> x" |
|
600 |
by (auto simp: image_iff Bex_def) |
|
601 |
from choice[OF this] guess f .. |
|
602 |
then show ?case |
|
603 |
by (auto intro!: bexI[of _ "(\<Union>x. f x)"] sigma_sets.Union |
|
604 |
simp add: image_iff) |
|
47694 | 605 |
qed (auto intro!: sigma_sets.intros(2-)) |
40859 | 606 |
qed |
607 |
||
47694 | 608 |
lemma sigma_sets_empty_eq: "sigma_sets A {} = {{}, A}" |
40859 | 609 |
proof (intro set_eqI iffI) |
47694 | 610 |
fix a assume "a \<in> sigma_sets A {}" then show "a \<in> {{}, A}" |
611 |
by induct blast+ |
|
612 |
qed (auto intro: sigma_sets.Empty sigma_sets_top) |
|
613 |
||
614 |
lemma sigma_sets_single[simp]: "sigma_sets A {A} = {{}, A}" |
|
615 |
proof (intro set_eqI iffI) |
|
616 |
fix x assume "x \<in> sigma_sets A {A}" |
|
617 |
then show "x \<in> {{}, A}" |
|
618 |
by induct blast+ |
|
40859 | 619 |
next |
47694 | 620 |
fix x assume "x \<in> {{}, A}" |
621 |
then show "x \<in> sigma_sets A {A}" |
|
40859 | 622 |
by (auto intro: sigma_sets.Empty sigma_sets_top) |
623 |
qed |
|
624 |
||
42987 | 625 |
lemma sigma_sets_sigma_sets_eq: |
626 |
"M \<subseteq> Pow S \<Longrightarrow> sigma_sets S (sigma_sets S M) = sigma_sets S M" |
|
47694 | 627 |
by (rule sigma_algebra.sigma_sets_eq[OF sigma_algebra_sigma_sets, of M S]) auto |
42987 | 628 |
|
42984 | 629 |
lemma sigma_sets_singleton: |
630 |
assumes "X \<subseteq> S" |
|
631 |
shows "sigma_sets S { X } = { {}, X, S - X, S }" |
|
632 |
proof - |
|
47694 | 633 |
interpret sigma_algebra S "{ {}, X, S - X, S }" |
42984 | 634 |
by (rule sigma_algebra_single_set) fact |
635 |
have "sigma_sets S { X } \<subseteq> sigma_sets S { {}, X, S - X, S }" |
|
636 |
by (rule sigma_sets_subseteq) simp |
|
637 |
moreover have "\<dots> = { {}, X, S - X, S }" |
|
47694 | 638 |
using sigma_sets_eq by simp |
42984 | 639 |
moreover |
640 |
{ fix A assume "A \<in> { {}, X, S - X, S }" |
|
641 |
then have "A \<in> sigma_sets S { X }" |
|
47694 | 642 |
by (auto intro: sigma_sets.intros(2-) sigma_sets_top) } |
42984 | 643 |
ultimately have "sigma_sets S { X } = sigma_sets S { {}, X, S - X, S }" |
644 |
by (intro antisym) auto |
|
47694 | 645 |
with sigma_sets_eq show ?thesis by simp |
42984 | 646 |
qed |
647 |
||
42863 | 648 |
lemma restricted_sigma: |
47694 | 649 |
assumes S: "S \<in> sigma_sets \<Omega> M" and M: "M \<subseteq> Pow \<Omega>" |
650 |
shows "algebra.restricted_space (sigma_sets \<Omega> M) S = |
|
651 |
sigma_sets S (algebra.restricted_space M S)" |
|
42863 | 652 |
proof - |
653 |
from S sigma_sets_into_sp[OF M] |
|
47694 | 654 |
have "S \<in> sigma_sets \<Omega> M" "S \<subseteq> \<Omega>" by auto |
42863 | 655 |
from sigma_sets_Int[OF this] |
47694 | 656 |
show ?thesis by simp |
42863 | 657 |
qed |
658 |
||
42987 | 659 |
lemma sigma_sets_vimage_commute: |
47694 | 660 |
assumes X: "X \<in> \<Omega> \<rightarrow> \<Omega>'" |
661 |
shows "{X -` A \<inter> \<Omega> |A. A \<in> sigma_sets \<Omega>' M'} |
|
662 |
= sigma_sets \<Omega> {X -` A \<inter> \<Omega> |A. A \<in> M'}" (is "?L = ?R") |
|
42987 | 663 |
proof |
664 |
show "?L \<subseteq> ?R" |
|
665 |
proof clarify |
|
47694 | 666 |
fix A assume "A \<in> sigma_sets \<Omega>' M'" |
667 |
then show "X -` A \<inter> \<Omega> \<in> ?R" |
|
42987 | 668 |
proof induct |
669 |
case Empty then show ?case |
|
670 |
by (auto intro!: sigma_sets.Empty) |
|
671 |
next |
|
672 |
case (Compl B) |
|
47694 | 673 |
have [simp]: "X -` (\<Omega>' - B) \<inter> \<Omega> = \<Omega> - (X -` B \<inter> \<Omega>)" |
42987 | 674 |
by (auto simp add: funcset_mem [OF X]) |
675 |
with Compl show ?case |
|
676 |
by (auto intro!: sigma_sets.Compl) |
|
677 |
next |
|
678 |
case (Union F) |
|
679 |
then show ?case |
|
680 |
by (auto simp add: vimage_UN UN_extend_simps(4) simp del: UN_simps |
|
681 |
intro!: sigma_sets.Union) |
|
47694 | 682 |
qed auto |
42987 | 683 |
qed |
684 |
show "?R \<subseteq> ?L" |
|
685 |
proof clarify |
|
686 |
fix A assume "A \<in> ?R" |
|
47694 | 687 |
then show "\<exists>B. A = X -` B \<inter> \<Omega> \<and> B \<in> sigma_sets \<Omega>' M'" |
42987 | 688 |
proof induct |
689 |
case (Basic B) then show ?case by auto |
|
690 |
next |
|
691 |
case Empty then show ?case |
|
47694 | 692 |
by (auto intro!: sigma_sets.Empty exI[of _ "{}"]) |
42987 | 693 |
next |
694 |
case (Compl B) |
|
47694 | 695 |
then obtain A where A: "B = X -` A \<inter> \<Omega>" "A \<in> sigma_sets \<Omega>' M'" by auto |
696 |
then have [simp]: "\<Omega> - B = X -` (\<Omega>' - A) \<inter> \<Omega>" |
|
42987 | 697 |
by (auto simp add: funcset_mem [OF X]) |
698 |
with A(2) show ?case |
|
47694 | 699 |
by (auto intro: sigma_sets.Compl) |
42987 | 700 |
next |
701 |
case (Union F) |
|
47694 | 702 |
then have "\<forall>i. \<exists>B. F i = X -` B \<inter> \<Omega> \<and> B \<in> sigma_sets \<Omega>' M'" by auto |
42987 | 703 |
from choice[OF this] guess A .. note A = this |
704 |
with A show ?case |
|
47694 | 705 |
by (auto simp: vimage_UN[symmetric] intro: sigma_sets.Union) |
42987 | 706 |
qed |
707 |
qed |
|
708 |
qed |
|
709 |
||
38656 | 710 |
section "Disjoint families" |
711 |
||
712 |
definition |
|
713 |
disjoint_family_on where |
|
714 |
"disjoint_family_on A S \<longleftrightarrow> (\<forall>m\<in>S. \<forall>n\<in>S. m \<noteq> n \<longrightarrow> A m \<inter> A n = {})" |
|
715 |
||
716 |
abbreviation |
|
717 |
"disjoint_family A \<equiv> disjoint_family_on A UNIV" |
|
718 |
||
719 |
lemma range_subsetD: "range f \<subseteq> B \<Longrightarrow> f i \<in> B" |
|
720 |
by blast |
|
721 |
||
722 |
lemma Int_Diff_disjoint: "A \<inter> B \<inter> (A - B) = {}" |
|
723 |
by blast |
|
724 |
||
725 |
lemma Int_Diff_Un: "A \<inter> B \<union> (A - B) = A" |
|
726 |
by blast |
|
727 |
||
728 |
lemma disjoint_family_subset: |
|
729 |
"disjoint_family A \<Longrightarrow> (!!x. B x \<subseteq> A x) \<Longrightarrow> disjoint_family B" |
|
730 |
by (force simp add: disjoint_family_on_def) |
|
731 |
||
40859 | 732 |
lemma disjoint_family_on_bisimulation: |
733 |
assumes "disjoint_family_on f S" |
|
734 |
and "\<And>n m. n \<in> S \<Longrightarrow> m \<in> S \<Longrightarrow> n \<noteq> m \<Longrightarrow> f n \<inter> f m = {} \<Longrightarrow> g n \<inter> g m = {}" |
|
735 |
shows "disjoint_family_on g S" |
|
736 |
using assms unfolding disjoint_family_on_def by auto |
|
737 |
||
38656 | 738 |
lemma disjoint_family_on_mono: |
739 |
"A \<subseteq> B \<Longrightarrow> disjoint_family_on f B \<Longrightarrow> disjoint_family_on f A" |
|
740 |
unfolding disjoint_family_on_def by auto |
|
741 |
||
742 |
lemma disjoint_family_Suc: |
|
743 |
assumes Suc: "!!n. A n \<subseteq> A (Suc n)" |
|
744 |
shows "disjoint_family (\<lambda>i. A (Suc i) - A i)" |
|
745 |
proof - |
|
746 |
{ |
|
747 |
fix m |
|
748 |
have "!!n. A n \<subseteq> A (m+n)" |
|
749 |
proof (induct m) |
|
750 |
case 0 show ?case by simp |
|
751 |
next |
|
752 |
case (Suc m) thus ?case |
|
753 |
by (metis Suc_eq_plus1 assms nat_add_commute nat_add_left_commute subset_trans) |
|
754 |
qed |
|
755 |
} |
|
756 |
hence "!!m n. m < n \<Longrightarrow> A m \<subseteq> A n" |
|
757 |
by (metis add_commute le_add_diff_inverse nat_less_le) |
|
758 |
thus ?thesis |
|
759 |
by (auto simp add: disjoint_family_on_def) |
|
760 |
(metis insert_absorb insert_subset le_SucE le_antisym not_leE) |
|
761 |
qed |
|
762 |
||
39092 | 763 |
lemma setsum_indicator_disjoint_family: |
764 |
fixes f :: "'d \<Rightarrow> 'e::semiring_1" |
|
765 |
assumes d: "disjoint_family_on A P" and "x \<in> A j" and "finite P" and "j \<in> P" |
|
766 |
shows "(\<Sum>i\<in>P. f i * indicator (A i) x) = f j" |
|
767 |
proof - |
|
768 |
have "P \<inter> {i. x \<in> A i} = {j}" |
|
769 |
using d `x \<in> A j` `j \<in> P` unfolding disjoint_family_on_def |
|
770 |
by auto |
|
771 |
thus ?thesis |
|
772 |
unfolding indicator_def |
|
773 |
by (simp add: if_distrib setsum_cases[OF `finite P`]) |
|
774 |
qed |
|
775 |
||
38656 | 776 |
definition disjointed :: "(nat \<Rightarrow> 'a set) \<Rightarrow> nat \<Rightarrow> 'a set " |
777 |
where "disjointed A n = A n - (\<Union>i\<in>{0..<n}. A i)" |
|
778 |
||
779 |
lemma finite_UN_disjointed_eq: "(\<Union>i\<in>{0..<n}. disjointed A i) = (\<Union>i\<in>{0..<n}. A i)" |
|
780 |
proof (induct n) |
|
781 |
case 0 show ?case by simp |
|
782 |
next |
|
783 |
case (Suc n) |
|
784 |
thus ?case by (simp add: atLeastLessThanSuc disjointed_def) |
|
785 |
qed |
|
786 |
||
787 |
lemma UN_disjointed_eq: "(\<Union>i. disjointed A i) = (\<Union>i. A i)" |
|
788 |
apply (rule UN_finite2_eq [where k=0]) |
|
789 |
apply (simp add: finite_UN_disjointed_eq) |
|
790 |
done |
|
791 |
||
792 |
lemma less_disjoint_disjointed: "m<n \<Longrightarrow> disjointed A m \<inter> disjointed A n = {}" |
|
793 |
by (auto simp add: disjointed_def) |
|
794 |
||
795 |
lemma disjoint_family_disjointed: "disjoint_family (disjointed A)" |
|
796 |
by (simp add: disjoint_family_on_def) |
|
797 |
(metis neq_iff Int_commute less_disjoint_disjointed) |
|
798 |
||
799 |
lemma disjointed_subset: "disjointed A n \<subseteq> A n" |
|
800 |
by (auto simp add: disjointed_def) |
|
801 |
||
42065
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
parents:
41983
diff
changeset
|
802 |
lemma (in ring_of_sets) UNION_in_sets: |
38656 | 803 |
fixes A:: "nat \<Rightarrow> 'a set" |
47694 | 804 |
assumes A: "range A \<subseteq> M" |
805 |
shows "(\<Union>i\<in>{0..<n}. A i) \<in> M" |
|
38656 | 806 |
proof (induct n) |
807 |
case 0 show ?case by simp |
|
808 |
next |
|
809 |
case (Suc n) |
|
810 |
thus ?case |
|
811 |
by (simp add: atLeastLessThanSuc) (metis A Un UNIV_I image_subset_iff) |
|
812 |
qed |
|
813 |
||
42065
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
parents:
41983
diff
changeset
|
814 |
lemma (in ring_of_sets) range_disjointed_sets: |
47694 | 815 |
assumes A: "range A \<subseteq> M" |
816 |
shows "range (disjointed A) \<subseteq> M" |
|
38656 | 817 |
proof (auto simp add: disjointed_def) |
818 |
fix n |
|
47694 | 819 |
show "A n - (\<Union>i\<in>{0..<n}. A i) \<in> M" using UNION_in_sets |
38656 | 820 |
by (metis A Diff UNIV_I image_subset_iff) |
821 |
qed |
|
822 |
||
42065
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
parents:
41983
diff
changeset
|
823 |
lemma (in algebra) range_disjointed_sets': |
47694 | 824 |
"range A \<subseteq> M \<Longrightarrow> range (disjointed A) \<subseteq> M" |
42065
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
parents:
41983
diff
changeset
|
825 |
using range_disjointed_sets . |
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
parents:
41983
diff
changeset
|
826 |
|
42145 | 827 |
lemma disjointed_0[simp]: "disjointed A 0 = A 0" |
828 |
by (simp add: disjointed_def) |
|
829 |
||
830 |
lemma incseq_Un: |
|
831 |
"incseq A \<Longrightarrow> (\<Union>i\<le>n. A i) = A n" |
|
832 |
unfolding incseq_def by auto |
|
833 |
||
834 |
lemma disjointed_incseq: |
|
835 |
"incseq A \<Longrightarrow> disjointed A (Suc n) = A (Suc n) - A n" |
|
836 |
using incseq_Un[of A] |
|
837 |
by (simp add: disjointed_def atLeastLessThanSuc_atLeastAtMost atLeast0AtMost) |
|
838 |
||
38656 | 839 |
lemma sigma_algebra_disjoint_iff: |
47694 | 840 |
"sigma_algebra \<Omega> M \<longleftrightarrow> algebra \<Omega> M \<and> |
841 |
(\<forall>A. range A \<subseteq> M \<longrightarrow> disjoint_family A \<longrightarrow> (\<Union>i::nat. A i) \<in> M)" |
|
38656 | 842 |
proof (auto simp add: sigma_algebra_iff) |
843 |
fix A :: "nat \<Rightarrow> 'a set" |
|
47694 | 844 |
assume M: "algebra \<Omega> M" |
845 |
and A: "range A \<subseteq> M" |
|
846 |
and UnA: "\<forall>A. range A \<subseteq> M \<longrightarrow> disjoint_family A \<longrightarrow> (\<Union>i::nat. A i) \<in> M" |
|
847 |
hence "range (disjointed A) \<subseteq> M \<longrightarrow> |
|
38656 | 848 |
disjoint_family (disjointed A) \<longrightarrow> |
47694 | 849 |
(\<Union>i. disjointed A i) \<in> M" by blast |
850 |
hence "(\<Union>i. disjointed A i) \<in> M" |
|
851 |
by (simp add: algebra.range_disjointed_sets'[of \<Omega>] M A disjoint_family_disjointed) |
|
852 |
thus "(\<Union>i::nat. A i) \<in> M" by (simp add: UN_disjointed_eq) |
|
853 |
qed |
|
854 |
||
47762 | 855 |
lemma disjoint_family_on_disjoint_image: |
856 |
"disjoint_family_on A I \<Longrightarrow> disjoint (A ` I)" |
|
857 |
unfolding disjoint_family_on_def disjoint_def by force |
|
858 |
||
859 |
lemma disjoint_image_disjoint_family_on: |
|
860 |
assumes d: "disjoint (A ` I)" and i: "inj_on A I" |
|
861 |
shows "disjoint_family_on A I" |
|
862 |
unfolding disjoint_family_on_def |
|
863 |
proof (intro ballI impI) |
|
864 |
fix n m assume nm: "m \<in> I" "n \<in> I" and "n \<noteq> m" |
|
865 |
with i[THEN inj_onD, of n m] show "A n \<inter> A m = {}" |
|
866 |
by (intro disjointD[OF d]) auto |
|
867 |
qed |
|
868 |
||
869 |
section {* Ring generated by a semiring *} |
|
870 |
||
871 |
definition (in semiring_of_sets) |
|
872 |
"generated_ring = { \<Union>C | C. C \<subseteq> M \<and> finite C \<and> disjoint C }" |
|
873 |
||
874 |
lemma (in semiring_of_sets) generated_ringE[elim?]: |
|
875 |
assumes "a \<in> generated_ring" |
|
876 |
obtains C where "finite C" "disjoint C" "C \<subseteq> M" "a = \<Union>C" |
|
877 |
using assms unfolding generated_ring_def by auto |
|
878 |
||
879 |
lemma (in semiring_of_sets) generated_ringI[intro?]: |
|
880 |
assumes "finite C" "disjoint C" "C \<subseteq> M" "a = \<Union>C" |
|
881 |
shows "a \<in> generated_ring" |
|
882 |
using assms unfolding generated_ring_def by auto |
|
883 |
||
884 |
lemma (in semiring_of_sets) generated_ringI_Basic: |
|
885 |
"A \<in> M \<Longrightarrow> A \<in> generated_ring" |
|
886 |
by (rule generated_ringI[of "{A}"]) (auto simp: disjoint_def) |
|
887 |
||
888 |
lemma (in semiring_of_sets) generated_ring_disjoint_Un[intro]: |
|
889 |
assumes a: "a \<in> generated_ring" and b: "b \<in> generated_ring" |
|
890 |
and "a \<inter> b = {}" |
|
891 |
shows "a \<union> b \<in> generated_ring" |
|
892 |
proof - |
|
893 |
from a guess Ca .. note Ca = this |
|
894 |
from b guess Cb .. note Cb = this |
|
895 |
show ?thesis |
|
896 |
proof |
|
897 |
show "disjoint (Ca \<union> Cb)" |
|
898 |
using `a \<inter> b = {}` Ca Cb by (auto intro!: disjoint_union) |
|
899 |
qed (insert Ca Cb, auto) |
|
900 |
qed |
|
901 |
||
902 |
lemma (in semiring_of_sets) generated_ring_empty: "{} \<in> generated_ring" |
|
903 |
by (auto simp: generated_ring_def disjoint_def) |
|
904 |
||
905 |
lemma (in semiring_of_sets) generated_ring_disjoint_Union: |
|
906 |
assumes "finite A" shows "A \<subseteq> generated_ring \<Longrightarrow> disjoint A \<Longrightarrow> \<Union>A \<in> generated_ring" |
|
907 |
using assms by (induct A) (auto simp: disjoint_def intro!: generated_ring_disjoint_Un generated_ring_empty) |
|
908 |
||
909 |
lemma (in semiring_of_sets) generated_ring_disjoint_UNION: |
|
910 |
"finite I \<Longrightarrow> disjoint (A ` I) \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> A i \<in> generated_ring) \<Longrightarrow> UNION I A \<in> generated_ring" |
|
911 |
unfolding SUP_def by (intro generated_ring_disjoint_Union) auto |
|
912 |
||
913 |
lemma (in semiring_of_sets) generated_ring_Int: |
|
914 |
assumes a: "a \<in> generated_ring" and b: "b \<in> generated_ring" |
|
915 |
shows "a \<inter> b \<in> generated_ring" |
|
916 |
proof - |
|
917 |
from a guess Ca .. note Ca = this |
|
918 |
from b guess Cb .. note Cb = this |
|
919 |
def C \<equiv> "(\<lambda>(a,b). a \<inter> b)` (Ca\<times>Cb)" |
|
920 |
show ?thesis |
|
921 |
proof |
|
922 |
show "disjoint C" |
|
923 |
proof (simp add: disjoint_def C_def, intro ballI impI) |
|
924 |
fix a1 b1 a2 b2 assume sets: "a1 \<in> Ca" "b1 \<in> Cb" "a2 \<in> Ca" "b2 \<in> Cb" |
|
925 |
assume "a1 \<inter> b1 \<noteq> a2 \<inter> b2" |
|
926 |
then have "a1 \<noteq> a2 \<or> b1 \<noteq> b2" by auto |
|
927 |
then show "(a1 \<inter> b1) \<inter> (a2 \<inter> b2) = {}" |
|
928 |
proof |
|
929 |
assume "a1 \<noteq> a2" |
|
930 |
with sets Ca have "a1 \<inter> a2 = {}" |
|
931 |
by (auto simp: disjoint_def) |
|
932 |
then show ?thesis by auto |
|
933 |
next |
|
934 |
assume "b1 \<noteq> b2" |
|
935 |
with sets Cb have "b1 \<inter> b2 = {}" |
|
936 |
by (auto simp: disjoint_def) |
|
937 |
then show ?thesis by auto |
|
938 |
qed |
|
939 |
qed |
|
940 |
qed (insert Ca Cb, auto simp: C_def) |
|
941 |
qed |
|
942 |
||
943 |
lemma (in semiring_of_sets) generated_ring_Inter: |
|
944 |
assumes "finite A" "A \<noteq> {}" shows "A \<subseteq> generated_ring \<Longrightarrow> \<Inter>A \<in> generated_ring" |
|
945 |
using assms by (induct A rule: finite_ne_induct) (auto intro: generated_ring_Int) |
|
946 |
||
947 |
lemma (in semiring_of_sets) generated_ring_INTER: |
|
948 |
"finite I \<Longrightarrow> I \<noteq> {} \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> A i \<in> generated_ring) \<Longrightarrow> INTER I A \<in> generated_ring" |
|
949 |
unfolding INF_def by (intro generated_ring_Inter) auto |
|
950 |
||
951 |
lemma (in semiring_of_sets) generating_ring: |
|
952 |
"ring_of_sets \<Omega> generated_ring" |
|
953 |
proof (rule ring_of_setsI) |
|
954 |
let ?R = generated_ring |
|
955 |
show "?R \<subseteq> Pow \<Omega>" |
|
956 |
