src/HOL/Limits.thy
 author huffman Mon Aug 08 19:26:53 2011 -0700 (2011-08-08) changeset 44081 730f7cced3a6 parent 44079 bcc60791b7b9 child 44194 0639898074ae permissions -rw-r--r--
rename type 'a net to 'a filter, following standard mathematical terminology
```     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 (open) '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 A)"
```
```    29   using Rep_filter [of A] 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 A \<longleftrightarrow> Rep_filter A 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 "A = B \<longleftrightarrow> (\<forall>P. eventually P A = eventually P B)"
```
```    47   unfolding Rep_filter_inject [symmetric] fun_eq_iff eventually_def ..
```
```    48
```
```    49 lemma eventually_True [simp]: "eventually (\<lambda>x. True) A"
```
```    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 A"
```
```    54 proof -
```
```    55   assume "\<forall>x. P x" hence "P = (\<lambda>x. True)" by (simp add: ext)
```
```    56   thus "eventually P A" by simp
```
```    57 qed
```
```    58
```
```    59 lemma eventually_mono:
```
```    60   "(\<forall>x. P x \<longrightarrow> Q x) \<Longrightarrow> eventually P A \<Longrightarrow> eventually Q A"
```
```    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) A"
```
```    66   assumes Q: "eventually (\<lambda>x. Q x) A"
```
```    67   shows "eventually (\<lambda>x. P x \<and> Q x) A"
```
```    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) A"
```
```    73   assumes "eventually (\<lambda>x. P x) A"
```
```    74   shows "eventually (\<lambda>x. Q x) A"
```
```    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) A"
```
```    78     using assms by (rule eventually_conj)
```
```    79 qed
```
```    80
```
```    81 lemma eventually_rev_mp:
```
```    82   assumes "eventually (\<lambda>x. P x) A"
```
```    83   assumes "eventually (\<lambda>x. P x \<longrightarrow> Q x) A"
```
```    84   shows "eventually (\<lambda>x. Q x) A"
```
```    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) A \<longleftrightarrow> eventually P A \<and> eventually Q A"
```
```    89   by (auto intro: eventually_conj elim: eventually_rev_mp)
```
```    90
```
```    91 lemma eventually_elim1:
```
```    92   assumes "eventually (\<lambda>i. P i) A"
```
```    93   assumes "\<And>i. P i \<Longrightarrow> Q i"
```
```    94   shows "eventually (\<lambda>i. Q i) A"
```
```    95   using assms by (auto elim!: eventually_rev_mp)
```
```    96
```
```    97 lemma eventually_elim2:
```
```    98   assumes "eventually (\<lambda>i. P i) A"
```
```    99   assumes "eventually (\<lambda>i. Q i) A"
```
```   100   assumes "\<And>i. P i \<Longrightarrow> Q i \<Longrightarrow> R i"
```
```   101   shows "eventually (\<lambda>i. R i) A"
```
```   102   using assms by (auto elim!: eventually_rev_mp)
```
```   103
```
```   104 subsection {* Finer-than relation *}
```
```   105
```
```   106 text {* @{term "A \<le> B"} means that filter @{term A} is finer than
```
```   107 filter @{term B}. *}
```
```   108
```
```   109 instantiation filter :: (type) complete_lattice
```
```   110 begin
```
```   111
```
```   112 definition le_filter_def:
```
```   113   "A \<le> B \<longleftrightarrow> (\<forall>P. eventually P B \<longrightarrow> eventually P A)"
```
```   114
```
```   115 definition
```
```   116   "(A :: 'a filter) < B \<longleftrightarrow> A \<le> B \<and> \<not> B \<le> A"
```
```   117
```
```   118 definition
```
```   119   "top = Abs_filter (\<lambda>P. \<forall>x. P x)"
```
```   120
```
```   121 definition
```
```   122   "bot = Abs_filter (\<lambda>P. True)"
```
```   123
```
```   124 definition
```
```   125   "sup A B = Abs_filter (\<lambda>P. eventually P A \<and> eventually P B)"
```
```   126
```
```   127 definition
```
```   128   "inf A B = Abs_filter
```
```   129       (\<lambda>P. \<exists>Q R. eventually Q A \<and> eventually R B \<and> (\<forall>x. Q x \<and> R x \<longrightarrow> P x))"
```
```   130
```
```   131 definition
```
```   132   "Sup S = Abs_filter (\<lambda>P. \<forall>A\<in>S. eventually P A)"
```
```   133
```
```   134 definition
```
```   135   "Inf S = Sup {A::'a filter. \<forall>B\<in>S. A \<le> B}"
```
```   136
```
```   137 lemma eventually_top [simp]: "eventually P top \<longleftrightarrow> (\<forall>x. P x)"
```
```   138   unfolding top_filter_def
```
```   139   by (rule eventually_Abs_filter, rule is_filter.intro, auto)
```
```   140
```
```   141 lemma eventually_bot [simp]: "eventually P bot"
```
```   142   unfolding bot_filter_def
```
```   143   by (subst eventually_Abs_filter, rule is_filter.intro, auto)
```
```   144
```
```   145 lemma eventually_sup:
```
```   146   "eventually P (sup A B) \<longleftrightarrow> eventually P A \<and> eventually P B"
```
```   147   unfolding sup_filter_def
```
```   148   by (rule eventually_Abs_filter, rule is_filter.intro)
```
```   149      (auto elim!: eventually_rev_mp)
```
```   150
```
```   151 lemma eventually_inf:
```
```   152   "eventually P (inf A B) \<longleftrightarrow>
```
```   153    (\<exists>Q R. eventually Q A \<and> eventually R B \<and> (\<forall>x. Q x \<and> R x \<longrightarrow> P x))"
```
```   154   unfolding inf_filter_def
```
```   155   apply (rule eventually_Abs_filter, rule is_filter.intro)
```
```   156   apply (fast intro: eventually_True)
```
```   157   apply clarify
```
```   158   apply (intro exI conjI)
```
```   159   apply (erule (1) eventually_conj)
```
```   160   apply (erule (1) eventually_conj)
```
```   161   apply simp
```
```   162   apply auto
```
```   163   done
```
```   164
```
```   165 lemma eventually_Sup:
```
