Reorganised HOL-Complex_Analysis
authorManuel Eberl <eberlm@in.tum.de>
Mon, 02 Dec 2019 17:51:54 +0100
changeset 71403 6617fb368a06
parent 71402 3548d54ce3ee
child 71404 fc57ee24deac
child 71422 35e465677a26
Reorganised HOL-Complex_Analysis
src/HOL/Complex_Analysis/Cauchy_Integral_Formula.thy
src/HOL/Complex_Analysis/Cauchy_Integral_Theorem.thy
src/HOL/Complex_Analysis/Complex_Analysis.thy
src/HOL/Complex_Analysis/Complex_Residues.thy
src/HOL/Complex_Analysis/Complex_Singularities.thy
src/HOL/Complex_Analysis/Conformal_Mappings.thy
src/HOL/Complex_Analysis/Contour_Integration.thy
src/HOL/Complex_Analysis/Great_Picard.thy
src/HOL/Complex_Analysis/Residue_Theorem.thy
src/HOL/Complex_Analysis/Riemann_Mapping.thy
src/HOL/Complex_Analysis/Winding_Numbers.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Complex_Analysis/Cauchy_Integral_Formula.thy	Mon Dec 02 17:51:54 2019 +0100
@@ -0,0 +1,2787 @@
+section \<open>Cauchy's Integral Formula\<close>
+theory Cauchy_Integral_Formula
+  imports Winding_Numbers
+begin
+
+subsection\<open>Proof\<close>
+
+lemma Cauchy_integral_formula_weak:
+    assumes s: "convex s" and "finite k" and conf: "continuous_on s f"
+        and fcd: "(\<And>x. x \<in> interior s - k \<Longrightarrow> f field_differentiable at x)"
+        and z: "z \<in> interior s - k" and vpg: "valid_path \<gamma>"
+        and pasz: "path_image \<gamma> \<subseteq> s - {z}" and loop: "pathfinish \<gamma> = pathstart \<gamma>"
+      shows "((\<lambda>w. f w / (w - z)) has_contour_integral (2*pi * \<i> * winding_number \<gamma> z * f z)) \<gamma>"
+proof -
+  obtain f' where f': "(f has_field_derivative f') (at z)"
+    using fcd [OF z] by (auto simp: field_differentiable_def)
+  have pas: "path_image \<gamma> \<subseteq> s" and znotin: "z \<notin> path_image \<gamma>" using pasz by blast+
+  have c: "continuous (at x within s) (\<lambda>w. if w = z then f' else (f w - f z) / (w - z))" if "x \<in> s" for x
+  proof (cases "x = z")
+    case True then show ?thesis
+      apply (simp add: continuous_within)
+      apply (rule Lim_transform_away_within [of _ "z+1" _ "\<lambda>w::complex. (f w - f z)/(w - z)"])
+      using has_field_derivative_at_within has_field_derivative_iff f'
+      apply (fastforce simp add:)+
+      done
+  next
+    case False
+    then have dxz: "dist x z > 0" by auto
+    have cf: "continuous (at x within s) f"
+      using conf continuous_on_eq_continuous_within that by blast
+    have "continuous (at x within s) (\<lambda>w. (f w - f z) / (w - z))"
+      by (rule cf continuous_intros | simp add: False)+
+    then show ?thesis
+      apply (rule continuous_transform_within [OF _ dxz that, of "\<lambda>w::complex. (f w - f z)/(w - z)"])
+      apply (force simp: dist_commute)
+      done
+  qed
+  have fink': "finite (insert z k)" using \<open>finite k\<close> by blast
+  have *: "((\<lambda>w. if w = z then f' else (f w - f z) / (w - z)) has_contour_integral 0) \<gamma>"
+    apply (rule Cauchy_theorem_convex [OF _ s fink' _ vpg pas loop])
+    using c apply (force simp: continuous_on_eq_continuous_within)
+    apply (rename_tac w)
+    apply (rule_tac d="dist w z" and f = "\<lambda>w. (f w - f z)/(w - z)" in field_differentiable_transform_within)
+    apply (simp_all add: dist_pos_lt dist_commute)
+    apply (metis less_irrefl)
+    apply (rule derivative_intros fcd | simp)+
+    done
+  show ?thesis
+    apply (rule has_contour_integral_eq)
+    using znotin has_contour_integral_add [OF has_contour_integral_lmul [OF has_contour_integral_winding_number [OF vpg znotin], of "f z"] *]
+    apply (auto simp: ac_simps divide_simps)
+    done
+qed
+
+theorem Cauchy_integral_formula_convex_simple:
+    "\<lbrakk>convex s; f holomorphic_on s; z \<in> interior s; valid_path \<gamma>; path_image \<gamma> \<subseteq> s - {z};
+      pathfinish \<gamma> = pathstart \<gamma>\<rbrakk>
+     \<Longrightarrow> ((\<lambda>w. f w / (w - z)) has_contour_integral (2*pi * \<i> * winding_number \<gamma> z * f z)) \<gamma>"
+  apply (rule Cauchy_integral_formula_weak [where k = "{}"])
+  using holomorphic_on_imp_continuous_on
+  by auto (metis at_within_interior holomorphic_on_def interiorE subsetCE)
+
+text\<open> Hence the Cauchy formula for points inside a circle.\<close>
+
+theorem Cauchy_integral_circlepath:
+  assumes contf: "continuous_on (cball z r) f" and holf: "f holomorphic_on (ball z r)" and wz: "norm(w - z) < r"
+  shows "((\<lambda>u. f u/(u - w)) has_contour_integral (2 * of_real pi * \<i> * f w))
+         (circlepath z r)"
+proof -
+  have "r > 0"
+    using assms le_less_trans norm_ge_zero by blast
+  have "((\<lambda>u. f u / (u - w)) has_contour_integral (2 * pi) * \<i> * winding_number (circlepath z r) w * f w)
+        (circlepath z r)"
+  proof (rule Cauchy_integral_formula_weak [where s = "cball z r" and k = "{}"])
+    show "\<And>x. x \<in> interior (cball z r) - {} \<Longrightarrow>
+         f field_differentiable at x"
+      using holf holomorphic_on_imp_differentiable_at by auto
+    have "w \<notin> sphere z r"
+      by simp (metis dist_commute dist_norm not_le order_refl wz)
+    then show "path_image (circlepath z r) \<subseteq> cball z r - {w}"
+      using \<open>r > 0\<close> by (auto simp add: cball_def sphere_def)
+  qed (use wz in \<open>simp_all add: dist_norm norm_minus_commute contf\<close>)
+  then show ?thesis
+    by (simp add: winding_number_circlepath assms)
+qed
+
+corollary\<^marker>\<open>tag unimportant\<close> Cauchy_integral_circlepath_simple:
+  assumes "f holomorphic_on cball z r" "norm(w - z) < r"
+  shows "((\<lambda>u. f u/(u - w)) has_contour_integral (2 * of_real pi * \<i> * f w))
+         (circlepath z r)"
+using assms by (force simp: holomorphic_on_imp_continuous_on holomorphic_on_subset Cauchy_integral_circlepath)
+
+subsection\<^marker>\<open>tag unimportant\<close> \<open>General stepping result for derivative formulas\<close>
+
+lemma Cauchy_next_derivative:
+  assumes "continuous_on (path_image \<gamma>) f'"
+      and leB: "\<And>t. t \<in> {0..1} \<Longrightarrow> norm (vector_derivative \<gamma> (at t)) \<le> B"
+      and int: "\<And>w. w \<in> s - path_image \<gamma> \<Longrightarrow> ((\<lambda>u. f' u / (u - w)^k) has_contour_integral f w) \<gamma>"
+      and k: "k \<noteq> 0"
+      and "open s"
+      and \<gamma>: "valid_path \<gamma>"
+      and w: "w \<in> s - path_image \<gamma>"
+    shows "(\<lambda>u. f' u / (u - w)^(Suc k)) contour_integrable_on \<gamma>"
+      and "(f has_field_derivative (k * contour_integral \<gamma> (\<lambda>u. f' u/(u - w)^(Suc k))))
+           (at w)"  (is "?thes2")
+proof -
+  have "open (s - path_image \<gamma>)" using \<open>open s\<close> closed_valid_path_image \<gamma> by blast
+  then obtain d where "d>0" and d: "ball w d \<subseteq> s - path_image \<gamma>" using w
+    using open_contains_ball by blast
+  have [simp]: "\<And>n. cmod (1 + of_nat n) = 1 + of_nat n"
+    by (metis norm_of_nat of_nat_Suc)
+  have cint: "\<And>x. \<lbrakk>x \<noteq> w; cmod (x - w) < d\<rbrakk>
+         \<Longrightarrow> (\<lambda>z. (f' z / (z - x) ^ k - f' z / (z - w) ^ k) / (x * k - w * k)) contour_integrable_on \<gamma>"
+    apply (rule contour_integrable_div [OF contour_integrable_diff])
+    using int w d
+    by (force simp: dist_norm norm_minus_commute intro!: has_contour_integral_integrable)+
+  have 1: "\<forall>\<^sub>F n in at w. (\<lambda>x. f' x * (inverse (x - n) ^ k - inverse (x - w) ^ k) / (n - w) / of_nat k)
+                         contour_integrable_on \<gamma>"
+    unfolding eventually_at
+    apply (rule_tac x=d in exI)
+    apply (simp add: \<open>d > 0\<close> dist_norm field_simps cint)
+    done
+  have bim_g: "bounded (image f' (path_image \<gamma>))"
+    by (simp add: compact_imp_bounded compact_continuous_image compact_valid_path_image assms)
+  then obtain C where "C > 0" and C: "\<And>x. \<lbrakk>0 \<le> x; x \<le> 1\<rbrakk> \<Longrightarrow> cmod (f' (\<gamma> x)) \<le> C"
+    by (force simp: bounded_pos path_image_def)
+  have twom: "\<forall>\<^sub>F n in at w.
+               \<forall>x\<in>path_image \<gamma>.
+                cmod ((inverse (x - n) ^ k - inverse (x - w) ^ k) / (n - w) / k - inverse (x - w) ^ Suc k) < e"
+         if "0 < e" for e
+  proof -
+    have *: "cmod ((inverse (x - u) ^ k - inverse (x - w) ^ k) / ((u - w) * k) - inverse (x - w) ^ Suc k)   < e"
+            if x: "x \<in> path_image \<gamma>" and "u \<noteq> w" and uwd: "cmod (u - w) < d/2"
+                and uw_less: "cmod (u - w) < e * (d/2) ^ (k+2) / (1 + real k)"
+            for u x
+    proof -
+      define ff where [abs_def]:
+        "ff n w =
+          (if n = 0 then inverse(x - w)^k
+           else if n = 1 then k / (x - w)^(Suc k)
+           else (k * of_real(Suc k)) / (x - w)^(k + 2))" for n :: nat and w
+      have km1: "\<And>z::complex. z \<noteq> 0 \<Longrightarrow> z ^ (k - Suc 0) = z ^ k / z"
+        by (simp add: field_simps) (metis Suc_pred \<open>k \<noteq> 0\<close> neq0_conv power_Suc)
+      have ff1: "(ff i has_field_derivative ff (Suc i) z) (at z within ball w (d/2))"
+              if "z \<in> ball w (d/2)" "i \<le> 1" for i z
+      proof -
+        have "z \<notin> path_image \<gamma>"
+          using \<open>x \<in> path_image \<gamma>\<close> d that ball_divide_subset_numeral by blast
+        then have xz[simp]: "x \<noteq> z" using \<open>x \<in> path_image \<gamma>\<close> by blast
+        then have neq: "x * x + z * z \<noteq> x * (z * 2)"
+          by (blast intro: dest!: sum_sqs_eq)
+        with xz have "\<And>v. v \<noteq> 0 \<Longrightarrow> (x * x + z * z) * v \<noteq> (x * (z * 2) * v)" by auto
+        then have neqq: "\<And>v. v \<noteq> 0 \<Longrightarrow> x * (x * v) + z * (z * v) \<noteq> x * (z * (2 * v))"
+          by (simp add: algebra_simps)
+        show ?thesis using \<open>i \<le> 1\<close>
+          apply (simp add: ff_def dist_norm Nat.le_Suc_eq km1, safe)
+          apply (rule derivative_eq_intros | simp add: km1 | simp add: field_simps neq neqq)+
+          done
+      qed
+      { fix a::real and b::real assume ab: "a > 0" "b > 0"
+        then have "k * (1 + real k) * (1 / a) \<le> k * (1 + real k) * (4 / b) \<longleftrightarrow> b \<le> 4 * a"
+          by (subst mult_le_cancel_left_pos)
+            (use \<open>k \<noteq> 0\<close> in \<open>auto simp: divide_simps\<close>)
+        with ab have "real k * (1 + real k) / a \<le> (real k * 4 + real k * real k * 4) / b \<longleftrightarrow> b \<le> 4 * a"
+          by (simp add: field_simps)
+      } note canc = this
+      have ff2: "cmod (ff (Suc 1) v) \<le> real (k * (k + 1)) / (d/2) ^ (k + 2)"
+                if "v \<in> ball w (d/2)" for v
+      proof -
+        have lessd: "\<And>z. cmod (\<gamma> z - v) < d/2 \<Longrightarrow> cmod (w - \<gamma> z) < d"
+          by (metis that norm_minus_commute norm_triangle_half_r dist_norm mem_ball)
+        have "d/2 \<le> cmod (x - v)" using d x that
+          using lessd d x
+          by (auto simp add: dist_norm path_image_def ball_def not_less [symmetric] del: divide_const_simps)
+        then have "d \<le> cmod (x - v) * 2"
+          by (simp add: field_split_simps)
+        then have dpow_le: "d ^ (k+2) \<le> (cmod (x - v) * 2) ^ (k+2)"
+          using \<open>0 < d\<close> order_less_imp_le power_mono by blast
+        have "x \<noteq> v" using that
+          using \<open>x \<in> path_image \<gamma>\<close> ball_divide_subset_numeral d by fastforce
+        then show ?thesis
+        using \<open>d > 0\<close> apply (simp add: ff_def norm_mult norm_divide norm_power dist_norm canc)
+        using dpow_le apply (simp add: field_split_simps)
+        done
+      qed
+      have ub: "u \<in> ball w (d/2)"
+        using uwd by (simp add: dist_commute dist_norm)
+      have "cmod (inverse (x - u) ^ k - (inverse (x - w) ^ k + of_nat k * (u - w) / ((x - w) * (x - w) ^ k)))
+                  \<le> (real k * 4 + real k * real k * 4) * (cmod (u - w) * cmod (u - w)) / (d * (d * (d/2) ^ k))"
+        using complex_Taylor [OF _ ff1 ff2 _ ub, of w, simplified]
+        by (simp add: ff_def \<open>0 < d\<close>)
+      then have "cmod (inverse (x - u) ^ k - (inverse (x - w) ^ k + of_nat k * (u - w) / ((x - w) * (x - w) ^ k)))
+                  \<le> (cmod (u - w) * real k) * (1 + real k) * cmod (u - w) / (d/2) ^ (k+2)"
+        by (simp add: field_simps)
+      then have "cmod (inverse (x - u) ^ k - (inverse (x - w) ^ k + of_nat k * (u - w) / ((x - w) * (x - w) ^ k)))
+                 / (cmod (u - w) * real k)
+                  \<le> (1 + real k) * cmod (u - w) / (d/2) ^ (k+2)"
+        using \<open>k \<noteq> 0\<close> \<open>u \<noteq> w\<close> by (simp add: mult_ac zero_less_mult_iff pos_divide_le_eq)
+      also have "\<dots> < e"
+        using uw_less \<open>0 < d\<close> by (simp add: mult_ac divide_simps)
+      finally have e: "cmod (inverse (x-u)^k - (inverse (x-w)^k + of_nat k * (u-w) / ((x-w) * (x-w)^k)))
+                        / cmod ((u - w) * real k)   <   e"
+        by (simp add: norm_mult)
+      have "x \<noteq> u"
+        using uwd \<open>0 < d\<close> x d by (force simp: dist_norm ball_def norm_minus_commute)
+      show ?thesis
+        apply (rule le_less_trans [OF _ e])
+        using \<open>k \<noteq> 0\<close> \<open>x \<noteq> u\<close> \<open>u \<noteq> w\<close>
+        apply (simp add: field_simps norm_divide [symmetric])
+        done
+    qed
+    show ?thesis
+      unfolding eventually_at
+      apply (rule_tac x = "min (d/2) ((e*(d/2)^(k + 2))/(Suc k))" in exI)
+      apply (force simp: \<open>d > 0\<close> dist_norm that simp del: power_Suc intro: *)
+      done
+  qed
+  have 2: "uniform_limit (path_image \<gamma>) (\<lambda>n x. f' x * (inverse (x - n) ^ k - inverse (x - w) ^ k) / (n - w) / of_nat k) (\<lambda>x. f' x / (x - w) ^ Suc k) (at w)"
+    unfolding uniform_limit_iff dist_norm
+  proof clarify
+    fix e::real
+    assume "0 < e"
+    have *: "cmod (f' (\<gamma> x) * (inverse (\<gamma> x - u) ^ k - inverse (\<gamma> x - w) ^ k) / ((u - w) * k) -
+                        f' (\<gamma> x) / ((\<gamma> x - w) * (\<gamma> x - w) ^ k)) < e"
+              if ec: "cmod ((inverse (\<gamma> x - u) ^ k - inverse (\<gamma> x - w) ^ k) / ((u - w) * k) -
+                      inverse (\<gamma> x - w) * inverse (\<gamma> x - w) ^ k) < e / C"
+                 and x: "0 \<le> x" "x \<le> 1"
+              for u x
+    proof (cases "(f' (\<gamma> x)) = 0")
+      case True then show ?thesis by (simp add: \<open>0 < e\<close>)
+    next
+      case False
+      have "cmod (f' (\<gamma> x) * (inverse (\<gamma> x - u) ^ k - inverse (\<gamma> x - w) ^ k) / ((u - w) * k) -
+                        f' (\<gamma> x) / ((\<gamma> x - w) * (\<gamma> x - w) ^ k)) =
+            cmod (f' (\<gamma> x) * ((inverse (\<gamma> x - u) ^ k - inverse (\<gamma> x - w) ^ k) / ((u - w) * k) -
+                             inverse (\<gamma> x - w) * inverse (\<gamma> x - w) ^ k))"
+        by (simp add: field_simps)
+      also have "\<dots> = cmod (f' (\<gamma> x)) *
+                       cmod ((inverse (\<gamma> x - u) ^ k - inverse (\<gamma> x - w) ^ k) / ((u - w) * k) -
+                             inverse (\<gamma> x - w) * inverse (\<gamma> x - w) ^ k)"
+        by (simp add: norm_mult)
+      also have "\<dots> < cmod (f' (\<gamma> x)) * (e/C)"
+        using False mult_strict_left_mono [OF ec] by force
+      also have "\<dots> \<le> e" using C
+        by (metis False \<open>0 < e\<close> frac_le less_eq_real_def mult.commute pos_le_divide_eq x zero_less_norm_iff)
+      finally show ?thesis .
+    qed
+    show "\<forall>\<^sub>F n in at w.
+              \<forall>x\<in>path_image \<gamma>.
+               cmod (f' x * (inverse (x - n) ^ k - inverse (x - w) ^ k) / (n - w) / of_nat k - f' x / (x - w) ^ Suc k) < e"
+      using twom [OF divide_pos_pos [OF \<open>0 < e\<close> \<open>C > 0\<close>]]   unfolding path_image_def
+      by (force intro: * elim: eventually_mono)
+  qed
+  show "(\<lambda>u. f' u / (u - w) ^ (Suc k)) contour_integrable_on \<gamma>"
+    by (rule contour_integral_uniform_limit [OF 1 2 leB \<gamma>]) auto
+  have *: "(\<lambda>n. contour_integral \<gamma> (\<lambda>x. f' x * (inverse (x - n) ^ k - inverse (x - w) ^ k) / (n - w) / k))
+           \<midarrow>w\<rightarrow> contour_integral \<gamma> (\<lambda>u. f' u / (u - w) ^ (Suc k))"
+    by (rule contour_integral_uniform_limit [OF 1 2 leB \<gamma>]) auto
+  have **: "contour_integral \<gamma> (\<lambda>x. f' x * (inverse (x - u) ^ k - inverse (x - w) ^ k) / ((u - w) * k)) =
+              (f u - f w) / (u - w) / k"
+    if "dist u w < d" for u
+  proof -
+    have u: "u \<in> s - path_image \<gamma>"
+      by (metis subsetD d dist_commute mem_ball that)
+    show ?thesis
+      apply (rule contour_integral_unique)
+      apply (simp add: diff_divide_distrib algebra_simps)
+      apply (intro has_contour_integral_diff has_contour_integral_div)
+      using u w apply (simp_all add: field_simps int)
+      done
+  qed
+  show ?thes2
+    apply (simp add: has_field_derivative_iff del: power_Suc)
+    apply (rule Lim_transform_within [OF tendsto_mult_left [OF *] \<open>0 < d\<close> ])
+    apply (simp add: \<open>k \<noteq> 0\<close> **)
+    done
+qed
+
+lemma Cauchy_next_derivative_circlepath:
+  assumes contf: "continuous_on (path_image (circlepath z r)) f"
+      and int: "\<And>w. w \<in> ball z r \<Longrightarrow> ((\<lambda>u. f u / (u - w)^k) has_contour_integral g w) (circlepath z r)"
+      and k: "k \<noteq> 0"
+      and w: "w \<in> ball z r"
+    shows "(\<lambda>u. f u / (u - w)^(Suc k)) contour_integrable_on (circlepath z r)"
+           (is "?thes1")
+      and "(g has_field_derivative (k * contour_integral (circlepath z r) (\<lambda>u. f u/(u - w)^(Suc k)))) (at w)"
+           (is "?thes2")
+proof -
+  have "r > 0" using w
+    using ball_eq_empty by fastforce
+  have wim: "w \<in> ball z r - path_image (circlepath z r)"
+    using w by (auto simp: dist_norm)
+  show ?thes1 ?thes2
+    by (rule Cauchy_next_derivative [OF contf _ int k open_ball valid_path_circlepath wim, where B = "2 * pi * \<bar>r\<bar>"];
+        auto simp: vector_derivative_circlepath norm_mult)+
+qed
+
+
+text\<open> In particular, the first derivative formula.\<close>
+
+lemma Cauchy_derivative_integral_circlepath:
+  assumes contf: "continuous_on (cball z r) f"
+      and holf: "f holomorphic_on ball z r"
+      and w: "w \<in> ball z r"
+    shows "(\<lambda>u. f u/(u - w)^2) contour_integrable_on (circlepath z r)"
+           (is "?thes1")
+      and "(f has_field_derivative (1 / (2 * of_real pi * \<i>) * contour_integral(circlepath z r) (\<lambda>u. f u / (u - w)^2))) (at w)"
+           (is "?thes2")
+proof -
+  have [simp]: "r \<ge> 0" using w
+    using ball_eq_empty by fastforce
+  have f: "continuous_on (path_image (circlepath z r)) f"
+    by (rule continuous_on_subset [OF contf]) (force simp: cball_def sphere_def)
+  have int: "\<And>w. dist z w < r \<Longrightarrow>
+                 ((\<lambda>u. f u / (u - w)) has_contour_integral (\<lambda>x. 2 * of_real pi * \<i> * f x) w) (circlepath z r)"
+    by (rule Cauchy_integral_circlepath [OF contf holf]) (simp add: dist_norm norm_minus_commute)
+  show ?thes1
+    apply (simp add: power2_eq_square)
+    apply (rule Cauchy_next_derivative_circlepath [OF f _ _ w, where k=1, simplified])
+    apply (blast intro: int)
+    done
+  have "((\<lambda>x. 2 * of_real pi * \<i> * f x) has_field_derivative contour_integral (circlepath z r) (\<lambda>u. f u / (u - w)^2)) (at w)"
+    apply (simp add: power2_eq_square)
+    apply (rule Cauchy_next_derivative_circlepath [OF f _ _ w, where k=1 and g = "\<lambda>x. 2 * of_real pi * \<i> * f x", simplified])
+    apply (blast intro: int)
+    done
+  then have fder: "(f has_field_derivative contour_integral (circlepath z r) (\<lambda>u. f u / (u - w)^2) / (2 * of_real pi * \<i>)) (at w)"
+    by (rule DERIV_cdivide [where f = "\<lambda>x. 2 * of_real pi * \<i> * f x" and c = "2 * of_real pi * \<i>", simplified])
+  show ?thes2
+    by simp (rule fder)
+qed
+
+subsection\<open>Existence of all higher derivatives\<close>
+
+proposition derivative_is_holomorphic:
+  assumes "open S"
+      and fder: "\<And>z. z \<in> S \<Longrightarrow> (f has_field_derivative f' z) (at z)"
+    shows "f' holomorphic_on S"
+proof -
+  have *: "\<exists>h. (f' has_field_derivative h) (at z)" if "z \<in> S" for z
+  proof -
+    obtain r where "r > 0" and r: "cball z r \<subseteq> S"
+      using open_contains_cball \<open>z \<in> S\<close> \<open>open S\<close> by blast
+    then have holf_cball: "f holomorphic_on cball z r"
+      apply (simp add: holomorphic_on_def)
+      using field_differentiable_at_within field_differentiable_def fder by blast
+    then have "continuous_on (path_image (circlepath z r)) f"
+      using \<open>r > 0\<close> by (force elim: holomorphic_on_subset [THEN holomorphic_on_imp_continuous_on])
+    then have contfpi: "continuous_on (path_image (circlepath z r)) (\<lambda>x. 1/(2 * of_real pi*\<i>) * f x)"
+      by (auto intro: continuous_intros)+
+    have contf_cball: "continuous_on (cball z r) f" using holf_cball
+      by (simp add: holomorphic_on_imp_continuous_on holomorphic_on_subset)
+    have holf_ball: "f holomorphic_on ball z r" using holf_cball
+      using ball_subset_cball holomorphic_on_subset by blast
+    { fix w  assume w: "w \<in> ball z r"
+      have intf: "(\<lambda>u. f u / (u - w)\<^sup>2) contour_integrable_on circlepath z r"
+        by (blast intro: w Cauchy_derivative_integral_circlepath [OF contf_cball holf_ball])
+      have fder': "(f has_field_derivative 1 / (2 * of_real pi * \<i>) * contour_integral (circlepath z r) (\<lambda>u. f u / (u - w)\<^sup>2))
+                  (at w)"
+        by (blast intro: w Cauchy_derivative_integral_circlepath [OF contf_cball holf_ball])
+      have f'_eq: "f' w = contour_integral (circlepath z r) (\<lambda>u. f u / (u - w)\<^sup>2) / (2 * of_real pi * \<i>)"
+        using fder' ball_subset_cball r w by (force intro: DERIV_unique [OF fder])
+      have "((\<lambda>u. f u / (u - w)\<^sup>2 / (2 * of_real pi * \<i>)) has_contour_integral
+                contour_integral (circlepath z r) (\<lambda>u. f u / (u - w)\<^sup>2) / (2 * of_real pi * \<i>))
+                (circlepath z r)"
+        by (rule has_contour_integral_div [OF has_contour_integral_integral [OF intf]])
+      then have "((\<lambda>u. f u / (2 * of_real pi * \<i> * (u - w)\<^sup>2)) has_contour_integral
+                contour_integral (circlepath z r) (\<lambda>u. f u / (u - w)\<^sup>2) / (2 * of_real pi * \<i>))
+                (circlepath z r)"
+        by (simp add: algebra_simps)
+      then have "((\<lambda>u. f u / (2 * of_real pi * \<i> * (u - w)\<^sup>2)) has_contour_integral f' w) (circlepath z r)"
+        by (simp add: f'_eq)
+    } note * = this
+    show ?thesis
+      apply (rule exI)
+      apply (rule Cauchy_next_derivative_circlepath [OF contfpi, of 2 f', simplified])
+      apply (simp_all add: \<open>0 < r\<close> * dist_norm)
+      done
+  qed
+  show ?thesis
+    by (simp add: holomorphic_on_open [OF \<open>open S\<close>] *)
+qed
+
+lemma holomorphic_deriv [holomorphic_intros]:
+    "\<lbrakk>f holomorphic_on S; open S\<rbrakk> \<Longrightarrow> (deriv f) holomorphic_on S"
+by (metis DERIV_deriv_iff_field_differentiable at_within_open derivative_is_holomorphic holomorphic_on_def)
+
+lemma analytic_deriv [analytic_intros]: "f analytic_on S \<Longrightarrow> (deriv f) analytic_on S"
+  using analytic_on_holomorphic holomorphic_deriv by auto
+
+lemma holomorphic_higher_deriv [holomorphic_intros]: "\<lbrakk>f holomorphic_on S; open S\<rbrakk> \<Longrightarrow> (deriv ^^ n) f holomorphic_on S"
+  by (induction n) (auto simp: holomorphic_deriv)
+
+lemma analytic_higher_deriv [analytic_intros]: "f analytic_on S \<Longrightarrow> (deriv ^^ n) f analytic_on S"
+  unfolding analytic_on_def using holomorphic_higher_deriv by blast
+
+lemma has_field_derivative_higher_deriv:
+     "\<lbrakk>f holomorphic_on S; open S; x \<in> S\<rbrakk>
+      \<Longrightarrow> ((deriv ^^ n) f has_field_derivative (deriv ^^ (Suc n)) f x) (at x)"
+by (metis (no_types, hide_lams) DERIV_deriv_iff_field_differentiable at_within_open comp_apply
+         funpow.simps(2) holomorphic_higher_deriv holomorphic_on_def)
+
+lemma valid_path_compose_holomorphic:
+  assumes "valid_path g" and holo:"f holomorphic_on S" and "open S" "path_image g \<subseteq> S"
+  shows "valid_path (f \<circ> g)"
+proof (rule valid_path_compose[OF \<open>valid_path g\<close>])
+  fix x assume "x \<in> path_image g"
+  then show "f field_differentiable at x"
+    using analytic_on_imp_differentiable_at analytic_on_open assms holo by blast
+next
+  have "deriv f holomorphic_on S"
+    using holomorphic_deriv holo \<open>open S\<close> by auto
+  then show "continuous_on (path_image g) (deriv f)"
+    using assms(4) holomorphic_on_imp_continuous_on holomorphic_on_subset by auto
+qed
+
+subsection\<open>Morera's theorem\<close>
+
+lemma Morera_local_triangle_ball:
+  assumes "\<And>z. z \<in> S
+          \<Longrightarrow> \<exists>e a. 0 < e \<and> z \<in> ball a e \<and> continuous_on (ball a e) f \<and>
+                    (\<forall>b c. closed_segment b c \<subseteq> ball a e
+                           \<longrightarrow> contour_integral (linepath a b) f +
+                               contour_integral (linepath b c) f +
+                               contour_integral (linepath c a) f = 0)"
+  shows "f analytic_on S"
+proof -
+  { fix z  assume "z \<in> S"
+    with assms obtain e a where
+            "0 < e" and z: "z \<in> ball a e" and contf: "continuous_on (ball a e) f"
+        and 0: "\<And>b c. closed_segment b c \<subseteq> ball a e
+                      \<Longrightarrow> contour_integral (linepath a b) f +
+                          contour_integral (linepath b c) f +
+                          contour_integral (linepath c a) f = 0"
+      by blast
+    have az: "dist a z < e" using mem_ball z by blast
+    have sb_ball: "ball z (e - dist a z) \<subseteq> ball a e"
+      by (simp add: dist_commute ball_subset_ball_iff)
+    have "\<exists>e>0. f holomorphic_on ball z e"
+    proof (intro exI conjI)
+      have sub_ball: "\<And>y. dist a y < e \<Longrightarrow> closed_segment a y \<subseteq> ball a e"
+        by (meson \<open>0 < e\<close> centre_in_ball convex_ball convex_contains_segment mem_ball)
+      show "f holomorphic_on ball z (e - dist a z)"
+        apply (rule holomorphic_on_subset [OF _ sb_ball])
+        apply (rule derivative_is_holomorphic[OF open_ball])
+        apply (rule triangle_contour_integrals_starlike_primitive [OF contf _ open_ball, of a])
+           apply (simp_all add: 0 \<open>0 < e\<close> sub_ball)
+        done
+    qed (simp add: az)
+  }
+  then show ?thesis
+    by (simp add: analytic_on_def)
+qed
+
+lemma Morera_local_triangle:
+  assumes "\<And>z. z \<in> S
+          \<Longrightarrow> \<exists>t. open t \<and> z \<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)"
+  shows "f analytic_on S"
+proof -
+  { fix z  assume "z \<in> S"
+    with assms obtain t where
+            "open t" and z: "z \<in> t" and contf: "continuous_on t f"
+        and 0: "\<And>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"
+      by force
+    then obtain e where "e>0" and e: "ball z e \<subseteq> t"
+      using open_contains_ball by blast
+    have [simp]: "continuous_on (ball z e) f" using contf
+      using continuous_on_subset e by blast
+    have eq0: "\<And>b c. closed_segment b c \<subseteq> ball z e \<Longrightarrow>
+                         contour_integral (linepath z b) f +
+                         contour_integral (linepath b c) f +
+                         contour_integral (linepath c z) f = 0"
+      by (meson 0 z \<open>0 < e\<close> centre_in_ball closed_segment_subset convex_ball dual_order.trans e starlike_convex_subset)
+    have "\<exists>e a. 0 < e \<and> z \<in> ball a e \<and> continuous_on (ball a e) f \<and>
+                (\<forall>b c. closed_segment b c \<subseteq> ball a e \<longrightarrow>
+                       contour_integral (linepath a b) f + contour_integral (linepath b c) f + contour_integral (linepath c a) f = 0)"
+      using \<open>e > 0\<close> eq0 by force
+  }
+  then show ?thesis
+    by (simp add: Morera_local_triangle_ball)
+qed
+
+proposition Morera_triangle:
+    "\<lbrakk>continuous_on S f; open S;
+      \<And>a b c. convex hull {a,b,c} \<subseteq> S
+              \<longrightarrow> contour_integral (linepath a b) f +
+                  contour_integral (linepath b c) f +
+                  contour_integral (linepath c a) f = 0\<rbrakk>
+     \<Longrightarrow> f analytic_on S"
+  using Morera_local_triangle by blast
+
+subsection\<open>Combining theorems for higher derivatives including Leibniz rule\<close>
+
+lemma higher_deriv_linear [simp]:
+    "(deriv ^^ n) (\<lambda>w. c*w) = (\<lambda>z. if n = 0 then c*z else if n = 1 then c else 0)"
+  by (induction n) auto
+
+lemma higher_deriv_const [simp]: "(deriv ^^ n) (\<lambda>w. c) = (\<lambda>w. if n=0 then c else 0)"
+  by (induction n) auto
+
+lemma higher_deriv_ident [simp]:
+     "(deriv ^^ n) (\<lambda>w. w) z = (if n = 0 then z else if n = 1 then 1 else 0)"
+  apply (induction n, simp)
+  apply (metis higher_deriv_linear lambda_one)
+  done
+
+lemma higher_deriv_id [simp]:
+     "(deriv ^^ n) id z = (if n = 0 then z else if n = 1 then 1 else 0)"
+  by (simp add: id_def)
+
+lemma has_complex_derivative_funpow_1:
+     "\<lbrakk>(f has_field_derivative 1) (at z); f z = z\<rbrakk> \<Longrightarrow> (f^^n has_field_derivative 1) (at z)"
+  apply (induction n, auto)
+  apply (simp add: id_def)
+  by (metis DERIV_chain comp_funpow comp_id funpow_swap1 mult.right_neutral)
+
+lemma higher_deriv_uminus:
+  assumes "f holomorphic_on S" "open S" and z: "z \<in> S"
+    shows "(deriv ^^ n) (\<lambda>w. -(f w)) z = - ((deriv ^^ n) f z)"
+using z
+proof (induction n arbitrary: z)
+  case 0 then show ?case by simp
+next
+  case (Suc n z)
+  have *: "((deriv ^^ n) f has_field_derivative deriv ((deriv ^^ n) f) z) (at z)"
+    using Suc.prems assms has_field_derivative_higher_deriv by auto
+  have "((deriv ^^ n) (\<lambda>w. - f w) has_field_derivative - deriv ((deriv ^^ n) f) z) (at z)"
+    apply (rule has_field_derivative_transform_within_open [of "\<lambda>w. -((deriv ^^ n) f w)"])
+       apply (rule derivative_eq_intros | rule * refl assms)+
+     apply (auto simp add: Suc)
+    done
+  then show ?case
+    by (simp add: DERIV_imp_deriv)
+qed
+
+lemma higher_deriv_add:
+  fixes z::complex
+  assumes "f holomorphic_on S" "g holomorphic_on S" "open S" and z: "z \<in> S"
+    shows "(deriv ^^ n) (\<lambda>w. f w + g w) z = (deriv ^^ n) f z + (deriv ^^ n) g z"
+using z
+proof (induction n arbitrary: z)
+  case 0 then show ?case by simp
+next
+  case (Suc n z)
+  have *: "((deriv ^^ n) f has_field_derivative deriv ((deriv ^^ n) f) z) (at z)"
+          "((deriv ^^ n) g has_field_derivative deriv ((deriv ^^ n) g) z) (at z)"
+    using Suc.prems assms has_field_derivative_higher_deriv by auto
+  have "((deriv ^^ n) (\<lambda>w. f w + g w) has_field_derivative
+        deriv ((deriv ^^ n) f) z + deriv ((deriv ^^ n) g) z) (at z)"
+    apply (rule has_field_derivative_transform_within_open [of "\<lambda>w. (deriv ^^ n) f w + (deriv ^^ n) g w"])
+       apply (rule derivative_eq_intros | rule * refl assms)+
+     apply (auto simp add: Suc)
+    done
+  then show ?case
+    by (simp add: DERIV_imp_deriv)
+qed
+
+lemma higher_deriv_diff:
+  fixes z::complex
+  assumes "f holomorphic_on S" "g holomorphic_on S" "open S" and z: "z \<in> S"
+    shows "(deriv ^^ n) (\<lambda>w. f w - g w) z = (deriv ^^ n) f z - (deriv ^^ n) g z"
+  apply (simp only: Groups.group_add_class.diff_conv_add_uminus higher_deriv_add)
+  apply (subst higher_deriv_add)
+  using assms holomorphic_on_minus apply (auto simp: higher_deriv_uminus)
+  done
+
+lemma bb: "Suc n choose k = (n choose k) + (if k = 0 then 0 else (n choose (k - 1)))"
+  by (cases k) simp_all
+
+lemma higher_deriv_mult:
+  fixes z::complex
+  assumes "f holomorphic_on S" "g holomorphic_on S" "open S" and z: "z \<in> S"
+    shows "(deriv ^^ n) (\<lambda>w. f w * g w) z =
+           (\<Sum>i = 0..n. of_nat (n choose i) * (deriv ^^ i) f z * (deriv ^^ (n - i)) g z)"
+using z
+proof (induction n arbitrary: z)
+  case 0 then show ?case by simp
+next
+  case (Suc n z)
+  have *: "\<And>n. ((deriv ^^ n) f has_field_derivative deriv ((deriv ^^ n) f) z) (at z)"
+          "\<And>n. ((deriv ^^ n) g has_field_derivative deriv ((deriv ^^ n) g) z) (at z)"
+    using Suc.prems assms has_field_derivative_higher_deriv by auto
+  have sumeq: "(\<Sum>i = 0..n.
+               of_nat (n choose i) * (deriv ((deriv ^^ i) f) z * (deriv ^^ (n - i)) g z + deriv ((deriv ^^ (n - i)) g) z * (deriv ^^ i) f z)) =
+            g z * deriv ((deriv ^^ n) f) z + (\<Sum>i = 0..n. (deriv ^^ i) f z * (of_nat (Suc n choose i) * (deriv ^^ (Suc n - i)) g z))"
+    apply (simp add: bb algebra_simps sum.distrib)
+    apply (subst (4) sum_Suc_reindex)
+    apply (auto simp: algebra_simps Suc_diff_le intro: sum.cong)
+    done
+  have "((deriv ^^ n) (\<lambda>w. f w * g w) has_field_derivative
+         (\<Sum>i = 0..Suc n. (Suc n choose i) * (deriv ^^ i) f z * (deriv ^^ (Suc n - i)) g z))
+        (at z)"
+    apply (rule has_field_derivative_transform_within_open
+        [of "\<lambda>w. (\<Sum>i = 0..n. of_nat (n choose i) * (deriv ^^ i) f w * (deriv ^^ (n - i)) g w)"])
+       apply (simp add: algebra_simps)
+       apply (rule DERIV_cong [OF DERIV_sum])
+        apply (rule DERIV_cmult)
+        apply (auto intro: DERIV_mult * sumeq \<open>open S\<close> Suc.prems Suc.IH [symmetric])
+    done
+  then show ?case
+    unfolding funpow.simps o_apply
+    by (simp add: DERIV_imp_deriv)
+qed
+
+lemma higher_deriv_transform_within_open:
+  fixes z::complex
+  assumes "f holomorphic_on S" "g holomorphic_on S" "open S" and z: "z \<in> S"
+      and fg: "\<And>w. w \<in> S \<Longrightarrow> f w = g w"
+    shows "(deriv ^^ i) f z = (deriv ^^ i) g z"
+using z
+by (induction i arbitrary: z)
+   (auto simp: fg intro: complex_derivative_transform_within_open holomorphic_higher_deriv assms)
+
+lemma higher_deriv_compose_linear:
+  fixes z::complex
+  assumes f: "f holomorphic_on T" and S: "open S" and T: "open T" and z: "z \<in> S"
+      and fg: "\<And>w. w \<in> S \<Longrightarrow> u * w \<in> T"
+    shows "(deriv ^^ n) (\<lambda>w. f (u * w)) z = u^n * (deriv ^^ n) f (u * z)"
+using z
+proof (induction n arbitrary: z)
+  case 0 then show ?case by simp
+next
+  case (Suc n z)
+  have holo0: "f holomorphic_on (*) u ` S"
+    by (meson fg f holomorphic_on_subset image_subset_iff)
+  have holo2: "(deriv ^^ n) f holomorphic_on (*) u ` S"
+    by (meson f fg holomorphic_higher_deriv holomorphic_on_subset image_subset_iff T)
+  have holo3: "(\<lambda>z. u ^ n * (deriv ^^ n) f (u * z)) holomorphic_on S"
+    by (intro holo2 holomorphic_on_compose [where g="(deriv ^^ n) f", unfolded o_def] holomorphic_intros)
+  have holo1: "(\<lambda>w. f (u * w)) holomorphic_on S"
+    apply (rule holomorphic_on_compose [where g=f, unfolded o_def])
+    apply (rule holo0 holomorphic_intros)+
+    done
+  have "deriv ((deriv ^^ n) (\<lambda>w. f (u * w))) z = deriv (\<lambda>z. u^n * (deriv ^^ n) f (u*z)) z"
+    apply (rule complex_derivative_transform_within_open [OF _ holo3 S Suc.prems])
+    apply (rule holomorphic_higher_deriv [OF holo1 S])
+    apply (simp add: Suc.IH)
+    done
+  also have "\<dots> = u^n * deriv (\<lambda>z. (deriv ^^ n) f (u * z)) z"
+    apply (rule deriv_cmult)
+    apply (rule analytic_on_imp_differentiable_at [OF _ Suc.prems])
+    apply (rule analytic_on_compose_gen [where g="(deriv ^^ n) f" and T=T, unfolded o_def])
+      apply (simp)
+     apply (simp add: analytic_on_open f holomorphic_higher_deriv T)
+    apply (blast intro: fg)
+    done
+  also have "\<dots> = u * u ^ n * deriv ((deriv ^^ n) f) (u * z)"
+      apply (subst deriv_chain [where g = "(deriv ^^ n) f" and f = "(*) u", unfolded o_def])
+      apply (rule derivative_intros)
+      using Suc.prems field_differentiable_def f fg has_field_derivative_higher_deriv T apply blast
+      apply (simp)
+      done
+  finally show ?case
+    by simp
+qed
+
+lemma higher_deriv_add_at:
+  assumes "f analytic_on {z}" "g analytic_on {z}"
+    shows "(deriv ^^ n) (\<lambda>w. f w + g w) z = (deriv ^^ n) f z + (deriv ^^ n) g z"
+proof -
+  have "f analytic_on {z} \<and> g analytic_on {z}"
+    using assms by blast
+  with higher_deriv_add show ?thesis
+    by (auto simp: analytic_at_two)
+qed
+
+lemma higher_deriv_diff_at:
+  assumes "f analytic_on {z}" "g analytic_on {z}"
+    shows "(deriv ^^ n) (\<lambda>w. f w - g w) z = (deriv ^^ n) f z - (deriv ^^ n) g z"
+proof -
+  have "f analytic_on {z} \<and> g analytic_on {z}"
+    using assms by blast
+  with higher_deriv_diff show ?thesis
+    by (auto simp: analytic_at_two)
+qed
+
+lemma higher_deriv_uminus_at:
+   "f analytic_on {z}  \<Longrightarrow> (deriv ^^ n) (\<lambda>w. -(f w)) z = - ((deriv ^^ n) f z)"
+  using higher_deriv_uminus
+    by (auto simp: analytic_at)
+
+lemma higher_deriv_mult_at:
+  assumes "f analytic_on {z}" "g analytic_on {z}"
+    shows "(deriv ^^ n) (\<lambda>w. f w * g w) z =
+           (\<Sum>i = 0..n. of_nat (n choose i) * (deriv ^^ i) f z * (deriv ^^ (n - i)) g z)"
+proof -
+  have "f analytic_on {z} \<and> g analytic_on {z}"
+    using assms by blast
+  with higher_deriv_mult show ?thesis
+    by (auto simp: analytic_at_two)
+qed
+
+
+text\<open> Nonexistence of isolated singularities and a stronger integral formula.\<close>
+
+proposition no_isolated_singularity:
+  fixes z::complex
+  assumes f: "continuous_on S f" and holf: "f holomorphic_on (S - K)" and S: "open S" and K: "finite K"
+    shows "f holomorphic_on S"
+proof -
+  { fix z
+    assume "z \<in> S" and cdf: "\<And>x. x \<in> S - K \<Longrightarrow> f field_differentiable at x"
+    have "f field_differentiable at z"
+    proof (cases "z \<in> K")
+      case False then show ?thesis by (blast intro: cdf \<open>z \<in> S\<close>)
+    next
+      case True
+      with finite_set_avoid [OF K, of z]
+      obtain d where "d>0" and d: "\<And>x. \<lbrakk>x\<in>K; x \<noteq> z\<rbrakk> \<Longrightarrow> d \<le> dist z x"
+        by blast
+      obtain e where "e>0" and e: "ball z e \<subseteq> S"
+        using  S \<open>z \<in> S\<close> by (force simp: open_contains_ball)
+      have fde: "continuous_on (ball z (min d e)) f"
+        by (metis Int_iff ball_min_Int continuous_on_subset e f subsetI)
+      have cont: "{a,b,c} \<subseteq> ball z (min d e) \<Longrightarrow> continuous_on (convex hull {a, b, c}) f" for a b c
+        by (simp add: hull_minimal continuous_on_subset [OF fde])
+      have fd: "\<lbrakk>{a,b,c} \<subseteq> ball z (min d e); x \<in> interior (convex hull {a, b, c}) - K\<rbrakk>
+            \<Longrightarrow> f field_differentiable at x" for a b c x
+        by (metis cdf Diff_iff Int_iff ball_min_Int subsetD convex_ball e interior_mono interior_subset subset_hull)
+      obtain g where "\<And>w. w \<in> ball z (min d e) \<Longrightarrow> (g has_field_derivative f w) (at w within ball z (min d e))"
+        apply (rule contour_integral_convex_primitive
+                     [OF convex_ball fde Cauchy_theorem_triangle_cofinite [OF _ K]])
+        using cont fd by auto
+      then have "f holomorphic_on ball z (min d e)"
+        by (metis open_ball at_within_open derivative_is_holomorphic)
+      then show ?thesis
+        unfolding holomorphic_on_def
+        by (metis open_ball \<open>0 < d\<close> \<open>0 < e\<close> at_within_open centre_in_ball min_less_iff_conj)
+    qed
+  }
+  with holf S K show ?thesis
+    by (simp add: holomorphic_on_open open_Diff finite_imp_closed field_differentiable_def [symmetric])
+qed
+
+lemma no_isolated_singularity':
+  fixes z::complex
+  assumes f: "\<And>z. z \<in> K \<Longrightarrow> (f \<longlongrightarrow> f z) (at z within S)"
+      and holf: "f holomorphic_on (S - K)" and S: "open S" and K: "finite K"
+    shows "f holomorphic_on S"
+proof (rule no_isolated_singularity[OF _ assms(2-)])
+  show "continuous_on S f" unfolding continuous_on_def
+  proof
+    fix z assume z: "z \<in> S"
+    show "(f \<longlongrightarrow> f z) (at z within S)"
+    proof (cases "z \<in> K")
+      case False
+      from holf have "continuous_on (S - K) f"
+        by (rule holomorphic_on_imp_continuous_on)
+      with z False have "(f \<longlongrightarrow> f z) (at z within (S - K))"
+        by (simp add: continuous_on_def)
+      also from z K S False have "at z within (S - K) = at z within S"
+        by (subst (1 2) at_within_open) (auto intro: finite_imp_closed)
+      finally show "(f \<longlongrightarrow> f z) (at z within S)" .
+    qed (insert assms z, simp_all)
+  qed
+qed
+
+proposition Cauchy_integral_formula_convex:
+  assumes S: "convex S" and K: "finite K" and contf: "continuous_on S f"
+    and fcd: "(\<And>x. x \<in> interior S - K \<Longrightarrow> f field_differentiable at x)"
+    and z: "z \<in> interior S" and vpg: "valid_path \<gamma>"
+    and pasz: "path_image \<gamma> \<subseteq> S - {z}" and loop: "pathfinish \<gamma> = pathstart \<gamma>"
+  shows "((\<lambda>w. f w / (w - z)) has_contour_integral (2*pi * \<i> * winding_number \<gamma> z * f z)) \<gamma>"
+proof -
+  have *: "\<And>x. x \<in> interior S \<Longrightarrow> f field_differentiable at x"
+    unfolding holomorphic_on_open [symmetric] field_differentiable_def
+    using no_isolated_singularity [where S = "interior S"]
+    by (meson K contf continuous_at_imp_continuous_on continuous_on_interior fcd
+          field_differentiable_at_within field_differentiable_def holomorphic_onI
+          holomorphic_on_imp_differentiable_at open_interior)
+  show ?thesis
+    by (rule Cauchy_integral_formula_weak [OF S finite.emptyI contf]) (use * assms in auto)
+qed
+
+text\<open> Formula for higher derivatives.\<close>
+
+lemma Cauchy_has_contour_integral_higher_derivative_circlepath:
+  assumes contf: "continuous_on (cball z r) f"
+      and holf: "f holomorphic_on ball z r"
+      and w: "w \<in> ball z r"
+    shows "((\<lambda>u. f u / (u - w) ^ (Suc k)) has_contour_integral ((2 * pi * \<i>) / (fact k) * (deriv ^^ k) f w))
+           (circlepath z r)"
+using w
+proof (induction k arbitrary: w)
+  case 0 then show ?case
+    using assms by (auto simp: Cauchy_integral_circlepath dist_commute dist_norm)
+next
+  case (Suc k)
+  have [simp]: "r > 0" using w
+    using ball_eq_empty by fastforce
+  have f: "continuous_on (path_image (circlepath z r)) f"
+    by (rule continuous_on_subset [OF contf]) (force simp: cball_def sphere_def less_imp_le)
+  obtain X where X: "((\<lambda>u. f u / (u - w) ^ Suc (Suc k)) has_contour_integral X) (circlepath z r)"
+    using Cauchy_next_derivative_circlepath(1) [OF f Suc.IH _ Suc.prems]
+    by (auto simp: contour_integrable_on_def)
+  then have con: "contour_integral (circlepath z r) ((\<lambda>u. f u / (u - w) ^ Suc (Suc k))) = X"
+    by (rule contour_integral_unique)
+  have "\<And>n. ((deriv ^^ n) f has_field_derivative deriv ((deriv ^^ n) f) w) (at w)"
+    using Suc.prems assms has_field_derivative_higher_deriv by auto
+  then have dnf_diff: "\<And>n. (deriv ^^ n) f field_differentiable (at w)"
+    by (force simp: field_differentiable_def)
+  have "deriv (\<lambda>w. complex_of_real (2 * pi) * \<i> / (fact k) * (deriv ^^ k) f w) w =
+          of_nat (Suc k) * contour_integral (circlepath z r) (\<lambda>u. f u / (u - w) ^ Suc (Suc k))"
+    by (force intro!: DERIV_imp_deriv Cauchy_next_derivative_circlepath [OF f Suc.IH _ Suc.prems])
+  also have "\<dots> = of_nat (Suc k) * X"
+    by (simp only: con)
+  finally have "deriv (\<lambda>w. ((2 * pi) * \<i> / (fact k)) * (deriv ^^ k) f w) w = of_nat (Suc k) * X" .
+  then have "((2 * pi) * \<i> / (fact k)) * deriv (\<lambda>w. (deriv ^^ k) f w) w = of_nat (Suc k) * X"
+    by (metis deriv_cmult dnf_diff)
+  then have "deriv (\<lambda>w. (deriv ^^ k) f w) w = of_nat (Suc k) * X / ((2 * pi) * \<i> / (fact k))"
+    by (simp add: field_simps)
+  then show ?case
+  using of_nat_eq_0_iff X by fastforce
+qed
+
+lemma Cauchy_higher_derivative_integral_circlepath:
+  assumes contf: "continuous_on (cball z r) f"
+      and holf: "f holomorphic_on ball z r"
+      and w: "w \<in> ball z r"
+    shows "(\<lambda>u. f u / (u - w)^(Suc k)) contour_integrable_on (circlepath z r)"
+           (is "?thes1")
+      and "(deriv ^^ k) f w = (fact k) / (2 * pi * \<i>) * contour_integral(circlepath z r) (\<lambda>u. f u/(u - w)^(Suc k))"
+           (is "?thes2")
+proof -
+  have *: "((\<lambda>u. f u / (u - w) ^ Suc k) has_contour_integral (2 * pi) * \<i> / (fact k) * (deriv ^^ k) f w)
+           (circlepath z r)"
+    using Cauchy_has_contour_integral_higher_derivative_circlepath [OF assms]
+    by simp
+  show ?thes1 using *
+    using contour_integrable_on_def by blast
+  show ?thes2
+    unfolding contour_integral_unique [OF *] by (simp add: field_split_simps)
+qed
+
+corollary Cauchy_contour_integral_circlepath:
+  assumes "continuous_on (cball z r) f" "f holomorphic_on ball z r" "w \<in> ball z r"
+    shows "contour_integral(circlepath z r) (\<lambda>u. f u/(u - w)^(Suc k)) = (2 * pi * \<i>) * (deriv ^^ k) f w / (fact k)"
+by (simp add: Cauchy_higher_derivative_integral_circlepath [OF assms])
+
+lemma Cauchy_contour_integral_circlepath_2:
+  assumes "continuous_on (cball z r) f" "f holomorphic_on ball z r" "w \<in> ball z r"
+    shows "contour_integral(circlepath z r) (\<lambda>u. f u/(u - w)^2) = (2 * pi * \<i>) * deriv f w"
+  using Cauchy_contour_integral_circlepath [OF assms, of 1]
+  by (simp add: power2_eq_square)
+
+
+subsection\<open>A holomorphic function is analytic, i.e. has local power series\<close>
+
+theorem holomorphic_power_series:
+  assumes holf: "f holomorphic_on ball z r"
+      and w: "w \<in> ball z r"
+    shows "((\<lambda>n. (deriv ^^ n) f z / (fact n) * (w - z)^n) sums f w)"
+proof -
+  \<comment> \<open>Replacing \<^term>\<open>r\<close> and the original (weak) premises with stronger ones\<close>
+  obtain r where "r > 0" and holfc: "f holomorphic_on cball z r" and w: "w \<in> ball z r"
+  proof
+    have "cball z ((r + dist w z) / 2) \<subseteq> ball z r"
+      using w by (simp add: dist_commute field_sum_of_halves subset_eq)
+    then show "f holomorphic_on cball z ((r + dist w z) / 2)"
+      by (rule holomorphic_on_subset [OF holf])
+    have "r > 0"
+      using w by clarsimp (metis dist_norm le_less_trans norm_ge_zero)
+    then show "0 < (r + dist w z) / 2"
+      by simp (use zero_le_dist [of w z] in linarith)
+  qed (use w in \<open>auto simp: dist_commute\<close>)
+  then have holf: "f holomorphic_on ball z r"
+    using ball_subset_cball holomorphic_on_subset by blast
+  have contf: "continuous_on (cball z r) f"
+    by (simp add: holfc holomorphic_on_imp_continuous_on)
+  have cint: "\<And>k. (\<lambda>u. f u / (u - z) ^ Suc k) contour_integrable_on circlepath z r"
+    by (rule Cauchy_higher_derivative_integral_circlepath [OF contf holf]) (simp add: \<open>0 < r\<close>)
+  obtain B where "0 < B" and B: "\<And>u. u \<in> cball z r \<Longrightarrow> norm(f u) \<le> B"
+    by (metis (no_types) bounded_pos compact_cball compact_continuous_image compact_imp_bounded contf image_eqI)
+  obtain k where k: "0 < k" "k \<le> r" and wz_eq: "norm(w - z) = r - k"
+             and kle: "\<And>u. norm(u - z) = r \<Longrightarrow> k \<le> norm(u - w)"
+  proof
+    show "\<And>u. cmod (u - z) = r \<Longrightarrow> r - dist z w \<le> cmod (u - w)"
+      by (metis add_diff_eq diff_add_cancel dist_norm norm_diff_ineq)
+  qed (use w in \<open>auto simp: dist_norm norm_minus_commute\<close>)
+  have ul: "uniform_limit (sphere z r) (\<lambda>n x. (\<Sum>k<n. (w - z) ^ k * (f x / (x - z) ^ Suc k))) (\<lambda>x. f x / (x - w)) sequentially"
+    unfolding uniform_limit_iff dist_norm
+  proof clarify
+    fix e::real
+    assume "0 < e"
+    have rr: "0 \<le> (r - k) / r" "(r - k) / r < 1" using  k by auto
+    obtain n where n: "((r - k) / r) ^ n < e / B * k"
+      using real_arch_pow_inv [of "e/B*k" "(r - k)/r"] \<open>0 < e\<close> \<open>0 < B\<close> k by force
+    have "norm ((\<Sum>k<N. (w - z) ^ k * f u / (u - z) ^ Suc k) - f u / (u - w)) < e"
+         if "n \<le> N" and r: "r = dist z u"  for N u
+    proof -
+      have N: "((r - k) / r) ^ N < e / B * k"
+        apply (rule le_less_trans [OF power_decreasing n])
+        using  \<open>n \<le> N\<close> k by auto
+      have u [simp]: "(u \<noteq> z) \<and> (u \<noteq> w)"
+        using \<open>0 < r\<close> r w by auto
+      have wzu_not1: "(w - z) / (u - z) \<noteq> 1"
+        by (metis (no_types) dist_norm divide_eq_1_iff less_irrefl mem_ball norm_minus_commute r w)
+      have "norm ((\<Sum>k<N. (w - z) ^ k * f u / (u - z) ^ Suc k) * (u - w) - f u)
+            = norm ((\<Sum>k<N. (((w - z) / (u - z)) ^ k)) * f u * (u - w) / (u - z) - f u)"
+        unfolding sum_distrib_right sum_divide_distrib power_divide by (simp add: algebra_simps)
+      also have "\<dots> = norm ((((w - z) / (u - z)) ^ N - 1) * (u - w) / (((w - z) / (u - z) - 1) * (u - z)) - 1) * norm (f u)"
+        using \<open>0 < B\<close>
+        apply (auto simp: geometric_sum [OF wzu_not1])
+        apply (simp add: field_simps norm_mult [symmetric])
+        done
+      also have "\<dots> = norm ((u-z) ^ N * (w - u) - ((w - z) ^ N - (u-z) ^ N) * (u-w)) / (r ^ N * norm (u-w)) * norm (f u)"
+        using \<open>0 < r\<close> r by (simp add: divide_simps norm_mult norm_divide norm_power dist_norm norm_minus_commute)
+      also have "\<dots> = norm ((w - z) ^ N * (w - u)) / (r ^ N * norm (u - w)) * norm (f u)"
+        by (simp add: algebra_simps)
+      also have "\<dots> = norm (w - z) ^ N * norm (f u) / r ^ N"
+        by (simp add: norm_mult norm_power norm_minus_commute)
+      also have "\<dots> \<le> (((r - k)/r)^N) * B"
+        using \<open>0 < r\<close> w k
+        apply (simp add: divide_simps)
+        apply (rule mult_mono [OF power_mono])
+        apply (auto simp: norm_divide wz_eq norm_power dist_norm norm_minus_commute B r)
+        done
+      also have "\<dots> < e * k"
+        using \<open>0 < B\<close> N by (simp add: divide_simps)
+      also have "\<dots> \<le> e * norm (u - w)"
+        using r kle \<open>0 < e\<close> by (simp add: dist_commute dist_norm)
+      finally show ?thesis
+        by (simp add: field_split_simps norm_divide del: power_Suc)
+    qed
+    with \<open>0 < r\<close> show "\<forall>\<^sub>F n in sequentially. \<forall>x\<in>sphere z r.
