src/HOL/Complex_Analysis/Residue_Theorem.thy
author wenzelm
Mon, 11 Sep 2023 19:30:48 +0200
changeset 78659 b5f3d1051b13
parent 77322 9c295f84d55f
child 78517 28c1f4f5335f
permissions -rw-r--r--
tuned;
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
71201
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
     1
section \<open>The Residue Theorem, the Argument Principle and Rouch\'{e}'s Theorem\<close>
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
     2
theory Residue_Theorem
73048
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
     3
  imports Complex_Residues "HOL-Library.Landau_Symbols"
71201
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
     4
begin
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
     5
77322
9c295f84d55f Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents: 77277
diff changeset
     6
text \<open>Several theorems that could be moved up, 
9c295f84d55f Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents: 77277
diff changeset
     7
  IF there were a previous theory importing both Landau Symbols and Elementary Metric Spaces\<close>
9c295f84d55f Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents: 77277
diff changeset
     8
77277
c6b50597abbc More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents: 77223
diff changeset
     9
lemma continuous_bounded_at_infinity_imp_bounded:
c6b50597abbc More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents: 77223
diff changeset
    10
  fixes f :: "real \<Rightarrow> 'a :: real_normed_field"
c6b50597abbc More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents: 77223
diff changeset
    11
  assumes "f \<in> O[at_bot](\<lambda>_. 1)"
c6b50597abbc More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents: 77223
diff changeset
    12
  assumes "f \<in> O[at_top](\<lambda>_. 1)"
c6b50597abbc More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents: 77223
diff changeset
    13
  assumes "continuous_on UNIV f"
c6b50597abbc More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents: 77223
diff changeset
    14
  shows   "bounded (range f)"
c6b50597abbc More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents: 77223
diff changeset
    15
proof -
c6b50597abbc More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents: 77223
diff changeset
    16
  from assms(1) obtain c1 where "eventually (\<lambda>x. norm (f x) \<le> c1) at_bot"
c6b50597abbc More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents: 77223
diff changeset
    17
    by (auto elim!: landau_o.bigE)
c6b50597abbc More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents: 77223
diff changeset
    18
  then obtain x1 where x1: "\<And>x. x \<le> x1 \<Longrightarrow> norm (f x) \<le> c1"
c6b50597abbc More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents: 77223
diff changeset
    19
    by (auto simp: eventually_at_bot_linorder)
c6b50597abbc More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents: 77223
diff changeset
    20
  from assms(2) obtain c2 where "eventually (\<lambda>x. norm (f x) \<le> c2) at_top"
c6b50597abbc More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents: 77223
diff changeset
    21
    by (auto elim!: landau_o.bigE)
c6b50597abbc More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents: 77223
diff changeset
    22
  then obtain x2 where x2: "\<And>x. x \<ge> x2 \<Longrightarrow> norm (f x) \<le> c2"
c6b50597abbc More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents: 77223
diff changeset
    23
    by (auto simp: eventually_at_top_linorder)
c6b50597abbc More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents: 77223
diff changeset
    24
  have "compact (f ` {x1..x2})"
c6b50597abbc More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents: 77223
diff changeset
    25
    by (intro compact_continuous_image continuous_on_subset[OF assms(3)]) auto
c6b50597abbc More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents: 77223
diff changeset
    26
  hence "bounded (f ` {x1..x2})"
c6b50597abbc More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents: 77223
diff changeset
    27
    by (rule compact_imp_bounded)
c6b50597abbc More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents: 77223
diff changeset
    28
  then obtain c3 where c3: "\<And>x. x \<in> {x1..x2} \<Longrightarrow> norm (f x) \<le> c3"
c6b50597abbc More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents: 77223
diff changeset
    29
    unfolding bounded_iff by fast
c6b50597abbc More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents: 77223
diff changeset
    30
  have "norm (f x) \<le> Max {c1, c2, c3}" for x
c6b50597abbc More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents: 77223
diff changeset
    31
    by (cases "x \<le> x1"; cases "x \<ge> x2") (use x1 x2 c3 in \<open>auto simp: le_max_iff_disj\<close>)
c6b50597abbc More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents: 77223
diff changeset
    32
  thus ?thesis
c6b50597abbc More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents: 77223
diff changeset
    33
    unfolding bounded_iff by blast
c6b50597abbc More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents: 77223
diff changeset
    34
qed
c6b50597abbc More of Eberl's contributions: memomorphic functions
paulson <lp15@cam.ac.uk>
parents: 77223
diff changeset
    35
77322
9c295f84d55f Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents: 77277
diff changeset
    36
lemma holomorphic_on_extend:
9c295f84d55f Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents: 77277
diff changeset
    37
  assumes "f holomorphic_on S - {\<xi>}" "\<xi> \<in> interior S" "f \<in> O[at \<xi>](\<lambda>_. 1)"
9c295f84d55f Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents: 77277
diff changeset
    38
  shows   "(\<exists>g. g holomorphic_on S \<and> (\<forall>z\<in>S - {\<xi>}. g z = f z))"
9c295f84d55f Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents: 77277
diff changeset
    39
  by (subst holomorphic_on_extend_bounded) (insert assms, auto elim!: landau_o.bigE)
9c295f84d55f Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents: 77277
diff changeset
    40
9c295f84d55f Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents: 77277
diff changeset
    41
lemma removable_singularities:
9c295f84d55f Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents: 77277
diff changeset
    42
  assumes "finite X" "X \<subseteq> interior S" "f holomorphic_on (S - X)"
9c295f84d55f Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents: 77277
diff changeset
    43
  assumes "\<And>z. z \<in> X \<Longrightarrow> f \<in> O[at z](\<lambda>_. 1)"
9c295f84d55f Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents: 77277
diff changeset
    44
  shows   "\<exists>g. g holomorphic_on S \<and> (\<forall>z\<in>S-X. g z = f z)"
9c295f84d55f Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents: 77277
diff changeset
    45
  using assms
9c295f84d55f Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents: 77277
diff changeset
    46
proof (induction arbitrary: f rule: finite_induct)
9c295f84d55f Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents: 77277
diff changeset
    47
  case empty
9c295f84d55f Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents: 77277
diff changeset
    48
  thus ?case by auto
9c295f84d55f Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents: 77277
diff changeset
    49
next
9c295f84d55f Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents: 77277
diff changeset
    50
  case (insert z0 X f)
9c295f84d55f Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents: 77277
diff changeset
    51
  from insert.prems and insert.hyps have z0: "z0 \<in> interior (S - X)"
9c295f84d55f Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents: 77277
diff changeset
    52
    by (auto simp: interior_diff finite_imp_closed)
9c295f84d55f Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents: 77277
diff changeset
    53
  hence "\<exists>g. g holomorphic_on (S - X) \<and> (\<forall>z\<in>S - X - {z0}. g z = f z)"
9c295f84d55f Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents: 77277
diff changeset
    54
    using insert.prems insert.hyps by (intro holomorphic_on_extend) auto
9c295f84d55f Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents: 77277
diff changeset
    55
  then obtain g where g: "g holomorphic_on (S - X)" "\<forall>z\<in>S - X - {z0}. g z = f z" by blast
9c295f84d55f Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents: 77277
diff changeset
    56
  have "\<exists>h. h holomorphic_on S \<and> (\<forall>z\<in>S - X. h z = g z)"
9c295f84d55f Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents: 77277
diff changeset
    57
  proof (rule insert.IH)
9c295f84d55f Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents: 77277
diff changeset
    58
    fix z0' assume z0': "z0' \<in> X"
9c295f84d55f Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents: 77277
diff changeset
    59
    hence "eventually (\<lambda>z. z \<in> interior S - (X - {z0'}) - {z0}) (nhds z0')"
9c295f84d55f Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents: 77277
diff changeset
    60
      using insert.prems insert.hyps
9c295f84d55f Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents: 77277
diff changeset
    61
      by (intro eventually_nhds_in_open open_Diff finite_imp_closed) auto
9c295f84d55f Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents: 77277
diff changeset
    62
    hence ev: "eventually (\<lambda>z. z \<in> S - X - {z0}) (at z0')"
9c295f84d55f Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents: 77277
diff changeset
    63
      unfolding eventually_at_filter 
9c295f84d55f Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents: 77277
diff changeset
    64
      by eventually_elim (insert z0' insert.hyps interior_subset[of S], auto)
9c295f84d55f Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents: 77277
diff changeset
    65
    have "g \<in> \<Theta>[at z0'](f)"
9c295f84d55f Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents: 77277
diff changeset
    66
      by (intro bigthetaI_cong eventually_mono[OF ev]) (insert g, auto)
9c295f84d55f Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents: 77277
diff changeset
    67
    also have "f \<in> O[at z0'](\<lambda>_. 1)"
9c295f84d55f Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents: 77277
diff changeset
    68
      using z0' by (intro insert.prems) auto
9c295f84d55f Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents: 77277
diff changeset
    69
    finally show "g \<in> \<dots>" .