using sets_into_space by (auto simp: generated_ring_def generated_ring_empty) |
|
957 |
show "{} \<in> ?R" by (rule generated_ring_empty) |
|
958 |
||
959 |
{ fix a assume a: "a \<in> ?R" then guess Ca .. note Ca = this |
|
960 |
fix b assume b: "b \<in> ?R" then guess Cb .. note Cb = this |
|
961 |
||
962 |
show "a - b \<in> ?R" |
|
963 |
proof cases |
|
964 |
assume "Cb = {}" with Cb `a \<in> ?R` show ?thesis |
|
965 |
by simp |
|
966 |
next |
|
967 |
assume "Cb \<noteq> {}" |
|
968 |
with Ca Cb have "a - b = (\<Union>a'\<in>Ca. \<Inter>b'\<in>Cb. a' - b')" by auto |
|
969 |
also have "\<dots> \<in> ?R" |
|
970 |
proof (intro generated_ring_INTER generated_ring_disjoint_UNION) |
|
971 |
fix a b assume "a \<in> Ca" "b \<in> Cb" |
|
972 |
with Ca Cb Diff_cover[of a b] show "a - b \<in> ?R" |
|
973 |
by (auto simp add: generated_ring_def) |
|
974 |
next |
|
975 |
show "disjoint ((\<lambda>a'. \<Inter>b'\<in>Cb. a' - b')`Ca)" |
|
976 |
using Ca by (auto simp add: disjoint_def `Cb \<noteq> {}`) |
|
977 |
next |
|
978 |
show "finite Ca" "finite Cb" "Cb \<noteq> {}" by fact+ |
|
979 |
qed |
|
980 |
finally show "a - b \<in> ?R" . |
|
981 |
qed } |
|
982 |
note Diff = this |
|
983 |
||
984 |
fix a b assume sets: "a \<in> ?R" "b \<in> ?R" |
|
985 |
have "a \<union> b = (a - b) \<union> (a \<inter> b) \<union> (b - a)" by auto |
|
986 |
also have "\<dots> \<in> ?R" |
|
987 |
by (intro sets generated_ring_disjoint_Un generated_ring_Int Diff) auto |
|
988 |
finally show "a \<union> b \<in> ?R" . |
|
989 |
qed |
|
990 |
||
991 |
lemma (in semiring_of_sets) sigma_sets_generated_ring_eq: "sigma_sets \<Omega> generated_ring = sigma_sets \<Omega> M" |
|
992 |
proof |
|
993 |
interpret M: sigma_algebra \<Omega> "sigma_sets \<Omega> M" |
|
994 |
using space_closed by (rule sigma_algebra_sigma_sets) |
|
995 |
show "sigma_sets \<Omega> generated_ring \<subseteq> sigma_sets \<Omega> M" |
|
996 |
by (blast intro!: sigma_sets_mono elim: generated_ringE) |
|
997 |
qed (auto intro!: generated_ringI_Basic sigma_sets_mono) |
|
998 |
||
47694 | 999 |
section {* Measure type *} |
1000 |
||
1001 |
definition positive :: "'a set set \<Rightarrow> ('a set \<Rightarrow> ereal) \<Rightarrow> bool" where |
|
1002 |
"positive M \<mu> \<longleftrightarrow> \<mu> {} = 0 \<and> (\<forall>A\<in>M. 0 \<le> \<mu> A)" |
|
1003 |
||
1004 |
definition countably_additive :: "'a set set \<Rightarrow> ('a set \<Rightarrow> ereal) \<Rightarrow> bool" where |
|
1005 |
"countably_additive M f \<longleftrightarrow> (\<forall>A. range A \<subseteq> M \<longrightarrow> disjoint_family A \<longrightarrow> (\<Union>i. A i) \<in> M \<longrightarrow> |
|
1006 |
(\<Sum>i. f (A i)) = f (\<Union>i. A i))" |
|
1007 |
||
1008 |
definition measure_space :: "'a set \<Rightarrow> 'a set set \<Rightarrow> ('a set \<Rightarrow> ereal) \<Rightarrow> bool" where |
|
1009 |
"measure_space \<Omega> A \<mu> \<longleftrightarrow> sigma_algebra \<Omega> A \<and> positive A \<mu> \<and> countably_additive A \<mu>" |
|
1010 |
||
49834 | 1011 |
typedef 'a measure = "{(\<Omega>::'a set, A, \<mu>). (\<forall>a\<in>-A. \<mu> a = 0) \<and> measure_space \<Omega> A \<mu> }" |
47694 | 1012 |
proof |
1013 |
have "sigma_algebra UNIV {{}, UNIV}" |
|
47762 | 1014 |
by (auto simp: sigma_algebra_iff2) |
47694 | 1015 |
then show "(UNIV, {{}, UNIV}, \<lambda>A. 0) \<in> {(\<Omega>, A, \<mu>). (\<forall>a\<in>-A. \<mu> a = 0) \<and> measure_space \<Omega> A \<mu>} " |
1016 |
by (auto simp: measure_space_def positive_def countably_additive_def) |
|
1017 |
qed |
|
1018 |
||
1019 |
definition space :: "'a measure \<Rightarrow> 'a set" where |
|
1020 |
"space M = fst (Rep_measure M)" |
|
1021 |
||
1022 |
definition sets :: "'a measure \<Rightarrow> 'a set set" where |
|
1023 |
"sets M = fst (snd (Rep_measure M))" |
|
1024 |
||
1025 |
definition emeasure :: "'a measure \<Rightarrow> 'a set \<Rightarrow> ereal" where |
|
1026 |
"emeasure M = snd (snd (Rep_measure M))" |
|
1027 |
||
1028 |
definition measure :: "'a measure \<Rightarrow> 'a set \<Rightarrow> real" where |
|
1029 |
"measure M A = real (emeasure M A)" |
|
1030 |
||
1031 |
declare [[coercion sets]] |
|
1032 |
||
1033 |
declare [[coercion measure]] |
|
1034 |
||
1035 |
declare [[coercion emeasure]] |
|
1036 |
||
1037 |
lemma measure_space: "measure_space (space M) (sets M) (emeasure M)" |
|
1038 |
by (cases M) (auto simp: space_def sets_def emeasure_def Abs_measure_inverse) |
|
1039 |
||
50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50096
diff
changeset
|
1040 |
interpretation sets!: sigma_algebra "space M" "sets M" for M :: "'a measure" |
47694 | 1041 |
using measure_space[of M] by (auto simp: measure_space_def) |
1042 |
||
1043 |
definition measure_of :: "'a set \<Rightarrow> 'a set set \<Rightarrow> ('a set \<Rightarrow> ereal) \<Rightarrow> 'a measure" where |
|
1044 |
"measure_of \<Omega> A \<mu> = Abs_measure (\<Omega>, sigma_sets \<Omega> A, |
|
1045 |
\<lambda>a. if a \<in> sigma_sets \<Omega> A \<and> measure_space \<Omega> (sigma_sets \<Omega> A) \<mu> then \<mu> a else 0)" |
|
1046 |
||
1047 |
abbreviation "sigma \<Omega> A \<equiv> measure_of \<Omega> A (\<lambda>x. 0)" |
|
1048 |
||
1049 |
lemma measure_space_0: "A \<subseteq> Pow \<Omega> \<Longrightarrow> measure_space \<Omega> (sigma_sets \<Omega> A) (\<lambda>x. 0)" |
|
1050 |
unfolding measure_space_def |
|
1051 |
by (auto intro!: sigma_algebra_sigma_sets simp: positive_def countably_additive_def) |
|
1052 |
||
1053 |
lemma (in ring_of_sets) positive_cong_eq: |
|
1054 |
"(\<And>a. a \<in> M \<Longrightarrow> \<mu>' a = \<mu> a) \<Longrightarrow> positive M \<mu>' = positive M \<mu>" |
|
1055 |
by (auto simp add: positive_def) |
|
1056 |
||
1057 |
lemma (in sigma_algebra) countably_additive_eq: |
|
1058 |
"(\<And>a. a \<in> M \<Longrightarrow> \<mu>' a = \<mu> a) \<Longrightarrow> countably_additive M \<mu>' = countably_additive M \<mu>" |
|
1059 |
unfolding countably_additive_def |
|
1060 |
by (intro arg_cong[where f=All] ext) (auto simp add: countably_additive_def subset_eq) |
|
1061 |
||
1062 |
lemma measure_space_eq: |
|
1063 |
assumes closed: "A \<subseteq> Pow \<Omega>" and eq: "\<And>a. a \<in> sigma_sets \<Omega> A \<Longrightarrow> \<mu> a = \<mu>' a" |
|
1064 |
shows "measure_space \<Omega> (sigma_sets \<Omega> A) \<mu> = measure_space \<Omega> (sigma_sets \<Omega> A) \<mu>'" |
|
1065 |
proof - |
|
1066 |
interpret sigma_algebra \<Omega> "sigma_sets \<Omega> A" using closed by (rule sigma_algebra_sigma_sets) |
|
1067 |
from positive_cong_eq[OF eq, of "\<lambda>i. i"] countably_additive_eq[OF eq, of "\<lambda>i. i"] show ?thesis |
|
1068 |
by (auto simp: measure_space_def) |
|
1069 |
qed |
|
1070 |
||
1071 |
lemma measure_of_eq: |
|
1072 |
assumes closed: "A \<subseteq> Pow \<Omega>" and eq: "(\<And>a. a \<in> sigma_sets \<Omega> A \<Longrightarrow> \<mu> a = \<mu>' a)" |
|
1073 |
shows "measure_of \<Omega> A \<mu> = measure_of \<Omega> A \<mu>'" |
|
1074 |
proof - |
|
1075 |
have "measure_space \<Omega> (sigma_sets \<Omega> A) \<mu> = measure_space \<Omega> (sigma_sets \<Omega> A) \<mu>'" |
|
1076 |
using assms by (rule measure_space_eq) |
|
1077 |
with eq show ?thesis |
|
1078 |
by (auto simp add: measure_of_def intro!: arg_cong[where f=Abs_measure]) |
|
1079 |
qed |
|
1080 |
||
1081 |
lemma |
|
1082 |
assumes A: "A \<subseteq> Pow \<Omega>" |
|
1083 |
shows sets_measure_of[simp]: "sets (measure_of \<Omega> A \<mu>) = sigma_sets \<Omega> A" (is ?sets) |
|
1084 |
and space_measure_of[simp]: "space (measure_of \<Omega> A \<mu>) = \<Omega>" (is ?space) |
|
1085 |
proof - |
|
1086 |
have "?sets \<and> ?space" |
|
1087 |
proof cases |
|
1088 |
assume "measure_space \<Omega> (sigma_sets \<Omega> A) \<mu>" |
|
1089 |
moreover have "measure_space \<Omega> (sigma_sets \<Omega> A) \<mu> = measure_space \<Omega> (sigma_sets \<Omega> A) |
|
1090 |
(\<lambda>a. if a \<in> sigma_sets \<Omega> A then \<mu> a else 0)" |
|
1091 |
using A by (rule measure_space_eq) auto |
|
1092 |
ultimately show "?sets \<and> ?space" |
|
1093 |
by (auto simp: Abs_measure_inverse measure_of_def sets_def space_def) |
|
1094 |
next |
|
1095 |
assume "\<not> measure_space \<Omega> (sigma_sets \<Omega> A) \<mu>" |
|
1096 |
with A show "?sets \<and> ?space" |
|
1097 |
by (auto simp: Abs_measure_inverse measure_of_def sets_def space_def measure_space_0) |
|
1098 |
qed |
|
1099 |
then show ?sets ?space by auto |
|
1100 |
qed |
|
1101 |
||
1102 |
lemma (in sigma_algebra) sets_measure_of_eq[simp]: |
|
1103 |
"sets (measure_of \<Omega> M \<mu>) = M" |
|
1104 |
using space_closed by (auto intro!: sigma_sets_eq) |
|
1105 |
||
1106 |
lemma (in sigma_algebra) space_measure_of_eq[simp]: |
|
1107 |
"space (measure_of \<Omega> M \<mu>) = \<Omega>" |
|
1108 |
using space_closed by (auto intro!: sigma_sets_eq) |
|
1109 |
||
1110 |
lemma measure_of_subset: |
|
1111 |
"M \<subseteq> Pow \<Omega> \<Longrightarrow> M' \<subseteq> M \<Longrightarrow> sets (measure_of \<Omega> M' \<mu>) \<subseteq> sets (measure_of \<Omega> M \<mu>')" |
|
1112 |
by (auto intro!: sigma_sets_subseteq) |
|
1113 |
||
50245
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset
|
1114 |
lemma sigma_sets_mono'': |
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset
|
1115 |
assumes "A \<in> sigma_sets C D" |
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset
|
1116 |
assumes "B \<subseteq> D" |
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset
|
1117 |
assumes "D \<subseteq> Pow C" |
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset
|
1118 |
shows "sigma_sets A B \<subseteq> sigma_sets C D" |
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset
|
1119 |
proof |
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset
|
1120 |
fix x assume "x \<in> sigma_sets A B" |
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset
|
1121 |
thus "x \<in> sigma_sets C D" |
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset
|
1122 |
proof induct |
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset
|
1123 |
case (Basic a) with assms have "a \<in> D" by auto |
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset
|
1124 |
thus ?case .. |
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset
|
1125 |
next |
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset
|
1126 |
case Empty show ?case by (rule sigma_sets.Empty) |
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset
|
1127 |
next |
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset
|
1128 |
from assms have "A \<in> sets (sigma C D)" by (subst sets_measure_of[OF `D \<subseteq> Pow C`]) |
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset
|
1129 |
moreover case (Compl a) hence "a \<in> sets (sigma C D)" by (subst sets_measure_of[OF `D \<subseteq> Pow C`]) |
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset
|
1130 |
ultimately have "A - a \<in> sets (sigma C D)" .. |
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset
|
1131 |
thus ?case by (subst (asm) sets_measure_of[OF `D \<subseteq> Pow C`]) |
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset
|
1132 |
next |
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset
|
1133 |
case (Union a) |
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset
|
1134 |
thus ?case by (intro sigma_sets.Union) |
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset
|
1135 |
qed |
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset
|
1136 |
qed |
dea9363887a6
based countable topological basis on Countable_Set
immler
parents:
50244
diff
changeset
|
1137 |
|
47756 | 1138 |
lemma in_measure_of[intro, simp]: "M \<subseteq> Pow \<Omega> \<Longrightarrow> A \<in> M \<Longrightarrow> A \<in> sets (measure_of \<Omega> M \<mu>)" |
47694 | 1139 |
by auto |
1140 |
||
1141 |
section {* Constructing simple @{typ "'a measure"} *} |
|
1142 |
||
1143 |
lemma emeasure_measure_of: |
|
1144 |
assumes M: "M = measure_of \<Omega> A \<mu>" |
|
1145 |
assumes ms: "A \<subseteq> Pow \<Omega>" "positive (sets M) \<mu>" "countably_additive (sets M) \<mu>" |
|
1146 |
assumes X: "X \<in> sets M" |
|
1147 |
shows "emeasure M X = \<mu> X" |
|
1148 |
proof - |
|
1149 |
interpret sigma_algebra \<Omega> "sigma_sets \<Omega> A" by (rule sigma_algebra_sigma_sets) fact |
|
1150 |
have "measure_space \<Omega> (sigma_sets \<Omega> A) \<mu>" |
|
1151 |
using ms M by (simp add: measure_space_def sigma_algebra_sigma_sets) |
|
1152 |
moreover have "measure_space \<Omega> (sigma_sets \<Omega> A) (\<lambda>a. if a \<in> sigma_sets \<Omega> A then \<mu> a else 0) |
|
1153 |
= measure_space \<Omega> (sigma_sets \<Omega> A) \<mu>" |
|
1154 |
using ms(1) by (rule measure_space_eq) auto |
|
1155 |
moreover have "X \<in> sigma_sets \<Omega> A" |
|
1156 |
using X M ms by simp |
|
1157 |
ultimately show ?thesis |
|
1158 |
unfolding emeasure_def measure_of_def M |
|
1159 |
by (subst Abs_measure_inverse) (simp_all add: sigma_sets_eq) |
|
1160 |
qed |
|
1161 |
||
1162 |
lemma emeasure_measure_of_sigma: |
|
1163 |
assumes ms: "sigma_algebra \<Omega> M" "positive M \<mu>" "countably_additive M \<mu>" |
|
1164 |
assumes A: "A \<in> M" |
|
1165 |
shows "emeasure (measure_of \<Omega> M \<mu>) A = \<mu> A" |
|
1166 |
proof - |
|
1167 |
interpret sigma_algebra \<Omega> M by fact |
|
1168 |
have "measure_space \<Omega> (sigma_sets \<Omega> M) \<mu>" |
|
1169 |
using ms sigma_sets_eq by (simp add: measure_space_def) |
|
1170 |
moreover have "measure_space \<Omega> (sigma_sets \<Omega> M) (\<lambda>a. if a \<in> sigma_sets \<Omega> M then \<mu> a else 0) |
|
1171 |
= measure_space \<Omega> (sigma_sets \<Omega> M) \<mu>" |
|
1172 |
using space_closed by (rule measure_space_eq) auto |
|
1173 |
ultimately show ?thesis using A |
|
1174 |
unfolding emeasure_def measure_of_def |
|
1175 |
by (subst Abs_measure_inverse) (simp_all add: sigma_sets_eq) |
|
1176 |
qed |
|
1177 |
||
1178 |
lemma measure_cases[cases type: measure]: |
|
1179 |
obtains (measure) \<Omega> A \<mu> where "x = Abs_measure (\<Omega>, A, \<mu>)" "\<forall>a\<in>-A. \<mu> a = 0" "measure_space \<Omega> A \<mu>" |
|
1180 |
by atomize_elim (cases x, auto) |
|
1181 |
||
1182 |
lemma sets_eq_imp_space_eq: |
|
1183 |
"sets M = sets M' \<Longrightarrow> space M = space M'" |
|
50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50096
diff
changeset
|
1184 |
using sets.top[of M] sets.top[of M'] sets.space_closed[of M] sets.space_closed[of M'] |
47694 | 1185 |
by blast |
1186 |
||
1187 |
lemma emeasure_notin_sets: "A \<notin> sets M \<Longrightarrow> emeasure M A = 0" |
|
1188 |
by (cases M) (auto simp: sets_def emeasure_def Abs_measure_inverse measure_space_def) |
|
1189 |
||
1190 |
lemma measure_notin_sets: "A \<notin> sets M \<Longrightarrow> measure M A = 0" |
|
1191 |
by (simp add: measure_def emeasure_notin_sets) |
|
1192 |
||
1193 |
lemma measure_eqI: |
|
1194 |
fixes M N :: "'a measure" |
|
1195 |
assumes "sets M = sets N" and eq: "\<And>A. A \<in> sets M \<Longrightarrow> emeasure M A = emeasure N A" |
|
1196 |
shows "M = N" |
|
1197 |
proof (cases M N rule: measure_cases[case_product measure_cases]) |
|
1198 |
case (measure_measure \<Omega> A \<mu> \<Omega>' A' \<mu>') |
|
1199 |
interpret M: sigma_algebra \<Omega> A using measure_measure by (auto simp: measure_space_def) |
|
1200 |
interpret N: sigma_algebra \<Omega>' A' using measure_measure by (auto simp: measure_space_def) |
|
1201 |
have "A = sets M" "A' = sets N" |
|
1202 |
using measure_measure by (simp_all add: sets_def Abs_measure_inverse) |
|
1203 |
with `sets M = sets N` have "A = A'" by simp |
|
1204 |
moreover with M.top N.top M.space_closed N.space_closed have "\<Omega> = \<Omega>'" by auto |
|
1205 |
moreover { fix B have "\<mu> B = \<mu>' B" |
|
1206 |
proof cases |
|
1207 |
assume "B \<in> A" |
|
1208 |
with eq `A = sets M` have "emeasure M B = emeasure N B" by simp |
|
1209 |
with measure_measure show "\<mu> B = \<mu>' B" |
|
1210 |
by (simp add: emeasure_def Abs_measure_inverse) |
|
1211 |
next |
|
1212 |
assume "B \<notin> A" |
|
1213 |
with `A = sets M` `A' = sets N` `A = A'` have "B \<notin> sets M" "B \<notin> sets N" |
|
1214 |
by auto |
|
1215 |
then have "emeasure M B = 0" "emeasure N B = 0" |
|
1216 |
by (simp_all add: emeasure_notin_sets) |
|
1217 |
with measure_measure show "\<mu> B = \<mu>' B" |
|
1218 |
by (simp add: emeasure_def Abs_measure_inverse) |
|
1219 |
qed } |
|
1220 |
then have "\<mu> = \<mu>'" by auto |
|
1221 |
ultimately show "M = N" |
|
1222 |
by (simp add: measure_measure) |
|
38656 | 1223 |
qed |
1224 |
||
47694 | 1225 |
lemma emeasure_sigma: "A \<subseteq> Pow \<Omega> \<Longrightarrow> emeasure (sigma \<Omega> A) = (\<lambda>_. 0)" |
1226 |
using measure_space_0[of A \<Omega>] |
|
1227 |
by (simp add: measure_of_def emeasure_def Abs_measure_inverse) |
|
1228 |
||
1229 |
lemma sigma_eqI: |
|
1230 |
assumes [simp]: "M \<subseteq> Pow \<Omega>" "N \<subseteq> Pow \<Omega>" "sigma_sets \<Omega> M = sigma_sets \<Omega> N" |
|
1231 |
shows "sigma \<Omega> M = sigma \<Omega> N" |
|
1232 |
by (rule measure_eqI) (simp_all add: emeasure_sigma) |
|
1233 |
||
1234 |
section {* Measurable functions *} |
|
1235 |
||
1236 |
definition measurable :: "'a measure \<Rightarrow> 'b measure \<Rightarrow> ('a \<Rightarrow> 'b) set" where |