```   166   "eventually P (Sup S) \<longleftrightarrow> (\<forall>A\<in>S. eventually P A)"
```
```   167   unfolding Sup_filter_def
```
```   168   apply (rule eventually_Abs_filter, rule is_filter.intro)
```
```   169   apply (auto intro: eventually_conj elim!: eventually_rev_mp)
```
```   170   done
```
```   171
```
```   172 instance proof
```
```   173   fix A B :: "'a filter" show "A < B \<longleftrightarrow> A \<le> B \<and> \<not> B \<le> A"
```
```   174     by (rule less_filter_def)
```
```   175 next
```
```   176   fix A :: "'a filter" show "A \<le> A"
```
```   177     unfolding le_filter_def by simp
```
```   178 next
```
```   179   fix A B C :: "'a filter" assume "A \<le> B" and "B \<le> C" thus "A \<le> C"
```
```   180     unfolding le_filter_def by simp
```
```   181 next
```
```   182   fix A B :: "'a filter" assume "A \<le> B" and "B \<le> A" thus "A = B"
```
```   183     unfolding le_filter_def filter_eq_iff by fast
```
```   184 next
```
```   185   fix A :: "'a filter" show "A \<le> top"
```
```   186     unfolding le_filter_def eventually_top by (simp add: always_eventually)
```
```   187 next
```
```   188   fix A :: "'a filter" show "bot \<le> A"
```
```   189     unfolding le_filter_def by simp
```
```   190 next
```
```   191   fix A B :: "'a filter" show "A \<le> sup A B" and "B \<le> sup A B"
```
```   192     unfolding le_filter_def eventually_sup by simp_all
```
```   193 next
```
```   194   fix A B C :: "'a filter" assume "A \<le> C" and "B \<le> C" thus "sup A B \<le> C"
```
```   195     unfolding le_filter_def eventually_sup by simp
```
```   196 next
```
```   197   fix A B :: "'a filter" show "inf A B \<le> A" and "inf A B \<le> B"
```
```   198     unfolding le_filter_def eventually_inf by (auto intro: eventually_True)
```
```   199 next
```
```   200   fix A B C :: "'a filter" assume "A \<le> B" and "A \<le> C" thus "A \<le> inf B C"
```
```   201     unfolding le_filter_def eventually_inf
```
```   202     by (auto elim!: eventually_mono intro: eventually_conj)
```
```   203 next
```
```   204   fix A :: "'a filter" and S assume "A \<in> S" thus "A \<le> Sup S"
```
```   205     unfolding le_filter_def eventually_Sup by simp
```
```   206 next
```
```   207   fix S and B :: "'a filter" assume "\<And>A. A \<in> S \<Longrightarrow> A \<le> B" thus "Sup S \<le> B"
```
```   208     unfolding le_filter_def eventually_Sup by simp
```
```   209 next
```
```   210   fix C :: "'a filter" and S assume "C \<in> S" thus "Inf S \<le> C"
```
```   211     unfolding le_filter_def Inf_filter_def eventually_Sup Ball_def by simp
```
```   212 next
```
```   213   fix S and A :: "'a filter" assume "\<And>B. B \<in> S \<Longrightarrow> A \<le> B" thus "A \<le> Inf S"
```
```   214     unfolding le_filter_def Inf_filter_def eventually_Sup Ball_def by simp
```
```   215 qed
```
```   216
```
```   217 end
```
```   218
```
```   219 lemma filter_leD:
```
```   220   "A \<le> B \<Longrightarrow> eventually P B \<Longrightarrow> eventually P A"
```
```   221   unfolding le_filter_def by simp
```
```   222
```
```   223 lemma filter_leI:
```
```   224   "(\<And>P. eventually P B \<Longrightarrow> eventually P A) \<Longrightarrow> A \<le> B"
```
```   225   unfolding le_filter_def by simp
```
```   226
```
```   227 lemma eventually_False:
```
```   228   "eventually (\<lambda>x. False) A \<longleftrightarrow> A = bot"
```
```   229   unfolding filter_eq_iff by (auto elim: eventually_rev_mp)
```
```   230
```
```   231 subsection {* Map function for filters *}
```
```   232
```
```   233 definition filtermap :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a filter \<Rightarrow> 'b filter"
```
```   234   where "filtermap f A = Abs_filter (\<lambda>P. eventually (\<lambda>x. P (f x)) A)"
```
```   235
```
```   236 lemma eventually_filtermap:
```
```   237   "eventually P (filtermap f A) = eventually (\<lambda>x. P (f x)) A"
```
```   238   unfolding filtermap_def
```
```   239   apply (rule eventually_Abs_filter)
```
```   240   apply (rule is_filter.intro)
```
```   241   apply (auto elim!: eventually_rev_mp)
```
```   242   done
```
```   243
```
```   244 lemma filtermap_ident: "filtermap (\<lambda>x. x) A = A"
```
```   245   by (simp add: filter_eq_iff eventually_filtermap)
```
```   246
```
```   247 lemma filtermap_filtermap:
```
```   248   "filtermap f (filtermap g A) = filtermap (\<lambda>x. f (g x)) A"
```
```   249   by (simp add: filter_eq_iff eventually_filtermap)
```
```   250
```
```   251 lemma filtermap_mono: "A \<le> B \<Longrightarrow> filtermap f A \<le> filtermap f B"
```
```   252   unfolding le_filter_def eventually_filtermap by simp
```
```   253
```
```   254 lemma filtermap_bot [simp]: "filtermap f bot = bot"
```
```   255   by (simp add: filter_eq_iff eventually_filtermap)
```
```   256
```
```   257
```
```   258 subsection {* Sequentially *}
```
```   259
```
```   260 definition sequentially :: "nat filter"
```
```   261   where "sequentially = Abs_filter (\<lambda>P. \<exists>k. \<forall>n\<ge>k. P n)"
```
```   262
```
```   263 lemma eventually_sequentially:
```
```   264   "eventually P sequentially \<longleftrightarrow> (\<exists>N. \<forall>n\<ge>N. P n)"
```
```   265 unfolding sequentially_def
```
```   266 proof (rule eventually_Abs_filter, rule is_filter.intro)
```
```   267   fix P Q :: "nat \<Rightarrow> bool"
```
```   268   assume "\<exists>i. \<forall>n\<ge>i. P n" and "\<exists>j. \<forall>n\<ge>j. Q n"
```
```   269   then obtain i j where "\<forall>n\<ge>i. P n" and "\<forall>n\<ge>j. Q n" by auto
```
```   270   then have "\<forall>n\<ge>max i j. P n \<and> Q n" by simp
```
```   271   then show "\<exists>k. \<forall>n\<ge>k. P n \<and> Q n" ..
