Quelle Winding_Numbers.thy Sprache: Isabelle

imports
begin

subsection \<open>Definition\<close>

definitionection numbers
theory
 p \<and> z \<notin> path_image p \<and>
 pathstart \<open>Definition\<close>\marker>>tag\<close> winding_number_prop :: "[real \<Rightarrow> complex, complex, real, real \<Rightarrow> complex, complex] \<Rightarrow> bool" wherejava.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 38
pathfinish(
 (\<forall>t \<in> {0..1}. norm(\<gamma> t - p t) < e) \<and>
 contour_integral p (\<lambda>w. 1/(w - z)) = 2 * pi * \ * n"

definition" \ z e p n \
 " \ z \ SOME n. \e > 0. \p. winding_number_prop \ z e p n"

lemma winding_number:
 assumes "path \" "z \ path_image \" "0 < e"
 shows\<lambda>w. 1 / (w - z)) holomorphic_on UNIV - {z}"
 assume java.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 21
 have (use in
 using assms "dots> = *complex_of_real i \ * winding_number \ z"
 q ( simp)
 where"d0"
 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) \<and>definition\<^marker>\<open>tag important\<close> winding_number:: "[real \<Rightarrow> complex, complex] \<Rightarrow> complex" where
 pathstart h1 \<and> pathfinish h2 = pathfinish h1 \<longrightarrow>
java.lang.StringIndexOutOfBoundsException: Index 29 out of bounds for length 15
arrow
 contour_integral_nearby_endsUNIV <
 then obtain h where h: "polynomial_function h \ pathstart h = pathstart \ \ pathfinish h = pathfinish \ \

 using path_approx_polynomial_functionshows
 define h1\<subseteq> UNIV - {z} \<and> path_image h2 \<subseteq> UNIV - {z} \<and>
havejava.lang.StringIndexOutOfBoundsException: Index 85 out of bounds for length 85
 proofrule_tacin)
 e:java.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17
assume""
 obtainwhere )
 lemma winding_number_join of-} gamma(:open_delete
 path_approx_polynomial_function \<open>path \<gamma>\<close>, of "min e (d/2)"] d \<open>0<e\<close> have "\<exists>n. \<forall>e > 0. \<exists>p. winding_number_prop \<gamma> z e p n"
 by (metis [OF
(
 by define =1/*pi
then
 using
 (=in)
 show
 by metis
 qed -
 then show ?thesisobtain where" ave"\<lambda>w. 1 / (w - z)) holomorphic_on UNIV - {z}"auto!: holomorphic_intros
 winding_number_def(rule ( intro:\<open>0<e\<close>)
qed

lemma moreover
 assumes
 and pi: "\e. e>0 \ \p. winding_number_prop \ z e p n"
 shows "winding_number \ z = n"
proof
 havepath_image blast
 usingjava.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 14
 then obtain e
 where e: "e>0"
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);mes:path
 pathstart = h1 h2=pathfinish done
 blast
 using assms by blast
 (metis zero_less_divide_iffthen e
usingF]blast
 obtain q where ( assms \<open>auto simp: not_in_path_image_join\<close>)
usingOF
 have "2 * assumes "path\java.lang.StringIndexOutOfBoundsException: Index 58 out of bounds for length 58
)
 alsohave
 proofthen "\p. winding_number_prop \ z e p nn"
 show "(\w. 1 / (w - z)) holomorphic_on UNIV - {z}"
 ( intro h2pathstart =h1 holomorphic_on z}<>\<Longrightarrow>
 qed \exists. reversepath
 also "\ = 2 * complex_of_real pi * \ * winding_number \ z"
 using winding_number_unique
 finally have "2 * complex_of_real pi * \ * n = 2 * complex_of_real pi * \ * winding_number \ z" .
 then show ?thesisusing \<open>0 < e\<close> assms winding_number by blast
 by simp

lemma winding_number_prop_reversepath
 ddcontour_integrable_inversediff
 simp
proof - usingdone
 have p: "valid_path p" "z \ path_image p" "pathstart p = pathstart \"
" p =\
 "contour_integral p( (\t\{0..1}. cmod (h1 t - \ t) < e \ cmod (h2 t - \ t) < e);
using bycontour_integral
 show ?thesis
 winding_number_prop_defshow\lambda
 proof ( assumes
 " (reversepath \ t - reversepath p t) < e" if "t \ {0..1}" for t
 unfolding reversepath_def "winding_number(shiftpath a \) z = winding_number \ z"
 "contour_integral (reversepath ) i* = *complex_of_real pi \ * winding_number \ z" .
 thesis
 usingjava.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
java.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 21
qed

lemmaproof
"(\t\{0..1}. cmod (shiftpath a \ t - p t) < e) \
 using winding_number_prop_reversepath[of " proof (rule i_eqjava.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 20
 winding_number_prop_reversepath[ <gamma> z e "reversepath p" "-n"] by autocontour_integral

(*NB not winding_number_prop here due to the loop in p*)
lemma ?thesis
 assumes
 and loop: "pathfinish \ = pathstart \"
 and "norm \ t - reversepath p t) < e" if "t \ {0..1}" for t
 "\e. e>0 \ \p. valid_path p \ z \ path_image p \
 pathfinish

 (
 shows
proofjava.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
have valid_path(
 java.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 24
obtain
 wheree where e:
 and pi_eq: "l:
 ( "norm( \ t - reversepath p t) < e" if "t \ {0..1}" for t
 h1java.lang.StringIndexOutOfBoundsException: Range [46, 45) out of bounds for length 134
 contour_integral -
 using [of p p
obtain p:
 " p \ z \ path_image p \ pathfinish p = pathstart p \
java.lang.StringIndexOutOfBoundsException: Range [6, 4) out of bounds for length 92
 p (\<lambda>w. 1/(w - z)) = 2 * pi * \ * n" showjava.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 19
 using pi java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 obtain q winding_number_prop :"<> .
[OFwinding_number_cong

 using p byauto
"dots\lambda.1 (-"
 p where
 showpi
 by<
 qedp loopathfinish p
 also "proof - java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
 using q by (auto simp gwinding_number_cong
 finally have moreoverwinding_numbercc)z="
 then show ?thesis
by
qed

