src/HOL/Analysis/Complex_Analysis_Basics.thy
 author nipkow Mon Oct 17 11:46:22 2016 +0200 (2016-10-17) changeset 64267 b9a1486e79be parent 63941 f353674c2528 child 64394 141e1ed8d5a0 permissions -rw-r--r--
setsum -> sum
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 *)
5 section \<open>Complex Analysis Basics\<close>
7 theory Complex_Analysis_Basics
8 imports Equivalence_Lebesgue_Henstock_Integration "~~/src/HOL/Library/Nonpos_Ints"
9 begin
12 subsection\<open>General lemmas\<close>
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)
17 lemma has_derivative_mult_right:
18   fixes c:: "'a :: real_normed_algebra"
19   shows "((op * c) has_derivative (op * c)) F"
20 by (rule has_derivative_mult_right [OF has_derivative_id])
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] .
26 lemma has_vector_derivative_real_complex:
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 "op * f'"]
29   by (simp add: scaleR_conv_of_real ac_simps has_vector_derivative_def has_field_derivative_def)
31 lemma fact_cancel:
32   fixes c :: "'a::real_field"
33   shows "of_nat (Suc n) * c / (fact (Suc n)) = c / (fact n)"
34   by (simp add: of_nat_mult del: of_nat_Suc times_nat.simps)
36 lemma bilinear_times:
37   fixes c::"'a::real_algebra" shows "bilinear (\<lambda>x y::'a. x*y)"
38   by (auto simp: bilinear_def distrib_left distrib_right intro!: linearI)
40 lemma linear_cnj: "linear cnj"
41   using bounded_linear.linear[OF bounded_linear_cnj] .
43 lemma tendsto_Re_upper:
44   assumes "~ (trivial_limit F)"
45           "(f \<longlongrightarrow> l) F"
46           "eventually (\<lambda>x. Re(f x) \<le> b) F"
47     shows  "Re(l) \<le> b"
48   by (metis assms tendsto_le [OF _ tendsto_const]  tendsto_Re)
50 lemma tendsto_Re_lower:
51   assumes "~ (trivial_limit F)"
52           "(f \<longlongrightarrow> l) F"
53           "eventually (\<lambda>x. b \<le> Re(f x)) F"
54     shows  "b \<le> Re(l)"
55   by (metis assms tendsto_le [OF _ _ tendsto_const]  tendsto_Re)
57 lemma tendsto_Im_upper:
58   assumes "~ (trivial_limit F)"
59           "(f \<longlongrightarrow> l) F"
60           "eventually (\<lambda>x. Im(f x) \<le> b) F"
61     shows  "Im(l) \<le> b"
62   by (metis assms tendsto_le [OF _ tendsto_const]  tendsto_Im)
64 lemma tendsto_Im_lower:
65   assumes "~ (trivial_limit F)"
66           "(f \<longlongrightarrow> l) F"
67           "eventually (\<lambda>x. b \<le> Im(f x)) F"
68     shows  "b \<le> Im(l)"
69   by (metis assms tendsto_le [OF _ _ tendsto_const]  tendsto_Im)
71 lemma lambda_zero: "(\<lambda>h::'a::mult_zero. 0) = op * 0"
72   by auto
74 lemma lambda_one: "(\<lambda>x::'a::monoid_mult. x) = op * 1"
75   by auto
77 lemma continuous_mult_left:
78   fixes c::"'a::real_normed_algebra"
79   shows "continuous F f \<Longrightarrow> continuous F (\<lambda>x. c * f x)"
80 by (rule continuous_mult [OF continuous_const])
82 lemma continuous_mult_right:
83   fixes c::"'a::real_normed_algebra"
84   shows "continuous F f \<Longrightarrow> continuous F (\<lambda>x. f x * c)"
85 by (rule continuous_mult [OF _ continuous_const])
87 lemma continuous_on_mult_left:
88   fixes c::"'a::real_normed_algebra"
89   shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. c * f x)"
90 by (rule continuous_on_mult [OF continuous_on_const])
92 lemma continuous_on_mult_right:
93   fixes c::"'a::real_normed_algebra"
94   shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. f x * c)"
95 by (rule continuous_on_mult [OF _ continuous_on_const])
97 lemma uniformly_continuous_on_cmul_right [continuous_intros]:
98   fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_algebra"
99   shows "uniformly_continuous_on s f \<Longrightarrow> uniformly_continuous_on s (\<lambda>x. f x * c)"
100   using bounded_linear.uniformly_continuous_on[OF bounded_linear_mult_left] .
102 lemma uniformly_continuous_on_cmul_left[continuous_intros]:
103   fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_algebra"
104   assumes "uniformly_continuous_on s f"
105     shows "uniformly_continuous_on s (\<lambda>x. c * f x)"
106 by (metis assms bounded_linear.uniformly_continuous_on bounded_linear_mult_right)
108 lemma continuous_within_norm_id [continuous_intros]: "continuous (at x within S) norm"
109   by (rule continuous_norm [OF continuous_ident])
111 lemma continuous_on_norm_id [continuous_intros]: "continuous_on S norm"
112   by (intro continuous_on_id continuous_on_norm)
114 subsection\<open>DERIV stuff\<close>
116 lemma DERIV_zero_connected_constant:
117   fixes f :: "'a::{real_normed_field,euclidean_space} \<Rightarrow> 'a"
118   assumes "connected s"
119       and "open s"
120       and "finite k"
121       and "continuous_on s f"
122       and "\<forall>x\<in>(s - k). DERIV f x :> 0"
123     obtains c where "\<And>x. x \<in> s \<Longrightarrow> f(x) = c"
124 using has_derivative_zero_connected_constant [OF assms(1-4)] assms
125 by (metis DERIV_const has_derivative_const Diff_iff at_within_open frechet_derivative_at has_field_derivative_def)
127 lemma DERIV_zero_constant:
128   fixes f :: "'a::{real_normed_field, real_inner} \<Rightarrow> 'a"
129   shows    "\<lbrakk>convex s;
130              \<And>x. x\<in>s \<Longrightarrow> (f has_field_derivative 0) (at x within s)\<rbrakk>
131              \<Longrightarrow> \<exists>c. \<forall>x \<in> s. f(x) = c"
132   by (auto simp: has_field_derivative_def lambda_zero intro: has_derivative_zero_constant)
134 lemma DERIV_zero_unique:
135   fixes f :: "'a::{real_normed_field, real_inner} \<Rightarrow> 'a"
136   assumes "convex s"
137       and d0: "\<And>x. x\<in>s \<Longrightarrow> (f has_field_derivative 0) (at x within s)"
138       and "a \<in> s"
139       and "x \<in> s"
140     shows "f x = f a"
141   by (rule has_derivative_zero_unique [OF assms(1) _ assms(4,3)])
142      (metis d0 has_field_derivative_imp_has_derivative lambda_zero)
144 lemma DERIV_zero_connected_unique:
145   fixes f :: "'a::{real_normed_field, real_inner} \<Rightarrow> 'a"
146   assumes "connected s"
147       and "open s"
148       and d0: "\<And>x. x\<in>s \<Longrightarrow> DERIV f x :> 0"
149       and "a \<in> s"
150       and "x \<in> s"
151     shows "f x = f a"
152     by (rule has_derivative_zero_unique_connected [OF assms(2,1) _ assms(5,4)])
153        (metis has_field_derivative_def lambda_zero d0)
155 lemma DERIV_transform_within:
156   assumes "(f has_field_derivative f') (at a within s)"
157       and "0 < d" "a \<in> s"
158       and "\<And>x. x\<in>s \<Longrightarrow> dist x a < d \<Longrightarrow> f x = g x"
159     shows "(g has_field_derivative f') (at a within s)"
160   using assms unfolding has_field_derivative_def
161   by (blast intro: has_derivative_transform_within)
163 lemma DERIV_transform_within_open:
164   assumes "DERIV f a :> f'"
165       and "open s" "a \<in> s"
166       and "\<And>x. x\<in>s \<Longrightarrow> f x = g x"
167     shows "DERIV g a :> f'"
168   using assms unfolding has_field_derivative_def
169 by (metis has_derivative_transform_within_open)
171 lemma DERIV_transform_at:
172   assumes "DERIV f a :> f'"
173       and "0 < d"
174       and "\<And>x. dist x a < d \<Longrightarrow> f x = g x"
175     shows "DERIV g a :> f'"
176   by (blast intro: assms DERIV_transform_within)
178 (*generalising DERIV_isconst_all, which requires type real (using the ordering)*)
179 lemma DERIV_zero_UNIV_unique:
180   fixes f :: "'a::{real_normed_field, real_inner} \<Rightarrow> 'a"
181   shows "(\<And>x. DERIV f x :> 0) \<Longrightarrow> f x = f a"
