src/HOL/Limits.thy
 author hoelzl Mon Mar 14 14:37:35 2011 +0100 (2011-03-14) changeset 41970 47d6e13d1710 parent 39302 d7728f65b353 child 44079 bcc60791b7b9 permissions -rw-r--r--
generalize infinite sums
```     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 {* Nets *}
```
```    12
```
```    13 text {*
```
```    14   A net is now defined simply as a filter on a set.
```
```    15   The definition also allows non-proper filters.
```
```    16 *}
```
```    17
```
```    18 locale is_filter =
```
```    19   fixes net :: "('a \<Rightarrow> bool) \<Rightarrow> bool"
```
```    20   assumes True: "net (\<lambda>x. True)"
```
```    21   assumes conj: "net (\<lambda>x. P x) \<Longrightarrow> net (\<lambda>x. Q x) \<Longrightarrow> net (\<lambda>x. P x \<and> Q x)"
```
```    22   assumes mono: "\<forall>x. P x \<longrightarrow> Q x \<Longrightarrow> net (\<lambda>x. P x) \<Longrightarrow> net (\<lambda>x. Q x)"
```
```    23
```
```    24 typedef (open) 'a net =
```
```    25   "{net :: ('a \<Rightarrow> bool) \<Rightarrow> bool. is_filter net}"
```
```    26 proof
```
```    27   show "(\<lambda>x. True) \<in> ?net" by (auto intro: is_filter.intro)
```
```    28 qed
```
```    29
```
```    30 lemma is_filter_Rep_net: "is_filter (Rep_net net)"
```
```    31 using Rep_net [of net] by simp
```
```    32
```
```    33 lemma Abs_net_inverse':
```
```    34   assumes "is_filter net" shows "Rep_net (Abs_net net) = net"
```
```    35 using assms by (simp add: Abs_net_inverse)
```
```    36
```
```    37
```
```    38 subsection {* Eventually *}
```
```    39
```
```    40 definition eventually :: "('a \<Rightarrow> bool) \<Rightarrow> 'a net \<Rightarrow> bool" where
```
```    41   "eventually P net \<longleftrightarrow> Rep_net net P"
```
```    42
```
```    43 lemma eventually_Abs_net:
```
```    44   assumes "is_filter net" shows "eventually P (Abs_net net) = net P"
```
```    45 unfolding eventually_def using assms by (simp add: Abs_net_inverse)
```
```    46
```
```    47 lemma expand_net_eq:
```
```    48   shows "net = net' \<longleftrightarrow> (\<forall>P. eventually P net = eventually P net')"
```
```    49 unfolding Rep_net_inject [symmetric] fun_eq_iff eventually_def ..
```
```    50
```
```    51 lemma eventually_True [simp]: "eventually (\<lambda>x. True) net"
```
```    52 unfolding eventually_def
```
```    53 by (rule is_filter.True [OF is_filter_Rep_net])
```
```    54
```
```    55 lemma always_eventually: "\<forall>x. P x \<Longrightarrow> eventually P net"
```
```    56 proof -
```
```    57   assume "\<forall>x. P x" hence "P = (\<lambda>x. True)" by (simp add: ext)
```
```    58   thus "eventually P net" by simp
```
```    59 qed
```
```    60
```
```    61 lemma eventually_mono:
```
```    62   "(\<forall>x. P x \<longrightarrow> Q x) \<Longrightarrow> eventually P net \<Longrightarrow> eventually Q net"
```
```    63 unfolding eventually_def
```
```    64 by (rule is_filter.mono [OF is_filter_Rep_net])
```
```    65
```
```    66 lemma eventually_conj:
```
```    67   assumes P: "eventually (\<lambda>x. P x) net"
```
```    68   assumes Q: "eventually (\<lambda>x. Q x) net"
```
```    69   shows "eventually (\<lambda>x. P x \<and> Q x) net"
```
```    70 using assms unfolding eventually_def
```
```    71 by (rule is_filter.conj [OF is_filter_Rep_net])
```
```    72
```
```    73 lemma eventually_mp:
```
```    74   assumes "eventually (\<lambda>x. P x \<longrightarrow> Q x) net"
```
```    75   assumes "eventually (\<lambda>x. P x) net"
```
```    76   shows "eventually (\<lambda>x. Q x) net"
```
```    77 proof (rule eventually_mono)
```
```    78   show "\<forall>x. (P x \<longrightarrow> Q x) \<and> P x \<longrightarrow> Q x" by simp
```
```    79   show "eventually (\<lambda>x. (P x \<longrightarrow> Q x) \<and> P x) net"
```
```    80     using assms by (rule eventually_conj)
```
```    81 qed
```
```    82
```
```    83 lemma eventually_rev_mp:
```
```    84   assumes "eventually (\<lambda>x. P x) net"
```
```    85   assumes "eventually (\<lambda>x. P x \<longrightarrow> Q x) net"
```
```    86   shows "eventually (\<lambda>x. Q x) net"
```
```    87 using assms(2) assms(1) by (rule eventually_mp)
```
```    88
```
```    89 lemma eventually_conj_iff:
```
```    90   "eventually (\<lambda>x. P x \<and> Q x) net \<longleftrightarrow> eventually P net \<and> eventually Q net"
```
```    91 by (auto intro: eventually_conj elim: eventually_rev_mp)
```
```    92
```
```    93 lemma eventually_elim1:
```
```    94   assumes "eventually (\<lambda>i. P i) net"
```
```    95   assumes "\<And>i. P i \<Longrightarrow> Q i"
```
```    96   shows "eventually (\<lambda>i. Q i) net"
```
```    97 using assms by (auto elim!: eventually_rev_mp)
```
```    98
```
```    99 lemma eventually_elim2:
```
```   100   assumes "eventually (\<lambda>i. P i) net"
```
```   101   assumes "eventually (\<lambda>i. Q i) net"
```
```   102   assumes "\<And>i. P i \<Longrightarrow> Q i \<Longrightarrow> R i"
```
```   103   shows "eventually (\<lambda>i. R i) net"
```
```   104 using assms by (auto elim!: eventually_rev_mp)
```
```   105
```
