src/HOL/Limits.thy
 author bulwahn Wed Jan 20 11:56:45 2010 +0100 (2010-01-20) changeset 34948 2d5f2a9f7601 parent 31902 862ae16a799d child 36358 246493d61204 permissions -rw-r--r--
```     1 (*  Title       : Limits.thy
```
```     2     Author      : Brian Huffman
```
```     3 *)
```
```     4
```
```     5 header {* Filters and Limits *}
```
```     6
```
```     7 theory Limits
```
```     8 imports RealVector RComplete
```
```     9 begin
```
```    10
```
```    11 subsection {* Nets *}
```
```    12
```
```    13 text {*
```
```    14   A net is now defined as a filter base.
```
```    15   The definition also allows non-proper filter bases.
```
```    16 *}
```
```    17
```
```    18 typedef (open) 'a net =
```
```    19   "{net :: 'a set set. (\<exists>A. A \<in> net)
```
```    20     \<and> (\<forall>A\<in>net. \<forall>B\<in>net. \<exists>C\<in>net. C \<subseteq> A \<and> C \<subseteq> B)}"
```
```    21 proof
```
```    22   show "UNIV \<in> ?net" by auto
```
```    23 qed
```
```    24
```
```    25 lemma Rep_net_nonempty: "\<exists>A. A \<in> Rep_net net"
```
```    26 using Rep_net [of net] by simp
```
```    27
```
```    28 lemma Rep_net_directed:
```
```    29   "A \<in> Rep_net net \<Longrightarrow> B \<in> Rep_net net \<Longrightarrow> \<exists>C\<in>Rep_net net. C \<subseteq> A \<and> C \<subseteq> B"
```
```    30 using Rep_net [of net] by simp
```
```    31
```
```    32 lemma Abs_net_inverse':
```
```    33   assumes "\<exists>A. A \<in> net"
```
```    34   assumes "\<And>A B. A \<in> net \<Longrightarrow> B \<in> net \<Longrightarrow> \<exists>C\<in>net. C \<subseteq> A \<and> C \<subseteq> B"
```
```    35   shows "Rep_net (Abs_net net) = net"
```
```    36 using assms by (simp add: Abs_net_inverse)
```
```    37
```
```    38 lemma image_nonempty: "\<exists>x. x \<in> A \<Longrightarrow> \<exists>x. x \<in> f ` A"
```
```    39 by auto
```
```    40
```
```    41
```
```    42 subsection {* Eventually *}
```
```    43
```
```    44 definition
```
```    45   eventually :: "('a \<Rightarrow> bool) \<Rightarrow> 'a net \<Rightarrow> bool" where
```
```    46   [code del]: "eventually P net \<longleftrightarrow> (\<exists>A\<in>Rep_net net. \<forall>x\<in>A. P x)"
```
```    47
```
```    48 lemma eventually_True [simp]: "eventually (\<lambda>x. True) net"
```
```    49 unfolding eventually_def using Rep_net_nonempty [of net] by fast
```
```    50
```
```    51 lemma eventually_mono:
```
```    52   "(\<forall>x. P x \<longrightarrow> Q x) \<Longrightarrow> eventually P net \<Longrightarrow> eventually Q net"
```
```    53 unfolding eventually_def by blast
```
```    54
```
```    55 lemma eventually_conj:
```
```    56   assumes P: "eventually (\<lambda>x. P x) net"
```
```    57   assumes Q: "eventually (\<lambda>x. Q x) net"
```
```    58   shows "eventually (\<lambda>x. P x \<and> Q x) net"
```
```    59 proof -
```
```    60   obtain A where A: "A \<in> Rep_net net" "\<forall>x\<in>A. P x"
```
```    61     using P unfolding eventually_def by fast
```
```    62   obtain B where B: "B \<in> Rep_net net" "\<forall>x\<in>B. Q x"
```
```    63     using Q unfolding eventually_def by fast
```
```    64   obtain C where C: "C \<in> Rep_net net" "C \<subseteq> A" "C \<subseteq> B"
```
```    65     using Rep_net_directed [OF A(1) B(1)] by fast
```
```    66   then have "\<forall>x\<in>C. P x \<and> Q x" "C \<in> Rep_net net"
```
```    67     using A(2) B(2) by auto
```
```    68   then show ?thesis unfolding eventually_def ..