```
```   272 qed auto
```
```   273
```
```   274 lemma sequentially_bot [simp]: "sequentially \<noteq> bot"
```
```   275   unfolding filter_eq_iff eventually_sequentially by auto
```
```   276
```
```   277 lemma eventually_False_sequentially [simp]:
```
```   278   "\<not> eventually (\<lambda>n. False) sequentially"
```
```   279   by (simp add: eventually_False)
```
```   280
```
```   281 lemma le_sequentially:
```
```   282   "A \<le> sequentially \<longleftrightarrow> (\<forall>N. eventually (\<lambda>n. N \<le> n) A)"
```
```   283   unfolding le_filter_def eventually_sequentially
```
```   284   by (safe, fast, drule_tac x=N in spec, auto elim: eventually_rev_mp)
```
```   285
```
```   286
```
```   287 definition trivial_limit :: "'a filter \<Rightarrow> bool"
```
```   288   where "trivial_limit A \<longleftrightarrow> eventually (\<lambda>x. False) A"
```
```   289
```
```   290 lemma trivial_limit_sequentially [intro]: "\<not> trivial_limit sequentially"
```
```   291   by (auto simp add: trivial_limit_def eventually_sequentially)
```
```   292
```
```   293 subsection {* Standard filters *}
```
```   294
```
```   295 definition within :: "'a filter \<Rightarrow> 'a set \<Rightarrow> 'a filter" (infixr "within" 70)
```
```   296   where "A within S = Abs_filter (\<lambda>P. eventually (\<lambda>x. x \<in> S \<longrightarrow> P x) A)"
```
```   297
```
```   298 definition nhds :: "'a::topological_space \<Rightarrow> 'a filter"
```
```   299   where "nhds a = Abs_filter (\<lambda>P. \<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x))"
```
```   300
```
```   301 definition at :: "'a::topological_space \<Rightarrow> 'a filter"
```
```   302   where "at a = nhds a within - {a}"
```
```   303
```
```   304 lemma eventually_within:
```
```   305   "eventually P (A within S) = eventually (\<lambda>x. x \<in> S \<longrightarrow> P x) A"
```
```   306   unfolding within_def
```
```   307   by (rule eventually_Abs_filter, rule is_filter.intro)
```
```   308      (auto elim!: eventually_rev_mp)
```
```   309
```
```   310 lemma within_UNIV: "A within UNIV = A"
```
```   311   unfolding filter_eq_iff eventually_within by simp
```
```   312
```
```   313 lemma eventually_nhds:
```
```   314   "eventually P (nhds a) \<longleftrightarrow> (\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x))"
```
```   315 unfolding nhds_def
```
```   316 proof (rule eventually_Abs_filter, rule is_filter.intro)
```
```   317   have "open UNIV \<and> a \<in> UNIV \<and> (\<forall>x\<in>UNIV. True)" by simp
```
```   318   thus "\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. True)" by - rule
```
```   319 next
```
```   320   fix P Q
```
```   321   assume "\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x)"
```
```   322      and "\<exists>T. open T \<and> a \<in> T \<and> (\<forall>x\<in>T. Q x)"
```
```   323   then obtain S T where
```
```   324     "open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x)"
```
```   325     "open T \<and> a \<in> T \<and> (\<forall>x\<in>T. Q x)" by auto
```
```   326   hence "open (S \<inter> T) \<and> a \<in> S \<inter> T \<and> (\<forall>x\<in>(S \<inter> T). P x \<and> Q x)"
```
```   327     by (simp add: open_Int)
```
```   328   thus "\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x \<and> Q x)" by - rule
```
```   329 qed auto
```
```   330
```
```   331 lemma eventually_nhds_metric:
```
```   332   "eventually P (nhds a) \<longleftrightarrow> (\<exists>d>0. \<forall>x. dist x a < d \<longrightarrow> P x)"
```
```   333 unfolding eventually_nhds open_dist
```
```   334 apply safe
```
```   335 apply fast
```
```   336 apply (rule_tac x="{x. dist x a < d}" in exI, simp)
```
```   337 apply clarsimp
```
```   338 apply (rule_tac x="d - dist x a" in exI, clarsimp)
```
```   339 apply (simp only: less_diff_eq)
```
```   340 apply (erule le_less_trans [OF dist_triangle])
```
```   341 done
```
```   342
```
```   343 lemma eventually_at_topological:
```
```   344   "eventually P (at a) \<longleftrightarrow> (\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. x \<noteq> a \<longrightarrow> P x))"
```
```   345 unfolding at_def eventually_within eventually_nhds by simp
```
```   346
```
```   347 lemma eventually_at:
```
```   348   fixes a :: "'a::metric_space"
```
```   349   shows "eventually P (at a) \<longleftrightarrow> (\<exists>d>0. \<forall>x. x \<noteq> a \<and> dist x a < d \<longrightarrow> P x)"
```