```   106 subsection {* Finer-than relation *}
```
```   107
```
```   108 text {* @{term "net \<le> net'"} means that @{term net} is finer than
```
```   109 @{term net'}. *}
```
```   110
```
```   111 instantiation net :: (type) complete_lattice
```
```   112 begin
```
```   113
```
```   114 definition
```
```   115   le_net_def: "net \<le> net' \<longleftrightarrow> (\<forall>P. eventually P net' \<longrightarrow> eventually P net)"
```
```   116
```
```   117 definition
```
```   118   less_net_def: "(net :: 'a net) < net' \<longleftrightarrow> net \<le> net' \<and> \<not> net' \<le> net"
```
```   119
```
```   120 definition
```
```   121   top_net_def: "top = Abs_net (\<lambda>P. \<forall>x. P x)"
```
```   122
```
```   123 definition
```
```   124   bot_net_def: "bot = Abs_net (\<lambda>P. True)"
```
```   125
```
```   126 definition
```
```   127   sup_net_def: "sup net net' = Abs_net (\<lambda>P. eventually P net \<and> eventually P net')"
```
```   128
```
```   129 definition
```
```   130   inf_net_def: "inf a b = Abs_net
```
```   131       (\<lambda>P. \<exists>Q R. eventually Q a \<and> eventually R b \<and> (\<forall>x. Q x \<and> R x \<longrightarrow> P x))"
```
```   132
```
```   133 definition
```
```   134   Sup_net_def: "Sup A = Abs_net (\<lambda>P. \<forall>net\<in>A. eventually P net)"
```
```   135
```
```   136 definition
```
```   137   Inf_net_def: "Inf A = Sup {x::'a net. \<forall>y\<in>A. x \<le> y}"
```
```   138
```
```   139 lemma eventually_top [simp]: "eventually P top \<longleftrightarrow> (\<forall>x. P x)"
```
```   140 unfolding top_net_def
```
```   141 by (rule eventually_Abs_net, rule is_filter.intro, auto)
```
```   142
```
```   143 lemma eventually_bot [simp]: "eventually P bot"
```
```   144 unfolding bot_net_def
```
```   145 by (subst eventually_Abs_net, rule is_filter.intro, auto)
```
```   146
```
```   147 lemma eventually_sup:
```
```   148   "eventually P (sup net net') \<longleftrightarrow> eventually P net \<and> eventually P net'"
```
```   149 unfolding sup_net_def
```
```   150 by (rule eventually_Abs_net, rule is_filter.intro)
```
```   151    (auto elim!: eventually_rev_mp)
```
```   152
```
```   153 lemma eventually_inf:
```
```   154   "eventually P (inf a b) \<longleftrightarrow>
```
```   155    (\<exists>Q R. eventually Q a \<and> eventually R b \<and> (\<forall>x. Q x \<and> R x \<longrightarrow> P x))"
```
```   156 unfolding inf_net_def
```
```   157 apply (rule eventually_Abs_net, rule is_filter.intro)
```
```   158 apply (fast intro: eventually_True)
```
```   159 apply clarify
```
```   160 apply (intro exI conjI)
```
```   161 apply (erule (1) eventually_conj)
```
```   162 apply (erule (1) eventually_conj)
```
```   163 apply simp
```
```   164 apply auto
```
```   165 done
```
```   166
```
```   167 lemma eventually_Sup:
```
```   168   "eventually P (Sup A) \<longleftrightarrow> (\<forall>net\<in>A. eventually P net)"
```
```   169 unfolding Sup_net_def
```
```   170 apply (rule eventually_Abs_net, rule is_filter.intro)
```
```   171 apply (auto intro: eventually_conj elim!: eventually_rev_mp)
```
```   172 done
```
```   173
```
```   174 instance proof
```
```   175   fix x y :: "'a net" show "x < y \<longleftrightarrow> x \<le> y \<and> \<not> y \<le> x"
```
```   176     by (rule less_net_def)
```
```   177 next
```
```   178   fix x :: "'a net" show "x \<le> x"
```
```   179     unfolding le_net_def by simp
```
```   180 next
```
```   181   fix x y z :: "'a net" assume "x \<le> y" and "y \<le> z" thus "x \<le> z"
```
```   182     unfolding le_net_def by simp
```
```   183 next
```
```   184   fix x y :: "'a net" assume "x \<le> y" and "y \<le> x" thus "x = y"
```
```   185     unfolding le_net_def expand_net_eq by fast
```
```   186 next
```
```   187   fix x :: "'a net" show "x \<le> top"
```
```   188     unfolding le_net_def eventually_top by (simp add: always_eventually)
```
```   189 next
```
```   190   fix x :: "'a net" show "bot \<le> x"
```
```   191     unfolding le_net_def by simp
```
```   192 next
```
```   193   fix x y :: "'a net" show "x \<le> sup x y" and "y \<le> sup x y"
```
```   194     unfolding le_net_def eventually_sup by simp_all
```
```   195 next
```
```   196   fix x y z :: "'a net" assume "x \<le> z" and "y \<le> z" thus "sup x y \<le> z"
```
```   197     unfolding le_net_def eventually_sup by simp
```
```   198 next
```
```   199   fix x y :: "'a net" show "inf x y \<le> x" and "inf x y \<le> y"
```
```   200     unfolding le_net_def eventually_inf by (auto intro: eventually_True)
```
```   201 next
```
```   202   fix x y z :: "'a net" assume "x \<le> y" and "x \<le> z" thus "x \<le> inf y z"
```
```   203     unfolding le_net_def eventually_inf
```
```   204     by (auto elim!: eventually_mono intro: eventually_conj)
```
```   205 next
```
```   206   fix x :: "'a net" and A assume "x \<in> A" thus "x \<le> Sup A"
```
```   207     unfolding le_net_def eventually_Sup by simp
```
```   208 next
```
```   209   fix A and y :: "'a net" assume "\<And>x. x \<in> A \<Longrightarrow> x \<le> y" thus "Sup A \<le> y"
```
