src/HOL/Filter.thy
author wenzelm
Tue Sep 26 20:54:40 2017 +0200 (22 months ago)
changeset 66695 91500c024c7f
parent 66171 454abfe923fe
child 67399 eab6ce8368fa
permissions -rw-r--r--
tuned;
hoelzl@60036
     1
(*  Title:      HOL/Filter.thy
hoelzl@60036
     2
    Author:     Brian Huffman
hoelzl@60036
     3
    Author:     Johannes Hölzl
hoelzl@60036
     4
*)
hoelzl@60036
     5
wenzelm@60758
     6
section \<open>Filters on predicates\<close>
hoelzl@60036
     7
hoelzl@60036
     8
theory Filter
hoelzl@60036
     9
imports Set_Interval Lifting_Set
hoelzl@60036
    10
begin
hoelzl@60036
    11
wenzelm@60758
    12
subsection \<open>Filters\<close>
hoelzl@60036
    13
wenzelm@60758
    14
text \<open>
hoelzl@60036
    15
  This definition also allows non-proper filters.
wenzelm@60758
    16
\<close>
hoelzl@60036
    17
hoelzl@60036
    18
locale is_filter =
hoelzl@60036
    19
  fixes F :: "('a \<Rightarrow> bool) \<Rightarrow> bool"
hoelzl@60036
    20
  assumes True: "F (\<lambda>x. True)"
hoelzl@60036
    21
  assumes conj: "F (\<lambda>x. P x) \<Longrightarrow> F (\<lambda>x. Q x) \<Longrightarrow> F (\<lambda>x. P x \<and> Q x)"
hoelzl@60036
    22
  assumes mono: "\<forall>x. P x \<longrightarrow> Q x \<Longrightarrow> F (\<lambda>x. P x) \<Longrightarrow> F (\<lambda>x. Q x)"
hoelzl@60036
    23
hoelzl@60036
    24
typedef 'a filter = "{F :: ('a \<Rightarrow> bool) \<Rightarrow> bool. is_filter F}"
hoelzl@60036
    25
proof
hoelzl@60036
    26
  show "(\<lambda>x. True) \<in> ?filter" by (auto intro: is_filter.intro)
hoelzl@60036
    27
qed
hoelzl@60036
    28
hoelzl@60036
    29
lemma is_filter_Rep_filter: "is_filter (Rep_filter F)"
hoelzl@60036
    30
  using Rep_filter [of F] by simp
hoelzl@60036
    31
hoelzl@60036
    32
lemma Abs_filter_inverse':
hoelzl@60036
    33
  assumes "is_filter F" shows "Rep_filter (Abs_filter F) = F"
hoelzl@60036
    34
  using assms by (simp add: Abs_filter_inverse)
hoelzl@60036
    35
hoelzl@60036
    36
wenzelm@60758
    37
subsubsection \<open>Eventually\<close>
hoelzl@60036
    38
hoelzl@60036
    39
definition eventually :: "('a \<Rightarrow> bool) \<Rightarrow> 'a filter \<Rightarrow> bool"
hoelzl@60036
    40
  where "eventually P F \<longleftrightarrow> Rep_filter F P"
hoelzl@60036
    41
wenzelm@61953
    42
syntax
wenzelm@61953
    43
  "_eventually" :: "pttrn => 'a filter => bool => bool"  ("(3\<forall>\<^sub>F _ in _./ _)" [0, 0, 10] 10)
hoelzl@60037
    44
translations
hoelzl@60038
    45
  "\<forall>\<^sub>Fx in F. P" == "CONST eventually (\<lambda>x. P) F"
hoelzl@60037
    46
hoelzl@60036
    47
lemma eventually_Abs_filter:
hoelzl@60036
    48
  assumes "is_filter F" shows "eventually P (Abs_filter F) = F P"
hoelzl@60036
    49
  unfolding eventually_def using assms by (simp add: Abs_filter_inverse)
hoelzl@60036
    50
hoelzl@60036
    51
lemma filter_eq_iff:
hoelzl@60036
    52
  shows "F = F' \<longleftrightarrow> (\<forall>P. eventually P F = eventually P F')"
hoelzl@60036
    53
  unfolding Rep_filter_inject [symmetric] fun_eq_iff eventually_def ..
hoelzl@60036
    54
hoelzl@60036
    55
lemma eventually_True [simp]: "eventually (\<lambda>x. True) F"
hoelzl@60036
    56
  unfolding eventually_def
hoelzl@60036
    57
  by (rule is_filter.True [OF is_filter_Rep_filter])
hoelzl@60036
    58
hoelzl@60036
    59
lemma always_eventually: "\<forall>x. P x \<Longrightarrow> eventually P F"
hoelzl@60036
    60
proof -
hoelzl@60036
    61
  assume "\<forall>x. P x" hence "P = (\<lambda>x. True)" by (simp add: ext)
hoelzl@60036
    62
  thus "eventually P F" by simp
hoelzl@60036
    63
qed
hoelzl@60036
    64
hoelzl@60040
    65
lemma eventuallyI: "(\<And>x. P x) \<Longrightarrow> eventually P F"
hoelzl@60040
    66
  by (auto intro: always_eventually)
hoelzl@60040
    67
hoelzl@60036
    68
lemma eventually_mono:
lp15@61806
    69
  "\<lbrakk>eventually P F; \<And>x. P x \<Longrightarrow> Q x\<rbrakk> \<Longrightarrow> eventually Q F"
lp15@61806
    70
  unfolding eventually_def
lp15@61806
    71
  by (blast intro: is_filter.mono [OF is_filter_Rep_filter])
lp15@61806
    72
hoelzl@60036
    73
lemma eventually_conj:
hoelzl@60036
    74
  assumes P: "eventually (\<lambda>x. P x) F"
hoelzl@60036
    75
  assumes Q: "eventually (\<lambda>x. Q x) F"
hoelzl@60036
    76
  shows "eventually (\<lambda>x. P x \<and> Q x) F"
hoelzl@60036
    77
  using assms unfolding eventually_def
hoelzl@60036
    78
  by (rule is_filter.conj [OF is_filter_Rep_filter])
hoelzl@60036
    79
hoelzl@60036
    80
lemma eventually_mp:
hoelzl@60036
    81
  assumes "eventually (\<lambda>x. P x \<longrightarrow> Q x) F"
hoelzl@60036
    82
  assumes "eventually (\<lambda>x. P x) F"
hoelzl@60036
    83
  shows "eventually (\<lambda>x. Q x) F"
lp15@61806
    84
proof -
lp15@61806
    85
  have "eventually (\<lambda>x. (P x \<longrightarrow> Q x) \<and> P x) F"
hoelzl@60036
    86
    using assms by (rule eventually_conj)
lp15@61806
    87
  then show ?thesis
lp15@61810
    88
    by (blast intro: eventually_mono)
hoelzl@60036
    89
qed
hoelzl@60036
    90
hoelzl@60036
    91
lemma eventually_rev_mp:
hoelzl@60036
    92
  assumes "eventually (\<lambda>x. P x) F"
hoelzl@60036
    93
  assumes "eventually (\<lambda>x. P x \<longrightarrow> Q x) F"
hoelzl@60036
    94
  shows "eventually (\<lambda>x. Q x) F"
hoelzl@60036
    95
using assms(2) assms(1) by (rule eventually_mp)
hoelzl@60036
    96
hoelzl@60036
    97
lemma eventually_conj_iff:
hoelzl@60036
    98
  "eventually (\<lambda>x. P x \<and> Q x) F \<longleftrightarrow> eventually P F \<and> eventually Q F"
hoelzl@60036
    99
  by (auto intro: eventually_conj elim: eventually_rev_mp)
hoelzl@60036
   100
hoelzl@60036
   101
lemma eventually_elim2:
hoelzl@60036
   102
  assumes "eventually (\<lambda>i. P i) F"
hoelzl@60036
   103
  assumes "eventually (\<lambda>i. Q i) F"
hoelzl@60036
   104
  assumes "\<And>i. P i \<Longrightarrow> Q i \<Longrightarrow> R i"
hoelzl@60036
   105
  shows "eventually (\<lambda>i. R i) F"
hoelzl@60036
   106
  using assms by (auto elim!: eventually_rev_mp)
hoelzl@60036
   107
hoelzl@60040
   108
lemma eventually_ball_finite_distrib:
hoelzl@60040
   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)"
hoelzl@60040
   110
  by (induction A rule: finite_induct) (auto simp: eventually_conj_iff)
hoelzl@60040
   111
hoelzl@60040
   112
lemma eventually_ball_finite:
hoelzl@60040
   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"
hoelzl@60040
   114
  by (auto simp: eventually_ball_finite_distrib)
hoelzl@60040
   115
hoelzl@60040
   116
lemma eventually_all_finite:
hoelzl@60040
   117
  fixes P :: "'a \<Rightarrow> 'b::finite \<Rightarrow> bool"
hoelzl@60040
   118
  assumes "\<And>y. eventually (\<lambda>x. P x y) net"
hoelzl@60040
   119
  shows "eventually (\<lambda>x. \<forall>y. P x y) net"
hoelzl@60040
   120
using eventually_ball_finite [of UNIV P] assms by simp
hoelzl@60040
   121
hoelzl@60040
   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))"
hoelzl@60040
   123
proof
hoelzl@60040
   124
  assume "\<forall>\<^sub>Fx in F. \<exists>y. P x y"
hoelzl@60040
   125
  then have "\<forall>\<^sub>Fx in F. P x (SOME y. P x y)"
lp15@61810
   126
    by (auto intro: someI_ex eventually_mono)
hoelzl@60040
   127
  then show "\<exists>Y. \<forall>\<^sub>Fx in F. P x (Y x)"
hoelzl@60040
   128
    by auto
lp15@61810
   129
qed (auto intro: eventually_mono)
hoelzl@60040
   130
hoelzl@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"
hoelzl@60036
   132
  by (auto intro: eventually_mp)
hoelzl@60036
   133
hoelzl@60036
   134
lemma not_eventuallyD: "\<not> eventually P F \<Longrightarrow> \<exists>x. \<not> P x"
hoelzl@60036
   135
  by (metis always_eventually)
hoelzl@60036
   136
hoelzl@60036
   137
lemma eventually_subst:
hoelzl@60036
   138
  assumes "eventually (\<lambda>n. P n = Q n) F"
hoelzl@60036
   139
  shows "eventually P F = eventually Q F" (is "?L = ?R")
hoelzl@60036
   140
proof -
hoelzl@60036
   141
  from assms have "eventually (\<lambda>x. P x \<longrightarrow> Q x) F"
hoelzl@60036
   142
      and "eventually (\<lambda>x. Q x \<longrightarrow> P x) F"
lp15@61810
   143
    by (auto elim: eventually_mono)
hoelzl@60036
   144
  then show ?thesis by (auto elim: eventually_elim2)
hoelzl@60036
   145
qed
hoelzl@60036
   146
hoelzl@60040
   147
subsection \<open> Frequently as dual to eventually \<close>
hoelzl@60040
   148
hoelzl@60040
   149
definition frequently :: "('a \<Rightarrow> bool) \<Rightarrow> 'a filter \<Rightarrow> bool"
hoelzl@60040
   150
  where "frequently P F \<longleftrightarrow> \<not> eventually (\<lambda>x. \<not> P x) F"
hoelzl@60040
   151
wenzelm@61953
   152
syntax
wenzelm@61953
   153
  "_frequently" :: "pttrn \<Rightarrow> 'a filter \<Rightarrow> bool \<Rightarrow> bool"  ("(3\<exists>\<^sub>F _ in _./ _)" [0, 0, 10] 10)
hoelzl@60040
   154
translations
hoelzl@60040
   155
  "\<exists>\<^sub>Fx in F. P" == "CONST frequently (\<lambda>x. P) F"
hoelzl@60040
   156
hoelzl@60040
   157
lemma not_frequently_False [simp]: "\<not> (\<exists>\<^sub>Fx in F. False)"
hoelzl@60040
   158
  by (simp add: frequently_def)
hoelzl@60040
   159
hoelzl@60040
   160
lemma frequently_ex: "\<exists>\<^sub>Fx in F. P x \<Longrightarrow> \<exists>x. P x"
hoelzl@60040
   161
  by (auto simp: frequently_def dest: not_eventuallyD)
hoelzl@60040
   162
hoelzl@60040
   163
lemma frequentlyE: assumes "frequently P F" obtains x where "P x"
hoelzl@60040
   164
  using frequently_ex[OF assms] by auto
hoelzl@60040
   165
hoelzl@60040
   166
lemma frequently_mp:
hoelzl@60040
   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"
lp15@61806
   168
proof -
hoelzl@60040
   169
  from ev have "eventually (\<lambda>x. \<not> Q x \<longrightarrow> \<not> P x) F"
hoelzl@60040
   170
    by (rule eventually_rev_mp) (auto intro!: always_eventually)
hoelzl@60040
   171
  from eventually_mp[OF this] P show ?thesis
hoelzl@60040
   172
    by (auto simp: frequently_def)
hoelzl@60040
   173
qed
hoelzl@60040
   174
hoelzl@60040
   175
lemma frequently_rev_mp:
hoelzl@60040
   176
  assumes "\<exists>\<^sub>Fx in F. P x"
hoelzl@60040
   177
  assumes "\<forall>\<^sub>Fx in F. P x \<longrightarrow> Q x"
hoelzl@60040
   178
  shows "\<exists>\<^sub>Fx in F. Q x"
hoelzl@60040
   179
using assms(2) assms(1) by (rule frequently_mp)
hoelzl@60040
   180
hoelzl@60040
   181
lemma frequently_mono: "(\<forall>x. P x \<longrightarrow> Q x) \<Longrightarrow> frequently P F \<Longrightarrow> frequently Q F"
hoelzl@60040
   182
  using frequently_mp[of P Q] by (simp add: always_eventually)
hoelzl@60040
   183
hoelzl@60040
   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"
hoelzl@60040
   185
  by (metis frequently_mono)
hoelzl@60040
   186
hoelzl@60040
   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)"
hoelzl@60040
   188
  by (simp add: frequently_def eventually_conj_iff)
hoelzl@60040
   189
hoelzl@60040
   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"
hoelzl@60040
   191
  by (simp add: frequently_disj_iff)
hoelzl@60040
   192
hoelzl@60040
   193
lemma frequently_bex_finite_distrib:
hoelzl@60040
   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)"
hoelzl@60040
   195
  using assms by induction (auto simp: frequently_disj_iff)
hoelzl@60040
   196
hoelzl@60040
   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"
hoelzl@60040
   198
  by (simp add: frequently_bex_finite_distrib)
hoelzl@60040
   199
hoelzl@60040
   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))"
hoelzl@60040
   201
  using eventually_ex[of "\<lambda>x y. \<not> P x y" F] by (simp add: frequently_def)
hoelzl@60040
   202
hoelzl@60040
   203
lemma
hoelzl@60040
   204
  shows not_eventually: "\<not> eventually P F \<longleftrightarrow> (\<exists>\<^sub>Fx in F. \<not> P x)"
hoelzl@60040
   205
    and not_frequently: "\<not> frequently P F \<longleftrightarrow> (\<forall>\<^sub>Fx in F. \<not> P x)"
hoelzl@60040
   206
  by (auto simp: frequently_def)
hoelzl@60040
   207
hoelzl@60040
   208
lemma frequently_imp_iff:
hoelzl@60040
   209
  "(\<exists>\<^sub>Fx in F. P x \<longrightarrow> Q x) \<longleftrightarrow> (eventually P F \<longrightarrow> frequently Q F)"
hoelzl@60040
   210
  unfolding imp_conv_disj frequently_disj_iff not_eventually[symmetric] ..
