src/HOL/Lim.thy
 author huffman Fri Aug 19 14:17:28 2011 -0700 (2011-08-19) changeset 44311 42c5cbf68052 parent 44310 ba3d031b5dbb child 44312 471ff02a8574 permissions -rw-r--r--
Transcendental.thy: add tendsto_intros lemmas;
new isCont theorems;
simplify some proofs.
```     1 (*  Title       : Lim.thy
```
```     2     Author      : Jacques D. Fleuriot
```
```     3     Copyright   : 1998  University of Cambridge
```
```     4     Conversion to Isar and new proofs by Lawrence C Paulson, 2004
```
```     5 *)
```
```     6
```
```     7 header{* Limits and Continuity *}
```
```     8
```
```     9 theory Lim
```
```    10 imports SEQ
```
```    11 begin
```
```    12
```
```    13 text{*Standard Definitions*}
```
```    14
```
```    15 abbreviation
```
```    16   LIM :: "['a::topological_space \<Rightarrow> 'b::topological_space, 'a, 'b] \<Rightarrow> bool"
```
```    17         ("((_)/ -- (_)/ --> (_))" [60, 0, 60] 60) where
```
```    18   "f -- a --> L \<equiv> (f ---> L) (at a)"
```
```    19
```
```    20 definition
```
```    21   isCont :: "['a::topological_space \<Rightarrow> 'b::topological_space, 'a] \<Rightarrow> bool" where
```
```    22   "isCont f a = (f -- a --> (f a))"
```
```    23
```
```    24 definition
```
```    25   isUCont :: "['a::metric_space \<Rightarrow> 'b::metric_space] \<Rightarrow> bool" where
```
```    26   "isUCont f = (\<forall>r>0. \<exists>s>0. \<forall>x y. dist x y < s \<longrightarrow> dist (f x) (f y) < r)"
```
```    27
```
```    28 subsection {* Limits of Functions *}
```
```    29
```
```    30 lemma LIM_def: "f -- a --> L =
```
```    31      (\<forall>r > 0. \<exists>s > 0. \<forall>x. x \<noteq> a & dist x a < s
```
```    32         --> dist (f x) L < r)"
```
```    33 unfolding tendsto_iff eventually_at ..
```
```    34
```
```    35 lemma metric_LIM_I:
```
```    36   "(\<And>r. 0 < r \<Longrightarrow> \<exists>s>0. \<forall>x. x \<noteq> a \<and> dist x a < s \<longrightarrow> dist (f x) L < r)
```
```    37     \<Longrightarrow> f -- a --> L"
```
```    38 by (simp add: LIM_def)
```
```    39
```
```    40 lemma metric_LIM_D:
```
```    41   "\<lbrakk>f -- a --> L; 0 < r\<rbrakk>
```
```    42     \<Longrightarrow> \<exists>s>0. \<forall>x. x \<noteq> a \<and> dist x a < s \<longrightarrow> dist (f x) L < r"
```
```    43 by (simp add: LIM_def)
```
```    44
```
```    45 lemma LIM_eq:
```
```    46   fixes a :: "'a::real_normed_vector" and L :: "'b::real_normed_vector"
```
```    47   shows "f -- a --> L =
```
```    48      (\<forall>r>0.\<exists>s>0.\<forall>x. x \<noteq> a & norm (x-a) < s --> norm (f x - L) < r)"
```
```    49 by (simp add: LIM_def dist_norm)
```
```    50
```
```    51 lemma LIM_I:
```
```    52   fixes a :: "'a::real_normed_vector" and L :: "'b::real_normed_vector"
```
```    53   shows "(!!r. 0<r ==> \<exists>s>0.\<forall>x. x \<noteq> a & norm (x-a) < s --> norm (f x - L) < r)
```
```    54       ==> f -- a --> L"
```
```    55 by (simp add: LIM_eq)
```
```    56
```
```    57 lemma LIM_D:
```
```    58   fixes a :: "'a::real_normed_vector" and L :: "'b::real_normed_vector"
```
```    59   shows "[| f -- a --> L; 0<r |]
```
```    60       ==> \<exists>s>0.\<forall>x. x \<noteq> a & norm (x-a) < s --> norm (f x - L) < r"
```
```    61 by (simp add: LIM_eq)
```
```    62
```
```    63 lemma LIM_offset:
```
```    64   fixes a :: "'a::real_normed_vector"
```
```    65   shows "f -- a --> L \<Longrightarrow> (\<lambda>x. f (x + k)) -- a - k --> L"
```
```    66 apply (rule topological_tendstoI)
```
```    67 apply (drule (2) topological_tendstoD)
```
```    68 apply (simp only: eventually_at dist_norm)
```
```    69 apply (clarify, rule_tac x=d in exI, safe)
```
```    70 apply (drule_tac x="x + k" in spec)
```
```    71 apply (simp add: algebra_simps)
```
```    72 done
```
