Quelle  Cauchy_Integral_Formula.thy

section ‹Cauchy's Integral Formula›
theory Cauchy_Integral_Formula
  imports Winding_Numbers
begin

subsection‹Proof›

lemma Cauchy_integral_formula_weak:
    assumes S: "convex S" and "finite k" and conf: "continuous_on S f"
        and fcd: "(∧x. x ∈ interior S - k ==> f field_differentiable at x)"
        and z: "z ∈ interior S - k" and vpg: "valid_path γ"
        and pasz: "path_image γ ⊆ S - {z}" and loop: "pathfinish γ = pathstart γ"
      shows "((λw. f w / (w-z)) has_contour_integral (2*pi * i * winding_number γ z * f z)) γ"
proof -
  let ?fz = "λw. (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 γ ⊆ S" and znotin: "z ∉ path_image γ" using pasz by blast+
  have c: "continuous (at x within S) (λw. if w = z then f' else (f w - f z) / (w-z))" if "x ∈ S" for x
  proof (cases "x = z")
    case True then show ?thesis
      using LIM_equal [of "z" ?fz "λw. 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) (λw. (f w - f z) / (w-z))"
      by (rule cf continuous_intros | simp add: False)+
    then show ?thesis
      using continuous_transform_within [OF _ dxz that] by (force simp: dist_commute)
  qed
  have fink': "finite (insert z k)" using ‹finite k› by blast
  have *: "((λw. if w = z then f' else ?fz w) has_contour_integral 0) γ"
  proof (rule Cauchy_theorem_convex [OF _ S fink' _ vpg pas loop])
    show "(λw. if w = z then f' else ?fz w) field_differentiable at w" 
      if "w ∈ interior S - insert z k" for w
    proof (rule field_differentiable_transform_within)
      show "(λw. ?fz w) field_differentiable at w"
        using that by (intro derivative_intros fcd; simp)
    qed (use that in ‹auto simp add: dist_pos_lt dist_commute›)
  qed (use c in ‹force simp: continuous_on_eq_continuous_within›)
  note ** = has_contour_integral_add [OF has_contour_integral_lmul [OF has_contour_integral_winding_number [OF vpg znotin], of "f z"] *]
  show ?thesis
    apply (rule has_contour_integral_eq)
    using znotin ** 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 ∈ interior S" "valid_path γ" "path_image γ ⊆ S - {z}"
      "pathfinish γ = pathstart γ"
    shows "((λw. f w / (w-z)) has_contour_integral (2*pi * i * winding_number γ z * f z)) γ"
proof -
  have "∧x. x ∈ interior S ==> 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‹ Hence the Cauchy formula for points inside a circle.›

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 "((λu. 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 "((λu. 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 "∧x. x ∈ interior (cball z r) - {} ==>
         f field_differentiable at x"
      using holf holomorphic_on_imp_differentiable_at by auto
    have "w ∉ sphere z r"
      by simp (metis dist_commute dist_norm not_le order_refl wz)
    then show "path_image (circlepath z r) ⊆ cball z r - {w}"
      using ‹r > 0› by (auto simp add: cball_def sphere_def)
  qed (use wz in ‹simp_all add: dist_norm norm_minus_commute contf›)
  then show ?thesis
    by (simp add: winding_number_circlepath assms)
qed

corollary🍋‹tag unimportant› Cauchy_integral_circlepath_simple:
  assumes "f holomorphic_on cball z r" "norm(w-z) < r"
  shows "((λu. 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🍋‹tag unimportant› ‹General stepping result for derivative formulas›

lemma Cauchy_next_derivative:
  fixes f' :: "complex ==> complex"
  defines "h ≡ λk w u. f' u / (u-w)^k"
  assumes "continuous_on (path_image γ) f'"
      and leB: "∧t. t ∈ {0..1} ==> norm (vector_derivative γ (at t)) ≤ B"
      and int: "∧w. w ∈ S - path_image γ ==> (h k w has_contour_integral f w) γ"
      and k: "k ≠ 0"
      and "open S"
      and γ: "valid_path γ"
      and w: "w ∈ S - path_image γ"
    shows "h (Suc k) w contour_integrable_on γ"
      and "(f has_field_derivative (k * contour_integral γ (h (Suc k) w))) (at w)"  (is "?thes2")
proof -
  have "open (S - path_image γ)" using ‹open S› closed_valid_path_image γ by blast
  then obtain d where "d>0" and d: "ball w d ⊆ S - path_image γ" using w
    using open_contains_ball by blast
  have [simp]: "∧n. cmod (1 + of_nat n) = 1 + of_nat n"
    by (metis norm_of_nat of_nat_Suc)
  have cint: "(λz. (h k x z - h k w z) / (x * k - w * k)) contour_integrable_on γ"
    if "x ≠ w" "cmod (x-w) < d" for x::complex
  proof -
    have "x ∈ S - path_image γ"
      by (metis d dist_commute dist_norm mem_ball subsetD that(2))
    then show ?thesis
      using contour_integrable_diff contour_integrable_div contour_integrable_on_def int w
      by meson
  qed
  then have 1: "∀🪙F x in at w. (λz. (h k x z - h k w z) / (x-w) / of_nat k)
                         contour_integrable_on γ"
    unfolding eventually_at 
    by (force intro: exI [where x=d] simp add: ‹d > 0› dist_norm field_simps)
  have bim_g: "bounded (image f' (path_image γ))"
    by (simp add: compact_imp_bounded compact_continuous_image compact_valid_path_image assms)
  then obtain C where "C > 0" and C: "∧x. [0 ≤ x; x ≤ 1] ==> cmod (f' (γ x)) ≤ C"
    by (force simp: bounded_pos path_image_def)
  have twom: "∀🪙F n in at w.
               ∀x∈path_image γ.
                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 ∈ path_image γ" and "u ≠ 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: "∧z::complex. z ≠ 0 ==> z ^ (k - Suc 0) = z ^ k / z"
        by (simp add: field_simps) (metis Suc_pred ‹k ≠ 0› neq0_conv power_Suc)
      have ff1: "(ff i has_field_derivative ff (Suc i) z) (at z within ball w (d/2))"
              if "z ∈ ball w (d/2)" "i ≤ 1" for i z
      proof -
        have "z ∉ path_image γ"
          using ‹x ∈ path_image γ› d that ball_divide_subset_numeral by blast
        then have xz[simp]: "x ≠ z" using ‹x ∈ path_image γ› by blast
        then have neq: "x * x + z * z ≠ x * (z * 2)"
          by (blast intro: dest!: sum_sqs_eq)
        with xz have "∧v. v ≠ 0 ==> (x * x + z * z) * v ≠ (x * (z * 2) * v)" by auto
        then have neqq: "∧v. v ≠ 0 ==> x * (x * v) + z * (z * v) ≠ x * (z * (2 * v))"
          by (simp add: algebra_simps)
        show ?thesis using ‹i ≤ 1›
          apply (simp add: ff_def dist_norm Nat.le_Suc_eq, 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) ≤ k * (1 + real k) * (4 / b) ⟷ b ≤ 4 * a"
          by (subst mult_le_cancel_left_pos)
            (use ‹k ≠ 0› in ‹auto simp: divide_simps›)
        with ab have "real k * (1 + real k) / a ≤ (real k * 4 + real k * real k * 4) / b ⟷ b ≤ 4 * a"
          by (simp add: field_simps)
      } note canc = this
      have ff2: "cmod (ff (Suc 1) v) ≤ real (k * (k + 1)) / (d/2) ^ (k + 2)"
                if "v ∈ ball w (d/2)" for v
      proof -
        have lessd: "∧z. cmod (γ z - v) < d/2 ==> cmod (w - γ z) < d"
          by (metis that norm_minus_commute norm_triangle_half_r dist_norm mem_ball)
        have "d/2 ≤ cmod (x-v)" using d x that
          using lessd d x unfolding path_image_def
          by (smt (verit, best) dist_norm imageE insert_Diff mem_ball subset_Diff_insert)
        then have "d ≤ cmod (x-v) * 2"
          by (simp add: field_split_simps)
        then have dpow_le: "d ^ (k+2) ≤ (cmod (x-v) * 2) ^ (k+2)"
          using ‹0 🚫› order_less_imp_le power_mono by blast
        have "x ≠ v" using that
          using ‹x ∈ path_image γ› ball_divide_subset_numeral d by fastforce
        then show ?thesis
        using ‹d > 0› 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 ∈ 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)))
                  ≤ (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 ‹0 🚫›)
      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) * (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)
                  ≤ (1 + real k) * cmod (u-w) / (d/2) ^ (k+2)"
        using ‹k ≠ 0› ‹u ≠ w› by (simp add: mult_ac zero_less_mult_iff pos_divide_le_eq)
      also have "… < e"
        using uw_less ‹0 🚫› 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 ≠ u"
        using uwd ‹0 🚫› x d by (force simp: dist_norm ball_def norm_minus_commute)
      show ?thesis
        using ‹k ≠ 0› ‹x ≠ u› ‹u ≠ w› le_less_trans [OF _ e]
        by (simp add: field_simps flip: norm_divide)
    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: ‹d > 0› dist_norm that simp del: power_Suc intro: *)
      done
  qed
  have 2: "uniform_limit (path_image γ) (λx z. (h k x z - h k w z) / (x-w) / k) (h (Suc k) w) (at w)"
    unfolding uniform_limit_iff dist_norm
  proof clarify
    fix e::real
    assume "0 < e"
    have *: "cmod ((h k x (γ u) - h k w (γ u)) / ((x-w) * k) - h (Suc k) w (γ u)) < e"
          if ec: "cmod ((inverse (γ u - x) ^ k - inverse (γ u - w) ^ k) / ((x-w) * k) -
                  inverse (γ u - w) * inverse (γ u - w) ^ k) < e / C"
                 and x: "0 ≤ u" "u ≤ 1"
               for x u
    proof (cases "(f' (γ u)) = 0")
      case True then show ?thesis by (simp add: ‹0 🚫› h_def)
    next
      case False
      have "cmod ((h k x (γ u) - h k w (γ u)) / ((x-w) * k) - h (Suc k) w (γ u)) =
            cmod (f' (γ u) * ((inverse (γ u - x) ^ k - inverse (γ u - w) ^ k) / ((x-w) * k) -
                             inverse (γ u - w) * inverse (γ u - w) ^ k))"
        by (simp add: h_def field_simps)
      also have "… = cmod (f' (γ u)) *
                       cmod ((inverse (γ u - x) ^ k - inverse (γ u - w) ^ k) / ((x-w) * k) -
                             inverse (γ u - w) * inverse (γ u - w) ^ k)"
        by (simp add: norm_mult)
      also have "… < cmod (f' (γ u)) * (e/C)"
        using False mult_strict_left_mono [OF ec] by force
      also have "… ≤ e" using C
        by (metis False ‹0 🚫› frac_le less_eq_real_def mult.commute pos_le_divide_eq x zero_less_norm_iff)
      finally show ?thesis .
    qed
    show "∀🪙F u in at w.
              ∀x∈path_image γ.
               cmod ((h k u x - h k w x) / (u-w) / of_nat k - h (Suc k) w x) < e"
      using twom [OF divide_pos_pos [OF ‹0 🚫› ‹C > 0›]] *
      unfolding path_image_def h_def
      by (force elim: eventually_mono)
  qed
  show "h (Suc k) w contour_integrable_on γ"
    using contour_integral_uniform_limit [OF 1 2 leB γ] by (simp add: h_def)
  have *: "(λu. contour_integral γ (λx. (h k u x - h k w x) / (u-w) / k))
           ←-w→ contour_integral γ (h (Suc k) w)"
    by (rule contour_integral_uniform_limit [OF 1 2 leB γ]) auto
  have **: "contour_integral γ (λx. (h k u x - h k w x) / ((u-w) * k)) =
              (f u - f w) / (u-w) / k"
    if "dist u w < d" for u
  proof -
    have "u ∈ S - path_image γ"
      by (metis subsetD d dist_commute mem_ball that)
    then have "(h k u has_contour_integral f u) γ" "(h k w has_contour_integral f w) γ"
      using w by (simp_all add: field_simps int)
    then show ?thesis
      using contour_integral_unique has_contour_integral_diff
        has_contour_integral_div by force
  qed
  show ?thes2
    unfolding has_field_derivative_iff
    by (simp add: ‹k ≠ 0› ** Lim_transform_within [OF tendsto_mult_left [OF *] ‹0 🚫›])
qed

lemma Cauchy_next_derivative_circlepath:
  assumes contf: "continuous_on (path_image (circlepath z r)) f"
      and int: "∧w. w ∈ ball z r ==> ((λu. f u / (u-w)^k) has_contour_integral g w) (circlepath z r)"
      and k: "k ≠ 0"
      and w: "w ∈ ball z r"
    shows "(λu. 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) (λu. 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 ∈ 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 * ∣r∣"];
        auto simp: vector_derivative_circlepath norm_mult)+
qed


text‹ In particular, the first derivative formula.›

lemma Cauchy_derivative_integral_circlepath:
  assumes contf: "continuous_on (cball z r) f"
      and holf: "f holomorphic_on ball z r"
      and w: "w ∈ ball z r"
    shows "(λu. 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) (λu. f u / (u-w)^2))) (at w)"
           (is "?thes2")
proof -
  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)
  have int: "∧w. dist z w < r ==>
                 ((λu. f u / (u-w)) has_contour_integral (λx. 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
    unfolding power2_eq_square
    using int Cauchy_next_derivative_circlepath [OF f _ _ w, where k=1]
    by fastforce
  have "((λx. 2 * of_real pi * i * f x) has_field_derivative contour_integral (circlepath z r) (λu. f u / (u-w)^2)) (at w)"
    unfolding power2_eq_square
    using int Cauchy_next_derivative_circlepath [OF f _ _ w, where k=1 and g = "λx. 2 * of_real pi * i * f x"]
    by fastforce
  then have fder: "(f has_field_derivative contour_integral (circlepath z r) (λu. f u / (u-w)^2) / (2 * of_real pi * i)) (at w)"
    by (rule DERIV_cdivide [where f = "λx. 2 * of_real pi * i * f x" and c = "2 * of_real pi * i", simplified])
  show ?thes2
    by simp (rule fder)
qed

subsection‹Existence of all higher derivatives›

proposition derivative_is_holomorphic:
  assumes "open S"
      and fder: "∧z. z ∈ S ==> (f has_field_derivative f' z) (at z)"
    shows "f' holomorphic_on S"
proof -
  have *: "∃h. (f' has_field_derivative h) (at z)" if "z ∈ S" for z
  proof -
    obtain r where "r > 0" and r: "cball z r ⊆ S"
      using open_contains_cball ‹z ∈ S› ‹open S› 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 ‹r > 0› by (force elim: holomorphic_on_subset [THEN holomorphic_on_imp_continuous_on])
    then have contfpi: "continuous_on (path_image (circlepath z r)) (λx. 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 ∈ ball z r"
      have intf: "(λu. f u / (u-w)🪙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) (λu. f u / (u-w)🪙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) (λu. f u / (u-w)🪙2) / (2 * of_real pi * i)"
        using fder' ball_subset_cball r w by (force intro: DERIV_unique [OF fder])
      have "((λu. f u / (u-w)🪙2 / (2 * of_real pi * i)) has_contour_integral
                contour_integral (circlepath z r) (λu. f u / (u-w)🪙2) / (2 * of_real pi * i))
                (circlepath z r)"
        by (rule has_contour_integral_div [OF has_contour_integral_integral [OF intf]])
      then have "((λu. f u / (2 * of_real pi * i * (u-w)🪙2)) has_contour_integral
                contour_integral (circlepath z r) (λu. f u / (u-w)🪙2) / (2 * of_real pi * i))
                (circlepath z r)"
        by (simp add: algebra_simps)
      then have "((λu. f u / (2 * of_real pi * i * (u-w)🪙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'] ‹0 🚫› *
      using centre_in_ball mem_ball by force
  qed
  show ?thesis
    by (simp add: holomorphic_on_open [OF ‹open S›] *)
qed