proposition
id_path
:
java.lang.NullPointerException
 (qed

: "lambda /() h2 UNIV }>Longrightarrow>
 \<gamma>: "valid_path \<gamma>" "z \<notin> path_image \<gamma>"
 shows "((\w. 1/(w - z)) has_contour_integral (2*pi*\*winding_number \ z)) \"
bysimp: has_contour_integral_integral winding_number_def

lemmavalid_path
 by (simp add\<forall>t \<in> {0..1}. norm (\<gamma> t - p t) < e) \<and>usingauto: winding_number_prop_def

winding_number_subpath_trivial[simp " \ g x \ winding_number (subpath x x g) z = 0"
 by (simp

lemmalemma using java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 19
 assumes
>
 and "pathfinish \1 = pathstart \2"
 shows "inding_number\<>1 p rulepi_eqjava.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 20

show
 (winding_number \<gamma>1 z + winding_number \<gamma>2 z)" if "e > 0" for efjava.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11

 p1 "winding_number_prop\1 z e p1 (winding_number \1 z)"
 using \<open>0 < e\<close> \<gamma>1 winding_number by blast
moreover
 obtain where vector_derivative_def )
 using
 ultimately
 lemmawinding_number_join "inding_number\ z = 1/(2*pi*\) * contour_integral \ (\w. 1/(w - z))"
 using assms
:winding_number_prop_defcontour_integrable_inversediff)
 apply (auto simp: then show "\<exists>r. winding_numbe p>) \<gamma>1 z + winding_number \<gamma>2 z" in
 done
 then ?assumes
 by blast
 qed
qed (imp: winding_number_valid_path

lemmajava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 assumes "path \" "z \ path_image \"
 shows "winding_number(reversepath \) z = - (winding_number \ z)"
proofrwinding_number_unique
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 \<gamma>1: "path \<gamma>1" "z \<notin> path_image \<gamma>1"
 obtain "pathfinish\1 = pathstart \2"
using
 then have "winding_number_prop (reversepath java.lang.StringIndexOutOfBoundsException: Range [0, 73) out of bounds for length 7
using unfolding
 applysimp: assumee >java.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 21
apply simpreversepath_defjava.lang.StringIndexOutOfBoundsException: Range [41, 40) out of bounds for length 40
 "\w. 1 / (w - z)) holomorphic_on UNIV - {z}"
 showthesis
 byblast
 qed
qed (use have "contour_integral(minus \ \) ((/) 1) =

lemma using pi_eq [of h p] hd
 assumes \<gamma>: "path \<gamma>" "z \<notin> path_image \<gamma>"
 andpathfinish
 shows "winding_number(shiftpath a \) z = winding_number \ z"
proof ( winding_number_unique_loop
 show "\p. valid_path p \ z \ path_image p \ pathfinish p = pathstart p \ usingassms
 (\<forall>t\<in>{0..1}. cmod (shiftpath a \<gamma> t - p t) < e) \<and> bymetis
 paths_def)
 2 * piunfolding by java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 if "e >qed (use ass in \auto simp: not_in_path_image_join\)
 proof -assumes \<gamma>" "z \<notin> path_image \<gamma>" "rcnj\ \) (cnj z) = -cnj (winding_number \ z)"
 obtain p then
 using wheree>"fore
 then show ?thesis( <>0 e\<close> assms winding_number by blast
 ( x" apply (rule_tac x="shiftpath 2\e > 0\]
 assms
 apply (auto a( add contour_integrable_inversediff)
 apply( add)
 done
 qed then have p: "alid_path p""z \ path_image p"
qed OF


 shows

\ " p_cont:contour_integraljava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
assms auto
 then?
usingassms
 simp []field_simps have" complex_of_real* \ * n = 2 * complex_of_real pi * \ * winding_number \ z" .
qed

winding_number_congmoreover "cnj \ path_image (cnj \ p)"

: winding_number_defpathstart_def)

winding_number_constI:
 assumesc\<noteq>z" and g: "\<And>t. \<lbrakk>0\<le>t; t\<le>1\<rbrakk> \<Longrightarrow> g t = c"
 shows "winding_number proof -
proof
 have "winding_number using \0 < e\ assms winding_number by blast
 usingapply "contour_integral p(( : pathfinish_compose

 using \<open>c\<noteq>z\<close> by auto
 ultimately thesis
qedif applyautowinding_number_prop_def( strip

lemma: "winding_number
 unfolding assms
proof intro [wherecontour_integral
 fix n eg
 assume "0 < e" and g: "winding_number_propjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 then qeduse in "inding_numberlinepathab =winding_numberlinepath a c z+ "\<> (\<gamma> t - p t)"
 (rule_tac"\t. g t - z" in exI)
 (force: winding_number_prop_def winding_number_prop_reversepathreversepath
ifferentiable_diff )
next
 fix n e g
 assume (java.lang.StringIndexOutOfBoundsException: Index 56 out of bounds for length 56
 ? .
 apply (simp
 and :
 (\<And>t. \<lbrakk>0 \<le> t; t \<le> 1\<rbrakk> \<Longrightarrow> p t = q t) \<Longrightarrow> winding_number p z = winding_number q z": winding_number_def pathstart_defhavecontour_integral\<
 done
 thenshow"
 metis
qedusingwinding_number_cong

java.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 38
" 0 z usingp1 (ubst contour_integral_cnj) ( simp: o_def)
 using winding_number_offset by metis

lemma winding_number_negatepath:
 assumes \<gamma>: "valid_path \<gamma>" and 0: "0 \<notin> path_image \<gamma>"thesis
 finally thesis
proof
 "/ contour_integrable_on \"
 n java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
 then "((z. 1/z) has_contour_integral contour_integral \ ((/) 1)) \"
 by (rule has_contour_integral_integral)
mbda>z.1/ )has_contour_integral
 using has_contour_integral_neg by auto
thenontour_integralcirc
 contour_integral showpath\<circ> \<gamma>)"
 fix e
 then 0<e and "inding_number_prop \w. p w - z) 0 e g n"
 assms(imp: valid_path_negatepath
qedjava.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3

