src/HOL/Multivariate_Analysis/Cauchy_Integral_Thm.thy
author paulson <lp15@cam.ac.uk>
Mon Nov 23 16:57:54 2015 +0000 (2015-11-23)
changeset 61738 c4f6031f1310
parent 61711 21d7910d6816
child 61762 d50b993b4fb9
permissions -rw-r--r--
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
lp15@60809
     1
section \<open>Complex path integrals and Cauchy's integral theorem\<close>
lp15@60809
     2
lp15@61711
     3
text\<open>By John Harrison et al.  Ported from HOL Light by L C Paulson (2015)\<close>
lp15@61711
     4
lp15@60809
     5
theory Cauchy_Integral_Thm
lp15@61738
     6
imports Complex_Transcendental Weierstrass Ordered_Euclidean_Space
lp15@60809
     7
begin
lp15@60809
     8
lp15@61190
     9
subsection \<open>Piecewise differentiable functions\<close>
lp15@60809
    10
lp15@60809
    11
definition piecewise_differentiable_on
lp15@60809
    12
           (infixr "piecewise'_differentiable'_on" 50)
lp15@60809
    13
  where "f piecewise_differentiable_on i  \<equiv>
lp15@60809
    14
           continuous_on i f \<and>
lp15@61190
    15
           (\<exists>s. finite s \<and> (\<forall>x \<in> i - s. f differentiable (at x within i)))"
lp15@60809
    16
lp15@60809
    17
lemma piecewise_differentiable_on_imp_continuous_on:
lp15@60809
    18
    "f piecewise_differentiable_on s \<Longrightarrow> continuous_on s f"
lp15@60809
    19
by (simp add: piecewise_differentiable_on_def)
lp15@60809
    20
lp15@60809
    21
lemma piecewise_differentiable_on_subset:
lp15@60809
    22
    "f piecewise_differentiable_on s \<Longrightarrow> t \<le> s \<Longrightarrow> f piecewise_differentiable_on t"
lp15@60809
    23
  using continuous_on_subset
lp15@61190
    24
  unfolding piecewise_differentiable_on_def
lp15@61190
    25
  apply safe
lp15@61190
    26
  apply (blast intro: elim: continuous_on_subset)
lp15@61190
    27
  by (meson Diff_iff differentiable_within_subset subsetCE)
lp15@60809
    28
lp15@60809
    29
lemma differentiable_on_imp_piecewise_differentiable:
lp15@60809
    30
  fixes a:: "'a::{linorder_topology,real_normed_vector}"
lp15@60809
    31
  shows "f differentiable_on {a..b} \<Longrightarrow> f piecewise_differentiable_on {a..b}"
lp15@60809
    32
  apply (simp add: piecewise_differentiable_on_def differentiable_imp_continuous_on)
lp15@61190
    33
  apply (rule_tac x="{a,b}" in exI, simp add: differentiable_on_def)
lp15@61190
    34
  done
lp15@60809
    35
lp15@60809
    36
lemma differentiable_imp_piecewise_differentiable:
lp15@61190
    37
    "(\<And>x. x \<in> s \<Longrightarrow> f differentiable (at x within s))
lp15@60809
    38
         \<Longrightarrow> f piecewise_differentiable_on s"
lp15@61190
    39
by (auto simp: piecewise_differentiable_on_def differentiable_imp_continuous_on differentiable_on_def
lp15@61190
    40
         intro: differentiable_within_subset)
lp15@60809
    41
paulson@61204
    42
lemma piecewise_differentiable_const [iff]: "(\<lambda>x. z) piecewise_differentiable_on s"
paulson@61204
    43
  by (simp add: differentiable_imp_piecewise_differentiable)
paulson@61204
    44
lp15@60809
    45
lemma piecewise_differentiable_compose:
lp15@60809
    46
    "\<lbrakk>f piecewise_differentiable_on s; g piecewise_differentiable_on (f ` s);
lp15@60809
    47
      \<And>x. finite (s \<inter> f-`{x})\<rbrakk>
lp15@60809
    48
      \<Longrightarrow> (g o f) piecewise_differentiable_on s"
lp15@60809
    49
  apply (simp add: piecewise_differentiable_on_def, safe)
lp15@60809
    50
  apply (blast intro: continuous_on_compose2)
lp15@60809
    51
  apply (rename_tac A B)
lp15@60809
    52
  apply (rule_tac x="A \<union> (\<Union>x\<in>B. s \<inter> f-`{x})" in exI)
lp15@61190
    53
  apply (blast intro: differentiable_chain_within)
lp15@61190
    54
  done
lp15@60809
    55
lp15@60809
    56
lemma piecewise_differentiable_affine:
lp15@60809
    57
  fixes m::real
lp15@60809
    58
  assumes "f piecewise_differentiable_on ((\<lambda>x. m *\<^sub>R x + c) ` s)"
lp15@60809
    59
  shows "(f o (\<lambda>x. m *\<^sub>R x + c)) piecewise_differentiable_on s"
lp15@60809
    60
proof (cases "m = 0")
lp15@60809
    61
  case True
lp15@60809
    62
  then show ?thesis
lp15@60809
    63
    unfolding o_def
lp15@60809
    64
    by (force intro: differentiable_imp_piecewise_differentiable differentiable_const)
lp15@60809
    65
next
lp15@60809
    66
  case False
lp15@60809
    67
  show ?thesis
lp15@60809
    68
    apply (rule piecewise_differentiable_compose [OF differentiable_imp_piecewise_differentiable])
lp15@60809
    69
    apply (rule assms derivative_intros | simp add: False vimage_def real_vector_affinity_eq)+
lp15@60809
    70
    done
lp15@60809
    71
qed
lp15@60809
    72
lp15@60809
    73
lemma piecewise_differentiable_cases:
lp15@60809
    74
  fixes c::real
lp15@60809
    75
  assumes "f piecewise_differentiable_on {a..c}"
lp15@60809
    76
          "g piecewise_differentiable_on {c..b}"
lp15@60809
    77
           "a \<le> c" "c \<le> b" "f c = g c"
lp15@60809
    78
  shows "(\<lambda>x. if x \<le> c then f x else g x) piecewise_differentiable_on {a..b}"
lp15@60809
    79
proof -
lp15@60809
    80
  obtain s t where st: "finite s" "finite t"
lp15@61190
    81
                       "\<forall>x\<in>{a..c} - s. f differentiable at x within {a..c}"
lp15@61190
    82
                       "\<forall>x\<in>{c..b} - t. g differentiable at x within {c..b}"
lp15@60809
    83
    using assms
lp15@60809
    84
    by (auto simp: piecewise_differentiable_on_def)
lp15@61190
    85
  have finabc: "finite ({a,b,c} \<union> (s \<union> t))"
wenzelm@61222
    86
    by (metis \<open>finite s\<close> \<open>finite t\<close> finite_Un finite_insert finite.emptyI)
lp15@60809
    87
  have "continuous_on {a..c} f" "continuous_on {c..b} g"
lp15@60809
    88
    using assms piecewise_differentiable_on_def by auto
lp15@60809
    89
  then have "continuous_on {a..b} (\<lambda>x. if x \<le> c then f x else g x)"
lp15@60809
    90
    using continuous_on_cases [OF closed_real_atLeastAtMost [of a c],
lp15@60809
    91
                               OF closed_real_atLeastAtMost [of c b],
lp15@60809
    92
                               of f g "\<lambda>x. x\<le>c"]  assms
lp15@60809
    93
    by (force simp: ivl_disj_un_two_touch)
lp15@60809
    94
  moreover
lp15@60809
    95
  { fix x
lp15@61190
    96
    assume x: "x \<in> {a..b} - ({a,b,c} \<union> (s \<union> t))"
lp15@61190
    97
    have "(\<lambda>x. if x \<le> c then f x else g x) differentiable at x within {a..b}" (is "?diff_fg")
lp15@60809
    98
    proof (cases x c rule: le_cases)
lp15@60809
    99
      case le show ?diff_fg
lp15@61190
   100
        apply (rule differentiable_transform_within [where d = "dist x c" and f = f])
lp15@61190
   101
        using x le st
lp15@61190
   102
        apply (simp_all add: dist_real_def dist_nz [symmetric])
lp15@61190
   103
        apply (rule differentiable_at_withinI)
lp15@61190
   104
        apply (rule differentiable_within_open [where s = "{a<..<c} - s", THEN iffD1], simp_all)
lp15@61190
   105
        apply (blast intro: open_greaterThanLessThan finite_imp_closed)
lp15@61190
   106
        apply (force elim!: differentiable_subset)
lp15@60809
   107
        done
lp15@60809
   108
    next
lp15@60809
   109
      case ge show ?diff_fg
lp15@61190
   110
        apply (rule differentiable_transform_within [where d = "dist x c" and f = g])
lp15@61190
   111
        using x ge st
lp15@61190
   112
        apply (simp_all add: dist_real_def dist_nz [symmetric])
lp15@61190
   113
        apply (rule differentiable_at_withinI)
lp15@61190
   114
        apply (rule differentiable_within_open [where s = "{c<..<b} - t", THEN iffD1], simp_all)
lp15@61190
   115
        apply (blast intro: open_greaterThanLessThan finite_imp_closed)
lp15@61190
   116
        apply (force elim!: differentiable_subset)
lp15@60809
   117
        done
lp15@60809
   118
    qed
lp15@60809
   119
  }
lp15@61190
   120
  then have "\<exists>s. finite s \<and>
lp15@61190
   121
                 (\<forall>x\<in>{a..b} - s. (\<lambda>x. if x \<le> c then f x else g x) differentiable at x within {a..b})"
lp15@61190
   122
    by (meson finabc)
lp15@60809
   123
  ultimately show ?thesis
lp15@60809
   124
    by (simp add: piecewise_differentiable_on_def)
lp15@60809
   125
qed
lp15@60809
   126
lp15@60809
   127
lemma piecewise_differentiable_neg:
lp15@60809
   128
    "f piecewise_differentiable_on s \<Longrightarrow> (\<lambda>x. -(f x)) piecewise_differentiable_on s"
lp15@60809
   129
  by (auto simp: piecewise_differentiable_on_def continuous_on_minus)
lp15@60809
   130
lp15@60809
   131
lemma piecewise_differentiable_add:
lp15@60809
   132
  assumes "f piecewise_differentiable_on i"
lp15@60809
   133
          "g piecewise_differentiable_on i"
lp15@60809
   134
    shows "(\<lambda>x. f x + g x) piecewise_differentiable_on i"
lp15@60809
   135
proof -
lp15@60809
   136
  obtain s t where st: "finite s" "finite t"
lp15@61190
   137
                       "\<forall>x\<in>i - s. f differentiable at x within i"
lp15@61190
   138
                       "\<forall>x\<in>i - t. g differentiable at x within i"
lp15@60809
   139
    using assms by (auto simp: piecewise_differentiable_on_def)
lp15@61190
   140
  then have "finite (s \<union> t) \<and> (\<forall>x\<in>i - (s \<union> t). (\<lambda>x. f x + g x) differentiable at x within i)"
lp15@60809
   141
    by auto
lp15@60809
   142
  moreover have "continuous_on i f" "continuous_on i g"
lp15@60809
   143
    using assms piecewise_differentiable_on_def by auto
lp15@60809
   144
  ultimately show ?thesis
lp15@60809
   145
    by (auto simp: piecewise_differentiable_on_def continuous_on_add)
lp15@60809
   146
qed
lp15@60809
   147
lp15@60809
   148
lemma piecewise_differentiable_diff:
lp15@60809
   149
    "\<lbrakk>f piecewise_differentiable_on s;  g piecewise_differentiable_on s\<rbrakk>
lp15@60809
   150
     \<Longrightarrow> (\<lambda>x. f x - g x) piecewise_differentiable_on s"
lp15@60809
   151
  unfolding diff_conv_add_uminus
lp15@60809
   152
  by (metis piecewise_differentiable_add piecewise_differentiable_neg)
lp15@60809
   153
lp15@61190
   154
lemma continuous_on_joinpaths_D1:
lp15@61190
   155
    "continuous_on {0..1} (g1 +++ g2) \<Longrightarrow> continuous_on {0..1} g1"
lp15@61190
   156
  apply (rule continuous_on_eq [of _ "(g1 +++ g2) o (op*(inverse 2))"])
lp15@61190
   157
  apply (rule continuous_intros | simp)+
lp15@61190
   158
  apply (auto elim!: continuous_on_subset simp: joinpaths_def)
lp15@61190
   159
  done
lp15@61190
   160
lp15@61190
   161
lemma continuous_on_joinpaths_D2:
lp15@61190
   162
    "\<lbrakk>continuous_on {0..1} (g1 +++ g2); pathfinish g1 = pathstart g2\<rbrakk> \<Longrightarrow> continuous_on {0..1} g2"
lp15@61190
   163
  apply (rule continuous_on_eq [of _ "(g1 +++ g2) o (\<lambda>x. inverse 2*x + 1/2)"])
lp15@61190
   164
  apply (rule continuous_intros | simp)+
lp15@61190
   165
  apply (auto elim!: continuous_on_subset simp add: joinpaths_def pathfinish_def pathstart_def Ball_def)
lp15@61190
   166
  done
lp15@61190
   167
lp15@61190
   168
lemma piecewise_differentiable_D1:
lp15@61190
   169
    "(g1 +++ g2) piecewise_differentiable_on {0..1} \<Longrightarrow> g1 piecewise_differentiable_on {0..1}"
lp15@61190
   170
  apply (clarsimp simp add: piecewise_differentiable_on_def dest!: continuous_on_joinpaths_D1)
lp15@61190
   171
  apply (rule_tac x="insert 1 ((op*2)`s)" in exI)
lp15@61190
   172
  apply simp
lp15@61190
   173
  apply (intro ballI)
lp15@61190
   174
  apply (rule_tac d="dist (x/2) (1/2)" and f = "(g1 +++ g2) o (op*(inverse 2))"
lp15@61190
   175
       in differentiable_transform_within)
lp15@61190
   176
  apply (auto simp: dist_real_def joinpaths_def)
lp15@61190
   177
  apply (rule differentiable_chain_within derivative_intros | simp)+
lp15@61190
   178
  apply (rule differentiable_subset)
lp15@61190
   179
  apply (force simp:)+
lp15@61190
   180
  done
lp15@61190
   181
lp15@61190
   182
lemma piecewise_differentiable_D2:
lp15@61190
   183
    "\<lbrakk>(g1 +++ g2) piecewise_differentiable_on {0..1}; pathfinish g1 = pathstart g2\<rbrakk>
lp15@61190
   184
    \<Longrightarrow> g2 piecewise_differentiable_on {0..1}"
lp15@61190
   185
  apply (clarsimp simp add: piecewise_differentiable_on_def dest!: continuous_on_joinpaths_D2)
lp15@61190
   186
  apply (rule_tac x="insert 0 ((\<lambda>x. 2*x-1)`s)" in exI)
lp15@61190
   187
  apply simp
lp15@61190
   188
  apply (intro ballI)
lp15@61190
   189
  apply (rule_tac d="dist ((x+1)/2) (1/2)" and f = "(g1 +++ g2) o (\<lambda>x. (x+1)/2)"
lp15@61190
   190
          in differentiable_transform_within)
lp15@61190
   191
  apply (auto simp: dist_real_def joinpaths_def abs_if field_simps split: split_if_asm)
lp15@61190
   192
  apply (rule differentiable_chain_within derivative_intros | simp)+
lp15@61190
   193
  apply (rule differentiable_subset)
lp15@61190
   194
  apply (force simp: divide_simps)+
lp15@61190
   195
  done
lp15@61190
   196
lp15@61190
   197
lp15@61190
   198
subsubsection\<open>The concept of continuously differentiable\<close>
lp15@61190
   199
lp15@61190
   200
definition C1_differentiable_on :: "(real \<Rightarrow> 'a::real_normed_vector) \<Rightarrow> real set \<Rightarrow> bool"
lp15@61190
   201
           (infix "C1'_differentiable'_on" 50)
lp15@61190
   202
  where
lp15@61190
   203
  "f C1_differentiable_on s \<longleftrightarrow>
lp15@61190
   204
   (\<exists>D. (\<forall>x \<in> s. (f has_vector_derivative (D x)) (at x)) \<and> continuous_on s D)"
lp15@61190
   205
lp15@61190
   206
lemma C1_differentiable_on_eq:
lp15@61190
   207
    "f C1_differentiable_on s \<longleftrightarrow>
lp15@61190
   208
     (\<forall>x \<in> s. f differentiable at x) \<and> continuous_on s (\<lambda>x. vector_derivative f (at x))"
lp15@61190
   209
  unfolding C1_differentiable_on_def
lp15@61190
   210
  apply safe
lp15@61190
   211
  using differentiable_def has_vector_derivative_def apply blast
lp15@61190
   212
  apply (erule continuous_on_eq)
lp15@61190
   213
  using vector_derivative_at apply fastforce
lp15@61190
   214
  using vector_derivative_works apply fastforce
lp15@61190
   215
  done
lp15@61190
   216
lp15@61190
   217
lemma C1_differentiable_on_subset:
lp15@61190
   218
  "f C1_differentiable_on t \<Longrightarrow> s \<subseteq> t \<Longrightarrow> f C1_differentiable_on s"
lp15@61190
   219
  unfolding C1_differentiable_on_def  continuous_on_eq_continuous_within
lp15@61190
   220
  by (blast intro:  continuous_within_subset)
lp15@61190
   221
lp15@61190
   222
lemma C1_differentiable_compose:
lp15@61190
   223
    "\<lbrakk>f C1_differentiable_on s; g C1_differentiable_on (f ` s);
lp15@61190
   224
      \<And>x. finite (s \<inter> f-`{x})\<rbrakk>
lp15@61190
   225
      \<Longrightarrow> (g o f) C1_differentiable_on s"
lp15@61190
   226
  apply (simp add: C1_differentiable_on_eq, safe)
lp15@61190
   227
   using differentiable_chain_at apply blast
lp15@61190
   228
  apply (rule continuous_on_eq [of _ "\<lambda>x. vector_derivative f (at x) *\<^sub>R vector_derivative g (at (f x))"])
lp15@61190
   229
   apply (rule Limits.continuous_on_scaleR, assumption)
lp15@61190
   230
   apply (metis (mono_tags, lifting) continuous_on_eq continuous_at_imp_continuous_on continuous_on_compose differentiable_imp_continuous_within o_def)
lp15@61190
   231
  by (simp add: vector_derivative_chain_at)
lp15@61190
   232
lp15@61190
   233
lemma C1_diff_imp_diff: "f C1_differentiable_on s \<Longrightarrow> f differentiable_on s"
lp15@61190
   234
  by (simp add: C1_differentiable_on_eq differentiable_at_imp_differentiable_on)
lp15@61190
   235
lp15@61190
   236
lemma C1_differentiable_on_ident [simp, derivative_intros]: "(\<lambda>x. x) C1_differentiable_on s"
lp15@61190
   237
  by (auto simp: C1_differentiable_on_eq continuous_on_const)
lp15@61190
   238
lp15@61190
   239
lemma C1_differentiable_on_const [simp, derivative_intros]: "(\<lambda>z. a) C1_differentiable_on s"
lp15@61190
   240
  by (auto simp: C1_differentiable_on_eq continuous_on_const)
lp15@61190
   241
lp15@61190
   242
lemma C1_differentiable_on_add [simp, derivative_intros]:
lp15@61190
   243
  "f C1_differentiable_on s \<Longrightarrow> g C1_differentiable_on s \<Longrightarrow> (\<lambda>x. f x + g x) C1_differentiable_on s"
lp15@61190
   244
  unfolding C1_differentiable_on_eq  by (auto intro: continuous_intros)
lp15@61190
   245
lp15@61190
   246
lemma C1_differentiable_on_minus [simp, derivative_intros]:
lp15@61190
   247
  "f C1_differentiable_on s \<Longrightarrow> (\<lambda>x. - f x) C1_differentiable_on s"
lp15@61190
   248
  unfolding C1_differentiable_on_eq  by (auto intro: continuous_intros)
lp15@61190
   249
lp15@61190
   250
lemma C1_differentiable_on_diff [simp, derivative_intros]:
lp15@61190
   251
  "f C1_differentiable_on s \<Longrightarrow> g C1_differentiable_on s \<Longrightarrow> (\<lambda>x. f x - g x) C1_differentiable_on s"
lp15@61190
   252
  unfolding C1_differentiable_on_eq  by (auto intro: continuous_intros)
lp15@61190
   253
lp15@61190
   254
lemma C1_differentiable_on_mult [simp, derivative_intros]:
lp15@61190
   255
  fixes f g :: "real \<Rightarrow> 'a :: real_normed_algebra"
lp15@61190
   256
  shows "f C1_differentiable_on s \<Longrightarrow> g C1_differentiable_on s \<Longrightarrow> (\<lambda>x. f x * g x) C1_differentiable_on s"
lp15@61190
   257
  unfolding C1_differentiable_on_eq
lp15@61190
   258
  by (auto simp: continuous_on_add continuous_on_mult continuous_at_imp_continuous_on differentiable_imp_continuous_within)
lp15@61190
   259
lp15@61190
   260
lemma C1_differentiable_on_scaleR [simp, derivative_intros]:
lp15@61190
   261
  "f C1_differentiable_on s \<Longrightarrow> g C1_differentiable_on s \<Longrightarrow> (\<lambda>x. f x *\<^sub>R g x) C1_differentiable_on s"
lp15@61190
   262
  unfolding C1_differentiable_on_eq
lp15@61190
   263
  by (rule continuous_intros | simp add: continuous_at_imp_continuous_on differentiable_imp_continuous_within)+
lp15@61190
   264
lp15@61190
   265
lp15@61190
   266
definition piecewise_C1_differentiable_on
lp15@61190
   267
           (infixr "piecewise'_C1'_differentiable'_on" 50)
lp15@61190
   268
  where "f piecewise_C1_differentiable_on i  \<equiv>
lp15@61190
   269
           continuous_on i f \<and>
lp15@61190
   270
           (\<exists>s. finite s \<and> (f C1_differentiable_on (i - s)))"
lp15@61190
   271
lp15@61190
   272
lemma C1_differentiable_imp_piecewise:
lp15@61190
   273
    "f C1_differentiable_on s \<Longrightarrow> f piecewise_C1_differentiable_on s"
lp15@61190
   274
  by (auto simp: piecewise_C1_differentiable_on_def C1_differentiable_on_eq continuous_at_imp_continuous_on differentiable_imp_continuous_within)
lp15@61190
   275
lp15@61190
   276
lemma piecewise_C1_imp_differentiable:
lp15@61190
   277
    "f piecewise_C1_differentiable_on i \<Longrightarrow> f piecewise_differentiable_on i"
lp15@61190
   278
  by (auto simp: piecewise_C1_differentiable_on_def piecewise_differentiable_on_def
lp15@61190
   279
