--- a/src/HOL/Analysis/Winding_Numbers.thy Wed Nov 27 16:54:33 2019 +0000
+++ b/src/HOL/Analysis/Winding_Numbers.thy Sun Dec 01 19:10:57 2019 +0000
@@ -1,1211 +1,1330 @@
section \<open>Winding Numbers\<close>
-
-text\<open>By John Harrison et al. Ported from HOL Light by L C Paulson (2017)\<close>
-
-theory Winding_Numbers
-imports
- Polytope
- Jordan_Curve
- Riemann_Mapping
+theory Winding_Numbers
+ imports Cauchy_Integral_Theorem
begin
-lemma simply_connected_inside_simple_path:
- fixes p :: "real \<Rightarrow> complex"
- shows "simple_path p \<Longrightarrow> simply_connected(inside(path_image p))"
- using Jordan_inside_outside connected_simple_path_image inside_simple_curve_imp_closed simply_connected_eq_frontier_properties
- by fastforce
+text\<open>We can treat even non-rectifiable paths as having a "length" for bounds on analytic functions in open sets.\<close>
+
+subsection \<open>Basic Winding Numbers\<close>
-lemma simply_connected_Int:
- fixes S :: "complex set"
- assumes "open S" "open T" "simply_connected S" "simply_connected T" "connected (S \<inter> T)"
- shows "simply_connected (S \<inter> T)"
- using assms by (force simp: simply_connected_eq_winding_number_zero open_Int)
+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"
-subsection\<open>Winding number for a triangle\<close>
+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 wn_triangle1:
- assumes "0 \<in> interior(convex hull {a,b,c})"
- shows "\<not> (Im(a/b) \<le> 0 \<and> 0 \<le> Im(b/c))"
+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 -
- { 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)
- }
- with False 0 have "convex hull {a,b,c} \<le> {z. Im z * Re b \<le> Im b * Re z}"
- apply (simp add: Complex.Im_divide divide_simps complex_neq_0 [symmetric])
- apply (simp add: algebra_simps)
- apply (rule hull_minimal)
- apply (auto simp: algebra_simps convex_alt)
+ 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 auto
+ 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 auto
+ have "(\<lambda>w. 1 / (w - z)) holomorphic_on UNIV - {z}"
+ by (auto simp: intro!: holomorphic_intros)
+ then show "\<exists>p. winding_number_prop \<gamma> z e p nn"
+ apply (rule_tac x=p in exI)
+ using pi_eq [of h p] h p d
+ apply (auto simp: valid_path_polynomial_function norm_minus_commute nn_def winding_number_prop_def)
done
- 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 "z \<in> ball 0 e"
- using \<open>e>0\<close>
- apply (simp add: z_def dist_norm)
- apply (rule le_less_trans [OF norm_triangle_ineq4])
- apply (simp add: norm_mult abs_sgn_eq)
- done
- then have "z \<in> {z. Im z * Re b \<le> Im b * Re z}"
- using e by blast
- then show False
- using \<open>e>0\<close> \<open>b \<noteq> 0\<close>
- apply (simp add: z_def dist_norm sgn_if less_eq_real_def mult_less_0_iff complex.expand split: if_split_asm)
- apply (auto simp: algebra_simps)
- apply (meson less_asym less_trans mult_pos_pos neg_less_0_iff_less)
- by (metis less_asym mult_pos_pos neg_less_0_iff_less)
- qed
- ultimately show ?thesis
- using interior_mono by blast
qed
- } with assms show ?thesis by blast
+ then show ?thesis
+ unfolding winding_number_def by (rule someI2_ex) (blast intro: \<open>0<e\<close>)
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))"
+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 [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)
+ 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
-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))"
+(*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 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]
+ 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
-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"
+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
+ apply (simp add: winding_number_prop_def 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)) =
+ complex_of_real (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 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)"
+ 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_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
- apply (fastforce simp add: winding_number_join path_image_join)
+ 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>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"
+ apply (rule winding_number_cong)
+ using assms unfolding linepath_def by auto
+ moreover have "winding_number (linepath c c) z =0"
+ apply (rule winding_number_trivial)
+ using assms 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 show "\<exists>r. winding_number_prop p z e r n"
+ apply (rule_tac x="\<lambda>t. g t + z" in exI)
+ 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
- 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)"
+subsubsection\<^marker>\<open>tag unimportant\<close> \<open>Some lemmas about negating a path\<close>
+
+lemma valid_path_negatepath: "valid_path \<gamma> \<Longrightarrow> valid_path (uminus \<circ> \<gamma>)"
+ unfolding o_def using piecewise_C1_differentiable_neg valid_path_def by blast
+
+lemma has_contour_integral_negatepath:
+ assumes \<gamma>: "valid_path \<gamma>" and cint: "((\<lambda>z. f (- z)) has_contour_integral - i) \<gamma>"
+ shows "(f has_contour_integral i) (uminus \<circ> \<gamma>)"
+proof -
+ obtain S where cont: "continuous_on {0..1} \<gamma>" and "finite S" and diff: "\<gamma> C1_differentiable_on {0..1} - S"
+ using \<gamma> by (auto simp: valid_path_def piecewise_C1_differentiable_on_def)
+ have "((\<lambda>x. - (f (- \<gamma> x) * vector_derivative \<gamma> (at x within {0..1}))) has_integral i) {0..1}"
+ using cint by (auto simp: has_contour_integral_def dest: has_integral_neg)
+ then
+ have "((\<lambda>x. f (- \<gamma> x) * vector_derivative (uminus \<circ> \<gamma>) (at x within {0..1})) has_integral i) {0..1}"
+ proof (rule rev_iffD1 [OF _ has_integral_spike_eq])
+ show "negligible S"
+ by (simp add: \<open>finite S\<close> negligible_finite)
+ show "f (- \<gamma> x) * vector_derivative (uminus \<circ> \<gamma>) (at x within {0..1}) =
+ - (f (- \<gamma> x) * vector_derivative \<gamma> (at x within {0..1}))"
+ if "x \<in> {0..1} - S" for x
+ proof -
+ have "vector_derivative (uminus \<circ> \<gamma>) (at x within cbox 0 1) = - vector_derivative \<gamma> (at x within cbox 0 1)"
+ proof (rule vector_derivative_within_cbox)
+ show "(uminus \<circ> \<gamma> has_vector_derivative - vector_derivative \<gamma> (at x within cbox 0 1)) (at x within cbox 0 1)"
+ using that unfolding o_def
+ by (metis C1_differentiable_on_eq UNIV_I diff differentiable_subset has_vector_derivative_minus subsetI that vector_derivative_works)
+ qed (use that in auto)
+ then show ?thesis
+ by simp
