src/HOL/Lim.thy
 author huffman Tue Aug 16 09:31:23 2011 -0700 (2011-08-16) changeset 44233 aa74ce315bae parent 44218 f0e442e24816 child 44251 d101ed3177b6 permissions -rw-r--r--
```     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 apply (rule tendstoI, drule tendstoD [OF f])
```
```   136 apply (simp add: eventually_at_topological, safe)
```
```   137 apply (rule_tac x="S" in exI, safe)
```
```   138 apply (drule_tac x="x" in bspec, safe)
```
```   139 apply (erule (1) order_le_less_trans [OF le])
```
```   140 done
```
```   141
```
```   142 lemma LIM_imp_LIM:
```
```   143   fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
```
```   144   fixes g :: "'a::topological_space \<Rightarrow> 'c::real_normed_vector"
```
```   145   assumes f: "f -- a --> l"
```
```   146   assumes le: "\<And>x. x \<noteq> a \<Longrightarrow> norm (g x - m) \<le> norm (f x - l)"
```
```   147   shows "g -- a --> m"
```
```   148 apply (rule metric_LIM_imp_LIM [OF f])
```
```   149 apply (simp add: dist_norm le)
```
```   150 done
```
```   151
```
```   152 lemma LIM_norm:
```
```   153   fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
```
```   154   shows "f -- a --> l \<Longrightarrow> (\<lambda>x. norm (f x)) -- a --> norm l"
```
```   155 by (rule tendsto_norm)
```
```   156
```
```   157 lemma LIM_norm_zero:
```
```   158   fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
```
```   159   shows "f -- a --> 0 \<Longrightarrow> (\<lambda>x. norm (f x)) -- a --> 0"
```
```   160 by (rule tendsto_norm_zero)
```
```   161
```
```   162 lemma LIM_norm_zero_cancel:
```
```   163   fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
```
```   164   shows "(\<lambda>x. norm (f x)) -- a --> 0 \<Longrightarrow> f -- a --> 0"
```
```   165 by (rule tendsto_norm_zero_cancel)
```
```   166
```
```   167 lemma LIM_norm_zero_iff:
```
```   168   fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
```
```   169   shows "(\<lambda>x. norm (f x)) -- a --> 0 = f -- a --> 0"
```
```   170 by (rule tendsto_norm_zero_iff)
```
```   171
```
```   172 lemma LIM_rabs: "f -- a --> (l::real) \<Longrightarrow> (\<lambda>x. \<bar>f x\<bar>) -- a --> \<bar>l\<bar>"
```
```   173   by (rule tendsto_rabs)
```
```   174
```
```   175 lemma LIM_rabs_zero: "f -- a --> (0::real) \<Longrightarrow> (\<lambda>x. \<bar>f x\<bar>) -- a --> 0"
```
```   176   by (rule tendsto_rabs_zero)
```
```   177
```
```   178 lemma LIM_rabs_zero_cancel: "(\<lambda>x. \<bar>f x\<bar>) -- a --> (0::real) \<Longrightarrow> f -- a --> 0"
```
```   179   by (rule tendsto_rabs_zero_cancel)
```
```   180
```
```   181 lemma LIM_rabs_zero_iff: "(\<lambda>x. \<bar>f x\<bar>) -- a --> (0::real) = f -- a --> 0"
```
```   182   by (rule tendsto_rabs_zero_iff)
```
```   183
```
```   184 lemma trivial_limit_at:
```
```   185   fixes a :: "'a::real_normed_algebra_1"
```
```   186   shows "\<not> trivial_limit (at a)"  -- {* TODO: find a more appropriate class *}
```
```   187 unfolding trivial_limit_def
```
```   188 unfolding eventually_at dist_norm
```
```   189 by (clarsimp, rule_tac x="a + of_real (d/2)" in exI, simp)
```
```   190
```
```   191 lemma LIM_const_not_eq:
```
```   192   fixes a :: "'a::real_normed_algebra_1"
```
```   193   fixes k L :: "'b::t2_space"
```
```   194   shows "k \<noteq> L \<Longrightarrow> \<not> (\<lambda>x. k) -- a --> L"
```
```   195 by (simp add: tendsto_const_iff trivial_limit_at)
```
```   196
```
```   197 lemmas LIM_not_zero = LIM_const_not_eq [where L = 0]
```
```   198
```
```   199 lemma LIM_const_eq:
```
```   200   fixes a :: "'a::real_normed_algebra_1"
```
```   201   fixes k L :: "'b::t2_space"
```
```   202   shows "(\<lambda>x. k) -- a --> L \<Longrightarrow> k = L"
```
```   203   by (simp add: tendsto_const_iff trivial_limit_at)
```
```   204
```
```   205 lemma LIM_unique:
```
