--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Multivariate_Analysis/Conformal_Mappings.thy Thu Feb 25 13:58:48 2016 +0000
@@ -0,0 +1,1728 @@
+section \<open>Conformal Mappings. Consequences of Cauchy's integral theorem.\<close>
+
+text\<open>By John Harrison et al. Ported from HOL Light by L C Paulson (2016)\<close>
+
+theory Conformal_Mappings
+imports "~~/src/HOL/Multivariate_Analysis/Cauchy_Integral_Thm"
+
+begin
+
+subsection\<open>Cauchy's inequality and more versions of Liouville\<close>
+
+lemma Cauchy_higher_deriv_bound:
+ assumes holf: "f holomorphic_on (ball z r)"
+ and contf: "continuous_on (cball z r) f"
+ and "0 < r" and "0 < n"
+ and fin : "\<And>w. w \<in> ball z r \<Longrightarrow> f w \<in> ball y B0"
+ shows "norm ((deriv ^^ n) f z) \<le> (fact n) * B0 / r^n"
+proof -
+ have "0 < B0" using \<open>0 < r\<close> fin [of z]
+ by (metis ball_eq_empty ex_in_conv fin not_less)
+ have le_B0: "\<And>w. cmod (w - z) \<le> r \<Longrightarrow> cmod (f w - y) \<le> B0"
+ apply (rule continuous_on_closure_norm_le [of "ball z r" "\<lambda>w. f w - y"])
+ apply (auto simp: \<open>0 < r\<close> dist_norm norm_minus_commute)
+ apply (rule continuous_intros contf)+
+ using fin apply (simp add: dist_commute dist_norm less_eq_real_def)
+ done
+ have "(deriv ^^ n) f z = (deriv ^^ n) (\<lambda>w. f w) z - (deriv ^^ n) (\<lambda>w. y) z"
+ using \<open>0 < n\<close> by simp
+ also have "... = (deriv ^^ n) (\<lambda>w. f w - y) z"
+ by (rule higher_deriv_diff [OF holf, symmetric]) (auto simp: \<open>0 < r\<close>)
+ finally have "(deriv ^^ n) f z = (deriv ^^ n) (\<lambda>w. f w - y) z" .
+ have contf': "continuous_on (cball z r) (\<lambda>u. f u - y)"
+ by (rule contf continuous_intros)+
+ have holf': "(\<lambda>u. (f u - y)) holomorphic_on (ball z r)"
+ by (simp add: holf holomorphic_on_diff)
+ def a \<equiv> "(2 * pi)/(fact n)"
+ have "0 < a" by (simp add: a_def)
+ have "B0/r^(Suc n)*2 * pi * r = a*((fact n)*B0/r^n)"
+ using \<open>0 < r\<close> by (simp add: a_def divide_simps)
+ have der_dif: "(deriv ^^ n) (\<lambda>w. f w - y) z = (deriv ^^ n) f z"
+ using \<open>0 < r\<close> \<open>0 < n\<close>
+ by (auto simp: higher_deriv_diff [OF holf holomorphic_on_const])
+ have "norm ((2 * of_real pi * ii)/(fact n) * (deriv ^^ n) (\<lambda>w. f w - y) z)
+ \<le> (B0/r^(Suc n)) * (2 * pi * r)"
+ apply (rule has_contour_integral_bound_circlepath [of "(\<lambda>u. (f u - y)/(u - z)^(Suc n))" _ z])
+ using Cauchy_has_contour_integral_higher_derivative_circlepath [OF contf' holf']
+ using \<open>0 < B0\<close> \<open>0 < r\<close>
+ apply (auto simp: norm_divide norm_mult norm_power divide_simps le_B0)
+ done
+ then show ?thesis
+ using \<open>0 < r\<close>
+ by (auto simp: norm_divide norm_mult norm_power field_simps der_dif le_B0)
+qed
+
+proposition Cauchy_inequality:
+ assumes holf: "f holomorphic_on (ball \<xi> r)"
+ and contf: "continuous_on (cball \<xi> r) f"
+ and "0 < r"
+ and nof: "\<And>x. norm(\<xi>-x) = r \<Longrightarrow> norm(f x) \<le> B"
+ shows "norm ((deriv ^^ n) f \<xi>) \<le> (fact n) * B / r^n"
+proof -
+ obtain x where "norm (\<xi>-x) = r"
+ by (metis abs_of_nonneg add_diff_cancel_left' \<open>0 < r\<close> diff_add_cancel
+ dual_order.strict_implies_order norm_of_real)
+ then have "0 \<le> B"
+ by (metis nof norm_not_less_zero not_le order_trans)
+ have "((\<lambda>u. f u / (u - \<xi>) ^ Suc n) has_contour_integral (2 * pi) * \<i> / fact n * (deriv ^^ n) f \<xi>)
+ (circlepath \<xi> r)"
+ apply (rule Cauchy_has_contour_integral_higher_derivative_circlepath [OF contf holf])
+ using \<open>0 < r\<close> by simp
+ then have "norm ((2 * pi * ii)/(fact n) * (deriv ^^ n) f \<xi>) \<le> (B / r^(Suc n)) * (2 * pi * r)"
+ apply (rule has_contour_integral_bound_circlepath)
+ using \<open>0 \<le> B\<close> \<open>0 < r\<close>
+ apply (simp_all add: norm_divide norm_power nof frac_le norm_minus_commute del: power_Suc)
+ done
+ then show ?thesis using \<open>0 < r\<close>
+ by (simp add: norm_divide norm_mult field_simps)
+qed
+
+proposition Liouville_polynomial:
+ assumes holf: "f holomorphic_on UNIV"
+ and nof: "\<And>z. A \<le> norm z \<Longrightarrow> norm(f z) \<le> B * norm z ^ n"
+ shows "f \<xi> = (\<Sum>k\<le>n. (deriv^^k) f 0 / fact k * \<xi> ^ k)"
+proof (cases rule: le_less_linear [THEN disjE])
+ assume "B \<le> 0"
+ then have "\<And>z. A \<le> norm z \<Longrightarrow> norm(f z) = 0"
+ by (metis nof less_le_trans zero_less_mult_iff neqE norm_not_less_zero norm_power not_le)
+ then have f0: "(f \<longlongrightarrow> 0) at_infinity"
+ using Lim_at_infinity by force
+ then have [simp]: "f = (\<lambda>w. 0)"
+ using Liouville_weak [OF holf, of 0]
+ by (simp add: eventually_at_infinity f0) meson
+ show ?thesis by simp
+next
+ assume "0 < B"
+ have "((\<lambda>k. (deriv ^^ k) f 0 / (fact k) * (\<xi> - 0)^k) sums f \<xi>)"
+ apply (rule holomorphic_power_series [where r = "norm \<xi> + 1"])
+ using holf holomorphic_on_subset apply auto
+ done
+ then have sumsf: "((\<lambda>k. (deriv ^^ k) f 0 / (fact k) * \<xi>^k) sums f \<xi>)" by simp
+ have "(deriv ^^ k) f 0 / fact k * \<xi> ^ k = 0" if "k>n" for k
+ proof (cases "(deriv ^^ k) f 0 = 0")
+ case True then show ?thesis by simp
+ next
+ case False
+ def w \<equiv> "complex_of_real (fact k * B / cmod ((deriv ^^ k) f 0) + (\<bar>A\<bar> + 1))"
+ have "1 \<le> abs (fact k * B / cmod ((deriv ^^ k) f 0) + (\<bar>A\<bar> + 1))"
+ using \<open>0 < B\<close> by simp
+ then have wge1: "1 \<le> norm w"
+ by (metis norm_of_real w_def)
+ then have "w \<noteq> 0" by auto
+ have kB: "0 < fact k * B"
+ using \<open>0 < B\<close> by simp
+ then have "0 \<le> fact k * B / cmod ((deriv ^^ k) f 0)"
+ by simp
+ then have wgeA: "A \<le> cmod w"
+ by (simp only: w_def norm_of_real)
+ have "fact k * B / cmod ((deriv ^^ k) f 0) < abs (fact k * B / cmod ((deriv ^^ k) f 0) + (\<bar>A\<bar> + 1))"
+ using \<open>0 < B\<close> by simp
+ then have wge: "fact k * B / cmod ((deriv ^^ k) f 0) < norm w"
+ by (metis norm_of_real w_def)
+ then have "fact k * B / norm w < cmod ((deriv ^^ k) f 0)"
+ using False by (simp add: divide_simps mult.commute split: if_split_asm)
+ also have "... \<le> fact k * (B * norm w ^ n) / norm w ^ k"
+ apply (rule Cauchy_inequality)
+ using holf holomorphic_on_subset apply force
+ using holf holomorphic_on_imp_continuous_on holomorphic_on_subset apply blast
+ using \<open>w \<noteq> 0\<close> apply (simp add:)
+ by (metis nof wgeA dist_0_norm dist_norm)
+ also have "... = fact k * (B * 1 / cmod w ^ (k-n))"
+ apply (simp only: mult_cancel_left times_divide_eq_right [symmetric])
+ using \<open>k>n\<close> \<open>w \<noteq> 0\<close> \<open>0 < B\<close> apply (simp add: divide_simps semiring_normalization_rules)
+ done
+ also have "... = fact k * B / cmod w ^ (k-n)"
+ by simp
+ finally have "fact k * B / cmod w < fact k * B / cmod w ^ (k - n)" .
+ then have "1 / cmod w < 1 / cmod w ^ (k - n)"
+ by (metis kB divide_inverse inverse_eq_divide mult_less_cancel_left_pos)
+ then have "cmod w ^ (k - n) < cmod w"
+ by (metis frac_le le_less_trans norm_ge_zero norm_one not_less order_refl wge1 zero_less_one)
+ with self_le_power [OF wge1] have False
+ by (meson diff_is_0_eq not_gr0 not_le that)
+ then show ?thesis by blast
+ qed
+ then have "(deriv ^^ (k + Suc n)) f 0 / fact (k + Suc n) * \<xi> ^ (k + Suc n) = 0" for k
+ using not_less_eq by blast
+ then have "(\<lambda>i. (deriv ^^ (i + Suc n)) f 0 / fact (i + Suc n) * \<xi> ^ (i + Suc n)) sums 0"
+ by (rule sums_0)
+ with sums_split_initial_segment [OF sumsf, where n = "Suc n"]
+ show ?thesis
+ using atLeast0AtMost lessThan_Suc_atMost sums_unique2 by fastforce
+qed
+
+text\<open>Every bounded entire function is a constant function.\<close>
+theorem Liouville_theorem:
+ assumes holf: "f holomorphic_on UNIV"
+ and bf: "bounded (range f)"
+ obtains c where "\<And>z. f z = c"
+proof -
+ obtain B where "\<And>z. cmod (f z) \<le> B"
+ by (meson bf bounded_pos rangeI)
+ then show ?thesis
+ using Liouville_polynomial [OF holf, of 0 B 0, simplified] that by blast
+qed
+
+
+
+text\<open>A holomorphic function f has only isolated zeros unless f is 0.\<close>
+
+proposition powser_0_nonzero:
+ fixes a :: "nat \<Rightarrow> 'a::{real_normed_field,banach}"
+ assumes r: "0 < r"
+ and sm: "\<And>x. norm (x - \<xi>) < r \<Longrightarrow> (\<lambda>n. a n * (x - \<xi>) ^ n) sums (f x)"
+ and [simp]: "f \<xi> = 0"
+ and m0: "a m \<noteq> 0" and "m>0"
+ obtains s where "0 < s" and "\<And>z. z \<in> cball \<xi> s - {\<xi>} \<Longrightarrow> f z \<noteq> 0"
+proof -
+ have "r \<le> conv_radius a"
+ using sm sums_summable by (auto simp: le_conv_radius_iff [where \<xi>=\<xi>])
+ obtain m where am: "a m \<noteq> 0" and az [simp]: "(\<And>n. n<m \<Longrightarrow> a n = 0)"
+ apply (rule_tac m = "LEAST n. a n \<noteq> 0" in that)
+ using m0
+ apply (rule LeastI2)
+ apply (fastforce intro: dest!: not_less_Least)+
+ done
+ def b \<equiv> "\<lambda>i. a (i+m) / a m"
+ def g \<equiv> "\<lambda>x. suminf (\<lambda>i. b i * (x - \<xi>) ^ i)"
+ have [simp]: "b 0 = 1"
+ by (simp add: am b_def)
+ { fix x::'a
+ assume "norm (x - \<xi>) < r"
+ then have "(\<lambda>n. (a m * (x - \<xi>)^m) * (b n * (x - \<xi>)^n)) sums (f x)"
+ using am az sm sums_zero_iff_shift [of m "(\<lambda>n. a n * (x - \<xi>) ^ n)" "f x"]
+ by (simp add: b_def monoid_mult_class.power_add algebra_simps)
+ then have "x \<noteq> \<xi> \<Longrightarrow> (\<lambda>n. b n * (x - \<xi>)^n) sums (f x / (a m * (x - \<xi>)^m))"
+ using am by (simp add: sums_mult_D)
+ } note bsums = this
+ then have "norm (x - \<xi>) < r \<Longrightarrow> summable (\<lambda>n. b n * (x - \<xi>)^n)" for x
+ using sums_summable by (cases "x=\<xi>") auto
+ then have "r \<le> conv_radius b"
+ by (simp add: le_conv_radius_iff [where \<xi>=\<xi>])
+ then have "r/2 < conv_radius b"
+ using not_le order_trans r by fastforce
+ then have "continuous_on (cball \<xi> (r/2)) g"
+ using powser_continuous_suminf [of "r/2" b \<xi>] by (simp add: g_def)
+ then obtain s where "s>0" "\<And>x. \<lbrakk>norm (x - \<xi>) \<le> s; norm (x - \<xi>) \<le> r/2\<rbrakk> \<Longrightarrow> dist (g x) (g \<xi>) < 1/2"
+ apply (rule continuous_onE [where x=\<xi> and e = "1/2"])
+ using r apply (auto simp: norm_minus_commute dist_norm)
+ done
+ moreover have "g \<xi> = 1"
+ by (simp add: g_def)
+ ultimately have gnz: "\<And>x. \<lbrakk>norm (x - \<xi>) \<le> s; norm (x - \<xi>) \<le> r/2\<rbrakk> \<Longrightarrow> (g x) \<noteq> 0"
+ by fastforce
+ have "f x \<noteq> 0" if "x \<noteq> \<xi>" "norm (x - \<xi>) \<le> s" "norm (x - \<xi>) \<le> r/2" for x
+ using bsums [of x] that gnz [of x]
+ apply (auto simp: g_def)
+ using r sums_iff by fastforce
+ then show ?thesis
+ apply (rule_tac s="min s (r/2)" in that)
