src/HOL/Analysis/Conformal_Mappings.thy
 author wenzelm Sun Sep 18 20:33:48 2016 +0200 (2016-09-18) changeset 63915 bab633745c7f parent 63627 6ddb43c6b711 child 63918 6bf55e6e0b75 permissions -rw-r--r--
tuned proofs;
 lp15@62408 ` 1` ```section \Conformal Mappings. Consequences of Cauchy's integral theorem.\ ``` lp15@62408 ` 2` lp15@62408 ` 3` ```text\By John Harrison et al. Ported from HOL Light by L C Paulson (2016)\ ``` lp15@62408 ` 4` lp15@62540 ` 5` ```text\Also Cauchy's residue theorem by Wenda Li (2016)\ ``` lp15@62540 ` 6` lp15@62408 ` 7` ```theory Conformal_Mappings ``` hoelzl@63627 ` 8` ```imports "~~/src/HOL/Analysis/Cauchy_Integral_Theorem" ``` lp15@62408 ` 9` lp15@62408 ` 10` ```begin ``` lp15@62408 ` 11` lp15@62408 ` 12` ```subsection\Cauchy's inequality and more versions of Liouville\ ``` lp15@62408 ` 13` lp15@62408 ` 14` ```lemma Cauchy_higher_deriv_bound: ``` lp15@62408 ` 15` ``` assumes holf: "f holomorphic_on (ball z r)" ``` lp15@62408 ` 16` ``` and contf: "continuous_on (cball z r) f" ``` lp15@62408 ` 17` ``` and "0 < r" and "0 < n" ``` lp15@62408 ` 18` ``` and fin : "\w. w \ ball z r \ f w \ ball y B0" ``` lp15@62408 ` 19` ``` shows "norm ((deriv ^^ n) f z) \ (fact n) * B0 / r^n" ``` lp15@62408 ` 20` ```proof - ``` lp15@62408 ` 21` ``` have "0 < B0" using \0 < r\ fin [of z] ``` lp15@62408 ` 22` ``` by (metis ball_eq_empty ex_in_conv fin not_less) ``` lp15@62408 ` 23` ``` have le_B0: "\w. cmod (w - z) \ r \ cmod (f w - y) \ B0" ``` lp15@62408 ` 24` ``` apply (rule continuous_on_closure_norm_le [of "ball z r" "\w. f w - y"]) ``` lp15@62408 ` 25` ``` apply (auto simp: \0 < r\ dist_norm norm_minus_commute) ``` lp15@62408 ` 26` ``` apply (rule continuous_intros contf)+ ``` lp15@62408 ` 27` ``` using fin apply (simp add: dist_commute dist_norm less_eq_real_def) ``` lp15@62408 ` 28` ``` done ``` lp15@62408 ` 29` ``` have "(deriv ^^ n) f z = (deriv ^^ n) (\w. f w) z - (deriv ^^ n) (\w. y) z" ``` lp15@62408 ` 30` ``` using \0 < n\ by simp ``` lp15@62408 ` 31` ``` also have "... = (deriv ^^ n) (\w. f w - y) z" ``` lp15@62408 ` 32` ``` by (rule higher_deriv_diff [OF holf, symmetric]) (auto simp: \0 < r\) ``` lp15@62408 ` 33` ``` finally have "(deriv ^^ n) f z = (deriv ^^ n) (\w. f w - y) z" . ``` lp15@62408 ` 34` ``` have contf': "continuous_on (cball z r) (\u. f u - y)" ``` lp15@62408 ` 35` ``` by (rule contf continuous_intros)+ ``` lp15@62408 ` 36` ``` have holf': "(\u. (f u - y)) holomorphic_on (ball z r)" ``` lp15@62408 ` 37` ``` by (simp add: holf holomorphic_on_diff) ``` wenzelm@63040 ` 38` ``` define a where "a = (2 * pi)/(fact n)" ``` lp15@62408 ` 39` ``` have "0 < a" by (simp add: a_def) ``` lp15@62408 ` 40` ``` have "B0/r^(Suc n)*2 * pi * r = a*((fact n)*B0/r^n)" ``` lp15@62408 ` 41` ``` using \0 < r\ by (simp add: a_def divide_simps) ``` lp15@62408 ` 42` ``` have der_dif: "(deriv ^^ n) (\w. f w - y) z = (deriv ^^ n) f z" ``` lp15@62408 ` 43` ``` using \0 < r\ \0 < n\ ``` lp15@62408 ` 44` ``` by (auto simp: higher_deriv_diff [OF holf holomorphic_on_const]) ``` wenzelm@63589 ` 45` ``` have "norm ((2 * of_real pi * \)/(fact n) * (deriv ^^ n) (\w. f w - y) z) ``` lp15@62408 ` 46` ``` \ (B0/r^(Suc n)) * (2 * pi * r)" ``` lp15@62408 ` 47` ``` apply (rule has_contour_integral_bound_circlepath [of "(\u. (f u - y)/(u - z)^(Suc n))" _ z]) ``` lp15@62408 ` 48` ``` using Cauchy_has_contour_integral_higher_derivative_circlepath [OF contf' holf'] ``` lp15@62408 ` 49` ``` using \0 < B0\ \0 < r\ ``` lp15@62408 ` 50` ``` apply (auto simp: norm_divide norm_mult norm_power divide_simps le_B0) ``` lp15@62408 ` 51` ``` done ``` lp15@62408 ` 52` ``` then show ?thesis ``` lp15@62408 ` 53` ``` using \0 < r\ ``` lp15@62408 ` 54` ``` by (auto simp: norm_divide norm_mult norm_power field_simps der_dif le_B0) ``` lp15@62408 ` 55` ```qed ``` lp15@62408 ` 56` lp15@62408 ` 57` ```proposition Cauchy_inequality: ``` lp15@62408 ` 58` ``` assumes holf: "f holomorphic_on (ball \ r)" ``` lp15@62408 ` 59` ``` and contf: "continuous_on (cball \ r) f" ``` lp15@62408 ` 60` ``` and "0 < r" ``` lp15@62408 ` 61` ``` and nof: "\x. norm(\-x) = r \ norm(f x) \ B" ``` lp15@62408 ` 62` ``` shows "norm ((deriv ^^ n) f \) \ (fact n) * B / r^n" ``` lp15@62408 ` 63` ```proof - ``` lp15@62408 ` 64` ``` obtain x where "norm (\-x) = r" ``` lp15@62408 ` 65` ``` by (metis abs_of_nonneg add_diff_cancel_left' \0 < r\ diff_add_cancel ``` lp15@62408 ` 66` ``` dual_order.strict_implies_order norm_of_real) ``` lp15@62408 ` 67` ``` then have "0 \ B" ``` lp15@62408 ` 68` ``` by (metis nof norm_not_less_zero not_le order_trans) ``` lp15@62408 ` 69` ``` have "((\u. f u / (u - \) ^ Suc n) has_contour_integral (2 * pi) * \ / fact n * (deriv ^^ n) f \) ``` lp15@62408 ` 70` ``` (circlepath \ r)" ``` lp15@62408 ` 71` ``` apply (rule Cauchy_has_contour_integral_higher_derivative_circlepath [OF contf holf]) ``` lp15@62408 ` 72` ``` using \0 < r\ by simp ``` wenzelm@63589 ` 73` ``` then have "norm ((2 * pi * \)/(fact n) * (deriv ^^ n) f \) \ (B / r^(Suc n)) * (2 * pi * r)" ``` lp15@62408 ` 74` ``` apply (rule has_contour_integral_bound_circlepath) ``` lp15@62408 ` 75` ``` using \0 \ B\ \0 < r\ ``` lp15@62408 ` 76` ``` apply (simp_all add: norm_divide norm_power nof frac_le norm_minus_commute del: power_Suc) ``` lp15@62408 ` 77` ``` done ``` lp15@62408 ` 78` ``` then show ?thesis using \0 < r\ ``` lp15@62408 ` 79` ``` by (simp add: norm_divide norm_mult field_simps) ``` lp15@62408 ` 80` ```qed ``` lp15@62408 ` 81` lp15@62408 ` 82` ```proposition Liouville_polynomial: ``` lp15@62408 ` 83` ``` assumes holf: "f holomorphic_on UNIV" ``` lp15@62408 ` 84` ``` and nof: "\z. A \ norm z \ norm(f z) \ B * norm z ^ n" ``` lp15@62408 ` 85` ``` shows "f \ = (\k\n. (deriv^^k) f 0 / fact k * \ ^ k)" ``` lp15@62408 ` 86` ```proof (cases rule: le_less_linear [THEN disjE]) ``` lp15@62408 ` 87` ``` assume "B \ 0" ``` lp15@62408 ` 88` ``` then have "\z. A \ norm z \ norm(f z) = 0" ``` lp15@62408 ` 89` ``` by (metis nof less_le_trans zero_less_mult_iff neqE norm_not_less_zero norm_power not_le) ``` lp15@62408 ` 90` ``` then have f0: "(f \ 0) at_infinity" ``` lp15@62408 ` 91` ``` using Lim_at_infinity by force ``` lp15@62408 ` 92` ``` then have [simp]: "f = (\w. 0)" ``` lp15@62408 ` 93` ``` using Liouville_weak [OF holf, of 0] ``` lp15@62408 ` 94` ``` by (simp add: eventually_at_infinity f0) meson ``` lp15@62408 ` 95` ``` show ?thesis by simp ``` lp15@62408 ` 96` ```next ``` lp15@62408 ` 97` ``` assume "0 < B" ``` lp15@62408 ` 98` ``` have "((\k. (deriv ^^ k) f 0 / (fact k) * (\ - 0)^k) sums f \)" ``` lp15@62408 ` 99` ``` apply (rule holomorphic_power_series [where r = "norm \ + 1"]) ``` lp15@62408 ` 100` ``` using holf holomorphic_on_subset apply auto ``` lp15@62408 ` 101` ``` done ``` lp15@62408 ` 102` ``` then have sumsf: "((\k. (deriv ^^ k) f 0 / (fact k) * \^k) sums f \)" by simp ``` lp15@62408 ` 103` ``` have "(deriv ^^ k) f 0 / fact k * \ ^ k = 0" if "k>n" for k ``` lp15@62408 ` 104` ``` proof (cases "(deriv ^^ k) f 0 = 0") ``` lp15@62408 ` 105` ``` case True then show ?thesis by simp ``` lp15@62408 ` 106` ``` next ``` lp15@62408 ` 107` ``` case False ``` wenzelm@63040 ` 108` ``` define w where "w = complex_of_real (fact k * B / cmod ((deriv ^^ k) f 0) + (\A\ + 1))" ``` wenzelm@63040 ` 109` ``` have "1 \ abs (fact k * B / cmod ((deriv ^^ k) f 0) + (\A\ + 1))" ``` wenzelm@63040 ` 110` ``` using \0 < B\ by simp ``` wenzelm@63040 ` 111` ``` then have wge1: "1 \ norm w" ``` wenzelm@63040 ` 112` ``` by (metis norm_of_real w_def) ``` wenzelm@63040 ` 113` ``` then have "w \ 0" by auto ``` wenzelm@63040 ` 114` ``` have kB: "0 < fact k * B" ``` wenzelm@63040 ` 115` ``` using \0 < B\ by simp ``` wenzelm@63040 ` 116` ``` then have "0 \ fact k * B / cmod ((deriv ^^ k) f 0)" ``` wenzelm@63040 ` 117` ``` by simp ``` wenzelm@63040 ` 118` ``` then have wgeA: "A \ cmod w" ``` wenzelm@63040 ` 119` ``` by (simp only: w_def norm_of_real) ``` wenzelm@63040 ` 120` ``` have "fact k * B / cmod ((deriv ^^ k) f 0) < abs (fact k * B / cmod ((deriv ^^ k) f 0) + (\A\ + 1))" ``` wenzelm@63040 ` 121` ``` using \0 < B\ by simp ``` wenzelm@63040 ` 122` ``` then have wge: "fact k * B / cmod ((deriv ^^ k) f 0) < norm w" ``` wenzelm@63040 ` 123` ``` by (metis norm_of_real w_def) ``` wenzelm@63040 ` 124` ``` then have "fact k * B / norm w < cmod ((deriv ^^ k) f 0)" ``` wenzelm@63040 ` 125` ``` using False by (simp add: divide_simps mult.commute split: if_split_asm) ``` wenzelm@63040 ` 126` ``` also have "... \ fact k * (B * norm w ^ n) / norm w ^ k" ``` wenzelm@63040 ` 127` ``` apply (rule Cauchy_inequality) ``` wenzelm@63040 ` 128` ``` using holf holomorphic_on_subset apply force ``` wenzelm@63040 ` 129` ``` using holf holomorphic_on_imp_continuous_on holomorphic_on_subset apply blast ``` wenzelm@63040 ` 130` ``` using \w \ 0\ apply (simp add:) ``` wenzelm@63040 ` 131` ``` by (metis nof wgeA dist_0_norm dist_norm) ``` wenzelm@63040 ` 132` ``` also have "... = fact k * (B * 1 / cmod w ^ (k-n))" ``` wenzelm@63040 ` 133` ``` apply (simp only: mult_cancel_left times_divide_eq_right [symmetric]) ``` wenzelm@63040 ` 134` ``` using \k>n\ \w \ 0\ \0 < B\ apply (simp add: divide_simps semiring_normalization_rules) ``` wenzelm@63040 ` 135` ``` done ``` wenzelm@63040 ` 136` ``` also have "... = fact k * B / cmod w ^ (k-n)" ``` wenzelm@63040 ` 137` ``` by simp ``` wenzelm@63040 ` 138` ``` finally have "fact k * B / cmod w < fact k * B / cmod w ^ (k - n)" . ``` wenzelm@63040 ` 139` ``` then have "1 / cmod w < 1 / cmod w ^ (k - n)" ``` wenzelm@63040 ` 140` ``` by (metis kB divide_inverse inverse_eq_divide mult_less_cancel_left_pos) ``` wenzelm@63040 ` 141` ``` then have "cmod w ^ (k - n) < cmod w" ``` wenzelm@63040 ` 142` ``` by (metis frac_le le_less_trans norm_ge_zero norm_one not_less order_refl wge1 zero_less_one) ``` wenzelm@63040 ` 143` ``` with self_le_power [OF wge1] have False ``` wenzelm@63040 ` 144` ``` by (meson diff_is_0_eq not_gr0 not_le that) ``` wenzelm@63040 ` 145` ``` then show ?thesis by blast ``` lp15@62408 ` 146` ``` qed ``` lp15@62408 ` 147` ``` then have "(deriv ^^ (k + Suc n)) f 0 / fact (k + Suc n) * \ ^ (k + Suc n) = 0" for k ``` lp15@62408 ` 148` ``` using not_less_eq by blast ``` lp15@62408 ` 149` ``` then have "(\i. (deriv ^^ (i + Suc n)) f 0 / fact (i + Suc n) * \ ^ (i + Suc n)) sums 0" ``` lp15@62408 ` 150` ``` by (rule sums_0) ``` lp15@62408 ` 151` ``` with sums_split_initial_segment [OF sumsf, where n = "Suc n"] ``` lp15@62408 ` 152` ``` show ?thesis ``` lp15@62408 ` 153` ``` using atLeast0AtMost lessThan_Suc_atMost sums_unique2 by fastforce ``` lp15@62408 ` 154` ```qed ``` lp15@62408 ` 155` lp15@62408 ` 156` ```text\Every bounded entire function is a constant function.\ ``` lp15@62408 ` 157` ```theorem Liouville_theorem: ``` lp15@62408 ` 158` ``` assumes holf: "f holomorphic_on UNIV" ``` lp15@62408 ` 159` ``` and bf: "bounded (range f)" ``` lp15@62408 ` 160` ``` obtains c where "\z. f z = c" ``` lp15@62408 ` 161` ```proof - ``` lp15@62408 ` 162` ``` obtain B where "\z. cmod (f z) \ B" ``` lp15@62408 ` 163` ``` by (meson bf bounded_pos rangeI) ``` lp15@62408 ` 164` ``` then show ?thesis ``` lp15@62408 ` 165` ``` using Liouville_polynomial [OF holf, of 0 B 0, simplified] that by blast ``` lp15@62408 ` 166` ```qed ``` lp15@62408 ` 167` lp15@62408 ` 168` lp15@62408 ` 169` lp15@62408 ` 170` ```text\A holomorphic function f has only isolated zeros unless f is 0.\ ``` lp15@62408 ` 171` lp15@62408 ` 172` ```proposition powser_0_nonzero: ``` lp15@62408 ` 173` ``` fixes a :: "nat \ 'a::{real_normed_field,banach}" ``` lp15@62408 ` 174` ``` assumes r: "0 < r" ``` lp15@62408 ` 175` ``` and sm: "\x. norm (x - \) < r \ (\n. a n * (x - \) ^ n) sums (f x)" ``` lp15@62408 ` 176` ``` and [simp]: "f \ = 0" ``` lp15@62408 ` 177` ``` and m0: "a m \ 0" and "m>0" ``` lp15@62408 ` 178` ``` obtains s where "0 < s" and "\z. z \ cball \ s - {\} \ f z \ 0" ``` lp15@62408 ` 179` ```proof - ``` lp15@62408 ` 180` ``` have "r \ conv_radius a" ``` lp15@62408 ` 181` ``` using sm sums_summable by (auto simp: le_conv_radius_iff [where \=\]) ``` lp15@62408 ` 182` ``` obtain m where am: "a m \ 0" and az [simp]: "(\n. n a n = 0)" ``` lp15@62408 ` 183` ``` apply (rule_tac m = "LEAST n. a n \ 0" in that) ``` lp15@62408 ` 184` ``` using m0 ``` lp15@62408 ` 185` ``` apply (rule LeastI2) ``` lp15@62408 ` 186` ``` apply (fastforce intro: dest!: not_less_Least)+ ``` lp15@62408 ` 187` ``` done ``` wenzelm@63040 ` 188` ``` define b where "b i = a (i+m) / a m" for i ``` wenzelm@63040 ` 189` ``` define g where "g x = suminf (\i. b i * (x - \) ^ i)" for x ``` lp15@62408 ` 190` ``` have [simp]: "b 0 = 1" ``` lp15@62408 ` 191` ``` by (simp add: am b_def) ``` lp15@62408 ` 192` ``` { fix x::'a ``` lp15@62408 ` 193` ``` assume "norm (x - \) < r" ``` lp15@62408 ` 194` ``` then have "(\n. (a m * (x - \)^m) * (b n * (x - \)^n)) sums (f x)" ``` lp15@62408 ` 195` ``` using am az sm sums_zero_iff_shift [of m "(\n. a n * (x - \) ^ n)" "f x"] ``` lp15@62408 ` 196` ``` by (simp add: b_def monoid_mult_class.power_add algebra_simps) ``` lp15@62408 ` 197` ``` then have "x \ \ \ (\n. b n * (x - \)^n) sums (f x / (a m * (x - \)^m))" ``` lp15@62408 ` 198` ``` using am by (simp add: sums_mult_D) ``` lp15@62408 ` 199` ``` } note bsums = this ``` lp15@62408 ` 200` ``` then have "norm (x - \) < r \ summable (\n. b n * (x - \)^n)" for x ``` lp15@62408 ` 201` ``` using sums_summable by (cases "x=\") auto ``` lp15@62408 ` 202` ``` then have "r \ conv_radius b" ``` lp15@62408 ` 203` ``` by (simp add: le_conv_radius_iff [where \=\]) ``` lp15@62408 ` 204` ``` then have "r/2 < conv_radius b" ``` lp15@62408 ` 205` ``` using not_le order_trans r by fastforce ``` lp15@62408 ` 206` ``` then have "continuous_on (cball \ (r/2)) g" ``` lp15@62408 ` 207` ``` using powser_continuous_suminf [of "r/2" b \] by (simp add: g_def) ``` lp15@62408 ` 208` ``` then obtain s where "s>0" "\x. \norm (x - \) \ s; norm (x - \) \ r/2\ \ dist (g x) (g \) < 1/2" ``` lp15@62408 ` 209` ``` apply (rule continuous_onE [where x=\ and e = "1/2"]) ``` lp15@62408 ` 210` ``` using r apply (auto simp: norm_minus_commute dist_norm) ``` lp15@62408 ` 211` ``` done ``` lp15@62408 ` 212` ``` moreover have "g \ = 1" ``` lp15@62408 ` 213` ``` by (simp add: g_def) ``` lp15@62408 ` 214` ``` ultimately have gnz: "\x. \norm (x - \) \ s; norm (x - \) \ r/2\ \ (g x) \ 0" ``` lp15@62408 ` 215` ``` by fastforce ``` lp15@62408 ` 216` ``` have "f x \ 0" if "x \ \" "norm (x - \) \ s" "norm (x - \) \ r/2" for x ``` lp15@62408 ` 217` ``` using bsums [of x] that gnz [of x] ``` lp15@62408 ` 218` ``` apply (auto simp: g_def) ``` lp15@62408 ` 219` ``` using r sums_iff by fastforce ``` lp15@62408 ` 220` ``` then show ?thesis ``` lp15@62408 ` 221` ``` apply (rule_tac s="min s (r/2)" in that) ``` lp15@62408 ` 222` ``` using \0 < r\ \0 < s\ by (auto simp: dist_commute dist_norm) ``` lp15@62408 ` 223` ```qed ``` lp15@62408 ` 224` lp15@62408 ` 225` ```proposition isolated_zeros: ``` lp15@62408 ` 226` ``` assumes holf: "f holomorphic_on S" ``` lp15@62408 ` 227` ``` and "open S" "connected S" "\ \ S" "f \ = 0" "\ \ S" "f \ \ 0" ``` lp15@62408 ` 228` ``` obtains r where "0 < r" "ball \ r \ S" "\z. z \ ball \ r - {\} \ f z \ 0" ``` lp15@62408 ` 229` ```proof - ``` lp15@62408 ` 230` ``` obtain r where "0 < r" and r: "ball \ r \ S" ``` lp15@62408 ` 231` ``` using \open S\ \\ \ S\ open_contains_ball_eq by blast ``` lp15@62408 ` 232` ``` have powf: "((\n. (deriv ^^ n) f \ / (fact n) * (z - \)^n) sums f z)" if "z \ ball \ r" for z ``` lp15@62408 ` 233` ``` apply (rule holomorphic_power_series [OF _ that]) ``` lp15@62408 ` 234` ``` apply (rule holomorphic_on_subset [OF holf r]) ``` lp15@62408 ` 235` ``` done ``` lp15@62408 ` 236` ``` obtain m where m: "(deriv ^^ m) f \ / (fact m) \ 0" ``` lp15@62408 ` 237` ``` using holomorphic_fun_eq_0_on_connected [OF holf \open S\ \connected S\ _ \\ \ S\ \\ \ S\] \f \ \ 0\ ``` lp15@62408 ` 238` ``` by auto ``` lp15@62408 ` 239` ``` then have "m \ 0" using assms(5) funpow_0 by fastforce ``` lp15@62408 ` 240` ``` obtain s where "0 < s" and s: "\z. z \ cball \ s - {\} \ f z \ 0" ``` lp15@62408 ` 241` ``` apply (rule powser_0_nonzero [OF \0 < r\ powf \f \ = 0\ m]) ``` lp15@62408 ` 242` ``` using \m \ 0\ by (auto simp: dist_commute dist_norm) ``` lp15@62408 ` 243` ``` have "0 < min r s" by (simp add: \0 < r\ \0 < s\) ``` lp15@62408 ` 244` ``` then show ?thesis ``` lp15@62408 ` 245` ``` apply (rule that) ``` lp15@62408 ` 246` ``` using r s by auto ``` lp15@62408 ` 247` ```qed ``` lp15@62408 ` 248` lp15@62408 ` 249` lp15@62408 ` 250` ```proposition analytic_continuation: ``` lp15@62408 ` 251` ``` assumes holf: "f holomorphic_on S" ``` lp15@62408 ` 252` ``` and S: "open S" "connected S" ``` lp15@62408 ` 253` ``` and "U \ S" "\ \ S" ``` lp15@62408 ` 254` ``` and "\ islimpt U" ``` lp15@62408 ` 255` ``` and fU0 [simp]: "\z. z \ U \ f z = 0" ``` lp15@62408 ` 256` ``` and "w \ S" ``` lp15@62408 ` 257` ``` shows "f w = 0" ``` lp15@62408 ` 258` ```proof - ``` lp15@62408 ` 259` ``` obtain e where "0 < e" and e: "cball \ e \ S" ``` lp15@62408 ` 260` ``` using \open S\ \\ \ S\ open_contains_cball_eq by blast ``` wenzelm@63040 ` 261` ``` define T where "T = cball \ e \ U" ``` lp15@62408 ` 262` ``` have contf: "continuous_on (closure T) f" ``` lp15@62408 ` 263` ``` by (metis T_def closed_cball closure_minimal e holf holomorphic_on_imp_continuous_on ``` lp15@62408 ` 264` ``` holomorphic_on_subset inf.cobounded1) ``` lp15@62408 ` 265` ``` have fT0 [simp]: "\x. x \ T \ f x = 0" ``` lp15@62408 ` 266` ``` by (simp add: T_def) ``` lp15@62408 ` 267` ``` have "\r. \\e>0. \x'\U. x' \ \ \ dist x' \ < e; 0 < r\ \ \x'\cball \ e \ U. x' \ \ \ dist x' \ < r" ``` lp15@62408 ` 268` ``` by (metis \0 < e\ IntI dist_commute less_eq_real_def mem_cball min_less_iff_conj) ``` lp15@62408 ` 269` ``` then have "\ islimpt T" using \\ islimpt U\ ``` lp15@62408 ` 270` ``` by (auto simp: T_def islimpt_approachable) ``` lp15@62408 ` 271` ``` then have "\ \ closure T" ``` lp15@62408 ` 272` ``` by (simp add: closure_def) ``` lp15@62408 ` 273` ``` then have "f \ = 0" ``` lp15@62408 ` 274` ``` by (auto simp: continuous_constant_on_closure [OF contf]) ``` lp15@62408 ` 275` ``` show ?thesis ``` lp15@62408 ` 276` ``` apply (rule ccontr) ``` lp15@62408 ` 277` ``` apply (rule isolated_zeros [OF holf \open S\ \connected S\ \\ \ S\ \f \ = 0\ \w \ S\], assumption) ``` lp15@62408 ` 278` ``` by (metis open_ball \\ islimpt T\ centre_in_ball fT0 insertE insert_Diff islimptE) ``` lp15@62408 ` 279` ```qed ``` lp15@62408 ` 280` lp15@62408 ` 281` lp15@62408 ` 282` ```subsection\Open mapping theorem\ ``` lp15@62408 ` 283` lp15@62408 ` 284` ```lemma holomorphic_contract_to_zero: ``` lp15@62408 ` 285` ``` assumes contf: "continuous_on (cball \ r) f" ``` lp15@62408 ` 286` ``` and holf: "f holomorphic_on ball \ r" ``` lp15@62408 ` 287` ``` and "0 < r" ``` lp15@62408 ` 288` ``` and norm_less: "\z. norm(\ - z) = r \ norm(f \) < norm(f z)" ``` lp15@62408 ` 289` ``` obtains z where "z \ ball \ r" "f z = 0" ``` lp15@62408 ` 290` ```proof - ``` lp15@62408 ` 291` ``` { assume fnz: "\w. w \ ball \ r \ f w \ 0" ``` lp15@62408 ` 292` ``` then have "0 < norm (f \)" ``` lp15@62408 ` 293` ``` by (simp add: \0 < r\) ``` lp15@62408 ` 294` ``` have fnz': "\w. w \ cball \ r \ f w \ 0" ``` lp15@62408 ` 295` ``` by (metis norm_less dist_norm fnz less_eq_real_def mem_ball mem_cball norm_not_less_zero norm_zero) ``` lp15@62408 ` 296` ``` have "frontier(cball \ r) \ {}" ``` lp15@62408 ` 297` ``` using \0 < r\ by simp ``` wenzelm@63040 ` 298` ``` define g where [abs_def]: "g z = inverse (f z)" for z ``` lp15@62408 ` 299` ``` have contg: "continuous_on (cball \ r) g" ``` lp15@62408 ` 300` ``` unfolding g_def using contf continuous_on_inverse fnz' by blast ``` lp15@62408 ` 301` ``` have holg: "g holomorphic_on ball \ r" ``` lp15@62408 ` 302` ``` unfolding g_def using fnz holf holomorphic_on_inverse by blast ``` lp15@62408 ` 303` ``` have "frontier (cball \ r) \ cball \ r" ``` lp15@62408 ` 304` ``` by (simp add: subset_iff) ``` lp15@62408 ` 305` ``` then have contf': "continuous_on (frontier (cball \ r)) f" ``` lp15@62408 ` 306` ``` and contg': "continuous_on (frontier (cball \ r)) g" ``` lp15@62408 ` 307` ``` by (blast intro: contf contg continuous_on_subset)+ ``` lp15@62408 ` 308` ``` have froc: "frontier(cball \ r) \ {}" ``` lp15@62408 ` 309` ``` using \0 < r\ by simp ``` lp15@62408 ` 310` ``` moreover have "continuous_on (frontier (cball \ r)) (norm o f)" ``` lp15@62408 ` 311` ``` using contf' continuous_on_compose continuous_on_norm_id by blast ``` lp15@62408 ` 312` ``` ultimately obtain w where w: "w \ frontier(cball \ r)" ``` lp15@62408 ` 313` ``` and now: "\x. x \ frontier(cball \ r) \ norm (f w) \ norm (f x)" ``` lp15@62408 ` 314` ``` apply (rule bexE [OF continuous_attains_inf [OF compact_frontier [OF compact_cball]]]) ``` lp15@62408 ` 315` ``` apply (simp add:) ``` lp15@62408 ` 316` ``` done ``` lp15@62408 ` 317` ``` then have fw: "0 < norm (f w)" ``` lp15@62408 ` 318` ``` by (simp add: fnz') ``` lp15@62408 ` 319` ``` have "continuous_on (frontier (cball \ r)) (norm o g)" ``` lp15@62408 ` 320` ``` using contg' continuous_on_compose continuous_on_norm_id by blast ``` lp15@62408 ` 321` ``` then obtain v where v: "v \ frontier(cball \ r)" ``` lp15@62408 ` 322` ``` and nov: "\x. x \ frontier(cball \ r) \ norm (g v) \ norm (g x)" ``` lp15@62408 ` 323` ``` apply (rule bexE [OF continuous_attains_sup [OF compact_frontier [OF compact_cball] froc]]) ``` lp15@62408 ` 324` ``` apply (simp add:) ``` lp15@62408 ` 325` ``` done ``` lp15@62408 ` 326` ``` then have fv: "0 < norm (f v)" ``` lp15@62408 ` 327` ``` by (simp add: fnz') ``` lp15@62408 ` 328` ``` have "norm ((deriv ^^ 0) g \) \ fact 0 * norm (g v) / r ^ 0" ``` lp15@62408 ` 329` ``` by (rule Cauchy_inequality [OF holg contg \0 < r\]) (simp add: dist_norm nov) ``` lp15@62408 ` 330` ``` then have "cmod (g \) \ norm (g v)" ``` lp15@62408 ` 331` ``` by simp ``` lp15@62408 ` 332` ``` with w have wr: "norm (\ - w) = r" and nfw: "norm (f w) \ norm (f \)" ``` lp15@62408 ` 333` ``` apply (simp_all add: dist_norm) ``` lp15@62408 ` 334` ``` by (metis \0 < cmod (f \)\ g_def less_imp_inverse_less norm_inverse not_le now order_trans v) ``` lp15@62408 ` 335` ``` with fw have False ``` lp15@62408 ` 336` ``` using norm_less by force ``` lp15@62408 ` 337` ``` } ``` lp15@62408 ` 338` ``` with that show ?thesis by blast ``` lp15@62408 ` 339` ```qed ``` lp15@62408 ` 340` lp15@62408 ` 341` lp15@62408 ` 342` ```theorem open_mapping_thm: ``` lp15@62408 ` 343` ``` assumes holf: "f holomorphic_on S" ``` lp15@62408 ` 344` ``` and S: "open S" "connected S" ``` lp15@62408 ` 345` ``` and "open U" "U \ S" ``` lp15@62408 ` 346` ``` and fne: "~ f constant_on S" ``` lp15@62408 ` 347` ``` shows "open (f ` U)" ``` lp15@62408 ` 348` ```proof - ``` lp15@62408 ` 349` ``` have *: "open (f ` U)" ``` lp15@62408 ` 350` ``` if "U \ {}" and U: "open U" "connected U" and "f holomorphic_on U" and fneU: "\x. \y \ U. f y \ x" ``` lp15@62408 ` 351` ``` for U ``` lp15@62408 ` 352` ``` proof (clarsimp simp: open_contains_ball) ``` lp15@62408 ` 353` ``` fix \ assume \: "\ \ U" ``` lp15@62408 ` 354` ``` show "\e>0. ball (f \) e \ f ` U" ``` lp15@62408 ` 355` ``` proof - ``` lp15@62408 ` 356` ``` have hol: "(\z. f z - f \) holomorphic_on U" ``` lp15@62408 ` 357` ``` by (rule holomorphic_intros that)+ ``` lp15@62408 ` 358` ``` obtain s where "0 < s" and sbU: "ball \ s \ U" ``` lp15@62408 ` 359` ``` and sne: "\z. z \ ball \ s - {\} \ (\z. f z - f \) z \ 0" ``` lp15@62408 ` 360` ``` using isolated_zeros [OF hol U \] by (metis fneU right_minus_eq) ``` lp15@62408 ` 361` ``` obtain r where "0 < r" and r: "cball \ r \ ball \ s" ``` lp15@62408 ` 362` ``` apply (rule_tac r="s/2" in that) ``` lp15@62408 ` 363` ``` using \0 < s\ by auto ``` lp15@62408 ` 364` ``` have "cball \ r \ U" ``` lp15@62408 ` 365` ``` using sbU r by blast ``` lp15@62408 ` 366` ``` then have frsbU: "frontier (cball \ r) \ U" ``` lp15@62408 ` 367` ``` using Diff_subset frontier_def order_trans by fastforce ``` lp15@62408 ` 368` ``` then have cof: "compact (frontier(cball \ r))" ``` lp15@62408 ` 369` ``` by blast ``` lp15@62408 ` 370` ``` have frne: "frontier (cball \ r) \ {}" ``` lp15@62408 ` 371` ``` using \0 < r\ by auto ``` lp15@62408 ` 372` ``` have contfr: "continuous_on (frontier (cball \ r)) (\z. norm (f z - f \))" ``` lp15@62408 ` 373` ``` apply (rule continuous_on_compose2 [OF Complex_Analysis_Basics.continuous_on_norm_id]) ``` lp15@62408 ` 374` ``` using hol frsbU holomorphic_on_imp_continuous_on holomorphic_on_subset by blast+ ``` lp15@62408 ` 375` ``` obtain w where "norm (\ - w) = r" ``` lp15@62408 ` 376` ``` and w: "(\z. norm (\ - z) = r \ norm (f w - f \) \ norm(f z - f \))" ``` lp15@62408 ` 377` ``` apply (rule bexE [OF continuous_attains_inf [OF cof frne contfr]]) ``` lp15@62408 ` 378` ``` apply (simp add: dist_norm) ``` lp15@62408 ` 379` ``` done ``` wenzelm@63040 ` 380` ``` moreover define \ where "\ \ norm (f w - f \) / 3" ``` lp15@62408 ` 381` ``` ultimately have "0 < \" ``` lp15@62408 ` 382` ``` using \0 < r\ dist_complex_def r sne by auto ``` lp15@62408 ` 383` ``` have "ball (f \) \ \ f ` U" ``` lp15@62408 ` 384` ``` proof ``` lp15@62408 ` 385` ``` fix \ ``` lp15@62408 ` 386` ``` assume \: "\ \ ball (f \) \" ``` lp15@62408 ` 387` ``` have *: "cmod (\ - f \) < cmod (\ - f z)" if "cmod (\ - z) = r" for z ``` lp15@62408 ` 388` ``` proof - ``` lp15@62408 ` 389` ``` have lt: "cmod (f w - f \) / 3 < cmod (\ - f z)" ``` lp15@62408 ` 390` ``` using w [OF that] \ ``` lp15@62408 ` 391` ``` using dist_triangle2 [of "f \" "\" "f z"] dist_triangle2 [of "f \" "f z" \] ``` lp15@62408 ` 392` ``` by (simp add: \_def dist_norm norm_minus_commute) ``` lp15@62408 ` 393` ``` show ?thesis ``` lp15@62408 ` 394` ``` by (metis \_def dist_commute dist_norm less_trans lt mem_ball \) ``` lp15@62408 ` 395` ``` qed ``` lp15@62408 ` 396` ``` have "continuous_on (cball \ r) (\z. \ - f z)" ``` lp15@62408 ` 397` ``` apply (rule continuous_intros)+ ``` lp15@62408 ` 398` ``` using \cball \ r \ U\ \f holomorphic_on U\ ``` lp15@62408 ` 399` ``` apply (blast intro: continuous_on_subset holomorphic_on_imp_continuous_on) ``` lp15@62408 ` 400` ``` done ``` lp15@62408 ` 401` ``` moreover have "(\z. \ - f z) holomorphic_on ball \ r" ``` lp15@62408 ` 402` ``` apply (rule holomorphic_intros)+ ``` lp15@62408 ` 403` ``` apply (metis \cball \ r \ U\ \f holomorphic_on U\ holomorphic_on_subset interior_cball interior_subset) ``` lp15@62408 ` 404` ``` done ``` lp15@62408 ` 405` ``` ultimately obtain z where "z \ ball \ r" "\ - f z = 0" ``` lp15@62408 ` 406` ``` apply (rule holomorphic_contract_to_zero) ``` lp15@62408 ` 407` ``` apply (blast intro!: \0 < r\ *)+ ``` lp15@62408 ` 408` ``` done ``` lp15@62408 ` 409` ``` then show "\ \ f ` U" ``` lp15@62408 ` 410` ``` using \cball \ r \ U\ by fastforce ``` lp15@62408 ` 411` ``` qed ``` lp15@62408 ` 412` ``` then show ?thesis using \0 < \\ by blast ``` lp15@62408 ` 413` ``` qed ``` lp15@62408 ` 414` ``` qed ``` lp15@62408 ` 415` ``` have "open (f ` X)" if "X \ components U" for X ``` lp15@62408 ` 416` ``` proof - ``` lp15@62408 ` 417` ``` have holfU: "f holomorphic_on U" ``` lp15@62408 ` 418` ``` using \U \ S\ holf holomorphic_on_subset by blast ``` lp15@62408 ` 419` ``` have "X \ {}" ``` lp15@62408 ` 420` ``` using that by (simp add: in_components_nonempty) ``` lp15@62408 ` 421` ``` moreover have "open X" ``` lp15@62408 ` 422` ``` using that \open U\ open_components by auto ``` lp15@62408 ` 423` ``` moreover have "connected X" ``` lp15@62408 ` 424` ``` using that in_components_maximal by blast ``` lp15@62408 ` 425` ``` moreover have "f holomorphic_on X" ``` lp15@62408 ` 426` ``` by (meson that holfU holomorphic_on_subset in_components_maximal) ``` lp15@62408 ` 427` ``` moreover have "\y\X. f y \ x" for x ``` lp15@62408 ` 428` ``` proof (rule ccontr) ``` lp15@62408 ` 429` ``` assume not: "\ (\y\X. f y \ x)" ``` lp15@62408 ` 430` ``` have "X \ S" ``` lp15@62408 ` 431` ``` using \U \ S\ in_components_subset that by blast ``` lp15@62408 ` 432` ``` obtain w where w: "w \ X" using \X \ {}\ by blast ``` lp15@62408 ` 433` ``` have wis: "w islimpt X" ``` lp15@62408 ` 434` ``` using w \open X\ interior_eq by auto ``` lp15@62408 ` 435` ``` have hol: "(\z. f z - x) holomorphic_on S" ``` lp15@62408 ` 436` ``` by (simp add: holf holomorphic_on_diff) ``` lp15@62408 ` 437` ``` with fne [unfolded constant_on_def] analytic_continuation [OF hol S \X \ S\ _ wis] ``` lp15@62408 ` 438` ``` not \X \ S\ w ``` lp15@62408 ` 439` ``` show False by auto ``` lp15@62408 ` 440` ``` qed ``` lp15@62408 ` 441` ``` ultimately show ?thesis ``` lp15@62408 ` 442` ``` by (rule *) ``` lp15@62408 ` 443` ``` qed ``` lp15@62843 ` 444` ``` then have "open (f ` \components U)" ``` lp15@62843 ` 445` ``` by (metis (no_types, lifting) imageE image_Union open_Union) ``` lp15@62408 ` 446` ``` then show ?thesis ``` lp15@62843 ` 447` ``` by force ``` lp15@62408 ` 448` ```qed ``` lp15@62408 ` 449` lp15@62408 ` 450` lp15@62408 ` 451` ```text\No need for @{term S} to be connected. But the nonconstant condition is stronger.\ ``` lp15@62408 ` 452` ```corollary open_mapping_thm2: ``` lp15@62408 ` 453` ``` assumes holf: "f holomorphic_on S" ``` lp15@62408 ` 454` ``` and S: "open S" ``` lp15@62408 ` 455` ``` and "open U" "U \ S" ``` lp15@62408 ` 456` ``` and fnc: "\X. \open X; X \ S; X \ {}\ \ ~ f constant_on X" ``` lp15@62408 ` 457` ``` shows "open (f ` U)" ``` lp15@62408 ` 458` ```proof - ``` lp15@62843 ` 459` ``` have "S = \(components S)" by simp ``` lp15@62408 ` 460` ``` with \U \ S\ have "U = (\C \ components S. C \ U)" by auto ``` lp15@62408 ` 461` ``` then have "f ` U = (\C \ components S. f ` (C \ U))" ``` lp15@62843 ` 462` ``` using image_UN by fastforce ``` lp15@62408 ` 463` ``` moreover ``` lp15@62408 ` 464` ``` { fix C assume "C \ components S" ``` lp15@62408 ` 465` ``` with S \C \ components S\ open_components in_components_connected ``` lp15@62408 ` 466` ``` have C: "open C" "connected C" by auto ``` lp15@62408 ` 467` ``` have "C \ S" ``` lp15@62408 ` 468` ``` by (metis \C \ components S\ in_components_maximal) ``` lp15@62408 ` 469` ``` have nf: "\ f constant_on C" ``` lp15@62408 ` 470` ``` apply (rule fnc) ``` lp15@62408 ` 471` ``` using C \C \ S\ \C \ components S\ in_components_nonempty by auto ``` lp15@62408 ` 472` ``` have "f holomorphic_on C" ``` lp15@62408 ` 473` ``` by (metis holf holomorphic_on_subset \C \ S\) ``` lp15@62408 ` 474` ``` then have "open (f ` (C \ U))" ``` lp15@62408 ` 475` ``` apply (rule open_mapping_thm [OF _ C _ _ nf]) ``` lp15@62408 ` 476` ``` apply (simp add: C \open U\ open_Int, blast) ``` lp15@62408 ` 477` ``` done ``` lp15@62408 ` 478` ``` } ultimately show ?thesis ``` lp15@62408 ` 479` ``` by force ``` lp15@62408 ` 480` ```qed ``` lp15@62408 ` 481` lp15@62408 ` 482` ```corollary open_mapping_thm3: ``` lp15@62408 ` 483` ``` assumes holf: "f holomorphic_on S" ``` lp15@62408 ` 484` ``` and "open S" and injf: "inj_on f S" ``` lp15@62408 ` 485` ``` shows "open (f ` S)" ``` lp15@62408 ` 486` ```apply (rule open_mapping_thm2 [OF holf]) ``` lp15@62408 ` 487` ```using assms ``` lp15@62408 ` 488` ```apply (simp_all add:) ``` lp15@62408 ` 489` ```using injective_not_constant subset_inj_on by blast ``` lp15@62408 ` 490` lp15@62408 ` 491` lp15@62408 ` 492` lp15@62408 ` 493` ```subsection\Maximum Modulus Principle\ ``` lp15@62408 ` 494` lp15@62408 ` 495` ```text\If @{term f} is holomorphic, then its norm (modulus) cannot exhibit a true local maximum that is ``` lp15@62408 ` 496` ``` properly within the domain of @{term f}.\ ``` lp15@62408 ` 497` lp15@62408 ` 498` ```proposition maximum_modulus_principle: ``` lp15@62408 ` 499` ``` assumes holf: "f holomorphic_on S" ``` lp15@62408 ` 500` ``` and S: "open S" "connected S" ``` lp15@62408 ` 501` ``` and "open U" "U \ S" "\ \ U" ``` lp15@62408 ` 502` ``` and no: "\z. z \ U \ norm(f z) \ norm(f \)" ``` lp15@62408 ` 503` ``` shows "f constant_on S" ``` lp15@62408 ` 504` ```proof (rule ccontr) ``` lp15@62408 ` 505` ``` assume "\ f constant_on S" ``` lp15@62408 ` 506` ``` then have "open (f ` U)" ``` lp15@62408 ` 507` ``` using open_mapping_thm assms by blast ``` lp15@62408 ` 508` ``` moreover have "~ open (f ` U)" ``` lp15@62408 ` 509` ``` proof - ``` lp15@62408 ` 510` ``` have "\t. cmod (f \ - t) < e \ t \ f ` U" if "0 < e" for e ``` lp15@62408 ` 511` ``` apply (rule_tac x="if 0 < Re(f \) then f \ + (e/2) else f \ - (e/2)" in exI) ``` lp15@62408 ` 512` ``` using that ``` lp15@62408 ` 513` ``` apply (simp add: dist_norm) ``` lp15@62408 ` 514` ``` apply (fastforce simp: cmod_Re_le_iff dest!: no dest: sym) ``` lp15@62408 ` 515` ``` done ``` lp15@62408 ` 516` ``` then show ?thesis ``` lp15@62408 ` 517` ``` unfolding open_contains_ball by (metis \\ \ U\ contra_subsetD dist_norm imageI mem_ball) ``` lp15@62408 ` 518` ``` qed ``` lp15@62408 ` 519` ``` ultimately show False ``` lp15@62408 ` 520` ``` by blast ``` lp15@62408 ` 521` ```qed ``` lp15@62408 ` 522` lp15@62408 ` 523` lp15@62408 ` 524` ```proposition maximum_modulus_frontier: ``` lp15@62408 ` 525` ``` assumes holf: "f holomorphic_on (interior S)" ``` lp15@62408 ` 526` ``` and contf: "continuous_on (closure S) f" ``` lp15@62408 ` 527` ``` and bos: "bounded S" ``` lp15@62408 ` 528` ``` and leB: "\z. z \ frontier S \ norm(f z) \ B" ``` lp15@62408 ` 529` ``` and "\ \ S" ``` lp15@62408 ` 530` ``` shows "norm(f \) \ B" ``` lp15@62408 ` 531` ```proof - ``` lp15@62408 ` 532` ``` have "compact (closure S)" using bos ``` lp15@62408 ` 533` ``` by (simp add: bounded_closure compact_eq_bounded_closed) ``` lp15@62408 ` 534` ``` moreover have "continuous_on (closure S) (cmod \ f)" ``` lp15@62408 ` 535` ``` using contf continuous_on_compose continuous_on_norm_id by blast ``` lp15@62408 ` 536` ``` ultimately obtain z where zin: "z \ closure S" and z: "\y. y \ closure S \ (cmod \ f) y \ (cmod \ f) z" ``` lp15@62408 ` 537` ``` using continuous_attains_sup [of "closure S" "norm o f"] \\ \ S\ by auto ``` lp15@62408 ` 538` ``` then consider "z \ frontier S" | "z \ interior S" using frontier_def by auto ``` lp15@62408 ` 539` ``` then have "norm(f z) \ B" ``` lp15@62408 ` 540` ``` proof cases ``` lp15@62408 ` 541` ``` case 1 then show ?thesis using leB by blast ``` lp15@62408 ` 542` ``` next ``` lp15@62408 ` 543` ``` case 2 ``` lp15@62408 ` 544` ``` have zin: "z \ connected_component_set (interior S) z" ``` lp15@62408 ` 545` ``` by (simp add: 2) ``` lp15@62408 ` 546` ``` have "f constant_on (connected_component_set (interior S) z)" ``` lp15@62408 ` 547` ``` apply (rule maximum_modulus_principle [OF _ _ _ _ _ zin]) ``` lp15@62408 ` 548` ``` apply (metis connected_component_subset holf holomorphic_on_subset) ``` lp15@62408 ` 549` ``` apply (simp_all add: open_connected_component) ``` lp15@62408 ` 550` ``` by (metis closure_subset comp_eq_dest_lhs interior_subset subsetCE z connected_component_in) ``` lp15@62408 ` 551` ``` then obtain c where c: "\w. w \ connected_component_set (interior S) z \ f w = c" ``` lp15@62408 ` 552` ``` by (auto simp: constant_on_def) ``` lp15@62408 ` 553` ``` have "f ` closure(connected_component_set (interior S) z) \ {c}" ``` lp15@62408 ` 554` ``` apply (rule image_closure_subset) ``` lp15@62408 ` 555` ``` apply (meson closure_mono