src/HOL/Limits.thy
author wenzelm
Thu May 24 17:25:53 2012 +0200 (2012-05-24)
changeset 47988 e4b69e10b990
parent 47432 e1576d13e933
child 49834 b27bbb021df1
permissions -rw-r--r--
tuned proofs;
huffman@31349
     1
(*  Title       : Limits.thy
huffman@31349
     2
    Author      : Brian Huffman
huffman@31349
     3
*)
huffman@31349
     4
huffman@31349
     5
header {* Filters and Limits *}
huffman@31349
     6
huffman@31349
     7
theory Limits
huffman@36822
     8
imports RealVector
huffman@31349
     9
begin
huffman@31349
    10
huffman@44081
    11
subsection {* Filters *}
huffman@31392
    12
huffman@31392
    13
text {*
huffman@44081
    14
  This definition also allows non-proper filters.
huffman@31392
    15
*}
huffman@31392
    16
huffman@36358
    17
locale is_filter =
huffman@44081
    18
  fixes F :: "('a \<Rightarrow> bool) \<Rightarrow> bool"
huffman@44081
    19
  assumes True: "F (\<lambda>x. True)"
huffman@44081
    20
  assumes conj: "F (\<lambda>x. P x) \<Longrightarrow> F (\<lambda>x. Q x) \<Longrightarrow> F (\<lambda>x. P x \<and> Q x)"
huffman@44081
    21
  assumes mono: "\<forall>x. P x \<longrightarrow> Q x \<Longrightarrow> F (\<lambda>x. P x) \<Longrightarrow> F (\<lambda>x. Q x)"
huffman@36358
    22
huffman@44081
    23
typedef (open) 'a filter = "{F :: ('a \<Rightarrow> bool) \<Rightarrow> bool. is_filter F}"
huffman@31392
    24
proof
huffman@44081
    25
  show "(\<lambda>x. True) \<in> ?filter" by (auto intro: is_filter.intro)
huffman@31392
    26
qed
huffman@31349
    27
huffman@44195
    28
lemma is_filter_Rep_filter: "is_filter (Rep_filter F)"
huffman@44195
    29
  using Rep_filter [of F] by simp
huffman@31392
    30
huffman@44081
    31
lemma Abs_filter_inverse':
huffman@44081
    32
  assumes "is_filter F" shows "Rep_filter (Abs_filter F) = F"
huffman@44081
    33
  using assms by (simp add: Abs_filter_inverse)
huffman@31392
    34
huffman@31392
    35
huffman@31392
    36
subsection {* Eventually *}
huffman@31349
    37
huffman@44081
    38
definition eventually :: "('a \<Rightarrow> bool) \<Rightarrow> 'a filter \<Rightarrow> bool"
huffman@44195
    39
  where "eventually P F \<longleftrightarrow> Rep_filter F P"
huffman@36358
    40
huffman@44081
    41
lemma eventually_Abs_filter:
huffman@44081
    42
  assumes "is_filter F" shows "eventually P (Abs_filter F) = F P"
huffman@44081
    43
  unfolding eventually_def using assms by (simp add: Abs_filter_inverse)
huffman@31349
    44
huffman@44081
    45
lemma filter_eq_iff:
huffman@44195
    46
  shows "F = F' \<longleftrightarrow> (\<forall>P. eventually P F = eventually P F')"
huffman@44081
    47
  unfolding Rep_filter_inject [symmetric] fun_eq_iff eventually_def ..
huffman@36360
    48
huffman@44195
    49
lemma eventually_True [simp]: "eventually (\<lambda>x. True) F"
huffman@44081
    50
  unfolding eventually_def
huffman@44081
    51
  by (rule is_filter.True [OF is_filter_Rep_filter])
huffman@31349
    52
huffman@44195
    53
lemma always_eventually: "\<forall>x. P x \<Longrightarrow> eventually P F"
huffman@36630
    54
proof -
huffman@36630
    55
  assume "\<forall>x. P x" hence "P = (\<lambda>x. True)" by (simp add: ext)
huffman@44195
    56
  thus "eventually P F" by simp
huffman@36630
    57
qed
huffman@36630
    58
huffman@31349
    59
lemma eventually_mono:
huffman@44195
    60
  "(\<forall>x. P x \<longrightarrow> Q x) \<Longrightarrow> eventually P F \<Longrightarrow> eventually Q F"
huffman@44081
    61
  unfolding eventually_def
huffman@44081
    62
  by (rule is_filter.mono [OF is_filter_Rep_filter])
huffman@31349
    63
huffman@31349
    64
lemma eventually_conj:
huffman@44195
    65
  assumes P: "eventually (\<lambda>x. P x) F"
huffman@44195
    66
  assumes Q: "eventually (\<lambda>x. Q x) F"
huffman@44195
    67
  shows "eventually (\<lambda>x. P x \<and> Q x) F"
huffman@44081
    68
  using assms unfolding eventually_def
huffman@44081
    69
  by (rule is_filter.conj [OF is_filter_Rep_filter])
huffman@31349
    70
huffman@31349
    71
lemma eventually_mp:
huffman@44195
    72
  assumes "eventually (\<lambda>x. P x \<longrightarrow> Q x) F"
huffman@44195
    73
  assumes "eventually (\<lambda>x. P x) F"
huffman@44195
    74
  shows "eventually (\<lambda>x. Q x) F"
huffman@31349
    75
proof (rule eventually_mono)
huffman@31349
    76
  show "\<forall>x. (P x \<longrightarrow> Q x) \<and> P x \<longrightarrow> Q x" by simp
huffman@44195
    77
  show "eventually (\<lambda>x. (P x \<longrightarrow> Q x) \<and> P x) F"
huffman@31349
    78
    using assms by (rule eventually_conj)
huffman@31349
    79
qed
huffman@31349
    80
huffman@31349
    81
lemma eventually_rev_mp:
huffman@44195
    82
  assumes "eventually (\<lambda>x. P x) F"
huffman@44195
    83
  assumes "eventually (\<lambda>x. P x \<longrightarrow> Q x) F"
huffman@44195
    84
  shows "eventually (\<lambda>x. Q x) F"
huffman@31349
    85
using assms(2) assms(1) by (rule eventually_mp)
huffman@31349
    86
huffman@31349
    87
lemma eventually_conj_iff:
huffman@44195
    88
  "eventually (\<lambda>x. P x \<and> Q x) F \<longleftrightarrow> eventually P F \<and> eventually Q F"
huffman@44081
    89
  by (auto intro: eventually_conj elim: eventually_rev_mp)
huffman@31349
    90
huffman@31349
    91
lemma eventually_elim1:
huffman@44195
    92
  assumes "eventually (\<lambda>i. P i) F"
huffman@31349
    93
  assumes "\<And>i. P i \<Longrightarrow> Q i"
huffman@44195
    94
  shows "eventually (\<lambda>i. Q i) F"
huffman@44081
    95
  using assms by (auto elim!: eventually_rev_mp)
huffman@31349
    96
huffman@31349
    97
lemma eventually_elim2:
huffman@44195
    98
  assumes "eventually (\<lambda>i. P i) F"
huffman@44195
    99
  assumes "eventually (\<lambda>i. Q i) F"
huffman@31349
   100
  assumes "\<And>i. P i \<Longrightarrow> Q i \<Longrightarrow> R i"
huffman@44195
   101
  shows "eventually (\<lambda>i. R i) F"
huffman@44081
   102
  using assms by (auto elim!: eventually_rev_mp)
huffman@31349
   103
noschinl@45892
   104
lemma eventually_subst:
noschinl@45892
   105
  assumes "eventually (\<lambda>n. P n = Q n) F"
noschinl@45892
   106
  shows "eventually P F = eventually Q F" (is "?L = ?R")
noschinl@45892
   107
proof -
noschinl@45892
   108
  from assms have "eventually (\<lambda>x. P x \<longrightarrow> Q x) F"
noschinl@45892
   109
      and "eventually (\<lambda>x. Q x \<longrightarrow> P x) F"
noschinl@45892
   110
    by (auto elim: eventually_elim1)
noschinl@45892
   111
  then show ?thesis by (auto elim: eventually_elim2)
noschinl@45892
   112
qed
noschinl@45892
   113
noschinl@46886
   114
ML {*
wenzelm@47432
   115
  fun eventually_elim_tac ctxt thms thm =
wenzelm@47432
   116
    let
noschinl@46886
   117
      val thy = Proof_Context.theory_of ctxt
noschinl@46886
   118
      val mp_thms = thms RL [@{thm eventually_rev_mp}]
noschinl@46886
   119
      val raw_elim_thm =
noschinl@46886
   120
        (@{thm allI} RS @{thm always_eventually})
noschinl@46886
   121
        |> fold (fn thm1 => fn thm2 => thm2 RS thm1) mp_thms
noschinl@46886
   122
        |> fold (fn _ => fn thm => @{thm impI} RS thm) thms
noschinl@46886
   123
      val cases_prop = prop_of (raw_elim_thm RS thm)
noschinl@46886
   124
      val cases = (Rule_Cases.make_common (thy, cases_prop) [(("elim", []), [])])
noschinl@46886
   125
    in
noschinl@46886
   126
      CASES cases (rtac raw_elim_thm 1) thm
noschinl@46886
   127
    end
noschinl@46886
   128
*}
noschinl@46886
   129
wenzelm@47432
   130
method_setup eventually_elim = {*
wenzelm@47432
   131
  Scan.succeed (fn ctxt => METHOD_CASES (eventually_elim_tac ctxt))
wenzelm@47432
   132
*} "elimination of eventually quantifiers"
noschinl@45892
   133
noschinl@45892
   134
huffman@36360
   135
subsection {* Finer-than relation *}
huffman@36360
   136
huffman@44195
   137
text {* @{term "F \<le> F'"} means that filter @{term F} is finer than
