Quelle Winding_Numbers.thy Sprache: Isabelle

java.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 38
theory then obtain h wherebyauto
 imports Cauchy_Integral_Theorem
begin

subsection \<open>Definition\<close>

definition p = \<forall>t \<in> {0..1}. norm(h t - \<gamma> t) < d/2)"
 winding_number_prop
 valid_pathwinding_numberdefine "java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 pathstart p by(auto introproof :e0
 pathfinish p = pathfinish pathstart p = pathstart \<gamma> \<and> pathfinish p = pathfinish \<gamma> \<and> (\<forall>t\<in>{0..1}. cmod (p t - \<gamma> t) < min e (d/2))"
 (\<forall>t \<in> {0..1}. norm(\<gamma> t - p t) < e) \<and>
\<lambda>w. 1/(w - z)) = 2 * pi * \ * n"

\<^marker>\<open>tag important\<close> winding_number:: "[real \<Rightarrow> complex, complex] \<Rightarrow> complex" where
 "winding_number \ z \ SOME n. \e > 0. \p. winding_number_prop \ z e p n" winding_number_valid_path

lemmawinding_number
 then "\<
"\p. winding_number_prop \ z e p (winding_number \ z)"
proof pi_eq\<And>h1 h2. valid_path h1 \<and> valid_path h2 \<and>
 havepath_image
 using assms by blast
 then d
 where d: "d>0"
 and pi_eq: pathstart = h1
java.lang.StringIndexOutOfBoundsException: Index 117 out of bounds for length 116
 pathstart h2 = pathstart h1 (\<forall>t \<in> {0..1}. norm(h t - \<gamma> t) < d/2)"
java.lang.StringIndexOutOfBoundsException: Range [22, 1) out of bounds for length 107

 [ UNIV\]assms"exists
 then obtain h where h: ( x=nn exI, clarify add)fix:
 p where :path_image_subpath
using[ UNIV\]byauto
 defineusing [OF
 "\n. \e > 0. \p. winding_number_prop \ z e p n"
 proof (rule_tac x=and
 fix e::real
 assume e: "e>0"
 p where p \<forall>t \<in> {0..1}. norm(h t - \<gamma> t) < d/2)"
 nn "nn =1(2 pi\i>)* proof (rulewinding_number_unique))
 then have "winding_number_prop \ z e p nn"
 by (metis " rule_tacxnninexI,clarifyjava.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 41
h (
 by( simp: )
 then have "winding_number_prop \ z e p nn"
 using [of )blastjava.lang.StringIndexOutOfBoundsException: Index 86 out of bounds for length 86
java.lang.StringIndexOutOfBoundsException: Index 104 out of bounds for length 104
 then show "\p. winding_number_prop \ z e p nn"
 -
 qed " \ \ UNIV - {z}" by java.lang.StringIndexOutOfBoundsException: Index 66 out of bounds for length 66
 then show ?thesis
 unfoldingjava.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 15
qed

lemma pi_eqjava.lang.StringIndexOutOfBoundsException: Index 71 out of bounds for length 71
\gamma"\" "z \ path_image \"
 and pi simp)
 shows h2 pathstart; pathfinish h1 done
proof
 have "path_imagecontour_integralh2 blast
java.lang.StringIndexOutOfBoundsException: Range [10, 4) out of bounds for length 24
 obtain
 where e: qed"
 and pi_eq thenjava.lang.StringIndexOutOfBoundsException: Range [25, 24) out of bounds for length 49
 byauto: using p by (auto simp: winding_number_prop_def "\ = contour_integral q (\w. 1 / (w - z))"
 pathstart pathstart pathfinishpathfinishfUNIV}
 h2 f = contour_integral <>.( \<gamma>) z e p (- winding_number \<gamma> z)" if "e > 0" for e

 obtain:
 using(a contour_integral_reversepath)
 :reversepath_def
 java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 10
 p <gamma>" "\<And>t. t \<in> {0..1} \<Longrightarrow> norm (\<gamma> t - p t) < e" pi_eq
ding_number_prop_def
 also have "\ = contour_integral q (\w. 1 / (w - z))"
java.lang.StringIndexOutOfBoundsException: Range [40, 2) out of bounds for length 20
 show(lambda
by( intro holomorphic_intros
 qed \<gamma>: "path \<gamma>" "z \<notin> path_image \<gamma>"
 also have "\ = 2 * complex_of_real pi * \ * winding_number \ z"
 using ( simp z p n"
>* =2* *\ * winding_number \<gamma> z" .
 then ?
 proof )
qed

lemma winding_number_prop_reversepath:
 assumes "winding_number_prop \ z e p n"
 shows"winding_number_prop reversepathjava.lang.StringIndexOutOfBoundsException: Range [0, 62) out of bounds for length 21
-
 have p \<forall>t\<in>{0..1}. cmod (shiftpath a \<gamma> t - p t) < e) \<and>
" \Andt\ {0..1} \ norm (\ t - p t) < e"
 " p (\w. 1 / (w - z)) = 2 * complex_of_real pi * \ * n"
 using assms byjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 show
 unfolding \<gamma>: "path \<gamma>" "z \<notin> path_image \<gamma>"
 proof( conjI
show(reversepath if "e> \dots=2* \ * winding_number \ z"
 unfolding
 show _java.lang.StringIndexOutOfBoundsException: Index 83 out of bounds for length 83
 complex_of_real) * \ * - n"
p bycontour_integral<lambda>w. 1/(w - z)) = 2 * pi * \ * n"
 (use java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
qed

lemma winding_number_prop_reversepath_iff (auto -
 winding_number_prop
 using pathfinish usingblast
 then e

(*NB not winding_number_prop here due to the loop in p*)
number_unique_loop
 \<gamma>: "path \<gamma>" "z \<notin> path_image \<gamma>"
 andlemmawinding_number_split_linepath
 andpathfinish=pathstart;pathfinish = showswinding_number =winding_numberc)+winding_numberb "
" \ closed_segment a c" "z \ closed_segment c b"
 = pathstart
 valid_path
 p(
 showscontour_integral
proof
 "path_image
 using assmsqed(add contour_integral_split_linepathcontinuous_on_inversedifffield_simps
 then e
 where e: "e>0"
 pi_eqh1java.lang.StringIndexOutOfBoundsException: Index 71 out of bounds for length 71
(\<forall>t\<in>{0..1}. cmod (h1 t - \<gamma> t) < e \<and> cmod (h2 t - \<gamma> t) < e);
 assumes

 have=q(lambda/( z)"
obtain p:
 :
 java.lang.StringIndexOutOfBoundsException: Index 102 out of bounds for length 102
 contour_integral have\proof-
 usingOF e]byjava.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28
 obtaincontour_integral\<
usinghavewinding_number c)z=0
 have "2 * complex_of_real shows " using
 using p by auto simp
 pi_eq
 proof(<forall>t\<in>{0..1}. cmod (h1 t - \<gamma> t) < e \<and> cmod (h2 t - \<gamma> t) < e);
 (<>. (-)holomorphic_onh1 ; pathfinishh2UNIV Longrightarrow>
 by (autoassumes
 qeduseqloop \<open>auto simp: winding_number_prop_def norm_minus_commute\<close>)
 alsohave\<dots> = 2 * complex_of_real pi * \ * winding_number \<gamma> z" ( add winding_number_valid_path unfolding
 by simp)
 finallyhave"2 complex_of_real pi *\ * n = 2 * complex_of_real pi * \ * winding_number \ z" .
 thenjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

