author | wenzelm |
Tue, 17 Jan 2017 13:59:10 +0100 | |
changeset 64911 | f0e07600de47 |
parent 64284 | f3b905b2eee2 |
child 66164 | 2d79288b042c |
permissions | -rw-r--r-- |
63627 | 1 |
(* Title: HOL/Analysis/Set_Integral.thy |
63329 | 2 |
Author: Jeremy Avigad (CMU), Johannes Hölzl (TUM), Luke Serafin (CMU) |
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HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
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|
3 |
Author: Sébastien Gouëzel sebastien.gouezel@univ-rennes1.fr |
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d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
4 |
|
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
5 |
Notation and useful facts for working with integrals over a set. |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
6 |
|
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
7 |
TODO: keep all these? Need unicode translations as well. |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
8 |
*) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
9 |
|
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
10 |
theory Set_Integral |
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979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
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|
11 |
imports Radon_Nikodym |
59092
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
12 |
begin |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
13 |
|
64284
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
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|
14 |
lemma bij_inv_eq_iff: "bij p \<Longrightarrow> x = inv p y \<longleftrightarrow> p x = y" (* COPIED FROM Permutations *) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
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|
15 |
using surj_f_inv_f[of p] by (auto simp add: bij_def) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
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16 |
|
64911 | 17 |
subsection \<open>Fun.thy\<close> |
64284
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
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|
18 |
|
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
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|
19 |
lemma inj_fn: |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
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|
20 |
fixes f::"'a \<Rightarrow> 'a" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
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|
21 |
assumes "inj f" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
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diff
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|
22 |
shows "inj (f^^n)" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
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|
23 |
proof (induction n) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
24 |
case (Suc n) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
25 |
have "inj (f o (f^^n))" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
26 |
using inj_comp[OF assms Suc.IH] by simp |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
27 |
then show "inj (f^^(Suc n))" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
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28 |
by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
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29 |
qed (auto) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
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30 |
|
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
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|
31 |
lemma surj_fn: |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
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diff
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|
32 |
fixes f::"'a \<Rightarrow> 'a" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
33 |
assumes "surj f" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
34 |
shows "surj (f^^n)" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
35 |
proof (induction n) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
36 |
case (Suc n) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
37 |
have "surj (f o (f^^n))" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
38 |
using assms Suc.IH by (simp add: comp_surj) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
39 |
then show "surj (f^^(Suc n))" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
40 |
by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
41 |
qed (auto) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
42 |
|
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
43 |
lemma bij_fn: |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
44 |
fixes f::"'a \<Rightarrow> 'a" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
45 |
assumes "bij f" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
46 |
shows "bij (f^^n)" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
47 |
by (rule bijI[OF inj_fn[OF bij_is_inj[OF assms]] surj_fn[OF bij_is_surj[OF assms]]]) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
48 |
|
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
49 |
lemma inv_fn_o_fn_is_id: |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
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diff
changeset
|
50 |
fixes f::"'a \<Rightarrow> 'a" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
51 |
assumes "bij f" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
52 |
shows "((inv f)^^n) o (f^^n) = (\<lambda>x. x)" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
53 |
proof - |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
54 |
have "((inv f)^^n)((f^^n) x) = x" for x n |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
55 |
proof (induction n) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
56 |
case (Suc n) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
57 |
have *: "(inv f) (f y) = y" for y |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
58 |
by (simp add: assms bij_is_inj) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
59 |
have "(inv f ^^ Suc n) ((f ^^ Suc n) x) = (inv f^^n) (inv f (f ((f^^n) x)))" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
60 |
by (simp add: funpow_swap1) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
61 |
also have "... = (inv f^^n) ((f^^n) x)" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
62 |
using * by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
63 |
also have "... = x" using Suc.IH by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
64 |
finally show ?case by simp |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
65 |
qed (auto) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
66 |
then show ?thesis unfolding o_def by blast |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
67 |
qed |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
68 |
|
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
69 |
lemma fn_o_inv_fn_is_id: |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
70 |
fixes f::"'a \<Rightarrow> 'a" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
71 |
assumes "bij f" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
72 |
shows "(f^^n) o ((inv f)^^n) = (\<lambda>x. x)" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
73 |
proof - |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
74 |
have "(f^^n) (((inv f)^^n) x) = x" for x n |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
75 |
proof (induction n) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
76 |
case (Suc n) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
77 |
have *: "f(inv f y) = y" for y |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
78 |
using assms by (meson bij_inv_eq_iff) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
79 |
have "(f ^^ Suc n) ((inv f ^^ Suc n) x) = (f^^n) (f (inv f ((inv f^^n) x)))" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
80 |
by (simp add: funpow_swap1) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
81 |
also have "... = (f^^n) ((inv f^^n) x)" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
82 |
using * by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
83 |
also have "... = x" using Suc.IH by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
84 |
finally show ?case by simp |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
85 |
qed (auto) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
86 |
then show ?thesis unfolding o_def by blast |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
87 |
qed |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
88 |
|
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
89 |
lemma inv_fn: |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
90 |
fixes f::"'a \<Rightarrow> 'a" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
91 |
assumes "bij f" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
92 |
shows "inv (f^^n) = ((inv f)^^n)" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
93 |
proof - |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
94 |
have "inv (f^^n) x = ((inv f)^^n) x" for x |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
95 |
apply (rule inv_into_f_eq, auto simp add: inj_fn[OF bij_is_inj[OF assms]]) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
96 |
using fn_o_inv_fn_is_id[OF assms, of n] by (metis comp_apply) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
97 |
then show ?thesis by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
98 |
qed |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
99 |
|
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
100 |
|
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
101 |
lemma mono_inv: |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
102 |
fixes f::"'a::linorder \<Rightarrow> 'b::linorder" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
103 |
assumes "mono f" "bij f" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
104 |
shows "mono (inv f)" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
105 |
proof |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
106 |
fix x y::'b assume "x \<le> y" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
107 |
then show "inv f x \<le> inv f y" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
108 |
by (metis (no_types, lifting) assms bij_is_surj eq_iff le_cases mono_def surj_f_inv_f) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
109 |
qed |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
110 |
|
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
111 |
lemma mono_bij_Inf: |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
112 |
fixes f :: "'a::complete_linorder \<Rightarrow> 'b::complete_linorder" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
113 |
assumes "mono f" "bij f" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
114 |
shows "f (Inf A) = Inf (f`A)" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
115 |
proof - |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
116 |
have "(inv f) (Inf (f`A)) \<le> Inf ((inv f)`(f`A))" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
117 |
using mono_Inf[OF mono_inv[OF assms], of "f`A"] by simp |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
118 |
then have "Inf (f`A) \<le> f (Inf ((inv f)`(f`A)))" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
119 |
by (metis (no_types, lifting) assms mono_def bij_inv_eq_iff) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
120 |
also have "... = f(Inf A)" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
121 |
using assms by (simp add: bij_is_inj) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
122 |
finally show ?thesis using mono_Inf[OF assms(1), of A] by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
123 |
qed |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
124 |
|
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
125 |
|
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
126 |
lemma Inf_nat_def1: |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
127 |
fixes K::"nat set" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
128 |
assumes "K \<noteq> {}" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
129 |
shows "Inf K \<in> K" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
130 |
by (auto simp add: Min_def Inf_nat_def) (meson LeastI assms bot.extremum_unique subsetI) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
131 |
|
64911 | 132 |
subsection \<open>Liminf-Limsup.thy\<close> |
64284
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
133 |
|
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
134 |
lemma limsup_shift: |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
135 |
"limsup (\<lambda>n. u (n+1)) = limsup u" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
136 |
proof - |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
137 |
have "(SUP m:{n+1..}. u m) = (SUP m:{n..}. u (m + 1))" for n |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
138 |
apply (rule SUP_eq) using Suc_le_D by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
139 |
then have a: "(INF n. SUP m:{n..}. u (m + 1)) = (INF n. (SUP m:{n+1..}. u m))" by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
140 |
have b: "(INF n. (SUP m:{n+1..}. u m)) = (INF n:{1..}. (SUP m:{n..}. u m))" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
141 |
apply (rule INF_eq) using Suc_le_D by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
142 |
have "(INF n:{1..}. v n) = (INF n. v n)" if "decseq v" for v::"nat \<Rightarrow> 'a" |
64911 | 143 |
apply (rule INF_eq) using \<open>decseq v\<close> decseq_Suc_iff by auto |
64284
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
144 |
moreover have "decseq (\<lambda>n. (SUP m:{n..}. u m))" by (simp add: SUP_subset_mono decseq_def) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
145 |
ultimately have c: "(INF n:{1..}. (SUP m:{n..}. u m)) = (INF n. (SUP m:{n..}. u m))" by simp |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
146 |
have "(INF n. SUPREMUM {n..} u) = (INF n. SUP m:{n..}. u (m + 1))" using a b c by simp |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
147 |
then show ?thesis by (auto cong: limsup_INF_SUP) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
148 |
qed |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
149 |
|
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
150 |
lemma limsup_shift_k: |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
151 |
"limsup (\<lambda>n. u (n+k)) = limsup u" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
152 |
proof (induction k) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
153 |
case (Suc k) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
154 |
have "limsup (\<lambda>n. u (n+k+1)) = limsup (\<lambda>n. u (n+k))" using limsup_shift[where ?u="\<lambda>n. u(n+k)"] by simp |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
155 |
then show ?case using Suc.IH by simp |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
156 |
qed (auto) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
157 |
|
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
158 |
lemma liminf_shift: |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
159 |
"liminf (\<lambda>n. u (n+1)) = liminf u" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
160 |
proof - |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
161 |
have "(INF m:{n+1..}. u m) = (INF m:{n..}. u (m + 1))" for n |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
162 |
apply (rule INF_eq) using Suc_le_D by (auto) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
163 |
then have a: "(SUP n. INF m:{n..}. u (m + 1)) = (SUP n. (INF m:{n+1..}. u m))" by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
164 |
have b: "(SUP n. (INF m:{n+1..}. u m)) = (SUP n:{1..}. (INF m:{n..}. u m))" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
165 |
apply (rule SUP_eq) using Suc_le_D by (auto) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
166 |
have "(SUP n:{1..}. v n) = (SUP n. v n)" if "incseq v" for v::"nat \<Rightarrow> 'a" |
64911 | 167 |
apply (rule SUP_eq) using \<open>incseq v\<close> incseq_Suc_iff by auto |
64284
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
168 |
moreover have "incseq (\<lambda>n. (INF m:{n..}. u m))" by (simp add: INF_superset_mono mono_def) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
169 |
ultimately have c: "(SUP n:{1..}. (INF m:{n..}. u m)) = (SUP n. (INF m:{n..}. u m))" by simp |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
170 |
have "(SUP n. INFIMUM {n..} u) = (SUP n. INF m:{n..}. u (m + 1))" using a b c by simp |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
171 |
then show ?thesis by (auto cong: liminf_SUP_INF) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
172 |
qed |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
173 |
|
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
174 |
lemma liminf_shift_k: |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
175 |
"liminf (\<lambda>n. u (n+k)) = liminf u" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
176 |
proof (induction k) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
177 |
case (Suc k) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
178 |
have "liminf (\<lambda>n. u (n+k+1)) = liminf (\<lambda>n. u (n+k))" using liminf_shift[where ?u="\<lambda>n. u(n+k)"] by simp |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
179 |
then show ?case using Suc.IH by simp |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
180 |
qed (auto) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
181 |
|
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
182 |
lemma Limsup_obtain: |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
183 |
fixes u::"_ \<Rightarrow> 'a :: complete_linorder" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
184 |
assumes "Limsup F u > c" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
185 |
shows "\<exists>i. u i > c" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
186 |
proof - |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
187 |
have "(INF P:{P. eventually P F}. SUP x:{x. P x}. u x) > c" using assms by (simp add: Limsup_def) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
188 |
then show ?thesis by (metis eventually_True mem_Collect_eq less_INF_D less_SUP_iff) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
189 |
qed |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
190 |
|
64911 | 191 |
text \<open>The next lemma is extremely useful, as it often makes it possible to reduce statements |
192 |
about limsups to statements about limits.\<close> |
|
64284
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
193 |
|
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
194 |
lemma limsup_subseq_lim: |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
195 |
fixes u::"nat \<Rightarrow> 'a :: {complete_linorder, linorder_topology}" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
196 |
shows "\<exists>r. subseq r \<and> (u o r) \<longlonglongrightarrow> limsup u" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
197 |
proof (cases) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
198 |
assume "\<forall>n. \<exists>p>n. \<forall>m\<ge>p. u m \<le> u p" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
199 |
then have "\<exists>r. \<forall>n. (\<forall>m\<ge>r n. u m \<le> u (r n)) \<and> r n < r (Suc n)" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
200 |
by (intro dependent_nat_choice) (auto simp: conj_commute) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
201 |
then obtain r where "subseq r" and mono: "\<And>n m. r n \<le> m \<Longrightarrow> u m \<le> u (r n)" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
202 |
by (auto simp: subseq_Suc_iff) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
203 |
define umax where "umax = (\<lambda>n. (SUP m:{n..}. u m))" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
204 |
have "decseq umax" unfolding umax_def by (simp add: SUP_subset_mono antimono_def) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
205 |
then have "umax \<longlonglongrightarrow> limsup u" unfolding umax_def by (metis LIMSEQ_INF limsup_INF_SUP) |
64911 | 206 |
then have *: "(umax o r) \<longlonglongrightarrow> limsup u" by (simp add: LIMSEQ_subseq_LIMSEQ \<open>subseq r\<close>) |
64284
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
207 |
have "\<And>n. umax(r n) = u(r n)" unfolding umax_def using mono |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
208 |
by (metis SUP_le_iff antisym atLeast_def mem_Collect_eq order_refl) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
209 |
then have "umax o r = u o r" unfolding o_def by simp |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
210 |
then have "(u o r) \<longlonglongrightarrow> limsup u" using * by simp |
64911 | 211 |
then show ?thesis using \<open>subseq r\<close> by blast |
64284
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
212 |
next |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
213 |
assume "\<not> (\<forall>n. \<exists>p>n. (\<forall>m\<ge>p. u m \<le> u p))" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
214 |
then obtain N where N: "\<And>p. p > N \<Longrightarrow> \<exists>m>p. u p < u m" by (force simp: not_le le_less) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
215 |
have "\<exists>r. \<forall>n. N < r n \<and> r n < r (Suc n) \<and> (\<forall>i\<in> {N<..r (Suc n)}. u i \<le> u (r (Suc n)))" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
216 |
proof (rule dependent_nat_choice) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
217 |
fix x assume "N < x" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
218 |
then have a: "finite {N<..x}" "{N<..x} \<noteq> {}" by simp_all |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
219 |
have "Max {u i |i. i \<in> {N<..x}} \<in> {u i |i. i \<in> {N<..x}}" apply (rule Max_in) using a by (auto) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
220 |
then obtain p where "p \<in> {N<..x}" and upmax: "u p = Max{u i |i. i \<in> {N<..x}}" by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
221 |
define U where "U = {m. m > p \<and> u p < u m}" |
64911 | 222 |
have "U \<noteq> {}" unfolding U_def using N[of p] \<open>p \<in> {N<..x}\<close> by auto |
64284
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
223 |
define y where "y = Inf U" |
64911 | 224 |
then have "y \<in> U" using \<open>U \<noteq> {}\<close> by (simp add: Inf_nat_def1) |
64284
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
225 |
have a: "\<And>i. i \<in> {N<..x} \<Longrightarrow> u i \<le> u p" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
226 |
proof - |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
227 |
fix i assume "i \<in> {N<..x}" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
228 |
then have "u i \<in> {u i |i. i \<in> {N<..x}}" by blast |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
229 |
then show "u i \<le> u p" using upmax by simp |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
230 |
qed |
64911 | 231 |
moreover have "u p < u y" using \<open>y \<in> U\<close> U_def by auto |
64284
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
232 |
ultimately have "y \<notin> {N<..x}" using not_le by blast |
64911 | 233 |
moreover have "y > N" using \<open>y \<in> U\<close> U_def \<open>p \<in> {N<..x}\<close> by auto |
64284
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
234 |
ultimately have "y > x" by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
235 |
|
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
236 |
have "\<And>i. i \<in> {N<..y} \<Longrightarrow> u i \<le> u y" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
237 |
proof - |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
238 |
fix i assume "i \<in> {N<..y}" show "u i \<le> u y" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
239 |
proof (cases) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
240 |
assume "i = y" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
241 |
then show ?thesis by simp |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
242 |
next |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
243 |
assume "\<not>(i=y)" |
64911 | 244 |
then have i:"i \<in> {N<..<y}" using \<open>i \<in> {N<..y}\<close> by simp |
64284
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
245 |
have "u i \<le> u p" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
246 |
proof (cases) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
247 |
assume "i \<le> x" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
248 |
then have "i \<in> {N<..x}" using i by simp |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
249 |
then show ?thesis using a by simp |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
250 |
next |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
251 |
assume "\<not>(i \<le> x)" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
252 |
then have "i > x" by simp |
64911 | 253 |
then have *: "i > p" using \<open>p \<in> {N<..x}\<close> by simp |
64284
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
254 |
have "i < Inf U" using i y_def by simp |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
255 |
then have "i \<notin> U" using Inf_nat_def not_less_Least by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
256 |
then show ?thesis using U_def * by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
257 |
qed |
64911 | 258 |
then show "u i \<le> u y" using \<open>u p < u y\<close> by auto |
64284
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
259 |
qed |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
260 |
qed |
64911 | 261 |
then have "N < y \<and> x < y \<and> (\<forall>i\<in>{N<..y}. u i \<le> u y)" using \<open>y > x\<close> \<open>y > N\<close> by auto |
64284
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
262 |
then show "\<exists>y>N. x < y \<and> (\<forall>i\<in>{N<..y}. u i \<le> u y)" by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
263 |
qed (auto) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
264 |
then obtain r where r: "\<forall>n. N < r n \<and> r n < r (Suc n) \<and> (\<forall>i\<in> {N<..r (Suc n)}. u i \<le> u (r (Suc n)))" by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
265 |
have "subseq r" using r by (auto simp: subseq_Suc_iff) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
266 |
have "incseq (u o r)" unfolding o_def using r by (simp add: incseq_SucI order.strict_implies_order) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
267 |
then have "(u o r) \<longlonglongrightarrow> (SUP n. (u o r) n)" using LIMSEQ_SUP by blast |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
268 |
then have "limsup (u o r) = (SUP n. (u o r) n)" by (simp add: lim_imp_Limsup) |
64911 | 269 |
moreover have "limsup (u o r) \<le> limsup u" using \<open>subseq r\<close> by (simp add: limsup_subseq_mono) |
64284
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
270 |
ultimately have "(SUP n. (u o r) n) \<le> limsup u" by simp |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
271 |
|
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
272 |
{ |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
273 |
fix i assume i: "i \<in> {N<..}" |
64911 | 274 |
obtain n where "i < r (Suc n)" using \<open>subseq r\<close> using Suc_le_eq seq_suble by blast |
64284
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
275 |
then have "i \<in> {N<..r(Suc n)}" using i by simp |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
276 |
then have "u i \<le> u (r(Suc n))" using r by simp |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
277 |
then have "u i \<le> (SUP n. (u o r) n)" unfolding o_def by (meson SUP_upper2 UNIV_I) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
278 |
} |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
279 |
then have "(SUP i:{N<..}. u i) \<le> (SUP n. (u o r) n)" using SUP_least by blast |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
280 |
then have "limsup u \<le> (SUP n. (u o r) n)" unfolding Limsup_def |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
281 |
by (metis (mono_tags, lifting) INF_lower2 atLeast_Suc_greaterThan atLeast_def eventually_ge_at_top mem_Collect_eq) |
64911 | 282 |
then have "limsup u = (SUP n. (u o r) n)" using \<open>(SUP n. (u o r) n) \<le> limsup u\<close> by simp |
283 |
then have "(u o r) \<longlonglongrightarrow> limsup u" using \<open>(u o r) \<longlonglongrightarrow> (SUP n. (u o r) n)\<close> by simp |
|
284 |
then show ?thesis using \<open>subseq r\<close> by auto |
|
64284
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
285 |
qed |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
286 |
|
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
287 |
lemma liminf_subseq_lim: |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
288 |
fixes u::"nat \<Rightarrow> 'a :: {complete_linorder, linorder_topology}" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
289 |
shows "\<exists>r. subseq r \<and> (u o r) \<longlonglongrightarrow> liminf u" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
290 |
proof (cases) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
291 |
assume "\<forall>n. \<exists>p>n. \<forall>m\<ge>p. u m \<ge> u p" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
292 |
then have "\<exists>r. \<forall>n. (\<forall>m\<ge>r n. u m \<ge> u (r n)) \<and> r n < r (Suc n)" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
293 |
by (intro dependent_nat_choice) (auto simp: conj_commute) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
294 |
then obtain r where "subseq r" and mono: "\<And>n m. r n \<le> m \<Longrightarrow> u m \<ge> u (r n)" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
295 |
by (auto simp: subseq_Suc_iff) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
296 |
define umin where "umin = (\<lambda>n. (INF m:{n..}. u m))" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
297 |
have "incseq umin" unfolding umin_def by (simp add: INF_superset_mono incseq_def) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
298 |
then have "umin \<longlonglongrightarrow> liminf u" unfolding umin_def by (metis LIMSEQ_SUP liminf_SUP_INF) |
64911 | 299 |
then have *: "(umin o r) \<longlonglongrightarrow> liminf u" by (simp add: LIMSEQ_subseq_LIMSEQ \<open>subseq r\<close>) |
