diff --git a/src/HOL/Analysis/Elementary_Normed_Spaces.thy b/src/HOL/Analysis/Elementary_Normed_Spaces.thy
--- a/src/HOL/Analysis/Elementary_Normed_Spaces.thy
+++ b/src/HOL/Analysis/Elementary_Normed_Spaces.thy
@@ -1,1755 +1,1789 @@
 (*  Author:     L C Paulson, University of Cambridge
     Author:     Amine Chaieb, University of Cambridge
     Author:     Robert Himmelmann, TU Muenchen
     Author:     Brian Huffman, Portland State University
 *)
 
 section \<open>Elementary Normed Vector Spaces\<close>
 
 theory Elementary_Normed_Spaces
   imports
   "HOL-Library.FuncSet"
   Elementary_Metric_Spaces Cartesian_Space
   Connected
 begin
 subsection \<open>Orthogonal Transformation of Balls\<close>
 
 subsection\<^marker>\<open>tag unimportant\<close> \<open>Various Lemmas Combining Imports\<close>
 
 lemma open_sums:
   fixes T :: "('b::real_normed_vector) set"
   assumes "open S \<or> open T"
   shows "open (\<Union>x\<in> S. \<Union>y \<in> T. {x + y})"
   using assms
 proof
   assume S: "open S"
   show ?thesis
   proof (clarsimp simp: open_dist)
     fix x y
     assume "x \<in> S" "y \<in> T"
     with S obtain e where "e > 0" and e: "\<And>x'. dist x' x < e \<Longrightarrow> x' \<in> S"
       by (auto simp: open_dist)
     then have "\<And>z. dist z (x + y) < e \<Longrightarrow> \<exists>x\<in>S. \<exists>y\<in>T. z = x + y"
       by (metis \<open>y \<in> T\<close> diff_add_cancel dist_add_cancel2)
     then show "\<exists>e>0. \<forall>z. dist z (x + y) < e \<longrightarrow> (\<exists>x\<in>S. \<exists>y\<in>T. z = x + y)"
       using \<open>0 < e\<close> \<open>x \<in> S\<close> by blast
   qed
 next
   assume T: "open T"
   show ?thesis
   proof (clarsimp simp: open_dist)
     fix x y
     assume "x \<in> S" "y \<in> T"
     with T obtain e where "e > 0" and e: "\<And>x'. dist x' y < e \<Longrightarrow> x' \<in> T"
       by (auto simp: open_dist)
     then have "\<And>z. dist z (x + y) < e \<Longrightarrow> \<exists>x\<in>S. \<exists>y\<in>T. z = x + y"
       by (metis \<open>x \<in> S\<close> add_diff_cancel_left' add_diff_eq diff_diff_add dist_norm)
     then show "\<exists>e>0. \<forall>z. dist z (x + y) < e \<longrightarrow> (\<exists>x\<in>S. \<exists>y\<in>T. z = x + y)"
       using \<open>0 < e\<close> \<open>y \<in> T\<close> by blast
   qed
 qed
 
 lemma image_orthogonal_transformation_ball:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'a"
   assumes "orthogonal_transformation f"
   shows "f ` ball x r = ball (f x) r"
 proof (intro equalityI subsetI)
   fix y assume "y \<in> f ` ball x r"
   with assms show "y \<in> ball (f x) r"
     by (auto simp: orthogonal_transformation_isometry)
 next
   fix y assume y: "y \<in> ball (f x) r"
   then obtain z where z: "y = f z"
     using assms orthogonal_transformation_surj by blast
   with y assms show "y \<in> f ` ball x r"
     by (auto simp: orthogonal_transformation_isometry)
 qed
 
 lemma image_orthogonal_transformation_cball:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'a"
   assumes "orthogonal_transformation f"
   shows "f ` cball x r = cball (f x) r"
 proof (intro equalityI subsetI)
   fix y assume "y \<in> f ` cball x r"
   with assms show "y \<in> cball (f x) r"
     by (auto simp: orthogonal_transformation_isometry)
 next
   fix y assume y: "y \<in> cball (f x) r"
   then obtain z where z: "y = f z"
     using assms orthogonal_transformation_surj by blast
   with y assms show "y \<in> f ` cball x r"
     by (auto simp: orthogonal_transformation_isometry)
 qed
 
 
 subsection \<open>Support\<close>
 
 definition (in monoid_add) support_on :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'b set"
   where "support_on S f = {x\<in>S. f x \<noteq> 0}"
 
 lemma in_support_on: "x \<in> support_on S f \<longleftrightarrow> x \<in> S \<and> f x \<noteq> 0"
   by (simp add: support_on_def)
 
 lemma support_on_simps[simp]:
   "support_on {} f = {}"
   "support_on (insert x S) f =
     (if f x = 0 then support_on S f else insert x (support_on S f))"
   "support_on (S \<union> T) f = support_on S f \<union> support_on T f"
   "support_on (S \<inter> T) f = support_on S f \<inter> support_on T f"
   "support_on (S - T) f = support_on S f - support_on T f"
   "support_on (f ` S) g = f ` (support_on S (g \<circ> f))"
   unfolding support_on_def by auto
 
 lemma support_on_cong:
   "(\<And>x. x \<in> S \<Longrightarrow> f x = 0 \<longleftrightarrow> g x = 0) \<Longrightarrow> support_on S f = support_on S g"
   by (auto simp: support_on_def)
 
 lemma support_on_if: "a \<noteq> 0 \<Longrightarrow> support_on A (\<lambda>x. if P x then a else 0) = {x\<in>A. P x}"
   by (auto simp: support_on_def)
 
 lemma support_on_if_subset: "support_on A (\<lambda>x. if P x then a else 0) \<subseteq> {x \<in> A. P x}"
   by (auto simp: support_on_def)
 
 lemma finite_support[intro]: "finite S \<Longrightarrow> finite (support_on S f)"
   unfolding support_on_def by auto
 
 (* TODO: is supp_sum really needed? TODO: Generalize to Finite_Set.fold *)
 definition (in comm_monoid_add) supp_sum :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b set \<Rightarrow> 'a"
   where "supp_sum f S = (\<Sum>x\<in>support_on S f. f x)"
 
 lemma supp_sum_empty[simp]: "supp_sum f {} = 0"
   unfolding supp_sum_def by auto
 
 lemma supp_sum_insert[simp]:
   "finite (support_on S f) \<Longrightarrow>
     supp_sum f (insert x S) = (if x \<in> S then supp_sum f S else f x + supp_sum f S)"
   by (simp add: supp_sum_def in_support_on insert_absorb)
 
 lemma supp_sum_divide_distrib: "supp_sum f A / (r::'a::field) = supp_sum (\<lambda>n. f n / r) A"
   by (cases "r = 0")
      (auto simp: supp_sum_def sum_divide_distrib intro!: sum.cong support_on_cong)
 
 
 subsection \<open>Intervals\<close>
 
 lemma image_affinity_interval:
   fixes c :: "'a::ordered_real_vector"
   shows "((\<lambda>x. m *\<^sub>R x + c) ` {a..b}) = 
            (if {a..b}={} then {}
             else if 0 \<le> m then {m *\<^sub>R a + c .. m  *\<^sub>R b + c}
             else {m *\<^sub>R b + c .. m *\<^sub>R a + c})"
          (is "?lhs = ?rhs")
 proof (cases "m=0")
   case True
   then show ?thesis
     by force
 next
   case False
   show ?thesis
   proof
     show "?lhs \<subseteq> ?rhs"
       by (auto simp: scaleR_left_mono scaleR_left_mono_neg)
     show "?rhs \<subseteq> ?lhs"
     proof (clarsimp, intro conjI impI subsetI)
       show "\<lbrakk>0 \<le> m; a \<le> b; x \<in> {m *\<^sub>R a + c..m *\<^sub>R b + c}\<rbrakk>
             \<Longrightarrow> x \<in> (\<lambda>x. m *\<^sub>R x + c) ` {a..b}" for x
         using False
         by (rule_tac x="inverse m *\<^sub>R (x-c)" in image_eqI)
            (auto simp: pos_le_divideR_eq pos_divideR_le_eq le_diff_eq diff_le_eq)
       show "\<lbrakk>\<not> 0 \<le> m; a \<le> b;  x \<in> {m *\<^sub>R b + c..m *\<^sub>R a + c}\<rbrakk>
             \<Longrightarrow> x \<in> (\<lambda>x. m *\<^sub>R x + c) ` {a..b}" for x
         by (rule_tac x="inverse m *\<^sub>R (x-c)" in image_eqI)
            (auto simp add: neg_le_divideR_eq neg_divideR_le_eq le_diff_eq diff_le_eq)
     qed
   qed
 qed
 
 subsection \<open>Limit Points\<close>
 
 lemma islimpt_ball:
   fixes x y :: "'a::{real_normed_vector,perfect_space}"
   shows "y islimpt ball x e \<longleftrightarrow> 0 < e \<and> y \<in> cball x e"
   (is "?lhs \<longleftrightarrow> ?rhs")
 proof
   show ?rhs if ?lhs
   proof
     {
       assume "e \<le> 0"
       then have *: "ball x e = {}"
         using ball_eq_empty[of x e] by auto
       have False using \<open>?lhs\<close>
         unfolding * using islimpt_EMPTY[of y] by auto
     }
     then show "e > 0" by (metis not_less)
     show "y \<in> cball x e"
       using closed_cball[of x e] islimpt_subset[of y "ball x e" "cball x e"]
         ball_subset_cball[of x e] \<open>?lhs\<close>
       unfolding closed_limpt by auto
   qed
   show ?lhs if ?rhs
   proof -
     from that have "e > 0" by auto
     {
       fix d :: real
       assume "d > 0"
       have "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
       proof (cases "d \<le> dist x y")
         case True
         then show ?thesis
         proof (cases "x = y")
           case True
           then have False
             using \<open>d \<le> dist x y\<close> \<open>d>0\<close> by auto
           then show ?thesis
             by auto
         next
           case False
           have "dist x (y - (d / (2 * dist y x)) *\<^sub>R (y - x)) =
             norm (x - y + (d / (2 * norm (y - x))) *\<^sub>R (y - x))"
             unfolding mem_cball mem_ball dist_norm diff_diff_eq2 diff_add_eq[symmetric]
             by auto
           also have "\<dots> = \<bar>- 1 + d / (2 * norm (x - y))\<bar> * norm (x - y)"
             using scaleR_left_distrib[of "- 1" "d / (2 * norm (y - x))", symmetric, of "y - x"]
             unfolding scaleR_minus_left scaleR_one
             by (auto simp: norm_minus_commute)
           also have "\<dots> = \<bar>- norm (x - y) + d / 2\<bar>"
             unfolding abs_mult_pos[of "norm (x - y)", OF norm_ge_zero[of "x - y"]]
             unfolding distrib_right using \<open>x\<noteq>y\<close>  by auto
           also have "\<dots> \<le> e - d/2" using \<open>d \<le> dist x y\<close> and \<open>d>0\<close> and \<open>?rhs\<close>
             by (auto simp: dist_norm)
           finally have "y - (d / (2 * dist y x)) *\<^sub>R (y - x) \<in> ball x e" using \<open>d>0\<close>
             by auto
           moreover
           have "(d / (2*dist y x)) *\<^sub>R (y - x) \<noteq> 0"
             using \<open>x\<noteq>y\<close>[unfolded dist_nz] \<open>d>0\<close> unfolding scaleR_eq_0_iff
             by (auto simp: dist_commute)
           moreover
           have "dist (y - (d / (2 * dist y x)) *\<^sub>R (y - x)) y < d"
             using \<open>0 < d\<close> by (fastforce simp: dist_norm)
           ultimately show ?thesis
             by (rule_tac x = "y - (d / (2*dist y x)) *\<^sub>R (y - x)" in bexI) auto
         qed
       next
         case False
         then have "d > dist x y" by auto
         show "\<exists>x' \<in> ball x e. x' \<noteq> y \<and> dist x' y < d"
         proof (cases "x = y")
           case True
           obtain z where z: "z \<noteq> y" "dist z y < min e d"
             using perfect_choose_dist[of "min e d" y]
             using \<open>d > 0\<close> \<open>e>0\<close> by auto
           show ?thesis
             by (metis True z dist_commute mem_ball min_less_iff_conj)
         next
           case False
           then show ?thesis
             using \<open>d>0\<close> \<open>d > dist x y\<close> \<open>?rhs\<close> by force
         qed
       qed
     }
     then show ?thesis
       unfolding mem_cball islimpt_approachable mem_ball by auto
   qed
 qed
 
 lemma closure_ball_lemma:
   fixes x y :: "'a::real_normed_vector"
   assumes "x \<noteq> y"
   shows "y islimpt ball x (dist x y)"
 proof (rule islimptI)
   fix T
   assume "y \<in> T" "open T"
   then obtain r where "0 < r" "\<forall>z. dist z y < r \<longrightarrow> z \<in> T"
     unfolding open_dist by fast
   \<comment>\<open>choose point between @{term x} and @{term y}, within distance @{term r} of @{term y}.\<close>
   define k where "k = min 1 (r / (2 * dist x y))"
   define z where "z = y + scaleR k (x - y)"
   have z_def2: "z = x + scaleR (1 - k) (y - x)"
     unfolding z_def by (simp add: algebra_simps)
   have "dist z y < r"
     unfolding z_def k_def using \<open>0 < r\<close>
     by (simp add: dist_norm min_def)
   then have "z \<in> T"
     using \<open>\<forall>z. dist z y < r \<longrightarrow> z \<in> T\<close> by simp
   have "dist x z < dist x y"
     using \<open>0 < r\<close> assms by (simp add: z_def2 k_def dist_norm norm_minus_commute) 
   then have "z \<in> ball x (dist x y)"
     by simp
   have "z \<noteq> y"
     unfolding z_def k_def using \<open>x \<noteq> y\<close> \<open>0 < r\<close>
     by (simp add: min_def)
   show "\<exists>z\<in>ball x (dist x y). z \<in> T \<and> z \<noteq> y"
     using \<open>z \<in> ball x (dist x y)\<close> \<open>z \<in> T\<close> \<open>z \<noteq> y\<close>
     by fast
 qed
 
 
 subsection \<open>Balls and Spheres in Normed Spaces\<close>
 
 lemma mem_ball_0 [simp]: "x \<in> ball 0 e \<longleftrightarrow> norm x < e"
   for x :: "'a::real_normed_vector"
   by simp
 
 lemma mem_cball_0 [simp]: "x \<in> cball 0 e \<longleftrightarrow> norm x \<le> e"
   for x :: "'a::real_normed_vector"
   by simp
 
 lemma closure_ball [simp]:
   fixes x :: "'a::real_normed_vector"
   assumes "0 < e"
   shows "closure (ball x e) = cball x e"
 proof
   show "closure (ball x e) \<subseteq> cball x e"
     using closed_cball closure_minimal by blast
   have "\<And>y. dist x y < e \<or> dist x y = e \<Longrightarrow> y \<in> closure (ball x e)"
     by (metis Un_iff assms closure_ball_lemma closure_def dist_eq_0_iff mem_Collect_eq mem_ball)
   then show "cball x e \<subseteq> closure (ball x e)"
     by force
 qed
 
 lemma mem_sphere_0 [simp]: "x \<in> sphere 0 e \<longleftrightarrow> norm x = e"
   for x :: "'a::real_normed_vector"
   by simp
 