```
```    69 qed
```
```    70
```
```    71 lemma eventually_mp:
```
```    72   assumes "eventually (\<lambda>x. P x \<longrightarrow> Q x) net"
```
```    73   assumes "eventually (\<lambda>x. P x) net"
```
```    74   shows "eventually (\<lambda>x. Q x) net"
```
```    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) net"
```
```    78     using assms by (rule eventually_conj)
```
```    79 qed
```
```    80
```
```    81 lemma eventually_rev_mp:
```
```    82   assumes "eventually (\<lambda>x. P x) net"
```
```    83   assumes "eventually (\<lambda>x. P x \<longrightarrow> Q x) net"
```
```    84   shows "eventually (\<lambda>x. Q x) net"
```
```    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) net \<longleftrightarrow> eventually P net \<and> eventually Q net"
```
```    89 by (auto intro: eventually_conj elim: eventually_rev_mp)
```
```    90
```
```    91 lemma eventually_elim1:
```
```    92   assumes "eventually (\<lambda>i. P i) net"
```
```    93   assumes "\<And>i. P i \<Longrightarrow> Q i"
```
```    94   shows "eventually (\<lambda>i. Q i) net"
```
```    95 using assms by (auto elim!: eventually_rev_mp)
```
```    96
```
```    97 lemma eventually_elim2:
```
```    98   assumes "eventually (\<lambda>i. P i) net"
```
```    99   assumes "eventually (\<lambda>i. Q i) net"
```
```   100   assumes "\<And>i. P i \<Longrightarrow> Q i \<Longrightarrow> R i"
```
```   101   shows "eventually (\<lambda>i. R i) net"
```
```   102 using assms by (auto elim!: eventually_rev_mp)
```
```   103
```
```   104
```
```   105 subsection {* Standard Nets *}
```
```   106
```
```   107 definition
```
```   108   sequentially :: "nat net" where
```
```   109   [code del]: "sequentially = Abs_net (range (\<lambda>n. {n..}))"
```
```   110
```
```   111 definition
```
```   112   within :: "'a net \<Rightarrow> 'a set \<Rightarrow> 'a net" (infixr "within" 70) where
```
```   113   [code del]: "net within S = Abs_net ((\<lambda>A. A \<inter> S) ` Rep_net net)"
```
```   114
```
```   115 definition
```
```   116   at :: "'a::topological_space \<Rightarrow> 'a net" where
```
```   117   [code del]: "at a = Abs_net ((\<lambda>S. S - {a}) ` {S. open S \<and> a \<in> S})"
```
```   118
```
```   119 lemma Rep_net_sequentially:
```
```   120   "Rep_net sequentially = range (\<lambda>n. {n..})"
```
```   121 unfolding sequentially_def
```
```   122 apply (rule Abs_net_inverse')
```
```   123 apply (rule image_nonempty, simp)
```
```   124 apply (clarsimp, rename_tac m n)
```
```   125 apply (rule_tac x="max m n" in exI, auto)
```
```   126 done
```
```   127
```
```   128 lemma Rep_net_within:
```
```   129   "Rep_net (net within S) = (\<lambda>A. A \<inter> S) ` Rep_net net"
```
```   130 unfolding within_def
```
```   131 apply (rule Abs_net_inverse')
```
```   132 apply (rule image_nonempty, rule Rep_net_nonempty)
```
```   133 apply (clarsimp, rename_tac A B)
```
```   134 apply (drule (1) Rep_net_directed)
```
```   135 apply (clarify, rule_tac x=C in bexI, auto)
```
```   136 done
```
```   137
```
```   138 lemma Rep_net_at:
```
```   139   "Rep_net (at a) = ((\<lambda>S. S - {a}) ` {S. open S \<and> a \<in> S})"
```
```   140 unfolding at_def
```
```   141 apply (rule Abs_net_inverse')
```
```   142 apply (rule image_nonempty)
```
```   143 apply (rule_tac x="UNIV" in exI, simp)
```
```   144 apply (clarsimp, rename_tac S T)
```
```   145 apply (rule_tac x="S \<inter> T" in exI, auto simp add: open_Int)
```
```   146 done
```
```   147
```
```   148 lemma eventually_sequentially:
```
```   149   "eventually P sequentially \<longleftrightarrow> (\<exists>N. \<forall>n\<ge>N. P n)"
```
```   150 unfolding eventually_def Rep_net_sequentially by auto
```
```   151
```
```   152 lemma eventually_within:
```
```   153   "eventually P (net within S) = eventually (\<lambda>x. x \<in> S \<longrightarrow> P x) net"
```
```   154 unfolding eventually_def Rep_net_within by auto
```
```   155
```
```   156 lemma eventually_at_topological:
```
```   157   "eventually P (at a) \<longleftrightarrow> (\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. x \<noteq> a \<longrightarrow> P x))"
```
```   158 unfolding eventually_def Rep_net_at by auto
