src/HOL/Limits.thy
author huffman
Sat May 01 09:43:40 2010 -0700 (2010-05-01)
changeset 36629 de62713aec6e
parent 36360 9d8f7efd9289
child 36630 aa1f8acdcc1c
permissions -rw-r--r--
swap ordering on nets, so x <= y means 'x is finer than y'
     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 simply as a filter.
    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
    41   eventually :: "('a \<Rightarrow> bool) \<Rightarrow> 'a net \<Rightarrow> bool" where
    42   [code del]: "eventually P net \<longleftrightarrow> Rep_net net P"
    43 
    44 lemma eventually_Abs_net:
    45   assumes "is_filter net" shows "eventually P (Abs_net net) = net P"
    46 unfolding eventually_def using assms by (simp add: Abs_net_inverse)
    47 
    48 lemma expand_net_eq:
    49   shows "net = net' \<longleftrightarrow> (\<forall>P. eventually P net = eventually P net')"
    50 unfolding Rep_net_inject [symmetric] expand_fun_eq eventually_def ..
    51 
    52 lemma eventually_True [simp]: "eventually (\<lambda>x. True) net"
    53 unfolding eventually_def
    54 by (rule is_filter.True [OF is_filter_Rep_net])
    55 
    56 lemma eventually_mono:
    57   "(\<forall>x. P x \<longrightarrow> Q x) \<Longrightarrow> eventually P net \<Longrightarrow> eventually Q net"
    58 unfolding eventually_def
    59 by (rule is_filter.mono [OF is_filter_Rep_net])
    60 
    61 lemma eventually_conj:
    62   assumes P: "eventually (\<lambda>x. P x) net"
    63   assumes Q: "eventually (\<lambda>x. Q x) net"
    64   shows "eventually (\<lambda>x. P x \<and> Q x) net"
    65 using assms unfolding eventually_def
    66 by (rule is_filter.conj [OF is_filter_Rep_net])
    67 
    68 lemma eventually_mp:
    69   assumes "eventually (\<lambda>x. P x \<longrightarrow> Q x) net"
    70   assumes "eventually (\<lambda>x. P x) net"
    71   shows "eventually (\<lambda>x. Q x) net"
    72 proof (rule eventually_mono)
    73   show "\<forall>x. (P x \<longrightarrow> Q x) \<and> P x \<longrightarrow> Q x" by simp
    74   show "eventually (\<lambda>x. (P x \<longrightarrow> Q x) \<and> P x) net"
    75     using assms by (rule eventually_conj)
    76 qed
    77 
    78 lemma eventually_rev_mp:
    79   assumes "eventually (\<lambda>x. P x) net"
    80   assumes "eventually (\<lambda>x. P x \<longrightarrow> Q x) net"
    81   shows "eventually (\<lambda>x. Q x) net"
    82 using assms(2) assms(1) by (rule eventually_mp)
    83 
    84 lemma eventually_conj_iff:
    85   "eventually (\<lambda>x. P x \<and> Q x) net \<longleftrightarrow> eventually P net \<and> eventually Q net"
    86 by (auto intro: eventually_conj elim: eventually_rev_mp)
    87 
    88 lemma eventually_elim1:
    89   assumes "eventually (\<lambda>i. P i) net"
    90   assumes "\<And>i. P i \<Longrightarrow> Q i"
    91   shows "eventually (\<lambda>i. Q i) net"
    92 using assms by (auto elim!: eventually_rev_mp)
    93 
    94 lemma eventually_elim2:
    95   assumes "eventually (\<lambda>i. P i) net"
    96   assumes "eventually (\<lambda>i. Q i) net"
    97   assumes "\<And>i. P i \<Longrightarrow> Q i \<Longrightarrow> R i"
    98   shows "eventually (\<lambda>i. R i) net"
    99 using assms by (auto elim!: eventually_rev_mp)
   100 
   101 
   102 subsection {* Finer-than relation *}