```   350 unfolding at_def eventually_within eventually_nhds_metric by auto
```
```   351
```
```   352
```
```   353 subsection {* Boundedness *}
```
```   354
```
```   355 definition Bfun :: "('a \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a filter \<Rightarrow> bool"
```
```   356   where "Bfun f A = (\<exists>K>0. eventually (\<lambda>x. norm (f x) \<le> K) A)"
```
```   357
```
```   358 lemma BfunI:
```
```   359   assumes K: "eventually (\<lambda>x. norm (f x) \<le> K) A" shows "Bfun f A"
```
```   360 unfolding Bfun_def
```
```   361 proof (intro exI conjI allI)
```
```   362   show "0 < max K 1" by simp
```
```   363 next
```
```   364   show "eventually (\<lambda>x. norm (f x) \<le> max K 1) A"
```
```   365     using K by (rule eventually_elim1, simp)
```
```   366 qed
```
```   367
```
```   368 lemma BfunE:
```
```   369   assumes "Bfun f A"
```
```   370   obtains B where "0 < B" and "eventually (\<lambda>x. norm (f x) \<le> B) A"
```
```   371 using assms unfolding Bfun_def by fast
```
```   372
```
```   373
```
```   374 subsection {* Convergence to Zero *}
```
```   375
```
```   376 definition Zfun :: "('a \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a filter \<Rightarrow> bool"
```
```   377   where "Zfun f A = (\<forall>r>0. eventually (\<lambda>x. norm (f x) < r) A)"
```
```   378
```
```   379 lemma ZfunI:
```
```   380   "(\<And>r. 0 < r \<Longrightarrow> eventually (\<lambda>x. norm (f x) < r) A) \<Longrightarrow> Zfun f A"
```
```   381   unfolding Zfun_def by simp
```
```   382
```
```   383 lemma ZfunD:
```
```   384   "\<lbrakk>Zfun f A; 0 < r\<rbrakk> \<Longrightarrow> eventually (\<lambda>x. norm (f x) < r) A"
```
```   385   unfolding Zfun_def by simp
```
```   386
```
```   387 lemma Zfun_ssubst:
```
```   388   "eventually (\<lambda>x. f x = g x) A \<Longrightarrow> Zfun g A \<Longrightarrow> Zfun f A"
```
```   389   unfolding Zfun_def by (auto elim!: eventually_rev_mp)
```
```   390
```
```   391 lemma Zfun_zero: "Zfun (\<lambda>x. 0) A"
```
```   392   unfolding Zfun_def by simp
```
```   393
```
```   394 lemma Zfun_norm_iff: "Zfun (\<lambda>x. norm (f x)) A = Zfun (\<lambda>x. f x) A"
```
```   395   unfolding Zfun_def by simp
```
```   396
```
```   397 lemma Zfun_imp_Zfun:
```
```   398   assumes f: "Zfun f A"
```
```   399   assumes g: "eventually (\<lambda>x. norm (g x) \<le> norm (f x) * K) A"
```
```   400   shows "Zfun (\<lambda>x. g x) A"
```
```   401 proof (cases)
```
```   402   assume K: "0 < K"
```
```   403   show ?thesis
```
```   404   proof (rule ZfunI)
```
```   405     fix r::real assume "0 < r"
```
```   406     hence "0 < r / K"
```
```   407       using K by (rule divide_pos_pos)
```
```   408     then have "eventually (\<lambda>x. norm (f x) < r / K) A"
```
```   409       using ZfunD [OF f] by fast
```
```   410     with g show "eventually (\<lambda>x. norm (g x) < r) A"
```
```   411     proof (rule eventually_elim2)
```
```   412       fix x
```
```   413       assume *: "norm (g x) \<le> norm (f x) * K"
```
```   414       assume "norm (f x) < r / K"
```
```   415       hence "norm (f x) * K < r"
```
```   416         by (simp add: pos_less_divide_eq K)
```
```   417       thus "norm (g x) < r"
```
```   418         by (simp add: order_le_less_trans [OF *])
```
```   419     qed
```
```   420   qed
```
```   421 next
```
```   422   assume "\<not> 0 < K"
```
```   423   hence K: "K \<le> 0" by (simp only: not_less)
```
```   424   show ?thesis
```
```   425   proof (rule ZfunI)
```
```   426     fix r :: real
```
```   427     assume "0 < r"
```
```   428     from g show "eventually (\<lambda>x. norm (g x) < r) A"
```
```   429     proof (rule eventually_elim1)
```
```   430       fix x
```
```   431       assume "norm (g x) \<le> norm (f x) * K"
```
```   432       also have "\<dots> \<le> norm (f x) * 0"
```
```   433         using K norm_ge_zero by (rule mult_left_mono)
```
```   434       finally show "norm (g x) < r"
```
```   435         using `0 < r` by simp
```
```   436     qed
```
```   437   qed
```
```   438 qed
```
```   439
```
```   440 lemma Zfun_le: "\<lbrakk>Zfun g A; \<forall>x. norm (f x) \<le> norm (g x)\<rbrakk> \<Longrightarrow> Zfun f A"
```
```   441   by (erule_tac K="1" in Zfun_imp_Zfun, simp)
```
```   442
```
```   443 lemma Zfun_add:
```
```   444   assumes f: "Zfun f A" and g: "Zfun g A"
```
```   445   shows "Zfun (\<lambda>x. f x + g x) A"
```
```   446 proof (rule ZfunI)
```
```   447   fix r::real assume "0 < r"
```
