section \<open>Winding numbers\<close>
theory Winding_Numbers
imports Cauchy_Integral_Theorem
begin
subsection \<open>Definition\<close>
definition\<^marker>\<open>tag important\<close> winding_number_prop :: "[real \<Rightarrow> complex, complex, real, real \<Rightarrow> complex, complex] \<Rightarrow> bool" where
"winding_number_prop \<gamma> z e p n \<equiv>
valid_path p \<and> z \<notin> path_image p \<and>
pathstart p = pathstart \<gamma> \<and>
pathfinish p = pathfinish \<gamma> \<and>
(\<forall>t \<in> {0..1}. norm(\<gamma> t - p t) < e) \<and>
contour_integral p (\<lambda>w. 1/(w - z)) = 2 * pi * \<i> * n"
definition\<^marker>\<open>tag important\<close> winding_number:: "[real \<Rightarrow> complex, complex] \<Rightarrow> complex" where
"winding_number \<gamma> z \<equiv> SOME n. \<forall>e > 0. \<exists>p. winding_number_prop \<gamma> z e p n"
lemma winding_number:
assumes "path \<gamma>" "z \<notin> path_image \<gamma>" "0 < e"
shows "\<exists>p. winding_number_prop \<gamma> z e p (winding_number \<gamma> z)"
proof -
have "path_image \<gamma> \<subseteq> UNIV - {z}"
using assms by blast
then obtain d
where d: "d>0"
and pi_eq: "\<And>h1 h2. valid_path h1 \<and> valid_path h2 \<and>
(\<forall>t\<in>{0..1}. cmod (h1 t - \<gamma> t) < d \<and> cmod (h2 t - \<gamma> t) < d) \<and>
pathstart h2 = pathstart h1 \<and> pathfinish h2 = pathfinish h1 \<longrightarrow>
path_image h1 \<subseteq> UNIV - {z} \<and> path_image h2 \<subseteq> UNIV - {z} \<and>
(\<forall>f. f holomorphic_on UNIV - {z} \<longrightarrow> contour_integral h2 f = contour_integral h1 f)"
using contour_integral_nearby_ends [of "UNIV - {z}" \<gamma>] assms by (auto simp: open_delete)
then obtain h where h: "polynomial_function h \<and> pathstart h = pathstart \<gamma> \<and> pathfinish h = pathfinish \<gamma> \<and>
(\<forall>t \<in> {0..1}. norm(h t - \<gamma> t) < d/2)"
using path_approx_polynomial_function [OF \<open>path \<gamma>\<close>, of "d/2"] d by (metis half_gt_zero_iff)
define nn where "nn = 1/(2* pi*\<i>) * contour_integral h (\<lambda>w. 1/(w - z))"
have "\<exists>n. \<forall>e > 0. \<exists>p. winding_number_prop \<gamma> z e p n"
proof (rule_tac x=nn in exI, clarify)
fix e::real
assume e: "e>0"
obtain p where p: "polynomial_function p \<and>
pathstart p = pathstart \<gamma> \<and> pathfinish p = pathfinish \<gamma> \<and> (\<forall>t\<in>{0..1}. cmod (p t - \<gamma> t) < min e (d/2))"
using path_approx_polynomial_function [OF \<open>path \<gamma>\<close>, of "min e (d/2)"] d \<open>0<e\<close>
by (metis min_less_iff_conj zero_less_divide_iff zero_less_numeral)
have "(\<lambda>w. 1 / (w - z)) holomorphic_on UNIV - {z}"
by (auto simp: intro!: holomorphic_intros)
then have "winding_number_prop \<gamma> z e p nn"
using pi_eq [of h p] h p d
by (auto simp: valid_path_polynomial_function norm_minus_commute nn_def winding_number_prop_def)
then show "\<exists>p. winding_number_prop \<gamma> z e p nn"
by metis
qed
then show ?thesis
unfolding winding_number_def by (rule someI2_ex) (blast intro: \<open>0<e\<close>)
qed
lemma winding_number_unique:
assumes \<gamma>: "path \<gamma>" "z \<notin> path_image \<gamma>"
and pi: "\<And>e. e>0 \<Longrightarrow> \<exists>p. winding_number_prop \<gamma> z e p n"
shows "winding_number \<gamma> z = n"
proof -
have "path_image \<gamma> \<subseteq> UNIV - {z}"
using assms by blast
then obtain e
where e: "e>0"
and pi_eq: "\<And>h1 h2 f. \<lbrakk>valid_path h1; valid_path h2;
(\<forall>t\<in>{0..1}. cmod (h1 t - \<gamma> t) < e \<and> cmod (h2 t - \<gamma> t) < e);
pathstart h2 = pathstart h1; pathfinish h2 = pathfinish h1; f holomorphic_on UNIV - {z}\<rbrakk> \<Longrightarrow>
contour_integral h2 f = contour_integral h1 f"
using contour_integral_nearby_ends [of "UNIV - {z}" \<gamma>] assms by (auto simp: open_delete)
obtain p where p: "winding_number_prop \<gamma> z e p n"
using pi [OF e] by blast
obtain q where q: "winding_number_prop \<gamma> z e q (winding_number \<gamma> z)"
using winding_number [OF \<gamma> e] by blast
have "2 * complex_of_real pi * \<i> * n = contour_integral p (\<lambda>w. 1 / (w - z))"
using p by (auto simp: winding_number_prop_def)
also have "\<dots> = contour_integral q (\<lambda>w. 1 / (w - z))"
proof (rule pi_eq)
show "(\<lambda>w. 1 / (w - z)) holomorphic_on UNIV - {z}"
by (auto intro!: holomorphic_intros)
qed (use p q in \<open>auto simp: winding_number_prop_def norm_minus_commute\<close>)
also have "\<dots> = 2 * complex_of_real pi * \<i> * winding_number \<gamma> z"
using q by (auto simp: winding_number_prop_def)
finally have "2 * complex_of_real pi * \<i> * n = 2 * complex_of_real pi * \<i> * winding_number \<gamma> z" .
then show ?thesis
by simp
qed
(*NB not winding_number_prop here due to the loop in p*)
lemma winding_number_unique_loop:
assumes \<gamma>: "path \<gamma>" "z \<notin> path_image \<gamma>"
and loop: "pathfinish \<gamma> = pathstart \<gamma>"
and pi:
"\<And>e. e>0 \<Longrightarrow> \<exists>p. valid_path p \<and> z \<notin> path_image p \<and>
pathfinish p = pathstart p \<and>
(\<forall>t \<in> {0..1}. norm (\<gamma> t - p t) < e) \<and>
contour_integral p (\<lambda>w. 1/(w - z)) = 2 * pi * \<i> * n"
shows "winding_number \<gamma> z = n"
proof -
have "path_image \<gamma> \<subseteq> UNIV - {z}"
using assms by blast
then obtain e
where e: "e>0"
and pi_eq: "\<And>h1 h2 f. \<lbrakk>valid_path h1; valid_path h2;
(\<forall>t\<in>{0..1}. cmod (h1 t - \<gamma> t) < e \<and> cmod (h2 t - \<gamma> t) < e);
pathfinish h1 = pathstart h1; pathfinish h2 = pathstart h2; f holomorphic_on UNIV - {z}\<rbrakk> \<Longrightarrow>
contour_integral h2 f = contour_integral h1 f"
using contour_integral_nearby_loops [of "UNIV - {z}" \<gamma>] assms by (auto simp: open_delete)
obtain p where p:
"valid_path p \<and> z \<notin> path_image p \<and> pathfinish p = pathstart p \<and>
(\<forall>t \<in> {0..1}. norm (\<gamma> t - p t) < e) \<and>
contour_integral p (\<lambda>w. 1/(w - z)) = 2 * pi * \<i> * n"
using pi [OF e] by blast
obtain q where q: "winding_number_prop \<gamma> z e q (winding_number \<gamma> z)"
using winding_number [OF \<gamma> e] by blast
have "2 * complex_of_real pi * \<i> * n = contour_integral p (\<lambda>w. 1 / (w - z))"
using p by auto
also have "\<dots> = contour_integral q (\<lambda>w. 1 / (w - z))"
proof (rule pi_eq)
show "(\<lambda>w. 1 / (w - z)) holomorphic_on UNIV - {z}"
by (auto intro!: holomorphic_intros)
qed (use p q loop in \<open>auto simp: winding_number_prop_def norm_minus_commute\<close>)
also have "\<dots> = 2 * complex_of_real pi * \<i> * winding_number \<gamma> z"
using q by (auto simp: winding_number_prop_def)
finally have "2 * complex_of_real pi * \<i> * n = 2 * complex_of_real pi * \<i> * winding_number \<gamma> z" .
then show ?thesis
by simp
qed
proposition winding_number_valid_path:
assumes "valid_path \<gamma>" "z \<notin> path_image \<gamma>"
shows "winding_number \<gamma> z = 1/(2*pi*\<i>) * contour_integral \<gamma> (\<lambda>w. 1/(w - z))"
by (rule winding_number_unique)
(use assms in \<open>auto simp: valid_path_imp_path winding_number_prop_def\<close>)
proposition has_contour_integral_winding_number:
assumes \<gamma>: "valid_path \<gamma>" "z \<notin> path_image \<gamma>"
shows "((\<lambda>w. 1/(w - z)) has_contour_integral (2*pi*\<i>*winding_number \<gamma> z)) \<gamma>"
by (simp add: winding_number_valid_path has_contour_integral_integral contour_integrable_inversediff assms)
lemma winding_number_trivial [simp]: "z \<noteq> a \<Longrightarrow> winding_number(linepath a a) z = 0"
by (simp add: winding_number_valid_path)
lemma winding_number_subpath_trivial [simp]: "z \<noteq> g x \<Longrightarrow> winding_number (subpath x x g) z = 0"
by (simp add: path_image_subpath winding_number_valid_path)
lemma winding_number_join:
assumes \<gamma>1: "path \<gamma>1" "z \<notin> path_image \<gamma>1"
and \<gamma>2: "path \<gamma>2" "z \<notin> path_image \<gamma>2"
and "pathfinish \<gamma>1 = pathstart \<gamma>2"
shows "winding_number(\<gamma>1 +++ \<gamma>2) z = winding_number \<gamma>1 z + winding_number \<gamma>2 z"
proof (rule winding_number_unique)
show "\<exists>p. winding_number_prop (\<gamma>1 +++ \<gamma>2) z e p
(winding_number \<gamma>1 z + winding_number \<gamma>2 z)" if "e > 0" for e
proof -
obtain p1 where "winding_number_prop \<gamma>1 z e p1 (winding_number \<gamma>1 z)"
using \<open>0 < e\<close> \<gamma>1 winding_number by blast
moreover
obtain p2 where "winding_number_prop \<gamma>2 z e p2 (winding_number \<gamma>2 z)"
using \<open>0 < e\<close> \<gamma>2 winding_number by blast
ultimately
have "winding_number_prop (\<gamma>1+++\<gamma>2) z e (p1+++p2) (winding_number \<gamma>1 z + winding_number \<gamma>2 z)"
using assms
apply (simp add: winding_number_prop_def not_in_path_image_join contour_integrable_inversediff algebra_simps)
apply (auto simp: joinpaths_def)
done
then show ?thesis
by blast
qed
qed (use assms in \<open>auto simp: not_in_path_image_join\<close>)
lemma winding_number_reversepath:
assumes "path \<gamma>" "z \<notin> path_image \<gamma>"
shows "winding_number(reversepath \<gamma>) z = - (winding_number \<gamma> z)"
proof (rule winding_number_unique)
show "\<exists>p. winding_number_prop (reversepath \<gamma>) z e p (- winding_number \<gamma> z)" if "e > 0" for e
proof -
obtain p where "winding_number_prop \<gamma> z e p (winding_number \<gamma> z)"
using \<open>0 < e\<close> assms winding_number by blast
then have "winding_number_prop (reversepath \<gamma>) z e (reversepath p) (- winding_number \<gamma> z)"
using assms unfolding winding_number_prop_def
apply (simp add: contour_integral_reversepath contour_integrable_inversediff valid_path_imp_reverse)
apply (auto simp: reversepath_def)
done
then show ?thesis
by blast
qed
qed (use assms in auto)
lemma winding_number_shiftpath:
assumes \<gamma>: "path \<gamma>" "z \<notin> path_image \<gamma>"
and "pathfinish \<gamma> = pathstart \<gamma>" "a \<in> {0..1}"
shows "winding_number(shiftpath a \<gamma>) z = winding_number \<gamma> z"
proof (rule winding_number_unique_loop)
show "\<exists>p. valid_path p \<and> z \<notin> path_image p \<and> pathfinish p = pathstart p \<and>
(\<forall>t\<in>{0..1}. cmod (shiftpath a \<gamma> t - p t) < e) \<and>
contour_integral p (\<lambda>w. 1 / (w - z)) =
2 * pi * \<i> * winding_number \<gamma> z"
if "e > 0" for e
proof -
obtain p where "winding_number_prop \<gamma> z e p (winding_number \<gamma> z)"
using \<open>0 < e\<close> assms winding_number by blast
then show ?thesis
apply (rule_tac x="shiftpath a p" in exI)
using assms that
apply (auto simp: winding_number_prop_def path_image_shiftpath pathfinish_shiftpath pathstart_shiftpath contour_integral_shiftpath)
apply (simp add: shiftpath_def)
done
qed
qed (use assms in \<open>auto simp: path_shiftpath path_image_shiftpath pathfinish_shiftpath pathstart_shiftpath\<close>)
lemma winding_number_split_linepath:
assumes "c \<in> closed_segment a b" "z \<notin> closed_segment a b"
shows "winding_number(linepath a b) z = winding_number(linepath a c) z + winding_number(linepath c b) z"
proof -
have "z \<notin> closed_segment a c" "z \<notin> closed_segment c b"
using assms by (meson convex_contains_segment convex_segment ends_in_segment subsetCE)+
then show ?thesis
using assms
by (simp add: winding_number_valid_path contour_integral_split_linepath [symmetric] continuous_on_inversediff field_simps)
qed
lemma winding_number_cong:
"(\<And>t. \<lbrakk>0 \<le> t; t \<le> 1\<rbrakk> \<Longrightarrow> p t = q t) \<Longrightarrow> winding_number p z = winding_number q z"
by (simp add: winding_number_def winding_number_prop_def pathstart_def pathfinish_def)
lemma winding_number_constI:
assumes "c\<noteq>z" and g: "\<And>t. \<lbrakk>0\<le>t; t\<le>1\<rbrakk> \<Longrightarrow> g t = c"
shows "winding_number g z = 0"
proof -
have "winding_number g z = winding_number (linepath c c) z"
using g winding_number_cong by fastforce
moreover have "winding_number (linepath c c) z = 0"
using \<open>c\<noteq>z\<close> by auto
ultimately show ?thesis by auto
qed
lemma winding_number_offset: "winding_number p z = winding_number (\<lambda>w. p w - z) 0"
unfolding winding_number_def
proof (intro ext arg_cong [where f = Eps] arg_cong [where f = All] imp_cong refl, safe)
fix n e g
assume "0 < e" and g: "winding_number_prop p z e g n"
then show "\<exists>r. winding_number_prop (\<lambda>w. p w - z) 0 e r n"
by (rule_tac x="\<lambda>t. g t - z" in exI)
(force simp: winding_number_prop_def contour_integral_integral valid_path_def path_defs
vector_derivative_def has_vector_derivative_diff_const piecewise_C1_differentiable_diff C1_differentiable_imp_piecewise)
next
fix n e g
assume "0 < e" and g: "winding_number_prop (\<lambda>w. p w - z) 0 e g n"
then have "winding_number_prop p z e (\<lambda>t. g t + z) n"
apply (simp add: winding_number_prop_def contour_integral_integral valid_path_def path_defs
piecewise_C1_differentiable_add vector_derivative_def has_vector_derivative_add_const C1_differentiable_imp_piecewise)
apply (force simp: algebra_simps)
done
then show "\<exists>r. winding_number_prop p z e r n"
by metis
qed
lemma winding_number_negatepath:
assumes \<gamma>: "valid_path \<gamma>" and 0: "0 \<notin> path_image \<gamma>"
shows "winding_number(uminus \<circ> \<gamma>) 0 = winding_number \<gamma> 0"
proof -
have "(/) 1 contour_integrable_on \<gamma>"
using "0" \<gamma> contour_integrable_inversediff by fastforce
then have "((\<lambda>z. 1/z) has_contour_integral contour_integral \<gamma> ((/) 1)) \<gamma>"
by (rule has_contour_integral_integral)
then have "((\<lambda>z. 1 / - z) has_contour_integral - contour_integral \<gamma> ((/) 1)) \<gamma>"
using has_contour_integral_neg by auto
then have "contour_integral (uminus \<circ> \<gamma>) ((/) 1) =
contour_integral \<gamma> ((/) 1)"
using \<gamma> by (simp add: contour_integral_unique has_contour_integral_negatepath)
then show ?thesis
using assms by (simp add: winding_number_valid_path valid_path_negatepath image_def path_defs)
qed
text \<open>A combined theorem deducing several things piecewise.\<close>
lemma winding_number_join_pos_combined:
"\<lbrakk>valid_path \<gamma>1; z \<notin> path_image \<gamma>1; 0 < Re(winding_number \<gamma>1 z);
valid_path \<gamma>2; z \<notin> path_image \<gamma>2; 0 < Re(winding_number \<gamma>2 z); pathfinish \<gamma>1 = pathstart \<gamma>2\<rbrakk>
\<Longrightarrow> valid_path(\<gamma>1 +++ \<gamma>2) \<and> z \<notin> path_image(\<gamma>1 +++ \<gamma>2) \<and> 0 < Re(winding_number(\<gamma>1 +++ \<gamma>2) z)"
by (simp add: valid_path_join path_image_join winding_number_join valid_path_imp_path)
subsubsection\<^marker>\<open>tag unimportant\<close> \<open>Useful sufficient conditions for the winding number to be positive\<close>
lemma Re_winding_number:
"\<lbrakk>valid_path \<gamma>; z \<notin> path_image \<gamma>\<rbrakk>
\<Longrightarrow> Re(winding_number \<gamma> z) = Im(contour_integral \<gamma> (\<lambda>w. 1/(w - z))) / (2*pi)"
by (simp add: winding_number_valid_path field_simps Re_divide power2_eq_square)
lemma winding_number_pos_le:
assumes \<gamma>: "valid_path \<gamma>" "z \<notin> path_image \<gamma>"
and ge: "\<And>x. \<lbrakk>0 < x; x < 1\<rbrakk> \<Longrightarrow> 0 \<le> Im (vector_derivative \<gamma> (at x) * cnj(\<gamma> x - z))"
shows "0 \<le> Re(winding_number \<gamma> z)"
proof -
have ge0: "0 \<le> Im (vector_derivative \<gamma> (at x) / (\<gamma> x - z))" if x: "0 < x" "x < 1" for x
using ge by (simp add: Complex.Im_divide algebra_simps x)
let ?vd = "\<lambda>x. 1 / (\<gamma> x - z) * vector_derivative \<gamma> (at x)"
let ?int = "\<lambda>z. contour_integral \<gamma> (\<lambda>w. 1 / (w - z))"
have "0 \<le> Im (?int z)"
proof (rule has_integral_component_nonneg [of \<i>, simplified])
show "\<And>x. x \<in> cbox 0 1 \<Longrightarrow> 0 \<le> Im (if 0 < x \<and> x < 1 then ?vd x else 0)"
by (force simp: ge0)
have "((\<lambda>a. 1 / (a - z)) has_contour_integral contour_integral \<gamma> (\<lambda>w. 1 / (w - z))) \<gamma>"
using \<gamma> by (simp flip: add: contour_integrable_inversediff has_contour_integral_integral)
then have hi: "(?vd has_integral ?int z) (cbox 0 1)"
using has_contour_integral by auto
show "((\<lambda>x. if 0 < x \<and> x < 1 then ?vd x else 0) has_integral ?int z) (cbox 0 1)"
by (rule has_integral_spike_interior [OF hi]) simp
qed
then show ?thesis
by (simp add: Re_winding_number [OF \<gamma>] field_simps)
qed
lemma winding_number_pos_lt_lemma:
assumes \<gamma>: "valid_path \<gamma>" "z \<notin> path_image \<gamma>"
and e: "0 < e"
and ge: "\<And>x. \<lbrakk>0 < x; x < 1\<rbrakk> \<Longrightarrow> e \<le> Im (vector_derivative \<gamma> (at x) / (\<gamma> x - z))"
shows "0 < Re(winding_number \<gamma> z)"
proof -
let ?vd = "\<lambda>x. 1 / (\<gamma> x - z) * vector_derivative \<gamma> (at x)"
let ?int = "\<lambda>z. contour_integral \<gamma> (\<lambda>w. 1 / (w - z))"
have "e \<le> Im (contour_integral \<gamma> (\<lambda>w. 1 / (w - z)))"
proof (rule has_integral_component_le [of \<i> "\<lambda>x. \<i>*e" "\<i>*e" "{0..1}", simplified])
have "((\<lambda>a. 1 / (a - z)) has_contour_integral contour_integral \<gamma> (\<lambda>w. 1 / (w - z))) \<gamma>"
thm has_integral_component_le [of \<i> "\<lambda>x. \<i>*e" "\<i>*e" "{0..1}", simplified]
using \<gamma> by (simp add: contour_integrable_inversediff has_contour_integral_integral)
then have hi: "(?vd has_integral ?int z) (cbox 0 1)"
using has_contour_integral by auto
show "((\<lambda>x. if 0 < x \<and> x < 1 then ?vd x else \<i> * e) has_integral ?int z) {0..1}"
by (rule has_integral_spike_interior [OF hi, simplified box_real]) (use e in simp)
show "\<And>x. 0 \<le> x \<and> x \<le> 1 \<Longrightarrow> e \<le> Im (if 0 < x \<and> x < 1 then ?vd x else \<i> * e)"
by (simp add: ge)
qed (use has_integral_const_real [of _ 0 1] in auto)
with e show ?thesis
by (simp add: Re_winding_number [OF \<gamma>] field_simps)
qed
lemma winding_number_pos_lt:
assumes \<gamma>: "valid_path \<gamma>" "z \<notin> path_image \<gamma>"
and e: "0 < e"
and ge: "\<And>x. \<lbrakk>0 < x; x < 1\<rbrakk> \<Longrightarrow> e \<le> Im (vector_derivative \<gamma> (at x) * cnj(\<gamma> x - z))"
shows "0 < Re (winding_number \<gamma> z)"
proof -
have bm: "bounded ((\<lambda>w. w - z) ` (path_image \<gamma>))"
using bounded_translation [of _ "-z"] \<gamma> by (simp add: bounded_valid_path_image)
then obtain B where B: "B > 0" and Bno: "\<And>x. x \<in> (\<lambda>w. w - z) ` (path_image \<gamma>) \<Longrightarrow> norm x \<le> B"
using bounded_pos [THEN iffD1, OF bm] by blast
{ fix x::real assume x: "0 < x" "x < 1"
then have B2: "cmod (\<gamma> x - z)^2 \<le> B^2" using Bno [of "\<gamma> x - z"]
by (simp add: path_image_def power2_eq_square mult_mono')
with x have "\<gamma> x \<noteq> z" using \<gamma>
using path_image_def by fastforce
then have "e / B\<^sup>2 \<le> e / (cmod (\<gamma> x - z))\<^sup>2"
using B B2 e by (auto simp: divide_left_mono)
also have "... \<le> Im (vector_derivative \<gamma> (at x) * cnj (\<gamma> x - z)) / (cmod (\<gamma> x - z))\<^sup>2"
using ge [OF x] by (auto simp: divide_right_mono)
finally have "e / B\<^sup>2 \<le> Im (vector_derivative \<gamma> (at x) * cnj (\<gamma> x - z)) / (cmod (\<gamma> x - z))\<^sup>2" .