           C1_differentiable_on_def differentiable_def has_vector_derivative_def
lp15@61190
   280
           intro: has_derivative_at_within)
lp15@61190
   281
lp15@61190
   282
lemma piecewise_C1_differentiable_compose:
lp15@61190
   283
    "\<lbrakk>f piecewise_C1_differentiable_on s; g piecewise_C1_differentiable_on (f ` s);
lp15@61190
   284
      \<And>x. finite (s \<inter> f-`{x})\<rbrakk>
lp15@61190
   285
      \<Longrightarrow> (g o f) piecewise_C1_differentiable_on s"
lp15@61190
   286
  apply (simp add: piecewise_C1_differentiable_on_def, safe)
lp15@61190
   287
  apply (blast intro: continuous_on_compose2)
lp15@61190
   288
  apply (rename_tac A B)
lp15@61190
   289
  apply (rule_tac x="A \<union> (\<Union>x\<in>B. s \<inter> f-`{x})" in exI)
lp15@61190
   290
  apply (rule conjI, blast)
lp15@61190
   291
  apply (rule C1_differentiable_compose)
lp15@61190
   292
  apply (blast intro: C1_differentiable_on_subset)
lp15@61190
   293
  apply (blast intro: C1_differentiable_on_subset)
lp15@61190
   294
  by (simp add: Diff_Int_distrib2)
lp15@61190
   295
lp15@61190
   296
lemma piecewise_C1_differentiable_on_subset:
lp15@61190
   297
    "f piecewise_C1_differentiable_on s \<Longrightarrow> t \<le> s \<Longrightarrow> f piecewise_C1_differentiable_on t"
lp15@61190
   298
  by (auto simp: piecewise_C1_differentiable_on_def elim!: continuous_on_subset C1_differentiable_on_subset)
lp15@61190
   299
lp15@61190
   300
lemma C1_differentiable_imp_continuous_on:
lp15@61190
   301
  "f C1_differentiable_on s \<Longrightarrow> continuous_on s f"
lp15@61190
   302
  unfolding C1_differentiable_on_eq continuous_on_eq_continuous_within
lp15@61190
   303
  using differentiable_at_withinI differentiable_imp_continuous_within by blast
lp15@61190
   304
lp15@61190
   305
lemma C1_differentiable_on_empty [iff]: "f C1_differentiable_on {}"
lp15@61190
   306
  unfolding C1_differentiable_on_def
lp15@61190
   307
  by auto
lp15@61190
   308
lp15@61190
   309
lemma piecewise_C1_differentiable_affine:
lp15@61190
   310
  fixes m::real
lp15@61190
   311
  assumes "f piecewise_C1_differentiable_on ((\<lambda>x. m * x + c) ` s)"
lp15@61190
   312
  shows "(f o (\<lambda>x. m *\<^sub>R x + c)) piecewise_C1_differentiable_on s"
lp15@61190
   313
proof (cases "m = 0")
lp15@61190
   314
  case True
lp15@61190
   315
  then show ?thesis
lp15@61190
   316
    unfolding o_def by (auto simp: piecewise_C1_differentiable_on_def continuous_on_const)
lp15@61190
   317
next
lp15@61190
   318
  case False
lp15@61190
   319
  show ?thesis
lp15@61190
   320
    apply (rule piecewise_C1_differentiable_compose [OF C1_differentiable_imp_piecewise])
lp15@61190
   321
    apply (rule assms derivative_intros | simp add: False vimage_def)+
lp15@61190
   322
    using real_vector_affinity_eq [OF False, where c=c, unfolded scaleR_conv_of_real]
lp15@61190
   323
    apply simp
lp15@61190
   324
    done
lp15@61190
   325
qed
lp15@61190
   326
lp15@61190
   327
lemma piecewise_C1_differentiable_cases:
lp15@61190
   328
  fixes c::real
lp15@61190
   329
  assumes "f piecewise_C1_differentiable_on {a..c}"
lp15@61190
   330
          "g piecewise_C1_differentiable_on {c..b}"
lp15@61190
   331
           "a \<le> c" "c \<le> b" "f c = g c"
lp15@61190
   332
  shows "(\<lambda>x. if x \<le> c then f x else g x) piecewise_C1_differentiable_on {a..b}"
lp15@61190
   333
proof -
lp15@61190
   334
  obtain s t where st: "f C1_differentiable_on ({a..c} - s)"
lp15@61190
   335
                       "g C1_differentiable_on ({c..b} - t)"
lp15@61190
   336
                       "finite s" "finite t"
lp15@61190
   337
    using assms
lp15@61190
   338
    by (force simp: piecewise_C1_differentiable_on_def)
lp15@61190
   339
  then have f_diff: "f differentiable_on {a..<c} - s"
lp15@61190
   340
        and g_diff: "g differentiable_on {c<..b} - t"
lp15@61190
   341
    by (simp_all add: C1_differentiable_on_eq differentiable_at_withinI differentiable_on_def)
lp15@61190
   342
  have "continuous_on {a..c} f" "continuous_on {c..b} g"
lp15@61190
   343
    using assms piecewise_C1_differentiable_on_def by auto
lp15@61190
   344
  then have cab: "continuous_on {a..b} (\<lambda>x. if x \<le> c then f x else g x)"
lp15@61190
   345
    using continuous_on_cases [OF closed_real_atLeastAtMost [of a c],
lp15@61190
   346
                               OF closed_real_atLeastAtMost [of c b],
lp15@61190
   347
                               of f g "\<lambda>x. x\<le>c"]  assms
lp15@61190
   348
    by (force simp: ivl_disj_un_two_touch)
lp15@61190
   349
  { fix x
lp15@61190
   350
    assume x: "x \<in> {a..b} - insert c (s \<union> t)"
lp15@61190
   351
    have "(\<lambda>x. if x \<le> c then f x else g x) differentiable at x" (is "?diff_fg")
lp15@61190
   352
    proof (cases x c rule: le_cases)
lp15@61190
   353
      case le show ?diff_fg
lp15@61190
   354
        apply (rule differentiable_transform_at [of "dist x c" _ f])
lp15@61190
   355
        using x dist_real_def le st by (auto simp: C1_differentiable_on_eq)
lp15@61190
   356
    next
lp15@61190
   357
      case ge show ?diff_fg
lp15@61190
   358
        apply (rule differentiable_transform_at [of "dist x c" _ g])
lp15@61190
   359
        using dist_nz x dist_real_def ge st x by (auto simp: C1_differentiable_on_eq)
lp15@61190
   360
    qed
lp15@61190
   361
  }
lp15@61190
   362
  then have "(\<forall>x \<in> {a..b} - insert c (s \<union> t). (\<lambda>x. if x \<le> c then f x else g x) differentiable at x)"
lp15@61190
   363
    by auto
lp15@61190
   364
  moreover
lp15@61190
   365
  { assume fcon: "continuous_on ({a<..<c} - s) (\<lambda>x. vector_derivative f (at x))"
lp15@61190
   366
       and gcon: "continuous_on ({c<..<b} - t) (\<lambda>x. vector_derivative g (at x))"
lp15@61190
   367
    have "open ({a<..<c} - s)"  "open ({c<..<b} - t)"
lp15@61190
   368
      using st by (simp_all add: open_Diff finite_imp_closed)
lp15@61190
   369
    moreover have "continuous_on ({a<..<c} - s) (\<lambda>x. vector_derivative (\<lambda>x. if x \<le> c then f x else g x) (at x))"
lp15@61190
   370
      apply (rule continuous_on_eq [OF fcon])
lp15@61190
   371
      apply (simp add:)
lp15@61190
   372
      apply (rule vector_derivative_at [symmetric])
lp15@61190
   373
      apply (rule_tac f=f and d="dist x c" in has_vector_derivative_transform_at)
lp15@61190
   374
      apply (simp_all add: dist_norm vector_derivative_works [symmetric])
lp15@61190
   375
      using f_diff
lp15@61190
   376
      by (meson C1_differentiable_on_eq Diff_iff atLeastAtMost_iff less_imp_le st(1))
lp15@61190
   377
    moreover have "continuous_on ({c<..<b} - t) (\<lambda>x. vector_derivative (\<lambda>x. if x \<le> c then f x else g x) (at x))"
lp15@61190
   378
      apply (rule continuous_on_eq [OF gcon])
lp15@61190
   379
      apply (simp add:)
lp15@61190
   380
      apply (rule vector_derivative_at [symmetric])
lp15@61190
   381
      apply (rule_tac f=g and d="dist x c" in has_vector_derivative_transform_at)
lp15@61190
   382
      apply (simp_all add: dist_norm vector_derivative_works [symmetric])
lp15@61190
   383
      using g_diff
lp15@61190
   384
      by (meson C1_differentiable_on_eq Diff_iff atLeastAtMost_iff less_imp_le st(2))
lp15@61190
   385
    ultimately have "continuous_on ({a<..<b} - insert c (s \<union> t))
lp15@61190
   386
        (\<lambda>x. vector_derivative (\<lambda>x. if x \<le> c then f x else g x) (at x))"
lp15@61190
   387
      apply (rule continuous_on_subset [OF continuous_on_open_Un], auto)
lp15@61190
   388
      done
lp15@61190
   389
  } note * = this
lp15@61190
   390
  have "continuous_on ({a<..<b} - insert c (s \<union> t)) (\<lambda>x. vector_derivative (\<lambda>x. if x \<le> c then f x else g x) (at x))"
lp15@61190
   391
    using st
lp15@61190
   392
    by (auto simp: C1_differentiable_on_eq elim!: continuous_on_subset intro: *)
lp15@61190
   393
  ultimately have "\<exists>s. finite s \<and> ((\<lambda>x. if x \<le> c then f x else g x) C1_differentiable_on {a..b} - s)"
lp15@61190
   394
    apply (rule_tac x="{a,b,c} \<union> s \<union> t" in exI)
lp15@61190
   395
    using st  by (auto simp: C1_differentiable_on_eq elim!: continuous_on_subset)
lp15@61190
   396
  with cab show ?thesis
lp15@61190
   397
    by (simp add: piecewise_C1_differentiable_on_def)
lp15@61190
   398
qed
lp15@61190
   399
lp15@61190
   400
lemma piecewise_C1_differentiable_neg:
lp15@61190
   401
    "f piecewise_C1_differentiable_on s \<Longrightarrow> (\<lambda>x. -(f x)) piecewise_C1_differentiable_on s"
lp15@61190
   402
  unfolding piecewise_C1_differentiable_on_def
lp15@61190
   403
  by (auto intro!: continuous_on_minus C1_differentiable_on_minus)
lp15@61190
   404
lp15@61190
   405
lemma piecewise_C1_differentiable_add:
lp15@61190
   406
  assumes "f piecewise_C1_differentiable_on i"
lp15@61190
   407
          "g piecewise_C1_differentiable_on i"
lp15@61190
   408
    shows "(\<lambda>x. f x + g x) piecewise_C1_differentiable_on i"
lp15@61190
   409
proof -
lp15@61190
   410
  obtain s t where st: "finite s" "finite t"
lp15@61190
   411
                       "f C1_differentiable_on (i-s)"
lp15@61190
   412
                       "g C1_differentiable_on (i-t)"
lp15@61190
   413
    using assms by (auto simp: piecewise_C1_differentiable_on_def)
lp15@61190
   414
  then have "finite (s \<union> t) \<and> (\<lambda>x. f x + g x) C1_differentiable_on i - (s \<union> t)"
lp15@61190
   415
    by (auto intro: C1_differentiable_on_add elim!: C1_differentiable_on_subset)
lp15@61190
   416
  moreover have "continuous_on i f" "continuous_on i g"
lp15@61190
   417
    using assms piecewise_C1_differentiable_on_def by auto
lp15@61190
   418
  ultimately show ?thesis
lp15@61190
   419
    by (auto simp: piecewise_C1_differentiable_on_def continuous_on_add)
lp15@61190
   420
qed
lp15@61190
   421
paulson@61204
   422
lemma piecewise_C1_differentiable_diff:
lp15@61190
   423
    "\<lbrakk>f piecewise_C1_differentiable_on s;  g piecewise_C1_differentiable_on s\<rbrakk>
lp15@61190
   424
     \<Longrightarrow> (\<lambda>x. f x - g x) piecewise_C1_differentiable_on s"
lp15@61190
   425
  unfolding diff_conv_add_uminus
lp15@61190
   426
  by (metis piecewise_C1_differentiable_add piecewise_C1_differentiable_neg)
lp15@61190
   427
lp15@61190
   428
lemma piecewise_C1_differentiable_D1:
lp15@61190
   429
  fixes g1 :: "real \<Rightarrow> 'a::real_normed_field"
lp15@61190
   430
  assumes "(g1 +++ g2) piecewise_C1_differentiable_on {0..1}"
lp15@61190
   431
    shows "g1 piecewise_C1_differentiable_on {0..1}"
lp15@61190
   432
proof -
lp15@61190
   433
  obtain s where "finite s"
lp15@61190
   434
             and co12: "continuous_on ({0..1} - s) (\<lambda>x. vector_derivative (g1 +++ g2) (at x))"
lp15@61190
   435
             and g12D: "\<forall>x\<in>{0..1} - s. g1 +++ g2 differentiable at x"
lp15@61190
   436
    using assms  by (auto simp: piecewise_C1_differentiable_on_def C1_differentiable_on_eq)
lp15@61190
   437
  then have g1D: "g1 differentiable at x" if "x \<in> {0..1} - insert 1 (op * 2 ` s)" for x
lp15@61190
   438
    apply (rule_tac d="dist (x/2) (1/2)" and f = "(g1 +++ g2) o (op*(inverse 2))" in differentiable_transform_at)
lp15@61190
   439
    using that
lp15@61190
   440
    apply (simp_all add: dist_real_def joinpaths_def)
lp15@61190
   441
    apply (rule differentiable_chain_at derivative_intros | force)+
lp15@61190
   442
    done
lp15@61190
   443
  have [simp]: "vector_derivative (g1 \<circ> op * 2) (at (x/2)) = 2 *\<^sub>R vector_derivative g1 (at x)"
lp15@61190
   444
               if "x \<in> {0..1} - insert 1 (op * 2 ` s)" for x
lp15@61190
   445
    apply (subst vector_derivative_chain_at)
lp15@61190
   446
    using that
lp15@61190
   447
    apply (rule derivative_eq_intros g1D | simp)+
lp15@61190
   448
    done
lp15@61190
   449
  have "continuous_on ({0..1/2} - insert (1/2) s) (\<lambda>x. vector_derivative (g1 +++ g2) (at x))"
lp15@61190
   450
    using co12 by (rule continuous_on_subset) force
lp15@61190
   451
  then have coDhalf: "continuous_on ({0..1/2} - insert (1/2) s) (\<lambda>x. vector_derivative (g1 o op*2) (at x))"
lp15@61190
   452
    apply (rule continuous_on_eq [OF _ vector_derivative_at])
lp15@61190
   453
    apply (rule_tac f="g1 o op*2" and d="dist x (1/2)" in has_vector_derivative_transform_at)
lp15@61190
   454
    apply (simp_all add: dist_norm joinpaths_def vector_derivative_works [symmetric])
lp15@61190
   455
    apply (force intro: g1D differentiable_chain_at)
lp15@61190
   456
    done
lp15@61190
   457
  have "continuous_on ({0..1} - insert 1 (op * 2 ` s))
lp15@61190
   458
                      ((\<lambda>x. 1/2 * vector_derivative (g1 o op*2) (at x)) o op*(1/2))"
lp15@61190
   459
    apply (rule continuous_intros)+
lp15@61190
   460
    using coDhalf
lp15@61190
   461
    apply (simp add: scaleR_conv_of_real image_set_diff image_image)
lp15@61190
   462
    done
lp15@61190
   463
  then have con_g1: "continuous_on ({0..1} - insert 1 (op * 2 ` s)) (\<lambda>x. vector_derivative g1 (at x))"
lp15@61190
   464
    by (rule continuous_on_eq) (simp add: scaleR_conv_of_real)
lp15@61190
   465
  have "continuous_on {0..1} g1"
lp15@61190
   466
    using continuous_on_joinpaths_D1 assms piecewise_C1_differentiable_on_def by blast
wenzelm@61222
   467
  with \<open>finite s\<close> show ?thesis
lp15@61190
   468
    apply (clarsimp simp add: piecewise_C1_differentiable_on_def C1_differentiable_on_eq)
lp15@61190
   469
    apply (rule_tac x="insert 1 ((op*2)`s)" in exI)
lp15@61190
   470
    apply (simp add: g1D con_g1)
lp15@61190
   471
  done
lp15@61190
   472
qed
lp15@61190
   473
lp15@61190
   474
lemma piecewise_C1_differentiable_D2:
lp15@61190
   475
  fixes g2 :: "real \<Rightarrow> 'a::real_normed_field"
lp15@61190
   476
  assumes "(g1 +++ g2) piecewise_C1_differentiable_on {0..1}" "pathfinish g1 = pathstart g2"
lp15@61190
   477
    shows "g2 piecewise_C1_differentiable_on {0..1}"
lp15@61190
   478
proof -
lp15@61190
   479
  obtain s where "finite s"
lp15@61190
   480
             and co12: "continuous_on ({0..1} - s) (\<lambda>x. vector_derivative (g1 +++ g2) (at x))"
lp15@61190
   481
             and g12D: "\<forall>x\<in>{0..1} - s. g1 +++ g2 differentiable at x"
lp15@61190
   482
    using assms  by (auto simp: piecewise_C1_differentiable_on_def C1_differentiable_on_eq)
lp15@61190
   483
  then have g2D: "g2 differentiable at x" if "x \<in> {0..1} - insert 0 ((\<lambda>x. 2*x-1) ` s)" for x
lp15@61190
   484
    apply (rule_tac d="dist ((x+1)/2) (1/2)" and f = "(g1 +++ g2) o (\<lambda>x. (x+1)/2)" in differentiable_transform_at)
lp15@61190
   485
    using that
lp15@61190
   486
    apply (simp_all add: dist_real_def joinpaths_def)
lp15@61190
   487
    apply (auto simp: dist_real_def joinpaths_def field_simps)
lp15@61190
   488
    apply (rule differentiable_chain_at derivative_intros | force)+
lp15@61190
   489
    apply (drule_tac x= "(x + 1) / 2" in bspec, force simp: divide_simps)
lp15@61190
   490
    apply assumption
lp15@61190
   491
    done
lp15@61190
   492
  have [simp]: "vector_derivative (g2 \<circ> (\<lambda>x. 2*x-1)) (at ((x+1)/2)) = 2 *\<^sub>R vector_derivative g2 (at x)"
lp15@61190
   493
               if "x \<in> {0..1} - insert 0 ((\<lambda>x. 2*x-1) ` s)" for x
lp15@61190
   494
    using that  by (auto simp: vector_derivative_chain_at divide_simps g2D)
lp15@61190
   495
  have "continuous_on ({1/2..1} - insert (1/2) s) (\<lambda>x. vector_derivative (g1 +++ g2) (at x))"
lp15@61190
   496
    using co12 by (rule continuous_on_subset) force
lp15@61190
   497
  then have coDhalf: "continuous_on ({1/2..1} - insert (1/2) s) (\<lambda>x. vector_derivative (g2 o (\<lambda>x. 2*x-1)) (at x))"
lp15@61190
   498
    apply (rule continuous_on_eq [OF _ vector_derivative_at])
lp15@61190
   499
    apply (rule_tac f="g2 o (\<lambda>x. 2*x-1)" and d="dist (3/4) ((x+1)/2)" in has_vector_derivative_transform_at)
lp15@61190
   500
    apply (auto simp: dist_real_def field_simps joinpaths_def vector_derivative_works [symmetric]
lp15@61190
   501
                intro!: g2D differentiable_chain_at)
lp15@61190
   502
    done
lp15@61190
   503
  have [simp]: "((\<lambda>x. (x + 1) / 2) ` ({0..1} - insert 0 ((\<lambda>x. 2 * x - 1) ` s))) = ({1/2..1} - insert (1/2) s)"
lp15@61190
   504
    apply (simp add: image_set_diff inj_on_def image_image)
lp15@61190
   505
    apply (auto simp: image_affinity_atLeastAtMost_div add_divide_distrib)
lp15@61190
   506
    done
lp15@61190
   507
  have "continuous_on ({0..1} - insert 0 ((\<lambda>x. 2*x-1) ` s))
lp15@61190
   508
                      ((\<lambda>x. 1/2 * vector_derivative (g2 \<circ> (\<lambda>x. 2*x-1)) (at x)) o (\<lambda>x. (x+1)/2))"
lp15@61190
   509
    by (rule continuous_intros | simp add:  coDhalf)+
lp15@61190
   510
  then have con_g2: "continuous_on ({0..1} - insert 0 ((\<lambda>x. 2*x-1) ` s)) (\<lambda>x. vector_derivative g2 (at x))"
lp15@61190
   511
    by (rule continuous_on_eq) (simp add: scaleR_conv_of_real)
lp15@61190
   512
  have "continuous_on {0..1} g2"
lp15@61190
   513
    using continuous_on_joinpaths_D2 assms piecewise_C1_differentiable_on_def by blast
wenzelm@61222
   514
  with \<open>finite s\<close> show ?thesis
lp15@61190
   515
    apply (clarsimp simp add: piecewise_C1_differentiable_on_def C1_differentiable_on_eq)
lp15@61190
   516
    apply (rule_tac x="insert 0 ((\<lambda>x. 2 * x - 1) ` s)" in exI)
lp15@61190
   517
    apply (simp add: g2D con_g2)
lp15@61190
   518
  done
lp15@61190
   519
qed
lp15@60809
   520
lp15@60809
   521
subsection \<open>Valid paths, and their start and finish\<close>
lp15@60809
   522
lp15@60809
   523
lemma Diff_Un_eq: "A - (B \<union> C) = A - B - C"
lp15@60809
   524
  by blast
lp15@60809
   525
lp15@60809
   526
definition valid_path :: "(real \<Rightarrow> 'a :: real_normed_vector) \<Rightarrow> bool"
lp15@61190
   527
  where "valid_path f \<equiv> f piecewise_C1_differentiable_on {0..1::real}"
lp15@60809
   528
lp15@60809
   529
definition closed_path :: "(real \<Rightarrow> 'a :: real_normed_vector) \<Rightarrow> bool"
lp15@60809
   530
  where "closed_path g \<equiv> g 0 = g 1"
lp15@60809
   531
lp15@60809
   532
subsubsection\<open>In particular, all results for paths apply\<close>
lp15@60809
   533
lp15@60809
   534
lemma valid_path_imp_path: "valid_path g \<Longrightarrow> path g"
lp15@61190
   535
by (simp add: path_def piecewise_C1_differentiable_on_def valid_path_def)
lp15@60809
   536
lp15@60809
   537
lemma connected_valid_path_image: "valid_path g \<Longrightarrow> connected(path_image g)"
lp15@60809
   538
  by (metis connected_path_image valid_path_imp_path)
lp15@60809
   539
lp15@60809
   540
lemma compact_valid_path_image: "valid_path g \<Longrightarrow> compact(path_image g)"
lp15@60809
   541
  by (metis compact_path_image valid_path_imp_path)
lp15@60809
   542
lp15@60809
   543
lemma bounded_valid_path_image: "valid_path g \<Longrightarrow> bounded(path_image g)"
lp15@60809
   544
  by (metis bounded_path_image valid_path_imp_path)