```
```   159
```
```   160 lemma eventually_at:
```
```   161   fixes a :: "'a::metric_space"
```
```   162   shows "eventually P (at a) \<longleftrightarrow> (\<exists>d>0. \<forall>x. x \<noteq> a \<and> dist x a < d \<longrightarrow> P x)"
```
```   163 unfolding eventually_at_topological open_dist
```
```   164 apply safe
```
```   165 apply fast
```
```   166 apply (rule_tac x="{x. dist x a < d}" in exI, simp)
```
```   167 apply clarsimp
```
```   168 apply (rule_tac x="d - dist x a" in exI, clarsimp)
```
```   169 apply (simp only: less_diff_eq)
```
```   170 apply (erule le_less_trans [OF dist_triangle])
```
```   171 done
```
```   172
```
```   173
```
```   174 subsection {* Boundedness *}
```
```   175
```
```   176 definition
```
```   177   Bfun :: "('a \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a net \<Rightarrow> bool" where
```
```   178   [code del]: "Bfun f net = (\<exists>K>0. eventually (\<lambda>x. norm (f x) \<le> K) net)"
```
```   179
```
```   180 lemma BfunI:
```
```   181   assumes K: "eventually (\<lambda>x. norm (f x) \<le> K) net" shows "Bfun f net"
```
```   182 unfolding Bfun_def
```
```   183 proof (intro exI conjI allI)
```
```   184   show "0 < max K 1" by simp
```
```   185 next
```
```   186   show "eventually (\<lambda>x. norm (f x) \<le> max K 1) net"
```
```   187     using K by (rule eventually_elim1, simp)
```
```   188 qed
```
```   189
```
```   190 lemma BfunE:
```
```   191   assumes "Bfun f net"
```
```   192   obtains B where "0 < B" and "eventually (\<lambda>x. norm (f x) \<le> B) net"
```
```   193 using assms unfolding Bfun_def by fast
```
```   194
```
```   195
```
```   196 subsection {* Convergence to Zero *}
```
```   197
```
```   198 definition
```
```   199   Zfun :: "('a \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a net \<Rightarrow> bool" where
```
```   200   [code del]: "Zfun f net = (\<forall>r>0. eventually (\<lambda>x. norm (f x) < r) net)"
```
```   201
```
```   202 lemma ZfunI:
```
```   203   "(\<And>r. 0 < r \<Longrightarrow> eventually (\<lambda>x. norm (f x) < r) net) \<Longrightarrow> Zfun f net"
```
```   204 unfolding Zfun_def by simp
```
```   205
```
```   206 lemma ZfunD:
```
```   207   "\<lbrakk>Zfun f net; 0 < r\<rbrakk> \<Longrightarrow> eventually (\<lambda>x. norm (f x) < r) net"
```
```   208 unfolding Zfun_def by simp
```
```   209
```
```   210 lemma Zfun_ssubst:
```
```   211   "eventually (\<lambda>x. f x = g x) net \<Longrightarrow> Zfun g net \<Longrightarrow> Zfun f net"
```
```   212 unfolding Zfun_def by (auto elim!: eventually_rev_mp)
```
```   213
```
```   214 lemma Zfun_zero: "Zfun (\<lambda>x. 0) net"
```
```   215 unfolding Zfun_def by simp
```
```   216
```
```   217 lemma Zfun_norm_iff: "Zfun (\<lambda>x. norm (f x)) net = Zfun (\<lambda>x. f x) net"
```
```   218 unfolding Zfun_def by simp
```
```   219
```
```   220 lemma Zfun_imp_Zfun:
```
```   221   assumes f: "Zfun f net"
```
```   222   assumes g: "eventually (\<lambda>x. norm (g x) \<le> norm (f x) * K) net"
```
```   223   shows "Zfun (\<lambda>x. g x) net"
```
```   224 proof (cases)
```
```   225   assume K: "0 < K"
```
```   226   show ?thesis
```
```   227   proof (rule ZfunI)
```
```   228     fix r::real assume "0 < r"
```
```   229     hence "0 < r / K"
```
```   230       using K by (rule divide_pos_pos)
```
```   231     then have "eventually (\<lambda>x. norm (f x) < r / K) net"
```
```   232       using ZfunD [OF f] by fast
```
```   233     with g show "eventually (\<lambda>x. norm (g x) < r) net"
```
```   234     proof (rule eventually_elim2)
```
```   235       fix x
```
```   236       assume *: "norm (g x) \<le> norm (f x) * K"
```
```   237       assume "norm (f x) < r / K"
```
```   238       hence "norm (f x) * K < r"
```
```   239         by (simp add: pos_less_divide_eq K)
```
```   240       thus "norm (g x) < r"
```
```   241         by (simp add: order_le_less_trans [OF *])
```
```   242     qed
```
```   243   qed
```
```   244 next
```
```   245   assume "\<not> 0 < K"
```
```   246   hence K: "K \<le> 0" by (simp only: not_less)
```
```   247   show ?thesis
```
```   248   proof (rule ZfunI)