   103 
   104 text {* @{term "net \<le> net'"} means that @{term net} is finer than
   105 @{term net'}. *}
   106 
   107 instantiation net :: (type) "{order,bot}"
   108 begin
   109 
   110 definition
   111   le_net_def [code del]:
   112     "net \<le> net' \<longleftrightarrow> (\<forall>P. eventually P net' \<longrightarrow> eventually P net)"
   113 
   114 definition
   115   less_net_def [code del]:
   116     "(net :: 'a net) < net' \<longleftrightarrow> net \<le> net' \<and> \<not> net' \<le> net"
   117 
   118 definition
   119   bot_net_def [code del]:
   120     "bot = Abs_net (\<lambda>P. True)"
   121 
   122 lemma eventually_bot [simp]: "eventually P bot"
   123 unfolding bot_net_def
   124 by (subst eventually_Abs_net, rule is_filter.intro, auto)
   125 
   126 instance proof
   127   fix x y :: "'a net" show "x < y \<longleftrightarrow> x \<le> y \<and> \<not> y \<le> x"
   128     by (rule less_net_def)
   129 next
   130   fix x :: "'a net" show "x \<le> x"
   131     unfolding le_net_def by simp
   132 next
   133   fix x y z :: "'a net" assume "x \<le> y" and "y \<le> z" thus "x \<le> z"
   134     unfolding le_net_def by simp
   135 next
   136   fix x y :: "'a net" assume "x \<le> y" and "y \<le> x" thus "x = y"
   137     unfolding le_net_def expand_net_eq by fast
   138 next
   139   fix x :: "'a net" show "bot \<le> x"
   140     unfolding le_net_def by simp
   141 qed
   142 
   143 end
   144 
   145 lemma net_leD:
   146   "net \<le> net' \<Longrightarrow> eventually P net' \<Longrightarrow> eventually P net"
   147 unfolding le_net_def by simp
   148 
   149 lemma net_leI:
   150   "(\<And>P. eventually P net' \<Longrightarrow> eventually P net) \<Longrightarrow> net \<le> net'"
   151 unfolding le_net_def by simp
   152 
   153 lemma eventually_False:
   154   "eventually (\<lambda>x. False) net \<longleftrightarrow> net = bot"
   155 unfolding expand_net_eq by (auto elim: eventually_rev_mp)
   156 
   157 
   158 subsection {* Standard Nets *}
   159 
   160 definition
   161   sequentially :: "nat net"
   162 where [code del]:
   163   "sequentially = Abs_net (\<lambda>P. \<exists>k. \<forall>n\<ge>k. P n)"
   164 
   165 definition
   166   within :: "'a net \<Rightarrow> 'a set \<Rightarrow> 'a net" (infixr "within" 70)
   167 where [code del]:
   168   "net within S = Abs_net (\<lambda>P. eventually (\<lambda>x. x \<in> S \<longrightarrow> P x) net)"
   169 
   170 definition
   171   at :: "'a::topological_space \<Rightarrow> 'a net"
   172 where [code del]:
   173   "at a = Abs_net (\<lambda>P. \<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. x \<noteq> a \<longrightarrow> P x))"
   174 
   175 lemma eventually_sequentially:
   176   "eventually P sequentially \<longleftrightarrow> (\<exists>N. \<forall>n\<ge>N. P n)"
   177 unfolding sequentially_def
   178 proof (rule eventually_Abs_net, rule is_filter.intro)
   179   fix P Q :: "nat \<Rightarrow> bool"
   180   assume "\<exists>i. \<forall>n\<ge>i. P n" and "\<exists>j. \<forall>n\<ge>j. Q n"
   181   then obtain i j where "\<forall>n\<ge>i. P n" and "\<forall>n\<ge>j. Q n" by auto
   182   then have "\<forall>n\<ge>max i j. P n \<and> Q n" by simp
   183   then show "\<exists>k. \<forall>n\<ge>k. P n \<and> Q n" ..
