Theory Great_Picard

theory Great_Picard
imports Conformal_Mappings Further_Topology
section‹The Great Picard Theorem and its Applications›

text‹Ported from HOL Light (cauchy.ml) by L C Paulson, 2017›

theory Great_Picard
  imports Conformal_Mappings Further_Topology

begin
  
subsection‹Schottky's theorem›

lemma Schottky_lemma0:
  assumes holf: "f holomorphic_on S" and cons: "contractible S" and "a ∈ S"
      and f: "⋀z. z ∈ S ⟹ f z ≠ 1 ∧ f z ≠ -1"
  obtains g where "g holomorphic_on S"
                  "norm(g a) ≤ 1 + norm(f a) / 3"
                  "⋀z. z ∈ S ⟹ f z = cos(of_real pi * g z)"
proof -
  obtain g where holg: "g holomorphic_on S" and g: "norm(g a) ≤ pi + norm(f a)"
             and f_eq_cos: "⋀z. z ∈ S ⟹ f z = cos(g z)"
    using contractible_imp_holomorphic_arccos_bounded [OF assms]
    by blast
  show ?thesis
  proof
    show "(λz. g z / pi) holomorphic_on S"
      by (auto intro: holomorphic_intros holg)
    have "3 ≤ pi"
      using pi_approx by force
    have "3 * norm(g a) ≤ 3 * (pi + norm(f a))"
      using g by auto
    also have "... ≤  pi * 3 + pi * cmod (f a)"
      using ‹3 ≤ pi› by (simp add: mult_right_mono algebra_simps)
    finally show "cmod (g a / complex_of_real pi) ≤ 1 + cmod (f a) / 3"
      by (simp add: field_simps norm_divide)
    show "⋀z. z ∈ S ⟹ f z = cos (complex_of_real pi * (g z / complex_of_real pi))"
      by (simp add: f_eq_cos)
  qed
qed


lemma Schottky_lemma1:
  fixes n::nat
  assumes "0 < n"
  shows "0 < n + sqrt(real n ^ 2 - 1)"
proof -
  have "(n-1)^2 ≤ n^2 - 1"
    using assms by (simp add: algebra_simps power2_eq_square)
  then have "real (n - 1) ≤ sqrt (real (n2 - 1))"
    by (metis of_nat_le_iff of_nat_power real_le_rsqrt)
  then have "n-1 ≤ sqrt(real n ^ 2 - 1)"
    by (simp add: Suc_leI assms of_nat_diff)
  then show ?thesis
    using assms by linarith
qed


lemma Schottky_lemma2:
  fixes x::real
  assumes "0 ≤ x"
  obtains n where "0 < n" "¦x - ln (real n + sqrt ((real n)2 - 1)) / pi¦ < 1/2"
proof -
  obtain n0::nat where "0 < n0" "ln(n0 + sqrt(real n0 ^ 2 - 1)) / pi ≤ x"
  proof
    show "ln(real 1 + sqrt(real 1 ^ 2 - 1))/pi ≤ x"
      by (auto simp: assms)
  qed auto
  moreover
  obtain M::nat where "⋀n. ⟦0 < n; ln(n + sqrt(real n ^ 2 - 1)) / pi ≤ x⟧ ⟹ n ≤ M"
  proof
    fix n::nat
    assume "0 < n" "ln (n + sqrt ((real n)2 - 1)) / pi ≤ x"
    then have "ln (n + sqrt ((real n)2 - 1)) ≤ x * pi"
      by (simp add: divide_simps)
    then have *: "exp (ln (n + sqrt ((real n)2 - 1))) ≤ exp (x * pi)"
      by blast
    have 0: "0 ≤ sqrt ((real n)2 - 1)"
      using ‹0 < n› by auto
    have "n + sqrt ((real n)2 - 1) = exp (ln (n + sqrt ((real n)2 - 1)))"
      by (simp add: Suc_leI ‹0 < n› add_pos_nonneg real_of_nat_ge_one_iff)
    also have "... ≤ exp (x * pi)"
      using "*" by blast
    finally have "real n ≤ exp (x * pi)"
      using 0 by linarith
    then show "n ≤ nat (ceiling (exp(x * pi)))"
      by linarith
  qed
  ultimately obtain n where
     "0 < n" and le_x: "ln(n + sqrt(real n ^ 2 - 1)) / pi ≤ x"
             and le_n: "⋀k. ⟦0 < k; ln(k + sqrt(real k ^ 2 - 1)) / pi ≤ x⟧ ⟹ k ≤ n"
    using bounded_Max_nat [of "λn. 0<n ∧ ln (n + sqrt ((real n)2 - 1)) / pi ≤ x"] by metis
  define a where "a ≡ ln(n + sqrt(real n ^ 2 - 1)) / pi"
  define b where "b ≡ ln (1 + real n + sqrt ((1 + real n)2 - 1)) / pi"
  have le_xa: "a ≤ x"
   and le_na: "⋀k. ⟦0 < k; ln(k + sqrt(real k ^ 2 - 1)) / pi ≤ x⟧ ⟹ k ≤ n"
    using le_x le_n by (auto simp: a_def)
  moreover have "x < b"
    using le_n [of "Suc n"] by (force simp: b_def)
  moreover have "b - a < 1"
  proof -
    have "ln (1 + real n + sqrt ((1 + real n)2 - 1)) - ln (real n + sqrt ((real n)2 - 1)) =
         ln ((1 + real n + sqrt ((1 + real n)2 - 1)) / (real n + sqrt ((real n)2 - 1)))"
      by (simp add: ‹0 < n› Schottky_lemma1 add_pos_nonneg ln_div [symmetric])
    also have "... ≤ 3"
    proof (cases "n = 1")
      case True
      have "sqrt 3 ≤ 2"
        by (simp add: real_le_lsqrt)
      then have "(2 + sqrt 3) ≤ 4"
        by simp
      also have "... ≤ exp 3"
        using exp_ge_add_one_self [of "3::real"] by simp
      finally have "ln (2 + sqrt 3) ≤ 3"
        by (metis add_nonneg_nonneg add_pos_nonneg dbl_def dbl_inc_def dbl_inc_simps(3)
            dbl_simps(3) exp_gt_zero ln_exp ln_le_cancel_iff real_sqrt_ge_0_iff zero_le_one zero_less_one)
      then show ?thesis
        by (simp add: True)
    next
      case False with ‹0 < n› have "1 < n" "2 ≤ n"
        by linarith+
      then have 1: "1 ≤ real n * real n"
        by (metis less_imp_le_nat one_le_power power2_eq_square real_of_nat_ge_one_iff)
      have *: "4 + (m+2) * 2 ≤ (m+2) * ((m+2) * 3)" for m::nat
        by simp
      have "4 + n * 2 ≤ n * (n * 3)"
        using * [of "n-2"]  ‹2 ≤ n›
        by (metis le_add_diff_inverse2)
      then have **: "4 + real n * 2 ≤ real n * (real n * 3)"
        by (metis (mono_tags, hide_lams) of_nat_le_iff of_nat_add of_nat_mult of_nat_numeral)
      have "sqrt ((1 + real n)2 - 1) ≤ 2 * sqrt ((real n)2 - 1)"
        by (auto simp: real_le_lsqrt power2_eq_square algebra_simps 1 **)
      then
      have "((1 + real n + sqrt ((1 + real n)2 - 1)) / (real n + sqrt ((real n)2 - 1))) ≤ 2"
        using Schottky_lemma1 ‹0 < n›  by (simp add: divide_simps)
      then have "ln ((1 + real n + sqrt ((1 + real n)2 - 1)) / (real n + sqrt ((real n)2 - 1))) ≤ ln 2"
        apply (subst ln_le_cancel_iff)
        using Schottky_lemma1 ‹0 < n› by auto (force simp: divide_simps)
      also have "... ≤ 3"
        using ln_add_one_self_le_self [of 1] by auto
      finally show ?thesis .
    qed
    also have "... < pi"
      using pi_approx by simp
    finally show ?thesis
      by (simp add: a_def b_def divide_simps)
  qed
  ultimately have "¦x - a¦ < 1/2 ∨ ¦x - b¦ < 1/2"
    by (auto simp: abs_if)
  then show thesis
  proof
    assume "¦x - a¦ < 1 / 2"
    then show ?thesis
      by (rule_tac n=n in that) (auto simp: a_def ‹0 < n›)
  next
    assume "¦x - b¦ < 1 / 2"
    then show ?thesis
      by (rule_tac n="Suc n" in that) (auto simp: b_def ‹0 < n›)
  qed
qed


lemma Schottky_lemma3:
  fixes z::complex
  assumes "z ∈ (⋃m ∈ Ints. ⋃n ∈ {0<..}. {Complex m (ln(n + sqrt(real n ^ 2 - 1)) / pi)})
             ∪ (⋃m ∈ Ints. ⋃n ∈ {0<..}. {Complex m (-ln(n + sqrt(real n ^ 2 - 1)) / pi)})"
  shows "cos(pi * cos(pi * z)) = 1 ∨ cos(pi * cos(pi * z)) = -1"
proof -
  have sqrt2 [simp]: "complex_of_real (sqrt x) * complex_of_real (sqrt x) = x" if "x ≥ 0" for x::real
    by (metis abs_of_nonneg of_real_mult real_sqrt_mult_self that)
  have 1: "∃k. exp (𝗂 * (of_int m * complex_of_real pi) -
                 (ln (real n + sqrt ((real n)2 - 1)))) +
            inverse
             (exp (𝗂 * (of_int m * complex_of_real pi) -
                    (ln (real n + sqrt ((real n)2 - 1))))) = of_int k * 2"
         if "n > 0" for m n
  proof -
    have eeq: "e ≠ 0 ⟹ e + inverse e = n*2 ⟷ inverse e^2 - 2 * n*inverse e + 1 = 0" for n e::complex
      by (auto simp: field_simps power2_eq_square)
    have [simp]: "1 ≤ real n * real n"
      by (metis One_nat_def add.commute nat_less_real_le of_nat_1 of_nat_Suc one_le_power power2_eq_square that)
    show ?thesis
      apply (simp add: eeq)
      using Schottky_lemma1 [OF that]
      apply (auto simp: eeq exp_diff exp_Euler exp_of_real algebra_simps sin_int_times_real cos_int_times_real)
       apply (rule_tac x="int n" in exI)
       apply (auto simp: power2_eq_square algebra_simps)
       apply (rule_tac x="- int n" in exI)
      apply (auto simp: power2_eq_square algebra_simps)
      done
  qed
  have 2: "∃k. exp (𝗂 * (of_int m * complex_of_real pi) +
                 (ln (real n + sqrt ((real n)2 - 1)))) +
            inverse
             (exp (𝗂 * (of_int m * complex_of_real pi) +
                    (ln (real n + sqrt ((real n)2 - 1))))) = of_int k * 2"
            if "n > 0" for m n
  proof -
    have eeq: "e ≠ 0 ⟹ e + inverse e = n*2 ⟷ e^2 - 2 * n*e + 1 = 0" for n e::complex
      by (auto simp: field_simps power2_eq_square)
    have [simp]: "1 ≤ real n * real n"
      by (metis One_nat_def add.commute nat_less_real_le of_nat_1 of_nat_Suc one_le_power power2_eq_square that)
    show ?thesis
      apply (simp add: eeq)
      using Schottky_lemma1 [OF that]
      apply (auto simp: exp_add exp_Euler exp_of_real algebra_simps sin_int_times_real cos_int_times_real)
       apply (rule_tac x="int n" in exI)
       apply (auto simp: power2_eq_square algebra_simps)
       apply (rule_tac x="- int n" in exI)
      apply (auto simp: power2_eq_square algebra_simps)
      done
  qed
  have "∃x. cos (complex_of_real pi * z) = of_int x"
    using assms
    apply safe
      apply (auto simp: Ints_def cos_exp_eq exp_minus Complex_eq)
     apply (auto simp: algebra_simps dest: 1 2)
      done
  then have "sin(pi * cos(pi * z)) ^ 2 = 0"
    by (simp add: Complex_Transcendental.sin_eq_0)
  then have "1 - cos(pi * cos(pi * z)) ^ 2 = 0"
    by (simp add: sin_squared_eq)
  then show ?thesis
    using power2_eq_1_iff by auto
qed