connected_component_subset contf continuous_on_subset interior_subset) ``` lp15@62408 ` 556` ``` using c ``` lp15@62408 ` 557` ``` apply auto ``` lp15@62408 ` 558` ``` done ``` lp15@62408 ` 559` ``` then have cc: "\w. w \ closure(connected_component_set (interior S) z) \ f w = c" by blast ``` lp15@62408 ` 560` ``` have "frontier(connected_component_set (interior S) z) \ {}" ``` lp15@62408 ` 561` ``` apply (simp add: frontier_eq_empty) ``` lp15@62408 ` 562` ``` by (metis "2" bos bounded_interior connected_component_eq_UNIV connected_component_refl not_bounded_UNIV) ``` lp15@62408 ` 563` ``` then obtain w where w: "w \ frontier(connected_component_set (interior S) z)" ``` lp15@62408 ` 564` ``` by auto ``` lp15@62408 ` 565` ``` then have "norm (f z) = norm (f w)" by (simp add: "2" c cc frontier_def) ``` lp15@62408 ` 566` ``` also have "... \ B" ``` lp15@62408 ` 567` ``` apply (rule leB) ``` lp15@62408 ` 568` ``` using w ``` lp15@62408 ` 569` ```using frontier_interior_subset frontier_of_connected_component_subset by blast ``` lp15@62408 ` 570` ``` finally show ?thesis . ``` lp15@62408 ` 571` ``` qed ``` lp15@62408 ` 572` ``` then show ?thesis ``` lp15@62408 ` 573` ``` using z \\ \ S\ closure_subset by fastforce ``` lp15@62408 ` 574` ```qed ``` lp15@62408 ` 575` lp15@62408 ` 576` ```corollary maximum_real_frontier: ``` lp15@62408 ` 577` ``` assumes holf: "f holomorphic_on (interior S)" ``` lp15@62408 ` 578` ``` and contf: "continuous_on (closure S) f" ``` lp15@62408 ` 579` ``` and bos: "bounded S" ``` lp15@62408 ` 580` ``` and leB: "\z. z \ frontier S \ Re(f z) \ B" ``` lp15@62408 ` 581` ``` and "\ \ S" ``` lp15@62408 ` 582` ``` shows "Re(f \) \ B" ``` lp15@62408 ` 583` ```using maximum_modulus_frontier [of "exp o f" S "exp B"] ``` lp15@62408 ` 584` ``` Transcendental.continuous_on_exp holomorphic_on_compose holomorphic_on_exp assms ``` lp15@62408 ` 585` ```by auto ``` lp15@62408 ` 586` lp15@62408 ` 587` lp15@62408 ` 588` ```subsection\Factoring out a zero according to its order\ ``` lp15@62408 ` 589` lp15@62408 ` 590` ```lemma holomorphic_factor_order_of_zero: ``` lp15@62408 ` 591` ``` assumes holf: "f holomorphic_on S" ``` lp15@62408 ` 592` ``` and os: "open S" ``` lp15@62408 ` 593` ``` and "\ \ S" "0 < n" ``` lp15@62408 ` 594` ``` and dnz: "(deriv ^^ n) f \ \ 0" ``` lp15@62408 ` 595` ``` and dfz: "\i. \0 < i; i < n\ \ (deriv ^^ i) f \ = 0" ``` lp15@62408 ` 596` ``` obtains g r where "0 < r" ``` lp15@62408 ` 597` ``` "g holomorphic_on ball \ r" ``` lp15@62408 ` 598` ``` "\w. w \ ball \ r \ f w - f \ = (w - \)^n * g w" ``` lp15@62408 ` 599` ``` "\w. w \ ball \ r \ g w \ 0" ``` lp15@62408 ` 600` ```proof - ``` lp15@62408 ` 601` ``` obtain r where "r>0" and r: "ball \ r \ S" using assms by (blast elim!: openE) ``` lp15@62408 ` 602` ``` then have holfb: "f holomorphic_on ball \ r" ``` lp15@62408 ` 603` ``` using holf holomorphic_on_subset by blast ``` wenzelm@63040 ` 604` ``` define g where "g w = suminf (\i. (deriv ^^ (i + n)) f \ / (fact(i + n)) * (w - \)^i)" for w ``` lp15@62408 ` 605` ``` have sumsg: "(\i. (deriv ^^ (i + n)) f \ / (fact(i + n)) * (w - \)^i) sums g w" ``` lp15@62408 ` 606` ``` and feq: "f w - f \ = (w - \)^n * g w" ``` lp15@62408 ` 607` ``` if w: "w \ ball \ r" for w ``` lp15@62408 ` 608` ``` proof - ``` wenzelm@63040 ` 609` ``` define powf where "powf = (\i. (deriv ^^ i) f \/(fact i) * (w - \)^i)" ``` lp15@62408 ` 610` ``` have sing: "{.. = 0 then {} else {0})" ``` lp15@62408 ` 611` ``` unfolding powf_def using \0 < n\ dfz by (auto simp: dfz; metis funpow_0 not_gr0) ``` lp15@62408 ` 612` ``` have "powf sums f w" ``` lp15@62408 ` 613` ``` unfolding powf_def by (rule holomorphic_power_series [OF holfb w]) ``` lp15@62408 ` 614` ``` moreover have "(\i" ``` lp15@62408 ` 615` ``` apply (subst Groups_Big.comm_monoid_add_class.setsum.setdiff_irrelevant [symmetric]) ``` lp15@62408 ` 616` ``` apply (simp add:) ``` lp15@62408 ` 617` ``` apply (simp only: dfz sing) ``` lp15@62408 ` 618` ``` apply (simp add: powf_def) ``` lp15@62408 ` 619` ``` done ``` lp15@62408 ` 620` ``` ultimately have fsums: "(\i. powf (i+n)) sums (f w - f \)" ``` lp15@62408 ` 621` ``` using w sums_iff_shift' by metis ``` lp15@62408 ` 622` ``` then have *: "summable (\i. (w - \) ^ n * ((deriv ^^ (i + n)) f \ * (w - \) ^ i / fact (i + n)))" ``` lp15@62408 ` 623` ``` unfolding powf_def using sums_summable ``` lp15@62408 ` 624` ``` by (auto simp: power_add mult_ac) ``` lp15@62408 ` 625` ``` have "summable (\i. (deriv ^^ (i + n)) f \ * (w - \) ^ i / fact (i + n))" ``` lp15@62408 ` 626` ``` proof (cases "w=\") ``` lp15@62408 ` 627` ``` case False then show ?thesis ``` lp15@62408 ` 628` ``` using summable_mult [OF *, of "1 / (w - \) ^ n"] by (simp add:) ``` lp15@62408 ` 629` ``` next ``` lp15@62408 ` 630` ``` case True then show ?thesis ``` lp15@62408 ` 631` ``` by (auto simp: Power.semiring_1_class.power_0_left intro!: summable_finite [of "{0}"] ``` lp15@62408 ` 632` ``` split: if_split_asm) ``` lp15@62408 ` 633` ``` qed ``` lp15@62408 ` 634` ``` then show sumsg: "(\i. (deriv ^^ (i + n)) f \ / (fact(i + n)) * (w - \)^i) sums g w" ``` lp15@62408 ` 635` ``` by (simp add: summable_sums_iff g_def) ``` lp15@62408 ` 636` ``` show "f w - f \ = (w - \)^n * g w" ``` lp15@62408 ` 637` ``` apply (rule sums_unique2) ``` lp15@62408 ` 638` ``` apply (rule fsums [unfolded powf_def]) ``` lp15@62408 ` 639` ``` using sums_mult [OF sumsg, of "(w - \) ^ n"] ``` lp15@62408 ` 640` ``` by (auto simp: power_add mult_ac) ``` lp15@62408 ` 641` ``` qed ``` lp15@62408 ` 642` ``` then have holg: "g holomorphic_on ball \ r" ``` lp15@62408 ` 643` ``` by (meson sumsg power_series_holomorphic) ``` lp15@62408 ` 644` ``` then have contg: "continuous_on (ball \ r) g" ``` lp15@62408 ` 645` ``` by (blast intro: holomorphic_on_imp_continuous_on) ``` lp15@62408 ` 646` ``` have "g \ \ 0" ``` lp15@62408 ` 647` ``` using dnz unfolding g_def ``` lp15@62408 ` 648` ``` by (subst suminf_finite [of "{0}"]) auto ``` lp15@62408 ` 649` ``` obtain d where "0 < d" and d: "\w. w \ ball \ d \ g w \ 0" ``` lp15@62408 ` 650` ``` apply (rule exE [OF continuous_on_avoid [OF contg _ \g \ \ 0\]]) ``` lp15@62408 ` 651` ``` using \0 < r\ ``` lp15@62408 ` 652` ``` apply force ``` lp15@62408 ` 653` ``` by (metis \0 < r\ less_trans mem_ball not_less_iff_gr_or_eq) ``` lp15@62408 ` 654` ``` show ?thesis ``` lp15@62408 ` 655` ``` apply (rule that [where g=g and r ="min r d"]) ``` lp15@62408 ` 656` ``` using \0 < r\ \0 < d\ holg ``` lp15@62408 ` 657` ``` apply (auto simp: feq holomorphic_on_subset subset_ball d) ``` lp15@62408 ` 658` ``` done ``` lp15@62408 ` 659` ```qed ``` lp15@62408 ` 660` lp15@62408 ` 661` lp15@62408 ` 662` ```lemma holomorphic_factor_order_of_zero_strong: ``` lp15@62408 ` 663` ``` assumes holf: "f holomorphic_on S" "open S" "\ \ S" "0 < n" ``` lp15@62408 ` 664` ``` and "(deriv ^^ n) f \ \ 0" ``` lp15@62408 ` 665` ``` and "\i. \0 < i; i < n\ \ (deriv ^^ i) f \ = 0" ``` lp15@62408 ` 666` ``` obtains g r where "0 < r" ``` lp15@62408 ` 667` ``` "g holomorphic_on ball \ r" ``` lp15@62408 ` 668` ``` "\w. w \ ball \ r \ f w - f \ = ((w - \) * g w) ^ n" ``` lp15@62408 ` 669` ``` "\w. w \ ball \ r \ g w \ 0" ``` lp15@62408 ` 670` ```proof - ``` lp15@62408 ` 671` ``` obtain g r where "0 < r" ``` lp15@62408 ` 672` ``` and holg: "g holomorphic_on ball \ r" ``` lp15@62408 ` 673` ``` and feq: "\w. w \ ball \ r \ f w - f \ = (w - \)^n * g w" ``` lp15@62408 ` 674` ``` and gne: "\w. w \ ball \ r \ g w \ 0" ``` lp15@62408 ` 675` ``` by (auto intro: holomorphic_factor_order_of_zero [OF assms]) ``` lp15@62408 ` 676` ``` have con: "continuous_on (ball \ r) (\z. deriv g z / g z)" ``` lp15@62408 ` 677` ``` by (rule continuous_intros) (auto simp: gne holg holomorphic_deriv holomorphic_on_imp_continuous_on) ``` lp15@62534 ` 678` ``` have cd: "\x. dist \ x < r \ (\z. deriv g z / g z) field_differentiable at x" ``` lp15@62408 ` 679` ``` apply (rule derivative_intros)+ ``` lp15@62408 ` 680` ``` using holg mem_ball apply (blast intro: holomorphic_deriv holomorphic_on_imp_differentiable_at) ``` lp15@62408 ` 681` ``` apply (metis Topology_Euclidean_Space.open_ball at_within_open holg holomorphic_on_def mem_ball) ``` lp15@62408 ` 682` ``` using gne mem_ball by blast ``` lp15@62408 ` 683` ``` obtain h where h: "\x. x \ ball \ r \ (h has_field_derivative deriv g x / g x) (at x)" ``` lp15@62408 ` 684` ``` apply (rule exE [OF holomorphic_convex_primitive [of "ball \ r" "{}" "\z. deriv g z / g z"]]) ``` lp15@62408 ` 685` ``` apply (auto simp: con cd) ``` lp15@62408 ` 686` ``` apply (metis open_ball at_within_open mem_ball) ``` lp15@62408 ` 687` ``` done ``` lp15@62408 ` 688` ``` then have "continuous_on (ball \ r) h" ``` lp15@62408 ` 689` ``` by (metis open_ball holomorphic_on_imp_continuous_on holomorphic_on_open) ``` lp15@62408 ` 690` ``` then have con: "continuous_on (ball \ r) (\x. exp (h x) / g x)" ``` lp15@62408 ` 691` ``` by (auto intro!: continuous_intros simp add: holg holomorphic_on_imp_continuous_on gne) ``` lp15@62408 ` 692` ``` have 0: "dist \ x < r \ ((\x. exp (h x) / g x) has_field_derivative 0) (at x)" for x ``` lp15@62408 ` 693` ``` apply (rule h derivative_eq_intros | simp)+ ``` lp15@62534 ` 694` ``` apply (rule DERIV_deriv_iff_field_differentiable [THEN iffD2]) ``` lp15@62408 ` 695` ``` using holg apply (auto simp: holomorphic_on_imp_differentiable_at gne h) ``` lp15@62408 ` 696` ``` done ``` lp15@62408 ` 697` ``` obtain c where c: "\x. x \ ball \ r \ exp (h x) / g x = c" ``` lp15@62408 ` 698` ``` by (rule DERIV_zero_connected_constant [of "ball \ r" "{}" "\x. exp(h x) / g x"]) (auto simp: con 0) ``` lp15@62408 ` 699` ``` have hol: "(\z. exp ((Ln (inverse c) + h z) / of_nat n)) holomorphic_on ball \ r" ``` lp15@62408 ` 700` ``` apply (rule holomorphic_on_compose [unfolded o_def, where g = exp]) ``` lp15@62408 ` 701` ``` apply (rule holomorphic_intros)+ ``` lp15@62408 ` 702` ``` using h holomorphic_on_open apply blast ``` lp15@62408 ` 703` ``` apply (rule holomorphic_intros)+ ``` lp15@62408 ` 704` ``` using \0 < n\ apply (simp add:) ``` lp15@62408 ` 705` ``` apply (rule holomorphic_intros)+ ``` lp15@62408 ` 706` ``` done ``` lp15@62408 ` 707` ``` show ?thesis ``` lp15@62408 ` 708` ``` apply (rule that [where g="\z. exp((Ln(inverse c) + h z)/n)" and r =r]) ``` lp15@62408 ` 709` ``` using \0 < r\ \0 < n\ ``` lp15@62408 ` 710` ``` apply (auto simp: feq power_mult_distrib exp_divide_power_eq c [symmetric]) ``` lp15@62408 ` 711` ``` apply (rule hol) ``` lp15@62408 ` 712` ``` apply (simp add: Transcendental.exp_add gne) ``` lp15@62408 ` 713` ``` done ``` lp15@62408 ` 714` ```qed ``` lp15@62408 ` 715` lp15@62408 ` 716` lp15@62408 ` 717` ```lemma ``` lp15@62408 ` 718` ``` fixes k :: "'a::wellorder" ``` lp15@62408 ` 719` ``` assumes a_def: "a == LEAST x. P x" and P: "P k" ``` lp15@62408 ` 720` ``` shows def_LeastI: "P a" and def_Least_le: "a \ k" ``` lp15@62408 ` 721` ```unfolding a_def ``` lp15@62408 ` 722` ```by (rule LeastI Least_le; rule P)+ ``` lp15@62408 ` 723` lp15@62408 ` 724` ```lemma holomorphic_factor_zero_nonconstant: ``` lp15@62408 ` 725` ``` assumes holf: "f holomorphic_on S" and S: "open S" "connected S" ``` lp15@62408 ` 726` ``` and "\ \ S" "f \ = 0" ``` lp15@62408 ` 727` ``` and nonconst: "\c. \z \ S. f z \ c" ``` lp15@62408 ` 728` ``` obtains g r n ``` lp15@62408 ` 729` ``` where "0 < n" "0 < r" "ball \ r \ S" ``` lp15@62408 ` 730` ``` "g holomorphic_on ball \ r" ``` lp15@62408 ` 731` ``` "\w. w \ ball \ r \ f w = (w - \)^n * g w" ``` lp15@62408 ` 732` ``` "\w. w \ ball \ r \ g w \ 0" ``` lp15@62408 ` 733` ```proof (cases "\n>0. (deriv ^^ n) f \ = 0") ``` lp15@62408 ` 734` ``` case True then show ?thesis ``` lp15@62408 ` 735` ``` using holomorphic_fun_eq_const_on_connected [OF holf S _ \\ \ S\] nonconst by auto ``` lp15@62408 ` 736` ```next ``` lp15@62408 ` 737` ``` case False ``` lp15@62408 ` 738` ``` then obtain n0 where "n0 > 0" and n0: "(deriv ^^ n0) f \ \ 0" by blast ``` lp15@62408 ` 739` ``` obtain r0 where "r0 > 0" "ball \ r0 \ S" using S openE \\ \ S\ by auto ``` wenzelm@63040 ` 740` ``` define n where "n \ LEAST n. (deriv ^^ n) f \ \ 0" ``` lp15@62408 ` 741` ``` have n_ne: "(deriv ^^ n) f \ \ 0" ``` lp15@62408 ` 742` ``` by (rule def_LeastI [OF n_def]) (rule n0) ``` lp15@62408 ` 743` ``` then have "0 < n" using \f \ = 0\ ``` lp15@62408 ` 744` ``` using funpow_0 by fastforce ``` lp15@62408 ` 745` ``` have n_min: "\k. k < n \ (deriv ^^ k) f \ = 0" ``` lp15@62408 ` 746` ``` using def_Least_le [OF n_def] not_le by blast ``` lp15@62408 ` 747` ``` then obtain g r1 ``` lp15@62408 ` 748` ``` where "0 < r1" "g holomorphic_on ball \ r1" ``` lp15@62408 ` 749` ``` "\w. w \ ball \ r1 \ f w = (w - \) ^ n * g w" ``` lp15@62408 ` 750` ``` "\w. w \ ball \ r1 \ g w \ 0" ``` lp15@62408 ` 751` ``` by (auto intro: holomorphic_factor_order_of_zero [OF holf \open S\ \\ \ S\ \n > 0\ n_ne] simp: \f \ = 0\) ``` lp15@62408 ` 752` ``` then show ?thesis ``` lp15@62408 ` 753` ``` apply (rule_tac g=g and r="min r0 r1" and n=n in that) ``` lp15@62408 ` 754` ``` using \0 < n\ \0 < r0\ \0 < r1\ \ball \ r0 \ S\ ``` lp15@62408 ` 755` ``` apply (auto simp: subset_ball intro: holomorphic_on_subset) ``` lp15@62408 ` 756` ``` done ``` lp15@62408 ` 757` ```qed ``` lp15@62408 ` 758` lp15@62408 ` 759` lp15@62408 ` 760` ```lemma holomorphic_lower_bound_difference: ``` lp15@62408 ` 761` ``` assumes holf: "f holomorphic_on S" and S: "open S" "connected S" ``` lp15@62408 ` 762` ``` and "\ \ S" and "\ \ S" ``` lp15@62408 ` 763` ``` and fne: "f \ \ f \" ``` lp15@62408 ` 764` ``` obtains k n r ``` lp15@62408 ` 765` ``` where "0 < k" "0 < r" ``` lp15@62408 ` 766` ``` "ball \ r \ S" ``` lp15@62408 ` 767` ``` "\w. w \ ball \ r \ k * norm(w - \)^n \ norm(f w - f \)" ``` lp15@62408 ` 768` ```proof - ``` wenzelm@63040 ` 769` ``` define n where "n = (LEAST n. 0 < n \ (deriv ^^ n) f \ \ 0)" ``` lp15@62408 ` 770` ``` obtain n0 where "0 < n0" and n0: "(deriv ^^ n0) f \ \ 0" ``` lp15@62408 ` 771` ``` using fne holomorphic_fun_eq_const_on_connected [OF holf S] \\ \ S\ \\ \ S\ by blast ``` lp15@62408 ` 772` ``` then have "0 < n" and n_ne: "(deriv ^^ n) f \ \ 0" ``` lp15@62408 ` 773` ``` unfolding n_def by (metis (mono_tags, lifting) LeastI)+ ``` lp15@62408 ` 774` ``` have n_min: "\k. \0 < k; k < n\ \ (deriv ^^ k) f \ = 0" ``` lp15@62408 ` 775` ``` unfolding n_def by (blast dest: not_less_Least) ``` lp15@62408 ` 776` ``` then obtain g r ``` lp15@62408 ` 777` ``` where "0 < r" and holg: "g holomorphic_on ball \ r" ``` lp15@62408 ` 778` ``` and fne: "\w. w \ ball \ r \ f w - f \ = (w - \) ^ n * g w" ``` lp15@62408 ` 779` ``` and gnz: "\w. w \ ball \ r \ g w \ 0" ``` lp15@62408 ` 780` ``` by (auto intro: holomorphic_factor_order_of_zero [OF holf \open S\ \\ \ S\ \n > 0\ n_ne]) ``` lp15@62408 ` 781` ``` obtain e where "e>0" and e: "ball \ e \ S" using assms by (blast elim!: openE) ``` lp15@62408 ` 782` ``` then have holfb: "f holomorphic_on ball \ e" ``` lp15@62408 ` 783` ``` using holf holomorphic_on_subset by blast ``` wenzelm@63040 ` 784` ``` define d where "d = (min e r) / 2" ``` lp15@62408 ` 785` ``` have "0 < d" using \0 < r\ \0 < e\ by (simp add: d_def) ``` lp15@62408 ` 786` ``` have "d < r" ``` lp15@62408 ` 787` ``` using \0 < r\ by (auto simp: d_def) ``` lp15@62408 ` 788` ``` then have cbb: "cball \ d \ ball \ r" ``` lp15@62408 ` 789` ``` by (auto simp: cball_subset_ball_iff) ``` lp15@62408 ` 790` ``` then have "g holomorphic_on cball \ d" ``` lp15@62408 ` 791` ``` by (rule holomorphic_on_subset [OF holg]) ``` lp15@62408 ` 792` ``` then have "closed (g ` cball \ d)" ``` lp15@62408 ` 793` ``` by (simp add: compact_imp_closed compact_continuous_image holomorphic_on_imp_continuous_on) ``` lp15@62408 ` 794` ``` moreover have "g ` cball \ d \ {}" ``` lp15@62408 ` 795` ``` using \0 < d\ by auto ``` lp15@62408 ` 796` ``` ultimately obtain x where x: "x \ g ` cball \ d" and "\y. y \ g ` cball \ d \ dist 0 x \ dist 0 y" ``` lp15@62408 ` 797` ``` by (rule distance_attains_inf) blast ``` lp15@62408 ` 798` ``` then have leg: "\w. w \ cball \ d \ norm x \ norm (g w)" ``` lp15@62408 ` 799` ``` by auto ``` lp15@62408 ` 800` ``` have "ball \ d \ cball \ d" by auto ``` lp15@62408 ` 801` ``` also have "... \ ball \ e" using \0 < d\ d_def by auto ``` lp15@62408 ` 802` ``` also have "... \ S" by (rule e) ``` lp15@62408 ` 803` ``` finally have dS: "ball \ d \ S" . ``` lp15@62408 ` 804` ``` moreover have "x \ 0" using gnz x \d < r\ by auto ``` lp15@62408 ` 805` ``` ultimately show ?thesis ``` lp15@62408 ` 806` ``` apply (rule_tac k="norm x" and n=n and r=d in that) ``` lp15@62408 ` 807` ``` using \d < r\ leg ``` lp15@62408 ` 808` ``` apply (auto simp: \0 < d\ fne norm_mult norm_power algebra_simps mult_right_mono) ``` lp15@62408 ` 809` ``` done ``` lp15@62408 ` 810` ```qed ``` lp15@62408 ` 811` lp15@62408 ` 812` ```lemma ``` lp15@62408 ` 813` ``` assumes holf: "f holomorphic_on (S - {\})" and \: "\ \ interior S" ``` lp15@62408 ` 814` ``` shows holomorphic_on_extend_lim: ``` lp15@62408 ` 815` ``` "(\g. g holomorphic_on S \ (\z \ S - {\}. g z = f z)) \ ``` lp15@62408 ` 816` ``` ((\z. (z - \) * f z) \ 0) (at \)" ``` lp15@62408 ` 817` ``` (is "?P = ?Q") ``` lp15@62408 ` 818` ``` and holomorphic_on_extend_bounded: ``` lp15@62408 ` 819` ``` "(\g. g holomorphic_on S \ (\z \ S - {\}. g z = f z)) \ ``` lp15@62408 ` 820` ``` (\B. eventually (\z. norm(f z) \ B) (at \))" ``` lp15@62408 ` 821` ``` (is "?P = ?R") ``` lp15@62408 ` 822` ```proof - ``` lp15@62408 ` 823` ``` obtain \ where "0 < \" and \: "ball \ \ \ S" ``` lp15@62408 ` 824` ``` using \ mem_interior by blast ``` lp15@62408 ` 825` ``` have "?R" if holg: "g holomorphic_on S" and gf: "\z. z \ S - {\} \ g