section\<open>The Great Picard Theorem and its Applications\<close>
text\<open>Ported from HOL Light (cauchy.ml) by L C Paulson, 2017\<close>
theory Great_Picard
imports Conformal_Mappings Further_Topology
begin
subsection\<open>Schottky's theorem\<close>
lemma Schottky_lemma0:
assumes holf: "f holomorphic_on S" and cons: "contractible S" and "a \<in> S"
and f: "\<And>z. z \<in> S \<Longrightarrow> f z \<noteq> 1 \<and> f z \<noteq> -1"
obtains g where "g holomorphic_on S"
"norm(g a) \<le> 1 + norm(f a) / 3"
"\<And>z. z \<in> S \<Longrightarrow> f z = cos(of_real pi * g z)"
proof -
obtain g where holg: "g holomorphic_on S" and g: "norm(g a) \<le> pi + norm(f a)"
and f_eq_cos: "\<And>z. z \<in> S \<Longrightarrow> f z = cos(g z)"
using contractible_imp_holomorphic_arccos_bounded [OF assms]
by blast
show ?thesis
proof
show "(\<lambda>z. g z / pi) holomorphic_on S"
by (auto intro: holomorphic_intros holg)
have "3 \<le> pi"
using pi_approx by force
have "3 * norm(g a) \<le> 3 * (pi + norm(f a))"
using g by auto
also have "... \<le> pi * 3 + pi * cmod (f a)"
using \<open>3 \<le> pi\<close> by (simp add: mult_right_mono algebra_simps)
finally show "cmod (g a / complex_of_real pi) \<le> 1 + cmod (f a) / 3"
by (simp add: field_simps norm_divide)
show "\<And>z. z \<in> S \<Longrightarrow> 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 \<le> n^2 - 1"
using assms by (simp add: algebra_simps power2_eq_square)
then have "real (n - 1) \<le> sqrt (real (n\<^sup>2 - 1))"
by (metis of_nat_le_iff of_nat_power real_le_rsqrt)
then have "n-1 \<le> 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 \<le> x"
obtains n where "0 < n" "\<bar>x - ln (real n + sqrt ((real n)\<^sup>2 - 1)) / pi\<bar> < 1/2"
proof -
obtain n0::nat where "0 < n0" "ln(n0 + sqrt(real n0 ^ 2 - 1)) / pi \<le> x"
proof
show "ln(real 1 + sqrt(real 1 ^ 2 - 1))/pi \<le> x"
by (auto simp: assms)
qed auto
moreover
obtain M::nat where "\<And>n. \<lbrakk>0 < n; ln(n + sqrt(real n ^ 2 - 1)) / pi \<le> x\<rbrakk> \<Longrightarrow> n \<le> M"
proof
fix n::nat
assume "0 < n" "ln (n + sqrt ((real n)\<^sup>2 - 1)) / pi \<le> x"
then have "ln (n + sqrt ((real n)\<^sup>2 - 1)) \<le> x * pi"
by (simp add: divide_simps)
then have *: "exp (ln (n + sqrt ((real n)\<^sup>2 - 1))) \<le> exp (x * pi)"
by blast
have 0: "0 \<le> sqrt ((real n)\<^sup>2 - 1)"
using \<open>0 < n\<close> by auto
have "n + sqrt ((real n)\<^sup>2 - 1) = exp (ln (n + sqrt ((real n)\<^sup>2 - 1)))"
by (simp add: Suc_leI \<open>0 < n\<close> add_pos_nonneg real_of_nat_ge_one_iff)
also have "... \<le> exp (x * pi)"
using "*" by blast
finally have "real n \<le> exp (x * pi)"
using 0 by linarith
then show "n \<le> 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 \<le> x"
and le_n: "\<And>k. \<lbrakk>0 < k; ln(k + sqrt(real k ^ 2 - 1)) / pi \<le> x\<rbrakk> \<Longrightarrow> k \<le> n"
using bounded_Max_nat [of "\<lambda>n. 0<n \<and> ln (n + sqrt ((real n)\<^sup>2 - 1)) / pi \<le> x"] by metis
define a where "a \<equiv> ln(n + sqrt(real n ^ 2 - 1)) / pi"
define b where "b \<equiv> ln (1 + real n + sqrt ((1 + real n)\<^sup>2 - 1)) / pi"
have le_xa: "a \<le> x"
and le_na: "\<And>k. \<lbrakk>0 < k; ln(k + sqrt(real k ^ 2 - 1)) / pi \<le> x\<rbrakk> \<Longrightarrow> k \<le> 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)\<^sup>2 - 1)) - ln (real n + sqrt ((real n)\<^sup>2 - 1)) =
ln ((1 + real n + sqrt ((1 + real n)\<^sup>2 - 1)) / (real n + sqrt ((real n)\<^sup>2 - 1)))"
by (simp add: \<open>0 < n\<close> Schottky_lemma1 add_pos_nonneg ln_div [symmetric])
also have "... \<le> 3"
proof (cases "n = 1")
case True
have "sqrt 3 \<le> 2"
by (simp add: real_le_lsqrt)
then have "(2 + sqrt 3) \<le> 4"
by simp
also have "... \<le> exp 3"
using exp_ge_add_one_self [of "3::real"] by simp
finally have "ln (2 + sqrt 3) \<le> 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 \<open>0 < n\<close> have "1 < n" "2 \<le> n"
by linarith+
then have 1: "1 \<le> 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 \<le> (m+2) * ((m+2) * 3)" for m::nat
by simp
have "4 + n * 2 \<le> n * (n * 3)"
using * [of "n-2"] \<open>2 \<le> n\<close>
by (metis le_add_diff_inverse2)
then have **: "4 + real n * 2 \<le> 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)\<^sup>2 - 1) \<le> 2 * sqrt ((real n)\<^sup>2 - 1)"
by (auto simp: real_le_lsqrt power2_eq_square algebra_simps 1 **)
then
have "((1 + real n + sqrt ((1 + real n)\<^sup>2 - 1)) / (real n + sqrt ((real n)\<^sup>2 - 1))) \<le> 2"
using Schottky_lemma1 \<open>0 < n\<close> by (simp add: divide_simps)
then have "ln ((1 + real n + sqrt ((1 + real n)\<^sup>2 - 1)) / (real n + sqrt ((real n)\<^sup>2 - 1))) \<le> ln 2"
apply (subst ln_le_cancel_iff)
using Schottky_lemma1 \<open>0 < n\<close> by auto (force simp: divide_simps)
also have "... \<le> 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 "\<bar>x - a\<bar> < 1/2 \<or> \<bar>x - b\<bar> < 1/2"
by (auto simp: abs_if)
then show thesis
proof
assume "\<bar>x - a\<bar> < 1 / 2"
then show ?thesis
by (rule_tac n=n in that) (auto simp: a_def \<open>0 < n\<close>)
next
assume "\<bar>x - b\<bar> < 1 / 2"
then show ?thesis
by (rule_tac n="Suc n" in that) (auto simp: b_def \<open>0 < n\<close>)
qed
qed
lemma Schottky_lemma3:
fixes z::complex
assumes "z \<in> (\<Union>m \<in> Ints. \<Union>n \<in> {0<..}. {Complex m (ln(n + sqrt(real n ^ 2 - 1)) / pi)})
\<union> (\<Union>m \<in> Ints. \<Union>n \<in> {0<..}. {Complex m (-ln(n + sqrt(real n ^ 2 - 1)) / pi)})"
shows "cos(pi * cos(pi * z)) = 1 \<or> cos(pi * cos(pi * z)) = -1"
proof -
have sqrt2 [simp]: "complex_of_real (sqrt x) * complex_of_real (sqrt x) = x" if "x \<ge> 0" for x::real
by (metis abs_of_nonneg of_real_mult real_sqrt_mult_self that)
have 1: "\<exists>k. exp (\<i> * (of_int m * complex_of_real pi) -
(ln (real n + sqrt ((real n)\<^sup>2 - 1)))) +
inverse
(exp (\<i> * (of_int m * complex_of_real pi) -
(ln (real n + sqrt ((real n)\<^sup>2 - 1))))) = of_int k * 2"
if "n > 0" for m n
proof -
have eeq: "e \<noteq> 0 \<Longrightarrow> e + inverse e = n*2 \<longleftrightarrow> inverse e^2 - 2 * n*inverse e + 1 = 0" for n e::complex
by (auto simp: field_simps power2_eq_square)
have [simp]: "1 \<le> 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: "\<exists>k. exp (\<i> * (of_int m * complex_of_real pi) +
(ln (real n + sqrt ((real n)\<^sup>2 - 1)))) +
inverse
(exp (\<i> * (of_int m * complex_of_real pi) +
(ln (real n + sqrt ((real n)\<^sup>2 - 1))))) = of_int k * 2"
if "n > 0" for m n
proof -
have eeq: "e \<noteq> 0 \<Longrightarrow> e + inverse e = n*2 \<longleftrightarrow> e^2 - 2 * n*e + 1 = 0" for n e::complex
by (auto simp: field_simps power2_eq_square)
have [simp]: "1 \<le> 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 "\<exists>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) \<le> r"
and not01: "\<And>z. z \<in> cball 0 1 \<Longrightarrow> \<not>(f z = 0 \<or> f z = 1)"
and "0 < t" "t < 1" "norm z \<le> t"
shows "norm(f z) \<le> 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) \<le> 1 + norm(2 * f 0 - 1) / 3"
and h: "\<And>z. z \<in> cball 0 1 \<Longrightarrow> 2 * f z - 1 = cos(of_real pi * h z)"
proof (rule Schottky_lemma0 [of "\<lambda>z. 2 * f z - 1" "cball 0 1" 0])
show "(\<lambda>z. 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 "\<And>z. z \<in> cball 0 1 \<Longrightarrow> 2 * f z - 1 \<noteq> 1 \<and> 2 * f z - 1 \<noteq> - 1"
using not01 by force
qed auto
obtain g where holg: "g holomorphic_on cball 0 1"
and ng0: "norm(g 0) \<le> 1 + norm(h 0) / 3"
and g: "\<And>z. z \<in> cball 0 1 \<Longrightarrow> h z = cos(of_real pi * g z)"
proof (rule Schottky_lemma0 [OF holf convex_imp_contractible, of 0])
show "\<And>z. z \<in> cball 0 1 \<Longrightarrow> h z \<noteq> 1 \<and> h z \<noteq> - 1"
using h not01 by fastforce+
qed auto
have g0_2_f0: "norm(g 0) \<le> 2 + norm(f 0)"
proof -
have "cmod (2 * f 0 - 1) \<le> cmod (2 * f 0) + 1"
by (metis norm_one norm_triangle_ineq4)
also have "... \<le> 2 + cmod (f 0) * 3"
by simp
finally have "1 + norm(2 * f 0 - 1) / 3 \<le> (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) \<le> (2 + norm(f 0) - 1) * 3"
by linarith
then have "1 + norm(h 0) / 3 \<le> 2 + norm(f 0)"
by simp
with ng0 show ?thesis
by auto
qed
have "z \<in> ball 0 1"
using assms by auto
have norm_g_12: "norm(g z - g 0) \<le> (12 * t) / (1 - t)"
proof -
obtain g' where g': "\<And>x. x \<in> cball 0 1 \<Longrightarrow> (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 \<open>z \<in> ball 0 1\<close> segment_bound1 by fastforce
have "cmod (g' w) \<le> 12 / (1 - t)" if "w \<in> closed_segment 0 z" for w
proof -
have w: "w \<in> ball 0 1"
using segment_bound [OF that] \<open>z \<in> ball 0 1\<close> by simp
have ttt: "\<And>z. z \<in> frontier (cball 0 1) \<Longrightarrow> 1 - t \<le> dist w z"
using \<open>norm z \<le> t\<close> segment_bound1 [OF \<open>w \<in> closed_segment 0 z\<close>]
apply (simp add: dist_complex_def)
by (metis diff_left_mono dist_commute dist_complex_def norm_triangle_ineq2 order_trans)
