src/HOL/Limits.thy
author huffman
Mon May 10 21:33:48 2010 -0700 (2010-05-10)
changeset 36822 38a480e0346f
parent 36662 621122eeb138
child 37767 a2b7a20d6ea3
permissions -rw-r--r--
minimize imports
     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
    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 always_eventually: "\<forall>x. P x \<Longrightarrow> eventually P net"
    57 proof -
    58   assume "\<forall>x. P x" hence "P = (\<lambda>x. True)" by (simp add: ext)
    59   thus "eventually P net" by simp
    60 qed
    61 
    62 lemma eventually_mono:
    63   "(\<forall>x. P x \<longrightarrow> Q x) \<Longrightarrow> eventually P net \<Longrightarrow> eventually Q net"
    64 unfolding eventually_def
    65 by (rule is_filter.mono [OF is_filter_Rep_net])
    66 
    67 lemma eventually_conj:
    68   assumes P: "eventually (\<lambda>x. P x) net"
    69   assumes Q: "eventually (\<lambda>x. Q x) net"
    70   shows "eventually (\<lambda>x. P x \<and> Q x) net"
    71 using assms unfolding eventually_def
    72 by (rule is_filter.conj [OF is_filter_Rep_net])
    73 
    74 lemma eventually_mp:
    75   assumes "eventually (\<lambda>x. P x \<longrightarrow> Q x) net"
    76   assumes "eventually (\<lambda>x. P x) net"
    77   shows "eventually (\<lambda>x. Q x) net"
    78 proof (rule eventually_mono)
    79   show "\<forall>x. (P x \<longrightarrow> Q x) \<and> P x \<longrightarrow> Q x" by simp
    80   show "eventually (\<lambda>x. (P x \<longrightarrow> Q x) \<and> P x) net"
    81     using assms by (rule eventually_conj)
    82 qed
    83 
    84 lemma eventually_rev_mp:
    85   assumes "eventually (\<lambda>x. P x) net"
    86   assumes "eventually (\<lambda>x. P x \<longrightarrow> Q x) net"
    87   shows "eventually (\<lambda>x. Q x) net"
    88 using assms(2) assms(1) by (rule eventually_mp)
    89 
    90 lemma eventually_conj_iff:
    91   "eventually (\<lambda>x. P x \<and> Q x) net \<longleftrightarrow> eventually P net \<and> eventually Q net"
    92 by (auto intro: eventually_conj elim: eventually_rev_mp)
    93 
    94 lemma eventually_elim1:
    95   assumes "eventually (\<lambda>i. P i) net"
    96   assumes "\<And>i. P i \<Longrightarrow> Q i"
    97   shows "eventually (\<lambda>i. Q i) net"
    98 using assms by (auto elim!: eventually_rev_mp)
    99 
   100 lemma eventually_elim2:
   101   assumes "eventually (\<lambda>i. P i) net"
   102   assumes "eventually (\<lambda>i. Q i) net"
   103   assumes "\<And>i. P i \<Longrightarrow> Q i \<Longrightarrow> R i"
   104   shows "eventually (\<lambda>i. R i) net"
   105 using assms by (auto elim!: eventually_rev_mp)
   106 
   107 
   108 subsection {* Finer-than relation *}
   109 
   110 text {* @{term "net \<le> net'"} means that @{term net} is finer than
   111 @{term net'}. *}
   112 
   113 instantiation net :: (type) complete_lattice
   114 begin
   115 
   116 definition
   117   le_net_def [code del]:
   118     "net \<le> net' \<longleftrightarrow> (\<forall>P. eventually P net' \<longrightarrow> eventually P net)"
   119 
   120 definition
   121   less_net_def [code del]:
   122     "(net :: 'a net) < net' \<longleftrightarrow> net \<le> net' \<and> \<not> net' \<le> net"
   123 
   124 definition
   125   top_net_def [code del]:
   126     "top = Abs_net (\<lambda>P. \<forall>x. P x)"
   127 
   128 definition
   129   bot_net_def [code del]:
   130     "bot = Abs_net (\<lambda>P. True)"
   131 
   132 definition
   133   sup_net_def [code del]:
   134     "sup net net' = Abs_net (\<lambda>P. eventually P net \<and> eventually P net')"
   135 
   136 definition
   137   inf_net_def [code del]:
   138     "inf a b = Abs_net
   139       (\<lambda>P. \<exists>Q R. eventually Q a \<and> eventually R b \<and> (\<forall>x. Q x \<and> R x \<longrightarrow> P x))"
   140 
   141 definition
   142   Sup_net_def [code del]:
   143     "Sup A = Abs_net (\<lambda>P. \<forall>net\<in>A. eventually P net)"
   144 
   145 definition
   146   Inf_net_def [code del]:
   147     "Inf A = Sup {x::'a net. \<forall>y\<in>A. x \<le> y}"
   148 