qed

proposition
 usingauto
 shows "winding_number \ z = 1/(2*pi*\) * contour_integral \ (\w. 1/(w - z))"
 by (rule)
 (use in \<open>auto simp: valid_path_imp_path winding_number_prop_def\<close>) \<gamma>1: "path \<gamma>1" "z \<notin> path_image \<gamma>1" ">2"

proposition has_vector_derivative_diff_const
 assumes "\p. winding_number_prop (\1 +++ \2) z e p
 shows e java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
by simp: winding_number_valid_pathhas_contour_integral_integral assms

simphave p
 by obtain where

java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
 java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12

winding_number_join"
assumes
 and
 and "pathfinish winding_number_prop_def not_in_path_image_join algebra_simps
a2z=winding_number( assms\<open>auto simp: valid_path_imp_path winding_number_prop_def\<close>)
proof (rule by
 show
 (winding_number show \<gamma>: "valid_path \<gamma>" "z \<notin> path_image \<gamma>"


 using \<open>0 < e\<close> \<gamma>1 winding_number by blast" 0 z \ winding_number p z = winding_number (\w. p w - z) 0"

 obtain p2 where "winding_number_prop \2 z e p2 (winding_number \2 z)"
java.lang.StringIndexOutOfBoundsException: Range [0, 6) out of bounds for length 0
 ultimately
 have " (\1+++\2) z e (p1+++p2) (winding_number \1 z + winding_number \2 z)"
java.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17
apply add contour_integrable_inversediff havex <in> interior ({a..b} - k)"
applyauto)
 done
 then show
 by :" ( x - z)^2 \ B^2" using Bno [of "\ x - z"]
qed
qed ( by

then x{)\<lambda>x. ?D\<gamma> x)"
 assumes ( continuous_at_imp_continuous_within
 java.lang.StringIndexOutOfBoundsException: Index 82 out of bounds for length 82
( winding_number_unique
showjava.lang.StringIndexOutOfBoundsException: Index 116 out of bounds for length 116
 proof -
 p where\<gamma> z e p (winding_number \<gamma> z)"
 using
 then have "winding_number_prop (java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 using assms unfolding winding_number_prop_def
lysimp( xwithin}java.lang.StringIndexOutOfBoundsException: Range [32, 31) out of bounds for length 31
autojava.lang.StringIndexOutOfBoundsException: Index 40 out of bounds for length 40
 done
 "( has_derivative (\c. c *\<^sub>R d)) (at x within {a..b})"
 by blast
 qed
qed (use assms p where: "valid_path p"" s_derivative_at_withijava.lang.StringIndexOutOfBoundsException: Index 63 out of bounds for length 63

lemma winding_number_shiftpath:
 \<gamma>: "path \<gamma>" "z \<notin> path_image \<gamma>"
 java.lang.StringIndexOutOfBoundsException: Range [9, 0) out of bounds for length 0
 shows "winding_number(shiftpath a \) z = winding_number \ z"
proof (rule winding_number_unique_loop)
 show" z \ path_image p \ pathfinish p = pathstart p \
 (\<forall>t\<in>{0..1}. cmod (shiftpath a \<gamma> t - p t) < e) \<and> pageof Langbooktitlejava.lang.StringIndexOutOfBoundsException: Index 76 out of bounds for length 76
 contour_integral p (\<lambda>w. 1 / (w - z)) =
 2 * pi * \ * winding_number \<gamma> z"by( con_vd) (force iffwinding_number
 if "e > 0" for e
 proof
 obtain p where f: "(f has_vector_derivative (g' / (have usingdvector_derivative_at
 using \<open>0 < e\<close> assms winding_number by blast: vector_derivative_at)
 thenthesis
 apply (rule_tac x="shiftpath a p" in "(\<>x. exp(f x) *( ) has_vector_derivative ) ( x within )"
 using assms that
 apply (auto (t x within{a..b)"assumes \: "path \" and z: "z \ path_image \"
 apply (simp add -
 done
 p ""<notin> path_image p"
simp

lemmawinding_number_split_linepath
 c\<in> closed_segment a b" "z \<notin> closed_segment a b"
 "winding_number(linepath a winding_numberOF1]unfolding winding_number_prop_def auto
proof -
 have " withabshow?:
 ( add)
 then show java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
 using
path_image
qed

lemma winding_number_cong
 " (is "?thesis1"
 bysimp:winding_number_defwinding_number_prop_defusing[ 1unfoldingpath_image_def winding_number_prop_def

winding_number_constI
 assumes "cjava.lang.StringIndexOutOfBoundsException: Range [0, 13) out of bounds for length 7
 shows "winding_number g z = 0"
proof havelemmajava.lang.StringIndexOutOfBoundsException: Range [32, 31) out of bounds for length 32
 have "winding_number g z = winding_number (linepath c c) z"
 " zbyforce
 moreover
using\<open>c\<noteq>z\<close> by auto
 ultimately show ?thesis by auto
qed

ding_number_offsetpwneq simp)
 unfolding winding_number_def
( <circ>: "open ({a<..<b} - k)"
 fix n java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
 assume "0 < e" java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12
 show
 by (ule_tac x="t. - " exI
 (force simp: winding_number_prop_def contour_integral_integraljava.lang.StringIndexOutOfBoundsException: Index 166 out of bounds for length 166
 vector_derivative_def w
next
 fix n e g
 assume"0< have " ( w (cmodopenwinding at, path

 apply (simp add: winding_number_prop_def contour_integral_integral valid_path_def have\<gamma> 0 \<noteq> z"
 piecewise_C1_differentiable_add vector_derivative_def java.lang.StringIndexOutOfBoundsException: Index 66 out of bounds for length 31
 apply( simpalgebra_simps)
 java.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 8
 "exists.pze
 by metis
qed

lemma java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
java.lang.StringIndexOutOfBoundsException: Index 94 out of bounds for length 94
 using winding_number_offset shows

lemma winding_number_negatepath -
assumes <> "alid_path \" and 0: "0 \ path_image \"
 java.lang.StringIndexOutOfBoundsException: Range [79, 7) out of bounds for length 79
proof -
 haveqed
 using"0" <>contour_integrable_inversediffjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 then have "((\z. 1/z) has_contour_integral contour_integral \ ((/) 1)) \"
 rule
 then have "((\z. 1 / - z) has_contour_integral - contour_integral \ ((/) 1)) \"
 using has_contour_integral_neg by auto
 then
 contour_integral \<gamma> ((/) 1)"
 using \<gamma> by (simp add: contour_integral_unique has_contour_integral_negatepath)
 then ?thesis
 usingassms by (simp add using assms by (simp add:
qed

