Quelle  Great_Picard.thy

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
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 "0 < n * n"
    by (simp add: assms)
  then show ?thesis
    by (metis add.commute add.right_neutral add_pos_nonneg assms diff_ge_0_iff_ge nat_less_real_le of_nat_0 of_nat_0_less_iff of_nat_power power2_eq_square real_sqrt_ge_0_iff)
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: field_split_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 🚫› by auto
    have "n + sqrt ((real n)🪙2 - 1) = exp (ln (n + sqrt ((real n)🪙2 - 1)))"
      by (simp add: Suc_leI ‹0 🚫› 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∧ 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 🚫› Schottky_lemma1 add_pos_nonneg ln_divide_pos [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 🚫› 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, opaque_lifting) 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 🚫›  by (simp add: field_split_simps)
      then have "ln ((1 + real n + sqrt ((1 + real n)🪙2 - 1)) / (real n + sqrt ((real n)🪙2 - 1))) ≤ ln 2"
        using Schottky_lemma1 [of n] ‹0 🚫›
        by (simp add: field_split_simps add_pos_nonneg)
      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 field_split_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 🚫›)
  next
    assume "∣x - b∣ < 1/2"
    then show ?thesis
      by (rule_tac n="Suc n" in that) (auto simp: b_def ‹0 🚫›)
  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)
  define plusi where "plusi (e::complex) ≡ e + inverse e" for e
  have 1: "∃k. plusi (exp (i * (of_int m * complex_of_real pi) - ln (real n + sqrt ((real n)🪙2 - 1)))) = of_int k * 2" 
           (is "∃k. ?Φ k")
         if "n > 0" for m n
  proof -
    have eeq: "e ≠ 0 ==> plusi e = n ⟷ (inverse e) ^ 2 = n/e - 1" for n e::complex
      by (auto simp: plusi_def field_simps power2_eq_square)
    have [simp]: "1 ≤ real n * real n"
      using nat_0_less_mult_iff nat_less_real_le that by force
    consider "odd m" | "even m"
      by blast
    then have "∃k. ?Φ k"
    proof cases
      case 1
      then have "?Φ (- n)"
        using Schottky_lemma1 [OF that]
        by (simp add: eeq) (simp add: power2_eq_square exp_diff exp_Euler exp_of_real algebra_simps sin_int_times_real cos_int_times_real)
      then show ?thesis ..
    next
      case 2
      then have "?Φ n"
        using Schottky_lemma1 [OF that]
        by (simp add: eeq) (simp add: power2_eq_square exp_diff exp_Euler exp_of_real algebra_simps)
      then show ?thesis ..
    qed
    then show ?thesis by blast
  qed
  have 2: "∃k. plusi (exp (i * (of_int m * complex_of_real pi) +
                      (ln (real n + sqrt ((real n)🪙2 - 1))))) = of_int k * 2"
           (is "∃k. ?Φ k")
            if "n > 0" for m n
  proof -
    have eeq: "e ≠ 0 ==> plusi e = n ⟷ e^2 - n*e + 1 = 0" for n e::complex
      by (auto simp: plusi_def 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)
    consider "odd m" | "even m"
      by blast
    then have "∃k. ?Φ k"
    proof cases
      case 1
      then have "?Φ (- n)"
        using Schottky_lemma1 [OF that]
        by (simp add: eeq) (simp add: power2_eq_square exp_add exp_Euler exp_of_real algebra_simps sin_int_times_real cos_int_times_real)
      then show ?thesis ..
    next
      case 2
      then have "?Φ n"
        using Schottky_lemma1 [OF that]
        by (simp add: eeq) (simp add: power2_eq_square exp_add exp_Euler exp_of_real algebra_simps)
      then show ?thesis ..