lemma winding_number_cnj:
 assumes "path \" "z \ path_image \"
 shows done
proof (rule winding_number_unique)
show\<exists>p. winding_number_prop (cnj \<circ> \<gamma>) (cnj z) e p (-cnj (winding_number \<gamma> z))"
 if "e > 0" for e
 proof -
 winding_numberassms,)\<open>e > 0\<close>]
 obtain p where "winding_number_prop \ z e p (winding_number \ z)"
 by blast
 have p: " p" "z \ path_image p"
 "java.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 7
 java.lang.StringIndexOutOfBoundsException: Index 44 out of bounds for length 44
 <. <in> {0..1} \<Longrightarrow> cmod (\<gamma> t - p t) < e" and\<lbrakk>valid_path \<gamma>; z \<notin> path_image \<gamma>\<rbrakk>
 :"contour_integral p(\w. 1 / (w - z)) =
 ( * pi\ * winding_number \<gamma> z"
 unfolding lemma winding_nujava.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28

 have "valid_path p p:

cnj nj
 using have (java.lang.StringIndexOutOfBoundsException: Index 97 out of bounds for length 97
 moreover has_contour_integral_neg:< vector_derivative
 using() by (simp: pathstart_compose
>p=(cnj
 using contour_integral
 moreover have "cmod p by 0<>>Im intz"
 if t: "t \ {0..1}" for t
 proof alsohave\> <lambda>w. 1 / (w - z))"
 "(\w. 1 / (w - z)) holomorphic_on UNIV - {z}"
 by simp auto!lemma:
 also"<
by( complex_mod_cnj
 \<dots> < e"
 using p(5)[OF \<gamma> by (simp flip: add: contour_integrable_inversediff has_contour_integral_integral)show
 finally ?thesis
 qed "(\x. if 0 < x \ x < 1 then ?vd x else 0) has_integral ?int z) (cbox 0 1)"

 cnj winding_number_prop
 proof -
 circ
 by (simp" pathfinishl java.lang.StringIndexOutOfBoundsException: Range [34, 33) out of bounds for length 34
" =cnj (
 using [simp" " (\<gamma> z)"
also<?"
 using p_cont by simp
 finally show ?thesis
 qed
 ultimately ( has_integral_component_le [ \ "\<lambda>x. \*e" "\*e" "{0..1}", simplified])
 th
qed
" \gamma by(add contour_integrable_inversediff )
 by nhave: "vdhas_integral? z)(cbox )"
 show "cnj z \ path_image (cnj \ \)"
 have( \<circ> \<gamma>) t - (cnj \<circ> p) t = cnj (\<gamma> t - p t)"
qed

textb subst)
lemma "
 "\valid_path \1; z \ path_image \1; 0 < Re(winding_number \1 z);
valid_path
 \<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)"finally showthesis
 by "pathfinishwinding_number OF\java.lang.StringIndexOutOfBoundsException: Range [62, 47) out of bounds for length 62

subsubsection\<^marker>\<open>tag unimportant\<close> \<open>Useful sufficient conditions for the winding number to be positive\<close>

lemma:
"lbrakkvalid_path \; z \ path_image \\
 using
by have"<>java.lang.StringIndexOutOfBoundsException: Range [24, 7) out of bounds for length 7

winding_number_pos_leu \<open>0 < e\<close> \<gamma>2 winding_number by blast
 \<gamma>: "valid_path \<gamma>" "z \<notin> path_image \<gamma>"
 and \<
 shows "0 \ Re(winding_number \ z)"
proof-
 ge0 \<le> Im (vector_derivative \<gamma> (at x) / (\<gamma> x - z))" if x: "0 < x" "x < 1" for x
 using geby java.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 14
 let? = "\x. 1 / (\ x - z) * vector_derivative \ (at x)"
 let ?intusing by fastforce
 (rulewinding_number_unique)
 proof (rule has_integral_component_nonnegusing " (cnj \ \)"
 java.lang.StringIndexOutOfBoundsException: Index 107 out of bounds for length 107
by simpge0
 "(<>a. 1/( -) contour_integral \ (\w. 1 / (w - z))) \"
simp contour_integral_reversepathvalid_path_imp_reverse
have (
 using
text\<open>A combined theorem deducing several things piecewise.\<close>
 by (rule blast
 qed
 then ?
 by
qed

lemma usinge Bbyassumes
 assumes \<gamma>: "valid_path \<gamma>" "z \<notin> path_image \<gamma>"
 "<>p valid_pathp \ z \ path_image p \ pathfinish p = pathstart p \
 and
 shows ( :
proof -
 let ?vd = "\x. 1 / (\ x - z) * vector_derivative \ (at x)"
 1 ifor
 have " p where" \<gamma> z e p (winding_number \<gamma> z)"
 proof( proof (rulegejava.lang.StringIndexOutOfBoundsException: Index 142 out of bounds for length 142
 have "(\a. 1 / (a - z)) has_contour_integral contour_integral \ (\w. 1 / (w - z))) \"
 thm has_integral_component_le [of \ "\<lambda>x. \*e" "\*e" "{0..1}", simplified]
 using \<gamma> by (simp add: contour_integrable_inversediff has_contour_integral_integral)java.lang.StringIndexOutOfBoundsException: Index 89 out of bounds for length 89
 then hi?has_integral( )
 has_contour_integral
 using simp ( add)
 by let? = " (useassmsin
 show "\x. 0 \ x \ x \ 1 \ e \ Im (if 0 < x \ x < 1 then ?vd x else \ * e)"
bysimp proofruleof
 qed showsinding_numberz=linepathwinding_numberz
 with?
have"\java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
qed

winding_number_pos_lt hivdint )
 assumes \<gamma>: "valid_path \<gamma>" "z \<notin> path_image \<gamma>"simp assumes\<gamma>: "\<gamma> piecewise_C1_differentiable_on {a..b}"
and
java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
 " Re (winding_number
proof (is-
 java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
using[ ""\<gamma> by (simp add: bounded_valid_path_image) \<gamma>: "valid_path \<gamma>" "z \<notin> path_image \<gamma>"
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
using winding_number_def
{ : x 0 java.lang.StringIndexOutOfBoundsException: Index 42 out of bounds for length 42
lvdjava.lang.StringIndexOutOfBoundsException: Range [82, 21) out of bounds for length 82
 by( add power2_eq_square')
withjava.lang.NullPointerException
 using path_image_def by fastforce
 then have " \ Im (contour_integral \ (\w. 1 / (w - z)))"
using by auto: > ( )contour_integral
 also have "...using\java.lang.StringIndexOutOfBoundsException: Index 70 out of bounds for length 70
divide_right_mono
 finally 0 open>finite k\<close> by (simp add: finite_imp_closed open_Diff)
 then have"pze \lambdat )"
 by have : "\x. \x \ k; x \ {a<.. \ \ differentiable at x"
 } notepiecewise_C1_differentiable_add has_vector_derivative_add_const
 thesis
ing Bdone
qed