+ using \<open>0 < r\<close> \<open>0 < s\<close> by (auto simp: dist_commute dist_norm)
+qed
+
+proposition isolated_zeros:
+ assumes holf: "f holomorphic_on S"
+ and "open S" "connected S" "\<xi> \<in> S" "f \<xi> = 0" "\<beta> \<in> S" "f \<beta> \<noteq> 0"
+ obtains r where "0 < r" "ball \<xi> r \<subseteq> S" "\<And>z. z \<in> ball \<xi> r - {\<xi>} \<Longrightarrow> f z \<noteq> 0"
+proof -
+ obtain r where "0 < r" and r: "ball \<xi> r \<subseteq> S"
+ using \<open>open S\<close> \<open>\<xi> \<in> S\<close> open_contains_ball_eq by blast
+ have powf: "((\<lambda>n. (deriv ^^ n) f \<xi> / (fact n) * (z - \<xi>)^n) sums f z)" if "z \<in> ball \<xi> r" for z
+ apply (rule holomorphic_power_series [OF _ that])
+ apply (rule holomorphic_on_subset [OF holf r])
+ done
+ obtain m where m: "(deriv ^^ m) f \<xi> / (fact m) \<noteq> 0"
+ using holomorphic_fun_eq_0_on_connected [OF holf \<open>open S\<close> \<open>connected S\<close> _ \<open>\<xi> \<in> S\<close> \<open>\<beta> \<in> S\<close>] \<open>f \<beta> \<noteq> 0\<close>
+ by auto
+ then have "m \<noteq> 0" using assms(5) funpow_0 by fastforce
+ obtain s where "0 < s" and s: "\<And>z. z \<in> cball \<xi> s - {\<xi>} \<Longrightarrow> f z \<noteq> 0"
+ apply (rule powser_0_nonzero [OF \<open>0 < r\<close> powf \<open>f \<xi> = 0\<close> m])
+ using \<open>m \<noteq> 0\<close> by (auto simp: dist_commute dist_norm)
+ have "0 < min r s" by (simp add: \<open>0 < r\<close> \<open>0 < s\<close>)
+ then show ?thesis
+ apply (rule that)
+ using r s by auto
+qed
+
+
+proposition analytic_continuation:
+ assumes holf: "f holomorphic_on S"
+ and S: "open S" "connected S"
+ and "U \<subseteq> S" "\<xi> \<in> S"
+ and "\<xi> islimpt U"
+ and fU0 [simp]: "\<And>z. z \<in> U \<Longrightarrow> f z = 0"
+ and "w \<in> S"
+ shows "f w = 0"
+proof -
+ obtain e where "0 < e" and e: "cball \<xi> e \<subseteq> S"
+ using \<open>open S\<close> \<open>\<xi> \<in> S\<close> open_contains_cball_eq by blast
+ def T \<equiv> "cball \<xi> e \<inter> U"
+ have contf: "continuous_on (closure T) f"
+ by (metis T_def closed_cball closure_minimal e holf holomorphic_on_imp_continuous_on
+ holomorphic_on_subset inf.cobounded1)
+ have fT0 [simp]: "\<And>x. x \<in> T \<Longrightarrow> f x = 0"
+ by (simp add: T_def)
+ have "\<And>r. \<lbrakk>\<forall>e>0. \<exists>x'\<in>U. x' \<noteq> \<xi> \<and> dist x' \<xi> < e; 0 < r\<rbrakk> \<Longrightarrow> \<exists>x'\<in>cball \<xi> e \<inter> U. x' \<noteq> \<xi> \<and> dist x' \<xi> < r"
+ by (metis \<open>0 < e\<close> IntI dist_commute less_eq_real_def mem_cball min_less_iff_conj)
+ then have "\<xi> islimpt T" using \<open>\<xi> islimpt U\<close>
+ by (auto simp: T_def islimpt_approachable)
+ then have "\<xi> \<in> closure T"
+ by (simp add: closure_def)
+ then have "f \<xi> = 0"
+ by (auto simp: continuous_constant_on_closure [OF contf])
+ show ?thesis
+ apply (rule ccontr)
+ apply (rule isolated_zeros [OF holf \<open>open S\<close> \<open>connected S\<close> \<open>\<xi> \<in> S\<close> \<open>f \<xi> = 0\<close> \<open>w \<in> S\<close>], assumption)
+ by (metis open_ball \<open>\<xi> islimpt T\<close> centre_in_ball fT0 insertE insert_Diff islimptE)
+qed
+
+
+subsection\<open>Open mapping theorem\<close>
+
+lemma holomorphic_contract_to_zero:
+ assumes contf: "continuous_on (cball \<xi> r) f"
+ and holf: "f holomorphic_on ball \<xi> r"
+ and "0 < r"
+ and norm_less: "\<And>z. norm(\<xi> - z) = r \<Longrightarrow> norm(f \<xi>) < norm(f z)"
+ obtains z where "z \<in> ball \<xi> r" "f z = 0"
+proof -
+ { assume fnz: "\<And>w. w \<in> ball \<xi> r \<Longrightarrow> f w \<noteq> 0"
+ then have "0 < norm (f \<xi>)"
+ by (simp add: \<open>0 < r\<close>)
+ have fnz': "\<And>w. w \<in> cball \<xi> r \<Longrightarrow> f w \<noteq> 0"
+ by (metis norm_less dist_norm fnz less_eq_real_def mem_ball mem_cball norm_not_less_zero norm_zero)
+ have "frontier(cball \<xi> r) \<noteq> {}"
+ using \<open>0 < r\<close> by simp
+ def g \<equiv> "\<lambda>z. inverse (f z)"
+ have contg: "continuous_on (cball \<xi> r) g"
+ unfolding g_def using contf continuous_on_inverse fnz' by blast
+ have holg: "g holomorphic_on ball \<xi> r"
+ unfolding g_def using fnz holf holomorphic_on_inverse by blast
+ have "frontier (cball \<xi> r) \<subseteq> cball \<xi> r"
+ by (simp add: subset_iff)
+ then have contf': "continuous_on (frontier (cball \<xi> r)) f"
+ and contg': "continuous_on (frontier (cball \<xi> r)) g"
+ by (blast intro: contf contg continuous_on_subset)+
+ have froc: "frontier(cball \<xi> r) \<noteq> {}"
+ using \<open>0 < r\<close> by simp
+ moreover have "continuous_on (frontier (cball \<xi> r)) (norm o f)"
+ using contf' continuous_on_compose continuous_on_norm_id by blast
+ ultimately obtain w where w: "w \<in> frontier(cball \<xi> r)"
+ and now: "\<And>x. x \<in> frontier(cball \<xi> r) \<Longrightarrow> norm (f w) \<le> norm (f x)"
+ apply (rule bexE [OF continuous_attains_inf [OF compact_frontier [OF compact_cball]]])
+ apply (simp add:)
+ done
+ then have fw: "0 < norm (f w)"
+ by (simp add: fnz')
+ have "continuous_on (frontier (cball \<xi> r)) (norm o g)"
+ using contg' continuous_on_compose continuous_on_norm_id by blast
+ then obtain v where v: "v \<in> frontier(cball \<xi> r)"
+ and nov: "\<And>x. x \<in> frontier(cball \<xi> r) \<Longrightarrow> norm (g v) \<ge> norm (g x)"
+ apply (rule bexE [OF continuous_attains_sup [OF compact_frontier [OF compact_cball] froc]])
+ apply (simp add:)
+ done
+ then have fv: "0 < norm (f v)"
+ by (simp add: fnz')
+ have "norm ((deriv ^^ 0) g \<xi>) \<le> fact 0 * norm (g v) / r ^ 0"
+ by (rule Cauchy_inequality [OF holg contg \<open>0 < r\<close>]) (simp add: dist_norm nov)
+ then have "cmod (g \<xi>) \<le> norm (g v)"
+ by simp
+ with w have wr: "norm (\<xi> - w) = r" and nfw: "norm (f w) \<le> norm (f \<xi>)"
+ apply (simp_all add: dist_norm)
+ by (metis \<open>0 < cmod (f \<xi>)\<close> g_def less_imp_inverse_less norm_inverse not_le now order_trans v)
+ with fw have False
+ using norm_less by force
+ }
+ with that show ?thesis by blast
+qed
+
+
+theorem open_mapping_thm:
+ assumes holf: "f holomorphic_on S"
+ and S: "open S" "connected S"
+ and "open U" "U \<subseteq> S"
+ and fne: "~ f constant_on S"
+ shows "open (f ` U)"
+proof -
+ have *: "open (f ` U)"
+ if "U \<noteq> {}" and U: "open U" "connected U" and "f holomorphic_on U" and fneU: "\<And>x. \<exists>y \<in> U. f y \<noteq> x"
+ for U
+ proof (clarsimp simp: open_contains_ball)
+ fix \<xi> assume \<xi>: "\<xi> \<in> U"
+ show "\<exists>e>0. ball (f \<xi>) e \<subseteq> f ` U"
+ proof -
+ have hol: "(\<lambda>z. f z - f \<xi>) holomorphic_on U"
+ by (rule holomorphic_intros that)+
+ obtain s where "0 < s" and sbU: "ball \<xi> s \<subseteq> U"
+ and sne: "\<And>z. z \<in> ball \<xi> s - {\<xi>} \<Longrightarrow> (\<lambda>z. f z - f \<xi>) z \<noteq> 0"
+ using isolated_zeros [OF hol U \<xi>] by (metis fneU right_minus_eq)
+ obtain r where "0 < r" and r: "cball \<xi> r \<subseteq> ball \<xi> s"
+ apply (rule_tac r="s/2" in that)
+ using \<open>0 < s\<close> by auto
+ have "cball \<xi> r \<subseteq> U"
+ using sbU r by blast
+ then have frsbU: "frontier (cball \<xi> r) \<subseteq> U"
+ using Diff_subset frontier_def order_trans by fastforce
+ then have cof: "compact (frontier(cball \<xi> r))"
+ by blast
+ have frne: "frontier (cball \<xi> r) \<noteq> {}"
+ using \<open>0 < r\<close> by auto
+ have contfr: "continuous_on (frontier (cball \<xi> r)) (\<lambda>z. norm (f z - f \<xi>))"
+ apply (rule continuous_on_compose2 [OF Complex_Analysis_Basics.continuous_on_norm_id])
+ using hol frsbU holomorphic_on_imp_continuous_on holomorphic_on_subset by blast+
+ obtain w where "norm (\<xi> - w) = r"
+ and w: "(\<And>z. norm (\<xi> - z) = r \<Longrightarrow> norm (f w - f \<xi>) \<le> norm(f z - f \<xi>))"
+ apply (rule bexE [OF continuous_attains_inf [OF cof frne contfr]])
+ apply (simp add: dist_norm)
+ done
+ moreover def \<epsilon> \<equiv> "norm (f w - f \<xi>) / 3"
+ ultimately have "0 < \<epsilon>"
+ using \<open>0 < r\<close> dist_complex_def r sne by auto
+ have "ball (f \<xi>) \<epsilon> \<subseteq> f ` U"
+ proof
+ fix \<gamma>
+ assume \<gamma>: "\<gamma> \<in> ball (f \<xi>) \<epsilon>"
+ have *: "cmod (\<gamma> - f \<xi>) < cmod (\<gamma> - f z)" if "cmod (\<xi> - z) = r" for z
+ proof -
+ have lt: "cmod (f w - f \<xi>) / 3 < cmod (\<gamma> - f z)"
+ using w [OF that] \<gamma>
+ using dist_triangle2 [of "f \<xi>" "\<gamma>" "f z"] dist_triangle2 [of "f \<xi>" "f z" \<gamma>]
+ by (simp add: \<epsilon>_def dist_norm norm_minus_commute)
+ show ?thesis
+ by (metis \<epsilon>_def dist_commute dist_norm less_trans lt mem_ball \<gamma>)
+ qed
+ have "continuous_on (cball \<xi> r) (\<lambda>z. \<gamma> - f z)"
+ apply (rule continuous_intros)+
+ using \<open>cball \<xi> r \<subseteq> U\<close> \<open>f holomorphic_on U\<close>
+ apply (blast intro: continuous_on_subset holomorphic_on_imp_continuous_on)
+ done
+ moreover have "(\<lambda>z. \<gamma> - f z) holomorphic_on ball \<xi> r"
+ apply (rule holomorphic_intros)+
+ apply (metis \<open>cball \<xi> r \<subseteq> U\<close> \<open>f holomorphic_on U\<close> holomorphic_on_subset interior_cball interior_subset)
+ done
+ ultimately obtain z where "z \<in> ball \<xi> r" "\<gamma> - f z = 0"
+ apply (rule holomorphic_contract_to_zero)
+ apply (blast intro!: \<open>0 < r\<close> *)+
+ done
+ then show "\<gamma> \<in> f ` U"
+ using \<open>cball \<xi> r \<subseteq> U\<close> by fastforce
+ qed
+ then show ?thesis using \<open>0 < \<epsilon>\<close> by blast
+ qed
+ qed
+ have "open (f ` X)" if "X \<in> components U" for X
+ proof -
+ have holfU: "f holomorphic_on U"
+ using \<open>U \<subseteq> S\<close> holf holomorphic_on_subset by blast
+ have "X \<noteq> {}"
+ using that by (simp add: in_components_nonempty)
+ moreover have "open X"
+ using that \<open>open U\<close> open_components by auto
+ moreover have "connected X"
+ using that in_components_maximal by blast
+ moreover have "f holomorphic_on X"
+ by (meson that holfU holomorphic_on_subset in_components_maximal)
+ moreover have "\<exists>y\<in>X. f y \<noteq> x" for x
+ proof (rule ccontr)
+ assume not: "\<not> (\<exists>y\<in>X. f y \<noteq> x)"
+ have "X \<subseteq> S"
+ using \<open>U \<subseteq> S\<close> in_components_subset that by blast
+ obtain w where w: "w \<in> X" using \<open>X \<noteq> {}\<close> by blast
+ have wis: "w islimpt X"
+ using w \<open>open X\<close> interior_eq by auto
+ have hol: "(\<lambda>z. f z - x) holomorphic_on S"
+ by (simp add: holf holomorphic_on_diff)
+ with fne [unfolded constant_on_def] analytic_continuation [OF hol S \<open>X \<subseteq> S\<close> _ wis]
+ not \<open>X \<subseteq> S\<close> w
+ show False by auto
+ qed