64284
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
300 |
have "\<And>n. umin(r n) = u(r n)" unfolding umin_def using mono |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
301 |
by (metis le_INF_iff antisym atLeast_def mem_Collect_eq order_refl) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
302 |
then have "umin o r = u o r" unfolding o_def by simp |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
303 |
then have "(u o r) \<longlonglongrightarrow> liminf u" using * by simp |
64911 | 304 |
then show ?thesis using \<open>subseq r\<close> by blast |
64284
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
305 |
next |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
306 |
assume "\<not> (\<forall>n. \<exists>p>n. (\<forall>m\<ge>p. u m \<ge> u p))" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
307 |
then obtain N where N: "\<And>p. p > N \<Longrightarrow> \<exists>m>p. u p > u m" by (force simp: not_le le_less) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
308 |
have "\<exists>r. \<forall>n. N < r n \<and> r n < r (Suc n) \<and> (\<forall>i\<in> {N<..r (Suc n)}. u i \<ge> u (r (Suc n)))" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
309 |
proof (rule dependent_nat_choice) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
310 |
fix x assume "N < x" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
311 |
then have a: "finite {N<..x}" "{N<..x} \<noteq> {}" by simp_all |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
312 |
have "Min {u i |i. i \<in> {N<..x}} \<in> {u i |i. i \<in> {N<..x}}" apply (rule Min_in) using a by (auto) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
313 |
then obtain p where "p \<in> {N<..x}" and upmin: "u p = Min{u i |i. i \<in> {N<..x}}" by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
314 |
define U where "U = {m. m > p \<and> u p > u m}" |
64911 | 315 |
have "U \<noteq> {}" unfolding U_def using N[of p] \<open>p \<in> {N<..x}\<close> by auto |
64284
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
316 |
define y where "y = Inf U" |
64911 | 317 |
then have "y \<in> U" using \<open>U \<noteq> {}\<close> by (simp add: Inf_nat_def1) |
64284
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
318 |
have a: "\<And>i. i \<in> {N<..x} \<Longrightarrow> u i \<ge> u p" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
319 |
proof - |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
320 |
fix i assume "i \<in> {N<..x}" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
321 |
then have "u i \<in> {u i |i. i \<in> {N<..x}}" by blast |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
322 |
then show "u i \<ge> u p" using upmin by simp |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
323 |
qed |
64911 | 324 |
moreover have "u p > u y" using \<open>y \<in> U\<close> U_def by auto |
64284
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
325 |
ultimately have "y \<notin> {N<..x}" using not_le by blast |
64911 | 326 |
moreover have "y > N" using \<open>y \<in> U\<close> U_def \<open>p \<in> {N<..x}\<close> by auto |
64284
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
327 |
ultimately have "y > x" by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
328 |
|
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
329 |
have "\<And>i. i \<in> {N<..y} \<Longrightarrow> u i \<ge> u y" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
330 |
proof - |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
331 |
fix i assume "i \<in> {N<..y}" show "u i \<ge> u y" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
332 |
proof (cases) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
333 |
assume "i = y" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
334 |
then show ?thesis by simp |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
335 |
next |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
336 |
assume "\<not>(i=y)" |
64911 | 337 |
then have i:"i \<in> {N<..<y}" using \<open>i \<in> {N<..y}\<close> by simp |
64284
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
338 |
have "u i \<ge> u p" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
339 |
proof (cases) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
340 |
assume "i \<le> x" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
341 |
then have "i \<in> {N<..x}" using i by simp |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
342 |
then show ?thesis using a by simp |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
343 |
next |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
344 |
assume "\<not>(i \<le> x)" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
345 |
then have "i > x" by simp |
64911 | 346 |
then have *: "i > p" using \<open>p \<in> {N<..x}\<close> by simp |
64284
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
347 |
have "i < Inf U" using i y_def by simp |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
348 |
then have "i \<notin> U" using Inf_nat_def not_less_Least by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
349 |
then show ?thesis using U_def * by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
350 |
qed |
64911 | 351 |
then show "u i \<ge> u y" using \<open>u p > u y\<close> by auto |
64284
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
352 |
qed |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
353 |
qed |
64911 | 354 |
then have "N < y \<and> x < y \<and> (\<forall>i\<in>{N<..y}. u i \<ge> u y)" using \<open>y > x\<close> \<open>y > N\<close> by auto |
64284
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
355 |
then show "\<exists>y>N. x < y \<and> (\<forall>i\<in>{N<..y}. u i \<ge> u y)" by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
356 |
qed (auto) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
357 |
then obtain r where r: "\<forall>n. N < r n \<and> r n < r (Suc n) \<and> (\<forall>i\<in> {N<..r (Suc n)}. u i \<ge> u (r (Suc n)))" by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
358 |
have "subseq r" using r by (auto simp: subseq_Suc_iff) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
359 |
have "decseq (u o r)" unfolding o_def using r by (simp add: decseq_SucI order.strict_implies_order) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
360 |
then have "(u o r) \<longlonglongrightarrow> (INF n. (u o r) n)" using LIMSEQ_INF by blast |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
361 |
then have "liminf (u o r) = (INF n. (u o r) n)" by (simp add: lim_imp_Liminf) |
64911 | 362 |
moreover have "liminf (u o r) \<ge> liminf u" using \<open>subseq r\<close> by (simp add: liminf_subseq_mono) |
64284
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
363 |
ultimately have "(INF n. (u o r) n) \<ge> liminf u" by simp |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
364 |
|
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
365 |
{ |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
366 |
fix i assume i: "i \<in> {N<..}" |
64911 | 367 |
obtain n where "i < r (Suc n)" using \<open>subseq r\<close> using Suc_le_eq seq_suble by blast |
64284
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
368 |
then have "i \<in> {N<..r(Suc n)}" using i by simp |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
369 |
then have "u i \<ge> u (r(Suc n))" using r by simp |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
370 |
then have "u i \<ge> (INF n. (u o r) n)" unfolding o_def by (meson INF_lower2 UNIV_I) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
371 |
} |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
372 |
then have "(INF i:{N<..}. u i) \<ge> (INF n. (u o r) n)" using INF_greatest by blast |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
373 |
then have "liminf u \<ge> (INF n. (u o r) n)" unfolding Liminf_def |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
374 |
by (metis (mono_tags, lifting) SUP_upper2 atLeast_Suc_greaterThan atLeast_def eventually_ge_at_top mem_Collect_eq) |
64911 | 375 |
then have "liminf u = (INF n. (u o r) n)" using \<open>(INF n. (u o r) n) \<ge> liminf u\<close> by simp |
376 |
then have "(u o r) \<longlonglongrightarrow> liminf u" using \<open>(u o r) \<longlonglongrightarrow> (INF n. (u o r) n)\<close> by simp |
|
377 |
then show ?thesis using \<open>subseq r\<close> by auto |
|
64284
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
378 |
qed |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
379 |
|
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
380 |
|
64911 | 381 |
subsection \<open>Extended-Real.thy\<close> |
64284
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
382 |
|
64911 | 383 |
text\<open>The proof of this one is copied from \verb+ereal_add_mono+.\<close> |
64284
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
384 |
lemma ereal_add_strict_mono2: |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
385 |
fixes a b c d :: ereal |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
386 |
assumes "a < b" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
387 |
and "c < d" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
388 |
shows "a + c < b + d" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
389 |
using assms |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
390 |
apply (cases a) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
391 |
apply (cases rule: ereal3_cases[of b c d], auto) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
392 |
apply (cases rule: ereal3_cases[of b c d], auto) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
393 |
done |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
394 |
|
64911 | 395 |
text \<open>The next ones are analogues of \verb+mult_mono+ and \verb+mult_mono'+ in ereal.\<close> |
64284
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
396 |
|
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
397 |
lemma ereal_mult_mono: |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
398 |
fixes a b c d::ereal |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
399 |
assumes "b \<ge> 0" "c \<ge> 0" "a \<le> b" "c \<le> d" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
400 |
shows "a * c \<le> b * d" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
401 |
by (metis ereal_mult_right_mono mult.commute order_trans assms) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
402 |
|
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
403 |
lemma ereal_mult_mono': |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
404 |
fixes a b c d::ereal |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
405 |
assumes "a \<ge> 0" "c \<ge> 0" "a \<le> b" "c \<le> d" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
406 |
shows "a * c \<le> b * d" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
407 |
by (metis ereal_mult_right_mono mult.commute order_trans assms) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
408 |
|
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
409 |
lemma ereal_mult_mono_strict: |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
410 |
fixes a b c d::ereal |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
411 |
assumes "b > 0" "c > 0" "a < b" "c < d" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
412 |
shows "a * c < b * d" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
413 |
proof - |
64911 | 414 |
have "c < \<infinity>" using \<open>c < d\<close> by auto |
64284
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
415 |
then have "a * c < b * c" by (metis ereal_mult_strict_left_mono[OF assms(3) assms(2)] mult.commute) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
416 |
moreover have "b * c \<le> b * d" using assms(2) assms(4) by (simp add: assms(1) ereal_mult_left_mono less_imp_le) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
417 |
ultimately show ?thesis by simp |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
418 |
qed |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
419 |
|
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
420 |
lemma ereal_mult_mono_strict': |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
421 |
fixes a b c d::ereal |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
422 |
assumes "a > 0" "c > 0" "a < b" "c < d" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
423 |
shows "a * c < b * d" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
424 |
apply (rule ereal_mult_mono_strict, auto simp add: assms) using assms by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
425 |
|
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
426 |
lemma ereal_abs_add: |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
427 |
fixes a b::ereal |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
428 |
shows "abs(a+b) \<le> abs a + abs b" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
429 |
by (cases rule: ereal2_cases[of a b]) (auto) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
430 |
|
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
431 |
lemma ereal_abs_diff: |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
432 |
fixes a b::ereal |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
433 |
shows "abs(a-b) \<le> abs a + abs b" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
434 |
by (cases rule: ereal2_cases[of a b]) (auto) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
435 |
|
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
436 |
lemma sum_constant_ereal: |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
437 |
fixes a::ereal |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
438 |
shows "(\<Sum>i\<in>I. a) = a * card I" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
439 |
apply (cases "finite I", induct set: finite, simp_all) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
440 |
apply (cases a, auto, metis (no_types, hide_lams) add.commute mult.commute semiring_normalization_rules(3)) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
441 |
done |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
442 |
|
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
443 |
lemma real_lim_then_eventually_real: |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
444 |
assumes "(u \<longlongrightarrow> ereal l) F" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
445 |
shows "eventually (\<lambda>n. u n = ereal(real_of_ereal(u n))) F" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
446 |
proof - |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
447 |
have "ereal l \<in> {-\<infinity><..<(\<infinity>::ereal)}" by simp |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
448 |
moreover have "open {-\<infinity><..<(\<infinity>::ereal)}" by simp |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
449 |
ultimately have "eventually (\<lambda>n. u n \<in> {-\<infinity><..<(\<infinity>::ereal)}) F" using assms tendsto_def by blast |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
450 |
moreover have "\<And>x. x \<in> {-\<infinity><..<(\<infinity>::ereal)} \<Longrightarrow> x = ereal(real_of_ereal x)" using ereal_real by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
451 |
ultimately show ?thesis by (metis (mono_tags, lifting) eventually_mono) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
452 |
qed |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
453 |
|
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
454 |
lemma ereal_Inf_cmult: |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
455 |
assumes "c>(0::real)" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
456 |
shows "Inf {ereal c * x |x. P x} = ereal c * Inf {x. P x}" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
457 |
proof - |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
458 |
have "(\<lambda>x::ereal. c * x) (Inf {x::ereal. P x}) = Inf ((\<lambda>x::ereal. c * x)`{x::ereal. P x})" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
459 |
apply (rule mono_bij_Inf) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
460 |
apply (simp add: assms ereal_mult_left_mono less_imp_le mono_def) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
461 |
apply (rule bij_betw_byWitness[of _ "\<lambda>x. (x::ereal) / c"], auto simp add: assms ereal_mult_divide) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
462 |
using assms ereal_divide_eq apply auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
463 |
done |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
464 |
then show ?thesis by (simp only: setcompr_eq_image[symmetric]) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
465 |
qed |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
466 |
|
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
467 |
|
64911 | 468 |
subsubsection \<open>Continuity of addition\<close> |
64284
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
469 |
|
64911 | 470 |
text \<open>The next few lemmas remove an unnecessary assumption in \verb+tendsto_add_ereal+, culminating |
64284
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
471 |
in \verb+tendsto_add_ereal_general+ which essentially says that the addition |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
472 |
is continuous on ereal times ereal, except at $(-\infty, \infty)$ and $(\infty, -\infty)$. |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
473 |
It is much more convenient in many situations, see for instance the proof of |
64911 | 474 |
\verb+tendsto_sum_ereal+ below.\<close> |
64284
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
475 |
|
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
476 |
lemma tendsto_add_ereal_PInf: |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
477 |
fixes y :: ereal |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
478 |
assumes y: "y \<noteq> -\<infinity>" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
479 |
assumes f: "(f \<longlongrightarrow> \<infinity>) F" and g: "(g \<longlongrightarrow> y) F" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
480 |
shows "((\<lambda>x. f x + g x) \<longlongrightarrow> \<infinity>) F" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
481 |
proof - |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
482 |
have "\<exists>C. eventually (\<lambda>x. g x > ereal C) F" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
483 |
proof (cases y) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
484 |
case (real r) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
485 |
have "y > y-1" using y real by (simp add: ereal_between(1)) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
486 |
then have "eventually (\<lambda>x. g x > y - 1) F" using g y order_tendsto_iff by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
487 |
moreover have "y-1 = ereal(real_of_ereal(y-1))" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
488 |
by (metis real ereal_eq_1(1) ereal_minus(1) real_of_ereal.simps(1)) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
489 |
ultimately have "eventually (\<lambda>x. g x > ereal(real_of_ereal(y - 1))) F" by simp |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
490 |
then show ?thesis by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
491 |
next |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
492 |
case (PInf) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
493 |
have "eventually (\<lambda>x. g x > ereal 0) F" using g PInf by (simp add: tendsto_PInfty) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
494 |
then show ?thesis by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
495 |
qed (simp add: y) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
496 |
then obtain C::real where ge: "eventually (\<lambda>x. g x > ereal C) F" by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
497 |
|
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
498 |
{ |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
499 |
fix M::real |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
500 |
have "eventually (\<lambda>x. f x > ereal(M - C)) F" using f by (simp add: tendsto_PInfty) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
501 |
then have "eventually (\<lambda>x. (f x > ereal (M-C)) \<and> (g x > ereal C)) F" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
502 |
by (auto simp add: ge eventually_conj_iff) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
503 |
moreover have "\<And>x. ((f x > ereal (M-C)) \<and> (g x > ereal C)) \<Longrightarrow> (f x + g x > ereal M)" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
504 |
using ereal_add_strict_mono2 by fastforce |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
505 |
ultimately have "eventually (\<lambda>x. f x + g x > ereal M) F" using eventually_mono by force |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
506 |
} |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
507 |
then show ?thesis by (simp add: tendsto_PInfty) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
508 |
qed |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
509 |
|
64911 | 510 |
text\<open>One would like to deduce the next lemma from the previous one, but the fact |
64284
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
511 |
that $-(x+y)$ is in general different from $(-x) + (-y)$ in ereal creates difficulties, |
64911 | 512 |
so it is more efficient to copy the previous proof.\<close> |
64284
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
513 |
|
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
514 |
lemma tendsto_add_ereal_MInf: |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
515 |
fixes y :: ereal |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
516 |
assumes y: "y \<noteq> \<infinity>" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
517 |
assumes f: "(f \<longlongrightarrow> -\<infinity>) F" and g: "(g \<longlongrightarrow> y) F" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
518 |
shows "((\<lambda>x. f x + g x) \<longlongrightarrow> -\<infinity>) F" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
519 |
proof - |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
520 |
have "\<exists>C. eventually (\<lambda>x. g x < ereal C) F" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
521 |
proof (cases y) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
522 |
case (real r) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
523 |
have "y < y+1" using y real by (simp add: ereal_between(1)) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
524 |
then have "eventually (\<lambda>x. g x < y + 1) F" using g y order_tendsto_iff by force |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
525 |
moreover have "y+1 = ereal(real_of_ereal (y+1))" by (simp add: real) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
526 |
ultimately have "eventually (\<lambda>x. g x < ereal(real_of_ereal(y + 1))) F" by simp |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
527 |
then show ?thesis by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
528 |
next |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
529 |
case (MInf) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
530 |
have "eventually (\<lambda>x. g x < ereal 0) F" using g MInf by (simp add: tendsto_MInfty) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
531 |
then show ?thesis by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
532 |
qed (simp add: y) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
533 |
then obtain C::real where ge: "eventually (\<lambda>x. g x < ereal C) F" by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
534 |
|
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
535 |
{ |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
536 |
fix M::real |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
537 |
have "eventually (\<lambda>x. f x < ereal(M - C)) F" using f by (simp add: tendsto_MInfty) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
538 |
then have "eventually (\<lambda>x. (f x < ereal (M- C)) \<and> (g x < ereal C)) F" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
539 |
by (auto simp add: ge eventually_conj_iff) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
540 |
moreover have "\<And>x. ((f x < ereal (M-C)) \<and> (g x < ereal C)) \<Longrightarrow> (f x + g x < ereal M)" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
541 |
using ereal_add_strict_mono2 by fastforce |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
542 |
ultimately have "eventually (\<lambda>x. f x + g x < ereal M) F" using eventually_mono by force |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
543 |
} |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
544 |
then show ?thesis by (simp add: tendsto_MInfty) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
545 |
qed |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
546 |
|
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
547 |
lemma tendsto_add_ereal_general1: |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
548 |
fixes x y :: ereal |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
549 |
assumes y: "\<bar>y\<bar> \<noteq> \<infinity>" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
550 |
assumes f: "(f \<longlongrightarrow> x) F" and g: "(g \<longlongrightarrow> y) F" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
551 |
shows "((\<lambda>x. f x + g x) \<longlongrightarrow> x + y) F" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
552 |
proof (cases x) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
553 |
case (real r) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
554 |
have a: "\<bar>x\<bar> \<noteq> \<infinity>" by (simp add: real) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
555 |
show ?thesis by (rule tendsto_add_ereal[OF a, OF y, OF f, OF g]) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
556 |
next |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
557 |
case PInf |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
558 |
then show ?thesis using tendsto_add_ereal_PInf assms by force |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
559 |
next |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
560 |
case MInf |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
561 |
then show ?thesis using tendsto_add_ereal_MInf assms |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
562 |
by (metis abs_ereal.simps(3) ereal_MInfty_eq_plus) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
563 |
qed |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
564 |
|
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
565 |
lemma tendsto_add_ereal_general2: |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
566 |
fixes x y :: ereal |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
567 |
assumes x: "\<bar>x\<bar> \<noteq> \<infinity>" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
568 |
and f: "(f \<longlongrightarrow> x) F" and g: "(g \<longlongrightarrow> y) F" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
569 |
shows "((\<lambda>x. f x + g x) \<longlongrightarrow> x + y) F" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
570 |
proof - |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
571 |
have "((\<lambda>x. g x + f x) \<longlongrightarrow> x + y) F" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
572 |
using tendsto_add_ereal_general1[OF x, OF g, OF f] add.commute[of "y", of "x"] by simp |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
573 |
moreover have "\<And>x. g x + f x = f x + g x" using add.commute by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
574 |
ultimately show ?thesis by simp |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
575 |
qed |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
576 |
|
64911 | 577 |
text \<open>The next lemma says that the addition is continuous on ereal, except at |
578 |
the pairs $(-\infty, \infty)$ and $(\infty, -\infty)$.\<close> |
|
64284
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
579 |
|
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
580 |
lemma tendsto_add_ereal_general [tendsto_intros]: |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
581 |
fixes x y :: ereal |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
582 |
assumes "\<not>((x=\<infinity> \<and> y=-\<infinity>) \<or> (x=-\<infinity> \<and> y=\<infinity>))" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
583 |
and f: "(f \<longlongrightarrow> x) F" and g: "(g \<longlongrightarrow> y) F" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
584 |
shows "((\<lambda>x. f x + g x) \<longlongrightarrow> x + y) F" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
585 |
proof (cases x) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
586 |
case (real r) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
587 |
show ?thesis |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
588 |
apply (rule tendsto_add_ereal_general2) using real assms by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
589 |
next |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
590 |
case (PInf) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
591 |
then have "y \<noteq> -\<infinity>" using assms by simp |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
592 |
then show ?thesis using tendsto_add_ereal_PInf PInf assms by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
593 |
next |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
594 |
case (MInf) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
595 |
then have "y \<noteq> \<infinity>" using assms by simp |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
596 |
then show ?thesis using tendsto_add_ereal_MInf MInf f g by (metis ereal_MInfty_eq_plus) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
597 |
qed |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
598 |
|
64911 | 599 |
subsubsection \<open>Continuity of multiplication\<close> |
64284
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
600 |
|
64911 | 601 |
text \<open>In the same way as for addition, we prove that the multiplication is continuous on |
64284
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
602 |
ereal times ereal, except at $(\infty, 0)$ and $(-\infty, 0)$ and $(0, \infty)$ and $(0, -\infty)$, |
64911 | 603 |
starting with specific situations.\<close> |
64284
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
604 |
|
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
605 |
lemma tendsto_mult_real_ereal: |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
606 |
