src/HOL/Lim.thy
 author nipkow Thu Dec 11 08:52:50 2008 +0100 (2008-12-11) changeset 29106 25e28a4070f3 parent 28952 15a4b2cf8c34 child 29197 6d4cb27ed19c permissions -rw-r--r--
Testfile for Stefan's code generator
```     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 "~~/src/HOL/Hyperreal/SEQ"
```
```    11 begin
```
```    12
```
```    13 text{*Standard Definitions*}
```
```    14
```
```    15 definition
```
```    16   LIM :: "['a::real_normed_vector => 'b::real_normed_vector, 'a, 'b] => bool"
```
```    17         ("((_)/ -- (_)/ --> (_))" [60, 0, 60] 60) where
```
```    18   [code del]: "f -- a --> L =
```
```    19      (\<forall>r > 0. \<exists>s > 0. \<forall>x. x \<noteq> a & norm (x - a) < s
```
```    20         --> norm (f x - L) < r)"
```
```    21
```
```    22 definition
```
```    23   isCont :: "['a::real_normed_vector => 'b::real_normed_vector, 'a] => bool" where
```
```    24   "isCont f a = (f -- a --> (f a))"
```
```    25
```
```    26 definition
```
```    27   isUCont :: "['a::real_normed_vector => 'b::real_normed_vector] => bool" where
```
```    28   [code del]: "isUCont f = (\<forall>r>0. \<exists>s>0. \<forall>x y. norm (x - y) < s \<longrightarrow> norm (f x - f y) < r)"
```
```    29
```
```    30
```
```    31 subsection {* Limits of Functions *}
```
```    32
```
```    33 subsubsection {* Purely standard proofs *}
```
```    34
```
```    35 lemma LIM_eq:
```
```    36      "f -- a --> L =
```
```    37      (\<forall>r>0.\<exists>s>0.\<forall>x. x \<noteq> a & norm (x-a) < s --> norm (f x - L) < r)"
```
```    38 by (simp add: LIM_def diff_minus)
```
```    39
```
```    40 lemma LIM_I:
```
```    41      "(!!r. 0<r ==> \<exists>s>0.\<forall>x. x \<noteq> a & norm (x-a) < s --> norm (f x - L) < r)
```
```    42       ==> f -- a --> L"
```
```    43 by (simp add: LIM_eq)
```
```    44
```
```    45 lemma LIM_D:
```
```    46      "[| f -- a --> L; 0<r |]
```
```    47       ==> \<exists>s>0.\<forall>x. x \<noteq> a & norm (x-a) < s --> norm (f x - L) < r"
```
```    48 by (simp add: LIM_eq)
```
```    49
```
```    50 lemma LIM_offset: "f -- a --> L \<Longrightarrow> (\<lambda>x. f (x + k)) -- a - k --> L"
```
```    51 apply (rule LIM_I)
```
```    52 apply (drule_tac r="r" in LIM_D, safe)
```
```    53 apply (rule_tac x="s" in exI, safe)
```
```    54 apply (drule_tac x="x + k" in spec)
```
```    55 apply (simp add: compare_rls)
```
```    56 done
```
```    57
```
```    58 lemma LIM_offset_zero: "f -- a --> L \<Longrightarrow> (\<lambda>h. f (a + h)) -- 0 --> L"
```
```    59 by (drule_tac k="a" in LIM_offset, simp add: add_commute)
```
```    60
```
```    61 lemma LIM_offset_zero_cancel: "(\<lambda>h. f (a + h)) -- 0 --> L \<Longrightarrow> f -- a --> L"
```
```    62 by (drule_tac k="- a" in LIM_offset, simp)
```
```    63
```
```    64 lemma LIM_const [simp]: "(%x. k) -- x --> k"
```
```    65 by (simp add: LIM_def)
```
```    66
```
```    67 lemma LIM_add:
```
```    68   fixes f g :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
```
```    69   assumes f: "f -- a --> L" and g: "g -- a --> M"
```
```    70   shows "(%x. f x + g(x)) -- a --> (L + M)"
```
```    71 proof (rule LIM_I)
```
```    72   fix r :: real
```
```    73   assume r: "0 < r"
```
```    74   from LIM_D [OF f half_gt_zero [OF r]]
```
```    75   obtain fs
```
```    76     where fs:    "0 < fs"
```
```    77       and fs_lt: "\<forall>x. x \<noteq> a & norm (x-a) < fs --> norm (f x - L) < r/2"
```
```    78   by blast
```
```    79   from LIM_D [OF g half_gt_zero [OF r]]
```
```    80   obtain gs
```
```    81     where gs:    "0 < gs"
```
```    82       and gs_lt: "\<forall>x. x \<noteq> a & norm (x-a) < gs --> norm (g x - M) < r/2"
```
```    83   by blast
```
