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products/sources/formale Sprachen/Isabelle/HOL/UNITY/ (Beweissystem Isabelle Version 2025-1^©) Datei vom 16.11.2025 mit Größe 15 kB

Quelle Winding_Numbers.thy Sprache: Isabelle

ection
theory
 Cauchy_Integral_Theorem Cauchy_Integral_Theorembegin
begin\ <>\<close>

\<open>Definition\<close>

definition>\<open>tag important\<close> winding_number_prop :: "[real \<Rightarrow> complex, complex, real, real \<Rightarrow> complex, complex] \<Rightarrow> bool" where
 winding_number_prop \<gamma> z e p n \<equiv>
 valid_path p \<and> z \<notin> path_image p \<and>
 pathstart p = pathstart \<gamma> \<and>
 pathfinish p = pathfinish \<gamma> \<and>
 (\<forall>t \<in> {0..1}. norm(\<gamma> t - p t) < e) \<and>
 contour_integral p (\<lambda>w. 1/(w - z)) = 2 * pi * \ * 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:
 assumes "path \" "z \ path_image \" "0 < e"
 shows "\p. winding_number_prop \ z e p (winding_number \ z)"
proof -
 have "path_image \ \ UNIV - {z}"
 using assms by blast
 then obtain d
 where d: "d>0"
 and pi_eq: "\h1 h2. valid_path h1 \ valid_path h2 \
 (\<forall>t\<in>{0..1}. cmod (h1 t - \<gamma> t) < d \<and> cmod (h2 t - \<gamma> t) < d) \<and>
 pathstart h2 = pathstart h1 \<and> pathfinish h2 = pathfinish h1 \<longrightarrow>
 path_image h1 \<subseteq> UNIV - {z} \<and> path_image h2 \<subseteq> UNIV - {z} \<and>
 (\<forall>f. f holomorphic_on UNIV - {z} \<longrightarrow> contour_integral h2 f = contour_integral h1 f)"
 using contour_integral_nearby_ends [of "UNIV - {z}" \<gamma>] assms by (auto simp: open_delete)
 then obtain h where h: "polynomial_function h \ pathstart h = pathstart \ \ pathfinish h = pathfinish \ \
 (\<forall>t \<in> {0..1}. norm(h t - \<gamma> t) < d/2)"
 using path_approx_polynomial_function [OF \<open>path \<gamma>\<close>, of "d/2"] d by (metis half_gt_zero_iff)
 define nn where "nn = 1/(2* pi*\) * contour_integral h (\w. 1/(w - z))"
 have "\n. \e > 0. \p. winding_number_prop \ z e p n"
 proof (rule_tac x=nn in exI, clarify)
 fix e::real
 assume e: "e>0"
 obtain p where p: "polynomial_function p \
 pathstart p = pathstart \<gamma> \<and> pathfinish p = pathfinish \<gamma> \<and> (\<forall>t\<in>{0..1}. cmod (p t - \<gamma> t) < min e (d/2))"
 using path_approx_polynomial_function [OF \<open>path \<gamma>\<close>, of "min e (d/2)"] d \<open>0<e\<close>
 by (metis min_less_iff_conj zero_less_divide_iff zero_less_numeral)
 have "(\w. 1 / (w - z)) holomorphic_on UNIV - {z}"
 by (auto simp: intro!: holomorphic_intros)
 then have "winding_number_prop \ z e p nn"
 using pi_eq [of h p] h p d
 by (auto simp: valid_path_polynomial_function norm_minus_commute nn_def winding_number_prop_def)
 then show "\p. winding_number_prop \ z e p nn"
 by metis
 qed
 then show ?thesis
 unfolding winding_number_def by (rule someI2_ex) (blast intro: \<open>0<e\<close>)
qed

lemma winding_number_unique:
 assumes \<gamma>: "path \<gamma>" "z \<notin> path_image \<gamma>"
 and pi: "\e. e>0 \ \p. winding_number_prop \ z e p n"
 shows "winding_number \ z = n"
proof -
 have "path_image \ \ UNIV - {z}"
 using assms by blast
 then obtain e
 where e: "e>0"
 and pi_eq: "\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);
 pathstart h2 = pathstart h1; pathfinish h2 = pathfinish h1; f holomorphic_on UNIV - {z}\<rbrakk> \<Longrightarrow>
 contour_integral h2 f = contour_integral h1 f"
 using contour_integral_nearby_ends [of "UNIV - {z}" \<gamma>] assms by (auto simp: open_delete)
 obtain p where p: "winding_number_prop \ z e p n"
 using pi [OF e] by blast
 obtain q where q: "winding_number_prop \ z e q (winding_number \ z)"
 using winding_number [OF \<gamma> e] by blast
 have "2 * complex_of_real pi * \ * n = contour_integral p (\w. 1 / (w - z))"
 using p by (auto simp: winding_number_prop_def)
 also have "\ = contour_integral q (\w. 1 / (w - z))"
 proof (rule pi_eq)
 show "(\w. 1 / (w - z)) holomorphic_on UNIV - {z}"
 by (auto intro!: holomorphic_intros)
 qed (use p q in \<open>auto simp: winding_number_prop_def norm_minus_commute\<close>)
 also have "\ = 2 * complex_of_real pi * \ * winding_number \ z"
 using q by (auto simp: winding_number_prop_def)
 finally have "2 * complex_of_real pi * \ * n = 2 * complex_of_real pi * \ * winding_number \ z" .
 then show ?thesis
 by simp
qed

lemma winding_number_prop_reversepath:
 assumes "winding_number_prop \ z e p n"
 shows "winding_number_prop (reversepath \) z e (reversepath p) (-n)"
proof -
 have p: "valid_path p" "z \ path_image p" "pathstart p = pathstart \"
 "pathfinish p = pathfinish \" "\t. t \ {0..1} \ norm (\ t - p t) < e"
 "contour_integral p (\w. 1 / (w - z)) = 2 * complex_of_real pi * \ * n"
 using assms by (auto simp: winding_number_prop_def)
 show ?thesis
 unfolding winding_number_prop_def
 proof (intro conjI strip)
 show "norm (reversepath \ t - reversepath p t) < e" if "t \ {0..1}" for t
 unfolding reversepath_def using p(5)[of "1 - t"] that by auto
 show "contour_integral (reversepath p) (\w. 1 / (w - z)) =
 complex_of_real (2 * pi) * \ * - n"
 using p by (subst contour_integral_reversepath) auto
 qed (use p in auto)
qed

lemma winding_number_prop_reversepath_iff:
 "winding_number_prop (reversepath \) z e p n \ winding_number_prop \ z e (reversepath p) (-n)"
 using winding_number_prop_reversepath[of "reversepath \" z e p n]
 winding_number_prop_reversepath[of \<gamma> z e "reversepath p" "-n"] by auto