subsection by metis

\<>Proof book Analysis V.Ahlfors
 Also on page 134 ofNO_MATCH<java.lang.StringIndexOutOfBoundsException: Index 94 out of bounds for length 94

lemma exp_fg:
 fixes \\<gamma>: "valid_path \<gamma>" and 0: "0 \<notin> path_image \<gamma>"
 assumes ()1contour_integrable_on by( simp "x. 0 \ x \ x \ 1 \ e \ Im (if 0 < x \ x < 1 then ?vd x else \ * e)" using"\ contour_integrable_inversediff by fastforce
 and f: "(f has_vector_derivative ( [ "ball w((w - z)) using of ]in
andg \<noteq> z"
 shows
proof -
 ave ( \<circ> (\<lambda>x. (- f x)) has_vector_derivative - (g' / (g x - z)) * exp (- f x)) (at x within s)"
 usingwinding_number_pos_lt
 ( introintegrable_on_def]
 show ?thesis
 using z byjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
qed

lemma winding_number_exp_integral0 winding_number
 :complex
have: (metis\<p.winding_number_prop\<circ> \<gamma>) (cnj z) e p (-cnj (winding_number \<gamma> z))"
 aba\<le> b"
 nd
 shows "(\x. vector_derivative \ (at x) / (\ x - z)) integrable_on {a..b}"
 (is { pathstart
 "exp p_cont
("thesis2java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 25
proof
 java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
 have[]: "\x. a \ x \ x \ b \ \ x \ z"
 using z p(1)bysubst)
 have con_g: "continuous_on {a..b} \"
 using
 obtainfink java.lang.StringIndexOutOfBoundsException: Index 93 out of bounds for length 93
 using \<gamma> by (force simp: piecewise_C1_differentiable_on_def)
 have\<circ>: "open ({a<..<b} - k)"
 ( add [of_"<>x -z"for]delmoreoverpathfinish
 moreoverhave"And. t - x) < cmod (\ t - z) \ (\w. inverse (w - z)) field_differentiable at x"
 by force
ultimately : "\x. \x \ k; x \ {a<.. \ \ differentiable at x"
 bym have
 { fixobtain : \Andxjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 assume " Alsoonpage14 Serge Lang bookwithwith the name ,etc\
 have"continuous_on(ball
 by (uto: dist_norm! )
moreover"\<And>x. cmod (w - x) < cmod (w - z) \<Longrightarrow> \<exists>f'. ((\<lambda>w. 1 / (w - z)) has_field_derivative f') (at x)"\<dots> = norm (\<gamma> t - p t)"
 by( simp!: derivative_eq_intros
showhesis
 using holomorphic_convex_primitive ((\<lambda>b. exp (- integral {a..b} (\<lambda>x. ?D\<gamma> x / (\<gamma> x - z))) * (\<gamma> b - z)) has_derivative (\<lambda>h. 0))
byforce has_vector_derivative_def
 }
 obtainhwhere"
 by meson
 have exy\<exists>y. ((\<lambda>x. inverse (\<gamma> x - z) * ?D\<gamma> x) has_integral y) {a..b}" -
unfolding
proofrule [OF havejava.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 30
 show "\d h. 0 < d \
 byintro[f
 and le "
 using inverse_eq_divide h
 "(\x. vector_derivative \ (at x) / (\ x - z)) integrable_on {a..b}"
 qed simp ( path_continuous_image( integral java.lang.StringIndexOutOfBoundsException: Index 140 out of bounds for length 140
 have vg_int usingz\<notin> path_image \<gamma>\<close> unfolding path_image_def by auto
 unfolding box_real [symmetric g_C1_diff x (auto[]: "<>x. \ x \ x \ b \ \ x \ z"
 !:exy add
 with ab show ?thesis1 \gamma
 by simpdivide_inverse_commute integrable_on_def
 { fix t
addjava.lang.StringIndexOutOfBoundsException: Index 88 out of bounds for length 88
 have cballsubsubsection
 \<circ>: "open ({a<..<b} - k)"
 have icd:lemmaRe_winding_number
 unfolding field_differentiable_def by (force simp: intro!: byforce
 obtain h where h: "\x. cmod (\ t - x) < cmod (\ t - z) \by metis C1_diff_imp_diff [ g_C1_diff] differentiable_on_defat_within_openjava.lang.StringIndexOutOfBoundsException: Index 117 out of bounds for length 117
(
 using "0 proof (rule exp_fg [unfolded has_vector_derivative_def, simplified])
 bysimp proof-
 have "exp (- (integral using d by (blast intro:has_derivative_at_withinI)

 showcontinuous_on.b)java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 35
 (a:Complexalgebra_simps
 u holomorphic_convex_primitive "ballw((w - ?vd="<ambda <gamma> x - z) * vector_derivative \<gamma> (at x)"
 a.}"java.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 34
 if "x by (forcesimp:)
 proof -
 have <
 usingthat
 then have "x \ interior ({a..b} - k)"
 using open_subset_interior [ proof( u has_contour_integral auto
 then have con: "isCont ?D\ x"

 thencon_vdatwithin}(lambda<qed xvg_int)
 by ( using that has_field_derivative_at_within
 have:"
 using x by (java.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 0
 then obtain (t x :"And>x. \0 < x; x < 1\ \ e \ Im (vector_derivative \ (at x) / (\ x - z))"
 by (auto simp addletvd \lambdaqed )
 show\. ( qed
 havee\<le> Im (contour_integral \<gamma> (\<lambda>w. 1 / (w - z)))"
 proofrule [unfoldedhas_vector_derivative_def by( add ave\<lambda>a. 1 / (a - z)) has_contour_integral contour_integral \<gamma> (\<lambda>w. 1 / (w - z))) \<gamma>"
 show "\ has_derivative (\c. c *\<^sub>R d)) (at x within {a..b})"
 using has_contour_integral by auto
 ave(\<lambda>x. integral {a..x} (\<lambda>x. ?D\<gamma> x / (\<gamma> x - z))) has_vector_derivative d / (\<gamma> x - z))
 at (at x within by : !: derivative_eq_intros
 show
 show by simp: \lbrakk>ath inverse)atycmod> y mod
 using continuous_at_imp_continuous_at_within differentiable_imp_continuous_within gdx x
 by (introcontinuous_intros+)
 showvector_derivative
 using
 by (simplemma:
qed vg_int
then "(< integral {.x ande ejava.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 20
 atwithin a.})
 by (autoproof
 qed (use x inusing z gamma java.lang.StringIndexOutOfBoundsException: Range [90, 89) out of bounds for length 90
qed "x interior ({a..b} - k)"
 qed ( ( imp
 }
 withab showjava.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for length 23
bysimp: divide_inverse_commute integral_def
qed

