# Theory Conformal_Mappings

theory Conformal_Mappings
imports Cauchy_Integral_Theorem
```section ‹Conformal Mappings and Consequences of Cauchy's integral theorem›

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

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

theory Conformal_Mappings
imports Cauchy_Integral_Theorem

begin

(* FIXME mv to Cauchy_Integral_Theorem.thy *)
subsection‹Cauchy's inequality and more versions of Liouville›

lemma Cauchy_higher_deriv_bound:
assumes holf: "f holomorphic_on (ball z r)"
and contf: "continuous_on (cball z r) f"
and fin : "⋀w. w ∈ ball z r ⟹ f w ∈ ball y B0"
and "0 < r" and "0 < n"
shows "norm ((deriv ^^ n) f z) ≤ (fact n) * B0 / r^n"
proof -
have "0 < B0" using ‹0 < r› fin [of z]
by (metis ball_eq_empty ex_in_conv fin not_less)
have le_B0: "⋀w. cmod (w - z) ≤ r ⟹ cmod (f w - y) ≤ B0"
apply (rule continuous_on_closure_norm_le [of "ball z r" "λw. f w - y"])
apply (auto simp: ‹0 < r›  dist_norm norm_minus_commute)
apply (rule continuous_intros contf)+
using fin apply (simp add: dist_commute dist_norm less_eq_real_def)
done
have "(deriv ^^ n) f z = (deriv ^^ n) (λw. f w) z - (deriv ^^ n) (λw. y) z"
using ‹0 < n› by simp
also have "... = (deriv ^^ n) (λw. f w - y) z"
by (rule higher_deriv_diff [OF holf, symmetric]) (auto simp: ‹0 < r›)
finally have "(deriv ^^ n) f z = (deriv ^^ n) (λw. f w - y) z" .
have contf': "continuous_on (cball z r) (λu. f u - y)"
by (rule contf continuous_intros)+
have holf': "(λu. (f u - y)) holomorphic_on (ball z r)"
by (simp add: holf holomorphic_on_diff)
define a where "a = (2 * pi)/(fact n)"
have "0 < a"  by (simp add: a_def)
have "B0/r^(Suc n)*2 * pi * r = a*((fact n)*B0/r^n)"
using ‹0 < r› by (simp add: a_def divide_simps)
have der_dif: "(deriv ^^ n) (λw. f w - y) z = (deriv ^^ n) f z"
using ‹0 < r› ‹0 < n›
by (auto simp: higher_deriv_diff [OF holf holomorphic_on_const])
have "norm ((2 * of_real pi * 𝗂)/(fact n) * (deriv ^^ n) (λw. f w - y) z)
≤ (B0/r^(Suc n)) * (2 * pi * r)"
apply (rule has_contour_integral_bound_circlepath [of "(λu. (f u - y)/(u - z)^(Suc n))" _ z])
using Cauchy_has_contour_integral_higher_derivative_circlepath [OF contf' holf']
using ‹0 < B0› ‹0 < r›
apply (auto simp: norm_divide norm_mult norm_power divide_simps le_B0)
done
then show ?thesis
using ‹0 < r›
by (auto simp: norm_divide norm_mult norm_power field_simps der_dif le_B0)
qed

lemma Cauchy_inequality:
assumes holf: "f holomorphic_on (ball ξ r)"
and contf: "continuous_on (cball ξ r) f"
and "0 < r"
and nof: "⋀x. norm(ξ-x) = r ⟹ norm(f x) ≤ B"
shows "norm ((deriv ^^ n) f ξ) ≤ (fact n) * B / r^n"
proof -
obtain x where "norm (ξ-x) = r"
by (metis abs_of_nonneg add_diff_cancel_left' ‹0 < r› diff_add_cancel
dual_order.strict_implies_order norm_of_real)
then have "0 ≤ B"
by (metis nof norm_not_less_zero not_le order_trans)
have  "((λu. f u / (u - ξ) ^ Suc n) has_contour_integral (2 * pi) * 𝗂 / fact n * (deriv ^^ n) f ξ)
(circlepath ξ r)"
apply (rule Cauchy_has_contour_integral_higher_derivative_circlepath [OF contf holf])
using ‹0 < r› by simp
then have "norm ((2 * pi * 𝗂)/(fact n) * (deriv ^^ n) f ξ) ≤ (B / r^(Suc n)) * (2 * pi * r)"
apply (rule has_contour_integral_bound_circlepath)
using ‹0 ≤ B› ‹0 < r›
apply (simp_all add: norm_divide norm_power nof frac_le norm_minus_commute del: power_Suc)
done
then show ?thesis using ‹0 < r›
by (simp add: norm_divide norm_mult field_simps)
qed

lemma Liouville_polynomial:
assumes holf: "f holomorphic_on UNIV"
and nof: "⋀z. A ≤ norm z ⟹ norm(f z) ≤ B * norm z ^ n"
shows "f ξ = (∑k≤n. (deriv^^k) f 0 / fact k * ξ ^ k)"
proof (cases rule: le_less_linear [THEN disjE])
assume "B ≤ 0"
then have "⋀z. A ≤ norm z ⟹ norm(f z) = 0"
by (metis nof less_le_trans zero_less_mult_iff neqE norm_not_less_zero norm_power not_le)
then have f0: "(f ⤏ 0) at_infinity"
using Lim_at_infinity by force
then have [simp]: "f = (λw. 0)"
using Liouville_weak [OF holf, of 0]
by (simp add: eventually_at_infinity f0) meson
show ?thesis by simp
next
assume "0 < B"
have "((λk. (deriv ^^ k) f 0 / (fact k) * (ξ - 0)^k) sums f ξ)"
apply (rule holomorphic_power_series [where r = "norm ξ + 1"])
using holf holomorphic_on_subset apply auto
done
then have sumsf: "((λk. (deriv ^^ k) f 0 / (fact k) * ξ^k) sums f ξ)" by simp
have "(deriv ^^ k) f 0 / fact k * ξ ^ k = 0" if "k>n" for k
proof (cases "(deriv ^^ k) f 0 = 0")
case True then show ?thesis by simp
next
case False
define w where "w = complex_of_real (fact k * B / cmod ((deriv ^^ k) f 0) + (¦A¦ + 1))"
have "1 ≤ abs (fact k * B / cmod ((deriv ^^ k) f 0) + (¦A¦ + 1))"
using ‹0 < B› by simp
then have wge1: "1 ≤ norm w"
by (metis norm_of_real w_def)
then have "w ≠ 0" by auto
have kB: "0 < fact k * B"
using ‹0 < B› by simp
then have "0 ≤ fact k * B / cmod ((deriv ^^ k) f 0)"
by simp
then have wgeA: "A ≤ cmod w"
by (simp only: w_def norm_of_real)
have "fact k * B / cmod ((deriv ^^ k) f 0) < abs (fact k * B / cmod ((deriv ^^ k) f 0) + (¦A¦ + 1))"
using ‹0 < B› by simp
then have wge: "fact k * B / cmod ((deriv ^^ k) f 0) < norm w"
by (metis norm_of_real w_def)
then have "fact k * B / norm w < cmod ((deriv ^^ k) f 0)"
using False by (simp add: divide_simps mult.commute split: if_split_asm)
also have "... ≤ fact k * (B * norm w ^ n) / norm w ^ k"
apply (rule Cauchy_inequality)
using holf holomorphic_on_subset apply force
using holf holomorphic_on_imp_continuous_on holomorphic_on_subset apply blast
using ‹w ≠ 0› apply simp
by (metis nof wgeA dist_0_norm dist_norm)
also have "... = fact k * (B * 1 / cmod w ^ (k-n))"
apply (simp only: mult_cancel_left times_divide_eq_right [symmetric])
using ‹k>n› ‹w ≠ 0› ‹0 < B› apply (simp add: divide_simps semiring_normalization_rules)
done
also have "... = fact k * B / cmod w ^ (k-n)"
by simp
finally have "fact k * B / cmod w < fact k * B / cmod w ^ (k - n)" .
then have "1 / cmod w < 1 / cmod w ^ (k - n)"
by (metis kB divide_inverse inverse_eq_divide mult_less_cancel_left_pos)
then have "cmod w ^ (k - n) < cmod w"
by (metis frac_le le_less_trans norm_ge_zero norm_one not_less order_refl wge1 zero_less_one)
with self_le_power [OF wge1] have False
by (meson diff_is_0_eq not_gr0 not_le that)
then show ?thesis by blast
qed
then have "(deriv ^^ (k + Suc n)) f 0 / fact (k + Suc n) * ξ ^ (k + Suc n) = 0" for k
using not_less_eq by blast
then have "(λi. (deriv ^^ (i + Suc n)) f 0 / fact (i + Suc n) * ξ ^ (i + Suc n)) sums 0"
by (rule sums_0)
with sums_split_initial_segment [OF sumsf, where n = "Suc n"]
show ?thesis
using atLeast0AtMost lessThan_Suc_atMost sums_unique2 by fastforce
qed

text‹Every bounded entire function is a constant function.›
theorem Liouville_theorem:
assumes holf: "f holomorphic_on UNIV"
and bf: "bounded (range f)"
obtains c where "⋀z. f z = c"
proof -
obtain B where "⋀z. cmod (f z) ≤ B"
by (meson bf bounded_pos rangeI)
then show ?thesis
using Liouville_polynomial [OF holf, of 0 B 0, simplified] that by blast
qed

text‹A holomorphic function f has only isolated zeros unless f is 0.›

lemma powser_0_nonzero:
fixes a :: "nat ⇒ 'a::{real_normed_field,banach}"
assumes r: "0 < r"
and sm: "⋀x. norm (x - ξ) < r ⟹ (λn. a n * (x - ξ) ^ n) sums (f x)"
and [simp]: "f ξ = 0"
and m0: "a m ≠ 0" and "m>0"
obtains s where "0 < s" and "⋀z. z ∈ cball ξ s - {ξ} ⟹ f z ≠ 0"
proof -
have "r ≤ conv_radius a"
using sm sums_summable by (auto simp: le_conv_radius_iff [where ξ=ξ])
obtain m where am: "a m ≠ 0" and az [simp]: "(⋀n. n<m ⟹ a n = 0)"
apply (rule_tac m = "LEAST n. a n ≠ 0" in that)
using m0
apply (rule LeastI2)
apply (fastforce intro:  dest!: not_less_Least)+
done
define b where "b i = a (i+m) / a m" for i
define g where "g x = suminf (λi. b i * (x - ξ) ^ i)" for x
have [simp]: "b 0 = 1"
by (simp add: am b_def)
{ fix x::'a
assume "norm (x - ξ) < r"
then have "(λn. (a m * (x - ξ)^m) * (b n * (x - ξ)^n)) sums (f x)"
using am az sm sums_zero_iff_shift [of m "(λn. a n * (x - ξ) ^ n)" "f x"]
by (simp add: b_def monoid_mult_class.power_add algebra_simps)
then have "x ≠ ξ ⟹ (λn. b n * (x - ξ)^n) sums (f x / (a m * (x - ξ)^m))"
using am by (simp add: sums_mult_D)
} note bsums = this
then have  "norm (x - ξ) < r ⟹ summable (λn. b n * (x - ξ)^n)" for x
using sums_summable by (cases "x=ξ") auto
then have "r ≤ conv_radius b"
by (simp add: le_conv_radius_iff [where ξ=ξ])
then have "r/2 < conv_radius b"
using not_le order_trans r by fastforce
then have "continuous_on (cball ξ (r/2)) g"
using powser_continuous_suminf [of "r/2" b ξ] by (simp add: g_def)
then obtain s where "s>0"  "⋀x. ⟦norm (x - ξ) ≤ s; norm (x - ξ) ≤ r/2⟧ ⟹ dist (g x) (g ξ) < 1/2"
apply (rule continuous_onE [where x=ξ and e = "1/2"])
using r apply (auto simp: norm_minus_commute dist_norm)
done
moreover have "g ξ = 1"
by (simp add: g_def)
ultimately have gnz: "⋀x. ⟦norm (x - ξ) ≤ s; norm (x - ξ) ≤ r/2⟧ ⟹ (g x) ≠ 0"
by fastforce
have "f x ≠ 0" if "x ≠ ξ" "norm (x - ξ) ≤ s" "norm (x - ξ) ≤ r/2" for x
using bsums [of x] that gnz [of x]
apply (auto simp: g_def)
using r sums_iff by fastforce
then show ?thesis
apply (rule_tac s="min s (r/2)" in that)
using ‹0 < r› ‹0 < s› by (auto simp: dist_commute dist_norm)
qed

subsection ‹Analytic continuation›

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

proposition analytic_continuation:
assumes holf: "f holomorphic_on S"
and "open S" and "connected S"
and "U ⊆ S" and "ξ ∈ S"
and "ξ islimpt U"
and fU0 [simp]: "⋀z. z ∈ U ⟹ f z = 0"
and "w ∈ S"
shows "f w = 0"
proof -
obtain e where "0 < e" and e: "cball ξ e ⊆ S"
using ‹open S› ‹ξ ∈ S› open_contains_cball_eq by blast
define T where "T = cball ξ e ∩ U"
have contf: "continuous_on (closure T) f"
by (metis T_def closed_cball closure_minimal e holf holomorphic_on_imp_continuous_on
holomorphic_on_subset inf.cobounded1)
have fT0 [simp]: "⋀x. x ∈ T ⟹ f x = 0"
by (simp add: T_def)
have "⋀r. ⟦∀e>0. ∃x'∈U. x' ≠ ξ ∧ dist x' ξ < e; 0 < r⟧ ⟹ ∃x'∈cball ξ e ∩ U. x' ≠ ξ ∧ dist x' ξ < r"
by (metis ‹0 < e› IntI dist_commute less_eq_real_def mem_cball min_less_iff_conj)
then have "ξ islimpt T" using ‹ξ islimpt U›
by (auto simp: T_def islimpt_approachable)
then have "ξ ∈ closure T"
by (simp add: closure_def)
then have "f ξ = 0"
by (auto simp: continuous_constant_on_closure [OF contf])
show ?thesis
apply (rule ccontr)
apply (rule isolated_zeros [OF holf ‹open S› ‹connected S› ‹ξ ∈ S› ‹f ξ = 0› ‹w ∈ S›], assumption)
by (metis open_ball ‹ξ islimpt T› centre_in_ball fT0 insertE insert_Diff islimptE)
qed

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

subsection‹Open mapping theorem›

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

theorem open_mapping_thm:
assumes holf: "f holomorphic_on S"
and S: "open S" and "connected S"
and "open U" and "U ⊆ S"
and fne: "~ f constant_on S"
shows "open (f ` U)"
proof -
have *: "open (f ` U)"
if "U ≠ {}" and U: "open U" "connected U" and "f holomorphic_on U" and fneU: "⋀x. ∃y ∈ U. f y ≠ x"
for U
proof (clarsimp simp: open_contains_ball)
fix ξ assume ξ: "ξ ∈ U"
show "∃e>0. ball (f ξ) e ⊆ f ` U"
proof -
have hol: "(λz. f z - f ξ) holomorphic_on U"
by (rule holomorphic_intros that)+
obtain s where "0 < s" and sbU: "ball ξ s ⊆ U"
and sne: "⋀z. z ∈ ball ξ s - {ξ} ⟹ (λz. f z - f ξ) z ≠ 0"
using isolated_zeros [OF hol U ξ]  by (metis fneU right_minus_eq)
obtain r where "0 < r" and r: "cball ξ r ⊆ ball ξ s"
apply (rule_tac r="s/2" in that)
using ‹0 < s› by auto
have "cball ξ r ⊆ U"
using sbU r by blast
then have frsbU: "frontier (cball ξ r) ⊆ U"
using Diff_subset frontier_def order_trans by fastforce
then have cof: "compact (frontier(cball ξ r))"
by blast
have frne: "frontier (cball ξ r) ≠ {}"
using ‹0 < r› by auto
have contfr: "continuous_on (frontier (cball ξ r)) (λz. norm (f z - f ξ))"
apply (rule continuous_on_compose2 [OF Complex_Analysis_Basics.continuous_on_norm_id])
using hol frsbU holomorphic_on_imp_continuous_on holomorphic_on_subset by blast+
obtain w where "norm (ξ - w) = r"
and w: "(⋀z. norm (ξ - z) = r ⟹ norm (f w - f ξ) ≤ norm(f z - f ξ))"
apply (rule bexE [OF continuous_attains_inf [OF cof frne contfr]])
apply (simp add: dist_norm)
done
moreover define ε where "ε ≡ norm (f w - f ξ) / 3"
ultimately have "0 < ε"
using ‹0 < r› dist_complex_def r sne by auto
have "ball (f ξ) ε ⊆ f ` U"
proof
fix γ
assume γ: "γ ∈ ball (f ξ) ε"
have *: "cmod (γ - f ξ) < cmod (γ - f z)" if "cmod (ξ - z) = r" for z
proof -
have lt: "cmod (f w - f ξ) / 3 < cmod (γ - f z)"
using w [OF that] γ
using dist_triangle2 [of "f ξ" "γ"  "f z"] dist_triangle2 [of "f ξ" "f z" γ]
by (simp add: ε_def dist_norm norm_minus_commute)