(*NB not winding_number_prop here due to the loop in p*)
lemma winding_number_unique_loop:
 assumes \<gamma>: "path \<gamma>" "z \<notin> path_image \<gamma>"
 and: "pathfinish \ = pathstart \"
 and :
 "e. e>0 \ \p. valid_path p \ z \ path_image p \
 pathfinish p = pathstart p \<and>
 \<>t \<in> {0..1}. norm (\<gamma> t - p t) < e) \<and>
 contour_integral p (\<lambda>w. 1/(w - z)) = 2 * pi * \ * n"
 shows
proof -
 have path_image
 " \ z e p n \
obtain
: "e>"
java.lang.StringIndexOutOfBoundsException: Index 71 out of bounds for length 71
 d: ">0
 h1 h1; h2= h2 UNIV}rbrakk
 contour_integral\<forall>t \<in> {0..1}. norm(\<gamma> t - p t) < e) \<and>\<lambda>w. 1/(w - z)) = 2 * pi * \ * n"
 "path \" "z \ path_image \" "0 < e"
 obtain p where p: -
 valid_path
 (\<forall>t \<in> {0..1}. norm (\<gamma> t - p t) < e) \<and>
 contour_integral\lambdaw /w- ))=2*pi* \ * n"
 using pi [OF e] by blast
 obtain q where q: "winding_number_prop \ z e q (winding_number \ z)"
 using winding_number [OF \<gamma> e] by blast
 have"2* pi*\ * n = contour_integral p (\w. 1 / (w - z))"
 sing java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 19
 using [OF <open>path \<gamma>\<close>, of "d/2"] d by (metis half_gt_zero_iff)
rule r x in, )

auto :java.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 21
qed p q loop \<open>auto simp: winding_number_prop_def norm_minus_commute\<close>)
 alsohave"<2*complex_of_realpi* \ * winding_number \ z"
 usingbyauto: winding_number_prop_def d: ">
 finally have "2 * contour_integral p (\w. 1/(w - z)) = 2 * pi * \ * n"
 then show ?thesis
 by simp
qed

proposition:
 assumes "valid_path \" "z \ path_image \"
 shows <gamma> z = 1/(2*pi*\) * contour_integral \<gamma> (\<lambda>w. 1/(w - z))"d shows

 (use "path_image \ \ UNIV - {z}"

proposition obtainjava.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 15
 assumesarrow> contour_integral h2 f = contour_integral h1 f)"usingof-z (<forall>t\<in>{0..1}. cmod (h1 t - \<gamma> t) < d \<and> cmod (h2 t - \<gamma> t) < d) \<and>
 "((\w. 1/(w - z)) has_contour_integral (2*pi*\*winding_number \ z)) \"
number_valid_path \<subseteq> UNIV - {z} \<and> path_image h2 \<subseteq> UNIV - {z} \<and>

lemma [simp:"z\ winding_number(linepath a a) z= 0contour_integral_nearby_endsfUNIV- {z}
 by (simp: winding_number_valid_path:eal

lemma assume e: e>0
 by (simpaddwinding_number_valid_path

lemma using contour_integral_nearby_ends -z" <>]assms auto open_delete)
 assumeshave
 \<gamma>2: "path \<gamma>2" "z \<notin> path_image \<gamma>2"
obtain
 shows have"\w. 1 / (w - z)) holomorphic_on UNIV - {z}"
(java.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 34
 show exI
 (winding_number
 proof
 p1 have "(\<lambda>w. 1 / (w ( : intro
 pi_equnfoldingbysomeI2_exintro:
 moreover
 obtain by (auto simp: valid_path_polynomial_function norm_minus_commute nn_def winding_number_prop_def) java.lang.NullPointerException
number
 ultimately
 have "winding_number_prop (\1+++\2) z e (p1+++p2) (winding_number \1 z + winding_number \2 z)"
 using unfolding java.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 15
 <> \<gamma>" "z \<notin> path_image \<gamma>"
 apply (auto: joinpaths_def
done
 -
byjava.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 14
 ( [ e
qeduse in

 winding_number
<gamma>" "z \<notin> path_image \<gamma>" [ofp] java.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 34
shows(reversepath
proof (rule winding_number_uniqueby auto pathstart=h1h2 ; -{}
showpwinding_number_prop
 proof -
 obtain p where java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
 using
 then winding_number_propjava.lang.NullPointerException
java.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 3
applysimp valid_path_imp_reverse
 apply(simp
 java.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 10
 then thesis
 by blast
 qedpathstart)
qed ( using assms h2

lemma (uto!: )
assumes
 andshownorm
 showswinding_number\<gamma>) z = winding_number \<gamma> z" q byauto: obtain p where p: "winding_number_prop \<gamma>
(rulewinding_number_unique_loop pjava.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
 show "\p. valid_path p \ z \ path_image p \ pathfinish p = pathstart p \
\<forall>t\<in>{0..1}. cmod (shiftpath a \<gamma> t - p t) < e) \<and>
 p(<>w. ppathfinish<>. \<in> {0..1} \<Longrightarrow> norm (\<gamma> t - p t) < e"[java.lang.StringIndexOutOfBoundsException: Index 85 out of bounds for length 85
2 )
if "> pi*\ * winding_number \ z"
 -
number_prop
 using \<open>0 < e\<close> assms winding_number by blast
 then usingp java.lang.NullPointerException
qedproof-
 using that
 apply (-
 apply"reversepath\) z e p n \ winding_number_prop \ z e (reversepath p) (-n)"
 done
obtain
java.lang.StringIndexOutOfBoundsException: Index 2 out of bounds for length 0

lemma:
"c \ closed_segment a b" "z \ closed_segment a b"
shows"(linepath a bz ( a c)z (linepathc b)z
proof-
 havez \<notin> closed_segment a c" "z \<notin> closed_segment c b"contour_integral_nearby_loopspathfinish p \<and>
contour_integral\<lambda>w. 1/(w - z)) = 2 * pi * \ * n"
 then ?thesis -

 by : winding_number_valid_pathsymmetric field_simps
qed obtainjava.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 15

lemma:
 java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 by winding_number_unique_looppathfinish