theorem Schottky:
  assumes holf: "f holomorphic_on cball 0 1"
      and nof0: "norm(f 0) ≤ r"
      and not01: "⋀z. z ∈ cball 0 1 ⟹ ¬(f z = 0 ∨ f z = 1)"
      and "0 < t" "t < 1" "norm z ≤ t"
    shows "norm(f z) ≤ exp(pi * exp(pi * (2 + 2 * r + 12 * t / (1 - t))))"
proof -
  obtain h where holf: "h holomorphic_on cball 0 1"
             and nh0: "norm (h 0) ≤ 1 + norm(2 * f 0 - 1) / 3"
             and h:   "⋀z. z ∈ cball 0 1 ⟹ 2 * f z - 1 = cos(of_real pi * h z)"
  proof (rule Schottky_lemma0 [of "λz. 2 * f z - 1" "cball 0 1" 0])
    show "(λz. 2 * f z - 1) holomorphic_on cball 0 1"
      by (intro holomorphic_intros holf)
    show "contractible (cball (0::complex) 1)"
      by (auto simp: convex_imp_contractible)
    show "⋀z. z ∈ cball 0 1 ⟹ 2 * f z - 1 ≠ 1 ∧ 2 * f z - 1 ≠ - 1"
      using not01 by force
  qed auto
  obtain g where holg: "g holomorphic_on cball 0 1"
             and ng0:  "norm(g 0) ≤ 1 + norm(h 0) / 3"
             and g:    "⋀z. z ∈ cball 0 1 ⟹ h z = cos(of_real pi * g z)"
  proof (rule Schottky_lemma0 [OF holf convex_imp_contractible, of 0])
    show "⋀z. z ∈ cball 0 1 ⟹ h z ≠ 1 ∧ h z ≠ - 1"
      using h not01 by fastforce+
  qed auto
  have g0_2_f0: "norm(g 0) ≤ 2 + norm(f 0)"
  proof -
    have "cmod (2 * f 0 - 1) ≤ cmod (2 * f 0) + 1"
      by (metis norm_one norm_triangle_ineq4)
    also have "... ≤ 2 + cmod (f 0) * 3"
      by simp
    finally have "1 + norm(2 * f 0 - 1) / 3 ≤ (2 + norm(f 0) - 1) * 3"
      apply (simp add: divide_simps)
      using norm_ge_zero [of "2 * f 0 - 1"]
      by linarith
    with nh0 have "norm(h 0) ≤ (2 + norm(f 0) - 1) * 3"
      by linarith
    then have "1 + norm(h 0) / 3 ≤ 2 + norm(f 0)"
      by simp
    with ng0 show ?thesis
      by auto
  qed
  have "z ∈ ball 0 1"
    using assms by auto
  have norm_g_12: "norm(g z - g 0) ≤ (12 * t) / (1 - t)"
  proof -
    obtain g' where g': "⋀x. x ∈ cball 0 1 ⟹ (g has_field_derivative g' x) (at x within cball 0 1)"
      using holg [unfolded holomorphic_on_def field_differentiable_def] by metis
    have int_g': "(g' has_contour_integral g z - g 0) (linepath 0 z)"
      using contour_integral_primitive [OF g' valid_path_linepath, of 0 z]
      using ‹z ∈ ball 0 1› segment_bound1 by fastforce
    have "cmod (g' w) ≤ 12 / (1 - t)" if "w ∈ closed_segment 0 z" for w
    proof -
      have w: "w ∈ ball 0 1"
        using segment_bound [OF that] ‹z ∈ ball 0 1› by simp
      have ttt: "⋀z. z ∈ frontier (cball 0 1) ⟹ 1 - t ≤ dist w z"
        using ‹norm z ≤ t› segment_bound1 [OF ‹w ∈ closed_segment 0 z›]
        apply (simp add: dist_complex_def)
        by (metis diff_left_mono dist_commute dist_complex_def norm_triangle_ineq2 order_trans)
      have *: "⟦⋀b. (∃w ∈ T ∪ U. w ∈ ball b 1); ⋀x. x ∈ D ⟹ g x ∉ T ∪ U⟧ ⟹ ∄b. ball b 1 ⊆ g ` D" for T U D
        by force
      have "∄b. ball b 1 ⊆ g ` cball 0 1"
      proof (rule *)
        show "(∃w ∈ (⋃m ∈ Ints. ⋃n ∈ {0<..}. {Complex m (ln(n + sqrt(real n ^ 2 - 1)) / pi)}) ∪
                    (⋃m ∈ Ints. ⋃n ∈ {0<..}. {Complex m (-ln(n + sqrt(real n ^ 2 - 1)) / pi)}). w ∈ ball b 1)" for b
        proof -
          obtain m where m: "m ∈ ℤ" "¦Re b - m¦ ≤ 1/2"
            by (metis Ints_of_int abs_minus_commute of_int_round_abs_le)
          show ?thesis
          proof (cases "0::real" "Im b" rule: le_cases)
            case le
            then obtain n where "0 < n" and n: "¦Im b - ln (n + sqrt ((real n)2 - 1)) / pi¦ < 1/2"
              using Schottky_lemma2 [of "Im b"] by blast
            have "dist b (Complex m (Im b)) ≤ 1/2"
              by (metis cancel_comm_monoid_add_class.diff_cancel cmod_eq_Re complex.sel(1) complex.sel(2) dist_norm m(2) minus_complex.code)
            moreover
            have "dist (Complex m (Im b)) (Complex m (ln (n + sqrt ((real n)2 - 1)) / pi)) < 1/2"
              using n by (simp add: complex_norm cmod_eq_Re complex_diff dist_norm del: Complex_eq)
            ultimately have "dist b (Complex m (ln (real n + sqrt ((real n)2 - 1)) / pi)) < 1"
              by (simp add: dist_triangle_lt [of b "Complex m (Im b)"] dist_commute)
            with le m ‹0 < n› show ?thesis
              apply (rule_tac x = "Complex m (ln (real n + sqrt ((real n)2 - 1)) / pi)" in bexI)
               apply (simp_all del: Complex_eq greaterThan_0)
              by blast
          next
            case ge
            then obtain n where "0 < n" and n: "¦- Im b - ln (real n + sqrt ((real n)2 - 1)) / pi¦ < 1/2"
              using Schottky_lemma2 [of "- Im b"] by auto
            have "dist b (Complex m (Im b)) ≤ 1/2"
              by (metis cancel_comm_monoid_add_class.diff_cancel cmod_eq_Re complex.sel(1) complex.sel(2) dist_norm m(2) minus_complex.code)
            moreover
            have "dist (Complex m (- ln (n + sqrt ((real n)2 - 1)) / pi)) (Complex m (Im b)) < 1/2"
              using n
              apply (simp add: complex_norm cmod_eq_Re complex_diff dist_norm del: Complex_eq)
              by (metis add.commute add_uminus_conv_diff)
            ultimately have "dist b (Complex m (- ln (real n + sqrt ((real n)2 - 1)) / pi)) < 1"
              by (simp add: dist_triangle_lt [of b "Complex m (Im b)"] dist_commute)
            with ge m ‹0 < n› show ?thesis
              apply (rule_tac x = "Complex m (- ln (real n + sqrt ((real n)2 - 1)) / pi)" in bexI)
               apply (simp_all del: Complex_eq greaterThan_0)
              by blast
          qed
        qed
        show "g v ∉ (⋃m ∈ Ints. ⋃n ∈ {0<..}. {Complex m (ln(n + sqrt(real n ^ 2 - 1)) / pi)}) ∪
                    (⋃m ∈ Ints. ⋃n ∈ {0<..}. {Complex m (-ln(n + sqrt(real n ^ 2 - 1)) / pi)})"
             if "v ∈ cball 0 1" for v
          using not01 [OF that]
          by (force simp: g [OF that, symmetric] h [OF that, symmetric] dest!: Schottky_lemma3 [of "g v"])
      qed
      then have 12: "(1 - t) * cmod (deriv g w) / 12 < 1"
        using Bloch_general [OF holg _ ttt, of 1] w by force
      have "g field_differentiable at w within cball 0 1"
        using holg w by (simp add: holomorphic_on_def)
      then have "g field_differentiable at w within ball 0 1"
        using ball_subset_cball field_differentiable_within_subset by blast
      with w have der_gw: "(g has_field_derivative deriv g w) (at w)"
        by (simp add: field_differentiable_within_open [of _ "ball 0 1"] field_differentiable_derivI)
      with DERIV_unique [OF der_gw] g' w have "deriv g w = g' w"
        by (metis open_ball at_within_open ball_subset_cball has_field_derivative_subset subsetCE)
      then show "cmod (g' w) ≤ 12 / (1 - t)"
        using g' 12 ‹t < 1› by (simp add: field_simps)
    qed
    then have "cmod (g z - g 0) ≤ 12 / (1 - t) * cmod z"
      using has_contour_integral_bound_linepath [OF int_g', of "12/(1 - t)"] assms
      by simp
    with ‹cmod z ≤ t› ‹t < 1› show ?thesis
      by (simp add: divide_simps)
  qed
  have fz: "f z = (1 + cos(of_real pi * h z)) / 2"
    using h ‹z ∈ ball 0 1› by (auto simp: field_simps)
  have "cmod (f z) ≤ exp (cmod (complex_of_real pi * h z))"
    by (simp add: fz mult.commute norm_cos_plus1_le)
  also have "... ≤ exp (pi * exp (pi * (2 + 2 * r + 12 * t / (1 - t))))"
  proof (simp add: norm_mult)
    have "cmod (g z - g 0) ≤ 12 * t / (1 - t)"
      using norm_g_12 ‹t < 1› by (simp add: norm_mult)
    then have "cmod (g z) - cmod (g 0) ≤ 12 * t / (1 - t)"
      using norm_triangle_ineq2 order_trans by blast
    then have *: "cmod (g z) ≤ 2 + 2 * r + 12 * t / (1 - t)"
      using g0_2_f0 norm_ge_zero [of "f 0"] nof0
        by linarith
    have "cmod (h z) ≤ exp (cmod (complex_of_real pi * g z))"
      using ‹z ∈ ball 0 1› by (simp add: g norm_cos_le)
    also have "... ≤ exp (pi * (2 + 2 * r + 12 * t / (1 - t)))"
      using ‹t < 1› nof0 * by (simp add: norm_mult)
    finally show "cmod (h z) ≤ exp (pi * (2 + 2 * r + 12 * t / (1 - t)))" .
  qed
  finally show ?thesis .
qed

  
subsection‹The Little Picard Theorem›

lemma Landau_Picard:
  obtains R
    where "⋀z. 0 < R z"
          "⋀f. ⟦f holomorphic_on cball 0 (R(f 0));
                 ⋀z. norm z ≤ R(f 0) ⟹ f z ≠ 0 ∧ f z ≠ 1⟧ ⟹ norm(deriv f 0) < 1"
proof -
  define R where "R ≡ λz. 3 * exp(pi * exp(pi*(2 + 2 * cmod z + 12)))"
  show ?thesis
  proof
    show Rpos: "⋀z. 0 < R z"
      by (auto simp: R_def)
    show "norm(deriv f 0) < 1"
         if holf: "f holomorphic_on cball 0 (R(f 0))"
         and Rf:  "⋀z. norm z ≤ R(f 0) ⟹ f z ≠ 0 ∧ f z ≠ 1" for f
    proof -
      let ?r = "R(f 0)"
      define g where "g ≡ f ∘ (λz. of_real ?r * z)"
      have "0 < ?r"
        using Rpos by blast
      have holg: "g holomorphic_on cball 0 1"
        unfolding g_def
        apply (intro holomorphic_intros holomorphic_on_compose holomorphic_on_subset [OF holf])
        using Rpos by (auto simp: less_imp_le norm_mult)
      have *: "norm(g z) ≤ exp(pi * exp(pi * (2 + 2 * norm (f 0) + 12 * t / (1 - t))))"
           if "0 < t" "t < 1" "norm z ≤ t" for t z
      proof (rule Schottky [OF holg])
        show "cmod (g 0) ≤ cmod (f 0)"
          by (simp add: g_def)
        show "⋀z. z ∈ cball 0 1 ⟹ ¬ (g z = 0 ∨ g z = 1)"
          using Rpos by (simp add: g_def less_imp_le norm_mult Rf)
      qed (auto simp: that)
      have C1: "g holomorphic_on ball 0 (1 / 2)"
        by (rule holomorphic_on_subset [OF holg]) auto
      have C2: "continuous_on (cball 0 (1 / 2)) g"
        by (meson cball_divide_subset_numeral holg holomorphic_on_imp_continuous_on holomorphic_on_subset)
      have C3: "cmod (g z) ≤ R (f 0) / 3" if "cmod (0 - z) = 1/2" for z
      proof -
        have "norm(g z) ≤ exp(pi * exp(pi * (2 + 2 * norm (f 0) + 12)))"
          using * [of "1/2"] that by simp
        also have "... = ?r / 3"
          by (simp add: R_def)
        finally show ?thesis .
      qed
      then have cmod_g'_le: "cmod (deriv g 0) * 3 ≤ R (f 0) * 2"
        using Cauchy_inequality [OF C1 C2 _ C3, of 1] by simp
      have holf': "f holomorphic_on ball 0 (R(f 0))"
        by (rule holomorphic_on_subset [OF holf]) auto
      then have fd0: "f field_differentiable at 0"
        by (rule holomorphic_on_imp_differentiable_at [OF _ open_ball])
           (auto simp: Rpos [of "f 0"])
      have g_eq: "deriv g 0 = of_real ?r * deriv f 0"
        apply (rule DERIV_imp_deriv)
        apply (simp add: g_def)
        by (metis DERIV_chain DERIV_cmult_Id fd0 field_differentiable_derivI mult.commute mult_zero_right)
      show ?thesis
        using cmod_g'_le Rpos [of "f 0"]  by (simp add: g_eq norm_mult)
    qed
  qed
qed

lemma little_Picard_01:
  assumes holf: "f holomorphic_on UNIV" and f01: "⋀z. f z ≠ 0 ∧ f z ≠ 1"
  obtains c where "f = (λx. c)"
proof -
  obtain R
    where Rpos: "⋀z. 0 < R z"
      and R:    "⋀h. ⟦h holomorphic_on cball 0 (R(h 0));
                      ⋀z. norm z ≤ R(h 0) ⟹ h z ≠ 0 ∧ h z ≠ 1⟧ ⟹ norm(deriv h 0) < 1"
    using Landau_Picard by metis
  have contf: "continuous_on UNIV f"
    by (simp add: holf holomorphic_on_imp_continuous_on)
  show ?thesis
  proof (cases "∀x. deriv f x = 0")
    case True
    obtain c where "⋀x. f(x) = c"
      apply (rule DERIV_zero_connected_constant [OF connected_UNIV open_UNIV finite.emptyI contf])
       apply (metis True DiffE holf holomorphic_derivI open_UNIV, auto)
      done
    then show ?thesis
      using that by auto
  next
    case False
    then obtain w where w: "deriv f w ≠ 0" by auto
    define fw where "fw ≡ (f ∘ (λz. w + z / deriv f w))"
    have norm_let1: "norm(deriv fw 0) < 1"
    proof (rule R)
      show "fw holomorphic_on cball 0 (R (fw 0))"
        unfolding fw_def
        by (intro holomorphic_intros holomorphic_on_compose w holomorphic_on_subset [OF holf] subset_UNIV)
      show "fw z ≠ 0 ∧ fw z ≠ 1" if "cmod z ≤ R (fw 0)" for z
        using f01 by (simp add: fw_def)
    qed
    have "(fw has_field_derivative deriv f w * inverse (deriv f w)) (at 0)"
      apply (simp add: fw_def)
      apply (rule DERIV_chain)
      using holf holomorphic_derivI apply force
      apply (intro derivative_eq_intros w)
          apply (auto simp: field_simps)
      done
    then show ?thesis
      using norm_let1 w by (simp add: DERIV_imp_deriv)
  qed
qed