```   210     unfolding le_net_def eventually_Sup by simp
```
```   211 next
```
```   212   fix z :: "'a net" and A assume "z \<in> A" thus "Inf A \<le> z"
```
```   213     unfolding le_net_def Inf_net_def eventually_Sup Ball_def by simp
```
```   214 next
```
```   215   fix A and x :: "'a net" assume "\<And>y. y \<in> A \<Longrightarrow> x \<le> y" thus "x \<le> Inf A"
```
```   216     unfolding le_net_def Inf_net_def eventually_Sup Ball_def by simp
```
```   217 qed
```
```   218
```
```   219 end
```
```   220
```
```   221 lemma net_leD:
```
```   222   "net \<le> net' \<Longrightarrow> eventually P net' \<Longrightarrow> eventually P net"
```
```   223 unfolding le_net_def by simp
```
```   224
```
```   225 lemma net_leI:
```
```   226   "(\<And>P. eventually P net' \<Longrightarrow> eventually P net) \<Longrightarrow> net \<le> net'"
```
```   227 unfolding le_net_def by simp
```
```   228
```
```   229 lemma eventually_False:
```
```   230   "eventually (\<lambda>x. False) net \<longleftrightarrow> net = bot"
```
```   231 unfolding expand_net_eq by (auto elim: eventually_rev_mp)
```
```   232
```
```   233 subsection {* Map function for nets *}
```
```   234
```
```   235 definition netmap :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a net \<Rightarrow> 'b net" where
```
```   236   "netmap f net = Abs_net (\<lambda>P. eventually (\<lambda>x. P (f x)) net)"
```
```   237
```
```   238 lemma eventually_netmap:
```
```   239   "eventually P (netmap f net) = eventually (\<lambda>x. P (f x)) net"
```
```   240 unfolding netmap_def
```
```   241 apply (rule eventually_Abs_net)
```
```   242 apply (rule is_filter.intro)
```
```   243 apply (auto elim!: eventually_rev_mp)
```
```   244 done
```
```   245
```
```   246 lemma netmap_ident: "netmap (\<lambda>x. x) net = net"
```
```   247 by (simp add: expand_net_eq eventually_netmap)
```
```   248
```
```   249 lemma netmap_netmap: "netmap f (netmap g net) = netmap (\<lambda>x. f (g x)) net"
```
```   250 by (simp add: expand_net_eq eventually_netmap)
```
```   251
```
```   252 lemma netmap_mono: "net \<le> net' \<Longrightarrow> netmap f net \<le> netmap f net'"
```
```   253 unfolding le_net_def eventually_netmap by simp
```
```   254
```
```   255 lemma netmap_bot [simp]: "netmap f bot = bot"
```
```   256 by (simp add: expand_net_eq eventually_netmap)
```
```   257
```
```   258
```
```   259 subsection {* Sequentially *}
```
```   260
```
```   261 definition sequentially :: "nat net" where
```
```   262   "sequentially = Abs_net (\<lambda>P. \<exists>k. \<forall>n\<ge>k. P n)"
```
```   263
```
```   264 lemma eventually_sequentially:
```
```   265   "eventually P sequentially \<longleftrightarrow> (\<exists>N. \<forall>n\<ge>N. P n)"
```
```   266 unfolding sequentially_def
```
```   267 proof (rule eventually_Abs_net, rule is_filter.intro)
```
```   268   fix P Q :: "nat \<Rightarrow> bool"
```
```   269   assume "\<exists>i. \<forall>n\<ge>i. P n" and "\<exists>j. \<forall>n\<ge>j. Q n"
```
```   270   then obtain i j where "\<forall>n\<ge>i. P n" and "\<forall>n\<ge>j. Q n" by auto
```
```   271   then have "\<forall>n\<ge>max i j. P n \<and> Q n" by simp
```
```   272   then show "\<exists>k. \<forall>n\<ge>k. P n \<and> Q n" ..
```
```   273 qed auto
```
```   274
```
```   275 lemma sequentially_bot [simp]: "sequentially \<noteq> bot"
```
```   276 unfolding expand_net_eq eventually_sequentially by auto
```
```   277
```
```   278 lemma eventually_False_sequentially [simp]:
```
```   279   "\<not> eventually (\<lambda>n. False) sequentially"
```
```   280 by (simp add: eventually_False)
```
```   281
```
```   282 lemma le_sequentially:
```
```   283   "net \<le> sequentially \<longleftrightarrow> (\<forall>N. eventually (\<lambda>n. N \<le> n) net)"
```
```   284 unfolding le_net_def eventually_sequentially
```
```   285 by (safe, fast, drule_tac x=N in spec, auto elim: eventually_rev_mp)
```
```   286
```
```   287
```
```   288 definition
```
```   289   trivial_limit :: "'a net \<Rightarrow> bool" where
```
```   290   "trivial_limit net \<longleftrightarrow> eventually (\<lambda>x. False) net"
```
```   291
```
```   292 lemma trivial_limit_sequentially[intro]: "\<not> trivial_limit sequentially"
```
```   293   by (auto simp add: trivial_limit_def eventually_sequentially)
```
```   294
```
```   295 subsection {* Standard Nets *}
```
```   296
```
```   297 definition within :: "'a net \<Rightarrow> 'a set \<Rightarrow> 'a net" (infixr "within" 70) where
```
```   298   "net within S = Abs_net (\<lambda>P. eventually (\<lambda>x. x \<in> S \<longrightarrow> P x) net)"
```
```   299
```
```   300 definition nhds :: "'a::topological_space \<Rightarrow> 'a net" where
```
```   301   "nhds a = Abs_net (\<lambda>P. \<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x))"
```
```   302
```
```   303 definition at :: "'a::topological_space \<Rightarrow> 'a net" where
```
```   304   "at a = nhds a within - {a}"
```
```   305
```
```   306 lemma eventually_within:
```
```   307   "eventually P (net within S) = eventually (\<lambda>x. x \<in> S \<longrightarrow> P x) net"
```
```   308 unfolding within_def