hoelzl@60040
   211
hoelzl@60040
   212
lemma eventually_frequently_const_simps:
hoelzl@60040
   213
  "(\<exists>\<^sub>Fx in F. P x \<and> C) \<longleftrightarrow> (\<exists>\<^sub>Fx in F. P x) \<and> C"
hoelzl@60040
   214
  "(\<exists>\<^sub>Fx in F. C \<and> P x) \<longleftrightarrow> C \<and> (\<exists>\<^sub>Fx in F. P x)"
hoelzl@60040
   215
  "(\<forall>\<^sub>Fx in F. P x \<or> C) \<longleftrightarrow> (\<forall>\<^sub>Fx in F. P x) \<or> C"
hoelzl@60040
   216
  "(\<forall>\<^sub>Fx in F. C \<or> P x) \<longleftrightarrow> C \<or> (\<forall>\<^sub>Fx in F. P x)"
hoelzl@60040
   217
  "(\<forall>\<^sub>Fx in F. P x \<longrightarrow> C) \<longleftrightarrow> ((\<exists>\<^sub>Fx in F. P x) \<longrightarrow> C)"
hoelzl@60040
   218
  "(\<forall>\<^sub>Fx in F. C \<longrightarrow> P x) \<longleftrightarrow> (C \<longrightarrow> (\<forall>\<^sub>Fx in F. P x))"
hoelzl@60040
   219
  by (cases C; simp add: not_frequently)+
hoelzl@60040
   220
lp15@61806
   221
lemmas eventually_frequently_simps =
hoelzl@60040
   222
  eventually_frequently_const_simps
hoelzl@60040
   223
  not_eventually
hoelzl@60040
   224
  eventually_conj_iff
hoelzl@60040
   225
  eventually_ball_finite_distrib
hoelzl@60040
   226
  eventually_ex
hoelzl@60040
   227
  not_frequently
hoelzl@60040
   228
  frequently_disj_iff
hoelzl@60040
   229
  frequently_bex_finite_distrib
hoelzl@60040
   230
  frequently_all
hoelzl@60040
   231
  frequently_imp_iff
hoelzl@60040
   232
wenzelm@60758
   233
ML \<open>
wenzelm@61841
   234
  fun eventually_elim_tac facts =
wenzelm@61841
   235
    CONTEXT_SUBGOAL (fn (goal, i) => fn (ctxt, st) =>
wenzelm@61841
   236
      let
wenzelm@61841
   237
        val mp_thms = facts RL @{thms eventually_rev_mp}
wenzelm@61841
   238
        val raw_elim_thm =
wenzelm@61841
   239
          (@{thm allI} RS @{thm always_eventually})
wenzelm@61841
   240
          |> fold (fn thm1 => fn thm2 => thm2 RS thm1) mp_thms
wenzelm@61841
   241
          |> fold (fn _ => fn thm => @{thm impI} RS thm) facts
wenzelm@61841
   242
        val cases_prop =
wenzelm@61841
   243
          Thm.prop_of
wenzelm@61841
   244
            (Rule_Cases.internalize_params (raw_elim_thm RS Goal.init (Thm.cterm_of ctxt goal)))
wenzelm@61841
   245
        val cases = Rule_Cases.make_common ctxt cases_prop [(("elim", []), [])]
wenzelm@61841
   246
      in CONTEXT_CASES cases (resolve_tac ctxt [raw_elim_thm] i) (ctxt, st) end)
wenzelm@60758
   247
\<close>
hoelzl@60036
   248
wenzelm@60758
   249
method_setup eventually_elim = \<open>
wenzelm@61841
   250
  Scan.succeed (fn _ => CONTEXT_METHOD (fn facts => eventually_elim_tac facts 1))
wenzelm@60758
   251
\<close> "elimination of eventually quantifiers"
hoelzl@60036
   252
wenzelm@60758
   253
subsubsection \<open>Finer-than relation\<close>
hoelzl@60036
   254
wenzelm@60758
   255
text \<open>@{term "F \<le> F'"} means that filter @{term F} is finer than
wenzelm@60758
   256
filter @{term F'}.\<close>
hoelzl@60036
   257
hoelzl@60036
   258
instantiation filter :: (type) complete_lattice
hoelzl@60036
   259
begin
hoelzl@60036
   260
hoelzl@60036
   261
definition le_filter_def:
hoelzl@60036
   262
  "F \<le> F' \<longleftrightarrow> (\<forall>P. eventually P F' \<longrightarrow> eventually P F)"
hoelzl@60036
   263
hoelzl@60036
   264
definition
hoelzl@60036
   265
  "(F :: 'a filter) < F' \<longleftrightarrow> F \<le> F' \<and> \<not> F' \<le> F"
hoelzl@60036
   266
hoelzl@60036
   267
definition
hoelzl@60036
   268
  "top = Abs_filter (\<lambda>P. \<forall>x. P x)"
hoelzl@60036
   269
hoelzl@60036
   270
definition
hoelzl@60036
   271
  "bot = Abs_filter (\<lambda>P. True)"
hoelzl@60036
   272
hoelzl@60036
   273
definition
hoelzl@60036
   274
  "sup F F' = Abs_filter (\<lambda>P. eventually P F \<and> eventually P F')"
hoelzl@60036
   275
hoelzl@60036
   276
definition
hoelzl@60036
   277
  "inf F F' = Abs_filter
hoelzl@60036
   278
      (\<lambda>P. \<exists>Q R. eventually Q F \<and> eventually R F' \<and> (\<forall>x. Q x \<and> R x \<longrightarrow> P x))"
hoelzl@60036
   279
hoelzl@60036
   280
definition
hoelzl@60036
   281
  "Sup S = Abs_filter (\<lambda>P. \<forall>F\<in>S. eventually P F)"
hoelzl@60036
   282
hoelzl@60036
   283
definition
hoelzl@60036
   284
  "Inf S = Sup {F::'a filter. \<forall>F'\<in>S. F \<le> F'}"
hoelzl@60036
   285
hoelzl@60036
   286
lemma eventually_top [simp]: "eventually P top \<longleftrightarrow> (\<forall>x. P x)"
hoelzl@60036
   287
  unfolding top_filter_def
hoelzl@60036
   288
  by (rule eventually_Abs_filter, rule is_filter.intro, auto)
hoelzl@60036
   289
hoelzl@60036
   290
lemma eventually_bot [simp]: "eventually P bot"
hoelzl@60036
   291
  unfolding bot_filter_def
hoelzl@60036
   292
  by (subst eventually_Abs_filter, rule is_filter.intro, auto)
hoelzl@60036
   293
hoelzl@60036
   294
lemma eventually_sup:
hoelzl@60036
   295
  "eventually P (sup F F') \<longleftrightarrow> eventually P F \<and> eventually P F'"
hoelzl@60036
   296
  unfolding sup_filter_def
hoelzl@60036
   297
  by (rule eventually_Abs_filter, rule is_filter.intro)
hoelzl@60036
   298
     (auto elim!: eventually_rev_mp)
hoelzl@60036
   299
hoelzl@60036
   300
lemma eventually_inf:
hoelzl@60036
   301
  "eventually P (inf F F') \<longleftrightarrow>
hoelzl@60036
   302
   (\<exists>Q R. eventually Q F \<and> eventually R F' \<and> (\<forall>x. Q x \<and> R x \<longrightarrow> P x))"
hoelzl@60036
   303
  unfolding inf_filter_def
hoelzl@60036
   304
  apply (rule eventually_Abs_filter, rule is_filter.intro)
hoelzl@60036
   305
  apply (fast intro: eventually_True)
hoelzl@60036
   306
  apply clarify
hoelzl@60036
   307
  apply (intro exI conjI)
hoelzl@60036
   308
  apply (erule (1) eventually_conj)
hoelzl@60036
   309
  apply (erule (1) eventually_conj)
hoelzl@60036
   310
  apply simp
hoelzl@60036
   311
  apply auto
hoelzl@60036
   312
  done
hoelzl@60036
   313
hoelzl@60036
   314
lemma eventually_Sup:
hoelzl@60036
   315
  "eventually P (Sup S) \<longleftrightarrow> (\<forall>F\<in>S. eventually P F)"
hoelzl@60036
   316
  unfolding Sup_filter_def
hoelzl@60036
   317
  apply (rule eventually_Abs_filter, rule is_filter.intro)
hoelzl@60036
   318
  apply (auto intro: eventually_conj elim!: eventually_rev_mp)
hoelzl@60036
   319
  done
hoelzl@60036
   320
hoelzl@60036
   321
instance proof
hoelzl@60036
   322
  fix F F' F'' :: "'a filter" and S :: "'a filter set"
hoelzl@60036
   323
  { show "F < F' \<longleftrightarrow> F \<le> F' \<and> \<not> F' \<le> F"
hoelzl@60036
   324
    by (rule less_filter_def) }
hoelzl@60036
   325
  { show "F \<le> F"
hoelzl@60036
   326
    unfolding le_filter_def by simp }
hoelzl@60036
   327
  { assume "F \<le> F'" and "F' \<le> F''" thus "F \<le> F''"
hoelzl@60036
   328
    unfolding le_filter_def by simp }
hoelzl@60036
   329
  { assume "F \<le> F'" and "F' \<le> F" thus "F = F'"
hoelzl@60036
   330
    unfolding le_filter_def filter_eq_iff by fast }
hoelzl@60036
   331
  { show "inf F F' \<le> F" and "inf F F' \<le> F'"
hoelzl@60036
   332
    unfolding le_filter_def eventually_inf by (auto intro: eventually_True) }
hoelzl@60036
   333
  { assume "F \<le> F'" and "F \<le> F''" thus "F \<le> inf F' F''"
hoelzl@60036
   334
    unfolding le_filter_def eventually_inf
lp15@61810
   335
    by (auto intro: eventually_mono [OF eventually_conj]) }
hoelzl@60036
   336
  { show "F \<le> sup F F'" and "F' \<le> sup F F'"
hoelzl@60036
   337
    unfolding le_filter_def eventually_sup by simp_all }
hoelzl@60036
   338
  { assume "F \<le> F''" and "F' \<le> F''" thus "sup F F' \<le> F''"
hoelzl@60036
   339
    unfolding le_filter_def eventually_sup by simp }
hoelzl@60036
   340
  { assume "F'' \<in> S" thus "Inf S \<le> F''"
hoelzl@60036
   341
    unfolding le_filter_def Inf_filter_def eventually_Sup Ball_def by simp }
hoelzl@60036
   342
  { assume "\<And>F'. F' \<in> S \<Longrightarrow> F \<le> F'" thus "F \<le> Inf S"
hoelzl@60036
   343
    unfolding le_filter_def Inf_filter_def eventually_Sup Ball_def by simp }
hoelzl@60036
   344
  { assume "F \<in> S" thus "F \<le> Sup S"
hoelzl@60036
   345
    unfolding le_filter_def eventually_Sup by simp }
hoelzl@60036
   346
  { assume "\<And>F. F \<in> S \<Longrightarrow> F \<le> F'" thus "Sup S \<le> F'"
hoelzl@60036
   347
    unfolding le_filter_def eventually_Sup by simp }
hoelzl@60036
   348
  { show "Inf {} = (top::'a filter)"
hoelzl@60036
   349
    by (auto simp: top_filter_def Inf_filter_def Sup_filter_def)
hoelzl@60036
   350
      (metis (full_types) top_filter_def always_eventually eventually_top) }
hoelzl@60036
   351
  { show "Sup {} = (bot::'a filter)"
hoelzl@60036
   352
    by (auto simp: bot_filter_def Sup_filter_def) }
hoelzl@60036
   353
qed
hoelzl@60036
   354
hoelzl@60036
   355
end
hoelzl@60036
   356
eberlm@66171
   357
instance filter :: (type) distrib_lattice
eberlm@66171
   358
proof
eberlm@66171
   359
  fix F G H :: "'a filter"
eberlm@66171
   360
  show "sup F (inf G H) = inf (sup F G) (sup F H)"
eberlm@66171
   361
  proof (rule order.antisym)
eberlm@66171
   362
    show "inf (sup F G) (sup F H) \<le> sup F (inf G H)" 
eberlm@66171
   363
      unfolding le_filter_def eventually_sup
eberlm@66171
   364
    proof safe
eberlm@66171
   365
      fix P assume 1: "eventually P F" and 2: "eventually P (inf G H)"
eberlm@66171
   366
      from 2 obtain Q R 
eberlm@66171
   367
        where QR: "eventually Q G" "eventually R H" "\<And>x. Q x \<Longrightarrow> R x \<Longrightarrow> P x"
eberlm@66171
   368
        by (auto simp: eventually_inf)
eberlm@66171
   369
      define Q' where "Q' = (\<lambda>x. Q x \<or> P x)"
eberlm@66171
   370
      define R' where "R' = (\<lambda>x. R x \<or> P x)"
eberlm@66171
   371
      from 1 have "eventually Q' F" 
eberlm@66171
   372
        by (elim eventually_mono) (auto simp: Q'_def)
eberlm@66171
   373
      moreover from 1 have "eventually R' F" 
eberlm@66171
   374
        by (elim eventually_mono) (auto simp: R'_def)
eberlm@66171
   375
      moreover from QR(1) have "eventually Q' G" 
eberlm@66171
   376
        by (elim eventually_mono) (auto simp: Q'_def)
eberlm@66171
   377
      moreover from QR(2) have "eventually R' H" 
eberlm@66171
   378
        by (elim eventually_mono)(auto simp: R'_def)
eberlm@66171
   379
      moreover from QR have "P x" if "Q' x" "R' x" for x 
eberlm@66171
   380
        using that by (auto simp: Q'_def R'_def)
eberlm@66171
   381
      ultimately show "eventually P (inf (sup F G) (sup F H))"
eberlm@66171
   382
        by (auto simp: eventually_inf eventually_sup)
eberlm@66171
   383
    qed
eberlm@66171
   384
  qed (auto intro: inf.coboundedI1 inf.coboundedI2)
eberlm@66171
   385
qed
eberlm@66171
   386
eberlm@66171
   387
hoelzl@60036
   388
lemma filter_leD:
hoelzl@60036
   389
  "F \<le> F' \<Longrightarrow> eventually P F' \<Longrightarrow> eventually P F"
hoelzl@60036
   390
  unfolding le_filter_def by simp
hoelzl@60036
   391
hoelzl@60036
   392
lemma filter_leI:
hoelzl@60036
   393
  "(\<And>P. eventually P F' \<Longrightarrow> eventually P F) \<Longrightarrow> F \<le> F'"
hoelzl@60036
   394
  unfolding le_filter_def by simp
hoelzl@60036
   395
hoelzl@60036
   396
lemma eventually_False:
hoelzl@60036
   397
  "eventually (\<lambda>x. False) F \<longleftrightarrow> F = bot"
hoelzl@60036
   398
  unfolding filter_eq_iff by (auto elim: eventually_rev_mp)
hoelzl@60036
   399
hoelzl@60040
   400
lemma eventually_frequently: "F \<noteq> bot \<Longrightarrow> eventually P F \<Longrightarrow> frequently P F"
hoelzl@60040
   401
  using eventually_conj[of P F "\<lambda>x. \<not> P x"]
hoelzl@60040
   402
  by (auto simp add: frequently_def eventually_False)
hoelzl@60040
   403
hoelzl@60040
   404
lemma eventually_const_iff: "eventually (\<lambda>x. P) F \<longleftrightarrow> P \<or> F = bot"
hoelzl@60040
   405
  by (cases P) (auto simp: eventually_False)
hoelzl@60040
   406
hoelzl@60040
   407
lemma eventually_const[simp]: "F \<noteq> bot \<Longrightarrow> eventually (\<lambda>x. P) F \<longleftrightarrow> P"
hoelzl@60040
   408
  by (simp add: eventually_const_iff)
hoelzl@60040
   409
hoelzl@60040
   410
lemma frequently_const_iff: "frequently (\<lambda>x. P) F \<longleftrightarrow> P \<and> F \<noteq> bot"
hoelzl@60040
   411
  by (simp add: frequently_def eventually_const_iff)
hoelzl@60040
   412
hoelzl@60040
   413
lemma frequently_const[simp]: "F \<noteq> bot \<Longrightarrow> frequently (\<lambda>x. P) F \<longleftrightarrow> P"
hoelzl@60040
   414
  by (simp add: frequently_const_iff)
hoelzl@60040
   415
hoelzl@61245
   416
lemma eventually_happens: "eventually P net \<Longrightarrow> net = bot \<or> (\<exists>x. P x)"
hoelzl@61245
   417
  by (metis frequentlyE eventually_frequently)
hoelzl@61245
   418
eberlm@61531
   419
lemma eventually_happens':
eberlm@61531
   420
  assumes "F \<noteq> bot" "eventually P F"
eberlm@61531
   421
  shows   "\<exists>x. P x"
eberlm@61531
   422
  using assms eventually_frequently frequentlyE by blast
eberlm@61531
   423
hoelzl@60036
   424
abbreviation (input) trivial_limit :: "'a filter \<Rightarrow> bool"
hoelzl@60036
   425
  where "trivial_limit F \<equiv> F = bot"
hoelzl@60036
   426
hoelzl@60036
   427
lemma trivial_limit_def: "trivial_limit F \<longleftrightarrow> eventually (\<lambda>x. False) F"
hoelzl@60036
   428
  by (rule eventually_False [symmetric])
hoelzl@60036
   429
lp15@61806
   430
lemma False_imp_not_eventually: "(\<forall>x. \<not> P x ) \<Longrightarrow> \<not> trivial_limit net \<Longrightarrow> \<not> eventually (\<lambda>x. P x) net"
lp15@61806
   431
  by (simp add: eventually_False)
lp15@61806
   432
hoelzl@60036
   433
lemma eventually_Inf: "eventually P (Inf B) \<longleftrightarrow> (\<exists>X\<subseteq>B. finite X \<and> eventually P (Inf X))"
hoelzl@60036
   434
proof -
hoelzl@60036
   435
  let ?F = "\<lambda>P. \<exists>X\<subseteq>B. finite X \<and> eventually P (Inf X)"
lp15@61806
   436
hoelzl@60036
   437
  { fix P have "eventually P (Abs_filter ?F) \<longleftrightarrow> ?F P"
hoelzl@60036
   438
    proof (rule eventually_Abs_filter is_filter.intro)+
hoelzl@60036
   439
      show "?F (\<lambda>x. True)"
hoelzl@60036
   440
        by (rule exI[of _ "{}"]) (simp add: le_fun_def)
hoelzl@60036
   441
    next
hoelzl@60036
   442
      fix P Q
hoelzl@60036
   443
      assume "?F P" then guess X ..
hoelzl@60036
   444
      moreover
hoelzl@60036
   445
      assume "?F Q" then guess Y ..
