src/HOL/Limits.thy
 author hoelzl Mon Jan 14 17:16:59 2013 +0100 (2013-01-14) changeset 50880 b22ecedde1c7 parent 50419 3177d0374701 child 50999 3de230ed0547 permissions -rw-r--r--
move eventually_Ball_finite to Limits
```     1 (*  Title       : Limits.thy
```
```     2     Author      : Brian Huffman
```
```     3 *)
```
```     4
```
```     5 header {* Filters and Limits *}
```
```     6
```
```     7 theory Limits
```
```     8 imports RealVector
```
```     9 begin
```
```    10
```
```    11 subsection {* Filters *}
```
```    12
```
```    13 text {*
```
```    14   This definition also allows non-proper filters.
```
```    15 *}
```
```    16
```
```    17 locale is_filter =
```
```    18   fixes F :: "('a \<Rightarrow> bool) \<Rightarrow> bool"
```
```    19   assumes True: "F (\<lambda>x. True)"
```
```    20   assumes conj: "F (\<lambda>x. P x) \<Longrightarrow> F (\<lambda>x. Q x) \<Longrightarrow> F (\<lambda>x. P x \<and> Q x)"
```
```    21   assumes mono: "\<forall>x. P x \<longrightarrow> Q x \<Longrightarrow> F (\<lambda>x. P x) \<Longrightarrow> F (\<lambda>x. Q x)"
```
```    22
```
```    23 typedef 'a filter = "{F :: ('a \<Rightarrow> bool) \<Rightarrow> bool. is_filter F}"
```
```    24 proof
```
```    25   show "(\<lambda>x. True) \<in> ?filter" by (auto intro: is_filter.intro)
```
```    26 qed
```
```    27
```
```    28 lemma is_filter_Rep_filter: "is_filter (Rep_filter F)"
```
```    29   using Rep_filter [of F] by simp
```
```    30
```
```    31 lemma Abs_filter_inverse':
```
```    32   assumes "is_filter F" shows "Rep_filter (Abs_filter F) = F"
```
```    33   using assms by (simp add: Abs_filter_inverse)
```
```    34
```
```    35
```
```    36 subsection {* Eventually *}
```
```    37
```
```    38 definition eventually :: "('a \<Rightarrow> bool) \<Rightarrow> 'a filter \<Rightarrow> bool"
```
```    39   where "eventually P F \<longleftrightarrow> Rep_filter F P"
```
```    40
```
```    41 lemma eventually_Abs_filter:
```
```    42   assumes "is_filter F" shows "eventually P (Abs_filter F) = F P"
```
```    43   unfolding eventually_def using assms by (simp add: Abs_filter_inverse)
```
```    44
```
```    45 lemma filter_eq_iff:
```
```    46   shows "F = F' \<longleftrightarrow> (\<forall>P. eventually P F = eventually P F')"
```
```    47   unfolding Rep_filter_inject [symmetric] fun_eq_iff eventually_def ..
```
```    48
```
```    49 lemma eventually_True [simp]: "eventually (\<lambda>x. True) F"
```
```    50   unfolding eventually_def
```
```    51   by (rule is_filter.True [OF is_filter_Rep_filter])
```
```    52
```
```    53 lemma always_eventually: "\<forall>x. P x \<Longrightarrow> eventually P F"
```
```    54 proof -
```
```    55   assume "\<forall>x. P x" hence "P = (\<lambda>x. True)" by (simp add: ext)
```
```    56   thus "eventually P F" by simp
```
```    57 qed
```
```    58
```
```    59 lemma eventually_mono:
```
```    60   "(\<forall>x. P x \<longrightarrow> Q x) \<Longrightarrow> eventually P F \<Longrightarrow> eventually Q F"
```
```    61   unfolding eventually_def
```
```    62   by (rule is_filter.mono [OF is_filter_Rep_filter])
```
```    63
```
```    64 lemma eventually_conj:
```
```    65   assumes P: "eventually (\<lambda>x. P x) F"
```
```    66   assumes Q: "eventually (\<lambda>x. Q x) F"
```
```    67   shows "eventually (\<lambda>x. P x \<and> Q x) F"
```
```    68   using assms unfolding eventually_def
```
```    69   by (rule is_filter.conj [OF is_filter_Rep_filter])
```
```    70
```
```    71 lemma eventually_Ball_finite:
```
```    72   assumes "finite A" and "\<forall>y\<in>A. eventually (\<lambda>x. P x y) net"
```
```    73   shows "eventually (\<lambda>x. \<forall>y\<in>A. P x y) net"
```
```    74 using assms by (induct set: finite, simp, simp add: eventually_conj)
```
```    75
```
```    76 lemma eventually_all_finite:
```
```    77   fixes P :: "'a \<Rightarrow> 'b::finite \<Rightarrow> bool"
```
```    78   assumes "\<And>y. eventually (\<lambda>x. P x y) net"
```
```    79   shows "eventually (\<lambda>x. \<forall>y. P x y) net"
```
```    80 using eventually_Ball_finite [of UNIV P] assms by simp
```
```    81
```
```    82 lemma eventually_mp:
```
```    83   assumes "eventually (\<lambda>x. P x \<longrightarrow> Q x) F"
```
```    84   assumes "eventually (\<lambda>x. P x) F"
```
```    85   shows "eventually (\<lambda>x. Q x) F"
```
```    86 proof (rule eventually_mono)
```
```    87   show "\<forall>x. (P x \<longrightarrow> Q x) \<and> P x \<longrightarrow> Q x" by simp
```
```    88   show "eventually (\<lambda>x. (P x \<longrightarrow> Q x) \<and> P x) F"
```
```    89     using assms by (rule eventually_conj)
```
```    90 qed
```
```    91
```
```    92 lemma eventually_rev_mp:
```
```    93   assumes "eventually (\<lambda>x. P x) F"
```
```    94   assumes "eventually (\<lambda>x. P x \<longrightarrow> Q x) F"
```
```    95   shows "eventually (\<lambda>x. Q x) F"
```
```    96 using assms(2) assms(1) by (rule eventually_mp)
```
```    97
```
```    98 lemma eventually_conj_iff:
```
```    99   "eventually (\<lambda>x. P x \<and> Q x) F \<longleftrightarrow> eventually P F \<and> eventually Q F"
```
```   100   by (auto intro: eventually_conj elim: eventually_rev_mp)
```
```   101
```
```   102 lemma eventually_elim1:
```
```   103   assumes "eventually (\<lambda>i. P i) F"
```
```   104   assumes "\<And>i. P i \<Longrightarrow> Q i"
```
```   105   shows "eventually (\<lambda>i. Q i) F"
```
```   106   using assms by (auto elim!: eventually_rev_mp)
```
```   107
```
```   108 lemma eventually_elim2:
```
```   109   assumes "eventually (\<lambda>i. P i) F"
```
```   110   assumes "eventually (\<lambda>i. Q i) F"
```
```   111   assumes "\<And>i. P i \<Longrightarrow> Q i \<Longrightarrow> R i"
```
```   112   shows "eventually (\<lambda>i. R i) F"
```
```   113   using assms by (auto elim!: eventually_rev_mp)
```
```   114
```
```   115 lemma eventually_subst:
```
```   116   assumes "eventually (\<lambda>n. P n = Q n) F"
```
```   117   shows "eventually P F = eventually Q F" (is "?L = ?R")
```
```   118 proof -
```
```   119   from assms have "eventually (\<lambda>x. P x \<longrightarrow> Q x) F"
```
```   120       and "eventually (\<lambda>x. Q x \<longrightarrow> P x) F"
```
```   121     by (auto elim: eventually_elim1)
```
```   122   then show ?thesis by (auto elim: eventually_elim2)
```
```   123 qed
```
```   124
```
```   125 ML {*
```
```   126   fun eventually_elim_tac ctxt thms thm =
```
```   127     let
```
```   128       val thy = Proof_Context.theory_of ctxt
```
```   129       val mp_thms = thms RL [@{thm eventually_rev_mp}]
```
```   130       val raw_elim_thm =
```
```   131         (@{thm allI} RS @{thm always_eventually})
```
```   132         |> fold (fn thm1 => fn thm2 => thm2 RS thm1) mp_thms
```
```   133         |> fold (fn _ => fn thm => @{thm impI} RS thm) thms
```
```   134       val cases_prop = prop_of (raw_elim_thm RS thm)
```
```   135       val cases = (Rule_Cases.make_common (thy, cases_prop) [(("elim", []), [])])
```
```   136     in
```
```   137       CASES cases (rtac raw_elim_thm 1) thm
```
```   138     end
```
```   139 *}
```
```   140
```
```   141 method_setup eventually_elim = {*
```
```   142   Scan.succeed (fn ctxt => METHOD_CASES (eventually_elim_tac ctxt))
```
```   143 *} "elimination of eventually quantifiers"
```
```   144
```
```   145
```
```   146 subsection {* Finer-than relation *}
```
```   147
```
```   148 text {* @{term "F \<le> F'"} means that filter @{term F} is finer than
```
```   149 filter @{term F'}. *}
```
```   150
```
```   151 instantiation filter :: (type) complete_lattice
```
```   152 begin
```
```   153
```
```   154 definition le_filter_def:
```
```   155   "F \<le> F' \<longleftrightarrow> (\<forall>P. eventually P F' \<longrightarrow> eventually P F)"
```
```   156
```
```   157 definition
```
```   158   "(F :: 'a filter) < F' \<longleftrightarrow> F \<le> F' \<and> \<not> F' \<le> F"
```
```   159
```
```   160 definition
```
```   161   "top = Abs_filter (\<lambda>P. \<forall>x. P x)"
```
```   162
```
```   163 definition
```
```   164   "bot = Abs_filter (\<lambda>P. True)"
```
```   165
```
```   166 definition
```
```   167   "sup F F' = Abs_filter (\<lambda>P. eventually P F \<and> eventually P F')"
```
```   168
```
```   169 definition
```
```   170   "inf F F' = Abs_filter
```
```   171       (\<lambda>P. \<exists>Q R. eventually Q F \<and> eventually R F' \<and> (\<forall>x. Q x \<and> R x \<longrightarrow> P x))"
```
```   172
```
```   173 definition
```
```   174   "Sup S = Abs_filter (\<lambda>P. \<forall>F\<in>S. eventually P F)"
```
```   175
```
```   176 definition
```
```   177   "Inf S = Sup {F::'a filter. \<forall>F'\<in>S. F \<le> F'}"
```
```   178
```
```   179 lemma eventually_top [simp]: "eventually P top \<longleftrightarrow> (\<forall>x. P x)"
```
```   180   unfolding top_filter_def
```
```   181   by (rule eventually_Abs_filter, rule is_filter.intro, auto)
```
```   182
```
```   183 lemma eventually_bot [simp]: "eventually P bot"
```
```   184   unfolding bot_filter_def
```
```   185   by (subst eventually_Abs_filter, rule is_filter.intro, auto)
```
```   186
```
```   187 lemma eventually_sup:
```
```   188   "eventually P (sup F F') \<longleftrightarrow> eventually P F \<and> eventually P F'"
```
```   189   unfolding sup_filter_def
```
```   190   by (rule eventually_Abs_filter, rule is_filter.intro)
```
```   191      (auto elim!: eventually_rev_mp)
```
```   192
```
```   193 lemma eventually_inf:
```
```   194   "eventually P (inf F F') \<longleftrightarrow>
```
```   195    (\<exists>Q R. eventually Q F \<and> eventually R F' \<and> (\<forall>x. Q x \<and> R x \<longrightarrow> P x))"
```
```   196   unfolding inf_filter_def
```
```   197   apply (rule eventually_Abs_filter, rule is_filter.intro)
```
```   198   apply (fast intro: eventually_True)
```
```   199   apply clarify
```
```   200   apply (intro exI conjI)
```
```   201   apply (erule (1) eventually_conj)
```
```   202   apply (erule (1) eventually_conj)
```
```   203   apply simp
```
```   204   apply auto
```
```   205   done
```
```   206
```
```   207 lemma eventually_Sup:
```
```   208   "eventually P (Sup S) \<longleftrightarrow> (\<forall>F\<in>S. eventually P F)"
```
```   209   unfolding Sup_filter_def
```
```   210   apply (rule eventually_Abs_filter, rule is_filter.intro)
```
```   211   apply (auto intro: eventually_conj elim!: eventually_rev_mp)
```
```   212   done
```
```   213
```
```   214 instance proof
```
```   215   fix F F' F'' :: "'a filter" and S :: "'a filter set"
```
```   216   { show "F < F' \<longleftrightarrow> F \<le> F' \<and> \<not> F' \<le> F"
```
```   217     by (rule less_filter_def) }
```
```   218   { show "F \<le> F"
```
```   219     unfolding le_filter_def by simp }
```
```   220   { assume "F \<le> F'" and "F' \<le> F''" thus "F \<le> F''"
```
```   221     unfolding le_filter_def by simp }
```
```   222   { assume "F \<le> F'" and "F' \<le> F" thus "F = F'"
```
```   223     unfolding le_filter_def filter_eq_iff by fast }
```
```   224   { show "F \<le> top"
```
```   225     unfolding le_filter_def eventually_top by (simp add: always_eventually) }
```
```   226   { show "bot \<le> F"
```
```   227     unfolding le_filter_def by simp }
```
```   228   { show "F \<le> sup F F'" and "F' \<le> sup F F'"
```
```   229     unfolding le_filter_def eventually_sup by simp_all }
```
```   230   { assume "F \<le> F''" and "F' \<le> F''" thus "sup F F' \<le> F''"
```
```   231     unfolding le_filter_def eventually_sup by simp }
```
```   232   { show "inf F F' \<le> F" and "inf F F' \<le> F'"
```
```   233     unfolding le_filter_def eventually_inf by (auto intro: eventually_True) }
```
```   234   { assume "F \<le> F'" and "F \<le> F''" thus "F \<le> inf F' F''"
```
```   235     unfolding le_filter_def eventually_inf
```
```   236     by (auto elim!: eventually_mono intro: eventually_conj) }
```
```   237   { assume "F \<in> S" thus "F \<le> Sup S"
```
```   238     unfolding le_filter_def eventually_Sup by simp }
```
```   239   { assume "\<And>F. F \<in> S \<Longrightarrow> F \<le> F'" thus "Sup S \<le> F'"
```
