# HG changeset patch
# User paulson <lp15@cam.ac.uk>
# Date 1456408728 0
# Node ID 86f27b264d3da58b3bda4aeee579b1e042d3a002
# Parent  8383b126b0a9e868629d2e85da6b7364aa299dad
Conformal_mappings: a big development in complex analysis (+ some lemmas)

diff -r 8383b126b0a9 -r 86f27b264d3d NEWS
--- a/NEWS	Thu Feb 25 00:36:44 2016 +0100
+++ b/NEWS	Thu Feb 25 13:58:48 2016 +0000
@@ -38,6 +38,9 @@
 * Compound constants INFIMUM and SUPREMUM are mere abbreviations now.
 INCOMPATIBILITY.
 
+* More complex analysis including Cauchy's inequality, Liouville theorem,
+open mapping theorem, maximum modulus principle, Schwarz Lemma.
+
 * "Gcd (f ` A)" and "Lcm (f ` A)" are printed with optional
 comprehension-like syntax analogously to "Inf (f ` A)" and "Sup (f ` A)".
 
diff -r 8383b126b0a9 -r 86f27b264d3d src/HOL/Multivariate_Analysis/Cauchy_Integral_Thm.thy
--- a/src/HOL/Multivariate_Analysis/Cauchy_Integral_Thm.thy	Thu Feb 25 00:36:44 2016 +0100
+++ b/src/HOL/Multivariate_Analysis/Cauchy_Integral_Thm.thy	Thu Feb 25 13:58:48 2016 +0000
@@ -197,6 +197,24 @@
 
 subsubsection\<open>The concept of continuously differentiable\<close>
 
+text \<open>
+John Harrison writes as follows:
+
+``The usual assumption in complex analysis texts is that a path \<gamma> should be piecewise
+continuously differentiable, which ensures that the path integral exists at least for any continuous
+f, since all piecewise continuous functions are integrable. However, our notion of validity is
+weaker, just piecewise differentiability... [namely] continuity plus differentiability except on a
+finite set ... [Our] underlying theory of integration is the Kurzweil-Henstock theory. In contrast to
+the Riemann or Lebesgue theory (but in common with a simple notion based on antiderivatives), this
+can integrate all derivatives.''
+
+"Formalizing basic complex analysis." From Insight to Proof: Festschrift in Honour of Andrzej Trybulec. 
+Studies in Logic, Grammar and Rhetoric 10.23 (2007): 151-165.
+
+And indeed he does not assume that his derivatives are continuous, but the penalty is unreasonably
+difficult proofs concerning winding numbers. We need a self-contained and straightforward theorem
+asserting that all derivatives can be integrated before we can adopt Harrison's choice.\<close>
+
 definition C1_differentiable_on :: "(real \<Rightarrow> 'a::real_normed_vector) \<Rightarrow> real set \<Rightarrow> bool"
            (infix "C1'_differentiable'_on" 50)
   where
@@ -549,6 +567,63 @@
 lemma closed_valid_path_image: "valid_path g \<Longrightarrow> closed(path_image g)"
   by (metis closed_path_image valid_path_imp_path)
 