assumes "(u \<longlongrightarrow> ereal l) F" "(v \<longlongrightarrow> ereal m) F" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
607 |
shows "((\<lambda>n. u n * v n) \<longlongrightarrow> ereal l * ereal m) F" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
608 |
proof - |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
609 |
have ureal: "eventually (\<lambda>n. u n = ereal(real_of_ereal(u n))) F" by (rule real_lim_then_eventually_real[OF assms(1)]) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
610 |
then have "((\<lambda>n. ereal(real_of_ereal(u n))) \<longlongrightarrow> ereal l) F" using assms by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
611 |
then have limu: "((\<lambda>n. real_of_ereal(u n)) \<longlongrightarrow> l) F" by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
612 |
have vreal: "eventually (\<lambda>n. v n = ereal(real_of_ereal(v n))) F" by (rule real_lim_then_eventually_real[OF assms(2)]) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
613 |
then have "((\<lambda>n. ereal(real_of_ereal(v n))) \<longlongrightarrow> ereal m) F" using assms by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
614 |
then have limv: "((\<lambda>n. real_of_ereal(v n)) \<longlongrightarrow> m) F" by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
615 |
|
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
616 |
{ |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
617 |
fix n assume "u n = ereal(real_of_ereal(u n))" "v n = ereal(real_of_ereal(v n))" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
618 |
then have "ereal(real_of_ereal(u n) * real_of_ereal(v n)) = u n * v n" by (metis times_ereal.simps(1)) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
619 |
} |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
620 |
then have *: "eventually (\<lambda>n. ereal(real_of_ereal(u n) * real_of_ereal(v n)) = u n * v n) F" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
621 |
using eventually_elim2[OF ureal vreal] by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
622 |
|
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
623 |
have "((\<lambda>n. real_of_ereal(u n) * real_of_ereal(v n)) \<longlongrightarrow> l * m) F" using tendsto_mult[OF limu limv] by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
624 |
then have "((\<lambda>n. ereal(real_of_ereal(u n)) * real_of_ereal(v n)) \<longlongrightarrow> ereal(l * m)) F" by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
625 |
then show ?thesis using * filterlim_cong by fastforce |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
626 |
qed |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
627 |
|
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
628 |
lemma tendsto_mult_ereal_PInf: |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
629 |
fixes f g::"_ \<Rightarrow> ereal" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
630 |
assumes "(f \<longlongrightarrow> l) F" "l>0" "(g \<longlongrightarrow> \<infinity>) F" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
631 |
shows "((\<lambda>x. f x * g x) \<longlongrightarrow> \<infinity>) F" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
632 |
proof - |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
633 |
obtain a::real where "0 < ereal a" "a < l" using assms(2) using ereal_dense2 by blast |
64911 | 634 |
have *: "eventually (\<lambda>x. f x > a) F" using \<open>a < l\<close> assms(1) by (simp add: order_tendsto_iff) |
64284
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
635 |
{ |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
636 |
fix K::real |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
637 |
define M where "M = max K 1" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
638 |
then have "M > 0" by simp |
64911 | 639 |
then have "ereal(M/a) > 0" using \<open>ereal a > 0\<close> by simp |
64284
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
640 |
then have "\<And>x. ((f x > a) \<and> (g x > M/a)) \<Longrightarrow> (f x * g x > ereal a * ereal(M/a))" |
64911 | 641 |
using ereal_mult_mono_strict'[where ?c = "M/a", OF \<open>0 < ereal a\<close>] by auto |
642 |
moreover have "ereal a * ereal(M/a) = M" using \<open>ereal a > 0\<close> by simp |
|
64284
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
643 |
ultimately have "\<And>x. ((f x > a) \<and> (g x > M/a)) \<Longrightarrow> (f x * g x > M)" by simp |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
644 |
moreover have "M \<ge> K" unfolding M_def by simp |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
645 |
ultimately have Imp: "\<And>x. ((f x > a) \<and> (g x > M/a)) \<Longrightarrow> (f x * g x > K)" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
646 |
using ereal_less_eq(3) le_less_trans by blast |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
647 |
|
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
648 |
have "eventually (\<lambda>x. g x > M/a) F" using assms(3) by (simp add: tendsto_PInfty) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
649 |
then have "eventually (\<lambda>x. (f x > a) \<and> (g x > M/a)) F" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
650 |
using * by (auto simp add: eventually_conj_iff) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
651 |
then have "eventually (\<lambda>x. f x * g x > K) F" using eventually_mono Imp by force |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
652 |
} |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
653 |
then show ?thesis by (auto simp add: tendsto_PInfty) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
654 |
qed |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
655 |
|
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
656 |
lemma tendsto_mult_ereal_pos: |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
657 |
fixes f g::"_ \<Rightarrow> ereal" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
658 |
assumes "(f \<longlongrightarrow> l) F" "(g \<longlongrightarrow> m) F" "l>0" "m>0" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
659 |
shows "((\<lambda>x. f x * g x) \<longlongrightarrow> l * m) F" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
660 |
proof (cases) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
661 |
assume *: "l = \<infinity> \<or> m = \<infinity>" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
662 |
then show ?thesis |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
663 |
proof (cases) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
664 |
assume "m = \<infinity>" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
665 |
then show ?thesis using tendsto_mult_ereal_PInf assms by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
666 |
next |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
667 |
assume "\<not>(m = \<infinity>)" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
668 |
then have "l = \<infinity>" using * by simp |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
669 |
then have "((\<lambda>x. g x * f x) \<longlongrightarrow> l * m) F" using tendsto_mult_ereal_PInf assms by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
670 |
moreover have "\<And>x. g x * f x = f x * g x" using mult.commute by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
671 |
ultimately show ?thesis by simp |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
672 |
qed |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
673 |
next |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
674 |
assume "\<not>(l = \<infinity> \<or> m = \<infinity>)" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
675 |
then have "l < \<infinity>" "m < \<infinity>" by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
676 |
then obtain lr mr where "l = ereal lr" "m = ereal mr" |
64911 | 677 |
using \<open>l>0\<close> \<open>m>0\<close> by (metis ereal_cases ereal_less(6) not_less_iff_gr_or_eq) |
64284
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
678 |
then show ?thesis using tendsto_mult_real_ereal assms by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
679 |
qed |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
680 |
|
64911 | 681 |
text \<open>We reduce the general situation to the positive case by multiplying by suitable signs. |
64284
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
682 |
Unfortunately, as ereal is not a ring, all the neat sign lemmas are not available there. We |
64911 | 683 |
give the bare minimum we need.\<close> |
64284
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
684 |
|
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
685 |
lemma ereal_sgn_abs: |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
686 |
fixes l::ereal |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
687 |
shows "sgn(l) * l = abs(l)" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
688 |
apply (cases l) by (auto simp add: sgn_if ereal_less_uminus_reorder) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
689 |
|
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
690 |
lemma sgn_squared_ereal: |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
691 |
assumes "l \<noteq> (0::ereal)" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
692 |
shows "sgn(l) * sgn(l) = 1" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
693 |
apply (cases l) using assms by (auto simp add: one_ereal_def sgn_if) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
694 |
|
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
695 |
lemma tendsto_mult_ereal [tendsto_intros]: |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
696 |
fixes f g::"_ \<Rightarrow> ereal" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
697 |
assumes "(f \<longlongrightarrow> l) F" "(g \<longlongrightarrow> m) F" "\<not>((l=0 \<and> abs(m) = \<infinity>) \<or> (m=0 \<and> abs(l) = \<infinity>))" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
698 |
shows "((\<lambda>x. f x * g x) \<longlongrightarrow> l * m) F" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
699 |
proof (cases) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
700 |
assume "l=0 \<or> m=0" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
701 |
then have "abs(l) \<noteq> \<infinity>" "abs(m) \<noteq> \<infinity>" using assms(3) by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
702 |
then obtain lr mr where "l = ereal lr" "m = ereal mr" by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
703 |
then show ?thesis using tendsto_mult_real_ereal assms by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
704 |
next |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
705 |
have sgn_finite: "\<And>a::ereal. abs(sgn a) \<noteq> \<infinity>" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
706 |
by (metis MInfty_neq_ereal(2) PInfty_neq_ereal(2) abs_eq_infinity_cases ereal_times(1) ereal_times(3) ereal_uminus_eq_reorder sgn_ereal.elims) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
707 |
then have sgn_finite2: "\<And>a b::ereal. abs(sgn a * sgn b) \<noteq> \<infinity>" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
708 |
by (metis abs_eq_infinity_cases abs_ereal.simps(2) abs_ereal.simps(3) ereal_mult_eq_MInfty ereal_mult_eq_PInfty) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
709 |
assume "\<not>(l=0 \<or> m=0)" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
710 |
then have "l \<noteq> 0" "m \<noteq> 0" by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
711 |
then have "abs(l) > 0" "abs(m) > 0" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
712 |
by (metis abs_ereal_ge0 abs_ereal_less0 abs_ereal_pos ereal_uminus_uminus ereal_uminus_zero less_le not_less)+ |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
713 |
then have "sgn(l) * l > 0" "sgn(m) * m > 0" using ereal_sgn_abs by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
714 |
moreover have "((\<lambda>x. sgn(l) * f x) \<longlongrightarrow> (sgn(l) * l)) F" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
715 |
by (rule tendsto_cmult_ereal, auto simp add: sgn_finite assms(1)) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
716 |
moreover have "((\<lambda>x. sgn(m) * g x) \<longlongrightarrow> (sgn(m) * m)) F" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
717 |
by (rule tendsto_cmult_ereal, auto simp add: sgn_finite assms(2)) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
718 |
ultimately have *: "((\<lambda>x. (sgn(l) * f x) * (sgn(m) * g x)) \<longlongrightarrow> (sgn(l) * l) * (sgn(m) * m)) F" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
719 |
using tendsto_mult_ereal_pos by force |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
720 |
have "((\<lambda>x. (sgn(l) * sgn(m)) * ((sgn(l) * f x) * (sgn(m) * g x))) \<longlongrightarrow> (sgn(l) * sgn(m)) * ((sgn(l) * l) * (sgn(m) * m))) F" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
721 |
by (rule tendsto_cmult_ereal, auto simp add: sgn_finite2 *) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
722 |
moreover have "\<And>x. (sgn(l) * sgn(m)) * ((sgn(l) * f x) * (sgn(m) * g x)) = f x * g x" |
64911 | 723 |
by (metis mult.left_neutral sgn_squared_ereal[OF \<open>l \<noteq> 0\<close>] sgn_squared_ereal[OF \<open>m \<noteq> 0\<close>] mult.assoc mult.commute) |
64284
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
724 |
moreover have "(sgn(l) * sgn(m)) * ((sgn(l) * l) * (sgn(m) * m)) = l * m" |
64911 | 725 |
by (metis mult.left_neutral sgn_squared_ereal[OF \<open>l \<noteq> 0\<close>] sgn_squared_ereal[OF \<open>m \<noteq> 0\<close>] mult.assoc mult.commute) |
64284
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
726 |
ultimately show ?thesis by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
727 |
qed |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
728 |
|
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
729 |
lemma tendsto_cmult_ereal_general [tendsto_intros]: |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
730 |
fixes f::"_ \<Rightarrow> ereal" and c::ereal |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
731 |
assumes "(f \<longlongrightarrow> l) F" "\<not> (l=0 \<and> abs(c) = \<infinity>)" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
732 |
shows "((\<lambda>x. c * f x) \<longlongrightarrow> c * l) F" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
733 |
by (cases "c = 0", auto simp add: assms tendsto_mult_ereal) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
734 |
|
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
735 |
|
64911 | 736 |
subsubsection \<open>Continuity of division\<close> |
64284
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
737 |
|
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
738 |
lemma tendsto_inverse_ereal_PInf: |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
739 |
fixes u::"_ \<Rightarrow> ereal" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
740 |
assumes "(u \<longlongrightarrow> \<infinity>) F" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
741 |
shows "((\<lambda>x. 1/ u x) \<longlongrightarrow> 0) F" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
742 |
proof - |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
743 |
{ |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
744 |
fix e::real assume "e>0" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
745 |
have "1/e < \<infinity>" by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
746 |
then have "eventually (\<lambda>n. u n > 1/e) F" using assms(1) by (simp add: tendsto_PInfty) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
747 |
moreover |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
748 |
{ |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
749 |
fix z::ereal assume "z>1/e" |
64911 | 750 |
then have "z>0" using \<open>e>0\<close> using less_le_trans not_le by fastforce |
64284
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
751 |
then have "1/z \<ge> 0" by auto |
64911 | 752 |
moreover have "1/z < e" using \<open>e>0\<close> \<open>z>1/e\<close> |
64284
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
753 |
apply (cases z) apply auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
754 |
by (metis (mono_tags, hide_lams) less_ereal.simps(2) less_ereal.simps(4) divide_less_eq ereal_divide_less_pos ereal_less(4) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
755 |
ereal_less_eq(4) less_le_trans mult_eq_0_iff not_le not_one_less_zero times_ereal.simps(1)) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
756 |
ultimately have "1/z \<ge> 0" "1/z < e" by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
757 |
} |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
758 |
ultimately have "eventually (\<lambda>n. 1/u n<e) F" "eventually (\<lambda>n. 1/u n\<ge>0) F" by (auto simp add: eventually_mono) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
759 |
} note * = this |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
760 |
show ?thesis |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
761 |
proof (subst order_tendsto_iff, auto) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
762 |
fix a::ereal assume "a<0" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
763 |
then show "eventually (\<lambda>n. 1/u n > a) F" using *(2) eventually_mono less_le_trans linordered_field_no_ub by fastforce |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
764 |
next |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
765 |
fix a::ereal assume "a>0" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
766 |
then obtain e::real where "e>0" "a>e" using ereal_dense2 ereal_less(2) by blast |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
767 |
then have "eventually (\<lambda>n. 1/u n < e) F" using *(1) by auto |
64911 | 768 |
then show "eventually (\<lambda>n. 1/u n < a) F" using \<open>a>e\<close> by (metis (mono_tags, lifting) eventually_mono less_trans) |
64284
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
769 |
qed |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
770 |
qed |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
771 |
|
64911 | 772 |
text \<open>The next lemma deserves to exist by itself, as it is so common and useful.\<close> |
64284
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
773 |
|
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
774 |
lemma tendsto_inverse_real [tendsto_intros]: |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
775 |
fixes u::"_ \<Rightarrow> real" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
776 |
shows "(u \<longlongrightarrow> l) F \<Longrightarrow> l \<noteq> 0 \<Longrightarrow> ((\<lambda>x. 1/ u x) \<longlongrightarrow> 1/l) F" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
777 |
using tendsto_inverse unfolding inverse_eq_divide . |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
778 |
|
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
779 |
lemma tendsto_inverse_ereal [tendsto_intros]: |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
780 |
fixes u::"_ \<Rightarrow> ereal" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
781 |
assumes "(u \<longlongrightarrow> l) F" "l \<noteq> 0" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
782 |
shows "((\<lambda>x. 1/ u x) \<longlongrightarrow> 1/l) F" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
783 |
proof (cases l) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
784 |
case (real r) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
785 |
then have "r \<noteq> 0" using assms(2) by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
786 |
then have "1/l = ereal(1/r)" using real by (simp add: one_ereal_def) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
787 |
define v where "v = (\<lambda>n. real_of_ereal(u n))" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
788 |
have ureal: "eventually (\<lambda>n. u n = ereal(v n)) F" unfolding v_def using real_lim_then_eventually_real assms(1) real by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
789 |
then have "((\<lambda>n. ereal(v n)) \<longlongrightarrow> ereal r) F" using assms real v_def by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
790 |
then have *: "((\<lambda>n. v n) \<longlongrightarrow> r) F" by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
791 |
then have "((\<lambda>n. 1/v n) \<longlongrightarrow> 1/r) F" using \<open>r \<noteq> 0\<close> tendsto_inverse_real by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
792 |
then have lim: "((\<lambda>n. ereal(1/v n)) \<longlongrightarrow> 1/l) F" using \<open>1/l = ereal(1/r)\<close> by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
793 |
|
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
794 |
have "r \<in> -{0}" "open (-{(0::real)})" using \<open>r \<noteq> 0\<close> by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
795 |
then have "eventually (\<lambda>n. v n \<in> -{0}) F" using * using topological_tendstoD by blast |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
796 |
then have "eventually (\<lambda>n. v n \<noteq> 0) F" by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
797 |
moreover |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
798 |
{ |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
799 |
fix n assume H: "v n \<noteq> 0" "u n = ereal(v n)" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
800 |
then have "ereal(1/v n) = 1/ereal(v n)" by (simp add: one_ereal_def) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
801 |
then have "ereal(1/v n) = 1/u n" using H(2) by simp |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
802 |
} |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
803 |
ultimately have "eventually (\<lambda>n. ereal(1/v n) = 1/u n) F" using ureal eventually_elim2 by force |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
804 |
with Lim_transform_eventually[OF this lim] show ?thesis by simp |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
805 |
next |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
806 |
case (PInf) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
807 |
then have "1/l = 0" by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
808 |
then show ?thesis using tendsto_inverse_ereal_PInf assms PInf by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
809 |
next |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
810 |
case (MInf) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
811 |
then have "1/l = 0" by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
812 |
have "1/z = -1/ -z" if "z < 0" for z::ereal |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
813 |
apply (cases z) using divide_ereal_def \<open> z < 0 \<close> by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
814 |
moreover have "eventually (\<lambda>n. u n < 0) F" by (metis (no_types) MInf assms(1) tendsto_MInfty zero_ereal_def) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
815 |
ultimately have *: "eventually (\<lambda>n. -1/-u n = 1/u n) F" by (simp add: eventually_mono) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
816 |
|
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
817 |
define v where "v = (\<lambda>n. - u n)" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
818 |
have "(v \<longlongrightarrow> \<infinity>) F" unfolding v_def using MInf assms(1) tendsto_uminus_ereal by fastforce |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
819 |
then have "((\<lambda>n. 1/v n) \<longlongrightarrow> 0) F" using tendsto_inverse_ereal_PInf by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
820 |
then have "((\<lambda>n. -1/v n) \<longlongrightarrow> 0) F" using tendsto_uminus_ereal by fastforce |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
821 |
then show ?thesis unfolding v_def using Lim_transform_eventually[OF *] \<open> 1/l = 0 \<close> by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
822 |
qed |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
823 |
|
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
824 |
lemma tendsto_divide_ereal [tendsto_intros]: |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
825 |
fixes f g::"_ \<Rightarrow> ereal" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
826 |
assumes "(f \<longlongrightarrow> l) F" "(g \<longlongrightarrow> m) F" "m \<noteq> 0" "\<not>(abs(l) = \<infinity> \<and> abs(m) = \<infinity>)" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
827 |
shows "((\<lambda>x. f x / g x) \<longlongrightarrow> l / m) F" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
828 |
proof - |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
829 |
define h where "h = (\<lambda>x. 1/ g x)" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
830 |
have *: "(h \<longlongrightarrow> 1/m) F" unfolding h_def using assms(2) assms(3) tendsto_inverse_ereal by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
831 |
have "((\<lambda>x. f x * h x) \<longlongrightarrow> l * (1/m)) F" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
832 |
apply (rule tendsto_mult_ereal[OF assms(1) *]) using assms(3) assms(4) by (auto simp add: divide_ereal_def) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
833 |
moreover have "f x * h x = f x / g x" for x unfolding h_def by (simp add: divide_ereal_def) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
834 |
moreover have "l * (1/m) = l/m" by (simp add: divide_ereal_def) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
835 |
ultimately show ?thesis unfolding h_def using Lim_transform_eventually by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
836 |
qed |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
837 |
|
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
838 |
|
64911 | 839 |
subsubsection \<open>Further limits\<close> |
64284
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
840 |
|
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
841 |
lemma id_nat_ereal_tendsto_PInf [tendsto_intros]: |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
842 |
"(\<lambda> n::nat. real n) \<longlonglongrightarrow> \<infinity>" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
843 |
by (simp add: filterlim_real_sequentially tendsto_PInfty_eq_at_top) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
844 |
|
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
845 |
lemma tendsto_at_top_pseudo_inverse [tendsto_intros]: |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
846 |
fixes u::"nat \<Rightarrow> nat" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
847 |
assumes "LIM n sequentially. u n :> at_top" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
848 |
shows "LIM n sequentially. Inf {N. u N \<ge> n} :> at_top" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
849 |
proof - |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
850 |
{ |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
851 |
fix C::nat |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
852 |
define M where "M = Max {u n| n. n \<le> C}+1" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
853 |
{ |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
854 |
fix n assume "n \<ge> M" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
855 |
have "eventually (\<lambda>N. u N \<ge> n) sequentially" using assms |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
856 |
by (simp add: filterlim_at_top) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
857 |
then have *: "{N. u N \<ge> n} \<noteq> {}" by force |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
858 |
|
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
859 |
have "N > C" if "u N \<ge> n" for N |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
860 |
proof (rule ccontr) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
861 |
assume "\<not>(N > C)" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
862 |
have "u N \<le> Max {u n| n. n \<le> C}" |
64911 | 863 |
apply (rule Max_ge) using \<open>\<not>(N > C)\<close> by auto |
864 |
then show False using \<open>u N \<ge> n\<close> \<open>n \<ge> M\<close> unfolding M_def by auto |
|
64284
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
865 |
qed |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
866 |
then have **: "{N. u N \<ge> n} \<subseteq> {C..}" by fastforce |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
867 |
have "Inf {N. u N \<ge> n} \<ge> C" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
868 |
by (metis "*" "**" Inf_nat_def1 atLeast_iff subset_eq) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
869 |
} |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
870 |
then have "eventually (\<lambda>n. Inf {N. u N \<ge> n} \<ge> C) sequentially" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
871 |
using eventually_sequentially by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
872 |
} |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
873 |
then show ?thesis using filterlim_at_top by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
874 |
qed |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
875 |
|
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
876 |
lemma pseudo_inverse_finite_set: |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
877 |
fixes u::"nat \<Rightarrow> nat" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
878 |
assumes "LIM n sequentially. u n :> at_top" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
879 |
shows "finite {N. u N \<le> n}" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
880 |
proof - |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
881 |
fix n |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
882 |
have "eventually (\<lambda>N. u N \<ge> n+1) sequentially" using assms |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
883 |
by (simp add: filterlim_at_top) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
884 |
then obtain N1 where N1: "\<And>N. N \<ge> N1 \<Longrightarrow> u N \<ge> n + 1" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
885 |
using eventually_sequentially by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
886 |
have "{N. u N \<le> n} \<subseteq> {..<N1}" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
887 |
apply auto using N1 by (metis Suc_eq_plus1 not_less not_less_eq_eq) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
888 |
then show "finite {N. u N \<le> n}" by (simp add: finite_subset) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
889 |
qed |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
890 |
|
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
891 |