```    84   show "\<exists>s>0.\<forall>x. x \<noteq> a \<and> norm (x-a) < s \<longrightarrow> norm (f x + g x - (L + M)) < r"
```
```    85   proof (intro exI conjI strip)
```
```    86     show "0 < min fs gs"  by (simp add: fs gs)
```
```    87     fix x :: 'a
```
```    88     assume "x \<noteq> a \<and> norm (x-a) < min fs gs"
```
```    89     hence "x \<noteq> a \<and> norm (x-a) < fs \<and> norm (x-a) < gs" by simp
```
```    90     with fs_lt gs_lt
```
```    91     have "norm (f x - L) < r/2" and "norm (g x - M) < r/2" by blast+
```
```    92     hence "norm (f x - L) + norm (g x - M) < r" by arith
```
```    93     thus "norm (f x + g x - (L + M)) < r"
```
```    94       by (blast intro: norm_diff_triangle_ineq order_le_less_trans)
```
```    95   qed
```
```    96 qed
```
```    97
```
```    98 lemma LIM_add_zero:
```
```    99   "\<lbrakk>f -- a --> 0; g -- a --> 0\<rbrakk> \<Longrightarrow> (\<lambda>x. f x + g x) -- a --> 0"
```
```   100 by (drule (1) LIM_add, simp)
```
```   101
```
```   102 lemma minus_diff_minus:
```
```   103   fixes a b :: "'a::ab_group_add"
```
```   104   shows "(- a) - (- b) = - (a - b)"
```
```   105 by simp
```
```   106
```
```   107 lemma LIM_minus: "f -- a --> L ==> (%x. -f(x)) -- a --> -L"
```
```   108 by (simp only: LIM_eq minus_diff_minus norm_minus_cancel)
```
```   109
```
```   110 lemma LIM_add_minus:
```
```   111     "[| f -- x --> l; g -- x --> m |] ==> (%x. f(x) + -g(x)) -- x --> (l + -m)"
```
```   112 by (intro LIM_add LIM_minus)
```
```   113
```
```   114 lemma LIM_diff:
```
```   115     "[| f -- x --> l; g -- x --> m |] ==> (%x. f(x) - g(x)) -- x --> l-m"
```
```   116 by (simp only: diff_minus LIM_add LIM_minus)
```
```   117
```
```   118 lemma LIM_zero: "f -- a --> l \<Longrightarrow> (\<lambda>x. f x - l) -- a --> 0"
```
```   119 by (simp add: LIM_def)
```
```   120
```
```   121 lemma LIM_zero_cancel: "(\<lambda>x. f x - l) -- a --> 0 \<Longrightarrow> f -- a --> l"
```
```   122 by (simp add: LIM_def)
```
```   123
```
```   124 lemma LIM_zero_iff: "(\<lambda>x. f x - l) -- a --> 0 = f -- a --> l"
```
```   125 by (simp add: LIM_def)
```
```   126
```
```   127 lemma LIM_imp_LIM:
```
```   128   assumes f: "f -- a --> l"
```
```   129   assumes le: "\<And>x. x \<noteq> a \<Longrightarrow> norm (g x - m) \<le> norm (f x - l)"
```
```   130   shows "g -- a --> m"
```
```   131 apply (rule LIM_I, drule LIM_D [OF f], safe)
```
```   132 apply (rule_tac x="s" in exI, safe)
```
```   133 apply (drule_tac x="x" in spec, safe)
```
```   134 apply (erule (1) order_le_less_trans [OF le])
```
```   135 done
```
```   136
```
```   137 lemma LIM_norm: "f -- a --> l \<Longrightarrow> (\<lambda>x. norm (f x)) -- a --> norm l"
```
```   138 by (erule LIM_imp_LIM, simp add: norm_triangle_ineq3)
```
```   139
```
```   140 lemma LIM_norm_zero: "f -- a --> 0 \<Longrightarrow> (\<lambda>x. norm (f x)) -- a --> 0"
```
```   141 by (drule LIM_norm, simp)
```
```   142
```
```   143 lemma LIM_norm_zero_cancel: "(\<lambda>x. norm (f x)) -- a --> 0 \<Longrightarrow> f -- a --> 0"
```
```   144 by (erule LIM_imp_LIM, simp)
```
```   145
```
```   146 lemma LIM_norm_zero_iff: "(\<lambda>x. norm (f x)) -- a --> 0 = f -- a --> 0"
```
```   147 by (rule iffI [OF LIM_norm_zero_cancel LIM_norm_zero])
```
```   148
```
```   149 lemma LIM_rabs: "f -- a --> (l::real) \<Longrightarrow> (\<lambda>x. \<bar>f x\<bar>) -- a --> \<bar>l\<bar>"
```
```   150 by (fold real_norm_def, rule LIM_norm)
```
```   151
```
```   152 lemma LIM_rabs_zero: "f -- a --> (0::real) \<Longrightarrow> (\<lambda>x. \<bar>f x\<bar>) -- a --> 0"
```
```   153 by (fold real_norm_def, rule LIM_norm_zero)
```
```   154
```
```   155 lemma LIM_rabs_zero_cancel: "(\<lambda>x. \<bar>f x\<bar>) -- a --> (0::real) \<Longrightarrow> f -- a --> 0"