```   240     unfolding le_filter_def eventually_Sup by simp }
```
```   241   { assume "F'' \<in> S" thus "Inf S \<le> F''"
```
```   242     unfolding le_filter_def Inf_filter_def eventually_Sup Ball_def by simp }
```
```   243   { assume "\<And>F'. F' \<in> S \<Longrightarrow> F \<le> F'" thus "F \<le> Inf S"
```
```   244     unfolding le_filter_def Inf_filter_def eventually_Sup Ball_def by simp }
```
```   245 qed
```
```   246
```
```   247 end
```
```   248
```
```   249 lemma filter_leD:
```
```   250   "F \<le> F' \<Longrightarrow> eventually P F' \<Longrightarrow> eventually P F"
```
```   251   unfolding le_filter_def by simp
```
```   252
```
```   253 lemma filter_leI:
```
```   254   "(\<And>P. eventually P F' \<Longrightarrow> eventually P F) \<Longrightarrow> F \<le> F'"
```
```   255   unfolding le_filter_def by simp
```
```   256
```
```   257 lemma eventually_False:
```
```   258   "eventually (\<lambda>x. False) F \<longleftrightarrow> F = bot"
```
```   259   unfolding filter_eq_iff by (auto elim: eventually_rev_mp)
```
```   260
```
```   261 abbreviation (input) trivial_limit :: "'a filter \<Rightarrow> bool"
```
```   262   where "trivial_limit F \<equiv> F = bot"
```
```   263
```
```   264 lemma trivial_limit_def: "trivial_limit F \<longleftrightarrow> eventually (\<lambda>x. False) F"
```
```   265   by (rule eventually_False [symmetric])
```
```   266
```
```   267
```
```   268 subsection {* Map function for filters *}
```
```   269
```
```   270 definition filtermap :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a filter \<Rightarrow> 'b filter"
```
```   271   where "filtermap f F = Abs_filter (\<lambda>P. eventually (\<lambda>x. P (f x)) F)"
```
```   272
```
```   273 lemma eventually_filtermap:
```
```   274   "eventually P (filtermap f F) = eventually (\<lambda>x. P (f x)) F"
```
```   275   unfolding filtermap_def
```
```   276   apply (rule eventually_Abs_filter)
```
```   277   apply (rule is_filter.intro)
```
```   278   apply (auto elim!: eventually_rev_mp)
```
```   279   done
```
```   280
```
```   281 lemma filtermap_ident: "filtermap (\<lambda>x. x) F = F"
```
```   282   by (simp add: filter_eq_iff eventually_filtermap)
```
```   283
```
```   284 lemma filtermap_filtermap:
```
```   285   "filtermap f (filtermap g F) = filtermap (\<lambda>x. f (g x)) F"
```
```   286   by (simp add: filter_eq_iff eventually_filtermap)
```
```   287
```
```   288 lemma filtermap_mono: "F \<le> F' \<Longrightarrow> filtermap f F \<le> filtermap f F'"
```
```   289   unfolding le_filter_def eventually_filtermap by simp
```
```   290
```
```   291 lemma filtermap_bot [simp]: "filtermap f bot = bot"
```
```   292   by (simp add: filter_eq_iff eventually_filtermap)
```
```   293
```
```   294 lemma filtermap_sup: "filtermap f (sup F1 F2) = sup (filtermap f F1) (filtermap f F2)"
```
```   295   by (auto simp: filter_eq_iff eventually_filtermap eventually_sup)
```
```   296
```
```   297 subsection {* Order filters *}
```
```   298
```
```   299 definition at_top :: "('a::order) filter"
```
```   300   where "at_top = Abs_filter (\<lambda>P. \<exists>k. \<forall>n\<ge>k. P n)"
```
```   301
```
```   302 lemma eventually_at_top_linorder: "eventually P at_top \<longleftrightarrow> (\<exists>N::'a::linorder. \<forall>n\<ge>N. P n)"
```
```   303   unfolding at_top_def
```
```   304 proof (rule eventually_Abs_filter, rule is_filter.intro)
```
```   305   fix P Q :: "'a \<Rightarrow> bool"
```
```   306   assume "\<exists>i. \<forall>n\<ge>i. P n" and "\<exists>j. \<forall>n\<ge>j. Q n"
```
```   307   then obtain i j where "\<forall>n\<ge>i. P n" and "\<forall>n\<ge>j. Q n" by auto
```
```   308   then have "\<forall>n\<ge>max i j. P n \<and> Q n" by simp
```
```   309   then show "\<exists>k. \<forall>n\<ge>k. P n \<and> Q n" ..
```
```   310 qed auto
```
```   311
```
```   312 lemma eventually_ge_at_top:
```
```   313   "eventually (\<lambda>x. (c::_::linorder) \<le> x) at_top"
```
```   314   unfolding eventually_at_top_linorder by auto
```
```   315
```
```   316 lemma eventually_at_top_dense: "eventually P at_top \<longleftrightarrow> (\<exists>N::'a::dense_linorder. \<forall>n>N. P n)"
```
```   317   unfolding eventually_at_top_linorder
```
```   318 proof safe
```
```   319   fix N assume "\<forall>n\<ge>N. P n" then show "\<exists>N. \<forall>n>N. P n" by (auto intro!: exI[of _ N])
```
```   320 next
```
```   321   fix N assume "\<forall>n>N. P n"
```
```   322   moreover from gt_ex[of N] guess y ..
```
```   323   ultimately show "\<exists>N. \<forall>n\<ge>N. P n" by (auto intro!: exI[of _ y])
```
```   324 qed
```
```   325
```
```   326 lemma eventually_gt_at_top:
```
```   327   "eventually (\<lambda>x. (c::_::dense_linorder) < x) at_top"
```
```   328   unfolding eventually_at_top_dense by auto
```
```   329
```
```   330 definition at_bot :: "('a::order) filter"
```
```   331   where "at_bot = Abs_filter (\<lambda>P. \<exists>k. \<forall>n\<le>k. P n)"
```
```   332
```
```   333 lemma eventually_at_bot_linorder:
```
```   334   fixes P :: "'a::linorder \<Rightarrow> bool" shows "eventually P at_bot \<longleftrightarrow> (\<exists>N. \<forall>n\<le>N. P n)"
```
```   335   unfolding at_bot_def
```
```   336 proof (rule eventually_Abs_filter, rule is_filter.intro)
```
```   337   fix P Q :: "'a \<Rightarrow> bool"
```
```   338   assume "\<exists>i. \<forall>n\<le>i. P n" and "\<exists>j. \<forall>n\<le>j. Q n"
```
```   339   then obtain i j where "\<forall>n\<le>i. P n" and "\<forall>n\<le>j. Q n" by auto
```
```   340   then have "\<forall>n\<le>min i j. P n \<and> Q n" by simp
```
```   341   then show "\<exists>k. \<forall>n\<le>k. P n \<and> Q n" ..
```
```   342 qed auto
```
```   343
```
```   344 lemma eventually_le_at_bot:
```
```   345   "eventually (\<lambda>x. x \<le> (c::_::linorder)) at_bot"
```
```   346   unfolding eventually_at_bot_linorder by auto
```
```   347
```
```   348 lemma eventually_at_bot_dense:
```
```   349   fixes P :: "'a::dense_linorder \<Rightarrow> bool" shows "eventually P at_bot \<longleftrightarrow> (\<exists>N. \<forall>n<N. P n)"
```
```   350   unfolding eventually_at_bot_linorder
```
```   351 proof safe
```
```   352   fix N assume "\<forall>n\<le>N. P n" then show "\<exists>N. \<forall>n<N. P n" by (auto intro!: exI[of _ N])
```
```   353 next
```
```   354   fix N assume "\<forall>n<N. P n"
```
```   355   moreover from lt_ex[of N] guess y ..
```
```   356   ultimately show "\<exists>N. \<forall>n\<le>N. P n" by (auto intro!: exI[of _ y])
```
```   357 qed
```
```   358
```
```   359 lemma eventually_gt_at_bot:
```
```   360   "eventually (\<lambda>x. x < (c::_::dense_linorder)) at_bot"
```
```   361   unfolding eventually_at_bot_dense by auto
```
```   362
```
```   363 subsection {* Sequentially *}
```
```   364
```
```   365 abbreviation sequentially :: "nat filter"
```
```   366   where "sequentially == at_top"
```
```   367
```
```   368 lemma sequentially_def: "sequentially = Abs_filter (\<lambda>P. \<exists>k. \<forall>n\<ge>k. P n)"
```
```   369   unfolding at_top_def by simp
```
```   370
```
```   371 lemma eventually_sequentially:
```
```   372   "eventually P sequentially \<longleftrightarrow> (\<exists>N. \<forall>n\<ge>N. P n)"
```
```   373   by (rule eventually_at_top_linorder)
```
```   374
```
```   375 lemma sequentially_bot [simp, intro]: "sequentially \<noteq> bot"
```
```   376   unfolding filter_eq_iff eventually_sequentially by auto
```
```   377
```
```   378 lemmas trivial_limit_sequentially = sequentially_bot
```
```   379
```
```   380 lemma eventually_False_sequentially [simp]:
```
```   381   "\<not> eventually (\<lambda>n. False) sequentially"
```
```   382   by (simp add: eventually_False)
```
```   383
```
```   384 lemma le_sequentially:
```
```   385   "F \<le> sequentially \<longleftrightarrow> (\<forall>N. eventually (\<lambda>n. N \<le> n) F)"
```
```   386   unfolding le_filter_def eventually_sequentially
```
```   387   by (safe, fast, drule_tac x=N in spec, auto elim: eventually_rev_mp)
```
```   388
```
```   389 lemma eventually_sequentiallyI:
```
```   390   assumes "\<And>x. c \<le> x \<Longrightarrow> P x"
```
```   391   shows "eventually P sequentially"
```
```   392 using assms by (auto simp: eventually_sequentially)
```
```   393
```
```   394
```
```   395 subsection {* Standard filters *}
```
```   396
```
```   397 definition within :: "'a filter \<Rightarrow> 'a set \<Rightarrow> 'a filter" (infixr "within" 70)
```
```   398   where "F within S = Abs_filter (\<lambda>P. eventually (\<lambda>x. x \<in> S \<longrightarrow> P x) F)"
```
```   399
```
```   400 definition (in topological_space) nhds :: "'a \<Rightarrow> 'a filter"
```
```   401   where "nhds a = Abs_filter (\<lambda>P. \<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x))"
```
```   402
```
```   403 definition (in topological_space) at :: "'a \<Rightarrow> 'a filter"
```
```   404   where "at a = nhds a within - {a}"
```
```   405
```
```   406 abbreviation at_right :: "'a\<Colon>{topological_space, order} \<Rightarrow> 'a filter" where
```
```   407   "at_right x \<equiv> at x within {x <..}"
```
```   408
```
```   409 abbreviation at_left :: "'a\<Colon>{topological_space, order} \<Rightarrow> 'a filter" where
```
```   410   "at_left x \<equiv> at x within {..< x}"
```
```   411
```
```   412 definition at_infinity :: "'a::real_normed_vector filter" where
```
```   413   "at_infinity = Abs_filter (\<lambda>P. \<exists>r. \<forall>x. r \<le> norm x \<longrightarrow> P x)"
```
```   414
```
```   415 lemma eventually_within:
```
```   416   "eventually P (F within S) = eventually (\<lambda>x. x \<in> S \<longrightarrow> P x) F"
```
```   417   unfolding within_def
```
```   418   by (rule eventually_Abs_filter, rule is_filter.intro)
```
```   419      (auto elim!: eventually_rev_mp)
```
```   420
```
```   421 lemma within_UNIV [simp]: "F within UNIV = F"
```
```   422   unfolding filter_eq_iff eventually_within by simp
```
```   423
```
```   424 lemma within_empty [simp]: "F within {} = bot"
```
```   425   unfolding filter_eq_iff eventually_within by simp
```
```   426
```
```   427 lemma within_within_eq: "(F within S) within T = F within (S \<inter> T)"
```
```   428   by (auto simp: filter_eq_iff eventually_within elim: eventually_elim1)
```
```   429
```
```   430 lemma at_within_eq: "at x within T = nhds x within (T - {x})"
```
```   431   unfolding at_def within_within_eq by (simp add: ac_simps Diff_eq)
```
```   432
```
```   433 lemma within_le: "F within S \<le> F"
```
```   434   unfolding le_filter_def eventually_within by (auto elim: eventually_elim1)
```
```   435
```
```   436 lemma le_withinI: "F \<le> F' \<Longrightarrow> eventually (\<lambda>x. x \<in> S) F \<Longrightarrow> F \<le> F' within S"
```
```   437   unfolding le_filter_def eventually_within by (auto elim: eventually_elim2)
```
```   438
```
```   439 lemma le_within_iff: "eventually (\<lambda>x. x \<in> S) F \<Longrightarrow> F \<le> F' within S \<longleftrightarrow> F \<le> F'"
```
```   440   by (blast intro: within_le le_withinI order_trans)
```
```   441
```
```   442 lemma eventually_nhds:
```
```   443   "eventually P (nhds a) \<longleftrightarrow> (\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x))"
```
```   444 unfolding nhds_def
```
```   445 proof (rule eventually_Abs_filter, rule is_filter.intro)
```
```   446   have "open UNIV \<and> a \<in> UNIV \<and> (\<forall>x\<in>UNIV. True)" by simp
```
```   447   thus "\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. True)" ..
```
```   448 next
```
```   449   fix P Q
```
```   450   assume "\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x)"
```
```   451      and "\<exists>T. open T \<and> a \<in> T \<and> (\<forall>x\<in>T. Q x)"
```
```   452   then obtain S T where
```
```   453     "open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x)"
```
```   454     "open T \<and> a \<in> T \<and> (\<forall>x\<in>T. Q x)" by auto
```
```   455   hence "open (S \<inter> T) \<and> a \<in> S \<inter> T \<and> (\<forall>x\<in>(S \<inter> T). P x \<and> Q x)"
```
```   456     by (simp add: open_Int)
```
```   457   thus "\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x \<and> Q x)" ..