182 by (metis DERIV_zero_unique UNIV_I convex_UNIV)
184 subsection \<open>Some limit theorems about real part of real series etc.\<close>
186 (*MOVE? But not to Finite_Cartesian_Product*)
187 lemma sums_vec_nth :
188   assumes "f sums a"
189   shows "(\<lambda>x. f x \$ i) sums a \$ i"
190 using assms unfolding sums_def
191 by (auto dest: tendsto_vec_nth [where i=i])
193 lemma summable_vec_nth :
194   assumes "summable f"
195   shows "summable (\<lambda>x. f x \$ i)"
196 using assms unfolding summable_def
197 by (blast intro: sums_vec_nth)
199 subsection \<open>Complex number lemmas\<close>
201 lemma
202   shows open_halfspace_Re_lt: "open {z. Re(z) < b}"
203     and open_halfspace_Re_gt: "open {z. Re(z) > b}"
204     and closed_halfspace_Re_ge: "closed {z. Re(z) \<ge> b}"
205     and closed_halfspace_Re_le: "closed {z. Re(z) \<le> b}"
206     and closed_halfspace_Re_eq: "closed {z. Re(z) = b}"
207     and open_halfspace_Im_lt: "open {z. Im(z) < b}"
208     and open_halfspace_Im_gt: "open {z. Im(z) > b}"
209     and closed_halfspace_Im_ge: "closed {z. Im(z) \<ge> b}"
210     and closed_halfspace_Im_le: "closed {z. Im(z) \<le> b}"
211     and closed_halfspace_Im_eq: "closed {z. Im(z) = b}"
212   by (intro open_Collect_less closed_Collect_le closed_Collect_eq continuous_on_Re
213             continuous_on_Im continuous_on_id continuous_on_const)+
215 lemma closed_complex_Reals: "closed (\<real> :: complex set)"
216 proof -
217   have "(\<real> :: complex set) = {z. Im z = 0}"
218     by (auto simp: complex_is_Real_iff)
219   then show ?thesis
220     by (metis closed_halfspace_Im_eq)
221 qed
223 lemma closed_Real_halfspace_Re_le: "closed (\<real> \<inter> {w. Re w \<le> x})"
224   by (simp add: closed_Int closed_complex_Reals closed_halfspace_Re_le)
226 corollary closed_nonpos_Reals_complex [simp]: "closed (\<real>\<^sub>\<le>\<^sub>0 :: complex set)"
227 proof -
228   have "\<real>\<^sub>\<le>\<^sub>0 = \<real> \<inter> {z. Re(z) \<le> 0}"
229     using complex_nonpos_Reals_iff complex_is_Real_iff by auto
230   then show ?thesis
231     by (metis closed_Real_halfspace_Re_le)
232 qed
234 lemma closed_Real_halfspace_Re_ge: "closed (\<real> \<inter> {w. x \<le> Re(w)})"
235   using closed_halfspace_Re_ge
236   by (simp add: closed_Int closed_complex_Reals)
238 corollary closed_nonneg_Reals_complex [simp]: "closed (\<real>\<^sub>\<ge>\<^sub>0 :: complex set)"
239 proof -
240   have "\<real>\<^sub>\<ge>\<^sub>0 = \<real> \<inter> {z. Re(z) \<ge> 0}"
241     using complex_nonneg_Reals_iff complex_is_Real_iff by auto
242   then show ?thesis
243     by (metis closed_Real_halfspace_Re_ge)
244 qed
246 lemma closed_real_abs_le: "closed {w \<in> \<real>. \<bar>Re w\<bar> \<le> r}"
247 proof -
248   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})"
249     by auto
250   then show "closed {w \<in> \<real>. \<bar>Re w\<bar> \<le> r}"
251     by (simp add: closed_Int closed_Real_halfspace_Re_ge closed_Real_halfspace_Re_le)
252 qed
254 lemma real_lim:
255   fixes l::complex
256   assumes "(f \<longlongrightarrow> l) F" and "~(trivial_limit F)" and "eventually P F" and "\<And>a. P a \<Longrightarrow> f a \<in> \<real>"
257   shows  "l \<in> \<real>"
258 proof (rule Lim_in_closed_set[OF closed_complex_Reals _ assms(2,1)])
259   show "eventually (\<lambda>x. f x \<in> \<real>) F"
260     using assms(3, 4) by (auto intro: eventually_mono)
261 qed
263 lemma real_lim_sequentially:
264   fixes l::complex
265   shows "(f \<longlongrightarrow> l) sequentially \<Longrightarrow> (\<exists>N. \<forall>n\<ge>N. f n \<in> \<real>) \<Longrightarrow> l \<in> \<real>"
266 by (rule real_lim [where F=sequentially]) (auto simp: eventually_sequentially)
268 lemma real_series:
269   fixes l::complex
270   shows "f sums l \<Longrightarrow> (\<And>n. f n \<in> \<real>) \<Longrightarrow> l \<in> \<real>"
271 unfolding sums_def
272 by (metis real_lim_sequentially sum_in_Reals)
274 lemma Lim_null_comparison_Re:
275   assumes "eventually (\<lambda>x. norm(f x) \<le> Re(g x)) F" "(g \<longlongrightarrow> 0) F" shows "(f \<longlongrightarrow> 0) F"
276   by (rule Lim_null_comparison[OF assms(1)] tendsto_eq_intros assms(2))+ simp
278 subsection\<open>Holomorphic functions\<close>
280 definition field_differentiable :: "['a \<Rightarrow> 'a::real_normed_field, 'a filter] \<Rightarrow> bool"
281            (infixr "(field'_differentiable)" 50)
282   where "f field_differentiable F \<equiv> \<exists>f'. (f has_field_derivative f') F"
284 lemma field_differentiable_derivI:
285     "f field_differentiable (at x) \<Longrightarrow> (f has_field_derivative deriv f x) (at x)"
286 by (simp add: field_differentiable_def DERIV_deriv_iff_has_field_derivative)
288 lemma field_differentiable_imp_continuous_at:
289     "f field_differentiable (at x within s) \<Longrightarrow> continuous (at x within s) f"
290   by (metis DERIV_continuous field_differentiable_def)
292 lemma field_differentiable_within_subset:
293     "\<lbrakk>f field_differentiable (at x within s); t \<subseteq> s\<rbrakk>
294      \<Longrightarrow> f field_differentiable (at x within t)"
295   by (metis DERIV_subset field_differentiable_def)
297 lemma field_differentiable_at_within:
298     "\<lbrakk>f field_differentiable (at x)\<rbrakk>
299      \<Longrightarrow> f field_differentiable (at x within s)"
300   unfolding field_differentiable_def
301   by (metis DERIV_subset top_greatest)
303 lemma field_differentiable_linear [simp,derivative_intros]: "(op * c) field_differentiable F"
304 proof -
305   show ?thesis
306     unfolding field_differentiable_def has_field_derivative_def mult_commute_abs
307     by (force intro: has_derivative_mult_right)
308 qed
310 lemma field_differentiable_const [simp,derivative_intros]: "(\<lambda>z. c) field_differentiable F"
311   unfolding field_differentiable_def has_field_derivative_def
312   by (rule exI [where x=0])
313      (metis has_derivative_const lambda_zero)
315 lemma field_differentiable_ident [simp,derivative_intros]: "(\<lambda>z. z) field_differentiable F"
316   unfolding field_differentiable_def has_field_derivative_def
317   by (rule exI [where x=1])
320 lemma field_differentiable_id [simp,derivative_intros]: "id field_differentiable F"
321   unfolding id_def by (rule field_differentiable_ident)
323 lemma field_differentiable_minus [derivative_intros]:
324   "f field_differentiable F \<Longrightarrow> (\<lambda>z. - (f z)) field_differentiable F"
325   unfolding field_differentiable_def
326   by (metis field_differentiable_minus)
329   assumes "f field_differentiable F" "g field_differentiable F"
330     shows "(\<lambda>z. f z + g z) field_differentiable F"
331   using assms unfolding field_differentiable_def
335      "op + c field_differentiable F"
338 lemma field_differentiable_sum [derivative_intros]:
339   "(\<And>i. i \<in> I \<Longrightarrow> (f i) field_differentiable F) \<Longrightarrow> (\<lambda>z. \<Sum>i\<in>I. f i z) field_differentiable F"
340   by (induct I rule: infinite_finite_induct)
343 lemma field_differentiable_diff [derivative_intros]:
344   assumes "f field_differentiable F" "g field_differentiable F"
345     shows "(\<lambda>z. f z - g z) field_differentiable F"
346   using assms unfolding field_differentiable_def
347   by (metis field_differentiable_diff)
349 lemma field_differentiable_inverse [derivative_intros]:
350   assumes "f field_differentiable (at a within s)" "f a \<noteq> 0"