    qed
    then show ?thesis by blast
  qed
  have "∃x. cos (complex_of_real pi * z) = of_int x"
    using assms
    apply (auto simp: Ints_def cos_exp_eq exp_minus Complex_eq simp flip: plusi_def)
     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 "... ≤ 6 + 9 * cmod (f 0)"
      by auto
    finally have "1 + norm(2 * f 0 - 1) / 3 ≤ (2 + norm(f 0) - 1) * 3"
      by (simp add: divide_simps)
    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 *: "[∧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 ttt: "1 - t ≤ dist w u" if "cmod u = 1" for u
        using ‹norm z ≤ t› segment_bound1 [OF ‹w ∈ closed_segment 0 z›] norm_triangle_ineq2 [of u w] that
        by (simp add: dist_norm norm_minus_commute)
      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 🚫› show ?thesis
              apply (rule_tac x = "Complex m (ln (real n + sqrt ((real n)🪙2 - 1)) / pi)" in bexI)
               by (force simp del: Complex_eq greaterThan_0)+
          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))
                = ∣ - Im b - ln (real n + sqrt ((real n)🪙2 - 1)) / pi∣"
              by (simp add: complex_norm dist_norm cmod_eq_Re complex_diff)
            ultimately have "dist b (Complex m (- ln (real n + sqrt ((real n)🪙2 - 1)) / pi)) < 1"
              using n by (simp add: dist_triangle_lt [of b "Complex m (Im b)"] dist_commute)
            with ge m ‹0 🚫› show ?thesis
              by (rule_tac x = "Complex m (- ln (real n + sqrt ((real n)🪙2 - 1)) / pi)" in bexI) auto
          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 🚫› 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 🚫› show ?thesis
      by (simp add: field_split_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 🚫› 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 🚫› 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›

theorem 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
      proof (intro holomorphic_intros holomorphic_on_compose holomorphic_on_subset [OF holf])
        show "(*) (complex_of_real (R (f 0))) ` cball 0 1 \ cball 0 (R (f 0))"
          using Rpos by (auto simp: less_imp_le norm_mult)
      qed
      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"
        unfolding g_def
        by (metis DERIV_imp_deriv 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
    have "(f has_field_derivative 0) (at x)" for x
      by (metis True UNIV_I holf holomorphic_derivI open_UNIV)
    then obtain c where "∧x. f(x) = c"
      by (meson UNIV_I DERIV_zero_connected_constant [OF connected_UNIV open_UNIV finite.emptyI contf])
    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)"
      unfolding fw_def
      apply (intro DERIV_chain derivative_eq_intros w)+
      using holf holomorphic_derivI by (force simp: field_simps)+
    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"
        using non_fp
        by (intro holomorphic_intros holf holomorphic_on_compose [unfolded o_def, OF holf]) auto
    qed auto
    then obtain "c ≠ 0" "c ≠ 1"
      by (metis (no_types, opaque_lifting) 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)"
        by (rule derivative_eq_intros holomorphic_derivI [OF holfU] 
                    DERIV_chain [unfolded o_def, where f=f and g=f] | simp add: algebra_simps)+
      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]
          by (simp add: algebra_simps) (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 Arzelà--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
        using seq_suble [OF sub_kk] strict_mono_less_eq [OF sub_rr]
        by (simp add: order_subst1)
    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)"
      unfolding strict_mono_Suc_iff
      by (simp add: Suc_le_lessD strict_monoD strict_mono_imp_increasing sub_kk sub_rr)
    have "∃j. i ≤ j ∧ rr (n+d) i = rr n j" for d n i
    proof (induction d arbitrary: i)
      case (Suc d)
      then show ?case
        using seq_suble [OF sub_kk] by simp (meson order_trans)
    qed auto
    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"] by (force simp: o_def)
    qed
    show "∃l. (λn. (f ∘ (r ∘ k2)) n (σ i)) <---- l" 
      if "∃l. (λn. (f ∘ (r ∘ k1)) n (σ i)) <---- l" "∧j. N ≤ j ==> ∃j'≥j. k2 j = k1 j'"
      for i N and r k1 k2 :: "nat==>nat"
      using that
      by (simp add: lim_sequentially) (metis (no_types, opaque_lifting) le_cases order_trans)
