src/HOL/Analysis/Complex_Analysis_Basics.thy
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removed dots at the end of (sub)titles
```     1 (*  Author: John Harrison, Marco Maggesi, Graziano Gentili, Gianni Ciolli, Valentina Bruno
```
```     2     Ported from "hol_light/Multivariate/canal.ml" by L C Paulson (2014)
```
```     3 *)
```
```     4
```
```     5 section \<open>Complex Analysis Basics\<close>
```
```     6
```
```     7 theory Complex_Analysis_Basics
```
```     8 imports Equivalence_Lebesgue_Henstock_Integration "HOL-Library.Nonpos_Ints"
```
```     9 begin
```
```    10
```
```    11
```
```    12 subsection\<open>General lemmas\<close>
```
```    13
```
```    14 lemma nonneg_Reals_cmod_eq_Re: "z \<in> \<real>\<^sub>\<ge>\<^sub>0 \<Longrightarrow> norm z = Re z"
```
```    15   by (simp add: complex_nonneg_Reals_iff cmod_eq_Re)
```
```    16
```
```    17 lemma has_derivative_mult_right:
```
```    18   fixes c:: "'a :: real_normed_algebra"
```
```    19   shows "((( * ) c) has_derivative (( * ) c)) F"
```
```    20 by (rule has_derivative_mult_right [OF has_derivative_id])
```
```    21
```
```    22 lemma has_derivative_of_real[derivative_intros, simp]:
```
```    23   "(f has_derivative f') F \<Longrightarrow> ((\<lambda>x. of_real (f x)) has_derivative (\<lambda>x. of_real (f' x))) F"
```
```    24   using bounded_linear.has_derivative[OF bounded_linear_of_real] .
```
```    25
```
```    26 lemma has_vector_derivative_real_field:
```
```    27   "DERIV f (of_real a) :> f' \<Longrightarrow> ((\<lambda>x. f (of_real x)) has_vector_derivative f') (at a within s)"
```
```    28   using has_derivative_compose[of of_real of_real a _ f "( * ) f'"]
```
```    29   by (simp add: scaleR_conv_of_real ac_simps has_vector_derivative_def has_field_derivative_def)
```
```    30 lemmas has_vector_derivative_real_complex = has_vector_derivative_real_field
```
```    31
```
```    32 lemma fact_cancel:
```
```    33   fixes c :: "'a::real_field"
```
```    34   shows "of_nat (Suc n) * c / (fact (Suc n)) = c / (fact n)"
```
```    35   by (simp add: of_nat_mult del: of_nat_Suc times_nat.simps)
```
```    36
```
```    37 lemma bilinear_times:
```
```    38   fixes c::"'a::real_algebra" shows "bilinear (\<lambda>x y::'a. x*y)"
```
```    39   by (auto simp: bilinear_def distrib_left distrib_right intro!: linearI)
```
```    40
```
```    41 lemma linear_cnj: "linear cnj"
```
```    42   using bounded_linear.linear[OF bounded_linear_cnj] .
```
```    43
```
```    44 lemma tendsto_Re_upper:
```
```    45   assumes "~ (trivial_limit F)"
```
```    46           "(f \<longlongrightarrow> l) F"
```
```    47           "eventually (\<lambda>x. Re(f x) \<le> b) F"
```
```    48     shows  "Re(l) \<le> b"
```
```    49   by (metis assms tendsto_le [OF _ tendsto_const]  tendsto_Re)
```
```    50
```
```    51 lemma tendsto_Re_lower:
```
```    52   assumes "~ (trivial_limit F)"
```
```    53           "(f \<longlongrightarrow> l) F"
```
```    54           "eventually (\<lambda>x. b \<le> Re(f x)) F"
```
```    55     shows  "b \<le> Re(l)"
```
```    56   by (metis assms tendsto_le [OF _ _ tendsto_const]  tendsto_Re)
```
```    57
```
```    58 lemma tendsto_Im_upper:
```
```    59   assumes "~ (trivial_limit F)"
```
```    60           "(f \<longlongrightarrow> l) F"
```
```    61           "eventually (\<lambda>x. Im(f x) \<le> b) F"
```
```    62     shows  "Im(l) \<le> b"
```
```    63   by (metis assms tendsto_le [OF _ tendsto_const]  tendsto_Im)
```
```    64
```
```    65 lemma tendsto_Im_lower:
```
```    66   assumes "~ (trivial_limit F)"
```
```    67           "(f \<longlongrightarrow> l) F"
```
```    68           "eventually (\<lambda>x. b \<le> Im(f x)) F"
```
```    69     shows  "b \<le> Im(l)"
```
```    70   by (metis assms tendsto_le [OF _ _ tendsto_const]  tendsto_Im)
```
```    71
```
```    72 lemma lambda_zero: "(\<lambda>h::'a::mult_zero. 0) = ( * ) 0"
```
```    73   by auto
```
```    74
```
```    75 lemma lambda_one: "(\<lambda>x::'a::monoid_mult. x) = ( * ) 1"
```
```    76   by auto
```
```    77
```
```    78 lemma continuous_mult_left:
```
```    79   fixes c::"'a::real_normed_algebra"
```
```    80   shows "continuous F f \<Longrightarrow> continuous F (\<lambda>x. c * f x)"
```
```    81 by (rule continuous_mult [OF continuous_const])
```
```    82
```
```    83 lemma continuous_mult_right:
```
```    84   fixes c::"'a::real_normed_algebra"
```
```    85   shows "continuous F f \<Longrightarrow> continuous F (\<lambda>x. f x * c)"
```
```    86 by (rule continuous_mult [OF _ continuous_const])
```
```    87
```
```    88 lemma continuous_on_mult_left:
```
```    89   fixes c::"'a::real_normed_algebra"
```
```    90   shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. c * f x)"
```
```    91 by (rule continuous_on_mult [OF continuous_on_const])
```
```    92
```
```    93 lemma continuous_on_mult_right:
```
```    94   fixes c::"'a::real_normed_algebra"
```
```    95   shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. f x * c)"
```
```    96 by (rule continuous_on_mult [OF _ continuous_on_const])
```
```    97
```
```    98 lemma uniformly_continuous_on_cmul_right [continuous_intros]:
```
```    99   fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_algebra"
```
```   100   shows "uniformly_continuous_on s f \<Longrightarrow> uniformly_continuous_on s (\<lambda>x. f x * c)"
```
```   101   using bounded_linear.uniformly_continuous_on[OF bounded_linear_mult_left] .
```
```   102
```
```   103 lemma uniformly_continuous_on_cmul_left[continuous_intros]:
```
```   104   fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_algebra"
```
```   105   assumes "uniformly_continuous_on s f"
```
```   106     shows "uniformly_continuous_on s (\<lambda>x. c * f x)"
```
```   107 by (metis assms bounded_linear.uniformly_continuous_on bounded_linear_mult_right)
```
```   108
```
```   109 lemma continuous_within_norm_id [continuous_intros]: "continuous (at x within S) norm"
```
```   110   by (rule continuous_norm [OF continuous_ident])
```
```   111
```
```   112 lemma continuous_on_norm_id [continuous_intros]: "continuous_on S norm"
```
```   113   by (intro continuous_on_id continuous_on_norm)
```
```   114
```
```   115 subsection\<open>DERIV stuff\<close>
```
```   116
```
```   117 lemma DERIV_zero_connected_constant:
```
```   118   fixes f :: "'a::{real_normed_field,euclidean_space} \<Rightarrow> 'a"
```
```   119   assumes "connected s"
```
```   120       and "open s"
```
```   121       and "finite k"
```
```   122       and "continuous_on s f"
```
```   123       and "\<forall>x\<in>(s - k). DERIV f x :> 0"
```
```   124     obtains c where "\<And>x. x \<in> s \<Longrightarrow> f(x) = c"
```
```   125 using has_derivative_zero_connected_constant [OF assms(1-4)] assms
```
```   126 by (metis DERIV_const has_derivative_const Diff_iff at_within_open frechet_derivative_at has_field_derivative_def)
```
```   127
```
```   128 lemmas DERIV_zero_constant = has_field_derivative_zero_constant
```
```   129
```
```   130 lemma DERIV_zero_unique:
```
```   131   assumes "convex s"
```
```   132       and d0: "\<And>x. x\<in>s \<Longrightarrow> (f has_field_derivative 0) (at x within s)"
```
```   133       and "a \<in> s"
```
```   134       and "x \<in> s"
```
```   135     shows "f x = f a"
```
```   136   by (rule has_derivative_zero_unique [OF assms(1) _ assms(4,3)])
```
```   137      (metis d0 has_field_derivative_imp_has_derivative lambda_zero)
```
```   138
```
```   139 lemma DERIV_zero_connected_unique:
```
```   140   assumes "connected s"
```
```   141       and "open s"
```
```   142       and d0: "\<And>x. x\<in>s \<Longrightarrow> DERIV f x :> 0"
```
```   143       and "a \<in> s"
```
```   144       and "x \<in> s"
```
```   145     shows "f x = f a"
```
```   146     by (rule has_derivative_zero_unique_connected [OF assms(2,1) _ assms(5,4)])
```
```   147        (metis has_field_derivative_def lambda_zero d0)
```
```   148
```
```   149 lemma DERIV_transform_within:
```
```   150   assumes "(f has_field_derivative f') (at a within s)"
```
```   151       and "0 < d" "a \<in> s"
```
```   152       and "\<And>x. x\<in>s \<Longrightarrow> dist x a < d \<Longrightarrow> f x = g x"
```
```   153     shows "(g has_field_derivative f') (at a within s)"
```
```   154   using assms unfolding has_field_derivative_def
```
```   155   by (blast intro: has_derivative_transform_within)
```
```   156
```
```   157 lemma DERIV_transform_within_open:
```
```   158   assumes "DERIV f a :> f'"
```
```   159       and "open s" "a \<in> s"
```
```   160       and "\<And>x. x\<in>s \<Longrightarrow> f x = g x"
```
```   161     shows "DERIV g a :> f'"
```
```   162   using assms unfolding has_field_derivative_def
```
```   163 by (metis has_derivative_transform_within_open)
```
```   164
```
```   165 lemma DERIV_transform_at:
```
```   166   assumes "DERIV f a :> f'"
```
```   167       and "0 < d"
```
```   168       and "\<And>x. dist x a < d \<Longrightarrow> f x = g x"
```
```   169     shows "DERIV g a :> f'"
```
```   170   by (blast intro: assms DERIV_transform_within)
```
```   171
```
```   172 (*generalising DERIV_isconst_all, which requires type real (using the ordering)*)
```
```   173 lemma DERIV_zero_UNIV_unique:
```
```   174   "(\<And>x. DERIV f x :> 0) \<Longrightarrow> f x = f a"
```
```   175   by (metis DERIV_zero_unique UNIV_I convex_UNIV)
```
```   176
```
```   177 subsection \<open>Some limit theorems about real part of real series etc\<close>
```
```   178
```
```   179 (*MOVE? But not to Finite_Cartesian_Product*)
```