```
```   458 qed auto
```
```   459
```
```   460 lemma eventually_nhds_metric:
```
```   461   "eventually P (nhds a) \<longleftrightarrow> (\<exists>d>0. \<forall>x. dist x a < d \<longrightarrow> P x)"
```
```   462 unfolding eventually_nhds open_dist
```
```   463 apply safe
```
```   464 apply fast
```
```   465 apply (rule_tac x="{x. dist x a < d}" in exI, simp)
```
```   466 apply clarsimp
```
```   467 apply (rule_tac x="d - dist x a" in exI, clarsimp)
```
```   468 apply (simp only: less_diff_eq)
```
```   469 apply (erule le_less_trans [OF dist_triangle])
```
```   470 done
```
```   471
```
```   472 lemma nhds_neq_bot [simp]: "nhds a \<noteq> bot"
```
```   473   unfolding trivial_limit_def eventually_nhds by simp
```
```   474
```
```   475 lemma eventually_at_topological:
```
```   476   "eventually P (at a) \<longleftrightarrow> (\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. x \<noteq> a \<longrightarrow> P x))"
```
```   477 unfolding at_def eventually_within eventually_nhds by simp
```
```   478
```
```   479 lemma eventually_at:
```
```   480   fixes a :: "'a::metric_space"
```
```   481   shows "eventually P (at a) \<longleftrightarrow> (\<exists>d>0. \<forall>x. x \<noteq> a \<and> dist x a < d \<longrightarrow> P x)"
```
```   482 unfolding at_def eventually_within eventually_nhds_metric by auto
```
```   483
```
```   484 lemma eventually_within_less: (* COPY FROM Topo/eventually_within *)
```
```   485   "eventually P (at a within S) \<longleftrightarrow> (\<exists>d>0. \<forall>x\<in>S. 0 < dist x a \<and> dist x a < d \<longrightarrow> P x)"
```
```   486   unfolding eventually_within eventually_at dist_nz by auto
```
```   487
```
```   488 lemma eventually_within_le: (* COPY FROM Topo/eventually_within_le *)
```
```   489   "eventually P (at a within S) \<longleftrightarrow> (\<exists>d>0. \<forall>x\<in>S. 0 < dist x a \<and> dist x a <= d \<longrightarrow> P x)"
```
```   490   unfolding eventually_within_less by auto (metis dense order_le_less_trans)
```
```   491
```
```   492 lemma at_eq_bot_iff: "at a = bot \<longleftrightarrow> open {a}"
```
```   493   unfolding trivial_limit_def eventually_at_topological
```
```   494   by (safe, case_tac "S = {a}", simp, fast, fast)
```
```   495
```
```   496 lemma at_neq_bot [simp]: "at (a::'a::perfect_space) \<noteq> bot"
```
```   497   by (simp add: at_eq_bot_iff not_open_singleton)
```
```   498
```
```   499 lemma trivial_limit_at_left_real [simp]: (* maybe generalize type *)
```
```   500   "\<not> trivial_limit (at_left (x::real))"
```
```   501   unfolding trivial_limit_def eventually_within_le
```
```   502   apply clarsimp
```
```   503   apply (rule_tac x="x - d/2" in bexI)
```
```   504   apply (auto simp: dist_real_def)
```
```   505   done
```
```   506
```
```   507 lemma trivial_limit_at_right_real [simp]: (* maybe generalize type *)
```
```   508   "\<not> trivial_limit (at_right (x::real))"
```
```   509   unfolding trivial_limit_def eventually_within_le
```
```   510   apply clarsimp
```
```   511   apply (rule_tac x="x + d/2" in bexI)
```
```   512   apply (auto simp: dist_real_def)
```
```   513   done
```
```   514
```
```   515 lemma eventually_at_infinity:
```
```   516   "eventually P at_infinity \<longleftrightarrow> (\<exists>b. \<forall>x. b \<le> norm x \<longrightarrow> P x)"
```
```   517 unfolding at_infinity_def
```
```   518 proof (rule eventually_Abs_filter, rule is_filter.intro)
```
```   519   fix P Q :: "'a \<Rightarrow> bool"
```
```   520   assume "\<exists>r. \<forall>x. r \<le> norm x \<longrightarrow> P x" and "\<exists>s. \<forall>x. s \<le> norm x \<longrightarrow> Q x"
```
```   521   then obtain r s where
```
```   522     "\<forall>x. r \<le> norm x \<longrightarrow> P x" and "\<forall>x. s \<le> norm x \<longrightarrow> Q x" by auto
```
```   523   then have "\<forall>x. max r s \<le> norm x \<longrightarrow> P x \<and> Q x" by simp
```
```   524   then show "\<exists>r. \<forall>x. r \<le> norm x \<longrightarrow> P x \<and> Q x" ..
```
```   525 qed auto
```
```   526
```
```   527 lemma at_infinity_eq_at_top_bot:
```
```   528   "(at_infinity \<Colon> real filter) = sup at_top at_bot"
```
```   529   unfolding sup_filter_def at_infinity_def eventually_at_top_linorder eventually_at_bot_linorder
```
```   530 proof (intro arg_cong[where f=Abs_filter] ext iffI)
```
```   531   fix P :: "real \<Rightarrow> bool" assume "\<exists>r. \<forall>x. r \<le> norm x \<longrightarrow> P x"
```
```   532   then guess r ..
```
```   533   then have "(\<forall>x\<ge>r. P x) \<and> (\<forall>x\<le>-r. P x)" by auto
```
```   534   then show "(\<exists>r. \<forall>x\<ge>r. P x) \<and> (\<exists>r. \<forall>x\<le>r. P x)" by auto
```
```   535 next
```
```   536   fix P :: "real \<Rightarrow> bool" assume "(\<exists>r. \<forall>x\<ge>r. P x) \<and> (\<exists>r. \<forall>x\<le>r. P x)"
```
```   537   then obtain p q where "\<forall>x\<ge>p. P x" "\<forall>x\<le>q. P x" by auto
```
```   538   then show "\<exists>r. \<forall>x. r \<le> norm x \<longrightarrow> P x"
```
```   539     by (intro exI[of _ "max p (-q)"])
```
```   540        (auto simp: abs_real_def)
```
```   541 qed
```
```   542
```
```   543 lemma at_top_le_at_infinity:
```
```   544   "at_top \<le> (at_infinity :: real filter)"
```
```   545   unfolding at_infinity_eq_at_top_bot by simp
```
```   546
```
```   547 lemma at_bot_le_at_infinity:
```
```   548   "at_bot \<le> (at_infinity :: real filter)"
```
```   549   unfolding at_infinity_eq_at_top_bot by simp
```
```   550
```
```   551 subsection {* Boundedness *}
```
```   552
```
```   553 definition Bfun :: "('a \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a filter \<Rightarrow> bool"
```
```   554   where "Bfun f F = (\<exists>K>0. eventually (\<lambda>x. norm (f x) \<le> K) F)"
```
```   555
```
```   556 lemma BfunI:
```
```   557   assumes K: "eventually (\<lambda>x. norm (f x) \<le> K) F" shows "Bfun f F"
```
```   558 unfolding Bfun_def
```
```   559 proof (intro exI conjI allI)
```
```   560   show "0 < max K 1" by simp
```
```   561 next
```
```   562   show "eventually (\<lambda>x. norm (f x) \<le> max K 1) F"
```
```   563     using K by (rule eventually_elim1, simp)
```
```   564 qed
```
```   565
```
```   566 lemma BfunE:
```
```   567   assumes "Bfun f F"
```
```   568   obtains B where "0 < B" and "eventually (\<lambda>x. norm (f x) \<le> B) F"
```
```   569 using assms unfolding Bfun_def by fast
```
```   570
```
```   571
```
```   572 subsection {* Convergence to Zero *}
```
```   573
```
```   574 definition Zfun :: "('a \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a filter \<Rightarrow> bool"
```
```   575   where "Zfun f F = (\<forall>r>0. eventually (\<lambda>x. norm (f x) < r) F)"
```
```   576
```
```   577 lemma ZfunI:
```
```   578   "(\<And>r. 0 < r \<Longrightarrow> eventually (\<lambda>x. norm (f x) < r) F) \<Longrightarrow> Zfun f F"
```
```   579   unfolding Zfun_def by simp
```
```   580
```
```   581 lemma ZfunD:
```
```   582   "\<lbrakk>Zfun f F; 0 < r\<rbrakk> \<Longrightarrow> eventually (\<lambda>x. norm (f x) < r) F"
```
```   583   unfolding Zfun_def by simp
```
```   584
```
```   585 lemma Zfun_ssubst:
```
```   586   "eventually (\<lambda>x. f x = g x) F \<Longrightarrow> Zfun g F \<Longrightarrow> Zfun f F"
```
```   587   unfolding Zfun_def by (auto elim!: eventually_rev_mp)
```
```   588
```
```   589 lemma Zfun_zero: "Zfun (\<lambda>x. 0) F"
```
```   590   unfolding Zfun_def by simp
```
```   591
```
```   592 lemma Zfun_norm_iff: "Zfun (\<lambda>x. norm (f x)) F = Zfun (\<lambda>x. f x) F"
```
```   593   unfolding Zfun_def by simp
```
```   594
```
```   595 lemma Zfun_imp_Zfun:
```
```   596   assumes f: "Zfun f F"
```
```   597   assumes g: "eventually (\<lambda>x. norm (g x) \<le> norm (f x) * K) F"
```
```   598   shows "Zfun (\<lambda>x. g x) F"
```
```   599 proof (cases)
```
```   600   assume K: "0 < K"
```
```   601   show ?thesis
```
```   602   proof (rule ZfunI)
```
```   603     fix r::real assume "0 < r"
```
```   604     hence "0 < r / K"
```
```   605       using K by (rule divide_pos_pos)
```
```   606     then have "eventually (\<lambda>x. norm (f x) < r / K) F"
```
```   607       using ZfunD [OF f] by fast
```
```   608     with g show "eventually (\<lambda>x. norm (g x) < r) F"
```
```   609     proof eventually_elim
```
```   610       case (elim x)
```
```   611       hence "norm (f x) * K < r"
```
```   612         by (simp add: pos_less_divide_eq K)
```
```   613       thus ?case
```
```   614         by (simp add: order_le_less_trans [OF elim(1)])
```
```   615     qed
```
```   616   qed
```
```   617 next
```
```   618   assume "\<not> 0 < K"
```
```   619   hence K: "K \<le> 0" by (simp only: not_less)
```
```   620   show ?thesis
```
```   621   proof (rule ZfunI)
```
```   622     fix r :: real
```
```   623     assume "0 < r"
```
```   624     from g show "eventually (\<lambda>x. norm (g x) < r) F"
```
```   625     proof eventually_elim
```
```   626       case (elim x)
```
```   627       also have "norm (f x) * K \<le> norm (f x) * 0"
```
```   628         using K norm_ge_zero by (rule mult_left_mono)
```
```   629       finally show ?case
```
```   630         using `0 < r` by simp
```
```   631     qed
```
```   632   qed
```
```   633 qed
```
```   634
```
```   635 lemma Zfun_le: "\<lbrakk>Zfun g F; \<forall>x. norm (f x) \<le> norm (g x)\<rbrakk> \<Longrightarrow> Zfun f F"
```
```   636   by (erule_tac K="1" in Zfun_imp_Zfun, simp)
```
```   637
```
```   638 lemma Zfun_add:
```
```   639   assumes f: "Zfun f F" and g: "Zfun g F"
```
```   640   shows "Zfun (\<lambda>x. f x + g x) F"
```
```   641 proof (rule ZfunI)
```
```   642   fix r::real assume "0 < r"
```
```   643   hence r: "0 < r / 2" by simp
```
```   644   have "eventually (\<lambda>x. norm (f x) < r/2) F"
```
```   645     using f r by (rule ZfunD)
```
```   646   moreover
```
```   647   have "eventually (\<lambda>x. norm (g x) < r/2) F"
```
```   648     using g r by (rule ZfunD)
```
```   649   ultimately
```
```   650   show "eventually (\<lambda>x. norm (f x + g x) < r) F"
```
```   651   proof eventually_elim
```
```   652     case (elim x)
```
```   653     have "norm (f x + g x) \<le> norm (f x) + norm (g x)"
```
```   654       by (rule norm_triangle_ineq)
```
```   655     also have "\<dots> < r/2 + r/2"
```
```   656       using elim by (rule add_strict_mono)
```
```   657     finally show ?case
```
```   658       by simp
```
```   659   qed
```
```   660 qed
```
```   661
```
```   662 lemma Zfun_minus: "Zfun f F \<Longrightarrow> Zfun (\<lambda>x. - f x) F"
```
```   663   unfolding Zfun_def by simp
```
```   664
```
```   665 lemma Zfun_diff: "\<lbrakk>Zfun f F; Zfun g F\<rbrakk> \<Longrightarrow> Zfun (\<lambda>x. f x - g x) F"
```
```   666   by (simp only: diff_minus Zfun_add Zfun_minus)
```
```   667
```
```   668 lemma (in bounded_linear) Zfun:
```
```   669   assumes g: "Zfun g F"
```
```   670   shows "Zfun (\<lambda>x. f (g x)) F"
```
```   671 proof -
```
```   672   obtain K where "\<And>x. norm (f x) \<le> norm x * K"
```
```   673     using bounded by fast
```
```   674   then have "eventually (\<lambda>x. norm (f (g x)) \<le> norm (g x) * K) F"
```
```   675     by simp
```
```   676   with g show ?thesis
```
```   677     by (rule Zfun_imp_Zfun)
```
```   678 qed
```
```   679
```
```   680 lemma (in bounded_bilinear) Zfun:
```
```   681   assumes f: "Zfun f F"
```
```   682   assumes g: "Zfun g F"
```
```   683   shows "Zfun (\<lambda>x. f x ** g x) F"
```
```   684 proof (rule ZfunI)
```
```   685   fix r::real assume r: "0 < r"
```
```   686   obtain K where K: "0 < K"
```
```   687     and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"
```
```   688     using pos_bounded by fast
```
```   689   from K have K': "0 < inverse K"
```
```   690     by (rule positive_imp_inverse_positive)
```
```   691   have "eventually (\<lambda>x. norm (f x) < r) F"
```
```   692     using f r by (rule ZfunD)
```
```   693   moreover
```
```   694   have "eventually (\<lambda>x. norm (g x) < inverse K) F"
```
```   695     using g K' by (rule ZfunD)
```
```   696   ultimately
```
```   697   show "eventually (\<lambda>x. norm (f x ** g x) < r) F"
```
```   698   proof eventually_elim
```
```   699     case (elim x)
```
```   700     have "norm (f x ** g x) \<le> norm (f x) * norm (g x) * K"
```
```   701       by (rule norm_le)
```
```   702     also have "norm (f x) * norm (g x) * K < r * inverse K * K"
```
```   703       by (intro mult_strict_right_mono mult_strict_mono' norm_ge_zero elim K)
```
```   704     also from K have "r * inverse K * K = r"
```
```   705       by simp
```
```   706     finally show ?case .