+                norm ((\<Sum>k<n. (w - z) ^ k * (f x / (x - z) ^ Suc k)) - f x / (x - w)) < e"
+      by (auto simp: mult_ac less_imp_le eventually_sequentially Ball_def)
+  qed
+  have eq: "\<forall>\<^sub>F x in sequentially.
+             contour_integral (circlepath z r) (\<lambda>u. \<Sum>k<x. (w - z) ^ k * (f u / (u - z) ^ Suc k)) =
+             (\<Sum>k<x. contour_integral (circlepath z r) (\<lambda>u. f u / (u - z) ^ Suc k) * (w - z) ^ k)"
+    apply (rule eventuallyI)
+    apply (subst contour_integral_sum, simp)
+    using contour_integrable_lmul [OF cint, of "(w - z) ^ a" for a] apply (simp add: field_simps)
+    apply (simp only: contour_integral_lmul cint algebra_simps)
+    done
+  have cic: "\<And>u. (\<lambda>y. \<Sum>k<u. (w - z) ^ k * (f y / (y - z) ^ Suc k)) contour_integrable_on circlepath z r"
+    apply (intro contour_integrable_sum contour_integrable_lmul, simp)
+    using \<open>0 < r\<close> by (force intro!: Cauchy_higher_derivative_integral_circlepath [OF contf holf])
+  have "(\<lambda>k. contour_integral (circlepath z r) (\<lambda>u. f u/(u - z)^(Suc k)) * (w - z)^k)
+        sums contour_integral (circlepath z r) (\<lambda>u. f u/(u - w))"
+    unfolding sums_def
+    apply (intro Lim_transform_eventually [OF _ eq] contour_integral_uniform_limit_circlepath [OF eventuallyI ul] cic)
+    using \<open>0 < r\<close> apply auto
+    done
+  then have "(\<lambda>k. contour_integral (circlepath z r) (\<lambda>u. f u/(u - z)^(Suc k)) * (w - z)^k)
+             sums (2 * of_real pi * \<i> * f w)"
+    using w by (auto simp: dist_commute dist_norm contour_integral_unique [OF Cauchy_integral_circlepath_simple [OF holfc]])
+  then have "(\<lambda>k. contour_integral (circlepath z r) (\<lambda>u. f u / (u - z) ^ Suc k) * (w - z)^k / (\<i> * (of_real pi * 2)))
+            sums ((2 * of_real pi * \<i> * f w) / (\<i> * (complex_of_real pi * 2)))"
+    by (rule sums_divide)
+  then have "(\<lambda>n. (w - z) ^ n * contour_integral (circlepath z r) (\<lambda>u. f u / (u - z) ^ Suc n) / (\<i> * (of_real pi * 2)))
+            sums f w"
+    by (simp add: field_simps)
+  then show ?thesis
+    by (simp add: field_simps \<open>0 < r\<close> Cauchy_higher_derivative_integral_circlepath [OF contf holf])
+qed
+
+subsection\<open>The Liouville theorem and the Fundamental Theorem of Algebra\<close>
+
+text\<open> These weak Liouville versions don't even need the derivative formula.\<close>
+
+lemma Liouville_weak_0:
+  assumes holf: "f holomorphic_on UNIV" and inf: "(f \<longlongrightarrow> 0) at_infinity"
+    shows "f z = 0"
+proof (rule ccontr)
+  assume fz: "f z \<noteq> 0"
+  with inf [unfolded Lim_at_infinity, rule_format, of "norm(f z)/2"]
+  obtain B where B: "\<And>x. B \<le> cmod x \<Longrightarrow> norm (f x) * 2 < cmod (f z)"
+    by (auto simp: dist_norm)
+  define R where "R = 1 + \<bar>B\<bar> + norm z"
+  have "R > 0" unfolding R_def
+  proof -
+    have "0 \<le> cmod z + \<bar>B\<bar>"
+      by (metis (full_types) add_nonneg_nonneg norm_ge_zero real_norm_def)
+    then show "0 < 1 + \<bar>B\<bar> + cmod z"
+      by linarith
+  qed
+  have *: "((\<lambda>u. f u / (u - z)) has_contour_integral 2 * complex_of_real pi * \<i> * f z) (circlepath z R)"
+    apply (rule Cauchy_integral_circlepath)
+    using \<open>R > 0\<close> apply (auto intro: holomorphic_on_subset [OF holf] holomorphic_on_imp_continuous_on)+
+    done
+  have "cmod (x - z) = R \<Longrightarrow> cmod (f x) * 2 < cmod (f z)" for x
+    unfolding R_def
+    by (rule B) (use norm_triangle_ineq4 [of x z] in auto)
+  with \<open>R > 0\<close> fz show False
+    using has_contour_integral_bound_circlepath [OF *, of "norm(f z)/2/R"]
+    by (auto simp: less_imp_le norm_mult norm_divide field_split_simps)
+qed
+
+proposition Liouville_weak:
+  assumes "f holomorphic_on UNIV" and "(f \<longlongrightarrow> l) at_infinity"
+    shows "f z = l"
+  using Liouville_weak_0 [of "\<lambda>z. f z - l"]
+  by (simp add: assms holomorphic_on_diff LIM_zero)
+
+proposition Liouville_weak_inverse:
+  assumes "f holomorphic_on UNIV" and unbounded: "\<And>B. eventually (\<lambda>x. norm (f x) \<ge> B) at_infinity"
+    obtains z where "f z = 0"
+proof -
+  { assume f: "\<And>z. f z \<noteq> 0"
+    have 1: "(\<lambda>x. 1 / f x) holomorphic_on UNIV"
+      by (simp add: holomorphic_on_divide assms f)
+    have 2: "((\<lambda>x. 1 / f x) \<longlongrightarrow> 0) at_infinity"
+      apply (rule tendstoI [OF eventually_mono])
+      apply (rule_tac B="2/e" in unbounded)
+      apply (simp add: dist_norm norm_divide field_split_simps)
+      done
+    have False
+      using Liouville_weak_0 [OF 1 2] f by simp
+  }
+  then show ?thesis
+    using that by blast
+qed
+
+text\<open> In particular we get the Fundamental Theorem of Algebra.\<close>
+
+theorem fundamental_theorem_of_algebra:
+    fixes a :: "nat \<Rightarrow> complex"
+  assumes "a 0 = 0 \<or> (\<exists>i \<in> {1..n}. a i \<noteq> 0)"
+  obtains z where "(\<Sum>i\<le>n. a i * z^i) = 0"
+using assms
+proof (elim disjE bexE)
+  assume "a 0 = 0" then show ?thesis
+    by (auto simp: that [of 0])
+next
+  fix i
+  assume i: "i \<in> {1..n}" and nz: "a i \<noteq> 0"
+  have 1: "(\<lambda>z. \<Sum>i\<le>n. a i * z^i) holomorphic_on UNIV"
+    by (rule holomorphic_intros)+
+  show thesis
+  proof (rule Liouville_weak_inverse [OF 1])
+    show "\<forall>\<^sub>F x in at_infinity. B \<le> cmod (\<Sum>i\<le>n. a i * x ^ i)" for B
+      using i nz by (intro polyfun_extremal exI[of _ i]) auto
+  qed (use that in auto)
+qed
+
+subsection\<open>Weierstrass convergence theorem\<close>
+
+lemma holomorphic_uniform_limit:
+  assumes cont: "eventually (\<lambda>n. continuous_on (cball z r) (f n) \<and> (f n) holomorphic_on ball z r) F"
+      and ulim: "uniform_limit (cball z r) f g F"
+      and F:  "\<not> trivial_limit F"
+  obtains "continuous_on (cball z r) g" "g holomorphic_on ball z r"
+proof (cases r "0::real" rule: linorder_cases)
+  case less then show ?thesis by (force simp: ball_empty less_imp_le continuous_on_def holomorphic_on_def intro: that)
+next
+  case equal then show ?thesis
+    by (force simp: holomorphic_on_def intro: that)
+next
+  case greater
+  have contg: "continuous_on (cball z r) g"
+    using cont uniform_limit_theorem [OF eventually_mono ulim F]  by blast
+  have "path_image (circlepath z r) \<subseteq> cball z r"
+    using \<open>0 < r\<close> by auto
+  then have 1: "continuous_on (path_image (circlepath z r)) (\<lambda>x. 1 / (2 * complex_of_real pi * \<i>) * g x)"
+    by (intro continuous_intros continuous_on_subset [OF contg])
+  have 2: "((\<lambda>u. 1 / (2 * of_real pi * \<i>) * g u / (u - w) ^ 1) has_contour_integral g w) (circlepath z r)"
+       if w: "w \<in> ball z r" for w
+  proof -
+    define d where "d = (r - norm(w - z))"
+    have "0 < d"  "d \<le> r" using w by (auto simp: norm_minus_commute d_def dist_norm)
+    have dle: "\<And>u. cmod (z - u) = r \<Longrightarrow> d \<le> cmod (u - w)"
+      unfolding d_def by (metis add_diff_eq diff_add_cancel norm_diff_ineq norm_minus_commute)
+    have ev_int: "\<forall>\<^sub>F n in F. (\<lambda>u. f n u / (u - w)) contour_integrable_on circlepath z r"
+      apply (rule eventually_mono [OF cont])
+      using w
+      apply (auto intro: Cauchy_higher_derivative_integral_circlepath [where k=0, simplified])
+      done
+    have ul_less: "uniform_limit (sphere z r) (\<lambda>n x. f n x / (x - w)) (\<lambda>x. g x / (x - w)) F"
+      using greater \<open>0 < d\<close>
+      apply (clarsimp simp add: uniform_limit_iff dist_norm norm_divide diff_divide_distrib [symmetric] divide_simps)
+      apply (rule_tac e1="e * d" in eventually_mono [OF uniform_limitD [OF ulim]])
+       apply (force simp: dist_norm intro: dle mult_left_mono less_le_trans)+
+      done
+    have g_cint: "(\<lambda>u. g u/(u - w)) contour_integrable_on circlepath z r"
+      by (rule contour_integral_uniform_limit_circlepath [OF ev_int ul_less F \<open>0 < r\<close>])
+    have cif_tends_cig: "((\<lambda>n. contour_integral(circlepath z r) (\<lambda>u. f n u / (u - w))) \<longlongrightarrow> contour_integral(circlepath z r) (\<lambda>u. g u/(u - w))) F"
+      by (rule contour_integral_uniform_limit_circlepath [OF ev_int ul_less F \<open>0 < r\<close>])
+    have f_tends_cig: "((\<lambda>n. 2 * of_real pi * \<i> * f n w) \<longlongrightarrow> contour_integral (circlepath z r) (\<lambda>u. g u / (u - w))) F"
+    proof (rule Lim_transform_eventually)
+      show "\<forall>\<^sub>F x in F. contour_integral (circlepath z r) (\<lambda>u. f x u / (u - w))
+                     = 2 * of_real pi * \<i> * f x w"
+        apply (rule eventually_mono [OF cont contour_integral_unique [OF Cauchy_integral_circlepath]])
+        using w\<open>0 < d\<close> d_def by auto
+    qed (auto simp: cif_tends_cig)
+    have "\<And>e. 0 < e \<Longrightarrow> \<forall>\<^sub>F n in F. dist (f n w) (g w) < e"
+      by (rule eventually_mono [OF uniform_limitD [OF ulim]]) (use w in auto)
+    then have "((\<lambda>n. 2 * of_real pi * \<i> * f n w) \<longlongrightarrow> 2 * of_real pi * \<i> * g w) F"
+      by (rule tendsto_mult_left [OF tendstoI])
+    then have "((\<lambda>u. g u / (u - w)) has_contour_integral 2 * of_real pi * \<i> * g w) (circlepath z r)"
+      using has_contour_integral_integral [OF g_cint] tendsto_unique [OF F f_tends_cig] w
+      by fastforce
+    then have "((\<lambda>u. g u / (2 * of_real pi * \<i> * (u - w))) has_contour_integral g w) (circlepath z r)"
+      using has_contour_integral_div [where c = "2 * of_real pi * \<i>"]
+      by (force simp: field_simps)
+    then show ?thesis
+      by (simp add: dist_norm)
+  qed
+  show ?thesis
+    using Cauchy_next_derivative_circlepath(2) [OF 1 2, simplified]
+    by (fastforce simp add: holomorphic_on_open contg intro: that)
+qed
+
+
+text\<open> Version showing that the limit is the limit of the derivatives.\<close>
+
+proposition has_complex_derivative_uniform_limit:
+  fixes z::complex
+  assumes cont: "eventually (\<lambda>n. continuous_on (cball z r) (f n) \<and>
+                               (\<forall>w \<in> ball z r. ((f n) has_field_derivative (f' n w)) (at w))) F"
+      and ulim: "uniform_limit (cball z r) f g F"
+      and F:  "\<not> trivial_limit F" and "0 < r"
+  obtains g' where
+      "continuous_on (cball z r) g"
+      "\<And>w. w \<in> ball z r \<Longrightarrow> (g has_field_derivative (g' w)) (at w) \<and> ((\<lambda>n. f' n w) \<longlongrightarrow> g' w) F"
+proof -
+  let ?conint = "contour_integral (circlepath z r)"
+  have g: "continuous_on (cball z r) g" "g holomorphic_on ball z r"
+    by (rule holomorphic_uniform_limit [OF eventually_mono [OF cont] ulim F];
+             auto simp: holomorphic_on_open field_differentiable_def)+
+  then obtain g' where g': "\<And>x. x \<in> ball z r \<Longrightarrow> (g has_field_derivative g' x) (at x)"
+    using DERIV_deriv_iff_has_field_derivative
+    by (fastforce simp add: holomorphic_on_open)
+  then have derg: "\<And>x. x \<in> ball z r \<Longrightarrow> deriv g x = g' x"
+    by (simp add: DERIV_imp_deriv)
+  have tends_f'n_g': "((\<lambda>n. f' n w) \<longlongrightarrow> g' w) F" if w: "w \<in> ball z r" for w
+  proof -
+    have eq_f': "?conint (\<lambda>x. f n x / (x - w)\<^sup>2) - ?conint (\<lambda>x. g x / (x - w)\<^sup>2) = (f' n w - g' w) * (2 * of_real pi * \<i>)"
+             if cont_fn: "continuous_on (cball z r) (f n)"
+             and fnd: "\<And>w. w \<in> ball z r \<Longrightarrow> (f n has_field_derivative f' n w) (at w)" for n
+    proof -
+      have hol_fn: "f n holomorphic_on ball z r"
+        using fnd by (force simp: holomorphic_on_open)
+      have "(f n has_field_derivative 1 / (2 * of_real pi * \<i>) * ?conint (\<lambda>u. f n u / (u - w)\<^sup>2)) (at w)"
+        by (rule Cauchy_derivative_integral_circlepath [OF cont_fn hol_fn w])
+      then have f': "f' n w = 1 / (2 * of_real pi * \<i>) * ?conint (\<lambda>u. f n u / (u - w)\<^sup>2)"
+        using DERIV_unique [OF fnd] w by blast
+      show ?thesis
+        by (simp add: f' Cauchy_contour_integral_circlepath_2 [OF g w] derg [OF w] field_split_simps)
+    qed
+    define d where "d = (r - norm(w - z))^2"
+    have "d > 0"
+      using w by (simp add: dist_commute dist_norm d_def)
+    have dle: "d \<le> cmod ((y - w)\<^sup>2)" if "r = cmod (z - y)" for y
+    proof -
+      have "w \<in> ball z (cmod (z - y))"
+        using that w by fastforce
+      then have "cmod (w - z) \<le> cmod (z - y)"
+        by (simp add: dist_complex_def norm_minus_commute)
+      moreover have "cmod (z - y) - cmod (w - z) \<le> cmod (y - w)"
+        by (metis diff_add_cancel diff_add_eq_diff_diff_swap norm_minus_commute norm_triangle_ineq2)
+      ultimately show ?thesis
+        using that by (simp add: d_def norm_power power_mono)
+    qed
+    have 1: "\<forall>\<^sub>F n in F. (\<lambda>x. f n x / (x - w)\<^sup>2) contour_integrable_on circlepath z r"
+      by (force simp: holomorphic_on_open intro: w Cauchy_derivative_integral_circlepath eventually_mono [OF cont])
+    have 2: "uniform_limit (sphere z r) (\<lambda>n x. f n x / (x - w)\<^sup>2) (\<lambda>x. g x / (x - w)\<^sup>2) F"
+      unfolding uniform_limit_iff
+    proof clarify
+      fix e::real
+      assume "0 < e"
+      with \<open>r > 0\<close> show "\<forall>\<^sub>F n in F. \<forall>x\<in>sphere z r. dist (f n x / (x - w)\<^sup>2) (g x / (x - w)\<^sup>2) < e"
+        apply (simp add: norm_divide field_split_simps sphere_def dist_norm)
+        apply (rule eventually_mono [OF uniform_limitD [OF ulim], of "e*d"])
+         apply (simp add: \<open>0 < d\<close>)
+        apply (force simp: dist_norm dle intro: less_le_trans)
+        done
+    qed
+    have "((\<lambda>n. contour_integral (circlepath z r) (\<lambda>x. f n x / (x - w)\<^sup>2))
+             \<longlongrightarrow> contour_integral (circlepath z r) ((\<lambda>x. g x / (x - w)\<^sup>2))) F"
+      by (rule contour_integral_uniform_limit_circlepath [OF 1 2 F \<open>0 < r\<close>])
+    then have tendsto_0: "((\<lambda>n. 1 / (2 * of_real pi * \<i>) * (?conint (\<lambda>x. f n x / (x - w)\<^sup>2) - ?conint (\<lambda>x. g x / (x - w)\<^sup>2))) \<longlongrightarrow> 0) F"
+      using Lim_null by (force intro!: tendsto_mult_right_zero)
+    have "((\<lambda>n. f' n w - g' w) \<longlongrightarrow> 0) F"
+      apply (rule Lim_transform_eventually [OF tendsto_0])
+      apply (force simp: divide_simps intro: eq_f' eventually_mono [OF cont])
+      done
+    then show ?thesis using Lim_null by blast
+  qed
+  obtain g' where "\<And>w. w \<in> ball z r \<Longrightarrow> (g has_field_derivative (g' w)) (at w) \<and> ((\<lambda>n. f' n w) \<longlongrightarrow> g' w) F"
+      by (blast intro: tends_f'n_g' g')
+  then show ?thesis using g
+    using that by blast
+qed
+
+
+subsection\<^marker>\<open>tag unimportant\<close> \<open>Some more simple/convenient versions for applications\<close>
+
+lemma holomorphic_uniform_sequence:
+  assumes S: "open S"
+      and hol_fn: "\<And>n. (f n) holomorphic_on S"
+      and ulim_g: "\<And>x. x \<in> S \<Longrightarrow> \<exists>d. 0 < d \<and> cball x d \<subseteq> S \<and> uniform_limit (cball x d) f g sequentially"
+  shows "g holomorphic_on S"
+proof -
+  have "\<exists>f'. (g has_field_derivative f') (at z)" if "z \<in> S" for z
+  proof -
+    obtain r where "0 < r" and r: "cball z r \<subseteq> S"
+               and ul: "uniform_limit (cball z r) f g sequentially"
+      using ulim_g [OF \<open>z \<in> S\<close>] by blast
+    have *: "\<forall>\<^sub>F n in sequentially. continuous_on (cball z r) (f n) \<and> f n holomorphic_on ball z r"
+    proof (intro eventuallyI conjI)
+      show "continuous_on (cball z r) (f x)" for x
+        using hol_fn holomorphic_on_imp_continuous_on holomorphic_on_subset r by blast
+      show "f x holomorphic_on ball z r" for x
+        by (metis hol_fn holomorphic_on_subset interior_cball interior_subset r)
+    qed
+    show ?thesis
+      apply (rule holomorphic_uniform_limit [OF *])
+      using \<open>0 < r\<close> centre_in_ball ul
+      apply (auto simp: holomorphic_on_open)
+      done
+  qed
+  with S show ?thesis
+    by (simp add: holomorphic_on_open)
+qed
+
+lemma has_complex_derivative_uniform_sequence:
+  fixes S :: "complex set"
+  assumes S: "open S"
+      and hfd: "\<And>n x. x \<in> S \<Longrightarrow> ((f n) has_field_derivative f' n x) (at x)"
+      and ulim_g: "\<And>x. x \<in> S
+             \<Longrightarrow> \<exists>d. 0 < d \<and> cball x d \<subseteq> S \<and> uniform_limit (cball x d) f g sequentially"
+  shows "\<exists>g'. \<forall>x \<in> S. (g has_field_derivative g' x) (at x) \<and> ((\<lambda>n. f' n x) \<longlongrightarrow> g' x) sequentially"
+proof -
+  have y: "\<exists>y. (g has_field_derivative y) (at z) \<and> (\<lambda>n. f' n z) \<longlonglongrightarrow> y" if "z \<in> S" for z
+  proof -
+    obtain r where "0 < r" and r: "cball z r \<subseteq> S"
+               and ul: "uniform_limit (cball z r) f g sequentially"
+      using ulim_g [OF \<open>z \<in> S\<close>] by blast
+    have *: "\<forall>\<^sub>F n in sequentially. continuous_on (cball z r) (f n) \<and>
+                                   (\<forall>w \<in> ball z r. ((f n) has_field_derivative (f' n w)) (at w))"
+    proof (intro eventuallyI conjI ballI)
+      show "continuous_on (cball z r) (f x)" for x
+        by (meson S continuous_on_subset hfd holomorphic_on_imp_continuous_on holomorphic_on_open r)
+      show "w \<in> ball z r \<Longrightarrow> (f x has_field_derivative f' x w) (at w)" for w x
+        using ball_subset_cball hfd r by blast
+    qed
+    show ?thesis
+      by (rule has_complex_derivative_uniform_limit [OF *, of g]) (use \<open>0 < r\<close> ul in \<open>force+\<close>)
+  qed
+  show ?thesis
+    by (rule bchoice) (blast intro: y)
+qed
+
+subsection\<open>On analytic functions defined by a series\<close>
+
+lemma series_and_derivative_comparison:
+  fixes S :: "complex set"
+  assumes S: "open S"
+      and h: "summable h"
+      and hfd: "\<And>n x. x \<in> S \<Longrightarrow> (f n has_field_derivative f' n x) (at x)"
+      and to_g: "\<forall>\<^sub>F n in sequentially. \<forall>x\<in>S. norm (f n x) \<le> h n"
+  obtains g g' where "\<forall>x \<in> S. ((\<lambda>n. f n x) sums g x) \<and> ((\<lambda>n. f' n x) sums g' x) \<and> (g has_field_derivative g' x) (at x)"
+proof -
+  obtain g where g: "uniform_limit S (\<lambda>n x. \<Sum>i<n. f i x) g sequentially"
+    using Weierstrass_m_test_ev [OF to_g h]  by force
+  have *: "\<exists>d>0. cball x d \<subseteq> S \<and> uniform_limit (cball x d) (\<lambda>n x. \<Sum>i<n. f i x) g sequentially"
+         if "x \<in> S" for x
+  proof -
+    obtain d where "d>0" and d: "cball x d \<subseteq> S"
+      using open_contains_cball [of "S"] \<open>x \<in> S\<close> S by blast
+    show ?thesis
+    proof (intro conjI exI)
+      show "uniform_limit (cball x d) (\<lambda>n x. \<Sum>i<n. f i x) g sequentially"
+        using d g uniform_limit_on_subset by (force simp: dist_norm eventually_sequentially)
+    qed (use \<open>d > 0\<close> d in auto)
+  qed
+  have "\<And>x. x \<in> S \<Longrightarrow> (\<lambda>n. \<Sum>i<n. f i x) \<longlonglongrightarrow> g x"
+    by (metis tendsto_uniform_limitI [OF g])
+  moreover have "\<exists>g'. \<forall>x\<in>S. (g has_field_derivative g' x) (at x) \<and> (\<lambda>n. \<Sum>i<n. f' i x) \<longlonglongrightarrow> g' x"
+    by (rule has_complex_derivative_uniform_sequence [OF S]) (auto intro: * hfd DERIV_sum)+
+  ultimately show ?thesis
+    by (metis sums_def that)
+qed
+
+text\<open>A version where we only have local uniform/comparative convergence.\<close>
+
+lemma series_and_derivative_comparison_local:
+  fixes S :: "complex set"
+  assumes S: "open S"
+      and hfd: "\<And>n x. x \<in> S \<Longrightarrow> (f n has_field_derivative f' n x) (at x)"
+      and to_g: "\<And>x. x \<in> S \<Longrightarrow> \<exists>d h. 0 < d \<and> summable h \<and> (\<forall>\<^sub>F n in sequentially. \<forall>y\<in>ball x d \<inter> S. norm (f n y) \<le> h n)"
+  shows "\<exists>g g'. \<forall>x \<in> S. ((\<lambda>n. f n x) sums g x) \<and> ((\<lambda>n. f' n x) sums g' x) \<and> (g has_field_derivative g' x) (at x)"
+proof -
+  have "\<exists>y. (\<lambda>n. f n z) sums (\<Sum>n. f n z) \<and> (\<lambda>n. f' n z) sums y \<and> ((\<lambda>x. \<Sum>n. f n x) has_field_derivative y) (at z)"
+       if "z \<in> S" for z
+  proof -
+    obtain d h where "0 < d" "summable h" and le_h: "\<forall>\<^sub>F n in sequentially. \<forall>y\<in>ball z d \<inter> S. norm (f n y) \<le> h n"
+      using to_g \<open>z \<in> S\<close> by meson
+    then obtain r where "r>0" and r: "ball z r \<subseteq> ball z d \<inter> S" using \<open>z \<in> S\<close> S
+      by (metis Int_iff open_ball centre_in_ball open_Int open_contains_ball_eq)
+    have 1: "open (ball z d \<inter> S)"
+      by (simp add: open_Int S)
+    have 2: "\<And>n x. x \<in> ball z d \<inter> S \<Longrightarrow> (f n has_field_derivative f' n x) (at x)"
+      by (auto simp: hfd)
+    obtain g g' where gg': "\<forall>x \<in> ball z d \<inter> S. ((\<lambda>n. f n x) sums g x) \<and>
+                                    ((\<lambda>n. f' n x) sums g' x) \<and> (g has_field_derivative g' x) (at x)"
+      by (auto intro: le_h series_and_derivative_comparison [OF 1 \<open>summable h\<close> hfd])
+    then have "(\<lambda>n. f' n z) sums g' z"
+      by (meson \<open>0 < r\<close> centre_in_ball contra_subsetD r)
+    moreover have "(\<lambda>n. f n z) sums (\<Sum>n. f n z)"
+      using  summable_sums centre_in_ball \<open>0 < d\<close> \<open>summable h\<close> le_h
+      by (metis (full_types) Int_iff gg' summable_def that)
+    moreover have "((\<lambda>x. \<Sum>n. f n x) has_field_derivative g' z) (at z)"
+    proof (rule has_field_derivative_transform_within)
+      show "\<And>x. dist x z < r \<Longrightarrow> g x = (\<Sum>n. f n x)"
+        by (metis subsetD dist_commute gg' mem_ball r sums_unique)
+    qed (use \<open>0 < r\<close> gg' \<open>z \<in> S\<close> \<open>0 < d\<close> in auto)
+    ultimately show ?thesis by auto
+  qed
+  then show ?thesis
+    by (rule_tac x="\<lambda>x. suminf (\<lambda>n. f n x)" in exI) meson
+qed
+
+
+text\<open>Sometimes convenient to compare with a complex series of positive reals. (?)\<close>
+
+lemma series_and_derivative_comparison_complex:
+  fixes S :: "complex set"
+  assumes S: "open S"
+      and hfd: "\<And>n x. x \<in> S \<Longrightarrow> (f n has_field_derivative f' n x) (at x)"
+      and to_g: "\<And>x. x \<in> S \<Longrightarrow> \<exists>d h. 0 < d \<and> summable h \<and> range h \<subseteq> \<real>\<^sub>\<ge>\<^sub>0 \<and> (\<forall>\<^sub>F n in sequentially. \<forall>y\<in>ball x d \<inter> S. cmod(f n y) \<le> cmod (h n))"
+  shows "\<exists>g g'. \<forall>x \<in> S. ((\<lambda>n. f n x) sums g x) \<and> ((\<lambda>n. f' n x) sums g' x) \<and> (g has_field_derivative g' x) (at x)"
+apply (rule series_and_derivative_comparison_local [OF S hfd], assumption)
+apply (rule ex_forward [OF to_g], assumption)
+apply (erule exE)
+apply (rule_tac x="Re \<circ> h" in exI)
+apply (force simp: summable_Re o_def nonneg_Reals_cmod_eq_Re image_subset_iff)
+done
+
+text\<open>Sometimes convenient to compare with a complex series of positive reals. (?)\<close>
+lemma series_differentiable_comparison_complex:
+  fixes S :: "complex set"
+  assumes S: "open S"
+    and hfd: "\<And>n x. x \<in> S \<Longrightarrow> f n field_differentiable (at x)"
+    and to_g: "\<And>x. x \<in> S \<Longrightarrow> \<exists>d h. 0 < d \<and> summable h \<and> range h \<subseteq> \<real>\<^sub>\<ge>\<^sub>0 \<and> (\<forall>\<^sub>F n in sequentially. \<forall>y\<in>ball x d \<inter> S. cmod(f n y) \<le> cmod (h n))"
+  obtains g where "\<forall>x \<in> S. ((\<lambda>n. f n x) sums g x) \<and> g field_differentiable (at x)"
+proof -
+  have hfd': "\<And>n x. x \<in> S \<Longrightarrow> (f n has_field_derivative deriv (f n) x) (at x)"
+    using hfd field_differentiable_derivI by blast
+  have "\<exists>g g'. \<forall>x \<in> S. ((\<lambda>n. f n x) sums g x) \<and> ((\<lambda>n. deriv (f n) x) sums g' x) \<and> (g has_field_derivative g' x) (at x)"
+    by (metis series_and_derivative_comparison_complex [OF S hfd' to_g])
+  then show ?thesis
+    using field_differentiable_def that by blast
+qed
+
+text\<open>In particular, a power series is analytic inside circle of convergence.\<close>
+
+lemma power_series_and_derivative_0:
+  fixes a :: "nat \<Rightarrow> complex" and r::real
+  assumes "summable (\<lambda>n. a n * r^n)"
+    shows "\<exists>g g'. \<forall>z. cmod z < r \<longrightarrow>
+             ((\<lambda>n. a n * z^n) sums g z) \<and> ((\<lambda>n. of_nat n * a n * z^(n - 1)) sums g' z) \<and> (g has_field_derivative g' z) (at z)"
+proof (cases "0 < r")
+  case True
+    have der: "\<And>n z. ((\<lambda>x. a n * x ^ n) has_field_derivative of_nat n * a n * z ^ (n - 1)) (at z)"
+      by (rule derivative_eq_intros | simp)+
+    have y_le: "\<lbrakk>cmod (z - y) * 2 < r - cmod z\<rbrakk> \<Longrightarrow> cmod y \<le> cmod (of_real r + of_real (cmod z)) / 2" for z y
+      using \<open>r > 0\<close>
+      apply (auto simp: algebra_simps norm_mult norm_divide norm_power simp flip: of_real_add)
+      using norm_triangle_ineq2 [of y z]
+      apply (simp only: diff_le_eq norm_minus_commute mult_2)
+      done
+    have "summable (\<lambda>n. a n * complex_of_real r ^ n)"
+      using assms \<open>r > 0\<close> by simp
+    moreover have "\<And>z. cmod z < r \<Longrightarrow> cmod ((of_real r + of_real (cmod z)) / 2) < cmod (of_real r)"
+      using \<open>r > 0\<close>
+      by (simp flip: of_real_add)
+    ultimately have sum: "\<And>z. cmod z < r \<Longrightarrow> summable (\<lambda>n. of_real (cmod (a n)) * ((of_real r + complex_of_real (cmod z)) / 2) ^ n)"
+      by (rule power_series_conv_imp_absconv_weak)
+    have "\<exists>g g'. \<forall>z \<in> ball 0 r. (\<lambda>n.  (a n) * z ^ n) sums g z \<and>
+               (\<lambda>n. of_nat n * (a n) * z ^ (n - 1)) sums g' z \<and> (g has_field_derivative g' z) (at z)"
+      apply (rule series_and_derivative_comparison_complex [OF open_ball der])
+      apply (rule_tac x="(r - norm z)/2" in exI)
+      apply (rule_tac x="\<lambda>n. of_real(norm(a n)*((r + norm z)/2)^n)" in exI)
+      using \<open>r > 0\<close>
+      apply (auto simp: sum eventually_sequentially norm_mult norm_power dist_norm intro!: mult_left_mono power_mono y_le)
+      done
+  then show ?thesis
+    by (simp add: ball_def)
+next
+  case False then show ?thesis
+    apply (simp add: not_less)
+    using less_le_trans norm_not_less_zero by blast
+qed
+
+proposition\<^marker>\<open>tag unimportant\<close> power_series_and_derivative:
+  fixes a :: "nat \<Rightarrow> complex" and r::real
+  assumes "summable (\<lambda>n. a n * r^n)"
+    obtains g g' where "\<forall>z \<in> ball w r.
+             ((\<lambda>n. a n * (z - w) ^ n) sums g z) \<and> ((\<lambda>n. of_nat n * a n * (z - w) ^ (n - 1)) sums g' z) \<and>
+              (g has_field_derivative g' z) (at z)"
+  using power_series_and_derivative_0 [OF assms]
+  apply clarify
+  apply (rule_tac g="(\<lambda>z. g(z - w))" in that)
+  using DERIV_shift [where z="-w"]
+  apply (auto simp: norm_minus_commute Ball_def dist_norm)
+  done
+
+proposition\<^marker>\<open>tag unimportant\<close> power_series_holomorphic:
+  assumes "\<And>w. w \<in> ball z r \<Longrightarrow> ((\<lambda>n. a n*(w - z)^n) sums f w)"
+    shows "f holomorphic_on ball z r"
+proof -
+  have "\<exists>f'. (f has_field_derivative f') (at w)" if w: "dist z w < r" for w
+  proof -
+    have inb: "z + complex_of_real ((dist z w + r) / 2) \<in> ball z r"
+    proof -
+      have wz: "cmod (w - z) < r" using w
+        by (auto simp: field_split_simps dist_norm norm_minus_commute)
+      then have "0 \<le> r"
+        by (meson less_eq_real_def norm_ge_zero order_trans)
+      show ?thesis
+        using w by (simp add: dist_norm \<open>0\<le>r\<close> flip: of_real_add)
+    qed
+    have sum: "summable (\<lambda>n. a n * of_real (((cmod (z - w) + r) / 2) ^ n))"
+      using assms [OF inb] by (force simp: summable_def dist_norm)
+    obtain g g' where gg': "\<And>u. u \<in> ball z ((cmod (z - w) + r) / 2) \<Longrightarrow>
+                               (\<lambda>n. a n * (u - z) ^ n) sums g u \<and>
+                               (\<lambda>n. of_nat n * a n * (u - z) ^ (n - 1)) sums g' u \<and> (g has_field_derivative g' u) (at u)"
+      by (rule power_series_and_derivative [OF sum, of z]) fastforce
+    have [simp]: "g u = f u" if "cmod (u - w) < (r - cmod (z - w)) / 2" for u
+    proof -
+      have less: "cmod (z - u) * 2 < cmod (z - w) + r"
+        using that dist_triangle2 [of z u w]
+        by (simp add: dist_norm [symmetric] algebra_simps)
+      show ?thesis
+        apply (rule sums_unique2 [of "\<lambda>n. a n*(u - z)^n"])
+        using gg' [of u] less w
+        apply (auto simp: assms dist_norm)
+        done
+    qed
+    have "(f has_field_derivative g' w) (at w)"
+      by (rule has_field_derivative_transform_within [where d="(r - norm(z - w))/2"])
+      (use w gg' [of w] in \<open>(force simp: dist_norm)+\<close>)
+    then show ?thesis ..
+  qed
+  then show ?thesis by (simp add: holomorphic_on_open)
+qed
+
+corollary holomorphic_iff_power_series:
+     "f holomorphic_on ball z r \<longleftrightarrow>
+      (\<forall>w \<in> ball z r. (\<lambda>n. (deriv ^^ n) f z / (fact n) * (w - z)^n) sums f w)"
+  apply (intro iffI ballI holomorphic_power_series, assumption+)
+  apply (force intro: power_series_holomorphic [where a = "\<lambda>n. (deriv ^^ n) f z / (fact n)"])
+  done
+
+lemma power_series_analytic:
+     "(\<And>w. w \<in> ball z r \<Longrightarrow> (\<lambda>n. a n*(w - z)^n) sums f w) \<Longrightarrow> f analytic_on ball z r"
+  by (force simp: analytic_on_open intro!: power_series_holomorphic)
+
+lemma analytic_iff_power_series:
+     "f analytic_on ball z r \<longleftrightarrow>
+      (\<forall>w \<in> ball z r. (\<lambda>n. (deriv ^^ n) f z / (fact n) * (w - z)^n) sums f w)"
+  by (simp add: analytic_on_open holomorphic_iff_power_series)
+
+subsection\<^marker>\<open>tag unimportant\<close> \<open>Equality between holomorphic functions, on open ball then connected set\<close>
+
+lemma holomorphic_fun_eq_on_ball:
+   "\<lbrakk>f holomorphic_on ball z r; g holomorphic_on ball z r;
+     w \<in> ball z r;
+     \<And>n. (deriv ^^ n) f z = (deriv ^^ n) g z\<rbrakk>
+     \<Longrightarrow> f w = g w"
+  apply (rule sums_unique2 [of "\<lambda>n. (deriv ^^ n) f z / (fact n) * (w - z)^n"])
+  apply (auto simp: holomorphic_iff_power_series)
+  done
+
+lemma holomorphic_fun_eq_0_on_ball:
+   "\<lbrakk>f holomorphic_on ball z r;  w \<in> ball z r;
+     \<And>n. (deriv ^^ n) f z = 0\<rbrakk>
+     \<Longrightarrow> f w = 0"
+  apply (rule sums_unique2 [of "\<lambda>n. (deriv ^^ n) f z / (fact n) * (w - z)^n"])
+  apply (auto simp: holomorphic_iff_power_series)
+  done
+
+lemma holomorphic_fun_eq_0_on_connected:
+  assumes holf: "f holomorphic_on S" and "open S"
+      and cons: "connected S"
+      and der: "\<And>n. (deriv ^^ n) f z = 0"
+      and "z \<in> S" "w \<in> S"
+    shows "f w = 0"
+proof -
+  have *: "ball x e \<subseteq> (\<Inter>n. {w \<in> S. (deriv ^^ n) f w = 0})"
+    if "\<forall>u. (deriv ^^ u) f x = 0" "ball x e \<subseteq> S" for x e
+  proof -
+    have "\<And>x' n. dist x x' < e \<Longrightarrow> (deriv ^^ n) f x' = 0"
+      apply (rule holomorphic_fun_eq_0_on_ball [OF holomorphic_higher_deriv])
+         apply (rule holomorphic_on_subset [OF holf])
+      using that apply simp_all
+      by (metis funpow_add o_apply)
+    with that show ?thesis by auto
+  qed
+  have 1: "openin (top_of_set S) (\<Inter>n. {w \<in> S. (deriv ^^ n) f w = 0})"
+    apply (rule open_subset, force)
+    using \<open>open S\<close>
+    apply (simp add: open_contains_ball Ball_def)
+    apply (erule all_forward)
+    using "*" by auto blast+
+  have 2: "closedin (top_of_set S) (\<Inter>n. {w \<in> S. (deriv ^^ n) f w = 0})"
+    using assms
+    by (auto intro: continuous_closedin_preimage_constant holomorphic_on_imp_continuous_on holomorphic_higher_deriv)
+  obtain e where "e>0" and e: "ball w e \<subseteq> S" using openE [OF \<open>open S\<close> \<open>w \<in> S\<close>] .
+  then have holfb: "f holomorphic_on ball w e"
+    using holf holomorphic_on_subset by blast
+  have 3: "(\<Inter>n. {w \<in> S. (deriv ^^ n) f w = 0}) = S \<Longrightarrow> f w = 0"
+    using \<open>e>0\<close> e by (force intro: holomorphic_fun_eq_0_on_ball [OF holfb])
+  show ?thesis
+    using cons der \<open>z \<in> S\<close>
+    apply (simp add: connected_clopen)
+    apply (drule_tac x="\<Inter>n. {w \<in> S. (deriv ^^ n) f w = 0}" in spec)
+    apply (auto simp: 1 2 3)
+    done
+qed
+
+lemma holomorphic_fun_eq_on_connected:
+  assumes "f holomorphic_on S" "g holomorphic_on S" and "open S"  "connected S"
+      and "\<And>n. (deriv ^^ n) f z = (deriv ^^ n) g z"
+      and "z \<in> S" "w \<in> S"
+    shows "f w = g w"
+proof (rule holomorphic_fun_eq_0_on_connected [of "\<lambda>x. f x - g x" S z, simplified])
+  show "(\<lambda>x. f x - g x) holomorphic_on S"
+    by (intro assms holomorphic_intros)
+  show "\<And>n. (deriv ^^ n) (\<lambda>x. f x - g x) z = 0"
+    using assms higher_deriv_diff by auto
+qed (use assms in auto)
+
+lemma holomorphic_fun_eq_const_on_connected:
+  assumes holf: "f holomorphic_on S" and "open S"
+      and cons: "connected S"
+      and der: "\<And>n. 0 < n \<Longrightarrow> (deriv ^^ n) f z = 0"
+      and "z \<in> S" "w \<in> S"
+    shows "f w = f z"
+proof (rule holomorphic_fun_eq_0_on_connected [of "\<lambda>w. f w - f z" S z, simplified])
+  show "(\<lambda>w. f w - f z) holomorphic_on S"
+    by (intro assms holomorphic_intros)
+  show "\<And>n. (deriv ^^ n) (\<lambda>w. f w - f z) z = 0"
+    by (subst higher_deriv_diff) (use assms in \<open>auto intro: holomorphic_intros\<close>)
+qed (use assms in auto)
+
+subsection\<^marker>\<open>tag unimportant\<close> \<open>Some basic lemmas about poles/singularities\<close>
+
+lemma pole_lemma:
+  assumes holf: "f holomorphic_on S" and a: "a \<in> interior S"
+    shows "(\<lambda>z. if z = a then deriv f a
+                 else (f z - f a) / (z - a)) holomorphic_on S" (is "?F holomorphic_on S")
+proof -
+  have F1: "?F field_differentiable (at u within S)" if "u \<in> S" "u \<noteq> a" for u
+  proof -
+    have fcd: "f field_differentiable at u within S"
+      using holf holomorphic_on_def by (simp add: \<open>u \<in> S\<close>)
+    have cd: "(\<lambda>z. (f z - f a) / (z - a)) field_differentiable at u within S"
+      by (rule fcd derivative_intros | simp add: that)+
+    have "0 < dist a u" using that dist_nz by blast
+    then show ?thesis
+      by (rule field_differentiable_transform_within [OF _ _ _ cd]) (auto simp: \<open>u \<in> S\<close>)
+  qed
+  have F2: "?F field_differentiable at a" if "0 < e" "ball a e \<subseteq> S" for e
+  proof -
+    have holfb: "f holomorphic_on ball a e"
+      by (rule holomorphic_on_subset [OF holf \<open>ball a e \<subseteq> S\<close>])
+    have 2: "?F holomorphic_on ball a e - {a}"
+      apply (simp add: holomorphic_on_def flip: field_differentiable_def)
+      using mem_ball that
+      apply (auto intro: F1 field_differentiable_within_subset)
+      done
+    have "isCont (\<lambda>z. if z = a then deriv f a else (f z - f a) / (z - a)) x"
+            if "dist a x < e" for x
+    proof (cases "x=a")
+      case True
+      then have "f field_differentiable at a"
+        using holfb \<open>0 < e\<close> holomorphic_on_imp_differentiable_at by auto
+      with True show ?thesis
+        by (auto simp: continuous_at has_field_derivative_iff simp flip: DERIV_deriv_iff_field_differentiable
+                elim: rev_iffD1 [OF _ LIM_equal])
+    next
+      case False with 2 that show ?thesis
+        by (force simp: holomorphic_on_open open_Diff field_differentiable_def [symmetric] field_differentiable_imp_continuous_at)
+    qed
+    then have 1: "continuous_on (ball a e) ?F"
+      by (clarsimp simp:  continuous_on_eq_continuous_at)
+    have "?F holomorphic_on ball a e"
+      by (auto intro: no_isolated_singularity [OF 1 2])
+    with that show ?thesis
+      by (simp add: holomorphic_on_open field_differentiable_def [symmetric]
+                    field_differentiable_at_within)
+  qed
+  show ?thesis
+  proof
+    fix x assume "x \<in> S" show "?F field_differentiable at x within S"
+    proof (cases "x=a")
+      case True then show ?thesis
+      using a by (auto simp: mem_interior intro: field_differentiable_at_within F2)
+    next
+      case False with F1 \<open>x \<in> S\<close>
+      show ?thesis by blast
+    qed
+  qed
+qed
+
+lemma pole_theorem:
+  assumes holg: "g holomorphic_on S" and a: "a \<in> interior S"
+      and eq: "\<And>z. z \<in> S - {a} \<Longrightarrow> g z = (z - a) * f z"
+    shows "(\<lambda>z. if z = a then deriv g a
+                 else f z - g a/(z - a)) holomorphic_on S"
+  using pole_lemma [OF holg a]
+  by (rule holomorphic_transform) (simp add: eq field_split_simps)
+
+lemma pole_lemma_open:
+  assumes "f holomorphic_on S" "open S"
+    shows "(\<lambda>z. if z = a then deriv f a else (f z - f a)/(z - a)) holomorphic_on S"
+proof (cases "a \<in> S")
+  case True with assms interior_eq pole_lemma
+    show ?thesis by fastforce
+next
+  case False with assms show ?thesis
+    apply (simp add: holomorphic_on_def field_differentiable_def [symmetric], clarify)
+    apply (rule field_differentiable_transform_within [where f = "\<lambda>z. (f z - f a)/(z - a)" and d = 1])
+    apply (rule derivative_intros | force)+
+    done
+qed
+
+lemma pole_theorem_open:
+  assumes holg: "g holomorphic_on S" and S: "open S"
+      and eq: "\<And>z. z \<in> S - {a} \<Longrightarrow> g z = (z - a) * f z"
+    shows "(\<lambda>z. if z = a then deriv g a
+                 else f z - g a/(z - a)) holomorphic_on S"
+  using pole_lemma_open [OF holg S]
+  by (rule holomorphic_transform) (auto simp: eq divide_simps)
+
+lemma pole_theorem_0:
+  assumes holg: "g holomorphic_on S" and a: "a \<in> interior S"
+      and eq: "\<And>z. z \<in> S - {a} \<Longrightarrow> g z = (z - a) * f z"
+      and [simp]: "f a = deriv g a" "g a = 0"
+    shows "f holomorphic_on S"
+  using pole_theorem [OF holg a eq]
+  by (rule holomorphic_transform) (auto simp: eq field_split_simps)
+
+lemma pole_theorem_open_0:
+  assumes holg: "g holomorphic_on S" and S: "open S"
+      and eq: "\<And>z. z \<in> S - {a} \<Longrightarrow> g z = (z - a) * f z"
+      and [simp]: "f a = deriv g a" "g a = 0"
+    shows "f holomorphic_on S"
+  using pole_theorem_open [OF holg S eq]
+  by (rule holomorphic_transform) (auto simp: eq field_split_simps)
+
+lemma pole_theorem_analytic:
+  assumes g: "g analytic_on S"
+      and eq: "\<And>z. z \<in> S
+             \<Longrightarrow> \<exists>d. 0 < d \<and> (\<forall>w \<in> ball z d - {a}. g w = (w - a) * f w)"
+    shows "(\<lambda>z. if z = a then deriv g a else f z - g a/(z - a)) analytic_on S" (is "?F analytic_on S")
+  unfolding analytic_on_def
+proof
+  fix x
+  assume "x \<in> S"
+  with g obtain e where "0 < e" and e: "g holomorphic_on ball x e"
+    by (auto simp add: analytic_on_def)
+  obtain d where "0 < d" and d: "\<And>w. w \<in> ball x d - {a} \<Longrightarrow> g w = (w - a) * f w"
+    using \<open>x \<in> S\<close> eq by blast
+  have "?F holomorphic_on ball x (min d e)"
+    using d e \<open>x \<in> S\<close> by (fastforce simp: holomorphic_on_subset subset_ball intro!: pole_theorem_open)
+  then show "\<exists>e>0. ?F holomorphic_on ball x e"
+    using \<open>0 < d\<close> \<open>0 < e\<close> not_le by fastforce
+qed
+
+lemma pole_theorem_analytic_0:
+  assumes g: "g analytic_on S"
+      and eq: "\<And>z. z \<in> S \<Longrightarrow> \<exists>d. 0 < d \<and> (\<forall>w \<in> ball z d - {a}. g w = (w - a) * f w)"
+      and [simp]: "f a = deriv g a" "g a = 0"
+    shows "f analytic_on S"
+proof -
+  have [simp]: "(\<lambda>z. if z = a then deriv g a else f z - g a / (z - a)) = f"
+    by auto
+  show ?thesis
+    using pole_theorem_analytic [OF g eq] by simp
+qed
+
+lemma pole_theorem_analytic_open_superset:
+  assumes g: "g analytic_on S" and "S \<subseteq> T" "open T"
+      and eq: "\<And>z. z \<in> T - {a} \<Longrightarrow> g z = (z - a) * f z"
+    shows "(\<lambda>z. if z = a then deriv g a
+                 else f z - g a/(z - a)) analytic_on S"
+proof (rule pole_theorem_analytic [OF g])
+  fix z
+  assume "z \<in> S"
+  then obtain e where "0 < e" and e: "ball z e \<subseteq> T"
+    using assms openE by blast
+  then show "\<exists>d>0. \<forall>w\<in>ball z d - {a}. g w = (w - a) * f w"
+    using eq by auto
+qed
+
+lemma pole_theorem_analytic_open_superset_0:
+  assumes g: "g analytic_on S" "S \<subseteq> T" "open T" "\<And>z. z \<in> T - {a} \<Longrightarrow> g z = (z - a) * f z"
+      and [simp]: "f a = deriv g a" "g a = 0"
+    shows "f analytic_on S"
+proof -
+  have [simp]: "(\<lambda>z. if z = a then deriv g a else f z - g a / (z - a)) = f"
+    by auto
+  have "(\<lambda>z. if z = a then deriv g a else f z - g a/(z - a)) analytic_on S"
+    by (rule pole_theorem_analytic_open_superset [OF g])
+  then show ?thesis by simp
+qed
+
+
+subsection\<open>General, homology form of Cauchy's theorem\<close>
+
+text\<open>Proof is based on Dixon's, as presented in Lang's "Complex Analysis" book (page 147).\<close>
+
+lemma contour_integral_continuous_on_linepath_2D:
+  assumes "open U" and cont_dw: "\<And>w. w \<in> U \<Longrightarrow> F w contour_integrable_on (linepath a b)"
+      and cond_uu: "continuous_on (U \<times> U) (\<lambda>(x,y). F x y)"
+      and abu: "closed_segment a b \<subseteq> U"
+    shows "continuous_on U (\<lambda>w. contour_integral (linepath a b) (F w))"
+proof -
+  have *: "\<exists>d>0. \<forall>x'\<in>U. dist x' w < d \<longrightarrow>
+                         dist (contour_integral (linepath a b) (F x'))
+                              (contour_integral (linepath a b) (F w)) \<le> \<epsilon>"
+          if "w \<in> U" "0 < \<epsilon>" "a \<noteq> b" for w \<epsilon>
+  proof -
+    obtain \<delta> where "\<delta>>0" and \<delta>: "cball w \<delta> \<subseteq> U" using open_contains_cball \<open>open U\<close> \<open>w \<in> U\<close> by force
+    let ?TZ = "cball w \<delta>  \<times> closed_segment a b"
+    have "uniformly_continuous_on ?TZ (\<lambda>(x,y). F x y)"
+    proof (rule compact_uniformly_continuous)
+      show "continuous_on ?TZ (\<lambda>(x,y). F x y)"
+        by (rule continuous_on_subset[OF cond_uu]) (use SigmaE \<delta> abu in blast)
+      show "compact ?TZ"
+        by (simp add: compact_Times)
+    qed
+    then obtain \<eta> where "\<eta>>0"
+        and \<eta>: "\<And>x x'. \<lbrakk>x\<in>?TZ; x'\<in>?TZ; dist x' x < \<eta>\<rbrakk> \<Longrightarrow>
+                         dist ((\<lambda>(x,y). F x y) x') ((\<lambda>(x,y). F x y) x) < \<epsilon>/norm(b - a)"
+      apply (rule uniformly_continuous_onE [where e = "\<epsilon>/norm(b - a)"])
+      using \<open>0 < \<epsilon>\<close> \<open>a \<noteq> b\<close> by auto
+    have \<eta>: "\<lbrakk>norm (w - x1) \<le> \<delta>;   x2 \<in> closed_segment a b;
+              norm (w - x1') \<le> \<delta>;  x2' \<in> closed_segment a b; norm ((x1', x2') - (x1, x2)) < \<eta>\<rbrakk>
+              \<Longrightarrow> norm (F x1' x2' - F x1 x2) \<le> \<epsilon> / cmod (b - a)"
+             for x1 x2 x1' x2'
+      using \<eta> [of "(x1,x2)" "(x1',x2')"] by (force simp: dist_norm)
+    have le_ee: "cmod (contour_integral (linepath a b) (\<lambda>x. F x' x - F w x)) \<le> \<epsilon>"
+                if "x' \<in> U" "cmod (x' - w) < \<delta>" "cmod (x' - w) < \<eta>"  for x'
+    proof -
+      have "(\<lambda>x. F x' x - F w x) contour_integrable_on linepath a b"
+        by (simp add: \<open>w \<in> U\<close> cont_dw contour_integrable_diff that)
+      then have "cmod (contour_integral (linepath a b) (\<lambda>x. F x' x - F w x)) \<le> \<epsilon>/norm(b - a) * norm(b - a)"
+        apply (rule has_contour_integral_bound_linepath [OF has_contour_integral_integral _ \<eta>])
+        using \<open>0 < \<epsilon>\<close> \<open>0 < \<delta>\<close> that apply (auto simp: norm_minus_commute)
+        done
+      also have "\<dots> = \<epsilon>" using \<open>a \<noteq> b\<close> by simp
+      finally show ?thesis .
+    qed
+    show ?thesis
+      apply (rule_tac x="min \<delta> \<eta>" in exI)
+      using \<open>0 < \<delta>\<close> \<open>0 < \<eta>\<close>
+      apply (auto simp: dist_norm contour_integral_diff [OF cont_dw cont_dw, symmetric] \<open>w \<in> U\<close> intro: le_ee)
+      done
+  qed
+  show ?thesis
+  proof (cases "a=b")
+    case True
+    then show ?thesis by simp
+  next
+    case False
+    show ?thesis
+      by (rule continuous_onI) (use False in \<open>auto intro: *\<close>)
+  qed
+qed
+
+text\<open>This version has \<^term>\<open>polynomial_function \<gamma>\<close> as an additional assumption.\<close>
+lemma Cauchy_integral_formula_global_weak:
+  assumes "open U" and holf: "f holomorphic_on U"
+        and z: "z \<in> U" and \<gamma>: "polynomial_function \<gamma>"
+        and pasz: "path_image \<gamma> \<subseteq> U - {z}" and loop: "pathfinish \<gamma> = pathstart \<gamma>"
+        and zero: "\<And>w. w \<notin> U \<Longrightarrow> winding_number \<gamma> w = 0"
+      shows "((\<lambda>w. f w / (w - z)) has_contour_integral (2*pi * \<i> * winding_number \<gamma> z * f z)) \<gamma>"
+proof -
+  obtain \<gamma>' where pf\<gamma>': "polynomial_function \<gamma>'" and \<gamma>': "\<And>x. (\<gamma> has_vector_derivative (\<gamma>' x)) (at x)"
+    using has_vector_derivative_polynomial_function [OF \<gamma>] by blast
+  then have "bounded(path_image \<gamma>')"
+    by (simp add: path_image_def compact_imp_bounded compact_continuous_image continuous_on_polymonial_function)
+  then obtain B where "B>0" and B: "\<And>x. x \<in> path_image \<gamma>' \<Longrightarrow> norm x \<le> B"
+    using bounded_pos by force
+  define d where [abs_def]: "d z w = (if w = z then deriv f z else (f w - f z)/(w - z))" for z w
+  define v where "v = {w. w \<notin> path_image \<gamma> \<and> winding_number \<gamma> w = 0}"
+  have "path \<gamma>" "valid_path \<gamma>" using \<gamma>
+    by (auto simp: path_polynomial_function valid_path_polynomial_function)
+  then have ov: "open v"
+    by (simp add: v_def open_winding_number_levelsets loop)
+  have uv_Un: "U \<union> v = UNIV"
+    using pasz zero by (auto simp: v_def)
+  have conf: "continuous_on U f"
+    by (metis holf holomorphic_on_imp_continuous_on)
+  have hol_d: "(d y) holomorphic_on U" if "y \<in> U" for y
+  proof -
+    have *: "(\<lambda>c. if c = y then deriv f y else (f c - f y) / (c - y)) holomorphic_on U"
+      by (simp add: holf pole_lemma_open \<open>open U\<close>)
+    then have "isCont (\<lambda>x. if x = y then deriv f y else (f x - f y) / (x - y)) y"
+      using at_within_open field_differentiable_imp_continuous_at holomorphic_on_def that \<open>open U\<close> by fastforce
+    then have "continuous_on U (d y)"
+      apply (simp add: d_def continuous_on_eq_continuous_at \<open>open U\<close>, clarify)
+      using * holomorphic_on_def
+      by (meson field_differentiable_within_open field_differentiable_imp_continuous_at \<open>open U\<close>)
+    moreover have "d y holomorphic_on U - {y}"
+    proof -
+      have "\<And>w. w \<in> U - {y} \<Longrightarrow>
+                 (\<lambda>w. if w = y then deriv f y else (f w - f y) / (w - y)) field_differentiable at w"
+        apply (rule_tac d="dist w y" and f = "\<lambda>w. (f w - f y)/(w - y)" in field_differentiable_transform_within)
+           apply (auto simp: dist_pos_lt dist_commute intro!: derivative_intros)
+        using \<open>open U\<close> holf holomorphic_on_imp_differentiable_at by blast
+      then show ?thesis
+        unfolding field_differentiable_def by (simp add: d_def holomorphic_on_open \<open>open U\<close> open_delete)
+    qed
+    ultimately show ?thesis
+      by (rule no_isolated_singularity) (auto simp: \<open>open U\<close>)
+  qed
+  have cint_fxy: "(\<lambda>x. (f x - f y) / (x - y)) contour_integrable_on \<gamma>" if "y \<notin> path_image \<gamma>" for y
+  proof (rule contour_integrable_holomorphic_simple [where S = "U-{y}"])
+    show "(\<lambda>x. (f x - f y) / (x - y)) holomorphic_on U - {y}"
+      by (force intro: holomorphic_intros holomorphic_on_subset [OF holf])
+    show "path_image \<gamma> \<subseteq> U - {y}"
+      using pasz that by blast
+  qed (auto simp: \<open>open U\<close> open_delete \<open>valid_path \<gamma>\<close>)
+  define h where
+    "h z = (if z \<in> U then contour_integral \<gamma> (d z) else contour_integral \<gamma> (\<lambda>w. f w/(w - z)))" for z
+  have U: "((d z) has_contour_integral h z) \<gamma>" if "z \<in> U" for z
+  proof -
+    have "d z holomorphic_on U"
+      by (simp add: hol_d that)
+    with that show ?thesis
+    apply (simp add: h_def)
+      by (meson Diff_subset \<open>open U\<close> \<open>valid_path \<gamma>\<close> contour_integrable_holomorphic_simple has_contour_integral_integral pasz subset_trans)
+  qed
+  have V: "((\<lambda>w. f w / (w - z)) has_contour_integral h z) \<gamma>" if z: "z \<in> v" for z
+  proof -
+    have 0: "0 = (f z) * 2 * of_real (2 * pi) * \<i> * winding_number \<gamma> z"
+      using v_def z by auto
+    then have "((\<lambda>x. 1 / (x - z)) has_contour_integral 0) \<gamma>"
+     using z v_def  has_contour_integral_winding_number [OF \<open>valid_path \<gamma>\<close>] by fastforce
+    then have "((\<lambda>x. f z * (1 / (x - z))) has_contour_integral 0) \<gamma>"
+      using has_contour_integral_lmul by fastforce
+    then have "((\<lambda>x. f z / (x - z)) has_contour_integral 0) \<gamma>"
+      by (simp add: field_split_simps)
+    moreover have "((\<lambda>x. (f x - f z) / (x - z)) has_contour_integral contour_integral \<gamma> (d z)) \<gamma>"
+      using z
+      apply (auto simp: v_def)
+      apply (metis (no_types, lifting) contour_integrable_eq d_def has_contour_integral_eq has_contour_integral_integral cint_fxy)
+      done
+    ultimately have *: "((\<lambda>x. f z / (x - z) + (f x - f z) / (x - z)) has_contour_integral (0 + contour_integral \<gamma> (d z))) \<gamma>"
+      by (rule has_contour_integral_add)
+    have "((\<lambda>w. f w / (w - z)) has_contour_integral contour_integral \<gamma> (d z)) \<gamma>"
+            if  "z \<in> U"
+      using * by (auto simp: divide_simps has_contour_integral_eq)
+    moreover have "((\<lambda>w. f w / (w - z)) has_contour_integral contour_integral \<gamma> (\<lambda>w. f w / (w - z))) \<gamma>"
+            if "z \<notin> U"
+      apply (rule has_contour_integral_integral [OF contour_integrable_holomorphic_simple [where S=U]])
+      using U pasz \<open>valid_path \<gamma>\<close> that
+      apply (auto intro: holomorphic_on_imp_continuous_on hol_d)
+       apply (rule continuous_intros conf holomorphic_intros holf assms | force)+
+      done
+    ultimately show ?thesis
+      using z by (simp add: h_def)
+  qed
+  have znot: "z \<notin> path_image \<gamma>"
+    using pasz by blast
+  obtain d0 where "d0>0" and d0: "\<And>x y. x \<in> path_image \<gamma> \<Longrightarrow> y \<in> - U \<Longrightarrow> d0 \<le> dist x y"
+    using separate_compact_closed [of "path_image \<gamma>" "-U"] pasz \<open>open U\<close> \<open>path \<gamma>\<close> compact_path_image
+    by blast    
+  obtain dd where "0 < dd" and dd: "{y + k | y k. y \<in> path_image \<gamma> \<and> k \<in> ball 0 dd} \<subseteq> U"
+    apply (rule that [of "d0/2"])
+    using \<open>0 < d0\<close>
+    apply (auto simp: dist_norm dest: d0)
+    done
+  have "\<And>x x'. \<lbrakk>x \<in> path_image \<gamma>; dist x x' * 2 < dd\<rbrakk> \<Longrightarrow> \<exists>y k. x' = y + k \<and> y \<in> path_image \<gamma> \<and> dist 0 k * 2 \<le> dd"
+    apply (rule_tac x=x in exI)
+    apply (rule_tac x="x'-x" in exI)
+    apply (force simp: dist_norm)
+    done
+  then have 1: "path_image \<gamma> \<subseteq> interior {y + k |y k. y \<in> path_image \<gamma> \<and> k \<in> cball 0 (dd / 2)}"
+    apply (clarsimp simp add: mem_interior)
+    using \<open>0 < dd\<close>
+    apply (rule_tac x="dd/2" in exI, auto)
+    done
+  obtain T where "compact T" and subt: "path_image \<gamma> \<subseteq> interior T" and T: "T \<subseteq> U"
+    apply (rule that [OF _ 1])
+    apply (fastforce simp add: \<open>valid_path \<gamma>\<close> compact_valid_path_image intro!: compact_sums)
+    apply (rule order_trans [OF _ dd])
+    using \<open>0 < dd\<close> by fastforce
+  obtain L where "L>0"
+           and L: "\<And>f B. \<lbrakk>f holomorphic_on interior T; \<And>z. z\<in>interior T \<Longrightarrow> cmod (f z) \<le> B\<rbrakk> \<Longrightarrow>
+                         cmod (contour_integral \<gamma> f) \<le> L * B"
+      using contour_integral_bound_exists [OF open_interior \<open>valid_path \<gamma>\<close> subt]
+      by blast
+  have "bounded(f ` T)"
+    by (meson \<open>compact T\<close> compact_continuous_image compact_imp_bounded conf continuous_on_subset T)
+  then obtain D where "D>0" and D: "\<And>x. x \<in> T \<Longrightarrow> norm (f x) \<le> D"
+    by (auto simp: bounded_pos)
+  obtain C where "C>0" and C: "\<And>x. x \<in> T \<Longrightarrow> norm x \<le> C"
+    using \<open>compact T\<close> bounded_pos compact_imp_bounded by force
+  have "dist (h y) 0 \<le> e" if "0 < e" and le: "D * L / e + C \<le> cmod y" for e y
+  proof -
+    have "D * L / e > 0"  using \<open>D>0\<close> \<open>L>0\<close> \<open>e>0\<close> by simp
+    with le have ybig: "norm y > C" by force
+    with C have "y \<notin> T"  by force
+    then have ynot: "y \<notin> path_image \<gamma>"
+      using subt interior_subset by blast
+    have [simp]: "winding_number \<gamma> y = 0"
+      apply (rule winding_number_zero_outside [of _ "cball 0 C"])
+      using ybig interior_subset subt
+      apply (force simp: loop \<open>path \<gamma>\<close> dist_norm intro!: C)+
+      done
+    have [simp]: "h y = contour_integral \<gamma> (\<lambda>w. f w/(w - y))"
+      by (rule contour_integral_unique [symmetric]) (simp add: v_def ynot V)
+    have holint: "(\<lambda>w. f w / (w - y)) holomorphic_on interior T"
+      apply (rule holomorphic_on_divide)
+      using holf holomorphic_on_subset interior_subset T apply blast
+      apply (rule holomorphic_intros)+
+      using \<open>y \<notin> T\<close> interior_subset by auto
+    have leD: "cmod (f z / (z - y)) \<le> D * (e / L / D)" if z: "z \<in> interior T" for z
+    proof -
+      have "D * L / e + cmod z \<le> cmod y"
+        using le C [of z] z using interior_subset by force
+      then have DL2: "D * L / e \<le> cmod (z - y)"
+        using norm_triangle_ineq2 [of y z] by (simp add: norm_minus_commute)
+      have "cmod (f z / (z - y)) = cmod (f z) * inverse (cmod (z - y))"
+        by (simp add: norm_mult norm_inverse Fields.field_class.field_divide_inverse)
+      also have "\<dots> \<le> D * (e / L / D)"
+        apply (rule mult_mono)
+        using that D interior_subset apply blast
+        using \<open>L>0\<close> \<open>e>0\<close> \<open>D>0\<close> DL2
+        apply (auto simp: norm_divide field_split_simps)
+        done
+      finally show ?thesis .