```
```   309 by (rule eventually_Abs_net, rule is_filter.intro)
```
```   310    (auto elim!: eventually_rev_mp)
```
```   311
```
```   312 lemma within_UNIV: "net within UNIV = net"
```
```   313   unfolding expand_net_eq eventually_within by simp
```
```   314
```
```   315 lemma eventually_nhds:
```
```   316   "eventually P (nhds a) \<longleftrightarrow> (\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x))"
```
```   317 unfolding nhds_def
```
```   318 proof (rule eventually_Abs_net, rule is_filter.intro)
```
```   319   have "open UNIV \<and> a \<in> UNIV \<and> (\<forall>x\<in>UNIV. True)" by simp
```
```   320   thus "\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. True)" by - rule
```
```   321 next
```
```   322   fix P Q
```
```   323   assume "\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x)"
```
```   324      and "\<exists>T. open T \<and> a \<in> T \<and> (\<forall>x\<in>T. Q x)"
```
```   325   then obtain S T where
```
```   326     "open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x)"
```
```   327     "open T \<and> a \<in> T \<and> (\<forall>x\<in>T. Q x)" by auto
```
```   328   hence "open (S \<inter> T) \<and> a \<in> S \<inter> T \<and> (\<forall>x\<in>(S \<inter> T). P x \<and> Q x)"
```
```   329     by (simp add: open_Int)
```
```   330   thus "\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x \<and> Q x)" by - rule
```
```   331 qed auto
```
```   332
```
```   333 lemma eventually_nhds_metric:
```
```   334   "eventually P (nhds a) \<longleftrightarrow> (\<exists>d>0. \<forall>x. dist x a < d \<longrightarrow> P x)"
```
```   335 unfolding eventually_nhds open_dist
```
```   336 apply safe
```
```   337 apply fast
```
```   338 apply (rule_tac x="{x. dist x a < d}" in exI, simp)
```
```   339 apply clarsimp
```
```   340 apply (rule_tac x="d - dist x a" in exI, clarsimp)
```
```   341 apply (simp only: less_diff_eq)
```
```   342 apply (erule le_less_trans [OF dist_triangle])
```
```   343 done
```
```   344
```
```   345 lemma eventually_at_topological:
```
```   346   "eventually P (at a) \<longleftrightarrow> (\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. x \<noteq> a \<longrightarrow> P x))"
```
```   347 unfolding at_def eventually_within eventually_nhds by simp
```
```   348
```
```   349 lemma eventually_at:
```
```   350   fixes a :: "'a::metric_space"
```
```   351   shows "eventually P (at a) \<longleftrightarrow> (\<exists>d>0. \<forall>x. x \<noteq> a \<and> dist x a < d \<longrightarrow> P x)"
```
```   352 unfolding at_def eventually_within eventually_nhds_metric by auto
```
```   353
```
```   354
```
```   355 subsection {* Boundedness *}
```
```   356
```
```   357 definition Bfun :: "('a \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a net \<Rightarrow> bool" where
```
```   358   "Bfun f net = (\<exists>K>0. eventually (\<lambda>x. norm (f x) \<le> K) net)"
```
```   359
```
```   360 lemma BfunI:
```
```   361   assumes K: "eventually (\<lambda>x. norm (f x) \<le> K) net" shows "Bfun f net"
```
```   362 unfolding Bfun_def
```
```   363 proof (intro exI conjI allI)
```
```   364   show "0 < max K 1" by simp
```
```   365 next
```
```   366   show "eventually (\<lambda>x. norm (f x) \<le> max K 1) net"
```
```   367     using K by (rule eventually_elim1, simp)
```
```   368 qed
```
```   369
```
```   370 lemma BfunE:
```
```   371   assumes "Bfun f net"
```
```   372   obtains B where "0 < B" and "eventually (\<lambda>x. norm (f x) \<le> B) net"
```
```   373 using assms unfolding Bfun_def by fast
```
```   374
```
```   375
```
```   376 subsection {* Convergence to Zero *}
```
```   377
```
```   378 definition Zfun :: "('a \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a net \<Rightarrow> bool" where
```
```   379   "Zfun f net = (\<forall>r>0. eventually (\<lambda>x. norm (f x) < r) net)"
```
```   380
```
```   381 lemma ZfunI:
```
```   382   "(\<And>r. 0 < r \<Longrightarrow> eventually (\<lambda>x. norm (f x) < r) net) \<Longrightarrow> Zfun f net"
```
```   383 unfolding Zfun_def by simp
```
```   384
```
```   385 lemma ZfunD:
```
```   386   "\<lbrakk>Zfun f net; 0 < r\<rbrakk> \<Longrightarrow> eventually (\<lambda>x. norm (f x) < r) net"
```
```   387 unfolding Zfun_def by simp
```
```   388
```
```   389 lemma Zfun_ssubst:
```
```   390   "eventually (\<lambda>x. f x = g x) net \<Longrightarrow> Zfun g net \<Longrightarrow> Zfun f net"
```
```   391 unfolding Zfun_def by (auto elim!: eventually_rev_mp)
```
```   392
```
```   393 lemma Zfun_zero: "Zfun (\<lambda>x. 0) net"
```
```   394 unfolding Zfun_def by simp
```
```   395
```
```   396 lemma Zfun_norm_iff: "Zfun (\<lambda>x. norm (f x)) net = Zfun (\<lambda>x. f x) net"
```
```   397 unfolding Zfun_def by simp
```
```   398
```
```   399 lemma Zfun_imp_Zfun:
```
```   400   assumes f: "Zfun f net"
```
```   401   assumes g: "eventually (\<lambda>x. norm (g x) \<le> norm (f x) * K) net"
```
```   402   shows "Zfun (\<lambda>x. g x) net"
```
```   403 proof (cases)
```
```   404   assume K: "0 < K"
```
```   405   show ?thesis
```
```   406   proof (rule ZfunI)
```