 (* In a trivial vector space, this fails for e = 0. *)
 lemma interior_cball [simp]:
   fixes x :: "'a::{real_normed_vector, perfect_space}"
   shows "interior (cball x e) = ball x e"
 proof (cases "e \<ge> 0")
   case False note cs = this
   from cs have null: "ball x e = {}"
     using ball_empty[of e x] by auto
   moreover
   have "cball x e = {}"
   proof (rule equals0I)
     fix y
     assume "y \<in> cball x e"
     then show False
       by (metis ball_eq_empty null cs dist_eq_0_iff dist_le_zero_iff empty_subsetI mem_cball
           subset_antisym subset_ball)
   qed
   then have "interior (cball x e) = {}"
     using interior_empty by auto
   ultimately show ?thesis by blast
 next
   case True note cs = this
   have "ball x e \<subseteq> cball x e"
     using ball_subset_cball by auto
   moreover
   {
     fix S y
     assume as: "S \<subseteq> cball x e" "open S" "y\<in>S"
     then obtain d where "d>0" and d: "\<forall>x'. dist x' y < d \<longrightarrow> x' \<in> S"
       unfolding open_dist by blast
     then obtain xa where xa_y: "xa \<noteq> y" and xa: "dist xa y < d"
       using perfect_choose_dist [of d] by auto
     have "xa \<in> S"
       using d[THEN spec[where x = xa]]
       using xa by (auto simp: dist_commute)
     then have xa_cball: "xa \<in> cball x e"
       using as(1) by auto
     then have "y \<in> ball x e"
     proof (cases "x = y")
       case True
       then have "e > 0" using cs order.order_iff_strict xa_cball xa_y by fastforce
       then show "y \<in> ball x e"
         using \<open>x = y \<close> by simp
     next
       case False
       have "dist (y + (d / 2 / dist y x) *\<^sub>R (y - x)) y < d"
         unfolding dist_norm
         using \<open>d>0\<close> norm_ge_zero[of "y - x"] \<open>x \<noteq> y\<close> by auto
       then have *: "y + (d / 2 / dist y x) *\<^sub>R (y - x) \<in> cball x e"
         using d as(1)[unfolded subset_eq] by blast
       have "y - x \<noteq> 0" using \<open>x \<noteq> y\<close> by auto
       hence **:"d / (2 * norm (y - x)) > 0"
         unfolding zero_less_norm_iff[symmetric] using \<open>d>0\<close> by auto
       have "dist (y + (d / 2 / dist y x) *\<^sub>R (y - x)) x =
         norm (y + (d / (2 * norm (y - x))) *\<^sub>R y - (d / (2 * norm (y - x))) *\<^sub>R x - x)"
         by (auto simp: dist_norm algebra_simps)
       also have "\<dots> = norm ((1 + d / (2 * norm (y - x))) *\<^sub>R (y - x))"
         by (auto simp: algebra_simps)
       also have "\<dots> = \<bar>1 + d / (2 * norm (y - x))\<bar> * norm (y - x)"
         using ** by auto
       also have "\<dots> = (dist y x) + d/2"
         using ** by (auto simp: distrib_right dist_norm)
       finally have "e \<ge> dist x y +d/2"
         using *[unfolded mem_cball] by (auto simp: dist_commute)
       then show "y \<in> ball x e"
         unfolding mem_ball using \<open>d>0\<close> by auto
     qed
   }
   then have "\<forall>S \<subseteq> cball x e. open S \<longrightarrow> S \<subseteq> ball x e"
     by auto
   ultimately show ?thesis
     using interior_unique[of "ball x e" "cball x e"]
     using open_ball[of x e]
     by auto
 qed
 
 lemma frontier_ball [simp]:
   fixes a :: "'a::real_normed_vector"
   shows "0 < e \<Longrightarrow> frontier (ball a e) = sphere a e"
   by (force simp: frontier_def)
 
 lemma frontier_cball [simp]:
   fixes a :: "'a::{real_normed_vector, perfect_space}"
   shows "frontier (cball a e) = sphere a e"
   by (force simp: frontier_def)
 
 corollary compact_sphere [simp]:
   fixes a :: "'a::{real_normed_vector,perfect_space,heine_borel}"
   shows "compact (sphere a r)"
 using compact_frontier [of "cball a r"] by simp
 
 corollary bounded_sphere [simp]:
   fixes a :: "'a::{real_normed_vector,perfect_space,heine_borel}"
   shows "bounded (sphere a r)"
 by (simp add: compact_imp_bounded)
 
 corollary closed_sphere  [simp]:
   fixes a :: "'a::{real_normed_vector,perfect_space,heine_borel}"
   shows "closed (sphere a r)"
 by (simp add: compact_imp_closed)
 
 lemma image_add_ball [simp]:
   fixes a :: "'a::real_normed_vector"
   shows "(+) b ` ball a r = ball (a+b) r"
 proof -
   { fix x :: 'a
     assume "dist (a + b) x < r"
     moreover
     have "b + (x - b) = x"
       by simp
     ultimately have "x \<in> (+) b ` ball a r"
       by (metis add.commute dist_add_cancel image_eqI mem_ball) }
   then show ?thesis
     by (auto simp: add.commute)
 qed
 
 lemma image_add_cball [simp]:
   fixes a :: "'a::real_normed_vector"
   shows "(+) b ` cball a r = cball (a+b) r"
 proof -
   have "\<And>x. dist (a + b) x \<le> r \<Longrightarrow> \<exists>y\<in>cball a r. x = b + y"
     by (metis (no_types) add.commute diff_add_cancel dist_add_cancel2 mem_cball)
   then show ?thesis
     by (force simp: add.commute)
 qed
 
 
 subsection\<^marker>\<open>tag unimportant\<close> \<open>Various Lemmas on Normed Algebras\<close>
 
 
 lemma closed_of_nat_image: "closed (of_nat ` A :: 'a::real_normed_algebra_1 set)"
   by (rule discrete_imp_closed[of 1]) (auto simp: dist_of_nat)
 
 lemma closed_of_int_image: "closed (of_int ` A :: 'a::real_normed_algebra_1 set)"
   by (rule discrete_imp_closed[of 1]) (auto simp: dist_of_int)
 
 lemma closed_Nats [simp]: "closed (\<nat> :: 'a :: real_normed_algebra_1 set)"
   unfolding Nats_def by (rule closed_of_nat_image)
 
 lemma closed_Ints [simp]: "closed (\<int> :: 'a :: real_normed_algebra_1 set)"
   unfolding Ints_def by (rule closed_of_int_image)
 
 lemma closed_subset_Ints:
   fixes A :: "'a :: real_normed_algebra_1 set"
   assumes "A \<subseteq> \<int>"
   shows   "closed A"
 proof (intro discrete_imp_closed[OF zero_less_one] ballI impI, goal_cases)
   case (1 x y)
   with assms have "x \<in> \<int>" and "y \<in> \<int>" by auto
   with \<open>dist y x < 1\<close> show "y = x"
     by (auto elim!: Ints_cases simp: dist_of_int)
 qed
 
 subsection \<open>Filters\<close>
 
 definition indirection :: "'a::real_normed_vector \<Rightarrow> 'a \<Rightarrow> 'a filter"  (infixr "indirection" 70)
   where "a indirection v = at a within {b. \<exists>c\<ge>0. b - a = scaleR c v}"
 
 
 subsection \<open>Trivial Limits\<close>
 
 lemma trivial_limit_at_infinity:
   "\<not> trivial_limit (at_infinity :: ('a::{real_normed_vector,perfect_space}) filter)"
 proof -
   obtain x::'a where "x \<noteq> 0"
     by (meson perfect_choose_dist zero_less_one)
   then have "b \<le> norm ((b / norm x) *\<^sub>R x)" for b
     by simp
   then show ?thesis
     unfolding trivial_limit_def eventually_at_infinity
     by blast
 qed
 
 lemma at_within_ball_bot_iff:
   fixes x y :: "'a::{real_normed_vector,perfect_space}"
   shows "at x within ball y r = bot \<longleftrightarrow> (r=0 \<or> x \<notin> cball y r)"
   unfolding trivial_limit_within
   by (metis (no_types) cball_empty equals0D islimpt_ball less_linear) 
 
 
 subsection \<open>Limits\<close>
 
 proposition Lim_at_infinity: "(f \<longlongrightarrow> l) at_infinity \<longleftrightarrow> (\<forall>e>0. \<exists>b. \<forall>x. norm x \<ge> b \<longrightarrow> dist (f x) l < e)"
   by (auto simp: tendsto_iff eventually_at_infinity)
 
 corollary Lim_at_infinityI [intro?]:
   assumes "\<And>e. e > 0 \<Longrightarrow> \<exists>B. \<forall>x. norm x \<ge> B \<longrightarrow> dist (f x) l \<le> e"
   shows "(f \<longlongrightarrow> l) at_infinity"
 proof -
   have "\<And>e. e > 0 \<Longrightarrow> \<exists>B. \<forall>x. norm x \<ge> B \<longrightarrow> dist (f x) l < e"
     by (meson assms dense le_less_trans)
   then show ?thesis
     using Lim_at_infinity by blast
 qed
 
 lemma Lim_transform_within_set_eq:
   fixes a :: "'a::metric_space" and l :: "'b::metric_space"
   shows "eventually (\<lambda>x. x \<in> S \<longleftrightarrow> x \<in> T) (at a)
          \<Longrightarrow> ((f \<longlongrightarrow> l) (at a within S) \<longleftrightarrow> (f \<longlongrightarrow> l) (at a within T))"
   by (force intro: Lim_transform_within_set elim: eventually_mono)
 
 lemma Lim_null:
   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
   shows "(f \<longlongrightarrow> l) net \<longleftrightarrow> ((\<lambda>x. f(x) - l) \<longlongrightarrow> 0) net"
   by (simp add: Lim dist_norm)
 
 lemma Lim_null_comparison:
   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
   assumes "eventually (\<lambda>x. norm (f x) \<le> g x) net" "(g \<longlongrightarrow> 0) net"
   shows "(f \<longlongrightarrow> 0) net"
   using assms(2)
 proof (rule metric_tendsto_imp_tendsto)
   show "eventually (\<lambda>x. dist (f x) 0 \<le> dist (g x) 0) net"
     using assms(1) by (rule eventually_mono) (simp add: dist_norm)
 qed
 
 lemma Lim_transform_bound:
   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
     and g :: "'a \<Rightarrow> 'c::real_normed_vector"
   assumes "eventually (\<lambda>n. norm (f n) \<le> norm (g n)) net"
     and "(g \<longlongrightarrow> 0) net"
   shows "(f \<longlongrightarrow> 0) net"
   using assms(1) tendsto_norm_zero [OF assms(2)]
   by (rule Lim_null_comparison)
 
 lemma lim_null_mult_right_bounded:
   fixes f :: "'a \<Rightarrow> 'b::real_normed_div_algebra"
   assumes f: "(f \<longlongrightarrow> 0) F" and g: "eventually (\<lambda>x. norm(g x) \<le> B) F"
     shows "((\<lambda>z. f z * g z) \<longlongrightarrow> 0) F"
 proof -
   have "((\<lambda>x. norm (f x) * norm (g x)) \<longlongrightarrow> 0) F"
   proof (rule Lim_null_comparison)
     show "\<forall>\<^sub>F x in F. norm (norm (f x) * norm (g x)) \<le> norm (f x) * B"
       by (simp add: eventually_mono [OF g] mult_left_mono)
     show "((\<lambda>x. norm (f x) * B) \<longlongrightarrow> 0) F"
       by (simp add: f tendsto_mult_left_zero tendsto_norm_zero)
   qed
   then show ?thesis
     by (subst tendsto_norm_zero_iff [symmetric]) (simp add: norm_mult)
 qed
 
 lemma lim_null_mult_left_bounded:
   fixes f :: "'a \<Rightarrow> 'b::real_normed_div_algebra"
   assumes g: "eventually (\<lambda>x. norm(g x) \<le> B) F" and f: "(f \<longlongrightarrow> 0) F"
     shows "((\<lambda>z. g z * f z) \<longlongrightarrow> 0) F"
 proof -
   have "((\<lambda>x. norm (g x) * norm (f x)) \<longlongrightarrow> 0) F"
   proof (rule Lim_null_comparison)
     show "\<forall>\<^sub>F x in F. norm (norm (g x) * norm (f x)) \<le> B * norm (f x)"
       by (simp add: eventually_mono [OF g] mult_right_mono)
     show "((\<lambda>x. B * norm (f x)) \<longlongrightarrow> 0) F"
       by (simp add: f tendsto_mult_right_zero tendsto_norm_zero)
   qed
   then show ?thesis
     by (subst tendsto_norm_zero_iff [symmetric]) (simp add: norm_mult)
 qed
 
 lemma lim_null_scaleR_bounded:
   assumes f: "(f \<longlongrightarrow> 0) net" and gB: "eventually (\<lambda>a. f a = 0 \<or> norm(g a) \<le> B) net"
     shows "((\<lambda>n. f n *\<^sub>R g n) \<longlongrightarrow> 0) net"
 proof
   fix \<epsilon>::real
   assume "0 < \<epsilon>"
   then have B: "0 < \<epsilon> / (abs B + 1)" by simp
   have *: "\<bar>f x\<bar> * norm (g x) < \<epsilon>" if f: "\<bar>f x\<bar> * (\<bar>B\<bar> + 1) < \<epsilon>" and g: "norm (g x) \<le> B" for x
   proof -
     have "\<bar>f x\<bar> * norm (g x) \<le> \<bar>f x\<bar> * B"
       by (simp add: mult_left_mono g)
     also have "\<dots> \<le> \<bar>f x\<bar> * (\<bar>B\<bar> + 1)"
       by (simp add: mult_left_mono)
     also have "\<dots> < \<epsilon>"
       by (rule f)
     finally show ?thesis .
   qed
   have "\<And>x. \<lbrakk>\<bar>f x\<bar> < \<epsilon> / (\<bar>B\<bar> + 1); norm (g x) \<le> B\<rbrakk> \<Longrightarrow> \<bar>f x\<bar> * norm (g x) < \<epsilon>"
     by (simp add: "*" pos_less_divide_eq)
   then show "\<forall>\<^sub>F x in net. dist (f x *\<^sub>R g x) 0 < \<epsilon>"
     using \<open>0 < \<epsilon>\<close> by (auto intro: eventually_mono [OF eventually_conj [OF tendstoD [OF f B] gB]])
 qed
 
 lemma Lim_norm_ubound:
   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
   assumes "\<not>(trivial_limit net)" "(f \<longlongrightarrow> l) net" "eventually (\<lambda>x. norm(f x) \<le> e) net"
   shows "norm(l) \<le> e"
   using assms by (fast intro: tendsto_le tendsto_intros)
 
 lemma Lim_norm_lbound:
   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
   assumes "\<not> trivial_limit net"
     and "(f \<longlongrightarrow> l) net"
     and "eventually (\<lambda>x. e \<le> norm (f x)) net"
   shows "e \<le> norm l"
   using assms by (fast intro: tendsto_le tendsto_intros)
 
 text\<open>Limit under bilinear function\<close>
 
 lemma Lim_bilinear:
   assumes "(f \<longlongrightarrow> l) net"
     and "(g \<longlongrightarrow> m) net"
     and "bounded_bilinear h"
   shows "((\<lambda>x. h (f x) (g x)) \<longlongrightarrow> (h l m)) net"
   using \<open>bounded_bilinear h\<close> \<open>(f \<longlongrightarrow> l) net\<close> \<open>(g \<longlongrightarrow> m) net\<close>
   by (rule bounded_bilinear.tendsto)
 
 lemma Lim_at_zero:
   fixes a :: "'a::real_normed_vector"
     and l :: "'b::topological_space"
   shows "(f \<longlongrightarrow> l) (at a) \<longleftrightarrow> ((\<lambda>x. f(a + x)) \<longlongrightarrow> l) (at 0)"
   using LIM_offset_zero LIM_offset_zero_cancel ..
 
