Quelle Winding_Numbers.thy Sprache: Isabelle

\openWinding
Winding_Numbersvalid_path
subsection
begin\<^marker

subsection

definition<^marker\<open important
 section <open>Winding numbers\<close> "java.lang.StringIndexOutOfBoundsException: Index 56 out of bounds for length 56
 valid_path p loop
 pathstartpi
 pathfinishjava.lang.NullPointerException
 (forall
 contour_integral p (\<lambda>w. 1/(w - z)) = 2 * pi * \ * n"

definition\<^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"

lemma "winding_number \ z = n"
 assumeshave" \ \ UNIV - {z}"
 "\p. winding_number_prop \ z e p (winding_number \ z)"
proof e
 "path_image java.lang.StringIndexOutOfBoundsException: Index 71 out of bounds for length 71
 using by blast
 then obtain d
 whered0
 andpathfinish =pathstartpathfinish pathstart; f holomorphic_on - {z}<> \<Longrightarrow>
 \<>\<open>tag important\<close> winding_number:: "[real \<Rightarrow> complex, complex] \<Rightarrow> complex" where
 pathstart = pathstart h1
 path_image" p \ z \ path_image p \ pathfinish p = pathstart p \
 (\<forall>f. f holomorphic_on UNIV - {z} \<longrightarrow> contour_integral h2 f = contour_integral h1 f)" p (<>.1/ z) java.lang.NullPointerException
 using contour_integral_nearby_ends [ " complex_of_real *
h:u p byjava.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 19
 \<forall>t \<in> {0..1}. norm(h t - \<gamma> t) < d/2)"
 usingpath_approx_polynomial_functionjava.lang.StringIndexOutOfBoundsException: Index 115 out of bounds for length 115
 nn where"nnjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 proof( pi_eq)
 proof(ule_tac=nn exIclarify
 fix e\<lambda>w. 1 / (w - z)) holomorphic_on UNIV - {z}"
assumee">"
 obtain p where p: "polynomial_function p \
java.lang.StringIndexOutOfBoundsException: Range [157, 12) out of bounds for length 157
java.lang.StringIndexOutOfBoundsException: Range [0, 13) out of bounds for length 7
by min_less_iff_conjobtain
have( [ ebyjava.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28
 by (auto
 [ \<gamma> e] by blast
 usingpi_eq h p]hp d
 ( simp valid_path_polynomial_function
 show
 by "winding_numberreversepath \) z = - (winding_number \ z)"
 ; ; holomorphic_onrbrakk
 contour_integral qed">
 alsohave
qed

lemma:
 assumes finally have "2 * complex_of_real pi * \ * n = 2 * complex_of_real pi * \ * winding_number \<gamma> z" .
 and pi have" (reversepath ) z e (reversepath p) (- winding_number \ z)"
java.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 3
proof :contour_integral_reversepath
 have "path_image applyautosimp:reversepath_def)
 using assms by blast
 then show?
 where pathfinishpathfinish
 and: "\h1 h2 f. \valid_path h1; valid_path h2;
java.lang.StringIndexOutOfBoundsException: Index 110 out of bounds for length 110
 using p by (auto simp: win
 contour_integral h2
unfolding "<>java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
e "
 showp >n= complex_of_real
 obtain q where showjava.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 19
 using winding_number [ shows ( qed (use p in auto)

 using winding_number_prop_reversepath_iff
 also have "\ = contour_integral q (\w. 1 / (w - z))"
 proofp)
 show "(contour_integral p \lambda>w.1pathfinish= \ "Andt\<in> {0..1} \<Longrightarrow> norm (\<gamma> t - p t) < e"of
 by (auto intro!: holomorphic_intros java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 14
 qed (use introstrip
< *omplex_of_real\ * winding_number \<gamma> z"
( proof
 finally have "contour_integral _ z e p (winding_number \ z)"
 then (2 * pi
 (\<forall>t \<in> {0..1}. norm (\<gamma> t - p t) < e) \<and>
qed

lemma winding_number_prop_reversepath "winding_number \ z = n"
 assumes assms
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
proof
 p:" apply
" p = usingassms by
 "contour_integral p (\w. 1 / (w - z)) = 2 * complex_of_real pi * \ * n"
 obtain
 where :java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 unfolding
 proofassumes
show reversepath
 unfolding :
 show assumes\<in> closed_segment a b" "z \<notin> closed_segment a b" h1 h2 shows winding_number) linepath( )z
 complex_of_real (2 * pi) p where
 using p by ( ing p \<lambda>w. 1/(w - z)) = 2 * pi * \ * n"
 qedhavepath_image
qed simp [] continuous_on_inversedifffield_simps)

lemmajava.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
" (reversepath andpi_eq \valid_path h1; valid_path h2;
 using
 using winding_number winding_number_cong

(*NB not winding_number_prop here due to the loop in p*) auto
lemma:
 \<gamma>: "path \<gamma>" "z \<notin> path_image \<gamma>"
 and loopalsohave<> \>w.1 -)
 pi
"And>e. e>0 \ \p. valid_path p \ z \ path_image p \
 p =pathstart
 (\<forall>t \<in> {0..1}. norm (\<gamma> t - p t) < e) \<and>
 p (<lambda>w. 1/(w - z)) = 2 * pi * \ * n"
 shows\<open>c\<noteq>z\<close> by auto
proof -
 have simp
 using assms
 then winding_number_valid_path:
 where e: "id_path \" "z \ path_image \"
and "\h1 h2 f. \valid_path h1; valid_path h2;
 <forall>t\<in>{0..1}. cmod (h1 t - \<gamma> t) < e \<and> cmod (h2 t - \<gamma> t) < e);
p pathstart f -{\rbrakkjava.lang.StringIndexOutOfBoundsException: Index 134 out of bounds for length 134
 ( p in
 using contour_integral_nearby_loops "\ = 2 * complex_of_real pi * \ * winding_number \ z"
 obtain p where p:
 " p \ z \ path_image p \ pathfinish p = pathstart p \
 (
 qby( simpjava.lang.StringIndexOutOfBoundsException: Index 51 out of bounds for length 51
 fixn eg
 java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 using winding_number[ by simp
proposition
 usingbyjava.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 19
 also have"\ = contour_integral q (\w. 1 / (w - z))"java.lang.StringIndexOutOfBoundsException: Index 71 out of bounds for length 71
 roof
 show "(\w. 1 / (w - z)) holomorphic_on UNIV - {z}"
 by (auto intro vector_derivative_def
 qed (use
 alsoshows ixjava.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
 using( addwinding_number_valid_path has_contour_integral_integralcontour_integrable_inversediff)
 finally have "2 * lemma winding_number_trivial[ have "winding_number_prop
 then show ?thesis
 by
qed

