author | hoelzl |
Fri, 08 Jan 2016 17:40:59 +0100 | |
changeset 62101 | 26c0a70f78a3 |
parent 61955 | e96292f32c3c |
child 62102 | 877463945ce9 |
permissions | -rw-r--r-- |
60036 | 1 |
(* Title: HOL/Filter.thy |
2 |
Author: Brian Huffman |
|
3 |
Author: Johannes Hölzl |
|
4 |
*) |
|
5 |
||
60758 | 6 |
section \<open>Filters on predicates\<close> |
60036 | 7 |
|
8 |
theory Filter |
|
9 |
imports Set_Interval Lifting_Set |
|
10 |
begin |
|
11 |
||
60758 | 12 |
subsection \<open>Filters\<close> |
60036 | 13 |
|
60758 | 14 |
text \<open> |
60036 | 15 |
This definition also allows non-proper filters. |
60758 | 16 |
\<close> |
60036 | 17 |
|
18 |
locale is_filter = |
|
19 |
fixes F :: "('a \<Rightarrow> bool) \<Rightarrow> bool" |
|
20 |
assumes True: "F (\<lambda>x. True)" |
|
21 |
assumes conj: "F (\<lambda>x. P x) \<Longrightarrow> F (\<lambda>x. Q x) \<Longrightarrow> F (\<lambda>x. P x \<and> Q x)" |
|
22 |
assumes mono: "\<forall>x. P x \<longrightarrow> Q x \<Longrightarrow> F (\<lambda>x. P x) \<Longrightarrow> F (\<lambda>x. Q x)" |
|
23 |
||
24 |
typedef 'a filter = "{F :: ('a \<Rightarrow> bool) \<Rightarrow> bool. is_filter F}" |
|
25 |
proof |
|
26 |
show "(\<lambda>x. True) \<in> ?filter" by (auto intro: is_filter.intro) |
|
27 |
qed |
|
28 |
||
29 |
lemma is_filter_Rep_filter: "is_filter (Rep_filter F)" |
|
30 |
using Rep_filter [of F] by simp |
|
31 |
||
32 |
lemma Abs_filter_inverse': |
|
33 |
assumes "is_filter F" shows "Rep_filter (Abs_filter F) = F" |
|
34 |
using assms by (simp add: Abs_filter_inverse) |
|
35 |
||
36 |
||
60758 | 37 |
subsubsection \<open>Eventually\<close> |
60036 | 38 |
|
39 |
definition eventually :: "('a \<Rightarrow> bool) \<Rightarrow> 'a filter \<Rightarrow> bool" |
|
40 |
where "eventually P F \<longleftrightarrow> Rep_filter F P" |
|
41 |
||
61953 | 42 |
syntax |
43 |
"_eventually" :: "pttrn => 'a filter => bool => bool" ("(3\<forall>\<^sub>F _ in _./ _)" [0, 0, 10] 10) |
|
60037 | 44 |
translations |
60038 | 45 |
"\<forall>\<^sub>Fx in F. P" == "CONST eventually (\<lambda>x. P) F" |
60037 | 46 |
|
60036 | 47 |
lemma eventually_Abs_filter: |
48 |
assumes "is_filter F" shows "eventually P (Abs_filter F) = F P" |
|
49 |
unfolding eventually_def using assms by (simp add: Abs_filter_inverse) |
|
50 |
||
51 |
lemma filter_eq_iff: |
|
52 |
shows "F = F' \<longleftrightarrow> (\<forall>P. eventually P F = eventually P F')" |
|
53 |
unfolding Rep_filter_inject [symmetric] fun_eq_iff eventually_def .. |
|
54 |
||
55 |
lemma eventually_True [simp]: "eventually (\<lambda>x. True) F" |
|
56 |
unfolding eventually_def |
|
57 |
by (rule is_filter.True [OF is_filter_Rep_filter]) |
|
58 |
||
59 |
lemma always_eventually: "\<forall>x. P x \<Longrightarrow> eventually P F" |
|
60 |
proof - |
|
61 |
assume "\<forall>x. P x" hence "P = (\<lambda>x. True)" by (simp add: ext) |
|
62 |
thus "eventually P F" by simp |
|
63 |
qed |
|
64 |
||
60040
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
65 |
lemma eventuallyI: "(\<And>x. P x) \<Longrightarrow> eventually P F" |
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
66 |
by (auto intro: always_eventually) |
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
67 |
|
60036 | 68 |
lemma eventually_mono: |
61806
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61531
diff
changeset
|
69 |
"\<lbrakk>eventually P F; \<And>x. P x \<Longrightarrow> Q x\<rbrakk> \<Longrightarrow> eventually Q F" |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61531
diff
changeset
|
70 |
unfolding eventually_def |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61531
diff
changeset
|
71 |
by (blast intro: is_filter.mono [OF is_filter_Rep_filter]) |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61531
diff
changeset
|
72 |
|
60036 | 73 |
lemma eventually_conj: |
74 |
assumes P: "eventually (\<lambda>x. P x) F" |
|
75 |
assumes Q: "eventually (\<lambda>x. Q x) F" |
|
76 |
shows "eventually (\<lambda>x. P x \<and> Q x) F" |
|
77 |
using assms unfolding eventually_def |
|
78 |
by (rule is_filter.conj [OF is_filter_Rep_filter]) |
|
79 |
||
80 |
lemma eventually_mp: |
|
81 |
assumes "eventually (\<lambda>x. P x \<longrightarrow> Q x) F" |
|
82 |
assumes "eventually (\<lambda>x. P x) F" |
|
83 |
shows "eventually (\<lambda>x. Q x) F" |
|
61806
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61531
diff
changeset
|
84 |
proof - |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61531
diff
changeset
|
85 |
have "eventually (\<lambda>x. (P x \<longrightarrow> Q x) \<and> P x) F" |
60036 | 86 |
using assms by (rule eventually_conj) |
61806
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61531
diff
changeset
|
87 |
then show ?thesis |
61810 | 88 |
by (blast intro: eventually_mono) |
60036 | 89 |
qed |
90 |
||
91 |
lemma eventually_rev_mp: |
|
92 |
assumes "eventually (\<lambda>x. P x) F" |
|
93 |
assumes "eventually (\<lambda>x. P x \<longrightarrow> Q x) F" |
|
94 |
shows "eventually (\<lambda>x. Q x) F" |
|
95 |
using assms(2) assms(1) by (rule eventually_mp) |
|
96 |
||
97 |
lemma eventually_conj_iff: |
|
98 |
"eventually (\<lambda>x. P x \<and> Q x) F \<longleftrightarrow> eventually P F \<and> eventually Q F" |
|
99 |
by (auto intro: eventually_conj elim: eventually_rev_mp) |
|
100 |
||
101 |
lemma eventually_elim2: |
|
102 |
assumes "eventually (\<lambda>i. P i) F" |
|
103 |
assumes "eventually (\<lambda>i. Q i) F" |
|
104 |
assumes "\<And>i. P i \<Longrightarrow> Q i \<Longrightarrow> R i" |
|
105 |
shows "eventually (\<lambda>i. R i) F" |
|
106 |
using assms by (auto elim!: eventually_rev_mp) |
|
107 |
||
60040
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
108 |
lemma eventually_ball_finite_distrib: |
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
109 |
"finite A \<Longrightarrow> (eventually (\<lambda>x. \<forall>y\<in>A. P x y) net) \<longleftrightarrow> (\<forall>y\<in>A. eventually (\<lambda>x. P x y) net)" |
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
110 |
by (induction A rule: finite_induct) (auto simp: eventually_conj_iff) |
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
111 |
|
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
112 |
lemma eventually_ball_finite: |
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
113 |
"finite A \<Longrightarrow> \<forall>y\<in>A. eventually (\<lambda>x. P x y) net \<Longrightarrow> eventually (\<lambda>x. \<forall>y\<in>A. P x y) net" |
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
114 |
by (auto simp: eventually_ball_finite_distrib) |
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
115 |
|
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
116 |
lemma eventually_all_finite: |
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
117 |
fixes P :: "'a \<Rightarrow> 'b::finite \<Rightarrow> bool" |
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
118 |
assumes "\<And>y. eventually (\<lambda>x. P x y) net" |
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
119 |
shows "eventually (\<lambda>x. \<forall>y. P x y) net" |
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
120 |
using eventually_ball_finite [of UNIV P] assms by simp |
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
121 |
|
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
122 |
lemma eventually_ex: "(\<forall>\<^sub>Fx in F. \<exists>y. P x y) \<longleftrightarrow> (\<exists>Y. \<forall>\<^sub>Fx in F. P x (Y x))" |
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
123 |
proof |
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
124 |
assume "\<forall>\<^sub>Fx in F. \<exists>y. P x y" |
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
125 |
then have "\<forall>\<^sub>Fx in F. P x (SOME y. P x y)" |
61810 | 126 |
by (auto intro: someI_ex eventually_mono) |
60040
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
127 |
then show "\<exists>Y. \<forall>\<^sub>Fx in F. P x (Y x)" |
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
128 |
by auto |
61810 | 129 |
qed (auto intro: eventually_mono) |
60040
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
130 |
|
60036 | 131 |
lemma not_eventually_impI: "eventually P F \<Longrightarrow> \<not> eventually Q F \<Longrightarrow> \<not> eventually (\<lambda>x. P x \<longrightarrow> Q x) F" |
132 |
by (auto intro: eventually_mp) |
|
133 |
||
134 |
lemma not_eventuallyD: "\<not> eventually P F \<Longrightarrow> \<exists>x. \<not> P x" |
|
135 |
by (metis always_eventually) |
|
136 |
||
137 |
lemma eventually_subst: |
|
138 |
assumes "eventually (\<lambda>n. P n = Q n) F" |
|
139 |
shows "eventually P F = eventually Q F" (is "?L = ?R") |
|
140 |
proof - |
|
141 |
from assms have "eventually (\<lambda>x. P x \<longrightarrow> Q x) F" |
|
142 |
and "eventually (\<lambda>x. Q x \<longrightarrow> P x) F" |
|
61810 | 143 |
by (auto elim: eventually_mono) |
60036 | 144 |
then show ?thesis by (auto elim: eventually_elim2) |
145 |
qed |
|
146 |
||
60040
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
147 |
subsection \<open> Frequently as dual to eventually \<close> |
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
148 |
|
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
149 |
definition frequently :: "('a \<Rightarrow> bool) \<Rightarrow> 'a filter \<Rightarrow> bool" |
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
150 |
where "frequently P F \<longleftrightarrow> \<not> eventually (\<lambda>x. \<not> P x) F" |
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
151 |
|
61953 | 152 |
syntax |
153 |
"_frequently" :: "pttrn \<Rightarrow> 'a filter \<Rightarrow> bool \<Rightarrow> bool" ("(3\<exists>\<^sub>F _ in _./ _)" [0, 0, 10] 10) |
|
60040
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
154 |
translations |
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
155 |
"\<exists>\<^sub>Fx in F. P" == "CONST frequently (\<lambda>x. P) F" |
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
156 |
|
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
157 |