+ qed
+ qed
+ then show ?thesis by (simp add: has_contour_integral_def)
+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 show ?thesis
+ using assms
+ apply (simp add: winding_number_valid_path valid_path_negatepath image_def path_defs)
+ apply (simp add: contour_integral_unique has_contour_integral_negatepath)
+ done
+qed
+
+lemma contour_integrable_negatepath:
+ assumes \<gamma>: "valid_path \<gamma>" and pi: "(\<lambda>z. f (- z)) contour_integrable_on \<gamma>"
+ shows "f contour_integrable_on (uminus \<circ> \<gamma>)"
+ by (metis \<gamma> add.inverse_inverse contour_integrable_on_def has_contour_integral_negatepath pi)
+
+(* A combined theorem deducing several things piecewise.*)
+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 [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)
+ 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 hi: "(?vd has_integral ?int z) (cbox 0 1)"
+ unfolding box_real
+ apply (subst has_contour_integral [symmetric])
+ using \<gamma> by (simp add: contour_integrable_inversediff has_contour_integral_integral)
+ 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)
+ 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 hi: "(?vd has_integral ?int z) (cbox 0 1)"
+ unfolding box_real
+ apply (subst has_contour_integral [symmetric])
+ using \<gamma> by (simp add: contour_integrable_inversediff has_contour_integral_integral)
+ 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])
+ show "((\<lambda>x. if 0 < x \<and> x < 1 then ?vd x else \<i> * complex_of_real 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> * complex_of_real 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> Im (vector_derivative \<gamma> (at x) * cnj (\<gamma> x - z)) / (cmod (\<gamma> x - z))\<^sup>2"
+ using B ge [OF x] B2 e
+ apply (rule_tac y="e / (cmod (\<gamma> x - z))\<^sup>2" in order_trans)
+ apply (auto simp: divide_left_mono divide_right_mono)
+ done
+ 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 wn_triangle2 [OF z] apply (rule disjE)
- apply (simp add: wn_triangle3 z)
- apply (simp add: path_image_join winding_number_reversepath * wn_triangle3 z)
+ 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
+ apply (rule has_vector_derivative_eq_rhs)
+ using z
+ apply (auto intro!: derivative_eq_intros * [unfolded o_def] g)
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"
+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 -
- 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
- have wn_cc2: "winding_number (c +++ reversepath c2) z = 0"
- apply (rule winding_number_zero_in_outside)
- apply (simp_all add: \<open>simple_path c2\<close> c c2 \<open>simple_path c\<close> simple_path_imp_path path_image_join)
- by (metis "0" ComplI UnE disjoint_iff_not_equal sup.commute union_with_inside z znot)
- 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
+ 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 cong: "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
+ apply (rule_tac x="norm(w - z)" in exI)
+ 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)
+ { fix x D
+ assume x: "x \<notin> k" "a < x" "x < b"
+ 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)
+ have "\<And>d. \<lbrakk>x \<notin> k; a < x; x < b;
+ (\<gamma> has_vector_derivative d) (at x); a \<le> t; t \<le> b\<rbrakk>
+ \<Longrightarrow> ((\<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})"
+ apply (rule has_vector_derivative_eq_rhs)
+ apply (rule integral_has_vector_derivative_continuous_at [where S = "{}", simplified])
+ apply (rule con_vd continuous_intros cong vg_int | simp add: continuous_at_imp_continuous_within has_vector_derivative_continuous vector_derivative_at)+
+ done
+ then have "((\<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})"
+ using x gdx t
+ apply (clarsimp simp add: differentiable_iff_scaleR)
+ apply (rule exp_fg [unfolded has_vector_derivative_def, simplified], blast intro: has_derivative_at_withinI)
+ apply (simp_all add: has_vector_derivative_def [symmetric])
+ done
+ } note * = this
+ have "exp (- (integral {a..t} (\<lambda>x. ?D\<gamma> x / (\<gamma> x - z)))) * (\<gamma> t - z) =\<gamma> a - z"
+ apply (rule has_derivative_zero_unique_strong_interval [of "{a,b} \<union> k" a b])
+ using t
+ apply (auto intro!: * continuous_intros fink cong indefinite_integral_continuous_1 [OF vg_int] simp add: ab)+
+ done
+ }
+ with ab show ?thesis2
+ by (simp add: divide_inverse_commute integral_def)
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))"
- 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 (p +++ linepath (a - e) (a + e)))"
- "cmod (winding_number (p +++ linepath (a - e) (a + e)) z) = 1"
+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 -
- 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)
- then have "a \<in> path_image(p +++ linepath (a - e) (a + e))"
- apply (simp add: assms path_image_join)
- by (metis midpoint_in_closed_segment)
- have "a \<in> frontier(inside (path_image(p +++ linepath (a - e) (a + e))))"
- apply (simp add: assms Jordan_inside_outside)
- apply (simp_all add: assms path_image_join)
- by (metis mid_eq_a midpoint_in_closed_segment)
- with \<open>0 < e\<close> obtain c where c: "c \<in> inside (path_image(p +++ linepath (a - e) (a + e)))"
- and dac: "dist a c < e"
- by (auto simp: frontier_straddle)
- then have "c \<notin> path_image(p +++ linepath (a - e) (a + e))"
- using inside_no_overlap by blast
- then have "c \<notin> path_image p"
- "c \<notin> closed_segment (a - of_real e) (a + of_real e)"
- by (simp_all add: assms path_image_join)
- with \<open>0 < e\<close> dac have "c \<notin> affine hull {a - of_real e, a + of_real 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}"
- apply (rule equalityI)
- apply (clarsimp simp: path_image_join)
- apply (metis pathstart_in_path_image psp pathfinish_in_path_image pfp)
- done
- 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, hide_lams) 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)))"
- apply (simp add: path_image_join)
- by (metis z inside_of_triangle DIM_complex Un_commute closed_segment_commute)
- 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 \<in> inside (path_image (p +++ linepath (a - e) (a + e)))"
- using zin by (simp add: assms path_image_join Un_commute closed_segment_commute)
- then have "cmod (winding_number (p +++ linepath (a - e) (a + e)) z) =
- cmod ((winding_number(reversepath (p +++ linepath (a - e) (a + e))) z))"
- apply (subst winding_number_reversepath)
- using simple_path_imp_path sp_pl apply blast
- apply (metis IntI emptyE inside_no_overlap)
- by (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 (p +++ linepath (a - e) (a + e)) z) = 1" .
