src/HOL/Limits.thy
author hoelzl
Mon Dec 03 18:19:01 2012 +0100 (2012-12-03)
changeset 50323 3764d4620fb3
parent 50322 b06b95a5fda2
child 50324 0a1242d5e7d4
permissions -rw-r--r--
add filterlim rules for unary minus and inverse
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
wenzelm@49834
    23
typedef '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
hoelzl@50247
   283
subsection {* Order filters *}
huffman@31392
   284
hoelzl@50247
   285
definition at_top :: "('a::order) filter"
hoelzl@50247
   286
  where "at_top = Abs_filter (\<lambda>P. \<exists>k. \<forall>n\<ge>k. P n)"
huffman@31392
   287
hoelzl@50247
   288
lemma eventually_at_top_linorder:
hoelzl@50247
   289
  fixes P :: "'a::linorder \<Rightarrow> bool" shows "eventually P at_top \<longleftrightarrow> (\<exists>N. \<forall>n\<ge>N. P n)"
hoelzl@50247
   290
  unfolding at_top_def
huffman@44081
   291
proof (rule eventually_Abs_filter, rule is_filter.intro)
hoelzl@50247
   292
  fix P Q :: "'a \<Rightarrow> bool"
huffman@36662
   293
  assume "\<exists>i. \<forall>n\<ge>i. P n" and "\<exists>j. \<forall>n\<ge>j. Q n"
huffman@36662
   294
  then obtain i j where "\<forall>n\<ge>i. P n" and "\<forall>n\<ge>j. Q n" by auto
huffman@36662
   295
  then have "\<forall>n\<ge>max i j. P n \<and> Q n" by simp
huffman@36662
   296
  then show "\<exists>k. \<forall>n\<ge>k. P n \<and> Q n" ..
huffman@36662
   297
qed auto
huffman@36662
   298
hoelzl@50247
   299
lemma eventually_at_top_dense:
hoelzl@50247
   300
  fixes P :: "'a::dense_linorder \<Rightarrow> bool" shows "eventually P at_top \<longleftrightarrow> (\<exists>N. \<forall>n>N. P n)"
hoelzl@50247
   301
  unfolding eventually_at_top_linorder
hoelzl@50247
   302
proof safe
hoelzl@50247
   303
  fix N assume "\<forall>n\<ge>N. P n" then show "\<exists>N. \<forall>n>N. P n" by (auto intro!: exI[of _ N])
hoelzl@50247
   304
next
hoelzl@50247
   305
  fix N assume "\<forall>n>N. P n" 
hoelzl@50247
   306
  moreover from gt_ex[of N] guess y ..
hoelzl@50247
   307
  ultimately show "\<exists>N. \<forall>n\<ge>N. P n" by (auto intro!: exI[of _ y])
hoelzl@50247
   308
qed
hoelzl@50247
   309
hoelzl@50247
   310
definition at_bot :: "('a::order) filter"
hoelzl@50247
   311
  where "at_bot = Abs_filter (\<lambda>P. \<exists>k. \<forall>n\<le>k. P n)"
hoelzl@50247
   312
hoelzl@50247
   313
lemma eventually_at_bot_linorder:
hoelzl@50247
   314
  fixes P :: "'a::linorder \<Rightarrow> bool" shows "eventually P at_bot \<longleftrightarrow> (\<exists>N. \<forall>n\<le>N. P n)"
hoelzl@50247
   315
  unfolding at_bot_def
hoelzl@50247
   316
proof (rule eventually_Abs_filter, rule is_filter.intro)
hoelzl@50247
   317
  fix P Q :: "'a \<Rightarrow> bool"
hoelzl@50247
   318
  assume "\<exists>i. \<forall>n\<le>i. P n" and "\<exists>j. \<forall>n\<le>j. Q n"
hoelzl@50247
   319
  then obtain i j where "\<forall>n\<le>i. P n" and "\<forall>n\<le>j. Q n" by auto
hoelzl@50247
   320
  then have "\<forall>n\<le>min i j. P n \<and> Q n" by simp
hoelzl@50247
   321
  then show "\<exists>k. \<forall>n\<le>k. P n \<and> Q n" ..
hoelzl@50247
   322
qed auto
hoelzl@50247
   323
hoelzl@50247
   324
lemma eventually_at_bot_dense:
hoelzl@50247
   325
  fixes P :: "'a::dense_linorder \<Rightarrow> bool" shows "eventually P at_bot \<longleftrightarrow> (\<exists>N. \<forall>n<N. P n)"
hoelzl@50247
   326
  unfolding eventually_at_bot_linorder
hoelzl@50247
   327
proof safe
hoelzl@50247
   328
  fix N assume "\<forall>n\<le>N. P n" then show "\<exists>N. \<forall>n<N. P n" by (auto intro!: exI[of _ N])
hoelzl@50247
   329
next
hoelzl@50247
   330
  fix N assume "\<forall>n<N. P n" 
hoelzl@50247
   331
  moreover from lt_ex[of N] guess y ..
hoelzl@50247
   332
  ultimately show "\<exists>N. \<forall>n\<le>N. P n" by (auto intro!: exI[of _ y])
hoelzl@50247
   333
qed
hoelzl@50247
   334
hoelzl@50247
   335
subsection {* Sequentially *}
hoelzl@50247
   336
hoelzl@50247
   337
abbreviation sequentially :: "nat filter"
hoelzl@50247
   338
  where "sequentially == at_top"
hoelzl@50247
   339
hoelzl@50247
   340
lemma sequentially_def: "sequentially = Abs_filter (\<lambda>P. \<exists>k. \<forall>n\<ge>k. P n)"
hoelzl@50247
   341
  unfolding at_top_def by simp
hoelzl@50247
   342
hoelzl@50247
   343
lemma eventually_sequentially:
hoelzl@50247
   344
  "eventually P sequentially \<longleftrightarrow> (\<exists>N. \<forall>n\<ge>N. P n)"
hoelzl@50247
   345
  by (rule eventually_at_top_linorder)
hoelzl@50247
   346
huffman@44342
   347
lemma sequentially_bot [simp, intro]: "sequentially \<noteq> bot"
huffman@44081
   348
  unfolding filter_eq_iff eventually_sequentially by auto
huffman@36662
   349
huffman@44342
   350
lemmas trivial_limit_sequentially = sequentially_bot
huffman@44342
   351
huffman@36662
   352
lemma eventually_False_sequentially [simp]:
huffman@36662
   353
  "\<not> eventually (\<lambda>n. False) sequentially"
huffman@44081
   354
  by (simp add: eventually_False)
huffman@36662
   355
huffman@36662
   356
lemma le_sequentially:
huffman@44195
   357
  "F \<le> sequentially \<longleftrightarrow> (\<forall>N. eventually (\<lambda>n. N \<le> n) F)"
huffman@44081
   358
  unfolding le_filter_def eventually_sequentially
huffman@44081
   359
  by (safe, fast, drule_tac x=N in spec, auto elim: eventually_rev_mp)
huffman@36662
   360
noschinl@45892
   361
lemma eventually_sequentiallyI:
noschinl@45892
   362
  assumes "\<And>x. c \<le> x \<Longrightarrow> P x"
noschinl@45892
   363
  shows "eventually P sequentially"
noschinl@45892
   364
using assms by (auto simp: eventually_sequentially)
noschinl@45892
   365
huffman@36662
   366
huffman@44081
   367
subsection {* Standard filters *}
huffman@36662
   368
huffman@44081
   369
definition within :: "'a filter \<Rightarrow> 'a set \<Rightarrow> 'a filter" (infixr "within" 70)
huffman@44195
   370
  where "F within S = Abs_filter (\<lambda>P. eventually (\<lambda>x. x \<in> S \<longrightarrow> P x) F)"
huffman@31392
   371
huffman@44206
   372
definition (in topological_space) nhds :: "'a \<Rightarrow> 'a filter"
huffman@44081
   373
  where "nhds a = Abs_filter (\<lambda>P. \<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x))"
huffman@36654
   374
huffman@44206
   375
definition (in topological_space) at :: "'a \<Rightarrow> 'a filter"
huffman@44081
   376
  where "at a = nhds a within - {a}"
huffman@31447
   377
huffman@31392