+lemma valid_path_compose:
+  assumes "valid_path g"
+      and der:"\<And>x. x \<in> path_image g \<Longrightarrow> \<exists>f'. (f has_field_derivative f') (at x)"
+      and con: "continuous_on {0..1} (\<lambda>x. deriv f (g x))"
+    shows "valid_path (f o g)"
+proof -
+  obtain s where "finite s" and g_diff: "g C1_differentiable_on {0..1} - s"
+    using `valid_path g` unfolding valid_path_def piecewise_C1_differentiable_on_def by auto
+  have "f \<circ> g differentiable at t" if "t \<in> {0..1} - s" for t
+    proof (rule differentiable_chain_at)
+      show "g differentiable at t" using `valid_path g`
+        by (meson C1_differentiable_on_eq \<open>g C1_differentiable_on {0..1} - s\<close> that)
+    next
+      have "g t\<in>path_image g" using that DiffD1 image_eqI path_image_def by metis
+      then obtain f' where "(f has_field_derivative f') (at (g t))"
+        using der by auto
+      then have " (f has_derivative op * f') (at (g t))"
+        using has_field_derivative_imp_has_derivative[of f f' "at (g t)"] by auto
+      then show "f differentiable at (g t)" using differentiableI by auto
+    qed
+  moreover have "continuous_on ({0..1} - s) (\<lambda>x. vector_derivative (f \<circ> g) (at x))"
+  proof (rule continuous_on_eq [where f = "\<lambda>x. vector_derivative g (at x) * deriv f (g x)"],
+      rule continuous_intros)
+    show "continuous_on ({0..1} - s) (\<lambda>x. vector_derivative g (at x))"
+      using g_diff C1_differentiable_on_eq by auto
+  next
+    show "continuous_on ({0..1} - s) (\<lambda>x. deriv f (g x))"
+      using con continuous_on_subset by blast
+  next
+    show "vector_derivative g (at t) * deriv f (g t) = vector_derivative (f \<circ> g) (at t)"
+         if "t \<in> {0..1} - s" for t
+    proof (rule vector_derivative_chain_at_general[symmetric])
+      show "g differentiable at t" by (meson C1_differentiable_on_eq g_diff that)
+    next
+      have "g t\<in>path_image g" using that DiffD1 image_eqI path_image_def by metis
+      then obtain f' where "(f has_field_derivative f') (at (g t))"
+        using der by auto
+      then show "\<exists>g'. (f has_field_derivative g') (at (g t))" by auto
+    qed
+  qed
+  ultimately have "f o g C1_differentiable_on {0..1} - s"
+    using C1_differentiable_on_eq by blast
+  moreover have "path (f o g)"
+  proof -
+    have "isCont f x" if "x \<in> path_image g" for x
+      proof -
+        obtain f' where "(f has_field_derivative f') (at x)"
+          using der `x\<in>path_image g` by auto
+        thus ?thesis using DERIV_isCont by auto
+      qed
+    then have "continuous_on (path_image g) f" using continuous_at_imp_continuous_on by auto
+    then show ?thesis using path_continuous_image `valid_path g` valid_path_imp_path by auto
+  qed
+  ultimately show ?thesis unfolding valid_path_def piecewise_C1_differentiable_on_def path_def
+    using `finite s` by auto
+qed
+
 
 subsection\<open>Contour Integrals along a path\<close>
 
diff -r 8383b126b0a9 -r 86f27b264d3d src/HOL/Multivariate_Analysis/Complex_Analysis_Basics.thy
--- a/src/HOL/Multivariate_Analysis/Complex_Analysis_Basics.thy	Thu Feb 25 00:36:44 2016 +0100
+++ b/src/HOL/Multivariate_Analysis/Complex_Analysis_Basics.thy	Thu Feb 25 13:58:48 2016 +0000
@@ -393,7 +393,7 @@
   unfolding complex_differentiable_def
   by (metis at_within_open)
 
-subsection\<open>Caratheodory characterization.\<close>
+subsection\<open>Caratheodory characterization\<close>
 
 lemma complex_differentiable_caratheodory_at:
   "f complex_differentiable (at z) \<longleftrightarrow>
@@ -750,7 +750,6 @@
     by (metis e e' min_less_iff_conj)
 qed
 
-
 lemma analytic_on_divide:
   assumes f: "f analytic_on s"
       and g: "g analytic_on s"
@@ -767,7 +766,30 @@
   "(\<And>i. i \<in> I \<Longrightarrow> (f i) analytic_on s) \<Longrightarrow> (\<lambda>x. setsum (\<lambda>i. f i x) I) analytic_on s"
   by (induct I rule: infinite_finite_induct) (auto simp: analytic_on_const analytic_on_add)
 
-subsection\<open>analyticity at a point.\<close>
+lemma deriv_left_inverse:
+  assumes "f holomorphic_on S" and "g holomorphic_on T"
+      and "open S" and "open T"
+      and "f ` S \<subseteq> T"
+      and [simp]: "\<And>z. z \<in> S \<Longrightarrow> g (f z) = z"
+      and "w \<in> S"
+    shows "deriv f w * deriv g (f w) = 1"
+proof -
+  have "deriv f w * deriv g (f w) = deriv g (f w) * deriv f w"
+    by (simp add: algebra_simps)
+  also have "... = deriv (g o f) w"
+    using assms
+    by (metis analytic_on_imp_differentiable_at analytic_on_open complex_derivative_chain image_subset_iff)
+  also have "... = deriv id w"
+    apply (rule complex_derivative_transform_within_open [where s=S])
+    apply (rule assms holomorphic_on_compose_gen holomorphic_intros)+
+    apply simp
+    done
+  also have "... = 1"
+    by simp
+  finally show ?thesis .
+qed
+
+subsection\<open>analyticity at a point\<close>
 
 lemma analytic_at_ball:
   "f analytic_on {z} \<longleftrightarrow> (\<exists>e. 0<e \<and> f holomorphic_on ball z e)"
@@ -972,7 +994,7 @@
   apply fact
   done
 