|
1237 |
"measurable A B = {f \<in> space A -> space B. \<forall>y \<in> sets B. f -` y \<inter> space A \<in> sets A}" |
|
1238 |
||
1239 |
lemma measurable_space: |
|
1240 |
"f \<in> measurable M A \<Longrightarrow> x \<in> space M \<Longrightarrow> f x \<in> space A" |
|
1241 |
unfolding measurable_def by auto |
|
1242 |
||
1243 |
lemma measurable_sets: |
|
1244 |
"f \<in> measurable M A \<Longrightarrow> S \<in> sets A \<Longrightarrow> f -` S \<inter> space M \<in> sets M" |
|
1245 |
unfolding measurable_def by auto |
|
1246 |
||
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1247 |
lemma measurable_sets_Collect: |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1248 |
assumes f: "f \<in> measurable M N" and P: "{x\<in>space N. P x} \<in> sets N" shows "{x\<in>space M. P (f x)} \<in> sets M" |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1249 |
proof - |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1250 |
have "f -` {x \<in> space N. P x} \<inter> space M = {x\<in>space M. P (f x)}" |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1251 |
using measurable_space[OF f] by auto |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1252 |
with measurable_sets[OF f P] show ?thesis |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1253 |
by simp |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1254 |
qed |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1255 |
|
47694 | 1256 |
lemma measurable_sigma_sets: |
1257 |
assumes B: "sets N = sigma_sets \<Omega> A" "A \<subseteq> Pow \<Omega>" |
|
1258 |
and f: "f \<in> space M \<rightarrow> \<Omega>" |
|
1259 |
and ba: "\<And>y. y \<in> A \<Longrightarrow> (f -` y) \<inter> space M \<in> sets M" |
|
1260 |
shows "f \<in> measurable M N" |
|
1261 |
proof - |
|
1262 |
interpret A: sigma_algebra \<Omega> "sigma_sets \<Omega> A" using B(2) by (rule sigma_algebra_sigma_sets) |
|
50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50096
diff
changeset
|
1263 |
from B sets.top[of N] A.top sets.space_closed[of N] A.space_closed have \<Omega>: "\<Omega> = space N" by force |
47694 | 1264 |
|
1265 |
{ fix X assume "X \<in> sigma_sets \<Omega> A" |
|
1266 |
then have "f -` X \<inter> space M \<in> sets M \<and> X \<subseteq> \<Omega>" |
|
1267 |
proof induct |
|
1268 |
case (Basic a) then show ?case |
|
1269 |
by (auto simp add: ba) (metis B(2) subsetD PowD) |
|
1270 |
next |
|
1271 |
case (Compl a) |
|
1272 |
have [simp]: "f -` \<Omega> \<inter> space M = space M" |
|
1273 |
by (auto simp add: funcset_mem [OF f]) |
|
1274 |
then show ?case |
|
50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50096
diff
changeset
|
1275 |
by (auto simp add: vimage_Diff Diff_Int_distrib2 sets.compl_sets Compl) |
47694 | 1276 |
next |
1277 |
case (Union a) |
|
1278 |
then show ?case |
|
1279 |
by (simp add: vimage_UN, simp only: UN_extend_simps(4)) blast |
|
1280 |
qed auto } |
|
1281 |
with f show ?thesis |
|
1282 |
by (auto simp add: measurable_def B \<Omega>) |
|
1283 |
qed |
|
1284 |
||
1285 |
lemma measurable_measure_of: |
|
1286 |
assumes B: "N \<subseteq> Pow \<Omega>" |
|
1287 |
and f: "f \<in> space M \<rightarrow> \<Omega>" |
|
1288 |
and ba: "\<And>y. y \<in> N \<Longrightarrow> (f -` y) \<inter> space M \<in> sets M" |
|
1289 |
shows "f \<in> measurable M (measure_of \<Omega> N \<mu>)" |
|
1290 |
proof - |
|
1291 |
have "sets (measure_of \<Omega> N \<mu>) = sigma_sets \<Omega> N" |
|
1292 |
using B by (rule sets_measure_of) |
|
1293 |
from this assms show ?thesis by (rule measurable_sigma_sets) |
|
1294 |
qed |
|
1295 |
||
1296 |
lemma measurable_iff_measure_of: |
|
1297 |
assumes "N \<subseteq> Pow \<Omega>" "f \<in> space M \<rightarrow> \<Omega>" |
|
1298 |
shows "f \<in> measurable M (measure_of \<Omega> N \<mu>) \<longleftrightarrow> (\<forall>A\<in>N. f -` A \<inter> space M \<in> sets M)" |
|
47756 | 1299 |
by (metis assms in_measure_of measurable_measure_of assms measurable_sets) |
47694 | 1300 |
|
50003 | 1301 |
lemma measurable_cong_sets: |
1302 |
assumes sets: "sets M = sets M'" "sets N = sets N'" |
|
1303 |
shows "measurable M N = measurable M' N'" |
|
1304 |
using sets[THEN sets_eq_imp_space_eq] sets by (simp add: measurable_def) |
|
1305 |
||
47694 | 1306 |
lemma measurable_cong: |
1307 |
assumes "\<And> w. w \<in> space M \<Longrightarrow> f w = g w" |
|
1308 |
shows "f \<in> measurable M M' \<longleftrightarrow> g \<in> measurable M M'" |
|
1309 |
unfolding measurable_def using assms |
|
1310 |
by (simp cong: vimage_inter_cong Pi_cong) |
|
1311 |
||
1312 |
lemma measurable_eqI: |
|
1313 |
"\<lbrakk> space m1 = space m1' ; space m2 = space m2' ; |
|
1314 |
sets m1 = sets m1' ; sets m2 = sets m2' \<rbrakk> |
|
1315 |
\<Longrightarrow> measurable m1 m2 = measurable m1' m2'" |
|
1316 |
by (simp add: measurable_def sigma_algebra_iff2) |
|
1317 |
||
50003 | 1318 |
lemma measurable_compose: |
1319 |
assumes f: "f \<in> measurable M N" and g: "g \<in> measurable N L" |
|
1320 |
shows "(\<lambda>x. g (f x)) \<in> measurable M L" |
|
1321 |
proof - |
|
1322 |
have "\<And>A. (\<lambda>x. g (f x)) -` A \<inter> space M = f -` (g -` A \<inter> space N) \<inter> space M" |
|
1323 |
using measurable_space[OF f] by auto |
|
1324 |
with measurable_space[OF f] measurable_space[OF g] show ?thesis |
|
1325 |
by (auto intro: measurable_sets[OF f] measurable_sets[OF g] |
|
1326 |
simp del: vimage_Int simp add: measurable_def) |
|
1327 |
qed |
|
1328 |
||
1329 |
lemma measurable_comp: |
|
1330 |
"f \<in> measurable M N \<Longrightarrow> g \<in> measurable N L \<Longrightarrow> g \<circ> f \<in> measurable M L" |
|
1331 |
using measurable_compose[of f M N g L] by (simp add: comp_def) |
|
1332 |
||
1333 |
lemma measurable_const: |
|
47694 | 1334 |
"c \<in> space M' \<Longrightarrow> (\<lambda>x. c) \<in> measurable M M'" |
1335 |
by (auto simp add: measurable_def) |
|
1336 |
||
1337 |
lemma measurable_If: |
|
1338 |
assumes measure: "f \<in> measurable M M'" "g \<in> measurable M M'" |
|
1339 |
assumes P: "{x\<in>space M. P x} \<in> sets M" |
|
1340 |
shows "(\<lambda>x. if P x then f x else g x) \<in> measurable M M'" |
|
1341 |
unfolding measurable_def |
|
1342 |
proof safe |
|
1343 |
fix x assume "x \<in> space M" |
|
1344 |
thus "(if P x then f x else g x) \<in> space M'" |
|
1345 |
using measure unfolding measurable_def by auto |
|
1346 |
next |
|
1347 |
fix A assume "A \<in> sets M'" |
|
1348 |
hence *: "(\<lambda>x. if P x then f x else g x) -` A \<inter> space M = |
|
1349 |
((f -` A \<inter> space M) \<inter> {x\<in>space M. P x}) \<union> |
|
1350 |
((g -` A \<inter> space M) \<inter> (space M - {x\<in>space M. P x}))" |
|
1351 |
using measure unfolding measurable_def by (auto split: split_if_asm) |
|
1352 |
show "(\<lambda>x. if P x then f x else g x) -` A \<inter> space M \<in> sets M" |
|
1353 |
using `A \<in> sets M'` measure P unfolding * measurable_def |
|
50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50096
diff
changeset
|
1354 |
by (auto intro!: sets.Un) |
47694 | 1355 |
qed |
1356 |
||
1357 |
lemma measurable_If_set: |
|
1358 |
assumes measure: "f \<in> measurable M M'" "g \<in> measurable M M'" |
|
49773
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
1359 |
assumes P: "A \<inter> space M \<in> sets M" |
47694 | 1360 |
shows "(\<lambda>x. if x \<in> A then f x else g x) \<in> measurable M M'" |
1361 |
proof (rule measurable_If[OF measure]) |
|
49773
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
1362 |
have "{x \<in> space M. x \<in> A} = A \<inter> space M" by auto |
16907431e477
tuned measurable_If; moved countably_additive equalities to Measure_Space; tuned proofs
hoelzl
parents:
47762
diff
changeset
|
1363 |
thus "{x \<in> space M. x \<in> A} \<in> sets M" using `A \<inter> space M \<in> sets M` by auto |
47694 | 1364 |
qed |
1365 |
||
50003 | 1366 |
lemma measurable_ident: "id \<in> measurable M M" |
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1367 |
by (auto simp add: measurable_def) |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1368 |
|
50021
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
1369 |
lemma measurable_ident_sets: |
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
1370 |
assumes eq: "sets M = sets M'" shows "(\<lambda>x. x) \<in> measurable M M'" |
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
1371 |
using measurable_ident[of M] |
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
1372 |
unfolding id_def measurable_def eq sets_eq_imp_space_eq[OF eq] . |
49782 | 1373 |
|
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1374 |
lemma sets_Least: |
47694 | 1375 |
assumes meas: "\<And>i::nat. {x\<in>space M. P i x} \<in> M" |
1376 |
shows "(\<lambda>x. LEAST j. P j x) -` A \<inter> space M \<in> sets M" |
|
1377 |
proof - |
|
1378 |
{ fix i have "(\<lambda>x. LEAST j. P j x) -` {i} \<inter> space M \<in> sets M" |
|
1379 |
proof cases |
|
1380 |
assume i: "(LEAST j. False) = i" |
|
1381 |
have "(\<lambda>x. LEAST j. P j x) -` {i} \<inter> space M = |
|
1382 |
{x\<in>space M. P i x} \<inter> (space M - (\<Union>j<i. {x\<in>space M. P j x})) \<union> (space M - (\<Union>i. {x\<in>space M. P i x}))" |
|
1383 |
by (simp add: set_eq_iff, safe) |
|
1384 |
(insert i, auto dest: Least_le intro: LeastI intro!: Least_equality) |
|
1385 |
with meas show ?thesis |
|
50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50096
diff
changeset
|
1386 |
by (auto intro!: sets.Int) |
47694 | 1387 |
next |
1388 |
assume i: "(LEAST j. False) \<noteq> i" |
|
1389 |
then have "(\<lambda>x. LEAST j. P j x) -` {i} \<inter> space M = |
|
1390 |
{x\<in>space M. P i x} \<inter> (space M - (\<Union>j<i. {x\<in>space M. P j x}))" |
|
1391 |
proof (simp add: set_eq_iff, safe) |
|
1392 |
fix x assume neq: "(LEAST j. False) \<noteq> (LEAST j. P j x)" |
|
1393 |
have "\<exists>j. P j x" |
|
1394 |
by (rule ccontr) (insert neq, auto) |
|
1395 |
then show "P (LEAST j. P j x) x" by (rule LeastI_ex) |
|
1396 |
qed (auto dest: Least_le intro!: Least_equality) |
|
1397 |
with meas show ?thesis |
|
1398 |
by auto |
|
1399 |
qed } |
|
1400 |
then have "(\<Union>i\<in>A. (\<lambda>x. LEAST j. P j x) -` {i} \<inter> space M) \<in> sets M" |
|
50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50096
diff
changeset
|
1401 |
by (intro sets.countable_UN) auto |
47694 | 1402 |
moreover have "(\<Union>i\<in>A. (\<lambda>x. LEAST j. P j x) -` {i} \<inter> space M) = |
1403 |
(\<lambda>x. LEAST j. P j x) -` A \<inter> space M" by auto |
|
1404 |
ultimately show ?thesis by auto |
|
1405 |
qed |
|
1406 |
||
1407 |
lemma measurable_strong: |
|
1408 |
fixes f :: "'a \<Rightarrow> 'b" and g :: "'b \<Rightarrow> 'c" |
|
1409 |
assumes f: "f \<in> measurable a b" and g: "g \<in> space b \<rightarrow> space c" |
|
1410 |
and t: "f ` (space a) \<subseteq> t" |
|
1411 |
and cb: "\<And>s. s \<in> sets c \<Longrightarrow> (g -` s) \<inter> t \<in> sets b" |
|
1412 |
shows "(g o f) \<in> measurable a c" |
|
1413 |
proof - |
|
1414 |
have fab: "f \<in> (space a -> space b)" |
|
1415 |
and ba: "\<And>y. y \<in> sets b \<Longrightarrow> (f -` y) \<inter> (space a) \<in> sets a" using f |
|
1416 |
by (auto simp add: measurable_def) |
|
1417 |
have eq: "\<And>y. f -` g -` y \<inter> space a = f -` (g -` y \<inter> t) \<inter> space a" using t |
|
1418 |
by force |
|
1419 |
show ?thesis |
|
1420 |
apply (auto simp add: measurable_def vimage_compose) |
|
1421 |
apply (metis funcset_mem fab g) |
|
1422 |
apply (subst eq, metis ba cb) |
|
1423 |
done |
|
1424 |
qed |
|
1425 |
||
1426 |
lemma measurable_mono1: |
|
1427 |
"M' \<subseteq> Pow \<Omega> \<Longrightarrow> M \<subseteq> M' \<Longrightarrow> |
|
1428 |
measurable (measure_of \<Omega> M \<mu>) N \<subseteq> measurable (measure_of \<Omega> M' \<mu>') N" |
|
1429 |
using measure_of_subset[of M' \<Omega> M] by (auto simp add: measurable_def) |
|
1430 |
||
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1431 |
section {* Counting space *} |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1432 |
|
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1433 |
definition count_space :: "'a set \<Rightarrow> 'a measure" where |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1434 |
"count_space \<Omega> = measure_of \<Omega> (Pow \<Omega>) (\<lambda>A. if finite A then ereal (card A) else \<infinity>)" |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1435 |
|
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1436 |
lemma |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1437 |
shows space_count_space[simp]: "space (count_space \<Omega>) = \<Omega>" |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1438 |
and sets_count_space[simp]: "sets (count_space \<Omega>) = Pow \<Omega>" |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1439 |
using sigma_sets_into_sp[of "Pow \<Omega>" \<Omega>] |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1440 |
by (auto simp: count_space_def) |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1441 |
|
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1442 |
lemma measurable_count_space_eq1[simp]: |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1443 |
"f \<in> measurable (count_space A) M \<longleftrightarrow> f \<in> A \<rightarrow> space M" |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1444 |
unfolding measurable_def by simp |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1445 |
|
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1446 |
lemma measurable_count_space_eq2: |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1447 |
assumes "finite A" |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1448 |
shows "f \<in> measurable M (count_space A) \<longleftrightarrow> (f \<in> space M \<rightarrow> A \<and> (\<forall>a\<in>A. f -` {a} \<inter> space M \<in> sets M))" |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1449 |
proof - |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1450 |
{ fix X assume "X \<subseteq> A" "f \<in> space M \<rightarrow> A" |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1451 |
with `finite A` have "f -` X \<inter> space M = (\<Union>a\<in>X. f -` {a} \<inter> space M)" "finite X" |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1452 |
by (auto dest: finite_subset) |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1453 |
moreover assume "\<forall>a\<in>A. f -` {a} \<inter> space M \<in> sets M" |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1454 |
ultimately have "f -` X \<inter> space M \<in> sets M" |
50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50096
diff
changeset
|
1455 |
using `X \<subseteq> A` by (auto intro!: sets.finite_UN simp del: UN_simps) } |
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1456 |
then show ?thesis |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1457 |
unfolding measurable_def by auto |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1458 |
qed |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1459 |
|
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1460 |
lemma measurable_compose_countable: |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1461 |
assumes f: "\<And>i::'i::countable. (\<lambda>x. f i x) \<in> measurable M N" and g: "g \<in> measurable M (count_space UNIV)" |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1462 |
shows "(\<lambda>x. f (g x) x) \<in> measurable M N" |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1463 |
unfolding measurable_def |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1464 |
proof safe |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1465 |
fix x assume "x \<in> space M" then show "f (g x) x \<in> space N" |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1466 |
using f[THEN measurable_space] g[THEN measurable_space] by auto |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1467 |
next |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1468 |
fix A assume A: "A \<in> sets N" |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1469 |
have "(\<lambda>x. f (g x) x) -` A \<inter> space M = (\<Union>i. (g -` {i} \<inter> space M) \<inter> (f i -` A \<inter> space M))" |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1470 |
by auto |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1471 |
also have "\<dots> \<in> sets M" using f[THEN measurable_sets, OF A] g[THEN measurable_sets] |
50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50096
diff
changeset
|
1472 |
by (auto intro!: sets.countable_UN measurable_sets) |
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1473 |
finally show "(\<lambda>x. f (g x) x) -` A \<inter> space M \<in> sets M" . |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1474 |
qed |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1475 |
|
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1476 |
subsection {* Measurable method *} |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1477 |
|
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1478 |
lemma (in algebra) sets_Collect_finite_All: |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1479 |
