author | nipkow |
Fri, 11 Dec 2020 17:58:01 +0100 | |
changeset 72884 | 50f18a822ee9 |
parent 72506 | 44468f28b2c3 |
child 73932 | fd21b4a93043 |
permissions | -rw-r--r-- |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1 |
section \<open>Winding numbers\<close> |
71189
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2 |
theory Winding_Numbers |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
3 |
imports Cauchy_Integral_Theorem |
71189
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
4 |
begin |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
5 |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
6 |
subsection \<open>Definition\<close> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
7 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
8 |
definition\<^marker>\<open>tag important\<close> winding_number_prop :: "[real \<Rightarrow> complex, complex, real, real \<Rightarrow> complex, complex] \<Rightarrow> bool" where |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
9 |
"winding_number_prop \<gamma> z e p n \<equiv> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
10 |
valid_path p \<and> z \<notin> path_image p \<and> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
11 |
pathstart p = pathstart \<gamma> \<and> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
12 |
pathfinish p = pathfinish \<gamma> \<and> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
13 |
(\<forall>t \<in> {0..1}. norm(\<gamma> t - p t) < e) \<and> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
14 |
contour_integral p (\<lambda>w. 1/(w - z)) = 2 * pi * \<i> * n" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
15 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
16 |
definition\<^marker>\<open>tag important\<close> winding_number:: "[real \<Rightarrow> complex, complex] \<Rightarrow> complex" where |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
17 |
"winding_number \<gamma> z \<equiv> SOME n. \<forall>e > 0. \<exists>p. winding_number_prop \<gamma> z e p n" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
18 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
19 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
20 |
lemma winding_number: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
21 |
assumes "path \<gamma>" "z \<notin> path_image \<gamma>" "0 < e" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
22 |
shows "\<exists>p. winding_number_prop \<gamma> z e p (winding_number \<gamma> z)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
23 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
24 |
have "path_image \<gamma> \<subseteq> UNIV - {z}" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
25 |
using assms by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
26 |
then obtain d |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
27 |
where d: "d>0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
28 |
and pi_eq: "\<And>h1 h2. valid_path h1 \<and> valid_path h2 \<and> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
29 |
(\<forall>t\<in>{0..1}. cmod (h1 t - \<gamma> t) < d \<and> cmod (h2 t - \<gamma> t) < d) \<and> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
30 |
pathstart h2 = pathstart h1 \<and> pathfinish h2 = pathfinish h1 \<longrightarrow> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
31 |
path_image h1 \<subseteq> UNIV - {z} \<and> path_image h2 \<subseteq> UNIV - {z} \<and> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
32 |
(\<forall>f. f holomorphic_on UNIV - {z} \<longrightarrow> contour_integral h2 f = contour_integral h1 f)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
33 |
using contour_integral_nearby_ends [of "UNIV - {z}" \<gamma>] assms by (auto simp: open_delete) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
34 |
then obtain h where h: "polynomial_function h \<and> pathstart h = pathstart \<gamma> \<and> pathfinish h = pathfinish \<gamma> \<and> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
35 |
(\<forall>t \<in> {0..1}. norm(h t - \<gamma> t) < d/2)" |
72379 | 36 |
using path_approx_polynomial_function [OF \<open>path \<gamma>\<close>, of "d/2"] d by (metis half_gt_zero_iff) |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
37 |
define nn where "nn = 1/(2* pi*\<i>) * contour_integral h (\<lambda>w. 1/(w - z))" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
38 |
have "\<exists>n. \<forall>e > 0. \<exists>p. winding_number_prop \<gamma> z e p n" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
39 |
proof (rule_tac x=nn in exI, clarify) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
40 |
fix e::real |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
41 |
assume e: "e>0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
42 |
obtain p where p: "polynomial_function p \<and> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
43 |
pathstart p = pathstart \<gamma> \<and> pathfinish p = pathfinish \<gamma> \<and> (\<forall>t\<in>{0..1}. cmod (p t - \<gamma> t) < min e (d/2))" |
72382 | 44 |
using path_approx_polynomial_function [OF \<open>path \<gamma>\<close>, of "min e (d/2)"] d \<open>0<e\<close> |
45 |
by (metis min_less_iff_conj zero_less_divide_iff zero_less_numeral) |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
46 |
have "(\<lambda>w. 1 / (w - z)) holomorphic_on UNIV - {z}" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
47 |
by (auto simp: intro!: holomorphic_intros) |
72379 | 48 |
then have "winding_number_prop \<gamma> z e p nn" |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
49 |
using pi_eq [of h p] h p d |
72379 | 50 |
by (auto simp: valid_path_polynomial_function norm_minus_commute nn_def winding_number_prop_def) |
51 |
then show "\<exists>p. winding_number_prop \<gamma> z e p nn" |
|
52 |
by metis |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
53 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
54 |
then show ?thesis |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
55 |
unfolding winding_number_def by (rule someI2_ex) (blast intro: \<open>0<e\<close>) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
56 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
57 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
58 |
lemma winding_number_unique: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
59 |
assumes \<gamma>: "path \<gamma>" "z \<notin> path_image \<gamma>" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
60 |
and pi: "\<And>e. e>0 \<Longrightarrow> \<exists>p. winding_number_prop \<gamma> z e p n" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
61 |
shows "winding_number \<gamma> z = n" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
62 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
63 |
have "path_image \<gamma> \<subseteq> UNIV - {z}" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
64 |
using assms by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
65 |
then obtain e |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
66 |
where e: "e>0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
67 |
and pi_eq: "\<And>h1 h2 f. \<lbrakk>valid_path h1; valid_path h2; |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
68 |
(\<forall>t\<in>{0..1}. cmod (h1 t - \<gamma> t) < e \<and> cmod (h2 t - \<gamma> t) < e); |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
69 |
pathstart h2 = pathstart h1; pathfinish h2 = pathfinish h1; f holomorphic_on UNIV - {z}\<rbrakk> \<Longrightarrow> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
70 |
contour_integral h2 f = contour_integral h1 f" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
71 |
using contour_integral_nearby_ends [of "UNIV - {z}" \<gamma>] assms by (auto simp: open_delete) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
72 |
obtain p where p: "winding_number_prop \<gamma> z e p n" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
73 |
using pi [OF e] by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
74 |
obtain q where q: "winding_number_prop \<gamma> z e q (winding_number \<gamma> z)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
75 |
using winding_number [OF \<gamma> e] by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
76 |
have "2 * complex_of_real pi * \<i> * n = contour_integral p (\<lambda>w. 1 / (w - z))" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
77 |
using p by (auto simp: winding_number_prop_def) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
78 |
also have "\<dots> = contour_integral q (\<lambda>w. 1 / (w - z))" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
79 |
proof (rule pi_eq) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
80 |
show "(\<lambda>w. 1 / (w - z)) holomorphic_on UNIV - {z}" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
81 |
by (auto intro!: holomorphic_intros) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
82 |
qed (use p q in \<open>auto simp: winding_number_prop_def norm_minus_commute\<close>) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
83 |
also have "\<dots> = 2 * complex_of_real pi * \<i> * winding_number \<gamma> z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
84 |
using q by (auto simp: winding_number_prop_def) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
85 |
finally have "2 * complex_of_real pi * \<i> * n = 2 * complex_of_real pi * \<i> * winding_number \<gamma> z" . |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
86 |
then show ?thesis |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
87 |
by simp |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
88 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
89 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
90 |
(*NB not winding_number_prop here due to the loop in p*) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
91 |
lemma winding_number_unique_loop: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
92 |
assumes \<gamma>: "path \<gamma>" "z \<notin> path_image \<gamma>" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
93 |
and loop: "pathfinish \<gamma> = pathstart \<gamma>" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
94 |
and pi: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
95 |
"\<And>e. e>0 \<Longrightarrow> \<exists>p. valid_path p \<and> z \<notin> path_image p \<and> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
96 |
pathfinish p = pathstart p \<and> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
97 |
(\<forall>t \<in> {0..1}. norm (\<gamma> t - p t) < e) \<and> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
98 |
contour_integral p (\<lambda>w. 1/(w - z)) = 2 * pi * \<i> * n" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
99 |
shows "winding_number \<gamma> z = n" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
100 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
101 |
have "path_image \<gamma> \<subseteq> UNIV - {z}" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
102 |
using assms by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
103 |
then obtain e |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
104 |
where e: "e>0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
105 |
and pi_eq: "\<And>h1 h2 f. \<lbrakk>valid_path h1; valid_path h2; |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
106 |
(\<forall>t\<in>{0..1}. cmod (h1 t - \<gamma> t) < e \<and> cmod (h2 t - \<gamma> t) < e); |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
107 |
pathfinish h1 = pathstart h1; pathfinish h2 = pathstart h2; f holomorphic_on UNIV - {z}\<rbrakk> \<Longrightarrow> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
108 |
contour_integral h2 f = contour_integral h1 f" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
109 |
using contour_integral_nearby_loops [of "UNIV - {z}" \<gamma>] assms by (auto simp: open_delete) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
110 |
obtain p where p: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
111 |
"valid_path p \<and> z \<notin> path_image p \<and> pathfinish p = pathstart p \<and> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
112 |
(\<forall>t \<in> {0..1}. norm (\<gamma> t - p t) < e) \<and> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
113 |
contour_integral p (\<lambda>w. 1/(w - z)) = 2 * pi * \<i> * n" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
114 |
using pi [OF e] by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
115 |
obtain q where q: "winding_number_prop \<gamma> z e q (winding_number \<gamma> z)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
116 |
using winding_number [OF \<gamma> e] by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
117 |
have "2 * complex_of_real pi * \<i> * n = contour_integral p (\<lambda>w. 1 / (w - z))" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
118 |
using p by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
119 |
also have "\<dots> = contour_integral q (\<lambda>w. 1 / (w - z))" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
120 |
proof (rule pi_eq) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
121 |
show "(\<lambda>w. 1 / (w - z)) holomorphic_on UNIV - {z}" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
122 |
by (auto intro!: holomorphic_intros) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
123 |
qed (use p q loop in \<open>auto simp: winding_number_prop_def norm_minus_commute\<close>) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
124 |
also have "\<dots> = 2 * complex_of_real pi * \<i> * winding_number \<gamma> z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
125 |
using q by (auto simp: winding_number_prop_def) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
126 |
finally have "2 * complex_of_real pi * \<i> * n = 2 * complex_of_real pi * \<i> * winding_number \<gamma> z" . |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
127 |
then show ?thesis |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
128 |
by simp |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
129 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
130 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
131 |
proposition winding_number_valid_path: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
132 |
assumes "valid_path \<gamma>" "z \<notin> path_image \<gamma>" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
133 |
shows "winding_number \<gamma> z = 1/(2*pi*\<i>) * contour_integral \<gamma> (\<lambda>w. 1/(w - z))" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
134 |
by (rule winding_number_unique) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
135 |
(use assms in \<open>auto simp: valid_path_imp_path winding_number_prop_def\<close>) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
136 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
137 |
proposition has_contour_integral_winding_number: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
138 |
assumes \<gamma>: "valid_path \<gamma>" "z \<notin> path_image \<gamma>" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
139 |
shows "((\<lambda>w. 1/(w - z)) has_contour_integral (2*pi*\<i>*winding_number \<gamma> z)) \<gamma>" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
140 |
by (simp add: winding_number_valid_path has_contour_integral_integral contour_integrable_inversediff assms) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
141 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
142 |
lemma winding_number_trivial [simp]: "z \<noteq> a \<Longrightarrow> winding_number(linepath a a) z = 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
143 |
by (simp add: winding_number_valid_path) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
144 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
145 |
lemma winding_number_subpath_trivial [simp]: "z \<noteq> g x \<Longrightarrow> winding_number (subpath x x g) z = 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
146 |
by (simp add: path_image_subpath winding_number_valid_path) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
147 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
148 |
lemma winding_number_join: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
149 |
assumes \<gamma>1: "path \<gamma>1" "z \<notin> path_image \<gamma>1" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
150 |
and \<gamma>2: "path \<gamma>2" "z \<notin> path_image \<gamma>2" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
151 |
and "pathfinish \<gamma>1 = pathstart \<gamma>2" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
152 |
shows "winding_number(\<gamma>1 +++ \<gamma>2) z = winding_number \<gamma>1 z + winding_number \<gamma>2 z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
153 |
proof (rule winding_number_unique) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
154 |
show "\<exists>p. winding_number_prop (\<gamma>1 +++ \<gamma>2) z e p |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
155 |
(winding_number \<gamma>1 z + winding_number \<gamma>2 z)" if "e > 0" for e |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
156 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
157 |
obtain p1 where "winding_number_prop \<gamma>1 z e p1 (winding_number \<gamma>1 z)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
158 |
using \<open>0 < e\<close> \<gamma>1 winding_number by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
159 |
moreover |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
160 |
obtain p2 where "winding_number_prop \<gamma>2 z e p2 (winding_number \<gamma>2 z)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
161 |
using \<open>0 < e\<close> \<gamma>2 winding_number by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
162 |
ultimately |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
163 |
have "winding_number_prop (\<gamma>1+++\<gamma>2) z e (p1+++p2) (winding_number \<gamma>1 z + winding_number \<gamma>2 z)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
164 |
using assms |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
165 |
apply (simp add: winding_number_prop_def not_in_path_image_join contour_integrable_inversediff algebra_simps) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
166 |
apply (auto simp: joinpaths_def) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
167 |
done |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
168 |
then show ?thesis |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
169 |
by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
170 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
171 |
qed (use assms in \<open>auto simp: not_in_path_image_join\<close>) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
172 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
173 |
lemma winding_number_reversepath: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
174 |
assumes "path \<gamma>" "z \<notin> path_image \<gamma>" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
175 |
shows "winding_number(reversepath \<gamma>) z = - (winding_number \<gamma> z)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
176 |
proof (rule winding_number_unique) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
177 |
show "\<exists>p. winding_number_prop (reversepath \<gamma>) z e p (- winding_number \<gamma> z)" if "e > 0" for e |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
178 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
179 |
obtain p where "winding_number_prop \<gamma> z e p (winding_number \<gamma> z)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
180 |
using \<open>0 < e\<close> assms winding_number by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
181 |
then have "winding_number_prop (reversepath \<gamma>) z e (reversepath p) (- winding_number \<gamma> z)" |
72506 | 182 |
using assms unfolding winding_number_prop_def |
183 |
apply (simp add: contour_integral_reversepath contour_integrable_inversediff valid_path_imp_reverse) |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
184 |
apply (auto simp: reversepath_def) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
185 |
done |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
186 |
then show ?thesis |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
187 |
by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
188 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
189 |
qed (use assms in auto) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
190 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
191 |
lemma winding_number_shiftpath: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
192 |
assumes \<gamma>: "path \<gamma>" "z \<notin> path_image \<gamma>" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
193 |
and "pathfinish \<gamma> = pathstart \<gamma>" "a \<in> {0..1}" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
194 |
shows "winding_number(shiftpath a \<gamma>) z = winding_number \<gamma> z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
195 |
proof (rule winding_number_unique_loop) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
196 |
show "\<exists>p. valid_path p \<and> z \<notin> path_image p \<and> pathfinish p = pathstart p \<and> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
197 |
(\<forall>t\<in>{0..1}. cmod (shiftpath a \<gamma> t - p t) < e) \<and> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
198 |
contour_integral p (\<lambda>w. 1 / (w - z)) = |
72506 | 199 |
2 * pi * \<i> * winding_number \<gamma> z" |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
200 |
if "e > 0" for e |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
201 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
202 |
obtain p where "winding_number_prop \<gamma> z e p (winding_number \<gamma> z)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
203 |
using \<open>0 < e\<close> assms winding_number by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
204 |
then show ?thesis |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
205 |
apply (rule_tac x="shiftpath a p" in exI) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
206 |
using assms that |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
207 |
apply (auto simp: winding_number_prop_def path_image_shiftpath pathfinish_shiftpath pathstart_shiftpath contour_integral_shiftpath) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
208 |
apply (simp add: shiftpath_def) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
209 |
done |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
210 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
211 |
qed (use assms in \<open>auto simp: path_shiftpath path_image_shiftpath pathfinish_shiftpath pathstart_shiftpath\<close>) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
212 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
213 |
lemma winding_number_split_linepath: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
214 |
assumes "c \<in> closed_segment a b" "z \<notin> closed_segment a b" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
215 |
shows "winding_number(linepath a b) z = winding_number(linepath a c) z + winding_number(linepath c b) z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
216 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
217 |
have "z \<notin> closed_segment a c" "z \<notin> closed_segment c b" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
218 |
using assms by (meson convex_contains_segment convex_segment ends_in_segment subsetCE)+ |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
219 |
then show ?thesis |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
220 |
using assms |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
221 |
by (simp add: winding_number_valid_path contour_integral_split_linepath [symmetric] continuous_on_inversediff field_simps) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
222 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
223 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
224 |
lemma winding_number_cong: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
225 |
"(\<And>t. \<lbrakk>0 \<le> t; t \<le> 1\<rbrakk> \<Longrightarrow> p t = q t) \<Longrightarrow> winding_number p z = winding_number q z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
226 |
by (simp add: winding_number_def winding_number_prop_def pathstart_def pathfinish_def) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
227 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
228 |
lemma winding_number_constI: |
72506 | 229 |
assumes "c\<noteq>z" and g: "\<And>t. \<lbrakk>0\<le>t; t\<le>1\<rbrakk> \<Longrightarrow> g t = c" |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
230 |
shows "winding_number g z = 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
231 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
232 |
have "winding_number g z = winding_number (linepath c c) z" |
72506 | 233 |
using g winding_number_cong by fastforce |
234 |
moreover have "winding_number (linepath c c) z = 0" |
|
235 |
using \<open>c\<noteq>z\<close> by auto |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
236 |
ultimately show ?thesis by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
237 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
238 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
239 |
lemma winding_number_offset: "winding_number p z = winding_number (\<lambda>w. p w - z) 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
240 |
unfolding winding_number_def |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
241 |
proof (intro ext arg_cong [where f = Eps] arg_cong [where f = All] imp_cong refl, safe) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
242 |
fix n e g |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
243 |
assume "0 < e" and g: "winding_number_prop p z e g n" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
244 |
then show "\<exists>r. winding_number_prop (\<lambda>w. p w - z) 0 e r n" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
245 |
by (rule_tac x="\<lambda>t. g t - z" in exI) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
246 |
(force simp: winding_number_prop_def contour_integral_integral valid_path_def path_defs |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
247 |
vector_derivative_def has_vector_derivative_diff_const piecewise_C1_differentiable_diff C1_differentiable_imp_piecewise) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
248 |
next |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
249 |
fix n e g |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
250 |
assume "0 < e" and g: "winding_number_prop (\<lambda>w. p w - z) 0 e g n" |
72506 | 251 |
then have "winding_number_prop p z e (\<lambda>t. g t + z) n" |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
252 |
apply (simp add: winding_number_prop_def contour_integral_integral valid_path_def path_defs |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
253 |
piecewise_C1_differentiable_add vector_derivative_def has_vector_derivative_add_const C1_differentiable_imp_piecewise) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
254 |
apply (force simp: algebra_simps) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
255 |
done |
72506 | 256 |
then show "\<exists>r. winding_number_prop p z e r n" |
257 |
by metis |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
258 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
259 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
260 |
lemma winding_number_negatepath: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
261 |
assumes \<gamma>: "valid_path \<gamma>" and 0: "0 \<notin> path_image \<gamma>" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
262 |
shows "winding_number(uminus \<circ> \<gamma>) 0 = winding_number \<gamma> 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
263 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
264 |
have "(/) 1 contour_integrable_on \<gamma>" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
265 |
using "0" \<gamma> contour_integrable_inversediff by fastforce |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
266 |
then have "((\<lambda>z. 1/z) has_contour_integral contour_integral \<gamma> ((/) 1)) \<gamma>" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