9c295f84d55f Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents: 77277
diff changeset
    70
  qed (insert insert.prems g, auto)
9c295f84d55f Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents: 77277
diff changeset
    71
  then obtain h where "h holomorphic_on S" "\<forall>z\<in>S - X. h z = g z" by blast
9c295f84d55f Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents: 77277
diff changeset
    72
  with g have "h holomorphic_on S" "\<forall>z\<in>S - insert z0 X. h z = f z" by auto
9c295f84d55f Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents: 77277
diff changeset
    73
  thus ?case by blast
9c295f84d55f Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents: 77277
diff changeset
    74
qed
9c295f84d55f Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents: 77277
diff changeset
    75
9c295f84d55f Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents: 77277
diff changeset
    76
lemma continuous_imp_bigo_1:
9c295f84d55f Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents: 77277
diff changeset
    77
  assumes "continuous (at x within A) f"
9c295f84d55f Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents: 77277
diff changeset
    78
  shows   "f \<in> O[at x within A](\<lambda>_. 1)"
9c295f84d55f Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents: 77277
diff changeset
    79
proof (rule bigoI_tendsto)
9c295f84d55f Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents: 77277
diff changeset
    80
  from assms show "((\<lambda>x. f x / 1) \<longlongrightarrow> f x) (at x within A)"
9c295f84d55f Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents: 77277
diff changeset
    81
    by (auto simp: continuous_within)
9c295f84d55f Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents: 77277
diff changeset
    82
qed auto
9c295f84d55f Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents: 77277
diff changeset
    83
9c295f84d55f Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents: 77277
diff changeset
    84
lemma taylor_bigo_linear:
9c295f84d55f Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents: 77277
diff changeset
    85
  assumes "f field_differentiable at x0 within A"
9c295f84d55f Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents: 77277
diff changeset
    86
  shows   "(\<lambda>x. f x - f x0) \<in> O[at x0 within A](\<lambda>x. x - x0)"
9c295f84d55f Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents: 77277
diff changeset
    87
proof -
9c295f84d55f Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents: 77277
diff changeset
    88
  from assms obtain f' where "(f has_field_derivative f') (at x0 within A)"
9c295f84d55f Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents: 77277
diff changeset
    89
    by (auto simp: field_differentiable_def)
9c295f84d55f Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents: 77277
diff changeset
    90
  hence "((\<lambda>x. (f x - f x0) / (x - x0)) \<longlongrightarrow> f') (at x0 within A)"
9c295f84d55f Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents: 77277
diff changeset
    91
    by (auto simp: has_field_derivative_iff)
9c295f84d55f Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents: 77277
diff changeset
    92
  thus ?thesis by (intro bigoI_tendsto[where c = f']) (auto simp: eventually_at_filter)
9c295f84d55f Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents: 77277
diff changeset
    93
qed
9c295f84d55f Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents: 77277
diff changeset
    94
9c295f84d55f Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents: 77277
diff changeset
    95
71201
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
    96
subsection \<open>Cauchy's residue theorem\<close>
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
    97
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
    98
lemma get_integrable_path:
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
    99
  assumes "open s" "connected (s-pts)" "finite pts" "f holomorphic_on (s-pts) " "a\<in>s-pts" "b\<in>s-pts"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   100
  obtains g where "valid_path g" "pathstart g = a" "pathfinish g = b"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   101
    "path_image g \<subseteq> s-pts" "f contour_integrable_on g" using assms
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   102
proof (induct arbitrary:s thesis a rule:finite_induct[OF \<open>finite pts\<close>])
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   103
  case 1
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   104
  obtain g where "valid_path g" "path_image g \<subseteq> s" "pathstart g = a" "pathfinish g = b"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   105
    using connected_open_polynomial_connected[OF \<open>open s\<close>,of a b ] \<open>connected (s - {})\<close>
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   106
      valid_path_polynomial_function "1.prems"(6) "1.prems"(7) by auto
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   107
  moreover have "f contour_integrable_on g"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   108
    using contour_integrable_holomorphic_simple[OF _ \<open>open s\<close> \<open>valid_path g\<close> \<open>path_image g \<subseteq> s\<close>,of f]
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   109
      \<open>f holomorphic_on s - {}\<close>
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   110
    by auto
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   111
  ultimately show ?case using "1"(1)[of g] by auto
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   112
next
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   113
  case idt:(2 p pts)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   114
  obtain e where "e>0" and e:"\<forall>w\<in>ball a e. w \<in> s \<and> (w \<noteq> a \<longrightarrow> w \<notin> insert p pts)"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   115
    using finite_ball_avoid[OF \<open>open s\<close> \<open>finite (insert p pts)\<close>, of a]
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   116
      \<open>a \<in> s - insert p pts\<close>
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   117
    by auto
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   118
  define a' where "a' \<equiv> a+e/2"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   119
  have "a'\<in>s-{p} -pts"  using e[rule_format,of "a+e/2"] \<open>e>0\<close>
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   120
    by (auto simp add:dist_complex_def a'_def)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   121
  then obtain g' where g'[simp]:"valid_path g'" "pathstart g' = a'" "pathfinish g' = b"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   122
    "path_image g' \<subseteq> s - {p} - pts" "f contour_integrable_on g'"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   123
    using idt.hyps(3)[of a' "s-{p}"] idt.prems idt.hyps(1)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   124
    by (metis Diff_insert2 open_delete)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   125
  define g where "g \<equiv> linepath a a' +++ g'"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   126
  have "valid_path g" unfolding g_def by (auto intro: valid_path_join)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   127
  moreover have "pathstart g = a" and  "pathfinish g = b" unfolding g_def by auto
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   128
  moreover have "path_image g \<subseteq> s - insert p pts" unfolding g_def
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   129
    proof (rule subset_path_image_join)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   130
      have "closed_segment a a' \<subseteq> ball a e" using \<open>e>0\<close>
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   131
        by (auto dest!:segment_bound1 simp:a'_def dist_complex_def norm_minus_commute)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   132
      then show "path_image (linepath a a') \<subseteq> s - insert p pts" using e idt(9)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   133
        by auto
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   134
    next
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   135
      show "path_image g' \<subseteq> s - insert p pts" using g'(4) by blast
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   136
    qed
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   137
  moreover have "f contour_integrable_on g"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   138
    proof -
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   139
      have "closed_segment a a' \<subseteq> ball a e" using \<open>e>0\<close>
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   140
        by (auto dest!:segment_bound1 simp:a'_def dist_complex_def norm_minus_commute)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   141
      then have "continuous_on (closed_segment a a') f"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   142
        using e idt.prems(6) holomorphic_on_imp_continuous_on[OF idt.prems(5)]
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   143
        apply (elim continuous_on_subset)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   144
        by auto
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   145
      then have "f contour_integrable_on linepath a a'"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   146
        using contour_integrable_continuous_linepath by auto
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   147
      then show ?thesis unfolding g_def
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   148
        apply (rule contour_integrable_joinI)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   149
        by (auto simp add: \<open>e>0\<close>)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   150
    qed
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   151
  ultimately show ?case using idt.prems(1)[of g] by auto
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   152
qed
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   153
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   154
lemma Cauchy_theorem_aux:
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   155
  assumes "open s" "connected (s-pts)" "finite pts" "pts \<subseteq> s" "f holomorphic_on s-pts"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   156
          "valid_path g" "pathfinish g = pathstart g" "path_image g \<subseteq> s-pts"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   157
          "\<forall>z. (z \<notin> s) \<longrightarrow> winding_number g z  = 0"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   158
          "\<forall>p\<in>s. h p>0 \<and> (\<forall>w\<in>cball p (h p). w\<in>s \<and> (w\<noteq>p \<longrightarrow> w \<notin> pts))"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   159
  shows "contour_integral g f = (\<Sum>p\<in>pts. winding_number g p * contour_integral (circlepath p (h p)) f)"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   160
    using assms
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   161
proof (induct arbitrary:s g rule:finite_induct[OF \<open>finite pts\<close>])
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   162
  case 1
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   163
  then show ?case by (simp add: Cauchy_theorem_global contour_integral_unique)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   164
next
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   165
  case (2 p pts)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   166
  note fin[simp] = \<open>finite (insert p pts)\<close>
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   167
    and connected = \<open>connected (s - insert p pts)\<close>
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   168
    and valid[simp] = \<open>valid_path g\<close>
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   169
    and g_loop[simp] = \<open>pathfinish g = pathstart g\<close>
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   170
    and holo[simp]= \<open>f holomorphic_on s - insert p pts\<close>
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   171
    and path_img = \<open>path_image g \<subseteq> s - insert p pts\<close>
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   172
    and winding = \<open>\<forall>z. z \<notin> s \<longrightarrow> winding_number g z = 0\<close>
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   173
    and h = \<open>\<forall>pa\<in>s. 0 < h pa \<and> (\<forall>w\<in>cball pa (h pa). w \<in> s \<and> (w \<noteq> pa \<longrightarrow> w \<notin> insert p pts))\<close>
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   174
  have "h p>0" and "p\<in>s"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   175
    and h_p: "\<forall>w\<in>cball p (h p). w \<in> s \<and> (w \<noteq> p \<longrightarrow> w \<notin> insert p pts)"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   176
    using h \<open>insert p pts \<subseteq> s\<close> by auto
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   177
  obtain pg where pg[simp]: "valid_path pg" "pathstart pg = pathstart g" "pathfinish pg=p+h p"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   178
      "path_image pg \<subseteq> s-insert p pts" "f contour_integrable_on pg"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   179
    proof -
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   180
      have "p + h p\<in>cball p (h p)" using h[rule_format,of p]
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   181
        by (simp add: \<open>p \<in> s\<close> dist_norm)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   182
      then have "p + h p \<in> s - insert p pts" using h[rule_format,of p] \<open>insert p pts \<subseteq> s\<close>
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   183
        by fastforce
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   184
      moreover have "pathstart g \<in> s - insert p pts " using path_img by auto
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   185
      ultimately show ?thesis
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   186
        using get_integrable_path[OF \<open>open s\<close> connected fin holo,of "pathstart g" "p+h p"] that
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   187
        by blast
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   188
    qed
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   189
  obtain n::int where "n=winding_number g p"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   190
    using integer_winding_number[OF _ g_loop,of p] valid path_img
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   191
    by (metis DiffD2 Ints_cases insertI1 subset_eq valid_path_imp_path)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   192
  define p_circ where "p_circ \<equiv> circlepath p (h p)"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   193
  define p_circ_pt where "p_circ_pt \<equiv> linepath (p+h p) (p+h p)"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   194
  define n_circ where "n_circ \<equiv> \<lambda>n. ((+++) p_circ ^^ n) p_circ_pt"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   195
  define cp where "cp \<equiv> if n\<ge>0 then reversepath (n_circ (nat n)) else n_circ (nat (- n))"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   196
  have n_circ:"valid_path (n_circ k)"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   197
      "winding_number (n_circ k) p = k"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   198
      "pathstart (n_circ k) = p + h p" "pathfinish (n_circ k) = p + h p"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   199
      "path_image (n_circ k) =  (if k=0 then {p + h p} else sphere p (h p))"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   200
      "p \<notin> path_image (n_circ k)"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   201
      "\<And>p'. p'\<notin>s - pts \<Longrightarrow> winding_number (n_circ k) p'=0 \<and> p'\<notin>path_image (n_circ k)"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   202
      "f contour_integrable_on (n_circ k)"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   203
      "contour_integral (n_circ k) f = k *  contour_integral p_circ f"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   204
      for k
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   205
    proof (induct k)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   206
      case 0
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   207
      show "valid_path (n_circ 0)"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   208
        and "path_image (n_circ 0) =  (if 0=0 then {p + h p} else sphere p (h p))"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   209
        and "winding_number (n_circ 0) p = of_nat 0"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   210
        and "pathstart (n_circ 0) = p + h p"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   211
        and "pathfinish (n_circ 0) = p + h p"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   212
        and "p \<notin> path_image (n_circ 0)"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   213
        unfolding n_circ_def p_circ_pt_def using \<open>h p > 0\<close>
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   214
        by (auto simp add: dist_norm)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   215
      show "winding_number (n_circ 0) p'=0 \<and> p'\<notin>path_image (n_circ 0)" when "p'\<notin>s- pts" for p'
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   216
        unfolding n_circ_def p_circ_pt_def
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   217
        apply (auto intro!:winding_number_trivial)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   218
        by (metis Diff_iff pathfinish_in_path_image pg(3) pg(4) subsetCE subset_insertI that)+
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   219
      show "f contour_integrable_on (n_circ 0)"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   220
        unfolding n_circ_def p_circ_pt_def
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   221
        by (auto intro!:contour_integrable_continuous_linepath simp add:continuous_on_sing)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   222
      show "contour_integral (n_circ 0) f = of_nat 0  *  contour_integral p_circ f"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   223
        unfolding n_circ_def p_circ_pt_def by auto
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   224
    next
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   225
      case (Suc k)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   226