lemma not_frequently_False [simp]: "\<not> (\<exists>\<^sub>Fx in F. False)" |
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
158 |
by (simp add: frequently_def) |
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
159 |
|
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
160 |
lemma frequently_ex: "\<exists>\<^sub>Fx in F. P x \<Longrightarrow> \<exists>x. P x" |
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
161 |
by (auto simp: frequently_def dest: not_eventuallyD) |
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
162 |
|
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
163 |
lemma frequentlyE: assumes "frequently P F" obtains x where "P x" |
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
164 |
using frequently_ex[OF assms] by auto |
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
165 |
|
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
166 |
lemma frequently_mp: |
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
167 |
assumes ev: "\<forall>\<^sub>Fx in F. P x \<longrightarrow> Q x" and P: "\<exists>\<^sub>Fx in F. P x" shows "\<exists>\<^sub>Fx in F. Q x" |
61806
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61531
diff
changeset
|
168 |
proof - |
60040
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
169 |
from ev have "eventually (\<lambda>x. \<not> Q x \<longrightarrow> \<not> P x) F" |
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
170 |
by (rule eventually_rev_mp) (auto intro!: always_eventually) |
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
171 |
from eventually_mp[OF this] P show ?thesis |
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
172 |
by (auto simp: frequently_def) |
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
173 |
qed |
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
174 |
|
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
175 |
lemma frequently_rev_mp: |
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
176 |
assumes "\<exists>\<^sub>Fx in F. P x" |
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
177 |
assumes "\<forall>\<^sub>Fx in F. P x \<longrightarrow> Q x" |
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
178 |
shows "\<exists>\<^sub>Fx in F. Q x" |
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
179 |
using assms(2) assms(1) by (rule frequently_mp) |
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
180 |
|
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
181 |
lemma frequently_mono: "(\<forall>x. P x \<longrightarrow> Q x) \<Longrightarrow> frequently P F \<Longrightarrow> frequently Q F" |
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
182 |
using frequently_mp[of P Q] by (simp add: always_eventually) |
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
183 |
|
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
184 |
lemma frequently_elim1: "\<exists>\<^sub>Fx in F. P x \<Longrightarrow> (\<And>i. P i \<Longrightarrow> Q i) \<Longrightarrow> \<exists>\<^sub>Fx in F. Q x" |
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
185 |
by (metis frequently_mono) |
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
186 |
|
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
187 |
lemma frequently_disj_iff: "(\<exists>\<^sub>Fx in F. P x \<or> Q x) \<longleftrightarrow> (\<exists>\<^sub>Fx in F. P x) \<or> (\<exists>\<^sub>Fx in F. Q x)" |
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
188 |
by (simp add: frequently_def eventually_conj_iff) |
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
189 |
|
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
190 |
lemma frequently_disj: "\<exists>\<^sub>Fx in F. P x \<Longrightarrow> \<exists>\<^sub>Fx in F. Q x \<Longrightarrow> \<exists>\<^sub>Fx in F. P x \<or> Q x" |
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
191 |
by (simp add: frequently_disj_iff) |
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
192 |
|
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
193 |
lemma frequently_bex_finite_distrib: |
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
194 |
assumes "finite A" shows "(\<exists>\<^sub>Fx in F. \<exists>y\<in>A. P x y) \<longleftrightarrow> (\<exists>y\<in>A. \<exists>\<^sub>Fx in F. P x y)" |
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
195 |
using assms by induction (auto simp: frequently_disj_iff) |
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
196 |
|
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
197 |
lemma frequently_bex_finite: "finite A \<Longrightarrow> \<exists>\<^sub>Fx in F. \<exists>y\<in>A. P x y \<Longrightarrow> \<exists>y\<in>A. \<exists>\<^sub>Fx in F. P x y" |
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
198 |
by (simp add: frequently_bex_finite_distrib) |
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
199 |
|
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
200 |
lemma frequently_all: "(\<exists>\<^sub>Fx in F. \<forall>y. P x y) \<longleftrightarrow> (\<forall>Y. \<exists>\<^sub>Fx in F. P x (Y x))" |
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
201 |
using eventually_ex[of "\<lambda>x y. \<not> P x y" F] by (simp add: frequently_def) |
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
202 |
|
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
203 |
lemma |
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
204 |
shows not_eventually: "\<not> eventually P F \<longleftrightarrow> (\<exists>\<^sub>Fx in F. \<not> P x)" |
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
205 |
and not_frequently: "\<not> frequently P F \<longleftrightarrow> (\<forall>\<^sub>Fx in F. \<not> P x)" |
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
206 |
by (auto simp: frequently_def) |
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
207 |
|
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
208 |
lemma frequently_imp_iff: |
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
209 |
"(\<exists>\<^sub>Fx in F. P x \<longrightarrow> Q x) \<longleftrightarrow> (eventually P F \<longrightarrow> frequently Q F)" |
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
210 |
unfolding imp_conv_disj frequently_disj_iff not_eventually[symmetric] .. |
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
211 |
|
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
212 |
lemma eventually_frequently_const_simps: |
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
213 |
"(\<exists>\<^sub>Fx in F. P x \<and> C) \<longleftrightarrow> (\<exists>\<^sub>Fx in F. P x) \<and> C" |
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
214 |
"(\<exists>\<^sub>Fx in F. C \<and> P x) \<longleftrightarrow> C \<and> (\<exists>\<^sub>Fx in F. P x)" |
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
215 |
"(\<forall>\<^sub>Fx in F. P x \<or> C) \<longleftrightarrow> (\<forall>\<^sub>Fx in F. P x) \<or> C" |
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
216 |
"(\<forall>\<^sub>Fx in F. C \<or> P x) \<longleftrightarrow> C \<or> (\<forall>\<^sub>Fx in F. P x)" |
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
217 |
"(\<forall>\<^sub>Fx in F. P x \<longrightarrow> C) \<longleftrightarrow> ((\<exists>\<^sub>Fx in F. P x) \<longrightarrow> C)" |
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
218 |
"(\<forall>\<^sub>Fx in F. C \<longrightarrow> P x) \<longleftrightarrow> (C \<longrightarrow> (\<forall>\<^sub>Fx in F. P x))" |
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
219 |
by (cases C; simp add: not_frequently)+ |
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
220 |
|
61806
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61531
diff
changeset
|
221 |
lemmas eventually_frequently_simps = |
60040
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
222 |
eventually_frequently_const_simps |
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
223 |
not_eventually |
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
224 |
eventually_conj_iff |
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
225 |
eventually_ball_finite_distrib |
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
226 |
eventually_ex |
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
227 |
not_frequently |
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
228 |
frequently_disj_iff |
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
229 |
frequently_bex_finite_distrib |
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
230 |
frequently_all |
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
231 |
frequently_imp_iff |
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
232 |
|
60758 | 233 |
ML \<open> |
61841
4d3527b94f2a
more general types Proof.method / context_tactic;
wenzelm
parents:
61810
diff
changeset
|
234 |
fun eventually_elim_tac facts = |
4d3527b94f2a
more general types Proof.method / context_tactic;
wenzelm
parents:
61810
diff
changeset
|
235 |
CONTEXT_SUBGOAL (fn (goal, i) => fn (ctxt, st) => |
4d3527b94f2a
more general types Proof.method / context_tactic;
wenzelm
parents:
61810
diff
changeset
|
236 |
let |
4d3527b94f2a
more general types Proof.method / context_tactic;
wenzelm
parents:
61810
diff
changeset
|
237 |
val mp_thms = facts RL @{thms eventually_rev_mp} |
4d3527b94f2a
more general types Proof.method / context_tactic;
wenzelm
parents:
61810
diff
changeset
|
238 |
val raw_elim_thm = |
4d3527b94f2a
more general types Proof.method / context_tactic;
wenzelm
parents:
61810
diff
changeset
|
239 |
(@{thm allI} RS @{thm always_eventually}) |
4d3527b94f2a
more general types Proof.method / context_tactic;
wenzelm
parents:
61810
diff
changeset
|
240 |
|> fold (fn thm1 => fn thm2 => thm2 RS thm1) mp_thms |
4d3527b94f2a
more general types Proof.method / context_tactic;
wenzelm
parents:
61810
diff
changeset
|
241 |
|> fold (fn _ => fn thm => @{thm impI} RS thm) facts |
4d3527b94f2a
more general types Proof.method / context_tactic;
wenzelm
parents:
61810
diff
changeset
|
242 |
val cases_prop = |
4d3527b94f2a
more general types Proof.method / context_tactic;