lemma holomorphic_deriv [holomorphic_intros]:
  "[f holomorphic_on S; open S] ==> (deriv f) holomorphic_on S"
  by (metis DERIV_deriv_iff_field_differentiable at_within_open derivative_is_holomorphic holomorphic_on_def)

lemma holomorphic_deriv_compose:
  assumes g: "g holomorphic_on B" and f: "f holomorphic_on A" and "f ` A ⊆ B" "open B"
  shows   "(λx. deriv g (f x)) holomorphic_on A"
  using holomorphic_on_compose_gen [OF f holomorphic_deriv[OF g]] assms
  by (auto simp: o_def)

lemma analytic_deriv [analytic_intros]: "f analytic_on S ==> (deriv f) analytic_on S"
  using analytic_on_holomorphic holomorphic_deriv by auto

lemma holomorphic_higher_deriv [holomorphic_intros]: "[f holomorphic_on S; open S] ==> (deriv ^^ n) f holomorphic_on S"
  by (induction n) (auto simp: holomorphic_deriv)

lemma analytic_higher_deriv [analytic_intros]: "f analytic_on S ==> (deriv ^^ n) f analytic_on S"
  unfolding analytic_on_def using holomorphic_higher_deriv by blast

lemma has_field_derivative_higher_deriv:
     "[f holomorphic_on S; open S; x ∈ S]
      ==> ((deriv ^^ n) f has_field_derivative (deriv ^^ (Suc n)) f x) (at x)"
  using holomorphic_derivI holomorphic_higher_deriv by fastforce
  
lemma higher_deriv_cmult:
  assumes "f holomorphic_on A" "x ∈ A" "open A"
  shows   "(deriv ^^ j) (λx. c * f x) x = c * (deriv ^^ j) f x"
  using assms
proof (induction j arbitrary: f x)
  case (Suc j f x)
  have "deriv ((deriv ^^ j) (λx. c * f x)) x = deriv (λx. c * (deriv ^^ j) f x) x"
    using eventually_nhds_in_open[of A x] assms(2,3) Suc.prems
    by (intro deriv_cong_ev refl) (auto elim!: eventually_mono simp: Suc.IH)
  also have "… = c * deriv ((deriv ^^ j) f) x" using Suc.prems assms(2,3)
    by (intro deriv_cmult holomorphic_on_imp_differentiable_at holomorphic_higher_deriv) auto
  finally show ?case by simp
qed simp_all

lemma higher_deriv_cmult':
  assumes "f analytic_on {x}"
  shows   "(deriv ^^ j) (λx. c * f x) x = c * (deriv ^^ j) f x"
  using assms higher_deriv_cmult[of f _ x j c] assms
  using analytic_at_two by blast

lemma deriv_cmult':
  assumes "f analytic_on {x}"
  shows   "deriv (λx. c * f x) x = c * deriv f x"
  using higher_deriv_cmult'[OF assms, of 1 c] by simp

lemma analytic_derivI:
  assumes "f analytic_on {z}"
  shows   "(f has_field_derivative (deriv f z)) (at z within A)"
  using assms holomorphic_derivI[of f _ z] analytic_at by blast

lemma deriv_compose_analytic:
  fixes f g :: "complex ==> complex"
  assumes "f analytic_on {g z}" "g analytic_on {z}"
  shows "deriv (λx. f (g x)) z = deriv f (g z) * deriv g z"
proof -
  have "((f ∘ g) has_field_derivative (deriv f (g z) * deriv g z)) (at z)"
    by (intro DERIV_chain analytic_derivI assms)
  thus ?thesis
    by (auto intro!: DERIV_imp_deriv simp add: o_def)
qed

lemma valid_path_compose_holomorphic:
  assumes "valid_path g" "f holomorphic_on S" and "open S" "path_image g ⊆ S"
  shows "valid_path (f ∘ g)"
  by (meson assms holomorphic_deriv holomorphic_on_imp_continuous_on holomorphic_on_imp_differentiable_at
      holomorphic_on_subset subsetD valid_path_compose)

lemma valid_path_compose_analytic:
  assumes "valid_path g" and holo:"f analytic_on S" and "path_image g ⊆ S"
  shows "valid_path (f ∘ g)"
proof (rule valid_path_compose[OF ‹valid_path g›])
  fix x assume "x ∈ path_image g"
  then show "f field_differentiable at x"
    using analytic_on_imp_differentiable_at analytic_on_open assms holo by blast
next
  show "continuous_on (path_image g) (deriv f)"
    by (intro holomorphic_on_imp_continuous_on analytic_imp_holomorphic analytic_intros
              analytic_on_subset[OF holo] assms)
qed

lemma analytic_on_deriv [analytic_intros]:
  assumes "f analytic_on g ` A"
  assumes "g analytic_on A"
  shows   "(λx. deriv f (g x)) analytic_on A"
proof -
  have "(deriv f ∘ g) analytic_on A"
    by (rule analytic_on_compose_gen[OF assms(2) analytic_deriv[OF assms(1)]]) auto
  thus ?thesis
    by (simp add: o_def)
qed

lemma contour_integral_comp_analyticW:
  assumes "f analytic_on s" "valid_path γ" "path_image γ ⊆ s"
  shows "contour_integral (f ∘ γ) g = contour_integral γ (λw. deriv f w * g (f w))"
proof -
  obtain spikes where "finite spikes" and γ_diff: "γ C1_differentiable_on {0..1} - spikes"
    using ‹valid_path γ› unfolding valid_path_def piecewise_C1_differentiable_on_def by auto  
  show "contour_integral (f ∘ γ) g
      = contour_integral γ (λw. deriv f w * g (f w))"
    unfolding contour_integral_integral
  proof (rule integral_spike[rule_format,OF negligible_finite[OF ‹finite spikes›]])
    fix t::real assume t:"t ∈ {0..1} - spikes"
    then have "γ differentiable at t" 
      using γ_diff unfolding C1_differentiable_on_eq by auto
    moreover have "f field_differentiable at (γ t)" 
    proof -
      have "γ t ∈ s" using t assms unfolding path_image_def by auto 
      thus ?thesis 
        using ‹f analytic_on s›  analytic_on_imp_differentiable_at by blast
    qed
    ultimately show "deriv f (γ t) * g (f (γ t)) * vector_derivative γ (at t) =
         g ((f ∘ γ) t) * vector_derivative (f ∘ γ) (at t)"
      by (subst vector_derivative_chain_at_general) (simp_all add:field_simps)
  qed
qed

subsection‹Morera's theorem›

lemma Morera_local_triangle_ball:
  assumes "∧z. z ∈ S
          ==> ∃e a. 0 < e ∧ z ∈ ball a e ∧ continuous_on (ball a e) f ∧
                    (∀b c. closed_segment b c ⊆ ball a e
                           ⟶ 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 ∈ S"
    with assms obtain e a where
            "0 < e" and z: "z ∈ ball a e" and contf: "continuous_on (ball a e) f"
        and 0: "∧b c. closed_segment b c ⊆ ball a e
                      ==> 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 "∃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) ⊆ ball a e"
          by (simp add: dist_commute ball_subset_ball_iff)
        have sub_ball: "∧y. dist a y < e ==> closed_segment a y ⊆ ball a e"
          by (meson ‹0 🚫› 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 ‹0 🚫› sub_ball)
      qed
    qed (simp add: az)
  }
  then show ?thesis
    by (simp add: analytic_on_def)
qed

lemma Morera_local_triangle:
  assumes "∧z. z ∈ S
          ==> ∃t. open t ∧ z ∈ t ∧ continuous_on t f ∧
                  (∀a b c. convex hull {a,b,c} ⊆ t
                              ⟶ 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 ∈ S"
    with assms obtain t where
            "open t" and z: "z ∈ t" and contf: "continuous_on t f"
        and 0: "∧a b c. convex hull {a,b,c} ⊆ t
                      ==> 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 ⊆ 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: "∧b c. closed_segment b c ⊆ ball z e ==>
                         contour_integral (linepath z b) f +
                         contour_integral (linepath b c) f +
                         contour_integral (linepath c z) f = 0"
      by (meson 0 z ‹0 🚫› centre_in_ball closed_segment_subset convex_ball dual_order.trans e starlike_convex_subset)
    have "∃e a. 0 < e ∧ z ∈ ball a e ∧ continuous_on (ball a e) f ∧
                (∀b c. closed_segment b c ⊆ ball a e ⟶
                       contour_integral (linepath a b) f + contour_integral (linepath b c) f + contour_integral (linepath c a) f = 0)"
      using ‹e > 0› eq0 by force
  }
  then show ?thesis
    by (simp add: Morera_local_triangle_ball)
qed

proposition Morera_triangle:
    "[continuous_on S f; open S;
      ∧a b c. convex hull {a,b,c} ⊆ S
              ⟶ contour_integral (linepath a b) f +
                  contour_integral (linepath b c) f +
                  contour_integral (linepath c a) f = 0]
     ==> f analytic_on S"
  using Morera_local_triangle by blast

subsection‹Combining theorems for higher derivatives including Leibniz rule›

lemma higher_deriv_linear [simp]:
    "(deriv ^^ n) (λw. c*w) = (λz. 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) (λw. c) = (λw. if n=0 then c else 0)"
  by (induction n) auto

lemma higher_deriv_ident [simp]:
     "(deriv ^^ n) (λw. 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:
     "[(f has_field_derivative 1) (at z); f z = z] ==> (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 ∈ S"
    shows "(deriv ^^ n) (λw. -(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 "∧x. x ∈ S ==> - (deriv ^^ n) f x = (deriv ^^ n) (λw. - f w) x"
    by (auto simp add: Suc)
  then have "((deriv ^^ n) (λw. - f w) has_field_derivative - deriv ((deriv ^^ n) f) z) (at z)"
    using  has_field_derivative_transform_within_open [of "λw. -((deriv ^^ n) f w)"]
    using "*" DERIV_minus Suc.prems ‹open S› 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 ∈ S"
    shows "(deriv ^^ n) (λw. 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 "∧x. x ∈ S ==> (deriv ^^ n) f x + (deriv ^^ n) g x = (deriv ^^ n) (λw. f w + g w) x"
    by (auto simp add: Suc)
  then have "((deriv ^^ n) (λw. 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 "λw. (deriv ^^ n) f w + (deriv ^^ n) g w"]
    using "*" Deriv.field_differentiable_add Suc.prems ‹open S› 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 ∈ S"
    shows "(deriv ^^ n) (λw. 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 Suc_choose: "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 ∈ S"
    shows "(deriv ^^ n) (λw. f w * g w) z =
           (∑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 *: "∧n. ((deriv ^^ n) f has_field_derivative deriv ((deriv ^^ n) f) z) (at z)"
          "∧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: "(∑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 + (∑i = 0..n. (deriv ^^ i) f z * (of_nat (Suc n choose i) * (deriv ^^ (Suc n - i)) g z))"
    apply (simp add: Suc_choose 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) (λw. f w * g w) has_field_derivative
         (∑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 "λw. (∑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 S› 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 ∈ S"
      and fg: "∧w. w ∈ S ==> 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 ∈ S"
      and fg: "∧w. w ∈ S ==> u*w + c ∈ T"
    shows "(deriv ^^ n) (λw. f (u*w + c)) z = u^n * (deriv ^^ n) f (u*z + c)"
using z
proof (induction n arbitrary: z)
  case 0 then show ?case by simp
next
  case (Suc n z)
  have holo0: "f holomorphic_on (λw. u * w+c) ` S"
    by (meson fg f holomorphic_on_subset image_subset_iff)
  have holo2: "(deriv ^^ n) f holomorphic_on (λw. u * w+c) ` S"
    by (meson f fg holomorphic_higher_deriv holomorphic_on_subset image_subset_iff T)
  have holo3: "(λz. u ^ n * (deriv ^^ n) f (u * z+c)) holomorphic_on S"
    by (intro holo2 holomorphic_on_compose [where g="(deriv ^^ n) f", unfolded o_def] holomorphic_intros)
  have "(λw. u * w+c) holomorphic_on S" "f holomorphic_on (λw. u * w+c) ` S"
    by (rule holo0 holomorphic_intros)+
  then have holo1: "(λw. f (u * w+c)) holomorphic_on S"
    by (rule holomorphic_on_compose [where g=f, unfolded o_def])
  have "deriv ((deriv ^^ n) (λw. f (u * w+c))) z = deriv (λz. u^n * (deriv ^^ n) f (u*z+c)) z"
  proof (rule complex_derivative_transform_within_open [OF _ holo3 S Suc.prems])
    show "(deriv ^^ n) (λw. f (u * w+c)) holomorphic_on S"
      by (rule holomorphic_higher_deriv [OF holo1 S])
  qed (simp add: Suc.IH)
  also have "… = u^n * deriv (λz. (deriv ^^ n) f (u * z+c)) z"
  proof -
    have "(deriv ^^ n) f analytic_on T"
      by (simp add: analytic_on_open f holomorphic_higher_deriv T)
    then have "(λw. (deriv ^^ n) f (u * w+c)) analytic_on S"
        using holomorphic_on_compose[OF _ holo2] ‹(λw. u * w+c) holomorphic_on S›
        by (simp add: S analytic_on_open o_def)
    then show ?thesis
      by (intro deriv_cmult analytic_on_imp_differentiable_at [OF _ Suc.prems])
  qed
  also have "… = u * u ^ n * deriv ((deriv ^^ n) f) (u * z+c)"
  proof -
    have "(deriv ^^ n) f field_differentiable at (u * z+c)"
      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_compose_linear:
  fixes z::complex
  assumes f: "f holomorphic_on T" and S: "open S" and T: "open T" and z: "z ∈ S"
      and fg: "∧w. w ∈ S ==> u * w ∈ T"
    shows "(deriv ^^ n) (λw. f (u * w)) z = u^n * (deriv ^^ n) f (u * z)"
  using higher_deriv_compose_linear' [where c=0] assms by simp

lemma higher_deriv_add_at:
  assumes "f analytic_on {z}" "g analytic_on {z}"
    shows "(deriv ^^ n) (λw. f w + g w) z = (deriv ^^ n) f z + (deriv ^^ n) g z"
  using analytic_at_two assms higher_deriv_add by blast

lemma higher_deriv_diff_at:
  assumes "f analytic_on {z}" "g analytic_on {z}"
    shows "(deriv ^^ n) (λw. f w - g w) z = (deriv ^^ n) f z - (deriv ^^ n) g z"
  using analytic_at_two assms higher_deriv_diff by blast

lemma higher_deriv_uminus_at:
   "f analytic_on {z} ==> (deriv ^^ n) (λw. -(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) (λw. f w * g w) z =
           (∑i = 0..n. of_nat (n choose i) * (deriv ^^ i) f z * (deriv ^^ (n-i)) g z)"
  using analytic_at_two assms higher_deriv_mult by blast