```   206   fixes a :: "'a::real_normed_algebra_1" -- {* TODO: find a more appropriate class *}
```
```   207   fixes L M :: "'b::t2_space"
```
```   208   shows "\<lbrakk>f -- a --> L; f -- a --> M\<rbrakk> \<Longrightarrow> L = M"
```
```   209   using trivial_limit_at by (rule tendsto_unique)
```
```   210
```
```   211 lemma LIM_ident [simp]: "(\<lambda>x. x) -- a --> a"
```
```   212 by (rule tendsto_ident_at)
```
```   213
```
```   214 text{*Limits are equal for functions equal except at limit point*}
```
```   215 lemma LIM_equal:
```
```   216      "[| \<forall>x. x \<noteq> a --> (f x = g x) |] ==> (f -- a --> l) = (g -- a --> l)"
```
```   217 unfolding tendsto_def eventually_at_topological by simp
```
```   218
```
```   219 lemma LIM_cong:
```
```   220   "\<lbrakk>a = b; \<And>x. x \<noteq> b \<Longrightarrow> f x = g x; l = m\<rbrakk>
```
```   221    \<Longrightarrow> ((\<lambda>x. f x) -- a --> l) = ((\<lambda>x. g x) -- b --> m)"
```
```   222 by (simp add: LIM_equal)
```
```   223
```
```   224 lemma metric_LIM_equal2:
```
```   225   assumes 1: "0 < R"
```
```   226   assumes 2: "\<And>x. \<lbrakk>x \<noteq> a; dist x a < R\<rbrakk> \<Longrightarrow> f x = g x"
```
```   227   shows "g -- a --> l \<Longrightarrow> f -- a --> l"
```
```   228 apply (rule topological_tendstoI)
```
```   229 apply (drule (2) topological_tendstoD)
```
```   230 apply (simp add: eventually_at, safe)
```
```   231 apply (rule_tac x="min d R" in exI, safe)
```
```   232 apply (simp add: 1)
```
```   233 apply (simp add: 2)
```
```   234 done
```
```   235
```
```   236 lemma LIM_equal2:
```
```   237   fixes f g :: "'a::real_normed_vector \<Rightarrow> 'b::topological_space"
```
```   238   assumes 1: "0 < R"
```
```   239   assumes 2: "\<And>x. \<lbrakk>x \<noteq> a; norm (x - a) < R\<rbrakk> \<Longrightarrow> f x = g x"
```
```   240   shows "g -- a --> l \<Longrightarrow> f -- a --> l"
```
```   241 by (rule metric_LIM_equal2 [OF 1 2], simp_all add: dist_norm)
```
```   242
```
```   243 text{*Two uses in Transcendental.ML*} (* BH: no longer true; delete? *)
```
```   244 lemma LIM_trans:
```
```   245   fixes f g :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
```
```   246   shows "[| (%x. f(x) + -g(x)) -- a --> 0;  g -- a --> l |] ==> f -- a --> l"
```
```   247 apply (drule LIM_add, assumption)
```
```   248 apply (auto simp add: add_assoc)
```
```   249 done
```
```   250
```
```   251 lemma LIM_compose:
```
```   252   assumes g: "g -- l --> g l"
```
```   253   assumes f: "f -- a --> l"
```
```   254   shows "(\<lambda>x. g (f x)) -- a --> g l"
```
```   255   using assms by (rule tendsto_compose)
```
```   256
```
```   257 lemma LIM_compose_eventually:
```
```   258   assumes f: "f -- a --> b"
```
```   259   assumes g: "g -- b --> c"
```
```   260   assumes inj: "eventually (\<lambda>x. f x \<noteq> b) (at a)"
```
```   261   shows "(\<lambda>x. g (f x)) -- a --> c"
```
```   262 proof (rule topological_tendstoI)
```
```   263   fix C assume C: "open C" "c \<in> C"
```
```   264   obtain B where B: "open B" "b \<in> B"
```
```   265     and gC: "\<And>y. y \<in> B \<Longrightarrow> y \<noteq> b \<Longrightarrow> g y \<in> C"
```
```   266     using topological_tendstoD [OF g C]
```
```   267     unfolding eventually_at_topological by fast
```
```   268   obtain A where A: "open A" "a \<in> A"
```
```   269     and fB: "\<And>x. x \<in> A \<Longrightarrow> x \<noteq> a \<Longrightarrow> f x \<in> B"
```
```   270     using topological_tendstoD [OF f B]
```
```   271     unfolding eventually_at_topological by fast
```
```   272   have "eventually (\<lambda>x. f x \<noteq> b \<longrightarrow> g (f x) \<in> C) (at a)"
```
```   273   unfolding eventually_at_topological
```
```   274   proof (intro exI conjI ballI impI)
```
```   275     show "open A" and "a \<in> A" using A .