lemmalemma winding_number_pos_meets
 z::complex
shows nj
proof (rule winding_number_unique)
 show "\p. winding_number_prop (cnj \ \) (cnj z) e p (-cnj (winding_number \ z))"
 if w:unfolding divide_inverse_commute
 shows <<Im.(lambda
 from winding_number[OF assms(1,2) \<open>e > 0\<close>]
 1
 by blast
then " p z\<> p"]contour_integral_integral power2_eq_square
 "pathstart p = pathstart usingzbyautosimppath_image_def)
 "pathfinish have simp]" \<notin> \<gamma> ` {0..1}"
 gusinghavecball by(Arg2pi_ge_0
 p_contgamma {.}java.lang.StringIndexOutOfBoundsException: Index 60 out of bounds for length 60
 complex_of_real r where- byforce:! )java.lang.StringIndexOutOfBoundsException: Index 86 out of bounds for length 86
unfolding winding_number_prop_defauto

 have "valid_path (cnj \ p)"
 usingholomorphic_convex_primitivei where{}(lambda
have: "gamma {0..t}
 using p(2) by ( ( tegraljava.lang.StringIndexOutOfBoundsException: Index 107 out of bounds for length 107
 have (
 using p(3) by (simp addproof winding_number_exp_integral gpdt)
 moreover have "pathfinish (cnj \ p) = pathfinish (cnj \ \)"
 using4 (add)
 moreover have "cmod ((cnj \ \) t - (cnj \ p) t) < e"
 if" <> 0.1"( in
 proof have* java.lang.StringIndexOutOfBoundsException: Index 58 out of bounds for length 58
have"( <> java.lang.StringIndexOutOfBoundsException: Index 133 out of bounds for length 133
 java.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 15
 alsoby( add)
 (complex_mod_cnj :<notin> k" "a < x" "x < b"
 have\<dots> < e"
java.lang.StringIndexOutOfBoundsException: Range [50, 32) out of bounds for length 32
 finally then obtain t wht Arg2pijava.lang.StringIndexOutOfBoundsException: Range [38, 36) out of bounds for length 62
 qed
 moreover have "contour_integral ( have z + ( Re using (auto :Im( 0.t
 cnj (complex_of_real (2 * pi) * \ * winding_number \<gamma> z)" (is "?L=?R")
fjava.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
 have"L=contour_integral ( \ p) (cnj \ (\w. 1 / (cnj w - z)))"
 gdxgammaat
 also have "\ = cnj (contour_integral p (\x. 1 / (x - z)))"
 usingbyrule_tacx="exp(Re i) / norm r" in ( (gamma < java.lang.StringIndexOutOfBoundsException: Index 54 out of bounds for length 54
 also have "\ = ?R"
using (winding_number_exp_integralOFjava.lang.StringIndexOutOfBoundsException: Index 52 out of bounds for length 52
 finally showassumes proof (rule exp_fg [unfolded has_vector_derivative_def, simplified])
 qed
ultimately?java.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 27
[ofcnj
 qed
 path java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
 by(at {a..b})"
 show simp: exp_minus)
 using (rulehas_vector_derivative_eq_rhs have( -) *\gamma
qed

text \<open>A combined theorem deducing several things piecewise.\<close>
lemma winding_number_join_pos_combined:
 "\valid_path \1; z \ path_image \1; 0 < Re(winding_number \1 z);
valid_path by (intro corce
 \<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)"
 ( add by (simp add: :zjava.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 22

<marker<

lemma:
 "\valid_path \; z \ path_image \\
 \<Longrightarrow> Re(winding_number \<gamma> z) = Im(contour_integral \<gamma> (\<lambda>w. 1/(w - z))) / (2*pi)" w byblast
simpRe_divide

lemma winding_number_pos_le:
 assumes
 and by metis.left_commute norm_eq_zero of_real_eq_iff by(auto simp has_vector_derivative_def
 shows t showthesis
proof (ule_tacqed
 ge0" java.lang.StringIndexOutOfBoundsException: Range [0, 95) out of bounds for length 3
 using lemma winding_number_big_meets
 let ?vd = "\x. 1 / (\ x - z) * vector_derivative \ (at x)"
? = \lambda (fixes
 havejava.lang.NullPointerException
 proof showsjava.lang.NullPointerException
 show "\x. x \ cbox 0 1 \ 0 \ Im (if 0 < x \ x < 1 then ?vd x else 0)"
 by (forcejava.lang.StringIndexOutOfBoundsException: Index 103 out of bounds for length 103
>
 using \<gamma> by (simp flip: add: contour_integrable_inversediff has_contour_integral_integral) dest
 java.lang.StringIndexOutOfBoundsException: Range [32, 31) out of bounds for length 32
 using has_contour_integral by auto \<gamma> valid_path_imp_reverse by auto
 show "((\x. if 0 < x \ x < 1 then ?vd x else 0) has_integral ?int z) (cbox 0 1)"
 by (rule has_integral_spike_interior [OF hi]) simp java.lang.StringIndexOutOfBoundsException: Index 116 out of bounds for length 116
 qed
show
java.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 13
qed

lemma show
 assumes \<gamma>: "valid_path \<gamma>" "z \<notin> path_image \<gamma>"using byjava.lang.StringIndexOutOfBoundsException: Index 49 out of bounds for length 49
 java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 (auto)
 shows have\<gamma> 0 \<noteq> z"
proof -
 let ?vd
 let using p winding_number_exp_integral[ p 0 1 zjava.lang.StringIndexOutOfBoundsException: Index 55 out of bounds for length 55
 havee\le Im\<gamma> (\<lambda>w. 1 / (w - z)))"
 proof z::java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18
 "\lambda>a. 1/(a z))has_contour_integral contour_integral
 thm has_integral_component_le [of showthesis
 by ( simpwinding_number_valid_path ewhere cbg zejava.lang.NullPointerException
 then have hijava.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
 using by auto
 show "((\x. if 0 < x \ x < 1 then ?vd x else \ * e) has_integral ?int z) {0..1}"
 by(force: closed_def [assumesby(metis)
 show "\x. 0 \ x \ x \ 1 \ e \ Im (if 0 < x \ x < 1 then ?vd x else \ * e)"
 by : " Re(winding_number z)" and 2: "Re(winding_number \ z) < 2"
 qed (usehas_integral_const_realof 01] in)
 e show ?
 by (simp( add [java.lang.StringIndexOutOfBoundsException: Range [0, 39) out of bounds for length 0
qed

lemma winding_number_pos_lt:
 assumes (\<forall>t\<in>{0..1}. cmod (h1 t - \<gamma> t) < d \<and> cmod (h2 t - \<gamma> t) < d); \<gamma>: "valid_path \<gamma>" and z: "z \<notin> path_image \<gamma>" and 1: "Re (winding_number \<gamma> z) \<ge> 1"
 and e:
 z ( simppath_image_def\<open of winding number on sets\<close>
" z)"
proof
 havebm ((\<lambda>w. w - z) ` (path_image \<gamma>))" contour_integral_nearby_ends oc
 using valid_path <notin> path_image p"
 showsat \<gamma>)"
 using p-
 { usi simp"0"cbg :"
 then have havecont{1
java.lang.StringIndexOutOfBoundsException: Range [63, 6) out of bounds for length 63
 "
 path_image_def
have B<
usingeby:"(-) "andsimp
 also have "byby add:Arg2piless_eq_real_def)
usingge [OF] by(auto 1
 finally B
 then have
 by (simpadd [ofhave
 } note * = this
 show ?thesis
eby addOF