show ?thesis
by (metis ε_def dist_commute dist_norm less_trans lt mem_ball γ)
qed
have "continuous_on (cball ξ r) (λz. γ - f z)"
apply (rule continuous_intros)+
using ‹cball ξ r ⊆ U› ‹f holomorphic_on U›
apply (blast intro: continuous_on_subset holomorphic_on_imp_continuous_on)
done
moreover have "(λz. γ - f z) holomorphic_on ball ξ r"
apply (rule holomorphic_intros)+
apply (metis ‹cball ξ r ⊆ U› ‹f holomorphic_on U› holomorphic_on_subset interior_cball interior_subset)
done
ultimately obtain z where "z ∈ ball ξ r" "γ - f z = 0"
apply (rule holomorphic_contract_to_zero)
apply (blast intro!: ‹0 < r› *)+
done
then show "γ ∈ f ` U"
using ‹cball ξ r ⊆ U› by fastforce
qed
then show ?thesis using  ‹0 < ε› by blast
qed
qed
have "open (f ` X)" if "X ∈ components U" for X
proof -
have holfU: "f holomorphic_on U"
using ‹U ⊆ S› holf holomorphic_on_subset by blast
have "X ≠ {}"
using that by (simp add: in_components_nonempty)
moreover have "open X"
using that ‹open U› open_components by auto
moreover have "connected X"
using that in_components_maximal by blast
moreover have "f holomorphic_on X"
by (meson that holfU holomorphic_on_subset in_components_maximal)
moreover have "∃y∈X. f y ≠ x" for x
proof (rule ccontr)
assume not: "¬ (∃y∈X. f y ≠ x)"
have "X ⊆ S"
using ‹U ⊆ S› in_components_subset that by blast
obtain w where w: "w ∈ X" using ‹X ≠ {}› by blast
have wis: "w islimpt X"
using w ‹open X› interior_eq by auto
have hol: "(λz. f z - x) holomorphic_on S"
by (simp add: holf holomorphic_on_diff)
with fne [unfolded constant_on_def]
analytic_continuation[OF hol S ‹connected S› ‹X ⊆ S› _ wis] not ‹X ⊆ S› w
show False by auto
qed
ultimately show ?thesis
by (rule *)
qed
then have "open (f ` ⋃components U)"
by (metis (no_types, lifting) imageE image_Union open_Union)
then show ?thesis
by force
qed

text‹No need for @{term S} to be connected. But the nonconstant condition is stronger.›
corollary%unimportant open_mapping_thm2:
assumes holf: "f holomorphic_on S"
and S: "open S"
and "open U" "U ⊆ S"
and fnc: "⋀X. ⟦open X; X ⊆ S; X ≠ {}⟧ ⟹ ~ f constant_on X"
shows "open (f ` U)"
proof -
have "S = ⋃(components S)" by simp
with ‹U ⊆ S› have "U = (⋃C ∈ components S. C ∩ U)" by auto
then have "f ` U = (⋃C ∈ components S. f ` (C ∩ U))"
using image_UN by fastforce
moreover
{ fix C assume "C ∈ components S"
with S ‹C ∈ components S› open_components in_components_connected
have C: "open C" "connected C" by auto
have "C ⊆ S"
by (metis ‹C ∈ components S› in_components_maximal)
have nf: "¬ f constant_on C"
apply (rule fnc)
using C ‹C ⊆ S› ‹C ∈ components S› in_components_nonempty by auto
have "f holomorphic_on C"
by (metis holf holomorphic_on_subset ‹C ⊆ S›)
then have "open (f ` (C ∩ U))"
apply (rule open_mapping_thm [OF _ C _ _ nf])
apply (simp add: C ‹open U› open_Int, blast)
done
} ultimately show ?thesis
by force
qed

corollary%unimportant open_mapping_thm3:
assumes holf: "f holomorphic_on S"
and "open S" and injf: "inj_on f S"
shows  "open (f ` S)"
apply (rule open_mapping_thm2 [OF holf])
using assms
apply (simp_all add:)
using injective_not_constant subset_inj_on by blast

subsection‹Maximum modulus principle›

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

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

proposition maximum_modulus_frontier:
assumes holf: "f holomorphic_on (interior S)"
and contf: "continuous_on (closure S) f"
and bos: "bounded S"
and leB: "⋀z. z ∈ frontier S ⟹ norm(f z) ≤ B"
and "ξ ∈ S"
shows "norm(f ξ) ≤ B"
proof -
have "compact (closure S)" using bos
by (simp add: bounded_closure compact_eq_bounded_closed)
moreover have "continuous_on (closure S) (cmod ∘ f)"
using contf continuous_on_compose continuous_on_norm_id by blast
ultimately obtain z where zin: "z ∈ closure S" and z: "⋀y. y ∈ closure S ⟹ (cmod ∘ f) y ≤ (cmod ∘ f) z"
using continuous_attains_sup [of "closure S" "norm o f"] ‹ξ ∈ S› by auto
then consider "z ∈ frontier S" | "z ∈ interior S" using frontier_def by auto
then have "norm(f z) ≤ B"
proof cases
case 1 then show ?thesis using leB by blast
next
case 2
have zin: "z ∈ connected_component_set (interior S) z"
by (simp add: 2)
have "f constant_on (connected_component_set (interior S) z)"
apply (rule maximum_modulus_principle [OF _ _ _ _ _ zin])
apply (metis connected_component_subset holf holomorphic_on_subset)
apply (simp_all add: open_connected_component)
by (metis closure_subset comp_eq_dest_lhs  interior_subset subsetCE z connected_component_in)
then obtain c where c: "⋀w. w ∈ connected_component_set (interior S) z ⟹ f w = c"
by (auto simp: constant_on_def)
have "f ` closure(connected_component_set (interior S) z) ⊆ {c}"
apply (rule image_closure_subset)
apply (meson closure_mono connected_component_subset contf continuous_on_subset interior_subset)
using c
apply auto
done
then have cc: "⋀w. w ∈ closure(connected_component_set (interior S) z) ⟹ f w = c" by blast
have "frontier(connected_component_set (interior S) z) ≠ {}"
apply (simp add: frontier_eq_empty)
by (metis "2" bos bounded_interior connected_component_eq_UNIV connected_component_refl not_bounded_UNIV)
then obtain w where w: "w ∈ frontier(connected_component_set (interior S) z)"
by auto
then have "norm (f z) = norm (f w)"  by (simp add: "2" c cc frontier_def)
also have "... ≤ B"
apply (rule leB)
using w
using frontier_interior_subset frontier_of_connected_component_subset by blast
finally show ?thesis .
qed
then show ?thesis
using z ‹ξ ∈ S› closure_subset by fastforce
qed

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

subsection%unimportant ‹Factoring out a zero according to its order›

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

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

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

lemma holomorphic_factor_zero_nonconstant:
assumes holf: "f holomorphic_on S" and S: "open S" "connected S"
and "ξ ∈ S" "f ξ = 0"
and nonconst: "~ f constant_on S"
obtains g r n
where "0 < n"  "0 < r"  "ball ξ r ⊆ S"
"g holomorphic_on ball ξ r"
"⋀w. w ∈ ball ξ r ⟹ f w = (w - ξ)^n * g w"
"⋀w. w ∈ ball ξ r ⟹ g w ≠ 0"
proof (cases "∀n>0. (deriv ^^ n) f ξ = 0")
case True then show ?thesis
using holomorphic_fun_eq_const_on_connected [OF holf S _ ‹ξ ∈ S›] nonconst by (simp add: constant_on_def)
next
case False
then obtain n0 where "n0 > 0" and n0: "(deriv ^^ n0) f ξ ≠ 0" by blast
obtain r0 where "r0 > 0" "ball ξ r0 ⊆ S" using S openE ‹ξ ∈ S› by auto
define n where "n ≡ LEAST n. (deriv ^^ n) f ξ ≠ 0"
have n_ne: "(deriv ^^ n) f ξ ≠ 0"
by (rule def_LeastI [OF n_def]) (rule n0)
then have "0 < n" using ‹f ξ = 0›
using funpow_0 by fastforce
have n_min: "⋀k. k < n ⟹ (deriv ^^ k) f ξ = 0"
using def_Least_le [OF n_def] not_le by blast
then obtain g r1
where  "0 < r1" "g holomorphic_on ball ξ r1"
"⋀w. w ∈ ball ξ r1 ⟹ f w = (w - ξ) ^ n * g w"
"⋀w. w ∈ ball ξ r1 ⟹ g w ≠ 0"
by (auto intro: holomorphic_factor_order_of_zero [OF holf ‹open S› ‹ξ ∈ S› ‹n > 0› n_ne] simp: ‹f ξ = 0›)
then show ?thesis
apply (rule_tac g=g and r="min r0 r1" and n=n in that)
using ‹0 < n› ‹0 < r0› ‹0 < r1› ‹ball ξ r0 ⊆ S›
apply (auto simp: subset_ball intro: holomorphic_on_subset)
done
qed

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

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

lemma pole_at_infinity:
assumes holf: "f holomorphic_on UNIV" and lim: "((inverse o f) ⤏ l) at_infinity"
obtains a n where "⋀z. f z = (∑i≤n. a i * z^i)"
proof (cases "l = 0")
case False
with tendsto_inverse [OF lim] show ?thesis
apply (rule_tac a="(λn. inverse l)" and n=0 in that)
apply (simp add: Liouville_weak [OF holf, of "inverse l"])
done
next
case True
then have [simp]: "l = 0" .
show ?thesis
proof (cases "∃r. 0 < r ∧ (∀z ∈ ball 0 r - {0}. f(inverse z) ≠ 0)")
case True
then obtain r where "0 < r" and r: "⋀z. z ∈ ball 0 r - {0} ⟹ f(inverse z) ≠ 0"
by auto
have 1: "inverse ∘ f ∘ inverse holomorphic_on ball 0 r - {0}"
by (rule holomorphic_on_compose holomorphic_intros holomorphic_on_subset [OF holf] | force simp: r)+
have 2: "0 ∈ interior (ball 0 r)"
using ‹0 < r› by simp
have "∃B. 0<B ∧ eventually (λz. cmod ((inverse ∘ f ∘ inverse) z) ≤ B) (at 0)"
apply (rule exI [where x=1])
apply simp
using tendstoD [OF lim [unfolded lim_at_infinity_0] zero_less_one]
apply (rule eventually_mono)
apply (simp add: dist_norm)
done
with holomorphic_on_extend_bounded [OF 1 2]
obtain g where holg: "g holomorphic_on ball 0 r"
and geq: "⋀z. z ∈ ball 0 r - {0} ⟹ g z = (inverse ∘ f ∘ inverse) z"
by meson
have ifi0: "(inverse ∘ f ∘ inverse) ─0→ 0"
using ‹l = 0› lim lim_at_infinity_0 by blast
have g2g0: "g ─0→ g 0"
using ‹0 < r› centre_in_ball continuous_at continuous_on_eq_continuous_at holg
by (blast intro: holomorphic_on_imp_continuous_on)
have g2g1: "g ─0→ 0"
apply (rule Lim_transform_within_open [OF ifi0 open_ball [of 0 r]])
using ‹0 < r› by (auto simp: geq)
have [simp]: "g 0 = 0"
by (rule tendsto_unique [OF _ g2g0 g2g1]) simp
have "ball 0 r - {0::complex} ≠ {}"
using ‹0 < r›
apply (clarsimp simp: ball_def dist_norm)
apply (drule_tac c="of_real r/2" in subsetD, auto)
done
then obtain w::complex where "w ≠ 0" and w: "norm w < r" by force
then have "g w ≠ 0" by (simp add: geq r)
obtain B n e where "0 < B" "0 < e" "e ≤ r"
and leg: "⋀w. norm w < e ⟹ B * cmod w ^ n ≤ cmod (g w)"
apply (rule holomorphic_lower_bound_difference [OF holg open_ball connected_ball, of 0 w])
using ‹0 < r› w ‹g w ≠ 0› by (auto simp: ball_subset_ball_iff)
have "cmod (f z) ≤ cmod z ^ n / B" if "2/e ≤ cmod z" for z
proof -
have ize: "inverse z ∈ ball 0 e - {0}" using that ‹0 < e›
by (auto simp: norm_divide divide_simps algebra_simps)
then have [simp]: "z ≠ 0" and izr: "inverse z ∈ ball 0 r - {0}" using  ‹e ≤ r›
by auto
then have [simp]: "f z ≠ 0"
using r [of "inverse z"] by simp
have [simp]: "f z = inverse (g (inverse z))"
using izr geq [of "inverse z"] by simp
show ?thesis using ize leg [of "inverse z"]  ‹0 < B›  ‹0 < e›
by (simp add: divide_simps norm_divide algebra_simps)
qed
then show ?thesis
apply (rule_tac a = "λk. (deriv ^^ k) f 0 / (fact k)" and n=n in that)
apply (rule_tac A = "2/e" and B = "1/B" in Liouville_polynomial [OF holf], simp)
done
next
case False
then have fi0: "⋀r. r > 0 ⟹ ∃z∈ball 0 r - {0}. f (inverse z) = 0"
by simp
have fz0: "f z = 0" if "0 < r" and lt1: "⋀x. x ≠ 0 ⟹ cmod x < r ⟹ inverse (cmod (f (inverse x))) < 1"
for z r
proof -
have f0: "(f ⤏ 0) at_infinity"
proof -
have DIM_complex[intro]: "2 ≤ DIM(complex)"  ― ‹should not be necessary!›
by simp
have "continuous_on (inverse ` (ball 0 r - {0})) f"
using continuous_on_subset holf holomorphic_on_imp_continuous_on by blast
then have "connected ((f ∘ inverse) ` (ball 0 r - {0}))"
apply (intro connected_continuous_image continuous_intros)
apply (force intro: connected_punctured_ball)+
done
then have "⟦w ≠ 0; cmod w < r⟧ ⟹ f (inverse w) = 0" for w
apply (rule disjE [OF connected_closedD [where A = "{0}" and B = "- ball 0 1"]], auto)
apply (metis (mono_tags, hide_lams) not_less_iff_gr_or_eq one_less_inverse lt1 zero_less_norm_iff)
using False ‹0 < r› apply fastforce
by (metis (no_types, hide_lams) Compl_iff IntI comp_apply empty_iff image_eqI insert_Diff_single insert_iff mem_ball_0 not_less_iff_gr_or_eq one_less_inverse that(2) zero_less_norm_iff)
then show ?thesis
apply (simp add: lim_at_infinity_0)
apply (rule Lim_eventually)
apply (simp add: eventually_at)
apply (rule_tac x=r in exI)
apply (simp add: ‹0 < r› dist_norm)
done
qed
obtain w where "w ∈ ball 0 r - {0}" and "f (inverse w) = 0"
using False ‹0 < r› by blast
then show ?thesis
by (auto simp: f0 Liouville_weak [OF holf, of 0])
qed
show ?thesis
apply (rule that [of "λn. 0" 0])
using lim [unfolded lim_at_infinity_0]
apply (simp add: Lim_at dist_norm norm_inverse)
apply (drule_tac x=1 in spec)
using fz0 apply auto
done
qed
qed

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

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

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

lemma proper_map_polyfun_eq:
assumes "f holomorphic_on UNIV"
shows "(∀k. compact k ⟶ compact {z. f z ∈ k}) ⟷
(∃c n. 0 < n ∧ (c n ≠ 0) ∧ f = (λz. ∑i≤n. c i * z^i))"
(is "?lhs = ?rhs")
proof
assume compf [rule_format]: ?lhs
have 2: "∃k. 0 < k ∧ a k ≠ 0 ∧ f = (λz. ∑i ≤ k. a i * z ^ i)"
if "⋀z. f z = (∑i≤n. a i * z ^ i)" for a n
proof (cases "∀i≤n. 0<i ⟶ a i = 0")
case True
then have [simp]: "⋀z. f z = a 0"
by (simp add: that sum_atMost_shift)
have False using compf [of "{a 0}"] by simp
then show ?thesis ..