267 |
by (rule has_contour_integral_integral) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
268 |
then have "((\<lambda>z. 1 / - z) has_contour_integral - contour_integral \<gamma> ((/) 1)) \<gamma>" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
269 |
using has_contour_integral_neg by auto |
72506 | 270 |
then have "contour_integral (uminus \<circ> \<gamma>) ((/) 1) = |
271 |
contour_integral \<gamma> ((/) 1)" |
|
272 |
using \<gamma> by (simp add: contour_integral_unique has_contour_integral_negatepath) |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
273 |
then show ?thesis |
72506 | 274 |
using assms by (simp add: winding_number_valid_path valid_path_negatepath image_def path_defs) |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
275 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
276 |
|
72506 | 277 |
text \<open>A combined theorem deducing several things piecewise.\<close> |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
278 |
lemma winding_number_join_pos_combined: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
279 |
"\<lbrakk>valid_path \<gamma>1; z \<notin> path_image \<gamma>1; 0 < Re(winding_number \<gamma>1 z); |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
280 |
valid_path \<gamma>2; z \<notin> path_image \<gamma>2; 0 < Re(winding_number \<gamma>2 z); pathfinish \<gamma>1 = pathstart \<gamma>2\<rbrakk> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
281 |
\<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)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
282 |
by (simp add: valid_path_join path_image_join winding_number_join valid_path_imp_path) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
283 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
284 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
285 |
subsubsection\<^marker>\<open>tag unimportant\<close> \<open>Useful sufficient conditions for the winding number to be positive\<close> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
286 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
287 |
lemma Re_winding_number: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
288 |
"\<lbrakk>valid_path \<gamma>; z \<notin> path_image \<gamma>\<rbrakk> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
289 |
\<Longrightarrow> Re(winding_number \<gamma> z) = Im(contour_integral \<gamma> (\<lambda>w. 1/(w - z))) / (2*pi)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
290 |
by (simp add: winding_number_valid_path field_simps Re_divide power2_eq_square) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
291 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
292 |
lemma winding_number_pos_le: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
293 |
assumes \<gamma>: "valid_path \<gamma>" "z \<notin> path_image \<gamma>" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
294 |
and ge: "\<And>x. \<lbrakk>0 < x; x < 1\<rbrakk> \<Longrightarrow> 0 \<le> Im (vector_derivative \<gamma> (at x) * cnj(\<gamma> x - z))" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
295 |
shows "0 \<le> Re(winding_number \<gamma> z)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
296 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
297 |
have ge0: "0 \<le> Im (vector_derivative \<gamma> (at x) / (\<gamma> x - z))" if x: "0 < x" "x < 1" for x |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
298 |
using ge by (simp add: Complex.Im_divide algebra_simps x) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
299 |
let ?vd = "\<lambda>x. 1 / (\<gamma> x - z) * vector_derivative \<gamma> (at x)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
300 |
let ?int = "\<lambda>z. contour_integral \<gamma> (\<lambda>w. 1 / (w - z))" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
301 |
have "0 \<le> Im (?int z)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
302 |
proof (rule has_integral_component_nonneg [of \<i>, simplified]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
303 |
show "\<And>x. x \<in> cbox 0 1 \<Longrightarrow> 0 \<le> Im (if 0 < x \<and> x < 1 then ?vd x else 0)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
304 |
by (force simp: ge0) |
72506 | 305 |
have "((\<lambda>a. 1 / (a - z)) has_contour_integral contour_integral \<gamma> (\<lambda>w. 1 / (w - z))) \<gamma>" |
306 |
using \<gamma> by (simp flip: add: contour_integrable_inversediff has_contour_integral_integral) |
|
307 |
then have hi: "(?vd has_integral ?int z) (cbox 0 1)" |
|
308 |
using has_contour_integral by auto |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
309 |
show "((\<lambda>x. if 0 < x \<and> x < 1 then ?vd x else 0) has_integral ?int z) (cbox 0 1)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
310 |
by (rule has_integral_spike_interior [OF hi]) simp |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
311 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
312 |
then show ?thesis |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
313 |
by (simp add: Re_winding_number [OF \<gamma>] field_simps) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
314 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
315 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
316 |
lemma winding_number_pos_lt_lemma: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
317 |
assumes \<gamma>: "valid_path \<gamma>" "z \<notin> path_image \<gamma>" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
318 |
and e: "0 < e" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
319 |
and ge: "\<And>x. \<lbrakk>0 < x; x < 1\<rbrakk> \<Longrightarrow> e \<le> Im (vector_derivative \<gamma> (at x) / (\<gamma> x - z))" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
320 |
shows "0 < Re(winding_number \<gamma> z)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
321 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
322 |
let ?vd = "\<lambda>x. 1 / (\<gamma> x - z) * vector_derivative \<gamma> (at x)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
323 |
let ?int = "\<lambda>z. contour_integral \<gamma> (\<lambda>w. 1 / (w - z))" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
324 |
have "e \<le> Im (contour_integral \<gamma> (\<lambda>w. 1 / (w - z)))" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
325 |
proof (rule has_integral_component_le [of \<i> "\<lambda>x. \<i>*e" "\<i>*e" "{0..1}", simplified]) |
72506 | 326 |
have "((\<lambda>a. 1 / (a - z)) has_contour_integral contour_integral \<gamma> (\<lambda>w. 1 / (w - z))) \<gamma>" |
327 |
thm has_integral_component_le [of \<i> "\<lambda>x. \<i>*e" "\<i>*e" "{0..1}", simplified] |
|
328 |
using \<gamma> by (simp add: contour_integrable_inversediff has_contour_integral_integral) |
|
329 |
then have hi: "(?vd has_integral ?int z) (cbox 0 1)" |
|
330 |
using has_contour_integral by auto |
|
331 |
show "((\<lambda>x. if 0 < x \<and> x < 1 then ?vd x else \<i> * e) has_integral ?int z) {0..1}" |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
332 |
by (rule has_integral_spike_interior [OF hi, simplified box_real]) (use e in simp) |
72506 | 333 |
show "\<And>x. 0 \<le> x \<and> x \<le> 1 \<Longrightarrow> e \<le> Im (if 0 < x \<and> x < 1 then ?vd x else \<i> * e)" |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
334 |
by (simp add: ge) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
335 |
qed (use has_integral_const_real [of _ 0 1] in auto) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
336 |
with e show ?thesis |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
337 |
by (simp add: Re_winding_number [OF \<gamma>] field_simps) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
338 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
339 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
340 |
lemma winding_number_pos_lt: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
341 |
assumes \<gamma>: "valid_path \<gamma>" "z \<notin> path_image \<gamma>" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
342 |
and e: "0 < e" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
343 |
and ge: "\<And>x. \<lbrakk>0 < x; x < 1\<rbrakk> \<Longrightarrow> e \<le> Im (vector_derivative \<gamma> (at x) * cnj(\<gamma> x - z))" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
344 |
shows "0 < Re (winding_number \<gamma> z)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
345 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
346 |
have bm: "bounded ((\<lambda>w. w - z) ` (path_image \<gamma>))" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
347 |
using bounded_translation [of _ "-z"] \<gamma> by (simp add: bounded_valid_path_image) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
348 |
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" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
349 |
using bounded_pos [THEN iffD1, OF bm] by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
350 |
{ fix x::real assume x: "0 < x" "x < 1" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
351 |
then have B2: "cmod (\<gamma> x - z)^2 \<le> B^2" using Bno [of "\<gamma> x - z"] |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
352 |
by (simp add: path_image_def power2_eq_square mult_mono') |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
353 |
with x have "\<gamma> x \<noteq> z" using \<gamma> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
354 |
using path_image_def by fastforce |
72506 | 355 |
then have "e / B\<^sup>2 \<le> e / (cmod (\<gamma> x - z))\<^sup>2" |
356 |
using B B2 e by (auto simp: divide_left_mono) |
|
357 |
also have "... \<le> Im (vector_derivative \<gamma> (at x) * cnj (\<gamma> x - z)) / (cmod (\<gamma> x - z))\<^sup>2" |
|
358 |
using ge [OF x] by (auto simp: divide_right_mono) |
|
359 |
finally have "e / B\<^sup>2 \<le> Im (vector_derivative \<gamma> (at x) * cnj (\<gamma> x - z)) / (cmod (\<gamma> x - z))\<^sup>2" . |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
360 |
then have "e / B\<^sup>2 \<le> Im (vector_derivative \<gamma> (at x) / (\<gamma> x - z))" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
361 |
by (simp add: complex_div_cnj [of _ "\<gamma> x - z" for x] del: complex_cnj_diff times_complex.sel) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
362 |
} note * = this |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
363 |
show ?thesis |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
364 |
using e B by (simp add: * winding_number_pos_lt_lemma [OF \<gamma>, of "e/B^2"]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
365 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
366 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
367 |
subsection\<open>The winding number is an integer\<close> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
368 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
369 |
text\<open>Proof from the book Complex Analysis by Lars V. Ahlfors, Chapter 4, section 2.1, |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
370 |
Also on page 134 of Serge Lang's book with the name title, etc.\<close> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
371 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
372 |
lemma exp_fg: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
373 |
fixes z::complex |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
374 |
assumes g: "(g has_vector_derivative g') (at x within s)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
375 |
and f: "(f has_vector_derivative (g' / (g x - z))) (at x within s)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
376 |
and z: "g x \<noteq> z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
377 |
shows "((\<lambda>x. exp(-f x) * (g x - z)) has_vector_derivative 0) (at x within s)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
378 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
379 |
have *: "(exp \<circ> (\<lambda>x. (- f x)) has_vector_derivative - (g' / (g x - z)) * exp (- f x)) (at x within s)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
380 |
using assms unfolding has_vector_derivative_def scaleR_conv_of_real |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
381 |
by (auto intro!: derivative_eq_intros) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
382 |
show ?thesis |
72506 | 383 |
using z by (auto intro!: derivative_eq_intros * [unfolded o_def] g) |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
384 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
385 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
386 |
lemma winding_number_exp_integral: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
387 |
fixes z::complex |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
388 |
assumes \<gamma>: "\<gamma> piecewise_C1_differentiable_on {a..b}" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
389 |
and ab: "a \<le> b" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
390 |
and z: "z \<notin> \<gamma> ` {a..b}" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
391 |
shows "(\<lambda>x. vector_derivative \<gamma> (at x) / (\<gamma> x - z)) integrable_on {a..b}" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
392 |
(is "?thesis1") |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
393 |
"exp (- (integral {a..b} (\<lambda>x. vector_derivative \<gamma> (at x) / (\<gamma> x - z)))) * (\<gamma> b - z) = \<gamma> a - z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
394 |
(is "?thesis2") |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
395 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
396 |
let ?D\<gamma> = "\<lambda>x. vector_derivative \<gamma> (at x)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
397 |
have [simp]: "\<And>x. a \<le> x \<Longrightarrow> x \<le> b \<Longrightarrow> \<gamma> x \<noteq> z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
398 |
using z by force |
72506 | 399 |
have con_g: "continuous_on {a..b} \<gamma>" |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
400 |
using \<gamma> by (simp add: piecewise_C1_differentiable_on_def) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
401 |
obtain k where fink: "finite k" and g_C1_diff: "\<gamma> C1_differentiable_on ({a..b} - k)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
402 |
using \<gamma> by (force simp: piecewise_C1_differentiable_on_def) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
403 |
have \<circ>: "open ({a<..<b} - k)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
404 |
using \<open>finite k\<close> by (simp add: finite_imp_closed open_Diff) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
405 |
moreover have "{a<..<b} - k \<subseteq> {a..b} - k" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
406 |
by force |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
407 |
ultimately have g_diff_at: "\<And>x. \<lbrakk>x \<notin> k; x \<in> {a<..<b}\<rbrakk> \<Longrightarrow> \<gamma> differentiable at x" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
408 |
by (metis Diff_iff differentiable_on_subset C1_diff_imp_diff [OF g_C1_diff] differentiable_on_def at_within_open) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
409 |
{ fix w |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
410 |
assume "w \<noteq> z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
411 |
have "continuous_on (ball w (cmod (w - z))) (\<lambda>w. 1 / (w - z))" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
412 |
by (auto simp: dist_norm intro!: continuous_intros) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
413 |
moreover have "\<And>x. cmod (w - x) < cmod (w - z) \<Longrightarrow> \<exists>f'. ((\<lambda>w. 1 / (w - z)) has_field_derivative f') (at x)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
414 |
by (auto simp: intro!: derivative_eq_intros) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
415 |
ultimately have "\<exists>h. \<forall>y. norm(y - w) < norm(w - z) \<longrightarrow> (h has_field_derivative 1/(y - z)) (at y)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
416 |
using holomorphic_convex_primitive [of "ball w (norm(w - z))" "{}" "\<lambda>w. 1/(w - z)"] |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
417 |
by (force simp: field_differentiable_def Ball_def dist_norm at_within_open_NO_MATCH norm_minus_commute) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
418 |
} |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
419 |
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)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
420 |
by meson |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
421 |
have exy: "\<exists>y. ((\<lambda>x. inverse (\<gamma> x - z) * ?D\<gamma> x) has_integral y) {a..b}" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
422 |
unfolding integrable_on_def [symmetric] |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
423 |
proof (rule contour_integral_local_primitive_any [OF piecewise_C1_imp_differentiable [OF \<gamma>]]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
424 |
show "\<exists>d h. 0 < d \<and> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
425 |
(\<forall>y. cmod (y - w) < d \<longrightarrow> (h has_field_derivative inverse (y - z))(at y within - {z}))" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
426 |
if "w \<in> - {z}" for w |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
427 |
using that inverse_eq_divide has_field_derivative_at_within h |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
428 |
by (metis Compl_insert DiffD2 insertCI right_minus_eq zero_less_norm_iff) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
429 |
qed simp |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
430 |
have vg_int: "(\<lambda>x. ?D\<gamma> x / (\<gamma> x - z)) integrable_on {a..b}" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
431 |
unfolding box_real [symmetric] divide_inverse_commute |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
432 |
by (auto intro!: exy integrable_subinterval simp add: integrable_on_def ab) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
433 |
with ab show ?thesis1 |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
434 |
by (simp add: divide_inverse_commute integral_def integrable_on_def) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
435 |
{ fix t |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
436 |
assume t: "t \<in> {a..b}" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
437 |
have cball: "continuous_on (ball (\<gamma> t) (dist (\<gamma> t) z)) (\<lambda>x. inverse (x - z))" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
438 |
using z by (auto intro!: continuous_intros simp: dist_norm) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
439 |
have icd: "\<And>x. cmod (\<gamma> t - x) < cmod (\<gamma> t - z) \<Longrightarrow> (\<lambda>w. inverse (w - z)) field_differentiable at x" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
440 |
unfolding field_differentiable_def by (force simp: intro!: derivative_eq_intros) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
441 |
obtain h where h: "\<And>x. cmod (\<gamma> t - x) < cmod (\<gamma> t - z) \<Longrightarrow> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
442 |
(h has_field_derivative inverse (x - z)) (at x within {y. cmod (\<gamma> t - y) < cmod (\<gamma> t - z)})" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
443 |
using holomorphic_convex_primitive [where f = "\<lambda>w. inverse(w - z)", OF convex_ball finite.emptyI cball icd] |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
444 |
by simp (auto simp: ball_def dist_norm that) |
72506 | 445 |
have "exp (- (integral {a..t} (\<lambda>x. ?D\<gamma> x / (\<gamma> x - z)))) * (\<gamma> t - z) =\<gamma> a - z" |
446 |
proof (rule has_derivative_zero_unique_strong_interval [of "{a,b} \<union> k" a b]) |
|
447 |
show "continuous_on {a..b} (\<lambda>b. exp (- integral {a..b} (\<lambda>x. ?D\<gamma> x / (\<gamma> x - z))) * (\<gamma> b - z))" |
|
448 |
by (auto intro!: continuous_intros con_g indefinite_integral_continuous_1 [OF vg_int]) |
|
449 |
show "((\<lambda>b. exp (- integral {a..b} (\<lambda>x. ?D\<gamma> x / (\<gamma> x - z))) * (\<gamma> b - z)) has_derivative (\<lambda>h. 0)) |
|
450 |
(at x within {a..b})" |
|
451 |
if "x \<in> {a..b} - ({a, b} \<union> k)" for x |
|
452 |
proof - |
|
453 |
have x: "x \<notin> k" "a < x" "x < b" |
|
454 |
using that by auto |
|
455 |
then have "x \<in> interior ({a..b} - k)" |
|
456 |
using open_subset_interior [OF \<circ>] by fastforce |
|
457 |
then have con: "isCont ?D\<gamma> x" |
|
458 |
using g_C1_diff x by (auto simp: C1_differentiable_on_eq intro: continuous_on_interior) |
|
459 |
then have con_vd: "continuous (at x within {a..b}) (\<lambda>x. ?D\<gamma> x)" |
|
460 |
by (rule continuous_at_imp_continuous_within) |
|
461 |
have gdx: "\<gamma> differentiable at x" |
|
462 |
using x by (simp add: g_diff_at) |
|
463 |
then obtain d where d: "(\<gamma> has_derivative (\<lambda>x. x *\<^sub>R d)) (at x)" |
|
464 |
by (auto simp add: differentiable_iff_scaleR) |
|
465 |
show "((\<lambda>c. exp (- integral {a..c} (\<lambda>x. ?D\<gamma> x / (\<gamma> x - z))) * (\<gamma> c - z)) has_derivative (\<lambda>h. 0)) |
|
466 |
(at x within {a..b})" |
|
467 |
proof (rule exp_fg [unfolded has_vector_derivative_def, simplified]) |
|
468 |
show "(\<gamma> has_derivative (\<lambda>c. c *\<^sub>R d)) (at x within {a..b})" |
|
469 |
using d by (blast intro: has_derivative_at_withinI) |
|
470 |
have "((\<lambda>x. integral {a..x} (\<lambda>x. ?D\<gamma> x / (\<gamma> x - z))) has_vector_derivative d / (\<gamma> x - z)) |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
471 |
(at x within {a..b})" |
72506 | 472 |
proof (rule has_vector_derivative_eq_rhs [OF integral_has_vector_derivative_continuous_at [where S = "{}", simplified]]) |
473 |
show "continuous (at x within {a..b}) (\<lambda>x. vector_derivative \<gamma> (at x) / (\<gamma> x - z))" |
|
474 |
using continuous_at_imp_continuous_at_within differentiable_imp_continuous_within gdx x |
|
475 |
by (intro con_vd continuous_intros) (force+) |
|
476 |
show "vector_derivative \<gamma> (at x) / (\<gamma> x - z) = d / (\<gamma> x - z)" |
|
477 |
using d vector_derivative_at |
|
478 |
by (simp add: vector_derivative_at has_vector_derivative_def) |
|
479 |
qed (use x vg_int in auto) |
|
480 |
then show "((\<lambda>x. integral {a..x} (\<lambda>x. ?D\<gamma> x / (\<gamma> x - z))) has_derivative (\<lambda>c. c *\<^sub>R (d / (\<gamma> x - z)))) |
|
481 |
(at x within {a..b})" |
|
482 |
by (auto simp: has_vector_derivative_def) |
|
483 |
qed (use x in auto) |
|
484 |
qed |
|
485 |
qed (use fink t in auto) |
|
486 |
} |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
487 |
with ab show ?thesis2 |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
488 |
by (simp add: divide_inverse_commute integral_def) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
489 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
490 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
491 |
lemma winding_number_exp_2pi: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
492 |
"\<lbrakk>path p; z \<notin> path_image p\<rbrakk> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
493 |
\<Longrightarrow> pathfinish p - z = exp (2 * pi * \<i> * winding_number p z) * (pathstart p - z)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
494 |
using winding_number [of p z 1] unfolding valid_path_def path_image_def pathstart_def pathfinish_def winding_number_prop_def |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
495 |
by (force dest: winding_number_exp_integral(2) [of _ 0 1 z] simp: field_simps contour_integral_integral exp_minus) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
496 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
497 |
lemma integer_winding_number_eq: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
498 |
assumes \<gamma>: "path \<gamma>" and z: "z \<notin> path_image \<gamma>" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
499 |
shows "winding_number \<gamma> z \<in> \<int> \<longleftrightarrow> pathfinish \<gamma> = pathstart \<gamma>" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
500 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
501 |
obtain p where p: "valid_path p" "z \<notin> path_image p" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
502 |
"pathstart p = pathstart \<gamma>" "pathfinish p = pathfinish \<gamma>" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
503 |
and eq: "contour_integral p (\<lambda>w. 1 / (w - z)) = complex_of_real (2 * pi) * \<i> * winding_number \<gamma> z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
504 |
using winding_number [OF assms, of 1] unfolding winding_number_prop_def by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
505 |
then have wneq: "winding_number \<gamma> z = winding_number p z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
506 |
using eq winding_number_valid_path by force |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
507 |
have iff: "(winding_number \<gamma> z \<in> \<int>) \<longleftrightarrow> (exp (contour_integral p (\<lambda>w. 1 / (w - z))) = 1)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
508 |
using eq by (simp add: exp_eq_1 complex_is_Int_iff) |
72506 | 509 |
have "\<gamma> 0 \<noteq> z" |
510 |
by (metis pathstart_def pathstart_in_path_image z) |
|
511 |
then have "exp (contour_integral p (\<lambda>w. 1 / (w - z))) = (\<gamma> 1 - z) / (\<gamma> 0 - z)" |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
512 |
using p winding_number_exp_integral(2) [of p 0 1 z] |
72506 | 513 |
by (simp add: valid_path_def path_defs contour_integral_integral exp_minus field_split_simps) |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
514 |
then have "winding_number p z \<in> \<int> \<longleftrightarrow> pathfinish p = pathstart p" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
515 |
using p wneq iff by (auto simp: path_defs) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
516 |
then show ?thesis using p eq |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
517 |
by (auto simp: winding_number_valid_path) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
518 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
519 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
520 |
theorem integer_winding_number: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
521 |
"\<lbrakk>path \<gamma>; pathfinish \<gamma> = pathstart \<gamma>; z \<notin> path_image \<gamma>\<rbrakk> \<Longrightarrow> winding_number \<gamma> z \<in> \<int>" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
522 |
by (metis integer_winding_number_eq) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
523 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
524 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
525 |
text\<open>If the winding number's magnitude is at least one, then the path must contain points in every direction.*) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
526 |
We can thus bound the winding number of a path that doesn't intersect a given ray. \<close> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
527 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
528 |
lemma winding_number_pos_meets: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
529 |
fixes z::complex |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
530 |
assumes \<gamma>: "valid_path \<gamma>" and z: "z \<notin> path_image \<gamma>" and 1: "Re (winding_number \<gamma> z) \<ge> 1" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
531 |
and w: "w \<noteq> z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
532 |
shows "\<exists>a::real. 0 < a \<and> z + a*(w - z) \<in> path_image \<gamma>" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
533 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
534 |
have [simp]: "\<And>x. 0 \<le> x \<Longrightarrow> x \<le> 1 \<Longrightarrow> \<gamma> x \<noteq> z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
535 |
using z by (auto simp: path_image_def) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
536 |
have [simp]: "z \<notin> \<gamma> ` {0..1}" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