   378
lemma eventually_within:
huffman@44195
   379
  "eventually P (F within S) = eventually (\<lambda>x. x \<in> S \<longrightarrow> P x) F"
huffman@44081
   380
  unfolding within_def
huffman@44081
   381
  by (rule eventually_Abs_filter, rule is_filter.intro)
huffman@44081
   382
     (auto elim!: eventually_rev_mp)
huffman@31392
   383
huffman@45031
   384
lemma within_UNIV [simp]: "F within UNIV = F"
huffman@45031
   385
  unfolding filter_eq_iff eventually_within by simp
huffman@45031
   386
huffman@45031
   387
lemma within_empty [simp]: "F within {} = bot"
huffman@44081
   388
  unfolding filter_eq_iff eventually_within by simp
huffman@36360
   389
hoelzl@50247
   390
lemma within_le: "F within S \<le> F"
hoelzl@50247
   391
  unfolding le_filter_def eventually_within by (auto elim: eventually_elim1)
hoelzl@50247
   392
hoelzl@50323
   393
lemma le_withinI: "F \<le> F' \<Longrightarrow> eventually (\<lambda>x. x \<in> S) F \<Longrightarrow> F \<le> F' within S"
hoelzl@50323
   394
  unfolding le_filter_def eventually_within by (auto elim: eventually_elim2)
hoelzl@50323
   395
hoelzl@50323
   396
lemma le_within_iff: "eventually (\<lambda>x. x \<in> S) F \<Longrightarrow> F \<le> F' within S \<longleftrightarrow> F \<le> F'"
hoelzl@50323
   397
  by (blast intro: within_le le_withinI order_trans)
hoelzl@50323
   398
huffman@36654
   399
lemma eventually_nhds:
huffman@36654
   400
  "eventually P (nhds a) \<longleftrightarrow> (\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x))"
huffman@36654
   401
unfolding nhds_def
huffman@44081
   402
proof (rule eventually_Abs_filter, rule is_filter.intro)
huffman@36654
   403
  have "open UNIV \<and> a \<in> UNIV \<and> (\<forall>x\<in>UNIV. True)" by simp
huffman@36654
   404
  thus "\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. True)" by - rule
huffman@36358
   405
next
huffman@36358
   406
  fix P Q
huffman@36654
   407
  assume "\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x)"
huffman@36654
   408
     and "\<exists>T. open T \<and> a \<in> T \<and> (\<forall>x\<in>T. Q x)"
huffman@36358
   409
  then obtain S T where
huffman@36654
   410
    "open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x)"
huffman@36654
   411
    "open T \<and> a \<in> T \<and> (\<forall>x\<in>T. Q x)" by auto
huffman@36654
   412
  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
   413
    by (simp add: open_Int)
huffman@36654
   414
  thus "\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x \<and> Q x)" by - rule
huffman@36358
   415
qed auto
huffman@31447
   416
huffman@36656
   417
lemma eventually_nhds_metric:
huffman@36656
   418
  "eventually P (nhds a) \<longleftrightarrow> (\<exists>d>0. \<forall>x. dist x a < d \<longrightarrow> P x)"
huffman@36656
   419
unfolding eventually_nhds open_dist
huffman@31447
   420
apply safe
huffman@31447
   421
apply fast
huffman@31492
   422
apply (rule_tac x="{x. dist x a < d}" in exI, simp)
huffman@31447
   423
apply clarsimp
huffman@31447
   424
apply (rule_tac x="d - dist x a" in exI, clarsimp)
huffman@31447
   425
apply (simp only: less_diff_eq)
huffman@31447
   426
apply (erule le_less_trans [OF dist_triangle])
huffman@31447
   427
done
huffman@31447
   428
huffman@44571
   429
lemma nhds_neq_bot [simp]: "nhds a \<noteq> bot"
huffman@44571
   430
  unfolding trivial_limit_def eventually_nhds by simp
huffman@44571
   431
huffman@36656
   432
lemma eventually_at_topological:
huffman@36656
   433
  "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
   434
unfolding at_def eventually_within eventually_nhds by simp
huffman@36656
   435
huffman@36656
   436
lemma eventually_at:
huffman@36656
   437
  fixes a :: "'a::metric_space"
huffman@36656
   438
  shows "eventually P (at a) \<longleftrightarrow> (\<exists>d>0. \<forall>x. x \<noteq> a \<and> dist x a < d \<longrightarrow> P x)"
huffman@36656
   439
unfolding at_def eventually_within eventually_nhds_metric by auto
huffman@36656
   440
huffman@44571
   441
lemma at_eq_bot_iff: "at a = bot \<longleftrightarrow> open {a}"
huffman@44571
   442
  unfolding trivial_limit_def eventually_at_topological
huffman@44571
   443
  by (safe, case_tac "S = {a}", simp, fast, fast)
huffman@44571
   444
huffman@44571
   445
lemma at_neq_bot [simp]: "at (a::'a::perfect_space) \<noteq> bot"
huffman@44571
   446
  by (simp add: at_eq_bot_iff not_open_singleton)
huffman@44571
   447
huffman@31392
   448
huffman@31355
   449
subsection {* Boundedness *}
huffman@31355
   450
huffman@44081
   451
definition Bfun :: "('a \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a filter \<Rightarrow> bool"
huffman@44195
   452
  where "Bfun f F = (\<exists>K>0. eventually (\<lambda>x. norm (f x) \<le> K) F)"
huffman@31355
   453
huffman@31487
   454
lemma BfunI:
huffman@44195
   455
  assumes K: "eventually (\<lambda>x. norm (f x) \<le> K) F" shows "Bfun f F"
huffman@31355
   456
unfolding Bfun_def
huffman@31355
   457
proof (intro exI conjI allI)
huffman@31355
   458
  show "0 < max K 1" by simp
huffman@31355
   459
next
huffman@44195
   460
  show "eventually (\<lambda>x. norm (f x) \<le> max K 1) F"
huffman@31355
   461
    using K by (rule eventually_elim1, simp)
huffman@31355
   462
qed
huffman@31355
   463
huffman@31355
   464
lemma BfunE:
huffman@44195
   465
  assumes "Bfun f F"
huffman@44195
   466
  obtains B where "0 < B" and "eventually (\<lambda>x. norm (f x) \<le> B) F"
huffman@31355
   467
using assms unfolding Bfun_def by fast
huffman@31355
   468
huffman@31355
   469
huffman@31349
   470
subsection {* Convergence to Zero *}
huffman@31349
   471
huffman@44081
   472
definition Zfun :: "('a \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a filter \<Rightarrow> bool"
huffman@44195
   473
  where "Zfun f F = (\<forall>r>0. eventually (\<lambda>x. norm (f x) < r) F)"
huffman@31349
   474
huffman@31349
   475
lemma ZfunI:
huffman@44195
   476
  "(\<And>r. 0 < r \<Longrightarrow> eventually (\<lambda>x. norm (f x) < r) F) \<Longrightarrow> Zfun f F"
huffman@44081
   477
  unfolding Zfun_def by simp
huffman@31349
   478
huffman@31349
   479
lemma ZfunD:
huffman@44195
   480
  "\<lbrakk>Zfun f F; 0 < r\<rbrakk> \<Longrightarrow> eventually (\<lambda>x. norm (f x) < r) F"
huffman@44081
   481
  unfolding Zfun_def by simp
huffman@31349
   482
huffman@31355
   483
lemma Zfun_ssubst:
huffman@44195
   484