next
case False
then obtain k where k: "0 < k" "k≤n" "a k ≠ 0" by force
define m where "m = (GREATEST k. k≤n ∧ a k ≠ 0)"
have m: "m≤n ∧ a m ≠ 0"
unfolding m_def
apply (rule GreatestI_nat [where b = n])
using k apply auto
done
have [simp]: "a i = 0" if "m < i" "i ≤ n" for i
using Greatest_le_nat [where b = "n" and P = "λk. k≤n ∧ a k ≠ 0"]
using m_def not_le that by auto
have "k ≤ m"
unfolding m_def
apply (rule Greatest_le_nat [where b = "n"])
using k apply auto
done
with k m show ?thesis
by (rule_tac x=m in exI) (auto simp: that comm_monoid_add_class.sum.mono_neutral_right)
qed
have "((inverse ∘ f) ⤏ 0) at_infinity"
proof (rule Lim_at_infinityI)
fix e::real assume "0 < e"
with compf [of "cball 0 (inverse e)"]
show "∃B. ∀x. B ≤ cmod x ⟶ dist ((inverse ∘ f) x) 0 ≤ e"
apply simp
apply (clarsimp simp add: compact_eq_bounded_closed bounded_pos norm_inverse)
apply (rule_tac x="b+1" in exI)
apply (metis inverse_inverse_eq less_add_same_cancel2 less_imp_inverse_less add.commute not_le not_less_iff_gr_or_eq order_trans zero_less_one)
done
qed
then show ?rhs
apply (rule pole_at_infinity [OF assms])
using 2 apply blast
done
next
assume ?rhs
then obtain c n where "0 < n" "c n ≠ 0" "f = (λz. ∑i≤n. c i * z ^ i)" by blast
then have "compact {z. f z ∈ k}" if "compact k" for k
by (auto intro: proper_map_polyfun_univ [OF that])
then show ?lhs by blast
qed

subsection ‹Relating invertibility and nonvanishing of derivative›

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

lemma has_complex_derivative_locally_invertible:
assumes holf: "f holomorphic_on S"
and S: "ξ ∈ S" "open S"
and dnz: "deriv f ξ ≠ 0"
obtains r where "r > 0" "ball ξ r ⊆ S" "open (f `  (ball ξ r))" "inj_on f (ball ξ r)"
proof -
obtain r where "r > 0" "ball ξ r ⊆ S" "inj_on f (ball ξ r)"
by (blast intro: that has_complex_derivative_locally_injective [OF assms])
then have ξ: "ξ ∈ ball ξ r" by simp
then have nc: "~ f constant_on ball ξ r"
using ‹inj_on f (ball ξ r)› injective_not_constant by fastforce
have holf': "f holomorphic_on ball ξ r"
using ‹ball ξ r ⊆ S› holf holomorphic_on_subset by blast
have "open (f ` ball ξ r)"
apply (rule open_mapping_thm [OF holf'])
using nc apply auto
done
then show ?thesis
using ‹0 < r› ‹ball ξ r ⊆ S› ‹inj_on f (ball ξ r)› that  by blast
qed

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

text‹Hence a nice clean inverse function theorem›

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

subsection‹The Schwarz Lemma›

lemma Schwarz1:
assumes holf: "f holomorphic_on S"
and contf: "continuous_on (closure S) f"
and S: "open S" "connected S"
and boS: "bounded S"
and "S ≠ {}"
obtains w where "w ∈ frontier S"
"⋀z. z ∈ closure S ⟹ norm (f z) ≤ norm (f w)"
proof -
have connf: "continuous_on (closure S) (norm o f)"
using contf continuous_on_compose continuous_on_norm_id by blast
have coc: "compact (closure S)"
by (simp add: ‹bounded S› bounded_closure compact_eq_bounded_closed)
then obtain x where x: "x ∈ closure S" and xmax: "⋀z. z ∈ closure S ⟹ norm(f z) ≤ norm(f x)"
apply (rule bexE [OF continuous_attains_sup [OF _ _ connf]])
using ‹S ≠ {}› apply auto
done
then show ?thesis
proof (cases "x ∈ frontier S")
case True
then show ?thesis using that xmax by blast
next
case False
then have "x ∈ S"
using ‹open S› frontier_def interior_eq x by auto
then have "f constant_on S"
apply (rule maximum_modulus_principle [OF holf S ‹open S› order_refl])
using closure_subset apply (blast intro: xmax)
done
then have "f constant_on (closure S)"
by (rule constant_on_closureI [OF _ contf])
then obtain c where c: "⋀x. x ∈ closure S ⟹ f x = c"
by (meson constant_on_def)
obtain w where "w ∈ frontier S"
by (metis coc all_not_in_conv assms(6) closure_UNIV frontier_eq_empty not_compact_UNIV)
then show ?thesis
by (simp add: c frontier_def that)
qed
qed

lemma Schwarz2:
"⟦f holomorphic_on ball 0 r;
0 < s; ball w s ⊆ ball 0 r;
⋀z. norm (w-z) < s ⟹ norm(f z) ≤ norm(f w)⟧
⟹ f constant_on ball 0 r"
by (rule maximum_modulus_principle [where U = "ball w s" and ξ = w]) (simp_all add: dist_norm)

lemma Schwarz3:
assumes holf: "f holomorphic_on (ball 0 r)" and [simp]: "f 0 = 0"
obtains h where "h holomorphic_on (ball 0 r)" and "⋀z. norm z < r ⟹ f z = z * (h z)" and "deriv f 0 = h 0"
proof -
define h where "h z = (if z = 0 then deriv f 0 else f z / z)" for z
have d0: "deriv f 0 = h 0"
by (simp add: h_def)
moreover have "h holomorphic_on (ball 0 r)"
by (rule pole_theorem_open_0 [OF holf, of 0]) (auto simp: h_def)
moreover have "norm z < r ⟹ f z = z * h z" for z
by (simp add: h_def)
ultimately show ?thesis
using that by blast
qed

proposition Schwarz_Lemma:
assumes holf: "f holomorphic_on (ball 0 1)" and [simp]: "f 0 = 0"
and no: "⋀z. norm z < 1 ⟹ norm (f z) < 1"
and ξ: "norm ξ < 1"
shows "norm (f ξ) ≤ norm ξ" and "norm(deriv f 0) ≤ 1"
and "((∃z. norm z < 1 ∧ z ≠ 0 ∧ norm(f z) = norm z)
∨ norm(deriv f 0) = 1)
⟹ ∃α. (∀z. norm z < 1 ⟶ f z = α * z) ∧ norm α = 1"
(is "?P ⟹ ?Q")
proof -
obtain h where holh: "h holomorphic_on (ball 0 1)"
and fz_eq: "⋀z. norm z < 1 ⟹ f z = z * (h z)" and df0: "deriv f 0 = h 0"
by (rule Schwarz3 [OF holf]) auto
have noh_le: "norm (h z) ≤ 1" if z: "norm z < 1" for z
proof -
have "norm (h z) < a" if a: "1 < a" for a
proof -
have "max (inverse a) (norm z) < 1"
using z a by (simp_all add: inverse_less_1_iff)
then obtain r where r: "max (inverse a) (norm z) < r" and "r < 1"
using Rats_dense_in_real by blast
then have nzr: "norm z < r" and ira: "inverse r < a"
using z a less_imp_inverse_less by force+
then have "0 < r"
by (meson norm_not_less_zero not_le order.strict_trans2)
have holh': "h holomorphic_on ball 0 r"
by (meson holh ‹r < 1› holomorphic_on_subset less_eq_real_def subset_ball)
have conth': "continuous_on (cball 0 r) h"
by (meson ‹r < 1› dual_order.trans holh holomorphic_on_imp_continuous_on holomorphic_on_subset mem_ball_0 mem_cball_0 not_less subsetI)
obtain w where w: "norm w = r" and lenw: "⋀z. norm z < r ⟹ norm(h z) ≤ norm(h w)"
apply (rule Schwarz1 [OF holh']) using conth' ‹0 < r› by auto
have "h w = f w / w" using fz_eq ‹r < 1› nzr w by auto
then have "cmod (h z) < inverse r"
by (metis ‹0 < r› ‹r < 1› divide_strict_right_mono inverse_eq_divide
le_less_trans lenw no norm_divide nzr w)
then show ?thesis using ira by linarith
qed
then show "norm (h z) ≤ 1"
using not_le by blast
qed
show "cmod (f ξ) ≤ cmod ξ"
proof (cases "ξ = 0")
case True then show ?thesis by auto
next
case False
then show ?thesis
by (simp add: noh_le fz_eq ξ mult_left_le norm_mult)
qed
show no_df0: "norm(deriv f 0) ≤ 1"
by (simp add: ‹⋀z. cmod z < 1 ⟹ cmod (h z) ≤ 1› df0)
show "?Q" if "?P"
using that
proof
assume "∃z. cmod z < 1 ∧ z ≠ 0 ∧ cmod (f z) = cmod z"
then obtain γ where γ: "cmod γ < 1" "γ ≠ 0" "cmod (f γ) = cmod γ" by blast
then have [simp]: "norm (h γ) = 1"
by (simp add: fz_eq norm_mult)
have "ball γ (1 - cmod γ) ⊆ ball 0 1"
by (simp add: ball_subset_ball_iff)
moreover have "⋀z. cmod (γ - z) < 1 - cmod γ ⟹ cmod (h z) ≤ cmod (h γ)"
apply (simp add: algebra_simps)
by (metis add_diff_cancel_left' diff_diff_eq2 le_less_trans noh_le norm_triangle_ineq4)
ultimately obtain c where c: "⋀z. norm z < 1 ⟹ h z = c"
using Schwarz2 [OF holh, of "1 - norm γ" γ, unfolded constant_on_def] γ by auto
then have "norm c = 1"
using γ by force
with c show ?thesis
using fz_eq by auto
next
assume [simp]: "cmod (deriv f 0) = 1"
then obtain c where c: "⋀z. norm z < 1 ⟹ h z = c"
using Schwarz2 [OF holh zero_less_one, of 0, unfolded constant_on_def] df0 noh_le
by auto
moreover have "norm c = 1"  using df0 c by auto
ultimately show ?thesis
using fz_eq by auto
qed
qed

corollary Schwarz_Lemma':
assumes holf: "f holomorphic_on (ball 0 1)" and [simp]: "f 0 = 0"
and no: "⋀z. norm z < 1 ⟹ norm (f z) < 1"
shows "((∀ξ. norm ξ < 1 ⟶ norm (f ξ) ≤ norm ξ)
∧ norm(deriv f 0) ≤ 1)
∧ (((∃z. norm z < 1 ∧ z ≠ 0 ∧ norm(f z) = norm z)
∨ norm(deriv f 0) = 1)
⟶ (∃α. (∀z. norm z < 1 ⟶ f z = α * z) ∧ norm α = 1))"
using Schwarz_Lemma [OF assms]
by (metis (no_types) norm_eq_zero zero_less_one)

subsection‹The Schwarz reflection principle›

lemma hol_pal_lem0:
assumes "d ∙ a ≤ k" "k ≤ d ∙ b"
obtains c where
"c ∈ closed_segment a b" "d ∙ c = k"
"⋀z. z ∈ closed_segment a c ⟹ d ∙ z ≤ k"
"⋀z. z ∈ closed_segment c b ⟹ k ≤ d ∙ z"
proof -
obtain c where cin: "c ∈ closed_segment a b" and keq: "k = d ∙ c"
using connected_ivt_hyperplane [of "closed_segment a b" a b d k]
by (auto simp: assms)
have "closed_segment a c ⊆ {z. d ∙ z ≤ k}"  "closed_segment c b ⊆ {z. k ≤ d ∙ z}"
unfolding segment_convex_hull using assms keq
by (auto simp: convex_halfspace_le convex_halfspace_ge hull_minimal)
then show ?thesis using cin that by fastforce
qed

lemma hol_pal_lem1:
assumes "convex S" "open S"
and abc: "a ∈ S" "b ∈ S" "c ∈ S"
"d ≠ 0" and lek: "d ∙ a ≤ k" "d ∙ b ≤ k" "d ∙ c ≤ k"
and holf1: "f holomorphic_on {z. z ∈ S ∧ d ∙ z < k}"
and contf: "continuous_on S f"
shows "contour_integral (linepath a b) f +
contour_integral (linepath b c) f +
contour_integral (linepath c a) f = 0"
proof -
have "interior (convex hull {a, b, c}) ⊆ interior(S ∩ {x. d ∙ x ≤ k})"
apply (rule interior_mono)
apply (rule hull_minimal)
apply (simp add: abc lek)
apply (rule convex_Int [OF ‹convex S› convex_halfspace_le])
done
also have "... ⊆ {z ∈ S. d ∙ z < k}"
by (force simp: interior_open [OF ‹open S›] ‹d ≠ 0›)
finally have *: "interior (convex hull {a, b, c}) ⊆ {z ∈ S. d ∙ z < k}" .
have "continuous_on (convex hull {a,b,c}) f"
using ‹convex S› contf abc continuous_on_subset subset_hull
by fastforce
moreover have "f holomorphic_on interior (convex hull {a,b,c})"
by (rule holomorphic_on_subset [OF holf1 *])
ultimately show ?thesis
using Cauchy_theorem_triangle_interior has_chain_integral_chain_integral3
by blast
qed

lemma hol_pal_lem2:
assumes S: "convex S" "open S"
and abc: "a ∈ S" "b ∈ S" "c ∈ S"
and "d ≠ 0" and lek: "d ∙ a ≤ k" "d ∙ b ≤ k"
and holf1: "f holomorphic_on {z. z ∈ S ∧ d ∙ z < k}"
and holf2: "f holomorphic_on {z. z ∈ S ∧ k < d ∙ z}"
and contf: "continuous_on S f"
shows "contour_integral (linepath a b) f +
contour_integral (linepath b c) f +
contour_integral (linepath c a) f = 0"
proof (cases "d ∙ c ≤ k")
case True show ?thesis
by (rule hol_pal_lem1 [OF S abc ‹d ≠ 0› lek True holf1 contf])
next
case False
then have "d ∙ c > k" by force
obtain a' where a': "a' ∈ closed_segment b c" and "d ∙ a' = k"
and ba': "⋀z. z ∈ closed_segment b a' ⟹ d ∙ z ≤ k"
and a'c: "⋀z. z ∈ closed_segment a' c ⟹ k ≤ d ∙ z"
apply (rule hol_pal_lem0 [of d b k c, OF ‹d ∙ b ≤ k›])
using False by auto
obtain b' where b': "b' ∈ closed_segment a c" and "d ∙ b' = k"
and ab': "⋀z. z ∈ closed_segment a b' ⟹ d ∙ z ≤ k"
and b'c: "⋀z. z ∈ closed_segment b' c ⟹ k ≤ d ∙ z"
apply (rule hol_pal_lem0 [of d a k c, OF ‹d ∙ a ≤ k›])
using False by auto
have a'b': "a' ∈ S ∧ b' ∈ S"
using a' abc b' convex_contains_segment ‹convex S› by auto
have "continuous_on (closed_segment c a) f"
by (meson abc contf continuous_on_subset convex_contains_segment ‹convex S›)
then have 1: "contour_integral (linepath c a) f =
contour_integral (linepath c b') f + contour_integral (linepath b' a) f"
apply (rule contour_integral_split_linepath)
using b' by (simp add: closed_segment_commute)
have "continuous_on (closed_segment b c) f"
by (meson abc contf continuous_on_subset convex_contains_segment ‹convex S›)
then have 2: "contour_integral (linepath b c) f =
contour_integral (linepath b a') f + contour_integral (linepath a' c) f"
by (rule contour_integral_split_linepath [OF _ a'])
have 3: "contour_integral (reversepath (linepath b' a')) f =
- contour_integral (linepath b' a') f"
by (rule contour_integral_reversepath [OF valid_path_linepath])
have fcd_le: "f field_differentiable at x"
if "x ∈ interior S ∧ x ∈ interior {x. d ∙ x ≤ k}" for x
proof -
have "f holomorphic_on S ∩ {c. d ∙ c < k}"
by (metis (no_types) Collect_conj_eq Collect_mem_eq holf1)
then have "∃C D. x ∈ interior C ∩ interior D ∧ f holomorphic_on interior C ∩ interior D"
using that
by (metis Collect_mem_eq Int_Collect ‹d ≠ 0› interior_halfspace_le interior_open ‹open S›)
then show "f field_differentiable at x"
by (metis at_within_interior holomorphic_on_def interior_Int interior_interior)
qed
have ab_le: "⋀x. x ∈ closed_segment a b ⟹ d ∙ x ≤ k"
proof -
fix x :: complex
assume "x ∈ closed_segment a b"
then have "⋀C. x ∈ C ∨ b ∉ C ∨ a ∉ C ∨ ¬ convex C"
by (meson contra_subsetD convex_contains_segment)
then show "d ∙ x ≤ k"
by (metis lek convex_halfspace_le mem_Collect_eq)
qed
have "continuous_on (S ∩ {x. d ∙ x ≤ k}) f" using contf
by (simp add: continuous_on_subset)
then have "(f has_contour_integral 0)
(linepath a b +++ linepath b a' +++ linepath a' b' +++ linepath b' a)"
apply (rule Cauchy_theorem_convex [where K = "{}"])
apply (simp_all add: path_image_join convex_Int convex_halfspace_le ‹convex S› fcd_le ab_le
closed_segment_subset abc a'b' ba')
by (metis ‹d ∙ a' = k› ‹d ∙ b' = k› convex_contains_segment convex_halfspace_le lek(1) mem_Collect_eq order_refl)
then have 4: "contour_integral (linepath a b) f +
contour_integral (linepath b a') f +
contour_integral (linepath a' b') f +
contour_integral (linepath b' a) f = 0"
by (rule has_chain_integral_chain_integral4)
have fcd_ge: "f field_differentiable at x"
if "x ∈ interior S ∧ x ∈ interior {x. k ≤ d ∙ x}" for x
proof -
have f2: "f holomorphic_on S ∩ {c. k < d ∙ c}"
