src/HOL/Limits.thy
 author haftmann Mon Nov 29 13:44:54 2010 +0100 (2010-11-29) changeset 40815 6e2d17cc0d1d parent 39302 d7728f65b353 child 41970 47d6e13d1710 permissions -rw-r--r--
equivI has replaced equiv.intro
```     1 (*  Title       : Limits.thy
```
```     2     Author      : Brian Huffman
```
```     3 *)
```
```     4
```
```     5 header {* Filters and Limits *}
```
```     6
```
```     7 theory Limits
```
```     8 imports RealVector
```
```     9 begin
```
```    10
```
```    11 subsection {* Nets *}
```
```    12
```
```    13 text {*
```
```    14   A net is now defined simply as a filter on a set.
```
```    15   The definition also allows non-proper filters.
```
```    16 *}
```
```    17
```
```    18 locale is_filter =
```
```    19   fixes net :: "('a \<Rightarrow> bool) \<Rightarrow> bool"
```
```    20   assumes True: "net (\<lambda>x. True)"
```
```    21   assumes conj: "net (\<lambda>x. P x) \<Longrightarrow> net (\<lambda>x. Q x) \<Longrightarrow> net (\<lambda>x. P x \<and> Q x)"
```
```    22   assumes mono: "\<forall>x. P x \<longrightarrow> Q x \<Longrightarrow> net (\<lambda>x. P x) \<Longrightarrow> net (\<lambda>x. Q x)"
```
```    23
```
```    24 typedef (open) 'a net =
```
```    25   "{net :: ('a \<Rightarrow> bool) \<Rightarrow> bool. is_filter net}"
```
```    26 proof
```
```    27   show "(\<lambda>x. True) \<in> ?net" by (auto intro: is_filter.intro)
```
```    28 qed
```
```    29
```
```    30 lemma is_filter_Rep_net: "is_filter (Rep_net net)"
```
```    31 using Rep_net [of net] by simp
```
```    32
```
```    33 lemma Abs_net_inverse':
```
```    34   assumes "is_filter net" shows "Rep_net (Abs_net net) = net"
```
```    35 using assms by (simp add: Abs_net_inverse)
```
```    36
```
```    37
```
```    38 subsection {* Eventually *}
```
```    39
```
```    40 definition eventually :: "('a \<Rightarrow> bool) \<Rightarrow> 'a net \<Rightarrow> bool" where
```
```    41   "eventually P net \<longleftrightarrow> Rep_net net P"
```
```    42
```
```    43 lemma eventually_Abs_net:
```
```    44   assumes "is_filter net" shows "eventually P (Abs_net net) = net P"
```
```    45 unfolding eventually_def using assms by (simp add: Abs_net_inverse)
```
```    46
```
```    47 lemma expand_net_eq:
```
```    48   shows "net = net' \<longleftrightarrow> (\<forall>P. eventually P net = eventually P net')"
```
```    49 unfolding Rep_net_inject [symmetric] fun_eq_iff eventually_def ..
```
```    50
```
```    51 lemma eventually_True [simp]: "eventually (\<lambda>x. True) net"
```
```    52 unfolding eventually_def
```
```    53 by (rule is_filter.True [OF is_filter_Rep_net])
```
```    54
```
```    55 lemma always_eventually: "\<forall>x. P x \<Longrightarrow> eventually P net"
```
```    56 proof -
```
```    57   assume "\<forall>x. P x" hence "P = (\<lambda>x. True)" by (simp add: ext)
```
```    58   thus "eventually P net" by simp
```
```    59 qed
```
```    60
```
```    61 lemma eventually_mono:
```
```    62   "(\<forall>x. P x \<longrightarrow> Q x) \<Longrightarrow> eventually P net \<Longrightarrow> eventually Q net"
```
```    63 unfolding eventually_def
```
```    64 by (rule is_filter.mono [OF is_filter_Rep_net])
```
```    65
```
```    66 lemma eventually_conj:
```
```    67   assumes P: "eventually (\<lambda>x. P x) net"
```
```    68   assumes Q: "eventually (\<lambda>x. Q x) net"
```
```    69   shows "eventually (\<lambda>x. P x \<and> Q x) net"
```
```    70 using assms unfolding eventually_def
```
```    71 by (rule is_filter.conj [OF is_filter_Rep_net])
```
```    72
```
```    73 lemma eventually_mp:
```
```    74   assumes "eventually (\<lambda>x. P x \<longrightarrow> Q x) net"
```
```    75   assumes "eventually (\<lambda>x. P x) net"
```
```    76   shows "eventually (\<lambda>x. Q x) net"
```
```    77 proof (rule eventually_mono)
```
```    78   show "\<forall>x. (P x \<longrightarrow> Q x) \<and> P x \<longrightarrow> Q x" by simp
```
```    79   show "eventually (\<lambda>x. (P x \<longrightarrow> Q x) \<and> P x) net"
```
```    80     using assms by (rule eventually_conj)
```
```    81 qed
```
```    82
```
```    83 lemma eventually_rev_mp:
```
```    84   assumes "eventually (\<lambda>x. P x) net"
```
```    85   assumes "eventually (\<lambda>x. P x \<longrightarrow> Q x) net"
```
```    86   shows "eventually (\<lambda>x. Q x) net"
```
```    87 using assms(2) assms(1) by (rule eventually_mp)
```
```    88
```
```    89 lemma eventually_conj_iff:
```
```    90   "eventually (\<lambda>x. P x \<and> Q x) net \<longleftrightarrow> eventually P net \<and> eventually Q net"
```
```    91 by (auto intro: eventually_conj elim: eventually_rev_mp)
```
```    92
```
```    93 lemma eventually_elim1:
```
```    94   assumes "eventually (\<lambda>i. P i) net"
```
```    95   assumes "\<And>i. P i \<Longrightarrow> Q i"
```
```    96   shows "eventually (\<lambda>i. Q i) net"
```
```    97 using assms by (auto elim!: eventually_rev_mp)
```
```    98
```
```    99 lemma eventually_elim2:
```
```   100   assumes "eventually (\<lambda>i. P i) net"
```