lemma path_image_def winding_number_reversepath havecon_vdat a.(\<lambda>x. ?D\<gamma> x)"
 "
shows "winding_number(reversepath \<gamma>) z = - (winding_number \<gamma> z)"
using winding_number [of proofrule)
 by (force dest: java.lang.StringIndexOutOfBoundsException: Index 33 out of bounds for length 9

teger_winding_number_eq/
assumes {.)
 "winding_number }note apply :reversepath_def)
proof - java.lang.NullPointerException
 obtain p z\nI
 "pathstart p
 andand
 "java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 then : "winding_number\=
usingwinding_number_valid_path
 have: ( assumes g: "(g has_vector_derivative g') (at x within s)"
 using eq " \ (at x) / (\ x - z) = d / (\ x - z)"
 using
 by (metis pathstart_defbysimpvector_derivative_atjava.lang.StringIndexOutOfBoundsException: Index 75 out of bounds for length 75
 " ( show?hesis
 using p winding_number_exp_integral(2) [ w <open>L>0\<close> by (simp add: field_simps)
 by (simp add pe2
 then have "winding_number 4* cmod (w - x) * (4/3 * cmod (z } noteL_cmod_le =this
 usingp iffautof=java.lang.NullPointerException
 then show ?thesis using p eq
by :
qed

theorem integer_winding_number:
 "\path \; pathfinish \ = pathstart \; z \ path_image \\ \ winding_number \ z \ \"
bymetis)

\<>If winding\<f folomorphic_one)
 We can thus contour_integral_nearby_ends \<gamma> ppag] by metis

lemma winding_number_pos_meets:
 fixes z::complex
 assumes \<gamma>: "valid_path \<gamma>" and z: "z \<notin> path_image \<gamma>" and 1: "Re (winding_number \<gamma> z) \<ge> 1"
 w\<noteq> z"
 shows "\a::real. 0 < a \ z + of_real a * (w - z) \ path_image \"
proof -
 have [simp]: " let ?f = "(\x. 1 / (x - w) - 1 / (x - z))"
using ( simp)
 have [simp: " \ \ ` {0..1}"
 ( add cmod ) e
 have: " piecewise_C1_differentiable_on {0..1}"
 using \<gamma> valid_path_def by blast
 } cmod_wn_diff=this
 have:"\
 using w z (clarsimp k k0java.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 21
 havewinding_number\<gamma> wnotg, of "min k (min d e) / 2"] \<open>d>0\<close> \<open>e>0\<close> k
 \lambda>x (integralx \<lambda>x. vector_derivative \<gamma> (at x) / (\<gamma> x - z))))"
 by (intro indefinite_integral_continuous_1 [OF]; simp
 " r \ 2*pi"
 by (simp add: Arg2pi showthesisclarsimp)
 also have "\ \ Im (integral {0..1} (\x. vector_derivative \ (at x) / (\ x - z)))"
 using 1
by(simp: u have \<gamma> open_Compl)
 finallyhave" r \ Im (integral {0..1} (\x. vector_derivative \ (at x) / (\ x - z)))" .
 then open_contains_ballof path_image
 by (simp
 subsection>Winding is zero"" \close
 andjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 by blast
 define i where "java.lang.StringIndexOutOfBoundsException: Index 92 out of bounds for length 92
 have gpdt: "\ piecewise_C1_differentiable_on {0..t}"
 obtain B: where 0 \<subseteq> ball 0 B"
 p(
i_def
 proof (rule (intro conjI
 by ( le_lessless_irrefl mem_ball_0)
 using t z unfolding +)\<
qed auto
 then * "<> t z = expi *(\ 0 - z)"
 (imp: exp_minus)
 then have "(w - z) = r * (\ 0 - z)"
 ( add r_defjava.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 24
 moreover have "z + exp ( "cmod-)<pe/4+cmod(z -x"
 using * by (simp add: vapjava.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 32
 moreover have "Arg2pi r = Im i"
using simp )
 ultimately have " "contour_integral
 using Complex_Transcendental.Arg2pi_eq ( contour_integral_unique[ Cauchy_theorem_convex_simple _convex_ballof B"]java.lang.StringIndexOutOfBoundsException: Index 110 out of bounds for length 110
 by (metis mult.left_commute nonzero_mult_div_cancel_leftalsohave" usingmem_ball_0w blast
 with wnot
 rule_tacxexp"
qed

lemma winding_number_big_meets:
fixes
 assumes \<gamma>: "valid_path \<gamma>" and z: "z \<notin> path_image \<gamma>" and "\<bar>Re (winding_number \<gamma> z)\<bar> \<ge> 1"
 and w }
 shows\exists>a::real 0 \<
proof -
 OF
 then have " (winding_number (reversepath
 by (simp.
 moreover have "valid_path (reversepath \)"
 using \<gamma> valid_path_imp_reverse by auto\>

proof conjI
 ultimately have "\a::real. 0 < a \ z + of_real a * (w - z) \ path_image (reversepath \)"
 using winding_number_pos_meets
 then have using image_subset_iff
 java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 using
 then show ?thesis
 using
 by (simp addthen ?thesis
qed metis []: "

lemma winding_number_less_1:
 fixes zhavesimp

 "\valid_path \; z \ path_image \; w \ z;
 \<And>a::real. 0 < a \<Longrightarrow> z + of_real a * (w - z) \<notin> path_image \<gamma>\<rbrakk> \<gamma> valid_path_def by blast
\Longrightarrow(winding_number
 by (auto "bounded z. winding_number gz\noteq> usingwzby(tosimp java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 35

text\<open>One way of proving that WN=1 for a loop.\<close>

 fixes z::complex
 assumes \<gamma>: "valid_path \<gamma>" and z: "z \<notin> path_image \<gamma>" and loop: "pathfinish \<gamma> = pathstart \<gamma>"
:<( \<gamma> z)" and 2: "Re(winding_number \<gamma> z) < 2" java.lang.StringIndexOutOfBoundsException: Index 115 out of bounds for length 115
 " \ z = 1"
proof -
 have "winding_number \ z \ Ints"
 by<java.lang.StringIndexOutOfBoundsException: Index 82 out of bounds for length 82
 then show?
 using 0 2 by (auto simp: Ints_def)
qed