theorem little_Picard:
  assumes holf: "f holomorphic_on UNIV"
      and "a ≠ b" "range f ∩ {a,b} = {}"
    obtains c where "f = (λx. c)"
proof -
  let ?g = "λx. 1/(b - a)*(f x - b) + 1"
  obtain c where "?g = (λx. c)"
  proof (rule little_Picard_01)
    show "?g holomorphic_on UNIV"
      by (intro holomorphic_intros holf)
    show "⋀z. ?g z ≠ 0 ∧ ?g z ≠ 1"
      using assms by (auto simp: field_simps)
  qed auto
  then have "?g x = c" for x
    by meson
  then have "f x = c * (b-a) + a" for x
    using assms by (auto simp: field_simps)
  then show ?thesis
    using that by blast
qed


text‹A couple of little applications of Little Picard›

lemma holomorphic_periodic_fixpoint:
  assumes holf: "f holomorphic_on UNIV"
      and "p ≠ 0" and per: "⋀z. f(z + p) = f z"
  obtains x where "f x = x"
proof -
  have False if non: "⋀x. f x ≠ x"
  proof -
    obtain c where "(λz. f z - z) = (λz. c)"
    proof (rule little_Picard)
      show "(λz. f z - z) holomorphic_on UNIV"
        by (simp add: holf holomorphic_on_diff)
      show "range (λz. f z - z) ∩ {p,0} = {}"
          using assms non by auto (metis add.commute diff_eq_eq)
      qed (auto simp: assms)
    with per show False
      by (metis add.commute add_cancel_left_left ‹p ≠ 0› diff_add_cancel)
  qed
  then show ?thesis
    using that by blast
qed


lemma holomorphic_involution_point:
  assumes holfU: "f holomorphic_on UNIV" and non: "⋀a. f ≠ (λx. a + x)"
  obtains x where "f(f x) = x"
proof -
  { assume non_ff [simp]: "⋀x. f(f x) ≠ x"
    then have non_fp [simp]: "f z ≠ z" for z
      by metis
    have holf: "f holomorphic_on X" for X
      using assms holomorphic_on_subset by blast
    obtain c where c: "(λx. (f(f x) - x)/(f x - x)) = (λx. c)"
    proof (rule little_Picard_01)
      show "(λx. (f(f x) - x)/(f x - x)) holomorphic_on UNIV"
        apply (intro holomorphic_intros holf holomorphic_on_compose [unfolded o_def, OF holf])
        using non_fp by auto
    qed auto
    then obtain "c ≠ 0" "c ≠ 1"
      by (metis (no_types, hide_lams) non_ff diff_zero divide_eq_0_iff right_inverse_eq)
    have eq: "f(f x) - c * f x = x*(1 - c)" for x
      using fun_cong [OF c, of x] by (simp add: field_simps)
    have df_times_dff: "deriv f z * (deriv f (f z) - c) = 1 * (1 - c)" for z
    proof (rule DERIV_unique)
      show "((λx. f (f x) - c * f x) has_field_derivative
              deriv f z * (deriv f (f z) - c)) (at z)"
        apply (intro derivative_eq_intros)
            apply (rule DERIV_chain [unfolded o_def, of f])
             apply (auto simp: algebra_simps intro!: holomorphic_derivI [OF holfU])
        done
      show "((λx. f (f x) - c * f x) has_field_derivative 1 * (1 - c)) (at z)"
        by (simp add: eq mult_commute_abs)
    qed
    { fix z::complex
      obtain k where k: "deriv f ∘ f = (λx. k)"
      proof (rule little_Picard)
        show "(deriv f ∘ f) holomorphic_on UNIV"
          by (meson holfU holomorphic_deriv holomorphic_on_compose holomorphic_on_subset open_UNIV subset_UNIV)
        obtain "deriv f (f x) ≠ 0" "deriv f (f x) ≠ c"  for x
          using df_times_dff ‹c ≠ 1› eq_iff_diff_eq_0
          by (metis lambda_one mult_zero_left mult_zero_right)
        then show "range (deriv f ∘ f) ∩ {0,c} = {}"
          by force
      qed (use ‹c ≠ 0› in auto)
      have "¬ f constant_on UNIV"
        by (meson UNIV_I non_ff constant_on_def)
      with holf open_mapping_thm have "open(range f)"
        by blast
      obtain l where l: "⋀x. f x - k * x = l"
      proof (rule DERIV_zero_connected_constant [of UNIV "{}" "λx. f x - k * x"], simp_all)
        have "deriv f w - k = 0" for w
        proof (rule analytic_continuation [OF _ open_UNIV connected_UNIV subset_UNIV, of "λz. deriv f z - k" "f z" "range f" w])
          show "(λz. deriv f z - k) holomorphic_on UNIV"
            by (intro holomorphic_intros holf open_UNIV)
          show "f z islimpt range f"
            by (metis (no_types, lifting) IntI UNIV_I ‹open (range f)› image_eqI inf.absorb_iff2 inf_aci(1) islimpt_UNIV islimpt_eq_acc_point open_Int top_greatest)
          show "⋀z. z ∈ range f ⟹ deriv f z - k = 0"
            by (metis comp_def diff_self image_iff k)
        qed auto
        moreover
        have "((λx. f x - k * x) has_field_derivative deriv f x - k) (at x)" for x
          by (metis DERIV_cmult_Id Deriv.field_differentiable_diff UNIV_I field_differentiable_derivI holf holomorphic_on_def)
        ultimately
        show "∀x. ((λx. f x - k * x) has_field_derivative 0) (at x)"
          by auto
        show "continuous_on UNIV (λx. f x - k * x)"
          by (simp add: continuous_on_diff holf holomorphic_on_imp_continuous_on)
      qed (auto simp: connected_UNIV)
      have False
      proof (cases "k=1")
        case True
        then have "∃x. k * x + l ≠ a + x" for a
          using l non [of a] ext [of f "(+) a"]
          by (metis add.commute diff_eq_eq)
        with True show ?thesis by auto
      next
        case False
        have "⋀x. (1 - k) * x ≠ f 0"
          using l [of 0] apply (simp add: algebra_simps)
          by (metis diff_add_cancel l mult.commute non_fp)
        then show False
          by (metis False eq_iff_diff_eq_0 mult.commute nonzero_mult_div_cancel_right times_divide_eq_right)
      qed
    }
  }
  then show thesis
    using that by blast
qed


subsection‹The ArzelĂ --Ascoli theorem›

lemma subsequence_diagonalization_lemma:
  fixes P :: "nat ⇒ (nat ⇒ 'a) ⇒ bool"
  assumes sub: "⋀i r. ∃k. strict_mono (k :: nat ⇒ nat) ∧ P i (r ∘ k)"
      and P_P:  "⋀i r::nat ⇒ 'a. ⋀k1 k2 N.
                   ⟦P i (r ∘ k1); ⋀j. N ≤ j ⟹ ∃j'. j ≤ j' ∧ k2 j = k1 j'⟧ ⟹ P i (r ∘ k2)"
   obtains k where "strict_mono (k :: nat ⇒ nat)" "⋀i. P i (r ∘ k)"
proof -
  obtain kk where "⋀i r. strict_mono (kk i r :: nat ⇒ nat) ∧ P i (r ∘ (kk i r))"
    using sub by metis
  then have sub_kk: "⋀i r. strict_mono (kk i r)" and P_kk: "⋀i r. P i (r ∘ (kk i r))"
    by auto
  define rr where "rr ≡ rec_nat (kk 0 r) (λn x. x ∘ kk (Suc n) (r ∘ x))"
  then have [simp]: "rr 0 = kk 0 r" "⋀n. rr(Suc n) = rr n ∘ kk (Suc n) (r ∘ rr n)"
    by auto
  show thesis
  proof
    have sub_rr: "strict_mono (rr i)" for i
      using sub_kk  by (induction i) (auto simp: strict_mono_def o_def)
    have P_rr: "P i (r ∘ rr i)" for i
      using P_kk  by (induction i) (auto simp: o_def)
    have "i ≤ i+d ⟹ rr i n ≤ rr (i+d) n" for d i n
    proof (induction d)
      case 0 then show ?case
        by simp
    next
      case (Suc d) then show ?case
        apply simp
          using seq_suble [OF sub_kk] order_trans strict_mono_less_eq [OF sub_rr] by blast
    qed
    then have "⋀i j n. i ≤ j ⟹ rr i n ≤ rr j n"
      by (metis le_iff_add)
    show "strict_mono (λn. rr n n)"
      apply (simp add: strict_mono_Suc_iff)
      by (meson lessI less_le_trans seq_suble strict_monoD sub_kk sub_rr)
    have "∃j. i ≤ j ∧ rr (n+d) i = rr n j" for d n i
      apply (induction d arbitrary: i, auto)
      by (meson order_trans seq_suble sub_kk)
    then have "⋀m n i. n ≤ m ⟹ ∃j. i ≤ j ∧ rr m i = rr n j"
      by (metis le_iff_add)
    then show "P i (r ∘ (λn. rr n n))" for i
      by (meson P_rr P_P)
  qed
qed

lemma function_convergent_subsequence:
  fixes f :: "[nat,'a] ⇒ 'b::{real_normed_vector,heine_borel}"
  assumes "countable S" and M: "⋀n::nat. ⋀x. x ∈ S ⟹ norm(f n x) ≤ M"
   obtains k where "strict_mono (k::nat⇒nat)" "⋀x. x ∈ S ⟹ ∃l. (λn. f (k n) x) ⇢ l"
proof (cases "S = {}")
  case True
  then show ?thesis
    using strict_mono_id that by fastforce
next
  case False
  with ‹countable S› obtain σ :: "nat ⇒ 'a" where σ: "S = range σ"
    using uncountable_def by blast
  obtain k where "strict_mono k" and k: "⋀i. ∃l. (λn. (f ∘ k) n (σ i)) ⇢ l"
  proof (rule subsequence_diagonalization_lemma
      [of "λi r. ∃l. ((λn. (f ∘ r) n (σ i)) ⤏ l) sequentially" id])
    show "∃k::nat⇒nat. strict_mono k ∧ (∃l. (λn. (f ∘ (r ∘ k)) n (σ i)) ⇢ l)" for i r
    proof -
      have "f (r n) (σ i) ∈ cball 0 M" for n
        by (simp add: σ M)
      then show ?thesis
        using compact_def [of "cball (0::'b) M"] apply simp
        apply (drule_tac x="(λn. f (r n) (σ i))" in spec)
        apply (force simp: o_def)
        done
    qed
    show "⋀i r k1 k2 N.
               ⟦∃l. (λn. (f ∘ (r ∘ k1)) n (σ i)) ⇢ l; ⋀j. N ≤ j ⟹ ∃j'≥j. k2 j = k1 j'⟧
               ⟹ ∃l. (λn. (f ∘ (r ∘ k2)) n (σ i)) ⇢ l"
      apply (simp add: lim_sequentially)
      apply (erule ex_forward all_forward imp_forward)+
        apply auto
      by (metis (no_types, hide_lams) le_cases order_trans)
  qed auto
  with σ that show ?thesis
    by force
qed


