--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Complex_Analysis/Riemann_Mapping.thy Sat Nov 30 13:47:33 2019 +0100
@@ -0,0 +1,1489 @@
+(* Title: HOL/Analysis/Riemann_Mapping.thy
+ Authors: LC Paulson, based on material from HOL Light
+*)
+
+section \<open>Moebius functions, Equivalents of Simply Connected Sets, Riemann Mapping Theorem\<close>
+
+theory Riemann_Mapping
+imports Great_Picard
+begin
+
+subsection\<open>Moebius functions are biholomorphisms of the unit disc\<close>
+
+definition\<^marker>\<open>tag important\<close> Moebius_function :: "[real,complex,complex] \<Rightarrow> complex" where
+ "Moebius_function \<equiv> \<lambda>t w z. exp(\<i> * of_real t) * (z - w) / (1 - cnj w * z)"
+
+lemma Moebius_function_simple:
+ "Moebius_function 0 w z = (z - w) / (1 - cnj w * z)"
+ by (simp add: Moebius_function_def)
+
+lemma Moebius_function_eq_zero:
+ "Moebius_function t w w = 0"
+ by (simp add: Moebius_function_def)
+
+lemma Moebius_function_of_zero:
+ "Moebius_function t w 0 = - exp(\<i> * of_real t) * w"
+ by (simp add: Moebius_function_def)
+
+lemma Moebius_function_norm_lt_1:
+ assumes w1: "norm w < 1" and z1: "norm z < 1"
+ shows "norm (Moebius_function t w z) < 1"
+proof -
+ have "1 - cnj w * z \<noteq> 0"
+ by (metis complex_cnj_cnj complex_mod_sqrt_Re_mult_cnj mult.commute mult_less_cancel_right1 norm_ge_zero norm_mult norm_one order.asym right_minus_eq w1 z1)
+ then have VV: "1 - w * cnj z \<noteq> 0"
+ by (metis complex_cnj_cnj complex_cnj_mult complex_cnj_one right_minus_eq)
+ then have "1 - norm (Moebius_function t w z) ^ 2 =
+ ((1 - norm w ^ 2) / (norm (1 - cnj w * z) ^ 2)) * (1 - norm z ^ 2)"
+ apply (cases w)
+ apply (cases z)
+ apply (simp add: Moebius_function_def divide_simps norm_divide norm_mult)
+ apply (simp add: complex_norm complex_diff complex_mult one_complex.code complex_cnj)
+ apply (auto simp: algebra_simps power2_eq_square)
+ done
+ then have "1 - (cmod (Moebius_function t w z))\<^sup>2 = (1 - cmod (w * w)) / (cmod (1 - cnj w * z))\<^sup>2 * (1 - cmod (z * z))"
+ by (simp add: norm_mult power2_eq_square)
+ moreover have "0 < 1 - cmod (z * z)"
+ by (metis (no_types) z1 diff_gt_0_iff_gt mult.left_neutral norm_mult_less)
+ ultimately have "0 < 1 - norm (Moebius_function t w z) ^ 2"
+ using \<open>1 - cnj w * z \<noteq> 0\<close> w1 norm_mult_less by fastforce
+ then show ?thesis
+ using linorder_not_less by fastforce
+qed
+
+lemma Moebius_function_holomorphic:
+ assumes "norm w < 1"
+ shows "Moebius_function t w holomorphic_on ball 0 1"
+proof -
+ have *: "1 - z * w \<noteq> 0" if "norm z < 1" for z
+ proof -
+ have "norm (1::complex) \<noteq> norm (z * w)"
+ using assms that norm_mult_less by fastforce
+ then show ?thesis by auto
+ qed
+ show ?thesis
+ apply (simp add: Moebius_function_def)
+ apply (intro holomorphic_intros)
+ using assms *
+ by (metis complex_cnj_cnj complex_cnj_mult complex_cnj_one complex_mod_cnj mem_ball_0 mult.commute right_minus_eq)
+qed
+
+lemma Moebius_function_compose:
+ assumes meq: "-w1 = w2" and "norm w1 < 1" "norm z < 1"
+ shows "Moebius_function 0 w1 (Moebius_function 0 w2 z) = z"
+proof -
+ have "norm w2 < 1"
+ using assms by auto
+ then have "-w1 = z" if "cnj w2 * z = 1"
+ by (metis assms(3) complex_mod_cnj less_irrefl mult.right_neutral norm_mult_less norm_one that)
+ moreover have "z=0" if "1 - cnj w2 * z = cnj w1 * (z - w2)"
+ proof -
+ have "w2 * cnj w2 = 1"
+ using that meq by (auto simp: algebra_simps)
+ then show "z = 0"
+ by (metis (no_types) \<open>cmod w2 < 1\<close> complex_mod_cnj less_irrefl mult.right_neutral norm_mult_less norm_one)
+ qed
+ moreover have "z - w2 - w1 * (1 - cnj w2 * z) = z * (1 - cnj w2 * z - cnj w1 * (z - w2))"
+ using meq by (fastforce simp: algebra_simps)
+ ultimately
+ show ?thesis
+ by (simp add: Moebius_function_def divide_simps norm_divide norm_mult)
+qed
+
+lemma ball_biholomorphism_exists:
+ assumes "a \<in> ball 0 1"
+ obtains f g where "f a = 0"
+ "f holomorphic_on ball 0 1" "f ` ball 0 1 \<subseteq> ball 0 1"
+ "g holomorphic_on ball 0 1" "g ` ball 0 1 \<subseteq> ball 0 1"
+ "\<And>z. z \<in> ball 0 1 \<Longrightarrow> f (g z) = z"
+ "\<And>z. z \<in> ball 0 1 \<Longrightarrow> g (f z) = z"
+proof
+ show "Moebius_function 0 a holomorphic_on ball 0 1" "Moebius_function 0 (-a) holomorphic_on ball 0 1"
+ using Moebius_function_holomorphic assms mem_ball_0 by auto
+ show "Moebius_function 0 a a = 0"
+ by (simp add: Moebius_function_eq_zero)
+ show "Moebius_function 0 a ` ball 0 1 \<subseteq> ball 0 1"
+ "Moebius_function 0 (- a) ` ball 0 1 \<subseteq> ball 0 1"
+ using Moebius_function_norm_lt_1 assms by auto
+ show "Moebius_function 0 a (Moebius_function 0 (- a) z) = z"
+ "Moebius_function 0 (- a) (Moebius_function 0 a z) = z" if "z \<in> ball 0 1" for z
+ using Moebius_function_compose assms that by auto
+qed
+
+
+subsection\<open>A big chain of equivalents of simple connectedness for an open set\<close>
+
+lemma biholomorphic_to_disc_aux:
+ assumes "open S" "connected S" "0 \<in> S" and S01: "S \<subseteq> ball 0 1"
+ and prev: "\<And>f. \<lbrakk>f holomorphic_on S; \<And>z. z \<in> S \<Longrightarrow> f z \<noteq> 0; inj_on f S\<rbrakk>
+ \<Longrightarrow> \<exists>g. g holomorphic_on S \<and> (\<forall>z \<in> S. f z = (g z)\<^sup>2)"
+ shows "\<exists>f g. f holomorphic_on S \<and> g holomorphic_on ball 0 1 \<and>
+ (\<forall>z \<in> S. f z \<in> ball 0 1 \<and> g(f z) = z) \<and>
+ (\<forall>z \<in> ball 0 1. g z \<in> S \<and> f(g z) = z)"
+proof -
+ define F where "F \<equiv> {h. h holomorphic_on S \<and> h ` S \<subseteq> ball 0 1 \<and> h 0 = 0 \<and> inj_on h S}"
+ have idF: "id \<in> F"
+ using S01 by (auto simp: F_def)
+ then have "F \<noteq> {}"
+ by blast
+ have imF_ne: "((\<lambda>h. norm(deriv h 0)) ` F) \<noteq> {}"
+ using idF by auto
+ have holF: "\<And>h. h \<in> F \<Longrightarrow> h holomorphic_on S"
+ by (auto simp: F_def)
+ obtain f where "f \<in> F" and normf: "\<And>h. h \<in> F \<Longrightarrow> norm(deriv h 0) \<le> norm(deriv f 0)"
+ proof -
+ obtain r where "r > 0" and r: "ball 0 r \<subseteq> S"
+ using \<open>open S\<close> \<open>0 \<in> S\<close> openE by auto
+ have bdd: "bdd_above ((\<lambda>h. norm(deriv h 0)) ` F)"
+ proof (intro bdd_aboveI exI ballI, clarify)
+ show "norm (deriv f 0) \<le> 1 / r" if "f \<in> F" for f
+ proof -
+ have r01: "(*) (complex_of_real r) ` ball 0 1 \<subseteq> S"
+ using that \<open>r > 0\<close> by (auto simp: norm_mult r [THEN subsetD])
+ then have "f holomorphic_on (*) (complex_of_real r) ` ball 0 1"
+ using holomorphic_on_subset [OF holF] by (simp add: that)
+ then have holf: "f \<circ> (\<lambda>z. (r * z)) holomorphic_on (ball 0 1)"
+ by (intro holomorphic_intros holomorphic_on_compose)
+ have f0: "(f \<circ> (*) (complex_of_real r)) 0 = 0"
+ using F_def that by auto
+ have "f ` S \<subseteq> ball 0 1"
+ using F_def that by blast
+ with r01 have fr1: "\<And>z. norm z < 1 \<Longrightarrow> norm ((f \<circ> (*)(of_real r))z) < 1"
+ by force
+ have *: "((\<lambda>w. f (r * w)) has_field_derivative deriv f (r * z) * r) (at z)"
+ if "z \<in> ball 0 1" for z::complex
+ proof (rule DERIV_chain' [where g=f])
+ show "(f has_field_derivative deriv f (of_real r * z)) (at (of_real r * z))"
+ apply (rule holomorphic_derivI [OF holF \<open>open S\<close>])
+ apply (rule \<open>f \<in> F\<close>)
+ by (meson imageI r01 subset_iff that)
+ qed simp
+ have df0: "((\<lambda>w. f (r * w)) has_field_derivative deriv f 0 * r) (at 0)"
+ using * [of 0] by simp
+ have deq: "deriv (\<lambda>x. f (complex_of_real r * x)) 0 = deriv f 0 * complex_of_real r"
+ using DERIV_imp_deriv df0 by blast
+ have "norm (deriv (f \<circ> (*) (complex_of_real r)) 0) \<le> 1"
+ by (auto intro: Schwarz_Lemma [OF holf f0 fr1, of 0])
+ with \<open>r > 0\<close> show ?thesis
+ by (simp add: deq norm_mult divide_simps o_def)
+ qed
+ qed
+ define l where "l \<equiv> SUP h\<in>F. norm (deriv h 0)"
+ have eql: "norm (deriv f 0) = l" if le: "l \<le> norm (deriv f 0)" and "f \<in> F" for f
+ apply (rule order_antisym [OF _ le])
+ using \<open>f \<in> F\<close> bdd cSUP_upper by (fastforce simp: l_def)
+ obtain \<F> where \<F>in: "\<And>n. \<F> n \<in> F" and \<F>lim: "(\<lambda>n. norm (deriv (\<F> n) 0)) \<longlonglongrightarrow> l"
+ proof -
+ have "\<exists>f. f \<in> F \<and> \<bar>norm (deriv f 0) - l\<bar> < 1 / (Suc n)" for n
+ proof -
+ obtain f where "f \<in> F" and f: "l < norm (deriv f 0) + 1/(Suc n)"
+ using cSup_least [OF imF_ne, of "l - 1/(Suc n)"] by (fastforce simp add: l_def)
+ then have "\<bar>norm (deriv f 0) - l\<bar> < 1 / (Suc n)"
+ by (fastforce simp add: abs_if not_less eql)
+ with \<open>f \<in> F\<close> show ?thesis
+ by blast
+ qed
+ then obtain \<F> where fF: "\<And>n. (\<F> n) \<in> F"
+ and fless: "\<And>n. \<bar>norm (deriv (\<F> n) 0) - l\<bar> < 1 / (Suc n)"
+ by metis
+ have "(\<lambda>n. norm (deriv (\<F> n) 0)) \<longlonglongrightarrow> l"
+ proof (rule metric_LIMSEQ_I)
+ fix e::real
+ assume "e > 0"
+ then obtain N::nat where N: "e > 1/(Suc N)"
+ using nat_approx_posE by blast
+ show "\<exists>N. \<forall>n\<ge>N. dist (norm (deriv (\<F> n) 0)) l < e"
+ proof (intro exI allI impI)
+ fix n assume "N \<le> n"
+ have "dist (norm (deriv (\<F> n) 0)) l < 1 / (Suc n)"
+ using fless by (simp add: dist_norm)
+ also have "... < e"
+ using N \<open>N \<le> n\<close> inverse_of_nat_le le_less_trans by blast
+ finally show "dist (norm (deriv (\<F> n) 0)) l < e" .