```
```   707   qed
```
```   708 qed
```
```   709
```
```   710 lemma (in bounded_bilinear) Zfun_left:
```
```   711   "Zfun f F \<Longrightarrow> Zfun (\<lambda>x. f x ** a) F"
```
```   712   by (rule bounded_linear_left [THEN bounded_linear.Zfun])
```
```   713
```
```   714 lemma (in bounded_bilinear) Zfun_right:
```
```   715   "Zfun f F \<Longrightarrow> Zfun (\<lambda>x. a ** f x) F"
```
```   716   by (rule bounded_linear_right [THEN bounded_linear.Zfun])
```
```   717
```
```   718 lemmas Zfun_mult = bounded_bilinear.Zfun [OF bounded_bilinear_mult]
```
```   719 lemmas Zfun_mult_right = bounded_bilinear.Zfun_right [OF bounded_bilinear_mult]
```
```   720 lemmas Zfun_mult_left = bounded_bilinear.Zfun_left [OF bounded_bilinear_mult]
```
```   721
```
```   722
```
```   723 subsection {* Limits *}
```
```   724
```
```   725 definition filterlim :: "('a \<Rightarrow> 'b) \<Rightarrow> 'b filter \<Rightarrow> 'a filter \<Rightarrow> bool" where
```
```   726   "filterlim f F2 F1 \<longleftrightarrow> filtermap f F1 \<le> F2"
```
```   727
```
```   728 syntax
```
```   729   "_LIM" :: "pttrns \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> bool" ("(3LIM (_)/ (_)./ (_) :> (_))" [1000, 10, 0, 10] 10)
```
```   730
```
```   731 translations
```
```   732   "LIM x F1. f :> F2"   == "CONST filterlim (%x. f) F2 F1"
```
```   733
```
```   734 lemma filterlim_iff:
```
```   735   "(LIM x F1. f x :> F2) \<longleftrightarrow> (\<forall>P. eventually P F2 \<longrightarrow> eventually (\<lambda>x. P (f x)) F1)"
```
```   736   unfolding filterlim_def le_filter_def eventually_filtermap ..
```
```   737
```
```   738 lemma filterlim_compose:
```
```   739   "filterlim g F3 F2 \<Longrightarrow> filterlim f F2 F1 \<Longrightarrow> filterlim (\<lambda>x. g (f x)) F3 F1"
```
```   740   unfolding filterlim_def filtermap_filtermap[symmetric] by (metis filtermap_mono order_trans)
```
```   741
```
```   742 lemma filterlim_mono:
```
```   743   "filterlim f F2 F1 \<Longrightarrow> F2 \<le> F2' \<Longrightarrow> F1' \<le> F1 \<Longrightarrow> filterlim f F2' F1'"
```
```   744   unfolding filterlim_def by (metis filtermap_mono order_trans)
```
```   745
```
```   746 lemma filterlim_ident: "LIM x F. x :> F"
```
```   747   by (simp add: filterlim_def filtermap_ident)
```
```   748
```
```   749 lemma filterlim_cong:
```
```   750   "F1 = F1' \<Longrightarrow> F2 = F2' \<Longrightarrow> eventually (\<lambda>x. f x = g x) F2 \<Longrightarrow> filterlim f F1 F2 = filterlim g F1' F2'"
```
```   751   by (auto simp: filterlim_def le_filter_def eventually_filtermap elim: eventually_elim2)
```
```   752
```
```   753 lemma filterlim_within:
```
```   754   "(LIM x F1. f x :> F2 within S) \<longleftrightarrow> (eventually (\<lambda>x. f x \<in> S) F1 \<and> (LIM x F1. f x :> F2))"
```
```   755   unfolding filterlim_def
```
```   756 proof safe
```
```   757   assume "filtermap f F1 \<le> F2 within S" then show "eventually (\<lambda>x. f x \<in> S) F1"
```
```   758     by (auto simp: le_filter_def eventually_filtermap eventually_within elim!: allE[of _ "\<lambda>x. x \<in> S"])
```
```   759 qed (auto intro: within_le order_trans simp: le_within_iff eventually_filtermap)
```
```   760
```
```   761 lemma filterlim_filtermap: "filterlim f F1 (filtermap g F2) = filterlim (\<lambda>x. f (g x)) F1 F2"
```
```   762   unfolding filterlim_def filtermap_filtermap ..
```
```   763
```
```   764 lemma filterlim_sup:
```
```   765   "filterlim f F F1 \<Longrightarrow> filterlim f F F2 \<Longrightarrow> filterlim f F (sup F1 F2)"
```
```   766   unfolding filterlim_def filtermap_sup by auto
```
```   767
```
```   768 lemma filterlim_Suc: "filterlim Suc sequentially sequentially"
```
```   769   by (simp add: filterlim_iff eventually_sequentially) (metis le_Suc_eq)
```
```   770
```
```   771 abbreviation (in topological_space)
```
```   772   tendsto :: "('b \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'b filter \<Rightarrow> bool" (infixr "--->" 55) where
```
```   773   "(f ---> l) F \<equiv> filterlim f (nhds l) F"
```
```   774
```
```   775 ML {*
```
```   776 structure Tendsto_Intros = Named_Thms
```
```   777 (
```
```   778   val name = @{binding tendsto_intros}
```
```   779   val description = "introduction rules for tendsto"
```
```   780 )
```
```   781 *}
```
```   782
```
```   783 setup Tendsto_Intros.setup
```
```   784
```
```   785 lemma tendsto_def: "(f ---> l) F \<longleftrightarrow> (\<forall>S. open S \<longrightarrow> l \<in> S \<longrightarrow> eventually (\<lambda>x. f x \<in> S) F)"
```
```   786   unfolding filterlim_def
```
```   787 proof safe
```
```   788   fix S assume "open S" "l \<in> S" "filtermap f F \<le> nhds l"
```
```   789   then show "eventually (\<lambda>x. f x \<in> S) F"
```
```   790     unfolding eventually_nhds eventually_filtermap le_filter_def
```
```   791     by (auto elim!: allE[of _ "\<lambda>x. x \<in> S"] eventually_rev_mp)
```
```   792 qed (auto elim!: eventually_rev_mp simp: eventually_nhds eventually_filtermap le_filter_def)
```
```   793
```
```   794 lemma filterlim_at:
```
```   795   "(LIM x F. f x :> at b) \<longleftrightarrow> (eventually (\<lambda>x. f x \<noteq> b) F \<and> (f ---> b) F)"
```
```   796   by (simp add: at_def filterlim_within)
```
```   797
```
```   798 lemma tendsto_mono: "F \<le> F' \<Longrightarrow> (f ---> l) F' \<Longrightarrow> (f ---> l) F"
```
```   799   unfolding tendsto_def le_filter_def by fast
```
```   800
```
```   801 lemma topological_tendstoI:
```
```   802   "(\<And>S. open S \<Longrightarrow> l \<in> S \<Longrightarrow> eventually (\<lambda>x. f x \<in> S) F)
```
```   803     \<Longrightarrow> (f ---> l) F"
```
```   804   unfolding tendsto_def by auto
```
```   805
```
```   806 lemma topological_tendstoD:
```
```   807   "(f ---> l) F \<Longrightarrow> open S \<Longrightarrow> l \<in> S \<Longrightarrow> eventually (\<lambda>x. f x \<in> S) F"
```
```   808   unfolding tendsto_def by auto
```
```   809
```
```   810 lemma tendstoI:
```
```   811   assumes "\<And>e. 0 < e \<Longrightarrow> eventually (\<lambda>x. dist (f x) l < e) F"
```
```   812   shows "(f ---> l) F"
```
```   813   apply (rule topological_tendstoI)
```
```   814   apply (simp add: open_dist)
```
```   815   apply (drule (1) bspec, clarify)
```
```   816   apply (drule assms)
```
```   817   apply (erule eventually_elim1, simp)
```
```   818   done
```
```   819
```
```   820 lemma tendstoD:
```
```   821   "(f ---> l) F \<Longrightarrow> 0 < e \<Longrightarrow> eventually (\<lambda>x. dist (f x) l < e) F"
```
```   822   apply (drule_tac S="{x. dist x l < e}" in topological_tendstoD)
```
```   823   apply (clarsimp simp add: open_dist)
```
```   824   apply (rule_tac x="e - dist x l" in exI, clarsimp)
```
```   825   apply (simp only: less_diff_eq)
```
```   826   apply (erule le_less_trans [OF dist_triangle])
```
```   827   apply simp
```
```   828   apply simp
```
```   829   done
```
```   830
```
```   831 lemma tendsto_iff:
```
```   832   "(f ---> l) F \<longleftrightarrow> (\<forall>e>0. eventually (\<lambda>x. dist (f x) l < e) F)"
```
```   833   using tendstoI tendstoD by fast
```
```   834
```
```   835 lemma tendsto_Zfun_iff: "(f ---> a) F = Zfun (\<lambda>x. f x - a) F"
```
```   836   by (simp only: tendsto_iff Zfun_def dist_norm)
```
```   837
```
```   838 lemma tendsto_bot [simp]: "(f ---> a) bot"
```
```   839   unfolding tendsto_def by simp
```
```   840
```
```   841 lemma tendsto_ident_at [tendsto_intros]: "((\<lambda>x. x) ---> a) (at a)"
```
```   842   unfolding tendsto_def eventually_at_topological by auto
```
```   843
```
```   844 lemma tendsto_ident_at_within [tendsto_intros]:
```
```   845   "((\<lambda>x. x) ---> a) (at a within S)"
```
```   846   unfolding tendsto_def eventually_within eventually_at_topological by auto
```
```   847
```
```   848 lemma tendsto_const [tendsto_intros]: "((\<lambda>x. k) ---> k) F"
```
```   849   by (simp add: tendsto_def)
```
```   850
```
```   851 lemma tendsto_unique:
```
```   852   fixes f :: "'a \<Rightarrow> 'b::t2_space"
```
```   853   assumes "\<not> trivial_limit F" and "(f ---> a) F" and "(f ---> b) F"
```
```   854   shows "a = b"
```
```   855 proof (rule ccontr)
```
```   856   assume "a \<noteq> b"
```
```   857   obtain U V where "open U" "open V" "a \<in> U" "b \<in> V" "U \<inter> V = {}"
```
```   858     using hausdorff [OF `a \<noteq> b`] by fast
```
```   859   have "eventually (\<lambda>x. f x \<in> U) F"
```
```   860     using `(f ---> a) F` `open U` `a \<in> U` by (rule topological_tendstoD)
```
```   861   moreover
```
```   862   have "eventually (\<lambda>x. f x \<in> V) F"
```
```   863     using `(f ---> b) F` `open V` `b \<in> V` by (rule topological_tendstoD)
```
```   864   ultimately
```
```   865   have "eventually (\<lambda>x. False) F"
```
```   866   proof eventually_elim
```
```   867     case (elim x)
```
```   868     hence "f x \<in> U \<inter> V" by simp
```
```   869     with `U \<inter> V = {}` show ?case by simp
```
```   870   qed
```
```   871   with `\<not> trivial_limit F` show "False"
```
```   872     by (simp add: trivial_limit_def)
```
```   873 qed
```
```   874
```
```   875 lemma tendsto_const_iff:
```
```   876   fixes a b :: "'a::t2_space"
```
```   877   assumes "\<not> trivial_limit F" shows "((\<lambda>x. a) ---> b) F \<longleftrightarrow> a = b"
```
```   878   by (safe intro!: tendsto_const tendsto_unique [OF assms tendsto_const])
```
```   879
```
```   880 lemma tendsto_at_iff_tendsto_nhds:
```
```   881   "(g ---> g l) (at l) \<longleftrightarrow> (g ---> g l) (nhds l)"
```
```   882   unfolding tendsto_def at_def eventually_within
```
```   883   by (intro ext all_cong imp_cong) (auto elim!: eventually_elim1)
```
```   884
```
```   885 lemma tendsto_compose:
```
```   886   "(g ---> g l) (at l) \<Longrightarrow> (f ---> l) F \<Longrightarrow> ((\<lambda>x. g (f x)) ---> g l) F"
```
```   887   unfolding tendsto_at_iff_tendsto_nhds by (rule filterlim_compose[of g])
```
```   888
```
```   889 lemma tendsto_compose_eventually:
```
```   890   "(g ---> m) (at l) \<Longrightarrow> (f ---> l) F \<Longrightarrow> eventually (\<lambda>x. f x \<noteq> l) F \<Longrightarrow> ((\<lambda>x. g (f x)) ---> m) F"
```
```   891   by (rule filterlim_compose[of g _ "at l"]) (auto simp add: filterlim_at)
```
```   892
```
```   893 lemma metric_tendsto_imp_tendsto:
```
```   894   assumes f: "(f ---> a) F"
```
```   895   assumes le: "eventually (\<lambda>x. dist (g x) b \<le> dist (f x) a) F"
```
```   896   shows "(g ---> b) F"
```
```   897 proof (rule tendstoI)
```
```   898   fix e :: real assume "0 < e"
```
```   899   with f have "eventually (\<lambda>x. dist (f x) a < e) F" by (rule tendstoD)
```
```   900   with le show "eventually (\<lambda>x. dist (g x) b < e) F"
```
```   901     using le_less_trans by (rule eventually_elim2)
```
```   902 qed
```
```   903
```
```   904 subsubsection {* Distance and norms *}
```
```   905
```
```   906 lemma tendsto_dist [tendsto_intros]:
```
```   907   assumes f: "(f ---> l) F" and g: "(g ---> m) F"
```
```   908   shows "((\<lambda>x. dist (f x) (g x)) ---> dist l m) F"
```
```   909 proof (rule tendstoI)
```
```   910   fix e :: real assume "0 < e"
```
```   911   hence e2: "0 < e/2" by simp
```
```   912   from tendstoD [OF f e2] tendstoD [OF g e2]
```
```   913   show "eventually (\<lambda>x. dist (dist (f x) (g x)) (dist l m) < e) F"