lp15@60809
   545
lp15@60809
   546
lemma closed_valid_path_image: "valid_path g \<Longrightarrow> closed(path_image g)"
lp15@60809
   547
  by (metis closed_path_image valid_path_imp_path)
lp15@60809
   548
lp15@60809
   549
lp15@60809
   550
subsection\<open>Contour Integrals along a path\<close>
lp15@60809
   551
lp15@60809
   552
text\<open>This definition is for complex numbers only, and does not generalise to line integrals in a vector field\<close>
lp15@60809
   553
lp15@61190
   554
text\<open>piecewise differentiable function on [0,1]\<close>
lp15@60809
   555
lp15@61738
   556
definition has_contour_integral :: "(complex \<Rightarrow> complex) \<Rightarrow> complex \<Rightarrow> (real \<Rightarrow> complex) \<Rightarrow> bool"
lp15@61738
   557
           (infixr "has'_contour'_integral" 50)
lp15@61738
   558
  where "(f has_contour_integral i) g \<equiv>
lp15@60809
   559
           ((\<lambda>x. f(g x) * vector_derivative g (at x within {0..1}))
lp15@60809
   560
            has_integral i) {0..1}"
lp15@60809
   561
lp15@61738
   562
definition contour_integrable_on
lp15@61738
   563
           (infixr "contour'_integrable'_on" 50)
lp15@61738
   564
  where "f contour_integrable_on g \<equiv> \<exists>i. (f has_contour_integral i) g"
lp15@61738
   565
lp15@61738
   566
definition contour_integral
lp15@61738
   567
  where "contour_integral g f \<equiv> @i. (f has_contour_integral i) g"
lp15@61738
   568
lp15@61738
   569
lemma contour_integral_unique: "(f has_contour_integral i)  g \<Longrightarrow> contour_integral g f = i"
lp15@61738
   570
  by (auto simp: contour_integral_def has_contour_integral_def integral_def [symmetric])
lp15@61738
   571
lp15@61738
   572
lemma has_contour_integral_integral:
lp15@61738
   573
    "f contour_integrable_on i \<Longrightarrow> (f has_contour_integral (contour_integral i f)) i"
lp15@61738
   574
  by (metis contour_integral_unique contour_integrable_on_def)
lp15@61738
   575
lp15@61738
   576
lemma has_contour_integral_unique:
lp15@61738
   577
    "(f has_contour_integral i) g \<Longrightarrow> (f has_contour_integral j) g \<Longrightarrow> i = j"
lp15@60809
   578
  using has_integral_unique
lp15@61738
   579
  by (auto simp: has_contour_integral_def)
lp15@61738
   580
lp15@61738
   581
lemma has_contour_integral_integrable: "(f has_contour_integral i) g \<Longrightarrow> f contour_integrable_on g"
lp15@61738
   582
  using contour_integrable_on_def by blast
lp15@60809
   583
lp15@60809
   584
(* Show that we can forget about the localized derivative.*)
lp15@60809
   585
lp15@60809
   586
lemma vector_derivative_within_interior:
lp15@60809
   587
     "\<lbrakk>x \<in> interior s; NO_MATCH UNIV s\<rbrakk>
lp15@60809
   588
      \<Longrightarrow> vector_derivative f (at x within s) = vector_derivative f (at x)"
lp15@60809
   589
  apply (simp add: vector_derivative_def has_vector_derivative_def has_derivative_def netlimit_within_interior)
lp15@60809
   590
  apply (subst lim_within_interior, auto)
lp15@60809
   591
  done
lp15@60809
   592
lp15@60809
   593
lemma has_integral_localized_vector_derivative:
lp15@60809
   594
    "((\<lambda>x. f (g x) * vector_derivative g (at x within {a..b})) has_integral i) {a..b} \<longleftrightarrow>
lp15@60809
   595
     ((\<lambda>x. f (g x) * vector_derivative g (at x)) has_integral i) {a..b}"
lp15@60809
   596
proof -
lp15@60809
   597
  have "{a..b} - {a,b} = interior {a..b}"
lp15@60809
   598
    by (simp add: atLeastAtMost_diff_ends)
lp15@60809
   599
  show ?thesis
lp15@60809
   600
    apply (rule has_integral_spike_eq [of "{a,b}"])
lp15@60809
   601
    apply (auto simp: vector_derivative_within_interior)
lp15@60809
   602
    done
lp15@60809
   603
qed
lp15@60809
   604
lp15@60809
   605
lemma integrable_on_localized_vector_derivative:
lp15@60809
   606
    "(\<lambda>x. f (g x) * vector_derivative g (at x within {a..b})) integrable_on {a..b} \<longleftrightarrow>
lp15@60809
   607
     (\<lambda>x. f (g x) * vector_derivative g (at x)) integrable_on {a..b}"
lp15@60809
   608
  by (simp add: integrable_on_def has_integral_localized_vector_derivative)
lp15@60809
   609
lp15@61738
   610
lemma has_contour_integral:
lp15@61738
   611
     "(f has_contour_integral i) g \<longleftrightarrow>
lp15@60809
   612
      ((\<lambda>x. f (g x) * vector_derivative g (at x)) has_integral i) {0..1}"
lp15@61738
   613
  by (simp add: has_integral_localized_vector_derivative has_contour_integral_def)
lp15@61738
   614
lp15@61738
   615
lemma contour_integrable_on:
lp15@61738
   616
     "f contour_integrable_on g \<longleftrightarrow>
lp15@60809
   617
      (\<lambda>t. f(g t) * vector_derivative g (at t)) integrable_on {0..1}"
lp15@61738
   618
  by (simp add: has_contour_integral integrable_on_def contour_integrable_on_def)
lp15@60809
   619
lp15@60809
   620
subsection\<open>Reversing a path\<close>
lp15@60809
   621
lp15@60809
   622
lemma valid_path_imp_reverse:
lp15@60809
   623
  assumes "valid_path g"
lp15@60809
   624
    shows "valid_path(reversepath g)"
lp15@60809
   625
proof -
lp15@61190
   626
  obtain s where "finite s" "g C1_differentiable_on ({0..1} - s)"
lp15@61190
   627
    using assms by (auto simp: valid_path_def piecewise_C1_differentiable_on_def)
lp15@61190
   628
  then have "finite (op - 1 ` s)" "(reversepath g C1_differentiable_on ({0..1} - op - 1 ` s))"
lp15@60809
   629
    apply (auto simp: reversepath_def)
lp15@61190
   630
    apply (rule C1_differentiable_compose [of "\<lambda>x::real. 1-x" _ g, unfolded o_def])
lp15@61190
   631
    apply (auto simp: C1_differentiable_on_eq)
lp15@61190
   632
    apply (rule continuous_intros, force)
lp15@61190
   633
    apply (force elim!: continuous_on_subset)
lp15@61190
   634
    apply (simp add: finite_vimageI inj_on_def)
lp15@60809
   635
    done
lp15@60809
   636
  then show ?thesis using assms
lp15@61190
   637
    by (auto simp: valid_path_def piecewise_C1_differentiable_on_def path_def [symmetric])
lp15@60809
   638
qed
lp15@60809
   639
lp15@60809
   640
lemma valid_path_reversepath: "valid_path(reversepath g) \<longleftrightarrow> valid_path g"
lp15@60809
   641
  using valid_path_imp_reverse by force
lp15@60809
   642
lp15@61738
   643
lemma has_contour_integral_reversepath:
lp15@61738
   644
  assumes "valid_path g" "(f has_contour_integral i) g"
lp15@61738
   645
    shows "(f has_contour_integral (-i)) (reversepath g)"
lp15@60809
   646
proof -
lp15@60809
   647
  { fix s x
lp15@61190
   648
    assume xs: "g C1_differentiable_on ({0..1} - s)" "x \<notin> op - 1 ` s" "0 \<le> x" "x \<le> 1"
lp15@60809
   649
      have "vector_derivative (\<lambda>x. g (1 - x)) (at x within {0..1}) =
lp15@60809
   650
            - vector_derivative g (at (1 - x) within {0..1})"
lp15@60809
   651
      proof -
lp15@60809
   652
        obtain f' where f': "(g has_vector_derivative f') (at (1 - x))"
lp15@60809
   653
          using xs
lp15@61190
   654
          by (force simp: has_vector_derivative_def C1_differentiable_on_def)
lp15@60809
   655
        have "(g o (\<lambda>x. 1 - x) has_vector_derivative -1 *\<^sub>R f') (at x)"
lp15@60809
   656
          apply (rule vector_diff_chain_within)
lp15@60809
   657
          apply (intro vector_diff_chain_within derivative_eq_intros | simp)+
lp15@60809
   658
          apply (rule has_vector_derivative_at_within [OF f'])
lp15@60809
   659
          done
lp15@60809
   660
        then have mf': "((\<lambda>x. g (1 - x)) has_vector_derivative -f') (at x)"
lp15@60809
   661
          by (simp add: o_def)
lp15@60809
   662
        show ?thesis
lp15@60809
   663
          using xs
lp15@60809
   664
          by (auto simp: vector_derivative_at_within_ivl [OF mf'] vector_derivative_at_within_ivl [OF f'])
lp15@60809
   665
      qed
lp15@60809
   666
  } note * = this
lp15@60809
   667
  have 01: "{0..1::real} = cbox 0 1"
lp15@60809
   668
    by simp
lp15@60809
   669
  show ?thesis using assms
lp15@61738
   670
    apply (auto simp: has_contour_integral_def)
lp15@60809
   671
    apply (drule has_integral_affinity01 [where m= "-1" and c=1])
lp15@61190
   672
    apply (auto simp: reversepath_def valid_path_def piecewise_C1_differentiable_on_def)
lp15@60809
   673
    apply (drule has_integral_neg)
lp15@60809
   674
    apply (rule_tac s = "(\<lambda>x. 1 - x) ` s" in has_integral_spike_finite)
lp15@60809
   675
    apply (auto simp: *)
lp15@60809
   676
    done
lp15@60809
   677
qed
lp15@60809
   678
lp15@61738
   679
lemma contour_integrable_reversepath:
lp15@61738
   680
    "valid_path g \<Longrightarrow> f contour_integrable_on g \<Longrightarrow> f contour_integrable_on (reversepath g)"
lp15@61738
   681
  using has_contour_integral_reversepath contour_integrable_on_def by blast
lp15@61738
   682
lp15@61738
   683
lemma contour_integrable_reversepath_eq:
lp15@61738
   684
    "valid_path g \<Longrightarrow> (f contour_integrable_on (reversepath g) \<longleftrightarrow> f contour_integrable_on g)"
lp15@61738
   685
  using contour_integrable_reversepath valid_path_reversepath by fastforce
lp15@61738
   686
lp15@61738
   687
lemma contour_integral_reversepath:
lp15@61738
   688
    "\<lbrakk>valid_path g; f contour_integrable_on g\<rbrakk> \<Longrightarrow> contour_integral (reversepath g) f = -(contour_integral g f)"
lp15@61738
   689
  using has_contour_integral_reversepath contour_integrable_on_def contour_integral_unique by blast
lp15@60809
   690
lp15@60809
   691
lp15@60809
   692
subsection\<open>Joining two paths together\<close>
lp15@60809
   693
lp15@60809
   694
lemma valid_path_join:
lp15@60809
   695
  assumes "valid_path g1" "valid_path g2" "pathfinish g1 = pathstart g2"
lp15@60809
   696
    shows "valid_path(g1 +++ g2)"
lp15@60809
   697
proof -
lp15@60809
   698
  have "g1 1 = g2 0"
lp15@60809
   699
    using assms by (auto simp: pathfinish_def pathstart_def)
lp15@61190
   700
  moreover have "(g1 o (\<lambda>x. 2*x)) piecewise_C1_differentiable_on {0..1/2}"
lp15@61190
   701
    apply (rule piecewise_C1_differentiable_compose)
lp15@60809
   702
    using assms
lp15@61190
   703
    apply (auto simp: valid_path_def piecewise_C1_differentiable_on_def continuous_on_joinpaths)
lp15@60809
   704
    apply (rule continuous_intros | simp)+
lp15@60809
   705
    apply (force intro: finite_vimageI [where h = "op*2"] inj_onI)
lp15@60809
   706
    done
lp15@61190
   707
  moreover have "(g2 o (\<lambda>x. 2*x-1)) piecewise_C1_differentiable_on {1/2..1}"
lp15@61190
   708
    apply (rule piecewise_C1_differentiable_compose)
lp15@61190
   709
    using assms unfolding valid_path_def piecewise_C1_differentiable_on_def
lp15@61190
   710
    by (auto intro!: continuous_intros finite_vimageI [where h = "(\<lambda>x. 2*x - 1)"] inj_onI
lp15@61190
   711
             simp: image_affinity_atLeastAtMost_diff continuous_on_joinpaths)
lp15@60809
   712
  ultimately show ?thesis
lp15@60809
   713
    apply (simp only: valid_path_def continuous_on_joinpaths joinpaths_def)
lp15@61190
   714
    apply (rule piecewise_C1_differentiable_cases)
lp15@60809
   715
    apply (auto simp: o_def)
lp15@60809
   716
    done
lp15@60809
   717
qed
lp15@60809
   718
lp15@61190
   719
lemma valid_path_join_D1:
lp15@61190
   720
  fixes g1 :: "real \<Rightarrow> 'a::real_normed_field"
lp15@61190
   721
  shows "valid_path (g1 +++ g2) \<Longrightarrow> valid_path g1"
lp15@61190
   722
  unfolding valid_path_def
lp15@61190
   723
  by (rule piecewise_C1_differentiable_D1)
lp15@60809
   724
lp15@61190
   725
lemma valid_path_join_D2:
lp15@61190
   726
  fixes g2 :: "real \<Rightarrow> 'a::real_normed_field"
lp15@61190
   727
  shows "\<lbrakk>valid_path (g1 +++ g2); pathfinish g1 = pathstart g2\<rbrakk> \<Longrightarrow> valid_path g2"
lp15@61190
   728
  unfolding valid_path_def
lp15@61190
   729
  by (rule piecewise_C1_differentiable_D2)
lp15@60809
   730
lp15@60809
   731
lemma valid_path_join_eq [simp]:
lp15@61190
   732
  fixes g2 :: "real \<Rightarrow> 'a::real_normed_field"
lp15@61190
   733
  shows "pathfinish g1 = pathstart g2 \<Longrightarrow> (valid_path(g1 +++ g2) \<longleftrightarrow> valid_path g1 \<and> valid_path g2)"
lp15@60809
   734
  using valid_path_join_D1 valid_path_join_D2 valid_path_join by blast
lp15@60809
   735
lp15@61738
   736
lemma has_contour_integral_join:
lp15@61738
   737
  assumes "(f has_contour_integral i1) g1" "(f has_contour_integral i2) g2"
lp15@60809
   738
          "valid_path g1" "valid_path g2"
lp15@61738
   739
    shows "(f has_contour_integral (i1 + i2)) (g1 +++ g2)"
lp15@60809
   740
proof -
lp15@60809
   741
  obtain s1 s2
lp15@60809
   742
    where s1: "finite s1" "\<forall>x\<in>{0..1} - s1. g1 differentiable at x"
lp15@60809
   743
      and s2: "finite s2" "\<forall>x\<in>{0..1} - s2. g2 differentiable at x"
lp15@60809
   744
    using assms
lp15@61190
   745
    by (auto simp: valid_path_def piecewise_C1_differentiable_on_def C1_differentiable_on_eq)
lp15@60809
   746
  have 1: "((\<lambda>x. f (g1 x) * vector_derivative g1 (at x)) has_integral i1) {0..1}"
lp15@60809
   747
   and 2: "((\<lambda>x. f (g2 x) * vector_derivative g2 (at x)) has_integral i2) {0..1}"
lp15@60809
   748
    using assms
lp15@61738
   749
    by (auto simp: has_contour_integral)
lp15@60809
   750
  have i1: "((\<lambda>x. (2*f (g1 (2*x))) * vector_derivative g1 (at (2*x))) has_integral i1) {0..1/2}"
lp15@60809
   751
   and i2: "((\<lambda>x. (2*f (g2 (2*x - 1))) * vector_derivative g2 (at (2*x - 1))) has_integral i2) {1/2..1}"
lp15@60809
   752
    using has_integral_affinity01 [OF 1, where m= 2 and c=0, THEN has_integral_cmul [where c=2]]
lp15@60809
   753
          has_integral_affinity01 [OF 2, where m= 2 and c="-1", THEN has_integral_cmul [where c=2]]
lp15@60809
   754
    by (simp_all only: image_affinity_atLeastAtMost_div_diff, simp_all add: scaleR_conv_of_real mult_ac)
lp15@60809
   755
  have g1: "\<lbrakk>0 \<le> z; z*2 < 1; z*2 \<notin> s1\<rbrakk> \<Longrightarrow>
lp15@60809
   756
            vector_derivative (\<lambda>x. if x*2 \<le> 1 then g1 (2*x) else g2 (2*x - 1)) (at z) =
lp15@60809
   757
            2 *\<^sub>R vector_derivative g1 (at (z*2))" for z
lp15@60809
   758
    apply (rule vector_derivative_at [OF has_vector_derivative_transform_at [of "\<bar>z - 1/2\<bar>" _ "(\<lambda>x. g1(2*x))"]])
lp15@60809
   759
    apply (simp_all add: dist_real_def abs_if split: split_if_asm)
lp15@60809
   760
    apply (rule vector_diff_chain_at [of "\<lambda>x. 2*x" 2 _ g1, simplified o_def])
lp15@60809
   761
    apply (simp add: has_vector_derivative_def has_derivative_def bounded_linear_mult_left)
lp15@60809
   762
    using s1
lp15@60809
   763
    apply (auto simp: algebra_simps vector_derivative_works)
lp15@60809
   764
    done
lp15@60809
   765
  have g2: "\<lbrakk>1 < z*2; z \<le> 1; z*2 - 1 \<notin> s2\<rbrakk> \<Longrightarrow>
lp15@60809
   766
            vector_derivative (\<lambda>x. if x*2 \<le> 1 then g1 (2*x) else g2 (2*x - 1)) (at z) =
lp15@60809
   767
            2 *\<^sub>R vector_derivative g2 (at (z*2 - 1))" for z
lp15@60809
   768
    apply (rule vector_derivative_at [OF has_vector_derivative_transform_at [of "\<bar>z - 1/2\<bar>" _ "(\<lambda>x. g2 (2*x - 1))"]])
lp15@60809
   769
    apply (simp_all add: dist_real_def abs_if split: split_if_asm)
lp15@60809
   770
    apply (rule vector_diff_chain_at [of "\<lambda>x. 2*x - 1" 2 _ g2, simplified o_def])
lp15@60809
   771
    apply (simp add: has_vector_derivative_def has_derivative_def bounded_linear_mult_left)
lp15@60809
   772
    using s2
lp15@60809
   773
    apply (auto simp: algebra_simps vector_derivative_works)
lp15@60809
   774
    done
lp15@60809
   775
  have "((\<lambda>x. f ((g1 +++ g2) x) * vector_derivative (g1 +++ g2) (at x)) has_integral i1) {0..1/2}"
lp15@60809
   776
    apply (rule has_integral_spike_finite [OF _ _ i1, of "insert (1/2) (op*2 -` s1)"])
lp15@60809
   777
    using s1
lp15@60809
   778
    apply (force intro: finite_vimageI [where h = "op*2"] inj_onI)
lp15@60809
   779
    apply (clarsimp simp add: joinpaths_def scaleR_conv_of_real mult_ac g1)
lp15@60809
   780
    done
lp15@60809
   781
  moreover have "((\<lambda>x. f ((g1 +++ g2) x) * vector_derivative (g1 +++ g2) (at x)) has_integral i2) {1/2..1}"
lp15@60809
   782
    apply (rule has_integral_spike_finite [OF _ _ i2, of "insert (1/2) ((\<lambda>x. 2*x-1) -` s2)"])
lp15@60809
   783
    using s2
lp15@60809
   784
    apply (force intro: finite_vimageI [where h = "\<lambda>x. 2*x-1"] inj_onI)
lp15@60809
   785
    apply (clarsimp simp add: joinpaths_def scaleR_conv_of_real mult_ac g2)
lp15@60809
   786
    done
lp15@60809
   787
  ultimately
lp15@60809
   788
  show ?thesis
lp15@61738
   789
    apply (simp add: has_contour_integral)
lp15@60809
   790
    apply (rule has_integral_combine [where c = "1/2"], auto)
lp15@60809
   791
    done
lp15@60809
   792
qed
lp15@60809
   793
lp15@61738
   794
lemma contour_integrable_joinI:
lp15@61738
   795
  assumes "f contour_integrable_on g1" "f contour_integrable_on g2"
lp15@60809
   796
          "valid_path g1" "valid_path g2"
lp15@61738
   797
    shows "f contour_integrable_on (g1 +++ g2)"
lp15@60809
   798
  using assms
lp15@61738
   799
  by (meson has_contour_integral_join contour_integrable_on_def)
lp15@61738
   800
lp15@61738
   801
lemma contour_integrable_joinD1:
lp15@61738
   802
  assumes "f contour_integrable_on (g1 +++ g2)" "valid_path g1"
lp15@61738
   803
    shows "f contour_integrable_on g1"
lp15@60809
   804
proof -
lp15@60809
   805
  obtain s1
lp15@60809
   806
    where s1: "finite s1" "\<forall>x\<in>{0..1} - s1. g1 differentiable at x"
lp15@61190
   807
    using assms by (auto simp: valid_path_def piecewise_C1_differentiable_on_def C1_differentiable_on_eq)
lp15@60809
   808
  have "(\<lambda>x. f ((g1 +++ g2) (x/2)) * vector_derivative (g1 +++ g2) (at (x/2))) integrable_on {0..1}"
lp15@60809
   809
    using assms
lp15@61738
   810
    apply (auto simp: contour_integrable_on)
lp15@60809
   811
    apply (drule integrable_on_subcbox [where a=0 and b="1/2"])
lp15@60809
   812
    apply (auto intro: integrable_affinity [of _ 0 "1/2::real" "1/2" 0, simplified])
lp15@60809
   813
    done
lp15@60809
   814
  then have *: "(\<lambda>x. (f ((g1 +++ g2) (x/2))/2) * vector_derivative (g1 +++ g2) (at (x/2))) integrable_on {0..1}"
lp15@61190
   815
    by (auto dest: integrable_cmul [where c="1/2"] simp: scaleR_conv_of_real)
lp15@60809
   816
  have g1: "\<lbrakk>0 < z; z < 1; z \<notin> s1\<rbrakk> \<Longrightarrow>
lp15@60809
   817
            vector_derivative (\<lambda>x. if x*2 \<le> 1 then g1 (2*x) else g2 (2*x - 1)) (at (z/2)) =
lp15@60809
   818
            2 *\<^sub>R vector_derivative g1 (at z)"  for z
lp15@60809
   819
    apply (rule vector_derivative_at [OF has_vector_derivative_transform_at [of "\<bar>(z-1)/2\<bar>" _ "(\<lambda>x. g1(2*x))"]])
lp15@60809
   820
    apply (simp_all add: field_simps dist_real_def abs_if split: split_if_asm)
lp15@60809
   821
    apply (rule vector_diff_chain_at [of "\<lambda>x. x*2" 2 _ g1, simplified o_def])
lp15@60809
   822
    using s1
lp15@60809
   823