  "eventually (\<lambda>x. f x = g x) F \<Longrightarrow> Zfun g F \<Longrightarrow> Zfun f F"
huffman@44081
   485
  unfolding Zfun_def by (auto elim!: eventually_rev_mp)
huffman@31355
   486
huffman@44195
   487
lemma Zfun_zero: "Zfun (\<lambda>x. 0) F"
huffman@44081
   488
  unfolding Zfun_def by simp
huffman@31349
   489
huffman@44195
   490
lemma Zfun_norm_iff: "Zfun (\<lambda>x. norm (f x)) F = Zfun (\<lambda>x. f x) F"
huffman@44081
   491
  unfolding Zfun_def by simp
huffman@31349
   492
huffman@31349
   493
lemma Zfun_imp_Zfun:
huffman@44195
   494
  assumes f: "Zfun f F"
huffman@44195
   495
  assumes g: "eventually (\<lambda>x. norm (g x) \<le> norm (f x) * K) F"
huffman@44195
   496
  shows "Zfun (\<lambda>x. g x) F"
huffman@31349
   497
proof (cases)
huffman@31349
   498
  assume K: "0 < K"
huffman@31349
   499
  show ?thesis
huffman@31349
   500
  proof (rule ZfunI)
huffman@31349
   501
    fix r::real assume "0 < r"
huffman@31349
   502
    hence "0 < r / K"
huffman@31349
   503
      using K by (rule divide_pos_pos)
huffman@44195
   504
    then have "eventually (\<lambda>x. norm (f x) < r / K) F"
huffman@31487
   505
      using ZfunD [OF f] by fast
huffman@44195
   506
    with g show "eventually (\<lambda>x. norm (g x) < r) F"
noschinl@46887
   507
    proof eventually_elim
noschinl@46887
   508
      case (elim x)
huffman@31487
   509
      hence "norm (f x) * K < r"
huffman@31349
   510
        by (simp add: pos_less_divide_eq K)
noschinl@46887
   511
      thus ?case
noschinl@46887
   512
        by (simp add: order_le_less_trans [OF elim(1)])
huffman@31349
   513
    qed
huffman@31349
   514
  qed
huffman@31349
   515
next
huffman@31349
   516
  assume "\<not> 0 < K"
huffman@31349
   517
  hence K: "K \<le> 0" by (simp only: not_less)
huffman@31355
   518
  show ?thesis
huffman@31355
   519
  proof (rule ZfunI)
huffman@31355
   520
    fix r :: real
huffman@31355
   521
    assume "0 < r"
huffman@44195
   522
    from g show "eventually (\<lambda>x. norm (g x) < r) F"
noschinl@46887
   523
    proof eventually_elim
noschinl@46887
   524
      case (elim x)
noschinl@46887
   525
      also have "norm (f x) * K \<le> norm (f x) * 0"
huffman@31355
   526
        using K norm_ge_zero by (rule mult_left_mono)
noschinl@46887
   527
      finally show ?case
huffman@31355
   528
        using `0 < r` by simp
huffman@31355
   529
    qed
huffman@31355
   530
  qed
huffman@31349
   531
qed
huffman@31349
   532
huffman@44195
   533
lemma Zfun_le: "\<lbrakk>Zfun g F; \<forall>x. norm (f x) \<le> norm (g x)\<rbrakk> \<Longrightarrow> Zfun f F"
huffman@44081
   534
  by (erule_tac K="1" in Zfun_imp_Zfun, simp)
huffman@31349
   535
huffman@31349
   536
lemma Zfun_add:
huffman@44195
   537
  assumes f: "Zfun f F" and g: "Zfun g F"
huffman@44195
   538
  shows "Zfun (\<lambda>x. f x + g x) F"
huffman@31349
   539
proof (rule ZfunI)
huffman@31349
   540
  fix r::real assume "0 < r"
huffman@31349
   541
  hence r: "0 < r / 2" by simp
huffman@44195
   542
  have "eventually (\<lambda>x. norm (f x) < r/2) F"
huffman@31487
   543
    using f r by (rule ZfunD)
huffman@31349
   544
  moreover
huffman@44195
   545
  have "eventually (\<lambda>x. norm (g x) < r/2) F"
huffman@31487
   546
    using g r by (rule ZfunD)
huffman@31349
   547
  ultimately
huffman@44195
   548
  show "eventually (\<lambda>x. norm (f x + g x) < r) F"
noschinl@46887
   549
  proof eventually_elim
noschinl@46887
   550
    case (elim x)
huffman@31487
   551
    have "norm (f x + g x) \<le> norm (f x) + norm (g x)"
huffman@31349
   552
      by (rule norm_triangle_ineq)
huffman@31349
   553
    also have "\<dots> < r/2 + r/2"
noschinl@46887
   554
      using elim by (rule add_strict_mono)
noschinl@46887
   555
    finally show ?case
huffman@31349
   556
      by simp
huffman@31349
   557
  qed
huffman@31349
   558
qed
huffman@31349
   559
huffman@44195
   560
lemma Zfun_minus: "Zfun f F \<Longrightarrow> Zfun (\<lambda>x. - f x) F"
huffman@44081
   561
  unfolding Zfun_def by simp
huffman@31349
   562
huffman@44195
   563
lemma Zfun_diff: "\<lbrakk>Zfun f F; Zfun g F\<rbrakk> \<Longrightarrow> Zfun (\<lambda>x. f x - g x) F"
huffman@44081
   564
  by (simp only: diff_minus Zfun_add Zfun_minus)
huffman@31349
   565
huffman@31349
   566
lemma (in bounded_linear) Zfun:
huffman@44195
   567
  assumes g: "Zfun g F"
huffman@44195
   568
  shows "Zfun (\<lambda>x. f (g x)) F"
huffman@31349
   569
proof -
huffman@31349
   570
  obtain K where "\<And>x. norm (f x) \<le> norm x * K"
huffman@31349
   571
    using bounded by fast
huffman@44195
   572
  then have "eventually (\<lambda>x. norm (f (g x)) \<le> norm (g x) * K) F"
huffman@31355
   573
    by simp
huffman@31487
   574
  with g show ?thesis
huffman@31349
   575
    by (rule Zfun_imp_Zfun)
huffman@31349
   576
qed
huffman@31349
   577
huffman@31349
   578
lemma (in bounded_bilinear) Zfun:
huffman@44195
   579
  assumes f: "Zfun f F"
huffman@44195
   580
  assumes g: "Zfun g F"
huffman@44195
   581
  shows "Zfun (\<lambda>x. f x ** g x) F"
huffman@31349
   582
proof (rule ZfunI)
huffman@31349
   583
  fix r::real assume r: "0 < r"
huffman@31349
   584
  obtain K where K: "0 < K"
huffman@31349
   585
    and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"
huffman@31349
   586
    using pos_bounded by fast
huffman@31349
   587
  from K have K': "0 < inverse K"
huffman@31349
   588
    by (rule positive_imp_inverse_positive)
huffman@44195
   589
  have "eventually (\<lambda>x. norm (f x) < r) F"
huffman@31487
   590
    using f r by (rule ZfunD)
huffman@31349
   591
  moreover
huffman@44195
   592
  have "eventually (\<lambda>x. norm (g x) < inverse K) F"
huffman@31487
   593
    using g K' by (rule ZfunD)
huffman@31349
   594
  ultimately
huffman@44195
   595
  show "eventually (\<lambda>x. norm (f x ** g x) < r) F"
noschinl@46887
   596
  proof eventually_elim
noschinl@46887
   597
    case (elim x)
huffman@31487
   598
    have "norm (f x ** g x) \<le> norm (f x) * norm (g x) * K"
huffman@31349
   599
      by (rule norm_le)
huffman@31487
   600
    also have "norm (f x) * norm (g x) * K < r * inverse K * K"
noschinl@46887
   601
      by (intro mult_strict_right_mono mult_strict_mono' norm_ge_zero elim K)
huffman@31349
   602
    also from K have "r * inverse K * K = r"
huffman@31349
   603
      by simp
noschinl@46887
   604
    finally show ?case .