  qed auto
  with σ that show ?thesis
    by force
qed


theorem Arzela_Ascoli:
  fixes F :: "[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"])
    have "continuous_on S (F (r n))" for n
      using UEQ by (force simp: continuous_on_iff)
    then show "∀🪙F n in sequentially. continuous_on S ((F ∘ r) n)"
      by (simp add: eventually_sequentially)
    show "uniform_limit S (F ∘ r) g sequentially"
      using that by (metis (mono_tags, opaque_lifting) comp_apply dist_norm uniform_limit_sequentially_iff)
  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"
    using ‹R ⊆ S› by (force intro: function_convergent_subsequence [OF ‹countable R› M])
  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 🚫› 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)"
        using ‹0 🚫› by (auto simp: closure_approachable)
      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 🚫› 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 obtain g where "∀e>0. ∃N. ∀n x. N ≤ n ∧ x ∈ S ⟶ norm ((F ∘ k) n x - g x) < e"
    using uniformly_convergent_eq_cauchy [of "λx. x ∈ S" "F ∘ k"] by (auto simp add: dist_norm)
  ultimately show thesis
    by (metis ‹strict_mono k› comp_apply that)
qed



subsubsection🍋‹tag important›\<open>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 F :: "[nat,complex] ==> complex"
  assumes "open S"
      and H: "∧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 F 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 🚫› 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: H ‹∧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 🚫› 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))"
           proof (rule *)
             show "dist y z < 2/3 * r"
               using that ‹0 🚫› by (simp only: dist_norm norm_minus_commute)
           qed
           have z: "((λw. F n w / (w - z)) has_contour_integral (2 * pi) * i * F n z)
                 (circlepath z (2/3 * r))"
             using ‹0 🚫› by (force intro!: *)
           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: field_split_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: field_split_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)))"
               using ‹r > 0› ‹M > 0› by (intro mult_mono [OF leM]) 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
               by (simp add: field_simps mult_less_0_iff norm_minus_commute)
             also have "... ≤ e/r"
               using ‹e > 0› ‹r > 0› r4_le_xy by (simp add: field_split_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))"

           proof (rule has_contour_integral_bound_circlepath [OF has_contour_integral_diff [OF y z]])
             show "∧x. cmod (x - z) = 2/3 * r ==> cmod (F n x / (x - y) - F n x / (x - z)) ≤ e / r"
               using le_er by auto
           qed (use ‹e > 0› ‹r > 0› in auto)
           also have "... = (2 * pi) * e * ((2/3))"
             using ‹r > 0› by (simp add: field_split_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 🚫› ‹0 🚫› ‹0 🚫› by simp
      qed
      show "∧n x. x ∈ K i ==> cmod (F n x) ≤ B i"
        using B ‹∧n. F n ∈ H› by blast
    next
      fix g :: "complex ==> complex" and k :: "nat ==> nat"
      assume *: "∧(g::complex==>complex) (k::nat==>nat). continuous_on (K i) g ==>
                  strict_mono k ==>
                  (∧e. 0 < e ==> ∃N. ∀n≥N. ∀x∈K i. cmod (F (k n) x - g x) < e) ==> thesis"
           "continuous_on (K i) g"
           "strict_mono k"
           "∧e. 0 < e ==> ∃N. ∀n x. N ≤ n ∧ x ∈ K i ⟶ cmod (F (k n) x - g x) < e"
      show ?thesis
        by (rule *(1)[OF *(2,3)], drule *(4)) auto
    qed (use comK in simp_all)
    then show ?thesis
      by auto
  qed
  define Φ where "Φ ≡ λg i r. λk::nat==>nat. ∀e>0. ∃N. ∀n≥N. ∀x∈K i. cmod ((F ∘ (r ∘ k)) n x - g x) < e"
  obtain k :: "nat ==> nat" where "strict_mono k" and k: "∧i. ∃g. Φ g i id k"
  proof (rule subsequence_diagonalization_lemma [where r=id])
    show "∃g. Φ g i id (r ∘ k2)" 
      if ex: "∃g. Φ g i id (r ∘ k1)" and "∧j. N ≤ j ==> ∃j'≥j. k2 j = k1 j'" 
      for i k1 k2 N and r::"nat==>nat"
    proof -
      obtain g where "Φ g i id (r ∘ k1)"
        using ex by blast
      then have "Φ g i id (r ∘ k2)"
        using that
        by (simp add: Φ_def) (metis (no_types, opaque_lifting) le_trans linear)
      then show ?thesis
        by metis
    qed
    have "∃k g. strict_mono (k::nat==>nat) ∧ Φ g i id (r ∘ k)" for i r