+ qed
+ qed
+ with fF show ?thesis
+ using that by blast
+ qed
+ have "\<And>K. \<lbrakk>compact K; K \<subseteq> S\<rbrakk> \<Longrightarrow> \<exists>B. \<forall>h\<in>F. \<forall>z\<in>K. norm (h z) \<le> B"
+ by (rule_tac x=1 in exI) (force simp: F_def)
+ moreover have "range \<F> \<subseteq> F"
+ using \<open>\<And>n. \<F> n \<in> F\<close> by blast
+ ultimately obtain f and r :: "nat \<Rightarrow> nat"
+ where holf: "f holomorphic_on S" and r: "strict_mono r"
+ and limf: "\<And>x. x \<in> S \<Longrightarrow> (\<lambda>n. \<F> (r n) x) \<longlonglongrightarrow> f x"
+ and ulimf: "\<And>K. \<lbrakk>compact K; K \<subseteq> S\<rbrakk> \<Longrightarrow> uniform_limit K (\<F> \<circ> r) f sequentially"
+ using Montel [of S F \<F>, OF \<open>open S\<close> holF] by auto+
+ have der: "\<And>n x. x \<in> S \<Longrightarrow> ((\<F> \<circ> r) n has_field_derivative ((\<lambda>n. deriv (\<F> n)) \<circ> r) n x) (at x)"
+ using \<open>\<And>n. \<F> n \<in> F\<close> \<open>open S\<close> holF holomorphic_derivI by fastforce
+ have ulim: "\<And>x. x \<in> S \<Longrightarrow> \<exists>d>0. cball x d \<subseteq> S \<and> uniform_limit (cball x d) (\<F> \<circ> r) f sequentially"
+ by (meson ulimf \<open>open S\<close> compact_cball open_contains_cball)
+ obtain f' :: "complex\<Rightarrow>complex" where f': "(f has_field_derivative f' 0) (at 0)"
+ and tof'0: "(\<lambda>n. ((\<lambda>n. deriv (\<F> n)) \<circ> r) n 0) \<longlonglongrightarrow> f' 0"
+ using has_complex_derivative_uniform_sequence [OF \<open>open S\<close> der ulim] \<open>0 \<in> S\<close> by metis
+ then have derf0: "deriv f 0 = f' 0"
+ by (simp add: DERIV_imp_deriv)
+ have "f field_differentiable (at 0)"
+ using field_differentiable_def f' by blast
+ have "(\<lambda>x. (norm (deriv (\<F> (r x)) 0))) \<longlonglongrightarrow> norm (deriv f 0)"
+ using isCont_tendsto_compose [OF continuous_norm [OF continuous_ident] tof'0] derf0 by auto
+ with LIMSEQ_subseq_LIMSEQ [OF \<F>lim r] have no_df0: "norm(deriv f 0) = l"
+ by (force simp: o_def intro: tendsto_unique)
+ have nonconstf: "\<not> f constant_on S"
+ proof -
+ have False if "\<And>x. x \<in> S \<Longrightarrow> f x = c" for c
+ proof -
+ have "deriv f 0 = 0"
+ by (metis that \<open>open S\<close> \<open>0 \<in> S\<close> DERIV_imp_deriv [OF has_field_derivative_transform_within_open [OF DERIV_const]])
+ with no_df0 have "l = 0"
+ by auto
+ with eql [OF _ idF] show False by auto
+ qed
+ then show ?thesis
+ by (meson constant_on_def)
+ qed
+ show ?thesis
+ proof
+ show "f \<in> F"
+ unfolding F_def
+ proof (intro CollectI conjI holf)
+ have "norm(f z) \<le> 1" if "z \<in> S" for z
+ proof (intro Lim_norm_ubound [OF _ limf] always_eventually allI that)
+ fix n
+ have "\<F> (r n) \<in> F"
+ by (simp add: \<F>in)
+ then show "norm (\<F> (r n) z) \<le> 1"
+ using that by (auto simp: F_def)
+ qed simp
+ then have fless1: "norm(f z) < 1" if "z \<in> S" for z
+ using maximum_modulus_principle [OF holf \<open>open S\<close> \<open>connected S\<close> \<open>open S\<close>] nonconstf that
+ by fastforce
+ then show "f ` S \<subseteq> ball 0 1"
+ by auto
+ have "(\<lambda>n. \<F> (r n) 0) \<longlonglongrightarrow> 0"
+ using \<F>in by (auto simp: F_def)
+ then show "f 0 = 0"
+ using tendsto_unique [OF _ limf ] \<open>0 \<in> S\<close> trivial_limit_sequentially by blast
+ show "inj_on f S"
+ proof (rule Hurwitz_injective [OF \<open>open S\<close> \<open>connected S\<close> _ holf])
+ show "\<And>n. (\<F> \<circ> r) n holomorphic_on S"
+ by (simp add: \<F>in holF)
+ show "\<And>K. \<lbrakk>compact K; K \<subseteq> S\<rbrakk> \<Longrightarrow> uniform_limit K(\<F> \<circ> r) f sequentially"
+ by (metis ulimf)
+ show "\<not> f constant_on S"
+ using nonconstf by auto
+ show "\<And>n. inj_on ((\<F> \<circ> r) n) S"
+ using \<F>in by (auto simp: F_def)
+ qed
+ qed
+ show "\<And>h. h \<in> F \<Longrightarrow> norm (deriv h 0) \<le> norm (deriv f 0)"
+ by (metis eql le_cases no_df0)
+ qed
+ qed
+ have holf: "f holomorphic_on S" and injf: "inj_on f S" and f01: "f ` S \<subseteq> ball 0 1"
+ using \<open>f \<in> F\<close> by (auto simp: F_def)
+ obtain g where holg: "g holomorphic_on (f ` S)"
+ and derg: "\<And>z. z \<in> S \<Longrightarrow> deriv f z * deriv g (f z) = 1"
+ and gf: "\<And>z. z \<in> S \<Longrightarrow> g(f z) = z"
+ using holomorphic_has_inverse [OF holf \<open>open S\<close> injf] by metis
+ have "ball 0 1 \<subseteq> f ` S"
+ proof
+ fix a::complex
+ assume a: "a \<in> ball 0 1"
+ have False if "\<And>x. x \<in> S \<Longrightarrow> f x \<noteq> a"
+ proof -
+ obtain h k where "h a = 0"
+ and holh: "h holomorphic_on ball 0 1" and h01: "h ` ball 0 1 \<subseteq> ball 0 1"
+ and holk: "k holomorphic_on ball 0 1" and k01: "k ` ball 0 1 \<subseteq> ball 0 1"
+ and hk: "\<And>z. z \<in> ball 0 1 \<Longrightarrow> h (k z) = z"
+ and kh: "\<And>z. z \<in> ball 0 1 \<Longrightarrow> k (h z) = z"
+ using ball_biholomorphism_exists [OF a] by blast
+ have nf1: "\<And>z. z \<in> S \<Longrightarrow> norm(f z) < 1"
+ using \<open>f \<in> F\<close> by (auto simp: F_def)
+ have 1: "h \<circ> f holomorphic_on S"
+ using F_def \<open>f \<in> F\<close> holh holomorphic_on_compose holomorphic_on_subset by blast
+ have 2: "\<And>z. z \<in> S \<Longrightarrow> (h \<circ> f) z \<noteq> 0"
+ by (metis \<open>h a = 0\<close> a comp_eq_dest_lhs nf1 kh mem_ball_0 that)
+ have 3: "inj_on (h \<circ> f) S"
+ by (metis (no_types, lifting) F_def \<open>f \<in> F\<close> comp_inj_on inj_on_inverseI injf kh mem_Collect_eq subset_inj_on)
+ obtain \<psi> where hol\<psi>: "\<psi> holomorphic_on ((h \<circ> f) ` S)"
+ and \<psi>2: "\<And>z. z \<in> S \<Longrightarrow> \<psi>(h (f z)) ^ 2 = h(f z)"
+ proof (rule exE [OF prev [OF 1 2 3]], safe)
+ fix \<theta>
+ assume hol\<theta>: "\<theta> holomorphic_on S" and \<theta>2: "(\<forall>z\<in>S. (h \<circ> f) z = (\<theta> z)\<^sup>2)"
+ show thesis
+ proof
+ show "(\<theta> \<circ> g \<circ> k) holomorphic_on (h \<circ> f) ` S"
+ proof (intro holomorphic_on_compose)
+ show "k holomorphic_on (h \<circ> f) ` S"
+ apply (rule holomorphic_on_subset [OF holk])
+ using f01 h01 by force
+ show "g holomorphic_on k ` (h \<circ> f) ` S"
+ apply (rule holomorphic_on_subset [OF holg])
+ by (auto simp: kh nf1)
+ show "\<theta> holomorphic_on g ` k ` (h \<circ> f) ` S"
+ apply (rule holomorphic_on_subset [OF hol\<theta>])
+ by (auto simp: gf kh nf1)
+ qed
+ show "((\<theta> \<circ> g \<circ> k) (h (f z)))\<^sup>2 = h (f z)" if "z \<in> S" for z
+ proof -
+ have "f z \<in> ball 0 1"
+ by (simp add: nf1 that)
+ then have "(\<theta> (g (k (h (f z)))))\<^sup>2 = (\<theta> (g (f z)))\<^sup>2"
+ by (metis kh)
+ also have "... = h (f z)"
+ using \<theta>2 gf that by auto
+ finally show ?thesis
+ by (simp add: o_def)
+ qed
+ qed
+ qed
+ have norm\<psi>1: "norm(\<psi> (h (f z))) < 1" if "z \<in> S" for z
+ proof -
+ have "norm (\<psi> (h (f z)) ^ 2) < 1"
+ by (metis (no_types) that DIM_complex \<psi>2 h01 image_subset_iff mem_ball_0 nf1)
+ then show ?thesis
+ by (metis le_less_trans mult_less_cancel_left2 norm_ge_zero norm_power not_le power2_eq_square)
+ qed
+ then have \<psi>01: "\<psi> (h (f 0)) \<in> ball 0 1"
+ by (simp add: \<open>0 \<in> S\<close>)
+ obtain p q where p0: "p (\<psi> (h (f 0))) = 0"
+ and holp: "p holomorphic_on ball 0 1" and p01: "p ` ball 0 1 \<subseteq> ball 0 1"
+ and holq: "q holomorphic_on ball 0 1" and q01: "q ` ball 0 1 \<subseteq> ball 0 1"
+ and pq: "\<And>z. z \<in> ball 0 1 \<Longrightarrow> p (q z) = z"
+ and qp: "\<And>z. z \<in> ball 0 1 \<Longrightarrow> q (p z) = z"
+ using ball_biholomorphism_exists [OF \<psi>01] by metis
+ have "p \<circ> \<psi> \<circ> h \<circ> f \<in> F"
+ unfolding F_def
+ proof (intro CollectI conjI holf)
+ show "p \<circ> \<psi> \<circ> h \<circ> f holomorphic_on S"
+ proof (intro holomorphic_on_compose holf)
+ show "h holomorphic_on f ` S"
+ apply (rule holomorphic_on_subset [OF holh])
+ using f01 by force
+ show "\<psi> holomorphic_on h ` f ` S"
+ apply (rule holomorphic_on_subset [OF hol\<psi>])
+ by auto
+ show "p holomorphic_on \<psi> ` h ` f ` S"
+ apply (rule holomorphic_on_subset [OF holp])
+ by (auto simp: norm\<psi>1)
+ qed
+ show "(p \<circ> \<psi> \<circ> h \<circ> f) ` S \<subseteq> ball 0 1"
+ apply clarsimp
+ by (meson norm\<psi>1 p01 image_subset_iff mem_ball_0)
+ show "(p \<circ> \<psi> \<circ> h \<circ> f) 0 = 0"
+ by (simp add: \<open>p (\<psi> (h (f 0))) = 0\<close>)
+ show "inj_on (p \<circ> \<psi> \<circ> h \<circ> f) S"
+ unfolding inj_on_def o_def
+ by (metis \<psi>2 dist_0_norm gf kh mem_ball nf1 norm\<psi>1 qp)
+ qed
+ then have le_norm_df0: "norm (deriv (p \<circ> \<psi> \<circ> h \<circ> f) 0) \<le> norm (deriv f 0)"
+ by (rule normf)