537 |
using path_image_def z by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
538 |
have gpd: "\<gamma> piecewise_C1_differentiable_on {0..1}" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
539 |
using \<gamma> valid_path_def by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
540 |
define r where "r = (w - z) / (\<gamma> 0 - z)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
541 |
have [simp]: "r \<noteq> 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
542 |
using w z by (auto simp: r_def) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
543 |
have cont: "continuous_on {0..1} |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
544 |
(\<lambda>x. Im (integral {0..x} (\<lambda>x. vector_derivative \<gamma> (at x) / (\<gamma> x - z))))" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
545 |
by (intro continuous_intros indefinite_integral_continuous_1 winding_number_exp_integral [OF gpd]; simp) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
546 |
have "Arg2pi r \<le> 2*pi" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
547 |
by (simp add: Arg2pi less_eq_real_def) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
548 |
also have "\<dots> \<le> Im (integral {0..1} (\<lambda>x. vector_derivative \<gamma> (at x) / (\<gamma> x - z)))" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
549 |
using 1 |
72506 | 550 |
by (simp add: winding_number_valid_path [OF \<gamma> z] contour_integral_integral Complex.Re_divide field_simps power2_eq_square) |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
551 |
finally have "Arg2pi r \<le> Im (integral {0..1} (\<lambda>x. vector_derivative \<gamma> (at x) / (\<gamma> x - z)))" . |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
552 |
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" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
553 |
by (simp add: Arg2pi_ge_0 cont IVT') |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
554 |
then obtain t where t: "t \<in> {0..1}" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
555 |
and eqArg: "Im (integral {0..t} (\<lambda>x. vector_derivative \<gamma> (at x)/(\<gamma> x - z))) = Arg2pi r" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
556 |
by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
557 |
define i where "i = integral {0..t} (\<lambda>x. vector_derivative \<gamma> (at x) / (\<gamma> x - z))" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
558 |
have gpdt: "\<gamma> piecewise_C1_differentiable_on {0..t}" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
559 |
by (metis atLeastAtMost_iff atLeastatMost_subset_iff order_refl piecewise_C1_differentiable_on_subset gpd t) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
560 |
have "exp (- i) * (\<gamma> t - z) = \<gamma> 0 - z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
561 |
unfolding i_def |
72506 | 562 |
proof (rule winding_number_exp_integral [OF gpdt]) |
563 |
show "z \<notin> \<gamma> ` {0..t}" |
|
564 |
using t z unfolding path_image_def by force |
|
565 |
qed (use t in auto) |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
566 |
then have *: "\<gamma> t - z = exp i * (\<gamma> 0 - z)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
567 |
by (simp add: exp_minus field_simps) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
568 |
then have "(w - z) = r * (\<gamma> 0 - z)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
569 |
by (simp add: r_def) |
72506 | 570 |
moreover have "z + exp (Re i) * (exp (\<i> * Im i) * (\<gamma> 0 - z)) = \<gamma> t" |
571 |
using * by (simp add: exp_eq_polar field_simps) |
|
572 |
moreover have "Arg2pi r = Im i" |
|
573 |
using eqArg by (simp add: i_def) |
|
574 |
ultimately have "z + complex_of_real (exp (Re i)) * (w - z) / complex_of_real (cmod r) = \<gamma> t" |
|
575 |
using Complex_Transcendental.Arg2pi_eq [of r] \<open>r \<noteq> 0\<close> |
|
576 |
by (metis mult.left_commute nonzero_mult_div_cancel_left norm_eq_zero of_real_0 of_real_eq_iff times_divide_eq_left) |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
577 |
with t show ?thesis |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
578 |
by (rule_tac x="exp(Re i) / norm r" in exI) (auto simp: path_image_def) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
579 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
580 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
581 |
lemma winding_number_big_meets: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
582 |
fixes z::complex |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
583 |
assumes \<gamma>: "valid_path \<gamma>" and z: "z \<notin> path_image \<gamma>" and "\<bar>Re (winding_number \<gamma> z)\<bar> \<ge> 1" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
584 |
and w: "w \<noteq> z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
585 |
shows "\<exists>a::real. 0 < a \<and> z + a*(w - z) \<in> path_image \<gamma>" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
586 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
587 |
{ assume "Re (winding_number \<gamma> z) \<le> - 1" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
588 |
then have "Re (winding_number (reversepath \<gamma>) z) \<ge> 1" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
589 |
by (simp add: \<gamma> valid_path_imp_path winding_number_reversepath z) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
590 |
moreover have "valid_path (reversepath \<gamma>)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
591 |
using \<gamma> valid_path_imp_reverse by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
592 |
moreover have "z \<notin> path_image (reversepath \<gamma>)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
593 |
by (simp add: z) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
594 |
ultimately have "\<exists>a::real. 0 < a \<and> z + a*(w - z) \<in> path_image (reversepath \<gamma>)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
595 |
using winding_number_pos_meets w by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
596 |
then have ?thesis |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
597 |
by simp |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
598 |
} |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
599 |
then show ?thesis |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
600 |
using assms |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
601 |
by (simp add: abs_if winding_number_pos_meets split: if_split_asm) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
602 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
603 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
604 |
lemma winding_number_less_1: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
605 |
fixes z::complex |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
606 |
shows |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
607 |
"\<lbrakk>valid_path \<gamma>; z \<notin> path_image \<gamma>; w \<noteq> z; |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
608 |
\<And>a::real. 0 < a \<Longrightarrow> z + a*(w - z) \<notin> path_image \<gamma>\<rbrakk> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
609 |
\<Longrightarrow> Re(winding_number \<gamma> z) < 1" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
610 |
by (auto simp: not_less dest: winding_number_big_meets) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
611 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
612 |
text\<open>One way of proving that WN=1 for a loop.\<close> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
613 |
lemma winding_number_eq_1: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
614 |
fixes z::complex |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
615 |
assumes \<gamma>: "valid_path \<gamma>" and z: "z \<notin> path_image \<gamma>" and loop: "pathfinish \<gamma> = pathstart \<gamma>" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
616 |
and 0: "0 < Re(winding_number \<gamma> z)" and 2: "Re(winding_number \<gamma> z) < 2" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
617 |
shows "winding_number \<gamma> z = 1" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
618 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
619 |
have "winding_number \<gamma> z \<in> Ints" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
620 |
by (simp add: \<gamma> integer_winding_number loop valid_path_imp_path z) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
621 |
then show ?thesis |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
622 |
using 0 2 by (auto simp: Ints_def) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
623 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
624 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
625 |
subsection\<open>Continuity of winding number and invariance on connected sets\<close> |
71189
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
626 |
|
72506 | 627 |
theorem continuous_at_winding_number: |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
628 |
fixes z::complex |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
629 |
assumes \<gamma>: "path \<gamma>" and z: "z \<notin> path_image \<gamma>" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
630 |
shows "continuous (at z) (winding_number \<gamma>)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
631 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
632 |
obtain e where "e>0" and cbg: "cball z e \<subseteq> - path_image \<gamma>" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
633 |
using open_contains_cball [of "- path_image \<gamma>"] z |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
634 |
by (force simp: closed_def [symmetric] closed_path_image [OF \<gamma>]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
635 |
then have ppag: "path_image \<gamma> \<subseteq> - cball z (e/2)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
636 |
by (force simp: cball_def dist_norm) |
72506 | 637 |
have oc: "open (- cball z (e/2))" |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
638 |
by (simp add: closed_def [symmetric]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
639 |
obtain d where "d>0" and pi_eq: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
640 |
"\<And>h1 h2. \<lbrakk>valid_path h1; valid_path h2; |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
641 |
(\<forall>t\<in>{0..1}. cmod (h1 t - \<gamma> t) < d \<and> cmod (h2 t - \<gamma> t) < d); |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
642 |
pathstart h2 = pathstart h1; pathfinish h2 = pathfinish h1\<rbrakk> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
643 |
\<Longrightarrow> |
72506 | 644 |
path_image h1 \<subseteq> - cball z (e/2) \<and> |
645 |
path_image h2 \<subseteq> - cball z (e/2) \<and> |
|
646 |
(\<forall>f. f holomorphic_on - cball z (e/2) \<longrightarrow> contour_integral h2 f = contour_integral h1 f)" |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
647 |
using contour_integral_nearby_ends [OF oc \<gamma> ppag] by metis |
72506 | 648 |
obtain p where "valid_path p" "z \<notin> path_image p" |
649 |
and p: "pathstart p = pathstart \<gamma>" "pathfinish p = pathfinish \<gamma>" |
|
650 |
and pg: "\<And>t. t\<in>{0..1} \<Longrightarrow> cmod (\<gamma> t - p t) < min d e/2" |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
651 |
and pi: "contour_integral p (\<lambda>x. 1 / (x - z)) = complex_of_real (2 * pi) * \<i> * winding_number \<gamma> z" |
72506 | 652 |
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) |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
653 |
{ fix w |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
654 |
assume d2: "cmod (w - z) < d/2" and e2: "cmod (w - z) < e/2" |
72506 | 655 |
have wnotp: "w \<notin> path_image p" |
656 |
proof (clarsimp simp add: path_image_def) |
|
657 |
show False if w: "w = p x" and "0 \<le> x" "x \<le> 1" for x |
|
658 |
proof - |
|
659 |
have "cmod (\<gamma> x - p x) < min d e/2" |
|
660 |
using pg that by auto |
|
661 |
then have "cmod (z - \<gamma> x) < e" |
|
662 |
by (metis e2 less_divide_eq_numeral1(1) min_less_iff_conj norm_minus_commute norm_triangle_half_l w) |
|
663 |
then show ?thesis |
|
664 |
using cbg that by (auto simp add: path_image_def cball_def dist_norm less_eq_real_def) |
|
665 |
qed |
|
666 |
qed |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
667 |
have wnotg: "w \<notin> path_image \<gamma>" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
668 |
using cbg e2 \<open>e>0\<close> by (force simp: dist_norm norm_minus_commute) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
669 |
{ fix k::real |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
670 |
assume k: "k>0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
671 |
then obtain q where q: "valid_path q" "w \<notin> path_image q" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
672 |
"pathstart q = pathstart \<gamma> \<and> pathfinish q = pathfinish \<gamma>" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
673 |
and qg: "\<And>t. t \<in> {0..1} \<Longrightarrow> cmod (\<gamma> t - q t) < min k (min d e) / 2" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
674 |
and qi: "contour_integral q (\<lambda>u. 1 / (u - w)) = complex_of_real (2 * pi) * \<i> * winding_number \<gamma> w" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
675 |
using winding_number [OF \<gamma> wnotg, of "min k (min d e) / 2"] \<open>d>0\<close> \<open>e>0\<close> k |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
676 |
by (force simp: min_divide_distrib_right winding_number_prop_def) |
72506 | 677 |
moreover have "\<And>t. t \<in> {0..1} \<Longrightarrow> cmod (q t - \<gamma> t) < d \<and> cmod (p t - \<gamma> t) < d" |
678 |
using pg qg \<open>0 < d\<close> by (fastforce simp add: norm_minus_commute) |
|
679 |
moreover have "(\<lambda>u. 1 / (u-w)) holomorphic_on - cball z (e/2)" |
|
680 |
using e2 by (auto simp: dist_norm norm_minus_commute intro!: holomorphic_intros) |
|
681 |
ultimately have "contour_integral p (\<lambda>u. 1 / (u - w)) = contour_integral q (\<lambda>u. 1 / (u - w))" |
|
682 |
by (metis p \<open>valid_path p\<close> pi_eq) |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
683 |
then have "contour_integral p (\<lambda>x. 1 / (x - w)) = complex_of_real (2 * pi) * \<i> * winding_number \<gamma> w" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
684 |
by (simp add: pi qi) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
685 |
} note pip = this |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
686 |
have "path p" |
72506 | 687 |
by (simp add: \<open>valid_path p\<close> valid_path_imp_path) |
688 |
moreover have "\<And>e. e > 0 \<Longrightarrow> winding_number_prop p w e p (winding_number \<gamma> w)" |
|
689 |
by (simp add: \<open>valid_path p\<close> pip winding_number_prop_def wnotp) |
|
690 |
ultimately have "winding_number p w = winding_number \<gamma> w" |
|
691 |
using winding_number_unique wnotp by blast |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
692 |
} note wnwn = this |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
693 |
obtain pe where "pe>0" and cbp: "cball z (3 / 4 * pe) \<subseteq> - path_image p" |
72506 | 694 |
using \<open>valid_path p\<close> \<open>z \<notin> path_image p\<close> open_contains_cball [of "- path_image p"] |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
695 |
by (force simp: closed_def [symmetric] closed_path_image [OF valid_path_imp_path]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
696 |
obtain L |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
697 |
where "L>0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
698 |
and L: "\<And>f B. \<lbrakk>f holomorphic_on - cball z (3 / 4 * pe); |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
699 |
\<forall>z \<in> - cball z (3 / 4 * pe). cmod (f z) \<le> B\<rbrakk> \<Longrightarrow> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
700 |
cmod (contour_integral p f) \<le> L * B" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
701 |
using contour_integral_bound_exists [of "- cball z (3/4*pe)" p] cbp \<open>valid_path p\<close> by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
702 |
{ fix e::real and w::complex |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
703 |
assume e: "0 < e" and w: "cmod (w - z) < pe/4" "cmod (w - z) < e * pe\<^sup>2 / (8 * L)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
704 |
then have [simp]: "w \<notin> path_image p" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
705 |
using cbp p(2) \<open>0 < pe\<close> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
706 |
by (force simp: dist_norm norm_minus_commute path_image_def cball_def) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
707 |
have [simp]: "contour_integral p (\<lambda>x. 1/(x - w)) - contour_integral p (\<lambda>x. 1/(x - z)) = |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
708 |
contour_integral p (\<lambda>x. 1/(x - w) - 1/(x - z))" |
72506 | 709 |
by (simp add: \<open>valid_path p\<close> \<open>z \<notin> path_image p\<close> contour_integrable_inversediff contour_integral_diff) |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
710 |
{ fix x |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
711 |
assume pe: "3/4 * pe < cmod (z - x)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
712 |
have "cmod (w - x) < pe/4 + cmod (z - x)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
713 |
by (meson add_less_cancel_right norm_diff_triangle_le order_refl order_trans_rules(21) w(1)) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
714 |
then have wx: "cmod (w - x) < 4/3 * cmod (z - x)" using pe by simp |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
715 |
have "cmod (z - x) \<le> cmod (z - w) + cmod (w - x)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
716 |
using norm_diff_triangle_le by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
717 |
also have "\<dots> < pe/4 + cmod (w - x)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
718 |
using w by (simp add: norm_minus_commute) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
719 |
finally have "pe/2 < cmod (w - x)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
720 |
using pe by auto |
72506 | 721 |
then have pe_less: "(pe/2)^2 < cmod (w - x) ^ 2" |
722 |
by (simp add: \<open>0 < pe\<close> less_eq_real_def power_strict_mono) |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
723 |
then have pe2: "pe^2 < 4 * cmod (w - x) ^ 2" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
724 |
by (simp add: power_divide) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
725 |
have "8 * L * cmod (w - z) < e * pe\<^sup>2" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
726 |
using w \<open>L>0\<close> by (simp add: field_simps) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
727 |
also have "\<dots> < e * 4 * cmod (w - x) * cmod (w - x)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
728 |
using pe2 \<open>e>0\<close> by (simp add: power2_eq_square) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
729 |
also have "\<dots> < e * 4 * cmod (w - x) * (4/3 * cmod (z - x))" |
72506 | 730 |
using \<open>0 < pe\<close> pe_less e less_eq_real_def wx by fastforce |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
731 |
finally have "L * cmod (w - z) < 2/3 * e * cmod (w - x) * cmod (z - x)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
732 |
by simp |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
733 |
also have "\<dots> \<le> e * cmod (w - x) * cmod (z - x)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
734 |
using e by simp |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
735 |
finally have Lwz: "L * cmod (w - z) < e * cmod (w - x) * cmod (z - x)" . |
72506 | 736 |
have "L * cmod (1 / (x - w) - 1 / (x - z)) \<le> e" |
737 |
proof (cases "x=z \<or> x=w") |
|
738 |
case True |
|
739 |
with pe \<open>pe>0\<close> w \<open>L>0\<close> |
|
740 |
show ?thesis |
|
741 |
by (force simp: norm_minus_commute) |
|
742 |
next |
|
743 |
case False |
|
744 |
with wx w(2) \<open>L>0\<close> pe pe2 Lwz |
|
745 |
show ?thesis |
|
746 |
by (auto simp: divide_simps mult_less_0_iff norm_minus_commute norm_divide norm_mult power2_eq_square) |
|
747 |
qed |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
748 |
} note L_cmod_le = this |
72506 | 749 |
let ?f = "(\<lambda>x. 1 / (x - w) - 1 / (x - z))" |
750 |
have "cmod (contour_integral p ?f) \<le> L * (e * pe\<^sup>2 / L / 4 * (inverse (pe / 2))\<^sup>2)" |
|
751 |
proof (rule L) |
|
752 |
show "?f holomorphic_on - cball z (3 / 4 * pe)" |
|
753 |
using \<open>pe>0\<close> w |
|
754 |
by (force simp: dist_norm norm_minus_commute intro!: holomorphic_intros) |
|
755 |
show " \<forall>u \<in>- cball z (3 / 4 * pe). cmod (?f u) \<le> e * pe\<^sup>2 / L / 4 * (inverse (pe / 2))\<^sup>2" |
|
756 |
using \<open>pe>0\<close> w \<open>L>0\<close> |
|
757 |
by (auto simp: cball_def dist_norm field_simps L_cmod_le simp del: less_divide_eq_numeral1 le_divide_eq_numeral1) |
|
758 |
qed |
|
759 |
also have "\<dots> < 2*e" |
|
760 |
using \<open>L>0\<close> e by (force simp: field_simps) |
|
761 |
finally have "cmod (winding_number p w - winding_number p z) < e" |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
762 |
using pi_ge_two e |
72506 | 763 |
by (force simp: winding_number_valid_path \<open>valid_path p\<close> \<open>z \<notin> path_image p\<close> field_simps norm_divide norm_mult intro: less_le_trans) |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
764 |
} note cmod_wn_diff = this |
72506 | 765 |
have "isCont (winding_number p) z" |
766 |
proof (clarsimp simp add: continuous_at_eps_delta) |
|
767 |
fix e::real assume "e>0" |
|
768 |
show "\<exists>d>0. \<forall>x'. dist x' z < d \<longrightarrow> dist (winding_number p x') (winding_number p z) < e" |
|
769 |
using \<open>pe>0\<close> \<open>L>0\<close> \<open>e>0\<close> |
|
770 |
by (rule_tac x="min (pe/4) (e/2*pe^2/L/4)" in exI) (simp add: dist_norm cmod_wn_diff) |
|
771 |
qed |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
772 |
then show ?thesis |
72506 | 773 |
apply (rule continuous_transform_within [where d = "min d e/2"]) |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
774 |
apply (auto simp: \<open>d>0\<close> \<open>e>0\<close> dist_norm wnwn) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
775 |
done |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
776 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
777 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
778 |
corollary continuous_on_winding_number: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
779 |
"path \<gamma> \<Longrightarrow> continuous_on (- path_image \<gamma>) (\<lambda>w. winding_number \<gamma> w)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
780 |
by (simp add: continuous_at_imp_continuous_on continuous_at_winding_number) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
781 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
782 |
subsection\<^marker>\<open>tag unimportant\<close> \<open>The winding number is constant on a connected region\<close> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
783 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
784 |
lemma winding_number_constant: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
785 |
assumes \<gamma>: "path \<gamma>" and loop: "pathfinish \<gamma> = pathstart \<gamma>" and cs: "connected S" and sg: "S \<inter> path_image \<gamma> = {}" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
786 |
shows "winding_number \<gamma> constant_on S" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
787 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
788 |
have *: "1 \<le> cmod (winding_number \<gamma> y - winding_number \<gamma> z)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
789 |
if ne: "winding_number \<gamma> y \<noteq> winding_number \<gamma> z" and "y \<in> S" "z \<in> S" for y z |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
790 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
791 |
have "winding_number \<gamma> y \<in> \<int>" "winding_number \<gamma> z \<in> \<int>" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
792 |
using that integer_winding_number [OF \<gamma> loop] sg \<open>y \<in> S\<close> by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
793 |
with ne show ?thesis |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
794 |
by (auto simp: Ints_def simp flip: of_int_diff) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
795 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
796 |
have cont: "continuous_on S (\<lambda>w. winding_number \<gamma> w)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
797 |
using continuous_on_winding_number [OF \<gamma>] sg |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
798 |
by (meson continuous_on_subset disjoint_eq_subset_Compl) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
799 |
show ?thesis |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
800 |
using "*" zero_less_one |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
801 |
by (blast intro: continuous_discrete_range_constant [OF cs cont]) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
802 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
803 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
804 |
lemma winding_number_eq: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
805 |
"\<lbrakk>path \<gamma>; pathfinish \<gamma> = pathstart \<gamma>; w \<in> S; z \<in> S; connected S; S \<inter> path_image \<gamma> = {}\<rbrakk> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
806 |
\<Longrightarrow> winding_number \<gamma> w = winding_number \<gamma> z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
807 |
using winding_number_constant by (metis constant_on_def) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
808 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
809 |
lemma open_winding_number_levelsets: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
810 |
assumes \<gamma>: "path \<gamma>" and loop: "pathfinish \<gamma> = pathstart \<gamma>" |
72506 | 811 |
shows "open {z. z \<notin> path_image \<gamma> \<and> winding_number \<gamma> z = k}" |
812 |
proof (clarsimp simp: open_dist) |
|
813 |
fix z assume z: "z \<notin> path_image \<gamma>" and k: "k = winding_number \<gamma> z" |
|
814 |
have "open (- path_image \<gamma>)" |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
815 |
by (simp add: closed_path_image \<gamma> open_Compl) |
72506 | 816 |
then obtain e where "e>0" "ball z e \<subseteq> - path_image \<gamma>" |
817 |
using open_contains_ball [of "- path_image \<gamma>"] z by blast |
|
818 |
then show "\<exists>e>0. \<forall>y. dist y z < e \<longrightarrow> y \<notin> path_image \<gamma> \<and> winding_number \<gamma> y = winding_number \<gamma> z" |
|
819 |
using \<open>e>0\<close> by (force simp: norm_minus_commute dist_norm intro: winding_number_eq [OF assms, where S = "ball z e"]) |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
820 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
821 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
822 |
subsection\<open>Winding number is zero "outside" a curve\<close> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
823 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
824 |
proposition winding_number_zero_in_outside: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
825 |
assumes \<gamma>: "path \<gamma>" and loop: "pathfinish \<gamma> = pathstart \<gamma>" and z: "z \<in> outside (path_image \<gamma>)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
826 |
shows "winding_number \<gamma> z = 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
827 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
828 |
obtain B::real where "0 < B" and B: "path_image \<gamma> \<subseteq> ball 0 B" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
829 |
using bounded_subset_ballD [OF bounded_path_image [OF \<gamma>]] by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
830 |
obtain w::complex where w: "w \<notin> ball 0 (B + 1)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
831 |