```   448   hence r: "0 < r / 2" by simp
```
```   449   have "eventually (\<lambda>x. norm (f x) < r/2) A"
```
```   450     using f r by (rule ZfunD)
```
```   451   moreover
```
```   452   have "eventually (\<lambda>x. norm (g x) < r/2) A"
```
```   453     using g r by (rule ZfunD)
```
```   454   ultimately
```
```   455   show "eventually (\<lambda>x. norm (f x + g x) < r) A"
```
```   456   proof (rule eventually_elim2)
```
```   457     fix x
```
```   458     assume *: "norm (f x) < r/2" "norm (g x) < r/2"
```
```   459     have "norm (f x + g x) \<le> norm (f x) + norm (g x)"
```
```   460       by (rule norm_triangle_ineq)
```
```   461     also have "\<dots> < r/2 + r/2"
```
```   462       using * by (rule add_strict_mono)
```
```   463     finally show "norm (f x + g x) < r"
```
```   464       by simp
```
```   465   qed
```
```   466 qed
```
```   467
```
```   468 lemma Zfun_minus: "Zfun f A \<Longrightarrow> Zfun (\<lambda>x. - f x) A"
```
```   469   unfolding Zfun_def by simp
```
```   470
```
```   471 lemma Zfun_diff: "\<lbrakk>Zfun f A; Zfun g A\<rbrakk> \<Longrightarrow> Zfun (\<lambda>x. f x - g x) A"
```
```   472   by (simp only: diff_minus Zfun_add Zfun_minus)
```
```   473
```
```   474 lemma (in bounded_linear) Zfun:
```
```   475   assumes g: "Zfun g A"
```
```   476   shows "Zfun (\<lambda>x. f (g x)) A"
```
```   477 proof -
```
```   478   obtain K where "\<And>x. norm (f x) \<le> norm x * K"
```
```   479     using bounded by fast
```
```   480   then have "eventually (\<lambda>x. norm (f (g x)) \<le> norm (g x) * K) A"
```
```   481     by simp
```
```   482   with g show ?thesis
```
```   483     by (rule Zfun_imp_Zfun)
```
```   484 qed
```
```   485
```
```   486 lemma (in bounded_bilinear) Zfun:
```
```   487   assumes f: "Zfun f A"
```
```   488   assumes g: "Zfun g A"
```
```   489   shows "Zfun (\<lambda>x. f x ** g x) A"
```
```   490 proof (rule ZfunI)
```
```   491   fix r::real assume r: "0 < r"
```
```   492   obtain K where K: "0 < K"
```
```   493     and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"
```
```   494     using pos_bounded by fast
```
```   495   from K have K': "0 < inverse K"
```
```   496     by (rule positive_imp_inverse_positive)
```
```   497   have "eventually (\<lambda>x. norm (f x) < r) A"
```
```   498     using f r by (rule ZfunD)
```
```   499   moreover
```
```   500   have "eventually (\<lambda>x. norm (g x) < inverse K) A"
```
```   501     using g K' by (rule ZfunD)
```
```   502   ultimately
```
```   503   show "eventually (\<lambda>x. norm (f x ** g x) < r) A"
```
```   504   proof (rule eventually_elim2)
```
```   505     fix x
```
```   506     assume *: "norm (f x) < r" "norm (g x) < inverse K"
```
```   507     have "norm (f x ** g x) \<le> norm (f x) * norm (g x) * K"
```
```   508       by (rule norm_le)
```
```   509     also have "norm (f x) * norm (g x) * K < r * inverse K * K"
```
```   510       by (intro mult_strict_right_mono mult_strict_mono' norm_ge_zero * K)
```
```   511     also from K have "r * inverse K * K = r"
```
```   512       by simp
```
```   513     finally show "norm (f x ** g x) < r" .
```
```   514   qed
```
```   515 qed
```
```   516
```
```   517 lemma (in bounded_bilinear) Zfun_left:
```
```   518   "Zfun f A \<Longrightarrow> Zfun (\<lambda>x. f x ** a) A"
```
```   519   by (rule bounded_linear_left [THEN bounded_linear.Zfun])
```
```   520
```
```   521 lemma (in bounded_bilinear) Zfun_right:
```
```   522   "Zfun f A \<Longrightarrow> Zfun (\<lambda>x. a ** f x) A"
```
```   523   by (rule bounded_linear_right [THEN bounded_linear.Zfun])
```
```   524
```
```   525 lemmas Zfun_mult = mult.Zfun
```
```   526 lemmas Zfun_mult_right = mult.Zfun_right
```
```   527 lemmas Zfun_mult_left = mult.Zfun_left
```
```   528
```
```   529
```
```   530 subsection {* Limits *}
```
```   531
```
```   532 definition tendsto :: "('a \<Rightarrow> 'b::topological_space) \<Rightarrow> 'b \<Rightarrow> 'a filter \<Rightarrow> bool"
```
```   533     (infixr "--->" 55) where
```
```   534   "(f ---> l) A \<longleftrightarrow> (\<forall>S. open S \<longrightarrow> l \<in> S \<longrightarrow> eventually (\<lambda>x. f x \<in> S) A)"
```
```   535
```
```   536 ML {*
```
```   537 structure Tendsto_Intros = Named_Thms
```
```   538 (
```
```   539   val name = "tendsto_intros"
```
```   540   val description = "introduction rules for tendsto"
```
```   541 )
```
```   542 *}
```
```   543
```