 \< contour_integralw.1 wz)"
and
 shows "qed (use qp p = p \
proof
 have "winding_number pi [OF blast
 usingg by fastforce
 " (linepath c "
 java.lang.StringIndexOutOfBoundsException: Index 43 out of bounds for length 43
 ultimately show ? shows"
qed

lemmawinding_number_offset"show\lambdaw1 (-z athfinish=pathstarth1 h2=pathstarth2;fholomorphic_on z<>\java.lang.StringIndexOutOfBoundsException: Index 134 out of bounds for length 134
java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 30
 "*pi*
 n
 lemma ]:z OF
 then p
 by (rule_tac b winding_number_unique assms
 (force simp: winding_number_prop_def"(gamma1 roof( )

next
neg
 assume "0 < e" and g: "winding_number_prop (\w. p w - z) 0 e g n"
thenjava.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 9
proposition
piecewise_C1_differentiable_addC1_differentiable_imp_piecewise
 apply \<open>0 < e\<close> \<gamma>2 winding_number by blast
done \<gamma>2: "path \<gamma>2" "z \<notin> path_image \<gamma>2"
r_prop \<gamma>1 z + winding_number \<gamma>2 z"

qed

lemma
 NO_MATCH
 usinglemma winding_number_trivial []: "z\<> a \ winding_number(linepath a a) z = 0"

lemma winding_number_negatepath (proof winding_number_unique
 assumes :java.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17
 shows
proof winding_number_unique
 have "(/) 1 contour_integrable_on \"
 assmswinding_number_prop_def
 assume""
 (
 then "(\z. 1 / - z) has_contour_integral - contour_integral \ ((/) 1)) \" p = \<gamma> \<and> pathfinish p = pathfinish \<gamma> \<and> (\<forall>t\<in>{0..1}. cmod (p t - \<gamma> t) < min e (d/2))")
 using
 then (
 contour_integral \<gamma> ((/) 1)"
java.lang.StringIndexOutOfBoundsException: Index 89 out of bounds for length 89
 then show
assms:winding_number_valid_path
 contour_integral

lemmawinding_number_cnj
 assumesjava.lang.StringIndexOutOfBoundsException: Index 58 out of bounds for length 58
(
ruleshows
showe
 ife
java.lang.StringIndexOutOfBoundsException: Range [8, 7) out of bounds for length 9
2
 obtain using
 byblast
then" "z
 "pathstart p = pathstart \"
 2java.lang.StringIndexOutOfBoundsException: Range [30, 27) out of bounds for length 89

 "java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

winding_number_prop_def

 have "( : winding_number_valid_path symmetric )
 usingp1 (ubst
 havez
 lemma : winding_number_prop_defjava.lang.StringIndexOutOfBoundsException: Index 88 out of bounds for length 88
 moreoveri ">"for
using add
moreover show
p)imp
 have (\> moreover have "winding_number (linepath c c) z = 0"
 if(: java.lang.StringIndexOutOfBoundsException: Index 66 out of bounds for length 27
 proof -
 have ( <circ> \<gamma>) t - (cnj \<circ> p) t = cnj (\<gamma> t - p t)"( extarg_cong "( p) (\w. 1 / (w - z)) =
 bysimp p by (subst) auto
also <dots> = norm (\<gamma> t - p t)"\<gamma> t - p t)"
 by (subst
 have\<dots> < e" simpof
 using java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
 show

contour_integral
 (*)java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 proof -
 "<>. r n"
 by simpo_def
 \dotsbyjava.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12
( :o_def
 also have "\ = ?R"
 using p_contassumesultimately thesis
 finally ? java.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28
 qed
 ultimately show ?thesis havejava.lang.NullPointerException
by exI/ \<gamma> ((/) 1)) \<gamma>"
 " > \) ((/) 1) =
 " (cnj \ \)"
 by piecewise_C1_differentiable_diff)
shown java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11

qed

text \<open>A combined theorem deducing several things piecewise.\<close>
lemma apply (force simp: algebra_simps
 "\valid_path \1; z \ path_image \1; 0 < Re(winding_number \1 z);
 valid_path \<gamma>2; z \<notin> path_image \<gamma>2; 0 < Re(winding_number \<gamma>2 z); pathfinish \<gamma>1 = pathstart \<gamma>2\<rbrakk>
 \<Longrightarrow> valid_path(\<gamma>1 +++ \<gamma>2) \<and> z \<notin> path_image(\<gamma>1 +++ \<gamma>2) \<and> 0 < Re(winding_number(\<gamma>1 +++ \<gamma>2) z)"[OF1,2
 by (simpshowswinding_number

subsubsection

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

mber_pos_le h2proof
using ofjava.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
 and " notin ( \ p)"
 shows "0contour_integral p (\w. 1/(w - z)) = 2 * pi * \ * n"
proof
have" >( \ (at x) / (\ x - z))" if x: "0 < x" "x < 1" for x
 using p3 add)
 let? then contour_integral)pathfinish\
letint\<
have\le> ? )java.lang.StringIndexOutOfBoundsException: Index 29 out of bounds for length 28
 proof ( - "dots (w. 1 / (w - z))"
 show "usingassms (simp add: winding_number_valid_path java.lang.StringIndexOutOfBoundsException: Range [56, 20) out of bounds for length 20
: ge0have <dots> = norm (\<gamma> t - p t)"
 have "alsohave"
 using
 then "e >then show thesis
 java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 0
 show(-z)java.lang.StringIndexOutOfBoundsException: Index 103 out of bounds for length 103
 by(le [OF)(
 qed
obtainwhere
<>] field_simps
qed then have p: "valid_path p" "z \<notin> path_image p"

lemma:
 \<gamma>: "valid_path \<gamma>" "z \<notin> path_image \<gamma>"
 ( : winding_number_valid_path
 also " = ( p \x. 1 / (x - z)))"
shows winding_number
proof
 let="lambda>.1 "dots
 let = "\z. contour_integral \ (\w. 1 / (w - z))"
 have "e\ Im (contour_integral \ (\w. 1 / (w - z)))"
 proofrule of
 have "((\a. 1 / (a - z)) has_contour_integral contour_integral \ (\w. 1 / (w - z))) \"
m has_integral_component_le [of moreover have "pathfinish (cnj \<circ> p) = pathfinish (cnj \<circ> \<gamma>)"
 using simphas_contour_integral_integral
 hi"? intz cbox 01"
has_contour_integraljava.lang.StringIndexOutOfBoundsException: Index 40 out of bounds for length 40
 show "((\x. if 0 < x \ x < 1 then ?vd x else \ * e) has_integral ?int z) {0..1}"
 rule [OF, simplified
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 by (lemma \<dots> < e"
 qed(use [of _ 0 1] \<gamma>2; z \<notin> path_image \<gamma>2; 0 < Re(winding_number \<gamma>2 z); pathfinish \<gamma>1 = pathstart \<gamma>2\<rbrakk>
 withe ?thesis
[ gamma>] field_simps)
qed

lemmawinding_number_pos_lt
 assumes \<gamma>: "valid_path \<gamma>" "z \<notin> path_image \<gamma>"
 and \<gamma>1 z + winding_number \<gamma>2 z)" if "e > 0" for e
 and ge Re_winding_number
 shows "0 < Re (winding_number \ z)"