```
```   156 by (fold real_norm_def, rule LIM_norm_zero_cancel)
```
```   157
```
```   158 lemma LIM_rabs_zero_iff: "(\<lambda>x. \<bar>f x\<bar>) -- a --> (0::real) = f -- a --> 0"
```
```   159 by (fold real_norm_def, rule LIM_norm_zero_iff)
```
```   160
```
```   161 lemma LIM_const_not_eq:
```
```   162   fixes a :: "'a::real_normed_algebra_1"
```
```   163   shows "k \<noteq> L \<Longrightarrow> \<not> (\<lambda>x. k) -- a --> L"
```
```   164 apply (simp add: LIM_eq)
```
```   165 apply (rule_tac x="norm (k - L)" in exI, simp, safe)
```
```   166 apply (rule_tac x="a + of_real (s/2)" in exI, simp add: norm_of_real)
```
```   167 done
```
```   168
```
```   169 lemmas LIM_not_zero = LIM_const_not_eq [where L = 0]
```
```   170
```
```   171 lemma LIM_const_eq:
```
```   172   fixes a :: "'a::real_normed_algebra_1"
```
```   173   shows "(\<lambda>x. k) -- a --> L \<Longrightarrow> k = L"
```
```   174 apply (rule ccontr)
```
```   175 apply (blast dest: LIM_const_not_eq)
```
```   176 done
```
```   177
```
```   178 lemma LIM_unique:
```
```   179   fixes a :: "'a::real_normed_algebra_1"
```
```   180   shows "\<lbrakk>f -- a --> L; f -- a --> M\<rbrakk> \<Longrightarrow> L = M"
```
```   181 apply (drule (1) LIM_diff)
```
```   182 apply (auto dest!: LIM_const_eq)
```
```   183 done
```
```   184
```
```   185 lemma LIM_ident [simp]: "(\<lambda>x. x) -- a --> a"
```
```   186 by (auto simp add: LIM_def)
```
```   187
```
```   188 text{*Limits are equal for functions equal except at limit point*}
```
```   189 lemma LIM_equal:
```
```   190      "[| \<forall>x. x \<noteq> a --> (f x = g x) |] ==> (f -- a --> l) = (g -- a --> l)"
```
```   191 by (simp add: LIM_def)
```
```   192
```
```   193 lemma LIM_cong:
```
```   194   "\<lbrakk>a = b; \<And>x. x \<noteq> b \<Longrightarrow> f x = g x; l = m\<rbrakk>
```
```   195    \<Longrightarrow> ((\<lambda>x. f x) -- a --> l) = ((\<lambda>x. g x) -- b --> m)"
```
```   196 by (simp add: LIM_def)
```
```   197
```
```   198 lemma LIM_equal2:
```
```   199   assumes 1: "0 < R"
```
```   200   assumes 2: "\<And>x. \<lbrakk>x \<noteq> a; norm (x - a) < R\<rbrakk> \<Longrightarrow> f x = g x"
```
```   201   shows "g -- a --> l \<Longrightarrow> f -- a --> l"
```
```   202 apply (unfold LIM_def, safe)
```
```   203 apply (drule_tac x="r" in spec, safe)
```
```   204 apply (rule_tac x="min s R" in exI, safe)
```
```   205 apply (simp add: 1)
```
```   206 apply (simp add: 2)
```
```   207 done
```
```   208
```
```   209 text{*Two uses in Hyperreal/Transcendental.ML*}
```
```   210 lemma LIM_trans:
```
```   211      "[| (%x. f(x) + -g(x)) -- a --> 0;  g -- a --> l |] ==> f -- a --> l"
```
```   212 apply (drule LIM_add, assumption)
```
```   213 apply (auto simp add: add_assoc)
```
```   214 done
```
```   215
```
```   216 lemma LIM_compose:
```
```   217   assumes g: "g -- l --> g l"
```
```   218   assumes f: "f -- a --> l"
```
```   219   shows "(\<lambda>x. g (f x)) -- a --> g l"
```
```   220 proof (rule LIM_I)
```
```   221   fix r::real assume r: "0 < r"
```
```   222   obtain s where s: "0 < s"
```
```   223     and less_r: "\<And>y. \<lbrakk>y \<noteq> l; norm (y - l) < s\<rbrakk> \<Longrightarrow> norm (g y - g l) < r"
```
```   224     using LIM_D [OF g r] by fast
```
```   225   obtain t where t: "0 < t"
```
```   226     and less_s: "\<And>x. \<lbrakk>x \<noteq> a; norm (x - a) < t\<rbrakk> \<Longrightarrow> norm (f x - l) < s"
```
```   227     using LIM_D [OF f s] by fast
```
```   228
```
```   229   show "\<exists>t>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < t \<longrightarrow> norm (g (f x) - g l) < r"
```
```   230   proof (rule exI, safe)
```
```   231     show "0 < t" using t .