have *: "\<lbrakk>\<And>b. (\<exists>w \<in> T \<union> U. w \<in> ball b 1); \<And>x. x \<in> D \<Longrightarrow> g x \<notin> T \<union> U\<rbrakk> \<Longrightarrow> \<nexists>b. ball b 1 \<subseteq> g ` D" for T U D
by force
have "\<nexists>b. ball b 1 \<subseteq> g ` cball 0 1"
proof (rule *)
show "(\<exists>w \<in> (\<Union>m \<in> Ints. \<Union>n \<in> {0<..}. {Complex m (ln(n + sqrt(real n ^ 2 - 1)) / pi)}) \<union>
(\<Union>m \<in> Ints. \<Union>n \<in> {0<..}. {Complex m (-ln(n + sqrt(real n ^ 2 - 1)) / pi)}). w \<in> ball b 1)" for b
proof -
obtain m where m: "m \<in> \<int>" "\<bar>Re b - m\<bar> \<le> 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: "\<bar>Im b - ln (n + sqrt ((real n)\<^sup>2 - 1)) / pi\<bar> < 1/2"
using Schottky_lemma2 [of "Im b"] by blast
have "dist b (Complex m (Im b)) \<le> 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)\<^sup>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)\<^sup>2 - 1)) / pi)) < 1"
by (simp add: dist_triangle_lt [of b "Complex m (Im b)"] dist_commute)
with le m \<open>0 < n\<close> show ?thesis
apply (rule_tac x = "Complex m (ln (real n + sqrt ((real n)\<^sup>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: "\<bar>- Im b - ln (real n + sqrt ((real n)\<^sup>2 - 1)) / pi\<bar> < 1/2"
using Schottky_lemma2 [of "- Im b"] by auto
have "dist b (Complex m (Im b)) \<le> 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)\<^sup>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)\<^sup>2 - 1)) / pi)) < 1"
by (simp add: dist_triangle_lt [of b "Complex m (Im b)"] dist_commute)
with ge m \<open>0 < n\<close> show ?thesis
apply (rule_tac x = "Complex m (- ln (real n + sqrt ((real n)\<^sup>2 - 1)) / pi)" in bexI)
apply (simp_all del: Complex_eq greaterThan_0)
by blast
qed
qed
show "g v \<notin> (\<Union>m \<in> Ints. \<Union>n \<in> {0<..}. {Complex m (ln(n + sqrt(real n ^ 2 - 1)) / pi)}) \<union>
(\<Union>m \<in> Ints. \<Union>n \<in> {0<..}. {Complex m (-ln(n + sqrt(real n ^ 2 - 1)) / pi)})"
if "v \<in> 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) \<le> 12 / (1 - t)"
using g' 12 \<open>t < 1\<close> by (simp add: field_simps)
qed
then have "cmod (g z - g 0) \<le> 12 / (1 - t) * cmod z"
using has_contour_integral_bound_linepath [OF int_g', of "12/(1 - t)"] assms
by simp
with \<open>cmod z \<le> t\<close> \<open>t < 1\<close> show ?thesis
by (simp add: divide_simps)
qed
have fz: "f z = (1 + cos(of_real pi * h z)) / 2"
using h \<open>z \<in> ball 0 1\<close> by (auto simp: field_simps)
have "cmod (f z) \<le> exp (cmod (complex_of_real pi * h z))"
by (simp add: fz mult.commute norm_cos_plus1_le)
also have "... \<le> exp (pi * exp (pi * (2 + 2 * r + 12 * t / (1 - t))))"
proof (simp add: norm_mult)
have "cmod (g z - g 0) \<le> 12 * t / (1 - t)"
using norm_g_12 \<open>t < 1\<close> by (simp add: norm_mult)
then have "cmod (g z) - cmod (g 0) \<le> 12 * t / (1 - t)"
using norm_triangle_ineq2 order_trans by blast
then have *: "cmod (g z) \<le> 2 + 2 * r + 12 * t / (1 - t)"
using g0_2_f0 norm_ge_zero [of "f 0"] nof0
by linarith
have "cmod (h z) \<le> exp (cmod (complex_of_real pi * g z))"
using \<open>z \<in> ball 0 1\<close> by (simp add: g norm_cos_le)
also have "... \<le> exp (pi * (2 + 2 * r + 12 * t / (1 - t)))"
using \<open>t < 1\<close> nof0 * by (simp add: norm_mult)
finally show "cmod (h z) \<le> exp (pi * (2 + 2 * r + 12 * t / (1 - t)))" .
qed
finally show ?thesis .
qed
subsection\<open>The Little Picard Theorem\<close>
lemma Landau_Picard:
obtains R
where "\<And>z. 0 < R z"
"\<And>f. \<lbrakk>f holomorphic_on cball 0 (R(f 0));
\<And>z. norm z \<le> R(f 0) \<Longrightarrow> f z \<noteq> 0 \<and> f z \<noteq> 1\<rbrakk> \<Longrightarrow> norm(deriv f 0) < 1"
proof -
define R where "R \<equiv> \<lambda>z. 3 * exp(pi * exp(pi*(2 + 2 * cmod z + 12)))"
show ?thesis
proof
show Rpos: "\<And>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: "\<And>z. norm z \<le> R(f 0) \<Longrightarrow> f z \<noteq> 0 \<and> f z \<noteq> 1" for f
proof -
let ?r = "R(f 0)"
define g where "g \<equiv> f \<circ> (\<lambda>z. 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) \<le> exp(pi * exp(pi * (2 + 2 * norm (f 0) + 12 * t / (1 - t))))"
if "0 < t" "t < 1" "norm z \<le> t" for t z
proof (rule Schottky [OF holg])
show "cmod (g 0) \<le> cmod (f 0)"
by (simp add: g_def)
show "\<And>z. z \<in> cball 0 1 \<Longrightarrow> \<not> (g z = 0 \<or> 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) \<le> R (f 0) / 3" if "cmod (0 - z) = 1/2" for z
proof -
have "norm(g z) \<le> 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 \<le> 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: "\<And>z. f z \<noteq> 0 \<and> f z \<noteq> 1"
obtains c where "f = (\<lambda>x. c)"
proof -
obtain R
where Rpos: "\<And>z. 0 < R z"
and R: "\<And>h. \<lbrakk>h holomorphic_on cball 0 (R(h 0));
\<And>z. norm z \<le> R(h 0) \<Longrightarrow> h z \<noteq> 0 \<and> h z \<noteq> 1\<rbrakk> \<Longrightarrow> 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 "\<forall>x. deriv f x = 0")
case True
obtain c where "\<And>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 \<noteq> 0" by auto
define fw where "fw \<equiv> (f \<circ> (\<lambda>z. 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 \<noteq> 0 \<and> fw z \<noteq> 1" if "cmod z \<le> 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 \<noteq> b" "range f \<inter> {a,b} = {}"
obtains c where "f = (\<lambda>x. c)"
proof -
let ?g = "\<lambda>x. 1/(b - a)*(f x - b) + 1"
obtain c where "?g = (\<lambda>x. c)"
proof (rule little_Picard_01)
show "?g holomorphic_on UNIV"
by (intro holomorphic_intros holf)
show "\<And>z. ?g z \<noteq> 0 \<and> ?g z \<noteq> 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\<open>A couple of little applications of Little Picard\<close>
lemma holomorphic_periodic_fixpoint:
assumes holf: "f holomorphic_on UNIV"
and "p \<noteq> 0" and per: "\<And>z. f(z + p) = f z"
obtains x where "f x = x"
proof -
have False if non: "\<And>x. f x \<noteq> x"
proof -
obtain c where "(\<lambda>z. f z - z) = (\<lambda>z. c)"
proof (rule little_Picard)
show "(\<lambda>z. f z - z) holomorphic_on UNIV"
by (simp add: holf holomorphic_on_diff)
show "range (\<lambda>z. f z - z) \<inter> {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 \<open>p \<noteq> 0\<close> diff_add_cancel)
qed
then show ?thesis
using that by blast
qed
lemma holomorphic_involution_point:
assumes holfU: "f holomorphic_on UNIV" and non: "\<And>a. f \<noteq> (\<lambda>x. a + x)"
obtains x where "f(f x) = x"
proof -
{ assume non_ff [simp]: "\<And>x. f(f x) \<noteq> x"
then have non_fp [simp]: "f z \<noteq> z" for z
by metis
have holf: "f holomorphic_on X" for X
using assms holomorphic_on_subset by blast
obtain c where c: "(\<lambda>x. (f(f x) - x)/(f x - x)) = (\<lambda>x. c)"
proof (rule little_Picard_01)
show "(\<lambda>x. (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 \<noteq> 0" "c \<noteq> 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 "((\<lambda>x. 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 "((\<lambda>x. 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 \<circ> f = (\<lambda>x. k)"
proof (rule little_Picard)
show "(deriv f \<circ> f) holomorphic_on UNIV"
by (meson holfU holomorphic_deriv holomorphic_on_compose holomorphic_on_subset open_UNIV subset_UNIV)
obtain "deriv f (f x) \<noteq> 0" "deriv f (f x) \<noteq> c" for x
using df_times_dff \<open>c \<noteq> 1\<close> eq_iff_diff_eq_0
by (metis lambda_one mult_zero_left mult_zero_right)
then show "range (deriv f \<circ> f) \<inter> {0,c} = {}"
by force
qed (use \<open>c \<noteq> 0\<close> in auto)
have "\<not> 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: "\<And>x. f x - k * x = l"
proof (rule DERIV_zero_connected_constant [of UNIV "{}" "\<lambda>x. 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 "\<lambda>z. deriv f z - k" "f z" "range f" w])
show "(\<lambda>z. 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>open (range f)\<close> image_eqI inf.absorb_iff2 inf_aci(1) islimpt_UNIV islimpt_eq_acc_point open_Int top_greatest)
show "\<And>z. z \<in> range f \<Longrightarrow> deriv f z - k = 0"
by (metis comp_def diff_self image_iff k)
qed auto
moreover
have "((\<lambda>x. 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 "\<forall>x. ((\<lambda>x. f x - k * x) has_field_derivative 0) (at x)"
by auto
show "continuous_on UNIV (\<lambda>x. 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 "\<exists>x. k * x + l \<noteq> 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 "\<And>x. (1 - k) * x \<noteq> 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\<open>The Arzelà--Ascoli theorem\<close>
lemma subsequence_diagonalization_lemma:
fixes P :: "nat \<Rightarrow> (nat \<Rightarrow> 'a) \<Rightarrow> bool"
assumes sub: "\<And>i r. \<exists>k. strict_mono (k :: nat \<Rightarrow> nat) \<and> P i (r \<circ> k)"
and P_P: "\<And>i r::nat \<Rightarrow> 'a. \<And>k1 k2 N.