by (metis (full_types) Collect_conj_eq Collect_mem_eq holf2)
have f3: "interior S = S"
by (simp add: interior_open ‹open S›)
then have "x ∈ S ∩ interior {c. k ≤ d ∙ c}"
using that by simp
then show "f field_differentiable at x"
using f3 f2 unfolding holomorphic_on_def
by (metis (no_types) ‹d ≠ 0› at_within_interior interior_Int interior_halfspace_ge interior_interior)
qed
have "continuous_on (S ∩ {x. k ≤ d ∙ x}) f" using contf
by (simp add: continuous_on_subset)
then have "(f has_contour_integral 0) (linepath a' c +++ linepath c b' +++ linepath b' a')"
apply (rule Cauchy_theorem_convex [where K = "{}"])
apply (simp_all add: path_image_join convex_Int convex_halfspace_ge ‹convex S›
fcd_ge closed_segment_subset abc a'b' a'c)
by (metis ‹d ∙ a' = k› b'c closed_segment_commute convex_contains_segment
convex_halfspace_ge ends_in_segment(2) mem_Collect_eq order_refl)
then have 5: "contour_integral (linepath a' c) f + contour_integral (linepath c b') f + contour_integral (linepath b' a') f = 0"
by (rule has_chain_integral_chain_integral3)
show ?thesis
using 1 2 3 4 5 by (metis add.assoc eq_neg_iff_add_eq_0 reversepath_linepath)
qed

lemma hol_pal_lem3:
assumes S: "convex S" "open S"
and abc: "a ∈ S" "b ∈ S" "c ∈ S"
and "d ≠ 0" and lek: "d ∙ a ≤ k"
and holf1: "f holomorphic_on {z. z ∈ S ∧ d ∙ z < k}"
and holf2: "f holomorphic_on {z. z ∈ S ∧ k < d ∙ z}"
and contf: "continuous_on S f"
shows "contour_integral (linepath a b) f +
contour_integral (linepath b c) f +
contour_integral (linepath c a) f = 0"
proof (cases "d ∙ b ≤ k")
case True show ?thesis
by (rule hol_pal_lem2 [OF S abc ‹d ≠ 0› lek True holf1 holf2 contf])
next
case False
show ?thesis
proof (cases "d ∙ c ≤ k")
case True
have "contour_integral (linepath c a) f +
contour_integral (linepath a b) f +
contour_integral (linepath b c) f = 0"
by (rule hol_pal_lem2 [OF S ‹c ∈ S› ‹a ∈ S› ‹b ∈ S› ‹d ≠ 0› ‹d ∙ c ≤ k› lek holf1 holf2 contf])
then show ?thesis
by (simp add: algebra_simps)
next
case False
have "contour_integral (linepath b c) f +
contour_integral (linepath c a) f +
contour_integral (linepath a b) f = 0"
apply (rule hol_pal_lem2 [OF S ‹b ∈ S› ‹c ∈ S› ‹a ∈ S›, of "-d" "-k"])
using ‹d ≠ 0› ‹¬ d ∙ b ≤ k› False by (simp_all add: holf1 holf2 contf)
then show ?thesis
by (simp add: algebra_simps)
qed
qed

lemma hol_pal_lem4:
assumes S: "convex S" "open S"
and abc: "a ∈ S" "b ∈ S" "c ∈ S" and "d ≠ 0"
and holf1: "f holomorphic_on {z. z ∈ S ∧ d ∙ z < k}"
and holf2: "f holomorphic_on {z. z ∈ S ∧ k < d ∙ z}"
and contf: "continuous_on S f"
shows "contour_integral (linepath a b) f +
contour_integral (linepath b c) f +
contour_integral (linepath c a) f = 0"
proof (cases "d ∙ a ≤ k")
case True show ?thesis
by (rule hol_pal_lem3 [OF S abc ‹d ≠ 0› True holf1 holf2 contf])
next
case False
show ?thesis
apply (rule hol_pal_lem3 [OF S abc, of "-d" "-k"])
using ‹d ≠ 0› False by (simp_all add: holf1 holf2 contf)
qed

lemma holomorphic_on_paste_across_line:
assumes S: "open S" and "d ≠ 0"
and holf1: "f holomorphic_on (S ∩ {z. d ∙ z < k})"
and holf2: "f holomorphic_on (S ∩ {z. k < d ∙ z})"
and contf: "continuous_on S f"
shows "f holomorphic_on S"
proof -
have *: "∃t. open t ∧ p ∈ t ∧ continuous_on t f ∧
(∀a b c. convex hull {a, b, c} ⊆ t ⟶
contour_integral (linepath a b) f +
contour_integral (linepath b c) f +
contour_integral (linepath c a) f = 0)"
if "p ∈ S" for p
proof -
obtain e where "e>0" and e: "ball p e ⊆ S"
using ‹p ∈ S› openE S by blast
then have "continuous_on (ball p e) f"
using contf continuous_on_subset by blast
moreover have "f holomorphic_on {z. dist p z < e ∧ d ∙ z < k}"
apply (rule holomorphic_on_subset [OF holf1])
using e by auto
moreover have "f holomorphic_on {z. dist p z < e ∧ k < d ∙ z}"
apply (rule holomorphic_on_subset [OF holf2])
using e by auto
ultimately show ?thesis
apply (rule_tac x="ball p e" in exI)
using ‹e > 0› e ‹d ≠ 0›
apply (simp add:, clarify)
apply (rule hol_pal_lem4 [of "ball p e" _ _ _ d _ k])
apply (auto simp: subset_hull)
done
qed
show ?thesis
by (blast intro: * Morera_local_triangle analytic_imp_holomorphic)
qed

proposition Schwarz_reflection:
assumes "open S" and cnjs: "cnj ` S ⊆ S"
and  holf: "f holomorphic_on (S ∩ {z. 0 < Im z})"
and contf: "continuous_on (S ∩ {z. 0 ≤ Im z}) f"
and f: "⋀z. ⟦z ∈ S; z ∈ ℝ⟧ ⟹ (f z) ∈ ℝ"
shows "(λz. if 0 ≤ Im z then f z else cnj(f(cnj z))) holomorphic_on S"
proof -
have 1: "(λz. if 0 ≤ Im z then f z else cnj (f (cnj z))) holomorphic_on (S ∩ {z. 0 < Im z})"
by (force intro: iffD1 [OF holomorphic_cong [OF refl] holf])
have cont_cfc: "continuous_on (S ∩ {z. Im z ≤ 0}) (cnj o f o cnj)"
apply (intro continuous_intros continuous_on_compose continuous_on_subset [OF contf])
using cnjs apply auto
done
have "cnj ∘ f ∘ cnj field_differentiable at x within S ∩ {z. Im z < 0}"
if "x ∈ S" "Im x < 0" "f field_differentiable at (cnj x) within S ∩ {z. 0 < Im z}" for x
using that
apply (simp add: field_differentiable_def has_field_derivative_iff Lim_within dist_norm, clarify)
apply (rule_tac x="cnj f'" in exI)
apply (elim all_forward ex_forward conj_forward imp_forward asm_rl, clarify)
apply (drule_tac x="cnj xa" in bspec)
using cnjs apply force
apply (metis complex_cnj_cnj complex_cnj_diff complex_cnj_divide complex_mod_cnj)
done
then have hol_cfc: "(cnj o f o cnj) holomorphic_on (S ∩ {z. Im z < 0})"
using holf cnjs
by (force simp: holomorphic_on_def)
have 2: "(λz. if 0 ≤ Im z then f z else cnj (f (cnj z))) holomorphic_on (S ∩ {z. Im z < 0})"
apply (rule iffD1 [OF holomorphic_cong [OF refl]])
using hol_cfc by auto
have [simp]: "(S ∩ {z. 0 ≤ Im z}) ∪ (S ∩ {z. Im z ≤ 0}) = S"
by force
have "continuous_on ((S ∩ {z. 0 ≤ Im z}) ∪ (S ∩ {z. Im z ≤ 0}))
(λz. if 0 ≤ Im z then f z else cnj (f (cnj z)))"
apply (rule continuous_on_cases_local)
using cont_cfc contf
apply (simp_all add: closedin_closed_Int closed_halfspace_Im_le closed_halfspace_Im_ge)
using f Reals_cnj_iff complex_is_Real_iff apply auto
done
then have 3: "continuous_on S (λz. if 0 ≤ Im z then f z else cnj (f (cnj z)))"
by force
show ?thesis
apply (rule holomorphic_on_paste_across_line [OF ‹open S›, of "- 𝗂" _ 0])
using 1 2 3
apply auto
done
qed

subsection‹Bloch's theorem›

lemma Bloch_lemma_0:
assumes holf: "f holomorphic_on cball 0 r" and "0 < r"
and [simp]: "f 0 = 0"
and le: "⋀z. norm z < r ⟹ norm(deriv f z) ≤ 2 * norm(deriv f 0)"
shows "ball 0 ((3 - 2 * sqrt 2) * r * norm(deriv f 0)) ⊆ f ` ball 0 r"
proof -
have "sqrt 2 < 3/2"
by (rule real_less_lsqrt) (auto simp: power2_eq_square)
then have sq3: "0 < 3 - 2 * sqrt 2" by simp
show ?thesis
proof (cases "deriv f 0 = 0")
case True then show ?thesis by simp
next
case False
define C where "C = 2 * norm(deriv f 0)"
have "0 < C" using False by (simp add: C_def)
have holf': "f holomorphic_on ball 0 r" using holf
using ball_subset_cball holomorphic_on_subset by blast
then have holdf': "deriv f holomorphic_on ball 0 r"
by (rule holomorphic_deriv [OF _ open_ball])
have "Le1": "norm(deriv f z - deriv f 0) ≤ norm z / (r - norm z) * C"
if "norm z < r" for z
proof -
have T1: "norm(deriv f z - deriv f 0) ≤ norm z / (R - norm z) * C"
if R: "norm z < R" "R < r" for R
proof -
have "0 < R" using R
by (metis less_trans norm_zero zero_less_norm_iff)
have df_le: "⋀x. norm x < r ⟹ norm (deriv f x) ≤ C"
using le by (simp add: C_def)
have hol_df: "deriv f holomorphic_on cball 0 R"
apply (rule holomorphic_on_subset) using R holdf' by auto
have *: "((λw. deriv f w / (w - z)) has_contour_integral 2 * pi * 𝗂 * deriv f z) (circlepath 0 R)"
if "norm z < R" for z
using ‹0 < R› that Cauchy_integral_formula_convex_simple [OF convex_cball hol_df, of _ "circlepath 0 R"]
by (force simp: winding_number_circlepath)
have **: "((λx. deriv f x / (x - z) - deriv f x / x) has_contour_integral
of_real (2 * pi) * 𝗂 * (deriv f z - deriv f 0))
(circlepath 0 R)"
using has_contour_integral_diff [OF * [of z] * [of 0]] ‹0 < R› that
by (simp add: algebra_simps)
have [simp]: "⋀x. norm x = R ⟹ x ≠ z"  using that(1) by blast
have "norm (deriv f x / (x - z) - deriv f x / x)
≤ C * norm z / (R * (R - norm z))"
if "norm x = R" for x
proof -
have [simp]: "norm (deriv f x * x - deriv f x * (x - z)) =
norm (deriv f x) * norm z"
by (simp add: norm_mult right_diff_distrib')
show ?thesis
using  ‹0 < R› ‹0 < C› R that
apply (simp add: norm_mult norm_divide divide_simps)
using df_le norm_triangle_ineq2 ‹0 < C› apply (auto intro!: mult_mono)
done
qed
then show ?thesis
using has_contour_integral_bound_circlepath
[OF **, of "C * norm z/(R*(R - norm z))"]
‹0 < R› ‹0 < C› R
apply (simp add: norm_mult norm_divide)
apply (simp add: divide_simps mult.commute)
done
qed
obtain r' where r': "norm z < r'" "r' < r"
using Rats_dense_in_real [of "norm z" r] ‹norm z < r› by blast
then have [simp]: "closure {r'<..<r} = {r'..r}" by simp
show ?thesis
apply (rule continuous_ge_on_closure
[where f = "λr. norm z / (r - norm z) * C" and s = "{r'<..<r}",
OF _ _ T1])
apply (intro continuous_intros)
using that r'
apply (auto simp: not_le)
done
qed
have "*": "(norm z - norm z^2/(r - norm z)) * norm(deriv f 0) ≤ norm(f z)"
if r: "norm z < r" for z
proof -
have 1: "⋀x. x ∈ ball 0 r ⟹
((λz. f z - deriv f 0 * z) has_field_derivative deriv f x - deriv f 0)
(at x within ball 0 r)"
by (rule derivative_eq_intros holomorphic_derivI holf' | simp)+
have 2: "closed_segment 0 z ⊆ ball 0 r"
by (metis ‹0 < r› convex_ball convex_contains_segment dist_self mem_ball mem_ball_0 that)
have 3: "(λt. (norm z)⇧2 * t / (r - norm z) * C) integrable_on {0..1}"
apply (rule integrable_on_cmult_right [where 'b=real, simplified])
apply (rule integrable_on_cdivide [where 'b=real, simplified])
apply (rule integrable_on_cmult_left [where 'b=real, simplified])
apply (rule ident_integrable_on)
done
have 4: "norm (deriv f (x *⇩R z) - deriv f 0) * norm z ≤ norm z * norm z * x * C / (r - norm z)"
if x: "0 ≤ x" "x ≤ 1" for x
proof -
have [simp]: "x * norm z < r"
using r x by (meson le_less_trans mult_le_cancel_right2 norm_not_less_zero)
have "norm (deriv f (x *⇩R z) - deriv f 0) ≤ norm (x *⇩R z) / (r - norm (x *⇩R z)) * C"
apply (rule Le1) using r x ‹0 < r› by simp
also have "... ≤ norm (x *⇩R z) / (r - norm z) * C"
using r x ‹0 < r›
apply (simp add: divide_simps)
by (simp add: ‹0 < C› mult.assoc mult_left_le_one_le ordered_comm_semiring_class.comm_mult_left_mono)
finally have "norm (deriv f (x *⇩R z) - deriv f 0) * norm z ≤ norm (x *⇩R z)  / (r - norm z) * C * norm z"
by (rule mult_right_mono) simp
with x show ?thesis by (simp add: algebra_simps)
qed
have le_norm: "abc ≤ norm d - e ⟹ norm(f - d) ≤ e ⟹ abc ≤ norm f" for abc d e and f::complex
by (metis add_diff_cancel_left' add_diff_eq diff_left_mono norm_diff_ineq order_trans)
have "norm (integral {0..1} (λx. (deriv f (x *⇩R z) - deriv f 0) * z))
≤ integral {0..1} (λt. (norm z)⇧2 * t / (r - norm z) * C)"
apply (rule integral_norm_bound_integral)
using contour_integral_primitive [OF 1, of "linepath 0 z"] 2
apply (simp add: has_contour_integral_linepath has_integral_integrable_integral)
apply (rule 3)
apply (simp add: norm_mult power2_eq_square 4)
done
then have int_le: "norm (f z - deriv f 0 * z) ≤ (norm z)⇧2 * norm(deriv f 0) / ((r - norm z))"
using contour_integral_primitive [OF 1, of "linepath 0 z"] 2
apply (simp add: has_contour_integral_linepath has_integral_integrable_integral C_def)
done
show ?thesis
apply (rule le_norm [OF _ int_le])
using ‹norm z < r›
apply (simp add: power2_eq_square divide_simps C_def norm_mult)
proof -
have "norm z * (norm (deriv f 0) * (r - norm z - norm z)) ≤ norm z * (norm (deriv f 0) * (r - norm z) - norm (deriv f 0) * norm z)"
by (simp add: linordered_field_class.sign_simps(38))
then show "(norm z * (r - norm z) - norm z * norm z) * norm (deriv f 0) ≤ norm (deriv f 0) * norm z * (r - norm z) - norm z * norm z * norm (deriv f 0)"
by (simp add: linordered_field_class.sign_simps(38) mult.commute mult.left_commute)
qed
qed
have sq201 [simp]: "0 < (1 - sqrt 2 / 2)" "(1 - sqrt 2 / 2)  < 1"
by (auto simp:  sqrt2_less_2)
have 1: "continuous_on (closure (ball 0 ((1 - sqrt 2 / 2) * r))) f"
apply (rule continuous_on_subset [OF holomorphic_on_imp_continuous_on [OF holf]])
apply (subst closure_ball)
using ‹0 < r› mult_pos_pos sq201
apply (auto simp: cball_subset_cball_iff)
done
have 2: "open (f ` interior (ball 0 ((1 - sqrt 2 / 2) * r)))"
apply (rule open_mapping_thm [OF holf' open_ball connected_ball], force)
using ‹0 < r› mult_pos_pos sq201 apply (simp add: ball_subset_ball_iff)
using False ‹0 < r› centre_in_ball holf' holomorphic_nonconstant by blast
have "ball 0 ((3 - 2 * sqrt 2) * r * norm (deriv f 0)) =
ball (f 0) ((3 - 2 * sqrt 2) * r * norm (deriv f 0))"
by simp
also have "...  ⊆ f ` ball 0 ((1 - sqrt 2 / 2) * r)"
proof -
have 3: "(3 - 2 * sqrt 2) * r * norm (deriv f 0) ≤ norm (f z)"
if "norm z = (1 - sqrt 2 / 2) * r" for z
apply (rule order_trans [OF _ *])
using  ‹0 < r›
apply (simp_all add: field_simps  power2_eq_square that)
apply (simp add: mult.assoc [symmetric])
done
show ?thesis
apply (rule ball_subset_open_map_image [OF 1 2 _ bounded_ball])
using ‹0 < r› sq201 3 apply simp_all
using C_def ‹0 < C› sq3 apply force
done
qed
also have "...  ⊆ f ` ball 0 r"
apply (rule image_subsetI [OF imageI], simp)
apply (erule less_le_trans)
using ‹0 < r› apply (auto simp: field_simps)
done
finally show ?thesis .