+ ultimately show ?thesis
+ by (rule *)
+ qed
+ then show ?thesis
+ by (subst Union_components [of U]) (auto simp: image_Union)
+qed
+
+
+text\<open>No need for @{term S} to be connected. But the nonconstant condition is stronger.\<close>
+corollary open_mapping_thm2:
+ assumes holf: "f holomorphic_on S"
+ and S: "open S"
+ and "open U" "U \<subseteq> S"
+ and fnc: "\<And>X. \<lbrakk>open X; X \<subseteq> S; X \<noteq> {}\<rbrakk> \<Longrightarrow> ~ f constant_on X"
+ shows "open (f ` U)"
+proof -
+ have "S = \<Union>(components S)" by (fact Union_components)
+ with \<open>U \<subseteq> S\<close> have "U = (\<Union>C \<in> components S. C \<inter> U)" by auto
+ then have "f ` U = (\<Union>C \<in> components S. f ` (C \<inter> U))"
+ by auto
+ moreover
+ { fix C assume "C \<in> components S"
+ with S \<open>C \<in> components S\<close> open_components in_components_connected
+ have C: "open C" "connected C" by auto
+ have "C \<subseteq> S"
+ by (metis \<open>C \<in> components S\<close> in_components_maximal)
+ have nf: "\<not> f constant_on C"
+ apply (rule fnc)
+ using C \<open>C \<subseteq> S\<close> \<open>C \<in> components S\<close> in_components_nonempty by auto
+ have "f holomorphic_on C"
+ by (metis holf holomorphic_on_subset \<open>C \<subseteq> S\<close>)
+ then have "open (f ` (C \<inter> U))"
+ apply (rule open_mapping_thm [OF _ C _ _ nf])
+ apply (simp add: C \<open>open U\<close> open_Int, blast)
+ done
+ } ultimately show ?thesis
+ by force
+qed
+
+corollary open_mapping_thm3:
+ assumes holf: "f holomorphic_on S"
+ and "open S" and injf: "inj_on f S"
+ shows "open (f ` S)"
+apply (rule open_mapping_thm2 [OF holf])
+using assms
+apply (simp_all add:)
+using injective_not_constant subset_inj_on by blast
+
+
+
+subsection\<open>Maximum Modulus Principle\<close>
+
+text\<open>If @{term f} is holomorphic, then its norm (modulus) cannot exhibit a true local maximum that is
+ properly within the domain of @{term f}.\<close>
+
+proposition maximum_modulus_principle:
+ assumes holf: "f holomorphic_on S"
+ and S: "open S" "connected S"
+ and "open U" "U \<subseteq> S" "\<xi> \<in> U"
+ and no: "\<And>z. z \<in> U \<Longrightarrow> norm(f z) \<le> norm(f \<xi>)"
+ shows "f constant_on S"
+proof (rule ccontr)
+ assume "\<not> f constant_on S"
+ then have "open (f ` U)"
+ using open_mapping_thm assms by blast
+ moreover have "~ open (f ` U)"
+ proof -
+ have "\<exists>t. cmod (f \<xi> - t) < e \<and> t \<notin> f ` U" if "0 < e" for e
+ apply (rule_tac x="if 0 < Re(f \<xi>) then f \<xi> + (e/2) else f \<xi> - (e/2)" in exI)
+ using that
+ apply (simp add: dist_norm)
+ apply (fastforce simp: cmod_Re_le_iff dest!: no dest: sym)
+ done
+ then show ?thesis
+ unfolding open_contains_ball by (metis \<open>\<xi> \<in> U\<close> contra_subsetD dist_norm imageI mem_ball)
+ qed
+ ultimately show False
+ by blast
+qed
+
+
+proposition maximum_modulus_frontier:
+ assumes holf: "f holomorphic_on (interior S)"
+ and contf: "continuous_on (closure S) f"
+ and bos: "bounded S"
+ and leB: "\<And>z. z \<in> frontier S \<Longrightarrow> norm(f z) \<le> B"
+ and "\<xi> \<in> S"
+ shows "norm(f \<xi>) \<le> B"
+proof -
+ have "compact (closure S)" using bos
+ by (simp add: bounded_closure compact_eq_bounded_closed)
+ moreover have "continuous_on (closure S) (cmod \<circ> f)"
+ using contf continuous_on_compose continuous_on_norm_id by blast
+ ultimately obtain z where zin: "z \<in> closure S" and z: "\<And>y. y \<in> closure S \<Longrightarrow> (cmod \<circ> f) y \<le> (cmod \<circ> f) z"
+ using continuous_attains_sup [of "closure S" "norm o f"] \<open>\<xi> \<in> S\<close> by auto
+ then consider "z \<in> frontier S" | "z \<in> interior S" using frontier_def by auto
+ then have "norm(f z) \<le> B"
+ proof cases
+ case 1 then show ?thesis using leB by blast
+ next
+ case 2
+ have zin: "z \<in> connected_component_set (interior S) z"
+ by (simp add: 2)
+ have "f constant_on (connected_component_set (interior S) z)"
+ apply (rule maximum_modulus_principle [OF _ _ _ _ _ zin])
+ apply (metis connected_component_subset holf holomorphic_on_subset)
+ apply (simp_all add: open_connected_component)
+ by (metis closure_subset comp_eq_dest_lhs interior_subset subsetCE z connected_component_in)
+ then obtain c where c: "\<And>w. w \<in> connected_component_set (interior S) z \<Longrightarrow> f w = c"
+ by (auto simp: constant_on_def)
+ have "f ` closure(connected_component_set (interior S) z) \<subseteq> {c}"
+ apply (rule image_closure_subset)
+ apply (meson closure_mono connected_component_subset contf continuous_on_subset interior_subset)
+ using c
+ apply auto
+ done
+ then have cc: "\<And>w. w \<in> closure(connected_component_set (interior S) z) \<Longrightarrow> f w = c" by blast
+ have "frontier(connected_component_set (interior S) z) \<noteq> {}"
+ apply (simp add: frontier_eq_empty)
+ by (metis "2" bos bounded_interior connected_component_eq_UNIV connected_component_refl not_bounded_UNIV)
+ then obtain w where w: "w \<in> frontier(connected_component_set (interior S) z)"
+ by auto
+ then have "norm (f z) = norm (f w)" by (simp add: "2" c cc frontier_def)
+ also have "... \<le> B"
+ apply (rule leB)
+ using w
+using frontier_interior_subset frontier_of_connected_component_subset by blast
+ finally show ?thesis .
+ qed
+ then show ?thesis
+ using z \<open>\<xi> \<in> S\<close> closure_subset by fastforce
+qed
+
+corollary maximum_real_frontier:
+ assumes holf: "f holomorphic_on (interior S)"
+ and contf: "continuous_on (closure S) f"
+ and bos: "bounded S"
+ and leB: "\<And>z. z \<in> frontier S \<Longrightarrow> Re(f z) \<le> B"
+ and "\<xi> \<in> S"
+ shows "Re(f \<xi>) \<le> B"
+using maximum_modulus_frontier [of "exp o f" S "exp B"]
+ Transcendental.continuous_on_exp holomorphic_on_compose holomorphic_on_exp assms
+by auto
+
+
+subsection\<open>Factoring out a zero according to its order\<close>
+
+lemma holomorphic_factor_order_of_zero:
+ assumes holf: "f holomorphic_on S"
+ and os: "open S"
+ and "\<xi> \<in> S" "0 < n"
+ and dnz: "(deriv ^^ n) f \<xi> \<noteq> 0"
+ and dfz: "\<And>i. \<lbrakk>0 < i; i < n\<rbrakk> \<Longrightarrow> (deriv ^^ i) f \<xi> = 0"
+ obtains g r where "0 < r"
+ "g holomorphic_on ball \<xi> r"
+ "\<And>w. w \<in> ball \<xi> r \<Longrightarrow> f w - f \<xi> = (w - \<xi>)^n * g w"
+ "\<And>w. w \<in> ball \<xi> r \<Longrightarrow> g w \<noteq> 0"
+proof -
+ obtain r where "r>0" and r: "ball \<xi> r \<subseteq> S" using assms by (blast elim!: openE)
+ then have holfb: "f holomorphic_on ball \<xi> r"
+ using holf holomorphic_on_subset by blast
+ def g \<equiv> "\<lambda>w. suminf (\<lambda>i. (deriv ^^ (i + n)) f \<xi> / (fact(i + n)) * (w - \<xi>)^i)"
+ have sumsg: "(\<lambda>i. (deriv ^^ (i + n)) f \<xi> / (fact(i + n)) * (w - \<xi>)^i) sums g w"
+ and feq: "f w - f \<xi> = (w - \<xi>)^n * g w"
+ if w: "w \<in> ball \<xi> r" for w
+ proof -
+ def powf \<equiv> "(\<lambda>i. (deriv ^^ i) f \<xi>/(fact i) * (w - \<xi>)^i)"
+ have sing: "{..<n} - {i. powf i = 0} = (if f \<xi> = 0 then {} else {0})"
+ unfolding powf_def using \<open>0 < n\<close> dfz by (auto simp: dfz; metis funpow_0 not_gr0)
+ have "powf sums f w"
+ unfolding powf_def by (rule holomorphic_power_series [OF holfb w])
+ moreover have "(\<Sum>i<n. powf i) = f \<xi>"
+ apply (subst Groups_Big.comm_monoid_add_class.setsum.setdiff_irrelevant [symmetric])
+ apply (simp add:)
+ apply (simp only: dfz sing)
+ apply (simp add: powf_def)
+ done
+ ultimately have fsums: "(\<lambda>i. powf (i+n)) sums (f w - f \<xi>)"
+ using w sums_iff_shift' by metis
+ then have *: "summable (\<lambda>i. (w - \<xi>) ^ n * ((deriv ^^ (i + n)) f \<xi> * (w - \<xi>) ^ i / fact (i + n)))"
+ unfolding powf_def using sums_summable
+ by (auto simp: power_add mult_ac)
+ have "summable (\<lambda>i. (deriv ^^ (i + n)) f \<xi> * (w - \<xi>) ^ i / fact (i + n))"
+ proof (cases "w=\<xi>")
+ case False then show ?thesis
+ using summable_mult [OF *, of "1 / (w - \<xi>) ^ n"] by (simp add:)
+ next
+ case True then show ?thesis
+ by (auto simp: Power.semiring_1_class.power_0_left intro!: summable_finite [of "{0}"]
+ split: if_split_asm)
+ qed
+ then show sumsg: "(\<lambda>i. (deriv ^^ (i + n)) f \<xi> / (fact(i + n)) * (w - \<xi>)^i) sums g w"
+ by (simp add: summable_sums_iff g_def)
+ show "f w - f \<xi> = (w - \<xi>)^n * g w"
+ apply (rule sums_unique2)
+ apply (rule fsums [unfolded powf_def])
+ using sums_mult [OF sumsg, of "(w - \<xi>) ^ n"]
+ by (auto simp: power_add mult_ac)
+ qed
+ then have holg: "g holomorphic_on ball \<xi> r"
+ by (meson sumsg power_series_holomorphic)
+ then have contg: "continuous_on (ball \<xi> r) g"
+ by (blast intro: holomorphic_on_imp_continuous_on)
+ have "g \<xi> \<noteq> 0"
+ using dnz unfolding g_def
+ by (subst suminf_finite [of "{0}"]) auto
+ obtain d where "0 < d" and d: "\<And>w. w \<in> ball \<xi> d \<Longrightarrow> g w \<noteq> 0"
+ apply (rule exE [OF continuous_on_avoid [OF contg _ \<open>g \<xi> \<noteq> 0\<close>]])
+ using \<open>0 < r\<close>
+ apply force
+ by (metis \<open>0 < r\<close> less_trans mem_ball not_less_iff_gr_or_eq)
+ show ?thesis
+ apply (rule that [where g=g and r ="min r d"])
+ using \<open>0 < r\<close> \<open>0 < d\<close> holg
+ apply (auto simp: feq holomorphic_on_subset subset_ball d)
+ done
+qed
+
+
+lemma holomorphic_factor_order_of_zero_strong:
+ assumes holf: "f holomorphic_on S" "open S" "\<xi> \<in> S" "0 < n"
+ and "(deriv ^^ n) f \<xi> \<noteq> 0"
+ and "\<And>i. \<lbrakk>0 < i; i < n\<rbrakk> \<Longrightarrow> (deriv ^^ i) f \<xi> = 0"
+ obtains g r where "0 < r"
+ "g holomorphic_on ball \<xi> r"
+ "\<And>w. w \<in> ball \<xi> r \<Longrightarrow> f w - f \<xi> = ((w - \<xi>) * g w) ^ n"
+ "\<And>w. w \<in> ball \<xi> r \<Longrightarrow> g w \<noteq> 0"
+proof -
+ obtain g r where "0 < r"
+ and holg: "g holomorphic_on ball \<xi> r"
+ and feq: "\<And>w. w \<in> ball \<xi> r \<Longrightarrow> f w - f \<xi> = (w - \<xi>)^n * g w"
+ and gne: "\<And>w. w \<in> ball \<xi> r \<Longrightarrow> g w \<noteq> 0"
+ by (auto intro: holomorphic_factor_order_of_zero [OF assms])
+ have con: "continuous_on (ball \<xi> r) (\<lambda>z. deriv g z / g z)"
+ by (rule continuous_intros) (auto simp: gne holg holomorphic_deriv holomorphic_on_imp_continuous_on)
+ have cd: "\<And>x. dist \<xi> x < r \<Longrightarrow> (\<lambda>z. deriv g z / g z) complex_differentiable at x"
+ apply (rule derivative_intros)+
+ using holg mem_ball apply (blast intro: holomorphic_deriv holomorphic_on_imp_differentiable_at)
+ apply (metis Topology_Euclidean_Space.open_ball at_within_open holg holomorphic_on_def mem_ball)
+ using gne mem_ball by blast
+ obtain h where h: "\<And>x. x \<in> ball \<xi> r \<Longrightarrow> (h has_field_derivative deriv g x / g x) (at x)"