\<lbrakk>P i (r \<circ> k1); \<And>j. N \<le> j \<Longrightarrow> \<exists>j'. j \<le> j' \<and> k2 j = k1 j'\<rbrakk> \<Longrightarrow> P i (r \<circ> k2)"
obtains k where "strict_mono (k :: nat \<Rightarrow> nat)" "\<And>i. P i (r \<circ> k)"
proof -
obtain kk where "\<And>i r. strict_mono (kk i r :: nat \<Rightarrow> nat) \<and> P i (r \<circ> (kk i r))"
using sub by metis
then have sub_kk: "\<And>i r. strict_mono (kk i r)" and P_kk: "\<And>i r. P i (r \<circ> (kk i r))"
by auto
define rr where "rr \<equiv> rec_nat (kk 0 r) (\<lambda>n x. x \<circ> kk (Suc n) (r \<circ> x))"
then have [simp]: "rr 0 = kk 0 r" "\<And>n. rr(Suc n) = rr n \<circ> kk (Suc n) (r \<circ> 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 \<circ> rr i)" for i
using P_kk by (induction i) (auto simp: o_def)
have "i \<le> i+d \<Longrightarrow> rr i n \<le> 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 "\<And>i j n. i \<le> j \<Longrightarrow> rr i n \<le> rr j n"
by (metis le_iff_add)
show "strict_mono (\<lambda>n. 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 "\<exists>j. i \<le> j \<and> 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 "\<And>m n i. n \<le> m \<Longrightarrow> \<exists>j. i \<le> j \<and> rr m i = rr n j"
by (metis le_iff_add)
then show "P i (r \<circ> (\<lambda>n. rr n n))" for i
by (meson P_rr P_P)
qed
qed
lemma function_convergent_subsequence:
fixes f :: "[nat,'a] \<Rightarrow> 'b::{real_normed_vector,heine_borel}"
assumes "countable S" and M: "\<And>n::nat. \<And>x. x \<in> S \<Longrightarrow> norm(f n x) \<le> M"
obtains k where "strict_mono (k::nat\<Rightarrow>nat)" "\<And>x. x \<in> S \<Longrightarrow> \<exists>l. (\<lambda>n. f (k n) x) \<longlonglongrightarrow> l"
proof (cases "S = {}")
case True
then show ?thesis
using strict_mono_id that by fastforce
next
case False
with \<open>countable S\<close> obtain \<sigma> :: "nat \<Rightarrow> 'a" where \<sigma>: "S = range \<sigma>"
using uncountable_def by blast
obtain k where "strict_mono k" and k: "\<And>i. \<exists>l. (\<lambda>n. (f \<circ> k) n (\<sigma> i)) \<longlonglongrightarrow> l"
proof (rule subsequence_diagonalization_lemma
[of "\<lambda>i r. \<exists>l. ((\<lambda>n. (f \<circ> r) n (\<sigma> i)) \<longlongrightarrow> l) sequentially" id])
show "\<exists>k::nat\<Rightarrow>nat. strict_mono k \<and> (\<exists>l. (\<lambda>n. (f \<circ> (r \<circ> k)) n (\<sigma> i)) \<longlonglongrightarrow> l)" for i r
proof -
have "f (r n) (\<sigma> i) \<in> cball 0 M" for n
by (simp add: \<sigma> M)
then show ?thesis
using compact_def [of "cball (0::'b) M"] apply simp
apply (drule_tac x="(\<lambda>n. f (r n) (\<sigma> i))" in spec)
apply (force simp: o_def)
done
qed
show "\<And>i r k1 k2 N.
\<lbrakk>\<exists>l. (\<lambda>n. (f \<circ> (r \<circ> k1)) n (\<sigma> i)) \<longlonglongrightarrow> l; \<And>j. N \<le> j \<Longrightarrow> \<exists>j'\<ge>j. k2 j = k1 j'\<rbrakk>
\<Longrightarrow> \<exists>l. (\<lambda>n. (f \<circ> (r \<circ> k2)) n (\<sigma> i)) \<longlonglongrightarrow> 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 \<sigma> that show ?thesis
by force
qed
theorem Arzela_Ascoli:
fixes \<F> :: "[nat,'a::euclidean_space] \<Rightarrow> 'b::{real_normed_vector,heine_borel}"
assumes "compact S"
and M: "\<And>n x. x \<in> S \<Longrightarrow> norm(\<F> n x) \<le> M"
and equicont:
"\<And>x e. \<lbrakk>x \<in> S; 0 < e\<rbrakk>
\<Longrightarrow> \<exists>d. 0 < d \<and> (\<forall>n y. y \<in> S \<and> norm(x - y) < d \<longrightarrow> norm(\<F> n x - \<F> n y) < e)"
obtains g k where "continuous_on S g" "strict_mono (k :: nat \<Rightarrow> nat)"
"\<And>e. 0 < e \<Longrightarrow> \<exists>N. \<forall>n x. n \<ge> N \<and> x \<in> S \<longrightarrow> norm(\<F>(k n) x - g x) < e"
proof -
have UEQ: "\<And>e. 0 < e \<Longrightarrow> \<exists>d. 0 < d \<and> (\<forall>n. \<forall>x \<in> S. \<forall>x' \<in> S. dist x' x < d \<longrightarrow> dist (\<F> n x') (\<F> n x) < e)"
apply (rule compact_uniformly_equicontinuous [OF \<open>compact S\<close>, of "range \<F>"])
using equicont by (force simp: dist_commute dist_norm)+
have "continuous_on S g"
if "\<And>e. 0 < e \<Longrightarrow> \<exists>N. \<forall>n x. n \<ge> N \<and> x \<in> S \<longrightarrow> norm(\<F>(r n) x - g x) < e"
for g:: "'a \<Rightarrow> 'b" and r :: "nat \<Rightarrow> nat"
proof (rule uniform_limit_theorem [of _ "\<F> \<circ> r"])
show "\<forall>\<^sub>F n in sequentially. continuous_on S ((\<F> \<circ> 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> \<circ> 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 \<subseteq> S" and SR: "S \<subseteq> closure R"
by (metis separable that)
obtain k where "strict_mono k" and k: "\<And>x. x \<in> R \<Longrightarrow> \<exists>l. (\<lambda>n. \<F> (k n) x) \<longlonglongrightarrow> l"
apply (rule function_convergent_subsequence [OF \<open>countable R\<close> M])
using \<open>R \<subseteq> S\<close> apply force+
done
then have Cauchy: "Cauchy ((\<lambda>n. \<F> (k n) x))" if "x \<in> R" for x
using convergent_eq_Cauchy that by blast
have "\<exists>N. \<forall>m n x. N \<le> m \<and> N \<le> n \<and> x \<in> S \<longrightarrow> dist ((\<F> \<circ> k) m x) ((\<F> \<circ> k) n x) < e"
if "0 < e" for e
proof -
obtain d where "0 < d"
and d: "\<And>n. \<forall>x \<in> S. \<forall>x' \<in> S. dist x' x < d \<longrightarrow> dist (\<F> n x') (\<F> n x) < e/3"
by (metis UEQ \<open>0 < e\<close> divide_pos_pos zero_less_numeral)
obtain T where "T \<subseteq> R" and "finite T" and T: "S \<subseteq> (\<Union>c\<in>T. ball c d)"
proof (rule compactE_image [OF \<open>compact S\<close>, of R "(\<lambda>x. ball x d)"])
have "closure R \<subseteq> (\<Union>c\<in>R. ball c d)"
apply clarsimp
using \<open>0 < d\<close> closure_approachable by blast
with SR show "S \<subseteq> (\<Union>c\<in>R. ball c d)"
by auto
qed auto
have "\<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (\<F> (k m) x) (\<F> (k n) x) < e/3" if "x \<in> R" for x
using Cauchy \<open>0 < e\<close> that unfolding Cauchy_def
by (metis less_divide_eq_numeral1(1) mult_zero_left)
then obtain MF where MF: "\<And>x m n. \<lbrakk>x \<in> R; m \<ge> MF x; n \<ge> MF x\<rbrakk> \<Longrightarrow> norm (\<F> (k m) x - \<F> (k n) x) < e/3"
using dist_norm by metis
have "dist ((\<F> \<circ> k) m x) ((\<F> \<circ> k) n x) < e"
if m: "Max (MF ` T) \<le> m" and n: "Max (MF ` T) \<le> n" "x \<in> S" for m n x
proof -
obtain t where "t \<in> T" and t: "x \<in> ball t d"
using \<open>x \<in> S\<close> T by auto
have "norm(\<F> (k m) t - \<F> (k m) x) < e / 3"
by (metis \<open>R \<subseteq> S\<close> \<open>T \<subseteq> R\<close> \<open>t \<in> T\<close> d dist_norm mem_ball subset_iff t \<open>x \<in> S\<close>)
moreover
have "norm(\<F> (k n) t - \<F> (k n) x) < e / 3"
by (metis \<open>R \<subseteq> S\<close> \<open>T \<subseteq> R\<close> \<open>t \<in> T\<close> subsetD d dist_norm mem_ball t \<open>x \<in> S\<close>)
moreover
have "norm(\<F> (k m) t - \<F> (k n) t) < e / 3"
proof (rule MF)
show "t \<in> R"
using \<open>T \<subseteq> R\<close> \<open>t \<in> T\<close> by blast
show "MF t \<le> m" "MF t \<le> n"
by (meson Max_ge \<open>finite T\<close> \<open>t \<in> T\<close> 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 "\<exists>g. \<forall>e>0. \<exists>N. \<forall>n\<ge>N. \<forall>x \<in> S. norm(\<F>(k n) x - g x) < e"
using uniformly_convergent_eq_cauchy [of "\<lambda>x. x \<in> S" "\<F> \<circ> k"]
apply (simp add: o_def dist_norm)
by meson
ultimately show thesis
by (metis that \<open>strict_mono k\<close>)
qed
subsubsection\<open>Montel's theorem\<close>
text\<open>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.\<close>
theorem Montel:
fixes \<F> :: "[nat,complex] \<Rightarrow> complex"
assumes "open S"
and \<H>: "\<And>h. h \<in> \<H> \<Longrightarrow> h holomorphic_on S"
and bounded: "\<And>K. \<lbrakk>compact K; K \<subseteq> S\<rbrakk> \<Longrightarrow> \<exists>B. \<forall>h \<in> \<H>. \<forall> z \<in> K. norm(h z) \<le> B"
and rng_f: "range \<F> \<subseteq> \<H>"
obtains g r
where "g holomorphic_on S" "strict_mono (r :: nat \<Rightarrow> nat)"
"\<And>x. x \<in> S \<Longrightarrow> ((\<lambda>n. \<F> (r n) x) \<longlongrightarrow> g x) sequentially"
"\<And>K. \<lbrakk>compact K; K \<subseteq> S\<rbrakk> \<Longrightarrow> uniform_limit K (\<F> \<circ> r) g sequentially"
proof -
obtain K where comK: "\<And>n. compact(K n)" and KS: "\<And>n::nat. K n \<subseteq> S"
and subK: "\<And>X. \<lbrakk>compact X; X \<subseteq> S\<rbrakk> \<Longrightarrow> \<exists>N. \<forall>n\<ge>N. X \<subseteq> K n"
using open_Union_compact_subsets [OF \<open>open S\<close>] by metis
then have "\<And>i. \<exists>B. \<forall>h \<in> \<H>. \<forall> z \<in> K i. norm(h z) \<le> B"
by (simp add: bounded)
then obtain B where B: "\<And>i h z. \<lbrakk>h \<in> \<H>; z \<in> K i\<rbrakk> \<Longrightarrow> norm(h z) \<le> B i"
by metis
have *: "\<exists>r g. strict_mono (r::nat\<Rightarrow>nat) \<and> (\<forall>e > 0. \<exists>N. \<forall>n\<ge>N. \<forall>x \<in> K i. norm((\<F> \<circ> r) n x - g x) < e)"
if "\<And>n. \<F> n \<in> \<H>" for \<F> i
proof -
obtain g k where "continuous_on (K i) g" "strict_mono (k::nat\<Rightarrow>nat)"
"\<And>e. 0 < e \<Longrightarrow> \<exists>N. \<forall>n\<ge>N. \<forall>x \<in> K i. norm(\<F>(k n) x - g x) < e"
proof (rule Arzela_Ascoli [of "K i" "\<F>" "B i"])
show "\<exists>d>0. \<forall>n y. y \<in> K i \<and> cmod (z - y) < d \<longrightarrow> cmod (\<F> n z - \<F> n y) < e"
if z: "z \<in> K i" and "0 < e" for z e
proof -
obtain r where "0 < r" and r: "cball z r \<subseteq> S"
using z KS [of i] \<open>open S\<close> by (force simp: open_contains_cball)
have "cball z (2 / 3 * r) \<subseteq> cball z r"
using \<open>0 < r\<close> by (simp add: cball_subset_cball_iff)
then have z23S: "cball z (2 / 3 * r) \<subseteq> S"
using r by blast
obtain M where "0 < M" and M: "\<And>n w. dist z w \<le> 2/3 * r \<Longrightarrow> norm(\<F> n w) \<le> M"
proof -
obtain N where N: "\<forall>n\<ge>N. cball z (2/3 * r) \<subseteq> K n"
using subK compact_cball [of z "(2 / 3 * r)"] z23S by force
have "cmod (\<F> n w) \<le> \<bar>B N\<bar> + 1" if "dist z w \<le> 2 / 3 * r" for n w
proof -
have "w \<in> K N"
using N mem_cball that by blast
then have "cmod (\<F> n w) \<le> B N"
using B \<open>\<And>n. \<F> n \<in> \<H>\<close> by blast
also have "... \<le> \<bar>B N\<bar> + 1"
by simp
finally show ?thesis .