```
```   914   proof (eventually_elim)
```
```   915     case (elim x)
```
```   916     then show "dist (dist (f x) (g x)) (dist l m) < e"
```
```   917       unfolding dist_real_def
```
```   918       using dist_triangle2 [of "f x" "g x" "l"]
```
```   919       using dist_triangle2 [of "g x" "l" "m"]
```
```   920       using dist_triangle3 [of "l" "m" "f x"]
```
```   921       using dist_triangle [of "f x" "m" "g x"]
```
```   922       by arith
```
```   923   qed
```
```   924 qed
```
```   925
```
```   926 lemma norm_conv_dist: "norm x = dist x 0"
```
```   927   unfolding dist_norm by simp
```
```   928
```
```   929 lemma tendsto_norm [tendsto_intros]:
```
```   930   "(f ---> a) F \<Longrightarrow> ((\<lambda>x. norm (f x)) ---> norm a) F"
```
```   931   unfolding norm_conv_dist by (intro tendsto_intros)
```
```   932
```
```   933 lemma tendsto_norm_zero:
```
```   934   "(f ---> 0) F \<Longrightarrow> ((\<lambda>x. norm (f x)) ---> 0) F"
```
```   935   by (drule tendsto_norm, simp)
```
```   936
```
```   937 lemma tendsto_norm_zero_cancel:
```
```   938   "((\<lambda>x. norm (f x)) ---> 0) F \<Longrightarrow> (f ---> 0) F"
```
```   939   unfolding tendsto_iff dist_norm by simp
```
```   940
```
```   941 lemma tendsto_norm_zero_iff:
```
```   942   "((\<lambda>x. norm (f x)) ---> 0) F \<longleftrightarrow> (f ---> 0) F"
```
```   943   unfolding tendsto_iff dist_norm by simp
```
```   944
```
```   945 lemma tendsto_rabs [tendsto_intros]:
```
```   946   "(f ---> (l::real)) F \<Longrightarrow> ((\<lambda>x. \<bar>f x\<bar>) ---> \<bar>l\<bar>) F"
```
```   947   by (fold real_norm_def, rule tendsto_norm)
```
```   948
```
```   949 lemma tendsto_rabs_zero:
```
```   950   "(f ---> (0::real)) F \<Longrightarrow> ((\<lambda>x. \<bar>f x\<bar>) ---> 0) F"
```
```   951   by (fold real_norm_def, rule tendsto_norm_zero)
```
```   952
```
```   953 lemma tendsto_rabs_zero_cancel:
```
```   954   "((\<lambda>x. \<bar>f x\<bar>) ---> (0::real)) F \<Longrightarrow> (f ---> 0) F"
```
```   955   by (fold real_norm_def, rule tendsto_norm_zero_cancel)
```
```   956
```
```   957 lemma tendsto_rabs_zero_iff:
```
```   958   "((\<lambda>x. \<bar>f x\<bar>) ---> (0::real)) F \<longleftrightarrow> (f ---> 0) F"
```
```   959   by (fold real_norm_def, rule tendsto_norm_zero_iff)
```
```   960
```
```   961 subsubsection {* Addition and subtraction *}
```
```   962
```
```   963 lemma tendsto_add [tendsto_intros]:
```
```   964   fixes a b :: "'a::real_normed_vector"
```
```   965   shows "\<lbrakk>(f ---> a) F; (g ---> b) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x + g x) ---> a + b) F"
```
```   966   by (simp only: tendsto_Zfun_iff add_diff_add Zfun_add)
```
```   967
```
```   968 lemma tendsto_add_zero:
```
```   969   fixes f g :: "'a::type \<Rightarrow> 'b::real_normed_vector"
```
```   970   shows "\<lbrakk>(f ---> 0) F; (g ---> 0) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x + g x) ---> 0) F"
```
```   971   by (drule (1) tendsto_add, simp)
```
```   972
```
```   973 lemma tendsto_minus [tendsto_intros]:
```
```   974   fixes a :: "'a::real_normed_vector"
```
```   975   shows "(f ---> a) F \<Longrightarrow> ((\<lambda>x. - f x) ---> - a) F"
```
```   976   by (simp only: tendsto_Zfun_iff minus_diff_minus Zfun_minus)
```
```   977
```
```   978 lemma tendsto_minus_cancel:
```
```   979   fixes a :: "'a::real_normed_vector"
```
```   980   shows "((\<lambda>x. - f x) ---> - a) F \<Longrightarrow> (f ---> a) F"
```
```   981   by (drule tendsto_minus, simp)
```
```   982
```
```   983 lemma tendsto_minus_cancel_left:
```
```   984     "(f ---> - (y::_::real_normed_vector)) F \<longleftrightarrow> ((\<lambda>x. - f x) ---> y) F"
```
```   985   using tendsto_minus_cancel[of f "- y" F]  tendsto_minus[of f "- y" F]
```
```   986   by auto
```
```   987
```
```   988 lemma tendsto_diff [tendsto_intros]:
```
```   989   fixes a b :: "'a::real_normed_vector"
```
```   990   shows "\<lbrakk>(f ---> a) F; (g ---> b) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x - g x) ---> a - b) F"
```
```   991   by (simp add: diff_minus tendsto_add tendsto_minus)
```
```   992
```
```   993 lemma tendsto_setsum [tendsto_intros]:
```
```   994   fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c::real_normed_vector"
```
```   995   assumes "\<And>i. i \<in> S \<Longrightarrow> (f i ---> a i) F"
```
```   996   shows "((\<lambda>x. \<Sum>i\<in>S. f i x) ---> (\<Sum>i\<in>S. a i)) F"
```
```   997 proof (cases "finite S")
```
```   998   assume "finite S" thus ?thesis using assms
```
```   999     by (induct, simp add: tendsto_const, simp add: tendsto_add)
```
```  1000 next
```
```  1001   assume "\<not> finite S" thus ?thesis
```
```  1002     by (simp add: tendsto_const)
```
```  1003 qed
```
```  1004
```
```  1005 lemma real_tendsto_sandwich:
```
```  1006   fixes f g h :: "'a \<Rightarrow> real"
```
```  1007   assumes ev: "eventually (\<lambda>n. f n \<le> g n) net" "eventually (\<lambda>n. g n \<le> h n) net"
```
```  1008   assumes lim: "(f ---> c) net" "(h ---> c) net"
```
```  1009   shows "(g ---> c) net"
```
```  1010 proof -
```
```  1011   have "((\<lambda>n. g n - f n) ---> 0) net"
```
```  1012   proof (rule metric_tendsto_imp_tendsto)
```
```  1013     show "eventually (\<lambda>n. dist (g n - f n) 0 \<le> dist (h n - f n) 0) net"
```
```  1014       using ev by (rule eventually_elim2) (simp add: dist_real_def)
```
```  1015     show "((\<lambda>n. h n - f n) ---> 0) net"
```
```  1016       using tendsto_diff[OF lim(2,1)] by simp
```
```  1017   qed
```
```  1018   from tendsto_add[OF this lim(1)] show ?thesis by simp
```
```  1019 qed
```
```  1020
```
```  1021 subsubsection {* Linear operators and multiplication *}
```
```  1022
```
```  1023 lemma (in bounded_linear) tendsto:
```
```  1024   "(g ---> a) F \<Longrightarrow> ((\<lambda>x. f (g x)) ---> f a) F"
```
```  1025   by (simp only: tendsto_Zfun_iff diff [symmetric] Zfun)
```
```  1026
```
```  1027 lemma (in bounded_linear) tendsto_zero:
```
```  1028   "(g ---> 0) F \<Longrightarrow> ((\<lambda>x. f (g x)) ---> 0) F"
```
```  1029   by (drule tendsto, simp only: zero)
```
```  1030
```
```  1031 lemma (in bounded_bilinear) tendsto:
```
```  1032   "\<lbrakk>(f ---> a) F; (g ---> b) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x ** g x) ---> a ** b) F"
```
```  1033   by (simp only: tendsto_Zfun_iff prod_diff_prod
```
```  1034                  Zfun_add Zfun Zfun_left Zfun_right)
```
```  1035
```
```  1036 lemma (in bounded_bilinear) tendsto_zero:
```
```  1037   assumes f: "(f ---> 0) F"
```
```  1038   assumes g: "(g ---> 0) F"
```
```  1039   shows "((\<lambda>x. f x ** g x) ---> 0) F"
```
```  1040   using tendsto [OF f g] by (simp add: zero_left)
```
```  1041
```
```  1042 lemma (in bounded_bilinear) tendsto_left_zero:
```
```  1043   "(f ---> 0) F \<Longrightarrow> ((\<lambda>x. f x ** c) ---> 0) F"
```
```  1044   by (rule bounded_linear.tendsto_zero [OF bounded_linear_left])
```
```  1045
```
```  1046 lemma (in bounded_bilinear) tendsto_right_zero:
```
```  1047   "(f ---> 0) F \<Longrightarrow> ((\<lambda>x. c ** f x) ---> 0) F"
```
```  1048   by (rule bounded_linear.tendsto_zero [OF bounded_linear_right])
```
```  1049
```
```  1050 lemmas tendsto_of_real [tendsto_intros] =
```
```  1051   bounded_linear.tendsto [OF bounded_linear_of_real]
```
```  1052
```
```  1053 lemmas tendsto_scaleR [tendsto_intros] =
```
```  1054   bounded_bilinear.tendsto [OF bounded_bilinear_scaleR]
```
```  1055
```
```  1056 lemmas tendsto_mult [tendsto_intros] =
```
```  1057   bounded_bilinear.tendsto [OF bounded_bilinear_mult]
```
```  1058
```
```  1059 lemmas tendsto_mult_zero =
```
```  1060   bounded_bilinear.tendsto_zero [OF bounded_bilinear_mult]
```
```  1061
```
```  1062 lemmas tendsto_mult_left_zero =
```
```  1063   bounded_bilinear.tendsto_left_zero [OF bounded_bilinear_mult]
```
```  1064
```
```  1065 lemmas tendsto_mult_right_zero =
```
```  1066   bounded_bilinear.tendsto_right_zero [OF bounded_bilinear_mult]
```
```  1067
```
```  1068 lemma tendsto_power [tendsto_intros]:
```
```  1069   fixes f :: "'a \<Rightarrow> 'b::{power,real_normed_algebra}"
```
```  1070   shows "(f ---> a) F \<Longrightarrow> ((\<lambda>x. f x ^ n) ---> a ^ n) F"
```
```  1071   by (induct n) (simp_all add: tendsto_const tendsto_mult)
```
```  1072
```
```  1073 lemma tendsto_setprod [tendsto_intros]:
```
```  1074   fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c::{real_normed_algebra,comm_ring_1}"
```
```  1075   assumes "\<And>i. i \<in> S \<Longrightarrow> (f i ---> L i) F"
```
```  1076   shows "((\<lambda>x. \<Prod>i\<in>S. f i x) ---> (\<Prod>i\<in>S. L i)) F"
```
```  1077 proof (cases "finite S")
```
```  1078   assume "finite S" thus ?thesis using assms
```
```  1079     by (induct, simp add: tendsto_const, simp add: tendsto_mult)
```
```  1080 next
```
```  1081   assume "\<not> finite S" thus ?thesis
```
```  1082     by (simp add: tendsto_const)
```
```  1083 qed
```
```  1084
```
```  1085 lemma tendsto_le_const:
```
```  1086   fixes f :: "_ \<Rightarrow> real"
```
```  1087   assumes F: "\<not> trivial_limit F"
```
```  1088   assumes x: "(f ---> x) F" and a: "eventually (\<lambda>x. a \<le> f x) F"
```
```  1089   shows "a \<le> x"
```
```  1090 proof (rule ccontr)
```
```  1091   assume "\<not> a \<le> x"
```
```  1092   with x have "eventually (\<lambda>x. f x < a) F"
```
```  1093     by (auto simp add: tendsto_def elim!: allE[of _ "{..< a}"])
```
```  1094   with a have "eventually (\<lambda>x. False) F"
```
```  1095     by eventually_elim auto
```
```  1096   with F show False
```
```  1097     by (simp add: eventually_False)
```
```  1098 qed
```
```  1099
```
```  1100 lemma tendsto_le:
```
```  1101   fixes f g :: "_ \<Rightarrow> real"
```
```  1102   assumes F: "\<not> trivial_limit F"
```
```  1103   assumes x: "(f ---> x) F" and y: "(g ---> y) F"
```
```  1104   assumes ev: "eventually (\<lambda>x. g x \<le> f x) F"
```
```  1105   shows "y \<le> x"
```
```  1106   using tendsto_le_const[OF F tendsto_diff[OF x y], of 0] ev
```
```  1107   by (simp add: sign_simps)
```
```  1108
```
```  1109 subsubsection {* Inverse and division *}
```
```  1110
```
```  1111 lemma (in bounded_bilinear) Zfun_prod_Bfun:
```
```  1112   assumes f: "Zfun f F"
```
```  1113   assumes g: "Bfun g F"
```
```  1114   shows "Zfun (\<lambda>x. f x ** g x) F"
```
```  1115 proof -
```
```  1116   obtain K where K: "0 \<le> K"
```
```  1117     and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"
```
```  1118     using nonneg_bounded by fast
```
```  1119   obtain B where B: "0 < B"
```
```  1120     and norm_g: "eventually (\<lambda>x. norm (g x) \<le> B) F"
```
```  1121     using g by (rule BfunE)
```
```  1122   have "eventually (\<lambda>x. norm (f x ** g x) \<le> norm (f x) * (B * K)) F"
```
```  1123   using norm_g proof eventually_elim
```
```  1124     case (elim x)
```
```  1125     have "norm (f x ** g x) \<le> norm (f x) * norm (g x) * K"
```
```  1126       by (rule norm_le)
```
```  1127     also have "\<dots> \<le> norm (f x) * B * K"
```
```  1128       by (intro mult_mono' order_refl norm_g norm_ge_zero
```
```  1129                 mult_nonneg_nonneg K elim)
```
```  1130     also have "\<dots> = norm (f x) * (B * K)"
```
```  1131       by (rule mult_assoc)
```
```  1132     finally show "norm (f x ** g x) \<le> norm (f x) * (B * K)" .