      have n_Suc:"n_circ (Suc k) = p_circ +++ n_circ k" unfolding n_circ_def by auto
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   227
      have pcirc:"p \<notin> path_image p_circ" "valid_path p_circ" "pathfinish p_circ = pathstart (n_circ k)"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   228
        using Suc(3) unfolding p_circ_def using \<open>h p > 0\<close> by (auto simp add: p_circ_def)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   229
      have pcirc_image:"path_image p_circ \<subseteq> s - insert p pts"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   230
        proof -
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   231
          have "path_image p_circ \<subseteq> cball p (h p)" using \<open>0 < h p\<close> p_circ_def by auto
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   232
          then show ?thesis using h_p pcirc(1) by auto
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   233
        qed
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   234
      have pcirc_integrable:"f contour_integrable_on p_circ"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   235
        by (auto simp add:p_circ_def intro!: pcirc_image[unfolded p_circ_def]
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   236
          contour_integrable_continuous_circlepath holomorphic_on_imp_continuous_on
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   237
          holomorphic_on_subset[OF holo])
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   238
      show "valid_path (n_circ (Suc k))"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   239
        using valid_path_join[OF pcirc(2) Suc(1) pcirc(3)] unfolding n_circ_def by auto
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   240
      show "path_image (n_circ (Suc k))
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   241
          = (if Suc k = 0 then {p + complex_of_real (h p)} else sphere p (h p))"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   242
        proof -
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   243
          have "path_image p_circ = sphere p (h p)"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   244
            unfolding p_circ_def using \<open>0 < h p\<close> by auto
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   245
          then show ?thesis unfolding n_Suc  using Suc.hyps(5)  \<open>h p>0\<close>
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   246
            by (auto simp add:  path_image_join[OF pcirc(3)]  dist_norm)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   247
        qed
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   248
      then show "p \<notin> path_image (n_circ (Suc k))" using \<open>h p>0\<close> by auto
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   249
      show "winding_number (n_circ (Suc k)) p = of_nat (Suc k)"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   250
        proof -
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   251
          have "winding_number p_circ p = 1"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   252
            by (simp add: \<open>h p > 0\<close> p_circ_def winding_number_circlepath_centre)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   253
          moreover have "p \<notin> path_image (n_circ k)" using Suc(5) \<open>h p>0\<close> by auto
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   254
          then have "winding_number (p_circ +++ n_circ k) p
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   255
              = winding_number p_circ p + winding_number (n_circ k) p"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   256
            using  valid_path_imp_path Suc.hyps(1) Suc.hyps(2) pcirc
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   257
            apply (intro winding_number_join)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   258
            by auto
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   259
          ultimately show ?thesis using Suc(2) unfolding n_circ_def
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   260
            by auto
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   261
        qed
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   262
      show "pathstart (n_circ (Suc k)) = p + h p"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   263
        by (simp add: n_circ_def p_circ_def)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   264
      show "pathfinish (n_circ (Suc k)) = p + h p"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   265
        using Suc(4) unfolding n_circ_def by auto
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   266
      show "winding_number (n_circ (Suc k)) p'=0 \<and>  p'\<notin>path_image (n_circ (Suc k))" when "p'\<notin>s-pts" for p'
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   267
        proof -
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   268
          have " p' \<notin> path_image p_circ" using \<open>p \<in> s\<close> h p_circ_def that using pcirc_image by blast
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   269
          moreover have "p' \<notin> path_image (n_circ k)"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   270
            using Suc.hyps(7) that by blast
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   271
          moreover have "winding_number p_circ p' = 0"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   272
            proof -
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   273
              have "path_image p_circ \<subseteq> cball p (h p)"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   274
                using h unfolding p_circ_def using \<open>p \<in> s\<close> by fastforce
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   275
              moreover have "p'\<notin>cball p (h p)" using \<open>p \<in> s\<close> h that "2.hyps"(2) by fastforce
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   276
              ultimately show ?thesis unfolding p_circ_def
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   277
                apply (intro winding_number_zero_outside)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   278
                by auto
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   279
            qed
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   280
          ultimately show ?thesis
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   281
            unfolding n_Suc
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   282
            apply (subst winding_number_join)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   283
            by (auto simp: valid_path_imp_path pcirc Suc that not_in_path_image_join Suc.hyps(7)[OF that])
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   284
        qed
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   285
      show "f contour_integrable_on (n_circ (Suc k))"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   286
        unfolding n_Suc
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   287
        by (rule contour_integrable_joinI[OF pcirc_integrable Suc(8) pcirc(2) Suc(1)])
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   288
      show "contour_integral (n_circ (Suc k)) f = (Suc k) *  contour_integral p_circ f"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   289
        unfolding n_Suc
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   290
        by (auto simp add:contour_integral_join[OF pcirc_integrable Suc(8) pcirc(2) Suc(1)]
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   291
          Suc(9) algebra_simps)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   292
    qed
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   293
  have cp[simp]:"pathstart cp = p + h p"  "pathfinish cp = p + h p"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   294
         "valid_path cp" "path_image cp \<subseteq> s - insert p pts"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   295
         "winding_number cp p = - n"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   296
         "\<And>p'. p'\<notin>s - pts \<Longrightarrow> winding_number cp p'=0 \<and> p' \<notin> path_image cp"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   297
         "f contour_integrable_on cp"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   298
         "contour_integral cp f = - n * contour_integral p_circ f"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   299
    proof -
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   300
      show "pathstart cp = p + h p" and "pathfinish cp = p + h p" and "valid_path cp"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   301
        using n_circ unfolding cp_def by auto
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   302
    next
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   303
      have "sphere p (h p) \<subseteq>  s - insert p pts"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   304
        using h[rule_format,of p] \<open>insert p pts \<subseteq> s\<close> by force
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   305
      moreover  have "p + complex_of_real (h p) \<in> s - insert p pts"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   306
        using pg(3) pg(4) by (metis pathfinish_in_path_image subsetCE)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   307
      ultimately show "path_image cp \<subseteq>  s - insert p pts" unfolding cp_def
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   308
        using n_circ(5)  by auto
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   309
    next
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   310
      show "winding_number cp p = - n"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   311
        unfolding cp_def using winding_number_reversepath n_circ \<open>h p>0\<close>
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   312
        by (auto simp: valid_path_imp_path)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   313
    next
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   314
      show "winding_number cp p'=0 \<and> p' \<notin> path_image cp" when "p'\<notin>s - pts" for p'
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   315
        unfolding cp_def
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   316
        apply (auto)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   317
        apply (subst winding_number_reversepath)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   318
        by (auto simp add: valid_path_imp_path n_circ(7)[OF that] n_circ(1))
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   319
    next
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   320
      show "f contour_integrable_on cp" unfolding cp_def
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   321
        using contour_integrable_reversepath_eq n_circ(1,8) by auto
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   322
    next
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   323
      show "contour_integral cp f = - n * contour_integral p_circ f"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   324
        unfolding cp_def using contour_integral_reversepath[OF n_circ(1)] n_circ(9)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   325
        by auto
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   326
    qed
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   327
  define g' where "g' \<equiv> g +++ pg +++ cp +++ (reversepath pg)"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   328
  have "contour_integral g' f = (\<Sum>p\<in>pts. winding_number g' p * contour_integral (circlepath p (h p)) f)"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   329
    proof (rule "2.hyps"(3)[of "s-{p}" "g'",OF _ _ \<open>finite pts\<close> ])
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   330
      show "connected (s - {p} - pts)" using connected by (metis Diff_insert2)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   331
      show "open (s - {p})" using \<open>open s\<close> by auto
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   332
      show " pts \<subseteq> s - {p}" using \<open>insert p pts \<subseteq> s\<close> \<open> p \<notin> pts\<close>  by blast
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   333
      show "f holomorphic_on s - {p} - pts" using holo \<open>p \<notin> pts\<close> by (metis Diff_insert2)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   334
      show "valid_path g'"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   335
        unfolding g'_def cp_def using n_circ valid pg g_loop
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   336
        by (auto intro!:valid_path_join )
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   337
      show "pathfinish g' = pathstart g'"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   338
        unfolding g'_def cp_def using pg(2) by simp
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   339
      show "path_image g' \<subseteq> s - {p} - pts"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   340
        proof -
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   341
          define s' where "s' \<equiv> s - {p} - pts"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   342
          have s':"s' = s-insert p pts " unfolding s'_def by auto
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   343
          then show ?thesis using path_img pg(4) cp(4)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   344
            unfolding g'_def
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   345
            apply (fold s'_def s')
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   346
            apply (intro subset_path_image_join)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   347
            by auto
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   348
        qed
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   349
      note path_join_imp[simp]
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   350
      show "\<forall>z. z \<notin> s - {p} \<longrightarrow> winding_number g' z = 0"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   351
        proof clarify
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   352
          fix z assume z:"z\<notin>s - {p}"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   353
          have "winding_number (g +++ pg +++ cp +++ reversepath pg) z = winding_number g z
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   354
              + winding_number (pg +++ cp +++ (reversepath pg)) z"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   355
            proof (rule winding_number_join)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   356
              show "path g" using \<open>valid_path g\<close> by (simp add: valid_path_imp_path)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   357
              show "z \<notin> path_image g" using z path_img by auto
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   358
              show "path (pg +++ cp +++ reversepath pg)" using pg(3) cp
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   359
                by (simp add: valid_path_imp_path)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   360
            next
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   361
              have "path_image (pg +++ cp +++ reversepath pg) \<subseteq> s - insert p pts"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   362
                using pg(4) cp(4) by (auto simp:subset_path_image_join)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   363
              then show "z \<notin> path_image (pg +++ cp +++ reversepath pg)" using z by auto
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   364
            next
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   365
              show "pathfinish g = pathstart (pg +++ cp +++ reversepath pg)" using g_loop by auto
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   366
            qed
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   367
          also have "... = winding_number g z + (winding_number pg z
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   368
              + winding_number (cp +++ (reversepath pg)) z)"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   369
            proof (subst add_left_cancel,rule winding_number_join)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   370
              show "path pg" and "path (cp +++ reversepath pg)"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   371
               and "pathfinish pg = pathstart (cp +++ reversepath pg)"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   372
                by (auto simp add: valid_path_imp_path)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   373
              show "z \<notin> path_image pg" using pg(4) z by blast
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   374
              show "z \<notin> path_image (cp +++ reversepath pg)" using z
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   375
                by (metis Diff_iff \<open>z \<notin> path_image pg\<close> contra_subsetD cp(4) insertI1
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   376
                  not_in_path_image_join path_image_reversepath singletonD)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   377
            qed
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   378
          also have "... = winding_number g z + (winding_number pg z
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   379
              + (winding_number cp z + winding_number (reversepath pg) z))"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   380
            apply (auto intro!:winding_number_join simp: valid_path_imp_path)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   381
            apply (metis Diff_iff contra_subsetD cp(4) insertI1 singletonD z)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   382
            by (metis Diff_insert2 Diff_subset contra_subsetD pg(4) z)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   383
          also have "... = winding_number g z + winding_number cp z"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   384
            apply (subst winding_number_reversepath)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   385
            apply (auto simp: valid_path_imp_path)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   386
            by (metis Diff_iff contra_subsetD insertI1 pg(4) singletonD z)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   387
          finally have "winding_number g' z = winding_number g z + winding_number cp z"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   388
            unfolding g'_def .