theorem Arzela_Ascoli:
  fixes  :: "[nat,'a::euclidean_space] ⇒ 'b::{real_normed_vector,heine_borel}"
  assumes "compact S"
      and M: "⋀n x. x ∈ S ⟹ norm(ℱ n x) ≤ M"
      and equicont:
          "⋀x e. ⟦x ∈ S; 0 < e⟧
                 ⟹ ∃d. 0 < d ∧ (∀n y. y ∈ S ∧ norm(x - y) < d ⟶ norm(ℱ n x - ℱ n y) < e)"
  obtains g k where "continuous_on S g" "strict_mono (k :: nat ⇒ nat)"
                    "⋀e. 0 < e ⟹ ∃N. ∀n x. n ≥ N ∧ x ∈ S ⟶ norm(ℱ(k n) x - g x) < e"
proof -
  have UEQ: "⋀e. 0 < e ⟹ ∃d. 0 < d ∧ (∀n. ∀x ∈ S. ∀x' ∈ S. dist x' x < d ⟶ dist (ℱ n x') (ℱ n x) < e)"
    apply (rule compact_uniformly_equicontinuous [OF ‹compact S›, of "range ℱ"])
    using equicont by (force simp: dist_commute dist_norm)+
  have "continuous_on S g"
       if "⋀e. 0 < e ⟹ ∃N. ∀n x. n ≥ N ∧ x ∈ S ⟶ norm(ℱ(r n) x - g x) < e"
       for g:: "'a ⇒ 'b" and r :: "nat ⇒ nat"
  proof (rule uniform_limit_theorem [of _ "ℱ ∘ r"])
    show "∀F n in sequentially. continuous_on S ((ℱ ∘ r) n)"
      apply (simp add: eventually_sequentially)
      apply (rule_tac x=0 in exI)
      using UEQ apply (force simp: continuous_on_iff)
      done
    show "uniform_limit S (ℱ ∘ r) g sequentially"
      apply (simp add: uniform_limit_iff eventually_sequentially)
        by (metis dist_norm that)
  qed auto
  moreover
  obtain R where "countable R" "R ⊆ S" and SR: "S ⊆ closure R"
    by (metis separable that)
  obtain k where "strict_mono k" and k: "⋀x. x ∈ R ⟹ ∃l. (λn. ℱ (k n) x) ⇢ l"
    apply (rule function_convergent_subsequence [OF ‹countable R› M])
    using ‹R ⊆ S› apply force+
    done
  then have Cauchy: "Cauchy ((λn. ℱ (k n) x))" if "x ∈ R" for x
    using convergent_eq_Cauchy that by blast
  have "∃N. ∀m n x. N ≤ m ∧ N ≤ n ∧ x ∈ S ⟶ dist ((ℱ ∘ k) m x) ((ℱ ∘ k) n x) < e"
    if "0 < e" for e
  proof -
    obtain d where "0 < d"
      and d: "⋀n. ∀x ∈ S. ∀x' ∈ S. dist x' x < d ⟶ dist (ℱ n x') (ℱ n x) < e/3"
      by (metis UEQ ‹0 < e› divide_pos_pos zero_less_numeral)
    obtain T where "T ⊆ R" and "finite T" and T: "S ⊆ (⋃c∈T. ball c d)"
    proof (rule compactE_image [OF  ‹compact S›, of R "(λx. ball x d)"])
      have "closure R ⊆ (⋃c∈R. ball c d)"
        apply clarsimp
        using ‹0 < d› closure_approachable by blast
      with SR show "S ⊆ (⋃c∈R. ball c d)"
        by auto
    qed auto
    have "∃M. ∀m≥M. ∀n≥M. dist (ℱ (k m) x) (ℱ (k n) x) < e/3" if "x ∈ R" for x
      using Cauchy ‹0 < e› that unfolding Cauchy_def
      by (metis less_divide_eq_numeral1(1) mult_zero_left)
    then obtain MF where MF: "⋀x m n. ⟦x ∈ R; m ≥ MF x; n ≥ MF x⟧ ⟹ norm (ℱ (k m) x - ℱ (k n) x) < e/3"
      using dist_norm by metis
    have "dist ((ℱ ∘ k) m x) ((ℱ ∘ k) n x) < e"
         if m: "Max (MF ` T) ≤ m" and n: "Max (MF ` T) ≤ n" "x ∈ S" for m n x
    proof -
      obtain t where "t ∈ T" and t: "x ∈ ball t d"
        using ‹x ∈ S› T by auto
      have "norm(ℱ (k m) t - ℱ (k m) x) < e / 3"
        by (metis ‹R ⊆ S› ‹T ⊆ R› ‹t ∈ T› d dist_norm mem_ball subset_iff t ‹x ∈ S›)
      moreover
      have "norm(ℱ (k n) t - ℱ (k n) x) < e / 3"
        by (metis ‹R ⊆ S› ‹T ⊆ R› ‹t ∈ T› subsetD d dist_norm mem_ball t ‹x ∈ S›)
      moreover
      have "norm(ℱ (k m) t - ℱ (k n) t) < e / 3"
      proof (rule MF)
        show "t ∈ R"
          using ‹T ⊆ R› ‹t ∈ T› by blast
        show "MF t ≤ m" "MF t ≤ n"
          by (meson Max_ge ‹finite T› ‹t ∈ T› finite_imageI imageI le_trans m n)+
      qed
      ultimately
      show ?thesis
        unfolding dist_norm [symmetric] o_def
          by (metis dist_triangle_third dist_commute)
    qed
    then show ?thesis
      by force
  qed
  then have "∃g. ∀e>0. ∃N. ∀n≥N. ∀x ∈ S. norm(ℱ(k n) x - g x) < e"
    using uniformly_convergent_eq_cauchy [of "λx. x ∈ S" "ℱ ∘ k"]
    apply (simp add: o_def dist_norm)
    by meson
  ultimately show thesis
    by (metis that ‹strict_mono k›)
qed



subsubsection‹Montel's theorem›

text‹a sequence of holomorphic functions uniformly bounded
on compact subsets of an open set S has a subsequence that converges to a
holomorphic function, and converges \emph{uniformly} on compact subsets of S.›


theorem Montel:
  fixes  :: "[nat,complex] ⇒ complex"
  assumes "open S"
      and : "⋀h. h ∈ ℋ ⟹ h holomorphic_on S"
      and bounded: "⋀K. ⟦compact K; K ⊆ S⟧ ⟹ ∃B. ∀h ∈ ℋ. ∀ z ∈ K. norm(h z) ≤ B"
      and rng_f: "range ℱ ⊆ ℋ"
  obtains g r
    where "g holomorphic_on S" "strict_mono (r :: nat ⇒ nat)"
          "⋀x. x ∈ S ⟹ ((λn. ℱ (r n) x) ⤏ g x) sequentially"
          "⋀K. ⟦compact K; K ⊆ S⟧ ⟹ uniform_limit K (ℱ ∘ r) g sequentially"        
proof -
  obtain K where comK: "⋀n. compact(K n)" and KS: "⋀n::nat. K n ⊆ S"
             and subK: "⋀X. ⟦compact X; X ⊆ S⟧ ⟹ ∃N. ∀n≥N. X ⊆ K n"
    using open_Union_compact_subsets [OF ‹open S›] by metis
  then have "⋀i. ∃B. ∀h ∈ ℋ. ∀ z ∈ K i. norm(h z) ≤ B"
    by (simp add: bounded)
  then obtain B where B: "⋀i h z. ⟦h ∈ ℋ; z ∈ K i⟧ ⟹ norm(h z) ≤ B i"
    by metis
  have *: "∃r g. strict_mono (r::nat⇒nat) ∧ (∀e > 0. ∃N. ∀n≥N. ∀x ∈ K i. norm((ℱ ∘ r) n x - g x) < e)"
        if "⋀n. ℱ n ∈ ℋ" for  i
  proof -
    obtain g k where "continuous_on (K i) g" "strict_mono (k::nat⇒nat)"
                    "⋀e. 0 < e ⟹ ∃N. ∀n≥N. ∀x ∈ K i. norm(ℱ(k n) x - g x) < e"
    proof (rule Arzela_Ascoli [of "K i" "ℱ" "B i"])
      show "∃d>0. ∀n y. y ∈ K i ∧ cmod (z - y) < d ⟶ cmod (ℱ n z - ℱ n y) < e"
             if z: "z ∈ K i" and "0 < e" for z e
      proof -
        obtain r where "0 < r" and r: "cball z r ⊆ S"
          using z KS [of i] ‹open S› by (force simp: open_contains_cball)
        have "cball z (2 / 3 * r) ⊆ cball z r"
          using ‹0 < r› by (simp add: cball_subset_cball_iff)
        then have z23S: "cball z (2 / 3 * r) ⊆ S"
          using r by blast
        obtain M where "0 < M" and M: "⋀n w. dist z w ≤ 2/3 * r ⟹ norm(ℱ n w) ≤ M"
        proof -
          obtain N where N: "∀n≥N. cball z (2/3 * r) ⊆ K n"
            using subK compact_cball [of z "(2 / 3 * r)"] z23S by force
          have "cmod (ℱ n w) ≤ ¦B N¦ + 1" if "dist z w ≤ 2 / 3 * r" for n w
          proof -
            have "w ∈ K N"
              using N mem_cball that by blast
            then have "cmod (ℱ n w) ≤ B N"
              using B ‹⋀n. ℱ n ∈ ℋ› by blast
            also have "... ≤ ¦B N¦ + 1"
              by simp
            finally show ?thesis .
          qed
          then show ?thesis
            by (rule_tac M="¦B N¦ + 1" in that) auto
        qed
        have "cmod (ℱ n z - ℱ n y) < e"
              if "y ∈ K i" and y_near_z: "cmod (z - y) < r/3" "cmod (z - y) < e * r / (6 * M)"
              for n y
        proof -
          have "((λw. ℱ n w / (w - ξ)) has_contour_integral
                    (2 * pi) * 𝗂 * winding_number (circlepath z (2 / 3 * r)) ξ * ℱ n ξ)
                (circlepath z (2 / 3 * r))"
             if "dist ξ z < (2 / 3 * r)" for ξ
          proof (rule Cauchy_integral_formula_convex_simple)
            have "ℱ n holomorphic_on S"
              by (simp add:  ‹⋀n. ℱ n ∈ ℋ›)
            with z23S show "ℱ n holomorphic_on cball z (2 / 3 * r)"
              using holomorphic_on_subset by blast
          qed (use that ‹0 < r› in ‹auto simp: dist_commute›)
          then have *: "((λw. ℱ n w / (w - ξ)) has_contour_integral (2 * pi) * 𝗂 * ℱ n ξ)
                     (circlepath z (2 / 3 * r))"
             if "dist ξ z < (2 / 3 * r)" for ξ
            using that by (simp add: winding_number_circlepath dist_norm)
           have y: "((λw. ℱ n w / (w - y)) has_contour_integral (2 * pi) * 𝗂 * ℱ n y)
                 (circlepath z (2 / 3 * r))"
             apply (rule *)
             using that ‹0 < r› by (simp only: dist_norm norm_minus_commute)
           have z: "((λw. ℱ n w / (w - z)) has_contour_integral (2 * pi) * 𝗂 * ℱ n z)
                 (circlepath z (2 / 3 * r))"
             apply (rule *)
             using ‹0 < r› by simp
           have le_er: "cmod (ℱ n x / (x - y) - ℱ n x / (x - z)) ≤ e / r"
                if "cmod (x - z) = r/3 + r/3" for x
           proof -
             have "~ (cmod (x - y) < r/3)"
               using y_near_z(1) that ‹M > 0› ‹r > 0›
               by (metis (full_types) norm_diff_triangle_less norm_minus_commute order_less_irrefl)
             then have r4_le_xy: "r/4 ≤ cmod (x - y)"
               using ‹r > 0› by simp
             then have neq: "x ≠ y" "x ≠ z"
               using that ‹r > 0› by (auto simp: divide_simps norm_minus_commute)
             have leM: "cmod (ℱ n x) ≤ M"
               by (simp add: M dist_commute dist_norm that)
             have "cmod (ℱ n x / (x - y) - ℱ n x / (x - z)) = cmod (ℱ n x) * cmod (1 / (x - y) - 1 / (x - z))"
               by (metis (no_types, lifting) divide_inverse mult.left_neutral norm_mult right_diff_distrib')
             also have "... = cmod (ℱ n x) * cmod ((y - z) / ((x - y) * (x - z)))"
               using neq by (simp add: divide_simps)
             also have "... = cmod (ℱ n x) * (cmod (y - z) / (cmod(x - y) * (2/3 * r)))"
               by (simp add: norm_mult norm_divide that)
             also have "... ≤ M * (cmod (y - z) / (cmod(x - y) * (2/3 * r)))"
               apply (rule mult_mono)
                  apply (rule leM)
                 using ‹r > 0› ‹M > 0› neq by auto
               also have "... < M * ((e * r / (6 * M)) / (cmod(x - y) * (2/3 * r)))"
                 unfolding mult_less_cancel_left
                 using y_near_z(2) ‹M > 0› ‹r > 0› neq
                 apply (simp add: field_simps mult_less_0_iff norm_minus_commute)
                 done
             also have "... ≤ e/r"
               using ‹e > 0› ‹r > 0› r4_le_xy by (simp add: divide_simps)
             finally show ?thesis by simp
           qed
           have "(2 * pi) * cmod (ℱ n y - ℱ n z) = cmod ((2 * pi) * 𝗂 * ℱ n y - (2 * pi) * 𝗂 * ℱ n z)"
             by (simp add: right_diff_distrib [symmetric] norm_mult)
           also have "cmod ((2 * pi) * 𝗂 * ℱ n y - (2 * pi) * 𝗂 * ℱ n z) ≤ e / r * (2 * pi * (2 / 3 * r))"
             apply (rule has_contour_integral_bound_circlepath [OF has_contour_integral_diff [OF y z], of "e/r"])
             using ‹e > 0› ‹r > 0› le_er by auto
           also have "... = (2 * pi) * e * ((2 / 3))"
             using ‹r > 0› by (simp add: divide_simps)
           finally have "cmod (ℱ n y - ℱ n z) ≤ e * (2 / 3)"
             by simp
           also have "... < e"
             using ‹e > 0› by simp
           finally show ?thesis by (simp add: norm_minus_commute)
        qed
        then show ?thesis
          apply (rule_tac x="min (r/3) ((e * r)/(6 * M))" in exI)
          using ‹0 < e› ‹0 < r› ‹0 < M› by simp
      qed
      show "⋀n x.  x ∈ K i ⟹ cmod (ℱ n x) ≤ B i"
        using B ‹⋀n. ℱ n ∈ ℋ› by blast
    qed (use comK in ‹fastforce+›)
    then show ?thesis
      by fastforce
  qed
  have "∃k g. strict_mono (k::nat⇒nat) ∧ (∀e > 0. ∃N. ∀n≥N. ∀x ∈ K i. norm((ℱ ∘ r ∘ k) n x - g x) < e)"
         for i r
    apply (rule *)
    using rng_f by auto
  then have **: "⋀i r. ∃k. strict_mono (k::nat⇒nat) ∧ (∃g. ∀e>0. ∃N. ∀n≥N. ∀x ∈ K i. norm((ℱ ∘ (r ∘ k)) n x - g x) < e)"
    by (force simp: o_assoc)
  obtain k :: "nat ⇒ nat" where "strict_mono k"
             and "⋀i. ∃g. ∀e>0. ∃N. ∀n≥N. ∀x∈K i. cmod ((ℱ ∘ (id ∘ k)) n x - g x) < e"
    apply (rule subsequence_diagonalization_lemma [OF **, of id])
     apply (erule ex_forward all_forward imp_forward)+
      apply auto
    apply (rule_tac x="max N Na" in exI, fastforce+)
    done
  then have lt_e: "⋀i. ∃g. ∀e>0. ∃N. ∀n≥N. ∀x∈K i. cmod ((ℱ ∘ k) n x - g x) < e"
    by simp
  have "∃l. ∀e>0. ∃N. ∀n≥N. norm(ℱ (k n) z - l) < e" if "z ∈ S" for z
  proof -
    obtain G where G: "⋀i e. e > 0 ⟹ ∃M. ∀n≥M. ∀x∈K i. cmod ((ℱ ∘ k) n x - G i x) < e"
      using lt_e by metis
    obtain N where "⋀n. n ≥ N ⟹ z ∈ K n"
      using subK [of "{z}"] that ‹z ∈ S› by auto
    moreover have "⋀e. e > 0 ⟹ ∃M. ∀n≥M. ∀x∈K N. cmod ((ℱ ∘ k) n x - G N x) < e"
      using G by auto
    ultimately show ?thesis
      by (metis comp_apply order_refl)
  qed
  then obtain g where g: "⋀z e. ⟦z ∈ S; e > 0⟧ ⟹ ∃N. ∀n≥N. norm(ℱ (k n) z - g z) < e"
    by metis
  show ?thesis
  proof
    show g_lim: "⋀x. x ∈ S ⟹ (λn. ℱ (k n) x) ⇢ g x"
      by (simp add: lim_sequentially g dist_norm)    
    have dg_le_e: "∃N. ∀n≥N. ∀x∈T. cmod (ℱ (k n) x - g x) < e"
      if T: "compact T" "T ⊆ S" and "0 < e" for T e
    proof -
      obtain N where N: "⋀n. n ≥ N ⟹ T ⊆ K n"
        using subK [OF T] by blast
      obtain h where h: "⋀e. e>0 ⟹ ∃M. ∀n≥M. ∀x∈K N. cmod ((ℱ ∘ k) n x - h x) < e"
        using lt_e by blast
      have geq: "g w = h w" if "w ∈ T" for w
        apply (rule LIMSEQ_unique [of "λn. ℱ(k n) w"])
        using ‹T ⊆ S› g_lim that apply blast
        using h N that by (force simp: lim_sequentially dist_norm)
      show ?thesis
        using T h N ‹0 < e› by (fastforce simp add: geq)
    qed
    then show "⋀K. ⟦compact K; K ⊆ S⟧
         ⟹ uniform_limit K (ℱ ∘ k) g sequentially"
      by (simp add: uniform_limit_iff dist_norm eventually_sequentially)
    show "g holomorphic_on S"
    proof (rule holomorphic_uniform_sequence [OF ‹open S› ])
      show "⋀n. (ℱ ∘ k) n ∈ ℋ"
        by (simp add: range_subsetD rng_f)
      show "∃d>0. cball z d ⊆ S ∧ uniform_limit (cball z d) (λn. (ℱ ∘ k) n) g sequentially"
        if "z ∈ S" for z
      proof -
        obtain d where d: "d>0" "cball z d ⊆ S"
          using ‹open S› ‹z ∈ S› open_contains_cball by blast
        then have "uniform_limit (cball z d) (ℱ ∘ k) g sequentially"
          using dg_le_e compact_cball by (auto simp: uniform_limit_iff eventually_sequentially dist_norm)
        with d show ?thesis by blast
      qed
    qed
  qed (auto simp: ‹strict_mono k›)
qed