then have "e / B\<^sup>2 \<le> Im (vector_derivative \<gamma> (at x) / (\<gamma> x - z))"
by (simp add: complex_div_cnj [of _ "\<gamma> x - z" for x] del: complex_cnj_diff times_complex.sel)
} note * = this
show ?thesis
using e B by (simp add: * winding_number_pos_lt_lemma [OF \<gamma>, of "e/B^2"])
qed
subsection\<open>The winding number is an integer\<close>
text\<open>Proof from the book Complex Analysis by Lars V. Ahlfors, Chapter 4, section 2.1,
Also on page 134 of Serge Lang's book with the name title, etc.\<close>
lemma exp_fg:
fixes z::complex
assumes g: "(g has_vector_derivative g') (at x within s)"
and f: "(f has_vector_derivative (g' / (g x - z))) (at x within s)"
and z: "g x \<noteq> z"
shows "((\<lambda>x. exp(-f x) * (g x - z)) has_vector_derivative 0) (at x within s)"
proof -
have *: "(exp \<circ> (\<lambda>x. (- f x)) has_vector_derivative - (g' / (g x - z)) * exp (- f x)) (at x within s)"
using assms unfolding has_vector_derivative_def scaleR_conv_of_real
by (auto intro!: derivative_eq_intros)
show ?thesis
using z by (auto intro!: derivative_eq_intros * [unfolded o_def] g)
qed
lemma winding_number_exp_integral:
fixes z::complex
assumes \<gamma>: "\<gamma> piecewise_C1_differentiable_on {a..b}"
and ab: "a \<le> b"
and z: "z \<notin> \<gamma> ` {a..b}"
shows "(\<lambda>x. vector_derivative \<gamma> (at x) / (\<gamma> x - z)) integrable_on {a..b}"
(is "?thesis1")
"exp (- (integral {a..b} (\<lambda>x. vector_derivative \<gamma> (at x) / (\<gamma> x - z)))) * (\<gamma> b - z) = \<gamma> a - z"
(is "?thesis2")
proof -
let ?D\<gamma> = "\<lambda>x. vector_derivative \<gamma> (at x)"
have [simp]: "\<And>x. a \<le> x \<Longrightarrow> x \<le> b \<Longrightarrow> \<gamma> x \<noteq> z"
using z by force
have con_g: "continuous_on {a..b} \<gamma>"
using \<gamma> by (simp add: piecewise_C1_differentiable_on_def)
obtain k where fink: "finite k" and g_C1_diff: "\<gamma> C1_differentiable_on ({a..b} - k)"
using \<gamma> by (force simp: piecewise_C1_differentiable_on_def)
have \<circ>: "open ({a<..<b} - k)"
using \<open>finite k\<close> by (simp add: finite_imp_closed open_Diff)
moreover have "{a<..<b} - k \<subseteq> {a..b} - k"
by force
ultimately have g_diff_at: "\<And>x. \<lbrakk>x \<notin> k; x \<in> {a<..<b}\<rbrakk> \<Longrightarrow> \<gamma> differentiable at x"
by (metis Diff_iff differentiable_on_subset C1_diff_imp_diff [OF g_C1_diff] differentiable_on_def at_within_open)
{ fix w
assume "w \<noteq> z"
have "continuous_on (ball w (cmod (w - z))) (\<lambda>w. 1 / (w - z))"
by (auto simp: dist_norm intro!: continuous_intros)
moreover have "\<And>x. cmod (w - x) < cmod (w - z) \<Longrightarrow> \<exists>f'. ((\<lambda>w. 1 / (w - z)) has_field_derivative f') (at x)"
by (auto simp: intro!: derivative_eq_intros)
ultimately have "\<exists>h. \<forall>y. norm(y - w) < norm(w - z) \<longrightarrow> (h has_field_derivative 1/(y - z)) (at y)"
using holomorphic_convex_primitive [of "ball w (norm(w - z))" "{}" "\<lambda>w. 1/(w - z)"]
by (force simp: field_differentiable_def Ball_def dist_norm at_within_open_NO_MATCH norm_minus_commute)
}
then obtain h where h: "\<And>w y. w \<noteq> z \<Longrightarrow> norm(y - w) < norm(w - z) \<Longrightarrow> (h w has_field_derivative 1/(y - z)) (at y)"
by meson
have exy: "\<exists>y. ((\<lambda>x. inverse (\<gamma> x - z) * ?D\<gamma> x) has_integral y) {a..b}"
unfolding integrable_on_def [symmetric]
proof (rule contour_integral_local_primitive_any [OF piecewise_C1_imp_differentiable [OF \<gamma>]])
show "\<exists>d h. 0 < d \<and>
(\<forall>y. cmod (y - w) < d \<longrightarrow> (h has_field_derivative inverse (y - z))(at y within - {z}))"
if "w \<in> - {z}" for w
using that inverse_eq_divide has_field_derivative_at_within h
by (metis Compl_insert DiffD2 insertCI right_minus_eq zero_less_norm_iff)
qed simp
have vg_int: "(\<lambda>x. ?D\<gamma> x / (\<gamma> x - z)) integrable_on {a..b}"
unfolding box_real [symmetric] divide_inverse_commute
by (auto intro!: exy integrable_subinterval simp add: integrable_on_def ab)
with ab show ?thesis1
by (simp add: divide_inverse_commute integral_def integrable_on_def)
{ fix t
assume t: "t \<in> {a..b}"
have cball: "continuous_on (ball (\<gamma> t) (dist (\<gamma> t) z)) (\<lambda>x. inverse (x - z))"
using z by (auto intro!: continuous_intros simp: dist_norm)
have icd: "\<And>x. cmod (\<gamma> t - x) < cmod (\<gamma> t - z) \<Longrightarrow> (\<lambda>w. inverse (w - z)) field_differentiable at x"
unfolding field_differentiable_def by (force simp: intro!: derivative_eq_intros)
obtain h where h: "\<And>x. cmod (\<gamma> t - x) < cmod (\<gamma> t - z) \<Longrightarrow>
(h has_field_derivative inverse (x - z)) (at x within {y. cmod (\<gamma> t - y) < cmod (\<gamma> t - z)})"
using holomorphic_convex_primitive [where f = "\<lambda>w. inverse(w - z)", OF convex_ball finite.emptyI cball icd]
by simp (auto simp: ball_def dist_norm that)
have "exp (- (integral {a..t} (\<lambda>x. ?D\<gamma> x / (\<gamma> x - z)))) * (\<gamma> t - z) =\<gamma> a - z"
proof (rule has_derivative_zero_unique_strong_interval [of "{a,b} \<union> k" a b])
show "continuous_on {a..b} (\<lambda>b. exp (- integral {a..b} (\<lambda>x. ?D\<gamma> x / (\<gamma> x - z))) * (\<gamma> b - z))"
by (auto intro!: continuous_intros con_g indefinite_integral_continuous_1 [OF vg_int])
show "((\<lambda>b. exp (- integral {a..b} (\<lambda>x. ?D\<gamma> x / (\<gamma> x - z))) * (\<gamma> b - z)) has_derivative (\<lambda>h. 0))
(at x within {a..b})"
if "x \<in> {a..b} - ({a, b} \<union> k)" for x
proof -
have x: "x \<notin> k" "a < x" "x < b"
using that by auto
then have "x \<in> interior ({a..b} - k)"
using open_subset_interior [OF \<circ>] by fastforce
then have con: "isCont ?D\<gamma> x"
using g_C1_diff x by (auto simp: C1_differentiable_on_eq intro: continuous_on_interior)
then have con_vd: "continuous (at x within {a..b}) (\<lambda>x. ?D\<gamma> x)"
by (rule continuous_at_imp_continuous_within)
have gdx: "\<gamma> differentiable at x"
using x by (simp add: g_diff_at)
then obtain d where d: "(\<gamma> has_derivative (\<lambda>x. x *\<^sub>R d)) (at x)"
by (auto simp add: differentiable_iff_scaleR)
show "((\<lambda>c. exp (- integral {a..c} (\<lambda>x. ?D\<gamma> x / (\<gamma> x - z))) * (\<gamma> c - z)) has_derivative (\<lambda>h. 0))
(at x within {a..b})"
proof (rule exp_fg [unfolded has_vector_derivative_def, simplified])
show "(\<gamma> has_derivative (\<lambda>c. c *\<^sub>R d)) (at x within {a..b})"
using d by (blast intro: has_derivative_at_withinI)
have "((\<lambda>x. integral {a..x} (\<lambda>x. ?D\<gamma> x / (\<gamma> x - z))) has_vector_derivative d / (\<gamma> x - z))
(at x within {a..b})"
proof (rule has_vector_derivative_eq_rhs [OF integral_has_vector_derivative_continuous_at [where S = "{}", simplified]])
show "continuous (at x within {a..b}) (\<lambda>x. vector_derivative \<gamma> (at x) / (\<gamma> x - z))"
using continuous_at_imp_continuous_at_within differentiable_imp_continuous_within gdx x
by (intro con_vd continuous_intros) (force+)
show "vector_derivative \<gamma> (at x) / (\<gamma> x - z) = d / (\<gamma> x - z)"
using d vector_derivative_at
by (simp add: vector_derivative_at has_vector_derivative_def)
qed (use x vg_int in auto)
then show "((\<lambda>x. integral {a..x} (\<lambda>x. ?D\<gamma> x / (\<gamma> x - z))) has_derivative (\<lambda>c. c *\<^sub>R (d / (\<gamma> x - z))))
(at x within {a..b})"
by (auto simp: has_vector_derivative_def)
qed (use x in auto)
qed
qed (use fink t in auto)
}
with ab show ?thesis2
by (simp add: divide_inverse_commute integral_def)
qed
lemma winding_number_exp_2pi:
"\<lbrakk>path p; z \<notin> path_image p\<rbrakk>
\<Longrightarrow> pathfinish p - z = exp (2 * pi * \<i> * winding_number p z) * (pathstart p - z)"
using winding_number [of p z 1] unfolding valid_path_def path_image_def pathstart_def pathfinish_def winding_number_prop_def
by (force dest: winding_number_exp_integral(2) [of _ 0 1 z] simp: field_simps contour_integral_integral exp_minus)
lemma integer_winding_number_eq:
assumes \<gamma>: "path \<gamma>" and z: "z \<notin> path_image \<gamma>"
shows "winding_number \<gamma> z \<in> \<int> \<longleftrightarrow> pathfinish \<gamma> = pathstart \<gamma>"
proof -
obtain p where p: "valid_path p" "z \<notin> path_image p"
"pathstart p = pathstart \<gamma>" "pathfinish p = pathfinish \<gamma>"
and eq: "contour_integral p (\<lambda>w. 1 / (w - z)) = complex_of_real (2 * pi) * \<i> * winding_number \<gamma> z"
using winding_number [OF assms, of 1] unfolding winding_number_prop_def by auto
then have wneq: "winding_number \<gamma> z = winding_number p z"
using eq winding_number_valid_path by force
have iff: "(winding_number \<gamma> z \<in> \<int>) \<longleftrightarrow> (exp (contour_integral p (\<lambda>w. 1 / (w - z))) = 1)"
using eq by (simp add: exp_eq_1 complex_is_Int_iff)
have "\<gamma> 0 \<noteq> z"
by (metis pathstart_def pathstart_in_path_image z)
then have "exp (contour_integral p (\<lambda>w. 1 / (w - z))) = (\<gamma> 1 - z) / (\<gamma> 0 - z)"
using p winding_number_exp_integral(2) [of p 0 1 z]
by (simp add: valid_path_def path_defs contour_integral_integral exp_minus field_split_simps)
then have "winding_number p z \<in> \<int> \<longleftrightarrow> pathfinish p = pathstart p"
using p wneq iff by (auto simp: path_defs)
then show ?thesis using p eq
by (auto simp: winding_number_valid_path)
qed
theorem integer_winding_number:
"\<lbrakk>path \<gamma>; pathfinish \<gamma> = pathstart \<gamma>; z \<notin> path_image \<gamma>\<rbrakk> \<Longrightarrow> winding_number \<gamma> z \<in> \<int>"
by (metis integer_winding_number_eq)
text\<open>If the winding number's magnitude is at least one, then the path must contain points in every direction.*)
We can thus bound the winding number of a path that doesn't intersect a given ray. \<close>
lemma winding_number_pos_meets:
fixes z::complex
assumes \<gamma>: "valid_path \<gamma>" and z: "z \<notin> path_image \<gamma>" and 1: "Re (winding_number \<gamma> z) \<ge> 1"
and w: "w \<noteq> z"
shows "\<exists>a::real. 0 < a \<and> z + a*(w - z) \<in> path_image \<gamma>"
proof -
have [simp]: "\<And>x. 0 \<le> x \<Longrightarrow> x \<le> 1 \<Longrightarrow> \<gamma> x \<noteq> z"
using z by (auto simp: path_image_def)
have [simp]: "z \<notin> \<gamma> ` {0..1}"
using path_image_def z by auto
have gpd: "\<gamma> piecewise_C1_differentiable_on {0..1}"
using \<gamma> valid_path_def by blast
define r where "r = (w - z) / (\<gamma> 0 - z)"
have [simp]: "r \<noteq> 0"
using w z by (auto simp: r_def)
have cont: "continuous_on {0..1}
(\<lambda>x. Im (integral {0..x} (\<lambda>x. vector_derivative \<gamma> (at x) / (\<gamma> x - z))))"
by (intro continuous_intros indefinite_integral_continuous_1 winding_number_exp_integral [OF gpd]; simp)
have "Arg2pi r \<le> 2*pi"
by (simp add: Arg2pi less_eq_real_def)
also have "\<dots> \<le> Im (integral {0..1} (\<lambda>x. vector_derivative \<gamma> (at x) / (\<gamma> x - z)))"
using 1
by (simp add: winding_number_valid_path [OF \<gamma> z] contour_integral_integral Complex.Re_divide field_simps power2_eq_square)
finally have "Arg2pi r \<le> Im (integral {0..1} (\<lambda>x. vector_derivative \<gamma> (at x) / (\<gamma> x - z)))" .
