src/HOL/Limits.thy
 author hoelzl Mon Dec 03 18:19:04 2012 +0100 (2012-12-03) changeset 50325 5e40ad9f212a parent 50324 0a1242d5e7d4 child 50326 b5afeccab2db permissions -rw-r--r--
add filterlim rules for inverse and at_infinity
```     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_mp:
```
```    72   assumes "eventually (\<lambda>x. P x \<longrightarrow> Q x) F"
```
```    73   assumes "eventually (\<lambda>x. P x) F"
```
```    74   shows "eventually (\<lambda>x. Q x) F"
```
```    75 proof (rule eventually_mono)
```
```    76   show "\<forall>x. (P x \<longrightarrow> Q x) \<and> P x \<longrightarrow> Q x" by simp
```
```    77   show "eventually (\<lambda>x. (P x \<longrightarrow> Q x) \<and> P x) F"
```
```    78     using assms by (rule eventually_conj)
```
```    79 qed
```
```    80
```
```    81 lemma eventually_rev_mp:
```
```    82   assumes "eventually (\<lambda>x. P x) F"
```
```    83   assumes "eventually (\<lambda>x. P x \<longrightarrow> Q x) F"
```
```    84   shows "eventually (\<lambda>x. Q x) F"
```
```    85 using assms(2) assms(1) by (rule eventually_mp)
```
```    86
```
```    87 lemma eventually_conj_iff:
```
```    88   "eventually (\<lambda>x. P x \<and> Q x) F \<longleftrightarrow> eventually P F \<and> eventually Q F"
```
```    89   by (auto intro: eventually_conj elim: eventually_rev_mp)
```
```    90
```
```    91 lemma eventually_elim1:
```
```    92   assumes "eventually (\<lambda>i. P i) F"
```
```    93   assumes "\<And>i. P i \<Longrightarrow> Q i"
```
```    94   shows "eventually (\<lambda>i. Q i) F"
```
```    95   using assms by (auto elim!: eventually_rev_mp)
```
```    96
```
```    97 lemma eventually_elim2:
```
```    98   assumes "eventually (\<lambda>i. P i) F"
```
```    99   assumes "eventually (\<lambda>i. Q i) F"
```
```   100   assumes "\<And>i. P i \<Longrightarrow> Q i \<Longrightarrow> R i"
```
```   101   shows "eventually (\<lambda>i. R i) F"
```
```   102   using assms by (auto elim!: eventually_rev_mp)
```
```   103
```
```   104 lemma eventually_subst:
```
```   105   assumes "eventually (\<lambda>n. P n = Q n) F"
```
```   106   shows "eventually P F = eventually Q F" (is "?L = ?R")
```
```   107 proof -
```
```   108   from assms have "eventually (\<lambda>x. P x \<longrightarrow> Q x) F"
```
```   109       and "eventually (\<lambda>x. Q x \<longrightarrow> P x) F"
```
```   110     by (auto elim: eventually_elim1)
```
```   111   then show ?thesis by (auto elim: eventually_elim2)
```
```   112 qed
```
```   113
```
```   114 ML {*
```
```   115   fun eventually_elim_tac ctxt thms thm =
```
```   116     let
```
```   117       val thy = Proof_Context.theory_of ctxt
```
```   118       val mp_thms = thms RL [@{thm eventually_rev_mp}]
```
```   119       val raw_elim_thm =
```
```   120         (@{thm allI} RS @{thm always_eventually})
```
```   121         |> fold (fn thm1 => fn thm2 => thm2 RS thm1) mp_thms
```
```   122         |> fold (fn _ => fn thm => @{thm impI} RS thm) thms
```
```   123       val cases_prop = prop_of (raw_elim_thm RS thm)
```
```   124       val cases = (Rule_Cases.make_common (thy, cases_prop) [(("elim", []), [])])
```
```   125     in
```
```   126       CASES cases (rtac raw_elim_thm 1) thm
```
```   127     end
```
```   128 *}
```
```   129
```
```   130 method_setup eventually_elim = {*
```
```   131   Scan.succeed (fn ctxt => METHOD_CASES (eventually_elim_tac ctxt))
```
```   132 *} "elimination of eventually quantifiers"
```
```   133
```
```   134
```
```   135 subsection {* Finer-than relation *}
```
```   136
```
```   137 text {* @{term "F \<le> F'"} means that filter @{term F} is finer than
```
```   138 filter @{term F'}. *}
```
```   139
```
```   140 instantiation filter :: (type) complete_lattice
```
```   141 begin
```
```   142
```
```   143 definition le_filter_def:
```
```   144   "F \<le> F' \<longleftrightarrow> (\<forall>P. eventually P F' \<longrightarrow> eventually P F)"
```
```   145
```
```   146 definition
```
```   147   "(F :: 'a filter) < F' \<longleftrightarrow> F \<le> F' \<and> \<not> F' \<le> F"
```
```   148
```
```   149 definition
```
```   150   "top = Abs_filter (\<lambda>P. \<forall>x. P x)"
```
```   151
```
```   152 definition
```
```   153   "bot = Abs_filter (\<lambda>P. True)"
```
```   154
```
```   155 definition
```
```   156   "sup F F' = Abs_filter (\<lambda>P. eventually P F \<and> eventually P F')"
```
```   157
```
```   158 definition
```
```   159   "inf F F' = Abs_filter
```
```   160       (\<lambda>P. \<exists>Q R. eventually Q F \<and> eventually R F' \<and> (\<forall>x. Q x \<and> R x \<longrightarrow> P x))"
```
```   161
```
```   162 definition
```
```   163   "Sup S = Abs_filter (\<lambda>P. \<forall>F\<in>S. eventually P F)"
```
```   164
```
```   165 definition
```
```   166   "Inf S = Sup {F::'a filter. \<forall>F'\<in>S. F \<le> F'}"
```
```   167
```
```   168 lemma eventually_top [simp]: "eventually P top \<longleftrightarrow> (\<forall>x. P x)"
```
```   169   unfolding top_filter_def
```
```   170   by (rule eventually_Abs_filter, rule is_filter.intro, auto)
```
```   171
```
```   172 lemma eventually_bot [simp]: "eventually P bot"
```
```   173   unfolding bot_filter_def
```
```   174   by (subst eventually_Abs_filter, rule is_filter.intro, auto)
```
```   175
```
```   176 lemma eventually_sup:
```
```   177   "eventually P (sup F F') \<longleftrightarrow> eventually P F \<and> eventually P F'"
```
```   178   unfolding sup_filter_def
```
```   179   by (rule eventually_Abs_filter, rule is_filter.intro)
```
```   180      (auto elim!: eventually_rev_mp)
```
```   181
```
```   182 lemma eventually_inf:
```
```   183   "eventually P (inf F F') \<longleftrightarrow>
```
```   184    (\<exists>Q R. eventually Q F \<and> eventually R F' \<and> (\<forall>x. Q x \<and> R x \<longrightarrow> P x))"
```
```   185   unfolding inf_filter_def
```
```   186   apply (rule eventually_Abs_filter, rule is_filter.intro)
```
```   187   apply (fast intro: eventually_True)
```
```   188   apply clarify
```
```   189   apply (intro exI conjI)
```
```   190   apply (erule (1) eventually_conj)
```
```   191   apply (erule (1) eventually_conj)
```
```   192   apply simp
```
```   193   apply auto
```
```   194   done
```
```   195
```
```   196 lemma eventually_Sup:
```
```   197   "eventually P (Sup S) \<longleftrightarrow> (\<forall>F\<in>S. eventually P F)"
```
```   198   unfolding Sup_filter_def
```
```   199   apply (rule eventually_Abs_filter, rule is_filter.intro)
```
```   200   apply (auto intro: eventually_conj elim!: eventually_rev_mp)
```
```   201   done
```
```   202
```
```   203 instance proof
```
```   204   fix F F' F'' :: "'a filter" and S :: "'a filter set"
```
```   205   { show "F < F' \<longleftrightarrow> F \<le> F' \<and> \<not> F' \<le> F"
```
```   206     by (rule less_filter_def) }
```
```   207   { show "F \<le> F"
```
```   208     unfolding le_filter_def by simp }
```
```   209   { assume "F \<le> F'" and "F' \<le> F''" thus "F \<le> F''"
```
```   210     unfolding le_filter_def by simp }
```
```   211   { assume "F \<le> F'" and "F' \<le> F" thus "F = F'"
```
```   212     unfolding le_filter_def filter_eq_iff by fast }
```
```   213   { show "F \<le> top"
```
```   214     unfolding le_filter_def eventually_top by (simp add: always_eventually) }
```
```   215   { show "bot \<le> F"
```
```   216     unfolding le_filter_def by simp }
```
```   217   { show "F \<le> sup F F'" and "F' \<le> sup F F'"
```
```   218     unfolding le_filter_def eventually_sup by simp_all }
```
```   219   { assume "F \<le> F''" and "F' \<le> F''" thus "sup F F' \<le> F''"
```
```   220     unfolding le_filter_def eventually_sup by simp }
```
```   221   { show "inf F F' \<le> F" and "inf F F' \<le> F'"
```
```   222     unfolding le_filter_def eventually_inf by (auto intro: eventually_True) }
```
```   223   { assume "F \<le> F'" and "F \<le> F''" thus "F \<le> inf F' F''"
```
```   224     unfolding le_filter_def eventually_inf
```
```   225     by (auto elim!: eventually_mono intro: eventually_conj) }
```
```   226   { assume "F \<in> S" thus "F \<le> Sup S"
```
```   227     unfolding le_filter_def eventually_Sup by simp }
```
```   228   { assume "\<And>F. F \<in> S \<Longrightarrow> F \<le> F'" thus "Sup S \<le> F'"
```
```   229     unfolding le_filter_def eventually_Sup by simp }
```
```   230   { assume "F'' \<in> S" thus "Inf S \<le> F''"
```
```   231     unfolding le_filter_def Inf_filter_def eventually_Sup Ball_def by simp }
```
```   232   { assume "\<And>F'. F' \<in> S \<Longrightarrow> F \<le> F'" thus "F \<le> Inf S"
```
```   233     unfolding le_filter_def Inf_filter_def eventually_Sup Ball_def by simp }
```
```   234 qed
```
```   235
```
```   236 end
```
```   237
```
```   238 lemma filter_leD:
```
```   239   "F \<le> F' \<Longrightarrow> eventually P F' \<Longrightarrow> eventually P F"
```
```   240   unfolding le_filter_def by simp
```
```   241
```
```   242 lemma filter_leI:
```
```   243   "(\<And>P. eventually P F' \<Longrightarrow> eventually P F) \<Longrightarrow> F \<le> F'"