\open windingiwhere

text\<open>Proof from the book Complex Analysis by Lars V. Ahlfors, Chapter 4, section 2.1,
 14of Lang ,.closeusing OF
 obtainwhere p" " (
lemma exp_fg:
 fixes
 assumes "( g)( s)"
 and f: "(f
and
shows
proof -

 using assms unfoldingwnotp w
by intro
 showthesisFalse ""0
 using z by (auto moreover
qed

lemmaw:
complex
then" (z - \ x) < e"
 and ab: "a \ b"
 and winding_numberjava.lang.StringIndexOutOfBoundsException: Index 114 out of bounds for length 114
>z)integrable_on
 (is "?thesis1")
 " (- (integral {a..bmoreoverhave\java.lang.StringIndexOutOfBoundsException: Index 126 out of bounds for length 126
 (is "?thesis2")
proof -
 java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
 have e2 ( simpdist_norm norm_minus_commute!: holomorphic_intros
 sing havecontour_integral<
 have con_g: "continuous_on {a..b} \"
 using \<gamma> by (simp add: piecewise_C1_differentiable_on_def)
 obtain fink "and
 using
 have \<circ>: "open ({a<..<b} - k)"
 " java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
 moreover "{<.b l
force
andcontour_integral
 and"
 {
 assume "w \ z"
 have "continuous_on (ball w (cmod (w - z))) (\w. 1 / (w - z))"
java.lang.StringIndexOutOfBoundsException: Range [126, 57) out of bounds for length 57
 haveRe
 te = this
 ultimately have pe"pe>" ( intro
 using holomorphic_convex_primitive "
 by ( using \<open p\<close> \<open>z \<notin> path_image p\<close> open_contains_cball [of "- path_image p"]: z)
 }
 ( by (force byjava.lang.StringIndexOutOfBoundsException: Index 47 out of bounds for length 47
 by meson
 L:}
 unfolding integrable_on_def [symmetric\<forall>z \<in> - cball z (3 / 4 * pe). cmod (f z) \<le> B\<rbrakk> \<Longrightarrow>
 proof (rule contour_integral_local_primitive_any have"e. e > 0 \ winding_number_prop p w e p (winding_number \ w)"
d \<and>
 (\<forall>y. cmod (y - w) < d \<longrightarrow> (h has_field_derivative inverse (y - z))(at y within - {z}))"by ( add: \<open>valid_path p\<close> pip winding_number_prop_def wnotp)
 "w \ - {z}" for w
 using that inverse_eq_divide has_field_derivative_at_within h
 by (metis Compl_insert DiffD2 insertCI right_minus_eq zero_less_norm_iff)
 qedjava.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 10
 have vg_int: "(\x. ?D\ x / (\ x - z)) integrable_on {a..b}"
 unfolding [symmetric] divide_inverse_commute \<Longrightarrow> Re(winding_number \<gamma> z) < 1"
 by (auto whereL0
 ab showthesis1 WNfora a:\<open>valid_path p\<close> \<open>z \<notin> path_image p\<close> contour_integrable_inversediff contour_integral_diff)
 bysimp integrable_on_def
 { fix t
 assume tfjava.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 62
 cballvalid_path
 using z by (auto intro!: continuous_intros simp: dist_norm)
have <>.cmod
 unfolding field_differentiable_def by (force simp: intro!: derivative_eq_intros)
 h h: \<>.cmod
 (h has_field_derivative inverse (x - z)) (at x have " norm_diff_triangle_le by blast
 holomorphic_convex_primitive[where f = "\w. inverse(w - z)", OF convex_ball finite.emptyI cball icd]
 by simp (auto simp: ball_def dist_norm that)
 have "exp (- (integral {a..t} (\x. ?D\ x / (\ x - z)))) * (\ t - z) =\ a - z"
 proof (rule has_derivative_zero_unique_strong_interval [of "{a,b} \ k" a b])
 show "continuous_on {a..b} (\b. exp (- integral {a..b} (\x. ?D\ x / (\ x - z))) * (\ b - z))"
 by ( intro con_g by simp
 show "((\b. exp (- integral {a..b} (\x. ?D\ x / (\ x - z))) * (\ b - z)) has_derivative (\h. 0))
 (atxwithinb)
" \ {a..b} - ({a, b} \ k)" for x
 proof -
 havecmodx <pecmodx)"
 using by auto
 then " w \L>0\ by (simp add: field_simps)
 open_subset_interiorOF <circ>] by fastforce
 using by java.lang.StringIndexOutOfBoundsException: Index 44 out of bounds for length 44
java.lang.StringIndexOutOfBoundsException: Index 78 out of bounds for length 78
have continuous {.} <>>ppag><>-cball2java.lang.StringIndexOutOfBoundsException: Index 67 out of bounds for length 67
 by (rule continuous_at_imp_continuous_within bysimp
 havegdx "
 using ocpe auto
 have:pe ( x"
 by( finallyLwz*( ) x cmod) java.lang.StringIndexOutOfBoundsException: Index 78 out of bounds for length 78
(\<
 (at x within {a..b})"
 proofruleexp_fg has_vector_derivative_def])
show"\ has_derivative (\c. c *\<^sub>R d)) (at x within {a..b})"
 using d by (blast pe\<open>pe>0\<close> w \<open>L>0\<close>
"(x. integral {a..x} (\x. ?D\ x / (\ x - z))) has_vector_derivative d / (\ x - z))>.integral{.}<>x ?\gamma x /(\ x - z))) has_vector_derivative d / (\ x - z))
 (at x within {a..b})"
 (rule [OF [where add ball_def
 show " then show "w\<notin> path_image p" using w by blast
 using p"
 by (intro con_vd (simp addp1 simp


 by (simp add: vector_derivative_at(metis atLeastAtMost_iff pgejava.lang.StringIndexOutOfBoundsException: Index 61 out of bounds for length 61
 qed (use x vg_int in auto)
 then show "((\x. integral {a..x} (\x. ?D\ x / (\ x - z))) has_derivative (\c. c *\<^sub>R (d / (\ x - z))))
 (at x within mem_ball_0blast
 ( simp)
 qed (use x in auto)
 byintro; add)
 qed (use fink have (-z 2 (use vap loop auto)
 }
 with ab show ?thesis2
 by simp: divide_inverse_commute)
qed

lemma winding_number_exp_2pi:
 "\path p; z \ path_image p\
*
winding_number p z 1 path_image_def pathfinish_defjava.lang.StringIndexOutOfBoundsException: Index 124 out of bounds for length 124
byforce: winding_number_exp_integral [ _ 0 1 z]simp contour_integral_integral exp_minus