```   544 setup Tendsto_Intros.setup
```
```   545
```
```   546 lemma tendsto_mono: "A \<le> A' \<Longrightarrow> (f ---> l) A' \<Longrightarrow> (f ---> l) A"
```
```   547   unfolding tendsto_def le_filter_def by fast
```
```   548
```
```   549 lemma topological_tendstoI:
```
```   550   "(\<And>S. open S \<Longrightarrow> l \<in> S \<Longrightarrow> eventually (\<lambda>x. f x \<in> S) A)
```
```   551     \<Longrightarrow> (f ---> l) A"
```
```   552   unfolding tendsto_def by auto
```
```   553
```
```   554 lemma topological_tendstoD:
```
```   555   "(f ---> l) A \<Longrightarrow> open S \<Longrightarrow> l \<in> S \<Longrightarrow> eventually (\<lambda>x. f x \<in> S) A"
```
```   556   unfolding tendsto_def by auto
```
```   557
```
```   558 lemma tendstoI:
```
```   559   assumes "\<And>e. 0 < e \<Longrightarrow> eventually (\<lambda>x. dist (f x) l < e) A"
```
```   560   shows "(f ---> l) A"
```
```   561   apply (rule topological_tendstoI)
```
```   562   apply (simp add: open_dist)
```
```   563   apply (drule (1) bspec, clarify)
```
```   564   apply (drule assms)
```
```   565   apply (erule eventually_elim1, simp)
```
```   566   done
```
```   567
```
```   568 lemma tendstoD:
```
```   569   "(f ---> l) A \<Longrightarrow> 0 < e \<Longrightarrow> eventually (\<lambda>x. dist (f x) l < e) A"
```
```   570   apply (drule_tac S="{x. dist x l < e}" in topological_tendstoD)
```
```   571   apply (clarsimp simp add: open_dist)
```
```   572   apply (rule_tac x="e - dist x l" in exI, clarsimp)
```
```   573   apply (simp only: less_diff_eq)
```
```   574   apply (erule le_less_trans [OF dist_triangle])
```
```   575   apply simp
```
```   576   apply simp
```
```   577   done
```
```   578
```
```   579 lemma tendsto_iff:
```
```   580   "(f ---> l) A \<longleftrightarrow> (\<forall>e>0. eventually (\<lambda>x. dist (f x) l < e) A)"
```
```   581   using tendstoI tendstoD by fast
```
```   582
```
```   583 lemma tendsto_Zfun_iff: "(f ---> a) A = Zfun (\<lambda>x. f x - a) A"
```
```   584   by (simp only: tendsto_iff Zfun_def dist_norm)
```
```   585
```
```   586 lemma tendsto_ident_at [tendsto_intros]: "((\<lambda>x. x) ---> a) (at a)"
```
```   587   unfolding tendsto_def eventually_at_topological by auto
```
```   588
```
```   589 lemma tendsto_ident_at_within [tendsto_intros]:
```
```   590   "((\<lambda>x. x) ---> a) (at a within S)"
```
```   591   unfolding tendsto_def eventually_within eventually_at_topological by auto
```
```   592
```
```   593 lemma tendsto_const [tendsto_intros]: "((\<lambda>x. k) ---> k) A"
```
```   594   by (simp add: tendsto_def)
```
```   595
```
```   596 lemma tendsto_const_iff:
```
```   597   fixes k l :: "'a::metric_space"
```
```   598   assumes "A \<noteq> bot" shows "((\<lambda>n. k) ---> l) A \<longleftrightarrow> k = l"
```
```   599   apply (safe intro!: tendsto_const)
```
```   600   apply (rule ccontr)
```
```   601   apply (drule_tac e="dist k l" in tendstoD)
```
```   602   apply (simp add: zero_less_dist_iff)
```
```   603   apply (simp add: eventually_False assms)
```
```   604   done
```
```   605
```
```   606 lemma tendsto_dist [tendsto_intros]:
```
```   607   assumes f: "(f ---> l) A" and g: "(g ---> m) A"
```
```   608   shows "((\<lambda>x. dist (f x) (g x)) ---> dist l m) A"
```
```   609 proof (rule tendstoI)
```
```   610   fix e :: real assume "0 < e"
```
```   611   hence e2: "0 < e/2" by simp
```
```   612   from tendstoD [OF f e2] tendstoD [OF g e2]
```
```   613   show "eventually (\<lambda>x. dist (dist (f x) (g x)) (dist l m) < e) A"
```
```   614   proof (rule eventually_elim2)
```
```   615     fix x assume "dist (f x) l < e/2" "dist (g x) m < e/2"
```
```   616     then show "dist (dist (f x) (g x)) (dist l m) < e"
```
```   617       unfolding dist_real_def
```
```   618       using dist_triangle2 [of "f x" "g x" "l"]
```
```   619       using dist_triangle2 [of "g x" "l" "m"]
```
```   620       using dist_triangle3 [of "l" "m" "f x"]
```
```   621       using dist_triangle [of "f x" "m" "g x"]
```
```   622       by arith
```
```   623   qed
```
```   624 qed
```
```   625
```
```   626 lemma norm_conv_dist: "norm x = dist x 0"
```
```   627   unfolding dist_norm by simp
```
```   628
```
```   629 lemma tendsto_norm [tendsto_intros]:
```
```   630   "(f ---> a) A \<Longrightarrow> ((\<lambda>x. norm (f x)) ---> norm a) A"
```
```   631   unfolding norm_conv_dist by (intro tendsto_intros)
```
```   632
```