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   389
          moreover have "winding_number g z + winding_number cp z = 0"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   390
            using winding z \<open>n=winding_number g p\<close> by auto
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   391
          ultimately show "winding_number g' z = 0" unfolding g'_def by auto
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   392
        qed
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   393
      show "\<forall>pa\<in>s - {p}. 0 < h pa \<and> (\<forall>w\<in>cball pa (h pa). w \<in> s - {p} \<and> (w \<noteq> pa \<longrightarrow> w \<notin> pts))"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   394
        using h by fastforce
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   395
    qed
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   396
  moreover have "contour_integral g' f = contour_integral g f
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   397
      - winding_number g p * contour_integral p_circ f"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   398
    proof -
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   399
      have "contour_integral g' f =  contour_integral g f
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   400
        + contour_integral (pg +++ cp +++ reversepath pg) f"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   401
        unfolding g'_def
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   402
        apply (subst contour_integral_join)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   403
        by (auto simp add:open_Diff[OF \<open>open s\<close>,OF finite_imp_closed[OF fin]]
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   404
          intro!: contour_integrable_holomorphic_simple[OF holo _ _ path_img]
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   405
          contour_integrable_reversepath)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   406
      also have "... = contour_integral g f + contour_integral pg f
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   407
          + contour_integral (cp +++ reversepath pg) f"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   408
        apply (subst contour_integral_join)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   409
        by (auto simp add:contour_integrable_reversepath)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   410
      also have "... = contour_integral g f + contour_integral pg f
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   411
          + contour_integral cp f + contour_integral (reversepath pg) f"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   412
        apply (subst contour_integral_join)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   413
        by (auto simp add:contour_integrable_reversepath)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   414
      also have "... = contour_integral g f + contour_integral cp f"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   415
        using contour_integral_reversepath
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   416
        by (auto simp add:contour_integrable_reversepath)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   417
      also have "... = contour_integral g f - winding_number g p * contour_integral p_circ f"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   418
        using \<open>n=winding_number g p\<close> by auto
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   419
      finally show ?thesis .
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   420
    qed
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   421
  moreover have "winding_number g' p' = winding_number g p'" when "p'\<in>pts" for p'
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   422
    proof -
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   423
      have [simp]: "p' \<notin> path_image g" "p' \<notin> path_image pg" "p'\<notin>path_image cp"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   424
        using "2.prems"(8) that
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   425
        apply blast
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   426
        apply (metis Diff_iff Diff_insert2 contra_subsetD pg(4) that)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   427
        by (meson DiffD2 cp(4) rev_subsetD subset_insertI that)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   428
      have "winding_number g' p' = winding_number g p'
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   429
          + winding_number (pg +++ cp +++ reversepath pg) p'" unfolding g'_def
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   430
        apply (subst winding_number_join)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   431
        apply (simp_all add: valid_path_imp_path)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   432
        apply (intro not_in_path_image_join)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   433
        by auto
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   434
      also have "... = winding_number g p' + winding_number pg p'
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   435
          + winding_number (cp +++ reversepath pg) p'"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   436
        apply (subst winding_number_join)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   437
        apply (simp_all add: valid_path_imp_path)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   438
        apply (intro not_in_path_image_join)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   439
        by auto
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   440
      also have "... = winding_number g p' + winding_number pg p'+ winding_number cp p'
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   441
          + winding_number (reversepath pg) p'"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   442
        apply (subst winding_number_join)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   443
        by (simp_all add: valid_path_imp_path)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   444
      also have "... = winding_number g p' + winding_number cp p'"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   445
        apply (subst winding_number_reversepath)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   446
        by (simp_all add: valid_path_imp_path)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   447
      also have "... = winding_number g p'" using that by auto
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   448
      finally show ?thesis .
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   449
    qed
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   450
  ultimately show ?case unfolding p_circ_def
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   451
    apply (subst (asm) sum.cong[OF refl,
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   452
        of pts _ "\<lambda>p. winding_number g p * contour_integral (circlepath p (h p)) f"])
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   453
    by (auto simp add:sum.insert[OF \<open>finite pts\<close> \<open>p\<notin>pts\<close>] algebra_simps)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   454
qed
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   455
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   456
lemma Cauchy_theorem_singularities:
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   457
  assumes "open s" "connected s" "finite pts" and
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   458
          holo:"f holomorphic_on s-pts" and
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   459
          "valid_path g" and
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   460
          loop:"pathfinish g = pathstart g" and
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   461
          "path_image g \<subseteq> s-pts" and
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   462
          homo:"\<forall>z. (z \<notin> s) \<longrightarrow> winding_number g z  = 0" and
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   463
          avoid:"\<forall>p\<in>s. h p>0 \<and> (\<forall>w\<in>cball p (h p). w\<in>s \<and> (w\<noteq>p \<longrightarrow> w \<notin> pts))"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   464
  shows "contour_integral g f = (\<Sum>p\<in>pts. winding_number g p * contour_integral (circlepath p (h p)) f)"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   465
    (is "?L=?R")
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   466
proof -
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   467
  define circ where "circ \<equiv> \<lambda>p. winding_number g p * contour_integral (circlepath p (h p)) f"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   468
  define pts1 where "pts1 \<equiv> pts \<inter> s"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   469
  define pts2 where "pts2 \<equiv> pts - pts1"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   470
  have "pts=pts1 \<union> pts2" "pts1 \<inter> pts2 = {}" "pts2 \<inter> s={}" "pts1\<subseteq>s"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   471
    unfolding pts1_def pts2_def by auto
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   472
  have "contour_integral g f =  (\<Sum>p\<in>pts1. circ p)" unfolding circ_def
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   473
    proof (rule Cauchy_theorem_aux[OF \<open>open s\<close> _ _ \<open>pts1\<subseteq>s\<close> _ \<open>valid_path g\<close> loop _ homo])
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   474
      have "finite pts1" unfolding pts1_def using \<open>finite pts\<close> by auto
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   475
      then show "connected (s - pts1)"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   476
        using \<open>open s\<close> \<open>connected s\<close> connected_open_delete_finite[of s] by auto
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   477
    next
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   478
      show "finite pts1" using \<open>pts = pts1 \<union> pts2\<close> assms(3) by auto
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   479
      show "f holomorphic_on s - pts1" by (metis Diff_Int2 Int_absorb holo pts1_def)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   480
      show "path_image g \<subseteq> s - pts1" using assms(7) pts1_def by auto
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   481
      show "\<forall>p\<in>s. 0 < h p \<and> (\<forall>w\<in>cball p (h p). w \<in> s \<and> (w \<noteq> p \<longrightarrow> w \<notin> pts1))"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   482
        by (simp add: avoid pts1_def)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   483
    qed
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   484
  moreover have "sum circ pts2=0"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   485
    proof -
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   486
      have "winding_number g p=0" when "p\<in>pts2" for p
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   487
        using  \<open>pts2 \<inter> s={}\<close> that homo[rule_format,of p] by auto
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   488
      thus ?thesis unfolding circ_def
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   489
        apply (intro sum.neutral)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   490
        by auto
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   491
    qed
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   492
  moreover have "?R=sum circ pts1 + sum circ pts2"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   493
    unfolding circ_def
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   494
    using sum.union_disjoint[OF _ _ \<open>pts1 \<inter> pts2 = {}\<close>] \<open>finite pts\<close> \<open>pts=pts1 \<union> pts2\<close>
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   495
    by blast
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   496
  ultimately show ?thesis
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   497
    apply (fold circ_def)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   498
    by auto
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   499
qed
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   500
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   501
theorem Residue_theorem:
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   502
  fixes s pts::"complex set" and f::"complex \<Rightarrow> complex"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   503
    and g::"real \<Rightarrow> complex"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   504
  assumes "open s" "connected s" "finite pts" and
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   505
          holo:"f holomorphic_on s-pts" and
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   506
          "valid_path g" and
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   507
          loop:"pathfinish g = pathstart g" and
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   508
          "path_image g \<subseteq> s-pts" and
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   509
          homo:"\<forall>z. (z \<notin> s) \<longrightarrow> winding_number g z  = 0"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   510
  shows "contour_integral g f = 2 * pi * \<i> *(\<Sum>p\<in>pts. winding_number g p * residue f p)"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   511
proof -
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   512
  define c where "c \<equiv>  2 * pi * \<i>"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   513
  obtain h where avoid:"\<forall>p\<in>s. h p>0 \<and> (\<forall>w\<in>cball p (h p). w\<in>s \<and> (w\<noteq>p \<longrightarrow> w \<notin> pts))"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   514
    using finite_cball_avoid[OF \<open>open s\<close> \<open>finite pts\<close>] by metis
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   515
  have "contour_integral g f
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   516
      = (\<Sum>p\<in>pts. winding_number g p * contour_integral (circlepath p (h p)) f)"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   517
    using Cauchy_theorem_singularities[OF assms avoid] .
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   518
  also have "... = (\<Sum>p\<in>pts.  c * winding_number g p * residue f p)"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   519
    proof (intro sum.cong)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   520
      show "pts = pts" by simp
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   521
    next
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   522
      fix x assume "x \<in> pts"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   523
      show "winding_number g x * contour_integral (circlepath x (h x)) f
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   524
          = c * winding_number g x * residue f x"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   525
        proof (cases "x\<in>s")
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   526
          case False
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   527
          then have "winding_number g x=0" using homo by auto
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   528
          thus ?thesis by auto
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   529
        next
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   530
          case True
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   531
          have "contour_integral (circlepath x (h x)) f = c* residue f x"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   532
            using \<open>x\<in>pts\<close> \<open>finite pts\<close> avoid[rule_format,OF True]
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   533
            apply (intro base_residue[of "s-(pts-{x})",THEN contour_integral_unique,folded c_def])
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   534
            by (auto intro:holomorphic_on_subset[OF holo] open_Diff[OF \<open>open s\<close> finite_imp_closed])
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   535
          then show ?thesis by auto
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   536
        qed
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   537
    qed
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   538
  also have "... = c * (\<Sum>p\<in>pts. winding_number g p * residue f p)"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   539
    by (simp add: sum_distrib_left algebra_simps)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   540
  finally show ?thesis unfolding c_def .