subsection‹Some simple but useful cases of Hurwitz's theorem›

proposition Hurwitz_no_zeros:
  assumes S: "open S" "connected S"
      and holf: "⋀n::nat. ℱ n holomorphic_on S"
      and holg: "g holomorphic_on S"
      and ul_g: "⋀K. ⟦compact K; K ⊆ S⟧ ⟹ uniform_limit K ℱ g sequentially"
      and nonconst: "~ g constant_on S"
      and nz: "⋀n z. z ∈ S ⟹ ℱ n z ≠ 0"
      and "z0 ∈ S"
      shows "g z0 ≠ 0"
proof
  assume g0: "g z0 = 0"
  obtain h r m
    where "0 < m" "0 < r" and subS: "ball z0 r ⊆ S"
      and holh: "h holomorphic_on ball z0 r"
      and geq:  "⋀w. w ∈ ball z0 r ⟹ g w = (w - z0)^m * h w"
      and hnz:  "⋀w. w ∈ ball z0 r ⟹ h w ≠ 0"
    by (blast intro: holomorphic_factor_zero_nonconstant [OF holg S ‹z0 ∈ S› g0 nonconst])
  then have holf0: "ℱ n holomorphic_on ball z0 r" for n
    by (meson holf holomorphic_on_subset)
  have *: "((λz. deriv (ℱ n) z / ℱ n z) has_contour_integral 0) (circlepath z0 (r/2))" for n
  proof (rule Cauchy_theorem_disc_simple [of _ z0 r])
    show "(λz. deriv (ℱ n) z / ℱ n z) holomorphic_on ball z0 r"
      apply (intro holomorphic_intros holomorphic_deriv holf holf0 open_ball nz)
      using ‹ball z0 r ⊆ S› by blast
  qed (use ‹0 < r› in auto)
  have hol_dg: "deriv g holomorphic_on S"
    by (simp add: ‹open S› holg holomorphic_deriv)
  have "continuous_on (sphere z0 (r/2)) (deriv g)"
    apply (intro holomorphic_on_imp_continuous_on holomorphic_on_subset [OF hol_dg])
    using ‹0 < r› subS by auto
  then have "compact (deriv g ` (sphere z0 (r/2)))"
    by (rule compact_continuous_image [OF _ compact_sphere])
  then have bo_dg: "bounded (deriv g ` (sphere z0 (r/2)))"
    using compact_imp_bounded by blast
  have "continuous_on (sphere z0 (r/2)) (cmod ∘ g)"
    apply (intro continuous_intros holomorphic_on_imp_continuous_on holomorphic_on_subset [OF holg])
    using ‹0 < r› subS by auto
  then have "compact ((cmod ∘ g) ` sphere z0 (r/2))"
    by (rule compact_continuous_image [OF _ compact_sphere])
  moreover have "(cmod ∘ g) ` sphere z0 (r/2) ≠ {}"
    using ‹0 < r› by auto
  ultimately obtain b where b: "b ∈ (cmod ∘ g) ` sphere z0 (r/2)"
                               "⋀t. t ∈ (cmod ∘ g) ` sphere z0 (r/2) ⟹ b ≤ t"
    using compact_attains_inf [of "(norm ∘ g) ` (sphere z0 (r/2))"] by blast
  have "(λn. contour_integral (circlepath z0 (r/2)) (λz. deriv (ℱ n) z / ℱ n z)) ⇢
        contour_integral (circlepath z0 (r/2)) (λz. deriv g z / g z)"
  proof (rule contour_integral_uniform_limit_circlepath)
    show "∀F n in sequentially. (λz. deriv (ℱ n) z / ℱ n z) contour_integrable_on circlepath z0 (r/2)"
      using * contour_integrable_on_def eventually_sequentiallyI by meson
    show "uniform_limit (sphere z0 (r/2)) (λn z. deriv (ℱ n) z / ℱ n z) (λz. deriv g z / g z) sequentially"
    proof (rule uniform_lim_divide [OF _ _ bo_dg])
      show "uniform_limit (sphere z0 (r/2)) (λa. deriv (ℱ a)) (deriv g) sequentially"
      proof (rule uniform_limitI)
        fix e::real
        assume "0 < e"
        have *: "dist (deriv (ℱ n) w) (deriv g w) < e"
          if e8: "⋀x. dist z0 x ≤ 3 * r / 4 ⟹ dist (ℱ n x) (g x) * 8 < r * e"
          and w: "dist w z0 = r/2"  for n w
        proof -
          have "ball w (r/4) ⊆ ball z0 r"  "cball w (r/4) ⊆ ball z0 r"
            using ‹0 < r› by (simp_all add: ball_subset_ball_iff cball_subset_ball_iff w)
          with subS have wr4_sub: "ball w (r/4) ⊆ S" "cball w (r/4) ⊆ S" by force+
          moreover
          have "(λz. ℱ n z - g z) holomorphic_on S"
            by (intro holomorphic_intros holf holg)
          ultimately have hol: "(λz. ℱ n z - g z) holomorphic_on ball w (r/4)"
            and cont: "continuous_on (cball w (r / 4)) (λz. ℱ n z - g z)"
            using holomorphic_on_subset by (blast intro: holomorphic_on_imp_continuous_on)+
          have "w ∈ S"
            using ‹0 < r› wr4_sub by auto
          have "⋀y. dist w y < r / 4 ⟹ dist z0 y ≤ 3 * r / 4"
            apply (rule dist_triangle_le [where z=w])
            using w by (simp add: dist_commute)
          with e8 have in_ball: "⋀y. y ∈ ball w (r/4) ⟹ ℱ n y - g y ∈ ball 0 (r/4 * e/2)"
            by (simp add: dist_norm [symmetric])
          have "ℱ n field_differentiable at w"
            by (metis holomorphic_on_imp_differentiable_at ‹w ∈ S› holf ‹open S›)
          moreover
          have "g field_differentiable at w"
            using ‹w ∈ S› ‹open S› holg holomorphic_on_imp_differentiable_at by auto
          moreover
          have "cmod (deriv (λw. ℱ n w - g w) w) * 2 ≤ e"
            apply (rule Cauchy_higher_deriv_bound [OF hol cont in_ball, of 1, simplified])
            using ‹r > 0› by auto
          ultimately have "dist (deriv (ℱ n) w) (deriv g w) ≤ e/2"
            by (simp add: dist_norm)
          then show ?thesis
            using ‹e > 0› by auto
        qed
        have "cball z0 (3 * r / 4) ⊆ ball z0 r"
          by (simp add: cball_subset_ball_iff ‹0 < r›)
        with subS have "uniform_limit (cball z0 (3 * r/4)) ℱ g sequentially"
          by (force intro: ul_g)
        then have "∀F n in sequentially. ∀x∈cball z0 (3 * r / 4). dist (ℱ n x) (g x) < r / 4 * e / 2"
          using ‹0 < e› ‹0 < r› by (force simp: intro!: uniform_limitD)
        then show "∀F n in sequentially. ∀x ∈ sphere z0 (r/2). dist (deriv (ℱ n) x) (deriv g x) < e"
          apply (simp add: eventually_sequentially)
          apply (elim ex_forward all_forward imp_forward asm_rl)
          using * apply (force simp: dist_commute)
          done
      qed
      show "uniform_limit (sphere z0 (r/2)) ℱ g sequentially"
      proof (rule uniform_limitI)
        fix e::real
        assume "0 < e"
        have "sphere z0 (r/2) ⊆ ball z0 r"
          using ‹0 < r› by auto
        with subS have "uniform_limit (sphere z0 (r/2)) ℱ g sequentially"
          by (force intro: ul_g)
        then show "∀F n in sequentially. ∀x ∈ sphere z0 (r/2). dist (ℱ n x) (g x) < e"
          apply (rule uniform_limitD)
          using ‹0 < e› by force
      qed
      show "b > 0" "⋀x. x ∈ sphere z0 (r/2) ⟹ b ≤ cmod (g x)"
        using b ‹0 < r› by (fastforce simp: geq hnz)+
    qed
  qed (use ‹0 < r› in auto)
  then have "(λn. 0) ⇢ contour_integral (circlepath z0 (r/2)) (λz. deriv g z / g z)"
    by (simp add: contour_integral_unique [OF *])
  then have "contour_integral (circlepath z0 (r/2)) (λz. deriv g z / g z) = 0"
    by (simp add: LIMSEQ_const_iff)
  moreover
  have "contour_integral (circlepath z0 (r/2)) (λz. deriv g z / g z) =
        contour_integral (circlepath z0 (r/2)) (λz. m / (z - z0) + deriv h z / h z)"
  proof (rule contour_integral_eq, use ‹0 < r› in simp)
    fix w
    assume w: "dist z0 w * 2 = r"
    then have w_inb: "w ∈ ball z0 r"
      using ‹0 < r› by auto
    have h_der: "(h has_field_derivative deriv h w) (at w)"
      using holh holomorphic_derivI w_inb by blast
    have "deriv g w = ((of_nat m * h w + deriv h w * (w - z0)) * (w - z0) ^ m) / (w - z0)"
         if "r = dist z0 w * 2" "w ≠ z0"
    proof -
      have "((λw. (w - z0) ^ m * h w) has_field_derivative
            (m * h w + deriv h w * (w - z0)) * (w - z0) ^ m / (w - z0)) (at w)"
        apply (rule derivative_eq_intros h_der refl)+
        using that ‹m > 0› ‹0 < r› apply (simp add: divide_simps distrib_right)
        apply (metis Suc_pred mult.commute power_Suc)
        done
      then show ?thesis
        apply (rule DERIV_imp_deriv [OF DERIV_transform_within_open [where S = "ball z0 r"]])
        using that ‹m > 0› ‹0 < r›
          apply (simp_all add: hnz geq)
        done
    qed
    with ‹0 < r› ‹0 < m› w w_inb show "deriv g w / g w = of_nat m / (w - z0) + deriv h w / h w"
      by (auto simp: geq divide_simps hnz)
  qed
  moreover
  have "contour_integral (circlepath z0 (r/2)) (λz. m / (z - z0) + deriv h z / h z) =
        2 * of_real pi * 𝗂 * m + 0"
  proof (rule contour_integral_unique [OF has_contour_integral_add])
    show "((λx. m / (x - z0)) has_contour_integral 2 * of_real pi * 𝗂 * m) (circlepath z0 (r/2))"
      by (force simp: ‹0 < r› intro: Cauchy_integral_circlepath_simple)
    show "((λx. deriv h x / h x) has_contour_integral 0) (circlepath z0 (r/2))"
      apply (rule Cauchy_theorem_disc_simple [of _ z0 r])
      using hnz holh holomorphic_deriv holomorphic_on_divide ‹0 < r›
         apply force+
      done
  qed
  ultimately show False using ‹0 < m› by auto
qed