```    73
```
```    74 lemma LIM_offset_zero:
```
```    75   fixes a :: "'a::real_normed_vector"
```
```    76   shows "f -- a --> L \<Longrightarrow> (\<lambda>h. f (a + h)) -- 0 --> L"
```
```    77 by (drule_tac k="a" in LIM_offset, simp add: add_commute)
```
```    78
```
```    79 lemma LIM_offset_zero_cancel:
```
```    80   fixes a :: "'a::real_normed_vector"
```
```    81   shows "(\<lambda>h. f (a + h)) -- 0 --> L \<Longrightarrow> f -- a --> L"
```
```    82 by (drule_tac k="- a" in LIM_offset, simp)
```
```    83
```
```    84 lemma LIM_const [simp]: "(%x. k) -- x --> k"
```
```    85 by (rule tendsto_const)
```
```    86
```
```    87 lemma LIM_cong_limit: "\<lbrakk> f -- x --> L ; K = L \<rbrakk> \<Longrightarrow> f -- x --> K" by simp
```
```    88
```
```    89 lemma LIM_add:
```
```    90   fixes f g :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
```
```    91   assumes f: "f -- a --> L" and g: "g -- a --> M"
```
```    92   shows "(\<lambda>x. f x + g x) -- a --> (L + M)"
```
```    93 using assms by (rule tendsto_add)
```
```    94
```
```    95 lemma LIM_add_zero:
```
```    96   fixes f g :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
```
```    97   shows "\<lbrakk>f -- a --> 0; g -- a --> 0\<rbrakk> \<Longrightarrow> (\<lambda>x. f x + g x) -- a --> 0"
```
```    98   by (rule tendsto_add_zero)
```
```    99
```
```   100 lemma LIM_minus:
```
```   101   fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
```
```   102   shows "f -- a --> L \<Longrightarrow> (\<lambda>x. - f x) -- a --> - L"
```
```   103 by (rule tendsto_minus)
```
```   104
```
```   105 (* TODO: delete *)
```
```   106 lemma LIM_add_minus:
```
```   107   fixes f g :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
```
```   108   shows "[| f -- x --> l; g -- x --> m |] ==> (%x. f(x) + -g(x)) -- x --> (l + -m)"
```
```   109 by (intro LIM_add LIM_minus)
```
```   110
```
```   111 lemma LIM_diff:
```
```   112   fixes f g :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
```
```   113   shows "\<lbrakk>f -- x --> l; g -- x --> m\<rbrakk> \<Longrightarrow> (\<lambda>x. f x - g x) -- x --> l - m"
```
```   114 by (rule tendsto_diff)
```
```   115
```
```   116 lemma LIM_zero:
```
```   117   fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
```
```   118   shows "f -- a --> l \<Longrightarrow> (\<lambda>x. f x - l) -- a --> 0"
```
```   119 unfolding tendsto_iff dist_norm by simp
```
```   120
```
```   121 lemma LIM_zero_cancel:
```
```   122   fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
```
```   123   shows "(\<lambda>x. f x - l) -- a --> 0 \<Longrightarrow> f -- a --> l"
```
```   124 unfolding tendsto_iff dist_norm by simp
```
```   125
```
```   126 lemma LIM_zero_iff:
```
```   127   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
```
```   128   shows "(\<lambda>x. f x - l) -- a --> 0 = f -- a --> l"
```
```   129 unfolding tendsto_iff dist_norm by simp
```
```   130
```
```   131 lemma metric_LIM_imp_LIM:
```
```   132   assumes f: "f -- a --> l"
```
```   133   assumes le: "\<And>x. x \<noteq> a \<Longrightarrow> dist (g x) m \<le> dist (f x) l"
```
```   134   shows "g -- a --> m"
```
```   135   by (rule metric_tendsto_imp_tendsto [OF f],
```
```   136     auto simp add: eventually_at_topological le)
```
```   137
```
```   138 lemma LIM_imp_LIM:
```
```   139   fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
```
```   140   fixes g :: "'a::topological_space \<Rightarrow> 'c::real_normed_vector"
```
```   141   assumes f: "f -- a --> l"
```
```   142   assumes le: "\<And>x. x \<noteq> a \<Longrightarrow> norm (g x - m) \<le> norm (f x - l)"
```
```   143   shows "g -- a --> m"
```
```   144   by (rule metric_LIM_imp_LIM [OF f],
```
```   145     simp add: dist_norm le)
```
```   146
```