hoelzl@60036
   446
      ultimately show "?F (\<lambda>x. P x \<and> Q x)"
hoelzl@60036
   447
        by (intro exI[of _ "X \<union> Y"])
hoelzl@60036
   448
           (auto simp: Inf_union_distrib eventually_inf)
hoelzl@60036
   449
    next
hoelzl@60036
   450
      fix P Q
hoelzl@60036
   451
      assume "?F P" then guess X ..
hoelzl@60036
   452
      moreover assume "\<forall>x. P x \<longrightarrow> Q x"
hoelzl@60036
   453
      ultimately show "?F Q"
lp15@61810
   454
        by (intro exI[of _ X]) (auto elim: eventually_mono)
hoelzl@60036
   455
    qed }
hoelzl@60036
   456
  note eventually_F = this
hoelzl@60036
   457
hoelzl@60036
   458
  have "Inf B = Abs_filter ?F"
hoelzl@60036
   459
  proof (intro antisym Inf_greatest)
hoelzl@60036
   460
    show "Inf B \<le> Abs_filter ?F"
hoelzl@60036
   461
      by (auto simp: le_filter_def eventually_F dest: Inf_superset_mono)
hoelzl@60036
   462
  next
hoelzl@60036
   463
    fix F assume "F \<in> B" then show "Abs_filter ?F \<le> F"
hoelzl@60036
   464
      by (auto simp add: le_filter_def eventually_F intro!: exI[of _ "{F}"])
hoelzl@60036
   465
  qed
hoelzl@60036
   466
  then show ?thesis
hoelzl@60036
   467
    by (simp add: eventually_F)
hoelzl@60036
   468
qed
hoelzl@60036
   469
hoelzl@60036
   470
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))"
haftmann@62343
   471
  unfolding eventually_Inf [of P "F`B"]
haftmann@62343
   472
  by (metis finite_imageI image_mono finite_subset_image)
hoelzl@60036
   473
hoelzl@60036
   474
lemma Inf_filter_not_bot:
hoelzl@60036
   475
  fixes B :: "'a filter set"
hoelzl@60036
   476
  shows "(\<And>X. X \<subseteq> B \<Longrightarrow> finite X \<Longrightarrow> Inf X \<noteq> bot) \<Longrightarrow> Inf B \<noteq> bot"
hoelzl@60036
   477
  unfolding trivial_limit_def eventually_Inf[of _ B]
hoelzl@60036
   478
    bot_bool_def [symmetric] bot_fun_def [symmetric] bot_unique by simp
hoelzl@60036
   479
hoelzl@60036
   480
lemma INF_filter_not_bot:
hoelzl@60036
   481
  fixes F :: "'i \<Rightarrow> 'a filter"
hoelzl@60036
   482
  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"
haftmann@62343
   483
  unfolding trivial_limit_def eventually_INF [of _ _ B]
hoelzl@60036
   484
    bot_bool_def [symmetric] bot_fun_def [symmetric] bot_unique by simp
hoelzl@60036
   485
hoelzl@60036
   486
lemma eventually_Inf_base:
hoelzl@60036
   487
  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"
hoelzl@60036
   488
  shows "eventually P (Inf B) \<longleftrightarrow> (\<exists>b\<in>B. eventually P b)"
hoelzl@60036
   489
proof (subst eventually_Inf, safe)
hoelzl@60036
   490
  fix X assume "finite X" "X \<subseteq> B"
hoelzl@60036
   491
  then have "\<exists>b\<in>B. \<forall>x\<in>X. b \<le> x"
hoelzl@60036
   492
  proof induct
hoelzl@60036
   493
    case empty then show ?case
wenzelm@60758
   494
      using \<open>B \<noteq> {}\<close> by auto
hoelzl@60036
   495
  next
hoelzl@60036
   496
    case (insert x X)
hoelzl@60036
   497
    then obtain b where "b \<in> B" "\<And>x. x \<in> X \<Longrightarrow> b \<le> x"
hoelzl@60036
   498
      by auto
wenzelm@60758
   499
    with \<open>insert x X \<subseteq> B\<close> base[of b x] show ?case
hoelzl@60036
   500
      by (auto intro: order_trans)
hoelzl@60036
   501
  qed
hoelzl@60036
   502
  then obtain b where "b \<in> B" "b \<le> Inf X"
hoelzl@60036
   503
    by (auto simp: le_Inf_iff)
hoelzl@60036
   504
  then show "eventually P (Inf X) \<Longrightarrow> Bex B (eventually P)"
hoelzl@60036
   505
    by (intro bexI[of _ b]) (auto simp: le_filter_def)
hoelzl@60036
   506
qed (auto intro!: exI[of _ "{x}" for x])
hoelzl@60036
   507
hoelzl@60036
   508
lemma eventually_INF_base:
hoelzl@60036
   509
  "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>
hoelzl@60036
   510
    eventually P (INF b:B. F b) \<longleftrightarrow> (\<exists>b\<in>B. eventually P (F b))"
haftmann@62343
   511
  by (subst eventually_Inf_base) auto
hoelzl@60036
   512
hoelzl@62369
   513
lemma eventually_INF1: "i \<in> I \<Longrightarrow> eventually P (F i) \<Longrightarrow> eventually P (INF i:I. F i)"
hoelzl@62369
   514
  using filter_leD[OF INF_lower] .
hoelzl@62369
   515
hoelzl@62367
   516
lemma eventually_INF_mono:
hoelzl@62367
   517
  assumes *: "\<forall>\<^sub>F x in \<Sqinter>i\<in>I. F i. P x"
hoelzl@62367
   518
  assumes T1: "\<And>Q R P. (\<And>x. Q x \<and> R x \<longrightarrow> P x) \<Longrightarrow> (\<And>x. T Q x \<Longrightarrow> T R x \<Longrightarrow> T P x)"
hoelzl@62367
   519
  assumes T2: "\<And>P. (\<And>x. P x) \<Longrightarrow> (\<And>x. T P x)"
hoelzl@62367
   520
  assumes **: "\<And>i P. i \<in> I \<Longrightarrow> \<forall>\<^sub>F x in F i. P x \<Longrightarrow> \<forall>\<^sub>F x in F' i. T P x"
hoelzl@62367
   521
  shows "\<forall>\<^sub>F x in \<Sqinter>i\<in>I. F' i. T P x"
hoelzl@62367
   522
proof -
wenzelm@63540
   523
  from * obtain X where X: "finite X" "X \<subseteq> I" "\<forall>\<^sub>F x in \<Sqinter>i\<in>X. F i. P x"
hoelzl@62378
   524
    unfolding eventually_INF[of _ _ I] by auto
wenzelm@63540
   525
  then have "eventually (T P) (INFIMUM X F')"
hoelzl@62367
   526
    apply (induction X arbitrary: P)
hoelzl@62367
   527
    apply (auto simp: eventually_inf T2)
hoelzl@62367
   528
    subgoal for x S P Q R
hoelzl@62367
   529
      apply (intro exI[of _ "T Q"])
hoelzl@62367
   530
      apply (auto intro!: **) []
hoelzl@62367
   531
      apply (intro exI[of _ "T R"])
hoelzl@62367
   532
      apply (auto intro: T1) []
hoelzl@62367
   533
      done
hoelzl@62367
   534
    done
wenzelm@63540
   535
  with X show "\<forall>\<^sub>F x in \<Sqinter>i\<in>I. F' i. T P x"
hoelzl@62367
   536
    by (subst eventually_INF) auto
hoelzl@62367
   537
qed
hoelzl@62367
   538
hoelzl@60036
   539
wenzelm@60758
   540
subsubsection \<open>Map function for filters\<close>
hoelzl@60036
   541
hoelzl@60036
   542
definition filtermap :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a filter \<Rightarrow> 'b filter"
hoelzl@60036
   543
  where "filtermap f F = Abs_filter (\<lambda>P. eventually (\<lambda>x. P (f x)) F)"
hoelzl@60036
   544
hoelzl@60036
   545
lemma eventually_filtermap:
hoelzl@60036
   546
  "eventually P (filtermap f F) = eventually (\<lambda>x. P (f x)) F"
hoelzl@60036
   547
  unfolding filtermap_def
hoelzl@60036
   548
  apply (rule eventually_Abs_filter)
hoelzl@60036
   549
  apply (rule is_filter.intro)
hoelzl@60036
   550
  apply (auto elim!: eventually_rev_mp)
hoelzl@60036
   551
  done
hoelzl@60036
   552
hoelzl@60036
   553
lemma filtermap_ident: "filtermap (\<lambda>x. x) F = F"
hoelzl@60036
   554
  by (simp add: filter_eq_iff eventually_filtermap)
hoelzl@60036
   555
hoelzl@60036
   556
lemma filtermap_filtermap:
hoelzl@60036
   557
  "filtermap f (filtermap g F) = filtermap (\<lambda>x. f (g x)) F"
hoelzl@60036
   558
  by (simp add: filter_eq_iff eventually_filtermap)
hoelzl@60036
   559
hoelzl@60036
   560
lemma filtermap_mono: "F \<le> F' \<Longrightarrow> filtermap f F \<le> filtermap f F'"
hoelzl@60036
   561
  unfolding le_filter_def eventually_filtermap by simp
hoelzl@60036
   562
hoelzl@60036
   563
lemma filtermap_bot [simp]: "filtermap f bot = bot"
hoelzl@60036
   564
  by (simp add: filter_eq_iff eventually_filtermap)
hoelzl@60036
   565
hoelzl@60036
   566
lemma filtermap_sup: "filtermap f (sup F1 F2) = sup (filtermap f F1) (filtermap f F2)"
hoelzl@60036
   567
  by (auto simp: filter_eq_iff eventually_filtermap eventually_sup)
hoelzl@60036
   568
hoelzl@60036
   569
lemma filtermap_inf: "filtermap f (inf F1 F2) \<le> inf (filtermap f F1) (filtermap f F2)"
hoelzl@60036
   570
  by (auto simp: le_filter_def eventually_filtermap eventually_inf)
hoelzl@60036
   571
hoelzl@60036
   572
lemma filtermap_INF: "filtermap f (INF b:B. F b) \<le> (INF b:B. filtermap f (F b))"
hoelzl@60036
   573
proof -
hoelzl@60036
   574
  { fix X :: "'c set" assume "finite X"
hoelzl@60036
   575
    then have "filtermap f (INFIMUM X F) \<le> (INF b:X. filtermap f (F b))"
hoelzl@60036
   576
    proof induct
hoelzl@60036
   577
      case (insert x X)
hoelzl@60036
   578
      have "filtermap f (INF a:insert x X. F a) \<le> inf (filtermap f (F x)) (filtermap f (INF a:X. F a))"
hoelzl@60036
   579
        by (rule order_trans[OF _ filtermap_inf]) simp
hoelzl@60036
   580
      also have "\<dots> \<le> inf (filtermap f (F x)) (INF a:X. filtermap f (F a))"
hoelzl@60036
   581
        by (intro inf_mono insert order_refl)
hoelzl@60036
   582
      finally show ?case
hoelzl@60036
   583
        by simp
hoelzl@60036
   584
    qed simp }
hoelzl@60036
   585
  then show ?thesis
hoelzl@60036
   586
    unfolding le_filter_def eventually_filtermap
hoelzl@60036
   587
    by (subst (1 2) eventually_INF) auto
hoelzl@60036
   588
qed
hoelzl@62101
   589
eberlm@66162
   590
eberlm@66162
   591
subsubsection \<open>Contravariant map function for filters\<close>
eberlm@66162
   592
eberlm@66162
   593
definition filtercomap :: "('a \<Rightarrow> 'b) \<Rightarrow> 'b filter \<Rightarrow> 'a filter" where
eberlm@66162
   594
  "filtercomap f F = Abs_filter (\<lambda>P. \<exists>Q. eventually Q F \<and> (\<forall>x. Q (f x) \<longrightarrow> P x))"
eberlm@66162
   595
eberlm@66162
   596
lemma eventually_filtercomap:
eberlm@66162
   597
  "eventually P (filtercomap f F) \<longleftrightarrow> (\<exists>Q. eventually Q F \<and> (\<forall>x. Q (f x) \<longrightarrow> P x))"
eberlm@66162
   598
  unfolding filtercomap_def
eberlm@66162
   599
proof (intro eventually_Abs_filter, unfold_locales, goal_cases)
eberlm@66162
   600
  case 1
eberlm@66162
   601
  show ?case by (auto intro!: exI[of _ "\<lambda>_. True"])
eberlm@66162
   602
next
eberlm@66162
   603
  case (2 P Q)
eberlm@66162
   604
  from 2(1) guess P' by (elim exE conjE) note P' = this
eberlm@66162
   605
  from 2(2) guess Q' by (elim exE conjE) note Q' = this
eberlm@66162
   606
  show ?case
eberlm@66162
   607
    by (rule exI[of _ "\<lambda>x. P' x \<and> Q' x"])
eberlm@66162
   608
       (insert P' Q', auto intro!: eventually_conj)
eberlm@66162
   609
next
eberlm@66162
   610
  case (3 P Q)
eberlm@66162
   611
  thus ?case by blast
eberlm@66162
   612
qed
eberlm@66162
   613
eberlm@66162
   614
lemma filtercomap_ident: "filtercomap (\<lambda>x. x) F = F"
eberlm@66162
   615
  by (auto simp: filter_eq_iff eventually_filtercomap elim!: eventually_mono)
eberlm@66162
   616
eberlm@66162
   617
lemma filtercomap_filtercomap: "filtercomap f (filtercomap g F) = filtercomap (\<lambda>x. g (f x)) F"
eberlm@66162
   618
  unfolding filter_eq_iff by (auto simp: eventually_filtercomap)
eberlm@66162
   619
  
eberlm@66162
   620
lemma filtercomap_mono: "F \<le> F' \<Longrightarrow> filtercomap f F \<le> filtercomap f F'"
eberlm@66162
   621
  by (auto simp: eventually_filtercomap le_filter_def)
eberlm@66162
   622
eberlm@66162
   623
lemma filtercomap_bot [simp]: "filtercomap f bot = bot"
eberlm@66162
   624
  by (auto simp: filter_eq_iff eventually_filtercomap)
eberlm@66162
   625
eberlm@66162
   626
lemma filtercomap_top [simp]: "filtercomap f top = top"
eberlm@66162
   627
  by (auto simp: filter_eq_iff eventually_filtercomap)
eberlm@66162
   628
eberlm@66162
   629
lemma filtercomap_inf: "filtercomap f (inf F1 F2) = inf (filtercomap f F1) (filtercomap f F2)"
eberlm@66162
   630
  unfolding filter_eq_iff
eberlm@66162
   631
proof safe
eberlm@66162
   632
  fix P
eberlm@66162
   633
  assume "eventually P (filtercomap f (F1 \<sqinter> F2))"
eberlm@66162
   634
  then obtain Q R S where *:
eberlm@66162
   635
    "eventually Q F1" "eventually R F2" "\<And>x. Q x \<Longrightarrow> R x \<Longrightarrow> S x" "\<And>x. S (f x) \<Longrightarrow> P x"
eberlm@66162
   636
    unfolding eventually_filtercomap eventually_inf by blast
eberlm@66162
   637
  from * have "eventually (\<lambda>x. Q (f x)) (filtercomap f F1)" 
eberlm@66162
   638
              "eventually (\<lambda>x. R (f x)) (filtercomap f F2)"
eberlm@66162
   639
    by (auto simp: eventually_filtercomap)
eberlm@66162
   640
  with * show "eventually P (filtercomap f F1 \<sqinter> filtercomap f F2)"
eberlm@66162
   641
    unfolding eventually_inf by blast
eberlm@66162
   642
next
eberlm@66162
   643
  fix P
eberlm@66162
   644
  assume "eventually P (inf (filtercomap f F1) (filtercomap f F2))"
eberlm@66162
   645
  then obtain Q Q' R R' where *:
eberlm@66162
   646
    "eventually Q F1" "eventually R F2" "\<And>x. Q (f x) \<Longrightarrow> Q' x" "\<And>x. R (f x) \<Longrightarrow> R' x" 
eberlm@66162
   647
    "\<And>x. Q' x \<Longrightarrow> R' x \<Longrightarrow> P x"
eberlm@66162
   648
    unfolding eventually_filtercomap eventually_inf by blast
eberlm@66162
   649
  from * have "eventually (\<lambda>x. Q x \<and> R x) (F1 \<sqinter> F2)" by (auto simp: eventually_inf)
eberlm@66162
   650
  with * show "eventually P (filtercomap f (F1 \<sqinter> F2))"
eberlm@66162
   651
    by (auto simp: eventually_filtercomap)
eberlm@66162
   652
qed
eberlm@66162
   653
eberlm@66162
   654
lemma filtercomap_sup: "filtercomap f (sup F1 F2) \<ge> sup (filtercomap f F1) (filtercomap f F2)"
eberlm@66162
   655
  unfolding le_filter_def
eberlm@66162
   656
proof safe
eberlm@66162
   657
  fix P
eberlm@66162
   658
  assume "eventually P (filtercomap f (sup F1 F2))"
eberlm@66162
   659
  thus "eventually P (sup (filtercomap f F1) (filtercomap f F2))"
eberlm@66162
   660
    by (auto simp: filter_eq_iff eventually_filtercomap eventually_sup)
eberlm@66162
   661
qed
eberlm@66162
   662
eberlm@66162
   663
lemma filtercomap_INF: "filtercomap f (INF b:B. F b) = (INF b:B. filtercomap f (F b))"
eberlm@66162
   664
proof -
eberlm@66162
   665
  have *: "filtercomap f (INF b:B. F b) = (INF b:B. filtercomap f (F b))" if "finite B" for B
eberlm@66162
   666
    using that by induction (simp_all add: filtercomap_inf)
eberlm@66162
   667
  show ?thesis unfolding filter_eq_iff
eberlm@66162
   668
  proof
eberlm@66162
   669
    fix P
eberlm@66162
   670
    have "eventually P (INF b:B. filtercomap f (F b)) \<longleftrightarrow> 
eberlm@66162
   671
            (\<exists>X. (X \<subseteq> B \<and> finite X) \<and> eventually P (\<Sqinter>b\<in>X. filtercomap f (F b)))"
eberlm@66162
   672
      by (subst eventually_INF) blast
eberlm@66162
   673
    also have "\<dots> \<longleftrightarrow> (\<exists>X. (X \<subseteq> B \<and> finite X) \<and> eventually P (filtercomap f (INF b:X. F b)))"
eberlm@66162
   674
      by (rule ex_cong) (simp add: *)
eberlm@66162
   675
    also have "\<dots> \<longleftrightarrow> eventually P (filtercomap f (INFIMUM B F))"
eberlm@66162
   676
      unfolding eventually_filtercomap by (subst eventually_INF) blast
eberlm@66162
   677
    finally show "eventually P (filtercomap f (INFIMUM B F)) = 
eberlm@66162
   678
                    eventually P (\<Sqinter>b\<in>B. filtercomap f (F b))" ..