wenzelm
parents:
61810
diff
changeset
|
243 |
Thm.prop_of |
4d3527b94f2a
more general types Proof.method / context_tactic;
wenzelm
parents:
61810
diff
changeset
|
244 |
(Rule_Cases.internalize_params (raw_elim_thm RS Goal.init (Thm.cterm_of ctxt goal))) |
4d3527b94f2a
more general types Proof.method / context_tactic;
wenzelm
parents:
61810
diff
changeset
|
245 |
val cases = Rule_Cases.make_common ctxt cases_prop [(("elim", []), [])] |
4d3527b94f2a
more general types Proof.method / context_tactic;
wenzelm
parents:
61810
diff
changeset
|
246 |
in CONTEXT_CASES cases (resolve_tac ctxt [raw_elim_thm] i) (ctxt, st) end) |
60758 | 247 |
\<close> |
60036 | 248 |
|
60758 | 249 |
method_setup eventually_elim = \<open> |
61841
4d3527b94f2a
more general types Proof.method / context_tactic;
wenzelm
parents:
61810
diff
changeset
|
250 |
Scan.succeed (fn _ => CONTEXT_METHOD (fn facts => eventually_elim_tac facts 1)) |
60758 | 251 |
\<close> "elimination of eventually quantifiers" |
60036 | 252 |
|
60758 | 253 |
subsubsection \<open>Finer-than relation\<close> |
60036 | 254 |
|
60758 | 255 |
text \<open>@{term "F \<le> F'"} means that filter @{term F} is finer than |
256 |
filter @{term F'}.\<close> |
|
60036 | 257 |
|
258 |
instantiation filter :: (type) complete_lattice |
|
259 |
begin |
|
260 |
||
261 |
definition le_filter_def: |
|
262 |
"F \<le> F' \<longleftrightarrow> (\<forall>P. eventually P F' \<longrightarrow> eventually P F)" |
|
263 |
||
264 |
definition |
|
265 |
"(F :: 'a filter) < F' \<longleftrightarrow> F \<le> F' \<and> \<not> F' \<le> F" |
|
266 |
||
267 |
definition |
|
268 |
"top = Abs_filter (\<lambda>P. \<forall>x. P x)" |
|
269 |
||
270 |
definition |
|
271 |
"bot = Abs_filter (\<lambda>P. True)" |
|
272 |
||
273 |
definition |
|
274 |
"sup F F' = Abs_filter (\<lambda>P. eventually P F \<and> eventually P F')" |
|
275 |
||
276 |
definition |
|
277 |
"inf F F' = Abs_filter |
|
278 |
(\<lambda>P. \<exists>Q R. eventually Q F \<and> eventually R F' \<and> (\<forall>x. Q x \<and> R x \<longrightarrow> P x))" |
|
279 |
||
280 |
definition |
|
281 |
"Sup S = Abs_filter (\<lambda>P. \<forall>F\<in>S. eventually P F)" |
|
282 |
||
283 |
definition |
|
284 |
"Inf S = Sup {F::'a filter. \<forall>F'\<in>S. F \<le> F'}" |
|
285 |
||
286 |
lemma eventually_top [simp]: "eventually P top \<longleftrightarrow> (\<forall>x. P x)" |
|
287 |
unfolding top_filter_def |
|
288 |
by (rule eventually_Abs_filter, rule is_filter.intro, auto) |
|
289 |
||
290 |
lemma eventually_bot [simp]: "eventually P bot" |
|
291 |
unfolding bot_filter_def |
|
292 |
by (subst eventually_Abs_filter, rule is_filter.intro, auto) |
|
293 |
||
294 |
lemma eventually_sup: |
|
295 |
"eventually P (sup F F') \<longleftrightarrow> eventually P F \<and> eventually P F'" |
|
296 |
unfolding sup_filter_def |
|
297 |
by (rule eventually_Abs_filter, rule is_filter.intro) |
|
298 |
(auto elim!: eventually_rev_mp) |
|
299 |
||
300 |
lemma eventually_inf: |
|
301 |
"eventually P (inf F F') \<longleftrightarrow> |
|
302 |
(\<exists>Q R. eventually Q F \<and> eventually R F' \<and> (\<forall>x. Q x \<and> R x \<longrightarrow> P x))" |
|
303 |
unfolding inf_filter_def |
|
304 |
apply (rule eventually_Abs_filter, rule is_filter.intro) |
|
305 |
apply (fast intro: eventually_True) |
|
306 |
apply clarify |
|
307 |
apply (intro exI conjI) |
|
308 |
apply (erule (1) eventually_conj) |
|
309 |
apply (erule (1) eventually_conj) |
|
310 |
apply simp |
|
311 |
apply auto |
|
312 |
done |
|
313 |
||
314 |
lemma eventually_Sup: |
|
315 |
"eventually P (Sup S) \<longleftrightarrow> (\<forall>F\<in>S. eventually P F)" |
|
316 |
unfolding Sup_filter_def |
|
317 |
apply (rule eventually_Abs_filter, rule is_filter.intro) |
|
318 |
apply (auto intro: eventually_conj elim!: eventually_rev_mp) |
|
319 |
done |
|
320 |
||
321 |
instance proof |
|
322 |
fix F F' F'' :: "'a filter" and S :: "'a filter set" |
|
323 |
{ show "F < F' \<longleftrightarrow> F \<le> F' \<and> \<not> F' \<le> F" |
|
324 |
by (rule less_filter_def) } |
|
325 |
{ show "F \<le> F" |
|
326 |
unfolding le_filter_def by simp } |
|
327 |
{ assume "F \<le> F'" and "F' \<le> F''" thus "F \<le> F''" |
|
328 |
unfolding le_filter_def by simp } |
|
329 |
{ assume "F \<le> F'" and "F' \<le> F" thus "F = F'" |
|
330 |
unfolding le_filter_def filter_eq_iff by fast } |
|
331 |
{ show "inf F F' \<le> F" and "inf F F' \<le> F'" |
|
332 |
unfolding le_filter_def eventually_inf by (auto intro: eventually_True) } |
|
333 |
{ assume "F \<le> F'" and "F \<le> F''" thus "F \<le> inf F' F''" |
|
334 |
unfolding le_filter_def eventually_inf |
|
61810 | 335 |
by (auto intro: eventually_mono [OF eventually_conj]) } |
60036 | 336 |
{ show "F \<le> sup F F'" and "F' \<le> sup F F'" |
337 |
unfolding le_filter_def eventually_sup by simp_all } |
|
338 |
{ assume "F \<le> F''" and "F' \<le> F''" thus "sup F F' \<le> F''" |
|
339 |
unfolding le_filter_def eventually_sup by simp } |
|
340 |
{ assume "F'' \<in> S" thus "Inf S \<le> F''" |
|
341 |
unfolding le_filter_def Inf_filter_def eventually_Sup Ball_def by simp } |
|
342 |
{ assume "\<And>F'. F' \<in> S \<Longrightarrow> F \<le> F'" thus "F \<le> Inf S" |
|
343 |
unfolding le_filter_def Inf_filter_def eventually_Sup Ball_def by simp } |
|
344 |
{ assume "F \<in> S" thus "F \<le> Sup S" |
|
345 |
unfolding le_filter_def eventually_Sup by simp } |
|
346 |
{ assume "\<And>F. F \<in> S \<Longrightarrow> F \<le> F'" thus "Sup S \<le> F'" |
|
347 |
unfolding le_filter_def eventually_Sup by simp } |
|
348 |
{ show "Inf {} = (top::'a filter)" |
|
349 |
by (auto simp: top_filter_def Inf_filter_def Sup_filter_def) |
|
350 |
(metis (full_types) top_filter_def always_eventually eventually_top) } |
|
351 |
{ show "Sup {} = (bot::'a filter)" |
|
352 |
by (auto simp: bot_filter_def Sup_filter_def) } |
|
353 |
qed |
|
354 |
||
355 |
end |
|
356 |
||
357 |
lemma filter_leD: |
|
358 |
"F \<le> F' \<Longrightarrow> eventually P F' \<Longrightarrow> eventually P F" |
|
359 |
unfolding le_filter_def by simp |
|
360 |
||
361 |
lemma filter_leI: |
|
362 |
"(\<And>P. eventually P F' \<Longrightarrow> eventually P F) \<Longrightarrow> F \<le> F'" |
|
363 |
unfolding le_filter_def by simp |
|
364 |
||
365 |
lemma eventually_False: |
|
366 |
"eventually (\<lambda>x. False) F \<longleftrightarrow> F = bot" |
|
367 |
unfolding filter_eq_iff by (auto elim: eventually_rev_mp) |
|
368 |
||
60040
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
369 |
lemma eventually_frequently: "F \<noteq> bot \<Longrightarrow> eventually P F \<Longrightarrow> frequently P F" |
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
370 |
using eventually_conj[of P F "\<lambda>x. \<not> P x"] |
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
371 |
by (auto simp add: frequently_def eventually_False) |
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
372 |
|
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
373 |
lemma eventually_const_iff: "eventually (\<lambda>x. P) F \<longleftrightarrow> P \<or> F = bot" |
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
374 |
by (cases P) (auto simp: eventually_False) |
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
375 |
|
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
376 |
lemma eventually_const[simp]: "F \<noteq> bot \<Longrightarrow> eventually (\<lambda>x. P) F \<longleftrightarrow> P" |
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
377 |
by (simp add: eventually_const_iff) |
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
378 |
|
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
379 |
lemma frequently_const_iff: "frequently (\<lambda>x. P) F \<longleftrightarrow> P \<and> F \<noteq> bot" |
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
380 |
by (simp add: frequently_def eventually_const_iff) |
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
381 |
|
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
382 |
lemma frequently_const[simp]: "F \<noteq> bot \<Longrightarrow> frequently (\<lambda>x. P) F \<longleftrightarrow> P" |
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
383 |
by (simp add: frequently_const_iff) |
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
384 |
|
61245 | 385 |
lemma eventually_happens: "eventually P net \<Longrightarrow> net = bot \<or> (\<exists>x. P x)" |
386 |
by (metis frequentlyE eventually_frequently) |
|
387 |
||
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61378
diff
changeset
|
388 |
lemma eventually_happens': |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61378
diff
changeset
|
389 |
assumes "F \<noteq> bot" "eventually P F" |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61378
diff
changeset
|
390 |
shows "\<exists>x. P x" |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61378
diff
changeset
|
391 |
using assms eventually_frequently frequentlyE by blast |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61378
diff
changeset
|
392 |
|
60036 | 393 |
abbreviation (input) trivial_limit :: "'a filter \<Rightarrow> bool" |
394 |
where "trivial_limit F \<equiv> F = bot" |
|
395 |
||
396 |
lemma trivial_limit_def: "trivial_limit F \<longleftrightarrow> eventually (\<lambda>x. False) F" |
|
397 |
by (rule eventually_False [symmetric]) |
|
398 |
||
61806
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61531
diff
changeset
|
399 |
lemma False_imp_not_eventually: "(\<forall>x. \<not> P x ) \<Longrightarrow> \<not> trivial_limit net \<Longrightarrow> \<not> eventually (\<lambda>x. P x) net" |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61531
diff
changeset
|
400 |
by (simp add: eventually_False) |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61531
diff
changeset
|
401 |
|
60036 | 402 |
lemma eventually_Inf: "eventually P (Inf B) \<longleftrightarrow> (\<exists>X\<subseteq>B. finite X \<and> eventually P (Inf X))" |
403 |
proof - |
|
404 |
let ?F = "\<lambda>P. \<exists>X\<subseteq>B. finite X \<and> eventually P (Inf X)" |