   149 lemma eventually_top [simp]: "eventually P top \<longleftrightarrow> (\<forall>x. P x)"
   150 unfolding top_net_def
   151 by (rule eventually_Abs_net, rule is_filter.intro, auto)
   152 
   153 lemma eventually_bot [simp]: "eventually P bot"
   154 unfolding bot_net_def
   155 by (subst eventually_Abs_net, rule is_filter.intro, auto)
   156 
   157 lemma eventually_sup:
   158   "eventually P (sup net net') \<longleftrightarrow> eventually P net \<and> eventually P net'"
   159 unfolding sup_net_def
   160 by (rule eventually_Abs_net, rule is_filter.intro)
   161    (auto elim!: eventually_rev_mp)
   162 
   163 lemma eventually_inf:
   164   "eventually P (inf a b) \<longleftrightarrow>
   165    (\<exists>Q R. eventually Q a \<and> eventually R b \<and> (\<forall>x. Q x \<and> R x \<longrightarrow> P x))"
   166 unfolding inf_net_def
   167 apply (rule eventually_Abs_net, rule is_filter.intro)
   168 apply (fast intro: eventually_True)
   169 apply clarify
   170 apply (intro exI conjI)
   171 apply (erule (1) eventually_conj)
   172 apply (erule (1) eventually_conj)
   173 apply simp
   174 apply auto
   175 done
   176 
   177 lemma eventually_Sup:
   178   "eventually P (Sup A) \<longleftrightarrow> (\<forall>net\<in>A. eventually P net)"
   179 unfolding Sup_net_def
   180 apply (rule eventually_Abs_net, rule is_filter.intro)
   181 apply (auto intro: eventually_conj elim!: eventually_rev_mp)
   182 done
   183 
   184 instance proof
   185   fix x y :: "'a net" show "x < y \<longleftrightarrow> x \<le> y \<and> \<not> y \<le> x"
   186     by (rule less_net_def)
   187 next
   188   fix x :: "'a net" show "x \<le> x"
   189     unfolding le_net_def by simp
   190 next
   191   fix x y z :: "'a net" assume "x \<le> y" and "y \<le> z" thus "x \<le> z"
   192     unfolding le_net_def by simp
   193 next
   194   fix x y :: "'a net" assume "x \<le> y" and "y \<le> x" thus "x = y"
   195     unfolding le_net_def expand_net_eq by fast
   196 next
   197   fix x :: "'a net" show "x \<le> top"
   198     unfolding le_net_def eventually_top by (simp add: always_eventually)
   199 next
   200   fix x :: "'a net" show "bot \<le> x"
   201     unfolding le_net_def by simp
   202 next
   203   fix x y :: "'a net" show "x \<le> sup x y" and "y \<le> sup x y"
   204     unfolding le_net_def eventually_sup by simp_all
   205 next
   206   fix x y z :: "'a net" assume "x \<le> z" and "y \<le> z" thus "sup x y \<le> z"
   207     unfolding le_net_def eventually_sup by simp
   208 next
   209   fix x y :: "'a net" show "inf x y \<le> x" and "inf x y \<le> y"
   210     unfolding le_net_def eventually_inf by (auto intro: eventually_True)
   211 next
   212   fix x y z :: "'a net" assume "x \<le> y" and "x \<le> z" thus "x \<le> inf y z"
   213     unfolding le_net_def eventually_inf
   214     by (auto elim!: eventually_mono intro: eventually_conj)
   215 next
   216   fix x :: "'a net" and A assume "x \<in> A" thus "x \<le> Sup A"
   217     unfolding le_net_def eventually_Sup by simp
   218 next
   219   fix A and y :: "'a net" assume "\<And>x. x \<in> A \<Longrightarrow> x \<le> y" thus "Sup A \<le> y"
   220     unfolding le_net_def eventually_Sup by simp
   221 next
   222   fix z :: "'a net" and A assume "z \<in> A" thus "Inf A \<le> z"
   223     unfolding le_net_def Inf_net_def eventually_Sup Ball_def by simp
   224 next
   225   fix A and x :: "'a net" assume "\<And>y. y \<in> A \<Longrightarrow> x \<le> y" thus "x \<le> Inf A"
   226     unfolding le_net_def Inf_net_def eventually_Sup Ball_def by simp
   227 qed
   228 
   229 end
   230 
   231 lemma net_leD:
   232   "net \<le> net' \<Longrightarrow> eventually P net' \<Longrightarrow> eventually P net"
   233 unfolding le_net_def by simp
   234 
   235 lemma net_leI:
   236   "(\<And>P. eventually P net' \<Longrightarrow> eventually P net) \<Longrightarrow> net \<le> net'"
   237 unfolding le_net_def by simp
   238 
   239 lemma eventually_False:
   240   "eventually (\<lambda>x. False) net \<longleftrightarrow> net = bot"
   241 unfolding expand_net_eq by (auto elim: eventually_rev_mp)