    apply (auto simp: vector_derivative_works has_vector_derivative_def has_derivative_def bounded_linear_mult_left)
lp15@60809
   824
    done
lp15@60809
   825
  show ?thesis
lp15@60809
   826
    using s1
lp15@61738
   827
    apply (auto simp: contour_integrable_on)
lp15@60809
   828
    apply (rule integrable_spike_finite [of "{0,1} \<union> s1", OF _ _ *])
lp15@60809
   829
    apply (auto simp: joinpaths_def scaleR_conv_of_real g1)
lp15@60809
   830
    done
lp15@60809
   831
qed
lp15@60809
   832
lp15@61738
   833
lemma contour_integrable_joinD2:
lp15@61738
   834
  assumes "f contour_integrable_on (g1 +++ g2)" "valid_path g2"
lp15@61738
   835
    shows "f contour_integrable_on g2"
lp15@60809
   836
proof -
lp15@60809
   837
  obtain s2
lp15@60809
   838
    where s2: "finite s2" "\<forall>x\<in>{0..1} - s2. g2 differentiable at x"
lp15@61190
   839
    using assms by (auto simp: valid_path_def piecewise_C1_differentiable_on_def C1_differentiable_on_eq)
lp15@60809
   840
  have "(\<lambda>x. f ((g1 +++ g2) (x/2 + 1/2)) * vector_derivative (g1 +++ g2) (at (x/2 + 1/2))) integrable_on {0..1}"
lp15@60809
   841
    using assms
lp15@61738
   842
    apply (auto simp: contour_integrable_on)
lp15@60809
   843
    apply (drule integrable_on_subcbox [where a="1/2" and b=1], auto)
lp15@60809
   844
    apply (drule integrable_affinity [of _ "1/2::real" 1 "1/2" "1/2", simplified])
lp15@60809
   845
    apply (simp add: image_affinity_atLeastAtMost_diff)
lp15@60809
   846
    done
lp15@60809
   847
  then have *: "(\<lambda>x. (f ((g1 +++ g2) (x/2 + 1/2))/2) * vector_derivative (g1 +++ g2) (at (x/2 + 1/2)))
lp15@60809
   848
                integrable_on {0..1}"
lp15@60809
   849
    by (auto dest: integrable_cmul [where c="1/2"] simp: scaleR_conv_of_real)
lp15@60809
   850
  have g2: "\<lbrakk>0 < z; z < 1; z \<notin> s2\<rbrakk> \<Longrightarrow>
lp15@60809
   851
            vector_derivative (\<lambda>x. if x*2 \<le> 1 then g1 (2*x) else g2 (2*x - 1)) (at (z/2+1/2)) =
lp15@60809
   852
            2 *\<^sub>R vector_derivative g2 (at z)" for z
lp15@60809
   853
    apply (rule vector_derivative_at [OF has_vector_derivative_transform_at [of "\<bar>z/2\<bar>" _ "(\<lambda>x. g2(2*x-1))"]])
lp15@60809
   854
    apply (simp_all add: field_simps dist_real_def abs_if split: split_if_asm)
lp15@60809
   855
    apply (rule vector_diff_chain_at [of "\<lambda>x. x*2-1" 2 _ g2, simplified o_def])
lp15@60809
   856
    using s2
lp15@60809
   857
    apply (auto simp: has_vector_derivative_def has_derivative_def bounded_linear_mult_left
lp15@60809
   858
                      vector_derivative_works add_divide_distrib)
lp15@60809
   859
    done
lp15@60809
   860
  show ?thesis
lp15@60809
   861
    using s2
lp15@61738
   862
    apply (auto simp: contour_integrable_on)
lp15@60809
   863
    apply (rule integrable_spike_finite [of "{0,1} \<union> s2", OF _ _ *])
lp15@60809
   864
    apply (auto simp: joinpaths_def scaleR_conv_of_real g2)
lp15@60809
   865
    done
lp15@60809
   866
qed
lp15@60809
   867
lp15@61738
   868
lemma contour_integrable_join [simp]:
lp15@60809
   869
  shows
lp15@60809
   870
    "\<lbrakk>valid_path g1; valid_path g2\<rbrakk>
lp15@61738
   871
     \<Longrightarrow> f contour_integrable_on (g1 +++ g2) \<longleftrightarrow> f contour_integrable_on g1 \<and> f contour_integrable_on g2"
lp15@61738
   872
using contour_integrable_joinD1 contour_integrable_joinD2 contour_integrable_joinI by blast
lp15@61738
   873
lp15@61738
   874
lemma contour_integral_join [simp]:
lp15@60809
   875
  shows
lp15@61738
   876
    "\<lbrakk>f contour_integrable_on g1; f contour_integrable_on g2; valid_path g1; valid_path g2\<rbrakk>
lp15@61738
   877
        \<Longrightarrow> contour_integral (g1 +++ g2) f = contour_integral g1 f + contour_integral g2 f"
lp15@61738
   878
  by (simp add: has_contour_integral_integral has_contour_integral_join contour_integral_unique)
lp15@60809
   879
lp15@60809
   880
lp15@60809
   881
subsection\<open>Shifting the starting point of a (closed) path\<close>
lp15@60809
   882
lp15@60809
   883
lemma shiftpath_alt_def: "shiftpath a f = (\<lambda>x. if x \<le> 1-a then f (a + x) else f (a + x - 1))"
lp15@60809
   884
  by (auto simp: shiftpath_def)
lp15@60809
   885
lp15@60809
   886
lemma valid_path_shiftpath [intro]:
lp15@60809
   887
  assumes "valid_path g" "pathfinish g = pathstart g" "a \<in> {0..1}"
lp15@60809
   888
    shows "valid_path(shiftpath a g)"
lp15@60809
   889
  using assms
lp15@60809
   890
  apply (auto simp: valid_path_def shiftpath_alt_def)
lp15@61190
   891
  apply (rule piecewise_C1_differentiable_cases)
lp15@60809
   892
  apply (auto simp: algebra_simps)
lp15@61190
   893
  apply (rule piecewise_C1_differentiable_affine [of g 1 a, simplified o_def scaleR_one])
lp15@61190
   894
  apply (auto simp: pathfinish_def pathstart_def elim: piecewise_C1_differentiable_on_subset)
lp15@61190
   895
  apply (rule piecewise_C1_differentiable_affine [of g 1 "a-1", simplified o_def scaleR_one algebra_simps])
lp15@61190
   896
  apply (auto simp: pathfinish_def pathstart_def elim: piecewise_C1_differentiable_on_subset)
lp15@60809
   897
  done
lp15@60809
   898
lp15@61738
   899
lemma has_contour_integral_shiftpath:
lp15@61738
   900
  assumes f: "(f has_contour_integral i) g" "valid_path g"
lp15@60809
   901
      and a: "a \<in> {0..1}"
lp15@61738
   902
    shows "(f has_contour_integral i) (shiftpath a g)"
lp15@60809
   903
proof -
lp15@60809
   904
  obtain s
lp15@60809
   905
    where s: "finite s" and g: "\<forall>x\<in>{0..1} - s. g differentiable at x"
lp15@61190
   906
    using assms by (auto simp: valid_path_def piecewise_C1_differentiable_on_def C1_differentiable_on_eq)
lp15@60809
   907
  have *: "((\<lambda>x. f (g x) * vector_derivative g (at x)) has_integral i) {0..1}"
lp15@61738
   908
    using assms by (auto simp: has_contour_integral)
lp15@60809
   909
  then have i: "i = integral {a..1} (\<lambda>x. f (g x) * vector_derivative g (at x)) +
lp15@60809
   910
                    integral {0..a} (\<lambda>x. f (g x) * vector_derivative g (at x))"
lp15@60809
   911
    apply (rule has_integral_unique)
lp15@60809
   912
    apply (subst add.commute)
lp15@60809
   913
    apply (subst Integration.integral_combine)
lp15@60809
   914
    using assms * integral_unique by auto
lp15@60809
   915
  { fix x
lp15@60809
   916
    have "0 \<le> x \<Longrightarrow> x + a < 1 \<Longrightarrow> x \<notin> (\<lambda>x. x - a) ` s \<Longrightarrow>
lp15@60809
   917
         vector_derivative (shiftpath a g) (at x) = vector_derivative g (at (x + a))"
lp15@60809
   918
      unfolding shiftpath_def
lp15@60809
   919
      apply (rule vector_derivative_at [OF has_vector_derivative_transform_at [of "dist(1-a) x" _ "(\<lambda>x. g(a+x))"]])
lp15@60809
   920
        apply (auto simp: field_simps dist_real_def abs_if split: split_if_asm)
lp15@60809
   921
      apply (rule vector_diff_chain_at [of "\<lambda>x. x+a" 1 _ g, simplified o_def scaleR_one])
lp15@60809
   922
       apply (intro derivative_eq_intros | simp)+
lp15@60809
   923
      using g
lp15@60809
   924
       apply (drule_tac x="x+a" in bspec)
lp15@60809
   925
      using a apply (auto simp: has_vector_derivative_def vector_derivative_works image_def add.commute)
lp15@60809
   926
      done
lp15@60809
   927
  } note vd1 = this
lp15@60809
   928
  { fix x
lp15@60809
   929
    have "1 < x + a \<Longrightarrow> x \<le> 1 \<Longrightarrow> x \<notin> (\<lambda>x. x - a + 1) ` s \<Longrightarrow>
lp15@60809
   930
          vector_derivative (shiftpath a g) (at x) = vector_derivative g (at (x + a - 1))"
lp15@60809
   931
      unfolding shiftpath_def
lp15@60809
   932
      apply (rule vector_derivative_at [OF has_vector_derivative_transform_at [of "dist (1-a) x" _ "(\<lambda>x. g(a+x-1))"]])
lp15@60809
   933
        apply (auto simp: field_simps dist_real_def abs_if split: split_if_asm)
lp15@60809
   934
      apply (rule vector_diff_chain_at [of "\<lambda>x. x+a-1" 1 _ g, simplified o_def scaleR_one])
lp15@60809
   935
       apply (intro derivative_eq_intros | simp)+
lp15@60809
   936
      using g
lp15@60809
   937
      apply (drule_tac x="x+a-1" in bspec)
lp15@60809
   938
      using a apply (auto simp: has_vector_derivative_def vector_derivative_works image_def add.commute)
lp15@60809
   939
      done
lp15@60809
   940
  } note vd2 = this
lp15@60809
   941
  have va1: "(\<lambda>x. f (g x) * vector_derivative g (at x)) integrable_on ({a..1})"
lp15@60809
   942
    using * a   by (fastforce intro: integrable_subinterval_real)
lp15@60809
   943
  have v0a: "(\<lambda>x. f (g x) * vector_derivative g (at x)) integrable_on ({0..a})"
lp15@60809
   944
    apply (rule integrable_subinterval_real)
lp15@60809
   945
    using * a by auto
lp15@60809
   946
  have "((\<lambda>x. f (shiftpath a g x) * vector_derivative (shiftpath a g) (at x))
lp15@60809
   947
        has_integral  integral {a..1} (\<lambda>x. f (g x) * vector_derivative g (at x)))  {0..1 - a}"
lp15@60809
   948
    apply (rule has_integral_spike_finite
lp15@60809
   949
             [where s = "{1-a} \<union> (\<lambda>x. x-a) ` s" and f = "\<lambda>x. f(g(a+x)) * vector_derivative g (at(a+x))"])
lp15@60809
   950
      using s apply blast
lp15@60809
   951
     using a apply (auto simp: algebra_simps vd1)
lp15@60809
   952
     apply (force simp: shiftpath_def add.commute)
lp15@60809
   953
    using has_integral_affinity [where m=1 and c=a, simplified, OF integrable_integral [OF va1]]
lp15@60809
   954
    apply (simp add: image_affinity_atLeastAtMost_diff [where m=1 and c=a, simplified] add.commute)
lp15@60809
   955
    done
lp15@60809
   956
  moreover
lp15@60809
   957
  have "((\<lambda>x. f (shiftpath a g x) * vector_derivative (shiftpath a g) (at x))
lp15@60809
   958
        has_integral  integral {0..a} (\<lambda>x. f (g x) * vector_derivative g (at x)))  {1 - a..1}"
lp15@60809
   959
    apply (rule has_integral_spike_finite
lp15@60809
   960
             [where s = "{1-a} \<union> (\<lambda>x. x-a+1) ` s" and f = "\<lambda>x. f(g(a+x-1)) * vector_derivative g (at(a+x-1))"])
lp15@60809
   961
      using s apply blast
lp15@60809
   962
     using a apply (auto simp: algebra_simps vd2)
lp15@60809
   963
     apply (force simp: shiftpath_def add.commute)
lp15@60809
   964
    using has_integral_affinity [where m=1 and c="a-1", simplified, OF integrable_integral [OF v0a]]
lp15@60809
   965
    apply (simp add: image_affinity_atLeastAtMost [where m=1 and c="1-a", simplified])
lp15@60809
   966
    apply (simp add: algebra_simps)
lp15@60809
   967
    done
lp15@60809
   968
  ultimately show ?thesis
lp15@60809
   969
    using a
lp15@61738
   970
    by (auto simp: i has_contour_integral intro: has_integral_combine [where c = "1-a"])
lp15@60809
   971
qed
lp15@60809
   972
lp15@61738
   973
lemma has_contour_integral_shiftpath_D:
lp15@61738
   974
  assumes "(f has_contour_integral i) (shiftpath a g)"
lp15@60809
   975
          "valid_path g" "pathfinish g = pathstart g" "a \<in> {0..1}"
lp15@61738
   976
    shows "(f has_contour_integral i) g"
lp15@60809
   977
proof -
lp15@60809
   978
  obtain s
lp15@60809
   979
    where s: "finite s" and g: "\<forall>x\<in>{0..1} - s. g differentiable at x"
lp15@61190
   980
    using assms by (auto simp: valid_path_def piecewise_C1_differentiable_on_def C1_differentiable_on_eq)
lp15@60809
   981
  { fix x
lp15@60809
   982
    assume x: "0 < x" "x < 1" "x \<notin> s"
lp15@60809
   983
    then have gx: "g differentiable at x"
lp15@60809
   984
      using g by auto
lp15@60809
   985
    have "vector_derivative g (at x within {0..1}) =
lp15@60809
   986
          vector_derivative (shiftpath (1 - a) (shiftpath a g)) (at x within {0..1})"
lp15@60809
   987
      apply (rule vector_derivative_at_within_ivl
lp15@60809
   988
                  [OF has_vector_derivative_transform_within_open
lp15@60809
   989
                      [of "{0<..<1}-s" _ "(shiftpath (1 - a) (shiftpath a g))"]])
lp15@60809
   990
      using s g assms x
lp15@60809
   991
      apply (auto simp: finite_imp_closed open_Diff shiftpath_shiftpath
lp15@60809
   992
                        vector_derivative_within_interior vector_derivative_works [symmetric])
lp15@60809
   993
      apply (rule Derivative.differentiable_transform_at [of "min x (1-x)", OF _ _ gx])
lp15@60809
   994
      apply (auto simp: dist_real_def shiftpath_shiftpath abs_if)
lp15@60809
   995
      done
lp15@60809
   996
  } note vd = this
lp15@61738
   997
  have fi: "(f has_contour_integral i) (shiftpath (1 - a) (shiftpath a g))"
lp15@61738
   998
    using assms  by (auto intro!: has_contour_integral_shiftpath)
lp15@60809
   999
  show ?thesis
lp15@61738
  1000
    apply (simp add: has_contour_integral_def)
lp15@61738
  1001
    apply (rule has_integral_spike_finite [of "{0,1} \<union> s", OF _ _  fi [unfolded has_contour_integral_def]])
lp15@60809
  1002
    using s assms vd
lp15@60809
  1003
    apply (auto simp: Path_Connected.shiftpath_shiftpath)
lp15@60809
  1004
    done
lp15@60809
  1005
qed
lp15@60809
  1006
lp15@61738
  1007
lemma has_contour_integral_shiftpath_eq:
lp15@60809
  1008
  assumes "valid_path g" "pathfinish g = pathstart g" "a \<in> {0..1}"
lp15@61738
  1009
    shows "(f has_contour_integral i) (shiftpath a g) \<longleftrightarrow> (f has_contour_integral i) g"
lp15@61738
  1010
  using assms has_contour_integral_shiftpath has_contour_integral_shiftpath_D by blast
lp15@61738
  1011
lp15@61738
  1012
lemma contour_integral_shiftpath:
lp15@60809
  1013
  assumes "valid_path g" "pathfinish g = pathstart g" "a \<in> {0..1}"
lp15@61738
  1014
    shows "contour_integral (shiftpath a g) f = contour_integral g f"
lp15@61738
  1015
   using assms by (simp add: contour_integral_def has_contour_integral_shiftpath_eq)
lp15@60809
  1016
lp15@60809
  1017
lp15@60809
  1018
subsection\<open>More about straight-line paths\<close>
lp15@60809
  1019
lp15@60809
  1020
lemma has_vector_derivative_linepath_within:
lp15@60809
  1021
    "(linepath a b has_vector_derivative (b - a)) (at x within s)"
lp15@60809
  1022
apply (simp add: linepath_def has_vector_derivative_def algebra_simps)
lp15@60809
  1023
apply (rule derivative_eq_intros | simp)+
lp15@60809
  1024
done
lp15@60809
  1025
lp15@60809
  1026
lemma vector_derivative_linepath_within:
lp15@60809
  1027
    "x \<in> {0..1} \<Longrightarrow> vector_derivative (linepath a b) (at x within {0..1}) = b - a"
lp15@60809
  1028
  apply (rule vector_derivative_within_closed_interval [of 0 "1::real", simplified])
lp15@60809
  1029
  apply (auto simp: has_vector_derivative_linepath_within)
lp15@60809
  1030
  done
lp15@60809
  1031
lp15@61190
  1032
lemma vector_derivative_linepath_at [simp]: "vector_derivative (linepath a b) (at x) = b - a"
lp15@60809
  1033
  by (simp add: has_vector_derivative_linepath_within vector_derivative_at)
lp15@60809
  1034
lp15@61190
  1035
lemma valid_path_linepath [iff]: "valid_path (linepath a b)"
lp15@61190
  1036
  apply (simp add: valid_path_def piecewise_C1_differentiable_on_def C1_differentiable_on_eq continuous_on_linepath)
lp15@61190
  1037
  apply (rule_tac x="{}" in exI)
lp15@61190
  1038
  apply (simp add: differentiable_on_def differentiable_def)
lp15@61190
  1039
  using has_vector_derivative_def has_vector_derivative_linepath_within
lp15@61190
  1040
  apply (fastforce simp add: continuous_on_eq_continuous_within)
lp15@61190
  1041
  done
lp15@61190
  1042
lp15@61738
  1043
lemma has_contour_integral_linepath:
lp15@61738
  1044
  shows "(f has_contour_integral i) (linepath a b) \<longleftrightarrow>
lp15@60809
  1045
         ((\<lambda>x. f(linepath a b x) * (b - a)) has_integral i) {0..1}"
lp15@61738
  1046
  by (simp add: has_contour_integral vector_derivative_linepath_at)
lp15@60809
  1047
lp15@60809
  1048
lemma linepath_in_path:
lp15@60809
  1049
  shows "x \<in> {0..1} \<Longrightarrow> linepath a b x \<in> closed_segment a b"
lp15@60809
  1050
  by (auto simp: segment linepath_def)
lp15@60809
  1051
lp15@60809
  1052
lemma linepath_image_01: "linepath a b ` {0..1} = closed_segment a b"
lp15@60809
  1053
  by (auto simp: segment linepath_def)
lp15@60809
  1054
lp15@60809
  1055
lemma linepath_in_convex_hull:
lp15@60809
  1056
    fixes x::real
lp15@60809
  1057
    assumes a: "a \<in> convex hull s"
lp15@60809
  1058
        and b: "b \<in> convex hull s"
lp15@60809
  1059
        and x: "0\<le>x" "x\<le>1"
lp15@60809
  1060
       shows "linepath a b x \<in> convex hull s"
lp15@60809
  1061
  apply (rule closed_segment_subset_convex_hull [OF a b, THEN subsetD])
lp15@60809
  1062
  using x
lp15@60809
  1063
  apply (auto simp: linepath_image_01 [symmetric])
lp15@60809
  1064
  done
lp15@60809
  1065
lp15@60809
  1066
lemma Re_linepath: "Re(linepath (of_real a) (of_real b) x) = (1 - x)*a + x*b"
lp15@60809
  1067
  by (simp add: linepath_def)
lp15@60809
  1068
lp15@60809
  1069
lemma Im_linepath: "Im(linepath (of_real a) (of_real b) x) = 0"
lp15@60809
  1070
  by (simp add: linepath_def)
lp15@60809
  1071
lp15@60809
  1072
lemma linepath_of_real: "(linepath (of_real a) (of_real b) x) = of_real ((1 - x)*a + x*b)"
lp15@60809
  1073
  by (simp add: scaleR_conv_of_real linepath_def)
lp15@60809
  1074
lp15@60809
  1075
lemma of_real_linepath: "of_real (linepath a b x) = linepath (of_real a) (of_real b) x"
lp15@60809
  1076
  by (metis linepath_of_real mult.right_neutral of_real_def real_scaleR_def)
lp15@60809
  1077
lp15@61738
  1078
lemma has_contour_integral_trivial [iff]: "(f has_contour_integral 0) (linepath a a)"
lp15@61738
  1079
  by (simp add: has_contour_integral_linepath)
lp15@61738
  1080
lp15@61738
  1081
lemma contour_integral_trivial [simp]: "contour_integral (linepath a a) f = 0"
lp15@61738
  1082
  using has_contour_integral_trivial contour_integral_unique by blast
lp15@60809
  1083
lp15@60809
  1084
lp15@60809
  1085
subsection\<open>Relation to subpath construction\<close>
lp15@60809
  1086
lp15@60809
  1087
lemma valid_path_subpath:
lp15@60809
  1088
  fixes g :: "real \<Rightarrow> 'a :: real_normed_vector"
lp15@60809
  1089
  assumes "valid_path g" "u \<in> {0..1}" "v \<in> {0..1}"
lp15@60809
  1090
    shows "valid_path(subpath u v g)"
lp15@60809
  1091
proof (cases "v=u")
lp15@60809
  1092
  case True
lp15@60809
  1093
  then show ?thesis
lp15@61190
  1094
    unfolding valid_path_def subpath_def
lp15@61190
  1095
    by (force intro: C1_differentiable_on_const C1_differentiable_imp_piecewise)
lp15@60809
  1096
next
lp15@60809
  1097
  case False
lp15@61190
  1098
  have "(g o (\<lambda>x. ((v-u) * x + u))) piecewise_C1_differentiable_on {0..1}"
lp15@61190
  1099
    apply (rule piecewise_C1_differentiable_compose)
lp15@61190
  1100
    apply (simp add: C1_differentiable_imp_piecewise)
lp15@60809
  1101
     apply (simp add: image_affinity_atLeastAtMost)
lp15@60809
  1102
    using assms False
lp15@61190
  1103
    apply (auto simp: algebra_simps valid_path_def piecewise_C1_differentiable_on_subset)
lp15@60809
  1104
    apply (subst Int_commute)
lp15@60809
  1105