- qed
+ 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 "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]
+ apply (simp add: valid_path_def path_defs contour_integral_integral exp_minus field_split_simps)
+ by (metis path_image_def pathstart_def pathstart_in_path_image)
+ 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
-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"
+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]: "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, hide_lams) 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}"
+ 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
+ apply (simp add: winding_number_valid_path [OF \<gamma> z] contour_integral_integral)
+ apply (simp add: Complex.Re_divide field_simps power2_eq_square)
+ done
+ 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
- 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 where u: "y = a + of_real u" and "0 < u"
- using y1 \<open>0 < kde\<close> \<open>kde < e\<close> \<open>0 < e\<close> apply (clarsimp simp: in_segment)
- apply (rule_tac u = "(1 - u)*kde + u*e" in that)
- apply (auto simp: scaleR_conv_of_real algebra_simps)
- by (meson le_less_trans less_add_same_cancel2 less_eq_real_def mult_left_mono)
- moreover
- obtain v where v: "y = a + of_real v" and "v \<le> 0"
- using y2 \<open>0 < kde\<close> \<open>0 < d\<close> \<open>0 < e\<close> apply (clarsimp simp: in_segment)
- apply (rule_tac v = "- ((1 - u)*d + u*kde)" in that)
- apply (force simp: scaleR_conv_of_real algebra_simps)
- by (meson less_eq_real_def neg_le_0_iff_le segment_bound_lemma)
- 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 \<in> closed_segment (a + kde) (a + e)" for x
- using that \<open>kde < e\<close> mult_le_cancel_left
- apply (auto simp: in_segment)
- apply (rule_tac x="(1-u)*kde + u*e" in exI)
- apply (fastforce simp: algebra_simps scaleR_conv_of_real)
+ define i where "i = integral {0..t} (\<lambda>x. vector_derivative \<gamma> (at x) / (\<gamma> x - z))"
+ have iArg: "Arg2pi r = Im i"
+ using eqArg by (simp add: i_def)
+ 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
+ apply (rule winding_number_exp_integral [OF gpdt])
+ using t z unfolding path_image_def by force+
+ 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)
+ then have "z + complex_of_real (exp (Re i)) * (w - z) / complex_of_real (cmod r) = \<gamma> t"
+ apply simp
+ apply (subst Complex_Transcendental.Arg2pi_eq [of r])
+ apply (simp add: iArg)
+ using * apply (simp add: exp_eq_polar field_simps)
done
- have ge_kde2: "\<exists>y. x = a + y \<and> y \<le> -kde" if "x \<in> closed_segment (a - d) (a - kde)" for x
- using that \<open>kde < d\<close> affine_ineq
- apply (auto simp: in_segment)
- apply (rule_tac x="- ((1-u)*d + u*kde)" in exI)
- apply (fastforce simp: algebra_simps scaleR_conv_of_real)
- done
- have notin_paq: "x \<notin> path_image ?q" if "dist a x < kde" for x
- using that using \<open>0 < kde\<close> \<open>kde < d\<close> \<open>kde < k\<close>
- apply (auto simp: path_image_join assms dist_norm dest!: ge_kde1 ge_kde2)
- by (meson k disjoint_iff_not_equal le_less_trans less_eq_real_def mem_ball that)
- 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_de: "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 "winding_number (linepath (a - d) (a + e)) z =
- winding_number (linepath (a - d) (a + kde)) z + winding_number (linepath (a + kde) (a + e)) z"
- apply (rule winding_number_split_linepath)
- apply (simp add: a_diff_kde)
- by (metis ComplD Un_iff inside_Un_outside path_image_join path_image_linepath pathstart_linepath pfp zzin)
- 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_de 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 ?q z + winding_number (linepath (a - kde) (a + kde)) z"
- using \<open>path p\<close> znotin assms zzin clsub1
- apply (subst winding_number_join, auto)
- apply (metis (no_types, hide_lams) ComplD Un_iff contra_subsetD inside_Un_outside path_image_join path_image_linepath pathstart_linepath)
- apply (metis Un_iff Un_closed_segment a_diff_kde' not_in_path_image_join path_image_linepath znotin_de)
- by (metis Un_iff Un_closed_segment a_diff_kde' path_image_linepath path_linepath pathstart_linepath winding_number_join znotin_de)
- also have "... = winding_number (?q +++ linepath (a - kde) (a + kde)) z"
- using \<open>path p\<close> assms zin
- apply (subst winding_number_join [symmetric], auto)
- apply (metis ComplD Un_iff path_image_join pathfinish_join pathfinish_linepath pathstart_linepath union_with_outside)
- by (metis Un_iff Un_closed_segment a_diff_kde' znotin_de)
- 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
+ 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
-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"
+subsection\<open>Continuity of winding number and invariance on connected sets\<close>
+
+lemma 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 -
- 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)"
- apply (rule ray_to_frontier [of "inside (path_image p)" a "-1"])
- using \<open>bounded (inside (path_image p))\<close> \<open>open (inside (path_image p))\<close> a interior_eq
- apply (auto simp: of_real_def)
- done
- 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)"
- apply (rule ray_to_frontier [of "inside (path_image p)" a 1])
- using \<open>bounded (inside (path_image p))\<close> \<open>open (inside (path_image p))\<close> a interior_eq
- apply (auto simp: of_real_def)
- done
- 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)"
- apply (simp add: algebra_simps)
- by (simp add: diff_add_eq of_real_def real_vector.scale_right_diff_distrib)
- 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)
- then have a_in_de: "a \<in> open_segment (a - d) (a + e)"
- using ad_not_ae \<open>0 < d\<close> \<open>0 < e\<close>
- apply (auto simp: in_segment algebra_simps scaleR_conv_of_real)
- apply (rule_tac x="d / (d+e)" in exI)
- apply (auto simp: field_simps)
+ 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 p: "valid_path p" "z \<notin> path_image p"
+ "pathstart p = pathstart \<gamma> \<and> 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"
+ then have wnotp: "w \<notin> path_image p"