```
```   276   next
```
```   277     fix x assume "x \<in> A" and "x \<noteq> a" and "f x \<noteq> b"
```
```   278     then show "g (f x) \<in> C" by (simp add: gC fB)
```
```   279   qed
```
```   280   with inj show "eventually (\<lambda>x. g (f x) \<in> C) (at a)"
```
```   281     by (rule eventually_rev_mp)
```
```   282 qed
```
```   283
```
```   284 lemma metric_LIM_compose2:
```
```   285   assumes f: "f -- a --> b"
```
```   286   assumes g: "g -- b --> c"
```
```   287   assumes inj: "\<exists>d>0. \<forall>x. x \<noteq> a \<and> dist x a < d \<longrightarrow> f x \<noteq> b"
```
```   288   shows "(\<lambda>x. g (f x)) -- a --> c"
```
```   289 using f g inj [folded eventually_at]
```
```   290 by (rule LIM_compose_eventually)
```
```   291
```
```   292 lemma LIM_compose2:
```
```   293   fixes a :: "'a::real_normed_vector"
```
```   294   assumes f: "f -- a --> b"
```
```   295   assumes g: "g -- b --> c"
```
```   296   assumes inj: "\<exists>d>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < d \<longrightarrow> f x \<noteq> b"
```
```   297   shows "(\<lambda>x. g (f x)) -- a --> c"
```
```   298 by (rule metric_LIM_compose2 [OF f g inj [folded dist_norm]])
```
```   299
```
```   300 lemma LIM_o: "\<lbrakk>g -- l --> g l; f -- a --> l\<rbrakk> \<Longrightarrow> (g \<circ> f) -- a --> g l"
```
```   301 unfolding o_def by (rule LIM_compose)
```
```   302
```
```   303 lemma real_LIM_sandwich_zero:
```
```   304   fixes f g :: "'a::topological_space \<Rightarrow> real"
```
```   305   assumes f: "f -- a --> 0"
```
```   306   assumes 1: "\<And>x. x \<noteq> a \<Longrightarrow> 0 \<le> g x"
```
```   307   assumes 2: "\<And>x. x \<noteq> a \<Longrightarrow> g x \<le> f x"
```
```   308   shows "g -- a --> 0"
```
```   309 proof (rule LIM_imp_LIM [OF f])
```
```   310   fix x assume x: "x \<noteq> a"
```
```   311   have "norm (g x - 0) = g x" by (simp add: 1 x)
```
```   312   also have "g x \<le> f x" by (rule 2 [OF x])
```
```   313   also have "f x \<le> \<bar>f x\<bar>" by (rule abs_ge_self)
```
```   314   also have "\<bar>f x\<bar> = norm (f x - 0)" by simp
```
```   315   finally show "norm (g x - 0) \<le> norm (f x - 0)" .