huffman@44195
   138
filter @{term F'}. *}
huffman@36360
   139
huffman@44081
   140
instantiation filter :: (type) complete_lattice
huffman@36360
   141
begin
huffman@36360
   142
huffman@44081
   143
definition le_filter_def:
huffman@44195
   144
  "F \<le> F' \<longleftrightarrow> (\<forall>P. eventually P F' \<longrightarrow> eventually P F)"
huffman@36360
   145
huffman@36360
   146
definition
huffman@44195
   147
  "(F :: 'a filter) < F' \<longleftrightarrow> F \<le> F' \<and> \<not> F' \<le> F"
huffman@36360
   148
huffman@36360
   149
definition
huffman@44081
   150
  "top = Abs_filter (\<lambda>P. \<forall>x. P x)"
huffman@36630
   151
huffman@36630
   152
definition
huffman@44081
   153
  "bot = Abs_filter (\<lambda>P. True)"
huffman@36360
   154
huffman@36630
   155
definition
huffman@44195
   156
  "sup F F' = Abs_filter (\<lambda>P. eventually P F \<and> eventually P F')"
huffman@36630
   157
huffman@36630
   158
definition
huffman@44195
   159
  "inf F F' = Abs_filter
huffman@44195
   160
      (\<lambda>P. \<exists>Q R. eventually Q F \<and> eventually R F' \<and> (\<forall>x. Q x \<and> R x \<longrightarrow> P x))"
huffman@36630
   161
huffman@36630
   162
definition
huffman@44195
   163
  "Sup S = Abs_filter (\<lambda>P. \<forall>F\<in>S. eventually P F)"
huffman@36630
   164
huffman@36630
   165
definition
huffman@44195
   166
  "Inf S = Sup {F::'a filter. \<forall>F'\<in>S. F \<le> F'}"
huffman@36630
   167
huffman@36630
   168
lemma eventually_top [simp]: "eventually P top \<longleftrightarrow> (\<forall>x. P x)"
huffman@44081
   169
  unfolding top_filter_def
huffman@44081
   170
  by (rule eventually_Abs_filter, rule is_filter.intro, auto)
huffman@36630
   171
huffman@36629
   172
lemma eventually_bot [simp]: "eventually P bot"
huffman@44081
   173
  unfolding bot_filter_def
huffman@44081
   174
  by (subst eventually_Abs_filter, rule is_filter.intro, auto)
huffman@36360
   175
huffman@36630
   176
lemma eventually_sup:
huffman@44195
   177
  "eventually P (sup F F') \<longleftrightarrow> eventually P F \<and> eventually P F'"
huffman@44081
   178
  unfolding sup_filter_def
huffman@44081
   179
  by (rule eventually_Abs_filter, rule is_filter.intro)
huffman@44081
   180
     (auto elim!: eventually_rev_mp)
huffman@36630
   181
huffman@36630
   182
lemma eventually_inf:
huffman@44195
   183
  "eventually P (inf F F') \<longleftrightarrow>
huffman@44195
   184
   (\<exists>Q R. eventually Q F \<and> eventually R F' \<and> (\<forall>x. Q x \<and> R x \<longrightarrow> P x))"
huffman@44081
   185
  unfolding inf_filter_def
huffman@44081
   186
  apply (rule eventually_Abs_filter, rule is_filter.intro)
huffman@44081
   187
  apply (fast intro: eventually_True)
huffman@44081
   188
  apply clarify
huffman@44081
   189
  apply (intro exI conjI)
huffman@44081
   190
  apply (erule (1) eventually_conj)
huffman@44081
   191
  apply (erule (1) eventually_conj)
huffman@44081
   192
  apply simp
huffman@44081
   193
  apply auto
huffman@44081
   194
  done
huffman@36630
   195
huffman@36630
   196
lemma eventually_Sup:
huffman@44195
   197
  "eventually P (Sup S) \<longleftrightarrow> (\<forall>F\<in>S. eventually P F)"
huffman@44081
   198
  unfolding Sup_filter_def
huffman@44081
   199
  apply (rule eventually_Abs_filter, rule is_filter.intro)
huffman@44081
   200
  apply (auto intro: eventually_conj elim!: eventually_rev_mp)
huffman@44081
   201
  done
huffman@36630
   202
huffman@36360
   203
instance proof
huffman@44195
   204
  fix F F' F'' :: "'a filter" and S :: "'a filter set"
huffman@44195
   205
  { show "F < F' \<longleftrightarrow> F \<le> F' \<and> \<not> F' \<le> F"
huffman@44195
   206
    by (rule less_filter_def) }
huffman@44195
   207
  { show "F \<le> F"
huffman@44195
   208
    unfolding le_filter_def by simp }
huffman@44195
   209
  { assume "F \<le> F'" and "F' \<le> F''" thus "F \<le> F''"
huffman@44195
   210
    unfolding le_filter_def by simp }
huffman@44195
   211
  { assume "F \<le> F'" and "F' \<le> F" thus "F = F'"
huffman@44195
   212
    unfolding le_filter_def filter_eq_iff by fast }
huffman@44195
   213
  { show "F \<le> top"
huffman@44195
   214
    unfolding le_filter_def eventually_top by (simp add: always_eventually) }
huffman@44195
   215
  { show "bot \<le> F"
huffman@44195
   216
    unfolding le_filter_def by simp }
huffman@44195
   217
  { show "F \<le> sup F F'" and "F' \<le> sup F F'"
huffman@44195
   218
    unfolding le_filter_def eventually_sup by simp_all }
huffman@44195
   219
  { assume "F \<le> F''" and "F' \<le> F''" thus "sup F F' \<le> F''"
huffman@44195
   220
    unfolding le_filter_def eventually_sup by simp }
huffman@44195
   221
  { show "inf F F' \<le> F" and "inf F F' \<le> F'"
huffman@44195
   222
    unfolding le_filter_def eventually_inf by (auto intro: eventually_True) }
huffman@44195
   223
  { assume "F \<le> F'" and "F \<le> F''" thus "F \<le> inf F' F''"
huffman@44081
   224
    unfolding le_filter_def eventually_inf
huffman@44195
   225
    by (auto elim!: eventually_mono intro: eventually_conj) }
huffman@44195
   226
  { assume "F \<in> S" thus "F \<le> Sup S"
huffman@44195
   227
    unfolding le_filter_def eventually_Sup by simp }
huffman@44195
   228
  { assume "\<And>F. F \<in> S \<Longrightarrow> F \<le> F'" thus "Sup S \<le> F'"
huffman@44195
   229
    unfolding le_filter_def eventually_Sup by simp }
huffman@44195
   230
  { assume "F'' \<in> S" thus "Inf S \<le> F''"
huffman@44195
   231
    unfolding le_filter_def Inf_filter_def eventually_Sup Ball_def by simp }
huffman@44195
   232
  { assume "\<And>F'. F' \<in> S \<Longrightarrow> F \<le> F'" thus "F \<le> Inf S"
huffman@44195
   233
    unfolding le_filter_def Inf_filter_def eventually_Sup Ball_def by simp }
huffman@36360
   234
qed
huffman@36360
   235
huffman@36360
   236
end
huffman@36360
   237
huffman@44081
   238
lemma filter_leD:
huffman@44195
   239
  "F \<le> F' \<Longrightarrow> eventually P F' \<Longrightarrow> eventually P F"
huffman@44081
   240
  unfolding le_filter_def by simp
huffman@36360
   241
huffman@44081
   242
lemma filter_leI:
huffman@44195
   243
  "(\<And>P. eventually P F' \<Longrightarrow> eventually P F) \<Longrightarrow> F \<le> F'"
huffman@44081
   244
  unfolding le_filter_def by simp
huffman@36360
   245
huffman@36360
   246
lemma eventually_False:
huffman@44195
   247
  "eventually (\<lambda>x. False) F \<longleftrightarrow> F = bot"
huffman@44081
   248
  unfolding filter_eq_iff by (auto elim: eventually_rev_mp)
huffman@36360
   249
huffman@44342
   250
abbreviation (input) trivial_limit :: "'a filter \<Rightarrow> bool"
huffman@44342
   251
  where "trivial_limit F \<equiv> F = bot"
huffman@44342
   252
huffman@44342
   253
lemma trivial_limit_def: "trivial_limit F \<longleftrightarrow> eventually (\<lambda>x. False) F"
huffman@44342
   254
  by (rule eventually_False [symmetric])
huffman@44342
   255
huffman@44342
   256
huffman@44081
   257
subsection {* Map function for filters *}
huffman@36654
   258
huffman@44081
   259
definition filtermap :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a filter \<Rightarrow> 'b filter"
huffman@44195
   260
  where "filtermap f F = Abs_filter (\<lambda>P. eventually (\<lambda>x. P (f x)) F)"
huffman@36654
   261
huffman@44081
   262
lemma eventually_filtermap:
huffman@44195
   263
  "eventually P (filtermap f F) = eventually (\<lambda>x. P (f x)) F"
huffman@44081
   264
  unfolding filtermap_def
huffman@44081
   265
  apply (rule eventually_Abs_filter)
huffman@44081
   266
  apply (rule is_filter.intro)
huffman@44081
   267
  apply (auto elim!: eventually_rev_mp)
huffman@44081
   268
  done
huffman@36654
   269
huffman@44195
   270
lemma filtermap_ident: "filtermap (\<lambda>x. x) F = F"
huffman@44081
   271
  by (simp add: filter_eq_iff eventually_filtermap)
huffman@36654
   272
huffman@44081
   273
lemma filtermap_filtermap:
huffman@44195
   274
  "filtermap f (filtermap g F) = filtermap (\<lambda>x. f (g x)) F"
huffman@44081
   275
  by (simp add: filter_eq_iff eventually_filtermap)
huffman@36654
   276
huffman@44195
   277
lemma filtermap_mono: "F \<le> F' \<Longrightarrow> filtermap f F \<le> filtermap f F'"
huffman@44081
   278
  unfolding le_filter_def eventually_filtermap by simp
huffman@44081
   279
huffman@44081
   280
lemma filtermap_bot [simp]: "filtermap f bot = bot"
huffman@44081
   281
  by (simp add: filter_eq_iff eventually_filtermap)
huffman@36654
   282
huffman@36654
   283
huffman@36662
   284
subsection {* Sequentially *}
huffman@31392
   285
huffman@44081
   286
definition sequentially :: "nat filter"
huffman@44081
   287
  where "sequentially = Abs_filter (\<lambda>P. \<exists>k. \<forall>n\<ge>k. P n)"
huffman@31392
   288
huffman@36662
   289
lemma eventually_sequentially:
huffman@36662
   290
  "eventually P sequentially \<longleftrightarrow> (\<exists>N. \<forall>n\<ge>N. P n)"
huffman@36662
   291
unfolding sequentially_def
huffman@44081
   292
proof (rule eventually_Abs_filter, rule is_filter.intro)
huffman@36662
   293
  fix P Q :: "nat \<Rightarrow> bool"
huffman@36662
   294
  assume "\<exists>i. \<forall>n\<ge>i. P n" and "\<exists>j. \<forall>n\<ge>j. Q n"
huffman@36662
   295
  then obtain i j where "\<forall>n\<ge>i. P n" and "\<forall>n\<ge>j. Q n" by auto
huffman@36662
   296
  then have "\<forall>n\<ge>max i j. P n \<and> Q n" by simp
huffman@36662
   297
  then show "\<exists>k. \<forall>n\<ge>k. P n \<and> Q n" ..
huffman@36662
   298
qed auto
huffman@36662
   299
huffman@44342
   300
lemma sequentially_bot [simp, intro]: "sequentially \<noteq> bot"
huffman@44081
   301
  unfolding filter_eq_iff eventually_sequentially by auto
huffman@36662
   302
huffman@44342
   303
lemmas trivial_limit_sequentially = sequentially_bot
huffman@44342
   304
huffman@36662
   305
lemma eventually_False_sequentially [simp]:
huffman@36662
   306
  "\<not> eventually (\<lambda>n. False) sequentially"
huffman@44081
   307
  by (simp add: eventually_False)
huffman@36662
   308
huffman@36662
   309
lemma le_sequentially:
huffman@44195
   310
  "F \<le> sequentially \<longleftrightarrow> (\<forall>N. eventually (\<lambda>n. N \<le> n) F)"
huffman@44081
   311
  unfolding le_filter_def eventually_sequentially
huffman@44081
   312
  by (safe, fast, drule_tac x=N in spec, auto elim: eventually_rev_mp)
huffman@36662
   313
noschinl@45892
   314
lemma eventually_sequentiallyI:
noschinl@45892
   315
  assumes "\<And>x. c \<le> x \<Longrightarrow> P x"
noschinl@45892
   316
  shows "eventually P sequentially"
noschinl@45892
   317
using assms by (auto simp: eventually_sequentially)
noschinl@45892
   318
huffman@36662
   319
huffman@44081
   320
subsection {* Standard filters *}
huffman@36662
   321
huffman@44081
   322
definition within :: "'a filter \<Rightarrow> 'a set \<Rightarrow> 'a filter" (infixr "within" 70)
huffman@44195
   323
  where "F within S = Abs_filter (\<lambda>P. eventually (\<lambda>x. x \<in> S \<longrightarrow> P x) F)"
huffman@31392
   324
huffman@44206
   325
definition (in topological_space) nhds :: "'a \<Rightarrow> 'a filter"
huffman@44081
   326
  where "nhds a = Abs_filter (\<lambda>P. \<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x))"
huffman@36654
   327
huffman@44206
   328
definition (in topological_space) at :: "'a \<Rightarrow> 'a filter"
huffman@44081
   329
  where "at a = nhds a within - {a}"
huffman@31447
   330
huffman@31392
   331
lemma eventually_within:
huffman@44195
   332
  "eventually P (F within S) = eventually (\<lambda>x. x \<in> S \<longrightarrow> P x) F"
huffman@44081
   333
  unfolding within_def
huffman@44081
   334
  by (rule eventually_Abs_filter, rule is_filter.intro)
huffman@44081
   335
     (auto elim!: eventually_rev_mp)
huffman@31392
   336
huffman@45031
   337
lemma within_UNIV [simp]: "F within UNIV = F"
huffman@45031
   338
  unfolding filter_eq_iff eventually_within by simp
huffman@45031
   339
huffman@45031
   340
lemma within_empty [simp]: "F within {} = bot"
huffman@44081
   341
  unfolding filter_eq_iff eventually_within by simp
huffman@36360
   342
huffman@36654
   343
lemma eventually_nhds:
huffman@36654
   344
  "eventually P (nhds a) \<longleftrightarrow> (\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x))"
huffman@36654
   345
unfolding nhds_def
huffman@44081
   346
proof (rule eventually_Abs_filter, rule is_filter.intro)
huffman@36654
   347
  have "open UNIV \<and> a \<in> UNIV \<and> (\<forall>x\<in>UNIV. True)" by simp
huffman@36654
   348
  thus "\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. True)" by - rule
huffman@36358
   349
next
huffman@36358
   350
  fix P Q
huffman@36654
   351
  assume "\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x)"
huffman@36654
   352
     and "\<exists>T. open T \<and> a \<in> T \<and> (\<forall>x\<in>T. Q x)"
huffman@36358
   353
  then obtain S T where
huffman@36654
   354
    "open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x)"
huffman@36654
   355
    "open T \<and> a \<in> T \<and> (\<forall>x\<in>T. Q x)" by auto
huffman@36654
   356
  hence "open (S \<inter> T) \<and> a \<in> S \<inter> T \<and> (\<forall>x\<in>(S \<inter> T). P x \<and> Q x)"
huffman@36358
   357
    by (simp add: open_Int)
huffman@36654
   358
  thus "\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x \<and> Q x)" by - rule
huffman@36358
   359
qed auto
huffman@31447
   360
huffman@36656
   361
lemma eventually_nhds_metric:
huffman@36656
   362
  "eventually P (nhds a) \<longleftrightarrow> (\<exists>d>0. \<forall>x. dist x a < d \<longrightarrow> P x)"
huffman@36656
   363
unfolding eventually_nhds open_dist
huffman@31447
   364
apply safe
huffman@31447
   365
apply fast
huffman@31492
   366
apply (rule_tac x="{x. dist x a < d}" in exI, simp)
huffman@31447
   367
apply clarsimp
huffman@31447
   368
apply (rule_tac x="d - dist x a" in exI, clarsimp)
huffman@31447
   369
apply (simp only: less_diff_eq)
huffman@31447
   370
apply (erule le_less_trans [OF dist_triangle])
huffman@31447
   371
done
huffman@31447
   372
huffman@44571
   373
lemma nhds_neq_bot [simp]: "nhds a \<noteq> bot"
huffman@44571
   374
  unfolding trivial_limit_def eventually_nhds by simp
huffman@44571
   375
huffman@36656
   376
lemma eventually_at_topological:
huffman@36656
   377
  "eventually P (at a) \<longleftrightarrow> (\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. x \<noteq> a \<longrightarrow> P x))"
huffman@36656
   378
unfolding at_def eventually_within eventually_nhds by simp
huffman@36656
   379
huffman@36656
   380
lemma eventually_at:
huffman@36656
   381
  fixes a :: "'a::metric_space"
huffman@36656
   382
  shows "eventually P (at a) \<longleftrightarrow> (\<exists>d>0. \<forall>x. x \<noteq> a \<and> dist x a < d \<longrightarrow> P x)"
huffman@36656
   383
unfolding at_def eventually_within eventually_nhds_metric by auto
huffman@36656
   384
huffman@44571
   385
lemma at_eq_bot_iff: "at a = bot \<longleftrightarrow> open {a}"
huffman@44571
   386
  unfolding trivial_limit_def eventually_at_topological
huffman@44571
   387
  by (safe, case_tac "S = {a}", simp, fast, fast)
huffman@44571
   388
huffman@44571
   389
lemma at_neq_bot [simp]: "at (a::'a::perfect_space) \<noteq> bot"
huffman@44571
   390
  by (simp add: at_eq_bot_iff not_open_singleton)
huffman@44571
   391
huffman@31392
   392
huffman@31355
   393
subsection {* Boundedness *}
huffman@31355
   394
huffman@44081
   395
definition Bfun :: "('a \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a filter \<Rightarrow> bool"
huffman@44195
   396
  where "Bfun f F = (\<exists>K>0. eventually (\<lambda>x. norm (f x) \<le> K) F)"
huffman@31355
   397
huffman@31487
   398
lemma BfunI:
huffman@44195
   399
  assumes K: "eventually (\<lambda>x. norm (f x) \<le> K) F" shows "Bfun f F"