+    qed
+    have "dist (h y) 0 = cmod (contour_integral \<gamma> (\<lambda>w. f w / (w - y)))"
+      by (simp add: dist_norm)
+    also have "\<dots> \<le> L * (D * (e / L / D))"
+      by (rule L [OF holint leD])
+    also have "\<dots> = e"
+      using  \<open>L>0\<close> \<open>0 < D\<close> by auto
+    finally show ?thesis .
+  qed
+  then have "(h \<longlongrightarrow> 0) at_infinity"
+    by (meson Lim_at_infinityI)
+  moreover have "h holomorphic_on UNIV"
+  proof -
+    have con_ff: "continuous (at (x,z)) (\<lambda>(x,y). (f y - f x) / (y - x))"
+                 if "x \<in> U" "z \<in> U" "x \<noteq> z" for x z
+      using that conf
+      apply (simp add: split_def continuous_on_eq_continuous_at \<open>open U\<close>)
+      apply (simp | rule continuous_intros continuous_within_compose2 [where g=f])+
+      done
+    have con_fstsnd: "continuous_on UNIV (\<lambda>x. (fst x - snd x) ::complex)"
+      by (rule continuous_intros)+
+    have open_uu_Id: "open (U \<times> U - Id)"
+      apply (rule open_Diff)
+      apply (simp add: open_Times \<open>open U\<close>)
+      using continuous_closed_preimage_constant [OF con_fstsnd closed_UNIV, of 0]
+      apply (auto simp: Id_fstsnd_eq algebra_simps)
+      done
+    have con_derf: "continuous (at z) (deriv f)" if "z \<in> U" for z
+      apply (rule continuous_on_interior [of U])
+      apply (simp add: holf holomorphic_deriv holomorphic_on_imp_continuous_on \<open>open U\<close>)
+      by (simp add: interior_open that \<open>open U\<close>)
+    have tendsto_f': "((\<lambda>(x,y). if y = x then deriv f (x)
+                                else (f (y) - f (x)) / (y - x)) \<longlongrightarrow> deriv f x)
+                      (at (x, x) within U \<times> U)" if "x \<in> U" for x
+    proof (rule Lim_withinI)
+      fix e::real assume "0 < e"
+      obtain k1 where "k1>0" and k1: "\<And>x'. norm (x' - x) \<le> k1 \<Longrightarrow> norm (deriv f x' - deriv f x) < e"
+        using \<open>0 < e\<close> continuous_within_E [OF con_derf [OF \<open>x \<in> U\<close>]]
+        by (metis UNIV_I dist_norm)
+      obtain k2 where "k2>0" and k2: "ball x k2 \<subseteq> U"
+        by (blast intro: openE [OF \<open>open U\<close>] \<open>x \<in> U\<close>)
+      have neq: "norm ((f z' - f x') / (z' - x') - deriv f x) \<le> e"
+                    if "z' \<noteq> x'" and less_k1: "norm (x'-x, z'-x) < k1" and less_k2: "norm (x'-x, z'-x) < k2"
+                 for x' z'
+      proof -
+        have cs_less: "w \<in> closed_segment x' z' \<Longrightarrow> cmod (w - x) \<le> norm (x'-x, z'-x)" for w
+          apply (drule segment_furthest_le [where y=x])
+          by (metis (no_types) dist_commute dist_norm norm_fst_le norm_snd_le order_trans)
+        have derf_le: "w \<in> closed_segment x' z' \<Longrightarrow> z' \<noteq> x' \<Longrightarrow> cmod (deriv f w - deriv f x) \<le> e" for w
+          by (blast intro: cs_less less_k1 k1 [unfolded divide_const_simps dist_norm] less_imp_le le_less_trans)
+        have f_has_der: "\<And>x. x \<in> U \<Longrightarrow> (f has_field_derivative deriv f x) (at x within U)"
+          by (metis DERIV_deriv_iff_field_differentiable at_within_open holf holomorphic_on_def \<open>open U\<close>)
+        have "closed_segment x' z' \<subseteq> U"
+          by (rule order_trans [OF _ k2]) (simp add: cs_less  le_less_trans [OF _ less_k2] dist_complex_def norm_minus_commute subset_iff)
+        then have cint_derf: "(deriv f has_contour_integral f z' - f x') (linepath x' z')"
+          using contour_integral_primitive [OF f_has_der valid_path_linepath] pasz  by simp
+        then have *: "((\<lambda>x. deriv f x / (z' - x')) has_contour_integral (f z' - f x') / (z' - x')) (linepath x' z')"
+          by (rule has_contour_integral_div)
+        have "norm ((f z' - f x') / (z' - x') - deriv f x) \<le> e/norm(z' - x') * norm(z' - x')"
+          apply (rule has_contour_integral_bound_linepath [OF has_contour_integral_diff [OF *]])
+          using has_contour_integral_div [where c = "z' - x'", OF has_contour_integral_const_linepath [of "deriv f x" z' x']]
+                 \<open>e > 0\<close>  \<open>z' \<noteq> x'\<close>
+          apply (auto simp: norm_divide divide_simps derf_le)
+          done
+        also have "\<dots> \<le> e" using \<open>0 < e\<close> by simp
+        finally show ?thesis .
+      qed
+      show "\<exists>d>0. \<forall>xa\<in>U \<times> U.
+                  0 < dist xa (x, x) \<and> dist xa (x, x) < d \<longrightarrow>
+                  dist (case xa of (x, y) \<Rightarrow> if y = x then deriv f x else (f y - f x) / (y - x)) (deriv f x) \<le> e"
+        apply (rule_tac x="min k1 k2" in exI)
+        using \<open>k1>0\<close> \<open>k2>0\<close> \<open>e>0\<close>
+        apply (force simp: dist_norm neq intro: dual_order.strict_trans2 k1 less_imp_le norm_fst_le)
+        done
+    qed
+    have con_pa_f: "continuous_on (path_image \<gamma>) f"
+      by (meson holf holomorphic_on_imp_continuous_on holomorphic_on_subset interior_subset subt T)
+    have le_B: "\<And>T. T \<in> {0..1} \<Longrightarrow> cmod (vector_derivative \<gamma> (at T)) \<le> B"
+      apply (rule B)
+      using \<gamma>' using path_image_def vector_derivative_at by fastforce
+    have f_has_cint: "\<And>w. w \<in> v - path_image \<gamma> \<Longrightarrow> ((\<lambda>u. f u / (u - w) ^ 1) has_contour_integral h w) \<gamma>"
+      by (simp add: V)
+    have cond_uu: "continuous_on (U \<times> U) (\<lambda>(x,y). d x y)"
+      apply (simp add: continuous_on_eq_continuous_within d_def continuous_within tendsto_f')
+      apply (simp add: tendsto_within_open_NO_MATCH open_Times \<open>open U\<close>, clarify)
+      apply (rule Lim_transform_within_open [OF _ open_uu_Id, where f = "(\<lambda>(x,y). (f y - f x) / (y - x))"])
+      using con_ff
+      apply (auto simp: continuous_within)
+      done
+    have hol_dw: "(\<lambda>z. d z w) holomorphic_on U" if "w \<in> U" for w
+    proof -
+      have "continuous_on U ((\<lambda>(x,y). d x y) \<circ> (\<lambda>z. (w,z)))"
+        by (rule continuous_on_compose continuous_intros continuous_on_subset [OF cond_uu] | force intro: that)+
+      then have *: "continuous_on U (\<lambda>z. if w = z then deriv f z else (f w - f z) / (w - z))"
+        by (rule rev_iffD1 [OF _ continuous_on_cong [OF refl]]) (simp add: d_def field_simps)
+      have **: "\<And>x. \<lbrakk>x \<in> U; x \<noteq> w\<rbrakk> \<Longrightarrow> (\<lambda>z. if w = z then deriv f z else (f w - f z) / (w - z)) field_differentiable at x"
+        apply (rule_tac f = "\<lambda>x. (f w - f x)/(w - x)" and d = "dist x w" in field_differentiable_transform_within)
+        apply (rule \<open>open U\<close> derivative_intros holomorphic_on_imp_differentiable_at [OF holf] | force simp: dist_commute)+
+        done
+      show ?thesis
+        unfolding d_def
+        apply (rule no_isolated_singularity [OF * _ \<open>open U\<close>, where K = "{w}"])
+        apply (auto simp: field_differentiable_def [symmetric] holomorphic_on_open open_Diff \<open>open U\<close> **)
+        done
+    qed
+    { fix a b
+      assume abu: "closed_segment a b \<subseteq> U"
+      then have "\<And>w. w \<in> U \<Longrightarrow> (\<lambda>z. d z w) contour_integrable_on (linepath a b)"
+        by (metis hol_dw continuous_on_subset contour_integrable_continuous_linepath holomorphic_on_imp_continuous_on)
+      then have cont_cint_d: "continuous_on U (\<lambda>w. contour_integral (linepath a b) (\<lambda>z. d z w))"
+        apply (rule contour_integral_continuous_on_linepath_2D [OF \<open>open U\<close> _ _ abu])
+        apply (auto intro: continuous_on_swap_args cond_uu)
+        done
+      have cont_cint_d\<gamma>: "continuous_on {0..1} ((\<lambda>w. contour_integral (linepath a b) (\<lambda>z. d z w)) \<circ> \<gamma>)"
+      proof (rule continuous_on_compose)
+        show "continuous_on {0..1} \<gamma>"
+          using \<open>path \<gamma>\<close> path_def by blast
+        show "continuous_on (\<gamma> ` {0..1}) (\<lambda>w. contour_integral (linepath a b) (\<lambda>z. d z w))"
+          using pasz unfolding path_image_def
+          by (auto intro!: continuous_on_subset [OF cont_cint_d])
+      qed
+      have cint_cint: "(\<lambda>w. contour_integral (linepath a b) (\<lambda>z. d z w)) contour_integrable_on \<gamma>"
+        apply (simp add: contour_integrable_on)
+        apply (rule integrable_continuous_real)
+        apply (rule continuous_on_mult [OF cont_cint_d\<gamma> [unfolded o_def]])
+        using pf\<gamma>'
+        by (simp add: continuous_on_polymonial_function vector_derivative_at [OF \<gamma>'])
+      have "contour_integral (linepath a b) h = contour_integral (linepath a b) (\<lambda>z. contour_integral \<gamma> (d z))"
+        using abu  by (force simp: h_def intro: contour_integral_eq)
+      also have "\<dots> =  contour_integral \<gamma> (\<lambda>w. contour_integral (linepath a b) (\<lambda>z. d z w))"
+        apply (rule contour_integral_swap)
+        apply (rule continuous_on_subset [OF cond_uu])
+        using abu pasz \<open>valid_path \<gamma>\<close>
+        apply (auto intro!: continuous_intros)
+        by (metis \<gamma>' continuous_on_eq path_def path_polynomial_function pf\<gamma>' vector_derivative_at)
+      finally have cint_h_eq:
+          "contour_integral (linepath a b) h =
+                    contour_integral \<gamma> (\<lambda>w. contour_integral (linepath a b) (\<lambda>z. d z w))" .
+      note cint_cint cint_h_eq
+    } note cint_h = this
+    have conthu: "continuous_on U h"
+    proof (simp add: continuous_on_sequentially, clarify)
+      fix a x
+      assume x: "x \<in> U" and au: "\<forall>n. a n \<in> U" and ax: "a \<longlonglongrightarrow> x"
+      then have A1: "\<forall>\<^sub>F n in sequentially. d (a n) contour_integrable_on \<gamma>"
+        by (meson U contour_integrable_on_def eventuallyI)
+      obtain dd where "dd>0" and dd: "cball x dd \<subseteq> U" using open_contains_cball \<open>open U\<close> x by force
+      have A2: "uniform_limit (path_image \<gamma>) (\<lambda>n. d (a n)) (d x) sequentially"
+        unfolding uniform_limit_iff dist_norm
+      proof clarify
+        fix ee::real
+        assume "0 < ee"
+        show "\<forall>\<^sub>F n in sequentially. \<forall>\<xi>\<in>path_image \<gamma>. cmod (d (a n) \<xi> - d x \<xi>) < ee"
+        proof -
+          let ?ddpa = "{(w,z) |w z. w \<in> cball x dd \<and> z \<in> path_image \<gamma>}"
+          have "uniformly_continuous_on ?ddpa (\<lambda>(x,y). d x y)"
+            apply (rule compact_uniformly_continuous [OF continuous_on_subset[OF cond_uu]])
+            using dd pasz \<open>valid_path \<gamma>\<close>
+             apply (auto simp: compact_Times compact_valid_path_image simp del: mem_cball)
+            done
+          then obtain kk where "kk>0"
+            and kk: "\<And>x x'. \<lbrakk>x \<in> ?ddpa; x' \<in> ?ddpa; dist x' x < kk\<rbrakk> \<Longrightarrow>
+                             dist ((\<lambda>(x,y). d x y) x') ((\<lambda>(x,y). d x y) x) < ee"
+            by (rule uniformly_continuous_onE [where e = ee]) (use \<open>0 < ee\<close> in auto)
+          have kk: "\<lbrakk>norm (w - x) \<le> dd; z \<in> path_image \<gamma>; norm ((w, z) - (x, z)) < kk\<rbrakk> \<Longrightarrow> norm (d w z - d x z) < ee"
+            for  w z
+            using \<open>dd>0\<close> kk [of "(x,z)" "(w,z)"] by (force simp: norm_minus_commute dist_norm)
+          show ?thesis
+            using ax unfolding lim_sequentially eventually_sequentially
+            apply (drule_tac x="min dd kk" in spec)
+            using \<open>dd > 0\<close> \<open>kk > 0\<close>
+            apply (fastforce simp: kk dist_norm)
+            done
+        qed
+      qed
+      have "(\<lambda>n. contour_integral \<gamma> (d (a n))) \<longlonglongrightarrow> contour_integral \<gamma> (d x)"
+        by (rule contour_integral_uniform_limit [OF A1 A2 le_B]) (auto simp: \<open>valid_path \<gamma>\<close>)
+      then have tendsto_hx: "(\<lambda>n. contour_integral \<gamma> (d (a n))) \<longlonglongrightarrow> h x"
+        by (simp add: h_def x)
+      then show "(h \<circ> a) \<longlonglongrightarrow> h x"
+        by (simp add: h_def x au o_def)
+    qed
+    show ?thesis
+    proof (simp add: holomorphic_on_open field_differentiable_def [symmetric], clarify)
+      fix z0
+      consider "z0 \<in> v" | "z0 \<in> U" using uv_Un by blast
+      then show "h field_differentiable at z0"
+      proof cases
+        assume "z0 \<in> v" then show ?thesis
+          using Cauchy_next_derivative [OF con_pa_f le_B f_has_cint _ ov] V f_has_cint \<open>valid_path \<gamma>\<close>
+          by (auto simp: field_differentiable_def v_def)
+      next
+        assume "z0 \<in> U" then
+        obtain e where "e>0" and e: "ball z0 e \<subseteq> U" by (blast intro: openE [OF \<open>open U\<close>])
+        have *: "contour_integral (linepath a b) h + contour_integral (linepath b c) h + contour_integral (linepath c a) h = 0"
+                if abc_subset: "convex hull {a, b, c} \<subseteq> ball z0 e"  for a b c
+        proof -
+          have *: "\<And>x1 x2 z. z \<in> U \<Longrightarrow> closed_segment x1 x2 \<subseteq> U \<Longrightarrow> (\<lambda>w. d w z) contour_integrable_on linepath x1 x2"
+            using  hol_dw holomorphic_on_imp_continuous_on \<open>open U\<close>
+            by (auto intro!: contour_integrable_holomorphic_simple)
+          have abc: "closed_segment a b \<subseteq> U"  "closed_segment b c \<subseteq> U"  "closed_segment c a \<subseteq> U"
+            using that e segments_subset_convex_hull by fastforce+
+          have eq0: "\<And>w. w \<in> U \<Longrightarrow> contour_integral (linepath a b +++ linepath b c +++ linepath c a) (\<lambda>z. d z w) = 0"
+            apply (rule contour_integral_unique [OF Cauchy_theorem_triangle])
+            apply (rule holomorphic_on_subset [OF hol_dw])
+            using e abc_subset by auto
+          have "contour_integral \<gamma>
+                   (\<lambda>x. contour_integral (linepath a b) (\<lambda>z. d z x) +
+                        (contour_integral (linepath b c) (\<lambda>z. d z x) +
+                         contour_integral (linepath c a) (\<lambda>z. d z x)))  =  0"
+            apply (rule contour_integral_eq_0)
+            using abc pasz U
+            apply (subst contour_integral_join [symmetric], auto intro: eq0 *)+
+            done
+          then show ?thesis
+            by (simp add: cint_h abc contour_integrable_add contour_integral_add [symmetric] add_ac)
+        qed
+        show ?thesis
+          using e \<open>e > 0\<close>
+          by (auto intro!: holomorphic_on_imp_differentiable_at [OF _ open_ball] analytic_imp_holomorphic
+                           Morera_triangle continuous_on_subset [OF conthu] *)
+      qed
+    qed
+  qed
+  ultimately have [simp]: "h z = 0" for z
+    by (meson Liouville_weak)
+  have "((\<lambda>w. 1 / (w - z)) has_contour_integral complex_of_real (2 * pi) * \<i> * winding_number \<gamma> z) \<gamma>"
+    by (rule has_contour_integral_winding_number [OF \<open>valid_path \<gamma>\<close> znot])
+  then have "((\<lambda>w. f z * (1 / (w - z))) has_contour_integral complex_of_real (2 * pi) * \<i> * winding_number \<gamma> z * f z) \<gamma>"
+    by (metis mult.commute has_contour_integral_lmul)
+  then have 1: "((\<lambda>w. f z / (w - z)) has_contour_integral complex_of_real (2 * pi) * \<i> * winding_number \<gamma> z * f z) \<gamma>"
+    by (simp add: field_split_simps)
+  moreover have 2: "((\<lambda>w. (f w - f z) / (w - z)) has_contour_integral 0) \<gamma>"
+    using U [OF z] pasz d_def by (force elim: has_contour_integral_eq [where g = "\<lambda>w. (f w - f z)/(w - z)"])
+  show ?thesis
+    using has_contour_integral_add [OF 1 2]  by (simp add: diff_divide_distrib)
+qed
+
+theorem Cauchy_integral_formula_global:
+    assumes S: "open S" and holf: "f holomorphic_on S"
+        and z: "z \<in> S" and vpg: "valid_path \<gamma>"
+        and pasz: "path_image \<gamma> \<subseteq> S - {z}" and loop: "pathfinish \<gamma> = pathstart \<gamma>"
+        and zero: "\<And>w. w \<notin> S \<Longrightarrow> winding_number \<gamma> w = 0"
+      shows "((\<lambda>w. f w / (w - z)) has_contour_integral (2*pi * \<i> * winding_number \<gamma> z * f z)) \<gamma>"
+proof -
+  have "path \<gamma>" using vpg by (blast intro: valid_path_imp_path)
+  have hols: "(\<lambda>w. f w / (w - z)) holomorphic_on S - {z}" "(\<lambda>w. 1 / (w - z)) holomorphic_on S - {z}"
+    by (rule holomorphic_intros holomorphic_on_subset [OF holf] | force)+
+  then have cint_fw: "(\<lambda>w. f w / (w - z)) contour_integrable_on \<gamma>"
+    by (meson contour_integrable_holomorphic_simple holomorphic_on_imp_continuous_on open_delete S vpg pasz)
+  obtain d where "d>0"
+      and d: "\<And>g h. \<lbrakk>valid_path g; valid_path h; \<forall>t\<in>{0..1}. cmod (g t - \<gamma> t) < d \<and> cmod (h t - \<gamma> t) < d;
+                     pathstart h = pathstart g \<and> pathfinish h = pathfinish g\<rbrakk>
+                     \<Longrightarrow> path_image h \<subseteq> S - {z} \<and> (\<forall>f. f holomorphic_on S - {z} \<longrightarrow> contour_integral h f = contour_integral g f)"
+    using contour_integral_nearby_ends [OF _ \<open>path \<gamma>\<close> pasz] S by (simp add: open_Diff) metis
+  obtain p where polyp: "polynomial_function p"
+             and ps: "pathstart p = pathstart \<gamma>" and pf: "pathfinish p = pathfinish \<gamma>" and led: "\<forall>t\<in>{0..1}. cmod (p t - \<gamma> t) < d"
+    using path_approx_polynomial_function [OF \<open>path \<gamma>\<close> \<open>d > 0\<close>] by blast
+  then have ploop: "pathfinish p = pathstart p" using loop by auto
+  have vpp: "valid_path p"  using polyp valid_path_polynomial_function by blast
+  have [simp]: "z \<notin> path_image \<gamma>" using pasz by blast
+  have paps: "path_image p \<subseteq> S - {z}" and cint_eq: "(\<And>f. f holomorphic_on S - {z} \<Longrightarrow> contour_integral p f = contour_integral \<gamma> f)"
+    using pf ps led d [OF vpg vpp] \<open>d > 0\<close> by auto
+  have wn_eq: "winding_number p z = winding_number \<gamma> z"
+    using vpp paps
+    by (simp add: subset_Diff_insert vpg valid_path_polynomial_function winding_number_valid_path cint_eq hols)
+  have "winding_number p w = winding_number \<gamma> w" if "w \<notin> S" for w
+  proof -
+    have hol: "(\<lambda>v. 1 / (v - w)) holomorphic_on S - {z}"
+      using that by (force intro: holomorphic_intros holomorphic_on_subset [OF holf])
+   have "w \<notin> path_image p" "w \<notin> path_image \<gamma>" using paps pasz that by auto
+   then show ?thesis
+    using vpp vpg by (simp add: subset_Diff_insert valid_path_polynomial_function winding_number_valid_path cint_eq [OF hol])
+  qed
+  then have wn0: "\<And>w. w \<notin> S \<Longrightarrow> winding_number p w = 0"
+    by (simp add: zero)
+  show ?thesis
+    using Cauchy_integral_formula_global_weak [OF S holf z polyp paps ploop wn0] hols
+    by (metis wn_eq cint_eq has_contour_integral_eqpath cint_fw cint_eq)
+qed
+
+theorem Cauchy_theorem_global:
+    assumes S: "open S" and holf: "f holomorphic_on S"
+        and vpg: "valid_path \<gamma>" and loop: "pathfinish \<gamma> = pathstart \<gamma>"
+        and pas: "path_image \<gamma> \<subseteq> S"
+        and zero: "\<And>w. w \<notin> S \<Longrightarrow> winding_number \<gamma> w = 0"
+      shows "(f has_contour_integral 0) \<gamma>"
+proof -
+  obtain z where "z \<in> S" and znot: "z \<notin> path_image \<gamma>"
+  proof -
+    have "compact (path_image \<gamma>)"
+      using compact_valid_path_image vpg by blast
+    then have "path_image \<gamma> \<noteq> S"
+      by (metis (no_types) compact_open path_image_nonempty S)
+    with pas show ?thesis by (blast intro: that)
+  qed
+  then have pasz: "path_image \<gamma> \<subseteq> S - {z}" using pas by blast
+  have hol: "(\<lambda>w. (w - z) * f w) holomorphic_on S"
+    by (rule holomorphic_intros holf)+
+  show ?thesis
+    using Cauchy_integral_formula_global [OF S hol \<open>z \<in> S\<close> vpg pasz loop zero]
+    by (auto simp: znot elim!: has_contour_integral_eq)
+qed
+
+corollary Cauchy_theorem_global_outside:
+    assumes "open S" "f holomorphic_on S" "valid_path \<gamma>"  "pathfinish \<gamma> = pathstart \<gamma>" "path_image \<gamma> \<subseteq> S"
+            "\<And>w. w \<notin> S \<Longrightarrow> w \<in> outside(path_image \<gamma>)"
+      shows "(f has_contour_integral 0) \<gamma>"
+by (metis Cauchy_theorem_global assms winding_number_zero_in_outside valid_path_imp_path)
+
+lemma simply_connected_imp_winding_number_zero:
+  assumes "simply_connected S" "path g"
+           "path_image g \<subseteq> S" "pathfinish g = pathstart g" "z \<notin> S"
+    shows "winding_number g z = 0"
+proof -
+  have hom: "homotopic_loops S g (linepath (pathstart g) (pathstart g))"
+    by (meson assms homotopic_paths_imp_homotopic_loops pathfinish_linepath simply_connected_eq_contractible_path)
+  then have "homotopic_paths (- {z}) g (linepath (pathstart g) (pathstart g))"
+    by (meson \<open>z \<notin> S\<close> homotopic_loops_imp_homotopic_paths_null homotopic_paths_subset subset_Compl_singleton)
+  then have "winding_number g z = winding_number(linepath (pathstart g) (pathstart g)) z"
+    by (rule winding_number_homotopic_paths)
+  also have "\<dots> = 0"
+    using assms by (force intro: winding_number_trivial)
+  finally show ?thesis .
+qed
+
+lemma Cauchy_theorem_simply_connected:
+  assumes "open S" "simply_connected S" "f holomorphic_on S" "valid_path g"
+           "path_image g \<subseteq> S" "pathfinish g = pathstart g"
+    shows "(f has_contour_integral 0) g"
+using assms
+apply (simp add: simply_connected_eq_contractible_path)
+apply (auto intro!: Cauchy_theorem_null_homotopic [where a = "pathstart g"]
+                         homotopic_paths_imp_homotopic_loops)
+using valid_path_imp_path by blast
+
+proposition\<^marker>\<open>tag unimportant\<close> holomorphic_logarithm_exists:
+  assumes A: "convex A" "open A"
+      and f: "f holomorphic_on A" "\<And>x. x \<in> A \<Longrightarrow> f x \<noteq> 0"
+      and z0: "z0 \<in> A"
+    obtains g where "g holomorphic_on A" and "\<And>x. x \<in> A \<Longrightarrow> exp (g x) = f x"
+proof -
+  note f' = holomorphic_derivI [OF f(1) A(2)]
+  obtain g where g: "\<And>x. x \<in> A \<Longrightarrow> (g has_field_derivative deriv f x / f x) (at x)"
+  proof (rule holomorphic_convex_primitive' [OF A])
+    show "(\<lambda>x. deriv f x / f x) holomorphic_on A"
+      by (intro holomorphic_intros f A)
+  qed (auto simp: A at_within_open[of _ A])
+  define h where "h = (\<lambda>x. -g z0 + ln (f z0) + g x)"
+  from g and A have g_holo: "g holomorphic_on A"
+    by (auto simp: holomorphic_on_def at_within_open[of _ A] field_differentiable_def)
+  hence h_holo: "h holomorphic_on A"
+    by (auto simp: h_def intro!: holomorphic_intros)
+  have "\<exists>c. \<forall>x\<in>A. f x / exp (h x) - 1 = c"
+  proof (rule has_field_derivative_zero_constant, goal_cases)
+    case (2 x)
+    note [simp] = at_within_open[OF _ \<open>open A\<close>]
+    from 2 and z0 and f show ?case
+      by (auto simp: h_def exp_diff field_simps intro!: derivative_eq_intros g f')
+  qed fact+
+  then obtain c where c: "\<And>x. x \<in> A \<Longrightarrow> f x / exp (h x) - 1 = c"
+    by blast
+  from c[OF z0] and z0 and f have "c = 0"
+    by (simp add: h_def)
+  with c have "\<And>x. x \<in> A \<Longrightarrow> exp (h x) = f x" by simp
+  from that[OF h_holo this] show ?thesis .
+qed
+
+
+(* FIXME mv to Cauchy_Integral_Theorem.thy *)
+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 fin : "\<And>w. w \<in> ball z r \<Longrightarrow> f w \<in> ball y B0"
+        and "0 < r" and "0 < n"
+      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)
+  define a where "a = (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 field_split_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 * \<i>)/(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
+
+lemma 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 * \<i>)/(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
+
+lemma 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
+    define w where "w = 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: field_split_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
+       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: field_split_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>
+
+lemma 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
+  define b where "b i = a (i+m) / a m" for i
+  define g where "g x = suminf (\<lambda>i. b i * (x - \<xi>) ^ i)" for x
+  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
+
+subsection \<open>Complex functions and power series\<close>
+
+text \<open>
+  The following defines the power series expansion of a complex function at a given point
+  (assuming that it is analytic at that point).
+\<close>
+definition\<^marker>\<open>tag important\<close> fps_expansion :: "(complex \<Rightarrow> complex) \<Rightarrow> complex \<Rightarrow> complex fps" where
+  "fps_expansion f z0 = Abs_fps (\<lambda>n. (deriv ^^ n) f z0 / fact n)"
+
+lemma
+  fixes r :: ereal
+  assumes "f holomorphic_on eball z0 r"
+  shows   conv_radius_fps_expansion: "fps_conv_radius (fps_expansion f z0) \<ge> r"
+    and   eval_fps_expansion: "\<And>z. z \<in> eball z0 r \<Longrightarrow> eval_fps (fps_expansion f z0) (z - z0) = f z"
+    and   eval_fps_expansion': "\<And>z. norm z < r \<Longrightarrow> eval_fps (fps_expansion f z0) z = f (z0 + z)"
+proof -
+  have "(\<lambda>n. fps_nth (fps_expansion f z0) n * (z - z0) ^ n) sums f z"
+    if "z \<in> ball z0 r'" "ereal r' < r" for z r'
+  proof -
+    from that(2) have "ereal r' \<le> r" by simp
+    from assms(1) and this have "f holomorphic_on ball z0 r'"
+      by (rule holomorphic_on_subset[OF _ ball_eball_mono])
+    from holomorphic_power_series [OF this that(1)] 
+      show ?thesis by (simp add: fps_expansion_def)
+  qed
+  hence *: "(\<lambda>n. fps_nth (fps_expansion f z0) n * (z - z0) ^ n) sums f z"
+    if "z \<in> eball z0 r" for z
+    using that by (subst (asm) eball_conv_UNION_balls) blast
+  show "fps_conv_radius (fps_expansion f z0) \<ge> r" unfolding fps_conv_radius_def
+  proof (rule conv_radius_geI_ex)
+    fix r' :: real assume r': "r' > 0" "ereal r' < r"
+    thus "\<exists>z. norm z = r' \<and> summable (\<lambda>n. fps_nth (fps_expansion f z0) n * z ^ n)"
+      using *[of "z0 + of_real r'"]
+      by (intro exI[of _ "of_real r'"]) (auto simp: summable_def dist_norm)
+  qed
+  show "eval_fps (fps_expansion f z0) (z - z0) = f z" if "z \<in> eball z0 r" for z
+    using *[OF that] by (simp add: eval_fps_def sums_iff)
+  show "eval_fps (fps_expansion f z0) z = f (z0 + z)" if "ereal (norm z) < r" for z
+    using *[of "z0 + z"] and that by (simp add: eval_fps_def sums_iff dist_norm)
+qed
+
+
+text \<open>
+  We can now show several more facts about power series expansions (at least in the complex case)
+  with relative ease that would have been trickier without complex analysis.
+\<close>
+lemma
+  fixes f :: "complex fps" and r :: ereal
+  assumes "\<And>z. ereal (norm z) < r \<Longrightarrow> eval_fps f z \<noteq> 0"
+  shows   fps_conv_radius_inverse: "fps_conv_radius (inverse f) \<ge> min r (fps_conv_radius f)"
+    and   eval_fps_inverse: "\<And>z. ereal (norm z) < fps_conv_radius f \<Longrightarrow> ereal (norm z) < r \<Longrightarrow> 
+                               eval_fps (inverse f) z = inverse (eval_fps f z)"
+proof -
+  define R where "R = min (fps_conv_radius f) r"
+  have *: "fps_conv_radius (inverse f) \<ge> min r (fps_conv_radius f) \<and> 
+          (\<forall>z\<in>eball 0 (min (fps_conv_radius f) r). eval_fps (inverse f) z = inverse (eval_fps f z))"
+  proof (cases "min r (fps_conv_radius f) > 0")
+    case True
+    define f' where "f' = fps_expansion (\<lambda>z. inverse (eval_fps f z)) 0"
+    have holo: "(\<lambda>z. inverse (eval_fps f z)) holomorphic_on eball 0 (min r (fps_conv_radius f))"
+      using assms by (intro holomorphic_intros) auto
+    from holo have radius: "fps_conv_radius f' \<ge> min r (fps_conv_radius f)"
+      unfolding f'_def by (rule conv_radius_fps_expansion)
+    have eval_f': "eval_fps f' z = inverse (eval_fps f z)" 
+      if "norm z < fps_conv_radius f" "norm z < r" for z
+      using that unfolding f'_def by (subst eval_fps_expansion'[OF holo]) auto
+  
+    have "f * f' = 1"
+    proof (rule eval_fps_eqD)
+      from radius and True have "0 < min (fps_conv_radius f) (fps_conv_radius f')"
+        by (auto simp: min_def split: if_splits)
+      also have "\<dots> \<le> fps_conv_radius (f * f')" by (rule fps_conv_radius_mult)
+      finally show "\<dots> > 0" .
+    next
+      from True have "R > 0" by (auto simp: R_def)
+      hence "eventually (\<lambda>z. z \<in> eball 0 R) (nhds 0)"
+        by (intro eventually_nhds_in_open) (auto simp: zero_ereal_def)
+      thus "eventually (\<lambda>z. eval_fps (f * f') z = eval_fps 1 z) (nhds 0)"
+      proof eventually_elim
+        case (elim z)
+        hence "eval_fps (f * f') z = eval_fps f z * eval_fps f' z"
+          using radius by (intro eval_fps_mult) 
+                          (auto simp: R_def min_def split: if_splits intro: less_trans)
+        also have "eval_fps f' z = inverse (eval_fps f z)"
+          using elim by (intro eval_f') (auto simp: R_def)
+        also from elim have "eval_fps f z \<noteq> 0"
+          by (intro assms) (auto simp: R_def)
+        hence "eval_fps f z * inverse (eval_fps f z) = eval_fps 1 z" 
+          by simp
+        finally show "eval_fps (f * f') z = eval_fps 1 z" .
+      qed
+    qed simp_all
+    hence "f' = inverse f"
+      by (intro fps_inverse_unique [symmetric]) (simp_all add: mult_ac)
+    with eval_f' and radius show ?thesis by simp
+  next
+    case False
+    hence *: "eball 0 R = {}" 
+      by (intro eball_empty) (auto simp: R_def min_def split: if_splits)
+    show ?thesis
+    proof safe
+      from False have "min r (fps_conv_radius f) \<le> 0"
+        by (simp add: min_def)
+      also have "0 \<le> fps_conv_radius (inverse f)"
+        by (simp add: fps_conv_radius_def conv_radius_nonneg)
+      finally show "min r (fps_conv_radius f) \<le> \<dots>" .
+    qed (unfold * [unfolded R_def], auto)
+  qed
+
+  from * show "fps_conv_radius (inverse f) \<ge> min r (fps_conv_radius f)" by blast
+  from * show "eval_fps (inverse f) z = inverse (eval_fps f z)" 
+    if "ereal (norm z) < fps_conv_radius f" "ereal (norm z) < r" for z
+    using that by auto
+qed
+
+lemma
+  fixes f g :: "complex fps" and r :: ereal
+  defines "R \<equiv> Min {r, fps_conv_radius f, fps_conv_radius g}"
+  assumes "fps_conv_radius f > 0" "fps_conv_radius g > 0" "r > 0"
+  assumes nz: "\<And>z. z \<in> eball 0 r \<Longrightarrow> eval_fps g z \<noteq> 0"
+  shows   fps_conv_radius_divide': "fps_conv_radius (f / g) \<ge> R"
+    and   eval_fps_divide':
+            "ereal (norm z) < R \<Longrightarrow> eval_fps (f / g) z = eval_fps f z / eval_fps g z"
+proof -
+  from nz[of 0] and \<open>r > 0\<close> have nz': "fps_nth g 0 \<noteq> 0" 
+    by (auto simp: eval_fps_at_0 zero_ereal_def)
+  have "R \<le> min r (fps_conv_radius g)"
+    by (auto simp: R_def intro: min.coboundedI2)
+  also have "min r (fps_conv_radius g) \<le> fps_conv_radius (inverse g)"
+    by (intro fps_conv_radius_inverse assms) (auto simp: zero_ereal_def)
+  finally have radius: "fps_conv_radius (inverse g) \<ge> R" .
+  have "R \<le> min (fps_conv_radius f) (fps_conv_radius (inverse g))"
+    by (intro radius min.boundedI) (auto simp: R_def intro: min.coboundedI1 min.coboundedI2)
+  also have "\<dots> \<le> fps_conv_radius (f * inverse g)"
+    by (rule fps_conv_radius_mult)
+  also have "f * inverse g = f / g"
+    by (intro fps_divide_unit [symmetric] nz')
+  finally show "fps_conv_radius (f / g) \<ge> R" .
+
+  assume z: "ereal (norm z) < R"
+  have "eval_fps (f * inverse g) z = eval_fps f z * eval_fps (inverse g) z"
+    using radius by (intro eval_fps_mult less_le_trans[OF z])
+                    (auto simp: R_def intro: min.coboundedI1 min.coboundedI2)
+  also have "eval_fps (inverse g) z = inverse (eval_fps g z)" using \<open>r > 0\<close>
+    by (intro eval_fps_inverse[where r = r] less_le_trans[OF z] nz)
+       (auto simp: R_def intro: min.coboundedI1 min.coboundedI2)
+  also have "f * inverse g = f / g" by fact
+  finally show "eval_fps (f / g) z = eval_fps f z / eval_fps g z" by (simp add: field_split_simps)
+qed
+
+lemma
+  fixes f g :: "complex fps" and r :: ereal
+  defines "R \<equiv> Min {r, fps_conv_radius f, fps_conv_radius g}"
+  assumes "subdegree g \<le> subdegree f"
+  assumes "fps_conv_radius f > 0" "fps_conv_radius g > 0" "r > 0"
+  assumes "\<And>z. z \<in> eball 0 r \<Longrightarrow> z \<noteq> 0 \<Longrightarrow> eval_fps g z \<noteq> 0"
+  shows   fps_conv_radius_divide: "fps_conv_radius (f / g) \<ge> R"
+    and   eval_fps_divide:
+            "ereal (norm z) < R \<Longrightarrow> c = fps_nth f (subdegree g) / fps_nth g (subdegree g) \<Longrightarrow>
+               eval_fps (f / g) z = (if z = 0 then c else eval_fps f z / eval_fps g z)"
+proof -
+  define f' g' where "f' = fps_shift (subdegree g) f" and "g' = fps_shift (subdegree g) g"
+  have f_eq: "f = f' * fps_X ^ subdegree g" and g_eq: "g = g' * fps_X ^ subdegree g"
+    unfolding f'_def g'_def by (rule subdegree_decompose' le_refl | fact)+
+  have subdegree: "subdegree f' = subdegree f - subdegree g" "subdegree g' = 0"
+    using assms(2) by (simp_all add: f'_def g'_def)
+  have [simp]: "fps_conv_radius f' = fps_conv_radius f" "fps_conv_radius g' = fps_conv_radius g"
+    by (simp_all add: f'_def g'_def)
+  have [simp]: "fps_nth f' 0 = fps_nth f (subdegree g)"
+               "fps_nth g' 0 = fps_nth g (subdegree g)" by (simp_all add: f'_def g'_def)
+  have g_nz: "g \<noteq> 0"
+  proof -
+    define z :: complex where "z = (if r = \<infinity> then 1 else of_real (real_of_ereal r / 2))"
+    from \<open>r > 0\<close> have "z \<in> eball 0 r"
+      by (cases r) (auto simp: z_def eball_def)
+    moreover have "z \<noteq> 0" using \<open>r > 0\<close> 
+      by (cases r) (auto simp: z_def)
+    ultimately have "eval_fps g z \<noteq> 0" by (rule assms(6))
+    thus "g \<noteq> 0" by auto
+  qed
+  have fg: "f / g = f' * inverse g'"
+    by (subst f_eq, subst (2) g_eq) (insert g_nz, simp add: fps_divide_unit)
+
+  have g'_nz: "eval_fps g' z \<noteq> 0" if z: "norm z < min r (fps_conv_radius g)" for z
+  proof (cases "z = 0")
+    case False
+    with assms and z have "eval_fps g z \<noteq> 0" by auto
+    also from z have "eval_fps g z = eval_fps g' z * z ^ subdegree g"
+      by (subst g_eq) (auto simp: eval_fps_mult)
+    finally show ?thesis by auto
+  qed (insert \<open>g \<noteq> 0\<close>, auto simp: g'_def eval_fps_at_0)
+
+  have "R \<le> min (min r (fps_conv_radius g)) (fps_conv_radius g')"
+    by (auto simp: R_def min.coboundedI1 min.coboundedI2)
+  also have "\<dots> \<le> fps_conv_radius (inverse g')"
+    using g'_nz by (rule fps_conv_radius_inverse)
+  finally have conv_radius_inv: "R \<le> fps_conv_radius (inverse g')" .
+  hence "R \<le> fps_conv_radius (f' * inverse g')"
+    by (intro order.trans[OF _ fps_conv_radius_mult])
+       (auto simp: R_def intro: min.coboundedI1 min.coboundedI2)
+  thus "fps_conv_radius (f / g) \<ge> R" by (simp add: fg)
+
+  fix z c :: complex assume z: "ereal (norm z) < R"
+  assume c: "c = fps_nth f (subdegree g) / fps_nth g (subdegree g)"
+  show "eval_fps (f / g) z = (if z = 0 then c else eval_fps f z / eval_fps g z)"
+  proof (cases "z = 0")
+    case False
+    from z and conv_radius_inv have "ereal (norm z) < fps_conv_radius (inverse g')"
+      by simp
+    with z have "eval_fps (f / g) z = eval_fps f' z * eval_fps (inverse g') z"
+      unfolding fg by (subst eval_fps_mult) (auto simp: R_def)
+    also have "eval_fps (inverse g') z = inverse (eval_fps g' z)"
+      using z by (intro eval_fps_inverse[of "min r (fps_conv_radius g')"] g'_nz) (auto simp: R_def)
+    also have "eval_fps f' z * \<dots> = eval_fps f z / eval_fps g z"
+      using z False assms(2) by (simp add: f'_def g'_def eval_fps_shift R_def)
+    finally show ?thesis using False by simp
+  qed (simp_all add: eval_fps_at_0 fg field_simps c)
+qed
+
+lemma has_fps_expansion_fps_expansion [intro]:
+  assumes "open A" "0 \<in> A" "f holomorphic_on A"
+  shows   "f has_fps_expansion fps_expansion f 0"
+proof -
+  from assms(1,2) obtain r where r: "r > 0 " "ball 0 r \<subseteq> A"
+    by (auto simp: open_contains_ball)
+  have holo: "f holomorphic_on eball 0 (ereal r)" 
+    using r(2) and assms(3) by auto
+  from r(1) have "0 < ereal r" by simp
+  also have "r \<le> fps_conv_radius (fps_expansion f 0)"
+    using holo by (intro conv_radius_fps_expansion) auto
+  finally have "\<dots> > 0" .