huffman@31349
   605
  qed
huffman@31349
   606
qed
huffman@31349
   607
huffman@31349
   608
lemma (in bounded_bilinear) Zfun_left:
huffman@44195
   609
  "Zfun f F \<Longrightarrow> Zfun (\<lambda>x. f x ** a) F"
huffman@44081
   610
  by (rule bounded_linear_left [THEN bounded_linear.Zfun])
huffman@31349
   611
huffman@31349
   612
lemma (in bounded_bilinear) Zfun_right:
huffman@44195
   613
  "Zfun f F \<Longrightarrow> Zfun (\<lambda>x. a ** f x) F"
huffman@44081
   614
  by (rule bounded_linear_right [THEN bounded_linear.Zfun])
huffman@31349
   615
huffman@44282
   616
lemmas Zfun_mult = bounded_bilinear.Zfun [OF bounded_bilinear_mult]
huffman@44282
   617
lemmas Zfun_mult_right = bounded_bilinear.Zfun_right [OF bounded_bilinear_mult]
huffman@44282
   618
lemmas Zfun_mult_left = bounded_bilinear.Zfun_left [OF bounded_bilinear_mult]
huffman@31349
   619
huffman@31349
   620
wenzelm@31902
   621
subsection {* Limits *}
huffman@31349
   622
hoelzl@50322
   623
definition filterlim :: "('a \<Rightarrow> 'b) \<Rightarrow> 'b filter \<Rightarrow> 'a filter \<Rightarrow> bool" where
hoelzl@50322
   624
  "filterlim f F2 F1 \<longleftrightarrow> filtermap f F1 \<le> F2"
hoelzl@50247
   625
hoelzl@50247
   626
syntax
hoelzl@50247
   627
  "_LIM" :: "pttrns \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> bool" ("(3LIM (_)/ (_)./ (_) :> (_))" [1000, 10, 0, 10] 10)
hoelzl@50247
   628
hoelzl@50247
   629
translations
hoelzl@50322
   630
  "LIM x F1. f :> F2"   == "CONST filterlim (%x. f) F2 F1"
hoelzl@50247
   631
hoelzl@50322
   632
lemma filterlimE: "(LIM x F1. f x :> F2) \<Longrightarrow> eventually P F2 \<Longrightarrow> eventually (\<lambda>x. P (f x)) F1"
hoelzl@50322
   633
  by (auto simp: filterlim_def eventually_filtermap le_filter_def)
hoelzl@50247
   634
hoelzl@50322
   635
lemma filterlimI: "(\<And>P. eventually P F2 \<Longrightarrow> eventually (\<lambda>x. P (f x)) F1) \<Longrightarrow> (LIM x F1. f x :> F2)"
hoelzl@50322
   636
  by (auto simp: filterlim_def eventually_filtermap le_filter_def)
hoelzl@50247
   637
hoelzl@50323
   638
lemma filterlim_compose: 
hoelzl@50323
   639
  "filterlim g F3 F2 \<Longrightarrow> filterlim f F2 F1 \<Longrightarrow> filterlim (\<lambda>x. g (f x)) F3 F1"
hoelzl@50323
   640
  unfolding filterlim_def filtermap_filtermap[symmetric] by (metis filtermap_mono order_trans)
hoelzl@50323
   641
hoelzl@50323
   642
lemma filterlim_mono: 
hoelzl@50323
   643
  "filterlim f F2 F1 \<Longrightarrow> F2 \<le> F2' \<Longrightarrow> F1' \<le> F1 \<Longrightarrow> filterlim f F2' F1'"
hoelzl@50323
   644
  unfolding filterlim_def by (metis filtermap_mono order_trans)
hoelzl@50323
   645
hoelzl@50247
   646
abbreviation (in topological_space)
huffman@44206
   647
  tendsto :: "('b \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'b filter \<Rightarrow> bool" (infixr "--->" 55) where
hoelzl@50322
   648
  "(f ---> l) F \<equiv> filterlim f (nhds l) F"
noschinl@45892
   649
wenzelm@31902
   650
ML {*
wenzelm@31902
   651
structure Tendsto_Intros = Named_Thms
wenzelm@31902
   652
(
wenzelm@45294
   653
  val name = @{binding tendsto_intros}
wenzelm@31902
   654
  val description = "introduction rules for tendsto"
wenzelm@31902
   655
)
huffman@31565
   656
*}
huffman@31565
   657
wenzelm@31902
   658
setup Tendsto_Intros.setup
huffman@31565
   659
hoelzl@50247
   660
lemma tendsto_def: "(f ---> l) F \<longleftrightarrow> (\<forall>S. open S \<longrightarrow> l \<in> S \<longrightarrow> eventually (\<lambda>x. f x \<in> S) F)"
hoelzl@50322
   661
  unfolding filterlim_def
hoelzl@50247
   662
proof safe
hoelzl@50247
   663
  fix S assume "open S" "l \<in> S" "filtermap f F \<le> nhds l"
hoelzl@50247
   664
  then show "eventually (\<lambda>x. f x \<in> S) F"
hoelzl@50247
   665
    unfolding eventually_nhds eventually_filtermap le_filter_def
hoelzl@50247
   666
    by (auto elim!: allE[of _ "\<lambda>x. x \<in> S"] eventually_rev_mp)
hoelzl@50247
   667
qed (auto elim!: eventually_rev_mp simp: eventually_nhds eventually_filtermap le_filter_def)
hoelzl@50247
   668
huffman@44195
   669
lemma tendsto_mono: "F \<le> F' \<Longrightarrow> (f ---> l) F' \<Longrightarrow> (f ---> l) F"
huffman@44081
   670
  unfolding tendsto_def le_filter_def by fast
huffman@36656
   671
huffman@31488
   672
lemma topological_tendstoI:
huffman@44195
   673
  "(\<And>S. open S \<Longrightarrow> l \<in> S \<Longrightarrow> eventually (\<lambda>x. f x \<in> S) F)
huffman@44195
   674
    \<Longrightarrow> (f ---> l) F"
huffman@31349
   675
  unfolding tendsto_def by auto
huffman@31349
   676
huffman@31488
   677
lemma topological_tendstoD:
huffman@44195
   678
  "(f ---> l) F \<Longrightarrow> open S \<Longrightarrow> l \<in> S \<Longrightarrow> eventually (\<lambda>x. f x \<in> S) F"
huffman@31488
   679
  unfolding tendsto_def by auto
huffman@31488
   680
huffman@31488
   681
lemma tendstoI:
huffman@44195
   682
  assumes "\<And>e. 0 < e \<Longrightarrow> eventually (\<lambda>x. dist (f x) l < e) F"
huffman@44195
   683
  shows "(f ---> l) F"
huffman@44081
   684
  apply (rule topological_tendstoI)
huffman@44081
   685
  apply (simp add: open_dist)
huffman@44081
   686
  apply (drule (1) bspec, clarify)
huffman@44081
   687
  apply (drule assms)
huffman@44081
   688
  apply (erule eventually_elim1, simp)
huffman@44081
   689
  done
huffman@31488
   690
huffman@31349
   691
lemma tendstoD:
huffman@44195
   692
  "(f ---> l) F \<Longrightarrow> 0 < e \<Longrightarrow> eventually (\<lambda>x. dist (f x) l < e) F"
huffman@44081
   693
  apply (drule_tac S="{x. dist x l < e}" in topological_tendstoD)
huffman@44081
   694
  apply (clarsimp simp add: open_dist)
huffman@44081
   695
  apply (rule_tac x="e - dist x l" in exI, clarsimp)
huffman@44081
   696
  apply (simp only: less_diff_eq)
huffman@44081
   697
  apply (erule le_less_trans [OF dist_triangle])
huffman@44081
   698
  apply simp
huffman@44081
   699
  apply simp
huffman@44081
   700
  done
huffman@31488
   701
huffman@31488
   702
lemma tendsto_iff:
huffman@44195
   703
  "(f ---> l) F \<longleftrightarrow> (\<forall>e>0. eventually (\<lambda>x. dist (f x) l < e) F)"
huffman@44081
   704
  using tendstoI tendstoD by fast
huffman@31349
   705
huffman@44195
   706
lemma tendsto_Zfun_iff: "(f ---> a) F = Zfun (\<lambda>x. f x - a) F"
huffman@44081
   707
  by (simp only: tendsto_iff Zfun_def dist_norm)
huffman@31349
   708
huffman@45031
   709
lemma tendsto_bot [simp]: "(f ---> a) bot"
huffman@45031
   710
  unfolding tendsto_def by simp
huffman@45031
   711
huffman@31565
   712
lemma tendsto_ident_at [tendsto_intros]: "((\<lambda>x. x) ---> a) (at a)"
huffman@44081
   713
  unfolding tendsto_def eventually_at_topological by auto
huffman@31565
   714
huffman@31565
   715
lemma tendsto_ident_at_within [tendsto_intros]:
huffman@36655
   716
  "((\<lambda>x. x) ---> a) (at a within S)"
huffman@44081
   717
  unfolding tendsto_def eventually_within eventually_at_topological by auto
huffman@31565
   718
huffman@44195
   719
lemma tendsto_const [tendsto_intros]: "((\<lambda>x. k) ---> k) F"
huffman@44081
   720
  by (simp add: tendsto_def)
huffman@31349
   721
huffman@44205
   722
lemma tendsto_unique:
huffman@44205
   723
  fixes f :: "'a \<Rightarrow> 'b::t2_space"
huffman@44205
   724
  assumes "\<not> trivial_limit F" and "(f ---> a) F" and "(f ---> b) F"
huffman@44205
   725
  shows "a = b"
huffman@44205
   726
proof (rule ccontr)
huffman@44205
   727
  assume "a \<noteq> b"
huffman@44205
   728
  obtain U V where "open U" "open V" "a \<in> U" "b \<in> V" "U \<inter> V = {}"
huffman@44205
   729
    using hausdorff [OF `a \<noteq> b`] by fast
huffman@44205
   730
  have "eventually (\<lambda>x. f x \<in> U) F"
huffman@44205
   731
    using `(f ---> a) F` `open U` `a \<in> U` by (rule topological_tendstoD)
huffman@44205
   732
  moreover
huffman@44205
   733
  have "eventually (\<lambda>x. f x \<in> V) F"
huffman@44205
   734
    using `(f ---> b) F` `open V` `b \<in> V` by (rule topological_tendstoD)
huffman@44205
   735
  ultimately
huffman@44205
   736
  have "eventually (\<lambda>x. False) F"
noschinl@46887
   737
  proof eventually_elim
noschinl@46887
   738
    case (elim x)
huffman@44205
   739
    hence "f x \<in> U \<inter> V" by simp
noschinl@46887
   740
    with `U \<inter> V = {}` show ?case by simp
huffman@44205
   741
  qed
huffman@44205
   742
  with `\<not> trivial_limit F` show "False"
huffman@44205
   743
    by (simp add: trivial_limit_def)
huffman@44205
   744
qed
huffman@44205
   745
huffman@36662
   746
lemma tendsto_const_iff:
huffman@44205
   747
  fixes a b :: "'a::t2_space"
huffman@44205
   748
  assumes "\<not> trivial_limit F" shows "((\<lambda>x. a) ---> b) F \<longleftrightarrow> a = b"
huffman@44205
   749
  by (safe intro!: tendsto_const tendsto_unique [OF assms tendsto_const])
huffman@44205
   750
hoelzl@50323
   751
lemma tendsto_at_iff_tendsto_nhds:
hoelzl@50323
   752
  "(g ---> g l) (at l) \<longleftrightarrow> (g ---> g l) (nhds l)"
hoelzl@50323
   753
  unfolding tendsto_def at_def eventually_within
hoelzl@50323
   754
  by (intro ext all_cong imp_cong) (auto elim!: eventually_elim1)
hoelzl@50323
   755
huffman@44218
   756
lemma tendsto_compose:
hoelzl@50323
   757
  "(g ---> g l) (at l) \<Longrightarrow> (f ---> l) F \<Longrightarrow> ((\<lambda>x. g (f x)) ---> g l) F"
hoelzl@50323
   758
  unfolding tendsto_at_iff_tendsto_nhds by (rule filterlim_compose[of g])
huffman@44218
   759
huffman@44253
   760
lemma tendsto_compose_eventually:
huffman@44253
   761
  assumes g: "(g ---> m) (at l)"
huffman@44253
   762
  assumes f: "(f ---> l) F"
huffman@44253
   763
  assumes inj: "eventually (\<lambda>x. f x \<noteq> l) F"
huffman@44253
   764
  shows "((\<lambda>x. g (f x)) ---> m) F"
hoelzl@50323
   765
proof -
hoelzl@50323
   766
  from f inj have "LIM x F. f x :> at l"
hoelzl@50323
   767
    unfolding filterlim_def at_def by (simp add: le_within_iff eventually_filtermap)
hoelzl@50323
   768
  from filterlim_compose[OF g this] show ?thesis .