huffman@31355
   400
unfolding Bfun_def
huffman@31355
   401
proof (intro exI conjI allI)
huffman@31355
   402
  show "0 < max K 1" by simp
huffman@31355
   403
next
huffman@44195
   404
  show "eventually (\<lambda>x. norm (f x) \<le> max K 1) F"
huffman@31355
   405
    using K by (rule eventually_elim1, simp)
huffman@31355
   406
qed
huffman@31355
   407
huffman@31355
   408
lemma BfunE:
huffman@44195
   409
  assumes "Bfun f F"
huffman@44195
   410
  obtains B where "0 < B" and "eventually (\<lambda>x. norm (f x) \<le> B) F"
huffman@31355
   411
using assms unfolding Bfun_def by fast
huffman@31355
   412
huffman@31355
   413
huffman@31349
   414
subsection {* Convergence to Zero *}
huffman@31349
   415
huffman@44081
   416
definition Zfun :: "('a \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a filter \<Rightarrow> bool"
huffman@44195
   417
  where "Zfun f F = (\<forall>r>0. eventually (\<lambda>x. norm (f x) < r) F)"
huffman@31349
   418
huffman@31349
   419
lemma ZfunI:
huffman@44195
   420
  "(\<And>r. 0 < r \<Longrightarrow> eventually (\<lambda>x. norm (f x) < r) F) \<Longrightarrow> Zfun f F"
huffman@44081
   421
  unfolding Zfun_def by simp
huffman@31349
   422
huffman@31349
   423
lemma ZfunD:
huffman@44195
   424
  "\<lbrakk>Zfun f F; 0 < r\<rbrakk> \<Longrightarrow> eventually (\<lambda>x. norm (f x) < r) F"
huffman@44081
   425
  unfolding Zfun_def by simp
huffman@31349
   426
huffman@31355
   427
lemma Zfun_ssubst:
huffman@44195
   428
  "eventually (\<lambda>x. f x = g x) F \<Longrightarrow> Zfun g F \<Longrightarrow> Zfun f F"
huffman@44081
   429
  unfolding Zfun_def by (auto elim!: eventually_rev_mp)
huffman@31355
   430
huffman@44195
   431
lemma Zfun_zero: "Zfun (\<lambda>x. 0) F"
huffman@44081
   432
  unfolding Zfun_def by simp
huffman@31349
   433
huffman@44195
   434
lemma Zfun_norm_iff: "Zfun (\<lambda>x. norm (f x)) F = Zfun (\<lambda>x. f x) F"
huffman@44081
   435
  unfolding Zfun_def by simp
huffman@31349
   436
huffman@31349
   437
lemma Zfun_imp_Zfun:
huffman@44195
   438
  assumes f: "Zfun f F"
huffman@44195
   439
  assumes g: "eventually (\<lambda>x. norm (g x) \<le> norm (f x) * K) F"
huffman@44195
   440
  shows "Zfun (\<lambda>x. g x) F"
huffman@31349
   441
proof (cases)
huffman@31349
   442
  assume K: "0 < K"
huffman@31349
   443
  show ?thesis
huffman@31349
   444
  proof (rule ZfunI)
huffman@31349
   445
    fix r::real assume "0 < r"
huffman@31349
   446
    hence "0 < r / K"
huffman@31349
   447
      using K by (rule divide_pos_pos)
huffman@44195
   448
    then have "eventually (\<lambda>x. norm (f x) < r / K) F"
huffman@31487
   449
      using ZfunD [OF f] by fast
huffman@44195
   450
    with g show "eventually (\<lambda>x. norm (g x) < r) F"
noschinl@46887
   451
    proof eventually_elim
noschinl@46887
   452
      case (elim x)
huffman@31487
   453
      hence "norm (f x) * K < r"
huffman@31349
   454
        by (simp add: pos_less_divide_eq K)
noschinl@46887
   455
      thus ?case
noschinl@46887
   456
        by (simp add: order_le_less_trans [OF elim(1)])
huffman@31349
   457
    qed
huffman@31349
   458
  qed
huffman@31349
   459
next
huffman@31349
   460
  assume "\<not> 0 < K"
huffman@31349
   461
  hence K: "K \<le> 0" by (simp only: not_less)
huffman@31355
   462
  show ?thesis
huffman@31355
   463
  proof (rule ZfunI)
huffman@31355
   464
    fix r :: real
huffman@31355
   465
    assume "0 < r"
huffman@44195
   466
    from g show "eventually (\<lambda>x. norm (g x) < r) F"
noschinl@46887
   467
    proof eventually_elim
noschinl@46887
   468
      case (elim x)
noschinl@46887
   469
      also have "norm (f x) * K \<le> norm (f x) * 0"
huffman@31355
   470
        using K norm_ge_zero by (rule mult_left_mono)
noschinl@46887
   471
      finally show ?case
huffman@31355
   472
        using `0 < r` by simp
huffman@31355
   473
    qed
huffman@31355
   474
  qed
huffman@31349
   475
qed
huffman@31349
   476
huffman@44195
   477
lemma Zfun_le: "\<lbrakk>Zfun g F; \<forall>x. norm (f x) \<le> norm (g x)\<rbrakk> \<Longrightarrow> Zfun f F"
huffman@44081
   478
  by (erule_tac K="1" in Zfun_imp_Zfun, simp)
huffman@31349
   479
huffman@31349
   480
lemma Zfun_add:
huffman@44195
   481
  assumes f: "Zfun f F" and g: "Zfun g F"
huffman@44195
   482
  shows "Zfun (\<lambda>x. f x + g x) F"
huffman@31349
   483
proof (rule ZfunI)
huffman@31349
   484
  fix r::real assume "0 < r"
huffman@31349
   485
  hence r: "0 < r / 2" by simp
huffman@44195
   486
  have "eventually (\<lambda>x. norm (f x) < r/2) F"
huffman@31487
   487
    using f r by (rule ZfunD)
huffman@31349
   488
  moreover
huffman@44195
   489
  have "eventually (\<lambda>x. norm (g x) < r/2) F"
huffman@31487
   490
    using g r by (rule ZfunD)
huffman@31349
   491
  ultimately
huffman@44195
   492
  show "eventually (\<lambda>x. norm (f x + g x) < r) F"
noschinl@46887
   493
  proof eventually_elim
noschinl@46887
   494
    case (elim x)
huffman@31487
   495
    have "norm (f x + g x) \<le> norm (f x) + norm (g x)"
huffman@31349
   496
      by (rule norm_triangle_ineq)
huffman@31349
   497
    also have "\<dots> < r/2 + r/2"
noschinl@46887
   498
      using elim by (rule add_strict_mono)
noschinl@46887
   499
    finally show ?case
huffman@31349
   500
      by simp
huffman@31349
   501
  qed
huffman@31349
   502
qed
huffman@31349
   503
huffman@44195
   504
lemma Zfun_minus: "Zfun f F \<Longrightarrow> Zfun (\<lambda>x. - f x) F"
huffman@44081
   505
  unfolding Zfun_def by simp
huffman@31349
   506
huffman@44195
   507
lemma Zfun_diff: "\<lbrakk>Zfun f F; Zfun g F\<rbrakk> \<Longrightarrow> Zfun (\<lambda>x. f x - g x) F"
huffman@44081
   508
  by (simp only: diff_minus Zfun_add Zfun_minus)
huffman@31349
   509
huffman@31349
   510
lemma (in bounded_linear) Zfun:
huffman@44195
   511
  assumes g: "Zfun g F"
huffman@44195
   512
  shows "Zfun (\<lambda>x. f (g x)) F"
huffman@31349
   513
proof -
huffman@31349
   514
  obtain K where "\<And>x. norm (f x) \<le> norm x * K"
huffman@31349
   515
    using bounded by fast
huffman@44195
   516
  then have "eventually (\<lambda>x. norm (f (g x)) \<le> norm (g x) * K) F"
huffman@31355
   517
    by simp
huffman@31487
   518
  with g show ?thesis
huffman@31349
   519
    by (rule Zfun_imp_Zfun)
huffman@31349
   520
qed
huffman@31349
   521
huffman@31349
   522
lemma (in bounded_bilinear) Zfun:
huffman@44195
   523
  assumes f: "Zfun f F"
huffman@44195
   524
  assumes g: "Zfun g F"
huffman@44195
   525
  shows "Zfun (\<lambda>x. f x ** g x) F"
huffman@31349
   526
proof (rule ZfunI)
huffman@31349
   527
  fix r::real assume r: "0 < r"
huffman@31349
   528
  obtain K where K: "0 < K"
huffman@31349
   529
    and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"
huffman@31349
   530
    using pos_bounded by fast
huffman@31349
   531
  from K have K': "0 < inverse K"
huffman@31349
   532
    by (rule positive_imp_inverse_positive)
huffman@44195
   533
  have "eventually (\<lambda>x. norm (f x) < r) F"
huffman@31487
   534
    using f r by (rule ZfunD)
huffman@31349
   535
  moreover
huffman@44195
   536
  have "eventually (\<lambda>x. norm (g x) < inverse K) F"
huffman@31487
   537
    using g K' by (rule ZfunD)
huffman@31349
   538
  ultimately
huffman@44195
   539
  show "eventually (\<lambda>x. norm (f x ** g x) < r) F"
noschinl@46887
   540
  proof eventually_elim
noschinl@46887
   541
    case (elim x)
huffman@31487
   542
    have "norm (f x ** g x) \<le> norm (f x) * norm (g x) * K"
huffman@31349
   543
      by (rule norm_le)
huffman@31487
   544
    also have "norm (f x) * norm (g x) * K < r * inverse K * K"
noschinl@46887
   545
      by (intro mult_strict_right_mono mult_strict_mono' norm_ge_zero elim K)
huffman@31349
   546
    also from K have "r * inverse K * K = r"
huffman@31349
   547
      by simp
noschinl@46887
   548
    finally show ?case .