qed
qed

lemma Bloch_lemma:
assumes holf: "f holomorphic_on cball a r" and "0 < r"
and le: "⋀z. z ∈ ball a r ⟹ norm(deriv f z) ≤ 2 * norm(deriv f a)"
shows "ball (f a) ((3 - 2 * sqrt 2) * r * norm(deriv f a)) ⊆ f ` ball a r"
proof -
have fz: "(λz. f (a + z)) = f o (λz. (a + z))"
by (simp add: o_def)
have hol0: "(λz. f (a + z)) holomorphic_on cball 0 r"
unfolding fz by (intro holomorphic_intros holf holomorphic_on_compose | simp)+
then have [simp]: "⋀x. norm x < r ⟹ (λz. f (a + z)) field_differentiable at x"
by (metis open_ball at_within_open ball_subset_cball diff_0 dist_norm holomorphic_on_def holomorphic_on_subset mem_ball norm_minus_cancel)
have [simp]: "⋀z. norm z < r ⟹ f field_differentiable at (a + z)"
by (metis holf open_ball add_diff_cancel_left' dist_complex_def holomorphic_on_imp_differentiable_at holomorphic_on_subset interior_cball interior_subset mem_ball norm_minus_commute)
then have [simp]: "f field_differentiable at a"
by (metis add.comm_neutral ‹0 < r› norm_eq_zero)
have hol1: "(λz. f (a + z) - f a) holomorphic_on cball 0 r"
by (intro holomorphic_intros hol0)
then have "ball 0 ((3 - 2 * sqrt 2) * r * norm (deriv (λz. f (a + z) - f a) 0))
⊆ (λz. f (a + z) - f a) ` ball 0 r"
apply (rule Bloch_lemma_0)
apply (simp_all add: ‹0 < r›)
apply (simp add: fz complex_derivative_chain)
apply (simp add: dist_norm le)
done
then show ?thesis
apply clarify
apply (drule_tac c="x - f a" in subsetD)
apply (force simp: fz ‹0 < r› dist_norm complex_derivative_chain field_differentiable_compose)+
done
qed

proposition Bloch_unit:
assumes holf: "f holomorphic_on ball a 1" and [simp]: "deriv f a = 1"
obtains b r where "1/12 < r" and "ball b r ⊆ f ` (ball a 1)"
proof -
define r :: real where "r = 249/256"
have "0 < r" "r < 1" by (auto simp: r_def)
define g where "g z = deriv f z * of_real(r - norm(z - a))" for z
have "deriv f holomorphic_on ball a 1"
by (rule holomorphic_deriv [OF holf open_ball])
then have "continuous_on (ball a 1) (deriv f)"
using holomorphic_on_imp_continuous_on by blast
then have "continuous_on (cball a r) (deriv f)"
by (rule continuous_on_subset) (simp add: cball_subset_ball_iff ‹r < 1›)
then have "continuous_on (cball a r) g"
by (simp add: g_def continuous_intros)
then have 1: "compact (g ` cball a r)"
by (rule compact_continuous_image [OF _ compact_cball])
have 2: "g ` cball a r ≠ {}"
using ‹r > 0› by auto
obtain p where pr: "p ∈ cball a r"
and pge: "⋀y. y ∈ cball a r ⟹ norm (g y) ≤ norm (g p)"
using distance_attains_sup [OF 1 2, of 0] by force
define t where "t = (r - norm(p - a)) / 2"
have "norm (p - a) ≠ r"
using pge [of a] ‹r > 0› by (auto simp: g_def norm_mult)
then have "norm (p - a) < r" using pr
by (simp add: norm_minus_commute dist_norm)
then have "0 < t"
by (simp add: t_def)
have cpt: "cball p t ⊆ ball a r"
using ‹0 < t› by (simp add: cball_subset_ball_iff dist_norm t_def field_simps)
have gen_le_dfp: "norm (deriv f y) * (r - norm (y - a)) / (r - norm (p - a)) ≤ norm (deriv f p)"
if "y ∈ cball a r" for y
proof -
have [simp]: "norm (y - a) ≤ r"
using that by (simp add: dist_norm norm_minus_commute)
have "norm (g y) ≤ norm (g p)"
using pge [OF that] by simp
then have "norm (deriv f y) * abs (r - norm (y - a)) ≤ norm (deriv f p) * abs (r - norm (p - a))"
by (simp only: dist_norm g_def norm_mult norm_of_real)
with that ‹norm (p - a) < r› show ?thesis
by (simp add: dist_norm divide_simps)
qed
have le_norm_dfp: "r / (r - norm (p - a)) ≤ norm (deriv f p)"
using gen_le_dfp [of a] ‹r > 0› by auto
have 1: "f holomorphic_on cball p t"
apply (rule holomorphic_on_subset [OF holf])
using cpt ‹r < 1› order_subst1 subset_ball by auto
have 2: "norm (deriv f z) ≤ 2 * norm (deriv f p)" if "z ∈ ball p t" for z
proof -
have z: "z ∈ cball a r"
by (meson ball_subset_cball subsetD cpt that)
then have "norm(z - a) < r"
by (metis ball_subset_cball contra_subsetD cpt dist_norm mem_ball norm_minus_commute that)
have "norm (deriv f z) * (r - norm (z - a)) / (r - norm (p - a)) ≤ norm (deriv f p)"
using gen_le_dfp [OF z] by simp
with ‹norm (z - a) < r› ‹norm (p - a) < r›
have "norm (deriv f z) ≤ (r - norm (p - a)) / (r - norm (z - a)) * norm (deriv f p)"
by (simp add: field_simps)
also have "... ≤ 2 * norm (deriv f p)"
apply (rule mult_right_mono)
using that ‹norm (p - a) < r› ‹norm(z - a) < r›
apply (simp_all add: field_simps t_def dist_norm [symmetric])
using dist_triangle3 [of z a p] by linarith
finally show ?thesis .
qed
have sqrt2: "sqrt 2 < 2113/1494"
by (rule real_less_lsqrt) (auto simp: power2_eq_square)
then have sq3: "0 < 3 - 2 * sqrt 2" by simp
have "1 / 12 / ((3 - 2 * sqrt 2) / 2) < r"
using sq3 sqrt2 by (auto simp: field_simps r_def)
also have "... ≤ cmod (deriv f p) * (r - cmod (p - a))"
using ‹norm (p - a) < r› le_norm_dfp   by (simp add: pos_divide_le_eq)
finally have "1 / 12 < cmod (deriv f p) * (r - cmod (p - a)) * ((3 - 2 * sqrt 2) / 2)"
using pos_divide_less_eq half_gt_zero_iff sq3 by blast
then have **: "1 / 12 < (3 - 2 * sqrt 2) * t * norm (deriv f p)"
using sq3 by (simp add: mult.commute t_def)
have "ball (f p) ((3 - 2 * sqrt 2) * t * norm (deriv f p)) ⊆ f ` ball p t"
by (rule Bloch_lemma [OF 1 ‹0 < t› 2])
also have "... ⊆ f ` ball a 1"
apply (rule image_mono)
apply (rule order_trans [OF ball_subset_cball])
apply (rule order_trans [OF cpt])
using ‹0 < t› ‹r < 1› apply (simp add: ball_subset_ball_iff dist_norm)
done
finally have "ball (f p) ((3 - 2 * sqrt 2) * t * norm (deriv f p)) ⊆ f ` ball a 1" .
with ** show ?thesis
by (rule that)
qed

theorem Bloch:
assumes holf: "f holomorphic_on ball a r" and "0 < r"
and r': "r' ≤ r * norm (deriv f a) / 12"
obtains b where "ball b r' ⊆ f ` (ball a r)"
proof (cases "deriv f a = 0")
case True with r' show ?thesis
using ball_eq_empty that by fastforce
next
case False
define C where "C = deriv f a"
have "0 < norm C" using False by (simp add: C_def)
have dfa: "f field_differentiable at a"
apply (rule holomorphic_on_imp_differentiable_at [OF holf])
using ‹0 < r› by auto
have fo: "(λz. f (a + of_real r * z)) = f o (λz. (a + of_real r * z))"
by (simp add: o_def)
have holf': "f holomorphic_on (λz. a + complex_of_real r * z) ` ball 0 1"
apply (rule holomorphic_on_subset [OF holf])
using ‹0 < r› apply (force simp: dist_norm norm_mult)
done
have 1: "(λz. f (a + r * z) / (C * r)) holomorphic_on ball 0 1"
apply (rule holomorphic_intros holomorphic_on_compose holf' | simp add: fo)+
using ‹0 < r› by (simp add: C_def False)
have "((λz. f (a + of_real r * z) / (C * of_real r)) has_field_derivative
(deriv f (a + of_real r * z) / C)) (at z)"
if "norm z < 1" for z
proof -
have *: "((λx. f (a + of_real r * x)) has_field_derivative
(deriv f (a + of_real r * z) * of_real r)) (at z)"
apply (simp add: fo)
apply (rule DERIV_chain [OF field_differentiable_derivI])
apply (rule holomorphic_on_imp_differentiable_at [OF holf], simp)
using ‹0 < r› apply (simp add: dist_norm norm_mult that)
apply (rule derivative_eq_intros | simp)+
done
show ?thesis
apply (rule derivative_eq_intros * | simp)+
using ‹0 < r› by (auto simp: C_def False)
qed
have 2: "deriv (λz. f (a + of_real r * z) / (C * of_real r)) 0 = 1"
apply (subst deriv_cdivide_right)
apply (simp add: field_differentiable_def fo)
apply (rule exI)
apply (rule DERIV_chain [OF field_differentiable_derivI])
apply (simp add: dfa)
apply (rule derivative_eq_intros | simp add: C_def False fo)+
using ‹0 < r›
apply (simp add: C_def False fo)
apply (simp add: derivative_intros dfa complex_derivative_chain)
done
have sb1: "( * ) (C * r) ` (λz. f (a + of_real r * z) / (C * r)) ` ball 0 1
⊆ f ` ball a r"
using ‹0 < r› by (auto simp: dist_norm norm_mult C_def False)
have sb2: "ball (C * r * b) r' ⊆ ( * ) (C * r) ` ball b t"
if "1 / 12 < t" for b t
proof -
have *: "r * cmod (deriv f a) / 12 ≤ r * (t * cmod (deriv f a))"
using that ‹0 < r› less_eq_real_def mult.commute mult.right_neutral mult_left_mono norm_ge_zero times_divide_eq_right
by auto
show ?thesis
apply clarify
apply (rule_tac x="x / (C * r)" in image_eqI)
using ‹0 < r›
apply (simp_all add: dist_norm norm_mult norm_divide C_def False field_simps)
apply (erule less_le_trans)
apply (rule order_trans [OF r' *])
done
qed
show ?thesis
apply (rule Bloch_unit [OF 1 2])
apply (rename_tac t)
apply (rule_tac b="(C * of_real r) * b" in that)
apply (drule image_mono [where f = "λz. (C * of_real r) * z"])
using sb1 sb2
apply force
done
qed

corollary Bloch_general:
assumes holf: "f holomorphic_on s" and "a ∈ s"
and tle: "⋀z. z ∈ frontier s ⟹ t ≤ dist a z"
and rle: "r ≤ t * norm(deriv f a) / 12"
obtains b where "ball b r ⊆ f ` s"
proof -
consider "r ≤ 0" | "0 < t * norm(deriv f a) / 12" using rle by force
then show ?thesis
proof cases
case 1 then show ?thesis
by (simp add: ball_empty that)
next
case 2
show ?thesis
proof (cases "deriv f a = 0")
case True then show ?thesis
using rle by (simp add: ball_empty that)
next
case False
then have "t > 0"
using 2 by (force simp: zero_less_mult_iff)
have "~ ball a t ⊆ s ⟹ ball a t ∩ frontier s ≠ {}"
apply (rule connected_Int_frontier [of "ball a t" s], simp_all)
using ‹0 < t› ‹a ∈ s› centre_in_ball apply blast
done
with tle have *: "ball a t ⊆ s" by fastforce
then have 1: "f holomorphic_on ball a t"
using holf using holomorphic_on_subset by blast
show ?thesis
apply (rule Bloch [OF 1 ‹t > 0› rle])
apply (rule_tac b=b in that)
using * apply force
done
qed
qed
qed

subsection ‹Cauchy's residue theorem›

text‹Wenda Li and LC Paulson (2016). A Formal Proof of Cauchy's Residue Theorem.
Interactive Theorem Proving›

definition%important residue :: "(complex ⇒ complex) ⇒ complex ⇒ complex" where
"residue f z = (SOME int. ∃e>0. ∀ε>0. ε<e
⟶ (f has_contour_integral 2*pi* 𝗂 *int) (circlepath z ε))"

lemma Eps_cong:
assumes "⋀x. P x = Q x"
shows   "Eps P = Eps Q"
using ext[of P Q, OF assms] by simp

lemma residue_cong:
assumes eq: "eventually (λz. f z = g z) (at z)" and "z = z'"
shows   "residue f z = residue g z'"
proof -
from assms have eq': "eventually (λz. g z = f z) (at z)"
by (simp add: eq_commute)
let ?P = "λf c e. (∀ε>0. ε < e ⟶
(f has_contour_integral of_real (2 * pi) * 𝗂 * c) (circlepath z ε))"
have "residue f z = residue g z" unfolding residue_def
proof (rule Eps_cong)
fix c :: complex
have "∃e>0. ?P g c e"
if "∃e>0. ?P f c e" and "eventually (λz. f z = g z) (at z)" for f g
proof -
from that(1) obtain e where e: "e > 0" "?P f c e"
by blast
from that(2) obtain e' where e': "e' > 0" "⋀z'. z' ≠ z ⟹ dist z' z < e' ⟹ f z' = g z'"
unfolding eventually_at by blast
have "?P g c (min e e')"
proof (intro allI exI impI, goal_cases)
case (1 ε)
hence "(f has_contour_integral of_real (2 * pi) * 𝗂 * c) (circlepath z ε)"
using e(2) by auto
thus ?case
proof (rule has_contour_integral_eq)
fix z' assume "z' ∈ path_image (circlepath z ε)"
hence "dist z' z < e'" and "z' ≠ z"
using 1 by (auto simp: dist_commute)
with e'(2)[of z'] show "f z' = g z'" by simp
qed
qed
moreover from e and e' have "min e e' > 0" by auto
ultimately show ?thesis by blast
qed
from this[OF _ eq] and this[OF _ eq']
show "(∃e>0. ?P f c e) ⟷ (∃e>0. ?P g c e)"
by blast
qed
with assms show ?thesis by simp
qed

lemma contour_integral_circlepath_eq:
assumes "open s" and f_holo:"f holomorphic_on (s-{z})" and "0<e1" "e1≤e2"
and e2_cball:"cball z e2 ⊆ s"
shows
"f contour_integrable_on circlepath z e1"
"f contour_integrable_on circlepath z e2"
"contour_integral (circlepath z e2) f = contour_integral (circlepath z e1) f"
proof -
define l where "l ≡ linepath (z+e2) (z+e1)"
have [simp]:"valid_path l" "pathstart l=z+e2" "pathfinish l=z+e1" unfolding l_def by auto
have "e2>0" using ‹e1>0› ‹e1≤e2› by auto
have zl_img:"z∉path_image l"
proof
assume "z ∈ path_image l"
then have "e2 ≤ cmod (e2 - e1)"
using segment_furthest_le[of z "z+e2" "z+e1" "z+e2",simplified] ‹e1>0› ‹e2>0› unfolding l_def
by (auto simp add:closed_segment_commute)
thus False using ‹e2>0› ‹e1>0› ‹e1≤e2›
apply (subst (asm) norm_of_real)
by auto
qed
define g where "g ≡ circlepath z e2 +++ l +++ reversepath (circlepath z e1) +++ reversepath l"
show [simp]: "f contour_integrable_on circlepath z e2" "f contour_integrable_on (circlepath z e1)"
proof -
show "f contour_integrable_on circlepath z e2"
apply (intro contour_integrable_continuous_circlepath[OF
continuous_on_subset[OF holomorphic_on_imp_continuous_on[OF f_holo]]])
using ‹e2>0› e2_cball by auto
show "f contour_integrable_on (circlepath z e1)"
apply (intro contour_integrable_continuous_circlepath[OF
continuous_on_subset[OF holomorphic_on_imp_continuous_on[OF f_holo]]])
using ‹e1>0› ‹e1≤e2› e2_cball by auto
qed
have [simp]:"f contour_integrable_on l"
proof -
have "closed_segment (z + e2) (z + e1) ⊆ cball z e2" using ‹e2>0› ‹e1>0› ‹e1≤e2›
by (intro closed_segment_subset,auto simp add:dist_norm)
hence "closed_segment (z + e2) (z + e1) ⊆ s - {z}" using zl_img e2_cball unfolding l_def
by auto
then show "f contour_integrable_on l" unfolding l_def
apply (intro contour_integrable_continuous_linepath[OF
continuous_on_subset[OF holomorphic_on_imp_continuous_on[OF f_holo]]])
by auto
qed
let ?ig="λg. contour_integral g f"
have "(f has_contour_integral 0) g"
proof (rule Cauchy_theorem_global[OF _ f_holo])
show "open (s - {z})" using ‹open s› by auto
show "valid_path g" unfolding g_def l_def by auto
show "pathfinish g = pathstart g" unfolding g_def l_def by auto
next
have path_img:"path_image g ⊆ cball z e2"
proof -
have "closed_segment (z + e2) (z + e1) ⊆ cball z e2" using ‹e2>0› ‹e1>0› ‹e1≤e2›
by (intro closed_segment_subset,auto simp add:dist_norm)
moreover have "sphere z ¦e1¦ ⊆ cball z e2" using ‹e2>0› ‹e1≤e2› ‹e1>0› by auto
ultimately show ?thesis unfolding g_def l_def using ‹e2>0›
by (simp add: path_image_join closed_segment_commute)
qed
show "path_image g ⊆ s - {z}"
proof -
have "z∉path_image g" using zl_img
unfolding g_def l_def by (auto simp add: path_image_join closed_segment_commute)
moreover note ‹cball z e2 ⊆ s› and path_img
ultimately show ?thesis by auto
qed
show "winding_number g w = 0" when"w ∉ s - {z}" for w
proof -
have "winding_number g w = 0" when "w∉s" using that e2_cball
apply (intro winding_number_zero_outside[OF _ _ _ _ path_img])
by (auto simp add:g_def l_def)
moreover have "winding_number g z=0"
proof -
let ?Wz="λg. winding_number g z"
have "?Wz g = ?Wz (circlepath z e2) + ?Wz l + ?Wz (reversepath (circlepath z e1))
+ ?Wz (reversepath l)"
using ‹e2>0› ‹e1>0› zl_img unfolding g_def l_def
by (subst winding_number_join,auto simp add:path_image_join closed_segment_commute)+
also have "... = ?Wz (circlepath z e2) + ?Wz (reversepath (circlepath z e1))"
using zl_img
apply (subst (2) winding_number_reversepath)
by (auto simp add:l_def closed_segment_commute)
also have "... = 0"
proof -
have "?Wz (circlepath z e2) = 1" using ‹e2>0›
by (auto intro: winding_number_circlepath_centre)
moreover have "?Wz (reversepath (circlepath z e1)) = -1" using ‹e1>0›
apply (subst winding_number_reversepath)
by (auto intro: winding_number_circlepath_centre)
ultimately show ?thesis by auto
qed
finally show ?thesis .