   242 
   243 
   244 subsection {* Map function for nets *}
   245 
   246 definition
   247   netmap :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a net \<Rightarrow> 'b net"
   248 where [code del]:
   249   "netmap f net = Abs_net (\<lambda>P. eventually (\<lambda>x. P (f x)) net)"
   250 
   251 lemma eventually_netmap:
   252   "eventually P (netmap f net) = eventually (\<lambda>x. P (f x)) net"
   253 unfolding netmap_def
   254 apply (rule eventually_Abs_net)
   255 apply (rule is_filter.intro)
   256 apply (auto elim!: eventually_rev_mp)
   257 done
   258 
   259 lemma netmap_ident: "netmap (\<lambda>x. x) net = net"
   260 by (simp add: expand_net_eq eventually_netmap)
   261 
   262 lemma netmap_netmap: "netmap f (netmap g net) = netmap (\<lambda>x. f (g x)) net"
   263 by (simp add: expand_net_eq eventually_netmap)
   264 
   265 lemma netmap_mono: "net \<le> net' \<Longrightarrow> netmap f net \<le> netmap f net'"
   266 unfolding le_net_def eventually_netmap by simp
   267 
   268 lemma netmap_bot [simp]: "netmap f bot = bot"
   269 by (simp add: expand_net_eq eventually_netmap)
   270 
   271 
   272 subsection {* Sequentially *}
   273 
   274 definition
   275   sequentially :: "nat net"
   276 where [code del]:
   277   "sequentially = Abs_net (\<lambda>P. \<exists>k. \<forall>n\<ge>k. P n)"
   278 
   279 lemma eventually_sequentially:
   280   "eventually P sequentially \<longleftrightarrow> (\<exists>N. \<forall>n\<ge>N. P n)"
   281 unfolding sequentially_def
   282 proof (rule eventually_Abs_net, rule is_filter.intro)
   283   fix P Q :: "nat \<Rightarrow> bool"
   284   assume "\<exists>i. \<forall>n\<ge>i. P n" and "\<exists>j. \<forall>n\<ge>j. Q n"
   285   then obtain i j where "\<forall>n\<ge>i. P n" and "\<forall>n\<ge>j. Q n" by auto
   286   then have "\<forall>n\<ge>max i j. P n \<and> Q n" by simp
   287   then show "\<exists>k. \<forall>n\<ge>k. P n \<and> Q n" ..
   288 qed auto
   289 
   290 lemma sequentially_bot [simp]: "sequentially \<noteq> bot"
   291 unfolding expand_net_eq eventually_sequentially by auto
   292 
   293 lemma eventually_False_sequentially [simp]:
   294   "\<not> eventually (\<lambda>n. False) sequentially"
   295 by (simp add: eventually_False)
   296 
   297 lemma le_sequentially:
   298   "net \<le> sequentially \<longleftrightarrow> (\<forall>N. eventually (\<lambda>n. N \<le> n) net)"
   299 unfolding le_net_def eventually_sequentially
   300 by (safe, fast, drule_tac x=N in spec, auto elim: eventually_rev_mp)
   301 
   302 
   303 subsection {* Standard Nets *}
   304 
   305 definition
   306   within :: "'a net \<Rightarrow> 'a set \<Rightarrow> 'a net" (infixr "within" 70)
   307 where [code del]:
   308   "net within S = Abs_net (\<lambda>P. eventually (\<lambda>x. x \<in> S \<longrightarrow> P x) net)"
   309 
   310 definition
   311   nhds :: "'a::topological_space \<Rightarrow> 'a net"
   312 where [code del]:
   313   "nhds a = Abs_net (\<lambda>P. \<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x))"
   314 
   315 definition
   316   at :: "'a::topological_space \<Rightarrow> 'a net"
   317 where [code del]:
   318   "at a = nhds a within - {a}"
   319 
   320 lemma eventually_within:
   321   "eventually P (net within S) = eventually (\<lambda>x. x \<in> S \<longrightarrow> P x) net"
   322 unfolding within_def
   323 by (rule eventually_Abs_net, rule is_filter.intro)
   324    (auto elim!: eventually_rev_mp)
   325 
   326 lemma within_UNIV: "net within UNIV = net"
   327   unfolding expand_net_eq eventually_within by simp
   328 
   329 lemma eventually_nhds:
   330   "eventually P (nhds a) \<longleftrightarrow> (\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x))"
   331 unfolding nhds_def
   332 proof (rule eventually_Abs_net, rule is_filter.intro)
   333   have "open UNIV \<and> a \<in> UNIV \<and> (\<forall>x\<in>UNIV. True)" by simp
   334   thus "\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. True)" by - rule
   335 next
   336   fix P Q
   337   assume "\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x)"