eberlm@66162
   679
  qed
eberlm@66162
   680
qed
eberlm@66162
   681
eberlm@66162
   682
lemma filtercomap_SUP_finite: 
eberlm@66162
   683
  "finite B \<Longrightarrow> filtercomap f (SUP b:B. F b) \<ge> (SUP b:B. filtercomap f (F b))"
eberlm@66162
   684
  by (induction B rule: finite_induct)
eberlm@66162
   685
     (auto intro: order_trans[OF _ order_trans[OF _ filtercomap_sup]] filtercomap_mono)
eberlm@66162
   686
     
eberlm@66162
   687
lemma eventually_filtercomapI [intro]:
eberlm@66162
   688
  assumes "eventually P F"
eberlm@66162
   689
  shows   "eventually (\<lambda>x. P (f x)) (filtercomap f F)"
eberlm@66162
   690
  using assms by (auto simp: eventually_filtercomap)
eberlm@66162
   691
eberlm@66162
   692
lemma filtermap_filtercomap: "filtermap f (filtercomap f F) \<le> F"
eberlm@66162
   693
  by (auto simp: le_filter_def eventually_filtermap eventually_filtercomap)
eberlm@66162
   694
    
eberlm@66162
   695
lemma filtercomap_filtermap: "filtercomap f (filtermap f F) \<ge> F"
eberlm@66162
   696
  unfolding le_filter_def eventually_filtermap eventually_filtercomap
eberlm@66162
   697
  by (auto elim!: eventually_mono)
eberlm@66162
   698
eberlm@66162
   699
wenzelm@60758
   700
subsubsection \<open>Standard filters\<close>
hoelzl@60036
   701
hoelzl@60036
   702
definition principal :: "'a set \<Rightarrow> 'a filter" where
hoelzl@60036
   703
  "principal S = Abs_filter (\<lambda>P. \<forall>x\<in>S. P x)"
hoelzl@60036
   704
hoelzl@60036
   705
lemma eventually_principal: "eventually P (principal S) \<longleftrightarrow> (\<forall>x\<in>S. P x)"
hoelzl@60036
   706
  unfolding principal_def
hoelzl@60036
   707
  by (rule eventually_Abs_filter, rule is_filter.intro) auto
hoelzl@60036
   708
hoelzl@60036
   709
lemma eventually_inf_principal: "eventually P (inf F (principal s)) \<longleftrightarrow> eventually (\<lambda>x. x \<in> s \<longrightarrow> P x) F"
lp15@61810
   710
  unfolding eventually_inf eventually_principal by (auto elim: eventually_mono)
hoelzl@60036
   711
hoelzl@60036
   712
lemma principal_UNIV[simp]: "principal UNIV = top"
hoelzl@60036
   713
  by (auto simp: filter_eq_iff eventually_principal)
hoelzl@60036
   714
hoelzl@60036
   715
lemma principal_empty[simp]: "principal {} = bot"
hoelzl@60036
   716
  by (auto simp: filter_eq_iff eventually_principal)
hoelzl@60036
   717
hoelzl@60036
   718
lemma principal_eq_bot_iff: "principal X = bot \<longleftrightarrow> X = {}"
hoelzl@60036
   719
  by (auto simp add: filter_eq_iff eventually_principal)
hoelzl@60036
   720
hoelzl@60036
   721
lemma principal_le_iff[iff]: "principal A \<le> principal B \<longleftrightarrow> A \<subseteq> B"
hoelzl@60036
   722
  by (auto simp: le_filter_def eventually_principal)
hoelzl@60036
   723
hoelzl@60036
   724
lemma le_principal: "F \<le> principal A \<longleftrightarrow> eventually (\<lambda>x. x \<in> A) F"
hoelzl@60036
   725
  unfolding le_filter_def eventually_principal
hoelzl@60036
   726
  apply safe
hoelzl@60036
   727
  apply (erule_tac x="\<lambda>x. x \<in> A" in allE)
lp15@61810
   728
  apply (auto elim: eventually_mono)
hoelzl@60036
   729
  done
hoelzl@60036
   730
hoelzl@60036
   731
lemma principal_inject[iff]: "principal A = principal B \<longleftrightarrow> A = B"
hoelzl@60036
   732
  unfolding eq_iff by simp
hoelzl@60036
   733
hoelzl@60036
   734
lemma sup_principal[simp]: "sup (principal A) (principal B) = principal (A \<union> B)"
hoelzl@60036
   735
  unfolding filter_eq_iff eventually_sup eventually_principal by auto
hoelzl@60036
   736
hoelzl@60036
   737
lemma inf_principal[simp]: "inf (principal A) (principal B) = principal (A \<inter> B)"
hoelzl@60036
   738
  unfolding filter_eq_iff eventually_inf eventually_principal
hoelzl@60036
   739
  by (auto intro: exI[of _ "\<lambda>x. x \<in> A"] exI[of _ "\<lambda>x. x \<in> B"])
hoelzl@60036
   740
hoelzl@60036
   741
lemma SUP_principal[simp]: "(SUP i : I. principal (A i)) = principal (\<Union>i\<in>I. A i)"
haftmann@62343
   742
  unfolding filter_eq_iff eventually_Sup by (auto simp: eventually_principal)
hoelzl@60036
   743
hoelzl@60036
   744
lemma INF_principal_finite: "finite X \<Longrightarrow> (INF x:X. principal (f x)) = principal (\<Inter>x\<in>X. f x)"
hoelzl@60036
   745
  by (induct X rule: finite_induct) auto
hoelzl@60036
   746
hoelzl@60036
   747
lemma filtermap_principal[simp]: "filtermap f (principal A) = principal (f ` A)"
hoelzl@60036
   748
  unfolding filter_eq_iff eventually_filtermap eventually_principal by simp
eberlm@66162
   749
    
eberlm@66162
   750
lemma filtercomap_principal[simp]: "filtercomap f (principal A) = principal (f -` A)"
eberlm@66162
   751
  unfolding filter_eq_iff eventually_filtercomap eventually_principal by fast
hoelzl@60036
   752
wenzelm@60758
   753
subsubsection \<open>Order filters\<close>
hoelzl@60036
   754
hoelzl@60036
   755
definition at_top :: "('a::order) filter"
hoelzl@60036
   756
  where "at_top = (INF k. principal {k ..})"
hoelzl@60036
   757
hoelzl@60036
   758
lemma at_top_sub: "at_top = (INF k:{c::'a::linorder..}. principal {k ..})"
hoelzl@60036
   759
  by (auto intro!: INF_eq max.cobounded1 max.cobounded2 simp: at_top_def)
hoelzl@60036
   760
hoelzl@60036
   761
lemma eventually_at_top_linorder: "eventually P at_top \<longleftrightarrow> (\<exists>N::'a::linorder. \<forall>n\<ge>N. P n)"
hoelzl@60036
   762
  unfolding at_top_def
hoelzl@60036
   763
  by (subst eventually_INF_base) (auto simp: eventually_principal intro: max.cobounded1 max.cobounded2)
hoelzl@60036
   764
eberlm@66162
   765
lemma eventually_filtercomap_at_top_linorder: 
eberlm@66162
   766
  "eventually P (filtercomap f at_top) \<longleftrightarrow> (\<exists>N::'a::linorder. \<forall>x. f x \<ge> N \<longrightarrow> P x)"
eberlm@66162
   767
  by (auto simp: eventually_filtercomap eventually_at_top_linorder)
eberlm@66162
   768
immler@63556
   769
lemma eventually_at_top_linorderI:
immler@63556
   770
  fixes c::"'a::linorder"
immler@63556
   771
  assumes "\<And>x. c \<le> x \<Longrightarrow> P x"
immler@63556
   772
  shows "eventually P at_top"
immler@63556
   773
  using assms by (auto simp: eventually_at_top_linorder)
immler@63556
   774
lp15@65578
   775
lemma eventually_ge_at_top [simp]:
hoelzl@60036
   776
  "eventually (\<lambda>x. (c::_::linorder) \<le> x) at_top"
hoelzl@60036
   777
  unfolding eventually_at_top_linorder by auto
hoelzl@60036
   778
hoelzl@60036
   779
lemma eventually_at_top_dense: "eventually P at_top \<longleftrightarrow> (\<exists>N::'a::{no_top, linorder}. \<forall>n>N. P n)"
hoelzl@60036
   780
proof -
hoelzl@60036
   781
  have "eventually P (INF k. principal {k <..}) \<longleftrightarrow> (\<exists>N::'a. \<forall>n>N. P n)"
hoelzl@60036
   782
    by (subst eventually_INF_base) (auto simp: eventually_principal intro: max.cobounded1 max.cobounded2)
hoelzl@60036
   783
  also have "(INF k. principal {k::'a <..}) = at_top"
lp15@61806
   784
    unfolding at_top_def
hoelzl@60036
   785
    by (intro INF_eq) (auto intro: less_imp_le simp: Ici_subset_Ioi_iff gt_ex)
hoelzl@60036
   786
  finally show ?thesis .
hoelzl@60036
   787
qed
eberlm@66162
   788
  
eberlm@66162
   789
lemma eventually_filtercomap_at_top_dense: 
eberlm@66162
   790
  "eventually P (filtercomap f at_top) \<longleftrightarrow> (\<exists>N::'a::{no_top, linorder}. \<forall>x. f x > N \<longrightarrow> P x)"
eberlm@66162
   791
  by (auto simp: eventually_filtercomap eventually_at_top_dense)
hoelzl@60036
   792
lp15@65578
   793
lemma eventually_at_top_not_equal [simp]: "eventually (\<lambda>x::'a::{no_top, linorder}. x \<noteq> c) at_top"
hoelzl@60721
   794
  unfolding eventually_at_top_dense by auto
hoelzl@60721
   795
lp15@65578
   796
lemma eventually_gt_at_top [simp]: "eventually (\<lambda>x. (c::_::{no_top, linorder}) < x) at_top"
hoelzl@60036
   797
  unfolding eventually_at_top_dense by auto
hoelzl@60036
   798
eberlm@61531
   799
lemma eventually_all_ge_at_top:
eberlm@61531
   800
  assumes "eventually P (at_top :: ('a :: linorder) filter)"
eberlm@61531
   801
  shows   "eventually (\<lambda>x. \<forall>y\<ge>x. P y) at_top"
eberlm@61531
   802
proof -
eberlm@61531
   803
  from assms obtain x where "\<And>y. y \<ge> x \<Longrightarrow> P y" by (auto simp: eventually_at_top_linorder)
eberlm@61531
   804
  hence "\<forall>z\<ge>y. P z" if "y \<ge> x" for y using that by simp
eberlm@61531
   805
  thus ?thesis by (auto simp: eventually_at_top_linorder)
eberlm@61531
   806
qed
eberlm@61531
   807
hoelzl@60036
   808
definition at_bot :: "('a::order) filter"
hoelzl@60036
   809
  where "at_bot = (INF k. principal {.. k})"
hoelzl@60036
   810
hoelzl@60036
   811
lemma at_bot_sub: "at_bot = (INF k:{.. c::'a::linorder}. principal {.. k})"
hoelzl@60036
   812
  by (auto intro!: INF_eq min.cobounded1 min.cobounded2 simp: at_bot_def)
hoelzl@60036
   813
hoelzl@60036
   814
lemma eventually_at_bot_linorder:
hoelzl@60036
   815
  fixes P :: "'a::linorder \<Rightarrow> bool" shows "eventually P at_bot \<longleftrightarrow> (\<exists>N. \<forall>n\<le>N. P n)"
hoelzl@60036
   816
  unfolding at_bot_def
hoelzl@60036
   817
  by (subst eventually_INF_base) (auto simp: eventually_principal intro: min.cobounded1 min.cobounded2)
hoelzl@60036
   818
eberlm@66162
   819
lemma eventually_filtercomap_at_bot_linorder: 
eberlm@66162
   820
  "eventually P (filtercomap f at_bot) \<longleftrightarrow> (\<exists>N::'a::linorder. \<forall>x. f x \<le> N \<longrightarrow> P x)"
eberlm@66162
   821
  by (auto simp: eventually_filtercomap eventually_at_bot_linorder)
eberlm@66162
   822
lp15@65578
   823
lemma eventually_le_at_bot [simp]:
hoelzl@60036
   824
  "eventually (\<lambda>x. x \<le> (c::_::linorder)) at_bot"
hoelzl@60036
   825
  unfolding eventually_at_bot_linorder by auto
hoelzl@60036
   826
hoelzl@60036
   827
lemma eventually_at_bot_dense: "eventually P at_bot \<longleftrightarrow> (\<exists>N::'a::{no_bot, linorder}. \<forall>n<N. P n)"
hoelzl@60036
   828
proof -
hoelzl@60036
   829
  have "eventually P (INF k. principal {..< k}) \<longleftrightarrow> (\<exists>N::'a. \<forall>n<N. P n)"
hoelzl@60036
   830
    by (subst eventually_INF_base) (auto simp: eventually_principal intro: min.cobounded1 min.cobounded2)
hoelzl@60036
   831
  also have "(INF k. principal {..< k::'a}) = at_bot"
lp15@61806
   832
    unfolding at_bot_def
hoelzl@60036
   833
    by (intro INF_eq) (auto intro: less_imp_le simp: Iic_subset_Iio_iff lt_ex)
hoelzl@60036
   834
  finally show ?thesis .