by (metis abs_of_nonneg le_less less_irrefl mem_ball_0 norm_of_real) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
832 |
have "- ball 0 (B + 1) \<subseteq> outside (path_image \<gamma>)" |
72506 | 833 |
using B subset_ball by (intro outside_subset_convex) auto |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
834 |
then have wout: "w \<in> outside (path_image \<gamma>)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
835 |
using w by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
836 |
moreover have "winding_number \<gamma> constant_on outside (path_image \<gamma>)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
837 |
using winding_number_constant [OF \<gamma> loop, of "outside(path_image \<gamma>)"] connected_outside |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
838 |
by (metis DIM_complex bounded_path_image dual_order.refl \<gamma> outside_no_overlap) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
839 |
ultimately have "winding_number \<gamma> z = winding_number \<gamma> w" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
840 |
by (metis (no_types, hide_lams) constant_on_def z) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
841 |
also have "\<dots> = 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
842 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
843 |
have wnot: "w \<notin> path_image \<gamma>" using wout by (simp add: outside_def) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
844 |
{ fix e::real assume "0<e" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
845 |
obtain p where p: "polynomial_function p" "pathstart p = pathstart \<gamma>" "pathfinish p = pathfinish \<gamma>" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
846 |
and pg1: "(\<And>t. \<lbrakk>0 \<le> t; t \<le> 1\<rbrakk> \<Longrightarrow> cmod (p t - \<gamma> t) < 1)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
847 |
and pge: "(\<And>t. \<lbrakk>0 \<le> t; t \<le> 1\<rbrakk> \<Longrightarrow> cmod (p t - \<gamma> t) < e)" |
72382 | 848 |
using path_approx_polynomial_function [OF \<gamma>, of "min 1 e"] \<open>e>0\<close> |
849 |
by (metis atLeastAtMost_iff min_less_iff_conj zero_less_one) |
|
72506 | 850 |
have "\<exists>p. valid_path p \<and> w \<notin> path_image p \<and> |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
851 |
pathstart p = pathstart \<gamma> \<and> pathfinish p = pathfinish \<gamma> \<and> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
852 |
(\<forall>t\<in>{0..1}. cmod (\<gamma> t - p t) < e) \<and> contour_integral p (\<lambda>wa. 1 / (wa - w)) = 0" |
72506 | 853 |
proof (intro exI conjI) |
854 |
have "\<And>x. \<lbrakk>0 \<le> x; x \<le> 1\<rbrakk> \<Longrightarrow> cmod (p x) < B + 1" |
|
855 |
using B unfolding image_subset_iff path_image_def |
|
856 |
by (meson add_strict_mono atLeastAtMost_iff le_less_trans mem_ball_0 norm_triangle_sub pg1) |
|
857 |
then have pip: "path_image p \<subseteq> ball 0 (B + 1)" |
|
858 |
by (auto simp add: path_image_def dist_norm ball_def) |
|
859 |
then show "w \<notin> path_image p" using w by blast |
|
860 |
show vap: "valid_path p" |
|
861 |
by (simp add: p(1) valid_path_polynomial_function) |
|
862 |
show "\<forall>t\<in>{0..1}. cmod (\<gamma> t - p t) < e" |
|
863 |
by (metis atLeastAtMost_iff norm_minus_commute pge) |
|
864 |
show "contour_integral p (\<lambda>wa. 1 / (wa - w)) = 0" |
|
865 |
proof (rule contour_integral_unique [OF Cauchy_theorem_convex_simple [OF _ convex_ball [of 0 "B+1"]]]) |
|
866 |
have "\<And>z. cmod z < B + 1 \<Longrightarrow> z \<noteq> w" |
|
867 |
using mem_ball_0 w by blast |
|
868 |
then show "(\<lambda>z. 1 / (z - w)) holomorphic_on ball 0 (B + 1)" |
|
869 |
by (intro holomorphic_intros; simp add: dist_norm) |
|
870 |
qed (use p vap pip loop in auto) |
|
871 |
qed (use p in auto) |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
872 |
} |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
873 |
then show ?thesis |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
874 |
by (auto intro: winding_number_unique [OF \<gamma>] simp add: winding_number_prop_def wnot) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
875 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
876 |
finally show ?thesis . |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
877 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
878 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
879 |
corollary\<^marker>\<open>tag unimportant\<close> winding_number_zero_const: "a \<noteq> z \<Longrightarrow> winding_number (\<lambda>t. a) z = 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
880 |
by (rule winding_number_zero_in_outside) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
881 |
(auto simp: pathfinish_def pathstart_def path_polynomial_function) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
882 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
883 |
corollary\<^marker>\<open>tag unimportant\<close> winding_number_zero_outside: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
884 |
"\<lbrakk>path \<gamma>; convex s; pathfinish \<gamma> = pathstart \<gamma>; z \<notin> s; path_image \<gamma> \<subseteq> s\<rbrakk> \<Longrightarrow> winding_number \<gamma> z = 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
885 |
by (meson convex_in_outside outside_mono subsetCE winding_number_zero_in_outside) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
886 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
887 |
lemma winding_number_zero_at_infinity: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
888 |
assumes \<gamma>: "path \<gamma>" and loop: "pathfinish \<gamma> = pathstart \<gamma>" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
889 |
shows "\<exists>B. \<forall>z. B \<le> norm z \<longrightarrow> winding_number \<gamma> z = 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
890 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
891 |
obtain B::real where "0 < B" and B: "path_image \<gamma> \<subseteq> ball 0 B" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
892 |
using bounded_subset_ballD [OF bounded_path_image [OF \<gamma>]] by auto |
72506 | 893 |
have "winding_number \<gamma> z = 0" if "B + 1 \<le> cmod z" for z |
894 |
proof (rule winding_number_zero_outside [OF \<gamma> convex_cball loop]) |
|
895 |
show "z \<notin> cball 0 B" |
|
896 |
using that by auto |
|
897 |
show "path_image \<gamma> \<subseteq> cball 0 B" |
|
898 |
using B order.trans by blast |
|
899 |
qed |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
900 |
then show ?thesis |
72506 | 901 |
by metis |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
902 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
903 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
904 |
lemma winding_number_zero_point: |
72506 | 905 |
"\<lbrakk>path \<gamma>; convex S; pathfinish \<gamma> = pathstart \<gamma>; open S; path_image \<gamma> \<subseteq> S\<rbrakk> |
906 |
\<Longrightarrow> \<exists>z. z \<in> S \<and> winding_number \<gamma> z = 0" |
|
907 |
using outside_compact_in_open [of "path_image \<gamma>" S] path_image_nonempty winding_number_zero_in_outside |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
908 |
by (fastforce simp add: compact_path_image) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
909 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
910 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
911 |
text\<open>If a path winds round a set, it winds rounds its inside.\<close> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
912 |
lemma winding_number_around_inside: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
913 |
assumes \<gamma>: "path \<gamma>" and loop: "pathfinish \<gamma> = pathstart \<gamma>" |
72506 | 914 |
and cls: "closed S" and cos: "connected S" and S_disj: "S \<inter> path_image \<gamma> = {}" |
915 |
and z: "z \<in> S" and wn_nz: "winding_number \<gamma> z \<noteq> 0" and w: "w \<in> S \<union> inside S" |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
916 |
shows "winding_number \<gamma> w = winding_number \<gamma> z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
917 |
proof - |
72506 | 918 |
have ssb: "S \<subseteq> inside(path_image \<gamma>)" |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
919 |
proof |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
920 |
fix x :: complex |
72506 | 921 |
assume "x \<in> S" |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
922 |
hence "x \<notin> path_image \<gamma>" |
72506 | 923 |
by (meson disjoint_iff_not_equal S_disj) |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
924 |
thus "x \<in> inside (path_image \<gamma>)" |
72506 | 925 |
by (metis Compl_iff S_disj UnE \<gamma> \<open>x \<in> S\<close> cos inside_outside loop winding_number_eq winding_number_zero_in_outside wn_nz z) |
926 |
qed |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
927 |
show ?thesis |
72506 | 928 |
proof (rule winding_number_eq [OF \<gamma> loop w]) |
929 |
show "z \<in> S \<union> inside S" |
|
930 |
using z by blast |
|
931 |
show "connected (S \<union> inside S)" |
|
932 |
by (simp add: cls connected_with_inside cos) |
|
933 |
show "(S \<union> inside S) \<inter> path_image \<gamma> = {}" |
|
934 |
unfolding disjoint_iff Un_iff |
|
935 |
by (metis ComplD UnI1 \<gamma> cls compact_path_image connected_path_image inside_Un_outside inside_inside_compact_connected ssb subsetD) |
|
936 |
qed |
|
937 |
qed |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
938 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
939 |
text\<open>Bounding a WN by 1/2 for a path and point in opposite halfspaces.\<close> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
940 |
lemma winding_number_subpath_continuous: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
941 |
assumes \<gamma>: "valid_path \<gamma>" and z: "z \<notin> path_image \<gamma>" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
942 |
shows "continuous_on {0..1} (\<lambda>x. winding_number(subpath 0 x \<gamma>) z)" |
72506 | 943 |
proof (rule continuous_on_eq) |
944 |
let ?f = "\<lambda>x. integral {0..x} (\<lambda>t. vector_derivative \<gamma> (at t) / (\<gamma> t - z))" |
|
945 |
show "continuous_on {0..1} (\<lambda>x. 1 / (2 * pi * \<i>) * ?f x)" |
|
946 |
proof (intro indefinite_integral_continuous_1 winding_number_exp_integral continuous_intros) |
|
947 |
show "\<gamma> piecewise_C1_differentiable_on {0..1}" |
|
948 |
using \<gamma> valid_path_def by blast |
|
949 |
qed (use path_image_def z in auto) |
|
950 |
show "1 / (2 * pi * \<i>) * ?f x = winding_number (subpath 0 x \<gamma>) z" |
|
951 |
if x: "x \<in> {0..1}" for x |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
952 |
proof - |
72506 | 953 |
have "1 / (2*pi*\<i>) * ?f x = 1 / (2*pi*\<i>) * contour_integral (subpath 0 x \<gamma>) (\<lambda>w. 1/(w - z))" |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
954 |
using assms x |
72506 | 955 |
by (simp add: contour_integral_subcontour_integral [OF contour_integrable_inversediff]) |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
956 |
also have "\<dots> = winding_number (subpath 0 x \<gamma>) z" |
72506 | 957 |
proof (subst winding_number_valid_path) |
958 |
show "z \<notin> path_image (subpath 0 x \<gamma>)" |
|
959 |
using assms x atLeastAtMost_iff path_image_subpath_subset by force |
|
960 |
qed (use assms x valid_path_subpath in \<open>force+\<close>) |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
961 |
finally show ?thesis . |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
962 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
963 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
964 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
965 |
lemma winding_number_ivt_pos: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
966 |
assumes \<gamma>: "valid_path \<gamma>" and z: "z \<notin> path_image \<gamma>" and "0 \<le> w" "w \<le> Re(winding_number \<gamma> z)" |
72506 | 967 |
shows "\<exists>t \<in> {0..1}. Re(winding_number(subpath 0 t \<gamma>) z) = w" |
968 |
proof - |
|
969 |
have "continuous_on {0..1} (\<lambda>x. winding_number (subpath 0 x \<gamma>) z)" |
|
970 |
using \<gamma> winding_number_subpath_continuous z by blast |
|
971 |
moreover have "Re (winding_number (subpath 0 0 \<gamma>) z) \<le> w" "w \<le> Re (winding_number (subpath 0 1 \<gamma>) z)" |
|
972 |
using assms by (auto simp: path_image_def image_def) |
|
973 |
ultimately show ?thesis |
|
974 |
using ivt_increasing_component_on_1[of 0 1, where ?k = "1"] by force |
|
975 |
qed |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
976 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
977 |
lemma winding_number_ivt_neg: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
978 |
assumes \<gamma>: "valid_path \<gamma>" and z: "z \<notin> path_image \<gamma>" and "Re(winding_number \<gamma> z) \<le> w" "w \<le> 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
979 |
shows "\<exists>t \<in> {0..1}. Re(winding_number(subpath 0 t \<gamma>) z) = w" |
72506 | 980 |
proof - |
981 |
have "continuous_on {0..1} (\<lambda>x. winding_number (subpath 0 x \<gamma>) z)" |
|
982 |
using \<gamma> winding_number_subpath_continuous z by blast |
|
983 |
moreover have "Re (winding_number (subpath 0 0 \<gamma>) z) \<ge> w" "w \<ge> Re (winding_number (subpath 0 1 \<gamma>) z)" |
|
984 |
using assms by (auto simp: path_image_def image_def) |
|
985 |
ultimately show ?thesis |
|
986 |
using ivt_decreasing_component_on_1[of 0 1, where ?k = "1"] by force |
|
987 |
qed |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
988 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
989 |
lemma winding_number_ivt_abs: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
990 |
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>" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
991 |
shows "\<exists>t \<in> {0..1}. \<bar>Re (winding_number (subpath 0 t \<gamma>) z)\<bar> = w" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
992 |
using assms winding_number_ivt_pos [of \<gamma> z w] winding_number_ivt_neg [of \<gamma> z "-w"] |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
993 |
by force |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
994 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
995 |
lemma winding_number_lt_half_lemma: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
996 |
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}" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
997 |
shows "Re(winding_number \<gamma> z) < 1/2" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
998 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
999 |
{ assume "Re(winding_number \<gamma> z) \<ge> 1/2" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1000 |
then obtain t::real where t: "0 \<le> t" "t \<le> 1" and sub12: "Re (winding_number (subpath 0 t \<gamma>) z) = 1/2" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1001 |
using winding_number_ivt_pos [OF \<gamma> z, of "1/2"] by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1002 |
have gt: "\<gamma> t - z = - (of_real (exp (- (2 * pi * Im (winding_number (subpath 0 t \<gamma>) z)))) * (\<gamma> 0 - z))" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1003 |
using winding_number_exp_2pi [of "subpath 0 t \<gamma>" z] |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1004 |
apply (simp add: t \<gamma> valid_path_imp_path) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1005 |
using closed_segment_eq_real_ivl path_image_def t z by (fastforce simp: path_image_subpath Euler sub12) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1006 |
have "b < a \<bullet> \<gamma> 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1007 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1008 |
have "\<gamma> 0 \<in> {c. b < a \<bullet> c}" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1009 |
by (metis (no_types) pag atLeastAtMost_iff image_subset_iff order_refl path_image_def zero_le_one) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1010 |
thus ?thesis |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1011 |
by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1012 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1013 |
moreover have "b < a \<bullet> \<gamma> t" |
72506 | 1014 |
by (metis atLeastAtMost_iff image_eqI mem_Collect_eq pag path_image_def subset_iff t) |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1015 |
ultimately have "0 < a \<bullet> (\<gamma> 0 - z)" "0 < a \<bullet> (\<gamma> t - z)" using az |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1016 |
by (simp add: inner_diff_right)+ |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1017 |
then have False |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1018 |
by (simp add: gt inner_mult_right mult_less_0_iff) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1019 |
} |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1020 |
then show ?thesis by force |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1021 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1022 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1023 |
lemma winding_number_lt_half: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1024 |
assumes "valid_path \<gamma>" "a \<bullet> z \<le> b" "path_image \<gamma> \<subseteq> {w. a \<bullet> w > b}" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1025 |
shows "\<bar>Re (winding_number \<gamma> z)\<bar> < 1/2" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1026 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1027 |
have "z \<notin> path_image \<gamma>" using assms by auto |
72506 | 1028 |
with assms have "Re (winding_number \<gamma> z) < 0 \<Longrightarrow> - Re (winding_number \<gamma> z) < 1/2" |
1029 |
by (metis complex_inner_1_right winding_number_lt_half_lemma [OF valid_path_imp_reverse, of \<gamma> z a b] |
|
1030 |
winding_number_reversepath valid_path_imp_path inner_minus_left path_image_reversepath) |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1031 |
with assms show ?thesis |
72506 | 1032 |
using \<open>z \<notin> path_image \<gamma>\<close> winding_number_lt_half_lemma by fastforce |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1033 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1034 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1035 |
lemma winding_number_le_half: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1036 |
assumes \<gamma>: "valid_path \<gamma>" and z: "z \<notin> path_image \<gamma>" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1037 |
and anz: "a \<noteq> 0" and azb: "a \<bullet> z \<le> b" and pag: "path_image \<gamma> \<subseteq> {w. a \<bullet> w \<ge> b}" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1038 |
shows "\<bar>Re (winding_number \<gamma> z)\<bar> \<le> 1/2" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1039 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1040 |
{ assume wnz_12: "\<bar>Re (winding_number \<gamma> z)\<bar> > 1/2" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1041 |
have "isCont (winding_number \<gamma>) z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1042 |
by (metis continuous_at_winding_number valid_path_imp_path \<gamma> z) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1043 |
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" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1044 |
using continuous_at_eps_delta wnz_12 diff_gt_0_iff_gt by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1045 |
define z' where "z' = z - (d / (2 * cmod a)) *\<^sub>R a" |
72506 | 1046 |
have "a \<bullet> z * 6 \<le> d * cmod a + b * 6" |
1047 |
by (metis \<open>0 < d\<close> add_increasing azb less_eq_real_def mult_nonneg_nonneg mult_right_mono norm_ge_zero norm_numeral) |
|
1048 |
with anz have *: "a \<bullet> z' \<le> b - d / 3 * cmod a" |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1049 |
unfolding z'_def inner_mult_right' divide_inverse |
72506 | 1050 |
by (simp add: field_split_simps algebra_simps dot_square_norm power2_eq_square) |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1051 |
have "cmod (winding_number \<gamma> z' - winding_number \<gamma> z) < \<bar>Re (winding_number \<gamma> z)\<bar> - 1/2" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1052 |
using d [of z'] anz \<open>d>0\<close> by (simp add: dist_norm z'_def) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1053 |
then have "1/2 < \<bar>Re (winding_number \<gamma> z)\<bar> - cmod (winding_number \<gamma> z' - winding_number \<gamma> z)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1054 |
by simp |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1055 |
then have "1/2 < \<bar>Re (winding_number \<gamma> z)\<bar> - \<bar>Re (winding_number \<gamma> z') - Re (winding_number \<gamma> z)\<bar>" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1056 |
using abs_Re_le_cmod [of "winding_number \<gamma> z' - winding_number \<gamma> z"] by simp |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1057 |
then have wnz_12': "\<bar>Re (winding_number \<gamma> z')\<bar> > 1/2" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1058 |
by linarith |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1059 |
moreover have "\<bar>Re (winding_number \<gamma> z')\<bar> < 1/2" |
72506 | 1060 |
proof (rule winding_number_lt_half [OF \<gamma> *]) |
1061 |
show "path_image \<gamma> \<subseteq> {w. b - d / 3 * cmod a < a \<bullet> w}" |
|
1062 |
using azb \<open>d>0\<close> pag by (auto simp: add_strict_increasing anz field_split_simps dest!: subsetD) |
|
1063 |
qed |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1064 |
ultimately have False |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1065 |
by simp |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1066 |
} |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1067 |
then show ?thesis by force |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1068 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1069 |
|
72506 | 1070 |
lemma winding_number_lt_half_linepath: |
1071 |
assumes "z \<notin> closed_segment a b" shows "\<bar>Re (winding_number (linepath a b) z)\<bar> < 1/2" |
|
1072 |
proof - |
|
1073 |
obtain u v where "u \<bullet> z \<le> v" and uv: "\<And>x. x \<in> closed_segment a b \<Longrightarrow> inner u x > v" |
|
1074 |
using separating_hyperplane_closed_point assms closed_segment convex_closed_segment less_eq_real_def by metis |
|
1075 |
moreover have "path_image (linepath a b) \<subseteq> {w. v < u \<bullet> w}" |
|
1076 |
using in_segment(1) uv by auto |
|
1077 |
ultimately show ?thesis |
|
1078 |
using winding_number_lt_half by auto |
|
1079 |
qed |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1080 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1081 |
text\<open> Positivity of WN for a linepath.\<close> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1082 |
lemma winding_number_linepath_pos_lt: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1083 |
assumes "0 < Im ((b - a) * cnj (b - z))" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1084 |
shows "0 < Re(winding_number(linepath a b) z)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1085 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1086 |
have z: "z \<notin> path_image (linepath a b)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1087 |
using assms |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1088 |
by (simp add: closed_segment_def) (force simp: algebra_simps) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1089 |
show ?thesis |
72506 | 1090 |
by (intro winding_number_pos_lt [OF valid_path_linepath z assms]) (simp add: linepath_def algebra_simps) |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1091 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1092 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1093 |
subsection\<^marker>\<open>tag unimportant\<close> \<open>More winding number properties\<close> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1094 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1095 |
text\<open>including the fact that it's +-1 inside a simple closed curve.\<close> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1096 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1097 |
lemma winding_number_homotopic_paths: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1098 |
assumes "homotopic_paths (-{z}) g h" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1099 |
shows "winding_number g z = winding_number h z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1100 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1101 |
have "path g" "path h" using homotopic_paths_imp_path [OF assms] by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1102 |
moreover have pag: "z \<notin> path_image g" and pah: "z \<notin> path_image h" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1103 |
using homotopic_paths_imp_subset [OF assms] by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1104 |
ultimately obtain d e where "d > 0" "e > 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1105 |
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> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1106 |
\<Longrightarrow> homotopic_paths (-{z}) g p" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1107 |
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> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1108 |
\<Longrightarrow> homotopic_paths (-{z}) h q" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1109 |
using homotopic_nearby_paths [of g "-{z}"] homotopic_nearby_paths [of h "-{z}"] by force |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1110 |
obtain p where p: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1111 |
"valid_path p" "z \<notin> path_image p" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1112 |
"pathstart p = pathstart g" "pathfinish p = pathfinish g" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1113 |
and gp_less:"\<forall>t\<in>{0..1}. cmod (g t - p t) < d" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1114 |
and pap: "contour_integral p (\<lambda>w. 1 / (w - z)) = complex_of_real (2 * pi) * \<i> * winding_number g z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1115 |
using winding_number [OF \<open>path g\<close> pag \<open>0 < d\<close>] unfolding winding_number_prop_def by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1116 |
obtain q where q: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1117 |
"valid_path q" "z \<notin> path_image q" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1118 |
"pathstart q = pathstart h" "pathfinish q = pathfinish h" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1119 |
and hq_less: "\<forall>t\<in>{0..1}. cmod (h t - q t) < e" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1120 |
and paq: "contour_integral q (\<lambda>w. 1 / (w - z)) = complex_of_real (2 * pi) * \<i> * winding_number h z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1121 |
using winding_number [OF \<open>path h\<close> pah \<open>0 < e\<close>] unfolding winding_number_prop_def by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1122 |
have "homotopic_paths (- {z}) g p" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1123 |
by (simp add: d p valid_path_imp_path norm_minus_commute gp_less) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1124 |
moreover have "homotopic_paths (- {z}) h q" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1125 |
by (simp add: e q valid_path_imp_path norm_minus_commute hq_less) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1126 |
ultimately have "homotopic_paths (- {z}) p q" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1127 |
by (blast intro: homotopic_paths_trans homotopic_paths_sym assms) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1128 |
then have "contour_integral p (\<lambda>w. 1/(w - z)) = contour_integral q (\<lambda>w. 1/(w - z))" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1129 |
by (rule Cauchy_theorem_homotopic_paths) (auto intro!: holomorphic_intros simp: p q) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1130 |
then show ?thesis |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1131 |