```   180 lemma sums_vec_nth :
```
```   181   assumes "f sums a"
```
```   182   shows "(\<lambda>x. f x \$ i) sums a \$ i"
```
```   183 using assms unfolding sums_def
```
```   184 by (auto dest: tendsto_vec_nth [where i=i])
```
```   185
```
```   186 lemma summable_vec_nth :
```
```   187   assumes "summable f"
```
```   188   shows "summable (\<lambda>x. f x \$ i)"
```
```   189 using assms unfolding summable_def
```
```   190 by (blast intro: sums_vec_nth)
```
```   191
```
```   192 subsection \<open>Complex number lemmas\<close>
```
```   193
```
```   194 lemma
```
```   195   shows open_halfspace_Re_lt: "open {z. Re(z) < b}"
```
```   196     and open_halfspace_Re_gt: "open {z. Re(z) > b}"
```
```   197     and closed_halfspace_Re_ge: "closed {z. Re(z) \<ge> b}"
```
```   198     and closed_halfspace_Re_le: "closed {z. Re(z) \<le> b}"
```
```   199     and closed_halfspace_Re_eq: "closed {z. Re(z) = b}"
```
```   200     and open_halfspace_Im_lt: "open {z. Im(z) < b}"
```
```   201     and open_halfspace_Im_gt: "open {z. Im(z) > b}"
```
```   202     and closed_halfspace_Im_ge: "closed {z. Im(z) \<ge> b}"
```
```   203     and closed_halfspace_Im_le: "closed {z. Im(z) \<le> b}"
```
```   204     and closed_halfspace_Im_eq: "closed {z. Im(z) = b}"
```
```   205   by (intro open_Collect_less closed_Collect_le closed_Collect_eq continuous_on_Re
```
```   206             continuous_on_Im continuous_on_id continuous_on_const)+
```
```   207
```
```   208 lemma closed_complex_Reals: "closed (\<real> :: complex set)"
```
```   209 proof -
```
```   210   have "(\<real> :: complex set) = {z. Im z = 0}"
```
```   211     by (auto simp: complex_is_Real_iff)
```
```   212   then show ?thesis
```
```   213     by (metis closed_halfspace_Im_eq)
```
```   214 qed
```
```   215
```
```   216 lemma closed_Real_halfspace_Re_le: "closed (\<real> \<inter> {w. Re w \<le> x})"
```
```   217   by (simp add: closed_Int closed_complex_Reals closed_halfspace_Re_le)
```
```   218
```
```   219 corollary closed_nonpos_Reals_complex [simp]: "closed (\<real>\<^sub>\<le>\<^sub>0 :: complex set)"
```
```   220 proof -
```
```   221   have "\<real>\<^sub>\<le>\<^sub>0 = \<real> \<inter> {z. Re(z) \<le> 0}"
```
```   222     using complex_nonpos_Reals_iff complex_is_Real_iff by auto
```
```   223   then show ?thesis
```
```   224     by (metis closed_Real_halfspace_Re_le)
```
```   225 qed
```
```   226
```
```   227 lemma closed_Real_halfspace_Re_ge: "closed (\<real> \<inter> {w. x \<le> Re(w)})"
```
```   228   using closed_halfspace_Re_ge
```
```   229   by (simp add: closed_Int closed_complex_Reals)
```
```   230
```
```   231 corollary closed_nonneg_Reals_complex [simp]: "closed (\<real>\<^sub>\<ge>\<^sub>0 :: complex set)"
```
```   232 proof -
```
```   233   have "\<real>\<^sub>\<ge>\<^sub>0 = \<real> \<inter> {z. Re(z) \<ge> 0}"
```
```   234     using complex_nonneg_Reals_iff complex_is_Real_iff by auto
```
```   235   then show ?thesis
```
```   236     by (metis closed_Real_halfspace_Re_ge)
```
```   237 qed
```
```   238
```
```   239 lemma closed_real_abs_le: "closed {w \<in> \<real>. \<bar>Re w\<bar> \<le> r}"
```
```   240 proof -
```
```   241   have "{w \<in> \<real>. \<bar>Re w\<bar> \<le> r} = (\<real> \<inter> {w. Re w \<le> r}) \<inter> (\<real> \<inter> {w. Re w \<ge> -r})"
```
```   242     by auto
```
```   243   then show "closed {w \<in> \<real>. \<bar>Re w\<bar> \<le> r}"
```
```   244     by (simp add: closed_Int closed_Real_halfspace_Re_ge closed_Real_halfspace_Re_le)
```
```   245 qed
```
```   246
```
```   247 lemma real_lim:
```
```   248   fixes l::complex
```
```   249   assumes "(f \<longlongrightarrow> l) F" and "~(trivial_limit F)" and "eventually P F" and "\<And>a. P a \<Longrightarrow> f a \<in> \<real>"
```
```   250   shows  "l \<in> \<real>"
```
```   251 proof (rule Lim_in_closed_set[OF closed_complex_Reals _ assms(2,1)])
```
```   252   show "eventually (\<lambda>x. f x \<in> \<real>) F"
```
```   253     using assms(3, 4) by (auto intro: eventually_mono)
```
```   254 qed
```
```   255
```
```   256 lemma real_lim_sequentially:
```
```   257   fixes l::complex
```
```   258   shows "(f \<longlongrightarrow> l) sequentially \<Longrightarrow> (\<exists>N. \<forall>n\<ge>N. f n \<in> \<real>) \<Longrightarrow> l \<in> \<real>"
```
```   259 by (rule real_lim [where F=sequentially]) (auto simp: eventually_sequentially)
```
```   260
```
```   261 lemma real_series:
```
```   262   fixes l::complex
```
```   263   shows "f sums l \<Longrightarrow> (\<And>n. f n \<in> \<real>) \<Longrightarrow> l \<in> \<real>"
```
```   264 unfolding sums_def
```
```   265 by (metis real_lim_sequentially sum_in_Reals)
```
```   266
```
```   267 lemma Lim_null_comparison_Re:
```
```   268   assumes "eventually (\<lambda>x. norm(f x) \<le> Re(g x)) F" "(g \<longlongrightarrow> 0) F" shows "(f \<longlongrightarrow> 0) F"
```
```   269   by (rule Lim_null_comparison[OF assms(1)] tendsto_eq_intros assms(2))+ simp
```
```   270
```
```   271 subsection\<open>Holomorphic functions\<close>
```
```   272
```
```   273 subsection\<open>Holomorphic functions\<close>
```
```   274
```
```   275 definition holomorphic_on :: "[complex \<Rightarrow> complex, complex set] \<Rightarrow> bool"
```
```   276            (infixl "(holomorphic'_on)" 50)
```
```   277   where "f holomorphic_on s \<equiv> \<forall>x\<in>s. f field_differentiable (at x within s)"
```
```   278
```
```   279 named_theorems holomorphic_intros "structural introduction rules for holomorphic_on"
```
```   280
```
```   281 lemma holomorphic_onI [intro?]: "(\<And>x. x \<in> s \<Longrightarrow> f field_differentiable (at x within s)) \<Longrightarrow> f holomorphic_on s"
```
```   282   by (simp add: holomorphic_on_def)
```
```   283
```
```   284 lemma holomorphic_onD [dest?]: "\<lbrakk>f holomorphic_on s; x \<in> s\<rbrakk> \<Longrightarrow> f field_differentiable (at x within s)"
```
```   285   by (simp add: holomorphic_on_def)
```
```   286
```
```   287 lemma holomorphic_on_imp_differentiable_on:
```
```   288     "f holomorphic_on s \<Longrightarrow> f differentiable_on s"
```
```   289   unfolding holomorphic_on_def differentiable_on_def
```
```   290   by (simp add: field_differentiable_imp_differentiable)
```
```   291
```
```   292 lemma holomorphic_on_imp_differentiable_at:
```
```   293    "\<lbrakk>f holomorphic_on s; open s; x \<in> s\<rbrakk> \<Longrightarrow> f field_differentiable (at x)"
```
```   294 using at_within_open holomorphic_on_def by fastforce
```
```   295
```
```   296 lemma holomorphic_on_empty [holomorphic_intros]: "f holomorphic_on {}"
```
```   297   by (simp add: holomorphic_on_def)
```
```   298
```
```   299 lemma holomorphic_on_open:
```
```   300     "open s \<Longrightarrow> f holomorphic_on s \<longleftrightarrow> (\<forall>x \<in> s. \<exists>f'. DERIV f x :> f')"
```
```   301   by (auto simp: holomorphic_on_def field_differentiable_def has_field_derivative_def at_within_open [of _ s])
```
```   302
```
```   303 lemma holomorphic_on_imp_continuous_on:
```
```   304     "f holomorphic_on s \<Longrightarrow> continuous_on s f"
```
```   305   by (metis field_differentiable_imp_continuous_at continuous_on_eq_continuous_within holomorphic_on_def)
```
```   306
```
```   307 lemma holomorphic_on_subset [elim]:
```
```   308     "f holomorphic_on s \<Longrightarrow> t \<subseteq> s \<Longrightarrow> f holomorphic_on t"
```
```   309   unfolding holomorphic_on_def
```
```   310   by (metis field_differentiable_within_subset subsetD)
```
```   311
```
```   312 lemma holomorphic_transform: "\<lbrakk>f holomorphic_on s; \<And>x. x \<in> s \<Longrightarrow> f x = g x\<rbrakk> \<Longrightarrow> g holomorphic_on s"
```
```   313   by (metis field_differentiable_transform_within linordered_field_no_ub holomorphic_on_def)
```
```   314
```
```   315 lemma holomorphic_cong: "s = t ==> (\<And>x. x \<in> s \<Longrightarrow> f x = g x) \<Longrightarrow> f holomorphic_on s \<longleftrightarrow> g holomorphic_on t"
```
```   316   by (metis holomorphic_transform)
```
```   317
```
```   318 lemma holomorphic_on_linear [simp, holomorphic_intros]: "(( * ) c) holomorphic_on s"
```
```   319   unfolding holomorphic_on_def by (metis field_differentiable_linear)
```
```   320
```
```   321 lemma holomorphic_on_const [simp, holomorphic_intros]: "(\<lambda>z. c) holomorphic_on s"
```
```   322   unfolding holomorphic_on_def by (metis field_differentiable_const)
```
```   323
```
```   324 lemma holomorphic_on_ident [simp, holomorphic_intros]: "(\<lambda>x. x) holomorphic_on s"
```
```   325   unfolding holomorphic_on_def by (metis field_differentiable_ident)
```
```   326
```
```   327 lemma holomorphic_on_id [simp, holomorphic_intros]: "id holomorphic_on s"
```
```   328   unfolding id_def by (rule holomorphic_on_ident)
```
```   329
```
```   330 lemma holomorphic_on_compose:
```
```   331   "f holomorphic_on s \<Longrightarrow> g holomorphic_on (f ` s) \<Longrightarrow> (g o f) holomorphic_on s"
```
```   332   using field_differentiable_compose_within[of f _ s g]
```
```   333   by (auto simp: holomorphic_on_def)
```
```   334
```
```   335 lemma holomorphic_on_compose_gen:
```
```   336   "f holomorphic_on s \<Longrightarrow> g holomorphic_on t \<Longrightarrow> f ` s \<subseteq> t \<Longrightarrow> (g o f) holomorphic_on s"
```
```   337   by (metis holomorphic_on_compose holomorphic_on_subset)
```
```   338
```
```   339 lemma holomorphic_on_minus [holomorphic_intros]: "f holomorphic_on s \<Longrightarrow> (\<lambda>z. -(f z)) holomorphic_on s"
```
```   340   by (metis field_differentiable_minus holomorphic_on_def)