    apply (auto simp: inj_on_def algebra_simps crossproduct_eq finite_vimage_IntI)
lp15@60809
  1106
    done
lp15@60809
  1107
  then show ?thesis
lp15@60809
  1108
    by (auto simp: o_def valid_path_def subpath_def)
lp15@60809
  1109
qed
lp15@60809
  1110
lp15@61738
  1111
lemma has_contour_integral_subpath_refl [iff]: "(f has_contour_integral 0) (subpath u u g)"
lp15@61738
  1112
  by (simp add: has_contour_integral subpath_def)
lp15@61738
  1113
lp15@61738
  1114
lemma contour_integrable_subpath_refl [iff]: "f contour_integrable_on (subpath u u g)"
lp15@61738
  1115
  using has_contour_integral_subpath_refl contour_integrable_on_def by blast
lp15@61738
  1116
lp15@61738
  1117
lemma contour_integral_subpath_refl [simp]: "contour_integral (subpath u u g) f = 0"
lp15@61738
  1118
  by (simp add: has_contour_integral_subpath_refl contour_integral_unique)
lp15@61738
  1119
lp15@61738
  1120
lemma has_contour_integral_subpath:
lp15@61738
  1121
  assumes f: "f contour_integrable_on g" and g: "valid_path g"
lp15@60809
  1122
      and uv: "u \<in> {0..1}" "v \<in> {0..1}" "u \<le> v"
lp15@61738
  1123
    shows "(f has_contour_integral  integral {u..v} (\<lambda>x. f(g x) * vector_derivative g (at x)))
lp15@60809
  1124
           (subpath u v g)"
lp15@60809
  1125
proof (cases "v=u")
lp15@60809
  1126
  case True
lp15@60809
  1127
  then show ?thesis
lp15@61738
  1128
    using f   by (simp add: contour_integrable_on_def subpath_def has_contour_integral)
lp15@60809
  1129
next
lp15@60809
  1130
  case False
lp15@60809
  1131
  obtain s where s: "\<And>x. x \<in> {0..1} - s \<Longrightarrow> g differentiable at x" and fs: "finite s"
lp15@61190
  1132
    using g unfolding piecewise_C1_differentiable_on_def C1_differentiable_on_eq valid_path_def by blast
lp15@60809
  1133
  have *: "((\<lambda>x. f (g ((v - u) * x + u)) * vector_derivative g (at ((v - u) * x + u)))
lp15@60809
  1134
            has_integral (1 / (v - u)) * integral {u..v} (\<lambda>t. f (g t) * vector_derivative g (at t)))
lp15@60809
  1135
           {0..1}"
lp15@60809
  1136
    using f uv
lp15@61738
  1137
    apply (simp add: contour_integrable_on subpath_def has_contour_integral)
lp15@60809
  1138
    apply (drule integrable_on_subcbox [where a=u and b=v, simplified])
lp15@60809
  1139
    apply (simp_all add: has_integral_integral)
lp15@60809
  1140
    apply (drule has_integral_affinity [where m="v-u" and c=u, simplified])
lp15@60809
  1141
    apply (simp_all add: False image_affinity_atLeastAtMost_div_diff scaleR_conv_of_real)
lp15@60809
  1142
    apply (simp add: divide_simps False)
lp15@60809
  1143
    done
lp15@60809
  1144
  { fix x
lp15@60809
  1145
    have "x \<in> {0..1} \<Longrightarrow>
lp15@60809
  1146
           x \<notin> (\<lambda>t. (v-u) *\<^sub>R t + u) -` s \<Longrightarrow>
lp15@60809
  1147
           vector_derivative (\<lambda>x. g ((v-u) * x + u)) (at x) = (v-u) *\<^sub>R vector_derivative g (at ((v-u) * x + u))"
lp15@60809
  1148
      apply (rule vector_derivative_at [OF vector_diff_chain_at [simplified o_def]])
lp15@60809
  1149
      apply (intro derivative_eq_intros | simp)+
lp15@60809
  1150
      apply (cut_tac s [of "(v - u) * x + u"])
lp15@60809
  1151
      using uv mult_left_le [of x "v-u"]
lp15@60809
  1152
      apply (auto simp:  vector_derivative_works)
lp15@60809
  1153
      done
lp15@60809
  1154
  } note vd = this
lp15@60809
  1155
  show ?thesis
lp15@60809
  1156
    apply (cut_tac has_integral_cmul [OF *, where c = "v-u"])
lp15@60809
  1157
    using fs assms
lp15@61738
  1158
    apply (simp add: False subpath_def has_contour_integral)
lp15@60809
  1159
    apply (rule_tac s = "(\<lambda>t. ((v-u) *\<^sub>R t + u)) -` s" in has_integral_spike_finite)
lp15@60809
  1160
    apply (auto simp: inj_on_def False finite_vimageI vd scaleR_conv_of_real)
lp15@60809
  1161
    done
lp15@60809
  1162
qed
lp15@60809
  1163
lp15@61738
  1164
lemma contour_integrable_subpath:
lp15@61738
  1165
  assumes "f contour_integrable_on g" "valid_path g" "u \<in> {0..1}" "v \<in> {0..1}"
lp15@61738
  1166
    shows "f contour_integrable_on (subpath u v g)"
lp15@60809
  1167
  apply (cases u v rule: linorder_class.le_cases)
lp15@61738
  1168
   apply (metis contour_integrable_on_def has_contour_integral_subpath [OF assms])
lp15@60809
  1169
  apply (subst reversepath_subpath [symmetric])
lp15@61738
  1170
  apply (rule contour_integrable_reversepath)
lp15@60809
  1171
   using assms apply (blast intro: valid_path_subpath)
lp15@61738
  1172
  apply (simp add: contour_integrable_on_def)
lp15@61738
  1173
  using assms apply (blast intro: has_contour_integral_subpath)
lp15@60809
  1174
  done
lp15@60809
  1175
lp15@60809
  1176
lemma has_integral_integrable_integral: "(f has_integral i) s \<longleftrightarrow> f integrable_on s \<and> integral s f = i"
lp15@60809
  1177
  by blast
lp15@60809
  1178
lp15@61738
  1179
lemma has_integral_contour_integral_subpath:
lp15@61738
  1180
  assumes "f contour_integrable_on g" "valid_path g" "u \<in> {0..1}" "v \<in> {0..1}" "u \<le> v"
lp15@60809
  1181
    shows "(((\<lambda>x. f(g x) * vector_derivative g (at x)))
lp15@61738
  1182
            has_integral  contour_integral (subpath u v g) f) {u..v}"
lp15@60809
  1183
  using assms
lp15@60809
  1184
  apply (auto simp: has_integral_integrable_integral)
lp15@60809
  1185
  apply (rule integrable_on_subcbox [where a=u and b=v and s = "{0..1}", simplified])
lp15@61738
  1186
  apply (auto simp: contour_integral_unique [OF has_contour_integral_subpath] contour_integrable_on)
lp15@60809
  1187
  done
lp15@60809
  1188
lp15@61738
  1189
lemma contour_integral_subcontour_integral:
lp15@61738
  1190
  assumes "f contour_integrable_on g" "valid_path g" "u \<in> {0..1}" "v \<in> {0..1}" "u \<le> v"
lp15@61738
  1191
    shows "contour_integral (subpath u v g) f =
lp15@60809
  1192
           integral {u..v} (\<lambda>x. f(g x) * vector_derivative g (at x))"
lp15@61738
  1193
  using assms has_contour_integral_subpath contour_integral_unique by blast
lp15@61738
  1194
lp15@61738
  1195
lemma contour_integral_subpath_combine_less:
lp15@61738
  1196
  assumes "f contour_integrable_on g" "valid_path g" "u \<in> {0..1}" "v \<in> {0..1}" "w \<in> {0..1}"
lp15@60809
  1197
          "u<v" "v<w"
lp15@61738
  1198
    shows "contour_integral (subpath u v g) f + contour_integral (subpath v w g) f =
lp15@61738
  1199
           contour_integral (subpath u w g) f"
lp15@61738
  1200
  using assms apply (auto simp: contour_integral_subcontour_integral)
lp15@60809
  1201
  apply (rule integral_combine, auto)
lp15@60809
  1202
  apply (rule integrable_on_subcbox [where a=u and b=w and s = "{0..1}", simplified])
lp15@61738
  1203
  apply (auto simp: contour_integrable_on)
lp15@60809
  1204
  done
lp15@60809
  1205
lp15@61738
  1206
lemma contour_integral_subpath_combine:
lp15@61738
  1207
  assumes "f contour_integrable_on g" "valid_path g" "u \<in> {0..1}" "v \<in> {0..1}" "w \<in> {0..1}"
lp15@61738
  1208
    shows "contour_integral (subpath u v g) f + contour_integral (subpath v w g) f =
lp15@61738
  1209
           contour_integral (subpath u w g) f"
lp15@60809
  1210
proof (cases "u\<noteq>v \<and> v\<noteq>w \<and> u\<noteq>w")
lp15@60809
  1211
  case True
lp15@60809
  1212
    have *: "subpath v u g = reversepath(subpath u v g) \<and>
lp15@60809
  1213
             subpath w u g = reversepath(subpath u w g) \<and>
lp15@60809
  1214
             subpath w v g = reversepath(subpath v w g)"
lp15@60809
  1215
      by (auto simp: reversepath_subpath)
lp15@60809
  1216
    have "u < v \<and> v < w \<or>
lp15@60809
  1217
          u < w \<and> w < v \<or>
lp15@60809
  1218
          v < u \<and> u < w \<or>
lp15@60809
  1219
          v < w \<and> w < u \<or>
lp15@60809
  1220
          w < u \<and> u < v \<or>
lp15@60809
  1221
          w < v \<and> v < u"
lp15@60809
  1222
      using True assms by linarith
lp15@60809
  1223
    with assms show ?thesis
lp15@61738
  1224
      using contour_integral_subpath_combine_less [of f g u v w]
lp15@61738
  1225
            contour_integral_subpath_combine_less [of f g u w v]
lp15@61738
  1226
            contour_integral_subpath_combine_less [of f g v u w]
lp15@61738
  1227
            contour_integral_subpath_combine_less [of f g v w u]
lp15@61738
  1228
            contour_integral_subpath_combine_less [of f g w u v]
lp15@61738
  1229
            contour_integral_subpath_combine_less [of f g w v u]
lp15@60809
  1230
      apply simp
lp15@60809
  1231
      apply (elim disjE)
lp15@61738
  1232
      apply (auto simp: * contour_integral_reversepath contour_integrable_subpath
lp15@60809
  1233
                   valid_path_reversepath valid_path_subpath algebra_simps)
lp15@60809
  1234
      done
lp15@60809
  1235
next
lp15@60809
  1236
  case False
lp15@60809
  1237
  then show ?thesis
lp15@61738
  1238
    apply (auto simp: contour_integral_subpath_refl)
lp15@60809
  1239
    using assms
lp15@61738
  1240
    by (metis eq_neg_iff_add_eq_0 contour_integrable_subpath contour_integral_reversepath reversepath_subpath valid_path_subpath)
lp15@60809
  1241
qed
lp15@60809
  1242
lp15@61738
  1243
lemma contour_integral_integral:
lp15@61738
  1244
  shows "contour_integral g f = integral {0..1} (\<lambda>x. f (g x) * vector_derivative g (at x))"
lp15@61738
  1245
  by (simp add: contour_integral_def integral_def has_contour_integral)
lp15@60809
  1246
lp15@60809
  1247
lp15@60809
  1248
subsection\<open>Segments via convex hulls\<close>
lp15@60809
  1249
lp15@60809
  1250
lemma segments_subset_convex_hull:
lp15@60809
  1251
    "closed_segment a b \<subseteq> (convex hull {a,b,c})"
lp15@60809
  1252
    "closed_segment a c \<subseteq> (convex hull {a,b,c})"
lp15@60809
  1253
    "closed_segment b c \<subseteq> (convex hull {a,b,c})"
lp15@60809
  1254
    "closed_segment b a \<subseteq> (convex hull {a,b,c})"
lp15@60809
  1255
    "closed_segment c a \<subseteq> (convex hull {a,b,c})"
lp15@60809
  1256
    "closed_segment c b \<subseteq> (convex hull {a,b,c})"
lp15@60809
  1257
by (auto simp: segment_convex_hull linepath_of_real  elim!: rev_subsetD [OF _ hull_mono])
lp15@60809
  1258
lp15@60809
  1259
lemma midpoints_in_convex_hull:
lp15@60809
  1260
  assumes "x \<in> convex hull s" "y \<in> convex hull s"
lp15@60809
  1261
    shows "midpoint x y \<in> convex hull s"
lp15@60809
  1262
proof -
lp15@60809
  1263
  have "(1 - inverse(2)) *\<^sub>R x + inverse(2) *\<^sub>R y \<in> convex hull s"
lp15@61426
  1264
    apply (rule convexD_alt)
lp15@60809
  1265
    using assms
lp15@60809
  1266
    apply (auto simp: convex_convex_hull)
lp15@60809
  1267
    done
lp15@60809
  1268
  then show ?thesis
lp15@60809
  1269
    by (simp add: midpoint_def algebra_simps)
lp15@60809
  1270
qed
lp15@60809
  1271
lp15@60809
  1272
lemma convex_hull_subset:
lp15@60809
  1273
    "s \<subseteq> convex hull t \<Longrightarrow> convex hull s \<subseteq> convex hull t"
lp15@60809
  1274
  by (simp add: convex_convex_hull subset_hull)
lp15@60809
  1275
lp15@60809
  1276
lemma not_in_interior_convex_hull_3:
lp15@60809
  1277
  fixes a :: "complex"
lp15@60809
  1278
  shows "a \<notin> interior(convex hull {a,b,c})"
lp15@60809
  1279
        "b \<notin> interior(convex hull {a,b,c})"
lp15@60809
  1280
        "c \<notin> interior(convex hull {a,b,c})"
lp15@60809
  1281
  by (auto simp: card_insert_le_m1 not_in_interior_convex_hull)
lp15@60809
  1282
lp15@60809
  1283
lp15@60809
  1284
text\<open>Cauchy's theorem where there's a primitive\<close>
lp15@60809
  1285
lp15@61738
  1286
lemma contour_integral_primitive_lemma:
lp15@60809
  1287
  fixes f :: "complex \<Rightarrow> complex" and g :: "real \<Rightarrow> complex"
lp15@60809
  1288
  assumes "a \<le> b"
lp15@60809
  1289
      and "\<And>x. x \<in> s \<Longrightarrow> (f has_field_derivative f' x) (at x within s)"
lp15@60809
  1290
      and "g piecewise_differentiable_on {a..b}"  "\<And>x. x \<in> {a..b} \<Longrightarrow> g x \<in> s"
lp15@60809
  1291
    shows "((\<lambda>x. f'(g x) * vector_derivative g (at x within {a..b}))
lp15@60809
  1292
             has_integral (f(g b) - f(g a))) {a..b}"
lp15@60809
  1293
proof -
lp15@61190
  1294
  obtain k where k: "finite k" "\<forall>x\<in>{a..b} - k. g differentiable (at x within {a..b})" and cg: "continuous_on {a..b} g"
lp15@60809
  1295
    using assms by (auto simp: piecewise_differentiable_on_def)
lp15@60809
  1296
  have cfg: "continuous_on {a..b} (\<lambda>x. f (g x))"
lp15@60809
  1297
    apply (rule continuous_on_compose [OF cg, unfolded o_def])
lp15@60809
  1298
    using assms
lp15@60809
  1299
    apply (metis complex_differentiable_def complex_differentiable_imp_continuous_at continuous_on_eq_continuous_within continuous_on_subset image_subset_iff)
lp15@60809
  1300
    done
lp15@60809
  1301
  { fix x::real
lp15@60809
  1302
    assume a: "a < x" and b: "x < b" and xk: "x \<notin> k"
lp15@60809
  1303
    then have "g differentiable at x within {a..b}"
lp15@60809
  1304
      using k by (simp add: differentiable_at_withinI)
lp15@60809
  1305
    then have "(g has_vector_derivative vector_derivative g (at x within {a..b})) (at x within {a..b})"
lp15@60809
  1306
      by (simp add: vector_derivative_works has_field_derivative_def scaleR_conv_of_real)
lp15@60809
  1307
    then have gdiff: "(g has_derivative (\<lambda>u. u * vector_derivative g (at x within {a..b}))) (at x within {a..b})"
lp15@60809
  1308
      by (simp add: has_vector_derivative_def scaleR_conv_of_real)
lp15@60809
  1309
    have "(f has_field_derivative (f' (g x))) (at (g x) within g ` {a..b})"
lp15@60809
  1310
      using assms by (metis a atLeastAtMost_iff b DERIV_subset image_subset_iff less_eq_real_def)
lp15@60809
  1311
    then have fdiff: "(f has_derivative op * (f' (g x))) (at (g x) within g ` {a..b})"
lp15@60809
  1312
      by (simp add: has_field_derivative_def)
lp15@60809
  1313
    have "((\<lambda>x. f (g x)) has_vector_derivative f' (g x) * vector_derivative g (at x within {a..b})) (at x within {a..b})"
lp15@60809
  1314
      using diff_chain_within [OF gdiff fdiff]
lp15@60809
  1315
      by (simp add: has_vector_derivative_def scaleR_conv_of_real o_def mult_ac)
lp15@60809
  1316
  } note * = this
lp15@60809
  1317
  show ?thesis
lp15@60809
  1318
    apply (rule fundamental_theorem_of_calculus_interior_strong)
lp15@60809
  1319
    using k assms cfg *
lp15@60809
  1320
    apply (auto simp: at_within_closed_interval)
lp15@60809
  1321
    done
lp15@60809
  1322
qed
lp15@60809
  1323
lp15@61738
  1324
lemma contour_integral_primitive:
lp15@60809
  1325
  assumes "\<And>x. x \<in> s \<Longrightarrow> (f has_field_derivative f' x) (at x within s)"
lp15@60809
  1326
      and "valid_path g" "path_image g \<subseteq> s"
lp15@61738
  1327
    shows "(f' has_contour_integral (f(pathfinish g) - f(pathstart g))) g"
lp15@60809
  1328
  using assms
lp15@61738
  1329
  apply (simp add: valid_path_def path_image_def pathfinish_def pathstart_def has_contour_integral_def)
lp15@61738
  1330
  apply (auto intro!: piecewise_C1_imp_differentiable contour_integral_primitive_lemma [of 0 1 s])
lp15@60809
  1331
  done
lp15@60809
  1332
lp15@60809
  1333
corollary Cauchy_theorem_primitive:
lp15@60809
  1334
  assumes "\<And>x. x \<in> s \<Longrightarrow> (f has_field_derivative f' x) (at x within s)"
lp15@60809
  1335
      and "valid_path g"  "path_image g \<subseteq> s" "pathfinish g = pathstart g"
lp15@61738
  1336
    shows "(f' has_contour_integral 0) g"
lp15@60809
  1337
  using assms
lp15@61738
  1338
  by (metis diff_self contour_integral_primitive)
lp15@60809
  1339
lp15@60809
  1340
lp15@60809
  1341
text\<open>Existence of path integral for continuous function\<close>
lp15@61738
  1342
lemma contour_integrable_continuous_linepath:
lp15@60809
  1343
  assumes "continuous_on (closed_segment a b) f"
lp15@61738
  1344
  shows "f contour_integrable_on (linepath a b)"
lp15@60809
  1345
proof -
lp15@60809
  1346
  have "continuous_on {0..1} ((\<lambda>x. f x * (b - a)) o linepath a b)"
lp15@60809
  1347
    apply (rule continuous_on_compose [OF continuous_on_linepath], simp add: linepath_image_01)
lp15@60809
  1348
    apply (rule continuous_intros | simp add: assms)+
lp15@60809
  1349
    done
lp15@60809
  1350
  then show ?thesis
lp15@61738
  1351
    apply (simp add: contour_integrable_on_def has_contour_integral_def integrable_on_def [symmetric])
lp15@60809
  1352
    apply (rule integrable_continuous [of 0 "1::real", simplified])
lp15@60809
  1353
    apply (rule continuous_on_eq [where f = "\<lambda>x. f(linepath a b x)*(b - a)"])
lp15@60809
  1354
    apply (auto simp: vector_derivative_linepath_within)
lp15@60809
  1355
    done
lp15@60809
  1356
qed
lp15@60809
  1357
lp15@60809
  1358
lemma has_field_der_id: "((\<lambda>x. x\<^sup>2 / 2) has_field_derivative x) (at x)"
lp15@60809
  1359
  by (rule has_derivative_imp_has_field_derivative)
lp15@60809
  1360
     (rule derivative_intros | simp)+
lp15@60809
  1361
lp15@61738
  1362
lemma contour_integral_id [simp]: "contour_integral (linepath a b) (\<lambda>y. y) = (b^2 - a^2)/2"
lp15@61738
  1363
  apply (rule contour_integral_unique)
lp15@61738
  1364
  using contour_integral_primitive [of UNIV "\<lambda>x. x^2/2" "\<lambda>x. x" "linepath a b"]
lp15@60809
  1365
  apply (auto simp: field_simps has_field_der_id)
lp15@60809
  1366
  done
lp15@60809
  1367
lp15@61738
  1368
lemma contour_integrable_on_const [iff]: "(\<lambda>x. c) contour_integrable_on (linepath a b)"
lp15@61738
  1369
  by (simp add: continuous_on_const contour_integrable_continuous_linepath)
lp15@61738
  1370
lp15@61738
  1371
lemma contour_integrable_on_id [iff]: "(\<lambda>x. x) contour_integrable_on (linepath a b)"
lp15@61738
  1372
  by (simp add: continuous_on_id contour_integrable_continuous_linepath)
lp15@60809
  1373
lp15@60809
  1374
lp15@60809
  1375
subsection\<open>Arithmetical combining theorems\<close>
lp15@60809
  1376
lp15@61738
  1377
lemma has_contour_integral_neg:
lp15@61738
  1378
    "(f has_contour_integral i) g \<Longrightarrow> ((\<lambda>x. -(f x)) has_contour_integral (-i)) g"
lp15@61738
  1379
  by (simp add: has_integral_neg has_contour_integral_def)
lp15@61738
  1380
lp15@61738
  1381
lemma has_contour_integral_add:
lp15@61738
  1382
    "\<lbrakk>(f1 has_contour_integral i1) g; (f2 has_contour_integral i2) g\<rbrakk>
lp15@61738
  1383
     \<Longrightarrow> ((\<lambda>x. f1 x + f2 x) has_contour_integral (i1 + i2)) g"
lp15@61738
  1384
  by (simp add: has_integral_add has_contour_integral_def algebra_simps)
lp15@61738
  1385
lp15@61738
  1386
lemma has_contour_integral_diff:
lp15@61738
  1387
  "\<lbrakk>(f1 has_contour_integral i1) g; (f2 has_contour_integral i2) g\<rbrakk>
lp15@61738
  1388
         \<Longrightarrow> ((\<lambda>x. f1 x - f2 x) has_contour_integral (i1 - i2)) g"
lp15@61738
  1389
  by (simp add: has_integral_sub has_contour_integral_def algebra_simps)
lp15@61738
  1390
lp15@61738
  1391
lemma has_contour_integral_lmul:
lp15@61738
  1392
  "(f has_contour_integral i) g \<Longrightarrow> ((\<lambda>x. c * (f x)) has_contour_integral (c*i)) g"