351   shows "(\<lambda>z. inverse (f z)) field_differentiable (at a within s)"
352   using assms unfolding field_differentiable_def
353   by (metis DERIV_inverse_fun)
355 lemma field_differentiable_mult [derivative_intros]:
356   assumes "f field_differentiable (at a within s)"
357           "g field_differentiable (at a within s)"
358     shows "(\<lambda>z. f z * g z) field_differentiable (at a within s)"
359   using assms unfolding field_differentiable_def
360   by (metis DERIV_mult [of f _ a s g])
362 lemma field_differentiable_divide [derivative_intros]:
363   assumes "f field_differentiable (at a within s)"
364           "g field_differentiable (at a within s)"
365           "g a \<noteq> 0"
366     shows "(\<lambda>z. f z / g z) field_differentiable (at a within s)"
367   using assms unfolding field_differentiable_def
368   by (metis DERIV_divide [of f _ a s g])
370 lemma field_differentiable_power [derivative_intros]:
371   assumes "f field_differentiable (at a within s)"
372     shows "(\<lambda>z. f z ^ n) field_differentiable (at a within s)"
373   using assms unfolding field_differentiable_def
374   by (metis DERIV_power)
376 lemma field_differentiable_transform_within:
377   "0 < d \<Longrightarrow>
378         x \<in> s \<Longrightarrow>
379         (\<And>x'. x' \<in> s \<Longrightarrow> dist x' x < d \<Longrightarrow> f x' = g x') \<Longrightarrow>
380         f field_differentiable (at x within s)
381         \<Longrightarrow> g field_differentiable (at x within s)"
382   unfolding field_differentiable_def has_field_derivative_def
383   by (blast intro: has_derivative_transform_within)
385 lemma field_differentiable_compose_within:
386   assumes "f field_differentiable (at a within s)"
387           "g field_differentiable (at (f a) within f`s)"
388     shows "(g o f) field_differentiable (at a within s)"
389   using assms unfolding field_differentiable_def
390   by (metis DERIV_image_chain)
392 lemma field_differentiable_compose:
393   "f field_differentiable at z \<Longrightarrow> g field_differentiable at (f z)
394           \<Longrightarrow> (g o f) field_differentiable at z"
395 by (metis field_differentiable_at_within field_differentiable_compose_within)
397 lemma field_differentiable_within_open:
398      "\<lbrakk>a \<in> s; open s\<rbrakk> \<Longrightarrow> f field_differentiable at a within s \<longleftrightarrow>
399                           f field_differentiable at a"
400   unfolding field_differentiable_def
401   by (metis at_within_open)
403 subsection\<open>Caratheodory characterization\<close>
405 lemma field_differentiable_caratheodory_at:
406   "f field_differentiable (at z) \<longleftrightarrow>
407          (\<exists>g. (\<forall>w. f(w) - f(z) = g(w) * (w - z)) \<and> continuous (at z) g)"
408   using CARAT_DERIV [of f]
409   by (simp add: field_differentiable_def has_field_derivative_def)
411 lemma field_differentiable_caratheodory_within:
412   "f field_differentiable (at z within s) \<longleftrightarrow>
413          (\<exists>g. (\<forall>w. f(w) - f(z) = g(w) * (w - z)) \<and> continuous (at z within s) g)"
414   using DERIV_caratheodory_within [of f]
415   by (simp add: field_differentiable_def has_field_derivative_def)
417 subsection\<open>Holomorphic\<close>
419 definition holomorphic_on :: "[complex \<Rightarrow> complex, complex set] \<Rightarrow> bool"
420            (infixl "(holomorphic'_on)" 50)
421   where "f holomorphic_on s \<equiv> \<forall>x\<in>s. f field_differentiable (at x within s)"
423 named_theorems holomorphic_intros "structural introduction rules for holomorphic_on"
425 lemma holomorphic_onI [intro?]: "(\<And>x. x \<in> s \<Longrightarrow> f field_differentiable (at x within s)) \<Longrightarrow> f holomorphic_on s"
428 lemma holomorphic_onD [dest?]: "\<lbrakk>f holomorphic_on s; x \<in> s\<rbrakk> \<Longrightarrow> f field_differentiable (at x within s)"
431 lemma holomorphic_on_imp_differentiable_at:
432    "\<lbrakk>f holomorphic_on s; open s; x \<in> s\<rbrakk> \<Longrightarrow> f field_differentiable (at x)"
433 using at_within_open holomorphic_on_def by fastforce
435 lemma holomorphic_on_empty [holomorphic_intros]: "f holomorphic_on {}"
438 lemma holomorphic_on_open:
439     "open s \<Longrightarrow> f holomorphic_on s \<longleftrightarrow> (\<forall>x \<in> s. \<exists>f'. DERIV f x :> f')"
440   by (auto simp: holomorphic_on_def field_differentiable_def has_field_derivative_def at_within_open [of _ s])
442 lemma holomorphic_on_imp_continuous_on:
443     "f holomorphic_on s \<Longrightarrow> continuous_on s f"
444   by (metis field_differentiable_imp_continuous_at continuous_on_eq_continuous_within holomorphic_on_def)
446 lemma holomorphic_on_subset [elim]:
447     "f holomorphic_on s \<Longrightarrow> t \<subseteq> s \<Longrightarrow> f holomorphic_on t"
448   unfolding holomorphic_on_def
449   by (metis field_differentiable_within_subset subsetD)
451 lemma holomorphic_transform: "\<lbrakk>f holomorphic_on s; \<And>x. x \<in> s \<Longrightarrow> f x = g x\<rbrakk> \<Longrightarrow> g holomorphic_on s"
452   by (metis field_differentiable_transform_within linordered_field_no_ub holomorphic_on_def)
454 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"
455   by (metis holomorphic_transform)
457 lemma holomorphic_on_linear [simp, holomorphic_intros]: "(op * c) holomorphic_on s"
458   unfolding holomorphic_on_def by (metis field_differentiable_linear)
460 lemma holomorphic_on_const [simp, holomorphic_intros]: "(\<lambda>z. c) holomorphic_on s"
461   unfolding holomorphic_on_def by (metis field_differentiable_const)
463 lemma holomorphic_on_ident [simp, holomorphic_intros]: "(\<lambda>x. x) holomorphic_on s"
464   unfolding holomorphic_on_def by (metis field_differentiable_ident)
466 lemma holomorphic_on_id [simp, holomorphic_intros]: "id holomorphic_on s"
467   unfolding id_def by (rule holomorphic_on_ident)
469 lemma holomorphic_on_compose:
470   "f holomorphic_on s \<Longrightarrow> g holomorphic_on (f ` s) \<Longrightarrow> (g o f) holomorphic_on s"
471   using field_differentiable_compose_within[of f _ s g]
472   by (auto simp: holomorphic_on_def)
474 lemma holomorphic_on_compose_gen:
475   "f holomorphic_on s \<Longrightarrow> g holomorphic_on t \<Longrightarrow> f ` s \<subseteq> t \<Longrightarrow> (g o f) holomorphic_on s"
476   by (metis holomorphic_on_compose holomorphic_on_subset)
478 lemma holomorphic_on_minus [holomorphic_intros]: "f holomorphic_on s \<Longrightarrow> (\<lambda>z. -(f z)) holomorphic_on s"
479   by (metis field_differentiable_minus holomorphic_on_def)
482   "\<lbrakk>f holomorphic_on s; g holomorphic_on s\<rbrakk> \<Longrightarrow> (\<lambda>z. f z + g z) holomorphic_on s"
483   unfolding holomorphic_on_def by (metis field_differentiable_add)
485 lemma holomorphic_on_diff [holomorphic_intros]:
486   "\<lbrakk>f holomorphic_on s; g holomorphic_on s\<rbrakk> \<Longrightarrow> (\<lambda>z. f z - g z) holomorphic_on s"
487   unfolding holomorphic_on_def by (metis field_differentiable_diff)
489 lemma holomorphic_on_mult [holomorphic_intros]:
490   "\<lbrakk>f holomorphic_on s; g holomorphic_on s\<rbrakk> \<Longrightarrow> (\<lambda>z. f z * g z) holomorphic_on s"
491   unfolding holomorphic_on_def by (metis field_differentiable_mult)
493 lemma holomorphic_on_inverse [holomorphic_intros]:
494   "\<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"
495   unfolding holomorphic_on_def by (metis field_differentiable_inverse)