```
```   341
```
```   342 lemma holomorphic_on_add [holomorphic_intros]:
```
```   343   "\<lbrakk>f holomorphic_on s; g holomorphic_on s\<rbrakk> \<Longrightarrow> (\<lambda>z. f z + g z) holomorphic_on s"
```
```   344   unfolding holomorphic_on_def by (metis field_differentiable_add)
```
```   345
```
```   346 lemma holomorphic_on_diff [holomorphic_intros]:
```
```   347   "\<lbrakk>f holomorphic_on s; g holomorphic_on s\<rbrakk> \<Longrightarrow> (\<lambda>z. f z - g z) holomorphic_on s"
```
```   348   unfolding holomorphic_on_def by (metis field_differentiable_diff)
```
```   349
```
```   350 lemma holomorphic_on_mult [holomorphic_intros]:
```
```   351   "\<lbrakk>f holomorphic_on s; g holomorphic_on s\<rbrakk> \<Longrightarrow> (\<lambda>z. f z * g z) holomorphic_on s"
```
```   352   unfolding holomorphic_on_def by (metis field_differentiable_mult)
```
```   353
```
```   354 lemma holomorphic_on_inverse [holomorphic_intros]:
```
```   355   "\<lbrakk>f holomorphic_on s; \<And>z. z \<in> s \<Longrightarrow> f z \<noteq> 0\<rbrakk> \<Longrightarrow> (\<lambda>z. inverse (f z)) holomorphic_on s"
```
```   356   unfolding holomorphic_on_def by (metis field_differentiable_inverse)
```
```   357
```
```   358 lemma holomorphic_on_divide [holomorphic_intros]:
```
```   359   "\<lbrakk>f holomorphic_on s; g holomorphic_on s; \<And>z. z \<in> s \<Longrightarrow> g z \<noteq> 0\<rbrakk> \<Longrightarrow> (\<lambda>z. f z / g z) holomorphic_on s"
```
```   360   unfolding holomorphic_on_def by (metis field_differentiable_divide)
```
```   361
```
```   362 lemma holomorphic_on_power [holomorphic_intros]:
```
```   363   "f holomorphic_on s \<Longrightarrow> (\<lambda>z. (f z)^n) holomorphic_on s"
```
```   364   unfolding holomorphic_on_def by (metis field_differentiable_power)
```
```   365
```
```   366 lemma holomorphic_on_sum [holomorphic_intros]:
```
```   367   "(\<And>i. i \<in> I \<Longrightarrow> (f i) holomorphic_on s) \<Longrightarrow> (\<lambda>x. sum (\<lambda>i. f i x) I) holomorphic_on s"
```
```   368   unfolding holomorphic_on_def by (metis field_differentiable_sum)
```
```   369
```
```   370 lemma holomorphic_on_prod [holomorphic_intros]:
```
```   371   "(\<And>i. i \<in> I \<Longrightarrow> (f i) holomorphic_on s) \<Longrightarrow> (\<lambda>x. prod (\<lambda>i. f i x) I) holomorphic_on s"
```
```   372   by (induction I rule: infinite_finite_induct) (auto intro: holomorphic_intros)
```
```   373
```
```   374 lemma holomorphic_pochhammer [holomorphic_intros]:
```
```   375   "f holomorphic_on A \<Longrightarrow> (\<lambda>s. pochhammer (f s) n) holomorphic_on A"
```
```   376   by (induction n) (auto intro!: holomorphic_intros simp: pochhammer_Suc)
```
```   377
```
```   378 lemma holomorphic_on_scaleR [holomorphic_intros]:
```
```   379   "f holomorphic_on A \<Longrightarrow> (\<lambda>x. c *\<^sub>R f x) holomorphic_on A"
```
```   380   by (auto simp: scaleR_conv_of_real intro!: holomorphic_intros)
```
```   381
```
```   382 lemma holomorphic_on_Un [holomorphic_intros]:
```
```   383   assumes "f holomorphic_on A" "f holomorphic_on B" "open A" "open B"
```
```   384   shows   "f holomorphic_on (A \<union> B)"
```
```   385   using assms by (auto simp: holomorphic_on_def  at_within_open[of _ A]
```
```   386                              at_within_open[of _ B]  at_within_open[of _ "A \<union> B"] open_Un)
```
```   387
```
```   388 lemma holomorphic_on_If_Un [holomorphic_intros]:
```
```   389   assumes "f holomorphic_on A" "g holomorphic_on B" "open A" "open B"
```
```   390   assumes "\<And>z. z \<in> A \<Longrightarrow> z \<in> B \<Longrightarrow> f z = g z"
```
```   391   shows   "(\<lambda>z. if z \<in> A then f z else g z) holomorphic_on (A \<union> B)" (is "?h holomorphic_on _")
```
```   392 proof (intro holomorphic_on_Un)
```
```   393   note \<open>f holomorphic_on A\<close>
```
```   394   also have "f holomorphic_on A \<longleftrightarrow> ?h holomorphic_on A"
```
```   395     by (intro holomorphic_cong) auto
```
```   396   finally show \<dots> .
```
```   397 next
```
```   398   note \<open>g holomorphic_on B\<close>
```
```   399   also have "g holomorphic_on B \<longleftrightarrow> ?h holomorphic_on B"
```
```   400     using assms by (intro holomorphic_cong) auto
```
```   401   finally show \<dots> .
```
```   402 qed (insert assms, auto)
```
```   403
```
```   404 lemma leibniz_rule_holomorphic:
```
```   405   fixes f::"complex \<Rightarrow> 'b::euclidean_space \<Rightarrow> complex"
```
```   406   assumes "\<And>x t. x \<in> U \<Longrightarrow> t \<in> cbox a b \<Longrightarrow> ((\<lambda>x. f x t) has_field_derivative fx x t) (at x within U)"
```
```   407   assumes "\<And>x. x \<in> U \<Longrightarrow> (f x) integrable_on cbox a b"
```
```   408   assumes "continuous_on (U \<times> (cbox a b)) (\<lambda>(x, t). fx x t)"
```
```   409   assumes "convex U"
```
```   410   shows "(\<lambda>x. integral (cbox a b) (f x)) holomorphic_on U"
```
```   411   using leibniz_rule_field_differentiable[OF assms(1-3) _ assms(4)]
```
```   412   by (auto simp: holomorphic_on_def)
```
```   413
```
```   414 lemma DERIV_deriv_iff_field_differentiable:
```
```   415   "DERIV f x :> deriv f x \<longleftrightarrow> f field_differentiable at x"
```
```   416   unfolding field_differentiable_def by (metis DERIV_imp_deriv)
```
```   417
```
```   418 lemma holomorphic_derivI:
```
```   419      "\<lbrakk>f holomorphic_on S; open S; x \<in> S\<rbrakk>
```
```   420       \<Longrightarrow> (f has_field_derivative deriv f x) (at x within T)"
```
```   421 by (metis DERIV_deriv_iff_field_differentiable at_within_open  holomorphic_on_def has_field_derivative_at_within)
```
```   422
```
```   423 lemma complex_derivative_chain:
```
```   424   "f field_differentiable at x \<Longrightarrow> g field_differentiable at (f x)
```
```   425     \<Longrightarrow> deriv (g o f) x = deriv g (f x) * deriv f x"
```
```   426   by (metis DERIV_deriv_iff_field_differentiable DERIV_chain DERIV_imp_deriv)
```
```   427
```
```   428 lemma deriv_linear [simp]: "deriv (\<lambda>w. c * w) = (\<lambda>z. c)"
```
```   429   by (metis DERIV_imp_deriv DERIV_cmult_Id)
```
```   430
```
```   431 lemma deriv_ident [simp]: "deriv (\<lambda>w. w) = (\<lambda>z. 1)"
```
```   432   by (metis DERIV_imp_deriv DERIV_ident)
```
```   433
```
```   434 lemma deriv_id [simp]: "deriv id = (\<lambda>z. 1)"
```
```   435   by (simp add: id_def)
```
```   436
```
```   437 lemma deriv_const [simp]: "deriv (\<lambda>w. c) = (\<lambda>z. 0)"
```
```   438   by (metis DERIV_imp_deriv DERIV_const)
```
```   439
```
```   440 lemma deriv_add [simp]:
```
```   441   "\<lbrakk>f field_differentiable at z; g field_differentiable at z\<rbrakk>
```
```   442    \<Longrightarrow> deriv (\<lambda>w. f w + g w) z = deriv f z + deriv g z"
```
```   443   unfolding DERIV_deriv_iff_field_differentiable[symmetric]
```
```   444   by (auto intro!: DERIV_imp_deriv derivative_intros)
```
```   445
```
```   446 lemma deriv_diff [simp]:
```
```   447   "\<lbrakk>f field_differentiable at z; g field_differentiable at z\<rbrakk>
```
```   448    \<Longrightarrow> deriv (\<lambda>w. f w - g w) z = deriv f z - deriv g z"
```
```   449   unfolding DERIV_deriv_iff_field_differentiable[symmetric]
```
```   450   by (auto intro!: DERIV_imp_deriv derivative_intros)
```
```   451
```
```   452 lemma deriv_mult [simp]:
```
```   453   "\<lbrakk>f field_differentiable at z; g field_differentiable at z\<rbrakk>
```
```   454    \<Longrightarrow> deriv (\<lambda>w. f w * g w) z = f z * deriv g z + deriv f z * g z"
```
```   455   unfolding DERIV_deriv_iff_field_differentiable[symmetric]
```
```   456   by (auto intro!: DERIV_imp_deriv derivative_eq_intros)
```
```   457
```
```   458 lemma deriv_cmult [simp]:
```
```   459   "f field_differentiable at z \<Longrightarrow> deriv (\<lambda>w. c * f w) z = c * deriv f z"
```
```   460   unfolding DERIV_deriv_iff_field_differentiable[symmetric]
```
```   461   by (auto intro!: DERIV_imp_deriv derivative_eq_intros)
```
```   462
```
```   463 lemma deriv_cmult_right [simp]:
```
```   464   "f field_differentiable at z \<Longrightarrow> deriv (\<lambda>w. f w * c) z = deriv f z * c"
```
```   465   unfolding DERIV_deriv_iff_field_differentiable[symmetric]
```
```   466   by (auto intro!: DERIV_imp_deriv derivative_eq_intros)
```
```   467
```
```   468 lemma deriv_cdivide_right [simp]:
```
```   469   "f field_differentiable at z \<Longrightarrow> deriv (\<lambda>w. f w / c) z = deriv f z / c"
```
```   470   unfolding Fields.field_class.field_divide_inverse
```
```   471   by (blast intro: deriv_cmult_right)
```
```   472
```
```   473 lemma complex_derivative_transform_within_open:
```
```   474   "\<lbrakk>f holomorphic_on s; g holomorphic_on s; open s; z \<in> s; \<And>w. w \<in> s \<Longrightarrow> f w = g w\<rbrakk>
```
```   475    \<Longrightarrow> deriv f z = deriv g z"
```
```   476   unfolding holomorphic_on_def
```
```   477   by (rule DERIV_imp_deriv)
```
```   478      (metis DERIV_deriv_iff_field_differentiable DERIV_transform_within_open at_within_open)
```
```   479
```
```   480 lemma deriv_compose_linear:
```
```   481   "f field_differentiable at (c * z) \<Longrightarrow> deriv (\<lambda>w. f (c * w)) z = c * deriv f (c * z)"
```
```   482 apply (rule DERIV_imp_deriv)
```
```   483 apply (simp add: DERIV_deriv_iff_field_differentiable [symmetric])
```
```   484 apply (drule DERIV_chain' [of "times c" c z UNIV f "deriv f (c * z)", OF DERIV_cmult_Id])
```
```   485 apply (simp add: algebra_simps)
```
```   486 done
```
```   487
```