lemma tendsto_at_top_pseudo_inverse2 [tendsto_intros]: |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
892 |
fixes u::"nat \<Rightarrow> nat" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
893 |
assumes "LIM n sequentially. u n :> at_top" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
894 |
shows "LIM n sequentially. Max {N. u N \<le> n} :> at_top" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
895 |
proof - |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
896 |
{ |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
897 |
fix N0::nat |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
898 |
have "N0 \<le> Max {N. u N \<le> n}" if "n \<ge> u N0" for n |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
899 |
apply (rule Max.coboundedI) using pseudo_inverse_finite_set[OF assms] that by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
900 |
then have "eventually (\<lambda>n. N0 \<le> Max {N. u N \<le> n}) sequentially" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
901 |
using eventually_sequentially by blast |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
902 |
} |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
903 |
then show ?thesis using filterlim_at_top by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
904 |
qed |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
905 |
|
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
906 |
lemma ereal_truncation_top [tendsto_intros]: |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
907 |
fixes x::ereal |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
908 |
shows "(\<lambda>n::nat. min x n) \<longlonglongrightarrow> x" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
909 |
proof (cases x) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
910 |
case (real r) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
911 |
then obtain K::nat where "K>0" "K > abs(r)" using reals_Archimedean2 gr0I by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
912 |
then have "min x n = x" if "n \<ge> K" for n apply (subst real, subst real, auto) using that eq_iff by fastforce |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
913 |
then have "eventually (\<lambda>n. min x n = x) sequentially" using eventually_at_top_linorder by blast |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
914 |
then show ?thesis by (simp add: Lim_eventually) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
915 |
next |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
916 |
case (PInf) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
917 |
then have "min x n = n" for n::nat by (auto simp add: min_def) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
918 |
then show ?thesis using id_nat_ereal_tendsto_PInf PInf by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
919 |
next |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
920 |
case (MInf) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
921 |
then have "min x n = x" for n::nat by (auto simp add: min_def) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
922 |
then show ?thesis by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
923 |
qed |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
924 |
|
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
925 |
lemma ereal_truncation_real_top [tendsto_intros]: |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
926 |
fixes x::ereal |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
927 |
assumes "x \<noteq> - \<infinity>" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
928 |
shows "(\<lambda>n::nat. real_of_ereal(min x n)) \<longlonglongrightarrow> x" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
929 |
proof (cases x) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
930 |
case (real r) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
931 |
then obtain K::nat where "K>0" "K > abs(r)" using reals_Archimedean2 gr0I by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
932 |
then have "min x n = x" if "n \<ge> K" for n apply (subst real, subst real, auto) using that eq_iff by fastforce |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
933 |
then have "real_of_ereal(min x n) = r" if "n \<ge> K" for n using real that by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
934 |
then have "eventually (\<lambda>n. real_of_ereal(min x n) = r) sequentially" using eventually_at_top_linorder by blast |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
935 |
then have "(\<lambda>n. real_of_ereal(min x n)) \<longlonglongrightarrow> r" by (simp add: Lim_eventually) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
936 |
then show ?thesis using real by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
937 |
next |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
938 |
case (PInf) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
939 |
then have "real_of_ereal(min x n) = n" for n::nat by (auto simp add: min_def) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
940 |
then show ?thesis using id_nat_ereal_tendsto_PInf PInf by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
941 |
qed (simp add: assms) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
942 |
|
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
943 |
lemma ereal_truncation_bottom [tendsto_intros]: |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
944 |
fixes x::ereal |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
945 |
shows "(\<lambda>n::nat. max x (- real n)) \<longlonglongrightarrow> x" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
946 |
proof (cases x) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
947 |
case (real r) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
948 |
then obtain K::nat where "K>0" "K > abs(r)" using reals_Archimedean2 gr0I by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
949 |
then have "max x (-real n) = x" if "n \<ge> K" for n apply (subst real, subst real, auto) using that eq_iff by fastforce |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
950 |
then have "eventually (\<lambda>n. max x (-real n) = x) sequentially" using eventually_at_top_linorder by blast |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
951 |
then show ?thesis by (simp add: Lim_eventually) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
952 |
next |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
953 |
case (MInf) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
954 |
then have "max x (-real n) = (-1)* ereal(real n)" for n::nat by (auto simp add: max_def) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
955 |
moreover have "(\<lambda>n. (-1)* ereal(real n)) \<longlonglongrightarrow> -\<infinity>" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
956 |
using tendsto_cmult_ereal[of "-1", OF _ id_nat_ereal_tendsto_PInf] by (simp add: one_ereal_def) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
957 |
ultimately show ?thesis using MInf by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
958 |
next |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
959 |
case (PInf) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
960 |
then have "max x (-real n) = x" for n::nat by (auto simp add: max_def) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
961 |
then show ?thesis by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
962 |
qed |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
963 |
|
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
964 |
lemma ereal_truncation_real_bottom [tendsto_intros]: |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
965 |
fixes x::ereal |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
966 |
assumes "x \<noteq> \<infinity>" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
967 |
shows "(\<lambda>n::nat. real_of_ereal(max x (- real n))) \<longlonglongrightarrow> x" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
968 |
proof (cases x) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
969 |
case (real r) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
970 |
then obtain K::nat where "K>0" "K > abs(r)" using reals_Archimedean2 gr0I by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
971 |
then have "max x (-real n) = x" if "n \<ge> K" for n apply (subst real, subst real, auto) using that eq_iff by fastforce |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
972 |
then have "real_of_ereal(max x (-real n)) = r" if "n \<ge> K" for n using real that by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
973 |
then have "eventually (\<lambda>n. real_of_ereal(max x (-real n)) = r) sequentially" using eventually_at_top_linorder by blast |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
974 |
then have "(\<lambda>n. real_of_ereal(max x (-real n))) \<longlonglongrightarrow> r" by (simp add: Lim_eventually) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
975 |
then show ?thesis using real by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
976 |
next |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
977 |
case (MInf) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
978 |
then have "real_of_ereal(max x (-real n)) = (-1)* ereal(real n)" for n::nat by (auto simp add: max_def) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
979 |
moreover have "(\<lambda>n. (-1)* ereal(real n)) \<longlonglongrightarrow> -\<infinity>" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
980 |
using tendsto_cmult_ereal[of "-1", OF _ id_nat_ereal_tendsto_PInf] by (simp add: one_ereal_def) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
981 |
ultimately show ?thesis using MInf by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
982 |
qed (simp add: assms) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
983 |
|
64911 | 984 |
text \<open>the next one is copied from \verb+tendsto_sum+.\<close> |
64284
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
985 |
lemma tendsto_sum_ereal [tendsto_intros]: |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
986 |
fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> ereal" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
987 |
assumes "\<And>i. i \<in> S \<Longrightarrow> (f i \<longlongrightarrow> a i) F" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
988 |
"\<And>i. abs(a i) \<noteq> \<infinity>" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
989 |
shows "((\<lambda>x. \<Sum>i\<in>S. f i x) \<longlongrightarrow> (\<Sum>i\<in>S. a i)) F" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
990 |
proof (cases "finite S") |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
991 |
assume "finite S" then show ?thesis using assms |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
992 |
by (induct, simp, simp add: tendsto_add_ereal_general2 assms) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
993 |
qed(simp) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
994 |
|
64911 | 995 |
subsubsection \<open>Limsups and liminfs\<close> |
64284
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
996 |
|
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
997 |
lemma limsup_finite_then_bounded: |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
998 |
fixes u::"nat \<Rightarrow> real" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
999 |
assumes "limsup u < \<infinity>" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1000 |
shows "\<exists>C. \<forall>n. u n \<le> C" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1001 |
proof - |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1002 |
obtain C where C: "limsup u < C" "C < \<infinity>" using assms ereal_dense2 by blast |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1003 |
then have "C = ereal(real_of_ereal C)" using ereal_real by force |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1004 |
have "eventually (\<lambda>n. u n < C) sequentially" using C(1) unfolding Limsup_def |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1005 |
apply (auto simp add: INF_less_iff) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1006 |
using SUP_lessD eventually_mono by fastforce |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1007 |
then obtain N where N: "\<And>n. n \<ge> N \<Longrightarrow> u n < C" using eventually_sequentially by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1008 |
define D where "D = max (real_of_ereal C) (Max {u n |n. n \<le> N})" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1009 |
have "\<And>n. u n \<le> D" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1010 |
proof - |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1011 |
fix n show "u n \<le> D" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1012 |
proof (cases) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1013 |
assume *: "n \<le> N" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1014 |
have "u n \<le> Max {u n |n. n \<le> N}" by (rule Max_ge, auto simp add: *) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1015 |
then show "u n \<le> D" unfolding D_def by linarith |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1016 |
next |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1017 |
assume "\<not>(n \<le> N)" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1018 |
then have "n \<ge> N" by simp |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1019 |
then have "u n < C" using N by auto |
64911 | 1020 |
then have "u n < real_of_ereal C" using \<open>C = ereal(real_of_ereal C)\<close> less_ereal.simps(1) by fastforce |
64284
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1021 |
then show "u n \<le> D" unfolding D_def by linarith |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1022 |
qed |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1023 |
qed |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1024 |
then show ?thesis by blast |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1025 |
qed |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1026 |
|
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1027 |
lemma liminf_finite_then_bounded_below: |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1028 |
fixes u::"nat \<Rightarrow> real" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1029 |
assumes "liminf u > -\<infinity>" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1030 |
shows "\<exists>C. \<forall>n. u n \<ge> C" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1031 |
proof - |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1032 |
obtain C where C: "liminf u > C" "C > -\<infinity>" using assms using ereal_dense2 by blast |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1033 |
then have "C = ereal(real_of_ereal C)" using ereal_real by force |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1034 |
have "eventually (\<lambda>n. u n > C) sequentially" using C(1) unfolding Liminf_def |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1035 |
apply (auto simp add: less_SUP_iff) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1036 |
using eventually_elim2 less_INF_D by fastforce |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1037 |
then obtain N where N: "\<And>n. n \<ge> N \<Longrightarrow> u n > C" using eventually_sequentially by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1038 |
define D where "D = min (real_of_ereal C) (Min {u n |n. n \<le> N})" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1039 |
have "\<And>n. u n \<ge> D" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1040 |
proof - |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1041 |
fix n show "u n \<ge> D" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1042 |
proof (cases) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1043 |
assume *: "n \<le> N" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1044 |
have "u n \<ge> Min {u n |n. n \<le> N}" by (rule Min_le, auto simp add: *) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1045 |
then show "u n \<ge> D" unfolding D_def by linarith |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1046 |
next |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1047 |
assume "\<not>(n \<le> N)" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1048 |
then have "n \<ge> N" by simp |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1049 |
then have "u n > C" using N by auto |
64911 | 1050 |
then have "u n > real_of_ereal C" using \<open>C = ereal(real_of_ereal C)\<close> less_ereal.simps(1) by fastforce |
64284
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1051 |
then show "u n \<ge> D" unfolding D_def by linarith |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1052 |
qed |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1053 |
qed |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1054 |
then show ?thesis by blast |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1055 |
qed |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1056 |
|
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1057 |
lemma liminf_upper_bound: |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1058 |
fixes u:: "nat \<Rightarrow> ereal" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1059 |
assumes "liminf u < l" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1060 |
shows "\<exists>N>k. u N < l" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1061 |
by (metis assms gt_ex less_le_trans liminf_bounded_iff not_less) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1062 |
|
64911 | 1063 |
text \<open>The following statement about limsups is reduced to a statement about limits using |
64284
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1064 |
subsequences thanks to \verb+limsup_subseq_lim+. The statement for limits follows for instance from |
64911 | 1065 |
\verb+tendsto_add_ereal_general+.\<close> |
64284
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1066 |
|
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1067 |
lemma ereal_limsup_add_mono: |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1068 |
fixes u v::"nat \<Rightarrow> ereal" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1069 |
shows "limsup (\<lambda>n. u n + v n) \<le> limsup u + limsup v" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1070 |
proof (cases) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1071 |
assume "(limsup u = \<infinity>) \<or> (limsup v = \<infinity>)" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1072 |
then have "limsup u + limsup v = \<infinity>" by simp |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1073 |
then show ?thesis by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1074 |
next |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1075 |
assume "\<not>((limsup u = \<infinity>) \<or> (limsup v = \<infinity>))" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1076 |
then have "limsup u < \<infinity>" "limsup v < \<infinity>" by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1077 |
|
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1078 |
define w where "w = (\<lambda>n. u n + v n)" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1079 |
obtain r where r: "subseq r" "(w o r) \<longlonglongrightarrow> limsup w" using limsup_subseq_lim by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1080 |
obtain s where s: "subseq s" "(u o r o s) \<longlonglongrightarrow> limsup (u o r)" using limsup_subseq_lim by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1081 |
obtain t where t: "subseq t" "(v o r o s o t) \<longlonglongrightarrow> limsup (v o r o s)" using limsup_subseq_lim by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1082 |
|
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1083 |
define a where "a = r o s o t" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1084 |
have "subseq a" using r s t by (simp add: a_def subseq_o) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1085 |
have l:"(w o a) \<longlonglongrightarrow> limsup w" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1086 |
"(u o a) \<longlonglongrightarrow> limsup (u o r)" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1087 |
"(v o a) \<longlonglongrightarrow> limsup (v o r o s)" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1088 |
apply (metis (no_types, lifting) r(2) s(1) t(1) LIMSEQ_subseq_LIMSEQ a_def comp_assoc) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1089 |
apply (metis (no_types, lifting) s(2) t(1) LIMSEQ_subseq_LIMSEQ a_def comp_assoc) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1090 |
apply (metis (no_types, lifting) t(2) a_def comp_assoc) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1091 |
done |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1092 |
|
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1093 |
have "limsup (u o r) \<le> limsup u" by (simp add: limsup_subseq_mono r(1)) |
64911 | 1094 |
then have a: "limsup (u o r) \<noteq> \<infinity>" using \<open>limsup u < \<infinity>\<close> by auto |
64284
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1095 |
have "limsup (v o r o s) \<le> limsup v" by (simp add: comp_assoc limsup_subseq_mono r(1) s(1) subseq_o) |
64911 | 1096 |
then have b: "limsup (v o r o s) \<noteq> \<infinity>" using \<open>limsup v < \<infinity>\<close> by auto |
64284
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1097 |
|
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1098 |
have "(\<lambda>n. (u o a) n + (v o a) n) \<longlonglongrightarrow> limsup (u o r) + limsup (v o r o s)" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1099 |
using l tendsto_add_ereal_general a b by fastforce |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1100 |
moreover have "(\<lambda>n. (u o a) n + (v o a) n) = (w o a)" unfolding w_def by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1101 |
ultimately have "(w o a) \<longlonglongrightarrow> limsup (u o r) + limsup (v o r o s)" by simp |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1102 |
then have "limsup w = limsup (u o r) + limsup (v o r o s)" using l(1) LIMSEQ_unique by blast |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1103 |
then have "limsup w \<le> limsup u + limsup v" |
64911 | 1104 |
using \<open>limsup (u o r) \<le> limsup u\<close> \<open>limsup (v o r o s) \<le> limsup v\<close> ereal_add_mono by simp |
64284
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1105 |
then show ?thesis unfolding w_def by simp |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1106 |
qed |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1107 |
|
64911 | 1108 |
text \<open>There is an asymmetry between liminfs and limsups in ereal, as $\infty + (-\infty) = \infty$. |
64284
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1109 |
This explains why there are more assumptions in the next lemma dealing with liminfs that in the |
64911 | 1110 |
previous one about limsups.\<close> |
64284
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1111 |
|
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1112 |
lemma ereal_liminf_add_mono: |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1113 |
fixes u v::"nat \<Rightarrow> ereal" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1114 |
assumes "\<not>((liminf u = \<infinity> \<and> liminf v = -\<infinity>) \<or> (liminf u = -\<infinity> \<and> liminf v = \<infinity>))" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1115 |
shows "liminf (\<lambda>n. u n + v n) \<ge> liminf u + liminf v" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1116 |
proof (cases) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1117 |
assume "(liminf u = -\<infinity>) \<or> (liminf v = -\<infinity>)" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1118 |
then have *: "liminf u + liminf v = -\<infinity>" using assms by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1119 |
show ?thesis by (simp add: *) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1120 |
next |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1121 |
assume "\<not>((liminf u = -\<infinity>) \<or> (liminf v = -\<infinity>))" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1122 |
then have "liminf u > -\<infinity>" "liminf v > -\<infinity>" by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1123 |
|
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1124 |
define w where "w = (\<lambda>n. u n + v n)" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1125 |
obtain r where r: "subseq r" "(w o r) \<longlonglongrightarrow> liminf w" using liminf_subseq_lim by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1126 |
obtain s where s: "subseq s" "(u o r o s) \<longlonglongrightarrow> liminf (u o r)" using liminf_subseq_lim by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1127 |
obtain t where t: "subseq t" "(v o r o s o t) \<longlonglongrightarrow> liminf (v o r o s)" using liminf_subseq_lim by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1128 |
|
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1129 |
define a where "a = r o s o t" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1130 |
have "subseq a" using r s t by (simp add: a_def subseq_o) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1131 |
have l:"(w o a) \<longlonglongrightarrow> liminf w" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1132 |
"(u o a) \<longlonglongrightarrow> liminf (u o r)" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1133 |
"(v o a) \<longlonglongrightarrow> liminf (v o r o s)" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1134 |
apply (metis (no_types, lifting) r(2) s(1) t(1) LIMSEQ_subseq_LIMSEQ a_def comp_assoc) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1135 |
apply (metis (no_types, lifting) s(2) t(1) LIMSEQ_subseq_LIMSEQ a_def comp_assoc) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1136 |
apply (metis (no_types, lifting) t(2) a_def comp_assoc) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1137 |
done |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1138 |
|
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1139 |
have "liminf (u o r) \<ge> liminf u" by (simp add: liminf_subseq_mono r(1)) |
64911 | 1140 |
then have a: "liminf (u o r) \<noteq> -\<infinity>" using \<open>liminf u > -\<infinity>\<close> by auto |
64284
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1141 |
have "liminf (v o r o s) \<ge> liminf v" by (simp add: comp_assoc liminf_subseq_mono r(1) s(1) subseq_o) |
64911 | 1142 |
then have b: "liminf (v o r o s) \<noteq> -\<infinity>" using \<open>liminf v > -\<infinity>\<close> by auto |
64284
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1143 |
|
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1144 |
have "(\<lambda>n. (u o a) n + (v o a) n) \<longlonglongrightarrow> liminf (u o r) + liminf (v o r o s)" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1145 |
using l tendsto_add_ereal_general a b by fastforce |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1146 |
moreover have "(\<lambda>n. (u o a) n + (v o a) n) = (w o a)" unfolding w_def by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1147 |
ultimately have "(w o a) \<longlonglongrightarrow> liminf (u o r) + liminf (v o r o s)" by simp |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1148 |
then have "liminf w = liminf (u o r) + liminf (v o r o s)" using l(1) LIMSEQ_unique by blast |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1149 |
then have "liminf w \<ge> liminf u + liminf v" |
64911 | 1150 |
using \<open>liminf (u o r) \<ge> liminf u\<close> \<open>liminf (v o r o s) \<ge> liminf v\<close> ereal_add_mono by simp |
64284
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1151 |
then show ?thesis unfolding w_def by simp |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1152 |
qed |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1153 |
|
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1154 |
lemma ereal_limsup_lim_add: |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1155 |
fixes u v::"nat \<Rightarrow> ereal" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1156 |
assumes "u \<longlonglongrightarrow> a" "abs(a) \<noteq> \<infinity>" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1157 |
shows "limsup (\<lambda>n. u n + v n) = a + limsup v" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1158 |
proof - |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1159 |
have "limsup u = a" using assms(1) using tendsto_iff_Liminf_eq_Limsup trivial_limit_at_top_linorder by blast |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1160 |
have "(\<lambda>n. -u n) \<longlonglongrightarrow> -a" using assms(1) by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1161 |
then have "limsup (\<lambda>n. -u n) = -a" using tendsto_iff_Liminf_eq_Limsup trivial_limit_at_top_linorder by blast |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1162 |
|
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1163 |
have "limsup (\<lambda>n. u n + v n) \<le> limsup u + limsup v" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1164 |
by (rule ereal_limsup_add_mono) |
64911 | 1165 |
then have up: "limsup (\<lambda>n. u n + v n) \<le> a + limsup v" using \<open>limsup u = a\<close> by simp |
64284
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1166 |
|
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1167 |
have a: "limsup (\<lambda>n. (u n + v n) + (-u n)) \<le> limsup (\<lambda>n. u n + v n) + limsup (\<lambda>n. -u n)" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1168 |
by (rule ereal_limsup_add_mono) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1169 |
have "eventually (\<lambda>n. u n = ereal(real_of_ereal(u n))) sequentially" using assms |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1170 |
real_lim_then_eventually_real by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1171 |
moreover have "\<And>x. x = ereal(real_of_ereal(x)) \<Longrightarrow> x + (-x) = 0" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1172 |
by (metis plus_ereal.simps(1) right_minus uminus_ereal.simps(1) zero_ereal_def) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1173 |
ultimately have "eventually (\<lambda>n. u n + (-u n) = 0) sequentially" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1174 |
by (metis (mono_tags, lifting) eventually_mono) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1175 |
moreover have "\<And>n. u n + (-u n) = 0 \<Longrightarrow> u n + v n + (-u n) = v n" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1176 |