+  moreover have "eventually (\<lambda>z. z \<in> ball 0 r) (nhds 0)"
+    using r(1) by (intro eventually_nhds_in_open) auto
+  hence "eventually (\<lambda>z. eval_fps (fps_expansion f 0) z = f z) (nhds 0)"
+    by eventually_elim (subst eval_fps_expansion'[OF holo], auto)
+  ultimately show ?thesis using r(1) by (auto simp: has_fps_expansion_def)
+qed
+
+lemma fps_conv_radius_tan:
+  fixes c :: complex
+  assumes "c \<noteq> 0"
+  shows   "fps_conv_radius (fps_tan c) \<ge> pi / (2 * norm c)"
+proof -
+  have "fps_conv_radius (fps_tan c) \<ge> 
+          Min {pi / (2 * norm c), fps_conv_radius (fps_sin c), fps_conv_radius (fps_cos c)}"
+    unfolding fps_tan_def
+  proof (rule fps_conv_radius_divide)
+    fix z :: complex assume "z \<in> eball 0 (pi / (2 * norm c))"
+    with cos_eq_zero_imp_norm_ge[of "c*z"] assms 
+      show "eval_fps (fps_cos  c) z \<noteq> 0" by (auto simp: norm_mult field_simps)
+  qed (insert assms, auto)
+  thus ?thesis by (simp add: min_def)
+qed
+
+lemma eval_fps_tan:
+  fixes c :: complex
+  assumes "norm z < pi / (2 * norm c)"
+  shows   "eval_fps (fps_tan c) z = tan (c * z)"
+proof (cases "c = 0")
+  case False
+  show ?thesis unfolding fps_tan_def
+  proof (subst eval_fps_divide'[where r = "pi / (2 * norm c)"])
+    fix z :: complex assume "z \<in> eball 0 (pi / (2 * norm c))"
+    with cos_eq_zero_imp_norm_ge[of "c*z"] assms 
+      show "eval_fps (fps_cos  c) z \<noteq> 0" using False by (auto simp: norm_mult field_simps)
+    qed (insert False assms, auto simp: field_simps tan_def)
+  qed simp_all
+
+end
--- a/src/HOL/Complex_Analysis/Cauchy_Integral_Theorem.thy	Mon Dec 02 22:40:16 2019 -0500
+++ b/src/HOL/Complex_Analysis/Cauchy_Integral_Theorem.thy	Mon Dec 02 17:51:54 2019 +0100
@@ -5,6 +5,7 @@
 theory Cauchy_Integral_Theorem
 imports
   "HOL-Analysis.Analysis"
+  Contour_Integration
 begin
 
 lemma leibniz_rule_holomorphic:
@@ -38,1005 +39,6 @@
   shows   "(\<lambda>x. f x powr g x :: complex) \<in> measurable M borel"
   using assms by (simp add: powr_def) 
 
-subsection\<open>Contour Integrals along a path\<close>
-
-text\<open>This definition is for complex numbers only, and does not generalise to line integrals in a vector field\<close>
-
-text\<open>piecewise differentiable function on [0,1]\<close>
-
-definition\<^marker>\<open>tag important\<close> has_contour_integral :: "(complex \<Rightarrow> complex) \<Rightarrow> complex \<Rightarrow> (real \<Rightarrow> complex) \<Rightarrow> bool"
-           (infixr "has'_contour'_integral" 50)
-  where "(f has_contour_integral i) g \<equiv>
-           ((\<lambda>x. f(g x) * vector_derivative g (at x within {0..1}))
-            has_integral i) {0..1}"
-
-definition\<^marker>\<open>tag important\<close> contour_integrable_on
-           (infixr "contour'_integrable'_on" 50)
-  where "f contour_integrable_on g \<equiv> \<exists>i. (f has_contour_integral i) g"
-
-definition\<^marker>\<open>tag important\<close> contour_integral
-  where "contour_integral g f \<equiv> SOME i. (f has_contour_integral i) g \<or> \<not> f contour_integrable_on g \<and> i=0"
-
-lemma not_integrable_contour_integral: "\<not> f contour_integrable_on g \<Longrightarrow> contour_integral g f = 0"
-  unfolding contour_integrable_on_def contour_integral_def by blast
-
-lemma contour_integral_unique: "(f has_contour_integral i) g \<Longrightarrow> contour_integral g f = i"
-  apply (simp add: contour_integral_def has_contour_integral_def contour_integrable_on_def)
-  using has_integral_unique by blast
-
-lemma has_contour_integral_eqpath:
-     "\<lbrakk>(f has_contour_integral y) p; f contour_integrable_on \<gamma>;
-       contour_integral p f = contour_integral \<gamma> f\<rbrakk>
-      \<Longrightarrow> (f has_contour_integral y) \<gamma>"
-using contour_integrable_on_def contour_integral_unique by auto
-
-lemma has_contour_integral_integral:
-    "f contour_integrable_on i \<Longrightarrow> (f has_contour_integral (contour_integral i f)) i"
-  by (metis contour_integral_unique contour_integrable_on_def)
-
-lemma has_contour_integral_unique:
-    "(f has_contour_integral i) g \<Longrightarrow> (f has_contour_integral j) g \<Longrightarrow> i = j"
-  using has_integral_unique
-  by (auto simp: has_contour_integral_def)
-
-lemma has_contour_integral_integrable: "(f has_contour_integral i) g \<Longrightarrow> f contour_integrable_on g"
-  using contour_integrable_on_def by blast
-
-text\<open>Show that we can forget about the localized derivative.\<close>
-
-lemma has_integral_localized_vector_derivative:
-    "((\<lambda>x. f (g x) * vector_derivative g (at x within {a..b})) has_integral i) {a..b} \<longleftrightarrow>
-     ((\<lambda>x. f (g x) * vector_derivative g (at x)) has_integral i) {a..b}"
-proof -
-  have *: "{a..b} - {a,b} = interior {a..b}"
-    by (simp add: atLeastAtMost_diff_ends)
-  show ?thesis
-    apply (rule has_integral_spike_eq [of "{a,b}"])
-    apply (auto simp: at_within_interior [of _ "{a..b}"])
-    done
-qed
-
-lemma integrable_on_localized_vector_derivative:
-    "(\<lambda>x. f (g x) * vector_derivative g (at x within {a..b})) integrable_on {a..b} \<longleftrightarrow>
-     (\<lambda>x. f (g x) * vector_derivative g (at x)) integrable_on {a..b}"
-  by (simp add: integrable_on_def has_integral_localized_vector_derivative)
-
-lemma has_contour_integral:
-     "(f has_contour_integral i) g \<longleftrightarrow>
-      ((\<lambda>x. f (g x) * vector_derivative g (at x)) has_integral i) {0..1}"
-  by (simp add: has_integral_localized_vector_derivative has_contour_integral_def)
-
-lemma contour_integrable_on:
-     "f contour_integrable_on g \<longleftrightarrow>
-      (\<lambda>t. f(g t) * vector_derivative g (at t)) integrable_on {0..1}"
-  by (simp add: has_contour_integral integrable_on_def contour_integrable_on_def)
-
-subsection\<^marker>\<open>tag unimportant\<close> \<open>Reversing a path\<close>
-
-
-
-lemma has_contour_integral_reversepath:
-  assumes "valid_path g" and f: "(f has_contour_integral i) g"
-    shows "(f has_contour_integral (-i)) (reversepath g)"
-proof -
-  { fix S x
-    assume xs: "g C1_differentiable_on ({0..1} - S)" "x \<notin> (-) 1 ` S" "0 \<le> x" "x \<le> 1"
-    have "vector_derivative (\<lambda>x. g (1 - x)) (at x within {0..1}) =
-            - vector_derivative g (at (1 - x) within {0..1})"
-    proof -
-      obtain f' where f': "(g has_vector_derivative f') (at (1 - x))"
-        using xs
-        by (force simp: has_vector_derivative_def C1_differentiable_on_def)
-      have "(g \<circ> (\<lambda>x. 1 - x) has_vector_derivative -1 *\<^sub>R f') (at x)"
-        by (intro vector_diff_chain_within has_vector_derivative_at_within [OF f'] derivative_eq_intros | simp)+
-      then have mf': "((\<lambda>x. g (1 - x)) has_vector_derivative -f') (at x)"
-        by (simp add: o_def)
-      show ?thesis
-        using xs
-        by (auto simp: vector_derivative_at_within_ivl [OF mf'] vector_derivative_at_within_ivl [OF f'])
-    qed
-  } note * = this
-  obtain S where S: "continuous_on {0..1} g" "finite S" "g C1_differentiable_on {0..1} - S"
-    using assms by (auto simp: valid_path_def piecewise_C1_differentiable_on_def)
-  have "((\<lambda>x. - (f (g (1 - x)) * vector_derivative g (at (1 - x) within {0..1}))) has_integral -i)
-       {0..1}"
-    using has_integral_affinity01 [where m= "-1" and c=1, OF f [unfolded has_contour_integral_def]]
-    by (simp add: has_integral_neg)
-  then show ?thesis
-    using S
-    apply (clarsimp simp: reversepath_def has_contour_integral_def)
-    apply (rule_tac S = "(\<lambda>x. 1 - x) ` S" in has_integral_spike_finite)
-      apply (auto simp: *)
-    done
-qed
-
-lemma contour_integrable_reversepath:
-    "valid_path g \<Longrightarrow> f contour_integrable_on g \<Longrightarrow> f contour_integrable_on (reversepath g)"
-  using has_contour_integral_reversepath contour_integrable_on_def by blast
-
-lemma contour_integrable_reversepath_eq:
-    "valid_path g \<Longrightarrow> (f contour_integrable_on (reversepath g) \<longleftrightarrow> f contour_integrable_on g)"
-  using contour_integrable_reversepath valid_path_reversepath by fastforce
-
-lemma contour_integral_reversepath:
-  assumes "valid_path g"
-    shows "contour_integral (reversepath g) f = - (contour_integral g f)"
-proof (cases "f contour_integrable_on g")
-  case True then show ?thesis
-    by (simp add: assms contour_integral_unique has_contour_integral_integral has_contour_integral_reversepath)
-next
-  case False then have "\<not> f contour_integrable_on (reversepath g)"
-    by (simp add: assms contour_integrable_reversepath_eq)
-  with False show ?thesis by (simp add: not_integrable_contour_integral)
-qed
-
-
-subsection\<^marker>\<open>tag unimportant\<close> \<open>Joining two paths together\<close>
-
-lemma has_contour_integral_join:
-  assumes "(f has_contour_integral i1) g1" "(f has_contour_integral i2) g2"
-          "valid_path g1" "valid_path g2"
-    shows "(f has_contour_integral (i1 + i2)) (g1 +++ g2)"
-proof -
-  obtain s1 s2
-    where s1: "finite s1" "\<forall>x\<in>{0..1} - s1. g1 differentiable at x"
-      and s2: "finite s2" "\<forall>x\<in>{0..1} - s2. g2 differentiable at x"
-    using assms
-    by (auto simp: valid_path_def piecewise_C1_differentiable_on_def C1_differentiable_on_eq)
-  have 1: "((\<lambda>x. f (g1 x) * vector_derivative g1 (at x)) has_integral i1) {0..1}"
-   and 2: "((\<lambda>x. f (g2 x) * vector_derivative g2 (at x)) has_integral i2) {0..1}"
-    using assms
-    by (auto simp: has_contour_integral)
-  have i1: "((\<lambda>x. (2*f (g1 (2*x))) * vector_derivative g1 (at (2*x))) has_integral i1) {0..1/2}"
-   and i2: "((\<lambda>x. (2*f (g2 (2*x - 1))) * vector_derivative g2 (at (2*x - 1))) has_integral i2) {1/2..1}"
-    using has_integral_affinity01 [OF 1, where m= 2 and c=0, THEN has_integral_cmul [where c=2]]
-          has_integral_affinity01 [OF 2, where m= 2 and c="-1", THEN has_integral_cmul [where c=2]]
-    by (simp_all only: image_affinity_atLeastAtMost_div_diff, simp_all add: scaleR_conv_of_real mult_ac)
-  have g1: "\<lbrakk>0 \<le> z; z*2 < 1; z*2 \<notin> s1\<rbrakk> \<Longrightarrow>
-            vector_derivative (\<lambda>x. if x*2 \<le> 1 then g1 (2*x) else g2 (2*x - 1)) (at z) =
-            2 *\<^sub>R vector_derivative g1 (at (z*2))" for z
-    apply (rule vector_derivative_at [OF has_vector_derivative_transform_within [where f = "(\<lambda>x. g1(2*x))" and d = "\<bar>z - 1/2\<bar>"]])
-    apply (simp_all add: dist_real_def abs_if split: if_split_asm)
-    apply (rule vector_diff_chain_at [of "\<lambda>x. 2*x" 2 _ g1, simplified o_def])
-    apply (simp add: has_vector_derivative_def has_derivative_def bounded_linear_mult_left)
-    using s1
-    apply (auto simp: algebra_simps vector_derivative_works)
-    done
-  have g2: "\<lbrakk>1 < z*2; z \<le> 1; z*2 - 1 \<notin> s2\<rbrakk> \<Longrightarrow>
-            vector_derivative (\<lambda>x. if x*2 \<le> 1 then g1 (2*x) else g2 (2*x - 1)) (at z) =
-            2 *\<^sub>R vector_derivative g2 (at (z*2 - 1))" for z
-    apply (rule vector_derivative_at [OF has_vector_derivative_transform_within [where f = "(\<lambda>x. g2 (2*x - 1))" and d = "\<bar>z - 1/2\<bar>"]])
-    apply (simp_all add: dist_real_def abs_if split: if_split_asm)
-    apply (rule vector_diff_chain_at [of "\<lambda>x. 2*x - 1" 2 _ g2, simplified o_def])
-    apply (simp add: has_vector_derivative_def has_derivative_def bounded_linear_mult_left)
-    using s2
-    apply (auto simp: algebra_simps vector_derivative_works)
-    done
-  have "((\<lambda>x. f ((g1 +++ g2) x) * vector_derivative (g1 +++ g2) (at x)) has_integral i1) {0..1/2}"
-    apply (rule has_integral_spike_finite [OF _ _ i1, of "insert (1/2) ((*)2 -` s1)"])
-    using s1
-    apply (force intro: finite_vimageI [where h = "(*)2"] inj_onI)
-    apply (clarsimp simp add: joinpaths_def scaleR_conv_of_real mult_ac g1)
-    done
-  moreover have "((\<lambda>x. f ((g1 +++ g2) x) * vector_derivative (g1 +++ g2) (at x)) has_integral i2) {1/2..1}"
-    apply (rule has_integral_spike_finite [OF _ _ i2, of "insert (1/2) ((\<lambda>x. 2*x-1) -` s2)"])
-    using s2
-    apply (force intro: finite_vimageI [where h = "\<lambda>x. 2*x-1"] inj_onI)
-    apply (clarsimp simp add: joinpaths_def scaleR_conv_of_real mult_ac g2)
-    done
-  ultimately
-  show ?thesis
-    apply (simp add: has_contour_integral)
-    apply (rule has_integral_combine [where c = "1/2"], auto)
-    done
-qed
-
-lemma contour_integrable_joinI:
-  assumes "f contour_integrable_on g1" "f contour_integrable_on g2"
-          "valid_path g1" "valid_path g2"
-    shows "f contour_integrable_on (g1 +++ g2)"
-  using assms
-  by (meson has_contour_integral_join contour_integrable_on_def)
-
-lemma contour_integrable_joinD1:
-  assumes "f contour_integrable_on (g1 +++ g2)" "valid_path g1"
-    shows "f contour_integrable_on g1"
-proof -
-  obtain s1
-    where s1: "finite s1" "\<forall>x\<in>{0..1} - s1. g1 differentiable at x"
-    using assms by (auto simp: valid_path_def piecewise_C1_differentiable_on_def C1_differentiable_on_eq)
-  have "(\<lambda>x. f ((g1 +++ g2) (x/2)) * vector_derivative (g1 +++ g2) (at (x/2))) integrable_on {0..1}"
-    using assms
-    apply (auto simp: contour_integrable_on)
-    apply (drule integrable_on_subcbox [where a=0 and b="1/2"])
-    apply (auto intro: integrable_affinity [of _ 0 "1/2::real" "1/2" 0, simplified])
-    done
-  then have *: "(\<lambda>x. (f ((g1 +++ g2) (x/2))/2) * vector_derivative (g1 +++ g2) (at (x/2))) integrable_on {0..1}"
-    by (auto dest: integrable_cmul [where c="1/2"] simp: scaleR_conv_of_real)
-  have g1: "\<lbrakk>0 < z; z < 1; z \<notin> s1\<rbrakk> \<Longrightarrow>
-            vector_derivative (\<lambda>x. if x*2 \<le> 1 then g1 (2*x) else g2 (2*x - 1)) (at (z/2)) =
-            2 *\<^sub>R vector_derivative g1 (at z)"  for z
-    apply (rule vector_derivative_at [OF has_vector_derivative_transform_within [where f = "(\<lambda>x. g1(2*x))" and d = "\<bar>(z-1)/2\<bar>"]])
-    apply (simp_all add: field_simps dist_real_def abs_if split: if_split_asm)
-    apply (rule vector_diff_chain_at [of "\<lambda>x. x*2" 2 _ g1, simplified o_def])
-    using s1
-    apply (auto simp: vector_derivative_works has_vector_derivative_def has_derivative_def bounded_linear_mult_left)
-    done
-  show ?thesis
-    using s1
-    apply (auto simp: contour_integrable_on)
-    apply (rule integrable_spike_finite [of "{0,1} \<union> s1", OF _ _ *])
-    apply (auto simp: joinpaths_def scaleR_conv_of_real g1)
-    done
-qed
-
-lemma contour_integrable_joinD2:
-  assumes "f contour_integrable_on (g1 +++ g2)" "valid_path g2"
-    shows "f contour_integrable_on g2"
-proof -
-  obtain s2
-    where s2: "finite s2" "\<forall>x\<in>{0..1} - s2. g2 differentiable at x"
-    using assms by (auto simp: valid_path_def piecewise_C1_differentiable_on_def C1_differentiable_on_eq)
-  have "(\<lambda>x. f ((g1 +++ g2) (x/2 + 1/2)) * vector_derivative (g1 +++ g2) (at (x/2 + 1/2))) integrable_on {0..1}"
-    using assms
-    apply (auto simp: contour_integrable_on)
-    apply (drule integrable_on_subcbox [where a="1/2" and b=1], auto)
-    apply (drule integrable_affinity [of _ "1/2::real" 1 "1/2" "1/2", simplified])
-    apply (simp add: image_affinity_atLeastAtMost_diff)
-    done
-  then have *: "(\<lambda>x. (f ((g1 +++ g2) (x/2 + 1/2))/2) * vector_derivative (g1 +++ g2) (at (x/2 + 1/2)))
-                integrable_on {0..1}"
-    by (auto dest: integrable_cmul [where c="1/2"] simp: scaleR_conv_of_real)
-  have g2: "\<lbrakk>0 < z; z < 1; z \<notin> s2\<rbrakk> \<Longrightarrow>
-            vector_derivative (\<lambda>x. if x*2 \<le> 1 then g1 (2*x) else g2 (2*x - 1)) (at (z/2+1/2)) =
-            2 *\<^sub>R vector_derivative g2 (at z)" for z
-    apply (rule vector_derivative_at [OF has_vector_derivative_transform_within [where f = "(\<lambda>x. g2(2*x-1))" and d = "\<bar>z/2\<bar>"]])
-    apply (simp_all add: field_simps dist_real_def abs_if split: if_split_asm)
-    apply (rule vector_diff_chain_at [of "\<lambda>x. x*2-1" 2 _ g2, simplified o_def])
-    using s2
-    apply (auto simp: has_vector_derivative_def has_derivative_def bounded_linear_mult_left
-                      vector_derivative_works add_divide_distrib)
-    done
-  show ?thesis
-    using s2
-    apply (auto simp: contour_integrable_on)
-    apply (rule integrable_spike_finite [of "{0,1} \<union> s2", OF _ _ *])
-    apply (auto simp: joinpaths_def scaleR_conv_of_real g2)
-    done
-qed
-
-lemma contour_integrable_join [simp]:
-  shows
-    "\<lbrakk>valid_path g1; valid_path g2\<rbrakk>
-     \<Longrightarrow> f contour_integrable_on (g1 +++ g2) \<longleftrightarrow> f contour_integrable_on g1 \<and> f contour_integrable_on g2"
-using contour_integrable_joinD1 contour_integrable_joinD2 contour_integrable_joinI by blast
-
-lemma contour_integral_join [simp]:
-  shows
-    "\<lbrakk>f contour_integrable_on g1; f contour_integrable_on g2; valid_path g1; valid_path g2\<rbrakk>
-        \<Longrightarrow> contour_integral (g1 +++ g2) f = contour_integral g1 f + contour_integral g2 f"
-  by (simp add: has_contour_integral_integral has_contour_integral_join contour_integral_unique)
-
-
-subsection\<^marker>\<open>tag unimportant\<close> \<open>Shifting the starting point of a (closed) path\<close>
-
-lemma has_contour_integral_shiftpath:
-  assumes f: "(f has_contour_integral i) g" "valid_path g"
-      and a: "a \<in> {0..1}"
-    shows "(f has_contour_integral i) (shiftpath a g)"
-proof -
-  obtain s
-    where s: "finite s" and g: "\<forall>x\<in>{0..1} - s. g differentiable at x"
-    using assms by (auto simp: valid_path_def piecewise_C1_differentiable_on_def C1_differentiable_on_eq)
-  have *: "((\<lambda>x. f (g x) * vector_derivative g (at x)) has_integral i) {0..1}"
-    using assms by (auto simp: has_contour_integral)
-  then have i: "i = integral {a..1} (\<lambda>x. f (g x) * vector_derivative g (at x)) +
-                    integral {0..a} (\<lambda>x. f (g x) * vector_derivative g (at x))"
-    apply (rule has_integral_unique)
-    apply (subst add.commute)
-    apply (subst Henstock_Kurzweil_Integration.integral_combine)
-    using assms * integral_unique by auto
-  { fix x
-    have "0 \<le> x \<Longrightarrow> x + a < 1 \<Longrightarrow> x \<notin> (\<lambda>x. x - a) ` s \<Longrightarrow>
-         vector_derivative (shiftpath a g) (at x) = vector_derivative g (at (x + a))"
-      unfolding shiftpath_def
-      apply (rule vector_derivative_at [OF has_vector_derivative_transform_within [where f = "(\<lambda>x. g(a+x))" and d = "dist(1-a) x"]])
-        apply (auto simp: field_simps dist_real_def abs_if split: if_split_asm)
-      apply (rule vector_diff_chain_at [of "\<lambda>x. x+a" 1 _ g, simplified o_def scaleR_one])
-       apply (intro derivative_eq_intros | simp)+
-      using g
-       apply (drule_tac x="x+a" in bspec)
-      using a apply (auto simp: has_vector_derivative_def vector_derivative_works image_def add.commute)
-      done
-  } note vd1 = this
-  { fix x
-    have "1 < x + a \<Longrightarrow> x \<le> 1 \<Longrightarrow> x \<notin> (\<lambda>x. x - a + 1) ` s \<Longrightarrow>
-          vector_derivative (shiftpath a g) (at x) = vector_derivative g (at (x + a - 1))"
-      unfolding shiftpath_def
-      apply (rule vector_derivative_at [OF has_vector_derivative_transform_within [where f = "(\<lambda>x. g(a+x-1))" and d = "dist (1-a) x"]])
-        apply (auto simp: field_simps dist_real_def abs_if split: if_split_asm)
-      apply (rule vector_diff_chain_at [of "\<lambda>x. x+a-1" 1 _ g, simplified o_def scaleR_one])
-       apply (intro derivative_eq_intros | simp)+
-      using g
-      apply (drule_tac x="x+a-1" in bspec)
-      using a apply (auto simp: has_vector_derivative_def vector_derivative_works image_def add.commute)
-      done
-  } note vd2 = this
-  have va1: "(\<lambda>x. f (g x) * vector_derivative g (at x)) integrable_on ({a..1})"
-    using * a   by (fastforce intro: integrable_subinterval_real)
-  have v0a: "(\<lambda>x. f (g x) * vector_derivative g (at x)) integrable_on ({0..a})"
-    apply (rule integrable_subinterval_real)
-    using * a by auto
-  have "((\<lambda>x. f (shiftpath a g x) * vector_derivative (shiftpath a g) (at x))
-        has_integral  integral {a..1} (\<lambda>x. f (g x) * vector_derivative g (at x)))  {0..1 - a}"
-    apply (rule has_integral_spike_finite
-             [where S = "{1-a} \<union> (\<lambda>x. x-a) ` s" and f = "\<lambda>x. f(g(a+x)) * vector_derivative g (at(a+x))"])
-      using s apply blast
-     using a apply (auto simp: algebra_simps vd1)
-     apply (force simp: shiftpath_def add.commute)
-    using has_integral_affinity [where m=1 and c=a, simplified, OF integrable_integral [OF va1]]
-    apply (simp add: image_affinity_atLeastAtMost_diff [where m=1 and c=a, simplified] add.commute)
-    done
-  moreover
-  have "((\<lambda>x. f (shiftpath a g x) * vector_derivative (shiftpath a g) (at x))
-        has_integral  integral {0..a} (\<lambda>x. f (g x) * vector_derivative g (at x)))  {1 - a..1}"
-    apply (rule has_integral_spike_finite
-             [where S = "{1-a} \<union> (\<lambda>x. x-a+1) ` s" and f = "\<lambda>x. f(g(a+x-1)) * vector_derivative g (at(a+x-1))"])
-      using s apply blast
-     using a apply (auto simp: algebra_simps vd2)
-     apply (force simp: shiftpath_def add.commute)
-    using has_integral_affinity [where m=1 and c="a-1", simplified, OF integrable_integral [OF v0a]]
-    apply (simp add: image_affinity_atLeastAtMost [where m=1 and c="1-a", simplified])
-    apply (simp add: algebra_simps)
-    done
-  ultimately show ?thesis
-    using a
-    by (auto simp: i has_contour_integral intro: has_integral_combine [where c = "1-a"])
-qed
-
-lemma has_contour_integral_shiftpath_D:
-  assumes "(f has_contour_integral i) (shiftpath a g)"
-          "valid_path g" "pathfinish g = pathstart g" "a \<in> {0..1}"
-    shows "(f has_contour_integral i) g"
-proof -
-  obtain s
-    where s: "finite s" and g: "\<forall>x\<in>{0..1} - s. g differentiable at x"
-    using assms by (auto simp: valid_path_def piecewise_C1_differentiable_on_def C1_differentiable_on_eq)
-  { fix x
-    assume x: "0 < x" "x < 1" "x \<notin> s"
-    then have gx: "g differentiable at x"
-      using g by auto
-    have "vector_derivative g (at x within {0..1}) =
-          vector_derivative (shiftpath (1 - a) (shiftpath a g)) (at x within {0..1})"
-      apply (rule vector_derivative_at_within_ivl
-                  [OF has_vector_derivative_transform_within_open
-                      [where f = "(shiftpath (1 - a) (shiftpath a g))" and S = "{0<..<1}-s"]])
-      using s g assms x
-      apply (auto simp: finite_imp_closed open_Diff shiftpath_shiftpath
-                        at_within_interior [of _ "{0..1}"] vector_derivative_works [symmetric])
-      apply (rule differentiable_transform_within [OF gx, of "min x (1-x)"])
-      apply (auto simp: dist_real_def shiftpath_shiftpath abs_if split: if_split_asm)
-      done
-  } note vd = this
-  have fi: "(f has_contour_integral i) (shiftpath (1 - a) (shiftpath a g))"
-    using assms  by (auto intro!: has_contour_integral_shiftpath)
-  show ?thesis
-    apply (simp add: has_contour_integral_def)
-    apply (rule has_integral_spike_finite [of "{0,1} \<union> s", OF _ _  fi [unfolded has_contour_integral_def]])
-    using s assms vd
-    apply (auto simp: Path_Connected.shiftpath_shiftpath)
-    done
-qed
-
-lemma has_contour_integral_shiftpath_eq:
-  assumes "valid_path g" "pathfinish g = pathstart g" "a \<in> {0..1}"
-    shows "(f has_contour_integral i) (shiftpath a g) \<longleftrightarrow> (f has_contour_integral i) g"
-  using assms has_contour_integral_shiftpath has_contour_integral_shiftpath_D by blast
-
-lemma contour_integrable_on_shiftpath_eq:
-  assumes "valid_path g" "pathfinish g = pathstart g" "a \<in> {0..1}"
-    shows "f contour_integrable_on (shiftpath a g) \<longleftrightarrow> f contour_integrable_on g"
-using assms contour_integrable_on_def has_contour_integral_shiftpath_eq by auto
-
-lemma contour_integral_shiftpath:
-  assumes "valid_path g" "pathfinish g = pathstart g" "a \<in> {0..1}"
-    shows "contour_integral (shiftpath a g) f = contour_integral g f"
-   using assms
-   by (simp add: contour_integral_def contour_integrable_on_def has_contour_integral_shiftpath_eq)
-
-
-subsection\<^marker>\<open>tag unimportant\<close> \<open>More about straight-line paths\<close>
-
-lemma has_contour_integral_linepath:
-  shows "(f has_contour_integral i) (linepath a b) \<longleftrightarrow>
-         ((\<lambda>x. f(linepath a b x) * (b - a)) has_integral i) {0..1}"
-  by (simp add: has_contour_integral)
-
-lemma has_contour_integral_trivial [iff]: "(f has_contour_integral 0) (linepath a a)"
-  by (simp add: has_contour_integral_linepath)
-
-lemma has_contour_integral_trivial_iff [simp]: "(f has_contour_integral i) (linepath a a) \<longleftrightarrow> i=0"
-  using has_contour_integral_unique by blast
-
-lemma contour_integral_trivial [simp]: "contour_integral (linepath a a) f = 0"
-  using has_contour_integral_trivial contour_integral_unique by blast
-
-
-subsection\<open>Relation to subpath construction\<close>
-
-lemma has_contour_integral_subpath_refl [iff]: "(f has_contour_integral 0) (subpath u u g)"
-  by (simp add: has_contour_integral subpath_def)
-
-lemma contour_integrable_subpath_refl [iff]: "f contour_integrable_on (subpath u u g)"
-  using has_contour_integral_subpath_refl contour_integrable_on_def by blast
-
-lemma contour_integral_subpath_refl [simp]: "contour_integral (subpath u u g) f = 0"
-  by (simp add: contour_integral_unique)
-
-lemma has_contour_integral_subpath:
-  assumes f: "f contour_integrable_on g" and g: "valid_path g"
-      and uv: "u \<in> {0..1}" "v \<in> {0..1}" "u \<le> v"
-    shows "(f has_contour_integral  integral {u..v} (\<lambda>x. f(g x) * vector_derivative g (at x)))
-           (subpath u v g)"
-proof (cases "v=u")
-  case True
-  then show ?thesis
-    using f   by (simp add: contour_integrable_on_def subpath_def has_contour_integral)
-next
-  case False
-  obtain s where s: "\<And>x. x \<in> {0..1} - s \<Longrightarrow> g differentiable at x" and fs: "finite s"
-    using g unfolding piecewise_C1_differentiable_on_def C1_differentiable_on_eq valid_path_def by blast
-  have *: "((\<lambda>x. f (g ((v - u) * x + u)) * vector_derivative g (at ((v - u) * x + u)))
-            has_integral (1 / (v - u)) * integral {u..v} (\<lambda>t. f (g t) * vector_derivative g (at t)))
-           {0..1}"
-    using f uv
-    apply (simp add: contour_integrable_on subpath_def has_contour_integral)
-    apply (drule integrable_on_subcbox [where a=u and b=v, simplified])
-    apply (simp_all add: has_integral_integral)
-    apply (drule has_integral_affinity [where m="v-u" and c=u, simplified])
-    apply (simp_all add: False image_affinity_atLeastAtMost_div_diff scaleR_conv_of_real)
-    apply (simp add: divide_simps False)
-    done
-  { fix x
-    have "x \<in> {0..1} \<Longrightarrow>
-           x \<notin> (\<lambda>t. (v-u) *\<^sub>R t + u) -` s \<Longrightarrow>
-           vector_derivative (\<lambda>x. g ((v-u) * x + u)) (at x) = (v-u) *\<^sub>R vector_derivative g (at ((v-u) * x + u))"
-      apply (rule vector_derivative_at [OF vector_diff_chain_at [simplified o_def]])
-      apply (intro derivative_eq_intros | simp)+
-      apply (cut_tac s [of "(v - u) * x + u"])
-      using uv mult_left_le [of x "v-u"]
-      apply (auto simp:  vector_derivative_works)
-      done
-  } note vd = this
-  show ?thesis
-    apply (cut_tac has_integral_cmul [OF *, where c = "v-u"])
-    using fs assms
-    apply (simp add: False subpath_def has_contour_integral)
-    apply (rule_tac S = "(\<lambda>t. ((v-u) *\<^sub>R t + u)) -` s" in has_integral_spike_finite)
-    apply (auto simp: inj_on_def False finite_vimageI vd scaleR_conv_of_real)
-    done
-qed
-
-lemma contour_integrable_subpath:
-  assumes "f contour_integrable_on g" "valid_path g" "u \<in> {0..1}" "v \<in> {0..1}"
-    shows "f contour_integrable_on (subpath u v g)"
-  apply (cases u v rule: linorder_class.le_cases)
-   apply (metis contour_integrable_on_def has_contour_integral_subpath [OF assms])
-  apply (subst reversepath_subpath [symmetric])
-  apply (rule contour_integrable_reversepath)
-   using assms apply (blast intro: valid_path_subpath)
-  apply (simp add: contour_integrable_on_def)
-  using assms apply (blast intro: has_contour_integral_subpath)
-  done
-
-lemma has_integral_contour_integral_subpath:
-  assumes "f contour_integrable_on g" "valid_path g" "u \<in> {0..1}" "v \<in> {0..1}" "u \<le> v"
-    shows "(((\<lambda>x. f(g x) * vector_derivative g (at x)))
-            has_integral  contour_integral (subpath u v g) f) {u..v}"
-  using assms
-  apply (auto simp: has_integral_integrable_integral)
-  apply (rule integrable_on_subcbox [where a=u and b=v and S = "{0..1}", simplified])
-  apply (auto simp: contour_integral_unique [OF has_contour_integral_subpath] contour_integrable_on)
-  done
-
-lemma contour_integral_subcontour_integral:
-  assumes "f contour_integrable_on g" "valid_path g" "u \<in> {0..1}" "v \<in> {0..1}" "u \<le> v"
-    shows "contour_integral (subpath u v g) f =
-           integral {u..v} (\<lambda>x. f(g x) * vector_derivative g (at x))"
-  using assms has_contour_integral_subpath contour_integral_unique by blast
-
-lemma contour_integral_subpath_combine_less:
-  assumes "f contour_integrable_on g" "valid_path g" "u \<in> {0..1}" "v \<in> {0..1}" "w \<in> {0..1}"
-          "u<v" "v<w"
-    shows "contour_integral (subpath u v g) f + contour_integral (subpath v w g) f =
-           contour_integral (subpath u w g) f"
-  using assms apply (auto simp: contour_integral_subcontour_integral)
-  apply (rule Henstock_Kurzweil_Integration.integral_combine, auto)
-  apply (rule integrable_on_subcbox [where a=u and b=w and S = "{0..1}", simplified])
-  apply (auto simp: contour_integrable_on)
-  done
-
-lemma contour_integral_subpath_combine:
-  assumes "f contour_integrable_on g" "valid_path g" "u \<in> {0..1}" "v \<in> {0..1}" "w \<in> {0..1}"
-    shows "contour_integral (subpath u v g) f + contour_integral (subpath v w g) f =
-           contour_integral (subpath u w g) f"
-proof (cases "u\<noteq>v \<and> v\<noteq>w \<and> u\<noteq>w")
-  case True
-    have *: "subpath v u g = reversepath(subpath u v g) \<and>
-             subpath w u g = reversepath(subpath u w g) \<and>
-             subpath w v g = reversepath(subpath v w g)"
-      by (auto simp: reversepath_subpath)
-    have "u < v \<and> v < w \<or>
-          u < w \<and> w < v \<or>
-          v < u \<and> u < w \<or>
-          v < w \<and> w < u \<or>
-          w < u \<and> u < v \<or>
-          w < v \<and> v < u"
-      using True assms by linarith
-    with assms show ?thesis
-      using contour_integral_subpath_combine_less [of f g u v w]
-            contour_integral_subpath_combine_less [of f g u w v]
-            contour_integral_subpath_combine_less [of f g v u w]
-            contour_integral_subpath_combine_less [of f g v w u]
-            contour_integral_subpath_combine_less [of f g w u v]
-            contour_integral_subpath_combine_less [of f g w v u]
-      apply simp
-      apply (elim disjE)
-      apply (auto simp: * contour_integral_reversepath contour_integrable_subpath
-               valid_path_subpath algebra_simps)
-      done
-next
-  case False
-  then show ?thesis
-    apply (auto)
-    using assms
-    by (metis eq_neg_iff_add_eq_0 contour_integral_reversepath reversepath_subpath valid_path_subpath)
-qed
-
-lemma contour_integral_integral:
-     "contour_integral g f = integral {0..1} (\<lambda>x. f (g x) * vector_derivative g (at x))"
-  by (simp add: contour_integral_def integral_def has_contour_integral contour_integrable_on)
-
-lemma contour_integral_cong:
-  assumes "g = g'" "\<And>x. x \<in> path_image g \<Longrightarrow> f x = f' x"
-  shows   "contour_integral g f = contour_integral g' f'"
-  unfolding contour_integral_integral using assms
-  by (intro integral_cong) (auto simp: path_image_def)
-
-
-text \<open>Contour integral along a segment on the real axis\<close>
-
-lemma has_contour_integral_linepath_Reals_iff:
-  fixes a b :: complex and f :: "complex \<Rightarrow> complex"
-  assumes "a \<in> Reals" "b \<in> Reals" "Re a < Re b"
-  shows   "(f has_contour_integral I) (linepath a b) \<longleftrightarrow>
-             ((\<lambda>x. f (of_real x)) has_integral I) {Re a..Re b}"
-proof -
-  from assms have [simp]: "of_real (Re a) = a" "of_real (Re b) = b"
-    by (simp_all add: complex_eq_iff)
-  from assms have "a \<noteq> b" by auto
-  have "((\<lambda>x. f (of_real x)) has_integral I) (cbox (Re a) (Re b)) \<longleftrightarrow>
-          ((\<lambda>x. f (a + b * of_real x - a * of_real x)) has_integral I /\<^sub>R (Re b - Re a)) {0..1}"
-    by (subst has_integral_affinity_iff [of "Re b - Re a" _ "Re a", symmetric])
-       (insert assms, simp_all add: field_simps scaleR_conv_of_real)
-  also have "(\<lambda>x. f (a + b * of_real x - a * of_real x)) =
-               (\<lambda>x. (f (a + b * of_real x - a * of_real x) * (b - a)) /\<^sub>R (Re b - Re a))"
-    using \<open>a \<noteq> b\<close> by (auto simp: field_simps fun_eq_iff scaleR_conv_of_real)
-  also have "(\<dots> has_integral I /\<^sub>R (Re b - Re a)) {0..1} \<longleftrightarrow> 
-               ((\<lambda>x. f (linepath a b x) * (b - a)) has_integral I) {0..1}" using assms
-    by (subst has_integral_cmul_iff) (auto simp: linepath_def scaleR_conv_of_real algebra_simps)
-  also have "\<dots> \<longleftrightarrow> (f has_contour_integral I) (linepath a b)" unfolding has_contour_integral_def
-    by (intro has_integral_cong) (simp add: vector_derivative_linepath_within)
-  finally show ?thesis by simp
-qed
-
-lemma contour_integrable_linepath_Reals_iff:
-  fixes a b :: complex and f :: "complex \<Rightarrow> complex"
-  assumes "a \<in> Reals" "b \<in> Reals" "Re a < Re b"
-  shows   "(f contour_integrable_on linepath a b) \<longleftrightarrow>
-             (\<lambda>x. f (of_real x)) integrable_on {Re a..Re b}"
-  using has_contour_integral_linepath_Reals_iff[OF assms, of f]
-  by (auto simp: contour_integrable_on_def integrable_on_def)
-
-lemma contour_integral_linepath_Reals_eq:
-  fixes a b :: complex and f :: "complex \<Rightarrow> complex"
-  assumes "a \<in> Reals" "b \<in> Reals" "Re a < Re b"
-  shows   "contour_integral (linepath a b) f = integral {Re a..Re b} (\<lambda>x. f (of_real x))"
-proof (cases "f contour_integrable_on linepath a b")
-  case True
-  thus ?thesis using has_contour_integral_linepath_Reals_iff[OF assms, of f]
-    using has_contour_integral_integral has_contour_integral_unique by blast
-next
-  case False
-  thus ?thesis using contour_integrable_linepath_Reals_iff[OF assms, of f]
-    by (simp add: not_integrable_contour_integral not_integrable_integral)
-qed
-
-
-
-text\<open>Cauchy's theorem where there's a primitive\<close>
-
-lemma contour_integral_primitive_lemma:
-  fixes f :: "complex \<Rightarrow> complex" and g :: "real \<Rightarrow> complex"
-  assumes "a \<le> b"
-      and "\<And>x. x \<in> s \<Longrightarrow> (f has_field_derivative f' x) (at x within s)"
-      and "g piecewise_differentiable_on {a..b}"  "\<And>x. x \<in> {a..b} \<Longrightarrow> g x \<in> s"
-    shows "((\<lambda>x. f'(g x) * vector_derivative g (at x within {a..b}))
-             has_integral (f(g b) - f(g a))) {a..b}"
-proof -
-  obtain k where k: "finite k" "\<forall>x\<in>{a..b} - k. g differentiable (at x within {a..b})" and cg: "continuous_on {a..b} g"
-    using assms by (auto simp: piecewise_differentiable_on_def)
-  have cfg: "continuous_on {a..b} (\<lambda>x. f (g x))"
-    apply (rule continuous_on_compose [OF cg, unfolded o_def])
-    using assms
-    apply (metis field_differentiable_def field_differentiable_imp_continuous_at continuous_on_eq_continuous_within continuous_on_subset image_subset_iff)
-    done
-  { fix x::real
-    assume a: "a < x" and b: "x < b" and xk: "x \<notin> k"
-    then have "g differentiable at x within {a..b}"
-      using k by (simp add: differentiable_at_withinI)
-    then have "(g has_vector_derivative vector_derivative g (at x within {a..b})) (at x within {a..b})"
-      by (simp add: vector_derivative_works has_field_derivative_def scaleR_conv_of_real)
-    then have gdiff: "(g has_derivative (\<lambda>u. u * vector_derivative g (at x within {a..b}))) (at x within {a..b})"
-      by (simp add: has_vector_derivative_def scaleR_conv_of_real)
-    have "(f has_field_derivative (f' (g x))) (at (g x) within g ` {a..b})"
-      using assms by (metis a atLeastAtMost_iff b DERIV_subset image_subset_iff less_eq_real_def)
-    then have fdiff: "(f has_derivative (*) (f' (g x))) (at (g x) within g ` {a..b})"
-      by (simp add: has_field_derivative_def)
-    have "((\<lambda>x. f (g x)) has_vector_derivative f' (g x) * vector_derivative g (at x within {a..b})) (at x within {a..b})"
-      using diff_chain_within [OF gdiff fdiff]
-      by (simp add: has_vector_derivative_def scaleR_conv_of_real o_def mult_ac)
-  } note * = this
-  show ?thesis
-    apply (rule fundamental_theorem_of_calculus_interior_strong)
-    using k assms cfg *
-    apply (auto simp: at_within_Icc_at)
-    done
-qed
-
-lemma contour_integral_primitive:
-  assumes "\<And>x. x \<in> s \<Longrightarrow> (f has_field_derivative f' x) (at x within s)"
-      and "valid_path g" "path_image g \<subseteq> s"
-    shows "(f' has_contour_integral (f(pathfinish g) - f(pathstart g))) g"
-  using assms
-  apply (simp add: valid_path_def path_image_def pathfinish_def pathstart_def has_contour_integral_def)
-  apply (auto intro!: piecewise_C1_imp_differentiable contour_integral_primitive_lemma [of 0 1 s])
-  done
-
-corollary Cauchy_theorem_primitive:
-  assumes "\<And>x. x \<in> s \<Longrightarrow> (f has_field_derivative f' x) (at x within s)"
-      and "valid_path g"  "path_image g \<subseteq> s" "pathfinish g = pathstart g"
-    shows "(f' has_contour_integral 0) g"
-  using assms
-  by (metis diff_self contour_integral_primitive)
-
-text\<open>Existence of path integral for continuous function\<close>
-lemma contour_integrable_continuous_linepath:
-  assumes "continuous_on (closed_segment a b) f"
-  shows "f contour_integrable_on (linepath a b)"
-proof -
-  have "continuous_on {0..1} ((\<lambda>x. f x * (b - a)) \<circ> linepath a b)"
-    apply (rule continuous_on_compose [OF continuous_on_linepath], simp add: linepath_image_01)
-    apply (rule continuous_intros | simp add: assms)+
-    done
-  then show ?thesis
-    apply (simp add: contour_integrable_on_def has_contour_integral_def integrable_on_def [symmetric])
-    apply (rule integrable_continuous [of 0 "1::real", simplified])
-    apply (rule continuous_on_eq [where f = "\<lambda>x. f(linepath a b x)*(b - a)"])
-    apply (auto simp: vector_derivative_linepath_within)
-    done
-qed
-
-lemma has_field_der_id: "((\<lambda>x. x\<^sup>2 / 2) has_field_derivative x) (at x)"
-  by (rule has_derivative_imp_has_field_derivative)
-     (rule derivative_intros | simp)+
-
-lemma contour_integral_id [simp]: "contour_integral (linepath a b) (\<lambda>y. y) = (b^2 - a^2)/2"
-  apply (rule contour_integral_unique)
-  using contour_integral_primitive [of UNIV "\<lambda>x. x^2/2" "\<lambda>x. x" "linepath a b"]
-  apply (auto simp: field_simps has_field_der_id)
-  done
-
-lemma contour_integrable_on_const [iff]: "(\<lambda>x. c) contour_integrable_on (linepath a b)"
-  by (simp add: contour_integrable_continuous_linepath)
-
-lemma contour_integrable_on_id [iff]: "(\<lambda>x. x) contour_integrable_on (linepath a b)"
-  by (simp add: contour_integrable_continuous_linepath)
-
-subsection\<^marker>\<open>tag unimportant\<close> \<open>Arithmetical combining theorems\<close>
-
-lemma has_contour_integral_neg:
-    "(f has_contour_integral i) g \<Longrightarrow> ((\<lambda>x. -(f x)) has_contour_integral (-i)) g"
-  by (simp add: has_integral_neg has_contour_integral_def)
-
-lemma has_contour_integral_add:
-    "\<lbrakk>(f1 has_contour_integral i1) g; (f2 has_contour_integral i2) g\<rbrakk>
-     \<Longrightarrow> ((\<lambda>x. f1 x + f2 x) has_contour_integral (i1 + i2)) g"
-  by (simp add: has_integral_add has_contour_integral_def algebra_simps)
-
-lemma has_contour_integral_diff:
-  "\<lbrakk>(f1 has_contour_integral i1) g; (f2 has_contour_integral i2) g\<rbrakk>
-         \<Longrightarrow> ((\<lambda>x. f1 x - f2 x) has_contour_integral (i1 - i2)) g"
-  by (simp add: has_integral_diff has_contour_integral_def algebra_simps)
-
-lemma has_contour_integral_lmul:
-  "(f has_contour_integral i) g \<Longrightarrow> ((\<lambda>x. c * (f x)) has_contour_integral (c*i)) g"
-apply (simp add: has_contour_integral_def)
-apply (drule has_integral_mult_right)
-apply (simp add: algebra_simps)
-done
-
-lemma has_contour_integral_rmul:
-  "(f has_contour_integral i) g \<Longrightarrow> ((\<lambda>x. (f x) * c) has_contour_integral (i*c)) g"
-apply (drule has_contour_integral_lmul)
-apply (simp add: mult.commute)
-done
-
-lemma has_contour_integral_div:
-  "(f has_contour_integral i) g \<Longrightarrow> ((\<lambda>x. f x/c) has_contour_integral (i/c)) g"
-  by (simp add: field_class.field_divide_inverse) (metis has_contour_integral_rmul)
-
-lemma has_contour_integral_eq:
-    "\<lbrakk>(f has_contour_integral y) p; \<And>x. x \<in> path_image p \<Longrightarrow> f x = g x\<rbrakk> \<Longrightarrow> (g has_contour_integral y) p"
-apply (simp add: path_image_def has_contour_integral_def)
-by (metis (no_types, lifting) image_eqI has_integral_eq)
-
-lemma has_contour_integral_bound_linepath:
-  assumes "(f has_contour_integral i) (linepath a b)"
-          "0 \<le> B" "\<And>x. x \<in> closed_segment a b \<Longrightarrow> norm(f x) \<le> B"
-    shows "norm i \<le> B * norm(b - a)"
-proof -
-  { fix x::real
-    assume x: "0 \<le> x" "x \<le> 1"
-  have "norm (f (linepath a b x)) *
-        norm (vector_derivative (linepath a b) (at x within {0..1})) \<le> B * norm (b - a)"
-    by (auto intro: mult_mono simp: assms linepath_in_path of_real_linepath vector_derivative_linepath_within x)
-  } note * = this
-  have "norm i \<le> (B * norm (b - a)) * content (cbox 0 (1::real))"
-    apply (rule has_integral_bound
-       [of _ "\<lambda>x. f (linepath a b x) * vector_derivative (linepath a b) (at x within {0..1})"])
-    using assms * unfolding has_contour_integral_def
-    apply (auto simp: norm_mult)
-    done
-  then show ?thesis
-    by (auto simp: content_real)
-qed
-
-(*UNUSED
-lemma has_contour_integral_bound_linepath_strong:
-  fixes a :: real and f :: "complex \<Rightarrow> real"
-  assumes "(f has_contour_integral i) (linepath a b)"
-          "finite k"
-          "0 \<le> B" "\<And>x::real. x \<in> closed_segment a b - k \<Longrightarrow> norm(f x) \<le> B"
-    shows "norm i \<le> B*norm(b - a)"
-*)
-
-lemma has_contour_integral_const_linepath: "((\<lambda>x. c) has_contour_integral c*(b - a))(linepath a b)"
-  unfolding has_contour_integral_linepath
-  by (metis content_real diff_0_right has_integral_const_real lambda_one of_real_1 scaleR_conv_of_real zero_le_one)
-
-lemma has_contour_integral_0: "((\<lambda>x. 0) has_contour_integral 0) g"
-  by (simp add: has_contour_integral_def)
-
-lemma has_contour_integral_is_0:
-    "(\<And>z. z \<in> path_image g \<Longrightarrow> f z = 0) \<Longrightarrow> (f has_contour_integral 0) g"
-  by (rule has_contour_integral_eq [OF has_contour_integral_0]) auto
-
-lemma has_contour_integral_sum:
-    "\<lbrakk>finite s; \<And>a. a \<in> s \<Longrightarrow> (f a has_contour_integral i a) p\<rbrakk>
-     \<Longrightarrow> ((\<lambda>x. sum (\<lambda>a. f a x) s) has_contour_integral sum i s) p"
-  by (induction s rule: finite_induct) (auto simp: has_contour_integral_0 has_contour_integral_add)
-
-subsection\<^marker>\<open>tag unimportant\<close> \<open>Operations on path integrals\<close>
-
-lemma contour_integral_const_linepath [simp]: "contour_integral (linepath a b) (\<lambda>x. c) = c*(b - a)"
-  by (rule contour_integral_unique [OF has_contour_integral_const_linepath])
-
-lemma contour_integral_neg:
-    "f contour_integrable_on g \<Longrightarrow> contour_integral g (\<lambda>x. -(f x)) = -(contour_integral g f)"
-  by (simp add: contour_integral_unique has_contour_integral_integral has_contour_integral_neg)
-
-lemma contour_integral_add:
-    "f1 contour_integrable_on g \<Longrightarrow> f2 contour_integrable_on g \<Longrightarrow> contour_integral g (\<lambda>x. f1 x + f2 x) =
-                contour_integral g f1 + contour_integral g f2"
-  by (simp add: contour_integral_unique has_contour_integral_integral has_contour_integral_add)
-
-lemma contour_integral_diff:
-    "f1 contour_integrable_on g \<Longrightarrow> f2 contour_integrable_on g \<Longrightarrow> contour_integral g (\<lambda>x. f1 x - f2 x) =
-                contour_integral g f1 - contour_integral g f2"
-  by (simp add: contour_integral_unique has_contour_integral_integral has_contour_integral_diff)
-
-lemma contour_integral_lmul:
-  shows "f contour_integrable_on g
-           \<Longrightarrow> contour_integral g (\<lambda>x. c * f x) = c*contour_integral g f"
-  by (simp add: contour_integral_unique has_contour_integral_integral has_contour_integral_lmul)
-
-lemma contour_integral_rmul:
-  shows "f contour_integrable_on g
-        \<Longrightarrow> contour_integral g (\<lambda>x. f x * c) = contour_integral g f * c"
-  by (simp add: contour_integral_unique has_contour_integral_integral has_contour_integral_rmul)
-
-lemma contour_integral_div:
-  shows "f contour_integrable_on g
-        \<Longrightarrow> contour_integral g (\<lambda>x. f x / c) = contour_integral g f / c"
-  by (simp add: contour_integral_unique has_contour_integral_integral has_contour_integral_div)
-
-lemma contour_integral_eq:
-    "(\<And>x. x \<in> path_image p \<Longrightarrow> f x = g x) \<Longrightarrow> contour_integral p f = contour_integral p g"
-  apply (simp add: contour_integral_def)
-  using has_contour_integral_eq
-  by (metis contour_integral_unique has_contour_integral_integrable has_contour_integral_integral)
-
-lemma contour_integral_eq_0:
-    "(\<And>z. z \<in> path_image g \<Longrightarrow> f z = 0) \<Longrightarrow> contour_integral g f = 0"
-  by (simp add: has_contour_integral_is_0 contour_integral_unique)
-
-lemma contour_integral_bound_linepath:
-  shows
-    "\<lbrakk>f contour_integrable_on (linepath a b);
-      0 \<le> B; \<And>x. x \<in> closed_segment a b \<Longrightarrow> norm(f x) \<le> B\<rbrakk>
-     \<Longrightarrow> norm(contour_integral (linepath a b) f) \<le> B*norm(b - a)"
-  apply (rule has_contour_integral_bound_linepath [of f])
-  apply (auto simp: has_contour_integral_integral)
-  done
-
-lemma contour_integral_0 [simp]: "contour_integral g (\<lambda>x. 0) = 0"
-  by (simp add: contour_integral_unique has_contour_integral_0)
-
-lemma contour_integral_sum:
-    "\<lbrakk>finite s; \<And>a. a \<in> s \<Longrightarrow> (f a) contour_integrable_on p\<rbrakk>
-     \<Longrightarrow> contour_integral p (\<lambda>x. sum (\<lambda>a. f a x) s) = sum (\<lambda>a. contour_integral p (f a)) s"
-  by (auto simp: contour_integral_unique has_contour_integral_sum has_contour_integral_integral)
-
-lemma contour_integrable_eq:
-    "\<lbrakk>f contour_integrable_on p; \<And>x. x \<in> path_image p \<Longrightarrow> f x = g x\<rbrakk> \<Longrightarrow> g contour_integrable_on p"
-  unfolding contour_integrable_on_def
-  by (metis has_contour_integral_eq)
-
-
-subsection\<^marker>\<open>tag unimportant\<close> \<open>Arithmetic theorems for path integrability\<close>
-
-lemma contour_integrable_neg:
-    "f contour_integrable_on g \<Longrightarrow> (\<lambda>x. -(f x)) contour_integrable_on g"
-  using has_contour_integral_neg contour_integrable_on_def by blast
-
-lemma contour_integrable_add:
-    "\<lbrakk>f1 contour_integrable_on g; f2 contour_integrable_on g\<rbrakk> \<Longrightarrow> (\<lambda>x. f1 x + f2 x) contour_integrable_on g"
-  using has_contour_integral_add contour_integrable_on_def
-  by fastforce
-
-lemma contour_integrable_diff:
-    "\<lbrakk>f1 contour_integrable_on g; f2 contour_integrable_on g\<rbrakk> \<Longrightarrow> (\<lambda>x. f1 x - f2 x) contour_integrable_on g"
-  using has_contour_integral_diff contour_integrable_on_def
-  by fastforce
-
-lemma contour_integrable_lmul:
-    "f contour_integrable_on g \<Longrightarrow> (\<lambda>x. c * f x) contour_integrable_on g"
-  using has_contour_integral_lmul contour_integrable_on_def
-  by fastforce
-
-lemma contour_integrable_rmul:
-    "f contour_integrable_on g \<Longrightarrow> (\<lambda>x. f x * c) contour_integrable_on g"
-  using has_contour_integral_rmul contour_integrable_on_def
-  by fastforce
-
-lemma contour_integrable_div:
-    "f contour_integrable_on g \<Longrightarrow> (\<lambda>x. f x / c) contour_integrable_on g"
-  using has_contour_integral_div contour_integrable_on_def
-  by fastforce
-
-lemma contour_integrable_sum:
-    "\<lbrakk>finite s; \<And>a. a \<in> s \<Longrightarrow> (f a) contour_integrable_on p\<rbrakk>
-     \<Longrightarrow> (\<lambda>x. sum (\<lambda>a. f a x) s) contour_integrable_on p"
-   unfolding contour_integrable_on_def
-   by (metis has_contour_integral_sum)
-
-
-subsection\<^marker>\<open>tag unimportant\<close> \<open>Reversing a path integral\<close>
-
-lemma has_contour_integral_reverse_linepath:
-    "(f has_contour_integral i) (linepath a b)
-     \<Longrightarrow> (f has_contour_integral (-i)) (linepath b a)"
-  using has_contour_integral_reversepath valid_path_linepath by fastforce
-
-lemma contour_integral_reverse_linepath:
-    "continuous_on (closed_segment a b) f
-     \<Longrightarrow> contour_integral (linepath a b) f = - (contour_integral(linepath b a) f)"
-apply (rule contour_integral_unique)
-apply (rule has_contour_integral_reverse_linepath)
-by (simp add: closed_segment_commute contour_integrable_continuous_linepath has_contour_integral_integral)
-
-
-(* Splitting a path integral in a flat way.*)
-
-lemma has_contour_integral_split:
-  assumes f: "(f has_contour_integral i) (linepath a c)" "(f has_contour_integral j) (linepath c b)"
-      and k: "0 \<le> k" "k \<le> 1"
-      and c: "c - a = k *\<^sub>R (b - a)"
-    shows "(f has_contour_integral (i + j)) (linepath a b)"
-proof (cases "k = 0 \<or> k = 1")
-  case True
-  then show ?thesis
-    using assms by auto
-next
-  case False
-  then have k: "0 < k" "k < 1" "complex_of_real k \<noteq> 1"
-    using assms by auto
-  have c': "c = k *\<^sub>R (b - a) + a"
-    by (metis diff_add_cancel c)
-  have bc: "(b - c) = (1 - k) *\<^sub>R (b - a)"
-    by (simp add: algebra_simps c')
-  { assume *: "((\<lambda>x. f ((1 - x) *\<^sub>R a + x *\<^sub>R c) * (c - a)) has_integral i) {0..1}"
-    have **: "\<And>x. ((k - x) / k) *\<^sub>R a + (x / k) *\<^sub>R c = (1 - x) *\<^sub>R a + x *\<^sub>R b"
-      using False apply (simp add: c' algebra_simps)
-      apply (simp add: real_vector.scale_left_distrib [symmetric] field_split_simps)
-      done
-    have "((\<lambda>x. f ((1 - x) *\<^sub>R a + x *\<^sub>R b) * (b - a)) has_integral i) {0..k}"
-      using k has_integral_affinity01 [OF *, of "inverse k" "0"]
-      apply (simp add: divide_simps mult.commute [of _ "k"] image_affinity_atLeastAtMost ** c)
-      apply (auto dest: has_integral_cmul [where c = "inverse k"])
-      done
-  } note fi = this
-  { assume *: "((\<lambda>x. f ((1 - x) *\<^sub>R c + x *\<^sub>R b) * (b - c)) has_integral j) {0..1}"
-    have **: "\<And>x. (((1 - x) / (1 - k)) *\<^sub>R c + ((x - k) / (1 - k)) *\<^sub>R b) = ((1 - x) *\<^sub>R a + x *\<^sub>R b)"
-      using k
-      apply (simp add: c' field_simps)
-      apply (simp add: scaleR_conv_of_real divide_simps)
-      apply (simp add: field_simps)
-      done
-    have "((\<lambda>x. f ((1 - x) *\<^sub>R a + x *\<^sub>R b) * (b - a)) has_integral j) {k..1}"
-      using k has_integral_affinity01 [OF *, of "inverse(1 - k)" "-(k/(1 - k))"]
-      apply (simp add: divide_simps mult.commute [of _ "1-k"] image_affinity_atLeastAtMost ** bc)
-      apply (auto dest: has_integral_cmul [where k = "(1 - k) *\<^sub>R j" and c = "inverse (1 - k)"])
-      done
-  } note fj = this
-  show ?thesis
-    using f k
-    apply (simp add: has_contour_integral_linepath)
-    apply (simp add: linepath_def)
-    apply (rule has_integral_combine [OF _ _ fi fj], simp_all)
-    done
-qed
-
-lemma continuous_on_closed_segment_transform:
-  assumes f: "continuous_on (closed_segment a b) f"
-      and k: "0 \<le> k" "k \<le> 1"
-      and c: "c - a = k *\<^sub>R (b - a)"
-    shows "continuous_on (closed_segment a c) f"
-proof -
-  have c': "c = (1 - k) *\<^sub>R a + k *\<^sub>R b"
-    using c by (simp add: algebra_simps)
-  have "closed_segment a c \<subseteq> closed_segment a b"
-    by (metis c' ends_in_segment(1) in_segment(1) k subset_closed_segment)
-  then show "continuous_on (closed_segment a c) f"
-    by (rule continuous_on_subset [OF f])
-qed
-
-lemma contour_integral_split:
-  assumes f: "continuous_on (closed_segment a b) f"
-      and k: "0 \<le> k" "k \<le> 1"
-      and c: "c - a = k *\<^sub>R (b - a)"
-    shows "contour_integral(linepath a b) f = contour_integral(linepath a c) f + contour_integral(linepath c b) f"
-proof -
-  have c': "c = (1 - k) *\<^sub>R a + k *\<^sub>R b"
-    using c by (simp add: algebra_simps)
-  have "closed_segment a c \<subseteq> closed_segment a b"
-    by (metis c' ends_in_segment(1) in_segment(1) k subset_closed_segment)
-  moreover have "closed_segment c b \<subseteq> closed_segment a b"
-    by (metis c' ends_in_segment(2) in_segment(1) k subset_closed_segment)
-  ultimately
-  have *: "continuous_on (closed_segment a c) f" "continuous_on (closed_segment c b) f"
-    by (auto intro: continuous_on_subset [OF f])
-  show ?thesis
-    by (rule contour_integral_unique) (meson "*" c contour_integrable_continuous_linepath has_contour_integral_integral has_contour_integral_split k)
-qed
-
-lemma contour_integral_split_linepath:
-  assumes f: "continuous_on (closed_segment a b) f"
-      and c: "c \<in> closed_segment a b"
-    shows "contour_integral(linepath a b) f = contour_integral(linepath a c) f + contour_integral(linepath c b) f"
-  using c by (auto simp: closed_segment_def algebra_simps intro!: contour_integral_split [OF f])
-
 text\<open>The special case of midpoints used in the main quadrisection\<close>
 
 lemma has_contour_integral_midpoint:
@@ -1056,7 +58,6 @@
   apply (auto simp: midpoint_def algebra_simps scaleR_conv_of_real)
   done
 
-
 text\<open>A couple of special case lemmas that are useful below\<close>
 
 lemma triangle_linear_has_chain_integral:
@@ -1081,87 +82,6 @@
   apply (simp add: valid_path_join)
   done
 
-subsection\<open>Reversing the order in a double path integral\<close>
-
-text\<open>The condition is stronger than needed but it's often true in typical situations\<close>
-
-lemma fst_im_cbox [simp]: "cbox c d \<noteq> {} \<Longrightarrow> (fst ` cbox (a,c) (b,d)) = cbox a b"
-  by (auto simp: cbox_Pair_eq)
-
-lemma snd_im_cbox [simp]: "cbox a b \<noteq> {} \<Longrightarrow> (snd ` cbox (a,c) (b,d)) = cbox c d"
-  by (auto simp: cbox_Pair_eq)
-
-proposition contour_integral_swap:
-  assumes fcon:  "continuous_on (path_image g \<times> path_image h) (\<lambda>(y1,y2). f y1 y2)"
-      and vp:    "valid_path g" "valid_path h"
-      and gvcon: "continuous_on {0..1} (\<lambda>t. vector_derivative g (at t))"
-      and hvcon: "continuous_on {0..1} (\<lambda>t. vector_derivative h (at t))"
-  shows "contour_integral g (\<lambda>w. contour_integral h (f w)) =
-         contour_integral h (\<lambda>z. contour_integral g (\<lambda>w. f w z))"
-proof -
-  have gcon: "continuous_on {0..1} g" and hcon: "continuous_on {0..1} h"
-    using assms by (auto simp: valid_path_def piecewise_C1_differentiable_on_def)
-  have fgh1: "\<And>x. (\<lambda>t. f (g x) (h t)) = (\<lambda>(y1,y2). f y1 y2) \<circ> (\<lambda>t. (g x, h t))"
-    by (rule ext) simp
-  have fgh2: "\<And>x. (\<lambda>t. f (g t) (h x)) = (\<lambda>(y1,y2). f y1 y2) \<circ> (\<lambda>t. (g t, h x))"
-    by (rule ext) simp
-  have fcon_im1: "\<And>x. 0 \<le> x \<Longrightarrow> x \<le> 1 \<Longrightarrow> continuous_on ((\<lambda>t. (g x, h t)) ` {0..1}) (\<lambda>(x, y). f x y)"
-    by (rule continuous_on_subset [OF fcon]) (auto simp: path_image_def)
-  have fcon_im2: "\<And>x. 0 \<le> x \<Longrightarrow> x \<le> 1 \<Longrightarrow> continuous_on ((\<lambda>t. (g t, h x)) ` {0..1}) (\<lambda>(x, y). f x y)"
-    by (rule continuous_on_subset [OF fcon]) (auto simp: path_image_def)
-  have "\<And>y. y \<in> {0..1} \<Longrightarrow> continuous_on {0..1} (\<lambda>x. f (g x) (h y))"
-    by (subst fgh2) (rule fcon_im2 gcon continuous_intros | simp)+
-  then have vdg: "\<And>y. y \<in> {0..1} \<Longrightarrow> (\<lambda>x. f (g x) (h y) * vector_derivative g (at x)) integrable_on {0..1}"
-    using continuous_on_mult gvcon integrable_continuous_real by blast
-  have "(\<lambda>z. vector_derivative g (at (fst z))) = (\<lambda>x. vector_derivative g (at x)) \<circ> fst"
-    by auto
-  then have gvcon': "continuous_on (cbox (0, 0) (1, 1::real)) (\<lambda>x. vector_derivative g (at (fst x)))"
-    apply (rule ssubst)
-    apply (rule continuous_intros | simp add: gvcon)+
-    done
-  have "(\<lambda>z. vector_derivative h (at (snd z))) = (\<lambda>x. vector_derivative h (at x)) \<circ> snd"
-    by auto
-  then have hvcon': "continuous_on (cbox (0, 0) (1::real, 1)) (\<lambda>x. vector_derivative h (at (snd x)))"
-    apply (rule ssubst)
-    apply (rule continuous_intros | simp add: hvcon)+
-    done
-  have "(\<lambda>x. f (g (fst x)) (h (snd x))) = (\<lambda>(y1,y2). f y1 y2) \<circ> (\<lambda>w. ((g \<circ> fst) w, (h \<circ> snd) w))"
-    by auto
-  then have fgh: "continuous_on (cbox (0, 0) (1, 1)) (\<lambda>x. f (g (fst x)) (h (snd x)))"
-    apply (rule ssubst)
-    apply (rule gcon hcon continuous_intros | simp)+
-    apply (auto simp: path_image_def intro: continuous_on_subset [OF fcon])
-    done
-  have "integral {0..1} (\<lambda>x. contour_integral h (f (g x)) * vector_derivative g (at x)) =
-        integral {0..1} (\<lambda>x. contour_integral h (\<lambda>y. f (g x) y * vector_derivative g (at x)))"
-  proof (rule integral_cong [OF contour_integral_rmul [symmetric]])
-    show "\<And>x. x \<in> {0..1} \<Longrightarrow> f (g x) contour_integrable_on h"
-      unfolding contour_integrable_on
-    apply (rule integrable_continuous_real)
-    apply (rule continuous_on_mult [OF _ hvcon])
-    apply (subst fgh1)
-    apply (rule fcon_im1 hcon continuous_intros | simp)+
-      done
-  qed
-  also have "\<dots> = integral {0..1}
-                     (\<lambda>y. contour_integral g (\<lambda>x. f x (h y) * vector_derivative h (at y)))"
-    unfolding contour_integral_integral
-    apply (subst integral_swap_continuous [where 'a = real and 'b = real, of 0 0 1 1, simplified])
-     apply (rule fgh gvcon' hvcon' continuous_intros | simp add: split_def)+
-    unfolding integral_mult_left [symmetric]
-    apply (simp only: mult_ac)
-    done
-  also have "\<dots> = contour_integral h (\<lambda>z. contour_integral g (\<lambda>w. f w z))"
-    unfolding contour_integral_integral
-    apply (rule integral_cong)
-    unfolding integral_mult_left [symmetric]
-    apply (simp add: algebra_simps)
-    done
-  finally show ?thesis
-    by (simp add: contour_integral_integral)
-qed
-
-
 subsection\<^marker>\<open>tag unimportant\<close> \<open>The key quadrisection step\<close>
 
 lemma norm_sum_half:
@@ -2431,21 +1351,6 @@
   using contour_integral_nearby [OF assms, where atends=False]
   unfolding linked_paths_def by simp_all
 
-lemma C1_differentiable_polynomial_function:
-  fixes p :: "real \<Rightarrow> 'a::euclidean_space"
-  shows "polynomial_function p \<Longrightarrow> p C1_differentiable_on S"
-  by (metis continuous_on_polymonial_function C1_differentiable_on_def  has_vector_derivative_polynomial_function)
-
-lemma valid_path_polynomial_function:
-  fixes p :: "real \<Rightarrow> 'a::euclidean_space"
-  shows "polynomial_function p \<Longrightarrow> valid_path p"
-by (force simp: valid_path_def piecewise_C1_differentiable_on_def continuous_on_polymonial_function C1_differentiable_polynomial_function)
-
-lemma valid_path_subpath_trivial [simp]:
-    fixes g :: "real \<Rightarrow> 'a::euclidean_space"
-    shows "z \<noteq> g x \<Longrightarrow> valid_path (subpath x x g)"
-  by (simp add: subpath_def valid_path_polynomial_function)
-
 lemma contour_integral_bound_exists:
 assumes S: "open S"
     and g: "valid_path g"
@@ -2497,1188 +1402,6 @@
     by (intro exI[of _ L]) auto
 qed
 
-text\<open>We can treat even non-rectifiable paths as having a "length" for bounds on analytic functions in open sets.\<close>
-
-subsection \<open>Winding Numbers\<close>
-
-definition\<^marker>\<open>tag important\<close> winding_number_prop :: "[real \<Rightarrow> complex, complex, real, real \<Rightarrow> complex, complex] \<Rightarrow> bool" where
-  "winding_number_prop \<gamma> z e p n \<equiv>
-      valid_path p \<and> z \<notin> path_image p \<and>
-      pathstart p = pathstart \<gamma> \<and>
-      pathfinish p = pathfinish \<gamma> \<and>
-      (\<forall>t \<in> {0..1}. norm(\<gamma> t - p t) < e) \<and>
-      contour_integral p (\<lambda>w. 1/(w - z)) = 2 * pi * \<i> * n"
-
-definition\<^marker>\<open>tag important\<close> winding_number:: "[real \<Rightarrow> complex, complex] \<Rightarrow> complex" where
-  "winding_number \<gamma> z \<equiv> SOME n. \<forall>e > 0. \<exists>p. winding_number_prop \<gamma> z e p n"
-
-
-lemma winding_number:
-  assumes "path \<gamma>" "z \<notin> path_image \<gamma>" "0 < e"
-    shows "\<exists>p. winding_number_prop \<gamma> z e p (winding_number \<gamma> z)"
-proof -
-  have "path_image \<gamma> \<subseteq> UNIV - {z}"
-    using assms by blast
-  then obtain d
-    where d: "d>0"
-      and pi_eq: "\<And>h1 h2. valid_path h1 \<and> valid_path h2 \<and>
-                    (\<forall>t\<in>{0..1}. cmod (h1 t - \<gamma> t) < d \<and> cmod (h2 t - \<gamma> t) < d) \<and>
-                    pathstart h2 = pathstart h1 \<and> pathfinish h2 = pathfinish h1 \<longrightarrow>
-                      path_image h1 \<subseteq> UNIV - {z} \<and> path_image h2 \<subseteq> UNIV - {z} \<and>
-                      (\<forall>f. f holomorphic_on UNIV - {z} \<longrightarrow> contour_integral h2 f = contour_integral h1 f)"
-    using contour_integral_nearby_ends [of "UNIV - {z}" \<gamma>] assms by (auto simp: open_delete)
-  then obtain h where h: "polynomial_function h \<and> pathstart h = pathstart \<gamma> \<and> pathfinish h = pathfinish \<gamma> \<and>
-                          (\<forall>t \<in> {0..1}. norm(h t - \<gamma> t) < d/2)"
-    using path_approx_polynomial_function [OF \<open>path \<gamma>\<close>, of "d/2"] d by auto
-  define nn where "nn = 1/(2* pi*\<i>) * contour_integral h (\<lambda>w. 1/(w - z))"
-  have "\<exists>n. \<forall>e > 0. \<exists>p. winding_number_prop \<gamma> z e p n"
-    proof (rule_tac x=nn in exI, clarify)
-      fix e::real
-      assume e: "e>0"
-      obtain p where p: "polynomial_function p \<and>
-            pathstart p = pathstart \<gamma> \<and> pathfinish p = pathfinish \<gamma> \<and> (\<forall>t\<in>{0..1}. cmod (p t - \<gamma> t) < min e (d/2))"
-        using path_approx_polynomial_function [OF \<open>path \<gamma>\<close>, of "min e (d/2)"] d \<open>0<e\<close> by auto
-      have "(\<lambda>w. 1 / (w - z)) holomorphic_on UNIV - {z}"
-        by (auto simp: intro!: holomorphic_intros)
-      then show "\<exists>p. winding_number_prop \<gamma> z e p nn"
-        apply (rule_tac x=p in exI)
-        using pi_eq [of h p] h p d
-        apply (auto simp: valid_path_polynomial_function norm_minus_commute nn_def winding_number_prop_def)
-        done
-    qed
-  then show ?thesis
-    unfolding winding_number_def by (rule someI2_ex) (blast intro: \<open>0<e\<close>)
-qed
-
-lemma winding_number_unique:
-  assumes \<gamma>: "path \<gamma>" "z \<notin> path_image \<gamma>"
-      and pi: "\<And>e. e>0 \<Longrightarrow> \<exists>p. winding_number_prop \<gamma> z e p n"
-   shows "winding_number \<gamma> z = n"
-proof -
-  have "path_image \<gamma> \<subseteq> UNIV - {z}"
-    using assms by blast
-  then obtain e
-    where e: "e>0"
-      and pi_eq: "\<And>h1 h2 f. \<lbrakk>valid_path h1; valid_path h2;
-                    (\<forall>t\<in>{0..1}. cmod (h1 t - \<gamma> t) < e \<and> cmod (h2 t - \<gamma> t) < e);
-                    pathstart h2 = pathstart h1; pathfinish h2 = pathfinish h1; f holomorphic_on UNIV - {z}\<rbrakk> \<Longrightarrow>
-                    contour_integral h2 f = contour_integral h1 f"
-    using contour_integral_nearby_ends [of "UNIV - {z}" \<gamma>] assms  by (auto simp: open_delete)
-  obtain p where p: "winding_number_prop \<gamma> z e p n"
-    using pi [OF e] by blast
-  obtain q where q: "winding_number_prop \<gamma> z e q (winding_number \<gamma> z)"
-    using winding_number [OF \<gamma> e] by blast
-  have "2 * complex_of_real pi * \<i> * n = contour_integral p (\<lambda>w. 1 / (w - z))"
-    using p by (auto simp: winding_number_prop_def)
-  also have "\<dots> = contour_integral q (\<lambda>w. 1 / (w - z))"
-  proof (rule pi_eq)
-    show "(\<lambda>w. 1 / (w - z)) holomorphic_on UNIV - {z}"
-      by (auto intro!: holomorphic_intros)
-  qed (use p q in \<open>auto simp: winding_number_prop_def norm_minus_commute\<close>)
-  also have "\<dots> = 2 * complex_of_real pi * \<i> * winding_number \<gamma> z"
-    using q by (auto simp: winding_number_prop_def)
-  finally have "2 * complex_of_real pi * \<i> * n = 2 * complex_of_real pi * \<i> * winding_number \<gamma> z" .