|
61806
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61531
diff
changeset
|
405 |
|
60036 | 406 |
{ fix P have "eventually P (Abs_filter ?F) \<longleftrightarrow> ?F P" |
407 |
proof (rule eventually_Abs_filter is_filter.intro)+ |
|
408 |
show "?F (\<lambda>x. True)" |
|
409 |
by (rule exI[of _ "{}"]) (simp add: le_fun_def) |
|
410 |
next |
|
411 |
fix P Q |
|
412 |
assume "?F P" then guess X .. |
|
413 |
moreover |
|
414 |
assume "?F Q" then guess Y .. |
|
415 |
ultimately show "?F (\<lambda>x. P x \<and> Q x)" |
|
416 |
by (intro exI[of _ "X \<union> Y"]) |
|
417 |
(auto simp: Inf_union_distrib eventually_inf) |
|
418 |
next |
|
419 |
fix P Q |
|
420 |
assume "?F P" then guess X .. |
|
421 |
moreover assume "\<forall>x. P x \<longrightarrow> Q x" |
|
422 |
ultimately show "?F Q" |
|
61810 | 423 |
by (intro exI[of _ X]) (auto elim: eventually_mono) |
60036 | 424 |
qed } |
425 |
note eventually_F = this |
|
426 |
||
427 |
have "Inf B = Abs_filter ?F" |
|
428 |
proof (intro antisym Inf_greatest) |
|
429 |
show "Inf B \<le> Abs_filter ?F" |
|
430 |
by (auto simp: le_filter_def eventually_F dest: Inf_superset_mono) |
|
431 |
next |
|
432 |
fix F assume "F \<in> B" then show "Abs_filter ?F \<le> F" |
|
433 |
by (auto simp add: le_filter_def eventually_F intro!: exI[of _ "{F}"]) |
|
434 |
qed |
|
435 |
then show ?thesis |
|
436 |
by (simp add: eventually_F) |
|
437 |
qed |
|
438 |
||
439 |
lemma eventually_INF: "eventually P (INF b:B. F b) \<longleftrightarrow> (\<exists>X\<subseteq>B. finite X \<and> eventually P (INF b:X. F b))" |
|
440 |
unfolding INF_def[of B] eventually_Inf[of P "F`B"] |
|
441 |
by (metis Inf_image_eq finite_imageI image_mono finite_subset_image) |
|
442 |
||
443 |
lemma Inf_filter_not_bot: |
|
444 |
fixes B :: "'a filter set" |
|
445 |
shows "(\<And>X. X \<subseteq> B \<Longrightarrow> finite X \<Longrightarrow> Inf X \<noteq> bot) \<Longrightarrow> Inf B \<noteq> bot" |
|
446 |
unfolding trivial_limit_def eventually_Inf[of _ B] |
|
447 |
bot_bool_def [symmetric] bot_fun_def [symmetric] bot_unique by simp |
|
448 |
||
449 |
lemma INF_filter_not_bot: |
|
450 |
fixes F :: "'i \<Rightarrow> 'a filter" |
|
451 |
shows "(\<And>X. X \<subseteq> B \<Longrightarrow> finite X \<Longrightarrow> (INF b:X. F b) \<noteq> bot) \<Longrightarrow> (INF b:B. F b) \<noteq> bot" |
|
452 |
unfolding trivial_limit_def eventually_INF[of _ B] |
|
453 |
bot_bool_def [symmetric] bot_fun_def [symmetric] bot_unique by simp |
|
454 |
||
455 |
lemma eventually_Inf_base: |
|
456 |
assumes "B \<noteq> {}" and base: "\<And>F G. F \<in> B \<Longrightarrow> G \<in> B \<Longrightarrow> \<exists>x\<in>B. x \<le> inf F G" |
|
457 |
shows "eventually P (Inf B) \<longleftrightarrow> (\<exists>b\<in>B. eventually P b)" |
|
458 |
proof (subst eventually_Inf, safe) |
|
459 |
fix X assume "finite X" "X \<subseteq> B" |
|
460 |
then have "\<exists>b\<in>B. \<forall>x\<in>X. b \<le> x" |
|
461 |
proof induct |
|
462 |
case empty then show ?case |
|
60758 | 463 |
using \<open>B \<noteq> {}\<close> by auto |
60036 | 464 |
next |
465 |
case (insert x X) |
|
466 |
then obtain b where "b \<in> B" "\<And>x. x \<in> X \<Longrightarrow> b \<le> x" |
|
467 |
by auto |
|
60758 | 468 |
with \<open>insert x X \<subseteq> B\<close> base[of b x] show ?case |
60036 | 469 |
by (auto intro: order_trans) |
470 |
qed |
|
471 |
then obtain b where "b \<in> B" "b \<le> Inf X" |
|
472 |
by (auto simp: le_Inf_iff) |
|
473 |
then show "eventually P (Inf X) \<Longrightarrow> Bex B (eventually P)" |
|
474 |
by (intro bexI[of _ b]) (auto simp: le_filter_def) |
|
475 |
qed (auto intro!: exI[of _ "{x}" for x]) |
|
476 |
||
477 |
lemma eventually_INF_base: |
|
478 |
"B \<noteq> {} \<Longrightarrow> (\<And>a b. a \<in> B \<Longrightarrow> b \<in> B \<Longrightarrow> \<exists>x\<in>B. F x \<le> inf (F a) (F b)) \<Longrightarrow> |
|
479 |
eventually P (INF b:B. F b) \<longleftrightarrow> (\<exists>b\<in>B. eventually P (F b))" |
|
480 |
unfolding INF_def by (subst eventually_Inf_base) auto |
|
481 |
||
482 |
||
60758 | 483 |
subsubsection \<open>Map function for filters\<close> |
60036 | 484 |
|
485 |
definition filtermap :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a filter \<Rightarrow> 'b filter" |
|
486 |
where "filtermap f F = Abs_filter (\<lambda>P. eventually (\<lambda>x. P (f x)) F)" |
|
487 |
||
488 |
lemma eventually_filtermap: |
|
489 |
"eventually P (filtermap f F) = eventually (\<lambda>x. P (f x)) F" |
|
490 |
unfolding filtermap_def |
|
491 |
apply (rule eventually_Abs_filter) |
|
492 |
apply (rule is_filter.intro) |
|
493 |
apply (auto elim!: eventually_rev_mp) |
|
494 |
done |
|
495 |
||
496 |
lemma filtermap_ident: "filtermap (\<lambda>x. x) F = F" |
|
497 |
by (simp add: filter_eq_iff eventually_filtermap) |
|
498 |
||
499 |
lemma filtermap_filtermap: |
|
500 |
"filtermap f (filtermap g F) = filtermap (\<lambda>x. f (g x)) F" |
|
501 |
by (simp add: filter_eq_iff eventually_filtermap) |
|
502 |
||
503 |
lemma filtermap_mono: "F \<le> F' \<Longrightarrow> filtermap f F \<le> filtermap f F'" |
|
504 |
unfolding le_filter_def eventually_filtermap by simp |
|
505 |
||
506 |
lemma filtermap_bot [simp]: "filtermap f bot = bot" |
|
507 |
by (simp add: filter_eq_iff eventually_filtermap) |
|
508 |
||
509 |
lemma filtermap_sup: "filtermap f (sup F1 F2) = sup (filtermap f F1) (filtermap f F2)" |
|
510 |
by (auto simp: filter_eq_iff eventually_filtermap eventually_sup) |
|
511 |
||
512 |
lemma filtermap_inf: "filtermap f (inf F1 F2) \<le> inf (filtermap f F1) (filtermap f F2)" |
|
513 |
by (auto simp: le_filter_def eventually_filtermap eventually_inf) |
|
514 |
||
515 |
lemma filtermap_INF: "filtermap f (INF b:B. F b) \<le> (INF b:B. filtermap f (F b))" |
|
516 |
proof - |
|
517 |
{ fix X :: "'c set" assume "finite X" |
|
518 |
then have "filtermap f (INFIMUM X F) \<le> (INF b:X. filtermap f (F b))" |
|
519 |
proof induct |
|
520 |
case (insert x X) |
|
521 |
have "filtermap f (INF a:insert x X. F a) \<le> inf (filtermap f (F x)) (filtermap f (INF a:X. F a))" |
|
522 |
by (rule order_trans[OF _ filtermap_inf]) simp |
|
523 |
also have "\<dots> \<le> inf (filtermap f (F x)) (INF a:X. filtermap f (F a))" |
|
524 |
by (intro inf_mono insert order_refl) |
|
525 |
finally show ?case |
|
526 |
by simp |
|
527 |
qed simp } |
|
528 |
then show ?thesis |
|
529 |
unfolding le_filter_def eventually_filtermap |
|
530 |
by (subst (1 2) eventually_INF) auto |
|
531 |
qed |
|
62101 | 532 |
|
533 |
||
60758 | 534 |
subsubsection \<open>Standard filters\<close> |
60036 | 535 |
|
536 |
definition principal :: "'a set \<Rightarrow> 'a filter" where |
|
537 |
"principal S = Abs_filter (\<lambda>P. \<forall>x\<in>S. P x)" |
|
538 |
||
539 |
lemma eventually_principal: "eventually P (principal S) \<longleftrightarrow> (\<forall>x\<in>S. P x)" |
|
540 |
unfolding principal_def |
|
541 |
by (rule eventually_Abs_filter, rule is_filter.intro) auto |
|
542 |
||
543 |
lemma eventually_inf_principal: "eventually P (inf F (principal s)) \<longleftrightarrow> eventually (\<lambda>x. x \<in> s \<longrightarrow> P x) F" |
|
61810 | 544 |
unfolding eventually_inf eventually_principal by (auto elim: eventually_mono) |
60036 | 545 |
|
546 |
lemma principal_UNIV[simp]: "principal UNIV = top" |
|
547 |
by (auto simp: filter_eq_iff eventually_principal) |
|
548 |
||
549 |
lemma principal_empty[simp]: "principal {} = bot" |
|
550 |
by (auto simp: filter_eq_iff eventually_principal) |
|
551 |
||
552 |
lemma principal_eq_bot_iff: "principal X = bot \<longleftrightarrow> X = {}" |
|
553 |
by (auto simp add: filter_eq_iff eventually_principal) |
|
554 |
||
555 |
lemma principal_le_iff[iff]: "principal A \<le> principal B \<longleftrightarrow> A \<subseteq> B" |
|
556 |
by (auto simp: le_filter_def eventually_principal) |
|
557 |
||
558 |
lemma le_principal: "F \<le> principal A \<longleftrightarrow> eventually (\<lambda>x. x \<in> A) F" |
|
559 |
unfolding le_filter_def eventually_principal |
|
560 |
apply safe |
|
561 |
apply (erule_tac x="\<lambda>x. x \<in> A" in allE) |
|
61810 | 562 |
apply (auto elim: eventually_mono) |
60036 | 563 |
done |
564 |
||
565 |
lemma principal_inject[iff]: "principal A = principal B \<longleftrightarrow> A = B" |
|
566 |
unfolding eq_iff by simp |
|
567 |
||
568 |
lemma sup_principal[simp]: "sup (principal A) (principal B) = principal (A \<union> B)" |
|
569 |
unfolding filter_eq_iff eventually_sup eventually_principal by auto |
|
570 |
||
571 |
lemma inf_principal[simp]: "inf (principal A) (principal B) = principal (A \<inter> B)" |
|
572 |
unfolding filter_eq_iff eventually_inf eventually_principal |
|
573 |
by (auto intro: exI[of _ "\<lambda>x. x \<in> A"] exI[of _ "\<lambda>x. x \<in> B"]) |
|
574 |
||
575 |
lemma SUP_principal[simp]: "(SUP i : I. principal (A i)) = principal (\<Union>i\<in>I. A i)" |
|
576 |
unfolding filter_eq_iff eventually_Sup SUP_def by (auto simp: eventually_principal) |
|
577 |
||
578 |
lemma INF_principal_finite: "finite X \<Longrightarrow> (INF x:X. principal (f x)) = principal (\<Inter>x\<in>X. f x)" |
|
579 |
by (induct X rule: finite_induct) auto |
|
580 |
||
581 |
lemma filtermap_principal[simp]: "filtermap f (principal A) = principal (f ` A)" |
|
582 |
unfolding filter_eq_iff eventually_filtermap eventually_principal by simp |
|
583 |
||
60758 | 584 |
subsubsection \<open>Order filters\<close> |
60036 | 585 |
|
586 |
definition at_top :: "('a::order) filter" |
|
587 |
where "at_top = (INF k. principal {k ..})" |
|
588 |
||
589 |
lemma at_top_sub: "at_top = (INF k:{c::'a::linorder..}. principal {k ..})" |
|
590 |
by (auto intro!: INF_eq max.cobounded1 max.cobounded2 simp: at_top_def) |
|
591 |
||
592 |
lemma eventually_at_top_linorder: "eventually P at_top \<longleftrightarrow> (\<exists>N::'a::linorder. \<forall>n\<ge>N. P n)" |
|
593 |
unfolding at_top_def |
|
594 |
by (subst eventually_INF_base) (auto simp: eventually_principal intro: max.cobounded1 max.cobounded2) |
|
595 |
||
596 |
lemma eventually_ge_at_top: |