by (metis add.commute add.left_commute add.left_neutral) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1177 |
ultimately have "eventually (\<lambda>n. u n + v n + (-u n) = v n) sequentially" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1178 |
using eventually_mono by force |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1179 |
then have "limsup v = limsup (\<lambda>n. u n + v n + (-u n))" using Limsup_eq by force |
64911 | 1180 |
then have "limsup v \<le> limsup (\<lambda>n. u n + v n) -a" using a \<open>limsup (\<lambda>n. -u n) = -a\<close> by (simp add: minus_ereal_def) |
64284
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1181 |
then have "limsup (\<lambda>n. u n + v n) \<ge> a + limsup v" using assms(2) by (metis add.commute ereal_le_minus) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1182 |
then show ?thesis using up by simp |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1183 |
qed |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1184 |
|
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1185 |
lemma ereal_limsup_lim_mult: |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1186 |
fixes u v::"nat \<Rightarrow> ereal" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1187 |
assumes "u \<longlonglongrightarrow> a" "a>0" "a \<noteq> \<infinity>" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1188 |
shows "limsup (\<lambda>n. u n * v n) = a * limsup v" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1189 |
proof - |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1190 |
define w where "w = (\<lambda>n. u n * v n)" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1191 |
obtain r where r: "subseq r" "(v o r) \<longlonglongrightarrow> limsup v" using limsup_subseq_lim by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1192 |
have "(u o r) \<longlonglongrightarrow> a" using assms(1) LIMSEQ_subseq_LIMSEQ r by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1193 |
with tendsto_mult_ereal[OF this r(2)] have "(\<lambda>n. (u o r) n * (v o r) n) \<longlonglongrightarrow> a * limsup v" using assms(2) assms(3) by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1194 |
moreover have "\<And>n. (w o r) n = (u o r) n * (v o r) n" unfolding w_def by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1195 |
ultimately have "(w o r) \<longlonglongrightarrow> a * limsup v" unfolding w_def by presburger |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1196 |
then have "limsup (w o r) = a * limsup v" by (simp add: tendsto_iff_Liminf_eq_Limsup) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1197 |
then have I: "limsup w \<ge> a * limsup v" by (metis limsup_subseq_mono r(1)) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1198 |
|
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1199 |
obtain s where s: "subseq s" "(w o s) \<longlonglongrightarrow> limsup w" using limsup_subseq_lim by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1200 |
have *: "(u o s) \<longlonglongrightarrow> a" using assms(1) LIMSEQ_subseq_LIMSEQ s by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1201 |
have "eventually (\<lambda>n. (u o s) n > 0) sequentially" using assms(2) * order_tendsto_iff by blast |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1202 |
moreover have "eventually (\<lambda>n. (u o s) n < \<infinity>) sequentially" using assms(3) * order_tendsto_iff by blast |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1203 |
moreover have "(w o s) n / (u o s) n = (v o s) n" if "(u o s) n > 0" "(u o s) n < \<infinity>" for n |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1204 |
unfolding w_def using that by (auto simp add: ereal_divide_eq) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1205 |
ultimately have "eventually (\<lambda>n. (w o s) n / (u o s) n = (v o s) n) sequentially" using eventually_elim2 by force |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1206 |
moreover have "(\<lambda>n. (w o s) n / (u o s) n) \<longlonglongrightarrow> (limsup w) / a" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1207 |
apply (rule tendsto_divide_ereal[OF s(2) *]) using assms(2) assms(3) by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1208 |
ultimately have "(v o s) \<longlonglongrightarrow> (limsup w) / a" using Lim_transform_eventually by fastforce |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1209 |
then have "limsup (v o s) = (limsup w) / a" by (simp add: tendsto_iff_Liminf_eq_Limsup) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1210 |
then have "limsup v \<ge> (limsup w) / a" by (metis limsup_subseq_mono s(1)) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1211 |
then have "a * limsup v \<ge> limsup w" using assms(2) assms(3) by (simp add: ereal_divide_le_pos) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1212 |
then show ?thesis using I unfolding w_def by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1213 |
qed |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1214 |
|
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1215 |
lemma ereal_liminf_lim_mult: |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1216 |
fixes u v::"nat \<Rightarrow> ereal" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1217 |
assumes "u \<longlonglongrightarrow> a" "a>0" "a \<noteq> \<infinity>" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1218 |
shows "liminf (\<lambda>n. u n * v n) = a * liminf v" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1219 |
proof - |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1220 |
define w where "w = (\<lambda>n. u n * v n)" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1221 |
obtain r where r: "subseq r" "(v o r) \<longlonglongrightarrow> liminf v" using liminf_subseq_lim by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1222 |
have "(u o r) \<longlonglongrightarrow> a" using assms(1) LIMSEQ_subseq_LIMSEQ r by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1223 |
with tendsto_mult_ereal[OF this r(2)] have "(\<lambda>n. (u o r) n * (v o r) n) \<longlonglongrightarrow> a * liminf v" using assms(2) assms(3) by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1224 |
moreover have "\<And>n. (w o r) n = (u o r) n * (v o r) n" unfolding w_def by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1225 |
ultimately have "(w o r) \<longlonglongrightarrow> a * liminf v" unfolding w_def by presburger |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1226 |
then have "liminf (w o r) = a * liminf v" by (simp add: tendsto_iff_Liminf_eq_Limsup) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1227 |
then have I: "liminf w \<le> a * liminf v" by (metis liminf_subseq_mono r(1)) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1228 |
|
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1229 |
obtain s where s: "subseq s" "(w o s) \<longlonglongrightarrow> liminf w" using liminf_subseq_lim by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1230 |
have *: "(u o s) \<longlonglongrightarrow> a" using assms(1) LIMSEQ_subseq_LIMSEQ s by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1231 |
have "eventually (\<lambda>n. (u o s) n > 0) sequentially" using assms(2) * order_tendsto_iff by blast |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1232 |
moreover have "eventually (\<lambda>n. (u o s) n < \<infinity>) sequentially" using assms(3) * order_tendsto_iff by blast |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1233 |
moreover have "(w o s) n / (u o s) n = (v o s) n" if "(u o s) n > 0" "(u o s) n < \<infinity>" for n |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1234 |
unfolding w_def using that by (auto simp add: ereal_divide_eq) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1235 |
ultimately have "eventually (\<lambda>n. (w o s) n / (u o s) n = (v o s) n) sequentially" using eventually_elim2 by force |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1236 |
moreover have "(\<lambda>n. (w o s) n / (u o s) n) \<longlonglongrightarrow> (liminf w) / a" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1237 |
apply (rule tendsto_divide_ereal[OF s(2) *]) using assms(2) assms(3) by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1238 |
ultimately have "(v o s) \<longlonglongrightarrow> (liminf w) / a" using Lim_transform_eventually by fastforce |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1239 |
then have "liminf (v o s) = (liminf w) / a" by (simp add: tendsto_iff_Liminf_eq_Limsup) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1240 |
then have "liminf v \<le> (liminf w) / a" by (metis liminf_subseq_mono s(1)) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1241 |
then have "a * liminf v \<le> liminf w" using assms(2) assms(3) by (simp add: ereal_le_divide_pos) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1242 |
then show ?thesis using I unfolding w_def by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1243 |
qed |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1244 |
|
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1245 |
lemma ereal_liminf_lim_add: |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1246 |
fixes u v::"nat \<Rightarrow> ereal" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1247 |
assumes "u \<longlonglongrightarrow> a" "abs(a) \<noteq> \<infinity>" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1248 |
shows "liminf (\<lambda>n. u n + v n) = a + liminf v" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1249 |
proof - |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1250 |
have "liminf u = a" using assms(1) tendsto_iff_Liminf_eq_Limsup trivial_limit_at_top_linorder by blast |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1251 |
then have *: "abs(liminf u) \<noteq> \<infinity>" using assms(2) by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1252 |
have "(\<lambda>n. -u n) \<longlonglongrightarrow> -a" using assms(1) by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1253 |
then have "liminf (\<lambda>n. -u n) = -a" using tendsto_iff_Liminf_eq_Limsup trivial_limit_at_top_linorder by blast |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1254 |
then have **: "abs(liminf (\<lambda>n. -u n)) \<noteq> \<infinity>" using assms(2) by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1255 |
|
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1256 |
have "liminf (\<lambda>n. u n + v n) \<ge> liminf u + liminf v" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1257 |
apply (rule ereal_liminf_add_mono) using * by auto |
64911 | 1258 |
then have up: "liminf (\<lambda>n. u n + v n) \<ge> a + liminf v" using \<open>liminf u = a\<close> by simp |
64284
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1259 |
|
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1260 |
have a: "liminf (\<lambda>n. (u n + v n) + (-u n)) \<ge> liminf (\<lambda>n. u n + v n) + liminf (\<lambda>n. -u n)" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1261 |
apply (rule ereal_liminf_add_mono) using ** by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1262 |
have "eventually (\<lambda>n. u n = ereal(real_of_ereal(u n))) sequentially" using assms |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1263 |
real_lim_then_eventually_real by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1264 |
moreover have "\<And>x. x = ereal(real_of_ereal(x)) \<Longrightarrow> x + (-x) = 0" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1265 |
by (metis plus_ereal.simps(1) right_minus uminus_ereal.simps(1) zero_ereal_def) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1266 |
ultimately have "eventually (\<lambda>n. u n + (-u n) = 0) sequentially" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1267 |
by (metis (mono_tags, lifting) eventually_mono) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1268 |
moreover have "\<And>n. u n + (-u n) = 0 \<Longrightarrow> u n + v n + (-u n) = v n" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1269 |
by (metis add.commute add.left_commute add.left_neutral) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1270 |
ultimately have "eventually (\<lambda>n. u n + v n + (-u n) = v n) sequentially" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1271 |
using eventually_mono by force |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1272 |
then have "liminf v = liminf (\<lambda>n. u n + v n + (-u n))" using Liminf_eq by force |
64911 | 1273 |
then have "liminf v \<ge> liminf (\<lambda>n. u n + v n) -a" using a \<open>liminf (\<lambda>n. -u n) = -a\<close> by (simp add: minus_ereal_def) |
64284
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1274 |
then have "liminf (\<lambda>n. u n + v n) \<le> a + liminf v" using assms(2) by (metis add.commute ereal_minus_le) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1275 |
then show ?thesis using up by simp |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1276 |
qed |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1277 |
|
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1278 |
lemma ereal_liminf_limsup_add: |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1279 |
fixes u v::"nat \<Rightarrow> ereal" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1280 |
shows "liminf (\<lambda>n. u n + v n) \<le> liminf u + limsup v" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1281 |
proof (cases) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1282 |
assume "limsup v = \<infinity> \<or> liminf u = \<infinity>" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1283 |
then show ?thesis by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1284 |
next |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1285 |
assume "\<not>(limsup v = \<infinity> \<or> liminf u = \<infinity>)" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1286 |
then have "limsup v < \<infinity>" "liminf u < \<infinity>" by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1287 |
|
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1288 |
define w where "w = (\<lambda>n. u n + v n)" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1289 |
obtain r where r: "subseq r" "(u o r) \<longlonglongrightarrow> liminf u" using liminf_subseq_lim by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1290 |
obtain s where s: "subseq s" "(w o r o s) \<longlonglongrightarrow> liminf (w o r)" using liminf_subseq_lim by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1291 |
obtain t where t: "subseq t" "(v o r o s o t) \<longlonglongrightarrow> limsup (v o r o s)" using limsup_subseq_lim by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1292 |
|
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1293 |
define a where "a = r o s o t" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1294 |
have "subseq a" using r s t by (simp add: a_def subseq_o) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1295 |
have l:"(u o a) \<longlonglongrightarrow> liminf u" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1296 |
"(w o a) \<longlonglongrightarrow> liminf (w o r)" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1297 |
"(v o a) \<longlonglongrightarrow> limsup (v o r o s)" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1298 |
apply (metis (no_types, lifting) r(2) s(1) t(1) LIMSEQ_subseq_LIMSEQ a_def comp_assoc) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1299 |
apply (metis (no_types, lifting) s(2) t(1) LIMSEQ_subseq_LIMSEQ a_def comp_assoc) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1300 |
apply (metis (no_types, lifting) t(2) a_def comp_assoc) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1301 |
done |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1302 |
|
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1303 |
have "liminf (w o r) \<ge> liminf w" by (simp add: liminf_subseq_mono r(1)) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1304 |
have "limsup (v o r o s) \<le> limsup v" by (simp add: comp_assoc limsup_subseq_mono r(1) s(1) subseq_o) |
64911 | 1305 |
then have b: "limsup (v o r o s) < \<infinity>" using \<open>limsup v < \<infinity>\<close> by auto |
64284
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1306 |
|
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1307 |
have "(\<lambda>n. (u o a) n + (v o a) n) \<longlonglongrightarrow> liminf u + limsup (v o r o s)" |
64911 | 1308 |
apply (rule tendsto_add_ereal_general) using b \<open>liminf u < \<infinity>\<close> l(1) l(3) by force+ |
64284
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1309 |
moreover have "(\<lambda>n. (u o a) n + (v o a) n) = (w o a)" unfolding w_def by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1310 |
ultimately have "(w o a) \<longlonglongrightarrow> liminf u + limsup (v o r o s)" by simp |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1311 |
then have "liminf (w o r) = liminf u + limsup (v o r o s)" using l(2) using LIMSEQ_unique by blast |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1312 |
then have "liminf w \<le> liminf u + limsup v" |
64911 | 1313 |
using \<open>liminf (w o r) \<ge> liminf w\<close> \<open>limsup (v o r o s) \<le> limsup v\<close> |
64284
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1314 |
by (metis add_mono_thms_linordered_semiring(2) le_less_trans not_less) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1315 |
then show ?thesis unfolding w_def by simp |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1316 |
qed |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1317 |
|
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1318 |
lemma ereal_liminf_limsup_minus: |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1319 |
fixes u v::"nat \<Rightarrow> ereal" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1320 |
shows "liminf (\<lambda>n. u n - v n) \<le> limsup u - limsup v" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1321 |
unfolding minus_ereal_def |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1322 |
apply (subst add.commute) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1323 |
apply (rule order_trans[OF ereal_liminf_limsup_add]) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1324 |
using ereal_Limsup_uminus[of sequentially "\<lambda>n. - v n"] |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1325 |
apply (simp add: add.commute) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1326 |
done |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1327 |
|
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1328 |
|
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1329 |
lemma liminf_minus_ennreal: |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1330 |
fixes u v::"nat \<Rightarrow> ennreal" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1331 |
shows "(\<And>n. v n \<le> u n) \<Longrightarrow> liminf (\<lambda>n. u n - v n) \<le> limsup u - limsup v" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1332 |
unfolding liminf_SUP_INF limsup_INF_SUP |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1333 |
including ennreal.lifting |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1334 |
proof (transfer, clarsimp) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1335 |
fix v u :: "nat \<Rightarrow> ereal" assume *: "\<forall>x. 0 \<le> v x" "\<forall>x. 0 \<le> u x" "\<And>n. v n \<le> u n" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1336 |
moreover have "0 \<le> limsup u - limsup v" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1337 |
using * by (intro ereal_diff_positive Limsup_mono always_eventually) simp |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1338 |
moreover have "0 \<le> (SUPREMUM {x..} v)" for x |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1339 |
using * by (intro SUP_upper2[of x]) auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1340 |
moreover have "0 \<le> (SUPREMUM {x..} u)" for x |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1341 |
using * by (intro SUP_upper2[of x]) auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1342 |
ultimately show "(SUP n. INF n:{n..}. max 0 (u n - v n)) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1343 |
\<le> max 0 ((INF x. max 0 (SUPREMUM {x..} u)) - (INF x. max 0 (SUPREMUM {x..} v)))" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1344 |
by (auto simp: * ereal_diff_positive max.absorb2 liminf_SUP_INF[symmetric] limsup_INF_SUP[symmetric] ereal_liminf_limsup_minus) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1345 |
qed |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
1346 |
|
60615
e5fa1d5d3952
Useful lemmas. The theorem concerning swapping the variables in a double integral.
paulson <lp15@cam.ac.uk>
parents:
59867
diff
changeset
|
1347 |
(* |
59092
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1348 |
Notation |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1349 |
*) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1350 |
|
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1351 |
abbreviation "set_borel_measurable M A f \<equiv> (\<lambda>x. indicator A x *\<^sub>R f x) \<in> borel_measurable M" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1352 |
|
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1353 |
abbreviation "set_integrable M A f \<equiv> integrable M (\<lambda>x. indicator A x *\<^sub>R f x)" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1354 |
|
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1355 |
abbreviation "set_lebesgue_integral M A f \<equiv> lebesgue_integral M (\<lambda>x. indicator A x *\<^sub>R f x)" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1356 |
|
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1357 |
syntax |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1358 |
"_ascii_set_lebesgue_integral" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'a measure \<Rightarrow> real \<Rightarrow> real" |
59358 | 1359 |
("(4LINT (_):(_)/|(_)./ _)" [0,60,110,61] 60) |
59092
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1360 |
|
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1361 |
translations |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1362 |
"LINT x:A|M. f" == "CONST set_lebesgue_integral M A (\<lambda>x. f)" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1363 |
|
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1364 |
(* |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1365 |
Notation for integration wrt lebesgue measure on the reals: |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1366 |
|
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1367 |
LBINT x. f |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1368 |
LBINT x : A. f |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1369 |
|
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1370 |
TODO: keep all these? Need unicode. |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1371 |
*) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1372 |
|
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1373 |
syntax |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1374 |
"_lebesgue_borel_integral" :: "pttrn \<Rightarrow> real \<Rightarrow> real" |
59358 | 1375 |
("(2LBINT _./ _)" [0,60] 60) |
59092
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1376 |
|
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1377 |
syntax |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1378 |
"_set_lebesgue_borel_integral" :: "pttrn \<Rightarrow> real set \<Rightarrow> real \<Rightarrow> real" |
59358 | 1379 |
("(3LBINT _:_./ _)" [0,60,61] 60) |
59092
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1380 |
|
60615
e5fa1d5d3952
Useful lemmas. The theorem concerning swapping the variables in a double integral.
paulson <lp15@cam.ac.uk>
parents:
59867
diff
changeset
|
1381 |
(* |
e5fa1d5d3952
Useful lemmas. The theorem concerning swapping the variables in a double integral.
paulson <lp15@cam.ac.uk>
parents:
59867
diff
changeset
|
1382 |
Basic properties |
59092
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1383 |
*) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1384 |
|
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1385 |
(* |
61945 | 1386 |
lemma indicator_abs_eq: "\<And>A x. \<bar>indicator A x\<bar> = ((indicator A x) :: real)" |
59092
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1387 |
by (auto simp add: indicator_def) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1388 |
*) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1389 |
|
62083 | 1390 |
lemma set_borel_measurable_sets: |
1391 |
fixes f :: "_ \<Rightarrow> _::real_normed_vector" |
|
1392 |
assumes "set_borel_measurable M X f" "B \<in> sets borel" "X \<in> sets M" |
|
1393 |
shows "f -` B \<inter> X \<in> sets M" |
|
1394 |
proof - |
|
1395 |
have "f \<in> borel_measurable (restrict_space M X)" |
|
1396 |
using assms by (subst borel_measurable_restrict_space_iff) auto |
|
1397 |
then have "f -` B \<inter> space (restrict_space M X) \<in> sets (restrict_space M X)" |
|
1398 |
by (rule measurable_sets) fact |
|
1399 |
with \<open>X \<in> sets M\<close> show ?thesis |
|
1400 |
by (subst (asm) sets_restrict_space_iff) (auto simp: space_restrict_space) |
|
1401 |
qed |
|
1402 |
||
59092
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1403 |
lemma set_lebesgue_integral_cong: |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1404 |
assumes "A \<in> sets M" and "\<forall>x. x \<in> A \<longrightarrow> f x = g x" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1405 |
shows "(LINT x:A|M. f x) = (LINT x:A|M. g x)" |
63886
685fb01256af
move Henstock-Kurzweil integration after Lebesgue_Measure; replace content by abbreviation measure lborel
hoelzl
parents:
63627
diff
changeset
|
1406 |
using assms by (auto intro!: Bochner_Integration.integral_cong split: split_indicator simp add: sets.sets_into_space) |
59092
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1407 |
|
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1408 |
lemma set_lebesgue_integral_cong_AE: |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1409 |
assumes [measurable]: "A \<in> sets M" "f \<in> borel_measurable M" "g \<in> borel_measurable M" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1410 |
assumes "AE x \<in> A in M. f x = g x" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1411 |
shows "LINT x:A|M. f x = LINT x:A|M. g x" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1412 |
proof- |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1413 |
have "AE x in M. indicator A x *\<^sub>R f x = indicator A x *\<^sub>R g x" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1414 |
using assms by auto |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1415 |
thus ?thesis by (intro integral_cong_AE) auto |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1416 |
qed |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1417 |
|
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1418 |
lemma set_integrable_cong_AE: |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1419 |
"f \<in> borel_measurable M \<Longrightarrow> g \<in> borel_measurable M \<Longrightarrow> |
60615
e5fa1d5d3952
Useful lemmas. The theorem concerning swapping the variables in a double integral.
paulson <lp15@cam.ac.uk>
parents:
59867
diff
changeset
|
1420 |
AE x \<in> A in M. f x = g x \<Longrightarrow> A \<in> sets M \<Longrightarrow> |
59092
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1421 |
set_integrable M A f = set_integrable M A g" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1422 |
by (rule integrable_cong_AE) auto |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1423 |
|
60615
e5fa1d5d3952
Useful lemmas. The theorem concerning swapping the variables in a double integral.
paulson <lp15@cam.ac.uk>
parents:
59867
diff
changeset
|
1424 |
lemma set_integrable_subset: |
59092
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1425 |
fixes M A B and f :: "_ \<Rightarrow> _ :: {banach, second_countable_topology}" |
60615
e5fa1d5d3952
Useful lemmas. The theorem concerning swapping the variables in a double integral.
paulson <lp15@cam.ac.uk>
parents:
59867
diff
changeset
|
1426 |
assumes "set_integrable M A f" "B \<in> sets M" "B \<subseteq> A" |
59092
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1427 |
shows "set_integrable M B f" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1428 |
proof - |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1429 |
have "set_integrable M B (\<lambda>x. indicator A x *\<^sub>R f x)" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1430 |
by (rule integrable_mult_indicator) fact+ |
61808 | 1431 |
with \<open>B \<subseteq> A\<close> show ?thesis |
59092
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1432 |
by (simp add: indicator_inter_arith[symmetric] Int_absorb2) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1433 |
qed |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1434 |
|
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1435 |
(* TODO: integral_cmul_indicator should be named set_integral_const *) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1436 |
(* TODO: borel_integrable_atLeastAtMost should be named something like set_integrable_Icc_isCont *) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1437 |
|
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1438 |
lemma set_integral_scaleR_right [simp]: "LINT t:A|M. a *\<^sub>R f t = a *\<^sub>R (LINT t:A|M. f t)" |
63886
685fb01256af
move Henstock-Kurzweil integration after Lebesgue_Measure; replace content by abbreviation measure lborel
hoelzl
parents:
63627
diff
changeset
|
1439 |
by (subst integral_scaleR_right[symmetric]) (auto intro!: Bochner_Integration.integral_cong) |
59092
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1440 |
|
60615
e5fa1d5d3952
Useful lemmas. The theorem concerning swapping the variables in a double integral.
paulson <lp15@cam.ac.uk>
parents:
59867
diff
changeset
|
1441 |
lemma set_integral_mult_right [simp]: |
59092
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1442 |
fixes a :: "'a::{real_normed_field, second_countable_topology}" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1443 |
shows "LINT t:A|M. a * f t = a * (LINT t:A|M. f t)" |
63886
685fb01256af
move Henstock-Kurzweil integration after Lebesgue_Measure; replace content by abbreviation measure lborel
hoelzl
parents:
63627
diff
changeset
|
1444 |
by (subst integral_mult_right_zero[symmetric]) auto |
59092
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1445 |
|
60615
e5fa1d5d3952
Useful lemmas. The theorem concerning swapping the variables in a double integral.