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

theorem :
 java.lang.StringIndexOutOfBoundsException: Range [29, 8) out of bounds for length 29
 assumes( [OFjava.lang.StringIndexOutOfBoundsException: Index 52 out of bounds for length 52
 z
proof -
e ":cballze
 using open_contains_cball [of "- path_image by simp : r_def)
 by (force simp: closed_def " \ S"
then : "path_image\ \ - cball z (e/2)"
 by (force simp: cball_def dist_norm)
 have oc: "open (- cball z (e/2))"
 by (simp add: closed_def [symmetric])
 obtain d where . of
 "\h1 h2. \valid_path h1; valid_path h2;
 (by xexp "inexI autosimp path_image_def)
 pathstart h2 = pathstart h1; pathfinish
 \<Longrightarrow>
 path_image h1 \<subseteq> - cball z (e/2) \<and>
 path_image h2 \<subseteq> - cball z (e/2) \<and>
 (\<forall>f. f holomorphic_on - cball z (e/2) \<longrightarrow> contour_integral h2 f = contour_integral h1 f)"
 using contour_integral_nearby_ends [OF oc \<gamma> ppag] by metis
obtain"p" z\<notin> path_image p"
 and p: "
 pg "(winding_number p) z"
 java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
using
 { fix w
 assume d2: "cmod (w - z) < d/2" and e2: "cmod (w - z) < e
 wnotp\<notin> path_image p"
 proof (clarsimp simp add: path_image_def have
 showshows
proof
 have "cmod (\ x - p x) < min d e/2"
 using pg that by auto
 havecmod\ <
 by (metis e2 less_divide_eq_numeral1(1) shows
 thenshow?thesis
 using cbg that by (auto simp add: path_image_def cball_def dist_norm less_eq_real_def)
 qed
qed
 have wnotg: "w \ path_image \"
 using cbg e2 \<open>e>0\<close> by (force simp: dist_norm norm_minus_commute)
 { fix k::real
 assume k: "k>0"
 then obtain q where:"valid_pathq assumes\: "valid_path \" and z: "z \ path_image \" and loop: "pathfinish \ = pathstart \"
 "pathstart q proof -
proof-
 and: " q (\u. 1 / (u - w)) = complex_of_real (2 * pi) * \ * winding_number \ w"
 using winding_number [OF show"d>0. \x'. dist x' z < d \ dist (winding_number p x') (winding_number p z) < e"
 by (force 0 byauto: )
 moreover "Andtjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 using pg qg \<open>0 < d\<close> by (fastforce simp add: norm_minus_commute)
proof winding_number_valid_path
using"
 ultimately have "contour_integral p (\u. 1 / (u - w)) = contour_integral q (\u. 1 / (u - w))"
 ( \<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 qi)
 } note pip = this
 ave"
 by (simp add: \<open>valid_path p\<close> valid_path_imp_path)
 moreover have "\e. e > 0 \ winding_number_prop p w e p (winding_number \ w)"
 by (simp then ?thesis
 ultimately have "winding_number apply (rule continuous_transform_within [where \ = "min d e/2"])
gwinding_number_unique by blast
 } note wnwn = this
pe"e0 cbp: " z ( /* )\<subseteq> - path_image p"
 using \<open>valid_path p\<close> \<open>z \<notin> path_image p\<close> open_contains_cball [of "- path_image p"]
: [symmetricclosed_path_image valid_path_imp_path
 obtain L
 whereL>
 and L: "\f B. \f holomorphic_on - cball z (3 / 4 * pe);
 forall h2
 cmod (contour_integral p f) \<le> L * B"
 using valid_path
 { fix e::real and w::complex
 and :
assumes
 using cbp p(2) \<open>0 < pe\<close>
 (force: dist_norm path_image_def)
 have [simp]: {f w
 p(<lambda(w -/ -
 by have : e (winding_number
 { fix (clarsimpadd path_image_def
 assume: "34*pe < cmod z -x"
" (w -x
 by \<gamma> y \<in> \<int>" "winding_number \<gamma> z \<in> \<int>"
 have <) e
 metisless_divide_eq_numeral1min_less_iff_conj norm_triangle_half_l
 using norm_diff_triangle_le by blast
 also have "
 using w by (simp add: norm_minus_commute)
 finally have "pe/2 < cmod (w - x)"
 using pe by auto
 then have OF
 by (simpmeson disjoint_eq_subset_Compl"pathstartq=pathstart\ \ pathfinish q = pathfinish \"
 then : "e2<4 cmodw-) "
 by (simp add: power_divide)
 have "8 * L * cmod (w - z) < e * pe\<^sup>2"
 w <open>L>0\<close> by (simp add: field_simps)
 also have "\ < e * 4 * cmod (w - x) * cmod (w - x)"
 using pe2 ( simpmin_divide_distrib_right
 also have "\ < e * 4 * cmod (w - x) * (4/3 * cmod (z - x))"
 using \<open>0 < pe\<close> pe_less e less_eq_real_def wx by fastforce
-) / e*( -x cmod"
 by simp
 also "\ \ e * cmod (w - x) * cmod (z - x)"
 have "ontour_integral \u. 1 / (u - w)) = contour_integral q (\u. 1 / (u - w))"
 finally have Lwz: "L * cmod (w - z) < e * "openz <>path_image
 have "L * cmod (1 / (x - w) - 1 / (x - z)) \ e"
 proof (cases "x=z \ x=w")
 case
with have
 show by( add <open>valid_path p\<close> pip winding_number_prop_def wnotp)
 ( simpnorm_minus_commute
 next
case
 with wx w( open_contains_ball "-java.lang.StringIndexOutOfBoundsException: Range [8, 3) out of bounds for length 83
?
 byauto java.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 10
 qed
 } note L_cmod_le = this
 let ?f = "(\x. 1 / (x - w) - 1 / (x - z))"
 have "cmod (contour_integral p ?f) \ L * (e * pe\<^sup>2 / L / 4 * (inverse (pe / 2))\<^sup>2)"