 
 subsection\<^marker>\<open>tag unimportant\<close> \<open>Limit Point of Filter\<close>
 
 lemma netlimit_at_vector:
   fixes a :: "'a::real_normed_vector"
   shows "netlimit (at a) = a"
 proof (cases "\<exists>x. x \<noteq> a")
   case True then obtain x where x: "x \<noteq> a" ..
   have "\<And>d. 0 < d \<Longrightarrow> \<exists>x. x \<noteq> a \<and> norm (x - a) < d"
     by (rule_tac x="a + scaleR (d / 2) (sgn (x - a))" in exI) (simp add: norm_sgn sgn_zero_iff x)
   then have "\<not> trivial_limit (at a)"
     by (auto simp: trivial_limit_def eventually_at dist_norm)
   then show ?thesis
     by (rule Lim_ident_at [of a UNIV])
 qed simp
 
 subsection \<open>Boundedness\<close>
 
 lemma continuous_on_closure_norm_le:
   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
   assumes "continuous_on (closure s) f"
     and "\<forall>y \<in> s. norm(f y) \<le> b"
     and "x \<in> (closure s)"
   shows "norm (f x) \<le> b"
 proof -
   have *: "f ` s \<subseteq> cball 0 b"
     using assms(2)[unfolded mem_cball_0[symmetric]] by auto
   show ?thesis
     by (meson "*" assms(1) assms(3) closed_cball image_closure_subset image_subset_iff mem_cball_0)
 qed
 
 lemma bounded_pos: "bounded S \<longleftrightarrow> (\<exists>b>0. \<forall>x\<in> S. norm x \<le> b)"
   unfolding bounded_iff 
   by (meson less_imp_le not_le order_trans zero_less_one)
 
 lemma bounded_pos_less: "bounded S \<longleftrightarrow> (\<exists>b>0. \<forall>x\<in> S. norm x < b)"
   by (metis bounded_pos le_less_trans less_imp_le linordered_field_no_ub)
 
 lemma Bseq_eq_bounded:
   fixes f :: "nat \<Rightarrow> 'a::real_normed_vector"
   shows "Bseq f \<longleftrightarrow> bounded (range f)"
   unfolding Bseq_def bounded_pos by auto
 
 lemma bounded_linear_image:
   assumes "bounded S"
     and "bounded_linear f"
   shows "bounded (f ` S)"
 proof -
   from assms(1) obtain b where "b > 0" and b: "\<forall>x\<in>S. norm x \<le> b"
     unfolding bounded_pos by auto
   from assms(2) obtain B where B: "B > 0" "\<forall>x. norm (f x) \<le> B * norm x"
     using bounded_linear.pos_bounded by (auto simp: ac_simps)
   show ?thesis
     unfolding bounded_pos
   proof (intro exI, safe)
     show "norm (f x) \<le> B * b" if "x \<in> S" for x
       by (meson B b less_imp_le mult_left_mono order_trans that)
   qed (use \<open>b > 0\<close> \<open>B > 0\<close> in auto)
 qed
 
 lemma bounded_scaling:
   fixes S :: "'a::real_normed_vector set"
   shows "bounded S \<Longrightarrow> bounded ((\<lambda>x. c *\<^sub>R x) ` S)"
   by (simp add: bounded_linear_image bounded_linear_scaleR_right)
 
 lemma bounded_scaleR_comp:
   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
   assumes "bounded (f ` S)"
   shows "bounded ((\<lambda>x. r *\<^sub>R f x) ` S)"
   using bounded_scaling[of "f ` S" r] assms
   by (auto simp: image_image)
 
 lemma bounded_translation:
   fixes S :: "'a::real_normed_vector set"
   assumes "bounded S"
   shows "bounded ((\<lambda>x. a + x) ` S)"
 proof -
   from assms obtain b where b: "b > 0" "\<forall>x\<in>S. norm x \<le> b"
     unfolding bounded_pos by auto
   {
     fix x
     assume "x \<in> S"
     then have "norm (a + x) \<le> b + norm a"
       using norm_triangle_ineq[of a x] b by auto
   }
   then show ?thesis
     unfolding bounded_pos
     using norm_ge_zero[of a] b(1) and add_strict_increasing[of b 0 "norm a"]
     by (auto intro!: exI[of _ "b + norm a"])
 qed
 
 lemma bounded_translation_minus:
   fixes S :: "'a::real_normed_vector set"
   shows "bounded S \<Longrightarrow> bounded ((\<lambda>x. x - a) ` S)"
 using bounded_translation [of S "-a"] by simp
 
 lemma bounded_uminus [simp]:
   fixes X :: "'a::real_normed_vector set"
   shows "bounded (uminus ` X) \<longleftrightarrow> bounded X"
 by (auto simp: bounded_def dist_norm; rule_tac x="-x" in exI; force simp: add.commute norm_minus_commute)
 
 lemma uminus_bounded_comp [simp]:
   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
   shows "bounded ((\<lambda>x. - f x) ` S) \<longleftrightarrow> bounded (f ` S)"
   using bounded_uminus[of "f ` S"]
   by (auto simp: image_image)
 
 lemma bounded_plus_comp:
   fixes f g::"'a \<Rightarrow> 'b::real_normed_vector"
   assumes "bounded (f ` S)"
   assumes "bounded (g ` S)"
   shows "bounded ((\<lambda>x. f x + g x) ` S)"
 proof -
   {
     fix B C
     assume "\<And>x. x\<in>S \<Longrightarrow> norm (f x) \<le> B" "\<And>x. x\<in>S \<Longrightarrow> norm (g x) \<le> C"
     then have "\<And>x. x \<in> S \<Longrightarrow> norm (f x + g x) \<le> B + C"
       by (auto intro!: norm_triangle_le add_mono)
   } then show ?thesis
     using assms by (fastforce simp: bounded_iff)
 qed
 
 lemma bounded_plus:
   fixes S ::"'a::real_normed_vector set"
   assumes "bounded S" "bounded T"
   shows "bounded ((\<lambda>(x,y). x + y) ` (S \<times> T))"
   using bounded_plus_comp [of fst "S \<times> T" snd] assms
   by (auto simp: split_def split: if_split_asm)
 
 lemma bounded_minus_comp:
   "bounded (f ` S) \<Longrightarrow> bounded (g ` S) \<Longrightarrow> bounded ((\<lambda>x. f x - g x) ` S)"
   for f g::"'a \<Rightarrow> 'b::real_normed_vector"
   using bounded_plus_comp[of "f" S "\<lambda>x. - g x"]
   by auto
 
 lemma bounded_minus:
   fixes S ::"'a::real_normed_vector set"
   assumes "bounded S" "bounded T"
   shows "bounded ((\<lambda>(x,y). x - y) ` (S \<times> T))"
   using bounded_minus_comp [of fst "S \<times> T" snd] assms
   by (auto simp: split_def split: if_split_asm)
 
+lemma bounded_sums:
+  fixes S :: "'a::real_normed_vector set"
+  assumes "bounded S" and "bounded T"
+  shows "bounded (\<Union>x\<in> S. \<Union>y \<in> T. {x + y})"
+  using assms by (simp add: bounded_iff) (meson norm_triangle_mono)
+
+lemma bounded_differences:
+  fixes S :: "'a::real_normed_vector set"
+  assumes "bounded S" and "bounded T"
+  shows "bounded (\<Union>x\<in> S. \<Union>y \<in> T. {x - y})"
+  using assms by (simp add: bounded_iff) (meson add_mono norm_triangle_le_diff)
+
 lemma not_bounded_UNIV[simp]:
   "\<not> bounded (UNIV :: 'a::{real_normed_vector, perfect_space} set)"
 proof (auto simp: bounded_pos not_le)
   obtain x :: 'a where "x \<noteq> 0"
     using perfect_choose_dist [OF zero_less_one] by fast
   fix b :: real
   assume b: "b >0"
   have b1: "b +1 \<ge> 0"
     using b by simp
   with \<open>x \<noteq> 0\<close> have "b < norm (scaleR (b + 1) (sgn x))"
     by (simp add: norm_sgn)
   then show "\<exists>x::'a. b < norm x" ..
 qed
 
 corollary cobounded_imp_unbounded:
     fixes S :: "'a::{real_normed_vector, perfect_space} set"
     shows "bounded (- S) \<Longrightarrow> \<not> bounded S"
   using bounded_Un [of S "-S"]  by (simp)
 
 subsection\<^marker>\<open>tag unimportant\<close>\<open>Relations among convergence and absolute convergence for power series\<close>
 
 lemma summable_imp_bounded:
   fixes f :: "nat \<Rightarrow> 'a::real_normed_vector"
   shows "summable f \<Longrightarrow> bounded (range f)"
 by (frule summable_LIMSEQ_zero) (simp add: convergent_imp_bounded)
 
 lemma summable_imp_sums_bounded:
    "summable f \<Longrightarrow> bounded (range (\<lambda>n. sum f {..<n}))"
 by (auto simp: summable_def sums_def dest: convergent_imp_bounded)
 
 lemma power_series_conv_imp_absconv_weak:
   fixes a:: "nat \<Rightarrow> 'a::{real_normed_div_algebra,banach}" and w :: 'a
   assumes sum: "summable (\<lambda>n. a n * z ^ n)" and no: "norm w < norm z"
     shows "summable (\<lambda>n. of_real(norm(a n)) * w ^ n)"
 proof -
   obtain M where M: "\<And>x. norm (a x * z ^ x) \<le> M"
     using summable_imp_bounded [OF sum] by (force simp: bounded_iff)
   show ?thesis
   proof (rule series_comparison_complex)
     have "\<And>n. norm (a n) * norm z ^ n \<le> M"
       by (metis (no_types) M norm_mult norm_power)
     then show "summable (\<lambda>n. complex_of_real (norm (a n) * norm w ^ n))"
       using Abel_lemma no norm_ge_zero summable_of_real by blast
   qed (auto simp: norm_mult norm_power)
 qed
 
 
 subsection \<open>Normed spaces with the Heine-Borel property\<close>
 
 lemma not_compact_UNIV[simp]:
   fixes s :: "'a::{real_normed_vector,perfect_space,heine_borel} set"
   shows "\<not> compact (UNIV::'a set)"
     by (simp add: compact_eq_bounded_closed)
 
 lemma not_compact_space_euclideanreal [simp]: "\<not> compact_space euclideanreal"
   by (simp add: compact_space_def)
 
 text\<open>Representing sets as the union of a chain of compact sets.\<close>
 lemma closed_Union_compact_subsets:
   fixes S :: "'a::{heine_borel,real_normed_vector} set"
   assumes "closed S"
   obtains F where "\<And>n. compact(F n)" "\<And>n. F n \<subseteq> S" "\<And>n. F n \<subseteq> F(Suc n)"
                   "(\<Union>n. F n) = S" "\<And>K. \<lbrakk>compact K; K \<subseteq> S\<rbrakk> \<Longrightarrow> \<exists>N. \<forall>n \<ge> N. K \<subseteq> F n"
 proof
   show "compact (S \<inter> cball 0 (of_nat n))" for n
     using assms compact_eq_bounded_closed by auto
 next
   show "(\<Union>n. S \<inter> cball 0 (real n)) = S"
     by (auto simp: real_arch_simple)
 next
   fix K :: "'a set"
   assume "compact K" "K \<subseteq> S"
   then obtain N where "K \<subseteq> cball 0 N"
     by (meson bounded_pos mem_cball_0 compact_imp_bounded subsetI)
   then show "\<exists>N. \<forall>n\<ge>N. K \<subseteq> S \<inter> cball 0 (real n)"
     by (metis of_nat_le_iff Int_subset_iff \<open>K \<subseteq> S\<close> real_arch_simple subset_cball subset_trans)
 qed auto
 
 subsection \<open>Intersecting chains of compact sets and the Baire property\<close>
 