z = f z" for g ``` lp15@62408 ` 826` ``` proof - ``` lp15@62408 ` 827` ``` have *: "\\<^sub>F z in at \. dist (g z) (g \) < 1 \ cmod (f z) \ cmod (g \) + 1" ``` lp15@62408 ` 828` ``` apply (simp add: eventually_at) ``` lp15@62408 ` 829` ``` apply (rule_tac x="\" in exI) ``` lp15@62408 ` 830` ``` using \ \0 < \\ ``` lp15@62408 ` 831` ``` apply (clarsimp simp:) ``` lp15@62408 ` 832` ``` apply (drule_tac c=x in subsetD) ``` lp15@62408 ` 833` ``` apply (simp add: dist_commute) ``` lp15@62408 ` 834` ``` by (metis DiffI add.commute diff_le_eq dist_norm gf le_less_trans less_eq_real_def norm_triangle_ineq2 singletonD) ``` lp15@62408 ` 835` ``` have "continuous_on (interior S) g" ``` lp15@62408 ` 836` ``` by (meson continuous_on_subset holg holomorphic_on_imp_continuous_on interior_subset) ``` lp15@62408 ` 837` ``` then have "\x. x \ interior S \ (g \ g x) (at x)" ``` lp15@62408 ` 838` ``` using continuous_on_interior continuous_within holg holomorphic_on_imp_continuous_on by blast ``` lp15@62408 ` 839` ``` then have "(g \ g \) (at \)" ``` lp15@62408 ` 840` ``` by (simp add: \) ``` lp15@62408 ` 841` ``` then show ?thesis ``` lp15@62408 ` 842` ``` apply (rule_tac x="norm(g \) + 1" in exI) ``` lp15@62408 ` 843` ``` apply (rule eventually_mp [OF * tendstoD [where e=1]], auto) ``` lp15@62408 ` 844` ``` done ``` lp15@62408 ` 845` ``` qed ``` lp15@62408 ` 846` ``` moreover have "?Q" if "\\<^sub>F z in at \. cmod (f z) \ B" for B ``` lp15@62408 ` 847` ``` by (rule lim_null_mult_right_bounded [OF _ that]) (simp add: LIM_zero) ``` lp15@62408 ` 848` ``` moreover have "?P" if "(\z. (z - \) * f z) \\\ 0" ``` lp15@62408 ` 849` ``` proof - ``` wenzelm@63040 ` 850` ``` define h where [abs_def]: "h z = (z - \)^2 * f z" for z ``` lp15@62408 ` 851` ``` have h0: "(h has_field_derivative 0) (at \)" ``` lp15@62408 ` 852` ``` apply (simp add: h_def Derivative.DERIV_within_iff) ``` lp15@62408 ` 853` ``` apply (rule Lim_transform_within [OF that, of 1]) ``` lp15@62408 ` 854` ``` apply (auto simp: divide_simps power2_eq_square) ``` lp15@62408 ` 855` ``` done ``` lp15@62408 ` 856` ``` have holh: "h holomorphic_on S" ``` lp15@62408 ` 857` ``` proof (simp add: holomorphic_on_def, clarify) ``` lp15@62408 ` 858` ``` fix z assume "z \ S" ``` lp15@62534 ` 859` ``` show "h field_differentiable at z within S" ``` lp15@62408 ` 860` ``` proof (cases "z = \") ``` lp15@62408 ` 861` ``` case True then show ?thesis ``` lp15@62534 ` 862` ``` using field_differentiable_at_within field_differentiable_def h0 by blast ``` lp15@62408 ` 863` ``` next ``` lp15@62408 ` 864` ``` case False ``` lp15@62534 ` 865` ``` then have "f field_differentiable at z within S" ``` lp15@62408 ` 866` ``` using holomorphic_onD [OF holf, of z] \z \ S\ ``` lp15@62534 ` 867` ``` unfolding field_differentiable_def DERIV_within_iff ``` lp15@62408 ` 868` ``` by (force intro: exI [where x="dist \ z"] elim: Lim_transform_within_set [unfolded eventually_at]) ``` lp15@62408 ` 869` ``` then show ?thesis ``` lp15@62408 ` 870` ``` by (simp add: h_def power2_eq_square derivative_intros) ``` lp15@62408 ` 871` ``` qed ``` lp15@62408 ` 872` ``` qed ``` wenzelm@63040 ` 873` ``` define g where [abs_def]: "g z = (if z = \ then deriv h \ else (h z - h \) / (z - \))" for z ``` lp15@62408 ` 874` ``` have holg: "g holomorphic_on S" ``` lp15@62408 ` 875` ``` unfolding g_def by (rule pole_lemma [OF holh \]) ``` lp15@62408 ` 876` ``` show ?thesis ``` lp15@62408 ` 877` ``` apply (rule_tac x="\z. if z = \ then deriv g \ else (g z - g \)/(z - \)" in exI) ``` lp15@62408 ` 878` ``` apply (rule conjI) ``` lp15@62408 ` 879` ``` apply (rule pole_lemma [OF holg \]) ``` lp15@62408 ` 880` ``` apply (auto simp: g_def power2_eq_square divide_simps) ``` lp15@62408 ` 881` ``` using h0 apply (simp add: h0 DERIV_imp_deriv h_def power2_eq_square) ``` lp15@62408 ` 882` ``` done ``` lp15@62408 ` 883` ``` qed ``` lp15@62408 ` 884` ``` ultimately show "?P = ?Q" and "?P = ?R" ``` lp15@62408 ` 885` ``` by meson+ ``` lp15@62408 ` 886` ```qed ``` lp15@62408 ` 887` lp15@62408 ` 888` lp15@62408 ` 889` ```proposition pole_at_infinity: ``` lp15@62408 ` 890` ``` assumes holf: "f holomorphic_on UNIV" and lim: "((inverse o f) \ l) at_infinity" ``` lp15@62408 ` 891` ``` obtains a n where "\z. f z = (\i\n. a i * z^i)" ``` lp15@62408 ` 892` ```proof (cases "l = 0") ``` lp15@62408 ` 893` ``` case False ``` lp15@62408 ` 894` ``` with tendsto_inverse [OF lim] show ?thesis ``` lp15@62408 ` 895` ``` apply (rule_tac a="(\n. inverse l)" and n=0 in that) ``` lp15@62408 ` 896` ``` apply (simp add: Liouville_weak [OF holf, of "inverse l"]) ``` lp15@62408 ` 897` ``` done ``` lp15@62408 ` 898` ```next ``` lp15@62408 ` 899` ``` case True ``` lp15@62408 ` 900` ``` then have [simp]: "l = 0" . ``` lp15@62408 ` 901` ``` show ?thesis ``` lp15@62408 ` 902` ``` proof (cases "\r. 0 < r \ (\z \ ball 0 r - {0}. f(inverse z) \ 0)") ``` lp15@62408 ` 903` ``` case True ``` lp15@62408 ` 904` ``` then obtain r where "0 < r" and r: "\z. z \ ball 0 r - {0} \ f(inverse z) \ 0" ``` lp15@62408 ` 905` ``` by auto ``` lp15@62408 ` 906` ``` have 1: "inverse \ f \ inverse holomorphic_on ball 0 r - {0}" ``` lp15@62408 ` 907` ``` by (rule holomorphic_on_compose holomorphic_intros holomorphic_on_subset [OF holf] | force simp: r)+ ``` lp15@62408 ` 908` ``` have 2: "0 \ interior (ball 0 r)" ``` lp15@62408 ` 909` ``` using \0 < r\ by simp ``` lp15@62408 ` 910` ``` have "\B. 0 eventually (\z. cmod ((inverse \ f \ inverse) z) \ B) (at 0)" ``` lp15@62408 ` 911` ``` apply (rule exI [where x=1]) ``` lp15@62408 ` 912` ``` apply (simp add:) ``` lp15@62408 ` 913` ``` using tendstoD [OF lim [unfolded lim_at_infinity_0] zero_less_one] ``` lp15@62408 ` 914` ``` apply (rule eventually_mono) ``` lp15@62408 ` 915` ``` apply (simp add: dist_norm) ``` lp15@62408 ` 916` ``` done ``` lp15@62408 ` 917` ``` with holomorphic_on_extend_bounded [OF 1 2] ``` lp15@62408 ` 918` ``` obtain g where holg: "g holomorphic_on ball 0 r" ``` lp15@62408 ` 919` ``` and geq: "\z. z \ ball 0 r - {0} \ g z = (inverse \ f \ inverse) z" ``` lp15@62408 ` 920` ``` by meson ``` lp15@62408 ` 921` ``` have ifi0: "(inverse \ f \ inverse) \0\ 0" ``` lp15@62408 ` 922` ``` using \l = 0\ lim lim_at_infinity_0 by blast ``` lp15@62408 ` 923` ``` have g2g0: "g \0\ g 0" ``` lp15@62408 ` 924` ``` using \0 < r\ centre_in_ball continuous_at continuous_on_eq_continuous_at holg ``` lp15@62408 ` 925` ``` by (blast intro: holomorphic_on_imp_continuous_on) ``` lp15@62408 ` 926` ``` have g2g1: "g \0\ 0" ``` lp15@62408 ` 927` ``` apply (rule Lim_transform_within_open [OF ifi0 open_ball [of 0 r]]) ``` lp15@62408 ` 928` ``` using \0 < r\ by (auto simp: geq) ``` lp15@62408 ` 929` ``` have [simp]: "g 0 = 0" ``` lp15@62408 ` 930` ``` by (rule tendsto_unique [OF _ g2g0 g2g1]) simp ``` lp15@62408 ` 931` ``` have "ball 0 r - {0::complex} \ {}" ``` lp15@62408 ` 932` ``` using \0 < r\ ``` lp15@62408 ` 933` ``` apply (clarsimp simp: ball_def dist_norm) ``` lp15@62408 ` 934` ``` apply (drule_tac c="of_real r/2" in subsetD, auto) ``` lp15@62408 ` 935` ``` done ``` lp15@62408 ` 936` ``` then obtain w::complex where "w \ 0" and w: "norm w < r" by force ``` lp15@62408 ` 937` ``` then have "g w \ 0" by (simp add: geq r) ``` lp15@62408 ` 938` ``` obtain B n e where "0 < B" "0 < e" "e \ r" ``` lp15@62408 ` 939` ``` and leg: "\w. norm w < e \ B * cmod w ^ n \ cmod (g w)" ``` lp15@62408 ` 940` ``` apply (rule holomorphic_lower_bound_difference [OF holg open_ball connected_ball, of 0 w]) ``` lp15@62408 ` 941` ``` using \0 < r\ w \g w \ 0\ by (auto simp: ball_subset_ball_iff) ``` lp15@62408 ` 942` ``` have "cmod (f z) \ cmod z ^ n / B" if "2/e \ cmod z" for z ``` lp15@62408 ` 943` ``` proof - ``` lp15@62408 ` 944` ``` have ize: "inverse z \ ball 0 e - {0}" using that \0 < e\ ``` lp15@62408 ` 945` ``` by (auto simp: norm_divide divide_simps algebra_simps) ``` lp15@62408 ` 946` ``` then have [simp]: "z \ 0" and izr: "inverse z \ ball 0 r - {0}" using \e \ r\ ``` lp15@62408 ` 947` ``` by auto ``` lp15@62408 ` 948` ``` then have [simp]: "f z \ 0" ``` lp15@62408 ` 949` ``` using r [of "inverse z"] by simp ``` lp15@62408 ` 950` ``` have [simp]: "f z = inverse (g (inverse z))" ``` lp15@62408 ` 951` ``` using izr geq [of "inverse z"] by simp ``` lp15@62408 ` 952` ``` show ?thesis using ize leg [of "inverse z"] \0 < B\ \0 < e\ ``` lp15@62408 ` 953` ``` by (simp add: divide_simps norm_divide algebra_simps) ``` lp15@62408 ` 954` ``` qed ``` lp15@62408 ` 955` ``` then show ?thesis ``` lp15@62408 ` 956` ``` apply (rule_tac a = "\k. (deriv ^^ k) f 0 / (fact k)" and n=n in that) ``` lp15@62408 ` 957` ``` apply (rule_tac A = "2/e" and B = "1/B" in Liouville_polynomial [OF holf]) ``` lp15@62408 ` 958` ``` apply (simp add:) ``` lp15@62408 ` 959` ``` done ``` lp15@62408 ` 960` ``` next ``` lp15@62408 ` 961` ``` case False ``` lp15@62408 ` 962` ``` then have fi0: "\r. r > 0 \ \z\ball 0 r - {0}. f (inverse z) = 0" ``` lp15@62408 ` 963` ``` by simp ``` lp15@62408 ` 964` ``` have fz0: "f z = 0" if "0 < r" and lt1: "\x. x \ 0 \ cmod x < r \ inverse (cmod (f (inverse x))) < 1" ``` lp15@62408 ` 965` ``` for z r ``` lp15@62408 ` 966` ``` proof - ``` lp15@62408 ` 967` ``` have f0: "(f \ 0) at_infinity" ``` lp15@62408 ` 968` ``` proof - ``` wenzelm@62837 ` 969` ``` have DIM_complex[intro]: "2 \ DIM(complex)" \\should not be necessary!\ ``` lp15@62408 ` 970` ``` by simp ``` lp15@62408 ` 971` ``` have "continuous_on (inverse ` (ball 0 r - {0})) f" ``` lp15@62408 ` 972` ``` using continuous_on_subset holf holomorphic_on_imp_continuous_on by blast ``` lp15@62408 ` 973` ``` then have "connected ((f \ inverse) ` (ball 0 r - {0}))" ``` lp15@62408 ` 974` ``` apply (intro connected_continuous_image continuous_intros) ``` lp15@62408 ` 975` ``` apply (force intro: connected_punctured_ball)+ ``` lp15@62408 ` 976` ``` done ``` lp15@62408 ` 977` ``` then have "\w \ 0; cmod w < r\ \ f (inverse w) = 0" for w ``` lp15@62408 ` 978` ``` apply (rule disjE [OF connected_closedD [where A = "{0}" and B = "- ball 0 1"]], auto) ``` lp15@62408 ` 979` ``` apply (metis (mono_tags, hide_lams) not_less_iff_gr_or_eq one_less_inverse lt1 zero_less_norm_iff) ``` lp15@62408 ` 980` ``` using False \0 < r\ apply fastforce ``` lp15@62408 ` 981` ``` by (metis (no_types, hide_lams) Compl_iff IntI comp_apply empty_iff image_eqI insert_Diff_single insert_iff mem_ball_0 not_less_iff_gr_or_eq one_less_inverse that(2) zero_less_norm_iff) ``` lp15@62408 ` 982` ``` then show ?thesis ``` lp15@62408 ` 983` ``` apply (simp add: lim_at_infinity_0) ``` lp15@62408 ` 984` ``` apply (rule Lim_eventually) ``` lp15@62408 ` 985` ``` apply (simp add: eventually_at) ``` lp15@62408 ` 986` ``` apply (rule_tac x=r in exI) ``` lp15@62408 ` 987` ``` apply (simp add: \0 < r\ dist_norm) ``` lp15@62408 ` 988` ``` done ``` lp15@62408 ` 989` ``` qed ``` lp15@62408 ` 990` ``` obtain w where "w \ ball 0 r - {0}" and "f (inverse w) = 0" ``` lp15@62408 ` 991` ``` using False \0 < r\ by blast ``` lp15@62408 ` 992` ``` then show ?thesis ``` lp15@62408 ` 993` ``` by (auto simp: f0 Liouville_weak [OF holf, of 0]) ``` lp15@62408 ` 994` ``` qed ``` lp15@62408 ` 995` ``` show ?thesis ``` lp15@62408 ` 996` ``` apply (rule that [of "\n. 0" 0]) ``` lp15@62408 ` 997` ``` using lim [unfolded lim_at_infinity_0] ``` lp15@62408 ` 998` ``` apply (simp add: Lim_at dist_norm norm_inverse) ``` lp15@62408 ` 999` ``` apply (drule_tac x=1 in spec) ``` lp15@62408 ` 1000` ``` using fz0 apply auto ``` lp15@62408 ` 1001` ``` done ``` lp15@62408 ` 1002` ``` qed ``` lp15@62408 ` 1003` ```qed ``` lp15@62408 ` 1004` lp15@62408 ` 1005` lp15@62408 ` 1006` ```subsection\Entire proper functions are precisely the non-trivial polynomials\ ``` lp15@62408 ` 1007` lp15@62408 ` 1008` ```proposition proper_map_polyfun: ``` lp15@62408 ` 1009` ``` fixes c :: "nat \ 'a::{real_normed_div_algebra,heine_borel}" ``` lp15@62408 ` 1010` ``` assumes "closed S" and "compact K" and c: "c i \ 0" "1 \ i" "i \ n" ``` lp15@62408 ` 1011` ``` shows "compact (S \ {z. (\i\n. c i * z^i) \ K})" ``` lp15@62408 ` 1012` ```proof - ``` lp15@62408 ` 1013` ``` obtain B where "B > 0" and B: "\x. x \ K \ norm x \ B" ``` lp15@62408 ` 1014` ``` by (metis compact_imp_bounded \compact K\ bounded_pos) ``` lp15@62408 ` 1015` ``` have *: "norm x \ b" ``` lp15@62408 ` 1016` ``` if "\x. b \ norm x \ B + 1 \ norm (\i\n. c i * x ^ i)" ``` lp15@62408 ` 1017` ``` "(\i\n. c i * x ^ i) \ K" for b x ``` lp15@62408 ` 1018` ``` proof - ``` lp15@62408 ` 1019` ``` have "norm (\i\n. c i * x ^ i) \ B" ``` lp15@62408 ` 1020` ``` using B that by blast ``` lp15@62408 ` 1021` ``` moreover have "\ B + 1 \ B" ``` lp15@62408 ` 1022` ``` by simp ``` lp15@62408 ` 1023` ``` ultimately show "norm x \ b" ``` lp15@62408 ` 1024` ``` using that by (metis (no_types) less_eq_real_def not_less order_trans) ``` lp15@62408 ` 1025` ``` qed ``` lp15@62408 ` 1026` ``` have "bounded {z. (\i\n. c i * z ^ i) \ K}" ``` lp15@62408 ` 1027` ``` using polyfun_extremal [where c=c and B="B+1", OF c] ``` lp15@62408 ` 1028` ``` by (auto simp: bounded_pos eventually_at_infinity_pos *) ``` lp15@62408 ` 1029` ``` moreover have "closed {z. (\i\n. c i * z ^ i) \ K}" ``` lp15@62408 ` 1030` ``` apply (rule allI continuous_closed_preimage_univ continuous_intros)+ ``` lp15@62408 ` 1031` ``` using \compact K\ compact_eq_bounded_closed by blast ``` lp15@62408 ` 1032` ``` ultimately show ?thesis ``` lp15@62843 ` 1033` ``` using closed_Int_compact [OF \closed S\] compact_eq_bounded_closed by blast ``` lp15@62408 ` 1034` ```qed ``` lp15@62408 ` 1035` lp15@62408 ` 1036` ```corollary proper_map_polyfun_univ: ``` lp15@62408 ` 1037` ``` fixes c :: "nat \ 'a::{real_normed_div_algebra,heine_borel}" ``` lp15@62408 ` 1038` ``` assumes "compact K" "c i \ 0" "1 \ i" "i \ n" ``` lp15@62408 ` 1039` ``` shows "compact ({z. (\i\n. c i * z^i) \ K})" ``` lp15@62408 ` 1040` ```using proper_map_polyfun [of UNIV K c i n] assms by simp ``` lp15@62408 ` 1041` lp15@62408 ` 1042` lp15@62408 ` 1043` ```proposition proper_map_polyfun_eq: ``` lp15@62408 ` 1044` ``` assumes "f holomorphic_on UNIV" ``` lp15@62408 ` 1045` ``` shows "(\k. compact k \ compact {z. f z \ k}) \ ``` lp15@62408 ` 1046` ``` (\c n. 0 < n \ (c n \ 0) \ f = (\z. \i\n. c i * z^i))" ``` lp15@62408 ` 1047` ``` (is "?lhs = ?rhs") ``` lp15@62408 ` 1048` ```proof ``` lp15@62408 ` 1049` ``` assume compf [rule_format]: ?lhs ``` lp15@62408 ` 1050` ``` have 2: "\k. 0 < k \ a k \ 0 \ f = (\z. \i \ k. a i * z ^ i)" ``` lp15@62408 ` 1051` ``` if "\z. f z = (\i\n. a i * z ^ i)" for a n ``` lp15@62408 ` 1052` ``` proof (cases "\i\n. 0 a i = 0") ``` lp15@62408 ` 1053` ``` case True ``` lp15@62408 ` 1054` ``` then have [simp]: "\z. f z = a 0" ``` lp15@62408 ` 1055` ``` by (simp add: that setsum_atMost_shift) ``` lp15@62408 ` 1056` ``` have False using compf [of "{a 0}"] by simp ``` lp15@62408 ` 1057` ``` then show ?thesis .. ``` lp15@62408 ` 1058` ``` next ``` lp15@62408 ` 1059` ``` case False ``` lp15@62408 ` 1060` ``` then obtain k where k: "0 < k" "k\n" "a k \ 0" by force ``` wenzelm@63040 ` 1061` ``` define m where "m = (GREATEST k. k\n \ a k \ 0)" ``` lp15@62408 ` 1062` ``` have m: "m\n \ a m \ 0" ``` lp15@62408 ` 1063` ``` unfolding m_def ``` lp15@62408 ` 1064` ``` apply (rule GreatestI [where b = "Suc n"]) ``` lp15@62408 ` 1065` ``` using k apply auto ``` lp15@62408 ` 1066` ``` done ``` lp15@62408 ` 1067` ``` have [simp]: "a i = 0" if "m < i" "i \ n" for i ``` lp15@62408 ` 1068` ``` using Greatest_le [where b = "Suc n" and P = "\k. k\n \ a k \ 0"] ``` lp15@62408 ` 1069` ``` using m_def not_le that by auto ``` lp15@62408 ` 1070` ``` have "k \ m" ``` lp15@62408 ` 1071` ``` unfolding m_def ``` lp15@62408 ` 1072` ``` apply (rule Greatest_le [where b = "Suc n"]) ``` lp15@62408 ` 1073` ``` using k apply auto ``` lp15@62408 ` 1074` ``` done ``` lp15@62408 ` 1075` ``` with k m show ?thesis ``` lp15@62408 ` 1076` ``` by (rule_tac x=m in exI) (auto simp: that comm_monoid_add_class.setsum.mono_neutral_right) ``` lp15@62408 ` 1077` ``` qed ``` lp15@62408 ` 1078` ``` have "((inverse \ f) \ 0) at_infinity" ``` lp15@62408 ` 1079` ``` proof (rule Lim_at_infinityI) ``` lp15@62408 ` 1080` ``` fix e::real assume "0 < e" ``` lp15@62408 ` 1081` ``` with compf [of "cball 0 (inverse e)"] ``` lp15@62408 ` 1082` ``` show "\B. \x. B \ cmod x \ dist ((inverse \ f) x) 0 \ e" ``` lp15@62408 ` 1083` ``` apply (simp add:) ``` lp15@62408 ` 1084` ``` apply (clarsimp simp add: compact_eq_bounded_closed bounded_pos norm_inverse) ``` lp15@62408 ` 1085` ``` apply (rule_tac x="b+1" in exI) ``` lp15@62408 ` 1086` ``` apply (metis inverse_inverse_eq less_add_same_cancel2 less_imp_inverse_less add.commute not_le not_less_iff_gr_or_eq order_trans zero_less_one) ``` lp15@62408 ` 1087` ``` done ``` lp15@62408 ` 1088` ``` qed ``` lp15@62408 ` 1089` ``` then show ?rhs ``` lp15@62408 ` 1090` ``` apply (rule pole_at_infinity [OF assms]) ``` lp15@62408 ` 1091` ``` using 2 apply blast ``` lp15@62408 ` 1092` ``` done ``` lp15@62408 ` 1093` ```next ``` lp15@62408 ` 1094` ``` assume ?rhs ``` lp15@62408 ` 1095` ``` then obtain c n where "0 < n" "c n \ 0" "f = (\z. \i\n. c i * z ^ i)" by blast ``` lp15@62408 ` 1096` ``` then have "compact {z. f z \ k}" if "compact k" for k ``` lp15@62408 ` 1097` ``` by (auto intro: proper_map_polyfun_univ [OF that]) ``` lp15@62408 ` 1098` ``` then show ?lhs by blast ``` lp15@62408 ` 1099` ```qed ``` lp15@62408 ` 1100` lp15@62408 ` 1101` lp15@62408 ` 1102` ```subsection\Relating invertibility and nonvanishing of derivative\ ``` lp15@62408 ` 1103` lp15@62408 ` 1104` ```proposition has_complex_derivative_locally_injective: ``` lp15@62408 ` 1105` ``` assumes holf: "f holomorphic_on