-  then show ?thesis
-    by simp
-qed
-
-(*NB not winding_number_prop here due to the loop in p*)
-lemma winding_number_unique_loop:
-  assumes \<gamma>: "path \<gamma>" "z \<notin> path_image \<gamma>"
-      and loop: "pathfinish \<gamma> = pathstart \<gamma>"
-      and pi:
-        "\<And>e. e>0 \<Longrightarrow> \<exists>p. valid_path p \<and> z \<notin> path_image p \<and>
-                           pathfinish p = pathstart p \<and>
-                           (\<forall>t \<in> {0..1}. norm (\<gamma> t - p t) < e) \<and>
-                           contour_integral p (\<lambda>w. 1/(w - z)) = 2 * pi * \<i> * n"
-   shows "winding_number \<gamma> z = n"
-proof -
-  have "path_image \<gamma> \<subseteq> UNIV - {z}"
-    using assms by blast
-  then obtain e
-    where e: "e>0"
-      and pi_eq: "\<And>h1 h2 f. \<lbrakk>valid_path h1; valid_path h2;
-                    (\<forall>t\<in>{0..1}. cmod (h1 t - \<gamma> t) < e \<and> cmod (h2 t - \<gamma> t) < e);
-                    pathfinish h1 = pathstart h1; pathfinish h2 = pathstart h2; f holomorphic_on UNIV - {z}\<rbrakk> \<Longrightarrow>
-                    contour_integral h2 f = contour_integral h1 f"
-    using contour_integral_nearby_loops [of "UNIV - {z}" \<gamma>] assms  by (auto simp: open_delete)
-  obtain p where p:
-     "valid_path p \<and> z \<notin> path_image p \<and> pathfinish p = pathstart p \<and>
-      (\<forall>t \<in> {0..1}. norm (\<gamma> t - p t) < e) \<and>
-      contour_integral p (\<lambda>w. 1/(w - z)) = 2 * pi * \<i> * n"
-    using pi [OF e] by blast
-  obtain q where q: "winding_number_prop \<gamma> z e q (winding_number \<gamma> z)"
-    using winding_number [OF \<gamma> e] by blast
-  have "2 * complex_of_real pi * \<i> * n = contour_integral p (\<lambda>w. 1 / (w - z))"
-    using p by auto
-  also have "\<dots> = contour_integral q (\<lambda>w. 1 / (w - z))"
-  proof (rule pi_eq)
-    show "(\<lambda>w. 1 / (w - z)) holomorphic_on UNIV - {z}"
-      by (auto intro!: holomorphic_intros)
-  qed (use p q loop in \<open>auto simp: winding_number_prop_def norm_minus_commute\<close>)
-  also have "\<dots> = 2 * complex_of_real pi * \<i> * winding_number \<gamma> z"
-    using q by (auto simp: winding_number_prop_def)
-  finally have "2 * complex_of_real pi * \<i> * n = 2 * complex_of_real pi * \<i> * winding_number \<gamma> z" .
-  then show ?thesis
-    by simp
-qed
-
-proposition winding_number_valid_path:
-  assumes "valid_path \<gamma>" "z \<notin> path_image \<gamma>"
-  shows "winding_number \<gamma> z = 1/(2*pi*\<i>) * contour_integral \<gamma> (\<lambda>w. 1/(w - z))"
-  by (rule winding_number_unique)
-  (use assms in \<open>auto simp: valid_path_imp_path winding_number_prop_def\<close>)
-
-proposition has_contour_integral_winding_number:
-  assumes \<gamma>: "valid_path \<gamma>" "z \<notin> path_image \<gamma>"
-    shows "((\<lambda>w. 1/(w - z)) has_contour_integral (2*pi*\<i>*winding_number \<gamma> z)) \<gamma>"
-by (simp add: winding_number_valid_path has_contour_integral_integral contour_integrable_inversediff assms)
-
-lemma winding_number_trivial [simp]: "z \<noteq> a \<Longrightarrow> winding_number(linepath a a) z = 0"
-  by (simp add: winding_number_valid_path)
-
-lemma winding_number_subpath_trivial [simp]: "z \<noteq> g x \<Longrightarrow> winding_number (subpath x x g) z = 0"
-  by (simp add: path_image_subpath winding_number_valid_path)
-
-lemma winding_number_join:
-  assumes \<gamma>1: "path \<gamma>1" "z \<notin> path_image \<gamma>1"
-      and \<gamma>2: "path \<gamma>2" "z \<notin> path_image \<gamma>2"
-      and "pathfinish \<gamma>1 = pathstart \<gamma>2"
-    shows "winding_number(\<gamma>1 +++ \<gamma>2) z = winding_number \<gamma>1 z + winding_number \<gamma>2 z"
-proof (rule winding_number_unique)
-  show "\<exists>p. winding_number_prop (\<gamma>1 +++ \<gamma>2) z e p
-              (winding_number \<gamma>1 z + winding_number \<gamma>2 z)" if "e > 0" for e
-  proof -
-    obtain p1 where "winding_number_prop \<gamma>1 z e p1 (winding_number \<gamma>1 z)"
-      using \<open>0 < e\<close> \<gamma>1 winding_number by blast
-    moreover
-    obtain p2 where "winding_number_prop \<gamma>2 z e p2 (winding_number \<gamma>2 z)"
-      using \<open>0 < e\<close> \<gamma>2 winding_number by blast
-    ultimately
-    have "winding_number_prop (\<gamma>1+++\<gamma>2) z e (p1+++p2) (winding_number \<gamma>1 z + winding_number \<gamma>2 z)"
-      using assms
-      apply (simp add: winding_number_prop_def not_in_path_image_join contour_integrable_inversediff algebra_simps)
-      apply (auto simp: joinpaths_def)
-      done
-    then show ?thesis
-      by blast
-  qed
-qed (use assms in \<open>auto simp: not_in_path_image_join\<close>)
-
-lemma winding_number_reversepath:
-  assumes "path \<gamma>" "z \<notin> path_image \<gamma>"
-    shows "winding_number(reversepath \<gamma>) z = - (winding_number \<gamma> z)"
-proof (rule winding_number_unique)
-  show "\<exists>p. winding_number_prop (reversepath \<gamma>) z e p (- winding_number \<gamma> z)" if "e > 0" for e
-  proof -
-    obtain p where "winding_number_prop \<gamma> z e p (winding_number \<gamma> z)"
-      using \<open>0 < e\<close> assms winding_number by blast
-    then have "winding_number_prop (reversepath \<gamma>) z e (reversepath p) (- winding_number \<gamma> z)"
-      using assms
-      apply (simp add: winding_number_prop_def contour_integral_reversepath contour_integrable_inversediff valid_path_imp_reverse)
-      apply (auto simp: reversepath_def)
-      done
-    then show ?thesis
-      by blast
-  qed
-qed (use assms in auto)
-
-lemma winding_number_shiftpath:
-  assumes \<gamma>: "path \<gamma>" "z \<notin> path_image \<gamma>"
-      and "pathfinish \<gamma> = pathstart \<gamma>" "a \<in> {0..1}"
-    shows "winding_number(shiftpath a \<gamma>) z = winding_number \<gamma> z"
-proof (rule winding_number_unique_loop)
-  show "\<exists>p. valid_path p \<and> z \<notin> path_image p \<and> pathfinish p = pathstart p \<and>
-            (\<forall>t\<in>{0..1}. cmod (shiftpath a \<gamma> t - p t) < e) \<and>
-            contour_integral p (\<lambda>w. 1 / (w - z)) =
-            complex_of_real (2 * pi) * \<i> * winding_number \<gamma> z"
-    if "e > 0" for e
-  proof -
-    obtain p where "winding_number_prop \<gamma> z e p (winding_number \<gamma> z)"
-      using \<open>0 < e\<close> assms winding_number by blast
-    then show ?thesis
-      apply (rule_tac x="shiftpath a p" in exI)
-      using assms that
-      apply (auto simp: winding_number_prop_def path_image_shiftpath pathfinish_shiftpath pathstart_shiftpath contour_integral_shiftpath)
-      apply (simp add: shiftpath_def)
-      done
-  qed
-qed (use assms in \<open>auto simp: path_shiftpath path_image_shiftpath pathfinish_shiftpath pathstart_shiftpath\<close>)
-
-lemma winding_number_split_linepath:
-  assumes "c \<in> closed_segment a b" "z \<notin> closed_segment a b"
-    shows "winding_number(linepath a b) z = winding_number(linepath a c) z + winding_number(linepath c b) z"
-proof -
-  have "z \<notin> closed_segment a c" "z \<notin> closed_segment c b"
-    using assms  by (meson convex_contains_segment convex_segment ends_in_segment subsetCE)+
-  then show ?thesis
-    using assms
-    by (simp add: winding_number_valid_path contour_integral_split_linepath [symmetric] continuous_on_inversediff field_simps)
-qed
-
-lemma winding_number_cong:
-   "(\<And>t. \<lbrakk>0 \<le> t; t \<le> 1\<rbrakk> \<Longrightarrow> p t = q t) \<Longrightarrow> winding_number p z = winding_number q z"
-  by (simp add: winding_number_def winding_number_prop_def pathstart_def pathfinish_def)
-
-lemma winding_number_constI:
-  assumes "c\<noteq>z" "\<And>t. \<lbrakk>0\<le>t; t\<le>1\<rbrakk> \<Longrightarrow> g t = c" 
-  shows "winding_number g z = 0"
-proof -
-  have "winding_number g z = winding_number (linepath c c) z"
-    apply (rule winding_number_cong)
-    using assms unfolding linepath_def by auto
-  moreover have "winding_number (linepath c c) z =0"
-    apply (rule winding_number_trivial)
-    using assms by auto
-  ultimately show ?thesis by auto
-qed
-
-lemma winding_number_offset: "winding_number p z = winding_number (\<lambda>w. p w - z) 0"
-  unfolding winding_number_def
-proof (intro ext arg_cong [where f = Eps] arg_cong [where f = All] imp_cong refl, safe)
-  fix n e g
-  assume "0 < e" and g: "winding_number_prop p z e g n"
-  then show "\<exists>r. winding_number_prop (\<lambda>w. p w - z) 0 e r n"
-    by (rule_tac x="\<lambda>t. g t - z" in exI)
-       (force simp: winding_number_prop_def contour_integral_integral valid_path_def path_defs
-                vector_derivative_def has_vector_derivative_diff_const piecewise_C1_differentiable_diff C1_differentiable_imp_piecewise)
-next
-  fix n e g
-  assume "0 < e" and g: "winding_number_prop (\<lambda>w. p w - z) 0 e g n"
-  then show "\<exists>r. winding_number_prop p z e r n"
-    apply (rule_tac x="\<lambda>t. g t + z" in exI)
-    apply (simp add: winding_number_prop_def contour_integral_integral valid_path_def path_defs
-        piecewise_C1_differentiable_add vector_derivative_def has_vector_derivative_add_const C1_differentiable_imp_piecewise)
-    apply (force simp: algebra_simps)
-    done
-qed
-
-subsubsection\<^marker>\<open>tag unimportant\<close> \<open>Some lemmas about negating a path\<close>
-
-lemma valid_path_negatepath: "valid_path \<gamma> \<Longrightarrow> valid_path (uminus \<circ> \<gamma>)"
-   unfolding o_def using piecewise_C1_differentiable_neg valid_path_def by blast
-
-lemma has_contour_integral_negatepath:
-  assumes \<gamma>: "valid_path \<gamma>" and cint: "((\<lambda>z. f (- z)) has_contour_integral - i) \<gamma>"
-  shows "(f has_contour_integral i) (uminus \<circ> \<gamma>)"
-proof -
-  obtain S where cont: "continuous_on {0..1} \<gamma>" and "finite S" and diff: "\<gamma> C1_differentiable_on {0..1} - S"
-    using \<gamma> by (auto simp: valid_path_def piecewise_C1_differentiable_on_def)
-  have "((\<lambda>x. - (f (- \<gamma> x) * vector_derivative \<gamma> (at x within {0..1}))) has_integral i) {0..1}"
-    using cint by (auto simp: has_contour_integral_def dest: has_integral_neg)
-  then
-  have "((\<lambda>x. f (- \<gamma> x) * vector_derivative (uminus \<circ> \<gamma>) (at x within {0..1})) has_integral i) {0..1}"
-  proof (rule rev_iffD1 [OF _ has_integral_spike_eq])
-    show "negligible S"
-      by (simp add: \<open>finite S\<close> negligible_finite)
-    show "f (- \<gamma> x) * vector_derivative (uminus \<circ> \<gamma>) (at x within {0..1}) =
-         - (f (- \<gamma> x) * vector_derivative \<gamma> (at x within {0..1}))"
-      if "x \<in> {0..1} - S" for x
-    proof -
-      have "vector_derivative (uminus \<circ> \<gamma>) (at x within cbox 0 1) = - vector_derivative \<gamma> (at x within cbox 0 1)"
-      proof (rule vector_derivative_within_cbox)
-        show "(uminus \<circ> \<gamma> has_vector_derivative - vector_derivative \<gamma> (at x within cbox 0 1)) (at x within cbox 0 1)"
-          using that unfolding o_def
-          by (metis C1_differentiable_on_eq UNIV_I diff differentiable_subset has_vector_derivative_minus subsetI that vector_derivative_works)
-      qed (use that in auto)
-      then show ?thesis
-        by simp
-    qed
-  qed
-  then show ?thesis by (simp add: has_contour_integral_def)
-qed
-
-lemma winding_number_negatepath:
-  assumes \<gamma>: "valid_path \<gamma>" and 0: "0 \<notin> path_image \<gamma>"
-  shows "winding_number(uminus \<circ> \<gamma>) 0 = winding_number \<gamma> 0"
-proof -
-  have "(/) 1 contour_integrable_on \<gamma>"
-    using "0" \<gamma> contour_integrable_inversediff by fastforce
-  then have "((\<lambda>z. 1/z) has_contour_integral contour_integral \<gamma> ((/) 1)) \<gamma>"
-    by (rule has_contour_integral_integral)
-  then have "((\<lambda>z. 1 / - z) has_contour_integral - contour_integral \<gamma> ((/) 1)) \<gamma>"
-    using has_contour_integral_neg by auto
-  then show ?thesis
-    using assms
-    apply (simp add: winding_number_valid_path valid_path_negatepath image_def path_defs)
-    apply (simp add: contour_integral_unique has_contour_integral_negatepath)
-    done
-qed
-
-lemma contour_integrable_negatepath:
-  assumes \<gamma>: "valid_path \<gamma>" and pi: "(\<lambda>z. f (- z)) contour_integrable_on \<gamma>"
-  shows "f contour_integrable_on (uminus \<circ> \<gamma>)"
-  by (metis \<gamma> add.inverse_inverse contour_integrable_on_def has_contour_integral_negatepath pi)
-
-(* A combined theorem deducing several things piecewise.*)
-lemma winding_number_join_pos_combined:
-     "\<lbrakk>valid_path \<gamma>1; z \<notin> path_image \<gamma>1; 0 < Re(winding_number \<gamma>1 z);
-       valid_path \<gamma>2; z \<notin> path_image \<gamma>2; 0 < Re(winding_number \<gamma>2 z); pathfinish \<gamma>1 = pathstart \<gamma>2\<rbrakk>
-      \<Longrightarrow> valid_path(\<gamma>1 +++ \<gamma>2) \<and> z \<notin> path_image(\<gamma>1 +++ \<gamma>2) \<and> 0 < Re(winding_number(\<gamma>1 +++ \<gamma>2) z)"
-  by (simp add: valid_path_join path_image_join winding_number_join valid_path_imp_path)
-
-
-subsubsection\<^marker>\<open>tag unimportant\<close> \<open>Useful sufficient conditions for the winding number to be positive\<close>
-
-lemma Re_winding_number:
-    "\<lbrakk>valid_path \<gamma>; z \<notin> path_image \<gamma>\<rbrakk>
-     \<Longrightarrow> Re(winding_number \<gamma> z) = Im(contour_integral \<gamma> (\<lambda>w. 1/(w - z))) / (2*pi)"
-by (simp add: winding_number_valid_path field_simps Re_divide power2_eq_square)
-
-lemma winding_number_pos_le:
-  assumes \<gamma>: "valid_path \<gamma>" "z \<notin> path_image \<gamma>"
-      and ge: "\<And>x. \<lbrakk>0 < x; x < 1\<rbrakk> \<Longrightarrow> 0 \<le> Im (vector_derivative \<gamma> (at x) * cnj(\<gamma> x - z))"
-    shows "0 \<le> Re(winding_number \<gamma> z)"
-proof -
-  have ge0: "0 \<le> Im (vector_derivative \<gamma> (at x) / (\<gamma> x - z))" if x: "0 < x" "x < 1" for x
-    using ge by (simp add: Complex.Im_divide algebra_simps x)
-  let ?vd = "\<lambda>x. 1 / (\<gamma> x - z) * vector_derivative \<gamma> (at x)"
-  let ?int = "\<lambda>z. contour_integral \<gamma> (\<lambda>w. 1 / (w - z))"
-  have hi: "(?vd has_integral ?int z) (cbox 0 1)"
-    unfolding box_real
-    apply (subst has_contour_integral [symmetric])
-    using \<gamma> by (simp add: contour_integrable_inversediff has_contour_integral_integral)
-  have "0 \<le> Im (?int z)"
-  proof (rule has_integral_component_nonneg [of \<i>, simplified])
-    show "\<And>x. x \<in> cbox 0 1 \<Longrightarrow> 0 \<le> Im (if 0 < x \<and> x < 1 then ?vd x else 0)"
-      by (force simp: ge0)
-    show "((\<lambda>x. if 0 < x \<and> x < 1 then ?vd x else 0) has_integral ?int z) (cbox 0 1)"
-      by (rule has_integral_spike_interior [OF hi]) simp
-  qed
-  then show ?thesis
-    by (simp add: Re_winding_number [OF \<gamma>] field_simps)
-qed
-
-lemma winding_number_pos_lt_lemma:
-  assumes \<gamma>: "valid_path \<gamma>" "z \<notin> path_image \<gamma>"
-      and e: "0 < e"
-      and ge: "\<And>x. \<lbrakk>0 < x; x < 1\<rbrakk> \<Longrightarrow> e \<le> Im (vector_derivative \<gamma> (at x) / (\<gamma> x - z))"
-    shows "0 < Re(winding_number \<gamma> z)"
-proof -
-  let ?vd = "\<lambda>x. 1 / (\<gamma> x - z) * vector_derivative \<gamma> (at x)"
-  let ?int = "\<lambda>z. contour_integral \<gamma> (\<lambda>w. 1 / (w - z))"
-  have hi: "(?vd has_integral ?int z) (cbox 0 1)"
-    unfolding box_real
-    apply (subst has_contour_integral [symmetric])
-    using \<gamma> by (simp add: contour_integrable_inversediff has_contour_integral_integral)
-  have "e \<le> Im (contour_integral \<gamma> (\<lambda>w. 1 / (w - z)))"
-  proof (rule has_integral_component_le [of \<i> "\<lambda>x. \<i>*e" "\<i>*e" "{0..1}", simplified])
-    show "((\<lambda>x. if 0 < x \<and> x < 1 then ?vd x else \<i> * complex_of_real e) has_integral ?int z) {0..1}"
-      by (rule has_integral_spike_interior [OF hi, simplified box_real]) (use e in simp)
-    show "\<And>x. 0 \<le> x \<and> x \<le> 1 \<Longrightarrow>
-              e \<le> Im (if 0 < x \<and> x < 1 then ?vd x else \<i> * complex_of_real e)"
-      by (simp add: ge)
-  qed (use has_integral_const_real [of _ 0 1] in auto)
-  with e show ?thesis
-    by (simp add: Re_winding_number [OF \<gamma>] field_simps)
-qed
-
-lemma winding_number_pos_lt:
-  assumes \<gamma>: "valid_path \<gamma>" "z \<notin> path_image \<gamma>"
-      and e: "0 < e"
-      and ge: "\<And>x. \<lbrakk>0 < x; x < 1\<rbrakk> \<Longrightarrow> e \<le> Im (vector_derivative \<gamma> (at x) * cnj(\<gamma> x - z))"
-    shows "0 < Re (winding_number \<gamma> z)"
-proof -
-  have bm: "bounded ((\<lambda>w. w - z) ` (path_image \<gamma>))"
-    using bounded_translation [of _ "-z"] \<gamma> by (simp add: bounded_valid_path_image)
-  then obtain B where B: "B > 0" and Bno: "\<And>x. x \<in> (\<lambda>w. w - z) ` (path_image \<gamma>) \<Longrightarrow> norm x \<le> B"
-    using bounded_pos [THEN iffD1, OF bm] by blast
-  { fix x::real  assume x: "0 < x" "x < 1"
-    then have B2: "cmod (\<gamma> x - z)^2 \<le> B^2" using Bno [of "\<gamma> x - z"]
-      by (simp add: path_image_def power2_eq_square mult_mono')
-    with x have "\<gamma> x \<noteq> z" using \<gamma>
-      using path_image_def by fastforce
-    then have "e / B\<^sup>2 \<le> Im (vector_derivative \<gamma> (at x) * cnj (\<gamma> x - z)) / (cmod (\<gamma> x - z))\<^sup>2"
-      using B ge [OF x] B2 e
-      apply (rule_tac y="e / (cmod (\<gamma> x - z))\<^sup>2" in order_trans)
-      apply (auto simp: divide_left_mono divide_right_mono)
-      done
-    then have "e / B\<^sup>2 \<le> Im (vector_derivative \<gamma> (at x) / (\<gamma> x - z))"
-      by (simp add: complex_div_cnj [of _ "\<gamma> x - z" for x] del: complex_cnj_diff times_complex.sel)
-  } note * = this
-  show ?thesis
-    using e B by (simp add: * winding_number_pos_lt_lemma [OF \<gamma>, of "e/B^2"])
-qed
-
-subsection\<open>The winding number is an integer\<close>
-
-text\<open>Proof from the book Complex Analysis by Lars V. Ahlfors, Chapter 4, section 2.1,
-     Also on page 134 of Serge Lang's book with the name title, etc.\<close>
-
-lemma exp_fg:
-  fixes z::complex
-  assumes g: "(g has_vector_derivative g') (at x within s)"
-      and f: "(f has_vector_derivative (g' / (g x - z))) (at x within s)"
-      and z: "g x \<noteq> z"
-    shows "((\<lambda>x. exp(-f x) * (g x - z)) has_vector_derivative 0) (at x within s)"
-proof -
-  have *: "(exp \<circ> (\<lambda>x. (- f x)) has_vector_derivative - (g' / (g x - z)) * exp (- f x)) (at x within s)"
-    using assms unfolding has_vector_derivative_def scaleR_conv_of_real
-    by (auto intro!: derivative_eq_intros)
-  show ?thesis
-    apply (rule has_vector_derivative_eq_rhs)
-    using z
-    apply (auto intro!: derivative_eq_intros * [unfolded o_def] g)
-    done
-qed
-
-lemma winding_number_exp_integral:
-  fixes z::complex
-  assumes \<gamma>: "\<gamma> piecewise_C1_differentiable_on {a..b}"
-      and ab: "a \<le> b"
-      and z: "z \<notin> \<gamma> ` {a..b}"
-    shows "(\<lambda>x. vector_derivative \<gamma> (at x) / (\<gamma> x - z)) integrable_on {a..b}"
-          (is "?thesis1")
-          "exp (- (integral {a..b} (\<lambda>x. vector_derivative \<gamma> (at x) / (\<gamma> x - z)))) * (\<gamma> b - z) = \<gamma> a - z"
-          (is "?thesis2")
-proof -
-  let ?D\<gamma> = "\<lambda>x. vector_derivative \<gamma> (at x)"
-  have [simp]: "\<And>x. a \<le> x \<Longrightarrow> x \<le> b \<Longrightarrow> \<gamma> x \<noteq> z"
-    using z by force
-  have cong: "continuous_on {a..b} \<gamma>"
-    using \<gamma> by (simp add: piecewise_C1_differentiable_on_def)
-  obtain k where fink: "finite k" and g_C1_diff: "\<gamma> C1_differentiable_on ({a..b} - k)"
-    using \<gamma> by (force simp: piecewise_C1_differentiable_on_def)
-  have \<circ>: "open ({a<..<b} - k)"
-    using \<open>finite k\<close> by (simp add: finite_imp_closed open_Diff)
-  moreover have "{a<..<b} - k \<subseteq> {a..b} - k"
-    by force
-  ultimately have g_diff_at: "\<And>x. \<lbrakk>x \<notin> k; x \<in> {a<..<b}\<rbrakk> \<Longrightarrow> \<gamma> differentiable at x"
-    by (metis Diff_iff differentiable_on_subset C1_diff_imp_diff [OF g_C1_diff] differentiable_on_def at_within_open)
-  { fix w
-    assume "w \<noteq> z"
-    have "continuous_on (ball w (cmod (w - z))) (\<lambda>w. 1 / (w - z))"
-      by (auto simp: dist_norm intro!: continuous_intros)
-    moreover have "\<And>x. cmod (w - x) < cmod (w - z) \<Longrightarrow> \<exists>f'. ((\<lambda>w. 1 / (w - z)) has_field_derivative f') (at x)"
-      by (auto simp: intro!: derivative_eq_intros)
-    ultimately have "\<exists>h. \<forall>y. norm(y - w) < norm(w - z) \<longrightarrow> (h has_field_derivative 1/(y - z)) (at y)"
-      using holomorphic_convex_primitive [of "ball w (norm(w - z))" "{}" "\<lambda>w. 1/(w - z)"]
-      by (force simp: field_differentiable_def Ball_def dist_norm at_within_open_NO_MATCH norm_minus_commute)
-  }
-  then obtain h where h: "\<And>w y. w \<noteq> z \<Longrightarrow> norm(y - w) < norm(w - z) \<Longrightarrow> (h w has_field_derivative 1/(y - z)) (at y)"
-    by meson
-  have exy: "\<exists>y. ((\<lambda>x. inverse (\<gamma> x - z) * ?D\<gamma> x) has_integral y) {a..b}"
-    unfolding integrable_on_def [symmetric]
-  proof (rule contour_integral_local_primitive_any [OF piecewise_C1_imp_differentiable [OF \<gamma>]])
-    show "\<exists>d h. 0 < d \<and>
-               (\<forall>y. cmod (y - w) < d \<longrightarrow> (h has_field_derivative inverse (y - z))(at y within - {z}))"
-          if "w \<in> - {z}" for w
-      apply (rule_tac x="norm(w - z)" in exI)
-      using that inverse_eq_divide has_field_derivative_at_within h
-      by (metis Compl_insert DiffD2 insertCI right_minus_eq zero_less_norm_iff)
-  qed simp
-  have vg_int: "(\<lambda>x. ?D\<gamma> x / (\<gamma> x - z)) integrable_on {a..b}"
-    unfolding box_real [symmetric] divide_inverse_commute
-    by (auto intro!: exy integrable_subinterval simp add: integrable_on_def ab)
-  with ab show ?thesis1
-    by (simp add: divide_inverse_commute integral_def integrable_on_def)
-  { fix t
-    assume t: "t \<in> {a..b}"
-    have cball: "continuous_on (ball (\<gamma> t) (dist (\<gamma> t) z)) (\<lambda>x. inverse (x - z))"
-        using z by (auto intro!: continuous_intros simp: dist_norm)
-    have icd: "\<And>x. cmod (\<gamma> t - x) < cmod (\<gamma> t - z) \<Longrightarrow> (\<lambda>w. inverse (w - z)) field_differentiable at x"
-      unfolding field_differentiable_def by (force simp: intro!: derivative_eq_intros)
-    obtain h where h: "\<And>x. cmod (\<gamma> t - x) < cmod (\<gamma> t - z) \<Longrightarrow>
-                       (h has_field_derivative inverse (x - z)) (at x within {y. cmod (\<gamma> t - y) < cmod (\<gamma> t - z)})"
-      using holomorphic_convex_primitive [where f = "\<lambda>w. inverse(w - z)", OF convex_ball finite.emptyI cball icd]
-      by simp (auto simp: ball_def dist_norm that)
-    { fix x D
-      assume x: "x \<notin> k" "a < x" "x < b"
-      then have "x \<in> interior ({a..b} - k)"
-        using open_subset_interior [OF \<circ>] by fastforce
-      then have con: "isCont ?D\<gamma> x"
-        using g_C1_diff x by (auto simp: C1_differentiable_on_eq intro: continuous_on_interior)
-      then have con_vd: "continuous (at x within {a..b}) (\<lambda>x. ?D\<gamma> x)"
-        by (rule continuous_at_imp_continuous_within)
-      have gdx: "\<gamma> differentiable at x"
-        using x by (simp add: g_diff_at)
-      have "\<And>d. \<lbrakk>x \<notin> k; a < x; x < b;
-          (\<gamma> has_vector_derivative d) (at x); a \<le> t; t \<le> b\<rbrakk>
-         \<Longrightarrow> ((\<lambda>x. integral {a..x}
-                     (\<lambda>x. ?D\<gamma> x /
-                           (\<gamma> x - z))) has_vector_derivative
-              d / (\<gamma> x - z))
-              (at x within {a..b})"
-        apply (rule has_vector_derivative_eq_rhs)
-         apply (rule integral_has_vector_derivative_continuous_at [where S = "{}", simplified])
-        apply (rule con_vd continuous_intros cong vg_int | simp add: continuous_at_imp_continuous_within has_vector_derivative_continuous vector_derivative_at)+
-        done
-      then have "((\<lambda>c. exp (- integral {a..c} (\<lambda>x. ?D\<gamma> x / (\<gamma> x - z))) * (\<gamma> c - z)) has_derivative (\<lambda>h. 0))
-          (at x within {a..b})"
-        using x gdx t
-        apply (clarsimp simp add: differentiable_iff_scaleR)
-        apply (rule exp_fg [unfolded has_vector_derivative_def, simplified], blast intro: has_derivative_at_withinI)
-        apply (simp_all add: has_vector_derivative_def [symmetric])
-        done
-      } note * = this
-    have "exp (- (integral {a..t} (\<lambda>x. ?D\<gamma> x / (\<gamma> x - z)))) * (\<gamma> t - z) =\<gamma> a - z"
-      apply (rule has_derivative_zero_unique_strong_interval [of "{a,b} \<union> k" a b])
-      using t
-      apply (auto intro!: * continuous_intros fink cong indefinite_integral_continuous_1 [OF vg_int]  simp add: ab)+
-      done
-   }
-  with ab show ?thesis2
-    by (simp add: divide_inverse_commute integral_def)
-qed
-
-lemma winding_number_exp_2pi:
-    "\<lbrakk>path p; z \<notin> path_image p\<rbrakk>
-     \<Longrightarrow> pathfinish p - z = exp (2 * pi * \<i> * winding_number p z) * (pathstart p - z)"
-using winding_number [of p z 1] unfolding valid_path_def path_image_def pathstart_def pathfinish_def winding_number_prop_def
-  by (force dest: winding_number_exp_integral(2) [of _ 0 1 z] simp: field_simps contour_integral_integral exp_minus)
-
-lemma integer_winding_number_eq:
-  assumes \<gamma>: "path \<gamma>" and z: "z \<notin> path_image \<gamma>"
-  shows "winding_number \<gamma> z \<in> \<int> \<longleftrightarrow> pathfinish \<gamma> = pathstart \<gamma>"
-proof -
-  obtain p where p: "valid_path p" "z \<notin> path_image p"
-                    "pathstart p = pathstart \<gamma>" "pathfinish p = pathfinish \<gamma>"
-           and eq: "contour_integral p (\<lambda>w. 1 / (w - z)) = complex_of_real (2 * pi) * \<i> * winding_number \<gamma> z"
-    using winding_number [OF assms, of 1] unfolding winding_number_prop_def by auto
-  then have wneq: "winding_number \<gamma> z = winding_number p z"
-      using eq winding_number_valid_path by force
-  have iff: "(winding_number \<gamma> z \<in> \<int>) \<longleftrightarrow> (exp (contour_integral p (\<lambda>w. 1 / (w - z))) = 1)"
-    using eq by (simp add: exp_eq_1 complex_is_Int_iff)
-  have "exp (contour_integral p (\<lambda>w. 1 / (w - z))) = (\<gamma> 1 - z) / (\<gamma> 0 - z)"
-    using p winding_number_exp_integral(2) [of p 0 1 z]
-    apply (simp add: valid_path_def path_defs contour_integral_integral exp_minus field_split_simps)
-    by (metis path_image_def pathstart_def pathstart_in_path_image)
-  then have "winding_number p z \<in> \<int> \<longleftrightarrow> pathfinish p = pathstart p"
-    using p wneq iff by (auto simp: path_defs)
-  then show ?thesis using p eq
-    by (auto simp: winding_number_valid_path)
-qed
-
-theorem integer_winding_number:
-  "\<lbrakk>path \<gamma>; pathfinish \<gamma> = pathstart \<gamma>; z \<notin> path_image \<gamma>\<rbrakk> \<Longrightarrow> winding_number \<gamma> z \<in> \<int>"
-by (metis integer_winding_number_eq)
-
-
-text\<open>If the winding number's magnitude is at least one, then the path must contain points in every direction.*)
-   We can thus bound the winding number of a path that doesn't intersect a given ray. \<close>
-
-lemma winding_number_pos_meets:
-  fixes z::complex
-  assumes \<gamma>: "valid_path \<gamma>" and z: "z \<notin> path_image \<gamma>" and 1: "Re (winding_number \<gamma> z) \<ge> 1"
-      and w: "w \<noteq> z"
-  shows "\<exists>a::real. 0 < a \<and> z + a*(w - z) \<in> path_image \<gamma>"
-proof -
-  have [simp]: "\<And>x. 0 \<le> x \<Longrightarrow> x \<le> 1 \<Longrightarrow> \<gamma> x \<noteq> z"
-    using z by (auto simp: path_image_def)
-  have [simp]: "z \<notin> \<gamma> ` {0..1}"
-    using path_image_def z by auto
-  have gpd: "\<gamma> piecewise_C1_differentiable_on {0..1}"
-    using \<gamma> valid_path_def by blast
-  define r where "r = (w - z) / (\<gamma> 0 - z)"
-  have [simp]: "r \<noteq> 0"
-    using w z by (auto simp: r_def)
-  have cont: "continuous_on {0..1}
-     (\<lambda>x. Im (integral {0..x} (\<lambda>x. vector_derivative \<gamma> (at x) / (\<gamma> x - z))))"
-    by (intro continuous_intros indefinite_integral_continuous_1 winding_number_exp_integral [OF gpd]; simp)
-  have "Arg2pi r \<le> 2*pi"
-    by (simp add: Arg2pi less_eq_real_def)
-  also have "\<dots> \<le> Im (integral {0..1} (\<lambda>x. vector_derivative \<gamma> (at x) / (\<gamma> x - z)))"
-    using 1
-    apply (simp add: winding_number_valid_path [OF \<gamma> z] contour_integral_integral)
-    apply (simp add: Complex.Re_divide field_simps power2_eq_square)
-    done
-  finally have "Arg2pi r \<le> Im (integral {0..1} (\<lambda>x. vector_derivative \<gamma> (at x) / (\<gamma> x - z)))" .
-  then have "\<exists>t. t \<in> {0..1} \<and> Im(integral {0..t} (\<lambda>x. vector_derivative \<gamma> (at x)/(\<gamma> x - z))) = Arg2pi r"
-    by (simp add: Arg2pi_ge_0 cont IVT')
-  then obtain t where t:     "t \<in> {0..1}"
-                  and eqArg: "Im (integral {0..t} (\<lambda>x. vector_derivative \<gamma> (at x)/(\<gamma> x - z))) = Arg2pi r"
-    by blast
-  define i where "i = integral {0..t} (\<lambda>x. vector_derivative \<gamma> (at x) / (\<gamma> x - z))"
-  have iArg: "Arg2pi r = Im i"
-    using eqArg by (simp add: i_def)
-  have gpdt: "\<gamma> piecewise_C1_differentiable_on {0..t}"
-    by (metis atLeastAtMost_iff atLeastatMost_subset_iff order_refl piecewise_C1_differentiable_on_subset gpd t)
-  have "exp (- i) * (\<gamma> t - z) = \<gamma> 0 - z"
-    unfolding i_def
-    apply (rule winding_number_exp_integral [OF gpdt])
-    using t z unfolding path_image_def by force+
-  then have *: "\<gamma> t - z = exp i * (\<gamma> 0 - z)"
-    by (simp add: exp_minus field_simps)
-  then have "(w - z) = r * (\<gamma> 0 - z)"
-    by (simp add: r_def)
-  then have "z + complex_of_real (exp (Re i)) * (w - z) / complex_of_real (cmod r) = \<gamma> t"
-    apply simp
-    apply (subst Complex_Transcendental.Arg2pi_eq [of r])
-    apply (simp add: iArg)
-    using * apply (simp add: exp_eq_polar field_simps)
-    done
-  with t show ?thesis
-    by (rule_tac x="exp(Re i) / norm r" in exI) (auto simp: path_image_def)
-qed
-
-lemma winding_number_big_meets:
-  fixes z::complex
-  assumes \<gamma>: "valid_path \<gamma>" and z: "z \<notin> path_image \<gamma>" and "\<bar>Re (winding_number \<gamma> z)\<bar> \<ge> 1"
-      and w: "w \<noteq> z"
-  shows "\<exists>a::real. 0 < a \<and> z + a*(w - z) \<in> path_image \<gamma>"
-proof -
-  { assume "Re (winding_number \<gamma> z) \<le> - 1"
-    then have "Re (winding_number (reversepath \<gamma>) z) \<ge> 1"
-      by (simp add: \<gamma> valid_path_imp_path winding_number_reversepath z)
-    moreover have "valid_path (reversepath \<gamma>)"
-      using \<gamma> valid_path_imp_reverse by auto
-    moreover have "z \<notin> path_image (reversepath \<gamma>)"
-      by (simp add: z)
-    ultimately have "\<exists>a::real. 0 < a \<and> z + a*(w - z) \<in> path_image (reversepath \<gamma>)"
-      using winding_number_pos_meets w by blast
-    then have ?thesis
-      by simp
-  }
-  then show ?thesis
-    using assms
-    by (simp add: abs_if winding_number_pos_meets split: if_split_asm)
-qed
-
-lemma winding_number_less_1:
-  fixes z::complex
-  shows
-  "\<lbrakk>valid_path \<gamma>; z \<notin> path_image \<gamma>; w \<noteq> z;
-    \<And>a::real. 0 < a \<Longrightarrow> z + a*(w - z) \<notin> path_image \<gamma>\<rbrakk>
-   \<Longrightarrow> Re(winding_number \<gamma> z) < 1"
-   by (auto simp: not_less dest: winding_number_big_meets)
-
-text\<open>One way of proving that WN=1 for a loop.\<close>
-lemma winding_number_eq_1:
-  fixes z::complex
-  assumes \<gamma>: "valid_path \<gamma>" and z: "z \<notin> path_image \<gamma>" and loop: "pathfinish \<gamma> = pathstart \<gamma>"
-      and 0: "0 < Re(winding_number \<gamma> z)" and 2: "Re(winding_number \<gamma> z) < 2"
-  shows "winding_number \<gamma> z = 1"
-proof -
-  have "winding_number \<gamma> z \<in> Ints"
-    by (simp add: \<gamma> integer_winding_number loop valid_path_imp_path z)
-  then show ?thesis
-    using 0 2 by (auto simp: Ints_def)
-qed
-
-subsection\<open>Continuity of winding number and invariance on connected sets\<close>
-
-lemma continuous_at_winding_number:
-  fixes z::complex
-  assumes \<gamma>: "path \<gamma>" and z: "z \<notin> path_image \<gamma>"
-  shows "continuous (at z) (winding_number \<gamma>)"
-proof -
-  obtain e where "e>0" and cbg: "cball z e \<subseteq> - path_image \<gamma>"
-    using open_contains_cball [of "- path_image \<gamma>"]  z
-    by (force simp: closed_def [symmetric] closed_path_image [OF \<gamma>])
-  then have ppag: "path_image \<gamma> \<subseteq> - cball z (e/2)"
-    by (force simp: cball_def dist_norm)
-  have oc: "open (- cball z (e / 2))"
-    by (simp add: closed_def [symmetric])
-  obtain d where "d>0" and pi_eq:
-    "\<And>h1 h2. \<lbrakk>valid_path h1; valid_path h2;
-              (\<forall>t\<in>{0..1}. cmod (h1 t - \<gamma> t) < d \<and> cmod (h2 t - \<gamma> t) < d);
-              pathstart h2 = pathstart h1; pathfinish h2 = pathfinish h1\<rbrakk>
-             \<Longrightarrow>
-               path_image h1 \<subseteq> - cball z (e / 2) \<and>
-               path_image h2 \<subseteq> - cball z (e / 2) \<and>
-               (\<forall>f. f holomorphic_on - cball z (e / 2) \<longrightarrow> contour_integral h2 f = contour_integral h1 f)"
-    using contour_integral_nearby_ends [OF oc \<gamma> ppag] by metis
-  obtain p where p: "valid_path p" "z \<notin> path_image p"
-                    "pathstart p = pathstart \<gamma> \<and> pathfinish p = pathfinish \<gamma>"
-              and pg: "\<And>t. t\<in>{0..1} \<Longrightarrow> cmod (\<gamma> t - p t) < min d e / 2"
-              and pi: "contour_integral p (\<lambda>x. 1 / (x - z)) = complex_of_real (2 * pi) * \<i> * winding_number \<gamma> z"
-    using winding_number [OF \<gamma> z, of "min d e / 2"] \<open>d>0\<close> \<open>e>0\<close> by (auto simp: winding_number_prop_def)
-  { fix w
-    assume d2: "cmod (w - z) < d/2" and e2: "cmod (w - z) < e/2"
-    then have wnotp: "w \<notin> path_image p"
-      using cbg \<open>d>0\<close> \<open>e>0\<close>
-      apply (simp add: path_image_def cball_def dist_norm, clarify)
-      apply (frule pg)
-      apply (drule_tac c="\<gamma> x" in subsetD)
-      apply (auto simp: less_eq_real_def norm_minus_commute norm_triangle_half_l)
-      done
-    have wnotg: "w \<notin> path_image \<gamma>"
-      using cbg e2 \<open>e>0\<close> by (force simp: dist_norm norm_minus_commute)
-    { fix k::real
-      assume k: "k>0"
-      then obtain q where q: "valid_path q" "w \<notin> path_image q"
-                             "pathstart q = pathstart \<gamma> \<and> pathfinish q = pathfinish \<gamma>"
-                    and qg: "\<And>t. t \<in> {0..1} \<Longrightarrow> cmod (\<gamma> t - q t) < min k (min d e) / 2"
-                    and qi: "contour_integral q (\<lambda>u. 1 / (u - w)) = complex_of_real (2 * pi) * \<i> * winding_number \<gamma> w"
-        using winding_number [OF \<gamma> wnotg, of "min k (min d e) / 2"] \<open>d>0\<close> \<open>e>0\<close> k
-        by (force simp: min_divide_distrib_right winding_number_prop_def)
-      have "contour_integral p (\<lambda>u. 1 / (u - w)) = contour_integral q (\<lambda>u. 1 / (u - w))"
-        apply (rule pi_eq [OF \<open>valid_path q\<close> \<open>valid_path p\<close>, THEN conjunct2, THEN conjunct2, rule_format])
-        apply (frule pg)
-        apply (frule qg)
-        using p q \<open>d>0\<close> e2
-        apply (auto simp: dist_norm norm_minus_commute intro!: holomorphic_intros)
-        done
-      then have "contour_integral p (\<lambda>x. 1 / (x - w)) = complex_of_real (2 * pi) * \<i> * winding_number \<gamma> w"
-        by (simp add: pi qi)
-    } note pip = this
-    have "path p"
-      using p by (simp add: valid_path_imp_path)
-    then have "winding_number p w = winding_number \<gamma> w"
-      apply (rule winding_number_unique [OF _ wnotp])
-      apply (rule_tac x=p in exI)
-      apply (simp add: p wnotp min_divide_distrib_right pip winding_number_prop_def)
-      done
-  } note wnwn = this
-  obtain pe where "pe>0" and cbp: "cball z (3 / 4 * pe) \<subseteq> - path_image p"
-    using p open_contains_cball [of "- path_image p"]
-    by (force simp: closed_def [symmetric] closed_path_image [OF valid_path_imp_path])
-  obtain L
-    where "L>0"
-      and L: "\<And>f B. \<lbrakk>f holomorphic_on - cball z (3 / 4 * pe);
-                      \<forall>z \<in> - cball z (3 / 4 * pe). cmod (f z) \<le> B\<rbrakk> \<Longrightarrow>
-                      cmod (contour_integral p f) \<le> L * B"
-    using contour_integral_bound_exists [of "- cball z (3/4*pe)" p] cbp \<open>valid_path p\<close> by blast
-  { fix e::real and w::complex
-    assume e: "0 < e" and w: "cmod (w - z) < pe/4" "cmod (w - z) < e * pe\<^sup>2 / (8 * L)"
-    then have [simp]: "w \<notin> path_image p"
-      using cbp p(2) \<open>0 < pe\<close>
-      by (force simp: dist_norm norm_minus_commute path_image_def cball_def)
-    have [simp]: "contour_integral p (\<lambda>x. 1/(x - w)) - contour_integral p (\<lambda>x. 1/(x - z)) =
-                  contour_integral p (\<lambda>x. 1/(x - w) - 1/(x - z))"
-      by (simp add: p contour_integrable_inversediff contour_integral_diff)
-    { fix x
-      assume pe: "3/4 * pe < cmod (z - x)"
-      have "cmod (w - x) < pe/4 + cmod (z - x)"
-        by (meson add_less_cancel_right norm_diff_triangle_le order_refl order_trans_rules(21) w(1))
-      then have wx: "cmod (w - x) < 4/3 * cmod (z - x)" using pe by simp
-      have "cmod (z - x) \<le> cmod (z - w) + cmod (w - x)"
-        using norm_diff_triangle_le by blast
-      also have "\<dots> < pe/4 + cmod (w - x)"
-        using w by (simp add: norm_minus_commute)
-      finally have "pe/2 < cmod (w - x)"
-        using pe by auto
-      then have "(pe/2)^2 < cmod (w - x) ^ 2"
-        apply (rule power_strict_mono)
-        using \<open>pe>0\<close> by auto
-      then have pe2: "pe^2 < 4 * cmod (w - x) ^ 2"
-        by (simp add: power_divide)
-      have "8 * L * cmod (w - z) < e * pe\<^sup>2"
-        using w \<open>L>0\<close> by (simp add: field_simps)
-      also have "\<dots> < e * 4 * cmod (w - x) * cmod (w - x)"
-        using pe2 \<open>e>0\<close> by (simp add: power2_eq_square)
-      also have "\<dots> < e * 4 * cmod (w - x) * (4/3 * cmod (z - x))"
-        using wx
-        apply (rule mult_strict_left_mono)
-        using pe2 e not_less_iff_gr_or_eq by fastforce
-      finally have "L * cmod (w - z) < 2/3 * e * cmod (w - x) * cmod (z - x)"
-        by simp
-      also have "\<dots> \<le> e * cmod (w - x) * cmod (z - x)"
-         using e by simp
-      finally have Lwz: "L * cmod (w - z) < e * cmod (w - x) * cmod (z - x)" .
-      have "L * cmod (1 / (x - w) - 1 / (x - z)) \<le> e"
-        apply (cases "x=z \<or> x=w")
-        using pe \<open>pe>0\<close> w \<open>L>0\<close>
-        apply (force simp: norm_minus_commute)
-        using wx w(2) \<open>L>0\<close> pe pe2 Lwz
-        apply (auto simp: divide_simps mult_less_0_iff norm_minus_commute norm_divide norm_mult power2_eq_square)
-        done
-    } note L_cmod_le = this
-    have *: "cmod (contour_integral p (\<lambda>x. 1 / (x - w) - 1 / (x - z))) \<le> L * (e * pe\<^sup>2 / L / 4 * (inverse (pe / 2))\<^sup>2)"
-      apply (rule L)
-      using \<open>pe>0\<close> w
-      apply (force simp: dist_norm norm_minus_commute intro!: holomorphic_intros)
-      using \<open>pe>0\<close> w \<open>L>0\<close>
-      apply (auto simp: cball_def dist_norm field_simps L_cmod_le  simp del: less_divide_eq_numeral1 le_divide_eq_numeral1)
-      done
-    have "cmod (contour_integral p (\<lambda>x. 1 / (x - w)) - contour_integral p (\<lambda>x. 1 / (x - z))) < 2*e"
-      apply simp
-      apply (rule le_less_trans [OF *])
-      using \<open>L>0\<close> e
-      apply (force simp: field_simps)
-      done
-    then have "cmod (winding_number p w - winding_number p z) < e"
-      using pi_ge_two e
-      by (force simp: winding_number_valid_path p field_simps norm_divide norm_mult intro: less_le_trans)
-  } note cmod_wn_diff = this
-  then have "isCont (winding_number p) z"
-    apply (simp add: continuous_at_eps_delta, clarify)
-    apply (rule_tac x="min (pe/4) (e/2*pe^2/L/4)" in exI)
-    using \<open>pe>0\<close> \<open>L>0\<close>
-    apply (simp add: dist_norm cmod_wn_diff)
-    done
-  then show ?thesis
-    apply (rule continuous_transform_within [where d = "min d e / 2"])
-    apply (auto simp: \<open>d>0\<close> \<open>e>0\<close> dist_norm wnwn)
-    done
-qed
-
-corollary continuous_on_winding_number:
-    "path \<gamma> \<Longrightarrow> continuous_on (- path_image \<gamma>) (\<lambda>w. winding_number \<gamma> w)"
-  by (simp add: continuous_at_imp_continuous_on continuous_at_winding_number)
-
-subsection\<^marker>\<open>tag unimportant\<close> \<open>The winding number is constant on a connected region\<close>
-
-lemma winding_number_constant:
-  assumes \<gamma>: "path \<gamma>" and loop: "pathfinish \<gamma> = pathstart \<gamma>" and cs: "connected S" and sg: "S \<inter> path_image \<gamma> = {}"
-  shows "winding_number \<gamma> constant_on S"
-proof -
-  have *: "1 \<le> cmod (winding_number \<gamma> y - winding_number \<gamma> z)"
-      if ne: "winding_number \<gamma> y \<noteq> winding_number \<gamma> z" and "y \<in> S" "z \<in> S" for y z
-  proof -
-    have "winding_number \<gamma> y \<in> \<int>"  "winding_number \<gamma> z \<in>  \<int>"
-      using that integer_winding_number [OF \<gamma> loop] sg \<open>y \<in> S\<close> by auto
-    with ne show ?thesis
-      by (auto simp: Ints_def simp flip: of_int_diff)
-  qed
-  have cont: "continuous_on S (\<lambda>w. winding_number \<gamma> w)"
-    using continuous_on_winding_number [OF \<gamma>] sg
-    by (meson continuous_on_subset disjoint_eq_subset_Compl)
-  show ?thesis
-    using "*" zero_less_one
-    by (blast intro: continuous_discrete_range_constant [OF cs cont])
-qed
-
-lemma winding_number_eq:
-     "\<lbrakk>path \<gamma>; pathfinish \<gamma> = pathstart \<gamma>; w \<in> S; z \<in> S; connected S; S \<inter> path_image \<gamma> = {}\<rbrakk>
-      \<Longrightarrow> winding_number \<gamma> w = winding_number \<gamma> z"
-  using winding_number_constant by (metis constant_on_def)
-
-lemma open_winding_number_levelsets:
-  assumes \<gamma>: "path \<gamma>" and loop: "pathfinish \<gamma> = pathstart \<gamma>"
-    shows "open {z. z \<notin> path_image \<gamma> \<and> winding_number \<gamma> z = k}"
-proof -
-  have opn: "open (- path_image \<gamma>)"
-    by (simp add: closed_path_image \<gamma> open_Compl)
-  { fix z assume z: "z \<notin> path_image \<gamma>" and k: "k = winding_number \<gamma> z"
-    obtain e where e: "e>0" "ball z e \<subseteq> - path_image \<gamma>"
-      using open_contains_ball [of "- path_image \<gamma>"] opn z
-      by blast
-    have "\<exists>e>0. \<forall>y. dist y z < e \<longrightarrow> y \<notin> path_image \<gamma> \<and> winding_number \<gamma> y = winding_number \<gamma> z"
-      apply (rule_tac x=e in exI)
-      using e apply (simp add: dist_norm ball_def norm_minus_commute)
-      apply (auto simp: dist_norm norm_minus_commute intro!: winding_number_eq [OF assms, where S = "ball z e"])
-      done
-  } then
-  show ?thesis
-    by (auto simp: open_dist)
-qed
-
-subsection\<open>Winding number is zero "outside" a curve\<close>
-
-proposition winding_number_zero_in_outside:
-  assumes \<gamma>: "path \<gamma>" and loop: "pathfinish \<gamma> = pathstart \<gamma>" and z: "z \<in> outside (path_image \<gamma>)"
-    shows "winding_number \<gamma> z = 0"
-proof -
-  obtain B::real where "0 < B" and B: "path_image \<gamma> \<subseteq> ball 0 B"
-    using bounded_subset_ballD [OF bounded_path_image [OF \<gamma>]] by auto
-  obtain w::complex where w: "w \<notin> ball 0 (B + 1)"
-    by (metis abs_of_nonneg le_less less_irrefl mem_ball_0 norm_of_real)
-  have "- ball 0 (B + 1) \<subseteq> outside (path_image \<gamma>)"
-    apply (rule outside_subset_convex)
-    using B subset_ball by auto
-  then have wout: "w \<in> outside (path_image \<gamma>)"
-    using w by blast
-  moreover have "winding_number \<gamma> constant_on outside (path_image \<gamma>)"
-    using winding_number_constant [OF \<gamma> loop, of "outside(path_image \<gamma>)"] connected_outside
-    by (metis DIM_complex bounded_path_image dual_order.refl \<gamma> outside_no_overlap)
-  ultimately have "winding_number \<gamma> z = winding_number \<gamma> w"
-    by (metis (no_types, hide_lams) constant_on_def z)
-  also have "\<dots> = 0"
-  proof -
-    have wnot: "w \<notin> path_image \<gamma>"  using wout by (simp add: outside_def)
-    { fix e::real assume "0<e"
-      obtain p where p: "polynomial_function p" "pathstart p = pathstart \<gamma>" "pathfinish p = pathfinish \<gamma>"
-                 and pg1: "(\<And>t. \<lbrakk>0 \<le> t; t \<le> 1\<rbrakk> \<Longrightarrow> cmod (p t - \<gamma> t) < 1)"
-                 and pge: "(\<And>t. \<lbrakk>0 \<le> t; t \<le> 1\<rbrakk> \<Longrightarrow> cmod (p t - \<gamma> t) < e)"
-        using path_approx_polynomial_function [OF \<gamma>, of "min 1 e"] \<open>e>0\<close> by force
-      have pip: "path_image p \<subseteq> ball 0 (B + 1)"
-        using B
-        apply (clarsimp simp add: path_image_def dist_norm ball_def)
-        apply (frule (1) pg1)
-        apply (fastforce dest: norm_add_less)
-        done
-      then have "w \<notin> path_image p"  using w by blast
-      then have "\<exists>p. valid_path p \<and> w \<notin> path_image p \<and>
-                     pathstart p = pathstart \<gamma> \<and> pathfinish p = pathfinish \<gamma> \<and>
-                     (\<forall>t\<in>{0..1}. cmod (\<gamma> t - p t) < e) \<and> contour_integral p (\<lambda>wa. 1 / (wa - w)) = 0"
-        apply (rule_tac x=p in exI)
-        apply (simp add: p valid_path_polynomial_function)
-        apply (intro conjI)
-        using pge apply (simp add: norm_minus_commute)
-        apply (rule contour_integral_unique [OF Cauchy_theorem_convex_simple [OF _ convex_ball [of 0 "B+1"]]])
-        apply (rule holomorphic_intros | simp add: dist_norm)+
-        using mem_ball_0 w apply blast
-        using p apply (simp_all add: valid_path_polynomial_function loop pip)
-        done
-    }
-    then show ?thesis
-      by (auto intro: winding_number_unique [OF \<gamma>] simp add: winding_number_prop_def wnot)
-  qed
-  finally show ?thesis .