+ apply (rule exE [OF holomorphic_convex_primitive [of "ball \<xi> r" "{}" "\<lambda>z. deriv g z / g z"]])
+ apply (auto simp: con cd)
+ apply (metis open_ball at_within_open mem_ball)
+ done
+ then have "continuous_on (ball \<xi> r) h"
+ by (metis open_ball holomorphic_on_imp_continuous_on holomorphic_on_open)
+ then have con: "continuous_on (ball \<xi> r) (\<lambda>x. exp (h x) / g x)"
+ by (auto intro!: continuous_intros simp add: holg holomorphic_on_imp_continuous_on gne)
+ have 0: "dist \<xi> x < r \<Longrightarrow> ((\<lambda>x. exp (h x) / g x) has_field_derivative 0) (at x)" for x
+ apply (rule h derivative_eq_intros | simp)+
+ apply (rule DERIV_deriv_iff_complex_differentiable [THEN iffD2])
+ using holg apply (auto simp: holomorphic_on_imp_differentiable_at gne h)
+ done
+ obtain c where c: "\<And>x. x \<in> ball \<xi> r \<Longrightarrow> exp (h x) / g x = c"
+ by (rule DERIV_zero_connected_constant [of "ball \<xi> r" "{}" "\<lambda>x. exp(h x) / g x"]) (auto simp: con 0)
+ have hol: "(\<lambda>z. exp ((Ln (inverse c) + h z) / of_nat n)) holomorphic_on ball \<xi> r"
+ apply (rule holomorphic_on_compose [unfolded o_def, where g = exp])
+ apply (rule holomorphic_intros)+
+ using h holomorphic_on_open apply blast
+ apply (rule holomorphic_intros)+
+ using \<open>0 < n\<close> apply (simp add:)
+ apply (rule holomorphic_intros)+
+ done
+ show ?thesis
+ apply (rule that [where g="\<lambda>z. exp((Ln(inverse c) + h z)/n)" and r =r])
+ using \<open>0 < r\<close> \<open>0 < n\<close>
+ apply (auto simp: feq power_mult_distrib exp_divide_power_eq c [symmetric])
+ apply (rule hol)
+ apply (simp add: Transcendental.exp_add gne)
+ done
+qed
+
+
+lemma
+ fixes k :: "'a::wellorder"
+ assumes a_def: "a == LEAST x. P x" and P: "P k"
+ shows def_LeastI: "P a" and def_Least_le: "a \<le> k"
+unfolding a_def
+by (rule LeastI Least_le; rule P)+
+
+lemma holomorphic_factor_zero_nonconstant:
+ assumes holf: "f holomorphic_on S" and S: "open S" "connected S"
+ and "\<xi> \<in> S" "f \<xi> = 0"
+ and nonconst: "\<And>c. \<exists>z \<in> S. f z \<noteq> c"
+ obtains g r n
+ where "0 < n" "0 < r" "ball \<xi> r \<subseteq> S"
+ "g holomorphic_on ball \<xi> r"
+ "\<And>w. w \<in> ball \<xi> r \<Longrightarrow> f w = (w - \<xi>)^n * g w"
+ "\<And>w. w \<in> ball \<xi> r \<Longrightarrow> g w \<noteq> 0"
+proof (cases "\<forall>n>0. (deriv ^^ n) f \<xi> = 0")
+ case True then show ?thesis
+ using holomorphic_fun_eq_const_on_connected [OF holf S _ \<open>\<xi> \<in> S\<close>] nonconst by auto
+next
+ case False
+ then obtain n0 where "n0 > 0" and n0: "(deriv ^^ n0) f \<xi> \<noteq> 0" by blast
+ obtain r0 where "r0 > 0" "ball \<xi> r0 \<subseteq> S" using S openE \<open>\<xi> \<in> S\<close> by auto
+ def n \<equiv> "LEAST n. (deriv ^^ n) f \<xi> \<noteq> 0"
+ have n_ne: "(deriv ^^ n) f \<xi> \<noteq> 0"
+ by (rule def_LeastI [OF n_def]) (rule n0)
+ then have "0 < n" using \<open>f \<xi> = 0\<close>
+ using funpow_0 by fastforce
+ have n_min: "\<And>k. k < n \<Longrightarrow> (deriv ^^ k) f \<xi> = 0"
+ using def_Least_le [OF n_def] not_le by blast
+ then obtain g r1
+ where "0 < r1" "g holomorphic_on ball \<xi> r1"
+ "\<And>w. w \<in> ball \<xi> r1 \<Longrightarrow> f w = (w - \<xi>) ^ n * g w"
+ "\<And>w. w \<in> ball \<xi> r1 \<Longrightarrow> g w \<noteq> 0"
+ by (auto intro: holomorphic_factor_order_of_zero [OF holf \<open>open S\<close> \<open>\<xi> \<in> S\<close> \<open>n > 0\<close> n_ne] simp: \<open>f \<xi> = 0\<close>)
+ then show ?thesis
+ apply (rule_tac g=g and r="min r0 r1" and n=n in that)
+ using \<open>0 < n\<close> \<open>0 < r0\<close> \<open>0 < r1\<close> \<open>ball \<xi> r0 \<subseteq> S\<close>
+ apply (auto simp: subset_ball intro: holomorphic_on_subset)
+ done
+qed
+
+
+lemma holomorphic_lower_bound_difference:
+ assumes holf: "f holomorphic_on S" and S: "open S" "connected S"
+ and "\<xi> \<in> S" and "\<phi> \<in> S"
+ and fne: "f \<phi> \<noteq> f \<xi>"
+ obtains k n r
+ where "0 < k" "0 < r"
+ "ball \<xi> r \<subseteq> S"
+ "\<And>w. w \<in> ball \<xi> r \<Longrightarrow> k * norm(w - \<xi>)^n \<le> norm(f w - f \<xi>)"
+proof -
+ def n \<equiv> "LEAST n. 0 < n \<and> (deriv ^^ n) f \<xi> \<noteq> 0"
+ obtain n0 where "0 < n0" and n0: "(deriv ^^ n0) f \<xi> \<noteq> 0"
+ using fne holomorphic_fun_eq_const_on_connected [OF holf S] \<open>\<xi> \<in> S\<close> \<open>\<phi> \<in> S\<close> by blast
+ then have "0 < n" and n_ne: "(deriv ^^ n) f \<xi> \<noteq> 0"
+ unfolding n_def by (metis (mono_tags, lifting) LeastI)+
+ have n_min: "\<And>k. \<lbrakk>0 < k; k < n\<rbrakk> \<Longrightarrow> (deriv ^^ k) f \<xi> = 0"
+ unfolding n_def by (blast dest: not_less_Least)
+ then obtain g r
+ where "0 < r" and holg: "g holomorphic_on ball \<xi> r"
+ and fne: "\<And>w. w \<in> ball \<xi> r \<Longrightarrow> f w - f \<xi> = (w - \<xi>) ^ n * g w"
+ and gnz: "\<And>w. w \<in> ball \<xi> r \<Longrightarrow> g w \<noteq> 0"
+ by (auto intro: holomorphic_factor_order_of_zero [OF holf \<open>open S\<close> \<open>\<xi> \<in> S\<close> \<open>n > 0\<close> n_ne])
+ obtain e where "e>0" and e: "ball \<xi> e \<subseteq> S" using assms by (blast elim!: openE)
+ then have holfb: "f holomorphic_on ball \<xi> e"
+ using holf holomorphic_on_subset by blast
+ def d \<equiv> "(min e r) / 2"
+ have "0 < d" using \<open>0 < r\<close> \<open>0 < e\<close> by (simp add: d_def)
+ have "d < r"
+ using \<open>0 < r\<close> by (auto simp: d_def)
+ then have cbb: "cball \<xi> d \<subseteq> ball \<xi> r"
+ by (auto simp: cball_subset_ball_iff)
+ then have "g holomorphic_on cball \<xi> d"
+ by (rule holomorphic_on_subset [OF holg])
+ then have "closed (g ` cball \<xi> d)"
+ by (simp add: compact_imp_closed compact_continuous_image holomorphic_on_imp_continuous_on)
+ moreover have "g ` cball \<xi> d \<noteq> {}"
+ using \<open>0 < d\<close> by auto
+ ultimately obtain x where x: "x \<in> g ` cball \<xi> d" and "\<And>y. y \<in> g ` cball \<xi> d \<Longrightarrow> dist 0 x \<le> dist 0 y"
+ by (rule distance_attains_inf) blast
+ then have leg: "\<And>w. w \<in> cball \<xi> d \<Longrightarrow> norm x \<le> norm (g w)"
+ by auto
+ have "ball \<xi> d \<subseteq> cball \<xi> d" by auto
+ also have "... \<subseteq> ball \<xi> e" using \<open>0 < d\<close> d_def by auto
+ also have "... \<subseteq> S" by (rule e)
+ finally have dS: "ball \<xi> d \<subseteq> S" .
+ moreover have "x \<noteq> 0" using gnz x \<open>d < r\<close> by auto
+ ultimately show ?thesis
+ apply (rule_tac k="norm x" and n=n and r=d in that)
+ using \<open>d < r\<close> leg
+ apply (auto simp: \<open>0 < d\<close> fne norm_mult norm_power algebra_simps mult_right_mono)
+ done
+qed
+
+lemma
+ assumes holf: "f holomorphic_on (S - {\<xi>})" and \<xi>: "\<xi> \<in> interior S"
+ shows holomorphic_on_extend_lim:
+ "(\<exists>g. g holomorphic_on S \<and> (\<forall>z \<in> S - {\<xi>}. g z = f z)) \<longleftrightarrow>
+ ((\<lambda>z. (z - \<xi>) * f z) \<longlongrightarrow> 0) (at \<xi>)"
+ (is "?P = ?Q")
+ and holomorphic_on_extend_bounded:
+ "(\<exists>g. g holomorphic_on S \<and> (\<forall>z \<in> S - {\<xi>}. g z = f z)) \<longleftrightarrow>
+ (\<exists>B. eventually (\<lambda>z. norm(f z) \<le> B) (at \<xi>))"
+ (is "?P = ?R")
+proof -
+ obtain \<delta> where "0 < \<delta>" and \<delta>: "ball \<xi> \<delta> \<subseteq> S"
+ using \<xi> mem_interior by blast
+ have "?R" if holg: "g holomorphic_on S" and gf: "\<And>z. z \<in> S - {\<xi>} \<Longrightarrow> g z = f z" for g
+ proof -
+ have *: "\<forall>\<^sub>F z in at \<xi>. dist (g z) (g \<xi>) < 1 \<longrightarrow> cmod (f z) \<le> cmod (g \<xi>) + 1"
+ apply (simp add: eventually_at)
+ apply (rule_tac x="\<delta>" in exI)
+ using \<delta> \<open>0 < \<delta>\<close>
+ apply (clarsimp simp:)
+ apply (drule_tac c=x in subsetD)
+ apply (simp add: dist_commute)
+ by (metis DiffI add.commute diff_le_eq dist_norm gf le_less_trans less_eq_real_def norm_triangle_ineq2 singletonD)
+ have "continuous_on (interior S) g"
+ by (meson continuous_on_subset holg holomorphic_on_imp_continuous_on interior_subset)
+ then have "\<And>x. x \<in> interior S \<Longrightarrow> (g \<longlongrightarrow> g x) (at x)"
+ using continuous_on_interior continuous_within holg holomorphic_on_imp_continuous_on by blast
+ then have "(g \<longlongrightarrow> g \<xi>) (at \<xi>)"
+ by (simp add: \<xi>)
+ then show ?thesis
+ apply (rule_tac x="norm(g \<xi>) + 1" in exI)
+ apply (rule eventually_mp [OF * tendstoD [where e=1]], auto)
+ done
+ qed
+ moreover have "?Q" if "\<forall>\<^sub>F z in at \<xi>. cmod (f z) \<le> B" for B
+ by (rule lim_null_mult_right_bounded [OF _ that]) (simp add: LIM_zero)
+ moreover have "?P" if "(\<lambda>z. (z - \<xi>) * f z) \<midarrow>\<xi>\<rightarrow> 0"
+ proof -
+ def h \<equiv> "\<lambda>z. (z - \<xi>)^2 * f z"
+ have h0: "(h has_field_derivative 0) (at \<xi>)"
+ apply (simp add: h_def Derivative.DERIV_within_iff)
+ apply (rule Lim_transform_within [OF that, of 1])
+ apply (auto simp: divide_simps power2_eq_square)
+ done
+ have holh: "h holomorphic_on S"
+ proof (simp add: holomorphic_on_def, clarify)
+ fix z assume "z \<in> S"
+ show "h complex_differentiable at z within S"
+ proof (cases "z = \<xi>")
+ case True then show ?thesis
+ using complex_differentiable_at_within complex_differentiable_def h0 by blast
+ next
+ case False
+ then have "f complex_differentiable at z within S"
+ using holomorphic_onD [OF holf, of z] \<open>z \<in> S\<close>
+ unfolding complex_differentiable_def DERIV_within_iff
+ by (force intro: exI [where x="dist \<xi> z"] elim: Lim_transform_within_set [unfolded eventually_at])
+ then show ?thesis
+ by (simp add: h_def power2_eq_square derivative_intros)
+ qed
+ qed
+ def g \<equiv> "\<lambda>z. if z = \<xi> then deriv h \<xi> else (h z - h \<xi>) / (z - \<xi>)"
+ have holg: "g holomorphic_on S"
+ unfolding g_def by (rule pole_lemma [OF holh \<xi>])
+ show ?thesis
+ apply (rule_tac x="\<lambda>z. if z = \<xi> then deriv g \<xi> else (g z - g \<xi>)/(z - \<xi>)" in exI)
+ apply (rule conjI)
+ apply (rule pole_lemma [OF holg \<xi>])
+ apply (auto simp: g_def power2_eq_square divide_simps)
+ using h0 apply (simp add: h0 DERIV_imp_deriv h_def power2_eq_square)
+ done
+ qed
+ ultimately show "?P = ?Q" and "?P = ?R"
+ by meson+
+qed
+
+
+proposition pole_at_infinity:
+ assumes holf: "f holomorphic_on UNIV" and lim: "((inverse o f) \<longlongrightarrow> l) at_infinity"
+ obtains a n where "\<And>z. f z = (\<Sum>i\<le>n. a i * z^i)"
+proof (cases "l = 0")
+ case False
+ with tendsto_inverse [OF lim] show ?thesis
+ apply (rule_tac a="(\<lambda>n. inverse l)" and n=0 in that)
+ apply (simp add: Liouville_weak [OF holf, of "inverse l"])
+ done
+next
+ case True
+ then have [simp]: "l = 0" .