```
```   249     fix r :: real
```
```   250     assume "0 < r"
```
```   251     from g show "eventually (\<lambda>x. norm (g x) < r) net"
```
```   252     proof (rule eventually_elim1)
```
```   253       fix x
```
```   254       assume "norm (g x) \<le> norm (f x) * K"
```
```   255       also have "\<dots> \<le> norm (f x) * 0"
```
```   256         using K norm_ge_zero by (rule mult_left_mono)
```
```   257       finally show "norm (g x) < r"
```
```   258         using `0 < r` by simp
```
```   259     qed
```
```   260   qed
```
```   261 qed
```
```   262
```
```   263 lemma Zfun_le: "\<lbrakk>Zfun g net; \<forall>x. norm (f x) \<le> norm (g x)\<rbrakk> \<Longrightarrow> Zfun f net"
```
```   264 by (erule_tac K="1" in Zfun_imp_Zfun, simp)
```
```   265
```
```   266 lemma Zfun_add:
```
```   267   assumes f: "Zfun f net" and g: "Zfun g net"
```
```   268   shows "Zfun (\<lambda>x. f x + g x) net"
```
```   269 proof (rule ZfunI)
```
```   270   fix r::real assume "0 < r"
```
```   271   hence r: "0 < r / 2" by simp
```
```   272   have "eventually (\<lambda>x. norm (f x) < r/2) net"
```
```   273     using f r by (rule ZfunD)
```
```   274   moreover
```
```   275   have "eventually (\<lambda>x. norm (g x) < r/2) net"
```
```   276     using g r by (rule ZfunD)
```
```   277   ultimately
```
```   278   show "eventually (\<lambda>x. norm (f x + g x) < r) net"
```
```   279   proof (rule eventually_elim2)
```
```   280     fix x
```
```   281     assume *: "norm (f x) < r/2" "norm (g x) < r/2"
```
```   282     have "norm (f x + g x) \<le> norm (f x) + norm (g x)"
```
```   283       by (rule norm_triangle_ineq)
```
```   284     also have "\<dots> < r/2 + r/2"
```
```   285       using * by (rule add_strict_mono)
```
```   286     finally show "norm (f x + g x) < r"
```
```   287       by simp
```
```   288   qed
```
```   289 qed
```
```   290
```
```   291 lemma Zfun_minus: "Zfun f net \<Longrightarrow> Zfun (\<lambda>x. - f x) net"
```
```   292 unfolding Zfun_def by simp
```
```   293
```
```   294 lemma Zfun_diff: "\<lbrakk>Zfun f net; Zfun g net\<rbrakk> \<Longrightarrow> Zfun (\<lambda>x. f x - g x) net"
```
```   295 by (simp only: diff_minus Zfun_add Zfun_minus)
```
```   296
```
```   297 lemma (in bounded_linear) Zfun:
```
```   298   assumes g: "Zfun g net"
```
```   299   shows "Zfun (\<lambda>x. f (g x)) net"
```
```   300 proof -
```
```   301   obtain K where "\<And>x. norm (f x) \<le> norm x * K"
```
```   302     using bounded by fast
```
```   303   then have "eventually (\<lambda>x. norm (f (g x)) \<le> norm (g x) * K) net"
```
```   304     by simp
```
```   305   with g show ?thesis
```
```   306     by (rule Zfun_imp_Zfun)
```
```   307 qed
```
```   308
```
```   309 lemma (in bounded_bilinear) Zfun:
```
```   310   assumes f: "Zfun f net"
```
```   311   assumes g: "Zfun g net"
```
```   312   shows "Zfun (\<lambda>x. f x ** g x) net"
```
```   313 proof (rule ZfunI)
```
```   314   fix r::real assume r: "0 < r"
```
```   315   obtain K where K: "0 < K"
```
```   316     and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"
```
```   317     using pos_bounded by fast
```
```   318   from K have K': "0 < inverse K"
```
```   319     by (rule positive_imp_inverse_positive)
```
```   320   have "eventually (\<lambda>x. norm (f x) < r) net"
```
```   321     using f r by (rule ZfunD)
```
```   322   moreover
```
```   323   have "eventually (\<lambda>x. norm (g x) < inverse K) net"
```
```   324     using g K' by (rule ZfunD)
```
```   325   ultimately
```
```   326   show "eventually (\<lambda>x. norm (f x ** g x) < r) net"
```
```   327   proof (rule eventually_elim2)
```
```   328     fix x
```
```   329     assume *: "norm (f x) < r" "norm (g x) < inverse K"
```
```   330     have "norm (f x ** g x) \<le> norm (f x) * norm (g x) * K"
```
```   331       by (rule norm_le)
```
```   332     also have "norm (f x) * norm (g x) * K < r * inverse K * K"
```
```   333       by (intro mult_strict_right_mono mult_strict_mono' norm_ge_zero * K)
```
```   334     also from K have "r * inverse K * K = r"
```
```   335       by simp
```
```   336     finally show "norm (f x ** g x) < r" .