-qed
-
-corollary\<^marker>\<open>tag unimportant\<close> winding_number_zero_const: "a \<noteq> z \<Longrightarrow> winding_number (\<lambda>t. a) z = 0"
-  by (rule winding_number_zero_in_outside)
-     (auto simp: pathfinish_def pathstart_def path_polynomial_function)
-
-corollary\<^marker>\<open>tag unimportant\<close> winding_number_zero_outside:
-    "\<lbrakk>path \<gamma>; convex s; pathfinish \<gamma> = pathstart \<gamma>; z \<notin> s; path_image \<gamma> \<subseteq> s\<rbrakk> \<Longrightarrow> winding_number \<gamma> z = 0"
-  by (meson convex_in_outside outside_mono subsetCE winding_number_zero_in_outside)
-
-lemma winding_number_zero_at_infinity:
-  assumes \<gamma>: "path \<gamma>" and loop: "pathfinish \<gamma> = pathstart \<gamma>"
-    shows "\<exists>B. \<forall>z. B \<le> norm z \<longrightarrow> winding_number \<gamma> z = 0"
-proof -
-  obtain B::real where "0 < B" and B: "path_image \<gamma> \<subseteq> ball 0 B"
-    using bounded_subset_ballD [OF bounded_path_image [OF \<gamma>]] by auto
-  then show ?thesis
-    apply (rule_tac x="B+1" in exI, clarify)
-    apply (rule winding_number_zero_outside [OF \<gamma> convex_cball [of 0 B] loop])
-    apply (meson less_add_one mem_cball_0 not_le order_trans)
-    using ball_subset_cball by blast
-qed
-
-lemma winding_number_zero_point:
-    "\<lbrakk>path \<gamma>; convex s; pathfinish \<gamma> = pathstart \<gamma>; open s; path_image \<gamma> \<subseteq> s\<rbrakk>
-     \<Longrightarrow> \<exists>z. z \<in> s \<and> winding_number \<gamma> z = 0"
-  using outside_compact_in_open [of "path_image \<gamma>" s] path_image_nonempty winding_number_zero_in_outside
-  by (fastforce simp add: compact_path_image)
-
-
-text\<open>If a path winds round a set, it winds rounds its inside.\<close>
-lemma winding_number_around_inside:
-  assumes \<gamma>: "path \<gamma>" and loop: "pathfinish \<gamma> = pathstart \<gamma>"
-      and cls: "closed s" and cos: "connected s" and s_disj: "s \<inter> path_image \<gamma> = {}"
-      and z: "z \<in> s" and wn_nz: "winding_number \<gamma> z \<noteq> 0" and w: "w \<in> s \<union> inside s"
-    shows "winding_number \<gamma> w = winding_number \<gamma> z"
-proof -
-  have ssb: "s \<subseteq> inside(path_image \<gamma>)"
-  proof
-    fix x :: complex
-    assume "x \<in> s"
-    hence "x \<notin> path_image \<gamma>"
-      by (meson disjoint_iff_not_equal s_disj)
-    thus "x \<in> inside (path_image \<gamma>)"
-      using \<open>x \<in> s\<close> by (metis (no_types) ComplI UnE cos \<gamma> loop s_disj union_with_outside winding_number_eq winding_number_zero_in_outside wn_nz z)
-qed
-  show ?thesis
-    apply (rule winding_number_eq [OF \<gamma> loop w])
-    using z apply blast
-    apply (simp add: cls connected_with_inside cos)
-    apply (simp add: Int_Un_distrib2 s_disj, safe)
-    by (meson ssb inside_inside_compact_connected [OF cls, of "path_image \<gamma>"] compact_path_image connected_path_image contra_subsetD disjoint_iff_not_equal \<gamma> inside_no_overlap)
- qed
-
-
-text\<open>Bounding a WN by 1/2 for a path and point in opposite halfspaces.\<close>
-lemma winding_number_subpath_continuous:
-  assumes \<gamma>: "valid_path \<gamma>" and z: "z \<notin> path_image \<gamma>"
-    shows "continuous_on {0..1} (\<lambda>x. winding_number(subpath 0 x \<gamma>) z)"
-proof -
-  have *: "integral {0..x} (\<lambda>t. vector_derivative \<gamma> (at t) / (\<gamma> t - z)) / (2 * of_real pi * \<i>) =
-         winding_number (subpath 0 x \<gamma>) z"
-         if x: "0 \<le> x" "x \<le> 1" for x
-  proof -
-    have "integral {0..x} (\<lambda>t. vector_derivative \<gamma> (at t) / (\<gamma> t - z)) / (2 * of_real pi * \<i>) =
-          1 / (2*pi*\<i>) * contour_integral (subpath 0 x \<gamma>) (\<lambda>w. 1/(w - z))"
-      using assms x
-      apply (simp add: contour_integral_subcontour_integral [OF contour_integrable_inversediff])
-      done
-    also have "\<dots> = winding_number (subpath 0 x \<gamma>) z"
-      apply (subst winding_number_valid_path)
-      using assms x
-      apply (simp_all add: path_image_subpath valid_path_subpath)
-      by (force simp: path_image_def)
-    finally show ?thesis .
-  qed
-  show ?thesis
-    apply (rule continuous_on_eq
-                 [where f = "\<lambda>x. 1 / (2*pi*\<i>) *
-                                 integral {0..x} (\<lambda>t. 1/(\<gamma> t - z) * vector_derivative \<gamma> (at t))"])
-    apply (rule continuous_intros)+
-    apply (rule indefinite_integral_continuous_1)
-    apply (rule contour_integrable_inversediff [OF assms, unfolded contour_integrable_on])
-      using assms
-    apply (simp add: *)
-    done
-qed
-
-lemma winding_number_ivt_pos:
-    assumes \<gamma>: "valid_path \<gamma>" and z: "z \<notin> path_image \<gamma>" and "0 \<le> w" "w \<le> Re(winding_number \<gamma> z)"
-      shows "\<exists>t \<in> {0..1}. Re(winding_number(subpath 0 t \<gamma>) z) = w"
-  apply (rule ivt_increasing_component_on_1 [of 0 1, where ?k = "1::complex", simplified complex_inner_1_right], simp)
-  apply (rule winding_number_subpath_continuous [OF \<gamma> z])
-  using assms
-  apply (auto simp: path_image_def image_def)
-  done
-
-lemma winding_number_ivt_neg:
-    assumes \<gamma>: "valid_path \<gamma>" and z: "z \<notin> path_image \<gamma>" and "Re(winding_number \<gamma> z) \<le> w" "w \<le> 0"
-      shows "\<exists>t \<in> {0..1}. Re(winding_number(subpath 0 t \<gamma>) z) = w"
-  apply (rule ivt_decreasing_component_on_1 [of 0 1, where ?k = "1::complex", simplified complex_inner_1_right], simp)
-  apply (rule winding_number_subpath_continuous [OF \<gamma> z])
-  using assms
-  apply (auto simp: path_image_def image_def)
-  done
-
-lemma winding_number_ivt_abs:
-    assumes \<gamma>: "valid_path \<gamma>" and z: "z \<notin> path_image \<gamma>" and "0 \<le> w" "w \<le> \<bar>Re(winding_number \<gamma> z)\<bar>"
-      shows "\<exists>t \<in> {0..1}. \<bar>Re (winding_number (subpath 0 t \<gamma>) z)\<bar> = w"
-  using assms winding_number_ivt_pos [of \<gamma> z w] winding_number_ivt_neg [of \<gamma> z "-w"]
-  by force
-
-lemma winding_number_lt_half_lemma:
-  assumes \<gamma>: "valid_path \<gamma>" and z: "z \<notin> path_image \<gamma>" and az: "a \<bullet> z \<le> b" and pag: "path_image \<gamma> \<subseteq> {w. a \<bullet> w > b}"
-    shows "Re(winding_number \<gamma> z) < 1/2"
-proof -
-  { assume "Re(winding_number \<gamma> z) \<ge> 1/2"
-    then obtain t::real where t: "0 \<le> t" "t \<le> 1" and sub12: "Re (winding_number (subpath 0 t \<gamma>) z) = 1/2"
-      using winding_number_ivt_pos [OF \<gamma> z, of "1/2"] by auto
-    have gt: "\<gamma> t - z = - (of_real (exp (- (2 * pi * Im (winding_number (subpath 0 t \<gamma>) z)))) * (\<gamma> 0 - z))"
-      using winding_number_exp_2pi [of "subpath 0 t \<gamma>" z]
-      apply (simp add: t \<gamma> valid_path_imp_path)
-      using closed_segment_eq_real_ivl path_image_def t z by (fastforce simp: path_image_subpath Euler sub12)
-    have "b < a \<bullet> \<gamma> 0"
-    proof -
-      have "\<gamma> 0 \<in> {c. b < a \<bullet> c}"
-        by (metis (no_types) pag atLeastAtMost_iff image_subset_iff order_refl path_image_def zero_le_one)
-      thus ?thesis
-        by blast
-    qed
-    moreover have "b < a \<bullet> \<gamma> t"
-    proof -
-      have "\<gamma> t \<in> {c. b < a \<bullet> c}"
-        by (metis (no_types) pag atLeastAtMost_iff image_subset_iff path_image_def t)
-      thus ?thesis
-        by blast
-    qed
-    ultimately have "0 < a \<bullet> (\<gamma> 0 - z)" "0 < a \<bullet> (\<gamma> t - z)" using az
-      by (simp add: inner_diff_right)+
-    then have False
-      by (simp add: gt inner_mult_right mult_less_0_iff)
-  }
-  then show ?thesis by force
-qed
-
-lemma winding_number_lt_half:
-  assumes "valid_path \<gamma>" "a \<bullet> z \<le> b" "path_image \<gamma> \<subseteq> {w. a \<bullet> w > b}"
-    shows "\<bar>Re (winding_number \<gamma> z)\<bar> < 1/2"
-proof -
-  have "z \<notin> path_image \<gamma>" using assms by auto
-  with assms show ?thesis
-    apply (simp add: winding_number_lt_half_lemma abs_if del: less_divide_eq_numeral1)
-    apply (metis complex_inner_1_right winding_number_lt_half_lemma [OF valid_path_imp_reverse, of \<gamma> z a b]
-                 winding_number_reversepath valid_path_imp_path inner_minus_left path_image_reversepath)
-    done
-qed
-
-lemma winding_number_le_half:
-  assumes \<gamma>: "valid_path \<gamma>" and z: "z \<notin> path_image \<gamma>"
-      and anz: "a \<noteq> 0" and azb: "a \<bullet> z \<le> b" and pag: "path_image \<gamma> \<subseteq> {w. a \<bullet> w \<ge> b}"
-    shows "\<bar>Re (winding_number \<gamma> z)\<bar> \<le> 1/2"
-proof -
-  { assume wnz_12: "\<bar>Re (winding_number \<gamma> z)\<bar> > 1/2"
-    have "isCont (winding_number \<gamma>) z"
-      by (metis continuous_at_winding_number valid_path_imp_path \<gamma> z)
-    then obtain d where "d>0" and d: "\<And>x'. dist x' z < d \<Longrightarrow> dist (winding_number \<gamma> x') (winding_number \<gamma> z) < \<bar>Re(winding_number \<gamma> z)\<bar> - 1/2"
-      using continuous_at_eps_delta wnz_12 diff_gt_0_iff_gt by blast
-    define z' where "z' = z - (d / (2 * cmod a)) *\<^sub>R a"
-    have *: "a \<bullet> z' \<le> b - d / 3 * cmod a"
-      unfolding z'_def inner_mult_right' divide_inverse
-      apply (simp add: field_split_simps algebra_simps dot_square_norm power2_eq_square anz)
-      apply (metis \<open>0 < d\<close> add_increasing azb less_eq_real_def mult_nonneg_nonneg mult_right_mono norm_ge_zero norm_numeral)
-      done
-    have "cmod (winding_number \<gamma> z' - winding_number \<gamma> z) < \<bar>Re (winding_number \<gamma> z)\<bar> - 1/2"
-      using d [of z'] anz \<open>d>0\<close> by (simp add: dist_norm z'_def)
-    then have "1/2 < \<bar>Re (winding_number \<gamma> z)\<bar> - cmod (winding_number \<gamma> z' - winding_number \<gamma> z)"
-      by simp
-    then have "1/2 < \<bar>Re (winding_number \<gamma> z)\<bar> - \<bar>Re (winding_number \<gamma> z') - Re (winding_number \<gamma> z)\<bar>"
-      using abs_Re_le_cmod [of "winding_number \<gamma> z' - winding_number \<gamma> z"] by simp
-    then have wnz_12': "\<bar>Re (winding_number \<gamma> z')\<bar> > 1/2"
-      by linarith
-    moreover have "\<bar>Re (winding_number \<gamma> z')\<bar> < 1/2"
-      apply (rule winding_number_lt_half [OF \<gamma> *])
-      using azb \<open>d>0\<close> pag
-      apply (auto simp: add_strict_increasing anz field_split_simps dest!: subsetD)
-      done
-    ultimately have False
-      by simp
-  }
-  then show ?thesis by force
-qed
-
-lemma winding_number_lt_half_linepath: "z \<notin> closed_segment a b \<Longrightarrow> \<bar>Re (winding_number (linepath a b) z)\<bar> < 1/2"
-  using separating_hyperplane_closed_point [of "closed_segment a b" z]
-  apply auto
-  apply (simp add: closed_segment_def)
-  apply (drule less_imp_le)
-  apply (frule winding_number_lt_half [OF valid_path_linepath [of a b]])
-  apply (auto simp: segment)
-  done
-
-
-text\<open> Positivity of WN for a linepath.\<close>
-lemma winding_number_linepath_pos_lt:
-    assumes "0 < Im ((b - a) * cnj (b - z))"
-      shows "0 < Re(winding_number(linepath a b) z)"
-proof -
-  have z: "z \<notin> path_image (linepath a b)"
-    using assms
-    by (simp add: closed_segment_def) (force simp: algebra_simps)
-  show ?thesis
-    apply (rule winding_number_pos_lt [OF valid_path_linepath z assms])
-    apply (simp add: linepath_def algebra_simps)
-    done
-qed
-
-
-subsection\<open>Cauchy's integral formula, again for a convex enclosing set\<close>
-
-lemma Cauchy_integral_formula_weak:
-    assumes s: "convex s" and "finite k" and conf: "continuous_on s f"
-        and fcd: "(\<And>x. x \<in> interior s - k \<Longrightarrow> f field_differentiable at x)"
-        and z: "z \<in> interior s - k" and vpg: "valid_path \<gamma>"
-        and pasz: "path_image \<gamma> \<subseteq> s - {z}" and loop: "pathfinish \<gamma> = pathstart \<gamma>"
-      shows "((\<lambda>w. f w / (w - z)) has_contour_integral (2*pi * \<i> * winding_number \<gamma> z * f z)) \<gamma>"
-proof -
-  obtain f' where f': "(f has_field_derivative f') (at z)"
-    using fcd [OF z] by (auto simp: field_differentiable_def)
-  have pas: "path_image \<gamma> \<subseteq> s" and znotin: "z \<notin> path_image \<gamma>" using pasz by blast+
-  have c: "continuous (at x within s) (\<lambda>w. if w = z then f' else (f w - f z) / (w - z))" if "x \<in> s" for x
-  proof (cases "x = z")
-    case True then show ?thesis
-      apply (simp add: continuous_within)
-      apply (rule Lim_transform_away_within [of _ "z+1" _ "\<lambda>w::complex. (f w - f z)/(w - z)"])
-      using has_field_derivative_at_within has_field_derivative_iff f'
-      apply (fastforce simp add:)+
-      done
-  next
-    case False
-    then have dxz: "dist x z > 0" by auto
-    have cf: "continuous (at x within s) f"
-      using conf continuous_on_eq_continuous_within that by blast
-    have "continuous (at x within s) (\<lambda>w. (f w - f z) / (w - z))"
-      by (rule cf continuous_intros | simp add: False)+
-    then show ?thesis
-      apply (rule continuous_transform_within [OF _ dxz that, of "\<lambda>w::complex. (f w - f z)/(w - z)"])
-      apply (force simp: dist_commute)
-      done
-  qed
-  have fink': "finite (insert z k)" using \<open>finite k\<close> by blast
-  have *: "((\<lambda>w. if w = z then f' else (f w - f z) / (w - z)) has_contour_integral 0) \<gamma>"
-    apply (rule Cauchy_theorem_convex [OF _ s fink' _ vpg pas loop])
-    using c apply (force simp: continuous_on_eq_continuous_within)
-    apply (rename_tac w)
-    apply (rule_tac d="dist w z" and f = "\<lambda>w. (f w - f z)/(w - z)" in field_differentiable_transform_within)
-    apply (simp_all add: dist_pos_lt dist_commute)
-    apply (metis less_irrefl)
-    apply (rule derivative_intros fcd | simp)+
-    done
-  show ?thesis
-    apply (rule has_contour_integral_eq)
-    using znotin has_contour_integral_add [OF has_contour_integral_lmul [OF has_contour_integral_winding_number [OF vpg znotin], of "f z"] *]
-    apply (auto simp: ac_simps divide_simps)
-    done
-qed
-
-theorem Cauchy_integral_formula_convex_simple:
-    "\<lbrakk>convex s; f holomorphic_on s; z \<in> interior s; valid_path \<gamma>; path_image \<gamma> \<subseteq> s - {z};
-      pathfinish \<gamma> = pathstart \<gamma>\<rbrakk>
-     \<Longrightarrow> ((\<lambda>w. f w / (w - z)) has_contour_integral (2*pi * \<i> * winding_number \<gamma> z * f z)) \<gamma>"
-  apply (rule Cauchy_integral_formula_weak [where k = "{}"])
-  using holomorphic_on_imp_continuous_on
-  by auto (metis at_within_interior holomorphic_on_def interiorE subsetCE)
 
 subsection\<open>Homotopy forms of Cauchy's theorem\<close>
 
@@ -3870,3290 +1593,4 @@
     apply (blast dest: holomorphic_on_imp_continuous_on homotopic_loops_imp_subset)
   by (simp add: Cauchy_theorem_homotopic_loops)
 
-subsection\<^marker>\<open>tag unimportant\<close> \<open>More winding number properties\<close>
-
-text\<open>including the fact that it's +-1 inside a simple closed curve.\<close>
-
-lemma winding_number_homotopic_paths:
-    assumes "homotopic_paths (-{z}) g h"
-      shows "winding_number g z = winding_number h z"
-proof -
-  have "path g" "path h" using homotopic_paths_imp_path [OF assms] by auto
-  moreover have pag: "z \<notin> path_image g" and pah: "z \<notin> path_image h"
-    using homotopic_paths_imp_subset [OF assms] by auto
-  ultimately obtain d e where "d > 0" "e > 0"
-      and d: "\<And>p. \<lbrakk>path p; pathstart p = pathstart g; pathfinish p = pathfinish g; \<forall>t\<in>{0..1}. norm (p t - g t) < d\<rbrakk>
-            \<Longrightarrow> homotopic_paths (-{z}) g p"
-      and e: "\<And>q. \<lbrakk>path q; pathstart q = pathstart h; pathfinish q = pathfinish h; \<forall>t\<in>{0..1}. norm (q t - h t) < e\<rbrakk>
-            \<Longrightarrow> homotopic_paths (-{z}) h q"
-    using homotopic_nearby_paths [of g "-{z}"] homotopic_nearby_paths [of h "-{z}"] by force
-  obtain p where p:
-       "valid_path p" "z \<notin> path_image p"
-       "pathstart p = pathstart g" "pathfinish p = pathfinish g"
-       and gp_less:"\<forall>t\<in>{0..1}. cmod (g t - p t) < d"
-       and pap: "contour_integral p (\<lambda>w. 1 / (w - z)) = complex_of_real (2 * pi) * \<i> * winding_number g z"
-    using winding_number [OF \<open>path g\<close> pag \<open>0 < d\<close>] unfolding winding_number_prop_def by blast
-  obtain q where q:
-       "valid_path q" "z \<notin> path_image q"
-       "pathstart q = pathstart h" "pathfinish q = pathfinish h"
-       and hq_less: "\<forall>t\<in>{0..1}. cmod (h t - q t) < e"
-       and paq:  "contour_integral q (\<lambda>w. 1 / (w - z)) = complex_of_real (2 * pi) * \<i> * winding_number h z"
-    using winding_number [OF \<open>path h\<close> pah \<open>0 < e\<close>] unfolding winding_number_prop_def by blast
-  have "homotopic_paths (- {z}) g p"
-    by (simp add: d p valid_path_imp_path norm_minus_commute gp_less)
-  moreover have "homotopic_paths (- {z}) h q"
-    by (simp add: e q valid_path_imp_path norm_minus_commute hq_less)
-  ultimately have "homotopic_paths (- {z}) p q"
-    by (blast intro: homotopic_paths_trans homotopic_paths_sym assms)
-  then have "contour_integral p (\<lambda>w. 1/(w - z)) = contour_integral q (\<lambda>w. 1/(w - z))"
-    by (rule Cauchy_theorem_homotopic_paths) (auto intro!: holomorphic_intros simp: p q)
-  then show ?thesis
-    by (simp add: pap paq)
-qed
-
-lemma winding_number_homotopic_loops:
-    assumes "homotopic_loops (-{z}) g h"
-      shows "winding_number g z = winding_number h z"
-proof -
-  have "path g" "path h" using homotopic_loops_imp_path [OF assms] by auto
-  moreover have pag: "z \<notin> path_image g" and pah: "z \<notin> path_image h"
-    using homotopic_loops_imp_subset [OF assms] by auto
-  moreover have gloop: "pathfinish g = pathstart g" and hloop: "pathfinish h = pathstart h"
-    using homotopic_loops_imp_loop [OF assms] by auto
-  ultimately obtain d e where "d > 0" "e > 0"
-      and d: "\<And>p. \<lbrakk>path p; pathfinish p = pathstart p; \<forall>t\<in>{0..1}. norm (p t - g t) < d\<rbrakk>
-            \<Longrightarrow> homotopic_loops (-{z}) g p"
-      and e: "\<And>q. \<lbrakk>path q; pathfinish q = pathstart q; \<forall>t\<in>{0..1}. norm (q t - h t) < e\<rbrakk>
-            \<Longrightarrow> homotopic_loops (-{z}) h q"
-    using homotopic_nearby_loops [of g "-{z}"] homotopic_nearby_loops [of h "-{z}"] by force
-  obtain p where p:
-       "valid_path p" "z \<notin> path_image p"
-       "pathstart p = pathstart g" "pathfinish p = pathfinish g"
-       and gp_less:"\<forall>t\<in>{0..1}. cmod (g t - p t) < d"
-       and pap: "contour_integral p (\<lambda>w. 1 / (w - z)) = complex_of_real (2 * pi) * \<i> * winding_number g z"
-    using winding_number [OF \<open>path g\<close> pag \<open>0 < d\<close>] unfolding winding_number_prop_def by blast
-  obtain q where q:
-       "valid_path q" "z \<notin> path_image q"
-       "pathstart q = pathstart h" "pathfinish q = pathfinish h"
-       and hq_less: "\<forall>t\<in>{0..1}. cmod (h t - q t) < e"
-       and paq:  "contour_integral q (\<lambda>w. 1 / (w - z)) = complex_of_real (2 * pi) * \<i> * winding_number h z"
-    using winding_number [OF \<open>path h\<close> pah \<open>0 < e\<close>] unfolding winding_number_prop_def by blast
-  have gp: "homotopic_loops (- {z}) g p"
-    by (simp add: gloop d gp_less norm_minus_commute p valid_path_imp_path)
-  have hq: "homotopic_loops (- {z}) h q"
-    by (simp add: e hloop hq_less norm_minus_commute q valid_path_imp_path)
-  have "contour_integral p (\<lambda>w. 1/(w - z)) = contour_integral q (\<lambda>w. 1/(w - z))"
-  proof (rule Cauchy_theorem_homotopic_loops)
-    show "homotopic_loops (- {z}) p q"
-      by (blast intro: homotopic_loops_trans homotopic_loops_sym gp hq assms)
-  qed (auto intro!: holomorphic_intros simp: p q)
-  then show ?thesis
-    by (simp add: pap paq)
-qed
-
-lemma winding_number_paths_linear_eq:
-  "\<lbrakk>path g; path h; pathstart h = pathstart g; pathfinish h = pathfinish g;
-    \<And>t. t \<in> {0..1} \<Longrightarrow> z \<notin> closed_segment (g t) (h t)\<rbrakk>
-        \<Longrightarrow> winding_number h z = winding_number g z"
-  by (blast intro: sym homotopic_paths_linear winding_number_homotopic_paths)
-
-lemma winding_number_loops_linear_eq:
-  "\<lbrakk>path g; path h; pathfinish g = pathstart g; pathfinish h = pathstart h;
-    \<And>t. t \<in> {0..1} \<Longrightarrow> z \<notin> closed_segment (g t) (h t)\<rbrakk>
-        \<Longrightarrow> winding_number h z = winding_number g z"
-  by (blast intro: sym homotopic_loops_linear winding_number_homotopic_loops)
-
-lemma winding_number_nearby_paths_eq:
-     "\<lbrakk>path g; path h; pathstart h = pathstart g; pathfinish h = pathfinish g;
-      \<And>t. t \<in> {0..1} \<Longrightarrow> norm(h t - g t) < norm(g t - z)\<rbrakk>
-      \<Longrightarrow> winding_number h z = winding_number g z"
-  by (metis segment_bound(2) norm_minus_commute not_le winding_number_paths_linear_eq)
-
-lemma winding_number_nearby_loops_eq:
-     "\<lbrakk>path g; path h; pathfinish g = pathstart g; pathfinish h = pathstart h;
-      \<And>t. t \<in> {0..1} \<Longrightarrow> norm(h t - g t) < norm(g t - z)\<rbrakk>
-      \<Longrightarrow> winding_number h z = winding_number g z"
-  by (metis segment_bound(2) norm_minus_commute not_le winding_number_loops_linear_eq)
-
-
-lemma winding_number_subpath_combine:
-    "\<lbrakk>path g; z \<notin> path_image g;
-      u \<in> {0..1}; v \<in> {0..1}; w \<in> {0..1}\<rbrakk>
-      \<Longrightarrow> winding_number (subpath u v g) z + winding_number (subpath v w g) z =
-          winding_number (subpath u w g) z"
-apply (rule trans [OF winding_number_join [THEN sym]
-                      winding_number_homotopic_paths [OF homotopic_join_subpaths]])
-  using path_image_subpath_subset by auto
-
-subsection\<open>Partial circle path\<close>
-
-definition\<^marker>\<open>tag important\<close> part_circlepath :: "[complex, real, real, real, real] \<Rightarrow> complex"
-  where "part_circlepath z r s t \<equiv> \<lambda>x. z + of_real r * exp (\<i> * of_real (linepath s t x))"
-
-lemma pathstart_part_circlepath [simp]:
-     "pathstart(part_circlepath z r s t) = z + r*exp(\<i> * s)"
-by (metis part_circlepath_def pathstart_def pathstart_linepath)
-
-lemma pathfinish_part_circlepath [simp]:
-     "pathfinish(part_circlepath z r s t) = z + r*exp(\<i>*t)"
-by (metis part_circlepath_def pathfinish_def pathfinish_linepath)
-
-lemma reversepath_part_circlepath[simp]:
-    "reversepath (part_circlepath z r s t) = part_circlepath z r t s"
-  unfolding part_circlepath_def reversepath_def linepath_def 
-  by (auto simp:algebra_simps)
-    
-lemma has_vector_derivative_part_circlepath [derivative_intros]:
-    "((part_circlepath z r s t) has_vector_derivative
-      (\<i> * r * (of_real t - of_real s) * exp(\<i> * linepath s t x)))
-     (at x within X)"
-  apply (simp add: part_circlepath_def linepath_def scaleR_conv_of_real)
-  apply (rule has_vector_derivative_real_field)
-  apply (rule derivative_eq_intros | simp)+
-  done
-
-lemma differentiable_part_circlepath:
-  "part_circlepath c r a b differentiable at x within A"
-  using has_vector_derivative_part_circlepath[of c r a b x A] differentiableI_vector by blast
-
-lemma vector_derivative_part_circlepath:
-    "vector_derivative (part_circlepath z r s t) (at x) =
-       \<i> * r * (of_real t - of_real s) * exp(\<i> * linepath s t x)"
-  using has_vector_derivative_part_circlepath vector_derivative_at by blast
-
-lemma vector_derivative_part_circlepath01:
-    "\<lbrakk>0 \<le> x; x \<le> 1\<rbrakk>
-     \<Longrightarrow> vector_derivative (part_circlepath z r s t) (at x within {0..1}) =
-          \<i> * r * (of_real t - of_real s) * exp(\<i> * linepath s t x)"
-  using has_vector_derivative_part_circlepath
-  by (auto simp: vector_derivative_at_within_ivl)
-
-lemma valid_path_part_circlepath [simp]: "valid_path (part_circlepath z r s t)"
-  apply (simp add: valid_path_def)
-  apply (rule C1_differentiable_imp_piecewise)
-  apply (auto simp: C1_differentiable_on_eq vector_derivative_works vector_derivative_part_circlepath has_vector_derivative_part_circlepath
-              intro!: continuous_intros)
-  done
-
-lemma path_part_circlepath [simp]: "path (part_circlepath z r s t)"
-  by (simp add: valid_path_imp_path)
-
-proposition path_image_part_circlepath:
-  assumes "s \<le> t"
-    shows "path_image (part_circlepath z r s t) = {z + r * exp(\<i> * of_real x) | x. s \<le> x \<and> x \<le> t}"
-proof -
-  { fix z::real
-    assume "0 \<le> z" "z \<le> 1"
-    with \<open>s \<le> t\<close> have "\<exists>x. (exp (\<i> * linepath s t z) = exp (\<i> * of_real x)) \<and> s \<le> x \<and> x \<le> t"
-      apply (rule_tac x="(1 - z) * s + z * t" in exI)
-      apply (simp add: linepath_def scaleR_conv_of_real algebra_simps)
-      apply (rule conjI)
-      using mult_right_mono apply blast
-      using affine_ineq  by (metis "mult.commute")
-  }
-  moreover
-  { fix z
-    assume "s \<le> z" "z \<le> t"
-    then have "z + of_real r * exp (\<i> * of_real z) \<in> (\<lambda>x. z + of_real r * exp (\<i> * linepath s t x)) ` {0..1}"
-      apply (rule_tac x="(z - s)/(t - s)" in image_eqI)
-      apply (simp add: linepath_def scaleR_conv_of_real divide_simps exp_eq)
-      apply (auto simp: field_split_simps)
-      done
-  }
-  ultimately show ?thesis
-    by (fastforce simp add: path_image_def part_circlepath_def)
-qed
-
-lemma path_image_part_circlepath':
-  "path_image (part_circlepath z r s t) = (\<lambda>x. z + r * cis x) ` closed_segment s t"
-proof -
-  have "path_image (part_circlepath z r s t) = 
-          (\<lambda>x. z + r * exp(\<i> * of_real x)) ` linepath s t ` {0..1}"
-    by (simp add: image_image path_image_def part_circlepath_def)
-  also have "linepath s t ` {0..1} = closed_segment s t"
-    by (rule linepath_image_01)
-  finally show ?thesis by (simp add: cis_conv_exp)
-qed
-
-lemma path_image_part_circlepath_subset:
-    "\<lbrakk>s \<le> t; 0 \<le> r\<rbrakk> \<Longrightarrow> path_image(part_circlepath z r s t) \<subseteq> sphere z r"
-by (auto simp: path_image_part_circlepath sphere_def dist_norm algebra_simps norm_mult)
-
-lemma in_path_image_part_circlepath:
-  assumes "w \<in> path_image(part_circlepath z r s t)" "s \<le> t" "0 \<le> r"
-    shows "norm(w - z) = r"
-proof -
-  have "w \<in> {c. dist z c = r}"
-    by (metis (no_types) path_image_part_circlepath_subset sphere_def subset_eq assms)
-  thus ?thesis
-    by (simp add: dist_norm norm_minus_commute)
-qed
-
-lemma path_image_part_circlepath_subset':
-  assumes "r \<ge> 0"
-  shows   "path_image (part_circlepath z r s t) \<subseteq> sphere z r"
-proof (cases "s \<le> t")
-  case True
-  thus ?thesis using path_image_part_circlepath_subset[of s t r z] assms by simp
-next
-  case False
-  thus ?thesis using path_image_part_circlepath_subset[of t s r z] assms
-    by (subst reversepath_part_circlepath [symmetric], subst path_image_reversepath) simp_all
-qed
-
-lemma part_circlepath_cnj: "cnj (part_circlepath c r a b x) = part_circlepath (cnj c) r (-a) (-b) x"
-  by (simp add: part_circlepath_def exp_cnj linepath_def algebra_simps)
-
-lemma contour_integral_bound_part_circlepath:
-  assumes "f contour_integrable_on part_circlepath c r a b"
-  assumes "B \<ge> 0" "r \<ge> 0" "\<And>x. x \<in> path_image (part_circlepath c r a b) \<Longrightarrow> norm (f x) \<le> B"
-  shows   "norm (contour_integral (part_circlepath c r a b) f) \<le> B * r * \<bar>b - a\<bar>"
-proof -
-  let ?I = "integral {0..1} (\<lambda>x. f (part_circlepath c r a b x) * \<i> * of_real (r * (b - a)) *
-              exp (\<i> * linepath a b x))"
-  have "norm ?I \<le> integral {0..1} (\<lambda>x::real. B * 1 * (r * \<bar>b - a\<bar>) * 1)"
-  proof (rule integral_norm_bound_integral, goal_cases)
-    case 1
-    with assms(1) show ?case
-      by (simp add: contour_integrable_on vector_derivative_part_circlepath mult_ac)
-  next
-    case (3 x)
-    with assms(2-) show ?case unfolding norm_mult norm_of_real abs_mult
-      by (intro mult_mono) (auto simp: path_image_def)
-  qed auto
-  also have "?I = contour_integral (part_circlepath c r a b) f"
-    by (simp add: contour_integral_integral vector_derivative_part_circlepath mult_ac)
-  finally show ?thesis by simp
-qed
-
-lemma has_contour_integral_part_circlepath_iff:
-  assumes "a < b"
-  shows "(f has_contour_integral I) (part_circlepath c r a b) \<longleftrightarrow>
-           ((\<lambda>t. f (c + r * cis t) * r * \<i> * cis t) has_integral I) {a..b}"
-proof -
-  have "(f has_contour_integral I) (part_circlepath c r a b) \<longleftrightarrow>
-          ((\<lambda>x. f (part_circlepath c r a b x) * vector_derivative (part_circlepath c r a b)
-           (at x within {0..1})) has_integral I) {0..1}"
-    unfolding has_contour_integral_def ..
-  also have "\<dots> \<longleftrightarrow> ((\<lambda>x. f (part_circlepath c r a b x) * r * (b - a) * \<i> *
-                            cis (linepath a b x)) has_integral I) {0..1}"
-    by (intro has_integral_cong, subst vector_derivative_part_circlepath01)
-       (simp_all add: cis_conv_exp)
-  also have "\<dots> \<longleftrightarrow> ((\<lambda>x. f (c + r * exp (\<i> * linepath (of_real a) (of_real b) x)) *
-                       r * \<i> * exp (\<i> * linepath (of_real a) (of_real b) x) *
-                       vector_derivative (linepath (of_real a) (of_real b)) 
-                         (at x within {0..1})) has_integral I) {0..1}"
-    by (intro has_integral_cong, subst vector_derivative_linepath_within)
-       (auto simp: part_circlepath_def cis_conv_exp of_real_linepath [symmetric])
-  also have "\<dots> \<longleftrightarrow> ((\<lambda>z. f (c + r * exp (\<i> * z)) * r * \<i> * exp (\<i> * z)) has_contour_integral I)
-                      (linepath (of_real a) (of_real b))"
-    by (simp add: has_contour_integral_def)
-  also have "\<dots> \<longleftrightarrow> ((\<lambda>t. f (c + r * cis t) * r * \<i> * cis t) has_integral I) {a..b}" using assms
-    by (subst has_contour_integral_linepath_Reals_iff) (simp_all add: cis_conv_exp)
-  finally show ?thesis .
-qed
-
-lemma contour_integrable_part_circlepath_iff:
-  assumes "a < b"
-  shows "f contour_integrable_on (part_circlepath c r a b) \<longleftrightarrow>
-           (\<lambda>t. f (c + r * cis t) * r * \<i> * cis t) integrable_on {a..b}"
-  using assms by (auto simp: contour_integrable_on_def integrable_on_def 
-                             has_contour_integral_part_circlepath_iff)
-
-lemma contour_integral_part_circlepath_eq:
-  assumes "a < b"
-  shows "contour_integral (part_circlepath c r a b) f =
-           integral {a..b} (\<lambda>t. f (c + r * cis t) * r * \<i> * cis t)"
-proof (cases "f contour_integrable_on part_circlepath c r a b")
-  case True
-  hence "(\<lambda>t. f (c + r * cis t) * r * \<i> * cis t) integrable_on {a..b}" 
-    using assms by (simp add: contour_integrable_part_circlepath_iff)
-  with True show ?thesis
-    using has_contour_integral_part_circlepath_iff[OF assms]
-          contour_integral_unique has_integral_integrable_integral by blast
-next
-  case False
-  hence "\<not>(\<lambda>t. f (c + r * cis t) * r * \<i> * cis t) integrable_on {a..b}" 
-    using assms by (simp add: contour_integrable_part_circlepath_iff)
-  with False show ?thesis
-    by (simp add: not_integrable_contour_integral not_integrable_integral)
-qed
-
-lemma contour_integral_part_circlepath_reverse:
-  "contour_integral (part_circlepath c r a b) f = -contour_integral (part_circlepath c r b a) f"
-  by (subst reversepath_part_circlepath [symmetric], subst contour_integral_reversepath) simp_all
-
-lemma contour_integral_part_circlepath_reverse':
-  "b < a \<Longrightarrow> contour_integral (part_circlepath c r a b) f = 
-               -contour_integral (part_circlepath c r b a) f"
-  by (rule contour_integral_part_circlepath_reverse)
-
-lemma finite_bounded_log: "finite {z::complex. norm z \<le> b \<and> exp z = w}"
-proof (cases "w = 0")
-  case True then show ?thesis by auto
-next
-  case False
-  have *: "finite {x. cmod (complex_of_real (2 * real_of_int x * pi) * \<i>) \<le> b + cmod (Ln w)}"
-    apply (simp add: norm_mult finite_int_iff_bounded_le)
-    apply (rule_tac x="\<lfloor>(b + cmod (Ln w)) / (2*pi)\<rfloor>" in exI)
-    apply (auto simp: field_split_simps le_floor_iff)
-    done
-  have [simp]: "\<And>P f. {z. P z \<and> (\<exists>n. z = f n)} = f ` {n. P (f n)}"
-    by blast
-  show ?thesis
-    apply (subst exp_Ln [OF False, symmetric])
-    apply (simp add: exp_eq)
-    using norm_add_leD apply (fastforce intro: finite_subset [OF _ *])
-    done
-qed
-
-lemma finite_bounded_log2:
-  fixes a::complex
-    assumes "a \<noteq> 0"
-    shows "finite {z. norm z \<le> b \<and> exp(a*z) = w}"
-proof -
-  have *: "finite ((\<lambda>z. z / a) ` {z. cmod z \<le> b * cmod a \<and> exp z = w})"
-    by (rule finite_imageI [OF finite_bounded_log])
-  show ?thesis
-    by (rule finite_subset [OF _ *]) (force simp: assms norm_mult)
-qed
-
-lemma has_contour_integral_bound_part_circlepath_strong:
-  assumes fi: "(f has_contour_integral i) (part_circlepath z r s t)"
-      and "finite k" and le: "0 \<le> B" "0 < r" "s \<le> t"
-      and B: "\<And>x. x \<in> path_image(part_circlepath z r s t) - k \<Longrightarrow> norm(f x) \<le> B"
-    shows "cmod i \<le> B * r * (t - s)"
-proof -
-  consider "s = t" | "s < t" using \<open>s \<le> t\<close> by linarith
-  then show ?thesis
-  proof cases
-    case 1 with fi [unfolded has_contour_integral]
-    have "i = 0"  by (simp add: vector_derivative_part_circlepath)
-    with assms show ?thesis by simp
-  next
-    case 2
-    have [simp]: "\<bar>r\<bar> = r" using \<open>r > 0\<close> by linarith
-    have [simp]: "cmod (complex_of_real t - complex_of_real s) = t-s"
-      by (metis "2" abs_of_pos diff_gt_0_iff_gt norm_of_real of_real_diff)
-    have "finite (part_circlepath z r s t -` {y} \<inter> {0..1})" if "y \<in> k" for y
-    proof -
-      define w where "w = (y - z)/of_real r / exp(\<i> * of_real s)"
-      have fin: "finite (of_real -` {z. cmod z \<le> 1 \<and> exp (\<i> * complex_of_real (t - s) * z) = w})"
-        apply (rule finite_vimageI [OF finite_bounded_log2])
-        using \<open>s < t\<close> apply (auto simp: inj_of_real)
-        done
-      show ?thesis
-        apply (simp add: part_circlepath_def linepath_def vimage_def)
-        apply (rule finite_subset [OF _ fin])
-        using le
-        apply (auto simp: w_def algebra_simps scaleR_conv_of_real exp_add exp_diff)
-        done
-    qed
-    then have fin01: "finite ((part_circlepath z r s t) -` k \<inter> {0..1})"
-      by (rule finite_finite_vimage_IntI [OF \<open>finite k\<close>])
-    have **: "((\<lambda>x. if (part_circlepath z r s t x) \<in> k then 0
-                    else f(part_circlepath z r s t x) *
-                       vector_derivative (part_circlepath z r s t) (at x)) has_integral i)  {0..1}"
-      by (rule has_integral_spike [OF negligible_finite [OF fin01]])  (use fi has_contour_integral in auto)
-    have *: "\<And>x. \<lbrakk>0 \<le> x; x \<le> 1; part_circlepath z r s t x \<notin> k\<rbrakk> \<Longrightarrow> cmod (f (part_circlepath z r s t x)) \<le> B"
-      by (auto intro!: B [unfolded path_image_def image_def, simplified])
-    show ?thesis
-      apply (rule has_integral_bound [where 'a=real, simplified, OF _ **, simplified])
-      using assms apply force
-      apply (simp add: norm_mult vector_derivative_part_circlepath)
-      using le * "2" \<open>r > 0\<close> by auto
-  qed
-qed
-
-lemma has_contour_integral_bound_part_circlepath:
-      "\<lbrakk>(f has_contour_integral i) (part_circlepath z r s t);
-        0 \<le> B; 0 < r; s \<le> t;
-        \<And>x. x \<in> path_image(part_circlepath z r s t) \<Longrightarrow> norm(f x) \<le> B\<rbrakk>
-       \<Longrightarrow> norm i \<le> B*r*(t - s)"
-  by (auto intro: has_contour_integral_bound_part_circlepath_strong)
-
-lemma contour_integrable_continuous_part_circlepath:
-     "continuous_on (path_image (part_circlepath z r s t)) f
-      \<Longrightarrow> f contour_integrable_on (part_circlepath z r s t)"
-  apply (simp add: contour_integrable_on has_contour_integral_def vector_derivative_part_circlepath path_image_def)
-  apply (rule integrable_continuous_real)
-  apply (fast intro: path_part_circlepath [unfolded path_def] continuous_intros continuous_on_compose2 [where g=f, OF _ _ order_refl])
-  done
-
-proposition winding_number_part_circlepath_pos_less:
-  assumes "s < t" and no: "norm(w - z) < r"
-    shows "0 < Re (winding_number(part_circlepath z r s t) w)"
-proof -
-  have "0 < r" by (meson no norm_not_less_zero not_le order.strict_trans2)
-  note valid_path_part_circlepath
-  moreover have " w \<notin> path_image (part_circlepath z r s t)"
-    using assms by (auto simp: path_image_def image_def part_circlepath_def norm_mult linepath_def)
-  moreover have "0 < r * (t - s) * (r - cmod (w - z))"
-    using assms by (metis \<open>0 < r\<close> diff_gt_0_iff_gt mult_pos_pos)
-  ultimately show ?thesis
-    apply (rule winding_number_pos_lt [where e = "r*(t - s)*(r - norm(w - z))"])
-    apply (simp add: vector_derivative_part_circlepath right_diff_distrib [symmetric] mult_ac)
-    apply (rule mult_left_mono)+
-    using Re_Im_le_cmod [of "w-z" "linepath s t x" for x]
-    apply (simp add: exp_Euler cos_of_real sin_of_real part_circlepath_def algebra_simps cos_squared_eq [unfolded power2_eq_square])
-    using assms \<open>0 < r\<close> by auto
-qed
-
-lemma simple_path_part_circlepath:
-    "simple_path(part_circlepath z r s t) \<longleftrightarrow> (r \<noteq> 0 \<and> s \<noteq> t \<and> \<bar>s - t\<bar> \<le> 2*pi)"
-proof (cases "r = 0 \<or> s = t")
-  case True
-  then show ?thesis
-    unfolding part_circlepath_def simple_path_def
-    by (rule disjE) (force intro: bexI [where x = "1/4"] bexI [where x = "1/3"])+
-next
-  case False then have "r \<noteq> 0" "s \<noteq> t" by auto
-  have *: "\<And>x y z s t. \<i>*((1 - x) * s + x * t) = \<i>*(((1 - y) * s + y * t)) + z  \<longleftrightarrow> \<i>*(x - y) * (t - s) = z"
-    by (simp add: algebra_simps)
-  have abs01: "\<And>x y::real. 0 \<le> x \<and> x \<le> 1 \<and> 0 \<le> y \<and> y \<le> 1
-                      \<Longrightarrow> (x = y \<or> x = 0 \<and> y = 1 \<or> x = 1 \<and> y = 0 \<longleftrightarrow> \<bar>x - y\<bar> \<in> {0,1})"
-    by auto
-  have **: "\<And>x y. (\<exists>n. (complex_of_real x - of_real y) * (of_real t - of_real s) = 2 * (of_int n * of_real pi)) \<longleftrightarrow>
-                  (\<exists>n. \<bar>x - y\<bar> * (t - s) = 2 * (of_int n * pi))"
-    by (force simp: algebra_simps abs_if dest: arg_cong [where f=Re] arg_cong [where f=complex_of_real]
-                    intro: exI [where x = "-n" for n])
-  have 1: "\<bar>s - t\<bar> \<le> 2 * pi"
-    if "\<And>x. 0 \<le> x \<and> x \<le> 1 \<Longrightarrow> (\<exists>n. x * (t - s) = 2 * (real_of_int n * pi)) \<longrightarrow> x = 0 \<or> x = 1"
-  proof (rule ccontr)
-    assume "\<not> \<bar>s - t\<bar> \<le> 2 * pi"
-    then have *: "\<And>n. t - s \<noteq> of_int n * \<bar>s - t\<bar>"
-      using False that [of "2*pi / \<bar>t - s\<bar>"]
-      by (simp add: abs_minus_commute divide_simps)
-    show False
-      using * [of 1] * [of "-1"] by auto
-  qed
-  have 2: "\<bar>s - t\<bar> = \<bar>2 * (real_of_int n * pi) / x\<bar>" if "x \<noteq> 0" "x * (t - s) = 2 * (real_of_int n * pi)" for x n
-  proof -
-    have "t-s = 2 * (real_of_int n * pi)/x"
-      using that by (simp add: field_simps)
-    then show ?thesis by (metis abs_minus_commute)
-  qed
-  have abs_away: "\<And>P. (\<forall>x\<in>{0..1}. \<forall>y\<in>{0..1}. P \<bar>x - y\<bar>) \<longleftrightarrow> (\<forall>x::real. 0 \<le> x \<and> x \<le> 1 \<longrightarrow> P x)"
-    by force
-  show ?thesis using False
-    apply (simp add: simple_path_def)
-    apply (simp add: part_circlepath_def linepath_def exp_eq  * ** abs01  del: Set.insert_iff)
-    apply (subst abs_away)
-    apply (auto simp: 1)
-    apply (rule ccontr)
-    apply (auto simp: 2 field_split_simps abs_mult dest: of_int_leD)
-    done
-qed
-
-lemma arc_part_circlepath:
-  assumes "r \<noteq> 0" "s \<noteq> t" "\<bar>s - t\<bar> < 2*pi"
-    shows "arc (part_circlepath z r s t)"
-proof -
-  have *: "x = y" if eq: "\<i> * (linepath s t x) = \<i> * (linepath s t y) + 2 * of_int n * complex_of_real pi * \<i>"
-    and x: "x \<in> {0..1}" and y: "y \<in> {0..1}" for x y n
-  proof (rule ccontr)
-    assume "x \<noteq> y"
-    have "(linepath s t x) = (linepath s t y) + 2 * of_int n * complex_of_real pi"
-      by (metis add_divide_eq_iff complex_i_not_zero mult.commute nonzero_mult_div_cancel_left eq)
-    then have "s*y + t*x = s*x + (t*y + of_int n * (pi * 2))"
-      by (force simp: algebra_simps linepath_def dest: arg_cong [where f=Re])
-    with \<open>x \<noteq> y\<close> have st: "s-t = (of_int n * (pi * 2) / (y-x))"
-      by (force simp: field_simps)
-    have "\<bar>real_of_int n\<bar> < \<bar>y - x\<bar>"
-      using assms \<open>x \<noteq> y\<close> by (simp add: st abs_mult field_simps)
-    then show False
-      using assms x y st by (auto dest: of_int_lessD)
-  qed
-  show ?thesis
-    using assms
-    apply (simp add: arc_def)
-    apply (simp add: part_circlepath_def inj_on_def exp_eq)
-    apply (blast intro: *)
-    done
-qed
-
-subsection\<open>Special case of one complete circle\<close>
-
-definition\<^marker>\<open>tag important\<close> circlepath :: "[complex, real, real] \<Rightarrow> complex"
-  where "circlepath z r \<equiv> part_circlepath z r 0 (2*pi)"
-
-lemma circlepath: "circlepath z r = (\<lambda>x. z + r * exp(2 * of_real pi * \<i> * of_real x))"
-  by (simp add: circlepath_def part_circlepath_def linepath_def algebra_simps)
-
-lemma pathstart_circlepath [simp]: "pathstart (circlepath z r) = z + r"
-  by (simp add: circlepath_def)
-
-lemma pathfinish_circlepath [simp]: "pathfinish (circlepath z r) = z + r"
-  by (simp add: circlepath_def) (metis exp_two_pi_i mult.commute)
-
-lemma circlepath_minus: "circlepath z (-r) x = circlepath z r (x + 1/2)"
-proof -
-  have "z + of_real r * exp (2 * pi * \<i> * (x + 1/2)) =
-        z + of_real r * exp (2 * pi * \<i> * x + pi * \<i>)"
-    by (simp add: divide_simps) (simp add: algebra_simps)
-  also have "\<dots> = z - r * exp (2 * pi * \<i> * x)"
-    by (simp add: exp_add)
-  finally show ?thesis
-    by (simp add: circlepath path_image_def sphere_def dist_norm)
-qed
-
-lemma circlepath_add1: "circlepath z r (x+1) = circlepath z r x"
-  using circlepath_minus [of z r "x+1/2"] circlepath_minus [of z "-r" x]
-  by (simp add: add.commute)
-
-lemma circlepath_add_half: "circlepath z r (x + 1/2) = circlepath z r (x - 1/2)"
-  using circlepath_add1 [of z r "x-1/2"]
-  by (simp add: add.commute)
-
-lemma path_image_circlepath_minus_subset:
-     "path_image (circlepath z (-r)) \<subseteq> path_image (circlepath z r)"
-  apply (simp add: path_image_def image_def circlepath_minus, clarify)
-  apply (case_tac "xa \<le> 1/2", force)
-  apply (force simp: circlepath_add_half)+
-  done
-
-lemma path_image_circlepath_minus: "path_image (circlepath z (-r)) = path_image (circlepath z r)"
-  using path_image_circlepath_minus_subset by fastforce
-
-lemma has_vector_derivative_circlepath [derivative_intros]:
- "((circlepath z r) has_vector_derivative (2 * pi * \<i> * r * exp (2 * of_real pi * \<i> * of_real x)))
-   (at x within X)"
-  apply (simp add: circlepath_def scaleR_conv_of_real)
-  apply (rule derivative_eq_intros)
-  apply (simp add: algebra_simps)
-  done
-
-lemma vector_derivative_circlepath:
-   "vector_derivative (circlepath z r) (at x) =
-    2 * pi * \<i> * r * exp(2 * of_real pi * \<i> * x)"
-using has_vector_derivative_circlepath vector_derivative_at by blast
-
-lemma vector_derivative_circlepath01:
-    "\<lbrakk>0 \<le> x; x \<le> 1\<rbrakk>
-     \<Longrightarrow> vector_derivative (circlepath z r) (at x within {0..1}) =
-          2 * pi * \<i> * r * exp(2 * of_real pi * \<i> * x)"
-  using has_vector_derivative_circlepath
-  by (auto simp: vector_derivative_at_within_ivl)
-
-lemma valid_path_circlepath [simp]: "valid_path (circlepath z r)"
-  by (simp add: circlepath_def)
-
-lemma path_circlepath [simp]: "path (circlepath z r)"
-  by (simp add: valid_path_imp_path)
-
-lemma path_image_circlepath_nonneg:
-  assumes "0 \<le> r" shows "path_image (circlepath z r) = sphere z r"
-proof -
-  have *: "x \<in> (\<lambda>u. z + (cmod (x - z)) * exp (\<i> * (of_real u * (of_real pi * 2)))) ` {0..1}" for x
-  proof (cases "x = z")
-    case True then show ?thesis by force
-  next
-    case False
-    define w where "w = x - z"
-    then have "w \<noteq> 0" by (simp add: False)
-    have **: "\<And>t. \<lbrakk>Re w = cos t * cmod w; Im w = sin t * cmod w\<rbrakk> \<Longrightarrow> w = of_real (cmod w) * exp (\<i> * t)"
-      using cis_conv_exp complex_eq_iff by auto
-    show ?thesis
-      apply (rule sincos_total_2pi [of "Re(w/of_real(norm w))" "Im(w/of_real(norm w))"])
-      apply (simp add: divide_simps \<open>w \<noteq> 0\<close> cmod_power2 [symmetric])
-      apply (rule_tac x="t / (2*pi)" in image_eqI)
-      apply (simp add: field_simps \<open>w \<noteq> 0\<close>)
-      using False **
-      apply (auto simp: w_def)
-      done
-  qed
-  show ?thesis
-    unfolding circlepath path_image_def sphere_def dist_norm
-    by (force simp: assms algebra_simps norm_mult norm_minus_commute intro: *)
-qed
-
-lemma path_image_circlepath [simp]:
-    "path_image (circlepath z r) = sphere z \<bar>r\<bar>"
-  using path_image_circlepath_minus
-  by (force simp: path_image_circlepath_nonneg abs_if)
-
-lemma has_contour_integral_bound_circlepath_strong:
-      "\<lbrakk>(f has_contour_integral i) (circlepath z r);
-        finite k; 0 \<le> B; 0 < r;
-        \<And>x. \<lbrakk>norm(x - z) = r; x \<notin> k\<rbrakk> \<Longrightarrow> norm(f x) \<le> B\<rbrakk>
-        \<Longrightarrow> norm i \<le> B*(2*pi*r)"
-  unfolding circlepath_def
-  by (auto simp: algebra_simps in_path_image_part_circlepath dest!: has_contour_integral_bound_part_circlepath_strong)
-
-lemma has_contour_integral_bound_circlepath:
-      "\<lbrakk>(f has_contour_integral i) (circlepath z r);
-        0 \<le> B; 0 < r; \<And>x. norm(x - z) = r \<Longrightarrow> norm(f x) \<le> B\<rbrakk>
-        \<Longrightarrow> norm i \<le> B*(2*pi*r)"
-  by (auto intro: has_contour_integral_bound_circlepath_strong)
-
-lemma contour_integrable_continuous_circlepath:
-    "continuous_on (path_image (circlepath z r)) f
-     \<Longrightarrow> f contour_integrable_on (circlepath z r)"
-  by (simp add: circlepath_def contour_integrable_continuous_part_circlepath)
-
-lemma simple_path_circlepath: "simple_path(circlepath z r) \<longleftrightarrow> (r \<noteq> 0)"
-  by (simp add: circlepath_def simple_path_part_circlepath)
-
-lemma notin_path_image_circlepath [simp]: "cmod (w - z) < r \<Longrightarrow> w \<notin> path_image (circlepath z r)"
-  by (simp add: sphere_def dist_norm norm_minus_commute)
-
-lemma contour_integral_circlepath:
-  assumes "r > 0"
-  shows "contour_integral (circlepath z r) (\<lambda>w. 1 / (w - z)) = 2 * complex_of_real pi * \<i>"
-proof (rule contour_integral_unique)
-  show "((\<lambda>w. 1 / (w - z)) has_contour_integral 2 * complex_of_real pi * \<i>) (circlepath z r)"
-    unfolding has_contour_integral_def using assms
-    apply (subst has_integral_cong)
-     apply (simp add: vector_derivative_circlepath01)
-    using has_integral_const_real [of _ 0 1] apply (force simp: circlepath)
-    done
-qed
-
-lemma winding_number_circlepath_centre: "0 < r \<Longrightarrow> winding_number (circlepath z r) z = 1"
-  apply (rule winding_number_unique_loop)
-  apply (simp_all add: sphere_def valid_path_imp_path)
-  apply (rule_tac x="circlepath z r" in exI)
-  apply (simp add: sphere_def contour_integral_circlepath)
-  done
-
-proposition winding_number_circlepath:
-  assumes "norm(w - z) < r" shows "winding_number(circlepath z r) w = 1"
-proof (cases "w = z")
-  case True then show ?thesis
-    using assms winding_number_circlepath_centre by auto
-next
-  case False
-  have [simp]: "r > 0"
-    using assms le_less_trans norm_ge_zero by blast
-  define r' where "r' = norm(w - z)"
-  have "r' < r"
-    by (simp add: assms r'_def)
-  have disjo: "cball z r' \<inter> sphere z r = {}"
-    using \<open>r' < r\<close> by (force simp: cball_def sphere_def)
-  have "winding_number(circlepath z r) w = winding_number(circlepath z r) z"
-  proof (rule winding_number_around_inside [where s = "cball z r'"])
-    show "winding_number (circlepath z r) z \<noteq> 0"
-      by (simp add: winding_number_circlepath_centre)
-    show "cball z r' \<inter> path_image (circlepath z r) = {}"
-      by (simp add: disjo less_eq_real_def)
-  qed (auto simp: r'_def dist_norm norm_minus_commute)
-  also have "\<dots> = 1"
-    by (simp add: winding_number_circlepath_centre)
-  finally show ?thesis .
-qed
-
-
-text\<open> Hence the Cauchy formula for points inside a circle.\<close>
-
-theorem Cauchy_integral_circlepath:
-  assumes contf: "continuous_on (cball z r) f" and holf: "f holomorphic_on (ball z r)" and wz: "norm(w - z) < r"
-  shows "((\<lambda>u. f u/(u - w)) has_contour_integral (2 * of_real pi * \<i> * f w))
-         (circlepath z r)"
-proof -
-  have "r > 0"
-    using assms le_less_trans norm_ge_zero by blast
-  have "((\<lambda>u. f u / (u - w)) has_contour_integral (2 * pi) * \<i> * winding_number (circlepath z r) w * f w)
-        (circlepath z r)"
-  proof (rule Cauchy_integral_formula_weak [where s = "cball z r" and k = "{}"])
-    show "\<And>x. x \<in> interior (cball z r) - {} \<Longrightarrow>
-         f field_differentiable at x"
-      using holf holomorphic_on_imp_differentiable_at by auto
-    have "w \<notin> sphere z r"
-      by simp (metis dist_commute dist_norm not_le order_refl wz)
-    then show "path_image (circlepath z r) \<subseteq> cball z r - {w}"
-      using \<open>r > 0\<close> by (auto simp add: cball_def sphere_def)
-  qed (use wz in \<open>simp_all add: dist_norm norm_minus_commute contf\<close>)
-  then show ?thesis
-    by (simp add: winding_number_circlepath assms)
-qed
-
-corollary\<^marker>\<open>tag unimportant\<close> Cauchy_integral_circlepath_simple:
-  assumes "f holomorphic_on cball z r" "norm(w - z) < r"
-  shows "((\<lambda>u. f u/(u - w)) has_contour_integral (2 * of_real pi * \<i> * f w))
-         (circlepath z r)"
-using assms by (force simp: holomorphic_on_imp_continuous_on holomorphic_on_subset Cauchy_integral_circlepath)
-
-
-lemma no_bounded_connected_component_imp_winding_number_zero:
-  assumes g: "path g" "path_image g \<subseteq> s" "pathfinish g = pathstart g" "z \<notin> s"
-      and nb: "\<And>z. bounded (connected_component_set (- s) z) \<longrightarrow> z \<in> s"
-  shows "winding_number g z = 0"
-apply (rule winding_number_zero_in_outside)
-apply (simp_all add: assms)
-by (metis nb [of z] \<open>path_image g \<subseteq> s\<close> \<open>z \<notin> s\<close> contra_subsetD mem_Collect_eq outside outside_mono)
-
-lemma no_bounded_path_component_imp_winding_number_zero:
-  assumes g: "path g" "path_image g \<subseteq> s" "pathfinish g = pathstart g" "z \<notin> s"
-      and nb: "\<And>z. bounded (path_component_set (- s) z) \<longrightarrow> z \<in> s"
-  shows "winding_number g z = 0"
-apply (rule no_bounded_connected_component_imp_winding_number_zero [OF g])
-by (simp add: bounded_subset nb path_component_subset_connected_component)
-
-
-subsection\<open> Uniform convergence of path integral\<close>
-
-text\<open>Uniform convergence when the derivative of the path is bounded, and in particular for the special case of a circle.\<close>
-
-proposition contour_integral_uniform_limit:
-  assumes ev_fint: "eventually (\<lambda>n::'a. (f n) contour_integrable_on \<gamma>) F"
-      and ul_f: "uniform_limit (path_image \<gamma>) f l F"
-      and noleB: "\<And>t. t \<in> {0..1} \<Longrightarrow> norm (vector_derivative \<gamma> (at t)) \<le> B"
-      and \<gamma>: "valid_path \<gamma>"
-      and [simp]: "\<not> trivial_limit F"
-  shows "l contour_integrable_on \<gamma>" "((\<lambda>n. contour_integral \<gamma> (f n)) \<longlongrightarrow> contour_integral \<gamma> l) F"
-proof -
-  have "0 \<le> B" by (meson noleB [of 0] atLeastAtMost_iff norm_ge_zero order_refl order_trans zero_le_one)
-  { fix e::real
-    assume "0 < e"
-    then have "0 < e / (\<bar>B\<bar> + 1)" by simp
-    then have "\<forall>\<^sub>F n in F. \<forall>x\<in>path_image \<gamma>. cmod (f n x - l x) < e / (\<bar>B\<bar> + 1)"
-      using ul_f [unfolded uniform_limit_iff dist_norm] by auto
-    with ev_fint
-    obtain a where fga: "\<And>x. x \<in> {0..1} \<Longrightarrow> cmod (f a (\<gamma> x) - l (\<gamma> x)) < e / (\<bar>B\<bar> + 1)"
-               and inta: "(\<lambda>t. f a (\<gamma> t) * vector_derivative \<gamma> (at t)) integrable_on {0..1}"
-      using eventually_happens [OF eventually_conj]
-      by (fastforce simp: contour_integrable_on path_image_def)
-    have Ble: "B * e / (\<bar>B\<bar> + 1) \<le> e"
-      using \<open>0 \<le> B\<close>  \<open>0 < e\<close> by (simp add: field_split_simps)
-    have "\<exists>h. (\<forall>x\<in>{0..1}. cmod (l (\<gamma> x) * vector_derivative \<gamma> (at x) - h x) \<le> e) \<and> h integrable_on {0..1}"
-    proof (intro exI conjI ballI)
-      show "cmod (l (\<gamma> x) * vector_derivative \<gamma> (at x) - f a (\<gamma> x) * vector_derivative \<gamma> (at x)) \<le> e"
-        if "x \<in> {0..1}" for x
-        apply (rule order_trans [OF _ Ble])
-        using noleB [OF that] fga [OF that] \<open>0 \<le> B\<close> \<open>0 < e\<close>
-        apply (simp add: norm_mult left_diff_distrib [symmetric] norm_minus_commute divide_simps)
-        apply (fastforce simp: mult_ac dest: mult_mono [OF less_imp_le])
-        done
-    qed (rule inta)
-  }
-  then show lintg: "l contour_integrable_on \<gamma>"
-    unfolding contour_integrable_on by (metis (mono_tags, lifting)integrable_uniform_limit_real)
-  { fix e::real
-    define B' where "B' = B + 1"
-    have B': "B' > 0" "B' > B" using  \<open>0 \<le> B\<close> by (auto simp: B'_def)
-    assume "0 < e"
-    then have ev_no': "\<forall>\<^sub>F n in F. \<forall>x\<in>path_image \<gamma>. 2 * cmod (f n x - l x) < e / B'"
-      using ul_f [unfolded uniform_limit_iff dist_norm, rule_format, of "e / B' / 2"] B'
-        by (simp add: field_simps)
-    have ie: "integral {0..1::real} (\<lambda>x. e / 2) < e" using \<open>0 < e\<close> by simp
-    have *: "cmod (f x (\<gamma> t) * vector_derivative \<gamma> (at t) - l (\<gamma> t) * vector_derivative \<gamma> (at t)) \<le> e / 2"
-             if t: "t\<in>{0..1}" and leB': "2 * cmod (f x (\<gamma> t) - l (\<gamma> t)) < e / B'" for x t
-    proof -
-      have "2 * cmod (f x (\<gamma> t) - l (\<gamma> t)) * cmod (vector_derivative \<gamma> (at t)) \<le> e * (B/ B')"
-        using mult_mono [OF less_imp_le [OF leB'] noleB] B' \<open>0 < e\<close> t by auto
-      also have "\<dots> < e"
-        by (simp add: B' \<open>0 < e\<close> mult_imp_div_pos_less)
-      finally have "2 * cmod (f x (\<gamma> t) - l (\<gamma> t)) * cmod (vector_derivative \<gamma> (at t)) < e" .