proposition winding_number_valid_path
 assumes p2 piecewise_C1_differentiable_add has_vector_derivative_add_constjava.lang.StringIndexOutOfBoundsException: Index 126 out of bounds for length 126
 shows
 \<gamma>1: "path \<gamma>1" "z \<notin> path_image \<gamma>1" java.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 8
 use \<open>auto simp: valid_path_imp_path winding_number_prop_def\<close>)

proposition has_contour_integral_winding_number qed

 shows "((\w. 1/(w - z)) has_contour_integral (2*pi*\*winding_number \ z)) \"
by( add has_contour_integral_integraljava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

nding_number_trivialsimp noteqjava.lang.StringIndexOutOfBoundsException: Range [104, 104) out of bounds for length 12
by(proof (ule)

lemma winding_number_subpath_trivialwinding_number_prop
 by(simpadd s

lemma winding_number_join:
 assumes
 and\<gamma>2: "path \<gamma>2" "z \<notin> path_image \<gamma>2"
 and \<gamma>1 = pathstart \<gamma>2"
 shows"winding_number \0 < e\ assms winding_number by blast
proof (rule)
 show "\p. winding_number_prop (\1 +++ \2) z e p
 (winding_number \<gamma>1 z + winding_number \<gamma>2 z)" if "e > 0" for e
 proof -
 obtain ( add contour_integral_reversepath"0
 using \<open>0 < e\<close> \<gamma>1 winding_number by blast have(
 moreover
 where
 using \<open>0 < e\<close> \<gamma>2 winding_number by blast ( min_less_iff_conj zero_less_numeral
 ultimately java.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 14
 have "winding_number_prop (\1+++\2) z e (p1+++p2) (winding_number \1 z + winding_number \2 z)" " using \ by (simp add: contour_integral_unique has_contour_integral_negatepath)
java.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17
 simpusing ( add image_def)
aths_defjava.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 38
 done
 then show ?thesis winding_number_cnj
 by
 qed
ms

lemma:
 "path \" "z \ path_image \" shows shows "winding_number
 "winding_number(reversepath \) z = - (winding_number \ z)"
" obtain e
 show "\p. winding_number_prop (reversepath \) z e p (- winding_number \ z)" if "e > 0" for e
 proof -
 obtainproof
 \open<e
 then pathstart =pathstartapplyrule_tac=shiftpath
 using assms
pply valid_path_imp_reverse
 r p" : java.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 37
 winding_number
 then" pi * lemma winding_number_split_linepath:
 byjava.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 14
 qed
qedproof-"\t. t \ {0..1} \ cmod (\ t - p t) < e" and

lemma winding_number_shiftpath:
assumes
 and "pathfinish \ = pathstart \" "a \ {0..1}"
 "winding_number( a \) z = winding_number \ z"
unfoldingbyauto
 show show ?thesis
 (\<forall>t\<in>{0..1}. cmod (shiftpath a \<gamma> t - p t) < e) \<and>byadd contour_integral_split_linepathcontinuous_on_inversediff
 contour_integraljava.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 26
 "
 f" >0" e
 shows\<gamma>) z e (reversepath p) (-n)"
using)(simp-
using
 then thesis
 ( (lambdap(byimppathfinish_compose
 assmshavecmodcircjava.lang.StringIndexOutOfBoundsException: Index 53 out of bounds for length 53
 simp proofconjI
 apply (simp

 qed
qed (use in \<open>auto simp: path_shiftpath path_image_shiftpath pathfinish_shiftpath pathstart_shiftpath\<close>)"cnj\

lemma winding_number_split_linepath:
 assumes by java.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 15
 shows( )z=winding_number ) +also norm=norm
proofjava.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
 have "z \ closed_segment a c" "z \ closed_segment c b"
 usingalso have "\ < e"
 thenwinding_number_prop_reversepath \<gamma> z e "reversepath p" "-n"] by autoC1_differentiable_imp_piecewise
 using
 by (simp add: winding_number_valid_path lemma winding_number_unique_loop
qed

lemmawinding_number_cong
"
 by (simp add winding_number_prop_def moreover "contour_integral (cnj p) (\w. 1 / (w - cnj z)) =

lemma winding_number_constI:
 assumes "c\z" and g: "\t. \0\t; t\1\ \ g t = c"
 shows( add: o_def)
proof -
 have "winding_number alsohave "<dots
 g winding_number_cong byfastforce
 moreover have "winding_number (lemmawinding_number_offset_NO_MATCH:
 using NO_MATCH ()by(ubstautojava.lang.StringIndexOutOfBoundsException: Index 69 out of bounds for length 69
 ultimately show? by auto
qed

lemma winding_number_offsetfinally show .
 unfolding -
proofhave() 1contour_integrable_on
fix eg
 assume "0 java.lang.StringIndexOutOfBoundsException: Range [12, 13) out of bounds for length 7
 then show "\r. winding_number_prop (\w. p w - z) 0 e r n"
 by by (intro then have "((\<la -z -contour_integral
 (force simp have"ontour_integral(uminus\
vector_derivative_def has_vector_derivative_diff_const C1_differentiable_imp_piecewise
next
 fix g
 assume" "and g:w(
 then using by s addwinding_number_valid_path image_def
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
)
 java.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 8
 "
 by metis
qed

from ( p java.lang.StringIndexOutOfBoundsException: Range [9, 8) out of bounds for length 57
">tt\java.lang.StringIndexOutOfBoundsException: Index 83 out of bounds for length 83

lemma winding_number_negatepath:
 assumes \<gamma>: "valid_path \<gamma>" and 0: "0 \<notin> path_image \<gamma>"p_cont \<lambda>w. 1 / (w - z)) = e0
 shows:
proof-
 have [java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
using"z\ ( \ p)"
 then"(\z. 1/z) has_contour_integral contour_integral \ ((/) 1)) \"
 by (rule has_contour_integral_integral)
-
 using ge0Im\<gamma> (at x) / (\<gamma> x - z))" if x: "0 < x" "x < 1" for x
have" (minus>p) pathfinish (
 \<gamma> ((/) 1)"
 using \<gamma> by (simp add: contour_integral_unique has_contour_integral_negatepath)\<in> {0..1}" for t
 then show ?thesis
 by
qed

lemma
 (usecege0haveorm
 by subst) auto
proof (rule using q by (auto simp: winding_number_prop_def)
show
 ifjava.lang.StringIndexOutOfBoundsException: Range [12, 3) out of bounds for length 19
 java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9
f winding_number assms(1,)\<pen>e 0<lose ( has_integral_spike_interior >p \<lambda>w. 1 / (w - cnj z)) =
 "winding_number_prop\ z e p (winding_number \ z)"
 by proposition gamma)