```   488 lemma nonzero_deriv_nonconstant:
```
```   489   assumes df: "DERIV f \<xi> :> df" and S: "open S" "\<xi> \<in> S" and "df \<noteq> 0"
```
```   490     shows "\<not> f constant_on S"
```
```   491 unfolding constant_on_def
```
```   492 by (metis \<open>df \<noteq> 0\<close> DERIV_transform_within_open [OF df S] DERIV_const DERIV_unique)
```
```   493
```
```   494 lemma holomorphic_nonconstant:
```
```   495   assumes holf: "f holomorphic_on S" and "open S" "\<xi> \<in> S" "deriv f \<xi> \<noteq> 0"
```
```   496     shows "\<not> f constant_on S"
```
```   497     apply (rule nonzero_deriv_nonconstant [of f "deriv f \<xi>" \<xi> S])
```
```   498     using assms
```
```   499     apply (auto simp: holomorphic_derivI)
```
```   500     done
```
```   501
```
```   502 subsection\<open>Caratheodory characterization\<close>
```
```   503
```
```   504 lemma field_differentiable_caratheodory_at:
```
```   505   "f field_differentiable (at z) \<longleftrightarrow>
```
```   506          (\<exists>g. (\<forall>w. f(w) - f(z) = g(w) * (w - z)) \<and> continuous (at z) g)"
```
```   507   using CARAT_DERIV [of f]
```
```   508   by (simp add: field_differentiable_def has_field_derivative_def)
```
```   509
```
```   510 lemma field_differentiable_caratheodory_within:
```
```   511   "f field_differentiable (at z within s) \<longleftrightarrow>
```
```   512          (\<exists>g. (\<forall>w. f(w) - f(z) = g(w) * (w - z)) \<and> continuous (at z within s) g)"
```
```   513   using DERIV_caratheodory_within [of f]
```
```   514   by (simp add: field_differentiable_def has_field_derivative_def)
```
```   515
```
```   516 subsection\<open>Analyticity on a set\<close>
```
```   517
```
```   518 definition analytic_on (infixl "(analytic'_on)" 50)
```
```   519   where
```
```   520    "f analytic_on s \<equiv> \<forall>x \<in> s. \<exists>e. 0 < e \<and> f holomorphic_on (ball x e)"
```
```   521
```
```   522 named_theorems analytic_intros "introduction rules for proving analyticity"
```
```   523
```
```   524 lemma analytic_imp_holomorphic: "f analytic_on s \<Longrightarrow> f holomorphic_on s"
```
```   525   by (simp add: at_within_open [OF _ open_ball] analytic_on_def holomorphic_on_def)
```
```   526      (metis centre_in_ball field_differentiable_at_within)
```
```   527
```
```   528 lemma analytic_on_open: "open s \<Longrightarrow> f analytic_on s \<longleftrightarrow> f holomorphic_on s"
```
```   529 apply (auto simp: analytic_imp_holomorphic)
```
```   530 apply (auto simp: analytic_on_def holomorphic_on_def)
```
```   531 by (metis holomorphic_on_def holomorphic_on_subset open_contains_ball)
```
```   532
```
```   533 lemma analytic_on_imp_differentiable_at:
```
```   534   "f analytic_on s \<Longrightarrow> x \<in> s \<Longrightarrow> f field_differentiable (at x)"
```
```   535  apply (auto simp: analytic_on_def holomorphic_on_def)
```
```   536 by (metis open_ball centre_in_ball field_differentiable_within_open)
```
```   537
```
```   538 lemma analytic_on_subset: "f analytic_on s \<Longrightarrow> t \<subseteq> s \<Longrightarrow> f analytic_on t"
```
```   539   by (auto simp: analytic_on_def)
```
```   540
```
```   541 lemma analytic_on_Un: "f analytic_on (s \<union> t) \<longleftrightarrow> f analytic_on s \<and> f analytic_on t"
```
```   542   by (auto simp: analytic_on_def)
```
```   543
```
```   544 lemma analytic_on_Union: "f analytic_on (\<Union>s) \<longleftrightarrow> (\<forall>t \<in> s. f analytic_on t)"
```
```   545   by (auto simp: analytic_on_def)
```
```   546
```
```   547 lemma analytic_on_UN: "f analytic_on (\<Union>i\<in>I. s i) \<longleftrightarrow> (\<forall>i\<in>I. f analytic_on (s i))"
```
```   548   by (auto simp: analytic_on_def)
```
```   549
```
```   550 lemma analytic_on_holomorphic:
```
```   551   "f analytic_on s \<longleftrightarrow> (\<exists>t. open t \<and> s \<subseteq> t \<and> f holomorphic_on t)"
```
```   552   (is "?lhs = ?rhs")
```
```   553 proof -
```
```   554   have "?lhs \<longleftrightarrow> (\<exists>t. open t \<and> s \<subseteq> t \<and> f analytic_on t)"
```
```   555   proof safe
```
```   556     assume "f analytic_on s"
```
```   557     then show "\<exists>t. open t \<and> s \<subseteq> t \<and> f analytic_on t"
```
```   558       apply (simp add: analytic_on_def)
```
```   559       apply (rule exI [where x="\<Union>{u. open u \<and> f analytic_on u}"], auto)
```
```   560       apply (metis open_ball analytic_on_open centre_in_ball)
```
```   561       by (metis analytic_on_def)
```
```   562   next
```
```   563     fix t
```
```   564     assume "open t" "s \<subseteq> t" "f analytic_on t"
```
```   565     then show "f analytic_on s"
```
```   566         by (metis analytic_on_subset)
```
```   567   qed
```
```   568   also have "... \<longleftrightarrow> ?rhs"
```
```   569     by (auto simp: analytic_on_open)
```
```   570   finally show ?thesis .
```
```   571 qed
```
```   572
```
```   573 lemma analytic_on_linear [analytic_intros,simp]: "(( * ) c) analytic_on s"
```
```   574   by (auto simp add: analytic_on_holomorphic)
```
```   575
```
```   576 lemma analytic_on_const [analytic_intros,simp]: "(\<lambda>z. c) analytic_on s"
```
```   577   by (metis analytic_on_def holomorphic_on_const zero_less_one)
```
```   578
```
```   579 lemma analytic_on_ident [analytic_intros,simp]: "(\<lambda>x. x) analytic_on s"
```
```   580   by (simp add: analytic_on_def gt_ex)
```
```   581
```
```   582 lemma analytic_on_id [analytic_intros]: "id analytic_on s"
```
```   583   unfolding id_def by (rule analytic_on_ident)
```
```   584
```
```   585 lemma analytic_on_compose:
```
```   586   assumes f: "f analytic_on s"
```
```   587       and g: "g analytic_on (f ` s)"
```
```   588     shows "(g o f) analytic_on s"
```
```   589 unfolding analytic_on_def
```
```   590 proof (intro ballI)
```
```   591   fix x
```
```   592   assume x: "x \<in> s"
```
```   593   then obtain e where e: "0 < e" and fh: "f holomorphic_on ball x e" using f
```
```   594     by (metis analytic_on_def)
```
```   595   obtain e' where e': "0 < e'" and gh: "g holomorphic_on ball (f x) e'" using g
```
```   596     by (metis analytic_on_def g image_eqI x)
```
```   597   have "isCont f x"
```
```   598     by (metis analytic_on_imp_differentiable_at field_differentiable_imp_continuous_at f x)
```
```   599   with e' obtain d where d: "0 < d" and fd: "f ` ball x d \<subseteq> ball (f x) e'"
```
```   600      by (auto simp: continuous_at_ball)
```
```   601   have "g \<circ> f holomorphic_on ball x (min d e)"
```
```   602     apply (rule holomorphic_on_compose)
```
```   603     apply (metis fh holomorphic_on_subset min.bounded_iff order_refl subset_ball)
```
```   604     by (metis fd gh holomorphic_on_subset image_mono min.cobounded1 subset_ball)
```
```   605   then show "\<exists>e>0. g \<circ> f holomorphic_on ball x e"
```
```   606     by (metis d e min_less_iff_conj)
```
```   607 qed
```
```   608
```
```   609 lemma analytic_on_compose_gen:
```
```   610   "f analytic_on s \<Longrightarrow> g analytic_on t \<Longrightarrow> (\<And>z. z \<in> s \<Longrightarrow> f z \<in> t)
```
```   611              \<Longrightarrow> g o f analytic_on s"
```
```   612 by (metis analytic_on_compose analytic_on_subset image_subset_iff)
```
```   613
```
```   614 lemma analytic_on_neg [analytic_intros]:
```
```   615   "f analytic_on s \<Longrightarrow> (\<lambda>z. -(f z)) analytic_on s"
```
```   616 by (metis analytic_on_holomorphic holomorphic_on_minus)
```
```   617
```
```   618 lemma analytic_on_add [analytic_intros]:
```
```   619   assumes f: "f analytic_on s"
```
```   620       and g: "g analytic_on s"
```
```   621     shows "(\<lambda>z. f z + g z) analytic_on s"
```
```   622 unfolding analytic_on_def
```
```   623 proof (intro ballI)
```
```   624   fix z
```
```   625   assume z: "z \<in> s"
```
```   626   then obtain e where e: "0 < e" and fh: "f holomorphic_on ball z e" using f
```
```   627     by (metis analytic_on_def)
```
```   628   obtain e' where e': "0 < e'" and gh: "g holomorphic_on ball z e'" using g
```
```   629     by (metis analytic_on_def g z)
```
```   630   have "(\<lambda>z. f z + g z) holomorphic_on ball z (min e e')"
```
```   631     apply (rule holomorphic_on_add)
```
```   632     apply (metis fh holomorphic_on_subset min.bounded_iff order_refl subset_ball)
```
```   633     by (metis gh holomorphic_on_subset min.bounded_iff order_refl subset_ball)
```
```   634   then show "\<exists>e>0. (\<lambda>z. f z + g z) holomorphic_on ball z e"
```
```   635     by (metis e e' min_less_iff_conj)
```
```   636 qed
```
```   637
```
```   638 lemma analytic_on_diff [analytic_intros]:
```
```   639   assumes f: "f analytic_on s"
```
```   640       and g: "g analytic_on s"
```
```   641     shows "(\<lambda>z. f z - g z) analytic_on s"
```
```   642 unfolding analytic_on_def
```
```   643 proof (intro ballI)
```
```   644   fix z
```
```   645   assume z: "z \<in> s"
```
```   646   then obtain e where e: "0 < e" and fh: "f holomorphic_on ball z e" using f
```
```   647     by (metis analytic_on_def)
```
```   648   obtain e' where e': "0 < e'" and gh: "g holomorphic_on ball z e'" using g
```
```   649     by (metis analytic_on_def g z)
```
```   650   have "(\<lambda>z. f z - g z) holomorphic_on ball z (min e e')"
```
```   651     apply (rule holomorphic_on_diff)
```
```   652     apply (metis fh holomorphic_on_subset min.bounded_iff order_refl subset_ball)
```
```   653     by (metis gh holomorphic_on_subset min.bounded_iff order_refl subset_ball)
```
```   654   then show "\<exists>e>0. (\<lambda>z. f z - g z) holomorphic_on ball z e"