then have "\<exists>t. t \<in> {0..1} \<and> Im(integral {0..t} (\<lambda>x. vector_derivative \<gamma> (at x)/(\<gamma> x - z))) = Arg2pi r"
by (simp add: Arg2pi_ge_0 cont IVT')
then obtain t where t: "t \<in> {0..1}"
and eqArg: "Im (integral {0..t} (\<lambda>x. vector_derivative \<gamma> (at x)/(\<gamma> x - z))) = Arg2pi r"
by blast
define i where "i = integral {0..t} (\<lambda>x. vector_derivative \<gamma> (at x) / (\<gamma> x - z))"
have gpdt: "\<gamma> piecewise_C1_differentiable_on {0..t}"
by (metis atLeastAtMost_iff atLeastatMost_subset_iff order_refl piecewise_C1_differentiable_on_subset gpd t)
have "exp (- i) * (\<gamma> t - z) = \<gamma> 0 - z"
unfolding i_def
proof (rule winding_number_exp_integral [OF gpdt])
show "z \<notin> \<gamma> ` {0..t}"
using t z unfolding path_image_def by force
qed (use t in auto)
then have *: "\<gamma> t - z = exp i * (\<gamma> 0 - z)"
by (simp add: exp_minus field_simps)
then have "(w - z) = r * (\<gamma> 0 - z)"
by (simp add: r_def)
moreover have "z + exp (Re i) * (exp (\<i> * Im i) * (\<gamma> 0 - z)) = \<gamma> t"
using * by (simp add: exp_eq_polar field_simps)
moreover have "Arg2pi r = Im i"
using eqArg by (simp add: i_def)
ultimately have "z + complex_of_real (exp (Re i)) * (w - z) / complex_of_real (cmod r) = \<gamma> t"
using Complex_Transcendental.Arg2pi_eq [of r] \<open>r \<noteq> 0\<close>
by (metis mult.left_commute nonzero_mult_div_cancel_left norm_eq_zero of_real_0 of_real_eq_iff times_divide_eq_left)
with t show ?thesis
by (rule_tac x="exp(Re i) / norm r" in exI) (auto simp: path_image_def)
qed
lemma winding_number_big_meets:
fixes z::complex
assumes \<gamma>: "valid_path \<gamma>" and z: "z \<notin> path_image \<gamma>" and "\<bar>Re (winding_number \<gamma> z)\<bar> \<ge> 1"
and w: "w \<noteq> z"
shows "\<exists>a::real. 0 < a \<and> z + a*(w - z) \<in> path_image \<gamma>"
proof -
{ assume "Re (winding_number \<gamma> z) \<le> - 1"
then have "Re (winding_number (reversepath \<gamma>) z) \<ge> 1"
by (simp add: \<gamma> valid_path_imp_path winding_number_reversepath z)
moreover have "valid_path (reversepath \<gamma>)"
using \<gamma> valid_path_imp_reverse by auto
moreover have "z \<notin> path_image (reversepath \<gamma>)"
by (simp add: z)
ultimately have "\<exists>a::real. 0 < a \<and> z + a*(w - z) \<in> path_image (reversepath \<gamma>)"
using winding_number_pos_meets w by blast
then have ?thesis
by simp
}
then show ?thesis
using assms
by (simp add: abs_if winding_number_pos_meets split: if_split_asm)
qed
lemma winding_number_less_1:
fixes z::complex
shows
"\<lbrakk>valid_path \<gamma>; z \<notin> path_image \<gamma>; w \<noteq> z;
\<And>a::real. 0 < a \<Longrightarrow> z + a*(w - z) \<notin> path_image \<gamma>\<rbrakk>
\<Longrightarrow> Re(winding_number \<gamma> z) < 1"
by (auto simp: not_less dest: winding_number_big_meets)
text\<open>One way of proving that WN=1 for a loop.\<close>
lemma winding_number_eq_1:
fixes z::complex
assumes \<gamma>: "valid_path \<gamma>" and z: "z \<notin> path_image \<gamma>" and loop: "pathfinish \<gamma> = pathstart \<gamma>"
and 0: "0 < Re(winding_number \<gamma> z)" and 2: "Re(winding_number \<gamma> z) < 2"
shows "winding_number \<gamma> z = 1"
proof -
have "winding_number \<gamma> z \<in> Ints"
by (simp add: \<gamma> integer_winding_number loop valid_path_imp_path z)
then show ?thesis
using 0 2 by (auto simp: Ints_def)
qed
subsection\<open>Continuity of winding number and invariance on connected sets\<close>
theorem continuous_at_winding_number:
fixes z::complex
assumes \<gamma>: "path \<gamma>" and z: "z \<notin> path_image \<gamma>"
shows "continuous (at z) (winding_number \<gamma>)"
proof -
obtain e where "e>0" and cbg: "cball z e \<subseteq> - path_image \<gamma>"
using open_contains_cball [of "- path_image \<gamma>"] z
by (force simp: closed_def [symmetric] closed_path_image [OF \<gamma>])
then have ppag: "path_image \<gamma> \<subseteq> - cball z (e/2)"
by (force simp: cball_def dist_norm)
have oc: "open (- cball z (e/2))"
by (simp add: closed_def [symmetric])
obtain d where "d>0" and pi_eq:
"\<And>h1 h2. \<lbrakk>valid_path h1; valid_path h2;
(\<forall>t\<in>{0..1}. cmod (h1 t - \<gamma> t) < d \<and> cmod (h2 t - \<gamma> t) < d);
pathstart h2 = pathstart h1; pathfinish h2 = pathfinish h1\<rbrakk>
\<Longrightarrow>
path_image h1 \<subseteq> - cball z (e/2) \<and>
path_image h2 \<subseteq> - cball z (e/2) \<and>
(\<forall>f. f holomorphic_on - cball z (e/2) \<longrightarrow> contour_integral h2 f = contour_integral h1 f)"
using contour_integral_nearby_ends [OF oc \<gamma> ppag] by metis
obtain p where "valid_path p" "z \<notin> path_image p"
and p: "pathstart p = pathstart \<gamma>" "pathfinish p = pathfinish \<gamma>"
and pg: "\<And>t. t\<in>{0..1} \<Longrightarrow> cmod (\<gamma> t - p t) < min d e/2"
and pi: "contour_integral p (\<lambda>x. 1 / (x - z)) = complex_of_real (2 * pi) * \<i> * winding_number \<gamma> z"
using winding_number [OF \<gamma> z, of "min d e/2"] \<open>d>0\<close> \<open>e>0\<close> by (auto simp: winding_number_prop_def)
{ fix w
assume d2: "cmod (w - z) < d/2" and e2: "cmod (w - z) < e/2"
have wnotp: "w \<notin> path_image p"
proof (clarsimp simp add: path_image_def)
show False if w: "w = p x" and "0 \<le> x" "x \<le> 1" for x
proof -
have "cmod (\<gamma> x - p x) < min d e/2"
using pg that by auto
then have "cmod (z - \<gamma> x) < e"
by (metis e2 less_divide_eq_numeral1(1) min_less_iff_conj norm_minus_commute norm_triangle_half_l w)
then show ?thesis
using cbg that by (auto simp add: path_image_def cball_def dist_norm less_eq_real_def)
qed
qed
have wnotg: "w \<notin> path_image \<gamma>"
using cbg e2 \<open>e>0\<close> by (force simp: dist_norm norm_minus_commute)
{ fix k::real
assume k: "k>0"
then obtain q where q: "valid_path q" "w \<notin> path_image q"
"pathstart q = pathstart \<gamma> \<and> pathfinish q = pathfinish \<gamma>"
and qg: "\<And>t. t \<in> {0..1} \<Longrightarrow> cmod (\<gamma> t - q t) < min k (min d e) / 2"
and qi: "contour_integral q (\<lambda>u. 1 / (u - w)) = complex_of_real (2 * pi) * \<i> * winding_number \<gamma> w"
using winding_number [OF \<gamma> wnotg, of "min k (min d e) / 2"] \<open>d>0\<close> \<open>e>0\<close> k
by (force simp: min_divide_distrib_right winding_number_prop_def)
moreover have "\<And>t. t \<in> {0..1} \<Longrightarrow> cmod (q t - \<gamma> t) < d \<and> cmod (p t - \<gamma> t) < d"
using pg qg \<open>0 < d\<close> by (fastforce simp add: norm_minus_commute)
moreover have "(\<lambda>u. 1 / (u-w)) holomorphic_on - cball z (e/2)"
using e2 by (auto simp: dist_norm norm_minus_commute intro!: holomorphic_intros)
ultimately have "contour_integral p (\<lambda>u. 1 / (u - w)) = contour_integral q (\<lambda>u. 1 / (u - w))"
by (metis p \<open>valid_path p\<close> pi_eq)
then have "contour_integral p (\<lambda>x. 1 / (x - w)) = complex_of_real (2 * pi) * \<i> * winding_number \<gamma> w"
by (simp add: pi qi)
} note pip = this
have "path p"
by (simp add: \<open>valid_path p\<close> valid_path_imp_path)
moreover have "\<And>e. e > 0 \<Longrightarrow> winding_number_prop p w e p (winding_number \<gamma> w)"
by (simp add: \<open>valid_path p\<close> pip winding_number_prop_def wnotp)
ultimately have "winding_number p w = winding_number \<gamma> w"
using winding_number_unique wnotp by blast
} note wnwn = this
obtain pe where "pe>0" and cbp: "cball z (3 / 4 * pe) \<subseteq> - path_image p"
using \<open>valid_path p\<close> \<open>z \<notin> path_image p\<close> open_contains_cball [of "- path_image p"]
by (force simp: closed_def [symmetric] closed_path_image [OF valid_path_imp_path])
obtain L
where "L>0"
and L: "\<And>f B. \<lbrakk>f holomorphic_on - cball z (3 / 4 * pe);
\<forall>z \<in> - cball z (3 / 4 * pe). cmod (f z) \<le> B\<rbrakk> \<Longrightarrow>
cmod (contour_integral p f) \<le> L * B"
using contour_integral_bound_exists [of "- cball z (3/4*pe)" p] cbp \<open>valid_path p\<close> by blast
{ fix e::real and w::complex
assume e: "0 < e" and w: "cmod (w - z) < pe/4" "cmod (w - z) < e * pe\<^sup>2 / (8 * L)"
then have [simp]: "w \<notin> path_image p"
using cbp p(2) \<open>0 < pe\<close>
by (force simp: dist_norm norm_minus_commute path_image_def cball_def)
have [simp]: "contour_integral p (\<lambda>x. 1/(x - w)) - contour_integral p (\<lambda>x. 1/(x - z)) =
contour_integral p (\<lambda>x. 1/(x - w) - 1/(x - z))"
by (simp add: \<open>valid_path p\<close> \<open>z \<notin> path_image p\<close> contour_integrable_inversediff contour_integral_diff)
{ fix x
assume pe: "3/4 * pe < cmod (z - x)"
have "cmod (w - x) < pe/4 + cmod (z - x)"
by (meson add_less_cancel_right norm_diff_triangle_le order_refl order_trans_rules(21) w(1))
then have wx: "cmod (w - x) < 4/3 * cmod (z - x)" using pe by simp
have "cmod (z - x) \<le> cmod (z - w) + cmod (w - x)"
using norm_diff_triangle_le by blast
also have "\<dots> < pe/4 + cmod (w - x)"
using w by (simp add: norm_minus_commute)
finally have "pe/2 < cmod (w - x)"
using pe by auto
then have pe_less: "(pe/2)^2 < cmod (w - x) ^ 2"
by (simp add: \<open>0 < pe\<close> less_eq_real_def power_strict_mono)
then have pe2: "pe^2 < 4 * cmod (w - x) ^ 2"
by (simp add: power_divide)
have "8 * L * cmod (w - z) < e * pe\<^sup>2"
using w \<open>L>0\<close> by (simp add: field_simps)
also have "\<dots> < e * 4 * cmod (w - x) * cmod (w - x)"
using pe2 \<open>e>0\<close> by (simp add: power2_eq_square)
also have "\<dots> < e * 4 * cmod (w - x) * (4/3 * cmod (z - x))"
using \<open>0 < pe\<close> pe_less e less_eq_real_def wx by fastforce
finally have "L * cmod (w - z) < 2/3 * e * cmod (w - x) * cmod (z - x)"
by simp
also have "\<dots> \<le> e * cmod (w - x) * cmod (z - x)"
using e by simp
finally have Lwz: "L * cmod (w - z) < e * cmod (w - x) * cmod (z - x)" .
have "L * cmod (1 / (x - w) - 1 / (x - z)) \<le> e"
proof (cases "x=z \<or> x=w")
case True
with pe \<open>pe>0\<close> w \<open>L>0\<close>
show ?thesis
by (force simp: norm_minus_commute)
next
case False
with wx w(2) \<open>L>0\<close> pe pe2 Lwz
show ?thesis
by (auto simp: divide_simps mult_less_0_iff norm_minus_commute norm_divide norm_mult power2_eq_square)
qed
} note L_cmod_le = this
let ?f = "(\<lambda>x. 1 / (x - w) - 1 / (x - z))"
have "cmod (contour_integral p ?f) \<le> L * (e * pe\<^sup>2 / L / 4 * (inverse (pe / 2))\<^sup>2)"
proof (rule L)
show "?f holomorphic_on - cball z (3 / 4 * pe)"
using \<open>pe>0\<close> w
by (force simp: dist_norm norm_minus_commute intro!: holomorphic_intros)
show " \<forall>u \<in>- cball z (3 / 4 * pe). cmod (?f u) \<le> e * pe\<^sup>2 / L / 4 * (inverse (pe / 2))\<^sup>2"
using \<open>pe>0\<close> w \<open>L>0\<close>
by (auto simp: cball_def dist_norm field_simps L_cmod_le simp del: less_divide_eq_numeral1 le_divide_eq_numeral1)
qed
also have "\<dots> < 2*e"
using \<open>L>0\<close> e by (force simp: field_simps)
finally have "cmod (winding_number p w - winding_number p z) < e"
using pi_ge_two e
by (force simp: winding_number_valid_path \<open>valid_path p\<close> \<open>z \<notin> path_image p\<close> field_simps norm_divide norm_mult intro: less_le_trans)
} note cmod_wn_diff = this
have "isCont (winding_number p) z"
proof (clarsimp simp add: continuous_at_eps_delta)
fix e::real assume "e>0"
show "\<exists>d>0. \<forall>x'. dist x' z < d \<longrightarrow> dist (winding_number p x') (winding_number p z) < e"
using \<open>pe>0\<close> \<open>L>0\<close> \<open>e>0\<close>
by (rule_tac x="min (pe/4) (e/2*pe^2/L/4)" in exI) (simp add: dist_norm cmod_wn_diff)
qed
then show ?thesis
apply (rule continuous_transform_within [where d = "min d e/2"])
apply (auto simp: \<open>d>0\<close> \<open>e>0\<close> dist_norm wnwn)
done
qed
corollary continuous_on_winding_number:
"path \<gamma> \<Longrightarrow> continuous_on (- path_image \<gamma>) (\<lambda>w. winding_number \<gamma> w)"
by (simp add: continuous_at_imp_continuous_on continuous_at_winding_number)
subsection\<^marker>\<open>tag unimportant\<close> \<open>The winding number is constant on a connected region\<close>
lemma winding_number_constant:
assumes \<gamma>: "path \<gamma>" and loop: "pathfinish \<gamma> = pathstart \<gamma>" and cs: "connected S" and sg: "S \<inter> path_image \<gamma> = {}"
shows "winding_number \<gamma> constant_on S"
proof -
have *: "1 \<le> cmod (winding_number \<gamma> y - winding_number \<gamma> z)"
if ne: "winding_number \<gamma> y \<noteq> winding_number \<gamma> z" and "y \<in> S" "z \<in> S" for y z
proof -
have "winding_number \<gamma> y \<in> \<int>" "winding_number \<gamma> z \<in> \<int>"
using that integer_winding_number [OF \<gamma> loop] sg \<open>y \<in> S\<close> by auto
with ne show ?thesis
by (auto simp: Ints_def simp flip: of_int_diff)
qed
have cont: "continuous_on S (\<lambda>w. winding_number \<gamma> w)"
using continuous_on_winding_number [OF \<gamma>] sg
by (meson continuous_on_subset disjoint_eq_subset_Compl)
show ?thesis
using "*" zero_less_one
by (blast intro: continuous_discrete_range_constant [OF cs cont])
qed
lemma winding_number_eq:
"\<lbrakk>path \<gamma>; pathfinish \<gamma> = pathstart \<gamma>; w \<in> S; z \<in> S; connected S; S \<inter> path_image \<gamma> = {}\<rbrakk>
\<Longrightarrow> winding_number \<gamma> w = winding_number \<gamma> z"
using winding_number_constant by (metis constant_on_def)
lemma open_winding_number_levelsets:
assumes \<gamma>: "path \<gamma>" and loop: "pathfinish \<gamma> = pathstart \<gamma>"
shows "open {z. z \<notin> path_image \<gamma> \<and> winding_number \<gamma> z = k}"
proof (clarsimp simp: open_dist)
fix z assume z: "z \<notin> path_image \<gamma>" and k: "k = winding_number \<gamma> z"
have "open (- path_image \<gamma>)"
by (simp add: closed_path_image \<gamma> open_Compl)
then obtain e where "e>0" "ball z e \<subseteq> - path_image \<gamma>"
using open_contains_ball [of "- path_image \<gamma>"] z by blast
then show "\<exists>e>0. \<forall>y. dist y z < e \<longrightarrow> y \<notin> path_image \<gamma> \<and> winding_number \<gamma> y = winding_number \<gamma> z"
using \<open>e>0\<close> by (force simp: norm_minus_commute dist_norm intro: winding_number_eq [OF assms, where S = "ball z e"])
qed
subsection\<open>Winding number is zero "outside" a curve\<close>
proposition winding_number_zero_in_outside:
assumes \<gamma>: "path \<gamma>" and loop: "pathfinish \<gamma> = pathstart \<gamma>" and z: "z \<in> outside (path_image \<gamma>)"
shows "winding_number \<gamma> z = 0"
proof -
obtain B::real where "0 < B" and B: "path_image \<gamma> \<subseteq> ball 0 B"
using bounded_subset_ballD [OF bounded_path_image [OF \<gamma>]] by auto
obtain w::complex where w: "w \<notin> ball 0 (B + 1)"
by (metis abs_of_nonneg le_less less_irrefl mem_ball_0 norm_of_real)
have "- ball 0 (B + 1) \<subseteq> outside (path_image \<gamma>)"
using B subset_ball by (intro outside_subset_convex) auto
then have wout: "w \<in> outside (path_image \<gamma>)"
using w by blast
moreover have "winding_number \<gamma> constant_on outside (path_image \<gamma>)"
using winding_number_constant [OF \<gamma> loop, of "outside(path_image \<gamma>)"] connected_outside
by (metis DIM_complex bounded_path_image dual_order.refl \<gamma> outside_no_overlap)
ultimately have "winding_number \<gamma> z = winding_number \<gamma> w"
by (metis (no_types, opaque_lifting) constant_on_def z)
also have "\<dots> = 0"
proof -
have wnot: "w \<notin> path_image \<gamma>" using wout by (simp add: outside_def)
{ fix e::real assume "0<e"
obtain p where p: "polynomial_function p" "pathstart p = pathstart \<gamma>" "pathfinish p = pathfinish \<gamma>"
and pg1: "(\<And>t. \<lbrakk>0 \<le> t; t \<le> 1\<rbrakk> \<Longrightarrow> cmod (p t - \<gamma> t) < 1)"
and pge: "(\<And>t. \<lbrakk>0 \<le> t; t \<le> 1\<rbrakk> \<Longrightarrow> cmod (p t - \<gamma> t) < e)"
using path_approx_polynomial_function [OF \<gamma>, of "min 1 e"] \<open>e>0\<close>
by (metis atLeastAtMost_iff min_less_iff_conj zero_less_one)
have "\<exists>p. valid_path p \<and> w \<notin> path_image p \<and>
pathstart p = pathstart \<gamma> \<and> pathfinish p = pathfinish \<gamma> \<and>
(\<forall>t\<in>{0..1}. cmod (\<gamma> t - p t) < e) \<and> contour_integral p (\<lambda>wa. 1 / (wa - w)) = 0"
proof (intro exI conjI)
have "\<And>x. \<lbrakk>0 \<le> x; x \<le> 1\<rbrakk> \<Longrightarrow> cmod (p x) < B + 1"
using B unfolding image_subset_iff path_image_def
by (meson add_strict_mono atLeastAtMost_iff le_less_trans mem_ball_0 norm_triangle_sub pg1)
then have pip: "path_image p \<subseteq> ball 0 (B + 1)"
by (auto simp add: path_image_def dist_norm ball_def)
then show "w \<notin> path_image p" using w by blast
show vap: "valid_path p"
by (simp add: p(1) valid_path_polynomial_function)
show "\<forall>t\<in>{0..1}. cmod (\<gamma> t - p t) < e"
by (metis atLeastAtMost_iff norm_minus_commute pge)
show "contour_integral p (\<lambda>wa. 1 / (wa - w)) = 0"
proof (rule contour_integral_unique [OF Cauchy_theorem_convex_simple [OF _ convex_ball [of 0 "B+1"]]])
have "\<And>z. cmod z < B + 1 \<Longrightarrow> z \<noteq> w"
using mem_ball_0 w by blast
then show "(\<lambda>z. 1 / (z - w)) holomorphic_on ball 0 (B + 1)"
by (intro holomorphic_intros; simp add: dist_norm)
qed (use p vap pip loop in auto)
qed (use p in auto)
}
then show ?thesis
by (auto intro: winding_number_unique [OF \<gamma>] simp add: winding_number_prop_def wnot)
qed
finally show ?thesis .