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   541
qed
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   542
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   543
subsection \<open>The argument principle\<close>
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   544
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   545
theorem argument_principle:
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   546
  fixes f::"complex \<Rightarrow> complex" and poles s:: "complex set"
77223
607e1e345e8f Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents: 73048
diff changeset
   547
  defines "pz \<equiv> {w\<in>s. f w = 0 \<or> w \<in> poles}" \<comment> \<open>\<^term>\<open>pz\<close> is the set of poles and zeros\<close>
607e1e345e8f Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents: 73048
diff changeset
   548
  assumes "open s" "connected s" and
71201
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   549
          f_holo:"f holomorphic_on s-poles" and
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   550
          h_holo:"h holomorphic_on s" and
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   551
          "valid_path g" and
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   552
          loop:"pathfinish g = pathstart g" and
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   553
          path_img:"path_image g \<subseteq> s - pz" and
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   554
          homo:"\<forall>z. (z \<notin> s) \<longrightarrow> winding_number g z = 0" and
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   555
          finite:"finite pz" and
77223
607e1e345e8f Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents: 73048
diff changeset
   556
          poles:"\<forall>p\<in>s\<inter>poles. is_pole f p"
71201
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   557
  shows "contour_integral g (\<lambda>x. deriv f x * h x / f x) = 2 * pi * \<i> *
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   558
          (\<Sum>p\<in>pz. winding_number g p * h p * zorder f p)"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   559
    (is "?L=?R")
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   560
proof -
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   561
  define c where "c \<equiv> 2 * complex_of_real pi * \<i> "
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   562
  define ff where "ff \<equiv> (\<lambda>x. deriv f x * h x / f x)"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   563
  define cont where "cont \<equiv> \<lambda>ff p e. (ff has_contour_integral c * zorder f p * h p ) (circlepath p e)"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   564
  define avoid where "avoid \<equiv> \<lambda>p e. \<forall>w\<in>cball p e. w \<in> s \<and> (w \<noteq> p \<longrightarrow> w \<notin> pz)"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   565
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   566
  have "\<exists>e>0. avoid p e \<and> (p\<in>pz \<longrightarrow> cont ff p e)" when "p\<in>s" for p
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   567
  proof -
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   568
    obtain e1 where "e1>0" and e1_avoid:"avoid p e1"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   569
      using finite_cball_avoid[OF \<open>open s\<close> finite] \<open>p\<in>s\<close> unfolding avoid_def by auto
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   570
    have "\<exists>e2>0. cball p e2 \<subseteq> ball p e1 \<and> cont ff p e2" when "p\<in>pz"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   571
    proof -
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   572
      define po where "po \<equiv> zorder f p"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   573
      define pp where "pp \<equiv> zor_poly f p"
77322
9c295f84d55f Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents: 77277
diff changeset
   574
      define f' where "f' \<equiv> \<lambda>w. pp w * (w - p) powi po"
71201
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   575
      define ff' where "ff' \<equiv> (\<lambda>x. deriv f' x * h x / f' x)"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   576
      obtain r where "pp p\<noteq>0" "r>0" and
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   577
          "r<e1" and
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   578
          pp_holo:"pp holomorphic_on cball p r" and
77322
9c295f84d55f Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents: 77277
diff changeset
   579
          pp_po:"(\<forall>w\<in>cball p r-{p}. f w = pp w * (w - p) powi po \<and> pp w \<noteq> 0)"
71201
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   580
      proof -
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   581
        have "isolated_singularity_at f p"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   582
        proof -
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   583
          have "f holomorphic_on ball p e1 - {p}"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   584
            apply (intro holomorphic_on_subset[OF f_holo])
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   585
            using e1_avoid \<open>p\<in>pz\<close> unfolding avoid_def pz_def by force
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   586
          then show ?thesis unfolding isolated_singularity_at_def
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   587
            using \<open>e1>0\<close> analytic_on_open open_delete by blast
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   588
        qed
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   589
        moreover have "not_essential f p"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   590
        proof (cases "is_pole f p")
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   591
          case True
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   592
          then show ?thesis unfolding not_essential_def by auto
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   593
        next
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   594
          case False
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   595
          then have "p\<in>s-poles" using \<open>p\<in>s\<close> poles unfolding pz_def by auto
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   596
          moreover have "open (s-poles)"
77223
607e1e345e8f Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents: 73048
diff changeset
   597
          proof -
607e1e345e8f Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents: 73048
diff changeset
   598
            have "closed (s \<inter> poles)"
607e1e345e8f Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents: 73048
diff changeset
   599
              using finite by (simp add: pz_def finite_imp_closed rev_finite_subset subset_eq)
607e1e345e8f Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents: 73048
diff changeset
   600
            then show ?thesis
607e1e345e8f Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents: 73048
diff changeset
   601
              by (metis Diff_Compl Diff_Diff_Int Diff_eq \<open>open s\<close> open_Diff) 
607e1e345e8f Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents: 73048
diff changeset
   602
          qed
71201
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   603
          ultimately have "isCont f p"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   604
            using holomorphic_on_imp_continuous_on[OF f_holo] continuous_on_eq_continuous_at
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   605
            by auto
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   606
          then show ?thesis unfolding isCont_def not_essential_def by auto
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   607
        qed
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   608
        moreover have "\<exists>\<^sub>F w in at p. f w \<noteq> 0 "
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   609
        proof (rule ccontr)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   610
          assume "\<not> (\<exists>\<^sub>F w in at p. f w \<noteq> 0)"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   611
          then have "\<forall>\<^sub>F w in at p. f w= 0" unfolding frequently_def by auto
77223
607e1e345e8f Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents: 73048
diff changeset
   612
          then obtain r1 where "r1>0" and r1:"\<forall>w\<in>ball p r1 - {p}. f w =0"
71201
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   613
            unfolding eventually_at by (auto simp add:dist_commute)
77223
607e1e345e8f Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents: 73048
diff changeset
   614
          obtain r2 where "r2>0" and r2: "ball p r2 \<subseteq> s"
607e1e345e8f Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents: 73048
diff changeset
   615
            using \<open>p\<in>s\<close> \<open>open s\<close> openE by blast
607e1e345e8f Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents: 73048
diff changeset
   616
          define rr where "rr=min r1 r2"
607e1e345e8f Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents: 73048
diff changeset
   617
607e1e345e8f Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents: 73048
diff changeset
   618
          from r1 r2
607e1e345e8f Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents: 73048
diff changeset
   619
          have "ball p rr - {p} \<subseteq> {w\<in> s \<inter> ball p rr-{p}. f w=0}"
607e1e345e8f Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents: 73048
diff changeset
   620
            unfolding rr_def by auto
607e1e345e8f Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents: 73048
diff changeset
   621
          moreover have "infinite (ball p rr - {p})" 
607e1e345e8f Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents: 73048
diff changeset
   622
            using \<open>r1>0\<close> \<open>r2>0\<close> finite_imp_not_open 
607e1e345e8f Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents: 73048
diff changeset
   623
            unfolding rr_def by fastforce
607e1e345e8f Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents: 73048
diff changeset
   624
          ultimately have "infinite {w\<in>s \<inter> ball p rr-{p}. f w=0}" using infinite_super by blast
71201
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   625
          then have "infinite pz"
77223
607e1e345e8f Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents: 73048
diff changeset
   626
            unfolding pz_def by (smt (verit) infinite_super Collect_mono_iff DiffE Int_iff)
71201
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   627
          then show False using \<open>finite pz\<close> by auto
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   628
        qed
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   629
        ultimately obtain r where "pp p \<noteq> 0" and r:"r>0" "pp holomorphic_on cball p r"
77322
9c295f84d55f Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents: 77277
diff changeset
   630
                  "(\<forall>w\<in>cball p r - {p}. f w = pp w * (w - p) powi po \<and> pp w \<noteq> 0)"
71201
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   631
          using zorder_exist[of f p,folded po_def pp_def] by auto
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   632
        define r1 where "r1=min r e1 / 2"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   633
        have "r1<e1" unfolding r1_def using \<open>e1>0\<close> \<open>r>0\<close> by auto
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   634
        moreover have "r1>0" "pp holomorphic_on cball p r1"
77322
9c295f84d55f Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents: 77277
diff changeset
   635
                  "(\<forall>w\<in>cball p r1 - {p}. f w = pp w * (w - p) powi po \<and> pp w \<noteq> 0)"
71201
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   636
          unfolding r1_def using \<open>e1>0\<close> r by auto
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   637
        ultimately show ?thesis using that \<open>pp p\<noteq>0\<close> by auto
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   638
      qed
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   639
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   640
      define e2 where "e2 \<equiv> r/2"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   641
      have "e2>0" using \<open>r>0\<close> unfolding e2_def by auto
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   642
      define anal where "anal \<equiv> \<lambda>w. deriv pp w * h w / pp w"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   643
      define prin where "prin \<equiv> \<lambda>w. po * h w / (w - p)"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   644
      have "((\<lambda>w.  prin w + anal w) has_contour_integral c * po * h p) (circlepath p e2)"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   645
      proof (rule has_contour_integral_add[of _ _ _ _ 0,simplified])
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   646
        have "ball p r \<subseteq> s"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   647
          using \<open>r<e1\<close> avoid_def ball_subset_cball e1_avoid by (simp add: subset_eq)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   648
        then have "cball p e2 \<subseteq> s"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   649
          using \<open>r>0\<close> unfolding e2_def by auto
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   650
        then have "(\<lambda>w. po * h w) holomorphic_on cball p e2"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   651
          using h_holo by (auto intro!: holomorphic_intros)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   652
        then show "(prin has_contour_integral c * po * h p ) (circlepath p e2)"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   653
          using Cauchy_integral_circlepath_simple[folded c_def, of "\<lambda>w. po * h w"] \<open>e2>0\<close>
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   654
          unfolding prin_def by (auto simp add: mult.assoc)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   655
        have "anal holomorphic_on ball p r" unfolding anal_def
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   656
          using pp_holo h_holo pp_po \<open>ball p r \<subseteq> s\<close> \<open>pp p\<noteq>0\<close>
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   657
          by (auto intro!: holomorphic_intros)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   658
        then show "(anal has_contour_integral 0) (circlepath p e2)"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   659
          using e2_def \<open>r>0\<close>
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   660
          by (auto elim!: Cauchy_theorem_disc_simple)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   661
      qed
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   662
      then have "cont ff' p e2" unfolding cont_def po_def
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   663
      proof (elim has_contour_integral_eq)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   664
        fix w assume "w \<in> path_image (circlepath p e2)"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   665
        then have "w\<in>ball p r" and "w\<noteq>p" unfolding e2_def using \<open>r>0\<close> by auto
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   666
        define wp where "wp \<equiv> w-p"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   667
        have "wp\<noteq>0" and "pp w \<noteq>0"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   668
          unfolding wp_def using \<open>w\<noteq>p\<close> \<open>w\<in>ball p r\<close> pp_po by auto
77322
9c295f84d55f Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents: 77277
diff changeset
   669
        moreover have der_f':"deriv f' w = po * pp w * (w-p) powi (po - 1) + deriv pp w * (w-p) powi po"
71201
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   670
        proof (rule DERIV_imp_deriv)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   671
          have "(pp has_field_derivative (deriv pp w)) (at w)"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   672
            using DERIV_deriv_iff_has_field_derivative pp_holo \<open>w\<noteq>p\<close>
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   673
            by (meson open_ball \<open>w \<in> ball p r\<close> ball_subset_cball holomorphic_derivI holomorphic_on_subset)
77322
9c295f84d55f Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents: 77277
diff changeset
   674
          then show " (f' has_field_derivative of_int po * pp w * (w - p) powi (po - 1)
9c295f84d55f Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents: 77277
diff changeset
   675
                  + deriv pp w * (w - p) powi po) (at w)"
71201
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   676
            unfolding f'_def using \<open>w\<noteq>p\<close>
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   677
            by (auto intro!: derivative_eq_intros DERIV_cong[OF has_field_derivative_powr_of_int])
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   678
        qed
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   679
        ultimately show "prin w + anal w = ff' w"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   680
          unfolding ff'_def prin_def anal_def
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   681
          apply simp