```
```   232   next
```
```   233     fix x assume "x \<noteq> a" and "norm (x - a) < t"
```
```   234     hence "norm (f x - l) < s" by (rule less_s)
```
```   235     thus "norm (g (f x) - g l) < r"
```
```   236       using r less_r by (case_tac "f x = l", simp_all)
```
```   237   qed
```
```   238 qed
```
```   239
```
```   240 lemma LIM_compose2:
```
```   241   assumes f: "f -- a --> b"
```
```   242   assumes g: "g -- b --> c"
```
```   243   assumes inj: "\<exists>d>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < d \<longrightarrow> f x \<noteq> b"
```
```   244   shows "(\<lambda>x. g (f x)) -- a --> c"
```
```   245 proof (rule LIM_I)
```
```   246   fix r :: real
```
```   247   assume r: "0 < r"
```
```   248   obtain s where s: "0 < s"
```
```   249     and less_r: "\<And>y. \<lbrakk>y \<noteq> b; norm (y - b) < s\<rbrakk> \<Longrightarrow> norm (g y - c) < r"
```
```   250     using LIM_D [OF g r] by fast
```
```   251   obtain t where t: "0 < t"
```
```   252     and less_s: "\<And>x. \<lbrakk>x \<noteq> a; norm (x - a) < t\<rbrakk> \<Longrightarrow> norm (f x - b) < s"
```
```   253     using LIM_D [OF f s] by fast
```
```   254   obtain d where d: "0 < d"
```
```   255     and neq_b: "\<And>x. \<lbrakk>x \<noteq> a; norm (x - a) < d\<rbrakk> \<Longrightarrow> f x \<noteq> b"
```
```   256     using inj by fast
```
```   257
```
```   258   show "\<exists>t>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < t \<longrightarrow> norm (g (f x) - c) < r"
```
```   259   proof (safe intro!: exI)
```
```   260     show "0 < min d t" using d t by simp
```
```   261   next
```
```   262     fix x
```
```   263     assume "x \<noteq> a" and "norm (x - a) < min d t"
```
```   264     hence "f x \<noteq> b" and "norm (f x - b) < s"
```
```   265       using neq_b less_s by simp_all
```
```   266     thus "norm (g (f x) - c) < r"
```
```   267       by (rule less_r)
```
```   268   qed
```
```   269 qed
```
```   270
```
```   271 lemma LIM_o: "\<lbrakk>g -- l --> g l; f -- a --> l\<rbrakk> \<Longrightarrow> (g \<circ> f) -- a --> g l"
```
```   272 unfolding o_def by (rule LIM_compose)
```
```   273
```
```   274 lemma real_LIM_sandwich_zero:
```
```   275   fixes f g :: "'a::real_normed_vector \<Rightarrow> real"
```
```   276   assumes f: "f -- a --> 0"
```
```   277   assumes 1: "\<And>x. x \<noteq> a \<Longrightarrow> 0 \<le> g x"
```
```   278   assumes 2: "\<And>x. x \<noteq> a \<Longrightarrow> g x \<le> f x"
```
```   279   shows "g -- a --> 0"
```
```   280 proof (rule LIM_imp_LIM [OF f])
```
```   281   fix x assume x: "x \<noteq> a"
```
```   282   have "norm (g x - 0) = g x" by (simp add: 1 x)
```
```   283   also have "g x \<le> f x" by (rule 2 [OF x])
```
```   284   also have "f x \<le> \<bar>f x\<bar>" by (rule abs_ge_self)
```
```   285   also have "\<bar>f x\<bar> = norm (f x - 0)" by simp
```
```   286   finally show "norm (g x - 0) \<le> norm (f x - 0)" .
```
```   287 qed
```
```   288
```
```   289 text {* Bounded Linear Operators *}
```
```   290
```
```   291 lemma (in bounded_linear) cont: "f -- a --> f a"
```
```   292 proof (rule LIM_I)
```
```   293   fix r::real assume r: "0 < r"
```
```   294   obtain K where K: "0 < K" and norm_le: "\<And>x. norm (f x) \<le> norm x * K"
```
```   295     using pos_bounded by fast
```
```   296   show "\<exists>s>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < s \<longrightarrow> norm (f x - f a) < r"
```
```   297   proof (rule exI, safe)
```
```   298     from r K show "0 < r / K" by (rule divide_pos_pos)
```
```   299   next
```
```   300     fix x assume x: "norm (x - a) < r / K"
```
```   301     have "norm (f x - f a) = norm (f (x - a))" by (simp only: diff)
```
```   302     also have "\<dots> \<le> norm (x - a) * K" by (rule norm_le)
```
```   303     also from K x have "\<dots> < r" by (simp only: pos_less_divide_eq)
```
```   304     finally show "norm (f x - f a) < r" .
```
```   305   qed
```
```   306 qed
```
```   307
```
```   308 lemma (in bounded_linear) LIM:
```
```   309   "g -- a --> l \<Longrightarrow> (\<lambda>x. f (g x)) -- a --> f l"
```
```   310 by (rule LIM_compose [OF cont])
```
```   311
```
```   312 lemma (in bounded_linear) LIM_zero:
```
```   313   "g -- a --> 0 \<Longrightarrow> (\<lambda>x. f (g x)) -- a --> 0"
```
```   314 by (drule LIM, simp only: zero)
```
```   315
```
```   316 text {* Bounded Bilinear Operators *}
```
```   317
```
```   318 lemma (in bounded_bilinear) LIM_prod_zero:
```
```   319   assumes f: "f -- a --> 0"
```
```   320   assumes g: "g -- a --> 0"
```
```   321   shows "(\<lambda>x. f x ** g x) -- a --> 0"
```
```   322 proof (rule LIM_I)