huffman@44253
   769
qed
huffman@44253
   770
huffman@44251
   771
lemma metric_tendsto_imp_tendsto:
huffman@44251
   772
  assumes f: "(f ---> a) F"
huffman@44251
   773
  assumes le: "eventually (\<lambda>x. dist (g x) b \<le> dist (f x) a) F"
huffman@44251
   774
  shows "(g ---> b) F"
huffman@44251
   775
proof (rule tendstoI)
huffman@44251
   776
  fix e :: real assume "0 < e"
huffman@44251
   777
  with f have "eventually (\<lambda>x. dist (f x) a < e) F" by (rule tendstoD)
huffman@44251
   778
  with le show "eventually (\<lambda>x. dist (g x) b < e) F"
huffman@44251
   779
    using le_less_trans by (rule eventually_elim2)
huffman@44251
   780
qed
huffman@44251
   781
huffman@44205
   782
subsubsection {* Distance and norms *}
huffman@36662
   783
huffman@31565
   784
lemma tendsto_dist [tendsto_intros]:
huffman@44195
   785
  assumes f: "(f ---> l) F" and g: "(g ---> m) F"
huffman@44195
   786
  shows "((\<lambda>x. dist (f x) (g x)) ---> dist l m) F"
huffman@31565
   787
proof (rule tendstoI)
huffman@31565
   788
  fix e :: real assume "0 < e"
huffman@31565
   789
  hence e2: "0 < e/2" by simp
huffman@31565
   790
  from tendstoD [OF f e2] tendstoD [OF g e2]
huffman@44195
   791
  show "eventually (\<lambda>x. dist (dist (f x) (g x)) (dist l m) < e) F"
noschinl@46887
   792
  proof (eventually_elim)
noschinl@46887
   793
    case (elim x)
huffman@31565
   794
    then show "dist (dist (f x) (g x)) (dist l m) < e"
huffman@31565
   795
      unfolding dist_real_def
huffman@31565
   796
      using dist_triangle2 [of "f x" "g x" "l"]
huffman@31565
   797
      using dist_triangle2 [of "g x" "l" "m"]
huffman@31565
   798
      using dist_triangle3 [of "l" "m" "f x"]
huffman@31565
   799
      using dist_triangle [of "f x" "m" "g x"]
huffman@31565
   800
      by arith
huffman@31565
   801
  qed
huffman@31565
   802
qed
huffman@31565
   803
huffman@36662
   804
lemma norm_conv_dist: "norm x = dist x 0"
huffman@44081
   805
  unfolding dist_norm by simp
huffman@36662
   806
huffman@31565
   807
lemma tendsto_norm [tendsto_intros]:
huffman@44195
   808
  "(f ---> a) F \<Longrightarrow> ((\<lambda>x. norm (f x)) ---> norm a) F"
huffman@44081
   809
  unfolding norm_conv_dist by (intro tendsto_intros)
huffman@36662
   810
huffman@36662
   811
lemma tendsto_norm_zero:
huffman@44195
   812
  "(f ---> 0) F \<Longrightarrow> ((\<lambda>x. norm (f x)) ---> 0) F"
huffman@44081
   813
  by (drule tendsto_norm, simp)
huffman@36662
   814
huffman@36662
   815
lemma tendsto_norm_zero_cancel:
huffman@44195
   816
  "((\<lambda>x. norm (f x)) ---> 0) F \<Longrightarrow> (f ---> 0) F"
huffman@44081
   817
  unfolding tendsto_iff dist_norm by simp
huffman@36662
   818
huffman@36662
   819
lemma tendsto_norm_zero_iff:
huffman@44195
   820
  "((\<lambda>x. norm (f x)) ---> 0) F \<longleftrightarrow> (f ---> 0) F"
huffman@44081
   821
  unfolding tendsto_iff dist_norm by simp
huffman@31349
   822
huffman@44194
   823
lemma tendsto_rabs [tendsto_intros]:
huffman@44195
   824
  "(f ---> (l::real)) F \<Longrightarrow> ((\<lambda>x. \<bar>f x\<bar>) ---> \<bar>l\<bar>) F"
huffman@44194
   825
  by (fold real_norm_def, rule tendsto_norm)
huffman@44194
   826
huffman@44194
   827
lemma tendsto_rabs_zero:
huffman@44195
   828
  "(f ---> (0::real)) F \<Longrightarrow> ((\<lambda>x. \<bar>f x\<bar>) ---> 0) F"
huffman@44194
   829
  by (fold real_norm_def, rule tendsto_norm_zero)
huffman@44194
   830
huffman@44194
   831
lemma tendsto_rabs_zero_cancel:
huffman@44195
   832
  "((\<lambda>x. \<bar>f x\<bar>) ---> (0::real)) F \<Longrightarrow> (f ---> 0) F"
huffman@44194
   833
  by (fold real_norm_def, rule tendsto_norm_zero_cancel)
huffman@44194
   834
huffman@44194
   835
lemma tendsto_rabs_zero_iff:
huffman@44195
   836
  "((\<lambda>x. \<bar>f x\<bar>) ---> (0::real)) F \<longleftrightarrow> (f ---> 0) F"
huffman@44194
   837
  by (fold real_norm_def, rule tendsto_norm_zero_iff)
huffman@44194
   838
huffman@44194
   839
subsubsection {* Addition and subtraction *}
huffman@44194
   840
huffman@31565
   841
lemma tendsto_add [tendsto_intros]:
huffman@31349
   842
  fixes a b :: "'a::real_normed_vector"
huffman@44195
   843
  shows "\<lbrakk>(f ---> a) F; (g ---> b) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x + g x) ---> a + b) F"
huffman@44081
   844
  by (simp only: tendsto_Zfun_iff add_diff_add Zfun_add)
huffman@31349
   845
huffman@44194
   846
lemma tendsto_add_zero:
huffman@44194
   847
  fixes f g :: "'a::type \<Rightarrow> 'b::real_normed_vector"
huffman@44195
   848
  shows "\<lbrakk>(f ---> 0) F; (g ---> 0) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x + g x) ---> 0) F"
huffman@44194
   849
  by (drule (1) tendsto_add, simp)
huffman@44194
   850
huffman@31565
   851
lemma tendsto_minus [tendsto_intros]:
huffman@31349
   852
  fixes a :: "'a::real_normed_vector"
huffman@44195
   853
  shows "(f ---> a) F \<Longrightarrow> ((\<lambda>x. - f x) ---> - a) F"
huffman@44081
   854
  by (simp only: tendsto_Zfun_iff minus_diff_minus Zfun_minus)
huffman@31349
   855
huffman@31349
   856
lemma tendsto_minus_cancel:
huffman@31349
   857
  fixes a :: "'a::real_normed_vector"
huffman@44195
   858
  shows "((\<lambda>x. - f x) ---> - a) F \<Longrightarrow> (f ---> a) F"
huffman@44081
   859
  by (drule tendsto_minus, simp)
huffman@31349
   860
huffman@31565
   861
lemma tendsto_diff [tendsto_intros]:
huffman@31349
   862
  fixes a b :: "'a::real_normed_vector"
huffman@44195
   863
  shows "\<lbrakk>(f ---> a) F; (g ---> b) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x - g x) ---> a - b) F"
huffman@44081
   864
  by (simp add: diff_minus tendsto_add tendsto_minus)
huffman@31349
   865
huffman@31588
   866
lemma tendsto_setsum [tendsto_intros]:
huffman@31588
   867
  fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c::real_normed_vector"
huffman@44195
   868
  assumes "\<And>i. i \<in> S \<Longrightarrow> (f i ---> a i) F"
huffman@44195
   869
  shows "((\<lambda>x. \<Sum>i\<in>S. f i x) ---> (\<Sum>i\<in>S. a i)) F"
huffman@31588
   870
proof (cases "finite S")
huffman@31588
   871
  assume "finite S" thus ?thesis using assms
huffman@44194
   872
    by (induct, simp add: tendsto_const, simp add: tendsto_add)
huffman@31588
   873
next
huffman@31588
   874
  assume "\<not> finite S" thus ?thesis
huffman@31588
   875