+ have 1: "k \<circ> power2 \<circ> q holomorphic_on ball 0 1"
+ proof (intro holomorphic_on_compose holq)
+ show "power2 holomorphic_on q ` ball 0 1"
+ using holomorphic_on_subset holomorphic_on_power
+ by (blast intro: holomorphic_on_ident)
+ show "k holomorphic_on power2 ` q ` ball 0 1"
+ apply (rule holomorphic_on_subset [OF holk])
+ using q01 by (auto simp: norm_power abs_square_less_1)
+ qed
+ have 2: "(k \<circ> power2 \<circ> q) 0 = 0"
+ using p0 F_def \<open>f \<in> F\<close> \<psi>01 \<psi>2 \<open>0 \<in> S\<close> kh qp by force
+ have 3: "norm ((k \<circ> power2 \<circ> q) z) < 1" if "norm z < 1" for z
+ proof -
+ have "norm ((power2 \<circ> q) z) < 1"
+ using that q01 by (force simp: norm_power abs_square_less_1)
+ with k01 show ?thesis
+ by fastforce
+ qed
+ have False if c: "\<forall>z. norm z < 1 \<longrightarrow> (k \<circ> power2 \<circ> q) z = c * z" and "norm c = 1" for c
+ proof -
+ have "c \<noteq> 0" using that by auto
+ have "norm (p(1/2)) < 1" "norm (p(-1/2)) < 1"
+ using p01 by force+
+ then have "(k \<circ> power2 \<circ> q) (p(1/2)) = c * p(1/2)" "(k \<circ> power2 \<circ> q) (p(-1/2)) = c * p(-1/2)"
+ using c by force+
+ then have "p (1/2) = p (- (1/2))"
+ by (auto simp: \<open>c \<noteq> 0\<close> qp o_def)
+ then have "q (p (1/2)) = q (p (- (1/2)))"
+ by simp
+ then have "1/2 = - (1/2::complex)"
+ by (auto simp: qp)
+ then show False
+ by simp
+ qed
+ moreover
+ have False if "norm (deriv (k \<circ> power2 \<circ> q) 0) \<noteq> 1" "norm (deriv (k \<circ> power2 \<circ> q) 0) \<le> 1"
+ and le: "\<And>\<xi>. norm \<xi> < 1 \<Longrightarrow> norm ((k \<circ> power2 \<circ> q) \<xi>) \<le> norm \<xi>"
+ proof -
+ have "norm (deriv (k \<circ> power2 \<circ> q) 0) < 1"
+ using that by simp
+ moreover have eq: "deriv f 0 = deriv (k \<circ> (\<lambda>z. z ^ 2) \<circ> q) 0 * deriv (p \<circ> \<psi> \<circ> h \<circ> f) 0"
+ proof (intro DERIV_imp_deriv has_field_derivative_transform_within_open [OF DERIV_chain])
+ show "(k \<circ> power2 \<circ> q has_field_derivative deriv (k \<circ> power2 \<circ> q) 0) (at ((p \<circ> \<psi> \<circ> h \<circ> f) 0))"
+ using "1" holomorphic_derivI p0 by auto
+ show "(p \<circ> \<psi> \<circ> h \<circ> f has_field_derivative deriv (p \<circ> \<psi> \<circ> h \<circ> f) 0) (at 0)"
+ using \<open>p \<circ> \<psi> \<circ> h \<circ> f \<in> F\<close> \<open>open S\<close> \<open>0 \<in> S\<close> holF holomorphic_derivI by blast
+ show "\<And>x. x \<in> S \<Longrightarrow> (k \<circ> power2 \<circ> q \<circ> (p \<circ> \<psi> \<circ> h \<circ> f)) x = f x"
+ using \<psi>2 f01 kh norm\<psi>1 qp by auto
+ qed (use assms in simp_all)
+ ultimately have "cmod (deriv (p \<circ> \<psi> \<circ> h \<circ> f) 0) \<le> 0"
+ using le_norm_df0
+ by (metis linorder_not_le mult.commute mult_less_cancel_left2 norm_mult)
+ moreover have "1 \<le> norm (deriv f 0)"
+ using normf [of id] by (simp add: idF)
+ ultimately show False
+ by (simp add: eq)
+ qed
+ ultimately show ?thesis
+ using Schwarz_Lemma [OF 1 2 3] norm_one by blast
+ qed
+ then show "a \<in> f ` S"
+ by blast
+ qed
+ then have "f ` S = ball 0 1"
+ using F_def \<open>f \<in> F\<close> by blast
+ then show ?thesis
+ apply (rule_tac x=f in exI)
+ apply (rule_tac x=g in exI)
+ using holf holg derg gf by safe force+
+qed
+
+
+locale SC_Chain =
+ fixes S :: "complex set"
+ assumes openS: "open S"
+begin
+
+lemma winding_number_zero:
+ assumes "simply_connected S"
+ shows "connected S \<and>
+ (\<forall>\<gamma> z. path \<gamma> \<and> path_image \<gamma> \<subseteq> S \<and>
+ pathfinish \<gamma> = pathstart \<gamma> \<and> z \<notin> S \<longrightarrow> winding_number \<gamma> z = 0)"
+ using assms
+ by (auto simp: simply_connected_imp_connected simply_connected_imp_winding_number_zero)
+
+lemma contour_integral_zero:
+ assumes "valid_path g" "path_image g \<subseteq> S" "pathfinish g = pathstart g" "f holomorphic_on S"
+ "\<And>\<gamma> z. \<lbrakk>path \<gamma>; path_image \<gamma> \<subseteq> S; pathfinish \<gamma> = pathstart \<gamma>; z \<notin> S\<rbrakk> \<Longrightarrow> winding_number \<gamma> z = 0"
+ shows "(f has_contour_integral 0) g"
+ using assms by (meson Cauchy_theorem_global openS valid_path_imp_path)
+
+lemma global_primitive:
+ assumes "connected S" and holf: "f holomorphic_on S"
+ and prev: "\<And>\<gamma> f. \<lbrakk>valid_path \<gamma>; path_image \<gamma> \<subseteq> S; pathfinish \<gamma> = pathstart \<gamma>; f holomorphic_on S\<rbrakk> \<Longrightarrow> (f has_contour_integral 0) \<gamma>"
+ shows "\<exists>h. \<forall>z \<in> S. (h has_field_derivative f z) (at z)"
+proof (cases "S = {}")
+case True then show ?thesis
+ by simp
+next
+ case False
+ then obtain a where "a \<in> S"
+ by blast
+ show ?thesis
+ proof (intro exI ballI)
+ fix x assume "x \<in> S"
+ then obtain d where "d > 0" and d: "cball x d \<subseteq> S"
+ using openS open_contains_cball_eq by blast
+ let ?g = "\<lambda>z. (SOME g. polynomial_function g \<and> path_image g \<subseteq> S \<and> pathstart g = a \<and> pathfinish g = z)"
+ show "((\<lambda>z. contour_integral (?g z) f) has_field_derivative f x)
+ (at x)"
+ proof (simp add: has_field_derivative_def has_derivative_at2 bounded_linear_mult_right, rule Lim_transform)
+ show "(\<lambda>y. inverse(norm(y - x)) *\<^sub>R (contour_integral(linepath x y) f - f x * (y - x))) \<midarrow>x\<rightarrow> 0"
+ proof (clarsimp simp add: Lim_at)
+ fix e::real assume "e > 0"
+ moreover have "continuous (at x) f"
+ using openS \<open>x \<in> S\<close> holf continuous_on_eq_continuous_at holomorphic_on_imp_continuous_on by auto
+ ultimately obtain d1 where "d1 > 0"
+ and d1: "\<And>x'. dist x' x < d1 \<Longrightarrow> dist (f x') (f x) < e/2"
+ unfolding continuous_at_eps_delta
+ by (metis less_divide_eq_numeral1(1) mult_zero_left)
+ obtain d2 where "d2 > 0" and d2: "ball x d2 \<subseteq> S"
+ using openS \<open>x \<in> S\<close> open_contains_ball_eq by blast
+ have "inverse (norm (y - x)) * norm (contour_integral (linepath x y) f - f x * (y - x)) < e"
+ if "0 < d1" "0 < d2" "y \<noteq> x" "dist y x < d1" "dist y x < d2" for y
+ proof -
+ have "f contour_integrable_on linepath x y"
+ proof (rule contour_integrable_continuous_linepath [OF continuous_on_subset])
+ show "continuous_on S f"
+ by (simp add: holf holomorphic_on_imp_continuous_on)
+ have "closed_segment x y \<subseteq> ball x d2"
+ by (meson dist_commute_lessI dist_in_closed_segment le_less_trans mem_ball subsetI that(5))
+ with d2 show "closed_segment x y \<subseteq> S"
+ by blast
+ qed
+ then obtain z where z: "(f has_contour_integral z) (linepath x y)"
+ by (force simp: contour_integrable_on_def)
+ have con: "((\<lambda>w. f x) has_contour_integral f x * (y - x)) (linepath x y)"
+ using has_contour_integral_const_linepath [of "f x" y x] by metis
+ have "norm (z - f x * (y - x)) \<le> (e/2) * norm (y - x)"
+ proof (rule has_contour_integral_bound_linepath)
+ show "((\<lambda>w. f w - f x) has_contour_integral z - f x * (y - x)) (linepath x y)"
+ by (rule has_contour_integral_diff [OF z con])
+ show "\<And>w. w \<in> closed_segment x y \<Longrightarrow> norm (f w - f x) \<le> e/2"
+ by (metis d1 dist_norm less_le_trans not_less not_less_iff_gr_or_eq segment_bound1 that(4))
+ qed (use \<open>e > 0\<close> in auto)
+ with \<open>e > 0\<close> have "inverse (norm (y - x)) * norm (z - f x * (y - x)) \<le> e/2"
+ by (simp add: field_split_simps)
+ also have "... < e"
+ using \<open>e > 0\<close> by simp
+ finally show ?thesis
+ by (simp add: contour_integral_unique [OF z])
+ qed
+ with \<open>d1 > 0\<close> \<open>d2 > 0\<close>
+ show "\<exists>d>0. \<forall>z. z \<noteq> x \<and> dist z x < d \<longrightarrow>
+ inverse (norm (z - x)) * norm (contour_integral (linepath x z) f - f x * (z - x)) < e"
+ by (rule_tac x="min d1 d2" in exI) auto
+ qed
+ next
+ have *: "(1 / norm (y - x)) *\<^sub>R (contour_integral (?g y) f -
+ (contour_integral (?g x) f + f x * (y - x))) =
+ (contour_integral (linepath x y) f - f x * (y - x)) /\<^sub>R norm (y - x)"
+ if "0 < d" "y \<noteq> x" and yx: "dist y x < d" for y
+ proof -
+ have "y \<in> S"
+ by (metis subsetD d dist_commute less_eq_real_def mem_cball yx)
+ have gxy: "polynomial_function (?g x) \<and> path_image (?g x) \<subseteq> S \<and> pathstart (?g x) = a \<and> pathfinish (?g x) = x"
+ "polynomial_function (?g y) \<and> path_image (?g y) \<subseteq> S \<and> pathstart (?g y) = a \<and> pathfinish (?g y) = y"
+ using someI_ex [OF connected_open_polynomial_connected [OF openS \<open>connected S\<close> \<open>a \<in> S\<close>]] \<open>x \<in> S\<close> \<open>y \<in> S\<close>
+ by meson+
+ then have vp: "valid_path (?g x)" "valid_path (?g y)"
+ by (simp_all add: valid_path_polynomial_function)