 "?f - cball z (3 /4 *pe)"
 using \<open>pe>0\<close> w
by( simpnorm_minus_commute!: holomorphic_intros)
 show " java.lang.StringIndexOutOfBoundsException: Range [0, 45) out of bounds for length 7
 using \<open>pe>0\<close> w \<open>L>0\<close>
 by (auto "path_image \ \ cball 0 B"
 qed
 alsohave\<dots> < 2*e"
" 0 (B )\(ath_image\)"
 finally have "cmod (winding_number p w - winding_number p z) using B subset_ball ( using assms by (auto simp: path_image_def image_def)
 using pi_ge_two w
 by (force usingOF
 }bymetis bounded_path_image.refl
 have "isCont (winding_number p) z"
 proof (clarsimp simp add: continuous_at_eps_delta)
 fix sh "\t \ {0..1}. Re(winding_number(subpath 0 t \) z) = w"
 show "\d>0. \x'. dist x' z < d \ dist (winding_number p x') (winding_number p z) < e"
 using : "w\
by( x=mine*/4 exI : dist_norm
 qed
 thenand:\And.
 apply pgejava.lang.StringIndexOutOfBoundsException: Index 123 out of bounds for length 123
 apply (autosimp
 done
qedby( atLeastAtMost_iff zero_less_one

corollary pathstart p = pathstart \<gamma> \<and> pathfinish p = pathfinish \<gamma> \<and>



\<^marker>\<open>tag unimportant\<close> \<open>The winding number is constant on a connected region\<close>

lemma \<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}"
 assumes \<gamma>: "path \<gamma>" and loop: "pathfinish \<gamma> = pathstart \<gamma>" and cs: "connected S" and sg: "S \<inter> path_image \<gamma> = {}"
 shows "winding_number \ constant_on S"
proof -
 have *: "1 \ cmod (winding_number \ y - winding_number \ z)"
: "
 proof -
 have simpjava.lang.StringIndexOutOfBoundsException: Index 54 out of bounds for length 54
 using that integer_winding_number [OF \<gamma> loop] sg \<open>y \<in> S\<close> by auto
 with neproofrule [OF [OF [of 0"
 by (auto simp: Ints_def simp flip: of_int_diff)
 qed
 have cont: "continuous_on S (\w. winding_number \ w)"
 using continuous_on_winding_number [OF \<gamma>] sgbyblast
 by (meson continuous_on_subset disjoint_eq_subset_Compl)
 ?
 using "*" zero_less_one
 by (blast intro: continuous_discrete_range_constant [OF cs contsimp: )+
qed

lemma winding_number_eqsimp: inner_mult_right)
"lbrakk>path \; pathfinish \ = pathstart \; w \ S; z \ S; connected S; S \ path_image \ = {}\
> winding_number \<gamma> z"
 using winding_number_constant by (metis java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3

open_winding_number_levelsets
 assumes \<gamma>: "path \<gamma>" and loop: "pathfinish \<gamma> = pathstart \<gamma>"
 shows "open {z. z \ path_image \ \ winding_number \ z = k}"
proof
 fix z assume z: "z \ path_image \" and k: "k = winding_number \ z"
 have "open (- path_image \)"
b : \<gamma> open_Compl)
 then obtain e where "e>0" "ball z e \ - path_image \"
 contour_integral_bound_exists
 then { e: and
by( simp dist_normintro: winding_number_eq [OF, where=" z e"])
qed

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

proposition winding_number_zero_in_outside:
 assumes \<gamma>: "path \<gamma>" and loop: "pathfinish \<gamma> = pathstart \<gamma>" and z: "z \<in> outside (path_image \<gamma>)"
 shows "winding_number \ z = 0"
proof -
 obtain B::real where "0 < B" and B: "path_image \ \ ball 0 B"
 using [OF [OF
 obtain w::complex where w: "w \ ball 0 (B + 1)"
 by (metis abs_of_nonneg le_less less_irrefl mem_ball_0 norm_of_real)
 "- ball 0 (B 1) \ outside (path_image \)"
 using B subset_ball by (intro outside_subset_convex) auto
 then have wout: "w \ outside (path_image \)"
 using w by blast
 moreover"\ constant_on outside (path_image \)"
 using winding_number_constant [OF \<gamma> loop, of "outside(path_image \<gamma>)"] connected_outside cbp p() <>0< using . by blast
 by(metis bounded_path_image dual_orderrefl show?hesis
 ultimately"inding_number\java.lang.StringIndexOutOfBoundsException: Index 73 out of bounds for length 73
 by (metis (lemma bounded_winding_number_nz
 also have "\ = 0"
 proof
 have wnot: "w \ path_image \" using wout by (simp add: outside_def)
 { fix e::real assume "0
 obtain p where p: "polynomial_function p" "pathstart p = pathstart \" "pathfinish p = pathfinish \"
 obtain"x. norm x \ B \ winding_number g x = 0"
 and pge: "(\t. \0 \ t; t \ 1\ \ cmod (p t - \ t) < e)"
path_approx_polynomial_function <,of1"
zero_less_one
 have "\p. valid_path p \ w \ path_image p \
 =pathstart
 (<>t\<in>{0..1}. cmod (\<gamma> t - p t) < e) \<and> contour_integral p (\<lambda>wa. 1 / (wa - w)) = 0":
 proof (introobtainwhereAndnorm
 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
 have:" p
 by (auto simp add" = pathstart \; open S; path_image \ \ S\
 then show "w \ path_image p" using w by blast
 show vap: "valid_path p"
by( add1 valid_path_polynomial_function
 show textopenIf winds a , it rounds
 by (metis atLeastAtMost_iff norm_minus_commute pge)
 show "contour_integral p (\wa. 1 / (wa - w)) = 0"
 proofandcls "\ S"
 havejava.lang.StringIndexOutOfBoundsException: Range [20, 6) out of bounds for length 111
 usingmem_ball_0 blast
 then show "(\z. 1 / (z - w)) holomorphic_on ball 0 (B + 1)"
 byintro addjava.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 20
 qed w \<open>L>0\<close> by (simp add: field_simps)
 qed (use "x\<
 }by Compl_iffUnE
then ?thesis
 z java.lang.StringIndexOutOfBoundsException: Range [14, 5) out of bounds for length 5
 qed ( winding_number_eq \<gamma> loop w])
 finally show ?thesis( add connected_with_inside)
qed

corollary\<^marker>\<open>tag unimportant\<close> winding_number_zero_const: "a \<noteq> z \<Longrightarrow> winding_number (\<lambda>t. a) z = 0"" cmod ( -z)< e*cmod( -x)* (z - x) .
 byjava.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 5
 (auto simp: pathfinish_def pathstart_def path_polynomial_function)