```
```   337   qed
```
```   338 qed
```
```   339
```
```   340 lemma (in bounded_bilinear) Zfun_left:
```
```   341   "Zfun f net \<Longrightarrow> Zfun (\<lambda>x. f x ** a) net"
```
```   342 by (rule bounded_linear_left [THEN bounded_linear.Zfun])
```
```   343
```
```   344 lemma (in bounded_bilinear) Zfun_right:
```
```   345   "Zfun f net \<Longrightarrow> Zfun (\<lambda>x. a ** f x) net"
```
```   346 by (rule bounded_linear_right [THEN bounded_linear.Zfun])
```
```   347
```
```   348 lemmas Zfun_mult = mult.Zfun
```
```   349 lemmas Zfun_mult_right = mult.Zfun_right
```
```   350 lemmas Zfun_mult_left = mult.Zfun_left
```
```   351
```
```   352
```
```   353 subsection {* Limits *}
```
```   354
```
```   355 definition
```
```   356   tendsto :: "('a \<Rightarrow> 'b::topological_space) \<Rightarrow> 'b \<Rightarrow> 'a net \<Rightarrow> bool"
```
```   357     (infixr "--->" 55)
```
```   358 where [code del]:
```
```   359   "(f ---> l) net \<longleftrightarrow> (\<forall>S. open S \<longrightarrow> l \<in> S \<longrightarrow> eventually (\<lambda>x. f x \<in> S) net)"
```
```   360
```
```   361 ML {*
```
```   362 structure Tendsto_Intros = Named_Thms
```
```   363 (
```
```   364   val name = "tendsto_intros"
```
```   365   val description = "introduction rules for tendsto"
```
```   366 )
```
```   367 *}
```
```   368
```
```   369 setup Tendsto_Intros.setup
```
```   370
```
```   371 lemma topological_tendstoI:
```
```   372   "(\<And>S. open S \<Longrightarrow> l \<in> S \<Longrightarrow> eventually (\<lambda>x. f x \<in> S) net)
```
```   373     \<Longrightarrow> (f ---> l) net"
```
```   374   unfolding tendsto_def by auto
```
```   375
```
```   376 lemma topological_tendstoD:
```
```   377   "(f ---> l) net \<Longrightarrow> open S \<Longrightarrow> l \<in> S \<Longrightarrow> eventually (\<lambda>x. f x \<in> S) net"
```
```   378   unfolding tendsto_def by auto
```
```   379
```
```   380 lemma tendstoI:
```
```   381   assumes "\<And>e. 0 < e \<Longrightarrow> eventually (\<lambda>x. dist (f x) l < e) net"
```
```   382   shows "(f ---> l) net"
```
```   383 apply (rule topological_tendstoI)
```
```   384 apply (simp add: open_dist)
```
```   385 apply (drule (1) bspec, clarify)
```
```   386 apply (drule assms)
```
```   387 apply (erule eventually_elim1, simp)
```
```   388 done
```
```   389
```
```   390 lemma tendstoD:
```
```   391   "(f ---> l) net \<Longrightarrow> 0 < e \<Longrightarrow> eventually (\<lambda>x. dist (f x) l < e) net"
```
```   392 apply (drule_tac S="{x. dist x l < e}" in topological_tendstoD)
```
```   393 apply (clarsimp simp add: open_dist)
```
```   394 apply (rule_tac x="e - dist x l" in exI, clarsimp)
```
```   395 apply (simp only: less_diff_eq)
```
```   396 apply (erule le_less_trans [OF dist_triangle])
```
```   397 apply simp
```
```   398 apply simp
```
```   399 done
```
```   400
```
```   401 lemma tendsto_iff:
```
```   402   "(f ---> l) net \<longleftrightarrow> (\<forall>e>0. eventually (\<lambda>x. dist (f x) l < e) net)"
```
```   403 using tendstoI tendstoD by fast
```
```   404
```
```   405 lemma tendsto_Zfun_iff: "(f ---> a) net = Zfun (\<lambda>x. f x - a) net"
```
```   406 by (simp only: tendsto_iff Zfun_def dist_norm)
```
```   407
```
```   408 lemma tendsto_ident_at [tendsto_intros]: "((\<lambda>x. x) ---> a) (at a)"
```
```   409 unfolding tendsto_def eventually_at_topological by auto
```
```   410
```
```   411 lemma tendsto_ident_at_within [tendsto_intros]:
```
```   412   "a \<in> S \<Longrightarrow> ((\<lambda>x. x) ---> a) (at a within S)"
```
```   413 unfolding tendsto_def eventually_within eventually_at_topological by auto