+ using cbg \<open>d>0\<close> \<open>e>0\<close>
+ apply (simp add: path_image_def cball_def dist_norm, clarify)
+ apply (frule pg)
+ apply (drule_tac c="\<gamma> x" in subsetD)
+ apply (auto simp: less_eq_real_def norm_minus_commute norm_triangle_half_l)
+ done
+ 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)
+ have "contour_integral p (\<lambda>u. 1 / (u - w)) = contour_integral q (\<lambda>u. 1 / (u - w))"
+ apply (rule pi_eq [OF \<open>valid_path q\<close> \<open>valid_path p\<close>, THEN conjunct2, THEN conjunct2, rule_format])
+ apply (frule pg)
+ apply (frule qg)
+ using p q \<open>d>0\<close> e2
+ apply (auto simp: dist_norm norm_minus_commute intro!: holomorphic_intros)
+ done
+ 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"
+ using p by (simp add: valid_path_imp_path)
+ then have "winding_number p w = winding_number \<gamma> w"
+ apply (rule winding_number_unique [OF _ wnotp])
+ apply (rule_tac x=p in exI)
+ apply (simp add: p wnotp min_divide_distrib_right pip winding_number_prop_def)
+ done
+ } note wnwn = this
+ obtain pe where "pe>0" and cbp: "cball z (3 / 4 * pe) \<subseteq> - path_image p"
+ using p 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 force
+ { 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: p 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/2)^2 < cmod (w - x) ^ 2"
+ apply (rule power_strict_mono)
+ using \<open>pe>0\<close> by auto
+ 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 wx
+ apply (rule mult_strict_left_mono)
+ using pe2 e not_less_iff_gr_or_eq 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"
+ apply (cases "x=z \<or> x=w")
+ using pe \<open>pe>0\<close> w \<open>L>0\<close>
+ apply (force simp: norm_minus_commute)
+ using wx w(2) \<open>L>0\<close> pe pe2 Lwz
+ apply (auto simp: divide_simps mult_less_0_iff norm_minus_commute norm_divide norm_mult power2_eq_square)
+ done
+ } note L_cmod_le = this
+ have *: "cmod (contour_integral p (\<lambda>x. 1 / (x - w) - 1 / (x - z))) \<le> L * (e * pe\<^sup>2 / L / 4 * (inverse (pe / 2))\<^sup>2)"
+ apply (rule L)
+ using \<open>pe>0\<close> w
+ apply (force simp: dist_norm norm_minus_commute intro!: holomorphic_intros)
+ using \<open>pe>0\<close> w \<open>L>0\<close>
+ apply (auto simp: cball_def dist_norm field_simps L_cmod_le simp del: less_divide_eq_numeral1 le_divide_eq_numeral1)
+ done
+ have "cmod (contour_integral p (\<lambda>x. 1 / (x - w)) - contour_integral p (\<lambda>x. 1 / (x - z))) < 2*e"
+ apply simp
+ apply (rule le_less_trans [OF *])
+ using \<open>L>0\<close> e
+ apply (force simp: field_simps)
+ done
+ then have "cmod (winding_number p w - winding_number p z) < e"
+ using pi_ge_two e
+ by (force simp: winding_number_valid_path p field_simps norm_divide norm_mult intro: less_le_trans)
+ } note cmod_wn_diff = this
+ then have "isCont (winding_number p) z"
+ apply (simp add: continuous_at_eps_delta, clarify)
+ apply (rule_tac x="min (pe/4) (e/2*pe^2/L/4)" in exI)
+ using \<open>pe>0\<close> \<open>L>0\<close>
+ apply (simp add: dist_norm cmod_wn_diff)
done
- 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, hide_lams) 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
+ 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
-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
- with z1 consider "winding_number \<gamma> z = 1" | "winding_number \<gamma> z = -1"
- apply (auto simp: Ints_def abs_if split: if_split_asm)
- by (metis of_int_1 of_int_eq_iff of_int_minus)
- 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
+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)
-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
+subsection\<^marker>\<open>tag unimportant\<close> \<open>The winding number is constant on a connected region\<close>
-lemma simple_closed_path_winding_number_cases:
- "\<lbrakk>simple_path \<gamma>; pathfinish \<gamma> = pathstart \<gamma>; z \<notin> path_image \<gamma>\<rbrakk> \<Longrightarrow> winding_number \<gamma> z \<in> {-1,0,1}"
-apply (simp add: inside_Un_outside [of "path_image \<gamma>", symmetric, unfolded set_eq_iff Set.Compl_iff] del: inside_Un_outside)
- apply (rule simple_closed_path_winding_number_inside)
- using simple_path_def winding_number_zero_in_outside by blast+
-
-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>
-
-definition\<^marker>\<open>tag important\<close> rectpath where
- "rectpath a1 a3 = (let a2 = Complex (Re a3) (Im a1); a4 = Complex (Re a1) (Im a3)
- in linepath a1 a2 +++ linepath a2 a3 +++ linepath a3 a4 +++ linepath a4 a1)"
-
-lemma path_rectpath [simp, intro]: "path (rectpath a b)"
- by (simp add: Let_def rectpath_def)
-
-lemma valid_path_rectpath [simp, intro]: "valid_path (rectpath a b)"
- by (simp add: Let_def rectpath_def)
-
-lemma pathstart_rectpath [simp]: "pathstart (rectpath a1 a3) = a1"
- by (simp add: rectpath_def Let_def)
-
-lemma pathfinish_rectpath [simp]: "pathfinish (rectpath a1 a3) = a1"
- by (simp add: rectpath_def Let_def)
-
-lemma simple_path_rectpath [simp, intro]:
- assumes "Re a1 \<noteq> Re a3" "Im a1 \<noteq> Im a3"
- shows "simple_path (rectpath a1 a3)"
- unfolding rectpath_def Let_def using assms
- by (intro simple_path_join_loop arc_join arc_linepath)
- (auto simp: complex_eq_iff path_image_join closed_segment_same_Re closed_segment_same_Im)
-
-lemma path_image_rectpath:
- assumes "Re a1 \<le> Re a3" "Im a1 \<le> Im a3"
- shows "path_image (rectpath a1 a3) =
- {z. Re z \<in> {Re a1, Re a3} \<and> Im z \<in> {Im a1..Im a3}} \<union>
- {z. Im z \<in> {Im a1, Im a3} \<and> Re z \<in> {Re a1..Re a3}}" (is "?lhs = ?rhs")
+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 -
- define a2 a4 where "a2 = Complex (Re a3) (Im a1)" and "a4 = Complex (Re a1) (Im a3)"
- have "?lhs = closed_segment a1 a2 \<union> closed_segment a2 a3 \<union>
- closed_segment a4 a3 \<union> closed_segment a1 a4"
- by (simp_all add: rectpath_def Let_def path_image_join closed_segment_commute
- a2_def a4_def Un_assoc)
- also have "\<dots> = ?rhs" using assms
- by (auto simp: rectpath_def Let_def path_image_join a2_def a4_def
- closed_segment_same_Re closed_segment_same_Im closed_segment_eq_real_ivl)
- finally show ?thesis .
