Theory Conformal_Mappings

section ‹Conformal Mappings and Consequences of Cauchy's Integral Theorem›

text‹By John Harrison et al.  Ported from HOL Light by L C Paulson (2016)›

text‹Also Cauchy's residue theorem by Wenda Li (2016)›

theory Conformal_Mappings
imports Cauchy_Integral_Formula

begin

subsection ‹Analytic continuation›

proposition isolated_zeros:
  assumes holf: "f holomorphic_on S"
      and "open S" "connected S" "ξ  S" "f ξ = 0" "β  S" "f β  0"
    obtains r where "0 < r" and "ball ξ r  S" and
        "z. z  ball ξ r - {ξ}  f z  0"
proof -
  obtain r where "0 < r" and r: "ball ξ r  S"
    using open S ξ  S open_contains_ball_eq by blast
  have powf: "((λn. (deriv ^^ n) f ξ / (fact n) * (z - ξ)^n) sums f z)" if "z  ball ξ r" for z
    by (intro holomorphic_power_series [OF _ that] holomorphic_on_subset [OF holf r])
  obtain m where m: "(deriv ^^ m) f ξ / (fact m)  0"
    using holomorphic_fun_eq_0_on_connected [OF holf open S connected S _ ξ  S β  S] f β  0
    by auto
  then have "m  0" using assms(5) funpow_0 by fastforce
  obtain s where "0 < s" and s: "z. z  cball ξ s - {ξ}  f z  0"
    using powser_0_nonzero [OF 0 < r powf f ξ = 0 m]
    by (metis m  0 dist_norm mem_ball norm_minus_commute not_gr_zero)
  have "0 < min r s"  by (simp add: 0 < r 0 < s)
  then show thesis
    apply (rule that)
    using r s by auto
qed

proposition analytic_continuation:
  assumes holf: "f holomorphic_on S"
      and "open S" and "connected S"
      and "U  S" and "ξ  S"
      and "ξ islimpt U"
      and fU0 [simp]: "z. z  U  f z = 0"
      and "w  S"
    shows "f w = 0"
proof -
  obtain e where "0 < e" and e: "cball ξ e  S"
    using open S ξ  S open_contains_cball_eq by blast
  define T where "T = cball ξ e  U"
  have contf: "continuous_on (closure T) f"
    by (metis T_def closed_cball closure_minimal e holf holomorphic_on_imp_continuous_on
              holomorphic_on_subset inf.cobounded1)
  have fT0 [simp]: "x. x  T  f x = 0"
    by (simp add: T_def)
  have "r. e>0. x'U. x'  ξ  dist x' ξ < e; 0 < r  x'cball ξ e  U. x'  ξ  dist x' ξ < r"
    by (metis 0 < e IntI dist_commute less_eq_real_def mem_cball min_less_iff_conj)
  then have "ξ islimpt T" using ξ islimpt U
    by (auto simp: T_def islimpt_approachable)
  then have "ξ  closure T"
    by (simp add: closure_def)
  then have "f ξ = 0"
    by (auto simp: continuous_constant_on_closure [OF contf])
  moreover have "r. 0 < r; z. z  ball ξ r - {ξ}  f z  0  False"
    by (metis open_ball ξ islimpt T centre_in_ball fT0 insertE insert_Diff islimptE)
  ultimately show ?thesis
    by (metis open S connected S ξ  S w  S holf isolated_zeros)
qed

corollary analytic_continuation_open:
  assumes "open s" and "open s'" and "s  {}" and "connected s'"
      and "s  s'"
  assumes "f holomorphic_on s'" and "g holomorphic_on s'"
      and "z. z  s  f z = g z"
  assumes "z  s'"
  shows   "f z = g z"
proof -
  from s  {} obtain ξ where "ξ  s" by auto
  with open s have ξ: "ξ islimpt s"
    by (intro interior_limit_point) (auto simp: interior_open)
  have "f z - g z = 0"
    by (rule analytic_continuation[of "λz. f z - g z" s' s ξ])
       (insert assms ξ  s ξ, auto intro: holomorphic_intros)
  thus ?thesis by simp
qed

corollary analytic_continuation':
  assumes "f holomorphic_on S" "open S" "connected S"
      and "U  S" "ξ  S" "ξ islimpt U"
      and "f constant_on U"
    shows "f constant_on S"
proof -
  obtain c where c: "x. x  U  f x - c = 0"
    by (metis f constant_on U constant_on_def diff_self)
  have "(λz. f z - c) holomorphic_on S"
    using assms by (intro holomorphic_intros)
  with c analytic_continuation assms have "x. x  S  f x - c = 0"
    by blast
  then show ?thesis
    unfolding constant_on_def by force
qed

lemma holomorphic_compact_finite_zeros:
  assumes S: "f holomorphic_on S" "open S" "connected S"
      and "compact K" "K  S"
      and "¬ f constant_on S"
    shows "finite {zK. f z = 0}"
proof (rule ccontr)
  assume "infinite {zK. f z = 0}"
  then obtain z where "z  K" and z: "z islimpt {zK. f z = 0}"
    using compact K by (auto simp: compact_eq_Bolzano_Weierstrass)
  moreover have "{zK. f z = 0}  S"
    using K  S by blast
    ultimately show False
    using assms analytic_continuation [OF S] unfolding constant_on_def
    by blast
qed

lemma holomorphic_countable_zeros:
  assumes S: "f holomorphic_on S" "open S" "connected S" and "fsigma S"
      and "¬ f constant_on S"
    shows "countable {zS. f z = 0}"
proof -
  obtain F::"nat  complex set" 
      where F: "range F  Collect compact" and Seq: "S = (i. F i)"
    using fsigma S by (meson fsigma_Union_compact)
  have fin: "finite {z  F i. f z = 0}" for i
    using holomorphic_compact_finite_zeros assms F Seq Union_iff by blast
  have "{z  S. f z = 0} = (i. {z  F i. f z = 0})"
    using Seq by auto
  with fin show ?thesis
    by (simp add: countable_finite)
qed

lemma holomorphic_countable_equal:
  assumes "f holomorphic_on S" "g holomorphic_on S" "open S" "connected S" and "fsigma S"
    and eq: "uncountable {zS. f z = g z}"
  shows "S  {zS. f z = g z}"
proof -
  obtain z where z: "zS" "f z = g z"
    using eq not_finite_existsD uncountable_infinite by blast
  have "(λx. f x - g x) holomorphic_on S"
    by (simp add: assms holomorphic_on_diff)
  then have "(λx. f x - g x) constant_on S"
    using holomorphic_countable_zeros assms by force
  with z have "x. xS  f x - g x = 0"
    unfolding constant_on_def by force
  then show ?thesis
    by auto
qed

lemma holomorphic_countable_equal_UNIV:
  assumes fg: "f holomorphic_on UNIV" "g holomorphic_on UNIV"
    and eq: "uncountable {z. f z = g z}"
  shows "f=g"
  using holomorphic_countable_equal [OF fg] eq by fastforce