```   101   assumes "eventually (\<lambda>i. Q i) net"
```
```   102   assumes "\<And>i. P i \<Longrightarrow> Q i \<Longrightarrow> R i"
```
```   103   shows "eventually (\<lambda>i. R i) net"
```
```   104 using assms by (auto elim!: eventually_rev_mp)
```
```   105
```
```   106
```
```   107 subsection {* Finer-than relation *}
```
```   108
```
```   109 text {* @{term "net \<le> net'"} means that @{term net} is finer than
```
```   110 @{term net'}. *}
```
```   111
```
```   112 instantiation net :: (type) complete_lattice
```
```   113 begin
```
```   114
```
```   115 definition
```
```   116   le_net_def: "net \<le> net' \<longleftrightarrow> (\<forall>P. eventually P net' \<longrightarrow> eventually P net)"
```
```   117
```
```   118 definition
```
```   119   less_net_def: "(net :: 'a net) < net' \<longleftrightarrow> net \<le> net' \<and> \<not> net' \<le> net"
```
```   120
```
```   121 definition
```
```   122   top_net_def: "top = Abs_net (\<lambda>P. \<forall>x. P x)"
```
```   123
```
```   124 definition
```
```   125   bot_net_def: "bot = Abs_net (\<lambda>P. True)"
```
```   126
```
```   127 definition
```
```   128   sup_net_def: "sup net net' = Abs_net (\<lambda>P. eventually P net \<and> eventually P net')"
```
```   129
```
```   130 definition
```
```   131   inf_net_def: "inf a b = Abs_net
```
```   132       (\<lambda>P. \<exists>Q R. eventually Q a \<and> eventually R b \<and> (\<forall>x. Q x \<and> R x \<longrightarrow> P x))"
```
```   133
```
```   134 definition
```
```   135   Sup_net_def: "Sup A = Abs_net (\<lambda>P. \<forall>net\<in>A. eventually P net)"
```
```   136
```
```   137 definition
```
```   138   Inf_net_def: "Inf A = Sup {x::'a net. \<forall>y\<in>A. x \<le> y}"
```
```   139
```
```   140 lemma eventually_top [simp]: "eventually P top \<longleftrightarrow> (\<forall>x. P x)"
```
```   141 unfolding top_net_def
```
```   142 by (rule eventually_Abs_net, rule is_filter.intro, auto)
```
```   143
```
```   144 lemma eventually_bot [simp]: "eventually P bot"
```
```   145 unfolding bot_net_def
```
```   146 by (subst eventually_Abs_net, rule is_filter.intro, auto)
```
```   147
```
```   148 lemma eventually_sup:
```
```   149   "eventually P (sup net net') \<longleftrightarrow> eventually P net \<and> eventually P net'"
```
```   150 unfolding sup_net_def
```
```   151 by (rule eventually_Abs_net, rule is_filter.intro)
```
```   152    (auto elim!: eventually_rev_mp)
```
```   153
```
```   154 lemma eventually_inf:
```
```   155   "eventually P (inf a b) \<longleftrightarrow>
```
```   156    (\<exists>Q R. eventually Q a \<and> eventually R b \<and> (\<forall>x. Q x \<and> R x \<longrightarrow> P x))"
```
```   157 unfolding inf_net_def
```
```   158 apply (rule eventually_Abs_net, rule is_filter.intro)
```
```   159 apply (fast intro: eventually_True)
```
```   160 apply clarify
```
```   161 apply (intro exI conjI)
```
```   162 apply (erule (1) eventually_conj)
```
```   163 apply (erule (1) eventually_conj)
```
```   164 apply simp
```
```   165 apply auto
```
```   166 done
```
```   167
```
```   168 lemma eventually_Sup:
```
```   169   "eventually P (Sup A) \<longleftrightarrow> (\<forall>net\<in>A. eventually P net)"
```
```   170 unfolding Sup_net_def
```
```   171 apply (rule eventually_Abs_net, rule is_filter.intro)
```
```   172 apply (auto intro: eventually_conj elim!: eventually_rev_mp)
```
```   173 done
```
```   174
```
```   175 instance proof
```
```   176   fix x y :: "'a net" show "x < y \<longleftrightarrow> x \<le> y \<and> \<not> y \<le> x"
```
```   177     by (rule less_net_def)
```
```   178 next
```
```   179   fix x :: "'a net" show "x \<le> x"
```
```   180     unfolding le_net_def by simp
```
```   181 next
```
```   182   fix x y z :: "'a net" assume "x \<le> y" and "y \<le> z" thus "x \<le> z"
```
```   183     unfolding le_net_def by simp
```
```   184 next
```
```   185   fix x y :: "'a net" assume "x \<le> y" and "y \<le> x" thus "x = y"
```
```   186     unfolding le_net_def expand_net_eq by fast
```
```   187 next
```
```   188   fix x :: "'a net" show "x \<le> top"
```
```   189     unfolding le_net_def eventually_top by (simp add: always_eventually)
```
```   190 next
```
```   191   fix x :: "'a net" show "bot \<le> x"
```
```   192     unfolding le_net_def by simp
```
```   193 next
```
```   194   fix x y :: "'a net" show "x \<le> sup x y" and "y \<le> sup x y"
```
```   195     unfolding le_net_def eventually_sup by simp_all
```
```   196 next
```
```   197   fix x y z :: "'a net" assume "x \<le> z" and "y \<le> z" thus "sup x y \<le> z"
```
```   198     unfolding le_net_def eventually_sup by simp
```
```   199 next
```
```   200   fix x y :: "'a net" show "inf x y \<le> x" and "inf x y \<le> y"
```
```   201     unfolding le_net_def eventually_inf by (auto intro: eventually_True)
```
```   202 next
```
```   203   fix x y z :: "'a net" assume "x \<le> y" and "x \<le> z" thus "x \<le> inf y z"
```
```   204     unfolding le_net_def eventually_inf
```
```   205     by (auto elim!: eventually_mono intro: eventually_conj)
```
```   206 next
```