    by (simp add: tendsto_const)
huffman@31588
   876
qed
huffman@31588
   877
noschinl@45892
   878
lemma real_tendsto_sandwich:
noschinl@45892
   879
  fixes f g h :: "'a \<Rightarrow> real"
noschinl@45892
   880
  assumes ev: "eventually (\<lambda>n. f n \<le> g n) net" "eventually (\<lambda>n. g n \<le> h n) net"
noschinl@45892
   881
  assumes lim: "(f ---> c) net" "(h ---> c) net"
noschinl@45892
   882
  shows "(g ---> c) net"
noschinl@45892
   883
proof -
noschinl@45892
   884
  have "((\<lambda>n. g n - f n) ---> 0) net"
noschinl@45892
   885
  proof (rule metric_tendsto_imp_tendsto)
noschinl@45892
   886
    show "eventually (\<lambda>n. dist (g n - f n) 0 \<le> dist (h n - f n) 0) net"
noschinl@45892
   887
      using ev by (rule eventually_elim2) (simp add: dist_real_def)
noschinl@45892
   888
    show "((\<lambda>n. h n - f n) ---> 0) net"
noschinl@45892
   889
      using tendsto_diff[OF lim(2,1)] by simp
noschinl@45892
   890
  qed
noschinl@45892
   891
  from tendsto_add[OF this lim(1)] show ?thesis by simp
noschinl@45892
   892
qed
noschinl@45892
   893
huffman@44194
   894
subsubsection {* Linear operators and multiplication *}
huffman@44194
   895
huffman@44282
   896
lemma (in bounded_linear) tendsto:
huffman@44195
   897
  "(g ---> a) F \<Longrightarrow> ((\<lambda>x. f (g x)) ---> f a) F"
huffman@44081
   898
  by (simp only: tendsto_Zfun_iff diff [symmetric] Zfun)
huffman@31349
   899
huffman@44194
   900
lemma (in bounded_linear) tendsto_zero:
huffman@44195
   901
  "(g ---> 0) F \<Longrightarrow> ((\<lambda>x. f (g x)) ---> 0) F"
huffman@44194
   902
  by (drule tendsto, simp only: zero)
huffman@44194
   903
huffman@44282
   904
lemma (in bounded_bilinear) tendsto:
huffman@44195
   905
  "\<lbrakk>(f ---> a) F; (g ---> b) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x ** g x) ---> a ** b) F"
huffman@44081
   906
  by (simp only: tendsto_Zfun_iff prod_diff_prod
huffman@44081
   907
                 Zfun_add Zfun Zfun_left Zfun_right)
huffman@31349
   908
huffman@44194
   909
lemma (in bounded_bilinear) tendsto_zero:
huffman@44195
   910
  assumes f: "(f ---> 0) F"
huffman@44195
   911
  assumes g: "(g ---> 0) F"
huffman@44195
   912
  shows "((\<lambda>x. f x ** g x) ---> 0) F"
huffman@44194
   913
  using tendsto [OF f g] by (simp add: zero_left)
huffman@31355
   914
huffman@44194
   915
lemma (in bounded_bilinear) tendsto_left_zero:
huffman@44195
   916
  "(f ---> 0) F \<Longrightarrow> ((\<lambda>x. f x ** c) ---> 0) F"
huffman@44194
   917
  by (rule bounded_linear.tendsto_zero [OF bounded_linear_left])
huffman@44194
   918
huffman@44194
   919
lemma (in bounded_bilinear) tendsto_right_zero:
huffman@44195
   920
  "(f ---> 0) F \<Longrightarrow> ((\<lambda>x. c ** f x) ---> 0) F"
huffman@44194
   921
  by (rule bounded_linear.tendsto_zero [OF bounded_linear_right])
huffman@44194
   922
huffman@44282
   923
lemmas tendsto_of_real [tendsto_intros] =
huffman@44282
   924
  bounded_linear.tendsto [OF bounded_linear_of_real]
huffman@44282
   925
huffman@44282
   926
lemmas tendsto_scaleR [tendsto_intros] =
huffman@44282
   927
  bounded_bilinear.tendsto [OF bounded_bilinear_scaleR]
huffman@44282
   928
huffman@44282
   929
lemmas tendsto_mult [tendsto_intros] =
huffman@44282
   930
  bounded_bilinear.tendsto [OF bounded_bilinear_mult]
huffman@44194
   931
huffman@44568
   932
lemmas tendsto_mult_zero =
huffman@44568
   933
  bounded_bilinear.tendsto_zero [OF bounded_bilinear_mult]
huffman@44568
   934
huffman@44568
   935
lemmas tendsto_mult_left_zero =
huffman@44568
   936
  bounded_bilinear.tendsto_left_zero [OF bounded_bilinear_mult]
huffman@44568
   937
huffman@44568
   938
lemmas tendsto_mult_right_zero =
huffman@44568
   939
  bounded_bilinear.tendsto_right_zero [OF bounded_bilinear_mult]
huffman@44568
   940
huffman@44194
   941
lemma tendsto_power [tendsto_intros]:
huffman@44194
   942
  fixes f :: "'a \<Rightarrow> 'b::{power,real_normed_algebra}"
huffman@44195
   943
  shows "(f ---> a) F \<Longrightarrow> ((\<lambda>x. f x ^ n) ---> a ^ n) F"
huffman@44194
   944
  by (induct n) (simp_all add: tendsto_const tendsto_mult)
huffman@44194
   945
huffman@44194
   946
lemma tendsto_setprod [tendsto_intros]:
huffman@44194
   947
  fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c::{real_normed_algebra,comm_ring_1}"
huffman@44195
   948
  assumes "\<And>i. i \<in> S \<Longrightarrow> (f i ---> L i) F"
huffman@44195
   949
  shows "((\<lambda>x. \<Prod>i\<in>S. f i x) ---> (\<Prod>i\<in>S. L i)) F"
huffman@44194
   950
proof (cases "finite S")
huffman@44194
   951
  assume "finite S" thus ?thesis using assms
huffman@44194
   952
    by (induct, simp add: tendsto_const, simp add: tendsto_mult)
huffman@44194
   953
next
huffman@44194
   954
  assume "\<not> finite S" thus ?thesis
huffman@44194
   955
    by (simp add: tendsto_const)
huffman@44194
   956
qed
huffman@44194
   957
huffman@44194
   958
subsubsection {* Inverse and division *}
huffman@31355
   959
huffman@31355
   960
lemma (in bounded_bilinear) Zfun_prod_Bfun:
huffman@44195
   961
  assumes f: "Zfun f F"
huffman@44195
   962
  assumes g: "Bfun g F"
huffman@44195
   963
  shows "Zfun (\<lambda>x. f x ** g x) F"
huffman@31355
   964
proof -
huffman@31355
   965
  obtain K where K: "0 \<le> K"
huffman@31355
   966
    and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"
huffman@31355
   967
    using nonneg_bounded by fast
huffman@31355
   968
  obtain B where B: "0 < B"
huffman@44195
   969
    and norm_g: "eventually (\<lambda>x. norm (g x) \<le> B) F"
huffman@31487
   970
    using g by (rule BfunE)
huffman@44195
   971
  have "eventually (\<lambda>x. norm (f x ** g x) \<le> norm (f x) * (B * K)) F"
noschinl@46887
   972
  using norm_g proof eventually_elim
noschinl@46887
   973
    case (elim x)
huffman@31487
   974
    have "norm (f x ** g x) \<le> norm (f x) * norm (g x) * K"
huffman@31355
   975
      by (rule norm_le)
huffman@31487
   976
    also have "\<dots> \<le> norm (f x) * B * K"
huffman@31487
   977
      by (intro mult_mono' order_refl norm_g norm_ge_zero
noschinl@46887
   978
                mult_nonneg_nonneg K elim)
huffman@31487
   979
    also have "\<dots> = norm (f x) * (B * K)"
huffman@31355
   980
      by (rule mult_assoc)
huffman@31487
   981
    finally show "norm (f x ** g x) \<le> norm (f x) * (B * K)" .