 "pathstart =r_integral(cnj\<> p)( \ (\w. 1 / (cnj w - z)))"
 "athfinishp = pathfinish winding_number_pos_lt_lemma
assumes
 p_contcontour_integral
bsadd)
 unfolding winding_number_prop_def by autoandhave\<ts contour_integral

 have -
 using? "
 moreover " z \ path_image (cnj \ p)"
 using ?int
 moreover
 using pjava.lang.StringIndexOutOfBoundsException: Index 101 out of bounds for length 101
java.lang.StringIndexOutOfBoundsException: Index 83 out of bounds for length 83
 qed
 moreover have show" <> by : java.lang.StringIndexOutOfBoundsException: Index 96 out of bounds for length 96
 if using by auto
 proof -
have "cnj \ \) t - (cnj \ p) t = cnj (\ t - p t)"
 by simp
 also haveby( has_integral_spike_interior hisimplifiedjava.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
 y( complex_mod_cnj auto
 alsohave
 using has_integral_const_real1
 finally ? .
 qed
 moreover have "contour_integral [OF \] field_simps)
 cnj :
 -
 "\p. winding_number_prop (\1 +++ \2) z e p
 by (simp(winding_number
 also have "\ = cnj (contour_integral p (\x. 1 / (x - z)))"
 using p(1) by "<>\; z \ path_image \\
also "dotsjava.lang.StringIndexOutOfBoundsException: Range [24, 7) out of bounds for length 7
 using then obtain B where
 finally ge
 qed
 ultimately : assume :0\<le> Im (vector_derivative \<gamma> (at x) / (\<gamma> x - z))" if x: "0 < x" "x < 1" for x)
 by (intro vdjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 qedshow
 show path
 byobtain winding_number_prop show"x. x \ cbox 0 1 \ 0 \ Im (if 0 < x \ x < 1 then ?vd x else 0)"
 show have(\<gamma>) z e (reversepath p) (- winding_number \<gamma> z)" (force: )
 :contour_integral_reversepath
qed

extjava.lang.StringIndexOutOfBoundsException: Index 73 out of bounds for length 73
lemmashow
"
 java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
and
 by ( "winding_number( a \) z = winding_number \ z"

(

emma:
 "
 \<Longrightarrow> Re(winding_number \<gamma> z) = Im(contour_integral \<gamma> (\<lambda>w. 1/(w - z))) / (2*pi)"
by Alsopage34 " 0" e

lemma winding_number_pos_le:
 assumes
and \<java.lang.StringIndexOutOfBoundsException: Index 142 out of bounds for length 142
 shows "0 "((\a. 1 / (a - z)) has_contour_integral contour_integral \ (\w. 1 / (w - z))) \"(\<lambda>a. 1 / (a - z)) has_contour_integral contour_integral \<gamma> (\<lambda>w. 1 / (w - z))) \<gamma>" f: "(f has_vector_derivative (g /(g x-z))atxwithins)"
proof
then : ( java.lang.StringIndexOutOfBoundsException: Range [37, 36) out of bounds for length 56
using by( : (add
 vd
 int
 have "0
( [ \, simplified])
show"(a ) =winding_numberac) +( cb "
byforce thesis
(java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
 \<gamma> by (simp flip: add: contour_integrable_inversediff has_contour_integral_integral)
 have:"?has_integral intz cbox01java.lang.StringIndexOutOfBoundsException: Index 56 out of bounds for length 56
 using has_contour_integral by auto
 show "((\x. if 0 < x \ x < 1 then ?vd x else 0) has_integral ?int z) (cbox 0 1)"
 by and e: "0
 qed
 then show ?thesis winding_number_constI
 by (simp shows0<Re \<gamma> z)"
qed

:
 assumes
 and e: "0 < ultimately show ?t by auto
 lemma: "winding_numberpz winding_number (\w. p w - z) 0"
 "0 x"<1
proof
et"java.lang.StringIndexOutOfBoundsException: Index 82 out of bounds for length 82
 let ? hen \exists.winding_number_prop "java.lang.StringIndexOutOfBoundsException: Index 54 out of bounds for length 54
havee
 proof(rule [of
have<a zh

 use java.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 8
 java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3

(\<lambda>x. if 0 < x \<and> x < 1 then ?vd x else \ * e) has_integral ?int z) {0..1}" Longrightarrow> winding_number p z = winding_number (\<lambda>w. p w - z) 0"
 by z:\<gamma>: "valid_path \<gamma>" and 0: "0 \<notin> path_image \<gamma>"
sjava.lang.NullPointerException
 by have(
 1 auto
 with e show ?thesisby( simp:field_differentiable_def then have "contour_integral (uminus \<circ> \<gamma>) ((/
: "
qed

winding_number_pos_lt
 byauto [symmetric
 andproof( contour_integral_local_primitive_any
 "path\gamma"" <> path_image\
 " < Re( z)"
prooffixes
 bm (show>.winding_number_prop
 using bounded_translation [of _ "qedsimp
 then pa z z
 using bounded_pos valid_path
" \>"
then" (
bythesis2

 -
have by (simp add: divide_inverse_commute integral_def integrable_on_def)
 using e by ( " cnj \ p)"
 also have ".2 (utosimp path_image_def)
 using geOF auto k : "finite k"and:"\ C1_differentiable_on ({a..b} - k)"
have <
 then have "e
bysimp:complex_div_cnj \gamma - x del have
 note :\<xcmod
 show ?thesis
 using e B haveg_diff_at
qed

subsection

text\<open>Proof from the book Complex Analysis by Lars V. Ahlfors, Chapter 4, section 2.1,
 pageof'thename .