```   407     fix r::real assume "0 < r"
```
```   408     hence "0 < r / K"
```
```   409       using K by (rule divide_pos_pos)
```
```   410     then have "eventually (\<lambda>x. norm (f x) < r / K) net"
```
```   411       using ZfunD [OF f] by fast
```
```   412     with g show "eventually (\<lambda>x. norm (g x) < r) net"
```
```   413     proof (rule eventually_elim2)
```
```   414       fix x
```
```   415       assume *: "norm (g x) \<le> norm (f x) * K"
```
```   416       assume "norm (f x) < r / K"
```
```   417       hence "norm (f x) * K < r"
```
```   418         by (simp add: pos_less_divide_eq K)
```
```   419       thus "norm (g x) < r"
```
```   420         by (simp add: order_le_less_trans [OF *])
```
```   421     qed
```
```   422   qed
```
```   423 next
```
```   424   assume "\<not> 0 < K"
```
```   425   hence K: "K \<le> 0" by (simp only: not_less)
```
```   426   show ?thesis
```
```   427   proof (rule ZfunI)
```
```   428     fix r :: real
```
```   429     assume "0 < r"
```
```   430     from g show "eventually (\<lambda>x. norm (g x) < r) net"
```
```   431     proof (rule eventually_elim1)
```
```   432       fix x
```
```   433       assume "norm (g x) \<le> norm (f x) * K"
```
```   434       also have "\<dots> \<le> norm (f x) * 0"
```
```   435         using K norm_ge_zero by (rule mult_left_mono)
```
```   436       finally show "norm (g x) < r"
```
```   437         using `0 < r` by simp
```
```   438     qed
```
```   439   qed
```
```   440 qed
```
```   441
```
```   442 lemma Zfun_le: "\<lbrakk>Zfun g net; \<forall>x. norm (f x) \<le> norm (g x)\<rbrakk> \<Longrightarrow> Zfun f net"
```
```   443 by (erule_tac K="1" in Zfun_imp_Zfun, simp)
```
```   444
```
```   445 lemma Zfun_add:
```
```   446   assumes f: "Zfun f net" and g: "Zfun g net"
```
```   447   shows "Zfun (\<lambda>x. f x + g x) net"
```
```   448 proof (rule ZfunI)
```
```   449   fix r::real assume "0 < r"
```
```   450   hence r: "0 < r / 2" by simp
```
```   451   have "eventually (\<lambda>x. norm (f x) < r/2) net"
```
```   452     using f r by (rule ZfunD)
```
```   453   moreover
```
```   454   have "eventually (\<lambda>x. norm (g x) < r/2) net"
```
```   455     using g r by (rule ZfunD)
```
```   456   ultimately
```
```   457   show "eventually (\<lambda>x. norm (f x + g x) < r) net"
```
```   458   proof (rule eventually_elim2)
```
```   459     fix x
```
```   460     assume *: "norm (f x) < r/2" "norm (g x) < r/2"
```
```   461     have "norm (f x + g x) \<le> norm (f x) + norm (g x)"
```
```   462       by (rule norm_triangle_ineq)
```
```   463     also have "\<dots> < r/2 + r/2"
```
```   464       using * by (rule add_strict_mono)
```
```   465     finally show "norm (f x + g x) < r"
```
```   466       by simp
```
```   467   qed
```
```   468 qed
```
```   469
```
```   470 lemma Zfun_minus: "Zfun f net \<Longrightarrow> Zfun (\<lambda>x. - f x) net"
```
```   471 unfolding Zfun_def by simp
```
```   472
```
```   473 lemma Zfun_diff: "\<lbrakk>Zfun f net; Zfun g net\<rbrakk> \<Longrightarrow> Zfun (\<lambda>x. f x - g x) net"
```
```   474 by (simp only: diff_minus Zfun_add Zfun_minus)
```
```   475
```
```   476 lemma (in bounded_linear) Zfun:
```
```   477   assumes g: "Zfun g net"
```
```   478   shows "Zfun (\<lambda>x. f (g x)) net"
```
```   479 proof -
```
```   480   obtain K where "\<And>x. norm (f x) \<le> norm x * K"
```
```   481     using bounded by fast
```
```   482   then have "eventually (\<lambda>x. norm (f (g x)) \<le> norm (g x) * K) net"
```
```   483     by simp
```
```   484   with g show ?thesis
```
```   485     by (rule Zfun_imp_Zfun)
```
```   486 qed
```
```   487
```
```   488 lemma (in bounded_bilinear) Zfun:
```
```   489   assumes f: "Zfun f net"
```
```   490   assumes g: "Zfun g net"
```
```   491   shows "Zfun (\<lambda>x. f x ** g x) net"
```
```   492 proof (rule ZfunI)
```
```   493   fix r::real assume r: "0 < r"
```
```   494   obtain K where K: "0 < K"
```
```   495     and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"
```
```   496     using pos_bounded by fast
```
```   497   from K have K': "0 < inverse K"
```
```   498     by (rule positive_imp_inverse_positive)
```
```   499   have "eventually (\<lambda>x. norm (f x) < r) net"
```
```   500     using f r by (rule ZfunD)
```
```   501   moreover
```
```   502   have "eventually (\<lambda>x. norm (g x) < inverse K) net"
```
```   503     using g K' by (rule ZfunD)
```
```   504   ultimately
```
```   505   show "eventually (\<lambda>x. norm (f x ** g x) < r) net"
```
```   506   proof (rule eventually_elim2)
```
```   507     fix x
```
```   508     assume *: "norm (f x) < r" "norm (g x) < inverse K"
```
```   509     have "norm (f x ** g x) \<le> norm (f x) * norm (g x) * K"
```
```   510       by (rule norm_le)
```
```   511     also have "norm (f x) * norm (g x) * K < r * inverse K * K"
```
```   512       by (intro mult_strict_right_mono mult_strict_mono' norm_ge_zero * K)
```
```   513     also from K have "r * inverse K * K = r"
```
```   514       by simp
```
```   515     finally show "norm (f x ** g x) < r" .