huffman@31355
   982
  qed
huffman@31487
   983
  with f show ?thesis
huffman@31487
   984
    by (rule Zfun_imp_Zfun)
huffman@31355
   985
qed
huffman@31355
   986
huffman@31355
   987
lemma (in bounded_bilinear) flip:
huffman@31355
   988
  "bounded_bilinear (\<lambda>x y. y ** x)"
huffman@44081
   989
  apply default
huffman@44081
   990
  apply (rule add_right)
huffman@44081
   991
  apply (rule add_left)
huffman@44081
   992
  apply (rule scaleR_right)
huffman@44081
   993
  apply (rule scaleR_left)
huffman@44081
   994
  apply (subst mult_commute)
huffman@44081
   995
  using bounded by fast
huffman@31355
   996
huffman@31355
   997
lemma (in bounded_bilinear) Bfun_prod_Zfun:
huffman@44195
   998
  assumes f: "Bfun f F"
huffman@44195
   999
  assumes g: "Zfun g F"
huffman@44195
  1000
  shows "Zfun (\<lambda>x. f x ** g x) F"
huffman@44081
  1001
  using flip g f by (rule bounded_bilinear.Zfun_prod_Bfun)
huffman@31355
  1002
huffman@31355
  1003
lemma Bfun_inverse_lemma:
huffman@31355
  1004
  fixes x :: "'a::real_normed_div_algebra"
huffman@31355
  1005
  shows "\<lbrakk>r \<le> norm x; 0 < r\<rbrakk> \<Longrightarrow> norm (inverse x) \<le> inverse r"
huffman@44081
  1006
  apply (subst nonzero_norm_inverse, clarsimp)
huffman@44081
  1007
  apply (erule (1) le_imp_inverse_le)
huffman@44081
  1008
  done
huffman@31355
  1009
huffman@31355
  1010
lemma Bfun_inverse:
huffman@31355
  1011
  fixes a :: "'a::real_normed_div_algebra"
huffman@44195
  1012
  assumes f: "(f ---> a) F"
huffman@31355
  1013
  assumes a: "a \<noteq> 0"
huffman@44195
  1014
  shows "Bfun (\<lambda>x. inverse (f x)) F"
huffman@31355
  1015
proof -
huffman@31355
  1016
  from a have "0 < norm a" by simp
huffman@31355
  1017
  hence "\<exists>r>0. r < norm a" by (rule dense)
huffman@31355
  1018
  then obtain r where r1: "0 < r" and r2: "r < norm a" by fast
huffman@44195
  1019
  have "eventually (\<lambda>x. dist (f x) a < r) F"
huffman@31487
  1020
    using tendstoD [OF f r1] by fast
huffman@44195
  1021
  hence "eventually (\<lambda>x. norm (inverse (f x)) \<le> inverse (norm a - r)) F"
noschinl@46887
  1022
  proof eventually_elim
noschinl@46887
  1023
    case (elim x)
huffman@31487
  1024
    hence 1: "norm (f x - a) < r"
huffman@31355
  1025
      by (simp add: dist_norm)
huffman@31487
  1026
    hence 2: "f x \<noteq> 0" using r2 by auto
huffman@31487
  1027
    hence "norm (inverse (f x)) = inverse (norm (f x))"
huffman@31355
  1028
      by (rule nonzero_norm_inverse)
huffman@31355
  1029
    also have "\<dots> \<le> inverse (norm a - r)"
huffman@31355
  1030
    proof (rule le_imp_inverse_le)
huffman@31355
  1031
      show "0 < norm a - r" using r2 by simp
huffman@31355
  1032
    next
huffman@31487
  1033
      have "norm a - norm (f x) \<le> norm (a - f x)"
huffman@31355
  1034
        by (rule norm_triangle_ineq2)
huffman@31487
  1035
      also have "\<dots> = norm (f x - a)"
huffman@31355
  1036
        by (rule norm_minus_commute)
huffman@31355
  1037
      also have "\<dots> < r" using 1 .
huffman@31487
  1038
      finally show "norm a - r \<le> norm (f x)" by simp
huffman@31355
  1039
    qed
huffman@31487
  1040
    finally show "norm (inverse (f x)) \<le> inverse (norm a - r)" .
huffman@31355
  1041
  qed
huffman@31355
  1042
  thus ?thesis by (rule BfunI)
huffman@31355
  1043
qed
huffman@31355
  1044
huffman@31565
  1045
lemma tendsto_inverse [tendsto_intros]:
huffman@31355
  1046
  fixes a :: "'a::real_normed_div_algebra"
huffman@44195
  1047
  assumes f: "(f ---> a) F"
huffman@31355
  1048
  assumes a: "a \<noteq> 0"
huffman@44195
  1049
  shows "((\<lambda>x. inverse (f x)) ---> inverse a) F"
huffman@31355
  1050
proof -
huffman@31355
  1051
  from a have "0 < norm a" by simp
huffman@44195
  1052
  with f have "eventually (\<lambda>x. dist (f x) a < norm a) F"
huffman@31355
  1053
    by (rule tendstoD)
huffman@44195
  1054
  then have "eventually (\<lambda>x. f x \<noteq> 0) F"
huffman@31355
  1055
    unfolding dist_norm by (auto elim!: eventually_elim1)
huffman@44627
  1056
  with a have "eventually (\<lambda>x. inverse (f x) - inverse a =
huffman@44627
  1057
    - (inverse (f x) * (f x - a) * inverse a)) F"
huffman@44627
  1058
    by (auto elim!: eventually_elim1 simp: inverse_diff_inverse)
huffman@44627
  1059
  moreover have "Zfun (\<lambda>x. - (inverse (f x) * (f x - a) * inverse a)) F"
huffman@44627
  1060
    by (intro Zfun_minus Zfun_mult_left
huffman@44627
  1061
      bounded_bilinear.Bfun_prod_Zfun [OF bounded_bilinear_mult]
huffman@44627
  1062
      Bfun_inverse [OF f a] f [unfolded tendsto_Zfun_iff])
huffman@44627
  1063
  ultimately show ?thesis
huffman@44627
  1064
    unfolding tendsto_Zfun_iff by (rule Zfun_ssubst)
huffman@31355
  1065
qed
huffman@31355
  1066
huffman@31565
  1067
lemma tendsto_divide [tendsto_intros]:
huffman@31355
  1068
  fixes a b :: "'a::real_normed_field"
huffman@44195
  1069
  shows "\<lbrakk>(f ---> a) F; (g ---> b) F; b \<noteq> 0\<rbrakk>
huffman@44195
  1070
    \<Longrightarrow> ((\<lambda>x. f x / g x) ---> a / b) F"
huffman@44282
  1071
  by (simp add: tendsto_mult tendsto_inverse divide_inverse)
huffman@31355
  1072
huffman@44194
  1073
lemma tendsto_sgn [tendsto_intros]:
huffman@44194
  1074
  fixes l :: "'a::real_normed_vector"
huffman@44195
  1075
  shows "\<lbrakk>(f ---> l) F; l \<noteq> 0\<rbrakk> \<Longrightarrow> ((\<lambda>x. sgn (f x)) ---> sgn l) F"
huffman@44194
  1076
  unfolding sgn_div_norm by (simp add: tendsto_intros)
huffman@44194
  1077
hoelzl@50247
  1078
subsection {* Limits to @{const at_top} and @{const at_bot} *}
hoelzl@50247
  1079
hoelzl@50322
  1080
lemma filterlim_at_top:
hoelzl@50247
  1081
  fixes f :: "'a \<Rightarrow> ('b::dense_linorder)"
hoelzl@50247
  1082
  shows "(LIM x F. f x :> at_top) \<longleftrightarrow> (\<forall>Z. eventually (\<lambda>x. Z < f x) F)"
hoelzl@50322
  1083
  by (safe elim!: filterlimE intro!: filterlimI)
hoelzl@50247
  1084
     (auto simp: eventually_at_top_dense elim!: eventually_elim1)
hoelzl@50247
  1085
hoelzl@50323
  1086
lemma filterlim_at_top_gt:
hoelzl@50323
  1087
  fixes f :: "'a \<Rightarrow> ('b::dense_linorder)" and c :: "'b"
hoelzl@50323
  1088
  shows "(LIM x F. f x :> at_top) \<longleftrightarrow> (\<forall>Z>c. eventually (\<lambda>x. Z < f x) F)"
hoelzl@50323
  1089
  unfolding filterlim_at_top
hoelzl@50323
  1090
proof safe
hoelzl@50323
  1091
  fix Z assume *: "\<forall>Z>c. eventually (\<lambda>x. Z < f x) F"
hoelzl@50323
  1092
  from gt_ex[of "max Z c"] guess x ..