text‹ Nonexistence of isolated singularities and a stronger integral formula.›

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 ∈ S" and cdf: "∧x. x ∈ S - K ==> f field_differentiable at x"
    have "f field_differentiable at z"
    proof (cases "z ∈ K")
      case False then show ?thesis by (blast intro: cdf ‹z ∈ S›)
    next
      case True
      with finite_set_avoid [OF K, of z]
      obtain d where "d>0" and d: "∧x. [x∈K; x ≠ z] ==> d ≤ dist z x"
        by blast
      obtain e where "e>0" and e: "ball z e ⊆ S"
        using  S ‹z ∈ S› 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} ⊆ ball z (min d e) ==> continuous_on (convex hull {a, b, c}) f" for a b c
        by (simp add: hull_minimal continuous_on_subset [OF fde])
      have fd: "[{a,b,c} ⊆ ball z (min d e); x ∈ interior (convex hull {a, b, c}) - K]
            ==> 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 "∧w. w ∈ ball z (min d e) ==> (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 ‹0 🚫› ‹0 🚫› 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: "∧z. z ∈ K ==> (f ---> 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-)])
  have "continuous_on (S-K) f"
    using holf holomorphic_on_imp_continuous_on by auto
  then show "continuous_on S f"
    by (metis Diff_iff K S at_within_open continuous_on_eq_continuous_at
        continuous_within f finite_imp_closed open_Diff) 
qed

proposition Cauchy_integral_formula_convex:
  assumes S: "convex S" and K: "finite K" and contf: "continuous_on S f"
    and fcd: "(∧x. x ∈ interior S - K ==> f field_differentiable at x)"
    and z: "z ∈ interior S" and vpg: "valid_path γ"
    and pasz: "path_image γ ⊆ S - {z}" and loop: "pathfinish γ = pathstart γ"
  shows "((λw. f w / (w-z)) has_contour_integral (2*pi * i * winding_number γ z * f z)) γ"
proof -
  have *: "∧x. x ∈ interior S ==> 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_on_subset fcd field_differentiable_def open_interior
        has_field_derivative_at_within holomorphic_derivI holomorphic_onI interior_subset)
  show ?thesis
    by (rule Cauchy_integral_formula_weak [OF S finite.emptyI contf]) (use * assms in auto)
qed

text‹ Formula for higher derivatives.›

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 ∈ ball z r"
    shows "((λu. 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: "((λu. 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) ((λu. f u / (u-w) ^ Suc (Suc k))) = X"
    by (rule contour_integral_unique)
  have "∧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: "∧n. (deriv ^^ n) f field_differentiable (at w)"
    by (force simp: field_differentiable_def)
  have "deriv (λw. complex_of_real (2 * pi) * i / (fact k) * (deriv ^^ k) f w) w =
          of_nat (Suc k) * contour_integral (circlepath z r) (λu. 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 "… = of_nat (Suc k) * X"
    by (simp only: con)
  finally have "deriv (λw. ((2 * pi) * i / (fact k)) * (deriv ^^ k) f w) w = of_nat (Suc k) * X" .
  then have "((2 * pi) * i / (fact k)) * deriv (λw. (deriv ^^ k) f w) w = of_nat (Suc k) * X"
    by (metis deriv_cmult dnf_diff)
  then have "deriv (λw. (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 ∈ ball z r"
    shows "(λu. 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) (λu. f u/(u-w)^(Suc k))"
           (is "?thes2")
proof -
  have *: "((λu. 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 ∈ ball z r"
  shows "contour_integral(circlepath z r) (λu. 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 ∈ ball z r"
    shows "contour_integral(circlepath z r) (λu. 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‹A holomorphic function is analytic, i.e. has local power series›

theorem holomorphic_power_series:
  assumes holf: "f holomorphic_on ball z r"
      and w: "w ∈ ball z r"
    shows "((λn. (deriv ^^ n) f z / (fact n) * (w-z)^n) sums f w)"
proof -
  🍋 ‹Replacing 🍋‹r› and the original (weak) premises with stronger ones›
  obtain r where "r > 0" and holfc: "f holomorphic_on cball z r" and w: "w ∈ ball z r"
  proof
    have "cball z ((r + dist w z) / 2) ⊆ 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"
      by (metis w dist_norm mem_ball norm_ge_zero not_less_iff_gr_or_eq order_less_le_trans)
    then show "0 < (r + dist w z) / 2"
      by simp (use zero_le_dist [of w z] in linarith)
  qed (use w in ‹auto simp: dist_commute›)
  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: "∧k. (λu. f u / (u-z) ^ Suc k) contour_integrable_on circlepath z r"
    by (rule Cauchy_higher_derivative_integral_circlepath [OF contf holf]) (simp add: ‹0 🚫›)
  obtain B where "0 < B" and B: "∧u. u ∈ cball z r ==> norm(f u) ≤ 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 ≤ r" and wz_eq: "norm(w-z) = r - k"
             and kle: "∧u. norm(u-z) = r ==> k ≤ norm(u-w)"
  proof
    show "∧u. cmod (u-z) = r ==> r - dist z w ≤ cmod (u-w)"
      by (metis add_diff_eq diff_add_cancel dist_norm norm_diff_ineq)
  qed (use w in ‹auto simp: dist_norm norm_minus_commute›)
  have ul: "uniform_limit (sphere z r) (λn x. (∑k
    unfolding uniform_limit_iff dist_norm
  proof clarify
    fix e::real
    assume "0 < e"
    have rr: "0 ≤ (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"] ‹0 🚫› ‹0 🚫› k by force
    have "norm ((∑k
         if "n ≤ 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 ‹n ≤ N› k by auto
      have u [simp]: "(u ≠ z) ∧ (u ≠ w)"
        using ‹0 🚫› r w by auto
      have wzu_not1: "(w-z) / (u-z) ≠ 1"
        by (metis (no_types) dist_norm divide_eq_1_iff less_irrefl mem_ball norm_minus_commute r w)
      have "norm ((∑k
            = norm ((∑k
        unfolding sum_distrib_right sum_divide_distrib power_divide by (simp add: algebra_simps)
      also have "… = norm ((((w-z) / (u-z)) ^ N - 1) * (u-w) / (((w-z) / (u-z) - 1) * (u-z)) - 1) * norm (f u)"
        using ‹0 🚫›
        apply (simp add: geometric_sum [OF wzu_not1] flip: norm_mult)
        apply (simp add: field_simps)
        done
      also have "… = norm ((u-z) ^ N * (w-u) - ((w-z) ^ N - (u-z) ^ N) * (u-w)) / (r ^ N * norm (u-w)) * norm (f u)"
        using ‹0 🚫› r by (simp add: divide_simps norm_mult norm_divide norm_power dist_norm norm_minus_commute)
      also have "… = norm ((w-z) ^ N * (w-u)) / (r ^ N * norm (u-w)) * norm (f u)"
        by (simp add: algebra_simps)
      also have "… = norm (w-z) ^ N * norm (f u) / r ^ N"
        by (simp add: norm_mult norm_power norm_minus_commute)
      also have "… ≤ (((r-k)/r)^N) * B"
        using ‹0 🚫› w k
        by (simp add: B divide_simps mult_mono r wz_eq)
      also have "… < e * k"
        using ‹0 🚫› N by (simp add: divide_simps)
      also have "… ≤ e * norm (u-w)"
        using r kle ‹0 🚫› by (simp add: dist_commute dist_norm)
      finally show ?thesis
        by (simp add: field_split_simps norm_divide del: power_Suc)
    qed
    with ‹0 🚫› show "∀🪙F n in sequentially. ∀x∈sphere z r.
                norm ((∑k
      by (auto simp: mult_ac less_imp_le eventually_sequentially Ball_def)
  qed
  have 🍋: "∧x k. k∈ {..==>
           (λu. (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: "∧n.
             (∑k
             contour_integral (circlepath z r) (λu. ∑k
    apply (subst contour_integral_sum)
    apply (simp_all only: 🍋 finite_lessThan contour_integral_lmul cint algebra_simps)
    done
  have "∧u k. k ∈ {..==> (λx. f x / (x-z) ^ Suc k) contour_integrable_on circlepath z r"
    using ‹0 🚫› by (force intro!: Cauchy_higher_derivative_integral_circlepath [OF contf holf])
  then have "∧u. (λy. ∑k
    by (intro contour_integrable_sum contour_integrable_lmul, simp)
  then have "(λk. contour_integral (circlepath z r) (λu. f u/(u-z)^(Suc k)) * (w-z)^k)
        sums contour_integral (circlepath z r) (λu. f u/(u-w))"
    unfolding sums_def eq 
    using ‹0 🚫› contour_integral_uniform_limit_circlepath [OF eventuallyI ul]    
    by fastforce
  then have "(λk. contour_integral (circlepath z r) (λu. 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 "(λk. contour_integral (circlepath z r) (λu. 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 "(λn. (w-z) ^ n * contour_integral (circlepath z r) (λu. 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 ‹0 🚫› Cauchy_higher_derivative_integral_circlepath [OF contf holf])
qed

subsection‹The Liouville theorem and the Fundamental Theorem of Algebra›

text‹ These weak Liouville versions don't even need the derivative formula.›

lemma Liouville_weak_0:
  assumes holf: "f holomorphic_on UNIV" and inf: "(f ---> 0) at_infinity"
    shows "f z = 0"
proof (rule ccontr)
  assume fz: "f z ≠ 0"
  with inf [unfolded Lim_at_infinity, rule_format, of "norm(f z)/2"]
  obtain B where B: "∧x. B ≤ cmod x ==> norm (f x) * 2 < cmod (f z)"
    by (auto simp: dist_norm)
  define R where "R = 1 + ∣B∣ + norm z"
  have "R > 0"
    unfolding R_def by (smt (verit) norm_ge_zero)
  have *: "((λu. 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 ‹0 🚫›
    by (force intro: holomorphic_on_imp_continuous_on Cauchy_integral_circlepath)
  have "cmod (x-z) = R ==> 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 ‹R > 0› 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 ---> l) at_infinity"
    shows "f z = l"
  using Liouville_weak_0 [of "λz. f z - l"]
  by (simp add: assms holomorphic_on_diff LIM_zero)

proposition Liouville_weak_inverse:
  assumes "f holomorphic_on UNIV" and unbounded: "∧B. eventually (λx. norm (f x) ≥ B) at_infinity"
    obtains z where "f z = 0"
proof -
  { assume f: "∧z. f z ≠ 0"
    have 1: "(λx. 1 / f x) holomorphic_on UNIV"
      by (simp add: holomorphic_on_divide assms f)
    have 2: "((λx. 1 / f x) ---> 0) at_infinity"
    proof (rule tendstoI [OF eventually_mono])
      fix e::real
      assume "e > 0"
      show "eventually (λx. 2/e ≤ 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‹ In particular we get the Fundamental Theorem of Algebra.›

theorem fundamental_theorem_of_algebra:
    fixes a :: "nat ==> complex"
  assumes "a 0 = 0 ∨ (∃i ∈ {1..n}. a i ≠ 0)"
  obtains z where "(∑i≤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 ∈ {1..n}" and nz: "a i ≠ 0"
  have 1: "(λz. ∑i≤n. a i * z^i) holomorphic_on UNIV"
    by (rule holomorphic_intros)+
  show thesis
  proof (rule Liouville_weak_inverse [OF 1])
    show "∀🪙F x in at_infinity. B ≤ cmod (∑i≤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‹Weierstrass convergence theorem›

lemma holomorphic_uniform_limit:
  assumes cont: "eventually (λn. continuous_on (cball z r) (f n) ∧ (f n) holomorphic_on ball z r) F"
      and ulim: "uniform_limit (cball z r) f g F"
      and F:  "¬ 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) ⊆ cball z r"
    using ‹0 🚫› by auto
  then have 1: "continuous_on (path_image (circlepath z r)) (λx. 1 / (2 * complex_of_real pi * i) * g x)"
    by (intro continuous_intros continuous_on_subset [OF contg])
  have 2: "((λu. 1 / (2 * of_real pi * i) * g u / (u-w) ^ 1) has_contour_integral g w) (circlepath z r)"
       if w: "w ∈ ball z r" for w
  proof -
    define d where "d = (r - norm(w-z))"
    have "0 < d"  "d ≤ r" using w by (auto simp: norm_minus_commute d_def dist_norm)
    have dle: "∧u. cmod (z-u) = r ==> d ≤ cmod (u-w)"
      unfolding d_def by (metis add_diff_eq diff_add_cancel norm_diff_ineq norm_minus_commute)
    have ev_int: "∀🪙F n in F. (λu. 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 "∧e. [0 < r; 0 < d; 0 < e]
         ==> ∀🪙F n in F.
                ∀x∈sphere z r.
                   x ≠ w ⟶
                   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) (λn x. f n x / (x-w)) (λx. g x / (x-w)) F"
      using greater ‹0 🚫›
      by (auto simp add: uniform_limit_iff dist_norm norm_divide diff_divide_distrib [symmetric] divide_simps)
    have g_cint: "(λu. g u/(u-w)) contour_integrable_on circlepath z r"
      by (rule contour_integral_uniform_limit_circlepath [OF ev_int ul_less F ‹0 🚫›])
    have cif_tends_cig: "((λn. contour_integral(circlepath z r) (λu. f n u / (u-w))) ---> contour_integral(circlepath z r) (λu. g u/(u-w))) F"
      by (rule contour_integral_uniform_limit_circlepath [OF ev_int ul_less F ‹0 🚫›])
    have f_tends_cig: "((λn. 2 * of_real pi * i * f n w) ---> contour_integral (circlepath z r) (λu. g u / (u-w))) F"
    proof (rule Lim_transform_eventually)
      show "∀🪙F x in F. contour_integral (circlepath z r) (λu. f x u / (u-w))
                     = 2 * of_real pi * i * f x w"
        using w‹0 🚫› d_def
        by (auto intro: eventually_mono [OF cont contour_integral_unique [OF Cauchy_integral_circlepath]])
    qed (auto simp: cif_tends_cig)
    have "∧e. 0 < e ==> ∀🪙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 "((λn. 2 * of_real pi * i * f n w) ---> 2 * of_real pi * i * g w) F"
      by (rule tendsto_mult_left [OF tendstoI])
    then have "((λu. 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 "((λu. 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

lemma higher_deriv_complex_uniform_limit:
  assumes ulim: "uniform_limit A f g F"
      and f_holo: "eventually (λn. f n holomorphic_on A) F"
      and F: "F ≠ bot"
      and A: "open A" "z ∈ A"
    shows "((λn. (deriv ^^ m) (f n) z) ---> (deriv ^^ m) g z) F"
proof -
  obtain r where r: "r > 0" "cball z r ⊆ A"
    using A by (meson open_contains_cball)
  have r': "ball z r ⊆ A"
    using r by auto
  define h where "h = (λn z. f n z - g z)"
  define c where "c = of_real (2*pi) * i / fact m"
  have [simp]: "c ≠ 0"
    by (simp add: c_def)
  have "g holomorphic_on ball z r ∧ continuous_on (cball z r) g"
  proof (rule holomorphic_uniform_limit)
    show "uniform_limit (cball z r) f g F"
      by (rule uniform_limit_on_subset[OF ulim r(2)])
    show "∀🪙F n in F. continuous_on (cball z r) (f n) ∧ f n holomorphic_on ball z r" using f_holo
      by eventually_elim
         (use holomorphic_on_subset[OF _ r(2)] holomorphic_on_subset[OF _ r'] 
          in  ‹auto intro!: holomorphic_on_imp_continuous_on›)
  qed (use F in auto)
  hence g_holo: "g holomorphic_on ball z r" and g_cont: "continuous_on (cball z r) g"
    by blast+