```
```   244   unfolding le_filter_def by simp
```
```   245
```
```   246 lemma eventually_False:
```
```   247   "eventually (\<lambda>x. False) F \<longleftrightarrow> F = bot"
```
```   248   unfolding filter_eq_iff by (auto elim: eventually_rev_mp)
```
```   249
```
```   250 abbreviation (input) trivial_limit :: "'a filter \<Rightarrow> bool"
```
```   251   where "trivial_limit F \<equiv> F = bot"
```
```   252
```
```   253 lemma trivial_limit_def: "trivial_limit F \<longleftrightarrow> eventually (\<lambda>x. False) F"
```
```   254   by (rule eventually_False [symmetric])
```
```   255
```
```   256
```
```   257 subsection {* Map function for filters *}
```
```   258
```
```   259 definition filtermap :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a filter \<Rightarrow> 'b filter"
```
```   260   where "filtermap f F = Abs_filter (\<lambda>P. eventually (\<lambda>x. P (f x)) F)"
```
```   261
```
```   262 lemma eventually_filtermap:
```
```   263   "eventually P (filtermap f F) = eventually (\<lambda>x. P (f x)) F"
```
```   264   unfolding filtermap_def
```
```   265   apply (rule eventually_Abs_filter)
```
```   266   apply (rule is_filter.intro)
```
```   267   apply (auto elim!: eventually_rev_mp)
```
```   268   done
```
```   269
```
```   270 lemma filtermap_ident: "filtermap (\<lambda>x. x) F = F"
```
```   271   by (simp add: filter_eq_iff eventually_filtermap)
```
```   272
```
```   273 lemma filtermap_filtermap:
```
```   274   "filtermap f (filtermap g F) = filtermap (\<lambda>x. f (g x)) F"
```
```   275   by (simp add: filter_eq_iff eventually_filtermap)
```
```   276
```
```   277 lemma filtermap_mono: "F \<le> F' \<Longrightarrow> filtermap f F \<le> filtermap f F'"
```
```   278   unfolding le_filter_def eventually_filtermap by simp
```
```   279
```
```   280 lemma filtermap_bot [simp]: "filtermap f bot = bot"
```
```   281   by (simp add: filter_eq_iff eventually_filtermap)
```
```   282
```
```   283 subsection {* Order filters *}
```
```   284
```
```   285 definition at_top :: "('a::order) filter"
```
```   286   where "at_top = Abs_filter (\<lambda>P. \<exists>k. \<forall>n\<ge>k. P n)"
```
```   287
```
```   288 lemma eventually_at_top_linorder: "eventually P at_top \<longleftrightarrow> (\<exists>N::'a::linorder. \<forall>n\<ge>N. P n)"
```
```   289   unfolding at_top_def
```
```   290 proof (rule eventually_Abs_filter, rule is_filter.intro)
```
```   291   fix P Q :: "'a \<Rightarrow> bool"
```
```   292   assume "\<exists>i. \<forall>n\<ge>i. P n" and "\<exists>j. \<forall>n\<ge>j. Q n"
```
```   293   then obtain i j where "\<forall>n\<ge>i. P n" and "\<forall>n\<ge>j. Q n" by auto
```
```   294   then have "\<forall>n\<ge>max i j. P n \<and> Q n" by simp
```
```   295   then show "\<exists>k. \<forall>n\<ge>k. P n \<and> Q n" ..
```
```   296 qed auto
```
```   297
```
```   298 lemma eventually_at_top_dense: "eventually P at_top \<longleftrightarrow> (\<exists>N::'a::dense_linorder. \<forall>n>N. P n)"
```
```   299   unfolding eventually_at_top_linorder
```
```   300 proof safe
```
```   301   fix N assume "\<forall>n\<ge>N. P n" then show "\<exists>N. \<forall>n>N. P n" by (auto intro!: exI[of _ N])
```
```   302 next
```
```   303   fix N assume "\<forall>n>N. P n"
```
```   304   moreover from gt_ex[of N] guess y ..
```
```   305   ultimately show "\<exists>N. \<forall>n\<ge>N. P n" by (auto intro!: exI[of _ y])
```
```   306 qed
```
```   307
```
```   308 definition at_bot :: "('a::order) filter"
```
```   309   where "at_bot = Abs_filter (\<lambda>P. \<exists>k. \<forall>n\<le>k. P n)"
```
```   310
```
```   311 lemma eventually_at_bot_linorder:
```
```   312   fixes P :: "'a::linorder \<Rightarrow> bool" shows "eventually P at_bot \<longleftrightarrow> (\<exists>N. \<forall>n\<le>N. P n)"
```
```   313   unfolding at_bot_def
```
```   314 proof (rule eventually_Abs_filter, rule is_filter.intro)
```
```   315   fix P Q :: "'a \<Rightarrow> bool"
```
```   316   assume "\<exists>i. \<forall>n\<le>i. P n" and "\<exists>j. \<forall>n\<le>j. Q n"
```
```   317   then obtain i j where "\<forall>n\<le>i. P n" and "\<forall>n\<le>j. Q n" by auto
```
```   318   then have "\<forall>n\<le>min i j. P n \<and> Q n" by simp
```
```   319   then show "\<exists>k. \<forall>n\<le>k. P n \<and> Q n" ..
```
```   320 qed auto
```
```   321
```
```   322 lemma eventually_at_bot_dense:
```
```   323   fixes P :: "'a::dense_linorder \<Rightarrow> bool" shows "eventually P at_bot \<longleftrightarrow> (\<exists>N. \<forall>n<N. P n)"
```
```   324   unfolding eventually_at_bot_linorder
```
```   325 proof safe
```
```   326   fix N assume "\<forall>n\<le>N. P n" then show "\<exists>N. \<forall>n<N. P n" by (auto intro!: exI[of _ N])
```
```   327 next
```
```   328   fix N assume "\<forall>n<N. P n"
```
```   329   moreover from lt_ex[of N] guess y ..
```
```   330   ultimately show "\<exists>N. \<forall>n\<le>N. P n" by (auto intro!: exI[of _ y])
```
```   331 qed
```
```   332
```
```   333 subsection {* Sequentially *}
```
```   334
```
```   335 abbreviation sequentially :: "nat filter"
```
```   336   where "sequentially == at_top"
```
```   337
```
```   338 lemma sequentially_def: "sequentially = Abs_filter (\<lambda>P. \<exists>k. \<forall>n\<ge>k. P n)"
```
```   339   unfolding at_top_def by simp
```
```   340
```
```   341 lemma eventually_sequentially:
```
```   342   "eventually P sequentially \<longleftrightarrow> (\<exists>N. \<forall>n\<ge>N. P n)"
```
```   343   by (rule eventually_at_top_linorder)
```
```   344
```
```   345 lemma sequentially_bot [simp, intro]: "sequentially \<noteq> bot"
```
```   346   unfolding filter_eq_iff eventually_sequentially by auto
```
```   347
```
```   348 lemmas trivial_limit_sequentially = sequentially_bot
```
```   349
```
```   350 lemma eventually_False_sequentially [simp]:
```
```   351   "\<not> eventually (\<lambda>n. False) sequentially"
```
```   352   by (simp add: eventually_False)
```
```   353
```
```   354 lemma le_sequentially:
```
```   355   "F \<le> sequentially \<longleftrightarrow> (\<forall>N. eventually (\<lambda>n. N \<le> n) F)"
```
```   356   unfolding le_filter_def eventually_sequentially
```
```   357   by (safe, fast, drule_tac x=N in spec, auto elim: eventually_rev_mp)
```
```   358
```
```   359 lemma eventually_sequentiallyI:
```
```   360   assumes "\<And>x. c \<le> x \<Longrightarrow> P x"
```
```   361   shows "eventually P sequentially"
```
```   362 using assms by (auto simp: eventually_sequentially)
```
```   363
```
```   364
```
```   365 subsection {* Standard filters *}
```
```   366
```
```   367 definition within :: "'a filter \<Rightarrow> 'a set \<Rightarrow> 'a filter" (infixr "within" 70)
```
```   368   where "F within S = Abs_filter (\<lambda>P. eventually (\<lambda>x. x \<in> S \<longrightarrow> P x) F)"
```
```   369
```
```   370 definition (in topological_space) nhds :: "'a \<Rightarrow> 'a filter"
```
```   371   where "nhds a = Abs_filter (\<lambda>P. \<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x))"
```
```   372
```
```   373 definition (in topological_space) at :: "'a \<Rightarrow> 'a filter"
```
```   374   where "at a = nhds a within - {a}"
```
```   375
```
```   376 definition at_infinity :: "'a::real_normed_vector filter" where
```
```   377   "at_infinity = Abs_filter (\<lambda>P. \<exists>r. \<forall>x. r \<le> norm x \<longrightarrow> P x)"
```
```   378
```
```   379 lemma eventually_within:
```
```   380   "eventually P (F within S) = eventually (\<lambda>x. x \<in> S \<longrightarrow> P x) F"
```
```   381   unfolding within_def
```
```   382   by (rule eventually_Abs_filter, rule is_filter.intro)
```
```   383      (auto elim!: eventually_rev_mp)
```
```   384
```
```   385 lemma within_UNIV [simp]: "F within UNIV = F"
```
```   386   unfolding filter_eq_iff eventually_within by simp
```
```   387
```
```   388 lemma within_empty [simp]: "F within {} = bot"
```
```   389   unfolding filter_eq_iff eventually_within by simp
```
```   390
```
```   391 lemma within_le: "F within S \<le> F"
```
```   392   unfolding le_filter_def eventually_within by (auto elim: eventually_elim1)
```
```   393
```
```   394 lemma le_withinI: "F \<le> F' \<Longrightarrow> eventually (\<lambda>x. x \<in> S) F \<Longrightarrow> F \<le> F' within S"
```
```   395   unfolding le_filter_def eventually_within by (auto elim: eventually_elim2)
```
```   396
```
```   397 lemma le_within_iff: "eventually (\<lambda>x. x \<in> S) F \<Longrightarrow> F \<le> F' within S \<longleftrightarrow> F \<le> F'"
```
```   398   by (blast intro: within_le le_withinI order_trans)
```
```   399
```
```   400 lemma eventually_nhds:
```
```   401   "eventually P (nhds a) \<longleftrightarrow> (\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x))"
```
```   402 unfolding nhds_def
```
```   403 proof (rule eventually_Abs_filter, rule is_filter.intro)
```
```   404   have "open UNIV \<and> a \<in> UNIV \<and> (\<forall>x\<in>UNIV. True)" by simp
```
```   405   thus "\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. True)" ..