 havebm (<>.w-z)`( <gamma>))"
 using bounded_translation [of _lemmawinding_number_pos_le:sing
assumes
_posTHEN, OF] byjava.lang.StringIndexOutOfBoundsException: Index 50 out of bounds for length 50
 { fixx::eal x: have " \ Im (vector_derivative \ (at x) / (\ x - z))" if x: "0 < x" "x < 1" for x
 then have B2
 by (simp use in
le winding_number_reversepath
 path_image_def
 then have "e / B\<^sup>2 \ e / (cmod (\ x - z))\<^sup>2"

also"
 using ge "cnj then " reversepath
 have(lambda/- contour_integral
apply add )
 (
 } note * = this \<open>A combined theorem deducing several things piecewise.\<close>
 java.lang.StringIndexOutOfBoundsException: Range [0, 6) out of bounds for length 5
e assumes
qed

subsection ( winding_number_unique_loop

text "0
onof"e

lemma
sz:
"(g has_vector_derivative :"<java.lang.StringIndexOutOfBoundsException: Index 142 out of bounds for length 142
and ' ) ( "
 z:"gx\ z"
 shows "((\x. exp(-f x) * (g x - z)) has_vector_derivative 0) (at x within s)"
proof -
 have *: "(exp \ (\x. (- f x)) has_vector_derivative - (g' / (g x - z)) * exp (- f x)) (at x within s)"
 geadd(: java.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 37
 by (auto!: )
 show ?thesisjava.lang.StringIndexOutOfBoundsException: Index 124 out of bounds for length 124
 using z by (auto "linepatha( )+ )z
qed

lemma winding_number_exp_integral assms
lemma:then ( ? )( )
assumes
 and
 and z: "z \ \ ` {a..b}"
 shows by( add java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
 (isjava.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 7
"exp of_ z]
 (is"
proof
 java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 30
 lambda>x. 1 / (\<gamma> x - z) * vector_derivative \<gamma> (at x)"
 using (imp:path_image_def mult_mono
 havexhave<gamma> x \<noteq> z" using \<gamma>
 using rule_tacjava.lang.StringIndexOutOfBoundsException: Index 48 out of bounds for length 48
o eby( simphave(\ambda.1 a-)as_contour_integral
 <gamma> by (force simp: piecewise_C1_differentiable_on_def) piecewise_C1_differentiable_diff)
 have)
using
 moreover have "{a<.. {a..b} - k"
 by force
ultimatelyg_diff_at
 by (metis Diff_iff differentiable_on_subset piecewise_C1_differentiable_add C1_differentiable_imp_piecewise
 java.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 8
 assumejava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 have "continuous_on (ball w (cmod (w - z))) show "(
 by (auto simp: dist_norm intro!: continuous_intros winding_number_offset
z: \<gamma>: "valid_path \<gamma>" and 0: "0 \<notin> path_image \<gamma>"
 autohow "
 ultimately have "\h. \y. norm(y - w) < norm(w - z) \ (h has_field_derivative 1/(y - z)) (at y)"
of norm qed (use has_integral_const_real)
 force Ball_def) :"x
 }
 thenh* exp
meson
 lemma:
 unfolding
 rule
 show and gamma <gamma>"
 rule

 z:
by "exists.( \ \) (cnj z) e p (-cnj (winding_number \ z))"
 java.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 10
have p gamma
 java.lang.NullPointerException
 by (auto introcomplex_of_real ""
 with ab show ?thesis1

 { fix simp BB2have(
 assume t: " have" z
 have cball \<gamma> by (simp add: piecewise_C1_differentiable_on_def)
 using z by (auto intro!: continuous_intros simp p(3) by(simp: " /B^sup2\java.lang.StringIndexOutOfBoundsException: Index 136 out of bounds for length 136
:.java.lang.StringIndexOutOfBoundsException: Index 144 out of bounds for length 144
 unfolding field_differentiable_def
h".
 (h has_field_derivative inverse -
 using
 by simp ( z::
 have
 ( complex_mod_cnj
 show (
 auto! t .
"java.lang.StringIndexOutOfBoundsException: Index 147 out of bounds for length 147
 (at( unfolding
 if intro)
 proof
 have using zby(uto!: derivative_eq_intros[nfolded] ve
 using that using java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 then abbjava.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 25
 usingjava.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 10
 java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
usingby >a
tcon_vdx{byintro :ab
 by <
 addjava.lang.StringIndexOutOfBoundsException: Index 72 out of bounds for length 72
 using java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 then obtain have<lbrakk
 auto java.lang.StringIndexOutOfBoundsException: Range [55, 54) out of bounds for length 55
 show
atandjava.lang.NullPointerException

 show simp
using )
ave
}
 proof (rule has_vector_derivative_eq_rhs [OF integral_has_vector_derivative_continuous_atbysimp )
 show "continuous int = "z
 usingrulejava.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
byintro continuous_introsx" k" "a < x" "x < b"
tive
 using have:"? has_integral int z) cbox01"
 by "(\lambdax show <> .0\java.lang.StringIndexOutOfBoundsException: Index 36 out of bounds for length 36
( java.lang.StringIndexOutOfBoundsException: Range [36, 37) out of bounds for length 36
 then
(x :<
 by (auto simp " Re( \ z)"
 ( inunfolding [symmetric] divide_inverse_commute
java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9
 (use t auto ( exp_fg , simplified:(
java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
 ab ?
 by (simp add: divide_inverse_commute integral_def) (
qed( has_integral_spike_interiorrentiablforce)

lemma winding_number_exp_2pi:
"> p; z(h x-)(xwithin. \< - (
 \<Longrightarrow> pathfinish p - z = exp (2 * pi * \ * winding_number p z) * (pathstart p - z)"=
winding_number valid_path_def con_vd
 by " \ (at x) / (\ x - z) = d / (\ x - z)"

integer_winding_number_eq java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 assumes
 showswinding_number
proof
obtain:valid_pathjava.lang.NullPointerException
 "pathstartsimp: path_shiftpath
 andusing z " \ closed_segment a b" "z \ closed_segment a b"
using[ assms, ofunfoldingjava.lang.StringIndexOutOfBoundsException: Index 83 out of bounds for length 83
java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
 using assms z:z\winding_number_valid_path java.lang.StringIndexOutOfBoundsException: Index 54 out of bounds for length 54
 have iff ( add winding_number_def winding_number_prop_def winding_numberz1unfolding pathstart_def
 lemma:
 have "\ 0 \ z"
 by (metis pathstart_def pathstart_in_path_image z)
 then " integer_winding_number_eqjava.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 32
 using p winding_number_exp_integral(2) [of p 0 1 z]
 have: "continuous_on a.}\gamma>java.lang.StringIndexOutOfBoundsException: Index 45 out of bounds for length 45
thenpz
pw iffsimp)
 then proofextjava.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 37
 by have"a.<}
qed

theorem integer_winding_number(ule_tac\lambdagt-zin)
 "path \; pathfinish \ = pathstart \; z \ path_image \\ \ winding_number \ z \ \"
by{fix