   184 qed auto
   185 
   186 lemma eventually_within:
   187   "eventually P (net within S) = eventually (\<lambda>x. x \<in> S \<longrightarrow> P x) net"
   188 unfolding within_def
   189 by (rule eventually_Abs_net, rule is_filter.intro)
   190    (auto elim!: eventually_rev_mp)
   191 
   192 lemma within_UNIV: "net within UNIV = net"
   193   unfolding expand_net_eq eventually_within by simp
   194 
   195 lemma eventually_at_topological:
   196   "eventually P (at a) \<longleftrightarrow> (\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. x \<noteq> a \<longrightarrow> P x))"
   197 unfolding at_def
   198 proof (rule eventually_Abs_net, rule is_filter.intro)
   199   have "open UNIV \<and> a \<in> UNIV \<and> (\<forall>x\<in>UNIV. x \<noteq> a \<longrightarrow> True)" by simp
   200   thus "\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. x \<noteq> a \<longrightarrow> True)" by - rule
   201 next
   202   fix P Q
   203   assume "\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. x \<noteq> a \<longrightarrow> P x)"
   204      and "\<exists>T. open T \<and> a \<in> T \<and> (\<forall>x\<in>T. x \<noteq> a \<longrightarrow> Q x)"
   205   then obtain S T where
   206     "open S \<and> a \<in> S \<and> (\<forall>x\<in>S. x \<noteq> a \<longrightarrow> P x)"
   207     "open T \<and> a \<in> T \<and> (\<forall>x\<in>T. x \<noteq> a \<longrightarrow> Q x)" by auto
   208   hence "open (S \<inter> T) \<and> a \<in> S \<inter> T \<and> (\<forall>x\<in>(S \<inter> T). x \<noteq> a \<longrightarrow> P x \<and> Q x)"
   209     by (simp add: open_Int)
   210   thus "\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. x \<noteq> a \<longrightarrow> P x \<and> Q x)" by - rule
   211 qed auto
   212 
   213 lemma eventually_at:
   214   fixes a :: "'a::metric_space"
   215   shows "eventually P (at a) \<longleftrightarrow> (\<exists>d>0. \<forall>x. x \<noteq> a \<and> dist x a < d \<longrightarrow> P x)"
   216 unfolding eventually_at_topological open_dist
   217 apply safe
   218 apply fast
   219 apply (rule_tac x="{x. dist x a < d}" in exI, simp)
   220 apply clarsimp
   221 apply (rule_tac x="d - dist x a" in exI, clarsimp)
   222 apply (simp only: less_diff_eq)
   223 apply (erule le_less_trans [OF dist_triangle])
   224 done
   225 
   226 
   227 subsection {* Boundedness *}
   228 
   229 definition
   230   Bfun :: "('a \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a net \<Rightarrow> bool" where
   231   [code del]: "Bfun f net = (\<exists>K>0. eventually (\<lambda>x. norm (f x) \<le> K) net)"
   232 
   233 lemma BfunI:
   234   assumes K: "eventually (\<lambda>x. norm (f x) \<le> K) net" shows "Bfun f net"
   235 unfolding Bfun_def
   236 proof (intro exI conjI allI)
   237   show "0 < max K 1" by simp
   238 next
   239   show "eventually (\<lambda>x. norm (f x) \<le> max K 1) net"
   240     using K by (rule eventually_elim1, simp)
   241 qed
   242 
   243 lemma BfunE:
   244   assumes "Bfun f net"
   245   obtains B where "0 < B" and "eventually (\<lambda>x. norm (f x) \<le> B) net"
   246 using assms unfolding Bfun_def by fast
   247 
   248 
   249 subsection {* Convergence to Zero *}
   250 
   251 definition
   252   Zfun :: "('a \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a net \<Rightarrow> bool" where
   253   [code del]: "Zfun f net = (\<forall>r>0. eventually (\<lambda>x. norm (f x) < r) net)"
   254 