```
```  1133   qed
```
```  1134   with f show ?thesis
```
```  1135     by (rule Zfun_imp_Zfun)
```
```  1136 qed
```
```  1137
```
```  1138 lemma (in bounded_bilinear) flip:
```
```  1139   "bounded_bilinear (\<lambda>x y. y ** x)"
```
```  1140   apply default
```
```  1141   apply (rule add_right)
```
```  1142   apply (rule add_left)
```
```  1143   apply (rule scaleR_right)
```
```  1144   apply (rule scaleR_left)
```
```  1145   apply (subst mult_commute)
```
```  1146   using bounded by fast
```
```  1147
```
```  1148 lemma (in bounded_bilinear) Bfun_prod_Zfun:
```
```  1149   assumes f: "Bfun f F"
```
```  1150   assumes g: "Zfun g F"
```
```  1151   shows "Zfun (\<lambda>x. f x ** g x) F"
```
```  1152   using flip g f by (rule bounded_bilinear.Zfun_prod_Bfun)
```
```  1153
```
```  1154 lemma Bfun_inverse_lemma:
```
```  1155   fixes x :: "'a::real_normed_div_algebra"
```
```  1156   shows "\<lbrakk>r \<le> norm x; 0 < r\<rbrakk> \<Longrightarrow> norm (inverse x) \<le> inverse r"
```
```  1157   apply (subst nonzero_norm_inverse, clarsimp)
```
```  1158   apply (erule (1) le_imp_inverse_le)
```
```  1159   done
```
```  1160
```
```  1161 lemma Bfun_inverse:
```
```  1162   fixes a :: "'a::real_normed_div_algebra"
```
```  1163   assumes f: "(f ---> a) F"
```
```  1164   assumes a: "a \<noteq> 0"
```
```  1165   shows "Bfun (\<lambda>x. inverse (f x)) F"
```
```  1166 proof -
```
```  1167   from a have "0 < norm a" by simp
```
```  1168   hence "\<exists>r>0. r < norm a" by (rule dense)
```
```  1169   then obtain r where r1: "0 < r" and r2: "r < norm a" by fast
```
```  1170   have "eventually (\<lambda>x. dist (f x) a < r) F"
```
```  1171     using tendstoD [OF f r1] by fast
```
```  1172   hence "eventually (\<lambda>x. norm (inverse (f x)) \<le> inverse (norm a - r)) F"
```
```  1173   proof eventually_elim
```
```  1174     case (elim x)
```
```  1175     hence 1: "norm (f x - a) < r"
```
```  1176       by (simp add: dist_norm)
```
```  1177     hence 2: "f x \<noteq> 0" using r2 by auto
```
```  1178     hence "norm (inverse (f x)) = inverse (norm (f x))"
```
```  1179       by (rule nonzero_norm_inverse)
```
```  1180     also have "\<dots> \<le> inverse (norm a - r)"
```
```  1181     proof (rule le_imp_inverse_le)
```
```  1182       show "0 < norm a - r" using r2 by simp
```
```  1183     next
```
```  1184       have "norm a - norm (f x) \<le> norm (a - f x)"
```
```  1185         by (rule norm_triangle_ineq2)
```
```  1186       also have "\<dots> = norm (f x - a)"
```
```  1187         by (rule norm_minus_commute)
```
```  1188       also have "\<dots> < r" using 1 .
```
```  1189       finally show "norm a - r \<le> norm (f x)" by simp
```
```  1190     qed
```
```  1191     finally show "norm (inverse (f x)) \<le> inverse (norm a - r)" .
```
```  1192   qed
```
```  1193   thus ?thesis by (rule BfunI)
```
```  1194 qed
```
```  1195
```
```  1196 lemma tendsto_inverse [tendsto_intros]:
```
```  1197   fixes a :: "'a::real_normed_div_algebra"
```
```  1198   assumes f: "(f ---> a) F"
```
```  1199   assumes a: "a \<noteq> 0"
```
```  1200   shows "((\<lambda>x. inverse (f x)) ---> inverse a) F"
```
```  1201 proof -
```
```  1202   from a have "0 < norm a" by simp
```
```  1203   with f have "eventually (\<lambda>x. dist (f x) a < norm a) F"
```
```  1204     by (rule tendstoD)
```
```  1205   then have "eventually (\<lambda>x. f x \<noteq> 0) F"
```
```  1206     unfolding dist_norm by (auto elim!: eventually_elim1)
```
```  1207   with a have "eventually (\<lambda>x. inverse (f x) - inverse a =
```
```  1208     - (inverse (f x) * (f x - a) * inverse a)) F"
```
```  1209     by (auto elim!: eventually_elim1 simp: inverse_diff_inverse)
```
```  1210   moreover have "Zfun (\<lambda>x. - (inverse (f x) * (f x - a) * inverse a)) F"
```
```  1211     by (intro Zfun_minus Zfun_mult_left
```
```  1212       bounded_bilinear.Bfun_prod_Zfun [OF bounded_bilinear_mult]
```
```  1213       Bfun_inverse [OF f a] f [unfolded tendsto_Zfun_iff])
```
```  1214   ultimately show ?thesis
```
```  1215     unfolding tendsto_Zfun_iff by (rule Zfun_ssubst)
```
```  1216 qed
```
```  1217
```
```  1218 lemma tendsto_divide [tendsto_intros]:
```
```  1219   fixes a b :: "'a::real_normed_field"
```
```  1220   shows "\<lbrakk>(f ---> a) F; (g ---> b) F; b \<noteq> 0\<rbrakk>
```
```  1221     \<Longrightarrow> ((\<lambda>x. f x / g x) ---> a / b) F"
```
```  1222   by (simp add: tendsto_mult tendsto_inverse divide_inverse)
```
```  1223
```
```  1224 lemma tendsto_sgn [tendsto_intros]:
```
```  1225   fixes l :: "'a::real_normed_vector"
```
```  1226   shows "\<lbrakk>(f ---> l) F; l \<noteq> 0\<rbrakk> \<Longrightarrow> ((\<lambda>x. sgn (f x)) ---> sgn l) F"
```
```  1227   unfolding sgn_div_norm by (simp add: tendsto_intros)
```
```  1228
```
```  1229 subsection {* Limits to @{const at_top} and @{const at_bot} *}
```
```  1230
```
```  1231 lemma filterlim_at_top:
```
```  1232   fixes f :: "'a \<Rightarrow> ('b::linorder)"
```
```  1233   shows "(LIM x F. f x :> at_top) \<longleftrightarrow> (\<forall>Z. eventually (\<lambda>x. Z \<le> f x) F)"
```
```  1234   by (auto simp: filterlim_iff eventually_at_top_linorder elim!: eventually_elim1)
```
```  1235
```
```  1236 lemma filterlim_at_top_dense:
```
```  1237   fixes f :: "'a \<Rightarrow> ('b::dense_linorder)"
```
```  1238   shows "(LIM x F. f x :> at_top) \<longleftrightarrow> (\<forall>Z. eventually (\<lambda>x. Z < f x) F)"
```
```  1239   by (metis eventually_elim1[of _ F] eventually_gt_at_top order_less_imp_le
```
```  1240             filterlim_at_top[of f F] filterlim_iff[of f at_top F])
```
```  1241
```
```  1242 lemma filterlim_at_top_ge:
```
```  1243   fixes f :: "'a \<Rightarrow> ('b::linorder)" and c :: "'b"
```
```  1244   shows "(LIM x F. f x :> at_top) \<longleftrightarrow> (\<forall>Z\<ge>c. eventually (\<lambda>x. Z \<le> f x) F)"
```
```  1245   unfolding filterlim_at_top
```
```  1246 proof safe
```
```  1247   fix Z assume *: "\<forall>Z\<ge>c. eventually (\<lambda>x. Z \<le> f x) F"
```
```  1248   with *[THEN spec, of "max Z c"] show "eventually (\<lambda>x. Z \<le> f x) F"
```
```  1249     by (auto elim!: eventually_elim1)
```
```  1250 qed simp
```
```  1251
```
```  1252 lemma filterlim_at_top_at_top:
```
```  1253   fixes f :: "'a::linorder \<Rightarrow> 'b::linorder"
```
```  1254   assumes mono: "\<And>x y. Q x \<Longrightarrow> Q y \<Longrightarrow> x \<le> y \<Longrightarrow> f x \<le> f y"
```
```  1255   assumes bij: "\<And>x. P x \<Longrightarrow> f (g x) = x" "\<And>x. P x \<Longrightarrow> Q (g x)"
```
```  1256   assumes Q: "eventually Q at_top"
```
```  1257   assumes P: "eventually P at_top"
```
```  1258   shows "filterlim f at_top at_top"
```
```  1259 proof -
```
```  1260   from P obtain x where x: "\<And>y. x \<le> y \<Longrightarrow> P y"
```
```  1261     unfolding eventually_at_top_linorder by auto
```
```  1262   show ?thesis
```
```  1263   proof (intro filterlim_at_top_ge[THEN iffD2] allI impI)
```
```  1264     fix z assume "x \<le> z"
```
```  1265     with x have "P z" by auto
```
```  1266     have "eventually (\<lambda>x. g z \<le> x) at_top"
```
```  1267       by (rule eventually_ge_at_top)
```
```  1268     with Q show "eventually (\<lambda>x. z \<le> f x) at_top"
```
```  1269       by eventually_elim (metis mono bij `P z`)
```
```  1270   qed
```
```  1271 qed
```
```  1272
```
```  1273 lemma filterlim_at_top_gt:
```
```  1274   fixes f :: "'a \<Rightarrow> ('b::dense_linorder)" and c :: "'b"
```
```  1275   shows "(LIM x F. f x :> at_top) \<longleftrightarrow> (\<forall>Z>c. eventually (\<lambda>x. Z \<le> f x) F)"
```
```  1276   by (metis filterlim_at_top order_less_le_trans gt_ex filterlim_at_top_ge)
```
```  1277
```
```  1278 lemma filterlim_at_bot:
```
```  1279   fixes f :: "'a \<Rightarrow> ('b::linorder)"
```
```  1280   shows "(LIM x F. f x :> at_bot) \<longleftrightarrow> (\<forall>Z. eventually (\<lambda>x. f x \<le> Z) F)"
```
```  1281   by (auto simp: filterlim_iff eventually_at_bot_linorder elim!: eventually_elim1)
```
```  1282
```
```  1283 lemma filterlim_at_bot_le:
```
```  1284   fixes f :: "'a \<Rightarrow> ('b::linorder)" and c :: "'b"
```
```  1285   shows "(LIM x F. f x :> at_bot) \<longleftrightarrow> (\<forall>Z\<le>c. eventually (\<lambda>x. Z \<ge> f x) F)"
```
```  1286   unfolding filterlim_at_bot
```
```  1287 proof safe
```
```  1288   fix Z assume *: "\<forall>Z\<le>c. eventually (\<lambda>x. Z \<ge> f x) F"
```
```  1289   with *[THEN spec, of "min Z c"] show "eventually (\<lambda>x. Z \<ge> f x) F"
```
```  1290     by (auto elim!: eventually_elim1)
```
```  1291 qed simp
```
```  1292
```
```  1293 lemma filterlim_at_bot_lt:
```
```  1294   fixes f :: "'a \<Rightarrow> ('b::dense_linorder)" and c :: "'b"
```
```  1295   shows "(LIM x F. f x :> at_bot) \<longleftrightarrow> (\<forall>Z<c. eventually (\<lambda>x. Z \<ge> f x) F)"
```
```  1296   by (metis filterlim_at_bot filterlim_at_bot_le lt_ex order_le_less_trans)
```
```  1297
```
```  1298 lemma filterlim_at_bot_at_right:
```
```  1299   fixes f :: "real \<Rightarrow> 'b::linorder"
```
```  1300   assumes mono: "\<And>x y. Q x \<Longrightarrow> Q y \<Longrightarrow> x \<le> y \<Longrightarrow> f x \<le> f y"
```
```  1301   assumes bij: "\<And>x. P x \<Longrightarrow> f (g x) = x" "\<And>x. P x \<Longrightarrow> Q (g x)"
```
```  1302   assumes Q: "eventually Q (at_right a)" and bound: "\<And>b. Q b \<Longrightarrow> a < b"
```
```  1303   assumes P: "eventually P at_bot"
```
```  1304   shows "filterlim f at_bot (at_right a)"
```
```  1305 proof -
```
```  1306   from P obtain x where x: "\<And>y. y \<le> x \<Longrightarrow> P y"
```
```  1307     unfolding eventually_at_bot_linorder by auto
```
```  1308   show ?thesis
```
```  1309   proof (intro filterlim_at_bot_le[THEN iffD2] allI impI)
```
```  1310     fix z assume "z \<le> x"
```
```  1311     with x have "P z" by auto
```
```  1312     have "eventually (\<lambda>x. x \<le> g z) (at_right a)"
```
```  1313       using bound[OF bij(2)[OF `P z`]]
```
```  1314       by (auto simp add: eventually_within_less dist_real_def intro!:  exI[of _ "g z - a"])
```
```  1315     with Q show "eventually (\<lambda>x. f x \<le> z) (at_right a)"
```
```  1316       by eventually_elim (metis bij `P z` mono)
```
```  1317   qed
```
```  1318 qed
```
```  1319
```
```  1320 lemma filterlim_at_top_at_left:
```
```  1321   fixes f :: "real \<Rightarrow> 'b::linorder"
```
```  1322   assumes mono: "\<And>x y. Q x \<Longrightarrow> Q y \<Longrightarrow> x \<le> y \<Longrightarrow> f x \<le> f y"
```
```  1323   assumes bij: "\<And>x. P x \<Longrightarrow> f (g x) = x" "\<And>x. P x \<Longrightarrow> Q (g x)"
```
```  1324   assumes Q: "eventually Q (at_left a)" and bound: "\<And>b. Q b \<Longrightarrow> b < a"
```
```  1325   assumes P: "eventually P at_top"
```
```  1326   shows "filterlim f at_top (at_left a)"
```
```  1327 proof -
```
```  1328   from P obtain x where x: "\<And>y. x \<le> y \<Longrightarrow> P y"
```
```  1329     unfolding eventually_at_top_linorder by auto
```
```  1330   show ?thesis
```
```  1331   proof (intro filterlim_at_top_ge[THEN iffD2] allI impI)
```
```  1332     fix z assume "x \<le> z"
```
```  1333     with x have "P z" by auto
```
```  1334     have "eventually (\<lambda>x. g z \<le> x) (at_left a)"
```
```  1335       using bound[OF bij(2)[OF `P z`]]
```
```  1336       by (auto simp add: eventually_within_less dist_real_def intro!:  exI[of _ "a - g z"])
```
```  1337     with Q show "eventually (\<lambda>x. z \<le> f x) (at_left a)"
```
```  1338       by eventually_elim (metis bij `P z` mono)
```
```  1339   qed
```
```  1340 qed
```
```  1341
```
```  1342 lemma filterlim_at_infinity:
```
```  1343   fixes f :: "_ \<Rightarrow> 'a\<Colon>real_normed_vector"
```
```  1344   assumes "0 \<le> c"
```
```  1345   shows "(LIM x F. f x :> at_infinity) \<longleftrightarrow> (\<forall>r>c. eventually (\<lambda>x. r \<le> norm (f x)) F)"
```
```  1346   unfolding filterlim_iff eventually_at_infinity
```
```  1347 proof safe
```
```  1348   fix P :: "'a \<Rightarrow> bool" and b
```
```  1349   assume *: "\<forall>r>c. eventually (\<lambda>x. r \<le> norm (f x)) F"
```
```  1350     and P: "\<forall>x. b \<le> norm x \<longrightarrow> P x"
```
```  1351   have "max b (c + 1) > c" by auto
```
```  1352   with * have "eventually (\<lambda>x. max b (c + 1) \<le> norm (f x)) F"
```
```  1353     by auto
```
```  1354   then show "eventually (\<lambda>x. P (f x)) F"
```
```  1355   proof eventually_elim
```
```  1356     fix x assume "max b (c + 1) \<le> norm (f x)"
```
```  1357     with P show "P (f x)" by auto
```
```  1358   qed
```
```  1359 qed force
```
```  1360
```
```  1361 lemma filterlim_real_sequentially: "LIM x sequentially. real x :> at_top"
```
```  1362   unfolding filterlim_at_top
```
```  1363   apply (intro allI)
```
```  1364   apply (rule_tac c="natceiling (Z + 1)" in eventually_sequentiallyI)
```
```  1365   apply (auto simp: natceiling_le_eq)
```
```  1366   done
```
```  1367
```
```  1368 subsection {* Relate @{const at}, @{const at_left} and @{const at_right} *}
```
```  1369
```
```  1370 text {*
```
```  1371
```
```  1372 This lemmas are useful for conversion between @{term "at x"} to @{term "at_left x"} and
```
```  1373 @{term "at_right x"} and also @{term "at_right 0"}.