```
```   655     by (metis e e' min_less_iff_conj)
```
```   656 qed
```
```   657
```
```   658 lemma analytic_on_mult [analytic_intros]:
```
```   659   assumes f: "f analytic_on s"
```
```   660       and g: "g analytic_on s"
```
```   661     shows "(\<lambda>z. f z * g z) analytic_on s"
```
```   662 unfolding analytic_on_def
```
```   663 proof (intro ballI)
```
```   664   fix z
```
```   665   assume z: "z \<in> s"
```
```   666   then obtain e where e: "0 < e" and fh: "f holomorphic_on ball z e" using f
```
```   667     by (metis analytic_on_def)
```
```   668   obtain e' where e': "0 < e'" and gh: "g holomorphic_on ball z e'" using g
```
```   669     by (metis analytic_on_def g z)
```
```   670   have "(\<lambda>z. f z * g z) holomorphic_on ball z (min e e')"
```
```   671     apply (rule holomorphic_on_mult)
```
```   672     apply (metis fh holomorphic_on_subset min.bounded_iff order_refl subset_ball)
```
```   673     by (metis gh holomorphic_on_subset min.bounded_iff order_refl subset_ball)
```
```   674   then show "\<exists>e>0. (\<lambda>z. f z * g z) holomorphic_on ball z e"
```
```   675     by (metis e e' min_less_iff_conj)
```
```   676 qed
```
```   677
```
```   678 lemma analytic_on_inverse [analytic_intros]:
```
```   679   assumes f: "f analytic_on s"
```
```   680       and nz: "(\<And>z. z \<in> s \<Longrightarrow> f z \<noteq> 0)"
```
```   681     shows "(\<lambda>z. inverse (f z)) analytic_on s"
```
```   682 unfolding analytic_on_def
```
```   683 proof (intro ballI)
```
```   684   fix z
```
```   685   assume z: "z \<in> s"
```
```   686   then obtain e where e: "0 < e" and fh: "f holomorphic_on ball z e" using f
```
```   687     by (metis analytic_on_def)
```
```   688   have "continuous_on (ball z e) f"
```
```   689     by (metis fh holomorphic_on_imp_continuous_on)
```
```   690   then obtain e' where e': "0 < e'" and nz': "\<And>y. dist z y < e' \<Longrightarrow> f y \<noteq> 0"
```
```   691     by (metis open_ball centre_in_ball continuous_on_open_avoid e z nz)
```
```   692   have "(\<lambda>z. inverse (f z)) holomorphic_on ball z (min e e')"
```
```   693     apply (rule holomorphic_on_inverse)
```
```   694     apply (metis fh holomorphic_on_subset min.cobounded2 min.commute subset_ball)
```
```   695     by (metis nz' mem_ball min_less_iff_conj)
```
```   696   then show "\<exists>e>0. (\<lambda>z. inverse (f z)) holomorphic_on ball z e"
```
```   697     by (metis e e' min_less_iff_conj)
```
```   698 qed
```
```   699
```
```   700 lemma analytic_on_divide [analytic_intros]:
```
```   701   assumes f: "f analytic_on s"
```
```   702       and g: "g analytic_on s"
```
```   703       and nz: "(\<And>z. z \<in> s \<Longrightarrow> g z \<noteq> 0)"
```
```   704     shows "(\<lambda>z. f z / g z) analytic_on s"
```
```   705 unfolding divide_inverse
```
```   706 by (metis analytic_on_inverse analytic_on_mult f g nz)
```
```   707
```
```   708 lemma analytic_on_power [analytic_intros]:
```
```   709   "f analytic_on s \<Longrightarrow> (\<lambda>z. (f z) ^ n) analytic_on s"
```
```   710 by (induct n) (auto simp: analytic_on_mult)
```
```   711
```
```   712 lemma analytic_on_sum [analytic_intros]:
```
```   713   "(\<And>i. i \<in> I \<Longrightarrow> (f i) analytic_on s) \<Longrightarrow> (\<lambda>x. sum (\<lambda>i. f i x) I) analytic_on s"
```
```   714   by (induct I rule: infinite_finite_induct) (auto simp: analytic_on_const analytic_on_add)
```
```   715
```
```   716 lemma deriv_left_inverse:
```
```   717   assumes "f holomorphic_on S" and "g holomorphic_on T"
```
```   718       and "open S" and "open T"
```
```   719       and "f ` S \<subseteq> T"
```
```   720       and [simp]: "\<And>z. z \<in> S \<Longrightarrow> g (f z) = z"
```
```   721       and "w \<in> S"
```
```   722     shows "deriv f w * deriv g (f w) = 1"
```
```   723 proof -
```
```   724   have "deriv f w * deriv g (f w) = deriv g (f w) * deriv f w"
```
```   725     by (simp add: algebra_simps)
```
```   726   also have "... = deriv (g o f) w"
```
```   727     using assms
```
```   728     by (metis analytic_on_imp_differentiable_at analytic_on_open complex_derivative_chain image_subset_iff)
```
```   729   also have "... = deriv id w"
```
```   730     apply (rule complex_derivative_transform_within_open [where s=S])
```
```   731     apply (rule assms holomorphic_on_compose_gen holomorphic_intros)+
```
```   732     apply simp
```
```   733     done
```
```   734   also have "... = 1"
```
```   735     by simp
```
```   736   finally show ?thesis .
```
```   737 qed
```
```   738
```
```   739 subsection\<open>analyticity at a point\<close>
```
```   740
```
```   741 lemma analytic_at_ball:
```
```   742   "f analytic_on {z} \<longleftrightarrow> (\<exists>e. 0<e \<and> f holomorphic_on ball z e)"
```
```   743 by (metis analytic_on_def singleton_iff)
```
```   744
```
```   745 lemma analytic_at:
```
```   746     "f analytic_on {z} \<longleftrightarrow> (\<exists>s. open s \<and> z \<in> s \<and> f holomorphic_on s)"
```
```   747 by (metis analytic_on_holomorphic empty_subsetI insert_subset)
```
```   748
```
```   749 lemma analytic_on_analytic_at:
```
```   750     "f analytic_on s \<longleftrightarrow> (\<forall>z \<in> s. f analytic_on {z})"
```
```   751 by (metis analytic_at_ball analytic_on_def)
```
```   752
```
```   753 lemma analytic_at_two:
```
```   754   "f analytic_on {z} \<and> g analytic_on {z} \<longleftrightarrow>
```
```   755    (\<exists>s. open s \<and> z \<in> s \<and> f holomorphic_on s \<and> g holomorphic_on s)"
```
```   756   (is "?lhs = ?rhs")
```
```   757 proof
```
```   758   assume ?lhs
```
```   759   then obtain s t
```
```   760     where st: "open s" "z \<in> s" "f holomorphic_on s"
```
```   761               "open t" "z \<in> t" "g holomorphic_on t"
```
```   762     by (auto simp: analytic_at)
```
```   763   show ?rhs
```
```   764     apply (rule_tac x="s \<inter> t" in exI)
```
```   765     using st
```
```   766     apply (auto simp: Diff_subset holomorphic_on_subset)
```
```   767     done
```
```   768 next
```
```   769   assume ?rhs
```
```   770   then show ?lhs
```
```   771     by (force simp add: analytic_at)
```
```   772 qed
```
```   773
```
```   774 subsection\<open>Combining theorems for derivative with ``analytic at'' hypotheses\<close>
```
```   775
```
```   776 lemma
```
```   777   assumes "f analytic_on {z}" "g analytic_on {z}"
```
```   778   shows complex_derivative_add_at: "deriv (\<lambda>w. f w + g w) z = deriv f z + deriv g z"
```
```   779     and complex_derivative_diff_at: "deriv (\<lambda>w. f w - g w) z = deriv f z - deriv g z"
```
```   780     and complex_derivative_mult_at: "deriv (\<lambda>w. f w * g w) z =
```
```   781            f z * deriv g z + deriv f z * g z"
```
```   782 proof -
```
```   783   obtain s where s: "open s" "z \<in> s" "f holomorphic_on s" "g holomorphic_on s"
```
```   784     using assms by (metis analytic_at_two)
```
```   785   show "deriv (\<lambda>w. f w + g w) z = deriv f z + deriv g z"
```
```   786     apply (rule DERIV_imp_deriv [OF DERIV_add])
```
```   787     using s
```
```   788     apply (auto simp: holomorphic_on_open field_differentiable_def DERIV_deriv_iff_field_differentiable)
```
```   789     done
```
```   790   show "deriv (\<lambda>w. f w - g w) z = deriv f z - deriv g z"
```
```   791     apply (rule DERIV_imp_deriv [OF DERIV_diff])
```
```   792     using s
```
```   793     apply (auto simp: holomorphic_on_open field_differentiable_def DERIV_deriv_iff_field_differentiable)
```
```   794     done
```
```   795   show "deriv (\<lambda>w. f w * g w) z = f z * deriv g z + deriv f z * g z"
```
```   796     apply (rule DERIV_imp_deriv [OF DERIV_mult'])
```
```   797     using s
```
```   798     apply (auto simp: holomorphic_on_open field_differentiable_def DERIV_deriv_iff_field_differentiable)
```
```   799     done
```
```   800 qed
```
```   801
```
```   802 lemma deriv_cmult_at:
```
```   803   "f analytic_on {z} \<Longrightarrow>  deriv (\<lambda>w. c * f w) z = c * deriv f z"
```
```   804 by (auto simp: complex_derivative_mult_at deriv_const analytic_on_const)
```
```   805
```
```   806 lemma deriv_cmult_right_at:
```
```   807   "f analytic_on {z} \<Longrightarrow>  deriv (\<lambda>w. f w * c) z = deriv f z * c"
```
```   808 by (auto simp: complex_derivative_mult_at deriv_const analytic_on_const)
```
```   809
```
```   810 subsection\<open>Complex differentiation of sequences and series\<close>
```
```   811
```
```   812 (* TODO: Could probably be simplified using Uniform_Limit *)
```
```   813 lemma has_complex_derivative_sequence:
```
```   814   fixes s :: "complex set"
```
```   815   assumes cvs: "convex s"
```
```   816       and df:  "\<And>n x. x \<in> s \<Longrightarrow> (f n has_field_derivative f' n x) (at x within s)"
```
```   817       and conv: "\<And>e. 0 < e \<Longrightarrow> \<exists>N. \<forall>n x. n \<ge> N \<longrightarrow> x \<in> s \<longrightarrow> norm (f' n x - g' x) \<le> e"
```
```   818       and "\<exists>x l. x \<in> s \<and> ((\<lambda>n. f n x) \<longlongrightarrow> l) sequentially"
```
```   819     shows "\<exists>g. \<forall>x \<in> s. ((\<lambda>n. f n x) \<longlongrightarrow> g x) sequentially \<and>
```
```   820                        (g has_field_derivative (g' x)) (at x within s)"
```
```   821 proof -
```
```   822   from assms obtain x l where x: "x \<in> s" and tf: "((\<lambda>n. f n x) \<longlongrightarrow> l) sequentially"
```
```   823     by blast
```
```   824   { fix e::real assume e: "e > 0"
```