huffman@31349
   549
  qed
huffman@31349
   550
qed
huffman@31349
   551
huffman@31349
   552
lemma (in bounded_bilinear) Zfun_left:
huffman@44195
   553
  "Zfun f F \<Longrightarrow> Zfun (\<lambda>x. f x ** a) F"
huffman@44081
   554
  by (rule bounded_linear_left [THEN bounded_linear.Zfun])
huffman@31349
   555
huffman@31349
   556
lemma (in bounded_bilinear) Zfun_right:
huffman@44195
   557
  "Zfun f F \<Longrightarrow> Zfun (\<lambda>x. a ** f x) F"
huffman@44081
   558
  by (rule bounded_linear_right [THEN bounded_linear.Zfun])
huffman@31349
   559
huffman@44282
   560
lemmas Zfun_mult = bounded_bilinear.Zfun [OF bounded_bilinear_mult]
huffman@44282
   561
lemmas Zfun_mult_right = bounded_bilinear.Zfun_right [OF bounded_bilinear_mult]
huffman@44282
   562
lemmas Zfun_mult_left = bounded_bilinear.Zfun_left [OF bounded_bilinear_mult]
huffman@31349
   563
huffman@31349
   564
wenzelm@31902
   565
subsection {* Limits *}
huffman@31349
   566
huffman@44206
   567
definition (in topological_space)
huffman@44206
   568
  tendsto :: "('b \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'b filter \<Rightarrow> bool" (infixr "--->" 55) where
huffman@44195
   569
  "(f ---> l) F \<longleftrightarrow> (\<forall>S. open S \<longrightarrow> l \<in> S \<longrightarrow> eventually (\<lambda>x. f x \<in> S) F)"
huffman@31349
   570
noschinl@45892
   571
definition real_tendsto_inf :: "('a \<Rightarrow> real) \<Rightarrow> 'a filter \<Rightarrow> bool" where
noschinl@45892
   572
  "real_tendsto_inf f F \<equiv> \<forall>x. eventually (\<lambda>y. x < f y) F"
noschinl@45892
   573
wenzelm@31902
   574
ML {*
wenzelm@31902
   575
structure Tendsto_Intros = Named_Thms
wenzelm@31902
   576
(
wenzelm@45294
   577
  val name = @{binding tendsto_intros}
wenzelm@31902
   578
  val description = "introduction rules for tendsto"
wenzelm@31902
   579
)
huffman@31565
   580
*}
huffman@31565
   581
wenzelm@31902
   582
setup Tendsto_Intros.setup
huffman@31565
   583
huffman@44195
   584
lemma tendsto_mono: "F \<le> F' \<Longrightarrow> (f ---> l) F' \<Longrightarrow> (f ---> l) F"
huffman@44081
   585
  unfolding tendsto_def le_filter_def by fast
huffman@36656
   586
huffman@31488
   587
lemma topological_tendstoI:
huffman@44195
   588
  "(\<And>S. open S \<Longrightarrow> l \<in> S \<Longrightarrow> eventually (\<lambda>x. f x \<in> S) F)
huffman@44195
   589
    \<Longrightarrow> (f ---> l) F"
huffman@31349
   590
  unfolding tendsto_def by auto
huffman@31349
   591
huffman@31488
   592
lemma topological_tendstoD:
huffman@44195
   593
  "(f ---> l) F \<Longrightarrow> open S \<Longrightarrow> l \<in> S \<Longrightarrow> eventually (\<lambda>x. f x \<in> S) F"
huffman@31488
   594
  unfolding tendsto_def by auto
huffman@31488
   595
huffman@31488
   596
lemma tendstoI:
huffman@44195
   597
  assumes "\<And>e. 0 < e \<Longrightarrow> eventually (\<lambda>x. dist (f x) l < e) F"
huffman@44195
   598
  shows "(f ---> l) F"
huffman@44081
   599
  apply (rule topological_tendstoI)
huffman@44081
   600
  apply (simp add: open_dist)
huffman@44081
   601
  apply (drule (1) bspec, clarify)
huffman@44081
   602
  apply (drule assms)
huffman@44081
   603
  apply (erule eventually_elim1, simp)
huffman@44081
   604
  done
huffman@31488
   605
huffman@31349
   606
lemma tendstoD:
huffman@44195
   607
  "(f ---> l) F \<Longrightarrow> 0 < e \<Longrightarrow> eventually (\<lambda>x. dist (f x) l < e) F"
huffman@44081
   608
  apply (drule_tac S="{x. dist x l < e}" in topological_tendstoD)
huffman@44081
   609
  apply (clarsimp simp add: open_dist)
huffman@44081
   610
  apply (rule_tac x="e - dist x l" in exI, clarsimp)
huffman@44081
   611
  apply (simp only: less_diff_eq)
huffman@44081
   612
  apply (erule le_less_trans [OF dist_triangle])
huffman@44081
   613
  apply simp
huffman@44081
   614
  apply simp
huffman@44081
   615
  done
huffman@31488
   616
huffman@31488
   617
lemma tendsto_iff:
huffman@44195
   618
  "(f ---> l) F \<longleftrightarrow> (\<forall>e>0. eventually (\<lambda>x. dist (f x) l < e) F)"
huffman@44081
   619
  using tendstoI tendstoD by fast
huffman@31349
   620
huffman@44195
   621
lemma tendsto_Zfun_iff: "(f ---> a) F = Zfun (\<lambda>x. f x - a) F"
huffman@44081
   622
  by (simp only: tendsto_iff Zfun_def dist_norm)
huffman@31349
   623
huffman@45031
   624
lemma tendsto_bot [simp]: "(f ---> a) bot"
huffman@45031
   625
  unfolding tendsto_def by simp
huffman@45031
   626
huffman@31565
   627
lemma tendsto_ident_at [tendsto_intros]: "((\<lambda>x. x) ---> a) (at a)"
huffman@44081
   628
  unfolding tendsto_def eventually_at_topological by auto
huffman@31565
   629
huffman@31565
   630
lemma tendsto_ident_at_within [tendsto_intros]:
huffman@36655
   631
  "((\<lambda>x. x) ---> a) (at a within S)"
huffman@44081
   632
  unfolding tendsto_def eventually_within eventually_at_topological by auto
huffman@31565
   633
huffman@44195
   634
lemma tendsto_const [tendsto_intros]: "((\<lambda>x. k) ---> k) F"
huffman@44081
   635
  by (simp add: tendsto_def)
huffman@31349
   636
huffman@44205
   637
lemma tendsto_unique:
huffman@44205
   638
  fixes f :: "'a \<Rightarrow> 'b::t2_space"
huffman@44205
   639
  assumes "\<not> trivial_limit F" and "(f ---> a) F" and "(f ---> b) F"
huffman@44205
   640
  shows "a = b"
huffman@44205
   641
proof (rule ccontr)
huffman@44205
   642
  assume "a \<noteq> b"
huffman@44205
   643
  obtain U V where "open U" "open V" "a \<in> U" "b \<in> V" "U \<inter> V = {}"
huffman@44205
   644
    using hausdorff [OF `a \<noteq> b`] by fast
huffman@44205
   645
  have "eventually (\<lambda>x. f x \<in> U) F"
huffman@44205
   646
    using `(f ---> a) F` `open U` `a \<in> U` by (rule topological_tendstoD)
huffman@44205
   647
  moreover
huffman@44205
   648
  have "eventually (\<lambda>x. f x \<in> V) F"
huffman@44205
   649
    using `(f ---> b) F` `open V` `b \<in> V` by (rule topological_tendstoD)
huffman@44205
   650
  ultimately
huffman@44205
   651
  have "eventually (\<lambda>x. False) F"
noschinl@46887
   652
  proof eventually_elim
noschinl@46887
   653
    case (elim x)
huffman@44205
   654
    hence "f x \<in> U \<inter> V" by simp
noschinl@46887
   655
    with `U \<inter> V = {}` show ?case by simp
huffman@44205
   656
  qed
huffman@44205
   657
  with `\<not> trivial_limit F` show "False"
huffman@44205
   658
    by (simp add: trivial_limit_def)
huffman@44205
   659
qed
huffman@44205
   660
huffman@36662
   661
lemma tendsto_const_iff:
huffman@44205
   662
  fixes a b :: "'a::t2_space"
huffman@44205
   663
  assumes "\<not> trivial_limit F" shows "((\<lambda>x. a) ---> b) F \<longleftrightarrow> a = b"
huffman@44205
   664
  by (safe intro!: tendsto_const tendsto_unique [OF assms tendsto_const])
huffman@44205
   665
huffman@44218
   666
lemma tendsto_compose:
huffman@44218
   667
  assumes g: "(g ---> g l) (at l)"
huffman@44218
   668
  assumes f: "(f ---> l) F"
huffman@44218
   669
  shows "((\<lambda>x. g (f x)) ---> g l) F"
huffman@44218
   670
proof (rule topological_tendstoI)
huffman@44218
   671
  fix B assume B: "open B" "g l \<in> B"
huffman@44218
   672
  obtain A where A: "open A" "l \<in> A"
huffman@44218
   673
    and gB: "\<forall>y. y \<in> A \<longrightarrow> g y \<in> B"
huffman@44218
   674