+ have f0: "(f has_contour_integral 0) ((?g x) +++ linepath x y +++ reversepath (?g y))"
+ proof (rule prev)
+ show "valid_path ((?g x) +++ linepath x y +++ reversepath (?g y))"
+ using gxy vp by (auto simp: valid_path_join)
+ have "closed_segment x y \<subseteq> cball x d"
+ using yx by (auto simp: dist_commute dest!: dist_in_closed_segment)
+ then have "closed_segment x y \<subseteq> S"
+ using d by blast
+ then show "path_image ((?g x) +++ linepath x y +++ reversepath (?g y)) \<subseteq> S"
+ using gxy by (auto simp: path_image_join)
+ qed (use gxy holf in auto)
+ then have fintxy: "f contour_integrable_on linepath x y"
+ by (metis (no_types, lifting) contour_integrable_joinD1 contour_integrable_joinD2 gxy(2) has_contour_integral_integrable pathfinish_linepath pathstart_reversepath valid_path_imp_reverse valid_path_join valid_path_linepath vp(2))
+ have fintgx: "f contour_integrable_on (?g x)" "f contour_integrable_on (?g y)"
+ using openS contour_integrable_holomorphic_simple gxy holf vp by blast+
+ show ?thesis
+ apply (clarsimp simp add: divide_simps)
+ using contour_integral_unique [OF f0]
+ apply (simp add: fintxy gxy contour_integrable_reversepath contour_integral_reversepath fintgx vp)
+ apply (simp add: algebra_simps)
+ done
+ qed
+ show "(\<lambda>z. (1 / norm (z - x)) *\<^sub>R
+ (contour_integral (?g z) f - (contour_integral (?g x) f + f x * (z - x))) -
+ (contour_integral (linepath x z) f - f x * (z - x)) /\<^sub>R norm (z - x))
+ \<midarrow>x\<rightarrow> 0"
+ apply (rule tendsto_eventually)
+ apply (simp add: eventually_at)
+ apply (rule_tac x=d in exI)
+ using \<open>d > 0\<close> * by simp
+ qed
+ qed
+qed
+
+lemma holomorphic_log:
+ assumes "connected S" and holf: "f holomorphic_on S" and nz: "\<And>z. z \<in> S \<Longrightarrow> f z \<noteq> 0"
+ and prev: "\<And>f. f holomorphic_on S \<Longrightarrow> \<exists>h. \<forall>z \<in> S. (h has_field_derivative f z) (at z)"
+ shows "\<exists>g. g holomorphic_on S \<and> (\<forall>z \<in> S. f z = exp(g z))"
+proof -
+ have "(\<lambda>z. deriv f z / f z) holomorphic_on S"
+ by (simp add: openS holf holomorphic_deriv holomorphic_on_divide nz)
+ then obtain g where g: "\<And>z. z \<in> S \<Longrightarrow> (g has_field_derivative deriv f z / f z) (at z)"
+ using prev [of "\<lambda>z. deriv f z / f z"] by metis
+ have hfd: "\<And>x. x \<in> S \<Longrightarrow> ((\<lambda>z. exp (g z) / f z) has_field_derivative 0) (at x)"
+ apply (rule derivative_eq_intros g| simp)+
+ apply (subst DERIV_deriv_iff_field_differentiable)
+ using openS holf holomorphic_on_imp_differentiable_at nz apply auto
+ done
+ obtain c where c: "\<And>x. x \<in> S \<Longrightarrow> exp (g x) / f x = c"
+ proof (rule DERIV_zero_connected_constant[OF \<open>connected S\<close> openS finite.emptyI])
+ show "continuous_on S (\<lambda>z. exp (g z) / f z)"
+ by (metis (full_types) openS g continuous_on_divide continuous_on_exp holf holomorphic_on_imp_continuous_on holomorphic_on_open nz)
+ then show "\<forall>x\<in>S - {}. ((\<lambda>z. exp (g z) / f z) has_field_derivative 0) (at x)"
+ using hfd by (blast intro: DERIV_zero_connected_constant [OF \<open>connected S\<close> openS finite.emptyI, of "\<lambda>z. exp(g z) / f z"])
+ qed auto
+ show ?thesis
+ proof (intro exI ballI conjI)
+ show "(\<lambda>z. Ln(inverse c) + g z) holomorphic_on S"
+ apply (intro holomorphic_intros)
+ using openS g holomorphic_on_open by blast
+ fix z :: complex
+ assume "z \<in> S"
+ then have "exp (g z) / c = f z"
+ by (metis c divide_divide_eq_right exp_not_eq_zero nonzero_mult_div_cancel_left)
+ moreover have "1 / c \<noteq> 0"
+ using \<open>z \<in> S\<close> c nz by fastforce
+ ultimately show "f z = exp (Ln (inverse c) + g z)"
+ by (simp add: exp_add inverse_eq_divide)
+ qed
+qed
+
+lemma holomorphic_sqrt:
+ assumes holf: "f holomorphic_on S" and nz: "\<And>z. z \<in> S \<Longrightarrow> f z \<noteq> 0"
+ and prev: "\<And>f. \<lbrakk>f holomorphic_on S; \<forall>z \<in> S. f z \<noteq> 0\<rbrakk> \<Longrightarrow> \<exists>g. g holomorphic_on S \<and> (\<forall>z \<in> S. f z = exp(g z))"
+ shows "\<exists>g. g holomorphic_on S \<and> (\<forall>z \<in> S. f z = (g z)\<^sup>2)"
+proof -
+ obtain g where holg: "g holomorphic_on S" and g: "\<And>z. z \<in> S \<Longrightarrow> f z = exp (g z)"
+ using prev [of f] holf nz by metis
+ show ?thesis
+ proof (intro exI ballI conjI)
+ show "(\<lambda>z. exp(g z/2)) holomorphic_on S"
+ by (intro holomorphic_intros) (auto simp: holg)
+ show "\<And>z. z \<in> S \<Longrightarrow> f z = (exp (g z/2))\<^sup>2"
+ by (metis (no_types) g exp_double nonzero_mult_div_cancel_left times_divide_eq_right zero_neq_numeral)
+ qed
+qed
+
+lemma biholomorphic_to_disc:
+ assumes "connected S" and S: "S \<noteq> {}" "S \<noteq> UNIV"
+ and prev: "\<And>f. \<lbrakk>f holomorphic_on S; \<forall>z \<in> S. f z \<noteq> 0\<rbrakk> \<Longrightarrow> \<exists>g. g holomorphic_on S \<and> (\<forall>z \<in> S. f z = (g z)\<^sup>2)"
+ shows "\<exists>f g. f holomorphic_on S \<and> g holomorphic_on ball 0 1 \<and>
+ (\<forall>z \<in> S. f z \<in> ball 0 1 \<and> g(f z) = z) \<and>
+ (\<forall>z \<in> ball 0 1. g z \<in> S \<and> f(g z) = z)"
+proof -
+ obtain a b where "a \<in> S" "b \<notin> S"
+ using S by blast
+ then obtain \<delta> where "\<delta> > 0" and \<delta>: "ball a \<delta> \<subseteq> S"
+ using openS openE by blast
+ obtain g where holg: "g holomorphic_on S" and eqg: "\<And>z. z \<in> S \<Longrightarrow> z - b = (g z)\<^sup>2"
+ proof (rule exE [OF prev [of "\<lambda>z. z - b"]])
+ show "(\<lambda>z. z - b) holomorphic_on S"
+ by (intro holomorphic_intros)
+ qed (use \<open>b \<notin> S\<close> in auto)
+ have "\<not> g constant_on S"
+ proof -
+ have "(a + \<delta>/2) \<in> ball a \<delta>" "a + (\<delta>/2) \<noteq> a"
+ using \<open>\<delta> > 0\<close> by (simp_all add: dist_norm)
+ then show ?thesis
+ unfolding constant_on_def
+ using eqg [of a] eqg [of "a + \<delta>/2"] \<open>a \<in> S\<close> \<delta>
+ by (metis diff_add_cancel subset_eq)
+ qed
+ then have "open (g ` ball a \<delta>)"
+ using open_mapping_thm [of g S "ball a \<delta>", OF holg openS \<open>connected S\<close>] \<delta> by blast
+ then obtain r where "r > 0" and r: "ball (g a) r \<subseteq> (g ` ball a \<delta>)"
+ by (metis \<open>0 < \<delta>\<close> centre_in_ball imageI openE)
+ have g_not_r: "g z \<notin> ball (-(g a)) r" if "z \<in> S" for z
+ proof
+ assume "g z \<in> ball (-(g a)) r"
+ then have "- g z \<in> ball (g a) r"
+ by (metis add.inverse_inverse dist_minus mem_ball)
+ with r have "- g z \<in> (g ` ball a \<delta>)"
+ by blast
+ then obtain w where w: "- g z = g w" "dist a w < \<delta>"
+ by auto
+ then have "w \<in> ball a \<delta>"
+ by simp
+ then have "w \<in> S"
+ using \<delta> by blast
+ then have "w = z"
+ by (metis diff_add_cancel eqg power_minus_Bit0 that w(1))
+ then have "g z = 0"
+ using \<open>- g z = g w\<close> by auto
+ with eqg [OF that] have "z = b"
+ by auto
+ with that \<open>b \<notin> S\<close> show False
+ by simp
+ qed
+ then have nz: "\<And>z. z \<in> S \<Longrightarrow> g z + g a \<noteq> 0"
+ by (metis \<open>0 < r\<close> add.commute add_diff_cancel_left' centre_in_ball diff_0)
+ let ?f = "\<lambda>z. (r/3) / (g z + g a) - (r/3) / (g a + g a)"
+ obtain h where holh: "h holomorphic_on S" and "h a = 0" and h01: "h ` S \<subseteq> ball 0 1" and "inj_on h S"
+ proof
+ show "?f holomorphic_on S"
+ by (intro holomorphic_intros holg nz)
+ have 3: "\<lbrakk>norm x \<le> 1/3; norm y \<le> 1/3\<rbrakk> \<Longrightarrow> norm(x - y) < 1" for x y::complex
+ using norm_triangle_ineq4 [of x y] by simp
+ have "norm ((r/3) / (g z + g a) - (r/3) / (g a + g a)) < 1" if "z \<in> S" for z
+ apply (rule 3)
+ unfolding norm_divide
+ using \<open>r > 0\<close> g_not_r [OF \<open>z \<in> S\<close>] g_not_r [OF \<open>a \<in> S\<close>]
+ by (simp_all add: field_split_simps dist_commute dist_norm)
+ then show "?f ` S \<subseteq> ball 0 1"
+ by auto
+ show "inj_on ?f S"
+ using \<open>r > 0\<close> eqg apply (clarsimp simp: inj_on_def)
+ by (metis diff_add_cancel)
+ qed auto
+ obtain k where holk: "k holomorphic_on (h ` S)"
+ and derk: "\<And>z. z \<in> S \<Longrightarrow> deriv h z * deriv k (h z) = 1"
+ and kh: "\<And>z. z \<in> S \<Longrightarrow> k(h z) = z"
+ using holomorphic_has_inverse [OF holh openS \<open>inj_on h S\<close>] by metis
+
+ have 1: "open (h ` S)"
+ by (simp add: \<open>inj_on h S\<close> holh openS open_mapping_thm3)
+ have 2: "connected (h ` S)"
+ by (simp add: connected_continuous_image \<open>connected S\<close> holh holomorphic_on_imp_continuous_on)
+ have 3: "0 \<in> h ` S"
+ using \<open>a \<in> S\<close> \<open>h a = 0\<close> by auto