497 lemma holomorphic_on_divide [holomorphic_intros]:
498   "\<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"
499   unfolding holomorphic_on_def by (metis field_differentiable_divide)
501 lemma holomorphic_on_power [holomorphic_intros]:
502   "f holomorphic_on s \<Longrightarrow> (\<lambda>z. (f z)^n) holomorphic_on s"
503   unfolding holomorphic_on_def by (metis field_differentiable_power)
505 lemma holomorphic_on_sum [holomorphic_intros]:
506   "(\<And>i. i \<in> I \<Longrightarrow> (f i) holomorphic_on s) \<Longrightarrow> (\<lambda>x. sum (\<lambda>i. f i x) I) holomorphic_on s"
507   unfolding holomorphic_on_def by (metis field_differentiable_sum)
509 lemma DERIV_deriv_iff_field_differentiable:
510   "DERIV f x :> deriv f x \<longleftrightarrow> f field_differentiable at x"
511   unfolding field_differentiable_def by (metis DERIV_imp_deriv)
513 lemma holomorphic_derivI:
514      "\<lbrakk>f holomorphic_on S; open S; x \<in> S\<rbrakk>
515       \<Longrightarrow> (f has_field_derivative deriv f x) (at x within T)"
516 by (metis DERIV_deriv_iff_field_differentiable at_within_open  holomorphic_on_def has_field_derivative_at_within)
518 lemma complex_derivative_chain:
519   "f field_differentiable at x \<Longrightarrow> g field_differentiable at (f x)
520     \<Longrightarrow> deriv (g o f) x = deriv g (f x) * deriv f x"
521   by (metis DERIV_deriv_iff_field_differentiable DERIV_chain DERIV_imp_deriv)
523 lemma deriv_linear [simp]: "deriv (\<lambda>w. c * w) = (\<lambda>z. c)"
524   by (metis DERIV_imp_deriv DERIV_cmult_Id)
526 lemma deriv_ident [simp]: "deriv (\<lambda>w. w) = (\<lambda>z. 1)"
527   by (metis DERIV_imp_deriv DERIV_ident)
529 lemma deriv_id [simp]: "deriv id = (\<lambda>z. 1)"
532 lemma deriv_const [simp]: "deriv (\<lambda>w. c) = (\<lambda>z. 0)"
533   by (metis DERIV_imp_deriv DERIV_const)
536   "\<lbrakk>f field_differentiable at z; g field_differentiable at z\<rbrakk>
537    \<Longrightarrow> deriv (\<lambda>w. f w + g w) z = deriv f z + deriv g z"
538   unfolding DERIV_deriv_iff_field_differentiable[symmetric]
539   by (auto intro!: DERIV_imp_deriv derivative_intros)
541 lemma deriv_diff [simp]:
542   "\<lbrakk>f field_differentiable at z; g field_differentiable at z\<rbrakk>
543    \<Longrightarrow> deriv (\<lambda>w. f w - g w) z = deriv f z - deriv g z"
544   unfolding DERIV_deriv_iff_field_differentiable[symmetric]
545   by (auto intro!: DERIV_imp_deriv derivative_intros)
547 lemma deriv_mult [simp]:
548   "\<lbrakk>f field_differentiable at z; g field_differentiable at z\<rbrakk>
549    \<Longrightarrow> deriv (\<lambda>w. f w * g w) z = f z * deriv g z + deriv f z * g z"
550   unfolding DERIV_deriv_iff_field_differentiable[symmetric]
551   by (auto intro!: DERIV_imp_deriv derivative_eq_intros)
553 lemma deriv_cmult [simp]:
554   "f field_differentiable at z \<Longrightarrow> deriv (\<lambda>w. c * f w) z = c * deriv f z"
555   unfolding DERIV_deriv_iff_field_differentiable[symmetric]
556   by (auto intro!: DERIV_imp_deriv derivative_eq_intros)
558 lemma deriv_cmult_right [simp]:
559   "f field_differentiable at z \<Longrightarrow> deriv (\<lambda>w. f w * c) z = deriv f z * c"
560   unfolding DERIV_deriv_iff_field_differentiable[symmetric]
561   by (auto intro!: DERIV_imp_deriv derivative_eq_intros)
563 lemma deriv_cdivide_right [simp]:
564   "f field_differentiable at z \<Longrightarrow> deriv (\<lambda>w. f w / c) z = deriv f z / c"
565   unfolding Fields.field_class.field_divide_inverse
566   by (blast intro: deriv_cmult_right)
568 lemma complex_derivative_transform_within_open:
569   "\<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>
570    \<Longrightarrow> deriv f z = deriv g z"
571   unfolding holomorphic_on_def
572   by (rule DERIV_imp_deriv)
573      (metis DERIV_deriv_iff_field_differentiable DERIV_transform_within_open at_within_open)
575 lemma deriv_compose_linear:
576   "f field_differentiable at (c * z) \<Longrightarrow> deriv (\<lambda>w. f (c * w)) z = c * deriv f (c * z)"
577 apply (rule DERIV_imp_deriv)
578 apply (simp add: DERIV_deriv_iff_field_differentiable [symmetric])
579 apply (drule DERIV_chain' [of "times c" c z UNIV f "deriv f (c * z)", OF DERIV_cmult_Id])
581 done
583 lemma nonzero_deriv_nonconstant:
584   assumes df: "DERIV f \<xi> :> df" and S: "open S" "\<xi> \<in> S" and "df \<noteq> 0"
585     shows "\<not> f constant_on S"
586 unfolding constant_on_def
587 by (metis \<open>df \<noteq> 0\<close> DERIV_transform_within_open [OF df S] DERIV_const DERIV_unique)
589 lemma holomorphic_nonconstant:
590   assumes holf: "f holomorphic_on S" and "open S" "\<xi> \<in> S" "deriv f \<xi> \<noteq> 0"
591     shows "\<not> f constant_on S"
592     apply (rule nonzero_deriv_nonconstant [of f "deriv f \<xi>" \<xi> S])
593     using assms
594     apply (auto simp: holomorphic_derivI)
595     done
597 subsection\<open>Analyticity on a set\<close>
599 definition analytic_on (infixl "(analytic'_on)" 50)
600   where
601    "f analytic_on s \<equiv> \<forall>x \<in> s. \<exists>e. 0 < e \<and> f holomorphic_on (ball x e)"
603 lemma analytic_imp_holomorphic: "f analytic_on s \<Longrightarrow> f holomorphic_on s"
604   by (simp add: at_within_open [OF _ open_ball] analytic_on_def holomorphic_on_def)
605      (metis centre_in_ball field_differentiable_at_within)
607 lemma analytic_on_open: "open s \<Longrightarrow> f analytic_on s \<longleftrightarrow> f holomorphic_on s"
608 apply (auto simp: analytic_imp_holomorphic)
609 apply (auto simp: analytic_on_def holomorphic_on_def)
610 by (metis holomorphic_on_def holomorphic_on_subset open_contains_ball)
612 lemma analytic_on_imp_differentiable_at:
613   "f analytic_on s \<Longrightarrow> x \<in> s \<Longrightarrow> f field_differentiable (at x)"
614  apply (auto simp: analytic_on_def holomorphic_on_def)
615 by (metis Topology_Euclidean_Space.open_ball centre_in_ball field_differentiable_within_open)
617 lemma analytic_on_subset: "f analytic_on s \<Longrightarrow> t \<subseteq> s \<Longrightarrow> f analytic_on t"
618   by (auto simp: analytic_on_def)
620 lemma analytic_on_Un: "f analytic_on (s \<union> t) \<longleftrightarrow> f analytic_on s \<and> f analytic_on t"
621   by (auto simp: analytic_on_def)
623 lemma analytic_on_Union: "f analytic_on (\<Union>s) \<longleftrightarrow> (\<forall>t \<in> s. f analytic_on t)"
624   by (auto simp: analytic_on_def)
626 lemma analytic_on_UN: "f analytic_on (\<Union>i\<in>I. s i) \<longleftrightarrow> (\<forall>i\<in>I. f analytic_on (s i))"
627   by (auto simp: analytic_on_def)
629 lemma analytic_on_holomorphic:
630   "f analytic_on s \<longleftrightarrow> (\<exists>t. open t \<and> s \<subseteq> t \<and> f holomorphic_on t)"
631   (is "?lhs = ?rhs")
632 proof -
633   have "?lhs \<longleftrightarrow> (\<exists>t. open t \<and> s \<subseteq> t \<and> f analytic_on t)"
634   proof safe
635     assume "f analytic_on s"
636     then show "\<exists>t. open t \<and> s \<subseteq> t \<and> f analytic_on t"
638       apply (rule exI [where x="\<Union>{u. open u \<and> f analytic_on u}"], auto)
639       apply (metis Topology_Euclidean_Space.open_ball analytic_on_open centre_in_ball)
640       by (metis analytic_on_def)
641   next
642     fix t
643     assume "open t" "s \<subseteq> t" "f analytic_on t"
644     then show "f analytic_on s"
645         by (metis analytic_on_subset)
646   qed
647   also have "... \<longleftrightarrow> ?rhs"
648     by (auto simp: analytic_on_open)
649   finally show ?thesis .