  have ulim': "uniform_limit (sphere z r) (λn x. h n x / (x - z) ^ (Suc m)) (λ_. 0) F"
  proof -
    have "uniform_limit (sphere z r) (λn x. f n x / (x - z) ^ Suc m) (λx. g x / (x - z) ^ Suc m) F"
    proof (intro uniform_lim_divide uniform_limit_intros uniform_limit_on_subset[OF ulim])
      have "compact (g ` sphere z r)"
        by (intro compact_continuous_image continuous_on_subset[OF g_cont]) auto
      thus "bounded (g ` sphere z r)"
        by (rule compact_imp_bounded)
      show "r ^ Suc m ≤ norm ((x - z) ^ Suc m)" if "x ∈ sphere z r" for x unfolding norm_power
        by (intro power_mono) (use that r(1) in ‹auto simp: dist_norm norm_minus_commute›)
    qed (use r in auto)
    hence "uniform_limit (sphere z r) (λn x. f n x / (x - z) ^ Suc m - g x / (x - z) ^ Suc m)
             (λx. g x / (x - z) ^ Suc m - g x / (x - z) ^ Suc m) F"
      by (intro uniform_limit_intros)
    thus ?thesis
      by (simp add: h_def diff_divide_distrib)
  qed

  have has_integral: "eventually (λn. ((λu. h n u / (u - z) ^ Suc m) has_contour_integral
                         c * (deriv ^^ m) (h n) z) (circlepath z r)) F"
    using f_holo
  proof eventually_elim
    case (elim n)
    show ?case
      unfolding c_def
    proof (rule Cauchy_has_contour_integral_higher_derivative_circlepath)
      show "continuous_on (cball z r) (h n)" unfolding h_def 
        by (intro continuous_intros g_cont holomorphic_on_imp_continuous_on
                  holomorphic_on_subset[OF elim] r)
      show "h n holomorphic_on ball z r"
        unfolding h_def by (intro holomorphic_intros g_holo holomorphic_on_subset[OF elim] r')
    qed (use r(1) in auto)
  qed

  have "((λn. contour_integral (circlepath z r) (λu. h n u / (u - z) ^ Suc m)) --->
         contour_integral (circlepath z r) (λu. 0 / (u - z) ^ Suc m)) F"
  proof (rule contour_integral_uniform_limit_circlepath)
    show "∀🪙F n in F. (λu. h n u / (u - z) ^ Suc m) contour_integrable_on circlepath z r"
      using has_integral by eventually_elim (blast intro: has_contour_integral_integrable)
  qed (use r(1) ‹F ≠ bot› ulim' in simp_all)
  hence "((λn. contour_integral (circlepath z r) (λu. h n u / (u - z) ^ Suc m)) ---> 0) F"
    by simp
  also have "?this ⟷ ((λn. c * (deriv ^^ m) (h n) z) ---> 0) F"
  proof (rule tendsto_cong)
    show "∀🪙F x in F. contour_integral (circlepath z r) (λu. h x u / (u - z) ^ Suc m) =
                       c * (deriv ^^ m) (h x) z"
      using has_integral by eventually_elim (simp add: contour_integral_unique)
  qed
  finally have "((λn. (deriv ^^ m) g z + c * (deriv ^^ m) (h n) z / c) --->
                  (deriv ^^ m) g z + 0 / c) F"
    by (intro tendsto_intros) auto
  also have "?this ⟷ ((λn. (deriv ^^ m) (f n) z) ---> (deriv ^^ m) g z) F"
  proof (intro filterlim_cong)
    show "∀🪙F n in F. (deriv ^^ m) g z + c * (deriv ^^ m) (h n) z / c = (deriv ^^ m) (f n) z"
      using f_holo
    proof eventually_elim
      case (elim n)
      have "(deriv ^^ m) (h n) z = (deriv ^^ m) (f n) z - (deriv ^^ m) g z" unfolding h_def
        by (rule higher_deriv_diff holomorphic_on_subset[OF elim r'] g_holo A)+ (use r(1) in auto)
      thus ?case
        by simp
    qed
  qed auto
  finally show ?thesis .
qed

lemma deriv_complex_uniform_limit:
  assumes ulim: "uniform_limit A f g F"
      and f_holo: "eventually (λn. f n holomorphic_on A) F"
      and F: "F ≠ bot"
      and A: "open A" "z ∈ A"
    shows "((λn. deriv (f n) z) ---> deriv g z) F"
  using higher_deriv_complex_uniform_limit[OF assms, of 1] by simp

lemma logderiv_prodinf_complex_uniform_limit:
  fixes f :: "nat ==> complex ==> complex"
  assumes lim: "uniform_limit A (λn x. ∏k
  assumes holo: "∧k. f k holomorphic_on A"
  assumes nz: "P z ≠ 0"
  assumes A: "open A" "z ∈ A"
  shows   "(λk. deriv (f k) z / f k z) sums (deriv P z / P z)"
proof -
  define f' where "f' = (λk. deriv (f k))"
  note [derivative_intros] = has_field_derivative_prod'
  have [derivative_intros]: 
    "(f k has_field_derivative f' k z) (at z within B)" if "z ∈ A" for B z k
    using that holomorphic_derivI[OF holo[of k],  of z B] A unfolding f'_def by auto
  have lim': "(λn. ∏k<---- P z"
    using lim by (rule tendsto_uniform_limitI) fact+

  have nz': "f k z ≠ 0" for k
  proof
    assume "f k z = 0"
    have "eventually (λn. (∏k
      using eventually_gt_at_top[of k] by eventually_elim (use ‹f k z = 0› in auto)
    hence "(λn. (∏k<---- 0"
      by (rule tendsto_eventually)
    with lim' have "P z = 0"
      using tendsto_unique sequentially_bot by blast
    with nz show False
      by simp
  qed

  from lim have "(λn. deriv (λx. ∏k<---- deriv P z"
    by (rule deriv_complex_uniform_limit)
       (use A in ‹auto intro!: always_eventually holomorphic_intros holo›)
  also have "(λn. deriv (λx. ∏k∏k∑k
    using ‹z ∈ A› by (auto intro!: ext DERIV_imp_deriv derivative_eq_intros simp: nz')
  finally have "(λn. (∏k∑k<---- deriv P z" .
  hence "(λn. (∏k∑k∏k<---- deriv P z / P z"
    by (intro tendsto_intros) (use nz lim' in auto)
  also have "(λn. (∏k∑k∏k
             (λn. (∑k
    by (simp add: nz')
  finally show "(λk. f' k z / f k z) sums (deriv P z / P z)"
    unfolding sums_def .
qed


text‹ Version showing that the limit is the limit of the derivatives.›

proposition has_complex_derivative_uniform_limit:
  fixes z::complex
  assumes cont: "eventually (λn. continuous_on (cball z r) (f n) ∧
                               (∀w ∈ 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:  "¬ trivial_limit F" and "0 < r"
  obtains g' where
      "continuous_on (cball z r) g"
      "∧w. w ∈ ball z r ==> (g has_field_derivative (g' w)) (at w) ∧ ((λn. f' n w) ---> 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': "∧x. x ∈ ball z r ==> (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: "∧x. x ∈ ball z r ==> deriv g x = g' x"
    by (simp add: DERIV_imp_deriv)
  have tends_f'n_g': "((λn. f' n w) ---> g' w) F" if w: "w ∈ ball z r" for w
  proof -
    have eq_f': "?conint (λx. f n x / (x-w)🪙2) - ?conint (λx. g x / (x-w)🪙2) = (f' n w - g' w) * (2 * of_real pi * i)"
             if cont_fn: "continuous_on (cball z r) (f n)"
             and fnd: "∧w. w ∈ ball z r ==> (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 (λu. f n u / (u-w)🪙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 (λu. f n u / (u-w)🪙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 ≤ cmod ((y-w)🪙2)" if "r = cmod (z-y)" for y
      by (smt (verit, best) d_def diff_add_cancel diff_diff_eq2 dist_norm mem_ball
          norm_minus_commute norm_power norm_triangle_ineq2 power_mono that w)
    have 1: "∀🪙F n in F. (λx. f n x / (x-w)🪙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) (λn x. f n x / (x-w)🪙2) (λx. g x / (x-w)🪙2) F"
      unfolding uniform_limit_iff
    proof clarify
      fix e::real
      assume "e > 0"
      with ‹r > 0›
      have "∀🪙F n in F. ∀x. x ≠ w ⟶ cmod (z-x) = r ⟶ cmod (f n x - g x) < e * cmod ((x-w)🪙2)"
        by (force simp: ‹0 🚫› dist_norm dle intro: less_le_trans eventually_mono [OF uniform_limitD [OF ulim], of "e*d"])
      with ‹r > 0› ‹e > 0›
      show "∀🪙F n in F. ∀x∈sphere z r. dist (f n x / (x-w)🪙2) (g x / (x-w)🪙2) < e"
        by (simp add: norm_divide field_split_simps sphere_def dist_norm)
    qed
    have "((λn. contour_integral (circlepath z r) (λx. f n x / (x-w)🪙2))
             ---> contour_integral (circlepath z r) ((λx. g x / (x-w)🪙2))) F"
      by (rule contour_integral_uniform_limit_circlepath [OF 1 2 F ‹0 🚫›])
    then have tendsto_0: "((λn. 1 / (2 * of_real pi * i) * (?conint (λx. f n x / (x-w)🪙2) - ?conint (λx. g x / (x-w)🪙2))) ---> 0) F"
      using Lim_null by (force intro!: tendsto_mult_right_zero)
    have "((λn. f' n w - g' w) ---> 0) F"
      by (force simp: divide_simps 
          intro: eq_f' eventually_mono [OF cont] Lim_transform_eventually [OF tendsto_0])
    then show ?thesis using Lim_null by blast
  qed
  obtain g' where "∧w. w ∈ ball z r ==> (g has_field_derivative (g' w)) (at w) ∧ ((λn. f' n w) ---> g' w) F"
      by (blast intro: tends_f'n_g' g')
  then show ?thesis using g
    using that by blast
qed


subsection🍋‹tag unimportant› ‹Some more simple/convenient versions for applications›

lemma holomorphic_uniform_sequence:
  assumes S: "open S"
      and hol_fn: "∧n. (f n) holomorphic_on S"
      and ulim_g: "∧x. x ∈ S ==> ∃d. 0 < d ∧ cball x d ⊆ S ∧ uniform_limit (cball x d) f g sequentially"
  shows "g holomorphic_on S"
proof -
  have "∃f'. (g has_field_derivative f') (at z)" if "z ∈ S" for z
  proof -
    obtain r where "0 < r" and r: "cball z r ⊆ S"
               and ul: "uniform_limit (cball z r) f g sequentially"
      using ulim_g [OF ‹z ∈ S›] by blast
    have *: "∀🪙F n in sequentially. continuous_on (cball z r) (f n) ∧ f n holomorphic_on ball z r"
      by (smt (verit, best) ball_subset_cball hol_fn holomorphic_on_imp_continuous_on 
          holomorphic_on_subset not_eventuallyD r)
    show ?thesis
      using ‹0 🚫› 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: "∧n x. x ∈ S ==> ((f n) has_field_derivative f' n x) (at x)"
      and ulim_g: "∧x. x ∈ S
             ==> ∃d. 0 < d ∧ cball x d ⊆ S ∧ uniform_limit (cball x d) f g sequentially"
  shows "∃g'. ∀x ∈ S. (g has_field_derivative g' x) (at x) ∧ ((λn. f' n x) ---> g' x) sequentially"
proof -
  have y: "∃y. (g has_field_derivative y) (at z) ∧ (λn. f' n z) <---- y" if "z ∈ S" for z
  proof -
    obtain r where "0 < r" and r: "cball z r ⊆ S"
               and ul: "uniform_limit (cball z r) f g sequentially"
      using ulim_g [OF ‹z ∈ S›] by blast
    have *: "∀🪙F n in sequentially. continuous_on (cball z r) (f n) ∧
                                   (∀w ∈ 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 ∈ ball z r ==> (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 ‹0 🚫› ul in ‹force+›)
  qed
  show ?thesis
    by (rule bchoice) (blast intro: y)
qed

subsection‹On analytic functions defined by a series›

lemma series_and_derivative_comparison:
  fixes S :: "complex set"
  assumes S: "open S"
      and h: "summable h"
      and hfd: "∧n x. x ∈ S ==> (f n has_field_derivative f' n x) (at x)"
      and to_g: "∀🪙F n in sequentially. ∀x∈S. norm (f n x) ≤ h n"
  obtains g g' where "∀x ∈ S. ((λn. f n x) sums g x) ∧ ((λn. f' n x) sums g' x) ∧ (g has_field_derivative g' x) (at x)"
proof -
  obtain g where g: "uniform_limit S (λn x. ∑i
    using Weierstrass_m_test_ev [OF to_g h]  by force
  have *: "∃d>0. cball x d ⊆ S ∧ uniform_limit (cball x d) (λn x. ∑i
    if "x ∈ S" for x
    using open_contains_cball [of "S"] ‹x ∈ S› S g uniform_limit_on_subset by blast
  have "∧x. x ∈ S ==> (λn. ∑i<---- g x"
    by (metis tendsto_uniform_limitI [OF g])
  moreover have "∃g'. ∀x∈S. (g has_field_derivative g' x) (at x) ∧ (λn. ∑i<---- 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‹A version where we only have local uniform/comparative convergence.›

lemma series_and_derivative_comparison_local:
  fixes S :: "complex set"
  assumes S: "open S"
      and hfd: "∧n x. x ∈ S ==> (f n has_field_derivative f' n x) (at x)"
      and to_g: "∧x. x ∈ S ==> ∃d h. 0 < d ∧ summable h ∧ (∀🪙F n in sequentially. ∀y∈ball x d ∩ S. norm (f n y) ≤ h n)"
  shows "∃g g'. ∀x ∈ S. ((λn. f n x) sums g x) ∧ ((λn. f' n x) sums g' x) ∧ (g has_field_derivative g' x) (at x)"
proof -
  have "∃y. (λn. f n z) sums (∑n. f n z) ∧ (λn. f' n z) sums y ∧ ((λx. ∑n. f n x) has_field_derivative y) (at z)"
       if "z ∈ S" for z
  proof -
    obtain d h where "0 < d" "summable h" and le_h: "∀🪙F n in sequentially. ∀y∈ball z d ∩ S. norm (f n y) ≤ h n"
      using to_g ‹z ∈ S› by meson
    then obtain r where "r>0" and r: "ball z r ⊆ ball z d ∩ S" using ‹z ∈ S› S
      by (metis Int_iff open_ball centre_in_ball open_Int open_contains_ball_eq)
    have 1: "open (ball z d ∩ S)"
      by (simp add: open_Int S)
    have 2: "∧n x. x ∈ ball z d ∩ S ==> (f n has_field_derivative f' n x) (at x)"
      by (auto simp: hfd)
    obtain g g' where gg': "∀x ∈ ball z d ∩ S. ((λn. f n x) sums g x) ∧
                                    ((λn. f' n x) sums g' x) ∧ (g has_field_derivative g' x) (at x)"
      by (auto intro: le_h series_and_derivative_comparison [OF 1 ‹summable h› hfd])
    then have "(λn. f' n z) sums g' z"
      by (meson ‹0 🚫› centre_in_ball contra_subsetD r)
    moreover have "(λn. f n z) sums (∑n. f n z)"
      using  summable_sums centre_in_ball ‹0 🚫› ‹summable h› le_h
      by (metis (full_types) Int_iff gg' summable_def that)
    moreover have "((λx. ∑n. f n x) has_field_derivative g' z) (at z)"
      by (metis (no_types, lifting) "1" r ‹0 🚫› gg' has_field_derivative_transform_within_open 
          open_contains_ball_eq sums_unique)
    ultimately show ?thesis by auto
  qed
  then show ?thesis
    by meson
qed