subsection‹Open mapping theorem›

lemma holomorphic_contract_to_zero:
  assumes contf: "continuous_on (cball ξ r) f"
      and holf: "f holomorphic_on ball ξ r"
      and "0 < r"
      and norm_less: "z. norm(ξ - z) = r  norm(f ξ) < norm(f z)"
  obtains z where "z  ball ξ r" "f z = 0"
proof -
  { assume fnz: "w. w  ball ξ r  f w  0"
    then have "0 < norm (f ξ)"
      by (simp add: 0 < r)
    have fnz': "w. w  cball ξ r  f w  0"
      by (metis norm_less dist_norm fnz less_eq_real_def mem_ball mem_cball norm_not_less_zero norm_zero)
    have "frontier(cball ξ r)  {}"
      using 0 < r by simp
    define g where [abs_def]: "g z = inverse (f z)" for z
    have contg: "continuous_on (cball ξ r) g"
      unfolding g_def using contf continuous_on_inverse fnz' by blast
    have holg: "g holomorphic_on ball ξ r"
      unfolding g_def using fnz holf holomorphic_on_inverse by blast
    have "frontier (cball ξ r)  cball ξ r"
      by (simp add: subset_iff)
    then have contf': "continuous_on (frontier (cball ξ r)) f"
          and contg': "continuous_on (frontier (cball ξ r)) g"
      by (blast intro: contf contg continuous_on_subset)+
    have froc: "frontier(cball ξ r)  {}"
      using 0 < r by simp
    moreover have "continuous_on (frontier (cball ξ r)) (norm o f)"
      using contf' continuous_on_compose continuous_on_norm_id by blast
    ultimately obtain w where w: "w  frontier(cball ξ r)"
                          and now: "x. x  frontier(cball ξ r)  norm (f w)  norm (f x)"
      using continuous_attains_inf [OF compact_frontier [OF compact_cball]]
      by (metis comp_apply)
    then have fw: "0 < norm (f w)"
      by (simp add: fnz')
    have "continuous_on (frontier (cball ξ r)) (norm o g)"
      using contg' continuous_on_compose continuous_on_norm_id by blast
    then obtain v where v: "v  frontier(cball ξ r)"
               and nov: "x. x  frontier(cball ξ r)  norm (g v)  norm (g x)"
      using continuous_attains_sup [OF compact_frontier [OF compact_cball] froc] by force
    then have fv: "0 < norm (f v)"
      by (simp add: fnz')
    have "norm ((deriv ^^ 0) g ξ)  fact 0 * norm (g v) / r ^ 0"
      by (rule Cauchy_inequality [OF holg contg 0 < r]) (simp add: dist_norm nov)
    then have "cmod (g ξ)  cmod (g v)"
      by simp
    moreover have "cmod (ξ - w) = r"
      by (metis (no_types) dist_norm frontier_cball mem_sphere w)
    ultimately obtain wr: "norm (ξ - w) = r" and nfw: "norm (f w)  norm (f ξ)"
      unfolding g_def
        by (metis (no_types) 0 < cmod (f ξ) less_imp_inverse_less norm_inverse not_le now order_trans v)
    with fw have False
      using norm_less by force
  }
  with that show ?thesis by blast
qed

theorem open_mapping_thm:
  assumes holf: "f holomorphic_on S"
      and S: "open S" and "connected S"
      and "open U" and "U  S"
      and fne: "¬ f constant_on S"
    shows "open (f ` U)"
proof -
  have *: "open (f ` U)"
          if "U  {}" and U: "open U" "connected U" and "f holomorphic_on U" and fneU: "x. y  U. f y  x"
          for U
  proof (clarsimp simp: open_contains_ball)
    fix ξ assume ξ: "ξ  U"
    show "e>0. ball (f ξ) e  f ` U"
    proof -
      have hol: "(λz. f z - f ξ) holomorphic_on U"
        by (rule holomorphic_intros that)+
      obtain s where "0 < s" and sbU: "ball ξ s  U"
                 and sne: "z. z  ball ξ s - {ξ}  (λz. f z - f ξ) z  0"
        using isolated_zeros [OF hol U ξ]  by (metis fneU right_minus_eq)
      obtain r where "0 < r" and r: "cball ξ r  ball ξ s"
        using 0 < s by (rule_tac r="s/2" in that) auto
      have "cball ξ r  U"
        using sbU r by blast
      then have frsbU: "frontier (cball ξ r)  U"
        using Diff_subset frontier_def order_trans by fastforce
      then have cof: "compact (frontier(cball ξ r))"
        by blast
      have frne: "frontier (cball ξ r)  {}"
        using 0 < r by auto
      have contfr: "continuous_on (frontier (cball ξ r)) (λz. norm (f z - f ξ))"
        by (metis continuous_on_norm continuous_on_subset frsbU hol holomorphic_on_imp_continuous_on)
      obtain w where "norm (ξ - w) = r"
                 and w: "(z. norm (ξ - z) = r  norm (f w - f ξ)  norm(f z - f ξ))"
        using continuous_attains_inf [OF cof frne contfr] by (auto simp: dist_norm)
      moreover define ε where "ε  norm (f w - f ξ) / 3"
      ultimately have "0 < ε"
        using 0 < r dist_complex_def r sne by auto
      have "ball (f ξ) ε  f ` U"
      proof
        fix γ
        assume γ: "γ  ball (f ξ) ε"
        have *: "cmod (γ - f ξ) < cmod (γ - f z)" if "cmod (ξ - z) = r" for z
        proof -
          have lt: "cmod (f w - f ξ) / 3 < cmod (γ - f z)"
            using w [OF that] γ
            using dist_triangle2 [of "f ξ" "γ"  "f z"] dist_triangle2 [of "f ξ" "f z" γ]
            by (simp add: ε_def dist_norm norm_minus_commute)
          show ?thesis
            by (metis ε_def dist_commute dist_norm less_trans lt mem_ball γ)
        qed
       have "continuous_on (cball ξ r) (λz. γ - f z)"
          using cball ξ r  U f holomorphic_on U
          by (force intro: continuous_intros continuous_on_subset holomorphic_on_imp_continuous_on)
        moreover have "(λz. γ - f z) holomorphic_on ball ξ r"
          using cball ξ r  U ball_subset_cball holomorphic_on_subset that(4) 
          by (intro holomorphic_intros) blast
        ultimately obtain z where "z  ball ξ r" "γ - f z = 0"
          using "*" 0 < r holomorphic_contract_to_zero by blast
        then show "γ  f ` U"
          using cball ξ r  U by fastforce
      qed
      then show ?thesis using  0 < ε by blast
    qed
  qed
  have "open (f ` X)" if "X  components U" for X
  proof -
    have holfU: "f holomorphic_on U"
      using U  S holf holomorphic_on_subset by blast
    have "X  {}"
      using that by (simp add: in_components_nonempty)
    moreover have "open X"
      using that open U open_components by auto
    moreover have "connected X"
      using that in_components_maximal by blast
    moreover have "f holomorphic_on X"
      by (meson that holfU holomorphic_on_subset in_components_maximal)
    moreover have "yX. f y  x" for x
    proof (rule ccontr)
      assume not: "¬ (yX. f y  x)"
      have "X  S"
        using U  S in_components_subset that by blast
      obtain w where w: "w  X" using X  {} by blast
      have wis: "w islimpt X"
        using w open X interior_eq by auto
      have hol: "(λz. f z - x) holomorphic_on S"
        by (simp add: holf holomorphic_on_diff)
      with fne [unfolded constant_on_def]
           analytic_continuation[OF hol S connected S X  S _ wis] not X  S w
      show False by auto
    qed
    ultimately show ?thesis
      by (rule *)
  qed
  then have "open (f ` (components U))"
    by (metis (no_types, lifting) imageE image_Union open_Union)
  then show ?thesis
    by force
qed