```
```   323   fix r::real assume r: "0 < r"
```
```   324   obtain K where K: "0 < K"
```
```   325     and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"
```
```   326     using pos_bounded by fast
```
```   327   from K have K': "0 < inverse K"
```
```   328     by (rule positive_imp_inverse_positive)
```
```   329   obtain s where s: "0 < s"
```
```   330     and norm_f: "\<And>x. \<lbrakk>x \<noteq> a; norm (x - a) < s\<rbrakk> \<Longrightarrow> norm (f x) < r"
```
```   331     using LIM_D [OF f r] by auto
```
```   332   obtain t where t: "0 < t"
```
```   333     and norm_g: "\<And>x. \<lbrakk>x \<noteq> a; norm (x - a) < t\<rbrakk> \<Longrightarrow> norm (g x) < inverse K"
```
```   334     using LIM_D [OF g K'] by auto
```
```   335   show "\<exists>s>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < s \<longrightarrow> norm (f x ** g x - 0) < r"
```
```   336   proof (rule exI, safe)
```
```   337     from s t show "0 < min s t" by simp
```
```   338   next
```
```   339     fix x assume x: "x \<noteq> a"
```
```   340     assume "norm (x - a) < min s t"
```
```   341     hence xs: "norm (x - a) < s" and xt: "norm (x - a) < t" by simp_all
```
```   342     from x xs have 1: "norm (f x) < r" by (rule norm_f)
```
```   343     from x xt have 2: "norm (g x) < inverse K" by (rule norm_g)
```
```   344     have "norm (f x ** g x) \<le> norm (f x) * norm (g x) * K" by (rule norm_le)
```
```   345     also from 1 2 K have "\<dots> < r * inverse K * K"
```
```   346       by (intro mult_strict_right_mono mult_strict_mono' norm_ge_zero)
```
```   347     also from K have "r * inverse K * K = r" by simp
```
```   348     finally show "norm (f x ** g x - 0) < r" by simp
```
```   349   qed
```
```   350 qed
```
```   351
```
```   352 lemma (in bounded_bilinear) LIM_left_zero:
```
```   353   "f -- a --> 0 \<Longrightarrow> (\<lambda>x. f x ** c) -- a --> 0"
```
```   354 by (rule bounded_linear.LIM_zero [OF bounded_linear_left])
```
```   355
```
```   356 lemma (in bounded_bilinear) LIM_right_zero:
```
```   357   "f -- a --> 0 \<Longrightarrow> (\<lambda>x. c ** f x) -- a --> 0"
```
```   358 by (rule bounded_linear.LIM_zero [OF bounded_linear_right])
```
```   359
```
```   360 lemma (in bounded_bilinear) LIM:
```
```   361   "\<lbrakk>f -- a --> L; g -- a --> M\<rbrakk> \<Longrightarrow> (\<lambda>x. f x ** g x) -- a --> L ** M"
```
```   362 apply (drule LIM_zero)
```
```   363 apply (drule LIM_zero)
```
```   364 apply (rule LIM_zero_cancel)
```
```   365 apply (subst prod_diff_prod)
```
```   366 apply (rule LIM_add_zero)
```
```   367 apply (rule LIM_add_zero)
```
```   368 apply (erule (1) LIM_prod_zero)
```
```   369 apply (erule LIM_left_zero)
```
```   370 apply (erule LIM_right_zero)
```
```   371 done
```
```   372
```
```   373 lemmas LIM_mult = mult.LIM
```
```   374
```
```   375 lemmas LIM_mult_zero = mult.LIM_prod_zero
```
```   376
```
```   377 lemmas LIM_mult_left_zero = mult.LIM_left_zero
```
```   378
```
```   379 lemmas LIM_mult_right_zero = mult.LIM_right_zero
```
```   380
```
```   381 lemmas LIM_scaleR = scaleR.LIM
```
```   382
```
```   383 lemmas LIM_of_real = of_real.LIM
```
```   384
```
```   385 lemma LIM_power:
```
```   386   fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::{recpower,real_normed_algebra}"
```
```   387   assumes f: "f -- a --> l"
```
```   388   shows "(\<lambda>x. f x ^ n) -- a --> l ^ n"
```
```   389 by (induct n, simp, simp add: power_Suc LIM_mult f)
```
```   390
```
```   391 subsubsection {* Derived theorems about @{term LIM} *}
```
```   392
```
```   393 lemma LIM_inverse_lemma:
```
```   394   fixes x :: "'a::real_normed_div_algebra"
```
```   395   assumes r: "0 < r"
```
```   396   assumes x: "norm (x - 1) < min (1/2) (r/2)"
```
```   397   shows "norm (inverse x - 1) < r"
```
```   398 proof -
```
```   399   from r have r2: "0 < r/2" by simp
```
```   400   from x have 0: "x \<noteq> 0" by clarsimp
```
```   401   from x have x': "norm (1 - x) < min (1/2) (r/2)"
```
```   402     by (simp only: norm_minus_commute)
```
```   403   hence less1: "norm (1 - x) < r/2" by simp
```
```   404   have "norm (1::'a) - norm x \<le> norm (1 - x)" by (rule norm_triangle_ineq2)
```
```   405   also from x' have "norm (1 - x) < 1/2" by simp
```
```   406   finally have "1/2 < norm x" by simp
```
```   407   hence "inverse (norm x) < inverse (1/2)"
```
```   408     by (rule less_imp_inverse_less, simp)
```
```   409   hence less2: "norm (inverse x) < 2"
```
```   410     by (simp add: nonzero_norm_inverse 0)
```
```   411   from less1 less2 r2 norm_ge_zero
```