corollary Hurwitz_injective:
  assumes S: "open S" "connected S"
      and holf: "⋀n::nat. ℱ n holomorphic_on S"
      and holg: "g holomorphic_on S"
      and ul_g: "⋀K. ⟦compact K; K ⊆ S⟧ ⟹ uniform_limit K ℱ g sequentially"
      and nonconst: "~ g constant_on S"
      and inj: "⋀n. inj_on (ℱ n) S"
    shows "inj_on g S"
proof -
  have False if z12: "z1 ∈ S" "z2 ∈ S" "z1 ≠ z2" "g z2 = g z1" for z1 z2
  proof -
    obtain z0 where "z0 ∈ S" and z0: "g z0 ≠ g z2"
      using constant_on_def nonconst by blast
    have "(λz. g z - g z1) holomorphic_on S"
      by (intro holomorphic_intros holg)
    then obtain r where "0 < r" "ball z2 r ⊆ S" "⋀z. dist z2 z < r ∧ z ≠ z2 ⟹ g z ≠ g z1"
      apply (rule isolated_zeros [of "λz. g z - g z1" S z2 z0])
      using S ‹z0 ∈ S› z0 z12 by auto
    have "g z2 - g z1 ≠ 0"
    proof (rule Hurwitz_no_zeros [of "S - {z1}" "λn z. ℱ n z - ℱ n z1" "λz. g z - g z1"])
      show "open (S - {z1})"
        by (simp add: S open_delete)
      show "connected (S - {z1})"
        by (simp add: connected_open_delete [OF S])
      show "⋀n. (λz. ℱ n z - ℱ n z1) holomorphic_on S - {z1}"
        by (intro holomorphic_intros holomorphic_on_subset [OF holf]) blast
      show "(λz. g z - g z1) holomorphic_on S - {z1}"
        by (intro holomorphic_intros holomorphic_on_subset [OF holg]) blast
      show "uniform_limit K (λn z. ℱ n z - ℱ n z1) (λz. g z - g z1) sequentially"
           if "compact K" "K ⊆ S - {z1}" for K
      proof (rule uniform_limitI)
        fix e::real
        assume "e > 0"
        have "uniform_limit K ℱ g sequentially"
          using that ul_g by fastforce
        then have K: "∀F n in sequentially. ∀x ∈ K. dist (ℱ n x) (g x) < e/2"
          using ‹0 < e› by (force simp: intro!: uniform_limitD)
        have "uniform_limit {z1} ℱ g sequentially"
          by (simp add: ul_g z12)
        then have "∀F n in sequentially. ∀x ∈ {z1}. dist (ℱ n x) (g x) < e/2"
          using ‹0 < e› by (force simp: intro!: uniform_limitD)
        then have z1: "∀F n in sequentially. dist (ℱ n z1) (g z1) < e/2"
          by simp
        have "∀F n in sequentially. ∀x∈K. dist (ℱ n x - ℱ n z1) (g x - g z1) < e/2 + e/2"
          apply (rule eventually_mono [OF eventually_conj [OF K z1]])
          apply (simp add: dist_norm algebra_simps del: divide_const_simps)
          by (metis add.commute dist_commute dist_norm dist_triangle_add_half)
        have "∀F n in sequentially. ∀x∈K. dist (ℱ n x - ℱ n z1) (g x - g z1) < e/2 + e/2"
          using eventually_conj [OF K z1]
          apply (rule eventually_mono)
          by (metis (no_types, hide_lams) diff_add_eq diff_diff_eq2 dist_commute dist_norm dist_triangle_add_half field_sum_of_halves)
        then
        show "∀F n in sequentially. ∀x∈K. dist (ℱ n x - ℱ n z1) (g x - g z1) < e"
          by simp
      qed
      show "¬ (λz. g z - g z1) constant_on S - {z1}"
        unfolding constant_on_def
        by (metis Diff_iff ‹z0 ∈ S› empty_iff insert_iff right_minus_eq z0 z12)
      show "⋀n z. z ∈ S - {z1} ⟹ ℱ n z - ℱ n z1 ≠ 0"
        by (metis DiffD1 DiffD2 eq_iff_diff_eq_0 inj inj_onD insertI1 ‹z1 ∈ S›)
      show "z2 ∈ S - {z1}"
        using ‹z2 ∈ S› ‹z1 ≠ z2› by auto
    qed
    with z12 show False by auto
  qed
  then show ?thesis by (auto simp: inj_on_def)
qed



subsection‹The Great Picard theorem›

lemma GPicard1:
  assumes S: "open S" "connected S" and "w ∈ S" "0 < r" "Y ⊆ X"
      and holX: "⋀h. h ∈ X ⟹ h holomorphic_on S"
      and X01:  "⋀h z. ⟦h ∈ X; z ∈ S⟧ ⟹ h z ≠ 0 ∧ h z ≠ 1"
      and r:    "⋀h. h ∈ Y ⟹ norm(h w) ≤ r"
  obtains B Z where "0 < B" "open Z" "w ∈ Z" "Z ⊆ S" "⋀h z. ⟦h ∈ Y; z ∈ Z⟧ ⟹ norm(h z) ≤ B"
proof -
  obtain e where "e > 0" and e: "cball w e ⊆ S"
    using assms open_contains_cball_eq by blast
  show ?thesis
  proof
    show "0 < exp(pi * exp(pi * (2 + 2 * r + 12)))"
      by simp
    show "ball w (e / 2) ⊆ S"
      using e ball_divide_subset_numeral ball_subset_cball by blast
    show "cmod (h z) ≤ exp (pi * exp (pi * (2 + 2 * r + 12)))"
         if "h ∈ Y" "z ∈ ball w (e / 2)" for h z
    proof -
      have "h ∈ X"
        using ‹Y ⊆ X› ‹h ∈ Y›  by blast
      with holX have "h holomorphic_on S" 
        by auto
      then have "h holomorphic_on cball w e"
        by (metis e holomorphic_on_subset)
      then have hol_h_o: "(h ∘ (λz. (w + of_real e * z))) holomorphic_on cball 0 1"
        apply (intro holomorphic_intros holomorphic_on_compose)
        apply (erule holomorphic_on_subset)
        using that ‹e > 0› by (auto simp: dist_norm norm_mult)
      have norm_le_r: "cmod ((h ∘ (λz. w + complex_of_real e * z)) 0) ≤ r"
        by (auto simp: r ‹h ∈ Y›)
      have le12: "norm (of_real(inverse e) * (z - w)) ≤ 1/2"
        using that ‹e > 0› by (simp add: inverse_eq_divide dist_norm norm_minus_commute norm_divide)
      have non01: "⋀z::complex. cmod z ≤ 1 ⟹ h (w + e * z) ≠ 0 ∧ h (w + e * z) ≠ 1"
        apply (rule X01 [OF ‹h ∈ X›])
          apply (rule subsetD [OF e])
        using ‹0 < e›  by (auto simp: dist_norm norm_mult)
      have "cmod (h z) ≤ cmod (h (w + of_real e * (inverse e * (z - w))))"
        using ‹0 < e› by (simp add: divide_simps)
      also have "... ≤ exp (pi * exp (pi * (14 + 2 * r)))"
        using r [OF ‹h ∈ Y›] Schottky [OF hol_h_o norm_le_r _ _ _ le12] non01 by auto
      finally
      show ?thesis by simp
    qed
  qed (use ‹e > 0› in auto)
qed 

lemma GPicard2:
  assumes "S ⊆ T" "connected T" "S ≠ {}" "open S" "⋀x. ⟦x islimpt S; x ∈ T⟧ ⟹ x ∈ S"
    shows "S = T"
  by (metis assms open_subset connected_clopen closedin_limpt)

    
lemma GPicard3:
  assumes S: "open S" "connected S" "w ∈ S" and "Y ⊆ X"
      and holX: "⋀h. h ∈ X ⟹ h holomorphic_on S"
      and X01:  "⋀h z. ⟦h ∈ X; z ∈ S⟧ ⟹ h z ≠ 0 ∧ h z ≠ 1"
      and no_hw_le1: "⋀h. h ∈ Y ⟹ norm(h w) ≤ 1"
      and "compact K" "K ⊆ S"
  obtains B where "⋀h z. ⟦h ∈ Y; z ∈ K⟧ ⟹ norm(h z) ≤ B"
proof -
  define U where "U ≡ {z ∈ S. ∃B Z. 0 < B ∧ open Z ∧ z ∈ Z ∧ Z ⊆ S ∧
                               (∀h z'. h ∈ Y ∧ z' ∈ Z ⟶ norm(h z') ≤ B)}"
  then have "U ⊆ S" by blast
  have "U = S"
  proof (rule GPicard2 [OF ‹U ⊆ S› ‹connected S›])
    show "U ≠ {}"
    proof -
      obtain B Z where "0 < B" "open Z" "w ∈ Z" "Z ⊆ S" 
        and  "⋀h z. ⟦h ∈ Y; z ∈ Z⟧ ⟹ norm(h z) ≤ B"
        apply (rule GPicard1 [OF S zero_less_one ‹Y ⊆ X› holX])
        using no_hw_le1 X01 by force+
      then show ?thesis
        unfolding U_def using ‹w ∈ S› by blast
    qed
    show "open U"
      unfolding open_subopen [of U] by (auto simp: U_def)
    fix v
    assume v: "v islimpt U" "v ∈ S"
    have "~ (∀r>0. ∃h∈Y. r < cmod (h v))"
    proof
      assume "∀r>0. ∃h∈Y. r < cmod (h v)"
      then have "∀n. ∃h∈Y. Suc n < cmod (h v)"
        by simp
      then obtain  where FY: "⋀n. ℱ n ∈ Y" and ltF: "⋀n. Suc n < cmod (ℱ n v)"
        by metis
      define 𝒢 where "𝒢 ≡ λn z. inverse(ℱ n z)"
      have hol𝒢: "𝒢 n holomorphic_on S" for n
        apply (simp add: 𝒢_def)
        using FY X01 ‹Y ⊆ X› holX apply (blast intro: holomorphic_on_inverse)
        done
      have 𝒢not0: "𝒢 n z ≠ 0" and 𝒢not1: "𝒢 n z ≠ 1" if "z ∈ S" for n z
        using FY X01 ‹Y ⊆ X› that by (force simp: 𝒢_def)+
      have 𝒢_le1: "cmod (𝒢 n v) ≤ 1" for n 
        using less_le_trans linear ltF 
        by (fastforce simp add: 𝒢_def norm_inverse inverse_le_1_iff)
      define W where "W ≡ {h. h holomorphic_on S ∧ (∀z ∈ S. h z ≠ 0 ∧ h z ≠ 1)}"
      obtain B Z where "0 < B" "open Z" "v ∈ Z" "Z ⊆ S" 
                   and B: "⋀h z. ⟦h ∈ range 𝒢; z ∈ Z⟧ ⟹ norm(h z) ≤ B"
        apply (rule GPicard1 [OF ‹open S› ‹connected S› ‹v ∈ S› zero_less_one, of "range 𝒢" W])
        using hol𝒢 𝒢not0 𝒢not1 𝒢_le1 by (force simp: W_def)+
      then obtain e where "e > 0" and e: "ball v e ⊆ Z"
        by (meson open_contains_ball)
      obtain h j where holh: "h holomorphic_on ball v e" and "strict_mono j"
                   and lim:  "⋀x. x ∈ ball v e ⟹ (λn. 𝒢 (j n) x) ⇢ h x"
                   and ulim: "⋀K. ⟦compact K; K ⊆ ball v e⟧
                                  ⟹ uniform_limit K (𝒢 ∘ j) h sequentially"
      proof (rule Montel)
        show "⋀h. h ∈ range 𝒢 ⟹ h holomorphic_on ball v e"
          by (metis ‹Z ⊆ S› e hol𝒢 holomorphic_on_subset imageE)
        show "⋀K. ⟦compact K; K ⊆ ball v e⟧ ⟹ ∃B. ∀h∈range 𝒢. ∀z∈K. cmod (h z) ≤ B"
          using B e by blast
      qed auto
      have "h v = 0"
      proof (rule LIMSEQ_unique)
        show "(λn. 𝒢 (j n) v) ⇢ h v"
          using ‹e > 0› lim by simp
        have lt_Fj: "real x ≤ cmod (ℱ (j x) v)" for x
          by (metis of_nat_Suc ltF ‹strict_mono j› add.commute less_eq_real_def less_le_trans nat_le_real_less seq_suble)
        show "(λn. 𝒢 (j n) v) ⇢ 0"
        proof (rule Lim_null_comparison [OF eventually_sequentiallyI lim_inverse_n])
          show "cmod (𝒢 (j x) v) ≤ inverse (real x)" if "1 ≤ x" for x
            using that by (simp add: 𝒢_def norm_inverse_le_norm [OF lt_Fj])
        qed        
      qed
      have "h v ≠ 0"
      proof (rule Hurwitz_no_zeros [of "ball v e" "𝒢 ∘ j" h])
        show "⋀n. (𝒢 ∘ j) n holomorphic_on ball v e"
          using ‹Z ⊆ S› e hol𝒢 by force
        show "⋀n z. z ∈ ball v e ⟹ (𝒢 ∘ j) n z ≠ 0"
          using 𝒢not0 ‹Z ⊆ S› e by fastforce
        show "¬ h constant_on ball v e"
        proof (clarsimp simp: constant_on_def)
          fix c
          have False if "⋀z. dist v z < e ⟹ h z = c"  
          proof -
            have "h v = c"
              by (simp add: ‹0 < e› that)
            obtain y where "y ∈ U" "y ≠ v" and y: "dist y v < e"
              using v ‹e > 0› by (auto simp: islimpt_approachable)
            then obtain C T where "y ∈ S" "C > 0" "open T" "y ∈ T" "T ⊆ S"
              and "⋀h z'. ⟦h ∈ Y; z' ∈ T⟧ ⟹ cmod (h z') ≤ C"
              using ‹y ∈ U› by (auto simp: U_def)
            then have le_C: "⋀n. cmod (ℱ n y) ≤ C"
              using FY by blast                
            have "∀F n in sequentially. dist (𝒢 (j n) y) (h y) < inverse C"
              using uniform_limitD [OF ulim [of "{y}"], of "inverse C"] ‹C > 0› y
              by (simp add: dist_commute)
            then obtain n where "dist (𝒢 (j n) y) (h y) < inverse C"
              by (meson eventually_at_top_linorder order_refl)
            moreover
            have "h y = h v"
              by (metis ‹h v = c› dist_commute that y)
            ultimately have "norm (𝒢 (j n) y) < inverse C"
              by (simp add: ‹h v = 0›)
            then have "C < norm (ℱ (j n) y)"
              apply (simp add: 𝒢_def)
              by (metis FY X01 ‹0 < C› ‹y ∈ S› ‹Y ⊆ X› inverse_less_iff_less norm_inverse subsetD zero_less_norm_iff)
            show False
              using ‹C < cmod (ℱ (j n) y)› le_C not_less by blast
          qed
          then show "∃x∈ball v e. h x ≠ c" by force
        qed
        show "h holomorphic_on ball v e"
          by (simp add: holh)
        show "⋀K. ⟦compact K; K ⊆ ball v e⟧ ⟹ uniform_limit K (𝒢 ∘ j) h sequentially"
          by (simp add: ulim)
      qed (use ‹e > 0› in auto)
      with ‹h v = 0› show False by blast
    qed
    then show "v ∈ U"
      apply (clarsimp simp add: U_def v)
      apply (rule GPicard1[OF ‹open S› ‹connected S› ‹v ∈ S› _ ‹Y ⊆ X› holX])
      using X01 no_hw_le1 apply (meson | force simp: not_less)+
      done
  qed
  have "⋀x. x ∈ K ⟶ x ∈ U"
    using ‹U = S› ‹K ⊆ S› by blast
  then have "⋀x. x ∈ K ⟶ (∃B Z. 0 < B ∧ open Z ∧ x ∈ Z ∧ 
                               (∀h z'. h ∈ Y ∧ z' ∈ Z ⟶ norm(h z') ≤ B))"
    unfolding U_def by blast
  then obtain F Z where F: "⋀x. x ∈ K ⟹ open (Z x) ∧ x ∈ Z x ∧ 
                               (∀h z'. h ∈ Y ∧ z' ∈ Z x ⟶ norm(h z') ≤ F x)"
    by metis
  then obtain L where "L ⊆ K" "finite L" and L: "K ⊆ (⋃c ∈ L. Z c)"
    by (auto intro: compactE_image [OF ‹compact K›, of K Z])
  then have *: "⋀x h z'. ⟦x ∈ L; h ∈ Y ∧ z' ∈ Z x⟧ ⟹ cmod (h z') ≤ F x"
    using F by blast
  have "∃B. ∀h z. h ∈ Y ∧ z ∈ K ⟶ norm(h z) ≤ B"
  proof (cases "L = {}")
    case True with L show ?thesis by simp
  next
    case False
    with ‹finite L› show ?thesis 
      apply (rule_tac x = "Max (F ` L)" in exI)
      apply (simp add: linorder_class.Max_ge_iff)
      using * F  by (metis L UN_E subsetD)
  qed
  with that show ?thesis by metis
qed