```
```   406 next
```
```   407   fix P Q
```
```   408   assume "\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x)"
```
```   409      and "\<exists>T. open T \<and> a \<in> T \<and> (\<forall>x\<in>T. Q x)"
```
```   410   then obtain S T where
```
```   411     "open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x)"
```
```   412     "open T \<and> a \<in> T \<and> (\<forall>x\<in>T. Q x)" by auto
```
```   413   hence "open (S \<inter> T) \<and> a \<in> S \<inter> T \<and> (\<forall>x\<in>(S \<inter> T). P x \<and> Q x)"
```
```   414     by (simp add: open_Int)
```
```   415   thus "\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x \<and> Q x)" ..
```
```   416 qed auto
```
```   417
```
```   418 lemma eventually_nhds_metric:
```
```   419   "eventually P (nhds a) \<longleftrightarrow> (\<exists>d>0. \<forall>x. dist x a < d \<longrightarrow> P x)"
```
```   420 unfolding eventually_nhds open_dist
```
```   421 apply safe
```
```   422 apply fast
```
```   423 apply (rule_tac x="{x. dist x a < d}" in exI, simp)
```
```   424 apply clarsimp
```
```   425 apply (rule_tac x="d - dist x a" in exI, clarsimp)
```
```   426 apply (simp only: less_diff_eq)
```
```   427 apply (erule le_less_trans [OF dist_triangle])
```
```   428 done
```
```   429
```
```   430 lemma nhds_neq_bot [simp]: "nhds a \<noteq> bot"
```
```   431   unfolding trivial_limit_def eventually_nhds by simp
```
```   432
```
```   433 lemma eventually_at_topological:
```
```   434   "eventually P (at a) \<longleftrightarrow> (\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. x \<noteq> a \<longrightarrow> P x))"
```
```   435 unfolding at_def eventually_within eventually_nhds by simp
```
```   436
```
```   437 lemma eventually_at:
```
```   438   fixes a :: "'a::metric_space"
```
```   439   shows "eventually P (at a) \<longleftrightarrow> (\<exists>d>0. \<forall>x. x \<noteq> a \<and> dist x a < d \<longrightarrow> P x)"
```
```   440 unfolding at_def eventually_within eventually_nhds_metric by auto
```
```   441
```
```   442 lemma at_eq_bot_iff: "at a = bot \<longleftrightarrow> open {a}"
```
```   443   unfolding trivial_limit_def eventually_at_topological
```
```   444   by (safe, case_tac "S = {a}", simp, fast, fast)
```
```   445
```
```   446 lemma at_neq_bot [simp]: "at (a::'a::perfect_space) \<noteq> bot"
```
```   447   by (simp add: at_eq_bot_iff not_open_singleton)
```
```   448
```
```   449 lemma eventually_at_infinity:
```
```   450   "eventually P at_infinity \<longleftrightarrow> (\<exists>b. \<forall>x. b \<le> norm x \<longrightarrow> P x)"
```
```   451 unfolding at_infinity_def
```
```   452 proof (rule eventually_Abs_filter, rule is_filter.intro)
```
```   453   fix P Q :: "'a \<Rightarrow> bool"
```
```   454   assume "\<exists>r. \<forall>x. r \<le> norm x \<longrightarrow> P x" and "\<exists>s. \<forall>x. s \<le> norm x \<longrightarrow> Q x"
```
```   455   then obtain r s where
```
```   456     "\<forall>x. r \<le> norm x \<longrightarrow> P x" and "\<forall>x. s \<le> norm x \<longrightarrow> Q x" by auto
```
```   457   then have "\<forall>x. max r s \<le> norm x \<longrightarrow> P x \<and> Q x" by simp
```
```   458   then show "\<exists>r. \<forall>x. r \<le> norm x \<longrightarrow> P x \<and> Q x" ..
```
```   459 qed auto
```
```   460
```
```   461 lemma at_infinity_eq_at_top_bot:
```
```   462   "(at_infinity \<Colon> real filter) = sup at_top at_bot"
```
```   463   unfolding sup_filter_def at_infinity_def eventually_at_top_linorder eventually_at_bot_linorder
```
```   464 proof (intro arg_cong[where f=Abs_filter] ext iffI)
```
```   465   fix P :: "real \<Rightarrow> bool" assume "\<exists>r. \<forall>x. r \<le> norm x \<longrightarrow> P x"
```
```   466   then guess r ..
```
```   467   then have "(\<forall>x\<ge>r. P x) \<and> (\<forall>x\<le>-r. P x)" by auto
```
```   468   then show "(\<exists>r. \<forall>x\<ge>r. P x) \<and> (\<exists>r. \<forall>x\<le>r. P x)" by auto
```
```   469 next
```
```   470   fix P :: "real \<Rightarrow> bool" assume "(\<exists>r. \<forall>x\<ge>r. P x) \<and> (\<exists>r. \<forall>x\<le>r. P x)"
```
```   471   then obtain p q where "\<forall>x\<ge>p. P x" "\<forall>x\<le>q. P x" by auto
```
```   472   then show "\<exists>r. \<forall>x. r \<le> norm x \<longrightarrow> P x"
```
```   473     by (intro exI[of _ "max p (-q)"])
```
```   474        (auto simp: abs_real_def)
```
```   475 qed
```
```   476
```
```   477 lemma at_top_le_at_infinity:
```
```   478   "at_top \<le> (at_infinity :: real filter)"
```
```   479   unfolding at_infinity_eq_at_top_bot by simp
```
```   480
```
```   481 lemma at_bot_le_at_infinity:
```
```   482   "at_bot \<le> (at_infinity :: real filter)"
```
```   483   unfolding at_infinity_eq_at_top_bot by simp
```
```   484
```
```   485 subsection {* Boundedness *}
```
```   486
```
```   487 definition Bfun :: "('a \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a filter \<Rightarrow> bool"
```
```   488   where "Bfun f F = (\<exists>K>0. eventually (\<lambda>x. norm (f x) \<le> K) F)"
```
```   489
```
```   490 lemma BfunI:
```
```   491   assumes K: "eventually (\<lambda>x. norm (f x) \<le> K) F" shows "Bfun f F"
```
```   492 unfolding Bfun_def
```
```   493 proof (intro exI conjI allI)
```
```   494   show "0 < max K 1" by simp
```
```   495 next
```
```   496   show "eventually (\<lambda>x. norm (f x) \<le> max K 1) F"
```
```   497     using K by (rule eventually_elim1, simp)
```
```   498 qed
```
```   499
```
```   500 lemma BfunE:
```
```   501   assumes "Bfun f F"
```
```   502   obtains B where "0 < B" and "eventually (\<lambda>x. norm (f x) \<le> B) F"
```
```   503 using assms unfolding Bfun_def by fast
```
```   504
```
```   505
```
```   506 subsection {* Convergence to Zero *}
```
```   507
```
```   508 definition Zfun :: "('a \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a filter \<Rightarrow> bool"
```
```   509   where "Zfun f F = (\<forall>r>0. eventually (\<lambda>x. norm (f x) < r) F)"
```
```   510
```
```   511 lemma ZfunI:
```
```   512   "(\<And>r. 0 < r \<Longrightarrow> eventually (\<lambda>x. norm (f x) < r) F) \<Longrightarrow> Zfun f F"
```
```   513   unfolding Zfun_def by simp
```
```   514
```
```   515 lemma ZfunD:
```
```   516   "\<lbrakk>Zfun f F; 0 < r\<rbrakk> \<Longrightarrow> eventually (\<lambda>x. norm (f x) < r) F"
```
```   517   unfolding Zfun_def by simp
```
```   518
```
```   519 lemma Zfun_ssubst:
```
```   520   "eventually (\<lambda>x. f x = g x) F \<Longrightarrow> Zfun g F \<Longrightarrow> Zfun f F"
```
```   521   unfolding Zfun_def by (auto elim!: eventually_rev_mp)
```
```   522
```
```   523 lemma Zfun_zero: "Zfun (\<lambda>x. 0) F"
```
```   524   unfolding Zfun_def by simp
```
```   525
```
```   526 lemma Zfun_norm_iff: "Zfun (\<lambda>x. norm (f x)) F = Zfun (\<lambda>x. f x) F"
```
```   527   unfolding Zfun_def by simp
```
```   528
```
```   529 lemma Zfun_imp_Zfun:
```
```   530   assumes f: "Zfun f F"
```
```   531   assumes g: "eventually (\<lambda>x. norm (g x) \<le> norm (f x) * K) F"
```
```   532   shows "Zfun (\<lambda>x. g x) F"
```
```   533 proof (cases)
```
```   534   assume K: "0 < K"
```
```   535   show ?thesis
```
```   536   proof (rule ZfunI)
```
```   537     fix r::real assume "0 < r"
```
```   538     hence "0 < r / K"
```
```   539       using K by (rule divide_pos_pos)
```
```   540     then have "eventually (\<lambda>x. norm (f x) < r / K) F"
```
```   541       using ZfunD [OF f] by fast
```
```   542     with g show "eventually (\<lambda>x. norm (g x) < r) F"
```
```   543     proof eventually_elim
```
```   544       case (elim x)
```
```   545       hence "norm (f x) * K < r"
```
```   546         by (simp add: pos_less_divide_eq K)
```
```   547       thus ?case
```
```   548         by (simp add: order_le_less_trans [OF elim(1)])
```
```   549     qed
```
```   550   qed
```
```   551 next
```
```   552   assume "\<not> 0 < K"
```
```   553   hence K: "K \<le> 0" by (simp only: not_less)
```
```   554   show ?thesis
```
```   555   proof (rule ZfunI)
```
```   556     fix r :: real
```
```   557     assume "0 < r"
```
```   558     from g show "eventually (\<lambda>x. norm (g x) < r) F"
```
```   559     proof eventually_elim
```
```   560       case (elim x)
```
```   561       also have "norm (f x) * K \<le> norm (f x) * 0"
```
```   562         using K norm_ge_zero by (rule mult_left_mono)
```
```   563       finally show ?case
```
```   564         using `0 < r` by simp
```
```   565     qed
```
```   566   qed
```
```   567 qed
```
```   568
```
```   569 lemma Zfun_le: "\<lbrakk>Zfun g F; \<forall>x. norm (f x) \<le> norm (g x)\<rbrakk> \<Longrightarrow> Zfun f F"
```