   255 lemma ZfunI:
   256   "(\<And>r. 0 < r \<Longrightarrow> eventually (\<lambda>x. norm (f x) < r) net) \<Longrightarrow> Zfun f net"
   257 unfolding Zfun_def by simp
   258 
   259 lemma ZfunD:
   260   "\<lbrakk>Zfun f net; 0 < r\<rbrakk> \<Longrightarrow> eventually (\<lambda>x. norm (f x) < r) net"
   261 unfolding Zfun_def by simp
   262 
   263 lemma Zfun_ssubst:
   264   "eventually (\<lambda>x. f x = g x) net \<Longrightarrow> Zfun g net \<Longrightarrow> Zfun f net"
   265 unfolding Zfun_def by (auto elim!: eventually_rev_mp)
   266 
   267 lemma Zfun_zero: "Zfun (\<lambda>x. 0) net"
   268 unfolding Zfun_def by simp
   269 
   270 lemma Zfun_norm_iff: "Zfun (\<lambda>x. norm (f x)) net = Zfun (\<lambda>x. f x) net"
   271 unfolding Zfun_def by simp
   272 
   273 lemma Zfun_imp_Zfun:
   274   assumes f: "Zfun f net"
   275   assumes g: "eventually (\<lambda>x. norm (g x) \<le> norm (f x) * K) net"
   276   shows "Zfun (\<lambda>x. g x) net"
   277 proof (cases)
   278   assume K: "0 < K"
   279   show ?thesis
   280   proof (rule ZfunI)
   281     fix r::real assume "0 < r"
   282     hence "0 < r / K"
   283       using K by (rule divide_pos_pos)
   284     then have "eventually (\<lambda>x. norm (f x) < r / K) net"
   285       using ZfunD [OF f] by fast
   286     with g show "eventually (\<lambda>x. norm (g x) < r) net"
   287     proof (rule eventually_elim2)
   288       fix x
   289       assume *: "norm (g x) \<le> norm (f x) * K"
   290       assume "norm (f x) < r / K"
   291       hence "norm (f x) * K < r"
   292         by (simp add: pos_less_divide_eq K)
   293       thus "norm (g x) < r"
   294         by (simp add: order_le_less_trans [OF *])
   295     qed
   296   qed
   297 next
   298   assume "\<not> 0 < K"
   299   hence K: "K \<le> 0" by (simp only: not_less)
   300   show ?thesis
   301   proof (rule ZfunI)
   302     fix r :: real
   303     assume "0 < r"
   304     from g show "eventually (\<lambda>x. norm (g x) < r) net"
   305     proof (rule eventually_elim1)
   306       fix x
   307       assume "norm (g x) \<le> norm (f x) * K"
   308       also have "\<dots> \<le> norm (f x) * 0"
   309         using K norm_ge_zero by (rule mult_left_mono)
   310       finally show "norm (g x) < r"
   311         using `0 < r` by simp
   312     qed
   313   qed
   314 qed
   315 
   316 lemma Zfun_le: "\<lbrakk>Zfun g net; \<forall>x. norm (f x) \<le> norm (g x)\<rbrakk> \<Longrightarrow> Zfun f net"
   317 by (erule_tac K="1" in Zfun_imp_Zfun, simp)
   318 
   319 lemma Zfun_add:
   320   assumes f: "Zfun f net" and g: "Zfun g net"
   321   shows "Zfun (\<lambda>x. f x + g x) net"
   322 proof (rule ZfunI)
   323   fix r::real assume "0 < r"
   324   hence r: "0 < r / 2" by simp
   325   have "eventually (\<lambda>x. norm (f x) < r/2) net"
   326     using f r by (rule ZfunD)
   327   moreover
   328   have "eventually (\<lambda>x. norm (g x) < r/2) net"
   329     using g r by (rule ZfunD)
   330   ultimately
   331   show "eventually (\<lambda>x. norm (f x + g x) < r) net"
   332   proof (rule eventually_elim2)
   333     fix x
   334     assume *: "norm (f x) < r/2" "norm (g x) < r/2"
   335     have "norm (f x + g x) \<le> norm (f x) + norm (g x)"
   336       by (rule norm_triangle_ineq)
   337     also have "\<dots> < r/2 + r/2"
   338       using * by (rule add_strict_mono)
   339     finally show "norm (f x + g x) < r"