```   147 lemma LIM_norm:
```
```   148   fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
```
```   149   shows "f -- a --> l \<Longrightarrow> (\<lambda>x. norm (f x)) -- a --> norm l"
```
```   150 by (rule tendsto_norm)
```
```   151
```
```   152 lemma LIM_norm_zero:
```
```   153   fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
```
```   154   shows "f -- a --> 0 \<Longrightarrow> (\<lambda>x. norm (f x)) -- a --> 0"
```
```   155 by (rule tendsto_norm_zero)
```
```   156
```
```   157 lemma LIM_norm_zero_cancel:
```
```   158   fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
```
```   159   shows "(\<lambda>x. norm (f x)) -- a --> 0 \<Longrightarrow> f -- a --> 0"
```
```   160 by (rule tendsto_norm_zero_cancel)
```
```   161
```
```   162 lemma LIM_norm_zero_iff:
```
```   163   fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
```
```   164   shows "(\<lambda>x. norm (f x)) -- a --> 0 = f -- a --> 0"
```
```   165 by (rule tendsto_norm_zero_iff)
```
```   166
```
```   167 lemma LIM_rabs: "f -- a --> (l::real) \<Longrightarrow> (\<lambda>x. \<bar>f x\<bar>) -- a --> \<bar>l\<bar>"
```
```   168   by (rule tendsto_rabs)
```
```   169
```
```   170 lemma LIM_rabs_zero: "f -- a --> (0::real) \<Longrightarrow> (\<lambda>x. \<bar>f x\<bar>) -- a --> 0"
```
```   171   by (rule tendsto_rabs_zero)
```
```   172
```
```   173 lemma LIM_rabs_zero_cancel: "(\<lambda>x. \<bar>f x\<bar>) -- a --> (0::real) \<Longrightarrow> f -- a --> 0"
```
```   174   by (rule tendsto_rabs_zero_cancel)
```
```   175
```
```   176 lemma LIM_rabs_zero_iff: "(\<lambda>x. \<bar>f x\<bar>) -- a --> (0::real) = f -- a --> 0"
```
```   177   by (rule tendsto_rabs_zero_iff)
```
```   178
```
```   179 lemma trivial_limit_at:
```
```   180   fixes a :: "'a::real_normed_algebra_1"
```
```   181   shows "\<not> trivial_limit (at a)"  -- {* TODO: find a more appropriate class *}
```
```   182 unfolding trivial_limit_def
```
```   183 unfolding eventually_at dist_norm
```
```   184 by (clarsimp, rule_tac x="a + of_real (d/2)" in exI, simp)
```
```   185
```
```   186 lemma LIM_const_not_eq:
```
```   187   fixes a :: "'a::real_normed_algebra_1"
```
```   188   fixes k L :: "'b::t2_space"
```
```   189   shows "k \<noteq> L \<Longrightarrow> \<not> (\<lambda>x. k) -- a --> L"
```
```   190 by (simp add: tendsto_const_iff trivial_limit_at)
```
```   191
```
```   192 lemmas LIM_not_zero = LIM_const_not_eq [where L = 0]
```
```   193
```
```   194 lemma LIM_const_eq:
```
```   195   fixes a :: "'a::real_normed_algebra_1"
```
```   196   fixes k L :: "'b::t2_space"
```
```   197   shows "(\<lambda>x. k) -- a --> L \<Longrightarrow> k = L"
```
```   198   by (simp add: tendsto_const_iff trivial_limit_at)
```
```   199
```
```   200 lemma LIM_unique:
```
```   201   fixes a :: "'a::real_normed_algebra_1" -- {* TODO: find a more appropriate class *}
```
```   202   fixes L M :: "'b::t2_space"
```
```   203   shows "\<lbrakk>f -- a --> L; f -- a --> M\<rbrakk> \<Longrightarrow> L = M"
```
```   204   using trivial_limit_at by (rule tendsto_unique)
```
```   205
```
```   206 lemma LIM_ident [simp]: "(\<lambda>x. x) -- a --> a"
```
```   207 by (rule tendsto_ident_at)
```
```   208
```
```   209 text{*Limits are equal for functions equal except at limit point*}
```
```   210 lemma LIM_equal:
```
```   211      "[| \<forall>x. x \<noteq> a --> (f x = g x) |] ==> (f -- a --> l) = (g -- a --> l)"
```
```   212 unfolding tendsto_def eventually_at_topological by simp
```
```   213
```
```   214 lemma LIM_cong:
```
```   215   "\<lbrakk>a = b; \<And>x. x \<noteq> b \<Longrightarrow> f x = g x; l = m\<rbrakk>
```
```   216    \<Longrightarrow> ((\<lambda>x. f x) -- a --> l) = ((\<lambda>x. g x) -- b --> m)"
```
```   217 by (simp add: LIM_equal)
```
```   218
```