by (simp add: pap paq) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1132 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1133 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1134 |
lemma winding_number_homotopic_loops: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1135 |
assumes "homotopic_loops (-{z}) g h" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1136 |
shows "winding_number g z = winding_number h z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1137 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1138 |
have "path g" "path h" using homotopic_loops_imp_path [OF assms] by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1139 |
moreover have pag: "z \<notin> path_image g" and pah: "z \<notin> path_image h" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1140 |
using homotopic_loops_imp_subset [OF assms] by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1141 |
moreover have gloop: "pathfinish g = pathstart g" and hloop: "pathfinish h = pathstart h" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1142 |
using homotopic_loops_imp_loop [OF assms] by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1143 |
ultimately obtain d e where "d > 0" "e > 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1144 |
and d: "\<And>p. \<lbrakk>path p; pathfinish p = pathstart p; \<forall>t\<in>{0..1}. norm (p t - g t) < d\<rbrakk> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1145 |
\<Longrightarrow> homotopic_loops (-{z}) g p" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1146 |
and e: "\<And>q. \<lbrakk>path q; pathfinish q = pathstart q; \<forall>t\<in>{0..1}. norm (q t - h t) < e\<rbrakk> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1147 |
\<Longrightarrow> homotopic_loops (-{z}) h q" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1148 |
using homotopic_nearby_loops [of g "-{z}"] homotopic_nearby_loops [of h "-{z}"] by force |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1149 |
obtain p where p: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1150 |
"valid_path p" "z \<notin> path_image p" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1151 |
"pathstart p = pathstart g" "pathfinish p = pathfinish g" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1152 |
and gp_less:"\<forall>t\<in>{0..1}. cmod (g t - p t) < d" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1153 |
and pap: "contour_integral p (\<lambda>w. 1 / (w - z)) = complex_of_real (2 * pi) * \<i> * winding_number g z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1154 |
using winding_number [OF \<open>path g\<close> pag \<open>0 < d\<close>] unfolding winding_number_prop_def by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1155 |
obtain q where q: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1156 |
"valid_path q" "z \<notin> path_image q" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1157 |
"pathstart q = pathstart h" "pathfinish q = pathfinish h" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1158 |
and hq_less: "\<forall>t\<in>{0..1}. cmod (h t - q t) < e" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1159 |
and paq: "contour_integral q (\<lambda>w. 1 / (w - z)) = complex_of_real (2 * pi) * \<i> * winding_number h z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1160 |
using winding_number [OF \<open>path h\<close> pah \<open>0 < e\<close>] unfolding winding_number_prop_def by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1161 |
have gp: "homotopic_loops (- {z}) g p" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1162 |
by (simp add: gloop d gp_less norm_minus_commute p valid_path_imp_path) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1163 |
have hq: "homotopic_loops (- {z}) h q" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1164 |
by (simp add: e hloop hq_less norm_minus_commute q valid_path_imp_path) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1165 |
have "contour_integral p (\<lambda>w. 1/(w - z)) = contour_integral q (\<lambda>w. 1/(w - z))" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1166 |
proof (rule Cauchy_theorem_homotopic_loops) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1167 |
show "homotopic_loops (- {z}) p q" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1168 |
by (blast intro: homotopic_loops_trans homotopic_loops_sym gp hq assms) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1169 |
qed (auto intro!: holomorphic_intros simp: p q) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1170 |
then show ?thesis |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1171 |
by (simp add: pap paq) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1172 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1173 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1174 |
lemma winding_number_paths_linear_eq: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1175 |
"\<lbrakk>path g; path h; pathstart h = pathstart g; pathfinish h = pathfinish g; |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1176 |
\<And>t. t \<in> {0..1} \<Longrightarrow> z \<notin> closed_segment (g t) (h t)\<rbrakk> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1177 |
\<Longrightarrow> winding_number h z = winding_number g z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1178 |
by (blast intro: sym homotopic_paths_linear winding_number_homotopic_paths) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1179 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1180 |
lemma winding_number_loops_linear_eq: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1181 |
"\<lbrakk>path g; path h; pathfinish g = pathstart g; pathfinish h = pathstart h; |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1182 |
\<And>t. t \<in> {0..1} \<Longrightarrow> z \<notin> closed_segment (g t) (h t)\<rbrakk> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1183 |
\<Longrightarrow> winding_number h z = winding_number g z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1184 |
by (blast intro: sym homotopic_loops_linear winding_number_homotopic_loops) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1185 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1186 |
lemma winding_number_nearby_paths_eq: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1187 |
"\<lbrakk>path g; path h; pathstart h = pathstart g; pathfinish h = pathfinish g; |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1188 |
\<And>t. t \<in> {0..1} \<Longrightarrow> norm(h t - g t) < norm(g t - z)\<rbrakk> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1189 |
\<Longrightarrow> winding_number h z = winding_number g z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1190 |
by (metis segment_bound(2) norm_minus_commute not_le winding_number_paths_linear_eq) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1191 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1192 |
lemma winding_number_nearby_loops_eq: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1193 |
"\<lbrakk>path g; path h; pathfinish g = pathstart g; pathfinish h = pathstart h; |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1194 |
\<And>t. t \<in> {0..1} \<Longrightarrow> norm(h t - g t) < norm(g t - z)\<rbrakk> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1195 |
\<Longrightarrow> winding_number h z = winding_number g z" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1196 |
by (metis segment_bound(2) norm_minus_commute not_le winding_number_loops_linear_eq) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1197 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1198 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1199 |
lemma winding_number_subpath_combine: |
72506 | 1200 |
assumes "path g" and notin: "z \<notin> path_image g" and "u \<in> {0..1}" "v \<in> {0..1}" "w \<in> {0..1}" |
1201 |
shows "winding_number (subpath u v g) z + winding_number (subpath v w g) z = |
|
1202 |
winding_number (subpath u w g) z" (is "?lhs = ?rhs") |
|
1203 |
proof - |
|
1204 |
have "?lhs = winding_number (subpath u v g +++ subpath v w g) z" |
|
1205 |
using assms |
|
1206 |
by (metis (no_types, lifting) path_image_subpath_subset path_subpath pathfinish_subpath pathstart_subpath subsetD winding_number_join) |
|
1207 |
also have "... = ?rhs" |
|
1208 |
by (meson assms homotopic_join_subpaths subset_Compl_singleton winding_number_homotopic_paths) |
|
1209 |
finally show ?thesis . |
|
1210 |
qed |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1211 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1212 |
text \<open>Winding numbers of circular contours\<close> |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1213 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1214 |
proposition winding_number_part_circlepath_pos_less: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1215 |
assumes "s < t" and no: "norm(w - z) < r" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1216 |
shows "0 < Re (winding_number(part_circlepath z r s t) w)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1217 |
proof - |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1218 |
have "0 < r" by (meson no norm_not_less_zero not_le order.strict_trans2) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1219 |
note valid_path_part_circlepath |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1220 |
moreover have " w \<notin> path_image (part_circlepath z r s t)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1221 |
using assms by (auto simp: path_image_def image_def part_circlepath_def norm_mult linepath_def) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1222 |
moreover have "0 < r * (t - s) * (r - cmod (w - z))" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1223 |
using assms by (metis \<open>0 < r\<close> diff_gt_0_iff_gt mult_pos_pos) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1224 |
ultimately show ?thesis |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1225 |
apply (rule winding_number_pos_lt [where e = "r*(t - s)*(r - norm(w - z))"]) |
72506 | 1226 |
apply (simp add: vector_derivative_part_circlepath right_diff_distrib [symmetric] mult_ac mult_le_cancel_left_pos assms \<open>0<r\<close>) |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1227 |
using Re_Im_le_cmod [of "w-z" "linepath s t x" for x] |
72506 | 1228 |
by (simp add: exp_Euler cos_of_real sin_of_real part_circlepath_def algebra_simps cos_squared_eq [unfolded power2_eq_square]) |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1229 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1230 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1231 |
lemma winding_number_circlepath_centre: "0 < r \<Longrightarrow> winding_number (circlepath z r) z = 1" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1232 |
apply (rule winding_number_unique_loop) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1233 |
apply (simp_all add: sphere_def valid_path_imp_path) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1234 |
apply (rule_tac x="circlepath z r" in exI) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1235 |
apply (simp add: sphere_def contour_integral_circlepath) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1236 |
done |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1237 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1238 |
proposition winding_number_circlepath: |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1239 |
assumes "norm(w - z) < r" shows "winding_number(circlepath z r) w = 1" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1240 |
proof (cases "w = z") |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1241 |
case True then show ?thesis |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1242 |
using assms winding_number_circlepath_centre by auto |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1243 |
next |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1244 |
case False |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1245 |
have [simp]: "r > 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1246 |
using assms le_less_trans norm_ge_zero by blast |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1247 |
define r' where "r' = norm(w - z)" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1248 |
have "r' < r" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1249 |
by (simp add: assms r'_def) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1250 |
have disjo: "cball z r' \<inter> sphere z r = {}" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1251 |
using \<open>r' < r\<close> by (force simp: cball_def sphere_def) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1252 |
have "winding_number(circlepath z r) w = winding_number(circlepath z r) z" |
72506 | 1253 |
proof (rule winding_number_around_inside [where S = "cball z r'"]) |
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1254 |
show "winding_number (circlepath z r) z \<noteq> 0" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1255 |
by (simp add: winding_number_circlepath_centre) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1256 |
show "cball z r' \<inter> path_image (circlepath z r) = {}" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1257 |
by (simp add: disjo less_eq_real_def) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1258 |
qed (auto simp: r'_def dist_norm norm_minus_commute) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1259 |
also have "\<dots> = 1" |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1260 |
by (simp add: winding_number_circlepath_centre) |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1261 |
finally show ?thesis . |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1262 |
qed |
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1263 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1264 |
lemma no_bounded_connected_component_imp_winding_number_zero: |
72506 | 1265 |
assumes g: "path g" "path_image g \<subseteq> S" "pathfinish g = pathstart g" "z \<notin> S" |
1266 |
and nb: "\<And>z. bounded (connected_component_set (- S) z) \<Longrightarrow> z \<in> S" |
|
1267 |
shows "winding_number g z = 0" |
|
1268 |
proof - |
|
1269 |
have "z \<in> outside (path_image g)" |
|
1270 |
by (metis nb [of z] \<open>path_image g \<subseteq> S\<close> \<open>z \<notin> S\<close> subsetD mem_Collect_eq outside outside_mono) |
|
1271 |
then show ?thesis |
|
1272 |
by (simp add: g winding_number_zero_in_outside) |
|
1273 |
qed |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1274 |
|
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1275 |
lemma no_bounded_path_component_imp_winding_number_zero: |
72506 | 1276 |
assumes g: "path g" "path_image g \<subseteq> S" "pathfinish g = pathstart g" "z \<notin> S" |
1277 |
and nb: "\<And>z. bounded (path_component_set (- S) z) \<Longrightarrow> z \<in> S" |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
1278 |
shows "winding_number g z = 0" |
72506 | 1279 |
by (simp add: bounded_subset nb path_component_subset_connected_component |
1280 |
no_bounded_connected_component_imp_winding_number_zero [OF g]) |
|
71189
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1281 |
|
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1282 |
subsection\<open>Winding number for a triangle\<close> |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1283 |
|
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1284 |
lemma wn_triangle1: |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1285 |
assumes "0 \<in> interior(convex hull {a,b,c})" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1286 |
shows "\<not> (Im(a/b) \<le> 0 \<and> 0 \<le> Im(b/c))" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1287 |
proof - |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1288 |
{ assume 0: "Im(a/b) \<le> 0" "0 \<le> Im(b/c)" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1289 |
have "0 \<notin> interior (convex hull {a,b,c})" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1290 |
proof (cases "a=0 \<or> b=0 \<or> c=0") |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1291 |
case True then show ?thesis |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1292 |
by (auto simp: not_in_interior_convex_hull_3) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1293 |
next |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1294 |
case False |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1295 |
then have "b \<noteq> 0" by blast |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1296 |
{ fix x y::complex and u::real |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1297 |
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" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1298 |
then have "((1 - u) * Im x) * Re b \<le> Im b * ((1 - u) * Re x)" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1299 |
by (simp add: mult_left_mono mult.assoc mult.left_commute [of "Im b"]) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1300 |
then have "((1 - u) * Im x + u * Im y) * Re b \<le> Im b * ((1 - u) * Re x + u * Re y)" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1301 |
using eq_f' ordered_comm_semiring_class.comm_mult_left_mono |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1302 |
by (fastforce simp add: algebra_simps) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1303 |
} |
72506 | 1304 |
then have "convex {z. Im z * Re b \<le> Im b * Re z}" |
1305 |
by (auto simp: algebra_simps convex_alt) |
|
71189
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1306 |
with False 0 have "convex hull {a,b,c} \<le> {z. Im z * Re b \<le> Im b * Re z}" |
72506 | 1307 |
by (simp add: subset_hull mult.commute Complex.Im_divide divide_simps complex_neq_0 [symmetric]) |
71189
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1308 |
moreover have "0 \<notin> interior({z. Im z * Re b \<le> Im b * Re z})" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1309 |
proof |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1310 |
assume "0 \<in> interior {z. Im z * Re b \<le> Im b * Re z}" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1311 |
then obtain e where "e>0" and e: "ball 0 e \<subseteq> {z. Im z * Re b \<le> Im b * Re z}" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1312 |
by (meson mem_interior) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1313 |
define z where "z \<equiv> - sgn (Im b) * (e/3) + sgn (Re b) * (e/3) * \<i>" |
72506 | 1314 |
have "cmod z = cmod (e/3) * cmod (- sgn (Im b) + sgn (Re b) * \<i>)" |
1315 |
unfolding z_def norm_mult [symmetric] by (simp add: algebra_simps) |
|
1316 |
also have "... < e" |
|
1317 |
using \<open>e>0\<close> by (auto simp: norm_mult intro: le_less_trans [OF norm_triangle_ineq4]) |
|
1318 |
finally have "z \<in> ball 0 e" |
|
1319 |
using \<open>e>0\<close> by (simp add: ) |
|
1320 |
then have "Im z * Re b \<le> Im b * Re z" |
|
71189
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1321 |
using e by blast |
72506 | 1322 |
then have b: "0 < Re b" "0 < Im b" and disj: "e * Re b < - (Im b * e) \<or> e * Re b = - (Im b * e)" |
71189
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1323 |
using \<open>e>0\<close> \<open>b \<noteq> 0\<close> |
72506 | 1324 |
by (auto simp add: z_def dist_norm sgn_if less_eq_real_def mult_less_0_iff complex.expand split: if_split_asm) |
1325 |
show False \<comment>\<open>or just one smt line\<close> |
|
1326 |
using disj |
|
1327 |
proof |
|
1328 |
assume "e * Re b < - (Im b * e)" |
|
1329 |
with b \<open>0 < e\<close> less_trans [of _ 0] show False |
|
1330 |
by (metis (no_types) mult_pos_pos neg_less_0_iff_less not_less_iff_gr_or_eq) |
|
1331 |
next |
|
1332 |
assume "e * Re b = - (Im b * e)" |
|
1333 |
with b \<open>0 < e\<close> show False |
|
1334 |
by (metis mult_pos_pos neg_less_0_iff_less not_less_iff_gr_or_eq) |
|
1335 |
qed |
|
71189
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1336 |
qed |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1337 |
ultimately show ?thesis |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1338 |
using interior_mono by blast |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1339 |
qed |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1340 |
} with assms show ?thesis by blast |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1341 |
qed |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1342 |
|
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1343 |
lemma wn_triangle2_0: |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1344 |
assumes "0 \<in> interior(convex hull {a,b,c})" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1345 |
shows |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1346 |
"0 < Im((b - a) * cnj (b)) \<and> |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1347 |
0 < Im((c - b) * cnj (c)) \<and> |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1348 |
0 < Im((a - c) * cnj (a)) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1349 |
\<or> |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1350 |
Im((b - a) * cnj (b)) < 0 \<and> |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1351 |
0 < Im((b - c) * cnj (b)) \<and> |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1352 |
0 < Im((a - b) * cnj (a)) \<and> |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1353 |
0 < Im((c - a) * cnj (c))" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1354 |
proof - |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1355 |
have [simp]: "{b,c,a} = {a,b,c}" "{c,a,b} = {a,b,c}" by auto |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1356 |
show ?thesis |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1357 |
using wn_triangle1 [OF assms] wn_triangle1 [of b c a] wn_triangle1 [of c a b] assms |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1358 |
by (auto simp: algebra_simps Im_complex_div_gt_0 Im_complex_div_lt_0 not_le not_less) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1359 |
qed |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1360 |
|
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1361 |
lemma wn_triangle2: |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1362 |
assumes "z \<in> interior(convex hull {a,b,c})" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1363 |
shows "0 < Im((b - a) * cnj (b - z)) \<and> |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1364 |
0 < Im((c - b) * cnj (c - z)) \<and> |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1365 |
0 < Im((a - c) * cnj (a - z)) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1366 |
\<or> |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1367 |
Im((b - a) * cnj (b - z)) < 0 \<and> |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1368 |
0 < Im((b - c) * cnj (b - z)) \<and> |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1369 |
0 < Im((a - b) * cnj (a - z)) \<and> |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1370 |
0 < Im((c - a) * cnj (c - z))" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1371 |
proof - |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1372 |
have 0: "0 \<in> interior(convex hull {a-z, b-z, c-z})" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1373 |
using assms convex_hull_translation [of "-z" "{a,b,c}"] |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1374 |
interior_translation [of "-z"] |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1375 |
by (simp cong: image_cong_simp) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1376 |
show ?thesis using wn_triangle2_0 [OF 0] |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1377 |
by simp |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1378 |
qed |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1379 |
|
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1380 |
lemma wn_triangle3: |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1381 |
assumes z: "z \<in> interior(convex hull {a,b,c})" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1382 |
and "0 < Im((b-a) * cnj (b-z))" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1383 |
"0 < Im((c-b) * cnj (c-z))" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1384 |
"0 < Im((a-c) * cnj (a-z))" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1385 |
shows "winding_number (linepath a b +++ linepath b c +++ linepath c a) z = 1" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1386 |
proof - |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1387 |
have znot[simp]: "z \<notin> closed_segment a b" "z \<notin> closed_segment b c" "z \<notin> closed_segment c a" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1388 |
using z interior_of_triangle [of a b c] |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1389 |
by (auto simp: closed_segment_def) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1390 |
have gt0: "0 < Re (winding_number (linepath a b +++ linepath b c +++ linepath c a) z)" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1391 |
using assms |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1392 |
by (simp add: winding_number_linepath_pos_lt path_image_join winding_number_join_pos_combined) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1393 |
have lt2: "Re (winding_number (linepath a b +++ linepath b c +++ linepath c a) z) < 2" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1394 |
using winding_number_lt_half_linepath [of _ a b] |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1395 |
using winding_number_lt_half_linepath [of _ b c] |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1396 |
using winding_number_lt_half_linepath [of _ c a] znot |
72506 | 1397 |
by (fastforce simp add: winding_number_join path_image_join) |
71189
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1398 |
show ?thesis |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1399 |
by (rule winding_number_eq_1) (simp_all add: path_image_join gt0 lt2) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1400 |
qed |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1401 |
|
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1402 |
proposition winding_number_triangle: |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1403 |
assumes z: "z \<in> interior(convex hull {a,b,c})" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1404 |
shows "winding_number(linepath a b +++ linepath b c +++ linepath c a) z = |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1405 |
(if 0 < Im((b - a) * cnj (b - z)) then 1 else -1)" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1406 |
proof - |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1407 |
have [simp]: "{a,c,b} = {a,b,c}" by auto |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1408 |
have znot[simp]: "z \<notin> closed_segment a b" "z \<notin> closed_segment b c" "z \<notin> closed_segment c a" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1409 |
using z interior_of_triangle [of a b c] |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1410 |
by (auto simp: closed_segment_def) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1411 |
then have [simp]: "z \<notin> closed_segment b a" "z \<notin> closed_segment c b" "z \<notin> closed_segment a c" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1412 |
using closed_segment_commute by blast+ |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1413 |
have *: "winding_number (linepath a b +++ linepath b c +++ linepath c a) z = |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1414 |
winding_number (reversepath (linepath a c +++ linepath c b +++ linepath b a)) z" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1415 |
by (simp add: reversepath_joinpaths winding_number_join not_in_path_image_join) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1416 |
show ?thesis |
72506 | 1417 |
apply (rule disjE [OF wn_triangle2 [OF z]]) |
1418 |
subgoal |
|
1419 |
by (simp add: wn_triangle3 z) |
|
1420 |
subgoal |
|
1421 |
by (simp add: path_image_join winding_number_reversepath * wn_triangle3 z) |
|
71189
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1422 |
done |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1423 |
qed |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1424 |
|
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1425 |