lemma GPicard4:
  assumes "0 < k" and holf: "f holomorphic_on (ball 0 k - {0})" 
      and AE: "⋀e. ⟦0 < e; e < k⟧ ⟹ ∃d. 0 < d ∧ d < e ∧ (∀z ∈ sphere 0 d. norm(f z) ≤ B)"
  obtains ε where "0 < ε" "ε < k" "⋀z. z ∈ ball 0 ε - {0} ⟹ norm(f z) ≤ B"
proof -
  obtain ε where "0 < ε" "ε < k/2" and ε: "⋀z. norm z = ε ⟹ norm(f z) ≤ B"
    using AE [of "k/2"] ‹0 < k› by auto
  show ?thesis
  proof
    show "ε < k"
      using ‹0 < k› ‹ε < k/2› by auto
    show "cmod (f ξ) ≤ B" if ξ: "ξ ∈ ball 0 ε - {0}" for ξ
    proof -
      obtain d where "0 < d" "d < norm ξ" and d: "⋀z. norm z = d ⟹ norm(f z) ≤ B"
        using AE [of "norm ξ"] ‹ε < k› ξ by auto
      have [simp]: "closure (cball 0 ε - ball 0 d) = cball 0 ε - ball 0 d"
        by (blast intro!: closure_closed)
      have [simp]: "interior (cball 0 ε - ball 0 d) = ball 0 ε - cball (0::complex) d"
        using ‹0 < ε› ‹0 < d› by (simp add: interior_diff)
      have *: "norm(f w) ≤ B" if "w ∈ cball 0 ε - ball 0 d" for w
      proof (rule maximum_modulus_frontier [of f "cball 0 ε - ball 0 d"])
        show "f holomorphic_on interior (cball 0 ε - ball 0 d)"
          apply (rule holomorphic_on_subset [OF holf])
          using ‹ε < k› ‹0 < d› that by auto
        show "continuous_on (closure (cball 0 ε - ball 0 d)) f"
          apply (rule holomorphic_on_imp_continuous_on)
          apply (rule holomorphic_on_subset [OF holf])
          using ‹0 < d› ‹ε < k› by auto
        show "⋀z. z ∈ frontier (cball 0 ε - ball 0 d) ⟹ cmod (f z) ≤ B"
          apply (simp add: frontier_def)
          using ε d less_eq_real_def by blast
      qed (use that in auto)
      show ?thesis
        using * ‹d < cmod ξ› that by auto
    qed
  qed (use ‹0 < ε› in auto)
qed
  

lemma GPicard5:
  assumes holf: "f holomorphic_on (ball 0 1 - {0})"
      and f01:  "⋀z. z ∈ ball 0 1 - {0} ⟹ f z ≠ 0 ∧ f z ≠ 1"
  obtains e B where "0 < e" "e < 1" "0 < B" 
                    "(∀z ∈ ball 0 e - {0}. norm(f z) ≤ B) ∨
                     (∀z ∈ ball 0 e - {0}. norm(f z) ≥ B)"
proof -
  have [simp]: "1 + of_nat n ≠ (0::complex)" for n
    using of_nat_eq_0_iff by fastforce
  have [simp]: "cmod (1 + of_nat n) = 1 + of_nat n" for n
    by (metis norm_of_nat of_nat_Suc)
  have *: "(λx::complex. x / of_nat (Suc n)) ` (ball 0 1 - {0}) ⊆ ball 0 1 - {0}" for n
    by (auto simp: norm_divide divide_simps split: if_split_asm)
  define h where "h ≡ λn z::complex. f (z / (Suc n))"
  have holh: "(h n) holomorphic_on ball 0 1 - {0}" for n
    unfolding h_def
  proof (rule holomorphic_on_compose_gen [unfolded o_def, OF _ holf *])
    show "(λx. x / of_nat (Suc n)) holomorphic_on ball 0 1 - {0}"
      by (intro holomorphic_intros) auto
  qed
  have h01: "⋀n z. z ∈ ball 0 1 - {0} ⟹ h n z ≠ 0 ∧ h n z ≠ 1" 
    unfolding h_def
    apply (rule f01)
    using * by force
  obtain w where w: "w ∈ ball 0 1 - {0::complex}"
    by (rule_tac w = "1/2" in that) auto
  consider "infinite {n. norm(h n w) ≤ 1}" | "infinite {n. 1 ≤ norm(h n w)}"
    by (metis (mono_tags, lifting) infinite_nat_iff_unbounded_le le_cases mem_Collect_eq)
  then show ?thesis
  proof cases
    case 1
    with infinite_enumerate obtain r :: "nat ⇒ nat" 
      where "strict_mono r" and r: "⋀n. r n ∈ {n. norm(h n w) ≤ 1}"
      by blast
    obtain B where B: "⋀j z. ⟦norm z = 1/2; j ∈ range (h ∘ r)⟧ ⟹ norm(j z) ≤ B"
    proof (rule GPicard3 [OF _ _ w, where K = "sphere 0 (1/2)"])  
      show "range (h ∘ r) ⊆ 
            {g. g holomorphic_on ball 0 1 - {0} ∧ (∀z∈ball 0 1 - {0}. g z ≠ 0 ∧ g z ≠ 1)}"
        apply clarsimp
        apply (intro conjI holomorphic_intros holomorphic_on_compose holh)
        using h01 apply auto
        done
      show "connected (ball 0 1 - {0::complex})"
        by (simp add: connected_open_delete)
    qed (use r in auto)        
    have normf_le_B: "cmod(f z) ≤ B" if "norm z = 1 / (2 * (1 + of_nat (r n)))" for z n
    proof -
      have *: "⋀w. norm w = 1/2 ⟹ cmod((f (w / (1 + of_nat (r n))))) ≤ B"
        using B by (auto simp: h_def o_def)
      have half: "norm (z * (1 + of_nat (r n))) = 1/2"
        by (simp add: norm_mult divide_simps that)
      show ?thesis
        using * [OF half] by simp
    qed
    obtain ε where "0 < ε" "ε < 1" "⋀z. z ∈ ball 0 ε - {0} ⟹ cmod(f z) ≤ B"
    proof (rule GPicard4 [OF zero_less_one holf, of B])
      fix e::real
      assume "0 < e" "e < 1"
      obtain n where "(1/e - 2) / 2 < real n"
        using reals_Archimedean2 by blast
      also have "... ≤ r n"
        using ‹strict_mono r› by (simp add: seq_suble)
      finally have "(1/e - 2) / 2 < real (r n)" .
      with ‹0 < e› have e: "e > 1 / (2 + 2 * real (r n))"
        by (simp add: field_simps)
      show "∃d>0. d < e ∧ (∀z∈sphere 0 d. cmod (f z) ≤ B)"
        apply (rule_tac x="1 / (2 * (1 + of_nat (r n)))" in exI)
        using normf_le_B by (simp add: e)
    qed blast
    then have ε: "cmod (f z) ≤ ¦B¦ + 1" if "cmod z < ε" "z ≠ 0" for z
      using that by fastforce
    have "0 < ¦B¦ + 1"
      by simp
    then show ?thesis
      apply (rule that [OF ‹0 < ε› ‹ε < 1›])
      using ε by auto 
  next
    case 2
    with infinite_enumerate obtain r :: "nat ⇒ nat" 
      where "strict_mono r" and r: "⋀n. r n ∈ {n. norm(h n w) ≥ 1}"
      by blast
    obtain B where B: "⋀j z. ⟦norm z = 1/2; j ∈ range (λn. inverse ∘ h (r n))⟧ ⟹ norm(j z) ≤ B"
    proof (rule GPicard3 [OF _ _ w, where K = "sphere 0 (1/2)"])  
      show "range (λn. inverse ∘ h (r n)) ⊆ 
            {g. g holomorphic_on ball 0 1 - {0} ∧ (∀z∈ball 0 1 - {0}. g z ≠ 0 ∧ g z ≠ 1)}"
        apply clarsimp
        apply (intro conjI holomorphic_intros holomorphic_on_compose_gen [unfolded o_def, OF _ holh] holomorphic_on_compose)
        using h01 apply auto
        done
      show "connected (ball 0 1 - {0::complex})"
        by (simp add: connected_open_delete)
      show "⋀j. j ∈ range (λn. inverse ∘ h (r n)) ⟹ cmod (j w) ≤ 1"
        using r norm_inverse_le_norm by fastforce
    qed (use r in auto)        
    have norm_if_le_B: "cmod(inverse (f z)) ≤ B" if "norm z = 1 / (2 * (1 + of_nat (r n)))" for z n
    proof -
      have *: "inverse (cmod((f (z / (1 + of_nat (r n)))))) ≤ B" if "norm z = 1/2" for z
        using B [OF that] by (force simp: norm_inverse h_def)
      have half: "norm (z * (1 + of_nat (r n))) = 1/2"
        by (simp add: norm_mult divide_simps that)
      show ?thesis
        using * [OF half] by (simp add: norm_inverse)
    qed
    have hol_if: "(inverse ∘ f) holomorphic_on (ball 0 1 - {0})"
      by (metis (no_types, lifting) holf comp_apply f01 holomorphic_on_inverse holomorphic_transform)
    obtain ε where "0 < ε" "ε < 1" and leB: "⋀z. z ∈ ball 0 ε - {0} ⟹ cmod((inverse ∘ f) z) ≤ B"
    proof (rule GPicard4 [OF zero_less_one hol_if, of B])
      fix e::real
      assume "0 < e" "e < 1"
      obtain n where "(1/e - 2) / 2 < real n"
        using reals_Archimedean2 by blast
      also have "... ≤ r n"
        using ‹strict_mono r› by (simp add: seq_suble)
      finally have "(1/e - 2) / 2 < real (r n)" .
      with ‹0 < e› have e: "e > 1 / (2 + 2 * real (r n))"
        by (simp add: field_simps)
      show "∃d>0. d < e ∧ (∀z∈sphere 0 d. cmod ((inverse ∘ f) z) ≤ B)"
        apply (rule_tac x="1 / (2 * (1 + of_nat (r n)))" in exI)
        using norm_if_le_B by (simp add: e)
    qed blast
    have ε: "cmod (f z) ≥ inverse B" and "B > 0" if "cmod z < ε" "z ≠ 0" for z
    proof -
      have "inverse (cmod (f z)) ≤ B"
        using leB that by (simp add: norm_inverse)
      moreover
      have "f z ≠ 0"
        using ‹ε < 1› f01 that by auto
      ultimately show "cmod (f z) ≥ inverse B"
        by (simp add: norm_inverse inverse_le_imp_le)
      show "B > 0"
        using ‹f z ≠ 0› ‹inverse (cmod (f z)) ≤ B› not_le order.trans by fastforce
    qed
    then have "B > 0"
      by (metis ‹0 < ε› dense leI order.asym vector_choose_size)
    then have "inverse B > 0"
      by (simp add: divide_simps)
    then show ?thesis
      apply (rule that [OF ‹0 < ε› ‹ε < 1›])
      using ε by auto 
  qed
qed