    using topological_tendstoD [OF g B] B(2)
huffman@44218
   675
    unfolding eventually_at_topological by fast
huffman@44218
   676
  hence "\<forall>x. f x \<in> A \<longrightarrow> g (f x) \<in> B" by simp
huffman@44218
   677
  from this topological_tendstoD [OF f A]
huffman@44218
   678
  show "eventually (\<lambda>x. g (f x) \<in> B) F"
huffman@44218
   679
    by (rule eventually_mono)
huffman@44218
   680
qed
huffman@44218
   681
huffman@44253
   682
lemma tendsto_compose_eventually:
huffman@44253
   683
  assumes g: "(g ---> m) (at l)"
huffman@44253
   684
  assumes f: "(f ---> l) F"
huffman@44253
   685
  assumes inj: "eventually (\<lambda>x. f x \<noteq> l) F"
huffman@44253
   686
  shows "((\<lambda>x. g (f x)) ---> m) F"
huffman@44253
   687
proof (rule topological_tendstoI)
huffman@44253
   688
  fix B assume B: "open B" "m \<in> B"
huffman@44253
   689
  obtain A where A: "open A" "l \<in> A"
huffman@44253
   690
    and gB: "\<And>y. y \<in> A \<Longrightarrow> y \<noteq> l \<Longrightarrow> g y \<in> B"
huffman@44253
   691
    using topological_tendstoD [OF g B]
huffman@44253
   692
    unfolding eventually_at_topological by fast
huffman@44253
   693
  show "eventually (\<lambda>x. g (f x) \<in> B) F"
huffman@44253
   694
    using topological_tendstoD [OF f A] inj
huffman@44253
   695
    by (rule eventually_elim2) (simp add: gB)
huffman@44253
   696
qed
huffman@44253
   697
huffman@44251
   698
lemma metric_tendsto_imp_tendsto:
huffman@44251
   699
  assumes f: "(f ---> a) F"
huffman@44251
   700
  assumes le: "eventually (\<lambda>x. dist (g x) b \<le> dist (f x) a) F"
huffman@44251
   701
  shows "(g ---> b) F"
huffman@44251
   702
proof (rule tendstoI)
huffman@44251
   703
  fix e :: real assume "0 < e"
huffman@44251
   704
  with f have "eventually (\<lambda>x. dist (f x) a < e) F" by (rule tendstoD)
huffman@44251
   705
  with le show "eventually (\<lambda>x. dist (g x) b < e) F"
huffman@44251
   706
    using le_less_trans by (rule eventually_elim2)
huffman@44251
   707
qed
huffman@44251
   708
noschinl@45892
   709
lemma real_tendsto_inf_real: "real_tendsto_inf real sequentially"
noschinl@45892
   710
proof (unfold real_tendsto_inf_def, rule allI)
noschinl@45892
   711
  fix x show "eventually (\<lambda>y. x < real y) sequentially"
noschinl@45892
   712
    by (rule eventually_sequentiallyI[of "natceiling (x + 1)"])
noschinl@45892
   713
        (simp add: natceiling_le_eq)
noschinl@45892
   714
qed
noschinl@45892
   715
noschinl@45892
   716
noschinl@45892
   717
huffman@44205
   718
subsubsection {* Distance and norms *}
huffman@36662
   719
huffman@31565
   720
lemma tendsto_dist [tendsto_intros]:
huffman@44195
   721
  assumes f: "(f ---> l) F" and g: "(g ---> m) F"
huffman@44195
   722
  shows "((\<lambda>x. dist (f x) (g x)) ---> dist l m) F"
huffman@31565
   723
proof (rule tendstoI)
huffman@31565
   724
  fix e :: real assume "0 < e"
huffman@31565
   725
  hence e2: "0 < e/2" by simp
huffman@31565
   726
  from tendstoD [OF f e2] tendstoD [OF g e2]
huffman@44195
   727
  show "eventually (\<lambda>x. dist (dist (f x) (g x)) (dist l m) < e) F"
noschinl@46887
   728
  proof (eventually_elim)
noschinl@46887
   729
    case (elim x)
huffman@31565
   730
    then show "dist (dist (f x) (g x)) (dist l m) < e"
huffman@31565
   731
      unfolding dist_real_def
huffman@31565
   732
      using dist_triangle2 [of "f x" "g x" "l"]
huffman@31565
   733
      using dist_triangle2 [of "g x" "l" "m"]
huffman@31565
   734
      using dist_triangle3 [of "l" "m" "f x"]
huffman@31565
   735
      using dist_triangle [of "f x" "m" "g x"]
huffman@31565
   736
      by arith
huffman@31565
   737
  qed
huffman@31565
   738
qed
huffman@31565
   739
huffman@36662
   740
lemma norm_conv_dist: "norm x = dist x 0"
huffman@44081
   741
  unfolding dist_norm by simp
huffman@36662
   742
huffman@31565
   743
lemma tendsto_norm [tendsto_intros]:
huffman@44195
   744
  "(f ---> a) F \<Longrightarrow> ((\<lambda>x. norm (f x)) ---> norm a) F"
huffman@44081
   745
  unfolding norm_conv_dist by (intro tendsto_intros)
huffman@36662
   746
huffman@36662
   747
lemma tendsto_norm_zero:
huffman@44195
   748
  "(f ---> 0) F \<Longrightarrow> ((\<lambda>x. norm (f x)) ---> 0) F"
huffman@44081
   749
  by (drule tendsto_norm, simp)
huffman@36662
   750
huffman@36662
   751
lemma tendsto_norm_zero_cancel:
huffman@44195
   752
  "((\<lambda>x. norm (f x)) ---> 0) F \<Longrightarrow> (f ---> 0) F"
huffman@44081
   753
  unfolding tendsto_iff dist_norm by simp
huffman@36662
   754
huffman@36662
   755
lemma tendsto_norm_zero_iff:
huffman@44195
   756
  "((\<lambda>x. norm (f x)) ---> 0) F \<longleftrightarrow> (f ---> 0) F"
huffman@44081
   757
  unfolding tendsto_iff dist_norm by simp
huffman@31349
   758
huffman@44194
   759
lemma tendsto_rabs [tendsto_intros]:
huffman@44195
   760
  "(f ---> (l::real)) F \<Longrightarrow> ((\<lambda>x. \<bar>f x\<bar>) ---> \<bar>l\<bar>) F"
huffman@44194
   761
  by (fold real_norm_def, rule tendsto_norm)
huffman@44194
   762
huffman@44194
   763
lemma tendsto_rabs_zero:
huffman@44195
   764
  "(f ---> (0::real)) F \<Longrightarrow> ((\<lambda>x. \<bar>f x\<bar>) ---> 0) F"
huffman@44194
   765
  by (fold real_norm_def, rule tendsto_norm_zero)
huffman@44194
   766
huffman@44194
   767
lemma tendsto_rabs_zero_cancel:
huffman@44195
   768
  "((\<lambda>x. \<bar>f x\<bar>) ---> (0::real)) F \<Longrightarrow> (f ---> 0) F"
huffman@44194
   769
  by (fold real_norm_def, rule tendsto_norm_zero_cancel)
huffman@44194
   770
huffman@44194
   771
lemma tendsto_rabs_zero_iff:
huffman@44195
   772
  "((\<lambda>x. \<bar>f x\<bar>) ---> (0::real)) F \<longleftrightarrow> (f ---> 0) F"
huffman@44194
   773
  by (fold real_norm_def, rule tendsto_norm_zero_iff)
huffman@44194
   774
huffman@44194
   775
subsubsection {* Addition and subtraction *}
huffman@44194
   776
huffman@31565
   777
lemma tendsto_add [tendsto_intros]:
huffman@31349
   778
  fixes a b :: "'a::real_normed_vector"
huffman@44195
   779
  shows "\<lbrakk>(f ---> a) F; (g ---> b) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x + g x) ---> a + b) F"
huffman@44081
   780
  by (simp only: tendsto_Zfun_iff add_diff_add Zfun_add)
huffman@31349
   781
huffman@44194
   782
lemma tendsto_add_zero:
huffman@44194
   783
  fixes f g :: "'a::type \<Rightarrow> 'b::real_normed_vector"
huffman@44195
   784
  shows "\<lbrakk>(f ---> 0) F; (g ---> 0) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x + g x) ---> 0) F"
huffman@44194
   785
  by (drule (1) tendsto_add, simp)
huffman@44194
   786
huffman@31565
   787
lemma tendsto_minus [tendsto_intros]:
huffman@31349
   788
  fixes a :: "'a::real_normed_vector"
huffman@44195
   789
  shows "(f ---> a) F \<Longrightarrow> ((\<lambda>x. - f x) ---> - a) F"
huffman@44081
   790
  by (simp only: tendsto_Zfun_iff minus_diff_minus Zfun_minus)
huffman@31349
   791
huffman@31349
   792
lemma tendsto_minus_cancel:
huffman@31349
   793
  fixes a :: "'a::real_normed_vector"
huffman@44195
   794
  shows "((\<lambda>x. - f x) ---> - a) F \<Longrightarrow> (f ---> a) F"
huffman@44081
   795
  by (drule tendsto_minus, simp)
huffman@31349
   796
huffman@31565
   797
lemma tendsto_diff [tendsto_intros]:
huffman@31349
   798