text‹No need for termS to be connected. But the nonconstant condition is stronger.›
corollarytag unimportant› open_mapping_thm2:
  assumes holf: "f holomorphic_on S"
      and S: "open S"
      and "open U" "U  S"
      and fnc: "X. open X; X  S; X  {}  ¬ f constant_on X"
    shows "open (f ` U)"
proof -
  have "S = (components S)" by simp
  with U  S have "U = (C  components S. C  U)" by auto
  then have "f ` U = (C  components S. f ` (C  U))"
    using image_UN by fastforce
  moreover
  { fix C assume "C  components S"
    with S C  components S open_components in_components_connected
    have C: "open C" "connected C" by auto
    have "C  S"
      by (metis C  components S in_components_maximal)
    have nf: "¬ f constant_on C"
      using open C C  components S C  S fnc in_components_nonempty by blast
    have "f holomorphic_on C"
      by (metis holf holomorphic_on_subset C  S)
    then have "open (f ` (C  U))"
      by (meson C open U inf_le1 nf open_Int open_mapping_thm)
  } ultimately show ?thesis
    by force
qed

corollarytag unimportant› open_mapping_thm3:
  assumes holf: "f holomorphic_on S"
      and "open S" and injf: "inj_on f S"
    shows  "open (f ` S)"
proof (rule open_mapping_thm2 [OF holf])
  show "X. open X; X  S; X  {}  ¬ f constant_on X"
    using inj_on_subset injective_not_constant injf by blast
qed (use assms in auto)

subsection‹Maximum modulus principle›

text‹If termf is holomorphic, then its norm (modulus) cannot exhibit a true local maximum that is
   properly within the domain of termf.›

proposition maximum_modulus_principle:
  assumes holf: "f holomorphic_on S"
      and S: "open S" and "connected S"
      and "open U" and "U  S" and "ξ  U"
      and no: "z. z  U  norm(f z)  norm(f ξ)"
    shows "f constant_on S"
proof (rule ccontr)
  assume "¬ f constant_on S"
  then have "open (f ` U)"
    using open_mapping_thm assms by blast
  moreover have "¬ open (f ` U)"
  proof -
    have "t. cmod (f ξ - t) < e  t  f ` U" if "0 < e" for e
      using that
      apply (rule_tac x="if 0 < Re(f ξ) then f ξ + (e/2) else f ξ - (e/2)" in exI)
      apply (simp add: dist_norm)
      apply (fastforce simp: cmod_Re_le_iff dest!: no dest: sym)
      done
    then show ?thesis
      unfolding open_contains_ball by (metis ξ  U contra_subsetD dist_norm imageI mem_ball)
  qed
  ultimately show False
    by blast
qed

proposition maximum_modulus_frontier:
  assumes holf: "f holomorphic_on (interior S)"
      and contf: "continuous_on (closure S) f"
      and bos: "bounded S"
      and leB: "z. z  frontier S  norm(f z)  B"
      and "ξ  S"
    shows "norm(f ξ)  B"
proof -
  have "compact (closure S)" using bos
    by (simp add: bounded_closure compact_eq_bounded_closed)
  moreover have "continuous_on (closure S) (cmod  f)"
    using contf continuous_on_compose continuous_on_norm_id by blast
  ultimately obtain z where "z  closure S" and z: "y. y  closure S  (cmod  f) y  (cmod  f) z"
    using continuous_attains_sup [of "closure S" "norm o f"] ξ  S by auto
  then consider "z  frontier S" | "z  interior S" using frontier_def by auto
  then have "norm(f z)  B"
  proof cases
    case 1 then show ?thesis using leB by blast
  next
    case 2
    have "f constant_on (connected_component_set (interior S) z)"
    proof (rule maximum_modulus_principle)
      show "f holomorphic_on connected_component_set (interior S) z"
        by (metis connected_component_subset holf holomorphic_on_subset)
      show zin: "z  connected_component_set (interior S) z"
        by (simp add: 2)
      show "W. W  connected_component_set (interior S) z  cmod (f W)  cmod (f z)"
        using closure_def connected_component_subset z by fastforce
    qed (auto simp: open_connected_component)
    then obtain c where c: "w. w  connected_component_set (interior S) z  f w = c"
      by (auto simp: constant_on_def)
    have "f ` closure(connected_component_set (interior S) z)  {c}"
    proof (rule image_closure_subset)
      show "continuous_on (closure (connected_component_set (interior S) z)) f"
        by (meson closure_mono connected_component_subset contf continuous_on_subset interior_subset)
    qed (use c in auto)
    then have cc: "w. w  closure(connected_component_set (interior S) z)  f w = c" by blast
    have "connected_component (interior S) z z"
      by (simp add: "2")
    moreover have "connected_component_set (interior S) z  UNIV"
      by (metis bos bounded_interior connected_component_eq_UNIV not_bounded_UNIV)
    ultimately have "frontier(connected_component_set (interior S) z)  {}"
      by (meson "2" connected_component_eq_empty frontier_not_empty)
    then obtain w where w: "w  frontier(connected_component_set (interior S) z)"
       by auto
    then have "norm (f z) = norm (f w)"  by (simp add: "2" c cc frontier_def)
    also have "...  B"
      using w frontier_interior_subset frontier_of_connected_component_subset 
      by (blast intro: leB)
    finally show ?thesis .
  qed
  then show ?thesis
    using z ξ  S closure_subset by fastforce
qed

corollarytag unimportant› maximum_real_frontier:
  assumes holf: "f holomorphic_on (interior S)"
      and contf: "continuous_on (closure S) f"
      and bos: "bounded S"
      and leB: "z. z  frontier S  Re(f z)  B"
      and "ξ  S"
    shows "Re(f ξ)  B"
using maximum_modulus_frontier [of "exp o f" S "exp B"]
      Transcendental.continuous_on_exp holomorphic_on_compose holomorphic_on_exp assms
by auto