```
```  1374
```
```  1375 *}
```
```  1376
```
```  1377 lemma at_eq_sup_left_right: "at (x::real) = sup (at_left x) (at_right x)"
```
```  1378   by (auto simp: eventually_within at_def filter_eq_iff eventually_sup
```
```  1379            elim: eventually_elim2 eventually_elim1)
```
```  1380
```
```  1381 lemma filterlim_split_at_real:
```
```  1382   "filterlim f F (at_left x) \<Longrightarrow> filterlim f F (at_right x) \<Longrightarrow> filterlim f F (at (x::real))"
```
```  1383   by (subst at_eq_sup_left_right) (rule filterlim_sup)
```
```  1384
```
```  1385 lemma filtermap_nhds_shift: "filtermap (\<lambda>x. x - d) (nhds a) = nhds (a - d::real)"
```
```  1386   unfolding filter_eq_iff eventually_filtermap eventually_nhds_metric
```
```  1387   by (intro allI ex_cong) (auto simp: dist_real_def field_simps)
```
```  1388
```
```  1389 lemma filtermap_nhds_minus: "filtermap (\<lambda>x. - x) (nhds a) = nhds (- a::real)"
```
```  1390   unfolding filter_eq_iff eventually_filtermap eventually_nhds_metric
```
```  1391   apply (intro allI ex_cong)
```
```  1392   apply (auto simp: dist_real_def field_simps)
```
```  1393   apply (erule_tac x="-x" in allE)
```
```  1394   apply simp
```
```  1395   done
```
```  1396
```
```  1397 lemma filtermap_at_shift: "filtermap (\<lambda>x. x - d) (at a) = at (a - d::real)"
```
```  1398   unfolding at_def filtermap_nhds_shift[symmetric]
```
```  1399   by (simp add: filter_eq_iff eventually_filtermap eventually_within)
```
```  1400
```
```  1401 lemma filtermap_at_right_shift: "filtermap (\<lambda>x. x - d) (at_right a) = at_right (a - d::real)"
```
```  1402   unfolding filtermap_at_shift[symmetric]
```
```  1403   by (simp add: filter_eq_iff eventually_filtermap eventually_within)
```
```  1404
```
```  1405 lemma at_right_to_0: "at_right (a::real) = filtermap (\<lambda>x. x + a) (at_right 0)"
```
```  1406   using filtermap_at_right_shift[of "-a" 0] by simp
```
```  1407
```
```  1408 lemma filterlim_at_right_to_0:
```
```  1409   "filterlim f F (at_right (a::real)) \<longleftrightarrow> filterlim (\<lambda>x. f (x + a)) F (at_right 0)"
```
```  1410   unfolding filterlim_def filtermap_filtermap at_right_to_0[of a] ..
```
```  1411
```
```  1412 lemma eventually_at_right_to_0:
```
```  1413   "eventually P (at_right (a::real)) \<longleftrightarrow> eventually (\<lambda>x. P (x + a)) (at_right 0)"
```
```  1414   unfolding at_right_to_0[of a] by (simp add: eventually_filtermap)
```
```  1415
```
```  1416 lemma filtermap_at_minus: "filtermap (\<lambda>x. - x) (at a) = at (- a::real)"
```
```  1417   unfolding at_def filtermap_nhds_minus[symmetric]
```
```  1418   by (simp add: filter_eq_iff eventually_filtermap eventually_within)
```
```  1419
```
```  1420 lemma at_left_minus: "at_left (a::real) = filtermap (\<lambda>x. - x) (at_right (- a))"
```
```  1421   by (simp add: filter_eq_iff eventually_filtermap eventually_within filtermap_at_minus[symmetric])
```
```  1422
```
```  1423 lemma at_right_minus: "at_right (a::real) = filtermap (\<lambda>x. - x) (at_left (- a))"
```
```  1424   by (simp add: filter_eq_iff eventually_filtermap eventually_within filtermap_at_minus[symmetric])
```
```  1425
```
```  1426 lemma filterlim_at_left_to_right:
```
```  1427   "filterlim f F (at_left (a::real)) \<longleftrightarrow> filterlim (\<lambda>x. f (- x)) F (at_right (-a))"
```
```  1428   unfolding filterlim_def filtermap_filtermap at_left_minus[of a] ..
```
```  1429
```
```  1430 lemma eventually_at_left_to_right:
```
```  1431   "eventually P (at_left (a::real)) \<longleftrightarrow> eventually (\<lambda>x. P (- x)) (at_right (-a))"
```
```  1432   unfolding at_left_minus[of a] by (simp add: eventually_filtermap)
```
```  1433
```
```  1434 lemma filterlim_at_split:
```
```  1435   "filterlim f F (at (x::real)) \<longleftrightarrow> filterlim f F (at_left x) \<and> filterlim f F (at_right x)"
```
```  1436   by (subst at_eq_sup_left_right) (simp add: filterlim_def filtermap_sup)
```
```  1437
```
```  1438 lemma eventually_at_split:
```
```  1439   "eventually P (at (x::real)) \<longleftrightarrow> eventually P (at_left x) \<and> eventually P (at_right x)"
```
```  1440   by (subst at_eq_sup_left_right) (simp add: eventually_sup)
```
```  1441
```
```  1442 lemma at_top_mirror: "at_top = filtermap uminus (at_bot :: real filter)"
```
```  1443   unfolding filter_eq_iff eventually_filtermap eventually_at_top_linorder eventually_at_bot_linorder
```
```  1444   by (metis le_minus_iff minus_minus)
```
```  1445
```
```  1446 lemma at_bot_mirror: "at_bot = filtermap uminus (at_top :: real filter)"
```
```  1447   unfolding at_top_mirror filtermap_filtermap by (simp add: filtermap_ident)
```
```  1448
```
```  1449 lemma filterlim_at_top_mirror: "(LIM x at_top. f x :> F) \<longleftrightarrow> (LIM x at_bot. f (-x::real) :> F)"
```
```  1450   unfolding filterlim_def at_top_mirror filtermap_filtermap ..
```
```  1451
```
```  1452 lemma filterlim_at_bot_mirror: "(LIM x at_bot. f x :> F) \<longleftrightarrow> (LIM x at_top. f (-x::real) :> F)"
```
```  1453   unfolding filterlim_def at_bot_mirror filtermap_filtermap ..
```
```  1454
```
```  1455 lemma filterlim_uminus_at_top_at_bot: "LIM x at_bot. - x :: real :> at_top"
```
```  1456   unfolding filterlim_at_top eventually_at_bot_dense
```
```  1457   by (metis leI minus_less_iff order_less_asym)
```
```  1458
```
```  1459 lemma filterlim_uminus_at_bot_at_top: "LIM x at_top. - x :: real :> at_bot"
```
```  1460   unfolding filterlim_at_bot eventually_at_top_dense
```
```  1461   by (metis leI less_minus_iff order_less_asym)
```
```  1462
```
```  1463 lemma filterlim_uminus_at_top: "(LIM x F. f x :> at_top) \<longleftrightarrow> (LIM x F. - (f x) :: real :> at_bot)"
```
```  1464   using filterlim_compose[OF filterlim_uminus_at_bot_at_top, of f F]
```
```  1465   using filterlim_compose[OF filterlim_uminus_at_top_at_bot, of "\<lambda>x. - f x" F]
```
```  1466   by auto
```
```  1467
```
```  1468 lemma filterlim_uminus_at_bot: "(LIM x F. f x :> at_bot) \<longleftrightarrow> (LIM x F. - (f x) :: real :> at_top)"
```
```  1469   unfolding filterlim_uminus_at_top by simp
```
```  1470
```
```  1471 lemma filterlim_inverse_at_top_right: "LIM x at_right (0::real). inverse x :> at_top"
```
```  1472   unfolding filterlim_at_top_gt[where c=0] eventually_within at_def
```
```  1473 proof safe
```
```  1474   fix Z :: real assume [arith]: "0 < Z"
```
```  1475   then have "eventually (\<lambda>x. x < inverse Z) (nhds 0)"
```
```  1476     by (auto simp add: eventually_nhds_metric dist_real_def intro!: exI[of _ "\<bar>inverse Z\<bar>"])
```
```  1477   then show "eventually (\<lambda>x. x \<in> - {0} \<longrightarrow> x \<in> {0<..} \<longrightarrow> Z \<le> inverse x) (nhds 0)"
```
```  1478     by (auto elim!: eventually_elim1 simp: inverse_eq_divide field_simps)
```
```  1479 qed
```
```  1480
```
```  1481 lemma filterlim_inverse_at_top:
```
```  1482   "(f ---> (0 :: real)) F \<Longrightarrow> eventually (\<lambda>x. 0 < f x) F \<Longrightarrow> LIM x F. inverse (f x) :> at_top"
```
```  1483   by (intro filterlim_compose[OF filterlim_inverse_at_top_right])
```
```  1484      (simp add: filterlim_def eventually_filtermap le_within_iff at_def eventually_elim1)
```
```  1485
```
```  1486 lemma filterlim_inverse_at_bot_neg:
```
```  1487   "LIM x (at_left (0::real)). inverse x :> at_bot"
```
```  1488   by (simp add: filterlim_inverse_at_top_right filterlim_uminus_at_bot filterlim_at_left_to_right)
```
```  1489
```
```  1490 lemma filterlim_inverse_at_bot:
```
```  1491   "(f ---> (0 :: real)) F \<Longrightarrow> eventually (\<lambda>x. f x < 0) F \<Longrightarrow> LIM x F. inverse (f x) :> at_bot"
```
```  1492   unfolding filterlim_uminus_at_bot inverse_minus_eq[symmetric]
```
```  1493   by (rule filterlim_inverse_at_top) (simp_all add: tendsto_minus_cancel_left[symmetric])
```
```  1494
```
```  1495 lemma tendsto_inverse_0:
```
```  1496   fixes x :: "_ \<Rightarrow> 'a\<Colon>real_normed_div_algebra"
```
```  1497   shows "(inverse ---> (0::'a)) at_infinity"
```
```  1498   unfolding tendsto_Zfun_iff diff_0_right Zfun_def eventually_at_infinity
```
```  1499 proof safe
```
```  1500   fix r :: real assume "0 < r"
```
```  1501   show "\<exists>b. \<forall>x. b \<le> norm x \<longrightarrow> norm (inverse x :: 'a) < r"
```
```  1502   proof (intro exI[of _ "inverse (r / 2)"] allI impI)
```
```  1503     fix x :: 'a
```
```  1504     from `0 < r` have "0 < inverse (r / 2)" by simp
```
```  1505     also assume *: "inverse (r / 2) \<le> norm x"
```
```  1506     finally show "norm (inverse x) < r"
```
```  1507       using * `0 < r` by (subst nonzero_norm_inverse) (simp_all add: inverse_eq_divide field_simps)
```
```  1508   qed
```
```  1509 qed
```
```  1510
```
```  1511 lemma at_right_to_top: "(at_right (0::real)) = filtermap inverse at_top"
```
```  1512 proof (rule antisym)
```
```  1513   have "(inverse ---> (0::real)) at_top"
```
```  1514     by (metis tendsto_inverse_0 filterlim_mono at_top_le_at_infinity order_refl)
```
```  1515   then show "filtermap inverse at_top \<le> at_right (0::real)"
```
```  1516     unfolding at_within_eq
```
```  1517     by (intro le_withinI) (simp_all add: eventually_filtermap eventually_gt_at_top filterlim_def)
```
```  1518 next
```
```  1519   have "filtermap inverse (filtermap inverse (at_right (0::real))) \<le> filtermap inverse at_top"
```
```  1520     using filterlim_inverse_at_top_right unfolding filterlim_def by (rule filtermap_mono)
```
```  1521   then show "at_right (0::real) \<le> filtermap inverse at_top"
```
```  1522     by (simp add: filtermap_ident filtermap_filtermap)
```
```  1523 qed
```
```  1524
```
```  1525 lemma eventually_at_right_to_top:
```
```  1526   "eventually P (at_right (0::real)) \<longleftrightarrow> eventually (\<lambda>x. P (inverse x)) at_top"
```
```  1527   unfolding at_right_to_top eventually_filtermap ..