```   570   by (erule_tac K="1" in Zfun_imp_Zfun, simp)
```
```   571
```
```   572 lemma Zfun_add:
```
```   573   assumes f: "Zfun f F" and g: "Zfun g F"
```
```   574   shows "Zfun (\<lambda>x. f x + g x) F"
```
```   575 proof (rule ZfunI)
```
```   576   fix r::real assume "0 < r"
```
```   577   hence r: "0 < r / 2" by simp
```
```   578   have "eventually (\<lambda>x. norm (f x) < r/2) F"
```
```   579     using f r by (rule ZfunD)
```
```   580   moreover
```
```   581   have "eventually (\<lambda>x. norm (g x) < r/2) F"
```
```   582     using g r by (rule ZfunD)
```
```   583   ultimately
```
```   584   show "eventually (\<lambda>x. norm (f x + g x) < r) F"
```
```   585   proof eventually_elim
```
```   586     case (elim x)
```
```   587     have "norm (f x + g x) \<le> norm (f x) + norm (g x)"
```
```   588       by (rule norm_triangle_ineq)
```
```   589     also have "\<dots> < r/2 + r/2"
```
```   590       using elim by (rule add_strict_mono)
```
```   591     finally show ?case
```
```   592       by simp
```
```   593   qed
```
```   594 qed
```
```   595
```
```   596 lemma Zfun_minus: "Zfun f F \<Longrightarrow> Zfun (\<lambda>x. - f x) F"
```
```   597   unfolding Zfun_def by simp
```
```   598
```
```   599 lemma Zfun_diff: "\<lbrakk>Zfun f F; Zfun g F\<rbrakk> \<Longrightarrow> Zfun (\<lambda>x. f x - g x) F"
```
```   600   by (simp only: diff_minus Zfun_add Zfun_minus)
```
```   601
```
```   602 lemma (in bounded_linear) Zfun:
```
```   603   assumes g: "Zfun g F"
```
```   604   shows "Zfun (\<lambda>x. f (g x)) F"
```
```   605 proof -
```
```   606   obtain K where "\<And>x. norm (f x) \<le> norm x * K"
```
```   607     using bounded by fast
```
```   608   then have "eventually (\<lambda>x. norm (f (g x)) \<le> norm (g x) * K) F"
```
```   609     by simp
```
```   610   with g show ?thesis
```
```   611     by (rule Zfun_imp_Zfun)
```
```   612 qed
```
```   613
```
```   614 lemma (in bounded_bilinear) Zfun:
```
```   615   assumes f: "Zfun f F"
```
```   616   assumes g: "Zfun g F"
```
```   617   shows "Zfun (\<lambda>x. f x ** g x) F"
```
```   618 proof (rule ZfunI)
```
```   619   fix r::real assume r: "0 < r"
```
```   620   obtain K where K: "0 < K"
```
```   621     and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"
```
```   622     using pos_bounded by fast
```
```   623   from K have K': "0 < inverse K"
```
```   624     by (rule positive_imp_inverse_positive)
```
```   625   have "eventually (\<lambda>x. norm (f x) < r) F"
```
```   626     using f r by (rule ZfunD)
```
```   627   moreover
```
```   628   have "eventually (\<lambda>x. norm (g x) < inverse K) F"
```
```   629     using g K' by (rule ZfunD)
```
```   630   ultimately
```
```   631   show "eventually (\<lambda>x. norm (f x ** g x) < r) F"
```
```   632   proof eventually_elim
```
```   633     case (elim x)
```
```   634     have "norm (f x ** g x) \<le> norm (f x) * norm (g x) * K"
```
```   635       by (rule norm_le)
```
```   636     also have "norm (f x) * norm (g x) * K < r * inverse K * K"
```
```   637       by (intro mult_strict_right_mono mult_strict_mono' norm_ge_zero elim K)
```
```   638     also from K have "r * inverse K * K = r"
```
```   639       by simp
```
```   640     finally show ?case .
```
```   641   qed
```
```   642 qed
```
```   643
```
```   644 lemma (in bounded_bilinear) Zfun_left:
```
```   645   "Zfun f F \<Longrightarrow> Zfun (\<lambda>x. f x ** a) F"
```
```   646   by (rule bounded_linear_left [THEN bounded_linear.Zfun])
```
```   647
```
```   648 lemma (in bounded_bilinear) Zfun_right:
```
```   649   "Zfun f F \<Longrightarrow> Zfun (\<lambda>x. a ** f x) F"
```
```   650   by (rule bounded_linear_right [THEN bounded_linear.Zfun])
```
```   651
```
```   652 lemmas Zfun_mult = bounded_bilinear.Zfun [OF bounded_bilinear_mult]
```
```   653 lemmas Zfun_mult_right = bounded_bilinear.Zfun_right [OF bounded_bilinear_mult]
```
```   654 lemmas Zfun_mult_left = bounded_bilinear.Zfun_left [OF bounded_bilinear_mult]
```
```   655
```
```   656
```
```   657 subsection {* Limits *}
```
```   658
```
```   659 definition filterlim :: "('a \<Rightarrow> 'b) \<Rightarrow> 'b filter \<Rightarrow> 'a filter \<Rightarrow> bool" where
```
```   660   "filterlim f F2 F1 \<longleftrightarrow> filtermap f F1 \<le> F2"
```
```   661
```
```   662 syntax
```
```   663   "_LIM" :: "pttrns \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> bool" ("(3LIM (_)/ (_)./ (_) :> (_))" [1000, 10, 0, 10] 10)
```
```   664
```
```   665 translations
```
```   666   "LIM x F1. f :> F2"   == "CONST filterlim (%x. f) F2 F1"
```
```   667
```
```   668 lemma filterlim_iff:
```
```   669   "(LIM x F1. f x :> F2) \<longleftrightarrow> (\<forall>P. eventually P F2 \<longrightarrow> eventually (\<lambda>x. P (f x)) F1)"
```
```   670   unfolding filterlim_def le_filter_def eventually_filtermap ..
```
```   671
```
```   672 lemma filterlim_compose:
```
```   673   "filterlim g F3 F2 \<Longrightarrow> filterlim f F2 F1 \<Longrightarrow> filterlim (\<lambda>x. g (f x)) F3 F1"
```
```   674   unfolding filterlim_def filtermap_filtermap[symmetric] by (metis filtermap_mono order_trans)
```
```   675
```
```   676 lemma filterlim_mono:
```
```   677   "filterlim f F2 F1 \<Longrightarrow> F2 \<le> F2' \<Longrightarrow> F1' \<le> F1 \<Longrightarrow> filterlim f F2' F1'"
```
```   678   unfolding filterlim_def by (metis filtermap_mono order_trans)
```
```   679
```
```   680 lemma filterlim_within:
```
```   681   "(LIM x F1. f x :> F2 within S) \<longleftrightarrow> (eventually (\<lambda>x. f x \<in> S) F1 \<and> (LIM x F1. f x :> F2))"
```
```   682   unfolding filterlim_def
```
```   683 proof safe
```
```   684   assume "filtermap f F1 \<le> F2 within S" then show "eventually (\<lambda>x. f x \<in> S) F1"
```
```   685     by (auto simp: le_filter_def eventually_filtermap eventually_within elim!: allE[of _ "\<lambda>x. x \<in> S"])
```
```   686 qed (auto intro: within_le order_trans simp: le_within_iff eventually_filtermap)
```
```   687
```
```   688 abbreviation (in topological_space)
```
```   689   tendsto :: "('b \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'b filter \<Rightarrow> bool" (infixr "--->" 55) where
```
```   690   "(f ---> l) F \<equiv> filterlim f (nhds l) F"
```
```   691
```
```   692 ML {*
```
```   693 structure Tendsto_Intros = Named_Thms
```
```   694 (
```
```   695   val name = @{binding tendsto_intros}
```
```   696   val description = "introduction rules for tendsto"
```
```   697 )
```
```   698 *}
```
```   699
```
```   700 setup Tendsto_Intros.setup
```
```   701
```
```   702 lemma tendsto_def: "(f ---> l) F \<longleftrightarrow> (\<forall>S. open S \<longrightarrow> l \<in> S \<longrightarrow> eventually (\<lambda>x. f x \<in> S) F)"
```
```   703   unfolding filterlim_def
```
```   704 proof safe
```
```   705   fix S assume "open S" "l \<in> S" "filtermap f F \<le> nhds l"
```
```   706   then show "eventually (\<lambda>x. f x \<in> S) F"
```
```   707     unfolding eventually_nhds eventually_filtermap le_filter_def
```
```   708     by (auto elim!: allE[of _ "\<lambda>x. x \<in> S"] eventually_rev_mp)
```
```   709 qed (auto elim!: eventually_rev_mp simp: eventually_nhds eventually_filtermap le_filter_def)
```
```   710
```
```   711 lemma filterlim_at:
```
```   712   "(LIM x F. f x :> at b) \<longleftrightarrow> (eventually (\<lambda>x. f x \<noteq> b) F \<and> (f ---> b) F)"
```
```   713   by (simp add: at_def filterlim_within)
```
```   714
```
```   715 lemma tendsto_mono: "F \<le> F' \<Longrightarrow> (f ---> l) F' \<Longrightarrow> (f ---> l) F"
```
```   716   unfolding tendsto_def le_filter_def by fast
```
```   717
```
```   718 lemma topological_tendstoI:
```
```   719   "(\<And>S. open S \<Longrightarrow> l \<in> S \<Longrightarrow> eventually (\<lambda>x. f x \<in> S) F)
```
```   720     \<Longrightarrow> (f ---> l) F"
```
```   721   unfolding tendsto_def by auto
```
```   722
```
```   723 lemma topological_tendstoD:
```
```   724   "(f ---> l) F \<Longrightarrow> open S \<Longrightarrow> l \<in> S \<Longrightarrow> eventually (\<lambda>x. f x \<in> S) F"
```
```   725   unfolding tendsto_def by auto
```
```   726
```
```   727 lemma tendstoI:
```
```   728   assumes "\<And>e. 0 < e \<Longrightarrow> eventually (\<lambda>x. dist (f x) l < e) F"
```
```   729   shows "(f ---> l) F"
```
```   730   apply (rule topological_tendstoI)
```
```   731   apply (simp add: open_dist)
```
```   732   apply (drule (1) bspec, clarify)
```
```   733   apply (drule assms)
```
```   734   apply (erule eventually_elim1, simp)
```
```   735   done
```
```   736
```
```   737 lemma tendstoD:
```
```   738   "(f ---> l) F \<Longrightarrow> 0 < e \<Longrightarrow> eventually (\<lambda>x. dist (f x) l < e) F"