qed
then show ?thesis
by (rule_tac M="\<bar>B N\<bar> + 1" in that) auto
qed
have "cmod (\<F> n z - \<F> n y) < e"
if "y \<in> K i" and y_near_z: "cmod (z - y) < r/3" "cmod (z - y) < e * r / (6 * M)"
for n y
proof -
have "((\<lambda>w. \<F> n w / (w - \<xi>)) has_contour_integral
(2 * pi) * \<i> * winding_number (circlepath z (2 / 3 * r)) \<xi> * \<F> n \<xi>)
(circlepath z (2 / 3 * r))"
if "dist \<xi> z < (2 / 3 * r)" for \<xi>
proof (rule Cauchy_integral_formula_convex_simple)
have "\<F> n holomorphic_on S"
by (simp add: \<H> \<open>\<And>n. \<F> n \<in> \<H>\<close>)
with z23S show "\<F> n holomorphic_on cball z (2 / 3 * r)"
using holomorphic_on_subset by blast
qed (use that \<open>0 < r\<close> in \<open>auto simp: dist_commute\<close>)
then have *: "((\<lambda>w. \<F> n w / (w - \<xi>)) has_contour_integral (2 * pi) * \<i> * \<F> n \<xi>)
(circlepath z (2 / 3 * r))"
if "dist \<xi> z < (2 / 3 * r)" for \<xi>
using that by (simp add: winding_number_circlepath dist_norm)
have y: "((\<lambda>w. \<F> n w / (w - y)) has_contour_integral (2 * pi) * \<i> * \<F> n y)
(circlepath z (2 / 3 * r))"
apply (rule *)
using that \<open>0 < r\<close> by (simp only: dist_norm norm_minus_commute)
have z: "((\<lambda>w. \<F> n w / (w - z)) has_contour_integral (2 * pi) * \<i> * \<F> n z)
(circlepath z (2 / 3 * r))"
apply (rule *)
using \<open>0 < r\<close> by simp
have le_er: "cmod (\<F> n x / (x - y) - \<F> n x / (x - z)) \<le> 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 \<open>M > 0\<close> \<open>r > 0\<close>
by (metis (full_types) norm_diff_triangle_less norm_minus_commute order_less_irrefl)
then have r4_le_xy: "r/4 \<le> cmod (x - y)"
using \<open>r > 0\<close> by simp
then have neq: "x \<noteq> y" "x \<noteq> z"
using that \<open>r > 0\<close> by (auto simp: divide_simps norm_minus_commute)
have leM: "cmod (\<F> n x) \<le> 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 "... \<le> M * (cmod (y - z) / (cmod(x - y) * (2/3 * r)))"
apply (rule mult_mono)
apply (rule leM)
using \<open>r > 0\<close> \<open>M > 0\<close> 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) \<open>M > 0\<close> \<open>r > 0\<close> neq
apply (simp add: field_simps mult_less_0_iff norm_minus_commute)
done
also have "... \<le> e/r"
using \<open>e > 0\<close> \<open>r > 0\<close> 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) \<le> 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 \<open>e > 0\<close> \<open>r > 0\<close> le_er by auto
also have "... = (2 * pi) * e * ((2 / 3))"
using \<open>r > 0\<close> by (simp add: divide_simps)
finally have "cmod (\<F> n y - \<F> n z) \<le> e * (2 / 3)"
by simp
also have "... < e"
using \<open>e > 0\<close> 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 \<open>0 < e\<close> \<open>0 < r\<close> \<open>0 < M\<close> by simp
qed
show "\<And>n x. x \<in> K i \<Longrightarrow> cmod (\<F> n x) \<le> B i"
using B \<open>\<And>n. \<F> n \<in> \<H>\<close> by blast
qed (use comK in \<open>fastforce+\<close>)
then show ?thesis
by fastforce
qed
have "\<exists>k g. strict_mono (k::nat\<Rightarrow>nat) \<and> (\<forall>e > 0. \<exists>N. \<forall>n\<ge>N. \<forall>x \<in> K i. norm((\<F> \<circ> r \<circ> k) n x - g x) < e)"
for i r
apply (rule *)
using rng_f by auto
then have **: "\<And>i r. \<exists>k. strict_mono (k::nat\<Rightarrow>nat) \<and> (\<exists>g. \<forall>e>0. \<exists>N. \<forall>n\<ge>N. \<forall>x \<in> K i. norm((\<F> \<circ> (r \<circ> k)) n x - g x) < e)"
by (force simp: o_assoc)
obtain k :: "nat \<Rightarrow> nat" where "strict_mono k"
and "\<And>i. \<exists>g. \<forall>e>0. \<exists>N. \<forall>n\<ge>N. \<forall>x\<in>K i. cmod ((\<F> \<circ> (id \<circ> 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: "\<And>i. \<exists>g. \<forall>e>0. \<exists>N. \<forall>n\<ge>N. \<forall>x\<in>K i. cmod ((\<F> \<circ> k) n x - g x) < e"
by simp
have "\<exists>l. \<forall>e>0. \<exists>N. \<forall>n\<ge>N. norm(\<F> (k n) z - l) < e" if "z \<in> S" for z
proof -
obtain G where G: "\<And>i e. e > 0 \<Longrightarrow> \<exists>M. \<forall>n\<ge>M. \<forall>x\<in>K i. cmod ((\<F> \<circ> k) n x - G i x) < e"
using lt_e by metis
obtain N where "\<And>n. n \<ge> N \<Longrightarrow> z \<in> K n"
using subK [of "{z}"] that \<open>z \<in> S\<close> by auto
moreover have "\<And>e. e > 0 \<Longrightarrow> \<exists>M. \<forall>n\<ge>M. \<forall>x\<in>K N. cmod ((\<F> \<circ> 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: "\<And>z e. \<lbrakk>z \<in> S; e > 0\<rbrakk> \<Longrightarrow> \<exists>N. \<forall>n\<ge>N. norm(\<F> (k n) z - g z) < e"
by metis
show ?thesis
proof
show g_lim: "\<And>x. x \<in> S \<Longrightarrow> (\<lambda>n. \<F> (k n) x) \<longlonglongrightarrow> g x"
by (simp add: lim_sequentially g dist_norm)
have dg_le_e: "\<exists>N. \<forall>n\<ge>N. \<forall>x\<in>T. cmod (\<F> (k n) x - g x) < e"
if T: "compact T" "T \<subseteq> S" and "0 < e" for T e
proof -
obtain N where N: "\<And>n. n \<ge> N \<Longrightarrow> T \<subseteq> K n"
using subK [OF T] by blast
obtain h where h: "\<And>e. e>0 \<Longrightarrow> \<exists>M. \<forall>n\<ge>M. \<forall>x\<in>K N. cmod ((\<F> \<circ> k) n x - h x) < e"
using lt_e by blast
have geq: "g w = h w" if "w \<in> T" for w
apply (rule LIMSEQ_unique [of "\<lambda>n. \<F>(k n) w"])
using \<open>T \<subseteq> S\<close> g_lim that apply blast
using h N that by (force simp: lim_sequentially dist_norm)
show ?thesis
using T h N \<open>0 < e\<close> by (fastforce simp add: geq)
qed
then show "\<And>K. \<lbrakk>compact K; K \<subseteq> S\<rbrakk>
\<Longrightarrow> uniform_limit K (\<F> \<circ> 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>open S\<close> \<H>])
show "\<And>n. (\<F> \<circ> k) n \<in> \<H>"
by (simp add: range_subsetD rng_f)
show "\<exists>d>0. cball z d \<subseteq> S \<and> uniform_limit (cball z d) (\<lambda>n. (\<F> \<circ> k) n) g sequentially"
if "z \<in> S" for z
proof -
obtain d where d: "d>0" "cball z d \<subseteq> S"
using \<open>open S\<close> \<open>z \<in> S\<close> open_contains_cball by blast
then have "uniform_limit (cball z d) (\<F> \<circ> 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: \<open>strict_mono k\<close>)
qed
subsection\<open>Some simple but useful cases of Hurwitz's theorem\<close>
proposition Hurwitz_no_zeros:
assumes S: "open S" "connected S"
and holf: "\<And>n::nat. \<F> n holomorphic_on S"
and holg: "g holomorphic_on S"
and ul_g: "\<And>K. \<lbrakk>compact K; K \<subseteq> S\<rbrakk> \<Longrightarrow> uniform_limit K \<F> g sequentially"
and nonconst: "~ g constant_on S"
and nz: "\<And>n z. z \<in> S \<Longrightarrow> \<F> n z \<noteq> 0"
and "z0 \<in> S"
shows "g z0 \<noteq> 0"
proof
assume g0: "g z0 = 0"
obtain h r m
where "0 < m" "0 < r" and subS: "ball z0 r \<subseteq> S"
and holh: "h holomorphic_on ball z0 r"
and geq: "\<And>w. w \<in> ball z0 r \<Longrightarrow> g w = (w - z0)^m * h w"
and hnz: "\<And>w. w \<in> ball z0 r \<Longrightarrow> h w \<noteq> 0"
by (blast intro: holomorphic_factor_zero_nonconstant [OF holg S \<open>z0 \<in> S\<close> g0 nonconst])
then have holf0: "\<F> n holomorphic_on ball z0 r" for n
by (meson holf holomorphic_on_subset)
have *: "((\<lambda>z. 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 "(\<lambda>z. deriv (\<F> n) z / \<F> n z) holomorphic_on ball z0 r"
apply (intro holomorphic_intros holomorphic_deriv holf holf0 open_ball nz)
using \<open>ball z0 r \<subseteq> S\<close> by blast
qed (use \<open>0 < r\<close> in auto)
have hol_dg: "deriv g holomorphic_on S"
by (simp add: \<open>open S\<close> 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 \<open>0 < r\<close> 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 \<circ> g)"
apply (intro continuous_intros holomorphic_on_imp_continuous_on holomorphic_on_subset [OF holg])
using \<open>0 < r\<close> subS by auto
then have "compact ((cmod \<circ> g) ` sphere z0 (r/2))"
by (rule compact_continuous_image [OF _ compact_sphere])