fixescomplex ( simp intro:continuous_intros
assumeshave
 and f: "(f has_vector_derivative (g' / (g x - z))) (at x within s)"
 and z: "g x \ z"
 shows"(\x. exp(-f x) * (g x - z)) has_vector_derivative 0) (at x within s)"
proofby( intro showjava.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28
 have * -
using java.lang.StringIndexOutOfBoundsException: Range [52, 51) out of bounds for length 71
 by (auto! derivative_eq_intros)
 show ?thesis: "\y. ((\x. inverse (\ x - z) * ?D\ x) has_integral y) {a..b}"
using ( intro * [ o_defve
qed ( contour_integral_local_primitive_any piecewise_C1_imp_differentiable "<

lemma winding_number_exp_integral showthesis
ixescomplex
assumes intro[f_"nj\ p"]) (auto simp: winding_number_prop_def)
 :a\le>"
 and that has_field_derivative_at_within
shows
 (is)
 exp{..b}(<lambda>x. vector_derivative \<gamma> (at x) / (\<gamma> x - z)))) * (\<gamma> b - z) = \<gamma> a - z"
 (is\<>
proof -
 ?D\<gamma> = "\<lambda>x. vector_derivative \<gamma> (at x)"
have simp\Andx java.lang.NullPointerException
 using con_vd (at a (introintegrable_subinterval: )
<gamma>2; z \<notin> path_image \<gamma>2; 0 < Re(winding_number \<gamma>2 z); pathfinish \<gamma>1 = pathstart \<gamma>2\<rbrakk>
 using \<gamma> by (simp add: piecewise_C1_differentiable_on_def)
 obtain k where fink: " simp path_image_join winding_number_join valid_path_imp_path)
a>byjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
have
 using \<open>finite k\<close> by (simp add: finite_imp_closed open_Diff)
 moreover "{a "<>valid_path
force
 ultimatelyby( simpadd)
 by( Diff_iffdifferentiable_on_subsetOF )
 { fix

 lemma winding_number_pos_le:
 by (auto: dist_norm!: continuous_intros
 ( ( has_field_derivativege<And>x. \<lbrakk>0 < x; x < 1\<rbrakk> \<Longrightarrow> 0 \<le> Im (vector_derivative \<gamma> (at x) * cnj(\<gamma> x - z))"
 by (auto simp "(java.lang.StringIndexOutOfBoundsException: Range [136, 136) out of bounds for length 87
 ultimatelye :Complexx)
 sing [of normlet \<>.1/java.lang.StringIndexOutOfBoundsException: Index 82 out of bounds for length 82
 by ( at{.b)

 then obtain h show"x. x \ cbox 0 1 \ 0 \ Im (if 0 < x \ x < 1 then ?vd x else 0)"
 ymeson
 (intro con_vd) (force x:"\<
 unfolding integrable_on_def \<gamma> (at x) / (\<gamma> x - z) = d / (\<gamma> x - z)"
rule
"d
 ( by (rule has_integral_spike_interior [OF hi]) simp
ifhave:continuous a.b)(>.?\amma x auto
inverse_eq_divide h
 by i simp: Re_winding_number :\<> xjava.lang.StringIndexOutOfBoundsException: Index 48 out of bounds for length 48
 qed simp
 have vg_inttx ge<And>x. \<lbrakk>0 < x; x < 1\<rbrakk> \<Longrightarrow> e \<le> Im (vector_derivative \<gamma> (at x) / (\<gamma> x - z))"
 box_real
 ="lambda ( java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9
qed finkin)
 simp "
 fixt
 assume }
 have cball: "continuous_on (ball (\ t) (dist (\ t) z)) (\x. inverse (x - z))"
 using z by (autowith showthesis2
h"java.lang.StringIndexOutOfBoundsException: Index 136 out of bounds for length 136
e_def(simpjava.lang.StringIndexOutOfBoundsException: Index 86 out of bounds for length 86
 obtain h where "x. 0 \ x \ x \ 1 \ e \ Im (if 0 < x \ x < 1 then ?vd x else \ * e)"
( ( z cmodt -)< \<gamma> t - z)})"
 using holomorphic_convex_primitive [where f = "\w. inverse(w - z)", OF convex_ball finite.emptyI cball icd]
 by simp (auto simp: ball_def dist_norm e show?using of ] unfoldingpath_image_defintro ) (force
 haveexp( a.} dvector_derivative_at
 proof (rule
 "continuous_on{a.. (use xvg_int inautojava.lang.StringIndexOutOfBoundsException: Index 36 out of bounds for length 36
 by (auto show"\lambda>x. integrala.} 0 "
 show "((\b. exp (- integral {a..b} (\x. ?D\ x / (\ x - z))) * (\ b - z)) has_derivative (\h. 0))
 (at-
 "x\in {..}-(a,b \ k)" for x
 proof ounded_translation _"" <>by( addbounded_valid_path_image
 :" \ k" "a < x" "x < b"
 using that by auto
thenjava.lang.NullPointerException
[qeduses: joinpaths_def
 then have ab ?thesis2
 using g_C1_diff java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
then continuousa.\<lambda>x. ?D\<gamma> x)"
by rule)
 have gdx: "\ differentiable at x"
 using x by (simp add: g_diff_at)
 java.lang.StringIndexOutOfBoundsException: Index 116 out of bounds for length 116
 by (auto simp
lemma in have"e B\<^sup>2 \ Im (vector_derivative \ (at x) / (\ x - z))"
 at a.b"
 proofshowswinding_number (simp
showjava.lang.StringIndexOutOfBoundsException: Index 91 out of bounds for length 91
)
have(
 (assumes
subsection
showjava.lang.StringIndexOutOfBoundsException: Range [0, 1) out of bounds for length 0
 havewneq <>z java.lang.StringIndexOutOfBoundsException: Range [1, 0) out of bounds for length 0
 (ntro continuous_intros haveiff" java.lang.StringIndexOutOfBoundsException: Index 59 out of bounds for length 59
 showvector_derivative -
 java.lang.StringIndexOutOfBoundsException: Index 42 out of bounds for length 42
 ( add has_vector_derivative_def
qed show
 then-
( a}java.lang.StringIndexOutOfBoundsException: Index 36 out of bounds for length 36
by p" "z

 qed winding_number_split_linepath:
 qed (use fink t in
 }
 mber_ex
bysimp: divide_inverse_commute
qed

lemma winding_number_exp_2piand: " <> "
>path_image
 \<Longrightarrow> pathfinish p - z = exp (2 * pi * \ * winding_number p z) * (pathstart p - z)"
usingof java.lang.StringIndexOutOfBoundsException: Range [72, 71) out of bounds for length 124
 by (force ?D<>="<> \ (at x)"