assumes "\<And>i. i \<in> S \<Longrightarrow> {x\<in>\<Omega>. P i x} \<in> M" "finite S" |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1480 |
shows "{x\<in>\<Omega>. \<forall>i\<in>S. P i x} \<in> M" |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1481 |
proof - |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1482 |
have "{x\<in>\<Omega>. \<forall>i\<in>S. P i x} = (if S = {} then \<Omega> else \<Inter>i\<in>S. {x\<in>\<Omega>. P i x})" |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1483 |
by auto |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1484 |
with assms show ?thesis by (auto intro!: sets_Collect_finite_All') |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1485 |
qed |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1486 |
|
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1487 |
abbreviation "pred M P \<equiv> P \<in> measurable M (count_space (UNIV::bool set))" |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1488 |
|
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1489 |
lemma pred_def: "pred M P \<longleftrightarrow> {x\<in>space M. P x} \<in> sets M" |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1490 |
proof |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1491 |
assume "pred M P" |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1492 |
then have "P -` {True} \<inter> space M \<in> sets M" |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1493 |
by (auto simp: measurable_count_space_eq2) |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1494 |
also have "P -` {True} \<inter> space M = {x\<in>space M. P x}" by auto |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1495 |
finally show "{x\<in>space M. P x} \<in> sets M" . |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1496 |
next |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1497 |
assume P: "{x\<in>space M. P x} \<in> sets M" |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1498 |
moreover |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1499 |
{ fix X |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1500 |
have "X \<in> Pow (UNIV :: bool set)" by simp |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1501 |
then have "P -` X \<inter> space M = {x\<in>space M. ((X = {True} \<longrightarrow> P x) \<and> (X = {False} \<longrightarrow> \<not> P x) \<and> X \<noteq> {})}" |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1502 |
unfolding UNIV_bool Pow_insert Pow_empty by auto |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1503 |
then have "P -` X \<inter> space M \<in> sets M" |
50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50096
diff
changeset
|
1504 |
by (auto intro!: sets.sets_Collect_neg sets.sets_Collect_imp sets.sets_Collect_conj sets.sets_Collect_const P) } |
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1505 |
then show "pred M P" |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1506 |
by (auto simp: measurable_def) |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1507 |
qed |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1508 |
|
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1509 |
lemma pred_sets1: "{x\<in>space M. P x} \<in> sets M \<Longrightarrow> f \<in> measurable N M \<Longrightarrow> pred N (\<lambda>x. P (f x))" |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1510 |
by (rule measurable_compose[where f=f and N=M]) (auto simp: pred_def) |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1511 |
|
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1512 |
lemma pred_sets2: "A \<in> sets N \<Longrightarrow> f \<in> measurable M N \<Longrightarrow> pred M (\<lambda>x. f x \<in> A)" |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1513 |
by (rule measurable_compose[where f=f and N=N]) (auto simp: pred_def Int_def[symmetric]) |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1514 |
|
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1515 |
lemma measurable_count_space_const: |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1516 |
"(\<lambda>x. c) \<in> measurable M (count_space UNIV)" |
50003 | 1517 |
by (simp add: measurable_const) |
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1518 |
|
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1519 |
lemma measurable_count_space: |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1520 |
"f \<in> measurable (count_space A) (count_space UNIV)" |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1521 |
by simp |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1522 |
|
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1523 |
lemma measurable_compose_rev: |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1524 |
assumes f: "f \<in> measurable L N" and g: "g \<in> measurable M L" |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1525 |
shows "(\<lambda>x. f (g x)) \<in> measurable M N" |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1526 |
using measurable_compose[OF g f] . |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1527 |
|
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1528 |
ML {* |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1529 |
|
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1530 |
structure Measurable = |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1531 |
struct |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1532 |
|
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1533 |
datatype level = Concrete | Generic; |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1534 |
|
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1535 |
structure Data = Generic_Data |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1536 |
( |
50021
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
1537 |
type T = { |
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
1538 |
concrete_thms : thm Item_Net.T, |
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
1539 |
generic_thms : thm Item_Net.T, |
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
1540 |
dest_thms : thm Item_Net.T, |
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
1541 |
app_thms : thm Item_Net.T } |
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
1542 |
val empty = { |
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
1543 |
concrete_thms = Thm.full_rules, |
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
1544 |
generic_thms = Thm.full_rules, |
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
1545 |
dest_thms = Thm.full_rules, |
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
1546 |
app_thms = Thm.full_rules}; |
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1547 |
val extend = I; |
50283
e79a8341dd6b
make SML/NJ happy (give names for all fields in a record)
hoelzl
parents:
50252
diff
changeset
|
1548 |
fun merge ({concrete_thms = ct1, generic_thms = gt1, dest_thms = dt1, app_thms = at1 }, |
e79a8341dd6b
make SML/NJ happy (give names for all fields in a record)
hoelzl
parents:
50252
diff
changeset
|
1549 |
{concrete_thms = ct2, generic_thms = gt2, dest_thms = dt2, app_thms = at2 }) = { |
e79a8341dd6b
make SML/NJ happy (give names for all fields in a record)
hoelzl
parents:
50252
diff
changeset
|
1550 |
concrete_thms = Item_Net.merge (ct1, ct2), |
e79a8341dd6b
make SML/NJ happy (give names for all fields in a record)
hoelzl
parents:
50252
diff
changeset
|
1551 |
generic_thms = Item_Net.merge (gt1, gt2), |
e79a8341dd6b
make SML/NJ happy (give names for all fields in a record)
hoelzl
parents:
50252
diff
changeset
|
1552 |
dest_thms = Item_Net.merge (dt1, dt2), |
e79a8341dd6b
make SML/NJ happy (give names for all fields in a record)
hoelzl
parents:
50252
diff
changeset
|
1553 |
app_thms = Item_Net.merge (at1, at2) }; |
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1554 |
); |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1555 |
|
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1556 |
val debug = |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1557 |
Attrib.setup_config_bool @{binding measurable_debug} (K false) |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1558 |
|
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1559 |
val backtrack = |
50021
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
1560 |
Attrib.setup_config_int @{binding measurable_backtrack} (K 20) |
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
1561 |
|
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
1562 |
val split = |
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
1563 |
Attrib.setup_config_bool @{binding measurable_split} (K true) |
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1564 |
|
50021
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
1565 |
fun TAKE n tac = Seq.take n o tac |
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
1566 |
|
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
1567 |
fun get lv = |
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
1568 |
rev o Item_Net.content o (case lv of Concrete => #concrete_thms | Generic => #generic_thms) o |
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
1569 |
Data.get o Context.Proof; |
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
1570 |
|
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1571 |
fun get_all ctxt = get Concrete ctxt @ get Generic ctxt; |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1572 |
|
50021
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
1573 |
fun map_data f1 f2 f3 f4 |
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
1574 |
{generic_thms = t1, concrete_thms = t2, dest_thms = t3, app_thms = t4} = |
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
1575 |
{generic_thms = f1 t1, concrete_thms = f2 t2, dest_thms = f3 t3, app_thms = f4 t4 } |
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
1576 |
|
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
1577 |
fun map_concrete_thms f = map_data f I I I |
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
1578 |
fun map_generic_thms f = map_data I f I I |
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
1579 |
fun map_dest_thms f = map_data I I f I |
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
1580 |
fun map_app_thms f = map_data I I I f |
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1581 |
|
50021
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
1582 |
fun update f lv = Data.map (case lv of Concrete => map_concrete_thms f | Generic => map_generic_thms f); |
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
1583 |
fun add thms' = update (fold Item_Net.update thms'); |
50003 | 1584 |
|
50021
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
1585 |
val get_dest = Item_Net.content o #dest_thms o Data.get; |
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
1586 |
val add_dest = Data.map o map_dest_thms o Item_Net.update; |
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
1587 |
|
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
1588 |
val get_app = Item_Net.content o #app_thms o Data.get; |
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
1589 |
val add_app = Data.map o map_app_thms o Item_Net.update; |
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1590 |
|
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1591 |
fun is_too_generic thm = |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1592 |
let |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1593 |
val concl = concl_of thm |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1594 |
val concl' = HOLogic.dest_Trueprop concl handle TERM _ => concl |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1595 |
in is_Var (head_of concl') end |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1596 |
|
50003 | 1597 |
fun import_theorem ctxt thm = if is_too_generic thm then [] else |
1598 |
[thm] @ map_filter (try (fn th' => thm RS th')) (get_dest ctxt); |
|
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1599 |
|
50003 | 1600 |
fun add_thm (raw, lv) thm ctxt = add (if raw then [thm] else import_theorem ctxt thm) lv ctxt; |
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1601 |
|
50021
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
1602 |
fun debug_tac ctxt msg f = if Config.get ctxt debug then print_tac (msg ()) THEN f else f |
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1603 |
|
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1604 |
fun nth_hol_goal thm i = |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1605 |
HOLogic.dest_Trueprop (Logic.strip_imp_concl (strip_all_body (nth (prems_of thm) (i - 1)))) |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1606 |
|
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1607 |
fun dest_measurable_fun t = |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1608 |
(case t of |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1609 |
(Const (@{const_name "Set.member"}, _) $ f $ (Const (@{const_name "measurable"}, _) $ _ $ _)) => f |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1610 |
| _ => raise (TERM ("not a measurability predicate", [t]))) |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1611 |
|
50021
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
1612 |
fun is_cond_formula n thm = if length (prems_of thm) < n then false else |
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
1613 |
(case nth_hol_goal thm n of |
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
1614 |
(Const (@{const_name "Set.member"}, _) $ _ $ (Const (@{const_name "sets"}, _) $ _)) => false |
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
1615 |
| (Const (@{const_name "Set.member"}, _) $ _ $ (Const (@{const_name "measurable"}, _) $ _ $ _)) => false |
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
1616 |
| _ => true) |
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
1617 |
handle TERM _ => true; |
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
1618 |
|
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1619 |
fun indep (Bound i) t b = i < b orelse t <= i |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1620 |
| indep (f $ t) top bot = indep f top bot andalso indep t top bot |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1621 |
| indep (Abs (_,_,t)) top bot = indep t (top + 1) (bot + 1) |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1622 |
| indep _ _ _ = true; |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1623 |
|
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1624 |
fun cnt_prefixes ctxt (Abs (n, T, t)) = let |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1625 |
fun is_countable t = Type.of_sort (Proof_Context.tsig_of ctxt) (t, @{sort countable}) |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1626 |
fun cnt_walk (Abs (ns, T, t)) Ts = |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1627 |
map (fn (t', t'') => (Abs (ns, T, t'), t'')) (cnt_walk t (T::Ts)) |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1628 |
| cnt_walk (f $ g) Ts = let |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1629 |
val n = length Ts - 1 |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1630 |
in |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1631 |
map (fn (f', t) => (f' $ g, t)) (cnt_walk f Ts) @ |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1632 |
map (fn (g', t) => (f $ g', t)) (cnt_walk g Ts) @ |
50021
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
1633 |
(if is_countable (type_of1 (Ts, g)) andalso loose_bvar1 (g, n) |
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1634 |
andalso indep g n 0 andalso g <> Bound n |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1635 |
then [(f $ Bound (n + 1), incr_boundvars (~ n) g)] |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1636 |
else []) |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1637 |
end |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1638 |
| cnt_walk _ _ = [] |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1639 |
in map (fn (t1, t2) => let |
50021
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
1640 |
val T1 = type_of1 ([T], t2) |
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
1641 |
val T2 = type_of1 ([T], t) |
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1642 |
in ([SOME (Abs (n, T1, Abs (n, T, t1))), NONE, NONE, SOME (Abs (n, T, t2))], |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1643 |
[SOME T1, SOME T, SOME T2]) |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1644 |
end) (cnt_walk t [T]) |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1645 |
end |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1646 |
| cnt_prefixes _ _ = [] |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1647 |
|
50021
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
1648 |
val split_countable_tac = |
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1649 |
Subgoal.FOCUS (fn {context = ctxt, ...} => SUBGOAL (fn (t, i) => |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1650 |
let |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1651 |
val f = dest_measurable_fun (HOLogic.dest_Trueprop t) |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1652 |