S" ``` lp15@62408 ` 1106` ``` and S: "\ \ S" "open S" ``` lp15@62408 ` 1107` ``` and dnz: "deriv f \ \ 0" ``` lp15@62408 ` 1108` ``` obtains r where "r > 0" "ball \ r \ S" "inj_on f (ball \ r)" ``` lp15@62408 ` 1109` ```proof - ``` lp15@62408 ` 1110` ``` have *: "\d>0. \x. dist \ x < d \ onorm (\v. deriv f x * v - deriv f \ * v) < e" if "e > 0" for e ``` lp15@62408 ` 1111` ``` proof - ``` lp15@62408 ` 1112` ``` have contdf: "continuous_on S (deriv f)" ``` lp15@62408 ` 1113` ``` by (simp add: holf holomorphic_deriv holomorphic_on_imp_continuous_on \open S\) ``` lp15@62408 ` 1114` ``` obtain \ where "\>0" and \: "\x. \x \ S; dist x \ \ \\ \ cmod (deriv f x - deriv f \) \ e/2" ``` lp15@62408 ` 1115` ``` using continuous_onE [OF contdf \\ \ S\, of "e/2"] \0 < e\ ``` lp15@62408 ` 1116` ``` by (metis dist_complex_def half_gt_zero less_imp_le) ``` lp15@62408 ` 1117` ``` obtain \ where "\>0" "ball \ \ \ S" ``` lp15@62408 ` 1118` ``` by (metis openE [OF \open S\ \\ \ S\]) ``` lp15@62408 ` 1119` ``` with \\>0\ have "\\>0. \x. dist \ x < \ \ onorm (\v. deriv f x * v - deriv f \ * v) \ e/2" ``` lp15@62408 ` 1120` ``` apply (rule_tac x="min \ \" in exI) ``` lp15@62408 ` 1121` ``` apply (intro conjI allI impI Operator_Norm.onorm_le) ``` lp15@62408 ` 1122` ``` apply (simp add:) ``` lp15@62408 ` 1123` ``` apply (simp only: Rings.ring_class.left_diff_distrib [symmetric] norm_mult) ``` lp15@62408 ` 1124` ``` apply (rule mult_right_mono [OF \]) ``` lp15@62408 ` 1125` ``` apply (auto simp: dist_commute Rings.ordered_semiring_class.mult_right_mono \) ``` lp15@62408 ` 1126` ``` done ``` lp15@62408 ` 1127` ``` with \e>0\ show ?thesis by force ``` lp15@62408 ` 1128` ``` qed ``` lp15@62408 ` 1129` ``` have "inj (op * (deriv f \))" ``` lp15@62408 ` 1130` ``` using dnz by simp ``` lp15@62408 ` 1131` ``` then obtain g' where g': "linear g'" "g' \ op * (deriv f \) = id" ``` lp15@62408 ` 1132` ``` using linear_injective_left_inverse [of "op * (deriv f \)"] ``` lp15@62408 ` 1133` ``` by (auto simp: linear_times) ``` lp15@62408 ` 1134` ``` show ?thesis ``` lp15@62408 ` 1135` ``` apply (rule has_derivative_locally_injective [OF S, where f=f and f' = "\z h. deriv f z * h" and g' = g']) ``` lp15@62408 ` 1136` ``` using g' * ``` lp15@62408 ` 1137` ``` apply (simp_all add: linear_conv_bounded_linear that) ``` lp15@62534 ` 1138` ``` using DERIV_deriv_iff_field_differentiable has_field_derivative_imp_has_derivative holf ``` lp15@62408 ` 1139` ``` holomorphic_on_imp_differentiable_at \open S\ apply blast ``` lp15@62408 ` 1140` ``` done ``` lp15@62408 ` 1141` ```qed ``` lp15@62408 ` 1142` lp15@62408 ` 1143` lp15@62408 ` 1144` ```proposition has_complex_derivative_locally_invertible: ``` lp15@62408 ` 1145` ``` assumes holf: "f holomorphic_on S" ``` lp15@62408 ` 1146` ``` and S: "\ \ S" "open S" ``` lp15@62408 ` 1147` ``` and dnz: "deriv f \ \ 0" ``` lp15@62408 ` 1148` ``` obtains r where "r > 0" "ball \ r \ S" "open (f ` (ball \ r))" "inj_on f (ball \ r)" ``` lp15@62408 ` 1149` ```proof - ``` lp15@62408 ` 1150` ``` obtain r where "r > 0" "ball \ r \ S" "inj_on f (ball \ r)" ``` lp15@62408 ` 1151` ``` by (blast intro: that has_complex_derivative_locally_injective [OF assms]) ``` lp15@62408 ` 1152` ``` then have \: "\ \ ball \ r" by simp ``` lp15@62408 ` 1153` ``` then have nc: "~ f constant_on ball \ r" ``` lp15@62408 ` 1154` ``` using \inj_on f (ball \ r)\ injective_not_constant by fastforce ``` lp15@62408 ` 1155` ``` have holf': "f holomorphic_on ball \ r" ``` lp15@62408 ` 1156` ``` using \ball \ r \ S\ holf holomorphic_on_subset by blast ``` lp15@62408 ` 1157` ``` have "open (f ` ball \ r)" ``` lp15@62408 ` 1158` ``` apply (rule open_mapping_thm [OF holf']) ``` lp15@62408 ` 1159` ``` using nc apply auto ``` lp15@62408 ` 1160` ``` done ``` lp15@62408 ` 1161` ``` then show ?thesis ``` lp15@62408 ` 1162` ``` using \0 < r\ \ball \ r \ S\ \inj_on f (ball \ r)\ that by blast ``` lp15@62408 ` 1163` ```qed ``` lp15@62408 ` 1164` lp15@62408 ` 1165` lp15@62408 ` 1166` ```proposition holomorphic_injective_imp_regular: ``` lp15@62408 ` 1167` ``` assumes holf: "f holomorphic_on S" ``` lp15@62408 ` 1168` ``` and "open S" and injf: "inj_on f S" ``` lp15@62408 ` 1169` ``` and "\ \ S" ``` lp15@62408 ` 1170` ``` shows "deriv f \ \ 0" ``` lp15@62408 ` 1171` ```proof - ``` lp15@62408 ` 1172` ``` obtain r where "r>0" and r: "ball \ r \ S" using assms by (blast elim!: openE) ``` lp15@62408 ` 1173` ``` have holf': "f holomorphic_on ball \ r" ``` lp15@62408 ` 1174` ``` using \ball \ r \ S\ holf holomorphic_on_subset by blast ``` lp15@62408 ` 1175` ``` show ?thesis ``` lp15@62408 ` 1176` ``` proof (cases "\n>0. (deriv ^^ n) f \ = 0") ``` lp15@62408 ` 1177` ``` case True ``` lp15@62408 ` 1178` ``` have fcon: "f w = f \" if "w \ ball \ r" for w ``` lp15@62408 ` 1179` ``` apply (rule holomorphic_fun_eq_const_on_connected [OF holf']) ``` lp15@62408 ` 1180` ``` using True \0 < r\ that by auto ``` lp15@62408 ` 1181` ``` have False ``` lp15@62408 ` 1182` ``` using fcon [of "\ + r/2"] \0 < r\ r injf unfolding inj_on_def ``` lp15@62408 ` 1183` ``` by (metis \\ \ S\ contra_subsetD dist_commute fcon mem_ball perfect_choose_dist) ``` lp15@62408 ` 1184` ``` then show ?thesis .. ``` lp15@62408 ` 1185` ``` next ``` lp15@62408 ` 1186` ``` case False ``` lp15@62408 ` 1187` ``` then obtain n0 where n0: "n0 > 0 \ (deriv ^^ n0) f \ \ 0" by blast ``` wenzelm@63040 ` 1188` ``` define n where [abs_def]: "n = (LEAST n. n > 0 \ (deriv ^^ n) f \ \ 0)" ``` lp15@62408 ` 1189` ``` have n_ne: "n > 0" "(deriv ^^ n) f \ \ 0" ``` lp15@62408 ` 1190` ``` using def_LeastI [OF n_def n0] by auto ``` lp15@62408 ` 1191` ``` have n_min: "\k. 0 < k \ k < n \ (deriv ^^ k) f \ = 0" ``` lp15@62408 ` 1192` ``` using def_Least_le [OF n_def] not_le by auto ``` lp15@62408 ` 1193` ``` obtain g \ where "0 < \" ``` lp15@62408 ` 1194` ``` and holg: "g holomorphic_on ball \ \" ``` lp15@62408 ` 1195` ``` and fd: "\w. w \ ball \ \ \ f w - f \ = ((w - \) * g w) ^ n" ``` lp15@62408 ` 1196` ``` and gnz: "\w. w \ ball \ \ \ g w \ 0" ``` lp15@62408 ` 1197` ``` apply (rule holomorphic_factor_order_of_zero_strong [OF holf \open S\ \\ \ S\ n_ne]) ``` lp15@62408 ` 1198` ``` apply (blast intro: n_min)+ ``` lp15@62408 ` 1199` ``` done ``` lp15@62408 ` 1200` ``` show ?thesis ``` lp15@62408 ` 1201` ``` proof (cases "n=1") ``` lp15@62408 ` 1202` ``` case True ``` lp15@62408 ` 1203` ``` with n_ne show ?thesis by auto ``` lp15@62408 ` 1204` ``` next ``` lp15@62408 ` 1205` ``` case False ``` lp15@62408 ` 1206` ``` have holgw: "(\w. (w - \) * g w) holomorphic_on ball \ (min r \)" ``` lp15@62408 ` 1207` ``` apply (rule holomorphic_intros)+ ``` lp15@62408 ` 1208` ``` using holg by (simp add: holomorphic_on_subset subset_ball) ``` lp15@62408 ` 1209` ``` have gd: "\w. dist \ w < \ \ (g has_field_derivative deriv g w) (at w)" ``` lp15@62408 ` 1210` ``` using holg ``` lp15@62534 ` 1211` ``` by (simp add: DERIV_deriv_iff_field_differentiable holomorphic_on_def at_within_open_NO_MATCH) ``` lp15@62408 ` 1212` ``` have *: "\w. w \ ball \ (min r \) ``` lp15@62408 ` 1213` ``` \ ((\w. (w - \) * g w) has_field_derivative ((w - \) * deriv g w + g w)) ``` lp15@62408 ` 1214` ``` (at w)" ``` lp15@62408 ` 1215` ``` by (rule gd derivative_eq_intros | simp)+ ``` lp15@62408 ` 1216` ``` have [simp]: "deriv (\w. (w - \) * g w) \ \ 0" ``` lp15@62408 ` 1217` ``` using * [of \] \0 < \\ \0 < r\ by (simp add: DERIV_imp_deriv gnz) ``` lp15@62408 ` 1218` ``` obtain T where "\ \ T" "open T" and Tsb: "T \ ball \ (min r \)" and oimT: "open ((\w. (w - \) * g w) ` T)" ``` lp15@62408 ` 1219` ``` apply (rule has_complex_derivative_locally_invertible [OF holgw, of \]) ``` lp15@62408 ` 1220` ``` using \0 < r\ \0 < \\ ``` lp15@62408 ` 1221` ``` apply (simp_all add:) ``` lp15@62408 ` 1222` ``` by (meson Topology_Euclidean_Space.open_ball centre_in_ball) ``` wenzelm@63040 ` 1223` ``` define U where "U = (\w. (w - \) * g w) ` T" ``` lp15@62408 ` 1224` ``` have "open U" by (metis oimT U_def) ``` lp15@62408 ` 1225` ``` have "0 \ U" ``` lp15@62408 ` 1226` ``` apply (auto simp: U_def) ``` lp15@62408 ` 1227` ``` apply (rule image_eqI [where x = \]) ``` lp15@62408 ` 1228` ``` apply (auto simp: \\ \ T\) ``` lp15@62408 ` 1229` ``` done ``` lp15@62408 ` 1230` ``` then obtain \ where "\>0" and \: "cball 0 \ \ U" ``` lp15@62408 ` 1231` ``` using \open U\ open_contains_cball by blast ``` wenzelm@63589 ` 1232` ``` then have "\ * exp(2 * of_real pi * \ * (0/n)) \ cball 0 \" ``` wenzelm@63589 ` 1233` ``` "\ * exp(2 * of_real pi * \ * (1/n)) \ cball 0 \" ``` lp15@62408 ` 1234` ``` by (auto simp: norm_mult) ``` wenzelm@63589 ` 1235` ``` with \ have "\ * exp(2 * of_real pi * \ * (0/n)) \ U" ``` wenzelm@63589 ` 1236` ``` "\ * exp(2 * of_real pi * \ * (1/n)) \ U" by blast+ ``` wenzelm@63589 ` 1237` ``` then obtain y0 y1 where "y0 \ T" and y0: "(y0 - \) * g y0 = \ * exp(2 * of_real pi * \ * (0/n))" ``` wenzelm@63589 ` 1238` ``` and "y1 \ T" and y1: "(y1 - \) * g y1 = \ * exp(2 * of_real pi * \ * (1/n))" ``` lp15@62408 ` 1239` ``` by (auto simp: U_def) ``` lp15@62408 ` 1240` ``` then have "y0 \ ball \ \" "y1 \ ball \ \" using Tsb by auto ``` lp15@62408 ` 1241` ``` moreover have "y0 \ y1" ``` lp15@62408 ` 1242` ``` using y0 y1 \\ > 0\ complex_root_unity_eq_1 [of n 1] \n > 0\ False by auto ``` lp15@62408 ` 1243` ``` moreover have "T \ S" ``` lp15@62408 ` 1244` ``` by (meson Tsb min.cobounded1 order_trans r subset_ball) ``` lp15@62408 ` 1245` ``` ultimately have False ``` lp15@62408 ` 1246` ``` using inj_onD [OF injf, of y0 y1] \y0 \ T\ \y1 \ T\ ``` lp15@62408 ` 1247` ``` using fd [of y0] fd [of y1] complex_root_unity [of n 1] ``` lp15@62408 ` 1248` ``` apply (simp add: y0 y1 power_mult_distrib) ``` lp15@62408 ` 1249` ``` apply (force simp: algebra_simps) ``` lp15@62408 ` 1250` ``` done ``` lp15@62408 ` 1251` ``` then show ?thesis .. ``` lp15@62408 ` 1252` ``` qed ``` lp15@62408 ` 1253` ``` qed ``` lp15@62408 ` 1254` ```qed ``` lp15@62408 ` 1255` lp15@62408 ` 1256` lp15@62408 ` 1257` ```text\Hence a nice clean inverse function theorem\ ``` lp15@62408 ` 1258` lp15@62408 ` 1259` ```proposition holomorphic_has_inverse: ``` lp15@62408 ` 1260` ``` assumes holf: "f holomorphic_on S" ``` lp15@62408 ` 1261` ``` and "open S" and injf: "inj_on f S" ``` lp15@62408 ` 1262` ``` obtains g where "g holomorphic_on (f ` S)" ``` lp15@62408 ` 1263` ``` "\z. z \ S \ deriv f z * deriv g (f z) = 1" ``` lp15@62408 ` 1264` ``` "\z. z \ S \ g(f z) = z" ``` lp15@62408 ` 1265` ```proof - ``` lp15@62408 ` 1266` ``` have ofs: "open (f ` S)" ``` lp15@62408 ` 1267` ``` by (rule open_mapping_thm3 [OF assms]) ``` lp15@62408 ` 1268` ``` have contf: "continuous_on S f" ``` lp15@62408 ` 1269` ``` by (simp add: holf holomorphic_on_imp_continuous_on) ``` lp15@62408 ` 1270` ``` have *: "(the_inv_into S f has_field_derivative inverse (deriv f z)) (at (f z))" if "z \ S" for z ``` lp15@62408 ` 1271` ``` proof - ``` lp15@62408 ` 1272` ``` have 1: "(f has_field_derivative deriv f z) (at z)" ``` lp15@62534 ` 1273` ``` using DERIV_deriv_iff_field_differentiable \z \ S\ \open S\ holf holomorphic_on_imp_differentiable_at ``` lp15@62408 ` 1274` ``` by blast ``` lp15@62408 ` 1275` ``` have 2: "deriv f z \ 0" ``` lp15@62408 ` 1276` ``` using \z \ S\ \open S\ holf holomorphic_injective_imp_regular injf by blast ``` lp15@62408 ` 1277` ``` show ?thesis ``` lp15@62408 ` 1278` ``` apply (rule has_complex_derivative_inverse_strong [OF 1 2 \open S\ \z \ S\]) ``` lp15@62408 ` 1279` ``` apply (simp add: holf holomorphic_on_imp_continuous_on) ``` lp15@62408 ` 1280` ``` by (simp add: injf the_inv_into_f_f) ``` lp15@62408 ` 1281` ``` qed ``` lp15@62408 ` 1282` ``` show ?thesis ``` lp15@62408 ` 1283` ``` proof ``` lp15@62408 ` 1284` ``` show "the_inv_into S f holomorphic_on f ` S" ``` lp15@62408 ` 1285` ``` by (simp add: holomorphic_on_open ofs) (blast intro: *) ``` lp15@62408 ` 1286` ``` next ``` lp15@62408 ` 1287` ``` fix z assume "z \ S" ``` lp15@62408 ` 1288` ``` have "deriv f z \ 0" ``` lp15@62408 ` 1289` ``` using \z \ S\ \open S\ holf holomorphic_injective_imp_regular injf by blast ``` lp15@62408 ` 1290` ``` then show "deriv f z * deriv (the_inv_into S f) (f z) = 1" ``` lp15@62408 ` 1291` ``` using * [OF \z \ S\] by (simp add: DERIV_imp_deriv) ``` lp15@62408 ` 1292` ``` next ``` lp15@62408 ` 1293` ``` fix z assume "z \ S" ``` lp15@62408 ` 1294` ``` show "the_inv_into S f (f z) = z" ``` lp15@62408 ` 1295` ``` by (simp add: \z \ S\ injf the_inv_into_f_f) ``` lp15@62408 ` 1296` ``` qed ``` lp15@62408 ` 1297` ```qed ``` lp15@62408 ` 1298` lp15@62408 ` 1299` lp15@62408 ` 1300` ```subsection\The Schwarz Lemma\ ``` lp15@62408 ` 1301` lp15@62408 ` 1302` ```lemma Schwarz1: ``` lp15@62408 ` 1303` ``` assumes holf: "f holomorphic_on S" ``` lp15@62408 ` 1304` ``` and contf: "continuous_on (closure S) f" ``` lp15@62408 ` 1305` ``` and S: "open S" "connected S" ``` lp15@62408 ` 1306` ``` and boS: "bounded S" ``` lp15@62408 ` 1307` ``` and "S \ {}" ``` lp15@62408 ` 1308` ``` obtains w where "w \ frontier S" ``` lp15@62408 ` 1309` ``` "\z. z \ closure S \ norm (f z) \ norm (f w)" ``` lp15@62408 ` 1310` ```proof - ``` lp15@62408 ` 1311` ``` have connf: "continuous_on (closure S) (norm o f)" ``` lp15@62408 ` 1312` ``` using contf continuous_on_compose continuous_on_norm_id by blast ``` lp15@62408 ` 1313` ``` have coc: "compact (closure S)" ``` lp15@62408 ` 1314` ``` by (simp add: \bounded S\ bounded_closure compact_eq_bounded_closed) ``` lp15@62408 ` 1315` ``` then obtain x where x: "x \ closure S" and xmax: "\z. z \ closure S \ norm(f z) \ norm(f x)" ``` lp15@62408 ` 1316` ``` apply (rule bexE [OF continuous_attains_sup [OF _ _ connf]]) ``` lp15@62408 ` 1317` ``` using \S \ {}\ apply auto ``` lp15@62408 ` 1318` ``` done ``` lp15@62408 ` 1319` ``` then show ?thesis ``` lp15@62408 ` 1320` ``` proof (cases "x \ frontier S") ``` lp15@62408 ` 1321` ``` case True ``` lp15@62408 ` 1322` ``` then show ?thesis using that xmax by blast ``` lp15@62408 ` 1323` ``` next ``` lp15@62408 ` 1324` ``` case False ``` lp15@62408 ` 1325` ``` then have "x \ S" ``` lp15@62408 ` 1326` ``` using \open S\ frontier_def interior_eq x by auto ``` lp15@62408 ` 1327` ``` then have "f constant_on S" ``` lp15@62408 ` 1328` ``` apply (rule maximum_modulus_principle [OF holf S \open S\ order_refl]) ``` lp15@62408 ` 1329` ``` using closure_subset apply (blast intro: xmax) ``` lp15@62408 ` 1330` ``` done ``` lp15@62408 ` 1331` ``` then have "f constant_on (closure S)" ``` lp15@62408 ` 1332` ``` by (rule constant_on_closureI [OF _ contf]) ``` lp15@62408 ` 1333` ``` then obtain c where c: "\x. x \ closure S \ f x = c" ``` lp15@62408 ` 1334` ``` by (meson constant_on_def) ``` lp15@62408 ` 1335` ``` obtain w where "w \ frontier S" ``` lp15@62408 ` 1336` ``` by (metis coc all_not_in_conv assms(6) closure_UNIV frontier_eq_empty not_compact_UNIV) ``` lp15@62408 ` 1337` ``` then show ?thesis ``` lp15@62408 ` 1338` ``` by (simp add: c frontier_def that) ``` lp15@62408 ` 1339` ``` qed ``` lp15@62408 ` 1340` ```qed ``` lp15@62408 ` 1341` lp15@62408 ` 1342` ```lemma Schwarz2: ``` lp15@62408 ` 1343` ``` "\f holomorphic_on ball 0 r; ``` lp15@62408 ` 1344` ``` 0 < s; ball w s \ ball 0 r; ``` lp15@62408 ` 1345` ``` \z. norm (w-z) < s \ norm(f z) \ norm(f w)\ ``` lp15@62408 ` 1346` ``` \ f constant_on ball 0 r" ``` lp15@62408 ` 1347` ```by (rule maximum_modulus_principle [where U = "ball w s" and \ = w]) (simp_all add: dist_norm) ``` lp15@62408 ` 1348` lp15@62408 ` 1349` ```lemma Schwarz3: ``` lp15@62408 ` 1350` ``` assumes holf: "f holomorphic_on (ball 0 r)" and [simp]: "f 0 = 0" ``` lp15@62408 ` 1351` ``` obtains h where "h holomorphic_on (ball 0 r)" and "\z. norm z < r \ f z = z * (h z)" and "deriv f 0 = h 0" ``` lp15@62408 ` 1352` ```proof - ``` wenzelm@63040 ` 1353` ``` define h where "h z = (if z = 0 then deriv f 0 else f z / z)" for z ``` lp15@62408 ` 1354` ``` have d0: "deriv f 0 = h 0" ``` lp15@62408 ` 1355` ``` by (simp add: h_def) ``` lp15@62408 ` 1356` ``` moreover have "h holomorphic_on (ball 0 r)" ``` lp15@62408 ` 1357` ``` by (rule pole_theorem_open_0 [OF holf, of 0]) (auto simp: h_def) ``` lp15@62408 ` 1358` ``` moreover have "norm z < r \ f z = z * h z" for z ``` lp15@62408 ` 1359` ``` by (simp add: h_def) ``` lp15@62408 ` 1360` ``` ultimately show ?thesis ``` lp15@62408 ` 1361` ``` using that by blast ``` lp15@62408 ` 1362` ```qed ``` lp15@62408 ` 1363` lp15@62408 ` 1364` lp15@62408 ` 1365` ```proposition Schwarz_Lemma: ``` lp15@62408 ` 1366` ``` assumes holf: "f holomorphic_on (ball 0 1)" and [simp]: "f 0 = 0" ``` lp15@62408 ` 1367` ``` and no: "\z. norm z < 1 \ norm (f z) < 1" ``` lp15@62408 ` 1368` ``` and \: "norm \ < 1" ``` lp15@62408 ` 1369` ``` shows "norm (f \) \ norm \" and "norm(deriv f 0) \ 1" ``` lp15@62408 ` 1370` ``` and "((\z. norm z < 1 \ z \ 0 \ norm(f z) = norm z) \ norm(deriv f 0) = 1) ``` lp15@62408 ` 1371` ``` \ \\. (\z. norm z < 1 \ f z = \ * z) \ norm \ = 1" (is "?P \ ?Q") ``` lp15@62408 ` 1372` ```proof - ``` lp15@62408 ` 1373` ``` obtain h where holh: "h holomorphic_on (ball 0 1)" ``` lp15@62408 ` 1374` ``` and fz_eq: "\z. norm z < 1 \ f z = z * (h z)" and df0: "deriv f 0 = h 0" ``` lp15@62408 ` 1375` ``` by (rule Schwarz3 [OF holf]) auto ``` lp15@62408 ` 1376` ``` have noh_le: "norm (h z) \ 1" if z: "norm z < 1" for z ``` lp15@62408 ` 1377` ``` proof - ``` lp15@62408 ` 1378` ``` have "norm (h z) < a" if a: "1 < a" for a ``` lp15@62408 ` 1379` ``` proof - ``` lp15@62408 ` 1380` ``` have "max (inverse a) (norm z) < 1" ``` lp15@62408 ` 1381` ``` using z a by (simp_all add: inverse_less_1_iff) ``` lp15@62408 ` 1382` ``` then obtain r where r: "max (inverse a) (norm z) < r" and "r < 1" ``` lp15@62408 ` 1383` ``` using Rats_dense_in_real by blast ``` lp15@62408 ` 1384` ``` then have nzr: "norm z < r" and ira: "inverse r < a" ``` lp15@62408 ` 1385` ``` using z a less_imp_inverse_less by force+ ``` lp15@62408 ` 1386` ``` then have "0 < r" ``` lp15@62408 ` 1387` ``` by (meson norm_not_less_zero not_le order.strict_trans2) ``` lp15@62408 ` 1388` ``` have holh': "h holomorphic_on ball 0 r" ``` lp15@62408 ` 1389` ``` by (meson holh \r < 1\ holomorphic_on_subset less_eq_real_def subset_ball) ``` lp15@62408 ` 1390` ``` have conth': "continuous_on (cball 0 r) h" ``` lp15@62408 ` 1391` ``` by (meson \r < 1\ dual_order.trans holh holomorphic_on_imp_continuous_on holomorphic_on_subset mem_ball_0 mem_cball_0 not_less subsetI) ``` lp15@62408 ` 1392` ``` obtain w where w: "norm w = r" and lenw: "\z. norm z < r \ norm(h z) \ norm(h w)" ``` lp15@62408 ` 1393` ``` apply (rule Schwarz1 [OF holh']) using conth' \0 < r\ by auto ``` lp15@62408 ` 1394` ``` have "h w = f w / w" using fz_eq \r < 1\ nzr w by auto ``` lp15@62408 ` 1395` ``` then have "cmod (h z) < inverse r" ``` lp15@62408 ` 1396` ``` by (metis \0 < r\ \r < 1\ divide_strict_right_mono inverse_eq_divide ``` lp15@62408 ` 1397` ``` le_less_trans lenw no norm_divide nzr w) ``` lp15@62408 ` 1398` ``` then show ?thesis using ira by linarith ``` lp15@62408 ` 1399` ``` qed ``` lp15@62408 ` 1400` ``` then show "norm (h z) \ 1" ``` lp15@62408 ` 1401` ``` using not_le by blast ``` lp15@62408 ` 1402` ``` qed ``` lp15@62408 ` 1403` ``` show "cmod (f \) \ cmod \" ``` lp15@62408 ` 1404` ``` proof (cases "\ = 0") ``` lp15@62408 ` 1405` ``` case True then show ?thesis by auto ``` lp15@62408 ` 1406` ``` next ``` lp15@62408 ` 1407` ``` case False ``` lp15@62408 ` 1408` ``` then show ?thesis ``` lp15@62408 ` 1409` ``` by (simp add: noh_le fz_eq \ mult_left_le norm_mult) ``` lp15@62408 ` 1410` ``` qed ``` lp15@62408 ` 1411` ``` show no_df0: "norm(deriv f 0) \ 1" ``` lp15@62408 ` 1412` ``` by (simp add: \\z. cmod z < 1 \ cmod (h z) \ 1\ df0) ``` lp15@62408 ` 1413` ``` show "?Q" if "?P" ``` wenzelm@63540 ` 1414` ``` using that ``` lp15@62408 ` 1415` ``` proof ``` lp15@62408 ` 1416` ``` assume "\z. cmod z < 1 \ z \ 0 \ cmod (f z) = cmod z" ``` lp15@62408 ` 1417` ``` then obtain \ where \: "cmod \ < 1" "\ \ 0" "cmod (f \) = cmod \" by blast ``` lp15@62408 ` 1418` ``` then have [simp]: "norm (h \) = 1" ``` lp15@62408 ` 1419` ``` by (simp add: fz_eq norm_mult) ``` lp15@62408 ` 1420` ``` have "ball \ (1 - cmod \) \ ball 0 1" ``` lp15@62408 ` 1421` ``` by (simp add: ball_subset_ball_iff) ``` lp15@62408 ` 1422` ``` moreover have "\z. cmod (\ - z) < 1 - cmod \ \ cmod (h z) \ cmod (h \)" ``` lp15@62408 ` 1423` ``` apply (simp add: algebra_simps) ``` lp15@62408 ` 1424` ``` by (metis add_diff_cancel_left' diff_diff_eq2 le_less_trans noh_le norm_triangle_ineq4) ``` lp15@62408 ` 1425` ``` ultimately obtain c where c: "\z. norm z < 1 \ h z = c" ``` lp15@62408 ` 1426` ``` using Schwarz2 [OF holh, of "1 - norm \" \, unfolded constant_on_def] \ by auto ``` wenzelm@63540 ` 1427` ``` then have "norm c = 1" ``` lp15@62408 ` 1428` ``` using \ by force ``` wenzelm@63540 ` 1429` ``` with c show ?thesis ``` lp15@62408 ` 1430` ``` using fz_eq by auto ``` lp15@62408 ` 1431` ``` next ``` lp15@62408 ` 1432` ``` assume [simp]: "cmod (deriv f 0) = 1" ``` lp15@62408 ` 1433` ``` then obtain c where c: "\z. norm z < 1 \ h z = c" ``` lp15@62408 ` 1434` ``` using Schwarz2 [OF holh zero_less_one, of 0, unfolded constant_on_def] df0 noh_le ``` lp15@62408 ` 1435` ``` by auto ``` lp15@62408 ` 1436` ``` moreover have "norm c = 1" using df0 c by auto ``` lp15@62408 ` 1437` ``` ultimately show ?thesis ``` lp15@62408 ` 1438` ``` using fz_eq by auto ``` lp15@62408 ` 1439` ``` qed ``` lp15@62408 ` 1440` ```qed ``` lp15@62408 ` 1441` lp15@62408 ` 1442` ```subsection\The Schwarz reflection principle\ ``` lp15@62408 ` 1443` lp15@62408 ` 1444` ```lemma hol_pal_lem0: ``` lp15@62408 ` 1445` ``` assumes "d \ a \ k" "k \ d \ b" ``` lp15@62408 ` 1446` ``` obtains c where ``` lp15@62408 ` 1447` ``` "c \ closed_segment a b" "d \ c = k" ``` lp15@62408 ` 1448` ``` "\z. z \ closed_segment a c \ d \ z \ k" ``` lp15@62408 ` 1449` ``` "\z. z \ closed_segment c b \ k \ d \ z" ``` lp15@62408 ` 1450` ```proof - ``` lp15@62408 ` 1451` ``` obtain c where cin: "c \ closed_segment a b" and keq: "k = d \ c" ``` lp15@62408 ` 1452` ``` using connected_ivt_hyperplane [of "closed_segment a b" a b d k] ``` lp15@62408 ` 1453` ``` by (auto simp: assms) ``` lp15@62408 ` 1454` ``` have "closed_segment a c \ {z. d \ z \ k}" "closed_segment c b \ {z. k \ d \ z}" ``` lp15@62408 ` 1455` ``` unfolding segment_convex_hull using assms keq ``` lp15@62408 ` 1456` ``` by (auto simp: convex_halfspace_le convex_halfspace_ge hull_minimal) ``` lp15@62408 ` 1457` ``` then show ?thesis using cin that by fastforce ``` lp15@62408 ` 1458` ```qed ``` lp15@62408 ` 1459` lp15@62408 ` 1460` ```lemma hol_pal_lem1: ``` lp15@62408 ` 1461` ``` assumes "convex S" "open S" ``` lp15@62408 ` 1462` ``` and abc: "a \ S" "b \ S" "c \ S" ``` lp15@62408 ` 1463` ``` "d \ 0" and lek: "d \ a \ k" "d \ b \ k" "d \ c \ k" ``` lp15@62408 ` 1464` ``` and holf1: "f holomorphic_on {z. z \ S \ d \ z < k}" ``` lp15@62408 ` 1465` ``` and contf: "continuous_on S f" ``` lp15@62408 ` 1466` ``` shows "contour_integral (linepath a b) f + ``` lp15@62408 ` 1467` ``` contour_integral (linepath b c) f + ``` lp15@62408 ` 1468` ``` contour_integral (linepath c a) f = 0" ``` lp15@62408 ` 1469` ```proof - ``` lp15@62408 ` 1470` ``` have "interior (convex hull {a, b, c}) \ interior(S \ {x. d \ x \ k})" ``` lp15@62408 ` 1471` ``` apply (rule interior_mono) ``` lp15@62408 ` 1472` ``` apply (rule hull_minimal) ``` lp15@62408 ` 1473` ``` apply (simp add: abc lek) ``` lp15@62408 ` 1474` ``` apply (rule convex_Int [OF \convex S\ convex_halfspace_le]) ``` lp15@62408 ` 1475` ``` done ``` lp15@62408 ` 1476` ``` also have "... \ {z \ S. d \ z < k}" ``` lp15@62408 ` 1477` ``` by (force simp: interior_open [OF \open S\] \d \ 0\) ``` lp15@62408 ` 1478` ``` finally have *: "interior (convex hull {a, b, c}) \ {z \ S. d \ z < k}" . ``` lp15@62408 ` 1479` ``` have "continuous_on (convex hull {a,b,c}) f" ``` lp15@62408 ` 1480` ``` using \convex S\ contf abc continuous_on_subset subset_hull ``` lp15@62408 ` 1481` ``` by fastforce ``` lp15@62408 ` 1482` ``` moreover have "f holomorphic_on interior (convex hull {a,b,c})" ``` lp15@62408 ` 1483` ``` by (rule holomorphic_on_subset [OF holf1 *]) ``` lp15@62408 ` 1484` ``` ultimately show ?thesis ``` lp15@62408 ` 1485` ``` using Cauchy_theorem_triangle_interior has_chain_integral_chain_integral3 ``` lp15@62408 ` 1486` ``` by blast ``` lp15@62408 ` 1487` ```qed ``` lp15@62408 ` 1488` lp15@62408 ` 1489` ```lemma hol_pal_lem2: ``` lp15@62408 ` 1490` ``` assumes S: "convex S" "open S" ``` lp15@62408 ` 1491` ``` and abc: "a \ S" "b \ S" "c \ S" ``` lp15@62408 ` 1492` ``` and "d \ 0" and lek: "d \ a \ k" "d \ b \ k" ``` lp15@62408 ` 1493` ``` and holf1: "f holomorphic_on {z. z \ S \ d \ z < k}" ``` lp15@62408 ` 1494` ``` and holf2: "f holomorphic_on {z. z \ S \ k < d \ z}" ``` lp15@62408 ` 1495` ``` and contf: "continuous_on S f" ``` lp15@62408 ` 1496` ``` shows "contour_integral (linepath a b) f + ``` lp15@62408 ` 1497` ``` contour_integral (linepath b c) f + ``` lp15@62408 ` 1498` ``` contour_integral (linepath c a) f = 0" ``` lp15@62408 ` 1499` ```proof (cases "d \ c \ k") ``` lp15@62408 ` 1500` ``` case True show ?thesis ``` lp15@62408 ` 1501` ``` by (rule hol_pal_lem1 [OF S abc \d \ 0\ lek True holf1 contf]) ``` lp15@62408 ` 1502` ```next ``` lp15@62408 ` 1503` ``` case False ``` lp15@62408 ` 1504` ``` then have "d \ c > k" by force ``` lp15@62408 ` 1505` ``` obtain a' where a': "a' \ closed_segment b c" and "d \ a' = k" ``` lp15@62408 ` 1506` ``` and ba': "\z. z \ closed_segment b a' \ d \ z \ k" ``` lp15@62408 ` 1507` ``` and a'c: "\z. z \ closed_segment a' c \ k \ d \ z" ``` lp15@62408 ` 1508` ``` apply (rule hol_pal_lem0 [of d b k c, OF \d \ b \ k\]) ``` lp15@62408 ` 1509` ``` using False by auto ``` lp15@62408 ` 1510` ``` obtain b' where b': "b' \ closed_segment a c" and "d \ b' = k" ``` lp15@62408 ` 1511` ``` and ab': "\z. z \ closed_segment a b' \ d \ z \ k" ``` lp15@62408 ` 1512` ``` and b'c: "\z. z \ closed_segment b' c \ k \ d \ z" ``` lp15@62408 ` 1513` ``` apply (rule hol_pal_lem0 [of d a k c, OF \d \ a \ k\]) ``` lp15@62408 ` 1514` ``` using False by auto ``` lp15@62408 ` 1515` ``` have a'b': "a' \ S \ b' \ S" ``` lp15@62408 ` 1516` ``` using a' abc b' convex_contains_segment \convex S\ by auto ``` lp15@62408 ` 1517` ``` have "continuous_on (closed_segment c a) f" ``` lp15@62408 ` 1518` ``` by (meson abc contf continuous_on_subset convex_contains_segment \convex S\) ``` lp15@62408 ` 1519` ``` then have 1: "contour_integral (linepath c a) f = ``` lp15@62408 ` 1520` ``` contour_integral (linepath c b') f + contour_integral (linepath b' a) f" ``` lp15@62408 ` 1521` ``` apply (rule contour_integral_split_linepath) ``` lp15@62408 ` 1522` ``` using b' by (simp add: closed_segment_commute) ``` lp15@62408 ` 1523` ``` have "continuous_on (closed_segment b c) f" ``` lp15@62408 ` 1524` ``` by (meson abc contf continuous_on_subset convex_contains_segment \convex S\) ``` lp15@62408 ` 1525` ``` then have 2: "contour_integral (linepath b c) f = ``` lp15@62408 ` 1526` ``` contour_integral (linepath b a') f + contour_integral (linepath a' c) f" ``` lp15@62408 ` 1527` ``` by (rule contour_integral_split_linepath [OF _ a']) ``` lp15@62463 ` 1528` ``` have 3: "contour_integral (reversepath (linepath b' a')) f = ``` lp15@62408 ` 1529` ``` - contour_integral (linepath b' a') f" ``` lp15@62408 ` 1530` ``` by (rule contour_integral_reversepath [OF valid_path_linepath]) ``` lp15@62534 ` 1531` ``` have fcd_le: "f field_differentiable at x" ``` lp15@62408 ` 1532` ``` if "x \ interior S \ x \ interior {x. d \ x \ k}" for x ``` lp15@62408 ` 1533` ``` proof - ``` lp15@62408 ` 1534` ``` have "f holomorphic_on S \ {c. d \ c < k}" ``` lp15@62408 ` 1535` ``` by (metis (no_types) Collect_conj_eq Collect_mem_eq holf1) ``` lp15@62408 ` 1536` ``` then have "\C D. x \ interior C \ interior D \ f holomorphic_on interior C \ interior D" ``` lp15@62408 ` 1537` ``` using that ``` lp15@62408 ` 1538` ``` by (metis Collect_mem_eq Int_Collect \d \ 0\ interior_halfspace_le interior_open \open S\) ``` lp15@62534 ` 1539` ``` then show "f field_differentiable at x" ``` lp15@62408 ` 1540` ``` by (metis at_within_interior holomorphic_on_def interior_Int interior_interior) ``` lp15@62408 ` 1541` ``` qed ``` lp15@62408 ` 1542` ``` have ab_le: "\x. x \ closed_segment a b \ d \ x \ k" ``` lp15@62408 ` 1543` ``` proof - ``` lp15@62408 ` 1544` ``` fix x :: complex ``` lp15@62408 ` 1545` ``` assume "x \ closed_segment a b" ``` lp15@62408 ` 1546` ``` then have "\C. x \ C \ b \ C \ a \ C \ \ convex C" ``` lp15@62408 ` 1547` ``` by (meson contra_subsetD convex_contains_segment) ``` lp15@62408 ` 1548` ``` then show "d \ x \ k" ``` lp15@62408 ` 1549` ``` by (metis lek convex_halfspace_le mem_Collect_eq) ``` lp15@62408 ` 1550` ``` qed ``` lp15@62408 ` 1551` ``` have "continuous_on (S \ {x. d \ x \ k}) f" using contf ``` lp15@62408 ` 1552` ``` by (simp add: continuous_on_subset) ``` lp15@62408 ` 1553` ``` then have "(f has_contour_integral 0) ``` lp15@62408 ` 1554` ``` (linepath a b +++ linepath b a' +++ linepath a' b' +++ linepath b' a)" ``` lp15@62408 ` 1555` ``` apply (rule Cauchy_theorem_convex [where k = "{}"]) ``` lp15@62408 ` 1556` ``` apply (simp_all add: path_image_join convex_Int convex_halfspace_le \convex S\ fcd_le ab_le ``` lp15@62408 ` 1557` ``` closed_segment_subset abc a'b' ba') ``` lp15@62408 ` 1558` ``` by (metis \d \ a' = k\ \d \ b' = k\ convex_contains_segment convex_halfspace_le lek(1) mem_Collect_eq order_refl) ``` lp15@62408 ` 1559` ``` then have 4: "contour_integral (linepath a b) f + ``` lp15@62408 ` 1560` ``` contour_integral (linepath b a') f + ``` lp15@62408 ` 1561` ``` contour_integral (linepath a' b') f + ``` lp15@62408 ` 1562` ``` contour_integral (linepath b' a) f = 0" ``` lp15@62408 ` 1563` ``` by (rule has_chain_integral_chain_integral4) ``` lp15@62534 ` 1564` ``` have fcd_ge: "f field_differentiable at x" ``` lp15@62408 ` 1565` ``` if "x \ interior S \ x \ interior {x. k \ d \ x}" for x ``` lp15@62408 ` 1566` ``` proof - ``` lp15@62408 ` 1567` ``` have f2: "f holomorphic_on S \ {c. k < d \ c}" ``` lp15@62408 ` 1568` ``` by (metis (full_types) Collect_conj_eq Collect_mem_eq holf2) ``` lp15@62408 ` 1569` ``` have f3: "interior S = S" ``` lp15@62408 ` 1570` ``` by (simp add: interior_open \open S\) ``` lp15@62408 ` 1571` ``` then have "x \ S \ interior {c. k \ d \ c}" ``` lp15@62408 ` 1572` ``` using that by simp ``` lp15@62534 ` 1573` ``` then show "f field_differentiable at x" ``` lp15@62408 ` 1574` ``` using f3 f2 unfolding holomorphic_on_def ``` lp15@62408 ` 1575` ``` by (metis (no_types) \d \ 0\ at_within_interior interior_Int interior_halfspace_ge interior_interior) ``` lp15@62408 ` 1576` ``` qed ``` lp15@62408 ` 1577` ``` have "continuous_on (S \ {x. k \ d \ x}) f" using contf ``` lp15@62408 ` 1578` ``` by (simp add: continuous_on_subset) ``` lp15@62408 ` 1579` ``` then have "(f has_contour_integral 0) (linepath a' c +++ linepath c b' +++ linepath b' a')" ``` lp15@62408 ` 1580` ``` apply (rule Cauchy_theorem_convex [where k = "{}"]) ``` lp15@62408 ` 1581` ``` apply (simp_all add: path_image_join convex_Int convex_halfspace_ge \convex S\ ``` lp15@62408 ` 1582` ``` fcd_ge closed_segment_subset abc a'b' a'c) ``` lp15@62408 ` 1583` ``` by (metis \d \ a' = k\ b'c closed_segment_commute convex_contains_segment ``` lp15@62408 ` 1584` ``` convex_halfspace_ge ends_in_segment(2) mem_Collect_eq order_refl) ``` lp15@62408 ` 1585` ``` then have 5: "contour_integral (linepath a' c) f + contour_integral (linepath c b') f + contour_integral (linepath b' a') f = 0" ``` lp15@62408 ` 1586` ``` by (rule has_chain_integral_chain_integral3) ``` lp15@62408 ` 1587` ``` show ?thesis ``` lp15@62408 ` 1588` ``` using 1 2 3 4 5 by (metis add.assoc eq_neg_iff_add_eq_0 reversepath_linepath) ``` lp15@62408 ` 1589` ```qed ``` lp15@62408 ` 1590` lp15@62408 ` 1591` ```lemma hol_pal_lem3: ``` lp15@62408 ` 1592` ``` assumes S: "convex S" "open S" ``` lp15@62408 ` 1593` ``` and abc: "a \ S" "b \ S" "c \ S" ``` lp15@62408 ` 1594` ``` and "d \ 0" and lek: "d \ a \ k" ``` lp15@62408 ` 1595` ``` and holf1: "f holomorphic_on {z. z \ S \ d \ z < k}" ``` lp15@62408 ` 1596` ``` and holf2: "f holomorphic_on {z. z \ S \ k < d \ z}" ``` lp15@62408 ` 1597` ``` and contf: "continuous_on S f" ``` lp15@62408 ` 1598` ``` shows "contour_integral (linepath a b) f + ``` lp15@62408 ` 1599` ``` contour_integral (linepath b c) f + ``` lp15@62408 ` 1600` ``` contour_integral (linepath c a) f = 0" ``` lp15@62408 ` 1601` ```proof (cases "d \ b \ k") ``` lp15@62408 ` 1602` ``` case True show ?thesis ``` lp15@62408 ` 1603` ``` by (rule hol_pal_lem2 [OF S abc \d \ 0\ lek True holf1 holf2 contf]) ``` lp15@62408 ` 1604` ```next ``` lp15@62408 ` 1605` ``` case False ``` lp15@62408 ` 1606` ``` show ?thesis ``` lp15@62408 ` 1607` ``` proof (cases "d \ c \ k") ``` lp15@62408 ` 1608` ``` case True ``` lp15@62408 ` 1609` ``` have "contour_integral (linepath c a) f + ``` lp15@62408 ` 1610` ``` contour_integral (linepath a b) f + ``` lp15@62408 ` 1611` ``` contour_integral (linepath b c) f = 0" ``` lp15@62408 ` 1612` ``` by (rule hol_pal_lem2 [OF S \c \ S\ \a \ S\ \b \ S\ \d \ 0\ \d \ c \ k\ lek holf1 holf2 contf]) ``` lp15@62408 ` 1613` ``` then show ?thesis ``` lp15@62408 ` 1614` ``` by (simp add: algebra_simps) ``` lp15@62408 ` 1615` ``` next ``` lp15@62408 ` 1616` ``` case False ``` lp15@62408 ` 1617` ``` have "contour_integral (linepath b c) f + ``` lp15@62408 ` 1618` ``` contour_integral (linepath c a) f + ``` lp15@62408 ` 1619` ``` contour_integral (linepath a b) f = 0" ``` lp15@62408 ` 1620` ``` apply (rule hol_pal_lem2 [OF S \b \ S\ \c \ S\ \a \ S\, of "-d" "-k"]) ``` lp15@62408 ` 1621` ``` using \d \ 0\ \\ d \ b \ k\ False by (simp_all add: holf1 holf2 contf) ``` lp15@62408 ` 1622` ``` then show ?thesis ``` lp15@62408 ` 1623` ``` by (simp add: algebra_simps) ``` lp15@62408 ` 1624` ``` qed ``` lp15@62408 ` 1625` ```qed ``` lp15@62408 ` 1626` lp15@62408 ` 1627` ```lemma hol_pal_lem4: ``` lp15@62408 ` 1628` ``` assumes S: "convex S" "open S" ``` lp15@62408 ` 1629` ``` and abc: "a \ S" "b \ S" "c \ S" and "d \ 0" ``` lp15@62408 ` 1630` ``` and holf1: "f holomorphic_on {z. z \ S \ d \ z < k}" ``` lp15@62408 ` 1631` ``` and holf2: "f holomorphic_on {z. z \ S \ k < d \ z}" ``` lp15@62408 ` 1632` ``` and contf: "continuous_on S f" ``` lp15@62408 ` 1633` ``` shows "contour_integral (linepath a b) f + ``` lp15@62408 ` 1634` ``` contour_integral (linepath b c) f + ``` lp15@62408 ` 1635` ``` contour_integral (linepath c a) f = 0" ``` lp15@62408 ` 1636` ```proof (cases "d \ a \ k") ``` lp15@62408 ` 1637` ``` case True show ?thesis ``` lp15@62408 ` 1638` ``` by (rule hol_pal_lem3 [OF S abc \d \ 0\ True holf1 holf2 contf]) ``` lp15@62408 ` 1639` ```next ``` lp15@62408 ` 1640` ``` case False ``` lp15@62408 ` 1641` ``` show ?thesis ``` lp15@62408 ` 1642` ``` apply (rule hol_pal_lem3 [OF S abc, of "-d" "-k"]) ``` lp15@62408 ` 1643` ``` using \d \ 0\ False by (simp_all add: holf1 holf2 contf) ``` lp15@62408 ` 1644` ```qed ``` lp15@62408 ` 1645` lp15@62408 ` 1646` ```proposition holomorphic_on_paste_across_line: ``` lp15@62408 ` 1647` ``` assumes S: "open S" and "d \ 0" ``` lp15@62408 ` 1648` ``` and holf1: "f holomorphic_on (S \ {z. d \ z < k})" ``` lp15@62408 ` 1649` ``` and holf2: "f holomorphic_on (S \ {z. k < d \ z})" ``` lp15@62408 ` 1650` ``` and contf: "continuous_on S f" ``` lp15@62408 ` 1651` ``` shows "f holomorphic_on S" ``` lp15@62408 ` 1652` ```proof - ``` lp15@62408 ` 1653` ``` have *: "\t. open t \ p \ t \ continuous_on t f \ ``` lp15@62408 ` 1654` ``` (\a b c. convex hull {a, b, c} \ t \ ``` lp15@62408 ` 1655` ``` contour_integral (linepath a b) f + ``` lp15@62408 ` 1656` ``` contour_integral (linepath b c) f + ``` lp15@62408 ` 1657` ``` contour_integral (linepath c a) f = 0)" ``` lp15@62408 ` 1658` ``` if "p \ S" for p ``` lp15@62408 ` 1659` ``` proof - ``` lp15@62408 ` 1660` ``` obtain e where "e>0" and e: "ball p e \ S" ``` lp15@62408 ` 1661` ``` using \p \ S\ openE S by blast ``` lp15@62408 ` 1662` ``` then have "continuous_on (ball p e) f" ``` lp15@62408 ` 1663` ``` using contf continuous_on_subset by blast ``` lp15@62408 ` 1664` ``` moreover have "f holomorphic_on {z. dist p z < e \ d \ z < k}" ``` lp15@62408 ` 1665` ``` apply (rule holomorphic_on_subset [OF holf1]) ``` lp15@62408 ` 1666` ``` using e by auto ``` lp15@62408 ` 1667` ``` moreover have "f holomorphic_on {z. dist p z < e \ k < d \ z}" ``` lp15@62408 ` 1668` ``` apply (rule holomorphic_on_subset [OF holf2]) ``` lp15@62408 ` 1669` ``` using e by auto ``` lp15@62408 ` 1670` ``` ultimately show ?thesis ``` lp15@62408 ` 1671` ``` apply (rule_tac x="ball p e" in exI) ``` lp15@62408 ` 1672` ``` using \e > 0\ e \d \ 0\ ``` lp15@62408 ` 1673` ``` apply (simp add:, clarify) ``` lp15@62408 ` 1674` ``` apply (rule hol_pal_lem4 [of "ball p e" _ _ _ d _ k]) ``` lp15@62408 ` 1675` ``` apply (auto simp: subset_hull) ``` lp15@62408 ` 1676` ``` done ``` lp15@62408 ` 1677` ``` qed ``` lp15@62408 ` 1678` ``` show ?thesis ``` lp15@62408 ` 1679` ``` by (blast intro: * Morera_local_triangle analytic_imp_holomorphic) ``` lp15@62408 ` 1680` ```qed ``` lp15@62408 ` 1681` lp15@62408 ` 1682` ```proposition Schwarz_reflection: ``` lp15@62408 ` 1683` ``` assumes "open S" and cnjs: "cnj ` S \ S" ``` lp15@62408 ` 1684` ``` and holf: "f holomorphic_on (S \ {z. 0 < Im z})" ``` lp15@62408 ` 1685` ``` and contf: "continuous_on (S \ {z. 0 \ Im z}) f" ``` lp15@62408 ` 1686` ``` and f: "\z. \z \ S; z \ \\ \ (f z) \ \" ``` lp15@62408 ` 1687` ``` shows "(\z. if 0 \ Im z then f z else cnj(f(cnj z))) holomorphic_on S" ``` lp15@62408 ` 1688` ```proof - ``` lp15@62408 ` 1689` ``` have 1: "(\z. if 0 \ Im z then f z else cnj (f (cnj z))) holomorphic_on (S \ {z. 0 < Im z})" ``` lp15@62408 ` 1690` ``` by (force intro: iffD1 [OF holomorphic_cong [OF refl] holf]) ``` lp15@62408 ` 1691` ``` have cont_cfc: "continuous_on (S \ {z. Im z \ 0}) (cnj o f o cnj)" ``` lp15@62408 ` 1692` ``` apply (intro continuous_intros continuous_on_compose continuous_on_subset [OF contf]) ``` lp15@62408 ` 1693` ``` using cnjs apply auto ``` lp15@62408 ` 1694` ``` done ``` lp15@62534 ` 1695` ``` have "cnj \ f \ cnj field_differentiable at x within S \ {z. Im z < 0}" ``` lp15@62534 ` 1696` ``` if "x \ S" "Im x < 0" "f field_differentiable at (cnj x) within S \ {z. 0 < Im z}" for x ``` lp15@62408 ` 1697` ``` using that ``` lp15@62534 ` 1698` ``` apply (simp add: field_differentiable_def Derivative.DERIV_within_iff Lim_within dist_norm, clarify) ``` lp15@62408 ` 1699` ``` apply (rule_tac x="cnj f'" in exI) ``` lp15@62408 ` 1700` ``` apply (elim all_forward ex_forward conj_forward imp_forward asm_rl, clarify) ``` lp15@62408 ` 1701` ``` apply (drule_tac x="cnj xa" in bspec) ``` lp15@62408 ` 1702` ``` using cnjs apply force ``` lp15@62408 ` 1703` ``` apply (metis complex_cnj_cnj complex_cnj_diff complex_cnj_divide complex_mod_cnj) ``` lp15@62408 ` 1704` ``` done ``` lp15@62408 ` 1705` ``` then have hol_cfc: "(cnj o f o cnj) holomorphic_on (S \ {z. Im z < 0})" ``` lp15@62408 ` 1706` ``` using holf cnjs ``` lp15@62408 ` 1707` ``` by (force simp: holomorphic_on_def) ``` lp15@62408 ` 1708` ``` have 2: "(\z. if 0 \ Im z then f z else cnj (f (cnj z))) holomorphic_on (S \ {z. Im z < 0})" ``` lp15@62408 ` 1709` ``` apply (rule iffD1 [OF holomorphic_cong [OF refl]]) ``` lp15@62408 ` 1710` ``` using hol_cfc by auto ``` lp15@62408 ` 1711` ``` have [simp]: "(S \ {z. 0 \ Im z}) \ (S \ {z. Im z \ 0}) = S" ``` lp15@62408 ` 1712` ``` by force ``` lp15@62408 ` 1713` ``` have "continuous_on ((S \ {z. 0 \ Im z}) \ (S \ {z. Im z \ 0})) ``` lp15@62408 ` 1714` ``` (\z. if 0 \ Im z then f z else cnj (f (cnj z)))" ``` lp15@62408 ` 1715` ``` apply (rule continuous_on_cases_local) ``` lp15@62408 ` 1716` ``` using cont_cfc contf ``` lp15@62408 ` 1717` ``` apply (simp_all add: closedin_closed_Int closed_halfspace_Im_le closed_halfspace_Im_ge) ``` lp15@62408 ` 1718` ``` using f Reals_cnj_iff complex_is_Real_iff apply auto ``` lp15@62408 ` 1719` ``` done ``` lp15@62408 ` 1720` ``` then have 3: "continuous_on S (\z. if 0 \ Im z then f z else cnj (f (cnj z)))" ``` lp15@62408 ` 1721` ``` by force ``` lp15@62408 ` 1722` ``` show ?thesis ``` wenzelm@63589 ` 1723` ``` apply (rule holomorphic_on_paste_across_line [OF \open S\, of "- \" _ 0]) ``` lp15@62408 ` 1724` ``` using 1 2 3 ``` lp15@62408 ` 1725` ``` apply auto ``` lp15@62408 ` 1726` ``` done ``` lp15@62408 ` 1727` ```qed ``` lp15@62408 ` 1728` lp15@62533 ` 1729` ```subsection\Bloch's theorem\ ``` lp15@62533 ` 1730` lp15@62533 ` 1731` ```lemma Bloch_lemma_0: ``` lp15@62533 ` 1732` ``` assumes holf: "f holomorphic_on cball 0 r" and "0 < r" ``` lp15@62533 ` 1733` ``` and [simp]: "f 0 = 0" ``` lp15@62533 ` 1734` ``` and le: "\z. norm z < r \ norm(deriv f z) \ 2 * norm(deriv f 0)" ``` lp15@62533 ` 1735` ``` shows "ball 0 ((3 - 2 * sqrt 2) * r * norm(deriv f 0)) \ f ` ball 0 r" ``` lp15@62533 ` 1736` ```proof - ``` lp15@62533 ` 1737` ``` have "sqrt 2 < 3/2" ``` lp15@62533 ` 1738` ``` by (rule real_less_lsqrt) (auto simp: power2_eq_square) ``` lp15@62533 ` 1739` ``` then have sq3: "0 < 3 - 2 * sqrt 2" by simp ``` lp15@62533 ` 1740` ``` show ?thesis ``` lp15@62533 ` 1741` ``` proof (cases "deriv f 0 = 0") ``` lp15@62533 ` 1742` ``` case True then show ?thesis by simp ``` lp15@62533 ` 1743` ``` next ``` lp15@62533 ` 1744` ``` case False ``` wenzelm@63040 ` 1745` ``` define C where "C = 2 * norm(deriv f 0)" ``` lp15@62533 ` 1746` ``` have "0 < C" using False by (simp add: C_def) ``` lp15@62533 ` 1747` ``` have holf': "f holomorphic_on ball 0 r" using holf ``` lp15@62533 ` 1748` ``` using ball_subset_cball holomorphic_on_subset by blast ``` lp15@62533 ` 1749` ``` then have holdf': "deriv f holomorphic_on ball 0 r" ``` lp15@62533 ` 1750` ``` by (rule holomorphic_deriv [OF _ open_ball]) ``` lp15@62533 ` 1751` ``` have "Le1": "norm(deriv f z - deriv f 0) \ norm z / (r - norm z) * C" ``` lp15@62533 ` 1752` ``` if "norm z < r" for z ``` lp15@62533 ` 1753` ``` proof - ``` lp15@62533 ` 1754` ``` have T1: "norm(deriv f z - deriv f 0) \ norm z / (R - norm z) * C" ``` lp15@62533 ` 1755` ``` if R: "norm z < R" "R < r" for R ``` lp15@62533 ` 1756` ``` proof - ``` lp15@62533 ` 1757` ``` have "0 < R" using R ``` lp15@62533 ` 1758` ``` by (metis less_trans norm_zero zero_less_norm_iff) ``` lp15@62533 ` 1759` ``` have df_le: "\x. norm x < r \ norm (deriv f x) \ C" ``` lp15@62533 ` 1760` ``` using le by (simp add: C_def) ``` lp15@62533 ` 1761` ``` have hol_df: "deriv f holomorphic_on cball 0 R" ``` lp15@62533 ` 1762` ``` apply (rule holomorphic_on_subset) using R holdf' by auto ``` lp15@62533 ` 1763` ``` have *: "((\w. deriv f w / (w - z)) has_contour_integral 2 * pi * \ * deriv f z) (circlepath 0 R)" ``` lp15@62533 ` 1764` ``` if "norm z < R" for z ``` lp15@62533 ` 1765` ``` using \0 < R\ that Cauchy_integral_formula_convex_simple [OF convex_cball hol_df, of _ "circlepath 0 R"] ``` lp15@62533 ` 1766` ``` by (force simp: winding_number_circlepath) ``` lp15@62533 ` 1767` ``` have **: "((\x. deriv f x / (x - z) - deriv f x / x) has_contour_integral ``` lp15@62533 ` 1768` ``` of_real (2 * pi) * \ * (deriv f z - deriv f 0)) ``` lp15@62533 ` 1769` ``` (circlepath 0 R)" ``` lp15@62533 ` 1770` ``` using has_contour_integral_diff [OF * [of z] * [of 0]] \0 < R\ that ``` lp15@62533 ` 1771` ``` by (simp add: algebra_simps) ``` lp15@62533 ` 1772` ``` have [simp]: "\x. norm x = R \ x \ z" using that(1) by blast ``` lp15@62533 ` 1773` ``` have "norm (deriv f x / (x - z) - deriv f x / x) ``` lp15@62533 ` 1774` ``` \ C * norm z / (R * (R - norm z))" ``` lp15@62533 ` 1775` ``` if "norm x = R" for x ``` lp15@62533 ` 1776` ``` proof - ``` lp15@62533 ` 1777` ``` have [simp]: "norm (deriv f x * x - deriv f x * (x - z)) = ``` lp15@62533 ` 1778` ``` norm (deriv f x) * norm z" ``` lp15@62533 ` 1779` ``` by (simp add: norm_mult right_diff_distrib') ``` lp15@62533 ` 1780` ``` show ?thesis ``` lp15@62533 ` 1781` ``` using \0 < R\ \0 < C\ R that ``` lp15@62533 ` 1782` ``` apply (simp add: norm_mult norm_divide divide_simps) ``` lp15@62533 ` 1783` ``` using df_le norm_triangle_ineq2 \0 < C\ apply (auto intro!: mult_mono) ``` lp15@62533 ` 1784` ``` done ``` lp15@62533 ` 1785` ``` qed ``` lp15@62533 ` 1786` ``` then show ?thesis ``` lp15@62533 ` 1787` ``` using has_contour_integral_bound_circlepath ``` lp15@62533 ` 1788` ``` [OF **, of "C * norm z/(R*(R - norm z))"] ``` lp15@62533 ` 1789` ``` \0 < R\ \0 < C\ R ``` lp15@62533 ` 1790` ``` apply (simp add: norm_mult norm_divide) ``` lp15@62533 ` 1791` ``` apply (simp add: divide_simps mult.commute) ``` lp15@62533 ` 1792` ``` done ``` lp15@62533 ` 1793` ``` qed ``` lp15@62533 ` 1794` ``` obtain r' where r': "norm z < r'" "r' < r" ``` lp15@62533 ` 1795` ``` using Rats_dense_in_real [of "norm z" r] \norm z < r\ by blast ``` lp15@62533 ` 1796` ``` then have [simp]: "closure {r'<.. norm(f z)" ``` lp15@62533 ` 1807` ``` if r: "norm z < r" for z ``` lp15@62533 ` 1808` ``` proof - ``` lp15@62533 ` 1809` ``` have 1: "\x. x \ ball 0 r \ ``` lp15@62533 ` 1810` ``` ((\z. f z - deriv f 0 * z) has_field_derivative deriv f x - deriv f 0) ``` lp15@62533 ` 1811` ``` (at x within ball 0 r)" ``` lp15@62533 ` 1812` ``` by (rule derivative_eq_intros holomorphic_derivI holf' | simp)+ ``` lp15@62533 ` 1813` ``` have 2: "closed_segment 0 z \ ball 0 r" ``` lp15@62533 ` 1814` ``` by (metis \0 < r\ convex_ball convex_contains_segment dist_self mem_ball mem_ball_0 that) ``` lp15@62533 ` 1815` ``` have 3: "(\t. (norm z)\<^sup>2 * t / (r - norm z) * C) integrable_on {0..1}" ``` lp15@62533 ` 1816` ``` apply (rule integrable_on_cmult_right [where 'b=real, simplified]) ``` lp15@62533 ` 1817` ``` apply (rule integrable_on_cdivide [where 'b=real, simplified]) ``` lp15@62533 ` 1818` ``` apply (rule integrable_on_cmult_left [where 'b=real, simplified]) ``` lp15@62533 ` 1819` ``` apply (rule ident_integrable_on) ``` lp15@62533 ` 1820` ``` done ``` lp15@62533 ` 1821` ``` have 4: "norm (deriv f (x *\<^sub>R z) - deriv f 0) * norm z \ norm z * norm z * x * C / (r - norm z)" ``` lp15@62533 ` 1822` ``` if x: "0 \ x" "x \ 1" for x ``` lp15@62533 ` 1823` ``` proof - ``` lp15@62533 ` 1824` ``` have [simp]: "x * norm z < r" ``` lp15@62533 ` 1825` ``` using r x by (meson le_less_trans mult_le_cancel_right2 norm_not_less_zero) ``` lp15@62533 ` 1826` ``` have "norm (deriv f (x *\<^sub>R z) - deriv f 0) \ norm (x *\<^sub>R z) / (r - norm (x *\<^sub>R z)) * C" ``` lp15@62533 ` 1827` ``` apply (rule Le1) using r x \0 < r\ by simp ``` lp15@62533 ` 1828` ``` also have "... \ norm (x *\<^sub>R z) / (r - norm z) * C" ``` lp15@62533 ` 1829` ``` using r x \0 < r\ ``` lp15@62533 ` 1830` ``` apply (simp add: divide_simps) ``` lp15@62533 ` 1831` ``` by (simp add: \0 < C\ mult.assoc mult_left_le_one_le ordered_comm_semiring_class.comm_mult_left_mono) ``` lp15@62533 ` 1832` ``` finally have "norm (deriv f (x *\<^sub>R z) - deriv f 0) * norm z \ norm (x *\<^sub>R z) / (r - norm z) * C * norm z" ``` lp15@62533 ` 1833` ``` by (rule mult_right_mono) simp ``` lp15@62533 ` 1834` ``` with x show ?thesis by (simp add: algebra_simps) ``` lp15@62533 ` 1835` ``` qed ``` lp15@62533 ` 1836` ``` have le_norm: "abc \ norm d - e \ norm(f - d) \ e \ abc \ norm f" for abc d e and f::complex ``` lp15@62533 ` 1837` ``` by (metis add_diff_cancel_left' add_diff_eq diff_left_mono norm_diff_ineq order_trans) ``` lp15@62533 ` 1838` ``` have "norm (integral {0..1} (\x. (deriv f (x *\<^sub>R z) - deriv f 0) * z)) ``` lp15@62533 ` 1839` ``` \ integral {0..1} (\t. (norm z)\<^sup>2 * t / (r - norm z) * C)" ``` lp15@62533 ` 1840` ``` apply (rule integral_norm_bound_integral) ``` lp15@62533 ` 1841` ``` using contour_integral_primitive [OF 1, of "linepath 0 z"] 2 ``` lp15@62533 ` 1842` ``` apply (simp add: has_contour_integral_linepath has_integral_integrable_integral) ``` lp15@62533 ` 1843` ``` apply (rule 3) ``` lp15@62533 ` 1844` ``` apply (simp add: norm_mult power2_eq_square 4) ``` lp15@62533 ` 1845` ``` done ``` lp15@62533 ` 1846` ``` then have int_le: "norm (f z - deriv f 0 * z) \ (norm z)\<^sup>2 * norm(deriv f 0) / ((r - norm z))" ``` lp15@62533 ` 1847` ``` using contour_integral_primitive [OF 1, of "linepath 0 z"] 2 ``` lp15@62533 ` 1848` ``` apply (simp add: has_contour_integral_linepath has_integral_integrable_integral C_def) ``` lp15@62533 ` 1849` ``` done ``` lp15@62533 ` 1850` ``` show ?thesis ``` lp15@62533 ` 1851` ``` apply (rule le_norm [OF _ int_le]) ``` lp15@62533 ` 1852` ``` using \norm z < r\ ``` lp15@62533 ` 1853` ``` apply (simp add: power2_eq_square divide_simps C_def norm_mult) ``` lp15@62533 ` 1854` ``` proof - ``` lp15@62533 ` 1855` ``` have "norm z * (norm (deriv f 0) * (r - norm z - norm z)) \ norm z * (norm (deriv f 0) * (r - norm z) - norm (deriv f 0) * norm z)" ``` lp15@62533 ` 1856` ``` by (simp add: linordered_field_class.sign_simps(38)) ``` lp15@62533 ` 1857` ``` then show "(norm z * (r - norm z) - norm z * norm z) * norm (deriv f 0) \ norm (deriv f 0) * norm z * (r - norm z) - norm z * norm z * norm (deriv f 0)" ``` lp15@62533 ` 1858` ``` by (simp add: linordered_field_class.sign_simps(38) mult.commute mult.left_commute) ``` lp15@62533 ` 1859` ``` qed ``` lp15@62533 ` 1860` ``` qed ``` lp15@62533 ` 1861` ``` have sq201 [simp]: "0 < (1 - sqrt 2 / 2)" "(1 - sqrt 2 / 2) < 1" ``` lp15@62533 ` 1862` ``` by (auto simp: sqrt2_less_2) ``` lp15@62533 ` 1863` ``` have 1: "continuous_on (closure (ball 0 ((1 - sqrt 2 / 2) * r))) f" ``` lp15@62533 ` 1864` ``` apply (rule continuous_on_subset [OF holomorphic_on_imp_continuous_on [OF holf]]) ``` lp15@62533 ` 1865` ``` apply (subst closure_ball) ``` lp15@62533 ` 1866` ``` using \0 < r\ mult_pos_pos sq201 ``` lp15@62533 ` 1867` ``` apply (auto simp: cball_subset_cball_iff) ``` lp15@62533 ` 1868` ``` done ``` lp15@62533 ` 1869` ``` have 2: "open (f ` interior (ball 0 ((1 - sqrt 2 / 2) * r)))" ``` lp15@62533 ` 1870` ``` apply (rule open_mapping_thm [OF holf' open_ball connected_ball], force) ``` lp15@62533 ` 1871` ``` using \0 < r\ mult_pos_pos sq201 apply (simp add: ball_subset_ball_iff) ``` lp15@62533 ` 1872` ``` using False \0 < r\ centre_in_ball holf' holomorphic_nonconstant by blast ``` lp15@62533 ` 1873` ``` have "ball 0 ((3 - 2 * sqrt 2) * r * norm (deriv f 0)) = ``` lp15@62533 ` 1874` ``` ball (f 0) ((3 - 2 * sqrt 2) * r * norm (deriv f 0))" ``` lp15@62533 ` 1875` ``` by simp ``` lp15@62533 ` 1876` ``` also have "... \ f ` ball 0 ((1 - sqrt 2 / 2) * r)" ``` lp15@62533 ` 1877` ``` proof - ``` lp15@62533 ` 1878` ``` have 3: "(3 - 2 * sqrt 2) * r * norm (deriv f 0) \ norm (f z)" ``` lp15@62533 ` 1879` ``` if "norm z = (1 - sqrt 2 / 2) * r" for z ``` lp15@62533 ` 1880` ``` apply (rule order_trans [OF _ *]) ``` lp15@62533 ` 1881` ``` using \0 < r\ ``` lp15@62533 ` 1882` ``` apply (simp_all add: field_simps power2_eq_square that) ``` lp15@62533 ` 1883` ``` apply (simp add: mult.assoc [symmetric]) ``` lp15@62533 ` 1884` ``` done ``` lp15@62533 ` 1885` ``` show ?thesis ``` lp15@62533 ` 1886` ``` apply (rule ball_subset_open_map_image [OF 1 2 _ bounded_ball]) ``` lp15@62533 ` 1887` ``` using \0 < r\ sq201 3 apply simp_all ``` lp15@62533 ` 1888` ``` using C_def \0 < C\ sq3 apply force ``` lp15@62533 ` 1889` ``` done ``` lp15@62533 ` 1890` ``` qed ``` lp15@62533 ` 1891` ``` also have "... \ f ` ball 0 r" ``` lp15@62533 ` 1892` ``` apply (rule image_subsetI [OF imageI], simp) ``` lp15@62533 ` 1893` ``` apply (erule less_le_trans) ``` lp15@62533 ` 1894` ``` using \0 < r\ apply (auto simp: field_simps) ``` lp15@62533 ` 1895` ``` done ``` lp15@62533 ` 1896` ``` finally show ?thesis . ``` lp15@62533 ` 1897` ``` qed ``` lp15@62533 ` 1898` ```qed ``` lp15@62533 ` 1899` hoelzl@63594 ` 1900` ```lemma Bloch_lemma: ``` lp15@62533 ` 1901` ``` assumes holf: "f holomorphic_on cball a r" and "0 < r" ``` lp15@62533 ` 1902` ``` and le: "\z. z \ ball a r \ norm(deriv f z) \ 2 * norm(deriv f a)" ``` lp15@62533 ` 1903` ``` shows "ball (f a) ((3 - 2 * sqrt 2) * r * norm(deriv f a)) \ f ` ball a r" ``` lp15@62533 ` 1904` ```proof - ``` lp15@62533 ` 1905` ``` have fz: "(\z. f (a + z)) = f o (\z. (a + z))" ``` lp15@62533 ` 1906` ``` by (simp add: o_def) ``` lp15@62533 ` 1907` ``` have hol0: "(\z. f (a + z)) holomorphic_on cball 0 r" ``` lp15@62533 ` 1908` ``` unfolding fz by (intro holomorphic_intros holf holomorphic_on_compose | simp)+ ``` lp15@62534 ` 1909` ``` then have [simp]: "\x. norm x < r \ (\z. f (a + z)) field_differentiable at x" ``` lp15@62533 ` 1910` ``` by (metis Topology_Euclidean_Space.open_ball at_within_open ball_subset_cball diff_0 dist_norm holomorphic_on_def holomorphic_on_subset mem_ball norm_minus_cancel) ``` lp15@62534 ` 1911` ``` have [simp]: "\z. norm z < r \ f field_differentiable at (a + z)" ``` lp15@62533 ` 1912` ``` by (metis holf open_ball add_diff_cancel_left' dist_complex_def holomorphic_on_imp_differentiable_at holomorphic_on_subset interior_cball