|
597 |
"eventually (\<lambda>x. (c::_::linorder) \<le> x) at_top" |
|
598 |
unfolding eventually_at_top_linorder by auto |
|
599 |
||
600 |
lemma eventually_at_top_dense: "eventually P at_top \<longleftrightarrow> (\<exists>N::'a::{no_top, linorder}. \<forall>n>N. P n)" |
|
601 |
proof - |
|
602 |
have "eventually P (INF k. principal {k <..}) \<longleftrightarrow> (\<exists>N::'a. \<forall>n>N. P n)" |
|
603 |
by (subst eventually_INF_base) (auto simp: eventually_principal intro: max.cobounded1 max.cobounded2) |
|
604 |
also have "(INF k. principal {k::'a <..}) = at_top" |
|
61806
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61531
diff
changeset
|
605 |
unfolding at_top_def |
60036 | 606 |
by (intro INF_eq) (auto intro: less_imp_le simp: Ici_subset_Ioi_iff gt_ex) |
607 |
finally show ?thesis . |
|
608 |
qed |
|
609 |
||
60721
c1b7793c23a3
generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents:
60589
diff
changeset
|
610 |
lemma eventually_at_top_not_equal: "eventually (\<lambda>x::'a::{no_top, linorder}. x \<noteq> c) at_top" |
c1b7793c23a3
generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents:
60589
diff
changeset
|
611 |
unfolding eventually_at_top_dense by auto |
c1b7793c23a3
generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents:
60589
diff
changeset
|
612 |
|
c1b7793c23a3
generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents:
60589
diff
changeset
|
613 |
lemma eventually_gt_at_top: "eventually (\<lambda>x. (c::_::{no_top, linorder}) < x) at_top" |
60036 | 614 |
unfolding eventually_at_top_dense by auto |
615 |
||
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61378
diff
changeset
|
616 |
lemma eventually_all_ge_at_top: |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61378
diff
changeset
|
617 |
assumes "eventually P (at_top :: ('a :: linorder) filter)" |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61378
diff
changeset
|
618 |
shows "eventually (\<lambda>x. \<forall>y\<ge>x. P y) at_top" |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61378
diff
changeset
|
619 |
proof - |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61378
diff
changeset
|
620 |
from assms obtain x where "\<And>y. y \<ge> x \<Longrightarrow> P y" by (auto simp: eventually_at_top_linorder) |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61378
diff
changeset
|
621 |
hence "\<forall>z\<ge>y. P z" if "y \<ge> x" for y using that by simp |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61378
diff
changeset
|
622 |
thus ?thesis by (auto simp: eventually_at_top_linorder) |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61378
diff
changeset
|
623 |
qed |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61378
diff
changeset
|
624 |
|
60036 | 625 |
definition at_bot :: "('a::order) filter" |
626 |
where "at_bot = (INF k. principal {.. k})" |
|
627 |
||
628 |
lemma at_bot_sub: "at_bot = (INF k:{.. c::'a::linorder}. principal {.. k})" |
|
629 |
by (auto intro!: INF_eq min.cobounded1 min.cobounded2 simp: at_bot_def) |
|
630 |
||
631 |
lemma eventually_at_bot_linorder: |
|
632 |
fixes P :: "'a::linorder \<Rightarrow> bool" shows "eventually P at_bot \<longleftrightarrow> (\<exists>N. \<forall>n\<le>N. P n)" |
|
633 |
unfolding at_bot_def |
|
634 |
by (subst eventually_INF_base) (auto simp: eventually_principal intro: min.cobounded1 min.cobounded2) |
|
635 |
||
636 |
lemma eventually_le_at_bot: |
|
637 |
"eventually (\<lambda>x. x \<le> (c::_::linorder)) at_bot" |
|
638 |
unfolding eventually_at_bot_linorder by auto |
|
639 |
||
640 |
lemma eventually_at_bot_dense: "eventually P at_bot \<longleftrightarrow> (\<exists>N::'a::{no_bot, linorder}. \<forall>n<N. P n)" |
|
641 |
proof - |
|
642 |
have "eventually P (INF k. principal {..< k}) \<longleftrightarrow> (\<exists>N::'a. \<forall>n<N. P n)" |
|
643 |
by (subst eventually_INF_base) (auto simp: eventually_principal intro: min.cobounded1 min.cobounded2) |
|
644 |
also have "(INF k. principal {..< k::'a}) = at_bot" |
|
61806
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61531
diff
changeset
|
645 |
unfolding at_bot_def |
60036 | 646 |
by (intro INF_eq) (auto intro: less_imp_le simp: Iic_subset_Iio_iff lt_ex) |
647 |
finally show ?thesis . |
|
648 |
qed |
|
649 |
||
60721
c1b7793c23a3
generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents:
60589
diff
changeset
|
650 |
lemma eventually_at_bot_not_equal: "eventually (\<lambda>x::'a::{no_bot, linorder}. x \<noteq> c) at_bot" |
c1b7793c23a3
generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents:
60589
diff
changeset
|
651 |
unfolding eventually_at_bot_dense by auto |
c1b7793c23a3
generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents:
60589
diff
changeset
|
652 |
|
60036 | 653 |
lemma eventually_gt_at_bot: |
654 |
"eventually (\<lambda>x. x < (c::_::unbounded_dense_linorder)) at_bot" |
|
655 |
unfolding eventually_at_bot_dense by auto |
|
656 |
||
657 |
lemma trivial_limit_at_bot_linorder: "\<not> trivial_limit (at_bot ::('a::linorder) filter)" |
|
658 |
unfolding trivial_limit_def |
|
659 |
by (metis eventually_at_bot_linorder order_refl) |
|
660 |
||
661 |
lemma trivial_limit_at_top_linorder: "\<not> trivial_limit (at_top ::('a::linorder) filter)" |
|
662 |
unfolding trivial_limit_def |
|
663 |
by (metis eventually_at_top_linorder order_refl) |
|
664 |
||
60758 | 665 |
subsection \<open>Sequentially\<close> |
60036 | 666 |
|
667 |
abbreviation sequentially :: "nat filter" |
|
668 |
where "sequentially \<equiv> at_top" |
|
669 |
||
670 |
lemma eventually_sequentially: |
|
671 |
"eventually P sequentially \<longleftrightarrow> (\<exists>N. \<forall>n\<ge>N. P n)" |
|
672 |
by (rule eventually_at_top_linorder) |
|
673 |
||
674 |
lemma sequentially_bot [simp, intro]: "sequentially \<noteq> bot" |
|
675 |
unfolding filter_eq_iff eventually_sequentially by auto |
|
676 |
||
677 |
lemmas trivial_limit_sequentially = sequentially_bot |
|
678 |
||
679 |
lemma eventually_False_sequentially [simp]: |
|
680 |
"\<not> eventually (\<lambda>n. False) sequentially" |
|
681 |
by (simp add: eventually_False) |
|
682 |
||
683 |
lemma le_sequentially: |
|
684 |
"F \<le> sequentially \<longleftrightarrow> (\<forall>N. eventually (\<lambda>n. N \<le> n) F)" |
|
685 |
by (simp add: at_top_def le_INF_iff le_principal) |
|
686 |
||
60974
6a6f15d8fbc4
New material and fixes related to the forthcoming Stone-Weierstrass development
paulson <lp15@cam.ac.uk>
parents:
60758
diff
changeset
|
687 |
lemma eventually_sequentiallyI [intro?]: |
60036 | 688 |
assumes "\<And>x. c \<le> x \<Longrightarrow> P x" |
689 |
shows "eventually P sequentially" |
|
690 |
using assms by (auto simp: eventually_sequentially) |
|
691 |
||
60040
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
692 |
lemma eventually_sequentially_Suc: "eventually (\<lambda>i. P (Suc i)) sequentially \<longleftrightarrow> eventually P sequentially" |
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
693 |
unfolding eventually_sequentially by (metis Suc_le_D Suc_le_mono le_Suc_eq) |
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
694 |
|
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
695 |
lemma eventually_sequentially_seg: "eventually (\<lambda>n. P (n + k)) sequentially \<longleftrightarrow> eventually P sequentially" |
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
696 |
using eventually_sequentially_Suc[of "\<lambda>n. P (n + k)" for k] by (induction k) auto |
60036 | 697 |
|
61955
e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61953
diff
changeset
|
698 |
|
e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61953
diff
changeset
|
699 |
subsection \<open>The cofinite filter\<close> |
60039 | 700 |
|
701 |
definition "cofinite = Abs_filter (\<lambda>P. finite {x. \<not> P x})" |
|
702 |
||
61955
e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61953
diff
changeset
|
703 |
abbreviation Inf_many :: "('a \<Rightarrow> bool) \<Rightarrow> bool" (binder "\<exists>\<^sub>\<infinity>" 10) |
e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61953
diff
changeset
|
704 |
where "Inf_many P \<equiv> frequently P cofinite" |
60040
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
705 |
|
61955
e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61953
diff
changeset
|
706 |
abbreviation Alm_all :: "('a \<Rightarrow> bool) \<Rightarrow> bool" (binder "\<forall>\<^sub>\<infinity>" 10) |
e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61953
diff
changeset
|
707 |
where "Alm_all P \<equiv> eventually P cofinite" |
60040
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
708 |
|
61955
e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61953
diff
changeset
|
709 |
notation (ASCII) |
e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61953
diff
changeset
|
710 |
Inf_many (binder "INFM " 10) and |
e96292f32c3c
former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents:
61953
diff
changeset
|
711 |
Alm_all (binder "MOST " 10) |
60040
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
712 |
|
60039 | 713 |
lemma eventually_cofinite: "eventually P cofinite \<longleftrightarrow> finite {x. \<not> P x}" |
714 |
unfolding cofinite_def |
|
715 |
proof (rule eventually_Abs_filter, rule is_filter.intro) |
|
716 |
fix P Q :: "'a \<Rightarrow> bool" assume "finite {x. \<not> P x}" "finite {x. \<not> Q x}" |
|
717 |
from finite_UnI[OF this] show "finite {x. \<not> (P x \<and> Q x)}" |
|
718 |
by (rule rev_finite_subset) auto |
|
719 |
next |
|
720 |
fix P Q :: "'a \<Rightarrow> bool" assume P: "finite {x. \<not> P x}" and *: "\<forall>x. P x \<longrightarrow> Q x" |
|
721 |
from * show "finite {x. \<not> Q x}" |
|
722 |
by (intro finite_subset[OF _ P]) auto |
|
723 |
qed simp |
|
724 |
||
60040
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
725 |
lemma frequently_cofinite: "frequently P cofinite \<longleftrightarrow> \<not> finite {x. P x}" |
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
726 |
by (simp add: frequently_def eventually_cofinite) |