650 qed
652 lemma analytic_on_linear: "(op * c) analytic_on s"
653   by (auto simp add: analytic_on_holomorphic holomorphic_on_linear)
655 lemma analytic_on_const: "(\<lambda>z. c) analytic_on s"
656   by (metis analytic_on_def holomorphic_on_const zero_less_one)
658 lemma analytic_on_ident: "(\<lambda>x. x) analytic_on s"
659   by (simp add: analytic_on_def holomorphic_on_ident gt_ex)
661 lemma analytic_on_id: "id analytic_on s"
662   unfolding id_def by (rule analytic_on_ident)
664 lemma analytic_on_compose:
665   assumes f: "f analytic_on s"
666       and g: "g analytic_on (f ` s)"
667     shows "(g o f) analytic_on s"
668 unfolding analytic_on_def
669 proof (intro ballI)
670   fix x
671   assume x: "x \<in> s"
672   then obtain e where e: "0 < e" and fh: "f holomorphic_on ball x e" using f
673     by (metis analytic_on_def)
674   obtain e' where e': "0 < e'" and gh: "g holomorphic_on ball (f x) e'" using g
675     by (metis analytic_on_def g image_eqI x)
676   have "isCont f x"
677     by (metis analytic_on_imp_differentiable_at field_differentiable_imp_continuous_at f x)
678   with e' obtain d where d: "0 < d" and fd: "f ` ball x d \<subseteq> ball (f x) e'"
679      by (auto simp: continuous_at_ball)
680   have "g \<circ> f holomorphic_on ball x (min d e)"
681     apply (rule holomorphic_on_compose)
682     apply (metis fh holomorphic_on_subset min.bounded_iff order_refl subset_ball)
683     by (metis fd gh holomorphic_on_subset image_mono min.cobounded1 subset_ball)
684   then show "\<exists>e>0. g \<circ> f holomorphic_on ball x e"
685     by (metis d e min_less_iff_conj)
686 qed
688 lemma analytic_on_compose_gen:
689   "f analytic_on s \<Longrightarrow> g analytic_on t \<Longrightarrow> (\<And>z. z \<in> s \<Longrightarrow> f z \<in> t)
690              \<Longrightarrow> g o f analytic_on s"
691 by (metis analytic_on_compose analytic_on_subset image_subset_iff)
693 lemma analytic_on_neg:
694   "f analytic_on s \<Longrightarrow> (\<lambda>z. -(f z)) analytic_on s"
695 by (metis analytic_on_holomorphic holomorphic_on_minus)
698   assumes f: "f analytic_on s"
699       and g: "g analytic_on s"
700     shows "(\<lambda>z. f z + g z) analytic_on s"
701 unfolding analytic_on_def
702 proof (intro ballI)
703   fix z
704   assume z: "z \<in> s"
705   then obtain e where e: "0 < e" and fh: "f holomorphic_on ball z e" using f
706     by (metis analytic_on_def)
707   obtain e' where e': "0 < e'" and gh: "g holomorphic_on ball z e'" using g
708     by (metis analytic_on_def g z)
709   have "(\<lambda>z. f z + g z) holomorphic_on ball z (min e e')"
711     apply (metis fh holomorphic_on_subset min.bounded_iff order_refl subset_ball)
712     by (metis gh holomorphic_on_subset min.bounded_iff order_refl subset_ball)
713   then show "\<exists>e>0. (\<lambda>z. f z + g z) holomorphic_on ball z e"
714     by (metis e e' min_less_iff_conj)
715 qed
717 lemma analytic_on_diff:
718   assumes f: "f analytic_on s"
719       and g: "g analytic_on s"
720     shows "(\<lambda>z. f z - g z) analytic_on s"
721 unfolding analytic_on_def
722 proof (intro ballI)
723   fix z
724   assume z: "z \<in> s"
725   then obtain e where e: "0 < e" and fh: "f holomorphic_on ball z e" using f
726     by (metis analytic_on_def)
727   obtain e' where e': "0 < e'" and gh: "g holomorphic_on ball z e'" using g
728     by (metis analytic_on_def g z)
729   have "(\<lambda>z. f z - g z) holomorphic_on ball z (min e e')"
730     apply (rule holomorphic_on_diff)
731     apply (metis fh holomorphic_on_subset min.bounded_iff order_refl subset_ball)
732     by (metis gh holomorphic_on_subset min.bounded_iff order_refl subset_ball)
733   then show "\<exists>e>0. (\<lambda>z. f z - g z) holomorphic_on ball z e"
734     by (metis e e' min_less_iff_conj)
735 qed
737 lemma analytic_on_mult:
738   assumes f: "f analytic_on s"
739       and g: "g analytic_on s"
740     shows "(\<lambda>z. f z * g z) analytic_on s"
741 unfolding analytic_on_def
742 proof (intro ballI)
743   fix z
744   assume z: "z \<in> s"
745   then obtain e where e: "0 < e" and fh: "f holomorphic_on ball z e" using f
746     by (metis analytic_on_def)
747   obtain e' where e': "0 < e'" and gh: "g holomorphic_on ball z e'" using g
748     by (metis analytic_on_def g z)
749   have "(\<lambda>z. f z * g z) holomorphic_on ball z (min e e')"
750     apply (rule holomorphic_on_mult)
751     apply (metis fh holomorphic_on_subset min.bounded_iff order_refl subset_ball)
752     by (metis gh holomorphic_on_subset min.bounded_iff order_refl subset_ball)
753   then show "\<exists>e>0. (\<lambda>z. f z * g z) holomorphic_on ball z e"
754     by (metis e e' min_less_iff_conj)
755 qed
757 lemma analytic_on_inverse:
758   assumes f: "f analytic_on s"
759       and nz: "(\<And>z. z \<in> s \<Longrightarrow> f z \<noteq> 0)"
760     shows "(\<lambda>z. inverse (f z)) analytic_on s"
761 unfolding analytic_on_def
762 proof (intro ballI)
763   fix z
764   assume z: "z \<in> s"
765   then obtain e where e: "0 < e" and fh: "f holomorphic_on ball z e" using f
766     by (metis analytic_on_def)
767   have "continuous_on (ball z e) f"
768     by (metis fh holomorphic_on_imp_continuous_on)
769   then obtain e' where e': "0 < e'" and nz': "\<And>y. dist z y < e' \<Longrightarrow> f y \<noteq> 0"
770     by (metis Topology_Euclidean_Space.open_ball centre_in_ball continuous_on_open_avoid e z nz)
771   have "(\<lambda>z. inverse (f z)) holomorphic_on ball z (min e e')"
772     apply (rule holomorphic_on_inverse)
773     apply (metis fh holomorphic_on_subset min.cobounded2 min.commute subset_ball)
774     by (metis nz' mem_ball min_less_iff_conj)
775   then show "\<exists>e>0. (\<lambda>z. inverse (f z)) holomorphic_on ball z e"
776     by (metis e e' min_less_iff_conj)
777 qed
779 lemma analytic_on_divide:
780   assumes f: "f analytic_on s"
781       and g: "g analytic_on s"
782       and nz: "(\<And>z. z \<in> s \<Longrightarrow> g z \<noteq> 0)"
783     shows "(\<lambda>z. f z / g z) analytic_on s"
784 unfolding divide_inverse
785 by (metis analytic_on_inverse analytic_on_mult f g nz)
787 lemma analytic_on_power:
788   "f analytic_on s \<Longrightarrow> (\<lambda>z. (f z) ^ n) analytic_on s"
789 by (induct n) (auto simp: analytic_on_const analytic_on_mult)
791 lemma analytic_on_sum:
792   "(\<And>i. i \<in> I \<Longrightarrow> (f i) analytic_on s) \<Longrightarrow> (\<lambda>x. sum (\<lambda>i. f i x) I) analytic_on s"
793   by (induct I rule: infinite_finite_induct) (auto simp: analytic_on_const analytic_on_add)
795 lemma deriv_left_inverse:
796   assumes "f holomorphic_on S" and "g holomorphic_on T"
797       and "open S" and "open T"
798       and "f ` S \<subseteq> T"
799       and [simp]: "\<And>z. z \<in> S \<Longrightarrow> g (f z) = z"
800       and "w \<in> S"
801     shows "deriv f w * deriv g (f w) = 1"
802 proof -
803   have "deriv f w * deriv g (f w) = deriv g (f w) * deriv f w"
805   also have "... = deriv (g o f) w"
806     using assms
807     by (metis analytic_on_imp_differentiable_at analytic_on_open complex_derivative_chain image_subset_iff)
808   also have "... = deriv id w"
809     apply (rule complex_derivative_transform_within_open [where s=S])
810     apply (rule assms holomorphic_on_compose_gen holomorphic_intros)+
811     apply simp
812     done
813   also have "... = 1"
814     by simp
815   finally show ?thesis .