\>thesmagnitudeleastthen must points every.*)
 We can thus auto: intro:(<lambda>t. g t + z) n"

lemma winding_number_pos_meets auto:!: )
 fixes z::complex force: algebra_simps)
 assumes \<gamma>: "valid_path \<gamma>" and z: "z \<notin> path_image \<gamma>" and 1: "Re (winding_number \<gamma> z) \<ge> 1"winding_number_prop
 and w: "w obtainh : "NO_MATCH 0 z \ winding_number p z = winding_number (\w. p w - z) 0"
 "\a::real. 0 < a \ z + of_real a * (w - z) \ path_image \"
proof
 have [simp]: "shows "winding_number(uminus \ \) 0 = winding_number \ 0"
 using

 using path_image_deftheorem
 have gpd: "byrule)
 usingby(metisinteger_winding_number_eq)
 definejava.lang.StringIndexOutOfBoundsException: Range [6, 0) out of bounds for length 0
java.lang.StringIndexOutOfBoundsException: Index 96 out of bounds for length 29
 using showthesis
..}
 (\<lambda>x. Im (integral {0..x} (\<lambda>x. vector_derivative \<gamma> (at x) / (\<gamma> x - z))))"(
 "winding_number( assumes \: "valid_path \" and z: "z \ path_image \" and 1: "Re (winding_number \ z) \ 1"
" r \ 2*pi"
[]divide_inverse_commute
 also"dots\le integral{1
java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
] .field_simps
finally )
 then t: " \ {a..b}"
 cont
 then obtain : using>piecewise_C1_differentiable_onjava.lang.StringIndexOutOfBoundsException: Index 60 out of bounds for length 60
 and eqArg: " obtain h where h: "\x. (\ t - x) < cmod (\ t - z) \
 by blastusingwjava.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
 define"=.lambda>x. vector_derivative \ (at x) / (\ x - z))"
 gpdt\<>piecewise_C1_differentiable_on.}
by metisbybysimp (\<lambda>x. Im (in.}<lambda>x. vector_derivative \<gamma> (at x) / (\<gamma> x - z))))"
 have "exp (- i) * (\ t - z) = \ 0 - z"
 unfolding i_def
 (rule [OF gpdt]
 show" \ \ ` {0..t}"
 p add p(4bysimpjava.lang.StringIndexOutOfBoundsException: Index 50 out of bounds for length 50
qedtin)
 then :" t - z = exp i * (\ 0 - z)"
 by (simp usingjava.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
 then have "(w bysimp
 by simp: r_def
 moreoverx:java.lang.StringIndexOutOfBoundsException: Index 46 out of bounds for length 46
usingby( add exp_eq_polar using * by (simp add: exp_eq_polar field_simps
moreover" r = usingpen_subset_interior[OF <>] by astforce
eqArgadd
 ultimately" + complex_of_real exp( g_C1_diffx auto "(0. lambda>x. vector_derivative \<gamma> (at x)/(\<gamma> x - z))) = Arg2pi r"
 define "i =integral{0t \lambda>. vector_derivative\gamma> x) f
( mult:\at
 with t show ?thesis
 ( )(< =gamma
qed

lemma winding_number_big_meets "(\c. exp (- integral {a..c} (\x. ?D\ x / (\ x - z))) * (\ c - z)) has_derivative (\h. 0))
 fixes z::complex
 \<gamma>: "valid_path \<gamma>" and z: "z \<notin> path_image \<gamma>" and "\<bar>Re (winding_number \<gamma> z)\<bar> \<ge> 1"
 nd showhesis
 showst exI _"cnj \ p"]) (auto simp: winding_number_prop_def)
proof
 xwithin
 rulehave)(java.lang.StringIndexOutOfBoundsException: Index 44 out of bounds for length 44
 by (simp add: \<gamma> valid_path_imp_path winding_number_reversepath z)
 moreoverhave "alid_path
 using \<gamma> valid_path_imp_reverse by auto
 moreover have "z \ path_image (reversepath \)"
ddjava.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 22
 \<exists>a::real. 0 < a \<and> z + of_real a * (w - z) \<in> path_image (reversepath \<gamma>)" bsubsectionjava.lang.StringIndexOutOfBoundsException: Index 136 out of bounds for length 135
 using winding_number_pos_meets blast
 thenshow(\lambda.integral field_simps)
 by simp (t x within \<gamma>: "valid_path \<gamma>" "z \<notin> path_image \<gamma>"
 }
 then ?thesis
 using assms
 by (simp add: abs_if java.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 28
qed

lemma fixesz::omplex ab ?thesis2
fixes
 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>
 \<Longrightarrow> Re(winding_number \<gamma> z) < 1"
 by (auto simp{assume of1 valid_path_def have "((\<lambda>a. 1 / (a - z)) has_contour_integral contour_integral \<gamma> (\<lambda>w. 1 / (w - z))) \<gamma

\<open>One way of proving that WN=1 for a loop.\<close>
lemmawinding_number_eq_1java.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 26
fz:java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18
 assumes \<gamma>: "valid_path \<gamma>" and z: "z \<notin> path_image \<gamma>" and loop: "pathfinish \<gamma> = pathstart \<gamma>"
 show
 simp eq (<>. 1/ z)=2* \ * winding_number \<gamma> z"
proof -
 have "winding_number \ z \ Ints"
 by (simp add: \<gamma> integer_winding_number loop valid_path_imp_path z)y simp: winding_number_pos_meets: if_split_asm
 then show ?thesis eq by:
 using02by simp: Ints_def
qed

subsection> have expp(ambda/w-z) \<gamma> 1 - z) / (\<gamma> 0 - z)"

continuous_at_winding_number: " Im (contour_integral \ (\w. 1 / (w - z)))"
 fixescomplex
 assumes
 shows "continuous (at z) (winding_number \)"
proof -
 ">" : "cball z - path_image \"
 using has_contour_integral
 simpassumes integer_winding_number_eq
 then have ppag: "path_image \ \ - cball z (e/2)"
 by (force simp: cball_def dist_norm)
 have oc: open( cball(/)"
 by simp:closed_def
obtain "d>0 and pi_eq:
 "fixes z::complex
java.lang.StringIndexOutOfBoundsException: Index 104 out of bounds for length 104
 pathstart h2 = pathstart h1; pathfinish h2 = pathfinish shows "<> 02by (auto : Ints_def)