```
```   739   apply (drule_tac S="{x. dist x l < e}" in topological_tendstoD)
```
```   740   apply (clarsimp simp add: open_dist)
```
```   741   apply (rule_tac x="e - dist x l" in exI, clarsimp)
```
```   742   apply (simp only: less_diff_eq)
```
```   743   apply (erule le_less_trans [OF dist_triangle])
```
```   744   apply simp
```
```   745   apply simp
```
```   746   done
```
```   747
```
```   748 lemma tendsto_iff:
```
```   749   "(f ---> l) F \<longleftrightarrow> (\<forall>e>0. eventually (\<lambda>x. dist (f x) l < e) F)"
```
```   750   using tendstoI tendstoD by fast
```
```   751
```
```   752 lemma tendsto_Zfun_iff: "(f ---> a) F = Zfun (\<lambda>x. f x - a) F"
```
```   753   by (simp only: tendsto_iff Zfun_def dist_norm)
```
```   754
```
```   755 lemma tendsto_bot [simp]: "(f ---> a) bot"
```
```   756   unfolding tendsto_def by simp
```
```   757
```
```   758 lemma tendsto_ident_at [tendsto_intros]: "((\<lambda>x. x) ---> a) (at a)"
```
```   759   unfolding tendsto_def eventually_at_topological by auto
```
```   760
```
```   761 lemma tendsto_ident_at_within [tendsto_intros]:
```
```   762   "((\<lambda>x. x) ---> a) (at a within S)"
```
```   763   unfolding tendsto_def eventually_within eventually_at_topological by auto
```
```   764
```
```   765 lemma tendsto_const [tendsto_intros]: "((\<lambda>x. k) ---> k) F"
```
```   766   by (simp add: tendsto_def)
```
```   767
```
```   768 lemma tendsto_unique:
```
```   769   fixes f :: "'a \<Rightarrow> 'b::t2_space"
```
```   770   assumes "\<not> trivial_limit F" and "(f ---> a) F" and "(f ---> b) F"
```
```   771   shows "a = b"
```
```   772 proof (rule ccontr)
```
```   773   assume "a \<noteq> b"
```
```   774   obtain U V where "open U" "open V" "a \<in> U" "b \<in> V" "U \<inter> V = {}"
```
```   775     using hausdorff [OF `a \<noteq> b`] by fast
```
```   776   have "eventually (\<lambda>x. f x \<in> U) F"
```
```   777     using `(f ---> a) F` `open U` `a \<in> U` by (rule topological_tendstoD)
```
```   778   moreover
```
```   779   have "eventually (\<lambda>x. f x \<in> V) F"
```
```   780     using `(f ---> b) F` `open V` `b \<in> V` by (rule topological_tendstoD)
```
```   781   ultimately
```
```   782   have "eventually (\<lambda>x. False) F"
```
```   783   proof eventually_elim
```
```   784     case (elim x)
```
```   785     hence "f x \<in> U \<inter> V" by simp
```
```   786     with `U \<inter> V = {}` show ?case by simp
```
```   787   qed
```
```   788   with `\<not> trivial_limit F` show "False"
```
```   789     by (simp add: trivial_limit_def)
```
```   790 qed
```
```   791
```
```   792 lemma tendsto_const_iff:
```
```   793   fixes a b :: "'a::t2_space"
```
```   794   assumes "\<not> trivial_limit F" shows "((\<lambda>x. a) ---> b) F \<longleftrightarrow> a = b"
```
```   795   by (safe intro!: tendsto_const tendsto_unique [OF assms tendsto_const])
```
```   796
```
```   797 lemma tendsto_at_iff_tendsto_nhds:
```
```   798   "(g ---> g l) (at l) \<longleftrightarrow> (g ---> g l) (nhds l)"
```
```   799   unfolding tendsto_def at_def eventually_within
```
```   800   by (intro ext all_cong imp_cong) (auto elim!: eventually_elim1)
```
```   801
```
```   802 lemma tendsto_compose:
```
```   803   "(g ---> g l) (at l) \<Longrightarrow> (f ---> l) F \<Longrightarrow> ((\<lambda>x. g (f x)) ---> g l) F"
```
```   804   unfolding tendsto_at_iff_tendsto_nhds by (rule filterlim_compose[of g])
```
```   805
```
```   806 lemma tendsto_compose_eventually:
```
```   807   "(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"
```
```   808   by (rule filterlim_compose[of g _ "at l"]) (auto simp add: filterlim_at)
```
```   809
```
```   810 lemma metric_tendsto_imp_tendsto:
```
```   811   assumes f: "(f ---> a) F"
```
```   812   assumes le: "eventually (\<lambda>x. dist (g x) b \<le> dist (f x) a) F"
```
```   813   shows "(g ---> b) F"
```
```   814 proof (rule tendstoI)
```
```   815   fix e :: real assume "0 < e"
```
```   816   with f have "eventually (\<lambda>x. dist (f x) a < e) F" by (rule tendstoD)
```
```   817   with le show "eventually (\<lambda>x. dist (g x) b < e) F"
```
```   818     using le_less_trans by (rule eventually_elim2)
```
```   819 qed
```
```   820
```
```   821 subsubsection {* Distance and norms *}
```
```   822
```
```   823 lemma tendsto_dist [tendsto_intros]:
```
```   824   assumes f: "(f ---> l) F" and g: "(g ---> m) F"
```
```   825   shows "((\<lambda>x. dist (f x) (g x)) ---> dist l m) F"
```
```   826 proof (rule tendstoI)
```
```   827   fix e :: real assume "0 < e"
```
```   828   hence e2: "0 < e/2" by simp
```
```   829   from tendstoD [OF f e2] tendstoD [OF g e2]
```
```   830   show "eventually (\<lambda>x. dist (dist (f x) (g x)) (dist l m) < e) F"
```
```   831   proof (eventually_elim)
```
```   832     case (elim x)
```
```   833     then show "dist (dist (f x) (g x)) (dist l m) < e"
```
```   834       unfolding dist_real_def
```
```   835       using dist_triangle2 [of "f x" "g x" "l"]
```
```   836       using dist_triangle2 [of "g x" "l" "m"]
```
```   837       using dist_triangle3 [of "l" "m" "f x"]
```
```   838       using dist_triangle [of "f x" "m" "g x"]
```
```   839       by arith
```
```   840   qed
```
```   841 qed
```
```   842
```
```   843 lemma norm_conv_dist: "norm x = dist x 0"
```
```   844   unfolding dist_norm by simp
```
```   845
```
```   846 lemma tendsto_norm [tendsto_intros]:
```
```   847   "(f ---> a) F \<Longrightarrow> ((\<lambda>x. norm (f x)) ---> norm a) F"
```
```   848   unfolding norm_conv_dist by (intro tendsto_intros)
```
```   849
```
```   850 lemma tendsto_norm_zero:
```
```   851   "(f ---> 0) F \<Longrightarrow> ((\<lambda>x. norm (f x)) ---> 0) F"
```
```   852   by (drule tendsto_norm, simp)
```
```   853
```
```   854 lemma tendsto_norm_zero_cancel:
```
```   855   "((\<lambda>x. norm (f x)) ---> 0) F \<Longrightarrow> (f ---> 0) F"
```
```   856   unfolding tendsto_iff dist_norm by simp
```
```   857
```
```   858 lemma tendsto_norm_zero_iff:
```
```   859   "((\<lambda>x. norm (f x)) ---> 0) F \<longleftrightarrow> (f ---> 0) F"
```
```   860   unfolding tendsto_iff dist_norm by simp
```
```   861
```
```   862 lemma tendsto_rabs [tendsto_intros]:
```
```   863   "(f ---> (l::real)) F \<Longrightarrow> ((\<lambda>x. \<bar>f x\<bar>) ---> \<bar>l\<bar>) F"
```
```   864   by (fold real_norm_def, rule tendsto_norm)
```
```   865
```
```   866 lemma tendsto_rabs_zero:
```
```   867   "(f ---> (0::real)) F \<Longrightarrow> ((\<lambda>x. \<bar>f x\<bar>) ---> 0) F"
```
```   868   by (fold real_norm_def, rule tendsto_norm_zero)
```
```   869
```
```   870 lemma tendsto_rabs_zero_cancel:
```
```   871   "((\<lambda>x. \<bar>f x\<bar>) ---> (0::real)) F \<Longrightarrow> (f ---> 0) F"
```
```   872   by (fold real_norm_def, rule tendsto_norm_zero_cancel)
```
```   873
```
```   874 lemma tendsto_rabs_zero_iff:
```
```   875   "((\<lambda>x. \<bar>f x\<bar>) ---> (0::real)) F \<longleftrightarrow> (f ---> 0) F"
```
```   876   by (fold real_norm_def, rule tendsto_norm_zero_iff)
```
```   877
```
```   878 subsubsection {* Addition and subtraction *}
```
```   879
```
```   880 lemma tendsto_add [tendsto_intros]:
```
```   881   fixes a b :: "'a::real_normed_vector"
```
```   882   shows "\<lbrakk>(f ---> a) F; (g ---> b) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x + g x) ---> a + b) F"
```
```   883   by (simp only: tendsto_Zfun_iff add_diff_add Zfun_add)
```
```   884
```
```   885 lemma tendsto_add_zero:
```
```   886   fixes f g :: "'a::type \<Rightarrow> 'b::real_normed_vector"
```
```   887   shows "\<lbrakk>(f ---> 0) F; (g ---> 0) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x + g x) ---> 0) F"
```
```   888   by (drule (1) tendsto_add, simp)
```
```   889
```
```   890 lemma tendsto_minus [tendsto_intros]:
```
```   891   fixes a :: "'a::real_normed_vector"
```
```   892   shows "(f ---> a) F \<Longrightarrow> ((\<lambda>x. - f x) ---> - a) F"
```
```   893   by (simp only: tendsto_Zfun_iff minus_diff_minus Zfun_minus)
```
```   894
```
```   895 lemma tendsto_minus_cancel:
```
```   896   fixes a :: "'a::real_normed_vector"
```
```   897   shows "((\<lambda>x. - f x) ---> - a) F \<Longrightarrow> (f ---> a) F"
```
```   898   by (drule tendsto_minus, simp)
```
```   899
```
```   900 lemma tendsto_diff [tendsto_intros]:
```
```   901   fixes a b :: "'a::real_normed_vector"
```
```   902   shows "\<lbrakk>(f ---> a) F; (g ---> b) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x - g x) ---> a - b) F"
```
```   903   by (simp add: diff_minus tendsto_add tendsto_minus)
```
```   904
```
```   905 lemma tendsto_setsum [tendsto_intros]:
```
```   906   fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c::real_normed_vector"
```
```   907   assumes "\<And>i. i \<in> S \<Longrightarrow> (f i ---> a i) F"
```
```   908   shows "((\<lambda>x. \<Sum>i\<in>S. f i x) ---> (\<Sum>i\<in>S. a i)) F"
```
```   909 proof (cases "finite S")
```
```   910   assume "finite S" thus ?thesis using assms
```
```   911     by (induct, simp add: tendsto_const, simp add: tendsto_add)
```
```   912 next
```
```   913   assume "\<not> finite S" thus ?thesis
```
```   914     by (simp add: tendsto_const)
```
```   915 qed
```
```   916
```
```   917 lemma real_tendsto_sandwich:
```
```   918   fixes f g h :: "'a \<Rightarrow> real"
```
```   919   assumes ev: "eventually (\<lambda>n. f n \<le> g n) net" "eventually (\<lambda>n. g n \<le> h n) net"
```
```   920   assumes lim: "(f ---> c) net" "(h ---> c) net"
```
```   921   shows "(g ---> c) net"
```
```   922 proof -
```
```   923   have "((\<lambda>n. g n - f n) ---> 0) net"
```
```   924   proof (rule metric_tendsto_imp_tendsto)
```
```   925     show "eventually (\<lambda>n. dist (g n - f n) 0 \<le> dist (h n - f n) 0) net"
```
```   926       using ev by (rule eventually_elim2) (simp add: dist_real_def)
```
```   927     show "((\<lambda>n. h n - f n) ---> 0) net"
```
```   928       using tendsto_diff[OF lim(2,1)] by simp
```
```   929   qed
```
```   930   from tendsto_add[OF this lim(1)] show ?thesis by simp
```
```   931 qed
```
```   932
```
```   933 subsubsection {* Linear operators and multiplication *}
```
```   934
```
```   935 lemma (in bounded_linear) tendsto:
```
```   936   "(g ---> a) F \<Longrightarrow> ((\<lambda>x. f (g x)) ---> f a) F"
```
```   937   by (simp only: tendsto_Zfun_iff diff [symmetric] Zfun)
```
```   938
```
```   939 lemma (in bounded_linear) tendsto_zero:
```
```   940   "(g ---> 0) F \<Longrightarrow> ((\<lambda>x. f (g x)) ---> 0) F"
```
```   941   by (drule tendsto, simp only: zero)
```
```   942
```
```   943 lemma (in bounded_bilinear) tendsto:
```
```   944   "\<lbrakk>(f ---> a) F; (g ---> b) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x ** g x) ---> a ** b) F"
```
```   945   by (simp only: tendsto_Zfun_iff prod_diff_prod
```
```   946                  Zfun_add Zfun Zfun_left Zfun_right)
```
```   947
```
```   948 lemma (in bounded_bilinear) tendsto_zero:
```
```   949   assumes f: "(f ---> 0) F"
```
```   950   assumes g: "(g ---> 0) F"
```
```   951   shows "((\<lambda>x. f x ** g x) ---> 0) F"
```
```   952   using tendsto [OF f g] by (simp add: zero_left)
```
```   953
```
```   954 lemma (in bounded_bilinear) tendsto_left_zero:
```
```   955   "(f ---> 0) F \<Longrightarrow> ((\<lambda>x. f x ** c) ---> 0) F"
```
```   956   by (rule bounded_linear.tendsto_zero [OF bounded_linear_left])
```
```   957
```
```   958 lemma (in bounded_bilinear) tendsto_right_zero:
```
```   959   "(f ---> 0) F \<Longrightarrow> ((\<lambda>x. c ** f x) ---> 0) F"
```
```   960   by (rule bounded_linear.tendsto_zero [OF bounded_linear_right])
```
```   961
```
```   962 lemmas tendsto_of_real [tendsto_intros] =
```
```   963   bounded_linear.tendsto [OF bounded_linear_of_real]
```
```   964
```
```   965 lemmas tendsto_scaleR [tendsto_intros] =
```
```   966   bounded_bilinear.tendsto [OF bounded_bilinear_scaleR]
```
```   967
```
```   968 lemmas tendsto_mult [tendsto_intros] =
```
```   969   bounded_bilinear.tendsto [OF bounded_bilinear_mult]
```
```   970
```
```   971 lemmas tendsto_mult_zero =
```
```   972   bounded_bilinear.tendsto_zero [OF bounded_bilinear_mult]
```
```   973
```
```   974 lemmas tendsto_mult_left_zero =
```
```   975   bounded_bilinear.tendsto_left_zero [OF bounded_bilinear_mult]
```
```   976
```
```   977 lemmas tendsto_mult_right_zero =
```
```   978   bounded_bilinear.tendsto_right_zero [OF bounded_bilinear_mult]
```
```   979
```
```   980 lemma tendsto_power [tendsto_intros]:
```
```   981   fixes f :: "'a \<Rightarrow> 'b::{power,real_normed_algebra}"
```
```   982   shows "(f ---> a) F \<Longrightarrow> ((\<lambda>x. f x ^ n) ---> a ^ n) F"
```
```   983   by (induct n) (simp_all add: tendsto_const tendsto_mult)
```
```   984
```
```   985 lemma tendsto_setprod [tendsto_intros]:
```
```   986   fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c::{real_normed_algebra,comm_ring_1}"
```
```   987   assumes "\<And>i. i \<in> S \<Longrightarrow> (f i ---> L i) F"
```
```   988   shows "((\<lambda>x. \<Prod>i\<in>S. f i x) ---> (\<Prod>i\<in>S. L i)) F"
```
```   989 proof (cases "finite S")
```
```   990   assume "finite S" thus ?thesis using assms
```
```   991     by (induct, simp add: tendsto_const, simp add: tendsto_mult)
```
```   992 next
```
```   993   assume "\<not> finite S" thus ?thesis
```
```   994     by (simp add: tendsto_const)
```
```   995 qed
```
```   996
```
```   997 subsubsection {* Inverse and division *}
```
```   998
```
```   999 lemma (in bounded_bilinear) Zfun_prod_Bfun:
```
```  1000   assumes f: "Zfun f F"
```
```  1001   assumes g: "Bfun g F"
```
```  1002   shows "Zfun (\<lambda>x. f x ** g x) F"
```
```  1003 proof -
```
```  1004   obtain K where K: "0 \<le> K"
```
```  1005     and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"
```
```  1006     using nonneg_bounded by fast
```
```  1007   obtain B where B: "0 < B"
```
```  1008     and norm_g: "eventually (\<lambda>x. norm (g x) \<le> B) F"
```
```  1009     using g by (rule BfunE)
```
```  1010   have "eventually (\<lambda>x. norm (f x ** g x) \<le> norm (f x) * (B * K)) F"
```
```  1011   using norm_g proof eventually_elim
```
```  1012     case (elim x)
```
```  1013     have "norm (f x ** g x) \<le> norm (f x) * norm (g x) * K"
```
```  1014       by (rule norm_le)
```
```  1015     also have "\<dots> \<le> norm (f x) * B * K"
```
```  1016       by (intro mult_mono' order_refl norm_g norm_ge_zero
```
```  1017                 mult_nonneg_nonneg K elim)
```
```  1018     also have "\<dots> = norm (f x) * (B * K)"
```
```  1019       by (rule mult_assoc)
```
```  1020     finally show "norm (f x ** g x) \<le> norm (f x) * (B * K)" .
```
```  1021   qed
```
```  1022   with f show ?thesis
```
```  1023     by (rule Zfun_imp_Zfun)
```
```  1024 qed
```
```  1025
```
```  1026 lemma (in bounded_bilinear) flip:
```
```  1027   "bounded_bilinear (\<lambda>x y. y ** x)"
```
```  1028   apply default
```
```  1029   apply (rule add_right)
```
```  1030   apply (rule add_left)
```
```  1031   apply (rule scaleR_right)
```
```  1032   apply (rule scaleR_left)
```
```  1033   apply (subst mult_commute)
```
```  1034   using bounded by fast
```
```  1035
```
```  1036 lemma (in bounded_bilinear) Bfun_prod_Zfun:
```
```  1037   assumes f: "Bfun f F"
```
```  1038   assumes g: "Zfun g F"
```
```  1039   shows "Zfun (\<lambda>x. f x ** g x) F"
```
```  1040   using flip g f by (rule bounded_bilinear.Zfun_prod_Bfun)
```
```  1041
```
```  1042 lemma Bfun_inverse_lemma:
```
```  1043   fixes x :: "'a::real_normed_div_algebra"
```
```  1044   shows "\<lbrakk>r \<le> norm x; 0 < r\<rbrakk> \<Longrightarrow> norm (inverse x) \<le> inverse r"
```
```  1045   apply (subst nonzero_norm_inverse, clarsimp)
```
```  1046   apply (erule (1) le_imp_inverse_le)
```
```  1047   done
```
```  1048
```
```  1049 lemma Bfun_inverse:
```
```  1050   fixes a :: "'a::real_normed_div_algebra"
```
```  1051   assumes f: "(f ---> a) F"
```
```  1052   assumes a: "a \<noteq> 0"
```
```  1053   shows "Bfun (\<lambda>x. inverse (f x)) F"
```
```  1054 proof -
```
```  1055   from a have "0 < norm a" by simp
```
```  1056   hence "\<exists>r>0. r < norm a" by (rule dense)
```
```  1057   then obtain r where r1: "0 < r" and r2: "r < norm a" by fast
```
```  1058   have "eventually (\<lambda>x. dist (f x) a < r) F"
```
```  1059     using tendstoD [OF f r1] by fast
```
```  1060   hence "eventually (\<lambda>x. norm (inverse (f x)) \<le> inverse (norm a - r)) F"
```
```  1061   proof eventually_elim
```
```  1062     case (elim x)
```
```  1063     hence 1: "norm (f x - a) < r"
```
```  1064       by (simp add: dist_norm)
```
```  1065     hence 2: "f x \<noteq> 0" using r2 by auto
```
```  1066     hence "norm (inverse (f x)) = inverse (norm (f x))"
```
```  1067       by (rule nonzero_norm_inverse)
```
```  1068     also have "\<dots> \<le> inverse (norm a - r)"
```
```  1069     proof (rule le_imp_inverse_le)
```
```  1070       show "0 < norm a - r" using r2 by simp
```
```  1071     next
```
```  1072       have "norm a - norm (f x) \<le> norm (a - f x)"
```
```  1073         by (rule norm_triangle_ineq2)
```
```  1074       also have "\<dots> = norm (f x - a)"
```
```  1075         by (rule norm_minus_commute)
```
```  1076       also have "\<dots> < r" using 1 .