moreover have "(cmod \<circ> g) ` sphere z0 (r/2) \<noteq> {}"
using \<open>0 < r\<close> by auto
ultimately obtain b where b: "b \<in> (cmod \<circ> g) ` sphere z0 (r/2)"
"\<And>t. t \<in> (cmod \<circ> g) ` sphere z0 (r/2) \<Longrightarrow> b \<le> t"
using compact_attains_inf [of "(norm \<circ> g) ` (sphere z0 (r/2))"] by blast
have "(\<lambda>n. contour_integral (circlepath z0 (r/2)) (\<lambda>z. deriv (\<F> n) z / \<F> n z)) \<longlonglongrightarrow>
contour_integral (circlepath z0 (r/2)) (\<lambda>z. deriv g z / g z)"
proof (rule contour_integral_uniform_limit_circlepath)
show "\<forall>\<^sub>F n in sequentially. (\<lambda>z. 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)) (\<lambda>n z. deriv (\<F> n) z / \<F> n z) (\<lambda>z. deriv g z / g z) sequentially"
proof (rule uniform_lim_divide [OF _ _ bo_dg])
show "uniform_limit (sphere z0 (r/2)) (\<lambda>a. 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: "\<And>x. dist z0 x \<le> 3 * r / 4 \<Longrightarrow> 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) \<subseteq> ball z0 r" "cball w (r/4) \<subseteq> ball z0 r"
using \<open>0 < r\<close> by (simp_all add: ball_subset_ball_iff cball_subset_ball_iff w)
with subS have wr4_sub: "ball w (r/4) \<subseteq> S" "cball w (r/4) \<subseteq> S" by force+
moreover
have "(\<lambda>z. \<F> n z - g z) holomorphic_on S"
by (intro holomorphic_intros holf holg)
ultimately have hol: "(\<lambda>z. \<F> n z - g z) holomorphic_on ball w (r/4)"
and cont: "continuous_on (cball w (r / 4)) (\<lambda>z. \<F> n z - g z)"
using holomorphic_on_subset by (blast intro: holomorphic_on_imp_continuous_on)+
have "w \<in> S"
using \<open>0 < r\<close> wr4_sub by auto
have "\<And>y. dist w y < r / 4 \<Longrightarrow> dist z0 y \<le> 3 * r / 4"
apply (rule dist_triangle_le [where z=w])
using w by (simp add: dist_commute)
with e8 have in_ball: "\<And>y. y \<in> ball w (r/4) \<Longrightarrow> \<F> n y - g y \<in> 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 \<open>w \<in> S\<close> holf \<open>open S\<close>)
moreover
have "g field_differentiable at w"
using \<open>w \<in> S\<close> \<open>open S\<close> holg holomorphic_on_imp_differentiable_at by auto
moreover
have "cmod (deriv (\<lambda>w. \<F> n w - g w) w) * 2 \<le> e"
apply (rule Cauchy_higher_deriv_bound [OF hol cont in_ball, of 1, simplified])
using \<open>r > 0\<close> by auto
ultimately have "dist (deriv (\<F> n) w) (deriv g w) \<le> e/2"
by (simp add: dist_norm)
then show ?thesis
using \<open>e > 0\<close> by auto
qed
have "cball z0 (3 * r / 4) \<subseteq> ball z0 r"
by (simp add: cball_subset_ball_iff \<open>0 < r\<close>)
with subS have "uniform_limit (cball z0 (3 * r/4)) \<F> g sequentially"
by (force intro: ul_g)
then have "\<forall>\<^sub>F n in sequentially. \<forall>x\<in>cball z0 (3 * r / 4). dist (\<F> n x) (g x) < r / 4 * e / 2"
using \<open>0 < e\<close> \<open>0 < r\<close> by (force simp: intro!: uniform_limitD)
then show "\<forall>\<^sub>F n in sequentially. \<forall>x \<in> 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) \<subseteq> ball z0 r"
using \<open>0 < r\<close> by auto
with subS have "uniform_limit (sphere z0 (r/2)) \<F> g sequentially"
by (force intro: ul_g)
then show "\<forall>\<^sub>F n in sequentially. \<forall>x \<in> sphere z0 (r/2). dist (\<F> n x) (g x) < e"
apply (rule uniform_limitD)
using \<open>0 < e\<close> by force
qed
show "b > 0" "\<And>x. x \<in> sphere z0 (r/2) \<Longrightarrow> b \<le> cmod (g x)"
using b \<open>0 < r\<close> by (fastforce simp: geq hnz)+
qed
qed (use \<open>0 < r\<close> in auto)
then have "(\<lambda>n. 0) \<longlonglongrightarrow> contour_integral (circlepath z0 (r/2)) (\<lambda>z. deriv g z / g z)"
by (simp add: contour_integral_unique [OF *])
then have "contour_integral (circlepath z0 (r/2)) (\<lambda>z. deriv g z / g z) = 0"
by (simp add: LIMSEQ_const_iff)
moreover
have "contour_integral (circlepath z0 (r/2)) (\<lambda>z. deriv g z / g z) =
contour_integral (circlepath z0 (r/2)) (\<lambda>z. m / (z - z0) + deriv h z / h z)"
proof (rule contour_integral_eq, use \<open>0 < r\<close> in simp)
fix w
assume w: "dist z0 w * 2 = r"
then have w_inb: "w \<in> ball z0 r"
using \<open>0 < r\<close> 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 \<noteq> z0"
proof -
have "((\<lambda>w. (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 \<open>m > 0\<close> \<open>0 < r\<close> 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 \<open>m > 0\<close> \<open>0 < r\<close>
apply (simp_all add: hnz geq)
done
qed
with \<open>0 < r\<close> \<open>0 < m\<close> 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)) (\<lambda>z. 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 "((\<lambda>x. m / (x - z0)) has_contour_integral 2 * of_real pi * \<i> * m) (circlepath z0 (r/2))"
by (force simp: \<open>0 < r\<close> intro: Cauchy_integral_circlepath_simple)
show "((\<lambda>x. 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 \<open>0 < r\<close>
apply force+
done
qed
ultimately show False using \<open>0 < m\<close> by auto
qed
corollary Hurwitz_injective:
assumes S: "open S" "connected S"
and holf: "\<And>n::nat. \<F> n holomorphic_on S"
and holg: "g holomorphic_on S"
and ul_g: "\<And>K. \<lbrakk>compact K; K \<subseteq> S\<rbrakk> \<Longrightarrow> uniform_limit K \<F> g sequentially"
and nonconst: "~ g constant_on S"
and inj: "\<And>n. inj_on (\<F> n) S"
shows "inj_on g S"
proof -
have False if z12: "z1 \<in> S" "z2 \<in> S" "z1 \<noteq> z2" "g z2 = g z1" for z1 z2
proof -
obtain z0 where "z0 \<in> S" and z0: "g z0 \<noteq> g z2"
using constant_on_def nonconst by blast
have "(\<lambda>z. g z - g z1) holomorphic_on S"
by (intro holomorphic_intros holg)
then obtain r where "0 < r" "ball z2 r \<subseteq> S" "\<And>z. dist z2 z < r \<and> z \<noteq> z2 \<Longrightarrow> g z \<noteq> g z1"
apply (rule isolated_zeros [of "\<lambda>z. g z - g z1" S z2 z0])
using S \<open>z0 \<in> S\<close> z0 z12 by auto
have "g z2 - g z1 \<noteq> 0"
proof (rule Hurwitz_no_zeros [of "S - {z1}" "\<lambda>n z. \<F> n z - \<F> n z1" "\<lambda>z. 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 "\<And>n. (\<lambda>z. \<F> n z - \<F> n z1) holomorphic_on S - {z1}"
by (intro holomorphic_intros holomorphic_on_subset [OF holf]) blast
show "(\<lambda>z. g z - g z1) holomorphic_on S - {z1}"
by (intro holomorphic_intros holomorphic_on_subset [OF holg]) blast
show "uniform_limit K (\<lambda>n z. \<F> n z - \<F> n z1) (\<lambda>z. g z - g z1) sequentially"
if "compact K" "K \<subseteq> 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: "\<forall>\<^sub>F n in sequentially. \<forall>x \<in> K. dist (\<F> n x) (g x) < e/2"
using \<open>0 < e\<close> by (force simp: intro!: uniform_limitD)
have "uniform_limit {z1} \<F> g sequentially"
by (simp add: ul_g z12)
then have "\<forall>\<^sub>F n in sequentially. \<forall>x \<in> {z1}. dist (\<F> n x) (g x) < e/2"
using \<open>0 < e\<close> by (force simp: intro!: uniform_limitD)
then have z1: "\<forall>\<^sub>F n in sequentially. dist (\<F> n z1) (g z1) < e/2"
by simp
have "\<forall>\<^sub>F n in sequentially. \<forall>x\<in>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 "\<forall>\<^sub>F n in sequentially. \<forall>x\<in>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 real_sum_of_halves)
then
show "\<forall>\<^sub>F n in sequentially. \<forall>x\<in>K. dist (\<F> n x - \<F> n z1) (g x - g z1) < e"
by simp
qed
show "\<not> (\<lambda>z. g z - g z1) constant_on S - {z1}"
unfolding constant_on_def
by (metis Diff_iff \<open>z0 \<in> S\<close> empty_iff insert_iff right_minus_eq z0 z12)
show "\<And>n z. z \<in> S - {z1} \<Longrightarrow> \<F> n z - \<F> n z1 \<noteq> 0"
by (metis DiffD1 DiffD2 eq_iff_diff_eq_0 inj inj_onD insertI1 \<open>z1 \<in> S\<close>)
show "z2 \<in> S - {z1}"
using \<open>z2 \<in> S\<close> \<open>z1 \<noteq> z2\<close> by auto
qed
with z12 show False by auto
qed