   338      and "\<exists>T. open T \<and> a \<in> T \<and> (\<forall>x\<in>T. Q x)"
   339   then obtain S T where
   340     "open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x)"
   341     "open T \<and> a \<in> T \<and> (\<forall>x\<in>T. Q x)" by auto
   342   hence "open (S \<inter> T) \<and> a \<in> S \<inter> T \<and> (\<forall>x\<in>(S \<inter> T). P x \<and> Q x)"
   343     by (simp add: open_Int)
   344   thus "\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x \<and> Q x)" by - rule
   345 qed auto
   346 
   347 lemma eventually_nhds_metric:
   348   "eventually P (nhds a) \<longleftrightarrow> (\<exists>d>0. \<forall>x. dist x a < d \<longrightarrow> P x)"
   349 unfolding eventually_nhds open_dist
   350 apply safe
   351 apply fast
   352 apply (rule_tac x="{x. dist x a < d}" in exI, simp)
   353 apply clarsimp
   354 apply (rule_tac x="d - dist x a" in exI, clarsimp)
   355 apply (simp only: less_diff_eq)
   356 apply (erule le_less_trans [OF dist_triangle])
   357 done
   358 
   359 lemma eventually_at_topological:
   360   "eventually P (at a) \<longleftrightarrow> (\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. x \<noteq> a \<longrightarrow> P x))"
   361 unfolding at_def eventually_within eventually_nhds by simp
   362 
   363 lemma eventually_at:
   364   fixes a :: "'a::metric_space"
   365   shows "eventually P (at a) \<longleftrightarrow> (\<exists>d>0. \<forall>x. x \<noteq> a \<and> dist x a < d \<longrightarrow> P x)"
   366 unfolding at_def eventually_within eventually_nhds_metric by auto
   367 
   368 
   369 subsection {* Boundedness *}
   370 
   371 definition
   372   Bfun :: "('a \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a net \<Rightarrow> bool" where
   373   [code del]: "Bfun f net = (\<exists>K>0. eventually (\<lambda>x. norm (f x) \<le> K) net)"
   374 
   375 lemma BfunI:
   376   assumes K: "eventually (\<lambda>x. norm (f x) \<le> K) net" shows "Bfun f net"
   377 unfolding Bfun_def
   378 proof (intro exI conjI allI)
   379   show "0 < max K 1" by simp
   380 next
   381   show "eventually (\<lambda>x. norm (f x) \<le> max K 1) net"
   382     using K by (rule eventually_elim1, simp)
   383 qed
   384 
   385 lemma BfunE:
   386   assumes "Bfun f net"
   387   obtains B where "0 < B" and "eventually (\<lambda>x. norm (f x) \<le> B) net"
   388 using assms unfolding Bfun_def by fast
   389 
   390 
   391 subsection {* Convergence to Zero *}
   392 
   393 definition
   394   Zfun :: "('a \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a net \<Rightarrow> bool" where
   395   [code del]: "Zfun f net = (\<forall>r>0. eventually (\<lambda>x. norm (f x) < r) net)"
   396 
   397 lemma ZfunI:
   398   "(\<And>r. 0 < r \<Longrightarrow> eventually (\<lambda>x. norm (f x) < r) net) \<Longrightarrow> Zfun f net"
   399 unfolding Zfun_def by simp
   400 
   401 lemma ZfunD:
   402   "\<lbrakk>Zfun f net; 0 < r\<rbrakk> \<Longrightarrow> eventually (\<lambda>x. norm (f x) < r) net"
   403 unfolding Zfun_def by simp
   404 
   405 lemma Zfun_ssubst:
   406   "eventually (\<lambda>x. f x = g x) net \<Longrightarrow> Zfun g net \<Longrightarrow> Zfun f net"
   407 unfolding Zfun_def by (auto elim!: eventually_rev_mp)
   408 
   409 lemma Zfun_zero: "Zfun (\<lambda>x. 0) net"
   410 unfolding Zfun_def by simp
   411 
   412 lemma Zfun_norm_iff: "Zfun (\<lambda>x. norm (f x)) net = Zfun (\<lambda>x. f x) net"
   413 unfolding Zfun_def by simp
   414 
   415 lemma Zfun_imp_Zfun:
   416   assumes f: "Zfun f net"
   417   assumes g: "eventually (\<lambda>x. norm (g x) \<le> norm (f x) * K) net"
   418   shows "Zfun (\<lambda>x. g x) net"
   419 proof (cases)
   420   assume K: "0 < K"
   421   show ?thesis
   422   proof (rule ZfunI)
   423     fix r::real assume "0 < r"
   424     hence "0 < r / K"
   425       using K by (rule divide_pos_pos)
   426     then have "eventually (\<lambda>x. norm (f x) < r / K) net"
   427       using ZfunD [OF f] by fast
   428     with g show "eventually (\<lambda>x. norm (g x) < r) net"