lp15@61738
  1393
apply (simp add: has_contour_integral_def)
lp15@60809
  1394
apply (drule has_integral_mult_right)
lp15@60809
  1395
apply (simp add: algebra_simps)
lp15@60809
  1396
done
lp15@60809
  1397
lp15@61738
  1398
lemma has_contour_integral_rmul:
lp15@61738
  1399
  "(f has_contour_integral i) g \<Longrightarrow> ((\<lambda>x. (f x) * c) has_contour_integral (i*c)) g"
lp15@61738
  1400
apply (drule has_contour_integral_lmul)
lp15@60809
  1401
apply (simp add: mult.commute)
lp15@60809
  1402
done
lp15@60809
  1403
lp15@61738
  1404
lemma has_contour_integral_div:
lp15@61738
  1405
  "(f has_contour_integral i) g \<Longrightarrow> ((\<lambda>x. f x/c) has_contour_integral (i/c)) g"
lp15@61738
  1406
  by (simp add: field_class.field_divide_inverse) (metis has_contour_integral_rmul)
lp15@61738
  1407
lp15@61738
  1408
lemma has_contour_integral_eq:
lp15@61738
  1409
    "\<lbrakk>(f has_contour_integral y) p; \<And>x. x \<in> path_image p \<Longrightarrow> f x = g x\<rbrakk> \<Longrightarrow> (g has_contour_integral y) p"
lp15@61738
  1410
apply (simp add: path_image_def has_contour_integral_def)
lp15@60809
  1411
by (metis (no_types, lifting) image_eqI has_integral_eq)
lp15@60809
  1412
lp15@61738
  1413
lemma has_contour_integral_bound_linepath:
lp15@61738
  1414
  assumes "(f has_contour_integral i) (linepath a b)"
lp15@60809
  1415
          "0 \<le> B" "\<And>x. x \<in> closed_segment a b \<Longrightarrow> norm(f x) \<le> B"
lp15@60809
  1416
    shows "norm i \<le> B * norm(b - a)"
lp15@60809
  1417
proof -
lp15@60809
  1418
  { fix x::real
lp15@60809
  1419
    assume x: "0 \<le> x" "x \<le> 1"
lp15@60809
  1420
  have "norm (f (linepath a b x)) *
lp15@60809
  1421
        norm (vector_derivative (linepath a b) (at x within {0..1})) \<le> B * norm (b - a)"
lp15@60809
  1422
    by (auto intro: mult_mono simp: assms linepath_in_path of_real_linepath vector_derivative_linepath_within x)
lp15@60809
  1423
  } note * = this
lp15@60809
  1424
  have "norm i \<le> (B * norm (b - a)) * content (cbox 0 (1::real))"
lp15@60809
  1425
    apply (rule has_integral_bound
lp15@60809
  1426
       [of _ "\<lambda>x. f (linepath a b x) * vector_derivative (linepath a b) (at x within {0..1})"])
lp15@61738
  1427
    using assms * unfolding has_contour_integral_def
lp15@60809
  1428
    apply (auto simp: norm_mult)
lp15@60809
  1429
    done
lp15@60809
  1430
  then show ?thesis
lp15@60809
  1431
    by (auto simp: content_real)
lp15@60809
  1432
qed
lp15@60809
  1433
lp15@60809
  1434
(*UNUSED
lp15@61738
  1435
lemma has_contour_integral_bound_linepath_strong:
lp15@60809
  1436
  fixes a :: real and f :: "complex \<Rightarrow> real"
lp15@61738
  1437
  assumes "(f has_contour_integral i) (linepath a b)"
lp15@60809
  1438
          "finite k"
lp15@60809
  1439
          "0 \<le> B" "\<And>x::real. x \<in> closed_segment a b - k \<Longrightarrow> norm(f x) \<le> B"
lp15@60809
  1440
    shows "norm i \<le> B*norm(b - a)"
lp15@60809
  1441
*)
lp15@60809
  1442
lp15@61738
  1443
lemma has_contour_integral_const_linepath: "((\<lambda>x. c) has_contour_integral c*(b - a))(linepath a b)"
lp15@61738
  1444
  unfolding has_contour_integral_linepath
lp15@60809
  1445
  by (metis content_real diff_0_right has_integral_const_real lambda_one of_real_1 scaleR_conv_of_real zero_le_one)
lp15@60809
  1446
lp15@61738
  1447
lemma has_contour_integral_0: "((\<lambda>x. 0) has_contour_integral 0) g"
lp15@61738
  1448
  by (simp add: has_contour_integral_def)
lp15@61738
  1449
lp15@61738
  1450
lemma has_contour_integral_is_0:
lp15@61738
  1451
    "(\<And>z. z \<in> path_image g \<Longrightarrow> f z = 0) \<Longrightarrow> (f has_contour_integral 0) g"
lp15@61738
  1452
  by (rule has_contour_integral_eq [OF has_contour_integral_0]) auto
lp15@61738
  1453
lp15@61738
  1454
lemma has_contour_integral_setsum:
lp15@61738
  1455
    "\<lbrakk>finite s; \<And>a. a \<in> s \<Longrightarrow> (f a has_contour_integral i a) p\<rbrakk>
lp15@61738
  1456
     \<Longrightarrow> ((\<lambda>x. setsum (\<lambda>a. f a x) s) has_contour_integral setsum i s) p"
lp15@61738
  1457
  by (induction s rule: finite_induct) (auto simp: has_contour_integral_0 has_contour_integral_add)
lp15@60809
  1458
lp15@60809
  1459
lp15@60809
  1460
subsection \<open>Operations on path integrals\<close>
lp15@60809
  1461
lp15@61738
  1462
lemma contour_integral_const_linepath [simp]: "contour_integral (linepath a b) (\<lambda>x. c) = c*(b - a)"
lp15@61738
  1463
  by (rule contour_integral_unique [OF has_contour_integral_const_linepath])
lp15@61738
  1464
lp15@61738
  1465
lemma contour_integral_neg:
lp15@61738
  1466
    "f contour_integrable_on g \<Longrightarrow> contour_integral g (\<lambda>x. -(f x)) = -(contour_integral g f)"
lp15@61738
  1467
  by (simp add: contour_integral_unique has_contour_integral_integral has_contour_integral_neg)
lp15@61738
  1468
lp15@61738
  1469
lemma contour_integral_add:
lp15@61738
  1470
    "f1 contour_integrable_on g \<Longrightarrow> f2 contour_integrable_on g \<Longrightarrow> contour_integral g (\<lambda>x. f1 x + f2 x) =
lp15@61738
  1471
                contour_integral g f1 + contour_integral g f2"
lp15@61738
  1472
  by (simp add: contour_integral_unique has_contour_integral_integral has_contour_integral_add)
lp15@61738
  1473
lp15@61738
  1474
lemma contour_integral_diff:
lp15@61738
  1475
    "f1 contour_integrable_on g \<Longrightarrow> f2 contour_integrable_on g \<Longrightarrow> contour_integral g (\<lambda>x. f1 x - f2 x) =
lp15@61738
  1476
                contour_integral g f1 - contour_integral g f2"
lp15@61738
  1477
  by (simp add: contour_integral_unique has_contour_integral_integral has_contour_integral_diff)
lp15@61738
  1478
lp15@61738
  1479
lemma contour_integral_lmul:
lp15@61738
  1480
  shows "f contour_integrable_on g
lp15@61738
  1481
           \<Longrightarrow> contour_integral g (\<lambda>x. c * f x) = c*contour_integral g f"
lp15@61738
  1482
  by (simp add: contour_integral_unique has_contour_integral_integral has_contour_integral_lmul)
lp15@61738
  1483
lp15@61738
  1484
lemma contour_integral_rmul:
lp15@61738
  1485
  shows "f contour_integrable_on g
lp15@61738
  1486
        \<Longrightarrow> contour_integral g (\<lambda>x. f x * c) = contour_integral g f * c"
lp15@61738
  1487
  by (simp add: contour_integral_unique has_contour_integral_integral has_contour_integral_rmul)
lp15@61738
  1488
lp15@61738
  1489
lemma contour_integral_div:
lp15@61738
  1490
  shows "f contour_integrable_on g
lp15@61738
  1491
        \<Longrightarrow> contour_integral g (\<lambda>x. f x / c) = contour_integral g f / c"
lp15@61738
  1492
  by (simp add: contour_integral_unique has_contour_integral_integral has_contour_integral_div)
lp15@61738
  1493
lp15@61738
  1494
lemma contour_integral_eq:
lp15@61738
  1495
    "(\<And>x. x \<in> path_image p \<Longrightarrow> f x = g x) \<Longrightarrow> contour_integral p f = contour_integral p g"
lp15@61738
  1496
  by (simp add: contour_integral_def) (metis has_contour_integral_eq)
lp15@61738
  1497
lp15@61738
  1498
lemma contour_integral_eq_0:
lp15@61738
  1499
    "(\<And>z. z \<in> path_image g \<Longrightarrow> f z = 0) \<Longrightarrow> contour_integral g f = 0"
lp15@61738
  1500
  by (simp add: has_contour_integral_is_0 contour_integral_unique)
lp15@61738
  1501
lp15@61738
  1502
lemma contour_integral_bound_linepath:
lp15@60809
  1503
  shows
lp15@61738
  1504
    "\<lbrakk>f contour_integrable_on (linepath a b);
lp15@60809
  1505
      0 \<le> B; \<And>x. x \<in> closed_segment a b \<Longrightarrow> norm(f x) \<le> B\<rbrakk>
lp15@61738
  1506
     \<Longrightarrow> norm(contour_integral (linepath a b) f) \<le> B*norm(b - a)"
lp15@61738
  1507
  apply (rule has_contour_integral_bound_linepath [of f])
lp15@61738
  1508
  apply (auto simp: has_contour_integral_integral)
lp15@60809
  1509
  done
lp15@60809
  1510
lp15@61738
  1511
lemma contour_integral_0: "contour_integral g (\<lambda>x. 0) = 0"
lp15@61738
  1512
  by (simp add: contour_integral_unique has_contour_integral_0)
lp15@61738
  1513
lp15@61738
  1514
lemma contour_integral_setsum:
lp15@61738
  1515
    "\<lbrakk>finite s; \<And>a. a \<in> s \<Longrightarrow> (f a) contour_integrable_on p\<rbrakk>
lp15@61738
  1516
     \<Longrightarrow> contour_integral p (\<lambda>x. setsum (\<lambda>a. f a x) s) = setsum (\<lambda>a. contour_integral p (f a)) s"
lp15@61738
  1517
  by (auto simp: contour_integral_unique has_contour_integral_setsum has_contour_integral_integral)
lp15@61738
  1518
lp15@61738
  1519
lemma contour_integrable_eq:
lp15@61738
  1520
    "\<lbrakk>f contour_integrable_on p; \<And>x. x \<in> path_image p \<Longrightarrow> f x = g x\<rbrakk> \<Longrightarrow> g contour_integrable_on p"
lp15@61738
  1521
  unfolding contour_integrable_on_def
lp15@61738
  1522
  by (metis has_contour_integral_eq)
lp15@60809
  1523
lp15@60809
  1524
lp15@60809
  1525
subsection \<open>Arithmetic theorems for path integrability\<close>
lp15@60809
  1526
lp15@61738
  1527
lemma contour_integrable_neg:
lp15@61738
  1528
    "f contour_integrable_on g \<Longrightarrow> (\<lambda>x. -(f x)) contour_integrable_on g"
lp15@61738
  1529
  using has_contour_integral_neg contour_integrable_on_def by blast
lp15@61738
  1530
lp15@61738
  1531
lemma contour_integrable_add:
lp15@61738
  1532
    "\<lbrakk>f1 contour_integrable_on g; f2 contour_integrable_on g\<rbrakk> \<Longrightarrow> (\<lambda>x. f1 x + f2 x) contour_integrable_on g"
lp15@61738
  1533
  using has_contour_integral_add contour_integrable_on_def
lp15@60809
  1534
  by fastforce
lp15@60809
  1535
lp15@61738
  1536
lemma contour_integrable_diff:
lp15@61738
  1537
    "\<lbrakk>f1 contour_integrable_on g; f2 contour_integrable_on g\<rbrakk> \<Longrightarrow> (\<lambda>x. f1 x - f2 x) contour_integrable_on g"
lp15@61738
  1538
  using has_contour_integral_diff contour_integrable_on_def
lp15@60809
  1539
  by fastforce
lp15@60809
  1540
lp15@61738
  1541
lemma contour_integrable_lmul:
lp15@61738
  1542
    "f contour_integrable_on g \<Longrightarrow> (\<lambda>x. c * f x) contour_integrable_on g"
lp15@61738
  1543
  using has_contour_integral_lmul contour_integrable_on_def
lp15@60809
  1544
  by fastforce
lp15@60809
  1545
lp15@61738
  1546
lemma contour_integrable_rmul:
lp15@61738
  1547
    "f contour_integrable_on g \<Longrightarrow> (\<lambda>x. f x * c) contour_integrable_on g"
lp15@61738
  1548
  using has_contour_integral_rmul contour_integrable_on_def
lp15@60809
  1549
  by fastforce
lp15@60809
  1550
lp15@61738
  1551
lemma contour_integrable_div:
lp15@61738
  1552
    "f contour_integrable_on g \<Longrightarrow> (\<lambda>x. f x / c) contour_integrable_on g"
lp15@61738
  1553
  using has_contour_integral_div contour_integrable_on_def
lp15@60809
  1554
  by fastforce
lp15@60809
  1555
lp15@61738
  1556
lemma contour_integrable_setsum:
lp15@61738
  1557
    "\<lbrakk>finite s; \<And>a. a \<in> s \<Longrightarrow> (f a) contour_integrable_on p\<rbrakk>
lp15@61738
  1558
     \<Longrightarrow> (\<lambda>x. setsum (\<lambda>a. f a x) s) contour_integrable_on p"
lp15@61738
  1559
   unfolding contour_integrable_on_def
lp15@61738
  1560
   by (metis has_contour_integral_setsum)
lp15@60809
  1561
lp15@60809
  1562
lp15@60809
  1563
subsection\<open>Reversing a path integral\<close>
lp15@60809
  1564
lp15@61738
  1565
lemma has_contour_integral_reverse_linepath:
lp15@61738
  1566
    "(f has_contour_integral i) (linepath a b)
lp15@61738
  1567
     \<Longrightarrow> (f has_contour_integral (-i)) (linepath b a)"
lp15@61738
  1568
  using has_contour_integral_reversepath valid_path_linepath by fastforce
lp15@61738
  1569
lp15@61738
  1570
lemma contour_integral_reverse_linepath:
lp15@60809
  1571
    "continuous_on (closed_segment a b) f
lp15@61738
  1572
     \<Longrightarrow> contour_integral (linepath a b) f = - (contour_integral(linepath b a) f)"
lp15@61738
  1573
apply (rule contour_integral_unique)
lp15@61738
  1574
apply (rule has_contour_integral_reverse_linepath)
lp15@61738
  1575
by (simp add: closed_segment_commute contour_integrable_continuous_linepath has_contour_integral_integral)
lp15@60809
  1576
lp15@60809
  1577
lp15@60809
  1578
(* Splitting a path integral in a flat way.*)
lp15@60809
  1579
lp15@61738
  1580
lemma has_contour_integral_split:
lp15@61738
  1581
  assumes f: "(f has_contour_integral i) (linepath a c)" "(f has_contour_integral j) (linepath c b)"
lp15@60809
  1582
      and k: "0 \<le> k" "k \<le> 1"
lp15@60809
  1583
      and c: "c - a = k *\<^sub>R (b - a)"
lp15@61738
  1584
    shows "(f has_contour_integral (i + j)) (linepath a b)"
lp15@60809
  1585
proof (cases "k = 0 \<or> k = 1")
lp15@60809
  1586
  case True
lp15@60809
  1587
  then show ?thesis
lp15@60809
  1588
    using assms
lp15@60809
  1589
    apply auto
lp15@61738
  1590
    apply (metis add.left_neutral has_contour_integral_trivial has_contour_integral_unique)
lp15@61738
  1591
    apply (metis add.right_neutral has_contour_integral_trivial has_contour_integral_unique)
lp15@60809
  1592
    done
lp15@60809
  1593
next
lp15@60809
  1594
  case False
lp15@60809
  1595
  then have k: "0 < k" "k < 1" "complex_of_real k \<noteq> 1"
lp15@60809
  1596
    using assms apply auto
lp15@60809
  1597
    using of_real_eq_iff by fastforce
lp15@60809
  1598
  have c': "c = k *\<^sub>R (b - a) + a"
lp15@60809
  1599
    by (metis diff_add_cancel c)
lp15@60809
  1600
  have bc: "(b - c) = (1 - k) *\<^sub>R (b - a)"
lp15@60809
  1601
    by (simp add: algebra_simps c')
lp15@60809
  1602
  { assume *: "((\<lambda>x. f ((1 - x) *\<^sub>R a + x *\<^sub>R c) * (c - a)) has_integral i) {0..1}"
lp15@60809
  1603
    have **: "\<And>x. ((k - x) / k) *\<^sub>R a + (x / k) *\<^sub>R c = (1 - x) *\<^sub>R a + x *\<^sub>R b"
lp15@60809
  1604
      using False
lp15@60809
  1605
      apply (simp add: c' algebra_simps)
lp15@60809
  1606
      apply (simp add: real_vector.scale_left_distrib [symmetric] divide_simps)
lp15@60809
  1607
      done
lp15@60809
  1608
    have "((\<lambda>x. f ((1 - x) *\<^sub>R a + x *\<^sub>R b) * (b - a)) has_integral i) {0..k}"
lp15@60809
  1609
      using * k
lp15@60809
  1610
      apply -
lp15@60809
  1611
      apply (drule has_integral_affinity [of _ _ 0 "1::real" "inverse k" "0", simplified])
lp15@60809
  1612
      apply (simp_all add: divide_simps mult.commute [of _ "k"] image_affinity_atLeastAtMost ** c)
lp15@60809
  1613
      apply (drule Integration.has_integral_cmul [where c = "inverse k"])
lp15@60809
  1614
      apply (simp add: Integration.has_integral_cmul)
lp15@60809
  1615
      done
lp15@60809
  1616
  } note fi = this
lp15@60809
  1617
  { assume *: "((\<lambda>x. f ((1 - x) *\<^sub>R c + x *\<^sub>R b) * (b - c)) has_integral j) {0..1}"
lp15@60809
  1618
    have **: "\<And>x. (((1 - x) / (1 - k)) *\<^sub>R c + ((x - k) / (1 - k)) *\<^sub>R b) = ((1 - x) *\<^sub>R a + x *\<^sub>R b)"
lp15@60809
  1619
      using k
lp15@60809
  1620
      apply (simp add: c' field_simps)
lp15@60809
  1621
      apply (simp add: scaleR_conv_of_real divide_simps)
lp15@60809
  1622
      apply (simp add: field_simps)
lp15@60809
  1623
      done
lp15@60809
  1624
    have "((\<lambda>x. f ((1 - x) *\<^sub>R a + x *\<^sub>R b) * (b - a)) has_integral j) {k..1}"
lp15@60809
  1625
      using * k
lp15@60809
  1626
      apply -
lp15@60809
  1627
      apply (drule has_integral_affinity [of _ _ 0 "1::real" "inverse(1 - k)" "-(k/(1 - k))", simplified])
lp15@60809
  1628
      apply (simp_all add: divide_simps mult.commute [of _ "1-k"] image_affinity_atLeastAtMost ** bc)
lp15@60809
  1629
      apply (drule Integration.has_integral_cmul [where k = "(1 - k) *\<^sub>R j" and c = "inverse (1 - k)"])
lp15@60809
  1630
      apply (simp add: Integration.has_integral_cmul)
lp15@60809
  1631
      done
lp15@60809
  1632
  } note fj = this
lp15@60809
  1633
  show ?thesis
lp15@60809
  1634
    using f k
lp15@61738
  1635
    apply (simp add: has_contour_integral_linepath)
lp15@60809
  1636
    apply (simp add: linepath_def)
lp15@60809
  1637
    apply (rule has_integral_combine [OF _ _ fi fj], simp_all)
lp15@60809
  1638
    done
lp15@60809
  1639
qed
lp15@60809
  1640
lp15@60809
  1641
lemma continuous_on_closed_segment_transform:
lp15@60809
  1642
  assumes f: "continuous_on (closed_segment a b) f"
lp15@60809
  1643
      and k: "0 \<le> k" "k \<le> 1"
lp15@60809
  1644
      and c: "c - a = k *\<^sub>R (b - a)"
lp15@60809
  1645
    shows "continuous_on (closed_segment a c) f"
lp15@60809
  1646
proof -
lp15@60809
  1647
  have c': "c = (1 - k) *\<^sub>R a + k *\<^sub>R b"
lp15@60809
  1648
    using c by (simp add: algebra_simps)
lp15@60809
  1649
  show "continuous_on (closed_segment a c) f"
lp15@60809
  1650
    apply (rule continuous_on_subset [OF f])
lp15@60809
  1651
    apply (simp add: segment_convex_hull)
lp15@60809
  1652
    apply (rule convex_hull_subset)
lp15@60809
  1653
    using assms
lp15@61426
  1654
    apply (auto simp: hull_inc c' Convex.convexD_alt)
lp15@60809
  1655
    done
lp15@60809
  1656
qed
lp15@60809
  1657
lp15@61738
  1658
lemma contour_integral_split:
lp15@60809
  1659
  assumes f: "continuous_on (closed_segment a b) f"
lp15@60809
  1660
      and k: "0 \<le> k" "k \<le> 1"
lp15@60809
  1661
      and c: "c - a = k *\<^sub>R (b - a)"
lp15@61738
  1662
    shows "contour_integral(linepath a b) f = contour_integral(linepath a c) f + contour_integral(linepath c b) f"
lp15@60809
  1663
proof -
lp15@60809
  1664
  have c': "c = (1 - k) *\<^sub>R a + k *\<^sub>R b"
lp15@60809
  1665
    using c by (simp add: algebra_simps)
lp15@60809
  1666
  have *: "continuous_on (closed_segment a c) f" "continuous_on (closed_segment c b) f"
lp15@60809
  1667
    apply (rule_tac [!] continuous_on_subset [OF f])
lp15@60809
  1668
    apply (simp_all add: segment_convex_hull)
lp15@60809
  1669
    apply (rule_tac [!] convex_hull_subset)
lp15@60809
  1670
    using assms
lp15@61426
  1671
    apply (auto simp: hull_inc c' Convex.convexD_alt)
lp15@60809
  1672
    done
lp15@60809
  1673
  show ?thesis
lp15@61738
  1674
    apply (rule contour_integral_unique)
lp15@61738
  1675
    apply (rule has_contour_integral_split [OF has_contour_integral_integral has_contour_integral_integral k c])
lp15@61738
  1676
    apply (rule contour_integrable_continuous_linepath *)+
lp15@60809
  1677
    done
lp15@60809
  1678
qed
lp15@60809
  1679
lp15@61738
  1680
lemma contour_integral_split_linepath:
lp15@60809
  1681
  assumes f: "continuous_on (closed_segment a b) f"
lp15@60809
  1682
      and c: "c \<in> closed_segment a b"
lp15@61738
  1683
    shows "contour_integral(linepath a b) f = contour_integral(linepath a c) f + contour_integral(linepath c b) f"
lp15@60809
  1684
  using c
lp15@61738
  1685
  by (auto simp: closed_segment_def algebra_simps intro!: contour_integral_split [OF f])
lp15@60809
  1686
lp15@60809
  1687
(* The special case of midpoints used in the main quadrisection.*)
lp15@60809
  1688
lp15@61738
  1689
lemma has_contour_integral_midpoint:
lp15@61738
  1690