```   207   fix x :: "'a net" and A assume "x \<in> A" thus "x \<le> Sup A"
```
```   208     unfolding le_net_def eventually_Sup by simp
```
```   209 next
```
```   210   fix A and y :: "'a net" assume "\<And>x. x \<in> A \<Longrightarrow> x \<le> y" thus "Sup A \<le> y"
```
```   211     unfolding le_net_def eventually_Sup by simp
```
```   212 next
```
```   213   fix z :: "'a net" and A assume "z \<in> A" thus "Inf A \<le> z"
```
```   214     unfolding le_net_def Inf_net_def eventually_Sup Ball_def by simp
```
```   215 next
```
```   216   fix A and x :: "'a net" assume "\<And>y. y \<in> A \<Longrightarrow> x \<le> y" thus "x \<le> Inf A"
```
```   217     unfolding le_net_def Inf_net_def eventually_Sup Ball_def by simp
```
```   218 qed
```
```   219
```
```   220 end
```
```   221
```
```   222 lemma net_leD:
```
```   223   "net \<le> net' \<Longrightarrow> eventually P net' \<Longrightarrow> eventually P net"
```
```   224 unfolding le_net_def by simp
```
```   225
```
```   226 lemma net_leI:
```
```   227   "(\<And>P. eventually P net' \<Longrightarrow> eventually P net) \<Longrightarrow> net \<le> net'"
```
```   228 unfolding le_net_def by simp
```
```   229
```
```   230 lemma eventually_False:
```
```   231   "eventually (\<lambda>x. False) net \<longleftrightarrow> net = bot"
```
```   232 unfolding expand_net_eq by (auto elim: eventually_rev_mp)
```
```   233
```
```   234
```
```   235 subsection {* Map function for nets *}
```
```   236
```
```   237 definition netmap :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a net \<Rightarrow> 'b net" where
```
```   238   "netmap f net = Abs_net (\<lambda>P. eventually (\<lambda>x. P (f x)) net)"
```
```   239
```
```   240 lemma eventually_netmap:
```
```   241   "eventually P (netmap f net) = eventually (\<lambda>x. P (f x)) net"
```
```   242 unfolding netmap_def
```
```   243 apply (rule eventually_Abs_net)
```
```   244 apply (rule is_filter.intro)
```
```   245 apply (auto elim!: eventually_rev_mp)
```
```   246 done
```
```   247
```
```   248 lemma netmap_ident: "netmap (\<lambda>x. x) net = net"
```
```   249 by (simp add: expand_net_eq eventually_netmap)
```
```   250
```
```   251 lemma netmap_netmap: "netmap f (netmap g net) = netmap (\<lambda>x. f (g x)) net"
```
```   252 by (simp add: expand_net_eq eventually_netmap)
```
```   253
```
```   254 lemma netmap_mono: "net \<le> net' \<Longrightarrow> netmap f net \<le> netmap f net'"
```
```   255 unfolding le_net_def eventually_netmap by simp
```
```   256
```
```   257 lemma netmap_bot [simp]: "netmap f bot = bot"
```
```   258 by (simp add: expand_net_eq eventually_netmap)
```
```   259
```
```   260
```
```   261 subsection {* Sequentially *}
```
```   262
```
```   263 definition sequentially :: "nat net" where
```
```   264   "sequentially = Abs_net (\<lambda>P. \<exists>k. \<forall>n\<ge>k. P n)"
```
```   265
```
```   266 lemma eventually_sequentially:
```
```   267   "eventually P sequentially \<longleftrightarrow> (\<exists>N. \<forall>n\<ge>N. P n)"
```
```   268 unfolding sequentially_def
```
```   269 proof (rule eventually_Abs_net, rule is_filter.intro)
```
```   270   fix P Q :: "nat \<Rightarrow> bool"
```
```   271   assume "\<exists>i. \<forall>n\<ge>i. P n" and "\<exists>j. \<forall>n\<ge>j. Q n"
```
```   272   then obtain i j where "\<forall>n\<ge>i. P n" and "\<forall>n\<ge>j. Q n" by auto
```
```   273   then have "\<forall>n\<ge>max i j. P n \<and> Q n" by simp
```
```   274   then show "\<exists>k. \<forall>n\<ge>k. P n \<and> Q n" ..
```
```   275 qed auto
```
```   276
```
```   277 lemma sequentially_bot [simp]: "sequentially \<noteq> bot"
```
```   278 unfolding expand_net_eq eventually_sequentially by auto
```
```   279
```
```   280 lemma eventually_False_sequentially [simp]:
```
```   281   "\<not> eventually (\<lambda>n. False) sequentially"
```
```   282 by (simp add: eventually_False)
```
```   283
```
```   284 lemma le_sequentially:
```
```   285   "net \<le> sequentially \<longleftrightarrow> (\<forall>N. eventually (\<lambda>n. N \<le> n) net)"
```
```   286 unfolding le_net_def eventually_sequentially
```
```   287 by (safe, fast, drule_tac x=N in spec, auto elim: eventually_rev_mp)
```
```   288
```
```   289
```
```   290 subsection {* Standard Nets *}
```
```   291
```
```   292 definition within :: "'a net \<Rightarrow> 'a set \<Rightarrow> 'a net" (infixr "within" 70) where
```
```   293   "net within S = Abs_net (\<lambda>P. eventually (\<lambda>x. x \<in> S \<longrightarrow> P x) net)"
```
```   294
```
```   295 definition nhds :: "'a::topological_space \<Rightarrow> 'a net" where
```
```   296   "nhds a = Abs_net (\<lambda>P. \<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x))"
```
```   297
```
```   298 definition at :: "'a::topological_space \<Rightarrow> 'a net" where
```
```   299   "at a = nhds a within - {a}"
```
```   300
```
```   301 lemma eventually_within:
```
```   302   "eventually P (net within S) = eventually (\<lambda>x. x \<in> S \<longrightarrow> P x) net"
```
```   303 unfolding within_def
```
```   304 by (rule eventually_Abs_net, rule is_filter.intro)