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60039
diff
changeset
|
727 |
|
60039 | 728 |
lemma cofinite_bot[simp]: "cofinite = (bot::'a filter) \<longleftrightarrow> finite (UNIV :: 'a set)" |
729 |
unfolding trivial_limit_def eventually_cofinite by simp |
|
730 |
||
731 |
lemma cofinite_eq_sequentially: "cofinite = sequentially" |
|
732 |
unfolding filter_eq_iff eventually_sequentially eventually_cofinite |
|
733 |
proof safe |
|
734 |
fix P :: "nat \<Rightarrow> bool" assume [simp]: "finite {x. \<not> P x}" |
|
735 |
show "\<exists>N. \<forall>n\<ge>N. P n" |
|
736 |
proof cases |
|
737 |
assume "{x. \<not> P x} \<noteq> {}" then show ?thesis |
|
738 |
by (intro exI[of _ "Suc (Max {x. \<not> P x})"]) (auto simp: Suc_le_eq) |
|
739 |
qed auto |
|
740 |
next |
|
741 |
fix P :: "nat \<Rightarrow> bool" and N :: nat assume "\<forall>n\<ge>N. P n" |
|
742 |
then have "{x. \<not> P x} \<subseteq> {..< N}" |
|
743 |
by (auto simp: not_le) |
|
744 |
then show "finite {x. \<not> P x}" |
|
745 |
by (blast intro: finite_subset) |
|
746 |
qed |
|
60036 | 747 |
|
62101 | 748 |
subsubsection \<open>Product of filters\<close> |
749 |
||
750 |
lemma filtermap_sequentually_ne_bot: "filtermap f sequentially \<noteq> bot" |
|
751 |
by (auto simp add: filter_eq_iff eventually_filtermap eventually_sequentially) |
|
752 |
||
753 |
definition prod_filter :: "'a filter \<Rightarrow> 'b filter \<Rightarrow> ('a \<times> 'b) filter" (infixr "\<times>\<^sub>F" 80) where |
|
754 |
"prod_filter F G = |
|
755 |
(INF (P, Q):{(P, Q). eventually P F \<and> eventually Q G}. principal {(x, y). P x \<and> Q y})" |
|
756 |
||
757 |
lemma eventually_prod_filter: "eventually P (F \<times>\<^sub>F G) \<longleftrightarrow> |
|
758 |
(\<exists>Pf Pg. eventually Pf F \<and> eventually Pg G \<and> (\<forall>x y. Pf x \<longrightarrow> Pg y \<longrightarrow> P (x, y)))" |
|
759 |
unfolding prod_filter_def |
|
760 |
proof (subst eventually_INF_base, goal_cases) |
|
761 |
case 2 |
|
762 |
moreover have "eventually Pf F \<Longrightarrow> eventually Qf F \<Longrightarrow> eventually Pg G \<Longrightarrow> eventually Qg G \<Longrightarrow> |
|
763 |
\<exists>P Q. eventually P F \<and> eventually Q G \<and> |
|
764 |
Collect P \<times> Collect Q \<subseteq> Collect Pf \<times> Collect Pg \<inter> Collect Qf \<times> Collect Qg" for Pf Pg Qf Qg |
|
765 |
by (intro conjI exI[of _ "inf Pf Qf"] exI[of _ "inf Pg Qg"]) |
|
766 |
(auto simp: inf_fun_def eventually_conj) |
|
767 |
ultimately show ?case |
|
768 |
by auto |
|
769 |
qed (auto simp: eventually_principal intro: eventually_True) |
|
770 |
||
771 |
lemma prod_filter_mono: "F \<le> F' \<Longrightarrow> G \<le> G' \<Longrightarrow> F \<times>\<^sub>F G \<le> F' \<times>\<^sub>F G'" |
|
772 |
by (auto simp: le_filter_def eventually_prod_filter) |
|
773 |
||
774 |
lemma eventually_prod_same: "eventually P (F \<times>\<^sub>F F) \<longleftrightarrow> |
|
775 |
(\<exists>Q. eventually Q F \<and> (\<forall>x y. Q x \<longrightarrow> Q y \<longrightarrow> P (x, y)))" |
|
776 |
unfolding eventually_prod_filter |
|
777 |
apply safe |
|
778 |
apply (rule_tac x="inf Pf Pg" in exI) |
|
779 |
apply (auto simp: inf_fun_def intro!: eventually_conj) |
|
780 |
done |
|
781 |
||
782 |
lemma eventually_prod_sequentially: |
|
783 |
"eventually P (sequentially \<times>\<^sub>F sequentially) \<longleftrightarrow> (\<exists>N. \<forall>m \<ge> N. \<forall>n \<ge> N. P (n, m))" |
|
784 |
unfolding eventually_prod_same eventually_sequentially by auto |
|
785 |
||
786 |
lemma principal_prod_principal: "principal A \<times>\<^sub>F principal B = principal (A \<times> B)" |
|
787 |
apply (simp add: filter_eq_iff eventually_prod_filter eventually_principal) |
|
788 |
apply safe |
|
789 |
apply blast |
|
790 |
apply (intro conjI exI[of _ "\<lambda>x. x \<in> A"] exI[of _ "\<lambda>x. x \<in> B"]) |
|
791 |
apply auto |
|
792 |
done |
|
793 |
||
60758 | 794 |
subsection \<open>Limits\<close> |
60036 | 795 |
|
796 |
definition filterlim :: "('a \<Rightarrow> 'b) \<Rightarrow> 'b filter \<Rightarrow> 'a filter \<Rightarrow> bool" where |
|
797 |
"filterlim f F2 F1 \<longleftrightarrow> filtermap f F1 \<le> F2" |
|
798 |
||
799 |
syntax |
|
800 |
"_LIM" :: "pttrns \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> bool" ("(3LIM (_)/ (_)./ (_) :> (_))" [1000, 10, 0, 10] 10) |
|
801 |
||
802 |
translations |
|
803 |
"LIM x F1. f :> F2" == "CONST filterlim (%x. f) F2 F1" |
|
804 |
||
805 |
lemma filterlim_iff: |
|
806 |
"(LIM x F1. f x :> F2) \<longleftrightarrow> (\<forall>P. eventually P F2 \<longrightarrow> eventually (\<lambda>x. P (f x)) F1)" |
|
807 |
unfolding filterlim_def le_filter_def eventually_filtermap .. |
|
808 |
||
809 |
lemma filterlim_compose: |
|
810 |
"filterlim g F3 F2 \<Longrightarrow> filterlim f F2 F1 \<Longrightarrow> filterlim (\<lambda>x. g (f x)) F3 F1" |
|
811 |
unfolding filterlim_def filtermap_filtermap[symmetric] by (metis filtermap_mono order_trans) |
|
812 |
||
813 |
lemma filterlim_mono: |
|
814 |
"filterlim f F2 F1 \<Longrightarrow> F2 \<le> F2' \<Longrightarrow> F1' \<le> F1 \<Longrightarrow> filterlim f F2' F1'" |
|
815 |
unfolding filterlim_def by (metis filtermap_mono order_trans) |
|
816 |
||
817 |
lemma filterlim_ident: "LIM x F. x :> F" |
|
818 |
by (simp add: filterlim_def filtermap_ident) |
|
819 |
||
820 |
lemma filterlim_cong: |
|
821 |
"F1 = F1' \<Longrightarrow> F2 = F2' \<Longrightarrow> eventually (\<lambda>x. f x = g x) F2 \<Longrightarrow> filterlim f F1 F2 = filterlim g F1' F2'" |
|
822 |
by (auto simp: filterlim_def le_filter_def eventually_filtermap elim: eventually_elim2) |
|
823 |
||
824 |
lemma filterlim_mono_eventually: |
|
825 |
assumes "filterlim f F G" and ord: "F \<le> F'" "G' \<le> G" |
|
826 |
assumes eq: "eventually (\<lambda>x. f x = f' x) G'" |
|
827 |
shows "filterlim f' F' G'" |
|
828 |
apply (rule filterlim_cong[OF refl refl eq, THEN iffD1]) |
|
829 |
apply (rule filterlim_mono[OF _ ord]) |
|
830 |
apply fact |
|
831 |
done |
|
832 |
||
833 |
lemma filtermap_mono_strong: "inj f \<Longrightarrow> filtermap f F \<le> filtermap f G \<longleftrightarrow> F \<le> G" |
|
834 |
apply (auto intro!: filtermap_mono) [] |
|
835 |
apply (auto simp: le_filter_def eventually_filtermap) |
|
836 |
apply (erule_tac x="\<lambda>x. P (inv f x)" in allE) |
|
837 |
apply auto |
|
838 |
done |
|
839 |
||
840 |
lemma filtermap_eq_strong: "inj f \<Longrightarrow> filtermap f F = filtermap f G \<longleftrightarrow> F = G" |
|
841 |
by (simp add: filtermap_mono_strong eq_iff) |
|
842 |
||
60721
c1b7793c23a3
generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents:
60589
diff
changeset
|
843 |
lemma filtermap_fun_inverse: |
c1b7793c23a3
generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents:
60589
diff
changeset
|
844 |
assumes g: "filterlim g F G" |
c1b7793c23a3
generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents:
60589
diff
changeset
|
845 |
assumes f: "filterlim f G F" |
c1b7793c23a3
generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents:
60589
diff
changeset
|
846 |
assumes ev: "eventually (\<lambda>x. f (g x) = x) G" |
c1b7793c23a3
generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents:
60589
diff
changeset
|
847 |
shows "filtermap f F = G" |
c1b7793c23a3
generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents:
60589
diff
changeset
|
848 |
proof (rule antisym) |
c1b7793c23a3
generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents:
60589
diff
changeset
|
849 |
show "filtermap f F \<le> G" |
c1b7793c23a3
generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents:
60589
diff
changeset
|
850 |
using f unfolding filterlim_def . |
c1b7793c23a3
generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents:
60589
diff
changeset
|
851 |
have "G = filtermap f (filtermap g G)" |
c1b7793c23a3
generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents:
60589
diff
changeset
|
852 |
using ev by (auto elim: eventually_elim2 simp: filter_eq_iff eventually_filtermap) |
c1b7793c23a3
generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents:
60589
diff
changeset
|
853 |
also have "\<dots> \<le> filtermap f F" |
c1b7793c23a3
generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents:
60589
diff
changeset
|
854 |
using g by (intro filtermap_mono) (simp add: filterlim_def) |
c1b7793c23a3
generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents:
60589
diff
changeset
|
855 |
finally show "G \<le> filtermap f F" . |
c1b7793c23a3
generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents:
60589
diff
changeset
|
856 |
qed |
c1b7793c23a3
generalized filtermap_homeomorph to filtermap_fun_inverse; add eventually_at_top/bot_not_equal
hoelzl
parents:
60589
diff
changeset
|
857 |
|
60036 | 858 |
lemma filterlim_principal: |
859 |
"(LIM x F. f x :> principal S) \<longleftrightarrow> (eventually (\<lambda>x. f x \<in> S) F)" |
|
860 |
unfolding filterlim_def eventually_filtermap le_principal .. |
|
861 |
||
862 |
lemma filterlim_inf: |
|
863 |
"(LIM x F1. f x :> inf F2 F3) \<longleftrightarrow> ((LIM x F1. f x :> F2) \<and> (LIM x F1. f x :> F3))" |
|
864 |
unfolding filterlim_def by simp |
|
865 |
||
866 |
lemma filterlim_INF: |
|
867 |
"(LIM x F. f x :> (INF b:B. G b)) \<longleftrightarrow> (\<forall>b\<in>B. LIM x F. f x :> G b)" |
|
868 |
unfolding filterlim_def le_INF_iff .. |
|
869 |
||
870 |
lemma filterlim_INF_INF: |
|
871 |
"(\<And>m. m \<in> J \<Longrightarrow> \<exists>i\<in>I. filtermap f (F i) \<le> G m) \<Longrightarrow> LIM x (INF i:I. F i). f x :> (INF j:J. G j)" |
|
872 |
unfolding filterlim_def by (rule order_trans[OF filtermap_INF INF_mono]) |
|
873 |
||
874 |
lemma filterlim_base: |
|
61806
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61531
diff
changeset
|
875 |
"(\<And>m x. m \<in> J \<Longrightarrow> i m \<in> I) \<Longrightarrow> (\<And>m x. m \<in> J \<Longrightarrow> x \<in> F (i m) \<Longrightarrow> f x \<in> G m) \<Longrightarrow> |
60036 | 876 |
LIM x (INF i:I. principal (F i)). f x :> (INF j:J. principal (G j))" |