+ have 4: "\<exists>g. g holomorphic_on h ` S \<and> (\<forall>z\<in>h ` S. f z = (g z)\<^sup>2)"
+ if holf: "f holomorphic_on h ` S" and nz: "\<And>z. z \<in> h ` S \<Longrightarrow> f z \<noteq> 0" "inj_on f (h ` S)" for f
+ proof -
+ obtain g where holg: "g holomorphic_on S" and eqg: "\<And>z. z \<in> S \<Longrightarrow> (f \<circ> h) z = (g z)\<^sup>2"
+ proof -
+ have "f \<circ> h holomorphic_on S"
+ by (simp add: holh holomorphic_on_compose holf)
+ moreover have "\<forall>z\<in>S. (f \<circ> h) z \<noteq> 0"
+ by (simp add: nz)
+ ultimately show thesis
+ using prev that by blast
+ qed
+ show ?thesis
+ proof (intro exI conjI)
+ show "g \<circ> k holomorphic_on h ` S"
+ proof -
+ have "k ` h ` S \<subseteq> S"
+ by (simp add: \<open>\<And>z. z \<in> S \<Longrightarrow> k (h z) = z\<close> image_subset_iff)
+ then show ?thesis
+ by (meson holg holk holomorphic_on_compose holomorphic_on_subset)
+ qed
+ show "\<forall>z\<in>h ` S. f z = ((g \<circ> k) z)\<^sup>2"
+ using eqg kh by auto
+ qed
+ qed
+ obtain f g where f: "f holomorphic_on h ` S" and g: "g holomorphic_on ball 0 1"
+ and gf: "\<forall>z\<in>h ` S. f z \<in> ball 0 1 \<and> g (f z) = z" and fg:"\<forall>z\<in>ball 0 1. g z \<in> h ` S \<and> f (g z) = z"
+ using biholomorphic_to_disc_aux [OF 1 2 3 h01 4] by blast
+ show ?thesis
+ proof (intro exI conjI)
+ show "f \<circ> h holomorphic_on S"
+ by (simp add: f holh holomorphic_on_compose)
+ show "k \<circ> g holomorphic_on ball 0 1"
+ by (metis holomorphic_on_subset image_subset_iff fg holk g holomorphic_on_compose)
+ qed (use fg gf kh in auto)
+qed
+
+lemma homeomorphic_to_disc:
+ assumes S: "S \<noteq> {}"
+ and prev: "S = UNIV \<or>
+ (\<exists>f g. f holomorphic_on S \<and> g holomorphic_on ball 0 1 \<and>
+ (\<forall>z \<in> S. f z \<in> ball 0 1 \<and> g(f z) = z) \<and>
+ (\<forall>z \<in> ball 0 1. g z \<in> S \<and> f(g z) = z))" (is "_ \<or> ?P")
+ shows "S homeomorphic ball (0::complex) 1"
+ using prev
+proof
+ assume "S = UNIV" then show ?thesis
+ using homeomorphic_ball01_UNIV homeomorphic_sym by blast
+next
+ assume ?P
+ then show ?thesis
+ unfolding homeomorphic_minimal
+ using holomorphic_on_imp_continuous_on by blast
+qed
+
+lemma homeomorphic_to_disc_imp_simply_connected:
+ assumes "S = {} \<or> S homeomorphic ball (0::complex) 1"
+ shows "simply_connected S"
+ using assms homeomorphic_simply_connected_eq convex_imp_simply_connected by auto
+
+end
+
+proposition
+ assumes "open S"
+ shows simply_connected_eq_winding_number_zero:
+ "simply_connected S \<longleftrightarrow>
+ connected S \<and>
+ (\<forall>g z. path g \<and> path_image g \<subseteq> S \<and>
+ pathfinish g = pathstart g \<and> (z \<notin> S)
+ \<longrightarrow> winding_number g z = 0)" (is "?wn0")
+ and simply_connected_eq_contour_integral_zero:
+ "simply_connected S \<longleftrightarrow>
+ connected S \<and>
+ (\<forall>g f. valid_path g \<and> path_image g \<subseteq> S \<and>
+ pathfinish g = pathstart g \<and> f holomorphic_on S
+ \<longrightarrow> (f has_contour_integral 0) g)" (is "?ci0")
+ and simply_connected_eq_global_primitive:
+ "simply_connected S \<longleftrightarrow>
+ connected S \<and>
+ (\<forall>f. f holomorphic_on S \<longrightarrow>
+ (\<exists>h. \<forall>z. z \<in> S \<longrightarrow> (h has_field_derivative f z) (at z)))" (is "?gp")
+ and simply_connected_eq_holomorphic_log:
+ "simply_connected S \<longleftrightarrow>
+ connected S \<and>
+ (\<forall>f. f holomorphic_on S \<and> (\<forall>z \<in> S. f z \<noteq> 0)
+ \<longrightarrow> (\<exists>g. g holomorphic_on S \<and> (\<forall>z \<in> S. f z = exp(g z))))" (is "?log")
+ and simply_connected_eq_holomorphic_sqrt:
+ "simply_connected S \<longleftrightarrow>
+ connected S \<and>
+ (\<forall>f. f holomorphic_on S \<and> (\<forall>z \<in> S. f z \<noteq> 0)
+ \<longrightarrow> (\<exists>g. g holomorphic_on S \<and> (\<forall>z \<in> S. f z = (g z)\<^sup>2)))" (is "?sqrt")
+ and simply_connected_eq_biholomorphic_to_disc:
+ "simply_connected S \<longleftrightarrow>
+ S = {} \<or> S = UNIV \<or>
+ (\<exists>f g. f holomorphic_on S \<and> g holomorphic_on ball 0 1 \<and>
+ (\<forall>z \<in> S. f z \<in> ball 0 1 \<and> g(f z) = z) \<and>
+ (\<forall>z \<in> ball 0 1. g z \<in> S \<and> f(g z) = z))" (is "?bih")
+ and simply_connected_eq_homeomorphic_to_disc:
+ "simply_connected S \<longleftrightarrow> S = {} \<or> S homeomorphic ball (0::complex) 1" (is "?disc")
+proof -
+ interpret SC_Chain
+ using assms by (simp add: SC_Chain_def)
+ have "?wn0 \<and> ?ci0 \<and> ?gp \<and> ?log \<and> ?sqrt \<and> ?bih \<and> ?disc"
+proof -
+ have *: "\<lbrakk>\<alpha> \<Longrightarrow> \<beta>; \<beta> \<Longrightarrow> \<gamma>; \<gamma> \<Longrightarrow> \<delta>; \<delta> \<Longrightarrow> \<zeta>; \<zeta> \<Longrightarrow> \<eta>; \<eta> \<Longrightarrow> \<theta>; \<theta> \<Longrightarrow> \<xi>; \<xi> \<Longrightarrow> \<alpha>\<rbrakk>
+ \<Longrightarrow> (\<alpha> \<longleftrightarrow> \<beta>) \<and> (\<alpha> \<longleftrightarrow> \<gamma>) \<and> (\<alpha> \<longleftrightarrow> \<delta>) \<and> (\<alpha> \<longleftrightarrow> \<zeta>) \<and>
+ (\<alpha> \<longleftrightarrow> \<eta>) \<and> (\<alpha> \<longleftrightarrow> \<theta>) \<and> (\<alpha> \<longleftrightarrow> \<xi>)" for \<alpha> \<beta> \<gamma> \<delta> \<zeta> \<eta> \<theta> \<xi>
+ by blast
+ show ?thesis
+ apply (rule *)
+ using winding_number_zero apply metis
+ using contour_integral_zero apply metis
+ using global_primitive apply metis
+ using holomorphic_log apply metis
+ using holomorphic_sqrt apply simp
+ using biholomorphic_to_disc apply blast
+ using homeomorphic_to_disc apply blast
+ using homeomorphic_to_disc_imp_simply_connected apply blast
+ done
+qed
+ then show ?wn0 ?ci0 ?gp ?log ?sqrt ?bih ?disc
+ by safe
+qed
+
+corollary contractible_eq_simply_connected_2d:
+ fixes S :: "complex set"
+ shows "open S \<Longrightarrow> (contractible S \<longleftrightarrow> simply_connected S)"
+ apply safe
+ apply (simp add: contractible_imp_simply_connected)
+ using convex_imp_contractible homeomorphic_contractible_eq simply_connected_eq_homeomorphic_to_disc by auto
+
+subsection\<open>A further chain of equivalences about components of the complement of a simply connected set\<close>
+
+text\<open>(following 1.35 in Burckel'S book)\<close>
+
+context SC_Chain
+begin
+
+lemma frontier_properties:
+ assumes "simply_connected S"
+ shows "if bounded S then connected(frontier S)
+ else \<forall>C \<in> components(frontier S). \<not> bounded C"
+proof -
+ have "S = {} \<or> S homeomorphic ball (0::complex) 1"
+ using simply_connected_eq_homeomorphic_to_disc assms openS by blast
+ then show ?thesis
+ proof
+ assume "S = {}"
+ then have "bounded S"
+ by simp
+ with \<open>S = {}\<close> show ?thesis
+ by simp
+ next
+ assume S01: "S homeomorphic ball (0::complex) 1"
+ then obtain g f
+ where gim: "g ` S = ball 0 1" and fg: "\<And>x. x \<in> S \<Longrightarrow> f(g x) = x"
+ and fim: "f ` ball 0 1 = S" and gf: "\<And>y. cmod y < 1 \<Longrightarrow> g(f y) = y"
+ and contg: "continuous_on S g" and contf: "continuous_on (ball 0 1) f"
+ by (fastforce simp: homeomorphism_def homeomorphic_def)
+ define D where "D \<equiv> \<lambda>n. ball (0::complex) (1 - 1/(of_nat n + 2))"
+ define A where "A \<equiv> \<lambda>n. {z::complex. 1 - 1/(of_nat n + 2) < norm z \<and> norm z < 1}"
+ define X where "X \<equiv> \<lambda>n::nat. closure(f ` A n)"
+ have D01: "D n \<subseteq> ball 0 1" for n
+ by (simp add: D_def ball_subset_ball_iff)
+ have A01: "A n \<subseteq> ball 0 1" for n
+ by (auto simp: A_def)
+ have cloX: "closed(X n)" for n
+ by (simp add: X_def)
+ have Xsubclo: "X n \<subseteq> closure S" for n
+ unfolding X_def by (metis A01 closure_mono fim image_mono)
+ have connX: "connected(X n)" for n
+ unfolding X_def
+ apply (rule connected_imp_connected_closure)
+ apply (rule connected_continuous_image)
+ apply (simp add: continuous_on_subset [OF contf A01])
+ using connected_annulus [of _ "0::complex"] by (simp add: A_def)
+ have nestX: "X n \<subseteq> X m" if "m \<le> n" for m n
+ proof -
+ have "1 - 1 / (real m + 2) \<le> 1 - 1 / (real n + 2)"
+ using that by (auto simp: field_simps)
+ then show ?thesis
+ by (auto simp: X_def A_def intro!: closure_mono)
+ qed
+ have "closure S - S \<subseteq> (\<Inter>n. X n)"
+ proof
+ fix x
+ assume "x \<in> closure S - S"
+ then have "x \<in> closure S" "x \<notin> S" by auto
+ show "x \<in> (\<Inter>n. X n)"
+ proof
+ fix n
+ have "ball 0 1 = closure (D n) \<union> A n"
+ by (auto simp: D_def A_def le_less_trans)
+ with fim have Seq: "S = f ` (closure (D n)) \<union> f ` (A n)"
+ by (simp add: image_Un)
+ have "continuous_on (closure (D n)) f"
+ by (simp add: D_def cball_subset_ball_iff continuous_on_subset [OF contf])