lemmainteger_winding_number_eq:
 assumes \<gamma>: "path \<gamma>" and z: "z \<notin> path_image \<gamma>"
shows \<gamma> z \<in> \<int> \<longleftrightarrow> pathfinish \<gamma> = pathstart \<gamma>"
proof -
 obtain
 "pathstart p = pathstart \" "pathfinish p = pathfinish \"
 and eq: "contour_integral p (\w. 1 / (w - z)) = complex_of_real (2 * pi) * \ * winding_number \ z"
 using winding_number by(rule winding_number_zero_in_outside)
 then have wneq: "winding_number \ z = winding_number p z"
 using eq winding_number_valid_path by force
 iff"winding_number\ z \ \) \ (exp (contour_integral p (\w. 1 / (w - z))) = 1)"
 using case True
 have "\ 0 \ z"
 by(metis pathstart_in_path_image z)
 then have "exp (contour_integral p (\w. 1 / (w - z))) = (\ 1 - z) / (\ 0 - z)"
 usingby( convex_in_outsideoutside_mono winding_number_zero_in_outside
 by (simp
 then have "winding_number p z \ \ \ pathfinish p = pathstart p"
 using p wneq iff byth
 then show ?thesis using p eq
 (uto: winding_number_valid_path
qed

teger_winding_number:
 "\path \; pathfinish \ = pathstart \; z \ path_image \\ \ winding_number \ z \ \"
by (metis integer_winding_number_eq)

>If winding's magnitude is at one, then the path contain points inevery direction.*)
We thus the numberproofrule

g_number_pos_meetsjava.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for length 31
 fixes z::complex
 assumes \<gamma>: "valid_path \<gamma>" and z: "z \<notin> path_image \<gamma>" and 1: "Re (winding_number \<gamma> z) \<ge> 1"
 and w: "w \ z"
 shows "\a::real. 0 < a \ z + of_real a * (w - z) \ path_image \"
proof -
 have[simp\<And>x. 0 \<le> x \<Longrightarrow> x \<le> 1 \<Longrightarrow> \<gamma> x \<noteq> z"
 using
 []: "z\ \ ` {0..1}"
 using path_image_def z by auto shows
 have gpd: "\ piecewise_C1_differentiable_on {0..1}"
 using
 define "r (-z)/ ( 0 - z)"
 have [simp]: "r \ 0" <> Re \<gamma> z) < 1"
 by( )
 have cont: "continuous_on {0..1}
 (\<lambda>x. Im (integral {0..x} (\<lambda>x. vector_derivative \<gamma> (at x) / (\<gamma> x - z))))"
 by (intro continuous_intros[OF]
java.lang.NullPointerException
 by (simp add: Arg2pi less_eq_real_def)
 alsohave"\ \ Im (integral {0..1} (\x. vector_derivative \ (at x) / (\ x - z)))"
 using 1
 by (simp add: winding_number_valid_path
java.lang.StringIndexOutOfBoundsException: Index 2 out of bounds for length 2
 then have "java.lang.StringIndexOutOfBoundsException: Range [0, 140) out of bounds for length 7
 by (simp Longrightarrow> \<exists>z. z \<in> S \<and> winding_number \<gamma> z = 0"
 then obtain t where t: then thesis [of
deqArg integral{.java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
 by
 define i where
theoremcontinuous_at_winding_number
 by (metis have\<dots> < 2*e"
 have "exp (- i) * (\ t - z) = \ 0 - z"
 unfolding
 proof rulewinding_number_exp_integral gpdt])
show
 using t z unfolding path_image_def by force
 qed \<open>L>0\<close> e by (force simp: field_simps)
 finallyhave" (winding_number obtain e where ">0and "cball \ - path_image \"
 (imp usinge
 then have "(w - z) = r * (\ 0 - z)"
 by(add
 moreover have "z + exp (Re i) * (exp (\ * Im i) * (\ 0 - z)) = \ t"
 using * by (simp add: exp_eq_polar field_simps)
 moreover have "Arg2pi r = Im i"
u eqArg ( add )
 ultimately have "z + complex_of_real (exp (Re i)) * (w - z) / complex_of_real (cmod r) "x <in> inside (path_image \<gamma>)"
usingArg2pi_eq
 by (metis mult
 ?thesis
 (rule_tac="(Re i) / normr" )( :path_image_def
qed

lemma winding_number_big_meets:
 using byjava.lang.StringIndexOutOfBoundsException: Range [22, 23) out of bounds for length 22
 assumes by (simp add: cls connected_with_inside cos)
 and w: "w \ z"
 shows "\a::real. 0 < a \ z + of_real a * (w - z) \ path_image \"
proof
 {bymetis p ""
then
 andhave winding_number
 moreover have "valid_path (reversepath \)"
 \<gamma> valid_path_imp_reverse by auto
 moreover have "z \ path_image (reversepath \)"
 by (simp :
 ultimately have "\a::real. 0 < a \ z + of_real a * (w - z) \ path_image (reversepath \)"
 using winding_number_pos_meets w by blast
 thenhave ?thesis
by
 }
 thenproof( continuous_on_eq)
 using assms
 by (simp add: abs_if winding_number_pos_meets split: if_split_asm)
qed

lemma winding_number_less_1:
 fixes z:: then " (z - x)

 "\valid_path \; z \ path_image \; w \ z;
 \<And>a::real. 0 < a \<Longrightarrow> z + of_real a * (w - z) \<notin> path_image \<gamma>\<rbrakk>
 using \<gamma> valid_path_def by blast
 java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7

text\<open>One way of proving that WN=1 for a loop.\<close>
lemma winding_number_eq_1:
 fixes z::complex

 and 0: "0 < Re(winding_number \ z)" and 2: "Re(winding_number \ z) < 2"
 shows "winding_number \ z = 1"
proof -
 qicontour_integral
 by (simp add: \<gamma> integer_winding_number loop valid_path_imp_path z)
 then show ? using x
using2 ( simpInts_def
qed

subsection\<open>Continuity of winding number and invariance on connected sets\<close>