   429     proof (rule eventually_elim2)
   430       fix x
   431       assume *: "norm (g x) \<le> norm (f x) * K"
   432       assume "norm (f x) < r / K"
   433       hence "norm (f x) * K < r"
   434         by (simp add: pos_less_divide_eq K)
   435       thus "norm (g x) < r"
   436         by (simp add: order_le_less_trans [OF *])
   437     qed
   438   qed
   439 next
   440   assume "\<not> 0 < K"
   441   hence K: "K \<le> 0" by (simp only: not_less)
   442   show ?thesis
   443   proof (rule ZfunI)
   444     fix r :: real
   445     assume "0 < r"
   446     from g show "eventually (\<lambda>x. norm (g x) < r) net"
   447     proof (rule eventually_elim1)
   448       fix x
   449       assume "norm (g x) \<le> norm (f x) * K"
   450       also have "\<dots> \<le> norm (f x) * 0"
   451         using K norm_ge_zero by (rule mult_left_mono)
   452       finally show "norm (g x) < r"
   453         using `0 < r` by simp
   454     qed
   455   qed
   456 qed
   457 
   458 lemma Zfun_le: "\<lbrakk>Zfun g net; \<forall>x. norm (f x) \<le> norm (g x)\<rbrakk> \<Longrightarrow> Zfun f net"
   459 by (erule_tac K="1" in Zfun_imp_Zfun, simp)
   460 
   461 lemma Zfun_add:
   462   assumes f: "Zfun f net" and g: "Zfun g net"
   463   shows "Zfun (\<lambda>x. f x + g x) net"
   464 proof (rule ZfunI)
   465   fix r::real assume "0 < r"
   466   hence r: "0 < r / 2" by simp
   467   have "eventually (\<lambda>x. norm (f x) < r/2) net"
   468     using f r by (rule ZfunD)
   469   moreover
   470   have "eventually (\<lambda>x. norm (g x) < r/2) net"
   471     using g r by (rule ZfunD)
   472   ultimately
   473   show "eventually (\<lambda>x. norm (f x + g x) < r) net"
   474   proof (rule eventually_elim2)
   475     fix x
   476     assume *: "norm (f x) < r/2" "norm (g x) < r/2"
   477     have "norm (f x + g x) \<le> norm (f x) + norm (g x)"
   478       by (rule norm_triangle_ineq)
   479     also have "\<dots> < r/2 + r/2"
   480       using * by (rule add_strict_mono)
   481     finally show "norm (f x + g x) < r"
   482       by simp
   483   qed
   484 qed
   485 
   486 lemma Zfun_minus: "Zfun f net \<Longrightarrow> Zfun (\<lambda>x. - f x) net"
   487 unfolding Zfun_def by simp
   488 
   489 lemma Zfun_diff: "\<lbrakk>Zfun f net; Zfun g net\<rbrakk> \<Longrightarrow> Zfun (\<lambda>x. f x - g x) net"
   490 by (simp only: diff_minus Zfun_add Zfun_minus)
   491 
   492 lemma (in bounded_linear) Zfun:
   493   assumes g: "Zfun g net"
   494   shows "Zfun (\<lambda>x. f (g x)) net"
   495 proof -
   496   obtain K where "\<And>x. norm (f x) \<le> norm x * K"
   497     using bounded by fast
   498   then have "eventually (\<lambda>x. norm (f (g x)) \<le> norm (g x) * K) net"
   499     by simp
   500   with g show ?thesis
   501     by (rule Zfun_imp_Zfun)
   502 qed
   503 
   504 lemma (in bounded_bilinear) Zfun:
   505   assumes f: "Zfun f net"
   506   assumes g: "Zfun g net"
   507   shows "Zfun (\<lambda>x. f x ** g x) net"
   508 proof (rule ZfunI)
   509   fix r::real assume r: "0 < r"
   510   obtain K where K: "0 < K"
   511     and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"
   512     using pos_bounded by fast
   513   from K have K': "0 < inverse K"
   514     by (rule positive_imp_inverse_positive)
   515   have "eventually (\<lambda>x. norm (f x) < r) net"
   516     using f r by (rule ZfunD)
   517   moreover
   518   have "eventually (\<lambda>x. norm (g x) < inverse K) net"
   519     using g K' by (rule ZfunD)
   520   ultimately
   521   show "eventually (\<lambda>x. norm (f x ** g x) < r) net"
   522   proof (rule eventually_elim2)
   523     fix x
   524     assume *: "norm (f x) < r" "norm (g x) < inverse K"
   525     have "norm (f x ** g x) \<le> norm (f x) * norm (g x) * K"
   526       by (rule norm_le)
   527     also have "norm (f x) * norm (g x) * K < r * inverse K * K"
   528       by (intro mult_strict_right_mono mult_strict_mono' norm_ge_zero * K)
   529     also from K have "r * inverse K * K = r"
   530       by simp
   531     finally show "norm (f x ** g x) < r" .