qed
corollary\<^marker>\<open>tag unimportant\<close> winding_number_zero_const: "a \<noteq> z \<Longrightarrow> winding_number (\<lambda>t. a) z = 0"
by (rule winding_number_zero_in_outside)
(auto simp: pathfinish_def pathstart_def path_polynomial_function)
corollary\<^marker>\<open>tag unimportant\<close> winding_number_zero_outside:
"\<lbrakk>path \<gamma>; convex s; pathfinish \<gamma> = pathstart \<gamma>; z \<notin> s; path_image \<gamma> \<subseteq> s\<rbrakk> \<Longrightarrow> winding_number \<gamma> z = 0"
by (meson convex_in_outside outside_mono subsetCE winding_number_zero_in_outside)
lemma winding_number_zero_at_infinity:
assumes \<gamma>: "path \<gamma>" and loop: "pathfinish \<gamma> = pathstart \<gamma>"
shows "\<exists>B. \<forall>z. B \<le> norm z \<longrightarrow> winding_number \<gamma> z = 0"
proof -
obtain B::real where "0 < B" and B: "path_image \<gamma> \<subseteq> ball 0 B"
using bounded_subset_ballD [OF bounded_path_image [OF \<gamma>]] by auto
have "winding_number \<gamma> z = 0" if "B + 1 \<le> cmod z" for z
proof (rule winding_number_zero_outside [OF \<gamma> convex_cball loop])
show "z \<notin> cball 0 B"
using that by auto
show "path_image \<gamma> \<subseteq> cball 0 B"
using B order.trans by blast
qed
then show ?thesis
by metis
qed
lemma winding_number_zero_point:
"\<lbrakk>path \<gamma>; convex S; pathfinish \<gamma> = pathstart \<gamma>; open S; path_image \<gamma> \<subseteq> S\<rbrakk>
\<Longrightarrow> \<exists>z. z \<in> S \<and> winding_number \<gamma> z = 0"
using outside_compact_in_open [of "path_image \<gamma>" S] path_image_nonempty winding_number_zero_in_outside
by (fastforce simp add: compact_path_image)
text\<open>If a path winds round a set, it winds rounds its inside.\<close>
lemma winding_number_around_inside:
assumes \<gamma>: "path \<gamma>" and loop: "pathfinish \<gamma> = pathstart \<gamma>"
and cls: "closed S" and cos: "connected S" and S_disj: "S \<inter> path_image \<gamma> = {}"
and z: "z \<in> S" and wn_nz: "winding_number \<gamma> z \<noteq> 0" and w: "w \<in> S \<union> inside S"
shows "winding_number \<gamma> w = winding_number \<gamma> z"
proof -
have ssb: "S \<subseteq> inside(path_image \<gamma>)"
proof
fix x :: complex
assume "x \<in> S"
hence "x \<notin> path_image \<gamma>"
by (meson disjoint_iff_not_equal S_disj)
thus "x \<in> inside (path_image \<gamma>)"
by (metis Compl_iff S_disj UnE \<gamma> \<open>x \<in> S\<close> cos inside_outside loop winding_number_eq winding_number_zero_in_outside wn_nz z)
qed
show ?thesis
proof (rule winding_number_eq [OF \<gamma> loop w])
show "z \<in> S \<union> inside S"
using z by blast
show "connected (S \<union> inside S)"
by (simp add: cls connected_with_inside cos)
show "(S \<union> inside S) \<inter> path_image \<gamma> = {}"
unfolding disjoint_iff Un_iff
by (metis ComplD UnI1 \<gamma> cls compact_path_image connected_path_image inside_Un_outside inside_inside_compact_connected ssb subsetD)
qed
qed
text\<open>Bounding a WN by 1/2 for a path and point in opposite halfspaces.\<close>
lemma winding_number_subpath_continuous:
assumes \<gamma>: "valid_path \<gamma>" and z: "z \<notin> path_image \<gamma>"
shows "continuous_on {0..1} (\<lambda>x. winding_number(subpath 0 x \<gamma>) z)"
proof (rule continuous_on_eq)
let ?f = "\<lambda>x. integral {0..x} (\<lambda>t. vector_derivative \<gamma> (at t) / (\<gamma> t - z))"
show "continuous_on {0..1} (\<lambda>x. 1 / (2 * pi * \<i>) * ?f x)"
proof (intro indefinite_integral_continuous_1 winding_number_exp_integral continuous_intros)
show "\<gamma> piecewise_C1_differentiable_on {0..1}"
using \<gamma> valid_path_def by blast
qed (use path_image_def z in auto)
show "1 / (2 * pi * \<i>) * ?f x = winding_number (subpath 0 x \<gamma>) z"
if x: "x \<in> {0..1}" for x
proof -
have "1 / (2*pi*\<i>) * ?f x = 1 / (2*pi*\<i>) * contour_integral (subpath 0 x \<gamma>) (\<lambda>w. 1/(w - z))"
using assms x
by (simp add: contour_integral_subcontour_integral [OF contour_integrable_inversediff])
also have "\<dots> = winding_number (subpath 0 x \<gamma>) z"
proof (subst winding_number_valid_path)
show "z \<notin> path_image (subpath 0 x \<gamma>)"
using assms x atLeastAtMost_iff path_image_subpath_subset by force
qed (use assms x valid_path_subpath in \<open>force+\<close>)
finally show ?thesis .
qed
qed
lemma winding_number_ivt_pos:
assumes \<gamma>: "valid_path \<gamma>" and z: "z \<notin> path_image \<gamma>" and "0 \<le> w" "w \<le> Re(winding_number \<gamma> z)"
shows "\<exists>t \<in> {0..1}. Re(winding_number(subpath 0 t \<gamma>) z) = w"
proof -
have "continuous_on {0..1} (\<lambda>x. winding_number (subpath 0 x \<gamma>) z)"
using \<gamma> winding_number_subpath_continuous z by blast
moreover have "Re (winding_number (subpath 0 0 \<gamma>) z) \<le> w" "w \<le> Re (winding_number (subpath 0 1 \<gamma>) z)"
using assms by (auto simp: path_image_def image_def)
ultimately show ?thesis
using ivt_increasing_component_on_1[of 0 1, where ?k = "1"] by force
qed
lemma winding_number_ivt_neg:
assumes \<gamma>: "valid_path \<gamma>" and z: "z \<notin> path_image \<gamma>" and "Re(winding_number \<gamma> z) \<le> w" "w \<le> 0"
shows "\<exists>t \<in> {0..1}. Re(winding_number(subpath 0 t \<gamma>) z) = w"
proof -
have "continuous_on {0..1} (\<lambda>x. winding_number (subpath 0 x \<gamma>) z)"
using \<gamma> winding_number_subpath_continuous z by blast
moreover have "Re (winding_number (subpath 0 0 \<gamma>) z) \<ge> w" "w \<ge> Re (winding_number (subpath 0 1 \<gamma>) z)"
using assms by (auto simp: path_image_def image_def)
ultimately show ?thesis
using ivt_decreasing_component_on_1[of 0 1, where ?k = "1"] by force
qed
lemma winding_number_ivt_abs:
assumes \<gamma>: "valid_path \<gamma>" and z: "z \<notin> path_image \<gamma>" and "0 \<le> w" "w \<le> \<bar>Re(winding_number \<gamma> z)\<bar>"
shows "\<exists>t \<in> {0..1}. \<bar>Re (winding_number (subpath 0 t \<gamma>) z)\<bar> = w"
using assms winding_number_ivt_pos [of \<gamma> z w] winding_number_ivt_neg [of \<gamma> z "-w"]
by force
lemma winding_number_lt_half_lemma:
assumes \<gamma>: "valid_path \<gamma>" and z: "z \<notin> path_image \<gamma>" and az: "a \<bullet> z \<le> b" and pag: "path_image \<gamma> \<subseteq> {w. a \<bullet> w > b}"
shows "Re(winding_number \<gamma> z) < 1/2"
proof -
{ assume "Re(winding_number \<gamma> z) \<ge> 1/2"
then obtain t::real where t: "0 \<le> t" "t \<le> 1" and sub12: "Re (winding_number (subpath 0 t \<gamma>) z) = 1/2"
using winding_number_ivt_pos [OF \<gamma> z, of "1/2"] by auto
have gt: "\<gamma> t - z = - (of_real (exp (- (2 * pi * Im (winding_number (subpath 0 t \<gamma>) z)))) * (\<gamma> 0 - z))"
using winding_number_exp_2pi [of "subpath 0 t \<gamma>" z]
apply (simp add: t \<gamma> valid_path_imp_path)
using closed_segment_eq_real_ivl path_image_def t z by (fastforce simp: path_image_subpath Euler sub12)
have "b < a \<bullet> \<gamma> 0"
proof -
have "\<gamma> 0 \<in> {c. b < a \<bullet> c}"
by (metis (no_types) pag atLeastAtMost_iff image_subset_iff order_refl path_image_def zero_le_one)
thus ?thesis
by blast
qed
moreover have "b < a \<bullet> \<gamma> t"
by (metis atLeastAtMost_iff image_eqI mem_Collect_eq pag path_image_def subset_iff t)
ultimately have "0 < a \<bullet> (\<gamma> 0 - z)" "0 < a \<bullet> (\<gamma> t - z)" using az
by (simp add: inner_diff_right)+
then have False
by (simp add: gt inner_mult_right mult_less_0_iff)
}
then show ?thesis by force
qed
lemma winding_number_lt_half:
assumes "valid_path \<gamma>" "a \<bullet> z \<le> b" "path_image \<gamma> \<subseteq> {w. a \<bullet> w > b}"
shows "\<bar>Re (winding_number \<gamma> z)\<bar> < 1/2"
proof -
have "z \<notin> path_image \<gamma>" using assms by auto
with assms have "Re (winding_number \<gamma> z) < 0 \<Longrightarrow> - Re (winding_number \<gamma> z) < 1/2"
by (metis complex_inner_1_right winding_number_lt_half_lemma [OF valid_path_imp_reverse, of \<gamma> z a b]
winding_number_reversepath valid_path_imp_path inner_minus_left path_image_reversepath)
with assms show ?thesis
using \<open>z \<notin> path_image \<gamma>\<close> winding_number_lt_half_lemma by fastforce
qed
lemma winding_number_le_half:
assumes \<gamma>: "valid_path \<gamma>" and z: "z \<notin> path_image \<gamma>"
and anz: "a \<noteq> 0" and azb: "a \<bullet> z \<le> b" and pag: "path_image \<gamma> \<subseteq> {w. a \<bullet> w \<ge> b}"
shows "\<bar>Re (winding_number \<gamma> z)\<bar> \<le> 1/2"
proof -
{ assume wnz_12: "\<bar>Re (winding_number \<gamma> z)\<bar> > 1/2"
have "isCont (winding_number \<gamma>) z"
by (metis continuous_at_winding_number valid_path_imp_path \<gamma> z)
then obtain d where "d>0" and d: "\<And>x'. dist x' z < d \<Longrightarrow> dist (winding_number \<gamma> x') (winding_number \<gamma> z) < \<bar>Re(winding_number \<gamma> z)\<bar> - 1/2"
using continuous_at_eps_delta wnz_12 diff_gt_0_iff_gt by blast
define z' where "z' = z - (d / (2 * cmod a)) *\<^sub>R a"
have "a \<bullet> z * 6 \<le> d * cmod a + b * 6"
by (metis \<open>0 < d\<close> add_increasing azb less_eq_real_def mult_nonneg_nonneg mult_right_mono norm_ge_zero norm_numeral)
with anz have *: "a \<bullet> z' \<le> b - d / 3 * cmod a"
unfolding z'_def inner_mult_right' divide_inverse
by (simp add: field_split_simps algebra_simps dot_square_norm power2_eq_square)
have "cmod (winding_number \<gamma> z' - winding_number \<gamma> z) < \<bar>Re (winding_number \<gamma> z)\<bar> - 1/2"
using d [of z'] anz \<open>d>0\<close> by (simp add: dist_norm z'_def)
then have "1/2 < \<bar>Re (winding_number \<gamma> z)\<bar> - cmod (winding_number \<gamma> z' - winding_number \<gamma> z)"
by simp
then have "1/2 < \<bar>Re (winding_number \<gamma> z)\<bar> - \<bar>Re (winding_number \<gamma> z') - Re (winding_number \<gamma> z)\<bar>"
using abs_Re_le_cmod [of "winding_number \<gamma> z' - winding_number \<gamma> z"] by simp
then have wnz_12': "\<bar>Re (winding_number \<gamma> z')\<bar> > 1/2"
by linarith
moreover have "\<bar>Re (winding_number \<gamma> z')\<bar> < 1/2"
proof (rule winding_number_lt_half [OF \<gamma> *])
show "path_image \<gamma> \<subseteq> {w. b - d / 3 * cmod a < a \<bullet> w}"
using azb \<open>d>0\<close> pag by (auto simp: add_strict_increasing anz field_split_simps dest!: subsetD)
qed
ultimately have False
by simp
}
then show ?thesis by force
qed
lemma winding_number_lt_half_linepath:
assumes "z \<notin> closed_segment a b" shows "\<bar>Re (winding_number (linepath a b) z)\<bar> < 1/2"
proof -
obtain u v where "u \<bullet> z \<le> v" and uv: "\<And>x. x \<in> closed_segment a b \<Longrightarrow> inner u x > v"
using separating_hyperplane_closed_point assms closed_segment convex_closed_segment less_eq_real_def by metis
moreover have "path_image (linepath a b) \<subseteq> {w. v < u \<bullet> w}"
using in_segment(1) uv by auto
ultimately show ?thesis
using winding_number_lt_half by auto
qed
text\<open> Positivity of WN for a linepath.\<close>
lemma winding_number_linepath_pos_lt:
assumes "0 < Im ((b - a) * cnj (b - z))"
shows "0 < Re(winding_number(linepath a b) z)"
proof -
have z: "z \<notin> path_image (linepath a b)"
using assms
by (simp add: closed_segment_def) (force simp: algebra_simps)
show ?thesis
by (intro winding_number_pos_lt [OF valid_path_linepath z assms]) (simp add: linepath_def algebra_simps)
qed
subsection\<^marker>\<open>tag unimportant\<close> \<open>More winding number properties\<close>
text\<open>including the fact that it's +-1 inside a simple closed curve.\<close>
lemma winding_number_homotopic_paths:
assumes "homotopic_paths (-{z}) g h"
shows "winding_number g z = winding_number h z"
proof -
have "path g" "path h" using homotopic_paths_imp_path [OF assms] by auto
moreover have pag: "z \<notin> path_image g" and pah: "z \<notin> path_image h"
using homotopic_paths_imp_subset [OF assms] by auto
ultimately obtain d e where "d > 0" "e > 0"
and d: "\<And>p. \<lbrakk>path p; pathstart p = pathstart g; pathfinish p = pathfinish g; \<forall>t\<in>{0..1}. norm (p t - g t) < d\<rbrakk>
\<Longrightarrow> homotopic_paths (-{z}) g p"
and e: "\<And>q. \<lbrakk>path q; pathstart q = pathstart h; pathfinish q = pathfinish h; \<forall>t\<in>{0..1}. norm (q t - h t) < e\<rbrakk>
\<Longrightarrow> homotopic_paths (-{z}) h q"
using homotopic_nearby_paths [of g "-{z}"] homotopic_nearby_paths [of h "-{z}"] by force
obtain p where p:
"valid_path p" "z \<notin> path_image p"
"pathstart p = pathstart g" "pathfinish p = pathfinish g"
and gp_less:"\<forall>t\<in>{0..1}. cmod (g t - p t) < d"
and pap: "contour_integral p (\<lambda>w. 1 / (w - z)) = complex_of_real (2 * pi) * \<i> * winding_number g z"
using winding_number [OF \<open>path g\<close> pag \<open>0 < d\<close>] unfolding winding_number_prop_def by blast
obtain q where q:
"valid_path q" "z \<notin> path_image q"
"pathstart q = pathstart h" "pathfinish q = pathfinish h"
and hq_less: "\<forall>t\<in>{0..1}. cmod (h t - q t) < e"
and paq: "contour_integral q (\<lambda>w. 1 / (w - z)) = complex_of_real (2 * pi) * \<i> * winding_number h z"
using winding_number [OF \<open>path h\<close> pah \<open>0 < e\<close>] unfolding winding_number_prop_def by blast
have "homotopic_paths (- {z}) g p"
by (simp add: d p valid_path_imp_path norm_minus_commute gp_less)
moreover have "homotopic_paths (- {z}) h q"
by (simp add: e q valid_path_imp_path norm_minus_commute hq_less)
ultimately have "homotopic_paths (- {z}) p q"
by (blast intro: homotopic_paths_trans homotopic_paths_sym assms)
then have "contour_integral p (\<lambda>w. 1/(w - z)) = contour_integral q (\<lambda>w. 1/(w - z))"
by (rule Cauchy_theorem_homotopic_paths) (auto intro!: holomorphic_intros simp: p q)
then show ?thesis
by (simp add: pap paq)
qed
lemma winding_number_homotopic_loops:
assumes "homotopic_loops (-{z}) g h"
shows "winding_number g z = winding_number h z"
proof -
have "path g" "path h" using homotopic_loops_imp_path [OF assms] by auto
moreover have pag: "z \<notin> path_image g" and pah: "z \<notin> path_image h"
using homotopic_loops_imp_subset [OF assms] by auto
moreover have gloop: "pathfinish g = pathstart g" and hloop: "pathfinish h = pathstart h"
using homotopic_loops_imp_loop [OF assms] by auto
ultimately obtain d e where "d > 0" "e > 0"
and d: "\<And>p. \<lbrakk>path p; pathfinish p = pathstart p; \<forall>t\<in>{0..1}. norm (p t - g t) < d\<rbrakk>
\<Longrightarrow> homotopic_loops (-{z}) g p"
and e: "\<And>q. \<lbrakk>path q; pathfinish q = pathstart q; \<forall>t\<in>{0..1}. norm (q t - h t) < e\<rbrakk>
\<Longrightarrow> homotopic_loops (-{z}) h q"
using homotopic_nearby_loops [of g "-{z}"] homotopic_nearby_loops [of h "-{z}"] by force
obtain p where p:
"valid_path p" "z \<notin> path_image p"
"pathstart p = pathstart g" "pathfinish p = pathfinish g"
and gp_less:"\<forall>t\<in>{0..1}. cmod (g t - p t) < d"
and pap: "contour_integral p (\<lambda>w. 1 / (w - z)) = complex_of_real (2 * pi) * \<i> * winding_number g z"
using winding_number [OF \<open>path g\<close> pag \<open>0 < d\<close>] unfolding winding_number_prop_def by blast
obtain q where q:
"valid_path q" "z \<notin> path_image q"
"pathstart q = pathstart h" "pathfinish q = pathfinish h"
and hq_less: "\<forall>t\<in>{0..1}. cmod (h t - q t) < e"
and paq: "contour_integral q (\<lambda>w. 1 / (w - z)) = complex_of_real (2 * pi) * \<i> * winding_number h z"
using winding_number [OF \<open>path h\<close> pah \<open>0 < e\<close>] unfolding winding_number_prop_def by blast
have gp: "homotopic_loops (- {z}) g p"
by (simp add: gloop d gp_less norm_minus_commute p valid_path_imp_path)
have hq: "homotopic_loops (- {z}) h q"
by (simp add: e hloop hq_less norm_minus_commute q valid_path_imp_path)
have "contour_integral p (\<lambda>w. 1/(w - z)) = contour_integral q (\<lambda>w. 1/(w - z))"
proof (rule Cauchy_theorem_homotopic_loops)
show "homotopic_loops (- {z}) p q"
by (blast intro: homotopic_loops_trans homotopic_loops_sym gp hq assms)
qed (auto intro!: holomorphic_intros simp: p q)
then show ?thesis
by (simp add: pap paq)
qed
lemma winding_number_paths_linear_eq:
"\<lbrakk>path g; path h; pathstart h = pathstart g; pathfinish h = pathfinish g;
\<And>t. t \<in> {0..1} \<Longrightarrow> z \<notin> closed_segment (g t) (h t)\<rbrakk>
\<Longrightarrow> winding_number h z = winding_number g z"
by (blast intro: sym homotopic_paths_linear winding_number_homotopic_paths)
lemma winding_number_loops_linear_eq:
"\<lbrakk>path g; path h; pathfinish g = pathstart g; pathfinish h = pathstart h;
\<And>t. t \<in> {0..1} \<Longrightarrow> z \<notin> closed_segment (g t) (h t)\<rbrakk>
\<Longrightarrow> winding_number h z = winding_number g z"
by (blast intro: sym homotopic_loops_linear winding_number_homotopic_loops)
lemma winding_number_nearby_paths_eq:
"\<lbrakk>path g; path h; pathstart h = pathstart g; pathfinish h = pathfinish g;
\<And>t. t \<in> {0..1} \<Longrightarrow> norm(h t - g t) < norm(g t - z)\<rbrakk>
\<Longrightarrow> winding_number h z = winding_number g z"
by (metis segment_bound(2) norm_minus_commute not_le winding_number_paths_linear_eq)
lemma winding_number_nearby_loops_eq:
"\<lbrakk>path g; path h; pathfinish g = pathstart g; pathfinish h = pathstart h;
\<And>t. t \<in> {0..1} \<Longrightarrow> norm(h t - g t) < norm(g t - z)\<rbrakk>
\<Longrightarrow> winding_number h z = winding_number g z"
by (metis segment_bound(2) norm_minus_commute not_le winding_number_loops_linear_eq)
lemma winding_number_subpath_combine:
assumes "path g" and notin: "z \<notin> path_image g" and "u \<in> {0..1}" "v \<in> {0..1}" "w \<in> {0..1}"
shows "winding_number (subpath u v g) z + winding_number (subpath v w g) z =
winding_number (subpath u w g) z" (is "?lhs = ?rhs")
proof -
have "?lhs = winding_number (subpath u v g +++ subpath v w g) z"
using assms
by (metis (no_types, lifting) path_image_subpath_subset path_subpath pathfinish_subpath pathstart_subpath subsetD winding_number_join)
also have "... = ?rhs"
by (meson assms homotopic_join_subpaths subset_Compl_singleton winding_number_homotopic_paths)
finally show ?thesis .
qed
text \<open>Winding numbers of circular contours\<close>
proposition winding_number_part_circlepath_pos_less:
assumes "s < t" and no: "norm(w - z) < r"
shows "0 < Re (winding_number(part_circlepath z r s t) w)"
proof -
have "0 < r" by (meson no norm_not_less_zero not_le order.strict_trans2)
note valid_path_part_circlepath
moreover have " w \<notin> path_image (part_circlepath z r s t)"
using assms by (auto simp: path_image_def image_def part_circlepath_def norm_mult linepath_def)
moreover have "0 < r * (t - s) * (r - cmod (w - z))"
using assms by (metis \<open>0 < r\<close> diff_gt_0_iff_gt mult_pos_pos)
ultimately show ?thesis
apply (rule winding_number_pos_lt [where e = "r*(t - s)*(r - norm(w - z))"])
apply (simp add: vector_derivative_part_circlepath right_diff_distrib [symmetric] mult_ac mult_le_cancel_left_pos assms \<open>0<r\<close>)
using Re_Im_le_cmod [of "w-z" "linepath s t x" for x]
by (simp add: exp_Euler cos_of_real sin_of_real part_circlepath_def algebra_simps cos_squared_eq [unfolded power2_eq_square])
qed
lemma winding_number_circlepath_centre: "0 < r \<Longrightarrow> winding_number (circlepath z r) z = 1"
apply (rule winding_number_unique_loop)
apply (simp_all add: sphere_def valid_path_imp_path)
apply (rule_tac x="circlepath z r" in exI)
apply (simp add: sphere_def contour_integral_circlepath)
done
proposition winding_number_circlepath:
assumes "norm(w - z) < r" shows "winding_number(circlepath z r) w = 1"
proof (cases "w = z")
case True then show ?thesis
using assms winding_number_circlepath_centre by auto
next
case False
have [simp]: "r > 0"
using assms le_less_trans norm_ge_zero by blast
define r' where "r' = norm(w - z)"
have "r' < r"
by (simp add: assms r'_def)
have disjo: "cball z r' \<inter> sphere z r = {}"
using \<open>r' < r\<close> by (force simp: cball_def sphere_def)
have "winding_number(circlepath z r) w = winding_number(circlepath z r) z"
proof (rule winding_number_around_inside [where S = "cball z r'"])
show "winding_number (circlepath z r) z \<noteq> 0"
by (simp add: winding_number_circlepath_centre)
show "cball z r' \<inter> path_image (circlepath z r) = {}"
by (simp add: disjo less_eq_real_def)
qed (auto simp: r'_def dist_norm norm_minus_commute)
also have "\<dots> = 1"
by (simp add: winding_number_circlepath_centre)
finally show ?thesis .
qed
lemma no_bounded_connected_component_imp_winding_number_zero:
assumes g: "path g" "path_image g \<subseteq> S" "pathfinish g = pathstart g" "z \<notin> S"
and nb: "\<And>z. bounded (connected_component_set (- S) z) \<Longrightarrow> z \<in> S"
shows "winding_number g z = 0"
proof -
have "z \<in> outside (path_image g)"
by (metis nb [of z] \<open>path_image g \<subseteq> S\<close> \<open>z \<notin> S\<close> subsetD mem_Collect_eq outside outside_mono)
then show ?thesis
by (simp add: g winding_number_zero_in_outside)
qed
lemma no_bounded_path_component_imp_winding_number_zero:
assumes g: "path g" "path_image g \<subseteq> S" "pathfinish g = pathstart g" "z \<notin> S"
and nb: "\<And>z. bounded (path_component_set (- S) z) \<Longrightarrow> z \<in> S"
shows "winding_number g z = 0"
by (simp add: bounded_subset nb path_component_subset_connected_component
no_bounded_connected_component_imp_winding_number_zero [OF g])
subsection\<open>Winding number for a triangle\<close>
lemma wn_triangle1:
assumes "0 \<in> interior(convex hull {a,b,c})"
shows "\<not> (Im(a/b) \<le> 0 \<and> 0 \<le> Im(b/c))"
proof -
{ assume 0: "Im(a/b) \<le> 0" "0 \<le> Im(b/c)"
have "0 \<notin> interior (convex hull {a,b,c})"
proof (cases "a=0 \<or> b=0 \<or> c=0")
case True then show ?thesis
by (auto simp: not_in_interior_convex_hull_3)
next
case False
then have "b \<noteq> 0" by blast
{ fix x y::complex and u::real
assume eq_f': "Im x * Re b \<le> Im b * Re x" "Im y * Re b \<le> Im b * Re y" "0 \<le> u" "u \<le> 1"
then have "((1 - u) * Im x) * Re b \<le> Im b * ((1 - u) * Re x)"
by (simp add: mult_left_mono mult.assoc mult.left_commute [of "Im b"])
then have "((1 - u) * Im x + u * Im y) * Re b \<le> Im b * ((1 - u) * Re x + u * Re y)"
using eq_f' ordered_comm_semiring_class.comm_mult_left_mono
by (fastforce simp add: algebra_simps)
}
then have "convex {z. Im z * Re b \<le> Im b * Re z}"
by (auto simp: algebra_simps convex_alt)
with False 0 have "convex hull {a,b,c} \<le> {z. Im z * Re b \<le> Im b * Re z}"
by (simp add: subset_hull mult.commute Complex.Im_divide divide_simps complex_neq_0 [symmetric])
moreover have "0 \<notin> interior({z. Im z * Re b \<le> Im b * Re z})"
proof
assume "0 \<in> interior {z. Im z * Re b \<le> Im b * Re z}"
then obtain e where "e>0" and e: "ball 0 e \<subseteq> {z. Im z * Re b \<le> Im b * Re z}"
by (meson mem_interior)
define z where "z \<equiv> - sgn (Im b) * (e/3) + sgn (Re b) * (e/3) * \<i>"
have "cmod z = cmod (e/3) * cmod (- sgn (Im b) + sgn (Re b) * \<i>)"
unfolding z_def norm_mult [symmetric] by (simp add: algebra_simps)
also have "... < e"
using \<open>e>0\<close> by (auto simp: norm_mult intro: le_less_trans [OF norm_triangle_ineq4])
finally have "z \<in> ball 0 e"
using \<open>e>0\<close> by (simp add: )
then have "Im z * Re b \<le> Im b * Re z"
using e by blast
then have b: "0 < Re b" "0 < Im b" and disj: "e * Re b < - (Im b * e) \<or> e * Re b = - (Im b * e)"
using \<open>e>0\<close> \<open>b \<noteq> 0\<close>
by (auto simp add: z_def dist_norm sgn_if less_eq_real_def mult_less_0_iff complex.expand split: if_split_asm)
show False \<comment>\<open>or just one smt line\<close>
using disj
proof
assume "e * Re b < - (Im b * e)"
with b \<open>0 < e\<close> less_trans [of _ 0] show False
by (metis (no_types) mult_pos_pos neg_less_0_iff_less not_less_iff_gr_or_eq)
next
assume "e * Re b = - (Im b * e)"
with b \<open>0 < e\<close> show False
by (metis mult_pos_pos neg_less_0_iff_less not_less_iff_gr_or_eq)
qed
qed
ultimately show ?thesis
using interior_mono by blast
qed
} with assms show ?thesis by blast
qed
lemma wn_triangle2_0:
assumes "0 \<in> interior(convex hull {a,b,c})"
shows
"0 < Im((b - a) * cnj (b)) \<and>
0 < Im((c - b) * cnj (c)) \<and>
0 < Im((a - c) * cnj (a))
\<or>
Im((b - a) * cnj (b)) < 0 \<and>
0 < Im((b - c) * cnj (b)) \<and>
0 < Im((a - b) * cnj (a)) \<and>
0 < Im((c - a) * cnj (c))"
proof -
have [simp]: "{b,c,a} = {a,b,c}" "{c,a,b} = {a,b,c}" by auto
show ?thesis
using wn_triangle1 [OF assms] wn_triangle1 [of b c a] wn_triangle1 [of c a b] assms
by (auto simp: algebra_simps Im_complex_div_gt_0 Im_complex_div_lt_0 not_le not_less)
qed
lemma wn_triangle2:
assumes "z \<in> interior(convex hull {a,b,c})"
shows "0 < Im((b - a) * cnj (b - z)) \<and>
0 < Im((c - b) * cnj (c - z)) \<and>
0 < Im((a - c) * cnj (a - z))
\<or>
Im((b - a) * cnj (b - z)) < 0 \<and>
0 < Im((b - c) * cnj (b - z)) \<and>
0 < Im((a - b) * cnj (a - z)) \<and>
0 < Im((c - a) * cnj (c - z))"
proof -
have 0: "0 \<in> interior(convex hull {a-z, b-z, c-z})"
using assms convex_hull_translation [of "-z" "{a,b,c}"]
interior_translation [of "-z"]
by (simp cong: image_cong_simp)
show ?thesis using wn_triangle2_0 [OF 0]
by simp
qed
lemma wn_triangle3:
assumes z: "z \<in> interior(convex hull {a,b,c})"
and "0 < Im((b-a) * cnj (b-z))"
"0 < Im((c-b) * cnj (c-z))"
"0 < Im((a-c) * cnj (a-z))"
shows "winding_number (linepath a b +++ linepath b c +++ linepath c a) z = 1"
proof -
have znot[simp]: "z \<notin> closed_segment a b" "z \<notin> closed_segment b c" "z \<notin> closed_segment c a"
using z interior_of_triangle [of a b c]
by (auto simp: closed_segment_def)
have gt0: "0 < Re (winding_number (linepath a b +++ linepath b c +++ linepath c a) z)"