```
```   316 qed
```
```   317
```
```   318 text {* Bounded Linear Operators *}
```
```   319
```
```   320 lemma (in bounded_linear) LIM:
```
```   321   "g -- a --> l \<Longrightarrow> (\<lambda>x. f (g x)) -- a --> f l"
```
```   322 by (rule tendsto)
```
```   323
```
```   324 lemma (in bounded_linear) LIM_zero:
```
```   325   "g -- a --> 0 \<Longrightarrow> (\<lambda>x. f (g x)) -- a --> 0"
```
```   326   by (rule tendsto_zero)
```
```   327
```
```   328 text {* Bounded Bilinear Operators *}
```
```   329
```
```   330 lemma (in bounded_bilinear) LIM:
```
```   331   "\<lbrakk>f -- a --> L; g -- a --> M\<rbrakk> \<Longrightarrow> (\<lambda>x. f x ** g x) -- a --> L ** M"
```
```   332 by (rule tendsto)
```
```   333
```
```   334 lemma (in bounded_bilinear) LIM_prod_zero:
```
```   335   fixes a :: "'d::metric_space"
```
```   336   assumes f: "f -- a --> 0"
```
```   337   assumes g: "g -- a --> 0"
```
```   338   shows "(\<lambda>x. f x ** g x) -- a --> 0"
```
```   339   using f g by (rule tendsto_zero)
```
```   340
```
```   341 lemma (in bounded_bilinear) LIM_left_zero:
```
```   342   "f -- a --> 0 \<Longrightarrow> (\<lambda>x. f x ** c) -- a --> 0"
```
```   343   by (rule tendsto_left_zero)
```
```   344
```
```   345 lemma (in bounded_bilinear) LIM_right_zero:
```
```   346   "f -- a --> 0 \<Longrightarrow> (\<lambda>x. c ** f x) -- a --> 0"
```
```   347   by (rule tendsto_right_zero)
```
```   348
```
```   349 lemmas LIM_mult = mult.LIM
```
```   350
```
```   351 lemmas LIM_mult_zero = mult.LIM_prod_zero
```
```   352
```
```   353 lemmas LIM_mult_left_zero = mult.LIM_left_zero
```
```   354
```
```   355 lemmas LIM_mult_right_zero = mult.LIM_right_zero
```
```   356
```
```   357 lemmas LIM_scaleR = scaleR.LIM
```
```   358
```
```   359 lemmas LIM_of_real = of_real.LIM
```
```   360
```
```   361 lemma LIM_power:
```
```   362   fixes f :: "'a::topological_space \<Rightarrow> 'b::{power,real_normed_algebra}"
```
```   363   assumes f: "f -- a --> l"
```
```   364   shows "(\<lambda>x. f x ^ n) -- a --> l ^ n"
```
```   365   using assms by (rule tendsto_power)
```
```   366
```
```   367 lemma LIM_inverse:
```
```   368   fixes L :: "'a::real_normed_div_algebra"
```
```   369   shows "\<lbrakk>f -- a --> L; L \<noteq> 0\<rbrakk> \<Longrightarrow> (\<lambda>x. inverse (f x)) -- a --> inverse L"
```
```   370 by (rule tendsto_inverse)
```
```   371
```
```   372 lemma LIM_inverse_fun:
```
```   373   assumes a: "a \<noteq> (0::'a::real_normed_div_algebra)"
```
```   374   shows "inverse -- a --> inverse a"
```
```   375 by (rule LIM_inverse [OF LIM_ident a])
```
```   376
```
```   377 lemma LIM_sgn:
```
```   378   fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
```
```   379   shows "\<lbrakk>f -- a --> l; l \<noteq> 0\<rbrakk> \<Longrightarrow> (\<lambda>x. sgn (f x)) -- a --> sgn l"
```
```   380   by (rule tendsto_sgn)
```
```   381
```
```   382
```
```   383 subsection {* Continuity *}
```
```   384
```
```   385 lemma LIM_isCont_iff:
```
```   386   fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::topological_space"
```
```   387   shows "(f -- a --> f a) = ((\<lambda>h. f (a + h)) -- 0 --> f a)"
```
```   388 by (rule iffI [OF LIM_offset_zero LIM_offset_zero_cancel])
```
```   389
```
```   390 lemma isCont_iff:
```
```   391   fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::topological_space"
```
```   392   shows "isCont f x = (\<lambda>h. f (x + h)) -- 0 --> f x"
```
```   393 by (simp add: isCont_def LIM_isCont_iff)
```
```   394
```
```   395 lemma isCont_ident [simp]: "isCont (\<lambda>x. x) a"
```
```   396   unfolding isCont_def by (rule LIM_ident)
```
```   397
```
```   398 lemma isCont_const [simp]: "isCont (\<lambda>x. k) a"
```
```   399   unfolding isCont_def by (rule LIM_const)
```
```   400
```
```   401 lemma isCont_norm [simp]:
```
```   402   fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
```
```   403   shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. norm (f x)) a"
```
```   404   unfolding isCont_def by (rule LIM_norm)
```
```   405
```
```   406 lemma isCont_rabs [simp]:
```
```   407   fixes f :: "'a::topological_space \<Rightarrow> real"
```