paulson <lp15@cam.ac.uk>
parents:
59867
diff
changeset
|
1446 |
lemma set_integral_mult_left [simp]: |
59092
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1447 |
fixes a :: "'a::{real_normed_field, second_countable_topology}" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1448 |
shows "LINT t:A|M. f t * a = (LINT t:A|M. f t) * a" |
63886
685fb01256af
move Henstock-Kurzweil integration after Lebesgue_Measure; replace content by abbreviation measure lborel
hoelzl
parents:
63627
diff
changeset
|
1449 |
by (subst integral_mult_left_zero[symmetric]) auto |
59092
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1450 |
|
60615
e5fa1d5d3952
Useful lemmas. The theorem concerning swapping the variables in a double integral.
paulson <lp15@cam.ac.uk>
parents:
59867
diff
changeset
|
1451 |
lemma set_integral_divide_zero [simp]: |
59867
58043346ca64
given up separate type classes demanding `inverse 0 = 0`
haftmann
parents:
59358
diff
changeset
|
1452 |
fixes a :: "'a::{real_normed_field, field, second_countable_topology}" |
59092
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1453 |
shows "LINT t:A|M. f t / a = (LINT t:A|M. f t) / a" |
63886
685fb01256af
move Henstock-Kurzweil integration after Lebesgue_Measure; replace content by abbreviation measure lborel
hoelzl
parents:
63627
diff
changeset
|
1454 |
by (subst integral_divide_zero[symmetric], intro Bochner_Integration.integral_cong) |
59092
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1455 |
(auto split: split_indicator) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1456 |
|
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1457 |
lemma set_integrable_scaleR_right [simp, intro]: |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1458 |
shows "(a \<noteq> 0 \<Longrightarrow> set_integrable M A f) \<Longrightarrow> set_integrable M A (\<lambda>t. a *\<^sub>R f t)" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1459 |
unfolding scaleR_left_commute by (rule integrable_scaleR_right) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1460 |
|
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1461 |
lemma set_integrable_scaleR_left [simp, intro]: |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1462 |
fixes a :: "_ :: {banach, second_countable_topology}" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1463 |
shows "(a \<noteq> 0 \<Longrightarrow> set_integrable M A f) \<Longrightarrow> set_integrable M A (\<lambda>t. f t *\<^sub>R a)" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1464 |
using integrable_scaleR_left[of a M "\<lambda>x. indicator A x *\<^sub>R f x"] by simp |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1465 |
|
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1466 |
lemma set_integrable_mult_right [simp, intro]: |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1467 |
fixes a :: "'a::{real_normed_field, second_countable_topology}" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1468 |
shows "(a \<noteq> 0 \<Longrightarrow> set_integrable M A f) \<Longrightarrow> set_integrable M A (\<lambda>t. a * f t)" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1469 |
using integrable_mult_right[of a M "\<lambda>x. indicator A x *\<^sub>R f x"] by simp |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1470 |
|
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1471 |
lemma set_integrable_mult_left [simp, intro]: |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1472 |
fixes a :: "'a::{real_normed_field, second_countable_topology}" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1473 |
shows "(a \<noteq> 0 \<Longrightarrow> set_integrable M A f) \<Longrightarrow> set_integrable M A (\<lambda>t. f t * a)" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1474 |
using integrable_mult_left[of a M "\<lambda>x. indicator A x *\<^sub>R f x"] by simp |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1475 |
|
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1476 |
lemma set_integrable_divide [simp, intro]: |
59867
58043346ca64
given up separate type classes demanding `inverse 0 = 0`
haftmann
parents:
59358
diff
changeset
|
1477 |
fixes a :: "'a::{real_normed_field, field, second_countable_topology}" |
59092
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1478 |
assumes "a \<noteq> 0 \<Longrightarrow> set_integrable M A f" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1479 |
shows "set_integrable M A (\<lambda>t. f t / a)" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1480 |
proof - |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1481 |
have "integrable M (\<lambda>x. indicator A x *\<^sub>R f x / a)" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1482 |
using assms by (rule integrable_divide_zero) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1483 |
also have "(\<lambda>x. indicator A x *\<^sub>R f x / a) = (\<lambda>x. indicator A x *\<^sub>R (f x / a))" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1484 |
by (auto split: split_indicator) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1485 |
finally show ?thesis . |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1486 |
qed |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1487 |
|
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1488 |
lemma set_integral_add [simp, intro]: |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1489 |
fixes f g :: "_ \<Rightarrow> _ :: {banach, second_countable_topology}" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1490 |
assumes "set_integrable M A f" "set_integrable M A g" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1491 |
shows "set_integrable M A (\<lambda>x. f x + g x)" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1492 |
and "LINT x:A|M. f x + g x = (LINT x:A|M. f x) + (LINT x:A|M. g x)" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1493 |
using assms by (simp_all add: scaleR_add_right) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1494 |
|
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1495 |
lemma set_integral_diff [simp, intro]: |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1496 |
assumes "set_integrable M A f" "set_integrable M A g" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1497 |
shows "set_integrable M A (\<lambda>x. f x - g x)" and "LINT x:A|M. f x - g x = |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1498 |
(LINT x:A|M. f x) - (LINT x:A|M. g x)" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1499 |
using assms by (simp_all add: scaleR_diff_right) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1500 |
|
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1501 |
(* question: why do we have this for negation, but multiplication by a constant |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1502 |
requires an integrability assumption? *) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1503 |
lemma set_integral_uminus: "set_integrable M A f \<Longrightarrow> LINT x:A|M. - f x = - (LINT x:A|M. f x)" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1504 |
by (subst integral_minus[symmetric]) simp_all |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1505 |
|
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1506 |
lemma set_integral_complex_of_real: |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1507 |
"LINT x:A|M. complex_of_real (f x) = of_real (LINT x:A|M. f x)" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1508 |
by (subst integral_complex_of_real[symmetric]) |
63886
685fb01256af
move Henstock-Kurzweil integration after Lebesgue_Measure; replace content by abbreviation measure lborel
hoelzl
parents:
63627
diff
changeset
|
1509 |
(auto intro!: Bochner_Integration.integral_cong split: split_indicator) |
59092
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1510 |
|
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1511 |
lemma set_integral_mono: |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1512 |
fixes f g :: "_ \<Rightarrow> real" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1513 |
assumes "set_integrable M A f" "set_integrable M A g" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1514 |
"\<And>x. x \<in> A \<Longrightarrow> f x \<le> g x" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1515 |
shows "(LINT x:A|M. f x) \<le> (LINT x:A|M. g x)" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1516 |
using assms by (auto intro: integral_mono split: split_indicator) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1517 |
|
60615
e5fa1d5d3952
Useful lemmas. The theorem concerning swapping the variables in a double integral.
paulson <lp15@cam.ac.uk>
parents:
59867
diff
changeset
|
1518 |
lemma set_integral_mono_AE: |
59092
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1519 |
fixes f g :: "_ \<Rightarrow> real" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1520 |
assumes "set_integrable M A f" "set_integrable M A g" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1521 |
"AE x \<in> A in M. f x \<le> g x" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1522 |
shows "(LINT x:A|M. f x) \<le> (LINT x:A|M. g x)" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1523 |
using assms by (auto intro: integral_mono_AE split: split_indicator) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1524 |
|
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1525 |
lemma set_integrable_abs: "set_integrable M A f \<Longrightarrow> set_integrable M A (\<lambda>x. \<bar>f x\<bar> :: real)" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1526 |
using integrable_abs[of M "\<lambda>x. f x * indicator A x"] by (simp add: abs_mult ac_simps) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1527 |
|
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1528 |
lemma set_integrable_abs_iff: |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1529 |
fixes f :: "_ \<Rightarrow> real" |
60615
e5fa1d5d3952
Useful lemmas. The theorem concerning swapping the variables in a double integral.
paulson <lp15@cam.ac.uk>
parents:
59867
diff
changeset
|
1530 |
shows "set_borel_measurable M A f \<Longrightarrow> set_integrable M A (\<lambda>x. \<bar>f x\<bar>) = set_integrable M A f" |
59092
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1531 |
by (subst (2) integrable_abs_iff[symmetric]) (simp_all add: abs_mult ac_simps) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1532 |
|
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1533 |
lemma set_integrable_abs_iff': |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1534 |
fixes f :: "_ \<Rightarrow> real" |
60615
e5fa1d5d3952
Useful lemmas. The theorem concerning swapping the variables in a double integral.
paulson <lp15@cam.ac.uk>
parents:
59867
diff
changeset
|
1535 |
shows "f \<in> borel_measurable M \<Longrightarrow> A \<in> sets M \<Longrightarrow> |
59092
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1536 |
set_integrable M A (\<lambda>x. \<bar>f x\<bar>) = set_integrable M A f" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1537 |
by (intro set_integrable_abs_iff) auto |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1538 |
|
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1539 |
lemma set_integrable_discrete_difference: |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1540 |
fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1541 |
assumes "countable X" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1542 |
assumes diff: "(A - B) \<union> (B - A) \<subseteq> X" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1543 |
assumes "\<And>x. x \<in> X \<Longrightarrow> emeasure M {x} = 0" "\<And>x. x \<in> X \<Longrightarrow> {x} \<in> sets M" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1544 |
shows "set_integrable M A f \<longleftrightarrow> set_integrable M B f" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1545 |
proof (rule integrable_discrete_difference[where X=X]) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1546 |
show "\<And>x. x \<in> space M \<Longrightarrow> x \<notin> X \<Longrightarrow> indicator A x *\<^sub>R f x = indicator B x *\<^sub>R f x" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1547 |
using diff by (auto split: split_indicator) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1548 |
qed fact+ |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1549 |
|
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1550 |
lemma set_integral_discrete_difference: |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1551 |
fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1552 |
assumes "countable X" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1553 |
assumes diff: "(A - B) \<union> (B - A) \<subseteq> X" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1554 |
assumes "\<And>x. x \<in> X \<Longrightarrow> emeasure M {x} = 0" "\<And>x. x \<in> X \<Longrightarrow> {x} \<in> sets M" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1555 |
shows "set_lebesgue_integral M A f = set_lebesgue_integral M B f" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1556 |
proof (rule integral_discrete_difference[where X=X]) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1557 |
show "\<And>x. x \<in> space M \<Longrightarrow> x \<notin> X \<Longrightarrow> indicator A x *\<^sub>R f x = indicator B x *\<^sub>R f x" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1558 |
using diff by (auto split: split_indicator) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1559 |
qed fact+ |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1560 |
|
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1561 |
lemma set_integrable_Un: |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1562 |
fixes f g :: "_ \<Rightarrow> _ :: {banach, second_countable_topology}" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1563 |
assumes f_A: "set_integrable M A f" and f_B: "set_integrable M B f" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1564 |
and [measurable]: "A \<in> sets M" "B \<in> sets M" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1565 |
shows "set_integrable M (A \<union> B) f" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1566 |
proof - |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1567 |
have "set_integrable M (A - B) f" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1568 |
using f_A by (rule set_integrable_subset) auto |
63886
685fb01256af
move Henstock-Kurzweil integration after Lebesgue_Measure; replace content by abbreviation measure lborel
hoelzl
parents:
63627
diff
changeset
|
1569 |
from Bochner_Integration.integrable_add[OF this f_B] show ?thesis |
59092
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1570 |
by (rule integrable_cong[THEN iffD1, rotated 2]) (auto split: split_indicator) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1571 |
qed |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1572 |
|
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1573 |
lemma set_integrable_UN: |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1574 |
fixes f :: "_ \<Rightarrow> _ :: {banach, second_countable_topology}" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1575 |
assumes "finite I" "\<And>i. i\<in>I \<Longrightarrow> set_integrable M (A i) f" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1576 |
"\<And>i. i\<in>I \<Longrightarrow> A i \<in> sets M" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1577 |
shows "set_integrable M (\<Union>i\<in>I. A i) f" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1578 |
using assms by (induct I) (auto intro!: set_integrable_Un) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1579 |
|
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1580 |
lemma set_integral_Un: |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1581 |
fixes f :: "_ \<Rightarrow> _ :: {banach, second_countable_topology}" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1582 |
assumes "A \<inter> B = {}" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1583 |
and "set_integrable M A f" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1584 |
and "set_integrable M B f" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1585 |
shows "LINT x:A\<union>B|M. f x = (LINT x:A|M. f x) + (LINT x:B|M. f x)" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1586 |
by (auto simp add: indicator_union_arith indicator_inter_arith[symmetric] |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1587 |
scaleR_add_left assms) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1588 |
|
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1589 |
lemma set_integral_cong_set: |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1590 |
fixes f :: "_ \<Rightarrow> _ :: {banach, second_countable_topology}" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1591 |
assumes [measurable]: "set_borel_measurable M A f" "set_borel_measurable M B f" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1592 |
and ae: "AE x in M. x \<in> A \<longleftrightarrow> x \<in> B" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1593 |
shows "LINT x:B|M. f x = LINT x:A|M. f x" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1594 |
proof (rule integral_cong_AE) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1595 |
show "AE x in M. indicator B x *\<^sub>R f x = indicator A x *\<^sub>R f x" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1596 |
using ae by (auto simp: subset_eq split: split_indicator) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1597 |
qed fact+ |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1598 |
|
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1599 |
lemma set_borel_measurable_subset: |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1600 |
fixes f :: "_ \<Rightarrow> _ :: {banach, second_countable_topology}" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1601 |
assumes [measurable]: "set_borel_measurable M A f" "B \<in> sets M" and "B \<subseteq> A" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1602 |
shows "set_borel_measurable M B f" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1603 |
proof - |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1604 |
have "set_borel_measurable M B (\<lambda>x. indicator A x *\<^sub>R f x)" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1605 |
by measurable |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1606 |
also have "(\<lambda>x. indicator B x *\<^sub>R indicator A x *\<^sub>R f x) = (\<lambda>x. indicator B x *\<^sub>R f x)" |
61808 | 1607 |
using \<open>B \<subseteq> A\<close> by (auto simp: fun_eq_iff split: split_indicator) |
59092
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1608 |
finally show ?thesis . |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1609 |
qed |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1610 |
|
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1611 |
lemma set_integral_Un_AE: |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1612 |
fixes f :: "_ \<Rightarrow> _ :: {banach, second_countable_topology}" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1613 |
assumes ae: "AE x in M. \<not> (x \<in> A \<and> x \<in> B)" and [measurable]: "A \<in> sets M" "B \<in> sets M" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1614 |
and "set_integrable M A f" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1615 |
and "set_integrable M B f" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1616 |
shows "LINT x:A\<union>B|M. f x = (LINT x:A|M. f x) + (LINT x:B|M. f x)" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1617 |
proof - |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1618 |
have f: "set_integrable M (A \<union> B) f" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1619 |
by (intro set_integrable_Un assms) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1620 |
then have f': "set_borel_measurable M (A \<union> B) f" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1621 |
by (rule borel_measurable_integrable) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1622 |
have "LINT x:A\<union>B|M. f x = LINT x:(A - A \<inter> B) \<union> (B - A \<inter> B)|M. f x" |
60615
e5fa1d5d3952
Useful lemmas. The theorem concerning swapping the variables in a double integral.
paulson <lp15@cam.ac.uk>
parents:
59867
diff
changeset
|
1623 |
proof (rule set_integral_cong_set) |
59092
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1624 |
show "AE x in M. (x \<in> A - A \<inter> B \<union> (B - A \<inter> B)) = (x \<in> A \<union> B)" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1625 |
using ae by auto |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1626 |
show "set_borel_measurable M (A - A \<inter> B \<union> (B - A \<inter> B)) f" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1627 |
using f' by (rule set_borel_measurable_subset) auto |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1628 |
qed fact |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1629 |
also have "\<dots> = (LINT x:(A - A \<inter> B)|M. f x) + (LINT x:(B - A \<inter> B)|M. f x)" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1630 |
by (auto intro!: set_integral_Un set_integrable_subset[OF f]) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1631 |
also have "\<dots> = (LINT x:A|M. f x) + (LINT x:B|M. f x)" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1632 |
using ae |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1633 |
by (intro arg_cong2[where f="op+"] set_integral_cong_set) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1634 |
(auto intro!: set_borel_measurable_subset[OF f']) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1635 |
finally show ?thesis . |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1636 |
qed |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1637 |
|
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1638 |
lemma set_integral_finite_Union: |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1639 |
fixes f :: "_ \<Rightarrow> _ :: {banach, second_countable_topology}" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1640 |
assumes "finite I" "disjoint_family_on A I" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1641 |
and "\<And>i. i \<in> I \<Longrightarrow> set_integrable M (A i) f" "\<And>i. i \<in> I \<Longrightarrow> A i \<in> sets M" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1642 |
shows "(LINT x:(\<Union>i\<in>I. A i)|M. f x) = (\<Sum>i\<in>I. LINT x:A i|M. f x)" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1643 |
using assms |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1644 |
apply induct |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1645 |
apply (auto intro!: set_integral_Un set_integrable_Un set_integrable_UN simp: disjoint_family_on_def) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1646 |
by (subst set_integral_Un, auto intro: set_integrable_UN) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1647 |
|
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1648 |
(* TODO: find a better name? *) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1649 |
lemma pos_integrable_to_top: |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1650 |
fixes l::real |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1651 |
assumes "\<And>i. A i \<in> sets M" "mono A" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1652 |
assumes nneg: "\<And>x i. x \<in> A i \<Longrightarrow> 0 \<le> f x" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1653 |
and intgbl: "\<And>i::nat. set_integrable M (A i) f" |
61969 | 1654 |
and lim: "(\<lambda>i::nat. LINT x:A i|M. f x) \<longlonglongrightarrow> l" |
59092
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1655 |
shows "set_integrable M (\<Union>i. A i) f" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1656 |
apply (rule integrable_monotone_convergence[where f = "\<lambda>i::nat. \<lambda>x. indicator (A i) x *\<^sub>R f x" and x = l]) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1657 |
apply (rule intgbl) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1658 |
prefer 3 apply (rule lim) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1659 |
apply (rule AE_I2) |
61808 | 1660 |
using \<open>mono A\<close> apply (auto simp: mono_def nneg split: split_indicator) [] |
59092
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1661 |
proof (rule AE_I2) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1662 |
{ fix x assume "x \<in> space M" |
61969 | 1663 |
show "(\<lambda>i. indicator (A i) x *\<^sub>R f x) \<longlonglongrightarrow> indicator (\<Union>i. A i) x *\<^sub>R f x" |
59092
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1664 |
proof cases |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1665 |
assume "\<exists>i. x \<in> A i" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1666 |
then guess i .. |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1667 |
then have *: "eventually (\<lambda>i. x \<in> A i) sequentially" |
61808 | 1668 |
using \<open>x \<in> A i\<close> \<open>mono A\<close> by (auto simp: eventually_sequentially mono_def) |
59092
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1669 |
show ?thesis |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1670 |
apply (intro Lim_eventually) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1671 |
using * |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1672 |
apply eventually_elim |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1673 |
apply (auto split: split_indicator) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1674 |
done |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1675 |
qed auto } |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1676 |
then show "(\<lambda>x. indicator (\<Union>i. A i) x *\<^sub>R f x) \<in> borel_measurable M" |
62624
59ceeb6f3079
generalized some Borel measurable statements to support ennreal
hoelzl
parents:
62083
diff
changeset
|
1677 |
apply (rule borel_measurable_LIMSEQ_real) |
59092
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1678 |
apply assumption |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1679 |
apply (intro borel_measurable_integrable intgbl) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1680 |
done |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1681 |
qed |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1682 |
|
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1683 |
(* Proof from Royden Real Analysis, p. 91. *) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1684 |
lemma lebesgue_integral_countable_add: |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1685 |
fixes f :: "_ \<Rightarrow> 'a :: {banach, second_countable_topology}" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1686 |
assumes meas[intro]: "\<And>i::nat. A i \<in> sets M" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1687 |
and disj: "\<And>i j. i \<noteq> j \<Longrightarrow> A i \<inter> A j = {}" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1688 |
and intgbl: "set_integrable M (\<Union>i. A i) f" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1689 |
shows "LINT x:(\<Union>i. A i)|M. f x = (\<Sum>i. (LINT x:(A i)|M. f x))" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1690 |
proof (subst integral_suminf[symmetric]) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1691 |
show int_A: "\<And>i. set_integrable M (A i) f" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1692 |
using intgbl by (rule set_integrable_subset) auto |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1693 |
{ fix x assume "x \<in> space M" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1694 |
have "(\<lambda>i. indicator (A i) x *\<^sub>R f x) sums (indicator (\<Union>i. A i) x *\<^sub>R f x)" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1695 |
by (intro sums_scaleR_left indicator_sums) fact } |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1696 |
note sums = this |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1697 |
|
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1698 |
have norm_f: "\<And>i. set_integrable M (A i) (\<lambda>x. norm (f x))" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1699 |
using int_A[THEN integrable_norm] by auto |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1700 |
|
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1701 |
show "AE x in M. summable (\<lambda>i. norm (indicator (A i) x *\<^sub>R f x))" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1702 |
using disj by (intro AE_I2) (auto intro!: summable_mult2 sums_summable[OF indicator_sums]) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1703 |
|
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1704 |
show "summable (\<lambda>i. LINT x|M. norm (indicator (A i) x *\<^sub>R f x))" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1705 |
proof (rule summableI_nonneg_bounded) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1706 |
fix n |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1707 |
show "0 \<le> LINT x|M. norm (indicator (A n) x *\<^sub>R f x)" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1708 |
using norm_f by (auto intro!: integral_nonneg_AE) |
60615
e5fa1d5d3952
Useful lemmas. The theorem concerning swapping the variables in a double integral.