877 |
by (force intro!: filterlim_INF_INF simp: image_subset_iff) |
|
878 |
||
61806
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61531
diff
changeset
|
879 |
lemma filterlim_base_iff: |
60036 | 880 |
assumes "I \<noteq> {}" and chain: "\<And>i j. i \<in> I \<Longrightarrow> j \<in> I \<Longrightarrow> F i \<subseteq> F j \<or> F j \<subseteq> F i" |
881 |
shows "(LIM x (INF i:I. principal (F i)). f x :> INF j:J. principal (G j)) \<longleftrightarrow> |
|
882 |
(\<forall>j\<in>J. \<exists>i\<in>I. \<forall>x\<in>F i. f x \<in> G j)" |
|
883 |
unfolding filterlim_INF filterlim_principal |
|
884 |
proof (subst eventually_INF_base) |
|
885 |
fix i j assume "i \<in> I" "j \<in> I" |
|
886 |
with chain[OF this] show "\<exists>x\<in>I. principal (F x) \<le> inf (principal (F i)) (principal (F j))" |
|
887 |
by auto |
|
60758 | 888 |
qed (auto simp: eventually_principal \<open>I \<noteq> {}\<close>) |
60036 | 889 |
|
890 |
lemma filterlim_filtermap: "filterlim f F1 (filtermap g F2) = filterlim (\<lambda>x. f (g x)) F1 F2" |
|
891 |
unfolding filterlim_def filtermap_filtermap .. |
|
892 |
||
893 |
lemma filterlim_sup: |
|
894 |
"filterlim f F F1 \<Longrightarrow> filterlim f F F2 \<Longrightarrow> filterlim f F (sup F1 F2)" |
|
895 |
unfolding filterlim_def filtermap_sup by auto |
|
896 |
||
897 |
lemma filterlim_sequentially_Suc: |
|
898 |
"(LIM x sequentially. f (Suc x) :> F) \<longleftrightarrow> (LIM x sequentially. f x :> F)" |
|
899 |
unfolding filterlim_iff by (subst eventually_sequentially_Suc) simp |
|
900 |
||
901 |
lemma filterlim_Suc: "filterlim Suc sequentially sequentially" |
|
902 |
by (simp add: filterlim_iff eventually_sequentially) (metis le_Suc_eq) |
|
903 |
||
60182
e1ea5a6379c9
generalized tends over powr; added DERIV rule for powr
hoelzl
parents:
60040
diff
changeset
|
904 |
lemma filterlim_If: |
e1ea5a6379c9
generalized tends over powr; added DERIV rule for powr
hoelzl
parents:
60040
diff
changeset
|
905 |
"LIM x inf F (principal {x. P x}). f x :> G \<Longrightarrow> |
e1ea5a6379c9
generalized tends over powr; added DERIV rule for powr
hoelzl
parents:
60040
diff
changeset
|
906 |
LIM x inf F (principal {x. \<not> P x}). g x :> G \<Longrightarrow> |
e1ea5a6379c9
generalized tends over powr; added DERIV rule for powr
hoelzl
parents:
60040
diff
changeset
|
907 |
LIM x F. if P x then f x else g x :> G" |
e1ea5a6379c9
generalized tends over powr; added DERIV rule for powr
hoelzl
parents:
60040
diff
changeset
|
908 |
unfolding filterlim_iff eventually_inf_principal by (auto simp: eventually_conj_iff) |
60036 | 909 |
|
60758 | 910 |
subsection \<open>Limits to @{const at_top} and @{const at_bot}\<close> |
60036 | 911 |
|
912 |
lemma filterlim_at_top: |
|
913 |
fixes f :: "'a \<Rightarrow> ('b::linorder)" |
|
914 |
shows "(LIM x F. f x :> at_top) \<longleftrightarrow> (\<forall>Z. eventually (\<lambda>x. Z \<le> f x) F)" |
|
61810 | 915 |
by (auto simp: filterlim_iff eventually_at_top_linorder elim!: eventually_mono) |
60036 | 916 |
|
917 |
lemma filterlim_at_top_mono: |
|
918 |
"LIM x F. f x :> at_top \<Longrightarrow> eventually (\<lambda>x. f x \<le> (g x::'a::linorder)) F \<Longrightarrow> |
|
919 |
LIM x F. g x :> at_top" |
|
920 |
by (auto simp: filterlim_at_top elim: eventually_elim2 intro: order_trans) |
|
921 |
||
922 |
lemma filterlim_at_top_dense: |
|
923 |
fixes f :: "'a \<Rightarrow> ('b::unbounded_dense_linorder)" |
|
924 |
shows "(LIM x F. f x :> at_top) \<longleftrightarrow> (\<forall>Z. eventually (\<lambda>x. Z < f x) F)" |
|
61810 | 925 |
by (metis eventually_mono[of _ F] eventually_gt_at_top order_less_imp_le |
60036 | 926 |
filterlim_at_top[of f F] filterlim_iff[of f at_top F]) |
927 |
||
928 |
lemma filterlim_at_top_ge: |
|
929 |
fixes f :: "'a \<Rightarrow> ('b::linorder)" and c :: "'b" |
|
930 |
shows "(LIM x F. f x :> at_top) \<longleftrightarrow> (\<forall>Z\<ge>c. eventually (\<lambda>x. Z \<le> f x) F)" |
|
931 |
unfolding at_top_sub[of c] filterlim_INF by (auto simp add: filterlim_principal) |
|
932 |
||
933 |
lemma filterlim_at_top_at_top: |
|
934 |
fixes f :: "'a::linorder \<Rightarrow> 'b::linorder" |
|
935 |
assumes mono: "\<And>x y. Q x \<Longrightarrow> Q y \<Longrightarrow> x \<le> y \<Longrightarrow> f x \<le> f y" |
|
936 |
assumes bij: "\<And>x. P x \<Longrightarrow> f (g x) = x" "\<And>x. P x \<Longrightarrow> Q (g x)" |
|
937 |
assumes Q: "eventually Q at_top" |
|
938 |
assumes P: "eventually P at_top" |
|
939 |
shows "filterlim f at_top at_top" |
|
940 |
proof - |
|
941 |
from P obtain x where x: "\<And>y. x \<le> y \<Longrightarrow> P y" |
|
942 |
unfolding eventually_at_top_linorder by auto |
|
943 |
show ?thesis |
|
944 |
proof (intro filterlim_at_top_ge[THEN iffD2] allI impI) |
|
945 |
fix z assume "x \<le> z" |
|
946 |
with x have "P z" by auto |
|
947 |
have "eventually (\<lambda>x. g z \<le> x) at_top" |
|
948 |
by (rule eventually_ge_at_top) |
|
949 |
with Q show "eventually (\<lambda>x. z \<le> f x) at_top" |
|
60758 | 950 |
by eventually_elim (metis mono bij \<open>P z\<close>) |
60036 | 951 |
qed |
952 |
qed |
|
953 |
||
954 |
lemma filterlim_at_top_gt: |
|
955 |
fixes f :: "'a \<Rightarrow> ('b::unbounded_dense_linorder)" and c :: "'b" |
|
956 |
shows "(LIM x F. f x :> at_top) \<longleftrightarrow> (\<forall>Z>c. eventually (\<lambda>x. Z \<le> f x) F)" |
|
957 |
by (metis filterlim_at_top order_less_le_trans gt_ex filterlim_at_top_ge) |
|
958 |
||
61806
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61531
diff
changeset
|
959 |
lemma filterlim_at_bot: |
60036 | 960 |
fixes f :: "'a \<Rightarrow> ('b::linorder)" |
961 |
shows "(LIM x F. f x :> at_bot) \<longleftrightarrow> (\<forall>Z. eventually (\<lambda>x. f x \<le> Z) F)" |
|
61810 | 962 |
by (auto simp: filterlim_iff eventually_at_bot_linorder elim!: eventually_mono) |
60036 | 963 |
|
964 |
lemma filterlim_at_bot_dense: |
|
965 |
fixes f :: "'a \<Rightarrow> ('b::{dense_linorder, no_bot})" |
|
966 |
shows "(LIM x F. f x :> at_bot) \<longleftrightarrow> (\<forall>Z. eventually (\<lambda>x. f x < Z) F)" |
|
967 |
proof (auto simp add: filterlim_at_bot[of f F]) |
|
968 |
fix Z :: 'b |
|
969 |
from lt_ex [of Z] obtain Z' where 1: "Z' < Z" .. |
|
970 |
assume "\<forall>Z. eventually (\<lambda>x. f x \<le> Z) F" |
|
971 |
hence "eventually (\<lambda>x. f x \<le> Z') F" by auto |
|
972 |
thus "eventually (\<lambda>x. f x < Z) F" |
|
61810 | 973 |
apply (rule eventually_mono) |
60036 | 974 |
using 1 by auto |
61806
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61531
diff
changeset
|
975 |
next |
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61531
diff
changeset
|
976 |
fix Z :: 'b |
60036 | 977 |
show "\<forall>Z. eventually (\<lambda>x. f x < Z) F \<Longrightarrow> eventually (\<lambda>x. f x \<le> Z) F" |
61810 | 978 |
by (drule spec [of _ Z], erule eventually_mono, auto simp add: less_imp_le) |
60036 | 979 |
qed |
980 |
||
981 |
lemma filterlim_at_bot_le: |
|
982 |
fixes f :: "'a \<Rightarrow> ('b::linorder)" and c :: "'b" |
|
983 |
shows "(LIM x F. f x :> at_bot) \<longleftrightarrow> (\<forall>Z\<le>c. eventually (\<lambda>x. Z \<ge> f x) F)" |
|
984 |
unfolding filterlim_at_bot |
|
985 |
proof safe |
|
986 |
fix Z assume *: "\<forall>Z\<le>c. eventually (\<lambda>x. Z \<ge> f x) F" |
|
987 |
with *[THEN spec, of "min Z c"] show "eventually (\<lambda>x. Z \<ge> f x) F" |
|
61810 | 988 |
by (auto elim!: eventually_mono) |
60036 | 989 |
qed simp |
990 |
||
991 |
lemma filterlim_at_bot_lt: |
|
992 |
fixes f :: "'a \<Rightarrow> ('b::unbounded_dense_linorder)" and c :: "'b" |
|
993 |
shows "(LIM x F. f x :> at_bot) \<longleftrightarrow> (\<forall>Z<c. eventually (\<lambda>x. Z \<ge> f x) F)" |
|
994 |
by (metis filterlim_at_bot filterlim_at_bot_le lt_ex order_le_less_trans) |
|
995 |
||
996 |
||
60758 | 997 |
subsection \<open>Setup @{typ "'a filter"} for lifting and transfer\<close> |
60036 | 998 |
|
999 |
context begin interpretation lifting_syntax . |
|
1000 |
||
1001 |
definition rel_filter :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a filter \<Rightarrow> 'b filter \<Rightarrow> bool" |
|
1002 |
where "rel_filter R F G = ((R ===> op =) ===> op =) (Rep_filter F) (Rep_filter G)" |
|
1003 |
||
1004 |
lemma rel_filter_eventually: |
|
61806
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61531
diff
changeset
|
1005 |
"rel_filter R F G \<longleftrightarrow> |
60036 | 1006 |
((R ===> op =) ===> op =) (\<lambda>P. eventually P F) (\<lambda>P. eventually P G)" |
1007 |
by(simp add: rel_filter_def eventually_def) |
|
1008 |
||
1009 |
lemma filtermap_id [simp, id_simps]: "filtermap id = id" |
|
1010 |
by(simp add: fun_eq_iff id_def filtermap_ident) |
|
1011 |
||
1012 |
lemma filtermap_id' [simp]: "filtermap (\<lambda>x. x) = (\<lambda>F. F)" |
|
1013 |
using filtermap_id unfolding id_def . |
|
1014 |
||
1015 |
lemma Quotient_filter [quot_map]: |
|
1016 |
assumes Q: "Quotient R Abs Rep T" |
|
1017 |
shows "Quotient (rel_filter R) (filtermap Abs) (filtermap Rep) (rel_filter T)" |
|
1018 |
unfolding Quotient_alt_def |
|
1019 |
proof(intro conjI strip) |
|
1020 |
from Q have *: "\<And>x y. T x y \<Longrightarrow> Abs x = y" |
|
1021 |
unfolding Quotient_alt_def by blast |
|
1022 |
||
1023 |
fix F G |
|
1024 |
assume "rel_filter T F G" |
|
1025 |
thus "filtermap Abs F = G" unfolding filter_eq_iff |
|
1026 |
by(auto simp add: eventually_filtermap rel_filter_eventually * rel_funI del: iffI elim!: rel_funD) |
|
1027 |
next |
|
1028 |
from Q have *: "\<And>x. T (Rep x) x" unfolding Quotient_alt_def by blast |
|
1029 |
||
1030 |
fix F |
|
61806
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61531
diff
changeset
|
1031 |
show "rel_filter T (filtermap Rep F) F" |
60036 | 1032 |
by(auto elim: rel_funD intro: * intro!: ext arg_cong[where f="\<lambda>P. eventually P F"] rel_funI |
1033 |
del: iffI simp add: eventually_filtermap rel_filter_eventually) |
|
1034 |
qed(auto simp add: map_fun_def o_def eventually_filtermap filter_eq_iff fun_eq_iff rel_filter_eventually |