  assumes "(f has_contour_integral i) (linepath a (midpoint a b))"
lp15@61738
  1691
          "(f has_contour_integral j) (linepath (midpoint a b) b)"
lp15@61738
  1692
    shows "(f has_contour_integral (i + j)) (linepath a b)"
lp15@61738
  1693
  apply (rule has_contour_integral_split [where c = "midpoint a b" and k = "1/2"])
lp15@60809
  1694
  using assms
lp15@60809
  1695
  apply (auto simp: midpoint_def algebra_simps scaleR_conv_of_real)
lp15@60809
  1696
  done
lp15@60809
  1697
lp15@61738
  1698
lemma contour_integral_midpoint:
lp15@60809
  1699
   "continuous_on (closed_segment a b) f
lp15@61738
  1700
    \<Longrightarrow> contour_integral (linepath a b) f =
lp15@61738
  1701
        contour_integral (linepath a (midpoint a b)) f + contour_integral (linepath (midpoint a b) b) f"
lp15@61738
  1702
  apply (rule contour_integral_split [where c = "midpoint a b" and k = "1/2"])
lp15@60809
  1703
  using assms
lp15@60809
  1704
  apply (auto simp: midpoint_def algebra_simps scaleR_conv_of_real)
lp15@60809
  1705
  done
lp15@60809
  1706
lp15@60809
  1707
lp15@60809
  1708
text\<open>A couple of special case lemmas that are useful below\<close>
lp15@60809
  1709
lp15@60809
  1710
lemma triangle_linear_has_chain_integral:
lp15@61738
  1711
    "((\<lambda>x. m*x + d) has_contour_integral 0) (linepath a b +++ linepath b c +++ linepath c a)"
lp15@60809
  1712
  apply (rule Cauchy_theorem_primitive [of UNIV "\<lambda>x. m/2 * x^2 + d*x"])
lp15@60809
  1713
  apply (auto intro!: derivative_eq_intros)
lp15@60809
  1714
  done
lp15@60809
  1715
lp15@60809
  1716
lemma has_chain_integral_chain_integral3:
lp15@61738
  1717
     "(f has_contour_integral i) (linepath a b +++ linepath b c +++ linepath c d)
lp15@61738
  1718
      \<Longrightarrow> contour_integral (linepath a b) f + contour_integral (linepath b c) f + contour_integral (linepath c d) f = i"
lp15@61738
  1719
  apply (subst contour_integral_unique [symmetric], assumption)
lp15@61738
  1720
  apply (drule has_contour_integral_integrable)
lp15@60809
  1721
  apply (simp add: valid_path_join)
lp15@60809
  1722
  done
lp15@60809
  1723
lp15@60809
  1724
subsection\<open>Reversing the order in a double path integral\<close>
lp15@60809
  1725
lp15@60809
  1726
text\<open>The condition is stronger than needed but it's often true in typical situations\<close>
lp15@60809
  1727
lp15@60809
  1728
lemma fst_im_cbox [simp]: "cbox c d \<noteq> {} \<Longrightarrow> (fst ` cbox (a,c) (b,d)) = cbox a b"
lp15@60809
  1729
  by (auto simp: cbox_Pair_eq)
lp15@60809
  1730
lp15@60809
  1731
lemma snd_im_cbox [simp]: "cbox a b \<noteq> {} \<Longrightarrow> (snd ` cbox (a,c) (b,d)) = cbox c d"
lp15@60809
  1732
  by (auto simp: cbox_Pair_eq)
lp15@60809
  1733
lp15@61738
  1734
lemma contour_integral_swap:
lp15@60809
  1735
  assumes fcon:  "continuous_on (path_image g \<times> path_image h) (\<lambda>(y1,y2). f y1 y2)"
lp15@60809
  1736
      and vp:    "valid_path g" "valid_path h"
lp15@60809
  1737
      and gvcon: "continuous_on {0..1} (\<lambda>t. vector_derivative g (at t))"
lp15@60809
  1738
      and hvcon: "continuous_on {0..1} (\<lambda>t. vector_derivative h (at t))"
lp15@61738
  1739
  shows "contour_integral g (\<lambda>w. contour_integral h (f w)) =
lp15@61738
  1740
         contour_integral h (\<lambda>z. contour_integral g (\<lambda>w. f w z))"
lp15@60809
  1741
proof -
lp15@60809
  1742
  have gcon: "continuous_on {0..1} g" and hcon: "continuous_on {0..1} h"
lp15@61190
  1743
    using assms by (auto simp: valid_path_def piecewise_C1_differentiable_on_def)
lp15@60809
  1744
  have fgh1: "\<And>x. (\<lambda>t. f (g x) (h t)) = (\<lambda>(y1,y2). f y1 y2) o (\<lambda>t. (g x, h t))"
lp15@60809
  1745
    by (rule ext) simp
lp15@60809
  1746
  have fgh2: "\<And>x. (\<lambda>t. f (g t) (h x)) = (\<lambda>(y1,y2). f y1 y2) o (\<lambda>t. (g t, h x))"
lp15@60809
  1747
    by (rule ext) simp
lp15@60809
  1748
  have fcon_im1: "\<And>x. 0 \<le> x \<Longrightarrow> x \<le> 1 \<Longrightarrow> continuous_on ((\<lambda>t. (g x, h t)) ` {0..1}) (\<lambda>(x, y). f x y)"
lp15@60809
  1749
    by (rule continuous_on_subset [OF fcon]) (auto simp: path_image_def)
lp15@60809
  1750
  have fcon_im2: "\<And>x. 0 \<le> x \<Longrightarrow> x \<le> 1 \<Longrightarrow> continuous_on ((\<lambda>t. (g t, h x)) ` {0..1}) (\<lambda>(x, y). f x y)"
lp15@60809
  1751
    by (rule continuous_on_subset [OF fcon]) (auto simp: path_image_def)
lp15@60809
  1752
  have vdg: "\<And>y. y \<in> {0..1} \<Longrightarrow> (\<lambda>x. f (g x) (h y) * vector_derivative g (at x)) integrable_on {0..1}"
lp15@60809
  1753
    apply (rule integrable_continuous_real)
lp15@60809
  1754
    apply (rule continuous_on_mult [OF _ gvcon])
lp15@60809
  1755
    apply (subst fgh2)
lp15@60809
  1756
    apply (rule fcon_im2 gcon continuous_intros | simp)+
lp15@60809
  1757
    done
lp15@60809
  1758
  have "(\<lambda>z. vector_derivative g (at (fst z))) = (\<lambda>x. vector_derivative g (at x)) o fst"
lp15@60809
  1759
    by auto
lp15@60809
  1760
  then have gvcon': "continuous_on (cbox (0, 0) (1, 1::real)) (\<lambda>x. vector_derivative g (at (fst x)))"
lp15@60809
  1761
    apply (rule ssubst)
lp15@60809
  1762
    apply (rule continuous_intros | simp add: gvcon)+
lp15@60809
  1763
    done
lp15@60809
  1764
  have "(\<lambda>z. vector_derivative h (at (snd z))) = (\<lambda>x. vector_derivative h (at x)) o snd"
lp15@60809
  1765
    by auto
lp15@60809
  1766
  then have hvcon': "continuous_on (cbox (0, 0) (1::real, 1)) (\<lambda>x. vector_derivative h (at (snd x)))"
lp15@60809
  1767
    apply (rule ssubst)
lp15@60809
  1768
    apply (rule continuous_intros | simp add: hvcon)+
lp15@60809
  1769
    done
lp15@60809
  1770
  have "(\<lambda>x. f (g (fst x)) (h (snd x))) = (\<lambda>(y1,y2). f y1 y2) o (\<lambda>w. ((g o fst) w, (h o snd) w))"
lp15@60809
  1771
    by auto
lp15@60809
  1772
  then have fgh: "continuous_on (cbox (0, 0) (1, 1)) (\<lambda>x. f (g (fst x)) (h (snd x)))"
lp15@60809
  1773
    apply (rule ssubst)
lp15@60809
  1774
    apply (rule gcon hcon continuous_intros | simp)+
lp15@60809
  1775
    apply (auto simp: path_image_def intro: continuous_on_subset [OF fcon])
lp15@60809
  1776
    done
lp15@61738
  1777
  have "integral {0..1} (\<lambda>x. contour_integral h (f (g x)) * vector_derivative g (at x)) =
lp15@61738
  1778
        integral {0..1} (\<lambda>x. contour_integral h (\<lambda>y. f (g x) y * vector_derivative g (at x)))"
lp15@61738
  1779
    apply (rule integral_cong [OF contour_integral_rmul [symmetric]])
lp15@61738
  1780
    apply (clarsimp simp: contour_integrable_on)
lp15@60809
  1781
    apply (rule integrable_continuous_real)
lp15@60809
  1782
    apply (rule continuous_on_mult [OF _ hvcon])
lp15@60809
  1783
    apply (subst fgh1)
lp15@60809
  1784
    apply (rule fcon_im1 hcon continuous_intros | simp)+
lp15@60809
  1785
    done
lp15@60809
  1786
  also have "... = integral {0..1}
lp15@61738
  1787
                     (\<lambda>y. contour_integral g (\<lambda>x. f x (h y) * vector_derivative h (at y)))"
lp15@61738
  1788
    apply (simp add: contour_integral_integral)
lp15@60809
  1789
    apply (subst integral_swap_continuous [where 'a = real and 'b = real, of 0 0 1 1, simplified])
lp15@60809
  1790
    apply (rule fgh gvcon' hvcon' continuous_intros | simp add: split_def)+
lp15@60809
  1791
    apply (simp add: algebra_simps)
lp15@60809
  1792
    done
lp15@61738
  1793
  also have "... = contour_integral h (\<lambda>z. contour_integral g (\<lambda>w. f w z))"
lp15@61738
  1794
    apply (simp add: contour_integral_integral)
lp15@60809
  1795
    apply (rule integral_cong)
lp15@60809
  1796
    apply (subst integral_mult_left [symmetric])
lp15@60809
  1797
    apply (blast intro: vdg)
lp15@60809
  1798
    apply (simp add: algebra_simps)
lp15@60809
  1799
    done
lp15@60809
  1800
  finally show ?thesis
lp15@61738
  1801
    by (simp add: contour_integral_integral)
lp15@60809
  1802
qed
lp15@60809
  1803
lp15@60809
  1804
lp15@60809
  1805
subsection\<open>The key quadrisection step\<close>
lp15@60809
  1806
lp15@60809
  1807
lemma norm_sum_half:
lp15@60809
  1808
  assumes "norm(a + b) >= e"
lp15@60809
  1809
    shows "norm a >= e/2 \<or> norm b >= e/2"
lp15@60809
  1810
proof -
lp15@60809
  1811
  have "e \<le> norm (- a - b)"
lp15@60809
  1812
    by (simp add: add.commute assms norm_minus_commute)
lp15@60809
  1813
  thus ?thesis
lp15@60809
  1814
    using norm_triangle_ineq4 order_trans by fastforce
lp15@60809
  1815
qed
lp15@60809
  1816
lp15@60809
  1817
lemma norm_sum_lemma:
lp15@60809
  1818
  assumes "e \<le> norm (a + b + c + d)"
lp15@60809
  1819
    shows "e / 4 \<le> norm a \<or> e / 4 \<le> norm b \<or> e / 4 \<le> norm c \<or> e / 4 \<le> norm d"
lp15@60809
  1820
proof -
lp15@60809
  1821
  have "e \<le> norm ((a + b) + (c + d))" using assms
lp15@60809
  1822
    by (simp add: algebra_simps)
lp15@60809
  1823
  then show ?thesis
lp15@60809
  1824
    by (auto dest!: norm_sum_half)
lp15@60809
  1825
qed
lp15@60809
  1826
lp15@60809
  1827
lemma Cauchy_theorem_quadrisection:
lp15@60809
  1828
  assumes f: "continuous_on (convex hull {a,b,c}) f"
lp15@60809
  1829
      and dist: "dist a b \<le> K" "dist b c \<le> K" "dist c a \<le> K"
lp15@60809
  1830
      and e: "e * K^2 \<le>
lp15@61738
  1831
              norm (contour_integral(linepath a b) f + contour_integral(linepath b c) f + contour_integral(linepath c a) f)"
lp15@60809
  1832
  shows "\<exists>a' b' c'.
lp15@60809
  1833
           a' \<in> convex hull {a,b,c} \<and> b' \<in> convex hull {a,b,c} \<and> c' \<in> convex hull {a,b,c} \<and>
lp15@60809
  1834
           dist a' b' \<le> K/2  \<and>  dist b' c' \<le> K/2  \<and>  dist c' a' \<le> K/2  \<and>
lp15@61738
  1835
           e * (K/2)^2 \<le> norm(contour_integral(linepath a' b') f + contour_integral(linepath b' c') f + contour_integral(linepath c' a') f)"
lp15@60809
  1836
proof -
lp15@60809
  1837
  note divide_le_eq_numeral1 [simp del]
lp15@60809
  1838
  def a' \<equiv> "midpoint b c"
lp15@60809
  1839
  def b' \<equiv> "midpoint c a"
lp15@60809
  1840
  def c' \<equiv> "midpoint a b"
lp15@60809
  1841
  have fabc: "continuous_on (closed_segment a b) f" "continuous_on (closed_segment b c) f" "continuous_on (closed_segment c a) f"
lp15@60809
  1842
    using f continuous_on_subset segments_subset_convex_hull by metis+
lp15@60809
  1843
  have fcont': "continuous_on (closed_segment c' b') f"
lp15@60809
  1844
               "continuous_on (closed_segment a' c') f"
lp15@60809
  1845
               "continuous_on (closed_segment b' a') f"
lp15@60809
  1846
    unfolding a'_def b'_def c'_def
lp15@60809
  1847
    apply (rule continuous_on_subset [OF f],
lp15@60809
  1848
           metis midpoints_in_convex_hull convex_hull_subset hull_subset insert_subset segment_convex_hull)+
lp15@60809
  1849
    done
lp15@61738
  1850
  let ?pathint = "\<lambda>x y. contour_integral(linepath x y) f"
lp15@60809
  1851
  have *: "?pathint a b + ?pathint b c + ?pathint c a =
lp15@60809
  1852
          (?pathint a c' + ?pathint c' b' + ?pathint b' a) +
lp15@60809
  1853
          (?pathint a' c' + ?pathint c' b + ?pathint b a') +
lp15@60809
  1854
          (?pathint a' c + ?pathint c b' + ?pathint b' a') +
lp15@60809
  1855
          (?pathint a' b' + ?pathint b' c' + ?pathint c' a')"
lp15@61738
  1856
    apply (simp add: fcont' contour_integral_reverse_linepath)
lp15@61738
  1857
    apply (simp add: a'_def b'_def c'_def contour_integral_midpoint fabc)
lp15@60809
  1858
    done
lp15@60809
  1859
  have [simp]: "\<And>x y. cmod (x * 2 - y * 2) = cmod (x - y) * 2"
lp15@60809
  1860
    by (metis left_diff_distrib mult.commute norm_mult_numeral1)
lp15@60809
  1861
  have [simp]: "\<And>x y. cmod (x - y) = cmod (y - x)"
lp15@60809
  1862
    by (simp add: norm_minus_commute)
lp15@60809
  1863
  consider "e * K\<^sup>2 / 4 \<le> cmod (?pathint a c' + ?pathint c' b' + ?pathint b' a)" |
lp15@60809
  1864
           "e * K\<^sup>2 / 4 \<le> cmod (?pathint a' c' + ?pathint c' b + ?pathint b a')" |
lp15@60809
  1865
           "e * K\<^sup>2 / 4 \<le> cmod (?pathint a' c + ?pathint c b' + ?pathint b' a')" |
lp15@60809
  1866
           "e * K\<^sup>2 / 4 \<le> cmod (?pathint a' b' + ?pathint b' c' + ?pathint c' a')"
lp15@60809
  1867
    using assms
lp15@60809
  1868
    apply (simp only: *)
lp15@60809
  1869
    apply (blast intro: that dest!: norm_sum_lemma)
lp15@60809
  1870
    done
lp15@60809
  1871
  then show ?thesis
lp15@60809
  1872
  proof cases
lp15@60809
  1873
    case 1 then show ?thesis
lp15@60809
  1874
      apply (rule_tac x=a in exI)
lp15@60809
  1875
      apply (rule exI [where x=c'])
lp15@60809
  1876
      apply (rule exI [where x=b'])
lp15@60809
  1877
      using assms
lp15@60809
  1878
      apply (auto simp: a'_def c'_def b'_def midpoints_in_convex_hull hull_subset [THEN subsetD])
lp15@60809
  1879
      apply (auto simp: midpoint_def dist_norm scaleR_conv_of_real divide_simps)
lp15@60809
  1880
      done
lp15@60809
  1881
  next
lp15@60809
  1882
    case 2 then show ?thesis
lp15@60809
  1883
      apply (rule_tac x=a' in exI)
lp15@60809
  1884
      apply (rule exI [where x=c'])
lp15@60809
  1885
      apply (rule exI [where x=b])
lp15@60809
  1886
      using assms
lp15@60809
  1887
      apply (auto simp: a'_def c'_def b'_def midpoints_in_convex_hull hull_subset [THEN subsetD])
lp15@60809
  1888
      apply (auto simp: midpoint_def dist_norm scaleR_conv_of_real divide_simps)
lp15@60809
  1889
      done
lp15@60809
  1890
  next
lp15@60809
  1891
    case 3 then show ?thesis
lp15@60809
  1892
      apply (rule_tac x=a' in exI)
lp15@60809
  1893
      apply (rule exI [where x=c])
lp15@60809
  1894
      apply (rule exI [where x=b'])
lp15@60809
  1895
      using assms
lp15@60809
  1896
      apply (auto simp: a'_def c'_def b'_def midpoints_in_convex_hull hull_subset [THEN subsetD])
lp15@60809
  1897
      apply (auto simp: midpoint_def dist_norm scaleR_conv_of_real divide_simps)
lp15@60809
  1898
      done
lp15@60809
  1899
  next
lp15@60809
  1900
    case 4 then show ?thesis
lp15@60809
  1901
      apply (rule_tac x=a' in exI)
lp15@60809
  1902
      apply (rule exI [where x=b'])
lp15@60809
  1903
      apply (rule exI [where x=c'])
lp15@60809
  1904
      using assms
lp15@60809
  1905
      apply (auto simp: a'_def c'_def b'_def midpoints_in_convex_hull hull_subset [THEN subsetD])
lp15@60809
  1906
      apply (auto simp: midpoint_def dist_norm scaleR_conv_of_real divide_simps)
lp15@60809
  1907
      done
lp15@60809
  1908
  qed
lp15@60809
  1909
qed
lp15@60809
  1910
lp15@60809
  1911
subsection\<open>Cauchy's theorem for triangles\<close>
lp15@60809
  1912
lp15@60809
  1913
lemma triangle_points_closer:
lp15@60809
  1914
  fixes a::complex
lp15@60809
  1915
  shows "\<lbrakk>x \<in> convex hull {a,b,c};  y \<in> convex hull {a,b,c}\<rbrakk>
lp15@60809
  1916
         \<Longrightarrow> norm(x - y) \<le> norm(a - b) \<or>
lp15@60809
  1917
             norm(x - y) \<le> norm(b - c) \<or>
lp15@60809
  1918
             norm(x - y) \<le> norm(c - a)"
lp15@60809
  1919
  using simplex_extremal_le [of "{a,b,c}"]
lp15@60809
  1920
  by (auto simp: norm_minus_commute)
lp15@60809
  1921
lp15@60809
  1922
lemma holomorphic_point_small_triangle:
lp15@60809
  1923
  assumes x: "x \<in> s"
lp15@60809
  1924
      and f: "continuous_on s f"
lp15@60809
  1925
      and cd: "f complex_differentiable (at x within s)"
lp15@60809
  1926
      and e: "0 < e"
lp15@60809
  1927
    shows "\<exists>k>0. \<forall>a b c. dist a b \<le> k \<and> dist b c \<le> k \<and> dist c a \<le> k \<and>
lp15@60809
  1928
              x \<in> convex hull {a,b,c} \<and> convex hull {a,b,c} \<subseteq> s
lp15@61738
  1929
              \<longrightarrow> norm(contour_integral(linepath a b) f + contour_integral(linepath b c) f +
lp15@61738
  1930
                       contour_integral(linepath c a) f)
lp15@60809
  1931
                  \<le> e*(dist a b + dist b c + dist c a)^2"
lp15@60809
  1932
           (is "\<exists>k>0. \<forall>a b c. _ \<longrightarrow> ?normle a b c")
lp15@60809
  1933
proof -
lp15@60809
  1934
  have le_of_3: "\<And>a x y z. \<lbrakk>0 \<le> x*y; 0 \<le> x*z; 0 \<le> y*z; a \<le> (e*(x + y + z))*x + (e*(x + y + z))*y + (e*(x + y + z))*z\<rbrakk>
lp15@60809
  1935
                     \<Longrightarrow> a \<le> e*(x + y + z)^2"
lp15@60809
  1936
    by (simp add: algebra_simps power2_eq_square)
lp15@60809
  1937
  have disj_le: "\<lbrakk>x \<le> a \<or> x \<le> b \<or> x \<le> c; 0 \<le> a; 0 \<le> b; 0 \<le> c\<rbrakk> \<Longrightarrow> x \<le> a + b + c"
lp15@60809
  1938
             for x::real and a b c
lp15@60809
  1939
    by linarith
lp15@61738
  1940
  have fabc: "f contour_integrable_on linepath a b" "f contour_integrable_on linepath b c" "f contour_integrable_on linepath c a"
lp15@60809
  1941
              if "convex hull {a, b, c} \<subseteq> s" for a b c
lp15@60809
  1942
    using segments_subset_convex_hull that
lp15@61738
  1943
    by (metis continuous_on_subset f contour_integrable_continuous_linepath)+
lp15@61738
  1944
  note path_bound = has_contour_integral_bound_linepath [simplified norm_minus_commute, OF has_contour_integral_integral]
lp15@60809
  1945
  { fix f' a b c d
lp15@60809
  1946
    assume d: "0 < d"
lp15@60809
  1947
       and f': "\<And>y. \<lbrakk>cmod (y - x) \<le> d; y \<in> s\<rbrakk> \<Longrightarrow> cmod (f y - f x - f' * (y - x)) \<le> e * cmod (y - x)"
lp15@60809
  1948
       and le: "cmod (a - b) \<le> d" "cmod (b - c) \<le> d" "cmod (c - a) \<le> d"
lp15@60809
  1949
       and xc: "x \<in> convex hull {a, b, c}"
lp15@60809
  1950
       and s: "convex hull {a, b, c} \<subseteq> s"
lp15@61738
  1951
    have pa: "contour_integral (linepath a b) f + contour_integral (linepath b c) f + contour_integral (linepath c a) f =
lp15@61738
  1952
              contour_integral (linepath a b) (\<lambda>y. f y - f x - f'*(y - x)) +
lp15@61738
  1953
              contour_integral (linepath b c) (\<lambda>y. f y - f x - f'*(y - x)) +
lp15@61738
  1954
              contour_integral (linepath c a) (\<lambda>y. f y - f x - f'*(y - x))"
lp15@61738
  1955
      apply (simp add: contour_integral_diff contour_integral_lmul contour_integrable_lmul contour_integrable_diff fabc [OF s])
lp15@60809
  1956
      apply (simp add: field_simps)
lp15@60809
  1957
      done
lp15@60809
  1958
    { fix y
lp15@60809
  1959
      assume yc: "y \<in> convex hull {a,b,c}"
lp15@60809
  1960
      have "cmod (f y - f x - f' * (y - x)) \<le> e*norm(y - x)"
lp15@60809
  1961
        apply (rule f')
lp15@60809
  1962
        apply (metis triangle_points_closer [OF xc yc] le norm_minus_commute order_trans)
lp15@60809
  1963
        using s yc by blast
lp15@60809
  1964
      also have "... \<le> e * (cmod (a - b) + cmod (b - c) + cmod (c - a))"
lp15@60809
  1965
        by (simp add: yc e xc disj_le [OF triangle_points_closer])
lp15@60809
  1966
      finally have "cmod (f y - f x - f' * (y - x)) \<le> e * (cmod (a - b) + cmod (b - c) + cmod (c - a))" .