fun cert f = map (Option.map (f (Proof_Context.theory_of ctxt))) |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1653 |
fun inst t (ts, Ts) = Drule.instantiate' (cert ctyp_of Ts) (cert cterm_of ts) t |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1654 |
val cps = cnt_prefixes ctxt f |> map (inst @{thm measurable_compose_countable}) |
50096 | 1655 |
in if null cps then no_tac else debug_tac ctxt (K "split countable fun") (resolve_tac cps i) end |
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1656 |
handle TERM _ => no_tac) 1) |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1657 |
|
50021
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
1658 |
fun measurable_tac' ctxt ss facts = let |
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
1659 |
|
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
1660 |
val imported_thms = |
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
1661 |
(maps (import_theorem (Context.Proof ctxt) o Simplifier.norm_hhf) facts) @ get_all ctxt |
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
1662 |
|
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
1663 |
fun debug_facts msg () = |
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
1664 |
msg ^ " + " ^ Pretty.str_of (Pretty.list "[" "]" |
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
1665 |
(map (Syntax.pretty_term ctxt o prop_of) (maps (import_theorem (Context.Proof ctxt)) facts))); |
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
1666 |
|
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
1667 |
val splitter = if Config.get ctxt split then split_countable_tac ctxt else K no_tac |
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
1668 |
|
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
1669 |
val split_app_tac = |
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
1670 |
Subgoal.FOCUS (fn {context = ctxt, ...} => SUBGOAL (fn (t, i) => |
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
1671 |
let |
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
1672 |
fun app_prefixes (Abs (n, T, (f $ g))) = let |
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
1673 |
val ps = (if not (loose_bvar1 (g, 0)) then [(f, g)] else []) |
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
1674 |
in map (fn (f, c) => (Abs (n, T, f), c, T, type_of c, type_of1 ([T], f $ c))) ps end |
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
1675 |
| app_prefixes _ = [] |
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1676 |
|
50021
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
1677 |
fun dest_app (Abs (_, T, t as ((f $ Bound 0) $ c))) = (f, c, T, type_of c, type_of1 ([T], t)) |
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
1678 |
| dest_app t = raise (TERM ("not a measurability predicate of an application", [t])) |
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
1679 |
val thy = Proof_Context.theory_of ctxt |
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
1680 |
val tunify = Sign.typ_unify thy |
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
1681 |
val thms = map |
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
1682 |
(fn thm => (thm, dest_app (dest_measurable_fun (HOLogic.dest_Trueprop (concl_of thm))))) |
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
1683 |
(get_app (Context.Proof ctxt)) |
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
1684 |
fun cert f = map (fn (t, t') => (f thy t, f thy t')) |
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
1685 |
fun inst (f, c, T, Tc, Tf) (thm, (thmf, thmc, thmT, thmTc, thmTf)) = |
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
1686 |
let |
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
1687 |
val inst = |
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
1688 |
(Vartab.empty, ~1) |
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
1689 |
|> tunify (T, thmT) |
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
1690 |
|> tunify (Tf, thmTf) |
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
1691 |
|> tunify (Tc, thmTc) |
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
1692 |
|> Vartab.dest o fst |
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
1693 |
val subst = subst_TVars (map (apsnd snd) inst) |
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
1694 |
in |
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
1695 |
Thm.instantiate (cert ctyp_of (map (fn (n, (s, T)) => (TVar (n, s), T)) inst), |
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
1696 |
cert cterm_of [(subst thmf, f), (subst thmc, c)]) thm |
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
1697 |
end |
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
1698 |
val cps = map_product inst (app_prefixes (dest_measurable_fun (HOLogic.dest_Trueprop t))) thms |
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
1699 |
in if null cps then no_tac |
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
1700 |
else debug_tac ctxt (K ("split app fun")) (resolve_tac cps i) |
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
1701 |
ORELSE debug_tac ctxt (fn () => "FAILED") no_tac end |
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
1702 |
handle TERM t => debug_tac ctxt (fn () => "TERM " ^ fst t ^ Pretty.str_of (Pretty.list "[" "]" (map (Syntax.pretty_term ctxt) (snd t)))) no_tac |
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
1703 |
handle Type.TUNIFY => debug_tac ctxt (fn () => "TUNIFY") no_tac) 1) |
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1704 |
|
50021
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
1705 |
val depth_measurable_tac = REPEAT |
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
1706 |
(COND (is_cond_formula 1) |
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
1707 |
(debug_tac ctxt (K "simp") (SOLVED' (asm_full_simp_tac ss) 1)) |
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
1708 |
((debug_tac ctxt (K "single") (resolve_tac imported_thms 1)) APPEND |
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
1709 |
(split_app_tac ctxt 1) APPEND |
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
1710 |
(splitter 1))) |
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1711 |
|
50021
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
1712 |
in debug_tac ctxt (debug_facts "start") depth_measurable_tac end; |
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
1713 |
|
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
1714 |
fun measurable_tac ctxt facts = |
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
1715 |
TAKE (Config.get ctxt backtrack) (measurable_tac' ctxt (simpset_of ctxt) facts); |
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1716 |
|
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1717 |
val attr_add = Thm.declaration_attribute o add_thm; |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1718 |
|
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1719 |
val attr : attribute context_parser = |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1720 |
Scan.lift (Scan.optional (Args.parens (Scan.optional (Args.$$$ "raw" >> K true) false -- |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1721 |
Scan.optional (Args.$$$ "generic" >> K Generic) Concrete)) (false, Concrete) >> attr_add); |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1722 |
|
50003 | 1723 |
val dest_attr : attribute context_parser = |
1724 |
Scan.lift (Scan.succeed (Thm.declaration_attribute add_dest)); |
|
1725 |
||
50021
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
1726 |
val app_attr : attribute context_parser = |
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
1727 |
Scan.lift (Scan.succeed (Thm.declaration_attribute add_app)); |
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
1728 |
|
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1729 |
val method : (Proof.context -> Method.method) context_parser = |
50021
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
1730 |
Scan.lift (Scan.succeed (fn ctxt => METHOD (fn facts => measurable_tac ctxt facts))); |
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1731 |
|
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1732 |
fun simproc ss redex = let |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1733 |
val ctxt = Simplifier.the_context ss; |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1734 |
val t = HOLogic.mk_Trueprop (term_of redex); |
50283
e79a8341dd6b
make SML/NJ happy (give names for all fields in a record)
hoelzl
parents:
50252
diff
changeset
|
1735 |
fun tac {context = ctxt, prems = _ } = |
50021
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
1736 |
SOLVE (measurable_tac' ctxt ss (Simplifier.prems_of ss)); |
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1737 |
in try (fn () => Goal.prove ctxt [] [] t tac RS @{thm Eq_TrueI}) () end; |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1738 |
|
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1739 |
end |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1740 |
|
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1741 |
*} |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1742 |
|
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1743 |
attribute_setup measurable = {* Measurable.attr *} "declaration of measurability theorems" |
50003 | 1744 |
attribute_setup measurable_dest = {* Measurable.dest_attr *} "add dest rule for measurability prover" |
50021
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
1745 |
attribute_setup measurable_app = {* Measurable.app_attr *} "add application rule for measurability prover" |
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1746 |
method_setup measurable = {* Measurable.method *} "measurability prover" |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1747 |
simproc_setup measurable ("A \<in> sets M" | "f \<in> measurable M N") = {* K Measurable.simproc *} |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1748 |
|
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1749 |
declare |
50003 | 1750 |
measurable_compose_rev[measurable_dest] |
1751 |
pred_sets1[measurable_dest] |
|
1752 |
pred_sets2[measurable_dest] |
|
50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50096
diff
changeset
|
1753 |
sets.sets_into_space[measurable_dest] |
50003 | 1754 |
|
1755 |
declare |
|
50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50096
diff
changeset
|
1756 |
sets.top[measurable] |
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50096
diff
changeset
|
1757 |
sets.empty_sets[measurable (raw)] |
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50096
diff
changeset
|
1758 |
sets.Un[measurable (raw)] |
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50096
diff
changeset
|
1759 |
sets.Diff[measurable (raw)] |
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1760 |
|
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1761 |
declare |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1762 |
measurable_count_space[measurable (raw)] |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1763 |
measurable_ident[measurable (raw)] |
50021
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
1764 |
measurable_ident_sets[measurable (raw)] |
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1765 |
measurable_const[measurable (raw)] |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1766 |
measurable_If[measurable (raw)] |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1767 |
measurable_comp[measurable (raw)] |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1768 |
measurable_sets[measurable (raw)] |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1769 |
|
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1770 |
lemma predE[measurable (raw)]: |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1771 |
"pred M P \<Longrightarrow> {x\<in>space M. P x} \<in> sets M" |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1772 |
unfolding pred_def . |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1773 |
|
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1774 |
lemma pred_intros_imp'[measurable (raw)]: |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1775 |
"(K \<Longrightarrow> pred M (\<lambda>x. P x)) \<Longrightarrow> pred M (\<lambda>x. K \<longrightarrow> P x)" |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1776 |
by (cases K) auto |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1777 |
|
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1778 |
lemma pred_intros_conj1'[measurable (raw)]: |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1779 |
"(K \<Longrightarrow> pred M (\<lambda>x. P x)) \<Longrightarrow> pred M (\<lambda>x. K \<and> P x)" |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1780 |
by (cases K) auto |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1781 |
|
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1782 |
lemma pred_intros_conj2'[measurable (raw)]: |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1783 |
"(K \<Longrightarrow> pred M (\<lambda>x. P x)) \<Longrightarrow> pred M (\<lambda>x. P x \<and> K)" |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1784 |
by (cases K) auto |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1785 |
|
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1786 |
lemma pred_intros_disj1'[measurable (raw)]: |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1787 |
"(\<not> K \<Longrightarrow> pred M (\<lambda>x. P x)) \<Longrightarrow> pred M (\<lambda>x. K \<or> P x)" |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1788 |
by (cases K) auto |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1789 |
|
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1790 |
lemma pred_intros_disj2'[measurable (raw)]: |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1791 |
"(\<not> K \<Longrightarrow> pred M (\<lambda>x. P x)) \<Longrightarrow> pred M (\<lambda>x. P x \<or> K)" |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1792 |
by (cases K) auto |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1793 |
|
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1794 |
lemma pred_intros_logic[measurable (raw)]: |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1795 |
"pred M (\<lambda>x. x \<in> space M)" |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1796 |
"pred M (\<lambda>x. P x) \<Longrightarrow> pred M (\<lambda>x. \<not> P x)" |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1797 |
"pred M (\<lambda>x. Q x) \<Longrightarrow> pred M (\<lambda>x. P x) \<Longrightarrow> pred M (\<lambda>x. Q x \<and> P x)" |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1798 |
"pred M (\<lambda>x. Q x) \<Longrightarrow> pred M (\<lambda>x. P x) \<Longrightarrow> pred M (\<lambda>x. Q x \<longrightarrow> P x)" |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1799 |
"pred M (\<lambda>x. Q x) \<Longrightarrow> pred M (\<lambda>x. P x) \<Longrightarrow> pred M (\<lambda>x. Q x \<or> P x)" |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1800 |
"pred M (\<lambda>x. Q x) \<Longrightarrow> pred M (\<lambda>x. P x) \<Longrightarrow> pred M (\<lambda>x. Q x = P x)" |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1801 |
"pred M (\<lambda>x. f x \<in> UNIV)" |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1802 |
"pred M (\<lambda>x. f x \<in> {})" |
50096 | 1803 |
"pred M (\<lambda>x. P' (f x) x) \<Longrightarrow> pred M (\<lambda>x. f x \<in> {y. P' y x})" |
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1804 |
"pred M (\<lambda>x. f x \<in> (B x)) \<Longrightarrow> pred M (\<lambda>x. f x \<in> - (B x))" |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1805 |
"pred M (\<lambda>x. f x \<in> (A x)) \<Longrightarrow> pred M (\<lambda>x. f x \<in> (B x)) \<Longrightarrow> pred M (\<lambda>x. f x \<in> (A x) - (B x))" |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1806 |
"pred M (\<lambda>x. f x \<in> (A x)) \<Longrightarrow> pred M (\<lambda>x. f x \<in> (B x)) \<Longrightarrow> pred M (\<lambda>x. f x \<in> (A x) \<inter> (B x))" |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1807 |
"pred M (\<lambda>x. f x \<in> (A x)) \<Longrightarrow> pred M (\<lambda>x. f x \<in> (B x)) \<Longrightarrow> pred M (\<lambda>x. f x \<in> (A x) \<union> (B x))" |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1808 |
"pred M (\<lambda>x. g x (f x) \<in> (X x)) \<Longrightarrow> pred M (\<lambda>x. f x \<in> (g x) -` (X x))" |
50003 | 1809 |
by (auto simp: iff_conv_conj_imp pred_def) |
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1810 |
|
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1811 |
lemma pred_intros_countable[measurable (raw)]: |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1812 |
fixes P :: "'a \<Rightarrow> 'i :: countable \<Rightarrow> bool" |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1813 |
shows |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1814 |
"(\<And>i. pred M (\<lambda>x. P x i)) \<Longrightarrow> pred M (\<lambda>x. \<forall>i. P x i)" |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1815 |