```
```   414
```
```   415 lemma tendsto_const [tendsto_intros]: "((\<lambda>x. k) ---> k) net"
```
```   416 by (simp add: tendsto_def)
```
```   417
```
```   418 lemma tendsto_dist [tendsto_intros]:
```
```   419   assumes f: "(f ---> l) net" and g: "(g ---> m) net"
```
```   420   shows "((\<lambda>x. dist (f x) (g x)) ---> dist l m) net"
```
```   421 proof (rule tendstoI)
```
```   422   fix e :: real assume "0 < e"
```
```   423   hence e2: "0 < e/2" by simp
```
```   424   from tendstoD [OF f e2] tendstoD [OF g e2]
```
```   425   show "eventually (\<lambda>x. dist (dist (f x) (g x)) (dist l m) < e) net"
```
```   426   proof (rule eventually_elim2)
```
```   427     fix x assume "dist (f x) l < e/2" "dist (g x) m < e/2"
```
```   428     then show "dist (dist (f x) (g x)) (dist l m) < e"
```
```   429       unfolding dist_real_def
```
```   430       using dist_triangle2 [of "f x" "g x" "l"]
```
```   431       using dist_triangle2 [of "g x" "l" "m"]
```
```   432       using dist_triangle3 [of "l" "m" "f x"]
```
```   433       using dist_triangle [of "f x" "m" "g x"]
```
```   434       by arith
```
```   435   qed
```
```   436 qed
```
```   437
```
```   438 lemma tendsto_norm [tendsto_intros]:
```
```   439   "(f ---> a) net \<Longrightarrow> ((\<lambda>x. norm (f x)) ---> norm a) net"
```
```   440 apply (simp add: tendsto_iff dist_norm, safe)
```
```   441 apply (drule_tac x="e" in spec, safe)
```
```   442 apply (erule eventually_elim1)
```
```   443 apply (erule order_le_less_trans [OF norm_triangle_ineq3])
```
```   444 done
```
```   445
```
```   446 lemma add_diff_add:
```
```   447   fixes a b c d :: "'a::ab_group_add"
```
```   448   shows "(a + c) - (b + d) = (a - b) + (c - d)"
```
```   449 by simp
```
```   450
```
```   451 lemma minus_diff_minus:
```
```   452   fixes a b :: "'a::ab_group_add"
```
```   453   shows "(- a) - (- b) = - (a - b)"
```
```   454 by simp
```
```   455
```
```   456 lemma tendsto_add [tendsto_intros]:
```
```   457   fixes a b :: "'a::real_normed_vector"
```
```   458   shows "\<lbrakk>(f ---> a) net; (g ---> b) net\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x + g x) ---> a + b) net"
```
```   459 by (simp only: tendsto_Zfun_iff add_diff_add Zfun_add)
```
```   460
```
```   461 lemma tendsto_minus [tendsto_intros]:
```
```   462   fixes a :: "'a::real_normed_vector"
```
```   463   shows "(f ---> a) net \<Longrightarrow> ((\<lambda>x. - f x) ---> - a) net"
```
```   464 by (simp only: tendsto_Zfun_iff minus_diff_minus Zfun_minus)
```
```   465
```
```   466 lemma tendsto_minus_cancel:
```
```   467   fixes a :: "'a::real_normed_vector"
```
```   468   shows "((\<lambda>x. - f x) ---> - a) net \<Longrightarrow> (f ---> a) net"
```
```   469 by (drule tendsto_minus, simp)
```
```   470
```
```   471 lemma tendsto_diff [tendsto_intros]:
```
```   472   fixes a b :: "'a::real_normed_vector"
```
```   473   shows "\<lbrakk>(f ---> a) net; (g ---> b) net\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x - g x) ---> a - b) net"
```
```   474 by (simp add: diff_minus tendsto_add tendsto_minus)
```
```   475
```
```   476 lemma tendsto_setsum [tendsto_intros]:
```
```   477   fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c::real_normed_vector"
```
```   478   assumes "\<And>i. i \<in> S \<Longrightarrow> (f i ---> a i) net"
```
```   479   shows "((\<lambda>x. \<Sum>i\<in>S. f i x) ---> (\<Sum>i\<in>S. a i)) net"
```
```   480 proof (cases "finite S")
```
```   481   assume "finite S" thus ?thesis using assms
```
```   482   proof (induct set: finite)
```
```   483     case empty show ?case
```