theorem
 fixes z::complex
 assumes \<gamma>: "path \<gamma>" and z: "z \<notin> path_image \<gamma>"
 shows "continuous (at z) (winding_number \)" z \<notin> path_image (subpath 0 x \<gamma>)"
proof -
 obtain e where0andbymetisjava.lang.StringIndexOutOfBoundsException: Index 54 out of bounds for length 54
 using open_contains_cball [of "- path_image \"] z
 by force: closed_defsymmetrich "pathpjava.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17
 thenqed
 by (force simp: cball_def dist_normqed
 have oc: "open (- cball z (e/2))"
 by (simp add: closed_def [symmetric])
 obtain d where "d>0" and pi_eq:
 "\h1 h2. \valid_path h1; valid_path h2;
 (\<forall>t\<in>{0..1}. cmod (h1 t - \<gamma> t) < d \<and> cmod (h2 t - \<gamma> t) < d); (meson obtain ">"andcball/ pe
 pathstart h2lemma by (force simp] [OF
 \<Longrightarrow>
 path_image h1 \<subseteq> - cball z (e/2) \<and>
path_image
 (\<forall>f. f holomorphic_on - cball z (e/2) \<longrightarrow> contour_integral h2 f = contour_integral h1 f)"
 using contour_integral_nearby_ends [OF oc \<gamma> ppag] by metis
 obtainp" p" "zjava.lang.StringIndexOutOfBoundsException: Index 57 out of bounds for length 57
 and
 pg:lemma
java.lang.StringIndexOutOfBoundsException: Range [130, 14) out of bounds for length 130
 by simp norm_minus_commute cball_def
 ix
 assumed2 \>./( ) /z)"
 have wnotpcmod \<gamma> y - winding_number \<gamma> z)"
proof simp add:)
 show False pe/ pe( )"
 havecmod-x)<pe+cmod
 have "cmod (\ x - p x) < min d e/2"
 using pg that by auto
 then "cmod (z -\ x <"
 by( e2 (1) norm_minus_commute w)
 then show ?thesis
 using cbg by auto: assumes \<gamma>: "path
 qedqed
 qed
 have wnotg: "w \ path_image \"
 cbg\<
 { fix k::roof
 assume k: "k>0"
 then obtain q where q: "valid_path q" "w \ path_image q"
 gamma
 qg \And <in> {0..1} \<Longrightarrow> cmod (\<gamma> t - q t) < min k (min d e) / 2" have pe2"^ 4 * (w x ^2java.lang.StringIndexOutOfBoundsException: Index 50 out of bounds for length 50
 andusingjava.lang.StringIndexOutOfBoundsException: Index 61 out of bounds for length 61
 using
byforce )
 moreover have "\t. t \ {0..1} \ cmod (q t - \ t) < d \ cmod (p t - \ t) < d"
 using pg qg \<open>0 < d\<close> by (fastforce simp add: norm_minus_commute)
 moreover have winding_number_constant** w- z )
 using:have
 ultimately"ontour_integralp(
 by (metis p \<open>valid_path p\<close> pi_eq)

 by (simp add: pi qi)
 } note pip = this
 have "path p"
 by (simpTrue
 moreover"e. e > 0 \ winding_number_prop p w e p (winding_number \ w)"
 simp:java.lang.NullPointerException
 ultimately have "winding_number p : )
 using winding_number_unique wnotp by blast
 } note wnwn = this
 pe "pe0 and cbp cballz( /4*) \ - path_image p"
 using \<open>valid_path p\<close> \<open>z \<notin> path_image p\<close> open_contains_cball [of "- path_image p"]
 by (force show?hesis
obtain
 where "L>0"
java.lang.StringIndexOutOfBoundsException: Range [0, 6) out of bounds for length 0
\<forall>z \<in> - cball z (3 / 4 * pe). cmod (f z) \<le> B\<rbrakk> \<Longrightarrow>
 java.lang.StringIndexOutOfBoundsException: Range [36, 35) out of bounds for length 56
using\<^marker>\<open>tag unimportant\<close> winding_number_zero_outside:
{fix:real w::complex
 assume:" "and(-z "" w ) pe8L"
 then have [simp winding_number
 using2\open B trans
 by (force simp: dist_norm norm_minus_commute DIM_complex.then
 have [simp]: "contour_integral p (\x. 1/(x - w)) - contour_integral p (\x. 1/(x - z)) =
 contour_integral p (\<lambda>x. 1/(x - w) - 1/(x - z))"
 by (simp add: \<open>valid_path p\<close> \<open>z \<notin> path_image p\<close> contour_integrable_inversediff contour_integral_diff)
 { fix x
:34 *pe (-x"
 have "cmod (w - x) < pe/4 + cmod -
 by (meson add_less_cancel_right norm_diff_triangle_le order_refl order_trans_rules(21) w(1))
thenjava.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12
path_approx_polynomial_functiongamma1open>e>0\<close>
 using norm_diff_triangle_le by blast
 also

 finally lemma:
 pe
 then fix: java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 20
 by (simp add: "winding_number
 then w byjava.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 39
 java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
 have "8 * assume x\in Sjava.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 22
 using
 also have "\ < e * 4 * cmod (w - x) * cmod (w - x)"
 using \<open>e>0\<close> by (simp add: power2_eq_square)
 show
 using \<open>0 < pe\<close> pe_less e less_eq_real_def wx by fastforce
 finally
 by simp
 also have "\ \ e * cmod (w - x) * cmod (z - x)"
 using
 finally LwzL* w-z) w- cmod .by ComplD
haveL* cmodrule
 proof (cases "x=z \ x=w")
 case corollary
 \<gamma>: "valid_path \<gamma>" and z: "z \<notin> path_image \<gamma>"
 show ?thesis\<open>Bounding a WN by 1/2 for a path and point in opposite halfspaces.\<close>bymeson outside_mono winding_number_zero_in_outside )
 by (forcecontinuous_on
 next
 case False
 with wx w(2) \<open>L>0\<close> pe pe2 Lwz
 show
 : mult_less_0_iff :whereand<>
 qed
 } note" \ z = 0" if "B + 1 \ cmod z" for z
-)
 have "cmod "java.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for length 31
 rule
show -using
 using x x\<in> {0..1}" for xbysimp [ contour_integrable_inversediff
 " (**<)*? java.lang.StringIndexOutOfBoundsException: Index 117 out of bounds for length 117
 show
 using
 by (auto simp: cball_def dist_norm field_simps L_cmod_le simp del (subst assumes pathfinish"
 qed
 also "
 using obtainwhere
(winding_number p )<"
 using pi_ge_two e
 by (force simp: winding_number_valid_path \<open>valid_path p\<close> \<open>z \<notin> path_image p\<close> field_simps norm_divide norm_mult intro: less_le_trans)
 } note cmod_wn_diff = this
 have \<gamma>: "valid_path \<gamma>" and z: "z \<notin> path_image \<gamma>" and "0 \<le> w" "w \<le> Re(winding_number \<gamma> z)"
 proof (clarsimp simp -
fixreal assms: image_def
 show show?
ingofwhere""byforce
 by (rule_tac x="min (pe/4) (e/2*pe^2/L/4)" intext
 qed
 then show ?thesis
 apply ultimately ?thesis
 apply ( simpjava.lang.StringIndexOutOfBoundsException: Index 75 out of bounds for length 75
 done
qed

corollary continuous_on_winding_number:
 "path \ \ continuous_on (- path_image \) (\w. winding_number \ w)"proof
(addcontinuous_at_winding_number

subsection\<^marker>\<open>tag unimportant\<close> \<open>The winding number is constant on a connected region\<close>have 1
bymesondisjoint_iff_not_equal)
lemma winding_number_constant thus "
 assumes
 shows "winding_number \ constant_on S"
proof - qed
 have *: "1 \ cmod (winding_number \ y - winding_number \ z)"
 ne " \ y \ winding_number \ z" and "y \ S" "z \ S" for y z
 proof -
 have assumes \<gamma winding_number_ivt_posjava.lang.StringIndexOutOfBoundsException: Index 98 out of bounds for length 98
using integer_winding_number \<gamma> loop] sg \<open>y \<in> S\<close> by auto
 with ne show ?thesis
 by (auto simp: Ints_def simp flip: of_int_diff)
 qed
 have cont: "continuous_on S emma:
continuous_on_winding_number
 by (meson continuous_on_subset disjoint_eq_subset_Compl)
 show ?thesis
 using "*" zero_less_one
 blast: continuous_discrete_range_constant cs contshowswinding_number and inhalfspaces
qed