 proposition bounded_closed_chain:
   fixes \<F> :: "'a::heine_borel set set"
   assumes "B \<in> \<F>" "bounded B" and \<F>: "\<And>S. S \<in> \<F> \<Longrightarrow> closed S" and "{} \<notin> \<F>"
       and chain: "\<And>S T. S \<in> \<F> \<and> T \<in> \<F> \<Longrightarrow> S \<subseteq> T \<or> T \<subseteq> S"
     shows "\<Inter>\<F> \<noteq> {}"
 proof -
   have "B \<inter> \<Inter>\<F> \<noteq> {}"
   proof (rule compact_imp_fip)
     show "compact B" "\<And>T. T \<in> \<F> \<Longrightarrow> closed T"
       by (simp_all add: assms compact_eq_bounded_closed)
     show "\<lbrakk>finite \<G>; \<G> \<subseteq> \<F>\<rbrakk> \<Longrightarrow> B \<inter> \<Inter>\<G> \<noteq> {}" for \<G>
     proof (induction \<G> rule: finite_induct)
       case empty
       with assms show ?case by force
     next
       case (insert U \<G>)
       then have "U \<in> \<F>" and ne: "B \<inter> \<Inter>\<G> \<noteq> {}" by auto
       then consider "B \<subseteq> U" | "U \<subseteq> B"
           using \<open>B \<in> \<F>\<close> chain by blast
         then show ?case
         proof cases
           case 1
           then show ?thesis
             using Int_left_commute ne by auto
         next
           case 2
           have "U \<noteq> {}"
             using \<open>U \<in> \<F>\<close> \<open>{} \<notin> \<F>\<close> by blast
           moreover
           have False if "\<And>x. x \<in> U \<Longrightarrow> \<exists>Y\<in>\<G>. x \<notin> Y"
           proof -
             have "\<And>x. x \<in> U \<Longrightarrow> \<exists>Y\<in>\<G>. Y \<subseteq> U"
               by (metis chain contra_subsetD insert.prems insert_subset that)
             then obtain Y where "Y \<in> \<G>" "Y \<subseteq> U"
               by (metis all_not_in_conv \<open>U \<noteq> {}\<close>)
             moreover obtain x where "x \<in> \<Inter>\<G>"
               by (metis Int_emptyI ne)
             ultimately show ?thesis
               by (metis Inf_lower subset_eq that)
           qed
           with 2 show ?thesis
             by blast
         qed
       qed
   qed
   then show ?thesis by blast
 qed
 
 corollary compact_chain:
   fixes \<F> :: "'a::heine_borel set set"
   assumes "\<And>S. S \<in> \<F> \<Longrightarrow> compact S" "{} \<notin> \<F>"
           "\<And>S T. S \<in> \<F> \<and> T \<in> \<F> \<Longrightarrow> S \<subseteq> T \<or> T \<subseteq> S"
     shows "\<Inter> \<F> \<noteq> {}"
 proof (cases "\<F> = {}")
   case True
   then show ?thesis by auto
 next
   case False
   show ?thesis
     by (metis False all_not_in_conv assms compact_imp_bounded compact_imp_closed bounded_closed_chain)
 qed
 
 lemma compact_nest:
   fixes F :: "'a::linorder \<Rightarrow> 'b::heine_borel set"
   assumes F: "\<And>n. compact(F n)" "\<And>n. F n \<noteq> {}" and mono: "\<And>m n. m \<le> n \<Longrightarrow> F n \<subseteq> F m"
   shows "\<Inter>(range F) \<noteq> {}"
 proof -
   have *: "\<And>S T. S \<in> range F \<and> T \<in> range F \<Longrightarrow> S \<subseteq> T \<or> T \<subseteq> S"
     by (metis mono image_iff le_cases)
   show ?thesis
     using F by (intro compact_chain [OF _ _ *]; blast dest: *)
 qed
 
 text\<open>The Baire property of dense sets\<close>
 theorem Baire:
   fixes S::"'a::{real_normed_vector,heine_borel} set"
   assumes "closed S" "countable \<G>"
       and ope: "\<And>T. T \<in> \<G> \<Longrightarrow> openin (top_of_set S) T \<and> S \<subseteq> closure T"
  shows "S \<subseteq> closure(\<Inter>\<G>)"
 proof (cases "\<G> = {}")
   case True
   then show ?thesis
     using closure_subset by auto
 next
   let ?g = "from_nat_into \<G>"
   case False
   then have gin: "?g n \<in> \<G>" for n
     by (simp add: from_nat_into)
   show ?thesis
   proof (clarsimp simp: closure_approachable)
     fix x and e::real
     assume "x \<in> S" "0 < e"
     obtain TF where opeF: "\<And>n. openin (top_of_set S) (TF n)"
                and ne: "\<And>n. TF n \<noteq> {}"
                and subg: "\<And>n. S \<inter> closure(TF n) \<subseteq> ?g n"
                and subball: "\<And>n. closure(TF n) \<subseteq> ball x e"
                and decr: "\<And>n. TF(Suc n) \<subseteq> TF n"
     proof -
       have *: "\<exists>Y. (openin (top_of_set S) Y \<and> Y \<noteq> {} \<and>
                    S \<inter> closure Y \<subseteq> ?g n \<and> closure Y \<subseteq> ball x e) \<and> Y \<subseteq> U"
         if opeU: "openin (top_of_set S) U" and "U \<noteq> {}" and cloU: "closure U \<subseteq> ball x e" for U n
       proof -
         obtain T where T: "open T" "U = T \<inter> S"
           using \<open>openin (top_of_set S) U\<close> by (auto simp: openin_subtopology)
         with \<open>U \<noteq> {}\<close> have "T \<inter> closure (?g n) \<noteq> {}"
           using gin ope by fastforce
         then have "T \<inter> ?g n \<noteq> {}"
           using \<open>open T\<close> open_Int_closure_eq_empty by blast
         then obtain y where "y \<in> U" "y \<in> ?g n"
           using T ope [of "?g n", OF gin] by (blast dest:  openin_imp_subset)
         moreover have "openin (top_of_set S) (U \<inter> ?g n)"
           using gin ope opeU by blast
         ultimately obtain d where U: "U \<inter> ?g n \<subseteq> S" and "d > 0" and d: "ball y d \<inter> S \<subseteq> U \<inter> ?g n"
           by (force simp: openin_contains_ball)
         show ?thesis
         proof (intro exI conjI)
           show "openin (top_of_set S) (S \<inter> ball y (d/2))"
             by (simp add: openin_open_Int)
           show "S \<inter> ball y (d/2) \<noteq> {}"
             using \<open>0 < d\<close> \<open>y \<in> U\<close> opeU openin_imp_subset by fastforce
           have "S \<inter> closure (S \<inter> ball y (d/2)) \<subseteq> S \<inter> closure (ball y (d/2))"
             using closure_mono by blast
           also have "... \<subseteq> ?g n"
             using \<open>d > 0\<close> d by force
           finally show "S \<inter> closure (S \<inter> ball y (d/2)) \<subseteq> ?g n" .
           have "closure (S \<inter> ball y (d/2)) \<subseteq> S \<inter> ball y d"
           proof -
             have "closure (ball y (d/2)) \<subseteq> ball y d"
               using \<open>d > 0\<close> by auto
             then have "closure (S \<inter> ball y (d/2)) \<subseteq> ball y d"
               by (meson closure_mono inf.cobounded2 subset_trans)
             then show ?thesis
               by (simp add: \<open>closed S\<close> closure_minimal)
           qed
           also have "...  \<subseteq> ball x e"
             using cloU closure_subset d by blast
           finally show "closure (S \<inter> ball y (d/2)) \<subseteq> ball x e" .
           show "S \<inter> ball y (d/2) \<subseteq> U"
             using ball_divide_subset_numeral d by blast
         qed
       qed
       let ?\<Phi> = "\<lambda>n X. openin (top_of_set S) X \<and> X \<noteq> {} \<and>
                       S \<inter> closure X \<subseteq> ?g n \<and> closure X \<subseteq> ball x e"
       have "closure (S \<inter> ball x (e/2)) \<subseteq> closure(ball x (e/2))"
         by (simp add: closure_mono)
       also have "...  \<subseteq> ball x e"
         using \<open>e > 0\<close> by auto
       finally have "closure (S \<inter> ball x (e/2)) \<subseteq> ball x e" .
       moreover have"openin (top_of_set S) (S \<inter> ball x (e/2))" "S \<inter> ball x (e/2) \<noteq> {}"
         using \<open>0 < e\<close> \<open>x \<in> S\<close> by auto
       ultimately obtain Y where Y: "?\<Phi> 0 Y \<and> Y \<subseteq> S \<inter> ball x (e/2)"
             using * [of "S \<inter> ball x (e/2)" 0] by metis
       show thesis
       proof (rule exE [OF dependent_nat_choice])
         show "\<exists>x. ?\<Phi> 0 x"
           using Y by auto
         show "\<exists>Y. ?\<Phi> (Suc n) Y \<and> Y \<subseteq> X" if "?\<Phi> n X" for X n
           using that by (blast intro: *)
       qed (use that in metis)
     qed
     have "(\<Inter>n. S \<inter> closure (TF n)) \<noteq> {}"
     proof (rule compact_nest)
       show "\<And>n. compact (S \<inter> closure (TF n))"
         by (metis closed_closure subball bounded_subset_ballI compact_eq_bounded_closed closed_Int_compact [OF \<open>closed S\<close>])
       show "\<And>n. S \<inter> closure (TF n) \<noteq> {}"
         by (metis Int_absorb1 opeF \<open>closed S\<close> closure_eq_empty closure_minimal ne openin_imp_subset)
       show "\<And>m n. m \<le> n \<Longrightarrow> S \<inter> closure (TF n) \<subseteq> S \<inter> closure (TF m)"
         by (meson closure_mono decr dual_order.refl inf_mono lift_Suc_antimono_le)
     qed
     moreover have "(\<Inter>n. S \<inter> closure (TF n)) \<subseteq> {y \<in> \<Inter>\<G>. dist y x < e}"
     proof (clarsimp, intro conjI)
       fix y
       assume "y \<in> S" and y: "\<forall>n. y \<in> closure (TF n)"
       then show "\<forall>T\<in>\<G>. y \<in> T"
         by (metis Int_iff from_nat_into_surj [OF \<open>countable \<G>\<close>] subsetD subg)
       show "dist y x < e"
         by (metis y dist_commute mem_ball subball subsetCE)
     qed
     ultimately show "\<exists>y \<in> \<Inter>\<G>. dist y x < e"
       by auto
   qed
 qed
 
 
 subsection \<open>Continuity\<close>
 
 subsubsection\<^marker>\<open>tag unimportant\<close> \<open>Structural rules for uniform continuity\<close>
 
 lemma (in bounded_linear) uniformly_continuous_on[continuous_intros]:
   fixes g :: "_::metric_space \<Rightarrow> _"
   assumes "uniformly_continuous_on s g"
   shows "uniformly_continuous_on s (\<lambda>x. f (g x))"
   using assms unfolding uniformly_continuous_on_sequentially
   unfolding dist_norm tendsto_norm_zero_iff diff[symmetric]
   by (auto intro: tendsto_zero)
 
 lemma uniformly_continuous_on_dist[continuous_intros]:
   fixes f g :: "'a::metric_space \<Rightarrow> 'b::metric_space"
   assumes "uniformly_continuous_on s f"
     and "uniformly_continuous_on s g"
   shows "uniformly_continuous_on s (\<lambda>x. dist (f x) (g x))"
 proof -
   {
     fix a b c d :: 'b
     have "\<bar>dist a b - dist c d\<bar> \<le> dist a c + dist b d"
       using dist_triangle2 [of a b c] dist_triangle2 [of b c d]
       using dist_triangle3 [of c d a] dist_triangle [of a d b]
       by arith
   } note le = this
   {
     fix x y
     assume f: "(\<lambda>n. dist (f (x n)) (f (y n))) \<longlonglongrightarrow> 0"
     assume g: "(\<lambda>n. dist (g (x n)) (g (y n))) \<longlonglongrightarrow> 0"
     have "(\<lambda>n. \<bar>dist (f (x n)) (g (x n)) - dist (f (y n)) (g (y n))\<bar>) \<longlonglongrightarrow> 0"
       by (rule Lim_transform_bound [OF _ tendsto_add_zero [OF f g]],
         simp add: le)
   }
   then show ?thesis
     using assms unfolding uniformly_continuous_on_sequentially
     unfolding dist_real_def by simp
 qed
 
 lemma uniformly_continuous_on_cmul_right [continuous_intros]:
   fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_algebra"
   shows "uniformly_continuous_on s f \<Longrightarrow> uniformly_continuous_on s (\<lambda>x. f x * c)"
   using bounded_linear.uniformly_continuous_on[OF bounded_linear_mult_left] .
 
 lemma uniformly_continuous_on_cmul_left[continuous_intros]:
   fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_algebra"
   assumes "uniformly_continuous_on s f"
     shows "uniformly_continuous_on s (\<lambda>x. c * f x)"
 by (metis assms bounded_linear.uniformly_continuous_on bounded_linear_mult_right)
 
 lemma uniformly_continuous_on_norm[continuous_intros]:
   fixes f :: "'a :: metric_space \<Rightarrow> 'b :: real_normed_vector"
   assumes "uniformly_continuous_on s f"
   shows "uniformly_continuous_on s (\<lambda>x. norm (f x))"
   unfolding norm_conv_dist using assms
   by (intro uniformly_continuous_on_dist uniformly_continuous_on_const)
 
 lemma uniformly_continuous_on_cmul[continuous_intros]:
   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
   assumes "uniformly_continuous_on s f"
   shows "uniformly_continuous_on s (\<lambda>x. c *\<^sub>R f(x))"
   using bounded_linear_scaleR_right assms
   by (rule bounded_linear.uniformly_continuous_on)
 
 lemma dist_minus:
   fixes x y :: "'a::real_normed_vector"
   shows "dist (- x) (- y) = dist x y"
   unfolding dist_norm minus_diff_minus norm_minus_cancel ..
 
 lemma uniformly_continuous_on_minus[continuous_intros]:
   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
   shows "uniformly_continuous_on s f \<Longrightarrow> uniformly_continuous_on s (\<lambda>x. - f x)"
   unfolding uniformly_continuous_on_def dist_minus .
 