```
```   305    (auto elim!: eventually_rev_mp)
```
```   306
```
```   307 lemma within_UNIV: "net within UNIV = net"
```
```   308   unfolding expand_net_eq eventually_within by simp
```
```   309
```
```   310 lemma eventually_nhds:
```
```   311   "eventually P (nhds a) \<longleftrightarrow> (\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x))"
```
```   312 unfolding nhds_def
```
```   313 proof (rule eventually_Abs_net, rule is_filter.intro)
```
```   314   have "open UNIV \<and> a \<in> UNIV \<and> (\<forall>x\<in>UNIV. True)" by simp
```
```   315   thus "\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. True)" by - rule
```
```   316 next
```
```   317   fix P Q
```
```   318   assume "\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x)"
```
```   319      and "\<exists>T. open T \<and> a \<in> T \<and> (\<forall>x\<in>T. Q x)"
```
```   320   then obtain S T where
```
```   321     "open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x)"
```
```   322     "open T \<and> a \<in> T \<and> (\<forall>x\<in>T. Q x)" by auto
```
```   323   hence "open (S \<inter> T) \<and> a \<in> S \<inter> T \<and> (\<forall>x\<in>(S \<inter> T). P x \<and> Q x)"
```
```   324     by (simp add: open_Int)
```
```   325   thus "\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x \<and> Q x)" by - rule
```
```   326 qed auto
```
```   327
```
```   328 lemma eventually_nhds_metric:
```
```   329   "eventually P (nhds a) \<longleftrightarrow> (\<exists>d>0. \<forall>x. dist x a < d \<longrightarrow> P x)"
```
```   330 unfolding eventually_nhds open_dist
```
```   331 apply safe
```
```   332 apply fast
```
```   333 apply (rule_tac x="{x. dist x a < d}" in exI, simp)
```
```   334 apply clarsimp
```
```   335 apply (rule_tac x="d - dist x a" in exI, clarsimp)
```
```   336 apply (simp only: less_diff_eq)
```
```   337 apply (erule le_less_trans [OF dist_triangle])
```
```   338 done
```
```   339
```
```   340 lemma eventually_at_topological:
```
```   341   "eventually P (at a) \<longleftrightarrow> (\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. x \<noteq> a \<longrightarrow> P x))"
```
```   342 unfolding at_def eventually_within eventually_nhds by simp
```
```   343
```
```   344 lemma eventually_at:
```
```   345   fixes a :: "'a::metric_space"
```
```   346   shows "eventually P (at a) \<longleftrightarrow> (\<exists>d>0. \<forall>x. x \<noteq> a \<and> dist x a < d \<longrightarrow> P x)"
```
```   347 unfolding at_def eventually_within eventually_nhds_metric by auto
```
```   348
```
```   349
```
```   350 subsection {* Boundedness *}
```
```   351
```
```   352 definition Bfun :: "('a \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a net \<Rightarrow> bool" where
```
```   353   "Bfun f net = (\<exists>K>0. eventually (\<lambda>x. norm (f x) \<le> K) net)"
```
```   354
```
```   355 lemma BfunI:
```
```   356   assumes K: "eventually (\<lambda>x. norm (f x) \<le> K) net" shows "Bfun f net"
```
```   357 unfolding Bfun_def
```
```   358 proof (intro exI conjI allI)
```
```   359   show "0 < max K 1" by simp
```
```   360 next
```
```   361   show "eventually (\<lambda>x. norm (f x) \<le> max K 1) net"
```
```   362     using K by (rule eventually_elim1, simp)
```
```   363 qed
```
```   364
```
```   365 lemma BfunE:
```
```   366   assumes "Bfun f net"
```
```   367   obtains B where "0 < B" and "eventually (\<lambda>x. norm (f x) \<le> B) net"
```
```   368 using assms unfolding Bfun_def by fast
```
```   369
```
```   370
```
```   371 subsection {* Convergence to Zero *}
```
```   372
```
```   373 definition Zfun :: "('a \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a net \<Rightarrow> bool" where
```
```   374   "Zfun f net = (\<forall>r>0. eventually (\<lambda>x. norm (f x) < r) net)"
```
```   375
```
```   376 lemma ZfunI:
```
```   377   "(\<And>r. 0 < r \<Longrightarrow> eventually (\<lambda>x. norm (f x) < r) net) \<Longrightarrow> Zfun f net"
```
```   378 unfolding Zfun_def by simp
```
```   379
```
```   380 lemma ZfunD:
```
```   381   "\<lbrakk>Zfun f net; 0 < r\<rbrakk> \<Longrightarrow> eventually (\<lambda>x. norm (f x) < r) net"
```
```   382 unfolding Zfun_def by simp
```
```   383
```
```   384 lemma Zfun_ssubst:
```
```   385   "eventually (\<lambda>x. f x = g x) net \<Longrightarrow> Zfun g net \<Longrightarrow> Zfun f net"
```
```   386 unfolding Zfun_def by (auto elim!: eventually_rev_mp)
```
```   387
```
```   388 lemma Zfun_zero: "Zfun (\<lambda>x. 0) net"
```
```   389 unfolding Zfun_def by simp
```
```   390
```
```   391 lemma Zfun_norm_iff: "Zfun (\<lambda>x. norm (f x)) net = Zfun (\<lambda>x. f x) net"
```
```   392 unfolding Zfun_def by simp
```
```   393
```
```   394 lemma Zfun_imp_Zfun:
```
```   395   assumes f: "Zfun f net"
```
```   396   assumes g: "eventually (\<lambda>x. norm (g x) \<le> norm (f x) * K) net"
```
```   397   shows "Zfun (\<lambda>x. g x) net"
```
```   398 proof (cases)
```
```   399   assume K: "0 < K"
```
```   400   show ?thesis
```
```   401   proof (rule ZfunI)
```
```   402     fix r::real assume "0 < r"