paulson <lp15@cam.ac.uk>
parents:
59867
diff
changeset
|
1709 |
|
59092
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1710 |
have "(\<Sum>i<n. LINT x|M. norm (indicator (A i) x *\<^sub>R f x)) = |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1711 |
(\<Sum>i<n. set_lebesgue_integral M (A i) (\<lambda>x. norm (f x)))" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1712 |
by (simp add: abs_mult) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1713 |
also have "\<dots> = set_lebesgue_integral M (\<Union>i<n. A i) (\<lambda>x. norm (f x))" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1714 |
using norm_f |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1715 |
by (subst set_integral_finite_Union) (auto simp: disjoint_family_on_def disj) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1716 |
also have "\<dots> \<le> set_lebesgue_integral M (\<Union>i. A i) (\<lambda>x. norm (f x))" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1717 |
using intgbl[THEN integrable_norm] |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1718 |
by (intro integral_mono set_integrable_UN[of "{..<n}"] norm_f) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1719 |
(auto split: split_indicator) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1720 |
finally show "(\<Sum>i<n. LINT x|M. norm (indicator (A i) x *\<^sub>R f x)) \<le> |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1721 |
set_lebesgue_integral M (\<Union>i. A i) (\<lambda>x. norm (f x))" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1722 |
by simp |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1723 |
qed |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1724 |
show "set_lebesgue_integral M (UNION UNIV A) f = LINT x|M. (\<Sum>i. indicator (A i) x *\<^sub>R f x)" |
63886
685fb01256af
move Henstock-Kurzweil integration after Lebesgue_Measure; replace content by abbreviation measure lborel
hoelzl
parents:
63627
diff
changeset
|
1725 |
apply (rule Bochner_Integration.integral_cong[OF refl]) |
59092
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1726 |
apply (subst suminf_scaleR_left[OF sums_summable[OF indicator_sums, OF disj], symmetric]) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1727 |
using sums_unique[OF indicator_sums[OF disj]] |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1728 |
apply auto |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1729 |
done |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1730 |
qed |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1731 |
|
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1732 |
lemma set_integral_cont_up: |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1733 |
fixes f :: "_ \<Rightarrow> 'a :: {banach, second_countable_topology}" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1734 |
assumes [measurable]: "\<And>i. A i \<in> sets M" and A: "incseq A" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1735 |
and intgbl: "set_integrable M (\<Union>i. A i) f" |
61969 | 1736 |
shows "(\<lambda>i. LINT x:(A i)|M. f x) \<longlonglongrightarrow> LINT x:(\<Union>i. A i)|M. f x" |
59092
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1737 |
proof (intro integral_dominated_convergence[where w="\<lambda>x. indicator (\<Union>i. A i) x *\<^sub>R norm (f x)"]) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1738 |
have int_A: "\<And>i. set_integrable M (A i) f" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1739 |
using intgbl by (rule set_integrable_subset) auto |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1740 |
then show "\<And>i. set_borel_measurable M (A i) f" "set_borel_measurable M (\<Union>i. A i) f" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1741 |
"set_integrable M (\<Union>i. A i) (\<lambda>x. norm (f x))" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1742 |
using intgbl integrable_norm[OF intgbl] by auto |
60615
e5fa1d5d3952
Useful lemmas. The theorem concerning swapping the variables in a double integral.
paulson <lp15@cam.ac.uk>
parents:
59867
diff
changeset
|
1743 |
|
59092
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1744 |
{ fix x i assume "x \<in> A i" |
61969 | 1745 |
with A have "(\<lambda>xa. indicator (A xa) x::real) \<longlonglongrightarrow> 1 \<longleftrightarrow> (\<lambda>xa. 1::real) \<longlonglongrightarrow> 1" |
59092
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1746 |
by (intro filterlim_cong refl) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1747 |
(fastforce simp: eventually_sequentially incseq_def subset_eq intro!: exI[of _ i]) } |
61969 | 1748 |
then show "AE x in M. (\<lambda>i. indicator (A i) x *\<^sub>R f x) \<longlonglongrightarrow> indicator (\<Union>i. A i) x *\<^sub>R f x" |
59092
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1749 |
by (intro AE_I2 tendsto_intros) (auto split: split_indicator) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1750 |
qed (auto split: split_indicator) |
60615
e5fa1d5d3952
Useful lemmas. The theorem concerning swapping the variables in a double integral.
paulson <lp15@cam.ac.uk>
parents:
59867
diff
changeset
|
1751 |
|
59092
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1752 |
(* Can the int0 hypothesis be dropped? *) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1753 |
lemma set_integral_cont_down: |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1754 |
fixes f :: "_ \<Rightarrow> 'a :: {banach, second_countable_topology}" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1755 |
assumes [measurable]: "\<And>i. A i \<in> sets M" and A: "decseq A" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1756 |
and int0: "set_integrable M (A 0) f" |
61969 | 1757 |
shows "(\<lambda>i::nat. LINT x:(A i)|M. f x) \<longlonglongrightarrow> LINT x:(\<Inter>i. A i)|M. f x" |
59092
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1758 |
proof (rule integral_dominated_convergence) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1759 |
have int_A: "\<And>i. set_integrable M (A i) f" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1760 |
using int0 by (rule set_integrable_subset) (insert A, auto simp: decseq_def) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1761 |
show "set_integrable M (A 0) (\<lambda>x. norm (f x))" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1762 |
using int0[THEN integrable_norm] by simp |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1763 |
have "set_integrable M (\<Inter>i. A i) f" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1764 |
using int0 by (rule set_integrable_subset) (insert A, auto simp: decseq_def) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1765 |
with int_A show "set_borel_measurable M (\<Inter>i. A i) f" "\<And>i. set_borel_measurable M (A i) f" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1766 |
by auto |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1767 |
show "\<And>i. AE x in M. norm (indicator (A i) x *\<^sub>R f x) \<le> indicator (A 0) x *\<^sub>R norm (f x)" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1768 |
using A by (auto split: split_indicator simp: decseq_def) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1769 |
{ fix x i assume "x \<in> space M" "x \<notin> A i" |
61969 | 1770 |
with A have "(\<lambda>i. indicator (A i) x::real) \<longlonglongrightarrow> 0 \<longleftrightarrow> (\<lambda>i. 0::real) \<longlonglongrightarrow> 0" |
59092
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1771 |
by (intro filterlim_cong refl) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1772 |
(auto split: split_indicator simp: eventually_sequentially decseq_def intro!: exI[of _ i]) } |
61969 | 1773 |
then show "AE x in M. (\<lambda>i. indicator (A i) x *\<^sub>R f x) \<longlonglongrightarrow> indicator (\<Inter>i. A i) x *\<^sub>R f x" |
59092
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1774 |
by (intro AE_I2 tendsto_intros) (auto split: split_indicator) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1775 |
qed |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1776 |
|
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1777 |
lemma set_integral_at_point: |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1778 |
fixes a :: real |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1779 |
assumes "set_integrable M {a} f" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1780 |
and [simp]: "{a} \<in> sets M" and "(emeasure M) {a} \<noteq> \<infinity>" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1781 |
shows "(LINT x:{a} | M. f x) = f a * measure M {a}" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1782 |
proof- |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1783 |
have "set_lebesgue_integral M {a} f = set_lebesgue_integral M {a} (%x. f a)" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1784 |
by (intro set_lebesgue_integral_cong) simp_all |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1785 |
then show ?thesis using assms by simp |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1786 |
qed |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1787 |
|
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1788 |
|
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1789 |
abbreviation complex_integrable :: "'a measure \<Rightarrow> ('a \<Rightarrow> complex) \<Rightarrow> bool" where |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1790 |
"complex_integrable M f \<equiv> integrable M f" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1791 |
|
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1792 |
abbreviation complex_lebesgue_integral :: "'a measure \<Rightarrow> ('a \<Rightarrow> complex) \<Rightarrow> complex" ("integral\<^sup>C") where |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1793 |
"integral\<^sup>C M f == integral\<^sup>L M f" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1794 |
|
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1795 |
syntax |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1796 |
"_complex_lebesgue_integral" :: "pttrn \<Rightarrow> complex \<Rightarrow> 'a measure \<Rightarrow> complex" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1797 |
("\<integral>\<^sup>C _. _ \<partial>_" [60,61] 110) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1798 |
|
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1799 |
translations |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1800 |
"\<integral>\<^sup>Cx. f \<partial>M" == "CONST complex_lebesgue_integral M (\<lambda>x. f)" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1801 |
|
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1802 |
syntax |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1803 |
"_ascii_complex_lebesgue_integral" :: "pttrn \<Rightarrow> 'a measure \<Rightarrow> real \<Rightarrow> real" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1804 |
("(3CLINT _|_. _)" [0,110,60] 60) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1805 |
|
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1806 |
translations |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1807 |
"CLINT x|M. f" == "CONST complex_lebesgue_integral M (\<lambda>x. f)" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1808 |
|
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1809 |
lemma complex_integrable_cnj [simp]: |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1810 |
"complex_integrable M (\<lambda>x. cnj (f x)) \<longleftrightarrow> complex_integrable M f" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1811 |
proof |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1812 |
assume "complex_integrable M (\<lambda>x. cnj (f x))" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1813 |
then have "complex_integrable M (\<lambda>x. cnj (cnj (f x)))" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1814 |
by (rule integrable_cnj) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1815 |
then show "complex_integrable M f" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1816 |
by simp |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1817 |
qed simp |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1818 |
|
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1819 |
lemma complex_of_real_integrable_eq: |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1820 |
"complex_integrable M (\<lambda>x. complex_of_real (f x)) \<longleftrightarrow> integrable M f" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1821 |
proof |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1822 |
assume "complex_integrable M (\<lambda>x. complex_of_real (f x))" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1823 |
then have "integrable M (\<lambda>x. Re (complex_of_real (f x)))" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1824 |
by (rule integrable_Re) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1825 |
then show "integrable M f" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1826 |
by simp |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1827 |
qed simp |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1828 |
|
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1829 |
|
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1830 |
abbreviation complex_set_integrable :: "'a measure \<Rightarrow> 'a set \<Rightarrow> ('a \<Rightarrow> complex) \<Rightarrow> bool" where |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1831 |
"complex_set_integrable M A f \<equiv> set_integrable M A f" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1832 |
|
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1833 |
abbreviation complex_set_lebesgue_integral :: "'a measure \<Rightarrow> 'a set \<Rightarrow> ('a \<Rightarrow> complex) \<Rightarrow> complex" where |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1834 |
"complex_set_lebesgue_integral M A f \<equiv> set_lebesgue_integral M A f" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1835 |
|
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1836 |
syntax |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1837 |
"_ascii_complex_set_lebesgue_integral" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'a measure \<Rightarrow> real \<Rightarrow> real" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1838 |
("(4CLINT _:_|_. _)" [0,60,110,61] 60) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1839 |
|
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1840 |
translations |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1841 |
"CLINT x:A|M. f" == "CONST complex_set_lebesgue_integral M A (\<lambda>x. f)" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1842 |
|
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1843 |
lemma set_borel_measurable_continuous: |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1844 |
fixes f :: "_ \<Rightarrow> _::real_normed_vector" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1845 |
assumes "S \<in> sets borel" "continuous_on S f" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1846 |
shows "set_borel_measurable borel S f" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1847 |
proof - |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1848 |
have "(\<lambda>x. if x \<in> S then f x else 0) \<in> borel_measurable borel" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1849 |
by (intro assms borel_measurable_continuous_on_if continuous_on_const) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1850 |
also have "(\<lambda>x. if x \<in> S then f x else 0) = (\<lambda>x. indicator S x *\<^sub>R f x)" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1851 |
by auto |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1852 |
finally show ?thesis . |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1853 |
qed |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1854 |
|
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1855 |
lemma set_measurable_continuous_on_ivl: |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1856 |
assumes "continuous_on {a..b} (f :: real \<Rightarrow> real)" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1857 |
shows "set_borel_measurable borel {a..b} f" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1858 |
by (rule set_borel_measurable_continuous[OF _ assms]) simp |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
1859 |
|
64283
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1860 |
|
64911 | 1861 |
text\<open>This notation is from Sébastien Gouëzel: His use is not directly in line with the |
1862 |
notations in this file, they are more in line with sum, and more readable he thinks.\<close> |
|
64283
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1863 |
|
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1864 |
abbreviation "set_nn_integral M A f \<equiv> nn_integral M (\<lambda>x. f x * indicator A x)" |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1865 |
|
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1866 |
syntax |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1867 |
"_set_nn_integral" :: "pttrn => 'a set => 'a measure => ereal => ereal" |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1868 |
("(\<integral>\<^sup>+((_)\<in>(_)./ _)/\<partial>_)" [0,60,110,61] 60) |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1869 |
|
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1870 |
"_set_lebesgue_integral" :: "pttrn => 'a set => 'a measure => ereal => ereal" |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1871 |
("(\<integral>((_)\<in>(_)./ _)/\<partial>_)" [0,60,110,61] 60) |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1872 |
|
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1873 |
translations |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1874 |
"\<integral>\<^sup>+x \<in> A. f \<partial>M" == "CONST set_nn_integral M A (\<lambda>x. f)" |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1875 |
"\<integral>x \<in> A. f \<partial>M" == "CONST set_lebesgue_integral M A (\<lambda>x. f)" |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1876 |
|
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1877 |
lemma nn_integral_disjoint_pair: |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1878 |
assumes [measurable]: "f \<in> borel_measurable M" |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1879 |
"B \<in> sets M" "C \<in> sets M" |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1880 |
"B \<inter> C = {}" |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1881 |
shows "(\<integral>\<^sup>+x \<in> B \<union> C. f x \<partial>M) = (\<integral>\<^sup>+x \<in> B. f x \<partial>M) + (\<integral>\<^sup>+x \<in> C. f x \<partial>M)" |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1882 |
proof - |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1883 |
have mes: "\<And>D. D \<in> sets M \<Longrightarrow> (\<lambda>x. f x * indicator D x) \<in> borel_measurable M" by simp |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1884 |
have pos: "\<And>D. AE x in M. f x * indicator D x \<ge> 0" using assms(2) by auto |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1885 |
have "\<And>x. f x * indicator (B \<union> C) x = f x * indicator B x + f x * indicator C x" using assms(4) |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1886 |
by (auto split: split_indicator) |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1887 |
then have "(\<integral>\<^sup>+x. f x * indicator (B \<union> C) x \<partial>M) = (\<integral>\<^sup>+x. f x * indicator B x + f x * indicator C x \<partial>M)" |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1888 |
by simp |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1889 |
also have "... = (\<integral>\<^sup>+x. f x * indicator B x \<partial>M) + (\<integral>\<^sup>+x. f x * indicator C x \<partial>M)" |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1890 |
by (rule nn_integral_add) (auto simp add: assms mes pos) |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1891 |
finally show ?thesis by simp |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1892 |
qed |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1893 |
|
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1894 |
lemma nn_integral_disjoint_pair_countspace: |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1895 |
assumes "B \<inter> C = {}" |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1896 |
shows "(\<integral>\<^sup>+x \<in> B \<union> C. f x \<partial>count_space UNIV) = (\<integral>\<^sup>+x \<in> B. f x \<partial>count_space UNIV) + (\<integral>\<^sup>+x \<in> C. f x \<partial>count_space UNIV)" |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1897 |
by (rule nn_integral_disjoint_pair) (simp_all add: assms) |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1898 |
|
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1899 |
lemma nn_integral_null_delta: |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1900 |
assumes "A \<in> sets M" "B \<in> sets M" |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1901 |
"(A - B) \<union> (B - A) \<in> null_sets M" |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1902 |
shows "(\<integral>\<^sup>+x \<in> A. f x \<partial>M) = (\<integral>\<^sup>+x \<in> B. f x \<partial>M)" |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1903 |
proof (rule nn_integral_cong_AE, auto simp add: indicator_def) |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1904 |
have *: "AE a in M. a \<notin> (A - B) \<union> (B - A)" |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1905 |
using assms(3) AE_not_in by blast |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1906 |
then show "AE a in M. a \<notin> A \<longrightarrow> a \<in> B \<longrightarrow> f a = 0" |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1907 |
by auto |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1908 |
show "AE x\<in>A in M. x \<notin> B \<longrightarrow> f x = 0" |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1909 |
using * by auto |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1910 |
qed |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1911 |
|
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1912 |
lemma nn_integral_disjoint_family: |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1913 |
assumes [measurable]: "f \<in> borel_measurable M" "\<And>(n::nat). B n \<in> sets M" |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1914 |
and "disjoint_family B" |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1915 |
shows "(\<integral>\<^sup>+x \<in> (\<Union>n. B n). f x \<partial>M) = (\<Sum>n. (\<integral>\<^sup>+x \<in> B n. f x \<partial>M))" |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1916 |
proof - |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1917 |
have "(\<integral>\<^sup>+ x. (\<Sum>n. f x * indicator (B n) x) \<partial>M) = (\<Sum>n. (\<integral>\<^sup>+ x. f x * indicator (B n) x \<partial>M))" |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1918 |
by (rule nn_integral_suminf) simp |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1919 |
moreover have "(\<Sum>n. f x * indicator (B n) x) = f x * indicator (\<Union>n. B n) x" for x |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1920 |
proof (cases) |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1921 |
assume "x \<in> (\<Union>n. B n)" |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1922 |
then obtain n where "x \<in> B n" by blast |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1923 |
have a: "finite {n}" by simp |
64911 | 1924 |
have "\<And>i. i \<noteq> n \<Longrightarrow> x \<notin> B i" using \<open>x \<in> B n\<close> assms(3) disjoint_family_on_def |
64283
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1925 |
by (metis IntI UNIV_I empty_iff) |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1926 |
then have "\<And>i. i \<notin> {n} \<Longrightarrow> indicator (B i) x = (0::ennreal)" using indicator_def by simp |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1927 |
then have b: "\<And>i. i \<notin> {n} \<Longrightarrow> f x * indicator (B i) x = (0::ennreal)" by simp |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1928 |
|
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1929 |
define h where "h = (\<lambda>i. f x * indicator (B i) x)" |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1930 |
then have "\<And>i. i \<notin> {n} \<Longrightarrow> h i = 0" using b by simp |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1931 |
then have "(\<Sum>i. h i) = (\<Sum>i\<in>{n}. h i)" |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1932 |
by (metis sums_unique[OF sums_finite[OF a]]) |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1933 |
then have "(\<Sum>i. h i) = h n" by simp |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1934 |
then have "(\<Sum>n. f x * indicator (B n) x) = f x * indicator (B n) x" using h_def by simp |
64911 | 1935 |
then have "(\<Sum>n. f x * indicator (B n) x) = f x" using \<open>x \<in> B n\<close> indicator_def by simp |
1936 |
then show ?thesis using \<open>x \<in> (\<Union>n. B n)\<close> by auto |
|
64283
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1937 |
next |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1938 |
assume "x \<notin> (\<Union>n. B n)" |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1939 |
then have "\<And>n. f x * indicator (B n) x = 0" by simp |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1940 |
have "(\<Sum>n. f x * indicator (B n) x) = 0" |
64911 | 1941 |
by (simp add: \<open>\<And>n. f x * indicator (B n) x = 0\<close>) |
1942 |
then show ?thesis using \<open>x \<notin> (\<Union>n. B n)\<close> by auto |
|
64283
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1943 |
qed |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1944 |
ultimately show ?thesis by simp |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1945 |
qed |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1946 |
|
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1947 |
lemma nn_set_integral_add: |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1948 |
assumes [measurable]: "f \<in> borel_measurable M" "g \<in> borel_measurable M" |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1949 |
"A \<in> sets M" |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1950 |
shows "(\<integral>\<^sup>+x \<in> A. (f x + g x) \<partial>M) = (\<integral>\<^sup>+x \<in> A. f x \<partial>M) + (\<integral>\<^sup>+x \<in> A. g x \<partial>M)" |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1951 |
proof - |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1952 |
have "(\<integral>\<^sup>+x \<in> A. (f x + g x) \<partial>M) = (\<integral>\<^sup>+x. (f x * indicator A x + g x * indicator A x) \<partial>M)" |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1953 |
by (auto simp add: indicator_def intro!: nn_integral_cong) |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1954 |
also have "... = (\<integral>\<^sup>+x. f x * indicator A x \<partial>M) + (\<integral>\<^sup>+x. g x * indicator A x \<partial>M)" |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1955 |
apply (rule nn_integral_add) using assms(1) assms(2) by auto |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1956 |
finally show ?thesis by simp |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1957 |
qed |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1958 |
|
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1959 |
lemma nn_set_integral_cong: |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1960 |
assumes "AE x in M. f x = g x" |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1961 |
shows "(\<integral>\<^sup>+x \<in> A. f x \<partial>M) = (\<integral>\<^sup>+x \<in> A. g x \<partial>M)" |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1962 |
apply (rule nn_integral_cong_AE) using assms(1) by auto |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1963 |
|
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1964 |
lemma nn_set_integral_set_mono: |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1965 |
"A \<subseteq> B \<Longrightarrow> (\<integral>\<^sup>+ x \<in> A. f x \<partial>M) \<le> (\<integral>\<^sup>+ x \<in> B. f x \<partial>M)" |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1966 |
by (auto intro!: nn_integral_mono split: split_indicator) |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1967 |
|
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1968 |
lemma nn_set_integral_mono: |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1969 |
assumes [measurable]: "f \<in> borel_measurable M" "g \<in> borel_measurable M" |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1970 |
"A \<in> sets M" |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1971 |
and "AE x\<in>A in M. f x \<le> g x" |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1972 |
shows "(\<integral>\<^sup>+x \<in> A. f x \<partial>M) \<le> (\<integral>\<^sup>+x \<in> A. g x \<partial>M)" |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1973 |
by (auto intro!: nn_integral_mono_AE split: split_indicator simp: assms) |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1974 |
|
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1975 |