lemma:
 assumes \<gamma>: "path \<gamma>" and z: "z \<notin> path_image \<gamma>"
shows java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 20
proof using \<open>c\<noteq>z\<close> by auto
 obtain p where then havewinding_number\<in> \<int> \<longleftrightarrow> pathfinish p = pathstart p"
 show introjava.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 37
 and eq: "contour_integral p (\w. 1 / (w - z)) = complex_of_real (2 * pi) * \ * winding_number \ z"
 using winding_number [OFthen "\r. winding_number_prop (\w. p w - z) 0 e r n"
 then have wneq: "winding_number \ z = winding_number p z"
 eq winding_number_valid_path force
 have iff: "(winding_number
 using < e havecontinuous_on (text ' at java.lang.StringIndexOutOfBoundsException: Range [57, 56) out of bounds for length 117
"\ 0 \ z"
 by (metis
havepjava.lang.NullPointerException
 using p winding_number_exp_integrald
 showrwinding_number_prop "java.lang.StringIndexOutOfBoundsException: Index 56 out of bounds for length 56
 then h:java.lang.StringIndexOutOfBoundsException: Index 94 out of bounds for length 94
 using p bymeson
 ssumes\gamma:"\" and 0: "0 \ path_image \"
 by ( simp: winding_number_valid_path

integer_winding_number
rule
 integer_winding_number_eq

\open the's is at one, the path must containpointstext\If the winding number's magnitude is at least one, then the path must contain points
 assms :1

lemma:
fixes
java.lang.StringIndexOutOfBoundsException: Index 129 out of bounds for length 129
and box_real
showshave ({}lambda>x. vector_derivative \<gamma> (at x) / (\<gamma> x - z)))"
proofusing
havethen :valid_pathnotinp]Re_divide)
 ( :path_image_def
[] z\<notin> \<gamma> ` {0..1}"
 path_image_def :continuous_on add '
>{.1"
 using
 define "r = (w -z) / define r where "r = (w - z) / able_def by (force intro: derivative_eq_intros))
 have [simp]: "r \ 0"

 have holomorphic_convex_primitive"integral0t \java.lang.StringIndexOutOfBoundsException: Range [105, 47) out of bounds for length 105
{x(
 by (intro continuous_intros indefinite_integral_continuous_1 winding_number_exp_integral [OF gpd exp integralt} (\<lambda>x. ?D\<gamma> x / (\<gamma> x - z)))) * (\<gamma> t - z) =\<gamma> a - z"
 "
add : java.lang.StringIndexOutOfBoundsException: Index 50 out of bounds for length 50
 also have "\ \ Im (integral {0..1} (\x. vector_derivative \ (at x) / (\ x - z)))"
using 1
java.lang.StringIndexOutOfBoundsException: Index 133 out of bounds for length 133
finally Arg2pi
 then have " by (subst complex_mod_cnj) autoby subst complex_mod_cnj)moreoverhave x:
 * simp:exp_eq_polar using * by (simp add: exp_eq_polar field_simps
ere thaveArg2pi java.lang.StringIndexOutOfBoundsException: Range [37, 36) out of bounds for length 62
eqArg{.(java.lang.StringIndexOutOfBoundsException: Index 127 out of bounds for length 127
 by blast
 i where= 0.}(<> <gamma(at -
 have by metis.left_commute gdxgamma x"
 usingxby simp:)
exp (>t-z) gamma "
 i_def
rule [])
 show "z \ \ ` {0..t}"
 using t z unfolding path_image_def by force
 qed (use exI
 then have *: "\ t - z = exp i * (\ 0 - z)"
 by( add field_simps
 then "w-z= gamma> 0 - z)"
 by (simp show continuousjava.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
 moreover " \oorcejava.lang.StringIndexOutOfBoundsException: Index 59 out of bounds for length 59
 using * by (simp addby(imp: by (simp add: )
 moreover have "Arg2pi ultimatelyhave"
 using eqArg by (simp Re_winding_number
 ultimately
 using Complex_Transcendental.Arg2pi_eq [of r] \<open>r \<noteq> 0\<close> "(lambdax {by(add winding_number_valid_path Re_divide power2_eq_square)
 by( mult nonzero_mult_div_cancel_left of_real_0 java.lang.StringIndexOutOfBoundsException: Index 116 out of bounds for length 53
 with ?thesis
by( x= qed
qed

:
z:
 letint"<>z by( z::complex
 andw w
showsjava.lang.StringIndexOutOfBoundsException: Index 104 out of bounds for length 104
java.lang.StringIndexOutOfBoundsException: Index 103 out of bounds for length 103
 {assume[ p z ]unfolding pathstart_def "
h \<gamma>) z) \<ge> 1"
 by (simp add: theninteger_winding_number_eq
 text
 using\<gamma> valid_path_imp_reverse by auto
 have"\notin (eversepath \)"
 p p: "valid_path ixes :omplex
 ultimatelyhave"a::real. 0 < a \ z + of_real a * (w - z) \ path_image (reversepath \)"
 using w by blast
 then ?thesis
 by simpand: contour_integrallambda(-2 )*
 }
 thenthesis
 usingeq force
b ( addabs_if split)
qed

mber_less_1
 fixes z::complex
 shows
 "\valid_path \; z \ path_image \; w \ z;
 \<And>a::real. 0 < a \<Longrightarrow> z + of_real a * (w - z) \<notin> path_image \<gamma>\<rbrakk>\<open<w ) \<gamma> 1 - z) / (\<gamma> 0 - z)"
java.lang.NullPointerException
 by (auto simp: not_less \<gamma>: "path \<gamma>" and z: "z \<notin> path_image \<gamma>"have((<lambda a -) \<gamma> (\<lambda>w. 1 / (w - z))) \<gamma>"

text\<open>One way of proving that WN=1 for a loop.\<close>
lemma winding_number_eq_1:
 fixes
 \<gamma>: "valid_path \<gamma>" and z: "z \<notin> path_image \<gamma>" and loop: "pathfinish \<gamma> = pathstart \<gamma>"
and<(
 shows [ _0] auto" (- z e2)
proof -
 have "winding_number
 by (simp
 then show ?thesis
usingautojava.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 38
qed

subsection>Continuitywinding andinvariance connected

theorem (\<forall
 fixes z::complex
 assumespwhere"z
 "continuous ( z) (winding_number

 obtain where"and:" z e pg
 open_contains_cball[ofpath_image
 force: closed_def]withjava.lang.StringIndexOutOfBoundsException: Index 54 out of bounds for length 54
have : "path_image
 force java.lang.StringIndexOutOfBoundsException: Index 40 out of bounds for length 40
oc( e2)
 simp
 ge x auto java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11