+ moreover have "compact (closure (D n))"
+ by (simp add: D_def)
+ ultimately have clo_fim: "closed (f ` closure (D n))"
+ using compact_continuous_image compact_imp_closed by blast
+ have *: "(f ` cball 0 (1 - 1 / (real n + 2))) \<subseteq> S"
+ by (force simp: D_def Seq)
+ show "x \<in> X n"
+ using \<open>x \<in> closure S\<close> unfolding X_def Seq
+ using \<open>x \<notin> S\<close> * D_def clo_fim by auto
+ qed
+ qed
+ moreover have "(\<Inter>n. X n) \<subseteq> closure S - S"
+ proof -
+ have "(\<Inter>n. X n) \<subseteq> closure S"
+ proof -
+ have "(\<Inter>n. X n) \<subseteq> X 0"
+ by blast
+ also have "... \<subseteq> closure S"
+ apply (simp add: X_def fim [symmetric])
+ apply (rule closure_mono)
+ by (auto simp: A_def)
+ finally show "(\<Inter>n. X n) \<subseteq> closure S" .
+ qed
+ moreover have "(\<Inter>n. X n) \<inter> S \<subseteq> {}"
+ proof (clarify, clarsimp simp: X_def fim [symmetric])
+ fix x assume x [rule_format]: "\<forall>n. f x \<in> closure (f ` A n)" and "cmod x < 1"
+ then obtain n where n: "1 / (1 - norm x) < of_nat n"
+ using reals_Archimedean2 by blast
+ with \<open>cmod x < 1\<close> gr0I have XX: "1 / of_nat n < 1 - norm x" and "n > 0"
+ by (fastforce simp: field_split_simps algebra_simps)+
+ have "f x \<in> f ` (D n)"
+ using n \<open>cmod x < 1\<close> by (auto simp: field_split_simps algebra_simps D_def)
+ moreover have " f ` D n \<inter> closure (f ` A n) = {}"
+ proof -
+ have op_fDn: "open(f ` (D n))"
+ proof (rule invariance_of_domain)
+ show "continuous_on (D n) f"
+ by (rule continuous_on_subset [OF contf D01])
+ show "open (D n)"
+ by (simp add: D_def)
+ show "inj_on f (D n)"
+ unfolding inj_on_def using D01 by (metis gf mem_ball_0 subsetCE)
+ qed
+ have injf: "inj_on f (ball 0 1)"
+ by (metis mem_ball_0 inj_on_def gf)
+ have "D n \<union> A n \<subseteq> ball 0 1"
+ using D01 A01 by simp
+ moreover have "D n \<inter> A n = {}"
+ by (auto simp: D_def A_def)
+ ultimately have "f ` D n \<inter> f ` A n = {}"
+ by (metis A01 D01 image_is_empty inj_on_image_Int injf)
+ then show ?thesis
+ by (simp add: open_Int_closure_eq_empty [OF op_fDn])
+ qed
+ ultimately show False
+ using x [of n] by blast
+ qed
+ ultimately
+ show "(\<Inter>n. X n) \<subseteq> closure S - S"
+ using closure_subset disjoint_iff_not_equal by blast
+ qed
+ ultimately have "closure S - S = (\<Inter>n. X n)" by blast
+ then have frontierS: "frontier S = (\<Inter>n. X n)"
+ by (simp add: frontier_def openS interior_open)
+ show ?thesis
+ proof (cases "bounded S")
+ case True
+ have bouX: "bounded (X n)" for n
+ apply (simp add: X_def)
+ apply (rule bounded_closure)
+ by (metis A01 fim image_mono bounded_subset [OF True])
+ have compaX: "compact (X n)" for n
+ apply (simp add: compact_eq_bounded_closed bouX)
+ apply (auto simp: X_def)
+ done
+ have "connected (\<Inter>n. X n)"
+ by (metis nestX compaX connX connected_nest)
+ then show ?thesis
+ by (simp add: True \<open>frontier S = (\<Inter>n. X n)\<close>)
+ next
+ case False
+ have unboundedX: "\<not> bounded(X n)" for n
+ proof
+ assume bXn: "bounded(X n)"
+ have "continuous_on (cball 0 (1 - 1 / (2 + real n))) f"
+ by (simp add: cball_subset_ball_iff continuous_on_subset [OF contf])
+ then have "bounded (f ` cball 0 (1 - 1 / (2 + real n)))"
+ by (simp add: compact_imp_bounded [OF compact_continuous_image])
+ moreover have "bounded (f ` A n)"
+ by (auto simp: X_def closure_subset image_subset_iff bounded_subset [OF bXn])
+ ultimately have "bounded (f ` (cball 0 (1 - 1/(2 + real n)) \<union> A n))"
+ by (simp add: image_Un)
+ then have "bounded (f ` ball 0 1)"
+ apply (rule bounded_subset)
+ apply (auto simp: A_def algebra_simps)
+ done
+ then show False
+ using False by (simp add: fim [symmetric])
+ qed
+ have clo_INTX: "closed(\<Inter>(range X))"
+ by (metis cloX closed_INT)
+ then have lcX: "locally compact (\<Inter>(range X))"
+ by (metis closed_imp_locally_compact)
+ have False if C: "C \<in> components (frontier S)" and boC: "bounded C" for C
+ proof -
+ have "closed C"
+ by (metis C closed_components frontier_closed)
+ then have "compact C"
+ by (metis boC compact_eq_bounded_closed)
+ have Cco: "C \<in> components (\<Inter>(range X))"
+ by (metis frontierS C)
+ obtain K where "C \<subseteq> K" "compact K"
+ and Ksub: "K \<subseteq> \<Inter>(range X)" and clo: "closed(\<Inter>(range X) - K)"
+ proof (cases "{k. C \<subseteq> k \<and> compact k \<and> openin (top_of_set (\<Inter>(range X))) k} = {}")
+ case True
+ then show ?thesis
+ using Sura_Bura [OF lcX Cco \<open>compact C\<close>] boC
+ by (simp add: True)
+ next
+ case False
+ then obtain L where "compact L" "C \<subseteq> L" and K: "openin (top_of_set (\<Inter>x. X x)) L"
+ by blast
+ show ?thesis
+ proof
+ show "L \<subseteq> \<Inter>(range X)"
+ by (metis K openin_imp_subset)
+ show "closed (\<Inter>(range X) - L)"
+ by (metis closedin_diff closedin_self closedin_closed_trans [OF _ clo_INTX] K)
+ qed (use \<open>compact L\<close> \<open>C \<subseteq> L\<close> in auto)
+ qed
+ obtain U V where "open U" and "compact (closure U)" and "open V" "K \<subseteq> U"
+ and V: "\<Inter>(range X) - K \<subseteq> V" and "U \<inter> V = {}"
+ using separation_normal_compact [OF \<open>compact K\<close> clo] by blast
+ then have "U \<inter> (\<Inter> (range X) - K) = {}"
+ by blast
+ have "(closure U - U) \<inter> (\<Inter>n. X n \<inter> closure U) \<noteq> {}"
+ proof (rule compact_imp_fip)
+ show "compact (closure U - U)"
+ by (metis \<open>compact (closure U)\<close> \<open>open U\<close> compact_diff)
+ show "\<And>T. T \<in> range (\<lambda>n. X n \<inter> closure U) \<Longrightarrow> closed T"
+ by clarify (metis cloX closed_Int closed_closure)
+ show "(closure U - U) \<inter> \<Inter>\<F> \<noteq> {}"
+ if "finite \<F>" and \<F>: "\<F> \<subseteq> range (\<lambda>n. X n \<inter> closure U)" for \<F>
+ proof
+ assume empty: "(closure U - U) \<inter> \<Inter>\<F> = {}"
+ obtain J where "finite J" and J: "\<F> = (\<lambda>n. X n \<inter> closure U) ` J"
+ using finite_subset_image [OF \<open>finite \<F>\<close> \<F>] by auto
+ show False
+ proof (cases "J = {}")
+ case True
+ with J empty have "closed U"
+ by (simp add: closure_subset_eq)
+ have "C \<noteq> {}"
+ using C in_components_nonempty by blast
+ then have "U \<noteq> {}"
+ using \<open>K \<subseteq> U\<close> \<open>C \<subseteq> K\<close> by blast
+ moreover have "U \<noteq> UNIV"
+ using \<open>compact (closure U)\<close> by auto
+ ultimately show False
+ using \<open>open U\<close> \<open>closed U\<close> clopen by blast
+ next
+ case False
+ define j where "j \<equiv> Max J"
+ have "j \<in> J"
+ by (simp add: False \<open>finite J\<close> j_def)
+ have jmax: "\<And>m. m \<in> J \<Longrightarrow> m \<le> j"
+ by (simp add: j_def \<open>finite J\<close>)
+ have "\<Inter> ((\<lambda>n. X n \<inter> closure U) ` J) = X j \<inter> closure U"
+ using False jmax nestX \<open>j \<in> J\<close> by auto
+ then have "X j \<inter> closure U = X j \<inter> U"
+ apply safe
+ using DiffI J empty apply auto[1]
+ using closure_subset by blast
+ then have "openin (top_of_set (X j)) (X j \<inter> closure U)"
+ by (simp add: openin_open_Int \<open>open U\<close>)
+ moreover have "closedin (top_of_set (X j)) (X j \<inter> closure U)"
+ by (simp add: closedin_closed_Int)
+ moreover have "X j \<inter> closure U \<noteq> X j"
+ by (metis unboundedX \<open>compact (closure U)\<close> bounded_subset compact_eq_bounded_closed inf.order_iff)
+ moreover have "X j \<inter> closure U \<noteq> {}"
+ proof -
+ have "C \<noteq> {}"
+ using C in_components_nonempty by blast
+ moreover have "C \<subseteq> X j \<inter> closure U"
+ using \<open>C \<subseteq> K\<close> \<open>K \<subseteq> U\<close> Ksub closure_subset by blast
+ ultimately show ?thesis by blast
+ qed
+ ultimately show False
+ using connX [of j] by (force simp: connected_clopen)
+ qed
+ qed
+ qed
+ moreover have "(\<Inter>n. X n \<inter> closure U) = (\<Inter>n. X n) \<inter> closure U"
+ by blast
+ moreover have "x \<in> U" if "\<And>n. x \<in> X n" "x \<in> closure U" for x
+ proof -
+ have "x \<notin> V"
+ using \<open>U \<inter> V = {}\<close> \<open>open V\<close> closure_iff_nhds_not_empty that(2) by blast
+ then show ?thesis
+ by (metis (no_types) Diff_iff INT_I V \<open>K \<subseteq> U\<close> contra_subsetD that(1))
+ qed
+ ultimately show False
+ by (auto simp: open_Int_closure_eq_empty [OF \<open>open V\<close>, of U])
+ qed
+ then show ?thesis
+ by (auto simp: False)
+ qed
+ qed
+qed
+
+
+lemma unbounded_complement_components:
+ assumes C: "C \<in> components (- S)" and S: "connected S"
+ and prev: "if bounded S then connected(frontier S)
+ else \<forall>C \<in> components(frontier S). \<not> bounded C"