+ 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 path_image_rectpath_subset_cbox:
- assumes "Re a \<le> Re b" "Im a \<le> Im b"
- shows "path_image (rectpath a b) \<subseteq> cbox a b"
- using assms by (auto simp: path_image_rectpath in_cbox_complex_iff)
-
-lemma path_image_rectpath_inter_box:
- assumes "Re a \<le> Re b" "Im a \<le> Im b"
- shows "path_image (rectpath a b) \<inter> box a b = {}"
- using assms by (auto simp: path_image_rectpath in_box_complex_iff)
+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 path_image_rectpath_cbox_minus_box:
- assumes "Re a \<le> Re b" "Im a \<le> Im b"
- shows "path_image (rectpath a b) = cbox a b - box a b"
- using assms by (auto simp: path_image_rectpath in_cbox_complex_iff
- in_box_complex_iff)
-
-proposition winding_number_rectpath:
- assumes "z \<in> box a1 a3"
- shows "winding_number (rectpath a1 a3) z = 1"
+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 -
- 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)
+ have opn: "open (- path_image \<gamma>)"
+ by (simp add: closed_path_image \<gamma> open_Compl)
+ { fix z assume z: "z \<notin> path_image \<gamma>" and k: "k = winding_number \<gamma> z"
+ obtain e where e: "e>0" "ball z e \<subseteq> - path_image \<gamma>"
+ using open_contains_ball [of "- path_image \<gamma>"] opn z
+ by blast
+ have "\<exists>e>0. \<forall>y. dist y z < e \<longrightarrow> y \<notin> path_image \<gamma> \<and> winding_number \<gamma> y = winding_number \<gamma> z"
+ apply (rule_tac x=e in exI)
+ using e apply (simp add: dist_norm ball_def norm_minus_commute)
+ apply (auto simp: dist_norm norm_minus_commute intro!: winding_number_eq [OF assms, where S = "ball z e"])
+ done
+ } then
+ show ?thesis
+ by (auto simp: open_dist)
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>)"
+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 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))"
- apply (rule setdist_le_dist)
- using \<open>t \<in> {0..1}\<close> path_image_def by fastforce+
- 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 meson
- 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)"
- apply (rule field_vector_diff_chain_at)
- apply (metis has_vector_derivative_polynomial_function pfg vector_derivative_at)
- using DERIV_exp has_field_derivative_def apply blast
+ 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>)"
+ apply (rule outside_subset_convex)
+ using B subset_ball by 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, hide_lams) 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 force
+ have pip: "path_image p \<subseteq> ball 0 (B + 1)"
+ using B
+ apply (clarsimp simp add: path_image_def dist_norm ball_def)
+ apply (frule (1) pg1)
+ apply (fastforce dest: norm_add_less)
done
- 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}"
- apply (rule fundamental_theorem_of_calculus [OF zero_le_one])
- by (metis has_vector_derivative_at_within has_vector_derivative_polynomial_function pfg vector_derivative_at)
+ then have "w \<notin> path_image p" using w by blast
+ then 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"
+ apply (rule_tac x=p in exI)
+ apply (simp add: p valid_path_polynomial_function)
+ apply (intro conjI)
+ using pge apply (simp add: norm_minus_commute)
+ apply (rule contour_integral_unique [OF Cauchy_theorem_convex_simple [OF _ convex_ball [of 0 "B+1"]]])
+ apply (rule holomorphic_intros | simp add: dist_norm)+
+ using mem_ball_0 w apply blast
+ using p apply (simp_all add: valid_path_polynomial_function loop pip)
+ done
+ }
then show ?thesis
- apply (simp add: pathfinish_def pathstart_def)
- using \<open>g 0 = p 0\<close> \<open>g 1 = p 1\<close> by auto
+ by (auto intro: winding_number_unique [OF \<gamma>] simp add: winding_number_prop_def wnot)
qed
finally show ?thesis .
qed
-subsection\<^marker>\<open>tag unimportant\<close> \<open>The winding number defines a continuous logarithm for the path itself\<close>
+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
+ then show ?thesis
+ apply (rule_tac x="B+1" in exI, clarify)
+ apply (rule winding_number_zero_outside [OF \<gamma> convex_cball [of 0 B] loop])
+ apply (meson less_add_one mem_cball_0 not_le order_trans)
+ using ball_subset_cball by blast
+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>)"
+ using \<open>x \<in> s\<close> by (metis (no_types) ComplI UnE cos \<gamma> loop s_disj union_with_outside winding_number_eq winding_number_zero_in_outside wn_nz z)
+qed
+ show ?thesis
+ apply (rule winding_number_eq [OF \<gamma> loop w])
+ using z apply blast
+ apply (simp add: cls connected_with_inside cos)
+ apply (simp add: Int_Un_distrib2 s_disj, safe)
+ by (meson ssb inside_inside_compact_connected [OF cls, of "path_image \<gamma>"] compact_path_image connected_path_image contra_subsetD disjoint_iff_not_equal \<gamma> inside_no_overlap)
+ qed
+
+subsection \<open>The real part of winding numbers\<close>
-lemma winding_number_as_continuous_log:
- assumes "path p" and \<zeta>: "\<zeta> \<notin> path_image p"
- obtains q where "path q"
- "pathfinish q - pathstart q = 2 * of_real pi * \<i> * winding_number p \<zeta>"
- "\<And>t. t \<in> {0..1} \<Longrightarrow> p t = \<zeta> + exp(q t)"
+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 -
- let ?q = "\<lambda>t. 2 * of_real pi * \<i> * winding_number(subpath 0 t p) \<zeta> + Ln(pathstart p - \<zeta>)"
+ have *: "integral {0..x} (\<lambda>t. vector_derivative \<gamma> (at t) / (\<gamma> t - z)) / (2 * of_real pi * \<i>) =
+ winding_number (subpath 0 x \<gamma>) z"
+ if x: "0 \<le> x" "x \<le> 1" for x
+ proof -
+ have "integral {0..x} (\<lambda>t. vector_derivative \<gamma> (at t) / (\<gamma> t - z)) / (2 * of_real pi * \<i>) =
+ 1 / (2*pi*\<i>) * contour_integral (subpath 0 x \<gamma>) (\<lambda>w. 1/(w - z))"
+ using assms x
+ apply (simp add: contour_integral_subcontour_integral [OF contour_integrable_inversediff])
+ done
+ also have "\<dots> = winding_number (subpath 0 x \<gamma>) z"
+ apply (subst winding_number_valid_path)
+ using assms x
+ apply (simp_all add: path_image_subpath valid_path_subpath)
+ by (force simp: path_image_def)
+ finally show ?thesis .