  fixes a b :: "'a::real_normed_vector"
huffman@44195
   799
  shows "\<lbrakk>(f ---> a) F; (g ---> b) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x - g x) ---> a - b) F"
huffman@44081
   800
  by (simp add: diff_minus tendsto_add tendsto_minus)
huffman@31349
   801
huffman@31588
   802
lemma tendsto_setsum [tendsto_intros]:
huffman@31588
   803
  fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c::real_normed_vector"
huffman@44195
   804
  assumes "\<And>i. i \<in> S \<Longrightarrow> (f i ---> a i) F"
huffman@44195
   805
  shows "((\<lambda>x. \<Sum>i\<in>S. f i x) ---> (\<Sum>i\<in>S. a i)) F"
huffman@31588
   806
proof (cases "finite S")
huffman@31588
   807
  assume "finite S" thus ?thesis using assms
huffman@44194
   808
    by (induct, simp add: tendsto_const, simp add: tendsto_add)
huffman@31588
   809
next
huffman@31588
   810
  assume "\<not> finite S" thus ?thesis
huffman@31588
   811
    by (simp add: tendsto_const)
huffman@31588
   812
qed
huffman@31588
   813
noschinl@45892
   814
lemma real_tendsto_sandwich:
noschinl@45892
   815
  fixes f g h :: "'a \<Rightarrow> real"
noschinl@45892
   816
  assumes ev: "eventually (\<lambda>n. f n \<le> g n) net" "eventually (\<lambda>n. g n \<le> h n) net"
noschinl@45892
   817
  assumes lim: "(f ---> c) net" "(h ---> c) net"
noschinl@45892
   818
  shows "(g ---> c) net"
noschinl@45892
   819
proof -
noschinl@45892
   820
  have "((\<lambda>n. g n - f n) ---> 0) net"
noschinl@45892
   821
  proof (rule metric_tendsto_imp_tendsto)
noschinl@45892
   822
    show "eventually (\<lambda>n. dist (g n - f n) 0 \<le> dist (h n - f n) 0) net"
noschinl@45892
   823
      using ev by (rule eventually_elim2) (simp add: dist_real_def)
noschinl@45892
   824
    show "((\<lambda>n. h n - f n) ---> 0) net"
noschinl@45892
   825
      using tendsto_diff[OF lim(2,1)] by simp
noschinl@45892
   826
  qed
noschinl@45892
   827
  from tendsto_add[OF this lim(1)] show ?thesis by simp
noschinl@45892
   828
qed
noschinl@45892
   829
huffman@44194
   830
subsubsection {* Linear operators and multiplication *}
huffman@44194
   831
huffman@44282
   832
lemma (in bounded_linear) tendsto:
huffman@44195
   833
  "(g ---> a) F \<Longrightarrow> ((\<lambda>x. f (g x)) ---> f a) F"
huffman@44081
   834
  by (simp only: tendsto_Zfun_iff diff [symmetric] Zfun)
huffman@31349
   835
huffman@44194
   836
lemma (in bounded_linear) tendsto_zero:
huffman@44195
   837
  "(g ---> 0) F \<Longrightarrow> ((\<lambda>x. f (g x)) ---> 0) F"
huffman@44194
   838
  by (drule tendsto, simp only: zero)
huffman@44194
   839
huffman@44282
   840
lemma (in bounded_bilinear) tendsto:
huffman@44195
   841
  "\<lbrakk>(f ---> a) F; (g ---> b) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x ** g x) ---> a ** b) F"
huffman@44081
   842
  by (simp only: tendsto_Zfun_iff prod_diff_prod
huffman@44081
   843
                 Zfun_add Zfun Zfun_left Zfun_right)
huffman@31349
   844
huffman@44194
   845
lemma (in bounded_bilinear) tendsto_zero:
huffman@44195
   846
  assumes f: "(f ---> 0) F"
huffman@44195
   847
  assumes g: "(g ---> 0) F"
huffman@44195
   848
  shows "((\<lambda>x. f x ** g x) ---> 0) F"
huffman@44194
   849
  using tendsto [OF f g] by (simp add: zero_left)
huffman@31355
   850
huffman@44194
   851
lemma (in bounded_bilinear) tendsto_left_zero:
huffman@44195
   852
  "(f ---> 0) F \<Longrightarrow> ((\<lambda>x. f x ** c) ---> 0) F"
huffman@44194
   853
  by (rule bounded_linear.tendsto_zero [OF bounded_linear_left])
huffman@44194
   854
huffman@44194
   855
lemma (in bounded_bilinear) tendsto_right_zero:
huffman@44195
   856
  "(f ---> 0) F \<Longrightarrow> ((\<lambda>x. c ** f x) ---> 0) F"
huffman@44194
   857
  by (rule bounded_linear.tendsto_zero [OF bounded_linear_right])
huffman@44194
   858
huffman@44282
   859
lemmas tendsto_of_real [tendsto_intros] =
huffman@44282
   860
  bounded_linear.tendsto [OF bounded_linear_of_real]
huffman@44282
   861
huffman@44282
   862
lemmas tendsto_scaleR [tendsto_intros] =
huffman@44282
   863
  bounded_bilinear.tendsto [OF bounded_bilinear_scaleR]
huffman@44282
   864
huffman@44282
   865
lemmas tendsto_mult [tendsto_intros] =
huffman@44282
   866
  bounded_bilinear.tendsto [OF bounded_bilinear_mult]
huffman@44194
   867
huffman@44568
   868
lemmas tendsto_mult_zero =
huffman@44568
   869
  bounded_bilinear.tendsto_zero [OF bounded_bilinear_mult]
huffman@44568
   870
huffman@44568
   871
lemmas tendsto_mult_left_zero =
huffman@44568
   872
  bounded_bilinear.tendsto_left_zero [OF bounded_bilinear_mult]
huffman@44568
   873
huffman@44568
   874
lemmas tendsto_mult_right_zero =
huffman@44568
   875
  bounded_bilinear.tendsto_right_zero [OF bounded_bilinear_mult]
huffman@44568
   876
huffman@44194
   877
lemma tendsto_power [tendsto_intros]:
huffman@44194
   878
  fixes f :: "'a \<Rightarrow> 'b::{power,real_normed_algebra}"
huffman@44195
   879
  shows "(f ---> a) F \<Longrightarrow> ((\<lambda>x. f x ^ n) ---> a ^ n) F"
huffman@44194
   880
  by (induct n) (simp_all add: tendsto_const tendsto_mult)
huffman@44194
   881
huffman@44194
   882
lemma tendsto_setprod [tendsto_intros]:
huffman@44194
   883
  fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c::{real_normed_algebra,comm_ring_1}"
huffman@44195
   884
  assumes "\<And>i. i \<in> S \<Longrightarrow> (f i ---> L i) F"
huffman@44195
   885
  shows "((\<lambda>x. \<Prod>i\<in>S. f i x) ---> (\<Prod>i\<in>S. L i)) F"
huffman@44194
   886
proof (cases "finite S")
huffman@44194
   887
  assume "finite S" thus ?thesis using assms
huffman@44194
   888
    by (induct, simp add: tendsto_const, simp add: tendsto_mult)
huffman@44194
   889
next
huffman@44194
   890
  assume "\<not> finite S" thus ?thesis
huffman@44194
   891
    by (simp add: tendsto_const)
huffman@44194
   892
qed
huffman@44194
   893
huffman@44194
   894
subsubsection {* Inverse and division *}
huffman@31355
   895
huffman@31355
   896
lemma (in bounded_bilinear) Zfun_prod_Bfun:
huffman@44195
   897
  assumes f: "Zfun f F"
huffman@44195
   898
  assumes g: "Bfun g F"
huffman@44195
   899
  shows "Zfun (\<lambda>x. f x ** g x) F"
huffman@31355
   900
proof -
huffman@31355
   901
  obtain K where K: "0 \<le> K"
huffman@31355
   902
    and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"
huffman@31355
   903
    using nonneg_bounded by fast
huffman@31355
   904
  obtain B where B: "0 < B"
huffman@44195
   905
    and norm_g: "eventually (\<lambda>x. norm (g x) \<le> B) F"
huffman@31487
   906
    using g by (rule BfunE)
huffman@44195
   907
  have "eventually (\<lambda>x. norm (f x ** g x) \<le> norm (f x) * (B * K)) F"
noschinl@46887
   908
  using norm_g proof eventually_elim
noschinl@46887
   909
    case (elim x)
huffman@31487
   910
    have "norm (f x ** g x) \<le> norm (f x) * norm (g x) * K"
huffman@31355
   911
      by (rule norm_le)
huffman@31487
   912
    also have "\<dots> \<le> norm (f x) * B * K"
huffman@31487
   913
      by (intro mult_mono' order_refl norm_g norm_ge_zero
noschinl@46887
   914
                mult_nonneg_nonneg K elim)
huffman@31487
   915
    also have "\<dots> = norm (f x) * (B * K)"
huffman@31355
   916
      by (rule mult_assoc)
huffman@31487
   917
    finally show "norm (f x ** g x) \<le> norm (f x) * (B * K)" .