lp15@60809
  1967
    } note cm_le = this
lp15@60809
  1968
    have "?normle a b c"
lp15@60809
  1969
      apply (simp add: dist_norm pa)
lp15@60809
  1970
      apply (rule le_of_3)
lp15@60809
  1971
      using f' xc s e
lp15@60809
  1972
      apply simp_all
lp15@60809
  1973
      apply (intro norm_triangle_le add_mono path_bound)
lp15@61738
  1974
      apply (simp_all add: contour_integral_diff contour_integral_lmul contour_integrable_lmul contour_integrable_diff fabc)
lp15@60809
  1975
      apply (blast intro: cm_le elim: dest: segments_subset_convex_hull [THEN subsetD])+
lp15@60809
  1976
      done
lp15@60809
  1977
  } note * = this
lp15@60809
  1978
  show ?thesis
lp15@60809
  1979
    using cd e
lp15@60809
  1980
    apply (simp add: complex_differentiable_def has_field_derivative_def has_derivative_within_alt approachable_lt_le2 Ball_def)
lp15@60809
  1981
    apply (clarify dest!: spec mp)
lp15@60809
  1982
    using *
lp15@60809
  1983
    apply (simp add: dist_norm, blast)
lp15@60809
  1984
    done
lp15@60809
  1985
qed
lp15@60809
  1986
lp15@60809
  1987
lp15@60809
  1988
(* Hence the most basic theorem for a triangle.*)
lp15@60809
  1989
locale Chain =
lp15@60809
  1990
  fixes x0 At Follows
lp15@60809
  1991
  assumes At0: "At x0 0"
lp15@60809
  1992
      and AtSuc: "\<And>x n. At x n \<Longrightarrow> \<exists>x'. At x' (Suc n) \<and> Follows x' x"
lp15@60809
  1993
begin
lp15@60809
  1994
  primrec f where
lp15@60809
  1995
    "f 0 = x0"
lp15@60809
  1996
  | "f (Suc n) = (SOME x. At x (Suc n) \<and> Follows x (f n))"
lp15@60809
  1997
lp15@60809
  1998
  lemma At: "At (f n) n"
lp15@60809
  1999
  proof (induct n)
lp15@60809
  2000
    case 0 show ?case
lp15@60809
  2001
      by (simp add: At0)
lp15@60809
  2002
  next
lp15@60809
  2003
    case (Suc n) show ?case
lp15@60809
  2004
      by (metis (no_types, lifting) AtSuc [OF Suc] f.simps(2) someI_ex)
lp15@60809
  2005
  qed
lp15@60809
  2006
lp15@60809
  2007
  lemma Follows: "Follows (f(Suc n)) (f n)"
lp15@60809
  2008
    by (metis (no_types, lifting) AtSuc [OF At [of n]] f.simps(2) someI_ex)
lp15@60809
  2009
lp15@60809
  2010
  declare f.simps(2) [simp del]
lp15@60809
  2011
end
lp15@60809
  2012
lp15@60809
  2013
lemma Chain3:
lp15@60809
  2014
  assumes At0: "At x0 y0 z0 0"
lp15@60809
  2015
      and AtSuc: "\<And>x y z n. At x y z n \<Longrightarrow> \<exists>x' y' z'. At x' y' z' (Suc n) \<and> Follows x' y' z' x y z"
lp15@60809
  2016
  obtains f g h where
lp15@60809
  2017
    "f 0 = x0" "g 0 = y0" "h 0 = z0"
lp15@60809
  2018
                      "\<And>n. At (f n) (g n) (h n) n"
lp15@60809
  2019
                       "\<And>n. Follows (f(Suc n)) (g(Suc n)) (h(Suc n)) (f n) (g n) (h n)"
lp15@60809
  2020
proof -
lp15@60809
  2021
  interpret three: Chain "(x0,y0,z0)" "\<lambda>(x,y,z). At x y z" "\<lambda>(x',y',z'). \<lambda>(x,y,z). Follows x' y' z' x y z"
lp15@60809
  2022
    apply unfold_locales
lp15@60809
  2023
    using At0 AtSuc by auto
lp15@60809
  2024
  show ?thesis
lp15@60809
  2025
  apply (rule that [of "\<lambda>n. fst (three.f n)"  "\<lambda>n. fst (snd (three.f n))" "\<lambda>n. snd (snd (three.f n))"])
lp15@60809
  2026
  apply simp_all
lp15@60809
  2027
  using three.At three.Follows
lp15@60809
  2028
  apply (simp_all add: split_beta')
lp15@60809
  2029
  done
lp15@60809
  2030
qed
lp15@60809
  2031
lp15@60809
  2032
lemma Cauchy_theorem_triangle:
lp15@60809
  2033
  assumes "f holomorphic_on (convex hull {a,b,c})"
lp15@61738
  2034
    shows "(f has_contour_integral 0) (linepath a b +++ linepath b c +++ linepath c a)"
lp15@60809
  2035
proof -
lp15@60809
  2036
  have contf: "continuous_on (convex hull {a,b,c}) f"
lp15@60809
  2037
    by (metis assms holomorphic_on_imp_continuous_on)
lp15@61738
  2038
  let ?pathint = "\<lambda>x y. contour_integral(linepath x y) f"
lp15@60809
  2039
  { fix y::complex
lp15@61738
  2040
    assume fy: "(f has_contour_integral y) (linepath a b +++ linepath b c +++ linepath c a)"
lp15@60809
  2041
       and ynz: "y \<noteq> 0"
lp15@60809
  2042
    def K \<equiv> "1 + max (dist a b) (max (dist b c) (dist c a))"
lp15@60809
  2043
    def e \<equiv> "norm y / K^2"
lp15@60809
  2044
    have K1: "K \<ge> 1"  by (simp add: K_def max.coboundedI1)
lp15@60809
  2045
    then have K: "K > 0" by linarith
lp15@60809
  2046
    have [iff]: "dist a b \<le> K" "dist b c \<le> K" "dist c a \<le> K"
lp15@60809
  2047
      by (simp_all add: K_def)
lp15@60809
  2048
    have e: "e > 0"
lp15@60809
  2049
      unfolding e_def using ynz K1 by simp
lp15@60809
  2050
    def At \<equiv> "\<lambda>x y z n. convex hull {x,y,z} \<subseteq> convex hull {a,b,c} \<and>
lp15@60809
  2051
                         dist x y \<le> K/2^n \<and> dist y z \<le> K/2^n \<and> dist z x \<le> K/2^n \<and>
lp15@60809
  2052
                         norm(?pathint x y + ?pathint y z + ?pathint z x) \<ge> e*(K/2^n)^2"
lp15@60809
  2053
    have At0: "At a b c 0"
lp15@60809
  2054
      using fy
lp15@60809
  2055
      by (simp add: At_def e_def has_chain_integral_chain_integral3)
lp15@60809
  2056
    { fix x y z n
lp15@60809
  2057
      assume At: "At x y z n"
lp15@60809
  2058
      then have contf': "continuous_on (convex hull {x,y,z}) f"
lp15@60809
  2059
        using contf At_def continuous_on_subset by blast
lp15@60809
  2060
      have "\<exists>x' y' z'. At x' y' z' (Suc n) \<and> convex hull {x',y',z'} \<subseteq> convex hull {x,y,z}"
lp15@60809
  2061
        using At
lp15@60809
  2062
        apply (simp add: At_def)
lp15@60809
  2063
        using  Cauchy_theorem_quadrisection [OF contf', of "K/2^n" e]
lp15@60809
  2064
        apply clarsimp
lp15@60809
  2065
        apply (rule_tac x="a'" in exI)
lp15@60809
  2066
        apply (rule_tac x="b'" in exI)
lp15@60809
  2067
        apply (rule_tac x="c'" in exI)
lp15@60809
  2068
        apply (simp add: algebra_simps)
lp15@60809
  2069
        apply (meson convex_hull_subset empty_subsetI insert_subset subsetCE)
lp15@60809
  2070
        done
lp15@60809
  2071
    } note AtSuc = this
lp15@60809
  2072
    obtain fa fb fc
lp15@60809
  2073
      where f0 [simp]: "fa 0 = a" "fb 0 = b" "fc 0 = c"
lp15@60809
  2074
        and cosb: "\<And>n. convex hull {fa n, fb n, fc n} \<subseteq> convex hull {a,b,c}"
lp15@60809
  2075
        and dist: "\<And>n. dist (fa n) (fb n) \<le> K/2^n"
lp15@60809
  2076
                  "\<And>n. dist (fb n) (fc n) \<le> K/2^n"
lp15@60809
  2077
                  "\<And>n. dist (fc n) (fa n) \<le> K/2^n"
lp15@60809
  2078
        and no: "\<And>n. norm(?pathint (fa n) (fb n) +
lp15@60809
  2079
                           ?pathint (fb n) (fc n) +
lp15@60809
  2080
                           ?pathint (fc n) (fa n)) \<ge> e * (K/2^n)^2"
lp15@60809
  2081
        and conv_le: "\<And>n. convex hull {fa(Suc n), fb(Suc n), fc(Suc n)} \<subseteq> convex hull {fa n, fb n, fc n}"
lp15@60809
  2082
      apply (rule Chain3 [of At, OF At0 AtSuc])
lp15@60809
  2083
      apply (auto simp: At_def)
lp15@60809
  2084
      done
lp15@60809
  2085
    have "\<exists>x. \<forall>n. x \<in> convex hull {fa n, fb n, fc n}"
lp15@60809
  2086
      apply (rule bounded_closed_nest)
lp15@60809
  2087
      apply (simp_all add: compact_imp_closed finite_imp_compact_convex_hull finite_imp_bounded_convex_hull)
lp15@60809
  2088
      apply (rule allI)
lp15@60809
  2089
      apply (rule transitive_stepwise_le)
lp15@60809
  2090
      apply (auto simp: conv_le)
lp15@60809
  2091
      done
lp15@60809
  2092
    then obtain x where x: "\<And>n. x \<in> convex hull {fa n, fb n, fc n}" by auto
lp15@60809
  2093
    then have xin: "x \<in> convex hull {a,b,c}"
lp15@60809
  2094
      using assms f0 by blast
lp15@60809
  2095
    then have fx: "f complex_differentiable at x within (convex hull {a,b,c})"
lp15@60809
  2096
      using assms holomorphic_on_def by blast
lp15@60809
  2097
    { fix k n
lp15@60809
  2098
      assume k: "0 < k"
lp15@60809
  2099
         and le:
lp15@60809
  2100
            "\<And>x' y' z'.
lp15@60809
  2101
               \<lbrakk>dist x' y' \<le> k; dist y' z' \<le> k; dist z' x' \<le> k;
lp15@60809
  2102
                x \<in> convex hull {x',y',z'};
lp15@60809
  2103
                convex hull {x',y',z'} \<subseteq> convex hull {a,b,c}\<rbrakk>
lp15@60809
  2104
               \<Longrightarrow>
lp15@60809
  2105
               cmod (?pathint x' y' + ?pathint y' z' + ?pathint z' x') * 10
lp15@60809
  2106
                     \<le> e * (dist x' y' + dist y' z' + dist z' x')\<^sup>2"
lp15@60809
  2107
         and Kk: "K / k < 2 ^ n"
lp15@60809
  2108
      have "K / 2 ^ n < k" using Kk k
lp15@60809
  2109
        by (auto simp: field_simps)
lp15@60809
  2110
      then have DD: "dist (fa n) (fb n) \<le> k" "dist (fb n) (fc n) \<le> k" "dist (fc n) (fa n) \<le> k"
lp15@60809
  2111
        using dist [of n]  k
lp15@60809
  2112
        by linarith+
lp15@60809
  2113
      have dle: "(dist (fa n) (fb n) + dist (fb n) (fc n) + dist (fc n) (fa n))\<^sup>2
lp15@60809
  2114
               \<le> (3 * K / 2 ^ n)\<^sup>2"
lp15@60809
  2115
        using dist [of n] e K
lp15@60809
  2116
        by (simp add: abs_le_square_iff [symmetric])
lp15@60809
  2117
      have less10: "\<And>x y::real. 0 < x \<Longrightarrow> y \<le> 9*x \<Longrightarrow> y < x*10"
lp15@60809
  2118
        by linarith
lp15@60809
  2119
      have "e * (dist (fa n) (fb n) + dist (fb n) (fc n) + dist (fc n) (fa n))\<^sup>2 \<le> e * (3 * K / 2 ^ n)\<^sup>2"
lp15@60809
  2120
        using ynz dle e mult_le_cancel_left_pos by blast
lp15@60809
  2121
      also have "... <
lp15@60809
  2122
          cmod (?pathint (fa n) (fb n) + ?pathint (fb n) (fc n) + ?pathint (fc n) (fa n)) * 10"
lp15@60809
  2123
        using no [of n] e K
lp15@60809
  2124
        apply (simp add: e_def field_simps)
lp15@60809
  2125
        apply (simp only: zero_less_norm_iff [symmetric])
lp15@60809
  2126
        done
lp15@60809
  2127
      finally have False
lp15@60809
  2128
        using le [OF DD x cosb] by auto
lp15@60809
  2129
    } then
lp15@60809
  2130
    have ?thesis
lp15@60809
  2131
      using holomorphic_point_small_triangle [OF xin contf fx, of "e/10"] e
lp15@60809
  2132
      apply clarsimp
lp15@60809
  2133
      apply (rule_tac x1="K/k" in exE [OF real_arch_pow2], blast)
lp15@60809
  2134
      done
lp15@60809
  2135
  }
lp15@61738
  2136
  moreover have "f contour_integrable_on (linepath a b +++ linepath b c +++ linepath c a)"
lp15@61738
  2137
    by simp (meson contf continuous_on_subset contour_integrable_continuous_linepath segments_subset_convex_hull(1)
lp15@60809
  2138
                   segments_subset_convex_hull(3) segments_subset_convex_hull(5))
lp15@60809
  2139
  ultimately show ?thesis
lp15@61738
  2140
    using has_contour_integral_integral by fastforce
lp15@60809
  2141
qed
lp15@60809
  2142
lp15@60809
  2143
lp15@60809
  2144
subsection\<open>Version needing function holomorphic in interior only\<close>
lp15@60809
  2145
lp15@60809
  2146
lemma Cauchy_theorem_flat_lemma:
lp15@60809
  2147
  assumes f: "continuous_on (convex hull {a,b,c}) f"
lp15@60809
  2148
      and c: "c - a = k *\<^sub>R (b - a)"
lp15@60809
  2149
      and k: "0 \<le> k"
lp15@61738
  2150
    shows "contour_integral (linepath a b) f + contour_integral (linepath b c) f +
lp15@61738
  2151
          contour_integral (linepath c a) f = 0"
lp15@60809
  2152
proof -
lp15@60809
  2153
  have fabc: "continuous_on (closed_segment a b) f" "continuous_on (closed_segment b c) f" "continuous_on (closed_segment c a) f"
lp15@60809
  2154
    using f continuous_on_subset segments_subset_convex_hull by metis+
lp15@60809
  2155
  show ?thesis
lp15@60809
  2156
  proof (cases "k \<le> 1")
lp15@60809
  2157
    case True show ?thesis
lp15@61738
  2158
      by (simp add: contour_integral_split [OF fabc(1) k True c] contour_integral_reverse_linepath fabc)
lp15@60809
  2159
  next
lp15@60809
  2160
    case False then show ?thesis
lp15@60809
  2161
      using fabc c
lp15@61738
  2162
      apply (subst contour_integral_split [of a c f "1/k" b, symmetric])
lp15@60809
  2163
      apply (metis closed_segment_commute fabc(3))
lp15@61738
  2164
      apply (auto simp: k contour_integral_reverse_linepath)
lp15@60809
  2165
      done
lp15@60809
  2166
  qed
lp15@60809
  2167
qed
lp15@60809
  2168
lp15@60809
  2169
lemma Cauchy_theorem_flat:
lp15@60809
  2170
  assumes f: "continuous_on (convex hull {a,b,c}) f"
lp15@60809
  2171
      and c: "c - a = k *\<^sub>R (b - a)"
lp15@61738
  2172
    shows "contour_integral (linepath a b) f +
lp15@61738
  2173
           contour_integral (linepath b c) f +
lp15@61738
  2174
           contour_integral (linepath c a) f = 0"
lp15@60809
  2175
proof (cases "0 \<le> k")
lp15@60809
  2176
  case True with assms show ?thesis
lp15@60809
  2177
    by (blast intro: Cauchy_theorem_flat_lemma)
lp15@60809
  2178
next
lp15@60809
  2179
  case False
lp15@60809
  2180
  have "continuous_on (closed_segment a b) f" "continuous_on (closed_segment b c) f" "continuous_on (closed_segment c a) f"
lp15@60809
  2181
    using f continuous_on_subset segments_subset_convex_hull by metis+
lp15@61738
  2182
  moreover have "contour_integral (linepath b a) f + contour_integral (linepath a c) f +
lp15@61738
  2183
        contour_integral (linepath c b) f = 0"
lp15@60809
  2184
    apply (rule Cauchy_theorem_flat_lemma [of b a c f "1-k"])
lp15@60809
  2185
    using False c
lp15@60809
  2186
    apply (auto simp: f insert_commute scaleR_conv_of_real algebra_simps)
lp15@60809
  2187
    done
lp15@60809
  2188
  ultimately show ?thesis
lp15@61738
  2189
    apply (auto simp: contour_integral_reverse_linepath)
lp15@60809
  2190
    using add_eq_0_iff by force
lp15@60809
  2191
qed
lp15@60809
  2192
lp15@60809
  2193
lp15@60809
  2194
lemma Cauchy_theorem_triangle_interior:
lp15@60809
  2195
  assumes contf: "continuous_on (convex hull {a,b,c}) f"
lp15@60809
  2196
      and holf:  "f holomorphic_on interior (convex hull {a,b,c})"
lp15@61738
  2197
     shows "(f has_contour_integral 0) (linepath a b +++ linepath b c +++ linepath c a)"
lp15@60809
  2198
proof -
lp15@60809
  2199
  have fabc: "continuous_on (closed_segment a b) f" "continuous_on (closed_segment b c) f" "continuous_on (closed_segment c a) f"
lp15@60809
  2200
    using contf continuous_on_subset segments_subset_convex_hull by metis+
lp15@60809
  2201
  have "bounded (f ` (convex hull {a,b,c}))"
lp15@60809
  2202
    by (simp add: compact_continuous_image compact_convex_hull compact_imp_bounded contf)
lp15@60809
  2203
  then obtain B where "0 < B" and Bnf: "\<And>x. x \<in> convex hull {a,b,c} \<Longrightarrow> norm (f x) \<le> B"
lp15@60809
  2204
     by (auto simp: dest!: bounded_pos [THEN iffD1])
lp15@60809
  2205
  have "bounded (convex hull {a,b,c})"
lp15@60809
  2206
    by (simp add: bounded_convex_hull)
lp15@60809
  2207
  then obtain C where C: "0 < C" and Cno: "\<And>y. y \<in> convex hull {a,b,c} \<Longrightarrow> norm y < C"
lp15@60809
  2208
    using bounded_pos_less by blast
lp15@60809
  2209
  then have diff_2C: "norm(x - y) \<le> 2*C"
lp15@60809
  2210
           if x: "x \<in> convex hull {a, b, c}" and y: "y \<in> convex hull {a, b, c}" for x y
lp15@60809
  2211
  proof -
lp15@60809
  2212
    have "cmod x \<le> C"
lp15@60809
  2213
      using x by (meson Cno not_le not_less_iff_gr_or_eq)
lp15@60809
  2214
    hence "cmod (x - y) \<le> C + C"
lp15@60809
  2215
      using y by (meson Cno add_mono_thms_linordered_field(4) less_eq_real_def norm_triangle_ineq4 order_trans)
lp15@60809
  2216
    thus "cmod (x - y) \<le> 2 * C"
lp15@60809
  2217
      by (metis mult_2)
lp15@60809
  2218
  qed
lp15@60809
  2219
  have contf': "continuous_on (convex hull {b,a,c}) f"
lp15@60809
  2220
    using contf by (simp add: insert_commute)
lp15@60809
  2221
  { fix y::complex
lp15@61738
  2222
    assume fy: "(f has_contour_integral y) (linepath a b +++ linepath b c +++ linepath c a)"
lp15@60809
  2223
       and ynz: "y \<noteq> 0"
lp15@61738
  2224
    have pi_eq_y: "contour_integral (linepath a b) f + contour_integral (linepath b c) f + contour_integral (linepath c a) f = y"
lp15@60809
  2225
      by (rule has_chain_integral_chain_integral3 [OF fy])
lp15@60809
  2226
    have ?thesis
lp15@60809
  2227
    proof (cases "c=a \<or> a=b \<or> b=c")
lp15@60809
  2228
      case True then show ?thesis
lp15@60809
  2229
        using Cauchy_theorem_flat [OF contf, of 0]
lp15@60809
  2230
        using has_chain_integral_chain_integral3 [OF fy] ynz
lp15@61738
  2231
        by (force simp: fabc contour_integral_reverse_linepath)
lp15@60809
  2232
    next
lp15@60809
  2233
      case False
lp15@60809
  2234
      then have car3: "card {a, b, c} = Suc (DIM(complex))"