```   408   shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. \<bar>f x\<bar>) a"
```
```   409   unfolding isCont_def by (rule LIM_rabs)
```
```   410
```
```   411 lemma isCont_add [simp]:
```
```   412   fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
```
```   413   shows "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x + g x) a"
```
```   414   unfolding isCont_def by (rule LIM_add)
```
```   415
```
```   416 lemma isCont_minus [simp]:
```
```   417   fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
```
```   418   shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. - f x) a"
```
```   419   unfolding isCont_def by (rule LIM_minus)
```
```   420
```
```   421 lemma isCont_diff [simp]:
```
```   422   fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
```
```   423   shows "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x - g x) a"
```
```   424   unfolding isCont_def by (rule LIM_diff)
```
```   425
```
```   426 lemma isCont_mult [simp]:
```
```   427   fixes f g :: "'a::topological_space \<Rightarrow> 'b::real_normed_algebra"
```
```   428   shows "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x * g x) a"
```
```   429   unfolding isCont_def by (rule LIM_mult)
```
```   430
```
```   431 lemma isCont_inverse [simp]:
```
```   432   fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_div_algebra"
```
```   433   shows "\<lbrakk>isCont f a; f a \<noteq> 0\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. inverse (f x)) a"
```
```   434   unfolding isCont_def by (rule LIM_inverse)
```
```   435
```
```   436 lemma isCont_divide [simp]:
```
```   437   fixes f g :: "'a::topological_space \<Rightarrow> 'b::real_normed_field"
```
```   438   shows "\<lbrakk>isCont f a; isCont g a; g a \<noteq> 0\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x / g x) a"
```
```   439   unfolding isCont_def by (rule tendsto_divide)
```
```   440
```
```   441 lemma isCont_LIM_compose:
```
```   442   "\<lbrakk>isCont g l; f -- a --> l\<rbrakk> \<Longrightarrow> (\<lambda>x. g (f x)) -- a --> g l"
```
```   443   unfolding isCont_def by (rule LIM_compose)
```
```   444
```
```   445 lemma metric_isCont_LIM_compose2:
```
```   446   assumes f [unfolded isCont_def]: "isCont f a"
```
```   447   assumes g: "g -- f a --> l"
```
```   448   assumes inj: "\<exists>d>0. \<forall>x. x \<noteq> a \<and> dist x a < d \<longrightarrow> f x \<noteq> f a"
```
```   449   shows "(\<lambda>x. g (f x)) -- a --> l"
```
```   450 by (rule metric_LIM_compose2 [OF f g inj])
```
```   451
```
```   452 lemma isCont_LIM_compose2:
```
```   453   fixes a :: "'a::real_normed_vector"
```
```   454   assumes f [unfolded isCont_def]: "isCont f a"
```
```   455   assumes g: "g -- f a --> l"
```
```   456   assumes inj: "\<exists>d>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < d \<longrightarrow> f x \<noteq> f a"
```
```   457   shows "(\<lambda>x. g (f x)) -- a --> l"
```
```   458 by (rule LIM_compose2 [OF f g inj])
```
```   459
```
```   460 lemma isCont_o2: "\<lbrakk>isCont f a; isCont g (f a)\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. g (f x)) a"
```
```   461   unfolding isCont_def by (rule LIM_compose)
```
```   462
```
```   463 lemma isCont_o: "\<lbrakk>isCont f a; isCont g (f a)\<rbrakk> \<Longrightarrow> isCont (g o f) a"
```
```   464   unfolding o_def by (rule isCont_o2)
```
```   465
```
```   466 lemma (in bounded_linear) isCont:
```
```   467   "isCont g a \<Longrightarrow> isCont (\<lambda>x. f (g x)) a"
```
```   468   unfolding isCont_def by (rule LIM)
```
```   469
```
```   470 lemma (in bounded_bilinear) isCont:
```
```   471   "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x ** g x) a"
```
```   472   unfolding isCont_def by (rule LIM)
```
```   473
```
```   474 lemmas isCont_scaleR [simp] = scaleR.isCont
```
```   475
```
```   476 lemma isCont_of_real [simp]:
```
```   477   "isCont f a \<Longrightarrow> isCont (\<lambda>x. of_real (f x)::'b::real_normed_algebra_1) a"
```
```   478   by (rule of_real.isCont)
```
```   479
```
```   480 lemma isCont_power [simp]:
```
```   481   fixes f :: "'a::topological_space \<Rightarrow> 'b::{power,real_normed_algebra}"
```