```
```   516   qed
```
```   517 qed
```
```   518
```
```   519 lemma (in bounded_bilinear) Zfun_left:
```
```   520   "Zfun f net \<Longrightarrow> Zfun (\<lambda>x. f x ** a) net"
```
```   521 by (rule bounded_linear_left [THEN bounded_linear.Zfun])
```
```   522
```
```   523 lemma (in bounded_bilinear) Zfun_right:
```
```   524   "Zfun f net \<Longrightarrow> Zfun (\<lambda>x. a ** f x) net"
```
```   525 by (rule bounded_linear_right [THEN bounded_linear.Zfun])
```
```   526
```
```   527 lemmas Zfun_mult = mult.Zfun
```
```   528 lemmas Zfun_mult_right = mult.Zfun_right
```
```   529 lemmas Zfun_mult_left = mult.Zfun_left
```
```   530
```
```   531
```
```   532 subsection {* Limits *}
```
```   533
```
```   534 definition tendsto :: "('a \<Rightarrow> 'b::topological_space) \<Rightarrow> 'b \<Rightarrow> 'a net \<Rightarrow> bool"
```
```   535     (infixr "--->" 55) where
```
```   536   "(f ---> l) net \<longleftrightarrow> (\<forall>S. open S \<longrightarrow> l \<in> S \<longrightarrow> eventually (\<lambda>x. f x \<in> S) net)"
```
```   537
```
```   538 ML {*
```
```   539 structure Tendsto_Intros = Named_Thms
```
```   540 (
```
```   541   val name = "tendsto_intros"
```
```   542   val description = "introduction rules for tendsto"
```
```   543 )
```
```   544 *}
```
```   545
```
```   546 setup Tendsto_Intros.setup
```
```   547
```
```   548 lemma tendsto_mono: "net \<le> net' \<Longrightarrow> (f ---> l) net' \<Longrightarrow> (f ---> l) net"
```
```   549 unfolding tendsto_def le_net_def by fast
```
```   550
```
```   551 lemma topological_tendstoI:
```
```   552   "(\<And>S. open S \<Longrightarrow> l \<in> S \<Longrightarrow> eventually (\<lambda>x. f x \<in> S) net)
```
```   553     \<Longrightarrow> (f ---> l) net"
```
```   554   unfolding tendsto_def by auto
```
```   555
```
```   556 lemma topological_tendstoD:
```
```   557   "(f ---> l) net \<Longrightarrow> open S \<Longrightarrow> l \<in> S \<Longrightarrow> eventually (\<lambda>x. f x \<in> S) net"
```
```   558   unfolding tendsto_def by auto
```
```   559
```
```   560 lemma tendstoI:
```
```   561   assumes "\<And>e. 0 < e \<Longrightarrow> eventually (\<lambda>x. dist (f x) l < e) net"
```
```   562   shows "(f ---> l) net"
```
```   563 apply (rule topological_tendstoI)
```
```   564 apply (simp add: open_dist)
```
```   565 apply (drule (1) bspec, clarify)
```
```   566 apply (drule assms)
```
```   567 apply (erule eventually_elim1, simp)
```
```   568 done
```
```   569
```
```   570 lemma tendstoD:
```
```   571   "(f ---> l) net \<Longrightarrow> 0 < e \<Longrightarrow> eventually (\<lambda>x. dist (f x) l < e) net"
```
```   572 apply (drule_tac S="{x. dist x l < e}" in topological_tendstoD)
```
```   573 apply (clarsimp simp add: open_dist)
```
```   574 apply (rule_tac x="e - dist x l" in exI, clarsimp)
```
```   575 apply (simp only: less_diff_eq)
```
```   576 apply (erule le_less_trans [OF dist_triangle])
```
```   577 apply simp
```
```   578 apply simp
```
```   579 done
```
```   580
```
```   581 lemma tendsto_iff:
```
```   582   "(f ---> l) net \<longleftrightarrow> (\<forall>e>0. eventually (\<lambda>x. dist (f x) l < e) net)"
```
```   583 using tendstoI tendstoD by fast
```
```   584
```
```   585 lemma tendsto_Zfun_iff: "(f ---> a) net = Zfun (\<lambda>x. f x - a) net"
```
```   586 by (simp only: tendsto_iff Zfun_def dist_norm)
```
```   587
```
```   588 lemma tendsto_ident_at [tendsto_intros]: "((\<lambda>x. x) ---> a) (at a)"
```
```   589 unfolding tendsto_def eventually_at_topological by auto
```
```   590
```
```   591 lemma tendsto_ident_at_within [tendsto_intros]:
```
```   592   "((\<lambda>x. x) ---> a) (at a within S)"
```
```   593 unfolding tendsto_def eventually_within eventually_at_topological by auto
```
```   594
```
```   595 lemma tendsto_const [tendsto_intros]: "((\<lambda>x. k) ---> k) net"
```
```   596 by (simp add: tendsto_def)
```
```   597
```
```   598 lemma tendsto_const_iff:
```
```   599   fixes k l :: "'a::metric_space"
```
```   600   assumes "net \<noteq> bot" shows "((\<lambda>n. k) ---> l) net \<longleftrightarrow> k = l"
```
```   601 apply (safe intro!: tendsto_const)
```
```   602 apply (rule ccontr)
```
```   603 apply (drule_tac e="dist k l" in tendstoD)
```
```   604 apply (simp add: zero_less_dist_iff)
```
```   605 apply (simp add: eventually_False assms)
```
```   606 done
```
```   607
```
```   608 lemma tendsto_dist [tendsto_intros]:
```
```   609   assumes f: "(f ---> l) net" and g: "(g ---> m) net"
```
```   610   shows "((\<lambda>x. dist (f x) (g x)) ---> dist l m) net"
```
```   611 proof (rule tendstoI)
```
```   612   fix e :: real assume "0 < e"
```
```   613   hence e2: "0 < e/2" by simp
```
```   614   from tendstoD [OF f e2] tendstoD [OF g e2]
```
```   615   show "eventually (\<lambda>x. dist (dist (f x) (g x)) (dist l m) < e) net"
```
```   616   proof (rule eventually_elim2)
```
```   617     fix x assume "dist (f x) l < e/2" "dist (g x) m < e/2"
```
```   618     then show "dist (dist (f x) (g x)) (dist l m) < e"
```
```   619       unfolding dist_real_def
```
```   620       using dist_triangle2 [of "f x" "g x" "l"]
```
```   621       using dist_triangle2 [of "g x" "l" "m"]
```
```   622       using dist_triangle3 [of "l" "m" "f x"]
```