816 qed
818 subsection\<open>analyticity at a point\<close>
820 lemma analytic_at_ball:
821   "f analytic_on {z} \<longleftrightarrow> (\<exists>e. 0<e \<and> f holomorphic_on ball z e)"
822 by (metis analytic_on_def singleton_iff)
824 lemma analytic_at:
825     "f analytic_on {z} \<longleftrightarrow> (\<exists>s. open s \<and> z \<in> s \<and> f holomorphic_on s)"
826 by (metis analytic_on_holomorphic empty_subsetI insert_subset)
828 lemma analytic_on_analytic_at:
829     "f analytic_on s \<longleftrightarrow> (\<forall>z \<in> s. f analytic_on {z})"
830 by (metis analytic_at_ball analytic_on_def)
832 lemma analytic_at_two:
833   "f analytic_on {z} \<and> g analytic_on {z} \<longleftrightarrow>
834    (\<exists>s. open s \<and> z \<in> s \<and> f holomorphic_on s \<and> g holomorphic_on s)"
835   (is "?lhs = ?rhs")
836 proof
837   assume ?lhs
838   then obtain s t
839     where st: "open s" "z \<in> s" "f holomorphic_on s"
840               "open t" "z \<in> t" "g holomorphic_on t"
841     by (auto simp: analytic_at)
842   show ?rhs
843     apply (rule_tac x="s \<inter> t" in exI)
844     using st
845     apply (auto simp: Diff_subset holomorphic_on_subset)
846     done
847 next
848   assume ?rhs
849   then show ?lhs
850     by (force simp add: analytic_at)
851 qed
853 subsection\<open>Combining theorems for derivative with ``analytic at'' hypotheses\<close>
855 lemma
856   assumes "f analytic_on {z}" "g analytic_on {z}"
857   shows complex_derivative_add_at: "deriv (\<lambda>w. f w + g w) z = deriv f z + deriv g z"
858     and complex_derivative_diff_at: "deriv (\<lambda>w. f w - g w) z = deriv f z - deriv g z"
859     and complex_derivative_mult_at: "deriv (\<lambda>w. f w * g w) z =
860            f z * deriv g z + deriv f z * g z"
861 proof -
862   obtain s where s: "open s" "z \<in> s" "f holomorphic_on s" "g holomorphic_on s"
863     using assms by (metis analytic_at_two)
864   show "deriv (\<lambda>w. f w + g w) z = deriv f z + deriv g z"
865     apply (rule DERIV_imp_deriv [OF DERIV_add])
866     using s
867     apply (auto simp: holomorphic_on_open field_differentiable_def DERIV_deriv_iff_field_differentiable)
868     done
869   show "deriv (\<lambda>w. f w - g w) z = deriv f z - deriv g z"
870     apply (rule DERIV_imp_deriv [OF DERIV_diff])
871     using s
872     apply (auto simp: holomorphic_on_open field_differentiable_def DERIV_deriv_iff_field_differentiable)
873     done
874   show "deriv (\<lambda>w. f w * g w) z = f z * deriv g z + deriv f z * g z"
875     apply (rule DERIV_imp_deriv [OF DERIV_mult'])
876     using s
877     apply (auto simp: holomorphic_on_open field_differentiable_def DERIV_deriv_iff_field_differentiable)
878     done
879 qed
881 lemma deriv_cmult_at:
882   "f analytic_on {z} \<Longrightarrow>  deriv (\<lambda>w. c * f w) z = c * deriv f z"
883 by (auto simp: complex_derivative_mult_at deriv_const analytic_on_const)
885 lemma deriv_cmult_right_at:
886   "f analytic_on {z} \<Longrightarrow>  deriv (\<lambda>w. f w * c) z = deriv f z * c"
887 by (auto simp: complex_derivative_mult_at deriv_const analytic_on_const)
889 subsection\<open>Complex differentiation of sequences and series\<close>
891 (* TODO: Could probably be simplified using Uniform_Limit *)
892 lemma has_complex_derivative_sequence:
893   fixes s :: "complex set"
894   assumes cvs: "convex s"
895       and df:  "\<And>n x. x \<in> s \<Longrightarrow> (f n has_field_derivative f' n x) (at x within s)"
896       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"
897       and "\<exists>x l. x \<in> s \<and> ((\<lambda>n. f n x) \<longlongrightarrow> l) sequentially"
898     shows "\<exists>g. \<forall>x \<in> s. ((\<lambda>n. f n x) \<longlongrightarrow> g x) sequentially \<and>
899                        (g has_field_derivative (g' x)) (at x within s)"
900 proof -
901   from assms obtain x l where x: "x \<in> s" and tf: "((\<lambda>n. f n x) \<longlongrightarrow> l) sequentially"
902     by blast
903   { fix e::real assume e: "e > 0"
904     then obtain N where N: "\<forall>n\<ge>N. \<forall>x. x \<in> s \<longrightarrow> cmod (f' n x - g' x) \<le> e"
905       by (metis conv)
906     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"
907     proof (rule exI [of _ N], clarify)
908       fix n y h
909       assume "N \<le> n" "y \<in> s"
910       then have "cmod (f' n y - g' y) \<le> e"
911         by (metis N)
912       then have "cmod h * cmod (f' n y - g' y) \<le> cmod h * e"
913         by (auto simp: antisym_conv2 mult_le_cancel_left norm_triangle_ineq2)
914       then show "cmod (f' n y * h - g' y * h) \<le> e * cmod h"
915         by (simp add: norm_mult [symmetric] field_simps)
916     qed
917   } note ** = this
918   show ?thesis
919   unfolding has_field_derivative_def
920   proof (rule has_derivative_sequence [OF cvs _ _ x])
921     show "\<forall>n. \<forall>x\<in>s. (f n has_derivative (op * (f' n x))) (at x within s)"
922       by (metis has_field_derivative_def df)
923   next show "(\<lambda>n. f n x) \<longlonglongrightarrow> l"
924     by (rule tf)
925   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"
926     by (blast intro: **)
927   qed
928 qed
930 lemma has_complex_derivative_series:
931   fixes s :: "complex set"
932   assumes cvs: "convex s"
933       and df:  "\<And>n x. x \<in> s \<Longrightarrow> (f n has_field_derivative f' n x) (at x within s)"
934       and conv: "\<And>e. 0 < e \<Longrightarrow> \<exists>N. \<forall>n x. n \<ge> N \<longrightarrow> x \<in> s
935                 \<longrightarrow> cmod ((\<Sum>i<n. f' i x) - g' x) \<le> e"
936       and "\<exists>x l. x \<in> s \<and> ((\<lambda>n. f n x) sums l)"
937     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))"
938 proof -
939   from assms obtain x l where x: "x \<in> s" and sf: "((\<lambda>n. f n x) sums l)"
940     by blast
941   { fix e::real assume e: "e > 0"
942     then obtain N where N: "\<forall>n x. n \<ge> N \<longrightarrow> x \<in> s
943             \<longrightarrow> cmod ((\<Sum>i<n. f' i x) - g' x) \<le> e"
944       by (metis conv)
945     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"
946     proof (rule exI [of _ N], clarify)
947       fix n y h
948       assume "N \<le> n" "y \<in> s"
949       then have "cmod ((\<Sum>i<n. f' i y) - g' y) \<le> e"
950         by (metis N)
951       then have "cmod h * cmod ((\<Sum>i<n. f' i y) - g' y) \<le> cmod h * e"
952         by (auto simp: antisym_conv2 mult_le_cancel_left norm_triangle_ineq2)
953       then show "cmod ((\<Sum>i<n. h * f' i y) - g' y * h) \<le> e * cmod h"
954         by (simp add: norm_mult [symmetric] field_simps sum_distrib_left)
955     qed
956   } note ** = this
957   show ?thesis
958   unfolding has_field_derivative_def
959   proof (rule has_derivative_series [OF cvs _ _ x])
960     fix n x
961     assume "x \<in> s"
962     then show "((f n) has_derivative (\<lambda>z. z * f' n x)) (at x within s)"
963       by (metis df has_field_derivative_def mult_commute_abs)
964   next show " ((\<lambda>n. f n x) sums l)"
965     by (rule sf)
966   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"
967     by (blast intro: **)
968   qed
969 qed
972 lemma field_differentiable_series:
973   fixes f :: "nat \<Rightarrow> complex \<Rightarrow> complex"
974   assumes "convex s" "open s"
975   assumes "\<And>n x. x \<in> s \<Longrightarrow> (f n has_field_derivative f' n x) (at x)"
976   assumes "uniformly_convergent_on s (\<lambda>n x. \<Sum>i<n. f' i x)"
977   assumes "x0 \<in> s" "summable (\<lambda>n. f n x0)" and x: "x \<in> s"
978   shows   "summable (\<lambda>n. f n x)" and "(\<lambda>x. \<Sum>n. f n x) field_differentiable (at x)"
979 proof -
980   from assms(4) obtain g' where A: "uniform_limit s (\<lambda>n x. \<Sum>i<n. f' i x) g' sequentially"
981     unfolding uniformly_convergent_on_def by blast
982   from x and \<open>open s\<close> have s: "at x within s = at x" by (rule at_within_open)
983   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)"
984     by (intro has_field_derivative_series[of s f f' g' x0] assms A has_field_derivative_at_within)
985   then obtain g where g: "\<And>x. x \<in> s \<Longrightarrow> (\<lambda>n. f n x) sums g x"
986     "\<And>x. x \<in> s \<Longrightarrow> (g has_field_derivative g' x) (at x within s)" by blast
987   from g[OF x] show "summable (\<lambda>n. f n x)" by (auto simp: summable_def)
988   from g(2)[OF x] have g': "(g has_derivative op * (g' x)) (at x)"
989     by (simp add: has_field_derivative_def s)
990   have "((\<lambda>x. \<Sum>n. f n x) has_derivative op * (g' x)) (at x)"
991     by (rule has_derivative_transform_within_open[OF g' \<open>open s\<close> x])