```   219 lemma metric_LIM_equal2:
```
```   220   assumes 1: "0 < R"
```
```   221   assumes 2: "\<And>x. \<lbrakk>x \<noteq> a; dist x a < R\<rbrakk> \<Longrightarrow> f x = g x"
```
```   222   shows "g -- a --> l \<Longrightarrow> f -- a --> l"
```
```   223 apply (rule topological_tendstoI)
```
```   224 apply (drule (2) topological_tendstoD)
```
```   225 apply (simp add: eventually_at, safe)
```
```   226 apply (rule_tac x="min d R" in exI, safe)
```
```   227 apply (simp add: 1)
```
```   228 apply (simp add: 2)
```
```   229 done
```
```   230
```
```   231 lemma LIM_equal2:
```
```   232   fixes f g :: "'a::real_normed_vector \<Rightarrow> 'b::topological_space"
```
```   233   assumes 1: "0 < R"
```
```   234   assumes 2: "\<And>x. \<lbrakk>x \<noteq> a; norm (x - a) < R\<rbrakk> \<Longrightarrow> f x = g x"
```
```   235   shows "g -- a --> l \<Longrightarrow> f -- a --> l"
```
```   236 by (rule metric_LIM_equal2 [OF 1 2], simp_all add: dist_norm)
```
```   237
```
```   238 text{*Two uses in Transcendental.ML*} (* BH: no longer true; delete? *)
```
```   239 lemma LIM_trans:
```
```   240   fixes f g :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
```
```   241   shows "[| (%x. f(x) + -g(x)) -- a --> 0;  g -- a --> l |] ==> f -- a --> l"
```
```   242 apply (drule LIM_add, assumption)
```
```   243 apply (auto simp add: add_assoc)
```
```   244 done
```
```   245
```
```   246 lemma LIM_compose:
```
```   247   assumes g: "g -- l --> g l"
```
```   248   assumes f: "f -- a --> l"
```
```   249   shows "(\<lambda>x. g (f x)) -- a --> g l"
```
```   250   using assms by (rule tendsto_compose)
```
```   251
```
```   252 lemma LIM_compose_eventually:
```
```   253   assumes f: "f -- a --> b"
```
```   254   assumes g: "g -- b --> c"
```
```   255   assumes inj: "eventually (\<lambda>x. f x \<noteq> b) (at a)"
```
```   256   shows "(\<lambda>x. g (f x)) -- a --> c"
```
```   257   using g f inj by (rule tendsto_compose_eventually)
```
```   258
```
```   259 lemma metric_LIM_compose2:
```
```   260   assumes f: "f -- a --> b"
```
```   261   assumes g: "g -- b --> c"
```
```   262   assumes inj: "\<exists>d>0. \<forall>x. x \<noteq> a \<and> dist x a < d \<longrightarrow> f x \<noteq> b"
```
```   263   shows "(\<lambda>x. g (f x)) -- a --> c"
```
```   264 using f g inj [folded eventually_at]
```
```   265 by (rule LIM_compose_eventually)
```
```   266
```
```   267 lemma LIM_compose2:
```
```   268   fixes a :: "'a::real_normed_vector"
```
```   269   assumes f: "f -- a --> b"
```
```   270   assumes g: "g -- b --> c"
```
```   271   assumes inj: "\<exists>d>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < d \<longrightarrow> f x \<noteq> b"
```
```   272   shows "(\<lambda>x. g (f x)) -- a --> c"
```
```   273 by (rule metric_LIM_compose2 [OF f g inj [folded dist_norm]])
```
```   274
```
```   275 lemma LIM_o: "\<lbrakk>g -- l --> g l; f -- a --> l\<rbrakk> \<Longrightarrow> (g \<circ> f) -- a --> g l"
```
```   276 unfolding o_def by (rule LIM_compose)
```
```   277
```
```   278 lemma real_LIM_sandwich_zero:
```
```   279   fixes f g :: "'a::topological_space \<Rightarrow> real"
```
```   280   assumes f: "f -- a --> 0"
```
```   281   assumes 1: "\<And>x. x \<noteq> a \<Longrightarrow> 0 \<le> g x"
```
```   282   assumes 2: "\<And>x. x \<noteq> a \<Longrightarrow> g x \<le> f x"
```
```   283   shows "g -- a --> 0"
```
```   284 proof (rule LIM_imp_LIM [OF f])
```
```   285   fix x assume x: "x \<noteq> a"
```
```   286   have "norm (g x - 0) = g x" by (simp add: 1 x)
```
```   287   also have "g x \<le> f x" by (rule 2 [OF x])
```
```   288   also have "f x \<le> \<bar>f x\<bar>" by (rule abs_ge_self)
```
```   289   also have "\<bar>f x\<bar> = norm (f x - 0)" by simp
```
```   290   finally show "norm (g x - 0) \<le> norm (f x - 0)" .