subsectiontag unimportant› ‹Factoring out a zero according to its order›

lemma holomorphic_factor_order_of_zero:
  assumes holf: "f holomorphic_on S"
      and os: "open S"
      and "ξ  S" "0 < n"
      and dnz: "(deriv ^^ n) f ξ  0"
      and dfz: "i. 0 < i; i < n  (deriv ^^ i) f ξ = 0"
   obtains g r where "0 < r"
                "g holomorphic_on ball ξ r"
                "w. w  ball ξ r  f w - f ξ = (w - ξ)^n * g w"
                "w. w  ball ξ r  g w  0"
proof -
  obtain r where "r>0" and r: "ball ξ r  S" using assms by (blast elim!: openE)
  then have holfb: "f holomorphic_on ball ξ r"
    using holf holomorphic_on_subset by blast
  define g where "g w = suminf (λi. (deriv ^^ (i + n)) f ξ / (fact(i + n)) * (w - ξ)^i)" for w
  have sumsg: "(λi. (deriv ^^ (i + n)) f ξ / (fact(i + n)) * (w - ξ)^i) sums g w"
   and feq: "f w - f ξ = (w - ξ)^n * g w"
       if w: "w  ball ξ r" for w
  proof -
    define powf where "powf = (λi. (deriv ^^ i) f ξ/(fact i) * (w - ξ)^i)"
    have [simp]: "powf 0 = f ξ"
      by (simp add: powf_def)
    have sing: "{..<n} - {i. powf i = 0} = (if f ξ = 0 then {} else {0})"
      unfolding powf_def using 0 < n dfz by (auto simp: dfz; metis funpow_0 not_gr0)
    have "powf sums f w"
      unfolding powf_def by (rule holomorphic_power_series [OF holfb w])
    moreover have "(i<n. powf i) = f ξ"
      by (subst sum.setdiff_irrelevant [symmetric]; simp add: dfz sing)
    ultimately have fsums: "(λi. powf (i+n)) sums (f w - f ξ)"
      using w sums_iff_shift' by metis
    then have *: "summable (λi. (w - ξ) ^ n * ((deriv ^^ (i + n)) f ξ * (w - ξ) ^ i / fact (i + n)))"
      unfolding powf_def using sums_summable
      by (auto simp: power_add mult_ac)
    have "summable (λi. (deriv ^^ (i + n)) f ξ * (w - ξ) ^ i / fact (i + n))"
    proof (cases "w=ξ")
      case False then show ?thesis
        using summable_mult [OF *, of "1 / (w - ξ) ^ n"] by simp
    next
      case True then show ?thesis
        by (auto simp: Power.semiring_1_class.power_0_left intro!: summable_finite [of "{0}"]
                 split: if_split_asm)
    qed
    then show sumsg: "(λi. (deriv ^^ (i + n)) f ξ / (fact(i + n)) * (w - ξ)^i) sums g w"
      by (simp add: summable_sums_iff g_def)
    show "f w - f ξ = (w - ξ)^n * g w"
      using sums_mult [OF sumsg, of "(w - ξ) ^ n"]
      by (intro sums_unique2 [OF fsums]) (auto simp: power_add mult_ac powf_def)
  qed
  then have holg: "g holomorphic_on ball ξ r"
    by (meson sumsg power_series_holomorphic)
  then have contg: "continuous_on (ball ξ r) g"
    by (blast intro: holomorphic_on_imp_continuous_on)
  have "g ξ  0"
    using dnz unfolding g_def
    by (subst suminf_finite [of "{0}"]) auto
  obtain d where "0 < d" and d: "w. w  ball ξ d  g w  0"
    using 0 < r continuous_on_avoid [OF contg _ g ξ  0]
    by (metis centre_in_ball le_cases mem_ball mem_ball_leI) 
  show ?thesis
  proof
    show "g holomorphic_on ball ξ (min r d)"
      using holg by (auto simp: feq holomorphic_on_subset subset_ball d)
  qed (use 0 < r 0 < d in auto simp: feq d)
qed


lemma holomorphic_factor_order_of_zero_strong:
  assumes holf: "f holomorphic_on S" "open S"  "ξ  S" "0 < n"
      and "(deriv ^^ n) f ξ  0"
      and "i. 0 < i; i < n  (deriv ^^ i) f ξ = 0"
   obtains g r where "0 < r"
                "g holomorphic_on ball ξ r"
                "w. w  ball ξ r  f w - f ξ = ((w - ξ) * g w) ^ n"
                "w. w  ball ξ r  g w  0"
proof -
  obtain g r where "0 < r"
               and holg: "g holomorphic_on ball ξ r"
               and feq: "w. w  ball ξ r  f w - f ξ = (w - ξ)^n * g w"
               and gne: "w. w  ball ξ r  g w  0"
    by (auto intro: holomorphic_factor_order_of_zero [OF assms])
  have con: "continuous_on (ball ξ r) (λz. deriv g z / g z)"
    by (rule continuous_intros) (auto simp: gne holg holomorphic_deriv holomorphic_on_imp_continuous_on)
  have cd: "(λz. deriv g z / g z) field_differentiable at x" if "dist ξ x < r" for x
  proof (intro derivative_intros)
    show "deriv g field_differentiable at x"
      using that holg mem_ball by (blast intro: holomorphic_deriv holomorphic_on_imp_differentiable_at)
    show "g field_differentiable at x"
      by (metis that open_ball at_within_open holg holomorphic_on_def mem_ball)
    qed (simp add: gne that)
    obtain h where h: "x. x  ball ξ r  (h has_field_derivative deriv g x / g x) (at x)"
      using holomorphic_convex_primitive [of "ball ξ r" "{}" "λz. deriv g z / g z"]
      by (metis (no_types, lifting) Diff_empty at_within_interior cd con convex_ball infinite_imp_nonempty interior_ball mem_ball)
  then have "continuous_on (ball ξ r) h"
    by (metis open_ball holomorphic_on_imp_continuous_on holomorphic_on_open)
  then have con: "continuous_on (ball ξ r) (λx. exp (h x) / g x)"
    by (auto intro!: continuous_intros simp add: holg holomorphic_on_imp_continuous_on gne)
  have 0: "dist ξ x < r  ((λx. exp (h x) / g x) has_field_derivative 0) (at x)" for x
    apply (rule h derivative_eq_intros DERIV_deriv_iff_field_differentiable [THEN iffD2] | simp)+
    using holg by (auto simp: holomorphic_on_imp_differentiable_at gne h)
  obtain c where c: "x. x  ball ξ r  exp (h x) / g x = c"
    by (rule DERIV_zero_connected_constant [of "ball ξ r" "{}" "λx. exp(h x) / g x"]) (auto simp: con 0)
  have hol: "(λz. exp ((Ln (inverse c) + h z) / of_nat n)) holomorphic_on ball ξ r"
  proof (intro holomorphic_intros holomorphic_on_compose [unfolded o_def, where g = exp])
    show "h holomorphic_on ball ξ r"
      using h holomorphic_on_open by blast
  qed (use 0 < n in auto)
  show ?thesis
  proof
    show "w. w  ball ξ r  f w - f ξ = ((w - ξ) * exp ((Ln (inverse c) + h w) / of_nat n)) ^ n"
      using 0 < n
      by (auto simp: feq power_mult_distrib exp_divide_power_eq exp_add gne simp flip: c)
  qed (use hol 0 < r in auto)
qed


lemma
  fixes k :: "'a::wellorder"
  assumes a_def: "a == LEAST x. P x" and P: "P k"
  shows def_LeastI: "P a" and def_Least_le: "a  k"
unfolding a_def
by (rule LeastI Least_le; rule P)+

lemma holomorphic_factor_zero_nonconstant:
  assumes holf: "f holomorphic_on S" and S: "open S" "connected S"
      and "ξ  S" "f ξ = 0"
      and nonconst: "¬ f constant_on S"
   obtains g r n
      where "0 < n"  "0 < r"  "ball ξ r  S"
            "g holomorphic_on ball ξ r"
            "w. w  ball ξ r  f w = (w - ξ)^n * g w"
            "w. w  ball ξ r  g w  0"
proof (cases "n>0. (deriv ^^ n) f ξ = 0")
  case True then show ?thesis
    using holomorphic_fun_eq_const_on_connected [OF holf S _ ξ  S] nonconst by (simp add: constant_on_def)
next
  case False
  then obtain n0 where "n0 > 0" and n0: "(deriv ^^ n0) f ξ  0" by blast
  obtain r0 where "r0 > 0" "ball ξ r0  S" using S openE ξ  S by auto
  define n where "n  LEAST n. (deriv ^^ n) f ξ  0"
  have n_ne: "(deriv ^^ n) f ξ  0"
    by (rule def_LeastI [OF n_def]) (rule n0)
  then have "0 < n" using f ξ = 0
    using funpow_0 by fastforce
  have n_min: "k. k < n  (deriv ^^ k) f ξ = 0"
    using def_Least_le [OF n_def] not_le by blast
  then obtain g r1
    where g: "0 < r1" "g holomorphic_on ball ξ r1"
          and geq: "w. w  ball ξ r1  f w = (w - ξ) ^ n * g w"
          and g0: "w. w  ball ξ r1  g w  0"
    by (auto intro: holomorphic_factor_order_of_zero [OF holf open S ξ  S n > 0 n_ne] simp: f ξ = 0)
  show ?thesis
  proof
    show "g holomorphic_on ball ξ (min r0 r1)"
      using g by auto
    show "w. w  ball ξ (min r0 r1)  f w = (w - ξ) ^ n * g w"
      by (simp add: geq)
  qed (use 0 < n 0 < r0 0 < r1 ball ξ r0  S g0 in auto)
qed