text‹Sometimes convenient to compare with a complex series of positive reals. (?)›

lemma series_and_derivative_comparison_complex:
  fixes S :: "complex set"
  assumes S: "open S"
    and hfd: "∧n x. x ∈ S ==> (f n has_field_derivative f' n x) (at x)"
    and to_g: "∧x. x ∈ S ==> ∃d h. 0 < d ∧ summable h ∧ range h ⊆ ℝ🪙≥🪙0 ∧ (∀🪙F n in sequentially. ∀y∈ball x d ∩ S. cmod(f n y) ≤ cmod (h n))"
  shows "∃g g'. ∀x ∈ S. ((λn. f n x) sums g x) ∧ ((λn. f' n x) sums g' x) ∧ (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 ∘ h" in exI)
  apply (force simp: summable_Re o_def nonneg_Reals_cmod_eq_Re image_subset_iff)
  done

text‹Sometimes convenient to compare with a complex series of positive reals. (?)›
lemma series_differentiable_comparison_complex:
  fixes S :: "complex set"
  assumes S: "open S"
    and hfd: "∧n x. x ∈ S ==> f n field_differentiable (at x)"
    and to_g: "∧x. x ∈ S ==> ∃d h. 0 < d ∧ summable h ∧ range h ⊆ ℝ🪙≥🪙0 ∧ (∀🪙F n in sequentially. ∀y∈ball x d ∩ S. cmod(f n y) ≤ cmod (h n))"
  obtains g where "∀x ∈ S. ((λn. f n x) sums g x) ∧ g field_differentiable (at x)"
proof -
  have hfd': "∧n x. x ∈ S ==> (f n has_field_derivative deriv (f n) x) (at x)"
    using hfd field_differentiable_derivI by blast
  show ?thesis
    by (metis field_differentiable_def that series_and_derivative_comparison_complex [OF S hfd' to_g]) 
qed

text‹In particular, a power series is analytic inside circle of convergence.›

lemma power_series_and_derivative_0:
  fixes a :: "nat ==> complex" and r::real
  assumes "summable (λn. a n * r^n)"
    shows "∃g g'. ∀z. cmod z < r ⟶
             ((λn. a n * z^n) sums g z) ∧ ((λn. of_nat n * a n * z^(n-1)) sums g' z) ∧ (g has_field_derivative g' z) (at z)"
proof (cases "0 < r")
  case True
    have der: "∧n z. ((λx. 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 ≤ cmod (of_real r + of_real (cmod z)) / 2" 
      if "cmod (z-y) * 2 < r - cmod z" for z y
      by (smt (verit, best) field_sum_of_halves norm_minus_commute norm_of_real norm_triangle_ineq2 of_real_add that)
    have "summable (λn. a n * complex_of_real r ^ n)"
      using assms ‹r > 0› by simp
    moreover have "∧z. cmod z < r ==> cmod ((of_real r + of_real (cmod z)) / 2) < cmod (of_real r)"
      using ‹r > 0›
      by (simp flip: of_real_add)
    ultimately have sum: "∧z. cmod z < r ==> summable (λn. of_real (cmod (a n)) * ((of_real r + complex_of_real (cmod z)) / 2) ^ n)"
      by (rule power_series_conv_imp_absconv_weak)
    have "∃g g'. ∀z ∈ ball 0 r. (λn. (a n) * z ^ n) sums g z ∧
               (λn. of_nat n * (a n) * z ^ (n-1)) sums g' z ∧ (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="λn. of_real(norm(a n)*((r + norm z)/2)^n)" in exI)
      using ‹r > 0›
      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🍋‹tag unimportant› power_series_and_derivative:
  fixes a :: "nat ==> complex" and r::real
  assumes "summable (λn. a n * r^n)"
    obtains g g' where "∀z ∈ ball w r.
             ((λn. a n * (z-w) ^ n) sums g z) ∧ ((λn. of_nat n * a n * (z-w) ^ (n-1)) sums g' z) ∧
              (g has_field_derivative g' z) (at z)"
  using power_series_and_derivative_0 [OF assms]
  apply clarify
  apply (rule_tac g="(λz. g(z-w))" in that)
  using DERIV_shift [where z="-w"]
  apply (auto simp: norm_minus_commute Ball_def dist_norm)
  done

proposition🍋‹tag unimportant› power_series_holomorphic:
  assumes "∧w. w ∈ ball z r ==> ((λn. a n*(w-z)^n) sums f w)"
    shows "f holomorphic_on ball z r"
proof -
  have "∃f'. (f has_field_derivative f') (at w)" if w: "dist z w < r" for w
  proof -
    have wz: "cmod (w-z) < r" using w
      by (auto simp: field_split_simps dist_norm norm_minus_commute)
    then have "0 ≤ r"
      by (meson less_eq_real_def norm_ge_zero order_trans)
    have inb: "z + complex_of_real ((dist z w + r) / 2) ∈ ball z r"
      using w by (simp add: dist_norm ‹0≤r› flip: of_real_add)
    have sum: "summable (λn. 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': "∧u. u ∈ ball z ((cmod (z-w) + r) / 2) ==>
                               (λn. a n * (u-z) ^ n) sums g u ∧
                               (λn. of_nat n * a n * (u-z) ^ (n-1)) sums g' u ∧ (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 "(λn. a n * (u-z) ^ n) sums g u" "(λn. 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)"
    proof (rule has_field_derivative_transform_within [where d="(r - norm(z-w))/2"])
    qed (use w gg' [of w] in ‹(force simp: dist_norm)+›)
    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 ⟷
      (∀w ∈ ball z r. (λn. (deriv ^^ n) f z / (fact n) * (w-z)^n) sums f w)"
  using power_series_holomorphic [where a = "λn. (deriv ^^ n) f z / (fact n)"] holomorphic_power_series
  by blast

lemma power_series_analytic:
     "(∧w. w ∈ ball z r ==> (λn. a n*(w-z)^n) sums f w) ==> 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 ⟷
      (∀w ∈ ball z r. (λn. (deriv ^^ n) f z / (fact n) * (w-z)^n) sums f w)"
  by (simp add: analytic_on_open holomorphic_iff_power_series)

subsection🍋‹tag unimportant› ‹Equality between holomorphic functions, on open ball then connected set›

lemma holomorphic_fun_eq_on_ball:
   "[f holomorphic_on ball z r; g holomorphic_on ball z r;
     w ∈ ball z r;
     ∧n. (deriv ^^ n) f z = (deriv ^^ n) g z]
     ==> f w = g w"
  by (auto simp: holomorphic_iff_power_series sums_unique2 [of "λn. (deriv ^^ n) f z / (fact n) * (w-z)^n"])

lemma holomorphic_fun_eq_0_on_ball:
   "[f holomorphic_on ball z r; w ∈ ball z r;
     ∧n. (deriv ^^ n) f z = 0]
     ==> f w = 0"
  using holomorphic_fun_eq_on_ball [where g = "λz. 0"] by simp

lemma holomorphic_fun_eq_0_on_connected:
  assumes holf: "f holomorphic_on S" and "open S"
      and cons: "connected S"
      and der: "∧n. (deriv ^^ n) f z = 0"
      and "z ∈ S" "w ∈ S"
    shows "f w = 0"
proof -
  have *: "ball x e ⊆ (∩n. {w ∈ S. (deriv ^^ n) f w = 0})"
    if "∀u. (deriv ^^ u) f x = 0" "ball x e ⊆ 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 "∧x' n. dist x x' < e ==> (deriv ^^ n) f x' = 0"
      using ‹open S›
      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 ⊆ S" using openE [OF ‹open S› ‹w ∈ S›] .
  then have holfb: "f holomorphic_on ball w e"
    using holf holomorphic_on_subset by blast
  have "open (∩n. {w ∈ S. (deriv ^^ n) f w = 0})"
    using ‹open S›
    apply (simp add: open_contains_ball Ball_def image_iff)
    by (metis (mono_tags) "*" mem_Collect_eq)
  then have "openin (top_of_set S) (∩n. {w ∈ S. (deriv ^^ n) f w = 0})"
    by (force intro: open_subset)
  moreover have "closedin (top_of_set S) (∩n. {w ∈ 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 "(∩n. {w ∈ S. (deriv ^^ n) f w = 0}) = S ==> f w = 0"
    using ‹e>0› e by (force intro: holomorphic_fun_eq_0_on_ball [OF holfb])
  ultimately show ?thesis
    using cons der ‹z ∈ S›
    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 "∧n. (deriv ^^ n) f z = (deriv ^^ n) g z"
      and "z ∈ S" "w ∈ S"
    shows "f w = g w"
proof (rule holomorphic_fun_eq_0_on_connected [of "λx. f x - g x" S z, simplified])
  show "(λx. f x - g x) holomorphic_on S"
    by (intro assms holomorphic_intros)
  show "∧n. (deriv ^^ n) (λx. 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: "∧n. 0 < n ==> (deriv ^^ n) f z = 0"
      and "z ∈ S" "w ∈ S"
    shows "f w = f z"
proof (rule holomorphic_fun_eq_0_on_connected [of "λw. f w - f z" S z, simplified])
  show "(λw. f w - f z) holomorphic_on S"
    by (intro assms holomorphic_intros)
  show "∧n. (deriv ^^ n) (λw. f w - f z) z = 0"
    by (subst higher_deriv_diff) (use assms in ‹auto intro: holomorphic_intros›)
qed (use assms in auto)

subsection🍋‹tag unimportant› ‹Some basic lemmas about poles/singularities›

lemma pole_lemma:
  assumes holf: "f holomorphic_on S" and a: "a ∈ interior S"
    shows "(λz. if z = a then deriv f a
                else (f z - f a) / (z-a)) holomorphic_on S" (is "?F holomorphic_on S")
proof -
  have *: "?F field_differentiable (at u within S)" if "u ∈ S" "u ≠ a" for u
  proof -
    have fcd: "f field_differentiable at u within S"
      using holf holomorphic_on_def by (simp add: ‹u ∈ S›)
    have cd: "(λz. (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: ‹u ∈ S›)
  qed
  moreover
  have "?F field_differentiable at a" if "0 < e" "ball a e ⊆ S" for e
  proof -
    have holfb: "f holomorphic_on ball a e"
      by (rule holomorphic_on_subset [OF holf ‹ball a e ⊆ S›])
    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: * field_differentiable_within_subset)
    have "isCont (λz. 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 ‹0 🚫› holomorphic_on_imp_differentiable_at by auto
      with True show ?thesis
        by (smt (verit) DERIV_deriv_iff_field_differentiable LIM_equal continuous_at has_field_derivativeD)
    next
      case False with 2 that show ?thesis
        by (simp add: field_differentiable_imp_continuous_at holomorphic_on_imp_differentiable_at open_Diff)
    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_imp_differentiable_at)
  qed
  ultimately show ?thesis
    by (metis (lifting) a at_within_interior holomorphic_onI mem_interior)
qed

lemma pole_theorem:
  assumes holg: "g holomorphic_on S" and a: "a ∈ interior S"
      and eq: "∧z. z ∈ S - {a} ==> g z = (z-a) * f z"
    shows "(λz. 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 "(λz. if z = a then deriv f a else (f z - f a)/(z-a)) holomorphic_on S"
proof (cases "a ∈ S")
  case True with assms interior_eq pole_lemma
    show ?thesis by fastforce
next
  case False 
  then have "(λz. (f z - f a) / (z-a)) field_differentiable at x within S"
    if "x ∈ S" for x
    using assms that unfolding holomorphic_on_def
    by (intro derivative_intros) auto
  with False show ?thesis
    using holomorphic_on_def holomorphic_transform by presburger
qed

lemma pole_theorem_open:
  assumes holg: "g holomorphic_on S" and S: "open S"
      and eq: "∧z. z ∈ S - {a} ==> g z = (z-a) * f z"
    shows "(λz. 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 ∈ interior S"
      and eq: "∧z. z ∈ S - {a} ==> 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: "∧z. z ∈ S - {a} ==> 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: "∧z. z ∈ S
             ==> ∃d. 0 < d ∧ (∀w ∈ ball z d - {a}. g w = (w-a) * f w)"
    shows "(λz. 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 ∈ 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: "∧w. w ∈ ball x d - {a} ==> g w = (w-a) * f w"
    using ‹x ∈ S› eq by blast
  have "?F holomorphic_on ball x (min d e)"
    using d e ‹x ∈ S› by (fastforce simp: holomorphic_on_subset subset_ball intro!: pole_theorem_open)
  then show "∃e>0. ?F holomorphic_on ball x e"
    using ‹0 🚫› ‹0 🚫› not_le by fastforce
qed

lemma pole_theorem_analytic_0:
  assumes g: "g analytic_on S"
      and eq: "∧z. z ∈ S ==> ∃d. 0 < d ∧ (∀w ∈ 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]: "(λz. 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 ⊆ T" "open T"
      and eq: "∧z. z ∈ T - {a} ==> g z = (z-a) * f z"
    shows "(λz. 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 ∈ S"
  then obtain e where "0 < e" and e: "ball z e ⊆ T"
    using assms openE by blast
  then show "∃d>0. ∀w∈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 ⊆ T" "open T" "∧z. z ∈ T - {a} ==> g z = (z-a) * f z"
      and [simp]: "f a = deriv g a" "g a = 0"
    shows "f analytic_on S"
proof -
  have [simp]: "(λz. if z = a then deriv g a else f z - g a / (z-a)) = f"
    by auto
  have "(λz. 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‹General, homology form of Cauchy's theorem›

text‹Proof is based on Dixon's, as presented in Lang's "Complex Analysis" book (page 147).›