```   484       by (simp add: tendsto_const)
```
```   485   next
```
```   486     case (insert i F) thus ?case
```
```   487       by (simp add: tendsto_add)
```
```   488   qed
```
```   489 next
```
```   490   assume "\<not> finite S" thus ?thesis
```
```   491     by (simp add: tendsto_const)
```
```   492 qed
```
```   493
```
```   494 lemma (in bounded_linear) tendsto [tendsto_intros]:
```
```   495   "(g ---> a) net \<Longrightarrow> ((\<lambda>x. f (g x)) ---> f a) net"
```
```   496 by (simp only: tendsto_Zfun_iff diff [symmetric] Zfun)
```
```   497
```
```   498 lemma (in bounded_bilinear) tendsto [tendsto_intros]:
```
```   499   "\<lbrakk>(f ---> a) net; (g ---> b) net\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x ** g x) ---> a ** b) net"
```
```   500 by (simp only: tendsto_Zfun_iff prod_diff_prod
```
```   501                Zfun_add Zfun Zfun_left Zfun_right)
```
```   502
```
```   503
```
```   504 subsection {* Continuity of Inverse *}
```
```   505
```
```   506 lemma (in bounded_bilinear) Zfun_prod_Bfun:
```
```   507   assumes f: "Zfun f net"
```
```   508   assumes g: "Bfun g net"
```
```   509   shows "Zfun (\<lambda>x. f x ** g x) net"
```
```   510 proof -
```
```   511   obtain K where K: "0 \<le> K"
```
```   512     and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"
```
```   513     using nonneg_bounded by fast
```
```   514   obtain B where B: "0 < B"
```
```   515     and norm_g: "eventually (\<lambda>x. norm (g x) \<le> B) net"
```
```   516     using g by (rule BfunE)
```
```   517   have "eventually (\<lambda>x. norm (f x ** g x) \<le> norm (f x) * (B * K)) net"
```
```   518   using norm_g proof (rule eventually_elim1)
```
```   519     fix x
```
```   520     assume *: "norm (g x) \<le> B"
```
```   521     have "norm (f x ** g x) \<le> norm (f x) * norm (g x) * K"
```
```   522       by (rule norm_le)
```
```   523     also have "\<dots> \<le> norm (f x) * B * K"
```
```   524       by (intro mult_mono' order_refl norm_g norm_ge_zero
```
```   525                 mult_nonneg_nonneg K *)
```
```   526     also have "\<dots> = norm (f x) * (B * K)"
```
```   527       by (rule mult_assoc)
```
```   528     finally show "norm (f x ** g x) \<le> norm (f x) * (B * K)" .
```
```   529   qed
```
```   530   with f show ?thesis
```
```   531     by (rule Zfun_imp_Zfun)
```
```   532 qed
```
```   533
```
```   534 lemma (in bounded_bilinear) flip:
```
```   535   "bounded_bilinear (\<lambda>x y. y ** x)"
```
```   536 apply default
```
```   537 apply (rule add_right)
```
```   538 apply (rule add_left)
```
```   539 apply (rule scaleR_right)
```
```   540 apply (rule scaleR_left)
```
```   541 apply (subst mult_commute)
```
```   542 using bounded by fast
```
```   543
```
```   544 lemma (in bounded_bilinear) Bfun_prod_Zfun:
```
```   545   assumes f: "Bfun f net"
```
```   546   assumes g: "Zfun g net"
```
```   547   shows "Zfun (\<lambda>x. f x ** g x) net"
```
```   548 using flip g f by (rule bounded_bilinear.Zfun_prod_Bfun)
```
```   549
```
```   550 lemma inverse_diff_inverse:
```
```   551   "\<lbrakk>(a::'a::division_ring) \<noteq> 0; b \<noteq> 0\<rbrakk>
```
```   552    \<Longrightarrow> inverse a - inverse b = - (inverse a * (a - b) * inverse b)"
```
```   553 by (simp add: algebra_simps)
```
```   554
```
```   555 lemma Bfun_inverse_lemma:
```
```   556   fixes x :: "'a::real_normed_div_algebra"
```
```   557   shows "\<lbrakk>r \<le> norm x; 0 < r\<rbrakk> \<Longrightarrow> norm (inverse x) \<le> inverse r"
```
```   558 apply (subst nonzero_norm_inverse, clarsimp)
```
```   559 apply (erule (1) le_imp_inverse_le)
```
```   560 done
```