 lemma uniformly_continuous_on_add[continuous_intros]:
   fixes f g :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
   assumes "uniformly_continuous_on s f"
     and "uniformly_continuous_on s g"
   shows "uniformly_continuous_on s (\<lambda>x. f x + g x)"
   using assms
   unfolding uniformly_continuous_on_sequentially
   unfolding dist_norm tendsto_norm_zero_iff add_diff_add
   by (auto intro: tendsto_add_zero)
 
 lemma uniformly_continuous_on_diff[continuous_intros]:
   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
   assumes "uniformly_continuous_on s f"
     and "uniformly_continuous_on s g"
   shows "uniformly_continuous_on s (\<lambda>x. f x - g x)"
   using assms uniformly_continuous_on_add [of s f "- g"]
     by (simp add: fun_Compl_def uniformly_continuous_on_minus)
 
 
 subsection\<^marker>\<open>tag unimportant\<close> \<open>Arithmetic Preserves Topological Properties\<close>
 
 lemma open_scaling[intro]:
   fixes s :: "'a::real_normed_vector set"
   assumes "c \<noteq> 0"
     and "open s"
   shows "open((\<lambda>x. c *\<^sub>R x) ` s)"
 proof -
   {
     fix x
     assume "x \<in> s"
     then obtain e where "e>0"
       and e:"\<forall>x'. dist x' x < e \<longrightarrow> x' \<in> s" using assms(2)[unfolded open_dist, THEN bspec[where x=x]]
       by auto
     have "e * \<bar>c\<bar> > 0"
       using assms(1)[unfolded zero_less_abs_iff[symmetric]] \<open>e>0\<close> by auto
     moreover
     {
       fix y
       assume "dist y (c *\<^sub>R x) < e * \<bar>c\<bar>"
       then have "norm (c *\<^sub>R ((1 / c) *\<^sub>R y - x)) < e * norm c"
         by (simp add: \<open>c \<noteq> 0\<close> dist_norm scale_right_diff_distrib)
       then have "norm ((1 / c) *\<^sub>R y - x) < e"
         by (simp add: \<open>c \<noteq> 0\<close>)
       then have "y \<in> (*\<^sub>R) c ` s"
         using rev_image_eqI[of "(1 / c) *\<^sub>R y" s y "(*\<^sub>R) c"]
         by (simp add: \<open>c \<noteq> 0\<close> dist_norm e)
     }
     ultimately have "\<exists>e>0. \<forall>x'. dist x' (c *\<^sub>R x) < e \<longrightarrow> x' \<in> (*\<^sub>R) c ` s"
       by (rule_tac x="e * \<bar>c\<bar>" in exI, auto)
   }
   then show ?thesis unfolding open_dist by auto
 qed
 
 lemma minus_image_eq_vimage:
   fixes A :: "'a::ab_group_add set"
   shows "(\<lambda>x. - x) ` A = (\<lambda>x. - x) -` A"
   by (auto intro!: image_eqI [where f="\<lambda>x. - x"])
 
 lemma open_negations:
   fixes S :: "'a::real_normed_vector set"
   shows "open S \<Longrightarrow> open ((\<lambda>x. - x) ` S)"
   using open_scaling [of "- 1" S] by simp
 
 lemma open_translation:
   fixes S :: "'a::real_normed_vector set"
   assumes "open S"
   shows "open((\<lambda>x. a + x) ` S)"
 proof -
   {
     fix x
     have "continuous (at x) (\<lambda>x. x - a)"
       by (intro continuous_diff continuous_ident continuous_const)
   }
   moreover have "{x. x - a \<in> S} = (+) a ` S"
     by force
   ultimately show ?thesis
     by (metis assms continuous_open_vimage vimage_def)
 qed
 
 lemma open_translation_subtract:
   fixes S :: "'a::real_normed_vector set"
   assumes "open S"
   shows "open ((\<lambda>x. x - a) ` S)" 
   using assms open_translation [of S "- a"] by (simp cong: image_cong_simp)
 
 lemma open_neg_translation:
   fixes S :: "'a::real_normed_vector set"
   assumes "open S"
   shows "open((\<lambda>x. a - x) ` S)"
   using open_translation[OF open_negations[OF assms], of a]
   by (auto simp: image_image)
 
 lemma open_affinity:
   fixes S :: "'a::real_normed_vector set"
   assumes "open S"  "c \<noteq> 0"
   shows "open ((\<lambda>x. a + c *\<^sub>R x) ` S)"
 proof -
   have *: "(\<lambda>x. a + c *\<^sub>R x) = (\<lambda>x. a + x) \<circ> (\<lambda>x. c *\<^sub>R x)"
     unfolding o_def ..
   have "(+) a ` (*\<^sub>R) c ` S = ((+) a \<circ> (*\<^sub>R) c) ` S"
     by auto
   then show ?thesis
     using assms open_translation[of "(*\<^sub>R) c ` S" a]
     unfolding *
     by auto
 qed
 
 lemma interior_translation:
   "interior ((+) a ` S) = (+) a ` (interior S)" for S :: "'a::real_normed_vector set"
 proof (rule set_eqI, rule)
   fix x
   assume "x \<in> interior ((+) a ` S)"
   then obtain e where "e > 0" and e: "ball x e \<subseteq> (+) a ` S"
     unfolding mem_interior by auto
   then have "ball (x - a) e \<subseteq> S"
     unfolding subset_eq Ball_def mem_ball dist_norm
     by (auto simp: diff_diff_eq)
   then show "x \<in> (+) a ` interior S"
     unfolding image_iff
     by (metis \<open>0 < e\<close> add.commute diff_add_cancel mem_interior)
 next
   fix x
   assume "x \<in> (+) a ` interior S"
   then obtain y e where "e > 0" and e: "ball y e \<subseteq> S" and y: "x = a + y"
     unfolding image_iff Bex_def mem_interior by auto
   {
     fix z
     have *: "a + y - z = y + a - z" by auto
     assume "z \<in> ball x e"
     then have "z - a \<in> S"
       using e[unfolded subset_eq, THEN bspec[where x="z - a"]]
       unfolding mem_ball dist_norm y group_add_class.diff_diff_eq2 *
       by auto
     then have "z \<in> (+) a ` S"
       unfolding image_iff by (auto intro!: bexI[where x="z - a"])
   }
   then have "ball x e \<subseteq> (+) a ` S"
     unfolding subset_eq by auto
   then show "x \<in> interior ((+) a ` S)"
     unfolding mem_interior using \<open>e > 0\<close> by auto
 qed
 
 lemma interior_translation_subtract:
   "interior ((\<lambda>x. x - a) ` S) = (\<lambda>x. x - a) ` interior S" for S :: "'a::real_normed_vector set"
   using interior_translation [of "- a"] by (simp cong: image_cong_simp)
 
 
 lemma compact_scaling:
   fixes s :: "'a::real_normed_vector set"
   assumes "compact s"
   shows "compact ((\<lambda>x. c *\<^sub>R x) ` s)"
 proof -
   let ?f = "\<lambda>x. scaleR c x"
   have *: "bounded_linear ?f" by (rule bounded_linear_scaleR_right)
   show ?thesis
     using compact_continuous_image[of s ?f] continuous_at_imp_continuous_on[of s ?f]
     using linear_continuous_at[OF *] assms
     by auto
 qed
 
 lemma compact_negations:
   fixes s :: "'a::real_normed_vector set"
   assumes "compact s"
   shows "compact ((\<lambda>x. - x) ` s)"
   using compact_scaling [OF assms, of "- 1"] by auto
 
 lemma compact_sums:
   fixes s t :: "'a::real_normed_vector set"
   assumes "compact s"
     and "compact t"
   shows "compact {x + y | x y. x \<in> s \<and> y \<in> t}"
 proof -
   have *: "{x + y | x y. x \<in> s \<and> y \<in> t} = (\<lambda>z. fst z + snd z) ` (s \<times> t)"
     by (fastforce simp: image_iff)
   have "continuous_on (s \<times> t) (\<lambda>z. fst z + snd z)"
     unfolding continuous_on by (rule ballI) (intro tendsto_intros)
   then show ?thesis
     unfolding * using compact_continuous_image compact_Times [OF assms] by auto
 qed
 
 lemma compact_differences:
   fixes s t :: "'a::real_normed_vector set"
   assumes "compact s"
     and "compact t"
   shows "compact {x - y | x y. x \<in> s \<and> y \<in> t}"
 proof-
   have "{x - y | x y. x\<in>s \<and> y \<in> t} = {x + y | x y. x \<in> s \<and> y \<in> (uminus ` t)}"
     using diff_conv_add_uminus by force
   then show ?thesis
     using compact_sums[OF assms(1) compact_negations[OF assms(2)]] by auto
 qed
 
+lemma compact_sums':
+  fixes S :: "'a::real_normed_vector set"
+  assumes "compact S" and "compact T"
+  shows "compact (\<Union>x\<in> S. \<Union>y \<in> T. {x + y})"
+proof -
+  have "(\<Union>x\<in>S. \<Union>y\<in>T. {x + y}) = {x + y |x y. x \<in> S \<and> y \<in> T}"
+    by blast
+  then show ?thesis
+    using compact_sums [OF assms] by simp
+qed
+
+lemma compact_differences':
+  fixes S :: "'a::real_normed_vector set"
+  assumes "compact S" and "compact T"
+  shows "compact (\<Union>x\<in> S. \<Union>y \<in> T. {x - y})"
+proof -
+  have "(\<Union>x\<in>S. \<Union>y\<in>T. {x - y}) = {x - y |x y. x \<in> S \<and> y \<in> T}"
+    by blast
+  then show ?thesis
+    using compact_differences [OF assms] by simp
+qed
+
 lemma compact_translation:
   "compact ((+) a ` s)" if "compact s" for s :: "'a::real_normed_vector set"
 proof -
   have "{x + y |x y. x \<in> s \<and> y \<in> {a}} = (\<lambda>x. a + x) ` s"
     by auto
   then show ?thesis
     using compact_sums [OF that compact_sing [of a]] by auto
 qed
 
 lemma compact_translation_subtract:
   "compact ((\<lambda>x. x - a) ` s)" if "compact s" for s :: "'a::real_normed_vector set"
   using that compact_translation [of s "- a"] by (simp cong: image_cong_simp)
 
 lemma compact_affinity:
   fixes s :: "'a::real_normed_vector set"
   assumes "compact s"
   shows "compact ((\<lambda>x. a + c *\<^sub>R x) ` s)"
 proof -
   have "(+) a ` (*\<^sub>R) c ` s = (\<lambda>x. a + c *\<^sub>R x) ` s"
     by auto
   then show ?thesis
     using compact_translation[OF compact_scaling[OF assms], of a c] by auto
 qed
 
 lemma closed_scaling:
   fixes S :: "'a::real_normed_vector set"
   assumes "closed S"
   shows "closed ((\<lambda>x. c *\<^sub>R x) ` S)"
 proof (cases "c = 0")
   case True then show ?thesis
     by (auto simp: image_constant_conv)
 next
   case False
   from assms have "closed ((\<lambda>x. inverse c *\<^sub>R x) -` S)"
     by (simp add: continuous_closed_vimage)
   also have "(\<lambda>x. inverse c *\<^sub>R x) -` S = (\<lambda>x. c *\<^sub>R x) ` S"
     using \<open>c \<noteq> 0\<close> by (auto elim: image_eqI [rotated])
   finally show ?thesis .
 qed
 
 lemma closed_negations:
   fixes S :: "'a::real_normed_vector set"
   assumes "closed S"
   shows "closed ((\<lambda>x. -x) ` S)"
   using closed_scaling[OF assms, of "- 1"] by simp
 
 lemma compact_closed_sums:
   fixes S :: "'a::real_normed_vector set"
   assumes "compact S" and "closed T"
   shows "closed (\<Union>x\<in> S. \<Union>y \<in> T. {x + y})"
 proof -
   let ?S = "{x + y |x y. x \<in> S \<and> y \<in> T}"
   {
     fix x l
     assume as: "\<forall>n. x n \<in> ?S"  "(x \<longlongrightarrow> l) sequentially"
     from as(1) obtain f where f: "\<forall>n. x n = fst (f n) + snd (f n)"  "\<forall>n. fst (f n) \<in> S"  "\<forall>n. snd (f n) \<in> T"
       using choice[of "\<lambda>n y. x n = (fst y) + (snd y) \<and> fst y \<in> S \<and> snd y \<in> T"] by auto
     obtain l' r where "l'\<in>S" and r: "strict_mono r" and lr: "(((\<lambda>n. fst (f n)) \<circ> r) \<longlongrightarrow> l') sequentially"
       using assms(1)[unfolded compact_def, THEN spec[where x="\<lambda> n. fst (f n)"]] using f(2) by auto
     have "((\<lambda>n. snd (f (r n))) \<longlongrightarrow> l - l') sequentially"
       using tendsto_diff[OF LIMSEQ_subseq_LIMSEQ[OF as(2) r] lr] and f(1)
       unfolding o_def
       by auto
     then have "l - l' \<in> T"
       using assms(2)[unfolded closed_sequential_limits,
         THEN spec[where x="\<lambda> n. snd (f (r n))"],
         THEN spec[where x="l - l'"]]
       using f(3)
       by auto
     then have "l \<in> ?S"
       using \<open>l' \<in> S\<close> by force
   }
   moreover have "?S = (\<Union>x\<in> S. \<Union>y \<in> T. {x + y})"
     by force
   ultimately show ?thesis
     unfolding closed_sequential_limits
     by (metis (no_types, lifting))
 qed
 
 lemma closed_compact_sums:
   fixes S T :: "'a::real_normed_vector set"
   assumes "closed S" "compact T"
   shows "closed (\<Union>x\<in> S. \<Union>y \<in> T. {x + y})"
 proof -
   have "(\<Union>x\<in> T. \<Union>y \<in> S. {x + y}) = (\<Union>x\<in> S. \<Union>y \<in> T. {x + y})"
     by auto
   then show ?thesis
     using compact_closed_sums[OF assms(2,1)] by simp
 qed
 
 lemma compact_closed_differences:
   fixes S T :: "'a::real_normed_vector set"
   assumes "compact S" "closed T"
   shows "closed (\<Union>x\<in> S. \<Union>y \<in> T. {x - y})"
 proof -
   have "(\<Union>x\<in> S. \<Union>y \<in> uminus ` T. {x + y}) = (\<Union>x\<in> S. \<Union>y \<in> T. {x - y})"
     by force
   then show ?thesis
     by (metis assms closed_negations compact_closed_sums)
 qed
 
 lemma closed_compact_differences:
   fixes S T :: "'a::real_normed_vector set"
   assumes "closed S" "compact T"
   shows "closed (\<Union>x\<in> S. \<Union>y \<in> T. {x - y})"
 proof -
   have "(\<Union>x\<in> S. \<Union>y \<in> uminus ` T. {x + y}) = {x - y |x y. x \<in> S \<and> y \<in> T}"
     by auto
  then show ?thesis
   using closed_compact_sums[OF assms(1) compact_negations[OF assms(2)]] by simp
 qed
 
 lemma closed_translation:
   "closed ((+) a ` S)" if "closed S" for a :: "'a::real_normed_vector"
 proof -
   have "(\<Union>x\<in> {a}. \<Union>y \<in> S. {x + y}) = ((+) a ` S)" by auto
   then show ?thesis
     using compact_closed_sums [OF compact_sing [of a] that] by auto
 qed
 