  
lemma GPicard6:
  assumes "open M" "z ∈ M" "a ≠ 0" and holf: "f holomorphic_on (M - {z})"
      and f0a: "⋀w. w ∈ M - {z} ⟹ f w ≠ 0 ∧ f w ≠ a"
  obtains r where "0 < r" "ball z r ⊆ M" 
                  "bounded(f ` (ball z r - {z})) ∨
                   bounded((inverse ∘ f) ` (ball z r - {z}))"
proof -
  obtain r where "0 < r" and r: "ball z r ⊆ M"
    using assms openE by blast 
  let ?g = "λw. f (z + of_real r * w) / a"
  obtain e B where "0 < e" "e < 1" "0 < B" 
    and B: "(∀z ∈ ball 0 e - {0}. norm(?g z) ≤ B) ∨ (∀z ∈ ball 0 e - {0}. norm(?g z) ≥ B)"
  proof (rule GPicard5)
    show "?g holomorphic_on ball 0 1 - {0}"
      apply (intro holomorphic_intros holomorphic_on_compose_gen [unfolded o_def, OF _ holf])
      using ‹0 < r› ‹a ≠ 0› r
      by (auto simp: dist_norm norm_mult subset_eq)
    show "⋀w. w ∈ ball 0 1 - {0} ⟹ f (z + of_real r * w) / a ≠ 0 ∧ f (z + of_real r * w) / a ≠ 1"
      apply (simp add: divide_simps ‹a ≠ 0›)
      apply (rule f0a)
      using ‹0 < r› r by (auto simp: dist_norm norm_mult subset_eq)
  qed
  show ?thesis
  proof
    show "0 < e*r"
      by (simp add: ‹0 < e› ‹0 < r›)
    have "ball z (e * r) ⊆ ball z r"
      by (simp add: ‹0 < r› ‹e < 1› order.strict_implies_order subset_ball)
    then show "ball z (e * r) ⊆ M"
      using r by blast
    consider "⋀z. z ∈ ball 0 e - {0} ⟹ norm(?g z) ≤ B" | "⋀z. z ∈ ball 0 e - {0} ⟹ norm(?g z) ≥ B"
      using B by blast
    then show "bounded (f ` (ball z (e * r) - {z})) ∨
          bounded ((inverse ∘ f) ` (ball z (e * r) - {z}))"
    proof cases
      case 1
      have "⟦dist z w < e * r; w ≠ z⟧ ⟹ cmod (f w) ≤ B * norm a" for w
        using ‹a ≠ 0› ‹0 < r› 1 [of "(w - z) / r"]
        by (simp add: norm_divide dist_norm divide_simps)
      then show ?thesis
        by (force simp: intro!: boundedI)
    next
      case 2
      have "⟦dist z w < e * r; w ≠ z⟧ ⟹ cmod (f w) ≥ B * norm a" for w
        using ‹a ≠ 0› ‹0 < r› 2 [of "(w - z) / r"]
        by (simp add: norm_divide dist_norm divide_simps)
      then have "⟦dist z w < e * r; w ≠ z⟧ ⟹ inverse (cmod (f w)) ≤ inverse (B * norm a)" for w
        by (metis ‹0 < B› ‹a ≠ 0› mult_pos_pos norm_inverse norm_inverse_le_norm zero_less_norm_iff)
      then show ?thesis 
        by (force simp: norm_inverse intro!: boundedI)
    qed
  qed
qed
  

theorem great_Picard:
  assumes "open M" "z ∈ M" "a ≠ b" and holf: "f holomorphic_on (M - {z})"
      and fab: "⋀w. w ∈ M - {z} ⟹ f w ≠ a ∧ f w ≠ b"
  obtains l where "(f ⤏ l) (at z) ∨ ((inverse ∘ f) ⤏ l) (at z)"
proof -
  obtain r where "0 < r" and zrM: "ball z r ⊆ M" 
             and r: "bounded((λz. f z - a) ` (ball z r - {z})) ∨
                     bounded((inverse ∘ (λz. f z - a)) ` (ball z r - {z}))"
  proof (rule GPicard6 [OF ‹open M› ‹z ∈ M›])
    show "b - a ≠ 0"
      using assms by auto
    show "(λz. f z - a) holomorphic_on M - {z}"
      by (intro holomorphic_intros holf)
  qed (use fab in auto)
  have holfb: "f holomorphic_on ball z r - {z}"
    apply (rule holomorphic_on_subset [OF holf])
    using zrM by auto
  have holfb_i: "(λz. inverse(f z - a)) holomorphic_on ball z r - {z}"
    apply (intro holomorphic_intros holfb)
    using fab zrM by fastforce
  show ?thesis
    using r
  proof              
    assume "bounded ((λz. f z - a) ` (ball z r - {z}))"
    then obtain B where B: "⋀w. w ∈ (λz. f z - a) ` (ball z r - {z}) ⟹ norm w ≤ B"
      by (force simp: bounded_iff)
    have "∀F w in at z. cmod (f w - a) ≤ B"
      apply (simp add: eventually_at)
      apply (rule_tac x=r in exI)
      using ‹0 < r› by (auto simp: dist_commute intro!: B)
    then have "∃B. ∀F w in at z. cmod (f w) ≤ B"
      apply (rule_tac x="B + norm a" in exI)
        apply (erule eventually_mono)
      by (metis add.commute add_le_cancel_right norm_triangle_sub order.trans)
    then obtain g where holg: "g holomorphic_on ball z r" and gf: "⋀w. w ∈ ball z r - {z} ⟹ g w = f w"
      using ‹0 < r› holomorphic_on_extend_bounded [OF holfb] by auto
    then have "g ─z→ g z"
      apply (simp add: continuous_at [symmetric])
      using ‹0 < r› centre_in_ball field_differentiable_imp_continuous_at holomorphic_on_imp_differentiable_at by blast
    then have "(f ⤏ g z) (at z)"
      apply (rule Lim_transform_within_open [of g "g z" z UNIV "ball z r"])
      using  ‹0 < r› by (auto simp: gf)
    then show ?thesis
      using that by blast
  next
    assume "bounded((inverse ∘ (λz. f z - a)) ` (ball z r - {z}))"
    then obtain B where B: "⋀w. w ∈ (inverse ∘ (λz. f z - a)) ` (ball z r - {z}) ⟹ norm w ≤ B"
      by (force simp: bounded_iff)
    have "∀F w in at z. cmod (inverse (f w - a)) ≤ B"
      apply (simp add: eventually_at)
      apply (rule_tac x=r in exI)
      using ‹0 < r› by (auto simp: dist_commute intro!: B)
    then have "∃B. ∀F z in at z. cmod (inverse (f z - a)) ≤ B"
      by blast
    then obtain g where holg: "g holomorphic_on ball z r" and gf: "⋀w. w ∈ ball z r - {z} ⟹ g w = inverse (f w - a)"
      using ‹0 < r› holomorphic_on_extend_bounded [OF holfb_i] by auto
    then have gz: "g ─z→ g z"
      apply (simp add: continuous_at [symmetric])
      using ‹0 < r› centre_in_ball field_differentiable_imp_continuous_at holomorphic_on_imp_differentiable_at by blast
    have gnz: "⋀w. w ∈ ball z r - {z} ⟹ g w ≠ 0"
      using gf fab zrM by fastforce
    show ?thesis
    proof (cases "g z = 0")
      case True
      have *: "⟦g ≠ 0; inverse g = f - a⟧ ⟹ g / (1 + a * g) = inverse f" for f g::complex
        by (auto simp: field_simps)
      have "(inverse ∘ f) ─z→ 0"
      proof (rule Lim_transform_within_open [of "λw. g w / (1 + a * g w)" _ _ UNIV "ball z r"])
        show "(λw. g w / (1 + a * g w)) ─z→ 0"
          using True by (auto simp: intro!: tendsto_eq_intros gz)
        show "⋀x. ⟦x ∈ ball z r; x ≠ z⟧ ⟹ g x / (1 + a * g x) = (inverse ∘ f) x"
          using * gf gnz by simp
      qed (use ‹0 < r› in auto)
      with that show ?thesis by blast
    next
      case False
      show ?thesis
      proof (cases "1 + a * g z = 0")
        case True
        have "(f ⤏ 0) (at z)"
        proof (rule Lim_transform_within_open [of "λw. (1 + a * g w) / g w" _ _ _ "ball z r"])
          show "(λw. (1 + a * g w) / g w) ─z→ 0"
            apply (rule tendsto_eq_intros refl gz ‹g z ≠ 0›)+
            by (simp add: True)
          show "⋀x. ⟦x ∈ ball z r; x ≠ z⟧ ⟹ (1 + a * g x) / g x = f x"
            using fab fab zrM by (fastforce simp add: gf divide_simps)
        qed (use ‹0 < r› in auto)
        then show ?thesis
          using that by blast 
      next
        case False
        have *: "⟦g ≠ 0; inverse g = f - a⟧ ⟹ g / (1 + a * g) = inverse f" for f g::complex
          by (auto simp: field_simps)
        have "(inverse ∘ f) ─z→ g z / (1 + a * g z)"
        proof (rule Lim_transform_within_open [of "λw. g w / (1 + a * g w)" _ _ UNIV "ball z r"])
          show "(λw. g w / (1 + a * g w)) ─z→ g z / (1 + a * g z)"
            using False by (auto simp: False intro!: tendsto_eq_intros gz)
          show "⋀x. ⟦x ∈ ball z r; x ≠ z⟧ ⟹ g x / (1 + a * g x) = (inverse ∘ f) x"
            using * gf gnz by simp
        qed (use ‹0 < r› in auto)
        with that show ?thesis by blast
      qed
    qed 
  qed
qed


corollary great_Picard_alt:
  assumes M: "open M" "z ∈ M" and holf: "f holomorphic_on (M - {z})"
    and non: "⋀l. ¬ (f ⤏ l) (at z)" "⋀l. ¬ ((inverse ∘ f) ⤏ l) (at z)"
  obtains a where "- {a} ⊆ f ` (M - {z})"
  apply (simp add: subset_iff image_iff)
  by (metis great_Picard [OF M _ holf] non)
    

corollary great_Picard_infinite:
  assumes M: "open M" "z ∈ M" and holf: "f holomorphic_on (M - {z})"
    and non: "⋀l. ¬ (f ⤏ l) (at z)" "⋀l. ¬ ((inverse ∘ f) ⤏ l) (at z)"
  obtains a where "⋀w. w ≠ a ⟹ infinite {x. x ∈ M - {z} ∧ f x = w}"
proof -
  have False if "a ≠ b" and ab: "finite {x. x ∈ M - {z} ∧ f x = a}" "finite {x. x ∈ M - {z} ∧ f x = b}" for a b
  proof -
    have finab: "finite {x. x ∈ M - {z} ∧ f x ∈ {a,b}}"
      using finite_UnI [OF ab]  unfolding mem_Collect_eq insert_iff empty_iff
      by (simp add: conj_disj_distribL)
    obtain r where "0 < r" and zrM: "ball z r ⊆ M" and r: "⋀x. ⟦x ∈ M - {z}; f x ∈ {a,b}⟧ ⟹ x ∉ ball z r"
    proof -
      obtain e where "e > 0" and e: "ball z e ⊆ M"
        using assms openE by blast
      show ?thesis
      proof (cases "{x ∈ M - {z}. f x ∈ {a, b}} = {}")
        case True
        then show ?thesis
          apply (rule_tac r=e in that)
          using e ‹e > 0› by auto
      next
        case False
        let ?r = "min e (Min (dist z ` {x ∈ M - {z}. f x ∈ {a,b}}))"
        show ?thesis
        proof
          show "0 < ?r"
            using min_less_iff_conj Min_gr_iff finab False ‹0 < e› by auto
          have "ball z ?r ⊆ ball z e"
            by (simp add: subset_ball)
          with e show "ball z ?r ⊆ M" by blast
          show "⋀x. ⟦x ∈ M - {z}; f x ∈ {a, b}⟧ ⟹ x ∉ ball z ?r"
            using min_less_iff_conj Min_gr_iff finab False ‹0 < e› by auto
        qed
      qed
    qed
    have holfb: "f holomorphic_on (ball z r - {z})"
      apply (rule holomorphic_on_subset [OF holf])
       using zrM by auto
     show ?thesis
       apply (rule great_Picard [OF open_ball _ ‹a ≠ b› holfb])
      using non ‹0 < r› r zrM by auto
  qed
  with that show thesis
    by meson
qed

corollary Casorati_Weierstrass:
  assumes "open M" "z ∈ M" "f holomorphic_on (M - {z})"
      and "⋀l. ¬ (f ⤏ l) (at z)" "⋀l. ¬ ((inverse ∘ f) ⤏ l) (at z)"
  shows "closure(f ` (M - {z})) = UNIV"
proof -
  obtain a where a: "- {a} ⊆ f ` (M - {z})"
    using great_Picard_alt [OF assms] .
  have "UNIV = closure(- {a})"
    by (simp add: closure_interior)
  also have "... ⊆ closure(f ` (M - {z}))"
    by (simp add: a closure_mono)
  finally show ?thesis
    by blast 
qed
  
end