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   682
          apply (unfold f'_def)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   683
          apply (fold wp_def)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   684
          apply (auto simp add:field_simps)
77322
9c295f84d55f Replacing z powr of_int i by z powi i and adding new material from the AFP
paulson <lp15@cam.ac.uk>
parents: 77277
diff changeset
   685
          by (metis (no_types, lifting) mult.commute power_int_minus_mult)
71201
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   686
      qed
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   687
      then have "cont ff p e2" unfolding cont_def
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   688
      proof (elim has_contour_integral_eq)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   689
        fix w assume "w \<in> path_image (circlepath p e2)"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   690
        then have "w\<in>ball p r" and "w\<noteq>p" unfolding e2_def using \<open>r>0\<close> by auto
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   691
        have "deriv f' w =  deriv f w"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   692
        proof (rule complex_derivative_transform_within_open[where s="ball p r - {p}"])
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   693
          show "f' holomorphic_on ball p r - {p}" unfolding f'_def using pp_holo
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   694
            by (auto intro!: holomorphic_intros)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   695
        next
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   696
          have "ball p e1 - {p} \<subseteq> s - poles"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   697
            using ball_subset_cball e1_avoid[unfolded avoid_def] unfolding pz_def
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   698
            by auto
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   699
          then have "ball p r - {p} \<subseteq> s - poles"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   700
            apply (elim dual_order.trans)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   701
            using \<open>r<e1\<close> by auto
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   702
          then show "f holomorphic_on ball p r - {p}" using f_holo
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   703
            by auto
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   704
        next
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   705
          show "open (ball p r - {p})" by auto
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   706
          show "w \<in> ball p r - {p}" using \<open>w\<in>ball p r\<close> \<open>w\<noteq>p\<close> by auto
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   707
        next
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   708
          fix x assume "x \<in> ball p r - {p}"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   709
          then show "f' x = f x"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   710
            using pp_po unfolding f'_def by auto
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   711
        qed
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   712
        moreover have " f' w  =  f w "
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   713
          using \<open>w \<in> ball p r\<close> ball_subset_cball subset_iff pp_po \<open>w\<noteq>p\<close>
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   714
          unfolding f'_def by auto
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   715
        ultimately show "ff' w = ff w"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   716
          unfolding ff'_def ff_def by simp
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   717
      qed
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   718
      moreover have "cball p e2 \<subseteq> ball p e1"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   719
        using \<open>0 < r\<close> \<open>r<e1\<close> e2_def by auto
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   720
      ultimately show ?thesis using \<open>e2>0\<close> by auto
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   721
    qed
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   722
    then obtain e2 where e2:"p\<in>pz \<longrightarrow> e2>0 \<and> cball p e2 \<subseteq> ball p e1 \<and> cont ff p e2"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   723
      by auto
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   724
    define e4 where "e4 \<equiv> if p\<in>pz then e2 else  e1"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   725
    have "e4>0" using e2 \<open>e1>0\<close> unfolding e4_def by auto
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   726
    moreover have "avoid p e4" using e2 \<open>e1>0\<close> e1_avoid unfolding e4_def avoid_def by auto
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   727
    moreover have "p\<in>pz \<longrightarrow> cont ff p e4"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   728
      by (auto simp add: e2 e4_def)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   729
    ultimately show ?thesis by auto
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   730
  qed
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   731
  then obtain get_e where get_e:"\<forall>p\<in>s. get_e p>0 \<and> avoid p (get_e p)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   732
      \<and> (p\<in>pz \<longrightarrow> cont ff p (get_e p))"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   733
    by metis
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   734
  define ci where "ci \<equiv> \<lambda>p. contour_integral (circlepath p (get_e p)) ff"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   735
  define w where "w \<equiv> \<lambda>p. winding_number g p"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   736
  have "contour_integral g ff = (\<Sum>p\<in>pz. w p * ci p)" unfolding ci_def w_def
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   737
  proof (rule Cauchy_theorem_singularities[OF \<open>open s\<close> \<open>connected s\<close> finite _ \<open>valid_path g\<close> loop
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   738
        path_img homo])
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   739
    have "open (s - pz)" using open_Diff[OF _ finite_imp_closed[OF finite]] \<open>open s\<close> by auto
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   740
    then show "ff holomorphic_on s - pz" unfolding ff_def using f_holo h_holo
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   741
      by (auto intro!: holomorphic_intros simp add:pz_def)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   742
  next
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   743
    show "\<forall>p\<in>s. 0 < get_e p \<and> (\<forall>w\<in>cball p (get_e p). w \<in> s \<and> (w \<noteq> p \<longrightarrow> w \<notin> pz))"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   744
      using get_e using avoid_def by blast
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   745
  qed
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   746
  also have "... = (\<Sum>p\<in>pz. c * w p * h p * zorder f p)"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   747
  proof (rule sum.cong[of pz pz,simplified])
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   748
    fix p assume "p \<in> pz"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   749
    show "w p * ci p = c * w p * h p * (zorder f p)"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   750
    proof (cases "p\<in>s")
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   751
      assume "p \<in> s"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   752
      have "ci p = c * h p * (zorder f p)" unfolding ci_def
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   753
        apply (rule contour_integral_unique)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   754
        using get_e \<open>p\<in>s\<close> \<open>p\<in>pz\<close> unfolding cont_def by (metis mult.assoc mult.commute)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   755
      thus ?thesis by auto
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   756
    next
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   757
      assume "p\<notin>s"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   758
      then have "w p=0" using homo unfolding w_def by auto
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   759
      then show ?thesis by auto
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   760
    qed
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   761
  qed
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   762
  also have "... = c*(\<Sum>p\<in>pz. w p * h p * zorder f p)"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   763
    unfolding sum_distrib_left by (simp add:algebra_simps)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   764
  finally have "contour_integral g ff = c * (\<Sum>p\<in>pz. w p * h p * of_int (zorder f p))" .
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   765
  then show ?thesis unfolding ff_def c_def w_def by simp
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   766
qed
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   767
73048
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   768
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   769
subsection \<open>Coefficient asymptotics for generating functions\<close>
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   770
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   771
text \<open>
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   772
  For a formal power series that has a meromorphic continuation on some disc in the
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   773
  context plane, we can use the Residue Theorem to extract precise asymptotic information
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   774
  from the residues at the poles. This can be used to derive the asymptotic behaviour
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   775
  of the coefficients (\<open>a\<^sub>n \<sim> \<dots>\<close>). If the function is meromorphic on the entire 
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   776
  complex plane, one can even derive a full asymptotic expansion.
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   777
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   778
  We will first show the relationship between the coefficients and the sum over the residues
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   779
  with an explicit remainder term (the contour integral along the circle used in the
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   780
  Residue theorem).
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   781
\<close>
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   782
theorem
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   783
  fixes f :: "complex \<Rightarrow> complex" and n :: nat and r :: real
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   784
  defines "g \<equiv> (\<lambda>w. f w / w ^ Suc n)" and "\<gamma> \<equiv> circlepath 0 r"
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   785
  assumes "open A" "connected A" "cball 0 r \<subseteq> A" "r > 0" 
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   786
  assumes "f holomorphic_on A - S" "S \<subseteq> ball 0 r" "finite S" "0 \<notin> S"
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   787
  shows   fps_coeff_conv_residues:
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   788
            "(deriv ^^ n) f 0 / fact n = 
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   789
                contour_integral \<gamma> g / (2 * pi * \<i>) - (\<Sum>z\<in>S. residue g z)" (is ?thesis1)
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   790
    and   fps_coeff_residues_bound:
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   791
            "(\<And>z. norm z = r \<Longrightarrow> z \<notin> k \<Longrightarrow> norm (f z) \<le> C) \<Longrightarrow> C \<ge> 0 \<Longrightarrow> finite k \<Longrightarrow>
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   792
               norm ((deriv ^^ n) f 0 / fact n + (\<Sum>z\<in>S. residue g z)) \<le> C / r ^ n"
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   793
proof -
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   794
  have holo: "g holomorphic_on A - insert 0 S"
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   795
    unfolding g_def using assms by (auto intro!: holomorphic_intros)
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   796
  have "contour_integral \<gamma> g = 2 * pi * \<i> * (\<Sum>z\<in>insert 0 S. winding_number \<gamma> z * residue g z)"
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   797
  proof (rule Residue_theorem)
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   798
    show "g holomorphic_on A - insert 0 S" by fact
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   799
    from assms show "\<forall>z. z \<notin> A \<longrightarrow> winding_number \<gamma> z = 0"
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   800
      unfolding \<gamma>_def by (intro allI impI winding_number_zero_outside[of _ "cball 0 r"]) auto
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   801
  qed (insert assms, auto simp: \<gamma>_def)
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   802
  also have "winding_number \<gamma> z = 1" if "z \<in> insert 0 S" for z
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   803
    unfolding \<gamma>_def using assms that by (intro winding_number_circlepath) auto
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   804
  hence "(\<Sum>z\<in>insert 0 S. winding_number \<gamma> z * residue g z) = (\<Sum>z\<in>insert 0 S. residue g z)"
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   805
    by (intro sum.cong) simp_all
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   806
  also have "\<dots> = residue g 0 + (\<Sum>z\<in>S. residue g z)" 
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   807
    using \<open>0 \<notin> S\<close> and \<open>finite S\<close> by (subst sum.insert) auto
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   808
  also from \<open>r > 0\<close> have "0 \<in> cball 0 r" by simp
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   809
  with assms have "0 \<in> A - S" by blast
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   810
  with assms have "residue g 0 = (deriv ^^ n) f 0 / fact n"
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   811
    unfolding g_def by (subst residue_holomorphic_over_power'[of "A - S"])
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   812
                       (auto simp: finite_imp_closed)
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   813
  finally show ?thesis1
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   814
    by (simp add: field_simps)
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   815
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   816
  assume C: "\<And>z. norm z = r \<Longrightarrow> z \<notin> k \<Longrightarrow> norm (f z) \<le> C" "C \<ge> 0" and k: "finite k"
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   817
  have "(deriv ^^ n) f 0 / fact n + (\<Sum>z\<in>S. residue g z) = contour_integral \<gamma> g / (2 * pi * \<i>)"
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   818
    using \<open>?thesis1\<close> by (simp add: algebra_simps)
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   819
  also have "norm \<dots> = norm (contour_integral \<gamma> g) / (2 * pi)"
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   820
    by (simp add: norm_divide norm_mult)
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   821
  also have "norm (contour_integral \<gamma> g) \<le> C / r ^ Suc n * (2 * pi * r)"
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   822
  proof (rule has_contour_integral_bound_circlepath_strong)
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   823
    from \<open>open A\<close> and \<open>finite S\<close> have "open (A - insert 0 S)"
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   824
      by (blast intro: finite_imp_closed)
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   825
    with assms show "(g has_contour_integral contour_integral \<gamma> g) (circlepath 0 r)"
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   826
      unfolding \<gamma>_def
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   827
      by (intro has_contour_integral_integral contour_integrable_holomorphic_simple [OF holo]) auto
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   828
  next
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   829
    fix z assume z: "norm (z - 0) = r" "z \<notin> k"
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   830
    hence "norm (g z) = norm (f z) / r ^ Suc n"
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   831
      by (simp add: norm_divide g_def norm_mult norm_power)
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   832
    also have "\<dots> \<le> C / r ^ Suc n" 
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   833
      using k and \<open>r > 0\<close> and z by (intro divide_right_mono C zero_le_power) auto
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   834
    finally show "norm (g z) \<le> C / r ^ Suc n" .