lemma nn_set_integral_space [simp]: |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1976 |
shows "(\<integral>\<^sup>+ x \<in> space M. f x \<partial>M) = (\<integral>\<^sup>+x. f x \<partial>M)" |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1977 |
by (metis (mono_tags, lifting) indicator_simps(1) mult.right_neutral nn_integral_cong) |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1978 |
|
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1979 |
lemma nn_integral_count_compose_inj: |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1980 |
assumes "inj_on g A" |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1981 |
shows "(\<integral>\<^sup>+x \<in> A. f (g x) \<partial>count_space UNIV) = (\<integral>\<^sup>+y \<in> g`A. f y \<partial>count_space UNIV)" |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1982 |
proof - |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1983 |
have "(\<integral>\<^sup>+x \<in> A. f (g x) \<partial>count_space UNIV) = (\<integral>\<^sup>+x. f (g x) \<partial>count_space A)" |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1984 |
by (auto simp add: nn_integral_count_space_indicator[symmetric]) |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1985 |
also have "... = (\<integral>\<^sup>+y. f y \<partial>count_space (g`A))" |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1986 |
by (simp add: assms nn_integral_bij_count_space inj_on_imp_bij_betw) |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1987 |
also have "... = (\<integral>\<^sup>+y \<in> g`A. f y \<partial>count_space UNIV)" |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1988 |
by (auto simp add: nn_integral_count_space_indicator[symmetric]) |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1989 |
finally show ?thesis by simp |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1990 |
qed |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1991 |
|
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1992 |
lemma nn_integral_count_compose_bij: |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1993 |
assumes "bij_betw g A B" |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1994 |
shows "(\<integral>\<^sup>+x \<in> A. f (g x) \<partial>count_space UNIV) = (\<integral>\<^sup>+y \<in> B. f y \<partial>count_space UNIV)" |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1995 |
proof - |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1996 |
have "inj_on g A" using assms bij_betw_def by auto |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1997 |
then have "(\<integral>\<^sup>+x \<in> A. f (g x) \<partial>count_space UNIV) = (\<integral>\<^sup>+y \<in> g`A. f y \<partial>count_space UNIV)" |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1998 |
by (rule nn_integral_count_compose_inj) |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
1999 |
then show ?thesis using assms by (simp add: bij_betw_def) |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
2000 |
qed |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
2001 |
|
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
2002 |
lemma set_integral_null_delta: |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
2003 |
fixes f::"_ \<Rightarrow> _ :: {banach, second_countable_topology}" |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
2004 |
assumes [measurable]: "integrable M f" "A \<in> sets M" "B \<in> sets M" |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
2005 |
and "(A - B) \<union> (B - A) \<in> null_sets M" |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
2006 |
shows "(\<integral>x \<in> A. f x \<partial>M) = (\<integral>x \<in> B. f x \<partial>M)" |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
2007 |
proof (rule set_integral_cong_set, auto) |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
2008 |
have *: "AE a in M. a \<notin> (A - B) \<union> (B - A)" |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
2009 |
using assms(4) AE_not_in by blast |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
2010 |
then show "AE x in M. (x \<in> B) = (x \<in> A)" |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
2011 |
by auto |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
2012 |
qed |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
2013 |
|
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
2014 |
lemma set_integral_space: |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
2015 |
assumes "integrable M f" |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
2016 |
shows "(\<integral>x \<in> space M. f x \<partial>M) = (\<integral>x. f x \<partial>M)" |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
2017 |
by (metis (mono_tags, lifting) indicator_simps(1) Bochner_Integration.integral_cong real_vector.scale_one) |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
2018 |
|
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
2019 |
lemma null_if_pos_func_has_zero_nn_int: |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
2020 |
fixes f::"'a \<Rightarrow> ennreal" |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
2021 |
assumes [measurable]: "f \<in> borel_measurable M" "A \<in> sets M" |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
2022 |
and "AE x\<in>A in M. f x > 0" "(\<integral>\<^sup>+x\<in>A. f x \<partial>M) = 0" |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
2023 |
shows "A \<in> null_sets M" |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
2024 |
proof - |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
2025 |
have "AE x in M. f x * indicator A x = 0" |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
2026 |
by (subst nn_integral_0_iff_AE[symmetric], auto simp add: assms(4)) |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
2027 |
then have "AE x\<in>A in M. False" using assms(3) by auto |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
2028 |
then show "A \<in> null_sets M" using assms(2) by (simp add: AE_iff_null_sets) |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
2029 |
qed |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
2030 |
|
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
2031 |
lemma null_if_pos_func_has_zero_int: |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
2032 |
assumes [measurable]: "integrable M f" "A \<in> sets M" |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
2033 |
and "AE x\<in>A in M. f x > 0" "(\<integral>x\<in>A. f x \<partial>M) = (0::real)" |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
2034 |
shows "A \<in> null_sets M" |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
2035 |
proof - |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
2036 |
have "AE x in M. indicator A x * f x = 0" |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
2037 |
apply (subst integral_nonneg_eq_0_iff_AE[symmetric]) |
64911 | 2038 |
using assms integrable_mult_indicator[OF \<open>A \<in> sets M\<close> assms(1)] by auto |
64283
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
2039 |
then have "AE x\<in>A in M. f x = 0" by auto |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
2040 |
then have "AE x\<in>A in M. False" using assms(3) by auto |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
2041 |
then show "A \<in> null_sets M" using assms(2) by (simp add: AE_iff_null_sets) |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
2042 |
qed |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
2043 |
|
64911 | 2044 |
text\<open>The next lemma is a variant of \<open>density_unique\<close>. Note that it uses the notation |
2045 |
for nonnegative set integrals introduced earlier.\<close> |
|
64283
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
2046 |
|
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
2047 |
lemma (in sigma_finite_measure) density_unique2: |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
2048 |
assumes [measurable]: "f \<in> borel_measurable M" "f' \<in> borel_measurable M" |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
2049 |
assumes density_eq: "\<And>A. A \<in> sets M \<Longrightarrow> (\<integral>\<^sup>+ x \<in> A. f x \<partial>M) = (\<integral>\<^sup>+ x \<in> A. f' x \<partial>M)" |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
2050 |
shows "AE x in M. f x = f' x" |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
2051 |
proof (rule density_unique) |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
2052 |
show "density M f = density M f'" |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
2053 |
by (intro measure_eqI) (auto simp: emeasure_density intro!: density_eq) |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
2054 |
qed (auto simp add: assms) |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
2055 |
|
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
2056 |
|
64911 | 2057 |
text \<open>The next lemma implies the same statement for Banach-space valued functions |
64283
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
2058 |
using Hahn-Banach theorem and linear forms. Since they are not yet easily available, I |
64911 | 2059 |
only formulate it for real-valued functions.\<close> |
64283
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
2060 |
|
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
2061 |
lemma density_unique_real: |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
2062 |
fixes f f'::"_ \<Rightarrow> real" |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
2063 |
assumes [measurable]: "integrable M f" "integrable M f'" |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
2064 |
assumes density_eq: "\<And>A. A \<in> sets M \<Longrightarrow> (\<integral>x \<in> A. f x \<partial>M) = (\<integral>x \<in> A. f' x \<partial>M)" |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
2065 |
shows "AE x in M. f x = f' x" |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
2066 |
proof - |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
2067 |
define A where "A = {x \<in> space M. f x < f' x}" |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
2068 |
then have [measurable]: "A \<in> sets M" by simp |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
2069 |
have "(\<integral>x \<in> A. (f' x - f x) \<partial>M) = (\<integral>x \<in> A. f' x \<partial>M) - (\<integral>x \<in> A. f x \<partial>M)" |
64911 | 2070 |
using \<open>A \<in> sets M\<close> assms(1) assms(2) integrable_mult_indicator by blast |
64283
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
2071 |
then have "(\<integral>x \<in> A. (f' x - f x) \<partial>M) = 0" using assms(3) by simp |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
2072 |
then have "A \<in> null_sets M" |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
2073 |
using A_def null_if_pos_func_has_zero_int[where ?f = "\<lambda>x. f' x - f x" and ?A = A] assms by auto |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
2074 |
then have "AE x in M. x \<notin> A" by (simp add: AE_not_in) |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
2075 |
then have *: "AE x in M. f' x \<le> f x" unfolding A_def by auto |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
2076 |
|
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
2077 |
|
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
2078 |
define B where "B = {x \<in> space M. f' x < f x}" |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
2079 |
then have [measurable]: "B \<in> sets M" by simp |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
2080 |
have "(\<integral>x \<in> B. (f x - f' x) \<partial>M) = (\<integral>x \<in> B. f x \<partial>M) - (\<integral>x \<in> B. f' x \<partial>M)" |
64911 | 2081 |
using \<open>B \<in> sets M\<close> assms(1) assms(2) integrable_mult_indicator by blast |
64283
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
2082 |
then have "(\<integral>x \<in> B. (f x - f' x) \<partial>M) = 0" using assms(3) by simp |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
2083 |
then have "B \<in> null_sets M" |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
2084 |
using B_def null_if_pos_func_has_zero_int[where ?f = "\<lambda>x. f x - f' x" and ?A = B] assms by auto |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
2085 |
then have "AE x in M. x \<notin> B" by (simp add: AE_not_in) |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
2086 |
then have "AE x in M. f' x \<ge> f x" unfolding B_def by auto |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
2087 |
|
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
2088 |
then show ?thesis using * by auto |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
2089 |
qed |
979cdfdf7a79
HOL-Probability: move conditional expectation from AFP/Ergodic_Theory
hoelzl
parents:
63958
diff
changeset
|
2090 |
|
64911 | 2091 |
text \<open>The next lemma shows that $L^1$ convergence of a sequence of functions follows from almost |
64284
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2092 |
everywhere convergence and the weaker condition of the convergence of the integrated norms (or even |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2093 |
just the nontrivial inequality about them). Useful in a lot of contexts! This statement (or its |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2094 |
variations) are known as Scheffe lemma. |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2095 |
|
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2096 |
The formalization is more painful as one should jump back and forth between reals and ereals and justify |
64911 | 2097 |
all the time positivity or integrability (thankfully, measurability is handled more or less automatically).\<close> |
64284
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2098 |
|
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2099 |
lemma Scheffe_lemma1: |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2100 |
assumes "\<And>n. integrable M (F n)" "integrable M f" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2101 |
"AE x in M. (\<lambda>n. F n x) \<longlonglongrightarrow> f x" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2102 |
"limsup (\<lambda>n. \<integral>\<^sup>+ x. norm(F n x) \<partial>M) \<le> (\<integral>\<^sup>+ x. norm(f x) \<partial>M)" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2103 |
shows "(\<lambda>n. \<integral>\<^sup>+ x. norm(F n x - f x) \<partial>M) \<longlonglongrightarrow> 0" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2104 |
proof - |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2105 |
have [measurable]: "\<And>n. F n \<in> borel_measurable M" "f \<in> borel_measurable M" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2106 |
using assms(1) assms(2) by simp_all |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2107 |
define G where "G = (\<lambda>n x. norm(f x) + norm(F n x) - norm(F n x - f x))" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2108 |
have [measurable]: "\<And>n. G n \<in> borel_measurable M" unfolding G_def by simp |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2109 |
have G_pos[simp]: "\<And>n x. G n x \<ge> 0" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2110 |
unfolding G_def by (metis ge_iff_diff_ge_0 norm_minus_commute norm_triangle_ineq4) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2111 |
have finint: "(\<integral>\<^sup>+ x. norm(f x) \<partial>M) \<noteq> \<infinity>" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2112 |
using has_bochner_integral_implies_finite_norm[OF has_bochner_integral_integrable[OF \<open>integrable M f\<close>]] |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2113 |
by simp |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2114 |
then have fin2: "2 * (\<integral>\<^sup>+ x. norm(f x) \<partial>M) \<noteq> \<infinity>" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2115 |
by (auto simp: ennreal_mult_eq_top_iff) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2116 |
|
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2117 |
{ |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2118 |
fix x assume *: "(\<lambda>n. F n x) \<longlonglongrightarrow> f x" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2119 |
then have "(\<lambda>n. norm(F n x)) \<longlonglongrightarrow> norm(f x)" using tendsto_norm by blast |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2120 |
moreover have "(\<lambda>n. norm(F n x - f x)) \<longlonglongrightarrow> 0" using * Lim_null tendsto_norm_zero_iff by fastforce |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2121 |
ultimately have a: "(\<lambda>n. norm(F n x) - norm(F n x - f x)) \<longlonglongrightarrow> norm(f x)" using tendsto_diff by fastforce |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2122 |
have "(\<lambda>n. norm(f x) + (norm(F n x) - norm(F n x - f x))) \<longlonglongrightarrow> norm(f x) + norm(f x)" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2123 |
by (rule tendsto_add) (auto simp add: a) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2124 |
moreover have "\<And>n. G n x = norm(f x) + (norm(F n x) - norm(F n x - f x))" unfolding G_def by simp |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2125 |
ultimately have "(\<lambda>n. G n x) \<longlonglongrightarrow> 2 * norm(f x)" by simp |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2126 |
then have "(\<lambda>n. ennreal(G n x)) \<longlonglongrightarrow> ennreal(2 * norm(f x))" by simp |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2127 |
then have "liminf (\<lambda>n. ennreal(G n x)) = ennreal(2 * norm(f x))" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2128 |
using sequentially_bot tendsto_iff_Liminf_eq_Limsup by blast |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2129 |
} |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2130 |
then have "AE x in M. liminf (\<lambda>n. ennreal(G n x)) = ennreal(2 * norm(f x))" using assms(3) by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2131 |
then have "(\<integral>\<^sup>+ x. liminf (\<lambda>n. ennreal (G n x)) \<partial>M) = (\<integral>\<^sup>+ x. 2 * ennreal(norm(f x)) \<partial>M)" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2132 |
by (simp add: nn_integral_cong_AE ennreal_mult) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2133 |
also have "... = 2 * (\<integral>\<^sup>+ x. norm(f x) \<partial>M)" by (rule nn_integral_cmult) auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2134 |
finally have int_liminf: "(\<integral>\<^sup>+ x. liminf (\<lambda>n. ennreal (G n x)) \<partial>M) = 2 * (\<integral>\<^sup>+ x. norm(f x) \<partial>M)" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2135 |
by simp |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2136 |
|
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2137 |
have "(\<integral>\<^sup>+x. ennreal(norm(f x)) + ennreal(norm(F n x)) \<partial>M) = (\<integral>\<^sup>+x. norm(f x) \<partial>M) + (\<integral>\<^sup>+x. norm(F n x) \<partial>M)" for n |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2138 |
by (rule nn_integral_add) (auto simp add: assms) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2139 |
then have "limsup (\<lambda>n. (\<integral>\<^sup>+x. ennreal(norm(f x)) + ennreal(norm(F n x)) \<partial>M)) = |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2140 |
limsup (\<lambda>n. (\<integral>\<^sup>+x. norm(f x) \<partial>M) + (\<integral>\<^sup>+x. norm(F n x) \<partial>M))" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2141 |
by simp |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2142 |
also have "... = (\<integral>\<^sup>+x. norm(f x) \<partial>M) + limsup (\<lambda>n. (\<integral>\<^sup>+x. norm(F n x) \<partial>M))" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2143 |
by (rule Limsup_const_add, auto simp add: finint) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2144 |
also have "... \<le> (\<integral>\<^sup>+x. norm(f x) \<partial>M) + (\<integral>\<^sup>+x. norm(f x) \<partial>M)" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2145 |
using assms(4) by (simp add: add_left_mono) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2146 |
also have "... = 2 * (\<integral>\<^sup>+x. norm(f x) \<partial>M)" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2147 |
unfolding one_add_one[symmetric] distrib_right by simp |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2148 |
ultimately have a: "limsup (\<lambda>n. (\<integral>\<^sup>+x. ennreal(norm(f x)) + ennreal(norm(F n x)) \<partial>M)) \<le> |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2149 |
2 * (\<integral>\<^sup>+x. norm(f x) \<partial>M)" by simp |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2150 |
|
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2151 |
have le: "ennreal (norm (F n x - f x)) \<le> ennreal (norm (f x)) + ennreal (norm (F n x))" for n x |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2152 |
by (simp add: norm_minus_commute norm_triangle_ineq4 ennreal_plus[symmetric] ennreal_minus del: ennreal_plus) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2153 |
then have le2: "(\<integral>\<^sup>+ x. ennreal (norm (F n x - f x)) \<partial>M) \<le> (\<integral>\<^sup>+ x. ennreal (norm (f x)) + ennreal (norm (F n x)) \<partial>M)" for n |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2154 |
by (rule nn_integral_mono) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2155 |
|
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2156 |
have "2 * (\<integral>\<^sup>+ x. norm(f x) \<partial>M) = (\<integral>\<^sup>+ x. liminf (\<lambda>n. ennreal (G n x)) \<partial>M)" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2157 |
by (simp add: int_liminf) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2158 |
also have "\<dots> \<le> liminf (\<lambda>n. (\<integral>\<^sup>+x. G n x \<partial>M))" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2159 |
by (rule nn_integral_liminf) auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2160 |
also have "liminf (\<lambda>n. (\<integral>\<^sup>+x. G n x \<partial>M)) = |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2161 |
liminf (\<lambda>n. (\<integral>\<^sup>+x. ennreal(norm(f x)) + ennreal(norm(F n x)) \<partial>M) - (\<integral>\<^sup>+x. norm(F n x - f x) \<partial>M))" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2162 |
proof (intro arg_cong[where f=liminf] ext) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2163 |
fix n |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2164 |
have "\<And>x. ennreal(G n x) = ennreal(norm(f x)) + ennreal(norm(F n x)) - ennreal(norm(F n x - f x))" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2165 |
unfolding G_def by (simp add: ennreal_plus[symmetric] ennreal_minus del: ennreal_plus) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2166 |
moreover have "(\<integral>\<^sup>+x. ennreal(norm(f x)) + ennreal(norm(F n x)) - ennreal(norm(F n x - f x)) \<partial>M) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2167 |
= (\<integral>\<^sup>+x. ennreal(norm(f x)) + ennreal(norm(F n x)) \<partial>M) - (\<integral>\<^sup>+x. norm(F n x - f x) \<partial>M)" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2168 |
proof (rule nn_integral_diff) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2169 |
from le show "AE x in M. ennreal (norm (F n x - f x)) \<le> ennreal (norm (f x)) + ennreal (norm (F n x))" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2170 |
by simp |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2171 |
from le2 have "(\<integral>\<^sup>+x. ennreal (norm (F n x - f x)) \<partial>M) < \<infinity>" using assms(1) assms(2) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2172 |
by (metis has_bochner_integral_implies_finite_norm integrable.simps Bochner_Integration.integrable_diff) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2173 |
then show "(\<integral>\<^sup>+x. ennreal (norm (F n x - f x)) \<partial>M) \<noteq> \<infinity>" by simp |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2174 |
qed (auto simp add: assms) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2175 |
ultimately show "(\<integral>\<^sup>+x. G n x \<partial>M) = (\<integral>\<^sup>+x. ennreal(norm(f x)) + ennreal(norm(F n x)) \<partial>M) - (\<integral>\<^sup>+x. norm(F n x - f x) \<partial>M)" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2176 |
by simp |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2177 |
qed |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2178 |
finally have "2 * (\<integral>\<^sup>+ x. norm(f x) \<partial>M) + limsup (\<lambda>n. (\<integral>\<^sup>+x. norm(F n x - f x) \<partial>M)) \<le> |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2179 |
liminf (\<lambda>n. (\<integral>\<^sup>+x. ennreal(norm(f x)) + ennreal(norm(F n x)) \<partial>M) - (\<integral>\<^sup>+x. norm(F n x - f x) \<partial>M)) + |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2180 |
limsup (\<lambda>n. (\<integral>\<^sup>+x. norm(F n x - f x) \<partial>M))" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2181 |
by (intro add_mono) auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2182 |
also have "\<dots> \<le> (limsup (\<lambda>n. \<integral>\<^sup>+x. ennreal(norm(f x)) + ennreal(norm(F n x)) \<partial>M) - limsup (\<lambda>n. \<integral>\<^sup>+x. norm (F n x - f x) \<partial>M)) + |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2183 |
limsup (\<lambda>n. (\<integral>\<^sup>+x. norm(F n x - f x) \<partial>M))" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2184 |
by (intro add_mono liminf_minus_ennreal le2) auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2185 |
also have "\<dots> = limsup (\<lambda>n. (\<integral>\<^sup>+x. ennreal(norm(f x)) + ennreal(norm(F n x)) \<partial>M))" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2186 |
by (intro diff_add_cancel_ennreal Limsup_mono always_eventually allI le2) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2187 |
also have "\<dots> \<le> 2 * (\<integral>\<^sup>+x. norm(f x) \<partial>M)" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2188 |
by fact |
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HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
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parents:
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|
2189 |
finally have "limsup (\<lambda>n. (\<integral>\<^sup>+x. norm(F n x - f x) \<partial>M)) = 0" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
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changeset
|
2190 |
using fin2 by simp |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
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changeset
|
2191 |
then show ?thesis |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
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changeset
|
2192 |
by (rule tendsto_0_if_Limsup_eq_0_ennreal) |
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HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
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parents:
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changeset
|
2193 |
qed |
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HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2194 |
|
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
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changeset
|
2195 |
lemma Scheffe_lemma2: |
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HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
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changeset
|
2196 |
fixes F::"nat \<Rightarrow> 'a \<Rightarrow> 'b::{banach, second_countable_topology}" |
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HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2197 |
assumes "\<And> n::nat. F n \<in> borel_measurable M" "integrable M f" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2198 |
"AE x in M. (\<lambda>n. F n x) \<longlonglongrightarrow> f x" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2199 |
"\<And>n. (\<integral>\<^sup>+ x. norm(F n x) \<partial>M) \<le> (\<integral>\<^sup>+ x. norm(f x) \<partial>M)" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2200 |
shows "(\<lambda>n. \<integral>\<^sup>+ x. norm(F n x - f x) \<partial>M) \<longlonglongrightarrow> 0" |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2201 |
proof (rule Scheffe_lemma1) |
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HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2202 |
fix n::nat |
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HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2203 |
have "(\<integral>\<^sup>+ x. norm(f x) \<partial>M) < \<infinity>" using assms(2) by (metis has_bochner_integral_implies_finite_norm integrable.cases) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2204 |
then have "(\<integral>\<^sup>+ x. norm(F n x) \<partial>M) < \<infinity>" using assms(4)[of n] by auto |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2205 |
then show "integrable M (F n)" by (subst integrable_iff_bounded, simp add: assms(1)[of n]) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2206 |
qed (auto simp add: assms Limsup_bounded) |
f3b905b2eee2
HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents:
64283
diff
changeset
|
2207 |
|
59092
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
2208 |
end |