```
```   291 qed
```
```   292
```
```   293 text {* Bounded Linear Operators *}
```
```   294
```
```   295 lemma (in bounded_linear) LIM:
```
```   296   "g -- a --> l \<Longrightarrow> (\<lambda>x. f (g x)) -- a --> f l"
```
```   297 by (rule tendsto)
```
```   298
```
```   299 lemma (in bounded_linear) LIM_zero:
```
```   300   "g -- a --> 0 \<Longrightarrow> (\<lambda>x. f (g x)) -- a --> 0"
```
```   301   by (rule tendsto_zero)
```
```   302
```
```   303 text {* Bounded Bilinear Operators *}
```
```   304
```
```   305 lemma (in bounded_bilinear) LIM:
```
```   306   "\<lbrakk>f -- a --> L; g -- a --> M\<rbrakk> \<Longrightarrow> (\<lambda>x. f x ** g x) -- a --> L ** M"
```
```   307 by (rule tendsto)
```
```   308
```
```   309 lemma (in bounded_bilinear) LIM_prod_zero:
```
```   310   fixes a :: "'d::metric_space"
```
```   311   assumes f: "f -- a --> 0"
```
```   312   assumes g: "g -- a --> 0"
```
```   313   shows "(\<lambda>x. f x ** g x) -- a --> 0"
```
```   314   using f g by (rule tendsto_zero)
```
```   315
```
```   316 lemma (in bounded_bilinear) LIM_left_zero:
```
```   317   "f -- a --> 0 \<Longrightarrow> (\<lambda>x. f x ** c) -- a --> 0"
```
```   318   by (rule tendsto_left_zero)
```
```   319
```
```   320 lemma (in bounded_bilinear) LIM_right_zero:
```
```   321   "f -- a --> 0 \<Longrightarrow> (\<lambda>x. c ** f x) -- a --> 0"
```
```   322   by (rule tendsto_right_zero)
```
```   323
```
```   324 lemmas LIM_mult =
```
```   325   bounded_bilinear.LIM [OF bounded_bilinear_mult]
```
```   326
```
```   327 lemmas LIM_mult_zero =
```
```   328   bounded_bilinear.LIM_prod_zero [OF bounded_bilinear_mult]
```
```   329
```
```   330 lemmas LIM_mult_left_zero =
```
```   331   bounded_bilinear.LIM_left_zero [OF bounded_bilinear_mult]
```
```   332
```
```   333 lemmas LIM_mult_right_zero =
```
```   334   bounded_bilinear.LIM_right_zero [OF bounded_bilinear_mult]
```
```   335
```
```   336 lemmas LIM_scaleR =
```
```   337   bounded_bilinear.LIM [OF bounded_bilinear_scaleR]
```
```   338
```
```   339 lemmas LIM_of_real =
```
```   340   bounded_linear.LIM [OF bounded_linear_of_real]
```
```   341
```
```   342 lemma LIM_power:
```
```   343   fixes f :: "'a::topological_space \<Rightarrow> 'b::{power,real_normed_algebra}"
```
```   344   assumes f: "f -- a --> l"
```
```   345   shows "(\<lambda>x. f x ^ n) -- a --> l ^ n"
```
```   346   using assms by (rule tendsto_power)
```
```   347
```
```   348 lemma LIM_inverse:
```
```   349   fixes L :: "'a::real_normed_div_algebra"
```
```   350   shows "\<lbrakk>f -- a --> L; L \<noteq> 0\<rbrakk> \<Longrightarrow> (\<lambda>x. inverse (f x)) -- a --> inverse L"
```
```   351 by (rule tendsto_inverse)
```
```   352
```
```   353 lemma LIM_inverse_fun:
```
```   354   assumes a: "a \<noteq> (0::'a::real_normed_div_algebra)"
```
```   355   shows "inverse -- a --> inverse a"
```
```   356 by (rule LIM_inverse [OF LIM_ident a])
```
```   357
```
```   358 lemma LIM_sgn:
```
```   359   fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
```
```   360   shows "\<lbrakk>f -- a --> l; l \<noteq> 0\<rbrakk> \<Longrightarrow> (\<lambda>x. sgn (f x)) -- a --> sgn l"
```
```   361   by (rule tendsto_sgn)
```
```   362
```
```   363
```
```   364 subsection {* Continuity *}
```
```   365
```
```   366 lemma LIM_isCont_iff:
```
```   367   fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::topological_space"
```
```   368   shows "(f -- a --> f a) = ((\<lambda>h. f (a + h)) -- 0 --> f a)"
```
```   369 by (rule iffI [OF LIM_offset_zero LIM_offset_zero_cancel])
```
```   370
```
```   371 lemma isCont_iff:
```
```   372   fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::topological_space"
```
```   373   shows "isCont f x = (\<lambda>h. f (x + h)) -- 0 --> f x"
```
```   374 by (simp add: isCont_def LIM_isCont_iff)
```
```   375
```
```   376 lemma isCont_ident [simp]: "isCont (\<lambda>x. x) a"
```
```   377   unfolding isCont_def by (rule LIM_ident)
```
```   378
```
```   379 lemma isCont_const [simp]: "isCont (\<lambda>x. k) a"
```
```   380   unfolding isCont_def by (rule LIM_const)
```
```   381
```
```   382 lemma isCont_norm [simp]:
```
```   383   fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
```
```   384   shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. norm (f x)) a"
```
```   385   unfolding isCont_def by (rule LIM_norm)
```
```   386
```
```   387 lemma isCont_rabs [simp]:
```
```   388   fixes f :: "'a::topological_space \<Rightarrow> real"
```
```   389   shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. \<bar>f x\<bar>) a"
```
```   390   unfolding isCont_def by (rule LIM_rabs)
```
```   391
```
```   392 lemma isCont_add [simp]:
```
```   393   fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
```
```   394   shows "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x + g x) a"
```
```   395   unfolding isCont_def by (rule LIM_add)
```
```   396
```
```   397 lemma isCont_minus [simp]:
```
```   398   fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
```
```   399   shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. - f x) a"
```
```   400   unfolding isCont_def by (rule LIM_minus)
```
```   401
```
```   402 lemma isCont_diff [simp]:
```
```   403   fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
```
```   404   shows "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x - g x) a"
```
```   405   unfolding isCont_def by (rule LIM_diff)
```
```   406
```
```   407 lemma isCont_mult [simp]:
```
```   408   fixes f g :: "'a::topological_space \<Rightarrow> 'b::real_normed_algebra"
```
```   409   shows "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x * g x) a"
```
```   410   unfolding isCont_def by (rule LIM_mult)
```
```   411
```
```   412 lemma isCont_inverse [simp]:
```
```   413   fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_div_algebra"
```
```   414   shows "\<lbrakk>isCont f a; f a \<noteq> 0\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. inverse (f x)) a"
```
```   415   unfolding isCont_def by (rule LIM_inverse)
```
```   416
```
```   417 lemma isCont_divide [simp]:
```
```   418   fixes f g :: "'a::topological_space \<Rightarrow> 'b::real_normed_field"
```
```   419   shows "\<lbrakk>isCont f a; isCont g a; g a \<noteq> 0\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x / g x) a"
```
```   420   unfolding isCont_def by (rule tendsto_divide)
```
```   421
```
```   422 lemma isCont_tendsto_compose:
```
```   423   "\<lbrakk>isCont g l; (f ---> l) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. g (f x)) ---> g l) F"
```
```   424   unfolding isCont_def by (rule tendsto_compose)
```
```   425
```
```   426 lemma isCont_LIM_compose:
```
```   427   "\<lbrakk>isCont g l; f -- a --> l\<rbrakk> \<Longrightarrow> (\<lambda>x. g (f x)) -- a --> g l"
```
```   428   by (rule isCont_tendsto_compose) (* TODO: delete? *)
```
```   429
```
```   430 lemma metric_isCont_LIM_compose2:
```
```   431   assumes f [unfolded isCont_def]: "isCont f a"
```
```   432   assumes g: "g -- f a --> l"
```
```   433   assumes inj: "\<exists>d>0. \<forall>x. x \<noteq> a \<and> dist x a < d \<longrightarrow> f x \<noteq> f a"
```
```   434   shows "(\<lambda>x. g (f x)) -- a --> l"
```
```   435 by (rule metric_LIM_compose2 [OF f g inj])
```
```   436
```
```   437 lemma isCont_LIM_compose2:
```
```   438   fixes a :: "'a::real_normed_vector"
```
```   439   assumes f [unfolded isCont_def]: "isCont f a"
```
```   440   assumes g: "g -- f a --> l"
```
```   441   assumes inj: "\<exists>d>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < d \<longrightarrow> f x \<noteq> f a"
```
```   442   shows "(\<lambda>x. g (f x)) -- a --> l"
```
```   443 by (rule LIM_compose2 [OF f g inj])
```
```   444
```
```   445 lemma isCont_o2: "\<lbrakk>isCont f a; isCont g (f a)\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. g (f x)) a"
```
```   446   unfolding isCont_def by (rule LIM_compose)
```
```   447
```
```   448 lemma isCont_o: "\<lbrakk>isCont f a; isCont g (f a)\<rbrakk> \<Longrightarrow> isCont (g o f) a"
```
```   449   unfolding o_def by (rule isCont_o2)
```
```   450
```
```   451 lemma (in bounded_linear) isCont:
```
```   452   "isCont g a \<Longrightarrow> isCont (\<lambda>x. f (g x)) a"
```
```   453   unfolding isCont_def by (rule LIM)
```
```   454
```
```   455 lemma (in bounded_bilinear) isCont:
```
```   456   "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x ** g x) a"
```
```   457   unfolding isCont_def by (rule LIM)
```
```   458
```
```   459 lemmas isCont_scaleR [simp] =
```
```   460   bounded_bilinear.isCont [OF bounded_bilinear_scaleR]
```
```   461
```
```   462 lemmas isCont_of_real [simp] =
```
```   463   bounded_linear.isCont [OF bounded_linear_of_real]
```
```   464
```
```   465 lemma isCont_power [simp]:
```