+ shows "\<not> bounded C"
+proof (cases "bounded S")
+ case True
+ with prev have "S \<noteq> UNIV" and confr: "connected(frontier S)"
+ by auto
+ obtain w where C_ccsw: "C = connected_component_set (- S) w" and "w \<notin> S"
+ using C by (auto simp: components_def)
+ show ?thesis
+ proof (cases "S = {}")
+ case True with C show ?thesis by auto
+ next
+ case False
+ show ?thesis
+ proof
+ assume "bounded C"
+ then have "outside C \<noteq> {}"
+ using outside_bounded_nonempty by metis
+ then obtain z where z: "\<not> bounded (connected_component_set (- C) z)" and "z \<notin> C"
+ by (auto simp: outside_def)
+ have clo_ccs: "closed (connected_component_set (- S) x)" for x
+ by (simp add: closed_Compl closed_connected_component openS)
+ have "connected_component_set (- S) w = connected_component_set (- S) z"
+ proof (rule joinable_connected_component_eq [OF confr])
+ show "frontier S \<subseteq> - S"
+ using openS by (auto simp: frontier_def interior_open)
+ have False if "connected_component_set (- S) w \<inter> frontier (- S) = {}"
+ proof -
+ have "C \<inter> frontier S = {}"
+ using that by (simp add: C_ccsw)
+ then show False
+ by (metis C_ccsw ComplI Compl_eq_Compl_iff Diff_subset False \<open>w \<notin> S\<close> clo_ccs closure_closed compl_bot_eq connected_component_eq_UNIV connected_component_eq_empty empty_subsetI frontier_complement frontier_def frontier_not_empty frontier_of_connected_component_subset le_inf_iff subset_antisym)
+ qed
+ then show "connected_component_set (- S) w \<inter> frontier S \<noteq> {}"
+ by auto
+ have *: "\<lbrakk>frontier C \<subseteq> C; frontier C \<subseteq> F; frontier C \<noteq> {}\<rbrakk> \<Longrightarrow> C \<inter> F \<noteq> {}" for C F::"complex set"
+ by blast
+ have "connected_component_set (- S) z \<inter> frontier (- S) \<noteq> {}"
+ proof (rule *)
+ show "frontier (connected_component_set (- S) z) \<subseteq> connected_component_set (- S) z"
+ by (auto simp: closed_Compl closed_connected_component frontier_def openS)
+ show "frontier (connected_component_set (- S) z) \<subseteq> frontier (- S)"
+ using frontier_of_connected_component_subset by fastforce
+ have "\<not> bounded (-S)"
+ by (simp add: True cobounded_imp_unbounded)
+ then have "connected_component_set (- S) z \<noteq> {}"
+ apply (simp only: connected_component_eq_empty)
+ using confr openS \<open>bounded C\<close> \<open>w \<notin> S\<close>
+ apply (simp add: frontier_def interior_open C_ccsw)
+ by (metis ComplI Compl_eq_Diff_UNIV connected_UNIV closed_closure closure_subset connected_component_eq_self
+ connected_diff_open_from_closed subset_UNIV)
+ then show "frontier (connected_component_set (- S) z) \<noteq> {}"
+ apply (simp add: frontier_eq_empty connected_component_eq_UNIV)
+ apply (metis False compl_top_eq double_compl)
+ done
+ qed
+ then show "connected_component_set (- S) z \<inter> frontier S \<noteq> {}"
+ by auto
+ qed
+ then show False
+ by (metis C_ccsw Compl_iff \<open>w \<notin> S\<close> \<open>z \<notin> C\<close> connected_component_eq_empty connected_component_idemp)
+ qed
+ qed
+next
+ case False
+ obtain w where C_ccsw: "C = connected_component_set (- S) w" and "w \<notin> S"
+ using C by (auto simp: components_def)
+ have "frontier (connected_component_set (- S) w) \<subseteq> connected_component_set (- S) w"
+ by (simp add: closed_Compl closed_connected_component frontier_subset_eq openS)
+ moreover have "frontier (connected_component_set (- S) w) \<subseteq> frontier S"
+ using frontier_complement frontier_of_connected_component_subset by blast
+ moreover have "frontier (connected_component_set (- S) w) \<noteq> {}"
+ by (metis C C_ccsw False bounded_empty compl_top_eq connected_component_eq_UNIV double_compl frontier_not_empty in_components_nonempty)
+ ultimately obtain z where zin: "z \<in> frontier S" and z: "z \<in> connected_component_set (- S) w"
+ by blast
+ have *: "connected_component_set (frontier S) z \<in> components(frontier S)"
+ by (simp add: \<open>z \<in> frontier S\<close> componentsI)
+ with prev False have "\<not> bounded (connected_component_set (frontier S) z)"
+ by simp
+ moreover have "connected_component (- S) w = connected_component (- S) z"
+ using connected_component_eq [OF z] by force
+ ultimately show ?thesis
+ by (metis C_ccsw * zin bounded_subset closed_Compl closure_closed connected_component_maximal
+ connected_component_refl connected_connected_component frontier_closures in_components_subset le_inf_iff mem_Collect_eq openS)
+qed
+
+lemma empty_inside:
+ assumes "connected S" "\<And>C. C \<in> components (- S) \<Longrightarrow> \<not> bounded C"
+ shows "inside S = {}"
+ using assms by (auto simp: components_def inside_def)
+
+lemma empty_inside_imp_simply_connected:
+ "\<lbrakk>connected S; inside S = {}\<rbrakk> \<Longrightarrow> simply_connected S"
+ by (metis ComplI inside_Un_outside openS outside_mono simply_connected_eq_winding_number_zero subsetCE sup_bot.left_neutral winding_number_zero_in_outside)
+
+end
+
+proposition
+ fixes S :: "complex set"
+ assumes "open S"
+ shows simply_connected_eq_frontier_properties:
+ "simply_connected S \<longleftrightarrow>
+ connected S \<and>
+ (if bounded S then connected(frontier S)
+ else (\<forall>C \<in> components(frontier S). \<not>bounded C))" (is "?fp")
+ and simply_connected_eq_unbounded_complement_components:
+ "simply_connected S \<longleftrightarrow>
+ connected S \<and> (\<forall>C \<in> components(- S). \<not>bounded C)" (is "?ucc")
+ and simply_connected_eq_empty_inside:
+ "simply_connected S \<longleftrightarrow>
+ connected S \<and> inside S = {}" (is "?ei")
+proof -
+ interpret SC_Chain
+ using assms by (simp add: SC_Chain_def)
+ have "?fp \<and> ?ucc \<and> ?ei"
+proof -
+ have *: "\<lbrakk>\<alpha> \<Longrightarrow> \<beta>; \<beta> \<Longrightarrow> \<gamma>; \<gamma> \<Longrightarrow> \<delta>; \<delta> \<Longrightarrow> \<alpha>\<rbrakk>
+ \<Longrightarrow> (\<alpha> \<longleftrightarrow> \<beta>) \<and> (\<alpha> \<longleftrightarrow> \<gamma>) \<and> (\<alpha> \<longleftrightarrow> \<delta>)" for \<alpha> \<beta> \<gamma> \<delta>
+ by blast
+ show ?thesis
+ apply (rule *)
+ using frontier_properties simply_connected_imp_connected apply blast
+apply clarify
+ using unbounded_complement_components simply_connected_imp_connected apply blast
+ using empty_inside apply blast
+ using empty_inside_imp_simply_connected apply blast
+ done
+qed
+ then show ?fp ?ucc ?ei
+ by safe
+qed
+
+lemma simply_connected_iff_simple:
+ fixes S :: "complex set"
+ assumes "open S" "bounded S"
+ shows "simply_connected S \<longleftrightarrow> connected S \<and> connected(- S)"
+ apply (simp add: simply_connected_eq_unbounded_complement_components assms, safe)
+ apply (metis DIM_complex assms(2) cobounded_has_bounded_component double_compl order_refl)
+ by (meson assms inside_bounded_complement_connected_empty simply_connected_eq_empty_inside simply_connected_eq_unbounded_complement_components)
+
+subsection\<open>Further equivalences based on continuous logs and sqrts\<close>
+
+context SC_Chain
+begin
+
+lemma continuous_log:
+ fixes f :: "complex\<Rightarrow>complex"
+ assumes S: "simply_connected S"
+ and contf: "continuous_on S f" and nz: "\<And>z. z \<in> S \<Longrightarrow> f z \<noteq> 0"
+ shows "\<exists>g. continuous_on S g \<and> (\<forall>z \<in> S. f z = exp(g z))"
+proof -
+ consider "S = {}" | "S homeomorphic ball (0::complex) 1"
+ using simply_connected_eq_homeomorphic_to_disc [OF openS] S by metis
+ then show ?thesis
+ proof cases
+ case 1 then show ?thesis
+ by simp
+ next
+ case 2
+ then obtain h k :: "complex\<Rightarrow>complex"
+ where kh: "\<And>x. x \<in> S \<Longrightarrow> k(h x) = x" and him: "h ` S = ball 0 1"
+ and conth: "continuous_on S h"
+ and hk: "\<And>y. y \<in> ball 0 1 \<Longrightarrow> h(k y) = y" and kim: "k ` ball 0 1 = S"
+ and contk: "continuous_on (ball 0 1) k"
+ unfolding homeomorphism_def homeomorphic_def by metis
+ obtain g where contg: "continuous_on (ball 0 1) g"
+ and expg: "\<And>z. z \<in> ball 0 1 \<Longrightarrow> (f \<circ> k) z = exp (g z)"
+ proof (rule continuous_logarithm_on_ball)
+ show "continuous_on (ball 0 1) (f \<circ> k)"
+ apply (rule continuous_on_compose [OF contk])
+ using kim continuous_on_subset [OF contf]
+ by blast
+ show "\<And>z. z \<in> ball 0 1 \<Longrightarrow> (f \<circ> k) z \<noteq> 0"
+ using kim nz by auto
+ qed auto
+ then show ?thesis
+ by (metis comp_apply conth continuous_on_compose him imageI kh)
+ qed
+qed
+
+lemma continuous_sqrt:
+ fixes f :: "complex\<Rightarrow>complex"
+ assumes contf: "continuous_on S f" and nz: "\<And>z. z \<in> S \<Longrightarrow> f z \<noteq> 0"
+ and prev: "\<And>f::complex\<Rightarrow>complex.