using assms
by (simp add: winding_number_linepath_pos_lt path_image_join winding_number_join_pos_combined)
have lt2: "Re (winding_number (linepath a b +++ linepath b c +++ linepath c a) z) < 2"
using winding_number_lt_half_linepath [of _ a b]
using winding_number_lt_half_linepath [of _ b c]
using winding_number_lt_half_linepath [of _ c a] znot
by (fastforce simp add: winding_number_join path_image_join)
show ?thesis
by (rule winding_number_eq_1) (simp_all add: path_image_join gt0 lt2)
qed
proposition winding_number_triangle:
assumes z: "z \<in> interior(convex hull {a,b,c})"
shows "winding_number(linepath a b +++ linepath b c +++ linepath c a) z =
(if 0 < Im((b - a) * cnj (b - z)) then 1 else -1)"
proof -
have [simp]: "{a,c,b} = {a,b,c}" by auto
have znot[simp]: "z \<notin> closed_segment a b" "z \<notin> closed_segment b c" "z \<notin> closed_segment c a"
using z interior_of_triangle [of a b c]
by (auto simp: closed_segment_def)
then have [simp]: "z \<notin> closed_segment b a" "z \<notin> closed_segment c b" "z \<notin> closed_segment a c"
using closed_segment_commute by blast+
have *: "winding_number (linepath a b +++ linepath b c +++ linepath c a) z =
winding_number (reversepath (linepath a c +++ linepath c b +++ linepath b a)) z"
by (simp add: reversepath_joinpaths winding_number_join not_in_path_image_join)
show ?thesis
apply (rule disjE [OF wn_triangle2 [OF z]])
subgoal
by (simp add: wn_triangle3 z)
subgoal
by (simp add: path_image_join winding_number_reversepath * wn_triangle3 z)
done
qed
subsection\<open>Winding numbers for simple closed paths\<close>
lemma winding_number_from_innerpath:
assumes "simple_path c1" and c1: "pathstart c1 = a" "pathfinish c1 = b"
and "simple_path c2" and c2: "pathstart c2 = a" "pathfinish c2 = b"
and "simple_path c" and c: "pathstart c = a" "pathfinish c = b"
and c1c2: "path_image c1 \<inter> path_image c2 = {a,b}"
and c1c: "path_image c1 \<inter> path_image c = {a,b}"
and c2c: "path_image c2 \<inter> path_image c = {a,b}"
and ne_12: "path_image c \<inter> inside(path_image c1 \<union> path_image c2) \<noteq> {}"
and z: "z \<in> inside(path_image c1 \<union> path_image c)"
and wn_d: "winding_number (c1 +++ reversepath c) z = d"
and "a \<noteq> b" "d \<noteq> 0"
obtains "z \<in> inside(path_image c1 \<union> path_image c2)" "winding_number (c1 +++ reversepath c2) z = d"
proof -
obtain 0: "inside(path_image c1 \<union> path_image c) \<inter> inside(path_image c2 \<union> path_image c) = {}"
and 1: "inside(path_image c1 \<union> path_image c) \<union> inside(path_image c2 \<union> path_image c) \<union>
(path_image c - {a,b}) = inside(path_image c1 \<union> path_image c2)"
by (rule split_inside_simple_closed_curve
[OF \<open>simple_path c1\<close> c1 \<open>simple_path c2\<close> c2 \<open>simple_path c\<close> c \<open>a \<noteq> b\<close> c1c2 c1c c2c ne_12])
have znot: "z \<notin> path_image c" "z \<notin> path_image c1" "z \<notin> path_image c2"
using union_with_outside z 1 by auto
then have zout: "z \<in> outside (path_image c \<union> path_image c2)"
by (metis "0" ComplI UnE disjoint_iff_not_equal sup.commute union_with_inside z)
have wn_cc2: "winding_number (c +++ reversepath c2) z = 0"
by (rule winding_number_zero_in_outside; simp add: zout \<open>simple_path c2\<close> c c2 \<open>simple_path c\<close> simple_path_imp_path path_image_join)
show ?thesis
proof
show "z \<in> inside (path_image c1 \<union> path_image c2)"
using "1" z by blast
have "winding_number c1 z - winding_number c z = d "
using assms znot
by (metis wn_d winding_number_join simple_path_imp_path winding_number_reversepath add.commute path_image_reversepath path_reversepath pathstart_reversepath uminus_add_conv_diff)
then show "winding_number (c1 +++ reversepath c2) z = d"
using wn_cc2 by (simp add: winding_number_join simple_path_imp_path assms znot winding_number_reversepath)
qed
qed
lemma simple_closed_path_wn1:
fixes a::complex and e::real
assumes "0 < e"
and sp_pl: "simple_path(p +++ linepath (a - e) (a + e))" (is "simple_path ?pae")
and psp: "pathstart p = a + e"
and pfp: "pathfinish p = a - e"
and disj: "ball a e \<inter> path_image p = {}"
obtains z where "z \<in> inside (path_image ?pae)" "cmod (winding_number ?pae z) = 1"
proof -
have "arc p" and arc_lp: "arc (linepath (a - e) (a + e))"
and pap: "path_image p \<inter> path_image (linepath (a - e) (a + e)) \<subseteq> {pathstart p, a-e}"
using simple_path_join_loop_eq [of "linepath (a - e) (a + e)" p] assms by auto
have mid_eq_a: "midpoint (a - e) (a + e) = a"
by (simp add: midpoint_def)
with assms have "a \<in> path_image ?pae"
by (simp add: assms path_image_join) (metis midpoint_in_closed_segment)
then have "a \<in> frontier(inside (path_image ?pae))"
using assms by (simp add: Jordan_inside_outside )
with \<open>0 < e\<close> obtain c where c: "c \<in> inside (path_image ?pae)"
and dac: "dist a c < e"
by (auto simp: frontier_straddle)
then have "c \<notin> path_image ?pae"
using inside_no_overlap by blast
then have "c \<notin> path_image p" "c \<notin> closed_segment (a - e) (a + e)"
by (simp_all add: assms path_image_join)
with \<open>0 < e\<close> dac have "c \<notin> affine hull {a - e, a + e}"
by (simp add: segment_as_ball not_le)
with \<open>0 < e\<close> have *: "\<not> collinear {a - e, c,a + e}"
using collinear_3_affine_hull [of "a-e" "a+e"] by (auto simp: insert_commute)
have 13: "1/3 + 1/3 + 1/3 = (1::real)" by simp
have "(1/3) *\<^sub>R (a - of_real e) + (1/3) *\<^sub>R c + (1/3) *\<^sub>R (a + of_real e) \<in> interior(convex hull {a - e, c, a + e})"
using interior_convex_hull_3_minimal [OF * DIM_complex]
by clarsimp (metis 13 zero_less_divide_1_iff zero_less_numeral)
then obtain z where z: "z \<in> interior(convex hull {a - e, c, a + e})" by force
have [simp]: "z \<notin> closed_segment (a - e) c"
by (metis DIM_complex Diff_iff IntD2 inf_sup_absorb interior_of_triangle z)
have [simp]: "z \<notin> closed_segment (a + e) (a - e)"
by (metis DIM_complex DiffD2 Un_iff interior_of_triangle z)
have [simp]: "z \<notin> closed_segment c (a + e)"
by (metis (no_types, lifting) DIM_complex DiffD2 Un_insert_right inf_sup_aci(5) insertCI interior_of_triangle mk_disjoint_insert z)
show thesis
proof
have "norm (winding_number (linepath (a - e) c +++ linepath c (a + e) +++ linepath (a + e) (a - e)) z) = 1"
using winding_number_triangle [OF z] by simp
have zin: "z \<in> inside (path_image (linepath (a + e) (a - e)) \<union> path_image p)"
and zeq: "winding_number (linepath (a + e) (a - e) +++ reversepath p) z =
winding_number (linepath (a - e) c +++ linepath c (a + e) +++ linepath (a + e) (a - e)) z"
proof (rule winding_number_from_innerpath
[of "linepath (a + e) (a - e)" "a+e" "a-e" p
"linepath (a + e) c +++ linepath c (a - e)" z
"winding_number (linepath (a - e) c +++ linepath c (a + e) +++ linepath (a + e) (a - e)) z"])
show sp_aec: "simple_path (linepath (a + e) c +++ linepath c (a - e))"
proof (rule arc_imp_simple_path [OF arc_join])
show "arc (linepath (a + e) c)"
by (metis \<open>c \<notin> path_image p\<close> arc_linepath pathstart_in_path_image psp)
show "arc (linepath c (a - e))"
by (metis \<open>c \<notin> path_image p\<close> arc_linepath pathfinish_in_path_image pfp)
show "path_image (linepath (a + e) c) \<inter> path_image (linepath c (a - e)) \<subseteq> {pathstart (linepath c (a - e))}"
by clarsimp (metis "*" IntI Int_closed_segment closed_segment_commute singleton_iff)
qed auto
show "simple_path p"
using \<open>arc p\<close> arc_simple_path by blast
show sp_ae2: "simple_path (linepath (a + e) (a - e))"
using \<open>arc p\<close> arc_distinct_ends pfp psp by fastforce
show pa: "pathfinish (linepath (a + e) (a - e)) = a - e"
"pathstart (linepath (a + e) c +++ linepath c (a - e)) = a + e"
"pathfinish (linepath (a + e) c +++ linepath c (a - e)) = a - e"
"pathstart p = a + e" "pathfinish p = a - e"
"pathstart (linepath (a + e) (a - e)) = a + e"
by (simp_all add: assms)
show 1: "path_image (linepath (a + e) (a - e)) \<inter> path_image p = {a + e, a - e}"
proof
show "path_image (linepath (a + e) (a - e)) \<inter> path_image p \<subseteq> {a + e, a - e}"
using pap closed_segment_commute psp segment_convex_hull by fastforce
show "{a + e, a - e} \<subseteq> path_image (linepath (a + e) (a - e)) \<inter> path_image p"
using pap pathfinish_in_path_image pathstart_in_path_image pfp psp by fastforce
qed
show 2: "path_image (linepath (a + e) (a - e)) \<inter> path_image (linepath (a + e) c +++ linepath c (a - e)) =
{a + e, a - e}" (is "?lhs = ?rhs")
proof
have "\<not> collinear {c, a + e, a - e}"
using * by (simp add: insert_commute)
then have "convex hull {a + e, a - e} \<inter> convex hull {a + e, c} = {a + e}"
"convex hull {a + e, a - e} \<inter> convex hull {c, a - e} = {a - e}"
by (metis (full_types) Int_closed_segment insert_commute segment_convex_hull)+
then show "?lhs \<subseteq> ?rhs"
by (metis Int_Un_distrib equalityD1 insert_is_Un path_image_join path_image_linepath path_join_eq path_linepath segment_convex_hull simple_path_def sp_aec)
show "?rhs \<subseteq> ?lhs"
using segment_convex_hull by (simp add: path_image_join)
qed
have "path_image p \<inter> path_image (linepath (a + e) c) \<subseteq> {a + e}"
proof (clarsimp simp: path_image_join)
fix x
assume "x \<in> path_image p" and x_ac: "x \<in> closed_segment (a + e) c"
then have "dist x a \<ge> e"
by (metis IntI all_not_in_conv disj dist_commute mem_ball not_less)
with x_ac dac \<open>e > 0\<close> show "x = a + e"
by (auto simp: norm_minus_commute dist_norm closed_segment_eq_open dest: open_segment_furthest_le [where y=a])
qed
moreover
have "path_image p \<inter> path_image (linepath c (a - e)) \<subseteq> {a - e}"
proof (clarsimp simp: path_image_join)
fix x
assume "x \<in> path_image p" and x_ac: "x \<in> closed_segment c (a - e)"
then have "dist x a \<ge> e"
by (metis IntI all_not_in_conv disj dist_commute mem_ball not_less)
with x_ac dac \<open>e > 0\<close> show "x = a - e"
by (auto simp: norm_minus_commute dist_norm closed_segment_eq_open dest: open_segment_furthest_le [where y=a])
qed
ultimately
have "path_image p \<inter> path_image (linepath (a + e) c +++ linepath c (a - e)) \<subseteq> {a + e, a - e}"
by (force simp: path_image_join)
then show 3: "path_image p \<inter> path_image (linepath (a + e) c +++ linepath c (a - e)) = {a + e, a - e}"
using "1" "2" by blast
show 4: "path_image (linepath (a + e) c +++ linepath c (a - e)) \<inter>
inside (path_image (linepath (a + e) (a - e)) \<union> path_image p) \<noteq> {}"
apply (clarsimp simp: path_image_join segment_convex_hull disjoint_iff_not_equal)
by (metis (no_types, opaque_lifting) UnI1 Un_commute c closed_segment_commute ends_in_segment(1) path_image_join
path_image_linepath pathstart_linepath pfp segment_convex_hull)
show zin_inside: "z \<in> inside (path_image (linepath (a + e) (a - e)) \<union>
path_image (linepath (a + e) c +++ linepath c (a - e)))"
proof (simp add: path_image_join)
show "z \<in> inside (closed_segment (a + e) (a - e) \<union> (closed_segment (a + e) c \<union> closed_segment c (a - e)))"
by (metis z inside_of_triangle DIM_complex Un_commute closed_segment_commute)
qed
show 5: "winding_number
(linepath (a + e) (a - e) +++ reversepath (linepath (a + e) c +++ linepath c (a - e))) z =
winding_number (linepath (a - e) c +++ linepath c (a + e) +++ linepath (a + e) (a - e)) z"
by (simp add: reversepath_joinpaths path_image_join winding_number_join)
show 6: "winding_number (linepath (a - e) c +++ linepath c (a + e) +++ linepath (a + e) (a - e)) z \<noteq> 0"
by (simp add: winding_number_triangle z)
show "winding_number (linepath (a + e) (a - e) +++ reversepath p) z =
winding_number (linepath (a - e) c +++ linepath c (a + e) +++ linepath (a + e) (a - e)) z"
by (metis 1 2 3 4 5 6 pa sp_aec sp_ae2 \<open>arc p\<close> \<open>simple_path p\<close> arc_distinct_ends winding_number_from_innerpath zin_inside)
qed (use assms \<open>e > 0\<close> in auto)
show z_inside: "z \<in> inside (path_image ?pae)"
using zin by (simp add: assms path_image_join Un_commute closed_segment_commute)
have "cmod (winding_number ?pae z) = cmod ((winding_number(reversepath ?pae) z))"
proof (subst winding_number_reversepath)
show "path ?pae"
using simple_path_imp_path sp_pl by blast
show "z \<notin> path_image ?pae"
by (metis IntI emptyE inside_no_overlap z_inside)
qed (simp add: inside_def)
also have "... = cmod (winding_number(linepath (a + e) (a - e) +++ reversepath p) z)"
by (simp add: pfp reversepath_joinpaths)
also have "... = cmod (winding_number (linepath (a - e) c +++ linepath c (a + e) +++ linepath (a + e) (a - e)) z)"
by (simp add: zeq)
also have "... = 1"
using z by (simp add: interior_of_triangle winding_number_triangle)
finally show "cmod (winding_number ?pae z) = 1" .