```   466   fixes f :: "'a::topological_space \<Rightarrow> 'b::{power,real_normed_algebra}"
```
```   467   shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. f x ^ n) a"
```
```   468   unfolding isCont_def by (rule LIM_power)
```
```   469
```
```   470 lemma isCont_sgn [simp]:
```
```   471   fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
```
```   472   shows "\<lbrakk>isCont f a; f a \<noteq> 0\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. sgn (f x)) a"
```
```   473   unfolding isCont_def by (rule LIM_sgn)
```
```   474
```
```   475 lemma isCont_setsum [simp]:
```
```   476   fixes f :: "'a \<Rightarrow> 'b::topological_space \<Rightarrow> 'c::real_normed_vector"
```
```   477   fixes A :: "'a set"
```
```   478   shows "\<forall>i\<in>A. isCont (f i) a \<Longrightarrow> isCont (\<lambda>x. \<Sum>i\<in>A. f i x) a"
```
```   479   unfolding isCont_def by (simp add: tendsto_setsum)
```
```   480
```
```   481 lemmas isCont_intros =
```
```   482   isCont_ident isCont_const isCont_norm isCont_rabs isCont_add isCont_minus
```
```   483   isCont_diff isCont_mult isCont_inverse isCont_divide isCont_scaleR
```
```   484   isCont_of_real isCont_power isCont_sgn isCont_setsum
```
```   485
```
```   486 lemma LIM_less_bound: fixes f :: "real \<Rightarrow> real" assumes "b < x"
```
```   487   and all_le: "\<forall> x' \<in> { b <..< x}. 0 \<le> f x'" and isCont: "isCont f x"
```
```   488   shows "0 \<le> f x"
```
```   489 proof (rule ccontr)
```
```   490   assume "\<not> 0 \<le> f x" hence "f x < 0" by auto
```
```   491   hence "0 < - f x / 2" by auto
```
```   492   from isCont[unfolded isCont_def, THEN LIM_D, OF this]
```
```   493   obtain s where "s > 0" and s_D: "\<And>x'. \<lbrakk> x' \<noteq> x ; \<bar> x' - x \<bar> < s \<rbrakk> \<Longrightarrow> \<bar> f x' - f x \<bar> < - f x / 2" by auto
```
```   494
```
```   495   let ?x = "x - min (s / 2) ((x - b) / 2)"
```
```   496   have "?x < x" and "\<bar> ?x - x \<bar> < s"
```
```   497     using `b < x` and `0 < s` by auto
```
```   498   have "b < ?x"
```
```   499   proof (cases "s < x - b")
```
```   500     case True thus ?thesis using `0 < s` by auto
```
```   501   next
```
```   502     case False hence "s / 2 \<ge> (x - b) / 2" by auto
```
```   503     hence "?x = (x + b) / 2" by (simp add: field_simps min_max.inf_absorb2)
```
```   504     thus ?thesis using `b < x` by auto
```
```   505   qed
```
```   506   hence "0 \<le> f ?x" using all_le `?x < x` by auto
```
```   507   moreover have "\<bar>f ?x - f x\<bar> < - f x / 2"
```
```   508     using s_D[OF _ `\<bar> ?x - x \<bar> < s`] `?x < x` by auto
```
```   509   hence "f ?x - f x < - f x / 2" by auto
```
```   510   hence "f ?x < f x / 2" by auto
```
```   511   hence "f ?x < 0" using `f x < 0` by auto
```
```   512   thus False using `0 \<le> f ?x` by auto
```
```   513 qed
```
```   514
```
```   515
```
```   516 subsection {* Uniform Continuity *}
```
```   517
```
```   518 lemma isUCont_isCont: "isUCont f ==> isCont f x"
```
```   519 by (simp add: isUCont_def isCont_def LIM_def, force)
```
```   520
```
```   521 lemma isUCont_Cauchy:
```
```   522   "\<lbrakk>isUCont f; Cauchy X\<rbrakk> \<Longrightarrow> Cauchy (\<lambda>n. f (X n))"
```
```   523 unfolding isUCont_def
```
```   524 apply (rule metric_CauchyI)
```
```   525 apply (drule_tac x=e in spec, safe)
```
```   526 apply (drule_tac e=s in metric_CauchyD, safe)
```
```   527 apply (rule_tac x=M in exI, simp)
```
```   528 done
```
```   529
```
```   530 lemma (in bounded_linear) isUCont: "isUCont f"
```
```   531 unfolding isUCont_def dist_norm
```
```   532 proof (intro allI impI)
```
```   533   fix r::real assume r: "0 < r"
```
```   534   obtain K where K: "0 < K" and norm_le: "\<And>x. norm (f x) \<le> norm x * K"
```
```   535     using pos_bounded by fast
```
```   536   show "\<exists>s>0. \<forall>x y. norm (x - y) < s \<longrightarrow> norm (f x - f y) < r"
```
```   537   proof (rule exI, safe)
```
```   538     from r K show "0 < r / K" by (rule divide_pos_pos)
```
```   539   next
```
```   540     fix x y :: 'a
```
```   541     assume xy: "norm (x - y) < r / K"
```
```   542     have "norm (f x - f y) = norm (f (x - y))" by (simp only: diff)
```
```   543     also have "\<dots> \<le> norm (x - y) * K" by (rule norm_le)
```
```   544     also from K xy have "\<dots> < r" by (simp only: pos_less_divide_eq)
```
```   545     finally show "norm (f x - f y) < r" .