-      then show ?thesis
-        by (simp add: left_diff_distrib [symmetric] norm_mult)
-    qed
-    have le_e: "\<And>x. \<lbrakk>\<forall>xa\<in>{0..1}. 2 * cmod (f x (\<gamma> xa) - l (\<gamma> xa)) < e / B'; f x contour_integrable_on \<gamma>\<rbrakk>
-         \<Longrightarrow> cmod (integral {0..1}
-                    (\<lambda>u. f x (\<gamma> u) * vector_derivative \<gamma> (at u) - l (\<gamma> u) * vector_derivative \<gamma> (at u))) < e"
-      apply (rule le_less_trans [OF integral_norm_bound_integral ie])
-        apply (simp add: lintg integrable_diff contour_integrable_on [symmetric])
-       apply (blast intro: *)+
-      done
-    have "\<forall>\<^sub>F x in F. dist (contour_integral \<gamma> (f x)) (contour_integral \<gamma> l) < e"
-      apply (rule eventually_mono [OF eventually_conj [OF ev_no' ev_fint]])
-      apply (simp add: dist_norm contour_integrable_on path_image_def contour_integral_integral)
-      apply (simp add: lintg integral_diff [symmetric] contour_integrable_on [symmetric] le_e)
-      done
-  }
-  then show "((\<lambda>n. contour_integral \<gamma> (f n)) \<longlongrightarrow> contour_integral \<gamma> l) F"
-    by (rule tendstoI)
-qed
-
-corollary\<^marker>\<open>tag unimportant\<close> contour_integral_uniform_limit_circlepath:
-  assumes "\<forall>\<^sub>F n::'a in F. (f n) contour_integrable_on (circlepath z r)"
-      and "uniform_limit (sphere z r) f l F"
-      and "\<not> trivial_limit F" "0 < r"
-    shows "l contour_integrable_on (circlepath z r)"
-          "((\<lambda>n. contour_integral (circlepath z r) (f n)) \<longlongrightarrow> contour_integral (circlepath z r) l) F"
-  using assms by (auto simp: vector_derivative_circlepath norm_mult intro!: contour_integral_uniform_limit)
-
-
-subsection\<^marker>\<open>tag unimportant\<close> \<open>General stepping result for derivative formulas\<close>
-
-lemma Cauchy_next_derivative:
-  assumes "continuous_on (path_image \<gamma>) f'"
-      and leB: "\<And>t. t \<in> {0..1} \<Longrightarrow> norm (vector_derivative \<gamma> (at t)) \<le> B"
-      and int: "\<And>w. w \<in> s - path_image \<gamma> \<Longrightarrow> ((\<lambda>u. f' u / (u - w)^k) has_contour_integral f w) \<gamma>"
-      and k: "k \<noteq> 0"
-      and "open s"
-      and \<gamma>: "valid_path \<gamma>"
-      and w: "w \<in> s - path_image \<gamma>"
-    shows "(\<lambda>u. f' u / (u - w)^(Suc k)) contour_integrable_on \<gamma>"
-      and "(f has_field_derivative (k * contour_integral \<gamma> (\<lambda>u. f' u/(u - w)^(Suc k))))
-           (at w)"  (is "?thes2")
-proof -
-  have "open (s - path_image \<gamma>)" using \<open>open s\<close> closed_valid_path_image \<gamma> by blast
-  then obtain d where "d>0" and d: "ball w d \<subseteq> s - path_image \<gamma>" using w
-    using open_contains_ball by blast
-  have [simp]: "\<And>n. cmod (1 + of_nat n) = 1 + of_nat n"
-    by (metis norm_of_nat of_nat_Suc)
-  have cint: "\<And>x. \<lbrakk>x \<noteq> w; cmod (x - w) < d\<rbrakk>
-         \<Longrightarrow> (\<lambda>z. (f' z / (z - x) ^ k - f' z / (z - w) ^ k) / (x * k - w * k)) contour_integrable_on \<gamma>"
-    apply (rule contour_integrable_div [OF contour_integrable_diff])
-    using int w d
-    by (force simp: dist_norm norm_minus_commute intro!: has_contour_integral_integrable)+
-  have 1: "\<forall>\<^sub>F n in at w. (\<lambda>x. f' x * (inverse (x - n) ^ k - inverse (x - w) ^ k) / (n - w) / of_nat k)
-                         contour_integrable_on \<gamma>"
-    unfolding eventually_at
-    apply (rule_tac x=d in exI)
-    apply (simp add: \<open>d > 0\<close> dist_norm field_simps cint)
-    done
-  have bim_g: "bounded (image f' (path_image \<gamma>))"
-    by (simp add: compact_imp_bounded compact_continuous_image compact_valid_path_image assms)
-  then obtain C where "C > 0" and C: "\<And>x. \<lbrakk>0 \<le> x; x \<le> 1\<rbrakk> \<Longrightarrow> cmod (f' (\<gamma> x)) \<le> C"
-    by (force simp: bounded_pos path_image_def)
-  have twom: "\<forall>\<^sub>F n in at w.
-               \<forall>x\<in>path_image \<gamma>.
-                cmod ((inverse (x - n) ^ k - inverse (x - w) ^ k) / (n - w) / k - inverse (x - w) ^ Suc k) < e"
-         if "0 < e" for e
-  proof -
-    have *: "cmod ((inverse (x - u) ^ k - inverse (x - w) ^ k) / ((u - w) * k) - inverse (x - w) ^ Suc k)   < e"
-            if x: "x \<in> path_image \<gamma>" and "u \<noteq> w" and uwd: "cmod (u - w) < d/2"
-                and uw_less: "cmod (u - w) < e * (d/2) ^ (k+2) / (1 + real k)"
-            for u x
-    proof -
-      define ff where [abs_def]:
-        "ff n w =
-          (if n = 0 then inverse(x - w)^k
-           else if n = 1 then k / (x - w)^(Suc k)
-           else (k * of_real(Suc k)) / (x - w)^(k + 2))" for n :: nat and w
-      have km1: "\<And>z::complex. z \<noteq> 0 \<Longrightarrow> z ^ (k - Suc 0) = z ^ k / z"
-        by (simp add: field_simps) (metis Suc_pred \<open>k \<noteq> 0\<close> neq0_conv power_Suc)
-      have ff1: "(ff i has_field_derivative ff (Suc i) z) (at z within ball w (d/2))"
-              if "z \<in> ball w (d/2)" "i \<le> 1" for i z
-      proof -
-        have "z \<notin> path_image \<gamma>"
-          using \<open>x \<in> path_image \<gamma>\<close> d that ball_divide_subset_numeral by blast
-        then have xz[simp]: "x \<noteq> z" using \<open>x \<in> path_image \<gamma>\<close> by blast
-        then have neq: "x * x + z * z \<noteq> x * (z * 2)"
-          by (blast intro: dest!: sum_sqs_eq)
-        with xz have "\<And>v. v \<noteq> 0 \<Longrightarrow> (x * x + z * z) * v \<noteq> (x * (z * 2) * v)" by auto
-        then have neqq: "\<And>v. v \<noteq> 0 \<Longrightarrow> x * (x * v) + z * (z * v) \<noteq> x * (z * (2 * v))"
-          by (simp add: algebra_simps)
-        show ?thesis using \<open>i \<le> 1\<close>
-          apply (simp add: ff_def dist_norm Nat.le_Suc_eq km1, safe)
-          apply (rule derivative_eq_intros | simp add: km1 | simp add: field_simps neq neqq)+
-          done
-      qed
-      { fix a::real and b::real assume ab: "a > 0" "b > 0"
-        then have "k * (1 + real k) * (1 / a) \<le> k * (1 + real k) * (4 / b) \<longleftrightarrow> b \<le> 4 * a"
-          by (subst mult_le_cancel_left_pos)
-            (use \<open>k \<noteq> 0\<close> in \<open>auto simp: divide_simps\<close>)
-        with ab have "real k * (1 + real k) / a \<le> (real k * 4 + real k * real k * 4) / b \<longleftrightarrow> b \<le> 4 * a"
-          by (simp add: field_simps)
-      } note canc = this
-      have ff2: "cmod (ff (Suc 1) v) \<le> real (k * (k + 1)) / (d/2) ^ (k + 2)"
-                if "v \<in> ball w (d/2)" for v
-      proof -
-        have lessd: "\<And>z. cmod (\<gamma> z - v) < d/2 \<Longrightarrow> cmod (w - \<gamma> z) < d"
-          by (metis that norm_minus_commute norm_triangle_half_r dist_norm mem_ball)
-        have "d/2 \<le> cmod (x - v)" using d x that
-          using lessd d x
-          by (auto simp add: dist_norm path_image_def ball_def not_less [symmetric] del: divide_const_simps)
-        then have "d \<le> cmod (x - v) * 2"
-          by (simp add: field_split_simps)
-        then have dpow_le: "d ^ (k+2) \<le> (cmod (x - v) * 2) ^ (k+2)"
-          using \<open>0 < d\<close> order_less_imp_le power_mono by blast
-        have "x \<noteq> v" using that
-          using \<open>x \<in> path_image \<gamma>\<close> ball_divide_subset_numeral d by fastforce
-        then show ?thesis
-        using \<open>d > 0\<close> apply (simp add: ff_def norm_mult norm_divide norm_power dist_norm canc)
-        using dpow_le apply (simp add: field_split_simps)
-        done
-      qed
-      have ub: "u \<in> ball w (d/2)"
-        using uwd by (simp add: dist_commute dist_norm)
-      have "cmod (inverse (x - u) ^ k - (inverse (x - w) ^ k + of_nat k * (u - w) / ((x - w) * (x - w) ^ k)))
-                  \<le> (real k * 4 + real k * real k * 4) * (cmod (u - w) * cmod (u - w)) / (d * (d * (d/2) ^ k))"
-        using complex_Taylor [OF _ ff1 ff2 _ ub, of w, simplified]
-        by (simp add: ff_def \<open>0 < d\<close>)
-      then have "cmod (inverse (x - u) ^ k - (inverse (x - w) ^ k + of_nat k * (u - w) / ((x - w) * (x - w) ^ k)))
-                  \<le> (cmod (u - w) * real k) * (1 + real k) * cmod (u - w) / (d/2) ^ (k+2)"
-        by (simp add: field_simps)
-      then have "cmod (inverse (x - u) ^ k - (inverse (x - w) ^ k + of_nat k * (u - w) / ((x - w) * (x - w) ^ k)))
-                 / (cmod (u - w) * real k)
-                  \<le> (1 + real k) * cmod (u - w) / (d/2) ^ (k+2)"
-        using \<open>k \<noteq> 0\<close> \<open>u \<noteq> w\<close> by (simp add: mult_ac zero_less_mult_iff pos_divide_le_eq)
-      also have "\<dots> < e"
-        using uw_less \<open>0 < d\<close> by (simp add: mult_ac divide_simps)
-      finally have e: "cmod (inverse (x-u)^k - (inverse (x-w)^k + of_nat k * (u-w) / ((x-w) * (x-w)^k)))
-                        / cmod ((u - w) * real k)   <   e"
-        by (simp add: norm_mult)
-      have "x \<noteq> u"
-        using uwd \<open>0 < d\<close> x d by (force simp: dist_norm ball_def norm_minus_commute)
-      show ?thesis
-        apply (rule le_less_trans [OF _ e])
-        using \<open>k \<noteq> 0\<close> \<open>x \<noteq> u\<close> \<open>u \<noteq> w\<close>
-        apply (simp add: field_simps norm_divide [symmetric])
-        done
-    qed
-    show ?thesis
-      unfolding eventually_at
-      apply (rule_tac x = "min (d/2) ((e*(d/2)^(k + 2))/(Suc k))" in exI)
-      apply (force simp: \<open>d > 0\<close> dist_norm that simp del: power_Suc intro: *)
-      done
-  qed
-  have 2: "uniform_limit (path_image \<gamma>) (\<lambda>n x. f' x * (inverse (x - n) ^ k - inverse (x - w) ^ k) / (n - w) / of_nat k) (\<lambda>x. f' x / (x - w) ^ Suc k) (at w)"
-    unfolding uniform_limit_iff dist_norm
-  proof clarify
-    fix e::real
-    assume "0 < e"
-    have *: "cmod (f' (\<gamma> x) * (inverse (\<gamma> x - u) ^ k - inverse (\<gamma> x - w) ^ k) / ((u - w) * k) -
-                        f' (\<gamma> x) / ((\<gamma> x - w) * (\<gamma> x - w) ^ k)) < e"
-              if ec: "cmod ((inverse (\<gamma> x - u) ^ k - inverse (\<gamma> x - w) ^ k) / ((u - w) * k) -
-                      inverse (\<gamma> x - w) * inverse (\<gamma> x - w) ^ k) < e / C"
-                 and x: "0 \<le> x" "x \<le> 1"
-              for u x
-    proof (cases "(f' (\<gamma> x)) = 0")
-      case True then show ?thesis by (simp add: \<open>0 < e\<close>)
-    next
-      case False
-      have "cmod (f' (\<gamma> x) * (inverse (\<gamma> x - u) ^ k - inverse (\<gamma> x - w) ^ k) / ((u - w) * k) -
-                        f' (\<gamma> x) / ((\<gamma> x - w) * (\<gamma> x - w) ^ k)) =
-            cmod (f' (\<gamma> x) * ((inverse (\<gamma> x - u) ^ k - inverse (\<gamma> x - w) ^ k) / ((u - w) * k) -
-                             inverse (\<gamma> x - w) * inverse (\<gamma> x - w) ^ k))"
-        by (simp add: field_simps)
-      also have "\<dots> = cmod (f' (\<gamma> x)) *
-                       cmod ((inverse (\<gamma> x - u) ^ k - inverse (\<gamma> x - w) ^ k) / ((u - w) * k) -
-                             inverse (\<gamma> x - w) * inverse (\<gamma> x - w) ^ k)"
-        by (simp add: norm_mult)
-      also have "\<dots> < cmod (f' (\<gamma> x)) * (e/C)"
-        using False mult_strict_left_mono [OF ec] by force
-      also have "\<dots> \<le> e" using C
-        by (metis False \<open>0 < e\<close> frac_le less_eq_real_def mult.commute pos_le_divide_eq x zero_less_norm_iff)
-      finally show ?thesis .
-    qed
-    show "\<forall>\<^sub>F n in at w.
-              \<forall>x\<in>path_image \<gamma>.
-               cmod (f' x * (inverse (x - n) ^ k - inverse (x - w) ^ k) / (n - w) / of_nat k - f' x / (x - w) ^ Suc k) < e"
-      using twom [OF divide_pos_pos [OF \<open>0 < e\<close> \<open>C > 0\<close>]]   unfolding path_image_def
-      by (force intro: * elim: eventually_mono)
-  qed
-  show "(\<lambda>u. f' u / (u - w) ^ (Suc k)) contour_integrable_on \<gamma>"
-    by (rule contour_integral_uniform_limit [OF 1 2 leB \<gamma>]) auto
-  have *: "(\<lambda>n. contour_integral \<gamma> (\<lambda>x. f' x * (inverse (x - n) ^ k - inverse (x - w) ^ k) / (n - w) / k))
-           \<midarrow>w\<rightarrow> contour_integral \<gamma> (\<lambda>u. f' u / (u - w) ^ (Suc k))"
-    by (rule contour_integral_uniform_limit [OF 1 2 leB \<gamma>]) auto
-  have **: "contour_integral \<gamma> (\<lambda>x. f' x * (inverse (x - u) ^ k - inverse (x - w) ^ k) / ((u - w) * k)) =
-              (f u - f w) / (u - w) / k"
-    if "dist u w < d" for u
-  proof -
-    have u: "u \<in> s - path_image \<gamma>"
-      by (metis subsetD d dist_commute mem_ball that)
-    show ?thesis
-      apply (rule contour_integral_unique)
-      apply (simp add: diff_divide_distrib algebra_simps)
-      apply (intro has_contour_integral_diff has_contour_integral_div)
-      using u w apply (simp_all add: field_simps int)
-      done
-  qed
-  show ?thes2
-    apply (simp add: has_field_derivative_iff del: power_Suc)
-    apply (rule Lim_transform_within [OF tendsto_mult_left [OF *] \<open>0 < d\<close> ])
-    apply (simp add: \<open>k \<noteq> 0\<close> **)
-    done
-qed
-
-lemma Cauchy_next_derivative_circlepath:
-  assumes contf: "continuous_on (path_image (circlepath z r)) f"
-      and int: "\<And>w. w \<in> ball z r \<Longrightarrow> ((\<lambda>u. f u / (u - w)^k) has_contour_integral g w) (circlepath z r)"
-      and k: "k \<noteq> 0"
-      and w: "w \<in> ball z r"
-    shows "(\<lambda>u. f u / (u - w)^(Suc k)) contour_integrable_on (circlepath z r)"
-           (is "?thes1")
-      and "(g has_field_derivative (k * contour_integral (circlepath z r) (\<lambda>u. f u/(u - w)^(Suc k)))) (at w)"
-           (is "?thes2")
-proof -
-  have "r > 0" using w
-    using ball_eq_empty by fastforce
-  have wim: "w \<in> ball z r - path_image (circlepath z r)"
-    using w by (auto simp: dist_norm)
-  show ?thes1 ?thes2
-    by (rule Cauchy_next_derivative [OF contf _ int k open_ball valid_path_circlepath wim, where B = "2 * pi * \<bar>r\<bar>"];
-        auto simp: vector_derivative_circlepath norm_mult)+
-qed
-
-
-text\<open> In particular, the first derivative formula.\<close>
-
-lemma Cauchy_derivative_integral_circlepath:
-  assumes contf: "continuous_on (cball z r) f"
-      and holf: "f holomorphic_on ball z r"
-      and w: "w \<in> ball z r"
-    shows "(\<lambda>u. f u/(u - w)^2) contour_integrable_on (circlepath z r)"
-           (is "?thes1")
-      and "(f has_field_derivative (1 / (2 * of_real pi * \<i>) * contour_integral(circlepath z r) (\<lambda>u. f u / (u - w)^2))) (at w)"
-           (is "?thes2")
-proof -
-  have [simp]: "r \<ge> 0" using w
-    using ball_eq_empty by fastforce
-  have f: "continuous_on (path_image (circlepath z r)) f"
-    by (rule continuous_on_subset [OF contf]) (force simp: cball_def sphere_def)
-  have int: "\<And>w. dist z w < r \<Longrightarrow>
-                 ((\<lambda>u. f u / (u - w)) has_contour_integral (\<lambda>x. 2 * of_real pi * \<i> * f x) w) (circlepath z r)"
-    by (rule Cauchy_integral_circlepath [OF contf holf]) (simp add: dist_norm norm_minus_commute)
-  show ?thes1
-    apply (simp add: power2_eq_square)
-    apply (rule Cauchy_next_derivative_circlepath [OF f _ _ w, where k=1, simplified])
-    apply (blast intro: int)
-    done
-  have "((\<lambda>x. 2 * of_real pi * \<i> * f x) has_field_derivative contour_integral (circlepath z r) (\<lambda>u. f u / (u - w)^2)) (at w)"
-    apply (simp add: power2_eq_square)
-    apply (rule Cauchy_next_derivative_circlepath [OF f _ _ w, where k=1 and g = "\<lambda>x. 2 * of_real pi * \<i> * f x", simplified])
-    apply (blast intro: int)
-    done
-  then have fder: "(f has_field_derivative contour_integral (circlepath z r) (\<lambda>u. f u / (u - w)^2) / (2 * of_real pi * \<i>)) (at w)"
-    by (rule DERIV_cdivide [where f = "\<lambda>x. 2 * of_real pi * \<i> * f x" and c = "2 * of_real pi * \<i>", simplified])
-  show ?thes2
-    by simp (rule fder)
-qed
-
-subsection\<open>Existence of all higher derivatives\<close>
-
-proposition derivative_is_holomorphic:
-  assumes "open S"
-      and fder: "\<And>z. z \<in> S \<Longrightarrow> (f has_field_derivative f' z) (at z)"
-    shows "f' holomorphic_on S"
-proof -
-  have *: "\<exists>h. (f' has_field_derivative h) (at z)" if "z \<in> S" for z
-  proof -
-    obtain r where "r > 0" and r: "cball z r \<subseteq> S"
-      using open_contains_cball \<open>z \<in> S\<close> \<open>open S\<close> by blast
-    then have holf_cball: "f holomorphic_on cball z r"
-      apply (simp add: holomorphic_on_def)
-      using field_differentiable_at_within field_differentiable_def fder by blast
-    then have "continuous_on (path_image (circlepath z r)) f"
-      using \<open>r > 0\<close> by (force elim: holomorphic_on_subset [THEN holomorphic_on_imp_continuous_on])
-    then have contfpi: "continuous_on (path_image (circlepath z r)) (\<lambda>x. 1/(2 * of_real pi*\<i>) * f x)"
-      by (auto intro: continuous_intros)+
-    have contf_cball: "continuous_on (cball z r) f" using holf_cball
-      by (simp add: holomorphic_on_imp_continuous_on holomorphic_on_subset)
-    have holf_ball: "f holomorphic_on ball z r" using holf_cball
-      using ball_subset_cball holomorphic_on_subset by blast
-    { fix w  assume w: "w \<in> ball z r"
-      have intf: "(\<lambda>u. f u / (u - w)\<^sup>2) contour_integrable_on circlepath z r"
-        by (blast intro: w Cauchy_derivative_integral_circlepath [OF contf_cball holf_ball])
-      have fder': "(f has_field_derivative 1 / (2 * of_real pi * \<i>) * contour_integral (circlepath z r) (\<lambda>u. f u / (u - w)\<^sup>2))
-                  (at w)"
-        by (blast intro: w Cauchy_derivative_integral_circlepath [OF contf_cball holf_ball])
-      have f'_eq: "f' w = contour_integral (circlepath z r) (\<lambda>u. f u / (u - w)\<^sup>2) / (2 * of_real pi * \<i>)"
-        using fder' ball_subset_cball r w by (force intro: DERIV_unique [OF fder])
-      have "((\<lambda>u. f u / (u - w)\<^sup>2 / (2 * of_real pi * \<i>)) has_contour_integral
-                contour_integral (circlepath z r) (\<lambda>u. f u / (u - w)\<^sup>2) / (2 * of_real pi * \<i>))
-                (circlepath z r)"
-        by (rule has_contour_integral_div [OF has_contour_integral_integral [OF intf]])
-      then have "((\<lambda>u. f u / (2 * of_real pi * \<i> * (u - w)\<^sup>2)) has_contour_integral
-                contour_integral (circlepath z r) (\<lambda>u. f u / (u - w)\<^sup>2) / (2 * of_real pi * \<i>))
-                (circlepath z r)"
-        by (simp add: algebra_simps)
-      then have "((\<lambda>u. f u / (2 * of_real pi * \<i> * (u - w)\<^sup>2)) has_contour_integral f' w) (circlepath z r)"
-        by (simp add: f'_eq)
-    } note * = this
-    show ?thesis
-      apply (rule exI)
-      apply (rule Cauchy_next_derivative_circlepath [OF contfpi, of 2 f', simplified])
-      apply (simp_all add: \<open>0 < r\<close> * dist_norm)
-      done
-  qed
-  show ?thesis
-    by (simp add: holomorphic_on_open [OF \<open>open S\<close>] *)
-qed
-
-lemma holomorphic_deriv [holomorphic_intros]:
-    "\<lbrakk>f holomorphic_on S; open S\<rbrakk> \<Longrightarrow> (deriv f) holomorphic_on S"
-by (metis DERIV_deriv_iff_field_differentiable at_within_open derivative_is_holomorphic holomorphic_on_def)
-
-lemma analytic_deriv [analytic_intros]: "f analytic_on S \<Longrightarrow> (deriv f) analytic_on S"
-  using analytic_on_holomorphic holomorphic_deriv by auto
-
-lemma holomorphic_higher_deriv [holomorphic_intros]: "\<lbrakk>f holomorphic_on S; open S\<rbrakk> \<Longrightarrow> (deriv ^^ n) f holomorphic_on S"
-  by (induction n) (auto simp: holomorphic_deriv)
-
-lemma analytic_higher_deriv [analytic_intros]: "f analytic_on S \<Longrightarrow> (deriv ^^ n) f analytic_on S"
-  unfolding analytic_on_def using holomorphic_higher_deriv by blast
-
-lemma has_field_derivative_higher_deriv:
-     "\<lbrakk>f holomorphic_on S; open S; x \<in> S\<rbrakk>
-      \<Longrightarrow> ((deriv ^^ n) f has_field_derivative (deriv ^^ (Suc n)) f x) (at x)"
-by (metis (no_types, hide_lams) DERIV_deriv_iff_field_differentiable at_within_open comp_apply
-         funpow.simps(2) holomorphic_higher_deriv holomorphic_on_def)
-
-lemma valid_path_compose_holomorphic:
-  assumes "valid_path g" and holo:"f holomorphic_on S" and "open S" "path_image g \<subseteq> S"
-  shows "valid_path (f \<circ> g)"
-proof (rule valid_path_compose[OF \<open>valid_path g\<close>])
-  fix x assume "x \<in> path_image g"
-  then show "f field_differentiable at x"
-    using analytic_on_imp_differentiable_at analytic_on_open assms holo by blast
-next
-  have "deriv f holomorphic_on S"
-    using holomorphic_deriv holo \<open>open S\<close> by auto
-  then show "continuous_on (path_image g) (deriv f)"
-    using assms(4) holomorphic_on_imp_continuous_on holomorphic_on_subset by auto
-qed
-
-
-subsection\<open>Morera's theorem\<close>
-
-lemma Morera_local_triangle_ball:
-  assumes "\<And>z. z \<in> S
-          \<Longrightarrow> \<exists>e a. 0 < e \<and> z \<in> ball a e \<and> continuous_on (ball a e) f \<and>
-                    (\<forall>b c. closed_segment b c \<subseteq> ball a e
-                           \<longrightarrow> contour_integral (linepath a b) f +
-                               contour_integral (linepath b c) f +
-                               contour_integral (linepath c a) f = 0)"
-  shows "f analytic_on S"
-proof -
-  { fix z  assume "z \<in> S"
-    with assms obtain e a where
-            "0 < e" and z: "z \<in> ball a e" and contf: "continuous_on (ball a e) f"
-        and 0: "\<And>b c. closed_segment b c \<subseteq> ball a e
-                      \<Longrightarrow> contour_integral (linepath a b) f +
-                          contour_integral (linepath b c) f +
-                          contour_integral (linepath c a) f = 0"
-      by blast
-    have az: "dist a z < e" using mem_ball z by blast
-    have sb_ball: "ball z (e - dist a z) \<subseteq> ball a e"
-      by (simp add: dist_commute ball_subset_ball_iff)
-    have "\<exists>e>0. f holomorphic_on ball z e"
-    proof (intro exI conjI)
-      have sub_ball: "\<And>y. dist a y < e \<Longrightarrow> closed_segment a y \<subseteq> ball a e"
-        by (meson \<open>0 < e\<close> centre_in_ball convex_ball convex_contains_segment mem_ball)
-      show "f holomorphic_on ball z (e - dist a z)"
-        apply (rule holomorphic_on_subset [OF _ sb_ball])
-        apply (rule derivative_is_holomorphic[OF open_ball])
-        apply (rule triangle_contour_integrals_starlike_primitive [OF contf _ open_ball, of a])
-           apply (simp_all add: 0 \<open>0 < e\<close> sub_ball)
-        done
-    qed (simp add: az)
-  }
-  then show ?thesis
-    by (simp add: analytic_on_def)
-qed
-
-lemma Morera_local_triangle:
-  assumes "\<And>z. z \<in> S
-          \<Longrightarrow> \<exists>t. open t \<and> z \<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)"
-  shows "f analytic_on S"
-proof -
-  { fix z  assume "z \<in> S"
-    with assms obtain t where
-            "open t" and z: "z \<in> t" and contf: "continuous_on t f"
-        and 0: "\<And>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"
-      by force
-    then obtain e where "e>0" and e: "ball z e \<subseteq> t"
-      using open_contains_ball by blast
-    have [simp]: "continuous_on (ball z e) f" using contf
-      using continuous_on_subset e by blast
-    have eq0: "\<And>b c. closed_segment b c \<subseteq> ball z e \<Longrightarrow>
-                         contour_integral (linepath z b) f +
-                         contour_integral (linepath b c) f +
-                         contour_integral (linepath c z) f = 0"
-      by (meson 0 z \<open>0 < e\<close> centre_in_ball closed_segment_subset convex_ball dual_order.trans e starlike_convex_subset)
-    have "\<exists>e a. 0 < e \<and> z \<in> ball a e \<and> continuous_on (ball a e) f \<and>
-                (\<forall>b c. closed_segment b c \<subseteq> ball a e \<longrightarrow>
-                       contour_integral (linepath a b) f + contour_integral (linepath b c) f + contour_integral (linepath c a) f = 0)"
-      using \<open>e > 0\<close> eq0 by force
-  }
-  then show ?thesis
-    by (simp add: Morera_local_triangle_ball)
-qed
-
-proposition Morera_triangle:
-    "\<lbrakk>continuous_on S f; open S;
-      \<And>a b c. convex hull {a,b,c} \<subseteq> S
-              \<longrightarrow> contour_integral (linepath a b) f +
-                  contour_integral (linepath b c) f +
-                  contour_integral (linepath c a) f = 0\<rbrakk>
-     \<Longrightarrow> f analytic_on S"
-  using Morera_local_triangle by blast
-
-subsection\<open>Combining theorems for higher derivatives including Leibniz rule\<close>
-
-lemma higher_deriv_linear [simp]:
-    "(deriv ^^ n) (\<lambda>w. c*w) = (\<lambda>z. if n = 0 then c*z else if n = 1 then c else 0)"
-  by (induction n) auto
-
-lemma higher_deriv_const [simp]: "(deriv ^^ n) (\<lambda>w. c) = (\<lambda>w. if n=0 then c else 0)"
-  by (induction n) auto
-
-lemma higher_deriv_ident [simp]:
-     "(deriv ^^ n) (\<lambda>w. w) z = (if n = 0 then z else if n = 1 then 1 else 0)"
-  apply (induction n, simp)
-  apply (metis higher_deriv_linear lambda_one)
-  done
-
-lemma higher_deriv_id [simp]:
-     "(deriv ^^ n) id z = (if n = 0 then z else if n = 1 then 1 else 0)"
-  by (simp add: id_def)
-
-lemma has_complex_derivative_funpow_1:
-     "\<lbrakk>(f has_field_derivative 1) (at z); f z = z\<rbrakk> \<Longrightarrow> (f^^n has_field_derivative 1) (at z)"
-  apply (induction n, auto)
-  apply (simp add: id_def)
-  by (metis DERIV_chain comp_funpow comp_id funpow_swap1 mult.right_neutral)
-
-lemma higher_deriv_uminus:
-  assumes "f holomorphic_on S" "open S" and z: "z \<in> S"
-    shows "(deriv ^^ n) (\<lambda>w. -(f w)) z = - ((deriv ^^ n) f z)"
-using z
-proof (induction n arbitrary: z)
-  case 0 then show ?case by simp
-next
-  case (Suc n z)
-  have *: "((deriv ^^ n) f has_field_derivative deriv ((deriv ^^ n) f) z) (at z)"
-    using Suc.prems assms has_field_derivative_higher_deriv by auto
-  have "((deriv ^^ n) (\<lambda>w. - f w) has_field_derivative - deriv ((deriv ^^ n) f) z) (at z)"
-    apply (rule has_field_derivative_transform_within_open [of "\<lambda>w. -((deriv ^^ n) f w)"])
-       apply (rule derivative_eq_intros | rule * refl assms)+
-     apply (auto simp add: Suc)
-    done
-  then show ?case
-    by (simp add: DERIV_imp_deriv)
-qed
-
-lemma higher_deriv_add:
-  fixes z::complex
-  assumes "f holomorphic_on S" "g holomorphic_on S" "open S" and z: "z \<in> S"
-    shows "(deriv ^^ n) (\<lambda>w. f w + g w) z = (deriv ^^ n) f z + (deriv ^^ n) g z"
-using z
-proof (induction n arbitrary: z)
-  case 0 then show ?case by simp
-next
-  case (Suc n z)
-  have *: "((deriv ^^ n) f has_field_derivative deriv ((deriv ^^ n) f) z) (at z)"
-          "((deriv ^^ n) g has_field_derivative deriv ((deriv ^^ n) g) z) (at z)"
-    using Suc.prems assms has_field_derivative_higher_deriv by auto
-  have "((deriv ^^ n) (\<lambda>w. f w + g w) has_field_derivative
-        deriv ((deriv ^^ n) f) z + deriv ((deriv ^^ n) g) z) (at z)"
-    apply (rule has_field_derivative_transform_within_open [of "\<lambda>w. (deriv ^^ n) f w + (deriv ^^ n) g w"])
-       apply (rule derivative_eq_intros | rule * refl assms)+
-     apply (auto simp add: Suc)
-    done
-  then show ?case
-    by (simp add: DERIV_imp_deriv)
-qed
-
-lemma higher_deriv_diff:
-  fixes z::complex
-  assumes "f holomorphic_on S" "g holomorphic_on S" "open S" and z: "z \<in> S"
-    shows "(deriv ^^ n) (\<lambda>w. f w - g w) z = (deriv ^^ n) f z - (deriv ^^ n) g z"
-  apply (simp only: Groups.group_add_class.diff_conv_add_uminus higher_deriv_add)
-  apply (subst higher_deriv_add)
-  using assms holomorphic_on_minus apply (auto simp: higher_deriv_uminus)
-  done
-
-lemma bb: "Suc n choose k = (n choose k) + (if k = 0 then 0 else (n choose (k - 1)))"
-  by (cases k) simp_all
-
-lemma higher_deriv_mult:
-  fixes z::complex
-  assumes "f holomorphic_on S" "g holomorphic_on S" "open S" and z: "z \<in> S"
-    shows "(deriv ^^ n) (\<lambda>w. f w * g w) z =
-           (\<Sum>i = 0..n. of_nat (n choose i) * (deriv ^^ i) f z * (deriv ^^ (n - i)) g z)"
-using z
-proof (induction n arbitrary: z)
-  case 0 then show ?case by simp
-next
-  case (Suc n z)
-  have *: "\<And>n. ((deriv ^^ n) f has_field_derivative deriv ((deriv ^^ n) f) z) (at z)"
-          "\<And>n. ((deriv ^^ n) g has_field_derivative deriv ((deriv ^^ n) g) z) (at z)"
-    using Suc.prems assms has_field_derivative_higher_deriv by auto
-  have sumeq: "(\<Sum>i = 0..n.
-               of_nat (n choose i) * (deriv ((deriv ^^ i) f) z * (deriv ^^ (n - i)) g z + deriv ((deriv ^^ (n - i)) g) z * (deriv ^^ i) f z)) =
-            g z * deriv ((deriv ^^ n) f) z + (\<Sum>i = 0..n. (deriv ^^ i) f z * (of_nat (Suc n choose i) * (deriv ^^ (Suc n - i)) g z))"
-    apply (simp add: bb algebra_simps sum.distrib)
-    apply (subst (4) sum_Suc_reindex)
-    apply (auto simp: algebra_simps Suc_diff_le intro: sum.cong)
-    done
-  have "((deriv ^^ n) (\<lambda>w. f w * g w) has_field_derivative
-         (\<Sum>i = 0..Suc n. (Suc n choose i) * (deriv ^^ i) f z * (deriv ^^ (Suc n - i)) g z))
-        (at z)"
-    apply (rule has_field_derivative_transform_within_open
-        [of "\<lambda>w. (\<Sum>i = 0..n. of_nat (n choose i) * (deriv ^^ i) f w * (deriv ^^ (n - i)) g w)"])
-       apply (simp add: algebra_simps)
-       apply (rule DERIV_cong [OF DERIV_sum])
-        apply (rule DERIV_cmult)
-        apply (auto intro: DERIV_mult * sumeq \<open>open S\<close> Suc.prems Suc.IH [symmetric])
-    done
-  then show ?case
-    unfolding funpow.simps o_apply
-    by (simp add: DERIV_imp_deriv)
-qed
-
-lemma higher_deriv_transform_within_open:
-  fixes z::complex
-  assumes "f holomorphic_on S" "g holomorphic_on S" "open S" and z: "z \<in> S"
-      and fg: "\<And>w. w \<in> S \<Longrightarrow> f w = g w"
-    shows "(deriv ^^ i) f z = (deriv ^^ i) g z"
-using z
-by (induction i arbitrary: z)
-   (auto simp: fg intro: complex_derivative_transform_within_open holomorphic_higher_deriv assms)
-
-lemma higher_deriv_compose_linear:
-  fixes z::complex
-  assumes f: "f holomorphic_on T" and S: "open S" and T: "open T" and z: "z \<in> S"
-      and fg: "\<And>w. w \<in> S \<Longrightarrow> u * w \<in> T"
-    shows "(deriv ^^ n) (\<lambda>w. f (u * w)) z = u^n * (deriv ^^ n) f (u * z)"
-using z
-proof (induction n arbitrary: z)
-  case 0 then show ?case by simp
-next
-  case (Suc n z)
-  have holo0: "f holomorphic_on (*) u ` S"
-    by (meson fg f holomorphic_on_subset image_subset_iff)
-  have holo2: "(deriv ^^ n) f holomorphic_on (*) u ` S"
-    by (meson f fg holomorphic_higher_deriv holomorphic_on_subset image_subset_iff T)
-  have holo3: "(\<lambda>z. u ^ n * (deriv ^^ n) f (u * z)) holomorphic_on S"
-    by (intro holo2 holomorphic_on_compose [where g="(deriv ^^ n) f", unfolded o_def] holomorphic_intros)
-  have holo1: "(\<lambda>w. f (u * w)) holomorphic_on S"
-    apply (rule holomorphic_on_compose [where g=f, unfolded o_def])
-    apply (rule holo0 holomorphic_intros)+
-    done
-  have "deriv ((deriv ^^ n) (\<lambda>w. f (u * w))) z = deriv (\<lambda>z. u^n * (deriv ^^ n) f (u*z)) z"
-    apply (rule complex_derivative_transform_within_open [OF _ holo3 S Suc.prems])
-    apply (rule holomorphic_higher_deriv [OF holo1 S])
-    apply (simp add: Suc.IH)
-    done
-  also have "\<dots> = u^n * deriv (\<lambda>z. (deriv ^^ n) f (u * z)) z"
-    apply (rule deriv_cmult)
-    apply (rule analytic_on_imp_differentiable_at [OF _ Suc.prems])
-    apply (rule analytic_on_compose_gen [where g="(deriv ^^ n) f" and T=T, unfolded o_def])
-      apply (simp)
-     apply (simp add: analytic_on_open f holomorphic_higher_deriv T)
-    apply (blast intro: fg)
-    done
-  also have "\<dots> = u * u ^ n * deriv ((deriv ^^ n) f) (u * z)"
-      apply (subst deriv_chain [where g = "(deriv ^^ n) f" and f = "(*) u", unfolded o_def])
-      apply (rule derivative_intros)
-      using Suc.prems field_differentiable_def f fg has_field_derivative_higher_deriv T apply blast
-      apply (simp)
-      done
-  finally show ?case
-    by simp
-qed
-
-lemma higher_deriv_add_at:
-  assumes "f analytic_on {z}" "g analytic_on {z}"
-    shows "(deriv ^^ n) (\<lambda>w. f w + g w) z = (deriv ^^ n) f z + (deriv ^^ n) g z"
-proof -
-  have "f analytic_on {z} \<and> g analytic_on {z}"
-    using assms by blast
-  with higher_deriv_add show ?thesis
-    by (auto simp: analytic_at_two)
-qed
-
-lemma higher_deriv_diff_at:
-  assumes "f analytic_on {z}" "g analytic_on {z}"
-    shows "(deriv ^^ n) (\<lambda>w. f w - g w) z = (deriv ^^ n) f z - (deriv ^^ n) g z"
-proof -
-  have "f analytic_on {z} \<and> g analytic_on {z}"
-    using assms by blast
-  with higher_deriv_diff show ?thesis
-    by (auto simp: analytic_at_two)
-qed
-
-lemma higher_deriv_uminus_at:
-   "f analytic_on {z}  \<Longrightarrow> (deriv ^^ n) (\<lambda>w. -(f w)) z = - ((deriv ^^ n) f z)"
-  using higher_deriv_uminus
-    by (auto simp: analytic_at)
-
-lemma higher_deriv_mult_at:
-  assumes "f analytic_on {z}" "g analytic_on {z}"
-    shows "(deriv ^^ n) (\<lambda>w. f w * g w) z =
-           (\<Sum>i = 0..n. of_nat (n choose i) * (deriv ^^ i) f z * (deriv ^^ (n - i)) g z)"
-proof -
-  have "f analytic_on {z} \<and> g analytic_on {z}"
-    using assms by blast
-  with higher_deriv_mult show ?thesis
-    by (auto simp: analytic_at_two)
-qed
-
-
-text\<open> Nonexistence of isolated singularities and a stronger integral formula.\<close>
-
-proposition no_isolated_singularity:
-  fixes z::complex
-  assumes f: "continuous_on S f" and holf: "f holomorphic_on (S - K)" and S: "open S" and K: "finite K"
-    shows "f holomorphic_on S"
-proof -
-  { fix z
-    assume "z \<in> S" and cdf: "\<And>x. x \<in> S - K \<Longrightarrow> f field_differentiable at x"
-    have "f field_differentiable at z"
-    proof (cases "z \<in> K")
-      case False then show ?thesis by (blast intro: cdf \<open>z \<in> S\<close>)
-    next
-      case True
-      with finite_set_avoid [OF K, of z]
-      obtain d where "d>0" and d: "\<And>x. \<lbrakk>x\<in>K; x \<noteq> z\<rbrakk> \<Longrightarrow> d \<le> dist z x"
-        by blast
-      obtain e where "e>0" and e: "ball z e \<subseteq> S"
-        using  S \<open>z \<in> S\<close> by (force simp: open_contains_ball)
-      have fde: "continuous_on (ball z (min d e)) f"
-        by (metis Int_iff ball_min_Int continuous_on_subset e f subsetI)
-      have cont: "{a,b,c} \<subseteq> ball z (min d e) \<Longrightarrow> continuous_on (convex hull {a, b, c}) f" for a b c
-        by (simp add: hull_minimal continuous_on_subset [OF fde])
-      have fd: "\<lbrakk>{a,b,c} \<subseteq> ball z (min d e); x \<in> interior (convex hull {a, b, c}) - K\<rbrakk>
-            \<Longrightarrow> f field_differentiable at x" for a b c x
-        by (metis cdf Diff_iff Int_iff ball_min_Int subsetD convex_ball e interior_mono interior_subset subset_hull)
-      obtain g where "\<And>w. w \<in> ball z (min d e) \<Longrightarrow> (g has_field_derivative f w) (at w within ball z (min d e))"
-        apply (rule contour_integral_convex_primitive
-                     [OF convex_ball fde Cauchy_theorem_triangle_cofinite [OF _ K]])
-        using cont fd by auto
-      then have "f holomorphic_on ball z (min d e)"
-        by (metis open_ball at_within_open derivative_is_holomorphic)
-      then show ?thesis
-        unfolding holomorphic_on_def
-        by (metis open_ball \<open>0 < d\<close> \<open>0 < e\<close> at_within_open centre_in_ball min_less_iff_conj)
-    qed
-  }
-  with holf S K show ?thesis
-    by (simp add: holomorphic_on_open open_Diff finite_imp_closed field_differentiable_def [symmetric])
-qed
-
-lemma no_isolated_singularity':
-  fixes z::complex
-  assumes f: "\<And>z. z \<in> K \<Longrightarrow> (f \<longlongrightarrow> f z) (at z within S)"
-      and holf: "f holomorphic_on (S - K)" and S: "open S" and K: "finite K"
-    shows "f holomorphic_on S"
-proof (rule no_isolated_singularity[OF _ assms(2-)])
-  show "continuous_on S f" unfolding continuous_on_def
-  proof
-    fix z assume z: "z \<in> S"
-    show "(f \<longlongrightarrow> f z) (at z within S)"
-    proof (cases "z \<in> K")
-      case False
-      from holf have "continuous_on (S - K) f"
-        by (rule holomorphic_on_imp_continuous_on)
-      with z False have "(f \<longlongrightarrow> f z) (at z within (S - K))"
-        by (simp add: continuous_on_def)
-      also from z K S False have "at z within (S - K) = at z within S"
-        by (subst (1 2) at_within_open) (auto intro: finite_imp_closed)
-      finally show "(f \<longlongrightarrow> f z) (at z within S)" .
-    qed (insert assms z, simp_all)
-  qed
-qed
-
-proposition Cauchy_integral_formula_convex:
-  assumes S: "convex S" and K: "finite K" and contf: "continuous_on S f"
-    and fcd: "(\<And>x. x \<in> interior S - K \<Longrightarrow> f field_differentiable at x)"
-    and z: "z \<in> interior S" and vpg: "valid_path \<gamma>"
-    and pasz: "path_image \<gamma> \<subseteq> S - {z}" and loop: "pathfinish \<gamma> = pathstart \<gamma>"
-  shows "((\<lambda>w. f w / (w - z)) has_contour_integral (2*pi * \<i> * winding_number \<gamma> z * f z)) \<gamma>"
-proof -
-  have *: "\<And>x. x \<in> interior S \<Longrightarrow> f field_differentiable at x"
-    unfolding holomorphic_on_open [symmetric] field_differentiable_def
-    using no_isolated_singularity [where S = "interior S"]
-    by (meson K contf continuous_at_imp_continuous_on continuous_on_interior fcd
-          field_differentiable_at_within field_differentiable_def holomorphic_onI
-          holomorphic_on_imp_differentiable_at open_interior)
-  show ?thesis
-    by (rule Cauchy_integral_formula_weak [OF S finite.emptyI contf]) (use * assms in auto)
-qed
-
-text\<open> Formula for higher derivatives.\<close>
-
-lemma Cauchy_has_contour_integral_higher_derivative_circlepath:
-  assumes contf: "continuous_on (cball z r) f"
-      and holf: "f holomorphic_on ball z r"
-      and w: "w \<in> ball z r"
-    shows "((\<lambda>u. f u / (u - w) ^ (Suc k)) has_contour_integral ((2 * pi * \<i>) / (fact k) * (deriv ^^ k) f w))
-           (circlepath z r)"
-using w
-proof (induction k arbitrary: w)
-  case 0 then show ?case
-    using assms by (auto simp: Cauchy_integral_circlepath dist_commute dist_norm)
-next
-  case (Suc k)
-  have [simp]: "r > 0" using w
-    using ball_eq_empty by fastforce
-  have f: "continuous_on (path_image (circlepath z r)) f"
-    by (rule continuous_on_subset [OF contf]) (force simp: cball_def sphere_def less_imp_le)
-  obtain X where X: "((\<lambda>u. f u / (u - w) ^ Suc (Suc k)) has_contour_integral X) (circlepath z r)"
-    using Cauchy_next_derivative_circlepath(1) [OF f Suc.IH _ Suc.prems]
-    by (auto simp: contour_integrable_on_def)
-  then have con: "contour_integral (circlepath z r) ((\<lambda>u. f u / (u - w) ^ Suc (Suc k))) = X"
-    by (rule contour_integral_unique)
-  have "\<And>n. ((deriv ^^ n) f has_field_derivative deriv ((deriv ^^ n) f) w) (at w)"
-    using Suc.prems assms has_field_derivative_higher_deriv by auto
-  then have dnf_diff: "\<And>n. (deriv ^^ n) f field_differentiable (at w)"
-    by (force simp: field_differentiable_def)
-  have "deriv (\<lambda>w. complex_of_real (2 * pi) * \<i> / (fact k) * (deriv ^^ k) f w) w =
-          of_nat (Suc k) * contour_integral (circlepath z r) (\<lambda>u. f u / (u - w) ^ Suc (Suc k))"
-    by (force intro!: DERIV_imp_deriv Cauchy_next_derivative_circlepath [OF f Suc.IH _ Suc.prems])
-  also have "\<dots> = of_nat (Suc k) * X"
-    by (simp only: con)
-  finally have "deriv (\<lambda>w. ((2 * pi) * \<i> / (fact k)) * (deriv ^^ k) f w) w = of_nat (Suc k) * X" .
-  then have "((2 * pi) * \<i> / (fact k)) * deriv (\<lambda>w. (deriv ^^ k) f w) w = of_nat (Suc k) * X"
-    by (metis deriv_cmult dnf_diff)
-  then have "deriv (\<lambda>w. (deriv ^^ k) f w) w = of_nat (Suc k) * X / ((2 * pi) * \<i> / (fact k))"
-    by (simp add: field_simps)
-  then show ?case
-  using of_nat_eq_0_iff X by fastforce
-qed
-
-lemma Cauchy_higher_derivative_integral_circlepath:
-  assumes contf: "continuous_on (cball z r) f"
-      and holf: "f holomorphic_on ball z r"
-      and w: "w \<in> ball z r"
-    shows "(\<lambda>u. f u / (u - w)^(Suc k)) contour_integrable_on (circlepath z r)"
-           (is "?thes1")
-      and "(deriv ^^ k) f w = (fact k) / (2 * pi * \<i>) * contour_integral(circlepath z r) (\<lambda>u. f u/(u - w)^(Suc k))"
-           (is "?thes2")
-proof -
-  have *: "((\<lambda>u. f u / (u - w) ^ Suc k) has_contour_integral (2 * pi) * \<i> / (fact k) * (deriv ^^ k) f w)
-           (circlepath z r)"
-    using Cauchy_has_contour_integral_higher_derivative_circlepath [OF assms]
-    by simp
-  show ?thes1 using *
-    using contour_integrable_on_def by blast
-  show ?thes2
-    unfolding contour_integral_unique [OF *] by (simp add: field_split_simps)
-qed
-
-corollary Cauchy_contour_integral_circlepath:
-  assumes "continuous_on (cball z r) f" "f holomorphic_on ball z r" "w \<in> ball z r"
-    shows "contour_integral(circlepath z r) (\<lambda>u. f u/(u - w)^(Suc k)) = (2 * pi * \<i>) * (deriv ^^ k) f w / (fact k)"
-by (simp add: Cauchy_higher_derivative_integral_circlepath [OF assms])
-
-lemma Cauchy_contour_integral_circlepath_2:
-  assumes "continuous_on (cball z r) f" "f holomorphic_on ball z r" "w \<in> ball z r"
-    shows "contour_integral(circlepath z r) (\<lambda>u. f u/(u - w)^2) = (2 * pi * \<i>) * deriv f w"
-  using Cauchy_contour_integral_circlepath [OF assms, of 1]
-  by (simp add: power2_eq_square)
-
-
-subsection\<open>A holomorphic function is analytic, i.e. has local power series\<close>
-
-theorem holomorphic_power_series:
-  assumes holf: "f holomorphic_on ball z r"
-      and w: "w \<in> ball z r"
-    shows "((\<lambda>n. (deriv ^^ n) f z / (fact n) * (w - z)^n) sums f w)"
-proof -
-  \<comment> \<open>Replacing \<^term>\<open>r\<close> and the original (weak) premises with stronger ones\<close>
-  obtain r where "r > 0" and holfc: "f holomorphic_on cball z r" and w: "w \<in> ball z r"
-  proof
-    have "cball z ((r + dist w z) / 2) \<subseteq> ball z r"
-      using w by (simp add: dist_commute field_sum_of_halves subset_eq)
-    then show "f holomorphic_on cball z ((r + dist w z) / 2)"
-      by (rule holomorphic_on_subset [OF holf])
-    have "r > 0"
-      using w by clarsimp (metis dist_norm le_less_trans norm_ge_zero)
-    then show "0 < (r + dist w z) / 2"
-      by simp (use zero_le_dist [of w z] in linarith)
-  qed (use w in \<open>auto simp: dist_commute\<close>)
-  then have holf: "f holomorphic_on ball z r"
-    using ball_subset_cball holomorphic_on_subset by blast
-  have contf: "continuous_on (cball z r) f"
-    by (simp add: holfc holomorphic_on_imp_continuous_on)
-  have cint: "\<And>k. (\<lambda>u. f u / (u - z) ^ Suc k) contour_integrable_on circlepath z r"
-    by (rule Cauchy_higher_derivative_integral_circlepath [OF contf holf]) (simp add: \<open>0 < r\<close>)
-  obtain B where "0 < B" and B: "\<And>u. u \<in> cball z r \<Longrightarrow> norm(f u) \<le> B"
-    by (metis (no_types) bounded_pos compact_cball compact_continuous_image compact_imp_bounded contf image_eqI)
-  obtain k where k: "0 < k" "k \<le> r" and wz_eq: "norm(w - z) = r - k"
-             and kle: "\<And>u. norm(u - z) = r \<Longrightarrow> k \<le> norm(u - w)"
-  proof
-    show "\<And>u. cmod (u - z) = r \<Longrightarrow> r - dist z w \<le> cmod (u - w)"
-      by (metis add_diff_eq diff_add_cancel dist_norm norm_diff_ineq)
-  qed (use w in \<open>auto simp: dist_norm norm_minus_commute\<close>)
-  have ul: "uniform_limit (sphere z r) (\<lambda>n x. (\<Sum>k<n. (w - z) ^ k * (f x / (x - z) ^ Suc k))) (\<lambda>x. f x / (x - w)) sequentially"
-    unfolding uniform_limit_iff dist_norm
-  proof clarify
-    fix e::real
-    assume "0 < e"
-    have rr: "0 \<le> (r - k) / r" "(r - k) / r < 1" using  k by auto
-    obtain n where n: "((r - k) / r) ^ n < e / B * k"
-      using real_arch_pow_inv [of "e/B*k" "(r - k)/r"] \<open>0 < e\<close> \<open>0 < B\<close> k by force
-    have "norm ((\<Sum>k<N. (w - z) ^ k * f u / (u - z) ^ Suc k) - f u / (u - w)) < e"
-         if "n \<le> N" and r: "r = dist z u"  for N u
-    proof -
-      have N: "((r - k) / r) ^ N < e / B * k"
-        apply (rule le_less_trans [OF power_decreasing n])
-        using  \<open>n \<le> N\<close> k by auto
-      have u [simp]: "(u \<noteq> z) \<and> (u \<noteq> w)"
-        using \<open>0 < r\<close> r w by auto
-      have wzu_not1: "(w - z) / (u - z) \<noteq> 1"
-        by (metis (no_types) dist_norm divide_eq_1_iff less_irrefl mem_ball norm_minus_commute r w)
-      have "norm ((\<Sum>k<N. (w - z) ^ k * f u / (u - z) ^ Suc k) * (u - w) - f u)
-            = norm ((\<Sum>k<N. (((w - z) / (u - z)) ^ k)) * f u * (u - w) / (u - z) - f u)"
-        unfolding sum_distrib_right sum_divide_distrib power_divide by (simp add: algebra_simps)
-      also have "\<dots> = norm ((((w - z) / (u - z)) ^ N - 1) * (u - w) / (((w - z) / (u - z) - 1) * (u - z)) - 1) * norm (f u)"
-        using \<open>0 < B\<close>
-        apply (auto simp: geometric_sum [OF wzu_not1])
-        apply (simp add: field_simps norm_mult [symmetric])
-        done
-      also have "\<dots> = norm ((u-z) ^ N * (w - u) - ((w - z) ^ N - (u-z) ^ N) * (u-w)) / (r ^ N * norm (u-w)) * norm (f u)"
-        using \<open>0 < r\<close> r by (simp add: divide_simps norm_mult norm_divide norm_power dist_norm norm_minus_commute)
-      also have "\<dots> = norm ((w - z) ^ N * (w - u)) / (r ^ N * norm (u - w)) * norm (f u)"
-        by (simp add: algebra_simps)
-      also have "\<dots> = norm (w - z) ^ N * norm (f u) / r ^ N"
-        by (simp add: norm_mult norm_power norm_minus_commute)
-      also have "\<dots> \<le> (((r - k)/r)^N) * B"
-        using \<open>0 < r\<close> w k
-        apply (simp add: divide_simps)
-        apply (rule mult_mono [OF power_mono])
-        apply (auto simp: norm_divide wz_eq norm_power dist_norm norm_minus_commute B r)
-        done
-      also have "\<dots> < e * k"
-        using \<open>0 < B\<close> N by (simp add: divide_simps)
-      also have "\<dots> \<le> e * norm (u - w)"
-        using r kle \<open>0 < e\<close> by (simp add: dist_commute dist_norm)
-      finally show ?thesis
-        by (simp add: field_split_simps norm_divide del: power_Suc)
-    qed
-    with \<open>0 < r\<close> show "\<forall>\<^sub>F n in sequentially. \<forall>x\<in>sphere z r.
-                norm ((\<Sum>k<n. (w - z) ^ k * (f x / (x - z) ^ Suc k)) - f x / (x - w)) < e"
-      by (auto simp: mult_ac less_imp_le eventually_sequentially Ball_def)
-  qed
-  have eq: "\<forall>\<^sub>F x in sequentially.