"(\<And>i. pred M (\<lambda>x. P x i)) \<Longrightarrow> pred M (\<lambda>x. \<exists>i. P x i)" |
50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50096
diff
changeset
|
1816 |
by (auto intro!: sets.sets_Collect_countable_All sets.sets_Collect_countable_Ex simp: pred_def) |
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1817 |
|
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1818 |
lemma pred_intros_countable_bounded[measurable (raw)]: |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1819 |
fixes X :: "'i :: countable set" |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1820 |
shows |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1821 |
"(\<And>i. i \<in> X \<Longrightarrow> pred M (\<lambda>x. x \<in> N x i)) \<Longrightarrow> pred M (\<lambda>x. x \<in> (\<Inter>i\<in>X. N x i))" |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1822 |
"(\<And>i. i \<in> X \<Longrightarrow> pred M (\<lambda>x. x \<in> N x i)) \<Longrightarrow> pred M (\<lambda>x. x \<in> (\<Union>i\<in>X. N x i))" |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1823 |
"(\<And>i. i \<in> X \<Longrightarrow> pred M (\<lambda>x. P x i)) \<Longrightarrow> pred M (\<lambda>x. \<forall>i\<in>X. P x i)" |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1824 |
"(\<And>i. i \<in> X \<Longrightarrow> pred M (\<lambda>x. P x i)) \<Longrightarrow> pred M (\<lambda>x. \<exists>i\<in>X. P x i)" |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1825 |
by (auto simp: Bex_def Ball_def) |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1826 |
|
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1827 |
lemma pred_intros_finite[measurable (raw)]: |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1828 |
"finite I \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> pred M (\<lambda>x. x \<in> N x i)) \<Longrightarrow> pred M (\<lambda>x. x \<in> (\<Inter>i\<in>I. N x i))" |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1829 |
"finite I \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> pred M (\<lambda>x. x \<in> N x i)) \<Longrightarrow> pred M (\<lambda>x. x \<in> (\<Union>i\<in>I. N x i))" |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1830 |
"finite I \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> pred M (\<lambda>x. P x i)) \<Longrightarrow> pred M (\<lambda>x. \<forall>i\<in>I. P x i)" |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1831 |
"finite I \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> pred M (\<lambda>x. P x i)) \<Longrightarrow> pred M (\<lambda>x. \<exists>i\<in>I. P x i)" |
50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50096
diff
changeset
|
1832 |
by (auto intro!: sets.sets_Collect_finite_Ex sets.sets_Collect_finite_All simp: iff_conv_conj_imp pred_def) |
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1833 |
|
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1834 |
lemma countable_Un_Int[measurable (raw)]: |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1835 |
"(\<And>i :: 'i :: countable. i \<in> I \<Longrightarrow> N i \<in> sets M) \<Longrightarrow> (\<Union>i\<in>I. N i) \<in> sets M" |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1836 |
"I \<noteq> {} \<Longrightarrow> (\<And>i :: 'i :: countable. i \<in> I \<Longrightarrow> N i \<in> sets M) \<Longrightarrow> (\<Inter>i\<in>I. N i) \<in> sets M" |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1837 |
by auto |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1838 |
|
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1839 |
declare |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1840 |
finite_UN[measurable (raw)] |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1841 |
finite_INT[measurable (raw)] |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1842 |
|
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1843 |
lemma sets_Int_pred[measurable (raw)]: |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1844 |
assumes space: "A \<inter> B \<subseteq> space M" and [measurable]: "pred M (\<lambda>x. x \<in> A)" "pred M (\<lambda>x. x \<in> B)" |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1845 |
shows "A \<inter> B \<in> sets M" |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1846 |
proof - |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1847 |
have "{x\<in>space M. x \<in> A \<inter> B} \<in> sets M" by auto |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1848 |
also have "{x\<in>space M. x \<in> A \<inter> B} = A \<inter> B" |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1849 |
using space by auto |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1850 |
finally show ?thesis . |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1851 |
qed |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1852 |
|
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1853 |
lemma [measurable (raw generic)]: |
50096 | 1854 |
assumes f: "f \<in> measurable M N" and c: "c \<in> space N \<Longrightarrow> {c} \<in> sets N" |
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1855 |
shows pred_eq_const1: "pred M (\<lambda>x. f x = c)" |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1856 |
and pred_eq_const2: "pred M (\<lambda>x. c = f x)" |
50096 | 1857 |
proof - |
1858 |
show "pred M (\<lambda>x. f x = c)" |
|
1859 |
proof cases |
|
1860 |
assume "c \<in> space N" |
|
1861 |
with measurable_sets[OF f c] show ?thesis |
|
1862 |
by (auto simp: Int_def conj_commute pred_def) |
|
1863 |
next |
|
1864 |
assume "c \<notin> space N" |
|
1865 |
with f[THEN measurable_space] have "{x \<in> space M. f x = c} = {}" by auto |
|
1866 |
then show ?thesis by (auto simp: pred_def cong: conj_cong) |
|
1867 |
qed |
|
1868 |
then show "pred M (\<lambda>x. c = f x)" |
|
1869 |
by (simp add: eq_commute) |
|
1870 |
qed |
|
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1871 |
|
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1872 |
lemma pred_le_const[measurable (raw generic)]: |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1873 |
assumes f: "f \<in> measurable M N" and c: "{.. c} \<in> sets N" shows "pred M (\<lambda>x. f x \<le> c)" |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1874 |
using measurable_sets[OF f c] |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1875 |
by (auto simp: Int_def conj_commute eq_commute pred_def) |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1876 |
|
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1877 |
lemma pred_const_le[measurable (raw generic)]: |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1878 |
assumes f: "f \<in> measurable M N" and c: "{c ..} \<in> sets N" shows "pred M (\<lambda>x. c \<le> f x)" |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1879 |
using measurable_sets[OF f c] |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1880 |
by (auto simp: Int_def conj_commute eq_commute pred_def) |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1881 |
|
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1882 |
lemma pred_less_const[measurable (raw generic)]: |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1883 |
assumes f: "f \<in> measurable M N" and c: "{..< c} \<in> sets N" shows "pred M (\<lambda>x. f x < c)" |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1884 |
using measurable_sets[OF f c] |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1885 |
by (auto simp: Int_def conj_commute eq_commute pred_def) |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1886 |
|
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1887 |
lemma pred_const_less[measurable (raw generic)]: |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1888 |
assumes f: "f \<in> measurable M N" and c: "{c <..} \<in> sets N" shows "pred M (\<lambda>x. c < f x)" |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1889 |
using measurable_sets[OF f c] |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1890 |
by (auto simp: Int_def conj_commute eq_commute pred_def) |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1891 |
|
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1892 |
declare |
50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50096
diff
changeset
|
1893 |
sets.Int[measurable (raw)] |
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1894 |
|
50003 | 1895 |
lemma pred_in_If[measurable (raw)]: |
50021
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
1896 |
"(P \<Longrightarrow> pred M (\<lambda>x. x \<in> A x)) \<Longrightarrow> (\<not> P \<Longrightarrow> pred M (\<lambda>x. x \<in> B x)) \<Longrightarrow> |
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
1897 |
pred M (\<lambda>x. x \<in> (if P then A x else B x))" |
50003 | 1898 |
by auto |
1899 |
||
1900 |
lemma sets_range[measurable_dest]: |
|
1901 |
"A ` I \<subseteq> sets M \<Longrightarrow> i \<in> I \<Longrightarrow> A i \<in> sets M" |
|
1902 |
by auto |
|
1903 |
||
1904 |
lemma pred_sets_range[measurable_dest]: |
|
1905 |
"A ` I \<subseteq> sets N \<Longrightarrow> i \<in> I \<Longrightarrow> f \<in> measurable M N \<Longrightarrow> pred M (\<lambda>x. f x \<in> A i)" |
|
1906 |
using pred_sets2[OF sets_range] by auto |
|
1907 |
||
1908 |
lemma sets_All[measurable_dest]: |
|
1909 |
"\<forall>i. A i \<in> sets (M i) \<Longrightarrow> A i \<in> sets (M i)" |
|
1910 |
by auto |
|
1911 |
||
1912 |
lemma pred_sets_All[measurable_dest]: |
|
1913 |
"\<forall>i. A i \<in> sets (N i) \<Longrightarrow> f \<in> measurable M (N i) \<Longrightarrow> pred M (\<lambda>x. f x \<in> A i)" |
|
1914 |
using pred_sets2[OF sets_All, of A N f] by auto |
|
1915 |
||
1916 |
lemma sets_Ball[measurable_dest]: |
|
1917 |
"\<forall>i\<in>I. A i \<in> sets (M i) \<Longrightarrow> i\<in>I \<Longrightarrow> A i \<in> sets (M i)" |
|
1918 |
by auto |
|
1919 |
||
1920 |
lemma pred_sets_Ball[measurable_dest]: |
|
1921 |
"\<forall>i\<in>I. A i \<in> sets (N i) \<Longrightarrow> i\<in>I \<Longrightarrow> f \<in> measurable M (N i) \<Longrightarrow> pred M (\<lambda>x. f x \<in> A i)" |
|
1922 |
using pred_sets2[OF sets_Ball, of _ _ _ f] by auto |
|
1923 |
||
50096 | 1924 |
lemma measurable_finite[measurable (raw)]: |
1925 |
fixes S :: "'a \<Rightarrow> nat set" |
|
1926 |
assumes [measurable]: "\<And>i. {x\<in>space M. i \<in> S x} \<in> sets M" |
|
1927 |
shows "pred M (\<lambda>x. finite (S x))" |
|
1928 |
unfolding finite_nat_set_iff_bounded by (simp add: Ball_def) |
|
1929 |
||
1930 |
lemma measurable_Least[measurable]: |
|
1931 |
assumes [measurable]: "(\<And>i::nat. (\<lambda>x. P i x) \<in> measurable M (count_space UNIV))"q |
|
1932 |
shows "(\<lambda>x. LEAST i. P i x) \<in> measurable M (count_space UNIV)" |
|
1933 |
unfolding measurable_def by (safe intro!: sets_Least) simp_all |
|
1934 |
||
1935 |
lemma measurable_count_space_insert[measurable (raw)]: |
|
1936 |
"s \<in> S \<Longrightarrow> A \<in> sets (count_space S) \<Longrightarrow> insert s A \<in> sets (count_space S)" |
|
1937 |
by simp |
|
1938 |
||
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1939 |
hide_const (open) pred |
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
49834
diff
changeset
|
1940 |
|
47694 | 1941 |
subsection {* Extend measure *} |
1942 |
||
1943 |
definition "extend_measure \<Omega> I G \<mu> = |
|
1944 |
(if (\<exists>\<mu>'. (\<forall>i\<in>I. \<mu>' (G i) = \<mu> i) \<and> measure_space \<Omega> (sigma_sets \<Omega> (G`I)) \<mu>') \<and> \<not> (\<forall>i\<in>I. \<mu> i = 0) |
|
1945 |
then measure_of \<Omega> (G`I) (SOME \<mu>'. (\<forall>i\<in>I. \<mu>' (G i) = \<mu> i) \<and> measure_space \<Omega> (sigma_sets \<Omega> (G`I)) \<mu>') |
|
1946 |
else measure_of \<Omega> (G`I) (\<lambda>_. 0))" |
|
1947 |
||
1948 |
lemma space_extend_measure: "G ` I \<subseteq> Pow \<Omega> \<Longrightarrow> space (extend_measure \<Omega> I G \<mu>) = \<Omega>" |
|
1949 |
unfolding extend_measure_def by simp |
|
1950 |
||
1951 |
lemma sets_extend_measure: "G ` I \<subseteq> Pow \<Omega> \<Longrightarrow> sets (extend_measure \<Omega> I G \<mu>) = sigma_sets \<Omega> (G`I)" |
|
1952 |
unfolding extend_measure_def by simp |
|
1953 |
||
1954 |
lemma emeasure_extend_measure: |
|
1955 |
assumes M: "M = extend_measure \<Omega> I G \<mu>" |
|
1956 |
and eq: "\<And>i. i \<in> I \<Longrightarrow> \<mu>' (G i) = \<mu> i" |
|
1957 |
and ms: "G ` I \<subseteq> Pow \<Omega>" "positive (sets M) \<mu>'" "countably_additive (sets M) \<mu>'" |
|
1958 |
and "i \<in> I" |
|
1959 |
shows "emeasure M (G i) = \<mu> i" |
|
1960 |
proof cases |
|
1961 |
assume *: "(\<forall>i\<in>I. \<mu> i = 0)" |
|
1962 |
with M have M_eq: "M = measure_of \<Omega> (G`I) (\<lambda>_. 0)" |
|
1963 |
by (simp add: extend_measure_def) |
|
1964 |
from measure_space_0[OF ms(1)] ms `i\<in>I` |
|
1965 |
have "emeasure M (G i) = 0" |
|
1966 |
by (intro emeasure_measure_of[OF M_eq]) (auto simp add: M measure_space_def sets_extend_measure) |
|
1967 |
with `i\<in>I` * show ?thesis |
|
1968 |
by simp |
|
1969 |
next |
|
1970 |
def P \<equiv> "\<lambda>\<mu>'. (\<forall>i\<in>I. \<mu>' (G i) = \<mu> i) \<and> measure_space \<Omega> (sigma_sets \<Omega> (G`I)) \<mu>'" |
|
1971 |
assume "\<not> (\<forall>i\<in>I. \<mu> i = 0)" |
|
1972 |
moreover |
|
1973 |
have "measure_space (space M) (sets M) \<mu>'" |
|
1974 |
using ms unfolding measure_space_def by auto default |
|
1975 |
with ms eq have "\<exists>\<mu>'. P \<mu>'" |
|
1976 |
unfolding P_def |
|
1977 |
by (intro exI[of _ \<mu>']) (auto simp add: M space_extend_measure sets_extend_measure) |
|
1978 |
ultimately have M_eq: "M = measure_of \<Omega> (G`I) (Eps P)" |
|
1979 |
by (simp add: M extend_measure_def P_def[symmetric]) |
|
1980 |
||
1981 |
from `\<exists>\<mu>'. P \<mu>'` have P: "P (Eps P)" by (rule someI_ex) |
|
1982 |
show "emeasure M (G i) = \<mu> i" |
|
1983 |
proof (subst emeasure_measure_of[OF M_eq]) |
|
1984 |
have sets_M: "sets M = sigma_sets \<Omega> (G`I)" |
|
1985 |
using M_eq ms by (auto simp: sets_extend_measure) |
|
1986 |
then show "G i \<in> sets M" using `i \<in> I` by auto |
|
1987 |
show "positive (sets M) (Eps P)" "countably_additive (sets M) (Eps P)" "Eps P (G i) = \<mu> i" |
|
1988 |
using P `i\<in>I` by (auto simp add: sets_M measure_space_def P_def) |
|
1989 |
qed fact |
|
1990 |
qed |
|
1991 |
||
1992 |
lemma emeasure_extend_measure_Pair: |
|
1993 |
assumes M: "M = extend_measure \<Omega> {(i, j). I i j} (\<lambda>(i, j). G i j) (\<lambda>(i, j). \<mu> i j)" |
|
1994 |
and eq: "\<And>i j. I i j \<Longrightarrow> \<mu>' (G i j) = \<mu> i j" |
|
1995 |
and ms: "\<And>i j. I i j \<Longrightarrow> G i j \<in> Pow \<Omega>" "positive (sets M) \<mu>'" "countably_additive (sets M) \<mu>'" |
|
1996 |
and "I i j" |
|
1997 |
shows "emeasure M (G i j) = \<mu> i j" |
|
1998 |
using emeasure_extend_measure[OF M _ _ ms(2,3), of "(i,j)"] eq ms(1) `I i j` |
|
1999 |
by (auto simp: subset_eq) |
|
2000 |
||
39090
a2d38b8b693e
Introduced sigma algebra generated by function preimages.
hoelzl
parents:
38656
diff
changeset
|
2001 |
subsection {* Sigma algebra generated by function preimages *} |
a2d38b8b693e
Introduced sigma algebra generated by function preimages.
hoelzl
parents:
38656
diff
changeset
|
2002 |
|
47694 | 2003 |
definition |
2004 |
"vimage_algebra M S f = sigma S ((\<lambda>A. f -` A \<inter> S) ` sets M)" |
|
39090
a2d38b8b693e
Introduced sigma algebra generated by function preimages.
hoelzl
parents:
38656
diff
changeset
|
2005 |
|
47694 | 2006 |
lemma sigma_algebra_preimages: |
40859 | 2007 |
fixes f :: "'x \<Rightarrow> 'a" |
47694 | 2008 |
assumes "f \<in> S \<rightarrow> space M" |
2009 |
shows "sigma_algebra S ((\<lambda>A. f -` A \<inter> S) ` sets M)" |
|
2010 |
(is "sigma_algebra _ (?F ` sets M)") |
|
40859 | 2011 |
proof (simp add: sigma_algebra_iff2, safe) |
2012 |
show "{} \<in> ?F ` sets M" by blast |
|
2013 |
next |
|
47694 | 2014 |
fix A assume "A \<in> sets M" |
2015 |
moreover have "S - ?F A = ?F (space M - A)" |
|
40859 | 2016 |
using assms by auto |
47694 | 2017 |
ultimately show "S - ?F A \<in> ?F ` sets M" |
40859 | 2018 |
by blast |
2019 |
next |
|
47694 | 2020 |
fix A :: "nat \<Rightarrow> 'x set" assume *: "range A \<subseteq> ?F ` M" |
2021 |
have "\<forall>i. \<exists>b. b \<in> M \<and> A i = ?F b" |
|
40859 | 2022 |
proof safe |
2023 |
fix i |
|
47694 | 2024 |
have "A i \<in> ?F ` M" using * by auto |
2025 |
then show "\<exists>b. b \<in> M \<and> A i = ?F b" by auto |
|
40859 | 2026 |
qed |
47694 | 2027 |
from choice[OF this] obtain b where b: "range b \<subseteq> M" "\<And>i. A i = ?F (b i)" |
40859 | 2028 |
by auto |
47694 | 2029 |
then have "(\<Union>i. A i) = ?F (\<Union>i. b i)" by auto |
2030 |
then show "(\<Union>i. A i) \<in> ?F ` M" using b(1) by blast |
|
40859 | 2031 |
qed |
2032 |
||
47694 | 2033 |
lemma sets_vimage_algebra[simp]: |
2034 |
"f \<in> S \<rightarrow> space M \<Longrightarrow> sets (vimage_algebra M S f) = (\<lambda>A. f -` A \<inter> S) ` sets M" |
|
2035 |
using sigma_algebra.sets_measure_of_eq[OF sigma_algebra_preimages, of f S M] |
|
2036 |
by (simp add: vimage_algebra_def) |
|
2037 |
||
2038 |