```   825     then obtain N where N: "\<forall>n\<ge>N. \<forall>x. x \<in> s \<longrightarrow> cmod (f' n x - g' x) \<le> e"
```
```   826       by (metis conv)
```
```   827     have "\<exists>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>h. cmod (f' n x * h - g' x * h) \<le> e * cmod h"
```
```   828     proof (rule exI [of _ N], clarify)
```
```   829       fix n y h
```
```   830       assume "N \<le> n" "y \<in> s"
```
```   831       then have "cmod (f' n y - g' y) \<le> e"
```
```   832         by (metis N)
```
```   833       then have "cmod h * cmod (f' n y - g' y) \<le> cmod h * e"
```
```   834         by (auto simp: antisym_conv2 mult_le_cancel_left norm_triangle_ineq2)
```
```   835       then show "cmod (f' n y * h - g' y * h) \<le> e * cmod h"
```
```   836         by (simp add: norm_mult [symmetric] field_simps)
```
```   837     qed
```
```   838   } note ** = this
```
```   839   show ?thesis
```
```   840   unfolding has_field_derivative_def
```
```   841   proof (rule has_derivative_sequence [OF cvs _ _ x])
```
```   842     show "\<forall>n. \<forall>x\<in>s. (f n has_derivative (( * ) (f' n x))) (at x within s)"
```
```   843       by (metis has_field_derivative_def df)
```
```   844   next show "(\<lambda>n. f n x) \<longlonglongrightarrow> l"
```
```   845     by (rule tf)
```
```   846   next show "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>h. cmod (f' n x * h - g' x * h) \<le> e * cmod h"
```
```   847     by (blast intro: **)
```
```   848   qed
```
```   849 qed
```
```   850
```
```   851 lemma has_complex_derivative_series:
```
```   852   fixes s :: "complex set"
```
```   853   assumes cvs: "convex s"
```
```   854       and df:  "\<And>n x. x \<in> s \<Longrightarrow> (f n has_field_derivative f' n x) (at x within s)"
```
```   855       and conv: "\<And>e. 0 < e \<Longrightarrow> \<exists>N. \<forall>n x. n \<ge> N \<longrightarrow> x \<in> s
```
```   856                 \<longrightarrow> cmod ((\<Sum>i<n. f' i x) - g' x) \<le> e"
```
```   857       and "\<exists>x l. x \<in> s \<and> ((\<lambda>n. f n x) sums l)"
```
```   858     shows "\<exists>g. \<forall>x \<in> s. ((\<lambda>n. f n x) sums g x) \<and> ((g has_field_derivative g' x) (at x within s))"
```
```   859 proof -
```
```   860   from assms obtain x l where x: "x \<in> s" and sf: "((\<lambda>n. f n x) sums l)"
```
```   861     by blast
```
```   862   { fix e::real assume e: "e > 0"
```
```   863     then obtain N where N: "\<forall>n x. n \<ge> N \<longrightarrow> x \<in> s
```
```   864             \<longrightarrow> cmod ((\<Sum>i<n. f' i x) - g' x) \<le> e"
```
```   865       by (metis conv)
```
```   866     have "\<exists>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>h. cmod ((\<Sum>i<n. h * f' i x) - g' x * h) \<le> e * cmod h"
```
```   867     proof (rule exI [of _ N], clarify)
```
```   868       fix n y h
```
```   869       assume "N \<le> n" "y \<in> s"
```
```   870       then have "cmod ((\<Sum>i<n. f' i y) - g' y) \<le> e"
```
```   871         by (metis N)
```
```   872       then have "cmod h * cmod ((\<Sum>i<n. f' i y) - g' y) \<le> cmod h * e"
```
```   873         by (auto simp: antisym_conv2 mult_le_cancel_left norm_triangle_ineq2)
```
```   874       then show "cmod ((\<Sum>i<n. h * f' i y) - g' y * h) \<le> e * cmod h"
```
```   875         by (simp add: norm_mult [symmetric] field_simps sum_distrib_left)
```
```   876     qed
```
```   877   } note ** = this
```
```   878   show ?thesis
```
```   879   unfolding has_field_derivative_def
```
```   880   proof (rule has_derivative_series [OF cvs _ _ x])
```
```   881     fix n x
```
```   882     assume "x \<in> s"
```
```   883     then show "((f n) has_derivative (\<lambda>z. z * f' n x)) (at x within s)"
```
```   884       by (metis df has_field_derivative_def mult_commute_abs)
```
```   885   next show " ((\<lambda>n. f n x) sums l)"
```
```   886     by (rule sf)
```
```   887   next show "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>h. cmod ((\<Sum>i<n. h * f' i x) - g' x * h) \<le> e * cmod h"
```
```   888     by (blast intro: **)
```
```   889   qed
```
```   890 qed
```
```   891
```
```   892
```
```   893 lemma field_differentiable_series:
```
```   894   fixes f :: "nat \<Rightarrow> 'a::{real_normed_field,banach} \<Rightarrow> 'a"
```
```   895   assumes "convex s" "open s"
```
```   896   assumes "\<And>n x. x \<in> s \<Longrightarrow> (f n has_field_derivative f' n x) (at x)"
```
```   897   assumes "uniformly_convergent_on s (\<lambda>n x. \<Sum>i<n. f' i x)"
```
```   898   assumes "x0 \<in> s" "summable (\<lambda>n. f n x0)" and x: "x \<in> s"
```
```   899   shows   "summable (\<lambda>n. f n x)" and "(\<lambda>x. \<Sum>n. f n x) field_differentiable (at x)"
```
```   900 proof -
```
```   901   from assms(4) obtain g' where A: "uniform_limit s (\<lambda>n x. \<Sum>i<n. f' i x) g' sequentially"
```
```   902     unfolding uniformly_convergent_on_def by blast
```
```   903   from x and \<open>open s\<close> have s: "at x within s = at x" by (rule at_within_open)
```
```   904   have "\<exists>g. \<forall>x\<in>s. (\<lambda>n. f n x) sums g x \<and> (g has_field_derivative g' x) (at x within s)"
```
```   905     by (intro has_field_derivative_series[of s f f' g' x0] assms A has_field_derivative_at_within)
```
```   906   then obtain g where g: "\<And>x. x \<in> s \<Longrightarrow> (\<lambda>n. f n x) sums g x"
```
```   907     "\<And>x. x \<in> s \<Longrightarrow> (g has_field_derivative g' x) (at x within s)" by blast
```
```   908   from g[OF x] show "summable (\<lambda>n. f n x)" by (auto simp: summable_def)
```
```   909   from g(2)[OF x] have g': "(g has_derivative ( * ) (g' x)) (at x)"
```
```   910     by (simp add: has_field_derivative_def s)
```
```   911   have "((\<lambda>x. \<Sum>n. f n x) has_derivative ( * ) (g' x)) (at x)"
```
```   912     by (rule has_derivative_transform_within_open[OF g' \<open>open s\<close> x])
```
```   913        (insert g, auto simp: sums_iff)
```
```   914   thus "(\<lambda>x. \<Sum>n. f n x) field_differentiable (at x)" unfolding differentiable_def
```
```   915     by (auto simp: summable_def field_differentiable_def has_field_derivative_def)
```
```   916 qed
```
```   917
```
```   918 lemma field_differentiable_series':
```
```   919   fixes f :: "nat \<Rightarrow> 'a::{real_normed_field,banach} \<Rightarrow> 'a"
```
```   920   assumes "convex s" "open s"
```
```   921   assumes "\<And>n x. x \<in> s \<Longrightarrow> (f n has_field_derivative f' n x) (at x)"
```
```   922   assumes "uniformly_convergent_on s (\<lambda>n x. \<Sum>i<n. f' i x)"
```
```   923   assumes "x0 \<in> s" "summable (\<lambda>n. f n x0)"
```
```   924   shows   "(\<lambda>x. \<Sum>n. f n x) field_differentiable (at x0)"
```
```   925   using field_differentiable_series[OF assms, of x0] \<open>x0 \<in> s\<close> by blast+
```
```   926
```
```   927 subsection\<open>Bound theorem\<close>
```
```   928
```
```   929 lemma field_differentiable_bound:
```
```   930   fixes s :: "'a::real_normed_field set"
```
```   931   assumes cvs: "convex s"
```
```   932       and df:  "\<And>z. z \<in> s \<Longrightarrow> (f has_field_derivative f' z) (at z within s)"
```
```   933       and dn:  "\<And>z. z \<in> s \<Longrightarrow> norm (f' z) \<le> B"
```
```   934       and "x \<in> s"  "y \<in> s"
```
```   935     shows "norm(f x - f y) \<le> B * norm(x - y)"
```
```   936   apply (rule differentiable_bound [OF cvs])
```
```   937   apply (rule ballI, erule df [unfolded has_field_derivative_def])
```
```   938   apply (rule ballI, rule onorm_le, simp add: norm_mult mult_right_mono dn)
```
```   939   apply fact
```
```   940   apply fact
```
```   941   done
```
```   942
```
```   943 subsection\<open>Inverse function theorem for complex derivatives\<close>
```
```   944
```
```   945 lemma has_field_derivative_inverse_basic:
```
```   946   shows "DERIV f (g y) :> f' \<Longrightarrow>
```
```   947         f' \<noteq> 0 \<Longrightarrow>
```
```   948         continuous (at y) g \<Longrightarrow>
```
```   949         open t \<Longrightarrow>
```
```   950         y \<in> t \<Longrightarrow>
```
```   951         (\<And>z. z \<in> t \<Longrightarrow> f (g z) = z)
```
```   952         \<Longrightarrow> DERIV g y :> inverse (f')"
```
```   953   unfolding has_field_derivative_def
```
```   954   apply (rule has_derivative_inverse_basic)
```
```   955   apply (auto simp:  bounded_linear_mult_right)
```
```   956   done
```
```   957
```
```   958 lemmas has_complex_derivative_inverse_basic = has_field_derivative_inverse_basic
```
```   959
```
```   960 lemma has_field_derivative_inverse_strong:
```
```   961   fixes f :: "'a::{euclidean_space,real_normed_field} \<Rightarrow> 'a"
```
```   962   shows "DERIV f x :> f' \<Longrightarrow>
```
```   963          f' \<noteq> 0 \<Longrightarrow>
```
```   964          open s \<Longrightarrow>
```
```   965          x \<in> s \<Longrightarrow>
```
```   966          continuous_on s f \<Longrightarrow>
```
```   967          (\<And>z. z \<in> s \<Longrightarrow> g (f z) = z)
```
```   968          \<Longrightarrow> DERIV g (f x) :> inverse (f')"
```
```   969   unfolding has_field_derivative_def
```
```   970   apply (rule has_derivative_inverse_strong [of s x f g ])
```
```   971   by auto
```
```   972 lemmas has_complex_derivative_inverse_strong = has_field_derivative_inverse_strong
```
```   973
```
```   974 lemma has_field_derivative_inverse_strong_x:
```
```   975   fixes f :: "'a::{euclidean_space,real_normed_field} \<Rightarrow> 'a"
```
```   976   shows  "DERIV f (g y) :> f' \<Longrightarrow>
```
```   977           f' \<noteq> 0 \<Longrightarrow>