992        (insert g, auto simp: sums_iff)
993   thus "(\<lambda>x. \<Sum>n. f n x) field_differentiable (at x)" unfolding differentiable_def
994     by (auto simp: summable_def field_differentiable_def has_field_derivative_def)
995 qed
997 lemma field_differentiable_series':
998   fixes f :: "nat \<Rightarrow> complex \<Rightarrow> complex"
999   assumes "convex s" "open s"
1000   assumes "\<And>n x. x \<in> s \<Longrightarrow> (f n has_field_derivative f' n x) (at x)"
1001   assumes "uniformly_convergent_on s (\<lambda>n x. \<Sum>i<n. f' i x)"
1002   assumes "x0 \<in> s" "summable (\<lambda>n. f n x0)"
1003   shows   "(\<lambda>x. \<Sum>n. f n x) field_differentiable (at x0)"
1004   using field_differentiable_series[OF assms, of x0] \<open>x0 \<in> s\<close> by blast+
1006 subsection\<open>Bound theorem\<close>
1008 lemma field_differentiable_bound:
1009   fixes s :: "complex set"
1010   assumes cvs: "convex s"
1011       and df:  "\<And>z. z \<in> s \<Longrightarrow> (f has_field_derivative f' z) (at z within s)"
1012       and dn:  "\<And>z. z \<in> s \<Longrightarrow> norm (f' z) \<le> B"
1013       and "x \<in> s"  "y \<in> s"
1014     shows "norm(f x - f y) \<le> B * norm(x - y)"
1015   apply (rule differentiable_bound [OF cvs])
1016   apply (rule ballI, erule df [unfolded has_field_derivative_def])
1017   apply (rule ballI, rule onorm_le, simp add: norm_mult mult_right_mono dn)
1018   apply fact
1019   apply fact
1020   done
1022 subsection\<open>Inverse function theorem for complex derivatives\<close>
1024 lemma has_complex_derivative_inverse_basic:
1025   fixes f :: "complex \<Rightarrow> complex"
1026   shows "DERIV f (g y) :> f' \<Longrightarrow>
1027         f' \<noteq> 0 \<Longrightarrow>
1028         continuous (at y) g \<Longrightarrow>
1029         open t \<Longrightarrow>
1030         y \<in> t \<Longrightarrow>
1031         (\<And>z. z \<in> t \<Longrightarrow> f (g z) = z)
1032         \<Longrightarrow> DERIV g y :> inverse (f')"
1033   unfolding has_field_derivative_def
1034   apply (rule has_derivative_inverse_basic)
1035   apply (auto simp:  bounded_linear_mult_right)
1036   done
1038 (*Used only once, in Multivariate/cauchy.ml. *)
1039 lemma has_complex_derivative_inverse_strong:
1040   fixes f :: "complex \<Rightarrow> complex"
1041   shows "DERIV f x :> f' \<Longrightarrow>
1042          f' \<noteq> 0 \<Longrightarrow>
1043          open s \<Longrightarrow>
1044          x \<in> s \<Longrightarrow>
1045          continuous_on s f \<Longrightarrow>
1046          (\<And>z. z \<in> s \<Longrightarrow> g (f z) = z)
1047          \<Longrightarrow> DERIV g (f x) :> inverse (f')"
1048   unfolding has_field_derivative_def
1049   apply (rule has_derivative_inverse_strong [of s x f g ])
1050   by auto
1052 lemma has_complex_derivative_inverse_strong_x:
1053   fixes f :: "complex \<Rightarrow> complex"
1054   shows  "DERIV f (g y) :> f' \<Longrightarrow>
1055           f' \<noteq> 0 \<Longrightarrow>
1056           open s \<Longrightarrow>
1057           continuous_on s f \<Longrightarrow>
1058           g y \<in> s \<Longrightarrow> f(g y) = y \<Longrightarrow>
1059           (\<And>z. z \<in> s \<Longrightarrow> g (f z) = z)
1060           \<Longrightarrow> DERIV g y :> inverse (f')"
1061   unfolding has_field_derivative_def
1062   apply (rule has_derivative_inverse_strong_x [of s g y f])
1063   by auto
1065 subsection \<open>Taylor on Complex Numbers\<close>
1067 lemma sum_Suc_reindex:
1068   fixes f :: "nat \<Rightarrow> 'a::ab_group_add"
1069     shows  "sum f {0..n} = f 0 - f (Suc n) + sum (\<lambda>i. f (Suc i)) {0..n}"
1070 by (induct n) auto
1072 lemma complex_taylor:
1073   assumes s: "convex s"
1074       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)"
1075       and B: "\<And>x. x \<in> s \<Longrightarrow> cmod (f (Suc n) x) \<le> B"
1076       and w: "w \<in> s"
1077       and z: "z \<in> s"
1078     shows "cmod(f 0 z - (\<Sum>i\<le>n. f i w * (z-w) ^ i / (fact i)))
1079           \<le> B * cmod(z - w)^(Suc n) / fact n"
1080 proof -
1081   have wzs: "closed_segment w z \<subseteq> s" using assms
1082     by (metis convex_contains_segment)
1083   { fix u
1084     assume "u \<in> closed_segment w z"
1085     then have "u \<in> s"
1086       by (metis wzs subsetD)
1087     have "(\<Sum>i\<le>n. f i u * (- of_nat i * (z-u)^(i - 1)) / (fact i) +
1088                       f (Suc i) u * (z-u)^i / (fact i)) =
1089               f (Suc n) u * (z-u) ^ n / (fact n)"
1090     proof (induction n)
1091       case 0 show ?case by simp
1092     next
1093       case (Suc n)
1094       have "(\<Sum>i\<le>Suc n. f i u * (- of_nat i * (z-u) ^ (i - 1)) / (fact i) +
1095                              f (Suc i) u * (z-u) ^ i / (fact i)) =
1096            f (Suc n) u * (z-u) ^ n / (fact n) +
1097            f (Suc (Suc n)) u * ((z-u) * (z-u) ^ n) / (fact (Suc n)) -
1098            f (Suc n) u * ((1 + of_nat n) * (z-u) ^ n) / (fact (Suc n))"
1099         using Suc by simp
1100       also have "... = f (Suc (Suc n)) u * (z-u) ^ Suc n / (fact (Suc n))"
1101       proof -
1102         have "(fact(Suc n)) *
1103              (f(Suc n) u *(z-u) ^ n / (fact n) +
1104                f(Suc(Suc n)) u *((z-u) *(z-u) ^ n) / (fact(Suc n)) -
1105                f(Suc n) u *((1 + of_nat n) *(z-u) ^ n) / (fact(Suc n))) =
1106             ((fact(Suc n)) *(f(Suc n) u *(z-u) ^ n)) / (fact n) +
1107             ((fact(Suc n)) *(f(Suc(Suc n)) u *((z-u) *(z-u) ^ n)) / (fact(Suc n))) -
1108             ((fact(Suc n)) *(f(Suc n) u *(of_nat(Suc n) *(z-u) ^ n))) / (fact(Suc n))"
1109           by (simp add: algebra_simps del: fact_Suc)
1110         also have "... = ((fact (Suc n)) * (f (Suc n) u * (z-u) ^ n)) / (fact n) +
1111                          (f (Suc (Suc n)) u * ((z-u) * (z-u) ^ n)) -
1112                          (f (Suc n) u * ((1 + of_nat n) * (z-u) ^ n))"
1113           by (simp del: fact_Suc)
1114         also have "... = (of_nat (Suc n) * (f (Suc n) u * (z-u) ^ n)) +
1115                          (f (Suc (Suc n)) u * ((z-u) * (z-u) ^ n)) -
1116                          (f (Suc n) u * ((1 + of_nat n) * (z-u) ^ n))"
1117           by (simp only: fact_Suc of_nat_mult ac_simps) simp
1118         also have "... = f (Suc (Suc n)) u * ((z-u) * (z-u) ^ n)"
1120         finally show ?thesis
1121         by (simp add: mult_left_cancel [where c = "(fact (Suc n))", THEN iffD1] del: fact_Suc)
1122       qed
1123       finally show ?case .
1124     qed
1125     then have "((\<lambda>v. (\<Sum>i\<le>n. f i v * (z - v)^i / (fact i)))
1126                 has_field_derivative f (Suc n) u * (z-u) ^ n / (fact n))
1127                (at u within s)"
1128       apply (intro derivative_eq_intros)
1129       apply (blast intro: assms \<open>u \<in> s\<close>)
1130       apply (rule refl)+
1131       apply (auto simp: field_simps)
1132       done
1133   } note sum_deriv = this
1134   { fix u
1135     assume u: "u \<in> closed_segment w z"
1136     then have us: "u \<in> s"
1137       by (metis wzs subsetD)
1138     have "cmod (f (Suc n) u) * cmod (z - u) ^ n \<le> cmod (f (Suc n) u) * cmod (u - z) ^ n"
1139       by (metis norm_minus_commute order_refl)
1140     also have "... \<le> cmod (f (Suc n) u) * cmod (z - w) ^ n"
1141       by (metis mult_left_mono norm_ge_zero power_mono segment_bound [OF u])
1142     also have "... \<le> B * cmod (z - w) ^ n"
1143       by (metis norm_ge_zero zero_le_power mult_right_mono  B [OF us])
1144     finally have "cmod (f (Suc n) u) * cmod (z - u) ^ n \<le> B * cmod (z - w) ^ n" .
1145   } note cmod_bound = this
1146   have "(\<Sum>i\<le>n. f i z * (z - z) ^ i / (fact i)) = (\<Sum>i\<le>n. (f i z / (fact i)) * 0 ^ i)"
1147     by simp
1148   also have "\<dots> = f 0 z / (fact 0)"
1149     by (subst sum_zero_power) simp
1150   finally have "cmod (f 0 z - (\<Sum>i\<le>n. f i w * (z - w) ^ i / (fact i)))
1151                 \<le> cmod ((\<Sum>i\<le>n. f i w * (z - w) ^ i / (fact i)) -
1152                         (\<Sum>i\<le>n. f i z * (z - z) ^ i / (fact i)))"
1154   also have "... \<le> B * cmod (z - w) ^ n / (fact n) * cmod (w - z)"
1155     apply (rule field_differentiable_bound
1156       [where f' = "\<lambda>w. f (Suc n) w * (z - w)^n / (fact n)"
1157          and s = "closed_segment w z", OF convex_closed_segment])
1158     apply (auto simp: ends_in_segment DERIV_subset [OF sum_deriv wzs]
1159                   norm_divide norm_mult norm_power divide_le_cancel cmod_bound)
1160     done
1161   also have "...  \<le> B * cmod (z - w) ^ Suc n / (fact n)"
1162     by (simp add: algebra_simps norm_minus_commute)
1163   finally show ?thesis .