```
```   403     hence "0 < r / K"
```
```   404       using K by (rule divide_pos_pos)
```
```   405     then have "eventually (\<lambda>x. norm (f x) < r / K) net"
```
```   406       using ZfunD [OF f] by fast
```
```   407     with g show "eventually (\<lambda>x. norm (g x) < r) net"
```
```   408     proof (rule eventually_elim2)
```
```   409       fix x
```
```   410       assume *: "norm (g x) \<le> norm (f x) * K"
```
```   411       assume "norm (f x) < r / K"
```
```   412       hence "norm (f x) * K < r"
```
```   413         by (simp add: pos_less_divide_eq K)
```
```   414       thus "norm (g x) < r"
```
```   415         by (simp add: order_le_less_trans [OF *])
```
```   416     qed
```
```   417   qed
```
```   418 next
```
```   419   assume "\<not> 0 < K"
```
```   420   hence K: "K \<le> 0" by (simp only: not_less)
```
```   421   show ?thesis
```
```   422   proof (rule ZfunI)
```
```   423     fix r :: real
```
```   424     assume "0 < r"
```
```   425     from g show "eventually (\<lambda>x. norm (g x) < r) net"
```
```   426     proof (rule eventually_elim1)
```
```   427       fix x
```
```   428       assume "norm (g x) \<le> norm (f x) * K"
```
```   429       also have "\<dots> \<le> norm (f x) * 0"
```
```   430         using K norm_ge_zero by (rule mult_left_mono)
```
```   431       finally show "norm (g x) < r"
```
```   432         using `0 < r` by simp
```
```   433     qed
```
```   434   qed
```
```   435 qed
```
```   436
```
```   437 lemma Zfun_le: "\<lbrakk>Zfun g net; \<forall>x. norm (f x) \<le> norm (g x)\<rbrakk> \<Longrightarrow> Zfun f net"
```
```   438 by (erule_tac K="1" in Zfun_imp_Zfun, simp)
```
```   439
```
```   440 lemma Zfun_add:
```
```   441   assumes f: "Zfun f net" and g: "Zfun g net"
```
```   442   shows "Zfun (\<lambda>x. f x + g x) net"
```
```   443 proof (rule ZfunI)
```
```   444   fix r::real assume "0 < r"
```
```   445   hence r: "0 < r / 2" by simp
```
```   446   have "eventually (\<lambda>x. norm (f x) < r/2) net"
```
```   447     using f r by (rule ZfunD)
```
```   448   moreover
```
```   449   have "eventually (\<lambda>x. norm (g x) < r/2) net"
```
```   450     using g r by (rule ZfunD)
```
```   451   ultimately
```
```   452   show "eventually (\<lambda>x. norm (f x + g x) < r) net"
```
```   453   proof (rule eventually_elim2)
```
```   454     fix x
```
```   455     assume *: "norm (f x) < r/2" "norm (g x) < r/2"
```
```   456     have "norm (f x + g x) \<le> norm (f x) + norm (g x)"
```
```   457       by (rule norm_triangle_ineq)
```
```   458     also have "\<dots> < r/2 + r/2"
```
```   459       using * by (rule add_strict_mono)
```
```   460     finally show "norm (f x + g x) < r"
```
```   461       by simp
```
```   462   qed
```
```   463 qed
```
```   464
```
```   465 lemma Zfun_minus: "Zfun f net \<Longrightarrow> Zfun (\<lambda>x. - f x) net"
```
```   466 unfolding Zfun_def by simp
```
```   467
```
```   468 lemma Zfun_diff: "\<lbrakk>Zfun f net; Zfun g net\<rbrakk> \<Longrightarrow> Zfun (\<lambda>x. f x - g x) net"
```
```   469 by (simp only: diff_minus Zfun_add Zfun_minus)
```
```   470
```
```   471 lemma (in bounded_linear) Zfun:
```
```   472   assumes g: "Zfun g net"
```
```   473   shows "Zfun (\<lambda>x. f (g x)) net"
```
```   474 proof -
```
```   475   obtain K where "\<And>x. norm (f x) \<le> norm x * K"
```
```   476     using bounded by fast
```
```   477   then have "eventually (\<lambda>x. norm (f (g x)) \<le> norm (g x) * K) net"
```
```   478     by simp
```
```   479   with g show ?thesis
```
```   480     by (rule Zfun_imp_Zfun)
```
```   481 qed
```
```   482
```
```   483 lemma (in bounded_bilinear) Zfun:
```
```   484   assumes f: "Zfun f net"
```
```   485   assumes g: "Zfun g net"
```
```   486   shows "Zfun (\<lambda>x. f x ** g x) net"
```
```   487 proof (rule ZfunI)
```
```   488   fix r::real assume r: "0 < r"
```
```   489   obtain K where K: "0 < K"
```
```   490     and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"
```
```   491     using pos_bounded by fast
```
```   492   from K have K': "0 < inverse K"
```
```   493     by (rule positive_imp_inverse_positive)
```
```   494   have "eventually (\<lambda>x. norm (f x) < r) net"
```
```   495     using f r by (rule ZfunD)
```
```   496   moreover
```
```   497   have "eventually (\<lambda>x. norm (g x) < inverse K) net"
```
```   498     using g K' by (rule ZfunD)
```
```   499   ultimately
```
```   500   show "eventually (\<lambda>x. norm (f x ** g x) < r) net"
```
```   501   proof (rule eventually_elim2)
```
```   502     fix x
```
```   503     assume *: "norm (f x) < r" "norm (g x) < inverse K"
```
```   504     have "norm (f x ** g x) \<le> norm (f x) * norm (g x) * K"
```
```   505       by (rule norm_le)
```
```   506     also have "norm (f x) * norm (g x) * K < r * inverse K * K"
```
```   507       by (intro mult_strict_right_mono mult_strict_mono' norm_ge_zero * K)
```
```   508     also from K have "r * inverse K * K = r"
```
```   509       by simp
```
```   510     finally show "norm (f x ** g x) < r" .