   340       by simp
   341   qed
   342 qed
   343 
   344 lemma Zfun_minus: "Zfun f net \<Longrightarrow> Zfun (\<lambda>x. - f x) net"
   345 unfolding Zfun_def by simp
   346 
   347 lemma Zfun_diff: "\<lbrakk>Zfun f net; Zfun g net\<rbrakk> \<Longrightarrow> Zfun (\<lambda>x. f x - g x) net"
   348 by (simp only: diff_minus Zfun_add Zfun_minus)
   349 
   350 lemma (in bounded_linear) Zfun:
   351   assumes g: "Zfun g net"
   352   shows "Zfun (\<lambda>x. f (g x)) net"
   353 proof -
   354   obtain K where "\<And>x. norm (f x) \<le> norm x * K"
   355     using bounded by fast
   356   then have "eventually (\<lambda>x. norm (f (g x)) \<le> norm (g x) * K) net"
   357     by simp
   358   with g show ?thesis
   359     by (rule Zfun_imp_Zfun)
   360 qed
   361 
   362 lemma (in bounded_bilinear) Zfun:
   363   assumes f: "Zfun f net"
   364   assumes g: "Zfun g net"
   365   shows "Zfun (\<lambda>x. f x ** g x) net"
   366 proof (rule ZfunI)
   367   fix r::real assume r: "0 < r"
   368   obtain K where K: "0 < K"
   369     and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"
   370     using pos_bounded by fast
   371   from K have K': "0 < inverse K"
   372     by (rule positive_imp_inverse_positive)
   373   have "eventually (\<lambda>x. norm (f x) < r) net"
   374     using f r by (rule ZfunD)
   375   moreover
   376   have "eventually (\<lambda>x. norm (g x) < inverse K) net"
   377     using g K' by (rule ZfunD)
   378   ultimately
   379   show "eventually (\<lambda>x. norm (f x ** g x) < r) net"
   380   proof (rule eventually_elim2)
   381     fix x
   382     assume *: "norm (f x) < r" "norm (g x) < inverse K"
   383     have "norm (f x ** g x) \<le> norm (f x) * norm (g x) * K"
   384       by (rule norm_le)
   385     also have "norm (f x) * norm (g x) * K < r * inverse K * K"
   386       by (intro mult_strict_right_mono mult_strict_mono' norm_ge_zero * K)
   387     also from K have "r * inverse K * K = r"
   388       by simp
   389     finally show "norm (f x ** g x) < r" .
   390   qed
   391 qed
   392 
   393 lemma (in bounded_bilinear) Zfun_left:
   394   "Zfun f net \<Longrightarrow> Zfun (\<lambda>x. f x ** a) net"
   395 by (rule bounded_linear_left [THEN bounded_linear.Zfun])
   396 
   397 lemma (in bounded_bilinear) Zfun_right:
   398   "Zfun f net \<Longrightarrow> Zfun (\<lambda>x. a ** f x) net"
   399 by (rule bounded_linear_right [THEN bounded_linear.Zfun])
   400 
   401 lemmas Zfun_mult = mult.Zfun
   402 lemmas Zfun_mult_right = mult.Zfun_right
   403 lemmas Zfun_mult_left = mult.Zfun_left
   404 
   405 
   406 subsection {* Limits *}
   407 
   408 definition
   409   tendsto :: "('a \<Rightarrow> 'b::topological_space) \<Rightarrow> 'b \<Rightarrow> 'a net \<Rightarrow> bool"
   410     (infixr "--->" 55)
   411 where [code del]:
   412   "(f ---> l) net \<longleftrightarrow> (\<forall>S. open S \<longrightarrow> l \<in> S \<longrightarrow> eventually (\<lambda>x. f x \<in> S) net)"
   413 
   414 ML {*
   415 structure Tendsto_Intros = Named_Thms
   416 (
   417   val name = "tendsto_intros"
   418   val description = "introduction rules for tendsto"
   419 )
   420 *}
   421 
   422 setup Tendsto_Intros.setup
   423 
   424 lemma topological_tendstoI:
   425   "(\<And>S. open S \<Longrightarrow> l \<in> S \<Longrightarrow> eventually (\<lambda>x. f x \<in> S) net)
   426     \<Longrightarrow> (f ---> l) net"
   427   unfolding tendsto_def by auto