hoelzl@60036
   835
qed
hoelzl@60036
   836
eberlm@66162
   837
lemma eventually_filtercomap_at_bot_dense: 
eberlm@66162
   838
  "eventually P (filtercomap f at_bot) \<longleftrightarrow> (\<exists>N::'a::{no_bot, linorder}. \<forall>x. f x < N \<longrightarrow> P x)"
eberlm@66162
   839
  by (auto simp: eventually_filtercomap eventually_at_bot_dense)
eberlm@66162
   840
lp15@65578
   841
lemma eventually_at_bot_not_equal [simp]: "eventually (\<lambda>x::'a::{no_bot, linorder}. x \<noteq> c) at_bot"
hoelzl@60721
   842
  unfolding eventually_at_bot_dense by auto
hoelzl@60721
   843
lp15@65578
   844
lemma eventually_gt_at_bot [simp]:
hoelzl@60036
   845
  "eventually (\<lambda>x. x < (c::_::unbounded_dense_linorder)) at_bot"
hoelzl@60036
   846
  unfolding eventually_at_bot_dense by auto
hoelzl@60036
   847
lp15@63967
   848
lemma trivial_limit_at_bot_linorder [simp]: "\<not> trivial_limit (at_bot ::('a::linorder) filter)"
hoelzl@60036
   849
  unfolding trivial_limit_def
hoelzl@60036
   850
  by (metis eventually_at_bot_linorder order_refl)
hoelzl@60036
   851
lp15@63967
   852
lemma trivial_limit_at_top_linorder [simp]: "\<not> trivial_limit (at_top ::('a::linorder) filter)"
hoelzl@60036
   853
  unfolding trivial_limit_def
hoelzl@60036
   854
  by (metis eventually_at_top_linorder order_refl)
hoelzl@60036
   855
wenzelm@60758
   856
subsection \<open>Sequentially\<close>
hoelzl@60036
   857
hoelzl@60036
   858
abbreviation sequentially :: "nat filter"
hoelzl@60036
   859
  where "sequentially \<equiv> at_top"
hoelzl@60036
   860
hoelzl@60036
   861
lemma eventually_sequentially:
hoelzl@60036
   862
  "eventually P sequentially \<longleftrightarrow> (\<exists>N. \<forall>n\<ge>N. P n)"
hoelzl@60036
   863
  by (rule eventually_at_top_linorder)
hoelzl@60036
   864
hoelzl@60036
   865
lemma sequentially_bot [simp, intro]: "sequentially \<noteq> bot"
hoelzl@60036
   866
  unfolding filter_eq_iff eventually_sequentially by auto
hoelzl@60036
   867
hoelzl@60036
   868
lemmas trivial_limit_sequentially = sequentially_bot
hoelzl@60036
   869
hoelzl@60036
   870
lemma eventually_False_sequentially [simp]:
hoelzl@60036
   871
  "\<not> eventually (\<lambda>n. False) sequentially"
hoelzl@60036
   872
  by (simp add: eventually_False)
hoelzl@60036
   873
hoelzl@60036
   874
lemma le_sequentially:
hoelzl@60036
   875
  "F \<le> sequentially \<longleftrightarrow> (\<forall>N. eventually (\<lambda>n. N \<le> n) F)"
hoelzl@60036
   876
  by (simp add: at_top_def le_INF_iff le_principal)
hoelzl@60036
   877
lp15@60974
   878
lemma eventually_sequentiallyI [intro?]:
hoelzl@60036
   879
  assumes "\<And>x. c \<le> x \<Longrightarrow> P x"
hoelzl@60036
   880
  shows "eventually P sequentially"
hoelzl@60036
   881
using assms by (auto simp: eventually_sequentially)
hoelzl@60036
   882
lp15@63967
   883
lemma eventually_sequentially_Suc [simp]: "eventually (\<lambda>i. P (Suc i)) sequentially \<longleftrightarrow> eventually P sequentially"
hoelzl@60040
   884
  unfolding eventually_sequentially by (metis Suc_le_D Suc_le_mono le_Suc_eq)
hoelzl@60040
   885
lp15@63967
   886
lemma eventually_sequentially_seg [simp]: "eventually (\<lambda>n. P (n + k)) sequentially \<longleftrightarrow> eventually P sequentially"
hoelzl@60040
   887
  using eventually_sequentially_Suc[of "\<lambda>n. P (n + k)" for k] by (induction k) auto
hoelzl@60036
   888
wenzelm@61955
   889
wenzelm@61955
   890
subsection \<open>The cofinite filter\<close>
hoelzl@60039
   891
hoelzl@60039
   892
definition "cofinite = Abs_filter (\<lambda>P. finite {x. \<not> P x})"
hoelzl@60039
   893
wenzelm@61955
   894
abbreviation Inf_many :: "('a \<Rightarrow> bool) \<Rightarrow> bool"  (binder "\<exists>\<^sub>\<infinity>" 10)
wenzelm@61955
   895
  where "Inf_many P \<equiv> frequently P cofinite"
hoelzl@60040
   896
wenzelm@61955
   897
abbreviation Alm_all :: "('a \<Rightarrow> bool) \<Rightarrow> bool"  (binder "\<forall>\<^sub>\<infinity>" 10)
wenzelm@61955
   898
  where "Alm_all P \<equiv> eventually P cofinite"
hoelzl@60040
   899
wenzelm@61955
   900
notation (ASCII)
wenzelm@61955
   901
  Inf_many  (binder "INFM " 10) and
wenzelm@61955
   902
  Alm_all  (binder "MOST " 10)
hoelzl@60040
   903
hoelzl@60039
   904
lemma eventually_cofinite: "eventually P cofinite \<longleftrightarrow> finite {x. \<not> P x}"
hoelzl@60039
   905
  unfolding cofinite_def
hoelzl@60039
   906
proof (rule eventually_Abs_filter, rule is_filter.intro)
hoelzl@60039
   907
  fix P Q :: "'a \<Rightarrow> bool" assume "finite {x. \<not> P x}" "finite {x. \<not> Q x}"
hoelzl@60039
   908
  from finite_UnI[OF this] show "finite {x. \<not> (P x \<and> Q x)}"
hoelzl@60039
   909
    by (rule rev_finite_subset) auto
hoelzl@60039
   910
next
hoelzl@60039
   911
  fix P Q :: "'a \<Rightarrow> bool" assume P: "finite {x. \<not> P x}" and *: "\<forall>x. P x \<longrightarrow> Q x"
hoelzl@60039
   912
  from * show "finite {x. \<not> Q x}"
hoelzl@60039
   913
    by (intro finite_subset[OF _ P]) auto
hoelzl@60039
   914
qed simp
hoelzl@60039
   915
hoelzl@60040
   916
lemma frequently_cofinite: "frequently P cofinite \<longleftrightarrow> \<not> finite {x. P x}"
hoelzl@60040
   917
  by (simp add: frequently_def eventually_cofinite)
hoelzl@60040
   918
hoelzl@60039
   919
lemma cofinite_bot[simp]: "cofinite = (bot::'a filter) \<longleftrightarrow> finite (UNIV :: 'a set)"
hoelzl@60039
   920
  unfolding trivial_limit_def eventually_cofinite by simp
hoelzl@60039
   921
hoelzl@60039
   922
lemma cofinite_eq_sequentially: "cofinite = sequentially"
hoelzl@60039
   923
  unfolding filter_eq_iff eventually_sequentially eventually_cofinite
hoelzl@60039
   924
proof safe
hoelzl@60039
   925
  fix P :: "nat \<Rightarrow> bool" assume [simp]: "finite {x. \<not> P x}"
hoelzl@60039
   926
  show "\<exists>N. \<forall>n\<ge>N. P n"
hoelzl@60039
   927
  proof cases
hoelzl@60039
   928
    assume "{x. \<not> P x} \<noteq> {}" then show ?thesis
hoelzl@60039
   929
      by (intro exI[of _ "Suc (Max {x. \<not> P x})"]) (auto simp: Suc_le_eq)
hoelzl@60039
   930
  qed auto
hoelzl@60039
   931
next
hoelzl@60039
   932
  fix P :: "nat \<Rightarrow> bool" and N :: nat assume "\<forall>n\<ge>N. P n"
hoelzl@60039
   933
  then have "{x. \<not> P x} \<subseteq> {..< N}"
hoelzl@60039
   934
    by (auto simp: not_le)
hoelzl@60039
   935
  then show "finite {x. \<not> P x}"
hoelzl@60039
   936
    by (blast intro: finite_subset)
hoelzl@60039
   937
qed
hoelzl@60036
   938
hoelzl@62101
   939
subsubsection \<open>Product of filters\<close>
hoelzl@62101
   940
hoelzl@62101
   941
lemma filtermap_sequentually_ne_bot: "filtermap f sequentially \<noteq> bot"
hoelzl@62101
   942
  by (auto simp add: filter_eq_iff eventually_filtermap eventually_sequentially)
hoelzl@62101
   943
hoelzl@62101
   944
definition prod_filter :: "'a filter \<Rightarrow> 'b filter \<Rightarrow> ('a \<times> 'b) filter" (infixr "\<times>\<^sub>F" 80) where
hoelzl@62101
   945
  "prod_filter F G =
hoelzl@62101
   946
    (INF (P, Q):{(P, Q). eventually P F \<and> eventually Q G}. principal {(x, y). P x \<and> Q y})"
hoelzl@62101
   947
hoelzl@62101
   948
lemma eventually_prod_filter: "eventually P (F \<times>\<^sub>F G) \<longleftrightarrow>
hoelzl@62101
   949
  (\<exists>Pf Pg. eventually Pf F \<and> eventually Pg G \<and> (\<forall>x y. Pf x \<longrightarrow> Pg y \<longrightarrow> P (x, y)))"
hoelzl@62101
   950
  unfolding prod_filter_def
hoelzl@62101
   951
proof (subst eventually_INF_base, goal_cases)
hoelzl@62101
   952
  case 2
hoelzl@62101
   953
  moreover have "eventually Pf F \<Longrightarrow> eventually Qf F \<Longrightarrow> eventually Pg G \<Longrightarrow> eventually Qg G \<Longrightarrow>
hoelzl@62101
   954
    \<exists>P Q. eventually P F \<and> eventually Q G \<and>
hoelzl@62101
   955
      Collect P \<times> Collect Q \<subseteq> Collect Pf \<times> Collect Pg \<inter> Collect Qf \<times> Collect Qg" for Pf Pg Qf Qg
hoelzl@62101
   956
    by (intro conjI exI[of _ "inf Pf Qf"] exI[of _ "inf Pg Qg"])
hoelzl@62101
   957
       (auto simp: inf_fun_def eventually_conj)
hoelzl@62101
   958
  ultimately show ?case
hoelzl@62101
   959
    by auto
hoelzl@62101
   960
qed (auto simp: eventually_principal intro: eventually_True)
hoelzl@62101
   961
hoelzl@62367
   962
lemma eventually_prod1:
hoelzl@62367
   963
  assumes "B \<noteq> bot"
hoelzl@62367
   964
  shows "(\<forall>\<^sub>F (x, y) in A \<times>\<^sub>F B. P x) \<longleftrightarrow> (\<forall>\<^sub>F x in A. P x)"
hoelzl@62367
   965
  unfolding eventually_prod_filter
hoelzl@62367
   966
proof safe
wenzelm@63540
   967
  fix R Q
wenzelm@63540
   968
  assume *: "\<forall>\<^sub>F x in A. R x" "\<forall>\<^sub>F x in B. Q x" "\<forall>x y. R x \<longrightarrow> Q y \<longrightarrow> P x"
wenzelm@63540
   969
  with \<open>B \<noteq> bot\<close> obtain y where "Q y" by (auto dest: eventually_happens)
wenzelm@63540
   970
  with * show "eventually P A"
hoelzl@62367
   971
    by (force elim: eventually_mono)
hoelzl@62367
   972
next
hoelzl@62367
   973
  assume "eventually P A"
hoelzl@62367
   974
  then show "\<exists>Pf Pg. eventually Pf A \<and> eventually Pg B \<and> (\<forall>x y. Pf x \<longrightarrow> Pg y \<longrightarrow> P x)"
hoelzl@62367
   975
    by (intro exI[of _ P] exI[of _ "\<lambda>x. True"]) auto
hoelzl@62367
   976
qed
hoelzl@62367
   977
hoelzl@62367
   978
lemma eventually_prod2:
hoelzl@62367
   979
  assumes "A \<noteq> bot"
hoelzl@62367
   980
  shows "(\<forall>\<^sub>F (x, y) in A \<times>\<^sub>F B. P y) \<longleftrightarrow> (\<forall>\<^sub>F y in B. P y)"
hoelzl@62367
   981
  unfolding eventually_prod_filter
hoelzl@62367
   982
proof safe
wenzelm@63540
   983
  fix R Q
wenzelm@63540
   984
  assume *: "\<forall>\<^sub>F x in A. R x" "\<forall>\<^sub>F x in B. Q x" "\<forall>x y. R x \<longrightarrow> Q y \<longrightarrow> P y"
wenzelm@63540
   985
  with \<open>A \<noteq> bot\<close> obtain x where "R x" by (auto dest: eventually_happens)
wenzelm@63540
   986
  with * show "eventually P B"
hoelzl@62367
   987
    by (force elim: eventually_mono)
hoelzl@62367
   988
next
hoelzl@62367
   989
  assume "eventually P B"
hoelzl@62367
   990
  then show "\<exists>Pf Pg. eventually Pf A \<and> eventually Pg B \<and> (\<forall>x y. Pf x \<longrightarrow> Pg y \<longrightarrow> P y)"
hoelzl@62367
   991
    by (intro exI[of _ P] exI[of _ "\<lambda>x. True"]) auto
hoelzl@62367
   992
qed
hoelzl@62367
   993
hoelzl@62367
   994
lemma INF_filter_bot_base:
hoelzl@62367
   995
  fixes F :: "'a \<Rightarrow> 'b filter"
hoelzl@62367
   996
  assumes *: "\<And>i j. i \<in> I \<Longrightarrow> j \<in> I \<Longrightarrow> \<exists>k\<in>I. F k \<le> F i \<sqinter> F j"
hoelzl@62367
   997
  shows "(INF i:I. F i) = bot \<longleftrightarrow> (\<exists>i\<in>I. F i = bot)"
wenzelm@63540
   998
proof (cases "\<exists>i\<in>I. F i = bot")
wenzelm@63540
   999
  case True
wenzelm@63540
  1000
  then have "(INF i:I. F i) \<le> bot"
hoelzl@62367
  1001
    by (auto intro: INF_lower2)
wenzelm@63540
  1002
  with True show ?thesis
hoelzl@62367
  1003
    by (auto simp: bot_unique)
hoelzl@62367
  1004
next
wenzelm@63540
  1005
  case False
hoelzl@62367
  1006
  moreover have "(INF i:I. F i) \<noteq> bot"
wenzelm@63540
  1007
  proof (cases "I = {}")
wenzelm@63540
  1008
    case True
wenzelm@63540
  1009
    then show ?thesis
wenzelm@63540
  1010
      by (auto simp add: filter_eq_iff)
wenzelm@63540
  1011
  next
wenzelm@63540
  1012
    case False': False
hoelzl@62367
  1013
    show ?thesis
hoelzl@62367
  1014
    proof (rule INF_filter_not_bot)
wenzelm@63540
  1015
      fix J
wenzelm@63540
  1016
      assume "finite J" "J \<subseteq> I"
hoelzl@62367
  1017
      then have "\<exists>k\<in>I. F k \<le> (\<Sqinter>i\<in>J. F i)"
wenzelm@63540
  1018
      proof (induct J)
wenzelm@63540
  1019
        case empty
wenzelm@63540
  1020
        then show ?case
hoelzl@62367
  1021
          using \<open>I \<noteq> {}\<close> by auto
hoelzl@62367
  1022
      next
hoelzl@62367
  1023
        case (insert i J)
wenzelm@63540
  1024
        then obtain k where "k \<in> I" "F k \<le> (\<Sqinter>i\<in>J. F i)" by auto
wenzelm@63540
  1025
        with insert *[of i k] show ?case
hoelzl@62367
  1026
          by auto
hoelzl@62367
  1027
      qed
wenzelm@63540
  1028
      with False show "(\<Sqinter>i\<in>J. F i) \<noteq> \<bottom>"
hoelzl@62367
  1029
        by (auto simp: bot_unique)
hoelzl@62367
  1030
    qed
wenzelm@63540
  1031
  qed
hoelzl@62367
  1032
  ultimately show ?thesis
hoelzl@62367
  1033
    by auto
hoelzl@62367
  1034
qed
hoelzl@62367
  1035
hoelzl@62367
  1036
lemma Collect_empty_eq_bot: "Collect P = {} \<longleftrightarrow> P = \<bottom>"
hoelzl@62367
  1037
  by auto
hoelzl@62367
  1038
hoelzl@62367
  1039
lemma prod_filter_eq_bot: "A \<times>\<^sub>F B = bot \<longleftrightarrow> A = bot \<or> B = bot"
hoelzl@62367
  1040
  unfolding prod_filter_def
hoelzl@62367
  1041
proof (subst INF_filter_bot_base; clarsimp simp: principal_eq_bot_iff Collect_empty_eq_bot bot_fun_def simp del: Collect_empty_eq)
hoelzl@62367
  1042
  fix A1 A2 B1 B2 assume "\<forall>\<^sub>F x in A. A1 x" "\<forall>\<^sub>F x in A. A2 x" "\<forall>\<^sub>F x in B. B1 x" "\<forall>\<^sub>F x in B. B2 x"
hoelzl@62367
  1043