qed
ultimately show ?thesis using that by auto
qed
qed
then have "0 = ?ig g" using contour_integral_unique by simp
also have "... = ?ig (circlepath z e2) + ?ig l + ?ig (reversepath (circlepath z e1))
+ ?ig (reversepath l)"
unfolding g_def
by (auto simp add:contour_integrable_reversepath_eq)
also have "... = ?ig (circlepath z e2)  - ?ig (circlepath z e1)"
by (auto simp add:contour_integral_reversepath)
finally show "contour_integral (circlepath z e2) f = contour_integral (circlepath z e1) f"
by simp
qed

lemma base_residue:
assumes "open s" "z∈s" "r>0" and f_holo:"f holomorphic_on (s - {z})"
and r_cball:"cball z r ⊆ s"
shows "(f has_contour_integral 2 * pi * 𝗂 * (residue f z)) (circlepath z r)"
proof -
obtain e where "e>0" and e_cball:"cball z e ⊆ s"
using open_contains_cball[of s] ‹open s› ‹z∈s› by auto
define c where "c ≡ 2 * pi * 𝗂"
define i where "i ≡ contour_integral (circlepath z e) f / c"
have "(f has_contour_integral c*i) (circlepath z ε)" when "ε>0" "ε<e" for ε
proof -
have "contour_integral (circlepath z e) f = contour_integral (circlepath z ε) f"
"f contour_integrable_on circlepath z ε"
"f contour_integrable_on circlepath z e"
using ‹ε<e›
by (intro contour_integral_circlepath_eq[OF ‹open s› f_holo ‹ε>0› _ e_cball],auto)+
then show ?thesis unfolding i_def c_def
by (auto intro:has_contour_integral_integral)
qed
then have "∃e>0. ∀ε>0. ε<e ⟶ (f has_contour_integral c * (residue f z)) (circlepath z ε)"
unfolding residue_def c_def
apply (rule_tac someI[of _ i],intro  exI[where x=e])
by (auto simp add:‹e>0› c_def)
then obtain e' where "e'>0"
and e'_def:"∀ε>0. ε<e' ⟶ (f has_contour_integral c * (residue f z)) (circlepath z ε)"
by auto
let ?int="λe. contour_integral (circlepath z e) f"
define  ε where "ε ≡ Min {r,e'} / 2"
have "ε>0" "ε≤r" "ε<e'" using ‹r>0› ‹e'>0› unfolding ε_def by auto
have "(f has_contour_integral c * (residue f z)) (circlepath z ε)"
using e'_def[rule_format,OF ‹ε>0› ‹ε<e'›] .
then show ?thesis unfolding c_def
using contour_integral_circlepath_eq[OF ‹open s› f_holo ‹ε>0› ‹ε≤r› r_cball]
by (auto elim: has_contour_integral_eqpath[of _ _ "circlepath z ε" "circlepath z r"])
qed

lemma residue_holo:
assumes "open s" "z ∈ s" and f_holo: "f holomorphic_on s"
shows "residue f z = 0"
proof -
define c where "c ≡ 2 * pi * 𝗂"
obtain e where "e>0" and e_cball:"cball z e ⊆ s" using ‹open s› ‹z∈s›
using open_contains_cball_eq by blast
have "(f has_contour_integral c*residue f z) (circlepath z e)"
using f_holo
by (auto intro: base_residue[OF ‹open s› ‹z∈s› ‹e>0› _ e_cball,folded c_def])
moreover have "(f has_contour_integral 0) (circlepath z e)"
using f_holo e_cball ‹e>0›
by (auto intro: Cauchy_theorem_convex_simple[of _ "cball z e"])
ultimately have "c*residue f z =0"
using has_contour_integral_unique by blast
thus ?thesis unfolding c_def  by auto
qed

lemma residue_const:"residue (λ_. c) z = 0"
by (intro residue_holo[of "UNIV::complex set"],auto intro:holomorphic_intros)

lemma residue_add:
assumes "open s" "z ∈ s" and f_holo: "f holomorphic_on s - {z}"
and g_holo:"g holomorphic_on s - {z}"
shows "residue (λz. f z + g z) z= residue f z + residue g z"
proof -
define c where "c ≡ 2 * pi * 𝗂"
define fg where "fg ≡ (λz. f z+g z)"
obtain e where "e>0" and e_cball:"cball z e ⊆ s" using ‹open s› ‹z∈s›
using open_contains_cball_eq by blast
have "(fg has_contour_integral c * residue fg z) (circlepath z e)"
unfolding fg_def using f_holo g_holo
apply (intro base_residue[OF ‹open s› ‹z∈s› ‹e>0› _ e_cball,folded c_def])
by (auto intro:holomorphic_intros)
moreover have "(fg has_contour_integral c*residue f z + c* residue g z) (circlepath z e)"
unfolding fg_def using f_holo g_holo
by (auto intro: has_contour_integral_add base_residue[OF ‹open s› ‹z∈s› ‹e>0› _ e_cball,folded c_def])
ultimately have "c*(residue f z + residue g z) = c * residue fg z"
using has_contour_integral_unique by (auto simp add:distrib_left)
thus ?thesis unfolding fg_def
by (auto simp add:c_def)
qed

lemma residue_lmul:
assumes "open s" "z ∈ s" and f_holo: "f holomorphic_on s - {z}"
shows "residue (λz. c * (f z)) z= c * residue f z"
proof (cases "c=0")
case True
thus ?thesis using residue_const by auto
next
case False
define c' where "c' ≡ 2 * pi * 𝗂"
define f' where "f' ≡ (λz. c * (f z))"
obtain e where "e>0" and e_cball:"cball z e ⊆ s" using ‹open s› ‹z∈s›
using open_contains_cball_eq by blast
have "(f' has_contour_integral c' * residue f' z) (circlepath z e)"
unfolding f'_def using f_holo
apply (intro base_residue[OF ‹open s› ‹z∈s› ‹e>0› _ e_cball,folded c'_def])
by (auto intro:holomorphic_intros)
moreover have "(f' has_contour_integral c * (c' * residue f z)) (circlepath z e)"
unfolding f'_def using f_holo
by (auto intro: has_contour_integral_lmul
base_residue[OF ‹open s› ‹z∈s› ‹e>0› _ e_cball,folded c'_def])
ultimately have "c' * residue f' z  = c * (c' * residue f z)"
using has_contour_integral_unique by auto
thus ?thesis unfolding f'_def c'_def using False
by (auto simp add:field_simps)
qed

lemma residue_rmul:
assumes "open s" "z ∈ s" and f_holo: "f holomorphic_on s - {z}"
shows "residue (λz. (f z) * c) z= residue f z * c"
using residue_lmul[OF assms,of c] by (auto simp add:algebra_simps)

lemma residue_div:
assumes "open s" "z ∈ s" and f_holo: "f holomorphic_on s - {z}"
shows "residue (λz. (f z) / c) z= residue f z / c "
using residue_lmul[OF assms,of "1/c"] by (auto simp add:algebra_simps)

lemma residue_neg:
assumes "open s" "z ∈ s" and f_holo: "f holomorphic_on s - {z}"
shows "residue (λz. - (f z)) z= - residue f z"
using residue_lmul[OF assms,of "-1"] by auto

lemma residue_diff:
assumes "open s" "z ∈ s" and f_holo: "f holomorphic_on s - {z}"
and g_holo:"g holomorphic_on s - {z}"
shows "residue (λz. f z - g z) z= residue f z - residue g z"
using residue_add[OF assms(1,2,3),of "λz. - g z"] residue_neg[OF assms(1,2,4)]
by (auto intro:holomorphic_intros g_holo)

lemma residue_simple:
assumes "open s" "z∈s" and f_holo:"f holomorphic_on s"
shows "residue (λw. f w / (w - z)) z = f z"
proof -
define c where "c ≡ 2 * pi * 𝗂"
define f' where "f' ≡ λw. f w / (w - z)"
obtain e where "e>0" and e_cball:"cball z e ⊆ s" using ‹open s› ‹z∈s›
using open_contains_cball_eq by blast
have "(f' has_contour_integral c * f z) (circlepath z e)"
unfolding f'_def c_def using ‹e>0› f_holo e_cball
by (auto intro!: Cauchy_integral_circlepath_simple holomorphic_intros)
moreover have "(f' has_contour_integral c * residue f' z) (circlepath z e)"
unfolding f'_def using f_holo
apply (intro base_residue[OF ‹open s› ‹z∈s› ‹e>0› _ e_cball,folded c_def])
by (auto intro!:holomorphic_intros)
ultimately have "c * f z = c * residue f' z"
using has_contour_integral_unique by blast
thus ?thesis unfolding c_def f'_def  by auto
qed

lemma residue_simple':
assumes s: "open s" "z ∈ s" and holo: "f holomorphic_on (s - {z})"
and lim: "((λw. f w * (w - z)) ⤏ c) (at z)"
shows   "residue f z = c"
proof -
define g where "g = (λw. if w = z then c else f w * (w - z))"
from holo have "(λw. f w * (w - z)) holomorphic_on (s - {z})" (is "?P")
by (force intro: holomorphic_intros)
also have "?P ⟷ g holomorphic_on (s - {z})"
by (intro holomorphic_cong refl) (simp_all add: g_def)
finally have *: "g holomorphic_on (s - {z})" .

note lim
also have "(λw. f w * (w - z)) ─z→ c ⟷ g ─z→ g z"
by (intro filterlim_cong refl) (simp_all add: g_def [abs_def] eventually_at_filter)
finally have **: "g ─z→ g z" .

have g_holo: "g holomorphic_on s"
by (rule no_isolated_singularity'[where K = "{z}"])
(insert assms * **, simp_all add: at_within_open_NO_MATCH)
from s and this have "residue (λw. g w / (w - z)) z = g z"
by (rule residue_simple)
also have "∀⇩F za in at z. g za / (za - z) = f za"
unfolding eventually_at by (auto intro!: exI[of _ 1] simp: field_simps g_def)
hence "residue (λw. g w / (w - z)) z = residue f z"
by (intro residue_cong refl)
finally show ?thesis
by (simp add: g_def)
qed

lemma residue_holomorphic_over_power:
assumes "open A" "z0 ∈ A" "f holomorphic_on A"
shows   "residue (λz. f z / (z - z0) ^ Suc n) z0 = (deriv ^^ n) f z0 / fact n"
proof -
let ?f = "λz. f z / (z - z0) ^ Suc n"
from assms(1,2) obtain r where r: "r > 0" "cball z0 r ⊆ A"
by (auto simp: open_contains_cball)
have "(?f has_contour_integral 2 * pi * 𝗂 * residue ?f z0) (circlepath z0 r)"
using r assms by (intro base_residue[of A]) (auto intro!: holomorphic_intros)
moreover have "(?f has_contour_integral 2 * pi * 𝗂 / fact n * (deriv ^^ n) f z0) (circlepath z0 r)"
using assms r
by (intro Cauchy_has_contour_integral_higher_derivative_circlepath)
(auto intro!: holomorphic_on_subset[OF assms(3)] holomorphic_on_imp_continuous_on)
ultimately have "2 * pi * 𝗂 * residue ?f z0 = 2 * pi * 𝗂 / fact n * (deriv ^^ n) f z0"
by (rule has_contour_integral_unique)
thus ?thesis by (simp add: field_simps)
qed

lemma residue_holomorphic_over_power':
assumes "open A" "0 ∈ A" "f holomorphic_on A"
shows   "residue (λz. f z / z ^ Suc n) 0 = (deriv ^^ n) f 0 / fact n"
using residue_holomorphic_over_power[OF assms] by simp

lemma get_integrable_path:
assumes "open s" "connected (s-pts)" "finite pts" "f holomorphic_on (s-pts) " "a∈s-pts" "b∈s-pts"
obtains g where "valid_path g" "pathstart g = a" "pathfinish g = b"
"path_image g ⊆ s-pts" "f contour_integrable_on g" using assms
proof (induct arbitrary:s thesis a rule:finite_induct[OF ‹finite pts›])
case 1
obtain g where "valid_path g" "path_image g ⊆ s" "pathstart g = a" "pathfinish g = b"
using connected_open_polynomial_connected[OF ‹open s›,of a b ] ‹connected (s - {})›
valid_path_polynomial_function "1.prems"(6) "1.prems"(7) by auto
moreover have "f contour_integrable_on g"
using contour_integrable_holomorphic_simple[OF _ ‹open s› ‹valid_path g› ‹path_image g ⊆ s›,of f]
‹f holomorphic_on s - {}›
by auto
ultimately show ?case using "1"(1)[of g] by auto
next
case idt:(2 p pts)
obtain e where "e>0" and e:"∀w∈ball a e. w ∈ s ∧ (w ≠ a ⟶ w ∉ insert p pts)"
using finite_ball_avoid[OF ‹open s› ‹finite (insert p pts)›, of a]
‹a ∈ s - insert p pts›
by auto
define a' where "a' ≡ a+e/2"
have "a'∈s-{p} -pts"  using e[rule_format,of "a+e/2"] ‹e>0›
by (auto simp add:dist_complex_def a'_def)
then obtain g' where g'[simp]:"valid_path g'" "pathstart g' = a'" "pathfinish g' = b"
"path_image g' ⊆ s - {p} - pts" "f contour_integrable_on g'"
using idt.hyps(3)[of a' "s-{p}"] idt.prems idt.hyps(1)
by (metis Diff_insert2 open_delete)
define g where "g ≡ linepath a a' +++ g'"
have "valid_path g" unfolding g_def by (auto intro: valid_path_join)
moreover have "pathstart g = a" and  "pathfinish g = b" unfolding g_def by auto
moreover have "path_image g ⊆ s - insert p pts" unfolding g_def
proof (rule subset_path_image_join)
have "closed_segment a a' ⊆ ball a e" using ‹e>0›
by (auto dest!:segment_bound1 simp:a'_def dist_complex_def norm_minus_commute)
then show "path_image (linepath a a') ⊆ s - insert p pts" using e idt(9)
by auto
next
show "path_image g' ⊆ s - insert p pts" using g'(4) by blast
qed
moreover have "f contour_integrable_on g"
proof -
have "closed_segment a a' ⊆ ball a e" using ‹e>0›
by (auto dest!:segment_bound1 simp:a'_def dist_complex_def norm_minus_commute)
then have "continuous_on (closed_segment a a') f"
using e idt.prems(6) holomorphic_on_imp_continuous_on[OF idt.prems(5)]
apply (elim continuous_on_subset)
by auto
then have "f contour_integrable_on linepath a a'"
using contour_integrable_continuous_linepath by auto
then show ?thesis unfolding g_def
apply (rule contour_integrable_joinI)
by (auto simp add: ‹e>0›)
qed
ultimately show ?case using idt.prems(1)[of g] by auto
qed

lemma Cauchy_theorem_aux:
assumes "open s" "connected (s-pts)" "finite pts" "pts ⊆ s" "f holomorphic_on s-pts"
"valid_path g" "pathfinish g = pathstart g" "path_image g ⊆ s-pts"
"∀z. (z ∉ s) ⟶ winding_number g z  = 0"
"∀p∈s. h p>0 ∧ (∀w∈cball p (h p). w∈s ∧ (w≠p ⟶ w ∉ pts))"
shows "contour_integral g f = (∑p∈pts. winding_number g p * contour_integral (circlepath p (h p)) f)"
using assms
proof (induct arbitrary:s g rule:finite_induct[OF ‹finite pts›])
case 1
then show ?case by (simp add: Cauchy_theorem_global contour_integral_unique)
next
case (2 p pts)
note fin[simp] = ‹finite (insert p pts)›
and connected = ‹connected (s - insert p pts)›
and valid[simp] = ‹valid_path g›
and g_loop[simp] = ‹pathfinish g = pathstart g›
and holo[simp]= ‹f holomorphic_on s - insert p pts›
and path_img = ‹path_image g ⊆ s - insert p pts›
and winding = ‹∀z. z ∉ s ⟶ winding_number g z = 0›
and h = ‹∀pa∈s. 0 < h pa ∧ (∀w∈cball pa (h pa). w ∈ s ∧ (w ≠ pa ⟶ w ∉ insert p pts))›
have "h p>0" and "p∈s"
and h_p: "∀w∈cball p (h p). w ∈ s ∧ (w ≠ p ⟶ w ∉ insert p pts)"
using h ‹insert p pts ⊆ s› by auto
obtain pg where pg[simp]: "valid_path pg" "pathstart pg = pathstart g" "pathfinish pg=p+h p"
"path_image pg ⊆ s-insert p pts" "f contour_integrable_on pg"
proof -
have "p + h p∈cball p (h p)" using h[rule_format,of p]
by (simp add: ‹p ∈ s› dist_norm)
then have "p + h p ∈ s - insert p pts" using h[rule_format,of p] ‹insert p pts ⊆ s›
by fastforce
moreover have "pathstart g ∈ s - insert p pts " using path_img by auto
ultimately show ?thesis
using get_integrable_path[OF ‹open s› connected fin holo,of "pathstart g" "p+h p"] that
by blast
qed
obtain n::int where "n=winding_number g p"
using integer_winding_number[OF _ g_loop,of p] valid path_img
by (metis DiffD2 Ints_cases insertI1 subset_eq valid_path_imp_path)
define p_circ where "p_circ ≡ circlepath p (h p)"
define p_circ_pt where "p_circ_pt ≡ linepath (p+h p) (p+h p)"
define n_circ where "n_circ ≡ λn. ((+++) p_circ ^^ n) p_circ_pt"
define cp where "cp ≡ if n≥0 then reversepath (n_circ (nat n)) else n_circ (nat (- n))"
have n_circ:"valid_path (n_circ k)"
"winding_number (n_circ k) p = k"
"pathstart (n_circ k) = p + h p" "pathfinish (n_circ k) = p + h p"
"path_image (n_circ k) =  (if k=0 then {p + h p} else sphere p (h p))"