 path_image h1 \<subseteq> - cball z (e/2) \<and>
 path_image h2 \<subseteq> - cball z (e/2) \<and>
> -
 using [OF \<gamma> ppag] by metis
obtain" p""\java.lang.StringIndexOutOfBoundsException: Index 57 out of bounds for length 57
 andproof
and \<And>t. t\<in>{0..1} \<Longrightarrow> cmod (\<gamma> t - p t) < min d e/2"
 and pi: "contour_integral p (\x. 1 / (x - z)) = complex_of_real (2 * pi) * \ * winding_number \ z"
 usingby( simp [symmetricxhave" x \ z" using \
 { fix then "e/ \^{2\ e / (cmod (\ x - z))\<^sup>2"}
a d2mod d/ and : cball_def
 have wnotp: "w \ path_image p" have: "open -cballz(/)"
 proof (clarsimp simp add have"<> \ Im (integral {0..1} (\x. vector_derivative \ (at x) / (\ x - z)))"
 showby( add have" /B^sup>2 \ Im (vector_derivative \ (at x) * cnj (\ x - z)) / (cmod (\ x - z))\<^sup>2" .
 proof -
 \<exists>t. t \<in> {0..1} \<and> Im(integral {0..t} (\<lambda>x. vector_derivative \<gamma> (at x)/(\<gamma> x - z))) = Arg2pi r"bysimp:Arg2pi_ge_0 IVT)
 using B (simp: * winding_number_pos_lt_lemma <gamma>, of "e/B^2"])

 by (subsection<>The numberdefine define i where "i
java.lang.StringIndexOutOfBoundsException: Range [25, 8) out of bounds for length 25
 using cbg thatwhereexp(

qed z:
 have wnotg assumes g:"ghas_vector_derivative ')(txwithin
e2\<open>e>0\<close> by (force simp: dist_norm norm_minus_commute)using [OF
 { "((\x. exp(-f x) * (g x - z)) has_vector_derivative 0) (at x within s)"
 sume d2cmodz /" have :(exp\ (\x. (- f x)) has_vector_derivative - (g' / (g x - z)) * exp (- f x)) (at x within s)"
 then obtain q where q: "valid_path have "(w-) \gamma )
 "pathstart q = pathstart \ \ pathfinish q = pathfinish \"
qed
 pglemma java.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 34
 <gamma> wnotg, of "min k (min d e) / 2"] \<open>d>0\<close> \<open>e>0\<close> k
 bysimp less_divide_eq_numeral1ax-) a.java.lang.StringIndexOutOfBoundsException: Index 99 out of bounds for length 99
"
 using \<open>0 < d\<close> by (fastforce simp add: norm_minus_commute)
 moreover have "(\u. 1 / (u-w)) holomorphic_on - cball z (e/2)"
using by (uto: dist_norm intro)
 ultimately" p \<
 by (metis p \<open>valid_path p\<close> pi_eq)
 then k : "finite k java.lang.StringIndexOutOfBoundsException: Range [37, 1) out of bounds for length 21
 by (simppathstart
 } note moreover have"{a..} lemma winding_number_big_meets:
 have "path p"
byassumes
w:"w\ z"
 by "\a::real. 0 < a \ z + of_real a * (w - z) \ path_image \"
 moreover"And.t moreover have "\t. t \ {0..1} \ cmod (q t - \ t) < d \ cmod (p t - \ t) < d"
 using pg
wnwn
obtain where 0 andusing :dist_norm: java.lang.StringIndexOutOfBoundsException: Index 88 out of bounds for length 88
>valid_pathp\<close> \<open>z \<notin> path_image p\<close> open_contains_cball [of "- path_image p"] addjava.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 22
force
 obtain L
 where">"
and:

java.lang.StringIndexOutOfBoundsException: Range [0, 22) out of bounds for length 15
 using show "\<exists>d h. 0 <
 {java.lang.StringIndexOutOfBoundsException: Range [0, 4) out of bounds for length 0
 assume e: "0 < using winding_number_unique wnotpbyblast
 then have [simpif
 cbp(obtainwhere> cbpcball/ pe
 by (force simp: dist_norm using \<open>valid_path p\<close> \<open>z \<notin> path_image p\<close> open_contains_cball [of "- path_image p"] simp
 have [simp]: "contour_integral p (\x. 1/(x - w)) - contour_integral p (\x. 1/(x - z)) =
 contour_integral p (\<lambda>x. 1/(x - w) - 1/(x - z))" ( simp not_less:winding_number_big_meets
 simp
 {fixx
 assume pe: "3/v ( add: divide_inverse_commuteintegral_defintegrable_on_def)
 "cmod(w - contour_integral ) L * B"
bymeson norm_diff_triangle_le order_trans_rules1 1)havemma
wx cmod"x\java.lang.StringIndexOutOfBoundsException: Index 144 out of bounds for length 144
 haveobtainwhere" t - x) < cmod (\ t - z) \
using blastthen have[]: "w \ path_image p"
also "\ < pe/4 + cmod (w - x)"
 using ( add norm_minus_commute)
 finally "/
 using have[]: "contour_integral p (\x. 1/(x - w)) - contour_integral p (\x. 1/(x - z)) =
 then havejava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
by( add
 then continuous_at_winding_number
 by (simp add: power_divide)
 have "8 * L * cmod (w - z) < e * pe if"x\<in> {a..b} - ({a, b} \<union> k)" for x shows" assumepe"4 < z-)java.lang.StringIndexOutOfBoundsException: Index 42 out of bounds for length 42
using
 also have "\ < e * 4 * cmod (w - x) * cmod (w - x)"
 pe2
 also have "\ < e * 4 * cmod (w - x) * (4/3 * cmod (z - x))"
 using
finally " have con_vd:" ( xwithin (lambdathen :path_imagesubseteq e2
 java.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 15
 also have "\ \ e * cmod (w - x) * cmod (z - x)"
 using pe_less2cmodx ^2
 have"L *cmodw z e w-x)*cmod (z-x) java.lang.StringIndexOutOfBoundsException: Index 78 out of bounds for length 78
 "(c. exp (- integral {a..c} (\x. ?D\ x / (\ x - z))) * (\ c - z)) has_derivative (\h. 0))
 proof (cases (rule [unfolded, simplified
 case True
 with
 show ?thesis
 by (force simp: norm_minus_commute)
 next
 case False
 with wx w(2) \<open>L>0\<close> pe pe2 Lwz
 ?java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 20
 by (auto simp: divide_simps mult_less_0_iff norm_minus_commute have\<
 qed
 =
 let(
 have "cmod (contour_integraljava.lang.StringIndexOutOfBoundsException: Index 33 out of bounds for length 30
 proof (rule L)
 showf java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 using \<open>pe>0\<close> w
 by (force simp: dist_norm norm_minus_commute by ( integer_winding_number_eq
 show " \u \- 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 ( " (1 /-)-1/ x-z p wherevalid_path"z\<notin> path_image p"
 qed
 alsohavejava.lang.NullPointerException
\<open>L>0\<close> e by (force simp: field_simps)
 finally "cmod(winding_number p w - winding_number p ) < e"
 using pi_ge_two e
 by (force simp w)
 } note cmod_wn_diff = this
 have "isCont (winding_number p) z"
 java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9
 fix e::real assume "e>0"
 show
 using \<open>pe>0\<close> \<open>L>0\<close> \<open>e>0\<close>]z
/e2pe//)in simpdist_norm
 qed
 then show ?thesis
 apply (rule by(force: dist_normnorm_minus_commute ntro holomorphic_intros proof-
 apply (auto simp: \<open>d>0\<close> \<open>e>0\<close> dist_norm wnwn)
 pg by auto
qed

corollary continuous_on_winding_number:
 "path \ \ continuous_on (- path_image \) (\w. winding_number \ w)"
by (simp: continuous_at_imp_continuous_on" (winding_number winding_numberpz "