   428 
   429 lemma topological_tendstoD:
   430   "(f ---> l) net \<Longrightarrow> open S \<Longrightarrow> l \<in> S \<Longrightarrow> eventually (\<lambda>x. f x \<in> S) net"
   431   unfolding tendsto_def by auto
   432 
   433 lemma tendstoI:
   434   assumes "\<And>e. 0 < e \<Longrightarrow> eventually (\<lambda>x. dist (f x) l < e) net"
   435   shows "(f ---> l) net"
   436 apply (rule topological_tendstoI)
   437 apply (simp add: open_dist)
   438 apply (drule (1) bspec, clarify)
   439 apply (drule assms)
   440 apply (erule eventually_elim1, simp)
   441 done
   442 
   443 lemma tendstoD:
   444   "(f ---> l) net \<Longrightarrow> 0 < e \<Longrightarrow> eventually (\<lambda>x. dist (f x) l < e) net"
   445 apply (drule_tac S="{x. dist x l < e}" in topological_tendstoD)
   446 apply (clarsimp simp add: open_dist)
   447 apply (rule_tac x="e - dist x l" in exI, clarsimp)
   448 apply (simp only: less_diff_eq)
   449 apply (erule le_less_trans [OF dist_triangle])
   450 apply simp
   451 apply simp
   452 done
   453 
   454 lemma tendsto_iff:
   455   "(f ---> l) net \<longleftrightarrow> (\<forall>e>0. eventually (\<lambda>x. dist (f x) l < e) net)"
   456 using tendstoI tendstoD by fast
   457 
   458 lemma tendsto_Zfun_iff: "(f ---> a) net = Zfun (\<lambda>x. f x - a) net"
   459 by (simp only: tendsto_iff Zfun_def dist_norm)
   460 
   461 lemma tendsto_ident_at [tendsto_intros]: "((\<lambda>x. x) ---> a) (at a)"
   462 unfolding tendsto_def eventually_at_topological by auto
   463 
   464 lemma tendsto_ident_at_within [tendsto_intros]:
   465   "a \<in> S \<Longrightarrow> ((\<lambda>x. x) ---> a) (at a within S)"
   466 unfolding tendsto_def eventually_within eventually_at_topological by auto
   467 
   468 lemma tendsto_const [tendsto_intros]: "((\<lambda>x. k) ---> k) net"
   469 by (simp add: tendsto_def)
   470 
   471 lemma tendsto_dist [tendsto_intros]:
   472   assumes f: "(f ---> l) net" and g: "(g ---> m) net"
   473   shows "((\<lambda>x. dist (f x) (g x)) ---> dist l m) net"
   474 proof (rule tendstoI)
   475   fix e :: real assume "0 < e"
   476   hence e2: "0 < e/2" by simp
   477   from tendstoD [OF f e2] tendstoD [OF g e2]
   478   show "eventually (\<lambda>x. dist (dist (f x) (g x)) (dist l m) < e) net"
   479   proof (rule eventually_elim2)
   480     fix x assume "dist (f x) l < e/2" "dist (g x) m < e/2"
   481     then show "dist (dist (f x) (g x)) (dist l m) < e"
   482       unfolding dist_real_def
   483       using dist_triangle2 [of "f x" "g x" "l"]
   484       using dist_triangle2 [of "g x" "l" "m"]
   485       using dist_triangle3 [of "l" "m" "f x"]
   486       using dist_triangle [of "f x" "m" "g x"]
   487       by arith
   488   qed
   489 qed
   490 
   491 lemma tendsto_norm [tendsto_intros]:
   492   "(f ---> a) net \<Longrightarrow> ((\<lambda>x. norm (f x)) ---> norm a) net"
   493 apply (simp add: tendsto_iff dist_norm, safe)
   494 apply (drule_tac x="e" in spec, safe)
   495 apply (erule eventually_elim1)
   496 apply (erule order_le_less_trans [OF norm_triangle_ineq3])
   497 done
   498 
   499 lemma add_diff_add:
   500   fixes a b c d :: "'a::ab_group_add"
   501   shows "(a + c) - (b + d) = (a - b) + (c - d)"
   502 by simp
   503 
   504 lemma minus_diff_minus:
   505   fixes a b :: "'a::ab_group_add"
   506   shows "(- a) - (- b) = - (a - b)"
   507 by simp
   508 