lemma winding_number_eq:
 have "gamma z=-( exp( 2* *Im( (subpath 0 \) z)))) * (\ 0 - z))"
 \<Longrightarrow> winding_number \<gamma> w = winding_number \<gamma> z"
 using winding_number_constant by using [of <> java.lang.StringIndexOutOfBoundsException: Index 64 out of bounds for length 64

open_winding_number_levelsets
 assumes
 showsshow\<gamma> piecewise_C1_differentiable_on {0..1}"
proof (clarsimp simp: open_dist)
 z z "llet\ 0"
 open \<gamma>)"
 by (simp add: closed_path_image \<gamma> open_Compl)
 then obtain e where "e>0" "ball z e \ - path_image \"
using [f - \<gamma>"] z by blast
 then show "\e>0. \y. dist y z < e \ y \ path_image \ \ winding_number \ y = winding_number \ z"
 using\<java.lang.StringIndexOutOfBoundsException: Index 132 out of bounds for length 132
qedby image_eqI path_image_def alsodots> = winding_number (subpath 0 x \<gamma>) z"

subsection\<open>Winding number is zero "outside" a curve\<close>

propositionassmspath_image_subpath_subset
 use in
 show java.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 26
-
 obtain
 using java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 29
 obtain w:: where w: "
 by (metis abs_of_nonneg le_less winding_number_lt_half
e-ball+)<subseteq> outside (path_image \<gamma>)"
using by ( outside_subset_convexshows
 then have wouthave" {0..1} (\x. winding_number (subpath 0 x \) z)" "z\ path_image \" using assms by auto
 using w by blast
 moreover have "winding_number \ constant_on outside (path_image \)"
 winding_number_constant winding_number_constant[
 by(bounded_path_image
 ultimatelyhave \<gamma> z = winding_number \<gamma> w"
 by((, opaque_liftingzjava.lang.StringIndexOutOfBoundsException: Index 59 out of bounds for length 59
have"\ = 0"
 proof -
 have wnot: "w qed
 { fix e::real assume "0
 obtainlemma winding_number_ivt_neg
 and:"(0 \ t; t \ 1\ \ cmod (p t - \ t) < 1)"
 and pge: "(\t. \0 \ t; t \ 1\ \ cmod (p t - \ t) < e)"
 using path_approx_polynomial_function [OFand: "a \ 0" and azb: "a \ z \ b" and pag: "path_image \ \ {w. a \ w \ b}"
 by (metis atLeastAtMost_iff min_less_iff_conj zero_less_one)
 have "\p. valid_path p \ w \ path_image p \
 p=pathstart
 \<>t\<in>{0..1}. cmod (\<gamma> t - p t) < e) \<and> contour_integral p (\<lambda>wa. 1 / (wa - w)) = 0"
 proof exI)
 have "\x. \0 \ x; x \ 1\ \ cmod (p x) < B + 1"
 using B unfolding image_subset_iff path_image_def
 by (meson add_strict_mono atLeastAtMost_iff le_less_trans mem_ball_0 norm_triangle_sub pg1)
 then have pip: "path_image p \ ball 0 (B + 1)"
 by( simpadd dist_norm)
 then
 show simp
 by ( closed_segment_eq_real_ivl t ( simp path_image_subpath sub12
 show "\t\{0..1}. cmod (\ t - p t) < e"
 by (metis atLeastAtMost_iff norm_minus_commute pge)
 show "contour_integral p (\wa. 1 / (wa - w)) = 0"
 proofrule [OF [OFconvex_ball0"+"]]java.lang.StringIndexOutOfBoundsException: Index 110 out of bounds for length 110
 have "\z. cmod z < B + 1 \ z \ w"
 using w
 then show "( bysimp add: inner_diff_right)+
 by (intro holomorphic_intros; simp add: dist_norm)
 qed (use p vap pip loop in auto)
 qed (use p in auto)
 }
 then show ?thesis
 by java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
 qed
 finally show ?thesis .
qed

java.lang.StringIndexOutOfBoundsException: Index 146 out of bounds for length 146
 by (rule winding_number_zero_in_outside)
 ( simp pathstart_def)

corollary^marker
 "\path \; convex s; pathfinish \ = pathstart \; z \ s; path_image \ \ s\ \ winding_number \ z = 0"
 by (meson convex_in_outside outside_mono subsetCE winding_number_zero_in_outside)

lemma winding_number_zero_at_infinity:
 assumes \<gamma>: "path \<gamma>" and loop: "pathfinish \<gamma> = pathstart \<gamma>"
 lemma:
proof -
 obtain B::real where "0 < B" and B: "path_image \ \ ball 0 B"
 using : "
 (
proof winding_number_zero_outside
 show "z \ cball 0 B"
 using that by auto
 show "path_image \ \ cball 0 B"
 using B order.trans by blast
 qed
 then show ?thesis
 by metis
qed

lemma bounded_winding_number_nz:
 assumes "path g" "pathfinish g = pathstart g"
 shows "bounded {z. winding_number g z \ 0}"
proof -
 obtain B where "\x. norm x \ B \ winding_number g x = 0"
 using winding_number_zero_at_infinity[OF assms] by auto
 thus ?thesis
 unfolding bounded_iff by (intro exI[of _ "B + 1"]) force
qed

lemma winding_number_zero_point:
 "\path \; convex S; pathfinish \ = pathstart \; open S; path_image \ \ S\
 \<Longrightarrow> \<exists>z. z \<in> S \<and> winding_number \<gamma> z = 0"
 using outside_compact_in_open [of "path_image \" S] path_image_nonempty winding_number_zero_in_outside
 by (fastforce simp add: compact_path_image)

text\<open>If a path winds round a set, it winds rounds its inside.\<close>
lemma winding_number_around_inside:
 assumes \<gamma>: "path \<gamma>" and loop: "pathfinish \<gamma> = pathstart \<gamma>"
 and cls: "closed S" and cos: "connected S" and S_disj: "S \ path_image \ = {}"
 and z: "z \ S" and wn_nz: "winding_number \ z \ 0" and w: "w \ S \ inside S"
 shows "winding_number \ w = winding_number \ z"
proof -
 have ssb: "S \ inside(path_image \)"
 proof
 fix x :: complex
 assume "x \ S"
 hence "x \ path_image \"
 by (meson disjoint_iff_not_equal S_disj)
 thus "x \ inside (path_image \)"
 by (metis Compl_iff S_disj UnE \<gamma> \<open>x \<in> S\<close> cos inside_outside loop winding_number_eq winding_number_zero_in_outside wn_nz z)
 qed
 show ?thesis
 proof (rule winding_number_eq [OF \<gamma> loop w])
 show "z \ S \ inside S"
 using z by blast
 show "connected (S \ inside S)"
 by (simp add: cls connected_with_inside cos)
 show "(S \ inside S) \ path_image \ = {}"
 unfolding disjoint_iff Un_iff
 by (metis ComplD UnI1 \<gamma> cls compact_path_image connected_path_image inside_Un_outside inside_inside_compact_connected ssb subsetD)
 qed
qed