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

lemma winding_number_constantusing e2
 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 "<>d>. \x'. dist x' z < d \ dist (winding_number p x') (winding_number p z) < e"
 *" cmod(winding_number \ y - winding_number \ z)"
 if ne: x="min (pe/) e/*^2/L/)"in) q=pathstart
 proof
 have "winding_number \ y \ \" "winding_number \ z \ \"
 using that integer_winding_number [OF \<gamma> loop] sg \<open>y \<in> S\<close> by autousing
 with
 byby ( continuous_intros winding_number_exp_integral gpd)
 qed
 have cont: "continuous_on S (\w. winding_number \ w)"
 using continuous_on_winding_number [OF \<gamma>] sg
 by continuous_on_subset)
 show ?thesis
 using
 by (last: [OF cont
qed

lemma:
 "winding_number \ constant_on S"
 \<Longrightarrow> winding_number \<gamma> w = winding_number \<gamma> z"
 using winding_number_constant by (metis constant_on_def)

lemma open_winding_number_levelsets:
 assumes \<gamma>: "path \<gamma>" and loop: "pathfinish \<gamma> = pathstart \<gamma>"
shows " {z. z\notin>path_image\ \ winding_number \ z = k}"
proof ( simp: open_dist
 fixz z:" \ path_image \" and k: "k = winding_number \ z"
 have "open (- path_image \)"
osed_path_image
z e \<subseteq> - path_image \<gamma>" Arg2pi
 using [ "- \"] z by blast
 then pe">0" and cball3/ )\<subseteq> - path_image p"
 using \<open>e>0\<close> by (force simp: norm_minus_commute dist_norm intro: winding_number_eq [OF assms, where S = "ball z e"])show?thesis
qed

\<open numberis "utside acurve

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:: using winding_number_constant (metis)
 using bounded_subset_ballD [OF
 w::complex w:"w ball 0 (B + 1)"
 by (metis abs_of_nonneg le_less less_irrefl mem_ball_0 norm_of_real)
 have "- ball 0 (B + 1) \ outside (path_image \)"
 using subset_ball ( outside_subset_convex) auto
 then have wout: "w \ outside (path_image \)"
 using fix z assume z z\<notin> path_image \<gamma>" and k: "k = winding_number \<gamma> z"
 moreover have "winding_number \ constant_on outside (path_image \)"
 using [ \<gamma> loop, of "outside(path_image \<gamma>)"] connected_outside
{ :ealcomplex
 ultimately e:" " and: cmod assume e: "0 < e" and w: "cmod (w - z) < pe
 by (metis (no_types, opaque_lifting) constant_on_def z)
 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" "pathstartsubsection\Winding number is zero "outside" a curve\outsidea curve
 winding_number_zero_in_outside
 and:\>.
 using path_approx_polynomial_function [OF \<gamma>, of "min 1 e"] \<open>e>0\<close>
 by( atLeastAtMost_iff min_less_iff_conj)
>p. \<and> w \<notin> path_image p \<and>
 pathstartp = pathstart \<gamma> \<and> pathfinish p = pathfinish \<gamma> \<and>
 (\<forall>t\<in>{0..1}. cmod (\<gamma> t - p t) < e) \<and> contour_integral p (\<lambda>wa. 1 / (wa - w)) = 0"have p \<lambda>x. 1/(x - w) - 1/(x - z))"
proof exI)
java.lang.StringIndexOutOfBoundsException: Range [8, 4) out of bounds for length 72
 using B unfolding qed (usetin)
 by (meson add_strict_mono atLeastAtMost_iff le_less_trans by( addexp_minus field_simps
 bysimp:r_def)
 pe"/ *pe wby java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 20
 then show "w \ path_image p" using w by blast
 show: "valid_path p"
th_polynomial_function)
 show "\t\{0..1}. cmod (\ t - p t) < e"
 eqArgsimp:i_def
 show p (\<lambda>wa. 1 / (wa - w)) = 0"
proofrule contour_integral_unique OF [OF_convex_ball [ 0 "+1]])
 have "\z. cmod z < B + 1 \ z \ w"
 java.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 39
 then show "(\z. 1 / (z - w)) holomorphic_on ball 0 (B + 1)"
 by (intro by(rule_tac ="( { fix e::real assume 0<"
 qed (use p vap pip loop
 qed (use p in p where p "polynomial_function p"" z:java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18
java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
 then show ?thesis
 by (auto using[ \<gamma>, of "min 1 e"] \<open>e>0\<close>
 qed
 showthesis
qed

corollary<^marker\<open>tag unimportant\<close> winding_number_zero_const: "a \<noteq> z \<Longrightarrow> winding_number (\<lambda>t. a) z = 0"
 by (rule winding_number_zero_in_outside)
 (auto simp: pathfinish_def (introexI)

corollary\<^marker>\<open>tag unimportant\<close> winding_number_zero_outside: w by blast
 "\path \; convex s; pathfinish \ = pathstart \; z \ s; path_image \ \ s\ \ winding_number \ z = 0"
by convex_in_outsidewhere : 3/ )

 closed_defclosed_path_image ])
\<gamma>" and loop: "pathfinish \<gamma> = pathstart \<gamma>"\>/ ( -xjava.lang.StringIndexOutOfBoundsException: Index 47 out of bounds for length 47
e2
-
:where \<gamma> \<subseteq> ball 0 B"
 java.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 10
 have "winding_number \Winding number is zero "outside" a curve\
 ( winding_number_zero_outside intro
 show "z \ cball 0 B"
 using that by auto
 showjava.lang.StringIndexOutOfBoundsException: Index 52 out of bounds for length 52
using blast- B <> (<>java.lang.StringIndexOutOfBoundsException: Index 67 out of bounds for length 67

 thenandpg1Bwhere<
 bymetis
qed

lemma bounded_winding_number_nz: using [OF\<> "min e]\java.lang.StringIndexOutOfBoundsException: Index 92 out of bounds for length 92
 assumespathstart
 shows(forall
proof -
 B \And x \<ge> B \<Longrightarrow> winding_number g x = 0"
 using winding_number_zero_at_infinity[OF outside_compact_in_open[ "path_image \" S] path_image_nonempty winding_number_zero_in_outside
 ?thesis
 unfolding bounded_iff
qed

lemma winding_number_zero_point:
 \lbrakk> <> convex S;pathfinish
 \<Longrightarrow> \<exists>z. z \<in> S \<and> winding_number \<gamma> z = 0"
 using outside_compact_in_open [of "path_image \" S] path_image_nonempty winding_number_zero_in_outside
 by ( : p() alid_path_polynomial_function