      unfolding Φ_def o_assoc using rng_f by (force intro!: *)
    then show "∧i r. ∃k. strict_mono (k::nat==>nat) ∧ (∃g. Φ g i id (r ∘ k))"
      by force
  qed fastforce
  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 k unfolding Φ_def by (metis id_comp)
    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 k unfolding Φ_def by (metis id_comp)
      have geq: "g w = h w" if "w ∈ T" for w
      proof (rule LIMSEQ_unique)
        show "(λn. F (k n) w) <---- g w"
          using ‹T ⊆ S› g_lim that by blast
        show "(λn. F (k n) w) <---- h w"
          using h N that by (force simp: lim_sequentially dist_norm)
      qed
      show ?thesis
        using T h N ‹0 🚫› 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› H])
      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)
    show "(λz. deriv (F n) z / F n z) holomorphic_on ball z0 r"
      by (metis (no_types) ‹open S› holf holomorphic_deriv holomorphic_on_divide holomorphic_on_subset nz subS)
  qed (use ‹0 🚫› 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)"
    using ‹0 🚫› subS 
    by (intro holomorphic_on_imp_continuous_on holomorphic_on_subset [OF hol_dg]) 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)"
    using ‹0 🚫› subS 
    by (intro continuous_intros holomorphic_on_imp_continuous_on holomorphic_on_subset [OF holg]) 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 🚫› 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"

        show "∀🪙F n in sequentially. ∀x ∈ sphere z0 (r/2). dist (deriv (F n) x) (deriv g x) < e"
        proof -
          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: "w ∈ sphere z0 (r/2)"  for n w
          proof -
            have "ball w (r/4) ⊆ ball z0 r"  "cball w (r/4) ⊆ ball z0 r"
              using ‹0 🚫› w by (simp_all add: ball_subset_ball_iff cball_subset_ball_iff dist_commute)
            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 🚫› wr4_sub by auto
            have "dist z0 y ≤ 3 * r / 4" if "dist w y < r/4" for y
            proof (rule dist_triangle_le [where z=w])
              show "dist z0 w + dist y w ≤ 3 * r / 4"
                using w that by (simp add: dist_commute)
            qed
            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"
              using Cauchy_higher_deriv_bound [OF hol cont in_ball, of 1] ‹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
          moreover
          have "cball z0 (3 * r / 4) ⊆ ball z0 r"
            by (simp add: cball_subset_ball_iff ‹0 🚫›)
          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 🚫› ‹0 🚫› by (force simp: intro!: uniform_limitD)
          ultimately show ?thesis
            by (force simp add: eventually_sequentially)
        qed
      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 🚫› 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"
          using ‹0 🚫› uniform_limit_iff by blast
      qed
      show "b > 0" "∧x. x ∈ sphere z0 (r/2) ==> b ≤ cmod (g x)"
        using b ‹0 🚫› by (fastforce simp: geq hnz)+
    qed
  qed (use ‹0 🚫› 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 🚫› in simp)
    fix w
    assume w: "dist z0 w * 2 = r"
    then have w_inb: "w ∈ ball z0 r"
      using ‹0 🚫› 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 🚫› apply (simp add: divide_simps distrib_right)
        by (metis Suc_pred mult.commute power_Suc)
      then show ?thesis
      proof (rule DERIV_imp_deriv [OF has_field_derivative_transform_within_open])
        show "∧x. x ∈ ball z0 r ==> (x - z0) ^ m * h x = g x"
          by (simp add: hnz geq)
      qed (use that ‹m > 0› ‹0 🚫› in auto)
    qed
    with ‹0 🚫› ‹0 🚫› w w_inb show "deriv g w / g w = of_nat m / (w - z0) + deriv h w / h w"
      by (auto simp: geq field_split_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 🚫› intro: Cauchy_integral_circlepath_simple)
    show "((λx. deriv h x / h x) has_contour_integral 0) (circlepath z0 (r/2))"
      using hnz holh holomorphic_deriv holomorphic_on_divide ‹0 🚫›
      by (fastforce intro!: Cauchy_theorem_disc_simple [of _ z0 r])
  qed
  ultimately show False using ‹0 🚫› 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"
      using isolated_zeros [of "λz. g z - g z1" S z2 z0] 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 🚫› by (force simp: intro!: uniform_limitD)
        have "uniform_limit {z1} F g sequentially"