```
```  1077       finally show "norm a - r \<le> norm (f x)" by simp
```
```  1078     qed
```
```  1079     finally show "norm (inverse (f x)) \<le> inverse (norm a - r)" .
```
```  1080   qed
```
```  1081   thus ?thesis by (rule BfunI)
```
```  1082 qed
```
```  1083
```
```  1084 lemma tendsto_inverse [tendsto_intros]:
```
```  1085   fixes a :: "'a::real_normed_div_algebra"
```
```  1086   assumes f: "(f ---> a) F"
```
```  1087   assumes a: "a \<noteq> 0"
```
```  1088   shows "((\<lambda>x. inverse (f x)) ---> inverse a) F"
```
```  1089 proof -
```
```  1090   from a have "0 < norm a" by simp
```
```  1091   with f have "eventually (\<lambda>x. dist (f x) a < norm a) F"
```
```  1092     by (rule tendstoD)
```
```  1093   then have "eventually (\<lambda>x. f x \<noteq> 0) F"
```
```  1094     unfolding dist_norm by (auto elim!: eventually_elim1)
```
```  1095   with a have "eventually (\<lambda>x. inverse (f x) - inverse a =
```
```  1096     - (inverse (f x) * (f x - a) * inverse a)) F"
```
```  1097     by (auto elim!: eventually_elim1 simp: inverse_diff_inverse)
```
```  1098   moreover have "Zfun (\<lambda>x. - (inverse (f x) * (f x - a) * inverse a)) F"
```
```  1099     by (intro Zfun_minus Zfun_mult_left
```
```  1100       bounded_bilinear.Bfun_prod_Zfun [OF bounded_bilinear_mult]
```
```  1101       Bfun_inverse [OF f a] f [unfolded tendsto_Zfun_iff])
```
```  1102   ultimately show ?thesis
```
```  1103     unfolding tendsto_Zfun_iff by (rule Zfun_ssubst)
```
```  1104 qed
```
```  1105
```
```  1106 lemma tendsto_divide [tendsto_intros]:
```
```  1107   fixes a b :: "'a::real_normed_field"
```
```  1108   shows "\<lbrakk>(f ---> a) F; (g ---> b) F; b \<noteq> 0\<rbrakk>
```
```  1109     \<Longrightarrow> ((\<lambda>x. f x / g x) ---> a / b) F"
```
```  1110   by (simp add: tendsto_mult tendsto_inverse divide_inverse)
```
```  1111
```
```  1112 lemma tendsto_sgn [tendsto_intros]:
```
```  1113   fixes l :: "'a::real_normed_vector"
```
```  1114   shows "\<lbrakk>(f ---> l) F; l \<noteq> 0\<rbrakk> \<Longrightarrow> ((\<lambda>x. sgn (f x)) ---> sgn l) F"
```
```  1115   unfolding sgn_div_norm by (simp add: tendsto_intros)
```
```  1116
```
```  1117 subsection {* Limits to @{const at_top} and @{const at_bot} *}
```
```  1118
```
```  1119 lemma filterlim_at_top:
```
```  1120   fixes f :: "'a \<Rightarrow> ('b::dense_linorder)"
```
```  1121   shows "(LIM x F. f x :> at_top) \<longleftrightarrow> (\<forall>Z. eventually (\<lambda>x. Z < f x) F)"
```
```  1122   by (auto simp: filterlim_iff eventually_at_top_dense elim!: eventually_elim1)
```
```  1123
```
```  1124 lemma filterlim_at_top_gt:
```
```  1125   fixes f :: "'a \<Rightarrow> ('b::dense_linorder)" and c :: "'b"
```
```  1126   shows "(LIM x F. f x :> at_top) \<longleftrightarrow> (\<forall>Z>c. eventually (\<lambda>x. Z < f x) F)"
```
```  1127   unfolding filterlim_at_top
```
```  1128 proof safe
```
```  1129   fix Z assume *: "\<forall>Z>c. eventually (\<lambda>x. Z < f x) F"
```
```  1130   from gt_ex[of "max Z c"] guess x ..
```
```  1131   with *[THEN spec, of x] show "eventually (\<lambda>x. Z < f x) F"
```
```  1132     by (auto elim!: eventually_elim1)
```
```  1133 qed simp
```
```  1134
```
```  1135 lemma filterlim_at_bot:
```
```  1136   fixes f :: "'a \<Rightarrow> ('b::dense_linorder)"
```
```  1137   shows "(LIM x F. f x :> at_bot) \<longleftrightarrow> (\<forall>Z. eventually (\<lambda>x. f x < Z) F)"
```
```  1138   by (auto simp: filterlim_iff eventually_at_bot_dense elim!: eventually_elim1)
```
```  1139
```
```  1140 lemma filterlim_at_bot_lt:
```
```  1141   fixes f :: "'a \<Rightarrow> ('b::dense_linorder)" and c :: "'b"
```
```  1142   shows "(LIM x F. f x :> at_bot) \<longleftrightarrow> (\<forall>Z<c. eventually (\<lambda>x. Z > f x) F)"
```
```  1143   unfolding filterlim_at_bot
```
```  1144 proof safe
```
```  1145   fix Z assume *: "\<forall>Z<c. eventually (\<lambda>x. Z > f x) F"
```
```  1146   from lt_ex[of "min Z c"] guess x ..
```
```  1147   with *[THEN spec, of x] show "eventually (\<lambda>x. Z > f x) F"
```
```  1148     by (auto elim!: eventually_elim1)
```
```  1149 qed simp
```
```  1150
```
```  1151 lemma filterlim_at_infinity:
```
```  1152   fixes f :: "_ \<Rightarrow> 'a\<Colon>real_normed_vector"
```
```  1153   assumes "0 \<le> c"
```
```  1154   shows "(LIM x F. f x :> at_infinity) \<longleftrightarrow> (\<forall>r>c. eventually (\<lambda>x. r \<le> norm (f x)) F)"
```
```  1155   unfolding filterlim_iff eventually_at_infinity
```
```  1156 proof safe
```
```  1157   fix P :: "'a \<Rightarrow> bool" and b
```
```  1158   assume *: "\<forall>r>c. eventually (\<lambda>x. r \<le> norm (f x)) F"
```
```  1159     and P: "\<forall>x. b \<le> norm x \<longrightarrow> P x"
```
```  1160   have "max b (c + 1) > c" by auto
```
```  1161   with * have "eventually (\<lambda>x. max b (c + 1) \<le> norm (f x)) F"
```
```  1162     by auto
```
```  1163   then show "eventually (\<lambda>x. P (f x)) F"
```
```  1164   proof eventually_elim
```
```  1165     fix x assume "max b (c + 1) \<le> norm (f x)"
```
```  1166     with P show "P (f x)" by auto
```
```  1167   qed
```
```  1168 qed force
```
```  1169
```
```  1170 lemma filterlim_real_sequentially: "LIM x sequentially. real x :> at_top"
```
```  1171   unfolding filterlim_at_top
```
```  1172   apply (intro allI)
```
```  1173   apply (rule_tac c="natceiling (Z + 1)" in eventually_sequentiallyI)
```
```  1174   apply (auto simp: natceiling_le_eq)
```
```  1175   done
```
```  1176
```
```  1177 lemma filterlim_inverse_at_top_pos:
```
```  1178   "LIM x (nhds 0 within {0::real <..}). inverse x :> at_top"
```
```  1179   unfolding filterlim_at_top_gt[where c=0] eventually_within
```
```  1180 proof safe
```
```  1181   fix Z :: real assume [arith]: "0 < Z"
```
```  1182   then have "eventually (\<lambda>x. x < inverse Z) (nhds 0)"
```
```  1183     by (auto simp add: eventually_nhds_metric dist_real_def intro!: exI[of _ "\<bar>inverse Z\<bar>"])
```
```  1184   then show "eventually (\<lambda>x. x \<in> {0<..} \<longrightarrow> Z < inverse x) (nhds 0)"
```
```  1185     by (auto elim!: eventually_elim1 simp: inverse_eq_divide field_simps)
```
```  1186 qed
```
```  1187
```
```  1188 lemma filterlim_inverse_at_top:
```
```  1189   "(f ---> (0 :: real)) F \<Longrightarrow> eventually (\<lambda>x. 0 < f x) F \<Longrightarrow> LIM x F. inverse (f x) :> at_top"
```
```  1190   by (intro filterlim_compose[OF filterlim_inverse_at_top_pos])
```
```  1191      (simp add: filterlim_def eventually_filtermap le_within_iff)
```
```  1192
```
```  1193 lemma filterlim_inverse_at_bot_neg:
```
```  1194   "LIM x (nhds 0 within {..< 0::real}). inverse x :> at_bot"
```
```  1195   unfolding filterlim_at_bot_lt[where c=0] eventually_within
```
```  1196 proof safe
```
```  1197   fix Z :: real assume [arith]: "Z < 0"
```
```  1198   have "eventually (\<lambda>x. inverse Z < x) (nhds 0)"
```
```  1199     by (auto simp add: eventually_nhds_metric dist_real_def intro!: exI[of _ "\<bar>inverse Z\<bar>"])
```
```  1200   then show "eventually (\<lambda>x. x \<in> {..< 0} \<longrightarrow> inverse x < Z) (nhds 0)"
```
```  1201     by (auto elim!: eventually_elim1 simp: inverse_eq_divide field_simps)
```
```  1202 qed
```
```  1203
```
```  1204 lemma filterlim_inverse_at_bot:
```
```  1205   "(f ---> (0 :: real)) F \<Longrightarrow> eventually (\<lambda>x. f x < 0) F \<Longrightarrow> LIM x F. inverse (f x) :> at_bot"
```
```  1206   by (intro filterlim_compose[OF filterlim_inverse_at_bot_neg])
```
```  1207      (simp add: filterlim_def eventually_filtermap le_within_iff)
```
```  1208
```
```  1209 lemma filterlim_uminus_at_top_at_bot: "LIM x at_bot. - x :: real :> at_top"
```