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   835
  qed (insert C(2) k \<open>r > 0\<close>, auto)
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   836
  also from \<open>r > 0\<close> have "C / r ^ Suc n * (2 * pi * r) / (2 * pi) = C / r ^ n"
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   837
    by simp
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   838
  finally show "norm ((deriv ^^ n) f 0 / fact n + (\<Sum>z\<in>S. residue g z)) \<le> \<dots>"
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   839
    by - (simp_all add: divide_right_mono)
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   840
qed
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   841
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   842
text \<open>
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   843
  Since the circle is fixed, we can get an upper bound on the values of the generating
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   844
  function on the circle and therefore show that the integral is $O(r^{-n})$.
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   845
\<close>
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   846
corollary fps_coeff_residues_bigo:
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   847
  fixes f :: "complex \<Rightarrow> complex" and r :: real
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   848
  assumes "open A" "connected A" "cball 0 r \<subseteq> A" "r > 0" 
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   849
  assumes "f holomorphic_on A - S" "S \<subseteq> ball 0 r" "finite S" "0 \<notin> S"
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   850
  assumes g: "eventually (\<lambda>n. g n = -(\<Sum>z\<in>S. residue (\<lambda>z. f z / z ^ Suc n) z)) sequentially"
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   851
                (is "eventually (\<lambda>n. _ = -?g' n) _")
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   852
  shows   "(\<lambda>n. (deriv ^^ n) f 0 / fact n - g n) \<in> O(\<lambda>n. 1 / r ^ n)" (is "(\<lambda>n. ?c n - _) \<in> O(_)")
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   853
proof -
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   854
  from assms have "compact (f ` sphere 0 r)"
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   855
    by (intro compact_continuous_image holomorphic_on_imp_continuous_on
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   856
              holomorphic_on_subset[OF \<open>f holomorphic_on A - S\<close>]) auto
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   857
  hence "bounded (f ` sphere 0 r)" by (rule compact_imp_bounded)
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   858
  then obtain C where C: "\<And>z. z \<in> sphere 0 r \<Longrightarrow> norm (f z) \<le> C"
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   859
    by (auto simp: bounded_iff sphere_def)
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   860
  have "0 \<le> norm (f (of_real r))" by simp
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   861
  also from C[of "of_real r"] and \<open>r > 0\<close> have "\<dots> \<le> C" by simp
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   862
  finally have C_nonneg: "C \<ge> 0" .
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   863
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   864
  have "(\<lambda>n. ?c n + ?g' n) \<in> O(\<lambda>n. of_real (1 / r ^ n))"
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   865
  proof (intro bigoI[of _ C] always_eventually allI )
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   866
    fix n :: nat
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   867
    from assms and C and C_nonneg have "norm (?c n + ?g' n) \<le> C / r ^ n"
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   868
      by (intro fps_coeff_residues_bound[where A = A and k = "{}"]) auto
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   869
    also have "\<dots> = C * norm (complex_of_real (1 / r ^ n))" 
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   870
      using \<open>r > 0\<close> by (simp add: norm_divide norm_power)
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   871
    finally show "norm (?c n + ?g' n) \<le> \<dots>" .
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   872
  qed
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   873
  also have "?this \<longleftrightarrow> (\<lambda>n. ?c n - g n) \<in> O(\<lambda>n. of_real (1 / r ^ n))"
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   874
    by (intro landau_o.big.in_cong eventually_mono[OF g]) simp_all
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   875
  finally show ?thesis .
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   876
qed
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   877
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   878
corollary fps_coeff_residues_bigo':
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   879
  fixes f :: "complex \<Rightarrow> complex" and r :: real
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   880
  assumes exp: "f has_fps_expansion F"
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   881
  assumes "open A" "connected A" "cball 0 r \<subseteq> A" "r > 0" 
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   882
  assumes "f holomorphic_on A - S" "S \<subseteq> ball 0 r" "finite S" "0 \<notin> S"
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   883
  assumes "eventually (\<lambda>n. g n = -(\<Sum>z\<in>S. residue (\<lambda>z. f z / z ^ Suc n) z)) sequentially"
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   884
             (is "eventually (\<lambda>n. _ = -?g' n) _")
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   885
  shows   "(\<lambda>n. fps_nth F n - g n) \<in> O(\<lambda>n. 1 / r ^ n)" (is "(\<lambda>n. ?c n - _) \<in> O(_)")
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   886
proof -
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   887
  have "fps_nth F = (\<lambda>n. (deriv ^^ n) f 0 / fact n)"
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   888
    using fps_nth_fps_expansion[OF exp] by (intro ext) simp_all
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   889
  with fps_coeff_residues_bigo[OF assms(2-)] show ?thesis by simp
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   890
qed
7ad9f197ca7e HOL-Complex_Analysis: coefficient asymptotics for meromorphic functions
Manuel Eberl <eberlm@in.tum.de>
parents: 71201
diff changeset
   891
71201
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   892
subsection \<open>Rouche's theorem \<close>
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   893
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   894
theorem Rouche_theorem:
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   895
  fixes f g::"complex \<Rightarrow> complex" and s:: "complex set"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   896
  defines "fg\<equiv>(\<lambda>p. f p + g p)"
77223
607e1e345e8f Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents: 73048
diff changeset
   897
  defines "zeros_fg\<equiv>{p\<in>s. fg p = 0}" and "zeros_f\<equiv>{p\<in>s. f p = 0}"
71201
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   898
  assumes
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   899
    "open s" and "connected s" and
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   900
    "finite zeros_fg" and
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   901
    "finite zeros_f" and
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   902
    f_holo:"f holomorphic_on s" and
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   903
    g_holo:"g holomorphic_on s" and
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   904
    "valid_path \<gamma>" and
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   905
    loop:"pathfinish \<gamma> = pathstart \<gamma>" and
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   906
    path_img:"path_image \<gamma> \<subseteq> s " and
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   907
    path_less:"\<forall>z\<in>path_image \<gamma>. cmod(f z) > cmod(g z)" and
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   908
    homo:"\<forall>z. (z \<notin> s) \<longrightarrow> winding_number \<gamma> z = 0"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   909
  shows "(\<Sum>p\<in>zeros_fg. winding_number \<gamma> p * zorder fg p)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   910
          = (\<Sum>p\<in>zeros_f. winding_number \<gamma> p * zorder f p)"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   911
proof -
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   912
  have path_fg:"path_image \<gamma> \<subseteq> s - zeros_fg"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   913
  proof -
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   914
    have False when "z\<in>path_image \<gamma>" and "f z + g z=0" for z
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   915
    proof -
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   916
      have "cmod (f z) > cmod (g z)" using \<open>z\<in>path_image \<gamma>\<close> path_less by auto
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   917
      moreover have "f z = - g z"  using \<open>f z + g z =0\<close> by (simp add: eq_neg_iff_add_eq_0)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   918
      then have "cmod (f z) = cmod (g z)" by auto
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   919
      ultimately show False by auto
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   920
    qed
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   921
    then show ?thesis unfolding zeros_fg_def fg_def using path_img by auto
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   922
  qed
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   923
  have path_f:"path_image \<gamma> \<subseteq> s - zeros_f"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   924
  proof -
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   925
    have False when "z\<in>path_image \<gamma>" and "f z =0" for z
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   926
    proof -
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   927
      have "cmod (g z) < cmod (f z) " using \<open>z\<in>path_image \<gamma>\<close> path_less by auto
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   928
      then have "cmod (g z) < 0" using \<open>f z=0\<close> by auto
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   929
      then show False by auto
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   930
    qed
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   931
    then show ?thesis unfolding zeros_f_def using path_img by auto
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   932
  qed
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   933
  define w where "w \<equiv> \<lambda>p. winding_number \<gamma> p"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   934
  define c where "c \<equiv> 2 * complex_of_real pi * \<i>"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   935
  define h where "h \<equiv> \<lambda>p. g p / f p + 1"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   936
  obtain spikes
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   937
    where "finite spikes" and spikes: "\<forall>x\<in>{0..1} - spikes. \<gamma> differentiable at x"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   938
    using \<open>valid_path \<gamma>\<close>
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   939
    by (auto simp: valid_path_def piecewise_C1_differentiable_on_def C1_differentiable_on_eq)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   940
  have h_contour:"((\<lambda>x. deriv h x / h x) has_contour_integral 0) \<gamma>"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   941
  proof -
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   942
    have outside_img:"0 \<in> outside (path_image (h o \<gamma>))"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   943
    proof -
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   944
      have "h p \<in> ball 1 1" when "p\<in>path_image \<gamma>" for p
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   945
      proof -
77223
607e1e345e8f Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents: 73048
diff changeset
   946
        have "cmod (g p/f p) <1"
607e1e345e8f Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents: 73048
diff changeset
   947
          by (smt (verit) divide_less_eq_1_pos norm_divide norm_ge_zero path_less that)
71201
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   948
        then show ?thesis unfolding h_def by (auto simp add:dist_complex_def)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   949
      qed
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   950
      then have "path_image (h o \<gamma>) \<subseteq> ball 1 1"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   951
        by (simp add: image_subset_iff path_image_compose)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   952
      moreover have " (0::complex) \<notin> ball 1 1" by (simp add: dist_norm)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   953
      ultimately show "?thesis"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   954
        using  convex_in_outside[of "ball 1 1" 0] outside_mono by blast
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   955
    qed
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   956
    have valid_h:"valid_path (h \<circ> \<gamma>)"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   957
    proof (rule valid_path_compose_holomorphic[OF \<open>valid_path \<gamma>\<close> _ _ path_f])
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   958
      show "h holomorphic_on s - zeros_f"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   959
        unfolding h_def using f_holo g_holo
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   960
        by (auto intro!: holomorphic_intros simp add:zeros_f_def)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   961
    next
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   962
      show "open (s - zeros_f)" using \<open>finite zeros_f\<close> \<open>open s\<close> finite_imp_closed
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   963
        by auto
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   964
    qed
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   965
    have "((\<lambda>z. 1/z) has_contour_integral 0) (h \<circ> \<gamma>)"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   966
    proof -
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   967
      have "0 \<notin> path_image (h \<circ> \<gamma>)" using outside_img by (simp add: outside_def)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   968
      then have "((\<lambda>z. 1/z) has_contour_integral c * winding_number (h \<circ> \<gamma>) 0) (h \<circ> \<gamma>)"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   969
        using has_contour_integral_winding_number[of "h o \<gamma>" 0,simplified] valid_h
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   970
        unfolding c_def by auto
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   971
      moreover have "winding_number (h o \<gamma>) 0 = 0"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   972
      proof -
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   973
        have "0 \<in> outside (path_image (h \<circ> \<gamma>))" using outside_img .