then show ?thesis by (auto simp: inj_on_def)
qed
subsection\<open>The Great Picard theorem\<close>
lemma GPicard1:
assumes S: "open S" "connected S" and "w \<in> S" "0 < r" "Y \<subseteq> X"
and holX: "\<And>h. h \<in> X \<Longrightarrow> h holomorphic_on S"
and X01: "\<And>h z. \<lbrakk>h \<in> X; z \<in> S\<rbrakk> \<Longrightarrow> h z \<noteq> 0 \<and> h z \<noteq> 1"
and r: "\<And>h. h \<in> Y \<Longrightarrow> norm(h w) \<le> r"
obtains B Z where "0 < B" "open Z" "w \<in> Z" "Z \<subseteq> S" "\<And>h z. \<lbrakk>h \<in> Y; z \<in> Z\<rbrakk> \<Longrightarrow> norm(h z) \<le> B"
proof -
obtain e where "e > 0" and e: "cball w e \<subseteq> 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) \<subseteq> S"
using e ball_divide_subset_numeral ball_subset_cball by blast
show "cmod (h z) \<le> exp (pi * exp (pi * (2 + 2 * r + 12)))"
if "h \<in> Y" "z \<in> ball w (e / 2)" for h z
proof -
have "h \<in> X"
using \<open>Y \<subseteq> X\<close> \<open>h \<in> Y\<close> 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 \<circ> (\<lambda>z. (w + of_real e * z))) holomorphic_on cball 0 1"
apply (intro holomorphic_intros holomorphic_on_compose)
apply (erule holomorphic_on_subset)
using that \<open>e > 0\<close> by (auto simp: dist_norm norm_mult)
have norm_le_r: "cmod ((h \<circ> (\<lambda>z. w + complex_of_real e * z)) 0) \<le> r"
by (auto simp: r \<open>h \<in> Y\<close>)
have le12: "norm (of_real(inverse e) * (z - w)) \<le> 1/2"
using that \<open>e > 0\<close> by (simp add: inverse_eq_divide dist_norm norm_minus_commute norm_divide)
have non01: "\<And>z::complex. cmod z \<le> 1 \<Longrightarrow> h (w + e * z) \<noteq> 0 \<and> h (w + e * z) \<noteq> 1"
apply (rule X01 [OF \<open>h \<in> X\<close>])
apply (rule subsetD [OF e])
using \<open>0 < e\<close> by (auto simp: dist_norm norm_mult)
have "cmod (h z) \<le> cmod (h (w + of_real e * (inverse e * (z - w))))"
using \<open>0 < e\<close> by (simp add: divide_simps)
also have "... \<le> exp (pi * exp (pi * (14 + 2 * r)))"
using r [OF \<open>h \<in> Y\<close>] Schottky [OF hol_h_o norm_le_r _ _ _ le12] non01 by auto
finally
show ?thesis by simp
qed
qed (use \<open>e > 0\<close> in auto)
qed
lemma GPicard2:
assumes "S \<subseteq> T" "connected T" "S \<noteq> {}" "open S" "\<And>x. \<lbrakk>x islimpt S; x \<in> T\<rbrakk> \<Longrightarrow> x \<in> S"
shows "S = T"
by (metis assms open_subset connected_clopen closedin_limpt)
lemma GPicard3:
assumes S: "open S" "connected S" "w \<in> S" and "Y \<subseteq> X"
and holX: "\<And>h. h \<in> X \<Longrightarrow> h holomorphic_on S"
and X01: "\<And>h z. \<lbrakk>h \<in> X; z \<in> S\<rbrakk> \<Longrightarrow> h z \<noteq> 0 \<and> h z \<noteq> 1"
and no_hw_le1: "\<And>h. h \<in> Y \<Longrightarrow> norm(h w) \<le> 1"
and "compact K" "K \<subseteq> S"
obtains B where "\<And>h z. \<lbrakk>h \<in> Y; z \<in> K\<rbrakk> \<Longrightarrow> norm(h z) \<le> B"
proof -
define U where "U \<equiv> {z \<in> S. \<exists>B Z. 0 < B \<and> open Z \<and> z \<in> Z \<and> Z \<subseteq> S \<and>
(\<forall>h z'. h \<in> Y \<and> z' \<in> Z \<longrightarrow> norm(h z') \<le> B)}"
then have "U \<subseteq> S" by blast
have "U = S"
proof (rule GPicard2 [OF \<open>U \<subseteq> S\<close> \<open>connected S\<close>])
show "U \<noteq> {}"
proof -
obtain B Z where "0 < B" "open Z" "w \<in> Z" "Z \<subseteq> S"
and "\<And>h z. \<lbrakk>h \<in> Y; z \<in> Z\<rbrakk> \<Longrightarrow> norm(h z) \<le> B"
apply (rule GPicard1 [OF S zero_less_one \<open>Y \<subseteq> X\<close> holX])
using no_hw_le1 X01 by force+
then show ?thesis
unfolding U_def using \<open>w \<in> S\<close> by blast
qed
show "open U"
unfolding open_subopen [of U] by (auto simp: U_def)
fix v
assume v: "v islimpt U" "v \<in> S"
have "~ (\<forall>r>0. \<exists>h\<in>Y. r < cmod (h v))"
proof
assume "\<forall>r>0. \<exists>h\<in>Y. r < cmod (h v)"
then have "\<forall>n. \<exists>h\<in>Y. Suc n < cmod (h v)"
by simp
then obtain \<F> where FY: "\<And>n. \<F> n \<in> Y" and ltF: "\<And>n. Suc n < cmod (\<F> n v)"
by metis
define \<G> where "\<G> \<equiv> \<lambda>n z. inverse(\<F> n z)"
have hol\<G>: "\<G> n holomorphic_on S" for n
apply (simp add: \<G>_def)
using FY X01 \<open>Y \<subseteq> X\<close> holX apply (blast intro: holomorphic_on_inverse)
done
have \<G>not0: "\<G> n z \<noteq> 0" and \<G>not1: "\<G> n z \<noteq> 1" if "z \<in> S" for n z
using FY X01 \<open>Y \<subseteq> X\<close> that by (force simp: \<G>_def)+
have \<G>_le1: "cmod (\<G> n v) \<le> 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 \<equiv> {h. h holomorphic_on S \<and> (\<forall>z \<in> S. h z \<noteq> 0 \<and> h z \<noteq> 1)}"
obtain B Z where "0 < B" "open Z" "v \<in> Z" "Z \<subseteq> S"
and B: "\<And>h z. \<lbrakk>h \<in> range \<G>; z \<in> Z\<rbrakk> \<Longrightarrow> norm(h z) \<le> B"
apply (rule GPicard1 [OF \<open>open S\<close> \<open>connected S\<close> \<open>v \<in> S\<close> zero_less_one, of "range \<G>" W])
using hol\<G> \<G>not0 \<G>not1 \<G>_le1 by (force simp: W_def)+
then obtain e where "e > 0" and e: "ball v e \<subseteq> Z"
by (meson open_contains_ball)
obtain h j where holh: "h holomorphic_on ball v e" and "strict_mono j"
and lim: "\<And>x. x \<in> ball v e \<Longrightarrow> (\<lambda>n. \<G> (j n) x) \<longlonglongrightarrow> h x"
and ulim: "\<And>K. \<lbrakk>compact K; K \<subseteq> ball v e\<rbrakk>
\<Longrightarrow> uniform_limit K (\<G> \<circ> j) h sequentially"
proof (rule Montel)
show "\<And>h. h \<in> range \<G> \<Longrightarrow> h holomorphic_on ball v e"
by (metis \<open>Z \<subseteq> S\<close> e hol\<G> holomorphic_on_subset imageE)
show "\<And>K. \<lbrakk>compact K; K \<subseteq> ball v e\<rbrakk> \<Longrightarrow> \<exists>B. \<forall>h\<in>range \<G>. \<forall>z\<in>K. cmod (h z) \<le> B"
using B e by blast
qed auto
have "h v = 0"
proof (rule LIMSEQ_unique)
show "(\<lambda>n. \<G> (j n) v) \<longlonglongrightarrow> h v"
using \<open>e > 0\<close> lim by simp
have lt_Fj: "real x \<le> cmod (\<F> (j x) v)" for x
by (metis of_nat_Suc ltF \<open>strict_mono j\<close> add.commute less_eq_real_def less_le_trans nat_le_real_less seq_suble)
show "(\<lambda>n. \<G> (j n) v) \<longlonglongrightarrow> 0"
proof (rule Lim_null_comparison [OF eventually_sequentiallyI lim_inverse_n])
show "cmod (\<G> (j x) v) \<le> inverse (real x)" if "1 \<le> x" for x
using that by (simp add: \<G>_def norm_inverse_le_norm [OF lt_Fj])
qed
qed
have "h v \<noteq> 0"
proof (rule Hurwitz_no_zeros [of "ball v e" "\<G> \<circ> j" h])
show "\<And>n. (\<G> \<circ> j) n holomorphic_on ball v e"
using \<open>Z \<subseteq> S\<close> e hol\<G> by force
show "\<And>n z. z \<in> ball v e \<Longrightarrow> (\<G> \<circ> j) n z \<noteq> 0"
using \<G>not0 \<open>Z \<subseteq> S\<close> e by fastforce
show "\<not> h constant_on ball v e"
proof (clarsimp simp: constant_on_def)
fix c
have False if "\<And>z. dist v z < e \<Longrightarrow> h z = c"
proof -
have "h v = c"
by (simp add: \<open>0 < e\<close> that)
obtain y where "y \<in> U" "y \<noteq> v" and y: "dist y v < e"
using v \<open>e > 0\<close> by (auto simp: islimpt_approachable)
then obtain C T where "y \<in> S" "C > 0" "open T" "y \<in> T" "T \<subseteq> S"
and "\<And>h z'. \<lbrakk>h \<in> Y; z' \<in> T\<rbrakk> \<Longrightarrow> cmod (h z') \<le> C"
using \<open>y \<in> U\<close> by (auto simp: U_def)
then have le_C: "\<And>n. cmod (\<F> n y) \<le> C"
using FY by blast
have "\<forall>\<^sub>F n in sequentially. dist (\<G> (j n) y) (h y) < inverse C"
using uniform_limitD [OF ulim [of "{y}"], of "inverse C"] \<open>C > 0\<close> 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 \<open>h v = c\<close> dist_commute that y)
ultimately have "norm (\<G> (j n) y) < inverse C"
by (simp add: \<open>h v = 0\<close>)
then have "C < norm (\<F> (j n) y)"
apply (simp add: \<G>_def)
by (metis FY X01 \<open>0 < C\<close> \<open>y \<in> S\<close> \<open>Y \<subseteq> X\<close> inverse_less_iff_less norm_inverse subsetD zero_less_norm_iff)