+ show ?thesis
+ proof (cases "\<exists>r. 0 < r \<and> (\<forall>z \<in> ball 0 r - {0}. f(inverse z) \<noteq> 0)")
+ case True
+ then obtain r where "0 < r" and r: "\<And>z. z \<in> ball 0 r - {0} \<Longrightarrow> f(inverse z) \<noteq> 0"
+ by auto
+ have 1: "inverse \<circ> f \<circ> inverse holomorphic_on ball 0 r - {0}"
+ by (rule holomorphic_on_compose holomorphic_intros holomorphic_on_subset [OF holf] | force simp: r)+
+ have 2: "0 \<in> interior (ball 0 r)"
+ using \<open>0 < r\<close> by simp
+ have "\<exists>B. 0<B \<and> eventually (\<lambda>z. cmod ((inverse \<circ> f \<circ> inverse) z) \<le> B) (at 0)"
+ apply (rule exI [where x=1])
+ apply (simp add:)
+ using tendstoD [OF lim [unfolded lim_at_infinity_0] zero_less_one]
+ apply (rule eventually_mono)
+ apply (simp add: dist_norm)
+ done
+ with holomorphic_on_extend_bounded [OF 1 2]
+ obtain g where holg: "g holomorphic_on ball 0 r"
+ and geq: "\<And>z. z \<in> ball 0 r - {0} \<Longrightarrow> g z = (inverse \<circ> f \<circ> inverse) z"
+ by meson
+ have ifi0: "(inverse \<circ> f \<circ> inverse) \<midarrow>0\<rightarrow> 0"
+ using \<open>l = 0\<close> lim lim_at_infinity_0 by blast
+ have g2g0: "g \<midarrow>0\<rightarrow> g 0"
+ using \<open>0 < r\<close> centre_in_ball continuous_at continuous_on_eq_continuous_at holg
+ by (blast intro: holomorphic_on_imp_continuous_on)
+ have g2g1: "g \<midarrow>0\<rightarrow> 0"
+ apply (rule Lim_transform_within_open [OF ifi0 open_ball [of 0 r]])
+ using \<open>0 < r\<close> by (auto simp: geq)
+ have [simp]: "g 0 = 0"
+ by (rule tendsto_unique [OF _ g2g0 g2g1]) simp
+ have "ball 0 r - {0::complex} \<noteq> {}"
+ using \<open>0 < r\<close>
+ apply (clarsimp simp: ball_def dist_norm)
+ apply (drule_tac c="of_real r/2" in subsetD, auto)
+ done
+ then obtain w::complex where "w \<noteq> 0" and w: "norm w < r" by force
+ then have "g w \<noteq> 0" by (simp add: geq r)
+ obtain B n e where "0 < B" "0 < e" "e \<le> r"
+ and leg: "\<And>w. norm w < e \<Longrightarrow> B * cmod w ^ n \<le> cmod (g w)"
+ apply (rule holomorphic_lower_bound_difference [OF holg open_ball connected_ball, of 0 w])
+ using \<open>0 < r\<close> w \<open>g w \<noteq> 0\<close> by (auto simp: ball_subset_ball_iff)
+ have "cmod (f z) \<le> cmod z ^ n / B" if "2/e \<le> cmod z" for z
+ proof -
+ have ize: "inverse z \<in> ball 0 e - {0}" using that \<open>0 < e\<close>
+ by (auto simp: norm_divide divide_simps algebra_simps)
+ then have [simp]: "z \<noteq> 0" and izr: "inverse z \<in> ball 0 r - {0}" using \<open>e \<le> r\<close>
+ by auto
+ then have [simp]: "f z \<noteq> 0"
+ using r [of "inverse z"] by simp
+ have [simp]: "f z = inverse (g (inverse z))"
+ using izr geq [of "inverse z"] by simp
+ show ?thesis using ize leg [of "inverse z"] \<open>0 < B\<close> \<open>0 < e\<close>
+ by (simp add: divide_simps norm_divide algebra_simps)
+ qed
+ then show ?thesis
+ apply (rule_tac a = "\<lambda>k. (deriv ^^ k) f 0 / (fact k)" and n=n in that)
+ apply (rule_tac A = "2/e" and B = "1/B" in Liouville_polynomial [OF holf])
+ apply (simp add:)
+ done
+ next
+ case False
+ then have fi0: "\<And>r. r > 0 \<Longrightarrow> \<exists>z\<in>ball 0 r - {0}. f (inverse z) = 0"
+ by simp
+ have fz0: "f z = 0" if "0 < r" and lt1: "\<And>x. x \<noteq> 0 \<Longrightarrow> cmod x < r \<Longrightarrow> inverse (cmod (f (inverse x))) < 1"
+ for z r
+ proof -
+ have f0: "(f \<longlongrightarrow> 0) at_infinity"
+ proof -
+ have DIM_complex[intro]: "2 \<le> DIM(complex)" --\<open>should not be necessary!\<close>
+ by simp
+ have "continuous_on (inverse ` (ball 0 r - {0})) f"
+ using continuous_on_subset holf holomorphic_on_imp_continuous_on by blast
+ then have "connected ((f \<circ> inverse) ` (ball 0 r - {0}))"
+ apply (intro connected_continuous_image continuous_intros)
+ apply (force intro: connected_punctured_ball)+
+ done
+ then have "\<lbrakk>w \<noteq> 0; cmod w < r\<rbrakk> \<Longrightarrow> f (inverse w) = 0" for w
+ apply (rule disjE [OF connected_closedD [where A = "{0}" and B = "- ball 0 1"]], auto)
+ apply (metis (mono_tags, hide_lams) not_less_iff_gr_or_eq one_less_inverse lt1 zero_less_norm_iff)
+ using False \<open>0 < r\<close> apply fastforce
+ 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)
+ then show ?thesis
+ apply (simp add: lim_at_infinity_0)
+ apply (rule Lim_eventually)
+ apply (simp add: eventually_at)
+ apply (rule_tac x=r in exI)
+ apply (simp add: \<open>0 < r\<close> dist_norm)
+ done
+ qed
+ obtain w where "w \<in> ball 0 r - {0}" and "f (inverse w) = 0"
+ using False \<open>0 < r\<close> by blast
+ then show ?thesis
+ by (auto simp: f0 Liouville_weak [OF holf, of 0])
+ qed
+ show ?thesis
+ apply (rule that [of "\<lambda>n. 0" 0])
+ using lim [unfolded lim_at_infinity_0]
+ apply (simp add: Lim_at dist_norm norm_inverse)
+ apply (drule_tac x=1 in spec)
+ using fz0 apply auto
+ done
+ qed
+qed
+
+
+subsection\<open>Entire proper functions are precisely the non-trivial polynomials\<close>
+
+proposition proper_map_polyfun:
+ fixes c :: "nat \<Rightarrow> 'a::{real_normed_div_algebra,heine_borel}"
+ assumes "closed S" and "compact K" and c: "c i \<noteq> 0" "1 \<le> i" "i \<le> n"
+ shows "compact (S \<inter> {z. (\<Sum>i\<le>n. c i * z^i) \<in> K})"
+proof -
+ obtain B where "B > 0" and B: "\<And>x. x \<in> K \<Longrightarrow> norm x \<le> B"
+ by (metis compact_imp_bounded \<open>compact K\<close> bounded_pos)
+ have *: "norm x \<le> b"
+ if "\<And>x. b \<le> norm x \<Longrightarrow> B + 1 \<le> norm (\<Sum>i\<le>n. c i * x ^ i)"
+ "(\<Sum>i\<le>n. c i * x ^ i) \<in> K" for b x
+ proof -
+ have "norm (\<Sum>i\<le>n. c i * x ^ i) \<le> B"
+ using B that by blast
+ moreover have "\<not> B + 1 \<le> B"
+ by simp
+ ultimately show "norm x \<le> b"
+ using that by (metis (no_types) less_eq_real_def not_less order_trans)
+ qed
+ have "bounded {z. (\<Sum>i\<le>n. c i * z ^ i) \<in> K}"
+ using polyfun_extremal [where c=c and B="B+1", OF c]
+ by (auto simp: bounded_pos eventually_at_infinity_pos *)
+ moreover have "closed {z. (\<Sum>i\<le>n. c i * z ^ i) \<in> K}"
+ apply (rule allI continuous_closed_preimage_univ continuous_intros)+
+ using \<open>compact K\<close> compact_eq_bounded_closed by blast
+ ultimately show ?thesis
+ using closed_inter_compact [OF \<open>closed S\<close>] compact_eq_bounded_closed by blast
+qed
+
+corollary proper_map_polyfun_univ:
+ fixes c :: "nat \<Rightarrow> 'a::{real_normed_div_algebra,heine_borel}"
+ assumes "compact K" "c i \<noteq> 0" "1 \<le> i" "i \<le> n"
+ shows "compact ({z. (\<Sum>i\<le>n. c i * z^i) \<in> K})"
+using proper_map_polyfun [of UNIV K c i n] assms by simp
+
+
+proposition proper_map_polyfun_eq:
+ assumes "f holomorphic_on UNIV"
+ shows "(\<forall>k. compact k \<longrightarrow> compact {z. f z \<in> k}) \<longleftrightarrow>
+ (\<exists>c n. 0 < n \<and> (c n \<noteq> 0) \<and> f = (\<lambda>z. \<Sum>i\<le>n. c i * z^i))"
+ (is "?lhs = ?rhs")
+proof
+ assume compf [rule_format]: ?lhs
+ have 2: "\<exists>k. 0 < k \<and> a k \<noteq> 0 \<and> f = (\<lambda>z. \<Sum>i \<le> k. a i * z ^ i)"
+ if "\<And>z. f z = (\<Sum>i\<le>n. a i * z ^ i)" for a n
+ proof (cases "\<forall>i\<le>n. 0<i \<longrightarrow> a i = 0")
+ case True
+ then have [simp]: "\<And>z. f z = a 0"
+ by (simp add: that setsum_atMost_shift)
+ have False using compf [of "{a 0}"] by simp
+ then show ?thesis ..