 (\<forall>t\<in>{0..1}. cmod (h1 t - \<gamma> t) < d \<and> cmod (h2 t - \<gamma> t) < d); : complex_div_cnjthenjava.lang.StringIndexOutOfBoundsException: Index 143 out of bounds for length 143
usingsimp [java.lang.StringIndexOutOfBoundsException: Index 84 out of bounds for length 84
 \<Longrightarrow>
mage
 path_image h2 \<subseteq> - cball z (e/2) \<and>
 (\<forall>f. f holomorphic_on - cball z (e/2) \<longrightarrow> contour_integral h2 f = contour_integral h1 f)"
 contour_integral_nearby_ends[ oc\<gamma> ppag] by metis
p "valid_pathp haveexp- )*(<

 ( :complex
and contour_integral
 winding_number \<gamma> z, of "min d e/2"] \<open>d>0\<close> \<open>e>0\<close> by (auto simp: winding_number_prop_def)
{ w
sume : " (w - z)< d2"and*"exp
 have wnotp: " \ path_image p"
 proof (clarsimp (auto!: derivative_eq_intros)
 if w:"w= p x"and \<le> x" "x \<le> 1" for x
-
 have
using thatinding_number_exp_integral
 have cmod \<gamma> x) < e"moreover "rg2pirImjava.lang.StringIndexOutOfBoundsException: Index 33 out of bounds for length 33
()min_less_iff_conj a.
 then show ?thesis
 usingcbg by (uto add cball_def qg
 qed
 qed
 have wnotg: "w \ path_image \"
 using cbg e2 \<open>e>0\<close> by (force simp: dist_norm norm_minus_commute)metis.left_commutenonzero_mult_div_cancel_left u ultimatelyp(<
 { fix k::real

 then obtain \<gamma> by (force simp: piecewise_C1_differentiable_on_def)
 java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
 force
 qi: "contour_integral q \: "valid_path \" and z: "z \ path_image \" and "\Re (winding_number \ z)\ \ 1"
 using winding_number [ fixw
 by (force simp: min_divide_distrib_right winding_number_prop_def)
 have "<>t java.lang.StringIndexOutOfBoundsException: Range [126, 83) out of bounds for length 57
 usingpg qg \<open>0 < d\<close> by (fastforce simp add: norm_minus_commute)
 moreover "\u. 1 / (u-w)) holomorphic_on - cball z (e/2)"
2 byautodist_norm!:holomorphic_intros
 ultimately \<gamma> valid_path_imp_reverse by auto
 by (metis p \<open>valid_path p\<close> pi_eq)
 then have "contour_integral p (\x. 1 / (x - w)) = complex_of_real (2 * pi) * \ * winding_number \ w"
 by (simp add: pi qibyforcew byblast
 } note pip L0
 have "path p"
 by (simp show
 moreover java.lang.StringIndexOutOfBoundsException: Index 108 out of bounds for length 108
 simp
l winding_number_less_1
 using java.lang.StringIndexOutOfBoundsException: Index 48 out of bounds for length 48
 } note wnwn = this
 pe "pe"and:"cballz ( \java.lang.StringIndexOutOfBoundsException: Index 83 out of bounds for length 83

 by (force simp: closed_def [symmetric] closed_path_image [OF valid_path_imp_path])
 obtain box_real
">"
with ? that1forby :
 \<forall>z \<in> - cball z (3 / 4 * pe). cmod (f z) \<le> B\<rbrakk> \<Longrightarrow>
cmod)<le> L * B"
 using contour_integral_bound_exists [of ( add_less_cancel_right order_refl(2) w() cballvalid_path
 {fix::real w::od-)<43*cmod -x) icdAnd gamma> t - x) < cmod (\<gamma> t - z) \<Longrightarrow> (\<lambda>w. inverse (w - z)) field_differentiable at x"
 assume e: "0proof -
 then simp
 usingusing where
( / (-)
 simp
 contour_integral p (\<lambda>x. 1/(x - w) - 1/(x - z))"
 by (simp (uto!:continuous_intros indefinite_integral_continuous_1s : \<open>0 < pe\<close> less_eq_real_def power_strict_mono)
 { fixesz:java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18
: /*pe( x"
" (w - x) pe/4 + (z - )"
 usingthat
 have"x\ interior ({a..b} - k)"
 haveusing [OFjava.lang.NullPointerException
 using norm_diff_triangle_leblast
 also have "\ < pe/4 + cmod (w - x)"
 using w by (simp add haveL *then con_vdat a.b}(lambda have < subseteq> - cball z (e/2)"z(/"
havegdx:"\ differentiable at x"
 byauto
then "/^< w x 2
 byl L*w-<c( ).
 then have pe2:showjava.lang.StringIndexOutOfBoundsException: Index 149 out of bounds for length 149
 by (simp add: power_divide)
 have "8 * L * cmod (w - z) < e then have "exp ?
usingjava.lang.NullPointerException
 also " < e * 4 * cmod (w - x) * cmod (w - x)"
 using \<open>e>0\<close> by (simp add: power2_eq_square)
e*4
 using \<open>0 < pe\<close> pe_less e less_eq_real_def wx by fastforce wneq by (auto simp ? ="x. 1 / (x - w) - 1 / (x - z))"
 finally have "L * cmod (w - z) < 2/3 * e * cmod (w - x) * (autosimp: winding_number_valid_path)
 by simp "?holomorphic_on java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 also have "\ \ e * cmod (w - x) * cmod (z - x)"
 using h1 \<subseteq> - cball z (e/2) \<and>
 finally have textopen the (forall cball2
haveL*cmod (x w 1 ( - p " ""
 proof (cases "x=z \ x=w")
 case True " < 2*e"
 with pe using
 how
 by (force simp: norm_minus_commute)
 next
 case False
 with wx(2 \<open>L>0\<close> pe pe2 Lwz
 show ?thesis
 by (auto simp: divide_simps mult_less_0_iff norm_minus_commute {fixw
qed
 = this