```   412   have "norm (1 - x) * norm (inverse x) < (r/2) * 2"
```
```   413     by (rule mult_strict_mono)
```
```   414   thus "norm (inverse x - 1) < r"
```
```   415     by (simp only: norm_mult [symmetric] left_diff_distrib, simp add: 0)
```
```   416 qed
```
```   417
```
```   418 lemma LIM_inverse_fun:
```
```   419   assumes a: "a \<noteq> (0::'a::real_normed_div_algebra)"
```
```   420   shows "inverse -- a --> inverse a"
```
```   421 proof (rule LIM_equal2)
```
```   422   from a show "0 < norm a" by simp
```
```   423 next
```
```   424   fix x assume "norm (x - a) < norm a"
```
```   425   hence "x \<noteq> 0" by auto
```
```   426   with a show "inverse x = inverse (inverse a * x) * inverse a"
```
```   427     by (simp add: nonzero_inverse_mult_distrib
```
```   428                   nonzero_imp_inverse_nonzero
```
```   429                   nonzero_inverse_inverse_eq mult_assoc)
```
```   430 next
```
```   431   have 1: "inverse -- 1 --> inverse (1::'a)"
```
```   432     apply (rule LIM_I)
```
```   433     apply (rule_tac x="min (1/2) (r/2)" in exI)
```
```   434     apply (simp add: LIM_inverse_lemma)
```
```   435     done
```
```   436   have "(\<lambda>x. inverse a * x) -- a --> inverse a * a"
```
```   437     by (intro LIM_mult LIM_ident LIM_const)
```
```   438   hence "(\<lambda>x. inverse a * x) -- a --> 1"
```
```   439     by (simp add: a)
```
```   440   with 1 have "(\<lambda>x. inverse (inverse a * x)) -- a --> inverse 1"
```
```   441     by (rule LIM_compose)
```
```   442   hence "(\<lambda>x. inverse (inverse a * x)) -- a --> 1"
```
```   443     by simp
```
```   444   hence "(\<lambda>x. inverse (inverse a * x) * inverse a) -- a --> 1 * inverse a"
```
```   445     by (intro LIM_mult LIM_const)
```
```   446   thus "(\<lambda>x. inverse (inverse a * x) * inverse a) -- a --> inverse a"
```
```   447     by simp
```
```   448 qed
```
```   449
```
```   450 lemma LIM_inverse:
```
```   451   fixes L :: "'a::real_normed_div_algebra"
```
```   452   shows "\<lbrakk>f -- a --> L; L \<noteq> 0\<rbrakk> \<Longrightarrow> (\<lambda>x. inverse (f x)) -- a --> inverse L"
```
```   453 by (rule LIM_inverse_fun [THEN LIM_compose])
```
```   454
```
```   455
```
```   456 subsection {* Continuity *}
```
```   457
```
```   458 subsubsection {* Purely standard proofs *}
```
```   459
```
```   460 lemma LIM_isCont_iff: "(f -- a --> f a) = ((\<lambda>h. f (a + h)) -- 0 --> f a)"
```
```   461 by (rule iffI [OF LIM_offset_zero LIM_offset_zero_cancel])
```
```   462
```
```   463 lemma isCont_iff: "isCont f x = (\<lambda>h. f (x + h)) -- 0 --> f x"
```
```   464 by (simp add: isCont_def LIM_isCont_iff)
```
```   465
```
```   466 lemma isCont_ident [simp]: "isCont (\<lambda>x. x) a"
```
```   467   unfolding isCont_def by (rule LIM_ident)
```
```   468
```
```   469 lemma isCont_const [simp]: "isCont (\<lambda>x. k) a"
```
```   470   unfolding isCont_def by (rule LIM_const)
```
```   471
```
```   472 lemma isCont_norm: "isCont f a \<Longrightarrow> isCont (\<lambda>x. norm (f x)) a"
```
```   473   unfolding isCont_def by (rule LIM_norm)
```
```   474
```
```   475 lemma isCont_rabs: "isCont f a \<Longrightarrow> isCont (\<lambda>x. \<bar>f x :: real\<bar>) a"
```
```   476   unfolding isCont_def by (rule LIM_rabs)
```
```   477
```
```   478 lemma isCont_add: "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x + g x) a"
```
```   479   unfolding isCont_def by (rule LIM_add)
```
```   480
```
```   481 lemma isCont_minus: "isCont f a \<Longrightarrow> isCont (\<lambda>x. - f x) a"
```
```   482   unfolding isCont_def by (rule LIM_minus)
```
```   483
```
```   484 lemma isCont_diff: "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x - g x) a"
```
```   485   unfolding isCont_def by (rule LIM_diff)
```
```   486
```
```   487 lemma isCont_mult:
```
```   488   fixes f g :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_algebra"
```
```   489   shows "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x * g x) a"
```
```   490   unfolding isCont_def by (rule LIM_mult)
```
```   491
```
```   492 lemma isCont_inverse:
```
```   493   fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_div_algebra"
```
```   494   shows "\<lbrakk>isCont f a; f a \<noteq> 0\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. inverse (f x)) a"
```
```   495   unfolding isCont_def by (rule LIM_inverse)
```
```   496
```
```   497 lemma isCont_LIM_compose:
```
```   498   "\<lbrakk>isCont g l; f -- a --> l\<rbrakk> \<Longrightarrow> (\<lambda>x. g (f x)) -- a --> g l"
```
```   499   unfolding isCont_def by (rule LIM_compose)
```
```   500
```
```   501 lemma isCont_LIM_compose2:
```
```   502   assumes f [unfolded isCont_def]: "isCont f a"
```
```   503   assumes g: "g -- f a --> l"
```
```   504   assumes inj: "\<exists>d>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < d \<longrightarrow> f x \<noteq> f a"
```
```   505   shows "(\<lambda>x. g (f x)) -- a --> l"
```
```   506 by (rule LIM_compose2 [OF f g inj])
```
```   507
```
```   508 lemma isCont_o2: "\<lbrakk>isCont f a; isCont g (f a)\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. g (f x)) a"
```
```   509   unfolding isCont_def by (rule LIM_compose)
```
```   510
```
```   511 lemma isCont_o: "\<lbrakk>isCont f a; isCont g (f a)\<rbrakk> \<Longrightarrow> isCont (g o f) a"
```
```   512   unfolding o_def by (rule isCont_o2)
```
```   513
```
```   514 lemma (in bounded_linear) isCont: "isCont f a"
```
```   515   unfolding isCont_def by (rule cont)
```
```   516
```
```   517 lemma (in bounded_bilinear) isCont:
```
```   518   "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x ** g x) a"
```
```   519   unfolding isCont_def by (rule LIM)
```
```   520
```
```   521 lemmas isCont_scaleR = scaleR.isCont
```
```   522
```
```   523 lemma isCont_of_real:
```
```   524   "isCont f a \<Longrightarrow> isCont (\<lambda>x. of_real (f x)) a"
```
```   525   unfolding isCont_def by (rule LIM_of_real)
```
```   526
```
```   527 lemma isCont_power:
```
```   528   fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::{recpower,real_normed_algebra}"
```
```   529   shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. f x ^ n) a"
```
```   530   unfolding isCont_def by (rule LIM_power)
```
```   531
```
```   532 lemma isCont_abs [simp]: "isCont abs (a::real)"
```
```   533 by (rule isCont_rabs [OF isCont_ident])
```
```   534
```
```   535
```
```   536 subsection {* Uniform Continuity *}
```
```   537
```
```   538 lemma isUCont_isCont: "isUCont f ==> isCont f x"
```
```   539 by (simp add: isUCont_def isCont_def LIM_def, force)
```
```   540
```
```   541 lemma isUCont_Cauchy:
```
```   542   "\<lbrakk>isUCont f; Cauchy X\<rbrakk> \<Longrightarrow> Cauchy (\<lambda>n. f (X n))"
```
```   543 unfolding isUCont_def
```
```   544 apply (rule CauchyI)
```
```   545 apply (drule_tac x=e in spec, safe)
```
```   546 apply (drule_tac e=s in CauchyD, safe)
```
```   547 apply (rule_tac x=M in exI, simp)
```
```   548 done
```
```   549
```
```   550 lemma (in bounded_linear) isUCont: "isUCont f"
```
```   551 unfolding isUCont_def
```
```   552 proof (intro allI impI)
```
```   553   fix r::real assume r: "0 < r"
```
```   554   obtain K where K: "0 < K" and norm_le: "\<And>x. norm (f x) \<le> norm x * K"
```
```   555     using pos_bounded by fast
```
```   556   show "\<exists>s>0. \<forall>x y. norm (x - y) < s \<longrightarrow> norm (f x - f y) < r"
```
```   557   proof (rule exI, safe)
```
```   558     from r K show "0 < r / K" by (rule divide_pos_pos)
```
```   559   next
```
```   560     fix x y :: 'a
```
```   561     assume xy: "norm (x - y) < r / K"
```
```   562     have "norm (f x - f y) = norm (f (x - y))" by (simp only: diff)
```
```   563     also have "\<dots> \<le> norm (x - y) * K" by (rule norm_le)
```
```   564     also from K xy have "\<dots> < r" by (simp only: pos_less_divide_eq)
```
```   565     finally show "norm (f x - f y) < r" .
```
```   566   qed
```
```   567 qed
```
```   568
```
```   569 lemma (in bounded_linear) Cauchy: "Cauchy X \<Longrightarrow> Cauchy (\<lambda>n. f (X n))"
```
```   570 by (rule isUCont [THEN isUCont_Cauchy])
```
```   571
```
```   572
```
```   573 subsection {* Relation of LIM and LIMSEQ *}
```
```   574
```
```   575 lemma LIMSEQ_SEQ_conv1:
```
```   576   fixes a :: "'a::real_normed_vector"
```
```   577   assumes X: "X -- a --> L"
```
```   578   shows "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> a \<longrightarrow> (\<lambda>n. X (S n)) ----> L"
```
```   579 proof (safe intro!: LIMSEQ_I)
```
```   580   fix S :: "nat \<Rightarrow> 'a"
```
```   581   fix r :: real
```
```   582   assume rgz: "0 < r"
```
```   583   assume as: "\<forall>n. S n \<noteq> a"
```
```   584   assume S: "S ----> a"
```
```   585   from LIM_D [OF X rgz] obtain s
```
```   586     where sgz: "0 < s"
```
```   587     and aux: "\<And>x. \<lbrakk>x \<noteq> a; norm (x - a) < s\<rbrakk> \<Longrightarrow> norm (X x - L) < r"
```
```   588     by fast
```
```   589   from LIMSEQ_D [OF S sgz]
```
```   590   obtain no where "\<forall>n\<ge>no. norm (S n - a) < s" by blast
```
```   591   hence "\<forall>n\<ge>no. norm (X (S n) - L) < r" by (simp add: aux as)
```
```   592   thus "\<exists>no. \<forall>n\<ge>no. norm (X (S n) - L) < r" ..