subsection\<open>Winding numbers for simple closed paths\<close> |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1426 |
|
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1427 |
lemma winding_number_from_innerpath: |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1428 |
assumes "simple_path c1" and c1: "pathstart c1 = a" "pathfinish c1 = b" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1429 |
and "simple_path c2" and c2: "pathstart c2 = a" "pathfinish c2 = b" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1430 |
and "simple_path c" and c: "pathstart c = a" "pathfinish c = b" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1431 |
and c1c2: "path_image c1 \<inter> path_image c2 = {a,b}" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1432 |
and c1c: "path_image c1 \<inter> path_image c = {a,b}" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1433 |
and c2c: "path_image c2 \<inter> path_image c = {a,b}" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1434 |
and ne_12: "path_image c \<inter> inside(path_image c1 \<union> path_image c2) \<noteq> {}" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1435 |
and z: "z \<in> inside(path_image c1 \<union> path_image c)" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1436 |
and wn_d: "winding_number (c1 +++ reversepath c) z = d" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1437 |
and "a \<noteq> b" "d \<noteq> 0" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1438 |
obtains "z \<in> inside(path_image c1 \<union> path_image c2)" "winding_number (c1 +++ reversepath c2) z = d" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1439 |
proof - |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1440 |
obtain 0: "inside(path_image c1 \<union> path_image c) \<inter> inside(path_image c2 \<union> path_image c) = {}" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1441 |
and 1: "inside(path_image c1 \<union> path_image c) \<union> inside(path_image c2 \<union> path_image c) \<union> |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1442 |
(path_image c - {a,b}) = inside(path_image c1 \<union> path_image c2)" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1443 |
by (rule split_inside_simple_closed_curve |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1444 |
[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]) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1445 |
have znot: "z \<notin> path_image c" "z \<notin> path_image c1" "z \<notin> path_image c2" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1446 |
using union_with_outside z 1 by auto |
72506 | 1447 |
then have zout: "z \<in> outside (path_image c \<union> path_image c2)" |
1448 |
by (metis "0" ComplI UnE disjoint_iff_not_equal sup.commute union_with_inside z) |
|
71189
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1449 |
have wn_cc2: "winding_number (c +++ reversepath c2) z = 0" |
72506 | 1450 |
by (rule winding_number_zero_in_outside; simp add: zout \<open>simple_path c2\<close> c c2 \<open>simple_path c\<close> simple_path_imp_path path_image_join) |
71189
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1451 |
show ?thesis |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1452 |
proof |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1453 |
show "z \<in> inside (path_image c1 \<union> path_image c2)" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1454 |
using "1" z by blast |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1455 |
have "winding_number c1 z - winding_number c z = d " |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1456 |
using assms znot |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1457 |
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) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1458 |
then show "winding_number (c1 +++ reversepath c2) z = d" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1459 |
using wn_cc2 by (simp add: winding_number_join simple_path_imp_path assms znot winding_number_reversepath) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1460 |
qed |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1461 |
qed |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1462 |
|
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1463 |
lemma simple_closed_path_wn1: |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1464 |
fixes a::complex and e::real |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1465 |
assumes "0 < e" |
72506 | 1466 |
and sp_pl: "simple_path(p +++ linepath (a - e) (a + e))" (is "simple_path ?pae") |
71189
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1467 |
and psp: "pathstart p = a + e" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1468 |
and pfp: "pathfinish p = a - e" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1469 |
and disj: "ball a e \<inter> path_image p = {}" |
72506 | 1470 |
obtains z where "z \<in> inside (path_image ?pae)" "cmod (winding_number ?pae z) = 1" |
71189
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1471 |
proof - |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1472 |
have "arc p" and arc_lp: "arc (linepath (a - e) (a + e))" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1473 |
and pap: "path_image p \<inter> path_image (linepath (a - e) (a + e)) \<subseteq> {pathstart p, a-e}" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1474 |
using simple_path_join_loop_eq [of "linepath (a - e) (a + e)" p] assms by auto |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1475 |
have mid_eq_a: "midpoint (a - e) (a + e) = a" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1476 |
by (simp add: midpoint_def) |
72506 | 1477 |
with assms have "a \<in> path_image ?pae" |
1478 |
by (simp add: assms path_image_join) (metis midpoint_in_closed_segment) |
|
1479 |
then have "a \<in> frontier(inside (path_image ?pae))" |
|
1480 |
using assms by (simp add: Jordan_inside_outside ) |
|
1481 |
with \<open>0 < e\<close> obtain c where c: "c \<in> inside (path_image ?pae)" |
|
71189
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1482 |
and dac: "dist a c < e" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1483 |
by (auto simp: frontier_straddle) |
72506 | 1484 |
then have "c \<notin> path_image ?pae" |
71189
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1485 |
using inside_no_overlap by blast |
72506 | 1486 |
then have "c \<notin> path_image p" "c \<notin> closed_segment (a - e) (a + e)" |
71189
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1487 |
by (simp_all add: assms path_image_join) |
72506 | 1488 |
with \<open>0 < e\<close> dac have "c \<notin> affine hull {a - e, a + e}" |
71189
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1489 |
by (simp add: segment_as_ball not_le) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1490 |
with \<open>0 < e\<close> have *: "\<not> collinear {a - e, c,a + e}" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1491 |
using collinear_3_affine_hull [of "a-e" "a+e"] by (auto simp: insert_commute) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1492 |
have 13: "1/3 + 1/3 + 1/3 = (1::real)" by simp |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1493 |
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})" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1494 |
using interior_convex_hull_3_minimal [OF * DIM_complex] |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1495 |
by clarsimp (metis 13 zero_less_divide_1_iff zero_less_numeral) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1496 |
then obtain z where z: "z \<in> interior(convex hull {a - e, c, a + e})" by force |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1497 |
have [simp]: "z \<notin> closed_segment (a - e) c" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1498 |
by (metis DIM_complex Diff_iff IntD2 inf_sup_absorb interior_of_triangle z) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1499 |
have [simp]: "z \<notin> closed_segment (a + e) (a - e)" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1500 |
by (metis DIM_complex DiffD2 Un_iff interior_of_triangle z) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1501 |
have [simp]: "z \<notin> closed_segment c (a + e)" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1502 |
by (metis (no_types, lifting) DIM_complex DiffD2 Un_insert_right inf_sup_aci(5) insertCI interior_of_triangle mk_disjoint_insert z) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1503 |
show thesis |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1504 |
proof |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1505 |
have "norm (winding_number (linepath (a - e) c +++ linepath c (a + e) +++ linepath (a + e) (a - e)) z) = 1" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1506 |
using winding_number_triangle [OF z] by simp |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1507 |
have zin: "z \<in> inside (path_image (linepath (a + e) (a - e)) \<union> path_image p)" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1508 |
and zeq: "winding_number (linepath (a + e) (a - e) +++ reversepath p) z = |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1509 |
winding_number (linepath (a - e) c +++ linepath c (a + e) +++ linepath (a + e) (a - e)) z" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1510 |
proof (rule winding_number_from_innerpath |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1511 |
[of "linepath (a + e) (a - e)" "a+e" "a-e" p |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1512 |
"linepath (a + e) c +++ linepath c (a - e)" z |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1513 |
"winding_number (linepath (a - e) c +++ linepath c (a + e) +++ linepath (a + e) (a - e)) z"]) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1514 |
show sp_aec: "simple_path (linepath (a + e) c +++ linepath c (a - e))" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1515 |
proof (rule arc_imp_simple_path [OF arc_join]) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1516 |
show "arc (linepath (a + e) c)" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1517 |
by (metis \<open>c \<notin> path_image p\<close> arc_linepath pathstart_in_path_image psp) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1518 |
show "arc (linepath c (a - e))" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1519 |
by (metis \<open>c \<notin> path_image p\<close> arc_linepath pathfinish_in_path_image pfp) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1520 |
show "path_image (linepath (a + e) c) \<inter> path_image (linepath c (a - e)) \<subseteq> {pathstart (linepath c (a - e))}" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1521 |
by clarsimp (metis "*" IntI Int_closed_segment closed_segment_commute singleton_iff) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1522 |
qed auto |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1523 |
show "simple_path p" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1524 |
using \<open>arc p\<close> arc_simple_path by blast |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1525 |
show sp_ae2: "simple_path (linepath (a + e) (a - e))" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1526 |
using \<open>arc p\<close> arc_distinct_ends pfp psp by fastforce |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1527 |
show pa: "pathfinish (linepath (a + e) (a - e)) = a - e" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1528 |
"pathstart (linepath (a + e) c +++ linepath c (a - e)) = a + e" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1529 |
"pathfinish (linepath (a + e) c +++ linepath c (a - e)) = a - e" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1530 |
"pathstart p = a + e" "pathfinish p = a - e" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1531 |
"pathstart (linepath (a + e) (a - e)) = a + e" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1532 |
by (simp_all add: assms) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1533 |
show 1: "path_image (linepath (a + e) (a - e)) \<inter> path_image p = {a + e, a - e}" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1534 |
proof |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1535 |
show "path_image (linepath (a + e) (a - e)) \<inter> path_image p \<subseteq> {a + e, a - e}" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1536 |
using pap closed_segment_commute psp segment_convex_hull by fastforce |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1537 |
show "{a + e, a - e} \<subseteq> path_image (linepath (a + e) (a - e)) \<inter> path_image p" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1538 |
using pap pathfinish_in_path_image pathstart_in_path_image pfp psp by fastforce |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1539 |
qed |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1540 |
show 2: "path_image (linepath (a + e) (a - e)) \<inter> path_image (linepath (a + e) c +++ linepath c (a - e)) = |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1541 |
{a + e, a - e}" (is "?lhs = ?rhs") |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1542 |
proof |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1543 |
have "\<not> collinear {c, a + e, a - e}" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1544 |
using * by (simp add: insert_commute) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1545 |
then have "convex hull {a + e, a - e} \<inter> convex hull {a + e, c} = {a + e}" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1546 |
"convex hull {a + e, a - e} \<inter> convex hull {c, a - e} = {a - e}" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1547 |
by (metis (full_types) Int_closed_segment insert_commute segment_convex_hull)+ |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1548 |
then show "?lhs \<subseteq> ?rhs" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1549 |
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) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1550 |
show "?rhs \<subseteq> ?lhs" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1551 |
using segment_convex_hull by (simp add: path_image_join) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1552 |
qed |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1553 |
have "path_image p \<inter> path_image (linepath (a + e) c) \<subseteq> {a + e}" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1554 |
proof (clarsimp simp: path_image_join) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1555 |
fix x |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1556 |
assume "x \<in> path_image p" and x_ac: "x \<in> closed_segment (a + e) c" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1557 |
then have "dist x a \<ge> e" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1558 |
by (metis IntI all_not_in_conv disj dist_commute mem_ball not_less) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1559 |
with x_ac dac \<open>e > 0\<close> show "x = a + e" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1560 |
by (auto simp: norm_minus_commute dist_norm closed_segment_eq_open dest: open_segment_furthest_le [where y=a]) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1561 |
qed |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1562 |
moreover |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1563 |
have "path_image p \<inter> path_image (linepath c (a - e)) \<subseteq> {a - e}" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1564 |
proof (clarsimp simp: path_image_join) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1565 |
fix x |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1566 |
assume "x \<in> path_image p" and x_ac: "x \<in> closed_segment c (a - e)" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1567 |
then have "dist x a \<ge> e" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1568 |
by (metis IntI all_not_in_conv disj dist_commute mem_ball not_less) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1569 |
with x_ac dac \<open>e > 0\<close> show "x = a - e" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1570 |
by (auto simp: norm_minus_commute dist_norm closed_segment_eq_open dest: open_segment_furthest_le [where y=a]) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1571 |
qed |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1572 |
ultimately |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1573 |
have "path_image p \<inter> path_image (linepath (a + e) c +++ linepath c (a - e)) \<subseteq> {a + e, a - e}" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1574 |
by (force simp: path_image_join) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1575 |
then show 3: "path_image p \<inter> path_image (linepath (a + e) c +++ linepath c (a - e)) = {a + e, a - e}" |
72506 | 1576 |
using "1" "2" by blast |
71189
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1577 |
show 4: "path_image (linepath (a + e) c +++ linepath c (a - e)) \<inter> |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1578 |
inside (path_image (linepath (a + e) (a - e)) \<union> path_image p) \<noteq> {}" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1579 |
apply (clarsimp simp: path_image_join segment_convex_hull disjoint_iff_not_equal) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1580 |
by (metis (no_types, hide_lams) UnI1 Un_commute c closed_segment_commute ends_in_segment(1) path_image_join |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1581 |
path_image_linepath pathstart_linepath pfp segment_convex_hull) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1582 |
show zin_inside: "z \<in> inside (path_image (linepath (a + e) (a - e)) \<union> |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1583 |
path_image (linepath (a + e) c +++ linepath c (a - e)))" |
72506 | 1584 |
proof (simp add: path_image_join) |
1585 |
show "z \<in> inside (closed_segment (a + e) (a - e) \<union> (closed_segment (a + e) c \<union> closed_segment c (a - e)))" |
|
1586 |
by (metis z inside_of_triangle DIM_complex Un_commute closed_segment_commute) |
|
1587 |
qed |
|
71189
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1588 |
show 5: "winding_number |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1589 |
(linepath (a + e) (a - e) +++ reversepath (linepath (a + e) c +++ linepath c (a - e))) z = |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1590 |
winding_number (linepath (a - e) c +++ linepath c (a + e) +++ linepath (a + e) (a - e)) z" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1591 |
by (simp add: reversepath_joinpaths path_image_join winding_number_join) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1592 |
show 6: "winding_number (linepath (a - e) c +++ linepath c (a + e) +++ linepath (a + e) (a - e)) z \<noteq> 0" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1593 |
by (simp add: winding_number_triangle z) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1594 |
show "winding_number (linepath (a + e) (a - e) +++ reversepath p) z = |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1595 |
winding_number (linepath (a - e) c +++ linepath c (a + e) +++ linepath (a + e) (a - e)) z" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1596 |
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) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1597 |
qed (use assms \<open>e > 0\<close> in auto) |
72506 | 1598 |
show z_inside: "z \<in> inside (path_image ?pae)" |
71189
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1599 |
using zin by (simp add: assms path_image_join Un_commute closed_segment_commute) |
72506 | 1600 |
have "cmod (winding_number ?pae z) = cmod ((winding_number(reversepath ?pae) z))" |
1601 |
proof (subst winding_number_reversepath) |
|
1602 |
show "path ?pae" |
|
1603 |
using simple_path_imp_path sp_pl by blast |
|
1604 |
show "z \<notin> path_image ?pae" |
|
1605 |
by (metis IntI emptyE inside_no_overlap z_inside) |
|
1606 |
qed (simp add: inside_def) |
|
71189
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1607 |
also have "... = cmod (winding_number(linepath (a + e) (a - e) +++ reversepath p) z)" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1608 |
by (simp add: pfp reversepath_joinpaths) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1609 |
also have "... = cmod (winding_number (linepath (a - e) c +++ linepath c (a + e) +++ linepath (a + e) (a - e)) z)" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1610 |
by (simp add: zeq) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1611 |
also have "... = 1" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1612 |
using z by (simp add: interior_of_triangle winding_number_triangle) |
72506 | 1613 |
finally show "cmod (winding_number ?pae z) = 1" . |
71189
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1614 |
qed |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1615 |
qed |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1616 |
|
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1617 |
lemma simple_closed_path_wn2: |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1618 |
fixes a::complex and d e::real |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1619 |
assumes "0 < d" "0 < e" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1620 |
and sp_pl: "simple_path(p +++ linepath (a - d) (a + e))" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1621 |
and psp: "pathstart p = a + e" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1622 |
and pfp: "pathfinish p = a - d" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1623 |
obtains z where "z \<in> inside (path_image (p +++ linepath (a - d) (a + e)))" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1624 |
"cmod (winding_number (p +++ linepath (a - d) (a + e)) z) = 1" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1625 |
proof - |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1626 |
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 |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1627 |
using closed_segment_translation_eq [of a] |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1628 |
by (metis (no_types, hide_lams) add_uminus_conv_diff of_real_minus of_real_closed_segment) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1629 |
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 |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1630 |
by (metis closed_segment_translation_eq diff_conv_add_uminus of_real_closed_segment of_real_minus) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1631 |
have "arc p" and arc_lp: "arc (linepath (a - d) (a + e))" and "path p" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1632 |
and pap: "path_image p \<inter> closed_segment (a - d) (a + e) \<subseteq> {a+e, a-d}" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1633 |
using simple_path_join_loop_eq [of "linepath (a - d) (a + e)" p] assms arc_imp_path by auto |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1634 |
have "0 \<in> closed_segment (-d) e" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1635 |
using \<open>0 < d\<close> \<open>0 < e\<close> closed_segment_eq_real_ivl by auto |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1636 |
then have "a \<in> path_image (linepath (a - d) (a + e))" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1637 |
using of_real_closed_segment [THEN iffD2] |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1638 |
by (force dest: closed_segment_translation_eq [of a, THEN iffD2] simp del: of_real_closed_segment) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1639 |
then have "a \<notin> path_image p" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1640 |
using \<open>0 < d\<close> \<open>0 < e\<close> pap by auto |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1641 |
then obtain k where "0 < k" and k: "ball a k \<inter> (path_image p) = {}" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1642 |
using \<open>0 < e\<close> \<open>path p\<close> not_on_path_ball by blast |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1643 |
define kde where "kde \<equiv> (min k (min d e)) / 2" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1644 |
have "0 < kde" "kde < k" "kde < d" "kde < e" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1645 |
using \<open>0 < k\<close> \<open>0 < d\<close> \<open>0 < e\<close> by (auto simp: kde_def) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1646 |
let ?q = "linepath (a + kde) (a + e) +++ p +++ linepath (a - d) (a - kde)" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1647 |
have "- kde \<in> closed_segment (-d) e" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1648 |
using \<open>0 < kde\<close> \<open>kde < d\<close> \<open>kde < e\<close> closed_segment_eq_real_ivl by auto |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1649 |
then have a_diff_kde: "a - kde \<in> closed_segment (a - d) (a + e)" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1650 |
using of_real_closed_segment [THEN iffD2] |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1651 |
by (force dest: closed_segment_translation_eq [of a, THEN iffD2] simp del: of_real_closed_segment) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1652 |
then have clsub2: "closed_segment (a - d) (a - kde) \<subseteq> closed_segment (a - d) (a + e)" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1653 |
by (simp add: subset_closed_segment) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1654 |
then have "path_image p \<inter> closed_segment (a - d) (a - kde) \<subseteq> {a + e, a - d}" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1655 |
using pap by force |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1656 |
moreover |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1657 |
have "a + e \<notin> path_image p \<inter> closed_segment (a - d) (a - kde)" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1658 |
using \<open>0 < kde\<close> \<open>kde < d\<close> \<open>0 < e\<close> by (auto simp: closed_segment_eq_real_ivl) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1659 |
ultimately have sub_a_diff_d: "path_image p \<inter> closed_segment (a - d) (a - kde) \<subseteq> {a - d}" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1660 |
by blast |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1661 |
have "kde \<in> closed_segment (-d) e" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1662 |
using \<open>0 < kde\<close> \<open>kde < d\<close> \<open>kde < e\<close> closed_segment_eq_real_ivl by auto |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1663 |
then have a_diff_kde: "a + kde \<in> closed_segment (a - d) (a + e)" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1664 |
using of_real_closed_segment [THEN iffD2] |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1665 |
by (force dest: closed_segment_translation_eq [of "a", THEN iffD2] simp del: of_real_closed_segment) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1666 |
then have clsub1: "closed_segment (a + kde) (a + e) \<subseteq> closed_segment (a - d) (a + e)" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1667 |
by (simp add: subset_closed_segment) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1668 |
then have "closed_segment (a + kde) (a + e) \<inter> path_image p \<subseteq> {a + e, a - d}" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1669 |
using pap by force |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1670 |
moreover |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1671 |
have "closed_segment (a + kde) (a + e) \<inter> closed_segment (a - d) (a - kde) = {}" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1672 |
proof (clarsimp intro!: equals0I) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1673 |
fix y |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1674 |
assume y1: "y \<in> closed_segment (a + kde) (a + e)" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1675 |
and y2: "y \<in> closed_segment (a - d) (a - kde)" |
72506 | 1676 |
obtain u::real where u: "y = a + u" and "0 < u" |
1677 |
proof - |
|
1678 |
obtain \<xi> where \<xi>: "y = (1 - \<xi>) *\<^sub>R (a + kde) + \<xi> *\<^sub>R (a + e)" and "0 \<le> \<xi>" "\<xi> \<le> 1" |
|
1679 |