interior_subset mem_ball norm_minus_commute) ``` lp15@62534 ` 1913` ``` then have [simp]: "f field_differentiable at a" ``` lp15@62533 ` 1914` ``` by (metis add.comm_neutral \0 < r\ norm_eq_zero) ``` lp15@62533 ` 1915` ``` have hol1: "(\z. f (a + z) - f a) holomorphic_on cball 0 r" ``` lp15@62533 ` 1916` ``` by (intro holomorphic_intros hol0) ``` lp15@62533 ` 1917` ``` then have "ball 0 ((3 - 2 * sqrt 2) * r * norm (deriv (\z. f (a + z) - f a) 0)) ``` lp15@62533 ` 1918` ``` \ (\z. f (a + z) - f a) ` ball 0 r" ``` lp15@62533 ` 1919` ``` apply (rule Bloch_lemma_0) ``` lp15@62533 ` 1920` ``` apply (simp_all add: \0 < r\) ``` lp15@62533 ` 1921` ``` apply (simp add: fz complex_derivative_chain) ``` lp15@62533 ` 1922` ``` apply (simp add: dist_norm le) ``` lp15@62533 ` 1923` ``` done ``` lp15@62533 ` 1924` ``` then show ?thesis ``` lp15@62533 ` 1925` ``` apply clarify ``` lp15@62533 ` 1926` ``` apply (drule_tac c="x - f a" in subsetD) ``` lp15@62534 ` 1927` ``` apply (force simp: fz \0 < r\ dist_norm complex_derivative_chain field_differentiable_compose)+ ``` lp15@62533 ` 1928` ``` done ``` lp15@62533 ` 1929` ```qed ``` lp15@62533 ` 1930` hoelzl@63594 ` 1931` ```proposition Bloch_unit: ``` lp15@62533 ` 1932` ``` assumes holf: "f holomorphic_on ball a 1" and [simp]: "deriv f a = 1" ``` lp15@62533 ` 1933` ``` obtains b r where "1/12 < r" "ball b r \ f ` (ball a 1)" ``` lp15@62533 ` 1934` ```proof - ``` wenzelm@63040 ` 1935` ``` define r :: real where "r = 249/256" ``` lp15@62533 ` 1936` ``` have "0 < r" "r < 1" by (auto simp: r_def) ``` wenzelm@63040 ` 1937` ``` define g where "g z = deriv f z * of_real(r - norm(z - a))" for z ``` lp15@62533 ` 1938` ``` have "deriv f holomorphic_on ball a 1" ``` lp15@62533 ` 1939` ``` by (rule holomorphic_deriv [OF holf open_ball]) ``` lp15@62533 ` 1940` ``` then have "continuous_on (ball a 1) (deriv f)" ``` lp15@62533 ` 1941` ``` using holomorphic_on_imp_continuous_on by blast ``` lp15@62533 ` 1942` ``` then have "continuous_on (cball a r) (deriv f)" ``` lp15@62533 ` 1943` ``` by (rule continuous_on_subset) (simp add: cball_subset_ball_iff \r < 1\) ``` lp15@62533 ` 1944` ``` then have "continuous_on (cball a r) g" ``` lp15@62533 ` 1945` ``` by (simp add: g_def continuous_intros) ``` lp15@62533 ` 1946` ``` then have 1: "compact (g ` cball a r)" ``` lp15@62533 ` 1947` ``` by (rule compact_continuous_image [OF _ compact_cball]) ``` lp15@62533 ` 1948` ``` have 2: "g ` cball a r \ {}" ``` lp15@62533 ` 1949` ``` using \r > 0\ by auto ``` hoelzl@63594 ` 1950` ``` obtain p where pr: "p \ cball a r" ``` lp15@62533 ` 1951` ``` and pge: "\y. y \ cball a r \ norm (g y) \ norm (g p)" ``` lp15@62533 ` 1952` ``` using distance_attains_sup [OF 1 2, of 0] by force ``` wenzelm@63040 ` 1953` ``` define t where "t = (r - norm(p - a)) / 2" ``` lp15@62533 ` 1954` ``` have "norm (p - a) \ r" ``` lp15@62533 ` 1955` ``` using pge [of a] \r > 0\ by (auto simp: g_def norm_mult) ``` hoelzl@63594 ` 1956` ``` then have "norm (p - a) < r" using pr ``` lp15@62533 ` 1957` ``` by (simp add: norm_minus_commute dist_norm) ``` hoelzl@63594 ` 1958` ``` then have "0 < t" ``` lp15@62533 ` 1959` ``` by (simp add: t_def) ``` lp15@62533 ` 1960` ``` have cpt: "cball p t \ ball a r" ``` lp15@62533 ` 1961` ``` using \0 < t\ by (simp add: cball_subset_ball_iff dist_norm t_def field_simps) ``` hoelzl@63594 ` 1962` ``` have gen_le_dfp: "norm (deriv f y) * (r - norm (y - a)) / (r - norm (p - a)) \ norm (deriv f p)" ``` lp15@62533 ` 1963` ``` if "y \ cball a r" for y ``` lp15@62533 ` 1964` ``` proof - ``` lp15@62533 ` 1965` ``` have [simp]: "norm (y - a) \ r" ``` hoelzl@63594 ` 1966` ``` using that by (simp add: dist_norm norm_minus_commute) ``` lp15@62533 ` 1967` ``` have "norm (g y) \ norm (g p)" ``` lp15@62533 ` 1968` ``` using pge [OF that] by simp ``` lp15@62533 ` 1969` ``` then have "norm (deriv f y) * abs (r - norm (y - a)) \ norm (deriv f p) * abs (r - norm (p - a))" ``` lp15@62533 ` 1970` ``` by (simp only: dist_norm g_def norm_mult norm_of_real) ``` lp15@62533 ` 1971` ``` with that \norm (p - a) < r\ show ?thesis ``` lp15@62533 ` 1972` ``` by (simp add: dist_norm divide_simps) ``` lp15@62533 ` 1973` ``` qed ``` lp15@62533 ` 1974` ``` have le_norm_dfp: "r / (r - norm (p - a)) \ norm (deriv f p)" ``` lp15@62533 ` 1975` ``` using gen_le_dfp [of a] \r > 0\ by auto ``` lp15@62533 ` 1976` ``` have 1: "f holomorphic_on cball p t" ``` lp15@62533 ` 1977` ``` apply (rule holomorphic_on_subset [OF holf]) ``` lp15@62533 ` 1978` ``` using cpt \r < 1\ order_subst1 subset_ball by auto ``` lp15@62533 ` 1979` ``` have 2: "norm (deriv f z) \ 2 * norm (deriv f p)" if "z \ ball p t" for z ``` lp15@62533 ` 1980` ``` proof - ``` lp15@62533 ` 1981` ``` have z: "z \ cball a r" ``` lp15@62533 ` 1982` ``` by (meson ball_subset_cball subsetD cpt that) ``` lp15@62533 ` 1983` ``` then have "norm(z - a) < r" ``` lp15@62533 ` 1984` ``` by (metis ball_subset_cball contra_subsetD cpt dist_norm mem_ball norm_minus_commute that) ``` hoelzl@63594 ` 1985` ``` have "norm (deriv f z) * (r - norm (z - a)) / (r - norm (p - a)) \ norm (deriv f p)" ``` lp15@62533 ` 1986` ``` using gen_le_dfp [OF z] by simp ``` hoelzl@63594 ` 1987` ``` with \norm (z - a) < r\ \norm (p - a) < r\ ``` lp15@62533 ` 1988` ``` have "norm (deriv f z) \ (r - norm (p - a)) / (r - norm (z - a)) * norm (deriv f p)" ``` lp15@62533 ` 1989` ``` by (simp add: field_simps) ``` lp15@62533 ` 1990` ``` also have "... \ 2 * norm (deriv f p)" ``` lp15@62533 ` 1991` ``` apply (rule mult_right_mono) ``` hoelzl@63594 ` 1992` ``` using that \norm (p - a) < r\ \norm(z - a) < r\ ``` lp15@62533 ` 1993` ``` apply (simp_all add: field_simps t_def dist_norm [symmetric]) ``` lp15@62533 ` 1994` ``` using dist_triangle3 [of z a p] by linarith ``` lp15@62533 ` 1995` ``` finally show ?thesis . ``` lp15@62533 ` 1996` ``` qed ``` lp15@62533 ` 1997` ``` have sqrt2: "sqrt 2 < 2113/1494" ``` lp15@62533 ` 1998` ``` by (rule real_less_lsqrt) (auto simp: power2_eq_square) ``` lp15@62533 ` 1999` ``` then have sq3: "0 < 3 - 2 * sqrt 2" by simp ``` lp15@62533 ` 2000` ``` have "1 / 12 / ((3 - 2 * sqrt 2) / 2) < r" ``` hoelzl@63594 ` 2001` ``` using sq3 sqrt2 by (auto simp: field_simps r_def) ``` lp15@62533 ` 2002` ``` also have "... \ cmod (deriv f p) * (r - cmod (p - a))" ``` hoelzl@63594 ` 2003` ``` using \norm (p - a) < r\ le_norm_dfp by (simp add: pos_divide_le_eq) ``` hoelzl@63594 ` 2004` ``` finally have "1 / 12 < cmod (deriv f p) * (r - cmod (p - a)) * ((3 - 2 * sqrt 2) / 2)" ``` lp15@62533 ` 2005` ``` using pos_divide_less_eq half_gt_zero_iff sq3 by blast ``` lp15@62533 ` 2006` ``` then have **: "1 / 12 < (3 - 2 * sqrt 2) * t * norm (deriv f p)" ``` lp15@62533 ` 2007` ``` using sq3 by (simp add: mult.commute t_def) ``` lp15@62533 ` 2008` ``` have "ball (f p) ((3 - 2 * sqrt 2) * t * norm (deriv f p)) \ f ` ball p t" ``` lp15@62533 ` 2009` ``` by (rule Bloch_lemma [OF 1 \0 < t\ 2]) ``` lp15@62533 ` 2010` ``` also have "... \ f ` ball a 1" ``` lp15@62533 ` 2011` ``` apply (rule image_mono) ``` lp15@62533 ` 2012` ``` apply (rule order_trans [OF ball_subset_cball]) ``` lp15@62533 ` 2013` ``` apply (rule order_trans [OF cpt]) ``` lp15@62533 ` 2014` ``` using \0 < t\ \r < 1\ apply (simp add: ball_subset_ball_iff dist_norm) ``` lp15@62533 ` 2015` ``` done ``` lp15@62533 ` 2016` ``` finally have "ball (f p) ((3 - 2 * sqrt 2) * t * norm (deriv f p)) \ f ` ball a 1" . ``` lp15@62533 ` 2017` ``` with ** show ?thesis ``` lp15@62533 ` 2018` ``` by (rule that) ``` lp15@62533 ` 2019` ```qed ``` lp15@62533 ` 2020` lp15@62533 ` 2021` lp15@62533 ` 2022` ```theorem Bloch: ``` hoelzl@63594 ` 2023` ``` assumes holf: "f holomorphic_on ball a r" and "0 < r" ``` lp15@62533 ` 2024` ``` and r': "r' \ r * norm (deriv f a) / 12" ``` lp15@62533 ` 2025` ``` obtains b where "ball b r' \ f ` (ball a r)" ``` lp15@62533 ` 2026` ```proof (cases "deriv f a = 0") ``` lp15@62533 ` 2027` ``` case True with r' show ?thesis ``` lp15@62533 ` 2028` ``` using ball_eq_empty that by fastforce ``` lp15@62533 ` 2029` ```next ``` lp15@62533 ` 2030` ``` case False ``` wenzelm@63040 ` 2031` ``` define C where "C = deriv f a" ``` wenzelm@63040 ` 2032` ``` have "0 < norm C" using False by (simp add: C_def) ``` wenzelm@63040 ` 2033` ``` have dfa: "f field_differentiable at a" ``` wenzelm@63040 ` 2034` ``` apply (rule holomorphic_on_imp_differentiable_at [OF holf]) ``` wenzelm@63040 ` 2035` ``` using \0 < r\ by auto ``` wenzelm@63040 ` 2036` ``` have fo: "(\z. f (a + of_real r * z)) = f o (\z. (a + of_real r * z))" ``` wenzelm@63040 ` 2037` ``` by (simp add: o_def) ``` wenzelm@63040 ` 2038` ``` have holf': "f holomorphic_on (\z. a + complex_of_real r * z) ` ball 0 1" ``` wenzelm@63040 ` 2039` ``` apply (rule holomorphic_on_subset [OF holf]) ``` wenzelm@63040 ` 2040` ``` using \0 < r\ apply (force simp: dist_norm norm_mult) ``` wenzelm@63040 ` 2041` ``` done ``` wenzelm@63040 ` 2042` ``` have 1: "(\z. f (a + r * z) / (C * r)) holomorphic_on ball 0 1" ``` wenzelm@63040 ` 2043` ``` apply (rule holomorphic_intros holomorphic_on_compose holf' | simp add: fo)+ ``` wenzelm@63040 ` 2044` ``` using \0 < r\ by (simp add: C_def False) ``` wenzelm@63040 ` 2045` ``` have "((\z. f (a + of_real r * z) / (C * of_real r)) has_field_derivative ``` hoelzl@63594 ` 2046` ``` (deriv f (a + of_real r * z) / C)) (at z)" ``` wenzelm@63040 ` 2047` ``` if "norm z < 1" for z ``` wenzelm@63040 ` 2048` ``` proof - ``` lp15@62533 ` 2049` ``` have *: "((\x. f (a + of_real r * x)) has_field_derivative ``` wenzelm@63040 ` 2050` ``` (deriv f (a + of_real r * z) * of_real r)) (at z)" ``` wenzelm@63040 ` 2051` ``` apply (simp add: fo) ``` lp15@62534 ` 2052` ``` apply (rule DERIV_chain [OF field_differentiable_derivI]) ``` wenzelm@63040 ` 2053` ``` apply (rule holomorphic_on_imp_differentiable_at [OF holf], simp) ``` wenzelm@63040 ` 2054` ``` using \0 < r\ apply (simp add: dist_norm norm_mult that) ``` wenzelm@63040 ` 2055` ``` apply (rule derivative_eq_intros | simp)+ ``` lp15@62533 ` 2056` ``` done ``` lp15@62533 ` 2057` ``` show ?thesis ``` wenzelm@63040 ` 2058` ``` apply (rule derivative_eq_intros * | simp)+ ``` wenzelm@63040 ` 2059` ``` using \0 < r\ by (auto simp: C_def False) ``` wenzelm@63040 ` 2060` ``` qed ``` wenzelm@63040 ` 2061` ``` have 2: "deriv (\z. f (a + of_real r * z) / (C * of_real r)) 0 = 1" ``` wenzelm@63040 ` 2062` ``` apply (subst deriv_cdivide_right) ``` wenzelm@63040 ` 2063` ``` apply (simp add: field_differentiable_def fo) ``` wenzelm@63040 ` 2064` ``` apply (rule exI) ``` wenzelm@63040 ` 2065` ``` apply (rule DERIV_chain [OF field_differentiable_derivI]) ``` wenzelm@63040 ` 2066` ``` apply (simp add: dfa) ``` wenzelm@63040 ` 2067` ``` apply (rule derivative_eq_intros | simp add: C_def False fo)+ ``` hoelzl@63594 ` 2068` ``` using \0 < r\ ``` wenzelm@63040 ` 2069` ``` apply (simp add: C_def False fo) ``` wenzelm@63040 ` 2070` ``` apply (simp add: derivative_intros dfa complex_derivative_chain) ``` wenzelm@63040 ` 2071` ``` done ``` hoelzl@63594 ` 2072` ``` have sb1: "op * (C * r) ` (\z. f (a + of_real r * z) / (C * r)) ` ball 0 1 ``` wenzelm@63040 ` 2073` ``` \ f ` ball a r" ``` wenzelm@63040 ` 2074` ``` using \0 < r\ by (auto simp: dist_norm norm_mult C_def False) ``` hoelzl@63594 ` 2075` ``` have sb2: "ball (C * r * b) r' \ op * (C * r) ` ball b t" ``` wenzelm@63040 ` 2076` ``` if "1 / 12 < t" for b t ``` wenzelm@63040 ` 2077` ``` proof - ``` wenzelm@63040 ` 2078` ``` have *: "r * cmod (deriv f a) / 12 \ r * (t * cmod (deriv f a))" ``` hoelzl@63594 ` 2079` ``` using that \0 < r\ less_eq_real_def mult.commute mult.right_neutral mult_left_mono norm_ge_zero times_divide_eq_right ``` wenzelm@63040 ` 2080` ``` by auto ``` wenzelm@63040 ` 2081` ``` show ?thesis ``` wenzelm@63040 ` 2082` ``` apply clarify ``` wenzelm@63040 ` 2083` ``` apply (rule_tac x="x / (C * r)" in image_eqI) ``` hoelzl@63594 ` 2084` ``` using \0 < r\ ``` wenzelm@63040 ` 2085` ``` apply (simp_all add: dist_norm norm_mult norm_divide C_def False field_simps) ``` wenzelm@63040 ` 2086` ``` apply (erule less_le_trans) ``` wenzelm@63040 ` 2087` ``` apply (rule order_trans [OF r' *]) ``` lp15@62533 ` 2088` ``` done ``` wenzelm@63040 ` 2089` ``` qed ``` wenzelm@63040 ` 2090` ``` show ?thesis ``` wenzelm@63040 ` 2091` ``` apply (rule Bloch_unit [OF 1 2]) ``` wenzelm@63040 ` 2092` ``` apply (rename_tac t) ``` wenzelm@63040 ` 2093` ``` apply (rule_tac b="(C * of_real r) * b" in that) ``` wenzelm@63040 ` 2094` ``` apply (drule image_mono [where f = "\z. (C * of_real r) * z"]) ``` wenzelm@63040 ` 2095` ``` using sb1 sb2 ``` wenzelm@63040 ` 2096` ``` apply force ``` wenzelm@63040 ` 2097` ``` done ``` lp15@62533 ` 2098` ```qed ``` lp15@62533 ` 2099` lp15@62533 ` 2100` ```corollary Bloch_general: ``` hoelzl@63594 ` 2101` ``` assumes holf: "f holomorphic_on s" and "a \ s" ``` lp15@62533 ` 2102` ``` and tle: "\z. z \ frontier s \ t \ dist a z" ``` lp15@62533 ` 2103` ``` and rle: "r \ t * norm(deriv f a) / 12" ``` lp15@62533 ` 2104` ``` obtains b where "ball b r \ f ` s" ``` lp15@62533 ` 2105` ```proof - ``` lp15@62533 ` 2106` ``` consider "r \ 0" | "0 < t * norm(deriv f a) / 12" using rle by force ``` lp15@62533 ` 2107` ``` then show ?thesis ``` lp15@62533 ` 2108` ``` proof cases ``` lp15@62533 ` 2109` ``` case 1 then show ?thesis ``` lp15@62533 ` 2110` ``` by (simp add: Topology_Euclidean_Space.ball_empty that) ``` lp15@62533 ` 2111` ``` next ``` lp15@62533 ` 2112` ``` case 2 ``` lp15@62533 ` 2113` ``` show ?thesis ``` lp15@62533 ` 2114` ``` proof (cases "deriv f a = 0") ``` lp15@62533 ` 2115` ``` case True then show ?thesis ``` lp15@62533 ` 2116` ``` using rle by (simp add: Topology_Euclidean_Space.ball_empty that) ``` lp15@62533 ` 2117` ``` next ``` lp15@62533 ` 2118` ``` case False ``` lp15@62533 ` 2119` ``` then have "t > 0" ``` lp15@62533 ` 2120` ``` using 2 by (force simp: zero_less_mult_iff) ``` lp15@62533 ` 2121` ``` have "~ ball a t \ s \ ball a t \ frontier s \ {}" ``` lp15@62533 ` 2122` ``` apply (rule connected_Int_frontier [of "ball a t" s], simp_all) ``` lp15@62533 ` 2123` ``` using \0 < t\ \a \ s\ centre_in_ball apply blast ``` lp15@62533 ` 2124` ``` done ``` lp15@62533 ` 2125` ``` with tle have *: "ball a t \ s" by fastforce ``` lp15@62533 ` 2126` ``` then have 1: "f holomorphic_on ball a t" ``` lp15@62533 ` 2127` ``` using holf using holomorphic_on_subset by blast ``` lp15@62533 ` 2128` ``` show ?thesis ``` lp15@62533 ` 2129` ``` apply (rule Bloch [OF 1 \t > 0\ rle]) ``` lp15@62533 ` 2130` ``` apply (rule_tac b=b in that) ``` lp15@62533 ` 2131` ``` using * apply force ``` lp15@62533 ` 2132` ``` done ``` lp15@62533 ` 2133` ``` qed ``` lp15@62533 ` 2134` ``` qed ``` lp15@62533 ` 2135` ```qed ``` lp15@62533 ` 2136` lp15@63151 ` 2137` ```subsection \Foundations of Cauchy's residue theorem\ ``` lp15@62540 ` 2138` lp15@63151 ` 2139` ```text\Wenda Li and LC Paulson (2016). A Formal Proof of Cauchy's Residue Theorem. ``` lp15@63151 ` 2140` ``` Interactive Theorem Proving\ ``` lp15@62540 ` 2141` lp15@63151 ` 2142` ```definition residue :: "(complex \ complex) \ complex \ complex" where ``` hoelzl@63594 ` 2143` ``` "residue f z = (SOME int. \e>0. \\>0. \ (f has_contour_integral 2*pi* \ *int) (circlepath z \))" ``` lp15@62540 ` 2145` lp15@63151 ` 2146` ```lemma contour_integral_circlepath_eq: ``` hoelzl@63594 ` 2147` ``` assumes "open s" and f_holo:"f holomorphic_on (s-{z})" and "0e2" ``` lp15@63151 ` 2148` ``` and e2_cball:"cball z e2 \ s" ``` lp15@63151 ` 2149` ``` shows ``` lp15@63151 ` 2150` ``` "f contour_integrable_on circlepath z e1" ``` lp15@63151 ` 2151` ``` "f contour_integrable_on circlepath z e2" ``` lp15@63151 ` 2152` ``` "contour_integral (circlepath z e2) f = contour_integral (circlepath z e1) f" ``` lp15@62540 ` 2153` ```proof - ``` lp15@63151 ` 2154` ``` define l where "l \ linepath (z+e2) (z+e1)" ``` lp15@63151 ` 2155` ``` have [simp]:"valid_path l" "pathstart l=z+e2" "pathfinish l=z+e1" unfolding l_def by auto ``` lp15@63151 ` 2156` ``` have "e2>0" using \e1>0\ \e1\e2\ by auto ``` lp15@63151 ` 2157` ``` have zl_img:"z\path_image l" ``` hoelzl@63594 ` 2158` ``` proof ``` lp15@63151 ` 2159` ``` assume "z \ path_image l" ``` lp15@63151 ` 2160` ``` then have "e2 \ cmod (e2 - e1)" ``` lp15@63151 ` 2161` ``` using segment_furthest_le[of z "z+e2" "z+e1" "z+e2",simplified] \e1>0\ \e2>0\ unfolding l_def ``` lp15@63151 ` 2162` ``` by (auto simp add:closed_segment_commute) ``` hoelzl@63594 ` 2163` ``` thus False using \e2>0\ \e1>0\ \e1\e2\ ``` lp15@63151 ` 2164` ``` apply (subst (asm) norm_of_real) ``` lp15@63151 ` 2165` ``` by auto ``` lp15@63151 ` 2166` ``` qed ``` lp15@63151 ` 2167` ``` define g where "g \ circlepath z e2 +++ l +++ reversepath (circlepath z e1) +++ reversepath l" ``` hoelzl@63594 ` 2168` ``` show [simp]: "f contour_integrable_on circlepath z e2" "f contour_integrable_on (circlepath z e1)" ``` lp15@62540 ` 2169` ``` proof - ``` hoelzl@63594 ` 2170` ``` show "f contour_integrable_on circlepath z e2" ``` hoelzl@63594 ` 2171` ``` apply (intro contour_integrable_continuous_circlepath[OF ``` lp15@63151 ` 2172` ``` continuous_on_subset[OF holomorphic_on_imp_continuous_on[OF f_holo]]]) ``` lp15@63151 ` 2173` ``` using \e2>0\ e2_cball by auto ``` hoelzl@63594 ` 2174` ``` show "f contour_integrable_on (circlepath z e1)" ``` hoelzl@63594 ` 2175` ``` apply (intro contour_integrable_continuous_circlepath[OF ``` lp15@63151 ` 2176` ``` continuous_on_subset[OF holomorphic_on_imp_continuous_on[OF f_holo]]]) ``` lp15@63151 ` 2177` ``` using \e1>0\ \e1\e2\ e2_cball by auto ``` lp15@63151 ` 2178` ``` qed ``` lp15@63151 ` 2179` ``` have [simp]:"f contour_integrable_on l" ``` lp15@63151 ` 2180` ``` proof - ```