```
```   511   qed
```
```   512 qed
```
```   513
```
```   514 lemma (in bounded_bilinear) Zfun_left:
```
```   515   "Zfun f net \<Longrightarrow> Zfun (\<lambda>x. f x ** a) net"
```
```   516 by (rule bounded_linear_left [THEN bounded_linear.Zfun])
```
```   517
```
```   518 lemma (in bounded_bilinear) Zfun_right:
```
```   519   "Zfun f net \<Longrightarrow> Zfun (\<lambda>x. a ** f x) net"
```
```   520 by (rule bounded_linear_right [THEN bounded_linear.Zfun])
```
```   521
```
```   522 lemmas Zfun_mult = mult.Zfun
```
```   523 lemmas Zfun_mult_right = mult.Zfun_right
```
```   524 lemmas Zfun_mult_left = mult.Zfun_left
```
```   525
```
```   526
```
```   527 subsection {* Limits *}
```
```   528
```
```   529 definition tendsto :: "('a \<Rightarrow> 'b::topological_space) \<Rightarrow> 'b \<Rightarrow> 'a net \<Rightarrow> bool"
```
```   530     (infixr "--->" 55) where
```
```   531   "(f ---> l) net \<longleftrightarrow> (\<forall>S. open S \<longrightarrow> l \<in> S \<longrightarrow> eventually (\<lambda>x. f x \<in> S) net)"
```
```   532
```
```   533 ML {*
```
```   534 structure Tendsto_Intros = Named_Thms
```
```   535 (
```
```   536   val name = "tendsto_intros"
```
```   537   val description = "introduction rules for tendsto"
```
```   538 )
```
```   539 *}
```
```   540
```
```   541 setup Tendsto_Intros.setup
```
```   542
```
```   543 lemma tendsto_mono: "net \<le> net' \<Longrightarrow> (f ---> l) net' \<Longrightarrow> (f ---> l) net"
```
```   544 unfolding tendsto_def le_net_def by fast
```
```   545
```
```   546 lemma topological_tendstoI:
```
```   547   "(\<And>S. open S \<Longrightarrow> l \<in> S \<Longrightarrow> eventually (\<lambda>x. f x \<in> S) net)
```
```   548     \<Longrightarrow> (f ---> l) net"
```
```   549   unfolding tendsto_def by auto
```
```   550
```
```   551 lemma topological_tendstoD:
```
```   552   "(f ---> l) net \<Longrightarrow> open S \<Longrightarrow> l \<in> S \<Longrightarrow> eventually (\<lambda>x. f x \<in> S) net"
```
```   553   unfolding tendsto_def by auto
```
```   554
```
```   555 lemma tendstoI:
```
```   556   assumes "\<And>e. 0 < e \<Longrightarrow> eventually (\<lambda>x. dist (f x) l < e) net"
```
```   557   shows "(f ---> l) net"
```
```   558 apply (rule topological_tendstoI)
```
```   559 apply (simp add: open_dist)
```
```   560 apply (drule (1) bspec, clarify)
```
```   561 apply (drule assms)
```
```   562 apply (erule eventually_elim1, simp)
```
```   563 done
```
```   564
```
```   565 lemma tendstoD:
```
```   566   "(f ---> l) net \<Longrightarrow> 0 < e \<Longrightarrow> eventually (\<lambda>x. dist (f x) l < e) net"
```
```   567 apply (drule_tac S="{x. dist x l < e}" in topological_tendstoD)
```
```   568 apply (clarsimp simp add: open_dist)
```
```   569 apply (rule_tac x="e - dist x l" in exI, clarsimp)
```
```   570 apply (simp only: less_diff_eq)
```
```   571 apply (erule le_less_trans [OF dist_triangle])
```
```   572 apply simp
```
```   573 apply simp
```
```   574 done
```
```   575
```
```   576 lemma tendsto_iff:
```
```   577   "(f ---> l) net \<longleftrightarrow> (\<forall>e>0. eventually (\<lambda>x. dist (f x) l < e) net)"
```
```   578 using tendstoI tendstoD by fast
```
```   579
```
```   580 lemma tendsto_Zfun_iff: "(f ---> a) net = Zfun (\<lambda>x. f x - a) net"
```
```   581 by (simp only: tendsto_iff Zfun_def dist_norm)
```
```   582
```
```   583 lemma tendsto_ident_at [tendsto_intros]: "((\<lambda>x. x) ---> a) (at a)"
```
```   584 unfolding tendsto_def eventually_at_topological by auto
```
```   585
```
```   586 lemma tendsto_ident_at_within [tendsto_intros]:
```
```   587   "((\<lambda>x. x) ---> a) (at a within S)"
```
```   588 unfolding tendsto_def eventually_within eventually_at_topological by auto
```
```   589
```
```   590 lemma tendsto_const [tendsto_intros]: "((\<lambda>x. k) ---> k) net"
```
```   591 by (simp add: tendsto_def)
```
```   592
```
```   593 lemma tendsto_const_iff:
```
```   594   fixes k l :: "'a::metric_space"
```
```   595   assumes "net \<noteq> bot" shows "((\<lambda>n. k) ---> l) net \<longleftrightarrow> k = l"
```
```   596 apply (safe intro!: tendsto_const)
```
```   597 apply (rule ccontr)
```
```   598 apply (drule_tac e="dist k l" in tendstoD)
```
```   599 apply (simp add: zero_less_dist_iff)
```
```   600 apply (simp add: eventually_False assms)
```
```   601 done
```
```   602
```
```   603 lemma tendsto_dist [tendsto_intros]:
```
```   604   assumes f: "(f ---> l) net" and g: "(g ---> m) net"
```
```   605   shows "((\<lambda>x. dist (f x) (g x)) ---> dist l m) net"
```
```   606 proof (rule tendstoI)
```
```   607   fix e :: real assume "0 < e"
```
```   608   hence e2: "0 < e/2" by simp
```
```   609   from tendstoD [OF f e2] tendstoD [OF g e2]
```
```   610   show "eventually (\<lambda>x. dist (dist (f x) (g x)) (dist l m) < e) net"
```
```   611   proof (rule eventually_elim2)
```
```   612     fix x assume "dist (f x) l < e/2" "dist (g x) m < e/2"
```
```   613     then show "dist (dist (f x) (g x)) (dist l m) < e"
```
```   614       unfolding dist_real_def
```
```   615       using dist_triangle2 [of "f x" "g x" "l"]
```
```   616       using dist_triangle2 [of "g x" "l" "m"]
```
```   617       using dist_triangle3 [of "l" "m" "f x"]
```