+ next
+ case False
+ then obtain k where k: "0 < k" "k\<le>n" "a k \<noteq> 0" by force
+ def m \<equiv> "GREATEST k. k\<le>n \<and> a k \<noteq> 0"
+ have m: "m\<le>n \<and> a m \<noteq> 0"
+ unfolding m_def
+ apply (rule GreatestI [where b = "Suc n"])
+ using k apply auto
+ done
+ have [simp]: "a i = 0" if "m < i" "i \<le> n" for i
+ using Greatest_le [where b = "Suc n" and P = "\<lambda>k. k\<le>n \<and> a k \<noteq> 0"]
+ using m_def not_le that by auto
+ have "k \<le> m"
+ unfolding m_def
+ apply (rule Greatest_le [where b = "Suc n"])
+ using k apply auto
+ done
+ with k m show ?thesis
+ by (rule_tac x=m in exI) (auto simp: that comm_monoid_add_class.setsum.mono_neutral_right)
+ qed
+ have "((inverse \<circ> f) \<longlongrightarrow> 0) at_infinity"
+ proof (rule Lim_at_infinityI)
+ fix e::real assume "0 < e"
+ with compf [of "cball 0 (inverse e)"]
+ show "\<exists>B. \<forall>x. B \<le> cmod x \<longrightarrow> dist ((inverse \<circ> f) x) 0 \<le> e"
+ apply (simp add:)
+ apply (clarsimp simp add: compact_eq_bounded_closed bounded_pos norm_inverse)
+ apply (rule_tac x="b+1" in exI)
+ 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)
+ done
+ qed
+ then show ?rhs
+ apply (rule pole_at_infinity [OF assms])
+ using 2 apply blast
+ done
+next
+ assume ?rhs
+ then obtain c n where "0 < n" "c n \<noteq> 0" "f = (\<lambda>z. \<Sum>i\<le>n. c i * z ^ i)" by blast
+ then have "compact {z. f z \<in> k}" if "compact k" for k
+ by (auto intro: proper_map_polyfun_univ [OF that])
+ then show ?lhs by blast
+qed
+
+
+subsection\<open>Relating invertibility and nonvanishing of derivative\<close>
+
+proposition has_complex_derivative_locally_injective:
+ assumes holf: "f holomorphic_on S"
+ and S: "\<xi> \<in> S" "open S"
+ and dnz: "deriv f \<xi> \<noteq> 0"
+ obtains r where "r > 0" "ball \<xi> r \<subseteq> S" "inj_on f (ball \<xi> r)"
+proof -
+ have *: "\<exists>d>0. \<forall>x. dist \<xi> x < d \<longrightarrow> onorm (\<lambda>v. deriv f x * v - deriv f \<xi> * v) < e" if "e > 0" for e
+ proof -
+ have contdf: "continuous_on S (deriv f)"
+ by (simp add: holf holomorphic_deriv holomorphic_on_imp_continuous_on \<open>open S\<close>)
+ obtain \<delta> where "\<delta>>0" and \<delta>: "\<And>x. \<lbrakk>x \<in> S; dist x \<xi> \<le> \<delta>\<rbrakk> \<Longrightarrow> cmod (deriv f x - deriv f \<xi>) \<le> e/2"
+ using continuous_onE [OF contdf \<open>\<xi> \<in> S\<close>, of "e/2"] \<open>0 < e\<close>
+ by (metis dist_complex_def half_gt_zero less_imp_le)
+ obtain \<epsilon> where "\<epsilon>>0" "ball \<xi> \<epsilon> \<subseteq> S"
+ by (metis openE [OF \<open>open S\<close> \<open>\<xi> \<in> S\<close>])
+ with \<open>\<delta>>0\<close> have "\<exists>\<delta>>0. \<forall>x. dist \<xi> x < \<delta> \<longrightarrow> onorm (\<lambda>v. deriv f x * v - deriv f \<xi> * v) \<le> e/2"
+ apply (rule_tac x="min \<delta> \<epsilon>" in exI)
+ apply (intro conjI allI impI Operator_Norm.onorm_le)
+ apply (simp add:)
+ apply (simp only: Rings.ring_class.left_diff_distrib [symmetric] norm_mult)
+ apply (rule mult_right_mono [OF \<delta>])
+ apply (auto simp: dist_commute Rings.ordered_semiring_class.mult_right_mono \<delta>)
+ done
+ with \<open>e>0\<close> show ?thesis by force
+ qed
+ have "inj (op * (deriv f \<xi>))"
+ using dnz by simp
+ then obtain g' where g': "linear g'" "g' \<circ> op * (deriv f \<xi>) = id"
+ using linear_injective_left_inverse [of "op * (deriv f \<xi>)"]
+ by (auto simp: linear_times)
+ show ?thesis
+ apply (rule has_derivative_locally_injective [OF S, where f=f and f' = "\<lambda>z h. deriv f z * h" and g' = g'])
+ using g' *
+ apply (simp_all add: linear_conv_bounded_linear that)
+ using DERIV_deriv_iff_complex_differentiable has_field_derivative_imp_has_derivative holf
+ holomorphic_on_imp_differentiable_at \<open>open S\<close> apply blast
+ done
+qed
+
+
+proposition has_complex_derivative_locally_invertible:
+ assumes holf: "f holomorphic_on S"
+ and S: "\<xi> \<in> S" "open S"
+ and dnz: "deriv f \<xi> \<noteq> 0"
+ obtains r where "r > 0" "ball \<xi> r \<subseteq> S" "open (f ` (ball \<xi> r))" "inj_on f (ball \<xi> r)"
+proof -
+ obtain r where "r > 0" "ball \<xi> r \<subseteq> S" "inj_on f (ball \<xi> r)"
+ by (blast intro: that has_complex_derivative_locally_injective [OF assms])
+ then have \<xi>: "\<xi> \<in> ball \<xi> r" by simp
+ then have nc: "~ f constant_on ball \<xi> r"
+ using \<open>inj_on f (ball \<xi> r)\<close> injective_not_constant by fastforce
+ have holf': "f holomorphic_on ball \<xi> r"
+ using \<open>ball \<xi> r \<subseteq> S\<close> holf holomorphic_on_subset by blast
+ have "open (f ` ball \<xi> r)"
+ apply (rule open_mapping_thm [OF holf'])
+ using nc apply auto
+ done
+ then show ?thesis
+ using \<open>0 < r\<close> \<open>ball \<xi> r \<subseteq> S\<close> \<open>inj_on f (ball \<xi> r)\<close> that by blast
+qed
+
+
+proposition holomorphic_injective_imp_regular:
+ assumes holf: "f holomorphic_on S"
+ and "open S" and injf: "inj_on f S"
+ and "\<xi> \<in> S"
+ shows "deriv f \<xi> \<noteq> 0"
+proof -
+ obtain r where "r>0" and r: "ball \<xi> r \<subseteq> S" using assms by (blast elim!: openE)
+ have holf': "f holomorphic_on ball \<xi> r"
+ using \<open>ball \<xi> r \<subseteq> S\<close> holf holomorphic_on_subset by blast
+ show ?thesis
+ proof (cases "\<forall>n>0. (deriv ^^ n) f \<xi> = 0")
+ case True
+ have fcon: "f w = f \<xi>" if "w \<in> ball \<xi> r" for w
+ apply (rule holomorphic_fun_eq_const_on_connected [OF holf'])
+ using True \<open>0 < r\<close> that by auto
+ have False
+ using fcon [of "\<xi> + r/2"] \<open>0 < r\<close> r injf unfolding inj_on_def
+ by (metis \<open>\<xi> \<in> S\<close> contra_subsetD dist_commute fcon mem_ball perfect_choose_dist)
+ then show ?thesis ..
+ next
+ case False
+ then obtain n0 where n0: "n0 > 0 \<and> (deriv ^^ n0) f \<xi> \<noteq> 0" by blast
+ def n \<equiv> "LEAST n. n > 0 \<and> (deriv ^^ n) f \<xi> \<noteq> 0"
+ have n_ne: "n > 0" "(deriv ^^ n) f \<xi> \<noteq> 0"
+ using def_LeastI [OF n_def n0] by auto
+ have n_min: "\<And>k. 0 < k \<Longrightarrow> k < n \<Longrightarrow> (deriv ^^ k) f \<xi> = 0"
+ using def_Least_le [OF n_def] not_le by auto
+ obtain g \<delta> where "0 < \<delta>"
+ and holg: "g holomorphic_on ball \<xi> \<delta>"
+ and fd: "\<And>w. w \<in> ball \<xi> \<delta> \<Longrightarrow> f w - f \<xi> = ((w - \<xi>) * g w) ^ n"
+ and gnz: "\<And>w. w \<in> ball \<xi> \<delta> \<Longrightarrow> g w \<noteq> 0"
+ apply (rule holomorphic_factor_order_of_zero_strong [OF holf \<open>open S\<close> \<open>\<xi> \<in> S\<close> n_ne])
+ apply (blast intro: n_min)+
+ done
+ show ?thesis
+ proof (cases "n=1")
+ case True
+ with n_ne show ?thesis by auto
+ next
+ case False
+ have holgw: "(\<lambda>w. (w - \<xi>) * g w) holomorphic_on ball \<xi> (min r \<delta>)"
+ apply (rule holomorphic_intros)+
+ using holg by (simp add: holomorphic_on_subset subset_ball)
+ have gd: "\<And>w. dist \<xi> w < \<delta> \<Longrightarrow> (g has_field_derivative deriv g w) (at w)"
+ using holg
+ by (simp add: DERIV_deriv_iff_complex_differentiable holomorphic_on_def at_within_open_NO_MATCH)
+ have *: "\<And>w. w \<in> ball \<xi> (min r \<delta>)
+ \<Longrightarrow> ((\<lambda>w. (w - \<xi>) * g w) has_field_derivative ((w - \<xi>) * deriv g w + g w))
+ (at w)"
+ by (rule gd derivative_eq_intros | simp)+
+ have [simp]: "deriv (\<lambda>w. (w - \<xi>) * g w) \<xi> \<noteq> 0"
+ using * [of \<xi>] \<open>0 < \<delta>\<close> \<open>0 < r\<close> by (simp add: DERIV_imp_deriv gnz)
+ obtain T where "\<xi> \<in> T" "open T" and Tsb: "T \<subseteq> ball \<xi> (min r \<delta>)" and oimT: "open ((\<lambda>w. (w - \<xi>) * g w) ` T)"
+ apply (rule has_complex_derivative_locally_invertible [OF holgw, of \<xi>])
+ using \<open>0 < r\<close> \<open>0 < \<delta>\<close>
+ apply (simp_all add:)
+ by (meson Topology_Euclidean_Space.open_ball centre_in_ball)
+ def U \<equiv> "(\<lambda>w. (w - \<xi>) * g w) ` T"
+ have "open U" by (metis oimT U_def)
+ have "0 \<in> U"
+ apply (auto simp: U_def)
+ apply (rule image_eqI [where x = \<xi>])
+ apply (auto simp: \<open>\<xi> \<in> T\<close>)
+ done
+ then obtain \<epsilon> where "\<epsilon>>0" and \<epsilon>: "cball 0 \<epsilon> \<subseteq> U"
+ using \<open>open U\<close> open_contains_cball by blast
+ then have "\<epsilon> * exp(2 * of_real pi * ii * (0/n)) \<in> cball 0 \<epsilon>"
+ "\<epsilon> * exp(2 * of_real pi * ii * (1/n)) \<in> cball 0 \<epsilon>"
+ by (auto simp: norm_mult)
+ with \<epsilon> have "\<epsilon> * exp(2 * of_real pi * ii * (0/n)) \<in> U"
+ "\<epsilon> * exp(2 * of_real pi * ii * (1/n)) \<in> U" by blast+
+ then obtain y0 y1 where "y0 \<in> T" and y0: "(y0 - \<xi>) * g y0 = \<epsilon> * exp(2 * of_real pi * ii * (0/n))"
+ and "y1 \<in> T" and y1: "(y1 - \<xi>) * g y1 = \<epsilon> * exp(2 * of_real pi * ii * (1/n))"
+ by (auto simp: U_def)
+ then have "y0 \<in> ball \<xi> \<delta>" "y1 \<in> ball \<xi> \<delta>" using Tsb by auto
+ moreover have "y0 \<noteq> y1"
+ using y0 y1 \<open>\<epsilon> > 0\<close> complex_root_unity_eq_1 [of n 1] \<open>n > 0\<close> False by auto
+ moreover have "T \<subseteq> S"
+ by (meson Tsb min.cobounded1 order_trans r subset_ball)
+ ultimately have False
+ using inj_onD [OF injf, of y0 y1] \<open>y0 \<in> T\<close> \<open>y1 \<in> T\<close>
+ using fd [of y0] fd [of y1] complex_root_unity [of n 1]
+ apply (simp add: y0 y1 power_mult_distrib)
+ apply (force simp: algebra_simps)
+ done
+ then show ?thesis ..
