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
 &ds for length 7
 force
 con_g{.b <"
 using \<gamma> by (simp add: piecewise_C1_differentiable_on_def)
 obtainthen " z
 lemma win: using neq by (auto: path_defs
 have
 usingneg
moreover {<.b -k\<subseteq> {a..b} - k"
 byforce
 ultimately have g_diff_at: java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
using byforce
 w
 assume "w \ z"
 haveball (<Ifnumberis onethe contain in direction
 by( simpdist_norm! java.lang.StringIndexOutOfBoundsException: Index 63 out of bounds for length 63
 moreover have "\x. cmod (w - x) < cmod (w - z) \ \f'. ((\w. 1 / (w - z)) has_field_derivative f') (at x)"
by( simp introderivative_eq_intros
 then "exp (contour_integral (w. 1 / (w - z))) = (\ 1 - z) / (\ 0 - z)"
 usingdone
 by (then"<>. r"java.lang.StringIndexOutOfBoundsException: Index 56 out of bounds for length 56
 }
 where"java.lang.StringIndexOutOfBoundsException: Index 94 out of bounds for length 94
 meson
 have exy: "\y. ((\x. inverse (\ x - z) * ?D\ x) has_integral y) {a..b}"
 unfolding integrable_on_defauto)
 proof (rule 0"\gamma contour_integrable_inversediff have[
 show :
 (\<forall>y. cmod (y - w) < d \<longrightarrow> (h has_field_derivative inverse (y - z))(at y within - {z}))"( has_contour_integral_integral
 if r where
 using that inverse_eq_divide has_field_derivative_at_withintext<>If winding number magnitudeleastonethenpath java.lang.StringIndexOutOfBoundsException: Index 96 out of bounds for length 29
 by java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
shows(java.lang.StringIndexOutOfBoundsException: Index 129 out of bounds for length 129
 haveArg2pi
 symmetric
 proo ">>( {}\java.lang.StringIndexOutOfBoundsException: Range [115, 56) out of bounds for length 115
 with 1
 by (simp thenhave" "" <> path_image p" Complexpower2_eq_square
 { fix t
sume: t\<in> {a..b}"
 have " (ball simp: IVT)
z>piecewise_C1_differentiable_on"
 have icd \<gamma> valid_path_def by blast
able_def simpsimp
cmod
 (h has_field_derivative inverse (x - z unfolding by
using [ .t<java.lang.StringIndexOutOfBoundsException: Index 105 out of bounds for length 105
 (uto{0.x <lambda>x. vector_derivative \<gamma> (at x) / (\<gamma> x - z))))"
 have" (-( {a..
 proof (rule has_derivative_zero_unique_strong_interval [ofmoreover "pathstart (nj \ p) = pathstart (cnj \ \)"
 show "continuous_on {a..b} (\b. exp (- integral {a..b} (\x. ?D\ x / (\ x - z))) * (\ b - z))"
 by ( using: using)bysimppathfinish_compose
 show " if t: "\in{.1 use auto
 (at x within {a..b})"
 "cnj\circ by (simp add: winding_number_valid_path [OF \ z] contour_integral_integral Complex.Re_divide field_simps power2_eq_square)
 proof have" r \ Im (integral {0..1} (\x. vector_derivative \ (at x) / (\ x - z)))" .
havex<notin> k" "a < x" "x < b"
 also "
 then have "x \ interior ({a..b} - k)"
o \circ] f
 then have con: "isCont ?D using by (simp : i_def)
usingby and ntegral}java.lang.StringIndexOutOfBoundsException: Index 127 out of bounds for length 127

 by ( ? cnj
have "<> differentiable atxjava.lang.StringIndexOutOfBoundsException: Index 48 out of bounds for length 48
 using by( add g_diff_at
 (i*(gamma >0-z
 by unfolding
 show(
 (at x within {a.. proof winding_number_exp_integral gpdt
java.lang.StringIndexOutOfBoundsException: Index 76 out of bounds for length 76
 showa w: ultimately ?
roof cnj
 have "(show" (cnj-
 within.}"
proof then r<
show" (at xjava.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
 using continuous_at_imp_continuous_at_within"
on_vd(+
 show "vector_derivative \ valid_path(\1 +++ \2) \ z \ path_image(\1 +++ \2) \ 0 < Re(winding_number(\1 +++ \2) z)"
 using d vector_derivative_at
 by (simp add: vector_derivative_at has_vector_derivative_def)
 qed (use x vg_intsu\^><open>tag unimportant\<close> \<open>Useful sufficient conditions for the winding number to be positive\<close>
then\>by:winding_number_valid_path)
( assumes
 auto:)
 qed (use x in auto showthesis
qed
 qedhave: 0java.lang.StringIndexOutOfBoundsException: Range [0, 95) out of bounds for length 3
 }
 with showthesis2
simpcomplex
qed :"\ z"

lemma winding_number_exp_2pi:
 "\path p; z \ path_image p\
 <Longrightarrow> pathfinish p - z = exp (2 * pi * \ * winding_number p z) * (pathstart p - z)"
ing_numberp z ]unfolding path_image_def pathfinish_def"
 by (force: winding_number_exp_integral

lemma:
 assumes \<gamma>: "path \<gamma>" and z: "z \<notin> path_image \<gamma>"
 shows winding_number_eq_1:
proofmoreover " <>path_image(eversepath \)"
obtainwhere"valid_pathz:java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18
 winding_number_pos_meetsjava.lang.StringIndexOutOfBoundsException: Index 47 out of bounds for length 47
 "p\lambdaw /w ) =complex_of_real ( pi
 using
?thesis
 winding_number_valid_path
 ysimpwinding_number_pos_meets
using java.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28
"
 by (metis pathstart_def
 then" (contour_integral (<>w.1 ( -))=(
 using (2) ofz]
 by (simp add: valid_path_def path_defs contour_integral_integral

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

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

lemma winding_number_big_meets:
 using byjava.lang.StringIndexOutOfBoundsException: Range [22, 23) out of bounds for length 22
 assumes by (simp add: cls connected_with_inside cos)
 and w: "w \ z"
 shows 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)
 &olor:red'>using w
 then show "( bysimp add: inner_diff_right)+
 by (intro holomorphic_intros; simp add: dist_norm)
 qed (use p vap pip loop in auto)
 qed (use p in auto)
 }
 then show ?thesis
 by java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
 qed
 finally show ?thesis .
qed

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

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

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

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

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

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

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

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

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

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

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

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

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

--> maximum size reached

--> --------------------

quality99%
pan>:
 0( 1java.lang.StringIndexOutOfBoundsException: Index 67 out of bounds for length 67
 shows
proof -
 have " with assms Re(inding_number\ z) < 0 \ - Re (winding_number \ z) < 1/2"
 by( complex_inner_1_right [OF, ofusing\<gamma> loop, of "outside(path_image \<gamma>)"] connected_outside
 by ultimately winding_number
 valid_path_imp_path ( no_types constant_on_def
 withusing
 using \<open>z \<notin> path_image \<gamma>\<close> winding_number_lt_half_lemma by fastforce
qed

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

--> maximum size reached

--> --------------------

quality99%

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