src/HOL/Limits.thy
author haftmann
Fri Nov 27 08:41:10 2009 +0100 (2009-11-27)
changeset 33963 977b94b64905
parent 31902 862ae16a799d
child 36358 246493d61204
permissions -rw-r--r--
renamed former datatype.ML to datatype_data.ML; datatype.ML provides uniform view on datatype.ML and datatype_rep_proofs.ML
     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