+ qed
+ qed
+qed
+
+
+text\<open>Hence a nice clean inverse function theorem\<close>
+
+proposition holomorphic_has_inverse:
+ assumes holf: "f holomorphic_on S"
+ and "open S" and injf: "inj_on f S"
+ obtains g where "g holomorphic_on (f ` S)"
+ "\<And>z. z \<in> S \<Longrightarrow> deriv f z * deriv g (f z) = 1"
+ "\<And>z. z \<in> S \<Longrightarrow> g(f z) = z"
+proof -
+ have ofs: "open (f ` S)"
+ by (rule open_mapping_thm3 [OF assms])
+ have contf: "continuous_on S f"
+ by (simp add: holf holomorphic_on_imp_continuous_on)
+ have *: "(the_inv_into S f has_field_derivative inverse (deriv f z)) (at (f z))" if "z \<in> S" for z
+ proof -
+ have 1: "(f has_field_derivative deriv f z) (at z)"
+ using DERIV_deriv_iff_complex_differentiable \<open>z \<in> S\<close> \<open>open S\<close> holf holomorphic_on_imp_differentiable_at
+ by blast
+ have 2: "deriv f z \<noteq> 0"
+ using \<open>z \<in> S\<close> \<open>open S\<close> holf holomorphic_injective_imp_regular injf by blast
+ show ?thesis
+ apply (rule has_complex_derivative_inverse_strong [OF 1 2 \<open>open S\<close> \<open>z \<in> S\<close>])
+ apply (simp add: holf holomorphic_on_imp_continuous_on)
+ by (simp add: injf the_inv_into_f_f)
+ qed
+ show ?thesis
+ proof
+ show "the_inv_into S f holomorphic_on f ` S"
+ by (simp add: holomorphic_on_open ofs) (blast intro: *)
+ next
+ fix z assume "z \<in> S"
+ have "deriv f z \<noteq> 0"
+ using \<open>z \<in> S\<close> \<open>open S\<close> holf holomorphic_injective_imp_regular injf by blast
+ then show "deriv f z * deriv (the_inv_into S f) (f z) = 1"
+ using * [OF \<open>z \<in> S\<close>] by (simp add: DERIV_imp_deriv)
+ next
+ fix z assume "z \<in> S"
+ show "the_inv_into S f (f z) = z"
+ by (simp add: \<open>z \<in> S\<close> injf the_inv_into_f_f)
+ qed
+qed
+
+
+subsection\<open>The Schwarz Lemma\<close>
+
+lemma Schwarz1:
+ assumes holf: "f holomorphic_on S"
+ and contf: "continuous_on (closure S) f"
+ and S: "open S" "connected S"
+ and boS: "bounded S"
+ and "S \<noteq> {}"
+ obtains w where "w \<in> frontier S"
+ "\<And>z. z \<in> closure S \<Longrightarrow> norm (f z) \<le> norm (f w)"
+proof -
+ have connf: "continuous_on (closure S) (norm o f)"
+ using contf continuous_on_compose continuous_on_norm_id by blast
+ have coc: "compact (closure S)"
+ by (simp add: \<open>bounded S\<close> bounded_closure compact_eq_bounded_closed)
+ then obtain x where x: "x \<in> closure S" and xmax: "\<And>z. z \<in> closure S \<Longrightarrow> norm(f z) \<le> norm(f x)"
+ apply (rule bexE [OF continuous_attains_sup [OF _ _ connf]])
+ using \<open>S \<noteq> {}\<close> apply auto
+ done
+ then show ?thesis
+ proof (cases "x \<in> frontier S")
+ case True
+ then show ?thesis using that xmax by blast
+ next
+ case False
+ then have "x \<in> S"
+ using \<open>open S\<close> frontier_def interior_eq x by auto
+ then have "f constant_on S"
+ apply (rule maximum_modulus_principle [OF holf S \<open>open S\<close> order_refl])
+ using closure_subset apply (blast intro: xmax)
+ done
+ then have "f constant_on (closure S)"
+ by (rule constant_on_closureI [OF _ contf])
+ then obtain c where c: "\<And>x. x \<in> closure S \<Longrightarrow> f x = c"
+ by (meson constant_on_def)
+ obtain w where "w \<in> frontier S"
+ by (metis coc all_not_in_conv assms(6) closure_UNIV frontier_eq_empty not_compact_UNIV)
+ then show ?thesis
+ by (simp add: c frontier_def that)
+ qed
+qed
+
+lemma Schwarz2:
+ "\<lbrakk>f holomorphic_on ball 0 r;
+ 0 < s; ball w s \<subseteq> ball 0 r;
+ \<And>z. norm (w-z) < s \<Longrightarrow> norm(f z) \<le> norm(f w)\<rbrakk>
+ \<Longrightarrow> f constant_on ball 0 r"
+by (rule maximum_modulus_principle [where U = "ball w s" and \<xi> = w]) (simp_all add: dist_norm)
+
+lemma Schwarz3:
+ assumes holf: "f holomorphic_on (ball 0 r)" and [simp]: "f 0 = 0"
+ obtains h where "h holomorphic_on (ball 0 r)" and "\<And>z. norm z < r \<Longrightarrow> f z = z * (h z)" and "deriv f 0 = h 0"
+proof -
+ def h \<equiv> "\<lambda>z. if z = 0 then deriv f 0 else f z / z"
+ have d0: "deriv f 0 = h 0"
+ by (simp add: h_def)
+ moreover have "h holomorphic_on (ball 0 r)"
+ by (rule pole_theorem_open_0 [OF holf, of 0]) (auto simp: h_def)
+ moreover have "norm z < r \<Longrightarrow> f z = z * h z" for z
+ by (simp add: h_def)
+ ultimately show ?thesis
+ using that by blast
+qed
+
+
+proposition Schwarz_Lemma:
+ assumes holf: "f holomorphic_on (ball 0 1)" and [simp]: "f 0 = 0"
+ and no: "\<And>z. norm z < 1 \<Longrightarrow> norm (f z) < 1"
+ and \<xi>: "norm \<xi> < 1"
+ shows "norm (f \<xi>) \<le> norm \<xi>" and "norm(deriv f 0) \<le> 1"
+ and "((\<exists>z. norm z < 1 \<and> z \<noteq> 0 \<and> norm(f z) = norm z) \<or> norm(deriv f 0) = 1)
+ \<Longrightarrow> \<exists>\<alpha>. (\<forall>z. norm z < 1 \<longrightarrow> f z = \<alpha> * z) \<and> norm \<alpha> = 1" (is "?P \<Longrightarrow> ?Q")
+proof -
+ obtain h where holh: "h holomorphic_on (ball 0 1)"
+ and fz_eq: "\<And>z. norm z < 1 \<Longrightarrow> f z = z * (h z)" and df0: "deriv f 0 = h 0"
+ by (rule Schwarz3 [OF holf]) auto
+ have noh_le: "norm (h z) \<le> 1" if z: "norm z < 1" for z
+ proof -
+ have "norm (h z) < a" if a: "1 < a" for a
+ proof -
+ have "max (inverse a) (norm z) < 1"
+ using z a by (simp_all add: inverse_less_1_iff)
+ then obtain r where r: "max (inverse a) (norm z) < r" and "r < 1"
+ using Rats_dense_in_real by blast
+ then have nzr: "norm z < r" and ira: "inverse r < a"
+ using z a less_imp_inverse_less by force+
+ then have "0 < r"
+ by (meson norm_not_less_zero not_le order.strict_trans2)
+ have holh': "h holomorphic_on ball 0 r"
+ by (meson holh \<open>r < 1\<close> holomorphic_on_subset less_eq_real_def subset_ball)
+ have conth': "continuous_on (cball 0 r) h"
+ by (meson \<open>r < 1\<close> dual_order.trans holh holomorphic_on_imp_continuous_on holomorphic_on_subset mem_ball_0 mem_cball_0 not_less subsetI)
+ obtain w where w: "norm w = r" and lenw: "\<And>z. norm z < r \<Longrightarrow> norm(h z) \<le> norm(h w)"
+ apply (rule Schwarz1 [OF holh']) using conth' \<open>0 < r\<close> by auto
+ have "h w = f w / w" using fz_eq \<open>r < 1\<close> nzr w by auto
+ then have "cmod (h z) < inverse r"
+ by (metis \<open>0 < r\<close> \<open>r < 1\<close> divide_strict_right_mono inverse_eq_divide
+ le_less_trans lenw no norm_divide nzr w)
+ then show ?thesis using ira by linarith
+ qed
+ then show "norm (h z) \<le> 1"
+ using not_le by blast
+ qed
+ show "cmod (f \<xi>) \<le> cmod \<xi>"
+ proof (cases "\<xi> = 0")
+ case True then show ?thesis by auto
+ next
+ case False
+ then show ?thesis
+ by (simp add: noh_le fz_eq \<xi> mult_left_le norm_mult)
+ qed
+ show no_df0: "norm(deriv f 0) \<le> 1"
+ by (simp add: \<open>\<And>z. cmod z < 1 \<Longrightarrow> cmod (h z) \<le> 1\<close> df0)
+ show "?Q" if "?P"
+ using that
+ proof
+ assume "\<exists>z. cmod z < 1 \<and> z \<noteq> 0 \<and> cmod (f z) = cmod z"
+ then obtain \<gamma> where \<gamma>: "cmod \<gamma> < 1" "\<gamma> \<noteq> 0" "cmod (f \<gamma>) = cmod \<gamma>" by blast
+ then have [simp]: "norm (h \<gamma>) = 1"
+ by (simp add: fz_eq norm_mult)
+ have "ball \<gamma> (1 - cmod \<gamma>) \<subseteq> ball 0 1"
+ by (simp add: ball_subset_ball_iff)
+ moreover have "\<And>z. cmod (\<gamma> - z) < 1 - cmod \<gamma> \<Longrightarrow> cmod (h z) \<le> cmod (h \<gamma>)"
+ apply (simp add: algebra_simps)
+ by (metis add_diff_cancel_left' diff_diff_eq2 le_less_trans noh_le norm_triangle_ineq4)
+ ultimately obtain c where c: "\<And>z. norm z < 1 \<Longrightarrow> h z = c"
+ using Schwarz2 [OF holh, of "1 - norm \<gamma>" \<gamma>, unfolded constant_on_def] \<gamma> by auto
+ moreover then have "norm c = 1"
+ using \<gamma> by force
+ ultimately show ?thesis
+ using fz_eq by auto
+ next
+ assume [simp]: "cmod (deriv f 0) = 1"
+ then obtain c where c: "\<And>z. norm z < 1 \<Longrightarrow> h z = c"
+ using Schwarz2 [OF holh zero_less_one, of 0, unfolded constant_on_def] df0 noh_le
+ by auto
+ moreover have "norm c = 1" using df0 c by auto
+ ultimately show ?thesis
+ using fz_eq by auto
+ qed
+qed
+
+subsection\<open>The Schwarz reflection principle\<close>
+
+lemma hol_pal_lem0:
+ assumes "d \<bullet> a \<le> k" "k \<le> d \<bullet> b"
+ obtains c where
+ "c \<in> closed_segment a b" "d \<bullet> c = k"
+ "\<And>z. z \<in> closed_segment a c \<Longrightarrow> d \<bullet> z \<le> k"
+ "\<And>z. z \<in> closed_segment c b \<Longrightarrow> k \<le> d \<bullet> z"
+proof -
+ obtain c where cin: "c \<in> closed_segment a b" and keq: "k = d \<bullet> c"
+ using connected_ivt_hyperplane [of "closed_segment a b" a b d k]
+ by (auto simp: assms)
+ have "closed_segment a c \<subseteq> {z. d \<bullet> z \<le> k}" "closed_segment c b \<subseteq> {z. k \<le> d \<bullet> z}"
+ unfolding segment_convex_hull using assms keq
+ by (auto simp: convex_halfspace_le convex_halfspace_ge hull_minimal)
+ then show ?thesis using cin that by fastforce
+qed
+
+lemma hol_pal_lem1:
+ assumes "convex S" "open S"
+ and abc: "a \<in> S" "b \<in> S" "c \<in> S"
+ "d \<noteq> 0" and lek: "d \<bullet> a \<le> k" "d \<bullet> b \<le> k" "d \<bullet> c \<le> k"
+ and holf1: "f holomorphic_on {z. z \<in> S \<and> d \<bullet> z < k}"
+ and contf: "continuous_on S f"
+ shows "contour_integral (linepath a b) f +
+ contour_integral (linepath b c) f +
+ contour_integral (linepath c a) f = 0"
+proof -
+ have "interior (convex hull {a, b, c}) \<subseteq> interior(S \<inter> {x. d \<bullet> x \<le> k})"
+ apply (rule interior_mono)
+ apply (rule hull_minimal)
+ apply (simp add: abc lek)
+ apply (rule convex_Int [OF \<open>convex S\<close> convex_halfspace_le])
+ done
+ also have "... \<subseteq> {z \<in> S. d \<bullet> z < k}"
+ by (force simp: interior_open [OF \<open>open S\<close>] \<open>d \<noteq> 0\<close>)
+ finally have *: "interior (convex hull {a, b, c}) \<subseteq> {z \<in> S. d \<bullet> z < k}" .