lemma holomorphic_lower_bound_difference:
  assumes holf: "f holomorphic_on S" and S: "open S" "connected S"
      and "ξ  S" and "φ  S"
      and fne: "f φ  f ξ"
   obtains k n r
      where "0 < k"  "0 < r"
            "ball ξ r  S"
            "w. w  ball ξ r  k * norm(w - ξ)^n  norm(f w - f ξ)"
proof -
  define n where "n = (LEAST n. 0 < n  (deriv ^^ n) f ξ  0)"
  obtain n0 where "0 < n0" and n0: "(deriv ^^ n0) f ξ  0"
    using fne holomorphic_fun_eq_const_on_connected [OF holf S] ξ  S φ  S by blast
  then have "0 < n" and n_ne: "(deriv ^^ n) f ξ  0"
    unfolding n_def by (metis (mono_tags, lifting) LeastI)+
  have n_min: "k. 0 < k; k < n  (deriv ^^ k) f ξ = 0"
    unfolding n_def by (blast dest: not_less_Least)
  then obtain g r
    where "0 < r" and holg: "g holomorphic_on ball ξ r"
      and fne: "w. w  ball ξ r  f w - f ξ = (w - ξ) ^ n * g w"
      and gnz: "w. w  ball ξ r  g w  0"
      by (auto intro: holomorphic_factor_order_of_zero  [OF holf open S ξ  S n > 0 n_ne])
  obtain e where "e>0" and e: "ball ξ e  S" using assms by (blast elim!: openE)
  then have holfb: "f holomorphic_on ball ξ e"
    using holf holomorphic_on_subset by blast
  define d where "d = (min e r) / 2"
  have "0 < d" using 0 < r 0 < e by (simp add: d_def)
  have "d < r"
    using 0 < r by (auto simp: d_def)
  then have cbb: "cball ξ d  ball ξ r"
    by (auto simp: cball_subset_ball_iff)
  then have "g holomorphic_on cball ξ d"
    by (rule holomorphic_on_subset [OF holg])
  then have "closed (g ` cball ξ d)"
    by (simp add: compact_imp_closed compact_continuous_image holomorphic_on_imp_continuous_on)
  moreover have "g ` cball ξ d  {}"
    using 0 < d by auto
  ultimately obtain x where x: "x  g ` cball ξ d" and "y. y  g ` cball ξ d  dist 0 x  dist 0 y"
    by (rule distance_attains_inf) blast
  then have leg: "w. w  cball ξ d  norm x  norm (g w)"
    by auto
  have "ball ξ d  cball ξ d" by auto
  also have "...  ball ξ e" using 0 < d d_def by auto
  also have "...  S" by (rule e)
  finally have dS: "ball ξ d  S" .
  have "x  0" using gnz x d < r by auto
  show thesis
  proof
    show "w. w  ball ξ d  cmod x * cmod (w - ξ) ^ n  cmod (f w - f ξ)"
      using d < r leg by (auto simp: fne norm_mult norm_power algebra_simps mult_right_mono)
  qed (use dS x  0 d > 0 in auto)
qed

lemma
  assumes holf: "f holomorphic_on (S - {ξ})" and ξ: "ξ  interior S"
    shows holomorphic_on_extend_lim:
          "(g. g holomorphic_on S  (z  S - {ξ}. g z = f z)) 
           ((λz. (z - ξ) * f z)  0) (at ξ)"
          (is "?P = ?Q")
     and holomorphic_on_extend_bounded:
          "(g. g holomorphic_on S  (z  S - {ξ}. g z = f z)) 
           (B. eventually (λz. norm(f z)  B) (at ξ))"
          (is "?P = ?R")
proof -
  obtain δ where "0 < δ" and δ: "ball ξ δ  S"
    using ξ mem_interior by blast
  have "?R" if holg: "g holomorphic_on S" and gf: "z. z  S - {ξ}  g z = f z" for g
  proof -
    have §: "cmod (f x)  cmod (g ξ) + 1" if "x  ξ" "dist x ξ < δ" "dist (g x) (g ξ) < 1" for x
    proof -
      have "x  S"
        by (metis δ dist_commute mem_ball subsetD that(2))
      with that gf [of x] show ?thesis
        using norm_triangle_ineq2 [of "f x" "g ξ"] dist_complex_def by auto
    qed
    then have *: "F z in at ξ. dist (g z) (g ξ) < 1  cmod (f z)  cmod (g ξ) + 1"
      using 0 < δ eventually_at by blast
    have "continuous_on (interior S) g"
      by (meson continuous_on_subset holg holomorphic_on_imp_continuous_on interior_subset)
    then have "x. x  interior S  (g  g x) (at x)"
      using continuous_on_interior continuous_within holg holomorphic_on_imp_continuous_on by blast
    then have "(g  g ξ) (at ξ)"
      by (simp add: ξ)
    then show ?thesis
      apply (rule_tac x="norm(g ξ) + 1" in exI)
      apply (rule eventually_mp [OF * tendstoD [where e=1]], auto)
      done
  qed
  moreover have "?Q" if "F z in at ξ. cmod (f z)  B" for B
    by (rule lim_null_mult_right_bounded [OF _ that]) (simp add: LIM_zero)
  moreover have "?P" if "(λz. (z - ξ) * f z) ξ 0"
  proof -
    define h where [abs_def]: "h z = (z - ξ)^2 * f z" for z
    have h0: "(h has_field_derivative 0) (at ξ)"
      apply (simp add: h_def has_field_derivative_iff)
      apply (auto simp: field_split_simps power2_eq_square Lim_transform_within [OF that, of 1])
      done
    have holh: "h holomorphic_on S"
    proof (simp add: holomorphic_on_def, clarify)
      fix z assume "z  S"
      show "h field_differentiable at z within S"
      proof (cases "z = ξ")
        case True then show ?thesis
          using field_differentiable_at_within field_differentiable_def h0 by blast
      next
        case False
        then have "f field_differentiable at z within S"
          using holomorphic_onD [OF holf, of z] z  S
          unfolding field_differentiable_def has_field_derivative_iff
          by (force intro: exI [where x="dist ξ z"] elim: Lim_transform_within_set [unfolded eventually_at])
        then show ?thesis
          by (simp add: h_def power2_eq_square derivative_intros)
      qed
    qed
    define g where [abs_def]: "g z = (if z = ξ then deriv h ξ else (h z - h ξ) / (z - ξ))" for z
    have holg: "g holomorphic_on S"
      unfolding g_def by (rule pole_lemma [OF holh ξ])
    have §: "zS - {ξ}. (g z - g ξ) / (z - ξ) = f z"
      using h0 by (auto simp: g_def power2_eq_square divide_simps DERIV_imp_deriv h_def)
    show ?thesis
      apply (intro exI conjI)
       apply (rule pole_lemma [OF holg ξ])
      apply (simp add: §)
      done
  qed
  ultimately show "?P = ?Q" and "?P = ?R"
    by meson+
qed