huffman@31355
   918
  qed
huffman@31487
   919
  with f show ?thesis
huffman@31487
   920
    by (rule Zfun_imp_Zfun)
huffman@31355
   921
qed
huffman@31355
   922
huffman@31355
   923
lemma (in bounded_bilinear) flip:
huffman@31355
   924
  "bounded_bilinear (\<lambda>x y. y ** x)"
huffman@44081
   925
  apply default
huffman@44081
   926
  apply (rule add_right)
huffman@44081
   927
  apply (rule add_left)
huffman@44081
   928
  apply (rule scaleR_right)
huffman@44081
   929
  apply (rule scaleR_left)
huffman@44081
   930
  apply (subst mult_commute)
huffman@44081
   931
  using bounded by fast
huffman@31355
   932
huffman@31355
   933
lemma (in bounded_bilinear) Bfun_prod_Zfun:
huffman@44195
   934
  assumes f: "Bfun f F"
huffman@44195
   935
  assumes g: "Zfun g F"
huffman@44195
   936
  shows "Zfun (\<lambda>x. f x ** g x) F"
huffman@44081
   937
  using flip g f by (rule bounded_bilinear.Zfun_prod_Bfun)
huffman@31355
   938
huffman@31355
   939
lemma Bfun_inverse_lemma:
huffman@31355
   940
  fixes x :: "'a::real_normed_div_algebra"
huffman@31355
   941
  shows "\<lbrakk>r \<le> norm x; 0 < r\<rbrakk> \<Longrightarrow> norm (inverse x) \<le> inverse r"
huffman@44081
   942
  apply (subst nonzero_norm_inverse, clarsimp)
huffman@44081
   943
  apply (erule (1) le_imp_inverse_le)
huffman@44081
   944
  done
huffman@31355
   945
huffman@31355
   946
lemma Bfun_inverse:
huffman@31355
   947
  fixes a :: "'a::real_normed_div_algebra"
huffman@44195
   948
  assumes f: "(f ---> a) F"
huffman@31355
   949
  assumes a: "a \<noteq> 0"
huffman@44195
   950
  shows "Bfun (\<lambda>x. inverse (f x)) F"
huffman@31355
   951
proof -
huffman@31355
   952
  from a have "0 < norm a" by simp
huffman@31355
   953
  hence "\<exists>r>0. r < norm a" by (rule dense)
huffman@31355
   954
  then obtain r where r1: "0 < r" and r2: "r < norm a" by fast
huffman@44195
   955
  have "eventually (\<lambda>x. dist (f x) a < r) F"
huffman@31487
   956
    using tendstoD [OF f r1] by fast
huffman@44195
   957
  hence "eventually (\<lambda>x. norm (inverse (f x)) \<le> inverse (norm a - r)) F"
noschinl@46887
   958
  proof eventually_elim
noschinl@46887
   959
    case (elim x)
huffman@31487
   960
    hence 1: "norm (f x - a) < r"
huffman@31355
   961
      by (simp add: dist_norm)
huffman@31487
   962
    hence 2: "f x \<noteq> 0" using r2 by auto
huffman@31487
   963
    hence "norm (inverse (f x)) = inverse (norm (f x))"
huffman@31355
   964
      by (rule nonzero_norm_inverse)
huffman@31355
   965
    also have "\<dots> \<le> inverse (norm a - r)"
huffman@31355
   966
    proof (rule le_imp_inverse_le)
huffman@31355
   967
      show "0 < norm a - r" using r2 by simp
huffman@31355
   968
    next
huffman@31487
   969
      have "norm a - norm (f x) \<le> norm (a - f x)"
huffman@31355
   970
        by (rule norm_triangle_ineq2)
huffman@31487
   971
      also have "\<dots> = norm (f x - a)"
huffman@31355
   972
        by (rule norm_minus_commute)
huffman@31355
   973
      also have "\<dots> < r" using 1 .
huffman@31487
   974
      finally show "norm a - r \<le> norm (f x)" by simp
huffman@31355
   975
    qed
huffman@31487
   976
    finally show "norm (inverse (f x)) \<le> inverse (norm a - r)" .
huffman@31355
   977
  qed
huffman@31355
   978
  thus ?thesis by (rule BfunI)
huffman@31355
   979
qed
huffman@31355
   980
huffman@31565
   981
lemma tendsto_inverse [tendsto_intros]:
huffman@31355
   982
  fixes a :: "'a::real_normed_div_algebra"
huffman@44195
   983
  assumes f: "(f ---> a) F"
huffman@31355
   984
  assumes a: "a \<noteq> 0"
huffman@44195
   985
  shows "((\<lambda>x. inverse (f x)) ---> inverse a) F"
huffman@31355
   986
proof -
huffman@31355
   987
  from a have "0 < norm a" by simp
huffman@44195
   988
  with f have "eventually (\<lambda>x. dist (f x) a < norm a) F"
huffman@31355
   989
    by (rule tendstoD)
huffman@44195
   990
  then have "eventually (\<lambda>x. f x \<noteq> 0) F"
huffman@31355
   991
    unfolding dist_norm by (auto elim!: eventually_elim1)
huffman@44627
   992
  with a have "eventually (\<lambda>x. inverse (f x) - inverse a =
huffman@44627
   993
    - (inverse (f x) * (f x - a) * inverse a)) F"
huffman@44627
   994
    by (auto elim!: eventually_elim1 simp: inverse_diff_inverse)
huffman@44627
   995
  moreover have "Zfun (\<lambda>x. - (inverse (f x) * (f x - a) * inverse a)) F"
huffman@44627
   996
    by (intro Zfun_minus Zfun_mult_left
huffman@44627
   997
      bounded_bilinear.Bfun_prod_Zfun [OF bounded_bilinear_mult]
huffman@44627
   998
      Bfun_inverse [OF f a] f [unfolded tendsto_Zfun_iff])
huffman@44627
   999
  ultimately show ?thesis
huffman@44627
  1000
    unfolding tendsto_Zfun_iff by (rule Zfun_ssubst)
huffman@31355
  1001
qed
huffman@31355
  1002
huffman@31565
  1003
lemma tendsto_divide [tendsto_intros]:
huffman@31355
  1004
  fixes a b :: "'a::real_normed_field"
huffman@44195
  1005
  shows "\<lbrakk>(f ---> a) F; (g ---> b) F; b \<noteq> 0\<rbrakk>
huffman@44195
  1006
    \<Longrightarrow> ((\<lambda>x. f x / g x) ---> a / b) F"
huffman@44282
  1007
  by (simp add: tendsto_mult tendsto_inverse divide_inverse)
huffman@31355
  1008
huffman@44194
  1009
lemma tendsto_sgn [tendsto_intros]:
huffman@44194
  1010
  fixes l :: "'a::real_normed_vector"
huffman@44195
  1011
  shows "\<lbrakk>(f ---> l) F; l \<noteq> 0\<rbrakk> \<Longrightarrow> ((\<lambda>x. sgn (f x)) ---> sgn l) F"
huffman@44194
  1012
  unfolding sgn_div_norm by (simp add: tendsto_intros)
huffman@44194
  1013
huffman@31349
  1014
end