```   633 lemma tendsto_norm_zero:
```
```   634   "(f ---> 0) A \<Longrightarrow> ((\<lambda>x. norm (f x)) ---> 0) A"
```
```   635   by (drule tendsto_norm, simp)
```
```   636
```
```   637 lemma tendsto_norm_zero_cancel:
```
```   638   "((\<lambda>x. norm (f x)) ---> 0) A \<Longrightarrow> (f ---> 0) A"
```
```   639   unfolding tendsto_iff dist_norm by simp
```
```   640
```
```   641 lemma tendsto_norm_zero_iff:
```
```   642   "((\<lambda>x. norm (f x)) ---> 0) A \<longleftrightarrow> (f ---> 0) A"
```
```   643   unfolding tendsto_iff dist_norm by simp
```
```   644
```
```   645 lemma tendsto_add [tendsto_intros]:
```
```   646   fixes a b :: "'a::real_normed_vector"
```
```   647   shows "\<lbrakk>(f ---> a) A; (g ---> b) A\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x + g x) ---> a + b) A"
```
```   648   by (simp only: tendsto_Zfun_iff add_diff_add Zfun_add)
```
```   649
```
```   650 lemma tendsto_minus [tendsto_intros]:
```
```   651   fixes a :: "'a::real_normed_vector"
```
```   652   shows "(f ---> a) A \<Longrightarrow> ((\<lambda>x. - f x) ---> - a) A"
```
```   653   by (simp only: tendsto_Zfun_iff minus_diff_minus Zfun_minus)
```
```   654
```
```   655 lemma tendsto_minus_cancel:
```
```   656   fixes a :: "'a::real_normed_vector"
```
```   657   shows "((\<lambda>x. - f x) ---> - a) A \<Longrightarrow> (f ---> a) A"
```
```   658   by (drule tendsto_minus, simp)
```
```   659
```
```   660 lemma tendsto_diff [tendsto_intros]:
```
```   661   fixes a b :: "'a::real_normed_vector"
```
```   662   shows "\<lbrakk>(f ---> a) A; (g ---> b) A\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x - g x) ---> a - b) A"
```
```   663   by (simp add: diff_minus tendsto_add tendsto_minus)
```
```   664
```
```   665 lemma tendsto_setsum [tendsto_intros]:
```
```   666   fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c::real_normed_vector"
```
```   667   assumes "\<And>i. i \<in> S \<Longrightarrow> (f i ---> a i) A"
```
```   668   shows "((\<lambda>x. \<Sum>i\<in>S. f i x) ---> (\<Sum>i\<in>S. a i)) A"
```
```   669 proof (cases "finite S")
```
```   670   assume "finite S" thus ?thesis using assms
```
```   671   proof (induct set: finite)
```
```   672     case empty show ?case
```
```   673       by (simp add: tendsto_const)
```
```   674   next
```
```   675     case (insert i F) thus ?case
```
```   676       by (simp add: tendsto_add)
```
```   677   qed
```
```   678 next
```
```   679   assume "\<not> finite S" thus ?thesis
```
```   680     by (simp add: tendsto_const)
```
```   681 qed
```
```   682
```
```   683 lemma (in bounded_linear) tendsto [tendsto_intros]:
```
```   684   "(g ---> a) A \<Longrightarrow> ((\<lambda>x. f (g x)) ---> f a) A"
```
```   685   by (simp only: tendsto_Zfun_iff diff [symmetric] Zfun)
```
```   686
```
```   687 lemma (in bounded_bilinear) tendsto [tendsto_intros]:
```
```   688   "\<lbrakk>(f ---> a) A; (g ---> b) A\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x ** g x) ---> a ** b) A"
```
```   689   by (simp only: tendsto_Zfun_iff prod_diff_prod
```
```   690                  Zfun_add Zfun Zfun_left Zfun_right)
```
```   691
```
```   692
```
```   693 subsection {* Continuity of Inverse *}
```
```   694
```
```   695 lemma (in bounded_bilinear) Zfun_prod_Bfun:
```
```   696   assumes f: "Zfun f A"
```
```   697   assumes g: "Bfun g A"
```
```   698   shows "Zfun (\<lambda>x. f x ** g x) A"
```
```   699 proof -
```
```   700   obtain K where K: "0 \<le> K"
```
```   701     and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"
```
```   702     using nonneg_bounded by fast
```
```   703   obtain B where B: "0 < B"
```
```   704     and norm_g: "eventually (\<lambda>x. norm (g x) \<le> B) A"
```
```   705     using g by (rule BfunE)
```
```   706   have "eventually (\<lambda>x. norm (f x ** g x) \<le> norm (f x) * (B * K)) A"
```
```   707   using norm_g proof (rule eventually_elim1)
```
```   708     fix x
```
```   709     assume *: "norm (g x) \<le> B"
```
```   710     have "norm (f x ** g x) \<le> norm (f x) * norm (g x) * K"
```
```   711       by (rule norm_le)
```
```   712     also have "\<dots> \<le> norm (f x) * B * K"
```
```   713       by (intro mult_mono' order_refl norm_g norm_ge_zero
```
```   714                 mult_nonneg_nonneg K *)
```
```   715     also have "\<dots> = norm (f x) * (B * K)"
```
```   716       by (rule mult_assoc)
```
```   717     finally show "norm (f x ** g x) \<le> norm (f x) * (B * K)" .