lemma space_vimage_algebra[simp]: |
|
2039 |
"f \<in> S \<rightarrow> space M \<Longrightarrow> space (vimage_algebra M S f) = S" |
|
2040 |
using sigma_algebra.space_measure_of_eq[OF sigma_algebra_preimages, of f S M] |
|
2041 |
by (simp add: vimage_algebra_def) |
|
2042 |
||
2043 |
lemma in_vimage_algebra[simp]: |
|
2044 |
"f \<in> S \<rightarrow> space M \<Longrightarrow> A \<in> sets (vimage_algebra M S f) \<longleftrightarrow> (\<exists>B\<in>sets M. A = f -` B \<inter> S)" |
|
2045 |
by (simp add: image_iff) |
|
2046 |
||
2047 |
lemma measurable_vimage_algebra: |
|
39090
a2d38b8b693e
Introduced sigma algebra generated by function preimages.
hoelzl
parents:
38656
diff
changeset
|
2048 |
fixes S :: "'c set" assumes "f \<in> S \<rightarrow> space M" |
47694 | 2049 |
shows "f \<in> measurable (vimage_algebra M S f) M" |
2050 |
unfolding measurable_def using assms by force |
|
39090
a2d38b8b693e
Introduced sigma algebra generated by function preimages.
hoelzl
parents:
38656
diff
changeset
|
2051 |
|
47694 | 2052 |
lemma measurable_vimage: |
40859 | 2053 |
fixes g :: "'a \<Rightarrow> 'c" and f :: "'d \<Rightarrow> 'a" |
2054 |
assumes "g \<in> measurable M M2" "f \<in> S \<rightarrow> space M" |
|
47694 | 2055 |
shows "(\<lambda>x. g (f x)) \<in> measurable (vimage_algebra M S f) M2" |
40859 | 2056 |
proof - |
2057 |
note measurable_vimage_algebra[OF assms(2)] |
|
2058 |
from measurable_comp[OF this assms(1)] |
|
2059 |
show ?thesis by (simp add: comp_def) |
|
2060 |
qed |
|
2061 |
||
2062 |
lemma sigma_sets_vimage: |
|
2063 |
assumes "f \<in> S' \<rightarrow> S" and "A \<subseteq> Pow S" |
|
2064 |
shows "sigma_sets S' ((\<lambda>X. f -` X \<inter> S') ` A) = (\<lambda>X. f -` X \<inter> S') ` sigma_sets S A" |
|
2065 |
proof (intro set_eqI iffI) |
|
2066 |
let ?F = "\<lambda>X. f -` X \<inter> S'" |
|
2067 |
fix X assume "X \<in> sigma_sets S' (?F ` A)" |
|
2068 |
then show "X \<in> ?F ` sigma_sets S A" |
|
2069 |
proof induct |
|
2070 |
case (Basic X) then obtain X' where "X = ?F X'" "X' \<in> A" |
|
2071 |
by auto |
|
47694 | 2072 |
then show ?case by auto |
40859 | 2073 |
next |
2074 |
case Empty then show ?case |
|
2075 |
by (auto intro!: image_eqI[of _ _ "{}"] sigma_sets.Empty) |
|
2076 |
next |
|
2077 |
case (Compl X) then obtain X' where X: "X = ?F X'" and "X' \<in> sigma_sets S A" |
|
2078 |
by auto |
|
2079 |
then have "S - X' \<in> sigma_sets S A" |
|
2080 |
by (auto intro!: sigma_sets.Compl) |
|
2081 |
then show ?case |
|
2082 |
using X assms by (auto intro!: image_eqI[where x="S - X'"]) |
|
2083 |
next |
|
2084 |
case (Union F) |
|
2085 |
then have "\<forall>i. \<exists>F'. F' \<in> sigma_sets S A \<and> F i = f -` F' \<inter> S'" |
|
2086 |
by (auto simp: image_iff Bex_def) |
|
2087 |
from choice[OF this] obtain F' where |
|
2088 |
"\<And>i. F' i \<in> sigma_sets S A" and "\<And>i. F i = f -` F' i \<inter> S'" |
|
2089 |
by auto |
|
2090 |
then show ?case |
|
2091 |
by (auto intro!: sigma_sets.Union image_eqI[where x="\<Union>i. F' i"]) |
|
2092 |
qed |
|
2093 |
next |
|
2094 |
let ?F = "\<lambda>X. f -` X \<inter> S'" |
|
2095 |
fix X assume "X \<in> ?F ` sigma_sets S A" |
|
2096 |
then obtain X' where "X' \<in> sigma_sets S A" "X = ?F X'" by auto |
|
2097 |
then show "X \<in> sigma_sets S' (?F ` A)" |
|
2098 |
proof (induct arbitrary: X) |
|
2099 |
case Empty then show ?case by (auto intro: sigma_sets.Empty) |
|
2100 |
next |
|
2101 |
case (Compl X') |
|
2102 |
have "S' - (S' - X) \<in> sigma_sets S' (?F ` A)" |
|
2103 |
apply (rule sigma_sets.Compl) |
|
2104 |
using assms by (auto intro!: Compl.hyps simp: Compl.prems) |
|
2105 |
also have "S' - (S' - X) = X" |
|
2106 |
using assms Compl by auto |
|
2107 |
finally show ?case . |
|
2108 |
next |
|
2109 |
case (Union F) |
|
2110 |
have "(\<Union>i. f -` F i \<inter> S') \<in> sigma_sets S' (?F ` A)" |
|
2111 |
by (intro sigma_sets.Union Union.hyps) simp |
|
2112 |
also have "(\<Union>i. f -` F i \<inter> S') = X" |
|
2113 |
using assms Union by auto |
|
2114 |
finally show ?case . |
|
47694 | 2115 |
qed auto |
39092 | 2116 |
qed |
2117 |
||
38656 | 2118 |
subsection {* A Two-Element Series *} |
2119 |
||
2120 |
definition binaryset :: "'a set \<Rightarrow> 'a set \<Rightarrow> nat \<Rightarrow> 'a set " |
|
50252 | 2121 |
where "binaryset A B = (\<lambda>x. {})(0 := A, Suc 0 := B)" |
38656 | 2122 |
|
2123 |
lemma range_binaryset_eq: "range(binaryset A B) = {A,B,{}}" |
|
2124 |
apply (simp add: binaryset_def) |
|
39302
d7728f65b353
renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents:
39092
diff
changeset
|
2125 |
apply (rule set_eqI) |
38656 | 2126 |
apply (auto simp add: image_iff) |
2127 |
done |
|
2128 |
||
2129 |
lemma UN_binaryset_eq: "(\<Union>i. binaryset A B i) = A \<union> B" |
|
44106 | 2130 |
by (simp add: SUP_def range_binaryset_eq) |
38656 | 2131 |
|
2132 |
section {* Closed CDI *} |
|
2133 |
||
47694 | 2134 |
definition closed_cdi where |
2135 |
"closed_cdi \<Omega> M \<longleftrightarrow> |
|
2136 |
M \<subseteq> Pow \<Omega> & |
|
2137 |
(\<forall>s \<in> M. \<Omega> - s \<in> M) & |
|
2138 |
(\<forall>A. (range A \<subseteq> M) & (A 0 = {}) & (\<forall>n. A n \<subseteq> A (Suc n)) \<longrightarrow> |
|
2139 |
(\<Union>i. A i) \<in> M) & |
|
2140 |
(\<forall>A. (range A \<subseteq> M) & disjoint_family A \<longrightarrow> (\<Union>i::nat. A i) \<in> M)" |
|
38656 | 2141 |
|
2142 |
inductive_set |
|
47694 | 2143 |
smallest_ccdi_sets :: "'a set \<Rightarrow> 'a set set \<Rightarrow> 'a set set" |
2144 |
for \<Omega> M |
|
38656 | 2145 |
where |
2146 |
Basic [intro]: |
|
47694 | 2147 |
"a \<in> M \<Longrightarrow> a \<in> smallest_ccdi_sets \<Omega> M" |
38656 | 2148 |
| Compl [intro]: |
47694 | 2149 |
"a \<in> smallest_ccdi_sets \<Omega> M \<Longrightarrow> \<Omega> - a \<in> smallest_ccdi_sets \<Omega> M" |
38656 | 2150 |
| Inc: |
47694 | 2151 |
"range A \<in> Pow(smallest_ccdi_sets \<Omega> M) \<Longrightarrow> A 0 = {} \<Longrightarrow> (\<And>n. A n \<subseteq> A (Suc n)) |
2152 |
\<Longrightarrow> (\<Union>i. A i) \<in> smallest_ccdi_sets \<Omega> M" |
|
38656 | 2153 |
| Disj: |
47694 | 2154 |
"range A \<in> Pow(smallest_ccdi_sets \<Omega> M) \<Longrightarrow> disjoint_family A |
2155 |
\<Longrightarrow> (\<Union>i::nat. A i) \<in> smallest_ccdi_sets \<Omega> M" |
|
38656 | 2156 |
|
47694 | 2157 |
lemma (in subset_class) smallest_closed_cdi1: "M \<subseteq> smallest_ccdi_sets \<Omega> M" |
2158 |
by auto |
|
38656 | 2159 |
|
47694 | 2160 |
lemma (in subset_class) smallest_ccdi_sets: "smallest_ccdi_sets \<Omega> M \<subseteq> Pow \<Omega>" |
38656 | 2161 |
apply (rule subsetI) |
2162 |
apply (erule smallest_ccdi_sets.induct) |
|
2163 |
apply (auto intro: range_subsetD dest: sets_into_space) |
|
2164 |
done |
|
2165 |
||
47694 | 2166 |
lemma (in subset_class) smallest_closed_cdi2: "closed_cdi \<Omega> (smallest_ccdi_sets \<Omega> M)" |
2167 |
apply (auto simp add: closed_cdi_def smallest_ccdi_sets) |
|
38656 | 2168 |
apply (blast intro: smallest_ccdi_sets.Inc smallest_ccdi_sets.Disj) + |
2169 |
done |
|
2170 |
||
47694 | 2171 |
lemma closed_cdi_subset: "closed_cdi \<Omega> M \<Longrightarrow> M \<subseteq> Pow \<Omega>" |
38656 | 2172 |
by (simp add: closed_cdi_def) |
2173 |
||
47694 | 2174 |
lemma closed_cdi_Compl: "closed_cdi \<Omega> M \<Longrightarrow> s \<in> M \<Longrightarrow> \<Omega> - s \<in> M" |
38656 | 2175 |
by (simp add: closed_cdi_def) |
2176 |
||
2177 |
lemma closed_cdi_Inc: |
|
47694 | 2178 |
"closed_cdi \<Omega> M \<Longrightarrow> range A \<subseteq> M \<Longrightarrow> A 0 = {} \<Longrightarrow> (!!n. A n \<subseteq> A (Suc n)) \<Longrightarrow> (\<Union>i. A i) \<in> M" |
38656 | 2179 |
by (simp add: closed_cdi_def) |
2180 |
||
2181 |
lemma closed_cdi_Disj: |
|
47694 | 2182 |
"closed_cdi \<Omega> M \<Longrightarrow> range A \<subseteq> M \<Longrightarrow> disjoint_family A \<Longrightarrow> (\<Union>i::nat. A i) \<in> M" |
38656 | 2183 |
by (simp add: closed_cdi_def) |
2184 |
||
2185 |
lemma closed_cdi_Un: |
|
47694 | 2186 |
assumes cdi: "closed_cdi \<Omega> M" and empty: "{} \<in> M" |
2187 |
and A: "A \<in> M" and B: "B \<in> M" |
|
38656 | 2188 |
and disj: "A \<inter> B = {}" |
47694 | 2189 |
shows "A \<union> B \<in> M" |
38656 | 2190 |
proof - |
47694 | 2191 |
have ra: "range (binaryset A B) \<subseteq> M" |
38656 | 2192 |
by (simp add: range_binaryset_eq empty A B) |
2193 |
have di: "disjoint_family (binaryset A B)" using disj |
|
2194 |
by (simp add: disjoint_family_on_def binaryset_def Int_commute) |
|
2195 |
from closed_cdi_Disj [OF cdi ra di] |
|
2196 |
show ?thesis |
|
2197 |
by (simp add: UN_binaryset_eq) |
|
2198 |
qed |
|
2199 |
||
2200 |
lemma (in algebra) smallest_ccdi_sets_Un: |
|
47694 | 2201 |
assumes A: "A \<in> smallest_ccdi_sets \<Omega> M" and B: "B \<in> smallest_ccdi_sets \<Omega> M" |
38656 | 2202 |
and disj: "A \<inter> B = {}" |
47694 | 2203 |
shows "A \<union> B \<in> smallest_ccdi_sets \<Omega> M" |
38656 | 2204 |
proof - |
47694 | 2205 |
have ra: "range (binaryset A B) \<in> Pow (smallest_ccdi_sets \<Omega> M)" |
38656 | 2206 |
by (simp add: range_binaryset_eq A B smallest_ccdi_sets.Basic) |
2207 |
have di: "disjoint_family (binaryset A B)" using disj |
|
2208 |
by (simp add: disjoint_family_on_def binaryset_def Int_commute) |
|
2209 |
from Disj [OF ra di] |
|
2210 |
show ?thesis |
|
2211 |
by (simp add: UN_binaryset_eq) |
|
2212 |
qed |
|
2213 |
||
2214 |
lemma (in algebra) smallest_ccdi_sets_Int1: |
|
47694 | 2215 |
assumes a: "a \<in> M" |
2216 |
shows "b \<in> smallest_ccdi_sets \<Omega> M \<Longrightarrow> a \<inter> b \<in> smallest_ccdi_sets \<Omega> M" |
|
38656 | 2217 |
proof (induct rule: smallest_ccdi_sets.induct) |
2218 |
case (Basic x) |
|
2219 |
thus ?case |
|
2220 |
by (metis a Int smallest_ccdi_sets.Basic) |
|
2221 |
next |
|
2222 |
case (Compl x) |
|
47694 | 2223 |
have "a \<inter> (\<Omega> - x) = \<Omega> - ((\<Omega> - a) \<union> (a \<inter> x))" |
38656 | 2224 |
by blast |
47694 | 2225 |
also have "... \<in> smallest_ccdi_sets \<Omega> M" |
38656 | 2226 |
by (metis smallest_ccdi_sets.Compl a Compl(2) Diff_Int2 Diff_Int_distrib2 |
47694 | 2227 |
Diff_disjoint Int_Diff Int_empty_right smallest_ccdi_sets_Un |
2228 |
smallest_ccdi_sets.Basic smallest_ccdi_sets.Compl) |
|
38656 | 2229 |
finally show ?case . |
2230 |
next |
|
2231 |
case (Inc A) |
|
2232 |
have 1: "(\<Union>i. (\<lambda>i. a \<inter> A i) i) = a \<inter> (\<Union>i. A i)" |
|
2233 |
by blast |
|
47694 | 2234 |
have "range (\<lambda>i. a \<inter> A i) \<in> Pow(smallest_ccdi_sets \<Omega> M)" using Inc |
38656 | 2235 |
by blast |
2236 |
moreover have "(\<lambda>i. a \<inter> A i) 0 = {}" |
|
2237 |
by (simp add: Inc) |
|
2238 |
moreover have "!!n. (\<lambda>i. a \<inter> A i) n \<subseteq> (\<lambda>i. a \<inter> A i) (Suc n)" using Inc |
|
2239 |
by blast |
|
47694 | 2240 |
ultimately have 2: "(\<Union>i. (\<lambda>i. a \<inter> A i) i) \<in> smallest_ccdi_sets \<Omega> M" |
38656 | 2241 |
by (rule smallest_ccdi_sets.Inc) |
2242 |
show ?case |
|
2243 |
by (metis 1 2) |
|
2244 |
next |
|
2245 |
case (Disj A) |
|
2246 |
have 1: "(\<Union>i. (\<lambda>i. a \<inter> A i) i) = a \<inter> (\<Union>i. A i)" |
|
2247 |
by blast |
|
47694 | 2248 |
have "range (\<lambda>i. a \<inter> A i) \<in> Pow(smallest_ccdi_sets \<Omega> M)" using Disj |
38656 | 2249 |
by blast |
2250 |
moreover have "disjoint_family (\<lambda>i. a \<inter> A i)" using Disj |
|
2251 |
by (auto simp add: disjoint_family_on_def) |
|
47694 | 2252 |
ultimately have 2: "(\<Union>i. (\<lambda>i. a \<inter> A i) i) \<in> smallest_ccdi_sets \<Omega> M" |
38656 | 2253 |
by (rule smallest_ccdi_sets.Disj) |
2254 |
show ?case |
|
2255 |
by (metis 1 2) |
|
2256 |
qed |
|
2257 |
||
2258 |
||
2259 |
lemma (in algebra) smallest_ccdi_sets_Int: |
|
47694 | 2260 |
assumes b: "b \<in> smallest_ccdi_sets \<Omega> M" |
2261 |
shows "a \<in> smallest_ccdi_sets \<Omega> M \<Longrightarrow> a \<inter> b \<in> smallest_ccdi_sets \<Omega> M" |
|
38656 | 2262 |
proof (induct rule: smallest_ccdi_sets.induct) |
2263 |
case (Basic x) |
|
2264 |
thus ?case |
|
2265 |
by (metis b smallest_ccdi_sets_Int1) |
|
2266 |
next |
|
2267 |
case (Compl x) |
|
47694 | 2268 |
have "(\<Omega> - x) \<inter> b = \<Omega> - (x \<inter> b \<union> (\<Omega> - b))" |
38656 | 2269 |
by blast |
47694 | 2270 |
also have "... \<in> smallest_ccdi_sets \<Omega> M" |
38656 | 2271 |
by (metis Compl(2) Diff_disjoint Int_Diff Int_commute Int_empty_right b |
2272 |
smallest_ccdi_sets.Compl smallest_ccdi_sets_Un) |
|
2273 |
finally show ?case . |
|
2274 |
next |
|
2275 |
case (Inc A) |
|
2276 |
have 1: "(\<Union>i. (\<lambda>i. A i \<inter> b) i) = (\<Union>i. A i) \<inter> b" |
|
2277 |
by blast |
|
47694 | 2278 |
have "range (\<lambda>i. A i \<inter> b) \<in> Pow(smallest_ccdi_sets \<Omega> M)" using Inc |
38656 | 2279 |
by blast |
2280 |
moreover have "(\<lambda>i. A i \<inter> b) 0 = {}" |
|
2281 |
by (simp add: Inc) |
|
2282 |
moreover have "!!n. (\<lambda>i. A i \<inter> b) n \<subseteq> (\<lambda>i. A i \<inter> b) (Suc n)" using Inc |
|
2283 |
by blast |
|
47694 | 2284 |
ultimately have 2: "(\<Union>i. (\<lambda>i. A i \<inter> b) i) \<in> smallest_ccdi_sets \<Omega> M" |
38656 | 2285 |
by (rule smallest_ccdi_sets.Inc) |
2286 |
show ?case |
|
2287 |
by (metis 1 2) |
|
2288 |
next |
|
2289 |
case (Disj A) |
|
2290 |
have 1: "(\<Union>i. (\<lambda>i. A i \<inter> b) i) = (\<Union>i. A i) \<inter> b" |
|
2291 |
by blast |
|
47694 | 2292 |
have "range (\<lambda>i. A i \<inter> b) \<in> Pow(smallest_ccdi_sets \<Omega> M)" using Disj |
38656 | 2293 |
by blast |
2294 |
moreover have "disjoint_family (\<lambda>i. A i \<inter> b)" using Disj |
|
2295 |
by (auto simp add: disjoint_family_on_def) |
|
47694 | 2296 |
ultimately have 2: "(\<Union>i. (\<lambda>i. A i \<inter> b) i) \<in> smallest_ccdi_sets \<Omega> M" |
38656 | 2297 |
by (rule smallest_ccdi_sets.Disj) |
2298 |
show ?case |
|
2299 |
by (metis 1 2) |
|
2300 |
qed |
|
2301 |
||
2302 |
lemma (in algebra) sigma_property_disjoint_lemma: |
|
47694 | 2303 |
assumes sbC: "M \<subseteq> C" |
2304 |
and ccdi: "closed_cdi \<Omega> C" |
|
2305 |
shows "sigma_sets \<Omega> M \<subseteq> C" |
|
38656 | 2306 |
proof - |
47694 | 2307 |
have "smallest_ccdi_sets \<Omega> M \<in> {B . M \<subseteq> B \<and> sigma_algebra \<Omega> B}" |
38656 | 2308 |
apply (auto simp add: sigma_algebra_disjoint_iff algebra_iff_Int |
2309 |
smallest_ccdi_sets_Int) |
|
2310 |
apply (metis Union_Pow_eq Union_upper subsetD smallest_ccdi_sets) |
|
2311 |
apply (blast intro: smallest_ccdi_sets.Disj) |
|
2312 |
done |
|
47694 | 2313 |
hence "sigma_sets (\<Omega>) (M) \<subseteq> smallest_ccdi_sets \<Omega> M" |
38656 | 2314 |
by clarsimp |
47694 | 2315 |
(drule sigma_algebra.sigma_sets_subset [where a="M"], auto) |
38656 | 2316 |
also have "... \<subseteq> C" |
2317 |
proof |
|
2318 |
fix x |
|
47694 | 2319 |
assume x: "x \<in> smallest_ccdi_sets \<Omega> M" |
38656 | 2320 |
thus "x \<in> C" |
2321 |
proof (induct rule: smallest_ccdi_sets.induct) |
|
2322 |
case (Basic x) |
|
2323 |
thus ?case |
|
2324 |
by (metis Basic subsetD sbC) |
|
2325 |
next |
|
2326 |
case (Compl x) |
|
2327 |
thus ?case |
|
2328 |
by (blast intro: closed_cdi_Compl [OF ccdi, simplified]) |
|
2329 |
next |
|
2330 |
case (Inc A) |
|
2331 |
thus ?case |
|
2332 |
by (auto intro: closed_cdi_Inc [OF ccdi, simplified]) |
|
2333 |
next |
|
2334 |
case (Disj A) |
|
2335 |
thus ?case |
|
2336 |
by (auto intro: closed_cdi_Disj [OF ccdi, simplified]) |
|
2337 |
qed |
|
2338 |
qed |
|
2339 |
finally show ?thesis . |
|
2340 |
qed |
|
2341 |
||
2342 |
lemma (in algebra) sigma_property_disjoint: |
|
47694 | 2343 |
assumes sbC: "M \<subseteq> C" |
2344 |
and compl: "!!s. s \<in> C \<inter> sigma_sets (\<Omega>) (M) \<Longrightarrow> \<Omega> - s \<in> C" |
|
2345 |
and inc: "!!A. range A \<subseteq> C \<inter> sigma_sets (\<Omega>) (M) |
|
38656 | 2346 |
\<Longrightarrow> A 0 = {} \<Longrightarrow> (!!n. A n \<subseteq> A (Suc n)) |
2347 |
\<Longrightarrow> (\<Union>i. A i) \<in> C" |
|
47694 | 2348 |
and disj: "!!A. range A \<subseteq> C \<inter> sigma_sets (\<Omega>) (M) |
38656 | 2349 |
\<Longrightarrow> disjoint_family A \<Longrightarrow> (\<Union>i::nat. A i) \<in> C" |
47694 | 2350 |
shows "sigma_sets (\<Omega>) (M) \<subseteq> C" |
38656 | 2351 |
proof - |
47694 | 2352 |
have "sigma_sets (\<Omega>) (M) \<subseteq> C \<inter> sigma_sets (\<Omega>) (M)" |
38656 | 2353 |
proof (rule sigma_property_disjoint_lemma) |
47694 | 2354 |
show "M \<subseteq> C \<inter> sigma_sets (\<Omega>) (M)" |
38656 | 2355 |
by (metis Int_greatest Set.subsetI sbC sigma_sets.Basic) |
2356 |
next |
|
47694 | 2357 |
show "closed_cdi \<Omega> (C \<inter> sigma_sets (\<Omega>) (M))" |
38656 | 2358 |
by (simp add: closed_cdi_def compl inc disj) |
2359 |
(metis PowI Set.subsetI le_infI2 sigma_sets_into_sp space_closed |
|
2360 |
IntE sigma_sets.Compl range_subsetD sigma_sets.Union) |
|
2361 |
qed |
|
2362 |
thus ?thesis |
|
2363 |
by blast |
|
2364 |
qed |
|
2365 |
||
40859 | 2366 |
section {* Dynkin systems *} |
2367 |
||
42065
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
parents:
41983
diff
changeset
|