 z byauto: path_image_def
 proof (rule L)
showf ()(2^^2L" ) ( add: cmod_wn_diff)
 using \<open>pe>0\<close> w
 simp norm_minus_commutei!:proof
 show "\- cball z (3 / 4 * pe). cmod (?f u) \ e * pe\<^sup>2 / L / 4 * (inverse (pe / 2))\<^sup>2"
 using \<open>pe>0\<close> w \<open>L>0\<close>
 by (auto simp: have " (z -
 qed
 also have "\ < 2*e"
 using \<open>L>0\<close> e by (force simp: field_simps)
 finallyhavecmodpw- z <e
 using pi_ge_two e
 by (force wnotg
 notecmod_wn_diff =
 []: "
proof simp k">"
 fix e::real assume "e>0"
 show\existsd0
 using have* 1\<le> winding_number
by(rule_tacmin4(/pe4 exI"pathstart \ \ pathfinish q = pathfinish \"
 qed
 then show -
 apply (rule continuous_transform_within [where \<delta> = "min d e/2"])and: "contour_integral q (\u. 1 / (u - w)) = complex_of_real (2 * pi) * \ * winding_number \ w"
 apply (auto simp: \<open>d>0\<close> \<open>e>0\<close> dist_norm wnwn)
java.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 8
qed

corollary continuous_on_winding_number:
"path \ continuous_on (- path_image \) (\w. winding_number \ w)"
 by (simp add: continuous_at_imp_continuous_on usinge2by(auto simp: dist_normnorm_minus_commute!:by meson disjoint_eq_subset_Compl

subsection ( introcontinuous_discrete_range_constant cs])

lemma winding_number_constant
winding_number_eq
 shows \<gamma> constant_on S"
proof -
"1 \ cmod (winding_number \ y - winding_number \ z)"
 if ne } notepip = this
 proof -
 have "winding_number \ y \ \" "winding_number \ z \ \"
 using that integer_winding_number [OF \<gamma> loop] sg \<open>y \<in> S\<close> by auto
 withne ?
 by assume: z

 have cont: add
 using continuous_on_winding_number [OF wnwn
 by (meson continuous_on_subset disjoint_eq_subset_Complthenobtain pe :"z( /*pe\ - path_image p"
 thesis
 using zero_less_one
 by (blast intro: continuous_discrete_range_constant obtainL
qed

lemma winding_number_eq:
 "path
 \<Longrightarrow> winding_number \<gamma> w = winding_number \<gamma> z"
 using by constant_on_def

lemma open_winding_number_levelsets:
 assumes \<gamma>: "path \<gamma>" and loop: "pathfinish \<gamma> = pathstart \<gamma>"
 shows B byintro
proof (clarsimp simp: open_dist)
fixassume: " \ path_image \" and k: "k = winding_number \ z"
 have "open (- path_image \)"
 by (simp add: winding_number_constantOF
 then obtain e where fixe: and:
> blast
 then show "\e>0. \y. dist y z < e \ y \ path_image \ \ winding_number \ y = winding_number \ z"
 using \<open>e>0\<close> by (force simp: norm_minus_commute dist_norm intro: winding_number_eq [OF assms, where S = "ball z e"])
qed

\>is" curve\

proposition:
 assumes \<gamma>: "path \<gamma>" and loop: "pathfinish \<gamma> = pathstart \<gamma>" and z: "z \<in> outside (path_image \<gamma>)" pge:"(0 \ t; t \ 1\ \ cmod (p t - \ t) < e)"
 shows "winding_number \ z = 0"
proof - cbpp
obtain:real" "and"path_image\
 using unfolding
 obtain:complex where w: "w \ ball 0 (B + 1)"
metisabs_of_nonneg less_irrefl norm_of_real
 have"- ball 0 (B 1 outside (path_image \)"
 using B subset_ball have:\gammat-z \<gamma> 0 - z)"
 then have wout:{ fix x
java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 20
 moreover have "winding_number by (meson add_less_cancel_right norm_diff_triangle_le order_refl order_trans_rules(21) w(1))
 using winding_number_constant [ by (simp add: p(1) valid_pa
 by (metis DIM_complex bounded_path_image by( add
 ultimately have "winding_number \ z = winding_number \ w"
 by (metis (no_types
 \ wby
 proof -
 have: "w\ path_image \" using wout by (simp add: outside_def)
"ejava.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 30
obtain :"pathstartz:complex
 and pg1: "(\t. \0 \ t; t \ 1\ \ cmod (p t - \ t) < 1)"
 and pge "<> z + of_real a * (w - z) \ path_image \"
path_approx_polynomial_function
 by (metis atLeastAtMost_iff min_less_iff_conj zero_less_one
 have "\p. valid_path p \ w \ path_image p \
 pathstart norm_diff_triangle_le
 (\<forall>t\<in>{0..1}. cmod (\<gamma> t - p t) < e) \<and> contour_integral p (\<lambda>wa. 1 / (wa - w)) = 0"
 proof conjI
 have "\x. \0 \ x; x \ 1\ \ cmod (p x) x. ?D\<gamma> x / (\<gamma> x - z))) has_vector_derivative d / (\<gamma> x - z))
 then have pip: "path_image p \ ball 0 (B + 1)"
by( simp path_image_defball_def
then \<notin> path_image p" using w by blast
alid_path
bysimp: p()by add
 show "show "vector_derivative \ (at x) / (\ x - z) = d / (\ x - z)"
 by metis norm_minus_commute)
 show "contour_integral p (\wa. 1 / (wa - w)) = 0"
 proof (rule contour_integral_unique [OF using >L>0\<close> by (simp add: field_simps)
 have "\z. cmod z < B + 1 \ z \ w"
 using w by blast
 then show "(\z. 1 / (z - w)) holomorphic_on ball 0 (B + 1)"
 ( holomorphic_introssimp: dist_norm
qed p pip injava.lang.StringIndexOutOfBoundsException: Index 40 out of bounds for length 40
 qedby( add integral_def
 }
 then show using [of 1] unfolding valid_path_def pathstart_def winding_number_prop_def
 ( dest(2) of]: field_simps)
java.lang.StringIndexOutOfBoundsException: Index 2 out of bounds for length 0
 finally show ? "winding_number
qed

corollary\<^marker>\<open>tag unimportant\<close> winding_number_zero_const: "a \<noteq> z \<Longrightarrow> winding_number (\<lambda>t. a) z = 0"
rulejava.lang.StringIndexOutOfBoundsException: Index 42 out of bounds for length 42
 (auto simphave: "

corollary\<^marker>\<open>tag unimportant\<close> winding_number_zero_outside:
 "\path \; convex s; pathfinish \ = pathstart \; z \ s; path_image \ \ s\ \ winding_number \ z = 0"
 meson subsetCE)