```   561
```
```   562 lemma Bfun_inverse:
```
```   563   fixes a :: "'a::real_normed_div_algebra"
```
```   564   assumes f: "(f ---> a) net"
```
```   565   assumes a: "a \<noteq> 0"
```
```   566   shows "Bfun (\<lambda>x. inverse (f x)) net"
```
```   567 proof -
```
```   568   from a have "0 < norm a" by simp
```
```   569   hence "\<exists>r>0. r < norm a" by (rule dense)
```
```   570   then obtain r where r1: "0 < r" and r2: "r < norm a" by fast
```
```   571   have "eventually (\<lambda>x. dist (f x) a < r) net"
```
```   572     using tendstoD [OF f r1] by fast
```
```   573   hence "eventually (\<lambda>x. norm (inverse (f x)) \<le> inverse (norm a - r)) net"
```
```   574   proof (rule eventually_elim1)
```
```   575     fix x
```
```   576     assume "dist (f x) a < r"
```
```   577     hence 1: "norm (f x - a) < r"
```
```   578       by (simp add: dist_norm)
```
```   579     hence 2: "f x \<noteq> 0" using r2 by auto
```
```   580     hence "norm (inverse (f x)) = inverse (norm (f x))"
```
```   581       by (rule nonzero_norm_inverse)
```
```   582     also have "\<dots> \<le> inverse (norm a - r)"
```
```   583     proof (rule le_imp_inverse_le)
```
```   584       show "0 < norm a - r" using r2 by simp
```
```   585     next
```
```   586       have "norm a - norm (f x) \<le> norm (a - f x)"
```
```   587         by (rule norm_triangle_ineq2)
```
```   588       also have "\<dots> = norm (f x - a)"
```
```   589         by (rule norm_minus_commute)
```
```   590       also have "\<dots> < r" using 1 .
```
```   591       finally show "norm a - r \<le> norm (f x)" by simp
```
```   592     qed
```
```   593     finally show "norm (inverse (f x)) \<le> inverse (norm a - r)" .
```
```   594   qed
```
```   595   thus ?thesis by (rule BfunI)
```
```   596 qed
```
```   597
```
```   598 lemma tendsto_inverse_lemma:
```
```   599   fixes a :: "'a::real_normed_div_algebra"
```
```   600   shows "\<lbrakk>(f ---> a) net; a \<noteq> 0; eventually (\<lambda>x. f x \<noteq> 0) net\<rbrakk>
```
```   601          \<Longrightarrow> ((\<lambda>x. inverse (f x)) ---> inverse a) net"
```
```   602 apply (subst tendsto_Zfun_iff)
```
```   603 apply (rule Zfun_ssubst)
```
```   604 apply (erule eventually_elim1)
```
```   605 apply (erule (1) inverse_diff_inverse)
```
```   606 apply (rule Zfun_minus)
```
```   607 apply (rule Zfun_mult_left)
```
```   608 apply (rule mult.Bfun_prod_Zfun)
```
```   609 apply (erule (1) Bfun_inverse)
```
```   610 apply (simp add: tendsto_Zfun_iff)
```
```   611 done
```
```   612
```
```   613 lemma tendsto_inverse [tendsto_intros]:
```
```   614   fixes a :: "'a::real_normed_div_algebra"
```
```   615   assumes f: "(f ---> a) net"
```
```   616   assumes a: "a \<noteq> 0"
```
```   617   shows "((\<lambda>x. inverse (f x)) ---> inverse a) net"
```
```   618 proof -
```
```   619   from a have "0 < norm a" by simp
```
```   620   with f have "eventually (\<lambda>x. dist (f x) a < norm a) net"
```
```   621     by (rule tendstoD)
```
```   622   then have "eventually (\<lambda>x. f x \<noteq> 0) net"
```
```   623     unfolding dist_norm by (auto elim!: eventually_elim1)
```
```   624   with f a show ?thesis
```
```   625     by (rule tendsto_inverse_lemma)
```
```   626 qed
```
```   627
```
```   628 lemma tendsto_divide [tendsto_intros]:
```
```   629   fixes a b :: "'a::real_normed_field"
```
```   630   shows "\<lbrakk>(f ---> a) net; (g ---> b) net; b \<noteq> 0\<rbrakk>
```
```   631     \<Longrightarrow> ((\<lambda>x. f x / g x) ---> a / b) net"
```
```   632 by (simp add: mult.tendsto tendsto_inverse divide_inverse)
```
```   633
```
```   634 end
```