lemma pole_at_infinity:
  assumes holf: "f holomorphic_on UNIV" and lim: "((inverse o f)  l) at_infinity"
  obtains a n where "z. f z = (in. a i * z^i)"
proof (cases "l = 0")
  case False
  show thesis
  proof
    show "f z = (i0. inverse l * z ^ i)" for z
      using False tendsto_inverse [OF lim] by (simp add: Liouville_weak [OF holf])
  qed
next
  case True
  then have [simp]: "l = 0" .
  show ?thesis
  proof (cases "r. 0 < r  (z  ball 0 r - {0}. f(inverse z)  0)")
    case True
      then obtain r where "0 < r" and r: "z. z  ball 0 r - {0}  f(inverse z)  0"
             by auto
      have 1: "inverse  f  inverse holomorphic_on ball 0 r - {0}"
        by (rule holomorphic_on_compose holomorphic_intros holomorphic_on_subset [OF holf] | force simp: r)+
      have 2: "0  interior (ball 0 r)"
        using 0 < r by simp
      have "B. 0<B  eventually (λz. cmod ((inverse  f  inverse) z)  B) (at 0)"
        apply (rule exI [where x=1])
        using tendstoD [OF lim [unfolded lim_at_infinity_0] zero_less_one]
        by (simp add: eventually_mono)
      with holomorphic_on_extend_bounded [OF 1 2]
      obtain g where holg: "g holomorphic_on ball 0 r"
                 and geq: "z. z  ball 0 r - {0}  g z = (inverse  f  inverse) z"
        by meson
      have ifi0: "(inverse  f  inverse) 0 0"
        using l = 0 lim lim_at_infinity_0 by blast
      have g2g0: "g 0 g 0"
        using 0 < r centre_in_ball continuous_at continuous_on_eq_continuous_at holg
        by (blast intro: holomorphic_on_imp_continuous_on)
      have g2g1: "g 0 0"
        apply (rule Lim_transform_within_open [OF ifi0 open_ball [of 0 r]])
        using 0 < r by (auto simp: geq)
      have [simp]: "g 0 = 0"
        by (rule tendsto_unique [OF _ g2g0 g2g1]) simp
      have "ball 0 r - {0::complex}  {}"
        using 0 < r by (metis "2" Diff_iff insert_Diff interior_ball interior_singleton)
      then obtain w::complex where "w  0" and w: "norm w < r" by force
      then have "g w  0" by (simp add: geq r)
      obtain B n e where "0 < B" "0 < e" "e  r"
                     and leg: "w. norm w < e  B * cmod w ^ n  cmod (g w)"
      proof (rule holomorphic_lower_bound_difference [OF holg open_ball connected_ball])
        show "g w  g 0"
          by (simp add: g w  0)
        show "w  ball 0 r"
          using mem_ball_0 w by blast
      qed (use 0 < r in auto simp: ball_subset_ball_iff)
      have §: "cmod (f z)  cmod z ^ n / B" if "2/e  cmod z" for z
      proof -
        have ize: "inverse z  ball 0 e - {0}" using that 0 < e
          by (auto simp: norm_divide field_split_simps algebra_simps)
        then have [simp]: "z  0" and izr: "inverse z  ball 0 r - {0}" using  e  r
          by auto
        then have [simp]: "f z  0"
          using r [of "inverse z"] by simp
        have [simp]: "f z = inverse (g (inverse z))"
          using izr geq [of "inverse z"] by simp
        show ?thesis using ize leg [of "inverse z"]  0 < B  0 < e
          by (simp add: field_split_simps norm_divide algebra_simps)
      qed
      show thesis
      proof
        show "f z = (in. (deriv ^^ i) f 0 / fact i * z ^ i)" for z
          using § by (rule_tac A = "2/e" and B = "1/B" in Liouville_polynomial [OF holf], simp)
      qed
  next
    case False
    then have fi0: "r. r > 0  zball 0 r - {0}. f (inverse z) = 0"
      by simp
    have fz0: "f z = 0" if "0 < r" and lt1: "x. x  0  cmod x < r  inverse (cmod (f (inverse x))) < 1"
              for z r
    proof -
      have f0: "(f  0) at_infinity"
      proof -
        have DIM_complex[intro]: "2  DIM(complex)"  ― ‹should not be necessary!›
          by simp
        have "f (inverse x)  0  x  0  cmod x < r  1 < cmod (f (inverse x))" for x
          using lt1[of x] by (auto simp: field_simps)
        then have **: "cmod (f (inverse x))  1  x  0  cmod x < r  f (inverse x) = 0" for x
          by force
        then have *: "(f  inverse) ` (ball 0 r - {0})  {0}  - ball 0 1"
          by force
        have "continuous_on (inverse ` (ball 0 r - {0})) f"
          using continuous_on_subset holf holomorphic_on_imp_continuous_on by blast
        then have "connected ((f  inverse) ` (ball 0 r - {0}))"
          using connected_punctured_ball
          by (intro connected_continuous_image continuous_intros; force)
        then have "{0}  (f  inverse) ` (ball 0 r - {0}) = {}  - ball 0 1  (f  inverse) ` (ball 0 r - {0}) = {}"
          by (rule connected_closedD) (use * in auto)
        then have "f (inverse w) = 0" if "w  0" "cmod w < r" for w
          using **[of w] fi0 0 < r  that by force
        then show ?thesis
          unfolding lim_at_infinity_0
          using eventually_at r > 0 by (force simp add: intro: tendsto_eventually)
      qed
      obtain w where "w  ball 0 r - {0}" and "f (inverse w) = 0"
        using False 0 < r by blast
      then show ?thesis
        by (auto simp: f0 Liouville_weak [OF holf, of 0])
    qed
    show thesis
    proof
      show "z. f z = (i0. 0 * z ^ i)"
        using lim 
        apply (simp add: lim_at_infinity_0 Lim_at dist_norm norm_inverse)
        using fz0 zero_less_one by blast
    qed
  qed
qed

subsectiontag unimportant› ‹Entire proper functions are precisely the non-trivial polynomials›

lemma proper_map_polyfun:
    fixes c :: "nat  'a::{real_normed_div_algebra,heine_borel}"
  assumes "closed S" and "compact K" and c: "c i  0" "1  i" "i  n"
    shows "compact (S  {z. (in. c i * z^i)  K})"
proof -
  obtain B where "B > 0" and B: "x. x  K  norm x  B"
    by (metis compact_imp_bounded compact K bounded_pos)
  have *: "norm x  b"
            if "x. b  norm x  B + 1  norm (in. c i * x ^ i)"
               "(in. c i * x ^ i)  K"  for b x
  proof -
    have "norm (in. c i * x ^ i)  B"
      using B that by blast
    moreover have "¬ B + 1  B"
      by simp
    ultimately show "norm x  b"
      using that by (metis (no_types) less_eq_real_def not_less order_trans)
  qed
  have "bounded {z. (in. c i * z ^ i)  K}"
    using Limits.polyfun_extremal [where c=c and B="B+1", OF c]
    by (auto simp: bounded_pos eventually_at_infinity_pos *)
  moreover have "closed ((λz. (in. c i * z ^ i)) -` K)"
    using compact K compact_eq_bounded_closed
    by (intro allI continuous_closed_vimage continuous_intros; force)
  ultimately show ?thesis
    using closed_Int_compact [OF closed S] compact_eq_bounded_closed
    by (auto simp add: vimage_def)
qed