+ qed
show ?thesis
- proof
- have *: "continuous (at t within {0..1}) (\<lambda>x. winding_number (subpath 0 x p) \<zeta>)"
- if t: "t \<in> {0..1}" for t
+ apply (rule continuous_on_eq
+ [where f = "\<lambda>x. 1 / (2*pi*\<i>) *
+ integral {0..x} (\<lambda>t. 1/(\<gamma> t - z) * vector_derivative \<gamma> (at t))"])
+ apply (rule continuous_intros)+
+ apply (rule indefinite_integral_continuous_1)
+ apply (rule contour_integrable_inversediff [OF assms, unfolded contour_integrable_on])
+ using assms
+ apply (simp add: *)
+ done
+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"
+ apply (rule ivt_increasing_component_on_1 [of 0 1, where ?k = "1::complex", simplified complex_inner_1_right], simp)
+ apply (rule winding_number_subpath_continuous [OF \<gamma> z])
+ using assms
+ apply (auto simp: path_image_def image_def)
+ done
+
+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"
+ apply (rule ivt_decreasing_component_on_1 [of 0 1, where ?k = "1::complex", simplified complex_inner_1_right], simp)
+ apply (rule winding_number_subpath_continuous [OF \<gamma> z])
+ using assms
+ apply (auto simp: path_image_def image_def)
+ done
+
+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 -
- let ?B = "ball (p t) (norm(p t - \<zeta>))"
- have "p t \<noteq> \<zeta>"
- using path_image_def that \<zeta> by blast
- then have "simply_connected ?B"
- by (simp add: convex_imp_simply_connected)
- then have "\<And>f::complex\<Rightarrow>complex. continuous_on ?B f \<and> (\<forall>\<zeta> \<in> ?B. f \<zeta> \<noteq> 0)
- \<longrightarrow> (\<exists>g. continuous_on ?B g \<and> (\<forall>\<zeta> \<in> ?B. f \<zeta> = exp (g \<zeta>)))"
- by (simp add: simply_connected_eq_continuous_log)
- moreover have "continuous_on ?B (\<lambda>w. w - \<zeta>)"
- by (intro continuous_intros)
- moreover have "(\<forall>z \<in> ?B. z - \<zeta> \<noteq> 0)"
- by (auto simp: dist_norm)
- ultimately obtain g where contg: "continuous_on ?B g"
- and geq: "\<And>z. z \<in> ?B \<Longrightarrow> z - \<zeta> = exp (g z)" by blast
- obtain d where "0 < d" and d:
- "\<And>x. \<lbrakk>x \<in> {0..1}; dist x t < d\<rbrakk> \<Longrightarrow> dist (p x) (p t) < cmod (p t - \<zeta>)"
- using \<open>path p\<close> t unfolding path_def continuous_on_iff
- by (metis \<open>p t \<noteq> \<zeta>\<close> right_minus_eq zero_less_norm_iff)
- have "((\<lambda>x. winding_number (\<lambda>w. subpath 0 x p w - \<zeta>) 0 -
- winding_number (\<lambda>w. subpath 0 t p w - \<zeta>) 0) \<longlongrightarrow> 0)
- (at t within {0..1})"
- proof (rule Lim_transform_within [OF _ \<open>d > 0\<close>])
- have "continuous (at t within {0..1}) (g o p)"
- proof (rule continuous_within_compose)
- show "continuous (at t within {0..1}) p"
- using \<open>path p\<close> continuous_on_eq_continuous_within path_def that by blast
- show "continuous (at (p t) within p ` {0..1}) g"
- by (metis (no_types, lifting) open_ball UNIV_I \<open>p t \<noteq> \<zeta>\<close> centre_in_ball contg continuous_on_eq_continuous_at continuous_within_topological right_minus_eq zero_less_norm_iff)
- qed
- with LIM_zero have "((\<lambda>u. (g (subpath t u p 1) - g (subpath t u p 0))) \<longlongrightarrow> 0) (at t within {0..1})"
- by (auto simp: subpath_def continuous_within o_def)
- then show "((\<lambda>u. (g (subpath t u p 1) - g (subpath t u p 0)) / (2 * of_real pi * \<i>)) \<longlongrightarrow> 0)
- (at t within {0..1})"
- by (simp add: tendsto_divide_zero)
- show "(g (subpath t u p 1) - g (subpath t u p 0)) / (2 * of_real pi * \<i>) =
- winding_number (\<lambda>w. subpath 0 u p w - \<zeta>) 0 - winding_number (\<lambda>w. subpath 0 t p w - \<zeta>) 0"
- if "u \<in> {0..1}" "0 < dist u t" "dist u t < d" for u
- proof -
- have "closed_segment t u \<subseteq> {0..1}"
- using closed_segment_eq_real_ivl t that by auto
- then have piB: "path_image(subpath t u p) \<subseteq> ?B"
- apply (clarsimp simp add: path_image_subpath_gen)
- by (metis subsetD le_less_trans \<open>dist u t < d\<close> d dist_commute dist_in_closed_segment)
- have *: "path (g \<circ> subpath t u p)"
- apply (rule path_continuous_image)
- using \<open>path p\<close> t that apply auto[1]
- using piB contg continuous_on_subset by blast
- have "(g (subpath t u p 1) - g (subpath t u p 0)) / (2 * of_real pi * \<i>)
- = winding_number (exp \<circ> g \<circ> subpath t u p) 0"
- using winding_number_compose_exp [OF *]
- by (simp add: pathfinish_def pathstart_def o_assoc)
- also have "... = winding_number (\<lambda>w. subpath t u p w - \<zeta>) 0"
- proof (rule winding_number_cong)
- have "exp(g y) = y - \<zeta>" if "y \<in> (subpath t u p) ` {0..1}" for y
- by (metis that geq path_image_def piB subset_eq)
- then show "\<And>x. \<lbrakk>0 \<le> x; x \<le> 1\<rbrakk> \<Longrightarrow> (exp \<circ> g \<circ> subpath t u p) x = subpath t u p x - \<zeta>"
- by auto
- qed
- also have "... = winding_number (\<lambda>w. subpath 0 u p w - \<zeta>) 0 -
- winding_number (\<lambda>w. subpath 0 t p w - \<zeta>) 0"
- apply (simp add: winding_number_offset [symmetric])
- using winding_number_subpath_combine [OF \<open>path p\<close> \<zeta>, of 0 t u] \<open>t \<in> {0..1}\<close> \<open>u \<in> {0..1}\<close>
- by (simp add: add.commute eq_diff_eq)
- finally show ?thesis .