```   618       using dist_triangle [of "f x" "m" "g x"]
```
```   619       by arith
```
```   620   qed
```
```   621 qed
```
```   622
```
```   623 lemma norm_conv_dist: "norm x = dist x 0"
```
```   624 unfolding dist_norm by simp
```
```   625
```
```   626 lemma tendsto_norm [tendsto_intros]:
```
```   627   "(f ---> a) net \<Longrightarrow> ((\<lambda>x. norm (f x)) ---> norm a) net"
```
```   628 unfolding norm_conv_dist by (intro tendsto_intros)
```
```   629
```
```   630 lemma tendsto_norm_zero:
```
```   631   "(f ---> 0) net \<Longrightarrow> ((\<lambda>x. norm (f x)) ---> 0) net"
```
```   632 by (drule tendsto_norm, simp)
```
```   633
```
```   634 lemma tendsto_norm_zero_cancel:
```
```   635   "((\<lambda>x. norm (f x)) ---> 0) net \<Longrightarrow> (f ---> 0) net"
```
```   636 unfolding tendsto_iff dist_norm by simp
```
```   637
```
```   638 lemma tendsto_norm_zero_iff:
```
```   639   "((\<lambda>x. norm (f x)) ---> 0) net \<longleftrightarrow> (f ---> 0) net"
```
```   640 unfolding tendsto_iff dist_norm by simp
```
```   641
```
```   642 lemma add_diff_add:
```
```   643   fixes a b c d :: "'a::ab_group_add"
```
```   644   shows "(a + c) - (b + d) = (a - b) + (c - d)"
```
```   645 by simp
```
```   646
```
```   647 lemma minus_diff_minus:
```
```   648   fixes a b :: "'a::ab_group_add"
```
```   649   shows "(- a) - (- b) = - (a - b)"
```
```   650 by simp
```
```   651
```
```   652 lemma tendsto_add [tendsto_intros]:
```
```   653   fixes a b :: "'a::real_normed_vector"
```
```   654   shows "\<lbrakk>(f ---> a) net; (g ---> b) net\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x + g x) ---> a + b) net"
```
```   655 by (simp only: tendsto_Zfun_iff add_diff_add Zfun_add)
```
```   656
```
```   657 lemma tendsto_minus [tendsto_intros]:
```
```   658   fixes a :: "'a::real_normed_vector"
```
```   659   shows "(f ---> a) net \<Longrightarrow> ((\<lambda>x. - f x) ---> - a) net"
```
```   660 by (simp only: tendsto_Zfun_iff minus_diff_minus Zfun_minus)
```
```   661
```
```   662 lemma tendsto_minus_cancel:
```
```   663   fixes a :: "'a::real_normed_vector"
```
```   664   shows "((\<lambda>x. - f x) ---> - a) net \<Longrightarrow> (f ---> a) net"
```
```   665 by (drule tendsto_minus, simp)
```
```   666
```
```   667 lemma tendsto_diff [tendsto_intros]:
```
```   668   fixes a b :: "'a::real_normed_vector"
```
```   669   shows "\<lbrakk>(f ---> a) net; (g ---> b) net\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x - g x) ---> a - b) net"
```
```   670 by (simp add: diff_minus tendsto_add tendsto_minus)
```
```   671
```
```   672 lemma tendsto_setsum [tendsto_intros]:
```
```   673   fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c::real_normed_vector"
```
```   674   assumes "\<And>i. i \<in> S \<Longrightarrow> (f i ---> a i) net"
```
```   675   shows "((\<lambda>x. \<Sum>i\<in>S. f i x) ---> (\<Sum>i\<in>S. a i)) net"
```
```   676 proof (cases "finite S")
```
```   677   assume "finite S" thus ?thesis using assms
```
```   678   proof (induct set: finite)
```
```   679     case empty show ?case
```
```   680       by (simp add: tendsto_const)
```
```   681   next
```
```   682     case (insert i F) thus ?case
```
```   683       by (simp add: tendsto_add)
```
```   684   qed
```
```   685 next
```
```   686   assume "\<not> finite S" thus ?thesis
```
```   687     by (simp add: tendsto_const)
```
```   688 qed
```
```   689
```
```   690 lemma (in bounded_linear) tendsto [tendsto_intros]:
```
```   691   "(g ---> a) net \<Longrightarrow> ((\<lambda>x. f (g x)) ---> f a) net"
```
```   692 by (simp only: tendsto_Zfun_iff diff [symmetric] Zfun)
```
```   693
```
```   694 lemma (in bounded_bilinear) tendsto [tendsto_intros]:
```
```   695   "\<lbrakk>(f ---> a) net; (g ---> b) net\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x ** g x) ---> a ** b) net"
```
```   696 by (simp only: tendsto_Zfun_iff prod_diff_prod
```
```   697                Zfun_add Zfun Zfun_left Zfun_right)
```
```   698
```
```   699
```
```   700 subsection {* Continuity of Inverse *}
```
```   701
```
```   702 lemma (in bounded_bilinear) Zfun_prod_Bfun:
```
```   703   assumes f: "Zfun f net"
```
```   704   assumes g: "Bfun g net"
```
```   705   shows "Zfun (\<lambda>x. f x ** g x) net"
```
```   706 proof -
```
```   707   obtain K where K: "0 \<le> K"
```
```   708     and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"
```
```   709     using nonneg_bounded by fast
```
```   710   obtain B where B: "0 < B"
```
```   711     and norm_g: "eventually (\<lambda>x. norm (g x) \<le> B) net"
```
```   712     using g by (rule BfunE)
```
```   713   have "eventually (\<lambda>x. norm (f x ** g x) \<le> norm (f x) * (B * K)) net"
```
```   714   using norm_g proof (rule eventually_elim1)
```
```   715     fix x
```
```   716     assume *: "norm (g x) \<le> B"
```
```   717     have "norm (f x ** g x) \<le> norm (f x) * norm (g x) * K"
```
```   718       by (rule norm_le)
```
```   719     also have "\<dots> \<le> norm (f x) * B * K"
```
```   720       by (intro mult_mono' order_refl norm_g norm_ge_zero
```
```   721                 mult_nonneg_nonneg K *)
```
```   722     also have "\<dots> = norm (f x) * (B * K)"
```
```   723       by (rule mult_assoc)
```
```   724     finally show "norm (f x ** g x) \<le> norm (f x) * (B * K)" .