   509 lemma tendsto_add [tendsto_intros]:
   510   fixes a b :: "'a::real_normed_vector"
   511   shows "\<lbrakk>(f ---> a) net; (g ---> b) net\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x + g x) ---> a + b) net"
   512 by (simp only: tendsto_Zfun_iff add_diff_add Zfun_add)
   513 
   514 lemma tendsto_minus [tendsto_intros]:
   515   fixes a :: "'a::real_normed_vector"
   516   shows "(f ---> a) net \<Longrightarrow> ((\<lambda>x. - f x) ---> - a) net"
   517 by (simp only: tendsto_Zfun_iff minus_diff_minus Zfun_minus)
   518 
   519 lemma tendsto_minus_cancel:
   520   fixes a :: "'a::real_normed_vector"
   521   shows "((\<lambda>x. - f x) ---> - a) net \<Longrightarrow> (f ---> a) net"
   522 by (drule tendsto_minus, simp)
   523 
   524 lemma tendsto_diff [tendsto_intros]:
   525   fixes a b :: "'a::real_normed_vector"
   526   shows "\<lbrakk>(f ---> a) net; (g ---> b) net\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x - g x) ---> a - b) net"
   527 by (simp add: diff_minus tendsto_add tendsto_minus)
   528 
   529 lemma tendsto_setsum [tendsto_intros]:
   530   fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c::real_normed_vector"
   531   assumes "\<And>i. i \<in> S \<Longrightarrow> (f i ---> a i) net"
   532   shows "((\<lambda>x. \<Sum>i\<in>S. f i x) ---> (\<Sum>i\<in>S. a i)) net"
   533 proof (cases "finite S")
   534   assume "finite S" thus ?thesis using assms
   535   proof (induct set: finite)
   536     case empty show ?case
   537       by (simp add: tendsto_const)
   538   next
   539     case (insert i F) thus ?case
   540       by (simp add: tendsto_add)
   541   qed
   542 next
   543   assume "\<not> finite S" thus ?thesis
   544     by (simp add: tendsto_const)
   545 qed
   546 
   547 lemma (in bounded_linear) tendsto [tendsto_intros]:
   548   "(g ---> a) net \<Longrightarrow> ((\<lambda>x. f (g x)) ---> f a) net"
   549 by (simp only: tendsto_Zfun_iff diff [symmetric] Zfun)
   550 
   551 lemma (in bounded_bilinear) tendsto [tendsto_intros]:
   552   "\<lbrakk>(f ---> a) net; (g ---> b) net\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x ** g x) ---> a ** b) net"
   553 by (simp only: tendsto_Zfun_iff prod_diff_prod
   554                Zfun_add Zfun Zfun_left Zfun_right)
   555 
   556 
   557 subsection {* Continuity of Inverse *}
   558 
   559 lemma (in bounded_bilinear) Zfun_prod_Bfun:
   560   assumes f: "Zfun f net"
   561   assumes g: "Bfun g net"
   562   shows "Zfun (\<lambda>x. f x ** g x) net"
   563 proof -
   564   obtain K where K: "0 \<le> K"
   565     and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"
   566     using nonneg_bounded by fast
   567   obtain B where B: "0 < B"
   568     and norm_g: "eventually (\<lambda>x. norm (g x) \<le> B) net"
   569     using g by (rule BfunE)
   570   have "eventually (\<lambda>x. norm (f x ** g x) \<le> norm (f x) * (B * K)) net"
   571   using norm_g proof (rule eventually_elim1)
   572     fix x
   573     assume *: "norm (g x) \<le> B"
   574     have "norm (f x ** g x) \<le> norm (f x) * norm (g x) * K"
   575       by (rule norm_le)
   576     also have "\<dots> \<le> norm (f x) * B * K"
   577       by (intro mult_mono' order_refl norm_g norm_ge_zero
   578                 mult_nonneg_nonneg K *)
   579     also have "\<dots> = norm (f x) * (B * K)"
   580       by (rule mult_assoc)
   581     finally show "norm (f x ** g x) \<le> norm (f x) * (B * K)" .