  then show "\<exists>x. eventually x A \<and> (\<exists>y. eventually y B \<and> Collect x \<times> Collect y \<subseteq> Collect A1 \<times> Collect B1 \<and> Collect x \<times> Collect y \<subseteq> Collect A2 \<times> Collect B2)"
hoelzl@62367
  1044
    by (intro exI[of _ "\<lambda>x. A1 x \<and> A2 x"] exI[of _ "\<lambda>x. B1 x \<and> B2 x"] conjI)
hoelzl@62367
  1045
       (auto simp: eventually_conj_iff)
hoelzl@62367
  1046
next
hoelzl@62367
  1047
  show "(\<exists>x. eventually x A \<and> (\<exists>y. eventually y B \<and> (x = (\<lambda>x. False) \<or> y = (\<lambda>x. False)))) = (A = \<bottom> \<or> B = \<bottom>)"
hoelzl@62367
  1048
    by (auto simp: trivial_limit_def intro: eventually_True)
hoelzl@62367
  1049
qed
hoelzl@62367
  1050
hoelzl@62101
  1051
lemma prod_filter_mono: "F \<le> F' \<Longrightarrow> G \<le> G' \<Longrightarrow> F \<times>\<^sub>F G \<le> F' \<times>\<^sub>F G'"
hoelzl@62101
  1052
  by (auto simp: le_filter_def eventually_prod_filter)
hoelzl@62101
  1053
hoelzl@62367
  1054
lemma prod_filter_mono_iff:
hoelzl@62367
  1055
  assumes nAB: "A \<noteq> bot" "B \<noteq> bot"
hoelzl@62367
  1056
  shows "A \<times>\<^sub>F B \<le> C \<times>\<^sub>F D \<longleftrightarrow> A \<le> C \<and> B \<le> D"
hoelzl@62367
  1057
proof safe
hoelzl@62367
  1058
  assume *: "A \<times>\<^sub>F B \<le> C \<times>\<^sub>F D"
wenzelm@63540
  1059
  with assms have "A \<times>\<^sub>F B \<noteq> bot"
hoelzl@62367
  1060
    by (auto simp: bot_unique prod_filter_eq_bot)
wenzelm@63540
  1061
  with * have "C \<times>\<^sub>F D \<noteq> bot"
hoelzl@62367
  1062
    by (auto simp: bot_unique)
hoelzl@62367
  1063
  then have nCD: "C \<noteq> bot" "D \<noteq> bot"
hoelzl@62367
  1064
    by (auto simp: prod_filter_eq_bot)
hoelzl@62367
  1065
hoelzl@62367
  1066
  show "A \<le> C"
hoelzl@62367
  1067
  proof (rule filter_leI)
hoelzl@62367
  1068
    fix P assume "eventually P C" with *[THEN filter_leD, of "\<lambda>(x, y). P x"] show "eventually P A"
hoelzl@62367
  1069
      using nAB nCD by (simp add: eventually_prod1 eventually_prod2)
hoelzl@62367
  1070
  qed
hoelzl@62367
  1071
hoelzl@62367
  1072
  show "B \<le> D"
hoelzl@62367
  1073
  proof (rule filter_leI)
hoelzl@62367
  1074
    fix P assume "eventually P D" with *[THEN filter_leD, of "\<lambda>(x, y). P y"] show "eventually P B"
hoelzl@62367
  1075
      using nAB nCD by (simp add: eventually_prod1 eventually_prod2)
hoelzl@62367
  1076
  qed
hoelzl@62367
  1077
qed (intro prod_filter_mono)
hoelzl@62367
  1078
hoelzl@62101
  1079
lemma eventually_prod_same: "eventually P (F \<times>\<^sub>F F) \<longleftrightarrow>
hoelzl@62101
  1080
    (\<exists>Q. eventually Q F \<and> (\<forall>x y. Q x \<longrightarrow> Q y \<longrightarrow> P (x, y)))"
hoelzl@62101
  1081
  unfolding eventually_prod_filter
hoelzl@62101
  1082
  apply safe
hoelzl@62101
  1083
  apply (rule_tac x="inf Pf Pg" in exI)
hoelzl@62101
  1084
  apply (auto simp: inf_fun_def intro!: eventually_conj)
hoelzl@62101
  1085
  done
hoelzl@62101
  1086
hoelzl@62101
  1087
lemma eventually_prod_sequentially:
hoelzl@62101
  1088
  "eventually P (sequentially \<times>\<^sub>F sequentially) \<longleftrightarrow> (\<exists>N. \<forall>m \<ge> N. \<forall>n \<ge> N. P (n, m))"
hoelzl@62101
  1089
  unfolding eventually_prod_same eventually_sequentially by auto
hoelzl@62101
  1090
hoelzl@62101
  1091
lemma principal_prod_principal: "principal A \<times>\<^sub>F principal B = principal (A \<times> B)"
hoelzl@62101
  1092
  apply (simp add: filter_eq_iff eventually_prod_filter eventually_principal)
hoelzl@62101
  1093
  apply safe
hoelzl@62101
  1094
  apply blast
hoelzl@62101
  1095
  apply (intro conjI exI[of _ "\<lambda>x. x \<in> A"] exI[of _ "\<lambda>x. x \<in> B"])
hoelzl@62101
  1096
  apply auto
hoelzl@62101
  1097
  done
hoelzl@62101
  1098
hoelzl@62367
  1099
lemma prod_filter_INF:
hoelzl@62367
  1100
  assumes "I \<noteq> {}" "J \<noteq> {}"
hoelzl@62367
  1101
  shows "(INF i:I. A i) \<times>\<^sub>F (INF j:J. B j) = (INF i:I. INF j:J. A i \<times>\<^sub>F B j)"
hoelzl@62367
  1102
proof (safe intro!: antisym INF_greatest)
hoelzl@62367
  1103
  from \<open>I \<noteq> {}\<close> obtain i where "i \<in> I" by auto
hoelzl@62367
  1104
  from \<open>J \<noteq> {}\<close> obtain j where "j \<in> J" by auto
hoelzl@62367
  1105
hoelzl@62367
  1106
  show "(\<Sqinter>i\<in>I. \<Sqinter>j\<in>J. A i \<times>\<^sub>F B j) \<le> (\<Sqinter>i\<in>I. A i) \<times>\<^sub>F (\<Sqinter>j\<in>J. B j)"
hoelzl@62367
  1107
    unfolding prod_filter_def
hoelzl@62367
  1108
  proof (safe intro!: INF_greatest)
hoelzl@62367
  1109
    fix P Q assume P: "\<forall>\<^sub>F x in \<Sqinter>i\<in>I. A i. P x" and Q: "\<forall>\<^sub>F x in \<Sqinter>j\<in>J. B j. Q x"
hoelzl@62367
  1110
    let ?X = "(\<Sqinter>i\<in>I. \<Sqinter>j\<in>J. \<Sqinter>(P, Q)\<in>{(P, Q). (\<forall>\<^sub>F x in A i. P x) \<and> (\<forall>\<^sub>F x in B j. Q x)}. principal {(x, y). P x \<and> Q y})"
hoelzl@62367
  1111
    have "?X \<le> principal {x. P (fst x)} \<sqinter> principal {x. Q (snd x)}"
hoelzl@62367
  1112
    proof (intro inf_greatest)
hoelzl@62367
  1113
      have "?X \<le> (\<Sqinter>i\<in>I. \<Sqinter>P\<in>{P. eventually P (A i)}. principal {x. P (fst x)})"
hoelzl@62367
  1114
        by (auto intro!: INF_greatest INF_lower2[of j] INF_lower2 \<open>j\<in>J\<close> INF_lower2[of "(_, \<lambda>x. True)"])
hoelzl@62367
  1115
      also have "\<dots> \<le> principal {x. P (fst x)}"
hoelzl@62367
  1116
        unfolding le_principal
hoelzl@62367
  1117
      proof (rule eventually_INF_mono[OF P])
hoelzl@62367
  1118
        fix i P assume "i \<in> I" "eventually P (A i)"
hoelzl@62367
  1119
        then show "\<forall>\<^sub>F x in \<Sqinter>P\<in>{P. eventually P (A i)}. principal {x. P (fst x)}. x \<in> {x. P (fst x)}"
hoelzl@62367
  1120
          unfolding le_principal[symmetric] by (auto intro!: INF_lower)
hoelzl@62367
  1121
      qed auto
hoelzl@62367
  1122
      finally show "?X \<le> principal {x. P (fst x)}" .
hoelzl@62367
  1123
hoelzl@62367
  1124
      have "?X \<le> (\<Sqinter>i\<in>J. \<Sqinter>P\<in>{P. eventually P (B i)}. principal {x. P (snd x)})"
hoelzl@62367
  1125
        by (auto intro!: INF_greatest INF_lower2[of i] INF_lower2 \<open>i\<in>I\<close> INF_lower2[of "(\<lambda>x. True, _)"])
hoelzl@62367
  1126
      also have "\<dots> \<le> principal {x. Q (snd x)}"
hoelzl@62367
  1127
        unfolding le_principal
hoelzl@62367
  1128
      proof (rule eventually_INF_mono[OF Q])
hoelzl@62367
  1129
        fix j Q assume "j \<in> J" "eventually Q (B j)"
hoelzl@62367
  1130
        then show "\<forall>\<^sub>F x in \<Sqinter>P\<in>{P. eventually P (B j)}. principal {x. P (snd x)}. x \<in> {x. Q (snd x)}"
hoelzl@62367
  1131
          unfolding le_principal[symmetric] by (auto intro!: INF_lower)
hoelzl@62367
  1132
      qed auto
hoelzl@62367
  1133
      finally show "?X \<le> principal {x. Q (snd x)}" .
hoelzl@62367
  1134
    qed
hoelzl@62367
  1135
    also have "\<dots> = principal {(x, y). P x \<and> Q y}"
hoelzl@62367
  1136
      by auto
hoelzl@62367
  1137
    finally show "?X \<le> principal {(x, y). P x \<and> Q y}" .
hoelzl@62367
  1138
  qed
hoelzl@62367
  1139
qed (intro prod_filter_mono INF_lower)
hoelzl@62367
  1140
hoelzl@62367
  1141
lemma filtermap_Pair: "filtermap (\<lambda>x. (f x, g x)) F \<le> filtermap f F \<times>\<^sub>F filtermap g F"
hoelzl@62367
  1142
  by (simp add: le_filter_def eventually_filtermap eventually_prod_filter)
hoelzl@62367
  1143
     (auto elim: eventually_elim2)
hoelzl@62367
  1144
hoelzl@62369
  1145
lemma eventually_prodI: "eventually P F \<Longrightarrow> eventually Q G \<Longrightarrow> eventually (\<lambda>x. P (fst x) \<and> Q (snd x)) (F \<times>\<^sub>F G)"
hoelzl@62369
  1146
  unfolding prod_filter_def
hoelzl@62369
  1147
  by (intro eventually_INF1[of "(P, Q)"]) (auto simp: eventually_principal)
hoelzl@62369
  1148
hoelzl@62369
  1149
lemma prod_filter_INF1: "I \<noteq> {} \<Longrightarrow> (INF i:I. A i) \<times>\<^sub>F B = (INF i:I. A i \<times>\<^sub>F B)"
hoelzl@62369
  1150
  using prod_filter_INF[of I "{B}" A "\<lambda>x. x"] by simp
hoelzl@62369
  1151
hoelzl@62369
  1152
lemma prod_filter_INF2: "J \<noteq> {} \<Longrightarrow> A \<times>\<^sub>F (INF i:J. B i) = (INF i:J. A \<times>\<^sub>F B i)"
hoelzl@62369
  1153
  using prod_filter_INF[of "{A}" J "\<lambda>x. x" B] by simp
hoelzl@62369
  1154
wenzelm@60758
  1155
subsection \<open>Limits\<close>
hoelzl@60036
  1156
hoelzl@60036
  1157
definition filterlim :: "('a \<Rightarrow> 'b) \<Rightarrow> 'b filter \<Rightarrow> 'a filter \<Rightarrow> bool" where
hoelzl@60036
  1158
  "filterlim f F2 F1 \<longleftrightarrow> filtermap f F1 \<le> F2"
hoelzl@60036
  1159
hoelzl@60036
  1160
syntax
hoelzl@60036
  1161
  "_LIM" :: "pttrns \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> bool" ("(3LIM (_)/ (_)./ (_) :> (_))" [1000, 10, 0, 10] 10)
hoelzl@60036
  1162
hoelzl@60036
  1163
translations
hoelzl@62367
  1164
  "LIM x F1. f :> F2" == "CONST filterlim (\<lambda>x. f) F2 F1"
hoelzl@60036
  1165
lp15@62379
  1166
lemma filterlim_top [simp]: "filterlim f top F"
lp15@62379
  1167
  by (simp add: filterlim_def)
lp15@62379
  1168
hoelzl@60036
  1169
lemma filterlim_iff:
hoelzl@60036
  1170
  "(LIM x F1. f x :> F2) \<longleftrightarrow> (\<forall>P. eventually P F2 \<longrightarrow> eventually (\<lambda>x. P (f x)) F1)"
hoelzl@60036
  1171
  unfolding filterlim_def le_filter_def eventually_filtermap ..
hoelzl@60036
  1172
hoelzl@60036
  1173
lemma filterlim_compose:
hoelzl@60036
  1174
  "filterlim g F3 F2 \<Longrightarrow> filterlim f F2 F1 \<Longrightarrow> filterlim (\<lambda>x. g (f x)) F3 F1"
hoelzl@60036
  1175
  unfolding filterlim_def filtermap_filtermap[symmetric] by (metis filtermap_mono order_trans)
hoelzl@60036
  1176
hoelzl@60036
  1177
lemma filterlim_mono:
hoelzl@60036
  1178
  "filterlim f F2 F1 \<Longrightarrow> F2 \<le> F2' \<Longrightarrow> F1' \<le> F1 \<Longrightarrow> filterlim f F2' F1'"
hoelzl@60036
  1179
  unfolding filterlim_def by (metis filtermap_mono order_trans)
hoelzl@60036
  1180
hoelzl@60036
  1181
lemma filterlim_ident: "LIM x F. x :> F"
hoelzl@60036
  1182
  by (simp add: filterlim_def filtermap_ident)
hoelzl@60036
  1183
hoelzl@60036
  1184
lemma filterlim_cong:
hoelzl@60036
  1185
  "F1 = F1' \<Longrightarrow> F2 = F2' \<Longrightarrow> eventually (\<lambda>x. f x = g x) F2 \<Longrightarrow> filterlim f F1 F2 = filterlim g F1' F2'"
hoelzl@60036
  1186
  by (auto simp: filterlim_def le_filter_def eventually_filtermap elim: eventually_elim2)
hoelzl@60036
  1187
hoelzl@60036
  1188
lemma filterlim_mono_eventually:
hoelzl@60036
  1189
  assumes "filterlim f F G" and ord: "F \<le> F'" "G' \<le> G"
hoelzl@60036
  1190
  assumes eq: "eventually (\<lambda>x. f x = f' x) G'"
hoelzl@60036
  1191
  shows "filterlim f' F' G'"
hoelzl@60036
  1192
  apply (rule filterlim_cong[OF refl refl eq, THEN iffD1])
hoelzl@60036
  1193
  apply (rule filterlim_mono[OF _ ord])
hoelzl@60036
  1194
  apply fact
hoelzl@60036
  1195
  done
hoelzl@60036
  1196
hoelzl@60036
  1197
lemma filtermap_mono_strong: "inj f \<Longrightarrow> filtermap f F \<le> filtermap f G \<longleftrightarrow> F \<le> G"
hoelzl@60036
  1198
  apply (auto intro!: filtermap_mono) []
hoelzl@60036
  1199
  apply (auto simp: le_filter_def eventually_filtermap)
hoelzl@60036
  1200
  apply (erule_tac x="\<lambda>x. P (inv f x)" in allE)
hoelzl@60036
  1201
  apply auto
hoelzl@60036
  1202
  done
hoelzl@60036
  1203
hoelzl@60036
  1204
lemma filtermap_eq_strong: "inj f \<Longrightarrow> filtermap f F = filtermap f G \<longleftrightarrow> F = G"
hoelzl@60036
  1205
  by (simp add: filtermap_mono_strong eq_iff)
hoelzl@60036
  1206
hoelzl@60721
  1207
lemma filtermap_fun_inverse:
hoelzl@60721
  1208
  assumes g: "filterlim g F G"
hoelzl@60721
  1209
  assumes f: "filterlim f G F"
hoelzl@60721
  1210
  assumes ev: "eventually (\<lambda>x. f (g x) = x) G"
hoelzl@60721
  1211
  shows "filtermap f F = G"
hoelzl@60721
  1212
proof (rule antisym)
hoelzl@60721
  1213
  show "filtermap f F \<le> G"
hoelzl@60721
  1214
    using f unfolding filterlim_def .
hoelzl@60721
  1215
  have "G = filtermap f (filtermap g G)"
hoelzl@60721
  1216
    using ev by (auto elim: eventually_elim2 simp: filter_eq_iff eventually_filtermap)
hoelzl@60721
  1217
  also have "\<dots> \<le> filtermap f F"
hoelzl@60721
  1218
    using g by (intro filtermap_mono) (simp add: filterlim_def)
hoelzl@60721
  1219
  finally show "G \<le> filtermap f F" .
hoelzl@60721
  1220
qed
hoelzl@60721
  1221
hoelzl@60036
  1222
lemma filterlim_principal:
hoelzl@60036
  1223
  "(LIM x F. f x :> principal S) \<longleftrightarrow> (eventually (\<lambda>x. f x \<in> S) F)"
hoelzl@60036
  1224
  unfolding filterlim_def eventually_filtermap le_principal ..
hoelzl@60036
  1225
hoelzl@60036
  1226
lemma filterlim_inf:
hoelzl@60036
  1227
  "(LIM x F1. f x :> inf F2 F3) \<longleftrightarrow> ((LIM x F1. f x :> F2) \<and> (LIM x F1. f x :> F3))"
hoelzl@60036
  1228
  unfolding filterlim_def by simp
hoelzl@60036
  1229
hoelzl@60036
  1230
lemma filterlim_INF:
hoelzl@60036
  1231
  "(LIM x F. f x :> (INF b:B. G b)) \<longleftrightarrow> (\<forall>b\<in>B. LIM x F. f x :> G b)"
hoelzl@60036
  1232
  unfolding filterlim_def le_INF_iff ..