using y1 by (auto simp: in_segment) |
|
1680 |
show thesis |
|
1681 |
proof |
|
1682 |
show "y = a + ((1 - \<xi>)*kde + \<xi>*e)" |
|
1683 |
using \<xi> by (auto simp: scaleR_conv_of_real algebra_simps) |
|
1684 |
have "(1 - \<xi>)*kde + \<xi>*e \<ge> 0" |
|
1685 |
using \<open>0 < kde\<close> \<open>0 \<le> \<xi>\<close> \<open>\<xi> \<le> 1\<close> \<open>0 < e\<close> by auto |
|
1686 |
moreover have "(1 - \<xi>)*kde + \<xi>*e \<noteq> 0" |
|
1687 |
using \<open>0 < kde\<close> \<open>0 \<le> \<xi>\<close> \<open>\<xi> \<le> 1\<close> \<open>0 < e\<close> by (auto simp: add_nonneg_eq_0_iff) |
|
1688 |
ultimately show "(1 - \<xi>)*kde + \<xi>*e > 0" by simp |
|
1689 |
qed |
|
1690 |
qed |
|
71189
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1691 |
moreover |
72506 | 1692 |
obtain v::real where v: "y = a + v" and "v \<le> 0" |
1693 |
proof - |
|
1694 |
obtain \<xi> where \<xi>: "y = (1 - \<xi>) *\<^sub>R (a - d) + \<xi> *\<^sub>R (a - kde)" and "0 \<le> \<xi>" "\<xi> \<le> 1" |
|
1695 |
using y2 by (auto simp: in_segment) |
|
1696 |
show thesis |
|
1697 |
proof |
|
1698 |
show "y = a + (- ((1 - \<xi>)*d + \<xi>*kde))" |
|
1699 |
using \<xi> by (force simp: scaleR_conv_of_real algebra_simps) |
|
1700 |
show "- ((1 - \<xi>)*d + \<xi>*kde) \<le> 0" |
|
1701 |
using \<open>0 < kde\<close> \<open>0 \<le> \<xi>\<close> \<open>\<xi> \<le> 1\<close> \<open>0 < d\<close> |
|
1702 |
by (metis add.left_neutral add_nonneg_nonneg le_diff_eq less_eq_real_def neg_le_0_iff_le split_mult_pos_le) |
|
1703 |
||
1704 |
qed |
|
1705 |
qed |
|
71189
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1706 |
ultimately show False |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1707 |
by auto |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1708 |
qed |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1709 |
moreover have "a - d \<notin> closed_segment (a + kde) (a + e)" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1710 |
using \<open>0 < kde\<close> \<open>kde < d\<close> \<open>0 < e\<close> by (auto simp: closed_segment_eq_real_ivl) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1711 |
ultimately have sub_a_plus_e: |
72506 | 1712 |
"closed_segment (a + kde) (a + e) \<inter> (path_image p \<union> closed_segment (a - d) (a - kde)) \<subseteq> {a + e}" |
71189
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1713 |
by auto |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1714 |
have "kde \<in> closed_segment (-kde) e" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1715 |
using \<open>0 < kde\<close> \<open>kde < d\<close> \<open>kde < e\<close> closed_segment_eq_real_ivl by auto |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1716 |
then have a_add_kde: "a + kde \<in> closed_segment (a - kde) (a + e)" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1717 |
using of_real_closed_segment [THEN iffD2] |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1718 |
by (force dest: closed_segment_translation_eq [of "a", THEN iffD2] simp del: of_real_closed_segment) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1719 |
have "closed_segment (a - kde) (a + kde) \<inter> closed_segment (a + kde) (a + e) = {a + kde}" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1720 |
by (metis a_add_kde Int_closed_segment) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1721 |
moreover |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1722 |
have "path_image p \<inter> closed_segment (a - kde) (a + kde) = {}" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1723 |
proof (rule equals0I, clarify) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1724 |
fix y assume "y \<in> path_image p" "y \<in> closed_segment (a - kde) (a + kde)" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1725 |
with equals0D [OF k, of y] \<open>0 < kde\<close> \<open>kde < k\<close> show False |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1726 |
by (auto simp: dist_norm dest: dist_decreases_closed_segment [where c=a]) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1727 |
qed |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1728 |
moreover |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1729 |
have "- kde \<in> closed_segment (-d) kde" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1730 |
using \<open>0 < kde\<close> \<open>kde < d\<close> \<open>kde < e\<close> closed_segment_eq_real_ivl by auto |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1731 |
then have a_diff_kde': "a - kde \<in> closed_segment (a - d) (a + kde)" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1732 |
using of_real_closed_segment [THEN iffD2] |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1733 |
by (force dest: closed_segment_translation_eq [of a, THEN iffD2] simp del: of_real_closed_segment) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1734 |
then have "closed_segment (a - d) (a - kde) \<inter> closed_segment (a - kde) (a + kde) = {a - kde}" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1735 |
by (metis Int_closed_segment) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1736 |
ultimately |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1737 |
have pa_subset_pm_kde: "path_image ?q \<inter> closed_segment (a - kde) (a + kde) \<subseteq> {a - kde, a + kde}" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1738 |
by (auto simp: path_image_join assms) |
72506 | 1739 |
have ge_kde1: "\<exists>y. x = a + y \<and> y \<ge> kde" if x: "x \<in> closed_segment (a + kde) (a + e)" for x |
1740 |
proof - |
|
1741 |
obtain u where "0 \<le> u" "u \<le> 1" and u: "x = (1 - u) *\<^sub>R (a + kde) + u *\<^sub>R (a + e)" |
|
1742 |
using x by (auto simp: in_segment) |
|
1743 |
then have "kde \<le> (1 - u) * kde + u * e" |
|
1744 |
using \<open>kde < e\<close> segment_bound_lemma by auto |
|
1745 |
moreover have "x = a + ((1 - u) * kde + u * e)" |
|
1746 |
by (fastforce simp: u algebra_simps scaleR_conv_of_real) |
|
1747 |
ultimately |
|
1748 |
show ?thesis by blast |
|
1749 |
qed |
|
1750 |
have ge_kde2: "\<exists>y. x = a + y \<and> y \<le> -kde" if x: "x \<in> closed_segment (a - d) (a - kde)" for x |
|
1751 |
proof - |
|
1752 |
obtain u where "0 \<le> u" "u \<le> 1" and u: "x = (1 - u) *\<^sub>R (a - d) + u *\<^sub>R (a - kde)" |
|
1753 |
using x by (auto simp: in_segment) |
|
1754 |
then have "kde \<le> ((1-u)*d + u*kde)" |
|
1755 |
using \<open>kde < d\<close> segment_bound_lemma by auto |
|
1756 |
moreover have "x = a - ((1-u)*d + u*kde)" |
|
1757 |
by (fastforce simp: u algebra_simps scaleR_conv_of_real) |
|
1758 |
ultimately show ?thesis |
|
1759 |
by (metis add_uminus_conv_diff neg_le_iff_le of_real_minus) |
|
1760 |
qed |
|
71189
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1761 |
have notin_paq: "x \<notin> path_image ?q" if "dist a x < kde" for x |
72506 | 1762 |
proof - |
1763 |
have "x \<notin> path_image p" |
|
1764 |
using k \<open>kde < k\<close> that by force |
|
1765 |
then show ?thesis |
|
1766 |
using that assms by (auto simp: path_image_join dist_norm dest!: ge_kde1 ge_kde2) |
|
1767 |
qed |
|
71189
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1768 |
obtain z where zin: "z \<in> inside (path_image (?q +++ linepath (a - kde) (a + kde)))" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1769 |
and z1: "cmod (winding_number (?q +++ linepath (a - kde) (a + kde)) z) = 1" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1770 |
proof (rule simple_closed_path_wn1 [of kde ?q a]) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1771 |
show "simple_path (?q +++ linepath (a - kde) (a + kde))" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1772 |
proof (intro simple_path_join_loop conjI) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1773 |
show "arc ?q" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1774 |
proof (rule arc_join) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1775 |
show "arc (linepath (a + kde) (a + e))" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1776 |
using \<open>kde < e\<close> \<open>arc p\<close> by (force simp: pfp) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1777 |
show "arc (p +++ linepath (a - d) (a - kde))" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1778 |
using \<open>kde < d\<close> \<open>kde < e\<close> \<open>arc p\<close> sub_a_diff_d by (force simp: pfp intro: arc_join) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1779 |
qed (auto simp: psp pfp path_image_join sub_a_plus_e) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1780 |
show "arc (linepath (a - kde) (a + kde))" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1781 |
using \<open>0 < kde\<close> by auto |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1782 |
qed (use pa_subset_pm_kde in auto) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1783 |
qed (use \<open>0 < kde\<close> notin_paq in auto) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1784 |
have eq: "path_image (?q +++ linepath (a - kde) (a + kde)) = path_image (p +++ linepath (a - d) (a + e))" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1785 |
(is "?lhs = ?rhs") |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1786 |
proof |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1787 |
show "?lhs \<subseteq> ?rhs" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1788 |
using clsub1 clsub2 apply (auto simp: path_image_join assms) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1789 |
by (meson subsetCE subset_closed_segment) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1790 |
show "?rhs \<subseteq> ?lhs" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1791 |
apply (simp add: path_image_join assms Un_ac) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1792 |
by (metis Un_closed_segment Un_assoc a_diff_kde a_diff_kde' le_supI2 subset_refl) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1793 |
qed |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1794 |
show thesis |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1795 |
proof |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1796 |
show zzin: "z \<in> inside (path_image (p +++ linepath (a - d) (a + e)))" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1797 |
by (metis eq zin) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1798 |
then have znotin: "z \<notin> path_image p" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1799 |
by (metis ComplD Un_iff inside_Un_outside path_image_join pathfinish_linepath pathstart_reversepath pfp reversepath_linepath) |
72506 | 1800 |
have znotin_d_kde: "z \<notin> closed_segment (a - d) (a + kde)" |
71189
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1801 |
by (metis ComplD Un_iff Un_closed_segment a_diff_kde inside_Un_outside path_image_join path_image_linepath pathstart_linepath pfp zzin) |
72506 | 1802 |
have znotin_d_e: "z \<notin> closed_segment (a - d) (a + e)" |
1803 |
by (metis IntI UnCI emptyE inside_no_overlap path_image_join path_image_linepath pathstart_linepath pfp zzin) |
|
1804 |
have znotin_kde_e: "z \<notin> closed_segment (a + kde) (a + e)" and znotin_d_kde': "z \<notin> closed_segment (a - d) (a - kde)" |
|
1805 |
using clsub1 clsub2 zzin |
|
1806 |
by (metis (no_types, hide_lams) IntI Un_iff emptyE inside_no_overlap path_image_join path_image_linepath pathstart_linepath pfp subsetD)+ |
|
71189
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1807 |
have "winding_number (linepath (a - d) (a + e)) z = |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1808 |
winding_number (linepath (a - d) (a + kde)) z + winding_number (linepath (a + kde) (a + e)) z" |
72506 | 1809 |
proof (rule winding_number_split_linepath) |
1810 |
show "a + complex_of_real kde \<in> closed_segment (a - d) (a + e)" |
|
1811 |
by (simp add: a_diff_kde) |
|
1812 |
show "z \<notin> closed_segment (a - d) (a + e)" |
|
1813 |
by (metis ComplD Un_iff inside_Un_outside path_image_join path_image_linepath pathstart_linepath pfp zzin) |
|
1814 |
qed |
|
71189
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1815 |
also have "... = winding_number (linepath (a + kde) (a + e)) z + |
72506 | 1816 |
(winding_number (linepath (a - d) (a - kde)) z + winding_number (linepath (a - kde) (a + kde)) z)" |
1817 |
by (simp add: winding_number_split_linepath [of "a-kde", symmetric] znotin_d_kde a_diff_kde') |
|
71189
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1818 |
finally have "winding_number (p +++ linepath (a - d) (a + e)) z = |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1819 |
winding_number p z + winding_number (linepath (a + kde) (a + e)) z + |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1820 |
(winding_number (linepath (a - d) (a - kde)) z + |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1821 |
winding_number (linepath (a - kde) (a + kde)) z)" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1822 |
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) |
72506 | 1823 |
also have "... = winding_number (linepath (a + kde) (a + e)) z |
1824 |
+ winding_number (p +++ linepath (a - d) (a - kde)) z |
|
1825 |
+ winding_number (linepath (a - kde) (a + kde)) z" |
|
1826 |
using \<open>path p\<close> znotin assms |
|
1827 |
by simp (metis Un_iff Un_closed_segment a_diff_kde' path_image_linepath path_linepath pathstart_linepath winding_number_join znotin_d_kde) |
|
71189
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1828 |
also have "... = winding_number ?q z + winding_number (linepath (a - kde) (a + kde)) z" |
72506 | 1829 |
using \<open>path p\<close> znotin assms by (simp add: path_image_join winding_number_join znotin_kde_e znotin_d_kde') |
71189
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1830 |
also have "... = winding_number (?q +++ linepath (a - kde) (a + kde)) z" |
72506 | 1831 |
by (metis (mono_tags, lifting) ComplD UnCI \<open>path p\<close> eq inside_outside path_image_join path_join_eq path_linepath pathfinish_join pathfinish_linepath pathstart_join pathstart_linepath pfp psp winding_number_join zzin) |
71189
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1832 |
finally have "winding_number (p +++ linepath (a - d) (a + e)) z = |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1833 |
winding_number (?q +++ linepath (a - kde) (a + kde)) z" . |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1834 |
then show "cmod (winding_number (p +++ linepath (a - d) (a + e)) z) = 1" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1835 |
by (simp add: z1) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1836 |
qed |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1837 |
qed |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1838 |
|
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1839 |
lemma simple_closed_path_wn3: |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1840 |
fixes p :: "real \<Rightarrow> complex" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1841 |
assumes "simple_path p" and loop: "pathfinish p = pathstart p" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1842 |
obtains z where "z \<in> inside (path_image p)" "cmod (winding_number p z) = 1" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1843 |
proof - |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1844 |
have ins: "inside(path_image p) \<noteq> {}" "open(inside(path_image p))" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1845 |
"connected(inside(path_image p))" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1846 |
and out: "outside(path_image p) \<noteq> {}" "open(outside(path_image p))" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1847 |
"connected(outside(path_image p))" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1848 |
and bo: "bounded(inside(path_image p))" "\<not> bounded(outside(path_image p))" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1849 |
and ins_out: "inside(path_image p) \<inter> outside(path_image p) = {}" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1850 |
"inside(path_image p) \<union> outside(path_image p) = - path_image p" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1851 |
and fro: "frontier(inside(path_image p)) = path_image p" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1852 |
"frontier(outside(path_image p)) = path_image p" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1853 |
using Jordan_inside_outside [OF assms] by auto |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1854 |
obtain a where a: "a \<in> inside(path_image p)" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1855 |
using \<open>inside (path_image p) \<noteq> {}\<close> by blast |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1856 |
obtain d::real where "0 < d" and d_fro: "a - d \<in> frontier (inside (path_image p))" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1857 |
and d_int: "\<And>\<epsilon>. \<lbrakk>0 \<le> \<epsilon>; \<epsilon> < d\<rbrakk> \<Longrightarrow> (a - \<epsilon>) \<in> inside (path_image p)" |
72506 | 1858 |
using ray_to_frontier [of "inside (path_image p)" a "-1"] bo ins a interior_eq |
1859 |
by (metis ab_group_add_class.ab_diff_conv_add_uminus of_real_def scale_minus_right zero_neq_neg_one) |
|
71189
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1860 |
obtain e::real where "0 < e" and e_fro: "a + e \<in> frontier (inside (path_image p))" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1861 |
and e_int: "\<And>\<epsilon>. \<lbrakk>0 \<le> \<epsilon>; \<epsilon> < e\<rbrakk> \<Longrightarrow> (a + \<epsilon>) \<in> inside (path_image p)" |
72506 | 1862 |
using ray_to_frontier [of "inside (path_image p)" a 1] bo ins a interior_eq |
1863 |
by (metis of_real_def zero_neq_one) |
|
71189
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1864 |
obtain t0 where "0 \<le> t0" "t0 \<le> 1" and pt: "p t0 = a - d" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1865 |
using a d_fro fro by (auto simp: path_image_def) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1866 |
obtain q where "simple_path q" and q_ends: "pathstart q = a - d" "pathfinish q = a - d" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1867 |
and q_eq_p: "path_image q = path_image p" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1868 |
and wn_q_eq_wn_p: "\<And>z. z \<in> inside(path_image p) \<Longrightarrow> winding_number q z = winding_number p z" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1869 |
proof |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1870 |
show "simple_path (shiftpath t0 p)" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1871 |
by (simp add: pathstart_shiftpath pathfinish_shiftpath |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1872 |
simple_path_shiftpath path_image_shiftpath \<open>0 \<le> t0\<close> \<open>t0 \<le> 1\<close> assms) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1873 |
show "pathstart (shiftpath t0 p) = a - d" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1874 |
using pt by (simp add: \<open>t0 \<le> 1\<close> pathstart_shiftpath) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1875 |
show "pathfinish (shiftpath t0 p) = a - d" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1876 |
by (simp add: \<open>0 \<le> t0\<close> loop pathfinish_shiftpath pt) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1877 |
show "path_image (shiftpath t0 p) = path_image p" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1878 |
by (simp add: \<open>0 \<le> t0\<close> \<open>t0 \<le> 1\<close> loop path_image_shiftpath) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1879 |
show "winding_number (shiftpath t0 p) z = winding_number p z" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1880 |
if "z \<in> inside (path_image p)" for z |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1881 |
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 |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1882 |
loop simple_path_imp_path that winding_number_shiftpath) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1883 |
qed |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1884 |
have ad_not_ae: "a - d \<noteq> a + e" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1885 |
by (metis \<open>0 < d\<close> \<open>0 < e\<close> add.left_inverse add_left_cancel add_uminus_conv_diff |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1886 |
le_add_same_cancel2 less_eq_real_def not_less of_real_add of_real_def of_real_eq_0_iff pt) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1887 |
have ad_ae_q: "{a - d, a + e} \<subseteq> path_image q" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1888 |
using \<open>path_image q = path_image p\<close> d_fro e_fro fro(1) by auto |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1889 |
have ada: "open_segment (a - d) a \<subseteq> inside (path_image p)" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1890 |
proof (clarsimp simp: in_segment) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1891 |
fix u::real assume "0 < u" "u < 1" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1892 |
with d_int have "a - (1 - u) * d \<in> inside (path_image p)" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1893 |
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) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1894 |
then show "(1 - u) *\<^sub>R (a - d) + u *\<^sub>R a \<in> inside (path_image p)" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1895 |
by (simp add: diff_add_eq of_real_def real_vector.scale_right_diff_distrib) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1896 |
qed |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1897 |
have aae: "open_segment a (a + e) \<subseteq> inside (path_image p)" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1898 |
proof (clarsimp simp: in_segment) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1899 |
fix u::real assume "0 < u" "u < 1" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1900 |
with e_int have "a + u * e \<in> inside (path_image p)" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1901 |
by (meson \<open>0 < e\<close> less_eq_real_def mult_less_cancel_right2 not_less zero_less_mult_iff) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1902 |
then show "(1 - u) *\<^sub>R a + u *\<^sub>R (a + e) \<in> inside (path_image p)" |
72506 | 1903 |
by (metis (mono_tags, lifting) add.assoc of_real_mult pth_6 scaleR_collapse scaleR_conv_of_real) |
71189
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1904 |
qed |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1905 |
have "complex_of_real (d * d + (e * e + d * (e + e))) \<noteq> 0" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1906 |
using ad_not_ae |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1907 |
by (metis \<open>0 < d\<close> \<open>0 < e\<close> add_strict_left_mono less_add_same_cancel1 not_sum_squares_lt_zero |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1908 |
of_real_eq_0_iff zero_less_double_add_iff_zero_less_single_add zero_less_mult_iff) |
72506 | 1909 |
moreover have "\<exists>u>0. u < 1 \<and> d = u * d + u * e" |
1910 |
using \<open>0 < d\<close> \<open>0 < e\<close> |
|
1911 |
by (rule_tac x="d / (d+e)" in exI) (simp add: divide_simps scaleR_conv_of_real flip: distrib_left) |
|
1912 |
ultimately have a_in_de: "a \<in> open_segment (a - d) (a + e)" |
|
1913 |
using ad_not_ae by (simp add: in_segment algebra_simps scaleR_conv_of_real flip: of_real_add of_real_mult) |
|
71189
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1914 |
then have "open_segment (a - d) (a + e) \<subseteq> inside (path_image p)" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1915 |
using ada a aae Un_open_segment [of a "a-d" "a+e"] by blast |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1916 |
then have "path_image q \<inter> open_segment (a - d) (a + e) = {}" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1917 |
using inside_no_overlap by (fastforce simp: q_eq_p) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1918 |
with ad_ae_q have paq_Int_cs: "path_image q \<inter> closed_segment (a - d) (a + e) = {a - d, a + e}" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1919 |
by (simp add: closed_segment_eq_open) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1920 |
obtain t where "0 \<le> t" "t \<le> 1" and qt: "q t = a + e" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1921 |
using a e_fro fro ad_ae_q by (auto simp: path_defs) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1922 |
then have "t \<noteq> 0" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1923 |
by (metis ad_not_ae pathstart_def q_ends(1)) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1924 |
then have "t \<noteq> 1" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1925 |
by (metis ad_not_ae pathfinish_def q_ends(2) qt) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1926 |
have q01: "q 0 = a - d" "q 1 = a - d" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1927 |
using q_ends by (auto simp: pathstart_def pathfinish_def) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1928 |
obtain z where zin: "z \<in> inside (path_image (subpath t 0 q +++ linepath (a - d) (a + e)))" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1929 |
and z1: "cmod (winding_number (subpath t 0 q +++ linepath (a - d) (a + e)) z) = 1" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1930 |
proof (rule simple_closed_path_wn2 [of d e "subpath t 0 q" a], simp_all add: q01) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1931 |
show "simple_path (subpath t 0 q +++ linepath (a - d) (a + e))" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1932 |
proof (rule simple_path_join_loop, simp_all add: qt q01) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1933 |
have "inj_on q (closed_segment t 0)" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1934 |
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> |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1935 |
by (fastforce simp: simple_path_def inj_on_def closed_segment_eq_real_ivl) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1936 |
then show "arc (subpath t 0 q)" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1937 |
using \<open>0 \<le> t\<close> \<open>simple_path q\<close> \<open>t \<le> 1\<close> \<open>t \<noteq> 0\<close> |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1938 |
by (simp add: arc_subpath_eq simple_path_imp_path) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1939 |
show "arc (linepath (a - d) (a + e))" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1940 |
by (simp add: ad_not_ae) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1941 |
show "path_image (subpath t 0 q) \<inter> closed_segment (a - d) (a + e) \<subseteq> {a + e, a - d}" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1942 |
using qt paq_Int_cs \<open>simple_path q\<close> \<open>0 \<le> t\<close> \<open>t \<le> 1\<close> |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1943 |
by (force simp: dest: rev_subsetD [OF _ path_image_subpath_subset] intro: simple_path_imp_path) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1944 |
qed |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1945 |
qed (auto simp: \<open>0 < d\<close> \<open>0 < e\<close> qt) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1946 |
have pa01_Un: "path_image (subpath 0 t q) \<union> path_image (subpath 1 t q) = path_image q" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1947 |
unfolding path_image_subpath |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1948 |
using \<open>0 \<le> t\<close> \<open>t \<le> 1\<close> by (force simp: path_image_def image_iff) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1949 |
with paq_Int_cs have pa_01q: |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1950 |
"(path_image (subpath 0 t q) \<union> path_image (subpath 1 t q)) \<inter> closed_segment (a - d) (a + e) = {a - d, a + e}" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1951 |
by metis |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1952 |
have z_notin_ed: "z \<notin> closed_segment (a + e) (a - d)" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1953 |
using zin q01 by (simp add: path_image_join closed_segment_commute inside_def) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1954 |