- qed
- qed
- then show ?thesis
- by (subst winding_number_offset) (simp add: continuous_within LIM_zero_iff)
+ 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"
+ proof -
+ have "\<gamma> t \<in> {c. b < a \<bullet> c}"
+ by (metis (no_types) pag atLeastAtMost_iff image_subset_iff path_image_def t)
+ thus ?thesis
+ by blast
qed
- show "path ?q"
- unfolding path_def
- by (intro continuous_intros) (simp add: continuous_on_eq_continuous_within *)
+ 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 show ?thesis
+ apply (simp add: winding_number_lt_half_lemma abs_if del: less_divide_eq_numeral1)
+ apply (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)
+ done
+qed
- have "\<zeta> \<noteq> p 0"
- by (metis \<zeta> pathstart_def pathstart_in_path_image)
- then show "pathfinish ?q - pathstart ?q = 2 * of_real pi * \<i> * winding_number p \<zeta>"
- by (simp add: pathfinish_def pathstart_def)
- show "p t = \<zeta> + exp (?q t)" if "t \<in> {0..1}" for t
- proof -
- have "path (subpath 0 t p)"
- using \<open>path p\<close> that by auto
- moreover
- have "\<zeta> \<notin> path_image (subpath 0 t p)"
- using \<zeta> [unfolded path_image_def] that by (auto simp: path_image_subpath)
- ultimately show ?thesis
- using winding_number_exp_2pi [of "subpath 0 t p" \<zeta>] \<open>\<zeta> \<noteq> p 0\<close>
- by (auto simp: exp_add algebra_simps pathfinish_def pathstart_def subpath_def)
- qed
- 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' \<le> b - d / 3 * cmod a"
+ unfolding z'_def inner_mult_right' divide_inverse
+ apply (simp add: field_split_simps algebra_simps dot_square_norm power2_eq_square anz)
+ apply (metis \<open>0 < d\<close> add_increasing azb less_eq_real_def mult_nonneg_nonneg mult_right_mono norm_ge_zero norm_numeral)
+ done
+ 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"
+ apply (rule winding_number_lt_half [OF \<gamma> *])
+ using azb \<open>d>0\<close> pag
+ apply (auto simp: add_strict_increasing anz field_split_simps dest!: subsetD)
+ done
+ ultimately have False
+ by simp
+ }
+ then show ?thesis by force
qed
-subsection \<open>Winding number equality is the same as path/loop homotopy in C - {0}\<close>
+lemma winding_number_lt_half_linepath: "z \<notin> closed_segment a b \<Longrightarrow> \<bar>Re (winding_number (linepath a b) z)\<bar> < 1/2"
+ using separating_hyperplane_closed_point [of "closed_segment a b" z]
+ apply auto
+ apply (simp add: closed_segment_def)
+ apply (drule less_imp_le)
+ apply (frule winding_number_lt_half [OF valid_path_linepath [of a b]])
+ apply (auto simp: segment)
+ done
+
-lemma winding_number_homotopic_loops_null_eq:
- assumes "path p" and \<zeta>: "\<zeta> \<notin> path_image p"
- shows "winding_number p \<zeta> = 0 \<longleftrightarrow> (\<exists>a. homotopic_loops (-{\<zeta>}) p (\<lambda>t. a))"
- (is "?lhs = ?rhs")
-proof
- assume [simp]: ?lhs
- obtain q where "path q"
- and qeq: "pathfinish q - pathstart q = 2 * of_real pi * \<i> * winding_number p \<zeta>"
- and peq: "\<And>t. t \<in> {0..1} \<Longrightarrow> p t = \<zeta> + exp(q t)"
- using winding_number_as_continuous_log [OF assms] by blast
- have *: "homotopic_with_canon (\<lambda>r. pathfinish r = pathstart r)
- {0..1} (-{\<zeta>}) ((\<lambda>w. \<zeta> + exp w) \<circ> q) ((\<lambda>w. \<zeta> + exp w) \<circ> (\<lambda>t. 0))"
- proof (rule homotopic_with_compose_continuous_left)
- show "homotopic_with_canon (\<lambda>f. pathfinish ((\<lambda>w. \<zeta> + exp w) \<circ> f) = pathstart ((\<lambda>w. \<zeta> + exp w) \<circ> f))
- {0..1} UNIV q (\<lambda>t. 0)"
- proof (rule homotopic_with_mono, simp_all add: pathfinish_def pathstart_def)
- have "homotopic_loops UNIV q (\<lambda>t. 0)"
- by (rule homotopic_loops_linear) (use qeq \<open>path q\<close> in \<open>auto simp: path_defs\<close>)
- then have "homotopic_with (\<lambda>r. r 1 = r 0) (top_of_set {0..1}) euclidean q (\<lambda>t. 0)"
- by (simp add: homotopic_loops_def pathfinish_def pathstart_def)
- then show "homotopic_with (\<lambda>h. exp (h 1) = exp (h 0)) (top_of_set {0..1}) euclidean q (\<lambda>t. 0)"
- by (rule homotopic_with_mono) simp
- qed
- show "continuous_on UNIV (\<lambda>w. \<zeta> + exp w)"
- by (rule continuous_intros)+
- show "range (\<lambda>w. \<zeta> + exp w) \<subseteq> -{\<zeta>}"
- by auto
- qed
- then have "homotopic_with_canon (\<lambda>r. pathfinish r = pathstart r) {0..1} (-{\<zeta>}) p (\<lambda>x. \<zeta> + 1)"
- by (rule homotopic_with_eq) (auto simp: o_def peq pathfinish_def pathstart_def)
- then have "homotopic_loops (-{\<zeta>}) p (\<lambda>t. \<zeta> + 1)"
- by (simp add: homotopic_loops_def)
- then show ?rhs ..
-next
- assume ?rhs
- then obtain a where "homotopic_loops (-{\<zeta>}) p (\<lambda>t. a)" ..