```   623       using dist_triangle [of "f x" "m" "g x"]
```
```   624       by arith
```
```   625   qed
```
```   626 qed
```
```   627
```
```   628 lemma norm_conv_dist: "norm x = dist x 0"
```
```   629 unfolding dist_norm by simp
```
```   630
```
```   631 lemma tendsto_norm [tendsto_intros]:
```
```   632   "(f ---> a) net \<Longrightarrow> ((\<lambda>x. norm (f x)) ---> norm a) net"
```
```   633 unfolding norm_conv_dist by (intro tendsto_intros)
```
```   634
```
```   635 lemma tendsto_norm_zero:
```
```   636   "(f ---> 0) net \<Longrightarrow> ((\<lambda>x. norm (f x)) ---> 0) net"
```
```   637 by (drule tendsto_norm, simp)
```
```   638
```
```   639 lemma tendsto_norm_zero_cancel:
```
```   640   "((\<lambda>x. norm (f x)) ---> 0) net \<Longrightarrow> (f ---> 0) net"
```
```   641 unfolding tendsto_iff dist_norm by simp
```
```   642
```
```   643 lemma tendsto_norm_zero_iff:
```
```   644   "((\<lambda>x. norm (f x)) ---> 0) net \<longleftrightarrow> (f ---> 0) net"
```
```   645 unfolding tendsto_iff dist_norm by simp
```
```   646
```
```   647 lemma add_diff_add:
```
```   648   fixes a b c d :: "'a::ab_group_add"
```
```   649   shows "(a + c) - (b + d) = (a - b) + (c - d)"
```
```   650 by simp
```
```   651
```
```   652 lemma minus_diff_minus:
```
```   653   fixes a b :: "'a::ab_group_add"
```
```   654   shows "(- a) - (- b) = - (a - b)"
```
```   655 by simp
```
```   656
```
```   657 lemma tendsto_add [tendsto_intros]:
```
```   658   fixes a b :: "'a::real_normed_vector"
```
```   659   shows "\<lbrakk>(f ---> a) net; (g ---> b) net\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x + g x) ---> a + b) net"
```
```   660 by (simp only: tendsto_Zfun_iff add_diff_add Zfun_add)
```
```   661
```
```   662 lemma tendsto_minus [tendsto_intros]:
```
```   663   fixes a :: "'a::real_normed_vector"
```
```   664   shows "(f ---> a) net \<Longrightarrow> ((\<lambda>x. - f x) ---> - a) net"
```
```   665 by (simp only: tendsto_Zfun_iff minus_diff_minus Zfun_minus)
```
```   666
```
```   667 lemma tendsto_minus_cancel:
```
```   668   fixes a :: "'a::real_normed_vector"
```
```   669   shows "((\<lambda>x. - f x) ---> - a) net \<Longrightarrow> (f ---> a) net"
```
```   670 by (drule tendsto_minus, simp)
```
```   671
```
```   672 lemma tendsto_diff [tendsto_intros]:
```
```   673   fixes a b :: "'a::real_normed_vector"
```
```   674   shows "\<lbrakk>(f ---> a) net; (g ---> b) net\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x - g x) ---> a - b) net"
```
```   675 by (simp add: diff_minus tendsto_add tendsto_minus)
```
```   676
```
```   677 lemma tendsto_setsum [tendsto_intros]:
```
```   678   fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c::real_normed_vector"
```
```   679   assumes "\<And>i. i \<in> S \<Longrightarrow> (f i ---> a i) net"
```
```   680   shows "((\<lambda>x. \<Sum>i\<in>S. f i x) ---> (\<Sum>i\<in>S. a i)) net"
```
```   681 proof (cases "finite S")
```
```   682   assume "finite S" thus ?thesis using assms
```
```   683   proof (induct set: finite)
```
```   684     case empty show ?case
```
```   685       by (simp add: tendsto_const)
```
```   686   next
```
```   687     case (insert i F) thus ?case
```
```   688       by (simp add: tendsto_add)
```
```   689   qed
```
```   690 next
```
```   691   assume "\<not> finite S" thus ?thesis
```
```   692     by (simp add: tendsto_const)
```
```   693 qed
```
```   694
```
```   695 lemma (in bounded_linear) tendsto [tendsto_intros]:
```
```   696   "(g ---> a) net \<Longrightarrow> ((\<lambda>x. f (g x)) ---> f a) net"
```
```   697 by (simp only: tendsto_Zfun_iff diff [symmetric] Zfun)
```
```   698
```
```   699 lemma (in bounded_bilinear) tendsto [tendsto_intros]:
```
```   700   "\<lbrakk>(f ---> a) net; (g ---> b) net\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x ** g x) ---> a ** b) net"
```
```   701 by (simp only: tendsto_Zfun_iff prod_diff_prod
```
```   702                Zfun_add Zfun Zfun_left Zfun_right)
```
```   703
```
```   704
```
```   705 subsection {* Continuity of Inverse *}
```
```   706
```
```   707 lemma (in bounded_bilinear) Zfun_prod_Bfun:
```
```   708   assumes f: "Zfun f net"
```
```   709   assumes g: "Bfun g net"
```
```   710   shows "Zfun (\<lambda>x. f x ** g x) net"
```
```   711 proof -
```
```   712   obtain K where K: "0 \<le> K"
```
```   713     and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"
```
```   714     using nonneg_bounded by fast
```
```   715   obtain B where B: "0 < B"
```
```   716     and norm_g: "eventually (\<lambda>x. norm (g x) \<le> B) net"
```
```   717     using g by (rule BfunE)
```
```   718   have "eventually (\<lambda>x. norm (f x ** g x) \<le> norm (f x) * (B * K)) net"
```
```   719   using norm_g proof (rule eventually_elim1)
```
```   720     fix x
```
```   721     assume *: "norm (g x) \<le> B"
```
```   722     have "norm (f x ** g x) \<le> norm (f x) * norm (g x) * K"
```
```   723       by (rule norm_le)
```
```   724     also have "\<dots> \<le> norm (f x) * B * K"
```
```   725       by (intro mult_mono' order_refl norm_g norm_ge_zero
```
```   726                 mult_nonneg_nonneg K *)
```
```   727     also have "\<dots> = norm (f x) * (B * K)"
```
```   728       by (rule mult_assoc)
```
```   729     finally show "norm (f x ** g x) \<le> norm (f x) * (B * K)" .