hoelzl@60036
  1233
hoelzl@60036
  1234
lemma filterlim_INF_INF:
hoelzl@60036
  1235
  "(\<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)"
hoelzl@60036
  1236
  unfolding filterlim_def by (rule order_trans[OF filtermap_INF INF_mono])
hoelzl@60036
  1237
hoelzl@60036
  1238
lemma filterlim_base:
lp15@61806
  1239
  "(\<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>
hoelzl@60036
  1240
    LIM x (INF i:I. principal (F i)). f x :> (INF j:J. principal (G j))"
hoelzl@60036
  1241
  by (force intro!: filterlim_INF_INF simp: image_subset_iff)
hoelzl@60036
  1242
lp15@61806
  1243
lemma filterlim_base_iff:
hoelzl@60036
  1244
  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"
hoelzl@60036
  1245
  shows "(LIM x (INF i:I. principal (F i)). f x :> INF j:J. principal (G j)) \<longleftrightarrow>
hoelzl@60036
  1246
    (\<forall>j\<in>J. \<exists>i\<in>I. \<forall>x\<in>F i. f x \<in> G j)"
hoelzl@60036
  1247
  unfolding filterlim_INF filterlim_principal
hoelzl@60036
  1248
proof (subst eventually_INF_base)
hoelzl@60036
  1249
  fix i j assume "i \<in> I" "j \<in> I"
hoelzl@60036
  1250
  with chain[OF this] show "\<exists>x\<in>I. principal (F x) \<le> inf (principal (F i)) (principal (F j))"
hoelzl@60036
  1251
    by auto
wenzelm@60758
  1252
qed (auto simp: eventually_principal \<open>I \<noteq> {}\<close>)
hoelzl@60036
  1253
hoelzl@60036
  1254
lemma filterlim_filtermap: "filterlim f F1 (filtermap g F2) = filterlim (\<lambda>x. f (g x)) F1 F2"
hoelzl@60036
  1255
  unfolding filterlim_def filtermap_filtermap ..
hoelzl@60036
  1256
hoelzl@60036
  1257
lemma filterlim_sup:
hoelzl@60036
  1258
  "filterlim f F F1 \<Longrightarrow> filterlim f F F2 \<Longrightarrow> filterlim f F (sup F1 F2)"
hoelzl@60036
  1259
  unfolding filterlim_def filtermap_sup by auto
hoelzl@60036
  1260
hoelzl@60036
  1261
lemma filterlim_sequentially_Suc:
hoelzl@60036
  1262
  "(LIM x sequentially. f (Suc x) :> F) \<longleftrightarrow> (LIM x sequentially. f x :> F)"
hoelzl@60036
  1263
  unfolding filterlim_iff by (subst eventually_sequentially_Suc) simp
hoelzl@60036
  1264
hoelzl@60036
  1265
lemma filterlim_Suc: "filterlim Suc sequentially sequentially"
lp15@63967
  1266
  by (simp add: filterlim_iff eventually_sequentially)
hoelzl@60036
  1267
hoelzl@60182
  1268
lemma filterlim_If:
hoelzl@60182
  1269
  "LIM x inf F (principal {x. P x}). f x :> G \<Longrightarrow>
hoelzl@60182
  1270
    LIM x inf F (principal {x. \<not> P x}). g x :> G \<Longrightarrow>
hoelzl@60182
  1271
    LIM x F. if P x then f x else g x :> G"
hoelzl@60182
  1272
  unfolding filterlim_iff eventually_inf_principal by (auto simp: eventually_conj_iff)
hoelzl@60036
  1273
hoelzl@62367
  1274
lemma filterlim_Pair:
hoelzl@62367
  1275
  "LIM x F. f x :> G \<Longrightarrow> LIM x F. g x :> H \<Longrightarrow> LIM x F. (f x, g x) :> G \<times>\<^sub>F H"
hoelzl@62367
  1276
  unfolding filterlim_def
hoelzl@62367
  1277
  by (rule order_trans[OF filtermap_Pair prod_filter_mono])
hoelzl@62367
  1278
wenzelm@60758
  1279
subsection \<open>Limits to @{const at_top} and @{const at_bot}\<close>
hoelzl@60036
  1280
hoelzl@60036
  1281
lemma filterlim_at_top:
hoelzl@60036
  1282
  fixes f :: "'a \<Rightarrow> ('b::linorder)"
hoelzl@60036
  1283
  shows "(LIM x F. f x :> at_top) \<longleftrightarrow> (\<forall>Z. eventually (\<lambda>x. Z \<le> f x) F)"
lp15@61810
  1284
  by (auto simp: filterlim_iff eventually_at_top_linorder elim!: eventually_mono)
hoelzl@60036
  1285
hoelzl@60036
  1286
lemma filterlim_at_top_mono:
hoelzl@60036
  1287
  "LIM x F. f x :> at_top \<Longrightarrow> eventually (\<lambda>x. f x \<le> (g x::'a::linorder)) F \<Longrightarrow>
hoelzl@60036
  1288
    LIM x F. g x :> at_top"
hoelzl@60036
  1289
  by (auto simp: filterlim_at_top elim: eventually_elim2 intro: order_trans)
hoelzl@60036
  1290
hoelzl@60036
  1291
lemma filterlim_at_top_dense:
hoelzl@60036
  1292
  fixes f :: "'a \<Rightarrow> ('b::unbounded_dense_linorder)"
hoelzl@60036
  1293
  shows "(LIM x F. f x :> at_top) \<longleftrightarrow> (\<forall>Z. eventually (\<lambda>x. Z < f x) F)"
lp15@61810
  1294
  by (metis eventually_mono[of _ F] eventually_gt_at_top order_less_imp_le
hoelzl@60036
  1295
            filterlim_at_top[of f F] filterlim_iff[of f at_top F])
hoelzl@60036
  1296
hoelzl@60036
  1297
lemma filterlim_at_top_ge:
hoelzl@60036
  1298
  fixes f :: "'a \<Rightarrow> ('b::linorder)" and c :: "'b"
hoelzl@60036
  1299
  shows "(LIM x F. f x :> at_top) \<longleftrightarrow> (\<forall>Z\<ge>c. eventually (\<lambda>x. Z \<le> f x) F)"
hoelzl@60036
  1300
  unfolding at_top_sub[of c] filterlim_INF by (auto simp add: filterlim_principal)
hoelzl@60036
  1301
hoelzl@60036
  1302
lemma filterlim_at_top_at_top:
hoelzl@60036
  1303
  fixes f :: "'a::linorder \<Rightarrow> 'b::linorder"
hoelzl@60036
  1304
  assumes mono: "\<And>x y. Q x \<Longrightarrow> Q y \<Longrightarrow> x \<le> y \<Longrightarrow> f x \<le> f y"
hoelzl@60036
  1305
  assumes bij: "\<And>x. P x \<Longrightarrow> f (g x) = x" "\<And>x. P x \<Longrightarrow> Q (g x)"
hoelzl@60036
  1306
  assumes Q: "eventually Q at_top"
hoelzl@60036
  1307
  assumes P: "eventually P at_top"
hoelzl@60036
  1308
  shows "filterlim f at_top at_top"
hoelzl@60036
  1309
proof -
hoelzl@60036
  1310
  from P obtain x where x: "\<And>y. x \<le> y \<Longrightarrow> P y"
hoelzl@60036
  1311
    unfolding eventually_at_top_linorder by auto
hoelzl@60036
  1312
  show ?thesis
hoelzl@60036
  1313
  proof (intro filterlim_at_top_ge[THEN iffD2] allI impI)
hoelzl@60036
  1314
    fix z assume "x \<le> z"
hoelzl@60036
  1315
    with x have "P z" by auto
hoelzl@60036
  1316
    have "eventually (\<lambda>x. g z \<le> x) at_top"
hoelzl@60036
  1317
      by (rule eventually_ge_at_top)
hoelzl@60036
  1318
    with Q show "eventually (\<lambda>x. z \<le> f x) at_top"
wenzelm@60758
  1319
      by eventually_elim (metis mono bij \<open>P z\<close>)
hoelzl@60036
  1320
  qed
hoelzl@60036
  1321
qed
hoelzl@60036
  1322
hoelzl@60036
  1323
lemma filterlim_at_top_gt:
hoelzl@60036
  1324
  fixes f :: "'a \<Rightarrow> ('b::unbounded_dense_linorder)" and c :: "'b"
hoelzl@60036
  1325
  shows "(LIM x F. f x :> at_top) \<longleftrightarrow> (\<forall>Z>c. eventually (\<lambda>x. Z \<le> f x) F)"
hoelzl@60036
  1326
  by (metis filterlim_at_top order_less_le_trans gt_ex filterlim_at_top_ge)
hoelzl@60036
  1327
lp15@61806
  1328
lemma filterlim_at_bot:
hoelzl@60036
  1329
  fixes f :: "'a \<Rightarrow> ('b::linorder)"
hoelzl@60036
  1330
  shows "(LIM x F. f x :> at_bot) \<longleftrightarrow> (\<forall>Z. eventually (\<lambda>x. f x \<le> Z) F)"
lp15@61810
  1331
  by (auto simp: filterlim_iff eventually_at_bot_linorder elim!: eventually_mono)
hoelzl@60036
  1332
hoelzl@60036
  1333
lemma filterlim_at_bot_dense:
hoelzl@60036
  1334
  fixes f :: "'a \<Rightarrow> ('b::{dense_linorder, no_bot})"
hoelzl@60036
  1335
  shows "(LIM x F. f x :> at_bot) \<longleftrightarrow> (\<forall>Z. eventually (\<lambda>x. f x < Z) F)"
hoelzl@60036
  1336
proof (auto simp add: filterlim_at_bot[of f F])
hoelzl@60036
  1337
  fix Z :: 'b
hoelzl@60036
  1338
  from lt_ex [of Z] obtain Z' where 1: "Z' < Z" ..
hoelzl@60036
  1339
  assume "\<forall>Z. eventually (\<lambda>x. f x \<le> Z) F"
hoelzl@60036
  1340
  hence "eventually (\<lambda>x. f x \<le> Z') F" by auto
hoelzl@60036
  1341
  thus "eventually (\<lambda>x. f x < Z) F"
lp15@61810
  1342
    apply (rule eventually_mono)
hoelzl@60036
  1343
    using 1 by auto
lp15@61806
  1344
  next
lp15@61806
  1345
    fix Z :: 'b
hoelzl@60036
  1346
    show "\<forall>Z. eventually (\<lambda>x. f x < Z) F \<Longrightarrow> eventually (\<lambda>x. f x \<le> Z) F"
lp15@61810
  1347
      by (drule spec [of _ Z], erule eventually_mono, auto simp add: less_imp_le)
hoelzl@60036
  1348
qed
hoelzl@60036
  1349
hoelzl@60036
  1350
lemma filterlim_at_bot_le:
hoelzl@60036
  1351
  fixes f :: "'a \<Rightarrow> ('b::linorder)" and c :: "'b"
hoelzl@60036
  1352
  shows "(LIM x F. f x :> at_bot) \<longleftrightarrow> (\<forall>Z\<le>c. eventually (\<lambda>x. Z \<ge> f x) F)"
hoelzl@60036
  1353
  unfolding filterlim_at_bot
hoelzl@60036
  1354
proof safe
hoelzl@60036
  1355
  fix Z assume *: "\<forall>Z\<le>c. eventually (\<lambda>x. Z \<ge> f x) F"
hoelzl@60036
  1356
  with *[THEN spec, of "min Z c"] show "eventually (\<lambda>x. Z \<ge> f x) F"
lp15@61810
  1357
    by (auto elim!: eventually_mono)
hoelzl@60036
  1358
qed simp
hoelzl@60036
  1359
hoelzl@60036
  1360
lemma filterlim_at_bot_lt:
hoelzl@60036
  1361
  fixes f :: "'a \<Rightarrow> ('b::unbounded_dense_linorder)" and c :: "'b"
hoelzl@60036
  1362
  shows "(LIM x F. f x :> at_bot) \<longleftrightarrow> (\<forall>Z<c. eventually (\<lambda>x. Z \<ge> f x) F)"
hoelzl@60036
  1363
  by (metis filterlim_at_bot filterlim_at_bot_le lt_ex order_le_less_trans)
eberlm@66162
  1364
    
eberlm@66162
  1365
lemma filterlim_filtercomap [intro]: "filterlim f F (filtercomap f F)"
eberlm@66162
  1366
  unfolding filterlim_def by (rule filtermap_filtercomap)
hoelzl@60036
  1367
hoelzl@60036
  1368
wenzelm@60758
  1369
subsection \<open>Setup @{typ "'a filter"} for lifting and transfer\<close>
hoelzl@60036
  1370
wenzelm@63343
  1371
context includes lifting_syntax
wenzelm@63343
  1372
begin
hoelzl@60036
  1373
hoelzl@60036
  1374
definition rel_filter :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a filter \<Rightarrow> 'b filter \<Rightarrow> bool"
hoelzl@60036
  1375
where "rel_filter R F G = ((R ===> op =) ===> op =) (Rep_filter F) (Rep_filter G)"
hoelzl@60036
  1376
hoelzl@60036
  1377
lemma rel_filter_eventually:
lp15@61806
  1378
  "rel_filter R F G \<longleftrightarrow>
hoelzl@60036
  1379
  ((R ===> op =) ===> op =) (\<lambda>P. eventually P F) (\<lambda>P. eventually P G)"
hoelzl@60036
  1380
by(simp add: rel_filter_def eventually_def)
hoelzl@60036
  1381
hoelzl@60036
  1382
lemma filtermap_id [simp, id_simps]: "filtermap id = id"
hoelzl@60036
  1383
by(simp add: fun_eq_iff id_def filtermap_ident)
hoelzl@60036
  1384
hoelzl@60036
  1385
lemma filtermap_id' [simp]: "filtermap (\<lambda>x. x) = (\<lambda>F. F)"
hoelzl@60036
  1386
using filtermap_id unfolding id_def .