```
```  1528
```
```  1529 lemma filterlim_at_right_to_top:
```
```  1530   "filterlim f F (at_right (0::real)) \<longleftrightarrow> (LIM x at_top. f (inverse x) :> F)"
```
```  1531   unfolding filterlim_def at_right_to_top filtermap_filtermap ..
```
```  1532
```
```  1533 lemma at_top_to_right: "at_top = filtermap inverse (at_right (0::real))"
```
```  1534   unfolding at_right_to_top filtermap_filtermap inverse_inverse_eq filtermap_ident ..
```
```  1535
```
```  1536 lemma eventually_at_top_to_right:
```
```  1537   "eventually P at_top \<longleftrightarrow> eventually (\<lambda>x. P (inverse x)) (at_right (0::real))"
```
```  1538   unfolding at_top_to_right eventually_filtermap ..
```
```  1539
```
```  1540 lemma filterlim_at_top_to_right:
```
```  1541   "filterlim f F at_top \<longleftrightarrow> (LIM x (at_right (0::real)). f (inverse x) :> F)"
```
```  1542   unfolding filterlim_def at_top_to_right filtermap_filtermap ..
```
```  1543
```
```  1544 lemma filterlim_inverse_at_infinity:
```
```  1545   fixes x :: "_ \<Rightarrow> 'a\<Colon>{real_normed_div_algebra, division_ring_inverse_zero}"
```
```  1546   shows "filterlim inverse at_infinity (at (0::'a))"
```
```  1547   unfolding filterlim_at_infinity[OF order_refl]
```
```  1548 proof safe
```
```  1549   fix r :: real assume "0 < r"
```
```  1550   then show "eventually (\<lambda>x::'a. r \<le> norm (inverse x)) (at 0)"
```
```  1551     unfolding eventually_at norm_inverse
```
```  1552     by (intro exI[of _ "inverse r"])
```
```  1553        (auto simp: norm_conv_dist[symmetric] field_simps inverse_eq_divide)
```
```  1554 qed
```
```  1555
```
```  1556 lemma filterlim_inverse_at_iff:
```
```  1557   fixes g :: "'a \<Rightarrow> 'b\<Colon>{real_normed_div_algebra, division_ring_inverse_zero}"
```
```  1558   shows "(LIM x F. inverse (g x) :> at 0) \<longleftrightarrow> (LIM x F. g x :> at_infinity)"
```
```  1559   unfolding filterlim_def filtermap_filtermap[symmetric]
```
```  1560 proof
```
```  1561   assume "filtermap g F \<le> at_infinity"
```
```  1562   then have "filtermap inverse (filtermap g F) \<le> filtermap inverse at_infinity"
```
```  1563     by (rule filtermap_mono)
```
```  1564   also have "\<dots> \<le> at 0"
```
```  1565     using tendsto_inverse_0
```
```  1566     by (auto intro!: le_withinI exI[of _ 1]
```
```  1567              simp: eventually_filtermap eventually_at_infinity filterlim_def at_def)
```
```  1568   finally show "filtermap inverse (filtermap g F) \<le> at 0" .
```
```  1569 next
```
```  1570   assume "filtermap inverse (filtermap g F) \<le> at 0"
```
```  1571   then have "filtermap inverse (filtermap inverse (filtermap g F)) \<le> filtermap inverse (at 0)"
```
```  1572     by (rule filtermap_mono)
```
```  1573   with filterlim_inverse_at_infinity show "filtermap g F \<le> at_infinity"
```
```  1574     by (auto intro: order_trans simp: filterlim_def filtermap_filtermap)
```
```  1575 qed
```
```  1576
```
```  1577 lemma tendsto_inverse_0_at_top:
```
```  1578   "LIM x F. f x :> at_top \<Longrightarrow> ((\<lambda>x. inverse (f x) :: real) ---> 0) F"
```
```  1579  by (metis at_top_le_at_infinity filterlim_at filterlim_inverse_at_iff filterlim_mono order_refl)
```
```  1580
```
```  1581 text {*
```
```  1582
```
```  1583 We only show rules for multiplication and addition when the functions are either against a real
```
```  1584 value or against infinity. Further rules are easy to derive by using @{thm filterlim_uminus_at_top}.
```
```  1585
```
```  1586 *}
```
```  1587
```
```  1588 lemma filterlim_tendsto_pos_mult_at_top:
```
```  1589   assumes f: "(f ---> c) F" and c: "0 < c"
```
```  1590   assumes g: "LIM x F. g x :> at_top"
```
```  1591   shows "LIM x F. (f x * g x :: real) :> at_top"
```
```  1592   unfolding filterlim_at_top_gt[where c=0]
```
```  1593 proof safe
```
```  1594   fix Z :: real assume "0 < Z"
```
```  1595   from f `0 < c` have "eventually (\<lambda>x. c / 2 < f x) F"
```
```  1596     by (auto dest!: tendstoD[where e="c / 2"] elim!: eventually_elim1
```
```  1597              simp: dist_real_def abs_real_def split: split_if_asm)
```
```  1598   moreover from g have "eventually (\<lambda>x. (Z / c * 2) \<le> g x) F"
```
```  1599     unfolding filterlim_at_top by auto
```
```  1600   ultimately show "eventually (\<lambda>x. Z \<le> f x * g x) F"
```
```  1601   proof eventually_elim
```
```  1602     fix x assume "c / 2 < f x" "Z / c * 2 \<le> g x"
```
```  1603     with `0 < Z` `0 < c` have "c / 2 * (Z / c * 2) \<le> f x * g x"
```
```  1604       by (intro mult_mono) (auto simp: zero_le_divide_iff)
```
```  1605     with `0 < c` show "Z \<le> f x * g x"
```
```  1606        by simp
```
```  1607   qed
```
```  1608 qed
```
```  1609
```
```  1610 lemma filterlim_at_top_mult_at_top:
```
```  1611   assumes f: "LIM x F. f x :> at_top"
```
```  1612   assumes g: "LIM x F. g x :> at_top"
```
```  1613   shows "LIM x F. (f x * g x :: real) :> at_top"
```
```  1614   unfolding filterlim_at_top_gt[where c=0]
```
```  1615 proof safe
```
```  1616   fix Z :: real assume "0 < Z"
```
```  1617   from f have "eventually (\<lambda>x. 1 \<le> f x) F"
```
```  1618     unfolding filterlim_at_top by auto
```
```  1619   moreover from g have "eventually (\<lambda>x. Z \<le> g x) F"
```
```  1620     unfolding filterlim_at_top by auto
```
```  1621   ultimately show "eventually (\<lambda>x. Z \<le> f x * g x) F"
```
```  1622   proof eventually_elim
```
```  1623     fix x assume "1 \<le> f x" "Z \<le> g x"
```
```  1624     with `0 < Z` have "1 * Z \<le> f x * g x"
```
```  1625       by (intro mult_mono) (auto simp: zero_le_divide_iff)
```
```  1626     then show "Z \<le> f x * g x"
```
```  1627        by simp
```
```  1628   qed
```
```  1629 qed
```
```  1630
```
```  1631 lemma filterlim_tendsto_pos_mult_at_bot:
```
```  1632   assumes "(f ---> c) F" "0 < (c::real)" "filterlim g at_bot F"
```
```  1633   shows "LIM x F. f x * g x :> at_bot"
```
```  1634   using filterlim_tendsto_pos_mult_at_top[OF assms(1,2), of "\<lambda>x. - g x"] assms(3)
```
```  1635   unfolding filterlim_uminus_at_bot by simp
```
```  1636
```
```  1637 lemma filterlim_tendsto_add_at_top:
```
```  1638   assumes f: "(f ---> c) F"
```
```  1639   assumes g: "LIM x F. g x :> at_top"
```
```  1640   shows "LIM x F. (f x + g x :: real) :> at_top"
```
```  1641   unfolding filterlim_at_top_gt[where c=0]
```
```  1642 proof safe
```
```  1643   fix Z :: real assume "0 < Z"
```
```  1644   from f have "eventually (\<lambda>x. c - 1 < f x) F"
```
```  1645     by (auto dest!: tendstoD[where e=1] elim!: eventually_elim1 simp: dist_real_def)
```
```  1646   moreover from g have "eventually (\<lambda>x. Z - (c - 1) \<le> g x) F"
```
```  1647     unfolding filterlim_at_top by auto
```
```  1648   ultimately show "eventually (\<lambda>x. Z \<le> f x + g x) F"
```
```  1649     by eventually_elim simp
```
```  1650 qed
```
```  1651
```
```  1652 lemma LIM_at_top_divide:
```
```  1653   fixes f g :: "'a \<Rightarrow> real"
```
```  1654   assumes f: "(f ---> a) F" "0 < a"
```
```  1655   assumes g: "(g ---> 0) F" "eventually (\<lambda>x. 0 < g x) F"
```
```  1656   shows "LIM x F. f x / g x :> at_top"
```
```  1657   unfolding divide_inverse
```
```  1658   by (rule filterlim_tendsto_pos_mult_at_top[OF f]) (rule filterlim_inverse_at_top[OF g])
```
```  1659
```
```  1660 lemma filterlim_at_top_add_at_top:
```
```  1661   assumes f: "LIM x F. f x :> at_top"
```
```  1662   assumes g: "LIM x F. g x :> at_top"
```
```  1663   shows "LIM x F. (f x + g x :: real) :> at_top"
```
```  1664   unfolding filterlim_at_top_gt[where c=0]
```
```  1665 proof safe
```
```  1666   fix Z :: real assume "0 < Z"
```
```  1667   from f have "eventually (\<lambda>x. 0 \<le> f x) F"
```
```  1668     unfolding filterlim_at_top by auto
```
```  1669   moreover from g have "eventually (\<lambda>x. Z \<le> g x) F"
```
```  1670     unfolding filterlim_at_top by auto
```
```  1671   ultimately show "eventually (\<lambda>x. Z \<le> f x + g x) F"
```
```  1672     by eventually_elim simp
```
```  1673 qed
```
```  1674
```
```  1675 lemma tendsto_divide_0:
```
```  1676   fixes f :: "_ \<Rightarrow> 'a\<Colon>{real_normed_div_algebra, division_ring_inverse_zero}"
```
```  1677   assumes f: "(f ---> c) F"
```
```  1678   assumes g: "LIM x F. g x :> at_infinity"
```
```  1679   shows "((\<lambda>x. f x / g x) ---> 0) F"
```
```  1680   using tendsto_mult[OF f filterlim_compose[OF tendsto_inverse_0 g]] by (simp add: divide_inverse)
```
```  1681
```
```  1682 lemma linear_plus_1_le_power:
```
```  1683   fixes x :: real
```
```  1684   assumes x: "0 \<le> x"
```
```  1685   shows "real n * x + 1 \<le> (x + 1) ^ n"
```
```  1686 proof (induct n)
```
```  1687   case (Suc n)
```
```  1688   have "real (Suc n) * x + 1 \<le> (x + 1) * (real n * x + 1)"
```
```  1689     by (simp add: field_simps real_of_nat_Suc mult_nonneg_nonneg x)
```
```  1690   also have "\<dots> \<le> (x + 1)^Suc n"
```
```  1691     using Suc x by (simp add: mult_left_mono)
```
```  1692   finally show ?case .
```
```  1693 qed simp
```
```  1694
```
```  1695 lemma filterlim_realpow_sequentially_gt1:
```
```  1696   fixes x :: "'a :: real_normed_div_algebra"
```
```  1697   assumes x[arith]: "1 < norm x"
```
```  1698   shows "LIM n sequentially. x ^ n :> at_infinity"
```
```  1699 proof (intro filterlim_at_infinity[THEN iffD2] allI impI)
```
```  1700   fix y :: real assume "0 < y"
```
```  1701   have "0 < norm x - 1" by simp
```
```  1702   then obtain N::nat where "y < real N * (norm x - 1)" by (blast dest: reals_Archimedean3)
```
```  1703   also have "\<dots> \<le> real N * (norm x - 1) + 1" by simp
```
```  1704   also have "\<dots> \<le> (norm x - 1 + 1) ^ N" by (rule linear_plus_1_le_power) simp
```
```  1705   also have "\<dots> = norm x ^ N" by simp
```
```  1706   finally have "\<forall>n\<ge>N. y \<le> norm x ^ n"
```
```  1707     by (metis order_less_le_trans power_increasing order_less_imp_le x)
```
```  1708   then show "eventually (\<lambda>n. y \<le> norm (x ^ n)) sequentially"
```
```  1709     unfolding eventually_sequentially
```
```  1710     by (auto simp: norm_power)
```
```  1711 qed simp
```
```  1712
```
```  1713 end
```
```  1714
```