```
```   978           open s \<Longrightarrow>
```
```   979           continuous_on s f \<Longrightarrow>
```
```   980           g y \<in> s \<Longrightarrow> f(g y) = y \<Longrightarrow>
```
```   981           (\<And>z. z \<in> s \<Longrightarrow> g (f z) = z)
```
```   982           \<Longrightarrow> DERIV g y :> inverse (f')"
```
```   983   unfolding has_field_derivative_def
```
```   984   apply (rule has_derivative_inverse_strong_x [of s g y f])
```
```   985   by auto
```
```   986 lemmas has_complex_derivative_inverse_strong_x = has_field_derivative_inverse_strong_x
```
```   987
```
```   988 subsection \<open>Taylor on Complex Numbers\<close>
```
```   989
```
```   990 lemma sum_Suc_reindex:
```
```   991   fixes f :: "nat \<Rightarrow> 'a::ab_group_add"
```
```   992     shows  "sum f {0..n} = f 0 - f (Suc n) + sum (\<lambda>i. f (Suc i)) {0..n}"
```
```   993 by (induct n) auto
```
```   994
```
```   995 lemma field_taylor:
```
```   996   assumes s: "convex s"
```
```   997       and f: "\<And>i x. x \<in> s \<Longrightarrow> i \<le> n \<Longrightarrow> (f i has_field_derivative f (Suc i) x) (at x within s)"
```
```   998       and B: "\<And>x. x \<in> s \<Longrightarrow> norm (f (Suc n) x) \<le> B"
```
```   999       and w: "w \<in> s"
```
```  1000       and z: "z \<in> s"
```
```  1001     shows "norm(f 0 z - (\<Sum>i\<le>n. f i w * (z-w) ^ i / (fact i)))
```
```  1002           \<le> B * norm(z - w)^(Suc n) / fact n"
```
```  1003 proof -
```
```  1004   have wzs: "closed_segment w z \<subseteq> s" using assms
```
```  1005     by (metis convex_contains_segment)
```
```  1006   { fix u
```
```  1007     assume "u \<in> closed_segment w z"
```
```  1008     then have "u \<in> s"
```
```  1009       by (metis wzs subsetD)
```
```  1010     have "(\<Sum>i\<le>n. f i u * (- of_nat i * (z-u)^(i - 1)) / (fact i) +
```
```  1011                       f (Suc i) u * (z-u)^i / (fact i)) =
```
```  1012               f (Suc n) u * (z-u) ^ n / (fact n)"
```
```  1013     proof (induction n)
```
```  1014       case 0 show ?case by simp
```
```  1015     next
```
```  1016       case (Suc n)
```
```  1017       have "(\<Sum>i\<le>Suc n. f i u * (- of_nat i * (z-u) ^ (i - 1)) / (fact i) +
```
```  1018                              f (Suc i) u * (z-u) ^ i / (fact i)) =
```
```  1019            f (Suc n) u * (z-u) ^ n / (fact n) +
```
```  1020            f (Suc (Suc n)) u * ((z-u) * (z-u) ^ n) / (fact (Suc n)) -
```
```  1021            f (Suc n) u * ((1 + of_nat n) * (z-u) ^ n) / (fact (Suc n))"
```
```  1022         using Suc by simp
```
```  1023       also have "... = f (Suc (Suc n)) u * (z-u) ^ Suc n / (fact (Suc n))"
```
```  1024       proof -
```
```  1025         have "(fact(Suc n)) *
```
```  1026              (f(Suc n) u *(z-u) ^ n / (fact n) +
```
```  1027                f(Suc(Suc n)) u *((z-u) *(z-u) ^ n) / (fact(Suc n)) -
```
```  1028                f(Suc n) u *((1 + of_nat n) *(z-u) ^ n) / (fact(Suc n))) =
```
```  1029             ((fact(Suc n)) *(f(Suc n) u *(z-u) ^ n)) / (fact n) +
```
```  1030             ((fact(Suc n)) *(f(Suc(Suc n)) u *((z-u) *(z-u) ^ n)) / (fact(Suc n))) -
```
```  1031             ((fact(Suc n)) *(f(Suc n) u *(of_nat(Suc n) *(z-u) ^ n))) / (fact(Suc n))"
```
```  1032           by (simp add: algebra_simps del: fact_Suc)
```
```  1033         also have "... = ((fact (Suc n)) * (f (Suc n) u * (z-u) ^ n)) / (fact n) +
```
```  1034                          (f (Suc (Suc n)) u * ((z-u) * (z-u) ^ n)) -
```
```  1035                          (f (Suc n) u * ((1 + of_nat n) * (z-u) ^ n))"
```
```  1036           by (simp del: fact_Suc)
```
```  1037         also have "... = (of_nat (Suc n) * (f (Suc n) u * (z-u) ^ n)) +
```
```  1038                          (f (Suc (Suc n)) u * ((z-u) * (z-u) ^ n)) -
```
```  1039                          (f (Suc n) u * ((1 + of_nat n) * (z-u) ^ n))"
```
```  1040           by (simp only: fact_Suc of_nat_mult ac_simps) simp
```
```  1041         also have "... = f (Suc (Suc n)) u * ((z-u) * (z-u) ^ n)"
```
```  1042           by (simp add: algebra_simps)
```
```  1043         finally show ?thesis
```
```  1044         by (simp add: mult_left_cancel [where c = "(fact (Suc n))", THEN iffD1] del: fact_Suc)
```
```  1045       qed
```
```  1046       finally show ?case .
```
```  1047     qed
```
```  1048     then have "((\<lambda>v. (\<Sum>i\<le>n. f i v * (z - v)^i / (fact i)))
```
```  1049                 has_field_derivative f (Suc n) u * (z-u) ^ n / (fact n))
```
```  1050                (at u within s)"
```
```  1051       apply (intro derivative_eq_intros)
```
```  1052       apply (blast intro: assms \<open>u \<in> s\<close>)
```
```  1053       apply (rule refl)+
```
```  1054       apply (auto simp: field_simps)
```
```  1055       done
```
```  1056   } note sum_deriv = this
```
```  1057   { fix u
```
```  1058     assume u: "u \<in> closed_segment w z"
```
```  1059     then have us: "u \<in> s"
```
```  1060       by (metis wzs subsetD)
```
```  1061     have "norm (f (Suc n) u) * norm (z - u) ^ n \<le> norm (f (Suc n) u) * norm (u - z) ^ n"
```
```  1062       by (metis norm_minus_commute order_refl)
```
```  1063     also have "... \<le> norm (f (Suc n) u) * norm (z - w) ^ n"
```
```  1064       by (metis mult_left_mono norm_ge_zero power_mono segment_bound [OF u])
```
```  1065     also have "... \<le> B * norm (z - w) ^ n"
```
```  1066       by (metis norm_ge_zero zero_le_power mult_right_mono  B [OF us])
```
```  1067     finally have "norm (f (Suc n) u) * norm (z - u) ^ n \<le> B * norm (z - w) ^ n" .
```
```  1068   } note cmod_bound = this
```
```  1069   have "(\<Sum>i\<le>n. f i z * (z - z) ^ i / (fact i)) = (\<Sum>i\<le>n. (f i z / (fact i)) * 0 ^ i)"
```
```  1070     by simp
```
```  1071   also have "\<dots> = f 0 z / (fact 0)"
```
```  1072     by (subst sum_zero_power) simp
```
```  1073   finally have "norm (f 0 z - (\<Sum>i\<le>n. f i w * (z - w) ^ i / (fact i)))
```
```  1074                 \<le> norm ((\<Sum>i\<le>n. f i w * (z - w) ^ i / (fact i)) -
```
```  1075                         (\<Sum>i\<le>n. f i z * (z - z) ^ i / (fact i)))"
```
```  1076     by (simp add: norm_minus_commute)
```
```  1077   also have "... \<le> B * norm (z - w) ^ n / (fact n) * norm (w - z)"
```
```  1078     apply (rule field_differentiable_bound
```
```  1079       [where f' = "\<lambda>w. f (Suc n) w * (z - w)^n / (fact n)"
```
```  1080          and s = "closed_segment w z", OF convex_closed_segment])
```
```  1081     apply (auto simp: ends_in_segment DERIV_subset [OF sum_deriv wzs]
```
```  1082                   norm_divide norm_mult norm_power divide_le_cancel cmod_bound)
```
```  1083     done
```
```  1084   also have "...  \<le> B * norm (z - w) ^ Suc n / (fact n)"
```
```  1085     by (simp add: algebra_simps norm_minus_commute)
```
```  1086   finally show ?thesis .
```
```  1087 qed
```
```  1088
```
```  1089 lemma complex_taylor:
```
```  1090   assumes s: "convex s"
```
```  1091       and f: "\<And>i x. x \<in> s \<Longrightarrow> i \<le> n \<Longrightarrow> (f i has_field_derivative f (Suc i) x) (at x within s)"
```
```  1092       and B: "\<And>x. x \<in> s \<Longrightarrow> cmod (f (Suc n) x) \<le> B"
```
```  1093       and w: "w \<in> s"
```
```  1094       and z: "z \<in> s"
```
```  1095     shows "cmod(f 0 z - (\<Sum>i\<le>n. f i w * (z-w) ^ i / (fact i)))
```
```  1096           \<le> B * cmod(z - w)^(Suc n) / fact n"
```
```  1097   using assms by (rule field_taylor)
```
```  1098
```
```  1099
```
```  1100 text\<open>Something more like the traditional MVT for real components\<close>
```
```  1101
```
```  1102 lemma complex_mvt_line:
```
```  1103   assumes "\<And>u. u \<in> closed_segment w z \<Longrightarrow> (f has_field_derivative f'(u)) (at u)"
```
```  1104     shows "\<exists>u. u \<in> closed_segment w z \<and> Re(f z) - Re(f w) = Re(f'(u) * (z - w))"
```
```  1105 proof -
```
```  1106   have twz: "\<And>t. (1 - t) *\<^sub>R w + t *\<^sub>R z = w + t *\<^sub>R (z - w)"
```
```  1107     by (simp add: real_vector.scale_left_diff_distrib real_vector.scale_right_diff_distrib)
```
```  1108   note assms[unfolded has_field_derivative_def, derivative_intros]
```
```  1109   show ?thesis
```
```  1110     apply (cut_tac mvt_simple
```
```  1111                      [of 0 1 "Re o f o (\<lambda>t. (1 - t) *\<^sub>R w +  t *\<^sub>R z)"
```
```  1112                       "\<lambda>u. Re o (\<lambda>h. f'((1 - u) *\<^sub>R w + u *\<^sub>R z) * h) o (\<lambda>t. t *\<^sub>R (z - w))"])
```
```  1113     apply auto
```
```  1114     apply (rule_tac x="(1 - x) *\<^sub>R w + x *\<^sub>R z" in exI)
```
```  1115     apply (auto simp: closed_segment_def twz) []
```
```  1116     apply (intro derivative_eq_intros has_derivative_at_within, simp_all)
```
```  1117     apply (simp add: fun_eq_iff real_vector.scale_right_diff_distrib)
```
```  1118     apply (force simp: twz closed_segment_def)
```
```  1119     done
```
```  1120 qed
```
```  1121
```
```  1122 lemma complex_taylor_mvt:
```
```  1123   assumes "\<And>i x. \<lbrakk>x \<in> closed_segment w z; i \<le> n\<rbrakk> \<Longrightarrow> ((f i) has_field_derivative f (Suc i) x) (at x)"
```
```  1124     shows "\<exists>u. u \<in> closed_segment w z \<and>
```
```  1125             Re (f 0 z) =
```
```  1126             Re ((\<Sum>i = 0..n. f i w * (z - w) ^ i / (fact i)) +
```
```  1127                 (f (Suc n) u * (z-u)^n / (fact n)) * (z - w))"
```
```  1128 proof -
```
```  1129   { fix u
```
```  1130     assume u: "u \<in> closed_segment w z"
```
```  1131     have "(\<Sum>i = 0..n.