2368 |
locale dynkin_system = subset_class + |
47694 | 2369 |
assumes space: "\<Omega> \<in> M" |
2370 |
and compl[intro!]: "\<And>A. A \<in> M \<Longrightarrow> \<Omega> - A \<in> M" |
|
2371 |
and UN[intro!]: "\<And>A. disjoint_family A \<Longrightarrow> range A \<subseteq> M |
|
2372 |
\<Longrightarrow> (\<Union>i::nat. A i) \<in> M" |
|
40859 | 2373 |
|
47694 | 2374 |
lemma (in dynkin_system) empty[intro, simp]: "{} \<in> M" |
2375 |
using space compl[of "\<Omega>"] by simp |
|
40859 | 2376 |
|
2377 |
lemma (in dynkin_system) diff: |
|
47694 | 2378 |
assumes sets: "D \<in> M" "E \<in> M" and "D \<subseteq> E" |
2379 |
shows "E - D \<in> M" |
|
40859 | 2380 |
proof - |
47694 | 2381 |
let ?f = "\<lambda>x. if x = 0 then D else if x = Suc 0 then \<Omega> - E else {}" |
2382 |
have "range ?f = {D, \<Omega> - E, {}}" |
|
40859 | 2383 |
by (auto simp: image_iff) |
47694 | 2384 |
moreover have "D \<union> (\<Omega> - E) = (\<Union>i. ?f i)" |
40859 | 2385 |
by (auto simp: image_iff split: split_if_asm) |
2386 |
moreover |
|
2387 |
then have "disjoint_family ?f" unfolding disjoint_family_on_def |
|
47694 | 2388 |
using `D \<in> M`[THEN sets_into_space] `D \<subseteq> E` by auto |
2389 |
ultimately have "\<Omega> - (D \<union> (\<Omega> - E)) \<in> M" |
|
40859 | 2390 |
using sets by auto |
47694 | 2391 |
also have "\<Omega> - (D \<union> (\<Omega> - E)) = E - D" |
40859 | 2392 |
using assms sets_into_space by auto |
2393 |
finally show ?thesis . |
|
2394 |
qed |
|
2395 |
||
2396 |
lemma dynkin_systemI: |
|
47694 | 2397 |
assumes "\<And> A. A \<in> M \<Longrightarrow> A \<subseteq> \<Omega>" "\<Omega> \<in> M" |
2398 |
assumes "\<And> A. A \<in> M \<Longrightarrow> \<Omega> - A \<in> M" |
|
2399 |
assumes "\<And> A. disjoint_family A \<Longrightarrow> range A \<subseteq> M |
|
2400 |
\<Longrightarrow> (\<Union>i::nat. A i) \<in> M" |
|
2401 |
shows "dynkin_system \<Omega> M" |
|
42065
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
parents:
41983
diff
changeset
|
2402 |
using assms by (auto simp: dynkin_system_def dynkin_system_axioms_def subset_class_def) |
40859 | 2403 |
|
42988 | 2404 |
lemma dynkin_systemI': |
47694 | 2405 |
assumes 1: "\<And> A. A \<in> M \<Longrightarrow> A \<subseteq> \<Omega>" |
2406 |
assumes empty: "{} \<in> M" |
|
2407 |
assumes Diff: "\<And> A. A \<in> M \<Longrightarrow> \<Omega> - A \<in> M" |
|
2408 |
assumes 2: "\<And> A. disjoint_family A \<Longrightarrow> range A \<subseteq> M |
|
2409 |
\<Longrightarrow> (\<Union>i::nat. A i) \<in> M" |
|
2410 |
shows "dynkin_system \<Omega> M" |
|
42988 | 2411 |
proof - |
47694 | 2412 |
from Diff[OF empty] have "\<Omega> \<in> M" by auto |
42988 | 2413 |
from 1 this Diff 2 show ?thesis |
2414 |
by (intro dynkin_systemI) auto |
|
2415 |
qed |
|
2416 |
||
40859 | 2417 |
lemma dynkin_system_trivial: |
47694 | 2418 |
shows "dynkin_system A (Pow A)" |
40859 | 2419 |
by (rule dynkin_systemI) auto |
2420 |
||
2421 |
lemma sigma_algebra_imp_dynkin_system: |
|
47694 | 2422 |
assumes "sigma_algebra \<Omega> M" shows "dynkin_system \<Omega> M" |
40859 | 2423 |
proof - |
47694 | 2424 |
interpret sigma_algebra \<Omega> M by fact |
44890
22f665a2e91c
new fastforce replacing fastsimp - less confusing name
nipkow
parents:
44537
diff
changeset
|
2425 |
show ?thesis using sets_into_space by (fastforce intro!: dynkin_systemI) |
40859 | 2426 |
qed |
2427 |
||
2428 |
subsection "Intersection stable algebras" |
|
2429 |
||
47694 | 2430 |
definition "Int_stable M \<longleftrightarrow> (\<forall> a \<in> M. \<forall> b \<in> M. a \<inter> b \<in> M)" |
40859 | 2431 |
|
2432 |
lemma (in algebra) Int_stable: "Int_stable M" |
|
2433 |
unfolding Int_stable_def by auto |
|
2434 |
||
42981 | 2435 |
lemma Int_stableI: |
47694 | 2436 |
"(\<And>a b. a \<in> A \<Longrightarrow> b \<in> A \<Longrightarrow> a \<inter> b \<in> A) \<Longrightarrow> Int_stable A" |
42981 | 2437 |
unfolding Int_stable_def by auto |
2438 |
||
2439 |
lemma Int_stableD: |
|
47694 | 2440 |
"Int_stable M \<Longrightarrow> a \<in> M \<Longrightarrow> b \<in> M \<Longrightarrow> a \<inter> b \<in> M" |
42981 | 2441 |
unfolding Int_stable_def by auto |
2442 |
||
40859 | 2443 |
lemma (in dynkin_system) sigma_algebra_eq_Int_stable: |
47694 | 2444 |
"sigma_algebra \<Omega> M \<longleftrightarrow> Int_stable M" |
40859 | 2445 |
proof |
47694 | 2446 |
assume "sigma_algebra \<Omega> M" then show "Int_stable M" |
40859 | 2447 |
unfolding sigma_algebra_def using algebra.Int_stable by auto |
2448 |
next |
|
2449 |
assume "Int_stable M" |
|
47694 | 2450 |
show "sigma_algebra \<Omega> M" |
42065
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
parents:
41983
diff
changeset
|
2451 |
unfolding sigma_algebra_disjoint_iff algebra_iff_Un |
40859 | 2452 |
proof (intro conjI ballI allI impI) |
47694 | 2453 |
show "M \<subseteq> Pow (\<Omega>)" using sets_into_space by auto |
40859 | 2454 |
next |
47694 | 2455 |
fix A B assume "A \<in> M" "B \<in> M" |
2456 |
then have "A \<union> B = \<Omega> - ((\<Omega> - A) \<inter> (\<Omega> - B))" |
|
2457 |
"\<Omega> - A \<in> M" "\<Omega> - B \<in> M" |
|
40859 | 2458 |
using sets_into_space by auto |
47694 | 2459 |
then show "A \<union> B \<in> M" |
40859 | 2460 |
using `Int_stable M` unfolding Int_stable_def by auto |
2461 |
qed auto |
|
2462 |
qed |
|
2463 |
||
2464 |
subsection "Smallest Dynkin systems" |
|
2465 |
||
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41543
diff
changeset
|
2466 |
definition dynkin where |
47694 | 2467 |
"dynkin \<Omega> M = (\<Inter>{D. dynkin_system \<Omega> D \<and> M \<subseteq> D})" |
40859 | 2468 |
|
2469 |
lemma dynkin_system_dynkin: |
|
47694 | 2470 |
assumes "M \<subseteq> Pow (\<Omega>)" |
2471 |
shows "dynkin_system \<Omega> (dynkin \<Omega> M)" |
|
40859 | 2472 |
proof (rule dynkin_systemI) |
47694 | 2473 |
fix A assume "A \<in> dynkin \<Omega> M" |
40859 | 2474 |
moreover |
47694 | 2475 |
{ fix D assume "A \<in> D" and d: "dynkin_system \<Omega> D" |
2476 |
then have "A \<subseteq> \<Omega>" by (auto simp: dynkin_system_def subset_class_def) } |
|
2477 |
moreover have "{D. dynkin_system \<Omega> D \<and> M \<subseteq> D} \<noteq> {}" |
|
44890
22f665a2e91c
new fastforce replacing fastsimp - less confusing name
nipkow
parents:
44537
diff
changeset
|
2478 |
using assms dynkin_system_trivial by fastforce |
47694 | 2479 |
ultimately show "A \<subseteq> \<Omega>" |
40859 | 2480 |
unfolding dynkin_def using assms |
47694 | 2481 |
by auto |
40859 | 2482 |
next |
47694 | 2483 |
show "\<Omega> \<in> dynkin \<Omega> M" |
44890
22f665a2e91c
new fastforce replacing fastsimp - less confusing name
nipkow
parents:
44537
diff
changeset
|
2484 |
unfolding dynkin_def using dynkin_system.space by fastforce |
40859 | 2485 |
next |
47694 | 2486 |
fix A assume "A \<in> dynkin \<Omega> M" |
2487 |
then show "\<Omega> - A \<in> dynkin \<Omega> M" |
|
40859 | 2488 |
unfolding dynkin_def using dynkin_system.compl by force |
2489 |
next |
|
2490 |
fix A :: "nat \<Rightarrow> 'a set" |
|
47694 | 2491 |
assume A: "disjoint_family A" "range A \<subseteq> dynkin \<Omega> M" |
2492 |
show "(\<Union>i. A i) \<in> dynkin \<Omega> M" unfolding dynkin_def |
|
40859 | 2493 |
proof (simp, safe) |
47694 | 2494 |
fix D assume "dynkin_system \<Omega> D" "M \<subseteq> D" |
2495 |
with A have "(\<Union>i. A i) \<in> D" |
|
40859 | 2496 |
by (intro dynkin_system.UN) (auto simp: dynkin_def) |
2497 |
then show "(\<Union>i. A i) \<in> D" by auto |
|
2498 |
qed |
|
2499 |
qed |
|
2500 |
||
47694 | 2501 |
lemma dynkin_Basic[intro]: "A \<in> M \<Longrightarrow> A \<in> dynkin \<Omega> M" |
40859 | 2502 |
unfolding dynkin_def by auto |
2503 |
||
2504 |
lemma (in dynkin_system) restricted_dynkin_system: |
|
47694 | 2505 |
assumes "D \<in> M" |
2506 |
shows "dynkin_system \<Omega> {Q. Q \<subseteq> \<Omega> \<and> Q \<inter> D \<in> M}" |
|
40859 | 2507 |
proof (rule dynkin_systemI, simp_all) |
47694 | 2508 |
have "\<Omega> \<inter> D = D" |
2509 |
using `D \<in> M` sets_into_space by auto |
|
2510 |
then show "\<Omega> \<inter> D \<in> M" |
|
2511 |
using `D \<in> M` by auto |
|
40859 | 2512 |
next |
47694 | 2513 |
fix A assume "A \<subseteq> \<Omega> \<and> A \<inter> D \<in> M" |
2514 |
moreover have "(\<Omega> - A) \<inter> D = (\<Omega> - (A \<inter> D)) - (\<Omega> - D)" |
|
40859 | 2515 |
by auto |
47694 | 2516 |
ultimately show "\<Omega> - A \<subseteq> \<Omega> \<and> (\<Omega> - A) \<inter> D \<in> M" |
2517 |
using `D \<in> M` by (auto intro: diff) |
|
40859 | 2518 |
next |
2519 |
fix A :: "nat \<Rightarrow> 'a set" |
|
47694 | 2520 |
assume "disjoint_family A" "range A \<subseteq> {Q. Q \<subseteq> \<Omega> \<and> Q \<inter> D \<in> M}" |
2521 |
then have "\<And>i. A i \<subseteq> \<Omega>" "disjoint_family (\<lambda>i. A i \<inter> D)" |
|
2522 |
"range (\<lambda>i. A i \<inter> D) \<subseteq> M" "(\<Union>x. A x) \<inter> D = (\<Union>x. A x \<inter> D)" |
|
44890
22f665a2e91c
new fastforce replacing fastsimp - less confusing name
nipkow
parents:
44537
diff
changeset
|
2523 |
by ((fastforce simp: disjoint_family_on_def)+) |
47694 | 2524 |
then show "(\<Union>x. A x) \<subseteq> \<Omega> \<and> (\<Union>x. A x) \<inter> D \<in> M" |
40859 | 2525 |
by (auto simp del: UN_simps) |
2526 |
qed |
|
2527 |
||
2528 |
lemma (in dynkin_system) dynkin_subset: |
|
47694 | 2529 |
assumes "N \<subseteq> M" |
2530 |
shows "dynkin \<Omega> N \<subseteq> M" |
|
40859 | 2531 |
proof - |
47694 | 2532 |
have "dynkin_system \<Omega> M" by default |
2533 |
then have "dynkin_system \<Omega> M" |
|
42065
2b98b4c2e2f1
add ring_of_sets and subset_class as basis for algebra
hoelzl
parents:
41983
diff
changeset
|
2534 |
using assms unfolding dynkin_system_def dynkin_system_axioms_def subset_class_def by simp |
47694 | 2535 |
with `N \<subseteq> M` show ?thesis by (auto simp add: dynkin_def) |
40859 | 2536 |
qed |
2537 |
||
2538 |
lemma sigma_eq_dynkin: |
|
47694 | 2539 |
assumes sets: "M \<subseteq> Pow \<Omega>" |
40859 | 2540 |
assumes "Int_stable M" |
47694 | 2541 |
shows "sigma_sets \<Omega> M = dynkin \<Omega> M" |
40859 | 2542 |
proof - |
47694 | 2543 |
have "dynkin \<Omega> M \<subseteq> sigma_sets (\<Omega>) (M)" |
40859 | 2544 |
using sigma_algebra_imp_dynkin_system |
47694 | 2545 |
unfolding dynkin_def sigma_sets_least_sigma_algebra[OF sets] by auto |
40859 | 2546 |
moreover |
47694 | 2547 |
interpret dynkin_system \<Omega> "dynkin \<Omega> M" |
40859 | 2548 |
using dynkin_system_dynkin[OF sets] . |
47694 | 2549 |
have "sigma_algebra \<Omega> (dynkin \<Omega> M)" |
40859 | 2550 |
unfolding sigma_algebra_eq_Int_stable Int_stable_def |
2551 |
proof (intro ballI) |
|
47694 | 2552 |
fix A B assume "A \<in> dynkin \<Omega> M" "B \<in> dynkin \<Omega> M" |
2553 |
let ?D = "\<lambda>E. {Q. Q \<subseteq> \<Omega> \<and> Q \<inter> E \<in> dynkin \<Omega> M}" |
|
2554 |
have "M \<subseteq> ?D B" |
|
40859 | 2555 |
proof |
47694 | 2556 |
fix E assume "E \<in> M" |
2557 |
then have "M \<subseteq> ?D E" "E \<in> dynkin \<Omega> M" |
|
40859 | 2558 |
using sets_into_space `Int_stable M` by (auto simp: Int_stable_def) |
47694 | 2559 |
then have "dynkin \<Omega> M \<subseteq> ?D E" |
2560 |
using restricted_dynkin_system `E \<in> dynkin \<Omega> M` |
|
40859 | 2561 |
by (intro dynkin_system.dynkin_subset) simp_all |
47694 | 2562 |
then have "B \<in> ?D E" |
2563 |
using `B \<in> dynkin \<Omega> M` by auto |
|
2564 |
then have "E \<inter> B \<in> dynkin \<Omega> M" |
|
40859 | 2565 |
by (subst Int_commute) simp |
47694 | 2566 |
then show "E \<in> ?D B" |
2567 |
using sets `E \<in> M` by auto |
|
40859 | 2568 |
qed |
47694 | 2569 |
then have "dynkin \<Omega> M \<subseteq> ?D B" |
2570 |
using restricted_dynkin_system `B \<in> dynkin \<Omega> M` |
|
40859 | 2571 |
by (intro dynkin_system.dynkin_subset) simp_all |
47694 | 2572 |
then show "A \<inter> B \<in> dynkin \<Omega> M" |
2573 |
using `A \<in> dynkin \<Omega> M` sets_into_space by auto |
|
40859 | 2574 |
qed |
47694 | 2575 |
from sigma_algebra.sigma_sets_subset[OF this, of "M"] |
2576 |
have "sigma_sets (\<Omega>) (M) \<subseteq> dynkin \<Omega> M" by auto |
|
2577 |
ultimately have "sigma_sets (\<Omega>) (M) = dynkin \<Omega> M" by auto |
|
40859 | 2578 |
then show ?thesis |
47694 | 2579 |
by (auto simp: dynkin_def) |
40859 | 2580 |
qed |
2581 |
||
2582 |
lemma (in dynkin_system) dynkin_idem: |
|
47694 | 2583 |
"dynkin \<Omega> M = M" |
40859 | 2584 |
proof - |
47694 | 2585 |
have "dynkin \<Omega> M = M" |
40859 | 2586 |
proof |
47694 | 2587 |
show "M \<subseteq> dynkin \<Omega> M" |
40859 | 2588 |
using dynkin_Basic by auto |
47694 | 2589 |
show "dynkin \<Omega> M \<subseteq> M" |
40859 | 2590 |
by (intro dynkin_subset) auto |
2591 |
qed |
|
2592 |
then show ?thesis |
|
47694 | 2593 |
by (auto simp: dynkin_def) |
40859 | 2594 |
qed |
2595 |
||
2596 |
lemma (in dynkin_system) dynkin_lemma: |
|
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41543
diff
changeset
|
2597 |
assumes "Int_stable E" |
47694 | 2598 |
and E: "E \<subseteq> M" "M \<subseteq> sigma_sets \<Omega> E" |
2599 |
shows "sigma_sets \<Omega> E = M" |
|
40859 | 2600 |
proof - |
47694 | 2601 |
have "E \<subseteq> Pow \<Omega>" |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41543
diff
changeset
|
2602 |
using E sets_into_space by force |
47694 | 2603 |
then have "sigma_sets \<Omega> E = dynkin \<Omega> E" |
40859 | 2604 |
using `Int_stable E` by (rule sigma_eq_dynkin) |
47694 | 2605 |
moreover then have "dynkin \<Omega> E = M" |
2606 |
using assms dynkin_subset[OF E(1)] by simp |
|
40859 | 2607 |
ultimately show ?thesis |
47694 | 2608 |
using assms by (auto simp: dynkin_def) |
42864 | 2609 |
qed |
2610 |
||
49789
e0a4cb91a8a9
add induction rule for intersection-stable sigma-sets
hoelzl
parents:
49782
diff
changeset
|
2611 |
lemma sigma_sets_induct_disjoint[consumes 3, case_names basic empty compl union]: |
e0a4cb91a8a9
add induction rule for intersection-stable sigma-sets
hoelzl
parents:
49782
diff
changeset
|
2612 |
assumes "Int_stable G" |
e0a4cb91a8a9
add induction rule for intersection-stable sigma-sets
hoelzl
parents:
49782
diff
changeset
|
2613 |
and closed: "G \<subseteq> Pow \<Omega>" |
e0a4cb91a8a9
add induction rule for intersection-stable sigma-sets
hoelzl
parents:
49782
diff
changeset
|
2614 |
and A: "A \<in> sigma_sets \<Omega> G" |
e0a4cb91a8a9
add induction rule for intersection-stable sigma-sets
hoelzl
parents:
49782
diff
changeset
|
2615 |
assumes basic: "\<And>A. A \<in> G \<Longrightarrow> P A" |
e0a4cb91a8a9
add induction rule for intersection-stable sigma-sets
hoelzl
parents:
49782
diff
changeset
|
2616 |
and empty: "P {}" |
e0a4cb91a8a9
add induction rule for intersection-stable sigma-sets
hoelzl
parents:
49782
diff
changeset
|
2617 |
and compl: "\<And>A. A \<in> sigma_sets \<Omega> G \<Longrightarrow> P A \<Longrightarrow> P (\<Omega> - A)" |
e0a4cb91a8a9
add induction rule for intersection-stable sigma-sets
hoelzl
parents:
49782
diff
changeset
|
2618 |
and union: "\<And>A. disjoint_family A \<Longrightarrow> range A \<subseteq> sigma_sets \<Omega> G \<Longrightarrow> (\<And>i. P (A i)) \<Longrightarrow> P (\<Union>i::nat. A i)" |
e0a4cb91a8a9
add induction rule for intersection-stable sigma-sets
hoelzl
parents:
49782
diff
changeset
|
2619 |
shows "P A" |
e0a4cb91a8a9
add induction rule for intersection-stable sigma-sets
hoelzl
parents:
49782
diff
changeset
|
2620 |
proof - |
e0a4cb91a8a9
add induction rule for intersection-stable sigma-sets
hoelzl
parents:
49782
diff
changeset
|
2621 |
let ?D = "{ A \<in> sigma_sets \<Omega> G. P A }" |
e0a4cb91a8a9
add induction rule for intersection-stable sigma-sets
hoelzl
parents:
49782
diff
changeset
|
2622 |
interpret sigma_algebra \<Omega> "sigma_sets \<Omega> G" |
e0a4cb91a8a9
add induction rule for intersection-stable sigma-sets
hoelzl
parents:
49782
diff
changeset
|
2623 |
using closed by (rule sigma_algebra_sigma_sets) |
e0a4cb91a8a9
add induction rule for intersection-stable sigma-sets
hoelzl
parents:
49782
diff
changeset
|
2624 |
from compl[OF _ empty] closed have space: "P \<Omega>" by simp |
e0a4cb91a8a9
add induction rule for intersection-stable sigma-sets
hoelzl
parents:
49782
diff
changeset
|
2625 |
interpret dynkin_system \<Omega> ?D |
e0a4cb91a8a9
add induction rule for intersection-stable sigma-sets
hoelzl
parents:
49782
diff
changeset
|
2626 |
by default (auto dest: sets_into_space intro!: space compl union) |
e0a4cb91a8a9
add induction rule for intersection-stable sigma-sets
hoelzl
parents:
49782
diff
changeset
|
2627 |
have "sigma_sets \<Omega> G = ?D" |
e0a4cb91a8a9
add induction rule for intersection-stable sigma-sets
hoelzl
parents:
49782
diff
changeset
|
2628 |
by (rule dynkin_lemma) (auto simp: basic `Int_stable G`) |
e0a4cb91a8a9
add induction rule for intersection-stable sigma-sets
hoelzl
parents:
49782
diff
changeset
|
2629 |
with A show ?thesis by auto |
e0a4cb91a8a9
add induction rule for intersection-stable sigma-sets
hoelzl
parents:
49782
diff
changeset
|
2630 |
qed |
e0a4cb91a8a9
add induction rule for intersection-stable sigma-sets
hoelzl
parents:
49782
diff
changeset
|
2631 |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
diff
changeset
|
2632 |
end |