```  1210   unfolding filterlim_at_top eventually_at_bot_dense
```
```  1211   by (blast intro: less_minus_iff[THEN iffD1])
```
```  1212
```
```  1213 lemma filterlim_uminus_at_top: "LIM x F. f x :> at_bot \<Longrightarrow> LIM x F. - (f x) :: real :> at_top"
```
```  1214   by (rule filterlim_compose[OF filterlim_uminus_at_top_at_bot])
```
```  1215
```
```  1216 lemma filterlim_uminus_at_bot_at_top: "LIM x at_top. - x :: real :> at_bot"
```
```  1217   unfolding filterlim_at_bot eventually_at_top_dense
```
```  1218   by (blast intro: minus_less_iff[THEN iffD1])
```
```  1219
```
```  1220 lemma filterlim_uminus_at_bot: "LIM x F. f x :> at_top \<Longrightarrow> LIM x F. - (f x) :: real :> at_bot"
```
```  1221   by (rule filterlim_compose[OF filterlim_uminus_at_bot_at_top])
```
```  1222
```
```  1223 lemma tendsto_inverse_0:
```
```  1224   fixes x :: "_ \<Rightarrow> 'a\<Colon>real_normed_div_algebra"
```
```  1225   shows "(inverse ---> (0::'a)) at_infinity"
```
```  1226   unfolding tendsto_Zfun_iff diff_0_right Zfun_def eventually_at_infinity
```
```  1227 proof safe
```
```  1228   fix r :: real assume "0 < r"
```
```  1229   show "\<exists>b. \<forall>x. b \<le> norm x \<longrightarrow> norm (inverse x :: 'a) < r"
```
```  1230   proof (intro exI[of _ "inverse (r / 2)"] allI impI)
```
```  1231     fix x :: 'a
```
```  1232     from `0 < r` have "0 < inverse (r / 2)" by simp
```
```  1233     also assume *: "inverse (r / 2) \<le> norm x"
```
```  1234     finally show "norm (inverse x) < r"
```
```  1235       using * `0 < r` by (subst nonzero_norm_inverse) (simp_all add: inverse_eq_divide field_simps)
```
```  1236   qed
```
```  1237 qed
```
```  1238
```
```  1239 lemma filterlim_inverse_at_infinity:
```
```  1240   fixes x :: "_ \<Rightarrow> 'a\<Colon>{real_normed_div_algebra, division_ring_inverse_zero}"
```
```  1241   shows "filterlim inverse at_infinity (at (0::'a))"
```
```  1242   unfolding filterlim_at_infinity[OF order_refl]
```
```  1243 proof safe
```
```  1244   fix r :: real assume "0 < r"
```
```  1245   then show "eventually (\<lambda>x::'a. r \<le> norm (inverse x)) (at 0)"
```
```  1246     unfolding eventually_at norm_inverse
```
```  1247     by (intro exI[of _ "inverse r"])
```
```  1248        (auto simp: norm_conv_dist[symmetric] field_simps inverse_eq_divide)
```
```  1249 qed
```
```  1250
```
```  1251 lemma filterlim_inverse_at_iff:
```
```  1252   fixes g :: "'a \<Rightarrow> 'b\<Colon>{real_normed_div_algebra, division_ring_inverse_zero}"
```
```  1253   shows "(LIM x F. inverse (g x) :> at 0) \<longleftrightarrow> (LIM x F. g x :> at_infinity)"
```
```  1254   unfolding filterlim_def filtermap_filtermap[symmetric]
```
```  1255 proof
```
```  1256   assume "filtermap g F \<le> at_infinity"
```
```  1257   then have "filtermap inverse (filtermap g F) \<le> filtermap inverse at_infinity"
```
```  1258     by (rule filtermap_mono)
```
```  1259   also have "\<dots> \<le> at 0"
```
```  1260     using tendsto_inverse_0
```
```  1261     by (auto intro!: le_withinI exI[of _ 1]
```
```  1262              simp: eventually_filtermap eventually_at_infinity filterlim_def at_def)
```
```  1263   finally show "filtermap inverse (filtermap g F) \<le> at 0" .
```
```  1264 next
```
```  1265   assume "filtermap inverse (filtermap g F) \<le> at 0"
```
```  1266   then have "filtermap inverse (filtermap inverse (filtermap g F)) \<le> filtermap inverse (at 0)"
```
```  1267     by (rule filtermap_mono)
```
```  1268   with filterlim_inverse_at_infinity show "filtermap g F \<le> at_infinity"
```
```  1269     by (auto intro: order_trans simp: filterlim_def filtermap_filtermap)
```
```  1270 qed
```
```  1271
```
```  1272 text {*
```
```  1273
```
```  1274 We only show rules for multiplication and addition when the functions are either against a real
```
```  1275 value or against infinity. Further rules are easy to derive by using @{thm filterlim_uminus_at_top}.
```
```  1276
```
```  1277 *}
```
```  1278
```
```  1279 lemma filterlim_tendsto_pos_mult_at_top:
```
```  1280   assumes f: "(f ---> c) F" and c: "0 < c"
```
```  1281   assumes g: "LIM x F. g x :> at_top"
```
```  1282   shows "LIM x F. (f x * g x :: real) :> at_top"
```
```  1283   unfolding filterlim_at_top_gt[where c=0]
```
```  1284 proof safe
```
```  1285   fix Z :: real assume "0 < Z"
```
```  1286   from f `0 < c` have "eventually (\<lambda>x. c / 2 < f x) F"
```
```  1287     by (auto dest!: tendstoD[where e="c / 2"] elim!: eventually_elim1
```
```  1288              simp: dist_real_def abs_real_def split: split_if_asm)
```
```  1289   moreover from g have "eventually (\<lambda>x. (Z / c * 2) < g x) F"
```
```  1290     unfolding filterlim_at_top by auto
```
```  1291   ultimately show "eventually (\<lambda>x. Z < f x * g x) F"
```
```  1292   proof eventually_elim
```
```  1293     fix x assume "c / 2 < f x" "Z / c * 2 < g x"
```
```  1294     with `0 < Z` `0 < c` have "c / 2 * (Z / c * 2) < f x * g x"
```
```  1295       by (intro mult_strict_mono) (auto simp: zero_le_divide_iff)
```
```  1296     with `0 < c` show "Z < f x * g x"
```
```  1297        by simp
```
```  1298   qed
```
```  1299 qed
```
```  1300
```
```  1301 lemma filterlim_at_top_mult_at_top:
```
```  1302   assumes f: "LIM x F. f x :> at_top"
```
```  1303   assumes g: "LIM x F. g x :> at_top"
```
```  1304   shows "LIM x F. (f x * g x :: real) :> at_top"
```
```  1305   unfolding filterlim_at_top_gt[where c=0]
```
```  1306 proof safe
```
```  1307   fix Z :: real assume "0 < Z"
```
```  1308   from f have "eventually (\<lambda>x. 1 < f x) F"
```
```  1309     unfolding filterlim_at_top by auto
```
```  1310   moreover from g have "eventually (\<lambda>x. Z < g x) F"
```
```  1311     unfolding filterlim_at_top by auto
```
```  1312   ultimately show "eventually (\<lambda>x. Z < f x * g x) F"
```
```  1313   proof eventually_elim
```
```  1314     fix x assume "1 < f x" "Z < g x"
```
```  1315     with `0 < Z` have "1 * Z < f x * g x"
```
```  1316       by (intro mult_strict_mono) (auto simp: zero_le_divide_iff)
```
```  1317     then show "Z < f x * g x"
```
```  1318        by simp
```
```  1319   qed
```
```  1320 qed
```
```  1321
```
```  1322 lemma filterlim_tendsto_add_at_top:
```
```  1323   assumes f: "(f ---> c) F"
```
```  1324   assumes g: "LIM x F. g x :> at_top"
```
```  1325   shows "LIM x F. (f x + g x :: real) :> at_top"
```
```  1326   unfolding filterlim_at_top_gt[where c=0]
```
```  1327 proof safe
```
```  1328   fix Z :: real assume "0 < Z"
```
```  1329   from f have "eventually (\<lambda>x. c - 1 < f x) F"
```
```  1330     by (auto dest!: tendstoD[where e=1] elim!: eventually_elim1 simp: dist_real_def)
```
```  1331   moreover from g have "eventually (\<lambda>x. Z - (c - 1) < g x) F"
```
```  1332     unfolding filterlim_at_top by auto
```
```  1333   ultimately show "eventually (\<lambda>x. Z < f x + g x) F"
```
```  1334     by eventually_elim simp
```
```  1335 qed
```
```  1336
```
```  1337 lemma filterlim_at_top_add_at_top:
```
```  1338   assumes f: "LIM x F. f x :> at_top"
```
```  1339   assumes g: "LIM x F. g x :> at_top"
```
```  1340   shows "LIM x F. (f x + g x :: real) :> at_top"
```
```  1341   unfolding filterlim_at_top_gt[where c=0]
```
```  1342 proof safe
```
```  1343   fix Z :: real assume "0 < Z"
```
```  1344   from f have "eventually (\<lambda>x. 0 < f x) F"
```
```  1345     unfolding filterlim_at_top by auto
```
```  1346   moreover from g have "eventually (\<lambda>x. Z < g x) F"
```
```  1347     unfolding filterlim_at_top by auto
```
```  1348   ultimately show "eventually (\<lambda>x. Z < f x + g x) F"
```
```  1349     by eventually_elim simp
```
```  1350 qed
```
```  1351
```
```  1352 end
```
```  1353
```