hoelzl@60036
  1387
hoelzl@60036
  1388
lemma Quotient_filter [quot_map]:
hoelzl@60036
  1389
  assumes Q: "Quotient R Abs Rep T"
hoelzl@60036
  1390
  shows "Quotient (rel_filter R) (filtermap Abs) (filtermap Rep) (rel_filter T)"
hoelzl@60036
  1391
unfolding Quotient_alt_def
hoelzl@60036
  1392
proof(intro conjI strip)
hoelzl@60036
  1393
  from Q have *: "\<And>x y. T x y \<Longrightarrow> Abs x = y"
hoelzl@60036
  1394
    unfolding Quotient_alt_def by blast
hoelzl@60036
  1395
hoelzl@60036
  1396
  fix F G
hoelzl@60036
  1397
  assume "rel_filter T F G"
hoelzl@60036
  1398
  thus "filtermap Abs F = G" unfolding filter_eq_iff
hoelzl@60036
  1399
    by(auto simp add: eventually_filtermap rel_filter_eventually * rel_funI del: iffI elim!: rel_funD)
hoelzl@60036
  1400
next
hoelzl@60036
  1401
  from Q have *: "\<And>x. T (Rep x) x" unfolding Quotient_alt_def by blast
hoelzl@60036
  1402
hoelzl@60036
  1403
  fix F
lp15@61806
  1404
  show "rel_filter T (filtermap Rep F) F"
hoelzl@60036
  1405
    by(auto elim: rel_funD intro: * intro!: ext arg_cong[where f="\<lambda>P. eventually P F"] rel_funI
hoelzl@60036
  1406
            del: iffI simp add: eventually_filtermap rel_filter_eventually)
hoelzl@60036
  1407
qed(auto simp add: map_fun_def o_def eventually_filtermap filter_eq_iff fun_eq_iff rel_filter_eventually
hoelzl@60036
  1408
         fun_quotient[OF fun_quotient[OF Q identity_quotient] identity_quotient, unfolded Quotient_alt_def])
hoelzl@60036
  1409
hoelzl@60036
  1410
lemma eventually_parametric [transfer_rule]:
hoelzl@60036
  1411
  "((A ===> op =) ===> rel_filter A ===> op =) eventually eventually"
hoelzl@60036
  1412
by(simp add: rel_fun_def rel_filter_eventually)
hoelzl@60036
  1413
hoelzl@60038
  1414
lemma frequently_parametric [transfer_rule]:
hoelzl@60038
  1415
  "((A ===> op =) ===> rel_filter A ===> op =) frequently frequently"
hoelzl@60038
  1416
  unfolding frequently_def[abs_def] by transfer_prover
hoelzl@60038
  1417
hoelzl@60036
  1418
lemma rel_filter_eq [relator_eq]: "rel_filter op = = op ="
hoelzl@60036
  1419
by(auto simp add: rel_filter_eventually rel_fun_eq fun_eq_iff filter_eq_iff)
hoelzl@60036
  1420
hoelzl@60036
  1421
lemma rel_filter_mono [relator_mono]:
hoelzl@60036
  1422
  "A \<le> B \<Longrightarrow> rel_filter A \<le> rel_filter B"
hoelzl@60036
  1423
unfolding rel_filter_eventually[abs_def]
hoelzl@60036
  1424
by(rule le_funI)+(intro fun_mono fun_mono[THEN le_funD, THEN le_funD] order.refl)
hoelzl@60036
  1425
hoelzl@60036
  1426
lemma rel_filter_conversep [simp]: "rel_filter A\<inverse>\<inverse> = (rel_filter A)\<inverse>\<inverse>"
lp15@61233
  1427
apply (simp add: rel_filter_eventually fun_eq_iff rel_fun_def)
lp15@61233
  1428
apply (safe; metis)
lp15@61233
  1429
done
hoelzl@60036
  1430
hoelzl@60036
  1431
lemma is_filter_parametric_aux:
hoelzl@60036
  1432
  assumes "is_filter F"
hoelzl@60036
  1433
  assumes [transfer_rule]: "bi_total A" "bi_unique A"
hoelzl@60036
  1434
  and [transfer_rule]: "((A ===> op =) ===> op =) F G"
hoelzl@60036
  1435
  shows "is_filter G"
hoelzl@60036
  1436
proof -
hoelzl@60036
  1437
  interpret is_filter F by fact
hoelzl@60036
  1438
  show ?thesis
hoelzl@60036
  1439
  proof
hoelzl@60036
  1440
    have "F (\<lambda>_. True) = G (\<lambda>x. True)" by transfer_prover
hoelzl@60036
  1441
    thus "G (\<lambda>x. True)" by(simp add: True)
hoelzl@60036
  1442
  next
hoelzl@60036
  1443
    fix P' Q'
hoelzl@60036
  1444
    assume "G P'" "G Q'"
hoelzl@60036
  1445
    moreover
wenzelm@60758
  1446
    from bi_total_fun[OF \<open>bi_unique A\<close> bi_total_eq, unfolded bi_total_def]
hoelzl@60036
  1447
    obtain P Q where [transfer_rule]: "(A ===> op =) P P'" "(A ===> op =) Q Q'" by blast
hoelzl@60036
  1448
    have "F P = G P'" "F Q = G Q'" by transfer_prover+
hoelzl@60036
  1449
    ultimately have "F (\<lambda>x. P x \<and> Q x)" by(simp add: conj)
hoelzl@60036
  1450
    moreover have "F (\<lambda>x. P x \<and> Q x) = G (\<lambda>x. P' x \<and> Q' x)" by transfer_prover
hoelzl@60036
  1451
    ultimately show "G (\<lambda>x. P' x \<and> Q' x)" by simp
hoelzl@60036
  1452
  next
hoelzl@60036
  1453
    fix P' Q'
hoelzl@60036
  1454
    assume "\<forall>x. P' x \<longrightarrow> Q' x" "G P'"
hoelzl@60036
  1455
    moreover
wenzelm@60758
  1456
    from bi_total_fun[OF \<open>bi_unique A\<close> bi_total_eq, unfolded bi_total_def]
hoelzl@60036
  1457
    obtain P Q where [transfer_rule]: "(A ===> op =) P P'" "(A ===> op =) Q Q'" by blast
hoelzl@60036
  1458
    have "F P = G P'" by transfer_prover
hoelzl@60036
  1459
    moreover have "(\<forall>x. P x \<longrightarrow> Q x) \<longleftrightarrow> (\<forall>x. P' x \<longrightarrow> Q' x)" by transfer_prover
hoelzl@60036
  1460
    ultimately have "F Q" by(simp add: mono)
hoelzl@60036
  1461
    moreover have "F Q = G Q'" by transfer_prover
hoelzl@60036
  1462
    ultimately show "G Q'" by simp
hoelzl@60036
  1463
  qed
hoelzl@60036
  1464
qed
hoelzl@60036
  1465
hoelzl@60036
  1466
lemma is_filter_parametric [transfer_rule]:
hoelzl@60036
  1467
  "\<lbrakk> bi_total A; bi_unique A \<rbrakk>
hoelzl@60036
  1468
  \<Longrightarrow> (((A ===> op =) ===> op =) ===> op =) is_filter is_filter"
hoelzl@60036
  1469
apply(rule rel_funI)
hoelzl@60036
  1470
apply(rule iffI)
hoelzl@60036
  1471
 apply(erule (3) is_filter_parametric_aux)
hoelzl@60036
  1472
apply(erule is_filter_parametric_aux[where A="conversep A"])
lp15@61233
  1473
apply (simp_all add: rel_fun_def)
lp15@61233
  1474
apply metis
hoelzl@60036
  1475
done
hoelzl@60036
  1476
hoelzl@60036
  1477
lemma left_total_rel_filter [transfer_rule]:
hoelzl@60036
  1478
  assumes [transfer_rule]: "bi_total A" "bi_unique A"
hoelzl@60036
  1479
  shows "left_total (rel_filter A)"
hoelzl@60036
  1480
proof(rule left_totalI)
hoelzl@60036
  1481
  fix F :: "'a filter"
wenzelm@60758
  1482
  from bi_total_fun[OF bi_unique_fun[OF \<open>bi_total A\<close> bi_unique_eq] bi_total_eq]
lp15@61806
  1483
  obtain G where [transfer_rule]: "((A ===> op =) ===> op =) (\<lambda>P. eventually P F) G"
hoelzl@60036
  1484
    unfolding  bi_total_def by blast
hoelzl@60036
  1485
  moreover have "is_filter (\<lambda>P. eventually P F) \<longleftrightarrow> is_filter G" by transfer_prover
hoelzl@60036
  1486
  hence "is_filter G" by(simp add: eventually_def is_filter_Rep_filter)
hoelzl@60036
  1487
  ultimately have "rel_filter A F (Abs_filter G)"
hoelzl@60036
  1488
    by(simp add: rel_filter_eventually eventually_Abs_filter)
hoelzl@60036
  1489
  thus "\<exists>G. rel_filter A F G" ..
hoelzl@60036
  1490
qed
hoelzl@60036
  1491
hoelzl@60036
  1492
lemma right_total_rel_filter [transfer_rule]:
hoelzl@60036
  1493
  "\<lbrakk> bi_total A; bi_unique A \<rbrakk> \<Longrightarrow> right_total (rel_filter A)"
hoelzl@60036
  1494
using left_total_rel_filter[of "A\<inverse>\<inverse>"] by simp
hoelzl@60036
  1495
hoelzl@60036
  1496
lemma bi_total_rel_filter [transfer_rule]:
hoelzl@60036
  1497
  assumes "bi_total A" "bi_unique A"
hoelzl@60036
  1498
  shows "bi_total (rel_filter A)"
hoelzl@60036
  1499
unfolding bi_total_alt_def using assms
hoelzl@60036
  1500
by(simp add: left_total_rel_filter right_total_rel_filter)
hoelzl@60036
  1501
hoelzl@60036
  1502
lemma left_unique_rel_filter [transfer_rule]:
hoelzl@60036
  1503
  assumes "left_unique A"
hoelzl@60036
  1504
  shows "left_unique (rel_filter A)"
hoelzl@60036
  1505
proof(rule left_uniqueI)
hoelzl@60036
  1506
  fix F F' G
hoelzl@60036
  1507
  assume [transfer_rule]: "rel_filter A F G" "rel_filter A F' G"
hoelzl@60036
  1508
  show "F = F'"
hoelzl@60036
  1509
    unfolding filter_eq_iff
hoelzl@60036
  1510
  proof
hoelzl@60036
  1511
    fix P :: "'a \<Rightarrow> bool"
hoelzl@60036
  1512
    obtain P' where [transfer_rule]: "(A ===> op =) P P'"
hoelzl@60036
  1513
      using left_total_fun[OF assms left_total_eq] unfolding left_total_def by blast
lp15@61806
  1514
    have "eventually P F = eventually P' G"
hoelzl@60036
  1515
      and "eventually P F' = eventually P' G" by transfer_prover+
hoelzl@60036
  1516
    thus "eventually P F = eventually P F'" by simp
hoelzl@60036
  1517
  qed
hoelzl@60036
  1518
qed
hoelzl@60036
  1519
hoelzl@60036
  1520
lemma right_unique_rel_filter [transfer_rule]:
hoelzl@60036
  1521
  "right_unique A \<Longrightarrow> right_unique (rel_filter A)"
hoelzl@60036
  1522
using left_unique_rel_filter[of "A\<inverse>\<inverse>"] by simp
hoelzl@60036
  1523
hoelzl@60036
  1524
lemma bi_unique_rel_filter [transfer_rule]:
hoelzl@60036
  1525
  "bi_unique A \<Longrightarrow> bi_unique (rel_filter A)"
hoelzl@60036
  1526
by(simp add: bi_unique_alt_def left_unique_rel_filter right_unique_rel_filter)
hoelzl@60036
  1527
hoelzl@60036
  1528
lemma top_filter_parametric [transfer_rule]:
hoelzl@60036
  1529
  "bi_total A \<Longrightarrow> (rel_filter A) top top"
hoelzl@60036
  1530
by(simp add: rel_filter_eventually All_transfer)
hoelzl@60036
  1531
hoelzl@60036
  1532
lemma bot_filter_parametric [transfer_rule]: "(rel_filter A) bot bot"
hoelzl@60036
  1533
by(simp add: rel_filter_eventually rel_fun_def)
hoelzl@60036
  1534
hoelzl@60036
  1535
lemma sup_filter_parametric [transfer_rule]:
hoelzl@60036
  1536
  "(rel_filter A ===> rel_filter A ===> rel_filter A) sup sup"
hoelzl@60036
  1537
by(fastforce simp add: rel_filter_eventually[abs_def] eventually_sup dest: rel_funD)
hoelzl@60036
  1538
hoelzl@60036
  1539
lemma Sup_filter_parametric [transfer_rule]:
hoelzl@60036
  1540
  "(rel_set (rel_filter A) ===> rel_filter A) Sup Sup"
hoelzl@60036
  1541
proof(rule rel_funI)
hoelzl@60036
  1542
  fix S T
hoelzl@60036
  1543
  assume [transfer_rule]: "rel_set (rel_filter A) S T"
hoelzl@60036
  1544
  show "rel_filter A (Sup S) (Sup T)"
hoelzl@60036
  1545
    by(simp add: rel_filter_eventually eventually_Sup) transfer_prover
hoelzl@60036
  1546
qed
hoelzl@60036
  1547
hoelzl@60036
  1548
lemma principal_parametric [transfer_rule]:
hoelzl@60036
  1549
  "(rel_set A ===> rel_filter A) principal principal"
hoelzl@60036
  1550
proof(rule rel_funI)
hoelzl@60036
  1551
  fix S S'
hoelzl@60036
  1552
  assume [transfer_rule]: "rel_set A S S'"
hoelzl@60036
  1553
  show "rel_filter A (principal S) (principal S')"
hoelzl@60036
  1554
    by(simp add: rel_filter_eventually eventually_principal) transfer_prover
hoelzl@60036
  1555
qed
eberlm@66162
  1556
  
eberlm@66162
  1557
lemma filtermap_parametric [transfer_rule]:
eberlm@66162
  1558
  "((A ===> B) ===> rel_filter A ===> rel_filter B) filtermap filtermap"
eberlm@66162
  1559
proof (intro rel_funI)
eberlm@66162
  1560
  fix f g F G assume [transfer_rule]: "(A ===> B) f g" "rel_filter A F G"
eberlm@66162
  1561
  show "rel_filter B (filtermap f F) (filtermap g G)"
eberlm@66162
  1562
    unfolding rel_filter_eventually eventually_filtermap by transfer_prover
eberlm@66162
  1563
qed
eberlm@66162
  1564
eberlm@66162
  1565
(* TODO: Are those assumptions needed? *)
eberlm@66162
  1566
lemma filtercomap_parametric [transfer_rule]:
eberlm@66162
  1567
  assumes [transfer_rule]: "bi_unique B" "bi_total A"
eberlm@66162
  1568
  shows   "((A ===> B) ===> rel_filter B ===> rel_filter A) filtercomap filtercomap"
eberlm@66162
  1569
proof (intro rel_funI)
eberlm@66162
  1570
  fix f g F G assume [transfer_rule]: "(A ===> B) f g" "rel_filter B F G"
eberlm@66162
  1571
  show "rel_filter A (filtercomap f F) (filtercomap g G)"
eberlm@66162
  1572
    unfolding rel_filter_eventually eventually_filtercomap by transfer_prover
eberlm@66162
  1573
qed
eberlm@66162
  1574
    
hoelzl@60036
  1575
hoelzl@60036
  1576
context
hoelzl@60036
  1577
  fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool"
lp15@61806
  1578
  assumes [transfer_rule]: "bi_unique A"
hoelzl@60036
  1579
begin
hoelzl@60036
  1580
hoelzl@60036
  1581
lemma le_filter_parametric [transfer_rule]:
hoelzl@60036
  1582
  "(rel_filter A ===> rel_filter A ===> op =) op \<le> op \<le>"
hoelzl@60036
  1583
unfolding le_filter_def[abs_def] by transfer_prover
hoelzl@60036
  1584
hoelzl@60036
  1585
lemma less_filter_parametric [transfer_rule]:
hoelzl@60036
  1586
  "(rel_filter A ===> rel_filter A ===> op =) op < op <"
hoelzl@60036
  1587
unfolding less_filter_def[abs_def] by transfer_prover
hoelzl@60036
  1588
hoelzl@60036
  1589
context
hoelzl@60036
  1590
  assumes [transfer_rule]: "bi_total A"
hoelzl@60036
  1591
begin
hoelzl@60036
  1592
hoelzl@60036
  1593
lemma Inf_filter_parametric [transfer_rule]:
hoelzl@60036
  1594
  "(rel_set (rel_filter A) ===> rel_filter A) Inf Inf"
hoelzl@60036
  1595
unfolding Inf_filter_def[abs_def] by transfer_prover
hoelzl@60036
  1596
hoelzl@60036
  1597
lemma inf_filter_parametric [transfer_rule]:
hoelzl@60036
  1598
  "(rel_filter A ===> rel_filter A ===> rel_filter A) inf inf"
hoelzl@60036
  1599
proof(intro rel_funI)+
hoelzl@60036
  1600
  fix F F' G G'
hoelzl@60036
  1601
  assume [transfer_rule]: "rel_filter A F F'" "rel_filter A G G'"
hoelzl@60036
  1602
  have "rel_filter A (Inf {F, G}) (Inf {F', G'})" by transfer_prover
hoelzl@60036
  1603
  thus "rel_filter A (inf F G) (inf F' G')" by simp
hoelzl@60036
  1604
qed
hoelzl@60036
  1605
hoelzl@60036
  1606
end
hoelzl@60036
  1607
hoelzl@60036
  1608
end
hoelzl@60036
  1609
hoelzl@60036
  1610
end
hoelzl@60036
  1611
hoelzl@62123
  1612
text \<open>Code generation for filters\<close>
hoelzl@62123
  1613
hoelzl@62123
  1614
definition abstract_filter :: "(unit \<Rightarrow> 'a filter) \<Rightarrow> 'a filter"
hoelzl@62123
  1615
  where [simp]: "abstract_filter f = f ()"
hoelzl@62123
  1616
hoelzl@62123
  1617
code_datatype principal abstract_filter
hoelzl@62123
  1618
hoelzl@62123
  1619
hide_const (open) abstract_filter
hoelzl@62123
  1620
hoelzl@62123
  1621
declare [[code drop: filterlim prod_filter filtermap eventually
hoelzl@62123
  1622
  "inf :: _ filter \<Rightarrow> _" "sup :: _ filter \<Rightarrow> _" "less_eq :: _ filter \<Rightarrow> _"
hoelzl@62123
  1623
  Abs_filter]]
hoelzl@62123
  1624
hoelzl@62123
  1625
declare filterlim_principal [code]
hoelzl@62123
  1626
declare principal_prod_principal [code]
hoelzl@62123
  1627
declare filtermap_principal [code]
eberlm@66162
  1628
declare filtercomap_principal [code]
hoelzl@62123
  1629
declare eventually_principal [code]
hoelzl@62123
  1630
declare inf_principal [code]
hoelzl@62123
  1631
declare sup_principal [code]
hoelzl@62123
  1632
declare principal_le_iff [code]
hoelzl@62123
  1633
hoelzl@62123
  1634
lemma Rep_filter_iff_eventually [simp, code]:
hoelzl@62123
  1635
  "Rep_filter F P \<longleftrightarrow> eventually P F"
hoelzl@62123
  1636
  by (simp add: eventually_def)
hoelzl@62123
  1637
hoelzl@62123
  1638
lemma bot_eq_principal_empty [code]:
hoelzl@62123
  1639
  "bot = principal {}"
hoelzl@62123
  1640
  by simp
hoelzl@62123
  1641
hoelzl@62123
  1642
lemma top_eq_principal_UNIV [code]:
hoelzl@62123
  1643
  "top = principal UNIV"
hoelzl@62123
  1644
  by simp
hoelzl@62123
  1645
hoelzl@62123
  1646
instantiation filter :: (equal) equal
hoelzl@62123
  1647
begin
hoelzl@62123
  1648
hoelzl@62123
  1649
definition equal_filter :: "'a filter \<Rightarrow> 'a filter \<Rightarrow> bool"
hoelzl@62123
  1650
  where "equal_filter F F' \<longleftrightarrow> F = F'"
hoelzl@62123
  1651
hoelzl@62123
  1652
lemma equal_filter [code]:
hoelzl@62123
  1653
  "HOL.equal (principal A) (principal B) \<longleftrightarrow> A = B"
hoelzl@62123
  1654
  by (simp add: equal_filter_def)
hoelzl@62123
  1655
hoelzl@62123
  1656
instance
hoelzl@62123
  1657
  by standard (simp add: equal_filter_def)
hoelzl@62123
  1658
lp15@61806
  1659
end
hoelzl@62123
  1660
hoelzl@62123
  1661
end