   532   qed
   533 qed
   534 
   535 lemma (in bounded_bilinear) Zfun_left:
   536   "Zfun f net \<Longrightarrow> Zfun (\<lambda>x. f x ** a) net"
   537 by (rule bounded_linear_left [THEN bounded_linear.Zfun])
   538 
   539 lemma (in bounded_bilinear) Zfun_right:
   540   "Zfun f net \<Longrightarrow> Zfun (\<lambda>x. a ** f x) net"
   541 by (rule bounded_linear_right [THEN bounded_linear.Zfun])
   542 
   543 lemmas Zfun_mult = mult.Zfun
   544 lemmas Zfun_mult_right = mult.Zfun_right
   545 lemmas Zfun_mult_left = mult.Zfun_left
   546 
   547 
   548 subsection {* Limits *}
   549 
   550 definition
   551   tendsto :: "('a \<Rightarrow> 'b::topological_space) \<Rightarrow> 'b \<Rightarrow> 'a net \<Rightarrow> bool"
   552     (infixr "--->" 55)
   553 where [code del]:
   554   "(f ---> l) net \<longleftrightarrow> (\<forall>S. open S \<longrightarrow> l \<in> S \<longrightarrow> eventually (\<lambda>x. f x \<in> S) net)"
   555 
   556 ML {*
   557 structure Tendsto_Intros = Named_Thms
   558 (
   559   val name = "tendsto_intros"
   560   val description = "introduction rules for tendsto"
   561 )
   562 *}
   563 
   564 setup Tendsto_Intros.setup
   565 
   566 lemma tendsto_mono: "net \<le> net' \<Longrightarrow> (f ---> l) net' \<Longrightarrow> (f ---> l) net"
   567 unfolding tendsto_def le_net_def by fast
   568 
   569 lemma topological_tendstoI:
   570   "(\<And>S. open S \<Longrightarrow> l \<in> S \<Longrightarrow> eventually (\<lambda>x. f x \<in> S) net)
   571     \<Longrightarrow> (f ---> l) net"
   572   unfolding tendsto_def by auto
   573 
   574 lemma topological_tendstoD:
   575   "(f ---> l) net \<Longrightarrow> open S \<Longrightarrow> l \<in> S \<Longrightarrow> eventually (\<lambda>x. f x \<in> S) net"
   576   unfolding tendsto_def by auto
   577 
   578 lemma tendstoI:
   579   assumes "\<And>e. 0 < e \<Longrightarrow> eventually (\<lambda>x. dist (f x) l < e) net"
   580   shows "(f ---> l) net"
   581 apply (rule topological_tendstoI)
   582 apply (simp add: open_dist)
   583 apply (drule (1) bspec, clarify)
   584 apply (drule assms)
   585 apply (erule eventually_elim1, simp)
   586 done
   587 
   588 lemma tendstoD:
   589   "(f ---> l) net \<Longrightarrow> 0 < e \<Longrightarrow> eventually (\<lambda>x. dist (f x) l < e) net"
   590 apply (drule_tac S="{x. dist x l < e}" in topological_tendstoD)
   591 apply (clarsimp simp add: open_dist)
   592 apply (rule_tac x="e - dist x l" in exI, clarsimp)
   593 apply (simp only: less_diff_eq)
   594 apply (erule le_less_trans [OF dist_triangle])
   595 apply simp
   596 apply simp
   597 done
   598 
   599 lemma tendsto_iff:
   600   "(f ---> l) net \<longleftrightarrow> (\<forall>e>0. eventually (\<lambda>x. dist (f x) l < e) net)"
   601 using tendstoI tendstoD by fast
   602 
   603 lemma tendsto_Zfun_iff: "(f ---> a) net = Zfun (\<lambda>x. f x - a) net"
   604 by (simp only: tendsto_iff Zfun_def dist_norm)
   605 
   606 lemma tendsto_ident_at [tendsto_intros]: "((\<lambda>x. x) ---> a) (at a)"
   607 unfolding tendsto_def eventually_at_topological by auto
   608 
   609 lemma tendsto_ident_at_within [tendsto_intros]:
   610   "((\<lambda>x. x) ---> a) (at a within S)"
   611 unfolding tendsto_def eventually_within eventually_at_topological by auto
   612 
   613 lemma tendsto_const [tendsto_intros]: "((\<lambda>x. k) ---> k) net"
   614 by (simp add: tendsto_def)
   615 
   616 lemma tendsto_const_iff:
   617   fixes k l :: "'a::metric_space"
   618   assumes "net \<noteq> bot" shows "((\<lambda>n. k) ---> l) net \<longleftrightarrow> k = l"
   619 apply (safe intro!: tendsto_const)
   620 apply (rule ccontr)
   621 apply (drule_tac e="dist k l" in tendstoD)
   622 apply (simp add: zero_less_dist_iff)
   623 apply (simp add: eventually_False assms)
   624 done
   625 
   626 lemma tendsto_dist [tendsto_intros]:
   627   assumes f: "(f ---> l) net" and g: "(g ---> m) net"
   628   shows "((\<lambda>x. dist (f x) (g x)) ---> dist l m) net"
   629 proof (rule tendstoI)
   630   fix e :: real assume "0 < e"
   631   hence e2: "0 < e/2" by simp
   632   from tendstoD [OF f e2] tendstoD [OF g e2]
   633   show "eventually (\<lambda>x. dist (dist (f x) (g x)) (dist l m) < e) net"
   634   proof (rule eventually_elim2)
   635     fix x assume "dist (f x) l < e/2" "dist (g x) m < e/2"
   636     then show "dist (dist (f x) (g x)) (dist l m) < e"
   637       unfolding dist_real_def
   638       using dist_triangle2 [of "f x" "g x" "l"]
   639       using dist_triangle2 [of "g x" "l" "m"]
   640       using dist_triangle3 [of "l" "m" "f x"]