lemma contour_integral_continuous_on_linepath_2D:
  assumes "open U" and cont_dw: "∧w. w ∈ U ==> F w contour_integrable_on (linepath a b)"
      and cond_uu: "continuous_on (U × U) (λ(x,y). F x y)"
      and abu: "closed_segment a b ⊆ U"
    shows "continuous_on U (λw. contour_integral (linepath a b) (F w))"
proof -
  have "∃d>0. ∀x'∈U. dist x' w < d ⟶
                     dist (contour_integral (linepath a b) (F x'))
                          (contour_integral (linepath a b) (F w)) ≤ ε"
          if "w ∈ U" "0 < ε" "a ≠ b" for w ε
  proof -
    obtain δ where "δ>0" and δ: "cball w δ ⊆ U" using open_contains_cball ‹open U› ‹w ∈ U› by force
    let ?TZ = "cball w δ × closed_segment a b"
    have "uniformly_continuous_on ?TZ (λ(x,y). F x y)"
      by (metis Sigma_mono δ abu compact_Times compact_cball compact_segment compact_uniformly_continuous 
          cond_uu continuous_on_subset)
    then obtain 🪙 where "🪙>0"
        and 🪙: "∧x x'. [x∈?TZ; x'∈?TZ; dist x' x < 🪙] ==>
                         dist ((λ(x,y). F x y) x') ((λ(x,y). F x y) x) < ε/norm(b-a)"
      using ‹0 🚫ε› ‹a ≠ b›
      by (auto elim: uniformly_continuous_onE [where ε = "ε/norm(b-a)"])
    have 🪙: "[norm (w - x1) ≤ δ; x2 ∈ closed_segment a b;
              norm (w - x1') ≤ δ; x2' ∈ closed_segment a b; norm ((x1', x2') - (x1, x2)) < 🪙]
              ==> norm (F x1' x2' - F x1 x2) ≤ ε / cmod (b-a)"
             for x1 x2 x1' x2'
      using 🪙 [of "(x1,x2)" "(x1',x2')"] by (force simp: dist_norm)
    have le_ee: "cmod (contour_integral (linepath a b) (λx. F x' x - F w x)) ≤ ε"
                if "x' ∈ U" "cmod (x' - w) < δ" "cmod (x' - w) < 🪙"  for x'
    proof -
      have "(λx. F x' x - F w x) contour_integrable_on linepath a b"
        by (simp add: ‹w ∈ U› cont_dw contour_integrable_diff that)
      then have "cmod (contour_integral (linepath a b) (λx. F x' x - F w x)) ≤ ε/norm(b-a) * norm(b-a)"
        using has_contour_integral_bound_linepath [OF has_contour_integral_integral _ 🪙]
        using ‹0 🚫ε› ‹0 🚫δ› that by (force simp: norm_minus_commute)
      also have "… = ε" using ‹a ≠ b› by simp
      finally show ?thesis .
    qed
    show ?thesis
      using ‹0 🚫δ› ‹0 🚫🪙› ‹w ∈ U›
      apply (intro exI[where x="min δ 🪙"])
      by (auto simp: dist_norm contour_integral_diff [OF cont_dw cont_dw, symmetric]  intro: le_ee)
  qed
  then show ?thesis
    by (metis (no_types, lifting) continuous_onI continuous_on_iff
        contour_integral_trivial dist_self)
qed

text‹This version has 🍋‹polynomial_function γ› as an additional assumption.›
lemma Cauchy_integral_formula_global_weak:
  assumes "open U" and holf: "f holomorphic_on U"
        and z: "z ∈ U" and γ: "polynomial_function γ"
        and pasz: "path_image γ ⊆ U - {z}" and loop: "pathfinish γ = pathstart γ"
        and zero: "∧w. w ∉ U ==> winding_number γ w = 0"
      shows "((λw. f w / (w-z)) has_contour_integral (2*pi * i * winding_number γ z * f z)) γ"
proof -
  obtain γ' where pfγ': "polynomial_function γ'" and γ': "∧x. (γ has_vector_derivative (γ' x)) (at x)"
    using has_vector_derivative_polynomial_function [OF γ] by blast
  then have "bounded(path_image γ')"
    by (simp add: path_image_def compact_imp_bounded compact_continuous_image continuous_on_polymonial_function)
  then obtain B where "B>0" and B: "∧x. x ∈ path_image γ' ==> norm x ≤ 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 ∉ path_image γ ∧ winding_number γ w = 0}"
  have "path γ" "valid_path γ" using γ
    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 ∪ 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 ∈ U" for y
  proof -
    have *: "(λc. 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 U›)
    then have "isCont (λx. 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 U› 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 "(λw. if w = y then deriv f y else (f w - f y) / (w-y)) field_differentiable at w"
        if "w ∈ U - {y}" for w
      proof (rule field_differentiable_transform_within)
        show "(λw. (f w - f y) / (w-y)) field_differentiable at w"
          using that ‹open U› 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 U› open_delete)
    qed
    ultimately show ?thesis
      by (rule no_isolated_singularity) (auto simp: ‹open U›)
  qed
  have cint_fxy: "(λx. (f x - f y) / (x-y)) contour_integrable_on γ" if "y ∉ path_image γ" for y
  proof (rule contour_integrable_holomorphic_simple [where S = "U-{y}"])
    show "(λx. (f x - f y) / (x-y)) holomorphic_on U - {y}"
      by (force intro: holomorphic_intros holomorphic_on_subset [OF holf])
    show "path_image γ ⊆ U - {y}"
      using pasz that by blast
  qed (auto simp: ‹open U› open_delete ‹valid_path γ›)
  define h where
    "h z = (if z ∈ U then contour_integral γ (d z) else contour_integral γ (λw. f w/(w-z)))" for z
  have U: "((d z) has_contour_integral h z) γ" if "z ∈ U" for z
  proof -
    have "d z holomorphic_on U"
      by (simp add: hol_d that)
    with that show ?thesis
      by (metis Diff_subset ‹valid_path γ› ‹open U› contour_integrable_holomorphic_simple h_def 
          has_contour_integral_integral pasz subset_trans)
  qed
  have V: "((λw. f w / (w-z)) has_contour_integral h z) γ" if z: "z ∈ v" for z
  proof -
    have 0: "0 = (f z) * 2 * of_real (2 * pi) * i * winding_number γ z"
      using v_def z by auto
    then have "((λx. 1 / (x-z)) has_contour_integral 0) γ"
     using z v_def  has_contour_integral_winding_number [OF ‹valid_path γ›] by fastforce
    then have "((λx. f z * (1 / (x-z))) has_contour_integral 0) γ"
      using has_contour_integral_lmul by fastforce
    then have "((λx. f z / (x-z)) has_contour_integral 0) γ"
      by (simp add: field_split_simps)
    moreover have "((λx. (f x - f z) / (x-z)) has_contour_integral contour_integral γ (d z)) γ"
      by (metis (no_types, lifting) z cint_fxy contour_integral_eq d_def has_contour_integral_integral mem_Collect_eq v_def)
    ultimately have *: "((λx. f z / (x-z) + (f x - f z) / (x-z)) has_contour_integral (0 + contour_integral γ (d z))) γ"
      by (rule has_contour_integral_add)
    have "((λw. f w / (w-z)) has_contour_integral contour_integral γ (d z)) γ"
      if "z ∈ U"
      using * by (auto simp: divide_simps has_contour_integral_eq)
    moreover have "((λw. f w / (w-z)) has_contour_integral contour_integral γ (λw. f w / (w-z))) γ"
      if "z ∉ U"
    proof (rule has_contour_integral_integral [OF contour_integrable_holomorphic_simple [where S=U]])
      show "(λw. f w / (w-z)) holomorphic_on U"
        by (rule holomorphic_intros assms | use that in force)+
    qed (use ‹open U› pasz ‹valid_path γ› in auto)
    ultimately show ?thesis
      using z by (simp add: h_def)
  qed
  have znot: "z ∉ path_image γ"
    using pasz by blast
  obtain d0 where "d0>0" and d0: "∧x y. x ∈ path_image γ ==> y ∈ - U ==> d0 ≤ dist x y"
    using separate_compact_closed [of "path_image γ" "-U"] pasz ‹open U› ‹path γ› compact_path_image
    by blast    
  obtain dd where "0 < dd" and dd: "{y + k | y k. y ∈ path_image γ ∧ k ∈ ball 0 dd} ⊆ U"
  proof
    show "0 < d0 / 2" using ‹0 🚫› by auto
  qed (use ‹0 🚫› d0 in ‹force simp: dist_norm›)
  define T where "T ≡ {y + k |y k. y ∈ path_image γ ∧ k ∈ cball 0 (dd / 2)}"
  have "∧x x'. [x ∈ path_image γ; dist x x' * 2 < dd]
              ==> ∃y k. x' = y + k ∧ y ∈ path_image γ ∧ dist 0 k * 2 ≤ dd"
    by (metis add.commute diff_add_cancel dist_0_norm dist_commute dist_norm less_eq_real_def)
  then have subt: "path_image γ ⊆ interior T"
    using ‹0 🚫›
    apply (clarsimp simp add: mem_interior T_def)
    apply (rule_tac x="dd/2" in exI, auto)
    done
  have "compact T"
    unfolding T_def
    using ‹valid_path γ› compact_cball compact_sums compact_valid_path_image by blast
  have T: "T ⊆ U"
    unfolding T_def using ‹0 🚫› dd by fastforce
  obtain L where "L>0"
           and L: "∧f B. [f holomorphic_on interior T; ∧z. z∈interior T ==> cmod (f z) ≤ B] ==>
                         cmod (contour_integral γ f) ≤ L * B"
      using contour_integral_bound_exists [OF open_interior ‹valid_path γ› subt]
      by blast
  have "bounded(f ` T)"
    by (meson ‹compact T› compact_continuous_image compact_imp_bounded conf continuous_on_subset T)
  then obtain D where "D>0" and D: "∧x. x ∈ T ==> norm (f x) ≤ D"
    by (auto simp: bounded_pos)
  obtain C where "C>0" and C: "∧x. x ∈ T ==> norm x ≤ C"
    using ‹compact T› bounded_pos compact_imp_bounded by force
  have "dist (h y) 0 ≤ e" if "0 < e" and le: "D * L / e + C ≤ cmod y" for e y
  proof -
    have "D * L / e > 0"  using ‹D>0› ‹L>0› ‹e>0› by simp
    with le have ybig: "norm y > C" by force
    with C have "y ∉ T"  by force
    then have ynot: "y ∉ path_image γ"
      using subt interior_subset by blast
    have [simp]: "winding_number γ y = 0"
    proof (rule winding_number_zero_outside)
      show "path_image γ ⊆ cball 0 C"
        by (meson C interior_subset mem_cball_0 subset_eq subt)
    qed (use ybig loop ‹path γ› in auto)
    have [simp]: "h y = contour_integral γ (λw. f w/(w-y))"
      by (rule contour_integral_unique [symmetric]) (simp add: v_def ynot V)
    have holint: "(λw. 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 ‹y ∉ T› interior_subset in auto)
    have leD: "cmod (f z / (z-y)) ≤ D * (e / L / D)" if z: "z ∈ interior T" for z
    proof -
      have "D * L / e + cmod z ≤ cmod y"
        using le C [of z] z using interior_subset by force
      then have DL2: "D * L / e ≤ 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 "… ≤ D * (e / L / D)"
      proof (rule mult_mono)
        show "cmod (f z) ≤ D"
          using D interior_subset z by blast 
        show "inverse (cmod (z-y)) ≤ e / L / D" "D ≥ 0"
          using ‹L>0› ‹e>0› ‹D>0› DL2 by (auto simp: norm_divide field_split_simps)
      qed auto
      finally show ?thesis .
    qed
    have "dist (h y) 0 = cmod (contour_integral γ (λw. f w / (w-y)))"
      by (simp add: dist_norm)
    also have "… ≤ L * (D * (e / L / D))"
      by (rule L [OF holint leD])
    also have "… = e"
      using  ‹L>0› ‹0 🚫› by auto
    finally show ?thesis .
  qed
  then have "(h ---> 0) at_infinity"
    by (meson Lim_at_infinityI)
  moreover have "h holomorphic_on UNIV"
  proof -
    have con_ff: "continuous (at (x,z)) (λ(x,y). (f y - f x) / (y-x))"
                 if "x ∈ U" "z ∈ U" "x ≠ z" for x z
      using that conf
      apply (simp add: split_def continuous_on_eq_continuous_at ‹open U›)
      apply (simp | rule continuous_intros continuous_within_compose2 [where g=f])+
      done
    have con_fstsnd: "continuous_on UNIV (λx. (fst x - snd x) ::complex)"
      by (rule continuous_intros)+
    have open_uu_Id: "open (U × U - Id)"
    proof (rule open_Diff)
      show "open (U × U)"
        by (simp add: open_Times ‹open U›)
      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 ∈ U" for z
      by (meson analytic_at analytic_at_imp_isCont assms(1) holf holomorphic_deriv that)
    have tendsto_f': "((λ(x,y). if y = x then deriv f (x)
                                else (f (y) - f (x)) / (y-x)) ---> deriv f x)
                      (at (x, x) within U × U)" if "x ∈ U" for x
    proof (rule Lim_withinI)
      fix e::real assume "0 < e"
      obtain k1 where "k1>0" and k1: "∧x'. norm (x' - x) ≤ k1 ==> norm (deriv f x' - deriv f x) < e"
        using ‹0 🚫› continuous_within_E [OF con_derf [OF ‹x ∈ U›]]
        by (metis UNIV_I dist_norm)
      obtain k2 where "k2>0" and k2: "ball x k2 ⊆ U"
        by (blast intro: openE [OF ‹open U›] ‹x ∈ U›)
      have neq: "norm ((f z' - f x') / (z' - x') - deriv f x) ≤ e"
                    if "z' ≠ 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 ∈ closed_segment x' z' ==> cmod (w-x) ≤ 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 ∈ closed_segment x' z' ==> z' ≠ x' ==> cmod (deriv f w - deriv f x) ≤ 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: "∧x. x ∈ U ==> (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 U›)
        have "closed_segment x' z' ⊆ 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 *: "((λx. 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) ≤ 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']]
                 ‹e > 0›  ‹z' ≠ x'›
          apply (auto simp: norm_divide divide_simps derf_le)
          done
        also have "… ≤ e" using ‹0 🚫› by simp
        finally show ?thesis .
      qed
      show "∃d>0. ∀xa∈U × U.
                  0 < dist xa (x, x) ∧ dist xa (x, x) < d ⟶
                  dist (case xa of (x, y) ==> if y = x then deriv f x else (f y - f x) / (y-x)) (deriv f x) ≤ e"
        apply (rule_tac x="min k1 k2" in exI)
        using ‹k1>0› ‹k2>0› ‹e>0›
        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 γ) f"
      by (meson holf holomorphic_on_imp_continuous_on holomorphic_on_subset interior_subset subt T)
    have le_B: "∧T. T ∈ {0..1} ==> cmod (vector_derivative γ (at T)) ≤ B"
      using γ' B by (simp add: path_image_def vector_derivative_at rev_image_eqI)
    have f_has_cint: "∧w. w ∈ v - path_image γ ==> ((λu. f u / (u-w) ^ 1) has_contour_integral h w) γ"
      by (simp add: V)
    have "∧x y. [x ∈ U; y ∈ U; y ≠ x] ==> (λ(x, y). d x y) ←-(x, y)→ (f y - f x) / (y - x)"
      unfolding d_def
      apply (rule Lim_transform_within_open [OF _ open_uu_Id, where f = "(λ(x,y). (f y - f x) / (y-x))"])
      using con_ff by (auto simp: continuous_within)
    then have cond_uu: "continuous_on (U × U) (λ(x,y). d x y)"
      unfolding continuous_on_eq_continuous_within continuous_within d_def
      by (fastforce simp add: tendsto_f' intro: Lim_at_imp_Lim_at_within)
    have hol_dw: "(λz. d z w) holomorphic_on U" if "w ∈ U" for w
    proof -
      have "continuous_on U ((λ(x,y). d x y) ∘ (λz. (w,z)))"
        by (rule continuous_on_compose continuous_intros continuous_on_subset [OF cond_uu] | force intro: that)+
      then have *: "continuous_on U (λz. 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 **: "(λz. if w = z then deriv f z else (f w - f z) / (w-z)) field_differentiable at x"
        if "x ∈ U" "x ≠ w" for x
      proof (rule_tac f = "λx. (f w - f x)/(w-x)" and d = "dist x w" in field_differentiable_transform_within)
        show "(λx. (f w - f x) / (w-x)) field_differentiable at x"
          using that ‹open U›
          by (intro derivative_intros holomorphic_on_imp_differentiable_at [OF holf]; force)
      qed (use that ‹open U› in ‹auto simp: dist_commute›)
      show ?thesis
        unfolding d_def
      proof (rule no_isolated_singularity [OF * _ ‹open U›])
        show "(λz. 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 U› **)
      qed auto
    qed
    { fix a b
      assume abu: "closed_segment a b ⊆ U"
      have cont_cint_d: "continuous_on U (λw. contour_integral (linepath a b) (λz. d z w))"
      proof (rule contour_integral_continuous_on_linepath_2D [OF ‹open U› _ _ abu])
        show "∧w. w ∈ U ==> (λz. 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 × U) (λ(x, y). d y x)"
          by (auto intro: continuous_on_swap_args cond_uu)
      qed
      have cont_cint_dγ: "continuous_on {0..1} ((λw. contour_integral (linepath a b) (λz. d z w)) ∘ γ)"
        by (metis Diff_subset ‹path γ› cont_cint_d continuous_on_compose continuous_on_subset pasz path_def path_image_def)
      have "continuous_on {0..1} (λx. vector_derivative γ (at x))"
        using pfγ' by (simp add: continuous_on_polymonial_function vector_derivative_at [OF γ'])
      then have cint_cint: "(λw. contour_integral (linepath a b) (λz. d z w)) contour_integrable_on γ"
        apply (simp add: contour_integrable_on)
        apply (rule integrable_continuous_real)
        by (rule continuous_on_mult [OF cont_cint_dγ [unfolded o_def]])
      have "contour_integral (linepath a b) h = contour_integral (linepath a b) (λz. contour_integral γ (d z))"
        using abu  by (force simp: h_def intro: contour_integral_eq)
      also have "… = contour_integral γ (λw. contour_integral (linepath a b) (λz. d z w))"
      proof (rule contour_integral_swap)
        show "continuous_on (path_image (linepath a b) × path_image γ) (λ(y1, y2). d y1 y2)"
          using abu pasz by (auto intro: continuous_on_subset [OF cond_uu])
        show "continuous_on {0..1} (λt. vector_derivative (linepath a b) (at t))"
          by (auto intro!: continuous_intros)
        show "continuous_on {0..1} (λt. vector_derivative γ (at t))"
          by (metis γ' continuous_on_eq path_def path_polynomial_function pfγ' vector_derivative_at)
      qed (use ‹valid_path γ› in auto)
      finally have cint_h_eq:
          "contour_integral (linepath a b) h =
                    contour_integral γ (λw. contour_integral (linepath a b) (λz. 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 ∈ U" and au: "∀n. a n ∈ U" and ax: "a <---- x"
      then have A1: "∀🪙F n in sequentially. d (a n) contour_integrable_on γ"
        by (meson U contour_integrable_on_def eventuallyI)
      obtain dd where "dd>0" and dd: "cball x dd ⊆ U" using open_contains_cball ‹open U› x by force
      have A2: "uniform_limit (path_image γ) (λn. d (a n)) (d x) sequentially"
        unfolding uniform_limit_iff dist_norm
      proof clarify
        fix ee::real
        assume "0 < ee"
        show "∀🪙F n in sequentially. ∀ξ∈path_image γ. cmod (d (a n) ξ - d x ξ) < ee"
        proof -
          let ?ddpa = "{(w,z) |w z. w ∈ cball x dd ∧ z ∈ path_image γ}"
          have "uniformly_continuous_on ?ddpa (λ(x,y). d x y)"
          proof (rule compact_uniformly_continuous [OF continuous_on_subset[OF cond_uu]])
            show "compact {(w, z) |w z. w ∈ cball x dd ∧ z ∈ path_image γ}"
              using ‹valid_path γ›
              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: "∧x x'. [x ∈ ?ddpa; x' ∈ ?ddpa; dist x' x < kk] ==>
                             dist ((λ(x,y). d x y) x') ((λ(x,y). d x y) x) < ee"
            by (rule uniformly_continuous_onE [where ε = ee]) (use ‹0 🚫› in auto)
          have kk: "[norm (w-x) ≤ dd; z ∈ path_image γ; norm ((w, z) - (x, z)) < kk] ==> norm (d w z - d x z) < ee"
            for  w z
            using ‹dd>0› kk [of "(x,z)" "(w,z)"] by (force simp: norm_minus_commute dist_norm)
          obtain no where "∀n≥no. dist (a n) x < min dd kk"
            using ax unfolding lim_sequentially
            by (meson ‹0 🚫› ‹0 🚫› min_less_iff_conj)
          then show ?thesis
            using ‹dd > 0› ‹kk > 0› by (fastforce simp: eventually_sequentially kk dist_norm)
        qed
      qed
      have "(λn. contour_integral γ (d (a n))) <---- contour_integral γ (d x)"
        by (rule contour_integral_uniform_limit [OF A1 A2 le_B]) (auto simp: ‹valid_path γ›)
      then have tendsto_hx: "(λn. contour_integral γ (d (a n))) <---- h x"
        by (simp add: h_def x)
      then show "(h ∘ a) <---- 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 ∈ v" | "z0 ∈ U" using uv_Un by blast
      then show "h field_differentiable at z0"
      proof cases
        assume "z0 ∈ v" then show ?thesis
          using Cauchy_next_derivative [OF con_pa_f le_B f_has_cint _ ov] V f_has_cint ‹valid_path γ›
          by (auto simp: field_differentiable_def v_def)
      next
        assume "z0 ∈ U" then
        obtain e where "e>0" and e: "ball z0 e ⊆ U" by (blast intro: openE [OF ‹open U›])
        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} ⊆ ball z0 e"  for a b c
        proof -
          have *: "∧x1 x2 z. z ∈ U ==> closed_segment x1 x2 ⊆ U ==> (λw. d w z) contour_integrable_on linepath x1 x2"
            using  hol_dw holomorphic_on_imp_continuous_on ‹open U›
            by (auto intro!: contour_integrable_holomorphic_simple)
          have abc: "closed_segment a b ⊆ U"  "closed_segment b c ⊆ U"  "closed_segment c a ⊆ U"
            using that e segments_subset_convex_hull by fastforce+
          have eq0: "∧w. w ∈ U ==> contour_integral (linepath a b +++ linepath b c +++ linepath c a) (λz. d z w) = 0"
          proof (rule contour_integral_unique [OF Cauchy_theorem_triangle])
            show "∧w. w ∈ U ==> (λz. 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 "∧z. z ∈ path_image γ ==>
                         contour_integral (linepath a b) (λx. d x z) +
                         (contour_integral (linepath b c) (λx. d x z) +
                          contour_integral (linepath c a) (λx. d x z)) = 0"
            using abc pasz U "*" eq0 by auto
          then show ?thesis
            by (simp add: contour_integral_eq_0 cint_h abc contour_integrable_add contour_integral_add [symmetric] add_ac)
        qed
        show ?thesis
          using e ‹e > 0›
          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 "((λw. 1 / (w-z)) has_contour_integral complex_of_real (2 * pi) * i * winding_number γ z) γ"
    by (rule has_contour_integral_winding_number [OF ‹valid_path γ› znot])
  then have "((λw. f z * (1 / (w-z))) has_contour_integral complex_of_real (2 * pi) * i * winding_number γ z * f z) γ"
    by (metis mult.commute has_contour_integral_lmul)
  then have 1: "((λw. f z / (w-z)) has_contour_integral complex_of_real (2 * pi) * i * winding_number γ z * f z) γ"
    by (simp add: field_split_simps)
  moreover have 2: "((λw. (f w - f z) / (w-z)) has_contour_integral 0) γ"
    using U [OF z] pasz d_def by (force elim: has_contour_integral_eq [where g = "λw. (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 ∈ S" and vpg: "valid_path γ"
        and pasz: "path_image γ ⊆ S - {z}" and loop: "pathfinish γ = pathstart γ"
        and zero: "∧w. w ∉ S ==> winding_number γ w = 0"
      shows "((λw. f w / (w-z)) has_contour_integral (2*pi * i * winding_number γ z * f z)) γ"
proof -
  have "path γ" using vpg by (blast intro: valid_path_imp_path)
  have hols: "(λw. f w / (w-z)) holomorphic_on S - {z}" "(λw. 1 / (w-z)) holomorphic_on S - {z}"
    by (rule holomorphic_intros holomorphic_on_subset [OF holf] | force)+
  then have cint_fw: "(λw. f w / (w-z)) contour_integrable_on γ"
    by (meson contour_integrable_holomorphic_simple holomorphic_on_imp_continuous_on open_delete S vpg pasz)
  obtain d where "d>0"
      and d: "∧g h. [valid_path g; valid_path h; ∀t∈{0..1}. cmod (g t - γ t) < d ∧ cmod (h t - γ t) < d;
                     pathstart h = pathstart g ∧ pathfinish h = pathfinish g]
                     ==> path_image h ⊆ S - {z} ∧ (∀f. f holomorphic_on S - {z} ⟶ contour_integral h f = contour_integral g f)"
    using contour_integral_nearby_ends [OF _ ‹path γ› pasz] S by (simp add: open_Diff) metis
  obtain p where polyp: "polynomial_function p"
             and ps: "pathstart p = pathstart γ" and pf: "pathfinish p = pathfinish γ" and led: "∀t∈{0..1}. cmod (p t - γ t) < d"
    using path_approx_polynomial_function [OF ‹path γ› ‹d > 0›] 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 ∉ path_image γ" using pasz by blast
  have paps: "path_image p ⊆ S - {z}" and cint_eq: "(∧f. f holomorphic_on S - {z} ==> contour_integral p f = contour_integral γ f)"
    using pf ps led d [OF vpg vpp] ‹d > 0› by auto
  have wn_eq: "winding_number p z = winding_number γ 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 γ w" if "w ∉ S" for w
  proof -
    have hol: "(λv. 1 / (v-w)) holomorphic_on S - {z}"
      using that by (force intro: holomorphic_intros holomorphic_on_subset [OF holf])
   have "w ∉ path_image p" "w ∉ path_image γ" 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: "∧w. w ∉ S ==> 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 γ" and loop: "pathfinish γ = pathstart γ"
        and pas: "path_image γ ⊆ S"
        and zero: "∧w. w ∉ S ==> winding_number γ w = 0"
      shows "(f has_contour_integral 0) γ"
proof -
  have "path_image γ ≠ S"
    by (metis compact_valid_path_image vpg compact_open path_image_nonempty S)
  then obtain z where "z ∈ S" and znot: "z ∉ path_image γ" and pasz: "path_image γ ⊆ S - {z}"
    using pas by blast
  have hol: "(λw. (w-z) * f w) holomorphic_on S"
    by (rule holomorphic_intros holf)+
  show ?thesis
    using Cauchy_integral_formula_global [OF S hol ‹z ∈ S› 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 γ"  "pathfinish γ = pathstart γ" "path_image γ ⊆ S"
            "∧w. w ∉ S ==> w ∈ outside(path_image γ)"
      shows "(f has_contour_integral 0) γ"
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 ⊆ S" "pathfinish g = pathstart g" "z ∉ 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 ‹z ∉ S› 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 "… = 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 ⊆ S" "pathfinish g = pathstart g"
    shows "(f has_contour_integral 0) g"
  by (meson assms Cauchy_theorem_global simply_connected_imp_winding_number_zero valid_path_imp_path)