          by (intro ul_g) (auto simp: z12)
        then have "∀🪙F n in sequentially. ∀x ∈ {z1}. dist (F n x) (g x) < e/2"
          using ‹0 🚫› by (force simp: intro!: uniform_limitD)
        then have z1: "∀🪙F n in sequentially. dist (F n z1) (g z1) < e/2"
          by simp
        show "∀🪙F n in sequentially. ∀x∈K. dist (F n x - F n z1) (g x - g z1) < e" 
          apply (intro eventually_mono [OF  eventually_conj [OF K z1]])
          by (metis (no_types, opaque_lifting) diff_add_eq diff_diff_eq2 dist_commute dist_norm dist_triangle_add_half)
      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
      have hol_h_o: "(h ∘ (λz. (w + of_real e * z))) holomorphic_on cball 0 1"
      proof (intro holomorphic_intros holomorphic_on_compose)
        have "h holomorphic_on S" 
          using holX ‹h ∈ X› by auto
        then have "h holomorphic_on cball w e"
          by (metis e holomorphic_on_subset)
        moreover have "(λz. w + complex_of_real e * z) ` cball 0 1 ⊆ cball w e"
          using that ‹e > 0› by (auto simp: dist_norm norm_mult)
        ultimately show "h holomorphic_on (λz. w + complex_of_real e * z) ` cball 0 1"
          by (rule holomorphic_on_subset)
      qed
      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: "h (w + e * z) ≠ 0 ∧ h (w + e * z) ≠ 1" if "z ∈ cball 0 1" for z::complex
      proof (rule X01 [OF ‹h ∈ X›])
        have "w + complex_of_real e * z ∈ cball w e"
          using ‹0 🚫› that by (auto simp: dist_norm norm_mult)
        then show "w + complex_of_real e * z ∈ S"
          by (rule subsetD [OF e])
      qed
      have "cmod (h z) ≤ cmod (h (w + of_real e * (inverse e * (z - w))))"
        using ‹0 🚫› by (simp add: field_split_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"
        using GPicard1 [OF S zero_less_one ‹Y ⊆ X› holX] X01 no_hw_le1 by blast
      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 F 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
      proof (simp add: G_def)
        show "(λz. inverse (F n z)) holomorphic_on S"
          using FY X01 ‹Y ⊆ X› holX by (blast intro: holomorphic_on_inverse)
      qed
      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 🚫› 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 "cmod (inverse (F (j n) y)) < inverse C"
              by (simp add: ‹h v = 0› G_def)
            then have "C < norm (F (j n) y)"
              by (metis G_def Gnot0 ‹y ∈ S› inverse_less_imp_less inverse_zero norm_inverse zero_less_norm_iff)
            show False
              using ‹C 🚫 (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 obtain r where "0 < r" and r: "∧h. h ∈ Y ==> cmod (h v) ≤ r"
      by (metis not_le)
    moreover     
    obtain B Z where "0 < B" "open Z" "v ∈ Z" "Z ⊆ S" "∧h z. [h ∈ Y; z ∈ Z] ==> norm(h z) ≤ B"
      using X01 
      by (auto simp: r intro: GPicard1[OF ‹open S› ‹connected S› ‹v ∈ S› ‹r>0› ‹Y ⊆ X› holX] X01)
    ultimately show "v ∈ U"
      using v by (simp add: U_def) meson
  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
    then have "∀h z. h ∈ Y ∧ z ∈ K ⟶ (∃x∈L. cmod (h z) ≤ F x)"
      by (metis "*" L UN_E subset_iff)
    with False ‹finite L› show ?thesis 
      by (rule_tac x = "Max (F ` L)" in exI) (simp add: linorder_class.Max_ge_iff)
  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 🚫› by auto
  show ?thesis
  proof
    show "ε < k"
      using ‹0 🚫› ‹ε 🚫/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 🚫› 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 🚫› 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)"
          using ‹ε 🚫› ‹0 🚫› that by (auto intro:  holomorphic_on_subset [OF holf])
        show "continuous_on (closure (cball 0 ε - ball 0 d)) f"
        proof (intro holomorphic_on_imp_continuous_on holomorphic_on_subset [OF holf])
          show "closure (cball 0 ε - ball 0 d) ⊆ ball 0 k - {0}"
            using ‹0 🚫› ‹ε 🚫› by auto
        qed
        show "∧z. z ∈ frontier (cball 0 ε - ball 0 d) ==> cmod (f z) ≤ B"
          unfolding frontier_def
          using ε d less_eq_real_def by force
      qed (use that in auto)
      show ?thesis
        using * ‹d 🚫 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 field_split_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 
    using * by (force intro!: f01)
  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)}"