   641       using dist_triangle [of "f x" "m" "g x"]
   642       by arith
   643   qed
   644 qed
   645 
   646 lemma norm_conv_dist: "norm x = dist x 0"
   647 unfolding dist_norm by simp
   648 
   649 lemma tendsto_norm [tendsto_intros]:
   650   "(f ---> a) net \<Longrightarrow> ((\<lambda>x. norm (f x)) ---> norm a) net"
   651 unfolding norm_conv_dist by (intro tendsto_intros)
   652 
   653 lemma tendsto_norm_zero:
   654   "(f ---> 0) net \<Longrightarrow> ((\<lambda>x. norm (f x)) ---> 0) net"
   655 by (drule tendsto_norm, simp)
   656 
   657 lemma tendsto_norm_zero_cancel:
   658   "((\<lambda>x. norm (f x)) ---> 0) net \<Longrightarrow> (f ---> 0) net"
   659 unfolding tendsto_iff dist_norm by simp
   660 
   661 lemma tendsto_norm_zero_iff:
   662   "((\<lambda>x. norm (f x)) ---> 0) net \<longleftrightarrow> (f ---> 0) net"
   663 unfolding tendsto_iff dist_norm by simp
   664 
   665 lemma add_diff_add:
   666   fixes a b c d :: "'a::ab_group_add"
   667   shows "(a + c) - (b + d) = (a - b) + (c - d)"
   668 by simp
   669 
   670 lemma minus_diff_minus:
   671   fixes a b :: "'a::ab_group_add"
   672   shows "(- a) - (- b) = - (a - b)"
   673 by simp
   674 
   675 lemma tendsto_add [tendsto_intros]:
   676   fixes a b :: "'a::real_normed_vector"
   677   shows "\<lbrakk>(f ---> a) net; (g ---> b) net\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x + g x) ---> a + b) net"
   678 by (simp only: tendsto_Zfun_iff add_diff_add Zfun_add)
   679 
   680 lemma tendsto_minus [tendsto_intros]:
   681   fixes a :: "'a::real_normed_vector"
   682   shows "(f ---> a) net \<Longrightarrow> ((\<lambda>x. - f x) ---> - a) net"
   683 by (simp only: tendsto_Zfun_iff minus_diff_minus Zfun_minus)
   684 
   685 lemma tendsto_minus_cancel:
   686   fixes a :: "'a::real_normed_vector"
   687   shows "((\<lambda>x. - f x) ---> - a) net \<Longrightarrow> (f ---> a) net"
   688 by (drule tendsto_minus, simp)
   689 
   690 lemma tendsto_diff [tendsto_intros]:
   691   fixes a b :: "'a::real_normed_vector"
   692   shows "\<lbrakk>(f ---> a) net; (g ---> b) net\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x - g x) ---> a - b) net"
   693 by (simp add: diff_minus tendsto_add tendsto_minus)
   694 
   695 lemma tendsto_setsum [tendsto_intros]:
   696   fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c::real_normed_vector"
   697   assumes "\<And>i. i \<in> S \<Longrightarrow> (f i ---> a i) net"
   698   shows "((\<lambda>x. \<Sum>i\<in>S. f i x) ---> (\<Sum>i\<in>S. a i)) net"
   699 proof (cases "finite S")
   700   assume "finite S" thus ?thesis using assms
   701   proof (induct set: finite)
   702     case empty show ?case
   703       by (simp add: tendsto_const)
   704   next
   705     case (insert i F) thus ?case
   706       by (simp add: tendsto_add)
   707   qed
   708 next
   709   assume "\<not> finite S" thus ?thesis
   710     by (simp add: tendsto_const)
   711 qed
   712 
   713 lemma (in bounded_linear) tendsto [tendsto_intros]:
   714   "(g ---> a) net \<Longrightarrow> ((\<lambda>x. f (g x)) ---> f a) net"
   715 by (simp only: tendsto_Zfun_iff diff [symmetric] Zfun)
   716 
   717 lemma (in bounded_bilinear) tendsto [tendsto_intros]:
   718   "\<lbrakk>(f ---> a) net; (g ---> b) net\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x ** g x) ---> a ** b) net"
   719 by (simp only: tendsto_Zfun_iff prod_diff_prod
   720                Zfun_add Zfun Zfun_left Zfun_right)
   721 
   722 
   723 subsection {* Continuity of Inverse *}
   724 
   725 lemma (in bounded_bilinear) Zfun_prod_Bfun:
   726   assumes f: "Zfun f net"
   727   assumes g: "Bfun g net"
   728   shows "Zfun (\<lambda>x. f x ** g x) net"
   729 proof -
   730   obtain K where K: "0 \<le> K"
   731     and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"
   732     using nonneg_bounded by fast
   733   obtain B where B: "0 < B"
   734     and norm_g: "eventually (\<lambda>x. norm (g x) \<le> B) net"
   735     using g by (rule BfunE)
   736   have "eventually (\<lambda>x. norm (f x ** g x) \<le> norm (f x) * (B * K)) net"
   737   using norm_g proof (rule eventually_elim1)
   738     fix x
   739     assume *: "norm (g x) \<le> B"
   740     have "norm (f x ** g x) \<le> norm (f x) * norm (g x) * K"
   741       by (rule norm_le)
   742     also have "\<dots> \<le> norm (f x) * B * K"
   743       by (intro mult_mono' order_refl norm_g norm_ge_zero
   744                 mult_nonneg_nonneg K *)
   745     also have "\<dots> = norm (f x) * (B * K)"
   746       by (rule mult_assoc)
   747     finally show "norm (f x ** g x) \<le> norm (f x) * (B * K)" .