- then have "winding_number p \<zeta> = winding_number (\<lambda>t. a) \<zeta>" "a \<noteq> \<zeta>"
- using winding_number_homotopic_loops homotopic_loops_imp_subset by (force simp:)+
- moreover have "winding_number (\<lambda>t. a) \<zeta> = 0"
- by (metis winding_number_zero_const \<open>a \<noteq> \<zeta>\<close>)
- ultimately show ?lhs by metis
+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
+ apply (rule winding_number_pos_lt [OF valid_path_linepath z assms])
+ apply (simp add: linepath_def algebra_simps)
+ done
qed
-lemma winding_number_homotopic_paths_null_explicit_eq:
- assumes "path p" and \<zeta>: "\<zeta> \<notin> path_image p"
- shows "winding_number p \<zeta> = 0 \<longleftrightarrow> homotopic_paths (-{\<zeta>}) p (linepath (pathstart p) (pathstart p))"
- (is "?lhs = ?rhs")
-proof
- assume ?lhs
- then show ?rhs
- apply (auto simp: winding_number_homotopic_loops_null_eq [OF assms])
- apply (rule homotopic_loops_imp_homotopic_paths_null)
- apply (simp add: linepath_refl)
- done
-next
- assume ?rhs
- then show ?lhs
- by (metis \<zeta> pathstart_in_path_image winding_number_homotopic_paths winding_number_trivial)
+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)
+ apply (rule mult_left_mono)+
+ using Re_Im_le_cmod [of "w-z" "linepath s t x" for x]
+ apply (simp add: exp_Euler cos_of_real sin_of_real part_circlepath_def algebra_simps cos_squared_eq [unfolded power2_eq_square])
+ using assms \<open>0 < r\<close> by auto
+qed
+
+subsection \<open>Invariance of winding numbers under homotopy\<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_paths_null_eq:
- assumes "path p" and \<zeta>: "\<zeta> \<notin> path_image p"
- shows "winding_number p \<zeta> = 0 \<longleftrightarrow> (\<exists>a. homotopic_paths (-{\<zeta>}) p (\<lambda>t. a))"
- (is "?lhs = ?rhs")
-proof
- assume ?lhs
- then show ?rhs
- by (auto simp: winding_number_homotopic_paths_null_explicit_eq [OF assms] linepath_refl)
-next
- assume ?rhs
- then show ?lhs
- by (metis \<zeta> homotopic_paths_imp_pathfinish pathfinish_def pathfinish_in_path_image winding_number_homotopic_paths winding_number_zero_const)
+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
-proposition winding_number_homotopic_paths_eq:
- assumes "path p" and \<zeta>p: "\<zeta> \<notin> path_image p"
- and "path q" and \<zeta>q: "\<zeta> \<notin> path_image q"
- and qp: "pathstart q = pathstart p" "pathfinish q = pathfinish p"
- shows "winding_number p \<zeta> = winding_number q \<zeta> \<longleftrightarrow> homotopic_paths (-{\<zeta>}) p q"
- (is "?lhs = ?rhs")
-proof
- assume ?lhs
- then have "winding_number (p +++ reversepath q) \<zeta> = 0"
- using assms by (simp add: winding_number_join winding_number_reversepath)
- moreover
- have "path (p +++ reversepath q)" "\<zeta> \<notin> path_image (p +++ reversepath q)"
- using assms by (auto simp: not_in_path_image_join)
- ultimately obtain a where "homotopic_paths (- {\<zeta>}) (p +++ reversepath q) (linepath a a)"
- using winding_number_homotopic_paths_null_explicit_eq by blast
- then show ?rhs
- using homotopic_paths_imp_pathstart assms
- by (fastforce simp add: dest: homotopic_paths_imp_homotopic_loops homotopic_paths_loop_parts)
+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:
+ "\<lbrakk>path g; z \<notin> path_image g;
+ u \<in> {0..1}; v \<in> {0..1}; w \<in> {0..1}\<rbrakk>
+ \<Longrightarrow> winding_number (subpath u v g) z + winding_number (subpath v w g) z =
+ winding_number (subpath u w g) z"
+apply (rule trans [OF winding_number_join [THEN sym]
+ winding_number_homotopic_paths [OF homotopic_join_subpaths]])
+ using path_image_subpath_subset by auto
+
+subsection \<open>Winding numbers of some simple paths\<close>
+
+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
- assume ?rhs
- then show ?lhs
- by (simp add: winding_number_homotopic_paths)
+ 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 winding_number_homotopic_loops_eq:
- assumes "path p" and \<zeta>p: "\<zeta> \<notin> path_image p"
- and "path q" and \<zeta>q: "\<zeta> \<notin> path_image q"
- and loops: "pathfinish p = pathstart p" "pathfinish q = pathstart q"
- shows "winding_number p \<zeta> = winding_number q \<zeta> \<longleftrightarrow> homotopic_loops (-{\<zeta>}) p q"
- (is "?lhs = ?rhs")
-proof
- assume L: ?lhs
- have "pathstart p \<noteq> \<zeta>" "pathstart q \<noteq> \<zeta>"
- using \<zeta>p \<zeta>q by blast+
- moreover have "path_connected (-{\<zeta>})"
- by (simp add: path_connected_punctured_universe)
- ultimately obtain r where "path r" and rim: "path_image r \<subseteq> -{\<zeta>}"
- and pas: "pathstart r = pathstart p" and paf: "pathfinish r = pathstart q"
- by (auto simp: path_connected_def)
- then have "pathstart r \<noteq> \<zeta>" by blast
- have "homotopic_loops (- {\<zeta>}) p (r +++ q +++ reversepath r)"
- proof (rule homotopic_paths_imp_homotopic_loops)
- show "homotopic_paths (- {\<zeta>}) p (r +++ q +++ reversepath r)"
- by (metis (mono_tags, hide_lams) \<open>path r\<close> L \<zeta>p \<zeta>q \<open>path p\<close> \<open>path q\<close> homotopic_loops_conjugate loops not_in_path_image_join paf pas path_image_reversepath path_imp_reversepath path_join_eq pathfinish_join pathfinish_reversepath pathstart_join pathstart_reversepath rim subset_Compl_singleton winding_number_homotopic_loops winding_number_homotopic_paths_eq)
- qed (use loops pas in auto)
- moreover have "homotopic_loops (- {\<zeta>}) (r +++ q +++ reversepath r) q"
- using rim \<zeta>q by (auto simp: homotopic_loops_conjugate paf \<open>path q\<close> \<open>path r\<close> loops)
- ultimately show ?rhs
- using homotopic_loops_trans by metis
-next
- assume ?rhs
- then show ?lhs
- by (simp add: winding_number_homotopic_loops)
+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"
+apply (rule winding_number_zero_in_outside)
+apply (simp_all add: assms)
+ by (metis nb [of z] \<open>path_image g \<subseteq> s\<close> \<open>z \<notin> s\<close> contra_subsetD mem_Collect_eq outside outside_mono)
+
+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"
+apply (rule no_bounded_connected_component_imp_winding_number_zero [OF g])
+by (simp add: bounded_subset nb path_component_subset_connected_component)
+
+lemma simply_connected_imp_winding_number_zero:
+ assumes "simply_connected S" "path g"
+ "path_image g \<subseteq> S" "pathfinish g = pathstart g" "z \<notin> S"
+ shows "winding_number g z = 0"
+proof -
+ have hom: "homotopic_loops S g (linepath (pathstart g) (pathstart g))"
+ by (meson assms homotopic_paths_imp_homotopic_loops pathfinish_linepath simply_connected_eq_contractible_path)
+ then have "homotopic_paths (- {z}) g (linepath (pathstart g) (pathstart g))"
+ by (meson \<open>z \<notin> S\<close> homotopic_loops_imp_homotopic_paths_null homotopic_paths_subset subset_Compl_singleton)
+ then have "winding_number g z = winding_number(linepath (pathstart g) (pathstart g)) z"
+ by (rule winding_number_homotopic_paths)
+ also have "\<dots> = 0"
+ using assms by (force intro: winding_number_trivial)
+ finally show ?thesis .
qed
-end
-
+end
\ No newline at end of file