 lemma closed_translation_subtract:
   "closed ((\<lambda>x. x - a) ` S)" if "closed S" for a :: "'a::real_normed_vector"
   using that closed_translation [of S "- a"] by (simp cong: image_cong_simp)
 
 lemma closure_translation:
   "closure ((+) a ` s) = (+) a ` closure s" for a :: "'a::real_normed_vector"
 proof -
   have *: "(+) a ` (- s) = - (+) a ` s"
     by (auto intro!: image_eqI [where x = "x - a" for x])
   show ?thesis
     using interior_translation [of a "- s", symmetric]
     by (simp add: closure_interior translation_Compl *)
 qed
 
 lemma closure_translation_subtract:
   "closure ((\<lambda>x. x - a) ` s) = (\<lambda>x. x - a) ` closure s" for a :: "'a::real_normed_vector"
   using closure_translation [of "- a" s] by (simp cong: image_cong_simp)
 
 lemma frontier_translation:
   "frontier ((+) a ` s) = (+) a ` frontier s" for a :: "'a::real_normed_vector"
   by (auto simp add: frontier_def translation_diff interior_translation closure_translation)
 
 lemma frontier_translation_subtract:
   "frontier ((+) a ` s) = (+) a ` frontier s" for a :: "'a::real_normed_vector"
   by (auto simp add: frontier_def translation_diff interior_translation closure_translation)
 
 lemma sphere_translation:
   "sphere (a + c) r = (+) a ` sphere c r" for a :: "'n::real_normed_vector"
   by (auto simp: dist_norm algebra_simps intro!: image_eqI [where x = "x - a" for x])
 
 lemma sphere_translation_subtract:
   "sphere (c - a) r = (\<lambda>x. x - a) ` sphere c r" for a :: "'n::real_normed_vector"
   using sphere_translation [of "- a" c] by (simp cong: image_cong_simp)
 
 lemma cball_translation:
   "cball (a + c) r = (+) a ` cball c r" for a :: "'n::real_normed_vector"
   by (auto simp: dist_norm algebra_simps intro!: image_eqI [where x = "x - a" for x])
 
 lemma cball_translation_subtract:
   "cball (c - a) r = (\<lambda>x. x - a) ` cball c r" for a :: "'n::real_normed_vector"
   using cball_translation [of "- a" c] by (simp cong: image_cong_simp)
 
 lemma ball_translation:
   "ball (a + c) r = (+) a ` ball c r" for a :: "'n::real_normed_vector"
   by (auto simp: dist_norm algebra_simps intro!: image_eqI [where x = "x - a" for x])
 
 lemma ball_translation_subtract:
   "ball (c - a) r = (\<lambda>x. x - a) ` ball c r" for a :: "'n::real_normed_vector"
   using ball_translation [of "- a" c] by (simp cong: image_cong_simp)
 
 
 subsection\<^marker>\<open>tag unimportant\<close>\<open>Homeomorphisms\<close>
 
 lemma homeomorphic_scaling:
   fixes S :: "'a::real_normed_vector set"
   assumes "c \<noteq> 0"
   shows "S homeomorphic ((\<lambda>x. c *\<^sub>R x) ` S)"
   unfolding homeomorphic_minimal
   apply (rule_tac x="\<lambda>x. c *\<^sub>R x" in exI)
   apply (rule_tac x="\<lambda>x. (1 / c) *\<^sub>R x" in exI)
   using assms by (auto simp: continuous_intros)
 
 lemma homeomorphic_translation:
   fixes S :: "'a::real_normed_vector set"
   shows "S homeomorphic ((\<lambda>x. a + x) ` S)"
   unfolding homeomorphic_minimal
   apply (rule_tac x="\<lambda>x. a + x" in exI)
   apply (rule_tac x="\<lambda>x. -a + x" in exI)
   by (auto simp: continuous_intros)
 
 lemma homeomorphic_affinity:
   fixes S :: "'a::real_normed_vector set"
   assumes "c \<noteq> 0"
   shows "S homeomorphic ((\<lambda>x. a + c *\<^sub>R x) ` S)"
 proof -
   have *: "(+) a ` (*\<^sub>R) c ` S = (\<lambda>x. a + c *\<^sub>R x) ` S" by auto
   show ?thesis
     by (metis "*" assms homeomorphic_scaling homeomorphic_trans homeomorphic_translation)
 qed
 
 lemma homeomorphic_balls:
   fixes a b ::"'a::real_normed_vector"
   assumes "0 < d"  "0 < e"
   shows "(ball a d) homeomorphic  (ball b e)" (is ?th)
     and "(cball a d) homeomorphic (cball b e)" (is ?cth)
 proof -
   show ?th unfolding homeomorphic_minimal
     apply(rule_tac x="\<lambda>x. b + (e/d) *\<^sub>R (x - a)" in exI)
     apply(rule_tac x="\<lambda>x. a + (d/e) *\<^sub>R (x - b)" in exI)
     using assms
     by (auto intro!: continuous_intros simp: dist_commute dist_norm pos_divide_less_eq)
   show ?cth unfolding homeomorphic_minimal
     apply(rule_tac x="\<lambda>x. b + (e/d) *\<^sub>R (x - a)" in exI)
     apply(rule_tac x="\<lambda>x. a + (d/e) *\<^sub>R (x - b)" in exI)
     using assms
     by (auto intro!: continuous_intros simp: dist_commute dist_norm pos_divide_le_eq)
 qed
 
 lemma homeomorphic_spheres:
   fixes a b ::"'a::real_normed_vector"
   assumes "0 < d"  "0 < e"
   shows "(sphere a d) homeomorphic (sphere b e)"
 unfolding homeomorphic_minimal
     apply(rule_tac x="\<lambda>x. b + (e/d) *\<^sub>R (x - a)" in exI)
     apply(rule_tac x="\<lambda>x. a + (d/e) *\<^sub>R (x - b)" in exI)
     using assms
     by (auto intro!: continuous_intros simp: dist_commute dist_norm pos_divide_less_eq)
 
 lemma homeomorphic_ball01_UNIV:
   "ball (0::'a::real_normed_vector) 1 homeomorphic (UNIV:: 'a set)"
   (is "?B homeomorphic ?U")
 proof
   have "x \<in> (\<lambda>z. z /\<^sub>R (1 - norm z)) ` ball 0 1" for x::'a
     apply (rule_tac x="x /\<^sub>R (1 + norm x)" in image_eqI)
      apply (auto simp: field_split_simps)
     using norm_ge_zero [of x] apply linarith+
     done
   then show "(\<lambda>z::'a. z /\<^sub>R (1 - norm z)) ` ?B = ?U"
     by blast
   have "x \<in> range (\<lambda>z. (1 / (1 + norm z)) *\<^sub>R z)" if "norm x < 1" for x::'a
     using that
     by (rule_tac x="x /\<^sub>R (1 - norm x)" in image_eqI) (auto simp: field_split_simps)
   then show "(\<lambda>z::'a. z /\<^sub>R (1 + norm z)) ` ?U = ?B"
     by (force simp: field_split_simps dest: add_less_zeroD)
   show "continuous_on (ball 0 1) (\<lambda>z. z /\<^sub>R (1 - norm z))"
     by (rule continuous_intros | force)+
   have 0: "\<And>z. 1 + norm z \<noteq> 0"
     by (metis (no_types) le_add_same_cancel1 norm_ge_zero not_one_le_zero)
   then show "continuous_on UNIV (\<lambda>z. z /\<^sub>R (1 + norm z))"
     by (auto intro!: continuous_intros)
   show "\<And>x. x \<in> ball 0 1 \<Longrightarrow>
          x /\<^sub>R (1 - norm x) /\<^sub>R (1 + norm (x /\<^sub>R (1 - norm x))) = x"
     by (auto simp: field_split_simps)
   show "\<And>y. y /\<^sub>R (1 + norm y) /\<^sub>R (1 - norm (y /\<^sub>R (1 + norm y))) = y"
     using 0 by (auto simp: field_split_simps)
 qed
 
 proposition homeomorphic_ball_UNIV:
   fixes a ::"'a::real_normed_vector"
   assumes "0 < r" shows "ball a r homeomorphic (UNIV:: 'a set)"
   using assms homeomorphic_ball01_UNIV homeomorphic_balls(1) homeomorphic_trans zero_less_one by blast
 
 
 subsection\<^marker>\<open>tag unimportant\<close> \<open>Discrete\<close>
 
 lemma finite_implies_discrete:
   fixes S :: "'a::topological_space set"
   assumes "finite (f ` S)"
   shows "(\<forall>x \<in> S. \<exists>e>0. \<forall>y. y \<in> S \<and> f y \<noteq> f x \<longrightarrow> e \<le> norm (f y - f x))"
 proof -
   have "\<exists>e>0. \<forall>y. y \<in> S \<and> f y \<noteq> f x \<longrightarrow> e \<le> norm (f y - f x)" if "x \<in> S" for x
   proof (cases "f ` S - {f x} = {}")
     case True
     with zero_less_numeral show ?thesis
       by (fastforce simp add: Set.image_subset_iff cong: conj_cong)
   next
     case False
     then obtain z where "z \<in> S" "f z \<noteq> f x"
       by blast
     moreover have finn: "finite {norm (z - f x) |z. z \<in> f ` S - {f x}}"
       using assms by simp
     ultimately have *: "0 < Inf{norm(z - f x) | z. z \<in> f ` S - {f x}}"
       by (force intro: finite_imp_less_Inf)
     show ?thesis
       by (force intro!: * cInf_le_finite [OF finn])
   qed
   with assms show ?thesis
     by blast
 qed
 
 
 subsection\<^marker>\<open>tag unimportant\<close> \<open>Completeness of "Isometry" (up to constant bounds)\<close>
 
 lemma cauchy_isometric:\<comment> \<open>TODO: rename lemma to \<open>Cauchy_isometric\<close>\<close>
   assumes e: "e > 0"
     and s: "subspace s"
     and f: "bounded_linear f"
     and normf: "\<forall>x\<in>s. norm (f x) \<ge> e * norm x"
     and xs: "\<forall>n. x n \<in> s"
     and cf: "Cauchy (f \<circ> x)"
   shows "Cauchy x"
 proof -
   interpret f: bounded_linear f by fact
   have "\<exists>N. \<forall>n\<ge>N. norm (x n - x N) < d" if "d > 0" for d :: real
   proof -
     from that obtain N where N: "\<forall>n\<ge>N. norm (f (x n) - f (x N)) < e * d"
       using cf[unfolded Cauchy_def o_def dist_norm, THEN spec[where x="e*d"]] e
       by auto
     have "norm (x n - x N) < d" if "n \<ge> N" for n
     proof -
       have "e * norm (x n - x N) \<le> norm (f (x n - x N))"
         using subspace_diff[OF s, of "x n" "x N"]
         using xs[THEN spec[where x=N]] and xs[THEN spec[where x=n]]
         using normf[THEN bspec[where x="x n - x N"]]
         by auto
       also have "norm (f (x n - x N)) < e * d"
         using \<open>N \<le> n\<close> N unfolding f.diff[symmetric] by auto
       finally show ?thesis
         using \<open>e>0\<close> by simp
     qed
     then show ?thesis by auto
   qed
   then show ?thesis
     by (simp add: Cauchy_altdef2 dist_norm)
 qed
 
 lemma complete_isometric_image:
   assumes "0 < e"
     and s: "subspace s"
     and f: "bounded_linear f"
     and normf: "\<forall>x\<in>s. norm(f x) \<ge> e * norm(x)"
     and cs: "complete s"
   shows "complete (f ` s)"
 proof -
   have "\<exists>l\<in>f ` s. (g \<longlongrightarrow> l) sequentially"
     if as:"\<forall>n::nat. g n \<in> f ` s" and cfg:"Cauchy g" for g
   proof -
     from that obtain x where "\<forall>n. x n \<in> s \<and> g n = f (x n)"
       using choice[of "\<lambda> n xa. xa \<in> s \<and> g n = f xa"] by auto
     then have x: "\<forall>n. x n \<in> s" "\<forall>n. g n = f (x n)" by auto
     then have "f \<circ> x = g" by (simp add: fun_eq_iff)
     then obtain l where "l\<in>s" and l:"(x \<longlongrightarrow> l) sequentially"
       using cs[unfolded complete_def, THEN spec[where x=x]]
       using cauchy_isometric[OF \<open>0 < e\<close> s f normf] and cfg and x(1)
       by auto
     then show ?thesis
       using linear_continuous_at[OF f, unfolded continuous_at_sequentially, THEN spec[where x=x], of l]
       by (auto simp: \<open>f \<circ> x = g\<close>)
   qed
   then show ?thesis
     unfolding complete_def by auto
 qed
 
 subsection \<open>Connected Normed Spaces\<close>
 
 lemma compact_components:
   fixes s :: "'a::heine_borel set"
   shows "\<lbrakk>compact s; c \<in> components s\<rbrakk> \<Longrightarrow> compact c"
 by (meson bounded_subset closed_components in_components_subset compact_eq_bounded_closed)
 
 lemma discrete_subset_disconnected:
   fixes S :: "'a::topological_space set"
   fixes t :: "'b::real_normed_vector set"
   assumes conf: "continuous_on S f"
       and no: "\<And>x. x \<in> S \<Longrightarrow> \<exists>e>0. \<forall>y. y \<in> S \<and> f y \<noteq> f x \<longrightarrow> e \<le> norm (f y - f x)"
    shows "f ` S \<subseteq> {y. connected_component_set (f ` S) y = {y}}"
 proof -
   { fix x assume x: "x \<in> S"
     then obtain e where "e>0" and ele: "\<And>y. \<lbrakk>y \<in> S; f y \<noteq> f x\<rbrakk> \<Longrightarrow> e \<le> norm (f y - f x)"
       using conf no [OF x] by auto
     then have e2: "0 \<le> e/2"
       by simp
     define F where "F \<equiv> connected_component_set (f ` S) (f x)"
     have False if "y \<in> S" and ccs: "f y \<in> F" and not: "f y \<noteq> f x" for y 
     proof -
       define C where "C \<equiv> cball (f x) (e/2)"
       define D where "D \<equiv> - ball (f x) e"
       have disj: "C \<inter> D = {}"
         unfolding C_def D_def using \<open>0 < e\<close> by fastforce
       moreover have FCD: "F \<subseteq> C \<union> D"
       proof -
         have "t \<in> C \<or> t \<in> D" if "t \<in> F" for t
         proof -
           obtain y where "y \<in> S" "t = f y"
             using F_def \<open>t \<in> F\<close> connected_component_in by blast
           then show ?thesis
             by (metis C_def ComplI D_def centre_in_cball dist_norm e2 ele mem_ball norm_minus_commute not_le)
         qed
         then show ?thesis
           by auto
       qed
       ultimately have "C \<inter> F = {} \<or> D \<inter> F = {}"
         using connected_closed [of "F"] \<open>e>0\<close> not
         unfolding C_def D_def
         by (metis Elementary_Metric_Spaces.open_ball F_def closed_cball connected_connected_component inf_bot_left open_closed)
       moreover have "C \<inter> F \<noteq> {}"
         unfolding disjoint_iff
         by (metis FCD ComplD image_eqI mem_Collect_eq subsetD x  D_def F_def Un_iff \<open>0 < e\<close> centre_in_ball connected_component_refl_eq)
       moreover have "D \<inter> F \<noteq> {}"
         unfolding disjoint_iff
         by (metis ComplI D_def ccs dist_norm ele mem_ball norm_minus_commute not not_le that(1))
       ultimately show ?thesis by metis
     qed
     moreover have "connected_component_set (f ` S) (f x) \<subseteq> f ` S"
       by (auto simp: connected_component_in)
     ultimately have "connected_component_set (f ` S) (f x) = {f x}"
       by (auto simp: x F_def)
   }
   with assms show ?thesis
     by blast
 qed
 