   748   qed
   749   with f show ?thesis
   750     by (rule Zfun_imp_Zfun)
   751 qed
   752 
   753 lemma (in bounded_bilinear) flip:
   754   "bounded_bilinear (\<lambda>x y. y ** x)"
   755 apply default
   756 apply (rule add_right)
   757 apply (rule add_left)
   758 apply (rule scaleR_right)
   759 apply (rule scaleR_left)
   760 apply (subst mult_commute)
   761 using bounded by fast
   762 
   763 lemma (in bounded_bilinear) Bfun_prod_Zfun:
   764   assumes f: "Bfun f net"
   765   assumes g: "Zfun g net"
   766   shows "Zfun (\<lambda>x. f x ** g x) net"
   767 using flip g f by (rule bounded_bilinear.Zfun_prod_Bfun)
   768 
   769 lemma inverse_diff_inverse:
   770   "\<lbrakk>(a::'a::division_ring) \<noteq> 0; b \<noteq> 0\<rbrakk>
   771    \<Longrightarrow> inverse a - inverse b = - (inverse a * (a - b) * inverse b)"
   772 by (simp add: algebra_simps)
   773 
   774 lemma Bfun_inverse_lemma:
   775   fixes x :: "'a::real_normed_div_algebra"
   776   shows "\<lbrakk>r \<le> norm x; 0 < r\<rbrakk> \<Longrightarrow> norm (inverse x) \<le> inverse r"
   777 apply (subst nonzero_norm_inverse, clarsimp)
   778 apply (erule (1) le_imp_inverse_le)
   779 done
   780 
   781 lemma Bfun_inverse:
   782   fixes a :: "'a::real_normed_div_algebra"
   783   assumes f: "(f ---> a) net"
   784   assumes a: "a \<noteq> 0"
   785   shows "Bfun (\<lambda>x. inverse (f x)) net"
   786 proof -
   787   from a have "0 < norm a" by simp
   788   hence "\<exists>r>0. r < norm a" by (rule dense)
   789   then obtain r where r1: "0 < r" and r2: "r < norm a" by fast
   790   have "eventually (\<lambda>x. dist (f x) a < r) net"
   791     using tendstoD [OF f r1] by fast
   792   hence "eventually (\<lambda>x. norm (inverse (f x)) \<le> inverse (norm a - r)) net"
   793   proof (rule eventually_elim1)
   794     fix x
   795     assume "dist (f x) a < r"
   796     hence 1: "norm (f x - a) < r"
   797       by (simp add: dist_norm)
   798     hence 2: "f x \<noteq> 0" using r2 by auto
   799     hence "norm (inverse (f x)) = inverse (norm (f x))"
   800       by (rule nonzero_norm_inverse)
   801     also have "\<dots> \<le> inverse (norm a - r)"
   802     proof (rule le_imp_inverse_le)
   803       show "0 < norm a - r" using r2 by simp
   804     next
   805       have "norm a - norm (f x) \<le> norm (a - f x)"
   806         by (rule norm_triangle_ineq2)
   807       also have "\<dots> = norm (f x - a)"
   808         by (rule norm_minus_commute)
   809       also have "\<dots> < r" using 1 .
   810       finally show "norm a - r \<le> norm (f x)" by simp
   811     qed
   812     finally show "norm (inverse (f x)) \<le> inverse (norm a - r)" .
   813   qed
   814   thus ?thesis by (rule BfunI)
   815 qed
   816 
   817 lemma tendsto_inverse_lemma:
   818   fixes a :: "'a::real_normed_div_algebra"
   819   shows "\<lbrakk>(f ---> a) net; a \<noteq> 0; eventually (\<lambda>x. f x \<noteq> 0) net\<rbrakk>
   820          \<Longrightarrow> ((\<lambda>x. inverse (f x)) ---> inverse a) net"
   821 apply (subst tendsto_Zfun_iff)
   822 apply (rule Zfun_ssubst)
   823 apply (erule eventually_elim1)
   824 apply (erule (1) inverse_diff_inverse)
   825 apply (rule Zfun_minus)
   826 apply (rule Zfun_mult_left)
   827 apply (rule mult.Bfun_prod_Zfun)
   828 apply (erule (1) Bfun_inverse)
   829 apply (simp add: tendsto_Zfun_iff)
   830 done
   831 
   832 lemma tendsto_inverse [tendsto_intros]:
   833   fixes a :: "'a::real_normed_div_algebra"
   834   assumes f: "(f ---> a) net"
   835   assumes a: "a \<noteq> 0"
   836   shows "((\<lambda>x. inverse (f x)) ---> inverse a) net"
   837 proof -
   838   from a have "0 < norm a" by simp
   839   with f have "eventually (\<lambda>x. dist (f x) a < norm a) net"
   840     by (rule tendstoD)
   841   then have "eventually (\<lambda>x. f x \<noteq> 0) net"
   842     unfolding dist_norm by (auto elim!: eventually_elim1)
   843   with f a show ?thesis
   844     by (rule tendsto_inverse_lemma)
   845 qed
   846 
   847 lemma tendsto_divide [tendsto_intros]:
   848   fixes a b :: "'a::real_normed_field"
   849   shows "\<lbrakk>(f ---> a) net; (g ---> b) net; b \<noteq> 0\<rbrakk>
   850     \<Longrightarrow> ((\<lambda>x. f x / g x) ---> a / b) net"
   851 by (simp add: mult.tendsto tendsto_inverse divide_inverse)
   852 
   853 end