```
```   730   qed
```
```   731   with f show ?thesis
```
```   732     by (rule Zfun_imp_Zfun)
```
```   733 qed
```
```   734
```
```   735 lemma (in bounded_bilinear) flip:
```
```   736   "bounded_bilinear (\<lambda>x y. y ** x)"
```
```   737 apply default
```
```   738 apply (rule add_right)
```
```   739 apply (rule add_left)
```
```   740 apply (rule scaleR_right)
```
```   741 apply (rule scaleR_left)
```
```   742 apply (subst mult_commute)
```
```   743 using bounded by fast
```
```   744
```
```   745 lemma (in bounded_bilinear) Bfun_prod_Zfun:
```
```   746   assumes f: "Bfun f net"
```
```   747   assumes g: "Zfun g net"
```
```   748   shows "Zfun (\<lambda>x. f x ** g x) net"
```
```   749 using flip g f by (rule bounded_bilinear.Zfun_prod_Bfun)
```
```   750
```
```   751 lemma inverse_diff_inverse:
```
```   752   "\<lbrakk>(a::'a::division_ring) \<noteq> 0; b \<noteq> 0\<rbrakk>
```
```   753    \<Longrightarrow> inverse a - inverse b = - (inverse a * (a - b) * inverse b)"
```
```   754 by (simp add: algebra_simps)
```
```   755
```
```   756 lemma Bfun_inverse_lemma:
```
```   757   fixes x :: "'a::real_normed_div_algebra"
```
```   758   shows "\<lbrakk>r \<le> norm x; 0 < r\<rbrakk> \<Longrightarrow> norm (inverse x) \<le> inverse r"
```
```   759 apply (subst nonzero_norm_inverse, clarsimp)
```
```   760 apply (erule (1) le_imp_inverse_le)
```
```   761 done
```
```   762
```
```   763 lemma Bfun_inverse:
```
```   764   fixes a :: "'a::real_normed_div_algebra"
```
```   765   assumes f: "(f ---> a) net"
```
```   766   assumes a: "a \<noteq> 0"
```
```   767   shows "Bfun (\<lambda>x. inverse (f x)) net"
```
```   768 proof -
```
```   769   from a have "0 < norm a" by simp
```
```   770   hence "\<exists>r>0. r < norm a" by (rule dense)
```
```   771   then obtain r where r1: "0 < r" and r2: "r < norm a" by fast
```
```   772   have "eventually (\<lambda>x. dist (f x) a < r) net"
```
```   773     using tendstoD [OF f r1] by fast
```
```   774   hence "eventually (\<lambda>x. norm (inverse (f x)) \<le> inverse (norm a - r)) net"
```
```   775   proof (rule eventually_elim1)
```
```   776     fix x
```
```   777     assume "dist (f x) a < r"
```
```   778     hence 1: "norm (f x - a) < r"
```
```   779       by (simp add: dist_norm)
```
```   780     hence 2: "f x \<noteq> 0" using r2 by auto
```
```   781     hence "norm (inverse (f x)) = inverse (norm (f x))"
```
```   782       by (rule nonzero_norm_inverse)
```
```   783     also have "\<dots> \<le> inverse (norm a - r)"
```
```   784     proof (rule le_imp_inverse_le)
```
```   785       show "0 < norm a - r" using r2 by simp
```
```   786     next
```
```   787       have "norm a - norm (f x) \<le> norm (a - f x)"
```
```   788         by (rule norm_triangle_ineq2)
```
```   789       also have "\<dots> = norm (f x - a)"
```
```   790         by (rule norm_minus_commute)
```
```   791       also have "\<dots> < r" using 1 .
```
```   792       finally show "norm a - r \<le> norm (f x)" by simp
```
```   793     qed
```
```   794     finally show "norm (inverse (f x)) \<le> inverse (norm a - r)" .
```
```   795   qed
```
```   796   thus ?thesis by (rule BfunI)
```
```   797 qed
```
```   798
```
```   799 lemma tendsto_inverse_lemma:
```
```   800   fixes a :: "'a::real_normed_div_algebra"
```
```   801   shows "\<lbrakk>(f ---> a) net; a \<noteq> 0; eventually (\<lambda>x. f x \<noteq> 0) net\<rbrakk>
```
```   802          \<Longrightarrow> ((\<lambda>x. inverse (f x)) ---> inverse a) net"
```
```   803 apply (subst tendsto_Zfun_iff)
```
```   804 apply (rule Zfun_ssubst)
```
```   805 apply (erule eventually_elim1)
```
```   806 apply (erule (1) inverse_diff_inverse)
```
```   807 apply (rule Zfun_minus)
```
```   808 apply (rule Zfun_mult_left)
```
```   809 apply (rule mult.Bfun_prod_Zfun)
```
```   810 apply (erule (1) Bfun_inverse)
```
```   811 apply (simp add: tendsto_Zfun_iff)
```
```   812 done
```
```   813
```
```   814 lemma tendsto_inverse [tendsto_intros]:
```
```   815   fixes a :: "'a::real_normed_div_algebra"
```
```   816   assumes f: "(f ---> a) net"
```
```   817   assumes a: "a \<noteq> 0"
```
```   818   shows "((\<lambda>x. inverse (f x)) ---> inverse a) net"
```
```   819 proof -
```
```   820   from a have "0 < norm a" by simp
```
```   821   with f have "eventually (\<lambda>x. dist (f x) a < norm a) net"
```
```   822     by (rule tendstoD)
```
```   823   then have "eventually (\<lambda>x. f x \<noteq> 0) net"
```
```   824     unfolding dist_norm by (auto elim!: eventually_elim1)
```
```   825   with f a show ?thesis
```
```   826     by (rule tendsto_inverse_lemma)
```
```   827 qed
```
```   828
```
```   829 lemma tendsto_divide [tendsto_intros]:
```
```   830   fixes a b :: "'a::real_normed_field"
```
```   831   shows "\<lbrakk>(f ---> a) net; (g ---> b) net; b \<noteq> 0\<rbrakk>
```
```   832     \<Longrightarrow> ((\<lambda>x. f x / g x) ---> a / b) net"
```
```   833 by (simp add: mult.tendsto tendsto_inverse divide_inverse)
```
```   834
```
```   835 lemma tendsto_unique:
```
```   836   fixes f :: "'a \<Rightarrow> 'b::t2_space"
```
```   837   assumes "\<not> trivial_limit net"  "(f ---> l) net"  "(f ---> l') net"
```
```   838   shows "l = l'"
```
```   839 proof (rule ccontr)
```
```   840   assume "l \<noteq> l'"
```
```   841   obtain U V where "open U" "open V" "l \<in> U" "l' \<in> V" "U \<inter> V = {}"
```
```   842     using hausdorff [OF `l \<noteq> l'`] by fast
```
```   843   have "eventually (\<lambda>x. f x \<in> U) net"
```
```   844     using `(f ---> l) net` `open U` `l \<in> U` by (rule topological_tendstoD)
```
```   845   moreover
```
```   846   have "eventually (\<lambda>x. f x \<in> V) net"
```
```   847     using `(f ---> l') net` `open V` `l' \<in> V` by (rule topological_tendstoD)
```
```   848   ultimately
```
```   849   have "eventually (\<lambda>x. False) net"
```
```   850   proof (rule eventually_elim2)
```
```   851     fix x
```
```   852     assume "f x \<in> U" "f x \<in> V"
```
```   853     hence "f x \<in> U \<inter> V" by simp
```
```   854     with `U \<inter> V = {}` show "False" by simp
```
```   855   qed
```
```   856   with `\<not> trivial_limit net` show "False"
```
```   857     by (simp add: trivial_limit_def)
```
```   858 qed
```
```   859
```
```   860 end
```