 lemma continuous_disconnected_range_constant_eq:
       "(connected S \<longleftrightarrow>
            (\<forall>f::'a::topological_space \<Rightarrow> 'b::real_normed_algebra_1.
             \<forall>t. continuous_on S f \<and> f ` S \<subseteq> t \<and> (\<forall>y \<in> t. connected_component_set t y = {y})
             \<longrightarrow> f constant_on S))" (is ?thesis1)
   and continuous_discrete_range_constant_eq:
       "(connected S \<longleftrightarrow>
          (\<forall>f::'a::topological_space \<Rightarrow> 'b::real_normed_algebra_1.
           continuous_on S f \<and>
           (\<forall>x \<in> S. \<exists>e. 0 < e \<and> (\<forall>y. y \<in> S \<and> (f y \<noteq> f x) \<longrightarrow> e \<le> norm(f y - f x)))
           \<longrightarrow> f constant_on S))" (is ?thesis2)
   and continuous_finite_range_constant_eq:
       "(connected S \<longleftrightarrow>
          (\<forall>f::'a::topological_space \<Rightarrow> 'b::real_normed_algebra_1.
           continuous_on S f \<and> finite (f ` S)
           \<longrightarrow> f constant_on S))" (is ?thesis3)
 proof -
   have *: "\<And>s t u v. \<lbrakk>s \<Longrightarrow> t; t \<Longrightarrow> u; u \<Longrightarrow> v; v \<Longrightarrow> s\<rbrakk>
     \<Longrightarrow> (s \<longleftrightarrow> t) \<and> (s \<longleftrightarrow> u) \<and> (s \<longleftrightarrow> v)"
     by blast
   have "?thesis1 \<and> ?thesis2 \<and> ?thesis3"
     apply (rule *)
     using continuous_disconnected_range_constant apply metis
     apply clarify
     apply (frule discrete_subset_disconnected; blast)
     apply (blast dest: finite_implies_discrete)
     apply (blast intro!: finite_range_constant_imp_connected)
     done
   then show ?thesis1 ?thesis2 ?thesis3
     by blast+
 qed
 
 lemma continuous_discrete_range_constant:
   fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_algebra_1"
   assumes S: "connected S"
       and "continuous_on S f"
       and "\<And>x. x \<in> S \<Longrightarrow> \<exists>e>0. \<forall>y. y \<in> S \<and> f y \<noteq> f x \<longrightarrow> e \<le> norm (f y - f x)"
     shows "f constant_on S"
   using continuous_discrete_range_constant_eq [THEN iffD1, OF S] assms by blast
 
 lemma continuous_finite_range_constant:
   fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_algebra_1"
   assumes "connected S"
       and "continuous_on S f"
       and "finite (f ` S)"
     shows "f constant_on S"
   using assms continuous_finite_range_constant_eq  by blast
 
 end
diff --git a/src/HOL/Complex_Analysis/Cauchy_Integral_Formula.thy b/src/HOL/Complex_Analysis/Cauchy_Integral_Formula.thy
--- a/src/HOL/Complex_Analysis/Cauchy_Integral_Formula.thy
+++ b/src/HOL/Complex_Analysis/Cauchy_Integral_Formula.thy
@@ -1,2814 +1,2815 @@
 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 -
   let ?fz = "\<lambda>w. (f w - f z)/(w - z)"
   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
       using LIM_equal [of "z" ?fz "\<lambda>w. if w = z then f' else ?fz w"] has_field_derivativeD [OF f'] 
       by (force simp add: continuous_within Lim_at_imp_Lim_at_within)
   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 ?fz])
       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 ?fz w) has_contour_integral 0) \<gamma>"
   proof (rule Cauchy_theorem_convex [OF _ S fink' _ vpg pas loop])
     show "(\<lambda>w. if w = z then f' else ?fz w) field_differentiable at w" 
       if "w \<in> interior S - insert z k" for w
     proof (rule field_differentiable_transform_within)
       show "(\<lambda>w. ?fz w) field_differentiable at w"
         using that by (intro derivative_intros fcd; simp)
     qed (use that in \<open>auto simp add: dist_pos_lt dist_commute\<close>)
   qed (use c in \<open>force simp: continuous_on_eq_continuous_within\<close>)
   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:
   assumes "convex S" and holf: "f holomorphic_on S" and "z \<in> interior S" "valid_path \<gamma>" "path_image \<gamma> \<subseteq> S - {z}"
       "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"
     using holf at_within_interior holomorphic_onD interior_subset by fastforce
   then show ?thesis
     using assms
     by (intro Cauchy_integral_formula_weak [where k = "{}"]) (auto simp: holomorphic_on_imp_continuous_on)
 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)
 
 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>"
     using int w d
     apply (intro contour_integrable_div contour_integrable_diff has_contour_integral_integrable)
     by (force simp: dist_norm norm_minus_commute)
   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)
     have \<section>: "((\<lambda>x. f' x * inverse (x - u) ^ k) has_contour_integral f u) \<gamma>"
             "((\<lambda>x. f' x * inverse (x - w) ^ k) has_contour_integral f w) \<gamma>"
       using u w by (simp_all add: field_simps int)
     show ?thesis
       apply (rule contour_integral_unique)
       apply (simp add: diff_divide_distrib algebra_simps \<section> has_contour_integral_diff has_contour_integral_div)
       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"
       unfolding holomorphic_on_def
       using field_differentiable_at_within field_differentiable_def fder by fastforce
     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
       using Cauchy_next_derivative_circlepath [OF contfpi, of 2 f'] \<open>0 < r\<close> *
       using centre_in_ball mem_ball by force
   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 "\<exists>e>0. f holomorphic_on ball z e"
     proof (intro exI conjI)
       show "f holomorphic_on ball z (e - dist a z)"
       proof (rule holomorphic_on_subset)
         show "ball z (e - dist a z) \<subseteq> ball a e"
           by (simp add: dist_commute ball_subset_ball_iff)
         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 a e"
           using triangle_contour_integrals_starlike_primitive [OF contf _ open_ball, of a]
             derivative_is_holomorphic[OF open_ball]
           by (force simp add: 0 \<open>0 < e\<close> sub_ball)
       qed
     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)"
 proof (induction n)
   case (Suc n)
   then show ?case by (metis higher_deriv_linear lambda_one)
 qed auto
 
 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)"
 proof (induction n)
   case 0
   then show ?case
     by (simp add: id_def)
 next
   case (Suc n)
   then show ?case
     by (metis DERIV_chain funpow_Suc_right mult.right_neutral)
 qed
 
 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 "\<And>x. x \<in> S \<Longrightarrow> - (deriv ^^ n) f x = (deriv ^^ n) (\<lambda>w. - f w) x"
     by (auto simp add: Suc)
   then have "((deriv ^^ n) (\<lambda>w. - f w) has_field_derivative - deriv ((deriv ^^ n) f) z) (at z)"
     using  has_field_derivative_transform_within_open [of "\<lambda>w. -((deriv ^^ n) f w)"]
     using "*" DERIV_minus Suc.prems \<open>open S\<close> by blast
   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 "\<And>x. x \<in> S \<Longrightarrow> (deriv ^^ n) f x + (deriv ^^ n) g x = (deriv ^^ n) (\<lambda>w. f w + g w) x"
     by (auto simp add: Suc)
   then have "((deriv ^^ n) (\<lambda>w. f w + g w) has_field_derivative
         deriv ((deriv ^^ n) f) z + deriv ((deriv ^^ n) g) z) (at z)"
     using  has_field_derivative_transform_within_open [of "\<lambda>w. (deriv ^^ n) f w + (deriv ^^ n) g w"]
     using "*" Deriv.field_differentiable_add Suc.prems \<open>open S\<close> by blast
   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" "z \<in> S"
     shows "(deriv ^^ n) (\<lambda>w. f w - g w) z = (deriv ^^ n) f z - (deriv ^^ n) g z"
   unfolding diff_conv_add_uminus higher_deriv_add
   using assms higher_deriv_add higher_deriv_uminus holomorphic_on_minus by presburger
 
 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)" _ _ S])
        apply (simp add: algebra_simps)
        apply (rule derivative_eq_intros | simp)+
            apply (auto intro: DERIV_mult * \<open>open S\<close> Suc.prems Suc.IH [symmetric])
     by (metis (no_types, lifting) mult.commute sum.cong sumeq)
   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 "(*) u holomorphic_on S" "f holomorphic_on (*) u ` S"
     by (rule holo0 holomorphic_intros)+
   then have holo1: "(\<lambda>w. f (u * w)) holomorphic_on S"
     by (rule holomorphic_on_compose [where g=f, unfolded o_def])
   have "deriv ((deriv ^^ n) (\<lambda>w. f (u * w))) z = deriv (\<lambda>z. u^n * (deriv ^^ n) f (u*z)) z"
   proof (rule complex_derivative_transform_within_open [OF _ holo3 S Suc.prems])
     show "(deriv ^^ n) (\<lambda>w. f (u * w)) holomorphic_on S"
       by (rule holomorphic_higher_deriv [OF holo1 S])
   qed (simp add: Suc.IH)
   also have "\<dots> = u^n * deriv (\<lambda>z. (deriv ^^ n) f (u * z)) z"
   proof -
     have "(deriv ^^ n) f analytic_on T"
       by (simp add: analytic_on_open f holomorphic_higher_deriv T)
     then have "(\<lambda>w. (deriv ^^ n) f (u * w)) analytic_on S"
     proof -
       have "(deriv ^^ n) f \<circ> (*) u holomorphic_on S"
         by (simp add: holo2 holomorphic_on_compose)
       then show ?thesis
         by (simp add: S analytic_on_open o_def)
     qed
     then show ?thesis
       by (intro deriv_cmult analytic_on_imp_differentiable_at [OF _ Suc.prems])
   qed
   also have "\<dots> = u * u ^ n * deriv ((deriv ^^ n) f) (u * z)"
   proof -
     have "(deriv ^^ n) f field_differentiable at (u * z)"
       using Suc.prems T f fg holomorphic_higher_deriv holomorphic_on_imp_differentiable_at by blast
     then show ?thesis
       by (simp add: deriv_compose_linear)
   qed
   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"
         using 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
         by (simp add: B divide_simps mult_mono r wz_eq)
       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 \<section>: "\<And>x k. k\<in> {..<x} \<Longrightarrow>
            (\<lambda>u. (w - z) ^ k * (f u / (u - z) ^ Suc k)) contour_integrable_on circlepath z r"
     using contour_integrable_lmul [OF cint, of "(w - z) ^ a" for a] by (simp add: field_simps)
   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)
     apply (simp_all only: \<section> contour_integral_lmul cint algebra_simps)
     done
   have "\<And>u k. k \<in> {..<u} \<Longrightarrow> (\<lambda>x. f x / (x - z) ^ Suc k) contour_integrable_on circlepath z r"
     using \<open>0 < r\<close> by (force intro!: Cauchy_higher_derivative_integral_circlepath [OF contf holf])
   then have "\<And>u. (\<lambda>y. \<Sum>k<u. (w - z) ^ k * (f y / (y - z) ^ Suc k)) contour_integrable_on circlepath z r"
     by (intro contour_integrable_sum contour_integrable_lmul, simp)
   then 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 using \<open>0 < r\<close> 
     by (intro Lim_transform_eventually [OF _ eq] contour_integral_uniform_limit_circlepath [OF eventuallyI ul]) auto
   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)"
     using continuous_on_subset holf  holomorphic_on_subset \<open>0 < R\<close>
     by (force intro: holomorphic_on_imp_continuous_on Cauchy_integral_circlepath)
   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"
     proof (rule tendstoI [OF eventually_mono])
       fix e::real
       assume "e > 0"
       show "eventually (\<lambda>x. 2/e \<le> cmod (f x)) at_infinity"
         by (rule_tac B="2/e" in unbounded)
     qed (simp add: dist_norm norm_divide field_split_simps)
     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"
       using w
       by (auto intro: eventually_mono [OF cont] Cauchy_higher_derivative_integral_circlepath [where k=0, simplified])
     have "\<And>e. \<lbrakk>0 < r; 0 < d; 0 < e\<rbrakk>
          \<Longrightarrow> \<forall>\<^sub>F n in F.
                 \<forall>x\<in>sphere z r.
                    x \<noteq> w \<longrightarrow>
                    cmod (f n x - g x) < e * cmod (x - w)"
       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
     then 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>
       by (auto simp add: uniform_limit_iff dist_norm norm_divide diff_divide_distrib [symmetric] divide_simps)
     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"
         using w\<open>0 < d\<close> d_def
         by (auto intro: eventually_mono [OF cont contour_integral_unique [OF Cauchy_integral_circlepath]])
     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 "e > 0"
       with \<open>r > 0\<close> 
       have "\<forall>\<^sub>F n in F. \<forall>x. x \<noteq> w \<longrightarrow> cmod (z - x) = r \<longrightarrow> cmod (f n x - g x) < e * cmod ((x - w)\<^sup>2)"
         by (force simp: \<open>0 < d\<close> dist_norm dle intro: less_le_trans eventually_mono [OF uniform_limitD [OF ulim], of "e*d"])
       with \<open>r > 0\<close> \<open>e > 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"
         by (simp add: norm_divide field_split_simps sphere_def dist_norm)
     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
       using \<open>0 < r\<close> centre_in_ball ul
       by (auto simp: holomorphic_on_open intro: holomorphic_uniform_limit [OF *])
   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: "cmod y \<le> cmod (of_real r + of_real (cmod z)) / 2" 
       if "cmod (z - y) * 2 < r - cmod z" for z y
     proof -
       have "cmod y * 2 \<le> r + cmod z"
         using norm_triangle_ineq2 [of y z] that
         by (simp only: diff_le_eq norm_minus_commute mult_2)
       then show ?thesis
         using \<open>r > 0\<close> 
         by (auto simp: algebra_simps norm_mult norm_divide norm_power simp flip: of_real_add)
     qed
     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
     unfolding 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)
       have "(\<lambda>n. a n * (u - z) ^ n) sums g u" "(\<lambda>n. a n * (u - z) ^ n) sums f u"
         using gg' [of u] less w by (auto simp: assms dist_norm)
       then show ?thesis
         by (metis sums_unique2)
     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"
   by (auto simp: holomorphic_iff_power_series sums_unique2 [of "\<lambda>n. (deriv ^^ n) f z / (fact n) * (w - z)^n"])
 