-             contour_integral (circlepath z r) (\<lambda>u. \<Sum>k<x. (w - z) ^ k * (f u / (u - z) ^ Suc k)) =
-             (\<Sum>k<x. contour_integral (circlepath z r) (\<lambda>u. f u / (u - z) ^ Suc k) * (w - z) ^ k)"
-    apply (rule eventuallyI)
-    apply (subst contour_integral_sum, simp)
-    using contour_integrable_lmul [OF cint, of "(w - z) ^ a" for a] apply (simp add: field_simps)
-    apply (simp only: contour_integral_lmul cint algebra_simps)
-    done
-  have cic: "\<And>u. (\<lambda>y. \<Sum>k<u. (w - z) ^ k * (f y / (y - z) ^ Suc k)) contour_integrable_on circlepath z r"
-    apply (intro contour_integrable_sum contour_integrable_lmul, simp)
-    using \<open>0 < r\<close> by (force intro!: Cauchy_higher_derivative_integral_circlepath [OF contf holf])
-  have "(\<lambda>k. contour_integral (circlepath z r) (\<lambda>u. f u/(u - z)^(Suc k)) * (w - z)^k)
-        sums contour_integral (circlepath z r) (\<lambda>u. f u/(u - w))"
-    unfolding sums_def
-    apply (intro Lim_transform_eventually [OF _ eq] contour_integral_uniform_limit_circlepath [OF eventuallyI ul] cic)
-    using \<open>0 < r\<close> apply auto
-    done
-  then have "(\<lambda>k. contour_integral (circlepath z r) (\<lambda>u. f u/(u - z)^(Suc k)) * (w - z)^k)
-             sums (2 * of_real pi * \<i> * f w)"
-    using w by (auto simp: dist_commute dist_norm contour_integral_unique [OF Cauchy_integral_circlepath_simple [OF holfc]])
-  then have "(\<lambda>k. contour_integral (circlepath z r) (\<lambda>u. f u / (u - z) ^ Suc k) * (w - z)^k / (\<i> * (of_real pi * 2)))
-            sums ((2 * of_real pi * \<i> * f w) / (\<i> * (complex_of_real pi * 2)))"
-    by (rule sums_divide)
-  then have "(\<lambda>n. (w - z) ^ n * contour_integral (circlepath z r) (\<lambda>u. f u / (u - z) ^ Suc n) / (\<i> * (of_real pi * 2)))
-            sums f w"
-    by (simp add: field_simps)
-  then show ?thesis
-    by (simp add: field_simps \<open>0 < r\<close> Cauchy_higher_derivative_integral_circlepath [OF contf holf])
-qed
-
-
-subsection\<open>The Liouville theorem and the Fundamental Theorem of Algebra\<close>
-
-text\<open> These weak Liouville versions don't even need the derivative formula.\<close>
-
-lemma Liouville_weak_0:
-  assumes holf: "f holomorphic_on UNIV" and inf: "(f \<longlongrightarrow> 0) at_infinity"
-    shows "f z = 0"
-proof (rule ccontr)
-  assume fz: "f z \<noteq> 0"
-  with inf [unfolded Lim_at_infinity, rule_format, of "norm(f z)/2"]
-  obtain B where B: "\<And>x. B \<le> cmod x \<Longrightarrow> norm (f x) * 2 < cmod (f z)"
-    by (auto simp: dist_norm)
-  define R where "R = 1 + \<bar>B\<bar> + norm z"
-  have "R > 0" unfolding R_def
-  proof -
-    have "0 \<le> cmod z + \<bar>B\<bar>"
-      by (metis (full_types) add_nonneg_nonneg norm_ge_zero real_norm_def)
-    then show "0 < 1 + \<bar>B\<bar> + cmod z"
-      by linarith
-  qed
-  have *: "((\<lambda>u. f u / (u - z)) has_contour_integral 2 * complex_of_real pi * \<i> * f z) (circlepath z R)"
-    apply (rule Cauchy_integral_circlepath)
-    using \<open>R > 0\<close> apply (auto intro: holomorphic_on_subset [OF holf] holomorphic_on_imp_continuous_on)+
-    done
-  have "cmod (x - z) = R \<Longrightarrow> cmod (f x) * 2 < cmod (f z)" for x
-    unfolding R_def
-    by (rule B) (use norm_triangle_ineq4 [of x z] in auto)
-  with \<open>R > 0\<close> fz show False
-    using has_contour_integral_bound_circlepath [OF *, of "norm(f z)/2/R"]
-    by (auto simp: less_imp_le norm_mult norm_divide field_split_simps)
-qed
-
-proposition Liouville_weak:
-  assumes "f holomorphic_on UNIV" and "(f \<longlongrightarrow> l) at_infinity"
-    shows "f z = l"
-  using Liouville_weak_0 [of "\<lambda>z. f z - l"]
-  by (simp add: assms holomorphic_on_diff LIM_zero)
-
-proposition Liouville_weak_inverse:
-  assumes "f holomorphic_on UNIV" and unbounded: "\<And>B. eventually (\<lambda>x. norm (f x) \<ge> B) at_infinity"
-    obtains z where "f z = 0"
-proof -
-  { assume f: "\<And>z. f z \<noteq> 0"
-    have 1: "(\<lambda>x. 1 / f x) holomorphic_on UNIV"
-      by (simp add: holomorphic_on_divide assms f)
-    have 2: "((\<lambda>x. 1 / f x) \<longlongrightarrow> 0) at_infinity"
-      apply (rule tendstoI [OF eventually_mono])
-      apply (rule_tac B="2/e" in unbounded)
-      apply (simp add: dist_norm norm_divide field_split_simps)
-      done
-    have False
-      using Liouville_weak_0 [OF 1 2] f by simp
-  }
-  then show ?thesis
-    using that by blast
-qed
-
-text\<open> In particular we get the Fundamental Theorem of Algebra.\<close>
-
-theorem fundamental_theorem_of_algebra:
-    fixes a :: "nat \<Rightarrow> complex"
-  assumes "a 0 = 0 \<or> (\<exists>i \<in> {1..n}. a i \<noteq> 0)"
-  obtains z where "(\<Sum>i\<le>n. a i * z^i) = 0"
-using assms
-proof (elim disjE bexE)
-  assume "a 0 = 0" then show ?thesis
-    by (auto simp: that [of 0])
-next
-  fix i
-  assume i: "i \<in> {1..n}" and nz: "a i \<noteq> 0"
-  have 1: "(\<lambda>z. \<Sum>i\<le>n. a i * z^i) holomorphic_on UNIV"
-    by (rule holomorphic_intros)+
-  show thesis
-  proof (rule Liouville_weak_inverse [OF 1])
-    show "\<forall>\<^sub>F x in at_infinity. B \<le> cmod (\<Sum>i\<le>n. a i * x ^ i)" for B
-      using i nz by (intro polyfun_extremal exI[of _ i]) auto
-  qed (use that in auto)
-qed
-
-subsection\<open>Weierstrass convergence theorem\<close>
-
-lemma holomorphic_uniform_limit:
-  assumes cont: "eventually (\<lambda>n. continuous_on (cball z r) (f n) \<and> (f n) holomorphic_on ball z r) F"
-      and ulim: "uniform_limit (cball z r) f g F"
-      and F:  "\<not> trivial_limit F"
-  obtains "continuous_on (cball z r) g" "g holomorphic_on ball z r"
-proof (cases r "0::real" rule: linorder_cases)
-  case less then show ?thesis by (force simp: ball_empty less_imp_le continuous_on_def holomorphic_on_def intro: that)
-next
-  case equal then show ?thesis
-    by (force simp: holomorphic_on_def intro: that)
-next
-  case greater
-  have contg: "continuous_on (cball z r) g"
-    using cont uniform_limit_theorem [OF eventually_mono ulim F]  by blast
-  have "path_image (circlepath z r) \<subseteq> cball z r"
-    using \<open>0 < r\<close> by auto
-  then have 1: "continuous_on (path_image (circlepath z r)) (\<lambda>x. 1 / (2 * complex_of_real pi * \<i>) * g x)"
-    by (intro continuous_intros continuous_on_subset [OF contg])
-  have 2: "((\<lambda>u. 1 / (2 * of_real pi * \<i>) * g u / (u - w) ^ 1) has_contour_integral g w) (circlepath z r)"
-       if w: "w \<in> ball z r" for w
-  proof -
-    define d where "d = (r - norm(w - z))"
-    have "0 < d"  "d \<le> r" using w by (auto simp: norm_minus_commute d_def dist_norm)
-    have dle: "\<And>u. cmod (z - u) = r \<Longrightarrow> d \<le> cmod (u - w)"
-      unfolding d_def by (metis add_diff_eq diff_add_cancel norm_diff_ineq norm_minus_commute)
-    have ev_int: "\<forall>\<^sub>F n in F. (\<lambda>u. f n u / (u - w)) contour_integrable_on circlepath z r"
-      apply (rule eventually_mono [OF cont])
-      using w
-      apply (auto intro: Cauchy_higher_derivative_integral_circlepath [where k=0, simplified])
-      done
-    have ul_less: "uniform_limit (sphere z r) (\<lambda>n x. f n x / (x - w)) (\<lambda>x. g x / (x - w)) F"
-      using greater \<open>0 < d\<close>
-      apply (clarsimp simp add: uniform_limit_iff dist_norm norm_divide diff_divide_distrib [symmetric] divide_simps)
-      apply (rule_tac e1="e * d" in eventually_mono [OF uniform_limitD [OF ulim]])
-       apply (force simp: dist_norm intro: dle mult_left_mono less_le_trans)+
-      done
-    have g_cint: "(\<lambda>u. g u/(u - w)) contour_integrable_on circlepath z r"
-      by (rule contour_integral_uniform_limit_circlepath [OF ev_int ul_less F \<open>0 < r\<close>])
-    have cif_tends_cig: "((\<lambda>n. contour_integral(circlepath z r) (\<lambda>u. f n u / (u - w))) \<longlongrightarrow> contour_integral(circlepath z r) (\<lambda>u. g u/(u - w))) F"
-      by (rule contour_integral_uniform_limit_circlepath [OF ev_int ul_less F \<open>0 < r\<close>])
-    have f_tends_cig: "((\<lambda>n. 2 * of_real pi * \<i> * f n w) \<longlongrightarrow> contour_integral (circlepath z r) (\<lambda>u. g u / (u - w))) F"
-    proof (rule Lim_transform_eventually)
-      show "\<forall>\<^sub>F x in F. contour_integral (circlepath z r) (\<lambda>u. f x u / (u - w))
-                     = 2 * of_real pi * \<i> * f x w"
-        apply (rule eventually_mono [OF cont contour_integral_unique [OF Cauchy_integral_circlepath]])
-        using w\<open>0 < d\<close> d_def by auto
-    qed (auto simp: cif_tends_cig)
-    have "\<And>e. 0 < e \<Longrightarrow> \<forall>\<^sub>F n in F. dist (f n w) (g w) < e"
-      by (rule eventually_mono [OF uniform_limitD [OF ulim]]) (use w in auto)
-    then have "((\<lambda>n. 2 * of_real pi * \<i> * f n w) \<longlongrightarrow> 2 * of_real pi * \<i> * g w) F"
-      by (rule tendsto_mult_left [OF tendstoI])
-    then have "((\<lambda>u. g u / (u - w)) has_contour_integral 2 * of_real pi * \<i> * g w) (circlepath z r)"
-      using has_contour_integral_integral [OF g_cint] tendsto_unique [OF F f_tends_cig] w
-      by fastforce
-    then have "((\<lambda>u. g u / (2 * of_real pi * \<i> * (u - w))) has_contour_integral g w) (circlepath z r)"
-      using has_contour_integral_div [where c = "2 * of_real pi * \<i>"]
-      by (force simp: field_simps)
-    then show ?thesis
-      by (simp add: dist_norm)
-  qed
-  show ?thesis
-    using Cauchy_next_derivative_circlepath(2) [OF 1 2, simplified]
-    by (fastforce simp add: holomorphic_on_open contg intro: that)
-qed
-
-
-text\<open> Version showing that the limit is the limit of the derivatives.\<close>
-
-proposition has_complex_derivative_uniform_limit:
-  fixes z::complex
-  assumes cont: "eventually (\<lambda>n. continuous_on (cball z r) (f n) \<and>
-                               (\<forall>w \<in> ball z r. ((f n) has_field_derivative (f' n w)) (at w))) F"
-      and ulim: "uniform_limit (cball z r) f g F"
-      and F:  "\<not> trivial_limit F" and "0 < r"
-  obtains g' where
-      "continuous_on (cball z r) g"
-      "\<And>w. w \<in> ball z r \<Longrightarrow> (g has_field_derivative (g' w)) (at w) \<and> ((\<lambda>n. f' n w) \<longlongrightarrow> g' w) F"
-proof -
-  let ?conint = "contour_integral (circlepath z r)"
-  have g: "continuous_on (cball z r) g" "g holomorphic_on ball z r"
-    by (rule holomorphic_uniform_limit [OF eventually_mono [OF cont] ulim F];
-             auto simp: holomorphic_on_open field_differentiable_def)+
-  then obtain g' where g': "\<And>x. x \<in> ball z r \<Longrightarrow> (g has_field_derivative g' x) (at x)"
-    using DERIV_deriv_iff_has_field_derivative
-    by (fastforce simp add: holomorphic_on_open)
-  then have derg: "\<And>x. x \<in> ball z r \<Longrightarrow> deriv g x = g' x"
-    by (simp add: DERIV_imp_deriv)
-  have tends_f'n_g': "((\<lambda>n. f' n w) \<longlongrightarrow> g' w) F" if w: "w \<in> ball z r" for w
-  proof -
-    have eq_f': "?conint (\<lambda>x. f n x / (x - w)\<^sup>2) - ?conint (\<lambda>x. g x / (x - w)\<^sup>2) = (f' n w - g' w) * (2 * of_real pi * \<i>)"
-             if cont_fn: "continuous_on (cball z r) (f n)"
-             and fnd: "\<And>w. w \<in> ball z r \<Longrightarrow> (f n has_field_derivative f' n w) (at w)" for n
-    proof -
-      have hol_fn: "f n holomorphic_on ball z r"
-        using fnd by (force simp: holomorphic_on_open)
-      have "(f n has_field_derivative 1 / (2 * of_real pi * \<i>) * ?conint (\<lambda>u. f n u / (u - w)\<^sup>2)) (at w)"
-        by (rule Cauchy_derivative_integral_circlepath [OF cont_fn hol_fn w])
-      then have f': "f' n w = 1 / (2 * of_real pi * \<i>) * ?conint (\<lambda>u. f n u / (u - w)\<^sup>2)"
-        using DERIV_unique [OF fnd] w by blast
-      show ?thesis
-        by (simp add: f' Cauchy_contour_integral_circlepath_2 [OF g w] derg [OF w] field_split_simps)
-    qed
-    define d where "d = (r - norm(w - z))^2"
-    have "d > 0"
-      using w by (simp add: dist_commute dist_norm d_def)
-    have dle: "d \<le> cmod ((y - w)\<^sup>2)" if "r = cmod (z - y)" for y
-    proof -
-      have "w \<in> ball z (cmod (z - y))"
-        using that w by fastforce
-      then have "cmod (w - z) \<le> cmod (z - y)"
-        by (simp add: dist_complex_def norm_minus_commute)
-      moreover have "cmod (z - y) - cmod (w - z) \<le> cmod (y - w)"
-        by (metis diff_add_cancel diff_add_eq_diff_diff_swap norm_minus_commute norm_triangle_ineq2)
-      ultimately show ?thesis
-        using that by (simp add: d_def norm_power power_mono)
-    qed
-    have 1: "\<forall>\<^sub>F n in F. (\<lambda>x. f n x / (x - w)\<^sup>2) contour_integrable_on circlepath z r"
-      by (force simp: holomorphic_on_open intro: w Cauchy_derivative_integral_circlepath eventually_mono [OF cont])
-    have 2: "uniform_limit (sphere z r) (\<lambda>n x. f n x / (x - w)\<^sup>2) (\<lambda>x. g x / (x - w)\<^sup>2) F"
-      unfolding uniform_limit_iff
-    proof clarify
-      fix e::real
-      assume "0 < e"
-      with \<open>r > 0\<close> show "\<forall>\<^sub>F n in F. \<forall>x\<in>sphere z r. dist (f n x / (x - w)\<^sup>2) (g x / (x - w)\<^sup>2) < e"
-        apply (simp add: norm_divide field_split_simps sphere_def dist_norm)
-        apply (rule eventually_mono [OF uniform_limitD [OF ulim], of "e*d"])
-         apply (simp add: \<open>0 < d\<close>)
-        apply (force simp: dist_norm dle intro: less_le_trans)
-        done
-    qed
-    have "((\<lambda>n. contour_integral (circlepath z r) (\<lambda>x. f n x / (x - w)\<^sup>2))
-             \<longlongrightarrow> contour_integral (circlepath z r) ((\<lambda>x. g x / (x - w)\<^sup>2))) F"
-      by (rule contour_integral_uniform_limit_circlepath [OF 1 2 F \<open>0 < r\<close>])
-    then have tendsto_0: "((\<lambda>n. 1 / (2 * of_real pi * \<i>) * (?conint (\<lambda>x. f n x / (x - w)\<^sup>2) - ?conint (\<lambda>x. g x / (x - w)\<^sup>2))) \<longlongrightarrow> 0) F"
-      using Lim_null by (force intro!: tendsto_mult_right_zero)
-    have "((\<lambda>n. f' n w - g' w) \<longlongrightarrow> 0) F"
-      apply (rule Lim_transform_eventually [OF tendsto_0])
-      apply (force simp: divide_simps intro: eq_f' eventually_mono [OF cont])
-      done
-    then show ?thesis using Lim_null by blast
-  qed
-  obtain g' where "\<And>w. w \<in> ball z r \<Longrightarrow> (g has_field_derivative (g' w)) (at w) \<and> ((\<lambda>n. f' n w) \<longlongrightarrow> g' w) F"
-      by (blast intro: tends_f'n_g' g')
-  then show ?thesis using g
-    using that by blast
-qed
-
-
-subsection\<^marker>\<open>tag unimportant\<close> \<open>Some more simple/convenient versions for applications\<close>
-
-lemma holomorphic_uniform_sequence:
-  assumes S: "open S"
-      and hol_fn: "\<And>n. (f n) holomorphic_on S"
-      and ulim_g: "\<And>x. x \<in> S \<Longrightarrow> \<exists>d. 0 < d \<and> cball x d \<subseteq> S \<and> uniform_limit (cball x d) f g sequentially"
-  shows "g holomorphic_on S"
-proof -
-  have "\<exists>f'. (g has_field_derivative f') (at z)" if "z \<in> S" for z
-  proof -
-    obtain r where "0 < r" and r: "cball z r \<subseteq> S"
-               and ul: "uniform_limit (cball z r) f g sequentially"
-      using ulim_g [OF \<open>z \<in> S\<close>] by blast
-    have *: "\<forall>\<^sub>F n in sequentially. continuous_on (cball z r) (f n) \<and> f n holomorphic_on ball z r"
-    proof (intro eventuallyI conjI)
-      show "continuous_on (cball z r) (f x)" for x
-        using hol_fn holomorphic_on_imp_continuous_on holomorphic_on_subset r by blast
-      show "f x holomorphic_on ball z r" for x
-        by (metis hol_fn holomorphic_on_subset interior_cball interior_subset r)
-    qed
-    show ?thesis
-      apply (rule holomorphic_uniform_limit [OF *])
-      using \<open>0 < r\<close> centre_in_ball ul
-      apply (auto simp: holomorphic_on_open)
-      done
-  qed
-  with S show ?thesis
-    by (simp add: holomorphic_on_open)
-qed
-
-lemma has_complex_derivative_uniform_sequence:
-  fixes S :: "complex set"
-  assumes S: "open S"
-      and hfd: "\<And>n x. x \<in> S \<Longrightarrow> ((f n) has_field_derivative f' n x) (at x)"
-      and ulim_g: "\<And>x. x \<in> S
-             \<Longrightarrow> \<exists>d. 0 < d \<and> cball x d \<subseteq> S \<and> uniform_limit (cball x d) f g sequentially"
-  shows "\<exists>g'. \<forall>x \<in> S. (g has_field_derivative g' x) (at x) \<and> ((\<lambda>n. f' n x) \<longlongrightarrow> g' x) sequentially"
-proof -
-  have y: "\<exists>y. (g has_field_derivative y) (at z) \<and> (\<lambda>n. f' n z) \<longlonglongrightarrow> y" if "z \<in> S" for z
-  proof -
-    obtain r where "0 < r" and r: "cball z r \<subseteq> S"
-               and ul: "uniform_limit (cball z r) f g sequentially"
-      using ulim_g [OF \<open>z \<in> S\<close>] by blast
-    have *: "\<forall>\<^sub>F n in sequentially. continuous_on (cball z r) (f n) \<and>
-                                   (\<forall>w \<in> ball z r. ((f n) has_field_derivative (f' n w)) (at w))"
-    proof (intro eventuallyI conjI ballI)
-      show "continuous_on (cball z r) (f x)" for x
-        by (meson S continuous_on_subset hfd holomorphic_on_imp_continuous_on holomorphic_on_open r)
-      show "w \<in> ball z r \<Longrightarrow> (f x has_field_derivative f' x w) (at w)" for w x
-        using ball_subset_cball hfd r by blast
-    qed
-    show ?thesis
-      by (rule has_complex_derivative_uniform_limit [OF *, of g]) (use \<open>0 < r\<close> ul in \<open>force+\<close>)
-  qed
-  show ?thesis
-    by (rule bchoice) (blast intro: y)
-qed
-
-subsection\<open>On analytic functions defined by a series\<close>
-
-lemma series_and_derivative_comparison:
-  fixes S :: "complex set"
-  assumes S: "open S"
-      and h: "summable h"
-      and hfd: "\<And>n x. x \<in> S \<Longrightarrow> (f n has_field_derivative f' n x) (at x)"
-      and to_g: "\<forall>\<^sub>F n in sequentially. \<forall>x\<in>S. norm (f n x) \<le> h n"
-  obtains g g' where "\<forall>x \<in> S. ((\<lambda>n. f n x) sums g x) \<and> ((\<lambda>n. f' n x) sums g' x) \<and> (g has_field_derivative g' x) (at x)"
-proof -
-  obtain g where g: "uniform_limit S (\<lambda>n x. \<Sum>i<n. f i x) g sequentially"
-    using Weierstrass_m_test_ev [OF to_g h]  by force
-  have *: "\<exists>d>0. cball x d \<subseteq> S \<and> uniform_limit (cball x d) (\<lambda>n x. \<Sum>i<n. f i x) g sequentially"
-         if "x \<in> S" for x
-  proof -
-    obtain d where "d>0" and d: "cball x d \<subseteq> S"
-      using open_contains_cball [of "S"] \<open>x \<in> S\<close> S by blast
-    show ?thesis
-    proof (intro conjI exI)
-      show "uniform_limit (cball x d) (\<lambda>n x. \<Sum>i<n. f i x) g sequentially"
-        using d g uniform_limit_on_subset by (force simp: dist_norm eventually_sequentially)
-    qed (use \<open>d > 0\<close> d in auto)
-  qed
-  have "\<And>x. x \<in> S \<Longrightarrow> (\<lambda>n. \<Sum>i<n. f i x) \<longlonglongrightarrow> g x"
-    by (metis tendsto_uniform_limitI [OF g])
-  moreover have "\<exists>g'. \<forall>x\<in>S. (g has_field_derivative g' x) (at x) \<and> (\<lambda>n. \<Sum>i<n. f' i x) \<longlonglongrightarrow> g' x"
-    by (rule has_complex_derivative_uniform_sequence [OF S]) (auto intro: * hfd DERIV_sum)+
-  ultimately show ?thesis
-    by (metis sums_def that)
-qed
-
-text\<open>A version where we only have local uniform/comparative convergence.\<close>
-
-lemma series_and_derivative_comparison_local:
-  fixes S :: "complex set"
-  assumes S: "open S"
-      and hfd: "\<And>n x. x \<in> S \<Longrightarrow> (f n has_field_derivative f' n x) (at x)"
-      and to_g: "\<And>x. x \<in> S \<Longrightarrow> \<exists>d h. 0 < d \<and> summable h \<and> (\<forall>\<^sub>F n in sequentially. \<forall>y\<in>ball x d \<inter> S. norm (f n y) \<le> h n)"
-  shows "\<exists>g g'. \<forall>x \<in> S. ((\<lambda>n. f n x) sums g x) \<and> ((\<lambda>n. f' n x) sums g' x) \<and> (g has_field_derivative g' x) (at x)"
-proof -
-  have "\<exists>y. (\<lambda>n. f n z) sums (\<Sum>n. f n z) \<and> (\<lambda>n. f' n z) sums y \<and> ((\<lambda>x. \<Sum>n. f n x) has_field_derivative y) (at z)"
-       if "z \<in> S" for z
-  proof -
-    obtain d h where "0 < d" "summable h" and le_h: "\<forall>\<^sub>F n in sequentially. \<forall>y\<in>ball z d \<inter> S. norm (f n y) \<le> h n"
-      using to_g \<open>z \<in> S\<close> by meson
-    then obtain r where "r>0" and r: "ball z r \<subseteq> ball z d \<inter> S" using \<open>z \<in> S\<close> S
-      by (metis Int_iff open_ball centre_in_ball open_Int open_contains_ball_eq)
-    have 1: "open (ball z d \<inter> S)"
-      by (simp add: open_Int S)
-    have 2: "\<And>n x. x \<in> ball z d \<inter> S \<Longrightarrow> (f n has_field_derivative f' n x) (at x)"
-      by (auto simp: hfd)
-    obtain g g' where gg': "\<forall>x \<in> ball z d \<inter> S. ((\<lambda>n. f n x) sums g x) \<and>
-                                    ((\<lambda>n. f' n x) sums g' x) \<and> (g has_field_derivative g' x) (at x)"
-      by (auto intro: le_h series_and_derivative_comparison [OF 1 \<open>summable h\<close> hfd])
-    then have "(\<lambda>n. f' n z) sums g' z"
-      by (meson \<open>0 < r\<close> centre_in_ball contra_subsetD r)
-    moreover have "(\<lambda>n. f n z) sums (\<Sum>n. f n z)"
-      using  summable_sums centre_in_ball \<open>0 < d\<close> \<open>summable h\<close> le_h
-      by (metis (full_types) Int_iff gg' summable_def that)
-    moreover have "((\<lambda>x. \<Sum>n. f n x) has_field_derivative g' z) (at z)"
-    proof (rule has_field_derivative_transform_within)
-      show "\<And>x. dist x z < r \<Longrightarrow> g x = (\<Sum>n. f n x)"
-        by (metis subsetD dist_commute gg' mem_ball r sums_unique)
-    qed (use \<open>0 < r\<close> gg' \<open>z \<in> S\<close> \<open>0 < d\<close> in auto)
-    ultimately show ?thesis by auto
-  qed
-  then show ?thesis
-    by (rule_tac x="\<lambda>x. suminf (\<lambda>n. f n x)" in exI) meson
-qed
-
-
-text\<open>Sometimes convenient to compare with a complex series of positive reals. (?)\<close>
-
-lemma series_and_derivative_comparison_complex:
-  fixes S :: "complex set"
-  assumes S: "open S"
-      and hfd: "\<And>n x. x \<in> S \<Longrightarrow> (f n has_field_derivative f' n x) (at x)"
-      and to_g: "\<And>x. x \<in> S \<Longrightarrow> \<exists>d h. 0 < d \<and> summable h \<and> range h \<subseteq> \<real>\<^sub>\<ge>\<^sub>0 \<and> (\<forall>\<^sub>F n in sequentially. \<forall>y\<in>ball x d \<inter> S. cmod(f n y) \<le> cmod (h n))"
-  shows "\<exists>g g'. \<forall>x \<in> S. ((\<lambda>n. f n x) sums g x) \<and> ((\<lambda>n. f' n x) sums g' x) \<and> (g has_field_derivative g' x) (at x)"
-apply (rule series_and_derivative_comparison_local [OF S hfd], assumption)
-apply (rule ex_forward [OF to_g], assumption)
-apply (erule exE)
-apply (rule_tac x="Re \<circ> h" in exI)
-apply (force simp: summable_Re o_def nonneg_Reals_cmod_eq_Re image_subset_iff)
-done
-
-text\<open>Sometimes convenient to compare with a complex series of positive reals. (?)\<close>
-lemma series_differentiable_comparison_complex:
-  fixes S :: "complex set"
-  assumes S: "open S"
-    and hfd: "\<And>n x. x \<in> S \<Longrightarrow> f n field_differentiable (at x)"
-    and to_g: "\<And>x. x \<in> S \<Longrightarrow> \<exists>d h. 0 < d \<and> summable h \<and> range h \<subseteq> \<real>\<^sub>\<ge>\<^sub>0 \<and> (\<forall>\<^sub>F n in sequentially. \<forall>y\<in>ball x d \<inter> S. cmod(f n y) \<le> cmod (h n))"
-  obtains g where "\<forall>x \<in> S. ((\<lambda>n. f n x) sums g x) \<and> g field_differentiable (at x)"
-proof -
-  have hfd': "\<And>n x. x \<in> S \<Longrightarrow> (f n has_field_derivative deriv (f n) x) (at x)"
-    using hfd field_differentiable_derivI by blast
-  have "\<exists>g g'. \<forall>x \<in> S. ((\<lambda>n. f n x) sums g x) \<and> ((\<lambda>n. deriv (f n) x) sums g' x) \<and> (g has_field_derivative g' x) (at x)"
-    by (metis series_and_derivative_comparison_complex [OF S hfd' to_g])
-  then show ?thesis
-    using field_differentiable_def that by blast
-qed
-
-text\<open>In particular, a power series is analytic inside circle of convergence.\<close>
-
-lemma power_series_and_derivative_0:
-  fixes a :: "nat \<Rightarrow> complex" and r::real
-  assumes "summable (\<lambda>n. a n * r^n)"
-    shows "\<exists>g g'. \<forall>z. cmod z < r \<longrightarrow>
-             ((\<lambda>n. a n * z^n) sums g z) \<and> ((\<lambda>n. of_nat n * a n * z^(n - 1)) sums g' z) \<and> (g has_field_derivative g' z) (at z)"
-proof (cases "0 < r")
-  case True
-    have der: "\<And>n z. ((\<lambda>x. a n * x ^ n) has_field_derivative of_nat n * a n * z ^ (n - 1)) (at z)"
-      by (rule derivative_eq_intros | simp)+
-    have y_le: "\<lbrakk>cmod (z - y) * 2 < r - cmod z\<rbrakk> \<Longrightarrow> cmod y \<le> cmod (of_real r + of_real (cmod z)) / 2" for z y
-      using \<open>r > 0\<close>
-      apply (auto simp: algebra_simps norm_mult norm_divide norm_power simp flip: of_real_add)
-      using norm_triangle_ineq2 [of y z]
-      apply (simp only: diff_le_eq norm_minus_commute mult_2)
-      done
-    have "summable (\<lambda>n. a n * complex_of_real r ^ n)"
-      using assms \<open>r > 0\<close> by simp
-    moreover have "\<And>z. cmod z < r \<Longrightarrow> cmod ((of_real r + of_real (cmod z)) / 2) < cmod (of_real r)"
-      using \<open>r > 0\<close>
-      by (simp flip: of_real_add)
-    ultimately have sum: "\<And>z. cmod z < r \<Longrightarrow> summable (\<lambda>n. of_real (cmod (a n)) * ((of_real r + complex_of_real (cmod z)) / 2) ^ n)"
-      by (rule power_series_conv_imp_absconv_weak)
-    have "\<exists>g g'. \<forall>z \<in> ball 0 r. (\<lambda>n.  (a n) * z ^ n) sums g z \<and>
-               (\<lambda>n. of_nat n * (a n) * z ^ (n - 1)) sums g' z \<and> (g has_field_derivative g' z) (at z)"
-      apply (rule series_and_derivative_comparison_complex [OF open_ball der])
-      apply (rule_tac x="(r - norm z)/2" in exI)
-      apply (rule_tac x="\<lambda>n. of_real(norm(a n)*((r + norm z)/2)^n)" in exI)
-      using \<open>r > 0\<close>
-      apply (auto simp: sum eventually_sequentially norm_mult norm_power dist_norm intro!: mult_left_mono power_mono y_le)
-      done
-  then show ?thesis
-    by (simp add: ball_def)
-next
-  case False then show ?thesis
-    apply (simp add: not_less)
-    using less_le_trans norm_not_less_zero by blast
-qed
-
-proposition\<^marker>\<open>tag unimportant\<close> power_series_and_derivative:
-  fixes a :: "nat \<Rightarrow> complex" and r::real
-  assumes "summable (\<lambda>n. a n * r^n)"
-    obtains g g' where "\<forall>z \<in> ball w r.
-             ((\<lambda>n. a n * (z - w) ^ n) sums g z) \<and> ((\<lambda>n. of_nat n * a n * (z - w) ^ (n - 1)) sums g' z) \<and>
-              (g has_field_derivative g' z) (at z)"
-  using power_series_and_derivative_0 [OF assms]
-  apply clarify
-  apply (rule_tac g="(\<lambda>z. g(z - w))" in that)
-  using DERIV_shift [where z="-w"]
-  apply (auto simp: norm_minus_commute Ball_def dist_norm)
-  done
-
-proposition\<^marker>\<open>tag unimportant\<close> power_series_holomorphic:
-  assumes "\<And>w. w \<in> ball z r \<Longrightarrow> ((\<lambda>n. a n*(w - z)^n) sums f w)"
-    shows "f holomorphic_on ball z r"
-proof -
-  have "\<exists>f'. (f has_field_derivative f') (at w)" if w: "dist z w < r" for w
-  proof -
-    have inb: "z + complex_of_real ((dist z w + r) / 2) \<in> ball z r"
-    proof -
-      have wz: "cmod (w - z) < r" using w
-        by (auto simp: field_split_simps dist_norm norm_minus_commute)
-      then have "0 \<le> r"
-        by (meson less_eq_real_def norm_ge_zero order_trans)
-      show ?thesis
-        using w by (simp add: dist_norm \<open>0\<le>r\<close> flip: of_real_add)
-    qed
-    have sum: "summable (\<lambda>n. a n * of_real (((cmod (z - w) + r) / 2) ^ n))"
-      using assms [OF inb] by (force simp: summable_def dist_norm)
-    obtain g g' where gg': "\<And>u. u \<in> ball z ((cmod (z - w) + r) / 2) \<Longrightarrow>
-                               (\<lambda>n. a n * (u - z) ^ n) sums g u \<and>
-                               (\<lambda>n. of_nat n * a n * (u - z) ^ (n - 1)) sums g' u \<and> (g has_field_derivative g' u) (at u)"
-      by (rule power_series_and_derivative [OF sum, of z]) fastforce
-    have [simp]: "g u = f u" if "cmod (u - w) < (r - cmod (z - w)) / 2" for u
-    proof -
-      have less: "cmod (z - u) * 2 < cmod (z - w) + r"
-        using that dist_triangle2 [of z u w]
-        by (simp add: dist_norm [symmetric] algebra_simps)
-      show ?thesis
-        apply (rule sums_unique2 [of "\<lambda>n. a n*(u - z)^n"])
-        using gg' [of u] less w
-        apply (auto simp: assms dist_norm)
-        done
-    qed
-    have "(f has_field_derivative g' w) (at w)"
-      by (rule has_field_derivative_transform_within [where d="(r - norm(z - w))/2"])
-      (use w gg' [of w] in \<open>(force simp: dist_norm)+\<close>)
-    then show ?thesis ..
-  qed
-  then show ?thesis by (simp add: holomorphic_on_open)
-qed
-
-corollary holomorphic_iff_power_series:
-     "f holomorphic_on ball z r \<longleftrightarrow>
-      (\<forall>w \<in> ball z r. (\<lambda>n. (deriv ^^ n) f z / (fact n) * (w - z)^n) sums f w)"
-  apply (intro iffI ballI holomorphic_power_series, assumption+)
-  apply (force intro: power_series_holomorphic [where a = "\<lambda>n. (deriv ^^ n) f z / (fact n)"])
-  done
-
-lemma power_series_analytic:
-     "(\<And>w. w \<in> ball z r \<Longrightarrow> (\<lambda>n. a n*(w - z)^n) sums f w) \<Longrightarrow> f analytic_on ball z r"
-  by (force simp: analytic_on_open intro!: power_series_holomorphic)
-
-lemma analytic_iff_power_series:
-     "f analytic_on ball z r \<longleftrightarrow>
-      (\<forall>w \<in> ball z r. (\<lambda>n. (deriv ^^ n) f z / (fact n) * (w - z)^n) sums f w)"
-  by (simp add: analytic_on_open holomorphic_iff_power_series)
-
-subsection\<^marker>\<open>tag unimportant\<close> \<open>Equality between holomorphic functions, on open ball then connected set\<close>
-
-lemma holomorphic_fun_eq_on_ball:
-   "\<lbrakk>f holomorphic_on ball z r; g holomorphic_on ball z r;
-     w \<in> ball z r;
-     \<And>n. (deriv ^^ n) f z = (deriv ^^ n) g z\<rbrakk>
-     \<Longrightarrow> f w = g w"
-  apply (rule sums_unique2 [of "\<lambda>n. (deriv ^^ n) f z / (fact n) * (w - z)^n"])
-  apply (auto simp: holomorphic_iff_power_series)
-  done
-
-lemma holomorphic_fun_eq_0_on_ball:
-   "\<lbrakk>f holomorphic_on ball z r;  w \<in> ball z r;
-     \<And>n. (deriv ^^ n) f z = 0\<rbrakk>
-     \<Longrightarrow> f w = 0"
-  apply (rule sums_unique2 [of "\<lambda>n. (deriv ^^ n) f z / (fact n) * (w - z)^n"])
-  apply (auto simp: holomorphic_iff_power_series)
-  done
-
-lemma holomorphic_fun_eq_0_on_connected:
-  assumes holf: "f holomorphic_on S" and "open S"
-      and cons: "connected S"
-      and der: "\<And>n. (deriv ^^ n) f z = 0"
-      and "z \<in> S" "w \<in> S"
-    shows "f w = 0"
-proof -
-  have *: "ball x e \<subseteq> (\<Inter>n. {w \<in> S. (deriv ^^ n) f w = 0})"
-    if "\<forall>u. (deriv ^^ u) f x = 0" "ball x e \<subseteq> S" for x e
-  proof -
-    have "\<And>x' n. dist x x' < e \<Longrightarrow> (deriv ^^ n) f x' = 0"
-      apply (rule holomorphic_fun_eq_0_on_ball [OF holomorphic_higher_deriv])
-         apply (rule holomorphic_on_subset [OF holf])
-      using that apply simp_all
-      by (metis funpow_add o_apply)
-    with that show ?thesis by auto
-  qed
-  have 1: "openin (top_of_set S) (\<Inter>n. {w \<in> S. (deriv ^^ n) f w = 0})"
-    apply (rule open_subset, force)
-    using \<open>open S\<close>
-    apply (simp add: open_contains_ball Ball_def)
-    apply (erule all_forward)
-    using "*" by auto blast+
-  have 2: "closedin (top_of_set S) (\<Inter>n. {w \<in> S. (deriv ^^ n) f w = 0})"
-    using assms
-    by (auto intro: continuous_closedin_preimage_constant holomorphic_on_imp_continuous_on holomorphic_higher_deriv)
-  obtain e where "e>0" and e: "ball w e \<subseteq> S" using openE [OF \<open>open S\<close> \<open>w \<in> S\<close>] .
-  then have holfb: "f holomorphic_on ball w e"
-    using holf holomorphic_on_subset by blast
-  have 3: "(\<Inter>n. {w \<in> S. (deriv ^^ n) f w = 0}) = S \<Longrightarrow> f w = 0"
-    using \<open>e>0\<close> e by (force intro: holomorphic_fun_eq_0_on_ball [OF holfb])
-  show ?thesis
-    using cons der \<open>z \<in> S\<close>
-    apply (simp add: connected_clopen)
-    apply (drule_tac x="\<Inter>n. {w \<in> S. (deriv ^^ n) f w = 0}" in spec)
-    apply (auto simp: 1 2 3)
-    done
-qed
-
-lemma holomorphic_fun_eq_on_connected:
-  assumes "f holomorphic_on S" "g holomorphic_on S" and "open S"  "connected S"
-      and "\<And>n. (deriv ^^ n) f z = (deriv ^^ n) g z"
-      and "z \<in> S" "w \<in> S"
-    shows "f w = g w"
-proof (rule holomorphic_fun_eq_0_on_connected [of "\<lambda>x. f x - g x" S z, simplified])
-  show "(\<lambda>x. f x - g x) holomorphic_on S"
-    by (intro assms holomorphic_intros)
-  show "\<And>n. (deriv ^^ n) (\<lambda>x. f x - g x) z = 0"
-    using assms higher_deriv_diff by auto
-qed (use assms in auto)
-
-lemma holomorphic_fun_eq_const_on_connected:
-  assumes holf: "f holomorphic_on S" and "open S"
-      and cons: "connected S"
-      and der: "\<And>n. 0 < n \<Longrightarrow> (deriv ^^ n) f z = 0"
-      and "z \<in> S" "w \<in> S"
-    shows "f w = f z"
-proof (rule holomorphic_fun_eq_0_on_connected [of "\<lambda>w. f w - f z" S z, simplified])
-  show "(\<lambda>w. f w - f z) holomorphic_on S"
-    by (intro assms holomorphic_intros)
-  show "\<And>n. (deriv ^^ n) (\<lambda>w. f w - f z) z = 0"
-    by (subst higher_deriv_diff) (use assms in \<open>auto intro: holomorphic_intros\<close>)
-qed (use assms in auto)
-
-subsection\<^marker>\<open>tag unimportant\<close> \<open>Some basic lemmas about poles/singularities\<close>
-
-lemma pole_lemma:
-  assumes holf: "f holomorphic_on S" and a: "a \<in> interior S"
-    shows "(\<lambda>z. if z = a then deriv f a
-                 else (f z - f a) / (z - a)) holomorphic_on S" (is "?F holomorphic_on S")
-proof -
-  have F1: "?F field_differentiable (at u within S)" if "u \<in> S" "u \<noteq> a" for u
-  proof -
-    have fcd: "f field_differentiable at u within S"
-      using holf holomorphic_on_def by (simp add: \<open>u \<in> S\<close>)
-    have cd: "(\<lambda>z. (f z - f a) / (z - a)) field_differentiable at u within S"
-      by (rule fcd derivative_intros | simp add: that)+
-    have "0 < dist a u" using that dist_nz by blast
-    then show ?thesis
-      by (rule field_differentiable_transform_within [OF _ _ _ cd]) (auto simp: \<open>u \<in> S\<close>)
-  qed
-  have F2: "?F field_differentiable at a" if "0 < e" "ball a e \<subseteq> S" for e
-  proof -
-    have holfb: "f holomorphic_on ball a e"
-      by (rule holomorphic_on_subset [OF holf \<open>ball a e \<subseteq> S\<close>])
-    have 2: "?F holomorphic_on ball a e - {a}"
-      apply (simp add: holomorphic_on_def flip: field_differentiable_def)
-      using mem_ball that
-      apply (auto intro: F1 field_differentiable_within_subset)
-      done
-    have "isCont (\<lambda>z. if z = a then deriv f a else (f z - f a) / (z - a)) x"
-            if "dist a x < e" for x
-    proof (cases "x=a")
-      case True
-      then have "f field_differentiable at a"
-        using holfb \<open>0 < e\<close> holomorphic_on_imp_differentiable_at by auto
-      with True show ?thesis
-        by (auto simp: continuous_at has_field_derivative_iff simp flip: DERIV_deriv_iff_field_differentiable
-                elim: rev_iffD1 [OF _ LIM_equal])
-    next
-      case False with 2 that show ?thesis
-        by (force simp: holomorphic_on_open open_Diff field_differentiable_def [symmetric] field_differentiable_imp_continuous_at)
-    qed
-    then have 1: "continuous_on (ball a e) ?F"
-      by (clarsimp simp:  continuous_on_eq_continuous_at)
-    have "?F holomorphic_on ball a e"
-      by (auto intro: no_isolated_singularity [OF 1 2])
-    with that show ?thesis
-      by (simp add: holomorphic_on_open field_differentiable_def [symmetric]
-                    field_differentiable_at_within)
-  qed
-  show ?thesis
-  proof
-    fix x assume "x \<in> S" show "?F field_differentiable at x within S"
-    proof (cases "x=a")
-      case True then show ?thesis
-      using a by (auto simp: mem_interior intro: field_differentiable_at_within F2)
-    next
-      case False with F1 \<open>x \<in> S\<close>
-      show ?thesis by blast
-    qed
-  qed
-qed
-
-lemma pole_theorem:
-  assumes holg: "g holomorphic_on S" and a: "a \<in> interior S"
-      and eq: "\<And>z. z \<in> S - {a} \<Longrightarrow> g z = (z - a) * f z"
-    shows "(\<lambda>z. if z = a then deriv g a
-                 else f z - g a/(z - a)) holomorphic_on S"
-  using pole_lemma [OF holg a]
-  by (rule holomorphic_transform) (simp add: eq field_split_simps)
-
-lemma pole_lemma_open:
-  assumes "f holomorphic_on S" "open S"
-    shows "(\<lambda>z. if z = a then deriv f a else (f z - f a)/(z - a)) holomorphic_on S"
-proof (cases "a \<in> S")
-  case True with assms interior_eq pole_lemma
-    show ?thesis by fastforce
-next
-  case False with assms show ?thesis
-    apply (simp add: holomorphic_on_def field_differentiable_def [symmetric], clarify)
-    apply (rule field_differentiable_transform_within [where f = "\<lambda>z. (f z - f a)/(z - a)" and d = 1])
-    apply (rule derivative_intros | force)+
-    done
-qed
-
-lemma pole_theorem_open:
-  assumes holg: "g holomorphic_on S" and S: "open S"
-      and eq: "\<And>z. z \<in> S - {a} \<Longrightarrow> g z = (z - a) * f z"
-    shows "(\<lambda>z. if z = a then deriv g a
-                 else f z - g a/(z - a)) holomorphic_on S"
-  using pole_lemma_open [OF holg S]
-  by (rule holomorphic_transform) (auto simp: eq divide_simps)
-
-lemma pole_theorem_0:
-  assumes holg: "g holomorphic_on S" and a: "a \<in> interior S"
-      and eq: "\<And>z. z \<in> S - {a} \<Longrightarrow> g z = (z - a) * f z"
-      and [simp]: "f a = deriv g a" "g a = 0"
-    shows "f holomorphic_on S"
-  using pole_theorem [OF holg a eq]
-  by (rule holomorphic_transform) (auto simp: eq field_split_simps)
-
-lemma pole_theorem_open_0:
-  assumes holg: "g holomorphic_on S" and S: "open S"
-      and eq: "\<And>z. z \<in> S - {a} \<Longrightarrow> g z = (z - a) * f z"
-      and [simp]: "f a = deriv g a" "g a = 0"
-    shows "f holomorphic_on S"
-  using pole_theorem_open [OF holg S eq]
-  by (rule holomorphic_transform) (auto simp: eq field_split_simps)
-
-lemma pole_theorem_analytic:
-  assumes g: "g analytic_on S"
-      and eq: "\<And>z. z \<in> S
-             \<Longrightarrow> \<exists>d. 0 < d \<and> (\<forall>w \<in> ball z d - {a}. g w = (w - a) * f w)"
-    shows "(\<lambda>z. if z = a then deriv g a else f z - g a/(z - a)) analytic_on S" (is "?F analytic_on S")
-  unfolding analytic_on_def
-proof
-  fix x
-  assume "x \<in> S"
-  with g obtain e where "0 < e" and e: "g holomorphic_on ball x e"
-    by (auto simp add: analytic_on_def)
-  obtain d where "0 < d" and d: "\<And>w. w \<in> ball x d - {a} \<Longrightarrow> g w = (w - a) * f w"
-    using \<open>x \<in> S\<close> eq by blast
-  have "?F holomorphic_on ball x (min d e)"
-    using d e \<open>x \<in> S\<close> by (fastforce simp: holomorphic_on_subset subset_ball intro!: pole_theorem_open)
-  then show "\<exists>e>0. ?F holomorphic_on ball x e"
-    using \<open>0 < d\<close> \<open>0 < e\<close> not_le by fastforce
-qed
-
-lemma pole_theorem_analytic_0:
-  assumes g: "g analytic_on S"
-      and eq: "\<And>z. z \<in> S \<Longrightarrow> \<exists>d. 0 < d \<and> (\<forall>w \<in> ball z d - {a}. g w = (w - a) * f w)"
-      and [simp]: "f a = deriv g a" "g a = 0"
-    shows "f analytic_on S"
-proof -
-  have [simp]: "(\<lambda>z. if z = a then deriv g a else f z - g a / (z - a)) = f"
-    by auto
-  show ?thesis
-    using pole_theorem_analytic [OF g eq] by simp
-qed
-
-lemma pole_theorem_analytic_open_superset:
-  assumes g: "g analytic_on S" and "S \<subseteq> T" "open T"
-      and eq: "\<And>z. z \<in> T - {a} \<Longrightarrow> g z = (z - a) * f z"
-    shows "(\<lambda>z. if z = a then deriv g a
-                 else f z - g a/(z - a)) analytic_on S"
-proof (rule pole_theorem_analytic [OF g])
-  fix z
-  assume "z \<in> S"
-  then obtain e where "0 < e" and e: "ball z e \<subseteq> T"
-    using assms openE by blast
-  then show "\<exists>d>0. \<forall>w\<in>ball z d - {a}. g w = (w - a) * f w"
-    using eq by auto
-qed
-
-lemma pole_theorem_analytic_open_superset_0:
-  assumes g: "g analytic_on S" "S \<subseteq> T" "open T" "\<And>z. z \<in> T - {a} \<Longrightarrow> g z = (z - a) * f z"
-      and [simp]: "f a = deriv g a" "g a = 0"
-    shows "f analytic_on S"
-proof -
-  have [simp]: "(\<lambda>z. if z = a then deriv g a else f z - g a / (z - a)) = f"
-    by auto
-  have "(\<lambda>z. if z = a then deriv g a else f z - g a/(z - a)) analytic_on S"
-    by (rule pole_theorem_analytic_open_superset [OF g])
-  then show ?thesis by simp
-qed
-
-
-subsection\<open>General, homology form of Cauchy's theorem\<close>
-
-text\<open>Proof is based on Dixon's, as presented in Lang's "Complex Analysis" book (page 147).\<close>
-
-lemma contour_integral_continuous_on_linepath_2D:
-  assumes "open U" and cont_dw: "\<And>w. w \<in> U \<Longrightarrow> F w contour_integrable_on (linepath a b)"
-      and cond_uu: "continuous_on (U \<times> U) (\<lambda>(x,y). F x y)"
-      and abu: "closed_segment a b \<subseteq> U"
-    shows "continuous_on U (\<lambda>w. contour_integral (linepath a b) (F w))"
-proof -
-  have *: "\<exists>d>0. \<forall>x'\<in>U. dist x' w < d \<longrightarrow>
-                         dist (contour_integral (linepath a b) (F x'))
-                              (contour_integral (linepath a b) (F w)) \<le> \<epsilon>"
-          if "w \<in> U" "0 < \<epsilon>" "a \<noteq> b" for w \<epsilon>
-  proof -
-    obtain \<delta> where "\<delta>>0" and \<delta>: "cball w \<delta> \<subseteq> U" using open_contains_cball \<open>open U\<close> \<open>w \<in> U\<close> by force
-    let ?TZ = "cball w \<delta>  \<times> closed_segment a b"
-    have "uniformly_continuous_on ?TZ (\<lambda>(x,y). F x y)"
-    proof (rule compact_uniformly_continuous)
-      show "continuous_on ?TZ (\<lambda>(x,y). F x y)"
-        by (rule continuous_on_subset[OF cond_uu]) (use SigmaE \<delta> abu in blast)
-      show "compact ?TZ"
-        by (simp add: compact_Times)
-    qed
-    then obtain \<eta> where "\<eta>>0"
-        and \<eta>: "\<And>x x'. \<lbrakk>x\<in>?TZ; x'\<in>?TZ; dist x' x < \<eta>\<rbrakk> \<Longrightarrow>
-                         dist ((\<lambda>(x,y). F x y) x') ((\<lambda>(x,y). F x y) x) < \<epsilon>/norm(b - a)"
-      apply (rule uniformly_continuous_onE [where e = "\<epsilon>/norm(b - a)"])
-      using \<open>0 < \<epsilon>\<close> \<open>a \<noteq> b\<close> by auto
-    have \<eta>: "\<lbrakk>norm (w - x1) \<le> \<delta>;   x2 \<in> closed_segment a b;
-              norm (w - x1') \<le> \<delta>;  x2' \<in> closed_segment a b; norm ((x1', x2') - (x1, x2)) < \<eta>\<rbrakk>
-              \<Longrightarrow> norm (F x1' x2' - F x1 x2) \<le> \<epsilon> / cmod (b - a)"
-             for x1 x2 x1' x2'
-      using \<eta> [of "(x1,x2)" "(x1',x2')"] by (force simp: dist_norm)
-    have le_ee: "cmod (contour_integral (linepath a b) (\<lambda>x. F x' x - F w x)) \<le> \<epsilon>"
-                if "x' \<in> U" "cmod (x' - w) < \<delta>" "cmod (x' - w) < \<eta>"  for x'
-    proof -
-      have "(\<lambda>x. F x' x - F w x) contour_integrable_on linepath a b"
-        by (simp add: \<open>w \<in> U\<close> cont_dw contour_integrable_diff that)
-      then have "cmod (contour_integral (linepath a b) (\<lambda>x. F x' x - F w x)) \<le> \<epsilon>/norm(b - a) * norm(b - a)"
-        apply (rule has_contour_integral_bound_linepath [OF has_contour_integral_integral _ \<eta>])
-        using \<open>0 < \<epsilon>\<close> \<open>0 < \<delta>\<close> that apply (auto simp: norm_minus_commute)
-        done
-      also have "\<dots> = \<epsilon>" using \<open>a \<noteq> b\<close> by simp
-      finally show ?thesis .
-    qed
-    show ?thesis
-      apply (rule_tac x="min \<delta> \<eta>" in exI)
-      using \<open>0 < \<delta>\<close> \<open>0 < \<eta>\<close>
-      apply (auto simp: dist_norm contour_integral_diff [OF cont_dw cont_dw, symmetric] \<open>w \<in> U\<close> intro: le_ee)
-      done
-  qed
-  show ?thesis
-  proof (cases "a=b")
-    case True
-    then show ?thesis by simp
-  next
-    case False
-    show ?thesis
-      by (rule continuous_onI) (use False in \<open>auto intro: *\<close>)
-  qed
-qed
-
-text\<open>This version has \<^term>\<open>polynomial_function \<gamma>\<close> as an additional assumption.\<close>
-lemma Cauchy_integral_formula_global_weak:
-  assumes "open U" and holf: "f holomorphic_on U"
-        and z: "z \<in> U" and \<gamma>: "polynomial_function \<gamma>"
-        and pasz: "path_image \<gamma> \<subseteq> U - {z}" and loop: "pathfinish \<gamma> = pathstart \<gamma>"
-        and zero: "\<And>w. w \<notin> U \<Longrightarrow> winding_number \<gamma> w = 0"
-      shows "((\<lambda>w. f w / (w - z)) has_contour_integral (2*pi * \<i> * winding_number \<gamma> z * f z)) \<gamma>"
-proof -
-  obtain \<gamma>' where pf\<gamma>': "polynomial_function \<gamma>'" and \<gamma>': "\<And>x. (\<gamma> has_vector_derivative (\<gamma>' x)) (at x)"
-    using has_vector_derivative_polynomial_function [OF \<gamma>] by blast
-  then have "bounded(path_image \<gamma>')"
-    by (simp add: path_image_def compact_imp_bounded compact_continuous_image continuous_on_polymonial_function)
-  then obtain B where "B>0" and B: "\<And>x. x \<in> path_image \<gamma>' \<Longrightarrow> norm x \<le> B"
-    using bounded_pos by force
-  define d where [abs_def]: "d z w = (if w = z then deriv f z else (f w - f z)/(w - z))" for z w
-  define v where "v = {w. w \<notin> path_image \<gamma> \<and> winding_number \<gamma> w = 0}"
-  have "path \<gamma>" "valid_path \<gamma>" using \<gamma>
-    by (auto simp: path_polynomial_function valid_path_polynomial_function)
-  then have ov: "open v"
-    by (simp add: v_def open_winding_number_levelsets loop)
-  have uv_Un: "U \<union> v = UNIV"
-    using pasz zero by (auto simp: v_def)
-  have conf: "continuous_on U f"
-    by (metis holf holomorphic_on_imp_continuous_on)
-  have hol_d: "(d y) holomorphic_on U" if "y \<in> U" for y
-  proof -
-    have *: "(\<lambda>c. if c = y then deriv f y else (f c - f y) / (c - y)) holomorphic_on U"
-      by (simp add: holf pole_lemma_open \<open>open U\<close>)
-    then have "isCont (\<lambda>x. if x = y then deriv f y else (f x - f y) / (x - y)) y"
-      using at_within_open field_differentiable_imp_continuous_at holomorphic_on_def that \<open>open U\<close> by fastforce
-    then have "continuous_on U (d y)"
-      apply (simp add: d_def continuous_on_eq_continuous_at \<open>open U\<close>, clarify)
-      using * holomorphic_on_def
-      by (meson field_differentiable_within_open field_differentiable_imp_continuous_at \<open>open U\<close>)
-    moreover have "d y holomorphic_on U - {y}"
-    proof -
-      have "\<And>w. w \<in> U - {y} \<Longrightarrow>
-                 (\<lambda>w. if w = y then deriv f y else (f w - f y) / (w - y)) field_differentiable at w"
-        apply (rule_tac d="dist w y" and f = "\<lambda>w. (f w - f y)/(w - y)" in field_differentiable_transform_within)
-           apply (auto simp: dist_pos_lt dist_commute intro!: derivative_intros)
-        using \<open>open U\<close> holf holomorphic_on_imp_differentiable_at by blast
-      then show ?thesis
-        unfolding field_differentiable_def by (simp add: d_def holomorphic_on_open \<open>open U\<close> open_delete)
-    qed
-    ultimately show ?thesis
-      by (rule no_isolated_singularity) (auto simp: \<open>open U\<close>)
-  qed
-  have cint_fxy: "(\<lambda>x. (f x - f y) / (x - y)) contour_integrable_on \<gamma>" if "y \<notin> path_image \<gamma>" for y
-  proof (rule contour_integrable_holomorphic_simple [where S = "U-{y}"])
-    show "(\<lambda>x. (f x - f y) / (x - y)) holomorphic_on U - {y}"
-      by (force intro: holomorphic_intros holomorphic_on_subset [OF holf])
-    show "path_image \<gamma> \<subseteq> U - {y}"
-      using pasz that by blast
-  qed (auto simp: \<open>open U\<close> open_delete \<open>valid_path \<gamma>\<close>)
-  define h where
-    "h z = (if z \<in> U then contour_integral \<gamma> (d z) else contour_integral \<gamma> (\<lambda>w. f w/(w - z)))" for z
-  have U: "((d z) has_contour_integral h z) \<gamma>" if "z \<in> U" for z
-  proof -
-    have "d z holomorphic_on U"
-      by (simp add: hol_d that)
-    with that show ?thesis
-    apply (simp add: h_def)
-      by (meson Diff_subset \<open>open U\<close> \<open>valid_path \<gamma>\<close> contour_integrable_holomorphic_simple has_contour_integral_integral pasz subset_trans)
-  qed
-  have V: "((\<lambda>w. f w / (w - z)) has_contour_integral h z) \<gamma>" if z: "z \<in> v" for z
-  proof -
-    have 0: "0 = (f z) * 2 * of_real (2 * pi) * \<i> * winding_number \<gamma> z"
-      using v_def z by auto
-    then have "((\<lambda>x. 1 / (x - z)) has_contour_integral 0) \<gamma>"
-     using z v_def  has_contour_integral_winding_number [OF \<open>valid_path \<gamma>\<close>] by fastforce
-    then have "((\<lambda>x. f z * (1 / (x - z))) has_contour_integral 0) \<gamma>"
-      using has_contour_integral_lmul by fastforce
-    then have "((\<lambda>x. f z / (x - z)) has_contour_integral 0) \<gamma>"
-      by (simp add: field_split_simps)
-    moreover have "((\<lambda>x. (f x - f z) / (x - z)) has_contour_integral contour_integral \<gamma> (d z)) \<gamma>"
-      using z
-      apply (auto simp: v_def)
-      apply (metis (no_types, lifting) contour_integrable_eq d_def has_contour_integral_eq has_contour_integral_integral cint_fxy)
-      done
-    ultimately have *: "((\<lambda>x. f z / (x - z) + (f x - f z) / (x - z)) has_contour_integral (0 + contour_integral \<gamma> (d z))) \<gamma>"
-      by (rule has_contour_integral_add)
-    have "((\<lambda>w. f w / (w - z)) has_contour_integral contour_integral \<gamma> (d z)) \<gamma>"
-            if  "z \<in> U"
-      using * by (auto simp: divide_simps has_contour_integral_eq)
-    moreover have "((\<lambda>w. f w / (w - z)) has_contour_integral contour_integral \<gamma> (\<lambda>w. f w / (w - z))) \<gamma>"
-            if "z \<notin> U"
-      apply (rule has_contour_integral_integral [OF contour_integrable_holomorphic_simple [where S=U]])
-      using U pasz \<open>valid_path \<gamma>\<close> that
-      apply (auto intro: holomorphic_on_imp_continuous_on hol_d)
-       apply (rule continuous_intros conf holomorphic_intros holf assms | force)+
-      done
-    ultimately show ?thesis
-      using z by (simp add: h_def)
-  qed
-  have znot: "z \<notin> path_image \<gamma>"
-    using pasz by blast
-  obtain d0 where "d0>0" and d0: "\<And>x y. x \<in> path_image \<gamma> \<Longrightarrow> y \<in> - U \<Longrightarrow> d0 \<le> dist x y"
-    using separate_compact_closed [of "path_image \<gamma>" "-U"] pasz \<open>open U\<close> \<open>path \<gamma>\<close> compact_path_image
-    by blast    
-  obtain dd where "0 < dd" and dd: "{y + k | y k. y \<in> path_image \<gamma> \<and> k \<in> ball 0 dd} \<subseteq> U"
-    apply (rule that [of "d0/2"])
-    using \<open>0 < d0\<close>
-    apply (auto simp: dist_norm dest: d0)
-    done
-  have "\<And>x x'. \<lbrakk>x \<in> path_image \<gamma>; dist x x' * 2 < dd\<rbrakk> \<Longrightarrow> \<exists>y k. x' = y + k \<and> y \<in> path_image \<gamma> \<and> dist 0 k * 2 \<le> dd"
-    apply (rule_tac x=x in exI)
-    apply (rule_tac x="x'-x" in exI)
-    apply (force simp: dist_norm)
-    done
-  then have 1: "path_image \<gamma> \<subseteq> interior {y + k |y k. y \<in> path_image \<gamma> \<and> k \<in> cball 0 (dd / 2)}"