lemma proper_map_polyfun_univ:
    fixes c :: "nat  'a::{real_normed_div_algebra,heine_borel}"
  assumes "compact K" "c i  0" "1  i" "i  n"
    shows "compact ({z. (in. c i * z^i)  K})"
  using proper_map_polyfun [of UNIV K c i n] assms by simp

lemma proper_map_polyfun_eq:
  assumes "f holomorphic_on UNIV"
    shows "(k. compact k  compact {z. f z  k}) 
           (c n. 0 < n  (c n  0)  f = (λz. in. c i * z^i))"
          (is "?lhs = ?rhs")
proof
  assume compf [rule_format]: ?lhs
  have 2: "k. 0 < k  a k  0  f = (λz. i  k. a i * z ^ i)"
        if "z. f z = (in. a i * z ^ i)" for a n
  proof (cases "in. 0<i  a i = 0")
    case True
    then have [simp]: "z. f z = a 0"
      by (simp add: that sum.atMost_shift)
    have False using compf [of "{a 0}"] by simp
    then show ?thesis ..
  next
    case False
    then obtain k where k: "0 < k" "kn" "a k  0" by force
    define m where "m = (GREATEST k. kn  a k  0)"
    have m: "mn  a m  0"
      unfolding m_def
      using GreatestI_nat [where b = n] k by (metis (mono_tags, lifting))
    have [simp]: "a i = 0" if "m < i" "i  n" for i
      using Greatest_le_nat [where b = "n" and P = "λk. kn  a k  0"]
      using m_def not_le that by auto
    have "k  m"
      unfolding m_def
      using Greatest_le_nat [where b = n] k by (metis (mono_tags, lifting))
    with k m show ?thesis
      by (rule_tac x=m in exI) (auto simp: that comm_monoid_add_class.sum.mono_neutral_right)
  qed
  have §: "((inverse  f)  0) at_infinity"
  proof (rule Lim_at_infinityI)
    fix e::real assume "0 < e"
    with compf [of "cball 0 (inverse e)"]
    show "B. x. B  cmod x  dist ((inverse  f) x) 0  e"
      apply simp
      apply (clarsimp simp add: compact_eq_bounded_closed bounded_pos norm_inverse)
      by (metis (no_types, opaque_lifting) inverse_inverse_eq le_less_trans less_eq_real_def less_imp_inverse_less linordered_field_no_ub not_less)
  qed
  then obtain a n where "z. f z = (in. a i * z^i)"
    using assms pole_at_infinity by blast
  with § 2 show ?rhs by blast
next
  assume ?rhs
  then obtain c n where "0 < n" "c n  0" "f = (λz. in. c i * z ^ i)" by blast
  then have "compact {z. f z  k}" if "compact k" for k
    by (auto intro: proper_map_polyfun_univ [OF that])
  then show ?lhs by blast
qed

subsection ‹Relating invertibility and nonvanishing of derivative›

lemma has_complex_derivative_locally_injective:
  assumes holf: "f holomorphic_on S"
      and S: "ξ  S" "open S"
      and dnz: "deriv f ξ  0"
  obtains r where "r > 0" "ball ξ r  S" "inj_on f (ball ξ r)"
proof -
  have *: "d>0. x. dist ξ x < d  onorm (λv. deriv f x * v - deriv f ξ * v) < e" if "e > 0" for e
  proof -
    have contdf: "continuous_on S (deriv f)"
      by (simp add: holf holomorphic_deriv holomorphic_on_imp_continuous_on open S)
    obtain δ where "δ>0" and δ: "x. x  S; dist x ξ  δ  cmod (deriv f x - deriv f ξ)  e/2"
      using continuous_onE [OF contdf ξ  S, of "e/2"] 0 < e
      by (metis dist_complex_def half_gt_zero less_imp_le)
    have §: "ζ z. ζ  S; dist ξ ζ < δ  cmod (deriv f ζ - deriv f ξ) * cmod z  e/2 * cmod z"
      by (intro mult_right_mono [OF δ]) (auto simp: dist_commute)
    obtain ε where "ε>0" "ball ξ ε  S"
      by (metis openE [OF open S ξ  S])
    with δ>0 have "δ>0. x. dist ξ x < δ  onorm (λv. deriv f x * v - deriv f ξ * v)  e/2"
      using §
      apply (rule_tac x="min δ ε" in exI)
      apply (intro conjI allI impI Operator_Norm.onorm_le)
      apply (force simp: mult_right_mono norm_mult [symmetric] left_diff_distrib δ)+
      done
    with e>0 show ?thesis by force
  qed
  have "inj ((*) (deriv f ξ))"
    using dnz by simp
  then obtain g' where g': "linear g'" "g'  (*) (deriv f ξ) = id"
    using linear_injective_left_inverse [of "(*) (deriv f ξ)"] by auto
  show ?thesis
    apply (rule has_derivative_locally_injective [OF S, where f=f and f' = "λz h. deriv f z * h" and g' = g'])
    using g' * 
    apply (simp_all add: linear_conv_bounded_linear that)
    using open S has_field_derivative_imp_has_derivative holf holomorphic_derivI by blast
qed

lemma has_complex_derivative_locally_invertible:
  assumes holf: "f holomorphic_on S"
      and S: "ξ  S" "open S"
      and dnz: "deriv f ξ  0"
  obtains r where "r > 0" "ball ξ r  S" "open (f `  (ball ξ r))" "inj_on f (ball ξ r)"
proof -
  obtain r where "r > 0" "ball ξ r  S" "inj_on f (ball ξ r)"
    by (blast intro: that has_complex_derivative_locally_injective [OF assms])
  then have ξ: "ξ  ball ξ r" by simp
  then have nc: "¬ f constant_on ball ξ r"
    using inj_on f (ball ξ r) injective_not_constant by fastforce
  have holf': "f holomorphic_on ball ξ r"
    using ball ξ r  S holf holomorphic_on_subset by blast
  have "open (f ` ball ξ r)"
    by (simp add: inj_on f (ball ξ r) holf' open_mapping_thm3)
  then show ?thesis
    using 0 < r ball ξ r  S inj_on f (ball ξ r) that  by blast
qed