   582   qed
   583   with f show ?thesis
   584     by (rule Zfun_imp_Zfun)
   585 qed
   586 
   587 lemma (in bounded_bilinear) flip:
   588   "bounded_bilinear (\<lambda>x y. y ** x)"
   589 apply default
   590 apply (rule add_right)
   591 apply (rule add_left)
   592 apply (rule scaleR_right)
   593 apply (rule scaleR_left)
   594 apply (subst mult_commute)
   595 using bounded by fast
   596 
   597 lemma (in bounded_bilinear) Bfun_prod_Zfun:
   598   assumes f: "Bfun f net"
   599   assumes g: "Zfun g net"
   600   shows "Zfun (\<lambda>x. f x ** g x) net"
   601 using flip g f by (rule bounded_bilinear.Zfun_prod_Bfun)
   602 
   603 lemma inverse_diff_inverse:
   604   "\<lbrakk>(a::'a::division_ring) \<noteq> 0; b \<noteq> 0\<rbrakk>
   605    \<Longrightarrow> inverse a - inverse b = - (inverse a * (a - b) * inverse b)"
   606 by (simp add: algebra_simps)
   607 
   608 lemma Bfun_inverse_lemma:
   609   fixes x :: "'a::real_normed_div_algebra"
   610   shows "\<lbrakk>r \<le> norm x; 0 < r\<rbrakk> \<Longrightarrow> norm (inverse x) \<le> inverse r"
   611 apply (subst nonzero_norm_inverse, clarsimp)
   612 apply (erule (1) le_imp_inverse_le)
   613 done
   614 
   615 lemma Bfun_inverse:
   616   fixes a :: "'a::real_normed_div_algebra"
   617   assumes f: "(f ---> a) net"
   618   assumes a: "a \<noteq> 0"
   619   shows "Bfun (\<lambda>x. inverse (f x)) net"
   620 proof -
   621   from a have "0 < norm a" by simp
   622   hence "\<exists>r>0. r < norm a" by (rule dense)
   623   then obtain r where r1: "0 < r" and r2: "r < norm a" by fast
   624   have "eventually (\<lambda>x. dist (f x) a < r) net"
   625     using tendstoD [OF f r1] by fast
   626   hence "eventually (\<lambda>x. norm (inverse (f x)) \<le> inverse (norm a - r)) net"
   627   proof (rule eventually_elim1)
   628     fix x
   629     assume "dist (f x) a < r"
   630     hence 1: "norm (f x - a) < r"
   631       by (simp add: dist_norm)
   632     hence 2: "f x \<noteq> 0" using r2 by auto
   633     hence "norm (inverse (f x)) = inverse (norm (f x))"
   634       by (rule nonzero_norm_inverse)
   635     also have "\<dots> \<le> inverse (norm a - r)"
   636     proof (rule le_imp_inverse_le)
   637       show "0 < norm a - r" using r2 by simp
   638     next
   639       have "norm a - norm (f x) \<le> norm (a - f x)"
   640         by (rule norm_triangle_ineq2)
   641       also have "\<dots> = norm (f x - a)"
   642         by (rule norm_minus_commute)
   643       also have "\<dots> < r" using 1 .
   644       finally show "norm a - r \<le> norm (f x)" by simp
   645     qed
   646     finally show "norm (inverse (f x)) \<le> inverse (norm a - r)" .
   647   qed
   648   thus ?thesis by (rule BfunI)
   649 qed
   650 
   651 lemma tendsto_inverse_lemma:
   652   fixes a :: "'a::real_normed_div_algebra"
   653   shows "\<lbrakk>(f ---> a) net; a \<noteq> 0; eventually (\<lambda>x. f x \<noteq> 0) net\<rbrakk>
   654          \<Longrightarrow> ((\<lambda>x. inverse (f x)) ---> inverse a) net"
   655 apply (subst tendsto_Zfun_iff)
   656 apply (rule Zfun_ssubst)
   657 apply (erule eventually_elim1)
   658 apply (erule (1) inverse_diff_inverse)
   659 apply (rule Zfun_minus)
   660 apply (rule Zfun_mult_left)
   661 apply (rule mult.Bfun_prod_Zfun)
   662 apply (erule (1) Bfun_inverse)
   663 apply (simp add: tendsto_Zfun_iff)
   664 done
   665 
   666 lemma tendsto_inverse [tendsto_intros]:
   667   fixes a :: "'a::real_normed_div_algebra"
   668   assumes f: "(f ---> a) net"
   669   assumes a: "a \<noteq> 0"
   670   shows "((\<lambda>x. inverse (f x)) ---> inverse a) net"
   671 proof -
   672   from a have "0 < norm a" by simp
   673   with f have "eventually (\<lambda>x. dist (f x) a < norm a) net"
   674     by (rule tendstoD)
   675   then have "eventually (\<lambda>x. f x \<noteq> 0) net"
   676     unfolding dist_norm by (auto elim!: eventually_elim1)
   677   with f a show ?thesis
   678     by (rule tendsto_inverse_lemma)
   679 qed
   680 
   681 lemma tendsto_divide [tendsto_intros]:
   682   fixes a b :: "'a::real_normed_field"
   683   shows "\<lbrakk>(f ---> a) net; (g ---> b) net; b \<noteq> 0\<rbrakk>
   684     \<Longrightarrow> ((\<lambda>x. f x / g x) ---> a / b) net"
   685 by (simp add: mult.tendsto tendsto_inverse divide_inverse)
   686 
   687 end