"p ∉ path_image (n_circ k)"
"⋀p'. p'∉s - pts ⟹ winding_number (n_circ k) p'=0 ∧ p'∉path_image (n_circ k)"
"f contour_integrable_on (n_circ k)"
"contour_integral (n_circ k) f = k *  contour_integral p_circ f"
for k
proof (induct k)
case 0
show "valid_path (n_circ 0)"
and "path_image (n_circ 0) =  (if 0=0 then {p + h p} else sphere p (h p))"
and "winding_number (n_circ 0) p = of_nat 0"
and "pathstart (n_circ 0) = p + h p"
and "pathfinish (n_circ 0) = p + h p"
and "p ∉ path_image (n_circ 0)"
unfolding n_circ_def p_circ_pt_def using ‹h p > 0›
by (auto simp add: dist_norm)
show "winding_number (n_circ 0) p'=0 ∧ p'∉path_image (n_circ 0)" when "p'∉s- pts" for p'
unfolding n_circ_def p_circ_pt_def
apply (auto intro!:winding_number_trivial)
by (metis Diff_iff pathfinish_in_path_image pg(3) pg(4) subsetCE subset_insertI that)+
show "f contour_integrable_on (n_circ 0)"
unfolding n_circ_def p_circ_pt_def
by (auto intro!:contour_integrable_continuous_linepath simp add:continuous_on_sing)
show "contour_integral (n_circ 0) f = of_nat 0  *  contour_integral p_circ f"
unfolding n_circ_def p_circ_pt_def by auto
next
case (Suc k)
have n_Suc:"n_circ (Suc k) = p_circ +++ n_circ k" unfolding n_circ_def by auto
have pcirc:"p ∉ path_image p_circ" "valid_path p_circ" "pathfinish p_circ = pathstart (n_circ k)"
using Suc(3) unfolding p_circ_def using ‹h p > 0› by (auto simp add: p_circ_def)
have pcirc_image:"path_image p_circ ⊆ s - insert p pts"
proof -
have "path_image p_circ ⊆ cball p (h p)" using ‹0 < h p› p_circ_def by auto
then show ?thesis using h_p pcirc(1) by auto
qed
have pcirc_integrable:"f contour_integrable_on p_circ"
by (auto simp add:p_circ_def intro!: pcirc_image[unfolded p_circ_def]
contour_integrable_continuous_circlepath holomorphic_on_imp_continuous_on
holomorphic_on_subset[OF holo])
show "valid_path (n_circ (Suc k))"
using valid_path_join[OF pcirc(2) Suc(1) pcirc(3)] unfolding n_circ_def by auto
show "path_image (n_circ (Suc k))
= (if Suc k = 0 then {p + complex_of_real (h p)} else sphere p (h p))"
proof -
have "path_image p_circ = sphere p (h p)"
unfolding p_circ_def using ‹0 < h p› by auto
then show ?thesis unfolding n_Suc  using Suc.hyps(5)  ‹h p>0›
by (auto simp add:  path_image_join[OF pcirc(3)]  dist_norm)
qed
then show "p ∉ path_image (n_circ (Suc k))" using ‹h p>0› by auto
show "winding_number (n_circ (Suc k)) p = of_nat (Suc k)"
proof -
have "winding_number p_circ p = 1"
by (simp add: ‹h p > 0› p_circ_def winding_number_circlepath_centre)
moreover have "p ∉ path_image (n_circ k)" using Suc(5) ‹h p>0› by auto
then have "winding_number (p_circ +++ n_circ k) p
= winding_number p_circ p + winding_number (n_circ k) p"
using  valid_path_imp_path Suc.hyps(1) Suc.hyps(2) pcirc
apply (intro winding_number_join)
by auto
ultimately show ?thesis using Suc(2) unfolding n_circ_def
by auto
qed
show "pathstart (n_circ (Suc k)) = p + h p"
by (simp add: n_circ_def p_circ_def)
show "pathfinish (n_circ (Suc k)) = p + h p"
using Suc(4) unfolding n_circ_def by auto
show "winding_number (n_circ (Suc k)) p'=0 ∧  p'∉path_image (n_circ (Suc k))" when "p'∉s-pts" for p'
proof -
have " p' ∉ path_image p_circ" using ‹p ∈ s› h p_circ_def that using pcirc_image by blast
moreover have "p' ∉ path_image (n_circ k)"
using Suc.hyps(7) that by blast
moreover have "winding_number p_circ p' = 0"
proof -
have "path_image p_circ ⊆ cball p (h p)"
using h unfolding p_circ_def using ‹p ∈ s› by fastforce
moreover have "p'∉cball p (h p)" using ‹p ∈ s› h that "2.hyps"(2) by fastforce
ultimately show ?thesis unfolding p_circ_def
apply (intro winding_number_zero_outside)
by auto
qed
ultimately show ?thesis
unfolding n_Suc
apply (subst winding_number_join)
by (auto simp: valid_path_imp_path pcirc Suc that not_in_path_image_join Suc.hyps(7)[OF that])
qed
show "f contour_integrable_on (n_circ (Suc k))"
unfolding n_Suc
by (rule contour_integrable_joinI[OF pcirc_integrable Suc(8) pcirc(2) Suc(1)])
show "contour_integral (n_circ (Suc k)) f = (Suc k) *  contour_integral p_circ f"
unfolding n_Suc
by (auto simp add:contour_integral_join[OF pcirc_integrable Suc(8) pcirc(2) Suc(1)]
Suc(9) algebra_simps)
qed
have cp[simp]:"pathstart cp = p + h p"  "pathfinish cp = p + h p"
"valid_path cp" "path_image cp ⊆ s - insert p pts"
"winding_number cp p = - n"
"⋀p'. p'∉s - pts ⟹ winding_number cp p'=0 ∧ p' ∉ path_image cp"
"f contour_integrable_on cp"
"contour_integral cp f = - n * contour_integral p_circ f"
proof -
show "pathstart cp = p + h p" and "pathfinish cp = p + h p" and "valid_path cp"
using n_circ unfolding cp_def by auto
next
have "sphere p (h p) ⊆  s - insert p pts"
using h[rule_format,of p] ‹insert p pts ⊆ s› by force
moreover  have "p + complex_of_real (h p) ∈ s - insert p pts"
using pg(3) pg(4) by (metis pathfinish_in_path_image subsetCE)
ultimately show "path_image cp ⊆  s - insert p pts" unfolding cp_def
using n_circ(5)  by auto
next
show "winding_number cp p = - n"
unfolding cp_def using winding_number_reversepath n_circ ‹h p>0›
by (auto simp: valid_path_imp_path)
next
show "winding_number cp p'=0 ∧ p' ∉ path_image cp" when "p'∉s - pts" for p'
unfolding cp_def
apply (auto)
apply (subst winding_number_reversepath)
by (auto simp add: valid_path_imp_path n_circ(7)[OF that] n_circ(1))
next
show "f contour_integrable_on cp" unfolding cp_def
using contour_integrable_reversepath_eq n_circ(1,8) by auto
next
show "contour_integral cp f = - n * contour_integral p_circ f"
unfolding cp_def using contour_integral_reversepath[OF n_circ(1)] n_circ(9)
by auto
qed
define g' where "g' ≡ g +++ pg +++ cp +++ (reversepath pg)"
have "contour_integral g' f = (∑p∈pts. winding_number g' p * contour_integral (circlepath p (h p)) f)"
proof (rule "2.hyps"(3)[of "s-{p}" "g'",OF _ _ ‹finite pts› ])
show "connected (s - {p} - pts)" using connected by (metis Diff_insert2)
show "open (s - {p})" using ‹open s› by auto
show " pts ⊆ s - {p}" using ‹insert p pts ⊆ s› ‹ p ∉ pts›  by blast
show "f holomorphic_on s - {p} - pts" using holo ‹p ∉ pts› by (metis Diff_insert2)
show "valid_path g'"
unfolding g'_def cp_def using n_circ valid pg g_loop
by (auto intro!:valid_path_join )
show "pathfinish g' = pathstart g'"
unfolding g'_def cp_def using pg(2) by simp
show "path_image g' ⊆ s - {p} - pts"
proof -
define s' where "s' ≡ s - {p} - pts"
have s':"s' = s-insert p pts " unfolding s'_def by auto
then show ?thesis using path_img pg(4) cp(4)
unfolding g'_def
apply (fold s'_def s')
apply (intro subset_path_image_join)
by auto
qed
note path_join_imp[simp]
show "∀z. z ∉ s - {p} ⟶ winding_number g' z = 0"
proof clarify
fix z assume z:"z∉s - {p}"
have "winding_number (g +++ pg +++ cp +++ reversepath pg) z = winding_number g z
+ winding_number (pg +++ cp +++ (reversepath pg)) z"
proof (rule winding_number_join)
show "path g" using ‹valid_path g› by (simp add: valid_path_imp_path)
show "z ∉ path_image g" using z path_img by auto
show "path (pg +++ cp +++ reversepath pg)" using pg(3) cp
by (simp add: valid_path_imp_path)
next
have "path_image (pg +++ cp +++ reversepath pg) ⊆ s - insert p pts"
using pg(4) cp(4) by (auto simp:subset_path_image_join)
then show "z ∉ path_image (pg +++ cp +++ reversepath pg)" using z by auto
next
show "pathfinish g = pathstart (pg +++ cp +++ reversepath pg)" using g_loop by auto
qed
also have "... = winding_number g z + (winding_number pg z
+ winding_number (cp +++ (reversepath pg)) z)"
proof (subst add_left_cancel,rule winding_number_join)
show "path pg" and "path (cp +++ reversepath pg)"
and "pathfinish pg = pathstart (cp +++ reversepath pg)"
by (auto simp add: valid_path_imp_path)
show "z ∉ path_image pg" using pg(4) z by blast
show "z ∉ path_image (cp +++ reversepath pg)" using z
by (metis Diff_iff ‹z ∉ path_image pg› contra_subsetD cp(4) insertI1
not_in_path_image_join path_image_reversepath singletonD)
qed
also have "... = winding_number g z + (winding_number pg z
+ (winding_number cp z + winding_number (reversepath pg) z))"
apply (auto intro!:winding_number_join simp: valid_path_imp_path)
apply (metis Diff_iff contra_subsetD cp(4) insertI1 singletonD z)
by (metis Diff_insert2 Diff_subset contra_subsetD pg(4) z)
also have "... = winding_number g z + winding_number cp z"
apply (subst winding_number_reversepath)
apply (auto simp: valid_path_imp_path)
by (metis Diff_iff contra_subsetD insertI1 pg(4) singletonD z)
finally have "winding_number g' z = winding_number g z + winding_number cp z"
unfolding g'_def .
moreover have "winding_number g z + winding_number cp z = 0"
using winding z ‹n=winding_number g p› by auto
ultimately show "winding_number g' z = 0" unfolding g'_def by auto
qed
show "∀pa∈s - {p}. 0 < h pa ∧ (∀w∈cball pa (h pa). w ∈ s - {p} ∧ (w ≠ pa ⟶ w ∉ pts))"
using h by fastforce
qed
moreover have "contour_integral g' f = contour_integral g f
- winding_number g p * contour_integral p_circ f"
proof -
have "contour_integral g' f =  contour_integral g f
+ contour_integral (pg +++ cp +++ reversepath pg) f"
unfolding g'_def
apply (subst contour_integral_join)
by (auto simp add:open_Diff[OF ‹open s›,OF finite_imp_closed[OF fin]]
intro!: contour_integrable_holomorphic_simple[OF holo _ _ path_img]
contour_integrable_reversepath)
also have "... = contour_integral g f + contour_integral pg f
+ contour_integral (cp +++ reversepath pg) f"
apply (subst contour_integral_join)
by (auto simp add:contour_integrable_reversepath)
also have "... = contour_integral g f + contour_integral pg f
+ contour_integral cp f + contour_integral (reversepath pg) f"
apply (subst contour_integral_join)
by (auto simp add:contour_integrable_reversepath)
also have "... = contour_integral g f + contour_integral cp f"
using contour_integral_reversepath
by (auto simp add:contour_integrable_reversepath)
also have "... = contour_integral g f - winding_number g p * contour_integral p_circ f"
using ‹n=winding_number g p› by auto
finally show ?thesis .
qed
moreover have "winding_number g' p' = winding_number g p'" when "p'∈pts" for p'
proof -
have [simp]: "p' ∉ path_image g" "p' ∉ path_image pg" "p'∉path_image cp"
using "2.prems"(8) that
apply blast
apply (metis Diff_iff Diff_insert2 contra_subsetD pg(4) that)
by (meson DiffD2 cp(4) set_rev_mp subset_insertI that)
have "winding_number g' p' = winding_number g p'
+ winding_number (pg +++ cp +++ reversepath pg) p'" unfolding g'_def
apply (subst winding_number_join)
apply (simp_all add: valid_path_imp_path)
apply (intro not_in_path_image_join)
by auto
also have "... = winding_number g p' + winding_number pg p'
+ winding_number (cp +++ reversepath pg) p'"
apply (subst winding_number_join)
apply (simp_all add: valid_path_imp_path)
apply (intro not_in_path_image_join)
by auto
also have "... = winding_number g p' + winding_number pg p'+ winding_number cp p'
+ winding_number (reversepath pg) p'"
apply (subst winding_number_join)
by (simp_all add: valid_path_imp_path)
also have "... = winding_number g p' + winding_number cp p'"
apply (subst winding_number_reversepath)
by (simp_all add: valid_path_imp_path)
also have "... = winding_number g p'" using that by auto
finally show ?thesis .
qed
ultimately show ?case unfolding p_circ_def
apply (subst (asm) sum.cong[OF refl,
of pts _ "λp. winding_number g p * contour_integral (circlepath p (h p)) f"])
by (auto simp add:sum.insert[OF ‹finite pts› ‹p∉pts›] algebra_simps)
qed

lemma Cauchy_theorem_singularities:
assumes "open s" "connected s" "finite pts" and
holo:"f holomorphic_on s-pts" and
"valid_path g" and
loop:"pathfinish g = pathstart g" and
"path_image g ⊆ s-pts" and
homo:"∀z. (z ∉ s) ⟶ winding_number g z  = 0" and
avoid:"∀p∈s. h p>0 ∧ (∀w∈cball p (h p). w∈s ∧ (w≠p ⟶ w ∉ pts))"
shows "contour_integral g f = (∑p∈pts. winding_number g p * contour_integral (circlepath p (h p)) f)"
(is "?L=?R")
proof -
define circ where "circ ≡ λp. winding_number g p * contour_integral (circlepath p (h p)) f"
define pts1 where "pts1 ≡ pts ∩ s"
define pts2 where "pts2 ≡ pts - pts1"
have "pts=pts1 ∪ pts2" "pts1 ∩ pts2 = {}" "pts2 ∩ s={}" "pts1⊆s"
unfolding pts1_def pts2_def by auto
have "contour_integral g f =  (∑p∈pts1. circ p)" unfolding circ_def
proof (rule Cauchy_theorem_aux[OF ‹open s› _ _ ‹pts1⊆s› _ ‹valid_path g› loop _ homo])
have "finite pts1" unfolding pts1_def using ‹finite pts› by auto
then show "connected (s - pts1)"
using ‹open s› ‹connected s› connected_open_delete_finite[of s] by auto
next
show "finite pts1" using ‹pts = pts1 ∪ pts2› assms(3) by auto
show "f holomorphic_on s - pts1" by (metis Diff_Int2 Int_absorb holo pts1_def)
show "path_image g ⊆ s - pts1" using assms(7) pts1_def by auto
show "∀p∈s. 0 < h p ∧ (∀w∈cball p (h p). w ∈ s ∧ (w ≠ p ⟶ w ∉ pts1))"
by (simp add: avoid pts1_def)
qed
moreover have "sum circ pts2=0"
proof -
have "winding_number g p=0" when "p∈pts2" for p
using  ‹pts2 ∩ s={}› that homo[rule_format,of p] by auto
thus ?thesis unfolding circ_def
apply (intro sum.neutral)
by auto
qed
moreover have "?R=sum circ pts1 + sum circ pts2"
unfolding circ_def
using sum.union_disjoint[OF _ _ ‹pts1 ∩ pts2 = {}›] ‹finite pts› ‹pts=pts1 ∪ pts2›
by blast
ultimately show ?thesis
apply (fold circ_def)
by auto
qed

theorem Residue_theorem:
fixes s pts::"complex set" and f::"complex ⇒ complex"
and g::"real ⇒ complex"
assumes "open s" "connected s" "finite pts" and
holo:"f holomorphic_on s-pts" and
"valid_path g" and
loop:"pathfinish g = pathstart g" and
"path_image g ⊆ s-pts" and
homo:"∀z. (z ∉ s) ⟶ winding_number g z  = 0"
shows "contour_integral g f = 2 * pi * 𝗂 *(∑p∈pts. winding_number g p * residue f p)"
proof -
define c where "c ≡  2 * pi * 𝗂"
obtain h where avoid:"∀p∈s. h p>0 ∧ (∀w∈cball p (h p). w∈s ∧ (w≠p ⟶ w ∉ pts))"
using finite_cball_avoid[OF ‹open s› ‹finite pts›] by metis
have "contour_integral g f
= (∑p∈pts. winding_number g p * contour_integral (circlepath p (h p)) f)"
using Cauchy_theorem_singularities[OF assms avoid] .