have z_notin_0t: "z \<notin> path_image (subpath 0 t q)" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1955 |
by (metis (no_types, hide_lams) IntI Un_upper1 subsetD empty_iff inside_no_overlap path_image_join |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1956 |
path_image_subpath_commute pathfinish_subpath pathstart_def pathstart_linepath q_ends(1) qt subpath_trivial zin) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1957 |
have [simp]: "- winding_number (subpath t 0 q) z = winding_number (subpath 0 t q) z" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1958 |
by (metis \<open>0 \<le> t\<close> \<open>simple_path q\<close> \<open>t \<le> 1\<close> atLeastAtMost_iff zero_le_one |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1959 |
path_image_subpath_commute path_subpath real_eq_0_iff_le_ge_0 |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1960 |
reversepath_subpath simple_path_imp_path winding_number_reversepath z_notin_0t) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1961 |
obtain z_in_q: "z \<in> inside(path_image q)" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1962 |
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" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1963 |
proof (rule winding_number_from_innerpath |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1964 |
[of "subpath 0 t q" "a-d" "a+e" "subpath 1 t q" "linepath (a - d) (a + e)" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1965 |
z "- winding_number (subpath t 0 q +++ linepath (a - d) (a + e)) z"], |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1966 |
simp_all add: q01 qt pa01_Un reversepath_subpath) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1967 |
show "simple_path (subpath 0 t q)" "simple_path (subpath 1 t q)" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1968 |
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) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1969 |
show "simple_path (linepath (a - d) (a + e))" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1970 |
using ad_not_ae by blast |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1971 |
show "path_image (subpath 0 t q) \<inter> path_image (subpath 1 t q) = {a - d, a + e}" (is "?lhs = ?rhs") |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1972 |
proof |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1973 |
show "?lhs \<subseteq> ?rhs" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1974 |
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 |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1975 |
by (force simp: pathfinish_def qt simple_path_def path_image_subpath) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1976 |
show "?rhs \<subseteq> ?lhs" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1977 |
using \<open>0 \<le> t\<close> \<open>t \<le> 1\<close> q01 qt by (force simp: path_image_subpath) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1978 |
qed |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1979 |
show "path_image (subpath 0 t q) \<inter> closed_segment (a - d) (a + e) = {a - d, a + e}" (is "?lhs = ?rhs") |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1980 |
proof |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1981 |
show "?lhs \<subseteq> ?rhs" using paq_Int_cs pa01_Un by fastforce |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1982 |
show "?rhs \<subseteq> ?lhs" using \<open>0 \<le> t\<close> \<open>t \<le> 1\<close> q01 qt by (force simp: path_image_subpath) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1983 |
qed |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1984 |
show "path_image (subpath 1 t q) \<inter> closed_segment (a - d) (a + e) = {a - d, a + e}" (is "?lhs = ?rhs") |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1985 |
proof |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1986 |
show "?lhs \<subseteq> ?rhs" by (auto simp: pa_01q [symmetric]) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1987 |
show "?rhs \<subseteq> ?lhs" using \<open>0 \<le> t\<close> \<open>t \<le> 1\<close> q01 qt by (force simp: path_image_subpath) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1988 |
qed |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1989 |
show "closed_segment (a - d) (a + e) \<inter> inside (path_image q) \<noteq> {}" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1990 |
using a a_in_de open_closed_segment pa01_Un q_eq_p by fastforce |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1991 |
show "z \<in> inside (path_image (subpath 0 t q) \<union> closed_segment (a - d) (a + e))" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1992 |
by (metis path_image_join path_image_linepath path_image_subpath_commute pathfinish_subpath pathstart_linepath q01(1) zin) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1993 |
show "winding_number (subpath 0 t q +++ linepath (a + e) (a - d)) z = |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1994 |
- winding_number (subpath t 0 q +++ linepath (a - d) (a + e)) z" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1995 |
using z_notin_ed z_notin_0t \<open>0 \<le> t\<close> \<open>simple_path q\<close> \<open>t \<le> 1\<close> |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1996 |
by (simp add: simple_path_imp_path qt q01 path_image_subpath_commute closed_segment_commute winding_number_join winding_number_reversepath [symmetric]) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1997 |
show "- d \<noteq> e" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1998 |
using ad_not_ae by auto |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
1999 |
show "winding_number (subpath t 0 q +++ linepath (a - d) (a + e)) z \<noteq> 0" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2000 |
using z1 by auto |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2001 |
qed |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2002 |
show ?thesis |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2003 |
proof |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2004 |
show "z \<in> inside (path_image p)" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2005 |
using q_eq_p z_in_q by auto |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2006 |
then have [simp]: "z \<notin> path_image q" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2007 |
by (metis disjoint_iff_not_equal inside_no_overlap q_eq_p) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2008 |
have [simp]: "z \<notin> path_image (subpath 1 t q)" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2009 |
using inside_def pa01_Un z_in_q by fastforce |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2010 |
have "winding_number(subpath 0 t q +++ subpath t 1 q) z = winding_number(subpath 0 1 q) z" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2011 |
using z_notin_0t \<open>0 \<le> t\<close> \<open>simple_path q\<close> \<open>t \<le> 1\<close> |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2012 |
by (simp add: simple_path_imp_path qt path_image_subpath_commute winding_number_join winding_number_subpath_combine) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2013 |
with wn_q have "winding_number (subpath t 0 q +++ linepath (a - d) (a + e)) z = - winding_number q z" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2014 |
by auto |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2015 |
with z1 have "cmod (winding_number q z) = 1" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2016 |
by simp |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2017 |
with z1 wn_q_eq_wn_p show "cmod (winding_number p z) = 1" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2018 |
using z1 wn_q_eq_wn_p by (simp add: \<open>z \<in> inside (path_image p)\<close>) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2019 |
qed |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2020 |
qed |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2021 |
|
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2022 |
proposition simple_closed_path_winding_number_inside: |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2023 |
assumes "simple_path \<gamma>" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2024 |
obtains "\<And>z. z \<in> inside(path_image \<gamma>) \<Longrightarrow> winding_number \<gamma> z = 1" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2025 |
| "\<And>z. z \<in> inside(path_image \<gamma>) \<Longrightarrow> winding_number \<gamma> z = -1" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2026 |
proof (cases "pathfinish \<gamma> = pathstart \<gamma>") |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2027 |
case True |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2028 |
have "path \<gamma>" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2029 |
by (simp add: assms simple_path_imp_path) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2030 |
then have const: "winding_number \<gamma> constant_on inside(path_image \<gamma>)" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2031 |
proof (rule winding_number_constant) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2032 |
show "connected (inside(path_image \<gamma>))" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2033 |
by (simp add: Jordan_inside_outside True assms) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2034 |
qed (use inside_no_overlap True in auto) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2035 |
obtain z where zin: "z \<in> inside (path_image \<gamma>)" and z1: "cmod (winding_number \<gamma> z) = 1" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2036 |
using simple_closed_path_wn3 [of \<gamma>] True assms by blast |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2037 |
have "winding_number \<gamma> z \<in> \<int>" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2038 |
using zin integer_winding_number [OF \<open>path \<gamma>\<close> True] inside_def by blast |
72506 | 2039 |
moreover have "real_of_int x = - 1 \<longleftrightarrow> x = -1" for x |
2040 |
by linarith |
|
2041 |
ultimately consider "winding_number \<gamma> z = 1" | "winding_number \<gamma> z = -1" |
|
2042 |
using z1 by (auto simp: Ints_def abs_if split: if_split_asm) |
|
71189
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2043 |
with that const zin show ?thesis |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2044 |
unfolding constant_on_def by metis |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2045 |
next |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2046 |
case False |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2047 |
then show ?thesis |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2048 |
using inside_simple_curve_imp_closed assms that(2) by blast |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2049 |
qed |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2050 |
|
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2051 |
lemma simple_closed_path_abs_winding_number_inside: |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2052 |
assumes "simple_path \<gamma>" "z \<in> inside(path_image \<gamma>)" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2053 |
shows "\<bar>Re (winding_number \<gamma> z)\<bar> = 1" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2054 |
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)) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2055 |
|
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2056 |
lemma simple_closed_path_norm_winding_number_inside: |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2057 |
assumes "simple_path \<gamma>" "z \<in> inside(path_image \<gamma>)" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2058 |
shows "norm (winding_number \<gamma> z) = 1" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2059 |
proof - |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2060 |
have "pathfinish \<gamma> = pathstart \<gamma>" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2061 |
using assms inside_simple_curve_imp_closed by blast |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2062 |
with assms integer_winding_number have "winding_number \<gamma> z \<in> \<int>" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2063 |
by (simp add: inside_def simple_path_def) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2064 |
then show ?thesis |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2065 |
by (metis assms norm_minus_cancel norm_one simple_closed_path_winding_number_inside) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2066 |
qed |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2067 |
|
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2068 |
lemma simple_closed_path_winding_number_cases: |
72506 | 2069 |
assumes "simple_path \<gamma>" "pathfinish \<gamma> = pathstart \<gamma>" "z \<notin> path_image \<gamma>" |
2070 |
shows "winding_number \<gamma> z \<in> {-1,0,1}" |
|
2071 |
proof - |
|
2072 |
consider "z \<in> inside (path_image \<gamma>)" | "z \<in> outside (path_image \<gamma>)" |
|
2073 |
by (metis ComplI UnE assms(3) inside_Un_outside) |
|
2074 |
then show ?thesis |
|
2075 |
proof cases |
|
2076 |
case 1 |
|
2077 |
then show ?thesis |
|
2078 |
using assms simple_closed_path_winding_number_inside by auto |
|
2079 |
next |
|
2080 |
case 2 |
|
2081 |
then show ?thesis |
|
2082 |
using assms simple_path_def winding_number_zero_in_outside by blast |
|
2083 |
qed |
|
2084 |
qed |
|
71189
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2085 |
|
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2086 |
lemma simple_closed_path_winding_number_pos: |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2087 |
"\<lbrakk>simple_path \<gamma>; pathfinish \<gamma> = pathstart \<gamma>; z \<notin> path_image \<gamma>; 0 < Re(winding_number \<gamma> z)\<rbrakk> |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2088 |
\<Longrightarrow> winding_number \<gamma> z = 1" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2089 |
using simple_closed_path_winding_number_cases |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2090 |
by fastforce |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2091 |
|
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2092 |
subsection \<open>Winding number for rectangular paths\<close> |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2093 |
|
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2094 |
proposition winding_number_rectpath: |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2095 |
assumes "z \<in> box a1 a3" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2096 |
shows "winding_number (rectpath a1 a3) z = 1" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2097 |
proof - |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2098 |
from assms have less: "Re a1 < Re a3" "Im a1 < Im a3" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2099 |
by (auto simp: in_box_complex_iff) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2100 |
define a2 a4 where "a2 = Complex (Re a3) (Im a1)" and "a4 = Complex (Re a1) (Im a3)" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2101 |
let ?l1 = "linepath a1 a2" and ?l2 = "linepath a2 a3" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2102 |
and ?l3 = "linepath a3 a4" and ?l4 = "linepath a4 a1" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2103 |
from assms and less have "z \<notin> path_image (rectpath a1 a3)" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2104 |
by (auto simp: path_image_rectpath_cbox_minus_box) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2105 |
also have "path_image (rectpath a1 a3) = |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2106 |
path_image ?l1 \<union> path_image ?l2 \<union> path_image ?l3 \<union> path_image ?l4" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2107 |
by (simp add: rectpath_def Let_def path_image_join Un_assoc a2_def a4_def) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2108 |
finally have "z \<notin> \<dots>" . |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2109 |
moreover have "\<forall>l\<in>{?l1,?l2,?l3,?l4}. Re (winding_number l z) > 0" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2110 |
unfolding ball_simps HOL.simp_thms a2_def a4_def |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2111 |
by (intro conjI; (rule winding_number_linepath_pos_lt; |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2112 |
(insert assms, auto simp: a2_def a4_def in_box_complex_iff mult_neg_neg))+) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2113 |
ultimately have "Re (winding_number (rectpath a1 a3) z) > 0" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2114 |
by (simp add: winding_number_join path_image_join rectpath_def Let_def a2_def a4_def) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2115 |
thus "winding_number (rectpath a1 a3) z = 1" using assms less |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2116 |
by (intro simple_closed_path_winding_number_pos simple_path_rectpath) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2117 |
(auto simp: path_image_rectpath_cbox_minus_box) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2118 |
qed |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2119 |
|
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2120 |
proposition winding_number_rectpath_outside: |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2121 |
assumes "Re a1 \<le> Re a3" "Im a1 \<le> Im a3" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2122 |
assumes "z \<notin> cbox a1 a3" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2123 |
shows "winding_number (rectpath a1 a3) z = 0" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2124 |
using assms by (intro winding_number_zero_outside[OF _ _ _ assms(3)] |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2125 |
path_image_rectpath_subset_cbox) simp_all |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2126 |
|
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2127 |
text\<open>A per-function version for continuous logs, a kind of monodromy\<close> |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2128 |
proposition\<^marker>\<open>tag unimportant\<close> winding_number_compose_exp: |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2129 |
assumes "path p" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2130 |
shows "winding_number (exp \<circ> p) 0 = (pathfinish p - pathstart p) / (2 * of_real pi * \<i>)" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2131 |
proof - |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2132 |
obtain e where "0 < e" and e: "\<And>t. t \<in> {0..1} \<Longrightarrow> e \<le> norm(exp(p t))" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2133 |
proof |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2134 |
have "closed (path_image (exp \<circ> p))" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2135 |
by (simp add: assms closed_path_image holomorphic_on_exp holomorphic_on_imp_continuous_on path_continuous_image) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2136 |
then show "0 < setdist {0} (path_image (exp \<circ> p))" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2137 |
by (metis exp_not_eq_zero imageE image_comp infdist_eq_setdist infdist_pos_not_in_closed path_defs(4) path_image_nonempty) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2138 |
next |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2139 |
fix t::real |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2140 |
assume "t \<in> {0..1}" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2141 |
have "setdist {0} (path_image (exp \<circ> p)) \<le> dist 0 (exp (p t))" |
72506 | 2142 |
proof (rule setdist_le_dist) |
2143 |
show "exp (p t) \<in> path_image (exp \<circ> p)" |
|
2144 |
using \<open>t \<in> {0..1}\<close> path_image_def by fastforce+ |
|
2145 |
qed auto |
|
71189
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2146 |
then show "setdist {0} (path_image (exp \<circ> p)) \<le> cmod (exp (p t))" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2147 |
by simp |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2148 |
qed |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2149 |
have "bounded (path_image p)" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2150 |
by (simp add: assms bounded_path_image) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2151 |
then obtain B where "0 < B" and B: "path_image p \<subseteq> cball 0 B" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2152 |
by (meson bounded_pos mem_cball_0 subsetI) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2153 |
let ?B = "cball (0::complex) (B+1)" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2154 |
have "uniformly_continuous_on ?B exp" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2155 |
using holomorphic_on_exp holomorphic_on_imp_continuous_on |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2156 |
by (force intro: compact_uniformly_continuous) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2157 |
then obtain d where "d > 0" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2158 |
and d: "\<And>x x'. \<lbrakk>x\<in>?B; x'\<in>?B; dist x' x < d\<rbrakk> \<Longrightarrow> norm (exp x' - exp x) < e" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2159 |
using \<open>e > 0\<close> by (auto simp: uniformly_continuous_on_def dist_norm) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2160 |
then have "min 1 d > 0" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2161 |
by force |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2162 |
then obtain g where pfg: "polynomial_function g" and "g 0 = p 0" "g 1 = p 1" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2163 |
and gless: "\<And>t. t \<in> {0..1} \<Longrightarrow> norm(g t - p t) < min 1 d" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2164 |
using path_approx_polynomial_function [OF \<open>path p\<close>] \<open>d > 0\<close> \<open>0 < e\<close> |
72379 | 2165 |
unfolding pathfinish_def pathstart_def by blast |
71189
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2166 |
have "winding_number (exp \<circ> p) 0 = winding_number (exp \<circ> g) 0" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2167 |
proof (rule winding_number_nearby_paths_eq [symmetric]) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2168 |
show "path (exp \<circ> p)" "path (exp \<circ> g)" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2169 |
by (simp_all add: pfg assms holomorphic_on_exp holomorphic_on_imp_continuous_on path_continuous_image path_polynomial_function) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2170 |
next |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2171 |
fix t :: "real" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2172 |
assume t: "t \<in> {0..1}" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2173 |
with gless have "norm(g t - p t) < 1" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2174 |
using min_less_iff_conj by blast |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2175 |
moreover have ptB: "norm (p t) \<le> B" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2176 |
using B t by (force simp: path_image_def) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2177 |
ultimately have "cmod (g t) \<le> B + 1" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2178 |
by (meson add_mono_thms_linordered_field(4) le_less_trans less_imp_le norm_triangle_sub) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2179 |
with ptB gless t have "cmod ((exp \<circ> g) t - (exp \<circ> p) t) < e" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2180 |
by (auto simp: dist_norm d) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2181 |
with e t show "cmod ((exp \<circ> g) t - (exp \<circ> p) t) < cmod ((exp \<circ> p) t - 0)" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2182 |
by fastforce |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2183 |
qed (use \<open>g 0 = p 0\<close> \<open>g 1 = p 1\<close> in \<open>auto simp: pathfinish_def pathstart_def\<close>) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2184 |
also have "... = 1 / (of_real (2 * pi) * \<i>) * contour_integral (exp \<circ> g) (\<lambda>w. 1 / (w - 0))" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2185 |
proof (rule winding_number_valid_path) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2186 |
have "continuous_on (path_image g) (deriv exp)" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2187 |
by (metis DERIV_exp DERIV_imp_deriv continuous_on_cong holomorphic_on_exp holomorphic_on_imp_continuous_on) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2188 |
then show "valid_path (exp \<circ> g)" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2189 |
by (simp add: field_differentiable_within_exp pfg valid_path_compose valid_path_polynomial_function) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2190 |
show "0 \<notin> path_image (exp \<circ> g)" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2191 |
by (auto simp: path_image_def) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2192 |
qed |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2193 |
also have "... = 1 / (of_real (2 * pi) * \<i>) * integral {0..1} (\<lambda>x. vector_derivative g (at x))" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2194 |
proof (simp add: contour_integral_integral, rule integral_cong) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2195 |
fix t :: "real" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2196 |
assume t: "t \<in> {0..1}" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2197 |
show "vector_derivative (exp \<circ> g) (at t) / exp (g t) = vector_derivative g (at t)" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2198 |
proof - |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2199 |
have "(exp \<circ> g has_vector_derivative vector_derivative (exp \<circ> g) (at t)) (at t)" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2200 |
by (meson DERIV_exp differentiable_def field_vector_diff_chain_at has_vector_derivative_def |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2201 |
has_vector_derivative_polynomial_function pfg vector_derivative_works) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2202 |
moreover have "(exp \<circ> g has_vector_derivative vector_derivative g (at t) * exp (g t)) (at t)" |
72506 | 2203 |
by (metis DERIV_exp field_vector_diff_chain_at has_vector_derivative_polynomial_function pfg vector_derivative_at) |
71189
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2204 |
ultimately show ?thesis |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2205 |
by (simp add: divide_simps, rule vector_derivative_unique_at) |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2206 |
qed |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2207 |
qed |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2208 |
also have "... = (pathfinish p - pathstart p) / (2 * of_real pi * \<i>)" |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2209 |
proof - |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2210 |
have "((\<lambda>x. vector_derivative g (at x)) has_integral g 1 - g 0) {0..1}" |
72506 | 2211 |
by (meson differentiable_at_polynomial_function fundamental_theorem_of_calculus |
2212 |
has_vector_derivative_at_within pfg vector_derivative_works zero_le_one) |
|
71189
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2213 |
then show ?thesis |
72379 | 2214 |
unfolding pathfinish_def pathstart_def |
71189
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2215 |
using \<open>g 0 = p 0\<close> \<open>g 1 = p 1\<close> by auto |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2216 |
qed |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2217 |
finally show ?thesis . |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2218 |
qed |
954ee5acaae0
Split off new HOL-Complex_Analysis session from HOL-Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff
changeset
|
2219 |
|
71201
6617fb368a06
Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
71189
diff
changeset
|
2220 |
end |