```   482   shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. f x ^ n) a"
```
```   483   unfolding isCont_def by (rule LIM_power)
```
```   484
```
```   485 lemma isCont_sgn [simp]:
```
```   486   fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
```
```   487   shows "\<lbrakk>isCont f a; f a \<noteq> 0\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. sgn (f x)) a"
```
```   488   unfolding isCont_def by (rule LIM_sgn)
```
```   489
```
```   490 lemma isCont_setsum [simp]:
```
```   491   fixes f :: "'a \<Rightarrow> 'b::topological_space \<Rightarrow> 'c::real_normed_vector"
```
```   492   fixes A :: "'a set"
```
```   493   shows "\<forall>i\<in>A. isCont (f i) a \<Longrightarrow> isCont (\<lambda>x. \<Sum>i\<in>A. f i x) a"
```
```   494   unfolding isCont_def by (simp add: tendsto_setsum)
```
```   495
```
```   496 lemmas isCont_intros =
```
```   497   isCont_ident isCont_const isCont_norm isCont_rabs isCont_add isCont_minus
```
```   498   isCont_diff isCont_mult isCont_inverse isCont_divide isCont_scaleR
```
```   499   isCont_of_real isCont_power isCont_sgn isCont_setsum
```
```   500
```
```   501 lemma LIM_less_bound: fixes f :: "real \<Rightarrow> real" assumes "b < x"
```
```   502   and all_le: "\<forall> x' \<in> { b <..< x}. 0 \<le> f x'" and isCont: "isCont f x"
```
```   503   shows "0 \<le> f x"
```
```   504 proof (rule ccontr)
```
```   505   assume "\<not> 0 \<le> f x" hence "f x < 0" by auto
```
```   506   hence "0 < - f x / 2" by auto
```
```   507   from isCont[unfolded isCont_def, THEN LIM_D, OF this]
```
```   508   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
```
```   509
```
```   510   let ?x = "x - min (s / 2) ((x - b) / 2)"
```
```   511   have "?x < x" and "\<bar> ?x - x \<bar> < s"
```
```   512     using `b < x` and `0 < s` by auto
```
```   513   have "b < ?x"
```
```   514   proof (cases "s < x - b")
```
```   515     case True thus ?thesis using `0 < s` by auto
```
```   516   next
```
```   517     case False hence "s / 2 \<ge> (x - b) / 2" by auto
```
```   518     hence "?x = (x + b) / 2" by (simp add: field_simps min_max.inf_absorb2)
```
```   519     thus ?thesis using `b < x` by auto
```
```   520   qed
```
```   521   hence "0 \<le> f ?x" using all_le `?x < x` by auto
```
```   522   moreover have "\<bar>f ?x - f x\<bar> < - f x / 2"
```
```   523     using s_D[OF _ `\<bar> ?x - x \<bar> < s`] `?x < x` by auto
```
```   524   hence "f ?x - f x < - f x / 2" by auto
```
```   525   hence "f ?x < f x / 2" by auto
```
```   526   hence "f ?x < 0" using `f x < 0` by auto
```
```   527   thus False using `0 \<le> f ?x` by auto
```
```   528 qed
```
```   529
```
```   530
```
```   531 subsection {* Uniform Continuity *}
```
```   532
```
```   533 lemma isUCont_isCont: "isUCont f ==> isCont f x"
```
```   534 by (simp add: isUCont_def isCont_def LIM_def, force)
```
```   535
```
```   536 lemma isUCont_Cauchy:
```
```   537   "\<lbrakk>isUCont f; Cauchy X\<rbrakk> \<Longrightarrow> Cauchy (\<lambda>n. f (X n))"
```
```   538 unfolding isUCont_def
```
```   539 apply (rule metric_CauchyI)
```
```   540 apply (drule_tac x=e in spec, safe)
```
```   541 apply (drule_tac e=s in metric_CauchyD, safe)
```
```   542 apply (rule_tac x=M in exI, simp)
```
```   543 done
```
```   544
```
```   545 lemma (in bounded_linear) isUCont: "isUCont f"
```
```   546 unfolding isUCont_def dist_norm
```
```   547 proof (intro allI impI)
```
```   548   fix r::real assume r: "0 < r"
```
```   549   obtain K where K: "0 < K" and norm_le: "\<And>x. norm (f x) \<le> norm x * K"
```
```   550     using pos_bounded by fast
```
```   551   show "\<exists>s>0. \<forall>x y. norm (x - y) < s \<longrightarrow> norm (f x - f y) < r"
```
```   552   proof (rule exI, safe)
```
```   553     from r K show "0 < r / K" by (rule divide_pos_pos)
```
```   554   next
```
```   555     fix x y :: 'a
```
```   556     assume xy: "norm (x - y) < r / K"
```
```   557     have "norm (f x - f y) = norm (f (x - y))" by (simp only: diff)
```
```   558     also have "\<dots> \<le> norm (x - y) * K" by (rule norm_le)
```
```   559     also from K xy have "\<dots> < r" by (simp only: pos_less_divide_eq)
```
```   560     finally show "norm (f x - f y) < r" .