show False
using \<open>C < cmod (\<F> (j n) y)\<close> le_C not_less by blast
qed
then show "\<exists>x\<in>ball v e. h x \<noteq> c" by force
qed
show "h holomorphic_on ball v e"
by (simp add: holh)
show "\<And>K. \<lbrakk>compact K; K \<subseteq> ball v e\<rbrakk> \<Longrightarrow> uniform_limit K (\<G> \<circ> j) h sequentially"
by (simp add: ulim)
qed (use \<open>e > 0\<close> in auto)
with \<open>h v = 0\<close> show False by blast
qed
then show "v \<in> U"
apply (clarsimp simp add: U_def v)
apply (rule GPicard1[OF \<open>open S\<close> \<open>connected S\<close> \<open>v \<in> S\<close> _ \<open>Y \<subseteq> X\<close> holX])
using X01 no_hw_le1 apply (meson | force simp: not_less)+
done
qed
have "\<And>x. x \<in> K \<longrightarrow> x \<in> U"
using \<open>U = S\<close> \<open>K \<subseteq> S\<close> by blast
then have "\<And>x. x \<in> K \<longrightarrow> (\<exists>B Z. 0 < B \<and> open Z \<and> x \<in> Z \<and>
(\<forall>h z'. h \<in> Y \<and> z' \<in> Z \<longrightarrow> norm(h z') \<le> B))"
unfolding U_def by blast
then obtain F Z where F: "\<And>x. x \<in> K \<Longrightarrow> open (Z x) \<and> x \<in> Z x \<and>
(\<forall>h z'. h \<in> Y \<and> z' \<in> Z x \<longrightarrow> norm(h z') \<le> F x)"
by metis
then obtain L where "L \<subseteq> K" "finite L" and L: "K \<subseteq> (\<Union>c \<in> L. Z c)"
by (auto intro: compactE_image [OF \<open>compact K\<close>, of K Z])
then have *: "\<And>x h z'. \<lbrakk>x \<in> L; h \<in> Y \<and> z' \<in> Z x\<rbrakk> \<Longrightarrow> cmod (h z') \<le> F x"
using F by blast
have "\<exists>B. \<forall>h z. h \<in> Y \<and> z \<in> K \<longrightarrow> norm(h z) \<le> B"
proof (cases "L = {}")
case True with L show ?thesis by simp
next
case False
with \<open>finite L\<close> 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: "\<And>e. \<lbrakk>0 < e; e < k\<rbrakk> \<Longrightarrow> \<exists>d. 0 < d \<and> d < e \<and> (\<forall>z \<in> sphere 0 d. norm(f z) \<le> B)"
obtains \<epsilon> where "0 < \<epsilon>" "\<epsilon> < k" "\<And>z. z \<in> ball 0 \<epsilon> - {0} \<Longrightarrow> norm(f z) \<le> B"
proof -
obtain \<epsilon> where "0 < \<epsilon>" "\<epsilon> < k/2" and \<epsilon>: "\<And>z. norm z = \<epsilon> \<Longrightarrow> norm(f z) \<le> B"
using AE [of "k/2"] \<open>0 < k\<close> by auto
show ?thesis
proof
show "\<epsilon> < k"
using \<open>0 < k\<close> \<open>\<epsilon> < k/2\<close> by auto
show "cmod (f \<xi>) \<le> B" if \<xi>: "\<xi> \<in> ball 0 \<epsilon> - {0}" for \<xi>
proof -
obtain d where "0 < d" "d < norm \<xi>" and d: "\<And>z. norm z = d \<Longrightarrow> norm(f z) \<le> B"
using AE [of "norm \<xi>"] \<open>\<epsilon> < k\<close> \<xi> by auto
have [simp]: "closure (cball 0 \<epsilon> - ball 0 d) = cball 0 \<epsilon> - ball 0 d"
by (blast intro!: closure_closed)
have [simp]: "interior (cball 0 \<epsilon> - ball 0 d) = ball 0 \<epsilon> - cball (0::complex) d"
using \<open>0 < \<epsilon>\<close> \<open>0 < d\<close> by (simp add: interior_diff)
have *: "norm(f w) \<le> B" if "w \<in> cball 0 \<epsilon> - ball 0 d" for w
proof (rule maximum_modulus_frontier [of f "cball 0 \<epsilon> - ball 0 d"])
show "f holomorphic_on interior (cball 0 \<epsilon> - ball 0 d)"
apply (rule holomorphic_on_subset [OF holf])
using \<open>\<epsilon> < k\<close> \<open>0 < d\<close> that by auto
show "continuous_on (closure (cball 0 \<epsilon> - ball 0 d)) f"
apply (rule holomorphic_on_imp_continuous_on)
apply (rule holomorphic_on_subset [OF holf])
using \<open>0 < d\<close> \<open>\<epsilon> < k\<close> by auto
show "\<And>z. z \<in> frontier (cball 0 \<epsilon> - ball 0 d) \<Longrightarrow> cmod (f z) \<le> B"
apply (simp add: frontier_def)
using \<epsilon> d less_eq_real_def by blast
qed (use that in auto)
show ?thesis
using * \<open>d < cmod \<xi>\<close> that by auto
qed
qed (use \<open>0 < \<epsilon>\<close> in auto)
qed
lemma GPicard5:
assumes holf: "f holomorphic_on (ball 0 1 - {0})"
and f01: "\<And>z. z \<in> ball 0 1 - {0} \<Longrightarrow> f z \<noteq> 0 \<and> f z \<noteq> 1"
obtains e B where "0 < e" "e < 1" "0 < B"
"(\<forall>z \<in> ball 0 e - {0}. norm(f z) \<le> B) \<or>
(\<forall>z \<in> ball 0 e - {0}. norm(f z) \<ge> B)"
proof -
have [simp]: "1 + of_nat n \<noteq> (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 *: "(\<lambda>x::complex. x / of_nat (Suc n)) ` (ball 0 1 - {0}) \<subseteq> ball 0 1 - {0}" for n
by (auto simp: norm_divide divide_simps split: if_split_asm)
define h where "h \<equiv> \<lambda>n 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 "(\<lambda>x. x / of_nat (Suc n)) holomorphic_on ball 0 1 - {0}"
by (intro holomorphic_intros) auto
qed
have h01: "\<And>n z. z \<in> ball 0 1 - {0} \<Longrightarrow> h n z \<noteq> 0 \<and> h n z \<noteq> 1"
unfolding h_def
apply (rule f01)
using * by force
obtain w where w: "w \<in> ball 0 1 - {0::complex}"
by (rule_tac w = "1/2" in that) auto
consider "infinite {n. norm(h n w) \<le> 1}" | "infinite {n. 1 \<le> 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 \<Rightarrow> nat"
where "strict_mono r" and r: "\<And>n. r n \<in> {n. norm(h n w) \<le> 1}"
by blast
obtain B where B: "\<And>j z. \<lbrakk>norm z = 1/2; j \<in> range (h \<circ> r)\<rbrakk> \<Longrightarrow> norm(j z) \<le> B"
proof (rule GPicard3 [OF _ _ w, where K = "sphere 0 (1/2)"])
show "range (h \<circ> r) \<subseteq>
{g. g holomorphic_on ball 0 1 - {0} \<and> (\<forall>z\<in>ball 0 1 - {0}. g z \<noteq> 0 \<and> g z \<noteq> 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) \<le> B" if "norm z = 1 / (2 * (1 + of_nat (r n)))" for z n
proof -
have *: "\<And>w. norm w = 1/2 \<Longrightarrow> cmod((f (w / (1 + of_nat (r n))))) \<le> 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 \<epsilon> where "0 < \<epsilon>" "\<epsilon> < 1" "\<And>z. z \<in> ball 0 \<epsilon> - {0} \<Longrightarrow> cmod(f z) \<le> 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 "... \<le> r n"
using \<open>strict_mono r\<close> by (simp add: seq_suble)
finally have "(1/e - 2) / 2 < real (r n)" .
with \<open>0 < e\<close> have e: "e > 1 / (2 + 2 * real (r n))"
by (simp add: field_simps)
show "\<exists>d>0. d < e \<and> (\<forall>z\<in>sphere 0 d. cmod (f z) \<le> 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 \<epsilon>: "cmod (f z) \<le> \<bar>B\<bar> + 1" if "cmod z < \<epsilon>" "z \<noteq> 0" for z
using that by fastforce
have "0 < \<bar>B\<bar> + 1"
by simp
then show ?thesis
apply (rule that [OF \<open>0 < \<epsilon>\<close> \<open>\<epsilon> < 1\<close>])
using \<epsilon> by auto
next
case 2
with infinite_enumerate obtain r :: "nat \<Rightarrow> nat"
where "strict_mono r" and r: "\<And>n. r n \<in> {n. norm(h n w) \<ge> 1}"
by blast
obtain B where B: "\<And>j z. \<lbrakk>norm z = 1/2; j \<in> range (\<lambda>n. inverse \<circ> h (r n))\<rbrakk> \<Longrightarrow> norm(j z) \<le> B"
proof (rule GPicard3 [OF _ _ w, where K = "sphere 0 (1/2)"])
show "range (\<lambda>n. inverse \<circ> h (r n)) \<subseteq>
{g. g holomorphic_on ball 0 1 - {0} \<and> (\<forall>z\<in>ball 0 1 - {0}. g z \<noteq> 0 \<and> g z \<noteq> 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 "\<And>j. j \<in> range (\<lambda>n. inverse \<circ> h (r n)) \<Longrightarrow> cmod (j w) \<le> 1"
using r norm_inverse_le_norm by fastforce
qed (use r in auto)
have norm_if_le_B: "cmod(inverse (f z)) \<le> 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)))))) \<le> 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 \<circ> f) holomorphic_on (ball 0 1 - {0})"
by (metis (no_types, lifting) holf comp_apply f01 holomorphic_on_inverse holomorphic_transform)
obtain \<epsilon> where "0 < \<epsilon>" "\<epsilon> < 1" and leB: "\<And>z. z \<in> ball 0 \<epsilon> - {0} \<Longrightarrow> cmod((inverse \<circ> f) z) \<le> 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 "... \<le> r n"
using \<open>strict_mono r\<close> by (simp add: seq_suble)
finally have "(1/e - 2) / 2 < real (r n)" .