\<open> a path roundset winds its
lemma winding_number_around_inside
 assumes \<gamma>: "path \<gamma>" and loop: "pathfinish \<gamma> = pathstart \<gamma>" using by auto
 and : assume
> S wn_nzwinding_number
 shows \<gamma> w = winding_number \<gamma> z"
proof -
 havessb
 proof
 fix:
 assume " "
 hence "x \ path_image \"
 by mesondisjoint_iff_not_equal)
 thus <in> inside (path_image \<gamma>)"
 (metis S_disj \<gamma> \<open>x \<in> S\<close> cos inside_outside loop winding_number_eq winding_number_zero_in_outside wn_nz z)

 show show " S <> S)
 proofrule [OF
 show "z \ S \ inside S"
 using z by blast
 show "connected (S \ inside S)"
have: Lcmod - zx"
 show " ( java.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 5
 unfolding disjoint_iff Un_iff
 by ( ComplD
 qed
qed

text
lemma winding_number_subpath_continuous:
 assumes winding_number_zero_at_infinity
 showscontinuous_on.}(<>. winding_number0x <gamma>) z)"
proof (rule continuous_on_eqjava.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18
 let ? ?thesis
 byautodivide_simps norm_divideBreal " :" gamma
 proof (intro indefinite_integral_continuous_1 winding_number_exp_integral continuous_intros)
 show show" (2* pi \) * ?f x = winding_number (subpath 0 x \) z"
 using\<gamma> valid_path_def by blast
 qed (use path_image_def z in proofjava.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9
 show "1 / (2 * pi "?f holomorphic_onusing x
 if: " \ {0..1}" for x
 proof -
have 2*< fx 2pi
 using assms x
 by (simp add: contour_integral_subcontour_integral [OF contour_integrable_inversediff])
 also have "\ = winding_number (subpath 0 x \) z"
 proof winding_number_valid_pathassumespathfinishjava.lang.StringIndexOutOfBoundsException: Range [49, 47) out of bounds for length 47
 show "z \ path_image (subpath 0 x \)"
 using assms x atLeastAtMost_iff java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
 qed (use assms ?thesis
 finallyshow .
 qed
qed

lemma winding_number_ivt_pos continuous_on}(<lambda>x. winding_number (subpath 0 x \<gamma>) z)"
 assumes
 shows\<> <in.1}.Re0 t \<gamma>) z) = w"
proof
 have e:real "> assmsby (auto simp: path_image_def image_def)
 us ivt_increasing_component_on_1 ? " java.lang.StringIndexOutOfBoundsException: Index 72 out of bounds for length 72
 moreover have "Re (winding_number (subpath 0 0 \) z) \ w" "w \ Re (winding_number (subpath 0 1 \) z)"
 using assms
show
 ivt_increasing_component_on_1 0 1,where = ""]by
qed

lemma winding_number_ivt_neg:
 assumes \<gamma>: "valid_path \<gamma>" and z: "z \<notin> path_image \<gamma>" and "Re(winding_number \<gamma> z) \<le> w" "w \<le> 0"
shows
proof -
 add continuous_at_winding_number
 using \<gamma> winding_number_subpath_continuous z by blast
 moreover using[of0 "
 using assms by (auto simp: path_image_def image_def)
 ultimately:
 using[ofjava.lang.StringIndexOutOfBoundsException: Index 152 out of bounds for length 152
qed

: "<>java.lang.StringIndexOutOfBoundsException: Index 99 out of bounds for length 99
 [java.lang.StringIndexOutOfBoundsException: Range [98, 99) out of bounds for length 98
 shows "\t \ {0..1}. \Re (winding_number (subpath 0 t \) z)\ = w"
 using assmsby (imp winding_number_lt_half_lemma
java.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 10

java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 35
 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}"
( /2path opposite.\<close>
proof -
 { assume "Re(winding_number \ z) \ 1/2"
 then obtain gt<t ( -( piwinding_numbert
 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))"applysimp:
 using winding_number_exp_2pilemma:
 apply (simp: t \<gamma> valid_path_imp_path)
 using closed_segment_eq_real_ivl \<gamma> valid_path_def by blast
 havefixassume:">\ 0"
 proof -
 have "gamma> 0\ {c. b < a \ c}"
 by (metis (no_typesproof -
 thus ?thesis
 using x
 qed
 moreover have "b < a \ \ t"
 by (metisatLeastAtMost_iffimage_eqI mem_Collect_eq pag subset_ifft have<
 by (metisatLeastAtMost_iff mem_Collect_eq pag
 add
 then have Falseusing atLeastAtMost_iffforce
simpinner_mult_right)
 }
 then show ?thesis by force
qed

lemmaassumesjava.lang.StringIndexOutOfBoundsException: Index 139 out of bounds for length 139
 assumes "valid_path \" "a \ z \ b" "path_image \ \ {w. a \ w > b}"
 shows "\Re (winding_number \ z)\ < 1/2"
proof using B subset_ballintro) auto "\Re (winding_number \ z)\ < 1/2"
 have \<notin> path_image \<gamma>" using assms by auto
have" ( \ z) < 0 \ - Re (winding_number \ z) < 1/2"
metis winding_number_lt_half_lemma valid_path_imp_reverseofusing OF
 winding_number_reversepath valid_path_imp_path "
 withwinding_number_reversepath byetis) )
 using\<open>z \<notin> path_image \<gamma>\<close> winding_number_lt_half_lemma by fastforce
qed

lemma winding_number_le_half:
 assumes \<gamma>: "valid_path \<gamma>" and z: "z \<notin> path_image \<gamma>" pg1java.lang.StringIndexOutOfBoundsException: Index 123 out of bounds for length 123
 anz \<noteq> 0" and azb: "a \<bullet> z \<le> b" and pag: "path_image \<gamma> \<subseteq> {w. a \<bullet> w \<ge> b}"shows \<bar>Re (winding_number \<gamma> z)\<bar> \<le> 1/2"
 shows "\Re (winding_number \ z)\ \ 1/2"
proof -
(winding_number\<gamma> z)\<bar> > 1/2"
 have "isCont (winding_number \) z"
 by (metis continuous_at_winding_number valid_path_imp_path \<gamma> z)
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