lemma holomorphic_injective_imp_regular:
  assumes holf: "f holomorphic_on S"
      and "open S" and injf: "inj_on f S"
      and "ξ  S"
    shows "deriv f ξ  0"
proof -
  obtain r where "r>0" and r: "ball ξ r  S" using assms by (blast elim!: openE)
  have holf': "f holomorphic_on ball ξ r"
    using ball ξ r  S holf holomorphic_on_subset by blast
  show ?thesis
  proof (cases "n>0. (deriv ^^ n) f ξ = 0")
    case True
    have fcon: "f w = f ξ" if "w  ball ξ r" for w
      by (meson open_ball True 0 < r centre_in_ball connected_ball holf' 
                holomorphic_fun_eq_const_on_connected that)
    have False
      using fcon [of "ξ + r/2"] 0 < r r injf unfolding inj_on_def
      by (metis ξ  S contra_subsetD dist_commute fcon mem_ball perfect_choose_dist)
    then show ?thesis ..
  next
    case False
    then obtain n0 where n0: "n0 > 0  (deriv ^^ n0) f ξ  0" by blast
    define n where [abs_def]: "n = (LEAST n. n > 0  (deriv ^^ n) f ξ  0)"
    have n_ne: "n > 0" "(deriv ^^ n) f ξ  0"
      using def_LeastI [OF n_def n0] by auto
    have n_min: "k. 0 < k  k < n  (deriv ^^ k) f ξ = 0"
      using def_Least_le [OF n_def] not_le by auto
    obtain g δ where "0 < δ"
             and holg: "g holomorphic_on ball ξ δ"
             and fd: "w. w  ball ξ δ  f w - f ξ = ((w - ξ) * g w) ^ n"
             and gnz: "w. w  ball ξ δ  g w  0"
      by (blast intro: n_min holomorphic_factor_order_of_zero_strong [OF holf open S ξ  S n_ne])
    show ?thesis
    proof (cases "n=1")
      case True
      with n_ne show ?thesis by auto
    next
      case False
      have "g holomorphic_on ball ξ (min r δ)"
        using holg by (simp add: holomorphic_on_subset subset_ball)
      then have holgw: "(λw. (w - ξ) * g w) holomorphic_on ball ξ (min r δ)"
        by (intro holomorphic_intros)
      have gd: "w. dist ξ w < δ  (g has_field_derivative deriv g w) (at w)"
        using holg
        by (simp add: DERIV_deriv_iff_field_differentiable holomorphic_on_def at_within_open_NO_MATCH)
      have *: "w. w  ball ξ (min r δ)
             ((λw. (w - ξ) * g w) has_field_derivative ((w - ξ) * deriv g w + g w))
                (at w)"
        by (rule gd derivative_eq_intros | simp)+
      have [simp]: "deriv (λw. (w - ξ) * g w) ξ  0"
        using * [of ξ] 0 < δ 0 < r by (simp add: DERIV_imp_deriv gnz)
      obtain T where "ξ  T" "open T" and Tsb: "T  ball ξ (min r δ)" and oimT: "open ((λw. (w - ξ) * g w) ` T)"
        using 0 < r 0 < δ has_complex_derivative_locally_invertible [OF holgw, of ξ]
        by force
      define U where "U = (λw. (w - ξ) * g w) ` T"
      have "open U" by (metis oimT U_def)
      moreover have "0  U"
        using ξ  T by (auto simp: U_def intro: image_eqI [where x = ξ])
      ultimately obtain ε where "ε>0" and ε: "cball 0 ε  U"
        using open U open_contains_cball by blast
      then have "ε * exp(2 * of_real pi * 𝗂 * (0/n))  cball 0 ε"
                "ε * exp(2 * of_real pi * 𝗂 * (1/n))  cball 0 ε"
        by (auto simp: norm_mult)
      with ε have "ε * exp(2 * of_real pi * 𝗂 * (0/n))  U"
                  "ε * exp(2 * of_real pi * 𝗂 * (1/n))  U" by blast+
      then obtain y0 y1 where "y0  T" and y0: "(y0 - ξ) * g y0 = ε * exp(2 * of_real pi * 𝗂 * (0/n))"
                          and "y1  T" and y1: "(y1 - ξ) * g y1 = ε * exp(2 * of_real pi * 𝗂 * (1/n))"
        by (auto simp: U_def)
      then have "y0  ball ξ δ" "y1  ball ξ δ" using Tsb by auto
      then have "f y0 - f ξ = ((y0 - ξ) * g y0) ^ n" "f y1 - f ξ = ((y1 - ξ) * g y1) ^ n"
        using fd by blast+
      moreover have "y0  y1"
        using y0 y1 ε > 0 complex_root_unity_eq_1 [of n 1] n > 0 False by auto
      moreover have "T  S"
        by (meson Tsb min.cobounded1 order_trans r subset_ball)
      ultimately have False
        using inj_onD [OF injf, of y0 y1] y0  T y1  T
        using complex_root_unity [of n 1] 
        apply (simp add: y0 y1 power_mult_distrib)
        apply (force simp: algebra_simps)
        done
      then show ?thesis ..
    qed
  qed
qed

subsubsection ‹Hence a nice clean inverse function theorem›

lemma has_field_derivative_inverse_strong:
  fixes f :: "'a::{euclidean_space,real_normed_field}  'a"
  shows "DERIV f x :> f'; f'  0; open S; x  S; continuous_on S f;
         z. z  S  g (f z) = z
          DERIV g (f x) :> inverse (f')"
  unfolding has_field_derivative_def
  by (rule has_derivative_inverse_strong [of S x f g]) auto

lemma has_field_derivative_inverse_strong_x:
  fixes f :: "'a::{euclidean_space,real_normed_field}  'a"
  shows  "DERIV f (g y) :> f'; f'  0; open S; continuous_on S f; g y  S; f(g y) = y;
           z. z  S  g (f z) = z
           DERIV g y :> inverse (f')"
  unfolding has_field_derivative_def
  by (rule has_derivative_inverse_strong_x [of S g y f]) auto

proposition holomorphic_has_inverse:
  assumes holf: "f holomorphic_on S"
      and "open S" and injf: "inj_on f S"
  obtains g where "g holomorphic_on (f ` S)"
                  "z. z  S  deriv f z * deriv g (f z) = 1"
                  "z. z  S  g(f z) = z"
proof -
  have ofs: "open (f ` S)"
    by (rule open_mapping_thm3 [OF assms])
  have contf: "continuous_on S f"
    by (simp add: holf holomorphic_on_imp_continuous_on)
  have *: "(the_inv_into S f has_field_derivative inverse (deriv f z)) (at (f z))" if "z  S" for z
  proof -
    have 1: "(f has_field_derivative deriv f z) (at z)"
      using DERIV_deriv_iff_field_differentiable z  S open S holf holomorphic_on_imp_differentiable_at
      by blast
    have 2: "deriv f z  0"
      using z  S open S holf holomorphic_injective_imp_regular injf by blast
    show ?thesis
    proof (rule has_field_derivative_inverse_strong [OF 1 2 open S z  S])
      show "continuous_on S f"
        by (simp add: holf holomorphic_on_imp_continuous_on)
      show "z. z  S  the_inv_into S f (f z) = z"
        by (simp add: injf the_inv_into_f_f)
    qed
  qed
  show ?thesis
    proof
      show "the_inv_into S f holomorphic_on f ` S"
        by (simp add: holomorphic_on_open ofs) (blast intro: *)
    next
      fix z assume "z  S"
      have "deriv f z  0"
        using z  S open S holf holomorphic_injective_imp_regular injf by blast
      then show "deriv f z * deriv (the_inv_into S f) (f z) = 1"
        using * [OF z  S]  by (simp add: DERIV_imp_deriv)
    next
      fix z assume "z  S"
      show "the_inv_into S f (f z) = z"
        by (simp add: z  S injf the_inv_into_f_f)
  qed
qed

subsubsection ‹ Holomorphism of covering maps and lifts.›

lemma covering_space_lift_is_holomorphic:
  assumes cov: "covering_space C p S"
      and C: "open C" "p holomorphic_on C"
      and holf: "f holomorphic_on U" and fim: "f  U  S" and gim: "g  U  C"
      and contg: "continuous_on U g" and pg_f: "x. x  U  p(g x) = f x"
    shows "g holomorphic_on U"
  unfolding holomorphic_on_def
proof (intro strip)
  fix z
  assume "z  U"
  with fim have "f z  S" by blast
  then obtain T 𝒱 where "f z  T" and opeT: "openin (top_of_set S) T" 
        and UV: "𝒱 = C  p -` T" 
        and <