 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"
   by (auto simp: holomorphic_iff_power_series sums_unique2 [of "\<lambda>n. (deriv ^^ n) f z / (fact n) * (w - z)^n"])
 
 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 "(deriv ^^ m) ((deriv ^^ n) f) x = 0" for m n
       by (metis funpow_add o_apply that(1))
     then have "\<And>x' n. dist x x' < e \<Longrightarrow> (deriv ^^ n) f x' = 0"
       using \<open>open S\<close> 
       by (meson holf holomorphic_fun_eq_0_on_ball holomorphic_higher_deriv holomorphic_on_subset mem_ball that(2))
     with that show ?thesis by auto
   qed
   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 "open (\<Inter>n. {w \<in> S. (deriv ^^ n) f w = 0})"
     using \<open>open S\<close>
     apply (simp add: open_contains_ball Ball_def)
     apply (erule all_forward)
     using "*" by auto blast+
   then have "openin (top_of_set S) (\<Inter>n. {w \<in> S. (deriv ^^ n) f w = 0})"
     by (force intro: open_subset)
   moreover have "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)
   moreover have "(\<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])
   ultimately show ?thesis
     using cons der \<open>z \<in> S\<close>
     by (auto simp add: connected_clopen)
 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}"
       using mem_ball that
       by (auto simp add: holomorphic_on_def simp flip: field_differentiable_def intro: F1 field_differentiable_within_subset)
     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)"
       using \<open>0 < \<epsilon>\<close> \<open>a \<noteq> b\<close>
       by (auto elim: uniformly_continuous_onE [where e = "\<epsilon>/norm(b - a)"])
     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 by (auto simp: norm_minus_commute)
       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>
       by (auto simp: dist_norm contour_integral_diff [OF cont_dw cont_dw, symmetric] \<open>w \<in> U\<close> intro: le_ee)
   qed
   show ?thesis
   proof (cases "a=b")
     case False
     show ?thesis
       by (rule continuous_onI) (use False in \<open>auto intro: *\<close>)
   qed auto
 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)"
       using "*" d_def holomorphic_on_imp_continuous_on by auto
     moreover have "d y holomorphic_on U - {y}"
     proof -
       have "(\<lambda>w. if w = y then deriv f y else (f w - f y) / (w - y)) field_differentiable at w"
         if "w \<in> U - {y}" for w
       proof (rule field_differentiable_transform_within)
         show "(\<lambda>w. (f w - f y) / (w - y)) field_differentiable at w"
           using that \<open>open U\<close> holf 
           by (auto intro!: holomorphic_on_imp_differentiable_at derivative_intros)
         show "dist w y > 0"
           using that by auto
       qed (auto simp: dist_commute)
       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
       by (metis Diff_subset \<open>valid_path \<gamma>\<close> \<open>open U\<close> contour_integrable_holomorphic_simple h_def 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 (simp add: 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"
     proof (rule has_contour_integral_integral [OF contour_integrable_holomorphic_simple [where S=U]])
       show "(\<lambda>w. f w / (w - z)) holomorphic_on U"
         by (rule holomorphic_intros assms | use that in force)+
     qed (use \<open>open U\<close> pasz \<open>valid_path \<gamma>\<close> in auto)
     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"
   proof
     show "0 < d0 / 2" using \<open>0 < d0\<close> by auto
   qed (use \<open>0 < d0\<close> d0 in \<open>force simp: dist_norm\<close>)
   define T where "T \<equiv> {y + k |y k. y \<in> path_image \<gamma> \<and> k \<in> cball 0 (dd / 2)}"
   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 subt: "path_image \<gamma> \<subseteq> interior T"
     using \<open>0 < dd\<close> 
     apply (clarsimp simp add: mem_interior T_def)
     apply (rule_tac x="dd/2" in exI, auto)
     done
   have "compact T"
     unfolding T_def
     by (fastforce simp add: \<open>valid_path \<gamma>\<close> compact_valid_path_image intro!: compact_sums)
   have T: "T \<subseteq> U"
     unfolding T_def using \<open>0 < dd\<close> dd 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"
     proof (rule winding_number_zero_outside)
       show "path_image \<gamma> \<subseteq> cball 0 C"
         by (meson C interior_subset mem_cball_0 subset_eq subt)
     qed (use ybig loop \<open>path \<gamma>\<close> in auto)
     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"
     proof (intro holomorphic_intros)
       show "f holomorphic_on interior T"
         using holf holomorphic_on_subset interior_subset T by blast
     qed (use \<open>y \<notin> T\<close> interior_subset in 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)"
       proof (rule mult_mono)
         show "cmod (f z) \<le> D"
           using D interior_subset z by blast 
         show "inverse (cmod (z - y)) \<le> e / L / D" "D \<ge> 0"
           using \<open>L>0\<close> \<open>e>0\<close> \<open>D>0\<close> DL2 by (auto simp: norm_divide field_split_simps)
       qed auto
       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)"
     proof (rule open_Diff)
       show "open (U \<times> U)"
         by (simp add: open_Times \<open>open U\<close>)
       show "closed (Id :: complex rel)"
         using continuous_closed_preimage_constant [OF con_fstsnd closed_UNIV, of 0]
         by (auto simp: Id_fstsnd_eq algebra_simps)
     qed
     have con_derf: "continuous (at z) (deriv f)" if "z \<in> U" for z
     proof (rule continuous_on_interior)
       show "continuous_on U (deriv f)"
         by (simp add: holf holomorphic_deriv holomorphic_on_imp_continuous_on \<open>open U\<close>)
     qed (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
           using segment_furthest_le [of w x' z' 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>
         by (force simp: dist_norm neq intro: dual_order.strict_trans2 k1 less_imp_le norm_fst_le)
     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"
       using \<gamma>' B by (simp add: path_image_def vector_derivative_at rev_image_eqI)
     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 **: "(\<lambda>z. if w = z then deriv f z else (f w - f z) / (w - z)) field_differentiable at x"
         if "x \<in> U" "x \<noteq> w" for x
       proof (rule_tac f = "\<lambda>x. (f w - f x)/(w - x)" and d = "dist x w" in field_differentiable_transform_within)
         show "(\<lambda>x. (f w - f x) / (w - x)) field_differentiable at x"
           using that \<open>open U\<close>
           by (intro derivative_intros holomorphic_on_imp_differentiable_at [OF holf]; force)
       qed (use that \<open>open U\<close> in \<open>auto simp: dist_commute\<close>)
       show ?thesis
         unfolding d_def
       proof (rule no_isolated_singularity [OF * _ \<open>open U\<close>])
         show "(\<lambda>z. if w = z then deriv f z else (f w - f z) / (w - z)) holomorphic_on U - {w}"
           by (auto simp: field_differentiable_def [symmetric] holomorphic_on_open open_Diff \<open>open U\<close> **)
       qed auto
     qed
     { fix a b
       assume abu: "closed_segment a b \<subseteq> U"
       have cont_cint_d: "continuous_on U (\<lambda>w. contour_integral (linepath a b) (\<lambda>z. d z w))"
       proof (rule contour_integral_continuous_on_linepath_2D [OF \<open>open U\<close> _ _ abu])
         show "\<And>w. w \<in> U \<Longrightarrow> (\<lambda>z. d z w) contour_integrable_on (linepath a b)"
           by (metis abu hol_dw continuous_on_subset contour_integrable_continuous_linepath holomorphic_on_imp_continuous_on)
         show "continuous_on (U \<times> U) (\<lambda>(x, y). d y x)"
           by (auto intro: continuous_on_swap_args cond_uu)
       qed
       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 "continuous_on {0..1} (\<lambda>x. vector_derivative \<gamma> (at x))"
         using pf\<gamma>' by (simp add: continuous_on_polymonial_function vector_derivative_at [OF \<gamma>'])
       then      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)
         by (rule continuous_on_mult [OF cont_cint_d\<gamma> [unfolded o_def]])
       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))"
       proof (rule contour_integral_swap)
         show "continuous_on (path_image (linepath a b) \<times> path_image \<gamma>) (\<lambda>(y1, y2). d y1 y2)"
           using abu pasz by (auto intro: continuous_on_subset [OF cond_uu])
         show "continuous_on {0..1} (\<lambda>t. vector_derivative (linepath a b) (at t))"
           by (auto intro!: continuous_intros)
         show "continuous_on {0..1} (\<lambda>t. vector_derivative \<gamma> (at t))"
           by (metis \<gamma>' continuous_on_eq path_def path_polynomial_function pf\<gamma>' vector_derivative_at)
       qed (use \<open>valid_path \<gamma>\<close> in auto)
       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)"
           proof (rule compact_uniformly_continuous [OF continuous_on_subset[OF cond_uu]])
             show "compact {(w, z) |w z. w \<in> cball x dd \<and> z \<in> path_image \<gamma>}"
               using \<open>valid_path \<gamma>\<close>
               by (auto simp: compact_Times compact_valid_path_image simp del: mem_cball)
           qed (use dd pasz in auto)
           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)
           obtain no where "\<forall>n\<ge>no. dist (a n) x < min dd kk"
             using ax unfolding lim_sequentially
             by (meson \<open>0 < dd\<close> \<open>0 < kk\<close> min_less_iff_conj)
           then show ?thesis
             using \<open>dd > 0\<close> \<open>kk > 0\<close> by (fastforce simp: eventually_sequentially kk dist_norm)
         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"
           proof (rule contour_integral_unique [OF Cauchy_theorem_triangle])
             show "\<And>w. w \<in> U \<Longrightarrow> (\<lambda>z. d z w) holomorphic_on convex hull {a, b, c}"
               using e abc_subset by (auto intro: holomorphic_on_subset [OF hol_dw])
           qed
           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 metis
   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: "cmod (f w - y) \<le> B0" if "cmod (w - z) \<le> r" for w
   proof (rule continuous_on_closure_norm_le [of "ball z r" "\<lambda>w. f w - y"], use \<open>0 < r\<close> in simp_all)
     show "continuous_on (cball z r) (\<lambda>w. f w - y)"
       by (intro continuous_intros contf)
     show "dist z w \<le> r"
       by (simp add: dist_commute dist_norm that)
     qed (use fin in \<open>auto simp: dist_norm less_eq_real_def norm_minus_commute\<close>)
   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 "\<xi> \<in> ball \<xi> r"
     using \<open>0 < r\<close> by simp
   then have  "((\<lambda>u. f u / (u-\<xi>) ^ Suc n) has_contour_integral (2 * pi) * \<i> / fact n * (deriv ^^ n) f \<xi>)
          (circlepath \<xi> r)"
     by (rule Cauchy_has_contour_integral_higher_derivative_circlepath [OF contf holf])
   have "norm ((2 * pi * \<i>)/(fact n) * (deriv ^^ n) f \<xi>) \<le> (B / r^(Suc n)) * (2 * pi * r)"
   proof (rule has_contour_integral_bound_circlepath)
     have "\<xi> \<in> ball \<xi> r"
       using \<open>0 < r\<close> by simp
     then show  "((\<lambda>u. f u / (u-\<xi>) ^ Suc n) has_contour_integral (2 * pi) * \<i> / fact n * (deriv ^^ n) f \<xi>)
          (circlepath \<xi> r)"
       by (rule Cauchy_has_contour_integral_higher_derivative_circlepath [OF contf holf])
     show "\<And>x. cmod (x-\<xi>) = r \<Longrightarrow> cmod (f x / (x-\<xi>) ^ Suc n) \<le> B / r ^ Suc n"
       using \<open>0 \<le> B\<close> \<open>0 < r\<close>
       by (simp add: norm_divide norm_power nof frac_le norm_minus_commute del: power_Suc)
   qed (use \<open>0 \<le> B\<close> \<open>0 < r\<close> in auto)
   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>)"
   proof (rule holomorphic_power_series [where r = "norm \<xi> + 1"])
     show "f holomorphic_on ball 0 (cmod \<xi> + 1)" "\<xi> \<in> ball 0 (cmod \<xi> + 1)"
       using holf holomorphic_on_subset by auto
   qed
   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"
     proof (rule Cauchy_inequality)
       show "f holomorphic_on ball 0 (cmod w)"
         using holf holomorphic_on_subset by force
       show "continuous_on (cball 0 (cmod w)) f"
         using holf holomorphic_on_imp_continuous_on holomorphic_on_subset by blast
       show "\<And>x. cmod (0 - x) = cmod w \<Longrightarrow> cmod (f x) \<le> B * cmod w ^ n"
         by (metis nof wgeA dist_0_norm dist_norm)
     qed (use \<open>w \<noteq> 0\<close> in auto)
     also have "... = fact k * (B * 1 / cmod w ^ (k-n))"
     proof -
       have "cmod w ^ n / cmod w ^ k = 1 / cmod w ^ (k - n)"
         using \<open>k>n\<close> \<open>w \<noteq> 0\<close> \<open>0 < B\<close> by (simp add: field_split_simps semiring_normalization_rules)
       then show ?thesis
         by (metis times_divide_eq_right)
     qed
     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"
+      shows "f constant_on UNIV"
 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
+    using Liouville_polynomial [OF holf, of 0 B 0, simplified]
+    by (meson constant_on_def)
 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)"
   proof
     show "a (LEAST n. a n \<noteq> 0) \<noteq> 0"
       by (metis (mono_tags, lifting) m0 LeastI)
   qed (fastforce dest!: not_less_Least)
   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"
   proof (rule continuous_onE)
     show "\<xi> \<in> cball \<xi> (r / 2)" "1/2 > (0::real)"
       using r by auto
   qed (auto simp: dist_commute dist_norm)
   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] r sums_iff unfolding g_def 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 (use False assms in \<open>auto simp: field_simps tan_def\<close>)
 qed simp_all
 
 end