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   974
        moreover have "path (h o \<gamma>)"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   975
          using valid_h  by (simp add: valid_path_imp_path)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   976
        moreover have "pathfinish (h o \<gamma>) = pathstart (h o \<gamma>)"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   977
          by (simp add: loop pathfinish_compose pathstart_compose)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   978
        ultimately show ?thesis using winding_number_zero_in_outside by auto
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   979
      qed
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   980
      ultimately show ?thesis by auto
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   981
    qed
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   982
    moreover have "vector_derivative (h \<circ> \<gamma>) (at x) = vector_derivative \<gamma> (at x) * deriv h (\<gamma> x)"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   983
      when "x\<in>{0..1} - spikes" for x
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   984
    proof (rule vector_derivative_chain_at_general)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   985
      show "\<gamma> differentiable at x" using that \<open>valid_path \<gamma>\<close> spikes by auto
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   986
    next
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   987
      define der where "der \<equiv> \<lambda>p. (deriv g p * f p - g p * deriv f p)/(f p * f p)"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   988
      define t where "t \<equiv> \<gamma> x"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   989
      have "f t\<noteq>0" unfolding zeros_f_def t_def
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   990
        by (metis DiffD1 image_eqI norm_not_less_zero norm_zero path_defs(4) path_less that)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   991
      moreover have "t\<in>s"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   992
        using contra_subsetD path_image_def path_fg t_def that by fastforce
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   993
      ultimately have "(h has_field_derivative der t) (at t)"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   994
        unfolding h_def der_def using g_holo f_holo \<open>open s\<close>
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   995
        by (auto intro!: holomorphic_derivI derivative_eq_intros)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   996
      then show "h field_differentiable at (\<gamma> x)"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   997
        unfolding t_def field_differentiable_def by blast
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   998
    qed
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
   999
    then have " ((/) 1 has_contour_integral 0) (h \<circ> \<gamma>)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
  1000
                  = ((\<lambda>x. deriv h x / h x) has_contour_integral 0) \<gamma>"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
  1001
      unfolding has_contour_integral
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
  1002
      apply (intro has_integral_spike_eq[OF negligible_finite, OF \<open>finite spikes\<close>])
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
  1003
      by auto
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
  1004
    ultimately show ?thesis by auto
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
  1005
  qed
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
  1006
  then have "contour_integral \<gamma> (\<lambda>x. deriv h x / h x) = 0"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
  1007
    using  contour_integral_unique by simp
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
  1008
  moreover have "contour_integral \<gamma> (\<lambda>x. deriv fg x / fg x) = contour_integral \<gamma> (\<lambda>x. deriv f x / f x)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
  1009
      + contour_integral \<gamma> (\<lambda>p. deriv h p / h p)"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
  1010
  proof -
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
  1011
    have "(\<lambda>p. deriv f p / f p) contour_integrable_on \<gamma>"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
  1012
    proof (rule contour_integrable_holomorphic_simple[OF _ _ \<open>valid_path \<gamma>\<close> path_f])
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
  1013
      show "open (s - zeros_f)" using finite_imp_closed[OF \<open>finite zeros_f\<close>] \<open>open s\<close>
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
  1014
        by auto
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
  1015
      then show "(\<lambda>p. deriv f p / f p) holomorphic_on s - zeros_f"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
  1016
        using f_holo
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
  1017
        by (auto intro!: holomorphic_intros simp add:zeros_f_def)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
  1018
    qed
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
  1019
    moreover have "(\<lambda>p. deriv h p / h p) contour_integrable_on \<gamma>"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
  1020
      using h_contour
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
  1021
      by (simp add: has_contour_integral_integrable)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
  1022
    ultimately have "contour_integral \<gamma> (\<lambda>x. deriv f x / f x + deriv h x / h x) =
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
  1023
                        contour_integral \<gamma> (\<lambda>p. deriv f p / f p) + contour_integral \<gamma> (\<lambda>p. deriv h p / h p)"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
  1024
      using contour_integral_add[of "(\<lambda>p. deriv f p / f p)" \<gamma> "(\<lambda>p. deriv h p / h p)" ]
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
  1025
      by auto
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
  1026
    moreover have "deriv fg p / fg p =  deriv f p / f p + deriv h p / h p"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
  1027
                      when "p\<in> path_image \<gamma>" for p
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
  1028
    proof -
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
  1029
      have "fg p\<noteq>0" and "f p\<noteq>0" using path_f path_fg that unfolding zeros_f_def zeros_fg_def
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
  1030
        by auto
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
  1031
      have "h p\<noteq>0"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
  1032
      proof (rule ccontr)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
  1033
        assume "\<not> h p \<noteq> 0"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
  1034
        then have "g p / f p= -1" unfolding h_def by (simp add: add_eq_0_iff2)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
  1035
        then have "cmod (g p/f p) = 1" by auto
77223
607e1e345e8f Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents: 73048
diff changeset
  1036
        moreover have "cmod (g p/f p) <1"
607e1e345e8f Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents: 73048
diff changeset
  1037
          by (simp add: \<open>f p \<noteq> 0\<close> norm_divide path_less that)
71201
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
  1038
        ultimately show False by auto
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
  1039
      qed
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
  1040
      have der_fg:"deriv fg p =  deriv f p + deriv g p" unfolding fg_def
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
  1041
        using f_holo g_holo holomorphic_on_imp_differentiable_at[OF _  \<open>open s\<close>] path_img that
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
  1042
        by auto
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
  1043
      have der_h:"deriv h p = (deriv g p * f p - g p * deriv f p)/(f p * f p)"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
  1044
      proof -
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
  1045
        define der where "der \<equiv> \<lambda>p. (deriv g p * f p - g p * deriv f p)/(f p * f p)"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
  1046
        have "p\<in>s" using path_img that by auto
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
  1047
        then have "(h has_field_derivative der p) (at p)"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
  1048
          unfolding h_def der_def using g_holo f_holo \<open>open s\<close> \<open>f p\<noteq>0\<close>
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
  1049
          by (auto intro!: derivative_eq_intros holomorphic_derivI)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
  1050
        then show ?thesis unfolding der_def using DERIV_imp_deriv by auto
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
  1051
      qed
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
  1052
      show ?thesis
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
  1053
        apply (simp only:der_fg der_h)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
  1054
        apply (auto simp add:field_simps \<open>h p\<noteq>0\<close> \<open>f p\<noteq>0\<close> \<open>fg p\<noteq>0\<close>)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
  1055
        by (auto simp add:field_simps h_def \<open>f p\<noteq>0\<close> fg_def)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
  1056
    qed
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
  1057
    then have "contour_integral \<gamma> (\<lambda>p. deriv fg p / fg p)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
  1058
                  = contour_integral \<gamma> (\<lambda>p. deriv f p / f p + deriv h p / h p)"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
  1059
      by (elim contour_integral_eq)
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
  1060
    ultimately show ?thesis by auto
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
  1061
  qed
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
  1062
  moreover have "contour_integral \<gamma> (\<lambda>x. deriv fg x / fg x) = c * (\<Sum>p\<in>zeros_fg. w p * zorder fg p)"
77223
607e1e345e8f Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents: 73048
diff changeset
  1063
  proof -
607e1e345e8f Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents: 73048
diff changeset
  1064
    have "fg holomorphic_on s" unfolding fg_def using f_holo g_holo holomorphic_on_add by auto
607e1e345e8f Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents: 73048
diff changeset
  1065
    moreover
607e1e345e8f Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents: 73048
diff changeset
  1066
    have "path_image \<gamma> \<subseteq> s - {p\<in>s. fg p = 0}" using path_fg unfolding zeros_fg_def .
607e1e345e8f Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents: 73048
diff changeset
  1067
    moreover
607e1e345e8f Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents: 73048
diff changeset
  1068
    have " finite {p\<in>s. fg p = 0}" using \<open>finite zeros_fg\<close> unfolding zeros_fg_def .
607e1e345e8f Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents: 73048
diff changeset
  1069
    ultimately show ?thesis
607e1e345e8f Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents: 73048
diff changeset
  1070
      unfolding c_def zeros_fg_def w_def
607e1e345e8f Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents: 73048
diff changeset
  1071
      using argument_principle[OF \<open>open s\<close> \<open>connected s\<close> _ _ \<open>valid_path \<gamma>\<close> loop _ homo, of _ "{}" "\<lambda>_. 1"]
607e1e345e8f Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents: 73048
diff changeset
  1072
      by simp
71201
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
  1073
  qed
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
  1074
  moreover have "contour_integral \<gamma> (\<lambda>x. deriv f x / f x) = c * (\<Sum>p\<in>zeros_f. w p * zorder f p)"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
  1075
    unfolding c_def zeros_f_def w_def
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
  1076
  proof (rule argument_principle[OF \<open>open s\<close> \<open>connected s\<close> _ _ \<open>valid_path \<gamma>\<close> loop _ homo
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
  1077
        , of _ "{}" "\<lambda>_. 1",simplified])
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
  1078
    show "f holomorphic_on s" using f_holo g_holo holomorphic_on_add by auto
77223
607e1e345e8f Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents: 73048
diff changeset
  1079
    show "path_image \<gamma> \<subseteq> s - {p\<in>s. f p = 0}" using path_f unfolding zeros_f_def .
607e1e345e8f Lots of new material chiefly about complex analysis
paulson <lp15@cam.ac.uk>
parents: 73048
diff changeset
  1080
    show " finite {p\<in>s. f p = 0}" using \<open>finite zeros_f\<close> unfolding zeros_f_def .
71201
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
  1081
  qed
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
  1082
  ultimately have " c* (\<Sum>p\<in>zeros_fg. w p * (zorder fg p)) = c* (\<Sum>p\<in>zeros_f. w p * (zorder f p))"
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
  1083
    by auto
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
  1084
  then show ?thesis unfolding c_def using w_def by auto
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
  1085
qed
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
  1086
6617fb368a06 Reorganised HOL-Complex_Analysis
Manuel Eberl <eberlm@in.tum.de>
parents:
diff changeset
  1087
end