```
```   725   qed
```
```   726   with f show ?thesis
```
```   727     by (rule Zfun_imp_Zfun)
```
```   728 qed
```
```   729
```
```   730 lemma (in bounded_bilinear) flip:
```
```   731   "bounded_bilinear (\<lambda>x y. y ** x)"
```
```   732 apply default
```
```   733 apply (rule add_right)
```
```   734 apply (rule add_left)
```
```   735 apply (rule scaleR_right)
```
```   736 apply (rule scaleR_left)
```
```   737 apply (subst mult_commute)
```
```   738 using bounded by fast
```
```   739
```
```   740 lemma (in bounded_bilinear) Bfun_prod_Zfun:
```
```   741   assumes f: "Bfun f net"
```
```   742   assumes g: "Zfun g net"
```
```   743   shows "Zfun (\<lambda>x. f x ** g x) net"
```
```   744 using flip g f by (rule bounded_bilinear.Zfun_prod_Bfun)
```
```   745
```
```   746 lemma inverse_diff_inverse:
```
```   747   "\<lbrakk>(a::'a::division_ring) \<noteq> 0; b \<noteq> 0\<rbrakk>
```
```   748    \<Longrightarrow> inverse a - inverse b = - (inverse a * (a - b) * inverse b)"
```
```   749 by (simp add: algebra_simps)
```
```   750
```
```   751 lemma Bfun_inverse_lemma:
```
```   752   fixes x :: "'a::real_normed_div_algebra"
```
```   753   shows "\<lbrakk>r \<le> norm x; 0 < r\<rbrakk> \<Longrightarrow> norm (inverse x) \<le> inverse r"
```
```   754 apply (subst nonzero_norm_inverse, clarsimp)
```
```   755 apply (erule (1) le_imp_inverse_le)
```
```   756 done
```
```   757
```
```   758 lemma Bfun_inverse:
```
```   759   fixes a :: "'a::real_normed_div_algebra"
```
```   760   assumes f: "(f ---> a) net"
```
```   761   assumes a: "a \<noteq> 0"
```
```   762   shows "Bfun (\<lambda>x. inverse (f x)) net"
```
```   763 proof -
```
```   764   from a have "0 < norm a" by simp
```
```   765   hence "\<exists>r>0. r < norm a" by (rule dense)
```
```   766   then obtain r where r1: "0 < r" and r2: "r < norm a" by fast
```
```   767   have "eventually (\<lambda>x. dist (f x) a < r) net"
```
```   768     using tendstoD [OF f r1] by fast
```
```   769   hence "eventually (\<lambda>x. norm (inverse (f x)) \<le> inverse (norm a - r)) net"
```
```   770   proof (rule eventually_elim1)
```
```   771     fix x
```
```   772     assume "dist (f x) a < r"
```
```   773     hence 1: "norm (f x - a) < r"
```
```   774       by (simp add: dist_norm)
```
```   775     hence 2: "f x \<noteq> 0" using r2 by auto
```
```   776     hence "norm (inverse (f x)) = inverse (norm (f x))"
```
```   777       by (rule nonzero_norm_inverse)
```
```   778     also have "\<dots> \<le> inverse (norm a - r)"
```
```   779     proof (rule le_imp_inverse_le)
```
```   780       show "0 < norm a - r" using r2 by simp
```
```   781     next
```
```   782       have "norm a - norm (f x) \<le> norm (a - f x)"
```
```   783         by (rule norm_triangle_ineq2)
```
```   784       also have "\<dots> = norm (f x - a)"
```
```   785         by (rule norm_minus_commute)
```
```   786       also have "\<dots> < r" using 1 .
```
```   787       finally show "norm a - r \<le> norm (f x)" by simp
```
```   788     qed
```
```   789     finally show "norm (inverse (f x)) \<le> inverse (norm a - r)" .
```
```   790   qed
```
```   791   thus ?thesis by (rule BfunI)
```
```   792 qed
```
```   793
```
```   794 lemma tendsto_inverse_lemma:
```
```   795   fixes a :: "'a::real_normed_div_algebra"
```
```   796   shows "\<lbrakk>(f ---> a) net; a \<noteq> 0; eventually (\<lambda>x. f x \<noteq> 0) net\<rbrakk>
```
```   797          \<Longrightarrow> ((\<lambda>x. inverse (f x)) ---> inverse a) net"
```
```   798 apply (subst tendsto_Zfun_iff)
```
```   799 apply (rule Zfun_ssubst)
```
```   800 apply (erule eventually_elim1)
```
```   801 apply (erule (1) inverse_diff_inverse)
```
```   802 apply (rule Zfun_minus)
```
```   803 apply (rule Zfun_mult_left)
```
```   804 apply (rule mult.Bfun_prod_Zfun)
```
```   805 apply (erule (1) Bfun_inverse)
```
```   806 apply (simp add: tendsto_Zfun_iff)
```
```   807 done
```
```   808
```
```   809 lemma tendsto_inverse [tendsto_intros]:
```
```   810   fixes a :: "'a::real_normed_div_algebra"
```
```   811   assumes f: "(f ---> a) net"
```
```   812   assumes a: "a \<noteq> 0"
```
```   813   shows "((\<lambda>x. inverse (f x)) ---> inverse a) net"
```
```   814 proof -
```
```   815   from a have "0 < norm a" by simp
```
```   816   with f have "eventually (\<lambda>x. dist (f x) a < norm a) net"
```
```   817     by (rule tendstoD)
```
```   818   then have "eventually (\<lambda>x. f x \<noteq> 0) net"
```
```   819     unfolding dist_norm by (auto elim!: eventually_elim1)
```
```   820   with f a show ?thesis
```
```   821     by (rule tendsto_inverse_lemma)
```
```   822 qed
```
```   823
```
```   824 lemma tendsto_divide [tendsto_intros]:
```
```   825   fixes a b :: "'a::real_normed_field"
```
```   826   shows "\<lbrakk>(f ---> a) net; (g ---> b) net; b \<noteq> 0\<rbrakk>
```
```   827     \<Longrightarrow> ((\<lambda>x. f x / g x) ---> a / b) net"
```
```   828 by (simp add: mult.tendsto tendsto_inverse divide_inverse)
```
```   829
```
```   830 end
```