-subsection\<open>Inverse function theorem for complex derivatives.\<close>
+subsection\<open>Inverse function theorem for complex derivatives\<close>
 
 lemma has_complex_derivative_inverse_basic:
   fixes f :: "complex \<Rightarrow> complex"
@@ -1118,7 +1140,7 @@
   finally show ?thesis .
 qed
 
-text\<open>Something more like the traditional MVT for real components.\<close>
+text\<open>Something more like the traditional MVT for real components\<close>
 
 lemma complex_mvt_line:
   assumes "\<And>u. u \<in> closed_segment w z \<Longrightarrow> (f has_field_derivative f'(u)) (at u)"
diff -r 8383b126b0a9 -r 86f27b264d3d src/HOL/Multivariate_Analysis/Conformal_Mappings.thy
--- /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
diff -r 8383b126b0a9 -r 86f27b264d3d src/HOL/Multivariate_Analysis/Derivative.thy
--- a/src/HOL/Multivariate_Analysis/Derivative.thy	Thu Feb 25 00:36:44 2016 +0100
+++ b/src/HOL/Multivariate_Analysis/Derivative.thy	Thu Feb 25 13:58:48 2016 +0000
@@ -2506,6 +2506,22 @@
          vector_derivative f (at x) *\<^sub>R vector_derivative g (at (f x))"
 by (metis vector_diff_chain_at vector_derivative_at vector_derivative_works assms)
 
+lemma field_vector_diff_chain_at:  (*thanks to Wenda Li*)
+ assumes Df: "(f has_vector_derivative f') (at x)"
+     and Dg: "(g has_field_derivative g') (at (f x))"
+ shows "((g \<circ> f) has_vector_derivative (f' * g')) (at x)"
+using diff_chain_at[OF Df[unfolded has_vector_derivative_def]
+                       Dg [unfolded has_field_derivative_def]]
+ by (auto simp: o_def mult.commute has_vector_derivative_def)
+
+lemma vector_derivative_chain_at_general:  (*thanks to Wenda Li*)
+ assumes "f differentiable at x" "(\<exists>g'. (g has_field_derivative g') (at (f x)))"
+ shows "vector_derivative (g \<circ> f) (at x) =
+        vector_derivative f (at x) * deriv g (f x)"
+apply (rule vector_derivative_at [OF field_vector_diff_chain_at])
+using assms
+by (auto simp: vector_derivative_works DERIV_deriv_iff_has_field_derivative)
+
 
 subsection \<open>Relation between convexity and derivative\<close>
 
diff -r 8383b126b0a9 -r 86f27b264d3d src/HOL/Multivariate_Analysis/Determinants.thy
--- a/src/HOL/Multivariate_Analysis/Determinants.thy	Thu Feb 25 00:36:44 2016 +0100
+++ b/src/HOL/Multivariate_Analysis/Determinants.thy	Thu Feb 25 13:58:48 2016 +0000
@@ -577,7 +577,7 @@
 qed
 
 
-lemma eq_id_iff[simp]: "(\<forall>x. f x = x) \<longleftrightarrow> f = id"
+lemma eq_id_iff: "(\<forall>x. f x = x) \<longleftrightarrow> f = id"
   by auto
 
 lemma det_linear_rows_setsum_lemma:
@@ -592,7 +592,7 @@
   have th0: "\<And>x y. (\<chi> i. if i \<in> {} then x i else y i) = (\<chi> i. y i)"
     by vector
   from empty.prems show ?case
-    unfolding th0 by simp
+    unfolding th0 by (simp add: eq_id_iff)
 next
   case (insert z T a c)
   let ?F = "\<lambda>T. {f. (\<forall>i \<in> T. f i \<in> S) \<and> (\<forall>i. i \<notin> T \<longrightarrow> f i = i)}"
diff -r 8383b126b0a9 -r 86f27b264d3d src/HOL/Multivariate_Analysis/Multivariate_Analysis.thy
--- a/src/HOL/Multivariate_Analysis/Multivariate_Analysis.thy	Thu Feb 25 00:36:44 2016 +0100
+++ b/src/HOL/Multivariate_Analysis/Multivariate_Analysis.thy	Thu Feb 25 13:58:48 2016 +0000
@@ -4,10 +4,9 @@
   Extended_Real_Limits
   Determinants
   Ordered_Euclidean_Space
-  Complex_Analysis_Basics
   Bounded_Continuous_Function
   Weierstrass
-  Cauchy_Integral_Thm
+  Conformal_Mappings
   Generalised_Binomial_Theorem
   Gamma
 begin