        using h01 by (auto intro: holomorphic_intros holomorphic_on_compose holh)
      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 🚫› 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
      using ε by (force intro!: that [OF ‹0 🚫ε› ‹ε 🚫›])
  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)}"
        using h01 by (auto intro!: holomorphic_intros holomorphic_on_compose_gen [unfolded o_def, OF _ holh] holomorphic_on_compose)
      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 🚫› 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 🚫› 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: field_split_simps)
    then show ?thesis
      using ε that [OF ‹0 🚫ε› ‹ε 🚫›]
      by (metis Diff_iff dist_0_norm insert_iff mem_ball)
  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}"
    proof (intro holomorphic_intros holomorphic_on_compose_gen [unfolded o_def, OF _ holf])
      show "(λx. z + complex_of_real r * x) ` (ball 0 1 - {0}) ⊆ M - {z}"
        using ‹0 🚫› r
        by (auto simp: dist_norm norm_mult subset_eq)
    qed (use ‹a ≠ 0› in auto)
    show "∧w. w ∈ ball 0 1 - {0} ==> f (z + of_real r * w) / a ≠ 0 ∧ f (z + of_real r * w) / a ≠ 1"
      using f0a ‹0 🚫› ‹a ≠ 0› r
      by (auto simp: field_split_simps dist_norm norm_mult subset_eq)
  qed
  show ?thesis
  proof
    show "0 < e*r"
      by (simp add: ‹0 🚫› ‹0 🚫›)
    have "ball z (e * r) ⊆ ball z r"
      by (simp add: ‹0 🚫› ‹e 🚫› 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 🚫› 1 [of "(w - z) / r"]
        by (simp add: norm_divide dist_norm field_split_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 🚫› 2 [of "(w - z) / r"]
        by (simp add: norm_divide dist_norm field_split_simps)
      then have "[dist z w < e * r; w ≠ z] ==> inverse (cmod (f w)) ≤ inverse (B * norm a)" for w
        by (metis ‹0 🚫› ‹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}"
    using zrM by (auto intro: holomorphic_on_subset [OF holf])
  have holfb_i: "(λz. inverse(f z - a)) holomorphic_on ball z r - {z}"
    using fab zrM by (fastforce intro!: holomorphic_intros holfb)
  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)
    then have "∀x. x ≠ z ∧ dist x z < r ⟶ cmod (f x - a) ≤ B"
      by (simp add: dist_commute)
    with ‹0 🚫› have "∀🪙F w in at z. cmod (f w - a) ≤ B"
      by (force simp add: eventually_at)
    moreover have "∧x. cmod (f x - a) ≤ B ==> cmod (f x) ≤ B + cmod a"
      by (metis add.commute add_le_cancel_right norm_triangle_sub order.trans)
    ultimately have "∃B. ∀🪙F w in at z. cmod (f w) ≤ B"
      by (metis (mono_tags, lifting) eventually_at)
    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 🚫› holomorphic_on_extend_bounded [OF holfb] by auto
    then have "g ←-z→ g z"
      unfolding continuous_at [symmetric]
      using ‹0 🚫› centre_in_ball field_differentiable_imp_continuous_at 
            holomorphic_on_imp_differentiable_at by blast
    then have "(f ---> g z) (at z)"
      using Lim_transform_within_open [of g "g z" z]
      using ‹0 🚫› centre_in_ball gf by blast
    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)
    then have "∀x. x ≠ z ∧ dist x z < r ⟶ cmod (inverse (f x - a)) ≤ B"
      by (simp add: dist_commute)
    with ‹0 🚫› have "∀🪙F w in at z. cmod (inverse (f w - a)) ≤ B"
      by (auto simp add: eventually_at)
    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 🚫› holomorphic_on_extend_bounded [OF holfb_i] by auto
    then have gz: "g ←-z→ g z"
      unfolding continuous_at [symmetric]
      using ‹0 🚫› 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 🚫› 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"
            by (rule tendsto_eq_intros refl gz ‹g z ≠ 0› | 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 field_split_simps)
        qed (use ‹0 🚫› 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 🚫› 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})"
unfolding subset_iff image_iff
  by (metis great_Picard [OF M _ holf] non Compl_iff insertI1)
    

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
          using e ‹e > 0› that by fastforce
      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 🚫› 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 🚫› 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 zrM by auto
  qed
  with that show thesis
    by meson
qed

theorem 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