```
```   593 qed
```
```   594
```
```   595 lemma LIMSEQ_SEQ_conv2:
```
```   596   fixes a :: real
```
```   597   assumes "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> a \<longrightarrow> (\<lambda>n. X (S n)) ----> L"
```
```   598   shows "X -- a --> L"
```
```   599 proof (rule ccontr)
```
```   600   assume "\<not> (X -- a --> L)"
```
```   601   hence "\<not> (\<forall>r > 0. \<exists>s > 0. \<forall>x. x \<noteq> a & norm (x - a) < s --> norm (X x - L) < r)" by (unfold LIM_def)
```
```   602   hence "\<exists>r > 0. \<forall>s > 0. \<exists>x. \<not>(x \<noteq> a \<and> \<bar>x - a\<bar> < s --> norm (X x - L) < r)" by simp
```
```   603   hence "\<exists>r > 0. \<forall>s > 0. \<exists>x. (x \<noteq> a \<and> \<bar>x - a\<bar> < s \<and> norm (X x - L) \<ge> r)" by (simp add: linorder_not_less)
```
```   604   then obtain r where rdef: "r > 0 \<and> (\<forall>s > 0. \<exists>x. (x \<noteq> a \<and> \<bar>x - a\<bar> < s \<and> norm (X x - L) \<ge> r))" by auto
```
```   605
```
```   606   let ?F = "\<lambda>n::nat. SOME x. x\<noteq>a \<and> \<bar>x - a\<bar> < inverse (real (Suc n)) \<and> norm (X x - L) \<ge> r"
```
```   607   have "\<And>n. \<exists>x. x\<noteq>a \<and> \<bar>x - a\<bar> < inverse (real (Suc n)) \<and> norm (X x - L) \<ge> r"
```
```   608     using rdef by simp
```
```   609   hence F: "\<And>n. ?F n \<noteq> a \<and> \<bar>?F n - a\<bar> < inverse (real (Suc n)) \<and> norm (X (?F n) - L) \<ge> r"
```
```   610     by (rule someI_ex)
```
```   611   hence F1: "\<And>n. ?F n \<noteq> a"
```
```   612     and F2: "\<And>n. \<bar>?F n - a\<bar> < inverse (real (Suc n))"
```
```   613     and F3: "\<And>n. norm (X (?F n) - L) \<ge> r"
```
```   614     by fast+
```
```   615
```
```   616   have "?F ----> a"
```
```   617   proof (rule LIMSEQ_I, unfold real_norm_def)
```
```   618       fix e::real
```
```   619       assume "0 < e"
```
```   620         (* choose no such that inverse (real (Suc n)) < e *)
```
```   621       then have "\<exists>no. inverse (real (Suc no)) < e" by (rule reals_Archimedean)
```
```   622       then obtain m where nodef: "inverse (real (Suc m)) < e" by auto
```
```   623       show "\<exists>no. \<forall>n. no \<le> n --> \<bar>?F n - a\<bar> < e"
```
```   624       proof (intro exI allI impI)
```
```   625         fix n
```
```   626         assume mlen: "m \<le> n"
```
```   627         have "\<bar>?F n - a\<bar> < inverse (real (Suc n))"
```
```   628           by (rule F2)
```
```   629         also have "inverse (real (Suc n)) \<le> inverse (real (Suc m))"
```
```   630           using mlen by auto
```
```   631         also from nodef have
```
```   632           "inverse (real (Suc m)) < e" .
```
```   633         finally show "\<bar>?F n - a\<bar> < e" .
```
```   634       qed
```
```   635   qed
```
```   636
```
```   637   moreover have "\<forall>n. ?F n \<noteq> a"
```
```   638     by (rule allI) (rule F1)
```
```   639
```
```   640   moreover from prems have "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> a \<longrightarrow> (\<lambda>n. X (S n)) ----> L" by simp
```
```   641   ultimately have "(\<lambda>n. X (?F n)) ----> L" by simp
```
```   642
```
```   643   moreover have "\<not> ((\<lambda>n. X (?F n)) ----> L)"
```
```   644   proof -
```
```   645     {
```
```   646       fix no::nat
```
```   647       obtain n where "n = no + 1" by simp
```
```   648       then have nolen: "no \<le> n" by simp
```
```   649         (* We prove this by showing that for any m there is an n\<ge>m such that |X (?F n) - L| \<ge> r *)
```
```   650       have "norm (X (?F n) - L) \<ge> r"
```
```   651         by (rule F3)
```
```   652       with nolen have "\<exists>n. no \<le> n \<and> norm (X (?F n) - L) \<ge> r" by fast
```
```   653     }
```
```   654     then have "(\<forall>no. \<exists>n. no \<le> n \<and> norm (X (?F n) - L) \<ge> r)" by simp
```
```   655     with rdef have "\<exists>e>0. (\<forall>no. \<exists>n. no \<le> n \<and> norm (X (?F n) - L) \<ge> e)" by auto
```
```   656     thus ?thesis by (unfold LIMSEQ_def, auto simp add: linorder_not_less)
```
```   657   qed
```
```   658   ultimately show False by simp
```
```   659 qed
```
```   660
```
```   661 lemma LIMSEQ_SEQ_conv:
```
```   662   "(\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> (a::real) \<longrightarrow> (\<lambda>n. X (S n)) ----> L) =
```
```   663    (X -- a --> L)"
```
```   664 proof
```
```   665   assume "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> a \<longrightarrow> (\<lambda>n. X (S n)) ----> L"
```
```   666   thus "X -- a --> L" by (rule LIMSEQ_SEQ_conv2)
```
```   667 next
```
```   668   assume "(X -- a --> L)"
```
```   669   thus "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> a \<longrightarrow> (\<lambda>n. X (S n)) ----> L" by (rule LIMSEQ_SEQ_conv1)
```
```   670 qed
```
```   671
```
```   672 end
```