proposition🍋‹tag unimportant› holomorphic_logarithm_exists:
  assumes A: "convex A" "open A"
      and f: "f holomorphic_on A" "∧x. x ∈ A ==> f x ≠ 0"
      and z0: "z0 ∈ A"
    obtains g where "g holomorphic_on A" and "∧x. x ∈ A ==> exp (g x) = f x"
proof -
  note f' = holomorphic_derivI [OF f(1) A(2)]
  obtain g where g: "∧x. x ∈ A ==> (g has_field_derivative deriv f x / f x) (at x)"
  proof (rule holomorphic_convex_primitive' [OF A])
    show "(λx. 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 = (λx. -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)
    note [simp] = at_within_open[OF _ ‹open A›]
    have "∃c. ∀x∈A. f x / exp (h x) - 1 = c"
      using ‹convex A› z0 f 
      by (force simp: h_def exp_diff field_simps intro!: has_field_derivative_zero_constant derivative_eq_intros g f')
  then obtain c where c: "∧x. x ∈ A ==> 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 "∧x. x ∈ A ==> exp (h x) = f x" by simp
  from that[OF h_holo this] show ?thesis .
qed


(* FIXME mv to Cauchy_Integral_Theorem.thy *)
subsection‹Cauchy's inequality and more versions of Liouville›

lemma Cauchy_higher_deriv_bound:
    assumes holf: "f holomorphic_on (ball z r)"
        and contf: "continuous_on (cball z r) f"
        and fin : "∧w. w ∈ ball z r ==> f w ∈ ball y B0"
        and "0 < r" and "0 < n"
      shows "cmod ((deriv ^^ n) f z) ≤ (fact n) * B0 / r^n"
proof -
  have "0 < B0" 
    using ‹0 🚫› fin [of z] by (metis ball_eq_empty ex_in_conv fin not_less)
  have le_B0: "cmod (f w - y) ≤ B0" if "cmod (w-z) ≤ r" for w
  proof (rule continuous_on_closure_norm_le [of "ball z r" "λw. f w - y"], use ‹0 🚫› in simp_all)
    show "continuous_on (cball z r) (λw. f w - y)"
      by (intro continuous_intros contf)
    show "dist z w ≤ r"
      by (simp add: dist_commute dist_norm that)
    qed (use fin in ‹auto simp: dist_norm less_eq_real_def norm_minus_commute›)
  have "(deriv ^^ n) f z = (deriv ^^ n) (λw. f w) z - (deriv ^^ n) (λw. y) z"
    using ‹0 🚫› by simp
  also have "... = (deriv ^^ n) (λw. f w - y) z"
    using ‹0 🚫› higher_deriv_diff holf by auto
  finally have "(deriv ^^ n) f z = (deriv ^^ n) (λw. f w - y) z" .
  have contf': "continuous_on (cball z r) (λu. f u - y)"
    by (rule contf continuous_intros)+
  have holf': "(λu. (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 ‹0 🚫› by (simp add: a_def field_split_simps)
  have der_dif: "(deriv ^^ n) (λw. f w - y) z = (deriv ^^ n) f z"
    using ‹0 🚫› ‹0 🚫›
    by (auto simp: higher_deriv_diff [OF holf holomorphic_on_const])
  have "norm ((2 * of_real pi * i)/(fact n) * (deriv ^^ n) (λw. f w - y) z)
        ≤ (B0/r^(Suc n)) * (2 * pi * r)"
    apply (rule has_contour_integral_bound_circlepath [of "(λu. (f u - y)/(u-z)^(Suc n))" _ z])
    using Cauchy_has_contour_integral_higher_derivative_circlepath [OF contf' holf']
    using ‹0 🚫› ‹0 🚫›
    apply (auto simp: norm_divide norm_mult norm_power divide_simps le_B0)
    done
  then show ?thesis
    using ‹0 🚫›
    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 ξ r)"
        and contf: "continuous_on (cball ξ r) f"
        and "0 < r"
        and nof: "∧x. norm(ξ-x) = r ==> norm(f x) ≤ B"
      shows "norm ((deriv ^^ n) f ξ) ≤ (fact n) * B / r^n"
proof -
  obtain x where "norm (ξ-x) = r"
    by (metis ‹0 🚫› dist_norm order_less_imp_le vector_choose_dist)
  then have "0 ≤ B"
    by (metis nof norm_not_less_zero not_le order_trans)
  have "ξ ∈ ball ξ r"
    using ‹0 🚫› by simp
  then have  "((λu. f u / (u-ξ) ^ Suc n) has_contour_integral (2 * pi) * i / fact n * (deriv ^^ n) f ξ)
         (circlepath ξ r)"
    by (rule Cauchy_has_contour_integral_higher_derivative_circlepath [OF contf holf])
  have "norm ((2 * pi * i)/(fact n) * (deriv ^^ n) f ξ) ≤ (B / r^(Suc n)) * (2 * pi * r)"
  proof (rule has_contour_integral_bound_circlepath)
    have "ξ ∈ ball ξ r"
      using ‹0 🚫› by simp
    then show  "((λu. f u / (u-ξ) ^ Suc n) has_contour_integral (2 * pi) * i / fact n * (deriv ^^ n) f ξ)
         (circlepath ξ r)"
      by (rule Cauchy_has_contour_integral_higher_derivative_circlepath [OF contf holf])
    show "∧x. cmod (x-ξ) = r ==> cmod (f x / (x-ξ) ^ Suc n) ≤ B / r ^ Suc n"
      using ‹0 ≤ B› ‹0 🚫›
      by (simp add: norm_divide norm_power nof frac_le norm_minus_commute del: power_Suc)
  qed (use ‹0 ≤ B› ‹0 🚫› in auto)
  then show ?thesis using ‹0 🚫›
    by (simp add: norm_divide norm_mult field_simps)
qed