with \<open>0 < e\<close> have e: "e > 1 / (2 + 2 * real (r n))"
by (simp add: field_simps)
show "\<exists>d>0. d < e \<and> (\<forall>z\<in>sphere 0 d. cmod ((inverse \<circ> f) z) \<le> 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 \<epsilon>: "cmod (f z) \<ge> inverse B" and "B > 0" if "cmod z < \<epsilon>" "z \<noteq> 0" for z
proof -
have "inverse (cmod (f z)) \<le> B"
using leB that by (simp add: norm_inverse)
moreover
have "f z \<noteq> 0"
using \<open>\<epsilon> < 1\<close> f01 that by auto
ultimately show "cmod (f z) \<ge> inverse B"
by (simp add: norm_inverse inverse_le_imp_le)
show "B > 0"
using \<open>f z \<noteq> 0\<close> \<open>inverse (cmod (f z)) \<le> B\<close> not_le order.trans by fastforce
qed
then have "B > 0"
by (metis \<open>0 < \<epsilon>\<close> dense leI order.asym vector_choose_size)
then have "inverse B > 0"
by (simp add: divide_simps)
then show ?thesis
apply (rule that [OF \<open>0 < \<epsilon>\<close> \<open>\<epsilon> < 1\<close>])
using \<epsilon> by auto
qed
qed
lemma GPicard6:
assumes "open M" "z \<in> M" "a \<noteq> 0" and holf: "f holomorphic_on (M - {z})"
and f0a: "\<And>w. w \<in> M - {z} \<Longrightarrow> f w \<noteq> 0 \<and> f w \<noteq> a"
obtains r where "0 < r" "ball z r \<subseteq> M"
"bounded(f ` (ball z r - {z})) \<or>
bounded((inverse \<circ> f) ` (ball z r - {z}))"
proof -
obtain r where "0 < r" and r: "ball z r \<subseteq> M"
using assms openE by blast
let ?g = "\<lambda>w. f (z + of_real r * w) / a"
obtain e B where "0 < e" "e < 1" "0 < B"
and B: "(\<forall>z \<in> ball 0 e - {0}. norm(?g z) \<le> B) \<or> (\<forall>z \<in> ball 0 e - {0}. norm(?g z) \<ge> 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 \<open>0 < r\<close> \<open>a \<noteq> 0\<close> r
by (auto simp: dist_norm norm_mult subset_eq)
show "\<And>w. w \<in> ball 0 1 - {0} \<Longrightarrow> f (z + of_real r * w) / a \<noteq> 0 \<and> f (z + of_real r * w) / a \<noteq> 1"
apply (simp add: divide_simps \<open>a \<noteq> 0\<close>)
apply (rule f0a)
using \<open>0 < r\<close> r by (auto simp: dist_norm norm_mult subset_eq)
qed
show ?thesis
proof
show "0 < e*r"
by (simp add: \<open>0 < e\<close> \<open>0 < r\<close>)
have "ball z (e * r) \<subseteq> ball z r"
by (simp add: \<open>0 < r\<close> \<open>e < 1\<close> order.strict_implies_order subset_ball)
then show "ball z (e * r) \<subseteq> M"
using r by blast
consider "\<And>z. z \<in> ball 0 e - {0} \<Longrightarrow> norm(?g z) \<le> B" | "\<And>z. z \<in> ball 0 e - {0} \<Longrightarrow> norm(?g z) \<ge> B"
using B by blast
then show "bounded (f ` (ball z (e * r) - {z})) \<or>
bounded ((inverse \<circ> f) ` (ball z (e * r) - {z}))"
proof cases
case 1
have "\<lbrakk>dist z w < e * r; w \<noteq> z\<rbrakk> \<Longrightarrow> cmod (f w) \<le> B * norm a" for w
using \<open>a \<noteq> 0\<close> \<open>0 < r\<close> 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 "\<lbrakk>dist z w < e * r; w \<noteq> z\<rbrakk> \<Longrightarrow> cmod (f w) \<ge> B * norm a" for w
using \<open>a \<noteq> 0\<close> \<open>0 < r\<close> 2 [of "(w - z) / r"]
by (simp add: norm_divide dist_norm divide_simps)
then have "\<lbrakk>dist z w < e * r; w \<noteq> z\<rbrakk> \<Longrightarrow> inverse (cmod (f w)) \<le> inverse (B * norm a)" for w
by (metis \<open>0 < B\<close> \<open>a \<noteq> 0\<close> 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 \<in> M" "a \<noteq> b" and holf: "f holomorphic_on (M - {z})"
and fab: "\<And>w. w \<in> M - {z} \<Longrightarrow> f w \<noteq> a \<and> f w \<noteq> b"
obtains l where "(f \<longlongrightarrow> l) (at z) \<or> ((inverse \<circ> f) \<longlongrightarrow> l) (at z)"
proof -
obtain r where "0 < r" and zrM: "ball z r \<subseteq> M"
and r: "bounded((\<lambda>z. f z - a) ` (ball z r - {z})) \<or>
bounded((inverse \<circ> (\<lambda>z. f z - a)) ` (ball z r - {z}))"
proof (rule GPicard6 [OF \<open>open M\<close> \<open>z \<in> M\<close>])
show "b - a \<noteq> 0"
using assms by auto
show "(\<lambda>z. 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: "(\<lambda>z. 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 ((\<lambda>z. f z - a) ` (ball z r - {z}))"
then obtain B where B: "\<And>w. w \<in> (\<lambda>z. f z - a) ` (ball z r - {z}) \<Longrightarrow> norm w \<le> B"
by (force simp: bounded_iff)
have "\<forall>\<^sub>F w in at z. cmod (f w - a) \<le> B"
apply (simp add: eventually_at)
apply (rule_tac x=r in exI)
using \<open>0 < r\<close> by (auto simp: dist_commute intro!: B)
then have "\<exists>B. \<forall>\<^sub>F w in at z. cmod (f w) \<le> 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: "\<And>w. w \<in> ball z r - {z} \<Longrightarrow> g w = f w"
using \<open>0 < r\<close> holomorphic_on_extend_bounded [OF holfb] by auto
then have "g \<midarrow>z\<rightarrow> g z"
apply (simp add: continuous_at [symmetric])
using \<open>0 < r\<close> centre_in_ball field_differentiable_imp_continuous_at holomorphic_on_imp_differentiable_at by blast
then have "(f \<longlongrightarrow> g z) (at z)"
apply (rule Lim_transform_within_open [of g "g z" z UNIV "ball z r"])
using \<open>0 < r\<close> by (auto simp: gf)
then show ?thesis
using that by blast
next
assume "bounded((inverse \<circ> (\<lambda>z. f z - a)) ` (ball z r - {z}))"
then obtain B where B: "\<And>w. w \<in> (inverse \<circ> (\<lambda>z. f z - a)) ` (ball z r - {z}) \<Longrightarrow> norm w \<le> B"
by (force simp: bounded_iff)
have "\<forall>\<^sub>F w in at z. cmod (inverse (f w - a)) \<le> B"
apply (simp add: eventually_at)
apply (rule_tac x=r in exI)
using \<open>0 < r\<close> by (auto simp: dist_commute intro!: B)
then have "\<exists>B. \<forall>\<^sub>F z in at z. cmod (inverse (f z - a)) \<le> B"
by blast
then obtain g where holg: "g holomorphic_on ball z r" and gf: "\<And>w. w \<in> ball z r - {z} \<Longrightarrow> g w = inverse (f w - a)"
using \<open>0 < r\<close> holomorphic_on_extend_bounded [OF holfb_i] by auto
then have gz: "g \<midarrow>z\<rightarrow> g z"
apply (simp add: continuous_at [symmetric])
using \<open>0 < r\<close> centre_in_ball field_differentiable_imp_continuous_at holomorphic_on_imp_differentiable_at by blast
have gnz: "\<And>w. w \<in> ball z r - {z} \<Longrightarrow> g w \<noteq> 0"
using gf fab zrM by fastforce
show ?thesis
proof (cases "g z = 0")
case True
have *: "\<lbrakk>g \<noteq> 0; inverse g = f - a\<rbrakk> \<Longrightarrow> g / (1 + a * g) = inverse f" for f g::complex
by (auto simp: field_simps)
have "(inverse \<circ> f) \<midarrow>z\<rightarrow> 0"
proof (rule Lim_transform_within_open [of "\<lambda>w. g w / (1 + a * g w)" _ _ UNIV "ball z r"])
show "(\<lambda>w. g w / (1 + a * g w)) \<midarrow>z\<rightarrow> 0"
using True by (auto simp: intro!: tendsto_eq_intros gz)
show "\<And>x. \<lbrakk>x \<in> ball z r; x \<noteq> z\<rbrakk> \<Longrightarrow> g x / (1 + a * g x) = (inverse \<circ> f) x"
using * gf gnz by simp
qed (use \<open>0 < r\<close> in auto)
with that show ?thesis by blast
next
case False
show ?thesis
proof (cases "1 + a * g z = 0")
case True
have "(f \<longlongrightarrow> 0) (at z)"
proof (rule Lim_transform_within_open [of "\<lambda>w. (1 + a * g w) / g w" _ _ _ "ball z r"])
show "(\<lambda>w. (1 + a * g w) / g w) \<midarrow>z\<rightarrow> 0"
apply (rule tendsto_eq_intros refl gz \<open>g z \<noteq> 0\<close>)+
by (simp add: True)
show "\<And>x. \<lbrakk>x \<in> ball z r; x \<noteq> z\<rbrakk> \<Longrightarrow> (1 + a * g x) / g x = f x"
using fab fab zrM by (fastforce simp add: gf divide_simps)
qed (use \<open>0 < r\<close> in auto)
then show ?thesis
using that by blast
next
case False
have *: "\<lbrakk>g \<noteq> 0; inverse g = f - a\<rbrakk> \<Longrightarrow> g / (1 + a * g) = inverse f" for f g::complex
by (auto simp: field_simps)
have "(inverse \<circ> f) \<midarrow>z\<rightarrow> g z / (1 + a * g z)"
proof (rule Lim_transform_within_open [of "\<lambda>w. g w / (1 + a * g w)" _ _ UNIV "ball z r"])
show "(\<lambda>w. g w / (1 + a * g w)) \<midarrow>z\<rightarrow> g z / (1 + a * g z)"
using False by (auto simp: False intro!: tendsto_eq_intros gz)
show "\<And>x. \<lbrakk>x \<in> ball z r; x \<noteq> z\<rbrakk> \<Longrightarrow> g x / (1 + a * g x) = (inverse \<circ> f) x"
using * gf gnz by simp
qed (use \<open>0 < r\<close> in auto)
with that show ?thesis by blast
qed
qed
qed
qed
corollary great_Picard_alt:
assumes M: "open M" "z \<in> M" and holf: "f holomorphic_on (M - {z})"
and non: "\<And>l. \<not> (f \<longlongrightarrow> l) (at z)" "\<And>l. \<not> ((inverse \<circ> f) \<longlongrightarrow> l) (at z)"
obtains a where "- {a} \<subseteq> 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 \<in> M" and holf: "f holomorphic_on (M - {z})"
and non: "\<And>l. \<not> (f \<longlongrightarrow> l) (at z)" "\<And>l. \<not> ((inverse \<circ> f) \<longlongrightarrow> l) (at z)"
obtains a where "\<And>w. w \<noteq> a \<Longrightarrow> infinite {x. x \<in> M - {z} \<and> f x = w}"
proof -
have False if "a \<noteq> b" and ab: "finite {x. x \<in> M - {z} \<and> f x = a}" "finite {x. x \<in> M - {z} \<and> f x = b}" for a b
proof -
have finab: "finite {x. x \<in> M - {z} \<and> f x \<in> {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 \<subseteq> M" and r: "\<And>x. \<lbrakk>x \<in> M - {z}; f x \<in> {a,b}\<rbrakk> \<Longrightarrow> x \<notin> ball z r"
proof -
obtain e where "e > 0" and e: "ball z e \<subseteq> M"
using assms openE by blast
show ?thesis
proof (cases "{x \<in> M - {z}. f x \<in> {a, b}} = {}")
case True
then show ?thesis
apply (rule_tac r=e in that)
using e \<open>e > 0\<close> by auto
next
case False
let ?r = "min e (Min (dist z ` {x \<in> M - {z}. f x \<in> {a,b}}))"
show ?thesis
proof
show "0 < ?r"
using min_less_iff_conj Min_gr_iff finab False \<open>0 < e\<close> by auto
have "ball z ?r \<subseteq> ball z e"
by (simp add: subset_ball)
with e show "ball z ?r \<subseteq> M" by blast
show "\<And>x. \<lbrakk>x \<in> M - {z}; f x \<in> {a, b}\<rbrakk> \<Longrightarrow> x \<notin> ball z ?r"
using min_less_iff_conj Min_gr_iff finab False \<open>0 < e\<close> 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 _ \<open>a \<noteq> b\<close> holfb])
using non \<open>0 < r\<close> r zrM by auto
qed
with that show thesis
by meson
qed
corollary Casorati_Weierstrass:
assumes "open M" "z \<in> M" "f holomorphic_on (M - {z})"
and "\<And>l. \<not> (f \<longlongrightarrow> l) (at z)" "\<And>l. \<not> ((inverse \<circ> f) \<longlongrightarrow> l) (at z)"
shows "closure(f ` (M - {z})) = UNIV"
proof -
obtain a where a: "- {a} \<subseteq> f ` (M - {z})"
using great_Picard_alt [OF assms] .
have "UNIV = closure(- {a})"
by (simp add: closure_interior)
also have "... \<subseteq> closure(f ` (M - {z}))"
by (simp add: a closure_mono)
finally show ?thesis
by blast
qed
end