+ have "continuous_on (convex hull {a,b,c}) f"
+ using \<open>convex S\<close> contf abc continuous_on_subset subset_hull
+ by fastforce
+ moreover have "f holomorphic_on interior (convex hull {a,b,c})"
+ by (rule holomorphic_on_subset [OF holf1 *])
+ ultimately show ?thesis
+ using Cauchy_theorem_triangle_interior has_chain_integral_chain_integral3
+ by blast
+qed
+
+lemma hol_pal_lem2:
+ assumes S: "convex S" "open S"
+ and abc: "a \<in> S" "b \<in> S" "c \<in> S"
+ and "d \<noteq> 0" and lek: "d \<bullet> a \<le> k" "d \<bullet> b \<le> k"
+ and holf1: "f holomorphic_on {z. z \<in> S \<and> d \<bullet> z < k}"
+ and holf2: "f holomorphic_on {z. z \<in> S \<and> k < d \<bullet> z}"
+ and contf: "continuous_on S f"
+ shows "contour_integral (linepath a b) f +
+ contour_integral (linepath b c) f +
+ contour_integral (linepath c a) f = 0"
+proof (cases "d \<bullet> c \<le> k")
+ case True show ?thesis
+ by (rule hol_pal_lem1 [OF S abc \<open>d \<noteq> 0\<close> lek True holf1 contf])
+next
+ case False
+ then have "d \<bullet> c > k" by force
+ obtain a' where a': "a' \<in> closed_segment b c" and "d \<bullet> a' = k"
+ and ba': "\<And>z. z \<in> closed_segment b a' \<Longrightarrow> d \<bullet> z \<le> k"
+ and a'c: "\<And>z. z \<in> closed_segment a' c \<Longrightarrow> k \<le> d \<bullet> z"
+ apply (rule hol_pal_lem0 [of d b k c, OF \<open>d \<bullet> b \<le> k\<close>])
+ using False by auto
+ obtain b' where b': "b' \<in> closed_segment a c" and "d \<bullet> b' = k"
+ and ab': "\<And>z. z \<in> closed_segment a b' \<Longrightarrow> d \<bullet> z \<le> k"
+ and b'c: "\<And>z. z \<in> closed_segment b' c \<Longrightarrow> k \<le> d \<bullet> z"
+ apply (rule hol_pal_lem0 [of d a k c, OF \<open>d \<bullet> a \<le> k\<close>])
+ using False by auto
+ have a'b': "a' \<in> S \<and> b' \<in> S"
+ using a' abc b' convex_contains_segment \<open>convex S\<close> by auto
+ have "continuous_on (closed_segment c a) f"
+ by (meson abc contf continuous_on_subset convex_contains_segment \<open>convex S\<close>)
+ then have 1: "contour_integral (linepath c a) f =
+ contour_integral (linepath c b') f + contour_integral (linepath b' a) f"
+ apply (rule contour_integral_split_linepath)
+ using b' by (simp add: closed_segment_commute)
+ have "continuous_on (closed_segment b c) f"
+ by (meson abc contf continuous_on_subset convex_contains_segment \<open>convex S\<close>)
+ then have 2: "contour_integral (linepath b c) f =
+ contour_integral (linepath b a') f + contour_integral (linepath a' c) f"
+ by (rule contour_integral_split_linepath [OF _ a'])
+ have "f contour_integrable_on linepath b' a'"
+ by (meson a'b' contf continuous_on_subset contour_integrable_continuous_linepath
+ convex_contains_segment \<open>convex S\<close>)
+ then have 3: "contour_integral (reversepath (linepath b' a')) f =
+ - contour_integral (linepath b' a') f"
+ by (rule contour_integral_reversepath [OF valid_path_linepath])
+ have fcd_le: "f complex_differentiable at x"
+ if "x \<in> interior S \<and> x \<in> interior {x. d \<bullet> x \<le> k}" for x
+ proof -
+ have "f holomorphic_on S \<inter> {c. d \<bullet> c < k}"
+ by (metis (no_types) Collect_conj_eq Collect_mem_eq holf1)
+ then have "\<exists>C D. x \<in> interior C \<inter> interior D \<and> f holomorphic_on interior C \<inter> interior D"
+ using that
+ by (metis Collect_mem_eq Int_Collect \<open>d \<noteq> 0\<close> interior_halfspace_le interior_open \<open>open S\<close>)
+ then show "f complex_differentiable at x"
+ by (metis at_within_interior holomorphic_on_def interior_Int interior_interior)
+ qed
+ have ab_le: "\<And>x. x \<in> closed_segment a b \<Longrightarrow> d \<bullet> x \<le> k"
+ proof -
+ fix x :: complex
+ assume "x \<in> closed_segment a b"
+ then have "\<And>C. x \<in> C \<or> b \<notin> C \<or> a \<notin> C \<or> \<not> convex C"
+ by (meson contra_subsetD convex_contains_segment)
+ then show "d \<bullet> x \<le> k"
+ by (metis lek convex_halfspace_le mem_Collect_eq)
+ qed
+ have "continuous_on (S \<inter> {x. d \<bullet> x \<le> k}) f" using contf
+ by (simp add: continuous_on_subset)
+ then have "(f has_contour_integral 0)
+ (linepath a b +++ linepath b a' +++ linepath a' b' +++ linepath b' a)"
+ apply (rule Cauchy_theorem_convex [where k = "{}"])
+ apply (simp_all add: path_image_join convex_Int convex_halfspace_le \<open>convex S\<close> fcd_le ab_le
+ closed_segment_subset abc a'b' ba')
+ by (metis \<open>d \<bullet> a' = k\<close> \<open>d \<bullet> b' = k\<close> convex_contains_segment convex_halfspace_le lek(1) mem_Collect_eq order_refl)
+ then have 4: "contour_integral (linepath a b) f +
+ contour_integral (linepath b a') f +
+ contour_integral (linepath a' b') f +
+ contour_integral (linepath b' a) f = 0"
+ by (rule has_chain_integral_chain_integral4)
+ have fcd_ge: "f complex_differentiable at x"
+ if "x \<in> interior S \<and> x \<in> interior {x. k \<le> d \<bullet> x}" for x
+ proof -
+ have f2: "f holomorphic_on S \<inter> {c. k < d \<bullet> c}"
+ by (metis (full_types) Collect_conj_eq Collect_mem_eq holf2)
+ have f3: "interior S = S"
+ by (simp add: interior_open \<open>open S\<close>)
+ then have "x \<in> S \<inter> interior {c. k \<le> d \<bullet> c}"
+ using that by simp
+ then show "f complex_differentiable at x"
+ using f3 f2 unfolding holomorphic_on_def
+ by (metis (no_types) \<open>d \<noteq> 0\<close> at_within_interior interior_Int interior_halfspace_ge interior_interior)
+ qed
+ have "continuous_on (S \<inter> {x. k \<le> d \<bullet> x}) f" using contf
+ by (simp add: continuous_on_subset)
+ then have "(f has_contour_integral 0) (linepath a' c +++ linepath c b' +++ linepath b' a')"
+ apply (rule Cauchy_theorem_convex [where k = "{}"])
+ apply (simp_all add: path_image_join convex_Int convex_halfspace_ge \<open>convex S\<close>
+ fcd_ge closed_segment_subset abc a'b' a'c)
+ by (metis \<open>d \<bullet> a' = k\<close> b'c closed_segment_commute convex_contains_segment
+ convex_halfspace_ge ends_in_segment(2) mem_Collect_eq order_refl)
+ then have 5: "contour_integral (linepath a' c) f + contour_integral (linepath c b') f + contour_integral (linepath b' a') f = 0"
+ by (rule has_chain_integral_chain_integral3)
+ show ?thesis
+ using 1 2 3 4 5 by (metis add.assoc eq_neg_iff_add_eq_0 reversepath_linepath)
+qed
+
+lemma hol_pal_lem3:
+ assumes S: "convex S" "open S"
+ and abc: "a \<in> S" "b \<in> S" "c \<in> S"
+ and "d \<noteq> 0" and lek: "d \<bullet> a \<le> k"
+ and holf1: "f holomorphic_on {z. z \<in> S \<and> d \<bullet> z < k}"
+ and holf2: "f holomorphic_on {z. z \<in> S \<and> k < d \<bullet> z}"
+ and contf: "continuous_on S f"
+ shows "contour_integral (linepath a b) f +
+ contour_integral (linepath b c) f +
+ contour_integral (linepath c a) f = 0"
+proof (cases "d \<bullet> b \<le> k")
+ case True show ?thesis
+ by (rule hol_pal_lem2 [OF S abc \<open>d \<noteq> 0\<close> lek True holf1 holf2 contf])
+next
+ case False
+ show ?thesis
+ proof (cases "d \<bullet> c \<le> k")
+ case True
+ have "contour_integral (linepath c a) f +
+ contour_integral (linepath a b) f +
+ contour_integral (linepath b c) f = 0"
+ by (rule hol_pal_lem2 [OF S \<open>c \<in> S\<close> \<open>a \<in> S\<close> \<open>b \<in> S\<close> \<open>d \<noteq> 0\<close> \<open>d \<bullet> c \<le> k\<close> lek holf1 holf2 contf])
+ then show ?thesis
+ by (simp add: algebra_simps)
+ next
+ case False
+ have "contour_integral (linepath b c) f +
+ contour_integral (linepath c a) f +
+ contour_integral (linepath a b) f = 0"
+ apply (rule hol_pal_lem2 [OF S \<open>b \<in> S\<close> \<open>c \<in> S\<close> \<open>a \<in> S\<close>, of "-d" "-k"])
+ using \<open>d \<noteq> 0\<close> \<open>\<not> d \<bullet> b \<le> k\<close> False by (simp_all add: holf1 holf2 contf)
+ then show ?thesis
+ by (simp add: algebra_simps)
+ qed
+qed
+
+lemma hol_pal_lem4:
+ assumes S: "convex S" "open S"
+ and abc: "a \<in> S" "b \<in> S" "c \<in> S" and "d \<noteq> 0"
+ and holf1: "f holomorphic_on {z. z \<in> S \<and> d \<bullet> z < k}"
+ and holf2: "f holomorphic_on {z. z \<in> S \<and> k < d \<bullet> z}"
+ and contf: "continuous_on S f"
+ shows "contour_integral (linepath a b) f +
+ contour_integral (linepath b c) f +
+ contour_integral (linepath c a) f = 0"
+proof (cases "d \<bullet> a \<le> k")
+ case True show ?thesis
+ by (rule hol_pal_lem3 [OF S abc \<open>d \<noteq> 0\<close> True holf1 holf2 contf])
+next
+ case False
+ show ?thesis
+ apply (rule hol_pal_lem3 [OF S abc, of "-d" "-k"])
+ using \<open>d \<noteq> 0\<close> False by (simp_all add: holf1 holf2 contf)
+qed
+
+proposition holomorphic_on_paste_across_line:
+ assumes S: "open S" and "d \<noteq> 0"
+ and holf1: "f holomorphic_on (S \<inter> {z. d \<bullet> z < k})"
+ and holf2: "f holomorphic_on (S \<inter> {z. k < d \<bullet> z})"
+ and contf: "continuous_on S f"
+ shows "f holomorphic_on S"
+proof -
+ have *: "\<exists>t. open t \<and> p \<in> t \<and> continuous_on t f \<and>
+ (\<forall>a b c. convex hull {a, b, c} \<subseteq> t \<longrightarrow>
+ contour_integral (linepath a b) f +
+ contour_integral (linepath b c) f +
+ contour_integral (linepath c a) f = 0)"
+ if "p \<in> S" for p
+ proof -
+ obtain e where "e>0" and e: "ball p e \<subseteq> S"
+ using \<open>p \<in> S\<close> openE S by blast
+ then have "continuous_on (ball p e) f"
+ using contf continuous_on_subset by blast
+ moreover have "f holomorphic_on {z. dist p z < e \<and> d \<bullet> z < k}"
+ apply (rule holomorphic_on_subset [OF holf1])
+ using e by auto
+ moreover have "f holomorphic_on {z. dist p z < e \<and> k < d \<bullet> z}"
+ apply (rule holomorphic_on_subset [OF holf2])
+ using e by auto
+ ultimately show ?thesis
+ apply (rule_tac x="ball p e" in exI)
+ using \<open>e > 0\<close> e \<open>d \<noteq> 0\<close>
+ apply (simp add:, clarify)
+ apply (rule hol_pal_lem4 [of "ball p e" _ _ _ d _ k])
+ apply (auto simp: subset_hull)
+ done
+ qed
+ show ?thesis
+ by (blast intro: * Morera_local_triangle analytic_imp_holomorphic)
+qed
+
+proposition Schwarz_reflection:
+ assumes "open S" and cnjs: "cnj ` S \<subseteq> S"
+ and holf: "f holomorphic_on (S \<inter> {z. 0 < Im z})"
+ and contf: "continuous_on (S \<inter> {z. 0 \<le> Im z}) f"
+ and f: "\<And>z. \<lbrakk>z \<in> S; z \<in> \<real>\<rbrakk> \<Longrightarrow> (f z) \<in> \<real>"
+ shows "(\<lambda>z. if 0 \<le> Im z then f z else cnj(f(cnj z))) holomorphic_on S"
+proof -
+ have 1: "(\<lambda>z. if 0 \<le> Im z then f z else cnj (f (cnj z))) holomorphic_on (S \<inter> {z. 0 < Im z})"
+ by (force intro: iffD1 [OF holomorphic_cong [OF refl] holf])
+ have cont_cfc: "continuous_on (S \<inter> {z. Im z \<le> 0}) (cnj o f o cnj)"
+ apply (intro continuous_intros continuous_on_compose continuous_on_subset [OF contf])
+ using cnjs apply auto
+ done
+ have "cnj \<circ> f \<circ> cnj complex_differentiable at x within S \<inter> {z. Im z < 0}"
+ if "x \<in> S" "Im x < 0" "f complex_differentiable at (cnj x) within S \<inter> {z. 0 < Im z}" for x
+ using that
+ apply (simp add: complex_differentiable_def Derivative.DERIV_within_iff Lim_within dist_norm, clarify)
+ apply (rule_tac x="cnj f'" in exI)
+ apply (elim all_forward ex_forward conj_forward imp_forward asm_rl, clarify)
+ apply (drule_tac x="cnj xa" in bspec)
+ using cnjs apply force
+ apply (metis complex_cnj_cnj complex_cnj_diff complex_cnj_divide complex_mod_cnj)
+ done
+ then have hol_cfc: "(cnj o f o cnj) holomorphic_on (S \<inter> {z. Im z < 0})"
+ using holf cnjs
+ by (force simp: holomorphic_on_def)
+ have 2: "(\<lambda>z. if 0 \<le> Im z then f z else cnj (f (cnj z))) holomorphic_on (S \<inter> {z. Im z < 0})"
+ apply (rule iffD1 [OF holomorphic_cong [OF refl]])
+ using hol_cfc by auto
+ have [simp]: "(S \<inter> {z. 0 \<le> Im z}) \<union> (S \<inter> {z. Im z \<le> 0}) = S"
+ by force
+ have "continuous_on ((S \<inter> {z. 0 \<le> Im z}) \<union> (S \<inter> {z. Im z \<le> 0}))
+ (\<lambda>z. if 0 \<le> Im z then f z else cnj (f (cnj z)))"
+ apply (rule continuous_on_cases_local)
+ using cont_cfc contf
+ apply (simp_all add: closedin_closed_Int closed_halfspace_Im_le closed_halfspace_Im_ge)
+ using f Reals_cnj_iff complex_is_Real_iff apply auto
+ done
+ then have 3: "continuous_on S (\<lambda>z. if 0 \<le> Im z then f z else cnj (f (cnj z)))"
+ by force
+ show ?thesis
+ apply (rule holomorphic_on_paste_across_line [OF \<open>open S\<close>, of "-ii" _ 0])
+ using 1 2 3
+ apply auto
+ done
+qed
+
+end