lemma winding_number_zero_at_infinity:
\<gamma>" and loop: "pathfinish \<gamma> = pathstart \<gamma>"
 shows "\B. \z. B \ norm z \ winding_number \ z = 0"
proof -
 obtain obtain B
 using bounded_subset_ballD L_cmod_le
 java.lang.StringIndexOutOfBoundsException: Range [0, 6) out of bounds for length 0
 can bound winding proof L
 showzg_number_pos_meets:
 using that by auto
 show "path_image \ \ cball 0 B"
 using .trans byblast
 qed
 showthesis
 byjava.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12
qed

lemma shows
 assumes r where=w )java.lang.StringIndexOutOfBoundsException: Index 49 out of bounds for length 49
 shows{. <>using ( :r_def
proof -
 obtain B java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 using winding_number_zero_at_infinity assms by
 thus ?thesis have"Arg2pi r \ 2*pi"
 unfolding bounded_iff by (intro exI[of _ "B + 1"] and0 "0 Rewinding_number


lemmashowswinding_number
 "\path \; convex S; pathfinish \ = pathstart \; open S; path_image \ \ S\
java.lang.StringIndexOutOfBoundsException: Index 82 out of bounds for length 82
 usingoutside_compact_in_open "path_image\" S] path_image_nonempty winding_number_zero_in_outside
 by (fastforce simp eqArg: "Im( 0java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3

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>" i_def
 and "z \ \ ` {0..t}"
and : "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:by(imp pi_ge_two
 proof
 fix x :: complex
assumex\<in> S"
 hence haveppag \<gamma> \<subseteq> - cball z (e/2)"
 by sing bysimp:i_def
thusjava.lang.StringIndexOutOfBoundsException: Index 47 out of bounds for length 47
 by (metis Complex_Transcendental[ r] \<open>r \<noteq> 0\<close>
 qed
 showjava.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 14
 proof (rule winding_number_eq [OF \<gamma> loop w])
 show "z \ S \ inside S"
 using z blast
 show "connected (S \ inside S)"
java.lang.StringIndexOutOfBoundsException: Range [50, 6) out of bounds for length 50
 show (S\<union> inside S) \<inter> path_image \<gamma> = {}"
 unfolding
 ( ComplD wherevalid_path \<notin> path_image p"
java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 5
qed

\<>Bounding WN by1/for andpoint using winding_number [OF \<gamma> z, of "min d e/2"] \<open>d>0\<close>
lemmawinding_number_subpath_continuous
 assumes \<gamma>: "valid_path \<gamma>" and z: "z \<notin> path_image \<gamma>"have: "w
 simp
rule
 let -
 show "continuous_on {0..java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
 proof (intro indefinite_integral_continuous_1 winding_number_exp_integral continuous_intros)
 show java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 25

b
 show "1 / (2 * pi *java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 ifobtain q:
java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9
 have "1 / (2*pi*\) * ?f x = 1 / (2*pi*\) * contour_integral (subpath 0 x \) (\w. 1/(w - z))"
 assms
byrule_tac"java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 also have "\ = winding_number (subpath 0 x \) z"
 (subst)
 show\<notin> path_image (subpath 0 x \<gamma>)"
 using assms x atLeastAtMost_iff e "e>0 cbg p
 qed (use assms x valid_path_subpath in \<open>force+\<close>)
 finally show ?thesis( simp [" "

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)"show holomorphic_onz 3/4*pe
 shows "\t \ {0..1}. Re(winding_number(subpath 0 t \) z) = w"
proofobtain:real where 0 winding_number_subpath_continuous z by blastw: wherew
 moreover have "Re (winding_number B order.transbyblast
java.lang.StringIndexOutOfBoundsException: Index 56 out of bounds for length 56
 show
 using ivt_increasing_component_on_1[ofmoreover have "winding_number \ constant_on outside (path_image \)"
qed

lemma winding_number_ivt_neg:
 \<gamma>: "valid_path \<gamma>" and z: "z \<notin> path_image \<gamma>" and "Re(winding_number \<gamma> z) \<le> w" "w \<le> 0"
ows\<exists>t \<in> {0..1}. Re(winding_number(subpath 0 t \<gamma>) z) = w"
proof -
 havehave:ewnot notin
 using:eal "<"
 rule_tac" (pe/4) (/2pe^2L/)"in) (simpadd cmod_wn_diff)
 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>"pathstart
 by(simp add: continuous_at_imp_continuous_on continuous_at_winding_number)
 proofintro
 by subsection

lemma winding_number_lt_half_lemma:
 assumes
\<gamma> z) < 1/2"
-
{ Re
 then obtain t::real then "w\notin> p" usingwbyblast
 using winding_number_ivt_pos [OF \<gamma> z, of "1/2"] by auto
 ifwinding_number><noteq> winding_number \<gamma> z" and "y \<in> S" "z \<in> S" for y z
 using winding_number_exp_2pi [of "subpath 0 t \" z]
 apply( add: t \<gamma> valid_path_imp_path)
u closed_segment_eq_real_ivl t byfastforce: path_image_subpath Euler)
 have "b < a \ \ 0"
 proof -
java.lang.StringIndexOutOfBoundsException: Index 103 out of bounds for length 52
 by (metis (no_types) paghave"<>z.cmod z \ w"
 mem_ball_0 blast
 blast
 qed
 moreover have "b < a \ \ t"
 by (metis atLeastAtMost_iff image_eqI mem_Collect_eq pag path_image_def subset_iff t)
 ultimately " < \ (\ 0 - z)" "0 < a \ (\ t - z)" using az
 by ( add inner_diff_right
 then java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
 by ( add gt mult_less_0_iff
 }
 then showby(ghtarrow\gamma winding_number
qed

lemma winding_number_lt_half:
lemma:
 shows "\Re (winding_number \ z)\ < 1/2"
proof -
 have "z \ path_image \" using assms by auto
by(rule
 by (metis complex_inner_1_right winding_number_lt_half_lemma( simp pathfinish_def bysimpclosed_path_image
 winding_number_reversepath valid_path_imp_path inner_minus_left path_image_reversepath)
 "path \; convex s; pathfinish \ = pathstart \; z \ s; path_image \ \ s\ \ winding_number \ z = 0"
 using \<open>z \<notin> path_image \<gamma>\<close> winding_number_lt_half_lemma by fastforce e 0<e w: "cmod w - )2 / (8* )java.lang.StringIndexOutOfBoundsException: Index 92 out of bounds for length 92
qed

lemma :

 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 -
 { B: where " and path_image
 have "isCont (winding_number \) z"
 by (metis continuous_at_winding_number valid_path_imp_path \<gamma> z)
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