```
```   546   qed
```
```   547 qed
```
```   548
```
```   549 lemma (in bounded_linear) Cauchy: "Cauchy X \<Longrightarrow> Cauchy (\<lambda>n. f (X n))"
```
```   550 by (rule isUCont [THEN isUCont_Cauchy])
```
```   551
```
```   552
```
```   553 subsection {* Relation of LIM and LIMSEQ *}
```
```   554
```
```   555 lemma LIMSEQ_SEQ_conv1:
```
```   556   fixes f :: "'a::topological_space \<Rightarrow> 'b::topological_space"
```
```   557   assumes f: "f -- a --> l"
```
```   558   shows "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> a \<longrightarrow> (\<lambda>n. f (S n)) ----> l"
```
```   559   using tendsto_compose_eventually [OF f, where F=sequentially] by simp
```
```   560
```
```   561 lemma LIMSEQ_SEQ_conv2_lemma:
```
```   562   fixes a :: "'a::metric_space"
```
```   563   assumes "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> a \<longrightarrow> eventually (\<lambda>x. P (S x)) sequentially"
```
```   564   shows "eventually P (at a)"
```
```   565 proof (rule ccontr)
```
```   566   let ?I = "\<lambda>n. inverse (real (Suc n))"
```
```   567   let ?F = "\<lambda>n::nat. SOME x. x \<noteq> a \<and> dist x a < ?I n \<and> \<not> P x"
```
```   568   assume "\<not> eventually P (at a)"
```
```   569   hence P: "\<forall>d>0. \<exists>x. x \<noteq> a \<and> dist x a < d \<and> \<not> P x"
```
```   570     unfolding eventually_at by simp
```
```   571   hence "\<And>n. \<exists>x. x \<noteq> a \<and> dist x a < ?I n \<and> \<not> P x" by simp
```
```   572   hence F: "\<And>n. ?F n \<noteq> a \<and> dist (?F n) a < ?I n \<and> \<not> P (?F n)"
```
```   573     by (rule someI_ex)
```
```   574   hence F1: "\<And>n. ?F n \<noteq> a"
```
```   575     and F2: "\<And>n. dist (?F n) a < ?I n"
```
```   576     and F3: "\<And>n. \<not> P (?F n)"
```
```   577     by fast+
```
```   578   have "?F ----> a"
```
```   579     using LIMSEQ_inverse_real_of_nat
```
```   580     by (rule metric_tendsto_imp_tendsto,
```
```   581       simp add: dist_norm F2 [THEN less_imp_le])
```
```   582   moreover have "\<forall>n. ?F n \<noteq> a"
```
```   583     by (rule allI) (rule F1)
```
```   584   ultimately have "eventually (\<lambda>n. P (?F n)) sequentially"
```
```   585     using assms by simp
```
```   586   thus "False" by (simp add: F3)
```
```   587 qed
```
```   588
```
```   589 lemma LIMSEQ_SEQ_conv2:
```
```   590   fixes f :: "'a::metric_space \<Rightarrow> 'b::topological_space"
```
```   591   assumes "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> a \<longrightarrow> (\<lambda>n. f (S n)) ----> l"
```
```   592   shows "f -- a --> l"
```
```   593   using assms unfolding tendsto_def [where l=l]
```
```   594   by (simp add: LIMSEQ_SEQ_conv2_lemma)
```
```   595
```
```   596 lemma LIMSEQ_SEQ_conv:
```
```   597   "(\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> (a::'a::metric_space) \<longrightarrow> (\<lambda>n. X (S n)) ----> L) =
```
```   598    (X -- a --> (L::'b::topological_space))"
```
```   599   using LIMSEQ_SEQ_conv2 LIMSEQ_SEQ_conv1 ..
```
```   600
```
```   601 end
```