```
```  1132                (f (Suc i) u * (z-u) ^ i - of_nat i * (f i u * (z-u) ^ (i - Suc 0))) /
```
```  1133                (fact i)) =
```
```  1134           f (Suc 0) u -
```
```  1135              (f (Suc (Suc n)) u * ((z-u) ^ Suc n) - (of_nat (Suc n)) * (z-u) ^ n * f (Suc n) u) /
```
```  1136              (fact (Suc n)) +
```
```  1137              (\<Sum>i = 0..n.
```
```  1138                  (f (Suc (Suc i)) u * ((z-u) ^ Suc i) - of_nat (Suc i) * (f (Suc i) u * (z-u) ^ i)) /
```
```  1139                  (fact (Suc i)))"
```
```  1140        by (subst sum_Suc_reindex) simp
```
```  1141     also have "... = f (Suc 0) u -
```
```  1142              (f (Suc (Suc n)) u * ((z-u) ^ Suc n) - (of_nat (Suc n)) * (z-u) ^ n * f (Suc n) u) /
```
```  1143              (fact (Suc n)) +
```
```  1144              (\<Sum>i = 0..n.
```
```  1145                  f (Suc (Suc i)) u * ((z-u) ^ Suc i) / (fact (Suc i))  -
```
```  1146                  f (Suc i) u * (z-u) ^ i / (fact i))"
```
```  1147       by (simp only: diff_divide_distrib fact_cancel ac_simps)
```
```  1148     also have "... = f (Suc 0) u -
```
```  1149              (f (Suc (Suc n)) u * (z-u) ^ Suc n - of_nat (Suc n) * (z-u) ^ n * f (Suc n) u) /
```
```  1150              (fact (Suc n)) +
```
```  1151              f (Suc (Suc n)) u * (z-u) ^ Suc n / (fact (Suc n)) - f (Suc 0) u"
```
```  1152       by (subst sum_Suc_diff) auto
```
```  1153     also have "... = f (Suc n) u * (z-u) ^ n / (fact n)"
```
```  1154       by (simp only: algebra_simps diff_divide_distrib fact_cancel)
```
```  1155     finally have "(\<Sum>i = 0..n. (f (Suc i) u * (z - u) ^ i
```
```  1156                              - of_nat i * (f i u * (z-u) ^ (i - Suc 0))) / (fact i)) =
```
```  1157                   f (Suc n) u * (z - u) ^ n / (fact n)" .
```
```  1158     then have "((\<lambda>u. \<Sum>i = 0..n. f i u * (z - u) ^ i / (fact i)) has_field_derivative
```
```  1159                 f (Suc n) u * (z - u) ^ n / (fact n))  (at u)"
```
```  1160       apply (intro derivative_eq_intros)+
```
```  1161       apply (force intro: u assms)
```
```  1162       apply (rule refl)+
```
```  1163       apply (auto simp: ac_simps)
```
```  1164       done
```
```  1165   }
```
```  1166   then show ?thesis
```
```  1167     apply (cut_tac complex_mvt_line [of w z "\<lambda>u. \<Sum>i = 0..n. f i u * (z-u) ^ i / (fact i)"
```
```  1168                "\<lambda>u. (f (Suc n) u * (z-u)^n / (fact n))"])
```
```  1169     apply (auto simp add: intro: open_closed_segment)
```
```  1170     done
```
```  1171 qed
```
```  1172
```
```  1173
```
```  1174 subsection \<open>Polynomal function extremal theorem, from HOL Light\<close>
```
```  1175
```
```  1176 lemma polyfun_extremal_lemma: (*COMPLEX_POLYFUN_EXTREMAL_LEMMA in HOL Light*)
```
```  1177     fixes c :: "nat \<Rightarrow> 'a::real_normed_div_algebra"
```
```  1178   assumes "0 < e"
```
```  1179     shows "\<exists>M. \<forall>z. M \<le> norm(z) \<longrightarrow> norm (\<Sum>i\<le>n. c(i) * z^i) \<le> e * norm(z) ^ (Suc n)"
```
```  1180 proof (induct n)
```
```  1181   case 0 with assms
```
```  1182   show ?case
```
```  1183     apply (rule_tac x="norm (c 0) / e" in exI)
```
```  1184     apply (auto simp: field_simps)
```
```  1185     done
```
```  1186 next
```
```  1187   case (Suc n)
```
```  1188   obtain M where M: "\<And>z. M \<le> norm z \<Longrightarrow> norm (\<Sum>i\<le>n. c i * z^i) \<le> e * norm z ^ Suc n"
```
```  1189     using Suc assms by blast
```
```  1190   show ?case
```
```  1191   proof (rule exI [where x= "max M (1 + norm(c(Suc n)) / e)"], clarsimp simp del: power_Suc)
```
```  1192     fix z::'a
```
```  1193     assume z1: "M \<le> norm z" and "1 + norm (c (Suc n)) / e \<le> norm z"
```
```  1194     then have z2: "e + norm (c (Suc n)) \<le> e * norm z"
```
```  1195       using assms by (simp add: field_simps)
```
```  1196     have "norm (\<Sum>i\<le>n. c i * z^i) \<le> e * norm z ^ Suc n"
```
```  1197       using M [OF z1] by simp
```
```  1198     then have "norm (\<Sum>i\<le>n. c i * z^i) + norm (c (Suc n) * z ^ Suc n) \<le> e * norm z ^ Suc n + norm (c (Suc n) * z ^ Suc n)"
```
```  1199       by simp
```
```  1200     then have "norm ((\<Sum>i\<le>n. c i * z^i) + c (Suc n) * z ^ Suc n) \<le> e * norm z ^ Suc n + norm (c (Suc n) * z ^ Suc n)"
```
```  1201       by (blast intro: norm_triangle_le elim: )
```
```  1202     also have "... \<le> (e + norm (c (Suc n))) * norm z ^ Suc n"
```
```  1203       by (simp add: norm_power norm_mult algebra_simps)
```
```  1204     also have "... \<le> (e * norm z) * norm z ^ Suc n"
```
```  1205       by (metis z2 mult.commute mult_left_mono norm_ge_zero norm_power)
```
```  1206     finally show "norm ((\<Sum>i\<le>n. c i * z^i) + c (Suc n) * z ^ Suc n) \<le> e * norm z ^ Suc (Suc n)"
```
```  1207       by simp
```
```  1208   qed
```
```  1209 qed
```
```  1210
```
```  1211 lemma polyfun_extremal: (*COMPLEX_POLYFUN_EXTREMAL in HOL Light*)
```
```  1212     fixes c :: "nat \<Rightarrow> 'a::real_normed_div_algebra"
```
```  1213   assumes k: "c k \<noteq> 0" "1\<le>k" and kn: "k\<le>n"
```
```  1214     shows "eventually (\<lambda>z. norm (\<Sum>i\<le>n. c(i) * z^i) \<ge> B) at_infinity"
```
```  1215 using kn
```
```  1216 proof (induction n)
```
```  1217   case 0
```
```  1218   then show ?case
```
```  1219     using k  by simp
```
```  1220 next
```
```  1221   case (Suc m)
```
```  1222   let ?even = ?case
```
```  1223   show ?even
```
```  1224   proof (cases "c (Suc m) = 0")
```
```  1225     case True
```
```  1226     then show ?even using Suc k
```
```  1227       by auto (metis antisym_conv less_eq_Suc_le not_le)
```
```  1228   next
```
```  1229     case False
```
```  1230     then obtain M where M:
```
```  1231           "\<And>z. M \<le> norm z \<Longrightarrow> norm (\<Sum>i\<le>m. c i * z^i) \<le> norm (c (Suc m)) / 2 * norm z ^ Suc m"
```
```  1232       using polyfun_extremal_lemma [of "norm(c (Suc m)) / 2" c m] Suc
```
```  1233       by auto
```
```  1234     have "\<exists>b. \<forall>z. b \<le> norm z \<longrightarrow> B \<le> norm (\<Sum>i\<le>Suc m. c i * z^i)"
```
```  1235     proof (rule exI [where x="max M (max 1 (\<bar>B\<bar> / (norm(c (Suc m)) / 2)))"], clarsimp simp del: power_Suc)
```
```  1236       fix z::'a
```
```  1237       assume z1: "M \<le> norm z" "1 \<le> norm z"
```
```  1238          and "\<bar>B\<bar> * 2 / norm (c (Suc m)) \<le> norm z"
```
```  1239       then have z2: "\<bar>B\<bar> \<le> norm (c (Suc m)) * norm z / 2"
```
```  1240         using False by (simp add: field_simps)
```
```  1241       have nz: "norm z \<le> norm z ^ Suc m"
```
```  1242         by (metis \<open>1 \<le> norm z\<close> One_nat_def less_eq_Suc_le power_increasing power_one_right zero_less_Suc)
```
```  1243       have *: "\<And>y x. norm (c (Suc m)) * norm z / 2 \<le> norm y - norm x \<Longrightarrow> B \<le> norm (x + y)"
```
```  1244         by (metis abs_le_iff add.commute norm_diff_ineq order_trans z2)
```
```  1245       have "norm z * norm (c (Suc m)) + 2 * norm (\<Sum>i\<le>m. c i * z^i)
```
```  1246             \<le> norm (c (Suc m)) * norm z + norm (c (Suc m)) * norm z ^ Suc m"
```
```  1247         using M [of z] Suc z1  by auto
```
```  1248       also have "... \<le> 2 * (norm (c (Suc m)) * norm z ^ Suc m)"
```
```  1249         using nz by (simp add: mult_mono del: power_Suc)
```
```  1250       finally show "B \<le> norm ((\<Sum>i\<le>m. c i * z^i) + c (Suc m) * z ^ Suc m)"
```
```  1251         using Suc.IH
```
```  1252         apply (auto simp: eventually_at_infinity)
```
```  1253         apply (rule *)
```
```  1254         apply (simp add: field_simps norm_mult norm_power)
```
```  1255         done
```
```  1256     qed
```
```  1257     then show ?even
```
```  1258       by (simp add: eventually_at_infinity)
```
```  1259   qed
```
```  1260 qed
```
```  1261
```
```  1262 end
```