```
```   561   qed
```
```   562 qed
```
```   563
```
```   564 lemma (in bounded_linear) Cauchy: "Cauchy X \<Longrightarrow> Cauchy (\<lambda>n. f (X n))"
```
```   565 by (rule isUCont [THEN isUCont_Cauchy])
```
```   566
```
```   567
```
```   568 subsection {* Relation of LIM and LIMSEQ *}
```
```   569
```
```   570 lemma LIMSEQ_SEQ_conv1:
```
```   571   fixes a :: "'a::metric_space" and L :: "'b::metric_space"
```
```   572   assumes X: "X -- a --> L"
```
```   573   shows "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> a \<longrightarrow> (\<lambda>n. X (S n)) ----> L"
```
```   574 proof (safe intro!: metric_LIMSEQ_I)
```
```   575   fix S :: "nat \<Rightarrow> 'a"
```
```   576   fix r :: real
```
```   577   assume rgz: "0 < r"
```
```   578   assume as: "\<forall>n. S n \<noteq> a"
```
```   579   assume S: "S ----> a"
```
```   580   from metric_LIM_D [OF X rgz] obtain s
```
```   581     where sgz: "0 < s"
```
```   582     and aux: "\<And>x. \<lbrakk>x \<noteq> a; dist x a < s\<rbrakk> \<Longrightarrow> dist (X x) L < r"
```
```   583     by fast
```
```   584   from metric_LIMSEQ_D [OF S sgz]
```
```   585   obtain no where "\<forall>n\<ge>no. dist (S n) a < s" by blast
```
```   586   hence "\<forall>n\<ge>no. dist (X (S n)) L < r" by (simp add: aux as)
```
```   587   thus "\<exists>no. \<forall>n\<ge>no. dist (X (S n)) L < r" ..
```
```   588 qed
```
```   589
```
```   590
```
```   591 lemma LIMSEQ_SEQ_conv2:
```
```   592   fixes a :: real and L :: "'a::metric_space"
```
```   593   assumes "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> a \<longrightarrow> (\<lambda>n. X (S n)) ----> L"
```
```   594   shows "X -- a --> L"
```
```   595 proof (rule ccontr)
```
```   596   assume "\<not> (X -- a --> L)"
```
```   597   hence "\<not> (\<forall>r > 0. \<exists>s > 0. \<forall>x. x \<noteq> a & norm (x - a) < s --> dist (X x) L < r)"
```
```   598     unfolding LIM_def dist_norm .