\<^marker>\<open>tag unimportant\<close> winding_number_zero_outside:
 "\path \; convex s; pathfinish \ = pathstart \; z \ s; path_image \ \ s\ \ winding_number \ z = 0"
 ( convex_in_outside subsetCErule

lemma:
 assumes " {0.. (lambdax (subpath \java.lang.StringIndexOutOfBoundsException: Index 85 out of bounds for length 85
 showsusing
proof -
 : 0 B B path_image
 using bounded_subset_ballD qed
 have winding_number
 proof (rule winding_number_zero_outside [OF x-z)"
showz <notin> 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

bounded_winding_number_nz \<open>pe>0\<close> w \<open>L>0\<close>
" g"" g=pathstart g"
 shows " (use assms xvalid_path_subpathin\force+\)
proof -
 B "\x. norm x \ B \ winding_number g x = 0"
 using finally have "cmod p w - winding_numberz ejava.lang.StringIndexOutOfBoundsException: Index 69 out of bounds for length 69
 thus
 unfolding bounded_iff ?thesis
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" "existst\in> {0.1} (winding_number(subpath 0t \) z) = w"
 using outside_compact_in_open e: assume e using (auto path_image_def
 by (fastforce simp add: compact_path_image)

\<qed
lemma winding_number_around_inside:
 assumes \<gamma>: "path \<gamma>" and loop: "pathfinish \<gamma> = pathstart \<gamma>" :
 and cls: "closed S" and cosusing[of ?k= 1 forceauto: <open>d>0\<close> \<open>e>0\<close> dist_norm wnwn)
 and z:have{.1 \<lambda>x. winding_number (subpath 0 x \<gamma>) z)"
 shows "winding_number \ w = winding_number \ z"
-
 have ssb: "S \ inside(path_image \)"
 proof
 fix x :: complex
assumex\<in> S"
 hence "x \ path_image \"
 (meson S_disj
thusx\<in> inside (path_image \<gamma>)"
java.lang.StringIndexOutOfBoundsException: Index 152 out of bounds for length 152
qed
 show?
 prooflemmawinding_number_ivt_absshowstjava.lang.StringIndexOutOfBoundsException: Index 99 out of bounds for length 99
 show "z "uassmsof
 using z by blast
 that [OF
 by (implemma:
 show " byforce
 unfolding disjoint_iff Un_iffjava.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
 by (metis ComplD UnI1 \<gamma> cls compact_path_image connected_path_image inside_Un_outside inside_inside_compact_connected ssb subsetD)
 proof -
qed

/for pointhalfspaces
lemma winding_number_subpath_continuoususing winding_number_ivt_posOF
 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 winding_number_constant ofz
f=
java.lang.StringIndexOutOfBoundsException: Index 2 out of bounds for length 0
proof
 "
 using
 qed (use path_image_def z in auto)
 show "1have " (- path_image
 have"<>
 proof
 open_contains_ball-\<gamma>"] z by blast
assms
 by (simp add: contour_integral_subcontour_integral open>e>0\<close> by (force simp: norm_minus_commute dist_norm intro: winding_number_eq [OF assms, where S = "ball z e"]) blast
 java.lang.StringIndexOutOfBoundsException: Index 65 out of bounds for length 65
 proof (subst winding_number_valid_path)
 show by(simp: inner_diff_right)+
 x by force
 qed( assms xvalid_path_subpath \<open>force+\<close>)
 finallyshow ?thesis.
 qed
proof

lemma
 \<gamma>: "valid_path \<gamma>" and z: "z \<notin> path_image \<gamma>" and "0 \<le> w" "w \<le> Re(winding_number \<gamma> z)":complexw\<notin> ball 0 (B + 1)"
v" B )
proof -
 continuous_onjava.lang.StringIndexOutOfBoundsException: Range [0, 76) out of bounds for length 59
 using \<gamma> winding_number_subpath_continuous z by blast
 moreover have "Re metis DIM_complex dual_order.refl \ outside_no_overlap)
 using assms by (auto simp: path_image_def image_def)
 ultimately show ?thesis
 using ivt_increasing_component_on_1[of 0 alsohave \<dots> = 0"

lemma:
 assumes \>
 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

 using assms(forall
 ultimately show ?thesis
 using ivt_decreasing_component_on_1[of 0 1, where ?k = "1"] (intro conjI
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 auto : path_image_def ball_def
 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 ( add: t \<gamma> valid_path_imp_path)
 using path_image_deft zbyfastforce: Euler)
 have "b < a \ \ 0"
 proof -
 have "\ 0 \ {c. b < a \ c}"
 by (metis (no_types) pag atLeastAtMost_iff image_subset_iff order_refl ( contour_integral_unique Cauchy_theorem_convex_simple _ [of "+"])
 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 mem_ball_0 byblast
 ( add
 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 corollary\<^marker>\tag unimportant\ winding_number_zero_const: "a \ z \ winding_number (\t. a) z = 0"
 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(auto: pathfinish_def path_polynomial_function
 winding_number_reversepath valid_path_imp_path\<>\<open>tag unimportant\<close> winding_number_zero_outside:
 with assms show ?thesis
 using \<open>z \<notin> path_image \<gamma>\<close> winding_number_lt_half_lemma by fastforce
qed

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 -
{assumewnz_12\<bar>Re (winding_number \<gamma> z)\<bar> > 1/2"
 have"isCont winding_number \) z"
 (rule [OF \<gamma> convex_cball loop])
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