```
```   718   qed
```
```   719   with f show ?thesis
```
```   720     by (rule Zfun_imp_Zfun)
```
```   721 qed
```
```   722
```
```   723 lemma (in bounded_bilinear) flip:
```
```   724   "bounded_bilinear (\<lambda>x y. y ** x)"
```
```   725   apply default
```
```   726   apply (rule add_right)
```
```   727   apply (rule add_left)
```
```   728   apply (rule scaleR_right)
```
```   729   apply (rule scaleR_left)
```
```   730   apply (subst mult_commute)
```
```   731   using bounded by fast
```
```   732
```
```   733 lemma (in bounded_bilinear) Bfun_prod_Zfun:
```
```   734   assumes f: "Bfun f A"
```
```   735   assumes g: "Zfun g A"
```
```   736   shows "Zfun (\<lambda>x. f x ** g x) A"
```
```   737   using flip g f by (rule bounded_bilinear.Zfun_prod_Bfun)
```
```   738
```
```   739 lemma Bfun_inverse_lemma:
```
```   740   fixes x :: "'a::real_normed_div_algebra"
```
```   741   shows "\<lbrakk>r \<le> norm x; 0 < r\<rbrakk> \<Longrightarrow> norm (inverse x) \<le> inverse r"
```
```   742   apply (subst nonzero_norm_inverse, clarsimp)
```
```   743   apply (erule (1) le_imp_inverse_le)
```
```   744   done
```
```   745
```
```   746 lemma Bfun_inverse:
```
```   747   fixes a :: "'a::real_normed_div_algebra"
```
```   748   assumes f: "(f ---> a) A"
```
```   749   assumes a: "a \<noteq> 0"
```
```   750   shows "Bfun (\<lambda>x. inverse (f x)) A"
```
```   751 proof -
```
```   752   from a have "0 < norm a" by simp
```
```   753   hence "\<exists>r>0. r < norm a" by (rule dense)
```
```   754   then obtain r where r1: "0 < r" and r2: "r < norm a" by fast
```
```   755   have "eventually (\<lambda>x. dist (f x) a < r) A"
```
```   756     using tendstoD [OF f r1] by fast
```
```   757   hence "eventually (\<lambda>x. norm (inverse (f x)) \<le> inverse (norm a - r)) A"
```
```   758   proof (rule eventually_elim1)
```
```   759     fix x
```
```   760     assume "dist (f x) a < r"
```
```   761     hence 1: "norm (f x - a) < r"
```
```   762       by (simp add: dist_norm)
```
```   763     hence 2: "f x \<noteq> 0" using r2 by auto
```
```   764     hence "norm (inverse (f x)) = inverse (norm (f x))"
```
```   765       by (rule nonzero_norm_inverse)
```
```   766     also have "\<dots> \<le> inverse (norm a - r)"
```
```   767     proof (rule le_imp_inverse_le)
```
```   768       show "0 < norm a - r" using r2 by simp
```
```   769     next
```
```   770       have "norm a - norm (f x) \<le> norm (a - f x)"
```
```   771         by (rule norm_triangle_ineq2)
```
```   772       also have "\<dots> = norm (f x - a)"
```
```   773         by (rule norm_minus_commute)
```
```   774       also have "\<dots> < r" using 1 .
```
```   775       finally show "norm a - r \<le> norm (f x)" by simp
```
```   776     qed
```
```   777     finally show "norm (inverse (f x)) \<le> inverse (norm a - r)" .
```
```   778   qed
```
```   779   thus ?thesis by (rule BfunI)
```
```   780 qed
```
```   781
```
```   782 lemma tendsto_inverse_lemma:
```
```   783   fixes a :: "'a::real_normed_div_algebra"
```
```   784   shows "\<lbrakk>(f ---> a) A; a \<noteq> 0; eventually (\<lambda>x. f x \<noteq> 0) A\<rbrakk>
```
```   785          \<Longrightarrow> ((\<lambda>x. inverse (f x)) ---> inverse a) A"
```
```   786   apply (subst tendsto_Zfun_iff)
```
```   787   apply (rule Zfun_ssubst)
```
```   788   apply (erule eventually_elim1)
```
```   789   apply (erule (1) inverse_diff_inverse)
```
```   790   apply (rule Zfun_minus)
```
```   791   apply (rule Zfun_mult_left)
```
```   792   apply (rule mult.Bfun_prod_Zfun)
```
```   793   apply (erule (1) Bfun_inverse)
```
```   794   apply (simp add: tendsto_Zfun_iff)
```
```   795   done
```
```   796
```
```   797 lemma tendsto_inverse [tendsto_intros]:
```
```   798   fixes a :: "'a::real_normed_div_algebra"
```
```   799   assumes f: "(f ---> a) A"
```
```   800   assumes a: "a \<noteq> 0"
```
```   801   shows "((\<lambda>x. inverse (f x)) ---> inverse a) A"
```
```   802 proof -
```
```   803   from a have "0 < norm a" by simp
```
```   804   with f have "eventually (\<lambda>x. dist (f x) a < norm a) A"
```
```   805     by (rule tendstoD)
```
```   806   then have "eventually (\<lambda>x. f x \<noteq> 0) A"
```
```   807     unfolding dist_norm by (auto elim!: eventually_elim1)
```
```   808   with f a show ?thesis
```
```   809     by (rule tendsto_inverse_lemma)
```
```   810 qed
```
```   811
```
```   812 lemma tendsto_divide [tendsto_intros]:
```
```   813   fixes a b :: "'a::real_normed_field"
```
```   814   shows "\<lbrakk>(f ---> a) A; (g ---> b) A; b \<noteq> 0\<rbrakk>
```
```   815     \<Longrightarrow> ((\<lambda>x. f x / g x) ---> a / b) A"
```
```   816   by (simp add: mult.tendsto tendsto_inverse divide_inverse)
```
```   817
```
```   818 lemma tendsto_unique:
```
```   819   fixes f :: "'a \<Rightarrow> 'b::t2_space"
```
```   820   assumes "\<not> trivial_limit A"  "(f ---> l) A"  "(f ---> l') A"
```
```   821   shows "l = l'"
```
```   822 proof (rule ccontr)
```
```   823   assume "l \<noteq> l'"
```
```   824   obtain U V where "open U" "open V" "l \<in> U" "l' \<in> V" "U \<inter> V = {}"
```
```   825     using hausdorff [OF `l \<noteq> l'`] by fast
```
```   826   have "eventually (\<lambda>x. f x \<in> U) A"
```
```   827     using `(f ---> l) A` `open U` `l \<in> U` by (rule topological_tendstoD)
```
```   828   moreover
```
```   829   have "eventually (\<lambda>x. f x \<in> V) A"
```
```   830     using `(f ---> l') A` `open V` `l' \<in> V` by (rule topological_tendstoD)
```
```   831   ultimately
```
```   832   have "eventually (\<lambda>x. False) A"
```
```   833   proof (rule eventually_elim2)
```
```   834     fix x
```
```   835     assume "f x \<in> U" "f x \<in> V"
```
```   836     hence "f x \<in> U \<inter> V" by simp
```
```   837     with `U \<inter> V = {}` show "False" by simp
```
```   838   qed
```
```   839   with `\<not> trivial_limit A` show "False"
```
```   840     by (simp add: trivial_limit_def)
```
```   841 qed
```
```   842
```
```   843 end
```