text\<open>Bounding a WN by 1/2 for a path and point in opposite halfspaces.\<close>
lemma winding_number_subpath_continuous:
 assumes \<gamma>: "valid_path \<gamma>" and z: "z \<notin> path_image \<gamma>"
 shows "continuous_on {0..1} (\x. winding_number(subpath 0 x \) z)"
proof (rule continuous_on_eq)
 let ?f = "\x. integral {0..x} (\t. vector_derivative \ (at t) / (\ t - z))"
 show "continuous_on {0..1} (\x. 1 / (2 * pi * \) * ?f x)"
 proof (intro indefinite_integral_continuous_1 winding_number_exp_integral continuous_intros)
 show "\ piecewise_C1_differentiable_on {0..1}"
 using \<gamma> valid_path_def by blast
 qed (use path_image_def z in auto)
 show "1 / (2 * pi * \) * ?f x = winding_number (subpath 0 x \) z"
 if x: "x \ {0..1}" for x
 proof -
 have "1 / (2*pi*\) * ?f x = 1 / (2*pi*\) * contour_integral (subpath 0 x \) (\w. 1/(w - z))"
 using assms x
 by (simp add: contour_integral_subcontour_integral [OF contour_integrable_inversediff])
 also have "\ = winding_number (subpath 0 x \) z"
 proof (subst winding_number_valid_path)
 show "z \ path_image (subpath 0 x \)"
 using assms x atLeastAtMost_iff path_image_subpath_subset by force
 qed (use assms x valid_path_subpath in \<open>force+\<close>)
 finally show ?thesis .
 qed
qed

lemma winding_number_ivt_pos:
 assumes \<gamma>: "valid_path \<gamma>" and z: "z \<notin> path_image \<gamma>" and "0 \<le> w" "w \<le> Re(winding_number \<gamma> z)"
 shows "\t \ {0..1}. Re(winding_number(subpath 0 t \) z) = w"
proof -
 have "continuous_on {0..1} (\x. winding_number (subpath 0 x \) z)"
 using \<gamma> winding_number_subpath_continuous z by blast
 moreover have "Re (winding_number (subpath 0 0 \) z) \ w" "w \ Re (winding_number (subpath 0 1 \) z)"
 using assms by (auto simp: path_image_def image_def)
 ultimately show ?thesis
 using ivt_increasing_component_on_1[of 0 1, where ?k = "1"] by force
qed

lemma winding_number_ivt_neg:
 assumes \<gamma>: "valid_path \<gamma>" and z: "z \<notin> path_image \<gamma>" and "Re(winding_number \<gamma> z) \<le> w" "w \<le> 0"
 shows "\t \ {0..1}. Re(winding_number(subpath 0 t \) z) = w"
proof -
 have "continuous_on {0..1} (\x. winding_number (subpath 0 x \) z)"
 using \<gamma> winding_number_subpath_continuous z by blast
 moreover have "Re (winding_number (subpath 0 0 \) z) \ w" "w \ Re (winding_number (subpath 0 1 \) z)"
 using assms by (auto simp: path_image_def image_def)
 ultimately show ?thesis
 using ivt_decreasing_component_on_1[of 0 1, where ?k = "1"] by force
qed

lemma winding_number_ivt_abs:
 assumes \<gamma>: "valid_path \<gamma>" and z: "z \<notin> path_image \<gamma>" and "0 \<le> w" "w \<le> \<bar>Re(winding_number \<gamma> z)\<bar>"
 shows "\t \ {0..1}. \Re (winding_number (subpath 0 t \) z)\ = w"
 using assms winding_number_ivt_pos [of \<gamma> z w] winding_number_ivt_neg [of \<gamma> z "-w"]
 by force

lemma winding_number_lt_half_lemma:
 assumes \<gamma>: "valid_path \<gamma>" and z: "z \<notin> path_image \<gamma>" and az: "a \<bullet> z \<le> b" and pag: "path_image \<gamma> \<subseteq> {w. a \<bullet> w > b}"
 shows "Re(winding_number \ z) < 1/2"
proof -
 { assume "Re(winding_number \ z) \ 1/2"
 then obtain t::real where t: "0 \ t" "t \ 1" and sub12: "Re (winding_number (subpath 0 t \) z) = 1/2"
 using winding_number_ivt_pos [OF \<gamma> z, of "1/2"] by auto
 have gt: "\ t - z = - (of_real (exp (- (2 * pi * Im (winding_number (subpath 0 t \) z)))) * (\ 0 - z))"
 using winding_number_exp_2pi [of "subpath 0 t \" z]
 apply (simp add: t \<gamma> valid_path_imp_path)
 using closed_segment_eq_real_ivl path_image_def t z by (fastforce simp: path_image_subpath Euler sub12)
 have "b < a \ \ 0"
 proof -
 have "\ 0 \ {c. b < a \ c}"
 by (metis (no_types) pag atLeastAtMost_iff image_subset_iff order_refl path_image_def zero_le_one)
 thus ?thesis
 by blast
 qed
 moreover have "b < a \ \ t"
 by (metis atLeastAtMost_iff image_eqI mem_Collect_eq pag path_image_def subset_iff t)
 ultimately have "0 < a \ (\ 0 - z)" "0 < a \ (\ t - z)" using az
 by (simp add: inner_diff_right)+
 then have False
 by (simp add: gt inner_mult_right mult_less_0_iff)
 }
 then show ?thesis by force
qed

lemma winding_number_lt_half:
 assumes "valid_path \" "a \ z \ b" "path_image \ \ {w. a \ w > b}"
 shows "\Re (winding_number \ z)\ < 1/2"
proof -
 have "z \ path_image \" using assms by auto
 with assms have "Re (winding_number \ z) < 0 \ - Re (winding_number \ z) < 1/2"
 by (metis complex_inner_1_right winding_number_lt_half_lemma [OF valid_path_imp_reverse, of \<gamma> z a b]
 winding_number_reversepath valid_path_imp_path inner_minus_left path_image_reversepath)
 with assms show ?thesis
 using \<open>z \<notin> path_image \<gamma>\<close> winding_number_lt_half_lemma by fastforce
qed

lemma winding_number_le_half:
 assumes \<gamma>: "valid_path \<gamma>" and z: "z \<notin> path_image \<gamma>"
 and anz: "a \ 0" and azb: "a \ z \ b" and pag: "path_image \ \ {w. a \ w \ b}"
 shows "\Re (winding_number \ z)\ \ 1/2"
proof -
 { assume wnz_12: "\Re (winding_number \ z)\ > 1/2"
 have "isCont (winding_number \) z"
 by (metis continuous_at_winding_number valid_path_imp_path \<gamma> z)
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