|
1035 |
fun_quotient[OF fun_quotient[OF Q identity_quotient] identity_quotient, unfolded Quotient_alt_def]) |
|
1036 |
||
1037 |
lemma eventually_parametric [transfer_rule]: |
|
1038 |
"((A ===> op =) ===> rel_filter A ===> op =) eventually eventually" |
|
1039 |
by(simp add: rel_fun_def rel_filter_eventually) |
|
1040 |
||
60038 | 1041 |
lemma frequently_parametric [transfer_rule]: |
1042 |
"((A ===> op =) ===> rel_filter A ===> op =) frequently frequently" |
|
1043 |
unfolding frequently_def[abs_def] by transfer_prover |
|
1044 |
||
60036 | 1045 |
lemma rel_filter_eq [relator_eq]: "rel_filter op = = op =" |
1046 |
by(auto simp add: rel_filter_eventually rel_fun_eq fun_eq_iff filter_eq_iff) |
|
1047 |
||
1048 |
lemma rel_filter_mono [relator_mono]: |
|
1049 |
"A \<le> B \<Longrightarrow> rel_filter A \<le> rel_filter B" |
|
1050 |
unfolding rel_filter_eventually[abs_def] |
|
1051 |
by(rule le_funI)+(intro fun_mono fun_mono[THEN le_funD, THEN le_funD] order.refl) |
|
1052 |
||
1053 |
lemma rel_filter_conversep [simp]: "rel_filter A\<inverse>\<inverse> = (rel_filter A)\<inverse>\<inverse>" |
|
61233
1da01148d4b1
Prepared two non-terminating proofs; no obvious link with my changes
paulson <lp15@cam.ac.uk>
parents:
60974
diff
changeset
|
1054 |
apply (simp add: rel_filter_eventually fun_eq_iff rel_fun_def) |
1da01148d4b1
Prepared two non-terminating proofs; no obvious link with my changes
paulson <lp15@cam.ac.uk>
parents:
60974
diff
changeset
|
1055 |
apply (safe; metis) |
1da01148d4b1
Prepared two non-terminating proofs; no obvious link with my changes
paulson <lp15@cam.ac.uk>
parents:
60974
diff
changeset
|
1056 |
done |
60036 | 1057 |
|
1058 |
lemma is_filter_parametric_aux: |
|
1059 |
assumes "is_filter F" |
|
1060 |
assumes [transfer_rule]: "bi_total A" "bi_unique A" |
|
1061 |
and [transfer_rule]: "((A ===> op =) ===> op =) F G" |
|
1062 |
shows "is_filter G" |
|
1063 |
proof - |
|
1064 |
interpret is_filter F by fact |
|
1065 |
show ?thesis |
|
1066 |
proof |
|
1067 |
have "F (\<lambda>_. True) = G (\<lambda>x. True)" by transfer_prover |
|
1068 |
thus "G (\<lambda>x. True)" by(simp add: True) |
|
1069 |
next |
|
1070 |
fix P' Q' |
|
1071 |
assume "G P'" "G Q'" |
|
1072 |
moreover |
|
60758 | 1073 |
from bi_total_fun[OF \<open>bi_unique A\<close> bi_total_eq, unfolded bi_total_def] |
60036 | 1074 |
obtain P Q where [transfer_rule]: "(A ===> op =) P P'" "(A ===> op =) Q Q'" by blast |
1075 |
have "F P = G P'" "F Q = G Q'" by transfer_prover+ |
|
1076 |
ultimately have "F (\<lambda>x. P x \<and> Q x)" by(simp add: conj) |
|
1077 |
moreover have "F (\<lambda>x. P x \<and> Q x) = G (\<lambda>x. P' x \<and> Q' x)" by transfer_prover |
|
1078 |
ultimately show "G (\<lambda>x. P' x \<and> Q' x)" by simp |
|
1079 |
next |
|
1080 |
fix P' Q' |
|
1081 |
assume "\<forall>x. P' x \<longrightarrow> Q' x" "G P'" |
|
1082 |
moreover |
|
60758 | 1083 |
from bi_total_fun[OF \<open>bi_unique A\<close> bi_total_eq, unfolded bi_total_def] |
60036 | 1084 |
obtain P Q where [transfer_rule]: "(A ===> op =) P P'" "(A ===> op =) Q Q'" by blast |
1085 |
have "F P = G P'" by transfer_prover |
|
1086 |
moreover have "(\<forall>x. P x \<longrightarrow> Q x) \<longleftrightarrow> (\<forall>x. P' x \<longrightarrow> Q' x)" by transfer_prover |
|
1087 |
ultimately have "F Q" by(simp add: mono) |
|
1088 |
moreover have "F Q = G Q'" by transfer_prover |
|
1089 |
ultimately show "G Q'" by simp |
|
1090 |
qed |
|
1091 |
qed |
|
1092 |
||
1093 |
lemma is_filter_parametric [transfer_rule]: |
|
1094 |
"\<lbrakk> bi_total A; bi_unique A \<rbrakk> |
|
1095 |
\<Longrightarrow> (((A ===> op =) ===> op =) ===> op =) is_filter is_filter" |
|
1096 |
apply(rule rel_funI) |
|
1097 |
apply(rule iffI) |
|
1098 |
apply(erule (3) is_filter_parametric_aux) |
|
1099 |
apply(erule is_filter_parametric_aux[where A="conversep A"]) |
|
61233
1da01148d4b1
Prepared two non-terminating proofs; no obvious link with my changes
paulson <lp15@cam.ac.uk>
parents:
60974
diff
changeset
|
1100 |
apply (simp_all add: rel_fun_def) |
1da01148d4b1
Prepared two non-terminating proofs; no obvious link with my changes
paulson <lp15@cam.ac.uk>
parents:
60974
diff
changeset
|
1101 |
apply metis |
60036 | 1102 |
done |
1103 |
||
1104 |
lemma left_total_rel_filter [transfer_rule]: |
|
1105 |
assumes [transfer_rule]: "bi_total A" "bi_unique A" |
|
1106 |
shows "left_total (rel_filter A)" |
|
1107 |
proof(rule left_totalI) |
|
1108 |
fix F :: "'a filter" |
|
60758 | 1109 |
from bi_total_fun[OF bi_unique_fun[OF \<open>bi_total A\<close> bi_unique_eq] bi_total_eq] |
61806
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61531
diff
changeset
|
1110 |
obtain G where [transfer_rule]: "((A ===> op =) ===> op =) (\<lambda>P. eventually P F) G" |
60036 | 1111 |
unfolding bi_total_def by blast |
1112 |
moreover have "is_filter (\<lambda>P. eventually P F) \<longleftrightarrow> is_filter G" by transfer_prover |
|
1113 |
hence "is_filter G" by(simp add: eventually_def is_filter_Rep_filter) |
|
1114 |
ultimately have "rel_filter A F (Abs_filter G)" |
|
1115 |
by(simp add: rel_filter_eventually eventually_Abs_filter) |
|
1116 |
thus "\<exists>G. rel_filter A F G" .. |
|
1117 |
qed |
|
1118 |
||
1119 |
lemma right_total_rel_filter [transfer_rule]: |
|
1120 |
"\<lbrakk> bi_total A; bi_unique A \<rbrakk> \<Longrightarrow> right_total (rel_filter A)" |
|
1121 |
using left_total_rel_filter[of "A\<inverse>\<inverse>"] by simp |
|
1122 |
||
1123 |
lemma bi_total_rel_filter [transfer_rule]: |
|
1124 |
assumes "bi_total A" "bi_unique A" |
|
1125 |
shows "bi_total (rel_filter A)" |
|
1126 |
unfolding bi_total_alt_def using assms |
|
1127 |
by(simp add: left_total_rel_filter right_total_rel_filter) |
|
1128 |
||
1129 |
lemma left_unique_rel_filter [transfer_rule]: |
|
1130 |
assumes "left_unique A" |
|
1131 |
shows "left_unique (rel_filter A)" |
|
1132 |
proof(rule left_uniqueI) |
|
1133 |
fix F F' G |
|
1134 |
assume [transfer_rule]: "rel_filter A F G" "rel_filter A F' G" |
|
1135 |
show "F = F'" |
|
1136 |
unfolding filter_eq_iff |
|
1137 |
proof |
|
1138 |
fix P :: "'a \<Rightarrow> bool" |
|
1139 |
obtain P' where [transfer_rule]: "(A ===> op =) P P'" |
|
1140 |
using left_total_fun[OF assms left_total_eq] unfolding left_total_def by blast |
|
61806
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61531
diff
changeset
|
1141 |
have "eventually P F = eventually P' G" |
60036 | 1142 |
and "eventually P F' = eventually P' G" by transfer_prover+ |
1143 |
thus "eventually P F = eventually P F'" by simp |
|
1144 |
qed |
|
1145 |
qed |
|
1146 |
||
1147 |
lemma right_unique_rel_filter [transfer_rule]: |
|
1148 |
"right_unique A \<Longrightarrow> right_unique (rel_filter A)" |
|
1149 |
using left_unique_rel_filter[of "A\<inverse>\<inverse>"] by simp |
|
1150 |
||
1151 |
lemma bi_unique_rel_filter [transfer_rule]: |
|
1152 |
"bi_unique A \<Longrightarrow> bi_unique (rel_filter A)" |
|
1153 |
by(simp add: bi_unique_alt_def left_unique_rel_filter right_unique_rel_filter) |
|
1154 |
||
1155 |
lemma top_filter_parametric [transfer_rule]: |
|
1156 |
"bi_total A \<Longrightarrow> (rel_filter A) top top" |
|
1157 |
by(simp add: rel_filter_eventually All_transfer) |
|
1158 |
||
1159 |
lemma bot_filter_parametric [transfer_rule]: "(rel_filter A) bot bot" |
|
1160 |
by(simp add: rel_filter_eventually rel_fun_def) |
|
1161 |
||
1162 |
lemma sup_filter_parametric [transfer_rule]: |
|
1163 |
"(rel_filter A ===> rel_filter A ===> rel_filter A) sup sup" |
|
1164 |
by(fastforce simp add: rel_filter_eventually[abs_def] eventually_sup dest: rel_funD) |
|
1165 |
||
1166 |
lemma Sup_filter_parametric [transfer_rule]: |
|
1167 |
"(rel_set (rel_filter A) ===> rel_filter A) Sup Sup" |
|
1168 |
proof(rule rel_funI) |
|
1169 |
fix S T |
|
1170 |
assume [transfer_rule]: "rel_set (rel_filter A) S T" |
|
1171 |
show "rel_filter A (Sup S) (Sup T)" |
|
1172 |
by(simp add: rel_filter_eventually eventually_Sup) transfer_prover |
|
1173 |
qed |
|
1174 |
||
1175 |
lemma principal_parametric [transfer_rule]: |
|
1176 |
"(rel_set A ===> rel_filter A) principal principal" |
|
1177 |
proof(rule rel_funI) |
|
1178 |
fix S S' |
|
1179 |
assume [transfer_rule]: "rel_set A S S'" |
|
1180 |
show "rel_filter A (principal S) (principal S')" |
|
1181 |
by(simp add: rel_filter_eventually eventually_principal) transfer_prover |
|
1182 |
qed |
|
1183 |
||
1184 |
context |
|
1185 |
fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool" |
|
61806
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61531
diff
changeset
|
1186 |
assumes [transfer_rule]: "bi_unique A" |
60036 | 1187 |
begin |
1188 |
||
1189 |
lemma le_filter_parametric [transfer_rule]: |
|
1190 |
"(rel_filter A ===> rel_filter A ===> op =) op \<le> op \<le>" |
|
1191 |
unfolding le_filter_def[abs_def] by transfer_prover |
|
1192 |
||
1193 |
lemma less_filter_parametric [transfer_rule]: |
|
1194 |
"(rel_filter A ===> rel_filter A ===> op =) op < op <" |
|
1195 |
unfolding less_filter_def[abs_def] by transfer_prover |
|
1196 |
||
1197 |
context |
|
1198 |
assumes [transfer_rule]: "bi_total A" |
|
1199 |
begin |
|
1200 |
||
1201 |
lemma Inf_filter_parametric [transfer_rule]: |
|
1202 |
"(rel_set (rel_filter A) ===> rel_filter A) Inf Inf" |
|
1203 |
unfolding Inf_filter_def[abs_def] by transfer_prover |
|
1204 |
||
1205 |
lemma inf_filter_parametric [transfer_rule]: |
|
1206 |
"(rel_filter A ===> rel_filter A ===> rel_filter A) inf inf" |
|
1207 |
proof(intro rel_funI)+ |
|
1208 |
fix F F' G G' |
|
1209 |
assume [transfer_rule]: "rel_filter A F F'" "rel_filter A G G'" |
|
1210 |
have "rel_filter A (Inf {F, G}) (Inf {F', G'})" by transfer_prover |
|
1211 |
thus "rel_filter A (inf F G) (inf F' G')" by simp |
|
1212 |
qed |
|
1213 |
||
1214 |
end |
|
1215 |
||
1216 |
end |
|
1217 |
||
1218 |
end |
|
1219 |
||
61806
d2e62ae01cd8
Cauchy's integral formula for circles. Starting to fix eventually_mono.
paulson <lp15@cam.ac.uk>
parents:
61531
diff
changeset
|
1220 |
end |