also have "... = (∑p∈pts.  c * winding_number g p * residue f p)"
proof (intro sum.cong)
show "pts = pts" by simp
next
fix x assume "x ∈ pts"
show "winding_number g x * contour_integral (circlepath x (h x)) f
= c * winding_number g x * residue f x"
proof (cases "x∈s")
case False
then have "winding_number g x=0" using homo by auto
thus ?thesis by auto
next
case True
have "contour_integral (circlepath x (h x)) f = c* residue f x"
using ‹x∈pts› ‹finite pts› avoid[rule_format,OF True]
apply (intro base_residue[of "s-(pts-{x})",THEN contour_integral_unique,folded c_def])
by (auto intro:holomorphic_on_subset[OF holo] open_Diff[OF ‹open s› finite_imp_closed])
then show ?thesis by auto
qed
qed
also have "... = c * (∑p∈pts. winding_number g p * residue f p)"
by (simp add: sum_distrib_left algebra_simps)
finally show ?thesis unfolding c_def .
qed

subsection ‹Non-essential singular points›

definition%important is_pole ::
"('a::topological_space ⇒ 'b::real_normed_vector) ⇒ 'a ⇒ bool" where
"is_pole f a =  (LIM x (at a). f x :> at_infinity)"

lemma is_pole_cong:
assumes "eventually (λx. f x = g x) (at a)" "a=b"
shows "is_pole f a ⟷ is_pole g b"
unfolding is_pole_def using assms by (intro filterlim_cong,auto)

lemma is_pole_transform:
assumes "is_pole f a" "eventually (λx. f x = g x) (at a)" "a=b"
shows "is_pole g b"
using is_pole_cong assms by auto

lemma is_pole_tendsto:
fixes f::"('a::topological_space ⇒ 'b::real_normed_div_algebra)"
shows "is_pole f x ⟹ ((inverse o f) ⤏ 0) (at x)"
unfolding is_pole_def
by (auto simp add:filterlim_inverse_at_iff[symmetric] comp_def filterlim_at)

lemma is_pole_inverse_holomorphic:
assumes "open s"
and f_holo:"f holomorphic_on (s-{z})"
and pole:"is_pole f z"
and non_z:"∀x∈s-{z}. f x≠0"
shows "(λx. if x=z then 0 else inverse (f x)) holomorphic_on s"
proof -
define g where "g ≡ λx. if x=z then 0 else inverse (f x)"
have "isCont g z" unfolding isCont_def  using is_pole_tendsto[OF pole]
apply (subst Lim_cong_at[where b=z and y=0 and g="inverse ∘ f"])
by (simp_all add:g_def)
moreover have "continuous_on (s-{z}) f" using f_holo holomorphic_on_imp_continuous_on by auto
hence "continuous_on (s-{z}) (inverse o f)" unfolding comp_def
by (auto elim!:continuous_on_inverse simp add:non_z)
hence "continuous_on (s-{z}) g" unfolding g_def
apply (subst continuous_on_cong[where t="s-{z}" and g="inverse o f"])
by auto
ultimately have "continuous_on s g" using open_delete[OF ‹open s›] ‹open s›
by (auto simp add:continuous_on_eq_continuous_at)
moreover have "(inverse o f) holomorphic_on (s-{z})"
unfolding comp_def using f_holo
by (auto elim!:holomorphic_on_inverse simp add:non_z)
hence "g holomorphic_on (s-{z})"
apply (subst holomorphic_cong[where t="s-{z}" and g="inverse o f"])
by (auto simp add:g_def)
ultimately show ?thesis unfolding g_def using ‹open s›
by (auto elim!: no_isolated_singularity)
qed

lemma not_is_pole_holomorphic:
assumes "open A" "x ∈ A" "f holomorphic_on A"
shows   "¬is_pole f x"
proof -
have "continuous_on A f" by (intro holomorphic_on_imp_continuous_on) fact
with assms have "isCont f x" by (simp add: continuous_on_eq_continuous_at)
hence "f ─x→ f x" by (simp add: isCont_def)
thus "¬is_pole f x" unfolding is_pole_def
using not_tendsto_and_filterlim_at_infinity[of "at x" f "f x"] by auto
qed

lemma is_pole_inverse_power: "n > 0 ⟹ is_pole (λz::complex. 1 / (z - a) ^ n) a"
unfolding is_pole_def inverse_eq_divide [symmetric]
by (intro filterlim_compose[OF filterlim_inverse_at_infinity] tendsto_intros)
(auto simp: filterlim_at eventually_at intro!: exI[of _ 1] tendsto_eq_intros)

lemma is_pole_inverse: "is_pole (λz::complex. 1 / (z - a)) a"
using is_pole_inverse_power[of 1 a] by simp

lemma is_pole_divide:
fixes f :: "'a :: t2_space ⇒ 'b :: real_normed_field"
assumes "isCont f z" "filterlim g (at 0) (at z)" "f z ≠ 0"
shows   "is_pole (λz. f z / g z) z"
proof -
have "filterlim (λz. f z * inverse (g z)) at_infinity (at z)"
by (intro tendsto_mult_filterlim_at_infinity[of _ "f z"]
filterlim_compose[OF filterlim_inverse_at_infinity])+
(insert assms, auto simp: isCont_def)
thus ?thesis by (simp add: divide_simps is_pole_def)
qed

lemma is_pole_basic:
assumes "f holomorphic_on A" "open A" "z ∈ A" "f z ≠ 0" "n > 0"
shows   "is_pole (λw. f w / (w - z) ^ n) z"
proof (rule is_pole_divide)
have "continuous_on A f" by (rule holomorphic_on_imp_continuous_on) fact
with assms show "isCont f z" by (auto simp: continuous_on_eq_continuous_at)
have "filterlim (λw. (w - z) ^ n) (nhds 0) (at z)"
using assms by (auto intro!: tendsto_eq_intros)
thus "filterlim (λw. (w - z) ^ n) (at 0) (at z)"
by (intro filterlim_atI tendsto_eq_intros)
(insert assms, auto simp: eventually_at_filter)
qed fact+

lemma is_pole_basic':
assumes "f holomorphic_on A" "open A" "0 ∈ A" "f 0 ≠ 0" "n > 0"
shows   "is_pole (λw. f w / w ^ n) 0"
using is_pole_basic[of f A 0] assms by simp

text ‹The proposition
@{term "∃x. ((f::complex⇒complex) ⤏ x) (at z) ∨ is_pole f z"}
can be interpreted as the complex function @{term f} has a non-essential singularity at @{term z}
(i.e. the singularity is either removable or a pole).›
definition not_essential::"[complex ⇒ complex, complex] ⇒ bool" where
"not_essential f z = (∃x. f─z→x ∨ is_pole f z)"

definition isolated_singularity_at::"[complex ⇒ complex, complex] ⇒ bool" where
"isolated_singularity_at f z = (∃r>0. f analytic_on ball z r-{z})"

named_theorems singularity_intros "introduction rules for singularities"

lemma holomorphic_factor_unique:
fixes f::"complex ⇒ complex" and z::complex and r::real and m n::int
assumes "r>0" "g z≠0" "h z≠0"
and asm:"∀w∈ball z r-{z}. f w = g w * (w-z) powr n ∧ g w≠0 ∧ f w =  h w * (w - z) powr m ∧ h w≠0"
and g_holo:"g holomorphic_on ball z r" and h_holo:"h holomorphic_on ball z r"
shows "n=m"
proof -
have [simp]:"at z within ball z r ≠ bot" using ‹r>0›
by (auto simp add:at_within_ball_bot_iff)
have False when "n>m"
proof -
have "(h ⤏ 0) (at z within ball z r)"
proof (rule Lim_transform_within[OF _ ‹r>0›, where f="λw. (w - z) powr (n - m) * g w"])
have "∀w∈ball z r-{z}. h w = (w-z)powr(n-m) * g w"
using ‹n>m› asm ‹r>0›
apply (auto simp add:field_simps powr_diff)
by force
then show "⟦x' ∈ ball z r; 0 < dist x' z;dist x' z < r⟧
⟹ (x' - z) powr (n - m) * g x' = h x'" for x' by auto
next
define F where "F ≡ at z within ball z r"
define f' where "f' ≡ λx. (x - z) powr (n-m)"
have "f' z=0" using ‹n>m› unfolding f'_def by auto
moreover have "continuous F f'" unfolding f'_def F_def continuous_def
apply (subst netlimit_within)
using ‹n>m› by (auto intro!:tendsto_powr_complex_0 tendsto_eq_intros)
ultimately have "(f' ⤏ 0) F" unfolding F_def
by (simp add: continuous_within)
moreover have "(g ⤏ g z) F"
using holomorphic_on_imp_continuous_on[OF g_holo,unfolded continuous_on_def] ‹r>0›
unfolding F_def by auto
ultimately show " ((λw. f' w * g w) ⤏ 0) F" using tendsto_mult by fastforce
qed
moreover have "(h ⤏ h z) (at z within ball z r)"
using holomorphic_on_imp_continuous_on[OF h_holo]
by (auto simp add:continuous_on_def ‹r>0›)
ultimately have "h z=0" by (auto intro!: tendsto_unique)
thus False using ‹h z≠0› by auto
qed
moreover have False when "m>n"
proof -
have "(g ⤏ 0) (at z within ball z r)"
proof (rule Lim_transform_within[OF _ ‹r>0›, where f="λw. (w - z) powr (m - n) * h w"])
have "∀w∈ball z r -{z}. g w = (w-z) powr (m-n) * h w" using ‹m>n› asm
apply (auto simp add:field_simps powr_diff)
by force
then show "⟦x' ∈ ball z r; 0 < dist x' z;dist x' z < r⟧
⟹ (x' - z) powr (m - n) * h x' = g x'" for x' by auto
next
define F where "F ≡ at z within ball z r"
define f' where "f' ≡λx. (x - z) powr (m-n)"
have "f' z=0" using ‹m>n› unfolding f'_def by auto
moreover have "continuous F f'" unfolding f'_def F_def continuous_def
apply (subst netlimit_within)
using ‹m>n› by (auto intro!:tendsto_powr_complex_0 tendsto_eq_intros)
ultimately have "(f' ⤏ 0) F" unfolding F_def
by (simp add: continuous_within)
moreover have "(h ⤏ h z) F"
using holomorphic_on_imp_continuous_on[OF h_holo,unfolded continuous_on_def] ‹r>0›
unfolding F_def by auto
ultimately show " ((λw. f' w * h w) ⤏ 0) F" using tendsto_mult by fastforce
qed
moreover have "(g ⤏ g z) (at z within ball z r)"
using holomorphic_on_imp_continuous_on[OF g_holo]
by (auto simp add:continuous_on_def ‹r>0›)
ultimately have "g z=0" by (auto intro!: tendsto_unique)
thus False using ‹g z≠0› by auto
qed
ultimately show "n=m" by fastforce
qed

lemma holomorphic_factor_puncture:
assumes f_iso:"isolated_singularity_at f z"
and "not_essential f z" ― ‹@{term f} has either a removable singularity or a pole at @{term z}›
and non_zero:"∃⇩Fw in (at z). f w≠0" ― ‹@{term f} will not be constantly zero in a neighbour of @{term z}›
shows "∃!n::int. ∃g r. 0 < r ∧ g holomorphic_on cball z r ∧ g z≠0
∧ (∀w∈cball z r-{z}. f w = g w * (w-z) powr n ∧ g w≠0)"
proof -
define P where "P = (λf n g r. 0 < r ∧ g holomorphic_on cball z r ∧ g z≠0
∧ (∀w∈cball z r - {z}. f w = g w * (w-z) powr (of_int n)  ∧ g w≠0))"
have imp_unique:"∃!n::int. ∃g r. P f n g r" when "∃n g r. P f n g r"
proof (rule ex_ex1I[OF that])
fix n1 n2 :: int
assume g1_asm:"∃g1 r1. P f n1 g1 r1" and g2_asm:"∃g2 r2. P f n2 g2 r2"
define fac where "fac ≡ λn g r. ∀w∈cball z r-{z}. f w = g w * (w - z) powr (of_int n) ∧ g w ≠ 0"
obtain g1 r1 where "0 < r1" and g1_holo: "g1 holomorphic_on cball z r1" and "g1 z≠0"
and "fac n1 g1 r1" using g1_asm unfolding P_def fac_def by auto
obtain g2 r2 where "0 < r2" and g2_holo: "g2 holomorphic_on cball z r2" and "g2 z≠0"
and "fac n2 g2 r2" using g2_asm unfolding P_def fac_def by auto
define r where "r ≡ min r1 r2"
have "r>0" using ‹r1>0› ‹r2>0› unfolding r_def by auto
moreover have "∀w∈ball z r-{z}. f w = g1 w * (w-z) powr n1 ∧ g1 w≠0
∧ f w = g2 w * (w - z) powr n2  ∧ g2 w≠0"
using ‹fac n1 g1 r1› ‹fac n2 g2 r2›   unfolding fac_def r_def
by fastforce
ultimately show "n1=n2" using g1_holo g2_holo ‹g1 z≠0› ‹g2 z≠0›
apply (elim holomorphic_factor_unique)
by (auto simp add:r_def)
qed

have P_exist:"∃ n g r. P h n g r" when
"∃z'. (h ⤏ z') (at z)" "isolated_singularity_at h z"  "∃⇩Fw in (at z). h w≠0"
for h
proof -
from that(2) obtain r where "r>0" "h analytic_on ball z r - {z}"
unfolding isolated_singularity_at_def by auto
obtain z' where "(h ⤏ z') (at z)" using ‹∃z'. (h ⤏ z') (at z)› by auto
define h' where "h'=(λx. if x=z then z' else h x)"
have "h' holomorphic_on ball z r"
apply (rule no_isolated_singularity'[of "{z}"])
subgoal by (metis LIM_equal Lim_at_imp_Lim_at_within ‹h ─z→ z'› empty_iff h'_def insert_iff)
subgoal using ‹h analytic_on ball z r - {z}› analytic_imp_holomorphic h'_def holomorphic_transform
by fastforce
by auto
have ?thesis when "z'=0"
proof -
have "h' z=0" using that unfolding h'_def by auto
moreover have "¬ h' constant_on ball z r"
using ‹∃⇩Fw in (at z). h w≠0› unfolding constant_on_def frequently_def eventually_at h'_def
apply simp
by (metis ‹0 < r› centre_in_ball dist_commute mem_ball that)
moreover note ‹h' holomorphic_on ball z r›
ultimately obtain g r1 n where "0 < n" "0 < r1" "ball z r1 ⊆ ball z r" and
g:"g holomorphic_on ball z r1"
"⋀w. w ∈ ball z r1 ⟹ h' w = (w - z) ^ n * g w"
"⋀w. w ∈ ball z r1 ⟹ g w ≠ 0"
using holomorphic_factor_zero_nonconstant[of _ "ball z r" z thesis,simplified,
OF ‹h' holomorphic_on ball z r› ‹r>0› ‹h' z=0› ‹¬ h' constant_on ball z r›]
by (auto simp add:dist_commute)
define rr where "rr=r1/2"
have "P h' n g rr"
unfolding P_def rr_def
using ‹n>0› ‹r1>0› g by (auto simp add:powr_nat)
then have "P h n g rr"
unfolding h'_def P_def by auto
then show ?thesis unfolding P_def by blast
qed
moreover have ?thesis when "z'≠0"
proof -
have "h' z≠0" using that unfolding h'_def by auto
obtain r1 where "r1>0" "cball z r1 ⊆ ball z r" "∀x∈cball z r1. h' x≠0"
proof -
have "isCont h' z" "h' z≠0"
by (auto simp add: Lim_cong_within ‹h ─z→ z'› ‹z'≠0› continuous_at h'_def)
then obtain r2 where r2:"r2>0" "∀x∈ball z r2. h' x≠0"
using continuous_at_avoid[of z h' 0 ] unfolding ball_def by auto
define r1 where "r1=min r2 r / 2"
have "0 < r1" "cball z r1 ⊆ ball z r"
using ‹r2>0› ‹r>0› unfolding r1_def by auto
moreover have "∀x∈cball z r1. h' x ≠ 0"
using r2 unfolding r1_def by simp
ultimately show ?thesis using that by auto
qed
then have "P h' 0 h' r1" using ‹h' holomorphic_on ball z r› unfolding P_def by auto
then have "P h 0 h' r1" unfolding P_def h'_def by auto
then show ?thesis ```