qed
qed
lemma simple_closed_path_wn2:
fixes a::complex and d e::real
assumes "0 < d" "0 < e"
and sp_pl: "simple_path(p +++ linepath (a - d) (a + e))"
and psp: "pathstart p = a + e"
and pfp: "pathfinish p = a - d"
obtains z where "z \<in> inside (path_image (p +++ linepath (a - d) (a + e)))"
"cmod (winding_number (p +++ linepath (a - d) (a + e)) z) = 1"
proof -
have [simp]: "a + of_real x \<in> closed_segment (a - \<alpha>) (a - \<beta>) \<longleftrightarrow> x \<in> closed_segment (-\<alpha>) (-\<beta>)" for x \<alpha> \<beta>::real
using closed_segment_translation_eq [of a]
by (metis (no_types, opaque_lifting) add_uminus_conv_diff of_real_minus of_real_closed_segment)
have [simp]: "a - of_real x \<in> closed_segment (a + \<alpha>) (a + \<beta>) \<longleftrightarrow> -x \<in> closed_segment \<alpha> \<beta>" for x \<alpha> \<beta>::real
by (metis closed_segment_translation_eq diff_conv_add_uminus of_real_closed_segment of_real_minus)
have "arc p" and arc_lp: "arc (linepath (a - d) (a + e))" and "path p"
and pap: "path_image p \<inter> closed_segment (a - d) (a + e) \<subseteq> {a+e, a-d}"
using simple_path_join_loop_eq [of "linepath (a - d) (a + e)" p] assms arc_imp_path by auto
have "0 \<in> closed_segment (-d) e"
using \<open>0 < d\<close> \<open>0 < e\<close> closed_segment_eq_real_ivl by auto
then have "a \<in> path_image (linepath (a - d) (a + e))"
using of_real_closed_segment [THEN iffD2]
by (force dest: closed_segment_translation_eq [of a, THEN iffD2] simp del: of_real_closed_segment)
then have "a \<notin> path_image p"
using \<open>0 < d\<close> \<open>0 < e\<close> pap by auto
then obtain k where "0 < k" and k: "ball a k \<inter> (path_image p) = {}"
using \<open>0 < e\<close> \<open>path p\<close> not_on_path_ball by blast
define kde where "kde \<equiv> (min k (min d e)) / 2"
have "0 < kde" "kde < k" "kde < d" "kde < e"
using \<open>0 < k\<close> \<open>0 < d\<close> \<open>0 < e\<close> by (auto simp: kde_def)
let ?q = "linepath (a + kde) (a + e) +++ p +++ linepath (a - d) (a - kde)"
have "- kde \<in> closed_segment (-d) e"
using \<open>0 < kde\<close> \<open>kde < d\<close> \<open>kde < e\<close> closed_segment_eq_real_ivl by auto
then have a_diff_kde: "a - kde \<in> closed_segment (a - d) (a + e)"
using of_real_closed_segment [THEN iffD2]
by (force dest: closed_segment_translation_eq [of a, THEN iffD2] simp del: of_real_closed_segment)
then have clsub2: "closed_segment (a - d) (a - kde) \<subseteq> closed_segment (a - d) (a + e)"
by (simp add: subset_closed_segment)
then have "path_image p \<inter> closed_segment (a - d) (a - kde) \<subseteq> {a + e, a - d}"
using pap by force
moreover
have "a + e \<notin> path_image p \<inter> closed_segment (a - d) (a - kde)"
using \<open>0 < kde\<close> \<open>kde < d\<close> \<open>0 < e\<close> by (auto simp: closed_segment_eq_real_ivl)
ultimately have sub_a_diff_d: "path_image p \<inter> closed_segment (a - d) (a - kde) \<subseteq> {a - d}"
by blast
have "kde \<in> closed_segment (-d) e"
using \<open>0 < kde\<close> \<open>kde < d\<close> \<open>kde < e\<close> closed_segment_eq_real_ivl by auto
then have a_diff_kde: "a + kde \<in> closed_segment (a - d) (a + e)"
using of_real_closed_segment [THEN iffD2]
by (force dest: closed_segment_translation_eq [of "a", THEN iffD2] simp del: of_real_closed_segment)
then have clsub1: "closed_segment (a + kde) (a + e) \<subseteq> closed_segment (a - d) (a + e)"
by (simp add: subset_closed_segment)
then have "closed_segment (a + kde) (a + e) \<inter> path_image p \<subseteq> {a + e, a - d}"
using pap by force
moreover
have "closed_segment (a + kde) (a + e) \<inter> closed_segment (a - d) (a - kde) = {}"
proof (clarsimp intro!: equals0I)
fix y
assume y1: "y \<in> closed_segment (a + kde) (a + e)"
and y2: "y \<in> closed_segment (a - d) (a - kde)"
obtain u::real where u: "y = a + u" and "0 < u"
proof -
obtain \<xi> where \<xi>: "y = (1 - \<xi>) *\<^sub>R (a + kde) + \<xi> *\<^sub>R (a + e)" and "0 \<le> \<xi>" "\<xi> \<le> 1"
using y1 by (auto simp: in_segment)
show thesis
proof
show "y = a + ((1 - \<xi>)*kde + \<xi>*e)"
using \<xi> by (auto simp: scaleR_conv_of_real algebra_simps)
have "(1 - \<xi>)*kde + \<xi>*e \<ge> 0"
using \<open>0 < kde\<close> \<open>0 \<le> \<xi>\<close> \<open>\<xi> \<le> 1\<close> \<open>0 < e\<close> by auto
moreover have "(1 - \<xi>)*kde + \<xi>*e \<noteq> 0"
using \<open>0 < kde\<close> \<open>0 \<le> \<xi>\<close> \<open>\<xi> \<le> 1\<close> \<open>0 < e\<close> by (auto simp: add_nonneg_eq_0_iff)
ultimately show "(1 - \<xi>)*kde + \<xi>*e > 0" by simp
qed
qed
moreover
obtain v::real where v: "y = a + v" and "v \<le> 0"
proof -
obtain \<xi> where \<xi>: "y = (1 - \<xi>) *\<^sub>R (a - d) + \<xi> *\<^sub>R (a - kde)" and "0 \<le> \<xi>" "\<xi> \<le> 1"
using y2 by (auto simp: in_segment)
show thesis
proof
show "y = a + (- ((1 - \<xi>)*d + \<xi>*kde))"
using \<xi> by (force simp: scaleR_conv_of_real algebra_simps)
show "- ((1 - \<xi>)*d + \<xi>*kde) \<le> 0"
using \<open>0 < kde\<close> \<open>0 \<le> \<xi>\<close> \<open>\<xi> \<le> 1\<close> \<open>0 < d\<close>
by (metis add.left_neutral add_nonneg_nonneg le_diff_eq less_eq_real_def neg_le_0_iff_le split_mult_pos_le)
qed
qed
ultimately show False
by auto
qed
moreover have "a - d \<notin> closed_segment (a + kde) (a + e)"
using \<open>0 < kde\<close> \<open>kde < d\<close> \<open>0 < e\<close> by (auto simp: closed_segment_eq_real_ivl)
ultimately have sub_a_plus_e:
"closed_segment (a + kde) (a + e) \<inter> (path_image p \<union> closed_segment (a - d) (a - kde)) \<subseteq> {a + e}"
by auto
have "kde \<in> closed_segment (-kde) e"
using \<open>0 < kde\<close> \<open>kde < d\<close> \<open>kde < e\<close> closed_segment_eq_real_ivl by auto
then have a_add_kde: "a + kde \<in> closed_segment (a - kde) (a + e)"
using of_real_closed_segment [THEN iffD2]
by (force dest: closed_segment_translation_eq [of "a", THEN iffD2] simp del: of_real_closed_segment)
have "closed_segment (a - kde) (a + kde) \<inter> closed_segment (a + kde) (a + e) = {a + kde}"
by (metis a_add_kde Int_closed_segment)
moreover
have "path_image p \<inter> closed_segment (a - kde) (a + kde) = {}"
proof (rule equals0I, clarify)
fix y assume "y \<in> path_image p" "y \<in> closed_segment (a - kde) (a + kde)"
with equals0D [OF k, of y] \<open>0 < kde\<close> \<open>kde < k\<close> show False
by (auto simp: dist_norm dest: dist_decreases_closed_segment [where c=a])
qed
moreover
have "- kde \<in> closed_segment (-d) kde"
using \<open>0 < kde\<close> \<open>kde < d\<close> \<open>kde < e\<close> closed_segment_eq_real_ivl by auto
then have a_diff_kde': "a - kde \<in> closed_segment (a - d) (a + kde)"
using of_real_closed_segment [THEN iffD2]
by (force dest: closed_segment_translation_eq [of a, THEN iffD2] simp del: of_real_closed_segment)
then have "closed_segment (a - d) (a - kde) \<inter> closed_segment (a - kde) (a + kde) = {a - kde}"
by (metis Int_closed_segment)
ultimately
have pa_subset_pm_kde: "path_image ?q \<inter> closed_segment (a - kde) (a + kde) \<subseteq> {a - kde, a + kde}"
by (auto simp: path_image_join assms)
have ge_kde1: "\<exists>y. x = a + y \<and> y \<ge> kde" if x: "x \<in> closed_segment (a + kde) (a + e)" for x
proof -
obtain u where "0 \<le> u" "u \<le> 1" and u: "x = (1 - u) *\<^sub>R (a + kde) + u *\<^sub>R (a + e)"
using x by (auto simp: in_segment)
then have "kde \<le> (1 - u) * kde + u * e"
using \<open>kde < e\<close> segment_bound_lemma by auto
moreover have "x = a + ((1 - u) * kde + u * e)"
by (fastforce simp: u algebra_simps scaleR_conv_of_real)
ultimately
show ?thesis by blast
qed
have ge_kde2: "\<exists>y. x = a + y \<and> y \<le> -kde" if x: "x \<in> closed_segment (a - d) (a - kde)" for x
proof -
obtain u where "0 \<le> u" "u \<le> 1" and u: "x = (1 - u) *\<^sub>R (a - d) + u *\<^sub>R (a - kde)"
using x by (auto simp: in_segment)
then have "kde \<le> ((1-u)*d + u*kde)"
using \<open>kde < d\<close> segment_bound_lemma by auto
moreover have "x = a - ((1-u)*d + u*kde)"
by (fastforce simp: u algebra_simps scaleR_conv_of_real)
ultimately show ?thesis
by (metis add_uminus_conv_diff neg_le_iff_le of_real_minus)
qed
have notin_paq: "x \<notin> path_image ?q" if "dist a x < kde" for x
proof -
have "x \<notin> path_image p"
using k \<open>kde < k\<close> that by force
then show ?thesis
using that assms by (auto simp: path_image_join dist_norm dest!: ge_kde1 ge_kde2)
qed
obtain z where zin: "z \<in> inside (path_image (?q +++ linepath (a - kde) (a + kde)))"
and z1: "cmod (winding_number (?q +++ linepath (a - kde) (a + kde)) z) = 1"
proof (rule simple_closed_path_wn1 [of kde ?q a])
show "simple_path (?q +++ linepath (a - kde) (a + kde))"
proof (intro simple_path_join_loop conjI)
show "arc ?q"
proof (rule arc_join)
show "arc (linepath (a + kde) (a + e))"
using \<open>kde < e\<close> \<open>arc p\<close> by (force simp: pfp)
show "arc (p +++ linepath (a - d) (a - kde))"
using \<open>kde < d\<close> \<open>kde < e\<close> \<open>arc p\<close> sub_a_diff_d by (force simp: pfp intro: arc_join)
qed (auto simp: psp pfp path_image_join sub_a_plus_e)
show "arc (linepath (a - kde) (a + kde))"
using \<open>0 < kde\<close> by auto
qed (use pa_subset_pm_kde in auto)
qed (use \<open>0 < kde\<close> notin_paq in auto)
have eq: "path_image (?q +++ linepath (a - kde) (a + kde)) = path_image (p +++ linepath (a - d) (a + e))"
(is "?lhs = ?rhs")
proof
show "?lhs \<subseteq> ?rhs"
using clsub1 clsub2 apply (auto simp: path_image_join assms)
by (meson subsetCE subset_closed_segment)
show "?rhs \<subseteq> ?lhs"
apply (simp add: path_image_join assms Un_ac)
by (metis Un_closed_segment Un_assoc a_diff_kde a_diff_kde' le_supI2 subset_refl)
qed
show thesis
proof
show zzin: "z \<in> inside (path_image (p +++ linepath (a - d) (a + e)))"
by (metis eq zin)
then have znotin: "z \<notin> path_image p"
by (metis ComplD Un_iff inside_Un_outside path_image_join pathfinish_linepath pathstart_reversepath pfp reversepath_linepath)
have znotin_d_kde: "z \<notin> closed_segment (a - d) (a + kde)"
by (metis ComplD Un_iff Un_closed_segment a_diff_kde inside_Un_outside path_image_join path_image_linepath pathstart_linepath pfp zzin)
have znotin_d_e: "z \<notin> closed_segment (a - d) (a + e)"
by (metis IntI UnCI emptyE inside_no_overlap path_image_join path_image_linepath pathstart_linepath pfp zzin)
have znotin_kde_e: "z \<notin> closed_segment (a + kde) (a + e)" and znotin_d_kde': "z \<notin> closed_segment (a - d) (a - kde)"
using clsub1 clsub2 zzin
by (metis (no_types, opaque_lifting) IntI Un_iff emptyE inside_no_overlap path_image_join path_image_linepath pathstart_linepath pfp subsetD)+
have "winding_number (linepath (a - d) (a + e)) z =
winding_number (linepath (a - d) (a + kde)) z + winding_number (linepath (a + kde) (a + e)) z"
proof (rule winding_number_split_linepath)
show "a + complex_of_real kde \<in> closed_segment (a - d) (a + e)"
by (simp add: a_diff_kde)
show "z \<notin> closed_segment (a - d) (a + e)"
by (metis ComplD Un_iff inside_Un_outside path_image_join path_image_linepath pathstart_linepath pfp zzin)
qed
also have "... = winding_number (linepath (a + kde) (a + e)) z +
(winding_number (linepath (a - d) (a - kde)) z + winding_number (linepath (a - kde) (a + kde)) z)"
by (simp add: winding_number_split_linepath [of "a-kde", symmetric] znotin_d_kde a_diff_kde')
finally have "winding_number (p +++ linepath (a - d) (a + e)) z =
winding_number p z + winding_number (linepath (a + kde) (a + e)) z +
(winding_number (linepath (a - d) (a - kde)) z +
winding_number (linepath (a - kde) (a + kde)) z)"
by (metis (no_types, lifting) ComplD Un_iff \<open>arc p\<close> add.assoc arc_imp_path eq path_image_join path_join_path_ends path_linepath simple_path_imp_path sp_pl union_with_outside winding_number_join zin)
also have "... = winding_number (linepath (a + kde) (a + e)) z
+ winding_number (p +++ linepath (a - d) (a - kde)) z
+ winding_number (linepath (a - kde) (a + kde)) z"
using \<open>path p\<close> znotin assms
by simp (metis Un_iff Un_closed_segment a_diff_kde' path_image_linepath path_linepath pathstart_linepath winding_number_join znotin_d_kde)
also have "... = winding_number ?q z + winding_number (linepath (a - kde) (a + kde)) z"
using \<open>path p\<close> znotin assms by (simp add: path_image_join winding_number_join znotin_kde_e znotin_d_kde')
also have "... = winding_number (?q +++ linepath (a - kde) (a + kde)) z"
by (metis (mono_tags, lifting) ComplD UnCI \<open>path p\<close> eq inside_outside path_image_join path_join_eq path_linepath pathfinish_join pathfinish_linepath pathstart_join pathstart_linepath pfp psp winding_number_join zzin)
finally have "winding_number (p +++ linepath (a - d) (a + e)) z =
winding_number (?q +++ linepath (a - kde) (a + kde)) z" .
then show "cmod (winding_number (p +++ linepath (a - d) (a + e)) z) = 1"
by (simp add: z1)
qed
qed
lemma simple_closed_path_wn3:
fixes p :: "real \<Rightarrow> complex"
assumes "simple_path p" and loop: "pathfinish p = pathstart p"
obtains z where "z \<in> inside (path_image p)" "cmod (winding_number p z) = 1"
proof -
have ins: "inside(path_image p) \<noteq> {}" "open(inside(path_image p))"
"connected(inside(path_image p))"
and out: "outside(path_image p) \<noteq> {}" "open(outside(path_image p))"
"connected(outside(path_image p))"
and bo: "bounded(inside(path_image p))" "\<not> bounded(outside(path_image p))"
and ins_out: "inside(path_image p) \<inter> outside(path_image p) = {}"
"inside(path_image p) \<union> outside(path_image p) = - path_image p"
and fro: "frontier(inside(path_image p)) = path_image p"
"frontier(outside(path_image p)) = path_image p"
using Jordan_inside_outside [OF assms] by auto
obtain a where a: "a \<in> inside(path_image p)"
using \<open>inside (path_image p) \<noteq> {}\<close> by blast
obtain d::real where "0 < d" and d_fro: "a - d \<in> frontier (inside (path_image p))"
and d_int: "\<And>\<epsilon>. \<lbrakk>0 \<le> \<epsilon>; \<epsilon> < d\<rbrakk> \<Longrightarrow> (a - \<epsilon>) \<in> inside (path_image p)"
using ray_to_frontier [of "inside (path_image p)" a "-1"] bo ins a interior_eq
by (metis ab_group_add_class.ab_diff_conv_add_uminus of_real_def scale_minus_right zero_neq_neg_one)
obtain e::real where "0 < e" and e_fro: "a + e \<in> frontier (inside (path_image p))"
and e_int: "\<And>\<epsilon>. \<lbrakk>0 \<le> \<epsilon>; \<epsilon> < e\<rbrakk> \<Longrightarrow> (a + \<epsilon>) \<in> inside (path_image p)"
using ray_to_frontier [of "inside (path_image p)" a 1] bo ins a interior_eq
by (metis of_real_def zero_neq_one)
obtain t0 where "0 \<le> t0" "t0 \<le> 1" and pt: "p t0 = a - d"
using a d_fro fro by (auto simp: path_image_def)
obtain q where "simple_path q" and q_ends: "pathstart q = a - d" "pathfinish q = a - d"
and q_eq_p: "path_image q = path_image p"
and wn_q_eq_wn_p: "\<And>z. z \<in> inside(path_image p) \<Longrightarrow> winding_number q z = winding_number p z"
proof
show "simple_path (shiftpath t0 p)"
by (simp add: pathstart_shiftpath pathfinish_shiftpath
simple_path_shiftpath path_image_shiftpath \<open>0 \<le> t0\<close> \<open>t0 \<le> 1\<close> assms)
show "pathstart (shiftpath t0 p) = a - d"
using pt by (simp add: \<open>t0 \<le> 1\<close> pathstart_shiftpath)
show "pathfinish (shiftpath t0 p) = a - d"
by (simp add: \<open>0 \<le> t0\<close> loop pathfinish_shiftpath pt)
show "path_image (shiftpath t0 p) = path_image p"
by (simp add: \<open>0 \<le> t0\<close> \<open>t0 \<le> 1\<close> loop path_image_shiftpath)
show "winding_number (shiftpath t0 p) z = winding_number p z"
if "z \<in> inside (path_image p)" for z
by (metis ComplD Un_iff \<open>0 \<le> t0\<close> \<open>t0 \<le> 1\<close> \<open>simple_path p\<close> atLeastAtMost_iff inside_Un_outside
loop simple_path_imp_path that winding_number_shiftpath)
qed
have ad_not_ae: "a - d \<noteq> a + e"
by (metis \<open>0 < d\<close> \<open>0 < e\<close> add.left_inverse add_left_cancel add_uminus_conv_diff
le_add_same_cancel2 less_eq_real_def not_less of_real_add of_real_def of_real_eq_0_iff pt)
have ad_ae_q: "{a - d, a + e} \<subseteq> path_image q"
using \<open>path_image q = path_image p\<close> d_fro e_fro fro(1) by auto
have ada: "open_segment (a - d) a \<subseteq> inside (path_image p)"
proof (clarsimp simp: in_segment)
fix u::real assume "0 < u" "u < 1"
with d_int have "a - (1 - u) * d \<in> inside (path_image p)"
by (metis \<open>0 < d\<close> add.commute diff_add_cancel left_diff_distrib' less_add_same_cancel2 less_eq_real_def mult.left_neutral zero_less_mult_iff)
then show "(1 - u) *\<^sub>R (a - d) + u *\<^sub>R a \<in> inside (path_image p)"
by (simp add: diff_add_eq of_real_def real_vector.scale_right_diff_distrib)
qed
have aae: "open_segment a (a + e) \<subseteq> inside (path_image p)"
proof (clarsimp simp: in_segment)
fix u::real assume "0 < u" "u < 1"
with e_int have "a + u * e \<in> inside (path_image p)"
by (meson \<open>0 < e\<close> less_eq_real_def mult_less_cancel_right2 not_less zero_less_mult_iff)
then show "(1 - u) *\<^sub>R a + u *\<^sub>R (a + e) \<in> inside (path_image p)"
by (metis (mono_tags, lifting) add.assoc of_real_mult pth_6 scaleR_collapse scaleR_conv_of_real)