lemma Liouville_polynomial:
    assumes holf: "f holomorphic_on UNIV"
        and nof: "∧z. A ≤ norm z ==> norm(f z) ≤ B * norm z ^ n"
      shows "f ξ = (∑k≤n. (deriv^^k) f 0 / fact k * ξ ^ k)"
proof (cases rule: le_less_linear [THEN disjE])
  assume "B ≤ 0"
  then have "∧z. A ≤ norm z ==> 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 ---> 0) at_infinity"
    using Lim_at_infinity by force
  then have [simp]: "f = (λw. 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 "((λk. (deriv ^^ k) f 0 / (fact k) * (ξ - 0)^k) sums f ξ)"
  proof (rule holomorphic_power_series [where r = "norm ξ + 1"])
    show "f holomorphic_on ball 0 (cmod ξ + 1)" "ξ ∈ ball 0 (cmod ξ + 1)"
      using holf holomorphic_on_subset by auto
  qed
  then have sumsf: "((λk. (deriv ^^ k) f 0 / (fact k) * ξ^k) sums f ξ)" by simp
  have "(deriv ^^ k) f 0 / fact k * ξ ^ 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) + (∣A∣ + 1))"
    have "1 ≤ abs (fact k * B / cmod ((deriv ^^ k) f 0) + (∣A∣ + 1))"
      using ‹0 🚫› by simp
    then have wge1: "1 ≤ norm w"
      by (metis norm_of_real w_def)
    then have "w ≠ 0" by auto
    have kB: "0 < fact k * B"
      using ‹0 🚫› by simp
    then have "0 ≤ fact k * B / cmod ((deriv ^^ k) f 0)"
      by simp
    then have wgeA: "A ≤ 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) + (∣A∣ + 1))"
      using ‹0 🚫› 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 "... ≤ 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 "∧x. cmod (0-x) = cmod w ==> cmod (f x) ≤ B * cmod w ^ n"
        by (metis nof wgeA dist_0_norm dist_norm)
    qed (use ‹w ≠ 0› in auto)
    also have "... = fact k * B / cmod w ^ (k-n)"
      using ‹k>n› by (simp add: divide_simps flip: power_add)
    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 (smt (verit, best) ‹w ≠ 0› frac_le zero_less_norm_iff)
    with self_le_power [OF wge1] show ?thesis
      by (meson diff_is_0_eq not_gr0 not_le that)
  qed
  then have "(deriv ^^ (k + Suc n)) f 0 / fact (k + Suc n) * ξ ^ (k + Suc n) = 0" for k
    using not_less_eq by blast
  then have "(λi. (deriv ^^ (i + Suc n)) f 0 / fact (i + Suc n) * ξ ^ (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‹Every bounded entire function is a constant function.›
theorem Liouville_theorem:
  assumes holf: "f holomorphic_on UNIV"
    and bf: "bounded (range f)"
  shows "f constant_on UNIV"
  using Liouville_polynomial [OF holf, of 0 _ 0, simplified]
  by (metis bf bounded_iff constant_on_def rangeI)

text‹A holomorphic function f has only isolated zeros unless f is 0.›

lemma powser_0_nonzero:
  fixes a :: "nat ==> 'a::{real_normed_field,banach}"
  assumes r: "0 < r"
      and sm: "∧x. norm (x-ξ) < r ==> (λn. a n * (x-ξ) ^ n) sums (f x)"
      and [simp]: "f ξ = 0"
      and m0: "a m ≠ 0" and "m>0"
  obtains s where "0 < s" and "∧z. z ∈ cball ξ s - {ξ} ==> f z ≠ 0"
proof -
  have "r ≤ conv_radius a"
    using sm sums_summable by (auto simp: le_conv_radius_iff [where ξ=ξ])
  obtain m where am: "a m ≠ 0" and az [simp]: "(∧n. n==> a n = 0)"
  proof
    show "a (LEAST n. a n ≠ 0) ≠ 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 (λi. b i * (x-ξ) ^ i)" for x
  have [simp]: "b 0 = 1"
    by (simp add: am b_def)
  { fix x::'a
    assume "norm (x-ξ) < r"
    then have "(λn. (a m * (x-ξ)^m) * (b n * (x-ξ)^n)) sums (f x)"
      using am az sm sums_zero_iff_shift [of m "(λn. a n * (x-ξ) ^ n)" "f x"]
      by (simp add: b_def monoid_mult_class.power_add algebra_simps)
    then have "x ≠ ξ ==> (λn. b n * (x-ξ)^n) sums (f x / (a m * (x-ξ)^m))"
      using am by (simp add: sums_mult_D)
  } note bsums = this
  then have  "norm (x-ξ) < r ==> summable (λn. b n * (x-ξ)^n)" for x
    using sums_summable by (cases "x=ξ") auto
  then have "r ≤ conv_radius b"
    by (simp add: le_conv_radius_iff [where ξ=ξ])
  then have "r/2 < conv_radius b"
    using not_le order_trans r by fastforce
  then have "continuous_on (cball ξ (r/2)) g"
    using powser_continuous_suminf [of "r/2" b ξ] by (simp add: g_def)
  then obtain s where "s>0"  "∧x. [norm (x-ξ) ≤ s; norm (x-ξ) ≤ r/2] ==> dist (g x) (g ξ) < 1/2"
  proof (rule continuous_onE)
    show "ξ ∈ cball ξ (r / 2)" "1/2 > (0::real)"
      using r by auto
  qed (auto simp: dist_commute dist_norm)
  moreover have "g ξ = 1"
    by (simp add: g_def)
  ultimately have gnz: "∧x. [norm (x-ξ) ≤ s; norm (x-ξ) ≤ r/2] ==> (g x) ≠ 0"
    by fastforce
  show ?thesis
  proof
    have *: "f x ≠ 0" if "x ≠ ξ" "norm (x-ξ) ≤ s" "norm (x-ξ) ≤ r/2" for x
      using bsums [of x] that gnz [of x] r sums_iff unfolding g_def by fastforce
    show "∧z. z ∈ cball ξ (min s (r / 2)) - {ξ} ==> f z ≠ 0"
      by (simp add: "*" dist_norm norm_minus_commute)
  qed (use ‹0 🚫› ‹0 🚫› in auto)
qed

subsection ‹Complex functions and power series›

text ‹
  The following defines the power series expansion of a complex function at a given point
  (assuming that it is analytic at that point).
 ›
definition🍋‹tag important› fps_expansion :: "(complex ==> complex) ==> complex ==> complex fps" where
  "fps_expansion f z0 = Abs_fps (λn. (deriv ^^ n) f z0 / fact n)"

lemma fps_expansion_cong:
  assumes "∀🪙F w in nhds x. f w =g w"
  shows "fps_expansion f x = fps_expansion g x"
  unfolding fps_expansion_def using assms higher_deriv_cong_ev by fastforce 

lemma
  fixes r :: ereal
  assumes "f holomorphic_on eball z0 r"
  shows conv_radius_fps_expansion: "fps_conv_radius (fps_expansion f z0) ≥ r"
    and eval_fps_expansion:  "∧z. z ∈ eball z0 r ==> eval_fps (fps_expansion f z0) (z - z0) = f z"
    and eval_fps_expansion': "∧z. norm z < r ==> eval_fps (fps_expansion f z0) z = f (z0 + z)"
proof -
  have "(λn. fps_nth (fps_expansion f z0) n * (z - z0) ^ n) sums f z"
    if "z ∈ ball z0 r'" "ereal r' < r" for z r'
  proof -
    have "f holomorphic_on ball z0 r'"
      using holomorphic_on_subset[OF _ ball_eball_mono] assms that by force
    then show ?thesis
      using fps_expansion_def holomorphic_power_series that by auto
  qed
  hence *: "(λn. fps_nth (fps_expansion f z0) n * (z - z0) ^ n) sums f z"
    if "z ∈ eball z0 r" for z
    using that by (subst (asm) eball_conv_UNION_balls) blast
  show "fps_conv_radius (fps_expansion f z0) ≥ r" unfolding fps_conv_radius_def
  proof (rule conv_radius_geI_ex)
    fix r' :: real assume r': "r' > 0" "ereal r' < r"
    thus "∃z. norm z = r' ∧ summable (λn. 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 ∈ 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 ‹
  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.
 ›
lemma
  fixes f :: "complex fps" and r :: ereal
  assumes "∧z. ereal (norm z) < r ==> eval_fps f z ≠ 0"
  shows   fps_conv_radius_inverse: "fps_conv_radius (inverse f) ≥ min r (fps_conv_radius f)"
    and   eval_fps_inverse: "∧z. ereal (norm z) < fps_conv_radius f ==> ereal (norm z) < r ==>
                               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) ≥ min r (fps_conv_radius f) ∧
          (∀z∈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 (λz. inverse (eval_fps f z)) 0"
    have holo: "(λz. 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' ≥ 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 "… ≤ fps_conv_radius (f * f')" by (rule fps_conv_radius_mult)
      finally show "… > 0" .
    next
      from True have "R > 0" by (auto simp: R_def)
      hence "eventually (λz. z ∈ eball 0 R) (nhds 0)"
        by (intro eventually_nhds_in_open) (auto simp: zero_ereal_def)
      thus "eventually (λz. 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 ≠ 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 
      from False have "min r (fps_conv_radius f) ≤ 0"
        by (simp add: min_def)
      also have "0 ≤ fps_conv_radius (inverse f)"
        by (simp add: fps_conv_radius_def conv_radius_nonneg)
      finally show "min r (fps_conv_radius f) ≤ …" .
    qed (unfold * [unfolded R_def], auto)
  qed

  show "fps_conv_radius (inverse f) ≥ min r (fps_conv_radius f)"
    using * by blast
  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 ≡ 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: "∧z. z ∈ eball 0 r ==> eval_fps g z ≠ 0"
  shows   fps_conv_radius_divide': "fps_conv_radius (f / g) ≥ R"
    and   eval_fps_divide':
            "ereal (norm z) < R ==> eval_fps (f / g) z = eval_fps f z / eval_fps g z"
proof -
  from nz[of 0] and ‹r > 0› have nz': "fps_nth g 0 ≠ 0" 
    by (auto simp: eval_fps_at_0 zero_ereal_def)
  have "R ≤ min r (fps_conv_radius g)"
    by (auto simp: R_def intro: min.coboundedI2)
  also have "min r (fps_conv_radius g) ≤ 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) ≥ R" .
  have "R ≤ 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 "… ≤ 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) ≥ 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 ‹r > 0›
    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 ≡ Min {r, fps_conv_radius f, fps_conv_radius g}"
  assumes "subdegree g ≤ subdegree f"
  assumes "fps_conv_radius f > 0" "fps_conv_radius g > 0" "r > 0"
  assumes "∧z. z ∈ eball 0 r ==> z ≠ 0 ==> eval_fps g z ≠ 0"
  shows   fps_conv_radius_divide: "fps_conv_radius (f / g) ≥ R"
    and   eval_fps_divide:
            "ereal (norm z) < R ==> c = fps_nth f (subdegree g) / fps_nth g (subdegree g) ==>
               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 ≠ 0"
  proof -
    define z :: complex where "z = (if r = ∞ then 1 else of_real (real_of_ereal r / 2))"
    have "z ∈ eball 0 r"
      using ‹r > 0› ereal_less_real_iff z_def by fastforce
    moreover have "z ≠ 0" using ‹r > 0›
      by (cases r) (auto simp: z_def)
    ultimately have "eval_fps g z ≠ 0" by (rule assms(6))
    thus "g ≠ 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 ≠ 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 ≠ 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 (use ‹g ≠ 0› in ‹auto simp: g'_def eval_fps_at_0›)

  have "R ≤ min (min r (fps_conv_radius g)) (fps_conv_radius g')"
    by (auto simp: R_def min.coboundedI1 min.coboundedI2)
  also have "… ≤ fps_conv_radius (inverse g')"
    using g'_nz by (rule fps_conv_radius_inverse)
  finally have conv_radius_inv: "R ≤ fps_conv_radius (inverse g')" .
  hence "R ≤ 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) ≥ 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 * … = 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 ∈ A" "f holomorphic_on A"
  shows   "f has_fps_expansion fps_expansion f 0"
proof -
  from assms obtain r where "r > 0 " and r: "ball 0 r ⊆ A"
    by (auto simp: open_contains_ball)
  with assms have holo: "f holomorphic_on eball 0 (ereal r)" 
    by auto
  have "r ≤ fps_conv_radius (fps_expansion f 0)"
    using holo by (intro conv_radius_fps_expansion) auto
  then have "… > 0"
    by (simp add: ereal_le_less ‹r > 0› zero_ereal_def) 
  moreover have "eventually (λz. z ∈ ball 0 r) (nhds 0)"
    using ‹r > 0› by (intro eventually_nhds_in_open) auto
  hence "eventually (λz. 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 > 0› by (auto simp: has_fps_expansion_def)
qed

lemma fps_conv_radius_tan:
  fixes c :: complex
  assumes "c ≠ 0"
  shows   "fps_conv_radius (fps_tan c) ≥ pi / (2 * norm c)"
proof -
  have "fps_conv_radius (fps_tan c) ≥
          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 ∈ eball 0 (pi / (2 * norm c))"
    with cos_eq_zero_imp_norm_ge[of "c*z"] assms 
      show "eval_fps (fps_cos c) z ≠ 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 ∈ eball 0 (pi / (2 * norm c))"
    with cos_eq_zero_imp_norm_ge[of "c*z"] assms 
    show "eval_fps (fps_cos c) z ≠ 0" using False by (auto simp: norm_mult field_simps)
  qed (use False assms in ‹auto simp: field_simps tan_def›)
qed simp_all

end