hoelzl@50323
  1093
  with *[THEN spec, of x] show "eventually (\<lambda>x. Z < f x) F"
hoelzl@50323
  1094
    by (auto elim!: eventually_elim1)
hoelzl@50323
  1095
qed simp
hoelzl@50323
  1096
hoelzl@50322
  1097
lemma filterlim_at_bot: 
hoelzl@50247
  1098
  fixes f :: "'a \<Rightarrow> ('b::dense_linorder)"
hoelzl@50247
  1099
  shows "(LIM x F. f x :> at_bot) \<longleftrightarrow> (\<forall>Z. eventually (\<lambda>x. f x < Z) F)"
hoelzl@50322
  1100
  by (safe elim!: filterlimE intro!: filterlimI)
hoelzl@50247
  1101
     (auto simp: eventually_at_bot_dense elim!: eventually_elim1)
hoelzl@50247
  1102
hoelzl@50323
  1103
lemma filterlim_at_bot_lt:
hoelzl@50323
  1104
  fixes f :: "'a \<Rightarrow> ('b::dense_linorder)" and c :: "'b"
hoelzl@50323
  1105
  shows "(LIM x F. f x :> at_bot) \<longleftrightarrow> (\<forall>Z<c. eventually (\<lambda>x. Z > f x) F)"
hoelzl@50323
  1106
  unfolding filterlim_at_bot
hoelzl@50323
  1107
proof safe
hoelzl@50323
  1108
  fix Z assume *: "\<forall>Z<c. eventually (\<lambda>x. Z > f x) F"
hoelzl@50323
  1109
  from lt_ex[of "min Z c"] guess x ..
hoelzl@50323
  1110
  with *[THEN spec, of x] show "eventually (\<lambda>x. Z > f x) F"
hoelzl@50323
  1111
    by (auto elim!: eventually_elim1)
hoelzl@50323
  1112
qed simp
hoelzl@50323
  1113
hoelzl@50322
  1114
lemma filterlim_real_sequentially: "LIM x sequentially. real x :> at_top"
hoelzl@50322
  1115
  unfolding filterlim_at_top
hoelzl@50247
  1116
  apply (intro allI)
hoelzl@50247
  1117
  apply (rule_tac c="natceiling (Z + 1)" in eventually_sequentiallyI)
hoelzl@50247
  1118
  apply (auto simp: natceiling_le_eq)
hoelzl@50247
  1119
  done
hoelzl@50247
  1120
hoelzl@50323
  1121
lemma filterlim_inverse_at_top_pos:
hoelzl@50323
  1122
  "LIM x (nhds 0 within {0::real <..}). inverse x :> at_top"
hoelzl@50323
  1123
  unfolding filterlim_at_top_gt[where c=0] eventually_within
hoelzl@50323
  1124
proof safe
hoelzl@50323
  1125
  fix Z :: real assume [arith]: "0 < Z"
hoelzl@50323
  1126
  then have "eventually (\<lambda>x. x < inverse Z) (nhds 0)"
hoelzl@50323
  1127
    by (auto simp add: eventually_nhds_metric dist_real_def intro!: exI[of _ "\<bar>inverse Z\<bar>"])
hoelzl@50323
  1128
  then show "eventually (\<lambda>x. x \<in> {0<..} \<longrightarrow> Z < inverse x) (nhds 0)"
hoelzl@50323
  1129
    by (auto elim!: eventually_elim1 simp: inverse_eq_divide field_simps)
hoelzl@50323
  1130
qed
hoelzl@50323
  1131
hoelzl@50323
  1132
lemma filterlim_inverse_at_top:
hoelzl@50323
  1133
  "(f ---> (0 :: real)) F \<Longrightarrow> eventually (\<lambda>x. 0 < f x) F \<Longrightarrow> LIM x F. inverse (f x) :> at_top"
hoelzl@50323
  1134
  by (intro filterlim_compose[OF filterlim_inverse_at_top_pos])
hoelzl@50323
  1135
     (simp add: filterlim_def eventually_filtermap le_within_iff)
hoelzl@50323
  1136
hoelzl@50323
  1137
lemma filterlim_inverse_at_bot_neg:
hoelzl@50323
  1138
  "LIM x (nhds 0 within {..< 0::real}). inverse x :> at_bot"
hoelzl@50323
  1139
  unfolding filterlim_at_bot_lt[where c=0] eventually_within
hoelzl@50323
  1140
proof safe
hoelzl@50323
  1141
  fix Z :: real assume [arith]: "Z < 0"
hoelzl@50323
  1142
  have "eventually (\<lambda>x. inverse Z < x) (nhds 0)"
hoelzl@50323
  1143
    by (auto simp add: eventually_nhds_metric dist_real_def intro!: exI[of _ "\<bar>inverse Z\<bar>"])
hoelzl@50323
  1144
  then show "eventually (\<lambda>x. x \<in> {..< 0} \<longrightarrow> inverse x < Z) (nhds 0)"
hoelzl@50323
  1145
    by (auto elim!: eventually_elim1 simp: inverse_eq_divide field_simps)
hoelzl@50323
  1146
qed
hoelzl@50323
  1147
hoelzl@50323
  1148
lemma filterlim_inverse_at_bot:
hoelzl@50323
  1149
  "(f ---> (0 :: real)) F \<Longrightarrow> eventually (\<lambda>x. f x < 0) F \<Longrightarrow> LIM x F. inverse (f x) :> at_bot"
hoelzl@50323
  1150
  by (intro filterlim_compose[OF filterlim_inverse_at_bot_neg])
hoelzl@50323
  1151
     (simp add: filterlim_def eventually_filtermap le_within_iff)
hoelzl@50323
  1152
hoelzl@50323
  1153
lemma filterlim_uminus_at_top_at_bot: "LIM x at_bot. - x :: real :> at_top"
hoelzl@50323
  1154
  unfolding filterlim_at_top eventually_at_bot_dense
hoelzl@50323
  1155
  by (blast intro: less_minus_iff[THEN iffD1])
hoelzl@50323
  1156
hoelzl@50323
  1157
lemma filterlim_uminus_at_top: "LIM x F. f x :> at_bot \<Longrightarrow> LIM x F. - (f x) :: real :> at_top"
hoelzl@50323
  1158
  by (rule filterlim_compose[OF filterlim_uminus_at_top_at_bot])
hoelzl@50323
  1159
hoelzl@50323
  1160
lemma filterlim_uminus_at_bot_at_top: "LIM x at_top. - x :: real :> at_bot"
hoelzl@50323
  1161
  unfolding filterlim_at_bot eventually_at_top_dense
hoelzl@50323
  1162
  by (blast intro: minus_less_iff[THEN iffD1])
hoelzl@50323
  1163
hoelzl@50323
  1164
lemma filterlim_uminus_at_bot: "LIM x F. f x :> at_top \<Longrightarrow> LIM x F. - (f x) :: real :> at_bot"
hoelzl@50323
  1165
  by (rule filterlim_compose[OF filterlim_uminus_at_bot_at_top])
hoelzl@50323
  1166
huffman@31349
  1167
end