1164 qed
1166 text\<open>Something more like the traditional MVT for real components\<close>
1168 lemma complex_mvt_line:
1169   assumes "\<And>u. u \<in> closed_segment w z \<Longrightarrow> (f has_field_derivative f'(u)) (at u)"
1170     shows "\<exists>u. u \<in> closed_segment w z \<and> Re(f z) - Re(f w) = Re(f'(u) * (z - w))"
1171 proof -
1172   have twz: "\<And>t. (1 - t) *\<^sub>R w + t *\<^sub>R z = w + t *\<^sub>R (z - w)"
1173     by (simp add: real_vector.scale_left_diff_distrib real_vector.scale_right_diff_distrib)
1174   note assms[unfolded has_field_derivative_def, derivative_intros]
1175   show ?thesis
1176     apply (cut_tac mvt_simple
1177                      [of 0 1 "Re o f o (\<lambda>t. (1 - t) *\<^sub>R w +  t *\<^sub>R z)"
1178                       "\<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))"])
1179     apply auto
1180     apply (rule_tac x="(1 - x) *\<^sub>R w + x *\<^sub>R z" in exI)
1181     apply (auto simp: closed_segment_def twz) []
1182     apply (intro derivative_eq_intros has_derivative_at_within, simp_all)
1183     apply (simp add: fun_eq_iff real_vector.scale_right_diff_distrib)
1184     apply (force simp: twz closed_segment_def)
1185     done
1186 qed
1188 lemma complex_taylor_mvt:
1189   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)"
1190     shows "\<exists>u. u \<in> closed_segment w z \<and>
1191             Re (f 0 z) =
1192             Re ((\<Sum>i = 0..n. f i w * (z - w) ^ i / (fact i)) +
1193                 (f (Suc n) u * (z-u)^n / (fact n)) * (z - w))"
1194 proof -
1195   { fix u
1196     assume u: "u \<in> closed_segment w z"
1197     have "(\<Sum>i = 0..n.
1198                (f (Suc i) u * (z-u) ^ i - of_nat i * (f i u * (z-u) ^ (i - Suc 0))) /
1199                (fact i)) =
1200           f (Suc 0) u -
1201              (f (Suc (Suc n)) u * ((z-u) ^ Suc n) - (of_nat (Suc n)) * (z-u) ^ n * f (Suc n) u) /
1202              (fact (Suc n)) +
1203              (\<Sum>i = 0..n.
1204                  (f (Suc (Suc i)) u * ((z-u) ^ Suc i) - of_nat (Suc i) * (f (Suc i) u * (z-u) ^ i)) /
1205                  (fact (Suc i)))"
1206        by (subst sum_Suc_reindex) simp
1207     also have "... = f (Suc 0) u -
1208              (f (Suc (Suc n)) u * ((z-u) ^ Suc n) - (of_nat (Suc n)) * (z-u) ^ n * f (Suc n) u) /
1209              (fact (Suc n)) +
1210              (\<Sum>i = 0..n.
1211                  f (Suc (Suc i)) u * ((z-u) ^ Suc i) / (fact (Suc i))  -
1212                  f (Suc i) u * (z-u) ^ i / (fact i))"
1213       by (simp only: diff_divide_distrib fact_cancel ac_simps)
1214     also have "... = f (Suc 0) u -
1215              (f (Suc (Suc n)) u * (z-u) ^ Suc n - of_nat (Suc n) * (z-u) ^ n * f (Suc n) u) /
1216              (fact (Suc n)) +
1217              f (Suc (Suc n)) u * (z-u) ^ Suc n / (fact (Suc n)) - f (Suc 0) u"
1218       by (subst sum_Suc_diff) auto
1219     also have "... = f (Suc n) u * (z-u) ^ n / (fact n)"
1220       by (simp only: algebra_simps diff_divide_distrib fact_cancel)
1221     finally have "(\<Sum>i = 0..n. (f (Suc i) u * (z - u) ^ i
1222                              - of_nat i * (f i u * (z-u) ^ (i - Suc 0))) / (fact i)) =
1223                   f (Suc n) u * (z - u) ^ n / (fact n)" .
1224     then have "((\<lambda>u. \<Sum>i = 0..n. f i u * (z - u) ^ i / (fact i)) has_field_derivative
1225                 f (Suc n) u * (z - u) ^ n / (fact n))  (at u)"
1226       apply (intro derivative_eq_intros)+
1227       apply (force intro: u assms)
1228       apply (rule refl)+
1229       apply (auto simp: ac_simps)
1230       done
1231   }
1232   then show ?thesis
1233     apply (cut_tac complex_mvt_line [of w z "\<lambda>u. \<Sum>i = 0..n. f i u * (z-u) ^ i / (fact i)"
1234                "\<lambda>u. (f (Suc n) u * (z-u)^n / (fact n))"])
1235     apply (auto simp add: intro: open_closed_segment)
1236     done
1237 qed
1240 subsection \<open>Polynomal function extremal theorem, from HOL Light\<close>
1242 lemma polyfun_extremal_lemma: (*COMPLEX_POLYFUN_EXTREMAL_LEMMA in HOL Light*)
1243     fixes c :: "nat \<Rightarrow> 'a::real_normed_div_algebra"
1244   assumes "0 < e"
1245     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)"
1246 proof (induct n)
1247   case 0 with assms
1248   show ?case
1249     apply (rule_tac x="norm (c 0) / e" in exI)
1250     apply (auto simp: field_simps)
1251     done
1252 next
1253   case (Suc n)
1254   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"
1255     using Suc assms by blast
1256   show ?case
1257   proof (rule exI [where x= "max M (1 + norm(c(Suc n)) / e)"], clarsimp simp del: power_Suc)
1258     fix z::'a
1259     assume z1: "M \<le> norm z" and "1 + norm (c (Suc n)) / e \<le> norm z"
1260     then have z2: "e + norm (c (Suc n)) \<le> e * norm z"
1261       using assms by (simp add: field_simps)
1262     have "norm (\<Sum>i\<le>n. c i * z^i) \<le> e * norm z ^ Suc n"
1263       using M [OF z1] by simp
1264     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)"
1265       by simp
1266     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)"
1267       by (blast intro: norm_triangle_le elim: )
1268     also have "... \<le> (e + norm (c (Suc n))) * norm z ^ Suc n"
1269       by (simp add: norm_power norm_mult algebra_simps)
1270     also have "... \<le> (e * norm z) * norm z ^ Suc n"
1271       by (metis z2 mult.commute mult_left_mono norm_ge_zero norm_power)
1272     finally show "norm ((\<Sum>i\<le>n. c i * z^i) + c (Suc n) * z ^ Suc n) \<le> e * norm z ^ Suc (Suc n)"
1273       by simp
1274   qed
1275 qed
1277 lemma polyfun_extremal: (*COMPLEX_POLYFUN_EXTREMAL in HOL Light*)
1278     fixes c :: "nat \<Rightarrow> 'a::real_normed_div_algebra"
1279   assumes k: "c k \<noteq> 0" "1\<le>k" and kn: "k\<le>n"
1280     shows "eventually (\<lambda>z. norm (\<Sum>i\<le>n. c(i) * z^i) \<ge> B) at_infinity"
1281 using kn
1282 proof (induction n)
1283   case 0
1284   then show ?case
1285     using k  by simp
1286 next
1287   case (Suc m)
1288   let ?even = ?case
1289   show ?even
1290   proof (cases "c (Suc m) = 0")
1291     case True
1292     then show ?even using Suc k
1293       by auto (metis antisym_conv less_eq_Suc_le not_le)
1294   next
1295     case False
1296     then obtain M where M:
1297           "\<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"
1298       using polyfun_extremal_lemma [of "norm(c (Suc m)) / 2" c m] Suc
1299       by auto
1300     have "\<exists>b. \<forall>z. b \<le> norm z \<longrightarrow> B \<le> norm (\<Sum>i\<le>Suc m. c i * z^i)"
1301     proof (rule exI [where x="max M (max 1 (\<bar>B\<bar> / (norm(c (Suc m)) / 2)))"], clarsimp simp del: power_Suc)
1302       fix z::'a
1303       assume z1: "M \<le> norm z" "1 \<le> norm z"
1304          and "\<bar>B\<bar> * 2 / norm (c (Suc m)) \<le> norm z"
1305       then have z2: "\<bar>B\<bar> \<le> norm (c (Suc m)) * norm z / 2"
1306         using False by (simp add: field_simps)
1307       have nz: "norm z \<le> norm z ^ Suc m"
1308         by (metis \<open>1 \<le> norm z\<close> One_nat_def less_eq_Suc_le power_increasing power_one_right zero_less_Suc)
1309       have *: "\<And>y x. norm (c (Suc m)) * norm z / 2 \<le> norm y - norm x \<Longrightarrow> B \<le> norm (x + y)"
1310         by (metis abs_le_iff add.commute norm_diff_ineq order_trans z2)
1311       have "norm z * norm (c (Suc m)) + 2 * norm (\<Sum>i\<le>m. c i * z^i)
1312             \<le> norm (c (Suc m)) * norm z + norm (c (Suc m)) * norm z ^ Suc m"
1313         using M [of z] Suc z1  by auto
1314       also have "... \<le> 2 * (norm (c (Suc m)) * norm z ^ Suc m)"
1315         using nz by (simp add: mult_mono del: power_Suc)
1316       finally show "B \<le> norm ((\<Sum>i\<le>m. c i * z^i) + c (Suc m) * z ^ Suc m)"
1317         using Suc.IH
1318         apply (auto simp: eventually_at_infinity)
1319         apply (rule *)
1320         apply (simp add: field_simps norm_mult norm_power)
1321         done
1322     qed
1323     then show ?even