qed
have "complex_of_real (d * d + (e * e + d * (e + e))) \<noteq> 0"
using ad_not_ae
by (metis \<open>0 < d\<close> \<open>0 < e\<close> add_strict_left_mono less_add_same_cancel1 not_sum_squares_lt_zero
of_real_eq_0_iff zero_less_double_add_iff_zero_less_single_add zero_less_mult_iff)
moreover have "\<exists>u>0. u < 1 \<and> d = u * d + u * e"
using \<open>0 < d\<close> \<open>0 < e\<close>
by (rule_tac x="d / (d+e)" in exI) (simp add: divide_simps scaleR_conv_of_real flip: distrib_left)
ultimately have a_in_de: "a \<in> open_segment (a - d) (a + e)"
using ad_not_ae by (simp add: in_segment algebra_simps scaleR_conv_of_real flip: of_real_add of_real_mult)
then have "open_segment (a - d) (a + e) \<subseteq> inside (path_image p)"
using ada a aae Un_open_segment [of a "a-d" "a+e"] by blast
then have "path_image q \<inter> open_segment (a - d) (a + e) = {}"
using inside_no_overlap by (fastforce simp: q_eq_p)
with ad_ae_q have paq_Int_cs: "path_image q \<inter> closed_segment (a - d) (a + e) = {a - d, a + e}"
by (simp add: closed_segment_eq_open)
obtain t where "0 \<le> t" "t \<le> 1" and qt: "q t = a + e"
using a e_fro fro ad_ae_q by (auto simp: path_defs)
then have "t \<noteq> 0"
by (metis ad_not_ae pathstart_def q_ends(1))
then have "t \<noteq> 1"
by (metis ad_not_ae pathfinish_def q_ends(2) qt)
have q01: "q 0 = a - d" "q 1 = a - d"
using q_ends by (auto simp: pathstart_def pathfinish_def)
obtain z where zin: "z \<in> inside (path_image (subpath t 0 q +++ linepath (a - d) (a + e)))"
and z1: "cmod (winding_number (subpath t 0 q +++ linepath (a - d) (a + e)) z) = 1"
proof (rule simple_closed_path_wn2 [of d e "subpath t 0 q" a], simp_all add: q01)
show "simple_path (subpath t 0 q +++ linepath (a - d) (a + e))"
proof (rule simple_path_join_loop, simp_all add: qt q01)
have "inj_on q (closed_segment t 0)"
using \<open>0 \<le> t\<close> \<open>simple_path q\<close> \<open>t \<le> 1\<close> \<open>t \<noteq> 0\<close> \<open>t \<noteq> 1\<close>
by (fastforce simp: simple_path_def inj_on_def closed_segment_eq_real_ivl)
then show "arc (subpath t 0 q)"
using \<open>0 \<le> t\<close> \<open>simple_path q\<close> \<open>t \<le> 1\<close> \<open>t \<noteq> 0\<close>
by (simp add: arc_subpath_eq simple_path_imp_path)
show "arc (linepath (a - d) (a + e))"
by (simp add: ad_not_ae)
show "path_image (subpath t 0 q) \<inter> closed_segment (a - d) (a + e) \<subseteq> {a + e, a - d}"
using qt paq_Int_cs \<open>simple_path q\<close> \<open>0 \<le> t\<close> \<open>t \<le> 1\<close>
by (force simp: dest: rev_subsetD [OF _ path_image_subpath_subset] intro: simple_path_imp_path)
qed
qed (auto simp: \<open>0 < d\<close> \<open>0 < e\<close> qt)
have pa01_Un: "path_image (subpath 0 t q) \<union> path_image (subpath 1 t q) = path_image q"
unfolding path_image_subpath
using \<open>0 \<le> t\<close> \<open>t \<le> 1\<close> by (force simp: path_image_def image_iff)
with paq_Int_cs have pa_01q:
"(path_image (subpath 0 t q) \<union> path_image (subpath 1 t q)) \<inter> closed_segment (a - d) (a + e) = {a - d, a + e}"
by metis
have z_notin_ed: "z \<notin> closed_segment (a + e) (a - d)"
using zin q01 by (simp add: path_image_join closed_segment_commute inside_def)
have z_notin_0t: "z \<notin> path_image (subpath 0 t q)"
by (metis (no_types, opaque_lifting) IntI Un_upper1 subsetD empty_iff inside_no_overlap path_image_join
path_image_subpath_commute pathfinish_subpath pathstart_def pathstart_linepath q_ends(1) qt subpath_trivial zin)
have [simp]: "- winding_number (subpath t 0 q) z = winding_number (subpath 0 t q) z"
by (metis \<open>0 \<le> t\<close> \<open>simple_path q\<close> \<open>t \<le> 1\<close> atLeastAtMost_iff zero_le_one
path_image_subpath_commute path_subpath real_eq_0_iff_le_ge_0
reversepath_subpath simple_path_imp_path winding_number_reversepath z_notin_0t)
obtain z_in_q: "z \<in> inside(path_image q)"
and wn_q: "winding_number (subpath 0 t q +++ subpath t 1 q) z = - winding_number (subpath t 0 q +++ linepath (a - d) (a + e)) z"
proof (rule winding_number_from_innerpath
[of "subpath 0 t q" "a-d" "a+e" "subpath 1 t q" "linepath (a - d) (a + e)"
z "- winding_number (subpath t 0 q +++ linepath (a - d) (a + e)) z"],
simp_all add: q01 qt pa01_Un reversepath_subpath)
show "simple_path (subpath 0 t q)" "simple_path (subpath 1 t q)"
by (simp_all add: \<open>0 \<le> t\<close> \<open>simple_path q\<close> \<open>t \<le> 1\<close> \<open>t \<noteq> 0\<close> \<open>t \<noteq> 1\<close> simple_path_subpath)
show "simple_path (linepath (a - d) (a + e))"
using ad_not_ae by blast
show "path_image (subpath 0 t q) \<inter> path_image (subpath 1 t q) = {a - d, a + e}" (is "?lhs = ?rhs")
proof
show "?lhs \<subseteq> ?rhs"
using \<open>0 \<le> t\<close> \<open>simple_path q\<close> \<open>t \<le> 1\<close> \<open>t \<noteq> 1\<close> q_ends qt q01
by (force simp: pathfinish_def qt simple_path_def path_image_subpath)
show "?rhs \<subseteq> ?lhs"
using \<open>0 \<le> t\<close> \<open>t \<le> 1\<close> q01 qt by (force simp: path_image_subpath)
qed
show "path_image (subpath 0 t q) \<inter> closed_segment (a - d) (a + e) = {a - d, a + e}" (is "?lhs = ?rhs")
proof
show "?lhs \<subseteq> ?rhs" using paq_Int_cs pa01_Un by fastforce
show "?rhs \<subseteq> ?lhs" using \<open>0 \<le> t\<close> \<open>t \<le> 1\<close> q01 qt by (force simp: path_image_subpath)
qed
show "path_image (subpath 1 t q) \<inter> closed_segment (a - d) (a + e) = {a - d, a + e}" (is "?lhs = ?rhs")
proof
show "?lhs \<subseteq> ?rhs" by (auto simp: pa_01q [symmetric])
show "?rhs \<subseteq> ?lhs" using \<open>0 \<le> t\<close> \<open>t \<le> 1\<close> q01 qt by (force simp: path_image_subpath)
qed
show "closed_segment (a - d) (a + e) \<inter> inside (path_image q) \<noteq> {}"
using a a_in_de open_closed_segment pa01_Un q_eq_p by fastforce
show "z \<in> inside (path_image (subpath 0 t q) \<union> closed_segment (a - d) (a + e))"
by (metis path_image_join path_image_linepath path_image_subpath_commute pathfinish_subpath pathstart_linepath q01(1) zin)
show "winding_number (subpath 0 t q +++ linepath (a + e) (a - d)) z =
- winding_number (subpath t 0 q +++ linepath (a - d) (a + e)) z"
using z_notin_ed z_notin_0t \<open>0 \<le> t\<close> \<open>simple_path q\<close> \<open>t \<le> 1\<close>
by (simp add: simple_path_imp_path qt q01 path_image_subpath_commute closed_segment_commute winding_number_join winding_number_reversepath [symmetric])
show "- d \<noteq> e"
using ad_not_ae by auto
show "winding_number (subpath t 0 q +++ linepath (a - d) (a + e)) z \<noteq> 0"
using z1 by auto
qed
show ?thesis
proof
show "z \<in> inside (path_image p)"
using q_eq_p z_in_q by auto
then have [simp]: "z \<notin> path_image q"
by (metis disjoint_iff_not_equal inside_no_overlap q_eq_p)
have [simp]: "z \<notin> path_image (subpath 1 t q)"
using inside_def pa01_Un z_in_q by fastforce
have "winding_number(subpath 0 t q +++ subpath t 1 q) z = winding_number(subpath 0 1 q) z"
using z_notin_0t \<open>0 \<le> t\<close> \<open>simple_path q\<close> \<open>t \<le> 1\<close>
by (simp add: simple_path_imp_path qt path_image_subpath_commute winding_number_join winding_number_subpath_combine)
with wn_q have "winding_number (subpath t 0 q +++ linepath (a - d) (a + e)) z = - winding_number q z"
by auto
with z1 have "cmod (winding_number q z) = 1"
by simp
with z1 wn_q_eq_wn_p show "cmod (winding_number p z) = 1"
using z1 wn_q_eq_wn_p by (simp add: \<open>z \<in> inside (path_image p)\<close>)
qed
qed
proposition simple_closed_path_winding_number_inside:
assumes "simple_path \<gamma>"
obtains "\<And>z. z \<in> inside(path_image \<gamma>) \<Longrightarrow> winding_number \<gamma> z = 1"
| "\<And>z. z \<in> inside(path_image \<gamma>) \<Longrightarrow> winding_number \<gamma> z = -1"
proof (cases "pathfinish \<gamma> = pathstart \<gamma>")
case True
have "path \<gamma>"
by (simp add: assms simple_path_imp_path)
then have const: "winding_number \<gamma> constant_on inside(path_image \<gamma>)"
proof (rule winding_number_constant)
show "connected (inside(path_image \<gamma>))"
by (simp add: Jordan_inside_outside True assms)
qed (use inside_no_overlap True in auto)
obtain z where zin: "z \<in> inside (path_image \<gamma>)" and z1: "cmod (winding_number \<gamma> z) = 1"
using simple_closed_path_wn3 [of \<gamma>] True assms by blast
have "winding_number \<gamma> z \<in> \<int>"
using zin integer_winding_number [OF \<open>path \<gamma>\<close> True] inside_def by blast
moreover have "real_of_int x = - 1 \<longleftrightarrow> x = -1" for x
by linarith
ultimately consider "winding_number \<gamma> z = 1" | "winding_number \<gamma> z = -1"
using z1 by (auto simp: Ints_def abs_if split: if_split_asm)
with that const zin show ?thesis
unfolding constant_on_def by metis
next
case False
then show ?thesis
using inside_simple_curve_imp_closed assms that(2) by blast
qed
lemma simple_closed_path_abs_winding_number_inside:
assumes "simple_path \<gamma>" "z \<in> inside(path_image \<gamma>)"
shows "\<bar>Re (winding_number \<gamma> z)\<bar> = 1"
by (metis assms norm_minus_cancel norm_one one_complex.simps(1) real_norm_def simple_closed_path_winding_number_inside uminus_complex.simps(1))
lemma simple_closed_path_norm_winding_number_inside:
assumes "simple_path \<gamma>" "z \<in> inside(path_image \<gamma>)"
shows "norm (winding_number \<gamma> z) = 1"
proof -
have "pathfinish \<gamma> = pathstart \<gamma>"
using assms inside_simple_curve_imp_closed by blast
with assms integer_winding_number have "winding_number \<gamma> z \<in> \<int>"
by (simp add: inside_def simple_path_def)
then show ?thesis
by (metis assms norm_minus_cancel norm_one simple_closed_path_winding_number_inside)
qed
lemma simple_closed_path_winding_number_cases:
assumes "simple_path \<gamma>" "pathfinish \<gamma> = pathstart \<gamma>" "z \<notin> path_image \<gamma>"
shows "winding_number \<gamma> z \<in> {-1,0,1}"
proof -
consider "z \<in> inside (path_image \<gamma>)" | "z \<in> outside (path_image \<gamma>)"
by (metis ComplI UnE assms(3) inside_Un_outside)
then show ?thesis
proof cases
case 1
then show ?thesis
using assms simple_closed_path_winding_number_inside by auto
next
case 2
then show ?thesis
using assms simple_path_def winding_number_zero_in_outside by blast
qed
qed
lemma simple_closed_path_winding_number_pos:
"\<lbrakk>simple_path \<gamma>; pathfinish \<gamma> = pathstart \<gamma>; z \<notin> path_image \<gamma>; 0 < Re(winding_number \<gamma> z)\<rbrakk>
\<Longrightarrow> winding_number \<gamma> z = 1"
using simple_closed_path_winding_number_cases
by fastforce
subsection \<open>Winding number for rectangular paths\<close>
proposition winding_number_rectpath:
assumes "z \<in> box a1 a3"
shows "winding_number (rectpath a1 a3) z = 1"
proof -
from assms have less: "Re a1 < Re a3" "Im a1 < Im a3"
by (auto simp: in_box_complex_iff)
define a2 a4 where "a2 = Complex (Re a3) (Im a1)" and "a4 = Complex (Re a1) (Im a3)"
let ?l1 = "linepath a1 a2" and ?l2 = "linepath a2 a3"
and ?l3 = "linepath a3 a4" and ?l4 = "linepath a4 a1"
from assms and less have "z \<notin> path_image (rectpath a1 a3)"
by (auto simp: path_image_rectpath_cbox_minus_box)
also have "path_image (rectpath a1 a3) =
path_image ?l1 \<union> path_image ?l2 \<union> path_image ?l3 \<union> path_image ?l4"
by (simp add: rectpath_def Let_def path_image_join Un_assoc a2_def a4_def)
finally have "z \<notin> \<dots>" .
moreover have "\<forall>l\<in>{?l1,?l2,?l3,?l4}. Re (winding_number l z) > 0"
unfolding ball_simps HOL.simp_thms a2_def a4_def
by (intro conjI; (rule winding_number_linepath_pos_lt;
(insert assms, auto simp: a2_def a4_def in_box_complex_iff mult_neg_neg))+)
ultimately have "Re (winding_number (rectpath a1 a3) z) > 0"
by (simp add: winding_number_join path_image_join rectpath_def Let_def a2_def a4_def)
thus "winding_number (rectpath a1 a3) z = 1" using assms less
by (intro simple_closed_path_winding_number_pos simple_path_rectpath)
(auto simp: path_image_rectpath_cbox_minus_box)
qed
proposition winding_number_rectpath_outside:
assumes "Re a1 \<le> Re a3" "Im a1 \<le> Im a3"
assumes "z \<notin> cbox a1 a3"
shows "winding_number (rectpath a1 a3) z = 0"
using assms by (intro winding_number_zero_outside[OF _ _ _ assms(3)]
path_image_rectpath_subset_cbox) simp_all
text\<open>A per-function version for continuous logs, a kind of monodromy\<close>
proposition\<^marker>\<open>tag unimportant\<close> winding_number_compose_exp:
assumes "path p"
shows "winding_number (exp \<circ> p) 0 = (pathfinish p - pathstart p) / (2 * of_real pi * \<i>)"
proof -
obtain e where "0 < e" and e: "\<And>t. t \<in> {0..1} \<Longrightarrow> e \<le> norm(exp(p t))"
proof
have "closed (path_image (exp \<circ> p))"
by (simp add: assms closed_path_image holomorphic_on_exp holomorphic_on_imp_continuous_on path_continuous_image)
then show "0 < setdist {0} (path_image (exp \<circ> p))"
by (metis exp_not_eq_zero imageE image_comp infdist_eq_setdist infdist_pos_not_in_closed path_defs(4) path_image_nonempty)
next
fix t::real
assume "t \<in> {0..1}"
have "setdist {0} (path_image (exp \<circ> p)) \<le> dist 0 (exp (p t))"
proof (rule setdist_le_dist)
show "exp (p t) \<in> path_image (exp \<circ> p)"
using \<open>t \<in> {0..1}\<close> path_image_def by fastforce+
qed auto
then show "setdist {0} (path_image (exp \<circ> p)) \<le> cmod (exp (p t))"
by simp
qed
have "bounded (path_image p)"
by (simp add: assms bounded_path_image)
then obtain B where "0 < B" and B: "path_image p \<subseteq> cball 0 B"
by (meson bounded_pos mem_cball_0 subsetI)
let ?B = "cball (0::complex) (B+1)"
have "uniformly_continuous_on ?B exp"
using holomorphic_on_exp holomorphic_on_imp_continuous_on
by (force intro: compact_uniformly_continuous)
then obtain d where "d > 0"
and d: "\<And>x x'. \<lbrakk>x\<in>?B; x'\<in>?B; dist x' x < d\<rbrakk> \<Longrightarrow> norm (exp x' - exp x) < e"
using \<open>e > 0\<close> by (auto simp: uniformly_continuous_on_def dist_norm)
then have "min 1 d > 0"
by force
then obtain g where pfg: "polynomial_function g" and "g 0 = p 0" "g 1 = p 1"
and gless: "\<And>t. t \<in> {0..1} \<Longrightarrow> norm(g t - p t) < min 1 d"
using path_approx_polynomial_function [OF \<open>path p\<close>] \<open>d > 0\<close> \<open>0 < e\<close>
unfolding pathfinish_def pathstart_def by blast
have "winding_number (exp \<circ> p) 0 = winding_number (exp \<circ> g) 0"
proof (rule winding_number_nearby_paths_eq [symmetric])
show "path (exp \<circ> p)" "path (exp \<circ> g)"
by (simp_all add: pfg assms holomorphic_on_exp holomorphic_on_imp_continuous_on path_continuous_image path_polynomial_function)
next
fix t :: "real"
assume t: "t \<in> {0..1}"
with gless have "norm(g t - p t) < 1"
using min_less_iff_conj by blast
moreover have ptB: "norm (p t) \<le> B"
using B t by (force simp: path_image_def)
ultimately have "cmod (g t) \<le> B + 1"
by (meson add_mono_thms_linordered_field(4) le_less_trans less_imp_le norm_triangle_sub)
with ptB gless t have "cmod ((exp \<circ> g) t - (exp \<circ> p) t) < e"
by (auto simp: dist_norm d)
with e t show "cmod ((exp \<circ> g) t - (exp \<circ> p) t) < cmod ((exp \<circ> p) t - 0)"
by fastforce
qed (use \<open>g 0 = p 0\<close> \<open>g 1 = p 1\<close> in \<open>auto simp: pathfinish_def pathstart_def\<close>)
also have "... = 1 / (of_real (2 * pi) * \<i>) * contour_integral (exp \<circ> g) (\<lambda>w. 1 / (w - 0))"
proof (rule winding_number_valid_path)
have "continuous_on (path_image g) (deriv exp)"
by (metis DERIV_exp DERIV_imp_deriv continuous_on_cong holomorphic_on_exp holomorphic_on_imp_continuous_on)
then show "valid_path (exp \<circ> g)"
by (simp add: field_differentiable_within_exp pfg valid_path_compose valid_path_polynomial_function)
show "0 \<notin> path_image (exp \<circ> g)"
by (auto simp: path_image_def)
qed
also have "... = 1 / (of_real (2 * pi) * \<i>) * integral {0..1} (\<lambda>x. vector_derivative g (at x))"
proof (simp add: contour_integral_integral, rule integral_cong)
fix t :: "real"
assume t: "t \<in> {0..1}"
show "vector_derivative (exp \<circ> g) (at t) / exp (g t) = vector_derivative g (at t)"
proof -
have "(exp \<circ> g has_vector_derivative vector_derivative (exp \<circ> g) (at t)) (at t)"
by (meson DERIV_exp differentiable_def field_vector_diff_chain_at has_vector_derivative_def
has_vector_derivative_polynomial_function pfg vector_derivative_works)
moreover have "(exp \<circ> g has_vector_derivative vector_derivative g (at t) * exp (g t)) (at t)"
by (metis DERIV_exp field_vector_diff_chain_at has_vector_derivative_polynomial_function pfg vector_derivative_at)
ultimately show ?thesis
by (simp add: divide_simps, rule vector_derivative_unique_at)
qed
qed
also have "... = (pathfinish p - pathstart p) / (2 * of_real pi * \<i>)"
proof -
have "((\<lambda>x. vector_derivative g (at x)) has_integral g 1 - g 0) {0..1}"
by (meson differentiable_at_polynomial_function fundamental_theorem_of_calculus
has_vector_derivative_at_within pfg vector_derivative_works zero_le_one)
then show ?thesis
unfolding pathfinish_def pathstart_def
using \<open>g 0 = p 0\<close> \<open>g 1 = p 1\<close> by auto
qed
finally show ?thesis .
qed
end