+ \<lbrakk>continuous_on S f; \<And>z. z \<in> S \<Longrightarrow> f z \<noteq> 0\<rbrakk>
+ \<Longrightarrow> \<exists>g. continuous_on S g \<and> (\<forall>z \<in> S. f z = exp(g z))"
+ shows "\<exists>g. continuous_on S g \<and> (\<forall>z \<in> S. f z = (g z)\<^sup>2)"
+proof -
+ obtain g where contg: "continuous_on S g" and geq: "\<And>z. z \<in> S \<Longrightarrow> f z = exp(g z)"
+ using contf nz prev by metis
+ show ?thesis
+proof (intro exI ballI conjI)
+ show "continuous_on S (\<lambda>z. exp(g z/2))"
+ by (intro continuous_intros) (auto simp: contg)
+ show "\<And>z. z \<in> S \<Longrightarrow> f z = (exp (g z/2))\<^sup>2"
+ by (metis (no_types, lifting) divide_inverse exp_double geq mult.left_commute mult.right_neutral right_inverse zero_neq_numeral)
+ qed
+qed
+
+lemma continuous_sqrt_imp_simply_connected:
+ assumes "connected S"
+ and prev: "\<And>f::complex\<Rightarrow>complex. \<lbrakk>continuous_on S f; \<forall>z \<in> S. f z \<noteq> 0\<rbrakk>
+ \<Longrightarrow> \<exists>g. continuous_on S g \<and> (\<forall>z \<in> S. f z = (g z)\<^sup>2)"
+ shows "simply_connected S"
+proof (clarsimp simp add: simply_connected_eq_holomorphic_sqrt [OF openS] \<open>connected S\<close>)
+ fix f
+ assume "f holomorphic_on S" and nz: "\<forall>z\<in>S. f z \<noteq> 0"
+ then obtain g where contg: "continuous_on S g" and geq: "\<And>z. z \<in> S \<Longrightarrow> f z = (g z)\<^sup>2"
+ by (metis holomorphic_on_imp_continuous_on prev)
+ show "\<exists>g. g holomorphic_on S \<and> (\<forall>z\<in>S. f z = (g z)\<^sup>2)"
+ proof (intro exI ballI conjI)
+ show "g holomorphic_on S"
+ proof (clarsimp simp add: holomorphic_on_open [OF openS])
+ fix z
+ assume "z \<in> S"
+ with nz geq have "g z \<noteq> 0"
+ by auto
+ obtain \<delta> where "0 < \<delta>" "\<And>w. \<lbrakk>w \<in> S; dist w z < \<delta>\<rbrakk> \<Longrightarrow> dist (g w) (g z) < cmod (g z)"
+ using contg [unfolded continuous_on_iff] by (metis \<open>g z \<noteq> 0\<close> \<open>z \<in> S\<close> zero_less_norm_iff)
+ then have \<delta>: "\<And>w. \<lbrakk>w \<in> S; w \<in> ball z \<delta>\<rbrakk> \<Longrightarrow> g w + g z \<noteq> 0"
+ apply (clarsimp simp: dist_norm)
+ by (metis \<open>g z \<noteq> 0\<close> add_diff_cancel_left' diff_0_right norm_eq_zero norm_increases_online norm_minus_commute norm_not_less_zero not_less_iff_gr_or_eq)
+ have *: "(\<lambda>x. (f x - f z) / (x - z) / (g x + g z)) \<midarrow>z\<rightarrow> deriv f z / (g z + g z)"
+ apply (intro tendsto_intros)
+ using SC_Chain.openS SC_Chain_axioms \<open>f holomorphic_on S\<close> \<open>z \<in> S\<close> has_field_derivativeD holomorphic_derivI apply fastforce
+ using \<open>z \<in> S\<close> contg continuous_on_eq_continuous_at isCont_def openS apply blast
+ by (simp add: \<open>g z \<noteq> 0\<close>)
+ then have "(g has_field_derivative deriv f z / (g z + g z)) (at z)"
+ unfolding has_field_derivative_iff
+ proof (rule Lim_transform_within_open)
+ show "open (ball z \<delta> \<inter> S)"
+ by (simp add: openS open_Int)
+ show "z \<in> ball z \<delta> \<inter> S"
+ using \<open>z \<in> S\<close> \<open>0 < \<delta>\<close> by simp
+ show "\<And>x. \<lbrakk>x \<in> ball z \<delta> \<inter> S; x \<noteq> z\<rbrakk>
+ \<Longrightarrow> (f x - f z) / (x - z) / (g x + g z) = (g x - g z) / (x - z)"
+ using \<delta>
+ apply (simp add: geq \<open>z \<in> S\<close> divide_simps)
+ apply (auto simp: algebra_simps power2_eq_square)
+ done
+ qed
+ then show "\<exists>f'. (g has_field_derivative f') (at z)" ..
+ qed
+ qed (use geq in auto)
+qed
+
+end
+
+proposition
+ fixes S :: "complex set"
+ assumes "open S"
+ shows simply_connected_eq_continuous_log:
+ "simply_connected S \<longleftrightarrow>
+ connected S \<and>
+ (\<forall>f::complex\<Rightarrow>complex. continuous_on S f \<and> (\<forall>z \<in> S. f z \<noteq> 0)
+ \<longrightarrow> (\<exists>g. continuous_on S g \<and> (\<forall>z \<in> S. f z = exp (g z))))" (is "?log")
+ and simply_connected_eq_continuous_sqrt:
+ "simply_connected S \<longleftrightarrow>
+ connected S \<and>
+ (\<forall>f::complex\<Rightarrow>complex. continuous_on S f \<and> (\<forall>z \<in> S. f z \<noteq> 0)
+ \<longrightarrow> (\<exists>g. continuous_on S g \<and> (\<forall>z \<in> S. f z = (g z)\<^sup>2)))" (is "?sqrt")
+proof -
+ interpret SC_Chain
+ using assms by (simp add: SC_Chain_def)
+ have "?log \<and> ?sqrt"
+proof -
+ have *: "\<lbrakk>\<alpha> \<Longrightarrow> \<beta>; \<beta> \<Longrightarrow> \<gamma>; \<gamma> \<Longrightarrow> \<alpha>\<rbrakk>
+ \<Longrightarrow> (\<alpha> \<longleftrightarrow> \<beta>) \<and> (\<alpha> \<longleftrightarrow> \<gamma>)" for \<alpha> \<beta> \<gamma>
+ by blast
+ show ?thesis
+ apply (rule *)
+ apply (simp add: local.continuous_log winding_number_zero)
+ apply (simp add: continuous_sqrt)
+ apply (simp add: continuous_sqrt_imp_simply_connected)
+ done
+qed
+ then show ?log ?sqrt
+ by safe
+qed
+
+
+subsection\<^marker>\<open>tag unimportant\<close> \<open>More Borsukian results\<close>
+
+lemma Borsukian_componentwise_eq:
+ fixes S :: "'a::euclidean_space set"
+ assumes S: "locally connected S \<or> compact S"
+ shows "Borsukian S \<longleftrightarrow> (\<forall>C \<in> components S. Borsukian C)"
+proof -
+ have *: "ANR(-{0::complex})"
+ by (simp add: ANR_delete open_Compl open_imp_ANR)
+ show ?thesis
+ using cohomotopically_trivial_on_components [OF assms *] by (auto simp: Borsukian_alt)
+qed
+
+lemma Borsukian_componentwise:
+ fixes S :: "'a::euclidean_space set"
+ assumes "locally connected S \<or> compact S" "\<And>C. C \<in> components S \<Longrightarrow> Borsukian C"
+ shows "Borsukian S"
+ by (metis Borsukian_componentwise_eq assms)
+
+lemma simply_connected_eq_Borsukian:
+ fixes S :: "complex set"
+ shows "open S \<Longrightarrow> (simply_connected S \<longleftrightarrow> connected S \<and> Borsukian S)"
+ by (auto simp: simply_connected_eq_continuous_log Borsukian_continuous_logarithm)
+
+lemma Borsukian_eq_simply_connected:
+ fixes S :: "complex set"
+ shows "open S \<Longrightarrow> Borsukian S \<longleftrightarrow> (\<forall>C \<in> components S. simply_connected C)"
+apply (auto simp: Borsukian_componentwise_eq open_imp_locally_connected)
+ using in_components_connected open_components simply_connected_eq_Borsukian apply blast
+ using open_components simply_connected_eq_Borsukian by blast
+
+lemma Borsukian_separation_open_closed:
+ fixes S :: "complex set"
+ assumes S: "open S \<or> closed S" and "bounded S"
+ shows "Borsukian S \<longleftrightarrow> connected(- S)"
+ using S
+proof
+ assume "open S"
+ show ?thesis
+ unfolding Borsukian_eq_simply_connected [OF \<open>open S\<close>]
+ by (meson \<open>open S\<close> \<open>bounded S\<close> bounded_subset in_components_connected in_components_subset nonseparation_by_component_eq open_components simply_connected_iff_simple)
+next
+ assume "closed S"
+ with \<open>bounded S\<close> show ?thesis
+ by (simp add: Borsukian_separation_compact compact_eq_bounded_closed)
+qed
+
+
+subsection\<open>Finally, the Riemann Mapping Theorem\<close>
+
+theorem Riemann_mapping_theorem:
+ "open S \<and> simply_connected S \<longleftrightarrow>
+ S = {} \<or> S = UNIV \<or>
+ (\<exists>f g. f holomorphic_on S \<and> g holomorphic_on ball 0 1 \<and>
+ (\<forall>z \<in> S. f z \<in> ball 0 1 \<and> g(f z) = z) \<and>
+ (\<forall>z \<in> ball 0 1. g z \<in> S \<and> f(g z) = z))"
+ (is "_ = ?rhs")
+proof -
+ have "simply_connected S \<longleftrightarrow> ?rhs" if "open S"
+ by (simp add: simply_connected_eq_biholomorphic_to_disc that)
+ moreover have "open S" if "?rhs"
+ proof -
+ { fix f g
+ assume g: "g holomorphic_on ball 0 1" "\<forall>z\<in>ball 0 1. g z \<in> S \<and> f (g z) = z"
+ and "\<forall>z\<in>S. cmod (f z) < 1 \<and> g (f z) = z"
+ then have "S = g ` (ball 0 1)"
+ by (force simp:)
+ then have "open S"
+ by (metis open_ball g inj_on_def open_mapping_thm3)
+ }
+ with that show "open S" by auto
+ qed
+ ultimately show ?thesis by metis
+qed
+
+end