```
```   599   hence "\<exists>r > 0. \<forall>s > 0. \<exists>x. \<not>(x \<noteq> a \<and> \<bar>x - a\<bar> < s --> dist (X x) L < r)" by simp
```
```   600   hence "\<exists>r > 0. \<forall>s > 0. \<exists>x. (x \<noteq> a \<and> \<bar>x - a\<bar> < s \<and> dist (X x) L \<ge> r)" by (simp add: not_less)
```
```   601   then obtain r where rdef: "r > 0 \<and> (\<forall>s > 0. \<exists>x. (x \<noteq> a \<and> \<bar>x - a\<bar> < s \<and> dist (X x) L \<ge> r))" by auto
```
```   602
```
```   603   let ?F = "\<lambda>n::nat. SOME x. x\<noteq>a \<and> \<bar>x - a\<bar> < inverse (real (Suc n)) \<and> dist (X x) L \<ge> r"
```
```   604   have "\<And>n. \<exists>x. x\<noteq>a \<and> \<bar>x - a\<bar> < inverse (real (Suc n)) \<and> dist (X x) L \<ge> r"
```
```   605     using rdef by simp
```
```   606   hence F: "\<And>n. ?F n \<noteq> a \<and> \<bar>?F n - a\<bar> < inverse (real (Suc n)) \<and> dist (X (?F n)) L \<ge> r"
```
```   607     by (rule someI_ex)
```
```   608   hence F1: "\<And>n. ?F n \<noteq> a"
```
```   609     and F2: "\<And>n. \<bar>?F n - a\<bar> < inverse (real (Suc n))"
```
```   610     and F3: "\<And>n. dist (X (?F n)) L \<ge> r"
```
```   611     by fast+
```
```   612
```
```   613   have "?F ----> a"
```
```   614   proof (rule LIMSEQ_I, unfold real_norm_def)
```
```   615       fix e::real
```
```   616       assume "0 < e"
```
```   617         (* choose no such that inverse (real (Suc n)) < e *)
```
```   618       then have "\<exists>no. inverse (real (Suc no)) < e" by (rule reals_Archimedean)
```
```   619       then obtain m where nodef: "inverse (real (Suc m)) < e" by auto
```
```   620       show "\<exists>no. \<forall>n. no \<le> n --> \<bar>?F n - a\<bar> < e"
```
```   621       proof (intro exI allI impI)
```
```   622         fix n
```
```   623         assume mlen: "m \<le> n"
```
```   624         have "\<bar>?F n - a\<bar> < inverse (real (Suc n))"
```
```   625           by (rule F2)
```
```   626         also have "inverse (real (Suc n)) \<le> inverse (real (Suc m))"
```
```   627           using mlen by auto
```
```   628         also from nodef have
```
```   629           "inverse (real (Suc m)) < e" .
```
```   630         finally show "\<bar>?F n - a\<bar> < e" .
```
```   631       qed
```
```   632   qed
```
```   633
```
```   634   moreover have "\<forall>n. ?F n \<noteq> a"
```
```   635     by (rule allI) (rule F1)
```
```   636
```
```   637   moreover note `\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> a \<longrightarrow> (\<lambda>n. X (S n)) ----> L`
```
```   638   ultimately have "(\<lambda>n. X (?F n)) ----> L" by simp
```
```   639
```
```   640   moreover have "\<not> ((\<lambda>n. X (?F n)) ----> L)"
```
```   641   proof -
```
```   642     {
```
```   643       fix no::nat
```
```   644       obtain n where "n = no + 1" by simp
```
```   645       then have nolen: "no \<le> n" by simp
```
```   646         (* We prove this by showing that for any m there is an n\<ge>m such that |X (?F n) - L| \<ge> r *)
```
```   647       have "dist (X (?F n)) L \<ge> r"
```
```   648         by (rule F3)
```
```   649       with nolen have "\<exists>n. no \<le> n \<and> dist (X (?F n)) L \<ge> r" by fast
```
```   650     }
```
```   651     then have "(\<forall>no. \<exists>n. no \<le> n \<and> dist (X (?F n)) L \<ge> r)" by simp
```
```   652     with rdef have "\<exists>e>0. (\<forall>no. \<exists>n. no \<le> n \<and> dist (X (?F n)) L \<ge> e)" by auto
```
```   653     thus ?thesis by (unfold LIMSEQ_def, auto simp add: not_less)
```
```   654   qed
```
```   655   ultimately show False by simp
```
```   656 qed
```
```   657
```
```   658 lemma LIMSEQ_SEQ_conv:
```
```   659   "(\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> (a::real) \<longrightarrow> (\<lambda>n. X (S n)) ----> L) =
```
```   660    (X -- a --> (L::'a::metric_space))"
```
```   661 proof
```
```   662   assume "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> a \<longrightarrow> (\<lambda>n. X (S n)) ----> L"
```
```   663   thus "X -- a --> L" by (rule LIMSEQ_SEQ_conv2)
```
```   664 next
```
```   665   assume "(X -- a --> L)"
```
```   666   thus "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> a \<longrightarrow> (\<lambda>n. X (S n)) ----> L" by (rule LIMSEQ_SEQ_conv1)
```
```   667 qed
```
```   668
```
```   669 end
```